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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier</title>  <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <meta content="gravitational waves (678) – black holes (162) – interplanetary medium (825) – diffuse radiation (383)" lang="en" name="keywords"/> <base href="/html/2411.15072v1/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S1" title="In Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Introduction</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2" title="In Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Strain noise sources</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.SS1" title="In 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.1 </span>Clock noise and its mitigation</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.SS1.SSS0.Px1" title="In 2.1 Clock noise and its mitigation ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_title">Single-arm and an atomic clock</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.SS1.SSS0.Px2" title="In 2.1 Clock noise and its mitigation ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_title">Two-arm and time-delay interferometry</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.SS2" title="In 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.2 </span>Radiometer noise</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.SS3" title="In 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.3 </span>Acceleration noise</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.SS3.SSS1" title="In 2.3 Acceleration noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.3.1 </span>Solar irradiance</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.SS3.SSS2" title="In 2.3 Acceleration noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.3.2 </span>Solar wind drag</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.SS3.SSS3" title="In 2.3 Acceleration noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.3.3 </span>Dust</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.SS3.SSS4" title="In 2.3 Acceleration noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.3.4 </span>Spacecraft charging</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.SS3.SSS5" title="In 2.3 Acceleration noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.3.5 </span>Gravity from asteroids and larger bodies</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.SS3.SSS6" title="In 2.3 Acceleration noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.3.6 </span>Off-gassing and other spacecraft emissions</span></a></li> </ol> </li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S3" title="In Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Strain power sensitivity</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S3.SS1" title="In 3 Strain power sensitivity ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.1 </span>Two-arm time-delay interferometry forecasts</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S3.SS2" title="In 3 Strain power sensitivity ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.2 </span>Single-arm with atomic clock forecasts</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S3.SS3" title="In 3 Strain power sensitivity ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.3 </span>Doppler tracking forecasts</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S4" title="In Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Instrumental considerations</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S4.SS1" title="In 4 Instrumental considerations ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.1 </span>Laser design</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S4.SS2" title="In 4 Instrumental considerations ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.2 </span>Radio design</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S4.SS3" title="In 4 Instrumental considerations ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.3 </span>Outer Solar System considerations</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S5" title="In Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5 </span>Conclusions</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#A1" title="In Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">A </span>Dispersion</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#A2" title="In Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">B </span>Refraction and diffraction</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#A3" title="In Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">C </span>Data rates</span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_multiline"> <h1 class="ltx_title ltx_title_document">Outer Solar System spacecraft without drag-free control to probe the <math alttext="\mu" class="ltx_Math" display="inline" id="id1.m1.1"><semantics id="id1.m1.1b"><mi id="id1.m1.1.1" xref="id1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="id1.m1.1c"><ci id="id1.m1.1.1.cmml" xref="id1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="id1.m1.1d">\mu</annotation><annotation encoding="application/x-llamapun" id="id1.m1.1e">italic_μ</annotation></semantics></math>Hz gravitational wave frontier</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname"><a class="ltx_ref orcid" href="https://orcid.org/0000-0001-7961-9735" title="">Matthew McQuinn</a> </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_email"><a href="mailto:mcquinn@uw.edu">mcquinn@uw.edu</a> </span> <span class="ltx_contact ltx_role_affiliation">Department of Astronomy, University of Washington, 3910 15th Ave NE, Seattle, WA 98195, USA </span></span></span> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname"><a class="ltx_ref orcid" href="https://orcid.org/0000-0002-6155-3501" title="">Casey McGrath</a> </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation">Center for Space Sciences and Technology, University of Maryland, Baltimore County, Baltimore, MD 21250, USA </span> <span class="ltx_contact ltx_role_affiliation">Center for Research and Exploration in Space Science and Technology II, NASA/GSFC, Greenbelt, MD 20771, USA </span> <span class="ltx_contact ltx_role_affiliation">Gravitational Astrophysics Lab, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA </span></span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id12.8">The microhertz frequency band of gravitational waves probes the merger of supermassive black holes as well as many other gravitational wave phenomena. However, space-interferometry methods that use test masses would require substantial development of test-mass isolation systems to detect anticipated astrophysical events. We propose an approach that avoids inertial test masses by situating spacecraft in the low-acceleration environment of the outer Solar System. We show that for Earth-spacecraft and inter-spacecraft distances of <math alttext="\gtrsim 10\," class="ltx_Math" display="inline" id="id5.1.m1.1"><semantics id="id5.1.m1.1a"><mrow id="id5.1.m1.1.1" xref="id5.1.m1.1.1.cmml"><mi id="id5.1.m1.1.1.2" xref="id5.1.m1.1.1.2.cmml"></mi><mo id="id5.1.m1.1.1.1" xref="id5.1.m1.1.1.1.cmml">≳</mo><mn id="id5.1.m1.1.1.3" xref="id5.1.m1.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="id5.1.m1.1b"><apply id="id5.1.m1.1.1.cmml" xref="id5.1.m1.1.1"><csymbol cd="latexml" id="id5.1.m1.1.1.1.cmml" xref="id5.1.m1.1.1.1">greater-than-or-equivalent-to</csymbol><csymbol cd="latexml" id="id5.1.m1.1.1.2.cmml" xref="id5.1.m1.1.1.2">absent</csymbol><cn id="id5.1.m1.1.1.3.cmml" type="integer" xref="id5.1.m1.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="id5.1.m1.1c">\gtrsim 10\,</annotation><annotation encoding="application/x-llamapun" id="id5.1.m1.1d">≳ 10</annotation></semantics></math>AU, the accelerations on the spacecraft would be sufficiently small to potentially achieve sensitivities determined by stochastic gravitational wave backgrounds. We further argue, for arm lengths of <math alttext="10-30~{}" class="ltx_Math" display="inline" id="id6.2.m2.1"><semantics id="id6.2.m2.1a"><mrow id="id6.2.m2.1.1" xref="id6.2.m2.1.1.cmml"><mn id="id6.2.m2.1.1.2" xref="id6.2.m2.1.1.2.cmml">10</mn><mo id="id6.2.m2.1.1.1" xref="id6.2.m2.1.1.1.cmml">−</mo><mn id="id6.2.m2.1.1.3" xref="id6.2.m2.1.1.3.cmml">30</mn></mrow><annotation-xml encoding="MathML-Content" id="id6.2.m2.1b"><apply id="id6.2.m2.1.1.cmml" xref="id6.2.m2.1.1"><minus id="id6.2.m2.1.1.1.cmml" xref="id6.2.m2.1.1.1"></minus><cn id="id6.2.m2.1.1.2.cmml" type="integer" xref="id6.2.m2.1.1.2">10</cn><cn id="id6.2.m2.1.1.3.cmml" type="integer" xref="id6.2.m2.1.1.3">30</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="id6.2.m2.1c">10-30~{}</annotation><annotation encoding="application/x-llamapun" id="id6.2.m2.1d">10 - 30</annotation></semantics></math>AU and <math alttext="10\," class="ltx_Math" display="inline" id="id7.3.m3.1"><semantics id="id7.3.m3.1a"><mn id="id7.3.m3.1.1" xref="id7.3.m3.1.1.cmml">10</mn><annotation-xml encoding="MathML-Content" id="id7.3.m3.1b"><cn id="id7.3.m3.1.1.cmml" type="integer" xref="id7.3.m3.1.1">10</cn></annotation-xml><annotation encoding="application/x-tex" id="id7.3.m3.1c">10\,</annotation><annotation encoding="application/x-llamapun" id="id7.3.m3.1d">10</annotation></semantics></math>Watt transmissions, that stable phase locks should be achievable with 20 cm mirrors or <math alttext="5" class="ltx_Math" display="inline" id="id8.4.m4.1"><semantics id="id8.4.m4.1a"><mn id="id8.4.m4.1.1" xref="id8.4.m4.1.1.cmml">5</mn><annotation-xml encoding="MathML-Content" id="id8.4.m4.1b"><cn id="id8.4.m4.1.1.cmml" type="integer" xref="id8.4.m4.1.1">5</cn></annotation-xml><annotation encoding="application/x-tex" id="id8.4.m4.1c">5</annotation><annotation encoding="application/x-llamapun" id="id8.4.m4.1d">5</annotation></semantics></math> m radio dishes. We discuss designs that send both laser beams and radio waves between the spacecraft, finding that despite the <math alttext="\sim 10^{4}\times" class="ltx_math_unparsed" display="inline" id="id9.5.m5.1"><semantics id="id9.5.m5.1a"><mrow id="id9.5.m5.1b"><mo id="id9.5.m5.1.1">∼</mo><msup id="id9.5.m5.1.2"><mn id="id9.5.m5.1.2.2">10</mn><mn id="id9.5.m5.1.2.3">4</mn></msup><mo id="id9.5.m5.1.3" lspace="0.222em">×</mo></mrow><annotation encoding="application/x-tex" id="id9.5.m5.1c">\sim 10^{4}\times</annotation><annotation encoding="application/x-llamapun" id="id9.5.m5.1d">∼ 10 start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ×</annotation></semantics></math> longer wavelengths, even a design with radio transmissions could reach stochastic background-limited sensitivities at <math alttext="\lesssim 0.3\times 10^{-4}" class="ltx_Math" display="inline" id="id10.6.m6.1"><semantics id="id10.6.m6.1a"><mrow id="id10.6.m6.1.1" xref="id10.6.m6.1.1.cmml"><mi id="id10.6.m6.1.1.2" xref="id10.6.m6.1.1.2.cmml"></mi><mo id="id10.6.m6.1.1.1" xref="id10.6.m6.1.1.1.cmml">≲</mo><mrow id="id10.6.m6.1.1.3" xref="id10.6.m6.1.1.3.cmml"><mn id="id10.6.m6.1.1.3.2" xref="id10.6.m6.1.1.3.2.cmml">0.3</mn><mo id="id10.6.m6.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="id10.6.m6.1.1.3.1.cmml">×</mo><msup id="id10.6.m6.1.1.3.3" xref="id10.6.m6.1.1.3.3.cmml"><mn id="id10.6.m6.1.1.3.3.2" xref="id10.6.m6.1.1.3.3.2.cmml">10</mn><mrow id="id10.6.m6.1.1.3.3.3" xref="id10.6.m6.1.1.3.3.3.cmml"><mo id="id10.6.m6.1.1.3.3.3a" xref="id10.6.m6.1.1.3.3.3.cmml">−</mo><mn id="id10.6.m6.1.1.3.3.3.2" xref="id10.6.m6.1.1.3.3.3.2.cmml">4</mn></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="id10.6.m6.1b"><apply id="id10.6.m6.1.1.cmml" xref="id10.6.m6.1.1"><csymbol cd="latexml" id="id10.6.m6.1.1.1.cmml" xref="id10.6.m6.1.1.1">less-than-or-similar-to</csymbol><csymbol cd="latexml" id="id10.6.m6.1.1.2.cmml" xref="id10.6.m6.1.1.2">absent</csymbol><apply id="id10.6.m6.1.1.3.cmml" xref="id10.6.m6.1.1.3"><times id="id10.6.m6.1.1.3.1.cmml" xref="id10.6.m6.1.1.3.1"></times><cn id="id10.6.m6.1.1.3.2.cmml" type="float" xref="id10.6.m6.1.1.3.2">0.3</cn><apply id="id10.6.m6.1.1.3.3.cmml" xref="id10.6.m6.1.1.3.3"><csymbol cd="ambiguous" id="id10.6.m6.1.1.3.3.1.cmml" xref="id10.6.m6.1.1.3.3">superscript</csymbol><cn id="id10.6.m6.1.1.3.3.2.cmml" type="integer" xref="id10.6.m6.1.1.3.3.2">10</cn><apply id="id10.6.m6.1.1.3.3.3.cmml" xref="id10.6.m6.1.1.3.3.3"><minus id="id10.6.m6.1.1.3.3.3.1.cmml" xref="id10.6.m6.1.1.3.3.3"></minus><cn id="id10.6.m6.1.1.3.3.3.2.cmml" type="integer" xref="id10.6.m6.1.1.3.3.3.2">4</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id10.6.m6.1c">\lesssim 0.3\times 10^{-4}</annotation><annotation encoding="application/x-llamapun" id="id10.6.m6.1d">≲ 0.3 × 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT</annotation></semantics></math>Hz. Operating in the radio significantly reduces many spacecraft design tolerances. Our baseline concept requires two arms to do interferometry. However, if one spacecraft carries a clock with Allan deviations at <math alttext="10^{4}" class="ltx_Math" display="inline" id="id11.7.m7.1"><semantics id="id11.7.m7.1a"><msup id="id11.7.m7.1.1" xref="id11.7.m7.1.1.cmml"><mn id="id11.7.m7.1.1.2" xref="id11.7.m7.1.1.2.cmml">10</mn><mn id="id11.7.m7.1.1.3" xref="id11.7.m7.1.1.3.cmml">4</mn></msup><annotation-xml encoding="MathML-Content" id="id11.7.m7.1b"><apply id="id11.7.m7.1.1.cmml" xref="id11.7.m7.1.1"><csymbol cd="ambiguous" id="id11.7.m7.1.1.1.cmml" xref="id11.7.m7.1.1">superscript</csymbol><cn id="id11.7.m7.1.1.2.cmml" type="integer" xref="id11.7.m7.1.1.2">10</cn><cn id="id11.7.m7.1.1.3.cmml" type="integer" xref="id11.7.m7.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="id11.7.m7.1c">10^{4}</annotation><annotation encoding="application/x-llamapun" id="id11.7.m7.1d">10 start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT</annotation></semantics></math> seconds of <math alttext="10^{-17}" class="ltx_Math" display="inline" id="id12.8.m8.1"><semantics id="id12.8.m8.1a"><msup id="id12.8.m8.1.1" xref="id12.8.m8.1.1.cmml"><mn id="id12.8.m8.1.1.2" xref="id12.8.m8.1.1.2.cmml">10</mn><mrow id="id12.8.m8.1.1.3" xref="id12.8.m8.1.1.3.cmml"><mo id="id12.8.m8.1.1.3a" xref="id12.8.m8.1.1.3.cmml">−</mo><mn id="id12.8.m8.1.1.3.2" xref="id12.8.m8.1.1.3.2.cmml">17</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="id12.8.m8.1b"><apply id="id12.8.m8.1.1.cmml" xref="id12.8.m8.1.1"><csymbol cd="ambiguous" id="id12.8.m8.1.1.1.cmml" xref="id12.8.m8.1.1">superscript</csymbol><cn id="id12.8.m8.1.1.2.cmml" type="integer" xref="id12.8.m8.1.1.2">10</cn><apply id="id12.8.m8.1.1.3.cmml" xref="id12.8.m8.1.1.3"><minus id="id12.8.m8.1.1.3.1.cmml" xref="id12.8.m8.1.1.3"></minus><cn id="id12.8.m8.1.1.3.2.cmml" type="integer" xref="id12.8.m8.1.1.3.2">17</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id12.8.m8.1c">10^{-17}</annotation><annotation encoding="application/x-llamapun" id="id12.8.m8.1d">10 start_POSTSUPERSCRIPT - 17 end_POSTSUPERSCRIPT</annotation></semantics></math>, a comparable sensitivity could be achieved with a single arm. Finally, we discuss the feasibility of achieving similar gravitational wave sensitivities in a ‘Doppler tracking’ configuration where the single arm is anchored to Earth.</p> </div> <div class="ltx_keywords">gravitational waves (678) – black holes (162) – interplanetary medium (825) – diffuse radiation (383) </div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">1 </span>Introduction</h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.1">It has been less than a decade since the first direct detection of gravitational waves by the LIGO/Virgo collaboration <cite class="ltx_cite ltx_citemacro_citep">(Abbott et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib1" title="">2016</a>)</cite>. In subsequent years, the LIGO/Virgo collaboration has cataloged more than a hundred black hole merger events at kilohertz frequencies <cite class="ltx_cite ltx_citemacro_citep">(Abbott et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib3" title="">2023</a>)</cite>, as well as several neutron star merger candidates, including the famous 2017 multi-messenger event <cite class="ltx_cite ltx_citemacro_citep">(Abbott et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib2" title="">2017</a>)</cite>. Recently, the observed spectral range of gravitational waves has been extended to almost a nanohertz with the likely detection of a stochastic gravitational wave background using pulsar timing arrays <cite class="ltx_cite ltx_citemacro_citep">(Reardon et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib59" title="">2023</a>; Agazie et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib5" title="">2023</a>; EPTA Collaboration et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib24" title="">2023</a>)</cite>, a signal that probably owes to inspirals of the most massive supermassive black holes.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.2">A gravitational wave interferometer sensitive to significantly lower frequencies than LIGO/Virgo requires going to outer space because of seismic noise. The Laser Interferometer Space Antenna (LISA), scheduled for launch starting in 2035 <cite class="ltx_cite ltx_citemacro_citep">(Colpi et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib21" title="">2024</a>)</cite>, aims to fill in the <math alttext="10^{-4}-1~{}" class="ltx_Math" display="inline" id="S1.p2.1.m1.1"><semantics id="S1.p2.1.m1.1a"><mrow id="S1.p2.1.m1.1.1" xref="S1.p2.1.m1.1.1.cmml"><msup id="S1.p2.1.m1.1.1.2" xref="S1.p2.1.m1.1.1.2.cmml"><mn id="S1.p2.1.m1.1.1.2.2" xref="S1.p2.1.m1.1.1.2.2.cmml">10</mn><mrow id="S1.p2.1.m1.1.1.2.3" xref="S1.p2.1.m1.1.1.2.3.cmml"><mo id="S1.p2.1.m1.1.1.2.3a" xref="S1.p2.1.m1.1.1.2.3.cmml">−</mo><mn id="S1.p2.1.m1.1.1.2.3.2" xref="S1.p2.1.m1.1.1.2.3.2.cmml">4</mn></mrow></msup><mo id="S1.p2.1.m1.1.1.1" xref="S1.p2.1.m1.1.1.1.cmml">−</mo><mn id="S1.p2.1.m1.1.1.3" xref="S1.p2.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.1.m1.1b"><apply id="S1.p2.1.m1.1.1.cmml" xref="S1.p2.1.m1.1.1"><minus id="S1.p2.1.m1.1.1.1.cmml" xref="S1.p2.1.m1.1.1.1"></minus><apply id="S1.p2.1.m1.1.1.2.cmml" xref="S1.p2.1.m1.1.1.2"><csymbol cd="ambiguous" id="S1.p2.1.m1.1.1.2.1.cmml" xref="S1.p2.1.m1.1.1.2">superscript</csymbol><cn id="S1.p2.1.m1.1.1.2.2.cmml" type="integer" xref="S1.p2.1.m1.1.1.2.2">10</cn><apply id="S1.p2.1.m1.1.1.2.3.cmml" xref="S1.p2.1.m1.1.1.2.3"><minus id="S1.p2.1.m1.1.1.2.3.1.cmml" xref="S1.p2.1.m1.1.1.2.3"></minus><cn id="S1.p2.1.m1.1.1.2.3.2.cmml" type="integer" xref="S1.p2.1.m1.1.1.2.3.2">4</cn></apply></apply><cn id="S1.p2.1.m1.1.1.3.cmml" type="integer" xref="S1.p2.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.1.m1.1c">10^{-4}-1~{}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.1.m1.1d">10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT - 1</annotation></semantics></math>Hz waveband that is intermediate between the pulsar timing arrays and the ground-based efforts like LIGO/Virgo. LISA will send laser beams between three spacecraft in a triangle configuration with side lengths of <math alttext="0.017~{}" class="ltx_Math" display="inline" id="S1.p2.2.m2.1"><semantics id="S1.p2.2.m2.1a"><mn id="S1.p2.2.m2.1.1" xref="S1.p2.2.m2.1.1.cmml">0.017</mn><annotation-xml encoding="MathML-Content" id="S1.p2.2.m2.1b"><cn id="S1.p2.2.m2.1.1.cmml" type="float" xref="S1.p2.2.m2.1.1">0.017</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.2.m2.1c">0.017~{}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.2.m2.1d">0.017</annotation></semantics></math>AU. The lasers will work as multiple Michelson-like interferometers, with the aim of measuring phase changes that result from displacements as small as an angstrom. The reference for these precise displacement measurements must be sufficiently isolated from sources of acceleration (such as the Sun’s irradiance variations) to reach the sensitivities needed to detect known astrophysical gravitational wave sources. Each LISA spacecraft employs the most sensitive accelerometer ever built that works by monitoring a nearly drag-free test mass.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.6">Considerable effort has been directed towards finding detection strategies in other regions of the gravitational wave spectrum. The most exciting frontiers are the decihertz region between LISA and LIGO/Virgo <cite class="ltx_cite ltx_citemacro_citep">(Arca Sedda et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib7" title="">2020</a>)</cite>, and the microhertz <math alttext="10^{-7}-10^{-4}" class="ltx_Math" display="inline" id="S1.p3.1.m1.1"><semantics id="S1.p3.1.m1.1a"><mrow id="S1.p3.1.m1.1.1" xref="S1.p3.1.m1.1.1.cmml"><msup id="S1.p3.1.m1.1.1.2" xref="S1.p3.1.m1.1.1.2.cmml"><mn id="S1.p3.1.m1.1.1.2.2" xref="S1.p3.1.m1.1.1.2.2.cmml">10</mn><mrow id="S1.p3.1.m1.1.1.2.3" xref="S1.p3.1.m1.1.1.2.3.cmml"><mo id="S1.p3.1.m1.1.1.2.3a" xref="S1.p3.1.m1.1.1.2.3.cmml">−</mo><mn id="S1.p3.1.m1.1.1.2.3.2" xref="S1.p3.1.m1.1.1.2.3.2.cmml">7</mn></mrow></msup><mo id="S1.p3.1.m1.1.1.1" xref="S1.p3.1.m1.1.1.1.cmml">−</mo><msup id="S1.p3.1.m1.1.1.3" xref="S1.p3.1.m1.1.1.3.cmml"><mn id="S1.p3.1.m1.1.1.3.2" xref="S1.p3.1.m1.1.1.3.2.cmml">10</mn><mrow id="S1.p3.1.m1.1.1.3.3" xref="S1.p3.1.m1.1.1.3.3.cmml"><mo id="S1.p3.1.m1.1.1.3.3a" xref="S1.p3.1.m1.1.1.3.3.cmml">−</mo><mn id="S1.p3.1.m1.1.1.3.3.2" xref="S1.p3.1.m1.1.1.3.3.2.cmml">4</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.1.m1.1b"><apply id="S1.p3.1.m1.1.1.cmml" xref="S1.p3.1.m1.1.1"><minus id="S1.p3.1.m1.1.1.1.cmml" xref="S1.p3.1.m1.1.1.1"></minus><apply id="S1.p3.1.m1.1.1.2.cmml" xref="S1.p3.1.m1.1.1.2"><csymbol cd="ambiguous" id="S1.p3.1.m1.1.1.2.1.cmml" xref="S1.p3.1.m1.1.1.2">superscript</csymbol><cn id="S1.p3.1.m1.1.1.2.2.cmml" type="integer" xref="S1.p3.1.m1.1.1.2.2">10</cn><apply id="S1.p3.1.m1.1.1.2.3.cmml" xref="S1.p3.1.m1.1.1.2.3"><minus id="S1.p3.1.m1.1.1.2.3.1.cmml" xref="S1.p3.1.m1.1.1.2.3"></minus><cn id="S1.p3.1.m1.1.1.2.3.2.cmml" type="integer" xref="S1.p3.1.m1.1.1.2.3.2">7</cn></apply></apply><apply id="S1.p3.1.m1.1.1.3.cmml" xref="S1.p3.1.m1.1.1.3"><csymbol cd="ambiguous" id="S1.p3.1.m1.1.1.3.1.cmml" xref="S1.p3.1.m1.1.1.3">superscript</csymbol><cn id="S1.p3.1.m1.1.1.3.2.cmml" type="integer" xref="S1.p3.1.m1.1.1.3.2">10</cn><apply id="S1.p3.1.m1.1.1.3.3.cmml" xref="S1.p3.1.m1.1.1.3.3"><minus id="S1.p3.1.m1.1.1.3.3.1.cmml" xref="S1.p3.1.m1.1.1.3.3"></minus><cn id="S1.p3.1.m1.1.1.3.3.2.cmml" type="integer" xref="S1.p3.1.m1.1.1.3.3.2">4</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.1.m1.1c">10^{-7}-10^{-4}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.1.m1.1d">10 start_POSTSUPERSCRIPT - 7 end_POSTSUPERSCRIPT - 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT</annotation></semantics></math>Hz band that falls between the pulsar timing arrays and LISA. This <math alttext="10^{-7}-10^{-4}" class="ltx_Math" display="inline" id="S1.p3.2.m2.1"><semantics id="S1.p3.2.m2.1a"><mrow id="S1.p3.2.m2.1.1" xref="S1.p3.2.m2.1.1.cmml"><msup id="S1.p3.2.m2.1.1.2" xref="S1.p3.2.m2.1.1.2.cmml"><mn id="S1.p3.2.m2.1.1.2.2" xref="S1.p3.2.m2.1.1.2.2.cmml">10</mn><mrow id="S1.p3.2.m2.1.1.2.3" xref="S1.p3.2.m2.1.1.2.3.cmml"><mo id="S1.p3.2.m2.1.1.2.3a" xref="S1.p3.2.m2.1.1.2.3.cmml">−</mo><mn id="S1.p3.2.m2.1.1.2.3.2" xref="S1.p3.2.m2.1.1.2.3.2.cmml">7</mn></mrow></msup><mo id="S1.p3.2.m2.1.1.1" xref="S1.p3.2.m2.1.1.1.cmml">−</mo><msup id="S1.p3.2.m2.1.1.3" xref="S1.p3.2.m2.1.1.3.cmml"><mn id="S1.p3.2.m2.1.1.3.2" xref="S1.p3.2.m2.1.1.3.2.cmml">10</mn><mrow id="S1.p3.2.m2.1.1.3.3" xref="S1.p3.2.m2.1.1.3.3.cmml"><mo id="S1.p3.2.m2.1.1.3.3a" xref="S1.p3.2.m2.1.1.3.3.cmml">−</mo><mn id="S1.p3.2.m2.1.1.3.3.2" xref="S1.p3.2.m2.1.1.3.3.2.cmml">4</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.2.m2.1b"><apply id="S1.p3.2.m2.1.1.cmml" xref="S1.p3.2.m2.1.1"><minus id="S1.p3.2.m2.1.1.1.cmml" xref="S1.p3.2.m2.1.1.1"></minus><apply id="S1.p3.2.m2.1.1.2.cmml" xref="S1.p3.2.m2.1.1.2"><csymbol cd="ambiguous" id="S1.p3.2.m2.1.1.2.1.cmml" xref="S1.p3.2.m2.1.1.2">superscript</csymbol><cn id="S1.p3.2.m2.1.1.2.2.cmml" type="integer" xref="S1.p3.2.m2.1.1.2.2">10</cn><apply id="S1.p3.2.m2.1.1.2.3.cmml" xref="S1.p3.2.m2.1.1.2.3"><minus id="S1.p3.2.m2.1.1.2.3.1.cmml" xref="S1.p3.2.m2.1.1.2.3"></minus><cn id="S1.p3.2.m2.1.1.2.3.2.cmml" type="integer" xref="S1.p3.2.m2.1.1.2.3.2">7</cn></apply></apply><apply id="S1.p3.2.m2.1.1.3.cmml" xref="S1.p3.2.m2.1.1.3"><csymbol cd="ambiguous" id="S1.p3.2.m2.1.1.3.1.cmml" xref="S1.p3.2.m2.1.1.3">superscript</csymbol><cn id="S1.p3.2.m2.1.1.3.2.cmml" type="integer" xref="S1.p3.2.m2.1.1.3.2">10</cn><apply id="S1.p3.2.m2.1.1.3.3.cmml" xref="S1.p3.2.m2.1.1.3.3"><minus id="S1.p3.2.m2.1.1.3.3.1.cmml" xref="S1.p3.2.m2.1.1.3.3"></minus><cn id="S1.p3.2.m2.1.1.3.3.2.cmml" type="integer" xref="S1.p3.2.m2.1.1.3.3.2">4</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.2.m2.1c">10^{-7}-10^{-4}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.2.m2.1d">10 start_POSTSUPERSCRIPT - 7 end_POSTSUPERSCRIPT - 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT</annotation></semantics></math>Hz band probes the early inspiral of the <math alttext="\sim 10^{5}-10^{6.5}M_{\odot}" class="ltx_Math" display="inline" id="S1.p3.3.m3.1"><semantics id="S1.p3.3.m3.1a"><mrow id="S1.p3.3.m3.1.1" xref="S1.p3.3.m3.1.1.cmml"><mi id="S1.p3.3.m3.1.1.2" xref="S1.p3.3.m3.1.1.2.cmml"></mi><mo id="S1.p3.3.m3.1.1.1" xref="S1.p3.3.m3.1.1.1.cmml">∼</mo><mrow id="S1.p3.3.m3.1.1.3" xref="S1.p3.3.m3.1.1.3.cmml"><msup id="S1.p3.3.m3.1.1.3.2" xref="S1.p3.3.m3.1.1.3.2.cmml"><mn id="S1.p3.3.m3.1.1.3.2.2" xref="S1.p3.3.m3.1.1.3.2.2.cmml">10</mn><mn id="S1.p3.3.m3.1.1.3.2.3" xref="S1.p3.3.m3.1.1.3.2.3.cmml">5</mn></msup><mo id="S1.p3.3.m3.1.1.3.1" xref="S1.p3.3.m3.1.1.3.1.cmml">−</mo><mrow id="S1.p3.3.m3.1.1.3.3" xref="S1.p3.3.m3.1.1.3.3.cmml"><msup id="S1.p3.3.m3.1.1.3.3.2" xref="S1.p3.3.m3.1.1.3.3.2.cmml"><mn id="S1.p3.3.m3.1.1.3.3.2.2" xref="S1.p3.3.m3.1.1.3.3.2.2.cmml">10</mn><mn id="S1.p3.3.m3.1.1.3.3.2.3" xref="S1.p3.3.m3.1.1.3.3.2.3.cmml">6.5</mn></msup><mo id="S1.p3.3.m3.1.1.3.3.1" xref="S1.p3.3.m3.1.1.3.3.1.cmml">⁢</mo><msub id="S1.p3.3.m3.1.1.3.3.3" xref="S1.p3.3.m3.1.1.3.3.3.cmml"><mi id="S1.p3.3.m3.1.1.3.3.3.2" xref="S1.p3.3.m3.1.1.3.3.3.2.cmml">M</mi><mo id="S1.p3.3.m3.1.1.3.3.3.3" xref="S1.p3.3.m3.1.1.3.3.3.3.cmml">⊙</mo></msub></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.3.m3.1b"><apply id="S1.p3.3.m3.1.1.cmml" xref="S1.p3.3.m3.1.1"><csymbol cd="latexml" id="S1.p3.3.m3.1.1.1.cmml" xref="S1.p3.3.m3.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S1.p3.3.m3.1.1.2.cmml" xref="S1.p3.3.m3.1.1.2">absent</csymbol><apply id="S1.p3.3.m3.1.1.3.cmml" xref="S1.p3.3.m3.1.1.3"><minus id="S1.p3.3.m3.1.1.3.1.cmml" xref="S1.p3.3.m3.1.1.3.1"></minus><apply id="S1.p3.3.m3.1.1.3.2.cmml" xref="S1.p3.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="S1.p3.3.m3.1.1.3.2.1.cmml" xref="S1.p3.3.m3.1.1.3.2">superscript</csymbol><cn id="S1.p3.3.m3.1.1.3.2.2.cmml" type="integer" xref="S1.p3.3.m3.1.1.3.2.2">10</cn><cn id="S1.p3.3.m3.1.1.3.2.3.cmml" type="integer" xref="S1.p3.3.m3.1.1.3.2.3">5</cn></apply><apply id="S1.p3.3.m3.1.1.3.3.cmml" xref="S1.p3.3.m3.1.1.3.3"><times id="S1.p3.3.m3.1.1.3.3.1.cmml" xref="S1.p3.3.m3.1.1.3.3.1"></times><apply id="S1.p3.3.m3.1.1.3.3.2.cmml" xref="S1.p3.3.m3.1.1.3.3.2"><csymbol cd="ambiguous" id="S1.p3.3.m3.1.1.3.3.2.1.cmml" xref="S1.p3.3.m3.1.1.3.3.2">superscript</csymbol><cn id="S1.p3.3.m3.1.1.3.3.2.2.cmml" type="integer" xref="S1.p3.3.m3.1.1.3.3.2.2">10</cn><cn id="S1.p3.3.m3.1.1.3.3.2.3.cmml" type="float" xref="S1.p3.3.m3.1.1.3.3.2.3">6.5</cn></apply><apply id="S1.p3.3.m3.1.1.3.3.3.cmml" xref="S1.p3.3.m3.1.1.3.3.3"><csymbol cd="ambiguous" id="S1.p3.3.m3.1.1.3.3.3.1.cmml" xref="S1.p3.3.m3.1.1.3.3.3">subscript</csymbol><ci id="S1.p3.3.m3.1.1.3.3.3.2.cmml" xref="S1.p3.3.m3.1.1.3.3.3.2">𝑀</ci><csymbol cd="latexml" id="S1.p3.3.m3.1.1.3.3.3.3.cmml" xref="S1.p3.3.m3.1.1.3.3.3.3">direct-product</csymbol></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.3.m3.1c">\sim 10^{5}-10^{6.5}M_{\odot}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.3.m3.1d">∼ 10 start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT - 10 start_POSTSUPERSCRIPT 6.5 end_POSTSUPERSCRIPT italic_M start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT</annotation></semantics></math> black holes that LISA observes nearer to merger, as well as the inspiral and merger of <math alttext="10^{6.5}-10^{10}M_{\odot}" class="ltx_Math" display="inline" id="S1.p3.4.m4.1"><semantics id="S1.p3.4.m4.1a"><mrow id="S1.p3.4.m4.1.1" xref="S1.p3.4.m4.1.1.cmml"><msup id="S1.p3.4.m4.1.1.2" xref="S1.p3.4.m4.1.1.2.cmml"><mn id="S1.p3.4.m4.1.1.2.2" xref="S1.p3.4.m4.1.1.2.2.cmml">10</mn><mn id="S1.p3.4.m4.1.1.2.3" xref="S1.p3.4.m4.1.1.2.3.cmml">6.5</mn></msup><mo id="S1.p3.4.m4.1.1.1" xref="S1.p3.4.m4.1.1.1.cmml">−</mo><mrow id="S1.p3.4.m4.1.1.3" xref="S1.p3.4.m4.1.1.3.cmml"><msup id="S1.p3.4.m4.1.1.3.2" xref="S1.p3.4.m4.1.1.3.2.cmml"><mn id="S1.p3.4.m4.1.1.3.2.2" xref="S1.p3.4.m4.1.1.3.2.2.cmml">10</mn><mn id="S1.p3.4.m4.1.1.3.2.3" xref="S1.p3.4.m4.1.1.3.2.3.cmml">10</mn></msup><mo id="S1.p3.4.m4.1.1.3.1" xref="S1.p3.4.m4.1.1.3.1.cmml">⁢</mo><msub id="S1.p3.4.m4.1.1.3.3" xref="S1.p3.4.m4.1.1.3.3.cmml"><mi id="S1.p3.4.m4.1.1.3.3.2" xref="S1.p3.4.m4.1.1.3.3.2.cmml">M</mi><mo id="S1.p3.4.m4.1.1.3.3.3" xref="S1.p3.4.m4.1.1.3.3.3.cmml">⊙</mo></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.4.m4.1b"><apply id="S1.p3.4.m4.1.1.cmml" xref="S1.p3.4.m4.1.1"><minus id="S1.p3.4.m4.1.1.1.cmml" xref="S1.p3.4.m4.1.1.1"></minus><apply id="S1.p3.4.m4.1.1.2.cmml" xref="S1.p3.4.m4.1.1.2"><csymbol cd="ambiguous" id="S1.p3.4.m4.1.1.2.1.cmml" xref="S1.p3.4.m4.1.1.2">superscript</csymbol><cn id="S1.p3.4.m4.1.1.2.2.cmml" type="integer" xref="S1.p3.4.m4.1.1.2.2">10</cn><cn id="S1.p3.4.m4.1.1.2.3.cmml" type="float" xref="S1.p3.4.m4.1.1.2.3">6.5</cn></apply><apply id="S1.p3.4.m4.1.1.3.cmml" xref="S1.p3.4.m4.1.1.3"><times id="S1.p3.4.m4.1.1.3.1.cmml" xref="S1.p3.4.m4.1.1.3.1"></times><apply id="S1.p3.4.m4.1.1.3.2.cmml" xref="S1.p3.4.m4.1.1.3.2"><csymbol cd="ambiguous" id="S1.p3.4.m4.1.1.3.2.1.cmml" xref="S1.p3.4.m4.1.1.3.2">superscript</csymbol><cn id="S1.p3.4.m4.1.1.3.2.2.cmml" type="integer" xref="S1.p3.4.m4.1.1.3.2.2">10</cn><cn id="S1.p3.4.m4.1.1.3.2.3.cmml" type="integer" xref="S1.p3.4.m4.1.1.3.2.3">10</cn></apply><apply id="S1.p3.4.m4.1.1.3.3.cmml" xref="S1.p3.4.m4.1.1.3.3"><csymbol cd="ambiguous" id="S1.p3.4.m4.1.1.3.3.1.cmml" xref="S1.p3.4.m4.1.1.3.3">subscript</csymbol><ci id="S1.p3.4.m4.1.1.3.3.2.cmml" xref="S1.p3.4.m4.1.1.3.3.2">𝑀</ci><csymbol cd="latexml" id="S1.p3.4.m4.1.1.3.3.3.cmml" xref="S1.p3.4.m4.1.1.3.3.3">direct-product</csymbol></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.4.m4.1c">10^{6.5}-10^{10}M_{\odot}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.4.m4.1d">10 start_POSTSUPERSCRIPT 6.5 end_POSTSUPERSCRIPT - 10 start_POSTSUPERSCRIPT 10 end_POSTSUPERSCRIPT italic_M start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT</annotation></semantics></math> black holes – the class of black holes that are associated with quasars and may be more likely to yield an electromagnetic counterpart. There are a host of other astrophysical sources that fall in the <math alttext="10^{-7}-10^{-4}" class="ltx_Math" display="inline" id="S1.p3.5.m5.1"><semantics id="S1.p3.5.m5.1a"><mrow id="S1.p3.5.m5.1.1" xref="S1.p3.5.m5.1.1.cmml"><msup id="S1.p3.5.m5.1.1.2" xref="S1.p3.5.m5.1.1.2.cmml"><mn id="S1.p3.5.m5.1.1.2.2" xref="S1.p3.5.m5.1.1.2.2.cmml">10</mn><mrow id="S1.p3.5.m5.1.1.2.3" xref="S1.p3.5.m5.1.1.2.3.cmml"><mo id="S1.p3.5.m5.1.1.2.3a" xref="S1.p3.5.m5.1.1.2.3.cmml">−</mo><mn id="S1.p3.5.m5.1.1.2.3.2" xref="S1.p3.5.m5.1.1.2.3.2.cmml">7</mn></mrow></msup><mo id="S1.p3.5.m5.1.1.1" xref="S1.p3.5.m5.1.1.1.cmml">−</mo><msup id="S1.p3.5.m5.1.1.3" xref="S1.p3.5.m5.1.1.3.cmml"><mn id="S1.p3.5.m5.1.1.3.2" xref="S1.p3.5.m5.1.1.3.2.cmml">10</mn><mrow id="S1.p3.5.m5.1.1.3.3" xref="S1.p3.5.m5.1.1.3.3.cmml"><mo id="S1.p3.5.m5.1.1.3.3a" xref="S1.p3.5.m5.1.1.3.3.cmml">−</mo><mn id="S1.p3.5.m5.1.1.3.3.2" xref="S1.p3.5.m5.1.1.3.3.2.cmml">4</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.5.m5.1b"><apply id="S1.p3.5.m5.1.1.cmml" xref="S1.p3.5.m5.1.1"><minus id="S1.p3.5.m5.1.1.1.cmml" xref="S1.p3.5.m5.1.1.1"></minus><apply id="S1.p3.5.m5.1.1.2.cmml" xref="S1.p3.5.m5.1.1.2"><csymbol cd="ambiguous" id="S1.p3.5.m5.1.1.2.1.cmml" xref="S1.p3.5.m5.1.1.2">superscript</csymbol><cn id="S1.p3.5.m5.1.1.2.2.cmml" type="integer" xref="S1.p3.5.m5.1.1.2.2">10</cn><apply id="S1.p3.5.m5.1.1.2.3.cmml" xref="S1.p3.5.m5.1.1.2.3"><minus id="S1.p3.5.m5.1.1.2.3.1.cmml" xref="S1.p3.5.m5.1.1.2.3"></minus><cn id="S1.p3.5.m5.1.1.2.3.2.cmml" type="integer" xref="S1.p3.5.m5.1.1.2.3.2">7</cn></apply></apply><apply id="S1.p3.5.m5.1.1.3.cmml" xref="S1.p3.5.m5.1.1.3"><csymbol cd="ambiguous" id="S1.p3.5.m5.1.1.3.1.cmml" xref="S1.p3.5.m5.1.1.3">superscript</csymbol><cn id="S1.p3.5.m5.1.1.3.2.cmml" type="integer" xref="S1.p3.5.m5.1.1.3.2">10</cn><apply id="S1.p3.5.m5.1.1.3.3.cmml" xref="S1.p3.5.m5.1.1.3.3"><minus id="S1.p3.5.m5.1.1.3.3.1.cmml" xref="S1.p3.5.m5.1.1.3.3"></minus><cn id="S1.p3.5.m5.1.1.3.3.2.cmml" type="integer" xref="S1.p3.5.m5.1.1.3.3.2">4</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.5.m5.1c">10^{-7}-10^{-4}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.5.m5.1d">10 start_POSTSUPERSCRIPT - 7 end_POSTSUPERSCRIPT - 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT</annotation></semantics></math>Hz band <cite class="ltx_cite ltx_citemacro_citep">(e.g. Sesana et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib62" title="">2021</a>)</cite>. While many proposed methods in this waveband lack sufficient sensitivity for known astrophysical processes, they may still detect larger backgrounds, such as those produced in the early universe <cite class="ltx_cite ltx_citemacro_citep">(Neronov et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib53" title="">2021</a>; Bai et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib11" title="">2023</a>)</cite>. One idea is to use measurements of the lunar orbit by future laser ranging <cite class="ltx_cite ltx_citemacro_citep">(Blas & Jenkins, <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib17" title="">2022</a>)</cite>. Another is to use the very precise angular localizations of stars to constrain angular variations from passing gravitational waves, i.e. gravitational wave astrometry <cite class="ltx_cite ltx_citemacro_citep">(Wang et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib72" title="">2022</a>; Fedderke et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib29" title="">2022b</a>; Crosta et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib23" title="">2024</a>)</cite>. The proposals forecast to be the most sensitive follow in the spirit of an expanded LISA, where the three spacecraft are situated in an equilateral triangle tracing Earth’s orbit <cite class="ltx_cite ltx_citemacro_citep">(Folkner, <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib30" title="">2011</a>; Ni, <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib54" title="">2010</a>)</cite>. Two recent examples are the <math alttext="\mu" class="ltx_Math" display="inline" id="S1.p3.6.m6.1"><semantics id="S1.p3.6.m6.1a"><mi id="S1.p3.6.m6.1.1" xref="S1.p3.6.m6.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S1.p3.6.m6.1b"><ci id="S1.p3.6.m6.1.1.cmml" xref="S1.p3.6.m6.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.6.m6.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S1.p3.6.m6.1d">italic_μ</annotation></semantics></math>Ares and LISAmax concepts <cite class="ltx_cite ltx_citemacro_citep">(Sesana et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib62" title="">2021</a>; Martens et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib48" title="">2023</a>)</cite>.</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.16">The <math alttext="\mu" class="ltx_Math" display="inline" id="S1.p4.1.m1.1"><semantics id="S1.p4.1.m1.1a"><mi id="S1.p4.1.m1.1.1" xref="S1.p4.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S1.p4.1.m1.1b"><ci id="S1.p4.1.m1.1.1.cmml" xref="S1.p4.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S1.p4.1.m1.1d">italic_μ</annotation></semantics></math>Ares concept assumes acceleration isolation to <math alttext="10^{-15}" class="ltx_Math" display="inline" id="S1.p4.2.m2.1"><semantics id="S1.p4.2.m2.1a"><msup id="S1.p4.2.m2.1.1" xref="S1.p4.2.m2.1.1.cmml"><mn id="S1.p4.2.m2.1.1.2" xref="S1.p4.2.m2.1.1.2.cmml">10</mn><mrow id="S1.p4.2.m2.1.1.3" xref="S1.p4.2.m2.1.1.3.cmml"><mo id="S1.p4.2.m2.1.1.3a" xref="S1.p4.2.m2.1.1.3.cmml">−</mo><mn id="S1.p4.2.m2.1.1.3.2" xref="S1.p4.2.m2.1.1.3.2.cmml">15</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S1.p4.2.m2.1b"><apply id="S1.p4.2.m2.1.1.cmml" xref="S1.p4.2.m2.1.1"><csymbol cd="ambiguous" id="S1.p4.2.m2.1.1.1.cmml" xref="S1.p4.2.m2.1.1">superscript</csymbol><cn id="S1.p4.2.m2.1.1.2.cmml" type="integer" xref="S1.p4.2.m2.1.1.2">10</cn><apply id="S1.p4.2.m2.1.1.3.cmml" xref="S1.p4.2.m2.1.1.3"><minus id="S1.p4.2.m2.1.1.3.1.cmml" xref="S1.p4.2.m2.1.1.3"></minus><cn id="S1.p4.2.m2.1.1.3.2.cmml" type="integer" xref="S1.p4.2.m2.1.1.3.2">15</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.2.m2.1c">10^{-15}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.2.m2.1d">10 start_POSTSUPERSCRIPT - 15 end_POSTSUPERSCRIPT</annotation></semantics></math> m s<sup class="ltx_sup" id="S1.p4.16.1"><span class="ltx_text ltx_font_italic" id="S1.p4.16.1.1">-2</span></sup> Hz<sup class="ltx_sup" id="S1.p4.16.2"><span class="ltx_text ltx_font_italic" id="S1.p4.16.2.1">-1/2</span></sup> over its proposed frequency band of <math alttext="10^{-7}-1~{}" class="ltx_Math" display="inline" id="S1.p4.5.m5.1"><semantics id="S1.p4.5.m5.1a"><mrow id="S1.p4.5.m5.1.1" xref="S1.p4.5.m5.1.1.cmml"><msup id="S1.p4.5.m5.1.1.2" xref="S1.p4.5.m5.1.1.2.cmml"><mn id="S1.p4.5.m5.1.1.2.2" xref="S1.p4.5.m5.1.1.2.2.cmml">10</mn><mrow id="S1.p4.5.m5.1.1.2.3" xref="S1.p4.5.m5.1.1.2.3.cmml"><mo id="S1.p4.5.m5.1.1.2.3a" xref="S1.p4.5.m5.1.1.2.3.cmml">−</mo><mn id="S1.p4.5.m5.1.1.2.3.2" xref="S1.p4.5.m5.1.1.2.3.2.cmml">7</mn></mrow></msup><mo id="S1.p4.5.m5.1.1.1" xref="S1.p4.5.m5.1.1.1.cmml">−</mo><mn id="S1.p4.5.m5.1.1.3" xref="S1.p4.5.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.5.m5.1b"><apply id="S1.p4.5.m5.1.1.cmml" xref="S1.p4.5.m5.1.1"><minus id="S1.p4.5.m5.1.1.1.cmml" xref="S1.p4.5.m5.1.1.1"></minus><apply id="S1.p4.5.m5.1.1.2.cmml" xref="S1.p4.5.m5.1.1.2"><csymbol cd="ambiguous" id="S1.p4.5.m5.1.1.2.1.cmml" xref="S1.p4.5.m5.1.1.2">superscript</csymbol><cn id="S1.p4.5.m5.1.1.2.2.cmml" type="integer" xref="S1.p4.5.m5.1.1.2.2">10</cn><apply id="S1.p4.5.m5.1.1.2.3.cmml" xref="S1.p4.5.m5.1.1.2.3"><minus id="S1.p4.5.m5.1.1.2.3.1.cmml" xref="S1.p4.5.m5.1.1.2.3"></minus><cn id="S1.p4.5.m5.1.1.2.3.2.cmml" type="integer" xref="S1.p4.5.m5.1.1.2.3.2">7</cn></apply></apply><cn id="S1.p4.5.m5.1.1.3.cmml" type="integer" xref="S1.p4.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.5.m5.1c">10^{-7}-1~{}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.5.m5.1d">10 start_POSTSUPERSCRIPT - 7 end_POSTSUPERSCRIPT - 1</annotation></semantics></math>Hz, which contrasts with the LISA acceleration specification of <math alttext="10^{-14}" class="ltx_Math" display="inline" id="S1.p4.6.m6.1"><semantics id="S1.p4.6.m6.1a"><msup id="S1.p4.6.m6.1.1" xref="S1.p4.6.m6.1.1.cmml"><mn id="S1.p4.6.m6.1.1.2" xref="S1.p4.6.m6.1.1.2.cmml">10</mn><mrow id="S1.p4.6.m6.1.1.3" xref="S1.p4.6.m6.1.1.3.cmml"><mo id="S1.p4.6.m6.1.1.3a" xref="S1.p4.6.m6.1.1.3.cmml">−</mo><mn id="S1.p4.6.m6.1.1.3.2" xref="S1.p4.6.m6.1.1.3.2.cmml">14</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S1.p4.6.m6.1b"><apply id="S1.p4.6.m6.1.1.cmml" xref="S1.p4.6.m6.1.1"><csymbol cd="ambiguous" id="S1.p4.6.m6.1.1.1.cmml" xref="S1.p4.6.m6.1.1">superscript</csymbol><cn id="S1.p4.6.m6.1.1.2.cmml" type="integer" xref="S1.p4.6.m6.1.1.2">10</cn><apply id="S1.p4.6.m6.1.1.3.cmml" xref="S1.p4.6.m6.1.1.3"><minus id="S1.p4.6.m6.1.1.3.1.cmml" xref="S1.p4.6.m6.1.1.3"></minus><cn id="S1.p4.6.m6.1.1.3.2.cmml" type="integer" xref="S1.p4.6.m6.1.1.3.2">14</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.6.m6.1c">10^{-14}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.6.m6.1d">10 start_POSTSUPERSCRIPT - 14 end_POSTSUPERSCRIPT</annotation></semantics></math>m s<sup class="ltx_sup" id="S1.p4.16.3"><span class="ltx_text ltx_font_italic" id="S1.p4.16.3.1">-2</span></sup> Hz<sup class="ltx_sup" id="S1.p4.16.4"><span class="ltx_text ltx_font_italic" id="S1.p4.16.4.1">-1/2</span></sup> at the bottom of the LISA band of <math alttext="10^{-4}" class="ltx_Math" display="inline" id="S1.p4.9.m9.1"><semantics id="S1.p4.9.m9.1a"><msup id="S1.p4.9.m9.1.1" xref="S1.p4.9.m9.1.1.cmml"><mn id="S1.p4.9.m9.1.1.2" xref="S1.p4.9.m9.1.1.2.cmml">10</mn><mrow id="S1.p4.9.m9.1.1.3" xref="S1.p4.9.m9.1.1.3.cmml"><mo id="S1.p4.9.m9.1.1.3a" xref="S1.p4.9.m9.1.1.3.cmml">−</mo><mn id="S1.p4.9.m9.1.1.3.2" xref="S1.p4.9.m9.1.1.3.2.cmml">4</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S1.p4.9.m9.1b"><apply id="S1.p4.9.m9.1.1.cmml" xref="S1.p4.9.m9.1.1"><csymbol cd="ambiguous" id="S1.p4.9.m9.1.1.1.cmml" xref="S1.p4.9.m9.1.1">superscript</csymbol><cn id="S1.p4.9.m9.1.1.2.cmml" type="integer" xref="S1.p4.9.m9.1.1.2">10</cn><apply id="S1.p4.9.m9.1.1.3.cmml" xref="S1.p4.9.m9.1.1.3"><minus id="S1.p4.9.m9.1.1.3.1.cmml" xref="S1.p4.9.m9.1.1.3"></minus><cn id="S1.p4.9.m9.1.1.3.2.cmml" type="integer" xref="S1.p4.9.m9.1.1.3.2">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.9.m9.1c">10^{-4}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.9.m9.1d">10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT</annotation></semantics></math>Hz. LISAmax more conservatively takes the same acceleration control specifications as the LISA mission, allowing it to achieve <math alttext="100\times" class="ltx_math_unparsed" display="inline" id="S1.p4.10.m10.1"><semantics id="S1.p4.10.m10.1a"><mrow id="S1.p4.10.m10.1b"><mn id="S1.p4.10.m10.1.1">100</mn><mo id="S1.p4.10.m10.1.2" lspace="0.222em">×</mo></mrow><annotation encoding="application/x-tex" id="S1.p4.10.m10.1c">100\times</annotation><annotation encoding="application/x-llamapun" id="S1.p4.10.m10.1d">100 ×</annotation></semantics></math> improved sensitivity over LISA owing to the longer arms. LISAmax additionally extrapolates below the LISA band to <math alttext="1~{}\mu" class="ltx_Math" display="inline" id="S1.p4.11.m11.1"><semantics id="S1.p4.11.m11.1a"><mrow id="S1.p4.11.m11.1.1" xref="S1.p4.11.m11.1.1.cmml"><mn id="S1.p4.11.m11.1.1.2" xref="S1.p4.11.m11.1.1.2.cmml">1</mn><mo id="S1.p4.11.m11.1.1.1" lspace="0.330em" xref="S1.p4.11.m11.1.1.1.cmml">⁢</mo><mi id="S1.p4.11.m11.1.1.3" xref="S1.p4.11.m11.1.1.3.cmml">μ</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.11.m11.1b"><apply id="S1.p4.11.m11.1.1.cmml" xref="S1.p4.11.m11.1.1"><times id="S1.p4.11.m11.1.1.1.cmml" xref="S1.p4.11.m11.1.1.1"></times><cn id="S1.p4.11.m11.1.1.2.cmml" type="integer" xref="S1.p4.11.m11.1.1.2">1</cn><ci id="S1.p4.11.m11.1.1.3.cmml" xref="S1.p4.11.m11.1.1.3">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.11.m11.1c">1~{}\mu</annotation><annotation encoding="application/x-llamapun" id="S1.p4.11.m11.1d">1 italic_μ</annotation></semantics></math>Hz assuming <math alttext="f^{-2}" class="ltx_Math" display="inline" id="S1.p4.12.m12.1"><semantics id="S1.p4.12.m12.1a"><msup id="S1.p4.12.m12.1.1" xref="S1.p4.12.m12.1.1.cmml"><mi id="S1.p4.12.m12.1.1.2" xref="S1.p4.12.m12.1.1.2.cmml">f</mi><mrow id="S1.p4.12.m12.1.1.3" xref="S1.p4.12.m12.1.1.3.cmml"><mo id="S1.p4.12.m12.1.1.3a" xref="S1.p4.12.m12.1.1.3.cmml">−</mo><mn id="S1.p4.12.m12.1.1.3.2" xref="S1.p4.12.m12.1.1.3.2.cmml">2</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S1.p4.12.m12.1b"><apply id="S1.p4.12.m12.1.1.cmml" xref="S1.p4.12.m12.1.1"><csymbol cd="ambiguous" id="S1.p4.12.m12.1.1.1.cmml" xref="S1.p4.12.m12.1.1">superscript</csymbol><ci id="S1.p4.12.m12.1.1.2.cmml" xref="S1.p4.12.m12.1.1.2">𝑓</ci><apply id="S1.p4.12.m12.1.1.3.cmml" xref="S1.p4.12.m12.1.1.3"><minus id="S1.p4.12.m12.1.1.3.1.cmml" xref="S1.p4.12.m12.1.1.3"></minus><cn id="S1.p4.12.m12.1.1.3.2.cmml" type="integer" xref="S1.p4.12.m12.1.1.3.2">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.12.m12.1c">f^{-2}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.12.m12.1d">italic_f start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_citep">(Martens et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib48" title="">2023</a>)</cite>. However, there are some acceleration sources for the LISA accelerometer that become important at <math alttext="\sim 2\times 10^{-5}" class="ltx_Math" display="inline" id="S1.p4.13.m13.1"><semantics id="S1.p4.13.m13.1a"><mrow id="S1.p4.13.m13.1.1" xref="S1.p4.13.m13.1.1.cmml"><mi id="S1.p4.13.m13.1.1.2" xref="S1.p4.13.m13.1.1.2.cmml"></mi><mo id="S1.p4.13.m13.1.1.1" xref="S1.p4.13.m13.1.1.1.cmml">∼</mo><mrow id="S1.p4.13.m13.1.1.3" xref="S1.p4.13.m13.1.1.3.cmml"><mn id="S1.p4.13.m13.1.1.3.2" xref="S1.p4.13.m13.1.1.3.2.cmml">2</mn><mo id="S1.p4.13.m13.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S1.p4.13.m13.1.1.3.1.cmml">×</mo><msup id="S1.p4.13.m13.1.1.3.3" xref="S1.p4.13.m13.1.1.3.3.cmml"><mn id="S1.p4.13.m13.1.1.3.3.2" xref="S1.p4.13.m13.1.1.3.3.2.cmml">10</mn><mrow id="S1.p4.13.m13.1.1.3.3.3" xref="S1.p4.13.m13.1.1.3.3.3.cmml"><mo id="S1.p4.13.m13.1.1.3.3.3a" xref="S1.p4.13.m13.1.1.3.3.3.cmml">−</mo><mn id="S1.p4.13.m13.1.1.3.3.3.2" xref="S1.p4.13.m13.1.1.3.3.3.2.cmml">5</mn></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.13.m13.1b"><apply id="S1.p4.13.m13.1.1.cmml" xref="S1.p4.13.m13.1.1"><csymbol cd="latexml" id="S1.p4.13.m13.1.1.1.cmml" xref="S1.p4.13.m13.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S1.p4.13.m13.1.1.2.cmml" xref="S1.p4.13.m13.1.1.2">absent</csymbol><apply id="S1.p4.13.m13.1.1.3.cmml" xref="S1.p4.13.m13.1.1.3"><times id="S1.p4.13.m13.1.1.3.1.cmml" xref="S1.p4.13.m13.1.1.3.1"></times><cn id="S1.p4.13.m13.1.1.3.2.cmml" type="integer" xref="S1.p4.13.m13.1.1.3.2">2</cn><apply id="S1.p4.13.m13.1.1.3.3.cmml" xref="S1.p4.13.m13.1.1.3.3"><csymbol cd="ambiguous" id="S1.p4.13.m13.1.1.3.3.1.cmml" xref="S1.p4.13.m13.1.1.3.3">superscript</csymbol><cn id="S1.p4.13.m13.1.1.3.3.2.cmml" type="integer" xref="S1.p4.13.m13.1.1.3.3.2">10</cn><apply id="S1.p4.13.m13.1.1.3.3.3.cmml" xref="S1.p4.13.m13.1.1.3.3.3"><minus id="S1.p4.13.m13.1.1.3.3.3.1.cmml" xref="S1.p4.13.m13.1.1.3.3.3"></minus><cn id="S1.p4.13.m13.1.1.3.3.3.2.cmml" type="integer" xref="S1.p4.13.m13.1.1.3.3.3.2">5</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.13.m13.1c">\sim 2\times 10^{-5}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.13.m13.1d">∼ 2 × 10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT</annotation></semantics></math>Hz and that scale much more strongly than <math alttext="f^{-2}" class="ltx_Math" display="inline" id="S1.p4.14.m14.1"><semantics id="S1.p4.14.m14.1a"><msup id="S1.p4.14.m14.1.1" xref="S1.p4.14.m14.1.1.cmml"><mi id="S1.p4.14.m14.1.1.2" xref="S1.p4.14.m14.1.1.2.cmml">f</mi><mrow id="S1.p4.14.m14.1.1.3" xref="S1.p4.14.m14.1.1.3.cmml"><mo id="S1.p4.14.m14.1.1.3a" xref="S1.p4.14.m14.1.1.3.cmml">−</mo><mn id="S1.p4.14.m14.1.1.3.2" xref="S1.p4.14.m14.1.1.3.2.cmml">2</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S1.p4.14.m14.1b"><apply id="S1.p4.14.m14.1.1.cmml" xref="S1.p4.14.m14.1.1"><csymbol cd="ambiguous" id="S1.p4.14.m14.1.1.1.cmml" xref="S1.p4.14.m14.1.1">superscript</csymbol><ci id="S1.p4.14.m14.1.1.2.cmml" xref="S1.p4.14.m14.1.1.2">𝑓</ci><apply id="S1.p4.14.m14.1.1.3.cmml" xref="S1.p4.14.m14.1.1.3"><minus id="S1.p4.14.m14.1.1.3.1.cmml" xref="S1.p4.14.m14.1.1.3"></minus><cn id="S1.p4.14.m14.1.1.3.2.cmml" type="integer" xref="S1.p4.14.m14.1.1.3.2">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.14.m14.1c">f^{-2}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.14.m14.1d">italic_f start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT</annotation></semantics></math> to lower frequencies <cite class="ltx_cite ltx_citemacro_citep">(e.g. Mueller et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib51" title="">2019</a>)</cite>. Concerned that substantial development in acceleration control would be required for space-interferometers to probe the <math alttext="\mu" class="ltx_Math" display="inline" id="S1.p4.15.m15.1"><semantics id="S1.p4.15.m15.1a"><mi id="S1.p4.15.m15.1.1" xref="S1.p4.15.m15.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S1.p4.15.m15.1b"><ci id="S1.p4.15.m15.1.1.cmml" xref="S1.p4.15.m15.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.15.m15.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S1.p4.15.m15.1d">italic_μ</annotation></semantics></math>Hz band, <cite class="ltx_cite ltx_citemacro_citet">Fedderke et al. (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib28" title="">2022a</a>)</cite> considered the possibility of instead establishing stations on two asteroids with orbits around <math alttext="1" class="ltx_Math" display="inline" id="S1.p4.16.m16.1"><semantics id="S1.p4.16.m16.1a"><mn id="S1.p4.16.m16.1.1" xref="S1.p4.16.m16.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S1.p4.16.m16.1b"><cn id="S1.p4.16.m16.1.1.cmml" type="integer" xref="S1.p4.16.m16.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.16.m16.1c">1</annotation><annotation encoding="application/x-llamapun" id="S1.p4.16.m16.1d">1</annotation></semantics></math> AU and carefully measuring their relative distance. Because of their large masses, the asteroids would behave as excellent test masses, avoiding the need for precise acceleration control.</p> </div> <div class="ltx_para" id="S1.p5"> <p class="ltx_p" id="S1.p5.1">Here we consider another method to avoid on-board acceleration monitoring – employing spacecraft further out in the Solar System, reaching distances and inter-spacecraft separations of tens of astronomical units. Abandoning drag-free control was considered by <cite class="ltx_cite ltx_citemacro_citet">McKenzie et al. (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib49" title="">2011</a>)</cite> and <cite class="ltx_cite ltx_citemacro_citet">Folkner (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib30" title="">2011</a>)</cite> in the context of a LISA-like mission. The outer Solar System application we consider results in a potentially massive reduction in acceleration sources, as the solar irradiance variations and the solar wind fall off as <math alttext="r^{-2}" class="ltx_Math" display="inline" id="S1.p5.1.m1.1"><semantics id="S1.p5.1.m1.1a"><msup id="S1.p5.1.m1.1.1" xref="S1.p5.1.m1.1.1.cmml"><mi id="S1.p5.1.m1.1.1.2" xref="S1.p5.1.m1.1.1.2.cmml">r</mi><mrow id="S1.p5.1.m1.1.1.3" xref="S1.p5.1.m1.1.1.3.cmml"><mo id="S1.p5.1.m1.1.1.3a" xref="S1.p5.1.m1.1.1.3.cmml">−</mo><mn id="S1.p5.1.m1.1.1.3.2" xref="S1.p5.1.m1.1.1.3.2.cmml">2</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S1.p5.1.m1.1b"><apply id="S1.p5.1.m1.1.1.cmml" xref="S1.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S1.p5.1.m1.1.1.1.cmml" xref="S1.p5.1.m1.1.1">superscript</csymbol><ci id="S1.p5.1.m1.1.1.2.cmml" xref="S1.p5.1.m1.1.1.2">𝑟</ci><apply id="S1.p5.1.m1.1.1.3.cmml" xref="S1.p5.1.m1.1.1.3"><minus id="S1.p5.1.m1.1.1.3.1.cmml" xref="S1.p5.1.m1.1.1.3"></minus><cn id="S1.p5.1.m1.1.1.3.2.cmml" type="integer" xref="S1.p5.1.m1.1.1.3.2">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.1.m1.1c">r^{-2}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.1.m1.1d">italic_r start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT</annotation></semantics></math> with distance from the Sun. Radial accelerations could be further suppressed by placing the spacecraft such that their arm-links are nearly perpendicular to the radial direction between the Sun and the individual spacecraft. Arms over which the gravitational wave signal is measured can be oriented perpendicular to the spacecraft-Sun direction to further suppress these largely radial accelerations <cite class="ltx_cite ltx_citemacro_citep">(McKenzie et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib49" title="">2011</a>)</cite>. Finally, the longer baselines of our outer Solar System concept may relax other system specifications: When fixing the strain noise power due to shot noise at long wavelengths, the timing error, displacement error, and square of the angular pointing error are relaxed in proportion to the baseline distance.</p> </div> <div class="ltx_para" id="S1.p6"> <p class="ltx_p" id="S1.p6.2">A drawback of such long baselines is that the electromagnetic transmissions between spacecraft would be weak. However, we argue that even for spacecraft that are separated by several tens of astronomical units, Watt-scale electromagnetic transmissions are still sufficiently strong to achieve a stable phase lock. Another concern is that only meager <math alttext="\sim 10\," class="ltx_Math" display="inline" id="S1.p6.1.m1.1"><semantics id="S1.p6.1.m1.1a"><mrow id="S1.p6.1.m1.1.1" xref="S1.p6.1.m1.1.1.cmml"><mi id="S1.p6.1.m1.1.1.2" xref="S1.p6.1.m1.1.1.2.cmml"></mi><mo id="S1.p6.1.m1.1.1.1" xref="S1.p6.1.m1.1.1.1.cmml">∼</mo><mn id="S1.p6.1.m1.1.1.3" xref="S1.p6.1.m1.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.1.m1.1b"><apply id="S1.p6.1.m1.1.1.cmml" xref="S1.p6.1.m1.1.1"><csymbol cd="latexml" id="S1.p6.1.m1.1.1.1.cmml" xref="S1.p6.1.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S1.p6.1.m1.1.1.2.cmml" xref="S1.p6.1.m1.1.1.2">absent</csymbol><cn id="S1.p6.1.m1.1.1.3.cmml" type="integer" xref="S1.p6.1.m1.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.1.m1.1c">\sim 10\,</annotation><annotation encoding="application/x-llamapun" id="S1.p6.1.m1.1d">∼ 10</annotation></semantics></math>kbps downlinks have been achieved to spacecraft in the outer Solar System. Fortunately, only a single phase measurement for every hour of data may be required because of the low frequencies of interest such that an hour per month of <math alttext="\sim 10\," class="ltx_Math" display="inline" id="S1.p6.2.m2.1"><semantics id="S1.p6.2.m2.1a"><mrow id="S1.p6.2.m2.1.1" xref="S1.p6.2.m2.1.1.cmml"><mi id="S1.p6.2.m2.1.1.2" xref="S1.p6.2.m2.1.1.2.cmml"></mi><mo id="S1.p6.2.m2.1.1.1" xref="S1.p6.2.m2.1.1.1.cmml">∼</mo><mn id="S1.p6.2.m2.1.1.3" xref="S1.p6.2.m2.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.2.m2.1b"><apply id="S1.p6.2.m2.1.1.cmml" xref="S1.p6.2.m2.1.1"><csymbol cd="latexml" id="S1.p6.2.m2.1.1.1.cmml" xref="S1.p6.2.m2.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S1.p6.2.m2.1.1.2.cmml" xref="S1.p6.2.m2.1.1.2">absent</csymbol><cn id="S1.p6.2.m2.1.1.3.cmml" type="integer" xref="S1.p6.2.m2.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.2.m2.1c">\sim 10\,</annotation><annotation encoding="application/x-llamapun" id="S1.p6.2.m2.1d">∼ 10</annotation></semantics></math>kbps downlinks would likely be sufficient.</p> </div> <div class="ltx_para" id="S1.p7"> <p class="ltx_p" id="S1.p7.4">This paper also considers an additional optimization, using radio dishes rather than lasers to measure spacecraft separations. One difficulty with using lasers pertains to the spacecraft relative velocities: larger relative velocities mean larger differences in the interfering frequencies, <math alttext="\Delta f" class="ltx_Math" display="inline" id="S1.p7.1.m1.1"><semantics id="S1.p7.1.m1.1a"><mrow id="S1.p7.1.m1.1.1" xref="S1.p7.1.m1.1.1.cmml"><mi id="S1.p7.1.m1.1.1.2" mathvariant="normal" xref="S1.p7.1.m1.1.1.2.cmml">Δ</mi><mo id="S1.p7.1.m1.1.1.1" xref="S1.p7.1.m1.1.1.1.cmml">⁢</mo><mi id="S1.p7.1.m1.1.1.3" xref="S1.p7.1.m1.1.1.3.cmml">f</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p7.1.m1.1b"><apply id="S1.p7.1.m1.1.1.cmml" xref="S1.p7.1.m1.1.1"><times id="S1.p7.1.m1.1.1.1.cmml" xref="S1.p7.1.m1.1.1.1"></times><ci id="S1.p7.1.m1.1.1.2.cmml" xref="S1.p7.1.m1.1.1.2">Δ</ci><ci id="S1.p7.1.m1.1.1.3.cmml" xref="S1.p7.1.m1.1.1.3">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p7.1.m1.1c">\Delta f</annotation><annotation encoding="application/x-llamapun" id="S1.p7.1.m1.1d">roman_Δ italic_f</annotation></semantics></math>. As phase errors scale with timing errors <math alttext="\delta t" class="ltx_Math" display="inline" id="S1.p7.2.m2.1"><semantics id="S1.p7.2.m2.1a"><mrow id="S1.p7.2.m2.1.1" xref="S1.p7.2.m2.1.1.cmml"><mi id="S1.p7.2.m2.1.1.2" xref="S1.p7.2.m2.1.1.2.cmml">δ</mi><mo id="S1.p7.2.m2.1.1.1" xref="S1.p7.2.m2.1.1.1.cmml">⁢</mo><mi id="S1.p7.2.m2.1.1.3" xref="S1.p7.2.m2.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p7.2.m2.1b"><apply id="S1.p7.2.m2.1.1.cmml" xref="S1.p7.2.m2.1.1"><times id="S1.p7.2.m2.1.1.1.cmml" xref="S1.p7.2.m2.1.1.1"></times><ci id="S1.p7.2.m2.1.1.2.cmml" xref="S1.p7.2.m2.1.1.2">𝛿</ci><ci id="S1.p7.2.m2.1.1.3.cmml" xref="S1.p7.2.m2.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p7.2.m2.1c">\delta t</annotation><annotation encoding="application/x-llamapun" id="S1.p7.2.m2.1d">italic_δ italic_t</annotation></semantics></math> as <math alttext="\delta\phi\sim\Delta f\delta t" class="ltx_Math" display="inline" id="S1.p7.3.m3.1"><semantics id="S1.p7.3.m3.1a"><mrow id="S1.p7.3.m3.1.1" xref="S1.p7.3.m3.1.1.cmml"><mrow id="S1.p7.3.m3.1.1.2" xref="S1.p7.3.m3.1.1.2.cmml"><mi id="S1.p7.3.m3.1.1.2.2" xref="S1.p7.3.m3.1.1.2.2.cmml">δ</mi><mo id="S1.p7.3.m3.1.1.2.1" xref="S1.p7.3.m3.1.1.2.1.cmml">⁢</mo><mi id="S1.p7.3.m3.1.1.2.3" xref="S1.p7.3.m3.1.1.2.3.cmml">ϕ</mi></mrow><mo id="S1.p7.3.m3.1.1.1" xref="S1.p7.3.m3.1.1.1.cmml">∼</mo><mrow id="S1.p7.3.m3.1.1.3" xref="S1.p7.3.m3.1.1.3.cmml"><mi id="S1.p7.3.m3.1.1.3.2" mathvariant="normal" xref="S1.p7.3.m3.1.1.3.2.cmml">Δ</mi><mo id="S1.p7.3.m3.1.1.3.1" xref="S1.p7.3.m3.1.1.3.1.cmml">⁢</mo><mi id="S1.p7.3.m3.1.1.3.3" xref="S1.p7.3.m3.1.1.3.3.cmml">f</mi><mo id="S1.p7.3.m3.1.1.3.1a" xref="S1.p7.3.m3.1.1.3.1.cmml">⁢</mo><mi id="S1.p7.3.m3.1.1.3.4" xref="S1.p7.3.m3.1.1.3.4.cmml">δ</mi><mo id="S1.p7.3.m3.1.1.3.1b" xref="S1.p7.3.m3.1.1.3.1.cmml">⁢</mo><mi id="S1.p7.3.m3.1.1.3.5" xref="S1.p7.3.m3.1.1.3.5.cmml">t</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p7.3.m3.1b"><apply id="S1.p7.3.m3.1.1.cmml" xref="S1.p7.3.m3.1.1"><csymbol cd="latexml" id="S1.p7.3.m3.1.1.1.cmml" xref="S1.p7.3.m3.1.1.1">similar-to</csymbol><apply id="S1.p7.3.m3.1.1.2.cmml" xref="S1.p7.3.m3.1.1.2"><times id="S1.p7.3.m3.1.1.2.1.cmml" xref="S1.p7.3.m3.1.1.2.1"></times><ci id="S1.p7.3.m3.1.1.2.2.cmml" xref="S1.p7.3.m3.1.1.2.2">𝛿</ci><ci id="S1.p7.3.m3.1.1.2.3.cmml" xref="S1.p7.3.m3.1.1.2.3">italic-ϕ</ci></apply><apply id="S1.p7.3.m3.1.1.3.cmml" xref="S1.p7.3.m3.1.1.3"><times id="S1.p7.3.m3.1.1.3.1.cmml" xref="S1.p7.3.m3.1.1.3.1"></times><ci id="S1.p7.3.m3.1.1.3.2.cmml" xref="S1.p7.3.m3.1.1.3.2">Δ</ci><ci id="S1.p7.3.m3.1.1.3.3.cmml" xref="S1.p7.3.m3.1.1.3.3">𝑓</ci><ci id="S1.p7.3.m3.1.1.3.4.cmml" xref="S1.p7.3.m3.1.1.3.4">𝛿</ci><ci id="S1.p7.3.m3.1.1.3.5.cmml" xref="S1.p7.3.m3.1.1.3.5">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p7.3.m3.1c">\delta\phi\sim\Delta f\delta t</annotation><annotation encoding="application/x-llamapun" id="S1.p7.3.m3.1d">italic_δ italic_ϕ ∼ roman_Δ italic_f italic_δ italic_t</annotation></semantics></math>, many of the spacecraft design tolerances would be set by the magnitude of this frequency difference. While the changes in velocity may be slow enough that small frequency differentials can be achieved by periodically tuning the frequencies with small adjustments to the laser cavity proprieties, the radio avoids this difficulty completely. Other advantages of the radio include being insensitive to intensity variations in the transmission as well as relaxed pointing requirements. However, using radio broadcasts rather than lasers is potentially much less sensitive to gravitational waves due to the <math alttext="\sim 10^{4}" class="ltx_Math" display="inline" id="S1.p7.4.m4.1"><semantics id="S1.p7.4.m4.1a"><mrow id="S1.p7.4.m4.1.1" xref="S1.p7.4.m4.1.1.cmml"><mi id="S1.p7.4.m4.1.1.2" xref="S1.p7.4.m4.1.1.2.cmml"></mi><mo id="S1.p7.4.m4.1.1.1" xref="S1.p7.4.m4.1.1.1.cmml">∼</mo><msup id="S1.p7.4.m4.1.1.3" xref="S1.p7.4.m4.1.1.3.cmml"><mn id="S1.p7.4.m4.1.1.3.2" xref="S1.p7.4.m4.1.1.3.2.cmml">10</mn><mn id="S1.p7.4.m4.1.1.3.3" xref="S1.p7.4.m4.1.1.3.3.cmml">4</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.p7.4.m4.1b"><apply id="S1.p7.4.m4.1.1.cmml" xref="S1.p7.4.m4.1.1"><csymbol cd="latexml" id="S1.p7.4.m4.1.1.1.cmml" xref="S1.p7.4.m4.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S1.p7.4.m4.1.1.2.cmml" xref="S1.p7.4.m4.1.1.2">absent</csymbol><apply id="S1.p7.4.m4.1.1.3.cmml" xref="S1.p7.4.m4.1.1.3"><csymbol cd="ambiguous" id="S1.p7.4.m4.1.1.3.1.cmml" xref="S1.p7.4.m4.1.1.3">superscript</csymbol><cn id="S1.p7.4.m4.1.1.3.2.cmml" type="integer" xref="S1.p7.4.m4.1.1.3.2">10</cn><cn id="S1.p7.4.m4.1.1.3.3.cmml" type="integer" xref="S1.p7.4.m4.1.1.3.3">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p7.4.m4.1c">\sim 10^{4}</annotation><annotation encoding="application/x-llamapun" id="S1.p7.4.m4.1d">∼ 10 start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT</annotation></semantics></math> times longer wavelengths. We show that a radio instrument can still be sufficiently sensitive that acceleration noise (which is insensitive to the transmission wavelength) is dominant over much of the gravitational waveband of interest. Even at the higher gravitational wave frequencies at which lasers are more sensitive, we show that a radio dish design could be sufficiently sensitive that the noise is set by confusion from stochastic gravitational wave backgrounds rather than radiometer noise. Another concern with a radio effort is that plasma dispersion would contaminate the measured phases. We show that this noise can be essentially eliminated using two frequency channels and at only moderate cost to the sensitivity.</p> </div> <div class="ltx_para" id="S1.p8"> <p class="ltx_p" id="S1.p8.1">A radio design may be more easily added to other outer Solar System spacecraft, which often already include a relatively large high-gain antenna for telemetry.<span class="ltx_note ltx_role_footnote" id="footnote1"><sup class="ltx_note_mark">1</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">1</sup><span class="ltx_tag ltx_tag_note">1</span>The radio-dish design we propose to measure gravitational waves could plausibly also execute <math alttext="\sim 10\;" class="ltx_Math" display="inline" id="footnote1.m1.1"><semantics id="footnote1.m1.1b"><mrow id="footnote1.m1.1.1" xref="footnote1.m1.1.1.cmml"><mi id="footnote1.m1.1.1.2" xref="footnote1.m1.1.1.2.cmml"></mi><mo id="footnote1.m1.1.1.1" xref="footnote1.m1.1.1.1.cmml">∼</mo><mn id="footnote1.m1.1.1.3" xref="footnote1.m1.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="footnote1.m1.1c"><apply id="footnote1.m1.1.1.cmml" xref="footnote1.m1.1.1"><csymbol cd="latexml" id="footnote1.m1.1.1.1.cmml" xref="footnote1.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="footnote1.m1.1.1.2.cmml" xref="footnote1.m1.1.1.2">absent</csymbol><cn id="footnote1.m1.1.1.3.cmml" type="integer" xref="footnote1.m1.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote1.m1.1d">\sim 10\;</annotation><annotation encoding="application/x-llamapun" id="footnote1.m1.1e">∼ 10</annotation></semantics></math>AU very long baseline interferometry to radio sources (particularly fast radio bursts) as has been proposed for measuring cosmic distances and dark matter structure in <cite class="ltx_cite ltx_citemacro_citet">Boone & McQuinn (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib18" title="">2023</a>)</cite> and <cite class="ltx_cite ltx_citemacro_citet">Xiao et al. (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib74" title="">2024</a>)</cite>. Additionally, this concept may also be able to use the timing of Fast Radio Bursts repetitions to measure gravitational waves over frequencies similar to those considered here <cite class="ltx_cite ltx_citemacro_citep">(Lu et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib46" title="">2024</a>)</cite>.</span></span></span> Indeed, there is a long history of using radio broadcasts to track spacecraft’s velocities and probe gravitational waves <cite class="ltx_cite ltx_citemacro_citep">(for a review see Armstrong, <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib8" title="">2006a</a>)</cite>. There has been recent interest in Doppler tracking in the context of a future outer Solar System mission, where it has been suggested that a large improvement in sensitivity may be possible, pushing Doppler tracking into a regime where it can detect anticipated astrophysical sources <cite class="ltx_cite ltx_citemacro_citep">(Zwick et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib75" title="">2024</a>)</cite>. We show that reaching the sensitivity benchmarks in <cite class="ltx_cite ltx_citemacro_citet">Zwick et al. (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib75" title="">2024</a>)</cite> at <math alttext="f\lesssim 10^{-6}" class="ltx_Math" display="inline" id="S1.p8.1.m1.1"><semantics id="S1.p8.1.m1.1a"><mrow id="S1.p8.1.m1.1.1" xref="S1.p8.1.m1.1.1.cmml"><mi id="S1.p8.1.m1.1.1.2" xref="S1.p8.1.m1.1.1.2.cmml">f</mi><mo id="S1.p8.1.m1.1.1.1" xref="S1.p8.1.m1.1.1.1.cmml">≲</mo><msup id="S1.p8.1.m1.1.1.3" xref="S1.p8.1.m1.1.1.3.cmml"><mn id="S1.p8.1.m1.1.1.3.2" xref="S1.p8.1.m1.1.1.3.2.cmml">10</mn><mrow id="S1.p8.1.m1.1.1.3.3" xref="S1.p8.1.m1.1.1.3.3.cmml"><mo id="S1.p8.1.m1.1.1.3.3a" xref="S1.p8.1.m1.1.1.3.3.cmml">−</mo><mn id="S1.p8.1.m1.1.1.3.3.2" xref="S1.p8.1.m1.1.1.3.3.2.cmml">6</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.p8.1.m1.1b"><apply id="S1.p8.1.m1.1.1.cmml" xref="S1.p8.1.m1.1.1"><csymbol cd="latexml" id="S1.p8.1.m1.1.1.1.cmml" xref="S1.p8.1.m1.1.1.1">less-than-or-similar-to</csymbol><ci id="S1.p8.1.m1.1.1.2.cmml" xref="S1.p8.1.m1.1.1.2">𝑓</ci><apply id="S1.p8.1.m1.1.1.3.cmml" xref="S1.p8.1.m1.1.1.3"><csymbol cd="ambiguous" id="S1.p8.1.m1.1.1.3.1.cmml" xref="S1.p8.1.m1.1.1.3">superscript</csymbol><cn id="S1.p8.1.m1.1.1.3.2.cmml" type="integer" xref="S1.p8.1.m1.1.1.3.2">10</cn><apply id="S1.p8.1.m1.1.1.3.3.cmml" xref="S1.p8.1.m1.1.1.3.3"><minus id="S1.p8.1.m1.1.1.3.3.1.cmml" xref="S1.p8.1.m1.1.1.3.3"></minus><cn id="S1.p8.1.m1.1.1.3.3.2.cmml" type="integer" xref="S1.p8.1.m1.1.1.3.3.2">6</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p8.1.m1.1c">f\lesssim 10^{-6}</annotation><annotation encoding="application/x-llamapun" id="S1.p8.1.m1.1d">italic_f ≲ 10 start_POSTSUPERSCRIPT - 6 end_POSTSUPERSCRIPT</annotation></semantics></math>Hz would likely require on-board instrumentation that corrects for the accelerations from solar radiation and the interplanetary plasma.</p> </div> <div class="ltx_para" id="S1.p9"> <p class="ltx_p" id="S1.p9.1">This paper is organized as follows. Section <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2" title="2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">2</span></a> discusses the radiometer and acceleration noise sources that are likely to shape the sensitivity of the proposed concept. Section <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S3" title="3 Strain power sensitivity ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">3</span></a> uses these estimates to predict the concept’s gravitational wave sensitivity, where we consider the three general mission architectures illustrated in Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S1.F1" title="Figure 1 ‣ 1 Introduction ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">1</span></a>. Section <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S4" title="4 Instrumental considerations ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">4</span></a> elaborates on some of the instrumental considerations that are most relevant. The appendices discuss the effects of the interplanetary plasma on the phase timing of radio waves, considering dispersion (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#A1" title="Appendix A Dispersion ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">A</span></a>) and refraction (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#A2" title="Appendix B Refraction and diffraction ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">B</span></a>), and also consider the downlink data rates that the concept would require (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#A3" title="Appendix C Data rates ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">C</span></a>).</p> </div> <div class="ltx_para" id="S1.p10"> <p class="ltx_p" id="S1.p10.1">Unless stated otherwise, 1D power spectra are always half-bandwidth power spectra. As both electromagnetic and gravitational signal frequencies appear in our calculations, to distinguish them, we use wavelengths when referring to electromagnetic transmissions that are sent along arms, and we use frequencies when referring to gravitational wave signals and their potential noise sources. We use Gaussian conventions for electromagnetic quantities such as electron charge (although we do use volt rather than statvolt).</p> </div> <figure class="ltx_figure" id="S1.F1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="430" id="S1.F1.g1" src="x1.png" width="747"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 1: </span> Illustration of the three different architectures considered in this paper: (1) A two-arm design that relies on time-delay interferometry, (2) a single-arm design that relies on a precise atomic clock, and (3) an Earth-anchored Doppler tracking design. Our primary focus is on the two-arm time-delay interferometry concept, but we discuss the clock requirements to achieve a comparable sensitivity with a single arm. We contrast the sensitivity of these two designs with the traditional Doppler tracking to outer Solar System spacecraft in § <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S3.SS3" title="3.3 Doppler tracking forecasts ‣ 3 Strain power sensitivity ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">3.3</span></a>. For the calculations in this paper, we consider designs with orbits/baselines of 10 AU and 30 AU. As a conceptual reference comparison, Saturn orbits between approximately 9-10 AU, Uranus between approximately 18-20 AU, and Pluto between approximately 30-50 AU.</figcaption> </figure> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">2 </span>Strain noise sources</h2> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.1">This section considers the different noise sources that set the sensitivity. We first consider clock errors and their avoidance through time-delay interferometry (§ <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.SS1" title="2.1 Clock noise and its mitigation ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">2.1</span></a>), then discuss errors related to the strength of the electromagnetic beams (§ <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.SS2" title="2.2 Radiometer noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">2.2</span></a>), and finally discuss acceleration errors (§ <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.SS3" title="2.3 Acceleration noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">2.3</span></a>). These sources are then used to calculate the gravitational wave sensitivity of our concepts in § <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S3" title="3 Strain power sensitivity ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">3</span></a>.</p> </div> <section class="ltx_subsection" id="S2.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.1 </span>Clock noise and its mitigation</h3> <section class="ltx_paragraph" id="S2.SS1.SSS0.Px1"> <h5 class="ltx_title ltx_font_bold ltx_title_paragraph">Single-arm and an atomic clock</h5> <div class="ltx_para" id="S2.SS1.SSS0.Px1.p1"> <p class="ltx_p" id="S2.SS1.SSS0.Px1.p1.2">Let us first consider a one-arm configuration in which a monochromatic light wave with phase <math alttext="\phi_{\rm os}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.1.m1.1"><semantics id="S2.SS1.SSS0.Px1.p1.1.m1.1a"><msub id="S2.SS1.SSS0.Px1.p1.1.m1.1.1" xref="S2.SS1.SSS0.Px1.p1.1.m1.1.1.cmml"><mi id="S2.SS1.SSS0.Px1.p1.1.m1.1.1.2" xref="S2.SS1.SSS0.Px1.p1.1.m1.1.1.2.cmml">ϕ</mi><mi id="S2.SS1.SSS0.Px1.p1.1.m1.1.1.3" xref="S2.SS1.SSS0.Px1.p1.1.m1.1.1.3.cmml">os</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p1.1.m1.1b"><apply id="S2.SS1.SSS0.Px1.p1.1.m1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px1.p1.1.m1.1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.1.m1.1.1">subscript</csymbol><ci id="S2.SS1.SSS0.Px1.p1.1.m1.1.1.2.cmml" xref="S2.SS1.SSS0.Px1.p1.1.m1.1.1.2">italic-ϕ</ci><ci id="S2.SS1.SSS0.Px1.p1.1.m1.1.1.3.cmml" xref="S2.SS1.SSS0.Px1.p1.1.m1.1.1.3">os</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.1.m1.1c">\phi_{\rm os}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.1.m1.1d">italic_ϕ start_POSTSUBSCRIPT roman_os end_POSTSUBSCRIPT</annotation></semantics></math> tied to a local oscillator or atomic clock. This wave is sent from one spacecraft to another a distance <math alttext="L_{1}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.2.m2.1"><semantics id="S2.SS1.SSS0.Px1.p1.2.m2.1a"><msub id="S2.SS1.SSS0.Px1.p1.2.m2.1.1" xref="S2.SS1.SSS0.Px1.p1.2.m2.1.1.cmml"><mi id="S2.SS1.SSS0.Px1.p1.2.m2.1.1.2" xref="S2.SS1.SSS0.Px1.p1.2.m2.1.1.2.cmml">L</mi><mn id="S2.SS1.SSS0.Px1.p1.2.m2.1.1.3" xref="S2.SS1.SSS0.Px1.p1.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p1.2.m2.1b"><apply id="S2.SS1.SSS0.Px1.p1.2.m2.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px1.p1.2.m2.1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.2.m2.1.1">subscript</csymbol><ci id="S2.SS1.SSS0.Px1.p1.2.m2.1.1.2.cmml" xref="S2.SS1.SSS0.Px1.p1.2.m2.1.1.2">𝐿</ci><cn id="S2.SS1.SSS0.Px1.p1.2.m2.1.1.3.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.2.m2.1c">L_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.2.m2.1d">italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> away and then returns. 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xref="S2.E1.m1.5.5.1.1.1.1.1.3.2">italic-ϕ</ci><ci id="S2.E1.m1.5.5.1.1.1.1.1.3.3.cmml" xref="S2.E1.m1.5.5.1.1.1.1.1.3.3">os</ci></apply><apply id="S2.E1.m1.5.5.1.1.1.1.1.1.1.1.cmml" xref="S2.E1.m1.5.5.1.1.1.1.1.1.1"><minus id="S2.E1.m1.5.5.1.1.1.1.1.1.1.1.1.cmml" xref="S2.E1.m1.5.5.1.1.1.1.1.1.1.1.1"></minus><ci id="S2.E1.m1.5.5.1.1.1.1.1.1.1.1.2.cmml" xref="S2.E1.m1.5.5.1.1.1.1.1.1.1.1.2">𝑡</ci><apply id="S2.E1.m1.5.5.1.1.1.1.1.1.1.1.3.cmml" xref="S2.E1.m1.5.5.1.1.1.1.1.1.1.1.3"><times id="S2.E1.m1.5.5.1.1.1.1.1.1.1.1.3.1.cmml" xref="S2.E1.m1.5.5.1.1.1.1.1.1.1.1.3.1"></times><cn id="S2.E1.m1.5.5.1.1.1.1.1.1.1.1.3.2.cmml" type="integer" xref="S2.E1.m1.5.5.1.1.1.1.1.1.1.1.3.2">2</cn><apply id="S2.E1.m1.5.5.1.1.1.1.1.1.1.1.3.3.cmml" xref="S2.E1.m1.5.5.1.1.1.1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S2.E1.m1.5.5.1.1.1.1.1.1.1.1.3.3.1.cmml" xref="S2.E1.m1.5.5.1.1.1.1.1.1.1.1.3.3">subscript</csymbol><ci id="S2.E1.m1.5.5.1.1.1.1.1.1.1.1.3.3.2.cmml" xref="S2.E1.m1.5.5.1.1.1.1.1.1.1.1.3.3.2">𝐿</ci><cn id="S2.E1.m1.5.5.1.1.1.1.1.1.1.1.3.3.3.cmml" type="integer" xref="S2.E1.m1.5.5.1.1.1.1.1.1.1.1.3.3.3">1</cn></apply></apply></apply></apply><apply id="S2.E1.m1.5.5.1.1.1.1.3.cmml" xref="S2.E1.m1.5.5.1.1.1.1.3"><times id="S2.E1.m1.5.5.1.1.1.1.3.1.cmml" xref="S2.E1.m1.5.5.1.1.1.1.3.1"></times><apply id="S2.E1.m1.5.5.1.1.1.1.3.2.cmml" xref="S2.E1.m1.5.5.1.1.1.1.3.2"><csymbol cd="ambiguous" id="S2.E1.m1.5.5.1.1.1.1.3.2.1.cmml" xref="S2.E1.m1.5.5.1.1.1.1.3.2">subscript</csymbol><ci id="S2.E1.m1.5.5.1.1.1.1.3.2.2.cmml" xref="S2.E1.m1.5.5.1.1.1.1.3.2.2">italic-ϕ</ci><ci id="S2.E1.m1.5.5.1.1.1.1.3.2.3.cmml" xref="S2.E1.m1.5.5.1.1.1.1.3.2.3">os</ci></apply><ci id="S2.E1.m1.3.3.cmml" xref="S2.E1.m1.3.3">𝑡</ci></apply></apply><apply id="S2.E1.m1.5.5.1.1.1.3.cmml" xref="S2.E1.m1.5.5.1.1.1.3"><times id="S2.E1.m1.5.5.1.1.1.3.1.cmml" xref="S2.E1.m1.5.5.1.1.1.3.1"></times><apply id="S2.E1.m1.5.5.1.1.1.3.2.cmml" xref="S2.E1.m1.5.5.1.1.1.3.2"><divide id="S2.E1.m1.5.5.1.1.1.3.2.1.cmml" xref="S2.E1.m1.5.5.1.1.1.3.2"></divide><apply id="S2.E1.m1.5.5.1.1.1.3.2.2.cmml" xref="S2.E1.m1.5.5.1.1.1.3.2.2"><times id="S2.E1.m1.5.5.1.1.1.3.2.2.1.cmml" xref="S2.E1.m1.5.5.1.1.1.3.2.2.1"></times><cn id="S2.E1.m1.5.5.1.1.1.3.2.2.2.cmml" type="integer" xref="S2.E1.m1.5.5.1.1.1.3.2.2.2">2</cn><ci id="S2.E1.m1.5.5.1.1.1.3.2.2.3.cmml" xref="S2.E1.m1.5.5.1.1.1.3.2.2.3">𝜋</ci></apply><ci id="S2.E1.m1.5.5.1.1.1.3.2.3.cmml" xref="S2.E1.m1.5.5.1.1.1.3.2.3">𝜆</ci></apply><apply id="S2.E1.m1.5.5.1.1.1.3.3.cmml" xref="S2.E1.m1.5.5.1.1.1.3.3"><csymbol cd="ambiguous" id="S2.E1.m1.5.5.1.1.1.3.3.1.cmml" xref="S2.E1.m1.5.5.1.1.1.3.3">subscript</csymbol><ci id="S2.E1.m1.5.5.1.1.1.3.3.2.cmml" xref="S2.E1.m1.5.5.1.1.1.3.3.2">𝐿</ci><cn id="S2.E1.m1.5.5.1.1.1.3.3.3.cmml" type="integer" xref="S2.E1.m1.5.5.1.1.1.3.3.3">1</cn></apply><apply id="S2.E1.m1.5.5.1.1.1.3.4.cmml" xref="S2.E1.m1.5.5.1.1.1.3.4"><csymbol cd="ambiguous" id="S2.E1.m1.5.5.1.1.1.3.4.1.cmml" xref="S2.E1.m1.5.5.1.1.1.3.4">subscript</csymbol><ci id="S2.E1.m1.5.5.1.1.1.3.4.2.cmml" xref="S2.E1.m1.5.5.1.1.1.3.4.2">ℎ</ci><cn id="S2.E1.m1.5.5.1.1.1.3.4.3.cmml" type="integer" xref="S2.E1.m1.5.5.1.1.1.3.4.3">1</cn></apply><ci id="S2.E1.m1.4.4.cmml" xref="S2.E1.m1.4.4">𝑡</ci></apply><apply id="S2.E1.m1.5.5.1.1.1.4.cmml" xref="S2.E1.m1.5.5.1.1.1.4"><csymbol cd="ambiguous" id="S2.E1.m1.5.5.1.1.1.4.1.cmml" xref="S2.E1.m1.5.5.1.1.1.4">subscript</csymbol><ci id="S2.E1.m1.5.5.1.1.1.4.2.cmml" xref="S2.E1.m1.5.5.1.1.1.4.2">italic-ϕ</ci><list id="S2.E1.m1.2.2.2.3.cmml" xref="S2.E1.m1.2.2.2.4"><cn id="S2.E1.m1.1.1.1.1.cmml" type="integer" xref="S2.E1.m1.1.1.1.1">1</cn><ci id="S2.E1.m1.2.2.2.2.cmml" xref="S2.E1.m1.2.2.2.2">N</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E1.m1.5c">\phi_{1}=\phi_{\rm os}(t-2L_{1})-\phi_{\rm os}(t)+\frac{2\pi}{\lambda}L_{1}h_{% 1}(t)+\phi_{\rm 1,N},</annotation><annotation encoding="application/x-llamapun" id="S2.E1.m1.5d">italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_ϕ start_POSTSUBSCRIPT roman_os end_POSTSUBSCRIPT ( italic_t - 2 italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) - italic_ϕ start_POSTSUBSCRIPT roman_os end_POSTSUBSCRIPT ( italic_t ) + divide start_ARG 2 italic_π end_ARG start_ARG italic_λ end_ARG italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_h start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) + italic_ϕ start_POSTSUBSCRIPT 1 , roman_N end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.SSS0.Px1.p1.11">where <math alttext="{2\pi}/{\lambda}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.3.m1.1"><semantics id="S2.SS1.SSS0.Px1.p1.3.m1.1a"><mrow id="S2.SS1.SSS0.Px1.p1.3.m1.1.1" xref="S2.SS1.SSS0.Px1.p1.3.m1.1.1.cmml"><mrow id="S2.SS1.SSS0.Px1.p1.3.m1.1.1.2" xref="S2.SS1.SSS0.Px1.p1.3.m1.1.1.2.cmml"><mn id="S2.SS1.SSS0.Px1.p1.3.m1.1.1.2.2" xref="S2.SS1.SSS0.Px1.p1.3.m1.1.1.2.2.cmml">2</mn><mo id="S2.SS1.SSS0.Px1.p1.3.m1.1.1.2.1" xref="S2.SS1.SSS0.Px1.p1.3.m1.1.1.2.1.cmml">⁢</mo><mi id="S2.SS1.SSS0.Px1.p1.3.m1.1.1.2.3" xref="S2.SS1.SSS0.Px1.p1.3.m1.1.1.2.3.cmml">π</mi></mrow><mo id="S2.SS1.SSS0.Px1.p1.3.m1.1.1.1" xref="S2.SS1.SSS0.Px1.p1.3.m1.1.1.1.cmml">/</mo><mi id="S2.SS1.SSS0.Px1.p1.3.m1.1.1.3" xref="S2.SS1.SSS0.Px1.p1.3.m1.1.1.3.cmml">λ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p1.3.m1.1b"><apply id="S2.SS1.SSS0.Px1.p1.3.m1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.3.m1.1.1"><divide id="S2.SS1.SSS0.Px1.p1.3.m1.1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.3.m1.1.1.1"></divide><apply id="S2.SS1.SSS0.Px1.p1.3.m1.1.1.2.cmml" xref="S2.SS1.SSS0.Px1.p1.3.m1.1.1.2"><times id="S2.SS1.SSS0.Px1.p1.3.m1.1.1.2.1.cmml" xref="S2.SS1.SSS0.Px1.p1.3.m1.1.1.2.1"></times><cn id="S2.SS1.SSS0.Px1.p1.3.m1.1.1.2.2.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p1.3.m1.1.1.2.2">2</cn><ci id="S2.SS1.SSS0.Px1.p1.3.m1.1.1.2.3.cmml" xref="S2.SS1.SSS0.Px1.p1.3.m1.1.1.2.3">𝜋</ci></apply><ci id="S2.SS1.SSS0.Px1.p1.3.m1.1.1.3.cmml" xref="S2.SS1.SSS0.Px1.p1.3.m1.1.1.3">𝜆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.3.m1.1c">{2\pi}/{\lambda}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.3.m1.1d">2 italic_π / italic_λ</annotation></semantics></math> is the electromagnetic wave’s angular frequency, <math alttext="\phi_{\rm 1,N}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.4.m2.2"><semantics id="S2.SS1.SSS0.Px1.p1.4.m2.2a"><msub id="S2.SS1.SSS0.Px1.p1.4.m2.2.3" xref="S2.SS1.SSS0.Px1.p1.4.m2.2.3.cmml"><mi id="S2.SS1.SSS0.Px1.p1.4.m2.2.3.2" xref="S2.SS1.SSS0.Px1.p1.4.m2.2.3.2.cmml">ϕ</mi><mrow id="S2.SS1.SSS0.Px1.p1.4.m2.2.2.2.4" xref="S2.SS1.SSS0.Px1.p1.4.m2.2.2.2.3.cmml"><mn id="S2.SS1.SSS0.Px1.p1.4.m2.1.1.1.1" xref="S2.SS1.SSS0.Px1.p1.4.m2.1.1.1.1.cmml">1</mn><mo id="S2.SS1.SSS0.Px1.p1.4.m2.2.2.2.4.1" xref="S2.SS1.SSS0.Px1.p1.4.m2.2.2.2.3.cmml">,</mo><mi id="S2.SS1.SSS0.Px1.p1.4.m2.2.2.2.2" mathvariant="normal" xref="S2.SS1.SSS0.Px1.p1.4.m2.2.2.2.2.cmml">N</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p1.4.m2.2b"><apply id="S2.SS1.SSS0.Px1.p1.4.m2.2.3.cmml" xref="S2.SS1.SSS0.Px1.p1.4.m2.2.3"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px1.p1.4.m2.2.3.1.cmml" xref="S2.SS1.SSS0.Px1.p1.4.m2.2.3">subscript</csymbol><ci id="S2.SS1.SSS0.Px1.p1.4.m2.2.3.2.cmml" xref="S2.SS1.SSS0.Px1.p1.4.m2.2.3.2">italic-ϕ</ci><list id="S2.SS1.SSS0.Px1.p1.4.m2.2.2.2.3.cmml" xref="S2.SS1.SSS0.Px1.p1.4.m2.2.2.2.4"><cn id="S2.SS1.SSS0.Px1.p1.4.m2.1.1.1.1.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p1.4.m2.1.1.1.1">1</cn><ci id="S2.SS1.SSS0.Px1.p1.4.m2.2.2.2.2.cmml" xref="S2.SS1.SSS0.Px1.p1.4.m2.2.2.2.2">N</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.4.m2.2c">\phi_{\rm 1,N}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.4.m2.2d">italic_ϕ start_POSTSUBSCRIPT 1 , roman_N end_POSTSUBSCRIPT</annotation></semantics></math> is the noise, and <math alttext="{2\pi}/{\lambda}L_{1}h_{1}(t)" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.5.m3.1"><semantics id="S2.SS1.SSS0.Px1.p1.5.m3.1a"><mrow id="S2.SS1.SSS0.Px1.p1.5.m3.1.2" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.2.cmml"><mrow id="S2.SS1.SSS0.Px1.p1.5.m3.1.2.2" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.2.2.cmml"><mrow id="S2.SS1.SSS0.Px1.p1.5.m3.1.2.2.2" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.2.2.2.cmml"><mn id="S2.SS1.SSS0.Px1.p1.5.m3.1.2.2.2.2" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.2.2.2.2.cmml">2</mn><mo id="S2.SS1.SSS0.Px1.p1.5.m3.1.2.2.2.1" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.2.2.2.1.cmml">⁢</mo><mi id="S2.SS1.SSS0.Px1.p1.5.m3.1.2.2.2.3" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.2.2.2.3.cmml">π</mi></mrow><mo id="S2.SS1.SSS0.Px1.p1.5.m3.1.2.2.1" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.2.2.1.cmml">/</mo><mi id="S2.SS1.SSS0.Px1.p1.5.m3.1.2.2.3" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.2.2.3.cmml">λ</mi></mrow><mo id="S2.SS1.SSS0.Px1.p1.5.m3.1.2.1" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.2.1.cmml">⁢</mo><msub id="S2.SS1.SSS0.Px1.p1.5.m3.1.2.3" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.2.3.cmml"><mi id="S2.SS1.SSS0.Px1.p1.5.m3.1.2.3.2" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.2.3.2.cmml">L</mi><mn id="S2.SS1.SSS0.Px1.p1.5.m3.1.2.3.3" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.2.3.3.cmml">1</mn></msub><mo id="S2.SS1.SSS0.Px1.p1.5.m3.1.2.1a" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.2.1.cmml">⁢</mo><msub id="S2.SS1.SSS0.Px1.p1.5.m3.1.2.4" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.2.4.cmml"><mi id="S2.SS1.SSS0.Px1.p1.5.m3.1.2.4.2" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.2.4.2.cmml">h</mi><mn id="S2.SS1.SSS0.Px1.p1.5.m3.1.2.4.3" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.2.4.3.cmml">1</mn></msub><mo id="S2.SS1.SSS0.Px1.p1.5.m3.1.2.1b" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.SSS0.Px1.p1.5.m3.1.2.5.2" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.2.cmml"><mo id="S2.SS1.SSS0.Px1.p1.5.m3.1.2.5.2.1" stretchy="false" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.2.cmml">(</mo><mi id="S2.SS1.SSS0.Px1.p1.5.m3.1.1" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.1.cmml">t</mi><mo id="S2.SS1.SSS0.Px1.p1.5.m3.1.2.5.2.2" stretchy="false" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p1.5.m3.1b"><apply id="S2.SS1.SSS0.Px1.p1.5.m3.1.2.cmml" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.2"><times id="S2.SS1.SSS0.Px1.p1.5.m3.1.2.1.cmml" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.2.1"></times><apply id="S2.SS1.SSS0.Px1.p1.5.m3.1.2.2.cmml" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.2.2"><divide id="S2.SS1.SSS0.Px1.p1.5.m3.1.2.2.1.cmml" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.2.2.1"></divide><apply id="S2.SS1.SSS0.Px1.p1.5.m3.1.2.2.2.cmml" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.2.2.2"><times id="S2.SS1.SSS0.Px1.p1.5.m3.1.2.2.2.1.cmml" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.2.2.2.1"></times><cn id="S2.SS1.SSS0.Px1.p1.5.m3.1.2.2.2.2.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.2.2.2.2">2</cn><ci id="S2.SS1.SSS0.Px1.p1.5.m3.1.2.2.2.3.cmml" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.2.2.2.3">𝜋</ci></apply><ci id="S2.SS1.SSS0.Px1.p1.5.m3.1.2.2.3.cmml" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.2.2.3">𝜆</ci></apply><apply id="S2.SS1.SSS0.Px1.p1.5.m3.1.2.3.cmml" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.2.3"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px1.p1.5.m3.1.2.3.1.cmml" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.2.3">subscript</csymbol><ci id="S2.SS1.SSS0.Px1.p1.5.m3.1.2.3.2.cmml" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.2.3.2">𝐿</ci><cn id="S2.SS1.SSS0.Px1.p1.5.m3.1.2.3.3.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.2.3.3">1</cn></apply><apply id="S2.SS1.SSS0.Px1.p1.5.m3.1.2.4.cmml" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.2.4"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px1.p1.5.m3.1.2.4.1.cmml" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.2.4">subscript</csymbol><ci id="S2.SS1.SSS0.Px1.p1.5.m3.1.2.4.2.cmml" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.2.4.2">ℎ</ci><cn id="S2.SS1.SSS0.Px1.p1.5.m3.1.2.4.3.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.2.4.3">1</cn></apply><ci id="S2.SS1.SSS0.Px1.p1.5.m3.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.5.m3.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.5.m3.1c">{2\pi}/{\lambda}L_{1}h_{1}(t)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.5.m3.1d">2 italic_π / italic_λ italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_h start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> is the gravitational wave signal in the long wavelength limit <math alttext="c/f\gg L_{1}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.6.m4.1"><semantics id="S2.SS1.SSS0.Px1.p1.6.m4.1a"><mrow id="S2.SS1.SSS0.Px1.p1.6.m4.1.1" xref="S2.SS1.SSS0.Px1.p1.6.m4.1.1.cmml"><mrow id="S2.SS1.SSS0.Px1.p1.6.m4.1.1.2" xref="S2.SS1.SSS0.Px1.p1.6.m4.1.1.2.cmml"><mi id="S2.SS1.SSS0.Px1.p1.6.m4.1.1.2.2" xref="S2.SS1.SSS0.Px1.p1.6.m4.1.1.2.2.cmml">c</mi><mo id="S2.SS1.SSS0.Px1.p1.6.m4.1.1.2.1" xref="S2.SS1.SSS0.Px1.p1.6.m4.1.1.2.1.cmml">/</mo><mi id="S2.SS1.SSS0.Px1.p1.6.m4.1.1.2.3" xref="S2.SS1.SSS0.Px1.p1.6.m4.1.1.2.3.cmml">f</mi></mrow><mo id="S2.SS1.SSS0.Px1.p1.6.m4.1.1.1" xref="S2.SS1.SSS0.Px1.p1.6.m4.1.1.1.cmml">≫</mo><msub id="S2.SS1.SSS0.Px1.p1.6.m4.1.1.3" xref="S2.SS1.SSS0.Px1.p1.6.m4.1.1.3.cmml"><mi id="S2.SS1.SSS0.Px1.p1.6.m4.1.1.3.2" xref="S2.SS1.SSS0.Px1.p1.6.m4.1.1.3.2.cmml">L</mi><mn id="S2.SS1.SSS0.Px1.p1.6.m4.1.1.3.3" xref="S2.SS1.SSS0.Px1.p1.6.m4.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p1.6.m4.1b"><apply id="S2.SS1.SSS0.Px1.p1.6.m4.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.6.m4.1.1"><csymbol cd="latexml" id="S2.SS1.SSS0.Px1.p1.6.m4.1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.6.m4.1.1.1">much-greater-than</csymbol><apply id="S2.SS1.SSS0.Px1.p1.6.m4.1.1.2.cmml" xref="S2.SS1.SSS0.Px1.p1.6.m4.1.1.2"><divide id="S2.SS1.SSS0.Px1.p1.6.m4.1.1.2.1.cmml" xref="S2.SS1.SSS0.Px1.p1.6.m4.1.1.2.1"></divide><ci id="S2.SS1.SSS0.Px1.p1.6.m4.1.1.2.2.cmml" xref="S2.SS1.SSS0.Px1.p1.6.m4.1.1.2.2">𝑐</ci><ci id="S2.SS1.SSS0.Px1.p1.6.m4.1.1.2.3.cmml" xref="S2.SS1.SSS0.Px1.p1.6.m4.1.1.2.3">𝑓</ci></apply><apply id="S2.SS1.SSS0.Px1.p1.6.m4.1.1.3.cmml" xref="S2.SS1.SSS0.Px1.p1.6.m4.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px1.p1.6.m4.1.1.3.1.cmml" xref="S2.SS1.SSS0.Px1.p1.6.m4.1.1.3">subscript</csymbol><ci id="S2.SS1.SSS0.Px1.p1.6.m4.1.1.3.2.cmml" xref="S2.SS1.SSS0.Px1.p1.6.m4.1.1.3.2">𝐿</ci><cn id="S2.SS1.SSS0.Px1.p1.6.m4.1.1.3.3.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p1.6.m4.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.6.m4.1c">c/f\gg L_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.6.m4.1d">italic_c / italic_f ≫ italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. (This limit applies at our target of 1<math alttext="\mu" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.7.m5.1"><semantics id="S2.SS1.SSS0.Px1.p1.7.m5.1a"><mi id="S2.SS1.SSS0.Px1.p1.7.m5.1.1" xref="S2.SS1.SSS0.Px1.p1.7.m5.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p1.7.m5.1b"><ci id="S2.SS1.SSS0.Px1.p1.7.m5.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.7.m5.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.7.m5.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.7.m5.1d">italic_μ</annotation></semantics></math>Hz, where the gravitational wavelength is <math alttext="2000\text{ AU}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.8.m6.1"><semantics id="S2.SS1.SSS0.Px1.p1.8.m6.1a"><mrow id="S2.SS1.SSS0.Px1.p1.8.m6.1.1" xref="S2.SS1.SSS0.Px1.p1.8.m6.1.1.cmml"><mn id="S2.SS1.SSS0.Px1.p1.8.m6.1.1.2" xref="S2.SS1.SSS0.Px1.p1.8.m6.1.1.2.cmml">2000</mn><mo id="S2.SS1.SSS0.Px1.p1.8.m6.1.1.1" xref="S2.SS1.SSS0.Px1.p1.8.m6.1.1.1.cmml">⁢</mo><mtext id="S2.SS1.SSS0.Px1.p1.8.m6.1.1.3" xref="S2.SS1.SSS0.Px1.p1.8.m6.1.1.3a.cmml"> AU</mtext></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p1.8.m6.1b"><apply id="S2.SS1.SSS0.Px1.p1.8.m6.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.8.m6.1.1"><times id="S2.SS1.SSS0.Px1.p1.8.m6.1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.8.m6.1.1.1"></times><cn id="S2.SS1.SSS0.Px1.p1.8.m6.1.1.2.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p1.8.m6.1.1.2">2000</cn><ci id="S2.SS1.SSS0.Px1.p1.8.m6.1.1.3a.cmml" xref="S2.SS1.SSS0.Px1.p1.8.m6.1.1.3"><mtext id="S2.SS1.SSS0.Px1.p1.8.m6.1.1.3.cmml" xref="S2.SS1.SSS0.Px1.p1.8.m6.1.1.3"> AU</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.8.m6.1c">2000\text{ AU}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.8.m6.1d">2000 AU</annotation></semantics></math>.) The phase of the electromagnetic signal <math alttext="\phi_{\rm os}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.9.m7.1"><semantics id="S2.SS1.SSS0.Px1.p1.9.m7.1a"><msub id="S2.SS1.SSS0.Px1.p1.9.m7.1.1" xref="S2.SS1.SSS0.Px1.p1.9.m7.1.1.cmml"><mi id="S2.SS1.SSS0.Px1.p1.9.m7.1.1.2" xref="S2.SS1.SSS0.Px1.p1.9.m7.1.1.2.cmml">ϕ</mi><mi id="S2.SS1.SSS0.Px1.p1.9.m7.1.1.3" xref="S2.SS1.SSS0.Px1.p1.9.m7.1.1.3.cmml">os</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p1.9.m7.1b"><apply id="S2.SS1.SSS0.Px1.p1.9.m7.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.9.m7.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px1.p1.9.m7.1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.9.m7.1.1">subscript</csymbol><ci id="S2.SS1.SSS0.Px1.p1.9.m7.1.1.2.cmml" xref="S2.SS1.SSS0.Px1.p1.9.m7.1.1.2">italic-ϕ</ci><ci id="S2.SS1.SSS0.Px1.p1.9.m7.1.1.3.cmml" xref="S2.SS1.SSS0.Px1.p1.9.m7.1.1.3">os</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.9.m7.1c">\phi_{\rm os}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.9.m7.1d">italic_ϕ start_POSTSUBSCRIPT roman_os end_POSTSUBSCRIPT</annotation></semantics></math> has noise that can be related to the Allan deviation <math alttext="\sigma_{y}(\tau)" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.10.m8.1"><semantics id="S2.SS1.SSS0.Px1.p1.10.m8.1a"><mrow id="S2.SS1.SSS0.Px1.p1.10.m8.1.2" xref="S2.SS1.SSS0.Px1.p1.10.m8.1.2.cmml"><msub id="S2.SS1.SSS0.Px1.p1.10.m8.1.2.2" xref="S2.SS1.SSS0.Px1.p1.10.m8.1.2.2.cmml"><mi id="S2.SS1.SSS0.Px1.p1.10.m8.1.2.2.2" xref="S2.SS1.SSS0.Px1.p1.10.m8.1.2.2.2.cmml">σ</mi><mi id="S2.SS1.SSS0.Px1.p1.10.m8.1.2.2.3" xref="S2.SS1.SSS0.Px1.p1.10.m8.1.2.2.3.cmml">y</mi></msub><mo id="S2.SS1.SSS0.Px1.p1.10.m8.1.2.1" xref="S2.SS1.SSS0.Px1.p1.10.m8.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.SSS0.Px1.p1.10.m8.1.2.3.2" xref="S2.SS1.SSS0.Px1.p1.10.m8.1.2.cmml"><mo id="S2.SS1.SSS0.Px1.p1.10.m8.1.2.3.2.1" stretchy="false" xref="S2.SS1.SSS0.Px1.p1.10.m8.1.2.cmml">(</mo><mi id="S2.SS1.SSS0.Px1.p1.10.m8.1.1" xref="S2.SS1.SSS0.Px1.p1.10.m8.1.1.cmml">τ</mi><mo id="S2.SS1.SSS0.Px1.p1.10.m8.1.2.3.2.2" stretchy="false" xref="S2.SS1.SSS0.Px1.p1.10.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p1.10.m8.1b"><apply id="S2.SS1.SSS0.Px1.p1.10.m8.1.2.cmml" xref="S2.SS1.SSS0.Px1.p1.10.m8.1.2"><times id="S2.SS1.SSS0.Px1.p1.10.m8.1.2.1.cmml" xref="S2.SS1.SSS0.Px1.p1.10.m8.1.2.1"></times><apply id="S2.SS1.SSS0.Px1.p1.10.m8.1.2.2.cmml" xref="S2.SS1.SSS0.Px1.p1.10.m8.1.2.2"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px1.p1.10.m8.1.2.2.1.cmml" xref="S2.SS1.SSS0.Px1.p1.10.m8.1.2.2">subscript</csymbol><ci id="S2.SS1.SSS0.Px1.p1.10.m8.1.2.2.2.cmml" xref="S2.SS1.SSS0.Px1.p1.10.m8.1.2.2.2">𝜎</ci><ci id="S2.SS1.SSS0.Px1.p1.10.m8.1.2.2.3.cmml" xref="S2.SS1.SSS0.Px1.p1.10.m8.1.2.2.3">𝑦</ci></apply><ci id="S2.SS1.SSS0.Px1.p1.10.m8.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.10.m8.1.1">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.10.m8.1c">\sigma_{y}(\tau)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.10.m8.1d">italic_σ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ( italic_τ )</annotation></semantics></math> of the clock. If we assume that the clock’s frequency noise is white such that <math alttext="\sigma_{y}(\tau)^{2}\propto\tau^{-1}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.11.m9.1"><semantics id="S2.SS1.SSS0.Px1.p1.11.m9.1a"><mrow id="S2.SS1.SSS0.Px1.p1.11.m9.1.2" xref="S2.SS1.SSS0.Px1.p1.11.m9.1.2.cmml"><mrow id="S2.SS1.SSS0.Px1.p1.11.m9.1.2.2" xref="S2.SS1.SSS0.Px1.p1.11.m9.1.2.2.cmml"><msub id="S2.SS1.SSS0.Px1.p1.11.m9.1.2.2.2" xref="S2.SS1.SSS0.Px1.p1.11.m9.1.2.2.2.cmml"><mi id="S2.SS1.SSS0.Px1.p1.11.m9.1.2.2.2.2" xref="S2.SS1.SSS0.Px1.p1.11.m9.1.2.2.2.2.cmml">σ</mi><mi id="S2.SS1.SSS0.Px1.p1.11.m9.1.2.2.2.3" xref="S2.SS1.SSS0.Px1.p1.11.m9.1.2.2.2.3.cmml">y</mi></msub><mo id="S2.SS1.SSS0.Px1.p1.11.m9.1.2.2.1" xref="S2.SS1.SSS0.Px1.p1.11.m9.1.2.2.1.cmml">⁢</mo><msup id="S2.SS1.SSS0.Px1.p1.11.m9.1.2.2.3" xref="S2.SS1.SSS0.Px1.p1.11.m9.1.2.2.3.cmml"><mrow id="S2.SS1.SSS0.Px1.p1.11.m9.1.2.2.3.2.2" xref="S2.SS1.SSS0.Px1.p1.11.m9.1.2.2.3.cmml"><mo id="S2.SS1.SSS0.Px1.p1.11.m9.1.2.2.3.2.2.1" stretchy="false" xref="S2.SS1.SSS0.Px1.p1.11.m9.1.2.2.3.cmml">(</mo><mi id="S2.SS1.SSS0.Px1.p1.11.m9.1.1" xref="S2.SS1.SSS0.Px1.p1.11.m9.1.1.cmml">τ</mi><mo id="S2.SS1.SSS0.Px1.p1.11.m9.1.2.2.3.2.2.2" stretchy="false" xref="S2.SS1.SSS0.Px1.p1.11.m9.1.2.2.3.cmml">)</mo></mrow><mn id="S2.SS1.SSS0.Px1.p1.11.m9.1.2.2.3.3" xref="S2.SS1.SSS0.Px1.p1.11.m9.1.2.2.3.3.cmml">2</mn></msup></mrow><mo id="S2.SS1.SSS0.Px1.p1.11.m9.1.2.1" xref="S2.SS1.SSS0.Px1.p1.11.m9.1.2.1.cmml">∝</mo><msup id="S2.SS1.SSS0.Px1.p1.11.m9.1.2.3" xref="S2.SS1.SSS0.Px1.p1.11.m9.1.2.3.cmml"><mi id="S2.SS1.SSS0.Px1.p1.11.m9.1.2.3.2" xref="S2.SS1.SSS0.Px1.p1.11.m9.1.2.3.2.cmml">τ</mi><mrow id="S2.SS1.SSS0.Px1.p1.11.m9.1.2.3.3" xref="S2.SS1.SSS0.Px1.p1.11.m9.1.2.3.3.cmml"><mo id="S2.SS1.SSS0.Px1.p1.11.m9.1.2.3.3a" xref="S2.SS1.SSS0.Px1.p1.11.m9.1.2.3.3.cmml">−</mo><mn id="S2.SS1.SSS0.Px1.p1.11.m9.1.2.3.3.2" xref="S2.SS1.SSS0.Px1.p1.11.m9.1.2.3.3.2.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p1.11.m9.1b"><apply id="S2.SS1.SSS0.Px1.p1.11.m9.1.2.cmml" xref="S2.SS1.SSS0.Px1.p1.11.m9.1.2"><csymbol cd="latexml" id="S2.SS1.SSS0.Px1.p1.11.m9.1.2.1.cmml" xref="S2.SS1.SSS0.Px1.p1.11.m9.1.2.1">proportional-to</csymbol><apply id="S2.SS1.SSS0.Px1.p1.11.m9.1.2.2.cmml" xref="S2.SS1.SSS0.Px1.p1.11.m9.1.2.2"><times id="S2.SS1.SSS0.Px1.p1.11.m9.1.2.2.1.cmml" xref="S2.SS1.SSS0.Px1.p1.11.m9.1.2.2.1"></times><apply id="S2.SS1.SSS0.Px1.p1.11.m9.1.2.2.2.cmml" xref="S2.SS1.SSS0.Px1.p1.11.m9.1.2.2.2"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px1.p1.11.m9.1.2.2.2.1.cmml" xref="S2.SS1.SSS0.Px1.p1.11.m9.1.2.2.2">subscript</csymbol><ci id="S2.SS1.SSS0.Px1.p1.11.m9.1.2.2.2.2.cmml" xref="S2.SS1.SSS0.Px1.p1.11.m9.1.2.2.2.2">𝜎</ci><ci id="S2.SS1.SSS0.Px1.p1.11.m9.1.2.2.2.3.cmml" xref="S2.SS1.SSS0.Px1.p1.11.m9.1.2.2.2.3">𝑦</ci></apply><apply id="S2.SS1.SSS0.Px1.p1.11.m9.1.2.2.3.cmml" xref="S2.SS1.SSS0.Px1.p1.11.m9.1.2.2.3"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px1.p1.11.m9.1.2.2.3.1.cmml" xref="S2.SS1.SSS0.Px1.p1.11.m9.1.2.2.3">superscript</csymbol><ci id="S2.SS1.SSS0.Px1.p1.11.m9.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.11.m9.1.1">𝜏</ci><cn id="S2.SS1.SSS0.Px1.p1.11.m9.1.2.2.3.3.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p1.11.m9.1.2.2.3.3">2</cn></apply></apply><apply id="S2.SS1.SSS0.Px1.p1.11.m9.1.2.3.cmml" xref="S2.SS1.SSS0.Px1.p1.11.m9.1.2.3"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px1.p1.11.m9.1.2.3.1.cmml" xref="S2.SS1.SSS0.Px1.p1.11.m9.1.2.3">superscript</csymbol><ci id="S2.SS1.SSS0.Px1.p1.11.m9.1.2.3.2.cmml" xref="S2.SS1.SSS0.Px1.p1.11.m9.1.2.3.2">𝜏</ci><apply id="S2.SS1.SSS0.Px1.p1.11.m9.1.2.3.3.cmml" xref="S2.SS1.SSS0.Px1.p1.11.m9.1.2.3.3"><minus id="S2.SS1.SSS0.Px1.p1.11.m9.1.2.3.3.1.cmml" xref="S2.SS1.SSS0.Px1.p1.11.m9.1.2.3.3"></minus><cn id="S2.SS1.SSS0.Px1.p1.11.m9.1.2.3.3.2.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p1.11.m9.1.2.3.3.2">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.11.m9.1c">\sigma_{y}(\tau)^{2}\propto\tau^{-1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.11.m9.1d">italic_σ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ( italic_τ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∝ italic_τ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>, the phase noise power is <cite class="ltx_cite ltx_citemacro_citep">(IEEE Standards Association, <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib38" title="">2009</a>)</cite></p> <table class="ltx_equation ltx_eqn_table" id="S2.E2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="S_{{\rm os}}(f)\equiv 2\,T^{-1}\left\langle\widetilde{\phi}_{\rm os}^{2}\right% \rangle=\frac{2\,({2\pi}/{\lambda})^{2}\times\tau\times\sigma_{y}(\tau)^{2}}{(% 2\pi f)^{2}}," class="ltx_Math" display="block" id="S2.E2.m1.5"><semantics id="S2.E2.m1.5a"><mrow id="S2.E2.m1.5.5.1" xref="S2.E2.m1.5.5.1.1.cmml"><mrow id="S2.E2.m1.5.5.1.1" xref="S2.E2.m1.5.5.1.1.cmml"><mrow id="S2.E2.m1.5.5.1.1.3" xref="S2.E2.m1.5.5.1.1.3.cmml"><msub id="S2.E2.m1.5.5.1.1.3.2" xref="S2.E2.m1.5.5.1.1.3.2.cmml"><mi id="S2.E2.m1.5.5.1.1.3.2.2" xref="S2.E2.m1.5.5.1.1.3.2.2.cmml">S</mi><mi id="S2.E2.m1.5.5.1.1.3.2.3" xref="S2.E2.m1.5.5.1.1.3.2.3.cmml">os</mi></msub><mo id="S2.E2.m1.5.5.1.1.3.1" xref="S2.E2.m1.5.5.1.1.3.1.cmml">⁢</mo><mrow id="S2.E2.m1.5.5.1.1.3.3.2" xref="S2.E2.m1.5.5.1.1.3.cmml"><mo id="S2.E2.m1.5.5.1.1.3.3.2.1" stretchy="false" xref="S2.E2.m1.5.5.1.1.3.cmml">(</mo><mi id="S2.E2.m1.4.4" xref="S2.E2.m1.4.4.cmml">f</mi><mo id="S2.E2.m1.5.5.1.1.3.3.2.2" stretchy="false" xref="S2.E2.m1.5.5.1.1.3.cmml">)</mo></mrow></mrow><mo id="S2.E2.m1.5.5.1.1.4" 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xref="S2.E2.m1.2.2.2.4.3">2</cn></apply></apply><apply id="S2.E2.m1.3.3.3.cmml" xref="S2.E2.m1.3.3.3"><csymbol cd="ambiguous" id="S2.E2.m1.3.3.3.2.cmml" xref="S2.E2.m1.3.3.3">superscript</csymbol><apply id="S2.E2.m1.3.3.3.1.1.1.cmml" xref="S2.E2.m1.3.3.3.1.1"><times id="S2.E2.m1.3.3.3.1.1.1.1.cmml" xref="S2.E2.m1.3.3.3.1.1.1.1"></times><cn id="S2.E2.m1.3.3.3.1.1.1.2.cmml" type="integer" xref="S2.E2.m1.3.3.3.1.1.1.2">2</cn><ci id="S2.E2.m1.3.3.3.1.1.1.3.cmml" xref="S2.E2.m1.3.3.3.1.1.1.3">𝜋</ci><ci id="S2.E2.m1.3.3.3.1.1.1.4.cmml" xref="S2.E2.m1.3.3.3.1.1.1.4">𝑓</ci></apply><cn id="S2.E2.m1.3.3.3.3.cmml" type="integer" xref="S2.E2.m1.3.3.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E2.m1.5c">S_{{\rm os}}(f)\equiv 2\,T^{-1}\left\langle\widetilde{\phi}_{\rm os}^{2}\right% \rangle=\frac{2\,({2\pi}/{\lambda})^{2}\times\tau\times\sigma_{y}(\tau)^{2}}{(% 2\pi f)^{2}},</annotation><annotation encoding="application/x-llamapun" id="S2.E2.m1.5d">italic_S start_POSTSUBSCRIPT roman_os end_POSTSUBSCRIPT ( italic_f ) ≡ 2 italic_T start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ⟨ over~ start_ARG italic_ϕ end_ARG start_POSTSUBSCRIPT roman_os end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⟩ = divide start_ARG 2 ( 2 italic_π / italic_λ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT × italic_τ × italic_σ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ( italic_τ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ( 2 italic_π italic_f ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.SSS0.Px1.p1.16">where <math alttext="\widetilde{\phi}_{\rm os}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.12.m1.1"><semantics id="S2.SS1.SSS0.Px1.p1.12.m1.1a"><msub id="S2.SS1.SSS0.Px1.p1.12.m1.1.1" xref="S2.SS1.SSS0.Px1.p1.12.m1.1.1.cmml"><mover accent="true" id="S2.SS1.SSS0.Px1.p1.12.m1.1.1.2" xref="S2.SS1.SSS0.Px1.p1.12.m1.1.1.2.cmml"><mi id="S2.SS1.SSS0.Px1.p1.12.m1.1.1.2.2" xref="S2.SS1.SSS0.Px1.p1.12.m1.1.1.2.2.cmml">ϕ</mi><mo id="S2.SS1.SSS0.Px1.p1.12.m1.1.1.2.1" xref="S2.SS1.SSS0.Px1.p1.12.m1.1.1.2.1.cmml">~</mo></mover><mi id="S2.SS1.SSS0.Px1.p1.12.m1.1.1.3" xref="S2.SS1.SSS0.Px1.p1.12.m1.1.1.3.cmml">os</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p1.12.m1.1b"><apply id="S2.SS1.SSS0.Px1.p1.12.m1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.12.m1.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px1.p1.12.m1.1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.12.m1.1.1">subscript</csymbol><apply id="S2.SS1.SSS0.Px1.p1.12.m1.1.1.2.cmml" xref="S2.SS1.SSS0.Px1.p1.12.m1.1.1.2"><ci id="S2.SS1.SSS0.Px1.p1.12.m1.1.1.2.1.cmml" xref="S2.SS1.SSS0.Px1.p1.12.m1.1.1.2.1">~</ci><ci id="S2.SS1.SSS0.Px1.p1.12.m1.1.1.2.2.cmml" xref="S2.SS1.SSS0.Px1.p1.12.m1.1.1.2.2">italic-ϕ</ci></apply><ci id="S2.SS1.SSS0.Px1.p1.12.m1.1.1.3.cmml" xref="S2.SS1.SSS0.Px1.p1.12.m1.1.1.3">os</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.12.m1.1c">\widetilde{\phi}_{\rm os}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.12.m1.1d">over~ start_ARG italic_ϕ end_ARG start_POSTSUBSCRIPT roman_os end_POSTSUBSCRIPT</annotation></semantics></math> is the Fourier transform of <math alttext="\phi_{\rm os}(t)" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.13.m2.1"><semantics id="S2.SS1.SSS0.Px1.p1.13.m2.1a"><mrow id="S2.SS1.SSS0.Px1.p1.13.m2.1.2" xref="S2.SS1.SSS0.Px1.p1.13.m2.1.2.cmml"><msub id="S2.SS1.SSS0.Px1.p1.13.m2.1.2.2" xref="S2.SS1.SSS0.Px1.p1.13.m2.1.2.2.cmml"><mi id="S2.SS1.SSS0.Px1.p1.13.m2.1.2.2.2" xref="S2.SS1.SSS0.Px1.p1.13.m2.1.2.2.2.cmml">ϕ</mi><mi id="S2.SS1.SSS0.Px1.p1.13.m2.1.2.2.3" xref="S2.SS1.SSS0.Px1.p1.13.m2.1.2.2.3.cmml">os</mi></msub><mo id="S2.SS1.SSS0.Px1.p1.13.m2.1.2.1" xref="S2.SS1.SSS0.Px1.p1.13.m2.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.SSS0.Px1.p1.13.m2.1.2.3.2" xref="S2.SS1.SSS0.Px1.p1.13.m2.1.2.cmml"><mo id="S2.SS1.SSS0.Px1.p1.13.m2.1.2.3.2.1" stretchy="false" xref="S2.SS1.SSS0.Px1.p1.13.m2.1.2.cmml">(</mo><mi id="S2.SS1.SSS0.Px1.p1.13.m2.1.1" xref="S2.SS1.SSS0.Px1.p1.13.m2.1.1.cmml">t</mi><mo id="S2.SS1.SSS0.Px1.p1.13.m2.1.2.3.2.2" stretchy="false" xref="S2.SS1.SSS0.Px1.p1.13.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p1.13.m2.1b"><apply id="S2.SS1.SSS0.Px1.p1.13.m2.1.2.cmml" xref="S2.SS1.SSS0.Px1.p1.13.m2.1.2"><times id="S2.SS1.SSS0.Px1.p1.13.m2.1.2.1.cmml" xref="S2.SS1.SSS0.Px1.p1.13.m2.1.2.1"></times><apply id="S2.SS1.SSS0.Px1.p1.13.m2.1.2.2.cmml" xref="S2.SS1.SSS0.Px1.p1.13.m2.1.2.2"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px1.p1.13.m2.1.2.2.1.cmml" xref="S2.SS1.SSS0.Px1.p1.13.m2.1.2.2">subscript</csymbol><ci id="S2.SS1.SSS0.Px1.p1.13.m2.1.2.2.2.cmml" xref="S2.SS1.SSS0.Px1.p1.13.m2.1.2.2.2">italic-ϕ</ci><ci id="S2.SS1.SSS0.Px1.p1.13.m2.1.2.2.3.cmml" xref="S2.SS1.SSS0.Px1.p1.13.m2.1.2.2.3">os</ci></apply><ci id="S2.SS1.SSS0.Px1.p1.13.m2.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.13.m2.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.13.m2.1c">\phi_{\rm os}(t)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.13.m2.1d">italic_ϕ start_POSTSUBSCRIPT roman_os end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> over time interval <math alttext="T" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.14.m3.1"><semantics id="S2.SS1.SSS0.Px1.p1.14.m3.1a"><mi id="S2.SS1.SSS0.Px1.p1.14.m3.1.1" xref="S2.SS1.SSS0.Px1.p1.14.m3.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p1.14.m3.1b"><ci id="S2.SS1.SSS0.Px1.p1.14.m3.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.14.m3.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.14.m3.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.14.m3.1d">italic_T</annotation></semantics></math>. Since the phase difference given by equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.E1" title="In Single-arm and an atomic clock ‣ 2.1 Clock noise and its mitigation ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">1</span></a>) involves <math alttext="\phi_{\rm os}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.15.m4.1"><semantics id="S2.SS1.SSS0.Px1.p1.15.m4.1a"><msub id="S2.SS1.SSS0.Px1.p1.15.m4.1.1" xref="S2.SS1.SSS0.Px1.p1.15.m4.1.1.cmml"><mi id="S2.SS1.SSS0.Px1.p1.15.m4.1.1.2" xref="S2.SS1.SSS0.Px1.p1.15.m4.1.1.2.cmml">ϕ</mi><mi id="S2.SS1.SSS0.Px1.p1.15.m4.1.1.3" xref="S2.SS1.SSS0.Px1.p1.15.m4.1.1.3.cmml">os</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p1.15.m4.1b"><apply id="S2.SS1.SSS0.Px1.p1.15.m4.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.15.m4.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px1.p1.15.m4.1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.15.m4.1.1">subscript</csymbol><ci id="S2.SS1.SSS0.Px1.p1.15.m4.1.1.2.cmml" xref="S2.SS1.SSS0.Px1.p1.15.m4.1.1.2">italic-ϕ</ci><ci id="S2.SS1.SSS0.Px1.p1.15.m4.1.1.3.cmml" xref="S2.SS1.SSS0.Px1.p1.15.m4.1.1.3">os</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.15.m4.1c">\phi_{\rm os}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.15.m4.1d">italic_ϕ start_POSTSUBSCRIPT roman_os end_POSTSUBSCRIPT</annotation></semantics></math> at two times, the phase noise power relevant for constraining <math alttext="h_{1}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.16.m5.1"><semantics id="S2.SS1.SSS0.Px1.p1.16.m5.1a"><msub id="S2.SS1.SSS0.Px1.p1.16.m5.1.1" xref="S2.SS1.SSS0.Px1.p1.16.m5.1.1.cmml"><mi id="S2.SS1.SSS0.Px1.p1.16.m5.1.1.2" xref="S2.SS1.SSS0.Px1.p1.16.m5.1.1.2.cmml">h</mi><mn id="S2.SS1.SSS0.Px1.p1.16.m5.1.1.3" xref="S2.SS1.SSS0.Px1.p1.16.m5.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p1.16.m5.1b"><apply id="S2.SS1.SSS0.Px1.p1.16.m5.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.16.m5.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px1.p1.16.m5.1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.16.m5.1.1">subscript</csymbol><ci id="S2.SS1.SSS0.Px1.p1.16.m5.1.1.2.cmml" xref="S2.SS1.SSS0.Px1.p1.16.m5.1.1.2">ℎ</ci><cn id="S2.SS1.SSS0.Px1.p1.16.m5.1.1.3.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p1.16.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.16.m5.1c">h_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.16.m5.1d">italic_h start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> is</p> <table class="ltx_equation ltx_eqn_table" id="S2.E3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td 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id="S2.E3.m1.4.4.1.1.1.1.1.1.1.1.1.1.5.cmml" xref="S2.E3.m1.4.4.1.1.1.1.1.1.1.1.1.1.5">𝑓</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E3.m1.4c">S_{\rm os,tot}=2S_{{\rm os}}\left[1-\cos(4\pi L_{1}f)\right].</annotation><annotation encoding="application/x-llamapun" id="S2.E3.m1.4d">italic_S start_POSTSUBSCRIPT roman_os , roman_tot end_POSTSUBSCRIPT = 2 italic_S start_POSTSUBSCRIPT roman_os end_POSTSUBSCRIPT [ 1 - roman_cos ( 4 italic_π italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_f ) ] .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.SSS0.Px1.p1.17">Equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.E3" title="In Single-arm and an atomic clock ‣ 2.1 Clock noise and its mitigation ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">3</span></a>) allows us to calculate the long-wavelength strain noise power of our interferometer owing to white frequency modulation clock noise:</p> <table class="ltx_equation ltx_eqn_table" id="S2.E4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\widetilde{h}_{\rm os}^{\rm LW}=~{}~{}\frac{\sqrt{S_{{\rm os},\rm tot}}}{({2% \pi}/{\lambda})L_{1}}={2~{}(1~{}\rm s)^{1/2}}\times\sigma_{y}(1~{}\rm s)=2% \times 10^{-13}{\rm Hz}^{-1/2}\left(\frac{\sigma_{y}(1~{}\rm s)}{10^{-13}}% \right)." class="ltx_Math" display="block" id="S2.E4.m1.5"><semantics id="S2.E4.m1.5a"><mrow id="S2.E4.m1.5.5.1" xref="S2.E4.m1.5.5.1.1.cmml"><mrow id="S2.E4.m1.5.5.1.1" 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Hz}^{-1/2}\left(\frac{\sigma_{y}(1~{}\rm s)}{10^{-13}}% \right).</annotation><annotation encoding="application/x-llamapun" id="S2.E4.m1.5d">over~ start_ARG italic_h end_ARG start_POSTSUBSCRIPT roman_os end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_LW end_POSTSUPERSCRIPT = divide start_ARG square-root start_ARG italic_S start_POSTSUBSCRIPT roman_os , roman_tot end_POSTSUBSCRIPT end_ARG end_ARG start_ARG ( 2 italic_π / italic_λ ) italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG = 2 ( 1 roman_s ) start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT × italic_σ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ( 1 roman_s ) = 2 × 10 start_POSTSUPERSCRIPT - 13 end_POSTSUPERSCRIPT roman_Hz start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT ( divide start_ARG italic_σ start_POSTSUBSCRIPT roman_y end_POSTSUBSCRIPT ( 1 roman_s ) end_ARG start_ARG 10 start_POSTSUPERSCRIPT - 13 end_POSTSUPERSCRIPT end_ARG ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.SSS0.Px1.p1.18">where we have adopted the convention of evaluating the Allan deviation at a second.</p> </div> <div class="ltx_para" id="S2.SS1.SSS0.Px1.p2"> <p class="ltx_p" id="S2.SS1.SSS0.Px1.p2.5">The Deep Space Atomic Clock – an atomic clock launched in 2019 and a prototype for future interplanetary space missions – achieved <math alttext="\sigma_{y}(\tau)\approx 2\times 10^{-12}(\tau/1~{}{\rm s})^{-1/2}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p2.1.m1.2"><semantics id="S2.SS1.SSS0.Px1.p2.1.m1.2a"><mrow id="S2.SS1.SSS0.Px1.p2.1.m1.2.2" xref="S2.SS1.SSS0.Px1.p2.1.m1.2.2.cmml"><mrow id="S2.SS1.SSS0.Px1.p2.1.m1.2.2.3" xref="S2.SS1.SSS0.Px1.p2.1.m1.2.2.3.cmml"><msub id="S2.SS1.SSS0.Px1.p2.1.m1.2.2.3.2" 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xref="S2.SS1.SSS0.Px1.p2.1.m1.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px1.p2.1.m1.2.2.1.1.2.cmml" xref="S2.SS1.SSS0.Px1.p2.1.m1.2.2.1.1">superscript</csymbol><apply id="S2.SS1.SSS0.Px1.p2.1.m1.2.2.1.1.1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p2.1.m1.2.2.1.1.1.1"><times id="S2.SS1.SSS0.Px1.p2.1.m1.2.2.1.1.1.1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p2.1.m1.2.2.1.1.1.1.1.1"></times><apply id="S2.SS1.SSS0.Px1.p2.1.m1.2.2.1.1.1.1.1.2.cmml" xref="S2.SS1.SSS0.Px1.p2.1.m1.2.2.1.1.1.1.1.2"><divide id="S2.SS1.SSS0.Px1.p2.1.m1.2.2.1.1.1.1.1.2.1.cmml" xref="S2.SS1.SSS0.Px1.p2.1.m1.2.2.1.1.1.1.1.2.1"></divide><ci id="S2.SS1.SSS0.Px1.p2.1.m1.2.2.1.1.1.1.1.2.2.cmml" xref="S2.SS1.SSS0.Px1.p2.1.m1.2.2.1.1.1.1.1.2.2">𝜏</ci><cn id="S2.SS1.SSS0.Px1.p2.1.m1.2.2.1.1.1.1.1.2.3.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p2.1.m1.2.2.1.1.1.1.1.2.3">1</cn></apply><ci id="S2.SS1.SSS0.Px1.p2.1.m1.2.2.1.1.1.1.1.3.cmml" xref="S2.SS1.SSS0.Px1.p2.1.m1.2.2.1.1.1.1.1.3">s</ci></apply><apply id="S2.SS1.SSS0.Px1.p2.1.m1.2.2.1.1.3.cmml" xref="S2.SS1.SSS0.Px1.p2.1.m1.2.2.1.1.3"><minus id="S2.SS1.SSS0.Px1.p2.1.m1.2.2.1.1.3.1.cmml" xref="S2.SS1.SSS0.Px1.p2.1.m1.2.2.1.1.3"></minus><apply id="S2.SS1.SSS0.Px1.p2.1.m1.2.2.1.1.3.2.cmml" xref="S2.SS1.SSS0.Px1.p2.1.m1.2.2.1.1.3.2"><divide id="S2.SS1.SSS0.Px1.p2.1.m1.2.2.1.1.3.2.1.cmml" xref="S2.SS1.SSS0.Px1.p2.1.m1.2.2.1.1.3.2.1"></divide><cn id="S2.SS1.SSS0.Px1.p2.1.m1.2.2.1.1.3.2.2.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p2.1.m1.2.2.1.1.3.2.2">1</cn><cn id="S2.SS1.SSS0.Px1.p2.1.m1.2.2.1.1.3.2.3.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p2.1.m1.2.2.1.1.3.2.3">2</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p2.1.m1.2c">\sigma_{y}(\tau)\approx 2\times 10^{-12}(\tau/1~{}{\rm s})^{-1/2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p2.1.m1.2d">italic_σ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ( italic_τ ) ≈ 2 × 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT ( italic_τ / 1 roman_s ) start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT</annotation></semantics></math> for <math alttext="\tau\lesssim 10^{5}~{}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p2.2.m2.1"><semantics id="S2.SS1.SSS0.Px1.p2.2.m2.1a"><mrow id="S2.SS1.SSS0.Px1.p2.2.m2.1.1" xref="S2.SS1.SSS0.Px1.p2.2.m2.1.1.cmml"><mi id="S2.SS1.SSS0.Px1.p2.2.m2.1.1.2" xref="S2.SS1.SSS0.Px1.p2.2.m2.1.1.2.cmml">τ</mi><mo id="S2.SS1.SSS0.Px1.p2.2.m2.1.1.1" xref="S2.SS1.SSS0.Px1.p2.2.m2.1.1.1.cmml">≲</mo><msup id="S2.SS1.SSS0.Px1.p2.2.m2.1.1.3" xref="S2.SS1.SSS0.Px1.p2.2.m2.1.1.3.cmml"><mn id="S2.SS1.SSS0.Px1.p2.2.m2.1.1.3.2" xref="S2.SS1.SSS0.Px1.p2.2.m2.1.1.3.2.cmml">10</mn><mn id="S2.SS1.SSS0.Px1.p2.2.m2.1.1.3.3" xref="S2.SS1.SSS0.Px1.p2.2.m2.1.1.3.3.cmml">5</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p2.2.m2.1b"><apply id="S2.SS1.SSS0.Px1.p2.2.m2.1.1.cmml" xref="S2.SS1.SSS0.Px1.p2.2.m2.1.1"><csymbol cd="latexml" id="S2.SS1.SSS0.Px1.p2.2.m2.1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p2.2.m2.1.1.1">less-than-or-similar-to</csymbol><ci id="S2.SS1.SSS0.Px1.p2.2.m2.1.1.2.cmml" xref="S2.SS1.SSS0.Px1.p2.2.m2.1.1.2">𝜏</ci><apply id="S2.SS1.SSS0.Px1.p2.2.m2.1.1.3.cmml" xref="S2.SS1.SSS0.Px1.p2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px1.p2.2.m2.1.1.3.1.cmml" xref="S2.SS1.SSS0.Px1.p2.2.m2.1.1.3">superscript</csymbol><cn id="S2.SS1.SSS0.Px1.p2.2.m2.1.1.3.2.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p2.2.m2.1.1.3.2">10</cn><cn id="S2.SS1.SSS0.Px1.p2.2.m2.1.1.3.3.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p2.2.m2.1.1.3.3">5</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p2.2.m2.1c">\tau\lesssim 10^{5}~{}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p2.2.m2.1d">italic_τ ≲ 10 start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT</annotation></semantics></math>s in a 2019 launch to a geostationary orbit <cite class="ltx_cite ltx_citemacro_citep">(Burt et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib19" title="">2021</a>)</cite>. The outer Solar System would avoid Earth magnetic field variations and orbital accelerations, perhaps allowing the Deep Space Atomic Clock to achieve a noise level closer to its <math alttext="10\times" class="ltx_math_unparsed" display="inline" id="S2.SS1.SSS0.Px1.p2.3.m3.1"><semantics id="S2.SS1.SSS0.Px1.p2.3.m3.1a"><mrow id="S2.SS1.SSS0.Px1.p2.3.m3.1b"><mn id="S2.SS1.SSS0.Px1.p2.3.m3.1.1">10</mn><mo id="S2.SS1.SSS0.Px1.p2.3.m3.1.2" lspace="0.222em">×</mo></mrow><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p2.3.m3.1c">10\times</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p2.3.m3.1d">10 ×</annotation></semantics></math> improved performance in the laboratory <cite class="ltx_cite ltx_citemacro_citep">(Burt et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib19" title="">2021</a>)</cite>, a precision that has also been achieved by other space-certified clocks <cite class="ltx_cite ltx_citemacro_citep">(Wang et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib71" title="">2021</a>)</cite>. The best atomic clocks on Earth have achieved <math alttext="\sigma_{y}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p2.4.m4.1"><semantics id="S2.SS1.SSS0.Px1.p2.4.m4.1a"><msub id="S2.SS1.SSS0.Px1.p2.4.m4.1.1" xref="S2.SS1.SSS0.Px1.p2.4.m4.1.1.cmml"><mi id="S2.SS1.SSS0.Px1.p2.4.m4.1.1.2" xref="S2.SS1.SSS0.Px1.p2.4.m4.1.1.2.cmml">σ</mi><mi id="S2.SS1.SSS0.Px1.p2.4.m4.1.1.3" xref="S2.SS1.SSS0.Px1.p2.4.m4.1.1.3.cmml">y</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p2.4.m4.1b"><apply id="S2.SS1.SSS0.Px1.p2.4.m4.1.1.cmml" xref="S2.SS1.SSS0.Px1.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px1.p2.4.m4.1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p2.4.m4.1.1">subscript</csymbol><ci id="S2.SS1.SSS0.Px1.p2.4.m4.1.1.2.cmml" xref="S2.SS1.SSS0.Px1.p2.4.m4.1.1.2">𝜎</ci><ci id="S2.SS1.SSS0.Px1.p2.4.m4.1.1.3.cmml" xref="S2.SS1.SSS0.Px1.p2.4.m4.1.1.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p2.4.m4.1c">\sigma_{y}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p2.4.m4.1d">italic_σ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT</annotation></semantics></math> values of <math alttext="\mathcal{O}\left(10^{-19}\right)" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p2.5.m5.1"><semantics id="S2.SS1.SSS0.Px1.p2.5.m5.1a"><mrow id="S2.SS1.SSS0.Px1.p2.5.m5.1.1" xref="S2.SS1.SSS0.Px1.p2.5.m5.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.SSS0.Px1.p2.5.m5.1.1.3" xref="S2.SS1.SSS0.Px1.p2.5.m5.1.1.3.cmml">𝒪</mi><mo id="S2.SS1.SSS0.Px1.p2.5.m5.1.1.2" xref="S2.SS1.SSS0.Px1.p2.5.m5.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.SSS0.Px1.p2.5.m5.1.1.1.1" xref="S2.SS1.SSS0.Px1.p2.5.m5.1.1.1.1.1.cmml"><mo id="S2.SS1.SSS0.Px1.p2.5.m5.1.1.1.1.2" xref="S2.SS1.SSS0.Px1.p2.5.m5.1.1.1.1.1.cmml">(</mo><msup id="S2.SS1.SSS0.Px1.p2.5.m5.1.1.1.1.1" xref="S2.SS1.SSS0.Px1.p2.5.m5.1.1.1.1.1.cmml"><mn id="S2.SS1.SSS0.Px1.p2.5.m5.1.1.1.1.1.2" xref="S2.SS1.SSS0.Px1.p2.5.m5.1.1.1.1.1.2.cmml">10</mn><mrow id="S2.SS1.SSS0.Px1.p2.5.m5.1.1.1.1.1.3" xref="S2.SS1.SSS0.Px1.p2.5.m5.1.1.1.1.1.3.cmml"><mo id="S2.SS1.SSS0.Px1.p2.5.m5.1.1.1.1.1.3a" xref="S2.SS1.SSS0.Px1.p2.5.m5.1.1.1.1.1.3.cmml">−</mo><mn id="S2.SS1.SSS0.Px1.p2.5.m5.1.1.1.1.1.3.2" xref="S2.SS1.SSS0.Px1.p2.5.m5.1.1.1.1.1.3.2.cmml">19</mn></mrow></msup><mo id="S2.SS1.SSS0.Px1.p2.5.m5.1.1.1.1.3" xref="S2.SS1.SSS0.Px1.p2.5.m5.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p2.5.m5.1b"><apply id="S2.SS1.SSS0.Px1.p2.5.m5.1.1.cmml" xref="S2.SS1.SSS0.Px1.p2.5.m5.1.1"><times id="S2.SS1.SSS0.Px1.p2.5.m5.1.1.2.cmml" xref="S2.SS1.SSS0.Px1.p2.5.m5.1.1.2"></times><ci id="S2.SS1.SSS0.Px1.p2.5.m5.1.1.3.cmml" xref="S2.SS1.SSS0.Px1.p2.5.m5.1.1.3">𝒪</ci><apply id="S2.SS1.SSS0.Px1.p2.5.m5.1.1.1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p2.5.m5.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px1.p2.5.m5.1.1.1.1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p2.5.m5.1.1.1.1">superscript</csymbol><cn id="S2.SS1.SSS0.Px1.p2.5.m5.1.1.1.1.1.2.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p2.5.m5.1.1.1.1.1.2">10</cn><apply id="S2.SS1.SSS0.Px1.p2.5.m5.1.1.1.1.1.3.cmml" xref="S2.SS1.SSS0.Px1.p2.5.m5.1.1.1.1.1.3"><minus id="S2.SS1.SSS0.Px1.p2.5.m5.1.1.1.1.1.3.1.cmml" xref="S2.SS1.SSS0.Px1.p2.5.m5.1.1.1.1.1.3"></minus><cn id="S2.SS1.SSS0.Px1.p2.5.m5.1.1.1.1.1.3.2.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p2.5.m5.1.1.1.1.1.3.2">19</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p2.5.m5.1c">\mathcal{O}\left(10^{-19}\right)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p2.5.m5.1d">caligraphic_O ( 10 start_POSTSUPERSCRIPT - 19 end_POSTSUPERSCRIPT )</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_citep">(Oelker et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib57" title="">2019</a>; Aeppli et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib4" title="">2024</a>)</cite>, and some studies have considered space-based gravitational wave detectors with atomic clocks of comparable precision <cite class="ltx_cite ltx_citemacro_citep">(Loeb & Maoz, <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib45" title="">2015</a>; Kolkowitz et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib41" title="">2016</a>)</cite>.</p> </div> <div class="ltx_para" id="S2.SS1.SSS0.Px1.p3"> <p class="ltx_p" id="S2.SS1.SSS0.Px1.p3.3">The strain sensitivity given by equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.E4" title="In Single-arm and an atomic clock ‣ 2.1 Clock noise and its mitigation ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">4</span></a>) for timing precision of the best space qualified clocks is close to the sensitivity to detect the strain from equal mass <math alttext="\gtrsim 10^{8}M_{\odot}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p3.1.m1.1"><semantics id="S2.SS1.SSS0.Px1.p3.1.m1.1a"><mrow id="S2.SS1.SSS0.Px1.p3.1.m1.1.1" xref="S2.SS1.SSS0.Px1.p3.1.m1.1.1.cmml"><mi id="S2.SS1.SSS0.Px1.p3.1.m1.1.1.2" xref="S2.SS1.SSS0.Px1.p3.1.m1.1.1.2.cmml"></mi><mo id="S2.SS1.SSS0.Px1.p3.1.m1.1.1.1" xref="S2.SS1.SSS0.Px1.p3.1.m1.1.1.1.cmml">≳</mo><mrow id="S2.SS1.SSS0.Px1.p3.1.m1.1.1.3" xref="S2.SS1.SSS0.Px1.p3.1.m1.1.1.3.cmml"><msup id="S2.SS1.SSS0.Px1.p3.1.m1.1.1.3.2" xref="S2.SS1.SSS0.Px1.p3.1.m1.1.1.3.2.cmml"><mn id="S2.SS1.SSS0.Px1.p3.1.m1.1.1.3.2.2" xref="S2.SS1.SSS0.Px1.p3.1.m1.1.1.3.2.2.cmml">10</mn><mn id="S2.SS1.SSS0.Px1.p3.1.m1.1.1.3.2.3" xref="S2.SS1.SSS0.Px1.p3.1.m1.1.1.3.2.3.cmml">8</mn></msup><mo id="S2.SS1.SSS0.Px1.p3.1.m1.1.1.3.1" xref="S2.SS1.SSS0.Px1.p3.1.m1.1.1.3.1.cmml">⁢</mo><msub id="S2.SS1.SSS0.Px1.p3.1.m1.1.1.3.3" xref="S2.SS1.SSS0.Px1.p3.1.m1.1.1.3.3.cmml"><mi id="S2.SS1.SSS0.Px1.p3.1.m1.1.1.3.3.2" xref="S2.SS1.SSS0.Px1.p3.1.m1.1.1.3.3.2.cmml">M</mi><mo id="S2.SS1.SSS0.Px1.p3.1.m1.1.1.3.3.3" xref="S2.SS1.SSS0.Px1.p3.1.m1.1.1.3.3.3.cmml">⊙</mo></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p3.1.m1.1b"><apply id="S2.SS1.SSS0.Px1.p3.1.m1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p3.1.m1.1.1"><csymbol cd="latexml" id="S2.SS1.SSS0.Px1.p3.1.m1.1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p3.1.m1.1.1.1">greater-than-or-equivalent-to</csymbol><csymbol cd="latexml" id="S2.SS1.SSS0.Px1.p3.1.m1.1.1.2.cmml" xref="S2.SS1.SSS0.Px1.p3.1.m1.1.1.2">absent</csymbol><apply id="S2.SS1.SSS0.Px1.p3.1.m1.1.1.3.cmml" xref="S2.SS1.SSS0.Px1.p3.1.m1.1.1.3"><times id="S2.SS1.SSS0.Px1.p3.1.m1.1.1.3.1.cmml" xref="S2.SS1.SSS0.Px1.p3.1.m1.1.1.3.1"></times><apply id="S2.SS1.SSS0.Px1.p3.1.m1.1.1.3.2.cmml" xref="S2.SS1.SSS0.Px1.p3.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px1.p3.1.m1.1.1.3.2.1.cmml" xref="S2.SS1.SSS0.Px1.p3.1.m1.1.1.3.2">superscript</csymbol><cn id="S2.SS1.SSS0.Px1.p3.1.m1.1.1.3.2.2.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p3.1.m1.1.1.3.2.2">10</cn><cn id="S2.SS1.SSS0.Px1.p3.1.m1.1.1.3.2.3.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p3.1.m1.1.1.3.2.3">8</cn></apply><apply id="S2.SS1.SSS0.Px1.p3.1.m1.1.1.3.3.cmml" xref="S2.SS1.SSS0.Px1.p3.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px1.p3.1.m1.1.1.3.3.1.cmml" xref="S2.SS1.SSS0.Px1.p3.1.m1.1.1.3.3">subscript</csymbol><ci id="S2.SS1.SSS0.Px1.p3.1.m1.1.1.3.3.2.cmml" xref="S2.SS1.SSS0.Px1.p3.1.m1.1.1.3.3.2">𝑀</ci><csymbol cd="latexml" id="S2.SS1.SSS0.Px1.p3.1.m1.1.1.3.3.3.cmml" xref="S2.SS1.SSS0.Px1.p3.1.m1.1.1.3.3.3">direct-product</csymbol></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p3.1.m1.1c">\gtrsim 10^{8}M_{\odot}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p3.1.m1.1d">≳ 10 start_POSTSUPERSCRIPT 8 end_POSTSUPERSCRIPT italic_M start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT</annotation></semantics></math> supermassive black hole mergers (as will be discussed in § <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S3" title="3 Strain power sensitivity ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">3</span></a>). However, a couple orders of magnitude more precise atomic clocks are likely required to reach the strain sensitivities that could justify a spacecraft relying on atomic clocks. Due to the rapid development of terrestrial atomic clocks, a one-arm mission that relies on an atomic clock may become a possibility in the near future, and such a mission would be most motivated at <math alttext="f\sim 1\mu" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p3.2.m2.1"><semantics id="S2.SS1.SSS0.Px1.p3.2.m2.1a"><mrow id="S2.SS1.SSS0.Px1.p3.2.m2.1.1" xref="S2.SS1.SSS0.Px1.p3.2.m2.1.1.cmml"><mi id="S2.SS1.SSS0.Px1.p3.2.m2.1.1.2" xref="S2.SS1.SSS0.Px1.p3.2.m2.1.1.2.cmml">f</mi><mo id="S2.SS1.SSS0.Px1.p3.2.m2.1.1.1" xref="S2.SS1.SSS0.Px1.p3.2.m2.1.1.1.cmml">∼</mo><mrow id="S2.SS1.SSS0.Px1.p3.2.m2.1.1.3" xref="S2.SS1.SSS0.Px1.p3.2.m2.1.1.3.cmml"><mn id="S2.SS1.SSS0.Px1.p3.2.m2.1.1.3.2" xref="S2.SS1.SSS0.Px1.p3.2.m2.1.1.3.2.cmml">1</mn><mo id="S2.SS1.SSS0.Px1.p3.2.m2.1.1.3.1" xref="S2.SS1.SSS0.Px1.p3.2.m2.1.1.3.1.cmml">⁢</mo><mi id="S2.SS1.SSS0.Px1.p3.2.m2.1.1.3.3" xref="S2.SS1.SSS0.Px1.p3.2.m2.1.1.3.3.cmml">μ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p3.2.m2.1b"><apply id="S2.SS1.SSS0.Px1.p3.2.m2.1.1.cmml" xref="S2.SS1.SSS0.Px1.p3.2.m2.1.1"><csymbol cd="latexml" id="S2.SS1.SSS0.Px1.p3.2.m2.1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p3.2.m2.1.1.1">similar-to</csymbol><ci id="S2.SS1.SSS0.Px1.p3.2.m2.1.1.2.cmml" xref="S2.SS1.SSS0.Px1.p3.2.m2.1.1.2">𝑓</ci><apply id="S2.SS1.SSS0.Px1.p3.2.m2.1.1.3.cmml" xref="S2.SS1.SSS0.Px1.p3.2.m2.1.1.3"><times id="S2.SS1.SSS0.Px1.p3.2.m2.1.1.3.1.cmml" xref="S2.SS1.SSS0.Px1.p3.2.m2.1.1.3.1"></times><cn id="S2.SS1.SSS0.Px1.p3.2.m2.1.1.3.2.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p3.2.m2.1.1.3.2">1</cn><ci id="S2.SS1.SSS0.Px1.p3.2.m2.1.1.3.3.cmml" xref="S2.SS1.SSS0.Px1.p3.2.m2.1.1.3.3">𝜇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p3.2.m2.1c">f\sim 1\mu</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p3.2.m2.1d">italic_f ∼ 1 italic_μ</annotation></semantics></math>Hz owing to the larger strain amplitudes from anticipated astrophysical sources. We provide estimates for the mission sensitivity for different <math alttext="\sigma_{y}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p3.3.m3.1"><semantics id="S2.SS1.SSS0.Px1.p3.3.m3.1a"><msub id="S2.SS1.SSS0.Px1.p3.3.m3.1.1" xref="S2.SS1.SSS0.Px1.p3.3.m3.1.1.cmml"><mi id="S2.SS1.SSS0.Px1.p3.3.m3.1.1.2" xref="S2.SS1.SSS0.Px1.p3.3.m3.1.1.2.cmml">σ</mi><mi id="S2.SS1.SSS0.Px1.p3.3.m3.1.1.3" xref="S2.SS1.SSS0.Px1.p3.3.m3.1.1.3.cmml">y</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p3.3.m3.1b"><apply id="S2.SS1.SSS0.Px1.p3.3.m3.1.1.cmml" xref="S2.SS1.SSS0.Px1.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px1.p3.3.m3.1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p3.3.m3.1.1">subscript</csymbol><ci id="S2.SS1.SSS0.Px1.p3.3.m3.1.1.2.cmml" xref="S2.SS1.SSS0.Px1.p3.3.m3.1.1.2">𝜎</ci><ci id="S2.SS1.SSS0.Px1.p3.3.m3.1.1.3.cmml" xref="S2.SS1.SSS0.Px1.p3.3.m3.1.1.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p3.3.m3.1c">\sigma_{y}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p3.3.m3.1d">italic_σ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT</annotation></semantics></math> in § <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S3" title="3 Strain power sensitivity ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">3</span></a>. We next review time-delay interferometry, an approach that essentially eliminates clock noise.</p> </div> </section> <section class="ltx_paragraph" id="S2.SS1.SSS0.Px2"> <h5 class="ltx_title ltx_font_bold ltx_title_paragraph">Two-arm and time-delay interferometry</h5> <div class="ltx_para" id="S2.SS1.SSS0.Px2.p1"> <p class="ltx_p" id="S2.SS1.SSS0.Px2.p1.3">The clock noise can be greatly reduced with a <math alttext="\geq 2" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.1.m1.1"><semantics id="S2.SS1.SSS0.Px2.p1.1.m1.1a"><mrow id="S2.SS1.SSS0.Px2.p1.1.m1.1.1" xref="S2.SS1.SSS0.Px2.p1.1.m1.1.1.cmml"><mi id="S2.SS1.SSS0.Px2.p1.1.m1.1.1.2" xref="S2.SS1.SSS0.Px2.p1.1.m1.1.1.2.cmml"></mi><mo id="S2.SS1.SSS0.Px2.p1.1.m1.1.1.1" xref="S2.SS1.SSS0.Px2.p1.1.m1.1.1.1.cmml">≥</mo><mn id="S2.SS1.SSS0.Px2.p1.1.m1.1.1.3" xref="S2.SS1.SSS0.Px2.p1.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.1.m1.1b"><apply id="S2.SS1.SSS0.Px2.p1.1.m1.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.1.m1.1.1"><geq id="S2.SS1.SSS0.Px2.p1.1.m1.1.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.1.m1.1.1.1"></geq><csymbol cd="latexml" id="S2.SS1.SSS0.Px2.p1.1.m1.1.1.2.cmml" xref="S2.SS1.SSS0.Px2.p1.1.m1.1.1.2">absent</csymbol><cn id="S2.SS1.SSS0.Px2.p1.1.m1.1.1.3.cmml" type="integer" xref="S2.SS1.SSS0.Px2.p1.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.1.m1.1c">\geq 2</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.1.m1.1d">≥ 2</annotation></semantics></math>-arm interferometer using a technique called time-delay interferometry <cite class="ltx_cite ltx_citemacro_citep">(Faller et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib26" title="">1985</a>; Tinto & Dhurandhar, <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib66" title="">2014</a>)</cite>. We consider a two-arm configuration with a home spacecraft broadcasting monochromatic waves (either with lasers or high-gain antennas) to two other spacecraft, with these spacecraft a distance <math alttext="L_{1}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.2.m2.1"><semantics id="S2.SS1.SSS0.Px2.p1.2.m2.1a"><msub id="S2.SS1.SSS0.Px2.p1.2.m2.1.1" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.cmml"><mi id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.2" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.2.cmml">L</mi><mn id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.3" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.2.m2.1b"><apply id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1">subscript</csymbol><ci id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.2.cmml" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.2">𝐿</ci><cn id="S2.SS1.SSS0.Px2.p1.2.m2.1.1.3.cmml" type="integer" xref="S2.SS1.SSS0.Px2.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.2.m2.1c">L_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.2.m2.1d">italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="L_{2}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.3.m3.1"><semantics id="S2.SS1.SSS0.Px2.p1.3.m3.1a"><msub id="S2.SS1.SSS0.Px2.p1.3.m3.1.1" xref="S2.SS1.SSS0.Px2.p1.3.m3.1.1.cmml"><mi id="S2.SS1.SSS0.Px2.p1.3.m3.1.1.2" xref="S2.SS1.SSS0.Px2.p1.3.m3.1.1.2.cmml">L</mi><mn id="S2.SS1.SSS0.Px2.p1.3.m3.1.1.3" xref="S2.SS1.SSS0.Px2.p1.3.m3.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.3.m3.1b"><apply id="S2.SS1.SSS0.Px2.p1.3.m3.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.3.m3.1.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.3.m3.1.1">subscript</csymbol><ci id="S2.SS1.SSS0.Px2.p1.3.m3.1.1.2.cmml" xref="S2.SS1.SSS0.Px2.p1.3.m3.1.1.2">𝐿</ci><cn id="S2.SS1.SSS0.Px2.p1.3.m3.1.1.3.cmml" type="integer" xref="S2.SS1.SSS0.Px2.p1.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.3.m3.1c">L_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.3.m3.1d">italic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> away. This wave travels to the other spacecraft and then is sent back and compared to the reference wave at the home spacecraft. The phase differences measured from each of the two arms are</p> <table class="ltx_equation ltx_eqn_table" id="S2.E5"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\phi_{1}=\phi_{\rm os}(t-2L_{1})-\phi_{\rm os}(t)+\frac{2\pi}{\lambda}L_{1}h_{% 1}(t)+\phi_{\rm 1,N};~{}~{}~{}\phi_{2}=\phi_{\rm os}(t-2L_{2})-\phi_{\rm os}(t% )+\frac{2\pi}{\lambda}L_{2}h_{2}(t)+\phi_{\rm 2,N}," class="ltx_Math" display="block" id="S2.E5.m1.9"><semantics id="S2.E5.m1.9a"><mrow id="S2.E5.m1.9.9.1"><mrow id="S2.E5.m1.9.9.1.1.2" xref="S2.E5.m1.9.9.1.1.3.cmml"><mrow id="S2.E5.m1.9.9.1.1.1.1" xref="S2.E5.m1.9.9.1.1.1.1.cmml"><msub id="S2.E5.m1.9.9.1.1.1.1.3" xref="S2.E5.m1.9.9.1.1.1.1.3.cmml"><mi id="S2.E5.m1.9.9.1.1.1.1.3.2" xref="S2.E5.m1.9.9.1.1.1.1.3.2.cmml">ϕ</mi><mn id="S2.E5.m1.9.9.1.1.1.1.3.3" xref="S2.E5.m1.9.9.1.1.1.1.3.3.cmml">1</mn></msub><mo id="S2.E5.m1.9.9.1.1.1.1.2" 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end_POSTSUBSCRIPT ( italic_t - 2 italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) - italic_ϕ start_POSTSUBSCRIPT roman_os end_POSTSUBSCRIPT ( italic_t ) + divide start_ARG 2 italic_π end_ARG start_ARG italic_λ end_ARG italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_h start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) + italic_ϕ start_POSTSUBSCRIPT 1 , roman_N end_POSTSUBSCRIPT ; italic_ϕ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_ϕ start_POSTSUBSCRIPT roman_os end_POSTSUBSCRIPT ( italic_t - 2 italic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) - italic_ϕ start_POSTSUBSCRIPT roman_os end_POSTSUBSCRIPT ( italic_t ) + divide start_ARG 2 italic_π end_ARG start_ARG italic_λ end_ARG italic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_h start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) + italic_ϕ start_POSTSUBSCRIPT 2 , roman_N end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(5)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.SSS0.Px2.p1.13">following the same conventions as in equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.E1" title="In Single-arm and an atomic clock ‣ 2.1 Clock noise and its mitigation ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">1</span></a>). 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id="S2.E6.m1.4.4.1.1.2.2.1.1.3.2.2.cmml" xref="S2.E6.m1.4.4.1.1.2.2.1.1.3.2.2">italic-ϕ</ci><cn id="S2.E6.m1.4.4.1.1.2.2.1.1.3.2.3.cmml" type="integer" xref="S2.E6.m1.4.4.1.1.2.2.1.1.3.2.3">1</cn></apply><ci id="S2.E6.m1.3.3.cmml" xref="S2.E6.m1.3.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E6.m1.4c">X(t)=[\phi_{2}(t-2L_{1})-\phi_{2}(t)]-[\phi_{1}(t-2L_{2})-\phi_{1}(t)].</annotation><annotation encoding="application/x-llamapun" id="S2.E6.m1.4d">italic_X ( italic_t ) = [ italic_ϕ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t - 2 italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) - italic_ϕ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) ] - [ italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t - 2 italic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) - italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) ] .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.SSS0.Px2.p1.12">Remarkably, the clock noise – <math alttext="\phi_{\rm os}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.4.m1.1"><semantics id="S2.SS1.SSS0.Px2.p1.4.m1.1a"><msub id="S2.SS1.SSS0.Px2.p1.4.m1.1.1" xref="S2.SS1.SSS0.Px2.p1.4.m1.1.1.cmml"><mi id="S2.SS1.SSS0.Px2.p1.4.m1.1.1.2" xref="S2.SS1.SSS0.Px2.p1.4.m1.1.1.2.cmml">ϕ</mi><mi id="S2.SS1.SSS0.Px2.p1.4.m1.1.1.3" xref="S2.SS1.SSS0.Px2.p1.4.m1.1.1.3.cmml">os</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.4.m1.1b"><apply id="S2.SS1.SSS0.Px2.p1.4.m1.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.4.m1.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.4.m1.1.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.4.m1.1.1">subscript</csymbol><ci id="S2.SS1.SSS0.Px2.p1.4.m1.1.1.2.cmml" xref="S2.SS1.SSS0.Px2.p1.4.m1.1.1.2">italic-ϕ</ci><ci id="S2.SS1.SSS0.Px2.p1.4.m1.1.1.3.cmml" xref="S2.SS1.SSS0.Px2.p1.4.m1.1.1.3">os</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.4.m1.1c">\phi_{\rm os}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.4.m1.1d">italic_ϕ start_POSTSUBSCRIPT roman_os end_POSTSUBSCRIPT</annotation></semantics></math> – cancels in this expression – and, when considering a single frequency <math alttext="f" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.5.m2.1"><semantics id="S2.SS1.SSS0.Px2.p1.5.m2.1a"><mi id="S2.SS1.SSS0.Px2.p1.5.m2.1.1" xref="S2.SS1.SSS0.Px2.p1.5.m2.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.5.m2.1b"><ci id="S2.SS1.SSS0.Px2.p1.5.m2.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.5.m2.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.5.m2.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.5.m2.1d">italic_f</annotation></semantics></math> and the limit <math alttext="L_{1}=L_{2}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.6.m3.1"><semantics id="S2.SS1.SSS0.Px2.p1.6.m3.1a"><mrow id="S2.SS1.SSS0.Px2.p1.6.m3.1.1" xref="S2.SS1.SSS0.Px2.p1.6.m3.1.1.cmml"><msub id="S2.SS1.SSS0.Px2.p1.6.m3.1.1.2" xref="S2.SS1.SSS0.Px2.p1.6.m3.1.1.2.cmml"><mi id="S2.SS1.SSS0.Px2.p1.6.m3.1.1.2.2" xref="S2.SS1.SSS0.Px2.p1.6.m3.1.1.2.2.cmml">L</mi><mn id="S2.SS1.SSS0.Px2.p1.6.m3.1.1.2.3" xref="S2.SS1.SSS0.Px2.p1.6.m3.1.1.2.3.cmml">1</mn></msub><mo id="S2.SS1.SSS0.Px2.p1.6.m3.1.1.1" xref="S2.SS1.SSS0.Px2.p1.6.m3.1.1.1.cmml">=</mo><msub id="S2.SS1.SSS0.Px2.p1.6.m3.1.1.3" xref="S2.SS1.SSS0.Px2.p1.6.m3.1.1.3.cmml"><mi id="S2.SS1.SSS0.Px2.p1.6.m3.1.1.3.2" xref="S2.SS1.SSS0.Px2.p1.6.m3.1.1.3.2.cmml">L</mi><mn id="S2.SS1.SSS0.Px2.p1.6.m3.1.1.3.3" xref="S2.SS1.SSS0.Px2.p1.6.m3.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.6.m3.1b"><apply id="S2.SS1.SSS0.Px2.p1.6.m3.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.6.m3.1.1"><eq id="S2.SS1.SSS0.Px2.p1.6.m3.1.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.6.m3.1.1.1"></eq><apply id="S2.SS1.SSS0.Px2.p1.6.m3.1.1.2.cmml" xref="S2.SS1.SSS0.Px2.p1.6.m3.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.6.m3.1.1.2.1.cmml" xref="S2.SS1.SSS0.Px2.p1.6.m3.1.1.2">subscript</csymbol><ci id="S2.SS1.SSS0.Px2.p1.6.m3.1.1.2.2.cmml" xref="S2.SS1.SSS0.Px2.p1.6.m3.1.1.2.2">𝐿</ci><cn id="S2.SS1.SSS0.Px2.p1.6.m3.1.1.2.3.cmml" type="integer" xref="S2.SS1.SSS0.Px2.p1.6.m3.1.1.2.3">1</cn></apply><apply id="S2.SS1.SSS0.Px2.p1.6.m3.1.1.3.cmml" xref="S2.SS1.SSS0.Px2.p1.6.m3.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.6.m3.1.1.3.1.cmml" xref="S2.SS1.SSS0.Px2.p1.6.m3.1.1.3">subscript</csymbol><ci id="S2.SS1.SSS0.Px2.p1.6.m3.1.1.3.2.cmml" xref="S2.SS1.SSS0.Px2.p1.6.m3.1.1.3.2">𝐿</ci><cn id="S2.SS1.SSS0.Px2.p1.6.m3.1.1.3.3.cmml" type="integer" xref="S2.SS1.SSS0.Px2.p1.6.m3.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.6.m3.1c">L_{1}=L_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.6.m3.1d">italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, other noise sources are suppressed by the same factor as <math alttext="h_{1}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.7.m4.1"><semantics id="S2.SS1.SSS0.Px2.p1.7.m4.1a"><msub id="S2.SS1.SSS0.Px2.p1.7.m4.1.1" xref="S2.SS1.SSS0.Px2.p1.7.m4.1.1.cmml"><mi id="S2.SS1.SSS0.Px2.p1.7.m4.1.1.2" xref="S2.SS1.SSS0.Px2.p1.7.m4.1.1.2.cmml">h</mi><mn id="S2.SS1.SSS0.Px2.p1.7.m4.1.1.3" xref="S2.SS1.SSS0.Px2.p1.7.m4.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.7.m4.1b"><apply id="S2.SS1.SSS0.Px2.p1.7.m4.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.7.m4.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.7.m4.1.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.7.m4.1.1">subscript</csymbol><ci id="S2.SS1.SSS0.Px2.p1.7.m4.1.1.2.cmml" xref="S2.SS1.SSS0.Px2.p1.7.m4.1.1.2">ℎ</ci><cn id="S2.SS1.SSS0.Px2.p1.7.m4.1.1.3.cmml" type="integer" xref="S2.SS1.SSS0.Px2.p1.7.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.7.m4.1c">h_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.7.m4.1d">italic_h start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> in this estimator – meaning that <math alttext="X(t)" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.8.m5.1"><semantics id="S2.SS1.SSS0.Px2.p1.8.m5.1a"><mrow id="S2.SS1.SSS0.Px2.p1.8.m5.1.2" xref="S2.SS1.SSS0.Px2.p1.8.m5.1.2.cmml"><mi id="S2.SS1.SSS0.Px2.p1.8.m5.1.2.2" xref="S2.SS1.SSS0.Px2.p1.8.m5.1.2.2.cmml">X</mi><mo id="S2.SS1.SSS0.Px2.p1.8.m5.1.2.1" xref="S2.SS1.SSS0.Px2.p1.8.m5.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.SSS0.Px2.p1.8.m5.1.2.3.2" xref="S2.SS1.SSS0.Px2.p1.8.m5.1.2.cmml"><mo id="S2.SS1.SSS0.Px2.p1.8.m5.1.2.3.2.1" stretchy="false" xref="S2.SS1.SSS0.Px2.p1.8.m5.1.2.cmml">(</mo><mi id="S2.SS1.SSS0.Px2.p1.8.m5.1.1" xref="S2.SS1.SSS0.Px2.p1.8.m5.1.1.cmml">t</mi><mo id="S2.SS1.SSS0.Px2.p1.8.m5.1.2.3.2.2" stretchy="false" xref="S2.SS1.SSS0.Px2.p1.8.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.8.m5.1b"><apply id="S2.SS1.SSS0.Px2.p1.8.m5.1.2.cmml" xref="S2.SS1.SSS0.Px2.p1.8.m5.1.2"><times id="S2.SS1.SSS0.Px2.p1.8.m5.1.2.1.cmml" xref="S2.SS1.SSS0.Px2.p1.8.m5.1.2.1"></times><ci id="S2.SS1.SSS0.Px2.p1.8.m5.1.2.2.cmml" xref="S2.SS1.SSS0.Px2.p1.8.m5.1.2.2">𝑋</ci><ci id="S2.SS1.SSS0.Px2.p1.8.m5.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.8.m5.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.8.m5.1c">X(t)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.8.m5.1d">italic_X ( italic_t )</annotation></semantics></math>’s sensitivity to <math alttext="h_{1}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.9.m6.1"><semantics id="S2.SS1.SSS0.Px2.p1.9.m6.1a"><msub id="S2.SS1.SSS0.Px2.p1.9.m6.1.1" xref="S2.SS1.SSS0.Px2.p1.9.m6.1.1.cmml"><mi id="S2.SS1.SSS0.Px2.p1.9.m6.1.1.2" xref="S2.SS1.SSS0.Px2.p1.9.m6.1.1.2.cmml">h</mi><mn id="S2.SS1.SSS0.Px2.p1.9.m6.1.1.3" xref="S2.SS1.SSS0.Px2.p1.9.m6.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.9.m6.1b"><apply id="S2.SS1.SSS0.Px2.p1.9.m6.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.9.m6.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.9.m6.1.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.9.m6.1.1">subscript</csymbol><ci id="S2.SS1.SSS0.Px2.p1.9.m6.1.1.2.cmml" xref="S2.SS1.SSS0.Px2.p1.9.m6.1.1.2">ℎ</ci><cn id="S2.SS1.SSS0.Px2.p1.9.m6.1.1.3.cmml" type="integer" xref="S2.SS1.SSS0.Px2.p1.9.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.9.m6.1c">h_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.9.m6.1d">italic_h start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> is the same as that of <math alttext="\phi_{1}-\phi_{2}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.10.m7.1"><semantics id="S2.SS1.SSS0.Px2.p1.10.m7.1a"><mrow id="S2.SS1.SSS0.Px2.p1.10.m7.1.1" xref="S2.SS1.SSS0.Px2.p1.10.m7.1.1.cmml"><msub id="S2.SS1.SSS0.Px2.p1.10.m7.1.1.2" xref="S2.SS1.SSS0.Px2.p1.10.m7.1.1.2.cmml"><mi id="S2.SS1.SSS0.Px2.p1.10.m7.1.1.2.2" xref="S2.SS1.SSS0.Px2.p1.10.m7.1.1.2.2.cmml">ϕ</mi><mn id="S2.SS1.SSS0.Px2.p1.10.m7.1.1.2.3" xref="S2.SS1.SSS0.Px2.p1.10.m7.1.1.2.3.cmml">1</mn></msub><mo id="S2.SS1.SSS0.Px2.p1.10.m7.1.1.1" xref="S2.SS1.SSS0.Px2.p1.10.m7.1.1.1.cmml">−</mo><msub id="S2.SS1.SSS0.Px2.p1.10.m7.1.1.3" xref="S2.SS1.SSS0.Px2.p1.10.m7.1.1.3.cmml"><mi id="S2.SS1.SSS0.Px2.p1.10.m7.1.1.3.2" xref="S2.SS1.SSS0.Px2.p1.10.m7.1.1.3.2.cmml">ϕ</mi><mn id="S2.SS1.SSS0.Px2.p1.10.m7.1.1.3.3" xref="S2.SS1.SSS0.Px2.p1.10.m7.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.10.m7.1b"><apply id="S2.SS1.SSS0.Px2.p1.10.m7.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.10.m7.1.1"><minus id="S2.SS1.SSS0.Px2.p1.10.m7.1.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.10.m7.1.1.1"></minus><apply id="S2.SS1.SSS0.Px2.p1.10.m7.1.1.2.cmml" xref="S2.SS1.SSS0.Px2.p1.10.m7.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.10.m7.1.1.2.1.cmml" xref="S2.SS1.SSS0.Px2.p1.10.m7.1.1.2">subscript</csymbol><ci id="S2.SS1.SSS0.Px2.p1.10.m7.1.1.2.2.cmml" xref="S2.SS1.SSS0.Px2.p1.10.m7.1.1.2.2">italic-ϕ</ci><cn id="S2.SS1.SSS0.Px2.p1.10.m7.1.1.2.3.cmml" type="integer" xref="S2.SS1.SSS0.Px2.p1.10.m7.1.1.2.3">1</cn></apply><apply id="S2.SS1.SSS0.Px2.p1.10.m7.1.1.3.cmml" xref="S2.SS1.SSS0.Px2.p1.10.m7.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.10.m7.1.1.3.1.cmml" xref="S2.SS1.SSS0.Px2.p1.10.m7.1.1.3">subscript</csymbol><ci id="S2.SS1.SSS0.Px2.p1.10.m7.1.1.3.2.cmml" xref="S2.SS1.SSS0.Px2.p1.10.m7.1.1.3.2">italic-ϕ</ci><cn id="S2.SS1.SSS0.Px2.p1.10.m7.1.1.3.3.cmml" type="integer" xref="S2.SS1.SSS0.Px2.p1.10.m7.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.10.m7.1c">\phi_{1}-\phi_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.10.m7.1d">italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_ϕ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> but without clock noise.<span class="ltx_note ltx_role_footnote" id="footnote2"><sup class="ltx_note_mark">2</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">2</sup><span class="ltx_tag ltx_tag_note">2</span>In practice, the arm lengths are not perfectly known, leading to a noise like equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.E3" title="In Single-arm and an atomic clock ‣ 2.1 Clock noise and its mitigation ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">3</span></a>) that does not cancel in the time-delay observable but where <math alttext="L\rightarrow\delta L" class="ltx_Math" display="inline" id="footnote2.m1.1"><semantics id="footnote2.m1.1b"><mrow id="footnote2.m1.1.1" xref="footnote2.m1.1.1.cmml"><mi id="footnote2.m1.1.1.2" xref="footnote2.m1.1.1.2.cmml">L</mi><mo id="footnote2.m1.1.1.1" stretchy="false" xref="footnote2.m1.1.1.1.cmml">→</mo><mrow id="footnote2.m1.1.1.3" xref="footnote2.m1.1.1.3.cmml"><mi id="footnote2.m1.1.1.3.2" xref="footnote2.m1.1.1.3.2.cmml">δ</mi><mo id="footnote2.m1.1.1.3.1" xref="footnote2.m1.1.1.3.1.cmml">⁢</mo><mi id="footnote2.m1.1.1.3.3" xref="footnote2.m1.1.1.3.3.cmml">L</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote2.m1.1c"><apply id="footnote2.m1.1.1.cmml" xref="footnote2.m1.1.1"><ci id="footnote2.m1.1.1.1.cmml" xref="footnote2.m1.1.1.1">→</ci><ci id="footnote2.m1.1.1.2.cmml" xref="footnote2.m1.1.1.2">𝐿</ci><apply id="footnote2.m1.1.1.3.cmml" xref="footnote2.m1.1.1.3"><times id="footnote2.m1.1.1.3.1.cmml" xref="footnote2.m1.1.1.3.1"></times><ci id="footnote2.m1.1.1.3.2.cmml" xref="footnote2.m1.1.1.3.2">𝛿</ci><ci id="footnote2.m1.1.1.3.3.cmml" xref="footnote2.m1.1.1.3.3">𝐿</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote2.m1.1d">L\rightarrow\delta L</annotation><annotation encoding="application/x-llamapun" id="footnote2.m1.1e">italic_L → italic_δ italic_L</annotation></semantics></math>, where <math alttext="\delta L" class="ltx_Math" display="inline" id="footnote2.m2.1"><semantics id="footnote2.m2.1b"><mrow id="footnote2.m2.1.1" xref="footnote2.m2.1.1.cmml"><mi id="footnote2.m2.1.1.2" xref="footnote2.m2.1.1.2.cmml">δ</mi><mo id="footnote2.m2.1.1.1" xref="footnote2.m2.1.1.1.cmml">⁢</mo><mi id="footnote2.m2.1.1.3" xref="footnote2.m2.1.1.3.cmml">L</mi></mrow><annotation-xml encoding="MathML-Content" id="footnote2.m2.1c"><apply id="footnote2.m2.1.1.cmml" xref="footnote2.m2.1.1"><times id="footnote2.m2.1.1.1.cmml" xref="footnote2.m2.1.1.1"></times><ci id="footnote2.m2.1.1.2.cmml" xref="footnote2.m2.1.1.2">𝛿</ci><ci id="footnote2.m2.1.1.3.cmml" xref="footnote2.m2.1.1.3">𝐿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote2.m2.1d">\delta L</annotation><annotation encoding="application/x-llamapun" id="footnote2.m2.1e">italic_δ italic_L</annotation></semantics></math> is the uncertainty in the arm lengths <cite class="ltx_cite ltx_citemacro_citep">(Barke et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib14" title="">2015</a>)</cite>. This leads to the RMS strain error being suppressed by the factor <math alttext="\sim\delta L/L" class="ltx_Math" display="inline" id="footnote2.m3.1"><semantics id="footnote2.m3.1b"><mrow id="footnote2.m3.1.1" xref="footnote2.m3.1.1.cmml"><mi id="footnote2.m3.1.1.2" xref="footnote2.m3.1.1.2.cmml"></mi><mo id="footnote2.m3.1.1.1" xref="footnote2.m3.1.1.1.cmml">∼</mo><mrow id="footnote2.m3.1.1.3" xref="footnote2.m3.1.1.3.cmml"><mrow id="footnote2.m3.1.1.3.2" xref="footnote2.m3.1.1.3.2.cmml"><mi id="footnote2.m3.1.1.3.2.2" xref="footnote2.m3.1.1.3.2.2.cmml">δ</mi><mo id="footnote2.m3.1.1.3.2.1" xref="footnote2.m3.1.1.3.2.1.cmml">⁢</mo><mi id="footnote2.m3.1.1.3.2.3" xref="footnote2.m3.1.1.3.2.3.cmml">L</mi></mrow><mo id="footnote2.m3.1.1.3.1" xref="footnote2.m3.1.1.3.1.cmml">/</mo><mi id="footnote2.m3.1.1.3.3" xref="footnote2.m3.1.1.3.3.cmml">L</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote2.m3.1c"><apply id="footnote2.m3.1.1.cmml" xref="footnote2.m3.1.1"><csymbol cd="latexml" id="footnote2.m3.1.1.1.cmml" xref="footnote2.m3.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="footnote2.m3.1.1.2.cmml" xref="footnote2.m3.1.1.2">absent</csymbol><apply id="footnote2.m3.1.1.3.cmml" xref="footnote2.m3.1.1.3"><divide id="footnote2.m3.1.1.3.1.cmml" xref="footnote2.m3.1.1.3.1"></divide><apply id="footnote2.m3.1.1.3.2.cmml" xref="footnote2.m3.1.1.3.2"><times id="footnote2.m3.1.1.3.2.1.cmml" xref="footnote2.m3.1.1.3.2.1"></times><ci id="footnote2.m3.1.1.3.2.2.cmml" xref="footnote2.m3.1.1.3.2.2">𝛿</ci><ci id="footnote2.m3.1.1.3.2.3.cmml" xref="footnote2.m3.1.1.3.2.3">𝐿</ci></apply><ci id="footnote2.m3.1.1.3.3.cmml" xref="footnote2.m3.1.1.3.3">𝐿</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote2.m3.1d">\sim\delta L/L</annotation><annotation encoding="application/x-llamapun" id="footnote2.m3.1e">∼ italic_δ italic_L / italic_L</annotation></semantics></math> compared to in equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.E4" title="In Single-arm and an atomic clock ‣ 2.1 Clock noise and its mitigation ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">4</span></a>), where if the positions of the detectors are known to tens of centimeter precision as is reasonable for ranging on Solar System scales <cite class="ltx_cite ltx_citemacro_citep">(Boone & McQuinn, <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib18" title="">2023</a>)</cite>, this leads to negligible clock noise. Additionally, equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.E6" title="In Two-arm and time-delay interferometry ‣ 2.1 Clock noise and its mitigation ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">6</span></a>) is one example of a synthetic time-delay interferometry observable that can be constructed. More advanced combinations can be created that can, for example, further suppress the clock noise even when the arm lengths are changing <cite class="ltx_cite ltx_citemacro_citep">(see, for example, Shaddock et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib63" title="">2003</a>)</cite>. </span></span></span> For our calculations, the knowledge that <math alttext="X(t)" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.11.m8.1"><semantics id="S2.SS1.SSS0.Px2.p1.11.m8.1a"><mrow id="S2.SS1.SSS0.Px2.p1.11.m8.1.2" xref="S2.SS1.SSS0.Px2.p1.11.m8.1.2.cmml"><mi id="S2.SS1.SSS0.Px2.p1.11.m8.1.2.2" xref="S2.SS1.SSS0.Px2.p1.11.m8.1.2.2.cmml">X</mi><mo id="S2.SS1.SSS0.Px2.p1.11.m8.1.2.1" xref="S2.SS1.SSS0.Px2.p1.11.m8.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.SSS0.Px2.p1.11.m8.1.2.3.2" xref="S2.SS1.SSS0.Px2.p1.11.m8.1.2.cmml"><mo id="S2.SS1.SSS0.Px2.p1.11.m8.1.2.3.2.1" stretchy="false" xref="S2.SS1.SSS0.Px2.p1.11.m8.1.2.cmml">(</mo><mi id="S2.SS1.SSS0.Px2.p1.11.m8.1.1" xref="S2.SS1.SSS0.Px2.p1.11.m8.1.1.cmml">t</mi><mo id="S2.SS1.SSS0.Px2.p1.11.m8.1.2.3.2.2" stretchy="false" xref="S2.SS1.SSS0.Px2.p1.11.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.11.m8.1b"><apply id="S2.SS1.SSS0.Px2.p1.11.m8.1.2.cmml" xref="S2.SS1.SSS0.Px2.p1.11.m8.1.2"><times id="S2.SS1.SSS0.Px2.p1.11.m8.1.2.1.cmml" xref="S2.SS1.SSS0.Px2.p1.11.m8.1.2.1"></times><ci id="S2.SS1.SSS0.Px2.p1.11.m8.1.2.2.cmml" xref="S2.SS1.SSS0.Px2.p1.11.m8.1.2.2">𝑋</ci><ci id="S2.SS1.SSS0.Px2.p1.11.m8.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.11.m8.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.11.m8.1c">X(t)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.11.m8.1d">italic_X ( italic_t )</annotation></semantics></math> exists allows us to consider the interferometric observable <math alttext="\phi_{2}-\phi_{1}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px2.p1.12.m9.1"><semantics id="S2.SS1.SSS0.Px2.p1.12.m9.1a"><mrow id="S2.SS1.SSS0.Px2.p1.12.m9.1.1" xref="S2.SS1.SSS0.Px2.p1.12.m9.1.1.cmml"><msub id="S2.SS1.SSS0.Px2.p1.12.m9.1.1.2" xref="S2.SS1.SSS0.Px2.p1.12.m9.1.1.2.cmml"><mi id="S2.SS1.SSS0.Px2.p1.12.m9.1.1.2.2" xref="S2.SS1.SSS0.Px2.p1.12.m9.1.1.2.2.cmml">ϕ</mi><mn id="S2.SS1.SSS0.Px2.p1.12.m9.1.1.2.3" xref="S2.SS1.SSS0.Px2.p1.12.m9.1.1.2.3.cmml">2</mn></msub><mo id="S2.SS1.SSS0.Px2.p1.12.m9.1.1.1" xref="S2.SS1.SSS0.Px2.p1.12.m9.1.1.1.cmml">−</mo><msub id="S2.SS1.SSS0.Px2.p1.12.m9.1.1.3" xref="S2.SS1.SSS0.Px2.p1.12.m9.1.1.3.cmml"><mi id="S2.SS1.SSS0.Px2.p1.12.m9.1.1.3.2" xref="S2.SS1.SSS0.Px2.p1.12.m9.1.1.3.2.cmml">ϕ</mi><mn id="S2.SS1.SSS0.Px2.p1.12.m9.1.1.3.3" xref="S2.SS1.SSS0.Px2.p1.12.m9.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px2.p1.12.m9.1b"><apply id="S2.SS1.SSS0.Px2.p1.12.m9.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.12.m9.1.1"><minus id="S2.SS1.SSS0.Px2.p1.12.m9.1.1.1.cmml" xref="S2.SS1.SSS0.Px2.p1.12.m9.1.1.1"></minus><apply id="S2.SS1.SSS0.Px2.p1.12.m9.1.1.2.cmml" xref="S2.SS1.SSS0.Px2.p1.12.m9.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.12.m9.1.1.2.1.cmml" xref="S2.SS1.SSS0.Px2.p1.12.m9.1.1.2">subscript</csymbol><ci id="S2.SS1.SSS0.Px2.p1.12.m9.1.1.2.2.cmml" xref="S2.SS1.SSS0.Px2.p1.12.m9.1.1.2.2">italic-ϕ</ci><cn id="S2.SS1.SSS0.Px2.p1.12.m9.1.1.2.3.cmml" type="integer" xref="S2.SS1.SSS0.Px2.p1.12.m9.1.1.2.3">2</cn></apply><apply id="S2.SS1.SSS0.Px2.p1.12.m9.1.1.3.cmml" xref="S2.SS1.SSS0.Px2.p1.12.m9.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px2.p1.12.m9.1.1.3.1.cmml" xref="S2.SS1.SSS0.Px2.p1.12.m9.1.1.3">subscript</csymbol><ci id="S2.SS1.SSS0.Px2.p1.12.m9.1.1.3.2.cmml" xref="S2.SS1.SSS0.Px2.p1.12.m9.1.1.3.2">italic-ϕ</ci><cn id="S2.SS1.SSS0.Px2.p1.12.m9.1.1.3.3.cmml" type="integer" xref="S2.SS1.SSS0.Px2.p1.12.m9.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px2.p1.12.m9.1c">\phi_{2}-\phi_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px2.p1.12.m9.1d">italic_ϕ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> in the absence of clock noise.<span class="ltx_note ltx_role_footnote" id="footnote3"><sup class="ltx_note_mark">3</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">3</sup><span class="ltx_tag ltx_tag_note">3</span>For the case <math alttext="L_{1}<L_{2}" class="ltx_Math" display="inline" id="footnote3.m1.1"><semantics id="footnote3.m1.1b"><mrow id="footnote3.m1.1.1" xref="footnote3.m1.1.1.cmml"><msub id="footnote3.m1.1.1.2" xref="footnote3.m1.1.1.2.cmml"><mi id="footnote3.m1.1.1.2.2" xref="footnote3.m1.1.1.2.2.cmml">L</mi><mn id="footnote3.m1.1.1.2.3" xref="footnote3.m1.1.1.2.3.cmml">1</mn></msub><mo id="footnote3.m1.1.1.1" xref="footnote3.m1.1.1.1.cmml"><</mo><msub id="footnote3.m1.1.1.3" xref="footnote3.m1.1.1.3.cmml"><mi id="footnote3.m1.1.1.3.2" xref="footnote3.m1.1.1.3.2.cmml">L</mi><mn id="footnote3.m1.1.1.3.3" xref="footnote3.m1.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="footnote3.m1.1c"><apply id="footnote3.m1.1.1.cmml" xref="footnote3.m1.1.1"><lt id="footnote3.m1.1.1.1.cmml" xref="footnote3.m1.1.1.1"></lt><apply id="footnote3.m1.1.1.2.cmml" xref="footnote3.m1.1.1.2"><csymbol cd="ambiguous" id="footnote3.m1.1.1.2.1.cmml" xref="footnote3.m1.1.1.2">subscript</csymbol><ci id="footnote3.m1.1.1.2.2.cmml" xref="footnote3.m1.1.1.2.2">𝐿</ci><cn id="footnote3.m1.1.1.2.3.cmml" type="integer" xref="footnote3.m1.1.1.2.3">1</cn></apply><apply id="footnote3.m1.1.1.3.cmml" xref="footnote3.m1.1.1.3"><csymbol cd="ambiguous" id="footnote3.m1.1.1.3.1.cmml" xref="footnote3.m1.1.1.3">subscript</csymbol><ci id="footnote3.m1.1.1.3.2.cmml" xref="footnote3.m1.1.1.3.2">𝐿</ci><cn id="footnote3.m1.1.1.3.3.cmml" type="integer" xref="footnote3.m1.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote3.m1.1d">L_{1}<L_{2}</annotation><annotation encoding="application/x-llamapun" id="footnote3.m1.1e">italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT < italic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> and in the long-wavelength limit <math alttext="c/f\gg L_{1}" class="ltx_Math" display="inline" id="footnote3.m2.1"><semantics id="footnote3.m2.1b"><mrow id="footnote3.m2.1.1" xref="footnote3.m2.1.1.cmml"><mrow id="footnote3.m2.1.1.2" xref="footnote3.m2.1.1.2.cmml"><mi id="footnote3.m2.1.1.2.2" xref="footnote3.m2.1.1.2.2.cmml">c</mi><mo id="footnote3.m2.1.1.2.1" xref="footnote3.m2.1.1.2.1.cmml">/</mo><mi id="footnote3.m2.1.1.2.3" xref="footnote3.m2.1.1.2.3.cmml">f</mi></mrow><mo id="footnote3.m2.1.1.1" xref="footnote3.m2.1.1.1.cmml">≫</mo><msub id="footnote3.m2.1.1.3" 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xref="footnote3.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote3.m2.1d">c/f\gg L_{1}</annotation><annotation encoding="application/x-llamapun" id="footnote3.m2.1e">italic_c / italic_f ≫ italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> the noise of this time-delay interferometry observable is increased by <math alttext="L_{2}/L_{1}" class="ltx_Math" display="inline" id="footnote3.m3.1"><semantics id="footnote3.m3.1b"><mrow id="footnote3.m3.1.1" xref="footnote3.m3.1.1.cmml"><msub id="footnote3.m3.1.1.2" xref="footnote3.m3.1.1.2.cmml"><mi id="footnote3.m3.1.1.2.2" xref="footnote3.m3.1.1.2.2.cmml">L</mi><mn id="footnote3.m3.1.1.2.3" xref="footnote3.m3.1.1.2.3.cmml">2</mn></msub><mo id="footnote3.m3.1.1.1" xref="footnote3.m3.1.1.1.cmml">/</mo><msub id="footnote3.m3.1.1.3" xref="footnote3.m3.1.1.3.cmml"><mi id="footnote3.m3.1.1.3.2" xref="footnote3.m3.1.1.3.2.cmml">L</mi><mn id="footnote3.m3.1.1.3.3" xref="footnote3.m3.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="footnote3.m3.1c"><apply id="footnote3.m3.1.1.cmml" xref="footnote3.m3.1.1"><divide id="footnote3.m3.1.1.1.cmml" xref="footnote3.m3.1.1.1"></divide><apply id="footnote3.m3.1.1.2.cmml" xref="footnote3.m3.1.1.2"><csymbol cd="ambiguous" id="footnote3.m3.1.1.2.1.cmml" xref="footnote3.m3.1.1.2">subscript</csymbol><ci id="footnote3.m3.1.1.2.2.cmml" xref="footnote3.m3.1.1.2.2">𝐿</ci><cn id="footnote3.m3.1.1.2.3.cmml" type="integer" xref="footnote3.m3.1.1.2.3">2</cn></apply><apply id="footnote3.m3.1.1.3.cmml" xref="footnote3.m3.1.1.3"><csymbol cd="ambiguous" id="footnote3.m3.1.1.3.1.cmml" xref="footnote3.m3.1.1.3">subscript</csymbol><ci id="footnote3.m3.1.1.3.2.cmml" xref="footnote3.m3.1.1.3.2">𝐿</ci><cn id="footnote3.m3.1.1.3.3.cmml" type="integer" xref="footnote3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote3.m3.1d">L_{2}/L_{1}</annotation><annotation encoding="application/x-llamapun" id="footnote3.m3.1e">italic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT / italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> relative to what we would compute from the observable <math alttext="\phi_{1}-\phi_{2}" class="ltx_Math" display="inline" id="footnote3.m4.1"><semantics id="footnote3.m4.1b"><mrow id="footnote3.m4.1.1" xref="footnote3.m4.1.1.cmml"><msub id="footnote3.m4.1.1.2" xref="footnote3.m4.1.1.2.cmml"><mi id="footnote3.m4.1.1.2.2" xref="footnote3.m4.1.1.2.2.cmml">ϕ</mi><mn id="footnote3.m4.1.1.2.3" xref="footnote3.m4.1.1.2.3.cmml">1</mn></msub><mo id="footnote3.m4.1.1.1" xref="footnote3.m4.1.1.1.cmml">−</mo><msub id="footnote3.m4.1.1.3" xref="footnote3.m4.1.1.3.cmml"><mi id="footnote3.m4.1.1.3.2" xref="footnote3.m4.1.1.3.2.cmml">ϕ</mi><mn id="footnote3.m4.1.1.3.3" xref="footnote3.m4.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="footnote3.m4.1c"><apply id="footnote3.m4.1.1.cmml" xref="footnote3.m4.1.1"><minus id="footnote3.m4.1.1.1.cmml" xref="footnote3.m4.1.1.1"></minus><apply id="footnote3.m4.1.1.2.cmml" xref="footnote3.m4.1.1.2"><csymbol cd="ambiguous" id="footnote3.m4.1.1.2.1.cmml" xref="footnote3.m4.1.1.2">subscript</csymbol><ci id="footnote3.m4.1.1.2.2.cmml" xref="footnote3.m4.1.1.2.2">italic-ϕ</ci><cn id="footnote3.m4.1.1.2.3.cmml" type="integer" xref="footnote3.m4.1.1.2.3">1</cn></apply><apply id="footnote3.m4.1.1.3.cmml" xref="footnote3.m4.1.1.3"><csymbol cd="ambiguous" id="footnote3.m4.1.1.3.1.cmml" xref="footnote3.m4.1.1.3">subscript</csymbol><ci id="footnote3.m4.1.1.3.2.cmml" xref="footnote3.m4.1.1.3.2">italic-ϕ</ci><cn id="footnote3.m4.1.1.3.3.cmml" type="integer" xref="footnote3.m4.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote3.m4.1d">\phi_{1}-\phi_{2}</annotation><annotation encoding="application/x-llamapun" id="footnote3.m4.1e">italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_ϕ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> without clock noise and if both arms have length <math alttext="L_{2}" class="ltx_Math" display="inline" id="footnote3.m5.1"><semantics id="footnote3.m5.1b"><msub id="footnote3.m5.1.1" xref="footnote3.m5.1.1.cmml"><mi id="footnote3.m5.1.1.2" xref="footnote3.m5.1.1.2.cmml">L</mi><mn id="footnote3.m5.1.1.3" xref="footnote3.m5.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="footnote3.m5.1c"><apply id="footnote3.m5.1.1.cmml" xref="footnote3.m5.1.1"><csymbol cd="ambiguous" id="footnote3.m5.1.1.1.cmml" xref="footnote3.m5.1.1">subscript</csymbol><ci id="footnote3.m5.1.1.2.cmml" xref="footnote3.m5.1.1.2">𝐿</ci><cn id="footnote3.m5.1.1.3.cmml" type="integer" xref="footnote3.m5.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote3.m5.1d">L_{2}</annotation><annotation encoding="application/x-llamapun" id="footnote3.m5.1e">italic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_citep">(Larson et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib44" title="">2002</a>)</cite>.</span></span></span></p> </div> </section> </section> <section class="ltx_subsection" id="S2.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.2 </span>Radiometer noise</h3> <div class="ltx_para" id="S2.SS2.p1"> <p class="ltx_p" id="S2.SS2.p1.3">A spacecraft can measure the incoming phase using a phase-lock loop with a half-bandwidth error power spectrum of <cite class="ltx_cite ltx_citemacro_citep">(e.g. Misra & Enge, <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib50" title="">2012</a>)</cite></p> <table class="ltx_equation ltx_eqn_table" id="S2.E7"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td 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xref="S2.E7.m1.4.4.1.1.1.1.1.1.1.1.1.2.2">italic-ϕ</ci></apply><list id="S2.E7.m1.2.2.2.3.cmml" xref="S2.E7.m1.2.2.2.4"><cn id="S2.E7.m1.1.1.1.1.cmml" type="integer" xref="S2.E7.m1.1.1.1.1">1</cn><ci id="S2.E7.m1.2.2.2.2.cmml" xref="S2.E7.m1.2.2.2.2">𝑁</ci></list></apply></apply><cn id="S2.E7.m1.4.4.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.E7.m1.4.4.1.1.1.1.1.1.3">2</cn></apply></apply></apply></apply><apply id="S2.E7.m1.4.4.1.1c.cmml" xref="S2.E7.m1.4.4.1"><eq id="S2.E7.m1.4.4.1.1.6.cmml" xref="S2.E7.m1.4.4.1.1.6"></eq><share href="https://arxiv.org/html/2411.15072v1#S2.E7.m1.4.4.1.1.1.cmml" id="S2.E7.m1.4.4.1.1d.cmml" xref="S2.E7.m1.4.4.1"></share><apply id="S2.E7.m1.4.4.1.1.2.cmml" xref="S2.E7.m1.4.4.1.1.2"><csymbol cd="ambiguous" id="S2.E7.m1.4.4.1.1.2.2.cmml" xref="S2.E7.m1.4.4.1.1.2">superscript</csymbol><apply id="S2.E7.m1.4.4.1.1.2.1.1.1.cmml" xref="S2.E7.m1.4.4.1.1.2.1.1"><divide id="S2.E7.m1.4.4.1.1.2.1.1.1.1.cmml" xref="S2.E7.m1.4.4.1.1.2.1.1.1.1"></divide><ci 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start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_k ) ≡ 2 italic_T start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ⟨ | over~ start_ARG italic_ϕ end_ARG start_POSTSUBSCRIPT 1 , italic_N end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⟩ = ( italic_C / italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(7)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p1.2">where <math alttext="C/N_{0}" class="ltx_Math" display="inline" id="S2.SS2.p1.1.m1.1"><semantics id="S2.SS2.p1.1.m1.1a"><mrow id="S2.SS2.p1.1.m1.1.1" xref="S2.SS2.p1.1.m1.1.1.cmml"><mi id="S2.SS2.p1.1.m1.1.1.2" xref="S2.SS2.p1.1.m1.1.1.2.cmml">C</mi><mo id="S2.SS2.p1.1.m1.1.1.1" xref="S2.SS2.p1.1.m1.1.1.1.cmml">/</mo><msub id="S2.SS2.p1.1.m1.1.1.3" xref="S2.SS2.p1.1.m1.1.1.3.cmml"><mi id="S2.SS2.p1.1.m1.1.1.3.2" xref="S2.SS2.p1.1.m1.1.1.3.2.cmml">N</mi><mn id="S2.SS2.p1.1.m1.1.1.3.3" xref="S2.SS2.p1.1.m1.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.1.m1.1b"><apply id="S2.SS2.p1.1.m1.1.1.cmml" xref="S2.SS2.p1.1.m1.1.1"><divide id="S2.SS2.p1.1.m1.1.1.1.cmml" xref="S2.SS2.p1.1.m1.1.1.1"></divide><ci id="S2.SS2.p1.1.m1.1.1.2.cmml" xref="S2.SS2.p1.1.m1.1.1.2">𝐶</ci><apply id="S2.SS2.p1.1.m1.1.1.3.cmml" xref="S2.SS2.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p1.1.m1.1.1.3.1.cmml" xref="S2.SS2.p1.1.m1.1.1.3">subscript</csymbol><ci id="S2.SS2.p1.1.m1.1.1.3.2.cmml" xref="S2.SS2.p1.1.m1.1.1.3.2">𝑁</ci><cn id="S2.SS2.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S2.SS2.p1.1.m1.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.1.m1.1c">C/N_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.1.m1.1d">italic_C / italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is the ratio of the carrier power to the unit-frequency noise power. Tildes denote the Fourier dual such that <math alttext="\widetilde{\phi_{N}}" class="ltx_Math" display="inline" id="S2.SS2.p1.2.m2.1"><semantics id="S2.SS2.p1.2.m2.1a"><mover accent="true" id="S2.SS2.p1.2.m2.1.1" xref="S2.SS2.p1.2.m2.1.1.cmml"><msub id="S2.SS2.p1.2.m2.1.1.2" xref="S2.SS2.p1.2.m2.1.1.2.cmml"><mi id="S2.SS2.p1.2.m2.1.1.2.2" xref="S2.SS2.p1.2.m2.1.1.2.2.cmml">ϕ</mi><mi id="S2.SS2.p1.2.m2.1.1.2.3" xref="S2.SS2.p1.2.m2.1.1.2.3.cmml">N</mi></msub><mo id="S2.SS2.p1.2.m2.1.1.1" xref="S2.SS2.p1.2.m2.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.2.m2.1b"><apply id="S2.SS2.p1.2.m2.1.1.cmml" xref="S2.SS2.p1.2.m2.1.1"><ci id="S2.SS2.p1.2.m2.1.1.1.cmml" xref="S2.SS2.p1.2.m2.1.1.1">~</ci><apply id="S2.SS2.p1.2.m2.1.1.2.cmml" xref="S2.SS2.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.p1.2.m2.1.1.2.1.cmml" xref="S2.SS2.p1.2.m2.1.1.2">subscript</csymbol><ci id="S2.SS2.p1.2.m2.1.1.2.2.cmml" xref="S2.SS2.p1.2.m2.1.1.2.2">italic-ϕ</ci><ci id="S2.SS2.p1.2.m2.1.1.2.3.cmml" xref="S2.SS2.p1.2.m2.1.1.2.3">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.2.m2.1c">\widetilde{\phi_{N}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.2.m2.1d">over~ start_ARG italic_ϕ start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_ARG</annotation></semantics></math> is the Fourier dual of the phase noise in equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.E1" title="In Single-arm and an atomic clock ‣ 2.1 Clock noise and its mitigation ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">1</span></a>). Equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.E7" title="In 2.2 Radiometer noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">7</span></a>) holds for both an interferometric setup using optical lasers or one with radio dishes, although what sets the noise is different in the two cases.</p> </div> <div class="ltx_para" id="S2.SS2.p2"> <p class="ltx_p" id="S2.SS2.p2.1">In the optical, <math alttext="C/N_{0}" class="ltx_Math" display="inline" id="S2.SS2.p2.1.m1.1"><semantics id="S2.SS2.p2.1.m1.1a"><mrow id="S2.SS2.p2.1.m1.1.1" xref="S2.SS2.p2.1.m1.1.1.cmml"><mi id="S2.SS2.p2.1.m1.1.1.2" xref="S2.SS2.p2.1.m1.1.1.2.cmml">C</mi><mo id="S2.SS2.p2.1.m1.1.1.1" xref="S2.SS2.p2.1.m1.1.1.1.cmml">/</mo><msub id="S2.SS2.p2.1.m1.1.1.3" xref="S2.SS2.p2.1.m1.1.1.3.cmml"><mi id="S2.SS2.p2.1.m1.1.1.3.2" xref="S2.SS2.p2.1.m1.1.1.3.2.cmml">N</mi><mn id="S2.SS2.p2.1.m1.1.1.3.3" xref="S2.SS2.p2.1.m1.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.1.m1.1b"><apply id="S2.SS2.p2.1.m1.1.1.cmml" xref="S2.SS2.p2.1.m1.1.1"><divide id="S2.SS2.p2.1.m1.1.1.1.cmml" xref="S2.SS2.p2.1.m1.1.1.1"></divide><ci id="S2.SS2.p2.1.m1.1.1.2.cmml" xref="S2.SS2.p2.1.m1.1.1.2">𝐶</ci><apply id="S2.SS2.p2.1.m1.1.1.3.cmml" xref="S2.SS2.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p2.1.m1.1.1.3.1.cmml" xref="S2.SS2.p2.1.m1.1.1.3">subscript</csymbol><ci id="S2.SS2.p2.1.m1.1.1.3.2.cmml" xref="S2.SS2.p2.1.m1.1.1.3.2">𝑁</ci><cn id="S2.SS2.p2.1.m1.1.1.3.3.cmml" type="integer" xref="S2.SS2.p2.1.m1.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.1.m1.1c">C/N_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.1.m1.1d">italic_C / italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is set by the shot noise of the received laser <cite class="ltx_cite ltx_citemacro_citep">(Barke et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib14" title="">2015</a>)</cite>:</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="A3.EGx1"> <tbody id="S2.E8"> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle[C/N_{0}]_{\rm dB-Hz}^{\text{Optical}}" class="ltx_Math" display="inline" id="S2.Ex1.m1.1"><semantics id="S2.Ex1.m1.1a"><msubsup id="S2.Ex1.m1.1.1" xref="S2.Ex1.m1.1.1.cmml"><mrow id="S2.Ex1.m1.1.1.1.1.1" xref="S2.Ex1.m1.1.1.1.1.2.cmml"><mo id="S2.Ex1.m1.1.1.1.1.1.2" stretchy="false" xref="S2.Ex1.m1.1.1.1.1.2.1.cmml">[</mo><mrow id="S2.Ex1.m1.1.1.1.1.1.1" xref="S2.Ex1.m1.1.1.1.1.1.1.cmml"><mi id="S2.Ex1.m1.1.1.1.1.1.1.2" xref="S2.Ex1.m1.1.1.1.1.1.1.2.cmml">C</mi><mo id="S2.Ex1.m1.1.1.1.1.1.1.1" xref="S2.Ex1.m1.1.1.1.1.1.1.1.cmml">/</mo><msub id="S2.Ex1.m1.1.1.1.1.1.1.3" xref="S2.Ex1.m1.1.1.1.1.1.1.3.cmml"><mi id="S2.Ex1.m1.1.1.1.1.1.1.3.2" xref="S2.Ex1.m1.1.1.1.1.1.1.3.2.cmml">N</mi><mn id="S2.Ex1.m1.1.1.1.1.1.1.3.3" xref="S2.Ex1.m1.1.1.1.1.1.1.3.3.cmml">0</mn></msub></mrow><mo id="S2.Ex1.m1.1.1.1.1.1.3" stretchy="false" xref="S2.Ex1.m1.1.1.1.1.2.1.cmml">]</mo></mrow><mrow id="S2.Ex1.m1.1.1.1.3" xref="S2.Ex1.m1.1.1.1.3.cmml"><mi id="S2.Ex1.m1.1.1.1.3.2" xref="S2.Ex1.m1.1.1.1.3.2.cmml">dB</mi><mo id="S2.Ex1.m1.1.1.1.3.1" xref="S2.Ex1.m1.1.1.1.3.1.cmml">−</mo><mi id="S2.Ex1.m1.1.1.1.3.3" xref="S2.Ex1.m1.1.1.1.3.3.cmml">Hz</mi></mrow><mtext id="S2.Ex1.m1.1.1.3" xref="S2.Ex1.m1.1.1.3a.cmml">Optical</mtext></msubsup><annotation-xml encoding="MathML-Content" id="S2.Ex1.m1.1b"><apply id="S2.Ex1.m1.1.1.cmml" xref="S2.Ex1.m1.1.1"><csymbol cd="ambiguous" id="S2.Ex1.m1.1.1.2.cmml" xref="S2.Ex1.m1.1.1">superscript</csymbol><apply id="S2.Ex1.m1.1.1.1.cmml" xref="S2.Ex1.m1.1.1"><csymbol cd="ambiguous" id="S2.Ex1.m1.1.1.1.2.cmml" xref="S2.Ex1.m1.1.1">subscript</csymbol><apply id="S2.Ex1.m1.1.1.1.1.2.cmml" xref="S2.Ex1.m1.1.1.1.1.1"><csymbol cd="latexml" id="S2.Ex1.m1.1.1.1.1.2.1.cmml" xref="S2.Ex1.m1.1.1.1.1.1.2">delimited-[]</csymbol><apply id="S2.Ex1.m1.1.1.1.1.1.1.cmml" xref="S2.Ex1.m1.1.1.1.1.1.1"><divide id="S2.Ex1.m1.1.1.1.1.1.1.1.cmml" xref="S2.Ex1.m1.1.1.1.1.1.1.1"></divide><ci id="S2.Ex1.m1.1.1.1.1.1.1.2.cmml" xref="S2.Ex1.m1.1.1.1.1.1.1.2">𝐶</ci><apply id="S2.Ex1.m1.1.1.1.1.1.1.3.cmml" xref="S2.Ex1.m1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.Ex1.m1.1.1.1.1.1.1.3.1.cmml" xref="S2.Ex1.m1.1.1.1.1.1.1.3">subscript</csymbol><ci id="S2.Ex1.m1.1.1.1.1.1.1.3.2.cmml" xref="S2.Ex1.m1.1.1.1.1.1.1.3.2">𝑁</ci><cn id="S2.Ex1.m1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S2.Ex1.m1.1.1.1.1.1.1.3.3">0</cn></apply></apply></apply><apply id="S2.Ex1.m1.1.1.1.3.cmml" xref="S2.Ex1.m1.1.1.1.3"><minus id="S2.Ex1.m1.1.1.1.3.1.cmml" xref="S2.Ex1.m1.1.1.1.3.1"></minus><ci id="S2.Ex1.m1.1.1.1.3.2.cmml" xref="S2.Ex1.m1.1.1.1.3.2">dB</ci><ci id="S2.Ex1.m1.1.1.1.3.3.cmml" xref="S2.Ex1.m1.1.1.1.3.3">Hz</ci></apply></apply><ci id="S2.Ex1.m1.1.1.3a.cmml" xref="S2.Ex1.m1.1.1.3"><mtext id="S2.Ex1.m1.1.1.3.cmml" mathsize="70%" xref="S2.Ex1.m1.1.1.3">Optical</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex1.m1.1c">\displaystyle[C/N_{0}]_{\rm dB-Hz}^{\text{Optical}}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex1.m1.1d">[ italic_C / italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT roman_dB - roman_Hz end_POSTSUBSCRIPT start_POSTSUPERSCRIPT Optical end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S2.Ex1.m2.1"><semantics id="S2.Ex1.m2.1a"><mo id="S2.Ex1.m2.1.1" xref="S2.Ex1.m2.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S2.Ex1.m2.1b"><eq id="S2.Ex1.m2.1.1.cmml" xref="S2.Ex1.m2.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex1.m2.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="S2.Ex1.m2.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle 10\log_{10}\frac{P_{\rm rec}}{2\eta hc/\lambda}," class="ltx_Math" display="inline" id="S2.Ex1.m3.1"><semantics id="S2.Ex1.m3.1a"><mrow id="S2.Ex1.m3.1.1.1" xref="S2.Ex1.m3.1.1.1.1.cmml"><mrow id="S2.Ex1.m3.1.1.1.1" xref="S2.Ex1.m3.1.1.1.1.cmml"><mn id="S2.Ex1.m3.1.1.1.1.2" xref="S2.Ex1.m3.1.1.1.1.2.cmml">10</mn><mo id="S2.Ex1.m3.1.1.1.1.1" lspace="0.167em" xref="S2.Ex1.m3.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.Ex1.m3.1.1.1.1.3" xref="S2.Ex1.m3.1.1.1.1.3.cmml"><msub id="S2.Ex1.m3.1.1.1.1.3.1" xref="S2.Ex1.m3.1.1.1.1.3.1.cmml"><mi id="S2.Ex1.m3.1.1.1.1.3.1.2" xref="S2.Ex1.m3.1.1.1.1.3.1.2.cmml">log</mi><mn id="S2.Ex1.m3.1.1.1.1.3.1.3" xref="S2.Ex1.m3.1.1.1.1.3.1.3.cmml">10</mn></msub><mo id="S2.Ex1.m3.1.1.1.1.3a" lspace="0.167em" xref="S2.Ex1.m3.1.1.1.1.3.cmml">⁡</mo><mstyle displaystyle="true" id="S2.Ex1.m3.1.1.1.1.3.2" xref="S2.Ex1.m3.1.1.1.1.3.2.cmml"><mfrac id="S2.Ex1.m3.1.1.1.1.3.2a" xref="S2.Ex1.m3.1.1.1.1.3.2.cmml"><msub id="S2.Ex1.m3.1.1.1.1.3.2.2" xref="S2.Ex1.m3.1.1.1.1.3.2.2.cmml"><mi id="S2.Ex1.m3.1.1.1.1.3.2.2.2" xref="S2.Ex1.m3.1.1.1.1.3.2.2.2.cmml">P</mi><mi id="S2.Ex1.m3.1.1.1.1.3.2.2.3" xref="S2.Ex1.m3.1.1.1.1.3.2.2.3.cmml">rec</mi></msub><mrow id="S2.Ex1.m3.1.1.1.1.3.2.3" xref="S2.Ex1.m3.1.1.1.1.3.2.3.cmml"><mrow id="S2.Ex1.m3.1.1.1.1.3.2.3.2" xref="S2.Ex1.m3.1.1.1.1.3.2.3.2.cmml"><mn id="S2.Ex1.m3.1.1.1.1.3.2.3.2.2" xref="S2.Ex1.m3.1.1.1.1.3.2.3.2.2.cmml">2</mn><mo id="S2.Ex1.m3.1.1.1.1.3.2.3.2.1" xref="S2.Ex1.m3.1.1.1.1.3.2.3.2.1.cmml">⁢</mo><mi id="S2.Ex1.m3.1.1.1.1.3.2.3.2.3" xref="S2.Ex1.m3.1.1.1.1.3.2.3.2.3.cmml">η</mi><mo id="S2.Ex1.m3.1.1.1.1.3.2.3.2.1a" xref="S2.Ex1.m3.1.1.1.1.3.2.3.2.1.cmml">⁢</mo><mi id="S2.Ex1.m3.1.1.1.1.3.2.3.2.4" xref="S2.Ex1.m3.1.1.1.1.3.2.3.2.4.cmml">h</mi><mo id="S2.Ex1.m3.1.1.1.1.3.2.3.2.1b" xref="S2.Ex1.m3.1.1.1.1.3.2.3.2.1.cmml">⁢</mo><mi id="S2.Ex1.m3.1.1.1.1.3.2.3.2.5" xref="S2.Ex1.m3.1.1.1.1.3.2.3.2.5.cmml">c</mi></mrow><mo id="S2.Ex1.m3.1.1.1.1.3.2.3.1" xref="S2.Ex1.m3.1.1.1.1.3.2.3.1.cmml">/</mo><mi id="S2.Ex1.m3.1.1.1.1.3.2.3.3" xref="S2.Ex1.m3.1.1.1.1.3.2.3.3.cmml">λ</mi></mrow></mfrac></mstyle></mrow></mrow><mo id="S2.Ex1.m3.1.1.1.2" xref="S2.Ex1.m3.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex1.m3.1b"><apply id="S2.Ex1.m3.1.1.1.1.cmml" xref="S2.Ex1.m3.1.1.1"><times id="S2.Ex1.m3.1.1.1.1.1.cmml" xref="S2.Ex1.m3.1.1.1.1.1"></times><cn id="S2.Ex1.m3.1.1.1.1.2.cmml" type="integer" xref="S2.Ex1.m3.1.1.1.1.2">10</cn><apply id="S2.Ex1.m3.1.1.1.1.3.cmml" xref="S2.Ex1.m3.1.1.1.1.3"><apply id="S2.Ex1.m3.1.1.1.1.3.1.cmml" xref="S2.Ex1.m3.1.1.1.1.3.1"><csymbol cd="ambiguous" id="S2.Ex1.m3.1.1.1.1.3.1.1.cmml" xref="S2.Ex1.m3.1.1.1.1.3.1">subscript</csymbol><log id="S2.Ex1.m3.1.1.1.1.3.1.2.cmml" xref="S2.Ex1.m3.1.1.1.1.3.1.2"></log><cn id="S2.Ex1.m3.1.1.1.1.3.1.3.cmml" type="integer" xref="S2.Ex1.m3.1.1.1.1.3.1.3">10</cn></apply><apply id="S2.Ex1.m3.1.1.1.1.3.2.cmml" xref="S2.Ex1.m3.1.1.1.1.3.2"><divide id="S2.Ex1.m3.1.1.1.1.3.2.1.cmml" xref="S2.Ex1.m3.1.1.1.1.3.2"></divide><apply id="S2.Ex1.m3.1.1.1.1.3.2.2.cmml" xref="S2.Ex1.m3.1.1.1.1.3.2.2"><csymbol cd="ambiguous" id="S2.Ex1.m3.1.1.1.1.3.2.2.1.cmml" xref="S2.Ex1.m3.1.1.1.1.3.2.2">subscript</csymbol><ci id="S2.Ex1.m3.1.1.1.1.3.2.2.2.cmml" xref="S2.Ex1.m3.1.1.1.1.3.2.2.2">𝑃</ci><ci id="S2.Ex1.m3.1.1.1.1.3.2.2.3.cmml" xref="S2.Ex1.m3.1.1.1.1.3.2.2.3">rec</ci></apply><apply id="S2.Ex1.m3.1.1.1.1.3.2.3.cmml" xref="S2.Ex1.m3.1.1.1.1.3.2.3"><divide id="S2.Ex1.m3.1.1.1.1.3.2.3.1.cmml" xref="S2.Ex1.m3.1.1.1.1.3.2.3.1"></divide><apply id="S2.Ex1.m3.1.1.1.1.3.2.3.2.cmml" xref="S2.Ex1.m3.1.1.1.1.3.2.3.2"><times id="S2.Ex1.m3.1.1.1.1.3.2.3.2.1.cmml" xref="S2.Ex1.m3.1.1.1.1.3.2.3.2.1"></times><cn id="S2.Ex1.m3.1.1.1.1.3.2.3.2.2.cmml" type="integer" xref="S2.Ex1.m3.1.1.1.1.3.2.3.2.2">2</cn><ci id="S2.Ex1.m3.1.1.1.1.3.2.3.2.3.cmml" xref="S2.Ex1.m3.1.1.1.1.3.2.3.2.3">𝜂</ci><ci id="S2.Ex1.m3.1.1.1.1.3.2.3.2.4.cmml" xref="S2.Ex1.m3.1.1.1.1.3.2.3.2.4">ℎ</ci><ci id="S2.Ex1.m3.1.1.1.1.3.2.3.2.5.cmml" xref="S2.Ex1.m3.1.1.1.1.3.2.3.2.5">𝑐</ci></apply><ci id="S2.Ex1.m3.1.1.1.1.3.2.3.3.cmml" xref="S2.Ex1.m3.1.1.1.1.3.2.3.3">𝜆</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex1.m3.1c">\displaystyle 10\log_{10}\frac{P_{\rm rec}}{2\eta hc/\lambda},</annotation><annotation encoding="application/x-llamapun" id="S2.Ex1.m3.1d">10 roman_log start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT divide start_ARG italic_P start_POSTSUBSCRIPT roman_rec end_POSTSUBSCRIPT end_ARG start_ARG 2 italic_η italic_h italic_c / italic_λ end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="3"><span class="ltx_tag ltx_tag_equation ltx_align_right">(8)</span></td> </tr> <tr class="ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S2.Ex2.m1.1"><semantics id="S2.Ex2.m1.1a"><mo id="S2.Ex2.m1.1.1" xref="S2.Ex2.m1.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S2.Ex2.m1.1b"><eq id="S2.Ex2.m1.1.1.cmml" xref="S2.Ex2.m1.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex2.m1.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="S2.Ex2.m1.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle 33\,\text{dB-Hz}+10\log_{10}\left(\frac{P_{\rm em}}{10{\rm~{}W}}% \right)+20\log_{10}\left(\frac{D}{20{\rm~{}cm}}\right)-10\log_{10}\eta" class="ltx_Math" display="inline" id="S2.Ex2.m2.4"><semantics id="S2.Ex2.m2.4a"><mrow id="S2.Ex2.m2.4.4" xref="S2.Ex2.m2.4.4.cmml"><mrow id="S2.Ex2.m2.4.4.2" xref="S2.Ex2.m2.4.4.2.cmml"><mrow id="S2.Ex2.m2.4.4.2.4" xref="S2.Ex2.m2.4.4.2.4.cmml"><mn id="S2.Ex2.m2.4.4.2.4.2" xref="S2.Ex2.m2.4.4.2.4.2.cmml">33</mn><mo id="S2.Ex2.m2.4.4.2.4.1" lspace="0.170em" xref="S2.Ex2.m2.4.4.2.4.1.cmml">⁢</mo><mtext id="S2.Ex2.m2.4.4.2.4.3" xref="S2.Ex2.m2.4.4.2.4.3a.cmml">dB-Hz</mtext></mrow><mo id="S2.Ex2.m2.4.4.2.3" xref="S2.Ex2.m2.4.4.2.3.cmml">+</mo><mrow id="S2.Ex2.m2.3.3.1.1" xref="S2.Ex2.m2.3.3.1.1.cmml"><mn id="S2.Ex2.m2.3.3.1.1.3" xref="S2.Ex2.m2.3.3.1.1.3.cmml">10</mn><mo id="S2.Ex2.m2.3.3.1.1.2" lspace="0.167em" xref="S2.Ex2.m2.3.3.1.1.2.cmml">⁢</mo><mrow id="S2.Ex2.m2.3.3.1.1.1.1" xref="S2.Ex2.m2.3.3.1.1.1.2.cmml"><msub id="S2.Ex2.m2.3.3.1.1.1.1.1" xref="S2.Ex2.m2.3.3.1.1.1.1.1.cmml"><mi id="S2.Ex2.m2.3.3.1.1.1.1.1.2" xref="S2.Ex2.m2.3.3.1.1.1.1.1.2.cmml">log</mi><mn id="S2.Ex2.m2.3.3.1.1.1.1.1.3" xref="S2.Ex2.m2.3.3.1.1.1.1.1.3.cmml">10</mn></msub><mo id="S2.Ex2.m2.3.3.1.1.1.1a" xref="S2.Ex2.m2.3.3.1.1.1.2.cmml">⁡</mo><mrow id="S2.Ex2.m2.3.3.1.1.1.1.2" xref="S2.Ex2.m2.3.3.1.1.1.2.cmml"><mo id="S2.Ex2.m2.3.3.1.1.1.1.2.1" xref="S2.Ex2.m2.3.3.1.1.1.2.cmml">(</mo><mstyle displaystyle="true" id="S2.Ex2.m2.1.1" xref="S2.Ex2.m2.1.1.cmml"><mfrac id="S2.Ex2.m2.1.1a" xref="S2.Ex2.m2.1.1.cmml"><msub id="S2.Ex2.m2.1.1.2" xref="S2.Ex2.m2.1.1.2.cmml"><mi id="S2.Ex2.m2.1.1.2.2" xref="S2.Ex2.m2.1.1.2.2.cmml">P</mi><mi id="S2.Ex2.m2.1.1.2.3" xref="S2.Ex2.m2.1.1.2.3.cmml">em</mi></msub><mrow id="S2.Ex2.m2.1.1.3" xref="S2.Ex2.m2.1.1.3.cmml"><mn id="S2.Ex2.m2.1.1.3.2" xref="S2.Ex2.m2.1.1.3.2.cmml">10</mn><mo id="S2.Ex2.m2.1.1.3.1" lspace="0.330em" xref="S2.Ex2.m2.1.1.3.1.cmml">⁢</mo><mi id="S2.Ex2.m2.1.1.3.3" mathvariant="normal" 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id="S2.Ex2.m2.4.4.4.3.1.3" xref="S2.Ex2.m2.4.4.4.3.1.3.cmml">10</mn></msub><mo id="S2.Ex2.m2.4.4.4.3a" lspace="0.167em" xref="S2.Ex2.m2.4.4.4.3.cmml">⁡</mo><mi id="S2.Ex2.m2.4.4.4.3.2" xref="S2.Ex2.m2.4.4.4.3.2.cmml">η</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex2.m2.4b"><apply id="S2.Ex2.m2.4.4.cmml" xref="S2.Ex2.m2.4.4"><minus id="S2.Ex2.m2.4.4.3.cmml" xref="S2.Ex2.m2.4.4.3"></minus><apply id="S2.Ex2.m2.4.4.2.cmml" xref="S2.Ex2.m2.4.4.2"><plus id="S2.Ex2.m2.4.4.2.3.cmml" xref="S2.Ex2.m2.4.4.2.3"></plus><apply id="S2.Ex2.m2.4.4.2.4.cmml" xref="S2.Ex2.m2.4.4.2.4"><times id="S2.Ex2.m2.4.4.2.4.1.cmml" xref="S2.Ex2.m2.4.4.2.4.1"></times><cn id="S2.Ex2.m2.4.4.2.4.2.cmml" type="integer" xref="S2.Ex2.m2.4.4.2.4.2">33</cn><ci id="S2.Ex2.m2.4.4.2.4.3a.cmml" xref="S2.Ex2.m2.4.4.2.4.3"><mtext id="S2.Ex2.m2.4.4.2.4.3.cmml" xref="S2.Ex2.m2.4.4.2.4.3">dB-Hz</mtext></ci></apply><apply id="S2.Ex2.m2.3.3.1.1.cmml" xref="S2.Ex2.m2.3.3.1.1"><times 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id="S2.Ex2.m2.2.2.1.cmml" xref="S2.Ex2.m2.2.2"></divide><ci id="S2.Ex2.m2.2.2.2.cmml" xref="S2.Ex2.m2.2.2.2">𝐷</ci><apply id="S2.Ex2.m2.2.2.3.cmml" xref="S2.Ex2.m2.2.2.3"><times id="S2.Ex2.m2.2.2.3.1.cmml" xref="S2.Ex2.m2.2.2.3.1"></times><cn id="S2.Ex2.m2.2.2.3.2.cmml" type="integer" xref="S2.Ex2.m2.2.2.3.2">20</cn><ci id="S2.Ex2.m2.2.2.3.3.cmml" xref="S2.Ex2.m2.2.2.3.3">cm</ci></apply></apply></apply></apply></apply><apply id="S2.Ex2.m2.4.4.4.cmml" xref="S2.Ex2.m2.4.4.4"><times id="S2.Ex2.m2.4.4.4.1.cmml" xref="S2.Ex2.m2.4.4.4.1"></times><cn id="S2.Ex2.m2.4.4.4.2.cmml" type="integer" xref="S2.Ex2.m2.4.4.4.2">10</cn><apply id="S2.Ex2.m2.4.4.4.3.cmml" xref="S2.Ex2.m2.4.4.4.3"><apply id="S2.Ex2.m2.4.4.4.3.1.cmml" xref="S2.Ex2.m2.4.4.4.3.1"><csymbol cd="ambiguous" id="S2.Ex2.m2.4.4.4.3.1.1.cmml" xref="S2.Ex2.m2.4.4.4.3.1">subscript</csymbol><log id="S2.Ex2.m2.4.4.4.3.1.2.cmml" xref="S2.Ex2.m2.4.4.4.3.1.2"></log><cn id="S2.Ex2.m2.4.4.4.3.1.3.cmml" type="integer" xref="S2.Ex2.m2.4.4.4.3.1.3">10</cn></apply><ci id="S2.Ex2.m2.4.4.4.3.2.cmml" xref="S2.Ex2.m2.4.4.4.3.2">𝜂</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex2.m2.4c">\displaystyle 33\,\text{dB-Hz}+10\log_{10}\left(\frac{P_{\rm em}}{10{\rm~{}W}}% \right)+20\log_{10}\left(\frac{D}{20{\rm~{}cm}}\right)-10\log_{10}\eta</annotation><annotation encoding="application/x-llamapun" id="S2.Ex2.m2.4d">33 dB-Hz + 10 roman_log start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT ( divide start_ARG italic_P start_POSTSUBSCRIPT roman_em end_POSTSUBSCRIPT end_ARG start_ARG 10 roman_W end_ARG ) + 20 roman_log start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT ( divide start_ARG italic_D end_ARG start_ARG 20 roman_cm end_ARG ) - 10 roman_log start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT italic_η</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr class="ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-20\log_{10}\left(\frac{L}{30{\rm~{}AU}}\right)-10\log_{10}\left(% \frac{\lambda}{1\mu{\rm m}}\right)," class="ltx_Math" display="inline" id="S2.E8.m1.3"><semantics id="S2.E8.m1.3a"><mrow id="S2.E8.m1.3.3.1" xref="S2.E8.m1.3.3.1.1.cmml"><mrow id="S2.E8.m1.3.3.1.1" xref="S2.E8.m1.3.3.1.1.cmml"><mrow id="S2.E8.m1.3.3.1.1.1" xref="S2.E8.m1.3.3.1.1.1.cmml"><mo id="S2.E8.m1.3.3.1.1.1a" xref="S2.E8.m1.3.3.1.1.1.cmml">−</mo><mrow id="S2.E8.m1.3.3.1.1.1.1" xref="S2.E8.m1.3.3.1.1.1.1.cmml"><mn id="S2.E8.m1.3.3.1.1.1.1.3" xref="S2.E8.m1.3.3.1.1.1.1.3.cmml">20</mn><mo id="S2.E8.m1.3.3.1.1.1.1.2" lspace="0.167em" xref="S2.E8.m1.3.3.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E8.m1.3.3.1.1.1.1.1.1" xref="S2.E8.m1.3.3.1.1.1.1.1.2.cmml"><msub id="S2.E8.m1.3.3.1.1.1.1.1.1.1" xref="S2.E8.m1.3.3.1.1.1.1.1.1.1.cmml"><mi 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xref="S2.E8.m1.1.1.3.3">AU</ci></apply></apply></apply></apply></apply><apply id="S2.E8.m1.3.3.1.1.2.cmml" xref="S2.E8.m1.3.3.1.1.2"><times id="S2.E8.m1.3.3.1.1.2.2.cmml" xref="S2.E8.m1.3.3.1.1.2.2"></times><cn id="S2.E8.m1.3.3.1.1.2.3.cmml" type="integer" xref="S2.E8.m1.3.3.1.1.2.3">10</cn><apply id="S2.E8.m1.3.3.1.1.2.1.2.cmml" xref="S2.E8.m1.3.3.1.1.2.1.1"><apply id="S2.E8.m1.3.3.1.1.2.1.1.1.cmml" xref="S2.E8.m1.3.3.1.1.2.1.1.1"><csymbol cd="ambiguous" id="S2.E8.m1.3.3.1.1.2.1.1.1.1.cmml" xref="S2.E8.m1.3.3.1.1.2.1.1.1">subscript</csymbol><log id="S2.E8.m1.3.3.1.1.2.1.1.1.2.cmml" xref="S2.E8.m1.3.3.1.1.2.1.1.1.2"></log><cn id="S2.E8.m1.3.3.1.1.2.1.1.1.3.cmml" type="integer" xref="S2.E8.m1.3.3.1.1.2.1.1.1.3">10</cn></apply><apply id="S2.E8.m1.2.2.cmml" xref="S2.E8.m1.2.2"><divide id="S2.E8.m1.2.2.1.cmml" xref="S2.E8.m1.2.2"></divide><ci id="S2.E8.m1.2.2.2.cmml" xref="S2.E8.m1.2.2.2">𝜆</ci><apply id="S2.E8.m1.2.2.3.cmml" xref="S2.E8.m1.2.2.3"><times id="S2.E8.m1.2.2.3.1.cmml" xref="S2.E8.m1.2.2.3.1"></times><cn id="S2.E8.m1.2.2.3.2.cmml" type="integer" xref="S2.E8.m1.2.2.3.2">1</cn><ci id="S2.E8.m1.2.2.3.3.cmml" xref="S2.E8.m1.2.2.3.3">𝜇</ci><ci id="S2.E8.m1.2.2.3.4.cmml" xref="S2.E8.m1.2.2.3.4">m</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E8.m1.3c">\displaystyle-20\log_{10}\left(\frac{L}{30{\rm~{}AU}}\right)-10\log_{10}\left(% \frac{\lambda}{1\mu{\rm m}}\right),</annotation><annotation encoding="application/x-llamapun" id="S2.E8.m1.3d">- 20 roman_log start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT ( divide start_ARG italic_L end_ARG start_ARG 30 roman_AU end_ARG ) - 10 roman_log start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT ( divide start_ARG italic_λ end_ARG start_ARG 1 italic_μ roman_m end_ARG ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </tbody> </table> <p class="ltx_p" id="S2.SS2.p2.9">where <math alttext="P_{\rm rec}" class="ltx_Math" display="inline" id="S2.SS2.p2.2.m1.1"><semantics id="S2.SS2.p2.2.m1.1a"><msub id="S2.SS2.p2.2.m1.1.1" xref="S2.SS2.p2.2.m1.1.1.cmml"><mi id="S2.SS2.p2.2.m1.1.1.2" xref="S2.SS2.p2.2.m1.1.1.2.cmml">P</mi><mi id="S2.SS2.p2.2.m1.1.1.3" xref="S2.SS2.p2.2.m1.1.1.3.cmml">rec</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.2.m1.1b"><apply id="S2.SS2.p2.2.m1.1.1.cmml" xref="S2.SS2.p2.2.m1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.2.m1.1.1.1.cmml" xref="S2.SS2.p2.2.m1.1.1">subscript</csymbol><ci id="S2.SS2.p2.2.m1.1.1.2.cmml" xref="S2.SS2.p2.2.m1.1.1.2">𝑃</ci><ci id="S2.SS2.p2.2.m1.1.1.3.cmml" xref="S2.SS2.p2.2.m1.1.1.3">rec</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.2.m1.1c">P_{\rm rec}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.2.m1.1d">italic_P start_POSTSUBSCRIPT roman_rec end_POSTSUBSCRIPT</annotation></semantics></math> is the received power of a Gaussian laser optimized to maximize the received power, <math alttext="\eta" class="ltx_Math" display="inline" id="S2.SS2.p2.3.m2.1"><semantics id="S2.SS2.p2.3.m2.1a"><mi id="S2.SS2.p2.3.m2.1.1" xref="S2.SS2.p2.3.m2.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.3.m2.1b"><ci id="S2.SS2.p2.3.m2.1.1.cmml" xref="S2.SS2.p2.3.m2.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.3.m2.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.3.m2.1d">italic_η</annotation></semantics></math> accounts for efficiency of the photodiodes and the fraction of the laser power at the heterodyne frequency, <math alttext="D" class="ltx_Math" display="inline" id="S2.SS2.p2.4.m3.1"><semantics id="S2.SS2.p2.4.m3.1a"><mi id="S2.SS2.p2.4.m3.1.1" xref="S2.SS2.p2.4.m3.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.4.m3.1b"><ci id="S2.SS2.p2.4.m3.1.1.cmml" xref="S2.SS2.p2.4.m3.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.4.m3.1c">D</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.4.m3.1d">italic_D</annotation></semantics></math> is the mirror diameter, and we are using the engineering convention of characterizing <math alttext="C/N_{0}" class="ltx_Math" display="inline" id="S2.SS2.p2.5.m4.1"><semantics id="S2.SS2.p2.5.m4.1a"><mrow id="S2.SS2.p2.5.m4.1.1" xref="S2.SS2.p2.5.m4.1.1.cmml"><mi id="S2.SS2.p2.5.m4.1.1.2" xref="S2.SS2.p2.5.m4.1.1.2.cmml">C</mi><mo id="S2.SS2.p2.5.m4.1.1.1" xref="S2.SS2.p2.5.m4.1.1.1.cmml">/</mo><msub id="S2.SS2.p2.5.m4.1.1.3" xref="S2.SS2.p2.5.m4.1.1.3.cmml"><mi id="S2.SS2.p2.5.m4.1.1.3.2" xref="S2.SS2.p2.5.m4.1.1.3.2.cmml">N</mi><mn id="S2.SS2.p2.5.m4.1.1.3.3" xref="S2.SS2.p2.5.m4.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.5.m4.1b"><apply id="S2.SS2.p2.5.m4.1.1.cmml" xref="S2.SS2.p2.5.m4.1.1"><divide id="S2.SS2.p2.5.m4.1.1.1.cmml" xref="S2.SS2.p2.5.m4.1.1.1"></divide><ci id="S2.SS2.p2.5.m4.1.1.2.cmml" xref="S2.SS2.p2.5.m4.1.1.2">𝐶</ci><apply id="S2.SS2.p2.5.m4.1.1.3.cmml" xref="S2.SS2.p2.5.m4.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p2.5.m4.1.1.3.1.cmml" xref="S2.SS2.p2.5.m4.1.1.3">subscript</csymbol><ci id="S2.SS2.p2.5.m4.1.1.3.2.cmml" xref="S2.SS2.p2.5.m4.1.1.3.2">𝑁</ci><cn id="S2.SS2.p2.5.m4.1.1.3.3.cmml" type="integer" xref="S2.SS2.p2.5.m4.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.5.m4.1c">C/N_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.5.m4.1d">italic_C / italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> in dB-Hz where <math alttext="[C/N_{0}]_{\rm dB-Hz}\equiv 10\log_{10}(C/N_{0})" class="ltx_Math" display="inline" id="S2.SS2.p2.6.m5.3"><semantics id="S2.SS2.p2.6.m5.3a"><mrow id="S2.SS2.p2.6.m5.3.3" xref="S2.SS2.p2.6.m5.3.3.cmml"><msub id="S2.SS2.p2.6.m5.1.1.1" xref="S2.SS2.p2.6.m5.1.1.1.cmml"><mrow 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start_POSTSUBSCRIPT roman_dB - roman_Hz end_POSTSUBSCRIPT ≡ 10 roman_log start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT ( italic_C / italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT )</annotation></semantics></math>.<span class="ltx_note ltx_role_footnote" id="footnote4"><sup class="ltx_note_mark">4</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">4</sup><span class="ltx_tag ltx_tag_note">4</span>The full sensitivity also depends on the shot noise of the local laser (that is used as a phase reference), the intensity noise of the received laser and the electronics noise from converting the photons to a voltage. <cite class="ltx_cite ltx_citemacro_citet">Barke et al. (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib14" title="">2015</a>)</cite> combines all of these into the shot noise term. We find, using the online interface associated with <cite class="ltx_cite ltx_citemacro_citet">Barke et al. (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib14" title="">2015</a>)</cite>, that the local laser power can be tuned so that the shot noise level of the received laser does indeed set the noise for our fiducial specifications on <math alttext="P_{\rm em}" class="ltx_Math" display="inline" id="footnote4.m1.1"><semantics id="footnote4.m1.1b"><msub id="footnote4.m1.1.1" xref="footnote4.m1.1.1.cmml"><mi id="footnote4.m1.1.1.2" xref="footnote4.m1.1.1.2.cmml">P</mi><mi id="footnote4.m1.1.1.3" xref="footnote4.m1.1.1.3.cmml">em</mi></msub><annotation-xml encoding="MathML-Content" id="footnote4.m1.1c"><apply id="footnote4.m1.1.1.cmml" xref="footnote4.m1.1.1"><csymbol cd="ambiguous" id="footnote4.m1.1.1.1.cmml" xref="footnote4.m1.1.1">subscript</csymbol><ci id="footnote4.m1.1.1.2.cmml" xref="footnote4.m1.1.1.2">𝑃</ci><ci id="footnote4.m1.1.1.3.cmml" xref="footnote4.m1.1.1.3">em</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote4.m1.1d">P_{\rm em}</annotation><annotation encoding="application/x-llamapun" id="footnote4.m1.1e">italic_P start_POSTSUBSCRIPT roman_em end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="L" class="ltx_Math" display="inline" id="footnote4.m2.1"><semantics id="footnote4.m2.1b"><mi id="footnote4.m2.1.1" xref="footnote4.m2.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="footnote4.m2.1c"><ci id="footnote4.m2.1.1.cmml" xref="footnote4.m2.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote4.m2.1d">L</annotation><annotation encoding="application/x-llamapun" id="footnote4.m2.1e">italic_L</annotation></semantics></math> and <math alttext="D" class="ltx_Math" display="inline" id="footnote4.m3.1"><semantics id="footnote4.m3.1b"><mi id="footnote4.m3.1.1" xref="footnote4.m3.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="footnote4.m3.1c"><ci id="footnote4.m3.1.1.cmml" xref="footnote4.m3.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote4.m3.1d">D</annotation><annotation encoding="application/x-llamapun" id="footnote4.m3.1e">italic_D</annotation></semantics></math>. However, the laser intensity variation power must be the same (or better) specification as for LISA. Whereas, we find all other tolerances can be relaxed by a factor of <math alttext="\sim 10^{3}" class="ltx_Math" display="inline" id="footnote4.m4.1"><semantics id="footnote4.m4.1b"><mrow id="footnote4.m4.1.1" xref="footnote4.m4.1.1.cmml"><mi id="footnote4.m4.1.1.2" xref="footnote4.m4.1.1.2.cmml"></mi><mo id="footnote4.m4.1.1.1" xref="footnote4.m4.1.1.1.cmml">∼</mo><msup id="footnote4.m4.1.1.3" xref="footnote4.m4.1.1.3.cmml"><mn id="footnote4.m4.1.1.3.2" xref="footnote4.m4.1.1.3.2.cmml">10</mn><mn id="footnote4.m4.1.1.3.3" xref="footnote4.m4.1.1.3.3.cmml">3</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="footnote4.m4.1c"><apply id="footnote4.m4.1.1.cmml" xref="footnote4.m4.1.1"><csymbol cd="latexml" id="footnote4.m4.1.1.1.cmml" xref="footnote4.m4.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="footnote4.m4.1.1.2.cmml" xref="footnote4.m4.1.1.2">absent</csymbol><apply id="footnote4.m4.1.1.3.cmml" xref="footnote4.m4.1.1.3"><csymbol cd="ambiguous" id="footnote4.m4.1.1.3.1.cmml" xref="footnote4.m4.1.1.3">superscript</csymbol><cn id="footnote4.m4.1.1.3.2.cmml" type="integer" xref="footnote4.m4.1.1.3.2">10</cn><cn id="footnote4.m4.1.1.3.3.cmml" type="integer" xref="footnote4.m4.1.1.3.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote4.m4.1d">\sim 10^{3}</annotation><annotation encoding="application/x-llamapun" id="footnote4.m4.1e">∼ 10 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT</annotation></semantics></math> owing to the <math alttext="\sim 10^{3}\times" class="ltx_math_unparsed" display="inline" id="footnote4.m5.1"><semantics id="footnote4.m5.1b"><mrow id="footnote4.m5.1c"><mo id="footnote4.m5.1.1">∼</mo><msup id="footnote4.m5.1.2"><mn id="footnote4.m5.1.2.2">10</mn><mn id="footnote4.m5.1.2.3">3</mn></msup><mo id="footnote4.m5.1.3" lspace="0.222em">×</mo></mrow><annotation encoding="application/x-tex" id="footnote4.m5.1d">\sim 10^{3}\times</annotation><annotation encoding="application/x-llamapun" id="footnote4.m5.1e">∼ 10 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ×</annotation></semantics></math> larger shot noise of our concept for fiducial specifications relative to LISA. We revisit these considerations in § <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S4.SS1" title="4.1 Laser design ‣ 4 Instrumental considerations ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">4.1</span></a>.</span></span></span> This paper considers <math alttext="L=10" class="ltx_Math" display="inline" id="S2.SS2.p2.7.m6.1"><semantics id="S2.SS2.p2.7.m6.1a"><mrow id="S2.SS2.p2.7.m6.1.1" xref="S2.SS2.p2.7.m6.1.1.cmml"><mi id="S2.SS2.p2.7.m6.1.1.2" xref="S2.SS2.p2.7.m6.1.1.2.cmml">L</mi><mo id="S2.SS2.p2.7.m6.1.1.1" xref="S2.SS2.p2.7.m6.1.1.1.cmml">=</mo><mn id="S2.SS2.p2.7.m6.1.1.3" xref="S2.SS2.p2.7.m6.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.7.m6.1b"><apply id="S2.SS2.p2.7.m6.1.1.cmml" xref="S2.SS2.p2.7.m6.1.1"><eq id="S2.SS2.p2.7.m6.1.1.1.cmml" xref="S2.SS2.p2.7.m6.1.1.1"></eq><ci id="S2.SS2.p2.7.m6.1.1.2.cmml" xref="S2.SS2.p2.7.m6.1.1.2">𝐿</ci><cn id="S2.SS2.p2.7.m6.1.1.3.cmml" type="integer" xref="S2.SS2.p2.7.m6.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.7.m6.1c">L=10</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.7.m6.1d">italic_L = 10</annotation></semantics></math> and <math alttext="30" class="ltx_Math" display="inline" id="S2.SS2.p2.8.m7.1"><semantics id="S2.SS2.p2.8.m7.1a"><mn id="S2.SS2.p2.8.m7.1.1" xref="S2.SS2.p2.8.m7.1.1.cmml">30</mn><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.8.m7.1b"><cn id="S2.SS2.p2.8.m7.1.1.cmml" type="integer" xref="S2.SS2.p2.8.m7.1.1">30</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.8.m7.1c">30</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.8.m7.1d">30</annotation></semantics></math>AU, and a LISA-like <math alttext="\lambda=1\;\mu" class="ltx_Math" display="inline" id="S2.SS2.p2.9.m8.1"><semantics id="S2.SS2.p2.9.m8.1a"><mrow id="S2.SS2.p2.9.m8.1.1" xref="S2.SS2.p2.9.m8.1.1.cmml"><mi id="S2.SS2.p2.9.m8.1.1.2" xref="S2.SS2.p2.9.m8.1.1.2.cmml">λ</mi><mo id="S2.SS2.p2.9.m8.1.1.1" xref="S2.SS2.p2.9.m8.1.1.1.cmml">=</mo><mrow id="S2.SS2.p2.9.m8.1.1.3" xref="S2.SS2.p2.9.m8.1.1.3.cmml"><mn id="S2.SS2.p2.9.m8.1.1.3.2" xref="S2.SS2.p2.9.m8.1.1.3.2.cmml">1</mn><mo id="S2.SS2.p2.9.m8.1.1.3.1" lspace="0.280em" xref="S2.SS2.p2.9.m8.1.1.3.1.cmml">⁢</mo><mi id="S2.SS2.p2.9.m8.1.1.3.3" xref="S2.SS2.p2.9.m8.1.1.3.3.cmml">μ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.9.m8.1b"><apply id="S2.SS2.p2.9.m8.1.1.cmml" xref="S2.SS2.p2.9.m8.1.1"><eq id="S2.SS2.p2.9.m8.1.1.1.cmml" xref="S2.SS2.p2.9.m8.1.1.1"></eq><ci id="S2.SS2.p2.9.m8.1.1.2.cmml" xref="S2.SS2.p2.9.m8.1.1.2">𝜆</ci><apply id="S2.SS2.p2.9.m8.1.1.3.cmml" xref="S2.SS2.p2.9.m8.1.1.3"><times id="S2.SS2.p2.9.m8.1.1.3.1.cmml" xref="S2.SS2.p2.9.m8.1.1.3.1"></times><cn id="S2.SS2.p2.9.m8.1.1.3.2.cmml" type="integer" xref="S2.SS2.p2.9.m8.1.1.3.2">1</cn><ci id="S2.SS2.p2.9.m8.1.1.3.3.cmml" xref="S2.SS2.p2.9.m8.1.1.3.3">𝜇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.9.m8.1c">\lambda=1\;\mu</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.9.m8.1d">italic_λ = 1 italic_μ</annotation></semantics></math>m.</p> </div> <div class="ltx_para" id="S2.SS2.p3"> <p class="ltx_p" id="S2.SS2.p3.1">In the radio, the noise is characterized by the effective temperature of the system, <math alttext="T_{\rm sys}" class="ltx_Math" display="inline" id="S2.SS2.p3.1.m1.1"><semantics id="S2.SS2.p3.1.m1.1a"><msub id="S2.SS2.p3.1.m1.1.1" xref="S2.SS2.p3.1.m1.1.1.cmml"><mi id="S2.SS2.p3.1.m1.1.1.2" xref="S2.SS2.p3.1.m1.1.1.2.cmml">T</mi><mi id="S2.SS2.p3.1.m1.1.1.3" xref="S2.SS2.p3.1.m1.1.1.3.cmml">sys</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.1.m1.1b"><apply id="S2.SS2.p3.1.m1.1.1.cmml" xref="S2.SS2.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p3.1.m1.1.1.1.cmml" xref="S2.SS2.p3.1.m1.1.1">subscript</csymbol><ci id="S2.SS2.p3.1.m1.1.1.2.cmml" xref="S2.SS2.p3.1.m1.1.1.2">𝑇</ci><ci id="S2.SS2.p3.1.m1.1.1.3.cmml" xref="S2.SS2.p3.1.m1.1.1.3">sys</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.1.m1.1c">T_{\rm sys}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.1.m1.1d">italic_T start_POSTSUBSCRIPT roman_sys end_POSTSUBSCRIPT</annotation></semantics></math>, and given by</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="A3.EGx2"> <tbody id="S2.E9"> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle[C/N_{0}]_{\rm dB-Hz}^{\text{Radio}}" class="ltx_Math" display="inline" id="S2.Ex3.m1.1"><semantics id="S2.Ex3.m1.1a"><msubsup id="S2.Ex3.m1.1.1" xref="S2.Ex3.m1.1.1.cmml"><mrow id="S2.Ex3.m1.1.1.1.1.1" xref="S2.Ex3.m1.1.1.1.1.2.cmml"><mo id="S2.Ex3.m1.1.1.1.1.1.2" stretchy="false" xref="S2.Ex3.m1.1.1.1.1.2.1.cmml">[</mo><mrow id="S2.Ex3.m1.1.1.1.1.1.1" xref="S2.Ex3.m1.1.1.1.1.1.1.cmml"><mi id="S2.Ex3.m1.1.1.1.1.1.1.2" xref="S2.Ex3.m1.1.1.1.1.1.1.2.cmml">C</mi><mo id="S2.Ex3.m1.1.1.1.1.1.1.1" xref="S2.Ex3.m1.1.1.1.1.1.1.1.cmml">/</mo><msub id="S2.Ex3.m1.1.1.1.1.1.1.3" xref="S2.Ex3.m1.1.1.1.1.1.1.3.cmml"><mi id="S2.Ex3.m1.1.1.1.1.1.1.3.2" xref="S2.Ex3.m1.1.1.1.1.1.1.3.2.cmml">N</mi><mn id="S2.Ex3.m1.1.1.1.1.1.1.3.3" xref="S2.Ex3.m1.1.1.1.1.1.1.3.3.cmml">0</mn></msub></mrow><mo id="S2.Ex3.m1.1.1.1.1.1.3" stretchy="false" xref="S2.Ex3.m1.1.1.1.1.2.1.cmml">]</mo></mrow><mrow id="S2.Ex3.m1.1.1.1.3" xref="S2.Ex3.m1.1.1.1.3.cmml"><mi id="S2.Ex3.m1.1.1.1.3.2" xref="S2.Ex3.m1.1.1.1.3.2.cmml">dB</mi><mo id="S2.Ex3.m1.1.1.1.3.1" xref="S2.Ex3.m1.1.1.1.3.1.cmml">−</mo><mi id="S2.Ex3.m1.1.1.1.3.3" xref="S2.Ex3.m1.1.1.1.3.3.cmml">Hz</mi></mrow><mtext id="S2.Ex3.m1.1.1.3" xref="S2.Ex3.m1.1.1.3a.cmml">Radio</mtext></msubsup><annotation-xml encoding="MathML-Content" id="S2.Ex3.m1.1b"><apply id="S2.Ex3.m1.1.1.cmml" xref="S2.Ex3.m1.1.1"><csymbol cd="ambiguous" id="S2.Ex3.m1.1.1.2.cmml" xref="S2.Ex3.m1.1.1">superscript</csymbol><apply id="S2.Ex3.m1.1.1.1.cmml" xref="S2.Ex3.m1.1.1"><csymbol cd="ambiguous" id="S2.Ex3.m1.1.1.1.2.cmml" xref="S2.Ex3.m1.1.1">subscript</csymbol><apply id="S2.Ex3.m1.1.1.1.1.2.cmml" xref="S2.Ex3.m1.1.1.1.1.1"><csymbol cd="latexml" id="S2.Ex3.m1.1.1.1.1.2.1.cmml" xref="S2.Ex3.m1.1.1.1.1.1.2">delimited-[]</csymbol><apply id="S2.Ex3.m1.1.1.1.1.1.1.cmml" xref="S2.Ex3.m1.1.1.1.1.1.1"><divide id="S2.Ex3.m1.1.1.1.1.1.1.1.cmml" xref="S2.Ex3.m1.1.1.1.1.1.1.1"></divide><ci id="S2.Ex3.m1.1.1.1.1.1.1.2.cmml" xref="S2.Ex3.m1.1.1.1.1.1.1.2">𝐶</ci><apply id="S2.Ex3.m1.1.1.1.1.1.1.3.cmml" xref="S2.Ex3.m1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.Ex3.m1.1.1.1.1.1.1.3.1.cmml" xref="S2.Ex3.m1.1.1.1.1.1.1.3">subscript</csymbol><ci id="S2.Ex3.m1.1.1.1.1.1.1.3.2.cmml" xref="S2.Ex3.m1.1.1.1.1.1.1.3.2">𝑁</ci><cn id="S2.Ex3.m1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S2.Ex3.m1.1.1.1.1.1.1.3.3">0</cn></apply></apply></apply><apply id="S2.Ex3.m1.1.1.1.3.cmml" xref="S2.Ex3.m1.1.1.1.3"><minus id="S2.Ex3.m1.1.1.1.3.1.cmml" xref="S2.Ex3.m1.1.1.1.3.1"></minus><ci id="S2.Ex3.m1.1.1.1.3.2.cmml" xref="S2.Ex3.m1.1.1.1.3.2">dB</ci><ci id="S2.Ex3.m1.1.1.1.3.3.cmml" xref="S2.Ex3.m1.1.1.1.3.3">Hz</ci></apply></apply><ci id="S2.Ex3.m1.1.1.3a.cmml" xref="S2.Ex3.m1.1.1.3"><mtext id="S2.Ex3.m1.1.1.3.cmml" mathsize="70%" xref="S2.Ex3.m1.1.1.3">Radio</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex3.m1.1c">\displaystyle[C/N_{0}]_{\rm dB-Hz}^{\text{Radio}}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex3.m1.1d">[ italic_C / italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT roman_dB - roman_Hz end_POSTSUBSCRIPT start_POSTSUPERSCRIPT Radio end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S2.Ex3.m2.1"><semantics id="S2.Ex3.m2.1a"><mo id="S2.Ex3.m2.1.1" xref="S2.Ex3.m2.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S2.Ex3.m2.1b"><eq id="S2.Ex3.m2.1.1.cmml" xref="S2.Ex3.m2.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex3.m2.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="S2.Ex3.m2.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle 10\log_{10}\left(\frac{P_{\rm rec}}{kT_{\rm sys}}\right)," class="ltx_Math" display="inline" id="S2.Ex3.m3.2"><semantics id="S2.Ex3.m3.2a"><mrow id="S2.Ex3.m3.2.2.1" xref="S2.Ex3.m3.2.2.1.1.cmml"><mrow id="S2.Ex3.m3.2.2.1.1" xref="S2.Ex3.m3.2.2.1.1.cmml"><mn id="S2.Ex3.m3.2.2.1.1.3" xref="S2.Ex3.m3.2.2.1.1.3.cmml">10</mn><mo id="S2.Ex3.m3.2.2.1.1.2" lspace="0.167em" xref="S2.Ex3.m3.2.2.1.1.2.cmml">⁢</mo><mrow id="S2.Ex3.m3.2.2.1.1.1.1" xref="S2.Ex3.m3.2.2.1.1.1.2.cmml"><msub id="S2.Ex3.m3.2.2.1.1.1.1.1" xref="S2.Ex3.m3.2.2.1.1.1.1.1.cmml"><mi id="S2.Ex3.m3.2.2.1.1.1.1.1.2" xref="S2.Ex3.m3.2.2.1.1.1.1.1.2.cmml">log</mi><mn id="S2.Ex3.m3.2.2.1.1.1.1.1.3" xref="S2.Ex3.m3.2.2.1.1.1.1.1.3.cmml">10</mn></msub><mo id="S2.Ex3.m3.2.2.1.1.1.1a" xref="S2.Ex3.m3.2.2.1.1.1.2.cmml">⁡</mo><mrow id="S2.Ex3.m3.2.2.1.1.1.1.2" xref="S2.Ex3.m3.2.2.1.1.1.2.cmml"><mo id="S2.Ex3.m3.2.2.1.1.1.1.2.1" xref="S2.Ex3.m3.2.2.1.1.1.2.cmml">(</mo><mstyle displaystyle="true" id="S2.Ex3.m3.1.1" xref="S2.Ex3.m3.1.1.cmml"><mfrac id="S2.Ex3.m3.1.1a" xref="S2.Ex3.m3.1.1.cmml"><msub id="S2.Ex3.m3.1.1.2" xref="S2.Ex3.m3.1.1.2.cmml"><mi id="S2.Ex3.m3.1.1.2.2" xref="S2.Ex3.m3.1.1.2.2.cmml">P</mi><mi id="S2.Ex3.m3.1.1.2.3" xref="S2.Ex3.m3.1.1.2.3.cmml">rec</mi></msub><mrow id="S2.Ex3.m3.1.1.3" xref="S2.Ex3.m3.1.1.3.cmml"><mi id="S2.Ex3.m3.1.1.3.2" xref="S2.Ex3.m3.1.1.3.2.cmml">k</mi><mo id="S2.Ex3.m3.1.1.3.1" xref="S2.Ex3.m3.1.1.3.1.cmml">⁢</mo><msub id="S2.Ex3.m3.1.1.3.3" xref="S2.Ex3.m3.1.1.3.3.cmml"><mi id="S2.Ex3.m3.1.1.3.3.2" xref="S2.Ex3.m3.1.1.3.3.2.cmml">T</mi><mi id="S2.Ex3.m3.1.1.3.3.3" xref="S2.Ex3.m3.1.1.3.3.3.cmml">sys</mi></msub></mrow></mfrac></mstyle><mo id="S2.Ex3.m3.2.2.1.1.1.1.2.2" xref="S2.Ex3.m3.2.2.1.1.1.2.cmml">)</mo></mrow></mrow></mrow><mo id="S2.Ex3.m3.2.2.1.2" xref="S2.Ex3.m3.2.2.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex3.m3.2b"><apply id="S2.Ex3.m3.2.2.1.1.cmml" xref="S2.Ex3.m3.2.2.1"><times id="S2.Ex3.m3.2.2.1.1.2.cmml" xref="S2.Ex3.m3.2.2.1.1.2"></times><cn id="S2.Ex3.m3.2.2.1.1.3.cmml" type="integer" xref="S2.Ex3.m3.2.2.1.1.3">10</cn><apply id="S2.Ex3.m3.2.2.1.1.1.2.cmml" xref="S2.Ex3.m3.2.2.1.1.1.1"><apply id="S2.Ex3.m3.2.2.1.1.1.1.1.cmml" xref="S2.Ex3.m3.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.Ex3.m3.2.2.1.1.1.1.1.1.cmml" xref="S2.Ex3.m3.2.2.1.1.1.1.1">subscript</csymbol><log id="S2.Ex3.m3.2.2.1.1.1.1.1.2.cmml" xref="S2.Ex3.m3.2.2.1.1.1.1.1.2"></log><cn id="S2.Ex3.m3.2.2.1.1.1.1.1.3.cmml" type="integer" xref="S2.Ex3.m3.2.2.1.1.1.1.1.3">10</cn></apply><apply id="S2.Ex3.m3.1.1.cmml" xref="S2.Ex3.m3.1.1"><divide id="S2.Ex3.m3.1.1.1.cmml" xref="S2.Ex3.m3.1.1"></divide><apply id="S2.Ex3.m3.1.1.2.cmml" xref="S2.Ex3.m3.1.1.2"><csymbol cd="ambiguous" id="S2.Ex3.m3.1.1.2.1.cmml" xref="S2.Ex3.m3.1.1.2">subscript</csymbol><ci id="S2.Ex3.m3.1.1.2.2.cmml" xref="S2.Ex3.m3.1.1.2.2">𝑃</ci><ci id="S2.Ex3.m3.1.1.2.3.cmml" xref="S2.Ex3.m3.1.1.2.3">rec</ci></apply><apply id="S2.Ex3.m3.1.1.3.cmml" xref="S2.Ex3.m3.1.1.3"><times id="S2.Ex3.m3.1.1.3.1.cmml" xref="S2.Ex3.m3.1.1.3.1"></times><ci id="S2.Ex3.m3.1.1.3.2.cmml" xref="S2.Ex3.m3.1.1.3.2">𝑘</ci><apply id="S2.Ex3.m3.1.1.3.3.cmml" xref="S2.Ex3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S2.Ex3.m3.1.1.3.3.1.cmml" xref="S2.Ex3.m3.1.1.3.3">subscript</csymbol><ci id="S2.Ex3.m3.1.1.3.3.2.cmml" xref="S2.Ex3.m3.1.1.3.3.2">𝑇</ci><ci id="S2.Ex3.m3.1.1.3.3.3.cmml" xref="S2.Ex3.m3.1.1.3.3.3">sys</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex3.m3.2c">\displaystyle 10\log_{10}\left(\frac{P_{\rm rec}}{kT_{\rm sys}}\right),</annotation><annotation encoding="application/x-llamapun" id="S2.Ex3.m3.2d">10 roman_log start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT ( divide start_ARG italic_P start_POSTSUBSCRIPT roman_rec end_POSTSUBSCRIPT end_ARG start_ARG italic_k italic_T start_POSTSUBSCRIPT roman_sys end_POSTSUBSCRIPT end_ARG ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="3"><span class="ltx_tag ltx_tag_equation ltx_align_right">(9)</span></td> </tr> <tr class="ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S2.Ex4.m1.1"><semantics id="S2.Ex4.m1.1a"><mo id="S2.Ex4.m1.1.1" xref="S2.Ex4.m1.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S2.Ex4.m1.1b"><eq id="S2.Ex4.m1.1.1.cmml" xref="S2.Ex4.m1.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex4.m1.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="S2.Ex4.m1.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle 34\,\text{dB-Hz}+10\log_{10}\left(\frac{P_{\rm em}}{10{\rm~{}W}}% \right)+20\log_{10}\left(\frac{D_{\rm eff}}{5{\rm~{}m}}\right)-10\log_{10}% \left(\frac{T_{\rm sys}}{50{\rm~{}K}}\right)" class="ltx_Math" display="inline" id="S2.Ex4.m2.6"><semantics id="S2.Ex4.m2.6a"><mrow id="S2.Ex4.m2.6.6" xref="S2.Ex4.m2.6.6.cmml"><mrow id="S2.Ex4.m2.5.5.2" xref="S2.Ex4.m2.5.5.2.cmml"><mrow id="S2.Ex4.m2.5.5.2.4" xref="S2.Ex4.m2.5.5.2.4.cmml"><mn id="S2.Ex4.m2.5.5.2.4.2" xref="S2.Ex4.m2.5.5.2.4.2.cmml">34</mn><mo id="S2.Ex4.m2.5.5.2.4.1" lspace="0.170em" xref="S2.Ex4.m2.5.5.2.4.1.cmml">⁢</mo><mtext id="S2.Ex4.m2.5.5.2.4.3" xref="S2.Ex4.m2.5.5.2.4.3a.cmml">dB-Hz</mtext></mrow><mo id="S2.Ex4.m2.5.5.2.3" xref="S2.Ex4.m2.5.5.2.3.cmml">+</mo><mrow id="S2.Ex4.m2.4.4.1.1" xref="S2.Ex4.m2.4.4.1.1.cmml"><mn id="S2.Ex4.m2.4.4.1.1.3" xref="S2.Ex4.m2.4.4.1.1.3.cmml">10</mn><mo id="S2.Ex4.m2.4.4.1.1.2" lspace="0.167em" xref="S2.Ex4.m2.4.4.1.1.2.cmml">⁢</mo><mrow id="S2.Ex4.m2.4.4.1.1.1.1" xref="S2.Ex4.m2.4.4.1.1.1.2.cmml"><msub id="S2.Ex4.m2.4.4.1.1.1.1.1" xref="S2.Ex4.m2.4.4.1.1.1.1.1.cmml"><mi id="S2.Ex4.m2.4.4.1.1.1.1.1.2" xref="S2.Ex4.m2.4.4.1.1.1.1.1.2.cmml">log</mi><mn id="S2.Ex4.m2.4.4.1.1.1.1.1.3" xref="S2.Ex4.m2.4.4.1.1.1.1.1.3.cmml">10</mn></msub><mo id="S2.Ex4.m2.4.4.1.1.1.1a" xref="S2.Ex4.m2.4.4.1.1.1.2.cmml">⁡</mo><mrow 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id="S2.Ex4.m2.5.5.2.2.3" xref="S2.Ex4.m2.5.5.2.2.3.cmml">20</mn><mo id="S2.Ex4.m2.5.5.2.2.2" lspace="0.167em" xref="S2.Ex4.m2.5.5.2.2.2.cmml">⁢</mo><mrow id="S2.Ex4.m2.5.5.2.2.1.1" xref="S2.Ex4.m2.5.5.2.2.1.2.cmml"><msub id="S2.Ex4.m2.5.5.2.2.1.1.1" xref="S2.Ex4.m2.5.5.2.2.1.1.1.cmml"><mi id="S2.Ex4.m2.5.5.2.2.1.1.1.2" xref="S2.Ex4.m2.5.5.2.2.1.1.1.2.cmml">log</mi><mn id="S2.Ex4.m2.5.5.2.2.1.1.1.3" xref="S2.Ex4.m2.5.5.2.2.1.1.1.3.cmml">10</mn></msub><mo id="S2.Ex4.m2.5.5.2.2.1.1a" xref="S2.Ex4.m2.5.5.2.2.1.2.cmml">⁡</mo><mrow id="S2.Ex4.m2.5.5.2.2.1.1.2" xref="S2.Ex4.m2.5.5.2.2.1.2.cmml"><mo id="S2.Ex4.m2.5.5.2.2.1.1.2.1" xref="S2.Ex4.m2.5.5.2.2.1.2.cmml">(</mo><mstyle displaystyle="true" id="S2.Ex4.m2.2.2" xref="S2.Ex4.m2.2.2.cmml"><mfrac id="S2.Ex4.m2.2.2a" xref="S2.Ex4.m2.2.2.cmml"><msub id="S2.Ex4.m2.2.2.2" xref="S2.Ex4.m2.2.2.2.cmml"><mi id="S2.Ex4.m2.2.2.2.2" xref="S2.Ex4.m2.2.2.2.2.cmml">D</mi><mi id="S2.Ex4.m2.2.2.2.3" xref="S2.Ex4.m2.2.2.2.3.cmml">eff</mi></msub><mrow 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xref="S2.Ex4.m2.6.6.3.1.2.cmml">⁡</mo><mrow id="S2.Ex4.m2.6.6.3.1.1.2" xref="S2.Ex4.m2.6.6.3.1.2.cmml"><mo id="S2.Ex4.m2.6.6.3.1.1.2.1" xref="S2.Ex4.m2.6.6.3.1.2.cmml">(</mo><mstyle displaystyle="true" id="S2.Ex4.m2.3.3" xref="S2.Ex4.m2.3.3.cmml"><mfrac id="S2.Ex4.m2.3.3a" xref="S2.Ex4.m2.3.3.cmml"><msub id="S2.Ex4.m2.3.3.2" xref="S2.Ex4.m2.3.3.2.cmml"><mi id="S2.Ex4.m2.3.3.2.2" xref="S2.Ex4.m2.3.3.2.2.cmml">T</mi><mi id="S2.Ex4.m2.3.3.2.3" xref="S2.Ex4.m2.3.3.2.3.cmml">sys</mi></msub><mrow id="S2.Ex4.m2.3.3.3" xref="S2.Ex4.m2.3.3.3.cmml"><mn id="S2.Ex4.m2.3.3.3.2" xref="S2.Ex4.m2.3.3.3.2.cmml">50</mn><mo id="S2.Ex4.m2.3.3.3.1" lspace="0.330em" xref="S2.Ex4.m2.3.3.3.1.cmml">⁢</mo><mi id="S2.Ex4.m2.3.3.3.3" mathvariant="normal" xref="S2.Ex4.m2.3.3.3.3.cmml">K</mi></mrow></mfrac></mstyle><mo id="S2.Ex4.m2.6.6.3.1.1.2.2" xref="S2.Ex4.m2.6.6.3.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex4.m2.6b"><apply id="S2.Ex4.m2.6.6.cmml" 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xref="S2.Ex4.m2.3.3"></divide><apply id="S2.Ex4.m2.3.3.2.cmml" xref="S2.Ex4.m2.3.3.2"><csymbol cd="ambiguous" id="S2.Ex4.m2.3.3.2.1.cmml" xref="S2.Ex4.m2.3.3.2">subscript</csymbol><ci id="S2.Ex4.m2.3.3.2.2.cmml" xref="S2.Ex4.m2.3.3.2.2">𝑇</ci><ci id="S2.Ex4.m2.3.3.2.3.cmml" xref="S2.Ex4.m2.3.3.2.3">sys</ci></apply><apply id="S2.Ex4.m2.3.3.3.cmml" xref="S2.Ex4.m2.3.3.3"><times id="S2.Ex4.m2.3.3.3.1.cmml" xref="S2.Ex4.m2.3.3.3.1"></times><cn id="S2.Ex4.m2.3.3.3.2.cmml" type="integer" xref="S2.Ex4.m2.3.3.3.2">50</cn><ci id="S2.Ex4.m2.3.3.3.3.cmml" xref="S2.Ex4.m2.3.3.3.3">K</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex4.m2.6c">\displaystyle 34\,\text{dB-Hz}+10\log_{10}\left(\frac{P_{\rm em}}{10{\rm~{}W}}% \right)+20\log_{10}\left(\frac{D_{\rm eff}}{5{\rm~{}m}}\right)-10\log_{10}% \left(\frac{T_{\rm sys}}{50{\rm~{}K}}\right)</annotation><annotation encoding="application/x-llamapun" id="S2.Ex4.m2.6d">34 dB-Hz + 10 roman_log start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT ( divide start_ARG italic_P start_POSTSUBSCRIPT roman_em end_POSTSUBSCRIPT end_ARG start_ARG 10 roman_W end_ARG ) + 20 roman_log start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT ( divide start_ARG italic_D start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT end_ARG start_ARG 5 roman_m end_ARG ) - 10 roman_log start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT ( divide start_ARG italic_T start_POSTSUBSCRIPT roman_sys end_POSTSUBSCRIPT end_ARG start_ARG 50 roman_K end_ARG )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr class="ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-20\log_{10}\left(\frac{L}{30{\rm~{}AU}}\right)-20\log_{10}\left(% \frac{\lambda}{1{\rm~{}cm}}\right)," class="ltx_Math" display="inline" id="S2.E9.m1.3"><semantics id="S2.E9.m1.3a"><mrow id="S2.E9.m1.3.3.1" xref="S2.E9.m1.3.3.1.1.cmml"><mrow id="S2.E9.m1.3.3.1.1" xref="S2.E9.m1.3.3.1.1.cmml"><mrow id="S2.E9.m1.3.3.1.1.1" xref="S2.E9.m1.3.3.1.1.1.cmml"><mo id="S2.E9.m1.3.3.1.1.1a" xref="S2.E9.m1.3.3.1.1.1.cmml">−</mo><mrow id="S2.E9.m1.3.3.1.1.1.1" xref="S2.E9.m1.3.3.1.1.1.1.cmml"><mn id="S2.E9.m1.3.3.1.1.1.1.3" xref="S2.E9.m1.3.3.1.1.1.1.3.cmml">20</mn><mo id="S2.E9.m1.3.3.1.1.1.1.2" lspace="0.167em" xref="S2.E9.m1.3.3.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E9.m1.3.3.1.1.1.1.1.1" xref="S2.E9.m1.3.3.1.1.1.1.1.2.cmml"><msub id="S2.E9.m1.3.3.1.1.1.1.1.1.1" xref="S2.E9.m1.3.3.1.1.1.1.1.1.1.cmml"><mi id="S2.E9.m1.3.3.1.1.1.1.1.1.1.2" xref="S2.E9.m1.3.3.1.1.1.1.1.1.1.2.cmml">log</mi><mn id="S2.E9.m1.3.3.1.1.1.1.1.1.1.3" xref="S2.E9.m1.3.3.1.1.1.1.1.1.1.3.cmml">10</mn></msub><mo id="S2.E9.m1.3.3.1.1.1.1.1.1a" xref="S2.E9.m1.3.3.1.1.1.1.1.2.cmml">⁡</mo><mrow id="S2.E9.m1.3.3.1.1.1.1.1.1.2" xref="S2.E9.m1.3.3.1.1.1.1.1.2.cmml"><mo id="S2.E9.m1.3.3.1.1.1.1.1.1.2.1" xref="S2.E9.m1.3.3.1.1.1.1.1.2.cmml">(</mo><mstyle displaystyle="true" id="S2.E9.m1.1.1" xref="S2.E9.m1.1.1.cmml"><mfrac id="S2.E9.m1.1.1a" xref="S2.E9.m1.1.1.cmml"><mi id="S2.E9.m1.1.1.2" xref="S2.E9.m1.1.1.2.cmml">L</mi><mrow id="S2.E9.m1.1.1.3" xref="S2.E9.m1.1.1.3.cmml"><mn id="S2.E9.m1.1.1.3.2" xref="S2.E9.m1.1.1.3.2.cmml">30</mn><mo id="S2.E9.m1.1.1.3.1" lspace="0.330em" xref="S2.E9.m1.1.1.3.1.cmml">⁢</mo><mi id="S2.E9.m1.1.1.3.3" xref="S2.E9.m1.1.1.3.3.cmml">AU</mi></mrow></mfrac></mstyle><mo id="S2.E9.m1.3.3.1.1.1.1.1.1.2.2" xref="S2.E9.m1.3.3.1.1.1.1.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S2.E9.m1.3.3.1.1.3" xref="S2.E9.m1.3.3.1.1.3.cmml">−</mo><mrow id="S2.E9.m1.3.3.1.1.2" xref="S2.E9.m1.3.3.1.1.2.cmml"><mn id="S2.E9.m1.3.3.1.1.2.3" xref="S2.E9.m1.3.3.1.1.2.3.cmml">20</mn><mo id="S2.E9.m1.3.3.1.1.2.2" lspace="0.167em" xref="S2.E9.m1.3.3.1.1.2.2.cmml">⁢</mo><mrow id="S2.E9.m1.3.3.1.1.2.1.1" xref="S2.E9.m1.3.3.1.1.2.1.2.cmml"><msub 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xref="S2.E9.m1.3.3.1.1.2.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S2.E9.m1.3.3.1.2" xref="S2.E9.m1.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E9.m1.3b"><apply id="S2.E9.m1.3.3.1.1.cmml" xref="S2.E9.m1.3.3.1"><minus id="S2.E9.m1.3.3.1.1.3.cmml" xref="S2.E9.m1.3.3.1.1.3"></minus><apply id="S2.E9.m1.3.3.1.1.1.cmml" xref="S2.E9.m1.3.3.1.1.1"><minus id="S2.E9.m1.3.3.1.1.1.2.cmml" xref="S2.E9.m1.3.3.1.1.1"></minus><apply id="S2.E9.m1.3.3.1.1.1.1.cmml" xref="S2.E9.m1.3.3.1.1.1.1"><times id="S2.E9.m1.3.3.1.1.1.1.2.cmml" xref="S2.E9.m1.3.3.1.1.1.1.2"></times><cn id="S2.E9.m1.3.3.1.1.1.1.3.cmml" type="integer" xref="S2.E9.m1.3.3.1.1.1.1.3">20</cn><apply id="S2.E9.m1.3.3.1.1.1.1.1.2.cmml" xref="S2.E9.m1.3.3.1.1.1.1.1.1"><apply id="S2.E9.m1.3.3.1.1.1.1.1.1.1.cmml" xref="S2.E9.m1.3.3.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.E9.m1.3.3.1.1.1.1.1.1.1.1.cmml" xref="S2.E9.m1.3.3.1.1.1.1.1.1.1">subscript</csymbol><log id="S2.E9.m1.3.3.1.1.1.1.1.1.1.2.cmml" xref="S2.E9.m1.3.3.1.1.1.1.1.1.1.2"></log><cn id="S2.E9.m1.3.3.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.E9.m1.3.3.1.1.1.1.1.1.1.3">10</cn></apply><apply id="S2.E9.m1.1.1.cmml" xref="S2.E9.m1.1.1"><divide id="S2.E9.m1.1.1.1.cmml" xref="S2.E9.m1.1.1"></divide><ci id="S2.E9.m1.1.1.2.cmml" xref="S2.E9.m1.1.1.2">𝐿</ci><apply id="S2.E9.m1.1.1.3.cmml" xref="S2.E9.m1.1.1.3"><times id="S2.E9.m1.1.1.3.1.cmml" xref="S2.E9.m1.1.1.3.1"></times><cn id="S2.E9.m1.1.1.3.2.cmml" type="integer" xref="S2.E9.m1.1.1.3.2">30</cn><ci id="S2.E9.m1.1.1.3.3.cmml" xref="S2.E9.m1.1.1.3.3">AU</ci></apply></apply></apply></apply></apply><apply id="S2.E9.m1.3.3.1.1.2.cmml" xref="S2.E9.m1.3.3.1.1.2"><times id="S2.E9.m1.3.3.1.1.2.2.cmml" xref="S2.E9.m1.3.3.1.1.2.2"></times><cn id="S2.E9.m1.3.3.1.1.2.3.cmml" type="integer" xref="S2.E9.m1.3.3.1.1.2.3">20</cn><apply id="S2.E9.m1.3.3.1.1.2.1.2.cmml" xref="S2.E9.m1.3.3.1.1.2.1.1"><apply id="S2.E9.m1.3.3.1.1.2.1.1.1.cmml" xref="S2.E9.m1.3.3.1.1.2.1.1.1"><csymbol cd="ambiguous" id="S2.E9.m1.3.3.1.1.2.1.1.1.1.cmml" xref="S2.E9.m1.3.3.1.1.2.1.1.1">subscript</csymbol><log id="S2.E9.m1.3.3.1.1.2.1.1.1.2.cmml" xref="S2.E9.m1.3.3.1.1.2.1.1.1.2"></log><cn id="S2.E9.m1.3.3.1.1.2.1.1.1.3.cmml" type="integer" xref="S2.E9.m1.3.3.1.1.2.1.1.1.3">10</cn></apply><apply id="S2.E9.m1.2.2.cmml" xref="S2.E9.m1.2.2"><divide id="S2.E9.m1.2.2.1.cmml" xref="S2.E9.m1.2.2"></divide><ci id="S2.E9.m1.2.2.2.cmml" xref="S2.E9.m1.2.2.2">𝜆</ci><apply id="S2.E9.m1.2.2.3.cmml" xref="S2.E9.m1.2.2.3"><times id="S2.E9.m1.2.2.3.1.cmml" xref="S2.E9.m1.2.2.3.1"></times><cn id="S2.E9.m1.2.2.3.2.cmml" type="integer" xref="S2.E9.m1.2.2.3.2">1</cn><ci id="S2.E9.m1.2.2.3.3.cmml" xref="S2.E9.m1.2.2.3.3">cm</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E9.m1.3c">\displaystyle-20\log_{10}\left(\frac{L}{30{\rm~{}AU}}\right)-20\log_{10}\left(% \frac{\lambda}{1{\rm~{}cm}}\right),</annotation><annotation encoding="application/x-llamapun" id="S2.E9.m1.3d">- 20 roman_log start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT ( divide start_ARG italic_L end_ARG start_ARG 30 roman_AU end_ARG ) - 20 roman_log start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT ( divide start_ARG italic_λ end_ARG start_ARG 1 roman_cm end_ARG ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </tbody> </table> <p class="ltx_p" id="S2.SS2.p3.5">where <math alttext="P_{\rm em}" class="ltx_Math" display="inline" id="S2.SS2.p3.2.m1.1"><semantics id="S2.SS2.p3.2.m1.1a"><msub id="S2.SS2.p3.2.m1.1.1" xref="S2.SS2.p3.2.m1.1.1.cmml"><mi id="S2.SS2.p3.2.m1.1.1.2" xref="S2.SS2.p3.2.m1.1.1.2.cmml">P</mi><mi id="S2.SS2.p3.2.m1.1.1.3" xref="S2.SS2.p3.2.m1.1.1.3.cmml">em</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.2.m1.1b"><apply id="S2.SS2.p3.2.m1.1.1.cmml" xref="S2.SS2.p3.2.m1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p3.2.m1.1.1.1.cmml" xref="S2.SS2.p3.2.m1.1.1">subscript</csymbol><ci id="S2.SS2.p3.2.m1.1.1.2.cmml" xref="S2.SS2.p3.2.m1.1.1.2">𝑃</ci><ci id="S2.SS2.p3.2.m1.1.1.3.cmml" xref="S2.SS2.p3.2.m1.1.1.3">em</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.2.m1.1c">P_{\rm em}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.2.m1.1d">italic_P start_POSTSUBSCRIPT roman_em end_POSTSUBSCRIPT</annotation></semantics></math> is the emitted power and <math alttext="D_{\rm eff}" class="ltx_Math" display="inline" id="S2.SS2.p3.3.m2.1"><semantics id="S2.SS2.p3.3.m2.1a"><msub id="S2.SS2.p3.3.m2.1.1" xref="S2.SS2.p3.3.m2.1.1.cmml"><mi id="S2.SS2.p3.3.m2.1.1.2" xref="S2.SS2.p3.3.m2.1.1.2.cmml">D</mi><mi id="S2.SS2.p3.3.m2.1.1.3" xref="S2.SS2.p3.3.m2.1.1.3.cmml">eff</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.3.m2.1b"><apply id="S2.SS2.p3.3.m2.1.1.cmml" xref="S2.SS2.p3.3.m2.1.1"><csymbol cd="ambiguous" id="S2.SS2.p3.3.m2.1.1.1.cmml" xref="S2.SS2.p3.3.m2.1.1">subscript</csymbol><ci id="S2.SS2.p3.3.m2.1.1.2.cmml" xref="S2.SS2.p3.3.m2.1.1.2">𝐷</ci><ci id="S2.SS2.p3.3.m2.1.1.3.cmml" xref="S2.SS2.p3.3.m2.1.1.3">eff</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.3.m2.1c">D_{\rm eff}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.3.m2.1d">italic_D start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT</annotation></semantics></math> is the effective diameter of the radio dishes <cite class="ltx_cite ltx_citemacro_citep">(Misra & Enge, <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib50" title="">2012</a>)</cite>. We will use Ka band transmissions with <math alttext="\lambda=1~{}" class="ltx_Math" display="inline" id="S2.SS2.p3.4.m3.1"><semantics id="S2.SS2.p3.4.m3.1a"><mrow id="S2.SS2.p3.4.m3.1.1" xref="S2.SS2.p3.4.m3.1.1.cmml"><mi id="S2.SS2.p3.4.m3.1.1.2" xref="S2.SS2.p3.4.m3.1.1.2.cmml">λ</mi><mo id="S2.SS2.p3.4.m3.1.1.1" xref="S2.SS2.p3.4.m3.1.1.1.cmml">=</mo><mn id="S2.SS2.p3.4.m3.1.1.3" xref="S2.SS2.p3.4.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.4.m3.1b"><apply id="S2.SS2.p3.4.m3.1.1.cmml" xref="S2.SS2.p3.4.m3.1.1"><eq id="S2.SS2.p3.4.m3.1.1.1.cmml" xref="S2.SS2.p3.4.m3.1.1.1"></eq><ci id="S2.SS2.p3.4.m3.1.1.2.cmml" xref="S2.SS2.p3.4.m3.1.1.2">𝜆</ci><cn id="S2.SS2.p3.4.m3.1.1.3.cmml" type="integer" xref="S2.SS2.p3.4.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.4.m3.1c">\lambda=1~{}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.4.m3.1d">italic_λ = 1</annotation></semantics></math>cm for our estimates. For reference, the Voyager and Cassini probes had 4 meter diameter radio dishes, the SMAP spacecraft employed a fold-out 6 meter dish <cite class="ltx_cite ltx_citemacro_citep">(Njoku et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib56" title="">2014</a>)</cite>, the RadioAstron satellite that orbited to the Moon had a fold-out 10-meter dish <cite class="ltx_cite ltx_citemacro_citep">(Kardashev et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib39" title="">2012</a>)</cite>. Additionally, values of the receiver noise temperature of <math alttext="\sim 50\;" class="ltx_Math" display="inline" id="S2.SS2.p3.5.m4.1"><semantics id="S2.SS2.p3.5.m4.1a"><mrow id="S2.SS2.p3.5.m4.1.1" xref="S2.SS2.p3.5.m4.1.1.cmml"><mi id="S2.SS2.p3.5.m4.1.1.2" xref="S2.SS2.p3.5.m4.1.1.2.cmml"></mi><mo id="S2.SS2.p3.5.m4.1.1.1" xref="S2.SS2.p3.5.m4.1.1.1.cmml">∼</mo><mn id="S2.SS2.p3.5.m4.1.1.3" xref="S2.SS2.p3.5.m4.1.1.3.cmml">50</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.5.m4.1b"><apply id="S2.SS2.p3.5.m4.1.1.cmml" xref="S2.SS2.p3.5.m4.1.1"><csymbol cd="latexml" id="S2.SS2.p3.5.m4.1.1.1.cmml" xref="S2.SS2.p3.5.m4.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S2.SS2.p3.5.m4.1.1.2.cmml" xref="S2.SS2.p3.5.m4.1.1.2">absent</csymbol><cn id="S2.SS2.p3.5.m4.1.1.3.cmml" type="integer" xref="S2.SS2.p3.5.m4.1.1.3">50</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.5.m4.1c">\sim 50\;</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.5.m4.1d">∼ 50</annotation></semantics></math>K are typical for narrow band receivers at room temperature, even space-qualified ones <cite class="ltx_cite ltx_citemacro_citep">(High Frequency Electronics, <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib36" title="">2023</a>)</cite>.<span class="ltx_note ltx_role_footnote" id="footnote5"><sup class="ltx_note_mark">5</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">5</sup><span class="ltx_tag ltx_tag_note">5</span>The low noise amplifiers can be further positioned on a cold plate on the spacecraft to achieve lower <math alttext="T_{\rm sys}" class="ltx_Math" display="inline" id="footnote5.m1.1"><semantics id="footnote5.m1.1b"><msub id="footnote5.m1.1.1" xref="footnote5.m1.1.1.cmml"><mi id="footnote5.m1.1.1.2" xref="footnote5.m1.1.1.2.cmml">T</mi><mi id="footnote5.m1.1.1.3" xref="footnote5.m1.1.1.3.cmml">sys</mi></msub><annotation-xml encoding="MathML-Content" id="footnote5.m1.1c"><apply id="footnote5.m1.1.1.cmml" xref="footnote5.m1.1.1"><csymbol cd="ambiguous" id="footnote5.m1.1.1.1.cmml" xref="footnote5.m1.1.1">subscript</csymbol><ci id="footnote5.m1.1.1.2.cmml" xref="footnote5.m1.1.1.2">𝑇</ci><ci id="footnote5.m1.1.1.3.cmml" xref="footnote5.m1.1.1.3">sys</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote5.m1.1d">T_{\rm sys}</annotation><annotation encoding="application/x-llamapun" id="footnote5.m1.1e">italic_T start_POSTSUBSCRIPT roman_sys end_POSTSUBSCRIPT</annotation></semantics></math>.</span></span></span></p> </div> <div class="ltx_para" id="S2.SS2.p4"> <p class="ltx_p" id="S2.SS2.p4.9">To put the dB-Hz in equations (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.E8" title="In 2.2 Radiometer noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">8</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.E9" title="In 2.2 Radiometer noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">9</span></a>) in context, which evaluate to <math alttext="[C/N_{0}]_{\rm dB-Hz}\sim 30-50" class="ltx_Math" display="inline" id="S2.SS2.p4.1.m1.1"><semantics id="S2.SS2.p4.1.m1.1a"><mrow id="S2.SS2.p4.1.m1.1.1" xref="S2.SS2.p4.1.m1.1.1.cmml"><msub id="S2.SS2.p4.1.m1.1.1.1" xref="S2.SS2.p4.1.m1.1.1.1.cmml"><mrow id="S2.SS2.p4.1.m1.1.1.1.1.1" xref="S2.SS2.p4.1.m1.1.1.1.1.2.cmml"><mo id="S2.SS2.p4.1.m1.1.1.1.1.1.2" stretchy="false" xref="S2.SS2.p4.1.m1.1.1.1.1.2.1.cmml">[</mo><mrow id="S2.SS2.p4.1.m1.1.1.1.1.1.1" xref="S2.SS2.p4.1.m1.1.1.1.1.1.1.cmml"><mi id="S2.SS2.p4.1.m1.1.1.1.1.1.1.2" xref="S2.SS2.p4.1.m1.1.1.1.1.1.1.2.cmml">C</mi><mo id="S2.SS2.p4.1.m1.1.1.1.1.1.1.1" xref="S2.SS2.p4.1.m1.1.1.1.1.1.1.1.cmml">/</mo><msub id="S2.SS2.p4.1.m1.1.1.1.1.1.1.3" xref="S2.SS2.p4.1.m1.1.1.1.1.1.1.3.cmml"><mi id="S2.SS2.p4.1.m1.1.1.1.1.1.1.3.2" xref="S2.SS2.p4.1.m1.1.1.1.1.1.1.3.2.cmml">N</mi><mn id="S2.SS2.p4.1.m1.1.1.1.1.1.1.3.3" xref="S2.SS2.p4.1.m1.1.1.1.1.1.1.3.3.cmml">0</mn></msub></mrow><mo id="S2.SS2.p4.1.m1.1.1.1.1.1.3" stretchy="false" xref="S2.SS2.p4.1.m1.1.1.1.1.2.1.cmml">]</mo></mrow><mrow id="S2.SS2.p4.1.m1.1.1.1.3" xref="S2.SS2.p4.1.m1.1.1.1.3.cmml"><mi id="S2.SS2.p4.1.m1.1.1.1.3.2" xref="S2.SS2.p4.1.m1.1.1.1.3.2.cmml">dB</mi><mo id="S2.SS2.p4.1.m1.1.1.1.3.1" xref="S2.SS2.p4.1.m1.1.1.1.3.1.cmml">−</mo><mi id="S2.SS2.p4.1.m1.1.1.1.3.3" xref="S2.SS2.p4.1.m1.1.1.1.3.3.cmml">Hz</mi></mrow></msub><mo id="S2.SS2.p4.1.m1.1.1.2" xref="S2.SS2.p4.1.m1.1.1.2.cmml">∼</mo><mrow id="S2.SS2.p4.1.m1.1.1.3" xref="S2.SS2.p4.1.m1.1.1.3.cmml"><mn id="S2.SS2.p4.1.m1.1.1.3.2" xref="S2.SS2.p4.1.m1.1.1.3.2.cmml">30</mn><mo id="S2.SS2.p4.1.m1.1.1.3.1" xref="S2.SS2.p4.1.m1.1.1.3.1.cmml">−</mo><mn id="S2.SS2.p4.1.m1.1.1.3.3" xref="S2.SS2.p4.1.m1.1.1.3.3.cmml">50</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.1.m1.1b"><apply id="S2.SS2.p4.1.m1.1.1.cmml" xref="S2.SS2.p4.1.m1.1.1"><csymbol cd="latexml" id="S2.SS2.p4.1.m1.1.1.2.cmml" xref="S2.SS2.p4.1.m1.1.1.2">similar-to</csymbol><apply id="S2.SS2.p4.1.m1.1.1.1.cmml" xref="S2.SS2.p4.1.m1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.1.m1.1.1.1.2.cmml" xref="S2.SS2.p4.1.m1.1.1.1">subscript</csymbol><apply id="S2.SS2.p4.1.m1.1.1.1.1.2.cmml" xref="S2.SS2.p4.1.m1.1.1.1.1.1"><csymbol cd="latexml" id="S2.SS2.p4.1.m1.1.1.1.1.2.1.cmml" xref="S2.SS2.p4.1.m1.1.1.1.1.1.2">delimited-[]</csymbol><apply id="S2.SS2.p4.1.m1.1.1.1.1.1.1.cmml" xref="S2.SS2.p4.1.m1.1.1.1.1.1.1"><divide id="S2.SS2.p4.1.m1.1.1.1.1.1.1.1.cmml" xref="S2.SS2.p4.1.m1.1.1.1.1.1.1.1"></divide><ci id="S2.SS2.p4.1.m1.1.1.1.1.1.1.2.cmml" xref="S2.SS2.p4.1.m1.1.1.1.1.1.1.2">𝐶</ci><apply id="S2.SS2.p4.1.m1.1.1.1.1.1.1.3.cmml" xref="S2.SS2.p4.1.m1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p4.1.m1.1.1.1.1.1.1.3.1.cmml" xref="S2.SS2.p4.1.m1.1.1.1.1.1.1.3">subscript</csymbol><ci id="S2.SS2.p4.1.m1.1.1.1.1.1.1.3.2.cmml" xref="S2.SS2.p4.1.m1.1.1.1.1.1.1.3.2">𝑁</ci><cn id="S2.SS2.p4.1.m1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S2.SS2.p4.1.m1.1.1.1.1.1.1.3.3">0</cn></apply></apply></apply><apply id="S2.SS2.p4.1.m1.1.1.1.3.cmml" xref="S2.SS2.p4.1.m1.1.1.1.3"><minus id="S2.SS2.p4.1.m1.1.1.1.3.1.cmml" xref="S2.SS2.p4.1.m1.1.1.1.3.1"></minus><ci id="S2.SS2.p4.1.m1.1.1.1.3.2.cmml" xref="S2.SS2.p4.1.m1.1.1.1.3.2">dB</ci><ci id="S2.SS2.p4.1.m1.1.1.1.3.3.cmml" xref="S2.SS2.p4.1.m1.1.1.1.3.3">Hz</ci></apply></apply><apply id="S2.SS2.p4.1.m1.1.1.3.cmml" xref="S2.SS2.p4.1.m1.1.1.3"><minus id="S2.SS2.p4.1.m1.1.1.3.1.cmml" xref="S2.SS2.p4.1.m1.1.1.3.1"></minus><cn id="S2.SS2.p4.1.m1.1.1.3.2.cmml" type="integer" xref="S2.SS2.p4.1.m1.1.1.3.2">30</cn><cn id="S2.SS2.p4.1.m1.1.1.3.3.cmml" type="integer" xref="S2.SS2.p4.1.m1.1.1.3.3">50</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.1.m1.1c">[C/N_{0}]_{\rm dB-Hz}\sim 30-50</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.1.m1.1d">[ italic_C / italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT roman_dB - roman_Hz end_POSTSUBSCRIPT ∼ 30 - 50</annotation></semantics></math> for the parameter values discussed in this paper, LISA’s pilot tones, whose phase is measured and used to correct timing jitter in the analog-to-digital conversion, have <math alttext="[C/N_{0}]_{\rm dB-Hz}=75" class="ltx_Math" display="inline" id="S2.SS2.p4.2.m2.1"><semantics id="S2.SS2.p4.2.m2.1a"><mrow id="S2.SS2.p4.2.m2.1.1" xref="S2.SS2.p4.2.m2.1.1.cmml"><msub id="S2.SS2.p4.2.m2.1.1.1" xref="S2.SS2.p4.2.m2.1.1.1.cmml"><mrow id="S2.SS2.p4.2.m2.1.1.1.1.1" xref="S2.SS2.p4.2.m2.1.1.1.1.2.cmml"><mo id="S2.SS2.p4.2.m2.1.1.1.1.1.2" stretchy="false" xref="S2.SS2.p4.2.m2.1.1.1.1.2.1.cmml">[</mo><mrow id="S2.SS2.p4.2.m2.1.1.1.1.1.1" xref="S2.SS2.p4.2.m2.1.1.1.1.1.1.cmml"><mi id="S2.SS2.p4.2.m2.1.1.1.1.1.1.2" xref="S2.SS2.p4.2.m2.1.1.1.1.1.1.2.cmml">C</mi><mo id="S2.SS2.p4.2.m2.1.1.1.1.1.1.1" xref="S2.SS2.p4.2.m2.1.1.1.1.1.1.1.cmml">/</mo><msub id="S2.SS2.p4.2.m2.1.1.1.1.1.1.3" xref="S2.SS2.p4.2.m2.1.1.1.1.1.1.3.cmml"><mi id="S2.SS2.p4.2.m2.1.1.1.1.1.1.3.2" xref="S2.SS2.p4.2.m2.1.1.1.1.1.1.3.2.cmml">N</mi><mn id="S2.SS2.p4.2.m2.1.1.1.1.1.1.3.3" xref="S2.SS2.p4.2.m2.1.1.1.1.1.1.3.3.cmml">0</mn></msub></mrow><mo id="S2.SS2.p4.2.m2.1.1.1.1.1.3" stretchy="false" xref="S2.SS2.p4.2.m2.1.1.1.1.2.1.cmml">]</mo></mrow><mrow id="S2.SS2.p4.2.m2.1.1.1.3" xref="S2.SS2.p4.2.m2.1.1.1.3.cmml"><mi id="S2.SS2.p4.2.m2.1.1.1.3.2" xref="S2.SS2.p4.2.m2.1.1.1.3.2.cmml">dB</mi><mo id="S2.SS2.p4.2.m2.1.1.1.3.1" xref="S2.SS2.p4.2.m2.1.1.1.3.1.cmml">−</mo><mi id="S2.SS2.p4.2.m2.1.1.1.3.3" xref="S2.SS2.p4.2.m2.1.1.1.3.3.cmml">Hz</mi></mrow></msub><mo id="S2.SS2.p4.2.m2.1.1.2" xref="S2.SS2.p4.2.m2.1.1.2.cmml">=</mo><mn id="S2.SS2.p4.2.m2.1.1.3" xref="S2.SS2.p4.2.m2.1.1.3.cmml">75</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.2.m2.1b"><apply id="S2.SS2.p4.2.m2.1.1.cmml" xref="S2.SS2.p4.2.m2.1.1"><eq id="S2.SS2.p4.2.m2.1.1.2.cmml" xref="S2.SS2.p4.2.m2.1.1.2"></eq><apply id="S2.SS2.p4.2.m2.1.1.1.cmml" xref="S2.SS2.p4.2.m2.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.2.m2.1.1.1.2.cmml" xref="S2.SS2.p4.2.m2.1.1.1">subscript</csymbol><apply id="S2.SS2.p4.2.m2.1.1.1.1.2.cmml" xref="S2.SS2.p4.2.m2.1.1.1.1.1"><csymbol cd="latexml" id="S2.SS2.p4.2.m2.1.1.1.1.2.1.cmml" xref="S2.SS2.p4.2.m2.1.1.1.1.1.2">delimited-[]</csymbol><apply id="S2.SS2.p4.2.m2.1.1.1.1.1.1.cmml" xref="S2.SS2.p4.2.m2.1.1.1.1.1.1"><divide id="S2.SS2.p4.2.m2.1.1.1.1.1.1.1.cmml" xref="S2.SS2.p4.2.m2.1.1.1.1.1.1.1"></divide><ci id="S2.SS2.p4.2.m2.1.1.1.1.1.1.2.cmml" xref="S2.SS2.p4.2.m2.1.1.1.1.1.1.2">𝐶</ci><apply id="S2.SS2.p4.2.m2.1.1.1.1.1.1.3.cmml" xref="S2.SS2.p4.2.m2.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p4.2.m2.1.1.1.1.1.1.3.1.cmml" xref="S2.SS2.p4.2.m2.1.1.1.1.1.1.3">subscript</csymbol><ci id="S2.SS2.p4.2.m2.1.1.1.1.1.1.3.2.cmml" xref="S2.SS2.p4.2.m2.1.1.1.1.1.1.3.2">𝑁</ci><cn id="S2.SS2.p4.2.m2.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S2.SS2.p4.2.m2.1.1.1.1.1.1.3.3">0</cn></apply></apply></apply><apply id="S2.SS2.p4.2.m2.1.1.1.3.cmml" xref="S2.SS2.p4.2.m2.1.1.1.3"><minus id="S2.SS2.p4.2.m2.1.1.1.3.1.cmml" xref="S2.SS2.p4.2.m2.1.1.1.3.1"></minus><ci id="S2.SS2.p4.2.m2.1.1.1.3.2.cmml" xref="S2.SS2.p4.2.m2.1.1.1.3.2">dB</ci><ci id="S2.SS2.p4.2.m2.1.1.1.3.3.cmml" xref="S2.SS2.p4.2.m2.1.1.1.3.3">Hz</ci></apply></apply><cn id="S2.SS2.p4.2.m2.1.1.3.cmml" type="integer" xref="S2.SS2.p4.2.m2.1.1.3">75</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.2.m2.1c">[C/N_{0}]_{\rm dB-Hz}=75</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.2.m2.1d">[ italic_C / italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT roman_dB - roman_Hz end_POSTSUBSCRIPT = 75</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_citep">(Barke et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib13" title="">2014</a>)</cite>, a full four orders of magnitude larger <math alttext="C/N_{0}" class="ltx_Math" display="inline" id="S2.SS2.p4.3.m3.1"><semantics id="S2.SS2.p4.3.m3.1a"><mrow id="S2.SS2.p4.3.m3.1.1" xref="S2.SS2.p4.3.m3.1.1.cmml"><mi id="S2.SS2.p4.3.m3.1.1.2" xref="S2.SS2.p4.3.m3.1.1.2.cmml">C</mi><mo id="S2.SS2.p4.3.m3.1.1.1" xref="S2.SS2.p4.3.m3.1.1.1.cmml">/</mo><msub id="S2.SS2.p4.3.m3.1.1.3" xref="S2.SS2.p4.3.m3.1.1.3.cmml"><mi id="S2.SS2.p4.3.m3.1.1.3.2" xref="S2.SS2.p4.3.m3.1.1.3.2.cmml">N</mi><mn id="S2.SS2.p4.3.m3.1.1.3.3" xref="S2.SS2.p4.3.m3.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.3.m3.1b"><apply id="S2.SS2.p4.3.m3.1.1.cmml" xref="S2.SS2.p4.3.m3.1.1"><divide id="S2.SS2.p4.3.m3.1.1.1.cmml" xref="S2.SS2.p4.3.m3.1.1.1"></divide><ci id="S2.SS2.p4.3.m3.1.1.2.cmml" xref="S2.SS2.p4.3.m3.1.1.2">𝐶</ci><apply id="S2.SS2.p4.3.m3.1.1.3.cmml" xref="S2.SS2.p4.3.m3.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p4.3.m3.1.1.3.1.cmml" xref="S2.SS2.p4.3.m3.1.1.3">subscript</csymbol><ci id="S2.SS2.p4.3.m3.1.1.3.2.cmml" xref="S2.SS2.p4.3.m3.1.1.3.2">𝑁</ci><cn id="S2.SS2.p4.3.m3.1.1.3.3.cmml" type="integer" xref="S2.SS2.p4.3.m3.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.3.m3.1c">C/N_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.3.m3.1d">italic_C / italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> than the signal for fiducial values in equations (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.E8" title="In 2.2 Radiometer noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">8</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.E9" title="In 2.2 Radiometer noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">9</span></a>). Another point of reference is the <math alttext="X" class="ltx_Math" display="inline" id="S2.SS2.p4.4.m4.1"><semantics id="S2.SS2.p4.4.m4.1a"><mi id="S2.SS2.p4.4.m4.1.1" xref="S2.SS2.p4.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.4.m4.1b"><ci id="S2.SS2.p4.4.m4.1.1.cmml" xref="S2.SS2.p4.4.m4.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.4.m4.1d">italic_X</annotation></semantics></math>-band downlink of Cassini at <math alttext="10~{}" class="ltx_Math" display="inline" id="S2.SS2.p4.5.m5.1"><semantics id="S2.SS2.p4.5.m5.1a"><mn id="S2.SS2.p4.5.m5.1.1" xref="S2.SS2.p4.5.m5.1.1.cmml">10</mn><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.5.m5.1b"><cn id="S2.SS2.p4.5.m5.1.1.cmml" type="integer" xref="S2.SS2.p4.5.m5.1.1">10</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.5.m5.1c">10~{}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.5.m5.1d">10</annotation></semantics></math>AU to 34 m Deep Space Network antenna – used for the most precise Doppler tracking experiment – had <math alttext="[C/N_{0}]_{\rm dB-Hz}\approx 40-50" class="ltx_Math" display="inline" id="S2.SS2.p4.6.m6.1"><semantics id="S2.SS2.p4.6.m6.1a"><mrow id="S2.SS2.p4.6.m6.1.1" xref="S2.SS2.p4.6.m6.1.1.cmml"><msub id="S2.SS2.p4.6.m6.1.1.1" xref="S2.SS2.p4.6.m6.1.1.1.cmml"><mrow id="S2.SS2.p4.6.m6.1.1.1.1.1" xref="S2.SS2.p4.6.m6.1.1.1.1.2.cmml"><mo id="S2.SS2.p4.6.m6.1.1.1.1.1.2" stretchy="false" xref="S2.SS2.p4.6.m6.1.1.1.1.2.1.cmml">[</mo><mrow id="S2.SS2.p4.6.m6.1.1.1.1.1.1" xref="S2.SS2.p4.6.m6.1.1.1.1.1.1.cmml"><mi id="S2.SS2.p4.6.m6.1.1.1.1.1.1.2" xref="S2.SS2.p4.6.m6.1.1.1.1.1.1.2.cmml">C</mi><mo id="S2.SS2.p4.6.m6.1.1.1.1.1.1.1" xref="S2.SS2.p4.6.m6.1.1.1.1.1.1.1.cmml">/</mo><msub id="S2.SS2.p4.6.m6.1.1.1.1.1.1.3" xref="S2.SS2.p4.6.m6.1.1.1.1.1.1.3.cmml"><mi id="S2.SS2.p4.6.m6.1.1.1.1.1.1.3.2" xref="S2.SS2.p4.6.m6.1.1.1.1.1.1.3.2.cmml">N</mi><mn id="S2.SS2.p4.6.m6.1.1.1.1.1.1.3.3" xref="S2.SS2.p4.6.m6.1.1.1.1.1.1.3.3.cmml">0</mn></msub></mrow><mo id="S2.SS2.p4.6.m6.1.1.1.1.1.3" stretchy="false" xref="S2.SS2.p4.6.m6.1.1.1.1.2.1.cmml">]</mo></mrow><mrow id="S2.SS2.p4.6.m6.1.1.1.3" xref="S2.SS2.p4.6.m6.1.1.1.3.cmml"><mi id="S2.SS2.p4.6.m6.1.1.1.3.2" xref="S2.SS2.p4.6.m6.1.1.1.3.2.cmml">dB</mi><mo id="S2.SS2.p4.6.m6.1.1.1.3.1" xref="S2.SS2.p4.6.m6.1.1.1.3.1.cmml">−</mo><mi id="S2.SS2.p4.6.m6.1.1.1.3.3" xref="S2.SS2.p4.6.m6.1.1.1.3.3.cmml">Hz</mi></mrow></msub><mo id="S2.SS2.p4.6.m6.1.1.2" xref="S2.SS2.p4.6.m6.1.1.2.cmml">≈</mo><mrow id="S2.SS2.p4.6.m6.1.1.3" xref="S2.SS2.p4.6.m6.1.1.3.cmml"><mn id="S2.SS2.p4.6.m6.1.1.3.2" xref="S2.SS2.p4.6.m6.1.1.3.2.cmml">40</mn><mo id="S2.SS2.p4.6.m6.1.1.3.1" xref="S2.SS2.p4.6.m6.1.1.3.1.cmml">−</mo><mn id="S2.SS2.p4.6.m6.1.1.3.3" xref="S2.SS2.p4.6.m6.1.1.3.3.cmml">50</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.6.m6.1b"><apply id="S2.SS2.p4.6.m6.1.1.cmml" xref="S2.SS2.p4.6.m6.1.1"><approx id="S2.SS2.p4.6.m6.1.1.2.cmml" xref="S2.SS2.p4.6.m6.1.1.2"></approx><apply id="S2.SS2.p4.6.m6.1.1.1.cmml" xref="S2.SS2.p4.6.m6.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.6.m6.1.1.1.2.cmml" xref="S2.SS2.p4.6.m6.1.1.1">subscript</csymbol><apply id="S2.SS2.p4.6.m6.1.1.1.1.2.cmml" xref="S2.SS2.p4.6.m6.1.1.1.1.1"><csymbol cd="latexml" id="S2.SS2.p4.6.m6.1.1.1.1.2.1.cmml" xref="S2.SS2.p4.6.m6.1.1.1.1.1.2">delimited-[]</csymbol><apply id="S2.SS2.p4.6.m6.1.1.1.1.1.1.cmml" xref="S2.SS2.p4.6.m6.1.1.1.1.1.1"><divide id="S2.SS2.p4.6.m6.1.1.1.1.1.1.1.cmml" xref="S2.SS2.p4.6.m6.1.1.1.1.1.1.1"></divide><ci id="S2.SS2.p4.6.m6.1.1.1.1.1.1.2.cmml" xref="S2.SS2.p4.6.m6.1.1.1.1.1.1.2">𝐶</ci><apply id="S2.SS2.p4.6.m6.1.1.1.1.1.1.3.cmml" xref="S2.SS2.p4.6.m6.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p4.6.m6.1.1.1.1.1.1.3.1.cmml" xref="S2.SS2.p4.6.m6.1.1.1.1.1.1.3">subscript</csymbol><ci id="S2.SS2.p4.6.m6.1.1.1.1.1.1.3.2.cmml" xref="S2.SS2.p4.6.m6.1.1.1.1.1.1.3.2">𝑁</ci><cn id="S2.SS2.p4.6.m6.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S2.SS2.p4.6.m6.1.1.1.1.1.1.3.3">0</cn></apply></apply></apply><apply id="S2.SS2.p4.6.m6.1.1.1.3.cmml" xref="S2.SS2.p4.6.m6.1.1.1.3"><minus id="S2.SS2.p4.6.m6.1.1.1.3.1.cmml" xref="S2.SS2.p4.6.m6.1.1.1.3.1"></minus><ci id="S2.SS2.p4.6.m6.1.1.1.3.2.cmml" xref="S2.SS2.p4.6.m6.1.1.1.3.2">dB</ci><ci id="S2.SS2.p4.6.m6.1.1.1.3.3.cmml" xref="S2.SS2.p4.6.m6.1.1.1.3.3">Hz</ci></apply></apply><apply id="S2.SS2.p4.6.m6.1.1.3.cmml" xref="S2.SS2.p4.6.m6.1.1.3"><minus id="S2.SS2.p4.6.m6.1.1.3.1.cmml" xref="S2.SS2.p4.6.m6.1.1.3.1"></minus><cn id="S2.SS2.p4.6.m6.1.1.3.2.cmml" type="integer" xref="S2.SS2.p4.6.m6.1.1.3.2">40</cn><cn id="S2.SS2.p4.6.m6.1.1.3.3.cmml" type="integer" xref="S2.SS2.p4.6.m6.1.1.3.3">50</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.6.m6.1c">[C/N_{0}]_{\rm dB-Hz}\approx 40-50</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.6.m6.1d">[ italic_C / italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT roman_dB - roman_Hz end_POSTSUBSCRIPT ≈ 40 - 50</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_citep">(Wang et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib70" title="">2005</a>)</cite>. Finally, the Voyager spacecraft at <math alttext="140" class="ltx_Math" display="inline" id="S2.SS2.p4.7.m7.1"><semantics id="S2.SS2.p4.7.m7.1a"><mn id="S2.SS2.p4.7.m7.1.1" xref="S2.SS2.p4.7.m7.1.1.cmml">140</mn><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.7.m7.1b"><cn id="S2.SS2.p4.7.m7.1.1.cmml" type="integer" xref="S2.SS2.p4.7.m7.1.1">140</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.7.m7.1c">140</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.7.m7.1d">140</annotation></semantics></math>AU communicated with <math alttext="[C/N_{0}]_{\rm dB-Hz}=30" class="ltx_Math" display="inline" id="S2.SS2.p4.8.m8.1"><semantics id="S2.SS2.p4.8.m8.1a"><mrow id="S2.SS2.p4.8.m8.1.1" xref="S2.SS2.p4.8.m8.1.1.cmml"><msub id="S2.SS2.p4.8.m8.1.1.1" xref="S2.SS2.p4.8.m8.1.1.1.cmml"><mrow id="S2.SS2.p4.8.m8.1.1.1.1.1" xref="S2.SS2.p4.8.m8.1.1.1.1.2.cmml"><mo id="S2.SS2.p4.8.m8.1.1.1.1.1.2" stretchy="false" xref="S2.SS2.p4.8.m8.1.1.1.1.2.1.cmml">[</mo><mrow id="S2.SS2.p4.8.m8.1.1.1.1.1.1" xref="S2.SS2.p4.8.m8.1.1.1.1.1.1.cmml"><mi id="S2.SS2.p4.8.m8.1.1.1.1.1.1.2" xref="S2.SS2.p4.8.m8.1.1.1.1.1.1.2.cmml">C</mi><mo id="S2.SS2.p4.8.m8.1.1.1.1.1.1.1" xref="S2.SS2.p4.8.m8.1.1.1.1.1.1.1.cmml">/</mo><msub id="S2.SS2.p4.8.m8.1.1.1.1.1.1.3" xref="S2.SS2.p4.8.m8.1.1.1.1.1.1.3.cmml"><mi id="S2.SS2.p4.8.m8.1.1.1.1.1.1.3.2" xref="S2.SS2.p4.8.m8.1.1.1.1.1.1.3.2.cmml">N</mi><mn id="S2.SS2.p4.8.m8.1.1.1.1.1.1.3.3" xref="S2.SS2.p4.8.m8.1.1.1.1.1.1.3.3.cmml">0</mn></msub></mrow><mo id="S2.SS2.p4.8.m8.1.1.1.1.1.3" stretchy="false" xref="S2.SS2.p4.8.m8.1.1.1.1.2.1.cmml">]</mo></mrow><mrow id="S2.SS2.p4.8.m8.1.1.1.3" xref="S2.SS2.p4.8.m8.1.1.1.3.cmml"><mi id="S2.SS2.p4.8.m8.1.1.1.3.2" xref="S2.SS2.p4.8.m8.1.1.1.3.2.cmml">dB</mi><mo id="S2.SS2.p4.8.m8.1.1.1.3.1" xref="S2.SS2.p4.8.m8.1.1.1.3.1.cmml">−</mo><mi id="S2.SS2.p4.8.m8.1.1.1.3.3" xref="S2.SS2.p4.8.m8.1.1.1.3.3.cmml">Hz</mi></mrow></msub><mo id="S2.SS2.p4.8.m8.1.1.2" xref="S2.SS2.p4.8.m8.1.1.2.cmml">=</mo><mn id="S2.SS2.p4.8.m8.1.1.3" xref="S2.SS2.p4.8.m8.1.1.3.cmml">30</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.8.m8.1b"><apply id="S2.SS2.p4.8.m8.1.1.cmml" xref="S2.SS2.p4.8.m8.1.1"><eq id="S2.SS2.p4.8.m8.1.1.2.cmml" xref="S2.SS2.p4.8.m8.1.1.2"></eq><apply id="S2.SS2.p4.8.m8.1.1.1.cmml" xref="S2.SS2.p4.8.m8.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.8.m8.1.1.1.2.cmml" xref="S2.SS2.p4.8.m8.1.1.1">subscript</csymbol><apply id="S2.SS2.p4.8.m8.1.1.1.1.2.cmml" xref="S2.SS2.p4.8.m8.1.1.1.1.1"><csymbol cd="latexml" id="S2.SS2.p4.8.m8.1.1.1.1.2.1.cmml" xref="S2.SS2.p4.8.m8.1.1.1.1.1.2">delimited-[]</csymbol><apply id="S2.SS2.p4.8.m8.1.1.1.1.1.1.cmml" xref="S2.SS2.p4.8.m8.1.1.1.1.1.1"><divide id="S2.SS2.p4.8.m8.1.1.1.1.1.1.1.cmml" xref="S2.SS2.p4.8.m8.1.1.1.1.1.1.1"></divide><ci id="S2.SS2.p4.8.m8.1.1.1.1.1.1.2.cmml" xref="S2.SS2.p4.8.m8.1.1.1.1.1.1.2">𝐶</ci><apply id="S2.SS2.p4.8.m8.1.1.1.1.1.1.3.cmml" xref="S2.SS2.p4.8.m8.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p4.8.m8.1.1.1.1.1.1.3.1.cmml" xref="S2.SS2.p4.8.m8.1.1.1.1.1.1.3">subscript</csymbol><ci id="S2.SS2.p4.8.m8.1.1.1.1.1.1.3.2.cmml" xref="S2.SS2.p4.8.m8.1.1.1.1.1.1.3.2">𝑁</ci><cn id="S2.SS2.p4.8.m8.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S2.SS2.p4.8.m8.1.1.1.1.1.1.3.3">0</cn></apply></apply></apply><apply id="S2.SS2.p4.8.m8.1.1.1.3.cmml" xref="S2.SS2.p4.8.m8.1.1.1.3"><minus id="S2.SS2.p4.8.m8.1.1.1.3.1.cmml" xref="S2.SS2.p4.8.m8.1.1.1.3.1"></minus><ci id="S2.SS2.p4.8.m8.1.1.1.3.2.cmml" xref="S2.SS2.p4.8.m8.1.1.1.3.2">dB</ci><ci id="S2.SS2.p4.8.m8.1.1.1.3.3.cmml" xref="S2.SS2.p4.8.m8.1.1.1.3.3">Hz</ci></apply></apply><cn id="S2.SS2.p4.8.m8.1.1.3.cmml" type="integer" xref="S2.SS2.p4.8.m8.1.1.3">30</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.8.m8.1c">[C/N_{0}]_{\rm dB-Hz}=30</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.8.m8.1d">[ italic_C / italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT roman_dB - roman_Hz end_POSTSUBSCRIPT = 30</annotation></semantics></math> when communicating to a <math alttext="D=70\," class="ltx_Math" display="inline" id="S2.SS2.p4.9.m9.1"><semantics id="S2.SS2.p4.9.m9.1a"><mrow id="S2.SS2.p4.9.m9.1.1" xref="S2.SS2.p4.9.m9.1.1.cmml"><mi id="S2.SS2.p4.9.m9.1.1.2" xref="S2.SS2.p4.9.m9.1.1.2.cmml">D</mi><mo id="S2.SS2.p4.9.m9.1.1.1" xref="S2.SS2.p4.9.m9.1.1.1.cmml">=</mo><mn id="S2.SS2.p4.9.m9.1.1.3" xref="S2.SS2.p4.9.m9.1.1.3.cmml">70</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.9.m9.1b"><apply id="S2.SS2.p4.9.m9.1.1.cmml" xref="S2.SS2.p4.9.m9.1.1"><eq id="S2.SS2.p4.9.m9.1.1.1.cmml" xref="S2.SS2.p4.9.m9.1.1.1"></eq><ci id="S2.SS2.p4.9.m9.1.1.2.cmml" xref="S2.SS2.p4.9.m9.1.1.2">𝐷</ci><cn id="S2.SS2.p4.9.m9.1.1.3.cmml" type="integer" xref="S2.SS2.p4.9.m9.1.1.3">70</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.9.m9.1c">D=70\,</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.9.m9.1d">italic_D = 70</annotation></semantics></math>m Deep Space Network antenna <cite class="ltx_cite ltx_citemacro_citep">(Taylor, <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib65" title="">2016</a>)</cite>.</p> </div> <div class="ltx_para" id="S2.SS2.p5"> <p class="ltx_p" id="S2.SS2.p5.6">Despite requiring lower <math alttext="C/N_{0}" class="ltx_Math" display="inline" id="S2.SS2.p5.1.m1.1"><semantics id="S2.SS2.p5.1.m1.1a"><mrow id="S2.SS2.p5.1.m1.1.1" xref="S2.SS2.p5.1.m1.1.1.cmml"><mi id="S2.SS2.p5.1.m1.1.1.2" xref="S2.SS2.p5.1.m1.1.1.2.cmml">C</mi><mo id="S2.SS2.p5.1.m1.1.1.1" xref="S2.SS2.p5.1.m1.1.1.1.cmml">/</mo><msub id="S2.SS2.p5.1.m1.1.1.3" xref="S2.SS2.p5.1.m1.1.1.3.cmml"><mi id="S2.SS2.p5.1.m1.1.1.3.2" xref="S2.SS2.p5.1.m1.1.1.3.2.cmml">N</mi><mn id="S2.SS2.p5.1.m1.1.1.3.3" xref="S2.SS2.p5.1.m1.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.1.m1.1b"><apply id="S2.SS2.p5.1.m1.1.1.cmml" xref="S2.SS2.p5.1.m1.1.1"><divide id="S2.SS2.p5.1.m1.1.1.1.cmml" xref="S2.SS2.p5.1.m1.1.1.1"></divide><ci id="S2.SS2.p5.1.m1.1.1.2.cmml" xref="S2.SS2.p5.1.m1.1.1.2">𝐶</ci><apply id="S2.SS2.p5.1.m1.1.1.3.cmml" xref="S2.SS2.p5.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p5.1.m1.1.1.3.1.cmml" xref="S2.SS2.p5.1.m1.1.1.3">subscript</csymbol><ci id="S2.SS2.p5.1.m1.1.1.3.2.cmml" xref="S2.SS2.p5.1.m1.1.1.3.2">𝑁</ci><cn id="S2.SS2.p5.1.m1.1.1.3.3.cmml" type="integer" xref="S2.SS2.p5.1.m1.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.1.m1.1c">C/N_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.1.m1.1d">italic_C / italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> than previous space missions, there has been substantial success at ranging with such weak electromagnetic signals. Indeed, dB-Hz <math alttext="\approx 35" class="ltx_Math" display="inline" id="S2.SS2.p5.2.m2.1"><semantics id="S2.SS2.p5.2.m2.1a"><mrow id="S2.SS2.p5.2.m2.1.1" xref="S2.SS2.p5.2.m2.1.1.cmml"><mi id="S2.SS2.p5.2.m2.1.1.2" xref="S2.SS2.p5.2.m2.1.1.2.cmml"></mi><mo id="S2.SS2.p5.2.m2.1.1.1" xref="S2.SS2.p5.2.m2.1.1.1.cmml">≈</mo><mn id="S2.SS2.p5.2.m2.1.1.3" xref="S2.SS2.p5.2.m2.1.1.3.cmml">35</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.2.m2.1b"><apply id="S2.SS2.p5.2.m2.1.1.cmml" xref="S2.SS2.p5.2.m2.1.1"><approx id="S2.SS2.p5.2.m2.1.1.1.cmml" xref="S2.SS2.p5.2.m2.1.1.1"></approx><csymbol cd="latexml" id="S2.SS2.p5.2.m2.1.1.2.cmml" xref="S2.SS2.p5.2.m2.1.1.2">absent</csymbol><cn id="S2.SS2.p5.2.m2.1.1.3.cmml" type="integer" xref="S2.SS2.p5.2.m2.1.1.3">35</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.2.m2.1c">\approx 35</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.2.m2.1d">≈ 35</annotation></semantics></math> is known as the <span class="ltx_text ltx_font_italic" id="S2.SS2.p5.6.1">acquisition threshold</span> for a receiver locking onto the Global Positioning System ranging code – roughly the threshold where a delay-lock loop can acquire the frequency and delay of a signal within <math alttext="1~{}" class="ltx_Math" display="inline" id="S2.SS2.p5.3.m3.1"><semantics id="S2.SS2.p5.3.m3.1a"><mn id="S2.SS2.p5.3.m3.1.1" xref="S2.SS2.p5.3.m3.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.3.m3.1b"><cn id="S2.SS2.p5.3.m3.1.1.cmml" type="integer" xref="S2.SS2.p5.3.m3.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.3.m3.1c">1~{}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.3.m3.1d">1</annotation></semantics></math>ms by brute force search over a grid of delays and frequencies motivated by typical terrestrial uncertainties <cite class="ltx_cite ltx_citemacro_citep">(Misra & Enge, <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib50" title="">2012</a>)</cite>. As our concept’s velocities and positions would be extremely well constrained and the search can extend beyond <math alttext="1~{}" class="ltx_Math" display="inline" id="S2.SS2.p5.4.m4.1"><semantics id="S2.SS2.p5.4.m4.1a"><mn id="S2.SS2.p5.4.m4.1.1" xref="S2.SS2.p5.4.m4.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.4.m4.1b"><cn id="S2.SS2.p5.4.m4.1.1.cmml" type="integer" xref="S2.SS2.p5.4.m4.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.4.m4.1c">1~{}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.4.m4.1d">1</annotation></semantics></math>ms, the threshold for acquisition of a ranging code would be much lower. Indeed, the Deep Space Network regularly achieves lock for electromagnetic signals with <math alttext="[C/N_{0}]_{\rm dB-Hz}\approx 15" class="ltx_Math" display="inline" id="S2.SS2.p5.5.m5.1"><semantics id="S2.SS2.p5.5.m5.1a"><mrow id="S2.SS2.p5.5.m5.1.1" xref="S2.SS2.p5.5.m5.1.1.cmml"><msub id="S2.SS2.p5.5.m5.1.1.1" xref="S2.SS2.p5.5.m5.1.1.1.cmml"><mrow id="S2.SS2.p5.5.m5.1.1.1.1.1" xref="S2.SS2.p5.5.m5.1.1.1.1.2.cmml"><mo id="S2.SS2.p5.5.m5.1.1.1.1.1.2" stretchy="false" xref="S2.SS2.p5.5.m5.1.1.1.1.2.1.cmml">[</mo><mrow id="S2.SS2.p5.5.m5.1.1.1.1.1.1" xref="S2.SS2.p5.5.m5.1.1.1.1.1.1.cmml"><mi id="S2.SS2.p5.5.m5.1.1.1.1.1.1.2" xref="S2.SS2.p5.5.m5.1.1.1.1.1.1.2.cmml">C</mi><mo id="S2.SS2.p5.5.m5.1.1.1.1.1.1.1" xref="S2.SS2.p5.5.m5.1.1.1.1.1.1.1.cmml">/</mo><msub id="S2.SS2.p5.5.m5.1.1.1.1.1.1.3" xref="S2.SS2.p5.5.m5.1.1.1.1.1.1.3.cmml"><mi id="S2.SS2.p5.5.m5.1.1.1.1.1.1.3.2" xref="S2.SS2.p5.5.m5.1.1.1.1.1.1.3.2.cmml">N</mi><mn id="S2.SS2.p5.5.m5.1.1.1.1.1.1.3.3" xref="S2.SS2.p5.5.m5.1.1.1.1.1.1.3.3.cmml">0</mn></msub></mrow><mo id="S2.SS2.p5.5.m5.1.1.1.1.1.3" stretchy="false" xref="S2.SS2.p5.5.m5.1.1.1.1.2.1.cmml">]</mo></mrow><mrow id="S2.SS2.p5.5.m5.1.1.1.3" xref="S2.SS2.p5.5.m5.1.1.1.3.cmml"><mi id="S2.SS2.p5.5.m5.1.1.1.3.2" xref="S2.SS2.p5.5.m5.1.1.1.3.2.cmml">dB</mi><mo id="S2.SS2.p5.5.m5.1.1.1.3.1" xref="S2.SS2.p5.5.m5.1.1.1.3.1.cmml">−</mo><mi id="S2.SS2.p5.5.m5.1.1.1.3.3" xref="S2.SS2.p5.5.m5.1.1.1.3.3.cmml">Hz</mi></mrow></msub><mo id="S2.SS2.p5.5.m5.1.1.2" xref="S2.SS2.p5.5.m5.1.1.2.cmml">≈</mo><mn id="S2.SS2.p5.5.m5.1.1.3" xref="S2.SS2.p5.5.m5.1.1.3.cmml">15</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.5.m5.1b"><apply id="S2.SS2.p5.5.m5.1.1.cmml" xref="S2.SS2.p5.5.m5.1.1"><approx id="S2.SS2.p5.5.m5.1.1.2.cmml" xref="S2.SS2.p5.5.m5.1.1.2"></approx><apply id="S2.SS2.p5.5.m5.1.1.1.cmml" xref="S2.SS2.p5.5.m5.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p5.5.m5.1.1.1.2.cmml" xref="S2.SS2.p5.5.m5.1.1.1">subscript</csymbol><apply id="S2.SS2.p5.5.m5.1.1.1.1.2.cmml" xref="S2.SS2.p5.5.m5.1.1.1.1.1"><csymbol cd="latexml" id="S2.SS2.p5.5.m5.1.1.1.1.2.1.cmml" xref="S2.SS2.p5.5.m5.1.1.1.1.1.2">delimited-[]</csymbol><apply id="S2.SS2.p5.5.m5.1.1.1.1.1.1.cmml" xref="S2.SS2.p5.5.m5.1.1.1.1.1.1"><divide id="S2.SS2.p5.5.m5.1.1.1.1.1.1.1.cmml" xref="S2.SS2.p5.5.m5.1.1.1.1.1.1.1"></divide><ci id="S2.SS2.p5.5.m5.1.1.1.1.1.1.2.cmml" xref="S2.SS2.p5.5.m5.1.1.1.1.1.1.2">𝐶</ci><apply id="S2.SS2.p5.5.m5.1.1.1.1.1.1.3.cmml" xref="S2.SS2.p5.5.m5.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p5.5.m5.1.1.1.1.1.1.3.1.cmml" xref="S2.SS2.p5.5.m5.1.1.1.1.1.1.3">subscript</csymbol><ci id="S2.SS2.p5.5.m5.1.1.1.1.1.1.3.2.cmml" xref="S2.SS2.p5.5.m5.1.1.1.1.1.1.3.2">𝑁</ci><cn id="S2.SS2.p5.5.m5.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S2.SS2.p5.5.m5.1.1.1.1.1.1.3.3">0</cn></apply></apply></apply><apply id="S2.SS2.p5.5.m5.1.1.1.3.cmml" xref="S2.SS2.p5.5.m5.1.1.1.3"><minus id="S2.SS2.p5.5.m5.1.1.1.3.1.cmml" xref="S2.SS2.p5.5.m5.1.1.1.3.1"></minus><ci id="S2.SS2.p5.5.m5.1.1.1.3.2.cmml" xref="S2.SS2.p5.5.m5.1.1.1.3.2">dB</ci><ci id="S2.SS2.p5.5.m5.1.1.1.3.3.cmml" xref="S2.SS2.p5.5.m5.1.1.1.3.3">Hz</ci></apply></apply><cn id="S2.SS2.p5.5.m5.1.1.3.cmml" type="integer" xref="S2.SS2.p5.5.m5.1.1.3">15</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.5.m5.1c">[C/N_{0}]_{\rm dB-Hz}\approx 15</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.5.m5.1d">[ italic_C / italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT roman_dB - roman_Hz end_POSTSUBSCRIPT ≈ 15</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_citep">(Folkner, <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib30" title="">2011</a>)</cite>. Like for LISA, our concept would involve modulating the carrier with a pseudo-random code, transmitting it, and first acquiring a delay lock to that code after reception. Once a delay lock is established, the pseudo-random code decorrelates the signal from contaminating signals, such that we anticipate phase measurements that are limited by either shot or thermal noise will be possible despite the small <math alttext="C/N_{0}" class="ltx_Math" display="inline" id="S2.SS2.p5.6.m6.1"><semantics id="S2.SS2.p5.6.m6.1a"><mrow id="S2.SS2.p5.6.m6.1.1" xref="S2.SS2.p5.6.m6.1.1.cmml"><mi id="S2.SS2.p5.6.m6.1.1.2" xref="S2.SS2.p5.6.m6.1.1.2.cmml">C</mi><mo id="S2.SS2.p5.6.m6.1.1.1" xref="S2.SS2.p5.6.m6.1.1.1.cmml">/</mo><msub id="S2.SS2.p5.6.m6.1.1.3" xref="S2.SS2.p5.6.m6.1.1.3.cmml"><mi id="S2.SS2.p5.6.m6.1.1.3.2" xref="S2.SS2.p5.6.m6.1.1.3.2.cmml">N</mi><mn id="S2.SS2.p5.6.m6.1.1.3.3" xref="S2.SS2.p5.6.m6.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.6.m6.1b"><apply id="S2.SS2.p5.6.m6.1.1.cmml" xref="S2.SS2.p5.6.m6.1.1"><divide id="S2.SS2.p5.6.m6.1.1.1.cmml" xref="S2.SS2.p5.6.m6.1.1.1"></divide><ci id="S2.SS2.p5.6.m6.1.1.2.cmml" xref="S2.SS2.p5.6.m6.1.1.2">𝐶</ci><apply id="S2.SS2.p5.6.m6.1.1.3.cmml" xref="S2.SS2.p5.6.m6.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p5.6.m6.1.1.3.1.cmml" xref="S2.SS2.p5.6.m6.1.1.3">subscript</csymbol><ci id="S2.SS2.p5.6.m6.1.1.3.2.cmml" xref="S2.SS2.p5.6.m6.1.1.3.2">𝑁</ci><cn id="S2.SS2.p5.6.m6.1.1.3.3.cmml" type="integer" xref="S2.SS2.p5.6.m6.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.6.m6.1c">C/N_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.6.m6.1d">italic_C / italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS2.p6"> <p class="ltx_p" id="S2.SS2.p6.1">Translating <math alttext="C/N_{0}" class="ltx_Math" display="inline" id="S2.SS2.p6.1.m1.1"><semantics id="S2.SS2.p6.1.m1.1a"><mrow id="S2.SS2.p6.1.m1.1.1" xref="S2.SS2.p6.1.m1.1.1.cmml"><mi id="S2.SS2.p6.1.m1.1.1.2" xref="S2.SS2.p6.1.m1.1.1.2.cmml">C</mi><mo id="S2.SS2.p6.1.m1.1.1.1" xref="S2.SS2.p6.1.m1.1.1.1.cmml">/</mo><msub id="S2.SS2.p6.1.m1.1.1.3" xref="S2.SS2.p6.1.m1.1.1.3.cmml"><mi id="S2.SS2.p6.1.m1.1.1.3.2" xref="S2.SS2.p6.1.m1.1.1.3.2.cmml">N</mi><mn id="S2.SS2.p6.1.m1.1.1.3.3" xref="S2.SS2.p6.1.m1.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.1.m1.1b"><apply id="S2.SS2.p6.1.m1.1.1.cmml" xref="S2.SS2.p6.1.m1.1.1"><divide id="S2.SS2.p6.1.m1.1.1.1.cmml" xref="S2.SS2.p6.1.m1.1.1.1"></divide><ci id="S2.SS2.p6.1.m1.1.1.2.cmml" xref="S2.SS2.p6.1.m1.1.1.2">𝐶</ci><apply id="S2.SS2.p6.1.m1.1.1.3.cmml" xref="S2.SS2.p6.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p6.1.m1.1.1.3.1.cmml" xref="S2.SS2.p6.1.m1.1.1.3">subscript</csymbol><ci id="S2.SS2.p6.1.m1.1.1.3.2.cmml" xref="S2.SS2.p6.1.m1.1.1.3.2">𝑁</ci><cn id="S2.SS2.p6.1.m1.1.1.3.3.cmml" type="integer" xref="S2.SS2.p6.1.m1.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.1.m1.1c">C/N_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.1.m1.1d">italic_C / italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> to a strain noises via equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.E7" title="In 2.2 Radiometer noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">7</span></a>) yields</p> <table class="ltx_equation ltx_eqn_table" id="S2.E10"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\widetilde{\phi}_{\rm rms}=\sqrt{S_{n}(k)}=0.01\times 10^{-\left([C/N_{0}]_{% \rm dB-Hz}-40\right)/20}~{}~{}\text{rad Hz${}^{-1/2}$}," class="ltx_Math" display="block" id="S2.E10.m1.4"><semantics 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id="S2.E10.m1.1.1.1e.cmml" xref="S2.E10.m1.1.1.1"><mrow id="S2.E10.m1.1.1.1.cmml" xref="S2.E10.m1.1.1.1"><mtext id="S2.E10.m1.1.1.1a.cmml" xref="S2.E10.m1.1.1.1">rad Hz</mtext><mtext id="S2.E10.m1.1.1.1b.cmml" xref="S2.E10.m1.1.1.1"><sup class="ltx_sup" id="S2.E10.m1.1.1.1.2anest"><span class="ltx_text ltx_font_italic" id="S2.E10.m1.1.1.1.2.1anest">-1/2</span></sup></mtext></mrow></ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E10.m1.4c">\widetilde{\phi}_{\rm rms}=\sqrt{S_{n}(k)}=0.01\times 10^{-\left([C/N_{0}]_{% \rm dB-Hz}-40\right)/20}~{}~{}\text{rad Hz${}^{-1/2}$},</annotation><annotation encoding="application/x-llamapun" id="S2.E10.m1.4d">over~ start_ARG italic_ϕ end_ARG start_POSTSUBSCRIPT roman_rms end_POSTSUBSCRIPT = square-root start_ARG italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_k ) end_ARG = 0.01 × 10 start_POSTSUPERSCRIPT - ( [ italic_C / italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT roman_dB - roman_Hz end_POSTSUBSCRIPT - 40 ) / 20 end_POSTSUPERSCRIPT rad Hz ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(10)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p6.4">which must satisfy the requirement <math alttext="T_{\phi}^{-1/2}\widetilde{\phi}_{\rm rms}\lesssim 0.1" class="ltx_Math" display="inline" id="S2.SS2.p6.2.m1.1"><semantics id="S2.SS2.p6.2.m1.1a"><mrow id="S2.SS2.p6.2.m1.1.1" xref="S2.SS2.p6.2.m1.1.1.cmml"><mrow id="S2.SS2.p6.2.m1.1.1.2" xref="S2.SS2.p6.2.m1.1.1.2.cmml"><msubsup id="S2.SS2.p6.2.m1.1.1.2.2" xref="S2.SS2.p6.2.m1.1.1.2.2.cmml"><mi id="S2.SS2.p6.2.m1.1.1.2.2.2.2" xref="S2.SS2.p6.2.m1.1.1.2.2.2.2.cmml">T</mi><mi id="S2.SS2.p6.2.m1.1.1.2.2.2.3" xref="S2.SS2.p6.2.m1.1.1.2.2.2.3.cmml">ϕ</mi><mrow id="S2.SS2.p6.2.m1.1.1.2.2.3" 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xref="S2.SS2.p6.2.m1.1.1.2.2.2.3">italic-ϕ</ci></apply><apply id="S2.SS2.p6.2.m1.1.1.2.2.3.cmml" xref="S2.SS2.p6.2.m1.1.1.2.2.3"><minus id="S2.SS2.p6.2.m1.1.1.2.2.3.1.cmml" xref="S2.SS2.p6.2.m1.1.1.2.2.3"></minus><apply id="S2.SS2.p6.2.m1.1.1.2.2.3.2.cmml" xref="S2.SS2.p6.2.m1.1.1.2.2.3.2"><divide id="S2.SS2.p6.2.m1.1.1.2.2.3.2.1.cmml" xref="S2.SS2.p6.2.m1.1.1.2.2.3.2.1"></divide><cn id="S2.SS2.p6.2.m1.1.1.2.2.3.2.2.cmml" type="integer" xref="S2.SS2.p6.2.m1.1.1.2.2.3.2.2">1</cn><cn id="S2.SS2.p6.2.m1.1.1.2.2.3.2.3.cmml" type="integer" xref="S2.SS2.p6.2.m1.1.1.2.2.3.2.3">2</cn></apply></apply></apply><apply id="S2.SS2.p6.2.m1.1.1.2.3.cmml" xref="S2.SS2.p6.2.m1.1.1.2.3"><csymbol cd="ambiguous" id="S2.SS2.p6.2.m1.1.1.2.3.1.cmml" xref="S2.SS2.p6.2.m1.1.1.2.3">subscript</csymbol><apply id="S2.SS2.p6.2.m1.1.1.2.3.2.cmml" xref="S2.SS2.p6.2.m1.1.1.2.3.2"><ci id="S2.SS2.p6.2.m1.1.1.2.3.2.1.cmml" xref="S2.SS2.p6.2.m1.1.1.2.3.2.1">~</ci><ci id="S2.SS2.p6.2.m1.1.1.2.3.2.2.cmml" xref="S2.SS2.p6.2.m1.1.1.2.3.2.2">italic-ϕ</ci></apply><ci id="S2.SS2.p6.2.m1.1.1.2.3.3.cmml" xref="S2.SS2.p6.2.m1.1.1.2.3.3">rms</ci></apply></apply><cn id="S2.SS2.p6.2.m1.1.1.3.cmml" type="float" xref="S2.SS2.p6.2.m1.1.1.3">0.1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.2.m1.1c">T_{\phi}^{-1/2}\widetilde{\phi}_{\rm rms}\lesssim 0.1</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.2.m1.1d">italic_T start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT over~ start_ARG italic_ϕ end_ARG start_POSTSUBSCRIPT roman_rms end_POSTSUBSCRIPT ≲ 0.1</annotation></semantics></math> needed for a stable phase lock for typical parameters considered in this paper, where <math alttext="T_{\phi}" class="ltx_Math" display="inline" id="S2.SS2.p6.3.m2.1"><semantics id="S2.SS2.p6.3.m2.1a"><msub id="S2.SS2.p6.3.m2.1.1" xref="S2.SS2.p6.3.m2.1.1.cmml"><mi id="S2.SS2.p6.3.m2.1.1.2" xref="S2.SS2.p6.3.m2.1.1.2.cmml">T</mi><mi id="S2.SS2.p6.3.m2.1.1.3" xref="S2.SS2.p6.3.m2.1.1.3.cmml">ϕ</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.3.m2.1b"><apply id="S2.SS2.p6.3.m2.1.1.cmml" xref="S2.SS2.p6.3.m2.1.1"><csymbol cd="ambiguous" id="S2.SS2.p6.3.m2.1.1.1.cmml" xref="S2.SS2.p6.3.m2.1.1">subscript</csymbol><ci id="S2.SS2.p6.3.m2.1.1.2.cmml" xref="S2.SS2.p6.3.m2.1.1.2">𝑇</ci><ci id="S2.SS2.p6.3.m2.1.1.3.cmml" xref="S2.SS2.p6.3.m2.1.1.3">italic-ϕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.3.m2.1c">T_{\phi}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.3.m2.1d">italic_T start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT</annotation></semantics></math> is the time over which the phase changes by an order one value because of displacements from accelerations or timing errors from clock noise <cite class="ltx_cite ltx_citemacro_citep">(Misra & Enge, <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib50" title="">2012</a>, and <math alttext="T_{\phi}" class="ltx_Math" display="inline" id="S2.SS2.p6.4.1.m1.1"><semantics id="S2.SS2.p6.4.1.m1.1a"><msub id="S2.SS2.p6.4.1.m1.1.1" xref="S2.SS2.p6.4.1.m1.1.1.cmml"><mi id="S2.SS2.p6.4.1.m1.1.1.2" xref="S2.SS2.p6.4.1.m1.1.1.2.cmml">T</mi><mi id="S2.SS2.p6.4.1.m1.1.1.3" xref="S2.SS2.p6.4.1.m1.1.1.3.cmml">ϕ</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.4.1.m1.1b"><apply id="S2.SS2.p6.4.1.m1.1.1.cmml" xref="S2.SS2.p6.4.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p6.4.1.m1.1.1.1.cmml" xref="S2.SS2.p6.4.1.m1.1.1">subscript</csymbol><ci id="S2.SS2.p6.4.1.m1.1.1.2.cmml" xref="S2.SS2.p6.4.1.m1.1.1.2">𝑇</ci><ci id="S2.SS2.p6.4.1.m1.1.1.3.cmml" xref="S2.SS2.p6.4.1.m1.1.1.3">italic-ϕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.4.1.m1.1c">T_{\phi}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.4.1.m1.1d">italic_T start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT</annotation></semantics></math> sets the maximum time constant for the phase-lock loop)</cite>. We find that this condition is satisfied for parameters considered in this paper.<span class="ltx_note ltx_role_footnote" id="footnote6"><sup class="ltx_note_mark">6</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">6</sup><span class="ltx_tag ltx_tag_note">6</span>In the laser setup, the required <math alttext="T_{\phi}" class="ltx_Math" display="inline" id="footnote6.m1.1"><semantics id="footnote6.m1.1b"><msub id="footnote6.m1.1.1" xref="footnote6.m1.1.1.cmml"><mi id="footnote6.m1.1.1.2" xref="footnote6.m1.1.1.2.cmml">T</mi><mi id="footnote6.m1.1.1.3" xref="footnote6.m1.1.1.3.cmml">ϕ</mi></msub><annotation-xml encoding="MathML-Content" id="footnote6.m1.1c"><apply id="footnote6.m1.1.1.cmml" xref="footnote6.m1.1.1"><csymbol cd="ambiguous" id="footnote6.m1.1.1.1.cmml" xref="footnote6.m1.1.1">subscript</csymbol><ci id="footnote6.m1.1.1.2.cmml" xref="footnote6.m1.1.1.2">𝑇</ci><ci id="footnote6.m1.1.1.3.cmml" xref="footnote6.m1.1.1.3">italic-ϕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote6.m1.1d">T_{\phi}</annotation><annotation encoding="application/x-llamapun" id="footnote6.m1.1e">italic_T start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT</annotation></semantics></math> is much smaller than the radio setup. We find that <math alttext="T_{\phi}\sim 10\,(r/30{\rm AU})~{}" class="ltx_Math" display="inline" id="footnote6.m2.1"><semantics id="footnote6.m2.1b"><mrow id="footnote6.m2.1.1" xref="footnote6.m2.1.1.cmml"><msub id="footnote6.m2.1.1.3" xref="footnote6.m2.1.1.3.cmml"><mi id="footnote6.m2.1.1.3.2" xref="footnote6.m2.1.1.3.2.cmml">T</mi><mi id="footnote6.m2.1.1.3.3" xref="footnote6.m2.1.1.3.3.cmml">ϕ</mi></msub><mo id="footnote6.m2.1.1.2" xref="footnote6.m2.1.1.2.cmml">∼</mo><mrow id="footnote6.m2.1.1.1" xref="footnote6.m2.1.1.1.cmml"><mn id="footnote6.m2.1.1.1.3" xref="footnote6.m2.1.1.1.3.cmml">10</mn><mo id="footnote6.m2.1.1.1.2" lspace="0.170em" xref="footnote6.m2.1.1.1.2.cmml">⁢</mo><mrow id="footnote6.m2.1.1.1.1.1" xref="footnote6.m2.1.1.1.1.1.1.cmml"><mo id="footnote6.m2.1.1.1.1.1.2" stretchy="false" xref="footnote6.m2.1.1.1.1.1.1.cmml">(</mo><mrow id="footnote6.m2.1.1.1.1.1.1" xref="footnote6.m2.1.1.1.1.1.1.cmml"><mrow id="footnote6.m2.1.1.1.1.1.1.2" xref="footnote6.m2.1.1.1.1.1.1.2.cmml"><mi id="footnote6.m2.1.1.1.1.1.1.2.2" xref="footnote6.m2.1.1.1.1.1.1.2.2.cmml">r</mi><mo id="footnote6.m2.1.1.1.1.1.1.2.1" xref="footnote6.m2.1.1.1.1.1.1.2.1.cmml">/</mo><mn id="footnote6.m2.1.1.1.1.1.1.2.3" xref="footnote6.m2.1.1.1.1.1.1.2.3.cmml">30</mn></mrow><mo id="footnote6.m2.1.1.1.1.1.1.1" xref="footnote6.m2.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="footnote6.m2.1.1.1.1.1.1.3" mathvariant="normal" xref="footnote6.m2.1.1.1.1.1.1.3.cmml">A</mi><mo id="footnote6.m2.1.1.1.1.1.1.1b" xref="footnote6.m2.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="footnote6.m2.1.1.1.1.1.1.4" mathvariant="normal" xref="footnote6.m2.1.1.1.1.1.1.4.cmml">U</mi></mrow><mo id="footnote6.m2.1.1.1.1.1.3" stretchy="false" xref="footnote6.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote6.m2.1c"><apply id="footnote6.m2.1.1.cmml" xref="footnote6.m2.1.1"><csymbol cd="latexml" id="footnote6.m2.1.1.2.cmml" xref="footnote6.m2.1.1.2">similar-to</csymbol><apply id="footnote6.m2.1.1.3.cmml" xref="footnote6.m2.1.1.3"><csymbol cd="ambiguous" id="footnote6.m2.1.1.3.1.cmml" xref="footnote6.m2.1.1.3">subscript</csymbol><ci id="footnote6.m2.1.1.3.2.cmml" xref="footnote6.m2.1.1.3.2">𝑇</ci><ci id="footnote6.m2.1.1.3.3.cmml" xref="footnote6.m2.1.1.3.3">italic-ϕ</ci></apply><apply id="footnote6.m2.1.1.1.cmml" xref="footnote6.m2.1.1.1"><times id="footnote6.m2.1.1.1.2.cmml" xref="footnote6.m2.1.1.1.2"></times><cn id="footnote6.m2.1.1.1.3.cmml" type="integer" xref="footnote6.m2.1.1.1.3">10</cn><apply id="footnote6.m2.1.1.1.1.1.1.cmml" xref="footnote6.m2.1.1.1.1.1"><times id="footnote6.m2.1.1.1.1.1.1.1.cmml" xref="footnote6.m2.1.1.1.1.1.1.1"></times><apply id="footnote6.m2.1.1.1.1.1.1.2.cmml" xref="footnote6.m2.1.1.1.1.1.1.2"><divide id="footnote6.m2.1.1.1.1.1.1.2.1.cmml" xref="footnote6.m2.1.1.1.1.1.1.2.1"></divide><ci id="footnote6.m2.1.1.1.1.1.1.2.2.cmml" xref="footnote6.m2.1.1.1.1.1.1.2.2">𝑟</ci><cn id="footnote6.m2.1.1.1.1.1.1.2.3.cmml" type="integer" xref="footnote6.m2.1.1.1.1.1.1.2.3">30</cn></apply><ci id="footnote6.m2.1.1.1.1.1.1.3.cmml" xref="footnote6.m2.1.1.1.1.1.1.3">A</ci><ci id="footnote6.m2.1.1.1.1.1.1.4.cmml" xref="footnote6.m2.1.1.1.1.1.1.4">U</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote6.m2.1d">T_{\phi}\sim 10\,(r/30{\rm AU})~{}</annotation><annotation encoding="application/x-llamapun" id="footnote6.m2.1e">italic_T start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT ∼ 10 ( italic_r / 30 roman_A roman_U )</annotation></semantics></math>s, assuming an effective area-to-mass ratio of the spacecraft of <math alttext="10^{-2}" class="ltx_Math" display="inline" id="footnote6.m3.1"><semantics id="footnote6.m3.1b"><msup id="footnote6.m3.1.1" xref="footnote6.m3.1.1.cmml"><mn id="footnote6.m3.1.1.2" xref="footnote6.m3.1.1.2.cmml">10</mn><mrow id="footnote6.m3.1.1.3" xref="footnote6.m3.1.1.3.cmml"><mo id="footnote6.m3.1.1.3b" xref="footnote6.m3.1.1.3.cmml">−</mo><mn id="footnote6.m3.1.1.3.2" xref="footnote6.m3.1.1.3.2.cmml">2</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="footnote6.m3.1c"><apply id="footnote6.m3.1.1.cmml" xref="footnote6.m3.1.1"><csymbol cd="ambiguous" id="footnote6.m3.1.1.1.cmml" xref="footnote6.m3.1.1">superscript</csymbol><cn id="footnote6.m3.1.1.2.cmml" type="integer" xref="footnote6.m3.1.1.2">10</cn><apply id="footnote6.m3.1.1.3.cmml" xref="footnote6.m3.1.1.3"><minus id="footnote6.m3.1.1.3.1.cmml" xref="footnote6.m3.1.1.3"></minus><cn id="footnote6.m3.1.1.3.2.cmml" type="integer" xref="footnote6.m3.1.1.3.2">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote6.m3.1d">10^{-2}</annotation><annotation encoding="application/x-llamapun" id="footnote6.m3.1e">10 start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT</annotation></semantics></math>m<sup class="ltx_sup" id="footnote6.1">2</sup>/kg and considering the mean solar irradiance. While the <math alttext="1000\times" class="ltx_math_unparsed" display="inline" id="footnote6.m5.1"><semantics id="footnote6.m5.1b"><mrow id="footnote6.m5.1c"><mn id="footnote6.m5.1.1">1000</mn><mo id="footnote6.m5.1.2" lspace="0.222em">×</mo></mrow><annotation encoding="application/x-tex" id="footnote6.m5.1d">1000\times</annotation><annotation encoding="application/x-llamapun" id="footnote6.m5.1e">1000 ×</annotation></semantics></math> smaller inhomogeneous irradiance variations is more relevant since the mean acceleration can be corrected, since <math alttext="T_{\phi}^{-1/2}\widetilde{\phi}_{\rm rms}\lesssim 0.1" class="ltx_Math" display="inline" id="footnote6.m6.1"><semantics id="footnote6.m6.1b"><mrow id="footnote6.m6.1.1" xref="footnote6.m6.1.1.cmml"><mrow id="footnote6.m6.1.1.2" xref="footnote6.m6.1.1.2.cmml"><msubsup id="footnote6.m6.1.1.2.2" xref="footnote6.m6.1.1.2.2.cmml"><mi id="footnote6.m6.1.1.2.2.2.2" xref="footnote6.m6.1.1.2.2.2.2.cmml">T</mi><mi id="footnote6.m6.1.1.2.2.2.3" xref="footnote6.m6.1.1.2.2.2.3.cmml">ϕ</mi><mrow id="footnote6.m6.1.1.2.2.3" xref="footnote6.m6.1.1.2.2.3.cmml"><mo id="footnote6.m6.1.1.2.2.3b" xref="footnote6.m6.1.1.2.2.3.cmml">−</mo><mrow id="footnote6.m6.1.1.2.2.3.2" xref="footnote6.m6.1.1.2.2.3.2.cmml"><mn id="footnote6.m6.1.1.2.2.3.2.2" xref="footnote6.m6.1.1.2.2.3.2.2.cmml">1</mn><mo id="footnote6.m6.1.1.2.2.3.2.1" xref="footnote6.m6.1.1.2.2.3.2.1.cmml">/</mo><mn id="footnote6.m6.1.1.2.2.3.2.3" xref="footnote6.m6.1.1.2.2.3.2.3.cmml">2</mn></mrow></mrow></msubsup><mo id="footnote6.m6.1.1.2.1" xref="footnote6.m6.1.1.2.1.cmml">⁢</mo><msub id="footnote6.m6.1.1.2.3" xref="footnote6.m6.1.1.2.3.cmml"><mover accent="true" id="footnote6.m6.1.1.2.3.2" xref="footnote6.m6.1.1.2.3.2.cmml"><mi id="footnote6.m6.1.1.2.3.2.2" xref="footnote6.m6.1.1.2.3.2.2.cmml">ϕ</mi><mo id="footnote6.m6.1.1.2.3.2.1" xref="footnote6.m6.1.1.2.3.2.1.cmml">~</mo></mover><mi id="footnote6.m6.1.1.2.3.3" xref="footnote6.m6.1.1.2.3.3.cmml">rms</mi></msub></mrow><mo id="footnote6.m6.1.1.1" xref="footnote6.m6.1.1.1.cmml">≲</mo><mn id="footnote6.m6.1.1.3" xref="footnote6.m6.1.1.3.cmml">0.1</mn></mrow><annotation-xml encoding="MathML-Content" id="footnote6.m6.1c"><apply id="footnote6.m6.1.1.cmml" xref="footnote6.m6.1.1"><csymbol cd="latexml" id="footnote6.m6.1.1.1.cmml" xref="footnote6.m6.1.1.1">less-than-or-similar-to</csymbol><apply id="footnote6.m6.1.1.2.cmml" xref="footnote6.m6.1.1.2"><times id="footnote6.m6.1.1.2.1.cmml" xref="footnote6.m6.1.1.2.1"></times><apply id="footnote6.m6.1.1.2.2.cmml" xref="footnote6.m6.1.1.2.2"><csymbol cd="ambiguous" id="footnote6.m6.1.1.2.2.1.cmml" xref="footnote6.m6.1.1.2.2">superscript</csymbol><apply id="footnote6.m6.1.1.2.2.2.cmml" xref="footnote6.m6.1.1.2.2"><csymbol cd="ambiguous" id="footnote6.m6.1.1.2.2.2.1.cmml" xref="footnote6.m6.1.1.2.2">subscript</csymbol><ci id="footnote6.m6.1.1.2.2.2.2.cmml" xref="footnote6.m6.1.1.2.2.2.2">𝑇</ci><ci id="footnote6.m6.1.1.2.2.2.3.cmml" xref="footnote6.m6.1.1.2.2.2.3">italic-ϕ</ci></apply><apply id="footnote6.m6.1.1.2.2.3.cmml" xref="footnote6.m6.1.1.2.2.3"><minus id="footnote6.m6.1.1.2.2.3.1.cmml" xref="footnote6.m6.1.1.2.2.3"></minus><apply id="footnote6.m6.1.1.2.2.3.2.cmml" xref="footnote6.m6.1.1.2.2.3.2"><divide id="footnote6.m6.1.1.2.2.3.2.1.cmml" xref="footnote6.m6.1.1.2.2.3.2.1"></divide><cn id="footnote6.m6.1.1.2.2.3.2.2.cmml" type="integer" xref="footnote6.m6.1.1.2.2.3.2.2">1</cn><cn id="footnote6.m6.1.1.2.2.3.2.3.cmml" type="integer" xref="footnote6.m6.1.1.2.2.3.2.3">2</cn></apply></apply></apply><apply id="footnote6.m6.1.1.2.3.cmml" xref="footnote6.m6.1.1.2.3"><csymbol cd="ambiguous" id="footnote6.m6.1.1.2.3.1.cmml" xref="footnote6.m6.1.1.2.3">subscript</csymbol><apply id="footnote6.m6.1.1.2.3.2.cmml" xref="footnote6.m6.1.1.2.3.2"><ci id="footnote6.m6.1.1.2.3.2.1.cmml" xref="footnote6.m6.1.1.2.3.2.1">~</ci><ci id="footnote6.m6.1.1.2.3.2.2.cmml" xref="footnote6.m6.1.1.2.3.2.2">italic-ϕ</ci></apply><ci id="footnote6.m6.1.1.2.3.3.cmml" xref="footnote6.m6.1.1.2.3.3">rms</ci></apply></apply><cn id="footnote6.m6.1.1.3.cmml" type="float" xref="footnote6.m6.1.1.3">0.1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote6.m6.1d">T_{\phi}^{-1/2}\widetilde{\phi}_{\rm rms}\lesssim 0.1</annotation><annotation encoding="application/x-llamapun" id="footnote6.m6.1e">italic_T start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT over~ start_ARG italic_ϕ end_ARG start_POSTSUBSCRIPT roman_rms end_POSTSUBSCRIPT ≲ 0.1</annotation></semantics></math> is still satisfied for our fiducial <math alttext="C/N_{0}" class="ltx_Math" display="inline" id="footnote6.m7.1"><semantics id="footnote6.m7.1b"><mrow id="footnote6.m7.1.1" xref="footnote6.m7.1.1.cmml"><mi id="footnote6.m7.1.1.2" xref="footnote6.m7.1.1.2.cmml">C</mi><mo id="footnote6.m7.1.1.1" xref="footnote6.m7.1.1.1.cmml">/</mo><msub id="footnote6.m7.1.1.3" xref="footnote6.m7.1.1.3.cmml"><mi id="footnote6.m7.1.1.3.2" xref="footnote6.m7.1.1.3.2.cmml">N</mi><mn id="footnote6.m7.1.1.3.3" xref="footnote6.m7.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="footnote6.m7.1c"><apply id="footnote6.m7.1.1.cmml" xref="footnote6.m7.1.1"><divide id="footnote6.m7.1.1.1.cmml" xref="footnote6.m7.1.1.1"></divide><ci id="footnote6.m7.1.1.2.cmml" xref="footnote6.m7.1.1.2">𝐶</ci><apply id="footnote6.m7.1.1.3.cmml" xref="footnote6.m7.1.1.3"><csymbol cd="ambiguous" id="footnote6.m7.1.1.3.1.cmml" xref="footnote6.m7.1.1.3">subscript</csymbol><ci id="footnote6.m7.1.1.3.2.cmml" xref="footnote6.m7.1.1.3.2">𝑁</ci><cn id="footnote6.m7.1.1.3.3.cmml" type="integer" xref="footnote6.m7.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote6.m7.1d">C/N_{0}</annotation><annotation encoding="application/x-llamapun" id="footnote6.m7.1e">italic_C / italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> when considering the mean irradiance acceleration, this shows that phase locks might even be achieved in the extreme case where the mean acceleration is not corrected.</span></span></span> Translating this one step further into a displacement noise yields</p> <table class="ltx_equation ltx_eqn_table" id="S2.E11"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\widetilde{\Delta x}_{\rm rms}=\frac{\lambda\widetilde{\phi}_{\rm rms}}{2\pi}=% (\overbrace{1.6\times 10^{-3}}^{\lambda=1\,\text{cm}},~{}\overbrace{1.6\times 1% 0^{-7}}^{\lambda=1\,\text{$\mu$m}})~{}~{}~{}\text{cm~{}Hz${}^{-1/2}$~{}}\times 1% 0^{-\left([C/N_{0}]_{\rm dB-Hz}-40\right)/20}," class="ltx_Math" display="block" id="S2.E11.m1.4"><semantics id="S2.E11.m1.4a"><mrow id="S2.E11.m1.4.4.1" 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xref="S2.E11.m1.3.3.1.1.1.1.1.1.3.3">Hz</ci></apply></apply><cn id="S2.E11.m1.3.3.1.1.1.1.1.3.cmml" type="integer" xref="S2.E11.m1.3.3.1.1.1.1.1.3">40</cn></apply><cn id="S2.E11.m1.3.3.1.1.3.cmml" type="integer" xref="S2.E11.m1.3.3.1.1.3">20</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E11.m1.4c">\widetilde{\Delta x}_{\rm rms}=\frac{\lambda\widetilde{\phi}_{\rm rms}}{2\pi}=% (\overbrace{1.6\times 10^{-3}}^{\lambda=1\,\text{cm}},~{}\overbrace{1.6\times 1% 0^{-7}}^{\lambda=1\,\text{$\mu$m}})~{}~{}~{}\text{cm~{}Hz${}^{-1/2}$~{}}\times 1% 0^{-\left([C/N_{0}]_{\rm dB-Hz}-40\right)/20},</annotation><annotation encoding="application/x-llamapun" id="S2.E11.m1.4d">over~ start_ARG roman_Δ italic_x end_ARG start_POSTSUBSCRIPT roman_rms end_POSTSUBSCRIPT = divide start_ARG italic_λ over~ start_ARG italic_ϕ end_ARG start_POSTSUBSCRIPT roman_rms end_POSTSUBSCRIPT end_ARG start_ARG 2 italic_π end_ARG = ( over⏞ start_ARG 1.6 × 10 start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT end_ARG start_POSTSUPERSCRIPT italic_λ = 1 cm end_POSTSUPERSCRIPT , over⏞ start_ARG 1.6 × 10 start_POSTSUPERSCRIPT - 7 end_POSTSUPERSCRIPT end_ARG start_POSTSUPERSCRIPT italic_λ = 1 italic_μ m end_POSTSUPERSCRIPT ) cm Hz × 10 start_POSTSUPERSCRIPT - ( [ italic_C / italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT roman_dB - roman_Hz end_POSTSUBSCRIPT - 40 ) / 20 end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(11)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p6.10">where the two displacement noise values correspond to <math alttext="\lambda=1" class="ltx_Math" display="inline" id="S2.SS2.p6.5.m1.1"><semantics id="S2.SS2.p6.5.m1.1a"><mrow id="S2.SS2.p6.5.m1.1.1" xref="S2.SS2.p6.5.m1.1.1.cmml"><mi id="S2.SS2.p6.5.m1.1.1.2" xref="S2.SS2.p6.5.m1.1.1.2.cmml">λ</mi><mo id="S2.SS2.p6.5.m1.1.1.1" xref="S2.SS2.p6.5.m1.1.1.1.cmml">=</mo><mn id="S2.SS2.p6.5.m1.1.1.3" xref="S2.SS2.p6.5.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.5.m1.1b"><apply id="S2.SS2.p6.5.m1.1.1.cmml" xref="S2.SS2.p6.5.m1.1.1"><eq id="S2.SS2.p6.5.m1.1.1.1.cmml" xref="S2.SS2.p6.5.m1.1.1.1"></eq><ci id="S2.SS2.p6.5.m1.1.1.2.cmml" xref="S2.SS2.p6.5.m1.1.1.2">𝜆</ci><cn id="S2.SS2.p6.5.m1.1.1.3.cmml" type="integer" xref="S2.SS2.p6.5.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.5.m1.1c">\lambda=1</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.5.m1.1d">italic_λ = 1</annotation></semantics></math>cm – the Ka band radio regularly used for ranging – <math alttext="\lambda=1\mu" class="ltx_Math" display="inline" id="S2.SS2.p6.6.m2.1"><semantics id="S2.SS2.p6.6.m2.1a"><mrow id="S2.SS2.p6.6.m2.1.1" xref="S2.SS2.p6.6.m2.1.1.cmml"><mi id="S2.SS2.p6.6.m2.1.1.2" xref="S2.SS2.p6.6.m2.1.1.2.cmml">λ</mi><mo id="S2.SS2.p6.6.m2.1.1.1" xref="S2.SS2.p6.6.m2.1.1.1.cmml">=</mo><mrow id="S2.SS2.p6.6.m2.1.1.3" xref="S2.SS2.p6.6.m2.1.1.3.cmml"><mn id="S2.SS2.p6.6.m2.1.1.3.2" xref="S2.SS2.p6.6.m2.1.1.3.2.cmml">1</mn><mo id="S2.SS2.p6.6.m2.1.1.3.1" xref="S2.SS2.p6.6.m2.1.1.3.1.cmml">⁢</mo><mi id="S2.SS2.p6.6.m2.1.1.3.3" xref="S2.SS2.p6.6.m2.1.1.3.3.cmml">μ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.6.m2.1b"><apply id="S2.SS2.p6.6.m2.1.1.cmml" xref="S2.SS2.p6.6.m2.1.1"><eq id="S2.SS2.p6.6.m2.1.1.1.cmml" xref="S2.SS2.p6.6.m2.1.1.1"></eq><ci id="S2.SS2.p6.6.m2.1.1.2.cmml" xref="S2.SS2.p6.6.m2.1.1.2">𝜆</ci><apply id="S2.SS2.p6.6.m2.1.1.3.cmml" xref="S2.SS2.p6.6.m2.1.1.3"><times id="S2.SS2.p6.6.m2.1.1.3.1.cmml" xref="S2.SS2.p6.6.m2.1.1.3.1"></times><cn id="S2.SS2.p6.6.m2.1.1.3.2.cmml" type="integer" xref="S2.SS2.p6.6.m2.1.1.3.2">1</cn><ci id="S2.SS2.p6.6.m2.1.1.3.3.cmml" xref="S2.SS2.p6.6.m2.1.1.3.3">𝜇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.6.m2.1c">\lambda=1\mu</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.6.m2.1d">italic_λ = 1 italic_μ</annotation></semantics></math>m – a LISA-like near infrared laser. For reference, LISA aims to achieve <math alttext="\widetilde{\phi}_{\rm rms}=9\times 10^{-6}" class="ltx_Math" display="inline" id="S2.SS2.p6.7.m3.1"><semantics id="S2.SS2.p6.7.m3.1a"><mrow id="S2.SS2.p6.7.m3.1.1" xref="S2.SS2.p6.7.m3.1.1.cmml"><msub id="S2.SS2.p6.7.m3.1.1.2" xref="S2.SS2.p6.7.m3.1.1.2.cmml"><mover accent="true" id="S2.SS2.p6.7.m3.1.1.2.2" xref="S2.SS2.p6.7.m3.1.1.2.2.cmml"><mi id="S2.SS2.p6.7.m3.1.1.2.2.2" xref="S2.SS2.p6.7.m3.1.1.2.2.2.cmml">ϕ</mi><mo id="S2.SS2.p6.7.m3.1.1.2.2.1" xref="S2.SS2.p6.7.m3.1.1.2.2.1.cmml">~</mo></mover><mi id="S2.SS2.p6.7.m3.1.1.2.3" xref="S2.SS2.p6.7.m3.1.1.2.3.cmml">rms</mi></msub><mo id="S2.SS2.p6.7.m3.1.1.1" xref="S2.SS2.p6.7.m3.1.1.1.cmml">=</mo><mrow id="S2.SS2.p6.7.m3.1.1.3" xref="S2.SS2.p6.7.m3.1.1.3.cmml"><mn id="S2.SS2.p6.7.m3.1.1.3.2" xref="S2.SS2.p6.7.m3.1.1.3.2.cmml">9</mn><mo id="S2.SS2.p6.7.m3.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S2.SS2.p6.7.m3.1.1.3.1.cmml">×</mo><msup id="S2.SS2.p6.7.m3.1.1.3.3" xref="S2.SS2.p6.7.m3.1.1.3.3.cmml"><mn id="S2.SS2.p6.7.m3.1.1.3.3.2" xref="S2.SS2.p6.7.m3.1.1.3.3.2.cmml">10</mn><mrow id="S2.SS2.p6.7.m3.1.1.3.3.3" xref="S2.SS2.p6.7.m3.1.1.3.3.3.cmml"><mo id="S2.SS2.p6.7.m3.1.1.3.3.3a" xref="S2.SS2.p6.7.m3.1.1.3.3.3.cmml">−</mo><mn id="S2.SS2.p6.7.m3.1.1.3.3.3.2" xref="S2.SS2.p6.7.m3.1.1.3.3.3.2.cmml">6</mn></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.7.m3.1b"><apply id="S2.SS2.p6.7.m3.1.1.cmml" xref="S2.SS2.p6.7.m3.1.1"><eq id="S2.SS2.p6.7.m3.1.1.1.cmml" xref="S2.SS2.p6.7.m3.1.1.1"></eq><apply id="S2.SS2.p6.7.m3.1.1.2.cmml" xref="S2.SS2.p6.7.m3.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.p6.7.m3.1.1.2.1.cmml" xref="S2.SS2.p6.7.m3.1.1.2">subscript</csymbol><apply id="S2.SS2.p6.7.m3.1.1.2.2.cmml" xref="S2.SS2.p6.7.m3.1.1.2.2"><ci id="S2.SS2.p6.7.m3.1.1.2.2.1.cmml" xref="S2.SS2.p6.7.m3.1.1.2.2.1">~</ci><ci id="S2.SS2.p6.7.m3.1.1.2.2.2.cmml" xref="S2.SS2.p6.7.m3.1.1.2.2.2">italic-ϕ</ci></apply><ci id="S2.SS2.p6.7.m3.1.1.2.3.cmml" xref="S2.SS2.p6.7.m3.1.1.2.3">rms</ci></apply><apply id="S2.SS2.p6.7.m3.1.1.3.cmml" xref="S2.SS2.p6.7.m3.1.1.3"><times id="S2.SS2.p6.7.m3.1.1.3.1.cmml" xref="S2.SS2.p6.7.m3.1.1.3.1"></times><cn id="S2.SS2.p6.7.m3.1.1.3.2.cmml" type="integer" xref="S2.SS2.p6.7.m3.1.1.3.2">9</cn><apply id="S2.SS2.p6.7.m3.1.1.3.3.cmml" xref="S2.SS2.p6.7.m3.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS2.p6.7.m3.1.1.3.3.1.cmml" xref="S2.SS2.p6.7.m3.1.1.3.3">superscript</csymbol><cn id="S2.SS2.p6.7.m3.1.1.3.3.2.cmml" type="integer" xref="S2.SS2.p6.7.m3.1.1.3.3.2">10</cn><apply id="S2.SS2.p6.7.m3.1.1.3.3.3.cmml" xref="S2.SS2.p6.7.m3.1.1.3.3.3"><minus id="S2.SS2.p6.7.m3.1.1.3.3.3.1.cmml" xref="S2.SS2.p6.7.m3.1.1.3.3.3"></minus><cn id="S2.SS2.p6.7.m3.1.1.3.3.3.2.cmml" type="integer" xref="S2.SS2.p6.7.m3.1.1.3.3.3.2">6</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.7.m3.1c">\widetilde{\phi}_{\rm rms}=9\times 10^{-6}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.7.m3.1d">over~ start_ARG italic_ϕ end_ARG start_POSTSUBSCRIPT roman_rms end_POSTSUBSCRIPT = 9 × 10 start_POSTSUPERSCRIPT - 6 end_POSTSUPERSCRIPT</annotation></semantics></math>rad Hz<sup class="ltx_sup" id="S2.SS2.p6.10.1"><span class="ltx_text ltx_font_italic" id="S2.SS2.p6.10.1.1">-1/2</span></sup> and <math alttext="\widetilde{\Delta x}_{\rm rms}=15\times 10^{-10}" class="ltx_Math" display="inline" id="S2.SS2.p6.9.m5.1"><semantics id="S2.SS2.p6.9.m5.1a"><mrow id="S2.SS2.p6.9.m5.1.1" xref="S2.SS2.p6.9.m5.1.1.cmml"><msub id="S2.SS2.p6.9.m5.1.1.2" xref="S2.SS2.p6.9.m5.1.1.2.cmml"><mover accent="true" id="S2.SS2.p6.9.m5.1.1.2.2" xref="S2.SS2.p6.9.m5.1.1.2.2.cmml"><mrow id="S2.SS2.p6.9.m5.1.1.2.2.2" xref="S2.SS2.p6.9.m5.1.1.2.2.2.cmml"><mi id="S2.SS2.p6.9.m5.1.1.2.2.2.2" mathvariant="normal" xref="S2.SS2.p6.9.m5.1.1.2.2.2.2.cmml">Δ</mi><mo id="S2.SS2.p6.9.m5.1.1.2.2.2.1" xref="S2.SS2.p6.9.m5.1.1.2.2.2.1.cmml">⁢</mo><mi id="S2.SS2.p6.9.m5.1.1.2.2.2.3" xref="S2.SS2.p6.9.m5.1.1.2.2.2.3.cmml">x</mi></mrow><mo id="S2.SS2.p6.9.m5.1.1.2.2.1" xref="S2.SS2.p6.9.m5.1.1.2.2.1.cmml">~</mo></mover><mi id="S2.SS2.p6.9.m5.1.1.2.3" xref="S2.SS2.p6.9.m5.1.1.2.3.cmml">rms</mi></msub><mo id="S2.SS2.p6.9.m5.1.1.1" xref="S2.SS2.p6.9.m5.1.1.1.cmml">=</mo><mrow id="S2.SS2.p6.9.m5.1.1.3" xref="S2.SS2.p6.9.m5.1.1.3.cmml"><mn id="S2.SS2.p6.9.m5.1.1.3.2" xref="S2.SS2.p6.9.m5.1.1.3.2.cmml">15</mn><mo id="S2.SS2.p6.9.m5.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S2.SS2.p6.9.m5.1.1.3.1.cmml">×</mo><msup id="S2.SS2.p6.9.m5.1.1.3.3" xref="S2.SS2.p6.9.m5.1.1.3.3.cmml"><mn id="S2.SS2.p6.9.m5.1.1.3.3.2" xref="S2.SS2.p6.9.m5.1.1.3.3.2.cmml">10</mn><mrow id="S2.SS2.p6.9.m5.1.1.3.3.3" xref="S2.SS2.p6.9.m5.1.1.3.3.3.cmml"><mo id="S2.SS2.p6.9.m5.1.1.3.3.3a" xref="S2.SS2.p6.9.m5.1.1.3.3.3.cmml">−</mo><mn id="S2.SS2.p6.9.m5.1.1.3.3.3.2" xref="S2.SS2.p6.9.m5.1.1.3.3.3.2.cmml">10</mn></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.9.m5.1b"><apply id="S2.SS2.p6.9.m5.1.1.cmml" xref="S2.SS2.p6.9.m5.1.1"><eq id="S2.SS2.p6.9.m5.1.1.1.cmml" xref="S2.SS2.p6.9.m5.1.1.1"></eq><apply id="S2.SS2.p6.9.m5.1.1.2.cmml" xref="S2.SS2.p6.9.m5.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.p6.9.m5.1.1.2.1.cmml" xref="S2.SS2.p6.9.m5.1.1.2">subscript</csymbol><apply id="S2.SS2.p6.9.m5.1.1.2.2.cmml" xref="S2.SS2.p6.9.m5.1.1.2.2"><ci id="S2.SS2.p6.9.m5.1.1.2.2.1.cmml" xref="S2.SS2.p6.9.m5.1.1.2.2.1">~</ci><apply id="S2.SS2.p6.9.m5.1.1.2.2.2.cmml" xref="S2.SS2.p6.9.m5.1.1.2.2.2"><times id="S2.SS2.p6.9.m5.1.1.2.2.2.1.cmml" xref="S2.SS2.p6.9.m5.1.1.2.2.2.1"></times><ci id="S2.SS2.p6.9.m5.1.1.2.2.2.2.cmml" xref="S2.SS2.p6.9.m5.1.1.2.2.2.2">Δ</ci><ci id="S2.SS2.p6.9.m5.1.1.2.2.2.3.cmml" xref="S2.SS2.p6.9.m5.1.1.2.2.2.3">𝑥</ci></apply></apply><ci id="S2.SS2.p6.9.m5.1.1.2.3.cmml" xref="S2.SS2.p6.9.m5.1.1.2.3">rms</ci></apply><apply id="S2.SS2.p6.9.m5.1.1.3.cmml" xref="S2.SS2.p6.9.m5.1.1.3"><times id="S2.SS2.p6.9.m5.1.1.3.1.cmml" xref="S2.SS2.p6.9.m5.1.1.3.1"></times><cn id="S2.SS2.p6.9.m5.1.1.3.2.cmml" type="integer" xref="S2.SS2.p6.9.m5.1.1.3.2">15</cn><apply id="S2.SS2.p6.9.m5.1.1.3.3.cmml" xref="S2.SS2.p6.9.m5.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS2.p6.9.m5.1.1.3.3.1.cmml" xref="S2.SS2.p6.9.m5.1.1.3.3">superscript</csymbol><cn id="S2.SS2.p6.9.m5.1.1.3.3.2.cmml" type="integer" xref="S2.SS2.p6.9.m5.1.1.3.3.2">10</cn><apply id="S2.SS2.p6.9.m5.1.1.3.3.3.cmml" xref="S2.SS2.p6.9.m5.1.1.3.3.3"><minus id="S2.SS2.p6.9.m5.1.1.3.3.3.1.cmml" xref="S2.SS2.p6.9.m5.1.1.3.3.3"></minus><cn id="S2.SS2.p6.9.m5.1.1.3.3.3.2.cmml" type="integer" xref="S2.SS2.p6.9.m5.1.1.3.3.3.2">10</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.9.m5.1c">\widetilde{\Delta x}_{\rm rms}=15\times 10^{-10}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.9.m5.1d">over~ start_ARG roman_Δ italic_x end_ARG start_POSTSUBSCRIPT roman_rms end_POSTSUBSCRIPT = 15 × 10 start_POSTSUPERSCRIPT - 10 end_POSTSUPERSCRIPT</annotation></semantics></math> cm Hz<sup class="ltx_sup" id="S2.SS2.p6.10.2"><span class="ltx_text ltx_font_italic" id="S2.SS2.p6.10.2.1">-1/2</span></sup> <cite class="ltx_cite ltx_citemacro_citep">(Colpi et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib21" title="">2024</a>)</cite>.</p> </div> <div class="ltx_para" id="S2.SS2.p7"> <p class="ltx_p" id="S2.SS2.p7.9">Finally, these estimates translate into a precision for how well the gravitational wave strain can be measured. The long wavelength strain noise for a single arm is</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="A3.EGx3"> <tbody id="S2.E12"> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\widetilde{h}_{\rm rms}^{\rm LW}" class="ltx_Math" display="inline" id="S2.Ex5.m1.1"><semantics id="S2.Ex5.m1.1a"><msubsup id="S2.Ex5.m1.1.1" xref="S2.Ex5.m1.1.1.cmml"><mover accent="true" id="S2.Ex5.m1.1.1.2.2" xref="S2.Ex5.m1.1.1.2.2.cmml"><mi id="S2.Ex5.m1.1.1.2.2.2" xref="S2.Ex5.m1.1.1.2.2.2.cmml">h</mi><mo id="S2.Ex5.m1.1.1.2.2.1" xref="S2.Ex5.m1.1.1.2.2.1.cmml">~</mo></mover><mi id="S2.Ex5.m1.1.1.2.3" xref="S2.Ex5.m1.1.1.2.3.cmml">rms</mi><mi id="S2.Ex5.m1.1.1.3" xref="S2.Ex5.m1.1.1.3.cmml">LW</mi></msubsup><annotation-xml encoding="MathML-Content" id="S2.Ex5.m1.1b"><apply id="S2.Ex5.m1.1.1.cmml" xref="S2.Ex5.m1.1.1"><csymbol cd="ambiguous" id="S2.Ex5.m1.1.1.1.cmml" xref="S2.Ex5.m1.1.1">superscript</csymbol><apply id="S2.Ex5.m1.1.1.2.cmml" xref="S2.Ex5.m1.1.1"><csymbol cd="ambiguous" id="S2.Ex5.m1.1.1.2.1.cmml" xref="S2.Ex5.m1.1.1">subscript</csymbol><apply id="S2.Ex5.m1.1.1.2.2.cmml" xref="S2.Ex5.m1.1.1.2.2"><ci id="S2.Ex5.m1.1.1.2.2.1.cmml" xref="S2.Ex5.m1.1.1.2.2.1">~</ci><ci id="S2.Ex5.m1.1.1.2.2.2.cmml" xref="S2.Ex5.m1.1.1.2.2.2">ℎ</ci></apply><ci id="S2.Ex5.m1.1.1.2.3.cmml" xref="S2.Ex5.m1.1.1.2.3">rms</ci></apply><ci id="S2.Ex5.m1.1.1.3.cmml" xref="S2.Ex5.m1.1.1.3">LW</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex5.m1.1c">\displaystyle\widetilde{h}_{\rm rms}^{\rm LW}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex5.m1.1d">over~ start_ARG italic_h end_ARG start_POSTSUBSCRIPT roman_rms end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_LW end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S2.Ex5.m2.1"><semantics id="S2.Ex5.m2.1a"><mo id="S2.Ex5.m2.1.1" xref="S2.Ex5.m2.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S2.Ex5.m2.1b"><eq id="S2.Ex5.m2.1.1.cmml" xref="S2.Ex5.m2.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex5.m2.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="S2.Ex5.m2.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\frac{\sqrt{2}\widetilde{\Delta x}_{\rm rms}}{L}," class="ltx_Math" display="inline" id="S2.Ex5.m3.1"><semantics id="S2.Ex5.m3.1a"><mrow id="S2.Ex5.m3.1.2.2" xref="S2.Ex5.m3.1.1.cmml"><mstyle displaystyle="true" id="S2.Ex5.m3.1.1" xref="S2.Ex5.m3.1.1.cmml"><mfrac id="S2.Ex5.m3.1.1a" xref="S2.Ex5.m3.1.1.cmml"><mrow id="S2.Ex5.m3.1.1.2" xref="S2.Ex5.m3.1.1.2.cmml"><msqrt id="S2.Ex5.m3.1.1.2.2" xref="S2.Ex5.m3.1.1.2.2.cmml"><mn id="S2.Ex5.m3.1.1.2.2.2" xref="S2.Ex5.m3.1.1.2.2.2.cmml">2</mn></msqrt><mo id="S2.Ex5.m3.1.1.2.1" xref="S2.Ex5.m3.1.1.2.1.cmml">⁢</mo><msub id="S2.Ex5.m3.1.1.2.3" xref="S2.Ex5.m3.1.1.2.3.cmml"><mover accent="true" id="S2.Ex5.m3.1.1.2.3.2" xref="S2.Ex5.m3.1.1.2.3.2.cmml"><mrow id="S2.Ex5.m3.1.1.2.3.2.2" xref="S2.Ex5.m3.1.1.2.3.2.2.cmml"><mi id="S2.Ex5.m3.1.1.2.3.2.2.2" mathvariant="normal" xref="S2.Ex5.m3.1.1.2.3.2.2.2.cmml">Δ</mi><mo id="S2.Ex5.m3.1.1.2.3.2.2.1" xref="S2.Ex5.m3.1.1.2.3.2.2.1.cmml">⁢</mo><mi id="S2.Ex5.m3.1.1.2.3.2.2.3" xref="S2.Ex5.m3.1.1.2.3.2.2.3.cmml">x</mi></mrow><mo id="S2.Ex5.m3.1.1.2.3.2.1" xref="S2.Ex5.m3.1.1.2.3.2.1.cmml">~</mo></mover><mi id="S2.Ex5.m3.1.1.2.3.3" xref="S2.Ex5.m3.1.1.2.3.3.cmml">rms</mi></msub></mrow><mi id="S2.Ex5.m3.1.1.3" xref="S2.Ex5.m3.1.1.3.cmml">L</mi></mfrac></mstyle><mo id="S2.Ex5.m3.1.2.2.1" xref="S2.Ex5.m3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex5.m3.1b"><apply id="S2.Ex5.m3.1.1.cmml" 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xref="S2.Ex5.m3.1.1.2.3.2.2.3">𝑥</ci></apply></apply><ci id="S2.Ex5.m3.1.1.2.3.3.cmml" xref="S2.Ex5.m3.1.1.2.3.3">rms</ci></apply></apply><ci id="S2.Ex5.m3.1.1.3.cmml" xref="S2.Ex5.m3.1.1.3">𝐿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex5.m3.1c">\displaystyle\frac{\sqrt{2}\widetilde{\Delta x}_{\rm rms}}{L},</annotation><annotation encoding="application/x-llamapun" id="S2.Ex5.m3.1d">divide start_ARG square-root start_ARG 2 end_ARG over~ start_ARG roman_Δ italic_x end_ARG start_POSTSUBSCRIPT roman_rms end_POSTSUBSCRIPT end_ARG start_ARG italic_L end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="2"><span class="ltx_tag ltx_tag_equation ltx_align_right">(12)</span></td> </tr> <tr class="ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S2.E12.m1.1"><semantics id="S2.E12.m1.1a"><mo id="S2.E12.m1.1.1" xref="S2.E12.m1.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S2.E12.m1.1b"><eq id="S2.E12.m1.1.1.cmml" xref="S2.E12.m1.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S2.E12.m1.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="S2.E12.m1.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle(\overbrace{5\times 10^{-18}}^{\lambda=1\,\text{cm}},~{}% \overbrace{5\times 10^{-22}}^{\lambda=1\,\text{$\mu$m}})\text{~{}~{}~{}Hz${}^{% -1/2}$~{}}{\cal A}\left(\frac{L}{30{\rm~{}AU}}\right)^{-1}\times 10^{-\left([C% /N_{0}]_{\rm dB-Hz}-40\right)/20}." class="ltx_Math" display="inline" id="S2.E12.m2.5"><semantics id="S2.E12.m2.5a"><mrow id="S2.E12.m2.5.5.1" xref="S2.E12.m2.5.5.1.1.cmml"><mrow id="S2.E12.m2.5.5.1.1" xref="S2.E12.m2.5.5.1.1.cmml"><mrow id="S2.E12.m2.5.5.1.1.2" xref="S2.E12.m2.5.5.1.1.2.cmml"><mrow id="S2.E12.m2.5.5.1.1.2.2.2" xref="S2.E12.m2.5.5.1.1.2.2.3.cmml"><mo id="S2.E12.m2.5.5.1.1.2.2.2.3" stretchy="false" xref="S2.E12.m2.5.5.1.1.2.2.3.cmml">(</mo><mover id="S2.E12.m2.5.5.1.1.1.1.1.1" xref="S2.E12.m2.5.5.1.1.1.1.1.1.cmml"><mover accent="true" id="S2.E12.m2.5.5.1.1.1.1.1.1.2" xref="S2.E12.m2.5.5.1.1.1.1.1.1.2.cmml"><mrow id="S2.E12.m2.5.5.1.1.1.1.1.1.2.2" xref="S2.E12.m2.5.5.1.1.1.1.1.1.2.2.cmml"><mn id="S2.E12.m2.5.5.1.1.1.1.1.1.2.2.2" xref="S2.E12.m2.5.5.1.1.1.1.1.1.2.2.2.cmml">5</mn><mo id="S2.E12.m2.5.5.1.1.1.1.1.1.2.2.1" lspace="0.222em" rspace="0.222em" xref="S2.E12.m2.5.5.1.1.1.1.1.1.2.2.1.cmml">×</mo><msup id="S2.E12.m2.5.5.1.1.1.1.1.1.2.2.3" xref="S2.E12.m2.5.5.1.1.1.1.1.1.2.2.3.cmml"><mn id="S2.E12.m2.5.5.1.1.1.1.1.1.2.2.3.2" xref="S2.E12.m2.5.5.1.1.1.1.1.1.2.2.3.2.cmml">10</mn><mrow 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10^{-18}}^{\lambda=1\,\text{cm}},~{}% \overbrace{5\times 10^{-22}}^{\lambda=1\,\text{$\mu$m}})\text{~{}~{}~{}Hz${}^{% -1/2}$~{}}{\cal A}\left(\frac{L}{30{\rm~{}AU}}\right)^{-1}\times 10^{-\left([C% /N_{0}]_{\rm dB-Hz}-40\right)/20}.</annotation><annotation encoding="application/x-llamapun" id="S2.E12.m2.5d">( over⏞ start_ARG 5 × 10 start_POSTSUPERSCRIPT - 18 end_POSTSUPERSCRIPT end_ARG start_POSTSUPERSCRIPT italic_λ = 1 cm end_POSTSUPERSCRIPT , over⏞ start_ARG 5 × 10 start_POSTSUPERSCRIPT - 22 end_POSTSUPERSCRIPT end_ARG start_POSTSUPERSCRIPT italic_λ = 1 italic_μ m end_POSTSUPERSCRIPT ) Hz caligraphic_A ( divide start_ARG italic_L end_ARG start_ARG 30 roman_AU end_ARG ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT × 10 start_POSTSUPERSCRIPT - ( [ italic_C / italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT roman_dB - roman_Hz end_POSTSUBSCRIPT - 40 ) / 20 end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </tbody> </table> <p class="ltx_p" id="S2.SS2.p7.8">We will eventually use a transfer function that converts the single-arm noise in the low <math alttext="f" class="ltx_Math" display="inline" id="S2.SS2.p7.1.m1.1"><semantics id="S2.SS2.p7.1.m1.1a"><mi id="S2.SS2.p7.1.m1.1.1" xref="S2.SS2.p7.1.m1.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p7.1.m1.1b"><ci id="S2.SS2.p7.1.m1.1.1.cmml" xref="S2.SS2.p7.1.m1.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p7.1.m1.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p7.1.m1.1d">italic_f</annotation></semantics></math> limit to a two-arm time-delay interferometry measurement that is applicable at all <math alttext="f" class="ltx_Math" display="inline" id="S2.SS2.p7.2.m2.1"><semantics id="S2.SS2.p7.2.m2.1a"><mi id="S2.SS2.p7.2.m2.1.1" xref="S2.SS2.p7.2.m2.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p7.2.m2.1b"><ci id="S2.SS2.p7.2.m2.1.1.cmml" xref="S2.SS2.p7.2.m2.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p7.2.m2.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p7.2.m2.1d">italic_f</annotation></semantics></math> (§ <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S3" title="3 Strain power sensitivity ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">3</span></a>). The factor of <math alttext="\sqrt{2}" class="ltx_Math" display="inline" id="S2.SS2.p7.3.m3.1"><semantics id="S2.SS2.p7.3.m3.1a"><msqrt id="S2.SS2.p7.3.m3.1.1" xref="S2.SS2.p7.3.m3.1.1.cmml"><mn id="S2.SS2.p7.3.m3.1.1.2" xref="S2.SS2.p7.3.m3.1.1.2.cmml">2</mn></msqrt><annotation-xml encoding="MathML-Content" id="S2.SS2.p7.3.m3.1b"><apply id="S2.SS2.p7.3.m3.1.1.cmml" xref="S2.SS2.p7.3.m3.1.1"><root id="S2.SS2.p7.3.m3.1.1a.cmml" xref="S2.SS2.p7.3.m3.1.1"></root><cn id="S2.SS2.p7.3.m3.1.1.2.cmml" type="integer" xref="S2.SS2.p7.3.m3.1.1.2">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p7.3.m3.1c">\sqrt{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p7.3.m3.1d">square-root start_ARG 2 end_ARG</annotation></semantics></math> in the first expression is because there are measurements of phase at both spacecraft in an arm. Here, <math alttext="{\cal A}" class="ltx_Math" display="inline" id="S2.SS2.p7.4.m4.1"><semantics id="S2.SS2.p7.4.m4.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p7.4.m4.1.1" xref="S2.SS2.p7.4.m4.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p7.4.m4.1b"><ci id="S2.SS2.p7.4.m4.1.1.cmml" xref="S2.SS2.p7.4.m4.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p7.4.m4.1c">{\cal A}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p7.4.m4.1d">caligraphic_A</annotation></semantics></math> encapsulates the increase in the phase error from using phase measurements at two wavelengths to eliminate plasma dispersion. Since the plasma dispersion is negligible in the radio, for lasers a single wavelength would be used and <math alttext="{\cal A}=1" class="ltx_Math" display="inline" id="S2.SS2.p7.5.m5.1"><semantics id="S2.SS2.p7.5.m5.1a"><mrow id="S2.SS2.p7.5.m5.1.1" xref="S2.SS2.p7.5.m5.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p7.5.m5.1.1.2" xref="S2.SS2.p7.5.m5.1.1.2.cmml">𝒜</mi><mo id="S2.SS2.p7.5.m5.1.1.1" xref="S2.SS2.p7.5.m5.1.1.1.cmml">=</mo><mn id="S2.SS2.p7.5.m5.1.1.3" xref="S2.SS2.p7.5.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p7.5.m5.1b"><apply id="S2.SS2.p7.5.m5.1.1.cmml" xref="S2.SS2.p7.5.m5.1.1"><eq id="S2.SS2.p7.5.m5.1.1.1.cmml" xref="S2.SS2.p7.5.m5.1.1.1"></eq><ci id="S2.SS2.p7.5.m5.1.1.2.cmml" xref="S2.SS2.p7.5.m5.1.1.2">𝒜</ci><cn id="S2.SS2.p7.5.m5.1.1.3.cmml" type="integer" xref="S2.SS2.p7.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p7.5.m5.1c">{\cal A}=1</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p7.5.m5.1d">caligraphic_A = 1</annotation></semantics></math>. We show in Appendix <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#A1" title="Appendix A Dispersion ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">A</span></a> that <math alttext="{\cal A}\approx 1.5~{}(2)" class="ltx_Math" display="inline" id="S2.SS2.p7.6.m6.1"><semantics id="S2.SS2.p7.6.m6.1a"><mrow id="S2.SS2.p7.6.m6.1.2" xref="S2.SS2.p7.6.m6.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p7.6.m6.1.2.2" xref="S2.SS2.p7.6.m6.1.2.2.cmml">𝒜</mi><mo id="S2.SS2.p7.6.m6.1.2.1" xref="S2.SS2.p7.6.m6.1.2.1.cmml">≈</mo><mrow id="S2.SS2.p7.6.m6.1.2.3" xref="S2.SS2.p7.6.m6.1.2.3.cmml"><mn id="S2.SS2.p7.6.m6.1.2.3.2" xref="S2.SS2.p7.6.m6.1.2.3.2.cmml">1.5</mn><mo id="S2.SS2.p7.6.m6.1.2.3.1" lspace="0.330em" xref="S2.SS2.p7.6.m6.1.2.3.1.cmml">⁢</mo><mrow id="S2.SS2.p7.6.m6.1.2.3.3.2" xref="S2.SS2.p7.6.m6.1.2.3.cmml"><mo id="S2.SS2.p7.6.m6.1.2.3.3.2.1" stretchy="false" xref="S2.SS2.p7.6.m6.1.2.3.cmml">(</mo><mn id="S2.SS2.p7.6.m6.1.1" xref="S2.SS2.p7.6.m6.1.1.cmml">2</mn><mo id="S2.SS2.p7.6.m6.1.2.3.3.2.2" stretchy="false" xref="S2.SS2.p7.6.m6.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p7.6.m6.1b"><apply id="S2.SS2.p7.6.m6.1.2.cmml" xref="S2.SS2.p7.6.m6.1.2"><approx id="S2.SS2.p7.6.m6.1.2.1.cmml" xref="S2.SS2.p7.6.m6.1.2.1"></approx><ci id="S2.SS2.p7.6.m6.1.2.2.cmml" xref="S2.SS2.p7.6.m6.1.2.2">𝒜</ci><apply id="S2.SS2.p7.6.m6.1.2.3.cmml" xref="S2.SS2.p7.6.m6.1.2.3"><times id="S2.SS2.p7.6.m6.1.2.3.1.cmml" xref="S2.SS2.p7.6.m6.1.2.3.1"></times><cn id="S2.SS2.p7.6.m6.1.2.3.2.cmml" type="float" xref="S2.SS2.p7.6.m6.1.2.3.2">1.5</cn><cn id="S2.SS2.p7.6.m6.1.1.cmml" type="integer" xref="S2.SS2.p7.6.m6.1.1">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p7.6.m6.1c">{\cal A}\approx 1.5~{}(2)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p7.6.m6.1d">caligraphic_A ≈ 1.5 ( 2 )</annotation></semantics></math> if wavelengths differing by a factor of 4 (2) are used when referenced to the shorter wavelength, assuming <math alttext="T_{\rm sys}" class="ltx_Math" display="inline" id="S2.SS2.p7.7.m7.1"><semantics id="S2.SS2.p7.7.m7.1a"><msub id="S2.SS2.p7.7.m7.1.1" xref="S2.SS2.p7.7.m7.1.1.cmml"><mi id="S2.SS2.p7.7.m7.1.1.2" xref="S2.SS2.p7.7.m7.1.1.2.cmml">T</mi><mi id="S2.SS2.p7.7.m7.1.1.3" xref="S2.SS2.p7.7.m7.1.1.3.cmml">sys</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p7.7.m7.1b"><apply id="S2.SS2.p7.7.m7.1.1.cmml" xref="S2.SS2.p7.7.m7.1.1"><csymbol cd="ambiguous" id="S2.SS2.p7.7.m7.1.1.1.cmml" xref="S2.SS2.p7.7.m7.1.1">subscript</csymbol><ci id="S2.SS2.p7.7.m7.1.1.2.cmml" xref="S2.SS2.p7.7.m7.1.1.2">𝑇</ci><ci id="S2.SS2.p7.7.m7.1.1.3.cmml" xref="S2.SS2.p7.7.m7.1.1.3">sys</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p7.7.m7.1c">T_{\rm sys}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p7.7.m7.1d">italic_T start_POSTSUBSCRIPT roman_sys end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="D_{\rm eff}" class="ltx_Math" display="inline" id="S2.SS2.p7.8.m8.1"><semantics id="S2.SS2.p7.8.m8.1a"><msub id="S2.SS2.p7.8.m8.1.1" xref="S2.SS2.p7.8.m8.1.1.cmml"><mi id="S2.SS2.p7.8.m8.1.1.2" xref="S2.SS2.p7.8.m8.1.1.2.cmml">D</mi><mi id="S2.SS2.p7.8.m8.1.1.3" xref="S2.SS2.p7.8.m8.1.1.3.cmml">eff</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p7.8.m8.1b"><apply id="S2.SS2.p7.8.m8.1.1.cmml" xref="S2.SS2.p7.8.m8.1.1"><csymbol cd="ambiguous" id="S2.SS2.p7.8.m8.1.1.1.cmml" xref="S2.SS2.p7.8.m8.1.1">subscript</csymbol><ci id="S2.SS2.p7.8.m8.1.1.2.cmml" xref="S2.SS2.p7.8.m8.1.1.2">𝐷</ci><ci id="S2.SS2.p7.8.m8.1.1.3.cmml" xref="S2.SS2.p7.8.m8.1.1.3">eff</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p7.8.m8.1c">D_{\rm eff}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p7.8.m8.1d">italic_D start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT</annotation></semantics></math> are the same at both wavelengths.</p> </div> <div class="ltx_para" id="S2.SS2.p8"> <p class="ltx_p" id="S2.SS2.p8.4">Another possibility is to use nanosecond laser pulses as a clock, rather than the carrier phase timing considered so far. Laser pulses would be more noisy for the Solar System-scale baselines we consider since the signal can no longer be matched to a template phase, with</p> <table class="ltx_equation ltx_eqn_table" id="S2.E13"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\widetilde{h}_{\rm rms}^{\rm LW}\Big{|}_{\rm laser-pulses}\sim 2\times 10^{-14% }{\rm~{}Hz}^{-1/2}\left(\frac{L}{30\,{\rm AU}}\right)^{2}\left(\frac{D_{1}D_{2% }}{[1\,{\rm m}]^{2}}\right)^{-3}\left(\frac{P_{\rm em}}{10\,{\rm W}}\right)^{-% 3/2}," class="ltx_Math" display="block" id="S2.E13.m1.5"><semantics id="S2.E13.m1.5a"><mrow id="S2.E13.m1.5.5.1" xref="S2.E13.m1.5.5.1.1.cmml"><mrow id="S2.E13.m1.5.5.1.1" xref="S2.E13.m1.5.5.1.1.cmml"><msub id="S2.E13.m1.5.5.1.1.1.1" xref="S2.E13.m1.5.5.1.1.1.2.cmml"><mrow id="S2.E13.m1.5.5.1.1.1.1.1" xref="S2.E13.m1.5.5.1.1.1.2.cmml"><msubsup 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start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT ( divide start_ARG italic_L end_ARG start_ARG 30 roman_AU end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( divide start_ARG italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG start_ARG [ 1 roman_m ] start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT ( divide start_ARG italic_P start_POSTSUBSCRIPT roman_em end_POSTSUBSCRIPT end_ARG start_ARG 10 roman_W end_ARG ) start_POSTSUPERSCRIPT - 3 / 2 end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(13)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p8.3">where <math alttext="D_{1}" class="ltx_Math" display="inline" id="S2.SS2.p8.1.m1.1"><semantics id="S2.SS2.p8.1.m1.1a"><msub id="S2.SS2.p8.1.m1.1.1" xref="S2.SS2.p8.1.m1.1.1.cmml"><mi id="S2.SS2.p8.1.m1.1.1.2" xref="S2.SS2.p8.1.m1.1.1.2.cmml">D</mi><mn id="S2.SS2.p8.1.m1.1.1.3" xref="S2.SS2.p8.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p8.1.m1.1b"><apply id="S2.SS2.p8.1.m1.1.1.cmml" xref="S2.SS2.p8.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p8.1.m1.1.1.1.cmml" xref="S2.SS2.p8.1.m1.1.1">subscript</csymbol><ci id="S2.SS2.p8.1.m1.1.1.2.cmml" xref="S2.SS2.p8.1.m1.1.1.2">𝐷</ci><cn id="S2.SS2.p8.1.m1.1.1.3.cmml" type="integer" xref="S2.SS2.p8.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p8.1.m1.1c">D_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p8.1.m1.1d">italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="D_{2}" class="ltx_Math" display="inline" id="S2.SS2.p8.2.m2.1"><semantics id="S2.SS2.p8.2.m2.1a"><msub id="S2.SS2.p8.2.m2.1.1" xref="S2.SS2.p8.2.m2.1.1.cmml"><mi id="S2.SS2.p8.2.m2.1.1.2" xref="S2.SS2.p8.2.m2.1.1.2.cmml">D</mi><mn id="S2.SS2.p8.2.m2.1.1.3" xref="S2.SS2.p8.2.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p8.2.m2.1b"><apply id="S2.SS2.p8.2.m2.1.1.cmml" xref="S2.SS2.p8.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS2.p8.2.m2.1.1.1.cmml" xref="S2.SS2.p8.2.m2.1.1">subscript</csymbol><ci id="S2.SS2.p8.2.m2.1.1.2.cmml" xref="S2.SS2.p8.2.m2.1.1.2">𝐷</ci><cn id="S2.SS2.p8.2.m2.1.1.3.cmml" type="integer" xref="S2.SS2.p8.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p8.2.m2.1c">D_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p8.2.m2.1d">italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> are the sizes of the mirrors on the emitting and receiving telescopes <cite class="ltx_cite ltx_citemacro_citep">(Fedderke et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib28" title="">2022a</a>, cf. their eqn. 95)</cite>. This sensitivity is likely insufficient, but could become interesting for Earth-spacecraft ranging if the terrestrial station has <math alttext="D_{2}\sim 10\," class="ltx_Math" display="inline" id="S2.SS2.p8.3.m3.1"><semantics id="S2.SS2.p8.3.m3.1a"><mrow id="S2.SS2.p8.3.m3.1.1" xref="S2.SS2.p8.3.m3.1.1.cmml"><msub id="S2.SS2.p8.3.m3.1.1.2" xref="S2.SS2.p8.3.m3.1.1.2.cmml"><mi id="S2.SS2.p8.3.m3.1.1.2.2" xref="S2.SS2.p8.3.m3.1.1.2.2.cmml">D</mi><mn id="S2.SS2.p8.3.m3.1.1.2.3" xref="S2.SS2.p8.3.m3.1.1.2.3.cmml">2</mn></msub><mo id="S2.SS2.p8.3.m3.1.1.1" xref="S2.SS2.p8.3.m3.1.1.1.cmml">∼</mo><mn id="S2.SS2.p8.3.m3.1.1.3" xref="S2.SS2.p8.3.m3.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p8.3.m3.1b"><apply id="S2.SS2.p8.3.m3.1.1.cmml" xref="S2.SS2.p8.3.m3.1.1"><csymbol cd="latexml" id="S2.SS2.p8.3.m3.1.1.1.cmml" xref="S2.SS2.p8.3.m3.1.1.1">similar-to</csymbol><apply id="S2.SS2.p8.3.m3.1.1.2.cmml" xref="S2.SS2.p8.3.m3.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.p8.3.m3.1.1.2.1.cmml" xref="S2.SS2.p8.3.m3.1.1.2">subscript</csymbol><ci id="S2.SS2.p8.3.m3.1.1.2.2.cmml" xref="S2.SS2.p8.3.m3.1.1.2.2">𝐷</ci><cn id="S2.SS2.p8.3.m3.1.1.2.3.cmml" type="integer" xref="S2.SS2.p8.3.m3.1.1.2.3">2</cn></apply><cn id="S2.SS2.p8.3.m3.1.1.3.cmml" type="integer" xref="S2.SS2.p8.3.m3.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p8.3.m3.1c">D_{2}\sim 10\,</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p8.3.m3.1d">italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∼ 10</annotation></semantics></math>m (§ <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S3.SS3" title="3.3 Doppler tracking forecasts ‣ 3 Strain power sensitivity ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">3.3</span></a>).</p> </div> </section> <section class="ltx_subsection" id="S2.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.3 </span>Acceleration noise</h3> <div class="ltx_para" id="S2.SS3.p1"> <p class="ltx_p" id="S2.SS3.p1.10">In this section, we consider different sources of acceleration on the spacecraft. 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encoding="application/x-llamapun" id="S2.SS3.p1.3.m3.1d">italic_T</annotation></semantics></math>. Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.F2" title="Figure 2 ‣ 2.3 Acceleration noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">2</span></a> summarizes our estimates for the important acceleration sources, assuming a fiducial spacecraft effective area over mass of <math alttext="A_{\rm eff}/M=0.01" class="ltx_Math" display="inline" id="S2.SS3.p1.4.m4.1"><semantics id="S2.SS3.p1.4.m4.1a"><mrow id="S2.SS3.p1.4.m4.1.1" xref="S2.SS3.p1.4.m4.1.1.cmml"><mrow id="S2.SS3.p1.4.m4.1.1.2" xref="S2.SS3.p1.4.m4.1.1.2.cmml"><msub id="S2.SS3.p1.4.m4.1.1.2.2" xref="S2.SS3.p1.4.m4.1.1.2.2.cmml"><mi id="S2.SS3.p1.4.m4.1.1.2.2.2" xref="S2.SS3.p1.4.m4.1.1.2.2.2.cmml">A</mi><mi id="S2.SS3.p1.4.m4.1.1.2.2.3" xref="S2.SS3.p1.4.m4.1.1.2.2.3.cmml">eff</mi></msub><mo id="S2.SS3.p1.4.m4.1.1.2.1" xref="S2.SS3.p1.4.m4.1.1.2.1.cmml">/</mo><mi id="S2.SS3.p1.4.m4.1.1.2.3" xref="S2.SS3.p1.4.m4.1.1.2.3.cmml">M</mi></mrow><mo id="S2.SS3.p1.4.m4.1.1.1" xref="S2.SS3.p1.4.m4.1.1.1.cmml">=</mo><mn id="S2.SS3.p1.4.m4.1.1.3" xref="S2.SS3.p1.4.m4.1.1.3.cmml">0.01</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.4.m4.1b"><apply id="S2.SS3.p1.4.m4.1.1.cmml" xref="S2.SS3.p1.4.m4.1.1"><eq id="S2.SS3.p1.4.m4.1.1.1.cmml" xref="S2.SS3.p1.4.m4.1.1.1"></eq><apply id="S2.SS3.p1.4.m4.1.1.2.cmml" xref="S2.SS3.p1.4.m4.1.1.2"><divide id="S2.SS3.p1.4.m4.1.1.2.1.cmml" xref="S2.SS3.p1.4.m4.1.1.2.1"></divide><apply id="S2.SS3.p1.4.m4.1.1.2.2.cmml" xref="S2.SS3.p1.4.m4.1.1.2.2"><csymbol cd="ambiguous" id="S2.SS3.p1.4.m4.1.1.2.2.1.cmml" xref="S2.SS3.p1.4.m4.1.1.2.2">subscript</csymbol><ci id="S2.SS3.p1.4.m4.1.1.2.2.2.cmml" xref="S2.SS3.p1.4.m4.1.1.2.2.2">𝐴</ci><ci id="S2.SS3.p1.4.m4.1.1.2.2.3.cmml" xref="S2.SS3.p1.4.m4.1.1.2.2.3">eff</ci></apply><ci id="S2.SS3.p1.4.m4.1.1.2.3.cmml" xref="S2.SS3.p1.4.m4.1.1.2.3">𝑀</ci></apply><cn id="S2.SS3.p1.4.m4.1.1.3.cmml" type="float" xref="S2.SS3.p1.4.m4.1.1.3">0.01</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.4.m4.1c">A_{\rm eff}/M=0.01</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.4.m4.1d">italic_A start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT / italic_M = 0.01</annotation></semantics></math>m<sup class="ltx_sup" id="S2.SS3.p1.10.1">2</sup> kg<sup class="ltx_sup" id="S2.SS3.p1.10.2"><span class="ltx_text ltx_font_italic" id="S2.SS3.p1.10.2.1">-1</span></sup> – justified further in § <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S3.SS1" title="3.1 Two-arm time-delay interferometry forecasts ‣ 3 Strain power sensitivity ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">3.1</span></a> – and spacecraft at a heliocentric radius of <math alttext="r=30\," class="ltx_Math" display="inline" id="S2.SS3.p1.7.m7.1"><semantics id="S2.SS3.p1.7.m7.1a"><mrow id="S2.SS3.p1.7.m7.1.1" xref="S2.SS3.p1.7.m7.1.1.cmml"><mi id="S2.SS3.p1.7.m7.1.1.2" xref="S2.SS3.p1.7.m7.1.1.2.cmml">r</mi><mo id="S2.SS3.p1.7.m7.1.1.1" xref="S2.SS3.p1.7.m7.1.1.1.cmml">=</mo><mn id="S2.SS3.p1.7.m7.1.1.3" xref="S2.SS3.p1.7.m7.1.1.3.cmml">30</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.7.m7.1b"><apply id="S2.SS3.p1.7.m7.1.1.cmml" xref="S2.SS3.p1.7.m7.1.1"><eq id="S2.SS3.p1.7.m7.1.1.1.cmml" xref="S2.SS3.p1.7.m7.1.1.1"></eq><ci id="S2.SS3.p1.7.m7.1.1.2.cmml" xref="S2.SS3.p1.7.m7.1.1.2">𝑟</ci><cn id="S2.SS3.p1.7.m7.1.1.3.cmml" type="integer" xref="S2.SS3.p1.7.m7.1.1.3">30</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.7.m7.1c">r=30\,</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.7.m7.1d">italic_r = 30</annotation></semantics></math>AU. These include solar irradiance variations, drag from the solar wind, Lorentz forces on the spacecraft assuming the maximum possible spacecraft charge, and dust for two maximum dust masses (as explained later, the lower of the two dust curves is the more applicable). Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.F2" title="Figure 2 ‣ 2.3 Acceleration noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">2</span></a> also shows the acceleration control specification of the Gravitational Reference Sensor (GRS) on LISA <cite class="ltx_cite ltx_citemacro_citep">(Colpi et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib21" title="">2024</a>)</cite>. A LISA requirement is to achieve the acceleration control shown by this curve to frequencies as low as <math alttext="10^{-4}" class="ltx_Math" display="inline" id="S2.SS3.p1.8.m8.1"><semantics id="S2.SS3.p1.8.m8.1a"><msup id="S2.SS3.p1.8.m8.1.1" xref="S2.SS3.p1.8.m8.1.1.cmml"><mn id="S2.SS3.p1.8.m8.1.1.2" xref="S2.SS3.p1.8.m8.1.1.2.cmml">10</mn><mrow id="S2.SS3.p1.8.m8.1.1.3" xref="S2.SS3.p1.8.m8.1.1.3.cmml"><mo id="S2.SS3.p1.8.m8.1.1.3a" xref="S2.SS3.p1.8.m8.1.1.3.cmml">−</mo><mn id="S2.SS3.p1.8.m8.1.1.3.2" xref="S2.SS3.p1.8.m8.1.1.3.2.cmml">4</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.8.m8.1b"><apply id="S2.SS3.p1.8.m8.1.1.cmml" xref="S2.SS3.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S2.SS3.p1.8.m8.1.1.1.cmml" xref="S2.SS3.p1.8.m8.1.1">superscript</csymbol><cn id="S2.SS3.p1.8.m8.1.1.2.cmml" type="integer" xref="S2.SS3.p1.8.m8.1.1.2">10</cn><apply id="S2.SS3.p1.8.m8.1.1.3.cmml" xref="S2.SS3.p1.8.m8.1.1.3"><minus id="S2.SS3.p1.8.m8.1.1.3.1.cmml" xref="S2.SS3.p1.8.m8.1.1.3"></minus><cn id="S2.SS3.p1.8.m8.1.1.3.2.cmml" type="integer" xref="S2.SS3.p1.8.m8.1.1.3.2">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.8.m8.1c">10^{-4}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.8.m8.1d">10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT</annotation></semantics></math>Hz, with the goal to achieve this to <math alttext="2\times 10^{-5}" class="ltx_Math" display="inline" id="S2.SS3.p1.9.m9.1"><semantics id="S2.SS3.p1.9.m9.1a"><mrow id="S2.SS3.p1.9.m9.1.1" xref="S2.SS3.p1.9.m9.1.1.cmml"><mn id="S2.SS3.p1.9.m9.1.1.2" xref="S2.SS3.p1.9.m9.1.1.2.cmml">2</mn><mo id="S2.SS3.p1.9.m9.1.1.1" lspace="0.222em" rspace="0.222em" xref="S2.SS3.p1.9.m9.1.1.1.cmml">×</mo><msup id="S2.SS3.p1.9.m9.1.1.3" xref="S2.SS3.p1.9.m9.1.1.3.cmml"><mn id="S2.SS3.p1.9.m9.1.1.3.2" xref="S2.SS3.p1.9.m9.1.1.3.2.cmml">10</mn><mrow id="S2.SS3.p1.9.m9.1.1.3.3" xref="S2.SS3.p1.9.m9.1.1.3.3.cmml"><mo id="S2.SS3.p1.9.m9.1.1.3.3a" xref="S2.SS3.p1.9.m9.1.1.3.3.cmml">−</mo><mn id="S2.SS3.p1.9.m9.1.1.3.3.2" xref="S2.SS3.p1.9.m9.1.1.3.3.2.cmml">5</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.9.m9.1b"><apply id="S2.SS3.p1.9.m9.1.1.cmml" xref="S2.SS3.p1.9.m9.1.1"><times id="S2.SS3.p1.9.m9.1.1.1.cmml" xref="S2.SS3.p1.9.m9.1.1.1"></times><cn id="S2.SS3.p1.9.m9.1.1.2.cmml" type="integer" xref="S2.SS3.p1.9.m9.1.1.2">2</cn><apply id="S2.SS3.p1.9.m9.1.1.3.cmml" xref="S2.SS3.p1.9.m9.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.p1.9.m9.1.1.3.1.cmml" xref="S2.SS3.p1.9.m9.1.1.3">superscript</csymbol><cn id="S2.SS3.p1.9.m9.1.1.3.2.cmml" type="integer" xref="S2.SS3.p1.9.m9.1.1.3.2">10</cn><apply id="S2.SS3.p1.9.m9.1.1.3.3.cmml" xref="S2.SS3.p1.9.m9.1.1.3.3"><minus id="S2.SS3.p1.9.m9.1.1.3.3.1.cmml" xref="S2.SS3.p1.9.m9.1.1.3.3"></minus><cn id="S2.SS3.p1.9.m9.1.1.3.3.2.cmml" type="integer" xref="S2.SS3.p1.9.m9.1.1.3.3.2">5</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.9.m9.1c">2\times 10^{-5}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.9.m9.1d">2 × 10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT</annotation></semantics></math>Hz. This figure shows that at 30AU the different sources of acceleration are only an order of magnitude larger than the sensitivity of the GRS at <math alttext="10^{-4}\," class="ltx_Math" display="inline" id="S2.SS3.p1.10.m10.1"><semantics id="S2.SS3.p1.10.m10.1a"><msup id="S2.SS3.p1.10.m10.1.1" xref="S2.SS3.p1.10.m10.1.1.cmml"><mn id="S2.SS3.p1.10.m10.1.1.2" xref="S2.SS3.p1.10.m10.1.1.2.cmml">10</mn><mrow id="S2.SS3.p1.10.m10.1.1.3" xref="S2.SS3.p1.10.m10.1.1.3.cmml"><mo id="S2.SS3.p1.10.m10.1.1.3a" xref="S2.SS3.p1.10.m10.1.1.3.cmml">−</mo><mn id="S2.SS3.p1.10.m10.1.1.3.2" xref="S2.SS3.p1.10.m10.1.1.3.2.cmml">4</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.10.m10.1b"><apply id="S2.SS3.p1.10.m10.1.1.cmml" xref="S2.SS3.p1.10.m10.1.1"><csymbol cd="ambiguous" id="S2.SS3.p1.10.m10.1.1.1.cmml" xref="S2.SS3.p1.10.m10.1.1">superscript</csymbol><cn id="S2.SS3.p1.10.m10.1.1.2.cmml" type="integer" xref="S2.SS3.p1.10.m10.1.1.2">10</cn><apply id="S2.SS3.p1.10.m10.1.1.3.cmml" xref="S2.SS3.p1.10.m10.1.1.3"><minus id="S2.SS3.p1.10.m10.1.1.3.1.cmml" xref="S2.SS3.p1.10.m10.1.1.3"></minus><cn id="S2.SS3.p1.10.m10.1.1.3.2.cmml" type="integer" xref="S2.SS3.p1.10.m10.1.1.3.2">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.10.m10.1c">10^{-4}\,</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.10.m10.1d">10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT</annotation></semantics></math>Hz. Furthermore, since the dominant accelerations are radial with respect to the Sun, additional geometric cancellation is likely when optimizing the spacecraft orientations. This motivates our overall direction of considering an outer Solar System instrument without a precise accelerometer.</p> </div> <div class="ltx_para" id="S2.SS3.p2"> <p class="ltx_p" id="S2.SS3.p2.1">In what follows, we discuss each source of acceleration, ordered roughly by their importance.</p> </div> <figure class="ltx_figure" id="S2.F2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="332" id="S2.F2.g1" src="x2.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 2: </span>Square root of the power spectrum (the amplitude spectral density) of the most important acceleration sources. The calculations assume a spacecraft at a solar distance of <math alttext="r=10" class="ltx_Math" display="inline" id="S2.F2.7.m1.1"><semantics id="S2.F2.7.m1.1b"><mrow id="S2.F2.7.m1.1.1" xref="S2.F2.7.m1.1.1.cmml"><mi id="S2.F2.7.m1.1.1.2" xref="S2.F2.7.m1.1.1.2.cmml">r</mi><mo id="S2.F2.7.m1.1.1.1" xref="S2.F2.7.m1.1.1.1.cmml">=</mo><mn id="S2.F2.7.m1.1.1.3" xref="S2.F2.7.m1.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.F2.7.m1.1c"><apply id="S2.F2.7.m1.1.1.cmml" xref="S2.F2.7.m1.1.1"><eq id="S2.F2.7.m1.1.1.1.cmml" xref="S2.F2.7.m1.1.1.1"></eq><ci id="S2.F2.7.m1.1.1.2.cmml" xref="S2.F2.7.m1.1.1.2">𝑟</ci><cn id="S2.F2.7.m1.1.1.3.cmml" type="integer" xref="S2.F2.7.m1.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.7.m1.1d">r=10</annotation><annotation encoding="application/x-llamapun" id="S2.F2.7.m1.1e">italic_r = 10</annotation></semantics></math> AU (left panel) and <math alttext="r=30~{}" class="ltx_Math" display="inline" id="S2.F2.8.m2.1"><semantics id="S2.F2.8.m2.1b"><mrow id="S2.F2.8.m2.1.1" xref="S2.F2.8.m2.1.1.cmml"><mi id="S2.F2.8.m2.1.1.2" xref="S2.F2.8.m2.1.1.2.cmml">r</mi><mo id="S2.F2.8.m2.1.1.1" xref="S2.F2.8.m2.1.1.1.cmml">=</mo><mn id="S2.F2.8.m2.1.1.3" xref="S2.F2.8.m2.1.1.3.cmml">30</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.F2.8.m2.1c"><apply id="S2.F2.8.m2.1.1.cmml" xref="S2.F2.8.m2.1.1"><eq id="S2.F2.8.m2.1.1.1.cmml" xref="S2.F2.8.m2.1.1.1"></eq><ci id="S2.F2.8.m2.1.1.2.cmml" xref="S2.F2.8.m2.1.1.2">𝑟</ci><cn id="S2.F2.8.m2.1.1.3.cmml" type="integer" xref="S2.F2.8.m2.1.1.3">30</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.8.m2.1d">r=30~{}</annotation><annotation encoding="application/x-llamapun" id="S2.F2.8.m2.1e">italic_r = 30</annotation></semantics></math>AU (right panel) with a mass of <math alttext="M=10^{3}" class="ltx_Math" display="inline" id="S2.F2.9.m3.1"><semantics id="S2.F2.9.m3.1b"><mrow id="S2.F2.9.m3.1.1" xref="S2.F2.9.m3.1.1.cmml"><mi id="S2.F2.9.m3.1.1.2" xref="S2.F2.9.m3.1.1.2.cmml">M</mi><mo id="S2.F2.9.m3.1.1.1" xref="S2.F2.9.m3.1.1.1.cmml">=</mo><msup id="S2.F2.9.m3.1.1.3" xref="S2.F2.9.m3.1.1.3.cmml"><mn id="S2.F2.9.m3.1.1.3.2" xref="S2.F2.9.m3.1.1.3.2.cmml">10</mn><mn id="S2.F2.9.m3.1.1.3.3" xref="S2.F2.9.m3.1.1.3.3.cmml">3</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.F2.9.m3.1c"><apply id="S2.F2.9.m3.1.1.cmml" xref="S2.F2.9.m3.1.1"><eq id="S2.F2.9.m3.1.1.1.cmml" xref="S2.F2.9.m3.1.1.1"></eq><ci id="S2.F2.9.m3.1.1.2.cmml" xref="S2.F2.9.m3.1.1.2">𝑀</ci><apply id="S2.F2.9.m3.1.1.3.cmml" xref="S2.F2.9.m3.1.1.3"><csymbol cd="ambiguous" id="S2.F2.9.m3.1.1.3.1.cmml" xref="S2.F2.9.m3.1.1.3">superscript</csymbol><cn id="S2.F2.9.m3.1.1.3.2.cmml" type="integer" xref="S2.F2.9.m3.1.1.3.2">10</cn><cn id="S2.F2.9.m3.1.1.3.3.cmml" type="integer" xref="S2.F2.9.m3.1.1.3.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.9.m3.1d">M=10^{3}</annotation><annotation encoding="application/x-llamapun" id="S2.F2.9.m3.1e">italic_M = 10 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT</annotation></semantics></math>kg and an effective area of <math alttext="A_{\rm eff}=10\;" class="ltx_Math" display="inline" id="S2.F2.10.m4.1"><semantics id="S2.F2.10.m4.1b"><mrow id="S2.F2.10.m4.1.1" xref="S2.F2.10.m4.1.1.cmml"><msub id="S2.F2.10.m4.1.1.2" xref="S2.F2.10.m4.1.1.2.cmml"><mi id="S2.F2.10.m4.1.1.2.2" xref="S2.F2.10.m4.1.1.2.2.cmml">A</mi><mi id="S2.F2.10.m4.1.1.2.3" xref="S2.F2.10.m4.1.1.2.3.cmml">eff</mi></msub><mo id="S2.F2.10.m4.1.1.1" xref="S2.F2.10.m4.1.1.1.cmml">=</mo><mn id="S2.F2.10.m4.1.1.3" xref="S2.F2.10.m4.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.F2.10.m4.1c"><apply id="S2.F2.10.m4.1.1.cmml" xref="S2.F2.10.m4.1.1"><eq id="S2.F2.10.m4.1.1.1.cmml" xref="S2.F2.10.m4.1.1.1"></eq><apply id="S2.F2.10.m4.1.1.2.cmml" xref="S2.F2.10.m4.1.1.2"><csymbol cd="ambiguous" id="S2.F2.10.m4.1.1.2.1.cmml" xref="S2.F2.10.m4.1.1.2">subscript</csymbol><ci id="S2.F2.10.m4.1.1.2.2.cmml" xref="S2.F2.10.m4.1.1.2.2">𝐴</ci><ci id="S2.F2.10.m4.1.1.2.3.cmml" xref="S2.F2.10.m4.1.1.2.3">eff</ci></apply><cn id="S2.F2.10.m4.1.1.3.cmml" type="integer" xref="S2.F2.10.m4.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.10.m4.1d">A_{\rm eff}=10\;</annotation><annotation encoding="application/x-llamapun" id="S2.F2.10.m4.1e">italic_A start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT = 10</annotation></semantics></math>m<sup class="ltx_sup" id="S2.F2.14.1">2</sup>. As irradiance and solar wind are directed radially, a reduction in these forces can be achieved by orienting the arms to be more perpendicular to these radial flows. Two different maximum dust masses are shown, reflecting uncertainties in the measured distribution. The contamination of accelerations from <math alttext="m_{\rm d,max}>10^{-9}" class="ltx_Math" display="inline" id="S2.F2.12.m6.2"><semantics id="S2.F2.12.m6.2b"><mrow id="S2.F2.12.m6.2.3" xref="S2.F2.12.m6.2.3.cmml"><msub id="S2.F2.12.m6.2.3.2" xref="S2.F2.12.m6.2.3.2.cmml"><mi id="S2.F2.12.m6.2.3.2.2" xref="S2.F2.12.m6.2.3.2.2.cmml">m</mi><mrow id="S2.F2.12.m6.2.2.2.4" xref="S2.F2.12.m6.2.2.2.3.cmml"><mi id="S2.F2.12.m6.1.1.1.1" mathvariant="normal" xref="S2.F2.12.m6.1.1.1.1.cmml">d</mi><mo id="S2.F2.12.m6.2.2.2.4.1" xref="S2.F2.12.m6.2.2.2.3.cmml">,</mo><mi id="S2.F2.12.m6.2.2.2.2" xref="S2.F2.12.m6.2.2.2.2.cmml">max</mi></mrow></msub><mo id="S2.F2.12.m6.2.3.1" xref="S2.F2.12.m6.2.3.1.cmml">></mo><msup id="S2.F2.12.m6.2.3.3" xref="S2.F2.12.m6.2.3.3.cmml"><mn id="S2.F2.12.m6.2.3.3.2" xref="S2.F2.12.m6.2.3.3.2.cmml">10</mn><mrow id="S2.F2.12.m6.2.3.3.3" xref="S2.F2.12.m6.2.3.3.3.cmml"><mo id="S2.F2.12.m6.2.3.3.3b" xref="S2.F2.12.m6.2.3.3.3.cmml">−</mo><mn id="S2.F2.12.m6.2.3.3.3.2" xref="S2.F2.12.m6.2.3.3.3.2.cmml">9</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.F2.12.m6.2c"><apply id="S2.F2.12.m6.2.3.cmml" xref="S2.F2.12.m6.2.3"><gt id="S2.F2.12.m6.2.3.1.cmml" xref="S2.F2.12.m6.2.3.1"></gt><apply id="S2.F2.12.m6.2.3.2.cmml" xref="S2.F2.12.m6.2.3.2"><csymbol cd="ambiguous" id="S2.F2.12.m6.2.3.2.1.cmml" xref="S2.F2.12.m6.2.3.2">subscript</csymbol><ci id="S2.F2.12.m6.2.3.2.2.cmml" xref="S2.F2.12.m6.2.3.2.2">𝑚</ci><list id="S2.F2.12.m6.2.2.2.3.cmml" xref="S2.F2.12.m6.2.2.2.4"><ci id="S2.F2.12.m6.1.1.1.1.cmml" xref="S2.F2.12.m6.1.1.1.1">d</ci><ci id="S2.F2.12.m6.2.2.2.2.cmml" xref="S2.F2.12.m6.2.2.2.2">max</ci></list></apply><apply id="S2.F2.12.m6.2.3.3.cmml" xref="S2.F2.12.m6.2.3.3"><csymbol cd="ambiguous" id="S2.F2.12.m6.2.3.3.1.cmml" xref="S2.F2.12.m6.2.3.3">superscript</csymbol><cn id="S2.F2.12.m6.2.3.3.2.cmml" type="integer" xref="S2.F2.12.m6.2.3.3.2">10</cn><apply id="S2.F2.12.m6.2.3.3.3.cmml" xref="S2.F2.12.m6.2.3.3.3"><minus id="S2.F2.12.m6.2.3.3.3.1.cmml" xref="S2.F2.12.m6.2.3.3.3"></minus><cn id="S2.F2.12.m6.2.3.3.3.2.cmml" type="integer" xref="S2.F2.12.m6.2.3.3.3.2">9</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.12.m6.2d">m_{\rm d,max}>10^{-9}</annotation><annotation encoding="application/x-llamapun" id="S2.F2.12.m6.2e">italic_m start_POSTSUBSCRIPT roman_d , roman_max end_POSTSUBSCRIPT > 10 start_POSTSUPERSCRIPT - 9 end_POSTSUPERSCRIPT</annotation></semantics></math>g grains likely can be cleaned. Also shown are the acceleration goal of the LISA Gravitational Reference Sensor (GRS).</figcaption> </figure> <section class="ltx_subsubsection" id="S2.SS3.SSS1"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">2.3.1 </span>Solar irradiance</h4> <div class="ltx_para" id="S2.SS3.SSS1.p1"> <p class="ltx_p" id="S2.SS3.SSS1.p1.3">During active periods, the Sun shows 0.2% peak-to-peak irradiance variations on the timescale of its 27-day rotation period plus fluctuations over a broad range of timescales. Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.F2" title="Figure 2 ‣ 2.3 Acceleration noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">2</span></a> shows the power spectrum of irradiance variations measured using the Variability of Solar Irradiance and Gravity Oscillations instrument on the Solar and Heliospheric Observatory <cite class="ltx_cite ltx_citemacro_citep">(VIRGO/SOHO; Fröhlich & Lean, <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib32" title="">2004</a>)</cite>, converting the radiation force to spacecraft acceleration.<span class="ltx_note ltx_role_footnote" id="footnote7"><sup class="ltx_note_mark">7</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">7</sup><span class="ltx_tag ltx_tag_note">7</span>We maintain the major features in the power spectrum using the data from Figure 12 in <cite class="ltx_cite ltx_citemacro_citet">Fröhlich & Lean (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib32" title="">2004</a>)</cite>, but we omit much of the fine-grained structure in this power spectrum at <math alttext="f>10^{-5}" class="ltx_Math" display="inline" id="footnote7.m1.1"><semantics id="footnote7.m1.1b"><mrow id="footnote7.m1.1.1" xref="footnote7.m1.1.1.cmml"><mi id="footnote7.m1.1.1.2" xref="footnote7.m1.1.1.2.cmml">f</mi><mo id="footnote7.m1.1.1.1" xref="footnote7.m1.1.1.1.cmml">></mo><msup id="footnote7.m1.1.1.3" xref="footnote7.m1.1.1.3.cmml"><mn id="footnote7.m1.1.1.3.2" xref="footnote7.m1.1.1.3.2.cmml">10</mn><mrow id="footnote7.m1.1.1.3.3" xref="footnote7.m1.1.1.3.3.cmml"><mo id="footnote7.m1.1.1.3.3b" xref="footnote7.m1.1.1.3.3.cmml">−</mo><mn id="footnote7.m1.1.1.3.3.2" xref="footnote7.m1.1.1.3.3.2.cmml">5</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="footnote7.m1.1c"><apply id="footnote7.m1.1.1.cmml" xref="footnote7.m1.1.1"><gt id="footnote7.m1.1.1.1.cmml" xref="footnote7.m1.1.1.1"></gt><ci id="footnote7.m1.1.1.2.cmml" xref="footnote7.m1.1.1.2">𝑓</ci><apply id="footnote7.m1.1.1.3.cmml" xref="footnote7.m1.1.1.3"><csymbol cd="ambiguous" id="footnote7.m1.1.1.3.1.cmml" xref="footnote7.m1.1.1.3">superscript</csymbol><cn id="footnote7.m1.1.1.3.2.cmml" type="integer" xref="footnote7.m1.1.1.3.2">10</cn><apply id="footnote7.m1.1.1.3.3.cmml" xref="footnote7.m1.1.1.3.3"><minus id="footnote7.m1.1.1.3.3.1.cmml" xref="footnote7.m1.1.1.3.3"></minus><cn id="footnote7.m1.1.1.3.3.2.cmml" type="integer" xref="footnote7.m1.1.1.3.3.2">5</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote7.m1.1d">f>10^{-5}</annotation><annotation encoding="application/x-llamapun" id="footnote7.m1.1e">italic_f > 10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT</annotation></semantics></math>Hz. Note that the effective area used for the conversion from irradiance to acceleration (<math alttext="A_{\rm eff}=10" class="ltx_Math" display="inline" id="footnote7.m2.1"><semantics id="footnote7.m2.1b"><mrow id="footnote7.m2.1.1" xref="footnote7.m2.1.1.cmml"><msub id="footnote7.m2.1.1.2" xref="footnote7.m2.1.1.2.cmml"><mi id="footnote7.m2.1.1.2.2" xref="footnote7.m2.1.1.2.2.cmml">A</mi><mi id="footnote7.m2.1.1.2.3" xref="footnote7.m2.1.1.2.3.cmml">eff</mi></msub><mo id="footnote7.m2.1.1.1" xref="footnote7.m2.1.1.1.cmml">=</mo><mn id="footnote7.m2.1.1.3" xref="footnote7.m2.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="footnote7.m2.1c"><apply id="footnote7.m2.1.1.cmml" xref="footnote7.m2.1.1"><eq id="footnote7.m2.1.1.1.cmml" xref="footnote7.m2.1.1.1"></eq><apply id="footnote7.m2.1.1.2.cmml" xref="footnote7.m2.1.1.2"><csymbol cd="ambiguous" id="footnote7.m2.1.1.2.1.cmml" xref="footnote7.m2.1.1.2">subscript</csymbol><ci id="footnote7.m2.1.1.2.2.cmml" xref="footnote7.m2.1.1.2.2">𝐴</ci><ci id="footnote7.m2.1.1.2.3.cmml" xref="footnote7.m2.1.1.2.3">eff</ci></apply><cn id="footnote7.m2.1.1.3.cmml" type="integer" xref="footnote7.m2.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote7.m2.1d">A_{\rm eff}=10</annotation><annotation encoding="application/x-llamapun" id="footnote7.m2.1e">italic_A start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT = 10</annotation></semantics></math>m<sup class="ltx_sup" id="footnote7.1">2</sup>) would include the factor of two for a reflective surface directed at the Sun.</span></span></span> We use measurements over <math alttext="1996-1997" class="ltx_Math" display="inline" id="S2.SS3.SSS1.p1.1.m1.1"><semantics id="S2.SS3.SSS1.p1.1.m1.1a"><mrow id="S2.SS3.SSS1.p1.1.m1.1.1" xref="S2.SS3.SSS1.p1.1.m1.1.1.cmml"><mn id="S2.SS3.SSS1.p1.1.m1.1.1.2" xref="S2.SS3.SSS1.p1.1.m1.1.1.2.cmml">1996</mn><mo id="S2.SS3.SSS1.p1.1.m1.1.1.1" xref="S2.SS3.SSS1.p1.1.m1.1.1.1.cmml">−</mo><mn id="S2.SS3.SSS1.p1.1.m1.1.1.3" xref="S2.SS3.SSS1.p1.1.m1.1.1.3.cmml">1997</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS1.p1.1.m1.1b"><apply id="S2.SS3.SSS1.p1.1.m1.1.1.cmml" xref="S2.SS3.SSS1.p1.1.m1.1.1"><minus id="S2.SS3.SSS1.p1.1.m1.1.1.1.cmml" xref="S2.SS3.SSS1.p1.1.m1.1.1.1"></minus><cn id="S2.SS3.SSS1.p1.1.m1.1.1.2.cmml" type="integer" xref="S2.SS3.SSS1.p1.1.m1.1.1.2">1996</cn><cn id="S2.SS3.SSS1.p1.1.m1.1.1.3.cmml" type="integer" xref="S2.SS3.SSS1.p1.1.m1.1.1.3">1997</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS1.p1.1.m1.1c">1996-1997</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS1.p1.1.m1.1d">1996 - 1997</annotation></semantics></math> that were near the minimum of solar activity <cite class="ltx_cite ltx_citemacro_citep">(Fröhlich & Lean, <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib32" title="">2004</a>)</cite>. The amplitude of the square root of the irradiance power spectrum varies by a factor of three at <math alttext="0.1-10\,\mu" class="ltx_Math" display="inline" id="S2.SS3.SSS1.p1.2.m2.1"><semantics id="S2.SS3.SSS1.p1.2.m2.1a"><mrow id="S2.SS3.SSS1.p1.2.m2.1.1" xref="S2.SS3.SSS1.p1.2.m2.1.1.cmml"><mn id="S2.SS3.SSS1.p1.2.m2.1.1.2" xref="S2.SS3.SSS1.p1.2.m2.1.1.2.cmml">0.1</mn><mo id="S2.SS3.SSS1.p1.2.m2.1.1.1" xref="S2.SS3.SSS1.p1.2.m2.1.1.1.cmml">−</mo><mrow id="S2.SS3.SSS1.p1.2.m2.1.1.3" xref="S2.SS3.SSS1.p1.2.m2.1.1.3.cmml"><mn id="S2.SS3.SSS1.p1.2.m2.1.1.3.2" xref="S2.SS3.SSS1.p1.2.m2.1.1.3.2.cmml">10</mn><mo id="S2.SS3.SSS1.p1.2.m2.1.1.3.1" lspace="0.170em" xref="S2.SS3.SSS1.p1.2.m2.1.1.3.1.cmml">⁢</mo><mi id="S2.SS3.SSS1.p1.2.m2.1.1.3.3" xref="S2.SS3.SSS1.p1.2.m2.1.1.3.3.cmml">μ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS1.p1.2.m2.1b"><apply id="S2.SS3.SSS1.p1.2.m2.1.1.cmml" xref="S2.SS3.SSS1.p1.2.m2.1.1"><minus id="S2.SS3.SSS1.p1.2.m2.1.1.1.cmml" xref="S2.SS3.SSS1.p1.2.m2.1.1.1"></minus><cn id="S2.SS3.SSS1.p1.2.m2.1.1.2.cmml" type="float" xref="S2.SS3.SSS1.p1.2.m2.1.1.2">0.1</cn><apply id="S2.SS3.SSS1.p1.2.m2.1.1.3.cmml" xref="S2.SS3.SSS1.p1.2.m2.1.1.3"><times id="S2.SS3.SSS1.p1.2.m2.1.1.3.1.cmml" xref="S2.SS3.SSS1.p1.2.m2.1.1.3.1"></times><cn id="S2.SS3.SSS1.p1.2.m2.1.1.3.2.cmml" type="integer" xref="S2.SS3.SSS1.p1.2.m2.1.1.3.2">10</cn><ci id="S2.SS3.SSS1.p1.2.m2.1.1.3.3.cmml" xref="S2.SS3.SSS1.p1.2.m2.1.1.3.3">𝜇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS1.p1.2.m2.1c">0.1-10\,\mu</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS1.p1.2.m2.1d">0.1 - 10 italic_μ</annotation></semantics></math>Hz over the eleven-year solar cycle, with smaller variations at <math alttext=">10~{}\mu" class="ltx_Math" display="inline" id="S2.SS3.SSS1.p1.3.m3.1"><semantics id="S2.SS3.SSS1.p1.3.m3.1a"><mrow id="S2.SS3.SSS1.p1.3.m3.1.1" xref="S2.SS3.SSS1.p1.3.m3.1.1.cmml"><mi id="S2.SS3.SSS1.p1.3.m3.1.1.2" xref="S2.SS3.SSS1.p1.3.m3.1.1.2.cmml"></mi><mo id="S2.SS3.SSS1.p1.3.m3.1.1.1" xref="S2.SS3.SSS1.p1.3.m3.1.1.1.cmml">></mo><mrow id="S2.SS3.SSS1.p1.3.m3.1.1.3" xref="S2.SS3.SSS1.p1.3.m3.1.1.3.cmml"><mn id="S2.SS3.SSS1.p1.3.m3.1.1.3.2" xref="S2.SS3.SSS1.p1.3.m3.1.1.3.2.cmml">10</mn><mo id="S2.SS3.SSS1.p1.3.m3.1.1.3.1" lspace="0.330em" xref="S2.SS3.SSS1.p1.3.m3.1.1.3.1.cmml">⁢</mo><mi id="S2.SS3.SSS1.p1.3.m3.1.1.3.3" xref="S2.SS3.SSS1.p1.3.m3.1.1.3.3.cmml">μ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS1.p1.3.m3.1b"><apply id="S2.SS3.SSS1.p1.3.m3.1.1.cmml" xref="S2.SS3.SSS1.p1.3.m3.1.1"><gt id="S2.SS3.SSS1.p1.3.m3.1.1.1.cmml" xref="S2.SS3.SSS1.p1.3.m3.1.1.1"></gt><csymbol cd="latexml" id="S2.SS3.SSS1.p1.3.m3.1.1.2.cmml" xref="S2.SS3.SSS1.p1.3.m3.1.1.2">absent</csymbol><apply id="S2.SS3.SSS1.p1.3.m3.1.1.3.cmml" xref="S2.SS3.SSS1.p1.3.m3.1.1.3"><times id="S2.SS3.SSS1.p1.3.m3.1.1.3.1.cmml" xref="S2.SS3.SSS1.p1.3.m3.1.1.3.1"></times><cn id="S2.SS3.SSS1.p1.3.m3.1.1.3.2.cmml" type="integer" xref="S2.SS3.SSS1.p1.3.m3.1.1.3.2">10</cn><ci id="S2.SS3.SSS1.p1.3.m3.1.1.3.3.cmml" xref="S2.SS3.SSS1.p1.3.m3.1.1.3.3">𝜇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS1.p1.3.m3.1c">>10~{}\mu</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS1.p1.3.m3.1d">> 10 italic_μ</annotation></semantics></math>Hz.</p> </div> </section> <section class="ltx_subsubsection" id="S2.SS3.SSS2"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">2.3.2 </span>Solar wind drag</h4> <div class="ltx_para" id="S2.SS3.SSS2.p1"> <p class="ltx_p" id="S2.SS3.SSS2.p1.3">We find that drag from the solar wind is the other important source of accelerations in addition to irradiance variations. While the mean drag force scales as <math alttext="r^{-2}" class="ltx_Math" display="inline" id="S2.SS3.SSS2.p1.1.m1.1"><semantics id="S2.SS3.SSS2.p1.1.m1.1a"><msup id="S2.SS3.SSS2.p1.1.m1.1.1" xref="S2.SS3.SSS2.p1.1.m1.1.1.cmml"><mi id="S2.SS3.SSS2.p1.1.m1.1.1.2" xref="S2.SS3.SSS2.p1.1.m1.1.1.2.cmml">r</mi><mrow id="S2.SS3.SSS2.p1.1.m1.1.1.3" xref="S2.SS3.SSS2.p1.1.m1.1.1.3.cmml"><mo id="S2.SS3.SSS2.p1.1.m1.1.1.3a" xref="S2.SS3.SSS2.p1.1.m1.1.1.3.cmml">−</mo><mn id="S2.SS3.SSS2.p1.1.m1.1.1.3.2" xref="S2.SS3.SSS2.p1.1.m1.1.1.3.2.cmml">2</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS2.p1.1.m1.1b"><apply id="S2.SS3.SSS2.p1.1.m1.1.1.cmml" xref="S2.SS3.SSS2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS2.p1.1.m1.1.1.1.cmml" xref="S2.SS3.SSS2.p1.1.m1.1.1">superscript</csymbol><ci id="S2.SS3.SSS2.p1.1.m1.1.1.2.cmml" xref="S2.SS3.SSS2.p1.1.m1.1.1.2">𝑟</ci><apply id="S2.SS3.SSS2.p1.1.m1.1.1.3.cmml" xref="S2.SS3.SSS2.p1.1.m1.1.1.3"><minus id="S2.SS3.SSS2.p1.1.m1.1.1.3.1.cmml" xref="S2.SS3.SSS2.p1.1.m1.1.1.3"></minus><cn id="S2.SS3.SSS2.p1.1.m1.1.1.3.2.cmml" type="integer" xref="S2.SS3.SSS2.p1.1.m1.1.1.3.2">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS2.p1.1.m1.1c">r^{-2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS2.p1.1.m1.1d">italic_r start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT</annotation></semantics></math> with distance from the Sun, the inhomogeneous component falls off somewhat less quickly at <math alttext="r\gtrsim 10\," class="ltx_Math" display="inline" id="S2.SS3.SSS2.p1.2.m2.1"><semantics id="S2.SS3.SSS2.p1.2.m2.1a"><mrow id="S2.SS3.SSS2.p1.2.m2.1.1" xref="S2.SS3.SSS2.p1.2.m2.1.1.cmml"><mi id="S2.SS3.SSS2.p1.2.m2.1.1.2" xref="S2.SS3.SSS2.p1.2.m2.1.1.2.cmml">r</mi><mo id="S2.SS3.SSS2.p1.2.m2.1.1.1" xref="S2.SS3.SSS2.p1.2.m2.1.1.1.cmml">≳</mo><mn id="S2.SS3.SSS2.p1.2.m2.1.1.3" xref="S2.SS3.SSS2.p1.2.m2.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS2.p1.2.m2.1b"><apply id="S2.SS3.SSS2.p1.2.m2.1.1.cmml" xref="S2.SS3.SSS2.p1.2.m2.1.1"><csymbol cd="latexml" id="S2.SS3.SSS2.p1.2.m2.1.1.1.cmml" xref="S2.SS3.SSS2.p1.2.m2.1.1.1">greater-than-or-equivalent-to</csymbol><ci id="S2.SS3.SSS2.p1.2.m2.1.1.2.cmml" xref="S2.SS3.SSS2.p1.2.m2.1.1.2">𝑟</ci><cn id="S2.SS3.SSS2.p1.2.m2.1.1.3.cmml" type="integer" xref="S2.SS3.SSS2.p1.2.m2.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS2.p1.2.m2.1c">r\gtrsim 10\,</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS2.p1.2.m2.1d">italic_r ≳ 10</annotation></semantics></math>AU owing to the amplitude of the fractional density fluctuations growing with <math alttext="r" class="ltx_Math" display="inline" id="S2.SS3.SSS2.p1.3.m3.1"><semantics id="S2.SS3.SSS2.p1.3.m3.1a"><mi id="S2.SS3.SSS2.p1.3.m3.1.1" xref="S2.SS3.SSS2.p1.3.m3.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS2.p1.3.m3.1b"><ci id="S2.SS3.SSS2.p1.3.m3.1.1.cmml" xref="S2.SS3.SSS2.p1.3.m3.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS2.p1.3.m3.1c">r</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS2.p1.3.m3.1d">italic_r</annotation></semantics></math>. From Fourier transforming the time series along its trajectory, Voyager 2 finds a 1D power spectrum of electron density fluctuations with the approximate form</p> <table class="ltx_equation ltx_eqn_table" id="S2.E14"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="P_{e}^{\rm 1D}(f,r)\approx P_{e,0}^{\rm 1D}\left(\frac{f}{10^{-5}{\rm Hz}}% \right)^{-\beta}\left(\frac{r}{30\rm AU}\right)^{-3}," class="ltx_Math" display="block" id="S2.E14.m1.7"><semantics id="S2.E14.m1.7a"><mrow id="S2.E14.m1.7.7.1" xref="S2.E14.m1.7.7.1.1.cmml"><mrow id="S2.E14.m1.7.7.1.1" xref="S2.E14.m1.7.7.1.1.cmml"><mrow id="S2.E14.m1.7.7.1.1.2" xref="S2.E14.m1.7.7.1.1.2.cmml"><msubsup id="S2.E14.m1.7.7.1.1.2.2" xref="S2.E14.m1.7.7.1.1.2.2.cmml"><mi id="S2.E14.m1.7.7.1.1.2.2.2.2" xref="S2.E14.m1.7.7.1.1.2.2.2.2.cmml">P</mi><mi id="S2.E14.m1.7.7.1.1.2.2.2.3" 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id="S2.E14.m1.7.7.1.1.3.3.3.cmml" xref="S2.E14.m1.7.7.1.1.3.3.3"><minus id="S2.E14.m1.7.7.1.1.3.3.3.1.cmml" xref="S2.E14.m1.7.7.1.1.3.3.3"></minus><ci id="S2.E14.m1.7.7.1.1.3.3.3.2.cmml" xref="S2.E14.m1.7.7.1.1.3.3.3.2">𝛽</ci></apply></apply><apply id="S2.E14.m1.7.7.1.1.3.4.cmml" xref="S2.E14.m1.7.7.1.1.3.4"><csymbol cd="ambiguous" id="S2.E14.m1.7.7.1.1.3.4.1.cmml" xref="S2.E14.m1.7.7.1.1.3.4">superscript</csymbol><apply id="S2.E14.m1.6.6.cmml" xref="S2.E14.m1.7.7.1.1.3.4.2.2"><divide id="S2.E14.m1.6.6.1.cmml" xref="S2.E14.m1.7.7.1.1.3.4.2.2"></divide><ci id="S2.E14.m1.6.6.2.cmml" xref="S2.E14.m1.6.6.2">𝑟</ci><apply id="S2.E14.m1.6.6.3.cmml" xref="S2.E14.m1.6.6.3"><times id="S2.E14.m1.6.6.3.1.cmml" xref="S2.E14.m1.6.6.3.1"></times><cn id="S2.E14.m1.6.6.3.2.cmml" type="integer" xref="S2.E14.m1.6.6.3.2">30</cn><ci id="S2.E14.m1.6.6.3.3.cmml" xref="S2.E14.m1.6.6.3.3">A</ci><ci id="S2.E14.m1.6.6.3.4.cmml" xref="S2.E14.m1.6.6.3.4">U</ci></apply></apply><apply id="S2.E14.m1.7.7.1.1.3.4.3.cmml" xref="S2.E14.m1.7.7.1.1.3.4.3"><minus id="S2.E14.m1.7.7.1.1.3.4.3.1.cmml" xref="S2.E14.m1.7.7.1.1.3.4.3"></minus><cn id="S2.E14.m1.7.7.1.1.3.4.3.2.cmml" type="integer" xref="S2.E14.m1.7.7.1.1.3.4.3.2">3</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E14.m1.7c">P_{e}^{\rm 1D}(f,r)\approx P_{e,0}^{\rm 1D}\left(\frac{f}{10^{-5}{\rm Hz}}% \right)^{-\beta}\left(\frac{r}{30\rm AU}\right)^{-3},</annotation><annotation encoding="application/x-llamapun" id="S2.E14.m1.7d">italic_P start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 roman_D end_POSTSUPERSCRIPT ( italic_f , italic_r ) ≈ italic_P start_POSTSUBSCRIPT italic_e , 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 roman_D end_POSTSUPERSCRIPT ( divide start_ARG italic_f end_ARG start_ARG 10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT roman_Hz end_ARG ) start_POSTSUPERSCRIPT - italic_β end_POSTSUPERSCRIPT ( divide start_ARG italic_r end_ARG start_ARG 30 roman_A roman_U end_ARG ) start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(14)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS3.SSS2.p1.6">where <math alttext="P_{e,0}^{\rm 1D}\approx 10^{11.1}~{}\text{m${}^{-6}$ Hz${}^{-1}$}" class="ltx_Math" display="inline" id="S2.SS3.SSS2.p1.4.m1.4"><semantics id="S2.SS3.SSS2.p1.4.m1.4a"><mrow id="S2.SS3.SSS2.p1.4.m1.4.5" xref="S2.SS3.SSS2.p1.4.m1.4.5.cmml"><msubsup id="S2.SS3.SSS2.p1.4.m1.4.5.2" xref="S2.SS3.SSS2.p1.4.m1.4.5.2.cmml"><mi id="S2.SS3.SSS2.p1.4.m1.4.5.2.2.2" xref="S2.SS3.SSS2.p1.4.m1.4.5.2.2.2.cmml">P</mi><mrow id="S2.SS3.SSS2.p1.4.m1.4.4.2.4" xref="S2.SS3.SSS2.p1.4.m1.4.4.2.3.cmml"><mi id="S2.SS3.SSS2.p1.4.m1.3.3.1.1" xref="S2.SS3.SSS2.p1.4.m1.3.3.1.1.cmml">e</mi><mo id="S2.SS3.SSS2.p1.4.m1.4.4.2.4.1" xref="S2.SS3.SSS2.p1.4.m1.4.4.2.3.cmml">,</mo><mn id="S2.SS3.SSS2.p1.4.m1.4.4.2.2" xref="S2.SS3.SSS2.p1.4.m1.4.4.2.2.cmml">0</mn></mrow><mrow id="S2.SS3.SSS2.p1.4.m1.4.5.2.3" xref="S2.SS3.SSS2.p1.4.m1.4.5.2.3.cmml"><mn id="S2.SS3.SSS2.p1.4.m1.4.5.2.3.2" xref="S2.SS3.SSS2.p1.4.m1.4.5.2.3.2.cmml">1</mn><mo id="S2.SS3.SSS2.p1.4.m1.4.5.2.3.1" xref="S2.SS3.SSS2.p1.4.m1.4.5.2.3.1.cmml">⁢</mo><mi id="S2.SS3.SSS2.p1.4.m1.4.5.2.3.3" mathvariant="normal" xref="S2.SS3.SSS2.p1.4.m1.4.5.2.3.3.cmml">D</mi></mrow></msubsup><mo id="S2.SS3.SSS2.p1.4.m1.4.5.1" xref="S2.SS3.SSS2.p1.4.m1.4.5.1.cmml">≈</mo><mrow id="S2.SS3.SSS2.p1.4.m1.4.5.3" xref="S2.SS3.SSS2.p1.4.m1.4.5.3.cmml"><msup id="S2.SS3.SSS2.p1.4.m1.4.5.3.2" xref="S2.SS3.SSS2.p1.4.m1.4.5.3.2.cmml"><mn id="S2.SS3.SSS2.p1.4.m1.4.5.3.2.2" xref="S2.SS3.SSS2.p1.4.m1.4.5.3.2.2.cmml">10</mn><mn id="S2.SS3.SSS2.p1.4.m1.4.5.3.2.3" xref="S2.SS3.SSS2.p1.4.m1.4.5.3.2.3.cmml">11.1</mn></msup><mo id="S2.SS3.SSS2.p1.4.m1.4.5.3.1" xref="S2.SS3.SSS2.p1.4.m1.4.5.3.1.cmml">⁢</mo><mrow id="S2.SS3.SSS2.p1.4.m1.2.2.2" xref="S2.SS3.SSS2.p1.4.m1.2.2.2i.cmml"><mtext id="S2.SS3.SSS2.p1.4.m1.2.2.2a" xref="S2.SS3.SSS2.p1.4.m1.2.2.2i.cmml">m</mtext><mtext id="S2.SS3.SSS2.p1.4.m1.2.2.2b" xref="S2.SS3.SSS2.p1.4.m1.2.2.2i.cmml"><sup class="ltx_sup" id="S2.SS3.SSS2.p1.4.m1.2.2.2.3nest"><span class="ltx_text ltx_font_italic" id="S2.SS3.SSS2.p1.4.m1.2.2.2.3.1nest">-6</span></sup></mtext><mtext id="S2.SS3.SSS2.p1.4.m1.2.2.2e" xref="S2.SS3.SSS2.p1.4.m1.2.2.2i.cmml"> Hz</mtext><mtext id="S2.SS3.SSS2.p1.4.m1.2.2.2f" xref="S2.SS3.SSS2.p1.4.m1.2.2.2i.cmml"><sup class="ltx_sup" id="S2.SS3.SSS2.p1.4.m1.2.2.2.4nest"><span class="ltx_text ltx_font_italic" id="S2.SS3.SSS2.p1.4.m1.2.2.2.4.1nest">-1</span></sup></mtext></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS2.p1.4.m1.4b"><apply id="S2.SS3.SSS2.p1.4.m1.4.5.cmml" xref="S2.SS3.SSS2.p1.4.m1.4.5"><approx id="S2.SS3.SSS2.p1.4.m1.4.5.1.cmml" xref="S2.SS3.SSS2.p1.4.m1.4.5.1"></approx><apply id="S2.SS3.SSS2.p1.4.m1.4.5.2.cmml" xref="S2.SS3.SSS2.p1.4.m1.4.5.2"><csymbol cd="ambiguous" id="S2.SS3.SSS2.p1.4.m1.4.5.2.1.cmml" xref="S2.SS3.SSS2.p1.4.m1.4.5.2">superscript</csymbol><apply id="S2.SS3.SSS2.p1.4.m1.4.5.2.2.cmml" xref="S2.SS3.SSS2.p1.4.m1.4.5.2"><csymbol cd="ambiguous" id="S2.SS3.SSS2.p1.4.m1.4.5.2.2.1.cmml" xref="S2.SS3.SSS2.p1.4.m1.4.5.2">subscript</csymbol><ci id="S2.SS3.SSS2.p1.4.m1.4.5.2.2.2.cmml" xref="S2.SS3.SSS2.p1.4.m1.4.5.2.2.2">𝑃</ci><list id="S2.SS3.SSS2.p1.4.m1.4.4.2.3.cmml" xref="S2.SS3.SSS2.p1.4.m1.4.4.2.4"><ci id="S2.SS3.SSS2.p1.4.m1.3.3.1.1.cmml" xref="S2.SS3.SSS2.p1.4.m1.3.3.1.1">𝑒</ci><cn id="S2.SS3.SSS2.p1.4.m1.4.4.2.2.cmml" type="integer" xref="S2.SS3.SSS2.p1.4.m1.4.4.2.2">0</cn></list></apply><apply id="S2.SS3.SSS2.p1.4.m1.4.5.2.3.cmml" xref="S2.SS3.SSS2.p1.4.m1.4.5.2.3"><times id="S2.SS3.SSS2.p1.4.m1.4.5.2.3.1.cmml" xref="S2.SS3.SSS2.p1.4.m1.4.5.2.3.1"></times><cn id="S2.SS3.SSS2.p1.4.m1.4.5.2.3.2.cmml" type="integer" xref="S2.SS3.SSS2.p1.4.m1.4.5.2.3.2">1</cn><ci id="S2.SS3.SSS2.p1.4.m1.4.5.2.3.3.cmml" xref="S2.SS3.SSS2.p1.4.m1.4.5.2.3.3">D</ci></apply></apply><apply id="S2.SS3.SSS2.p1.4.m1.4.5.3.cmml" xref="S2.SS3.SSS2.p1.4.m1.4.5.3"><times id="S2.SS3.SSS2.p1.4.m1.4.5.3.1.cmml" xref="S2.SS3.SSS2.p1.4.m1.4.5.3.1"></times><apply id="S2.SS3.SSS2.p1.4.m1.4.5.3.2.cmml" xref="S2.SS3.SSS2.p1.4.m1.4.5.3.2"><csymbol cd="ambiguous" id="S2.SS3.SSS2.p1.4.m1.4.5.3.2.1.cmml" xref="S2.SS3.SSS2.p1.4.m1.4.5.3.2">superscript</csymbol><cn id="S2.SS3.SSS2.p1.4.m1.4.5.3.2.2.cmml" type="integer" xref="S2.SS3.SSS2.p1.4.m1.4.5.3.2.2">10</cn><cn id="S2.SS3.SSS2.p1.4.m1.4.5.3.2.3.cmml" type="float" xref="S2.SS3.SSS2.p1.4.m1.4.5.3.2.3">11.1</cn></apply><ci id="S2.SS3.SSS2.p1.4.m1.2.2.2i.cmml" xref="S2.SS3.SSS2.p1.4.m1.2.2.2"><mrow id="S2.SS3.SSS2.p1.4.m1.2.2.2.cmml" xref="S2.SS3.SSS2.p1.4.m1.2.2.2"><mtext id="S2.SS3.SSS2.p1.4.m1.2.2.2a.cmml" xref="S2.SS3.SSS2.p1.4.m1.2.2.2">m</mtext><mtext id="S2.SS3.SSS2.p1.4.m1.2.2.2b.cmml" xref="S2.SS3.SSS2.p1.4.m1.2.2.2"><sup class="ltx_sup" id="S2.SS3.SSS2.p1.4.m1.2.2.2.3anest"><span class="ltx_text ltx_font_italic" id="S2.SS3.SSS2.p1.4.m1.2.2.2.3.1anest">-6</span></sup></mtext><mtext id="S2.SS3.SSS2.p1.4.m1.2.2.2e.cmml" xref="S2.SS3.SSS2.p1.4.m1.2.2.2"> Hz</mtext><mtext id="S2.SS3.SSS2.p1.4.m1.2.2.2f.cmml" xref="S2.SS3.SSS2.p1.4.m1.2.2.2"><sup class="ltx_sup" id="S2.SS3.SSS2.p1.4.m1.2.2.2.4anest"><span class="ltx_text ltx_font_italic" id="S2.SS3.SSS2.p1.4.m1.2.2.2.4.1anest">-1</span></sup></mtext></mrow></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS2.p1.4.m1.4c">P_{e,0}^{\rm 1D}\approx 10^{11.1}~{}\text{m${}^{-6}$ Hz${}^{-1}$}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS2.p1.4.m1.4d">italic_P start_POSTSUBSCRIPT italic_e , 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 roman_D end_POSTSUPERSCRIPT ≈ 10 start_POSTSUPERSCRIPT 11.1 end_POSTSUPERSCRIPT m Hz</annotation></semantics></math> and the <math alttext="-3" class="ltx_Math" display="inline" id="S2.SS3.SSS2.p1.5.m2.1"><semantics id="S2.SS3.SSS2.p1.5.m2.1a"><mrow id="S2.SS3.SSS2.p1.5.m2.1.1" xref="S2.SS3.SSS2.p1.5.m2.1.1.cmml"><mo id="S2.SS3.SSS2.p1.5.m2.1.1a" xref="S2.SS3.SSS2.p1.5.m2.1.1.cmml">−</mo><mn id="S2.SS3.SSS2.p1.5.m2.1.1.2" xref="S2.SS3.SSS2.p1.5.m2.1.1.2.cmml">3</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS2.p1.5.m2.1b"><apply id="S2.SS3.SSS2.p1.5.m2.1.1.cmml" xref="S2.SS3.SSS2.p1.5.m2.1.1"><minus id="S2.SS3.SSS2.p1.5.m2.1.1.1.cmml" xref="S2.SS3.SSS2.p1.5.m2.1.1"></minus><cn id="S2.SS3.SSS2.p1.5.m2.1.1.2.cmml" type="integer" xref="S2.SS3.SSS2.p1.5.m2.1.1.2">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS2.p1.5.m2.1c">-3</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS2.p1.5.m2.1d">- 3</annotation></semantics></math> index approximates the radial scaling over <math alttext="10-50" class="ltx_Math" display="inline" id="S2.SS3.SSS2.p1.6.m3.1"><semantics id="S2.SS3.SSS2.p1.6.m3.1a"><mrow id="S2.SS3.SSS2.p1.6.m3.1.1" xref="S2.SS3.SSS2.p1.6.m3.1.1.cmml"><mn id="S2.SS3.SSS2.p1.6.m3.1.1.2" xref="S2.SS3.SSS2.p1.6.m3.1.1.2.cmml">10</mn><mo id="S2.SS3.SSS2.p1.6.m3.1.1.1" xref="S2.SS3.SSS2.p1.6.m3.1.1.1.cmml">−</mo><mn id="S2.SS3.SSS2.p1.6.m3.1.1.3" xref="S2.SS3.SSS2.p1.6.m3.1.1.3.cmml">50</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS2.p1.6.m3.1b"><apply id="S2.SS3.SSS2.p1.6.m3.1.1.cmml" xref="S2.SS3.SSS2.p1.6.m3.1.1"><minus id="S2.SS3.SSS2.p1.6.m3.1.1.1.cmml" xref="S2.SS3.SSS2.p1.6.m3.1.1.1"></minus><cn id="S2.SS3.SSS2.p1.6.m3.1.1.2.cmml" type="integer" xref="S2.SS3.SSS2.p1.6.m3.1.1.2">10</cn><cn id="S2.SS3.SSS2.p1.6.m3.1.1.3.cmml" type="integer" xref="S2.SS3.SSS2.p1.6.m3.1.1.3">50</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS2.p1.6.m3.1c">10-50</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS2.p1.6.m3.1d">10 - 50</annotation></semantics></math>AU <cite class="ltx_cite ltx_citemacro_citep">(Bellamy et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib16" title="">2005</a>)</cite>.</p> </div> <div class="ltx_para" id="S2.SS3.SSS2.p2"> <p class="ltx_p" id="S2.SS3.SSS2.p2.4">We can convert from an electron density power to a spacecraft acceleration using</p> <table class="ltx_equation ltx_eqn_table" id="S2.E15"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\sqrt{S_{a}}=\mu\,m_{p}\,v_{\rm sw}^{2}A_{\rm eff}M^{-1}\sqrt{P_{e}^{\rm 1D}(f% ,r)}." class="ltx_Math" display="block" id="S2.E15.m1.3"><semantics id="S2.E15.m1.3a"><mrow id="S2.E15.m1.3.3.1" xref="S2.E15.m1.3.3.1.1.cmml"><mrow id="S2.E15.m1.3.3.1.1" 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end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 roman_D end_POSTSUPERSCRIPT ( italic_f , italic_r ) end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(15)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS3.SSS2.p2.3">If we take our approximate form for the power given by equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.E14" title="In 2.3.2 Solar wind drag ‣ 2.3 Acceleration noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">14</span></a>), a mean molecular weight of <math alttext="\mu=1.2" class="ltx_Math" display="inline" id="S2.SS3.SSS2.p2.1.m1.1"><semantics id="S2.SS3.SSS2.p2.1.m1.1a"><mrow id="S2.SS3.SSS2.p2.1.m1.1.1" xref="S2.SS3.SSS2.p2.1.m1.1.1.cmml"><mi id="S2.SS3.SSS2.p2.1.m1.1.1.2" xref="S2.SS3.SSS2.p2.1.m1.1.1.2.cmml">μ</mi><mo id="S2.SS3.SSS2.p2.1.m1.1.1.1" xref="S2.SS3.SSS2.p2.1.m1.1.1.1.cmml">=</mo><mn id="S2.SS3.SSS2.p2.1.m1.1.1.3" xref="S2.SS3.SSS2.p2.1.m1.1.1.3.cmml">1.2</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS2.p2.1.m1.1b"><apply id="S2.SS3.SSS2.p2.1.m1.1.1.cmml" xref="S2.SS3.SSS2.p2.1.m1.1.1"><eq id="S2.SS3.SSS2.p2.1.m1.1.1.1.cmml" xref="S2.SS3.SSS2.p2.1.m1.1.1.1"></eq><ci id="S2.SS3.SSS2.p2.1.m1.1.1.2.cmml" xref="S2.SS3.SSS2.p2.1.m1.1.1.2">𝜇</ci><cn id="S2.SS3.SSS2.p2.1.m1.1.1.3.cmml" type="float" xref="S2.SS3.SSS2.p2.1.m1.1.1.3">1.2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS2.p2.1.m1.1c">\mu=1.2</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS2.p2.1.m1.1d">italic_μ = 1.2</annotation></semantics></math>, and <math alttext="v_{\rm sw}=500~{}" class="ltx_Math" display="inline" id="S2.SS3.SSS2.p2.2.m2.1"><semantics id="S2.SS3.SSS2.p2.2.m2.1a"><mrow id="S2.SS3.SSS2.p2.2.m2.1.1" xref="S2.SS3.SSS2.p2.2.m2.1.1.cmml"><msub id="S2.SS3.SSS2.p2.2.m2.1.1.2" xref="S2.SS3.SSS2.p2.2.m2.1.1.2.cmml"><mi id="S2.SS3.SSS2.p2.2.m2.1.1.2.2" xref="S2.SS3.SSS2.p2.2.m2.1.1.2.2.cmml">v</mi><mi id="S2.SS3.SSS2.p2.2.m2.1.1.2.3" xref="S2.SS3.SSS2.p2.2.m2.1.1.2.3.cmml">sw</mi></msub><mo id="S2.SS3.SSS2.p2.2.m2.1.1.1" xref="S2.SS3.SSS2.p2.2.m2.1.1.1.cmml">=</mo><mn id="S2.SS3.SSS2.p2.2.m2.1.1.3" xref="S2.SS3.SSS2.p2.2.m2.1.1.3.cmml">500</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS2.p2.2.m2.1b"><apply id="S2.SS3.SSS2.p2.2.m2.1.1.cmml" xref="S2.SS3.SSS2.p2.2.m2.1.1"><eq id="S2.SS3.SSS2.p2.2.m2.1.1.1.cmml" xref="S2.SS3.SSS2.p2.2.m2.1.1.1"></eq><apply id="S2.SS3.SSS2.p2.2.m2.1.1.2.cmml" xref="S2.SS3.SSS2.p2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S2.SS3.SSS2.p2.2.m2.1.1.2.1.cmml" xref="S2.SS3.SSS2.p2.2.m2.1.1.2">subscript</csymbol><ci id="S2.SS3.SSS2.p2.2.m2.1.1.2.2.cmml" xref="S2.SS3.SSS2.p2.2.m2.1.1.2.2">𝑣</ci><ci id="S2.SS3.SSS2.p2.2.m2.1.1.2.3.cmml" xref="S2.SS3.SSS2.p2.2.m2.1.1.2.3">sw</ci></apply><cn id="S2.SS3.SSS2.p2.2.m2.1.1.3.cmml" type="integer" xref="S2.SS3.SSS2.p2.2.m2.1.1.3">500</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS2.p2.2.m2.1c">v_{\rm sw}=500~{}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS2.p2.2.m2.1d">italic_v start_POSTSUBSCRIPT roman_sw end_POSTSUBSCRIPT = 500</annotation></semantics></math>km s<sup class="ltx_sup" id="S2.SS3.SSS2.p2.3.1"><span class="ltx_text ltx_font_italic" id="S2.SS3.SSS2.p2.3.1.1">-1</span></sup> for the velocity of the solar wind, we can rewrite equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.E15" title="In 2.3.2 Solar wind drag ‣ 2.3 Acceleration noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">15</span></a>) as</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="A3.EGx4"> <tbody id="S2.E16"> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\sqrt{S_{a}}" class="ltx_Math" display="inline" id="S2.Ex6.m1.1"><semantics id="S2.Ex6.m1.1a"><msqrt id="S2.Ex6.m1.1.1" xref="S2.Ex6.m1.1.1.cmml"><msub id="S2.Ex6.m1.1.1.2" xref="S2.Ex6.m1.1.1.2.cmml"><mi id="S2.Ex6.m1.1.1.2.2" xref="S2.Ex6.m1.1.1.2.2.cmml">S</mi><mi id="S2.Ex6.m1.1.1.2.3" xref="S2.Ex6.m1.1.1.2.3.cmml">a</mi></msub></msqrt><annotation-xml encoding="MathML-Content" id="S2.Ex6.m1.1b"><apply id="S2.Ex6.m1.1.1.cmml" xref="S2.Ex6.m1.1.1"><root id="S2.Ex6.m1.1.1a.cmml" xref="S2.Ex6.m1.1.1"></root><apply id="S2.Ex6.m1.1.1.2.cmml" xref="S2.Ex6.m1.1.1.2"><csymbol cd="ambiguous" id="S2.Ex6.m1.1.1.2.1.cmml" xref="S2.Ex6.m1.1.1.2">subscript</csymbol><ci id="S2.Ex6.m1.1.1.2.2.cmml" xref="S2.Ex6.m1.1.1.2.2">𝑆</ci><ci id="S2.Ex6.m1.1.1.2.3.cmml" xref="S2.Ex6.m1.1.1.2.3">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex6.m1.1c">\displaystyle\sqrt{S_{a}}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex6.m1.1d">square-root start_ARG italic_S start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT end_ARG</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S2.Ex6.m2.1"><semantics id="S2.Ex6.m2.1a"><mo id="S2.Ex6.m2.1.1" xref="S2.Ex6.m2.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S2.Ex6.m2.1b"><eq id="S2.Ex6.m2.1.1.cmml" xref="S2.Ex6.m2.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex6.m2.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="S2.Ex6.m2.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle 1.2\times 10^{-12}\text{m s${}^{-2}$ Hz}^{-1/2}\left(\frac{A_{% \rm eff}/M}{10^{-2}{\rm m^{2}\;kg^{-1}}}\right)^{1/2}\left(\frac{P_{e,0}^{\rm 1% D}}{10^{11.1}~{}\text{m${}^{-6}$ Hz${}^{-1}$}}\right)^{1/2}" class="ltx_Math" display="inline" id="S2.Ex6.m3.6"><semantics id="S2.Ex6.m3.6a"><mrow id="S2.Ex6.m3.6.7" xref="S2.Ex6.m3.6.7.cmml"><mrow id="S2.Ex6.m3.6.7.2" xref="S2.Ex6.m3.6.7.2.cmml"><mn id="S2.Ex6.m3.6.7.2.2" xref="S2.Ex6.m3.6.7.2.2.cmml">1.2</mn><mo id="S2.Ex6.m3.6.7.2.1" lspace="0.222em" rspace="0.222em" xref="S2.Ex6.m3.6.7.2.1.cmml">×</mo><msup id="S2.Ex6.m3.6.7.2.3" xref="S2.Ex6.m3.6.7.2.3.cmml"><mn id="S2.Ex6.m3.6.7.2.3.2" xref="S2.Ex6.m3.6.7.2.3.2.cmml">10</mn><mrow id="S2.Ex6.m3.6.7.2.3.3" xref="S2.Ex6.m3.6.7.2.3.3.cmml"><mo id="S2.Ex6.m3.6.7.2.3.3a" xref="S2.Ex6.m3.6.7.2.3.3.cmml">−</mo><mn id="S2.Ex6.m3.6.7.2.3.3.2" xref="S2.Ex6.m3.6.7.2.3.3.2.cmml">12</mn></mrow></msup></mrow><mo id="S2.Ex6.m3.6.7.1" xref="S2.Ex6.m3.6.7.1.cmml">⁢</mo><msup id="S2.Ex6.m3.6.7.3" xref="S2.Ex6.m3.6.7.3.cmml"><mrow id="S2.Ex6.m3.1.1.1a" xref="S2.Ex6.m3.1.1.1af.cmml"><mtext id="S2.Ex6.m3.1.1.1aa" xref="S2.Ex6.m3.1.1.1af.cmml">m s</mtext><mtext id="S2.Ex6.m3.1.1.1ab" xref="S2.Ex6.m3.1.1.1af.cmml"><sup class="ltx_sup" id="S2.Ex6.m3.1.1.1a.2nest"><span class="ltx_text ltx_font_italic" id="S2.Ex6.m3.1.1.1a.2.1nest">-2</span></sup></mtext><mtext id="S2.Ex6.m3.1.1.1ae" xref="S2.Ex6.m3.1.1.1af.cmml"> Hz</mtext></mrow><mrow id="S2.Ex6.m3.6.7.3.2" xref="S2.Ex6.m3.6.7.3.2.cmml"><mo id="S2.Ex6.m3.6.7.3.2a" xref="S2.Ex6.m3.6.7.3.2.cmml">−</mo><mrow id="S2.Ex6.m3.6.7.3.2.2" xref="S2.Ex6.m3.6.7.3.2.2.cmml"><mn id="S2.Ex6.m3.6.7.3.2.2.2" xref="S2.Ex6.m3.6.7.3.2.2.2.cmml">1</mn><mo id="S2.Ex6.m3.6.7.3.2.2.1" xref="S2.Ex6.m3.6.7.3.2.2.1.cmml">/</mo><mn id="S2.Ex6.m3.6.7.3.2.2.3" 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id="S2.Ex6.m3.6.7.4.3" xref="S2.Ex6.m3.6.7.4.3.cmml"><mn id="S2.Ex6.m3.6.7.4.3.2" xref="S2.Ex6.m3.6.7.4.3.2.cmml">1</mn><mo id="S2.Ex6.m3.6.7.4.3.1" xref="S2.Ex6.m3.6.7.4.3.1.cmml">/</mo><mn id="S2.Ex6.m3.6.7.4.3.3" xref="S2.Ex6.m3.6.7.4.3.3.cmml">2</mn></mrow></msup><mo id="S2.Ex6.m3.6.7.1b" xref="S2.Ex6.m3.6.7.1.cmml">⁢</mo><msup id="S2.Ex6.m3.6.7.5" xref="S2.Ex6.m3.6.7.5.cmml"><mrow id="S2.Ex6.m3.6.7.5.2.2" xref="S2.Ex6.m3.5.5.cmml"><mo id="S2.Ex6.m3.6.7.5.2.2.1" xref="S2.Ex6.m3.5.5.cmml">(</mo><mstyle displaystyle="true" id="S2.Ex6.m3.5.5" xref="S2.Ex6.m3.5.5.cmml"><mfrac id="S2.Ex6.m3.5.5a" xref="S2.Ex6.m3.5.5.cmml"><msubsup id="S2.Ex6.m3.5.5.4" xref="S2.Ex6.m3.5.5.4.cmml"><mi id="S2.Ex6.m3.5.5.4.4.2" xref="S2.Ex6.m3.5.5.4.4.2.cmml">P</mi><mrow id="S2.Ex6.m3.5.5.4.2.2.4" xref="S2.Ex6.m3.5.5.4.2.2.3.cmml"><mi id="S2.Ex6.m3.4.4.3.1.1.1" xref="S2.Ex6.m3.4.4.3.1.1.1.cmml">e</mi><mo id="S2.Ex6.m3.5.5.4.2.2.4.1" xref="S2.Ex6.m3.5.5.4.2.2.3.cmml">,</mo><mn id="S2.Ex6.m3.5.5.4.2.2.2" 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id="S2.Ex6.m3.3.3.2.2.2ai.cmml" xref="S2.Ex6.m3.3.3.2.2.2a"><mrow id="S2.Ex6.m3.3.3.2.2.2a.cmml" xref="S2.Ex6.m3.3.3.2.2.2a"><mtext id="S2.Ex6.m3.3.3.2.2.2aa.cmml" xref="S2.Ex6.m3.3.3.2.2.2a">m</mtext><mtext id="S2.Ex6.m3.3.3.2.2.2ab.cmml" xref="S2.Ex6.m3.3.3.2.2.2a"><sup class="ltx_sup" id="S2.Ex6.m3.3.3.2.2.2a.3anest"><span class="ltx_text ltx_font_italic" id="S2.Ex6.m3.3.3.2.2.2a.3.1anest">-6</span></sup></mtext><mtext id="S2.Ex6.m3.3.3.2.2.2ae.cmml" xref="S2.Ex6.m3.3.3.2.2.2a"> Hz</mtext><mtext id="S2.Ex6.m3.3.3.2.2.2af.cmml" xref="S2.Ex6.m3.3.3.2.2.2a"><sup class="ltx_sup" id="S2.Ex6.m3.3.3.2.2.2a.4anest"><span class="ltx_text ltx_font_italic" id="S2.Ex6.m3.3.3.2.2.2a.4.1anest">-1</span></sup></mtext></mrow></ci></apply></apply><apply id="S2.Ex6.m3.6.7.5.3.cmml" xref="S2.Ex6.m3.6.7.5.3"><divide id="S2.Ex6.m3.6.7.5.3.1.cmml" xref="S2.Ex6.m3.6.7.5.3.1"></divide><cn id="S2.Ex6.m3.6.7.5.3.2.cmml" type="integer" xref="S2.Ex6.m3.6.7.5.3.2">1</cn><cn id="S2.Ex6.m3.6.7.5.3.3.cmml" type="integer" xref="S2.Ex6.m3.6.7.5.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex6.m3.6c">\displaystyle 1.2\times 10^{-12}\text{m s${}^{-2}$ Hz}^{-1/2}\left(\frac{A_{% \rm eff}/M}{10^{-2}{\rm m^{2}\;kg^{-1}}}\right)^{1/2}\left(\frac{P_{e,0}^{\rm 1% D}}{10^{11.1}~{}\text{m${}^{-6}$ Hz${}^{-1}$}}\right)^{1/2}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex6.m3.6d">1.2 × 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT m s Hz start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT ( divide start_ARG italic_A start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT / italic_M end_ARG start_ARG 10 start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT roman_m start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_kg start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT end_ARG ) start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT ( divide start_ARG italic_P start_POSTSUBSCRIPT italic_e , 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 roman_D end_POSTSUPERSCRIPT end_ARG start_ARG 10 start_POSTSUPERSCRIPT 11.1 end_POSTSUPERSCRIPT m Hz end_ARG ) start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="2"><span class="ltx_tag ltx_tag_equation ltx_align_right">(16)</span></td> </tr> <tr class="ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\times\left(\frac{f}{10^{-5}{\rm Hz}}\right)^{-\beta/2}\left(% \frac{r}{30\rm AU}\right)^{-1.5}." class="ltx_Math" display="inline" id="S2.E16.m1.3"><semantics id="S2.E16.m1.3a"><mrow id="S2.E16.m1.3.3.1" xref="S2.E16.m1.3.3.1.1.cmml"><mrow id="S2.E16.m1.3.3.1.1" xref="S2.E16.m1.3.3.1.1.cmml"><mi id="S2.E16.m1.3.3.1.1.2" 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xref="S2.E16.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E16.m1.3b"><apply id="S2.E16.m1.3.3.1.1.cmml" xref="S2.E16.m1.3.3.1"><times id="S2.E16.m1.3.3.1.1.1.cmml" xref="S2.E16.m1.3.3.1.1.1"></times><csymbol cd="latexml" id="S2.E16.m1.3.3.1.1.2.cmml" xref="S2.E16.m1.3.3.1.1.2">absent</csymbol><apply id="S2.E16.m1.3.3.1.1.3.cmml" xref="S2.E16.m1.3.3.1.1.3"><times id="S2.E16.m1.3.3.1.1.3.1.cmml" xref="S2.E16.m1.3.3.1.1.3.1"></times><apply id="S2.E16.m1.3.3.1.1.3.2.cmml" xref="S2.E16.m1.3.3.1.1.3.2"><csymbol cd="ambiguous" id="S2.E16.m1.3.3.1.1.3.2.1.cmml" xref="S2.E16.m1.3.3.1.1.3.2">superscript</csymbol><apply id="S2.E16.m1.1.1.cmml" xref="S2.E16.m1.3.3.1.1.3.2.2.2"><divide id="S2.E16.m1.1.1.1.cmml" xref="S2.E16.m1.3.3.1.1.3.2.2.2"></divide><ci id="S2.E16.m1.1.1.2.cmml" xref="S2.E16.m1.1.1.2">𝑓</ci><apply id="S2.E16.m1.1.1.3.cmml" xref="S2.E16.m1.1.1.3"><times id="S2.E16.m1.1.1.3.1.cmml" xref="S2.E16.m1.1.1.3.1"></times><apply 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type="integer" xref="S2.E16.m1.3.3.1.1.3.2.3.2.3">2</cn></apply></apply></apply><apply id="S2.E16.m1.3.3.1.1.3.3.cmml" xref="S2.E16.m1.3.3.1.1.3.3"><csymbol cd="ambiguous" id="S2.E16.m1.3.3.1.1.3.3.1.cmml" xref="S2.E16.m1.3.3.1.1.3.3">superscript</csymbol><apply id="S2.E16.m1.2.2.cmml" xref="S2.E16.m1.3.3.1.1.3.3.2.2"><divide id="S2.E16.m1.2.2.1.cmml" xref="S2.E16.m1.3.3.1.1.3.3.2.2"></divide><ci id="S2.E16.m1.2.2.2.cmml" xref="S2.E16.m1.2.2.2">𝑟</ci><apply id="S2.E16.m1.2.2.3.cmml" xref="S2.E16.m1.2.2.3"><times id="S2.E16.m1.2.2.3.1.cmml" xref="S2.E16.m1.2.2.3.1"></times><cn id="S2.E16.m1.2.2.3.2.cmml" type="integer" xref="S2.E16.m1.2.2.3.2">30</cn><ci id="S2.E16.m1.2.2.3.3.cmml" xref="S2.E16.m1.2.2.3.3">A</ci><ci id="S2.E16.m1.2.2.3.4.cmml" xref="S2.E16.m1.2.2.3.4">U</ci></apply></apply><apply id="S2.E16.m1.3.3.1.1.3.3.3.cmml" xref="S2.E16.m1.3.3.1.1.3.3.3"><minus id="S2.E16.m1.3.3.1.1.3.3.3.1.cmml" xref="S2.E16.m1.3.3.1.1.3.3.3"></minus><cn id="S2.E16.m1.3.3.1.1.3.3.3.2.cmml" type="float" xref="S2.E16.m1.3.3.1.1.3.3.3.2">1.5</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E16.m1.3c">\displaystyle\times\left(\frac{f}{10^{-5}{\rm Hz}}\right)^{-\beta/2}\left(% \frac{r}{30\rm AU}\right)^{-1.5}.</annotation><annotation encoding="application/x-llamapun" id="S2.E16.m1.3d">× ( divide start_ARG italic_f end_ARG start_ARG 10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT roman_Hz end_ARG ) start_POSTSUPERSCRIPT - italic_β / 2 end_POSTSUPERSCRIPT ( divide start_ARG italic_r end_ARG start_ARG 30 roman_A roman_U end_ARG ) start_POSTSUPERSCRIPT - 1.5 end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </tbody> </table> </div> <div class="ltx_para" id="S2.SS3.SSS2.p3"> <p class="ltx_p" id="S2.SS3.SSS2.p3.10">However, rather than assuming some power-law spectrum as in equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.E16" title="In 2.3.2 Solar wind drag ‣ 2.3 Acceleration noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">16</span></a>), we can compute the drag using the actual electron power that Voyager 2 measured using the measurements presented in <cite class="ltx_cite ltx_citemacro_citep">(Bellamy et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib16" title="">2005</a>)</cite>. Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.F2" title="Figure 2 ‣ 2.3 Acceleration noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">2</span></a> shows just this at 30 AU assuming a spacecraft effective area to mass of <math alttext="A_{\rm eff}/M=0.01" class="ltx_Math" display="inline" id="S2.SS3.SSS2.p3.1.m1.1"><semantics id="S2.SS3.SSS2.p3.1.m1.1a"><mrow id="S2.SS3.SSS2.p3.1.m1.1.1" xref="S2.SS3.SSS2.p3.1.m1.1.1.cmml"><mrow id="S2.SS3.SSS2.p3.1.m1.1.1.2" xref="S2.SS3.SSS2.p3.1.m1.1.1.2.cmml"><msub id="S2.SS3.SSS2.p3.1.m1.1.1.2.2" xref="S2.SS3.SSS2.p3.1.m1.1.1.2.2.cmml"><mi id="S2.SS3.SSS2.p3.1.m1.1.1.2.2.2" xref="S2.SS3.SSS2.p3.1.m1.1.1.2.2.2.cmml">A</mi><mi id="S2.SS3.SSS2.p3.1.m1.1.1.2.2.3" xref="S2.SS3.SSS2.p3.1.m1.1.1.2.2.3.cmml">eff</mi></msub><mo id="S2.SS3.SSS2.p3.1.m1.1.1.2.1" xref="S2.SS3.SSS2.p3.1.m1.1.1.2.1.cmml">/</mo><mi id="S2.SS3.SSS2.p3.1.m1.1.1.2.3" xref="S2.SS3.SSS2.p3.1.m1.1.1.2.3.cmml">M</mi></mrow><mo id="S2.SS3.SSS2.p3.1.m1.1.1.1" xref="S2.SS3.SSS2.p3.1.m1.1.1.1.cmml">=</mo><mn id="S2.SS3.SSS2.p3.1.m1.1.1.3" xref="S2.SS3.SSS2.p3.1.m1.1.1.3.cmml">0.01</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS2.p3.1.m1.1b"><apply id="S2.SS3.SSS2.p3.1.m1.1.1.cmml" xref="S2.SS3.SSS2.p3.1.m1.1.1"><eq id="S2.SS3.SSS2.p3.1.m1.1.1.1.cmml" xref="S2.SS3.SSS2.p3.1.m1.1.1.1"></eq><apply id="S2.SS3.SSS2.p3.1.m1.1.1.2.cmml" xref="S2.SS3.SSS2.p3.1.m1.1.1.2"><divide id="S2.SS3.SSS2.p3.1.m1.1.1.2.1.cmml" xref="S2.SS3.SSS2.p3.1.m1.1.1.2.1"></divide><apply id="S2.SS3.SSS2.p3.1.m1.1.1.2.2.cmml" xref="S2.SS3.SSS2.p3.1.m1.1.1.2.2"><csymbol cd="ambiguous" id="S2.SS3.SSS2.p3.1.m1.1.1.2.2.1.cmml" xref="S2.SS3.SSS2.p3.1.m1.1.1.2.2">subscript</csymbol><ci id="S2.SS3.SSS2.p3.1.m1.1.1.2.2.2.cmml" xref="S2.SS3.SSS2.p3.1.m1.1.1.2.2.2">𝐴</ci><ci id="S2.SS3.SSS2.p3.1.m1.1.1.2.2.3.cmml" xref="S2.SS3.SSS2.p3.1.m1.1.1.2.2.3">eff</ci></apply><ci id="S2.SS3.SSS2.p3.1.m1.1.1.2.3.cmml" xref="S2.SS3.SSS2.p3.1.m1.1.1.2.3">𝑀</ci></apply><cn id="S2.SS3.SSS2.p3.1.m1.1.1.3.cmml" type="float" xref="S2.SS3.SSS2.p3.1.m1.1.1.3">0.01</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS2.p3.1.m1.1c">A_{\rm eff}/M=0.01</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS2.p3.1.m1.1d">italic_A start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT / italic_M = 0.01</annotation></semantics></math> m<sup class="ltx_sup" id="S2.SS3.SSS2.p3.10.1"><span class="ltx_text ltx_font_italic" id="S2.SS3.SSS2.p3.10.1.1">2</span></sup> kg<sup class="ltx_sup" id="S2.SS3.SSS2.p3.10.2"><span class="ltx_text ltx_font_italic" id="S2.SS3.SSS2.p3.10.2.1">-1</span></sup>, also using equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.E15" title="In 2.3.2 Solar wind drag ‣ 2.3 Acceleration noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">15</span></a>). Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.F2" title="Figure 2 ‣ 2.3 Acceleration noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">2</span></a> illustrates that Voyager 2’s electron power spectrum is not exactly a power law. Between <math alttext="10^{-4}-10^{-2}" class="ltx_Math" display="inline" id="S2.SS3.SSS2.p3.4.m4.1"><semantics id="S2.SS3.SSS2.p3.4.m4.1a"><mrow id="S2.SS3.SSS2.p3.4.m4.1.1" xref="S2.SS3.SSS2.p3.4.m4.1.1.cmml"><msup id="S2.SS3.SSS2.p3.4.m4.1.1.2" xref="S2.SS3.SSS2.p3.4.m4.1.1.2.cmml"><mn id="S2.SS3.SSS2.p3.4.m4.1.1.2.2" xref="S2.SS3.SSS2.p3.4.m4.1.1.2.2.cmml">10</mn><mrow id="S2.SS3.SSS2.p3.4.m4.1.1.2.3" xref="S2.SS3.SSS2.p3.4.m4.1.1.2.3.cmml"><mo id="S2.SS3.SSS2.p3.4.m4.1.1.2.3a" xref="S2.SS3.SSS2.p3.4.m4.1.1.2.3.cmml">−</mo><mn id="S2.SS3.SSS2.p3.4.m4.1.1.2.3.2" xref="S2.SS3.SSS2.p3.4.m4.1.1.2.3.2.cmml">4</mn></mrow></msup><mo id="S2.SS3.SSS2.p3.4.m4.1.1.1" xref="S2.SS3.SSS2.p3.4.m4.1.1.1.cmml">−</mo><msup id="S2.SS3.SSS2.p3.4.m4.1.1.3" xref="S2.SS3.SSS2.p3.4.m4.1.1.3.cmml"><mn id="S2.SS3.SSS2.p3.4.m4.1.1.3.2" xref="S2.SS3.SSS2.p3.4.m4.1.1.3.2.cmml">10</mn><mrow id="S2.SS3.SSS2.p3.4.m4.1.1.3.3" xref="S2.SS3.SSS2.p3.4.m4.1.1.3.3.cmml"><mo id="S2.SS3.SSS2.p3.4.m4.1.1.3.3a" xref="S2.SS3.SSS2.p3.4.m4.1.1.3.3.cmml">−</mo><mn id="S2.SS3.SSS2.p3.4.m4.1.1.3.3.2" xref="S2.SS3.SSS2.p3.4.m4.1.1.3.3.2.cmml">2</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS2.p3.4.m4.1b"><apply id="S2.SS3.SSS2.p3.4.m4.1.1.cmml" xref="S2.SS3.SSS2.p3.4.m4.1.1"><minus id="S2.SS3.SSS2.p3.4.m4.1.1.1.cmml" xref="S2.SS3.SSS2.p3.4.m4.1.1.1"></minus><apply id="S2.SS3.SSS2.p3.4.m4.1.1.2.cmml" xref="S2.SS3.SSS2.p3.4.m4.1.1.2"><csymbol cd="ambiguous" id="S2.SS3.SSS2.p3.4.m4.1.1.2.1.cmml" xref="S2.SS3.SSS2.p3.4.m4.1.1.2">superscript</csymbol><cn id="S2.SS3.SSS2.p3.4.m4.1.1.2.2.cmml" type="integer" xref="S2.SS3.SSS2.p3.4.m4.1.1.2.2">10</cn><apply id="S2.SS3.SSS2.p3.4.m4.1.1.2.3.cmml" xref="S2.SS3.SSS2.p3.4.m4.1.1.2.3"><minus id="S2.SS3.SSS2.p3.4.m4.1.1.2.3.1.cmml" xref="S2.SS3.SSS2.p3.4.m4.1.1.2.3"></minus><cn id="S2.SS3.SSS2.p3.4.m4.1.1.2.3.2.cmml" type="integer" xref="S2.SS3.SSS2.p3.4.m4.1.1.2.3.2">4</cn></apply></apply><apply id="S2.SS3.SSS2.p3.4.m4.1.1.3.cmml" xref="S2.SS3.SSS2.p3.4.m4.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.SSS2.p3.4.m4.1.1.3.1.cmml" xref="S2.SS3.SSS2.p3.4.m4.1.1.3">superscript</csymbol><cn id="S2.SS3.SSS2.p3.4.m4.1.1.3.2.cmml" type="integer" xref="S2.SS3.SSS2.p3.4.m4.1.1.3.2">10</cn><apply id="S2.SS3.SSS2.p3.4.m4.1.1.3.3.cmml" xref="S2.SS3.SSS2.p3.4.m4.1.1.3.3"><minus id="S2.SS3.SSS2.p3.4.m4.1.1.3.3.1.cmml" xref="S2.SS3.SSS2.p3.4.m4.1.1.3.3"></minus><cn id="S2.SS3.SSS2.p3.4.m4.1.1.3.3.2.cmml" type="integer" xref="S2.SS3.SSS2.p3.4.m4.1.1.3.3.2">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS2.p3.4.m4.1c">10^{-4}-10^{-2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS2.p3.4.m4.1d">10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT - 10 start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT</annotation></semantics></math> Hz, <cite class="ltx_cite ltx_citemacro_citet">Bellamy et al. (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib16" title="">2005</a>)</cite> finds <math alttext="\beta\approx 2" class="ltx_Math" display="inline" id="S2.SS3.SSS2.p3.5.m5.1"><semantics id="S2.SS3.SSS2.p3.5.m5.1a"><mrow id="S2.SS3.SSS2.p3.5.m5.1.1" xref="S2.SS3.SSS2.p3.5.m5.1.1.cmml"><mi id="S2.SS3.SSS2.p3.5.m5.1.1.2" xref="S2.SS3.SSS2.p3.5.m5.1.1.2.cmml">β</mi><mo id="S2.SS3.SSS2.p3.5.m5.1.1.1" xref="S2.SS3.SSS2.p3.5.m5.1.1.1.cmml">≈</mo><mn id="S2.SS3.SSS2.p3.5.m5.1.1.3" xref="S2.SS3.SSS2.p3.5.m5.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS2.p3.5.m5.1b"><apply id="S2.SS3.SSS2.p3.5.m5.1.1.cmml" xref="S2.SS3.SSS2.p3.5.m5.1.1"><approx id="S2.SS3.SSS2.p3.5.m5.1.1.1.cmml" xref="S2.SS3.SSS2.p3.5.m5.1.1.1"></approx><ci id="S2.SS3.SSS2.p3.5.m5.1.1.2.cmml" xref="S2.SS3.SSS2.p3.5.m5.1.1.2">𝛽</ci><cn id="S2.SS3.SSS2.p3.5.m5.1.1.3.cmml" type="integer" xref="S2.SS3.SSS2.p3.5.m5.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS2.p3.5.m5.1c">\beta\approx 2</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS2.p3.5.m5.1d">italic_β ≈ 2</annotation></semantics></math>. At higher frequencies, the spectrum is more consistent with a Kolmogorov-like <math alttext="\beta=5/3" class="ltx_Math" display="inline" id="S2.SS3.SSS2.p3.6.m6.1"><semantics id="S2.SS3.SSS2.p3.6.m6.1a"><mrow id="S2.SS3.SSS2.p3.6.m6.1.1" xref="S2.SS3.SSS2.p3.6.m6.1.1.cmml"><mi id="S2.SS3.SSS2.p3.6.m6.1.1.2" xref="S2.SS3.SSS2.p3.6.m6.1.1.2.cmml">β</mi><mo id="S2.SS3.SSS2.p3.6.m6.1.1.1" xref="S2.SS3.SSS2.p3.6.m6.1.1.1.cmml">=</mo><mrow id="S2.SS3.SSS2.p3.6.m6.1.1.3" xref="S2.SS3.SSS2.p3.6.m6.1.1.3.cmml"><mn id="S2.SS3.SSS2.p3.6.m6.1.1.3.2" xref="S2.SS3.SSS2.p3.6.m6.1.1.3.2.cmml">5</mn><mo id="S2.SS3.SSS2.p3.6.m6.1.1.3.1" xref="S2.SS3.SSS2.p3.6.m6.1.1.3.1.cmml">/</mo><mn id="S2.SS3.SSS2.p3.6.m6.1.1.3.3" xref="S2.SS3.SSS2.p3.6.m6.1.1.3.3.cmml">3</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS2.p3.6.m6.1b"><apply id="S2.SS3.SSS2.p3.6.m6.1.1.cmml" xref="S2.SS3.SSS2.p3.6.m6.1.1"><eq id="S2.SS3.SSS2.p3.6.m6.1.1.1.cmml" xref="S2.SS3.SSS2.p3.6.m6.1.1.1"></eq><ci id="S2.SS3.SSS2.p3.6.m6.1.1.2.cmml" xref="S2.SS3.SSS2.p3.6.m6.1.1.2">𝛽</ci><apply id="S2.SS3.SSS2.p3.6.m6.1.1.3.cmml" xref="S2.SS3.SSS2.p3.6.m6.1.1.3"><divide id="S2.SS3.SSS2.p3.6.m6.1.1.3.1.cmml" xref="S2.SS3.SSS2.p3.6.m6.1.1.3.1"></divide><cn id="S2.SS3.SSS2.p3.6.m6.1.1.3.2.cmml" type="integer" xref="S2.SS3.SSS2.p3.6.m6.1.1.3.2">5</cn><cn id="S2.SS3.SSS2.p3.6.m6.1.1.3.3.cmml" type="integer" xref="S2.SS3.SSS2.p3.6.m6.1.1.3.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS2.p3.6.m6.1c">\beta=5/3</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS2.p3.6.m6.1d">italic_β = 5 / 3</annotation></semantics></math> spectrum. There is the additional complication that <cite class="ltx_cite ltx_citemacro_citet">Bellamy et al. (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib16" title="">2005</a>)</cite> did not consider <math alttext="f<10^{-5}" class="ltx_Math" display="inline" id="S2.SS3.SSS2.p3.7.m7.1"><semantics id="S2.SS3.SSS2.p3.7.m7.1a"><mrow id="S2.SS3.SSS2.p3.7.m7.1.1" xref="S2.SS3.SSS2.p3.7.m7.1.1.cmml"><mi id="S2.SS3.SSS2.p3.7.m7.1.1.2" xref="S2.SS3.SSS2.p3.7.m7.1.1.2.cmml">f</mi><mo id="S2.SS3.SSS2.p3.7.m7.1.1.1" xref="S2.SS3.SSS2.p3.7.m7.1.1.1.cmml"><</mo><msup id="S2.SS3.SSS2.p3.7.m7.1.1.3" xref="S2.SS3.SSS2.p3.7.m7.1.1.3.cmml"><mn id="S2.SS3.SSS2.p3.7.m7.1.1.3.2" xref="S2.SS3.SSS2.p3.7.m7.1.1.3.2.cmml">10</mn><mrow id="S2.SS3.SSS2.p3.7.m7.1.1.3.3" xref="S2.SS3.SSS2.p3.7.m7.1.1.3.3.cmml"><mo id="S2.SS3.SSS2.p3.7.m7.1.1.3.3a" xref="S2.SS3.SSS2.p3.7.m7.1.1.3.3.cmml">−</mo><mn id="S2.SS3.SSS2.p3.7.m7.1.1.3.3.2" xref="S2.SS3.SSS2.p3.7.m7.1.1.3.3.2.cmml">5</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS2.p3.7.m7.1b"><apply id="S2.SS3.SSS2.p3.7.m7.1.1.cmml" xref="S2.SS3.SSS2.p3.7.m7.1.1"><lt id="S2.SS3.SSS2.p3.7.m7.1.1.1.cmml" xref="S2.SS3.SSS2.p3.7.m7.1.1.1"></lt><ci id="S2.SS3.SSS2.p3.7.m7.1.1.2.cmml" xref="S2.SS3.SSS2.p3.7.m7.1.1.2">𝑓</ci><apply id="S2.SS3.SSS2.p3.7.m7.1.1.3.cmml" xref="S2.SS3.SSS2.p3.7.m7.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.SSS2.p3.7.m7.1.1.3.1.cmml" xref="S2.SS3.SSS2.p3.7.m7.1.1.3">superscript</csymbol><cn id="S2.SS3.SSS2.p3.7.m7.1.1.3.2.cmml" type="integer" xref="S2.SS3.SSS2.p3.7.m7.1.1.3.2">10</cn><apply id="S2.SS3.SSS2.p3.7.m7.1.1.3.3.cmml" xref="S2.SS3.SSS2.p3.7.m7.1.1.3.3"><minus id="S2.SS3.SSS2.p3.7.m7.1.1.3.3.1.cmml" xref="S2.SS3.SSS2.p3.7.m7.1.1.3.3"></minus><cn id="S2.SS3.SSS2.p3.7.m7.1.1.3.3.2.cmml" type="integer" xref="S2.SS3.SSS2.p3.7.m7.1.1.3.3.2">5</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS2.p3.7.m7.1c">f<10^{-5}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS2.p3.7.m7.1d">italic_f < 10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT</annotation></semantics></math> Hz. We extrapolate with the index <math alttext="\beta=1.5" class="ltx_Math" display="inline" id="S2.SS3.SSS2.p3.8.m8.1"><semantics id="S2.SS3.SSS2.p3.8.m8.1a"><mrow id="S2.SS3.SSS2.p3.8.m8.1.1" xref="S2.SS3.SSS2.p3.8.m8.1.1.cmml"><mi id="S2.SS3.SSS2.p3.8.m8.1.1.2" xref="S2.SS3.SSS2.p3.8.m8.1.1.2.cmml">β</mi><mo id="S2.SS3.SSS2.p3.8.m8.1.1.1" xref="S2.SS3.SSS2.p3.8.m8.1.1.1.cmml">=</mo><mn id="S2.SS3.SSS2.p3.8.m8.1.1.3" xref="S2.SS3.SSS2.p3.8.m8.1.1.3.cmml">1.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS2.p3.8.m8.1b"><apply id="S2.SS3.SSS2.p3.8.m8.1.1.cmml" xref="S2.SS3.SSS2.p3.8.m8.1.1"><eq id="S2.SS3.SSS2.p3.8.m8.1.1.1.cmml" xref="S2.SS3.SSS2.p3.8.m8.1.1.1"></eq><ci id="S2.SS3.SSS2.p3.8.m8.1.1.2.cmml" xref="S2.SS3.SSS2.p3.8.m8.1.1.2">𝛽</ci><cn id="S2.SS3.SSS2.p3.8.m8.1.1.3.cmml" type="float" xref="S2.SS3.SSS2.p3.8.m8.1.1.3">1.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS2.p3.8.m8.1c">\beta=1.5</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS2.p3.8.m8.1d">italic_β = 1.5</annotation></semantics></math>. Extrapolating with a flatter scaling than that observed at higher frequencies is motivated by the power spectrum of the solar wind within several astronomical units, which tends to have a power spectrum of <math alttext="\beta\approx 1" class="ltx_Math" display="inline" id="S2.SS3.SSS2.p3.9.m9.1"><semantics id="S2.SS3.SSS2.p3.9.m9.1a"><mrow id="S2.SS3.SSS2.p3.9.m9.1.1" xref="S2.SS3.SSS2.p3.9.m9.1.1.cmml"><mi id="S2.SS3.SSS2.p3.9.m9.1.1.2" xref="S2.SS3.SSS2.p3.9.m9.1.1.2.cmml">β</mi><mo id="S2.SS3.SSS2.p3.9.m9.1.1.1" xref="S2.SS3.SSS2.p3.9.m9.1.1.1.cmml">≈</mo><mn id="S2.SS3.SSS2.p3.9.m9.1.1.3" xref="S2.SS3.SSS2.p3.9.m9.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS2.p3.9.m9.1b"><apply id="S2.SS3.SSS2.p3.9.m9.1.1.cmml" xref="S2.SS3.SSS2.p3.9.m9.1.1"><approx id="S2.SS3.SSS2.p3.9.m9.1.1.1.cmml" xref="S2.SS3.SSS2.p3.9.m9.1.1.1"></approx><ci id="S2.SS3.SSS2.p3.9.m9.1.1.2.cmml" xref="S2.SS3.SSS2.p3.9.m9.1.1.2">𝛽</ci><cn id="S2.SS3.SSS2.p3.9.m9.1.1.3.cmml" type="integer" xref="S2.SS3.SSS2.p3.9.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS2.p3.9.m9.1c">\beta\approx 1</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS2.p3.9.m9.1d">italic_β ≈ 1</annotation></semantics></math> at these low frequencies. (The properties of the solar wind are likely to be largely maintained as it advects into the outer Solar System.) Additionally, the power spectrum of density fluctuations is often similar to that of magnetic field fluctuations, and the power spectrum of the magnetic field amplitude is observed to flatten at sub-<math alttext="\mu" class="ltx_Math" display="inline" id="S2.SS3.SSS2.p3.10.m10.1"><semantics id="S2.SS3.SSS2.p3.10.m10.1a"><mi id="S2.SS3.SSS2.p3.10.m10.1.1" xref="S2.SS3.SSS2.p3.10.m10.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS2.p3.10.m10.1b"><ci id="S2.SS3.SSS2.p3.10.m10.1.1.cmml" xref="S2.SS3.SSS2.p3.10.m10.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS2.p3.10.m10.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS2.p3.10.m10.1d">italic_μ</annotation></semantics></math>Hz frequencies <cite class="ltx_cite ltx_citemacro_citep">(Smith, <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib64" title="">1989</a>)</cite>.<span class="ltx_note ltx_role_footnote" id="footnote8"><sup class="ltx_note_mark">8</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">8</sup><span class="ltx_tag ltx_tag_note">8</span>The power spectrum of the magnetic field is proportional to the accelerations from Lorentz forces, which is shown in Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.F2" title="Figure 2 ‣ 2.3 Acceleration noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">2</span></a> (cf. § <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.SS3.SSS4" title="2.3.4 Spacecraft charging ‣ 2.3 Acceleration noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">2.3.4</span></a>).</span></span></span></p> </div> </section> <section class="ltx_subsubsection" id="S2.SS3.SSS3"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">2.3.3 </span>Dust</h4> <div class="ltx_para" id="S2.SS3.SSS3.p1"> <p class="ltx_p" id="S2.SS3.SSS3.p1.20">Each collision with a dust grain results in a change of velocity of the spacecraft of <math alttext="\Delta v_{i}=m_{d,i}v_{d}/M" class="ltx_Math" display="inline" id="S2.SS3.SSS3.p1.1.m1.2"><semantics id="S2.SS3.SSS3.p1.1.m1.2a"><mrow id="S2.SS3.SSS3.p1.1.m1.2.3" xref="S2.SS3.SSS3.p1.1.m1.2.3.cmml"><mrow id="S2.SS3.SSS3.p1.1.m1.2.3.2" xref="S2.SS3.SSS3.p1.1.m1.2.3.2.cmml"><mi id="S2.SS3.SSS3.p1.1.m1.2.3.2.2" mathvariant="normal" xref="S2.SS3.SSS3.p1.1.m1.2.3.2.2.cmml">Δ</mi><mo id="S2.SS3.SSS3.p1.1.m1.2.3.2.1" xref="S2.SS3.SSS3.p1.1.m1.2.3.2.1.cmml">⁢</mo><msub id="S2.SS3.SSS3.p1.1.m1.2.3.2.3" xref="S2.SS3.SSS3.p1.1.m1.2.3.2.3.cmml"><mi id="S2.SS3.SSS3.p1.1.m1.2.3.2.3.2" xref="S2.SS3.SSS3.p1.1.m1.2.3.2.3.2.cmml">v</mi><mi id="S2.SS3.SSS3.p1.1.m1.2.3.2.3.3" xref="S2.SS3.SSS3.p1.1.m1.2.3.2.3.3.cmml">i</mi></msub></mrow><mo id="S2.SS3.SSS3.p1.1.m1.2.3.1" xref="S2.SS3.SSS3.p1.1.m1.2.3.1.cmml">=</mo><mrow id="S2.SS3.SSS3.p1.1.m1.2.3.3" xref="S2.SS3.SSS3.p1.1.m1.2.3.3.cmml"><mrow id="S2.SS3.SSS3.p1.1.m1.2.3.3.2" xref="S2.SS3.SSS3.p1.1.m1.2.3.3.2.cmml"><msub id="S2.SS3.SSS3.p1.1.m1.2.3.3.2.2" xref="S2.SS3.SSS3.p1.1.m1.2.3.3.2.2.cmml"><mi id="S2.SS3.SSS3.p1.1.m1.2.3.3.2.2.2" xref="S2.SS3.SSS3.p1.1.m1.2.3.3.2.2.2.cmml">m</mi><mrow id="S2.SS3.SSS3.p1.1.m1.2.2.2.4" xref="S2.SS3.SSS3.p1.1.m1.2.2.2.3.cmml"><mi id="S2.SS3.SSS3.p1.1.m1.1.1.1.1" xref="S2.SS3.SSS3.p1.1.m1.1.1.1.1.cmml">d</mi><mo id="S2.SS3.SSS3.p1.1.m1.2.2.2.4.1" xref="S2.SS3.SSS3.p1.1.m1.2.2.2.3.cmml">,</mo><mi id="S2.SS3.SSS3.p1.1.m1.2.2.2.2" xref="S2.SS3.SSS3.p1.1.m1.2.2.2.2.cmml">i</mi></mrow></msub><mo id="S2.SS3.SSS3.p1.1.m1.2.3.3.2.1" xref="S2.SS3.SSS3.p1.1.m1.2.3.3.2.1.cmml">⁢</mo><msub id="S2.SS3.SSS3.p1.1.m1.2.3.3.2.3" xref="S2.SS3.SSS3.p1.1.m1.2.3.3.2.3.cmml"><mi 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xref="S2.SS3.SSS3.p1.1.m1.2.3.2.3">subscript</csymbol><ci id="S2.SS3.SSS3.p1.1.m1.2.3.2.3.2.cmml" xref="S2.SS3.SSS3.p1.1.m1.2.3.2.3.2">𝑣</ci><ci id="S2.SS3.SSS3.p1.1.m1.2.3.2.3.3.cmml" xref="S2.SS3.SSS3.p1.1.m1.2.3.2.3.3">𝑖</ci></apply></apply><apply id="S2.SS3.SSS3.p1.1.m1.2.3.3.cmml" xref="S2.SS3.SSS3.p1.1.m1.2.3.3"><divide id="S2.SS3.SSS3.p1.1.m1.2.3.3.1.cmml" xref="S2.SS3.SSS3.p1.1.m1.2.3.3.1"></divide><apply id="S2.SS3.SSS3.p1.1.m1.2.3.3.2.cmml" xref="S2.SS3.SSS3.p1.1.m1.2.3.3.2"><times id="S2.SS3.SSS3.p1.1.m1.2.3.3.2.1.cmml" xref="S2.SS3.SSS3.p1.1.m1.2.3.3.2.1"></times><apply id="S2.SS3.SSS3.p1.1.m1.2.3.3.2.2.cmml" xref="S2.SS3.SSS3.p1.1.m1.2.3.3.2.2"><csymbol cd="ambiguous" id="S2.SS3.SSS3.p1.1.m1.2.3.3.2.2.1.cmml" xref="S2.SS3.SSS3.p1.1.m1.2.3.3.2.2">subscript</csymbol><ci id="S2.SS3.SSS3.p1.1.m1.2.3.3.2.2.2.cmml" xref="S2.SS3.SSS3.p1.1.m1.2.3.3.2.2.2">𝑚</ci><list id="S2.SS3.SSS3.p1.1.m1.2.2.2.3.cmml" xref="S2.SS3.SSS3.p1.1.m1.2.2.2.4"><ci id="S2.SS3.SSS3.p1.1.m1.1.1.1.1.cmml" xref="S2.SS3.SSS3.p1.1.m1.1.1.1.1">𝑑</ci><ci id="S2.SS3.SSS3.p1.1.m1.2.2.2.2.cmml" xref="S2.SS3.SSS3.p1.1.m1.2.2.2.2">𝑖</ci></list></apply><apply id="S2.SS3.SSS3.p1.1.m1.2.3.3.2.3.cmml" xref="S2.SS3.SSS3.p1.1.m1.2.3.3.2.3"><csymbol cd="ambiguous" id="S2.SS3.SSS3.p1.1.m1.2.3.3.2.3.1.cmml" xref="S2.SS3.SSS3.p1.1.m1.2.3.3.2.3">subscript</csymbol><ci id="S2.SS3.SSS3.p1.1.m1.2.3.3.2.3.2.cmml" xref="S2.SS3.SSS3.p1.1.m1.2.3.3.2.3.2">𝑣</ci><ci id="S2.SS3.SSS3.p1.1.m1.2.3.3.2.3.3.cmml" xref="S2.SS3.SSS3.p1.1.m1.2.3.3.2.3.3">𝑑</ci></apply></apply><ci id="S2.SS3.SSS3.p1.1.m1.2.3.3.3.cmml" xref="S2.SS3.SSS3.p1.1.m1.2.3.3.3">𝑀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS3.p1.1.m1.2c">\Delta v_{i}=m_{d,i}v_{d}/M</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS3.p1.1.m1.2d">roman_Δ italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_m start_POSTSUBSCRIPT italic_d , italic_i end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT / italic_M</annotation></semantics></math>, where <math alttext="m_{d,i}" class="ltx_Math" display="inline" id="S2.SS3.SSS3.p1.2.m2.2"><semantics id="S2.SS3.SSS3.p1.2.m2.2a"><msub id="S2.SS3.SSS3.p1.2.m2.2.3" xref="S2.SS3.SSS3.p1.2.m2.2.3.cmml"><mi id="S2.SS3.SSS3.p1.2.m2.2.3.2" xref="S2.SS3.SSS3.p1.2.m2.2.3.2.cmml">m</mi><mrow id="S2.SS3.SSS3.p1.2.m2.2.2.2.4" xref="S2.SS3.SSS3.p1.2.m2.2.2.2.3.cmml"><mi id="S2.SS3.SSS3.p1.2.m2.1.1.1.1" xref="S2.SS3.SSS3.p1.2.m2.1.1.1.1.cmml">d</mi><mo id="S2.SS3.SSS3.p1.2.m2.2.2.2.4.1" xref="S2.SS3.SSS3.p1.2.m2.2.2.2.3.cmml">,</mo><mi id="S2.SS3.SSS3.p1.2.m2.2.2.2.2" xref="S2.SS3.SSS3.p1.2.m2.2.2.2.2.cmml">i</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS3.p1.2.m2.2b"><apply id="S2.SS3.SSS3.p1.2.m2.2.3.cmml" xref="S2.SS3.SSS3.p1.2.m2.2.3"><csymbol cd="ambiguous" id="S2.SS3.SSS3.p1.2.m2.2.3.1.cmml" xref="S2.SS3.SSS3.p1.2.m2.2.3">subscript</csymbol><ci id="S2.SS3.SSS3.p1.2.m2.2.3.2.cmml" xref="S2.SS3.SSS3.p1.2.m2.2.3.2">𝑚</ci><list id="S2.SS3.SSS3.p1.2.m2.2.2.2.3.cmml" xref="S2.SS3.SSS3.p1.2.m2.2.2.2.4"><ci id="S2.SS3.SSS3.p1.2.m2.1.1.1.1.cmml" xref="S2.SS3.SSS3.p1.2.m2.1.1.1.1">𝑑</ci><ci id="S2.SS3.SSS3.p1.2.m2.2.2.2.2.cmml" xref="S2.SS3.SSS3.p1.2.m2.2.2.2.2">𝑖</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS3.p1.2.m2.2c">m_{d,i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS3.p1.2.m2.2d">italic_m start_POSTSUBSCRIPT italic_d , italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is the mass of the <math alttext="i^{\rm th}" class="ltx_Math" display="inline" id="S2.SS3.SSS3.p1.3.m3.1"><semantics id="S2.SS3.SSS3.p1.3.m3.1a"><msup id="S2.SS3.SSS3.p1.3.m3.1.1" xref="S2.SS3.SSS3.p1.3.m3.1.1.cmml"><mi id="S2.SS3.SSS3.p1.3.m3.1.1.2" xref="S2.SS3.SSS3.p1.3.m3.1.1.2.cmml">i</mi><mi id="S2.SS3.SSS3.p1.3.m3.1.1.3" xref="S2.SS3.SSS3.p1.3.m3.1.1.3.cmml">th</mi></msup><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS3.p1.3.m3.1b"><apply id="S2.SS3.SSS3.p1.3.m3.1.1.cmml" xref="S2.SS3.SSS3.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS3.p1.3.m3.1.1.1.cmml" xref="S2.SS3.SSS3.p1.3.m3.1.1">superscript</csymbol><ci id="S2.SS3.SSS3.p1.3.m3.1.1.2.cmml" xref="S2.SS3.SSS3.p1.3.m3.1.1.2">𝑖</ci><ci id="S2.SS3.SSS3.p1.3.m3.1.1.3.cmml" xref="S2.SS3.SSS3.p1.3.m3.1.1.3">th</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS3.p1.3.m3.1c">i^{\rm th}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS3.p1.3.m3.1d">italic_i start_POSTSUPERSCRIPT roman_th end_POSTSUPERSCRIPT</annotation></semantics></math> dust grain and <math alttext="M" class="ltx_Math" display="inline" id="S2.SS3.SSS3.p1.4.m4.1"><semantics id="S2.SS3.SSS3.p1.4.m4.1a"><mi id="S2.SS3.SSS3.p1.4.m4.1.1" xref="S2.SS3.SSS3.p1.4.m4.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS3.p1.4.m4.1b"><ci id="S2.SS3.SSS3.p1.4.m4.1.1.cmml" xref="S2.SS3.SSS3.p1.4.m4.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS3.p1.4.m4.1c">M</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS3.p1.4.m4.1d">italic_M</annotation></semantics></math> the mass of the spacecraft. We treat all grains as having the same velocity of <math alttext="v_{d}=20~{}" class="ltx_Math" display="inline" id="S2.SS3.SSS3.p1.5.m5.1"><semantics id="S2.SS3.SSS3.p1.5.m5.1a"><mrow id="S2.SS3.SSS3.p1.5.m5.1.1" xref="S2.SS3.SSS3.p1.5.m5.1.1.cmml"><msub id="S2.SS3.SSS3.p1.5.m5.1.1.2" xref="S2.SS3.SSS3.p1.5.m5.1.1.2.cmml"><mi id="S2.SS3.SSS3.p1.5.m5.1.1.2.2" xref="S2.SS3.SSS3.p1.5.m5.1.1.2.2.cmml">v</mi><mi id="S2.SS3.SSS3.p1.5.m5.1.1.2.3" xref="S2.SS3.SSS3.p1.5.m5.1.1.2.3.cmml">d</mi></msub><mo id="S2.SS3.SSS3.p1.5.m5.1.1.1" xref="S2.SS3.SSS3.p1.5.m5.1.1.1.cmml">=</mo><mn id="S2.SS3.SSS3.p1.5.m5.1.1.3" xref="S2.SS3.SSS3.p1.5.m5.1.1.3.cmml">20</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS3.p1.5.m5.1b"><apply id="S2.SS3.SSS3.p1.5.m5.1.1.cmml" xref="S2.SS3.SSS3.p1.5.m5.1.1"><eq id="S2.SS3.SSS3.p1.5.m5.1.1.1.cmml" xref="S2.SS3.SSS3.p1.5.m5.1.1.1"></eq><apply id="S2.SS3.SSS3.p1.5.m5.1.1.2.cmml" xref="S2.SS3.SSS3.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="S2.SS3.SSS3.p1.5.m5.1.1.2.1.cmml" xref="S2.SS3.SSS3.p1.5.m5.1.1.2">subscript</csymbol><ci id="S2.SS3.SSS3.p1.5.m5.1.1.2.2.cmml" xref="S2.SS3.SSS3.p1.5.m5.1.1.2.2">𝑣</ci><ci id="S2.SS3.SSS3.p1.5.m5.1.1.2.3.cmml" xref="S2.SS3.SSS3.p1.5.m5.1.1.2.3">𝑑</ci></apply><cn id="S2.SS3.SSS3.p1.5.m5.1.1.3.cmml" type="integer" xref="S2.SS3.SSS3.p1.5.m5.1.1.3">20</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS3.p1.5.m5.1c">v_{d}=20~{}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS3.p1.5.m5.1d">italic_v start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT = 20</annotation></semantics></math>km s<sup class="ltx_sup" id="S2.SS3.SSS3.p1.20.1"><span class="ltx_text ltx_font_italic" id="S2.SS3.SSS3.p1.20.1.1">-1</span></sup>, motivated by the interstellar flow of dust that is found to dominate the dust population in the outer Solar System <cite class="ltx_cite ltx_citemacro_citep">(Gruen et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib33" title="">1994</a>)</cite>. If the interferometer arm is not oriented in the flow direction of the interplanetary dust (which has been found to be moving in roughly the ecliptic with longitude <math alttext="250^{\circ}" class="ltx_Math" display="inline" id="S2.SS3.SSS3.p1.7.m7.1"><semantics id="S2.SS3.SSS3.p1.7.m7.1a"><msup id="S2.SS3.SSS3.p1.7.m7.1.1" xref="S2.SS3.SSS3.p1.7.m7.1.1.cmml"><mn id="S2.SS3.SSS3.p1.7.m7.1.1.2" xref="S2.SS3.SSS3.p1.7.m7.1.1.2.cmml">250</mn><mo id="S2.SS3.SSS3.p1.7.m7.1.1.3" xref="S2.SS3.SSS3.p1.7.m7.1.1.3.cmml">∘</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS3.p1.7.m7.1b"><apply id="S2.SS3.SSS3.p1.7.m7.1.1.cmml" xref="S2.SS3.SSS3.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS3.p1.7.m7.1.1.1.cmml" xref="S2.SS3.SSS3.p1.7.m7.1.1">superscript</csymbol><cn id="S2.SS3.SSS3.p1.7.m7.1.1.2.cmml" type="integer" xref="S2.SS3.SSS3.p1.7.m7.1.1.2">250</cn><compose id="S2.SS3.SSS3.p1.7.m7.1.1.3.cmml" xref="S2.SS3.SSS3.p1.7.m7.1.1.3"></compose></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS3.p1.7.m7.1c">250^{\circ}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS3.p1.7.m7.1d">250 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT</annotation></semantics></math>), there is a geometric suppression over our estimates. Since each dust collision can be approximated as causing a step function in the spacecraft velocity, the acceleration of the spacecraft <math alttext="dV/dt" class="ltx_Math" display="inline" id="S2.SS3.SSS3.p1.8.m8.1"><semantics id="S2.SS3.SSS3.p1.8.m8.1a"><mrow id="S2.SS3.SSS3.p1.8.m8.1.1" xref="S2.SS3.SSS3.p1.8.m8.1.1.cmml"><mrow id="S2.SS3.SSS3.p1.8.m8.1.1.2" xref="S2.SS3.SSS3.p1.8.m8.1.1.2.cmml"><mrow id="S2.SS3.SSS3.p1.8.m8.1.1.2.2" xref="S2.SS3.SSS3.p1.8.m8.1.1.2.2.cmml"><mi id="S2.SS3.SSS3.p1.8.m8.1.1.2.2.2" xref="S2.SS3.SSS3.p1.8.m8.1.1.2.2.2.cmml">d</mi><mo id="S2.SS3.SSS3.p1.8.m8.1.1.2.2.1" xref="S2.SS3.SSS3.p1.8.m8.1.1.2.2.1.cmml">⁢</mo><mi id="S2.SS3.SSS3.p1.8.m8.1.1.2.2.3" xref="S2.SS3.SSS3.p1.8.m8.1.1.2.2.3.cmml">V</mi></mrow><mo id="S2.SS3.SSS3.p1.8.m8.1.1.2.1" xref="S2.SS3.SSS3.p1.8.m8.1.1.2.1.cmml">/</mo><mi id="S2.SS3.SSS3.p1.8.m8.1.1.2.3" xref="S2.SS3.SSS3.p1.8.m8.1.1.2.3.cmml">d</mi></mrow><mo id="S2.SS3.SSS3.p1.8.m8.1.1.1" xref="S2.SS3.SSS3.p1.8.m8.1.1.1.cmml">⁢</mo><mi id="S2.SS3.SSS3.p1.8.m8.1.1.3" xref="S2.SS3.SSS3.p1.8.m8.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS3.p1.8.m8.1b"><apply id="S2.SS3.SSS3.p1.8.m8.1.1.cmml" xref="S2.SS3.SSS3.p1.8.m8.1.1"><times id="S2.SS3.SSS3.p1.8.m8.1.1.1.cmml" xref="S2.SS3.SSS3.p1.8.m8.1.1.1"></times><apply id="S2.SS3.SSS3.p1.8.m8.1.1.2.cmml" xref="S2.SS3.SSS3.p1.8.m8.1.1.2"><divide id="S2.SS3.SSS3.p1.8.m8.1.1.2.1.cmml" xref="S2.SS3.SSS3.p1.8.m8.1.1.2.1"></divide><apply id="S2.SS3.SSS3.p1.8.m8.1.1.2.2.cmml" xref="S2.SS3.SSS3.p1.8.m8.1.1.2.2"><times id="S2.SS3.SSS3.p1.8.m8.1.1.2.2.1.cmml" xref="S2.SS3.SSS3.p1.8.m8.1.1.2.2.1"></times><ci id="S2.SS3.SSS3.p1.8.m8.1.1.2.2.2.cmml" xref="S2.SS3.SSS3.p1.8.m8.1.1.2.2.2">𝑑</ci><ci id="S2.SS3.SSS3.p1.8.m8.1.1.2.2.3.cmml" xref="S2.SS3.SSS3.p1.8.m8.1.1.2.2.3">𝑉</ci></apply><ci id="S2.SS3.SSS3.p1.8.m8.1.1.2.3.cmml" xref="S2.SS3.SSS3.p1.8.m8.1.1.2.3">𝑑</ci></apply><ci id="S2.SS3.SSS3.p1.8.m8.1.1.3.cmml" xref="S2.SS3.SSS3.p1.8.m8.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS3.p1.8.m8.1c">dV/dt</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS3.p1.8.m8.1d">italic_d italic_V / italic_d italic_t</annotation></semantics></math> is then a sum of Dirac <math alttext="\delta" class="ltx_Math" display="inline" id="S2.SS3.SSS3.p1.9.m9.1"><semantics id="S2.SS3.SSS3.p1.9.m9.1a"><mi id="S2.SS3.SSS3.p1.9.m9.1.1" xref="S2.SS3.SSS3.p1.9.m9.1.1.cmml">δ</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS3.p1.9.m9.1b"><ci id="S2.SS3.SSS3.p1.9.m9.1.1.cmml" xref="S2.SS3.SSS3.p1.9.m9.1.1">𝛿</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS3.p1.9.m9.1c">\delta</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS3.p1.9.m9.1d">italic_δ</annotation></semantics></math>-functions, which in Fourier space is <math 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xref="S2.SS3.SSS3.p1.10.m10.4.4.1.1.1.1.1.1.1.1.1.1.3.3">𝑖</ci></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS3.p1.10.m10.4c">d\widetilde{V}/dt=\sum_{i}\Delta v_{d,i}\exp[-i\omega(t-t_{i})]</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS3.p1.10.m10.4d">italic_d over~ start_ARG italic_V end_ARG / italic_d italic_t = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_Δ italic_v start_POSTSUBSCRIPT italic_d , italic_i end_POSTSUBSCRIPT roman_exp [ - italic_i italic_ω ( italic_t - italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ]</annotation></semantics></math>. If we treat each grain as uncorrelated in time, then the acceleration power is <math alttext="S_{a}=2T^{-1}\langle|d\tilde{V}/dt|^{2}\rangle=2\int_{0}^{m_{\rm d,max}}d\ln m% _{d}\,d\rho_{d}/d\ln m_{d}\times[\Delta v(m_{d})]^{2}" class="ltx_Math" display="inline" id="S2.SS3.SSS3.p1.11.m11.4"><semantics id="S2.SS3.SSS3.p1.11.m11.4a"><mrow id="S2.SS3.SSS3.p1.11.m11.4.4" xref="S2.SS3.SSS3.p1.11.m11.4.4.cmml"><msub id="S2.SS3.SSS3.p1.11.m11.4.4.4" xref="S2.SS3.SSS3.p1.11.m11.4.4.4.cmml"><mi id="S2.SS3.SSS3.p1.11.m11.4.4.4.2" xref="S2.SS3.SSS3.p1.11.m11.4.4.4.2.cmml">S</mi><mi id="S2.SS3.SSS3.p1.11.m11.4.4.4.3" xref="S2.SS3.SSS3.p1.11.m11.4.4.4.3.cmml">a</mi></msub><mo id="S2.SS3.SSS3.p1.11.m11.4.4.5" xref="S2.SS3.SSS3.p1.11.m11.4.4.5.cmml">=</mo><mrow id="S2.SS3.SSS3.p1.11.m11.3.3.1" xref="S2.SS3.SSS3.p1.11.m11.3.3.1.cmml"><mn id="S2.SS3.SSS3.p1.11.m11.3.3.1.3" xref="S2.SS3.SSS3.p1.11.m11.3.3.1.3.cmml">2</mn><mo id="S2.SS3.SSS3.p1.11.m11.3.3.1.2" 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over~ start_ARG italic_V end_ARG / italic_d italic_t | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⟩ = 2 ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT roman_d , roman_max end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_d roman_ln italic_m start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT italic_d italic_ρ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT / italic_d roman_ln italic_m start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT × [ roman_Δ italic_v ( italic_m start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) ] start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>. The density distribution of interplanetary dust is found to be roughly constant per <math alttext="\log" class="ltx_Math" display="inline" id="S2.SS3.SSS3.p1.12.m12.1"><semantics id="S2.SS3.SSS3.p1.12.m12.1a"><mi id="S2.SS3.SSS3.p1.12.m12.1.1" xref="S2.SS3.SSS3.p1.12.m12.1.1.cmml">log</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS3.p1.12.m12.1b"><log id="S2.SS3.SSS3.p1.12.m12.1.1.cmml" xref="S2.SS3.SSS3.p1.12.m12.1.1"></log></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS3.p1.12.m12.1c">\log</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS3.p1.12.m12.1d">roman_log</annotation></semantics></math> mass between <math alttext="m_{d}=10^{-12}" class="ltx_Math" display="inline" id="S2.SS3.SSS3.p1.13.m13.1"><semantics id="S2.SS3.SSS3.p1.13.m13.1a"><mrow id="S2.SS3.SSS3.p1.13.m13.1.1" xref="S2.SS3.SSS3.p1.13.m13.1.1.cmml"><msub id="S2.SS3.SSS3.p1.13.m13.1.1.2" xref="S2.SS3.SSS3.p1.13.m13.1.1.2.cmml"><mi id="S2.SS3.SSS3.p1.13.m13.1.1.2.2" xref="S2.SS3.SSS3.p1.13.m13.1.1.2.2.cmml">m</mi><mi id="S2.SS3.SSS3.p1.13.m13.1.1.2.3" xref="S2.SS3.SSS3.p1.13.m13.1.1.2.3.cmml">d</mi></msub><mo id="S2.SS3.SSS3.p1.13.m13.1.1.1" xref="S2.SS3.SSS3.p1.13.m13.1.1.1.cmml">=</mo><msup id="S2.SS3.SSS3.p1.13.m13.1.1.3" xref="S2.SS3.SSS3.p1.13.m13.1.1.3.cmml"><mn id="S2.SS3.SSS3.p1.13.m13.1.1.3.2" xref="S2.SS3.SSS3.p1.13.m13.1.1.3.2.cmml">10</mn><mrow id="S2.SS3.SSS3.p1.13.m13.1.1.3.3" xref="S2.SS3.SSS3.p1.13.m13.1.1.3.3.cmml"><mo id="S2.SS3.SSS3.p1.13.m13.1.1.3.3a" xref="S2.SS3.SSS3.p1.13.m13.1.1.3.3.cmml">−</mo><mn id="S2.SS3.SSS3.p1.13.m13.1.1.3.3.2" xref="S2.SS3.SSS3.p1.13.m13.1.1.3.3.2.cmml">12</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS3.p1.13.m13.1b"><apply id="S2.SS3.SSS3.p1.13.m13.1.1.cmml" xref="S2.SS3.SSS3.p1.13.m13.1.1"><eq id="S2.SS3.SSS3.p1.13.m13.1.1.1.cmml" xref="S2.SS3.SSS3.p1.13.m13.1.1.1"></eq><apply id="S2.SS3.SSS3.p1.13.m13.1.1.2.cmml" 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encoding="application/x-tex" id="S2.SS3.SSS3.p1.13.m13.1c">m_{d}=10^{-12}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS3.p1.13.m13.1d">italic_m start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT = 10 start_POSTSUPERSCRIPT - 12 end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="10^{-9}" class="ltx_Math" display="inline" id="S2.SS3.SSS3.p1.14.m14.1"><semantics id="S2.SS3.SSS3.p1.14.m14.1a"><msup id="S2.SS3.SSS3.p1.14.m14.1.1" xref="S2.SS3.SSS3.p1.14.m14.1.1.cmml"><mn id="S2.SS3.SSS3.p1.14.m14.1.1.2" xref="S2.SS3.SSS3.p1.14.m14.1.1.2.cmml">10</mn><mrow id="S2.SS3.SSS3.p1.14.m14.1.1.3" xref="S2.SS3.SSS3.p1.14.m14.1.1.3.cmml"><mo id="S2.SS3.SSS3.p1.14.m14.1.1.3a" xref="S2.SS3.SSS3.p1.14.m14.1.1.3.cmml">−</mo><mn id="S2.SS3.SSS3.p1.14.m14.1.1.3.2" xref="S2.SS3.SSS3.p1.14.m14.1.1.3.2.cmml">9</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS3.p1.14.m14.1b"><apply id="S2.SS3.SSS3.p1.14.m14.1.1.cmml" xref="S2.SS3.SSS3.p1.14.m14.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS3.p1.14.m14.1.1.1.cmml" xref="S2.SS3.SSS3.p1.14.m14.1.1">superscript</csymbol><cn id="S2.SS3.SSS3.p1.14.m14.1.1.2.cmml" type="integer" xref="S2.SS3.SSS3.p1.14.m14.1.1.2">10</cn><apply id="S2.SS3.SSS3.p1.14.m14.1.1.3.cmml" xref="S2.SS3.SSS3.p1.14.m14.1.1.3"><minus id="S2.SS3.SSS3.p1.14.m14.1.1.3.1.cmml" xref="S2.SS3.SSS3.p1.14.m14.1.1.3"></minus><cn id="S2.SS3.SSS3.p1.14.m14.1.1.3.2.cmml" type="integer" xref="S2.SS3.SSS3.p1.14.m14.1.1.3.2">9</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS3.p1.14.m14.1c">10^{-9}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS3.p1.14.m14.1d">10 start_POSTSUPERSCRIPT - 9 end_POSTSUPERSCRIPT</annotation></semantics></math>g with <math alttext="d\rho_{d}/d\ln m_{d}=1.2\times 10^{-27}" class="ltx_Math" display="inline" id="S2.SS3.SSS3.p1.15.m15.1"><semantics id="S2.SS3.SSS3.p1.15.m15.1a"><mrow 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xref="S2.SS3.SSS3.p1.15.m15.1.1.2.2.3.cmml">d</mi></mrow><mo id="S2.SS3.SSS3.p1.15.m15.1.1.2.1" lspace="0.167em" xref="S2.SS3.SSS3.p1.15.m15.1.1.2.1.cmml">⁢</mo><mrow id="S2.SS3.SSS3.p1.15.m15.1.1.2.3" xref="S2.SS3.SSS3.p1.15.m15.1.1.2.3.cmml"><mi id="S2.SS3.SSS3.p1.15.m15.1.1.2.3.1" xref="S2.SS3.SSS3.p1.15.m15.1.1.2.3.1.cmml">ln</mi><mo id="S2.SS3.SSS3.p1.15.m15.1.1.2.3a" lspace="0.167em" xref="S2.SS3.SSS3.p1.15.m15.1.1.2.3.cmml">⁡</mo><msub id="S2.SS3.SSS3.p1.15.m15.1.1.2.3.2" xref="S2.SS3.SSS3.p1.15.m15.1.1.2.3.2.cmml"><mi id="S2.SS3.SSS3.p1.15.m15.1.1.2.3.2.2" xref="S2.SS3.SSS3.p1.15.m15.1.1.2.3.2.2.cmml">m</mi><mi id="S2.SS3.SSS3.p1.15.m15.1.1.2.3.2.3" xref="S2.SS3.SSS3.p1.15.m15.1.1.2.3.2.3.cmml">d</mi></msub></mrow></mrow><mo id="S2.SS3.SSS3.p1.15.m15.1.1.1" xref="S2.SS3.SSS3.p1.15.m15.1.1.1.cmml">=</mo><mrow id="S2.SS3.SSS3.p1.15.m15.1.1.3" xref="S2.SS3.SSS3.p1.15.m15.1.1.3.cmml"><mn id="S2.SS3.SSS3.p1.15.m15.1.1.3.2" xref="S2.SS3.SSS3.p1.15.m15.1.1.3.2.cmml">1.2</mn><mo id="S2.SS3.SSS3.p1.15.m15.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S2.SS3.SSS3.p1.15.m15.1.1.3.1.cmml">×</mo><msup id="S2.SS3.SSS3.p1.15.m15.1.1.3.3" xref="S2.SS3.SSS3.p1.15.m15.1.1.3.3.cmml"><mn id="S2.SS3.SSS3.p1.15.m15.1.1.3.3.2" xref="S2.SS3.SSS3.p1.15.m15.1.1.3.3.2.cmml">10</mn><mrow id="S2.SS3.SSS3.p1.15.m15.1.1.3.3.3" xref="S2.SS3.SSS3.p1.15.m15.1.1.3.3.3.cmml"><mo id="S2.SS3.SSS3.p1.15.m15.1.1.3.3.3a" xref="S2.SS3.SSS3.p1.15.m15.1.1.3.3.3.cmml">−</mo><mn id="S2.SS3.SSS3.p1.15.m15.1.1.3.3.3.2" xref="S2.SS3.SSS3.p1.15.m15.1.1.3.3.3.2.cmml">27</mn></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS3.p1.15.m15.1b"><apply id="S2.SS3.SSS3.p1.15.m15.1.1.cmml" xref="S2.SS3.SSS3.p1.15.m15.1.1"><eq id="S2.SS3.SSS3.p1.15.m15.1.1.1.cmml" xref="S2.SS3.SSS3.p1.15.m15.1.1.1"></eq><apply id="S2.SS3.SSS3.p1.15.m15.1.1.2.cmml" xref="S2.SS3.SSS3.p1.15.m15.1.1.2"><times id="S2.SS3.SSS3.p1.15.m15.1.1.2.1.cmml" xref="S2.SS3.SSS3.p1.15.m15.1.1.2.1"></times><apply id="S2.SS3.SSS3.p1.15.m15.1.1.2.2.cmml" xref="S2.SS3.SSS3.p1.15.m15.1.1.2.2"><divide id="S2.SS3.SSS3.p1.15.m15.1.1.2.2.1.cmml" xref="S2.SS3.SSS3.p1.15.m15.1.1.2.2.1"></divide><apply id="S2.SS3.SSS3.p1.15.m15.1.1.2.2.2.cmml" xref="S2.SS3.SSS3.p1.15.m15.1.1.2.2.2"><times id="S2.SS3.SSS3.p1.15.m15.1.1.2.2.2.1.cmml" xref="S2.SS3.SSS3.p1.15.m15.1.1.2.2.2.1"></times><ci id="S2.SS3.SSS3.p1.15.m15.1.1.2.2.2.2.cmml" xref="S2.SS3.SSS3.p1.15.m15.1.1.2.2.2.2">𝑑</ci><apply id="S2.SS3.SSS3.p1.15.m15.1.1.2.2.2.3.cmml" xref="S2.SS3.SSS3.p1.15.m15.1.1.2.2.2.3"><csymbol cd="ambiguous" id="S2.SS3.SSS3.p1.15.m15.1.1.2.2.2.3.1.cmml" xref="S2.SS3.SSS3.p1.15.m15.1.1.2.2.2.3">subscript</csymbol><ci id="S2.SS3.SSS3.p1.15.m15.1.1.2.2.2.3.2.cmml" xref="S2.SS3.SSS3.p1.15.m15.1.1.2.2.2.3.2">𝜌</ci><ci id="S2.SS3.SSS3.p1.15.m15.1.1.2.2.2.3.3.cmml" xref="S2.SS3.SSS3.p1.15.m15.1.1.2.2.2.3.3">𝑑</ci></apply></apply><ci id="S2.SS3.SSS3.p1.15.m15.1.1.2.2.3.cmml" xref="S2.SS3.SSS3.p1.15.m15.1.1.2.2.3">𝑑</ci></apply><apply id="S2.SS3.SSS3.p1.15.m15.1.1.2.3.cmml" xref="S2.SS3.SSS3.p1.15.m15.1.1.2.3"><ln id="S2.SS3.SSS3.p1.15.m15.1.1.2.3.1.cmml" xref="S2.SS3.SSS3.p1.15.m15.1.1.2.3.1"></ln><apply id="S2.SS3.SSS3.p1.15.m15.1.1.2.3.2.cmml" xref="S2.SS3.SSS3.p1.15.m15.1.1.2.3.2"><csymbol cd="ambiguous" id="S2.SS3.SSS3.p1.15.m15.1.1.2.3.2.1.cmml" xref="S2.SS3.SSS3.p1.15.m15.1.1.2.3.2">subscript</csymbol><ci id="S2.SS3.SSS3.p1.15.m15.1.1.2.3.2.2.cmml" xref="S2.SS3.SSS3.p1.15.m15.1.1.2.3.2.2">𝑚</ci><ci id="S2.SS3.SSS3.p1.15.m15.1.1.2.3.2.3.cmml" xref="S2.SS3.SSS3.p1.15.m15.1.1.2.3.2.3">𝑑</ci></apply></apply></apply><apply id="S2.SS3.SSS3.p1.15.m15.1.1.3.cmml" xref="S2.SS3.SSS3.p1.15.m15.1.1.3"><times id="S2.SS3.SSS3.p1.15.m15.1.1.3.1.cmml" xref="S2.SS3.SSS3.p1.15.m15.1.1.3.1"></times><cn id="S2.SS3.SSS3.p1.15.m15.1.1.3.2.cmml" type="float" xref="S2.SS3.SSS3.p1.15.m15.1.1.3.2">1.2</cn><apply id="S2.SS3.SSS3.p1.15.m15.1.1.3.3.cmml" xref="S2.SS3.SSS3.p1.15.m15.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS3.SSS3.p1.15.m15.1.1.3.3.1.cmml" xref="S2.SS3.SSS3.p1.15.m15.1.1.3.3">superscript</csymbol><cn id="S2.SS3.SSS3.p1.15.m15.1.1.3.3.2.cmml" type="integer" xref="S2.SS3.SSS3.p1.15.m15.1.1.3.3.2">10</cn><apply id="S2.SS3.SSS3.p1.15.m15.1.1.3.3.3.cmml" xref="S2.SS3.SSS3.p1.15.m15.1.1.3.3.3"><minus id="S2.SS3.SSS3.p1.15.m15.1.1.3.3.3.1.cmml" xref="S2.SS3.SSS3.p1.15.m15.1.1.3.3.3"></minus><cn id="S2.SS3.SSS3.p1.15.m15.1.1.3.3.3.2.cmml" type="integer" xref="S2.SS3.SSS3.p1.15.m15.1.1.3.3.3.2">27</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS3.p1.15.m15.1c">d\rho_{d}/d\ln m_{d}=1.2\times 10^{-27}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS3.p1.15.m15.1d">italic_d italic_ρ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT / italic_d roman_ln italic_m start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT = 1.2 × 10 start_POSTSUPERSCRIPT - 27 end_POSTSUPERSCRIPT</annotation></semantics></math>g cm<sup class="ltx_sup" id="S2.SS3.SSS3.p1.20.2"><span class="ltx_text ltx_font_italic" id="S2.SS3.SSS3.p1.20.2.1">-3</span></sup> <cite class="ltx_cite ltx_citemacro_citep">(Grün et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib34" title="">2000</a>)</cite>. The grain size distribution has not been measured above <math alttext="10^{-9}" class="ltx_Math" display="inline" id="S2.SS3.SSS3.p1.17.m17.1"><semantics id="S2.SS3.SSS3.p1.17.m17.1a"><msup id="S2.SS3.SSS3.p1.17.m17.1.1" xref="S2.SS3.SSS3.p1.17.m17.1.1.cmml"><mn id="S2.SS3.SSS3.p1.17.m17.1.1.2" xref="S2.SS3.SSS3.p1.17.m17.1.1.2.cmml">10</mn><mrow id="S2.SS3.SSS3.p1.17.m17.1.1.3" xref="S2.SS3.SSS3.p1.17.m17.1.1.3.cmml"><mo id="S2.SS3.SSS3.p1.17.m17.1.1.3a" xref="S2.SS3.SSS3.p1.17.m17.1.1.3.cmml">−</mo><mn id="S2.SS3.SSS3.p1.17.m17.1.1.3.2" xref="S2.SS3.SSS3.p1.17.m17.1.1.3.2.cmml">9</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS3.p1.17.m17.1b"><apply id="S2.SS3.SSS3.p1.17.m17.1.1.cmml" xref="S2.SS3.SSS3.p1.17.m17.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS3.p1.17.m17.1.1.1.cmml" xref="S2.SS3.SSS3.p1.17.m17.1.1">superscript</csymbol><cn id="S2.SS3.SSS3.p1.17.m17.1.1.2.cmml" type="integer" xref="S2.SS3.SSS3.p1.17.m17.1.1.2">10</cn><apply id="S2.SS3.SSS3.p1.17.m17.1.1.3.cmml" xref="S2.SS3.SSS3.p1.17.m17.1.1.3"><minus id="S2.SS3.SSS3.p1.17.m17.1.1.3.1.cmml" xref="S2.SS3.SSS3.p1.17.m17.1.1.3"></minus><cn id="S2.SS3.SSS3.p1.17.m17.1.1.3.2.cmml" type="integer" xref="S2.SS3.SSS3.p1.17.m17.1.1.3.2">9</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS3.p1.17.m17.1c">10^{-9}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS3.p1.17.m17.1d">10 start_POSTSUPERSCRIPT - 9 end_POSTSUPERSCRIPT</annotation></semantics></math>g. We assume <math alttext="d\rho_{d}/d\ln m_{d}=1.2\times 10^{-27}" class="ltx_Math" display="inline" id="S2.SS3.SSS3.p1.18.m18.1"><semantics id="S2.SS3.SSS3.p1.18.m18.1a"><mrow id="S2.SS3.SSS3.p1.18.m18.1.1" xref="S2.SS3.SSS3.p1.18.m18.1.1.cmml"><mrow id="S2.SS3.SSS3.p1.18.m18.1.1.2" xref="S2.SS3.SSS3.p1.18.m18.1.1.2.cmml"><mrow id="S2.SS3.SSS3.p1.18.m18.1.1.2.2" xref="S2.SS3.SSS3.p1.18.m18.1.1.2.2.cmml"><mrow id="S2.SS3.SSS3.p1.18.m18.1.1.2.2.2" xref="S2.SS3.SSS3.p1.18.m18.1.1.2.2.2.cmml"><mi id="S2.SS3.SSS3.p1.18.m18.1.1.2.2.2.2" xref="S2.SS3.SSS3.p1.18.m18.1.1.2.2.2.2.cmml">d</mi><mo id="S2.SS3.SSS3.p1.18.m18.1.1.2.2.2.1" xref="S2.SS3.SSS3.p1.18.m18.1.1.2.2.2.1.cmml">⁢</mo><msub id="S2.SS3.SSS3.p1.18.m18.1.1.2.2.2.3" xref="S2.SS3.SSS3.p1.18.m18.1.1.2.2.2.3.cmml"><mi id="S2.SS3.SSS3.p1.18.m18.1.1.2.2.2.3.2" xref="S2.SS3.SSS3.p1.18.m18.1.1.2.2.2.3.2.cmml">ρ</mi><mi id="S2.SS3.SSS3.p1.18.m18.1.1.2.2.2.3.3" xref="S2.SS3.SSS3.p1.18.m18.1.1.2.2.2.3.3.cmml">d</mi></msub></mrow><mo 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xref="S2.SS3.SSS3.p1.18.m18.1.1.3.cmml"><mn id="S2.SS3.SSS3.p1.18.m18.1.1.3.2" xref="S2.SS3.SSS3.p1.18.m18.1.1.3.2.cmml">1.2</mn><mo id="S2.SS3.SSS3.p1.18.m18.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S2.SS3.SSS3.p1.18.m18.1.1.3.1.cmml">×</mo><msup id="S2.SS3.SSS3.p1.18.m18.1.1.3.3" xref="S2.SS3.SSS3.p1.18.m18.1.1.3.3.cmml"><mn id="S2.SS3.SSS3.p1.18.m18.1.1.3.3.2" xref="S2.SS3.SSS3.p1.18.m18.1.1.3.3.2.cmml">10</mn><mrow id="S2.SS3.SSS3.p1.18.m18.1.1.3.3.3" xref="S2.SS3.SSS3.p1.18.m18.1.1.3.3.3.cmml"><mo id="S2.SS3.SSS3.p1.18.m18.1.1.3.3.3a" xref="S2.SS3.SSS3.p1.18.m18.1.1.3.3.3.cmml">−</mo><mn id="S2.SS3.SSS3.p1.18.m18.1.1.3.3.3.2" xref="S2.SS3.SSS3.p1.18.m18.1.1.3.3.3.2.cmml">27</mn></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS3.p1.18.m18.1b"><apply id="S2.SS3.SSS3.p1.18.m18.1.1.cmml" xref="S2.SS3.SSS3.p1.18.m18.1.1"><eq id="S2.SS3.SSS3.p1.18.m18.1.1.1.cmml" xref="S2.SS3.SSS3.p1.18.m18.1.1.1"></eq><apply 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xref="S2.SS3.SSS3.p1.18.m18.1.1.3.2">1.2</cn><apply id="S2.SS3.SSS3.p1.18.m18.1.1.3.3.cmml" xref="S2.SS3.SSS3.p1.18.m18.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS3.SSS3.p1.18.m18.1.1.3.3.1.cmml" xref="S2.SS3.SSS3.p1.18.m18.1.1.3.3">superscript</csymbol><cn id="S2.SS3.SSS3.p1.18.m18.1.1.3.3.2.cmml" type="integer" xref="S2.SS3.SSS3.p1.18.m18.1.1.3.3.2">10</cn><apply id="S2.SS3.SSS3.p1.18.m18.1.1.3.3.3.cmml" xref="S2.SS3.SSS3.p1.18.m18.1.1.3.3.3"><minus id="S2.SS3.SSS3.p1.18.m18.1.1.3.3.3.1.cmml" xref="S2.SS3.SSS3.p1.18.m18.1.1.3.3.3"></minus><cn id="S2.SS3.SSS3.p1.18.m18.1.1.3.3.3.2.cmml" type="integer" xref="S2.SS3.SSS3.p1.18.m18.1.1.3.3.3.2">27</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS3.p1.18.m18.1c">d\rho_{d}/d\ln m_{d}=1.2\times 10^{-27}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS3.p1.18.m18.1d">italic_d italic_ρ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT / italic_d roman_ln italic_m start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT = 1.2 × 10 start_POSTSUPERSCRIPT - 27 end_POSTSUPERSCRIPT</annotation></semantics></math>g cm<sup class="ltx_sup" id="S2.SS3.SSS3.p1.20.3"><span class="ltx_text ltx_font_italic" id="S2.SS3.SSS3.p1.20.3.1">-3</span></sup> up to <math alttext="m_{d,\rm max}" class="ltx_Math" display="inline" id="S2.SS3.SSS3.p1.20.m20.2"><semantics id="S2.SS3.SSS3.p1.20.m20.2a"><msub id="S2.SS3.SSS3.p1.20.m20.2.3" xref="S2.SS3.SSS3.p1.20.m20.2.3.cmml"><mi id="S2.SS3.SSS3.p1.20.m20.2.3.2" xref="S2.SS3.SSS3.p1.20.m20.2.3.2.cmml">m</mi><mrow id="S2.SS3.SSS3.p1.20.m20.2.2.2.4" xref="S2.SS3.SSS3.p1.20.m20.2.2.2.3.cmml"><mi id="S2.SS3.SSS3.p1.20.m20.1.1.1.1" xref="S2.SS3.SSS3.p1.20.m20.1.1.1.1.cmml">d</mi><mo id="S2.SS3.SSS3.p1.20.m20.2.2.2.4.1" xref="S2.SS3.SSS3.p1.20.m20.2.2.2.3.cmml">,</mo><mi id="S2.SS3.SSS3.p1.20.m20.2.2.2.2" xref="S2.SS3.SSS3.p1.20.m20.2.2.2.2.cmml">max</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS3.p1.20.m20.2b"><apply id="S2.SS3.SSS3.p1.20.m20.2.3.cmml" xref="S2.SS3.SSS3.p1.20.m20.2.3"><csymbol cd="ambiguous" id="S2.SS3.SSS3.p1.20.m20.2.3.1.cmml" xref="S2.SS3.SSS3.p1.20.m20.2.3">subscript</csymbol><ci id="S2.SS3.SSS3.p1.20.m20.2.3.2.cmml" xref="S2.SS3.SSS3.p1.20.m20.2.3.2">𝑚</ci><list id="S2.SS3.SSS3.p1.20.m20.2.2.2.3.cmml" xref="S2.SS3.SSS3.p1.20.m20.2.2.2.4"><ci id="S2.SS3.SSS3.p1.20.m20.1.1.1.1.cmml" xref="S2.SS3.SSS3.p1.20.m20.1.1.1.1">𝑑</ci><ci id="S2.SS3.SSS3.p1.20.m20.2.2.2.2.cmml" xref="S2.SS3.SSS3.p1.20.m20.2.2.2.2">max</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS3.p1.20.m20.2c">m_{d,\rm max}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS3.p1.20.m20.2d">italic_m start_POSTSUBSCRIPT italic_d , roman_max end_POSTSUBSCRIPT</annotation></semantics></math>, which yields an acceleration noise power of</p> <table class="ltx_equation ltx_eqn_table" id="S2.E17"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\sqrt{S_{a}}=9.8\times 10^{-15}\text{m s${}^{-2}$ Hz}^{-1/2}\left(\frac{m_{d,% \rm max}}{10^{-9}{\rm g}}\frac{d\rho_{d}/d\ln m_{d}}{1.2\times 10^{-27}{\rm\,% cm}^{-3}}\right)^{1/2}\left(\frac{M}{10^{3}{\rm kg}}\right)^{-1}\left(\frac{A}% {10{\rm m^{2}}}\right)^{1/2}." class="ltx_Math" display="block" id="S2.E17.m1.6"><semantics id="S2.E17.m1.6a"><mrow id="S2.E17.m1.6.6.1" xref="S2.E17.m1.6.6.1.1.cmml"><mrow id="S2.E17.m1.6.6.1.1" xref="S2.E17.m1.6.6.1.1.cmml"><msqrt id="S2.E17.m1.6.6.1.1.3" xref="S2.E17.m1.6.6.1.1.3.cmml"><msub id="S2.E17.m1.6.6.1.1.3.2" xref="S2.E17.m1.6.6.1.1.3.2.cmml"><mi id="S2.E17.m1.6.6.1.1.3.2.2" xref="S2.E17.m1.6.6.1.1.3.2.2.cmml">S</mi><mi id="S2.E17.m1.6.6.1.1.3.2.3" xref="S2.E17.m1.6.6.1.1.3.2.3.cmml">a</mi></msub></msqrt><mo id="S2.E17.m1.6.6.1.1.2" xref="S2.E17.m1.6.6.1.1.2.cmml">=</mo><mrow id="S2.E17.m1.6.6.1.1.1" xref="S2.E17.m1.6.6.1.1.1.cmml"><mrow id="S2.E17.m1.6.6.1.1.1.3" xref="S2.E17.m1.6.6.1.1.1.3.cmml"><mn id="S2.E17.m1.6.6.1.1.1.3.2" xref="S2.E17.m1.6.6.1.1.1.3.2.cmml">9.8</mn><mo id="S2.E17.m1.6.6.1.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S2.E17.m1.6.6.1.1.1.3.1.cmml">×</mo><msup id="S2.E17.m1.6.6.1.1.1.3.3" xref="S2.E17.m1.6.6.1.1.1.3.3.cmml"><mn id="S2.E17.m1.6.6.1.1.1.3.3.2" xref="S2.E17.m1.6.6.1.1.1.3.3.2.cmml">10</mn><mrow id="S2.E17.m1.6.6.1.1.1.3.3.3" xref="S2.E17.m1.6.6.1.1.1.3.3.3.cmml"><mo id="S2.E17.m1.6.6.1.1.1.3.3.3a" xref="S2.E17.m1.6.6.1.1.1.3.3.3.cmml">−</mo><mn id="S2.E17.m1.6.6.1.1.1.3.3.3.2" xref="S2.E17.m1.6.6.1.1.1.3.3.3.2.cmml">15</mn></mrow></msup></mrow><mo id="S2.E17.m1.6.6.1.1.1.2" xref="S2.E17.m1.6.6.1.1.1.2.cmml">⁢</mo><msup id="S2.E17.m1.6.6.1.1.1.4" xref="S2.E17.m1.6.6.1.1.1.4.cmml"><mrow id="S2.E17.m1.1.1.1" xref="S2.E17.m1.1.1.1f.cmml"><mtext id="S2.E17.m1.1.1.1a" xref="S2.E17.m1.1.1.1f.cmml">m s</mtext><mtext 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id="S2.E17.m1.6c">\sqrt{S_{a}}=9.8\times 10^{-15}\text{m s${}^{-2}$ Hz}^{-1/2}\left(\frac{m_{d,% \rm max}}{10^{-9}{\rm g}}\frac{d\rho_{d}/d\ln m_{d}}{1.2\times 10^{-27}{\rm\,% cm}^{-3}}\right)^{1/2}\left(\frac{M}{10^{3}{\rm kg}}\right)^{-1}\left(\frac{A}% {10{\rm m^{2}}}\right)^{1/2}.</annotation><annotation encoding="application/x-llamapun" id="S2.E17.m1.6d">square-root start_ARG italic_S start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT end_ARG = 9.8 × 10 start_POSTSUPERSCRIPT - 15 end_POSTSUPERSCRIPT m s Hz start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT ( divide start_ARG italic_m start_POSTSUBSCRIPT italic_d , roman_max end_POSTSUBSCRIPT end_ARG start_ARG 10 start_POSTSUPERSCRIPT - 9 end_POSTSUPERSCRIPT roman_g end_ARG divide start_ARG italic_d italic_ρ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT / italic_d roman_ln italic_m start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT end_ARG start_ARG 1.2 × 10 start_POSTSUPERSCRIPT - 27 end_POSTSUPERSCRIPT roman_cm start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT end_ARG ) start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT ( divide start_ARG italic_M end_ARG start_ARG 10 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT roman_kg end_ARG ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( divide start_ARG italic_A end_ARG start_ARG 10 roman_m start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(17)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS3.SSS3.p1.22">Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.F2" title="Figure 2 ‣ 2.3 Acceleration noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">2</span></a> shows that the dust acceleration power given by equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.E17" title="In 2.3.3 Dust ‣ 2.3 Acceleration noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">17</span></a>) is a subdominant source of acceleration for our fiducial spacecraft specifications. Only if we increase the maximum mass to <math alttext="10^{-7}{\rm g}" class="ltx_Math" display="inline" id="S2.SS3.SSS3.p1.21.m1.1"><semantics id="S2.SS3.SSS3.p1.21.m1.1a"><mrow id="S2.SS3.SSS3.p1.21.m1.1.1" xref="S2.SS3.SSS3.p1.21.m1.1.1.cmml"><msup id="S2.SS3.SSS3.p1.21.m1.1.1.2" xref="S2.SS3.SSS3.p1.21.m1.1.1.2.cmml"><mn id="S2.SS3.SSS3.p1.21.m1.1.1.2.2" xref="S2.SS3.SSS3.p1.21.m1.1.1.2.2.cmml">10</mn><mrow id="S2.SS3.SSS3.p1.21.m1.1.1.2.3" xref="S2.SS3.SSS3.p1.21.m1.1.1.2.3.cmml"><mo id="S2.SS3.SSS3.p1.21.m1.1.1.2.3a" xref="S2.SS3.SSS3.p1.21.m1.1.1.2.3.cmml">−</mo><mn id="S2.SS3.SSS3.p1.21.m1.1.1.2.3.2" xref="S2.SS3.SSS3.p1.21.m1.1.1.2.3.2.cmml">7</mn></mrow></msup><mo id="S2.SS3.SSS3.p1.21.m1.1.1.1" xref="S2.SS3.SSS3.p1.21.m1.1.1.1.cmml">⁢</mo><mi id="S2.SS3.SSS3.p1.21.m1.1.1.3" mathvariant="normal" xref="S2.SS3.SSS3.p1.21.m1.1.1.3.cmml">g</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS3.p1.21.m1.1b"><apply id="S2.SS3.SSS3.p1.21.m1.1.1.cmml" xref="S2.SS3.SSS3.p1.21.m1.1.1"><times id="S2.SS3.SSS3.p1.21.m1.1.1.1.cmml" xref="S2.SS3.SSS3.p1.21.m1.1.1.1"></times><apply id="S2.SS3.SSS3.p1.21.m1.1.1.2.cmml" xref="S2.SS3.SSS3.p1.21.m1.1.1.2"><csymbol cd="ambiguous" id="S2.SS3.SSS3.p1.21.m1.1.1.2.1.cmml" xref="S2.SS3.SSS3.p1.21.m1.1.1.2">superscript</csymbol><cn id="S2.SS3.SSS3.p1.21.m1.1.1.2.2.cmml" type="integer" xref="S2.SS3.SSS3.p1.21.m1.1.1.2.2">10</cn><apply id="S2.SS3.SSS3.p1.21.m1.1.1.2.3.cmml" xref="S2.SS3.SSS3.p1.21.m1.1.1.2.3"><minus id="S2.SS3.SSS3.p1.21.m1.1.1.2.3.1.cmml" xref="S2.SS3.SSS3.p1.21.m1.1.1.2.3"></minus><cn id="S2.SS3.SSS3.p1.21.m1.1.1.2.3.2.cmml" type="integer" xref="S2.SS3.SSS3.p1.21.m1.1.1.2.3.2">7</cn></apply></apply><ci id="S2.SS3.SSS3.p1.21.m1.1.1.3.cmml" xref="S2.SS3.SSS3.p1.21.m1.1.1.3">g</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS3.p1.21.m1.1c">10^{-7}{\rm g}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS3.p1.21.m1.1d">10 start_POSTSUPERSCRIPT - 7 end_POSTSUPERSCRIPT roman_g</annotation></semantics></math> does it become comparable to other sources of acceleration at <math alttext="r=30\," class="ltx_Math" display="inline" id="S2.SS3.SSS3.p1.22.m2.1"><semantics id="S2.SS3.SSS3.p1.22.m2.1a"><mrow id="S2.SS3.SSS3.p1.22.m2.1.1" xref="S2.SS3.SSS3.p1.22.m2.1.1.cmml"><mi id="S2.SS3.SSS3.p1.22.m2.1.1.2" xref="S2.SS3.SSS3.p1.22.m2.1.1.2.cmml">r</mi><mo id="S2.SS3.SSS3.p1.22.m2.1.1.1" xref="S2.SS3.SSS3.p1.22.m2.1.1.1.cmml">=</mo><mn id="S2.SS3.SSS3.p1.22.m2.1.1.3" xref="S2.SS3.SSS3.p1.22.m2.1.1.3.cmml">30</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS3.p1.22.m2.1b"><apply id="S2.SS3.SSS3.p1.22.m2.1.1.cmml" xref="S2.SS3.SSS3.p1.22.m2.1.1"><eq id="S2.SS3.SSS3.p1.22.m2.1.1.1.cmml" xref="S2.SS3.SSS3.p1.22.m2.1.1.1"></eq><ci id="S2.SS3.SSS3.p1.22.m2.1.1.2.cmml" xref="S2.SS3.SSS3.p1.22.m2.1.1.2">𝑟</ci><cn id="S2.SS3.SSS3.p1.22.m2.1.1.3.cmml" type="integer" xref="S2.SS3.SSS3.p1.22.m2.1.1.3">30</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS3.p1.22.m2.1c">r=30\,</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS3.p1.22.m2.1d">italic_r = 30</annotation></semantics></math>AU.</p> </div> <div class="ltx_para" id="S2.SS3.SSS3.p2"> <p class="ltx_p" id="S2.SS3.SSS3.p2.9">The accelerations from more massive dust particles can be fit and their contribution to the noise power removed. 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id="S2.SS3.SSS3.p2.1.m1.1.1.1.1.1.1.5.1.cmml" xref="S2.SS3.SSS3.p2.1.m1.1.1.1.1.1.1.5">subscript</csymbol><ci id="S2.SS3.SSS3.p2.1.m1.1.1.1.1.1.1.5.2.cmml" xref="S2.SS3.SSS3.p2.1.m1.1.1.1.1.1.1.5.2">𝑇</ci><ci id="S2.SS3.SSS3.p2.1.m1.1.1.1.1.1.1.5.3.cmml" xref="S2.SS3.SSS3.p2.1.m1.1.1.1.1.1.1.5.3">fit</ci></apply><ci id="S2.SS3.SSS3.p2.1.m1.1.1.1.1.1.1.6.cmml" xref="S2.SS3.SSS3.p2.1.m1.1.1.1.1.1.1.6">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS3.p2.1.m1.1c">h_{c}^{\rm dust}={\Delta v}/(2\pi^{2}f^{2}T_{\rm fit}x)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS3.p2.1.m1.1d">italic_h start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_dust end_POSTSUPERSCRIPT = roman_Δ italic_v / ( 2 italic_π start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_f start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_T start_POSTSUBSCRIPT roman_fit end_POSTSUBSCRIPT italic_x )</annotation></semantics></math>, where <math alttext="T_{\rm fit}" class="ltx_Math" display="inline" id="S2.SS3.SSS3.p2.2.m2.1"><semantics id="S2.SS3.SSS3.p2.2.m2.1a"><msub id="S2.SS3.SSS3.p2.2.m2.1.1" xref="S2.SS3.SSS3.p2.2.m2.1.1.cmml"><mi id="S2.SS3.SSS3.p2.2.m2.1.1.2" xref="S2.SS3.SSS3.p2.2.m2.1.1.2.cmml">T</mi><mi id="S2.SS3.SSS3.p2.2.m2.1.1.3" xref="S2.SS3.SSS3.p2.2.m2.1.1.3.cmml">fit</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS3.p2.2.m2.1b"><apply id="S2.SS3.SSS3.p2.2.m2.1.1.cmml" xref="S2.SS3.SSS3.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS3.p2.2.m2.1.1.1.cmml" xref="S2.SS3.SSS3.p2.2.m2.1.1">subscript</csymbol><ci id="S2.SS3.SSS3.p2.2.m2.1.1.2.cmml" xref="S2.SS3.SSS3.p2.2.m2.1.1.2">𝑇</ci><ci id="S2.SS3.SSS3.p2.2.m2.1.1.3.cmml" xref="S2.SS3.SSS3.p2.2.m2.1.1.3">fit</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS3.p2.2.m2.1c">T_{\rm fit}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS3.p2.2.m2.1d">italic_T start_POSTSUBSCRIPT roman_fit end_POSTSUBSCRIPT</annotation></semantics></math> is the period over which other dust collisions contribute less to the acceleration relative to the grain in question. 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8 end_POSTSUPERSCRIPT ( 10 start_POSTSUPERSCRIPT - 9 end_POSTSUPERSCRIPT )</annotation></semantics></math>g if <math alttext="T_{\rm fit}=10^{4}~{}(10^{6})" class="ltx_Math" display="inline" id="S2.SS3.SSS3.p2.4.m4.1"><semantics id="S2.SS3.SSS3.p2.4.m4.1a"><mrow id="S2.SS3.SSS3.p2.4.m4.1.1" xref="S2.SS3.SSS3.p2.4.m4.1.1.cmml"><msub id="S2.SS3.SSS3.p2.4.m4.1.1.3" xref="S2.SS3.SSS3.p2.4.m4.1.1.3.cmml"><mi id="S2.SS3.SSS3.p2.4.m4.1.1.3.2" xref="S2.SS3.SSS3.p2.4.m4.1.1.3.2.cmml">T</mi><mi id="S2.SS3.SSS3.p2.4.m4.1.1.3.3" xref="S2.SS3.SSS3.p2.4.m4.1.1.3.3.cmml">fit</mi></msub><mo id="S2.SS3.SSS3.p2.4.m4.1.1.2" xref="S2.SS3.SSS3.p2.4.m4.1.1.2.cmml">=</mo><mrow id="S2.SS3.SSS3.p2.4.m4.1.1.1" xref="S2.SS3.SSS3.p2.4.m4.1.1.1.cmml"><msup id="S2.SS3.SSS3.p2.4.m4.1.1.1.3" xref="S2.SS3.SSS3.p2.4.m4.1.1.1.3.cmml"><mn id="S2.SS3.SSS3.p2.4.m4.1.1.1.3.2" xref="S2.SS3.SSS3.p2.4.m4.1.1.1.3.2.cmml">10</mn><mn id="S2.SS3.SSS3.p2.4.m4.1.1.1.3.3" xref="S2.SS3.SSS3.p2.4.m4.1.1.1.3.3.cmml">4</mn></msup><mo id="S2.SS3.SSS3.p2.4.m4.1.1.1.2" xref="S2.SS3.SSS3.p2.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS3.SSS3.p2.4.m4.1.1.1.1.1" xref="S2.SS3.SSS3.p2.4.m4.1.1.1.1.1.1.cmml"><mo id="S2.SS3.SSS3.p2.4.m4.1.1.1.1.1.2" stretchy="false" xref="S2.SS3.SSS3.p2.4.m4.1.1.1.1.1.1.cmml">(</mo><msup id="S2.SS3.SSS3.p2.4.m4.1.1.1.1.1.1" xref="S2.SS3.SSS3.p2.4.m4.1.1.1.1.1.1.cmml"><mn id="S2.SS3.SSS3.p2.4.m4.1.1.1.1.1.1.2" xref="S2.SS3.SSS3.p2.4.m4.1.1.1.1.1.1.2.cmml">10</mn><mn id="S2.SS3.SSS3.p2.4.m4.1.1.1.1.1.1.3" xref="S2.SS3.SSS3.p2.4.m4.1.1.1.1.1.1.3.cmml">6</mn></msup><mo id="S2.SS3.SSS3.p2.4.m4.1.1.1.1.1.3" stretchy="false" xref="S2.SS3.SSS3.p2.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS3.p2.4.m4.1b"><apply id="S2.SS3.SSS3.p2.4.m4.1.1.cmml" xref="S2.SS3.SSS3.p2.4.m4.1.1"><eq id="S2.SS3.SSS3.p2.4.m4.1.1.2.cmml" xref="S2.SS3.SSS3.p2.4.m4.1.1.2"></eq><apply id="S2.SS3.SSS3.p2.4.m4.1.1.3.cmml" xref="S2.SS3.SSS3.p2.4.m4.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.SSS3.p2.4.m4.1.1.3.1.cmml" xref="S2.SS3.SSS3.p2.4.m4.1.1.3">subscript</csymbol><ci id="S2.SS3.SSS3.p2.4.m4.1.1.3.2.cmml" xref="S2.SS3.SSS3.p2.4.m4.1.1.3.2">𝑇</ci><ci id="S2.SS3.SSS3.p2.4.m4.1.1.3.3.cmml" xref="S2.SS3.SSS3.p2.4.m4.1.1.3.3">fit</ci></apply><apply id="S2.SS3.SSS3.p2.4.m4.1.1.1.cmml" xref="S2.SS3.SSS3.p2.4.m4.1.1.1"><times id="S2.SS3.SSS3.p2.4.m4.1.1.1.2.cmml" xref="S2.SS3.SSS3.p2.4.m4.1.1.1.2"></times><apply id="S2.SS3.SSS3.p2.4.m4.1.1.1.3.cmml" xref="S2.SS3.SSS3.p2.4.m4.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.SSS3.p2.4.m4.1.1.1.3.1.cmml" xref="S2.SS3.SSS3.p2.4.m4.1.1.1.3">superscript</csymbol><cn id="S2.SS3.SSS3.p2.4.m4.1.1.1.3.2.cmml" type="integer" xref="S2.SS3.SSS3.p2.4.m4.1.1.1.3.2">10</cn><cn id="S2.SS3.SSS3.p2.4.m4.1.1.1.3.3.cmml" type="integer" xref="S2.SS3.SSS3.p2.4.m4.1.1.1.3.3">4</cn></apply><apply id="S2.SS3.SSS3.p2.4.m4.1.1.1.1.1.1.cmml" xref="S2.SS3.SSS3.p2.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS3.p2.4.m4.1.1.1.1.1.1.1.cmml" xref="S2.SS3.SSS3.p2.4.m4.1.1.1.1.1">superscript</csymbol><cn id="S2.SS3.SSS3.p2.4.m4.1.1.1.1.1.1.2.cmml" type="integer" xref="S2.SS3.SSS3.p2.4.m4.1.1.1.1.1.1.2">10</cn><cn id="S2.SS3.SSS3.p2.4.m4.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS3.SSS3.p2.4.m4.1.1.1.1.1.1.3">6</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS3.p2.4.m4.1c">T_{\rm fit}=10^{4}~{}(10^{6})</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS3.p2.4.m4.1d">italic_T start_POSTSUBSCRIPT roman_fit end_POSTSUBSCRIPT = 10 start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ( 10 start_POSTSUPERSCRIPT 6 end_POSTSUPERSCRIPT )</annotation></semantics></math>s, the acceleration from the collisions with the dust grain can be fit for and removed. Grains of mass <math alttext="m_{d}" class="ltx_Math" display="inline" id="S2.SS3.SSS3.p2.5.m5.1"><semantics id="S2.SS3.SSS3.p2.5.m5.1a"><msub id="S2.SS3.SSS3.p2.5.m5.1.1" xref="S2.SS3.SSS3.p2.5.m5.1.1.cmml"><mi id="S2.SS3.SSS3.p2.5.m5.1.1.2" xref="S2.SS3.SSS3.p2.5.m5.1.1.2.cmml">m</mi><mi id="S2.SS3.SSS3.p2.5.m5.1.1.3" xref="S2.SS3.SSS3.p2.5.m5.1.1.3.cmml">d</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS3.p2.5.m5.1b"><apply id="S2.SS3.SSS3.p2.5.m5.1.1.cmml" xref="S2.SS3.SSS3.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS3.p2.5.m5.1.1.1.cmml" xref="S2.SS3.SSS3.p2.5.m5.1.1">subscript</csymbol><ci id="S2.SS3.SSS3.p2.5.m5.1.1.2.cmml" xref="S2.SS3.SSS3.p2.5.m5.1.1.2">𝑚</ci><ci id="S2.SS3.SSS3.p2.5.m5.1.1.3.cmml" xref="S2.SS3.SSS3.p2.5.m5.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS3.p2.5.m5.1c">m_{d}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS3.p2.5.m5.1d">italic_m start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT</annotation></semantics></math> strike the spacecraft every <math alttext="\sim 4\times 10^{6}{\rm~{}s~{}}(m_{d}/10^{-9}{\rm g})^{-1}(A_{\rm eff}/10{\rm m% }^{2})" class="ltx_Math" display="inline" id="S2.SS3.SSS3.p2.6.m6.2"><semantics id="S2.SS3.SSS3.p2.6.m6.2a"><mrow id="S2.SS3.SSS3.p2.6.m6.2.2" xref="S2.SS3.SSS3.p2.6.m6.2.2.cmml"><mi id="S2.SS3.SSS3.p2.6.m6.2.2.4" xref="S2.SS3.SSS3.p2.6.m6.2.2.4.cmml"></mi><mo id="S2.SS3.SSS3.p2.6.m6.2.2.3" xref="S2.SS3.SSS3.p2.6.m6.2.2.3.cmml">∼</mo><mrow id="S2.SS3.SSS3.p2.6.m6.2.2.2" xref="S2.SS3.SSS3.p2.6.m6.2.2.2.cmml"><mrow id="S2.SS3.SSS3.p2.6.m6.2.2.2.4" xref="S2.SS3.SSS3.p2.6.m6.2.2.2.4.cmml"><mn id="S2.SS3.SSS3.p2.6.m6.2.2.2.4.2" xref="S2.SS3.SSS3.p2.6.m6.2.2.2.4.2.cmml">4</mn><mo id="S2.SS3.SSS3.p2.6.m6.2.2.2.4.1" lspace="0.222em" rspace="0.222em" xref="S2.SS3.SSS3.p2.6.m6.2.2.2.4.1.cmml">×</mo><msup id="S2.SS3.SSS3.p2.6.m6.2.2.2.4.3" xref="S2.SS3.SSS3.p2.6.m6.2.2.2.4.3.cmml"><mn 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xref="S2.SS3.SSS3.p2.6.m6.2.2.2.2.1.1.3.2">m</ci><cn id="S2.SS3.SSS3.p2.6.m6.2.2.2.2.1.1.3.3.cmml" type="integer" xref="S2.SS3.SSS3.p2.6.m6.2.2.2.2.1.1.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS3.p2.6.m6.2c">\sim 4\times 10^{6}{\rm~{}s~{}}(m_{d}/10^{-9}{\rm g})^{-1}(A_{\rm eff}/10{\rm m% }^{2})</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS3.p2.6.m6.2d">∼ 4 × 10 start_POSTSUPERSCRIPT 6 end_POSTSUPERSCRIPT roman_s ( italic_m start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT / 10 start_POSTSUPERSCRIPT - 9 end_POSTSUPERSCRIPT roman_g ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_A start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT / 10 roman_m start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT )</annotation></semantics></math>, motivating these values for <math alttext="T_{\rm fit}" class="ltx_Math" display="inline" id="S2.SS3.SSS3.p2.7.m7.1"><semantics id="S2.SS3.SSS3.p2.7.m7.1a"><msub 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We find this statement is relatively independent of the radiometer noise for typical values considered in this paper, as <math alttext="h_{c}^{\rm dust}" class="ltx_Math" display="inline" id="S2.SS3.SSS3.p2.8.m8.1"><semantics id="S2.SS3.SSS3.p2.8.m8.1a"><msubsup id="S2.SS3.SSS3.p2.8.m8.1.1" xref="S2.SS3.SSS3.p2.8.m8.1.1.cmml"><mi id="S2.SS3.SSS3.p2.8.m8.1.1.2.2" xref="S2.SS3.SSS3.p2.8.m8.1.1.2.2.cmml">h</mi><mi id="S2.SS3.SSS3.p2.8.m8.1.1.2.3" xref="S2.SS3.SSS3.p2.8.m8.1.1.2.3.cmml">c</mi><mi id="S2.SS3.SSS3.p2.8.m8.1.1.3" xref="S2.SS3.SSS3.p2.8.m8.1.1.3.cmml">dust</mi></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS3.p2.8.m8.1b"><apply id="S2.SS3.SSS3.p2.8.m8.1.1.cmml" xref="S2.SS3.SSS3.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS3.p2.8.m8.1.1.1.cmml" xref="S2.SS3.SSS3.p2.8.m8.1.1">superscript</csymbol><apply id="S2.SS3.SSS3.p2.8.m8.1.1.2.cmml" xref="S2.SS3.SSS3.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS3.p2.8.m8.1.1.2.1.cmml" xref="S2.SS3.SSS3.p2.8.m8.1.1">subscript</csymbol><ci id="S2.SS3.SSS3.p2.8.m8.1.1.2.2.cmml" xref="S2.SS3.SSS3.p2.8.m8.1.1.2.2">ℎ</ci><ci id="S2.SS3.SSS3.p2.8.m8.1.1.2.3.cmml" xref="S2.SS3.SSS3.p2.8.m8.1.1.2.3">𝑐</ci></apply><ci id="S2.SS3.SSS3.p2.8.m8.1.1.3.cmml" xref="S2.SS3.SSS3.p2.8.m8.1.1.3">dust</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS3.p2.8.m8.1c">h_{c}^{\rm dust}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS3.p2.8.m8.1d">italic_h start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_dust end_POSTSUPERSCRIPT</annotation></semantics></math> has a similar scaling with <math alttext="f" class="ltx_Math" display="inline" id="S2.SS3.SSS3.p2.9.m9.1"><semantics id="S2.SS3.SSS3.p2.9.m9.1a"><mi id="S2.SS3.SSS3.p2.9.m9.1.1" xref="S2.SS3.SSS3.p2.9.m9.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS3.p2.9.m9.1b"><ci id="S2.SS3.SSS3.p2.9.m9.1.1.cmml" xref="S2.SS3.SSS3.p2.9.m9.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS3.p2.9.m9.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS3.p2.9.m9.1d">italic_f</annotation></semantics></math> to the total acceleration noise. This indicates that this source of anomalous accelerations would be removable for dust grain masses that could lead to significant acceleration noise relative to other acceleration sources.</p> </div> </section> <section class="ltx_subsubsection" id="S2.SS3.SSS4"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">2.3.4 </span>Spacecraft charging</h4> <div class="ltx_para" id="S2.SS3.SSS4.p1"> <p class="ltx_p" id="S2.SS3.SSS4.p1.9">A spacecraft will build up charge as it flows through the interplanetary plasma. The Lorentz force of the interplanetary magnetic field will then impart accelerations on the spacecraft. 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id="S2.SS3.SSS4.p1.1.m1.1d">italic_Z start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT</annotation></semantics></math> is roughly the charge able to repel solar wind protons from striking the spacecraft <math alttext="1/2m_{p}v_{\rm sw}^{2}=Z_{\rm max}e^{2}/R_{\rm sc}" class="ltx_Math" display="inline" id="S2.SS3.SSS4.p1.2.m2.1"><semantics id="S2.SS3.SSS4.p1.2.m2.1a"><mrow id="S2.SS3.SSS4.p1.2.m2.1.1" xref="S2.SS3.SSS4.p1.2.m2.1.1.cmml"><mrow id="S2.SS3.SSS4.p1.2.m2.1.1.2" xref="S2.SS3.SSS4.p1.2.m2.1.1.2.cmml"><mrow id="S2.SS3.SSS4.p1.2.m2.1.1.2.2" xref="S2.SS3.SSS4.p1.2.m2.1.1.2.2.cmml"><mn id="S2.SS3.SSS4.p1.2.m2.1.1.2.2.2" xref="S2.SS3.SSS4.p1.2.m2.1.1.2.2.2.cmml">1</mn><mo id="S2.SS3.SSS4.p1.2.m2.1.1.2.2.1" xref="S2.SS3.SSS4.p1.2.m2.1.1.2.2.1.cmml">/</mo><mn id="S2.SS3.SSS4.p1.2.m2.1.1.2.2.3" xref="S2.SS3.SSS4.p1.2.m2.1.1.2.2.3.cmml">2</mn></mrow><mo id="S2.SS3.SSS4.p1.2.m2.1.1.2.1" xref="S2.SS3.SSS4.p1.2.m2.1.1.2.1.cmml">⁢</mo><msub id="S2.SS3.SSS4.p1.2.m2.1.1.2.3" 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id="S2.SS3.SSS4.p1.2.m2.1.1.3.2.cmml" xref="S2.SS3.SSS4.p1.2.m2.1.1.3.2"><times id="S2.SS3.SSS4.p1.2.m2.1.1.3.2.1.cmml" xref="S2.SS3.SSS4.p1.2.m2.1.1.3.2.1"></times><apply id="S2.SS3.SSS4.p1.2.m2.1.1.3.2.2.cmml" xref="S2.SS3.SSS4.p1.2.m2.1.1.3.2.2"><csymbol cd="ambiguous" id="S2.SS3.SSS4.p1.2.m2.1.1.3.2.2.1.cmml" xref="S2.SS3.SSS4.p1.2.m2.1.1.3.2.2">subscript</csymbol><ci id="S2.SS3.SSS4.p1.2.m2.1.1.3.2.2.2.cmml" xref="S2.SS3.SSS4.p1.2.m2.1.1.3.2.2.2">𝑍</ci><ci id="S2.SS3.SSS4.p1.2.m2.1.1.3.2.2.3.cmml" xref="S2.SS3.SSS4.p1.2.m2.1.1.3.2.2.3">max</ci></apply><apply id="S2.SS3.SSS4.p1.2.m2.1.1.3.2.3.cmml" xref="S2.SS3.SSS4.p1.2.m2.1.1.3.2.3"><csymbol cd="ambiguous" id="S2.SS3.SSS4.p1.2.m2.1.1.3.2.3.1.cmml" xref="S2.SS3.SSS4.p1.2.m2.1.1.3.2.3">superscript</csymbol><ci id="S2.SS3.SSS4.p1.2.m2.1.1.3.2.3.2.cmml" xref="S2.SS3.SSS4.p1.2.m2.1.1.3.2.3.2">𝑒</ci><cn id="S2.SS3.SSS4.p1.2.m2.1.1.3.2.3.3.cmml" type="integer" xref="S2.SS3.SSS4.p1.2.m2.1.1.3.2.3.3">2</cn></apply></apply><apply id="S2.SS3.SSS4.p1.2.m2.1.1.3.3.cmml" xref="S2.SS3.SSS4.p1.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS3.SSS4.p1.2.m2.1.1.3.3.1.cmml" xref="S2.SS3.SSS4.p1.2.m2.1.1.3.3">subscript</csymbol><ci id="S2.SS3.SSS4.p1.2.m2.1.1.3.3.2.cmml" xref="S2.SS3.SSS4.p1.2.m2.1.1.3.3.2">𝑅</ci><ci id="S2.SS3.SSS4.p1.2.m2.1.1.3.3.3.cmml" xref="S2.SS3.SSS4.p1.2.m2.1.1.3.3.3">sc</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS4.p1.2.m2.1c">1/2m_{p}v_{\rm sw}^{2}=Z_{\rm max}e^{2}/R_{\rm sc}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS4.p1.2.m2.1d">1 / 2 italic_m start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT roman_sw end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_Z start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / italic_R start_POSTSUBSCRIPT roman_sc end_POSTSUBSCRIPT</annotation></semantics></math>, where <math alttext="e" class="ltx_Math" display="inline" id="S2.SS3.SSS4.p1.3.m3.1"><semantics id="S2.SS3.SSS4.p1.3.m3.1a"><mi id="S2.SS3.SSS4.p1.3.m3.1.1" xref="S2.SS3.SSS4.p1.3.m3.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS4.p1.3.m3.1b"><ci id="S2.SS3.SSS4.p1.3.m3.1.1.cmml" xref="S2.SS3.SSS4.p1.3.m3.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS4.p1.3.m3.1c">e</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS4.p1.3.m3.1d">italic_e</annotation></semantics></math> is the electron charge, <math alttext="m_{p}" class="ltx_Math" display="inline" id="S2.SS3.SSS4.p1.4.m4.1"><semantics id="S2.SS3.SSS4.p1.4.m4.1a"><msub id="S2.SS3.SSS4.p1.4.m4.1.1" xref="S2.SS3.SSS4.p1.4.m4.1.1.cmml"><mi id="S2.SS3.SSS4.p1.4.m4.1.1.2" xref="S2.SS3.SSS4.p1.4.m4.1.1.2.cmml">m</mi><mi id="S2.SS3.SSS4.p1.4.m4.1.1.3" xref="S2.SS3.SSS4.p1.4.m4.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS4.p1.4.m4.1b"><apply id="S2.SS3.SSS4.p1.4.m4.1.1.cmml" xref="S2.SS3.SSS4.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS4.p1.4.m4.1.1.1.cmml" xref="S2.SS3.SSS4.p1.4.m4.1.1">subscript</csymbol><ci id="S2.SS3.SSS4.p1.4.m4.1.1.2.cmml" xref="S2.SS3.SSS4.p1.4.m4.1.1.2">𝑚</ci><ci id="S2.SS3.SSS4.p1.4.m4.1.1.3.cmml" xref="S2.SS3.SSS4.p1.4.m4.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS4.p1.4.m4.1c">m_{p}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS4.p1.4.m4.1d">italic_m start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> the proton mass, <math alttext="v_{\rm sw}" class="ltx_Math" display="inline" id="S2.SS3.SSS4.p1.5.m5.1"><semantics id="S2.SS3.SSS4.p1.5.m5.1a"><msub id="S2.SS3.SSS4.p1.5.m5.1.1" xref="S2.SS3.SSS4.p1.5.m5.1.1.cmml"><mi id="S2.SS3.SSS4.p1.5.m5.1.1.2" xref="S2.SS3.SSS4.p1.5.m5.1.1.2.cmml">v</mi><mi id="S2.SS3.SSS4.p1.5.m5.1.1.3" xref="S2.SS3.SSS4.p1.5.m5.1.1.3.cmml">sw</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS4.p1.5.m5.1b"><apply id="S2.SS3.SSS4.p1.5.m5.1.1.cmml" xref="S2.SS3.SSS4.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS4.p1.5.m5.1.1.1.cmml" xref="S2.SS3.SSS4.p1.5.m5.1.1">subscript</csymbol><ci id="S2.SS3.SSS4.p1.5.m5.1.1.2.cmml" xref="S2.SS3.SSS4.p1.5.m5.1.1.2">𝑣</ci><ci id="S2.SS3.SSS4.p1.5.m5.1.1.3.cmml" xref="S2.SS3.SSS4.p1.5.m5.1.1.3">sw</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS4.p1.5.m5.1c">v_{\rm sw}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS4.p1.5.m5.1d">italic_v start_POSTSUBSCRIPT roman_sw end_POSTSUBSCRIPT</annotation></semantics></math> the velocity of the solar wind, and <math alttext="R_{\rm sc}" class="ltx_Math" display="inline" id="S2.SS3.SSS4.p1.6.m6.1"><semantics id="S2.SS3.SSS4.p1.6.m6.1a"><msub id="S2.SS3.SSS4.p1.6.m6.1.1" xref="S2.SS3.SSS4.p1.6.m6.1.1.cmml"><mi id="S2.SS3.SSS4.p1.6.m6.1.1.2" xref="S2.SS3.SSS4.p1.6.m6.1.1.2.cmml">R</mi><mi id="S2.SS3.SSS4.p1.6.m6.1.1.3" xref="S2.SS3.SSS4.p1.6.m6.1.1.3.cmml">sc</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS4.p1.6.m6.1b"><apply id="S2.SS3.SSS4.p1.6.m6.1.1.cmml" xref="S2.SS3.SSS4.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS4.p1.6.m6.1.1.1.cmml" xref="S2.SS3.SSS4.p1.6.m6.1.1">subscript</csymbol><ci id="S2.SS3.SSS4.p1.6.m6.1.1.2.cmml" xref="S2.SS3.SSS4.p1.6.m6.1.1.2">𝑅</ci><ci id="S2.SS3.SSS4.p1.6.m6.1.1.3.cmml" xref="S2.SS3.SSS4.p1.6.m6.1.1.3">sc</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS4.p1.6.m6.1c">R_{\rm sc}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS4.p1.6.m6.1d">italic_R start_POSTSUBSCRIPT roman_sc end_POSTSUBSCRIPT</annotation></semantics></math> the characteristic size of the spacecraft. We find for a velocity characteristic of the solar wind of <math alttext="v_{\rm sw}=500\;" class="ltx_Math" display="inline" id="S2.SS3.SSS4.p1.7.m7.1"><semantics id="S2.SS3.SSS4.p1.7.m7.1a"><mrow id="S2.SS3.SSS4.p1.7.m7.1.1" xref="S2.SS3.SSS4.p1.7.m7.1.1.cmml"><msub id="S2.SS3.SSS4.p1.7.m7.1.1.2" xref="S2.SS3.SSS4.p1.7.m7.1.1.2.cmml"><mi id="S2.SS3.SSS4.p1.7.m7.1.1.2.2" xref="S2.SS3.SSS4.p1.7.m7.1.1.2.2.cmml">v</mi><mi id="S2.SS3.SSS4.p1.7.m7.1.1.2.3" xref="S2.SS3.SSS4.p1.7.m7.1.1.2.3.cmml">sw</mi></msub><mo id="S2.SS3.SSS4.p1.7.m7.1.1.1" xref="S2.SS3.SSS4.p1.7.m7.1.1.1.cmml">=</mo><mn id="S2.SS3.SSS4.p1.7.m7.1.1.3" xref="S2.SS3.SSS4.p1.7.m7.1.1.3.cmml">500</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS4.p1.7.m7.1b"><apply id="S2.SS3.SSS4.p1.7.m7.1.1.cmml" xref="S2.SS3.SSS4.p1.7.m7.1.1"><eq id="S2.SS3.SSS4.p1.7.m7.1.1.1.cmml" xref="S2.SS3.SSS4.p1.7.m7.1.1.1"></eq><apply id="S2.SS3.SSS4.p1.7.m7.1.1.2.cmml" xref="S2.SS3.SSS4.p1.7.m7.1.1.2"><csymbol cd="ambiguous" id="S2.SS3.SSS4.p1.7.m7.1.1.2.1.cmml" xref="S2.SS3.SSS4.p1.7.m7.1.1.2">subscript</csymbol><ci id="S2.SS3.SSS4.p1.7.m7.1.1.2.2.cmml" xref="S2.SS3.SSS4.p1.7.m7.1.1.2.2">𝑣</ci><ci id="S2.SS3.SSS4.p1.7.m7.1.1.2.3.cmml" xref="S2.SS3.SSS4.p1.7.m7.1.1.2.3">sw</ci></apply><cn id="S2.SS3.SSS4.p1.7.m7.1.1.3.cmml" type="integer" xref="S2.SS3.SSS4.p1.7.m7.1.1.3">500</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS4.p1.7.m7.1c">v_{\rm sw}=500\;</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS4.p1.7.m7.1d">italic_v start_POSTSUBSCRIPT roman_sw end_POSTSUBSCRIPT = 500</annotation></semantics></math>km s<sup class="ltx_sup" id="S2.SS3.SSS4.p1.9.1"><span class="ltx_text ltx_font_italic" id="S2.SS3.SSS4.p1.9.1.1">-1</span></sup>, this results in a maximum voltage of <math alttext="V\sim Z_{\rm max}e/R_{\rm sc}=1300~{}" class="ltx_Math" display="inline" id="S2.SS3.SSS4.p1.9.m9.1"><semantics id="S2.SS3.SSS4.p1.9.m9.1a"><mrow id="S2.SS3.SSS4.p1.9.m9.1.1" xref="S2.SS3.SSS4.p1.9.m9.1.1.cmml"><mi id="S2.SS3.SSS4.p1.9.m9.1.1.2" xref="S2.SS3.SSS4.p1.9.m9.1.1.2.cmml">V</mi><mo id="S2.SS3.SSS4.p1.9.m9.1.1.3" xref="S2.SS3.SSS4.p1.9.m9.1.1.3.cmml">∼</mo><mrow id="S2.SS3.SSS4.p1.9.m9.1.1.4" xref="S2.SS3.SSS4.p1.9.m9.1.1.4.cmml"><mrow id="S2.SS3.SSS4.p1.9.m9.1.1.4.2" xref="S2.SS3.SSS4.p1.9.m9.1.1.4.2.cmml"><msub id="S2.SS3.SSS4.p1.9.m9.1.1.4.2.2" xref="S2.SS3.SSS4.p1.9.m9.1.1.4.2.2.cmml"><mi id="S2.SS3.SSS4.p1.9.m9.1.1.4.2.2.2" xref="S2.SS3.SSS4.p1.9.m9.1.1.4.2.2.2.cmml">Z</mi><mi id="S2.SS3.SSS4.p1.9.m9.1.1.4.2.2.3" xref="S2.SS3.SSS4.p1.9.m9.1.1.4.2.2.3.cmml">max</mi></msub><mo id="S2.SS3.SSS4.p1.9.m9.1.1.4.2.1" xref="S2.SS3.SSS4.p1.9.m9.1.1.4.2.1.cmml">⁢</mo><mi id="S2.SS3.SSS4.p1.9.m9.1.1.4.2.3" xref="S2.SS3.SSS4.p1.9.m9.1.1.4.2.3.cmml">e</mi></mrow><mo id="S2.SS3.SSS4.p1.9.m9.1.1.4.1" xref="S2.SS3.SSS4.p1.9.m9.1.1.4.1.cmml">/</mo><msub id="S2.SS3.SSS4.p1.9.m9.1.1.4.3" xref="S2.SS3.SSS4.p1.9.m9.1.1.4.3.cmml"><mi id="S2.SS3.SSS4.p1.9.m9.1.1.4.3.2" xref="S2.SS3.SSS4.p1.9.m9.1.1.4.3.2.cmml">R</mi><mi id="S2.SS3.SSS4.p1.9.m9.1.1.4.3.3" xref="S2.SS3.SSS4.p1.9.m9.1.1.4.3.3.cmml">sc</mi></msub></mrow><mo id="S2.SS3.SSS4.p1.9.m9.1.1.5" xref="S2.SS3.SSS4.p1.9.m9.1.1.5.cmml">=</mo><mn id="S2.SS3.SSS4.p1.9.m9.1.1.6" xref="S2.SS3.SSS4.p1.9.m9.1.1.6.cmml">1300</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS4.p1.9.m9.1b"><apply id="S2.SS3.SSS4.p1.9.m9.1.1.cmml" xref="S2.SS3.SSS4.p1.9.m9.1.1"><and id="S2.SS3.SSS4.p1.9.m9.1.1a.cmml" xref="S2.SS3.SSS4.p1.9.m9.1.1"></and><apply id="S2.SS3.SSS4.p1.9.m9.1.1b.cmml" xref="S2.SS3.SSS4.p1.9.m9.1.1"><csymbol cd="latexml" id="S2.SS3.SSS4.p1.9.m9.1.1.3.cmml" xref="S2.SS3.SSS4.p1.9.m9.1.1.3">similar-to</csymbol><ci id="S2.SS3.SSS4.p1.9.m9.1.1.2.cmml" xref="S2.SS3.SSS4.p1.9.m9.1.1.2">𝑉</ci><apply id="S2.SS3.SSS4.p1.9.m9.1.1.4.cmml" xref="S2.SS3.SSS4.p1.9.m9.1.1.4"><divide id="S2.SS3.SSS4.p1.9.m9.1.1.4.1.cmml" 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id="S2.SS3.SSS4.p1.9.m9.1.1.4.3.3.cmml" xref="S2.SS3.SSS4.p1.9.m9.1.1.4.3.3">sc</ci></apply></apply></apply><apply id="S2.SS3.SSS4.p1.9.m9.1.1c.cmml" xref="S2.SS3.SSS4.p1.9.m9.1.1"><eq id="S2.SS3.SSS4.p1.9.m9.1.1.5.cmml" xref="S2.SS3.SSS4.p1.9.m9.1.1.5"></eq><share href="https://arxiv.org/html/2411.15072v1#S2.SS3.SSS4.p1.9.m9.1.1.4.cmml" id="S2.SS3.SSS4.p1.9.m9.1.1d.cmml" xref="S2.SS3.SSS4.p1.9.m9.1.1"></share><cn id="S2.SS3.SSS4.p1.9.m9.1.1.6.cmml" type="integer" xref="S2.SS3.SSS4.p1.9.m9.1.1.6">1300</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS4.p1.9.m9.1c">V\sim Z_{\rm max}e/R_{\rm sc}=1300~{}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS4.p1.9.m9.1d">italic_V ∼ italic_Z start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT italic_e / italic_R start_POSTSUBSCRIPT roman_sc end_POSTSUBSCRIPT = 1300</annotation></semantics></math>V.</p> </div> <div class="ltx_para" id="S2.SS3.SSS4.p2"> <p class="ltx_p" id="S2.SS3.SSS4.p2.5">The magnitude of both the homogeneous and inhomogeneous magnetic fields in the outer Solar System are on the order of a <math alttext="\mu" class="ltx_Math" display="inline" id="S2.SS3.SSS4.p2.1.m1.1"><semantics id="S2.SS3.SSS4.p2.1.m1.1a"><mi id="S2.SS3.SSS4.p2.1.m1.1.1" xref="S2.SS3.SSS4.p2.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS4.p2.1.m1.1b"><ci id="S2.SS3.SSS4.p2.1.m1.1.1.cmml" xref="S2.SS3.SSS4.p2.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS4.p2.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS4.p2.1.m1.1d">italic_μ</annotation></semantics></math>Gauss, falling off by only a factor of two from 1AU to 20AU <cite class="ltx_cite ltx_citemacro_citep">(Smith, <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib64" title="">1989</a>)</cite>. The magnetic field power spectrum is also found to maintain a similar spectrum with distance from the Sun. This near constancy supports our approach of using the magnetic field power spectrum measured by the Voyager 1 spacecraft at <math alttext="6.1-8.9" class="ltx_Math" display="inline" id="S2.SS3.SSS4.p2.2.m2.1"><semantics id="S2.SS3.SSS4.p2.2.m2.1a"><mrow id="S2.SS3.SSS4.p2.2.m2.1.1" xref="S2.SS3.SSS4.p2.2.m2.1.1.cmml"><mn id="S2.SS3.SSS4.p2.2.m2.1.1.2" xref="S2.SS3.SSS4.p2.2.m2.1.1.2.cmml">6.1</mn><mo id="S2.SS3.SSS4.p2.2.m2.1.1.1" xref="S2.SS3.SSS4.p2.2.m2.1.1.1.cmml">−</mo><mn id="S2.SS3.SSS4.p2.2.m2.1.1.3" xref="S2.SS3.SSS4.p2.2.m2.1.1.3.cmml">8.9</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS4.p2.2.m2.1b"><apply id="S2.SS3.SSS4.p2.2.m2.1.1.cmml" xref="S2.SS3.SSS4.p2.2.m2.1.1"><minus id="S2.SS3.SSS4.p2.2.m2.1.1.1.cmml" xref="S2.SS3.SSS4.p2.2.m2.1.1.1"></minus><cn id="S2.SS3.SSS4.p2.2.m2.1.1.2.cmml" type="float" xref="S2.SS3.SSS4.p2.2.m2.1.1.2">6.1</cn><cn id="S2.SS3.SSS4.p2.2.m2.1.1.3.cmml" type="float" xref="S2.SS3.SSS4.p2.2.m2.1.1.3">8.9</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS4.p2.2.m2.1c">6.1-8.9</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS4.p2.2.m2.1d">6.1 - 8.9</annotation></semantics></math>AU to compute the Lorentz force on the spacecraft at all the solar radii considered. This acceleration power for the maximum charge is shown in Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.F2" title="Figure 2 ‣ 2.3 Acceleration noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">2</span></a> assuming a size typical of outer Solar System spacecraft with <math alttext="R_{\rm sc}=2\;" class="ltx_Math" display="inline" id="S2.SS3.SSS4.p2.3.m3.1"><semantics id="S2.SS3.SSS4.p2.3.m3.1a"><mrow id="S2.SS3.SSS4.p2.3.m3.1.1" xref="S2.SS3.SSS4.p2.3.m3.1.1.cmml"><msub id="S2.SS3.SSS4.p2.3.m3.1.1.2" xref="S2.SS3.SSS4.p2.3.m3.1.1.2.cmml"><mi id="S2.SS3.SSS4.p2.3.m3.1.1.2.2" xref="S2.SS3.SSS4.p2.3.m3.1.1.2.2.cmml">R</mi><mi id="S2.SS3.SSS4.p2.3.m3.1.1.2.3" xref="S2.SS3.SSS4.p2.3.m3.1.1.2.3.cmml">sc</mi></msub><mo id="S2.SS3.SSS4.p2.3.m3.1.1.1" xref="S2.SS3.SSS4.p2.3.m3.1.1.1.cmml">=</mo><mn id="S2.SS3.SSS4.p2.3.m3.1.1.3" xref="S2.SS3.SSS4.p2.3.m3.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS4.p2.3.m3.1b"><apply id="S2.SS3.SSS4.p2.3.m3.1.1.cmml" xref="S2.SS3.SSS4.p2.3.m3.1.1"><eq id="S2.SS3.SSS4.p2.3.m3.1.1.1.cmml" xref="S2.SS3.SSS4.p2.3.m3.1.1.1"></eq><apply id="S2.SS3.SSS4.p2.3.m3.1.1.2.cmml" xref="S2.SS3.SSS4.p2.3.m3.1.1.2"><csymbol cd="ambiguous" id="S2.SS3.SSS4.p2.3.m3.1.1.2.1.cmml" xref="S2.SS3.SSS4.p2.3.m3.1.1.2">subscript</csymbol><ci id="S2.SS3.SSS4.p2.3.m3.1.1.2.2.cmml" xref="S2.SS3.SSS4.p2.3.m3.1.1.2.2">𝑅</ci><ci id="S2.SS3.SSS4.p2.3.m3.1.1.2.3.cmml" xref="S2.SS3.SSS4.p2.3.m3.1.1.2.3">sc</ci></apply><cn id="S2.SS3.SSS4.p2.3.m3.1.1.3.cmml" type="integer" xref="S2.SS3.SSS4.p2.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS4.p2.3.m3.1c">R_{\rm sc}=2\;</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS4.p2.3.m3.1d">italic_R start_POSTSUBSCRIPT roman_sc end_POSTSUBSCRIPT = 2</annotation></semantics></math>m to calculate <math alttext="Z_{\rm max}" class="ltx_Math" display="inline" id="S2.SS3.SSS4.p2.4.m4.1"><semantics id="S2.SS3.SSS4.p2.4.m4.1a"><msub id="S2.SS3.SSS4.p2.4.m4.1.1" xref="S2.SS3.SSS4.p2.4.m4.1.1.cmml"><mi id="S2.SS3.SSS4.p2.4.m4.1.1.2" xref="S2.SS3.SSS4.p2.4.m4.1.1.2.cmml">Z</mi><mi id="S2.SS3.SSS4.p2.4.m4.1.1.3" xref="S2.SS3.SSS4.p2.4.m4.1.1.3.cmml">max</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS4.p2.4.m4.1b"><apply id="S2.SS3.SSS4.p2.4.m4.1.1.cmml" xref="S2.SS3.SSS4.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS4.p2.4.m4.1.1.1.cmml" xref="S2.SS3.SSS4.p2.4.m4.1.1">subscript</csymbol><ci id="S2.SS3.SSS4.p2.4.m4.1.1.2.cmml" xref="S2.SS3.SSS4.p2.4.m4.1.1.2">𝑍</ci><ci id="S2.SS3.SSS4.p2.4.m4.1.1.3.cmml" xref="S2.SS3.SSS4.p2.4.m4.1.1.3">max</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS4.p2.4.m4.1c">Z_{\rm max}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS4.p2.4.m4.1d">italic_Z start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT</annotation></semantics></math> and then the acceleration power along an arm is given by <math alttext="\sqrt{S_{a}}=Z_{\rm max}v_{\rm sw}\sqrt{P_{B}(f,r)}/(\sqrt{3}M)" class="ltx_Math" display="inline" id="S2.SS3.SSS4.p2.5.m5.3"><semantics id="S2.SS3.SSS4.p2.5.m5.3a"><mrow id="S2.SS3.SSS4.p2.5.m5.3.3" xref="S2.SS3.SSS4.p2.5.m5.3.3.cmml"><msqrt id="S2.SS3.SSS4.p2.5.m5.3.3.3" xref="S2.SS3.SSS4.p2.5.m5.3.3.3.cmml"><msub id="S2.SS3.SSS4.p2.5.m5.3.3.3.2" xref="S2.SS3.SSS4.p2.5.m5.3.3.3.2.cmml"><mi id="S2.SS3.SSS4.p2.5.m5.3.3.3.2.2" xref="S2.SS3.SSS4.p2.5.m5.3.3.3.2.2.cmml">S</mi><mi id="S2.SS3.SSS4.p2.5.m5.3.3.3.2.3" xref="S2.SS3.SSS4.p2.5.m5.3.3.3.2.3.cmml">a</mi></msub></msqrt><mo id="S2.SS3.SSS4.p2.5.m5.3.3.2" xref="S2.SS3.SSS4.p2.5.m5.3.3.2.cmml">=</mo><mrow id="S2.SS3.SSS4.p2.5.m5.3.3.1" xref="S2.SS3.SSS4.p2.5.m5.3.3.1.cmml"><mrow id="S2.SS3.SSS4.p2.5.m5.3.3.1.3" 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)</annotation></semantics></math>. However, this is likely a conservative estimate as interplanetary spacecraft are designed to have voltages that are likely to be closer to a tenth of our estimate for the maximum <cite class="ltx_cite ltx_citemacro_citep">(LAI, <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib42" title="">2012</a>)</cite>.</p> </div> </section> <section class="ltx_subsubsection" id="S2.SS3.SSS5"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">2.3.5 </span>Gravity from asteroids and larger bodies</h4> <div class="ltx_para" id="S2.SS3.SSS5.p1"> <p class="ltx_p" id="S2.SS3.SSS5.p1.10">The gravitational attraction of asteroids will be another source accelerations. Using the JPL Small-Body Database, <cite class="ltx_cite ltx_citemacro_citet">Fedderke et al. (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib27" title="">2021</a>)</cite> computed the acceleration power from asteroids for spacecraft at <math alttext="30" class="ltx_Math" display="inline" id="S2.SS3.SSS5.p1.1.m1.1"><semantics id="S2.SS3.SSS5.p1.1.m1.1a"><mn id="S2.SS3.SSS5.p1.1.m1.1.1" xref="S2.SS3.SSS5.p1.1.m1.1.1.cmml">30</mn><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS5.p1.1.m1.1b"><cn id="S2.SS3.SSS5.p1.1.m1.1.1.cmml" type="integer" xref="S2.SS3.SSS5.p1.1.m1.1.1">30</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS5.p1.1.m1.1c">30</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS5.p1.1.m1.1d">30</annotation></semantics></math>AU from the Sun, finding an acceleration power that is absolutely negligible (<math alttext="\sim 10^{-18}\text{m s${}^{-2}$ Hz}^{-1/2}" class="ltx_Math" display="inline" id="S2.SS3.SSS5.p1.2.m2.1"><semantics id="S2.SS3.SSS5.p1.2.m2.1a"><mrow id="S2.SS3.SSS5.p1.2.m2.1.2" 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id="S2.SS3.SSS5.p1.2.m2.1.2.3.2.3.2.cmml" type="integer" xref="S2.SS3.SSS5.p1.2.m2.1.2.3.2.3.2">18</cn></apply></apply><apply id="S2.SS3.SSS5.p1.2.m2.1.2.3.3.cmml" xref="S2.SS3.SSS5.p1.2.m2.1.2.3.3"><csymbol cd="ambiguous" id="S2.SS3.SSS5.p1.2.m2.1.2.3.3.1.cmml" xref="S2.SS3.SSS5.p1.2.m2.1.2.3.3">superscript</csymbol><ci id="S2.SS3.SSS5.p1.2.m2.1.1.1f.cmml" xref="S2.SS3.SSS5.p1.2.m2.1.1.1"><mrow id="S2.SS3.SSS5.p1.2.m2.1.1.1.cmml" xref="S2.SS3.SSS5.p1.2.m2.1.1.1"><mtext id="S2.SS3.SSS5.p1.2.m2.1.1.1a.cmml" xref="S2.SS3.SSS5.p1.2.m2.1.1.1">m s</mtext><mtext id="S2.SS3.SSS5.p1.2.m2.1.1.1b.cmml" xref="S2.SS3.SSS5.p1.2.m2.1.1.1"><sup class="ltx_sup" id="S2.SS3.SSS5.p1.2.m2.1.1.1.2anest"><span class="ltx_text ltx_font_italic" id="S2.SS3.SSS5.p1.2.m2.1.1.1.2.1anest">-2</span></sup></mtext><mtext id="S2.SS3.SSS5.p1.2.m2.1.1.1e.cmml" xref="S2.SS3.SSS5.p1.2.m2.1.1.1"> Hz</mtext></mrow></ci><apply id="S2.SS3.SSS5.p1.2.m2.1.2.3.3.2.cmml" xref="S2.SS3.SSS5.p1.2.m2.1.2.3.3.2"><minus id="S2.SS3.SSS5.p1.2.m2.1.2.3.3.2.1.cmml" xref="S2.SS3.SSS5.p1.2.m2.1.2.3.3.2"></minus><apply id="S2.SS3.SSS5.p1.2.m2.1.2.3.3.2.2.cmml" xref="S2.SS3.SSS5.p1.2.m2.1.2.3.3.2.2"><divide id="S2.SS3.SSS5.p1.2.m2.1.2.3.3.2.2.1.cmml" xref="S2.SS3.SSS5.p1.2.m2.1.2.3.3.2.2.1"></divide><cn id="S2.SS3.SSS5.p1.2.m2.1.2.3.3.2.2.2.cmml" type="integer" xref="S2.SS3.SSS5.p1.2.m2.1.2.3.3.2.2.2">1</cn><cn id="S2.SS3.SSS5.p1.2.m2.1.2.3.3.2.2.3.cmml" type="integer" xref="S2.SS3.SSS5.p1.2.m2.1.2.3.3.2.2.3">2</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS5.p1.2.m2.1c">\sim 10^{-18}\text{m s${}^{-2}$ Hz}^{-1/2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS5.p1.2.m2.1d">∼ 10 start_POSTSUPERSCRIPT - 18 end_POSTSUPERSCRIPT m s Hz start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT</annotation></semantics></math> at <math alttext="10^{-7}" class="ltx_Math" display="inline" id="S2.SS3.SSS5.p1.3.m3.1"><semantics id="S2.SS3.SSS5.p1.3.m3.1a"><msup id="S2.SS3.SSS5.p1.3.m3.1.1" xref="S2.SS3.SSS5.p1.3.m3.1.1.cmml"><mn id="S2.SS3.SSS5.p1.3.m3.1.1.2" xref="S2.SS3.SSS5.p1.3.m3.1.1.2.cmml">10</mn><mrow id="S2.SS3.SSS5.p1.3.m3.1.1.3" xref="S2.SS3.SSS5.p1.3.m3.1.1.3.cmml"><mo id="S2.SS3.SSS5.p1.3.m3.1.1.3a" xref="S2.SS3.SSS5.p1.3.m3.1.1.3.cmml">−</mo><mn id="S2.SS3.SSS5.p1.3.m3.1.1.3.2" xref="S2.SS3.SSS5.p1.3.m3.1.1.3.2.cmml">7</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS5.p1.3.m3.1b"><apply id="S2.SS3.SSS5.p1.3.m3.1.1.cmml" xref="S2.SS3.SSS5.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS5.p1.3.m3.1.1.1.cmml" xref="S2.SS3.SSS5.p1.3.m3.1.1">superscript</csymbol><cn id="S2.SS3.SSS5.p1.3.m3.1.1.2.cmml" type="integer" xref="S2.SS3.SSS5.p1.3.m3.1.1.2">10</cn><apply id="S2.SS3.SSS5.p1.3.m3.1.1.3.cmml" xref="S2.SS3.SSS5.p1.3.m3.1.1.3"><minus id="S2.SS3.SSS5.p1.3.m3.1.1.3.1.cmml" xref="S2.SS3.SSS5.p1.3.m3.1.1.3"></minus><cn id="S2.SS3.SSS5.p1.3.m3.1.1.3.2.cmml" type="integer" xref="S2.SS3.SSS5.p1.3.m3.1.1.3.2">7</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS5.p1.3.m3.1c">10^{-7}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS5.p1.3.m3.1d">10 start_POSTSUPERSCRIPT - 7 end_POSTSUPERSCRIPT</annotation></semantics></math>Hz and even smaller values at higher frequencies). However, this catalog of asteroids is wildly incomplete at <math alttext="30\;" class="ltx_Math" display="inline" id="S2.SS3.SSS5.p1.4.m4.1"><semantics id="S2.SS3.SSS5.p1.4.m4.1a"><mn id="S2.SS3.SSS5.p1.4.m4.1.1" xref="S2.SS3.SSS5.p1.4.m4.1.1.cmml">30</mn><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS5.p1.4.m4.1b"><cn id="S2.SS3.SSS5.p1.4.m4.1.1.cmml" type="integer" xref="S2.SS3.SSS5.p1.4.m4.1.1">30</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS5.p1.4.m4.1c">30\;</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS5.p1.4.m4.1d">30</annotation></semantics></math>AU. 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14 end_POSTSUPERSCRIPT m s Hz start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT</annotation></semantics></math> at <math alttext="4\times 10^{-7}" class="ltx_Math" display="inline" id="S2.SS3.SSS5.p1.9.m9.1"><semantics id="S2.SS3.SSS5.p1.9.m9.1a"><mrow id="S2.SS3.SSS5.p1.9.m9.1.1" xref="S2.SS3.SSS5.p1.9.m9.1.1.cmml"><mn id="S2.SS3.SSS5.p1.9.m9.1.1.2" xref="S2.SS3.SSS5.p1.9.m9.1.1.2.cmml">4</mn><mo id="S2.SS3.SSS5.p1.9.m9.1.1.1" lspace="0.222em" rspace="0.222em" xref="S2.SS3.SSS5.p1.9.m9.1.1.1.cmml">×</mo><msup id="S2.SS3.SSS5.p1.9.m9.1.1.3" xref="S2.SS3.SSS5.p1.9.m9.1.1.3.cmml"><mn id="S2.SS3.SSS5.p1.9.m9.1.1.3.2" xref="S2.SS3.SSS5.p1.9.m9.1.1.3.2.cmml">10</mn><mrow id="S2.SS3.SSS5.p1.9.m9.1.1.3.3" xref="S2.SS3.SSS5.p1.9.m9.1.1.3.3.cmml"><mo id="S2.SS3.SSS5.p1.9.m9.1.1.3.3a" xref="S2.SS3.SSS5.p1.9.m9.1.1.3.3.cmml">−</mo><mn id="S2.SS3.SSS5.p1.9.m9.1.1.3.3.2" xref="S2.SS3.SSS5.p1.9.m9.1.1.3.3.2.cmml">7</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS5.p1.9.m9.1b"><apply id="S2.SS3.SSS5.p1.9.m9.1.1.cmml" xref="S2.SS3.SSS5.p1.9.m9.1.1"><times id="S2.SS3.SSS5.p1.9.m9.1.1.1.cmml" xref="S2.SS3.SSS5.p1.9.m9.1.1.1"></times><cn id="S2.SS3.SSS5.p1.9.m9.1.1.2.cmml" type="integer" xref="S2.SS3.SSS5.p1.9.m9.1.1.2">4</cn><apply id="S2.SS3.SSS5.p1.9.m9.1.1.3.cmml" xref="S2.SS3.SSS5.p1.9.m9.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.SSS5.p1.9.m9.1.1.3.1.cmml" xref="S2.SS3.SSS5.p1.9.m9.1.1.3">superscript</csymbol><cn id="S2.SS3.SSS5.p1.9.m9.1.1.3.2.cmml" type="integer" xref="S2.SS3.SSS5.p1.9.m9.1.1.3.2">10</cn><apply id="S2.SS3.SSS5.p1.9.m9.1.1.3.3.cmml" xref="S2.SS3.SSS5.p1.9.m9.1.1.3.3"><minus id="S2.SS3.SSS5.p1.9.m9.1.1.3.3.1.cmml" xref="S2.SS3.SSS5.p1.9.m9.1.1.3.3"></minus><cn id="S2.SS3.SSS5.p1.9.m9.1.1.3.3.2.cmml" type="integer" xref="S2.SS3.SSS5.p1.9.m9.1.1.3.3.2">7</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS5.p1.9.m9.1c">4\times 10^{-7}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS5.p1.9.m9.1d">4 × 10 start_POSTSUPERSCRIPT - 7 end_POSTSUPERSCRIPT</annotation></semantics></math>Hz. Given that even these <math alttext="1~{}" class="ltx_Math" display="inline" id="S2.SS3.SSS5.p1.10.m10.1"><semantics id="S2.SS3.SSS5.p1.10.m10.1a"><mn id="S2.SS3.SSS5.p1.10.m10.1.1" xref="S2.SS3.SSS5.p1.10.m10.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS5.p1.10.m10.1b"><cn id="S2.SS3.SSS5.p1.10.m10.1.1.cmml" type="integer" xref="S2.SS3.SSS5.p1.10.m10.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS5.p1.10.m10.1c">1~{}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS5.p1.10.m10.1d">1</annotation></semantics></math>AU values are subdominant when compared to the other outer Solar System acceleration sources we compute, asteroids are unlikely to be an important acceleration source.</p> </div> <div class="ltx_para" id="S2.SS3.SSS5.p2"> <p class="ltx_p" id="S2.SS3.SSS5.p2.3">Displacements owing to gravitational pulls from the Sun and planets should be correctable, as the spacecraft ranging system would measure spacecraft separations to centimeter precision <cite class="ltx_cite ltx_citemacro_citep">(Boone & McQuinn, <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib18" title="">2023</a>)</cite>. Long-term trends from these pulls on the <math alttext="\gtrsim" class="ltx_Math" display="inline" id="S2.SS3.SSS5.p2.1.m1.1"><semantics id="S2.SS3.SSS5.p2.1.m1.1a"><mo id="S2.SS3.SSS5.p2.1.m1.1.1" xref="S2.SS3.SSS5.p2.1.m1.1.1.cmml">≳</mo><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS5.p2.1.m1.1b"><csymbol cd="latexml" id="S2.SS3.SSS5.p2.1.m1.1.1.cmml" xref="S2.SS3.SSS5.p2.1.m1.1.1">greater-than-or-equivalent-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS5.p2.1.m1.1c">\gtrsim</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS5.p2.1.m1.1d">≳</annotation></semantics></math> year orbital time scales of these bodies can be fit for and removed from contaminating the <math alttext="f\gg" class="ltx_Math" display="inline" id="S2.SS3.SSS5.p2.2.m2.1"><semantics id="S2.SS3.SSS5.p2.2.m2.1a"><mrow id="S2.SS3.SSS5.p2.2.m2.1.1" xref="S2.SS3.SSS5.p2.2.m2.1.1.cmml"><mi id="S2.SS3.SSS5.p2.2.m2.1.1.2" xref="S2.SS3.SSS5.p2.2.m2.1.1.2.cmml">f</mi><mo id="S2.SS3.SSS5.p2.2.m2.1.1.1" xref="S2.SS3.SSS5.p2.2.m2.1.1.1.cmml">≫</mo><mi id="S2.SS3.SSS5.p2.2.m2.1.1.3" xref="S2.SS3.SSS5.p2.2.m2.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS5.p2.2.m2.1b"><apply id="S2.SS3.SSS5.p2.2.m2.1.1.cmml" xref="S2.SS3.SSS5.p2.2.m2.1.1"><csymbol cd="latexml" id="S2.SS3.SSS5.p2.2.m2.1.1.1.cmml" xref="S2.SS3.SSS5.p2.2.m2.1.1.1">much-greater-than</csymbol><ci id="S2.SS3.SSS5.p2.2.m2.1.1.2.cmml" xref="S2.SS3.SSS5.p2.2.m2.1.1.2">𝑓</ci><csymbol cd="latexml" id="S2.SS3.SSS5.p2.2.m2.1.1.3.cmml" xref="S2.SS3.SSS5.p2.2.m2.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS5.p2.2.m2.1c">f\gg</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS5.p2.2.m2.1d">italic_f ≫</annotation></semantics></math>yr<sup class="ltx_sup" id="S2.SS3.SSS5.p2.3.1"><span class="ltx_text ltx_font_italic" id="S2.SS3.SSS5.p2.3.1.1">-1</span></sup> frequencies of interest. Of course these trends could also be used to analyze other bodies in the Solar System, such as the presence of a ninth planet (if one exists).</p> </div> </section> <section class="ltx_subsubsection" id="S2.SS3.SSS6"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">2.3.6 </span>Off-gassing and other spacecraft emissions</h4> <div class="ltx_para" id="S2.SS3.SSS6.p1"> <p class="ltx_p" id="S2.SS3.SSS6.p1.2">Because our spacecraft would be in such a low acceleration environment, a concern is that off-gassing either from thrusters or from other components in the spacecraft could drive a substantial acceleration. Since external accelerations are <math alttext="\sqrt{fS_{a}}\sim 10^{-14}" class="ltx_Math" display="inline" id="S2.SS3.SSS6.p1.1.m1.1"><semantics id="S2.SS3.SSS6.p1.1.m1.1a"><mrow id="S2.SS3.SSS6.p1.1.m1.1.1" xref="S2.SS3.SSS6.p1.1.m1.1.1.cmml"><msqrt id="S2.SS3.SSS6.p1.1.m1.1.1.2" xref="S2.SS3.SSS6.p1.1.m1.1.1.2.cmml"><mrow id="S2.SS3.SSS6.p1.1.m1.1.1.2.2" xref="S2.SS3.SSS6.p1.1.m1.1.1.2.2.cmml"><mi id="S2.SS3.SSS6.p1.1.m1.1.1.2.2.2" xref="S2.SS3.SSS6.p1.1.m1.1.1.2.2.2.cmml">f</mi><mo id="S2.SS3.SSS6.p1.1.m1.1.1.2.2.1" xref="S2.SS3.SSS6.p1.1.m1.1.1.2.2.1.cmml">⁢</mo><msub id="S2.SS3.SSS6.p1.1.m1.1.1.2.2.3" xref="S2.SS3.SSS6.p1.1.m1.1.1.2.2.3.cmml"><mi id="S2.SS3.SSS6.p1.1.m1.1.1.2.2.3.2" xref="S2.SS3.SSS6.p1.1.m1.1.1.2.2.3.2.cmml">S</mi><mi id="S2.SS3.SSS6.p1.1.m1.1.1.2.2.3.3" xref="S2.SS3.SSS6.p1.1.m1.1.1.2.2.3.3.cmml">a</mi></msub></mrow></msqrt><mo id="S2.SS3.SSS6.p1.1.m1.1.1.1" xref="S2.SS3.SSS6.p1.1.m1.1.1.1.cmml">∼</mo><msup id="S2.SS3.SSS6.p1.1.m1.1.1.3" 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type="integer" xref="S2.SS3.SSS6.p1.1.m1.1.1.3.3.2">14</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS6.p1.1.m1.1c">\sqrt{fS_{a}}\sim 10^{-14}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS6.p1.1.m1.1d">square-root start_ARG italic_f italic_S start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT end_ARG ∼ 10 start_POSTSUPERSCRIPT - 14 end_POSTSUPERSCRIPT</annotation></semantics></math> m s<sup class="ltx_sup" id="S2.SS3.SSS6.p1.2.1"><span class="ltx_text ltx_font_italic" id="S2.SS3.SSS6.p1.2.1.1">-2</span></sup> for the wave periods of interest, 10 picoNewton of thrust over these periods could result in increased noise.</p> </div> <div class="ltx_para" id="S2.SS3.SSS6.p2"> <p class="ltx_p" id="S2.SS3.SSS6.p2.6">Thrusters are necessary to dissipate angular momentum on the reaction wheels on week to month times,<span class="ltx_note ltx_role_footnote" id="footnote9"><sup class="ltx_note_mark">9</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">9</sup><span class="ltx_tag ltx_tag_note">9</span>For a center of solar radiation deposition differing by <math alttext="\delta x" class="ltx_Math" display="inline" id="footnote9.m1.1"><semantics id="footnote9.m1.1b"><mrow id="footnote9.m1.1.1" xref="footnote9.m1.1.1.cmml"><mi id="footnote9.m1.1.1.2" xref="footnote9.m1.1.1.2.cmml">δ</mi><mo id="footnote9.m1.1.1.1" xref="footnote9.m1.1.1.1.cmml">⁢</mo><mi id="footnote9.m1.1.1.3" xref="footnote9.m1.1.1.3.cmml">x</mi></mrow><annotation-xml encoding="MathML-Content" id="footnote9.m1.1c"><apply id="footnote9.m1.1.1.cmml" xref="footnote9.m1.1.1"><times id="footnote9.m1.1.1.1.cmml" xref="footnote9.m1.1.1.1"></times><ci id="footnote9.m1.1.1.2.cmml" xref="footnote9.m1.1.1.2">𝛿</ci><ci id="footnote9.m1.1.1.3.cmml" xref="footnote9.m1.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote9.m1.1d">\delta x</annotation><annotation 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id="footnote9.m2.5.5.5.5.1.1.2.1.cmml" xref="footnote9.m2.5.5.5.5.1.1.2.1"></divide><ci id="footnote9.m2.5.5.5.5.1.1.2.2.cmml" xref="footnote9.m2.5.5.5.5.1.1.2.2">𝑡</ci><cn id="footnote9.m2.5.5.5.5.1.1.2.3.cmml" type="integer" xref="footnote9.m2.5.5.5.5.1.1.2.3">1</cn></apply><ci id="footnote9.m2.5.5.5.5.1.1.3.cmml" xref="footnote9.m2.5.5.5.5.1.1.3">week</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote9.m2.5d">H\sim A_{\rm eff}\delta xF_{\odot}/c\times I\times t=30~{}(\delta x/1~{}{\rm m% })(A_{\rm eff}/10~{}{\rm m}^{2})(r/30{\rm AU})^{-2}(I/1000{\rm\;kg\,m^{2}})(t/% 1{\rm~{}week})~{}</annotation><annotation encoding="application/x-llamapun" id="footnote9.m2.5e">italic_H ∼ italic_A start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT italic_δ italic_x italic_F start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT / italic_c × italic_I × italic_t = 30 ( italic_δ italic_x / 1 roman_m ) ( italic_A start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT / 10 roman_m start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ( italic_r / 30 roman_A roman_U ) start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT ( italic_I / 1000 roman_kg roman_m start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ( italic_t / 1 roman_week )</annotation></semantics></math>N m s, where <math alttext="F_{\odot}" class="ltx_Math" display="inline" id="footnote9.m3.1"><semantics id="footnote9.m3.1b"><msub id="footnote9.m3.1.1" xref="footnote9.m3.1.1.cmml"><mi id="footnote9.m3.1.1.2" xref="footnote9.m3.1.1.2.cmml">F</mi><mo id="footnote9.m3.1.1.3" xref="footnote9.m3.1.1.3.cmml">⊙</mo></msub><annotation-xml encoding="MathML-Content" id="footnote9.m3.1c"><apply id="footnote9.m3.1.1.cmml" xref="footnote9.m3.1.1"><csymbol cd="ambiguous" id="footnote9.m3.1.1.1.cmml" xref="footnote9.m3.1.1">subscript</csymbol><ci id="footnote9.m3.1.1.2.cmml" xref="footnote9.m3.1.1.2">𝐹</ci><csymbol cd="latexml" id="footnote9.m3.1.1.3.cmml" xref="footnote9.m3.1.1.3">direct-product</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote9.m3.1d">F_{\odot}</annotation><annotation encoding="application/x-llamapun" id="footnote9.m3.1e">italic_F start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT</annotation></semantics></math> is the solar energy flux and where moment of inertia of <math alttext="I\sim 1000{\rm\;kg\,m^{2}}" class="ltx_Math" display="inline" id="footnote9.m4.1"><semantics id="footnote9.m4.1b"><mrow id="footnote9.m4.1.1" xref="footnote9.m4.1.1.cmml"><mi id="footnote9.m4.1.1.2" xref="footnote9.m4.1.1.2.cmml">I</mi><mo id="footnote9.m4.1.1.1" xref="footnote9.m4.1.1.1.cmml">∼</mo><mrow id="footnote9.m4.1.1.3" xref="footnote9.m4.1.1.3.cmml"><mn id="footnote9.m4.1.1.3.2" xref="footnote9.m4.1.1.3.2.cmml">1000</mn><mo id="footnote9.m4.1.1.3.1" lspace="0.280em" xref="footnote9.m4.1.1.3.1.cmml">⁢</mo><mi id="footnote9.m4.1.1.3.3" xref="footnote9.m4.1.1.3.3.cmml">kg</mi><mo id="footnote9.m4.1.1.3.1b" lspace="0.170em" xref="footnote9.m4.1.1.3.1.cmml">⁢</mo><msup id="footnote9.m4.1.1.3.4" xref="footnote9.m4.1.1.3.4.cmml"><mi id="footnote9.m4.1.1.3.4.2" mathvariant="normal" xref="footnote9.m4.1.1.3.4.2.cmml">m</mi><mn id="footnote9.m4.1.1.3.4.3" xref="footnote9.m4.1.1.3.4.3.cmml">2</mn></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote9.m4.1c"><apply id="footnote9.m4.1.1.cmml" xref="footnote9.m4.1.1"><csymbol cd="latexml" id="footnote9.m4.1.1.1.cmml" xref="footnote9.m4.1.1.1">similar-to</csymbol><ci id="footnote9.m4.1.1.2.cmml" xref="footnote9.m4.1.1.2">𝐼</ci><apply id="footnote9.m4.1.1.3.cmml" xref="footnote9.m4.1.1.3"><times id="footnote9.m4.1.1.3.1.cmml" xref="footnote9.m4.1.1.3.1"></times><cn id="footnote9.m4.1.1.3.2.cmml" type="integer" xref="footnote9.m4.1.1.3.2">1000</cn><ci id="footnote9.m4.1.1.3.3.cmml" xref="footnote9.m4.1.1.3.3">kg</ci><apply id="footnote9.m4.1.1.3.4.cmml" xref="footnote9.m4.1.1.3.4"><csymbol cd="ambiguous" id="footnote9.m4.1.1.3.4.1.cmml" xref="footnote9.m4.1.1.3.4">superscript</csymbol><ci id="footnote9.m4.1.1.3.4.2.cmml" xref="footnote9.m4.1.1.3.4.2">m</ci><cn id="footnote9.m4.1.1.3.4.3.cmml" type="integer" xref="footnote9.m4.1.1.3.4.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote9.m4.1d">I\sim 1000{\rm\;kg\,m^{2}}</annotation><annotation encoding="application/x-llamapun" id="footnote9.m4.1e">italic_I ∼ 1000 roman_kg roman_m start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> are reasonable for the radio dish configuration. We have found several examples of commercial reaction wheels max out at <math alttext="H_{\rm max}=1-10" class="ltx_Math" display="inline" id="footnote9.m5.1"><semantics id="footnote9.m5.1b"><mrow id="footnote9.m5.1.1" xref="footnote9.m5.1.1.cmml"><msub id="footnote9.m5.1.1.2" xref="footnote9.m5.1.1.2.cmml"><mi id="footnote9.m5.1.1.2.2" xref="footnote9.m5.1.1.2.2.cmml">H</mi><mi id="footnote9.m5.1.1.2.3" xref="footnote9.m5.1.1.2.3.cmml">max</mi></msub><mo id="footnote9.m5.1.1.1" xref="footnote9.m5.1.1.1.cmml">=</mo><mrow id="footnote9.m5.1.1.3" xref="footnote9.m5.1.1.3.cmml"><mn id="footnote9.m5.1.1.3.2" xref="footnote9.m5.1.1.3.2.cmml">1</mn><mo id="footnote9.m5.1.1.3.1" xref="footnote9.m5.1.1.3.1.cmml">−</mo><mn id="footnote9.m5.1.1.3.3" xref="footnote9.m5.1.1.3.3.cmml">10</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote9.m5.1c"><apply id="footnote9.m5.1.1.cmml" xref="footnote9.m5.1.1"><eq id="footnote9.m5.1.1.1.cmml" xref="footnote9.m5.1.1.1"></eq><apply id="footnote9.m5.1.1.2.cmml" xref="footnote9.m5.1.1.2"><csymbol cd="ambiguous" id="footnote9.m5.1.1.2.1.cmml" xref="footnote9.m5.1.1.2">subscript</csymbol><ci id="footnote9.m5.1.1.2.2.cmml" xref="footnote9.m5.1.1.2.2">𝐻</ci><ci id="footnote9.m5.1.1.2.3.cmml" xref="footnote9.m5.1.1.2.3">max</ci></apply><apply id="footnote9.m5.1.1.3.cmml" xref="footnote9.m5.1.1.3"><minus id="footnote9.m5.1.1.3.1.cmml" xref="footnote9.m5.1.1.3.1"></minus><cn id="footnote9.m5.1.1.3.2.cmml" type="integer" xref="footnote9.m5.1.1.3.2">1</cn><cn id="footnote9.m5.1.1.3.3.cmml" type="integer" xref="footnote9.m5.1.1.3.3">10</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote9.m5.1d">H_{\rm max}=1-10</annotation><annotation encoding="application/x-llamapun" id="footnote9.m5.1e">italic_H start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT = 1 - 10</annotation></semantics></math> N m s. They consume between 4 and 10 Watts during normal operation and weigh on the order of a kilogram, values that could potentially be accommodated by outer Solar System spacecraft. This suggests that dissipating the reaction wheels must be done every <math alttext="\sim 1~{}" class="ltx_Math" display="inline" id="footnote9.m6.1"><semantics id="footnote9.m6.1b"><mrow id="footnote9.m6.1.1" xref="footnote9.m6.1.1.cmml"><mi id="footnote9.m6.1.1.2" xref="footnote9.m6.1.1.2.cmml"></mi><mo id="footnote9.m6.1.1.1" xref="footnote9.m6.1.1.1.cmml">∼</mo><mn id="footnote9.m6.1.1.3" xref="footnote9.m6.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="footnote9.m6.1c"><apply id="footnote9.m6.1.1.cmml" xref="footnote9.m6.1.1"><csymbol cd="latexml" id="footnote9.m6.1.1.1.cmml" xref="footnote9.m6.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="footnote9.m6.1.1.2.cmml" xref="footnote9.m6.1.1.2">absent</csymbol><cn id="footnote9.m6.1.1.3.cmml" type="integer" xref="footnote9.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote9.m6.1d">\sim 1~{}</annotation><annotation encoding="application/x-llamapun" id="footnote9.m6.1e">∼ 1</annotation></semantics></math>week, although the design can surely be optimized to minimize this build-up time.</span></span></span> but for the science operation would be generally turned off. Micro-Newton thrusters must be able to have their thrust turned off at the part in <math alttext="10^{5}" class="ltx_Math" display="inline" id="S2.SS3.SSS6.p2.1.m1.1"><semantics id="S2.SS3.SSS6.p2.1.m1.1a"><msup id="S2.SS3.SSS6.p2.1.m1.1.1" xref="S2.SS3.SSS6.p2.1.m1.1.1.cmml"><mn id="S2.SS3.SSS6.p2.1.m1.1.1.2" xref="S2.SS3.SSS6.p2.1.m1.1.1.2.cmml">10</mn><mn id="S2.SS3.SSS6.p2.1.m1.1.1.3" xref="S2.SS3.SSS6.p2.1.m1.1.1.3.cmml">5</mn></msup><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS6.p2.1.m1.1b"><apply id="S2.SS3.SSS6.p2.1.m1.1.1.cmml" xref="S2.SS3.SSS6.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS6.p2.1.m1.1.1.1.cmml" xref="S2.SS3.SSS6.p2.1.m1.1.1">superscript</csymbol><cn id="S2.SS3.SSS6.p2.1.m1.1.1.2.cmml" type="integer" xref="S2.SS3.SSS6.p2.1.m1.1.1.2">10</cn><cn id="S2.SS3.SSS6.p2.1.m1.1.1.3.cmml" type="integer" xref="S2.SS3.SSS6.p2.1.m1.1.1.3">5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS6.p2.1.m1.1c">10^{5}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS6.p2.1.m1.1d">10 start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT</annotation></semantics></math> level to achieve this specification. Additionally, consider the off-gassing of <math alttext="\sim 280" class="ltx_Math" display="inline" id="S2.SS3.SSS6.p2.2.m2.1"><semantics id="S2.SS3.SSS6.p2.2.m2.1a"><mrow id="S2.SS3.SSS6.p2.2.m2.1.1" xref="S2.SS3.SSS6.p2.2.m2.1.1.cmml"><mi id="S2.SS3.SSS6.p2.2.m2.1.1.2" xref="S2.SS3.SSS6.p2.2.m2.1.1.2.cmml"></mi><mo id="S2.SS3.SSS6.p2.2.m2.1.1.1" xref="S2.SS3.SSS6.p2.2.m2.1.1.1.cmml">∼</mo><mn id="S2.SS3.SSS6.p2.2.m2.1.1.3" xref="S2.SS3.SSS6.p2.2.m2.1.1.3.cmml">280</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS6.p2.2.m2.1b"><apply id="S2.SS3.SSS6.p2.2.m2.1.1.cmml" xref="S2.SS3.SSS6.p2.2.m2.1.1"><csymbol cd="latexml" id="S2.SS3.SSS6.p2.2.m2.1.1.1.cmml" xref="S2.SS3.SSS6.p2.2.m2.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S2.SS3.SSS6.p2.2.m2.1.1.2.cmml" xref="S2.SS3.SSS6.p2.2.m2.1.1.2">absent</csymbol><cn id="S2.SS3.SSS6.p2.2.m2.1.1.3.cmml" type="integer" xref="S2.SS3.SSS6.p2.2.m2.1.1.3">280</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS6.p2.2.m2.1c">\sim 280</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS6.p2.2.m2.1d">∼ 280</annotation></semantics></math> K gas from within the thermally regulated spacecraft. If the spacecraft emits in a single direction, the acceleration would be <math alttext="a\sim\Delta mfv" class="ltx_Math" display="inline" id="S2.SS3.SSS6.p2.3.m3.1"><semantics id="S2.SS3.SSS6.p2.3.m3.1a"><mrow id="S2.SS3.SSS6.p2.3.m3.1.1" xref="S2.SS3.SSS6.p2.3.m3.1.1.cmml"><mi id="S2.SS3.SSS6.p2.3.m3.1.1.2" xref="S2.SS3.SSS6.p2.3.m3.1.1.2.cmml">a</mi><mo id="S2.SS3.SSS6.p2.3.m3.1.1.1" xref="S2.SS3.SSS6.p2.3.m3.1.1.1.cmml">∼</mo><mrow id="S2.SS3.SSS6.p2.3.m3.1.1.3" xref="S2.SS3.SSS6.p2.3.m3.1.1.3.cmml"><mi id="S2.SS3.SSS6.p2.3.m3.1.1.3.2" mathvariant="normal" xref="S2.SS3.SSS6.p2.3.m3.1.1.3.2.cmml">Δ</mi><mo id="S2.SS3.SSS6.p2.3.m3.1.1.3.1" xref="S2.SS3.SSS6.p2.3.m3.1.1.3.1.cmml">⁢</mo><mi id="S2.SS3.SSS6.p2.3.m3.1.1.3.3" xref="S2.SS3.SSS6.p2.3.m3.1.1.3.3.cmml">m</mi><mo id="S2.SS3.SSS6.p2.3.m3.1.1.3.1a" xref="S2.SS3.SSS6.p2.3.m3.1.1.3.1.cmml">⁢</mo><mi id="S2.SS3.SSS6.p2.3.m3.1.1.3.4" xref="S2.SS3.SSS6.p2.3.m3.1.1.3.4.cmml">f</mi><mo id="S2.SS3.SSS6.p2.3.m3.1.1.3.1b" xref="S2.SS3.SSS6.p2.3.m3.1.1.3.1.cmml">⁢</mo><mi id="S2.SS3.SSS6.p2.3.m3.1.1.3.5" xref="S2.SS3.SSS6.p2.3.m3.1.1.3.5.cmml">v</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS6.p2.3.m3.1b"><apply id="S2.SS3.SSS6.p2.3.m3.1.1.cmml" xref="S2.SS3.SSS6.p2.3.m3.1.1"><csymbol cd="latexml" id="S2.SS3.SSS6.p2.3.m3.1.1.1.cmml" xref="S2.SS3.SSS6.p2.3.m3.1.1.1">similar-to</csymbol><ci id="S2.SS3.SSS6.p2.3.m3.1.1.2.cmml" xref="S2.SS3.SSS6.p2.3.m3.1.1.2">𝑎</ci><apply id="S2.SS3.SSS6.p2.3.m3.1.1.3.cmml" xref="S2.SS3.SSS6.p2.3.m3.1.1.3"><times id="S2.SS3.SSS6.p2.3.m3.1.1.3.1.cmml" xref="S2.SS3.SSS6.p2.3.m3.1.1.3.1"></times><ci id="S2.SS3.SSS6.p2.3.m3.1.1.3.2.cmml" xref="S2.SS3.SSS6.p2.3.m3.1.1.3.2">Δ</ci><ci id="S2.SS3.SSS6.p2.3.m3.1.1.3.3.cmml" xref="S2.SS3.SSS6.p2.3.m3.1.1.3.3">𝑚</ci><ci id="S2.SS3.SSS6.p2.3.m3.1.1.3.4.cmml" xref="S2.SS3.SSS6.p2.3.m3.1.1.3.4">𝑓</ci><ci id="S2.SS3.SSS6.p2.3.m3.1.1.3.5.cmml" xref="S2.SS3.SSS6.p2.3.m3.1.1.3.5">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS6.p2.3.m3.1c">a\sim\Delta mfv</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS6.p2.3.m3.1d">italic_a ∼ roman_Δ italic_m italic_f italic_v</annotation></semantics></math> for <math alttext="\Delta m" class="ltx_Math" display="inline" id="S2.SS3.SSS6.p2.4.m4.1"><semantics id="S2.SS3.SSS6.p2.4.m4.1a"><mrow id="S2.SS3.SSS6.p2.4.m4.1.1" xref="S2.SS3.SSS6.p2.4.m4.1.1.cmml"><mi id="S2.SS3.SSS6.p2.4.m4.1.1.2" mathvariant="normal" xref="S2.SS3.SSS6.p2.4.m4.1.1.2.cmml">Δ</mi><mo id="S2.SS3.SSS6.p2.4.m4.1.1.1" xref="S2.SS3.SSS6.p2.4.m4.1.1.1.cmml">⁢</mo><mi id="S2.SS3.SSS6.p2.4.m4.1.1.3" xref="S2.SS3.SSS6.p2.4.m4.1.1.3.cmml">m</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS6.p2.4.m4.1b"><apply id="S2.SS3.SSS6.p2.4.m4.1.1.cmml" xref="S2.SS3.SSS6.p2.4.m4.1.1"><times id="S2.SS3.SSS6.p2.4.m4.1.1.1.cmml" xref="S2.SS3.SSS6.p2.4.m4.1.1.1"></times><ci id="S2.SS3.SSS6.p2.4.m4.1.1.2.cmml" xref="S2.SS3.SSS6.p2.4.m4.1.1.2">Δ</ci><ci id="S2.SS3.SSS6.p2.4.m4.1.1.3.cmml" xref="S2.SS3.SSS6.p2.4.m4.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS6.p2.4.m4.1c">\Delta m</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS6.p2.4.m4.1d">roman_Δ italic_m</annotation></semantics></math> emitted over time <math alttext="f^{-1}" class="ltx_Math" display="inline" id="S2.SS3.SSS6.p2.5.m5.1"><semantics id="S2.SS3.SSS6.p2.5.m5.1a"><msup id="S2.SS3.SSS6.p2.5.m5.1.1" xref="S2.SS3.SSS6.p2.5.m5.1.1.cmml"><mi id="S2.SS3.SSS6.p2.5.m5.1.1.2" xref="S2.SS3.SSS6.p2.5.m5.1.1.2.cmml">f</mi><mrow id="S2.SS3.SSS6.p2.5.m5.1.1.3" xref="S2.SS3.SSS6.p2.5.m5.1.1.3.cmml"><mo id="S2.SS3.SSS6.p2.5.m5.1.1.3a" xref="S2.SS3.SSS6.p2.5.m5.1.1.3.cmml">−</mo><mn id="S2.SS3.SSS6.p2.5.m5.1.1.3.2" xref="S2.SS3.SSS6.p2.5.m5.1.1.3.2.cmml">1</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS6.p2.5.m5.1b"><apply id="S2.SS3.SSS6.p2.5.m5.1.1.cmml" xref="S2.SS3.SSS6.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS6.p2.5.m5.1.1.1.cmml" xref="S2.SS3.SSS6.p2.5.m5.1.1">superscript</csymbol><ci id="S2.SS3.SSS6.p2.5.m5.1.1.2.cmml" xref="S2.SS3.SSS6.p2.5.m5.1.1.2">𝑓</ci><apply id="S2.SS3.SSS6.p2.5.m5.1.1.3.cmml" xref="S2.SS3.SSS6.p2.5.m5.1.1.3"><minus id="S2.SS3.SSS6.p2.5.m5.1.1.3.1.cmml" xref="S2.SS3.SSS6.p2.5.m5.1.1.3"></minus><cn id="S2.SS3.SSS6.p2.5.m5.1.1.3.2.cmml" type="integer" xref="S2.SS3.SSS6.p2.5.m5.1.1.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS6.p2.5.m5.1c">f^{-1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS6.p2.5.m5.1d">italic_f start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>, or <math alttext="a\sim 10^{-14}(\Delta m/10^{-4}{\rm g})(f/10^{-6}{\rm Hz})(M/10^{3}{\rm kg}){% \rm m~{}s}^{-2}" class="ltx_Math" 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id="S2.SS3.SSS6.p2.6.m6.1.1.1.1.1.1.2.3.2" xref="S2.SS3.SSS6.p2.6.m6.1.1.1.1.1.1.2.3.2.cmml">10</mn><mrow id="S2.SS3.SSS6.p2.6.m6.1.1.1.1.1.1.2.3.3" xref="S2.SS3.SSS6.p2.6.m6.1.1.1.1.1.1.2.3.3.cmml"><mo id="S2.SS3.SSS6.p2.6.m6.1.1.1.1.1.1.2.3.3a" xref="S2.SS3.SSS6.p2.6.m6.1.1.1.1.1.1.2.3.3.cmml">−</mo><mn id="S2.SS3.SSS6.p2.6.m6.1.1.1.1.1.1.2.3.3.2" xref="S2.SS3.SSS6.p2.6.m6.1.1.1.1.1.1.2.3.3.2.cmml">4</mn></mrow></msup></mrow><mo id="S2.SS3.SSS6.p2.6.m6.1.1.1.1.1.1.1" xref="S2.SS3.SSS6.p2.6.m6.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S2.SS3.SSS6.p2.6.m6.1.1.1.1.1.1.3" mathvariant="normal" xref="S2.SS3.SSS6.p2.6.m6.1.1.1.1.1.1.3.cmml">g</mi></mrow><mo id="S2.SS3.SSS6.p2.6.m6.1.1.1.1.1.3" stretchy="false" xref="S2.SS3.SSS6.p2.6.m6.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S2.SS3.SSS6.p2.6.m6.3.3.3.4a" xref="S2.SS3.SSS6.p2.6.m6.3.3.3.4.cmml">⁢</mo><mrow id="S2.SS3.SSS6.p2.6.m6.2.2.2.2.1" xref="S2.SS3.SSS6.p2.6.m6.2.2.2.2.1.1.cmml"><mo id="S2.SS3.SSS6.p2.6.m6.2.2.2.2.1.2" stretchy="false" 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xref="S2.SS3.SSS6.p2.6.m6.2.2.2.2.1.1.1.cmml">⁢</mo><mi id="S2.SS3.SSS6.p2.6.m6.2.2.2.2.1.1.3" xref="S2.SS3.SSS6.p2.6.m6.2.2.2.2.1.1.3.cmml">Hz</mi></mrow><mo id="S2.SS3.SSS6.p2.6.m6.2.2.2.2.1.3" stretchy="false" xref="S2.SS3.SSS6.p2.6.m6.2.2.2.2.1.1.cmml">)</mo></mrow><mo id="S2.SS3.SSS6.p2.6.m6.3.3.3.4b" xref="S2.SS3.SSS6.p2.6.m6.3.3.3.4.cmml">⁢</mo><mrow id="S2.SS3.SSS6.p2.6.m6.3.3.3.3.1" xref="S2.SS3.SSS6.p2.6.m6.3.3.3.3.1.1.cmml"><mo id="S2.SS3.SSS6.p2.6.m6.3.3.3.3.1.2" stretchy="false" xref="S2.SS3.SSS6.p2.6.m6.3.3.3.3.1.1.cmml">(</mo><mrow id="S2.SS3.SSS6.p2.6.m6.3.3.3.3.1.1" xref="S2.SS3.SSS6.p2.6.m6.3.3.3.3.1.1.cmml"><mrow id="S2.SS3.SSS6.p2.6.m6.3.3.3.3.1.1.2" xref="S2.SS3.SSS6.p2.6.m6.3.3.3.3.1.1.2.cmml"><mi id="S2.SS3.SSS6.p2.6.m6.3.3.3.3.1.1.2.2" xref="S2.SS3.SSS6.p2.6.m6.3.3.3.3.1.1.2.2.cmml">M</mi><mo id="S2.SS3.SSS6.p2.6.m6.3.3.3.3.1.1.2.1" xref="S2.SS3.SSS6.p2.6.m6.3.3.3.3.1.1.2.1.cmml">/</mo><msup id="S2.SS3.SSS6.p2.6.m6.3.3.3.3.1.1.2.3" 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xref="S2.SS3.SSS6.p2.6.m6.3.3.3.7.3"></minus><cn id="S2.SS3.SSS6.p2.6.m6.3.3.3.7.3.2.cmml" type="integer" xref="S2.SS3.SSS6.p2.6.m6.3.3.3.7.3.2">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS6.p2.6.m6.3c">a\sim 10^{-14}(\Delta m/10^{-4}{\rm g})(f/10^{-6}{\rm Hz})(M/10^{3}{\rm kg}){% \rm m~{}s}^{-2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS6.p2.6.m6.3d">italic_a ∼ 10 start_POSTSUPERSCRIPT - 14 end_POSTSUPERSCRIPT ( roman_Δ italic_m / 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT roman_g ) ( italic_f / 10 start_POSTSUPERSCRIPT - 6 end_POSTSUPERSCRIPT roman_Hz ) ( italic_M / 10 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT roman_kg ) roman_m roman_s start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT</annotation></semantics></math>. Ventilation and thermal control systems that are designed to off-gas perpendicular to the arms and maintain a stable environment over the gravitational wave periods of interest would likely be required for such precision.</p> </div> </section> </section> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">3 </span>Strain power sensitivity</h2> <figure class="ltx_figure" id="S3.F3"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="332" id="S3.F3.g1" src="x3.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 3: </span>Gravitational wave strain sensitivities for the <span class="ltx_text ltx_font_bold" id="S3.F3.29.1">first design architecture: the two-arm time-delay interferometry concept</span>, with <math alttext="r=L=10" class="ltx_Math" display="inline" id="S3.F3.13.m1.1"><semantics id="S3.F3.13.m1.1b"><mrow id="S3.F3.13.m1.1.1" xref="S3.F3.13.m1.1.1.cmml"><mi id="S3.F3.13.m1.1.1.2" xref="S3.F3.13.m1.1.1.2.cmml">r</mi><mo id="S3.F3.13.m1.1.1.3" xref="S3.F3.13.m1.1.1.3.cmml">=</mo><mi id="S3.F3.13.m1.1.1.4" xref="S3.F3.13.m1.1.1.4.cmml">L</mi><mo id="S3.F3.13.m1.1.1.5" xref="S3.F3.13.m1.1.1.5.cmml">=</mo><mn id="S3.F3.13.m1.1.1.6" xref="S3.F3.13.m1.1.1.6.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.F3.13.m1.1c"><apply id="S3.F3.13.m1.1.1.cmml" xref="S3.F3.13.m1.1.1"><and id="S3.F3.13.m1.1.1a.cmml" xref="S3.F3.13.m1.1.1"></and><apply id="S3.F3.13.m1.1.1b.cmml" xref="S3.F3.13.m1.1.1"><eq id="S3.F3.13.m1.1.1.3.cmml" xref="S3.F3.13.m1.1.1.3"></eq><ci id="S3.F3.13.m1.1.1.2.cmml" xref="S3.F3.13.m1.1.1.2">𝑟</ci><ci id="S3.F3.13.m1.1.1.4.cmml" xref="S3.F3.13.m1.1.1.4">𝐿</ci></apply><apply id="S3.F3.13.m1.1.1c.cmml" xref="S3.F3.13.m1.1.1"><eq id="S3.F3.13.m1.1.1.5.cmml" xref="S3.F3.13.m1.1.1.5"></eq><share href="https://arxiv.org/html/2411.15072v1#S3.F3.13.m1.1.1.4.cmml" id="S3.F3.13.m1.1.1d.cmml" xref="S3.F3.13.m1.1.1"></share><cn id="S3.F3.13.m1.1.1.6.cmml" type="integer" xref="S3.F3.13.m1.1.1.6">10</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.13.m1.1d">r=L=10</annotation><annotation encoding="application/x-llamapun" id="S3.F3.13.m1.1e">italic_r = italic_L = 10</annotation></semantics></math> AU (left panel) and <math alttext="r=L=30" class="ltx_Math" display="inline" id="S3.F3.14.m2.1"><semantics id="S3.F3.14.m2.1b"><mrow id="S3.F3.14.m2.1.1" xref="S3.F3.14.m2.1.1.cmml"><mi id="S3.F3.14.m2.1.1.2" xref="S3.F3.14.m2.1.1.2.cmml">r</mi><mo id="S3.F3.14.m2.1.1.3" xref="S3.F3.14.m2.1.1.3.cmml">=</mo><mi id="S3.F3.14.m2.1.1.4" xref="S3.F3.14.m2.1.1.4.cmml">L</mi><mo id="S3.F3.14.m2.1.1.5" xref="S3.F3.14.m2.1.1.5.cmml">=</mo><mn id="S3.F3.14.m2.1.1.6" xref="S3.F3.14.m2.1.1.6.cmml">30</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.F3.14.m2.1c"><apply id="S3.F3.14.m2.1.1.cmml" xref="S3.F3.14.m2.1.1"><and id="S3.F3.14.m2.1.1a.cmml" xref="S3.F3.14.m2.1.1"></and><apply id="S3.F3.14.m2.1.1b.cmml" xref="S3.F3.14.m2.1.1"><eq id="S3.F3.14.m2.1.1.3.cmml" xref="S3.F3.14.m2.1.1.3"></eq><ci id="S3.F3.14.m2.1.1.2.cmml" xref="S3.F3.14.m2.1.1.2">𝑟</ci><ci id="S3.F3.14.m2.1.1.4.cmml" xref="S3.F3.14.m2.1.1.4">𝐿</ci></apply><apply id="S3.F3.14.m2.1.1c.cmml" xref="S3.F3.14.m2.1.1"><eq id="S3.F3.14.m2.1.1.5.cmml" xref="S3.F3.14.m2.1.1.5"></eq><share href="https://arxiv.org/html/2411.15072v1#S3.F3.14.m2.1.1.4.cmml" id="S3.F3.14.m2.1.1d.cmml" xref="S3.F3.14.m2.1.1"></share><cn id="S3.F3.14.m2.1.1.6.cmml" type="integer" xref="S3.F3.14.m2.1.1.6">30</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.14.m2.1d">r=L=30</annotation><annotation encoding="application/x-llamapun" id="S3.F3.14.m2.1e">italic_r = italic_L = 30</annotation></semantics></math> AU (right panel). The calculations assume an effective area to mass ratio for the spacecraft of <math alttext="A_{\rm eff}/M=0.01~{}" class="ltx_Math" display="inline" id="S3.F3.15.m3.1"><semantics id="S3.F3.15.m3.1b"><mrow id="S3.F3.15.m3.1.1" xref="S3.F3.15.m3.1.1.cmml"><mrow id="S3.F3.15.m3.1.1.2" xref="S3.F3.15.m3.1.1.2.cmml"><msub id="S3.F3.15.m3.1.1.2.2" xref="S3.F3.15.m3.1.1.2.2.cmml"><mi id="S3.F3.15.m3.1.1.2.2.2" xref="S3.F3.15.m3.1.1.2.2.2.cmml">A</mi><mi id="S3.F3.15.m3.1.1.2.2.3" xref="S3.F3.15.m3.1.1.2.2.3.cmml">eff</mi></msub><mo id="S3.F3.15.m3.1.1.2.1" xref="S3.F3.15.m3.1.1.2.1.cmml">/</mo><mi id="S3.F3.15.m3.1.1.2.3" xref="S3.F3.15.m3.1.1.2.3.cmml">M</mi></mrow><mo id="S3.F3.15.m3.1.1.1" xref="S3.F3.15.m3.1.1.1.cmml">=</mo><mn id="S3.F3.15.m3.1.1.3" xref="S3.F3.15.m3.1.1.3.cmml">0.01</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.F3.15.m3.1c"><apply id="S3.F3.15.m3.1.1.cmml" xref="S3.F3.15.m3.1.1"><eq id="S3.F3.15.m3.1.1.1.cmml" xref="S3.F3.15.m3.1.1.1"></eq><apply id="S3.F3.15.m3.1.1.2.cmml" xref="S3.F3.15.m3.1.1.2"><divide id="S3.F3.15.m3.1.1.2.1.cmml" xref="S3.F3.15.m3.1.1.2.1"></divide><apply id="S3.F3.15.m3.1.1.2.2.cmml" xref="S3.F3.15.m3.1.1.2.2"><csymbol cd="ambiguous" id="S3.F3.15.m3.1.1.2.2.1.cmml" xref="S3.F3.15.m3.1.1.2.2">subscript</csymbol><ci id="S3.F3.15.m3.1.1.2.2.2.cmml" xref="S3.F3.15.m3.1.1.2.2.2">𝐴</ci><ci id="S3.F3.15.m3.1.1.2.2.3.cmml" xref="S3.F3.15.m3.1.1.2.2.3">eff</ci></apply><ci id="S3.F3.15.m3.1.1.2.3.cmml" xref="S3.F3.15.m3.1.1.2.3">𝑀</ci></apply><cn id="S3.F3.15.m3.1.1.3.cmml" type="float" xref="S3.F3.15.m3.1.1.3">0.01</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.15.m3.1d">A_{\rm eff}/M=0.01~{}</annotation><annotation encoding="application/x-llamapun" id="S3.F3.15.m3.1e">italic_A start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT / italic_M = 0.01</annotation></semantics></math>m<sup class="ltx_sup" id="S3.F3.30.2">2</sup> kg<sup class="ltx_sup" id="S3.F3.31.3"><span class="ltx_text ltx_font_italic" id="S3.F3.31.3.1">-1</span></sup> and a geometric suppression of <math alttext="0.5" class="ltx_Math" display="inline" id="S3.F3.18.m6.1"><semantics id="S3.F3.18.m6.1b"><mn id="S3.F3.18.m6.1.1" xref="S3.F3.18.m6.1.1.cmml">0.5</mn><annotation-xml encoding="MathML-Content" id="S3.F3.18.m6.1c"><cn id="S3.F3.18.m6.1.1.cmml" type="float" xref="S3.F3.18.m6.1.1">0.5</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.18.m6.1d">0.5</annotation><annotation encoding="application/x-llamapun" id="S3.F3.18.m6.1e">0.5</annotation></semantics></math>, as would occur if each interferometer arm was a side of an equilateral triangle with the other two sides the Sun-spacecraft distances. The curves show the contributions from accelerations owing to solar irradiance variations, solar wind drag, the maximum possible Lorentz force from spacecraft charging, and dust collisions assuming <math alttext="m_{d,\rm max}=10^{-9}" class="ltx_Math" display="inline" id="S3.F3.19.m7.2"><semantics id="S3.F3.19.m7.2b"><mrow id="S3.F3.19.m7.2.3" xref="S3.F3.19.m7.2.3.cmml"><msub id="S3.F3.19.m7.2.3.2" xref="S3.F3.19.m7.2.3.2.cmml"><mi id="S3.F3.19.m7.2.3.2.2" xref="S3.F3.19.m7.2.3.2.2.cmml">m</mi><mrow id="S3.F3.19.m7.2.2.2.4" xref="S3.F3.19.m7.2.2.2.3.cmml"><mi id="S3.F3.19.m7.1.1.1.1" xref="S3.F3.19.m7.1.1.1.1.cmml">d</mi><mo id="S3.F3.19.m7.2.2.2.4.1" xref="S3.F3.19.m7.2.2.2.3.cmml">,</mo><mi id="S3.F3.19.m7.2.2.2.2" xref="S3.F3.19.m7.2.2.2.2.cmml">max</mi></mrow></msub><mo id="S3.F3.19.m7.2.3.1" xref="S3.F3.19.m7.2.3.1.cmml">=</mo><msup id="S3.F3.19.m7.2.3.3" xref="S3.F3.19.m7.2.3.3.cmml"><mn id="S3.F3.19.m7.2.3.3.2" xref="S3.F3.19.m7.2.3.3.2.cmml">10</mn><mrow id="S3.F3.19.m7.2.3.3.3" xref="S3.F3.19.m7.2.3.3.3.cmml"><mo id="S3.F3.19.m7.2.3.3.3b" xref="S3.F3.19.m7.2.3.3.3.cmml">−</mo><mn id="S3.F3.19.m7.2.3.3.3.2" xref="S3.F3.19.m7.2.3.3.3.2.cmml">9</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.F3.19.m7.2c"><apply id="S3.F3.19.m7.2.3.cmml" xref="S3.F3.19.m7.2.3"><eq id="S3.F3.19.m7.2.3.1.cmml" xref="S3.F3.19.m7.2.3.1"></eq><apply id="S3.F3.19.m7.2.3.2.cmml" xref="S3.F3.19.m7.2.3.2"><csymbol cd="ambiguous" id="S3.F3.19.m7.2.3.2.1.cmml" xref="S3.F3.19.m7.2.3.2">subscript</csymbol><ci id="S3.F3.19.m7.2.3.2.2.cmml" xref="S3.F3.19.m7.2.3.2.2">𝑚</ci><list id="S3.F3.19.m7.2.2.2.3.cmml" xref="S3.F3.19.m7.2.2.2.4"><ci id="S3.F3.19.m7.1.1.1.1.cmml" xref="S3.F3.19.m7.1.1.1.1">𝑑</ci><ci id="S3.F3.19.m7.2.2.2.2.cmml" xref="S3.F3.19.m7.2.2.2.2">max</ci></list></apply><apply id="S3.F3.19.m7.2.3.3.cmml" xref="S3.F3.19.m7.2.3.3"><csymbol cd="ambiguous" id="S3.F3.19.m7.2.3.3.1.cmml" xref="S3.F3.19.m7.2.3.3">superscript</csymbol><cn id="S3.F3.19.m7.2.3.3.2.cmml" type="integer" xref="S3.F3.19.m7.2.3.3.2">10</cn><apply id="S3.F3.19.m7.2.3.3.3.cmml" xref="S3.F3.19.m7.2.3.3.3"><minus id="S3.F3.19.m7.2.3.3.3.1.cmml" xref="S3.F3.19.m7.2.3.3.3"></minus><cn id="S3.F3.19.m7.2.3.3.3.2.cmml" type="integer" xref="S3.F3.19.m7.2.3.3.3.2">9</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.19.m7.2d">m_{d,\rm max}=10^{-9}</annotation><annotation encoding="application/x-llamapun" id="S3.F3.19.m7.2e">italic_m start_POSTSUBSCRIPT italic_d , roman_max end_POSTSUBSCRIPT = 10 start_POSTSUPERSCRIPT - 9 end_POSTSUPERSCRIPT</annotation></semantics></math>g and <math alttext="A_{\rm eff}=10\;" class="ltx_Math" display="inline" id="S3.F3.20.m8.1"><semantics id="S3.F3.20.m8.1b"><mrow id="S3.F3.20.m8.1.1" xref="S3.F3.20.m8.1.1.cmml"><msub id="S3.F3.20.m8.1.1.2" xref="S3.F3.20.m8.1.1.2.cmml"><mi id="S3.F3.20.m8.1.1.2.2" xref="S3.F3.20.m8.1.1.2.2.cmml">A</mi><mi id="S3.F3.20.m8.1.1.2.3" xref="S3.F3.20.m8.1.1.2.3.cmml">eff</mi></msub><mo id="S3.F3.20.m8.1.1.1" xref="S3.F3.20.m8.1.1.1.cmml">=</mo><mn id="S3.F3.20.m8.1.1.3" xref="S3.F3.20.m8.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.F3.20.m8.1c"><apply id="S3.F3.20.m8.1.1.cmml" xref="S3.F3.20.m8.1.1"><eq id="S3.F3.20.m8.1.1.1.cmml" xref="S3.F3.20.m8.1.1.1"></eq><apply id="S3.F3.20.m8.1.1.2.cmml" xref="S3.F3.20.m8.1.1.2"><csymbol cd="ambiguous" id="S3.F3.20.m8.1.1.2.1.cmml" xref="S3.F3.20.m8.1.1.2">subscript</csymbol><ci id="S3.F3.20.m8.1.1.2.2.cmml" xref="S3.F3.20.m8.1.1.2.2">𝐴</ci><ci id="S3.F3.20.m8.1.1.2.3.cmml" xref="S3.F3.20.m8.1.1.2.3">eff</ci></apply><cn id="S3.F3.20.m8.1.1.3.cmml" type="integer" xref="S3.F3.20.m8.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.20.m8.1d">A_{\rm eff}=10\;</annotation><annotation encoding="application/x-llamapun" id="S3.F3.20.m8.1e">italic_A start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT = 10</annotation></semantics></math>m<sup class="ltx_sup" id="S3.F3.32.4">2</sup>. Also shown by the thick curves is the total strain error for radio and laser transmissions with different carrier-to-noise ratios (<math alttext="C/N_{0}" class="ltx_Math" display="inline" id="S3.F3.22.m10.1"><semantics id="S3.F3.22.m10.1b"><mrow id="S3.F3.22.m10.1.1" xref="S3.F3.22.m10.1.1.cmml"><mi id="S3.F3.22.m10.1.1.2" xref="S3.F3.22.m10.1.1.2.cmml">C</mi><mo id="S3.F3.22.m10.1.1.1" xref="S3.F3.22.m10.1.1.1.cmml">/</mo><msub id="S3.F3.22.m10.1.1.3" xref="S3.F3.22.m10.1.1.3.cmml"><mi id="S3.F3.22.m10.1.1.3.2" xref="S3.F3.22.m10.1.1.3.2.cmml">N</mi><mn id="S3.F3.22.m10.1.1.3.3" xref="S3.F3.22.m10.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.F3.22.m10.1c"><apply id="S3.F3.22.m10.1.1.cmml" xref="S3.F3.22.m10.1.1"><divide id="S3.F3.22.m10.1.1.1.cmml" xref="S3.F3.22.m10.1.1.1"></divide><ci id="S3.F3.22.m10.1.1.2.cmml" xref="S3.F3.22.m10.1.1.2">𝐶</ci><apply id="S3.F3.22.m10.1.1.3.cmml" xref="S3.F3.22.m10.1.1.3"><csymbol cd="ambiguous" id="S3.F3.22.m10.1.1.3.1.cmml" xref="S3.F3.22.m10.1.1.3">subscript</csymbol><ci id="S3.F3.22.m10.1.1.3.2.cmml" xref="S3.F3.22.m10.1.1.3.2">𝑁</ci><cn id="S3.F3.22.m10.1.1.3.3.cmml" type="integer" xref="S3.F3.22.m10.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.22.m10.1d">C/N_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.F3.22.m10.1e">italic_C / italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>), where we have excluded the smaller contributions to the accelerations from dust collisions and spacecraft charging. A requirement of the LISA mission is to achieve the sensitivity shown by the gray solid curve for <math alttext="f>10^{-4}" class="ltx_Math" display="inline" id="S3.F3.23.m11.1"><semantics id="S3.F3.23.m11.1b"><mrow id="S3.F3.23.m11.1.1" xref="S3.F3.23.m11.1.1.cmml"><mi id="S3.F3.23.m11.1.1.2" xref="S3.F3.23.m11.1.1.2.cmml">f</mi><mo id="S3.F3.23.m11.1.1.1" xref="S3.F3.23.m11.1.1.1.cmml">></mo><msup id="S3.F3.23.m11.1.1.3" xref="S3.F3.23.m11.1.1.3.cmml"><mn id="S3.F3.23.m11.1.1.3.2" xref="S3.F3.23.m11.1.1.3.2.cmml">10</mn><mrow id="S3.F3.23.m11.1.1.3.3" xref="S3.F3.23.m11.1.1.3.3.cmml"><mo id="S3.F3.23.m11.1.1.3.3b" xref="S3.F3.23.m11.1.1.3.3.cmml">−</mo><mn id="S3.F3.23.m11.1.1.3.3.2" xref="S3.F3.23.m11.1.1.3.3.2.cmml">4</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.F3.23.m11.1c"><apply id="S3.F3.23.m11.1.1.cmml" xref="S3.F3.23.m11.1.1"><gt id="S3.F3.23.m11.1.1.1.cmml" xref="S3.F3.23.m11.1.1.1"></gt><ci id="S3.F3.23.m11.1.1.2.cmml" xref="S3.F3.23.m11.1.1.2">𝑓</ci><apply id="S3.F3.23.m11.1.1.3.cmml" xref="S3.F3.23.m11.1.1.3"><csymbol cd="ambiguous" id="S3.F3.23.m11.1.1.3.1.cmml" xref="S3.F3.23.m11.1.1.3">superscript</csymbol><cn id="S3.F3.23.m11.1.1.3.2.cmml" type="integer" xref="S3.F3.23.m11.1.1.3.2">10</cn><apply id="S3.F3.23.m11.1.1.3.3.cmml" xref="S3.F3.23.m11.1.1.3.3"><minus id="S3.F3.23.m11.1.1.3.3.1.cmml" xref="S3.F3.23.m11.1.1.3.3"></minus><cn id="S3.F3.23.m11.1.1.3.3.2.cmml" type="integer" xref="S3.F3.23.m11.1.1.3.3.2">4</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.23.m11.1d">f>10^{-4}</annotation><annotation encoding="application/x-llamapun" id="S3.F3.23.m11.1e">italic_f > 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT</annotation></semantics></math>Hz, with the goal to be sensitive to <math alttext="f>2\times 10^{-5}" class="ltx_Math" display="inline" id="S3.F3.24.m12.1"><semantics id="S3.F3.24.m12.1b"><mrow id="S3.F3.24.m12.1.1" xref="S3.F3.24.m12.1.1.cmml"><mi id="S3.F3.24.m12.1.1.2" xref="S3.F3.24.m12.1.1.2.cmml">f</mi><mo id="S3.F3.24.m12.1.1.1" xref="S3.F3.24.m12.1.1.1.cmml">></mo><mrow id="S3.F3.24.m12.1.1.3" xref="S3.F3.24.m12.1.1.3.cmml"><mn id="S3.F3.24.m12.1.1.3.2" xref="S3.F3.24.m12.1.1.3.2.cmml">2</mn><mo id="S3.F3.24.m12.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S3.F3.24.m12.1.1.3.1.cmml">×</mo><msup id="S3.F3.24.m12.1.1.3.3" xref="S3.F3.24.m12.1.1.3.3.cmml"><mn id="S3.F3.24.m12.1.1.3.3.2" xref="S3.F3.24.m12.1.1.3.3.2.cmml">10</mn><mrow id="S3.F3.24.m12.1.1.3.3.3" xref="S3.F3.24.m12.1.1.3.3.3.cmml"><mo id="S3.F3.24.m12.1.1.3.3.3b" xref="S3.F3.24.m12.1.1.3.3.3.cmml">−</mo><mn id="S3.F3.24.m12.1.1.3.3.3.2" xref="S3.F3.24.m12.1.1.3.3.3.2.cmml">5</mn></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.F3.24.m12.1c"><apply id="S3.F3.24.m12.1.1.cmml" xref="S3.F3.24.m12.1.1"><gt id="S3.F3.24.m12.1.1.1.cmml" xref="S3.F3.24.m12.1.1.1"></gt><ci id="S3.F3.24.m12.1.1.2.cmml" xref="S3.F3.24.m12.1.1.2">𝑓</ci><apply id="S3.F3.24.m12.1.1.3.cmml" xref="S3.F3.24.m12.1.1.3"><times id="S3.F3.24.m12.1.1.3.1.cmml" xref="S3.F3.24.m12.1.1.3.1"></times><cn id="S3.F3.24.m12.1.1.3.2.cmml" type="integer" xref="S3.F3.24.m12.1.1.3.2">2</cn><apply id="S3.F3.24.m12.1.1.3.3.cmml" xref="S3.F3.24.m12.1.1.3.3"><csymbol cd="ambiguous" id="S3.F3.24.m12.1.1.3.3.1.cmml" xref="S3.F3.24.m12.1.1.3.3">superscript</csymbol><cn id="S3.F3.24.m12.1.1.3.3.2.cmml" type="integer" xref="S3.F3.24.m12.1.1.3.3.2">10</cn><apply id="S3.F3.24.m12.1.1.3.3.3.cmml" xref="S3.F3.24.m12.1.1.3.3.3"><minus id="S3.F3.24.m12.1.1.3.3.3.1.cmml" xref="S3.F3.24.m12.1.1.3.3.3"></minus><cn id="S3.F3.24.m12.1.1.3.3.3.2.cmml" type="integer" xref="S3.F3.24.m12.1.1.3.3.3.2">5</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.24.m12.1d">f>2\times 10^{-5}</annotation><annotation encoding="application/x-llamapun" id="S3.F3.24.m12.1e">italic_f > 2 × 10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT</annotation></semantics></math>Hz as illustrated by the dashed extension. The total strain sensitivity lines are repeated in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S3.F4" title="Figure 4 ‣ 3.1 Two-arm time-delay interferometry forecasts ‣ 3 Strain power sensitivity ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">4</span></a>.</figcaption> </figure> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.2">The carrier-to-noise ratio discussed in § <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.SS2" title="2.2 Radiometer noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">2.2</span></a> and the acceleration noise in § <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.SS3" title="2.3 Acceleration noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">2.3</span></a> allow us to estimate the gravitational wave strain sensitivity. Namely, the radiometer noise power, <math alttext="S_{n}" class="ltx_Math" display="inline" id="S3.p1.1.m1.1"><semantics id="S3.p1.1.m1.1a"><msub id="S3.p1.1.m1.1.1" xref="S3.p1.1.m1.1.1.cmml"><mi id="S3.p1.1.m1.1.1.2" xref="S3.p1.1.m1.1.1.2.cmml">S</mi><mi id="S3.p1.1.m1.1.1.3" xref="S3.p1.1.m1.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p1.1.m1.1b"><apply id="S3.p1.1.m1.1.1.cmml" xref="S3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.p1.1.m1.1.1.1.cmml" xref="S3.p1.1.m1.1.1">subscript</csymbol><ci id="S3.p1.1.m1.1.1.2.cmml" xref="S3.p1.1.m1.1.1.2">𝑆</ci><ci id="S3.p1.1.m1.1.1.3.cmml" xref="S3.p1.1.m1.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.1.m1.1c">S_{n}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.1.m1.1d">italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>, and acceleration power, <math alttext="S_{a}" class="ltx_Math" display="inline" id="S3.p1.2.m2.1"><semantics id="S3.p1.2.m2.1a"><msub id="S3.p1.2.m2.1.1" xref="S3.p1.2.m2.1.1.cmml"><mi id="S3.p1.2.m2.1.1.2" xref="S3.p1.2.m2.1.1.2.cmml">S</mi><mi id="S3.p1.2.m2.1.1.3" xref="S3.p1.2.m2.1.1.3.cmml">a</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p1.2.m2.1b"><apply id="S3.p1.2.m2.1.1.cmml" xref="S3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.p1.2.m2.1.1.1.cmml" xref="S3.p1.2.m2.1.1">subscript</csymbol><ci id="S3.p1.2.m2.1.1.2.cmml" xref="S3.p1.2.m2.1.1.2">𝑆</ci><ci id="S3.p1.2.m2.1.1.3.cmml" xref="S3.p1.2.m2.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.2.m2.1c">S_{a}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.2.m2.1d">italic_S start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math>, can be related to the interferometer’s sensitivity to the polarization- and sky-averaged gravitational wave strain power at a specific frequency (e.g. <cite class="ltx_cite ltx_citemacro_citet">Larson et al. <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib43" title="">2000</a>; Robson et al. <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib60" title="">2019</a></cite>). The gravitational wave instrument’s noise when performing the average that generates this strain power is</p> <table class="ltx_equation ltx_eqn_table" id="S3.E18"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="S_{h}(f)=\frac{2}{L^{2}R(f)}\left(\frac{2{\cal A}^{2}~{}S_{n}}{(2\pi f)^{2}}+% \frac{4\big{[}1+\cos^{2}(2\pi fL)\big{]}S_{a}}{(2\pi f)^{4}}\right)," class="ltx_Math" display="block" id="S3.E18.m1.6"><semantics id="S3.E18.m1.6a"><mrow id="S3.E18.m1.6.6.1" xref="S3.E18.m1.6.6.1.1.cmml"><mrow id="S3.E18.m1.6.6.1.1" xref="S3.E18.m1.6.6.1.1.cmml"><mrow id="S3.E18.m1.6.6.1.1.3" xref="S3.E18.m1.6.6.1.1.3.cmml"><msub id="S3.E18.m1.6.6.1.1.3.2" xref="S3.E18.m1.6.6.1.1.3.2.cmml"><mi id="S3.E18.m1.6.6.1.1.3.2.2" xref="S3.E18.m1.6.6.1.1.3.2.2.cmml">S</mi><mi id="S3.E18.m1.6.6.1.1.3.2.3" xref="S3.E18.m1.6.6.1.1.3.2.3.cmml">h</mi></msub><mo 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id="S3.E18.m1.4.4.2.3.cmml" type="integer" xref="S3.E18.m1.4.4.2.3">4</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E18.m1.6c">S_{h}(f)=\frac{2}{L^{2}R(f)}\left(\frac{2{\cal A}^{2}~{}S_{n}}{(2\pi f)^{2}}+% \frac{4\big{[}1+\cos^{2}(2\pi fL)\big{]}S_{a}}{(2\pi f)^{4}}\right),</annotation><annotation encoding="application/x-llamapun" id="S3.E18.m1.6d">italic_S start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( italic_f ) = divide start_ARG 2 end_ARG start_ARG italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_R ( italic_f ) end_ARG ( divide start_ARG 2 caligraphic_A start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG start_ARG ( 2 italic_π italic_f ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG + divide start_ARG 4 [ 1 + roman_cos start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 2 italic_π italic_f italic_L ) ] italic_S start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT end_ARG start_ARG ( 2 italic_π italic_f ) start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(18)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p1.9">where the factor of <math alttext="2{\cal A}^{2}" class="ltx_Math" display="inline" id="S3.p1.3.m1.1"><semantics id="S3.p1.3.m1.1a"><mrow id="S3.p1.3.m1.1.1" xref="S3.p1.3.m1.1.1.cmml"><mn id="S3.p1.3.m1.1.1.2" xref="S3.p1.3.m1.1.1.2.cmml">2</mn><mo id="S3.p1.3.m1.1.1.1" xref="S3.p1.3.m1.1.1.1.cmml">⁢</mo><msup id="S3.p1.3.m1.1.1.3" xref="S3.p1.3.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p1.3.m1.1.1.3.2" xref="S3.p1.3.m1.1.1.3.2.cmml">𝒜</mi><mn id="S3.p1.3.m1.1.1.3.3" xref="S3.p1.3.m1.1.1.3.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.3.m1.1b"><apply id="S3.p1.3.m1.1.1.cmml" xref="S3.p1.3.m1.1.1"><times id="S3.p1.3.m1.1.1.1.cmml" xref="S3.p1.3.m1.1.1.1"></times><cn id="S3.p1.3.m1.1.1.2.cmml" type="integer" xref="S3.p1.3.m1.1.1.2">2</cn><apply id="S3.p1.3.m1.1.1.3.cmml" xref="S3.p1.3.m1.1.1.3"><csymbol cd="ambiguous" id="S3.p1.3.m1.1.1.3.1.cmml" xref="S3.p1.3.m1.1.1.3">superscript</csymbol><ci id="S3.p1.3.m1.1.1.3.2.cmml" xref="S3.p1.3.m1.1.1.3.2">𝒜</ci><cn id="S3.p1.3.m1.1.1.3.3.cmml" type="integer" xref="S3.p1.3.m1.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.3.m1.1c">2{\cal A}^{2}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.3.m1.1d">2 caligraphic_A start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> is because there are two spacecraft in an arm with independent radiometer noise and <math alttext="{\cal A}" class="ltx_Math" display="inline" id="S3.p1.4.m2.1"><semantics id="S3.p1.4.m2.1a"><mi 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The factor <math alttext="4\big{[}1+\cos^{2}(2\pi fL)\big{]}" class="ltx_Math" display="inline" id="S3.p1.5.m3.1"><semantics id="S3.p1.5.m3.1a"><mrow id="S3.p1.5.m3.1.1" xref="S3.p1.5.m3.1.1.cmml"><mn id="S3.p1.5.m3.1.1.3" xref="S3.p1.5.m3.1.1.3.cmml">4</mn><mo id="S3.p1.5.m3.1.1.2" xref="S3.p1.5.m3.1.1.2.cmml">⁢</mo><mrow id="S3.p1.5.m3.1.1.1.1" xref="S3.p1.5.m3.1.1.1.2.cmml"><mo id="S3.p1.5.m3.1.1.1.1.2" maxsize="120%" minsize="120%" xref="S3.p1.5.m3.1.1.1.2.1.cmml">[</mo><mrow id="S3.p1.5.m3.1.1.1.1.1" xref="S3.p1.5.m3.1.1.1.1.1.cmml"><mn id="S3.p1.5.m3.1.1.1.1.1.4" xref="S3.p1.5.m3.1.1.1.1.1.4.cmml">1</mn><mo id="S3.p1.5.m3.1.1.1.1.1.3" xref="S3.p1.5.m3.1.1.1.1.1.3.cmml">+</mo><mrow id="S3.p1.5.m3.1.1.1.1.1.2.2" xref="S3.p1.5.m3.1.1.1.1.1.2.3.cmml"><msup id="S3.p1.5.m3.1.1.1.1.1.1.1.1" xref="S3.p1.5.m3.1.1.1.1.1.1.1.1.cmml"><mi id="S3.p1.5.m3.1.1.1.1.1.1.1.1.2" xref="S3.p1.5.m3.1.1.1.1.1.1.1.1.2.cmml">cos</mi><mn id="S3.p1.5.m3.1.1.1.1.1.1.1.1.3" xref="S3.p1.5.m3.1.1.1.1.1.1.1.1.3.cmml">2</mn></msup><mo id="S3.p1.5.m3.1.1.1.1.1.2.2a" xref="S3.p1.5.m3.1.1.1.1.1.2.3.cmml">⁡</mo><mrow id="S3.p1.5.m3.1.1.1.1.1.2.2.2" xref="S3.p1.5.m3.1.1.1.1.1.2.3.cmml"><mo id="S3.p1.5.m3.1.1.1.1.1.2.2.2.2" stretchy="false" xref="S3.p1.5.m3.1.1.1.1.1.2.3.cmml">(</mo><mrow id="S3.p1.5.m3.1.1.1.1.1.2.2.2.1" xref="S3.p1.5.m3.1.1.1.1.1.2.2.2.1.cmml"><mn id="S3.p1.5.m3.1.1.1.1.1.2.2.2.1.2" xref="S3.p1.5.m3.1.1.1.1.1.2.2.2.1.2.cmml">2</mn><mo id="S3.p1.5.m3.1.1.1.1.1.2.2.2.1.1" xref="S3.p1.5.m3.1.1.1.1.1.2.2.2.1.1.cmml">⁢</mo><mi id="S3.p1.5.m3.1.1.1.1.1.2.2.2.1.3" xref="S3.p1.5.m3.1.1.1.1.1.2.2.2.1.3.cmml">π</mi><mo id="S3.p1.5.m3.1.1.1.1.1.2.2.2.1.1a" xref="S3.p1.5.m3.1.1.1.1.1.2.2.2.1.1.cmml">⁢</mo><mi id="S3.p1.5.m3.1.1.1.1.1.2.2.2.1.4" xref="S3.p1.5.m3.1.1.1.1.1.2.2.2.1.4.cmml">f</mi><mo id="S3.p1.5.m3.1.1.1.1.1.2.2.2.1.1b" xref="S3.p1.5.m3.1.1.1.1.1.2.2.2.1.1.cmml">⁢</mo><mi id="S3.p1.5.m3.1.1.1.1.1.2.2.2.1.5" xref="S3.p1.5.m3.1.1.1.1.1.2.2.2.1.5.cmml">L</mi></mrow><mo id="S3.p1.5.m3.1.1.1.1.1.2.2.2.3" stretchy="false" xref="S3.p1.5.m3.1.1.1.1.1.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="S3.p1.5.m3.1.1.1.1.3" maxsize="120%" minsize="120%" xref="S3.p1.5.m3.1.1.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.5.m3.1b"><apply id="S3.p1.5.m3.1.1.cmml" xref="S3.p1.5.m3.1.1"><times id="S3.p1.5.m3.1.1.2.cmml" xref="S3.p1.5.m3.1.1.2"></times><cn id="S3.p1.5.m3.1.1.3.cmml" type="integer" xref="S3.p1.5.m3.1.1.3">4</cn><apply id="S3.p1.5.m3.1.1.1.2.cmml" xref="S3.p1.5.m3.1.1.1.1"><csymbol cd="latexml" id="S3.p1.5.m3.1.1.1.2.1.cmml" xref="S3.p1.5.m3.1.1.1.1.2">delimited-[]</csymbol><apply id="S3.p1.5.m3.1.1.1.1.1.cmml" xref="S3.p1.5.m3.1.1.1.1.1"><plus id="S3.p1.5.m3.1.1.1.1.1.3.cmml" xref="S3.p1.5.m3.1.1.1.1.1.3"></plus><cn id="S3.p1.5.m3.1.1.1.1.1.4.cmml" type="integer" xref="S3.p1.5.m3.1.1.1.1.1.4">1</cn><apply id="S3.p1.5.m3.1.1.1.1.1.2.3.cmml" xref="S3.p1.5.m3.1.1.1.1.1.2.2"><apply id="S3.p1.5.m3.1.1.1.1.1.1.1.1.cmml" xref="S3.p1.5.m3.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.p1.5.m3.1.1.1.1.1.1.1.1.1.cmml" xref="S3.p1.5.m3.1.1.1.1.1.1.1.1">superscript</csymbol><cos id="S3.p1.5.m3.1.1.1.1.1.1.1.1.2.cmml" xref="S3.p1.5.m3.1.1.1.1.1.1.1.1.2"></cos><cn id="S3.p1.5.m3.1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.p1.5.m3.1.1.1.1.1.1.1.1.3">2</cn></apply><apply id="S3.p1.5.m3.1.1.1.1.1.2.2.2.1.cmml" xref="S3.p1.5.m3.1.1.1.1.1.2.2.2.1"><times id="S3.p1.5.m3.1.1.1.1.1.2.2.2.1.1.cmml" xref="S3.p1.5.m3.1.1.1.1.1.2.2.2.1.1"></times><cn id="S3.p1.5.m3.1.1.1.1.1.2.2.2.1.2.cmml" type="integer" xref="S3.p1.5.m3.1.1.1.1.1.2.2.2.1.2">2</cn><ci id="S3.p1.5.m3.1.1.1.1.1.2.2.2.1.3.cmml" xref="S3.p1.5.m3.1.1.1.1.1.2.2.2.1.3">𝜋</ci><ci id="S3.p1.5.m3.1.1.1.1.1.2.2.2.1.4.cmml" xref="S3.p1.5.m3.1.1.1.1.1.2.2.2.1.4">𝑓</ci><ci id="S3.p1.5.m3.1.1.1.1.1.2.2.2.1.5.cmml" xref="S3.p1.5.m3.1.1.1.1.1.2.2.2.1.5">𝐿</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.5.m3.1c">4\big{[}1+\cos^{2}(2\pi fL)\big{]}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.5.m3.1d">4 [ 1 + roman_cos start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 2 italic_π italic_f italic_L ) ]</annotation></semantics></math> is because the acceleration of a spacecraft contributes to a displacement that coherently impacts the phase both in the incoming and outgoing directions. The function <math alttext="R(f)" class="ltx_Math" display="inline" id="S3.p1.6.m4.1"><semantics id="S3.p1.6.m4.1a"><mrow id="S3.p1.6.m4.1.2" xref="S3.p1.6.m4.1.2.cmml"><mi id="S3.p1.6.m4.1.2.2" xref="S3.p1.6.m4.1.2.2.cmml">R</mi><mo id="S3.p1.6.m4.1.2.1" xref="S3.p1.6.m4.1.2.1.cmml">⁢</mo><mrow id="S3.p1.6.m4.1.2.3.2" xref="S3.p1.6.m4.1.2.cmml"><mo id="S3.p1.6.m4.1.2.3.2.1" stretchy="false" xref="S3.p1.6.m4.1.2.cmml">(</mo><mi id="S3.p1.6.m4.1.1" xref="S3.p1.6.m4.1.1.cmml">f</mi><mo id="S3.p1.6.m4.1.2.3.2.2" stretchy="false" xref="S3.p1.6.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.6.m4.1b"><apply id="S3.p1.6.m4.1.2.cmml" xref="S3.p1.6.m4.1.2"><times id="S3.p1.6.m4.1.2.1.cmml" xref="S3.p1.6.m4.1.2.1"></times><ci id="S3.p1.6.m4.1.2.2.cmml" xref="S3.p1.6.m4.1.2.2">𝑅</ci><ci id="S3.p1.6.m4.1.1.cmml" xref="S3.p1.6.m4.1.1">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.6.m4.1c">R(f)</annotation><annotation encoding="application/x-llamapun" id="S3.p1.6.m4.1d">italic_R ( italic_f )</annotation></semantics></math> is a transfer function that transforms the single-arm noise in the long-wavelength limit (and in which the gravitational wave is propagating orthogonally) to the noise power seen when averaging the instrument’s strain response over all angles. We will first consider the instrument to be a two-arm (time-delay) interferometer and later consider one-arm configurations. The overall factor of two out front of equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S3.E18" title="In 3 Strain power sensitivity ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">18</span></a>) follows the conventions for time-delay interferometry to account for two independent arms, although we will define <math alttext="R(f)" class="ltx_Math" display="inline" id="S3.p1.7.m5.1"><semantics id="S3.p1.7.m5.1a"><mrow id="S3.p1.7.m5.1.2" xref="S3.p1.7.m5.1.2.cmml"><mi id="S3.p1.7.m5.1.2.2" xref="S3.p1.7.m5.1.2.2.cmml">R</mi><mo id="S3.p1.7.m5.1.2.1" xref="S3.p1.7.m5.1.2.1.cmml">⁢</mo><mrow id="S3.p1.7.m5.1.2.3.2" xref="S3.p1.7.m5.1.2.cmml"><mo id="S3.p1.7.m5.1.2.3.2.1" stretchy="false" xref="S3.p1.7.m5.1.2.cmml">(</mo><mi id="S3.p1.7.m5.1.1" xref="S3.p1.7.m5.1.1.cmml">f</mi><mo id="S3.p1.7.m5.1.2.3.2.2" stretchy="false" xref="S3.p1.7.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.7.m5.1b"><apply id="S3.p1.7.m5.1.2.cmml" xref="S3.p1.7.m5.1.2"><times id="S3.p1.7.m5.1.2.1.cmml" xref="S3.p1.7.m5.1.2.1"></times><ci id="S3.p1.7.m5.1.2.2.cmml" xref="S3.p1.7.m5.1.2.2">𝑅</ci><ci id="S3.p1.7.m5.1.1.cmml" xref="S3.p1.7.m5.1.1">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.7.m5.1c">R(f)</annotation><annotation encoding="application/x-llamapun" id="S3.p1.7.m5.1d">italic_R ( italic_f )</annotation></semantics></math> for one-arm configurations to cancel this factor of two so that this equation still applies. We use the analytic form for <math alttext="R(f)" class="ltx_Math" display="inline" id="S3.p1.8.m6.1"><semantics id="S3.p1.8.m6.1a"><mrow id="S3.p1.8.m6.1.2" xref="S3.p1.8.m6.1.2.cmml"><mi id="S3.p1.8.m6.1.2.2" xref="S3.p1.8.m6.1.2.2.cmml">R</mi><mo id="S3.p1.8.m6.1.2.1" xref="S3.p1.8.m6.1.2.1.cmml">⁢</mo><mrow id="S3.p1.8.m6.1.2.3.2" xref="S3.p1.8.m6.1.2.cmml"><mo id="S3.p1.8.m6.1.2.3.2.1" stretchy="false" xref="S3.p1.8.m6.1.2.cmml">(</mo><mi id="S3.p1.8.m6.1.1" xref="S3.p1.8.m6.1.1.cmml">f</mi><mo id="S3.p1.8.m6.1.2.3.2.2" stretchy="false" xref="S3.p1.8.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.8.m6.1b"><apply id="S3.p1.8.m6.1.2.cmml" xref="S3.p1.8.m6.1.2"><times id="S3.p1.8.m6.1.2.1.cmml" xref="S3.p1.8.m6.1.2.1"></times><ci id="S3.p1.8.m6.1.2.2.cmml" xref="S3.p1.8.m6.1.2.2">𝑅</ci><ci id="S3.p1.8.m6.1.1.cmml" xref="S3.p1.8.m6.1.1">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.8.m6.1c">R(f)</annotation><annotation encoding="application/x-llamapun" id="S3.p1.8.m6.1d">italic_R ( italic_f )</annotation></semantics></math> calculated in <cite class="ltx_cite ltx_citemacro_citet">Larson et al. (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib43" title="">2000</a>)</cite>, and for our time-delay interferometer configurations assume an angle between the arms of <math alttext="90^{\circ}" class="ltx_Math" display="inline" id="S3.p1.9.m7.1"><semantics id="S3.p1.9.m7.1a"><msup id="S3.p1.9.m7.1.1" xref="S3.p1.9.m7.1.1.cmml"><mn id="S3.p1.9.m7.1.1.2" xref="S3.p1.9.m7.1.1.2.cmml">90</mn><mo id="S3.p1.9.m7.1.1.3" xref="S3.p1.9.m7.1.1.3.cmml">∘</mo></msup><annotation-xml encoding="MathML-Content" id="S3.p1.9.m7.1b"><apply id="S3.p1.9.m7.1.1.cmml" xref="S3.p1.9.m7.1.1"><csymbol cd="ambiguous" id="S3.p1.9.m7.1.1.1.cmml" xref="S3.p1.9.m7.1.1">superscript</csymbol><cn id="S3.p1.9.m7.1.1.2.cmml" type="integer" xref="S3.p1.9.m7.1.1.2">90</cn><compose id="S3.p1.9.m7.1.1.3.cmml" xref="S3.p1.9.m7.1.1.3"></compose></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.9.m7.1c">90^{\circ}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.9.m7.1d">90 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT</annotation></semantics></math>.<span class="ltx_note ltx_role_footnote" id="footnote10"><sup class="ltx_note_mark">10</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">10</sup><span class="ltx_tag ltx_tag_note">10</span><math alttext="R(f)\rightarrow 4/5\sin^{2}(\gamma)" class="ltx_Math" display="inline" id="footnote10.m1.3"><semantics id="footnote10.m1.3b"><mrow id="footnote10.m1.3.3" xref="footnote10.m1.3.3.cmml"><mrow id="footnote10.m1.3.3.3" xref="footnote10.m1.3.3.3.cmml"><mi id="footnote10.m1.3.3.3.2" xref="footnote10.m1.3.3.3.2.cmml">R</mi><mo id="footnote10.m1.3.3.3.1" xref="footnote10.m1.3.3.3.1.cmml">⁢</mo><mrow id="footnote10.m1.3.3.3.3.2" xref="footnote10.m1.3.3.3.cmml"><mo id="footnote10.m1.3.3.3.3.2.1" stretchy="false" xref="footnote10.m1.3.3.3.cmml">(</mo><mi id="footnote10.m1.1.1" xref="footnote10.m1.1.1.cmml">f</mi><mo id="footnote10.m1.3.3.3.3.2.2" stretchy="false" xref="footnote10.m1.3.3.3.cmml">)</mo></mrow></mrow><mo id="footnote10.m1.3.3.2" stretchy="false" xref="footnote10.m1.3.3.2.cmml">→</mo><mrow id="footnote10.m1.3.3.1" xref="footnote10.m1.3.3.1.cmml"><mrow id="footnote10.m1.3.3.1.3" xref="footnote10.m1.3.3.1.3.cmml"><mn id="footnote10.m1.3.3.1.3.2" xref="footnote10.m1.3.3.1.3.2.cmml">4</mn><mo id="footnote10.m1.3.3.1.3.1" xref="footnote10.m1.3.3.1.3.1.cmml">/</mo><mn id="footnote10.m1.3.3.1.3.3" xref="footnote10.m1.3.3.1.3.3.cmml">5</mn></mrow><mo id="footnote10.m1.3.3.1.2" lspace="0.167em" xref="footnote10.m1.3.3.1.2.cmml">⁢</mo><mrow id="footnote10.m1.3.3.1.1.1" xref="footnote10.m1.3.3.1.1.2.cmml"><msup id="footnote10.m1.3.3.1.1.1.1" xref="footnote10.m1.3.3.1.1.1.1.cmml"><mi id="footnote10.m1.3.3.1.1.1.1.2" xref="footnote10.m1.3.3.1.1.1.1.2.cmml">sin</mi><mn id="footnote10.m1.3.3.1.1.1.1.3" xref="footnote10.m1.3.3.1.1.1.1.3.cmml">2</mn></msup><mo id="footnote10.m1.3.3.1.1.1b" xref="footnote10.m1.3.3.1.1.2.cmml">⁡</mo><mrow id="footnote10.m1.3.3.1.1.1.2" xref="footnote10.m1.3.3.1.1.2.cmml"><mo id="footnote10.m1.3.3.1.1.1.2.1" stretchy="false" xref="footnote10.m1.3.3.1.1.2.cmml">(</mo><mi id="footnote10.m1.2.2" xref="footnote10.m1.2.2.cmml">γ</mi><mo id="footnote10.m1.3.3.1.1.1.2.2" stretchy="false" xref="footnote10.m1.3.3.1.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote10.m1.3c"><apply id="footnote10.m1.3.3.cmml" xref="footnote10.m1.3.3"><ci id="footnote10.m1.3.3.2.cmml" xref="footnote10.m1.3.3.2">→</ci><apply id="footnote10.m1.3.3.3.cmml" xref="footnote10.m1.3.3.3"><times id="footnote10.m1.3.3.3.1.cmml" xref="footnote10.m1.3.3.3.1"></times><ci id="footnote10.m1.3.3.3.2.cmml" xref="footnote10.m1.3.3.3.2">𝑅</ci><ci id="footnote10.m1.1.1.cmml" xref="footnote10.m1.1.1">𝑓</ci></apply><apply id="footnote10.m1.3.3.1.cmml" xref="footnote10.m1.3.3.1"><times id="footnote10.m1.3.3.1.2.cmml" xref="footnote10.m1.3.3.1.2"></times><apply id="footnote10.m1.3.3.1.3.cmml" xref="footnote10.m1.3.3.1.3"><divide id="footnote10.m1.3.3.1.3.1.cmml" xref="footnote10.m1.3.3.1.3.1"></divide><cn id="footnote10.m1.3.3.1.3.2.cmml" type="integer" xref="footnote10.m1.3.3.1.3.2">4</cn><cn id="footnote10.m1.3.3.1.3.3.cmml" type="integer" xref="footnote10.m1.3.3.1.3.3">5</cn></apply><apply id="footnote10.m1.3.3.1.1.2.cmml" xref="footnote10.m1.3.3.1.1.1"><apply id="footnote10.m1.3.3.1.1.1.1.cmml" xref="footnote10.m1.3.3.1.1.1.1"><csymbol cd="ambiguous" id="footnote10.m1.3.3.1.1.1.1.1.cmml" xref="footnote10.m1.3.3.1.1.1.1">superscript</csymbol><sin id="footnote10.m1.3.3.1.1.1.1.2.cmml" xref="footnote10.m1.3.3.1.1.1.1.2"></sin><cn id="footnote10.m1.3.3.1.1.1.1.3.cmml" type="integer" xref="footnote10.m1.3.3.1.1.1.1.3">2</cn></apply><ci id="footnote10.m1.2.2.cmml" xref="footnote10.m1.2.2">𝛾</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote10.m1.3d">R(f)\rightarrow 4/5\sin^{2}(\gamma)</annotation><annotation encoding="application/x-llamapun" id="footnote10.m1.3e">italic_R ( italic_f ) → 4 / 5 roman_sin start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_γ )</annotation></semantics></math> at low frequencies, where <math alttext="\gamma" class="ltx_Math" display="inline" id="footnote10.m2.1"><semantics id="footnote10.m2.1b"><mi id="footnote10.m2.1.1" xref="footnote10.m2.1.1.cmml">γ</mi><annotation-xml encoding="MathML-Content" id="footnote10.m2.1c"><ci id="footnote10.m2.1.1.cmml" xref="footnote10.m2.1.1">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote10.m2.1d">\gamma</annotation><annotation encoding="application/x-llamapun" id="footnote10.m2.1e">italic_γ</annotation></semantics></math> is the angle between arms that we will take to be <math alttext="90^{\circ}" class="ltx_Math" display="inline" id="footnote10.m3.1"><semantics id="footnote10.m3.1b"><msup id="footnote10.m3.1.1" xref="footnote10.m3.1.1.cmml"><mn id="footnote10.m3.1.1.2" xref="footnote10.m3.1.1.2.cmml">90</mn><mo id="footnote10.m3.1.1.3" xref="footnote10.m3.1.1.3.cmml">∘</mo></msup><annotation-xml encoding="MathML-Content" id="footnote10.m3.1c"><apply id="footnote10.m3.1.1.cmml" xref="footnote10.m3.1.1"><csymbol cd="ambiguous" id="footnote10.m3.1.1.1.cmml" xref="footnote10.m3.1.1">superscript</csymbol><cn id="footnote10.m3.1.1.2.cmml" type="integer" xref="footnote10.m3.1.1.2">90</cn><compose id="footnote10.m3.1.1.3.cmml" xref="footnote10.m3.1.1.3"></compose></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote10.m3.1d">90^{\circ}</annotation><annotation encoding="application/x-llamapun" id="footnote10.m3.1e">90 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT</annotation></semantics></math>. This differs from the limit <math alttext="R(f)\rightarrow 6/5" class="ltx_Math" display="inline" id="footnote10.m4.1"><semantics id="footnote10.m4.1b"><mrow id="footnote10.m4.1.2" xref="footnote10.m4.1.2.cmml"><mrow id="footnote10.m4.1.2.2" xref="footnote10.m4.1.2.2.cmml"><mi id="footnote10.m4.1.2.2.2" xref="footnote10.m4.1.2.2.2.cmml">R</mi><mo id="footnote10.m4.1.2.2.1" xref="footnote10.m4.1.2.2.1.cmml">⁢</mo><mrow id="footnote10.m4.1.2.2.3.2" xref="footnote10.m4.1.2.2.cmml"><mo id="footnote10.m4.1.2.2.3.2.1" stretchy="false" xref="footnote10.m4.1.2.2.cmml">(</mo><mi id="footnote10.m4.1.1" xref="footnote10.m4.1.1.cmml">f</mi><mo id="footnote10.m4.1.2.2.3.2.2" stretchy="false" xref="footnote10.m4.1.2.2.cmml">)</mo></mrow></mrow><mo id="footnote10.m4.1.2.1" stretchy="false" xref="footnote10.m4.1.2.1.cmml">→</mo><mrow id="footnote10.m4.1.2.3" xref="footnote10.m4.1.2.3.cmml"><mn id="footnote10.m4.1.2.3.2" xref="footnote10.m4.1.2.3.2.cmml">6</mn><mo id="footnote10.m4.1.2.3.1" xref="footnote10.m4.1.2.3.1.cmml">/</mo><mn id="footnote10.m4.1.2.3.3" xref="footnote10.m4.1.2.3.3.cmml">5</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote10.m4.1c"><apply id="footnote10.m4.1.2.cmml" xref="footnote10.m4.1.2"><ci id="footnote10.m4.1.2.1.cmml" xref="footnote10.m4.1.2.1">→</ci><apply id="footnote10.m4.1.2.2.cmml" xref="footnote10.m4.1.2.2"><times id="footnote10.m4.1.2.2.1.cmml" xref="footnote10.m4.1.2.2.1"></times><ci id="footnote10.m4.1.2.2.2.cmml" xref="footnote10.m4.1.2.2.2">𝑅</ci><ci id="footnote10.m4.1.1.cmml" xref="footnote10.m4.1.1">𝑓</ci></apply><apply id="footnote10.m4.1.2.3.cmml" xref="footnote10.m4.1.2.3"><divide id="footnote10.m4.1.2.3.1.cmml" xref="footnote10.m4.1.2.3.1"></divide><cn id="footnote10.m4.1.2.3.2.cmml" type="integer" xref="footnote10.m4.1.2.3.2">6</cn><cn id="footnote10.m4.1.2.3.3.cmml" type="integer" xref="footnote10.m4.1.2.3.3">5</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote10.m4.1d">R(f)\rightarrow 6/5</annotation><annotation encoding="application/x-llamapun" id="footnote10.m4.1e">italic_R ( italic_f ) → 6 / 5</annotation></semantics></math> that applies for LISA by the factor of <math alttext="2\sin^{2}(60^{\circ})/\sin^{2}(\gamma)" class="ltx_Math" display="inline" id="footnote10.m5.4"><semantics id="footnote10.m5.4b"><mrow id="footnote10.m5.4.4" xref="footnote10.m5.4.4.cmml"><mrow id="footnote10.m5.3.3.2" xref="footnote10.m5.3.3.2.cmml"><mn id="footnote10.m5.3.3.2.4" xref="footnote10.m5.3.3.2.4.cmml">2</mn><mo id="footnote10.m5.3.3.2.3" lspace="0.167em" xref="footnote10.m5.3.3.2.3.cmml">⁢</mo><mrow id="footnote10.m5.3.3.2.2.2" xref="footnote10.m5.3.3.2.2.3.cmml"><msup id="footnote10.m5.2.2.1.1.1.1" xref="footnote10.m5.2.2.1.1.1.1.cmml"><mi id="footnote10.m5.2.2.1.1.1.1.2" xref="footnote10.m5.2.2.1.1.1.1.2.cmml">sin</mi><mn id="footnote10.m5.2.2.1.1.1.1.3" xref="footnote10.m5.2.2.1.1.1.1.3.cmml">2</mn></msup><mo id="footnote10.m5.3.3.2.2.2b" xref="footnote10.m5.3.3.2.2.3.cmml">⁡</mo><mrow id="footnote10.m5.3.3.2.2.2.2" xref="footnote10.m5.3.3.2.2.3.cmml"><mo id="footnote10.m5.3.3.2.2.2.2.2" stretchy="false" xref="footnote10.m5.3.3.2.2.3.cmml">(</mo><msup id="footnote10.m5.3.3.2.2.2.2.1" xref="footnote10.m5.3.3.2.2.2.2.1.cmml"><mn id="footnote10.m5.3.3.2.2.2.2.1.2" xref="footnote10.m5.3.3.2.2.2.2.1.2.cmml">60</mn><mo id="footnote10.m5.3.3.2.2.2.2.1.3" xref="footnote10.m5.3.3.2.2.2.2.1.3.cmml">∘</mo></msup><mo id="footnote10.m5.3.3.2.2.2.2.3" stretchy="false" xref="footnote10.m5.3.3.2.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="footnote10.m5.4.4.4" xref="footnote10.m5.4.4.4.cmml">/</mo><mrow id="footnote10.m5.4.4.3.1" xref="footnote10.m5.4.4.3.2.cmml"><msup id="footnote10.m5.4.4.3.1.1" xref="footnote10.m5.4.4.3.1.1.cmml"><mi id="footnote10.m5.4.4.3.1.1.2" xref="footnote10.m5.4.4.3.1.1.2.cmml">sin</mi><mn id="footnote10.m5.4.4.3.1.1.3" xref="footnote10.m5.4.4.3.1.1.3.cmml">2</mn></msup><mo id="footnote10.m5.4.4.3.1b" xref="footnote10.m5.4.4.3.2.cmml">⁡</mo><mrow id="footnote10.m5.4.4.3.1.2" xref="footnote10.m5.4.4.3.2.cmml"><mo id="footnote10.m5.4.4.3.1.2.1" stretchy="false" xref="footnote10.m5.4.4.3.2.cmml">(</mo><mi id="footnote10.m5.1.1" xref="footnote10.m5.1.1.cmml">γ</mi><mo id="footnote10.m5.4.4.3.1.2.2" stretchy="false" xref="footnote10.m5.4.4.3.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote10.m5.4c"><apply id="footnote10.m5.4.4.cmml" xref="footnote10.m5.4.4"><divide id="footnote10.m5.4.4.4.cmml" xref="footnote10.m5.4.4.4"></divide><apply id="footnote10.m5.3.3.2.cmml" xref="footnote10.m5.3.3.2"><times id="footnote10.m5.3.3.2.3.cmml" xref="footnote10.m5.3.3.2.3"></times><cn id="footnote10.m5.3.3.2.4.cmml" type="integer" xref="footnote10.m5.3.3.2.4">2</cn><apply id="footnote10.m5.3.3.2.2.3.cmml" xref="footnote10.m5.3.3.2.2.2"><apply id="footnote10.m5.2.2.1.1.1.1.cmml" xref="footnote10.m5.2.2.1.1.1.1"><csymbol cd="ambiguous" id="footnote10.m5.2.2.1.1.1.1.1.cmml" xref="footnote10.m5.2.2.1.1.1.1">superscript</csymbol><sin id="footnote10.m5.2.2.1.1.1.1.2.cmml" xref="footnote10.m5.2.2.1.1.1.1.2"></sin><cn id="footnote10.m5.2.2.1.1.1.1.3.cmml" type="integer" xref="footnote10.m5.2.2.1.1.1.1.3">2</cn></apply><apply id="footnote10.m5.3.3.2.2.2.2.1.cmml" xref="footnote10.m5.3.3.2.2.2.2.1"><csymbol cd="ambiguous" id="footnote10.m5.3.3.2.2.2.2.1.1.cmml" xref="footnote10.m5.3.3.2.2.2.2.1">superscript</csymbol><cn id="footnote10.m5.3.3.2.2.2.2.1.2.cmml" type="integer" xref="footnote10.m5.3.3.2.2.2.2.1.2">60</cn><compose id="footnote10.m5.3.3.2.2.2.2.1.3.cmml" xref="footnote10.m5.3.3.2.2.2.2.1.3"></compose></apply></apply></apply><apply id="footnote10.m5.4.4.3.2.cmml" xref="footnote10.m5.4.4.3.1"><apply id="footnote10.m5.4.4.3.1.1.cmml" xref="footnote10.m5.4.4.3.1.1"><csymbol cd="ambiguous" id="footnote10.m5.4.4.3.1.1.1.cmml" xref="footnote10.m5.4.4.3.1.1">superscript</csymbol><sin id="footnote10.m5.4.4.3.1.1.2.cmml" xref="footnote10.m5.4.4.3.1.1.2"></sin><cn id="footnote10.m5.4.4.3.1.1.3.cmml" type="integer" xref="footnote10.m5.4.4.3.1.1.3">2</cn></apply><ci id="footnote10.m5.1.1.cmml" xref="footnote10.m5.1.1">𝛾</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote10.m5.4d">2\sin^{2}(60^{\circ})/\sin^{2}(\gamma)</annotation><annotation encoding="application/x-llamapun" id="footnote10.m5.4e">2 roman_sin start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 60 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT ) / roman_sin start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_γ )</annotation></semantics></math> owing to LISA having two independent channels at low frequencies and <math alttext="60^{\circ}" class="ltx_Math" display="inline" id="footnote10.m6.1"><semantics id="footnote10.m6.1b"><msup id="footnote10.m6.1.1" xref="footnote10.m6.1.1.cmml"><mn id="footnote10.m6.1.1.2" xref="footnote10.m6.1.1.2.cmml">60</mn><mo id="footnote10.m6.1.1.3" xref="footnote10.m6.1.1.3.cmml">∘</mo></msup><annotation-xml encoding="MathML-Content" id="footnote10.m6.1c"><apply id="footnote10.m6.1.1.cmml" xref="footnote10.m6.1.1"><csymbol cd="ambiguous" id="footnote10.m6.1.1.1.cmml" xref="footnote10.m6.1.1">superscript</csymbol><cn id="footnote10.m6.1.1.2.cmml" type="integer" xref="footnote10.m6.1.1.2">60</cn><compose id="footnote10.m6.1.1.3.cmml" xref="footnote10.m6.1.1.3"></compose></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote10.m6.1d">60^{\circ}</annotation><annotation encoding="application/x-llamapun" id="footnote10.m6.1e">60 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT</annotation></semantics></math> orientations for the arms <cite class="ltx_cite ltx_citemacro_citep">(Robson et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib60" title="">2019</a>)</cite>. Equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S3.E18" title="In 3 Strain power sensitivity ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">18</span></a>) assumes the noise and accelerations are independent between spacecraft: This will not be true for the acceleration noise on the intermediate spacecraft that joins both arms in the time-delayed inteferometry configuration, as the radial accelerations from solar irradiance and the solar wind will project onto both interferometer arms.</span></span></span></p> </div> <div class="ltx_para" id="S3.p2"> <p class="ltx_p" id="S3.p2.1">In what follows, we present the sensitivity for both a two-arm time-delay interferometry setup (§ <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S3.SS1" title="3.1 Two-arm time-delay interferometry forecasts ‣ 3 Strain power sensitivity ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">3.1</span></a>) as well as a setup with a single-arm and atomic clock (§ <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S3.SS2" title="3.2 Single-arm with atomic clock forecasts ‣ 3 Strain power sensitivity ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">3.2</span></a>), and finally consider the ‘Doppler tracking’ architecture in which the single-arm is anchored to Earth (§ <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S3.SS3" title="3.3 Doppler tracking forecasts ‣ 3 Strain power sensitivity ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">3.3</span></a>).</p> </div> <section class="ltx_subsection" id="S3.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.1 </span>Two-arm time-delay interferometry forecasts</h3> <div class="ltx_para" id="S3.SS1.p1"> <p class="ltx_p" id="S3.SS1.p1.15">The left and right panels in Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S3.F3" title="Figure 3 ‣ 3 Strain power sensitivity ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">3</span></a> show respectively the gravitational wave strain sensitivities calculated from equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S3.E18" title="In 3 Strain power sensitivity ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">18</span></a>) for the proposed concept at a solar distance of <math alttext="r=10" class="ltx_Math" display="inline" id="S3.SS1.p1.1.m1.1"><semantics id="S3.SS1.p1.1.m1.1a"><mrow id="S3.SS1.p1.1.m1.1.1" xref="S3.SS1.p1.1.m1.1.1.cmml"><mi id="S3.SS1.p1.1.m1.1.1.2" xref="S3.SS1.p1.1.m1.1.1.2.cmml">r</mi><mo id="S3.SS1.p1.1.m1.1.1.1" xref="S3.SS1.p1.1.m1.1.1.1.cmml">=</mo><mn id="S3.SS1.p1.1.m1.1.1.3" xref="S3.SS1.p1.1.m1.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.1.m1.1b"><apply id="S3.SS1.p1.1.m1.1.1.cmml" xref="S3.SS1.p1.1.m1.1.1"><eq id="S3.SS1.p1.1.m1.1.1.1.cmml" xref="S3.SS1.p1.1.m1.1.1.1"></eq><ci id="S3.SS1.p1.1.m1.1.1.2.cmml" xref="S3.SS1.p1.1.m1.1.1.2">𝑟</ci><cn id="S3.SS1.p1.1.m1.1.1.3.cmml" type="integer" xref="S3.SS1.p1.1.m1.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.1.m1.1c">r=10</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.1.m1.1d">italic_r = 10</annotation></semantics></math> AU and <math alttext="r=30" class="ltx_Math" display="inline" id="S3.SS1.p1.2.m2.1"><semantics id="S3.SS1.p1.2.m2.1a"><mrow id="S3.SS1.p1.2.m2.1.1" xref="S3.SS1.p1.2.m2.1.1.cmml"><mi id="S3.SS1.p1.2.m2.1.1.2" xref="S3.SS1.p1.2.m2.1.1.2.cmml">r</mi><mo id="S3.SS1.p1.2.m2.1.1.1" xref="S3.SS1.p1.2.m2.1.1.1.cmml">=</mo><mn id="S3.SS1.p1.2.m2.1.1.3" xref="S3.SS1.p1.2.m2.1.1.3.cmml">30</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.2.m2.1b"><apply id="S3.SS1.p1.2.m2.1.1.cmml" xref="S3.SS1.p1.2.m2.1.1"><eq id="S3.SS1.p1.2.m2.1.1.1.cmml" xref="S3.SS1.p1.2.m2.1.1.1"></eq><ci id="S3.SS1.p1.2.m2.1.1.2.cmml" xref="S3.SS1.p1.2.m2.1.1.2">𝑟</ci><cn id="S3.SS1.p1.2.m2.1.1.3.cmml" type="integer" xref="S3.SS1.p1.2.m2.1.1.3">30</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.2.m2.1c">r=30</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.2.m2.1d">italic_r = 30</annotation></semantics></math> AU, assuming the same arm lengths as the solar distance, e.g. <math alttext="L=r" class="ltx_Math" display="inline" id="S3.SS1.p1.3.m3.1"><semantics id="S3.SS1.p1.3.m3.1a"><mrow id="S3.SS1.p1.3.m3.1.1" xref="S3.SS1.p1.3.m3.1.1.cmml"><mi id="S3.SS1.p1.3.m3.1.1.2" xref="S3.SS1.p1.3.m3.1.1.2.cmml">L</mi><mo id="S3.SS1.p1.3.m3.1.1.1" xref="S3.SS1.p1.3.m3.1.1.1.cmml">=</mo><mi id="S3.SS1.p1.3.m3.1.1.3" xref="S3.SS1.p1.3.m3.1.1.3.cmml">r</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.3.m3.1b"><apply id="S3.SS1.p1.3.m3.1.1.cmml" xref="S3.SS1.p1.3.m3.1.1"><eq id="S3.SS1.p1.3.m3.1.1.1.cmml" xref="S3.SS1.p1.3.m3.1.1.1"></eq><ci id="S3.SS1.p1.3.m3.1.1.2.cmml" xref="S3.SS1.p1.3.m3.1.1.2">𝐿</ci><ci id="S3.SS1.p1.3.m3.1.1.3.cmml" xref="S3.SS1.p1.3.m3.1.1.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.3.m3.1c">L=r</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.3.m3.1d">italic_L = italic_r</annotation></semantics></math>. The calculations assume a geometric suppression of the radial accelerations from the Sun’s radiation and drag forces by <math alttext="0.5" class="ltx_Math" display="inline" id="S3.SS1.p1.4.m4.1"><semantics id="S3.SS1.p1.4.m4.1a"><mn id="S3.SS1.p1.4.m4.1.1" xref="S3.SS1.p1.4.m4.1.1.cmml">0.5</mn><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.4.m4.1b"><cn id="S3.SS1.p1.4.m4.1.1.cmml" type="float" xref="S3.SS1.p1.4.m4.1.1">0.5</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.4.m4.1c">0.5</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.4.m4.1d">0.5</annotation></semantics></math>, as would occur if each interferometer arm was a side of an equilateral triangle with the other two sides the Sun-spacecraft distances. If instead the configuration were an isosceles triangle with two satellites at distance <math alttext="r" class="ltx_Math" display="inline" id="S3.SS1.p1.5.m5.1"><semantics id="S3.SS1.p1.5.m5.1a"><mi id="S3.SS1.p1.5.m5.1.1" xref="S3.SS1.p1.5.m5.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.5.m5.1b"><ci id="S3.SS1.p1.5.m5.1.1.cmml" xref="S3.SS1.p1.5.m5.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.5.m5.1c">r</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.5.m5.1d">italic_r</annotation></semantics></math> from the Sun, the suppression factor would be <math alttext="L/2r" class="ltx_Math" display="inline" id="S3.SS1.p1.6.m6.1"><semantics id="S3.SS1.p1.6.m6.1a"><mrow id="S3.SS1.p1.6.m6.1.1" xref="S3.SS1.p1.6.m6.1.1.cmml"><mrow id="S3.SS1.p1.6.m6.1.1.2" xref="S3.SS1.p1.6.m6.1.1.2.cmml"><mi id="S3.SS1.p1.6.m6.1.1.2.2" xref="S3.SS1.p1.6.m6.1.1.2.2.cmml">L</mi><mo id="S3.SS1.p1.6.m6.1.1.2.1" xref="S3.SS1.p1.6.m6.1.1.2.1.cmml">/</mo><mn id="S3.SS1.p1.6.m6.1.1.2.3" xref="S3.SS1.p1.6.m6.1.1.2.3.cmml">2</mn></mrow><mo id="S3.SS1.p1.6.m6.1.1.1" xref="S3.SS1.p1.6.m6.1.1.1.cmml">⁢</mo><mi id="S3.SS1.p1.6.m6.1.1.3" xref="S3.SS1.p1.6.m6.1.1.3.cmml">r</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.6.m6.1b"><apply id="S3.SS1.p1.6.m6.1.1.cmml" xref="S3.SS1.p1.6.m6.1.1"><times id="S3.SS1.p1.6.m6.1.1.1.cmml" xref="S3.SS1.p1.6.m6.1.1.1"></times><apply id="S3.SS1.p1.6.m6.1.1.2.cmml" xref="S3.SS1.p1.6.m6.1.1.2"><divide id="S3.SS1.p1.6.m6.1.1.2.1.cmml" xref="S3.SS1.p1.6.m6.1.1.2.1"></divide><ci id="S3.SS1.p1.6.m6.1.1.2.2.cmml" xref="S3.SS1.p1.6.m6.1.1.2.2">𝐿</ci><cn id="S3.SS1.p1.6.m6.1.1.2.3.cmml" type="integer" xref="S3.SS1.p1.6.m6.1.1.2.3">2</cn></apply><ci id="S3.SS1.p1.6.m6.1.1.3.cmml" xref="S3.SS1.p1.6.m6.1.1.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.6.m6.1c">L/2r</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.6.m6.1d">italic_L / 2 italic_r</annotation></semantics></math>. These calculations additionally assume an effective area-to-mass ratio for the spacecraft of <math alttext="A_{\rm eff}/M=0.01~{}" class="ltx_Math" display="inline" id="S3.SS1.p1.7.m7.1"><semantics id="S3.SS1.p1.7.m7.1a"><mrow id="S3.SS1.p1.7.m7.1.1" xref="S3.SS1.p1.7.m7.1.1.cmml"><mrow id="S3.SS1.p1.7.m7.1.1.2" xref="S3.SS1.p1.7.m7.1.1.2.cmml"><msub id="S3.SS1.p1.7.m7.1.1.2.2" xref="S3.SS1.p1.7.m7.1.1.2.2.cmml"><mi id="S3.SS1.p1.7.m7.1.1.2.2.2" xref="S3.SS1.p1.7.m7.1.1.2.2.2.cmml">A</mi><mi id="S3.SS1.p1.7.m7.1.1.2.2.3" xref="S3.SS1.p1.7.m7.1.1.2.2.3.cmml">eff</mi></msub><mo id="S3.SS1.p1.7.m7.1.1.2.1" xref="S3.SS1.p1.7.m7.1.1.2.1.cmml">/</mo><mi id="S3.SS1.p1.7.m7.1.1.2.3" xref="S3.SS1.p1.7.m7.1.1.2.3.cmml">M</mi></mrow><mo id="S3.SS1.p1.7.m7.1.1.1" xref="S3.SS1.p1.7.m7.1.1.1.cmml">=</mo><mn id="S3.SS1.p1.7.m7.1.1.3" xref="S3.SS1.p1.7.m7.1.1.3.cmml">0.01</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.7.m7.1b"><apply id="S3.SS1.p1.7.m7.1.1.cmml" xref="S3.SS1.p1.7.m7.1.1"><eq id="S3.SS1.p1.7.m7.1.1.1.cmml" xref="S3.SS1.p1.7.m7.1.1.1"></eq><apply id="S3.SS1.p1.7.m7.1.1.2.cmml" xref="S3.SS1.p1.7.m7.1.1.2"><divide id="S3.SS1.p1.7.m7.1.1.2.1.cmml" xref="S3.SS1.p1.7.m7.1.1.2.1"></divide><apply id="S3.SS1.p1.7.m7.1.1.2.2.cmml" xref="S3.SS1.p1.7.m7.1.1.2.2"><csymbol cd="ambiguous" id="S3.SS1.p1.7.m7.1.1.2.2.1.cmml" xref="S3.SS1.p1.7.m7.1.1.2.2">subscript</csymbol><ci id="S3.SS1.p1.7.m7.1.1.2.2.2.cmml" xref="S3.SS1.p1.7.m7.1.1.2.2.2">𝐴</ci><ci id="S3.SS1.p1.7.m7.1.1.2.2.3.cmml" xref="S3.SS1.p1.7.m7.1.1.2.2.3">eff</ci></apply><ci id="S3.SS1.p1.7.m7.1.1.2.3.cmml" xref="S3.SS1.p1.7.m7.1.1.2.3">𝑀</ci></apply><cn id="S3.SS1.p1.7.m7.1.1.3.cmml" type="float" xref="S3.SS1.p1.7.m7.1.1.3">0.01</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.7.m7.1c">A_{\rm eff}/M=0.01~{}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.7.m7.1d">italic_A start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT / italic_M = 0.01</annotation></semantics></math>m<sup class="ltx_sup" id="S3.SS1.p1.15.1">2</sup> kg<sup class="ltx_sup" id="S3.SS1.p1.15.2"><span class="ltx_text ltx_font_italic" id="S3.SS1.p1.15.2.1">-1</span></sup>, which for example could be achieved with a spacecraft with <math alttext="A_{\rm eff}=10" class="ltx_Math" display="inline" id="S3.SS1.p1.10.m10.1"><semantics id="S3.SS1.p1.10.m10.1a"><mrow id="S3.SS1.p1.10.m10.1.1" xref="S3.SS1.p1.10.m10.1.1.cmml"><msub id="S3.SS1.p1.10.m10.1.1.2" xref="S3.SS1.p1.10.m10.1.1.2.cmml"><mi id="S3.SS1.p1.10.m10.1.1.2.2" xref="S3.SS1.p1.10.m10.1.1.2.2.cmml">A</mi><mi id="S3.SS1.p1.10.m10.1.1.2.3" xref="S3.SS1.p1.10.m10.1.1.2.3.cmml">eff</mi></msub><mo id="S3.SS1.p1.10.m10.1.1.1" xref="S3.SS1.p1.10.m10.1.1.1.cmml">=</mo><mn id="S3.SS1.p1.10.m10.1.1.3" xref="S3.SS1.p1.10.m10.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.10.m10.1b"><apply id="S3.SS1.p1.10.m10.1.1.cmml" xref="S3.SS1.p1.10.m10.1.1"><eq id="S3.SS1.p1.10.m10.1.1.1.cmml" xref="S3.SS1.p1.10.m10.1.1.1"></eq><apply id="S3.SS1.p1.10.m10.1.1.2.cmml" xref="S3.SS1.p1.10.m10.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p1.10.m10.1.1.2.1.cmml" xref="S3.SS1.p1.10.m10.1.1.2">subscript</csymbol><ci id="S3.SS1.p1.10.m10.1.1.2.2.cmml" xref="S3.SS1.p1.10.m10.1.1.2.2">𝐴</ci><ci id="S3.SS1.p1.10.m10.1.1.2.3.cmml" xref="S3.SS1.p1.10.m10.1.1.2.3">eff</ci></apply><cn id="S3.SS1.p1.10.m10.1.1.3.cmml" type="integer" xref="S3.SS1.p1.10.m10.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.10.m10.1c">A_{\rm eff}=10</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.10.m10.1d">italic_A start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT = 10</annotation></semantics></math>m<sup class="ltx_sup" id="S3.SS1.p1.15.3">2</sup> and mass of <math alttext="M=1000~{}" class="ltx_Math" display="inline" id="S3.SS1.p1.12.m12.1"><semantics id="S3.SS1.p1.12.m12.1a"><mrow id="S3.SS1.p1.12.m12.1.1" xref="S3.SS1.p1.12.m12.1.1.cmml"><mi id="S3.SS1.p1.12.m12.1.1.2" xref="S3.SS1.p1.12.m12.1.1.2.cmml">M</mi><mo id="S3.SS1.p1.12.m12.1.1.1" xref="S3.SS1.p1.12.m12.1.1.1.cmml">=</mo><mn id="S3.SS1.p1.12.m12.1.1.3" xref="S3.SS1.p1.12.m12.1.1.3.cmml">1000</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.12.m12.1b"><apply id="S3.SS1.p1.12.m12.1.1.cmml" xref="S3.SS1.p1.12.m12.1.1"><eq id="S3.SS1.p1.12.m12.1.1.1.cmml" xref="S3.SS1.p1.12.m12.1.1.1"></eq><ci id="S3.SS1.p1.12.m12.1.1.2.cmml" xref="S3.SS1.p1.12.m12.1.1.2">𝑀</ci><cn id="S3.SS1.p1.12.m12.1.1.3.cmml" type="integer" xref="S3.SS1.p1.12.m12.1.1.3">1000</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.12.m12.1c">M=1000~{}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.12.m12.1d">italic_M = 1000</annotation></semantics></math>kg (perhaps more applicable for the radio dish case) or <math alttext="A_{\rm eff}=3~{}" class="ltx_Math" display="inline" id="S3.SS1.p1.13.m13.1"><semantics id="S3.SS1.p1.13.m13.1a"><mrow id="S3.SS1.p1.13.m13.1.1" xref="S3.SS1.p1.13.m13.1.1.cmml"><msub id="S3.SS1.p1.13.m13.1.1.2" xref="S3.SS1.p1.13.m13.1.1.2.cmml"><mi id="S3.SS1.p1.13.m13.1.1.2.2" xref="S3.SS1.p1.13.m13.1.1.2.2.cmml">A</mi><mi id="S3.SS1.p1.13.m13.1.1.2.3" xref="S3.SS1.p1.13.m13.1.1.2.3.cmml">eff</mi></msub><mo id="S3.SS1.p1.13.m13.1.1.1" xref="S3.SS1.p1.13.m13.1.1.1.cmml">=</mo><mn id="S3.SS1.p1.13.m13.1.1.3" xref="S3.SS1.p1.13.m13.1.1.3.cmml">3</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.13.m13.1b"><apply id="S3.SS1.p1.13.m13.1.1.cmml" xref="S3.SS1.p1.13.m13.1.1"><eq id="S3.SS1.p1.13.m13.1.1.1.cmml" xref="S3.SS1.p1.13.m13.1.1.1"></eq><apply id="S3.SS1.p1.13.m13.1.1.2.cmml" xref="S3.SS1.p1.13.m13.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p1.13.m13.1.1.2.1.cmml" xref="S3.SS1.p1.13.m13.1.1.2">subscript</csymbol><ci id="S3.SS1.p1.13.m13.1.1.2.2.cmml" xref="S3.SS1.p1.13.m13.1.1.2.2">𝐴</ci><ci id="S3.SS1.p1.13.m13.1.1.2.3.cmml" xref="S3.SS1.p1.13.m13.1.1.2.3">eff</ci></apply><cn id="S3.SS1.p1.13.m13.1.1.3.cmml" type="integer" xref="S3.SS1.p1.13.m13.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.13.m13.1c">A_{\rm eff}=3~{}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.13.m13.1d">italic_A start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT = 3</annotation></semantics></math>m<sup class="ltx_sup" id="S3.SS1.p1.15.4">2</sup> and <math alttext="M=300~{}" class="ltx_Math" display="inline" id="S3.SS1.p1.15.m15.1"><semantics id="S3.SS1.p1.15.m15.1a"><mrow id="S3.SS1.p1.15.m15.1.1" xref="S3.SS1.p1.15.m15.1.1.cmml"><mi id="S3.SS1.p1.15.m15.1.1.2" xref="S3.SS1.p1.15.m15.1.1.2.cmml">M</mi><mo id="S3.SS1.p1.15.m15.1.1.1" xref="S3.SS1.p1.15.m15.1.1.1.cmml">=</mo><mn id="S3.SS1.p1.15.m15.1.1.3" xref="S3.SS1.p1.15.m15.1.1.3.cmml">300</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.15.m15.1b"><apply id="S3.SS1.p1.15.m15.1.1.cmml" xref="S3.SS1.p1.15.m15.1.1"><eq id="S3.SS1.p1.15.m15.1.1.1.cmml" xref="S3.SS1.p1.15.m15.1.1.1"></eq><ci id="S3.SS1.p1.15.m15.1.1.2.cmml" xref="S3.SS1.p1.15.m15.1.1.2">𝑀</ci><cn id="S3.SS1.p1.15.m15.1.1.3.cmml" type="integer" xref="S3.SS1.p1.15.m15.1.1.3">300</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.15.m15.1c">M=300~{}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.15.m15.1d">italic_M = 300</annotation></semantics></math>kg (perhaps applicable to the laser case and similar to each LISA spacecraft once the solar panels are removed).</p> </div> <div class="ltx_para" id="S3.SS1.p2"> <p class="ltx_p" id="S3.SS1.p2.14">The curves in Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S3.F3" title="Figure 3 ‣ 3 Strain power sensitivity ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">3</span></a> show the contributions to the noise from accelerations owing to solar irradiance variations, the solar wind, the maximum possible Lorentz force from spacecraft charging, and dust collisions up to a dust particle mass assuming <math alttext="m_{d,\rm max}=10^{-9}" class="ltx_Math" display="inline" id="S3.SS1.p2.1.m1.2"><semantics id="S3.SS1.p2.1.m1.2a"><mrow id="S3.SS1.p2.1.m1.2.3" xref="S3.SS1.p2.1.m1.2.3.cmml"><msub id="S3.SS1.p2.1.m1.2.3.2" xref="S3.SS1.p2.1.m1.2.3.2.cmml"><mi id="S3.SS1.p2.1.m1.2.3.2.2" xref="S3.SS1.p2.1.m1.2.3.2.2.cmml">m</mi><mrow id="S3.SS1.p2.1.m1.2.2.2.4" xref="S3.SS1.p2.1.m1.2.2.2.3.cmml"><mi id="S3.SS1.p2.1.m1.1.1.1.1" xref="S3.SS1.p2.1.m1.1.1.1.1.cmml">d</mi><mo id="S3.SS1.p2.1.m1.2.2.2.4.1" xref="S3.SS1.p2.1.m1.2.2.2.3.cmml">,</mo><mi id="S3.SS1.p2.1.m1.2.2.2.2" xref="S3.SS1.p2.1.m1.2.2.2.2.cmml">max</mi></mrow></msub><mo id="S3.SS1.p2.1.m1.2.3.1" xref="S3.SS1.p2.1.m1.2.3.1.cmml">=</mo><msup id="S3.SS1.p2.1.m1.2.3.3" xref="S3.SS1.p2.1.m1.2.3.3.cmml"><mn id="S3.SS1.p2.1.m1.2.3.3.2" xref="S3.SS1.p2.1.m1.2.3.3.2.cmml">10</mn><mrow id="S3.SS1.p2.1.m1.2.3.3.3" xref="S3.SS1.p2.1.m1.2.3.3.3.cmml"><mo id="S3.SS1.p2.1.m1.2.3.3.3a" xref="S3.SS1.p2.1.m1.2.3.3.3.cmml">−</mo><mn id="S3.SS1.p2.1.m1.2.3.3.3.2" xref="S3.SS1.p2.1.m1.2.3.3.3.2.cmml">9</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.1.m1.2b"><apply id="S3.SS1.p2.1.m1.2.3.cmml" xref="S3.SS1.p2.1.m1.2.3"><eq id="S3.SS1.p2.1.m1.2.3.1.cmml" xref="S3.SS1.p2.1.m1.2.3.1"></eq><apply id="S3.SS1.p2.1.m1.2.3.2.cmml" xref="S3.SS1.p2.1.m1.2.3.2"><csymbol cd="ambiguous" id="S3.SS1.p2.1.m1.2.3.2.1.cmml" xref="S3.SS1.p2.1.m1.2.3.2">subscript</csymbol><ci id="S3.SS1.p2.1.m1.2.3.2.2.cmml" xref="S3.SS1.p2.1.m1.2.3.2.2">𝑚</ci><list id="S3.SS1.p2.1.m1.2.2.2.3.cmml" xref="S3.SS1.p2.1.m1.2.2.2.4"><ci id="S3.SS1.p2.1.m1.1.1.1.1.cmml" xref="S3.SS1.p2.1.m1.1.1.1.1">𝑑</ci><ci id="S3.SS1.p2.1.m1.2.2.2.2.cmml" xref="S3.SS1.p2.1.m1.2.2.2.2">max</ci></list></apply><apply id="S3.SS1.p2.1.m1.2.3.3.cmml" xref="S3.SS1.p2.1.m1.2.3.3"><csymbol cd="ambiguous" id="S3.SS1.p2.1.m1.2.3.3.1.cmml" xref="S3.SS1.p2.1.m1.2.3.3">superscript</csymbol><cn id="S3.SS1.p2.1.m1.2.3.3.2.cmml" type="integer" xref="S3.SS1.p2.1.m1.2.3.3.2">10</cn><apply id="S3.SS1.p2.1.m1.2.3.3.3.cmml" xref="S3.SS1.p2.1.m1.2.3.3.3"><minus id="S3.SS1.p2.1.m1.2.3.3.3.1.cmml" xref="S3.SS1.p2.1.m1.2.3.3.3"></minus><cn id="S3.SS1.p2.1.m1.2.3.3.3.2.cmml" type="integer" xref="S3.SS1.p2.1.m1.2.3.3.3.2">9</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.1.m1.2c">m_{d,\rm max}=10^{-9}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.1.m1.2d">italic_m start_POSTSUBSCRIPT italic_d , roman_max end_POSTSUBSCRIPT = 10 start_POSTSUPERSCRIPT - 9 end_POSTSUPERSCRIPT</annotation></semantics></math>g and <math alttext="A_{\rm eff}=10\;" class="ltx_Math" display="inline" id="S3.SS1.p2.2.m2.1"><semantics id="S3.SS1.p2.2.m2.1a"><mrow id="S3.SS1.p2.2.m2.1.1" xref="S3.SS1.p2.2.m2.1.1.cmml"><msub id="S3.SS1.p2.2.m2.1.1.2" xref="S3.SS1.p2.2.m2.1.1.2.cmml"><mi id="S3.SS1.p2.2.m2.1.1.2.2" xref="S3.SS1.p2.2.m2.1.1.2.2.cmml">A</mi><mi id="S3.SS1.p2.2.m2.1.1.2.3" xref="S3.SS1.p2.2.m2.1.1.2.3.cmml">eff</mi></msub><mo id="S3.SS1.p2.2.m2.1.1.1" xref="S3.SS1.p2.2.m2.1.1.1.cmml">=</mo><mn id="S3.SS1.p2.2.m2.1.1.3" xref="S3.SS1.p2.2.m2.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.2.m2.1b"><apply id="S3.SS1.p2.2.m2.1.1.cmml" xref="S3.SS1.p2.2.m2.1.1"><eq id="S3.SS1.p2.2.m2.1.1.1.cmml" xref="S3.SS1.p2.2.m2.1.1.1"></eq><apply id="S3.SS1.p2.2.m2.1.1.2.cmml" xref="S3.SS1.p2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p2.2.m2.1.1.2.1.cmml" xref="S3.SS1.p2.2.m2.1.1.2">subscript</csymbol><ci id="S3.SS1.p2.2.m2.1.1.2.2.cmml" xref="S3.SS1.p2.2.m2.1.1.2.2">𝐴</ci><ci id="S3.SS1.p2.2.m2.1.1.2.3.cmml" xref="S3.SS1.p2.2.m2.1.1.2.3">eff</ci></apply><cn id="S3.SS1.p2.2.m2.1.1.3.cmml" type="integer" xref="S3.SS1.p2.2.m2.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.2.m2.1c">A_{\rm eff}=10\;</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.2.m2.1d">italic_A start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT = 10</annotation></semantics></math>m<sup class="ltx_sup" id="S3.SS1.p2.14.1">2</sup>. Also shown by the thick curves is the total strain error for radio and laser transmissions with different carrier-to-noise ratios (<math alttext="C/N_{0}" class="ltx_Math" display="inline" id="S3.SS1.p2.4.m4.1"><semantics id="S3.SS1.p2.4.m4.1a"><mrow id="S3.SS1.p2.4.m4.1.1" xref="S3.SS1.p2.4.m4.1.1.cmml"><mi id="S3.SS1.p2.4.m4.1.1.2" xref="S3.SS1.p2.4.m4.1.1.2.cmml">C</mi><mo id="S3.SS1.p2.4.m4.1.1.1" xref="S3.SS1.p2.4.m4.1.1.1.cmml">/</mo><msub id="S3.SS1.p2.4.m4.1.1.3" xref="S3.SS1.p2.4.m4.1.1.3.cmml"><mi id="S3.SS1.p2.4.m4.1.1.3.2" xref="S3.SS1.p2.4.m4.1.1.3.2.cmml">N</mi><mn id="S3.SS1.p2.4.m4.1.1.3.3" xref="S3.SS1.p2.4.m4.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.4.m4.1b"><apply id="S3.SS1.p2.4.m4.1.1.cmml" xref="S3.SS1.p2.4.m4.1.1"><divide id="S3.SS1.p2.4.m4.1.1.1.cmml" xref="S3.SS1.p2.4.m4.1.1.1"></divide><ci id="S3.SS1.p2.4.m4.1.1.2.cmml" xref="S3.SS1.p2.4.m4.1.1.2">𝐶</ci><apply id="S3.SS1.p2.4.m4.1.1.3.cmml" xref="S3.SS1.p2.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p2.4.m4.1.1.3.1.cmml" xref="S3.SS1.p2.4.m4.1.1.3">subscript</csymbol><ci id="S3.SS1.p2.4.m4.1.1.3.2.cmml" xref="S3.SS1.p2.4.m4.1.1.3.2">𝑁</ci><cn id="S3.SS1.p2.4.m4.1.1.3.3.cmml" type="integer" xref="S3.SS1.p2.4.m4.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.4.m4.1c">C/N_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.4.m4.1d">italic_C / italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>), where we have excluded the smaller contributions to the accelerations from dust collisions and spacecraft charging. Those that are for <math alttext="\lambda=1\," class="ltx_Math" display="inline" id="S3.SS1.p2.5.m5.1"><semantics id="S3.SS1.p2.5.m5.1a"><mrow id="S3.SS1.p2.5.m5.1.1" xref="S3.SS1.p2.5.m5.1.1.cmml"><mi id="S3.SS1.p2.5.m5.1.1.2" xref="S3.SS1.p2.5.m5.1.1.2.cmml">λ</mi><mo id="S3.SS1.p2.5.m5.1.1.1" xref="S3.SS1.p2.5.m5.1.1.1.cmml">=</mo><mn id="S3.SS1.p2.5.m5.1.1.3" xref="S3.SS1.p2.5.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.5.m5.1b"><apply id="S3.SS1.p2.5.m5.1.1.cmml" xref="S3.SS1.p2.5.m5.1.1"><eq id="S3.SS1.p2.5.m5.1.1.1.cmml" xref="S3.SS1.p2.5.m5.1.1.1"></eq><ci id="S3.SS1.p2.5.m5.1.1.2.cmml" xref="S3.SS1.p2.5.m5.1.1.2">𝜆</ci><cn id="S3.SS1.p2.5.m5.1.1.3.cmml" type="integer" xref="S3.SS1.p2.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.5.m5.1c">\lambda=1\,</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.5.m5.1d">italic_λ = 1</annotation></semantics></math>cm assume carrier-to-noise ratios (<math alttext="C/N_{0}" class="ltx_Math" display="inline" id="S3.SS1.p2.6.m6.1"><semantics id="S3.SS1.p2.6.m6.1a"><mrow id="S3.SS1.p2.6.m6.1.1" xref="S3.SS1.p2.6.m6.1.1.cmml"><mi id="S3.SS1.p2.6.m6.1.1.2" xref="S3.SS1.p2.6.m6.1.1.2.cmml">C</mi><mo id="S3.SS1.p2.6.m6.1.1.1" xref="S3.SS1.p2.6.m6.1.1.1.cmml">/</mo><msub id="S3.SS1.p2.6.m6.1.1.3" xref="S3.SS1.p2.6.m6.1.1.3.cmml"><mi id="S3.SS1.p2.6.m6.1.1.3.2" xref="S3.SS1.p2.6.m6.1.1.3.2.cmml">N</mi><mn id="S3.SS1.p2.6.m6.1.1.3.3" xref="S3.SS1.p2.6.m6.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.6.m6.1b"><apply id="S3.SS1.p2.6.m6.1.1.cmml" xref="S3.SS1.p2.6.m6.1.1"><divide id="S3.SS1.p2.6.m6.1.1.1.cmml" xref="S3.SS1.p2.6.m6.1.1.1"></divide><ci id="S3.SS1.p2.6.m6.1.1.2.cmml" xref="S3.SS1.p2.6.m6.1.1.2">𝐶</ci><apply id="S3.SS1.p2.6.m6.1.1.3.cmml" xref="S3.SS1.p2.6.m6.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p2.6.m6.1.1.3.1.cmml" xref="S3.SS1.p2.6.m6.1.1.3">subscript</csymbol><ci id="S3.SS1.p2.6.m6.1.1.3.2.cmml" xref="S3.SS1.p2.6.m6.1.1.3.2">𝑁</ci><cn id="S3.SS1.p2.6.m6.1.1.3.3.cmml" type="integer" xref="S3.SS1.p2.6.m6.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.6.m6.1c">C/N_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.6.m6.1d">italic_C / italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>) of <math alttext="30~{}" class="ltx_Math" display="inline" id="S3.SS1.p2.7.m7.1"><semantics id="S3.SS1.p2.7.m7.1a"><mn id="S3.SS1.p2.7.m7.1.1" xref="S3.SS1.p2.7.m7.1.1.cmml">30</mn><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.7.m7.1b"><cn id="S3.SS1.p2.7.m7.1.1.cmml" type="integer" xref="S3.SS1.p2.7.m7.1.1">30</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.7.m7.1c">30~{}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.7.m7.1d">30</annotation></semantics></math>dB-Hz and <math alttext="50~{}" class="ltx_Math" display="inline" id="S3.SS1.p2.8.m8.1"><semantics id="S3.SS1.p2.8.m8.1a"><mn id="S3.SS1.p2.8.m8.1.1" xref="S3.SS1.p2.8.m8.1.1.cmml">50</mn><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.8.m8.1b"><cn id="S3.SS1.p2.8.m8.1.1.cmml" type="integer" xref="S3.SS1.p2.8.m8.1.1">50</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.8.m8.1c">50~{}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.8.m8.1d">50</annotation></semantics></math>dB-Hz (§ <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.SS2" title="2.2 Radiometer noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">2.2</span></a>) and <math alttext="{\cal A}=1.5" class="ltx_Math" display="inline" id="S3.SS1.p2.9.m9.1"><semantics id="S3.SS1.p2.9.m9.1a"><mrow id="S3.SS1.p2.9.m9.1.1" xref="S3.SS1.p2.9.m9.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p2.9.m9.1.1.2" xref="S3.SS1.p2.9.m9.1.1.2.cmml">𝒜</mi><mo id="S3.SS1.p2.9.m9.1.1.1" xref="S3.SS1.p2.9.m9.1.1.1.cmml">=</mo><mn id="S3.SS1.p2.9.m9.1.1.3" xref="S3.SS1.p2.9.m9.1.1.3.cmml">1.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.9.m9.1b"><apply id="S3.SS1.p2.9.m9.1.1.cmml" xref="S3.SS1.p2.9.m9.1.1"><eq id="S3.SS1.p2.9.m9.1.1.1.cmml" xref="S3.SS1.p2.9.m9.1.1.1"></eq><ci id="S3.SS1.p2.9.m9.1.1.2.cmml" xref="S3.SS1.p2.9.m9.1.1.2">𝒜</ci><cn id="S3.SS1.p2.9.m9.1.1.3.cmml" type="float" xref="S3.SS1.p2.9.m9.1.1.3">1.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.9.m9.1c">{\cal A}=1.5</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.9.m9.1d">caligraphic_A = 1.5</annotation></semantics></math> (Appendix <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#A1" title="Appendix A Dispersion ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">A</span></a>). Also shown is a laser effort with <math alttext="\lambda=1\mu" class="ltx_Math" display="inline" id="S3.SS1.p2.10.m10.1"><semantics id="S3.SS1.p2.10.m10.1a"><mrow id="S3.SS1.p2.10.m10.1.1" xref="S3.SS1.p2.10.m10.1.1.cmml"><mi id="S3.SS1.p2.10.m10.1.1.2" xref="S3.SS1.p2.10.m10.1.1.2.cmml">λ</mi><mo id="S3.SS1.p2.10.m10.1.1.1" xref="S3.SS1.p2.10.m10.1.1.1.cmml">=</mo><mrow id="S3.SS1.p2.10.m10.1.1.3" xref="S3.SS1.p2.10.m10.1.1.3.cmml"><mn id="S3.SS1.p2.10.m10.1.1.3.2" xref="S3.SS1.p2.10.m10.1.1.3.2.cmml">1</mn><mo id="S3.SS1.p2.10.m10.1.1.3.1" xref="S3.SS1.p2.10.m10.1.1.3.1.cmml">⁢</mo><mi id="S3.SS1.p2.10.m10.1.1.3.3" xref="S3.SS1.p2.10.m10.1.1.3.3.cmml">μ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.10.m10.1b"><apply id="S3.SS1.p2.10.m10.1.1.cmml" xref="S3.SS1.p2.10.m10.1.1"><eq id="S3.SS1.p2.10.m10.1.1.1.cmml" xref="S3.SS1.p2.10.m10.1.1.1"></eq><ci id="S3.SS1.p2.10.m10.1.1.2.cmml" xref="S3.SS1.p2.10.m10.1.1.2">𝜆</ci><apply id="S3.SS1.p2.10.m10.1.1.3.cmml" xref="S3.SS1.p2.10.m10.1.1.3"><times id="S3.SS1.p2.10.m10.1.1.3.1.cmml" xref="S3.SS1.p2.10.m10.1.1.3.1"></times><cn id="S3.SS1.p2.10.m10.1.1.3.2.cmml" type="integer" xref="S3.SS1.p2.10.m10.1.1.3.2">1</cn><ci id="S3.SS1.p2.10.m10.1.1.3.3.cmml" xref="S3.SS1.p2.10.m10.1.1.3.3">𝜇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.10.m10.1c">\lambda=1\mu</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.10.m10.1d">italic_λ = 1 italic_μ</annotation></semantics></math>m and <math alttext="30~{}" class="ltx_Math" display="inline" id="S3.SS1.p2.11.m11.1"><semantics id="S3.SS1.p2.11.m11.1a"><mn id="S3.SS1.p2.11.m11.1.1" xref="S3.SS1.p2.11.m11.1.1.cmml">30</mn><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.11.m11.1b"><cn id="S3.SS1.p2.11.m11.1.1.cmml" type="integer" xref="S3.SS1.p2.11.m11.1.1">30</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.11.m11.1c">30~{}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.11.m11.1d">30</annotation></semantics></math>dB-Hz. The noise is smaller for the laser compared to the radio at <math alttext="f\gtrsim 5\times 10^{-5}" class="ltx_Math" display="inline" id="S3.SS1.p2.12.m12.1"><semantics id="S3.SS1.p2.12.m12.1a"><mrow id="S3.SS1.p2.12.m12.1.1" xref="S3.SS1.p2.12.m12.1.1.cmml"><mi id="S3.SS1.p2.12.m12.1.1.2" xref="S3.SS1.p2.12.m12.1.1.2.cmml">f</mi><mo id="S3.SS1.p2.12.m12.1.1.1" xref="S3.SS1.p2.12.m12.1.1.1.cmml">≳</mo><mrow id="S3.SS1.p2.12.m12.1.1.3" xref="S3.SS1.p2.12.m12.1.1.3.cmml"><mn id="S3.SS1.p2.12.m12.1.1.3.2" xref="S3.SS1.p2.12.m12.1.1.3.2.cmml">5</mn><mo id="S3.SS1.p2.12.m12.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S3.SS1.p2.12.m12.1.1.3.1.cmml">×</mo><msup id="S3.SS1.p2.12.m12.1.1.3.3" xref="S3.SS1.p2.12.m12.1.1.3.3.cmml"><mn id="S3.SS1.p2.12.m12.1.1.3.3.2" xref="S3.SS1.p2.12.m12.1.1.3.3.2.cmml">10</mn><mrow id="S3.SS1.p2.12.m12.1.1.3.3.3" xref="S3.SS1.p2.12.m12.1.1.3.3.3.cmml"><mo id="S3.SS1.p2.12.m12.1.1.3.3.3a" xref="S3.SS1.p2.12.m12.1.1.3.3.3.cmml">−</mo><mn id="S3.SS1.p2.12.m12.1.1.3.3.3.2" xref="S3.SS1.p2.12.m12.1.1.3.3.3.2.cmml">5</mn></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.12.m12.1b"><apply id="S3.SS1.p2.12.m12.1.1.cmml" xref="S3.SS1.p2.12.m12.1.1"><csymbol cd="latexml" id="S3.SS1.p2.12.m12.1.1.1.cmml" xref="S3.SS1.p2.12.m12.1.1.1">greater-than-or-equivalent-to</csymbol><ci id="S3.SS1.p2.12.m12.1.1.2.cmml" xref="S3.SS1.p2.12.m12.1.1.2">𝑓</ci><apply id="S3.SS1.p2.12.m12.1.1.3.cmml" xref="S3.SS1.p2.12.m12.1.1.3"><times id="S3.SS1.p2.12.m12.1.1.3.1.cmml" xref="S3.SS1.p2.12.m12.1.1.3.1"></times><cn id="S3.SS1.p2.12.m12.1.1.3.2.cmml" type="integer" xref="S3.SS1.p2.12.m12.1.1.3.2">5</cn><apply id="S3.SS1.p2.12.m12.1.1.3.3.cmml" xref="S3.SS1.p2.12.m12.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS1.p2.12.m12.1.1.3.3.1.cmml" xref="S3.SS1.p2.12.m12.1.1.3.3">superscript</csymbol><cn id="S3.SS1.p2.12.m12.1.1.3.3.2.cmml" type="integer" xref="S3.SS1.p2.12.m12.1.1.3.3.2">10</cn><apply id="S3.SS1.p2.12.m12.1.1.3.3.3.cmml" xref="S3.SS1.p2.12.m12.1.1.3.3.3"><minus id="S3.SS1.p2.12.m12.1.1.3.3.3.1.cmml" xref="S3.SS1.p2.12.m12.1.1.3.3.3"></minus><cn id="S3.SS1.p2.12.m12.1.1.3.3.3.2.cmml" type="integer" xref="S3.SS1.p2.12.m12.1.1.3.3.3.2">5</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.12.m12.1c">f\gtrsim 5\times 10^{-5}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.12.m12.1d">italic_f ≳ 5 × 10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT</annotation></semantics></math>Hz, whereas at lower frequencies the sensitives are similar as acceleration noise dominates. A major goal of the proposed concepts would be to fill in the portion of the gravitational wave spectrum not probed by LISA. The LISA mission is required to achieve the sensitivity shown by the gray solid curve for <math alttext="f>10^{-4}\," class="ltx_Math" display="inline" id="S3.SS1.p2.13.m13.1"><semantics id="S3.SS1.p2.13.m13.1a"><mrow id="S3.SS1.p2.13.m13.1.1" xref="S3.SS1.p2.13.m13.1.1.cmml"><mi id="S3.SS1.p2.13.m13.1.1.2" xref="S3.SS1.p2.13.m13.1.1.2.cmml">f</mi><mo id="S3.SS1.p2.13.m13.1.1.1" xref="S3.SS1.p2.13.m13.1.1.1.cmml">></mo><msup id="S3.SS1.p2.13.m13.1.1.3" xref="S3.SS1.p2.13.m13.1.1.3.cmml"><mn id="S3.SS1.p2.13.m13.1.1.3.2" xref="S3.SS1.p2.13.m13.1.1.3.2.cmml">10</mn><mrow id="S3.SS1.p2.13.m13.1.1.3.3" xref="S3.SS1.p2.13.m13.1.1.3.3.cmml"><mo id="S3.SS1.p2.13.m13.1.1.3.3a" xref="S3.SS1.p2.13.m13.1.1.3.3.cmml">−</mo><mn id="S3.SS1.p2.13.m13.1.1.3.3.2" xref="S3.SS1.p2.13.m13.1.1.3.3.2.cmml">4</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.13.m13.1b"><apply id="S3.SS1.p2.13.m13.1.1.cmml" xref="S3.SS1.p2.13.m13.1.1"><gt id="S3.SS1.p2.13.m13.1.1.1.cmml" xref="S3.SS1.p2.13.m13.1.1.1"></gt><ci id="S3.SS1.p2.13.m13.1.1.2.cmml" xref="S3.SS1.p2.13.m13.1.1.2">𝑓</ci><apply id="S3.SS1.p2.13.m13.1.1.3.cmml" xref="S3.SS1.p2.13.m13.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p2.13.m13.1.1.3.1.cmml" xref="S3.SS1.p2.13.m13.1.1.3">superscript</csymbol><cn id="S3.SS1.p2.13.m13.1.1.3.2.cmml" type="integer" xref="S3.SS1.p2.13.m13.1.1.3.2">10</cn><apply id="S3.SS1.p2.13.m13.1.1.3.3.cmml" xref="S3.SS1.p2.13.m13.1.1.3.3"><minus id="S3.SS1.p2.13.m13.1.1.3.3.1.cmml" xref="S3.SS1.p2.13.m13.1.1.3.3"></minus><cn id="S3.SS1.p2.13.m13.1.1.3.3.2.cmml" type="integer" xref="S3.SS1.p2.13.m13.1.1.3.3.2">4</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.13.m13.1c">f>10^{-4}\,</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.13.m13.1d">italic_f > 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT</annotation></semantics></math>Hz, with the goal for LISA to be sensitive to <math alttext="f>2\times 10^{-5}" class="ltx_Math" display="inline" id="S3.SS1.p2.14.m14.1"><semantics id="S3.SS1.p2.14.m14.1a"><mrow id="S3.SS1.p2.14.m14.1.1" xref="S3.SS1.p2.14.m14.1.1.cmml"><mi id="S3.SS1.p2.14.m14.1.1.2" xref="S3.SS1.p2.14.m14.1.1.2.cmml">f</mi><mo id="S3.SS1.p2.14.m14.1.1.1" xref="S3.SS1.p2.14.m14.1.1.1.cmml">></mo><mrow id="S3.SS1.p2.14.m14.1.1.3" xref="S3.SS1.p2.14.m14.1.1.3.cmml"><mn id="S3.SS1.p2.14.m14.1.1.3.2" xref="S3.SS1.p2.14.m14.1.1.3.2.cmml">2</mn><mo id="S3.SS1.p2.14.m14.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S3.SS1.p2.14.m14.1.1.3.1.cmml">×</mo><msup id="S3.SS1.p2.14.m14.1.1.3.3" xref="S3.SS1.p2.14.m14.1.1.3.3.cmml"><mn id="S3.SS1.p2.14.m14.1.1.3.3.2" xref="S3.SS1.p2.14.m14.1.1.3.3.2.cmml">10</mn><mrow id="S3.SS1.p2.14.m14.1.1.3.3.3" xref="S3.SS1.p2.14.m14.1.1.3.3.3.cmml"><mo id="S3.SS1.p2.14.m14.1.1.3.3.3a" xref="S3.SS1.p2.14.m14.1.1.3.3.3.cmml">−</mo><mn id="S3.SS1.p2.14.m14.1.1.3.3.3.2" xref="S3.SS1.p2.14.m14.1.1.3.3.3.2.cmml">5</mn></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.14.m14.1b"><apply id="S3.SS1.p2.14.m14.1.1.cmml" xref="S3.SS1.p2.14.m14.1.1"><gt id="S3.SS1.p2.14.m14.1.1.1.cmml" xref="S3.SS1.p2.14.m14.1.1.1"></gt><ci id="S3.SS1.p2.14.m14.1.1.2.cmml" xref="S3.SS1.p2.14.m14.1.1.2">𝑓</ci><apply id="S3.SS1.p2.14.m14.1.1.3.cmml" xref="S3.SS1.p2.14.m14.1.1.3"><times id="S3.SS1.p2.14.m14.1.1.3.1.cmml" xref="S3.SS1.p2.14.m14.1.1.3.1"></times><cn id="S3.SS1.p2.14.m14.1.1.3.2.cmml" type="integer" xref="S3.SS1.p2.14.m14.1.1.3.2">2</cn><apply id="S3.SS1.p2.14.m14.1.1.3.3.cmml" xref="S3.SS1.p2.14.m14.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS1.p2.14.m14.1.1.3.3.1.cmml" xref="S3.SS1.p2.14.m14.1.1.3.3">superscript</csymbol><cn id="S3.SS1.p2.14.m14.1.1.3.3.2.cmml" type="integer" xref="S3.SS1.p2.14.m14.1.1.3.3.2">10</cn><apply id="S3.SS1.p2.14.m14.1.1.3.3.3.cmml" xref="S3.SS1.p2.14.m14.1.1.3.3.3"><minus id="S3.SS1.p2.14.m14.1.1.3.3.3.1.cmml" xref="S3.SS1.p2.14.m14.1.1.3.3.3"></minus><cn id="S3.SS1.p2.14.m14.1.1.3.3.3.2.cmml" type="integer" xref="S3.SS1.p2.14.m14.1.1.3.3.3.2">5</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.14.m14.1c">f>2\times 10^{-5}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.14.m14.1d">italic_f > 2 × 10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT</annotation></semantics></math>Hz as illustrated by the dashed extension.</p> </div> <div class="ltx_para" id="S3.SS1.p3"> <p class="ltx_p" id="S3.SS1.p3.6">Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S3.F4" title="Figure 4 ‣ 3.1 Two-arm time-delay interferometry forecasts ‣ 3 Strain power sensitivity ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">4</span></a> shows how these sensitivity projections compare to the gravitational wave signals from astrophysical sources, where we have taken <math alttext="50\;" class="ltx_Math" display="inline" id="S3.SS1.p3.1.m1.1"><semantics id="S3.SS1.p3.1.m1.1a"><mn id="S3.SS1.p3.1.m1.1.1" xref="S3.SS1.p3.1.m1.1.1.cmml">50</mn><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.1.m1.1b"><cn id="S3.SS1.p3.1.m1.1.1.cmml" type="integer" xref="S3.SS1.p3.1.m1.1.1">50</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.1.m1.1c">50\;</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.1.m1.1d">50</annotation></semantics></math>dB-Hz for the <math alttext="r=L=10\;" class="ltx_Math" display="inline" id="S3.SS1.p3.2.m2.1"><semantics id="S3.SS1.p3.2.m2.1a"><mrow id="S3.SS1.p3.2.m2.1.1" xref="S3.SS1.p3.2.m2.1.1.cmml"><mi id="S3.SS1.p3.2.m2.1.1.2" xref="S3.SS1.p3.2.m2.1.1.2.cmml">r</mi><mo id="S3.SS1.p3.2.m2.1.1.3" xref="S3.SS1.p3.2.m2.1.1.3.cmml">=</mo><mi id="S3.SS1.p3.2.m2.1.1.4" xref="S3.SS1.p3.2.m2.1.1.4.cmml">L</mi><mo id="S3.SS1.p3.2.m2.1.1.5" xref="S3.SS1.p3.2.m2.1.1.5.cmml">=</mo><mn id="S3.SS1.p3.2.m2.1.1.6" xref="S3.SS1.p3.2.m2.1.1.6.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.2.m2.1b"><apply id="S3.SS1.p3.2.m2.1.1.cmml" xref="S3.SS1.p3.2.m2.1.1"><and id="S3.SS1.p3.2.m2.1.1a.cmml" xref="S3.SS1.p3.2.m2.1.1"></and><apply id="S3.SS1.p3.2.m2.1.1b.cmml" xref="S3.SS1.p3.2.m2.1.1"><eq id="S3.SS1.p3.2.m2.1.1.3.cmml" xref="S3.SS1.p3.2.m2.1.1.3"></eq><ci id="S3.SS1.p3.2.m2.1.1.2.cmml" xref="S3.SS1.p3.2.m2.1.1.2">𝑟</ci><ci id="S3.SS1.p3.2.m2.1.1.4.cmml" xref="S3.SS1.p3.2.m2.1.1.4">𝐿</ci></apply><apply id="S3.SS1.p3.2.m2.1.1c.cmml" xref="S3.SS1.p3.2.m2.1.1"><eq id="S3.SS1.p3.2.m2.1.1.5.cmml" xref="S3.SS1.p3.2.m2.1.1.5"></eq><share href="https://arxiv.org/html/2411.15072v1#S3.SS1.p3.2.m2.1.1.4.cmml" id="S3.SS1.p3.2.m2.1.1d.cmml" xref="S3.SS1.p3.2.m2.1.1"></share><cn id="S3.SS1.p3.2.m2.1.1.6.cmml" type="integer" xref="S3.SS1.p3.2.m2.1.1.6">10</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.2.m2.1c">r=L=10\;</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.2.m2.1d">italic_r = italic_L = 10</annotation></semantics></math>AU case (left panel) and <math alttext="30\;" class="ltx_Math" display="inline" id="S3.SS1.p3.3.m3.1"><semantics id="S3.SS1.p3.3.m3.1a"><mn id="S3.SS1.p3.3.m3.1.1" xref="S3.SS1.p3.3.m3.1.1.cmml">30</mn><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.3.m3.1b"><cn id="S3.SS1.p3.3.m3.1.1.cmml" type="integer" xref="S3.SS1.p3.3.m3.1.1">30</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.3.m3.1c">30\;</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.3.m3.1d">30</annotation></semantics></math>dB-Hz for the <math alttext="r=L=30\;" class="ltx_Math" display="inline" id="S3.SS1.p3.4.m4.1"><semantics id="S3.SS1.p3.4.m4.1a"><mrow id="S3.SS1.p3.4.m4.1.1" xref="S3.SS1.p3.4.m4.1.1.cmml"><mi id="S3.SS1.p3.4.m4.1.1.2" xref="S3.SS1.p3.4.m4.1.1.2.cmml">r</mi><mo id="S3.SS1.p3.4.m4.1.1.3" xref="S3.SS1.p3.4.m4.1.1.3.cmml">=</mo><mi id="S3.SS1.p3.4.m4.1.1.4" xref="S3.SS1.p3.4.m4.1.1.4.cmml">L</mi><mo id="S3.SS1.p3.4.m4.1.1.5" xref="S3.SS1.p3.4.m4.1.1.5.cmml">=</mo><mn id="S3.SS1.p3.4.m4.1.1.6" xref="S3.SS1.p3.4.m4.1.1.6.cmml">30</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.4.m4.1b"><apply id="S3.SS1.p3.4.m4.1.1.cmml" xref="S3.SS1.p3.4.m4.1.1"><and id="S3.SS1.p3.4.m4.1.1a.cmml" xref="S3.SS1.p3.4.m4.1.1"></and><apply id="S3.SS1.p3.4.m4.1.1b.cmml" xref="S3.SS1.p3.4.m4.1.1"><eq id="S3.SS1.p3.4.m4.1.1.3.cmml" xref="S3.SS1.p3.4.m4.1.1.3"></eq><ci id="S3.SS1.p3.4.m4.1.1.2.cmml" xref="S3.SS1.p3.4.m4.1.1.2">𝑟</ci><ci id="S3.SS1.p3.4.m4.1.1.4.cmml" xref="S3.SS1.p3.4.m4.1.1.4">𝐿</ci></apply><apply id="S3.SS1.p3.4.m4.1.1c.cmml" xref="S3.SS1.p3.4.m4.1.1"><eq id="S3.SS1.p3.4.m4.1.1.5.cmml" xref="S3.SS1.p3.4.m4.1.1.5"></eq><share href="https://arxiv.org/html/2411.15072v1#S3.SS1.p3.4.m4.1.1.4.cmml" id="S3.SS1.p3.4.m4.1.1d.cmml" xref="S3.SS1.p3.4.m4.1.1"></share><cn id="S3.SS1.p3.4.m4.1.1.6.cmml" type="integer" xref="S3.SS1.p3.4.m4.1.1.6">30</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.4.m4.1c">r=L=30\;</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.4.m4.1d">italic_r = italic_L = 30</annotation></semantics></math>AU one (right panel). The green curves show the characteristic strain of equal-mass supermassive black hole mergers at <math alttext="z=3" class="ltx_Math" display="inline" id="S3.SS1.p3.5.m5.1"><semantics id="S3.SS1.p3.5.m5.1a"><mrow id="S3.SS1.p3.5.m5.1.1" xref="S3.SS1.p3.5.m5.1.1.cmml"><mi id="S3.SS1.p3.5.m5.1.1.2" xref="S3.SS1.p3.5.m5.1.1.2.cmml">z</mi><mo id="S3.SS1.p3.5.m5.1.1.1" xref="S3.SS1.p3.5.m5.1.1.1.cmml">=</mo><mn id="S3.SS1.p3.5.m5.1.1.3" xref="S3.SS1.p3.5.m5.1.1.3.cmml">3</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.5.m5.1b"><apply id="S3.SS1.p3.5.m5.1.1.cmml" xref="S3.SS1.p3.5.m5.1.1"><eq id="S3.SS1.p3.5.m5.1.1.1.cmml" xref="S3.SS1.p3.5.m5.1.1.1"></eq><ci id="S3.SS1.p3.5.m5.1.1.2.cmml" xref="S3.SS1.p3.5.m5.1.1.2">𝑧</ci><cn id="S3.SS1.p3.5.m5.1.1.3.cmml" type="integer" xref="S3.SS1.p3.5.m5.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.5.m5.1c">z=3</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.5.m5.1d">italic_z = 3</annotation></semantics></math> for the annotated initial mass of each black hole, showing the five years before merger <cite class="ltx_cite ltx_citemacro_citep">(Ajith et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib6" title="">2007</a>)</cite>. The characteristic strain is defined such that the integral over <math alttext="\log f" class="ltx_Math" display="inline" id="S3.SS1.p3.6.m6.1"><semantics id="S3.SS1.p3.6.m6.1a"><mrow id="S3.SS1.p3.6.m6.1.1" xref="S3.SS1.p3.6.m6.1.1.cmml"><mi id="S3.SS1.p3.6.m6.1.1.1" xref="S3.SS1.p3.6.m6.1.1.1.cmml">log</mi><mo id="S3.SS1.p3.6.m6.1.1a" lspace="0.167em" xref="S3.SS1.p3.6.m6.1.1.cmml">⁡</mo><mi id="S3.SS1.p3.6.m6.1.1.2" xref="S3.SS1.p3.6.m6.1.1.2.cmml">f</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.6.m6.1b"><apply id="S3.SS1.p3.6.m6.1.1.cmml" xref="S3.SS1.p3.6.m6.1.1"><log id="S3.SS1.p3.6.m6.1.1.1.cmml" xref="S3.SS1.p3.6.m6.1.1.1"></log><ci id="S3.SS1.p3.6.m6.1.1.2.cmml" xref="S3.SS1.p3.6.m6.1.1.2">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.6.m6.1c">\log f</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.6.m6.1d">roman_log italic_f</annotation></semantics></math> of the ratio of the characteristic strain to the concept’s noise power equals the square of the signal-to-noise ratio for detection.</p> </div> <div class="ltx_para" id="S3.SS1.p4"> <p class="ltx_p" id="S3.SS1.p4.3">Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S3.F4" title="Figure 4 ‣ 3.1 Two-arm time-delay interferometry forecasts ‣ 3 Strain power sensitivity ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">4</span></a> also shows the estimate for the Galactic white dwarf binary stochastic gravitational wave background (WDB GWB) from <cite class="ltx_cite ltx_citemacro_citet">Cornish & Robson (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib22" title="">2017</a>)</cite>, which we find to be a factor of two lower than the estimate of <cite class="ltx_cite ltx_citemacro_citet">Nissanke et al. (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib55" title="">2012</a>)</cite>. The stochastic background is defined as the background where the density of sources is too high in each spectral bin over a <math alttext="\sim 5\;" class="ltx_Math" display="inline" id="S3.SS1.p4.1.m1.1"><semantics id="S3.SS1.p4.1.m1.1a"><mrow id="S3.SS1.p4.1.m1.1.1" xref="S3.SS1.p4.1.m1.1.1.cmml"><mi id="S3.SS1.p4.1.m1.1.1.2" xref="S3.SS1.p4.1.m1.1.1.2.cmml"></mi><mo id="S3.SS1.p4.1.m1.1.1.1" xref="S3.SS1.p4.1.m1.1.1.1.cmml">∼</mo><mn id="S3.SS1.p4.1.m1.1.1.3" xref="S3.SS1.p4.1.m1.1.1.3.cmml">5</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.1.m1.1b"><apply id="S3.SS1.p4.1.m1.1.1.cmml" xref="S3.SS1.p4.1.m1.1.1"><csymbol cd="latexml" id="S3.SS1.p4.1.m1.1.1.1.cmml" xref="S3.SS1.p4.1.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S3.SS1.p4.1.m1.1.1.2.cmml" xref="S3.SS1.p4.1.m1.1.1.2">absent</csymbol><cn id="S3.SS1.p4.1.m1.1.1.3.cmml" type="integer" xref="S3.SS1.p4.1.m1.1.1.3">5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.1.m1.1c">\sim 5\;</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.1.m1.1d">∼ 5</annotation></semantics></math>yr observing period for cleaning to be effective and hence this represents an irreducible noise <cite class="ltx_cite ltx_citemacro_citep">(Cornish & Robson, <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib22" title="">2017</a>)</cite>. Frequencies of <math alttext="\sim 10^{-4}~{}" class="ltx_Math" display="inline" id="S3.SS1.p4.2.m2.1"><semantics id="S3.SS1.p4.2.m2.1a"><mrow id="S3.SS1.p4.2.m2.1.1" xref="S3.SS1.p4.2.m2.1.1.cmml"><mi id="S3.SS1.p4.2.m2.1.1.2" xref="S3.SS1.p4.2.m2.1.1.2.cmml"></mi><mo id="S3.SS1.p4.2.m2.1.1.1" xref="S3.SS1.p4.2.m2.1.1.1.cmml">∼</mo><msup id="S3.SS1.p4.2.m2.1.1.3" xref="S3.SS1.p4.2.m2.1.1.3.cmml"><mn id="S3.SS1.p4.2.m2.1.1.3.2" xref="S3.SS1.p4.2.m2.1.1.3.2.cmml">10</mn><mrow id="S3.SS1.p4.2.m2.1.1.3.3" xref="S3.SS1.p4.2.m2.1.1.3.3.cmml"><mo id="S3.SS1.p4.2.m2.1.1.3.3a" xref="S3.SS1.p4.2.m2.1.1.3.3.cmml">−</mo><mn id="S3.SS1.p4.2.m2.1.1.3.3.2" xref="S3.SS1.p4.2.m2.1.1.3.3.2.cmml">4</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.2.m2.1b"><apply id="S3.SS1.p4.2.m2.1.1.cmml" xref="S3.SS1.p4.2.m2.1.1"><csymbol cd="latexml" id="S3.SS1.p4.2.m2.1.1.1.cmml" xref="S3.SS1.p4.2.m2.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S3.SS1.p4.2.m2.1.1.2.cmml" xref="S3.SS1.p4.2.m2.1.1.2">absent</csymbol><apply id="S3.SS1.p4.2.m2.1.1.3.cmml" xref="S3.SS1.p4.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p4.2.m2.1.1.3.1.cmml" xref="S3.SS1.p4.2.m2.1.1.3">superscript</csymbol><cn id="S3.SS1.p4.2.m2.1.1.3.2.cmml" type="integer" xref="S3.SS1.p4.2.m2.1.1.3.2">10</cn><apply id="S3.SS1.p4.2.m2.1.1.3.3.cmml" xref="S3.SS1.p4.2.m2.1.1.3.3"><minus id="S3.SS1.p4.2.m2.1.1.3.3.1.cmml" xref="S3.SS1.p4.2.m2.1.1.3.3"></minus><cn id="S3.SS1.p4.2.m2.1.1.3.3.2.cmml" type="integer" xref="S3.SS1.p4.2.m2.1.1.3.3.2">4</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.2.m2.1c">\sim 10^{-4}~{}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.2.m2.1d">∼ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT</annotation></semantics></math>Hz correspond to orbital periods for which gravitational waves in stellar mass systems are not able to drive coalescence in <math alttext="\sim 1\;" class="ltx_Math" display="inline" id="S3.SS1.p4.3.m3.1"><semantics id="S3.SS1.p4.3.m3.1a"><mrow id="S3.SS1.p4.3.m3.1.1" xref="S3.SS1.p4.3.m3.1.1.cmml"><mi id="S3.SS1.p4.3.m3.1.1.2" xref="S3.SS1.p4.3.m3.1.1.2.cmml"></mi><mo id="S3.SS1.p4.3.m3.1.1.1" xref="S3.SS1.p4.3.m3.1.1.1.cmml">∼</mo><mn id="S3.SS1.p4.3.m3.1.1.3" xref="S3.SS1.p4.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.3.m3.1b"><apply id="S3.SS1.p4.3.m3.1.1.cmml" xref="S3.SS1.p4.3.m3.1.1"><csymbol cd="latexml" id="S3.SS1.p4.3.m3.1.1.1.cmml" xref="S3.SS1.p4.3.m3.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S3.SS1.p4.3.m3.1.1.2.cmml" xref="S3.SS1.p4.3.m3.1.1.2">absent</csymbol><cn id="S3.SS1.p4.3.m3.1.1.3.cmml" type="integer" xref="S3.SS1.p4.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.3.m3.1c">\sim 1\;</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.3.m3.1d">∼ 1</annotation></semantics></math>Gyr. Since this is approaching the maximum age of stellar systems, at lower frequencies (where the coalescence time is even longer) the spectrum of these gravitational waves will be shaped by the distribution of stellar binary properties and not just gravitational radiation as these curves assume.</p> </div> <div class="ltx_para" id="S3.SS1.p5"> <p class="ltx_p" id="S3.SS1.p5.4">Below <math alttext="10^{-4}" class="ltx_Math" display="inline" id="S3.SS1.p5.1.m1.1"><semantics id="S3.SS1.p5.1.m1.1a"><msup id="S3.SS1.p5.1.m1.1.1" xref="S3.SS1.p5.1.m1.1.1.cmml"><mn id="S3.SS1.p5.1.m1.1.1.2" xref="S3.SS1.p5.1.m1.1.1.2.cmml">10</mn><mrow id="S3.SS1.p5.1.m1.1.1.3" xref="S3.SS1.p5.1.m1.1.1.3.cmml"><mo id="S3.SS1.p5.1.m1.1.1.3a" xref="S3.SS1.p5.1.m1.1.1.3.cmml">−</mo><mn id="S3.SS1.p5.1.m1.1.1.3.2" xref="S3.SS1.p5.1.m1.1.1.3.2.cmml">4</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.1.m1.1b"><apply id="S3.SS1.p5.1.m1.1.1.cmml" xref="S3.SS1.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p5.1.m1.1.1.1.cmml" xref="S3.SS1.p5.1.m1.1.1">superscript</csymbol><cn id="S3.SS1.p5.1.m1.1.1.2.cmml" type="integer" xref="S3.SS1.p5.1.m1.1.1.2">10</cn><apply id="S3.SS1.p5.1.m1.1.1.3.cmml" xref="S3.SS1.p5.1.m1.1.1.3"><minus id="S3.SS1.p5.1.m1.1.1.3.1.cmml" xref="S3.SS1.p5.1.m1.1.1.3"></minus><cn id="S3.SS1.p5.1.m1.1.1.3.2.cmml" type="integer" xref="S3.SS1.p5.1.m1.1.1.3.2">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.1.m1.1c">10^{-4}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.1.m1.1d">10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT</annotation></semantics></math> the unresolvable massive black hole binary stochastic gravitational wave background (MBHB GWB) at the centers of galaxies likely exceeds the background from stellar binaries. The ‘most likely’ estimate for the MBHB GWB from <cite class="ltx_cite ltx_citemacro_citet">Rosado (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib61" title="">2011</a>)</cite> is shown by the darker grey solid – and we find the total background in this estimate is in agreement with pulsar timing array observations at <math alttext="f=3\times 10^{-8}" class="ltx_Math" display="inline" id="S3.SS1.p5.2.m2.1"><semantics id="S3.SS1.p5.2.m2.1a"><mrow id="S3.SS1.p5.2.m2.1.1" xref="S3.SS1.p5.2.m2.1.1.cmml"><mi id="S3.SS1.p5.2.m2.1.1.2" xref="S3.SS1.p5.2.m2.1.1.2.cmml">f</mi><mo id="S3.SS1.p5.2.m2.1.1.1" xref="S3.SS1.p5.2.m2.1.1.1.cmml">=</mo><mrow id="S3.SS1.p5.2.m2.1.1.3" xref="S3.SS1.p5.2.m2.1.1.3.cmml"><mn id="S3.SS1.p5.2.m2.1.1.3.2" xref="S3.SS1.p5.2.m2.1.1.3.2.cmml">3</mn><mo id="S3.SS1.p5.2.m2.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S3.SS1.p5.2.m2.1.1.3.1.cmml">×</mo><msup id="S3.SS1.p5.2.m2.1.1.3.3" xref="S3.SS1.p5.2.m2.1.1.3.3.cmml"><mn id="S3.SS1.p5.2.m2.1.1.3.3.2" xref="S3.SS1.p5.2.m2.1.1.3.3.2.cmml">10</mn><mrow id="S3.SS1.p5.2.m2.1.1.3.3.3" xref="S3.SS1.p5.2.m2.1.1.3.3.3.cmml"><mo id="S3.SS1.p5.2.m2.1.1.3.3.3a" xref="S3.SS1.p5.2.m2.1.1.3.3.3.cmml">−</mo><mn id="S3.SS1.p5.2.m2.1.1.3.3.3.2" xref="S3.SS1.p5.2.m2.1.1.3.3.3.2.cmml">8</mn></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.2.m2.1b"><apply id="S3.SS1.p5.2.m2.1.1.cmml" xref="S3.SS1.p5.2.m2.1.1"><eq id="S3.SS1.p5.2.m2.1.1.1.cmml" xref="S3.SS1.p5.2.m2.1.1.1"></eq><ci id="S3.SS1.p5.2.m2.1.1.2.cmml" xref="S3.SS1.p5.2.m2.1.1.2">𝑓</ci><apply id="S3.SS1.p5.2.m2.1.1.3.cmml" xref="S3.SS1.p5.2.m2.1.1.3"><times id="S3.SS1.p5.2.m2.1.1.3.1.cmml" xref="S3.SS1.p5.2.m2.1.1.3.1"></times><cn id="S3.SS1.p5.2.m2.1.1.3.2.cmml" type="integer" xref="S3.SS1.p5.2.m2.1.1.3.2">3</cn><apply id="S3.SS1.p5.2.m2.1.1.3.3.cmml" xref="S3.SS1.p5.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS1.p5.2.m2.1.1.3.3.1.cmml" xref="S3.SS1.p5.2.m2.1.1.3.3">superscript</csymbol><cn id="S3.SS1.p5.2.m2.1.1.3.3.2.cmml" type="integer" xref="S3.SS1.p5.2.m2.1.1.3.3.2">10</cn><apply id="S3.SS1.p5.2.m2.1.1.3.3.3.cmml" xref="S3.SS1.p5.2.m2.1.1.3.3.3"><minus id="S3.SS1.p5.2.m2.1.1.3.3.3.1.cmml" xref="S3.SS1.p5.2.m2.1.1.3.3.3"></minus><cn id="S3.SS1.p5.2.m2.1.1.3.3.3.2.cmml" type="integer" xref="S3.SS1.p5.2.m2.1.1.3.3.3.2">8</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.2.m2.1c">f=3\times 10^{-8}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.2.m2.1d">italic_f = 3 × 10 start_POSTSUPERSCRIPT - 8 end_POSTSUPERSCRIPT</annotation></semantics></math>Hz <cite class="ltx_cite ltx_citemacro_citep">(e.g. Reardon et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib59" title="">2023</a>)</cite>. <cite class="ltx_cite ltx_citemacro_citet">Rosado (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib61" title="">2011</a>)</cite> also provided ‘maximum’ and ‘minimum’ bounds on the MBHB GWB, with each bound shifting the amplitude of <math alttext="\sqrt{S_{h}}" class="ltx_Math" display="inline" id="S3.SS1.p5.3.m3.1"><semantics id="S3.SS1.p5.3.m3.1a"><msqrt id="S3.SS1.p5.3.m3.1.1" xref="S3.SS1.p5.3.m3.1.1.cmml"><msub id="S3.SS1.p5.3.m3.1.1.2" xref="S3.SS1.p5.3.m3.1.1.2.cmml"><mi id="S3.SS1.p5.3.m3.1.1.2.2" xref="S3.SS1.p5.3.m3.1.1.2.2.cmml">S</mi><mi id="S3.SS1.p5.3.m3.1.1.2.3" xref="S3.SS1.p5.3.m3.1.1.2.3.cmml">h</mi></msub></msqrt><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.3.m3.1b"><apply id="S3.SS1.p5.3.m3.1.1.cmml" xref="S3.SS1.p5.3.m3.1.1"><root id="S3.SS1.p5.3.m3.1.1a.cmml" xref="S3.SS1.p5.3.m3.1.1"></root><apply id="S3.SS1.p5.3.m3.1.1.2.cmml" xref="S3.SS1.p5.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p5.3.m3.1.1.2.1.cmml" xref="S3.SS1.p5.3.m3.1.1.2">subscript</csymbol><ci id="S3.SS1.p5.3.m3.1.1.2.2.cmml" xref="S3.SS1.p5.3.m3.1.1.2.2">𝑆</ci><ci id="S3.SS1.p5.3.m3.1.1.2.3.cmml" xref="S3.SS1.p5.3.m3.1.1.2.3">ℎ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.3.m3.1c">\sqrt{S_{h}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.3.m3.1d">square-root start_ARG italic_S start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT end_ARG</annotation></semantics></math> by a factor of <math alttext="\sim 5" class="ltx_Math" display="inline" id="S3.SS1.p5.4.m4.1"><semantics id="S3.SS1.p5.4.m4.1a"><mrow id="S3.SS1.p5.4.m4.1.1" xref="S3.SS1.p5.4.m4.1.1.cmml"><mi id="S3.SS1.p5.4.m4.1.1.2" xref="S3.SS1.p5.4.m4.1.1.2.cmml"></mi><mo id="S3.SS1.p5.4.m4.1.1.1" xref="S3.SS1.p5.4.m4.1.1.1.cmml">∼</mo><mn id="S3.SS1.p5.4.m4.1.1.3" xref="S3.SS1.p5.4.m4.1.1.3.cmml">5</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.4.m4.1b"><apply id="S3.SS1.p5.4.m4.1.1.cmml" xref="S3.SS1.p5.4.m4.1.1"><csymbol cd="latexml" id="S3.SS1.p5.4.m4.1.1.1.cmml" xref="S3.SS1.p5.4.m4.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S3.SS1.p5.4.m4.1.1.2.cmml" xref="S3.SS1.p5.4.m4.1.1.2">absent</csymbol><cn id="S3.SS1.p5.4.m4.1.1.3.cmml" type="integer" xref="S3.SS1.p5.4.m4.1.1.3">5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.4.m4.1c">\sim 5</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.4.m4.1d">∼ 5</annotation></semantics></math> relative to the ‘most likely’ model.</p> </div> <div class="ltx_para" id="S3.SS1.p6"> <p class="ltx_p" id="S3.SS1.p6.8">We can also forecast the number of astrophysical sources to which this concept would be sensitive, using the fact that the strain sensitivities we are forecasting at <math alttext="f\lesssim 0.5\times 10^{-4}" class="ltx_Math" display="inline" id="S3.SS1.p6.1.m1.1"><semantics id="S3.SS1.p6.1.m1.1a"><mrow id="S3.SS1.p6.1.m1.1.1" xref="S3.SS1.p6.1.m1.1.1.cmml"><mi id="S3.SS1.p6.1.m1.1.1.2" xref="S3.SS1.p6.1.m1.1.1.2.cmml">f</mi><mo id="S3.SS1.p6.1.m1.1.1.1" xref="S3.SS1.p6.1.m1.1.1.1.cmml">≲</mo><mrow id="S3.SS1.p6.1.m1.1.1.3" xref="S3.SS1.p6.1.m1.1.1.3.cmml"><mn id="S3.SS1.p6.1.m1.1.1.3.2" xref="S3.SS1.p6.1.m1.1.1.3.2.cmml">0.5</mn><mo id="S3.SS1.p6.1.m1.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S3.SS1.p6.1.m1.1.1.3.1.cmml">×</mo><msup id="S3.SS1.p6.1.m1.1.1.3.3" xref="S3.SS1.p6.1.m1.1.1.3.3.cmml"><mn id="S3.SS1.p6.1.m1.1.1.3.3.2" xref="S3.SS1.p6.1.m1.1.1.3.3.2.cmml">10</mn><mrow id="S3.SS1.p6.1.m1.1.1.3.3.3" xref="S3.SS1.p6.1.m1.1.1.3.3.3.cmml"><mo id="S3.SS1.p6.1.m1.1.1.3.3.3a" xref="S3.SS1.p6.1.m1.1.1.3.3.3.cmml">−</mo><mn id="S3.SS1.p6.1.m1.1.1.3.3.3.2" xref="S3.SS1.p6.1.m1.1.1.3.3.3.2.cmml">4</mn></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p6.1.m1.1b"><apply id="S3.SS1.p6.1.m1.1.1.cmml" xref="S3.SS1.p6.1.m1.1.1"><csymbol cd="latexml" id="S3.SS1.p6.1.m1.1.1.1.cmml" xref="S3.SS1.p6.1.m1.1.1.1">less-than-or-similar-to</csymbol><ci id="S3.SS1.p6.1.m1.1.1.2.cmml" xref="S3.SS1.p6.1.m1.1.1.2">𝑓</ci><apply id="S3.SS1.p6.1.m1.1.1.3.cmml" xref="S3.SS1.p6.1.m1.1.1.3"><times id="S3.SS1.p6.1.m1.1.1.3.1.cmml" xref="S3.SS1.p6.1.m1.1.1.3.1"></times><cn id="S3.SS1.p6.1.m1.1.1.3.2.cmml" type="float" xref="S3.SS1.p6.1.m1.1.1.3.2">0.5</cn><apply id="S3.SS1.p6.1.m1.1.1.3.3.cmml" xref="S3.SS1.p6.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS1.p6.1.m1.1.1.3.3.1.cmml" xref="S3.SS1.p6.1.m1.1.1.3.3">superscript</csymbol><cn id="S3.SS1.p6.1.m1.1.1.3.3.2.cmml" type="integer" xref="S3.SS1.p6.1.m1.1.1.3.3.2">10</cn><apply id="S3.SS1.p6.1.m1.1.1.3.3.3.cmml" xref="S3.SS1.p6.1.m1.1.1.3.3.3"><minus id="S3.SS1.p6.1.m1.1.1.3.3.3.1.cmml" xref="S3.SS1.p6.1.m1.1.1.3.3.3"></minus><cn id="S3.SS1.p6.1.m1.1.1.3.3.3.2.cmml" type="integer" xref="S3.SS1.p6.1.m1.1.1.3.3.3.2">4</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p6.1.m1.1c">f\lesssim 0.5\times 10^{-4}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p6.1.m1.1d">italic_f ≲ 0.5 × 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT</annotation></semantics></math>Hz are somewhat similar to the <math alttext="\mu" class="ltx_Math" display="inline" id="S3.SS1.p6.2.m2.1"><semantics id="S3.SS1.p6.2.m2.1a"><mi id="S3.SS1.p6.2.m2.1.1" xref="S3.SS1.p6.2.m2.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p6.2.m2.1b"><ci id="S3.SS1.p6.2.m2.1.1.cmml" xref="S3.SS1.p6.2.m2.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p6.2.m2.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p6.2.m2.1d">italic_μ</annotation></semantics></math>-Ares concept <cite class="ltx_cite ltx_citemacro_citep">(Sesana et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib62" title="">2021</a>)</cite>, particularly for our <math alttext="r=L=30~{}" class="ltx_Math" display="inline" id="S3.SS1.p6.3.m3.1"><semantics id="S3.SS1.p6.3.m3.1a"><mrow id="S3.SS1.p6.3.m3.1.1" xref="S3.SS1.p6.3.m3.1.1.cmml"><mi id="S3.SS1.p6.3.m3.1.1.2" xref="S3.SS1.p6.3.m3.1.1.2.cmml">r</mi><mo id="S3.SS1.p6.3.m3.1.1.3" xref="S3.SS1.p6.3.m3.1.1.3.cmml">=</mo><mi id="S3.SS1.p6.3.m3.1.1.4" xref="S3.SS1.p6.3.m3.1.1.4.cmml">L</mi><mo id="S3.SS1.p6.3.m3.1.1.5" xref="S3.SS1.p6.3.m3.1.1.5.cmml">=</mo><mn id="S3.SS1.p6.3.m3.1.1.6" xref="S3.SS1.p6.3.m3.1.1.6.cmml">30</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p6.3.m3.1b"><apply id="S3.SS1.p6.3.m3.1.1.cmml" xref="S3.SS1.p6.3.m3.1.1"><and id="S3.SS1.p6.3.m3.1.1a.cmml" xref="S3.SS1.p6.3.m3.1.1"></and><apply id="S3.SS1.p6.3.m3.1.1b.cmml" xref="S3.SS1.p6.3.m3.1.1"><eq id="S3.SS1.p6.3.m3.1.1.3.cmml" xref="S3.SS1.p6.3.m3.1.1.3"></eq><ci id="S3.SS1.p6.3.m3.1.1.2.cmml" xref="S3.SS1.p6.3.m3.1.1.2">𝑟</ci><ci id="S3.SS1.p6.3.m3.1.1.4.cmml" xref="S3.SS1.p6.3.m3.1.1.4">𝐿</ci></apply><apply id="S3.SS1.p6.3.m3.1.1c.cmml" xref="S3.SS1.p6.3.m3.1.1"><eq id="S3.SS1.p6.3.m3.1.1.5.cmml" xref="S3.SS1.p6.3.m3.1.1.5"></eq><share href="https://arxiv.org/html/2411.15072v1#S3.SS1.p6.3.m3.1.1.4.cmml" id="S3.SS1.p6.3.m3.1.1d.cmml" xref="S3.SS1.p6.3.m3.1.1"></share><cn id="S3.SS1.p6.3.m3.1.1.6.cmml" type="integer" xref="S3.SS1.p6.3.m3.1.1.6">30</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p6.3.m3.1c">r=L=30~{}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p6.3.m3.1d">italic_r = italic_L = 30</annotation></semantics></math>AU configuration. Over a ten year period, <math alttext="\mu" class="ltx_Math" display="inline" id="S3.SS1.p6.4.m4.1"><semantics id="S3.SS1.p6.4.m4.1a"><mi id="S3.SS1.p6.4.m4.1.1" xref="S3.SS1.p6.4.m4.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p6.4.m4.1b"><ci id="S3.SS1.p6.4.m4.1.1.cmml" xref="S3.SS1.p6.4.m4.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p6.4.m4.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p6.4.m4.1d">italic_μ</annotation></semantics></math>-Ares forecasts detecting <math alttext="{\cal O}(1000)" class="ltx_Math" display="inline" id="S3.SS1.p6.5.m5.1"><semantics id="S3.SS1.p6.5.m5.1a"><mrow id="S3.SS1.p6.5.m5.1.2" xref="S3.SS1.p6.5.m5.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p6.5.m5.1.2.2" xref="S3.SS1.p6.5.m5.1.2.2.cmml">𝒪</mi><mo id="S3.SS1.p6.5.m5.1.2.1" xref="S3.SS1.p6.5.m5.1.2.1.cmml">⁢</mo><mrow id="S3.SS1.p6.5.m5.1.2.3.2" xref="S3.SS1.p6.5.m5.1.2.cmml"><mo id="S3.SS1.p6.5.m5.1.2.3.2.1" stretchy="false" xref="S3.SS1.p6.5.m5.1.2.cmml">(</mo><mn id="S3.SS1.p6.5.m5.1.1" xref="S3.SS1.p6.5.m5.1.1.cmml">1000</mn><mo id="S3.SS1.p6.5.m5.1.2.3.2.2" stretchy="false" xref="S3.SS1.p6.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p6.5.m5.1b"><apply id="S3.SS1.p6.5.m5.1.2.cmml" xref="S3.SS1.p6.5.m5.1.2"><times id="S3.SS1.p6.5.m5.1.2.1.cmml" xref="S3.SS1.p6.5.m5.1.2.1"></times><ci id="S3.SS1.p6.5.m5.1.2.2.cmml" xref="S3.SS1.p6.5.m5.1.2.2">𝒪</ci><cn id="S3.SS1.p6.5.m5.1.1.cmml" type="integer" xref="S3.SS1.p6.5.m5.1.1">1000</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p6.5.m5.1c">{\cal O}(1000)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p6.5.m5.1d">caligraphic_O ( 1000 )</annotation></semantics></math> inspiralling supermassive black holes, finding <math alttext="{\cal O}(100)" class="ltx_Math" display="inline" id="S3.SS1.p6.6.m6.1"><semantics id="S3.SS1.p6.6.m6.1a"><mrow id="S3.SS1.p6.6.m6.1.2" xref="S3.SS1.p6.6.m6.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p6.6.m6.1.2.2" xref="S3.SS1.p6.6.m6.1.2.2.cmml">𝒪</mi><mo id="S3.SS1.p6.6.m6.1.2.1" xref="S3.SS1.p6.6.m6.1.2.1.cmml">⁢</mo><mrow id="S3.SS1.p6.6.m6.1.2.3.2" xref="S3.SS1.p6.6.m6.1.2.cmml"><mo id="S3.SS1.p6.6.m6.1.2.3.2.1" stretchy="false" xref="S3.SS1.p6.6.m6.1.2.cmml">(</mo><mn id="S3.SS1.p6.6.m6.1.1" xref="S3.SS1.p6.6.m6.1.1.cmml">100</mn><mo id="S3.SS1.p6.6.m6.1.2.3.2.2" stretchy="false" xref="S3.SS1.p6.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p6.6.m6.1b"><apply id="S3.SS1.p6.6.m6.1.2.cmml" xref="S3.SS1.p6.6.m6.1.2"><times id="S3.SS1.p6.6.m6.1.2.1.cmml" xref="S3.SS1.p6.6.m6.1.2.1"></times><ci id="S3.SS1.p6.6.m6.1.2.2.cmml" xref="S3.SS1.p6.6.m6.1.2.2">𝒪</ci><cn id="S3.SS1.p6.6.m6.1.1.cmml" type="integer" xref="S3.SS1.p6.6.m6.1.1">100</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p6.6.m6.1c">{\cal O}(100)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p6.6.m6.1d">caligraphic_O ( 100 )</annotation></semantics></math> black hole binaries in the Milky Way, and observing all stars and compact objects that would merge with Sag A<sup class="ltx_sup" id="S3.SS1.p6.8.1">∗</sup> in <math alttext="10^{6}-10^{8}" class="ltx_Math" display="inline" id="S3.SS1.p6.8.m8.1"><semantics id="S3.SS1.p6.8.m8.1a"><mrow id="S3.SS1.p6.8.m8.1.1" xref="S3.SS1.p6.8.m8.1.1.cmml"><msup id="S3.SS1.p6.8.m8.1.1.2" xref="S3.SS1.p6.8.m8.1.1.2.cmml"><mn id="S3.SS1.p6.8.m8.1.1.2.2" xref="S3.SS1.p6.8.m8.1.1.2.2.cmml">10</mn><mn id="S3.SS1.p6.8.m8.1.1.2.3" xref="S3.SS1.p6.8.m8.1.1.2.3.cmml">6</mn></msup><mo id="S3.SS1.p6.8.m8.1.1.1" xref="S3.SS1.p6.8.m8.1.1.1.cmml">−</mo><msup id="S3.SS1.p6.8.m8.1.1.3" xref="S3.SS1.p6.8.m8.1.1.3.cmml"><mn id="S3.SS1.p6.8.m8.1.1.3.2" xref="S3.SS1.p6.8.m8.1.1.3.2.cmml">10</mn><mn id="S3.SS1.p6.8.m8.1.1.3.3" xref="S3.SS1.p6.8.m8.1.1.3.3.cmml">8</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p6.8.m8.1b"><apply id="S3.SS1.p6.8.m8.1.1.cmml" xref="S3.SS1.p6.8.m8.1.1"><minus id="S3.SS1.p6.8.m8.1.1.1.cmml" xref="S3.SS1.p6.8.m8.1.1.1"></minus><apply id="S3.SS1.p6.8.m8.1.1.2.cmml" xref="S3.SS1.p6.8.m8.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p6.8.m8.1.1.2.1.cmml" xref="S3.SS1.p6.8.m8.1.1.2">superscript</csymbol><cn id="S3.SS1.p6.8.m8.1.1.2.2.cmml" type="integer" xref="S3.SS1.p6.8.m8.1.1.2.2">10</cn><cn id="S3.SS1.p6.8.m8.1.1.2.3.cmml" type="integer" xref="S3.SS1.p6.8.m8.1.1.2.3">6</cn></apply><apply id="S3.SS1.p6.8.m8.1.1.3.cmml" xref="S3.SS1.p6.8.m8.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p6.8.m8.1.1.3.1.cmml" xref="S3.SS1.p6.8.m8.1.1.3">superscript</csymbol><cn id="S3.SS1.p6.8.m8.1.1.3.2.cmml" type="integer" xref="S3.SS1.p6.8.m8.1.1.3.2">10</cn><cn id="S3.SS1.p6.8.m8.1.1.3.3.cmml" type="integer" xref="S3.SS1.p6.8.m8.1.1.3.3">8</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p6.8.m8.1c">10^{6}-10^{8}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p6.8.m8.1d">10 start_POSTSUPERSCRIPT 6 end_POSTSUPERSCRIPT - 10 start_POSTSUPERSCRIPT 8 end_POSTSUPERSCRIPT</annotation></semantics></math> year. Our most sensitive designs would be capable of similar returns.</p> </div> <div class="ltx_para" id="S3.SS1.p7"> <p class="ltx_p" id="S3.SS1.p7.3">Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S3.F4" title="Figure 4 ‣ 3.1 Two-arm time-delay interferometry forecasts ‣ 3 Strain power sensitivity ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">4</span></a> also shows curves for a more aggressive suppression factor of <math alttext="0.05" class="ltx_Math" display="inline" id="S3.SS1.p7.1.m1.1"><semantics id="S3.SS1.p7.1.m1.1a"><mn id="S3.SS1.p7.1.m1.1.1" xref="S3.SS1.p7.1.m1.1.1.cmml">0.05</mn><annotation-xml encoding="MathML-Content" id="S3.SS1.p7.1.m1.1b"><cn id="S3.SS1.p7.1.m1.1.1.cmml" type="float" xref="S3.SS1.p7.1.m1.1.1">0.05</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p7.1.m1.1c">0.05</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p7.1.m1.1d">0.05</annotation></semantics></math>. Such a suppression would likely require monitoring and then correcting the radiation from the Sun and the solar plasma with additional onboard instrumentation. In fact, the Lagrange concept, proposed as a potentially cheaper replacement to LISA by abandoning drag-free control, found that such instrumentation could potentially produce a factor of <math alttext="100" class="ltx_Math" display="inline" id="S3.SS1.p7.2.m2.1"><semantics id="S3.SS1.p7.2.m2.1a"><mn id="S3.SS1.p7.2.m2.1.1" xref="S3.SS1.p7.2.m2.1.1.cmml">100</mn><annotation-xml encoding="MathML-Content" id="S3.SS1.p7.2.m2.1b"><cn id="S3.SS1.p7.2.m2.1.1.cmml" type="integer" xref="S3.SS1.p7.2.m2.1.1">100</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p7.2.m2.1c">100</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p7.2.m2.1d">100</annotation></semantics></math> suppression beyond the factor of <math alttext="\sim 10" class="ltx_Math" display="inline" id="S3.SS1.p7.3.m3.1"><semantics id="S3.SS1.p7.3.m3.1a"><mrow id="S3.SS1.p7.3.m3.1.1" xref="S3.SS1.p7.3.m3.1.1.cmml"><mi id="S3.SS1.p7.3.m3.1.1.2" xref="S3.SS1.p7.3.m3.1.1.2.cmml"></mi><mo id="S3.SS1.p7.3.m3.1.1.1" xref="S3.SS1.p7.3.m3.1.1.1.cmml">∼</mo><mn id="S3.SS1.p7.3.m3.1.1.3" xref="S3.SS1.p7.3.m3.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p7.3.m3.1b"><apply id="S3.SS1.p7.3.m3.1.1.cmml" xref="S3.SS1.p7.3.m3.1.1"><csymbol cd="latexml" id="S3.SS1.p7.3.m3.1.1.1.cmml" xref="S3.SS1.p7.3.m3.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S3.SS1.p7.3.m3.1.1.2.cmml" xref="S3.SS1.p7.3.m3.1.1.2">absent</csymbol><cn id="S3.SS1.p7.3.m3.1.1.3.cmml" type="integer" xref="S3.SS1.p7.3.m3.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p7.3.m3.1c">\sim 10</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p7.3.m3.1d">∼ 10</annotation></semantics></math> they aimed to achieve geometrically <cite class="ltx_cite ltx_citemacro_citep">(McKenzie et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib49" title="">2011</a>)</cite>.</p> </div> <figure class="ltx_figure" id="S3.F4"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="332" id="S3.F4.g1" src="x4.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 4: </span>Gravitational wave sensitivity for the same time-delay interferometry architectures as shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S3.F3" title="Figure 3 ‣ 3 Strain power sensitivity ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">3</span></a> with <math alttext="r=L=10\," class="ltx_Math" display="inline" id="S3.F4.7.m1.1"><semantics id="S3.F4.7.m1.1b"><mrow id="S3.F4.7.m1.1.1" xref="S3.F4.7.m1.1.1.cmml"><mi id="S3.F4.7.m1.1.1.2" xref="S3.F4.7.m1.1.1.2.cmml">r</mi><mo id="S3.F4.7.m1.1.1.3" xref="S3.F4.7.m1.1.1.3.cmml">=</mo><mi id="S3.F4.7.m1.1.1.4" xref="S3.F4.7.m1.1.1.4.cmml">L</mi><mo id="S3.F4.7.m1.1.1.5" xref="S3.F4.7.m1.1.1.5.cmml">=</mo><mn id="S3.F4.7.m1.1.1.6" xref="S3.F4.7.m1.1.1.6.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.F4.7.m1.1c"><apply id="S3.F4.7.m1.1.1.cmml" xref="S3.F4.7.m1.1.1"><and id="S3.F4.7.m1.1.1a.cmml" xref="S3.F4.7.m1.1.1"></and><apply id="S3.F4.7.m1.1.1b.cmml" xref="S3.F4.7.m1.1.1"><eq id="S3.F4.7.m1.1.1.3.cmml" xref="S3.F4.7.m1.1.1.3"></eq><ci id="S3.F4.7.m1.1.1.2.cmml" xref="S3.F4.7.m1.1.1.2">𝑟</ci><ci id="S3.F4.7.m1.1.1.4.cmml" xref="S3.F4.7.m1.1.1.4">𝐿</ci></apply><apply id="S3.F4.7.m1.1.1c.cmml" xref="S3.F4.7.m1.1.1"><eq id="S3.F4.7.m1.1.1.5.cmml" xref="S3.F4.7.m1.1.1.5"></eq><share href="https://arxiv.org/html/2411.15072v1#S3.F4.7.m1.1.1.4.cmml" id="S3.F4.7.m1.1.1d.cmml" xref="S3.F4.7.m1.1.1"></share><cn id="S3.F4.7.m1.1.1.6.cmml" type="integer" xref="S3.F4.7.m1.1.1.6">10</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.7.m1.1d">r=L=10\,</annotation><annotation encoding="application/x-llamapun" id="S3.F4.7.m1.1e">italic_r = italic_L = 10</annotation></semantics></math>AU (left panel) and <math alttext="r=L=30\," class="ltx_Math" display="inline" id="S3.F4.8.m2.1"><semantics id="S3.F4.8.m2.1b"><mrow id="S3.F4.8.m2.1.1" xref="S3.F4.8.m2.1.1.cmml"><mi id="S3.F4.8.m2.1.1.2" xref="S3.F4.8.m2.1.1.2.cmml">r</mi><mo id="S3.F4.8.m2.1.1.3" xref="S3.F4.8.m2.1.1.3.cmml">=</mo><mi id="S3.F4.8.m2.1.1.4" xref="S3.F4.8.m2.1.1.4.cmml">L</mi><mo id="S3.F4.8.m2.1.1.5" xref="S3.F4.8.m2.1.1.5.cmml">=</mo><mn id="S3.F4.8.m2.1.1.6" xref="S3.F4.8.m2.1.1.6.cmml">30</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.F4.8.m2.1c"><apply id="S3.F4.8.m2.1.1.cmml" xref="S3.F4.8.m2.1.1"><and id="S3.F4.8.m2.1.1a.cmml" xref="S3.F4.8.m2.1.1"></and><apply id="S3.F4.8.m2.1.1b.cmml" xref="S3.F4.8.m2.1.1"><eq id="S3.F4.8.m2.1.1.3.cmml" xref="S3.F4.8.m2.1.1.3"></eq><ci id="S3.F4.8.m2.1.1.2.cmml" xref="S3.F4.8.m2.1.1.2">𝑟</ci><ci id="S3.F4.8.m2.1.1.4.cmml" xref="S3.F4.8.m2.1.1.4">𝐿</ci></apply><apply id="S3.F4.8.m2.1.1c.cmml" xref="S3.F4.8.m2.1.1"><eq id="S3.F4.8.m2.1.1.5.cmml" xref="S3.F4.8.m2.1.1.5"></eq><share href="https://arxiv.org/html/2411.15072v1#S3.F4.8.m2.1.1.4.cmml" id="S3.F4.8.m2.1.1d.cmml" xref="S3.F4.8.m2.1.1"></share><cn id="S3.F4.8.m2.1.1.6.cmml" type="integer" xref="S3.F4.8.m2.1.1.6">30</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.8.m2.1d">r=L=30\,</annotation><annotation encoding="application/x-llamapun" id="S3.F4.8.m2.1e">italic_r = italic_L = 30</annotation></semantics></math>AU (right panel) and a geometric suppression of <math alttext="0.5" class="ltx_Math" display="inline" id="S3.F4.9.m3.1"><semantics id="S3.F4.9.m3.1b"><mn id="S3.F4.9.m3.1.1" xref="S3.F4.9.m3.1.1.cmml">0.5</mn><annotation-xml encoding="MathML-Content" id="S3.F4.9.m3.1c"><cn id="S3.F4.9.m3.1.1.cmml" type="float" xref="S3.F4.9.m3.1.1">0.5</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.9.m3.1d">0.5</annotation><annotation encoding="application/x-llamapun" id="S3.F4.9.m3.1e">0.5</annotation></semantics></math>, plus the same cases but assuming a more aggressive <math alttext="0.05" class="ltx_Math" display="inline" id="S3.F4.10.m4.1"><semantics id="S3.F4.10.m4.1b"><mn id="S3.F4.10.m4.1.1" xref="S3.F4.10.m4.1.1.cmml">0.05</mn><annotation-xml encoding="MathML-Content" id="S3.F4.10.m4.1c"><cn id="S3.F4.10.m4.1.1.cmml" type="float" xref="S3.F4.10.m4.1.1">0.05</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.10.m4.1d">0.05</annotation><annotation encoding="application/x-llamapun" id="S3.F4.10.m4.1e">0.05</annotation></semantics></math> suppression factor. Also shown are different astrophysical sources: estimates for the galactic white dwarf binary stochastic gravitational wave background (WDB GWB; <cite class="ltx_cite ltx_citemacro_citet">Cornish & Robson <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib22" title="">2017</a></cite>), the massive black hole binary stochastic gravitational wave background (MBHB GWB; <cite class="ltx_cite ltx_citemacro_citet">Rosado <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib61" title="">2011</a></cite>), and the characteristic strain within <math alttext="5~{}" class="ltx_Math" display="inline" id="S3.F4.11.m5.1"><semantics id="S3.F4.11.m5.1b"><mn id="S3.F4.11.m5.1.1" xref="S3.F4.11.m5.1.1.cmml">5</mn><annotation-xml encoding="MathML-Content" id="S3.F4.11.m5.1c"><cn id="S3.F4.11.m5.1.1.cmml" type="integer" xref="S3.F4.11.m5.1.1">5</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.11.m5.1d">5~{}</annotation><annotation encoding="application/x-llamapun" id="S3.F4.11.m5.1e">5</annotation></semantics></math>yr of coalescence from equal-mass black hole mergers at <math alttext="z=3" class="ltx_Math" display="inline" id="S3.F4.12.m6.1"><semantics id="S3.F4.12.m6.1b"><mrow id="S3.F4.12.m6.1.1" xref="S3.F4.12.m6.1.1.cmml"><mi id="S3.F4.12.m6.1.1.2" xref="S3.F4.12.m6.1.1.2.cmml">z</mi><mo id="S3.F4.12.m6.1.1.1" xref="S3.F4.12.m6.1.1.1.cmml">=</mo><mn id="S3.F4.12.m6.1.1.3" xref="S3.F4.12.m6.1.1.3.cmml">3</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.F4.12.m6.1c"><apply id="S3.F4.12.m6.1.1.cmml" xref="S3.F4.12.m6.1.1"><eq id="S3.F4.12.m6.1.1.1.cmml" xref="S3.F4.12.m6.1.1.1"></eq><ci id="S3.F4.12.m6.1.1.2.cmml" xref="S3.F4.12.m6.1.1.2">𝑧</ci><cn id="S3.F4.12.m6.1.1.3.cmml" type="integer" xref="S3.F4.12.m6.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.12.m6.1d">z=3</annotation><annotation encoding="application/x-llamapun" id="S3.F4.12.m6.1e">italic_z = 3</annotation></semantics></math>, where the pre-merger black hole masses are annotated.</figcaption> </figure> <figure class="ltx_figure" id="S3.F5"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="332" id="S3.F5.g1" src="x5.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 5: </span>Gravitational wave strain sensitivities for the <span class="ltx_text ltx_font_bold" id="S3.F5.18.1">second design architecture: the single-arm atomic clock concept</span> (similar to Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S3.F4" title="Figure 4 ‣ 3.1 Two-arm time-delay interferometry forecasts ‣ 3 Strain power sensitivity ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">4</span></a>). Note one time-delay interferometry curve is included for comparison. The curves consider the <math alttext="\lambda=1\," class="ltx_Math" display="inline" id="S3.F5.9.m1.1"><semantics id="S3.F5.9.m1.1b"><mrow id="S3.F5.9.m1.1.1" xref="S3.F5.9.m1.1.1.cmml"><mi id="S3.F5.9.m1.1.1.2" xref="S3.F5.9.m1.1.1.2.cmml">λ</mi><mo id="S3.F5.9.m1.1.1.1" xref="S3.F5.9.m1.1.1.1.cmml">=</mo><mn id="S3.F5.9.m1.1.1.3" xref="S3.F5.9.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.F5.9.m1.1c"><apply id="S3.F5.9.m1.1.1.cmml" xref="S3.F5.9.m1.1.1"><eq id="S3.F5.9.m1.1.1.1.cmml" xref="S3.F5.9.m1.1.1.1"></eq><ci id="S3.F5.9.m1.1.1.2.cmml" xref="S3.F5.9.m1.1.1.2">𝜆</ci><cn id="S3.F5.9.m1.1.1.3.cmml" type="integer" xref="S3.F5.9.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.9.m1.1d">\lambda=1\,</annotation><annotation encoding="application/x-llamapun" id="S3.F5.9.m1.1e">italic_λ = 1</annotation></semantics></math>cm radio case, although the sensitivity would not be different for the laser case to the extent that clock noise limits the sensitivity. Allan deviations for <math alttext="\tau=1\," class="ltx_Math" display="inline" id="S3.F5.10.m2.1"><semantics id="S3.F5.10.m2.1b"><mrow id="S3.F5.10.m2.1.1" xref="S3.F5.10.m2.1.1.cmml"><mi id="S3.F5.10.m2.1.1.2" xref="S3.F5.10.m2.1.1.2.cmml">τ</mi><mo id="S3.F5.10.m2.1.1.1" xref="S3.F5.10.m2.1.1.1.cmml">=</mo><mn id="S3.F5.10.m2.1.1.3" xref="S3.F5.10.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.F5.10.m2.1c"><apply id="S3.F5.10.m2.1.1.cmml" xref="S3.F5.10.m2.1.1"><eq id="S3.F5.10.m2.1.1.1.cmml" xref="S3.F5.10.m2.1.1.1"></eq><ci id="S3.F5.10.m2.1.1.2.cmml" xref="S3.F5.10.m2.1.1.2">𝜏</ci><cn id="S3.F5.10.m2.1.1.3.cmml" type="integer" xref="S3.F5.10.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.10.m2.1d">\tau=1\,</annotation><annotation encoding="application/x-llamapun" id="S3.F5.10.m2.1e">italic_τ = 1</annotation></semantics></math>s of <math alttext="\sigma_{y}=10^{-13}" class="ltx_Math" display="inline" id="S3.F5.11.m3.1"><semantics id="S3.F5.11.m3.1b"><mrow id="S3.F5.11.m3.1.1" xref="S3.F5.11.m3.1.1.cmml"><msub id="S3.F5.11.m3.1.1.2" xref="S3.F5.11.m3.1.1.2.cmml"><mi id="S3.F5.11.m3.1.1.2.2" xref="S3.F5.11.m3.1.1.2.2.cmml">σ</mi><mi id="S3.F5.11.m3.1.1.2.3" xref="S3.F5.11.m3.1.1.2.3.cmml">y</mi></msub><mo id="S3.F5.11.m3.1.1.1" xref="S3.F5.11.m3.1.1.1.cmml">=</mo><msup id="S3.F5.11.m3.1.1.3" xref="S3.F5.11.m3.1.1.3.cmml"><mn id="S3.F5.11.m3.1.1.3.2" xref="S3.F5.11.m3.1.1.3.2.cmml">10</mn><mrow id="S3.F5.11.m3.1.1.3.3" xref="S3.F5.11.m3.1.1.3.3.cmml"><mo id="S3.F5.11.m3.1.1.3.3b" xref="S3.F5.11.m3.1.1.3.3.cmml">−</mo><mn id="S3.F5.11.m3.1.1.3.3.2" xref="S3.F5.11.m3.1.1.3.3.2.cmml">13</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.F5.11.m3.1c"><apply id="S3.F5.11.m3.1.1.cmml" xref="S3.F5.11.m3.1.1"><eq id="S3.F5.11.m3.1.1.1.cmml" xref="S3.F5.11.m3.1.1.1"></eq><apply id="S3.F5.11.m3.1.1.2.cmml" xref="S3.F5.11.m3.1.1.2"><csymbol cd="ambiguous" id="S3.F5.11.m3.1.1.2.1.cmml" xref="S3.F5.11.m3.1.1.2">subscript</csymbol><ci id="S3.F5.11.m3.1.1.2.2.cmml" xref="S3.F5.11.m3.1.1.2.2">𝜎</ci><ci id="S3.F5.11.m3.1.1.2.3.cmml" xref="S3.F5.11.m3.1.1.2.3">𝑦</ci></apply><apply id="S3.F5.11.m3.1.1.3.cmml" xref="S3.F5.11.m3.1.1.3"><csymbol cd="ambiguous" id="S3.F5.11.m3.1.1.3.1.cmml" xref="S3.F5.11.m3.1.1.3">superscript</csymbol><cn id="S3.F5.11.m3.1.1.3.2.cmml" type="integer" xref="S3.F5.11.m3.1.1.3.2">10</cn><apply id="S3.F5.11.m3.1.1.3.3.cmml" xref="S3.F5.11.m3.1.1.3.3"><minus id="S3.F5.11.m3.1.1.3.3.1.cmml" xref="S3.F5.11.m3.1.1.3.3"></minus><cn id="S3.F5.11.m3.1.1.3.3.2.cmml" type="integer" xref="S3.F5.11.m3.1.1.3.3.2">13</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.11.m3.1d">\sigma_{y}=10^{-13}</annotation><annotation encoding="application/x-llamapun" id="S3.F5.11.m3.1e">italic_σ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT = 10 start_POSTSUPERSCRIPT - 13 end_POSTSUPERSCRIPT</annotation></semantics></math> are similar to the Deep Space Atomic Clock, whereas the most precise atomic clocks on Earth achieve <math alttext="\sigma_{y}\sim 10^{-19}" class="ltx_Math" display="inline" id="S3.F5.12.m4.1"><semantics id="S3.F5.12.m4.1b"><mrow id="S3.F5.12.m4.1.1" xref="S3.F5.12.m4.1.1.cmml"><msub id="S3.F5.12.m4.1.1.2" xref="S3.F5.12.m4.1.1.2.cmml"><mi id="S3.F5.12.m4.1.1.2.2" xref="S3.F5.12.m4.1.1.2.2.cmml">σ</mi><mi id="S3.F5.12.m4.1.1.2.3" xref="S3.F5.12.m4.1.1.2.3.cmml">y</mi></msub><mo id="S3.F5.12.m4.1.1.1" xref="S3.F5.12.m4.1.1.1.cmml">∼</mo><msup id="S3.F5.12.m4.1.1.3" xref="S3.F5.12.m4.1.1.3.cmml"><mn id="S3.F5.12.m4.1.1.3.2" xref="S3.F5.12.m4.1.1.3.2.cmml">10</mn><mrow id="S3.F5.12.m4.1.1.3.3" xref="S3.F5.12.m4.1.1.3.3.cmml"><mo id="S3.F5.12.m4.1.1.3.3b" xref="S3.F5.12.m4.1.1.3.3.cmml">−</mo><mn id="S3.F5.12.m4.1.1.3.3.2" xref="S3.F5.12.m4.1.1.3.3.2.cmml">19</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.F5.12.m4.1c"><apply id="S3.F5.12.m4.1.1.cmml" xref="S3.F5.12.m4.1.1"><csymbol cd="latexml" id="S3.F5.12.m4.1.1.1.cmml" xref="S3.F5.12.m4.1.1.1">similar-to</csymbol><apply id="S3.F5.12.m4.1.1.2.cmml" xref="S3.F5.12.m4.1.1.2"><csymbol cd="ambiguous" id="S3.F5.12.m4.1.1.2.1.cmml" xref="S3.F5.12.m4.1.1.2">subscript</csymbol><ci id="S3.F5.12.m4.1.1.2.2.cmml" xref="S3.F5.12.m4.1.1.2.2">𝜎</ci><ci id="S3.F5.12.m4.1.1.2.3.cmml" xref="S3.F5.12.m4.1.1.2.3">𝑦</ci></apply><apply id="S3.F5.12.m4.1.1.3.cmml" xref="S3.F5.12.m4.1.1.3"><csymbol cd="ambiguous" id="S3.F5.12.m4.1.1.3.1.cmml" xref="S3.F5.12.m4.1.1.3">superscript</csymbol><cn id="S3.F5.12.m4.1.1.3.2.cmml" type="integer" xref="S3.F5.12.m4.1.1.3.2">10</cn><apply id="S3.F5.12.m4.1.1.3.3.cmml" xref="S3.F5.12.m4.1.1.3.3"><minus id="S3.F5.12.m4.1.1.3.3.1.cmml" xref="S3.F5.12.m4.1.1.3.3"></minus><cn id="S3.F5.12.m4.1.1.3.3.2.cmml" type="integer" xref="S3.F5.12.m4.1.1.3.3.2">19</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.12.m4.1d">\sigma_{y}\sim 10^{-19}</annotation><annotation encoding="application/x-llamapun" id="S3.F5.12.m4.1e">italic_σ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 19 end_POSTSUPERSCRIPT</annotation></semantics></math>. The dashed pink curve shows what happens to the <math alttext="\sigma_{y}(\tau=1{\rm s})=10^{-15}" class="ltx_Math" display="inline" id="S3.F5.13.m5.1"><semantics id="S3.F5.13.m5.1b"><mrow id="S3.F5.13.m5.1.1" xref="S3.F5.13.m5.1.1.cmml"><mrow id="S3.F5.13.m5.1.1.1" xref="S3.F5.13.m5.1.1.1.cmml"><msub id="S3.F5.13.m5.1.1.1.3" xref="S3.F5.13.m5.1.1.1.3.cmml"><mi id="S3.F5.13.m5.1.1.1.3.2" xref="S3.F5.13.m5.1.1.1.3.2.cmml">σ</mi><mi id="S3.F5.13.m5.1.1.1.3.3" xref="S3.F5.13.m5.1.1.1.3.3.cmml">y</mi></msub><mo id="S3.F5.13.m5.1.1.1.2" xref="S3.F5.13.m5.1.1.1.2.cmml">⁢</mo><mrow id="S3.F5.13.m5.1.1.1.1.1" xref="S3.F5.13.m5.1.1.1.1.1.1.cmml"><mo id="S3.F5.13.m5.1.1.1.1.1.2" stretchy="false" xref="S3.F5.13.m5.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.F5.13.m5.1.1.1.1.1.1" xref="S3.F5.13.m5.1.1.1.1.1.1.cmml"><mi id="S3.F5.13.m5.1.1.1.1.1.1.2" xref="S3.F5.13.m5.1.1.1.1.1.1.2.cmml">τ</mi><mo id="S3.F5.13.m5.1.1.1.1.1.1.1" xref="S3.F5.13.m5.1.1.1.1.1.1.1.cmml">=</mo><mrow id="S3.F5.13.m5.1.1.1.1.1.1.3" xref="S3.F5.13.m5.1.1.1.1.1.1.3.cmml"><mn id="S3.F5.13.m5.1.1.1.1.1.1.3.2" xref="S3.F5.13.m5.1.1.1.1.1.1.3.2.cmml">1</mn><mo id="S3.F5.13.m5.1.1.1.1.1.1.3.1" xref="S3.F5.13.m5.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S3.F5.13.m5.1.1.1.1.1.1.3.3" mathvariant="normal" xref="S3.F5.13.m5.1.1.1.1.1.1.3.3.cmml">s</mi></mrow></mrow><mo id="S3.F5.13.m5.1.1.1.1.1.3" stretchy="false" xref="S3.F5.13.m5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.F5.13.m5.1.1.2" xref="S3.F5.13.m5.1.1.2.cmml">=</mo><msup id="S3.F5.13.m5.1.1.3" xref="S3.F5.13.m5.1.1.3.cmml"><mn id="S3.F5.13.m5.1.1.3.2" xref="S3.F5.13.m5.1.1.3.2.cmml">10</mn><mrow id="S3.F5.13.m5.1.1.3.3" xref="S3.F5.13.m5.1.1.3.3.cmml"><mo id="S3.F5.13.m5.1.1.3.3b" xref="S3.F5.13.m5.1.1.3.3.cmml">−</mo><mn id="S3.F5.13.m5.1.1.3.3.2" xref="S3.F5.13.m5.1.1.3.3.2.cmml">15</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.F5.13.m5.1c"><apply id="S3.F5.13.m5.1.1.cmml" xref="S3.F5.13.m5.1.1"><eq id="S3.F5.13.m5.1.1.2.cmml" xref="S3.F5.13.m5.1.1.2"></eq><apply id="S3.F5.13.m5.1.1.1.cmml" xref="S3.F5.13.m5.1.1.1"><times id="S3.F5.13.m5.1.1.1.2.cmml" xref="S3.F5.13.m5.1.1.1.2"></times><apply id="S3.F5.13.m5.1.1.1.3.cmml" xref="S3.F5.13.m5.1.1.1.3"><csymbol cd="ambiguous" id="S3.F5.13.m5.1.1.1.3.1.cmml" xref="S3.F5.13.m5.1.1.1.3">subscript</csymbol><ci id="S3.F5.13.m5.1.1.1.3.2.cmml" xref="S3.F5.13.m5.1.1.1.3.2">𝜎</ci><ci id="S3.F5.13.m5.1.1.1.3.3.cmml" xref="S3.F5.13.m5.1.1.1.3.3">𝑦</ci></apply><apply id="S3.F5.13.m5.1.1.1.1.1.1.cmml" xref="S3.F5.13.m5.1.1.1.1.1"><eq id="S3.F5.13.m5.1.1.1.1.1.1.1.cmml" xref="S3.F5.13.m5.1.1.1.1.1.1.1"></eq><ci id="S3.F5.13.m5.1.1.1.1.1.1.2.cmml" xref="S3.F5.13.m5.1.1.1.1.1.1.2">𝜏</ci><apply id="S3.F5.13.m5.1.1.1.1.1.1.3.cmml" xref="S3.F5.13.m5.1.1.1.1.1.1.3"><times id="S3.F5.13.m5.1.1.1.1.1.1.3.1.cmml" xref="S3.F5.13.m5.1.1.1.1.1.1.3.1"></times><cn id="S3.F5.13.m5.1.1.1.1.1.1.3.2.cmml" type="integer" xref="S3.F5.13.m5.1.1.1.1.1.1.3.2">1</cn><ci id="S3.F5.13.m5.1.1.1.1.1.1.3.3.cmml" xref="S3.F5.13.m5.1.1.1.1.1.1.3.3">s</ci></apply></apply></apply><apply id="S3.F5.13.m5.1.1.3.cmml" xref="S3.F5.13.m5.1.1.3"><csymbol cd="ambiguous" id="S3.F5.13.m5.1.1.3.1.cmml" xref="S3.F5.13.m5.1.1.3">superscript</csymbol><cn id="S3.F5.13.m5.1.1.3.2.cmml" type="integer" xref="S3.F5.13.m5.1.1.3.2">10</cn><apply id="S3.F5.13.m5.1.1.3.3.cmml" xref="S3.F5.13.m5.1.1.3.3"><minus id="S3.F5.13.m5.1.1.3.3.1.cmml" xref="S3.F5.13.m5.1.1.3.3"></minus><cn id="S3.F5.13.m5.1.1.3.3.2.cmml" type="integer" xref="S3.F5.13.m5.1.1.3.3.2">15</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.13.m5.1d">\sigma_{y}(\tau=1{\rm s})=10^{-15}</annotation><annotation encoding="application/x-llamapun" id="S3.F5.13.m5.1e">italic_σ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ( italic_τ = 1 roman_s ) = 10 start_POSTSUPERSCRIPT - 15 end_POSTSUPERSCRIPT</annotation></semantics></math> case if the clock enters the flicker frequency noise regime at <math alttext="f<10^{-4}" class="ltx_Math" display="inline" id="S3.F5.14.m6.1"><semantics id="S3.F5.14.m6.1b"><mrow id="S3.F5.14.m6.1.1" xref="S3.F5.14.m6.1.1.cmml"><mi id="S3.F5.14.m6.1.1.2" xref="S3.F5.14.m6.1.1.2.cmml">f</mi><mo id="S3.F5.14.m6.1.1.1" xref="S3.F5.14.m6.1.1.1.cmml"><</mo><msup id="S3.F5.14.m6.1.1.3" xref="S3.F5.14.m6.1.1.3.cmml"><mn id="S3.F5.14.m6.1.1.3.2" xref="S3.F5.14.m6.1.1.3.2.cmml">10</mn><mrow id="S3.F5.14.m6.1.1.3.3" xref="S3.F5.14.m6.1.1.3.3.cmml"><mo id="S3.F5.14.m6.1.1.3.3b" xref="S3.F5.14.m6.1.1.3.3.cmml">−</mo><mn id="S3.F5.14.m6.1.1.3.3.2" xref="S3.F5.14.m6.1.1.3.3.2.cmml">4</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.F5.14.m6.1c"><apply id="S3.F5.14.m6.1.1.cmml" xref="S3.F5.14.m6.1.1"><lt id="S3.F5.14.m6.1.1.1.cmml" xref="S3.F5.14.m6.1.1.1"></lt><ci id="S3.F5.14.m6.1.1.2.cmml" xref="S3.F5.14.m6.1.1.2">𝑓</ci><apply id="S3.F5.14.m6.1.1.3.cmml" xref="S3.F5.14.m6.1.1.3"><csymbol cd="ambiguous" id="S3.F5.14.m6.1.1.3.1.cmml" xref="S3.F5.14.m6.1.1.3">superscript</csymbol><cn id="S3.F5.14.m6.1.1.3.2.cmml" type="integer" xref="S3.F5.14.m6.1.1.3.2">10</cn><apply id="S3.F5.14.m6.1.1.3.3.cmml" xref="S3.F5.14.m6.1.1.3.3"><minus id="S3.F5.14.m6.1.1.3.3.1.cmml" xref="S3.F5.14.m6.1.1.3.3"></minus><cn id="S3.F5.14.m6.1.1.3.3.2.cmml" type="integer" xref="S3.F5.14.m6.1.1.3.3.2">4</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.14.m6.1d">f<10^{-4}</annotation><annotation encoding="application/x-llamapun" id="S3.F5.14.m6.1e">italic_f < 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT</annotation></semantics></math>Hz (rather than white frequency noise as otherwise assumed). Phase measurements referenced to a local clock in each arm would achieve nearly the sensitivity of time-delay interferometry with two arms if the clock has <math alttext="\sigma_{y}\lesssim 10^{-16}" class="ltx_Math" display="inline" id="S3.F5.15.m7.1"><semantics id="S3.F5.15.m7.1b"><mrow id="S3.F5.15.m7.1.1" xref="S3.F5.15.m7.1.1.cmml"><msub id="S3.F5.15.m7.1.1.2" xref="S3.F5.15.m7.1.1.2.cmml"><mi id="S3.F5.15.m7.1.1.2.2" xref="S3.F5.15.m7.1.1.2.2.cmml">σ</mi><mi id="S3.F5.15.m7.1.1.2.3" xref="S3.F5.15.m7.1.1.2.3.cmml">y</mi></msub><mo id="S3.F5.15.m7.1.1.1" xref="S3.F5.15.m7.1.1.1.cmml">≲</mo><msup id="S3.F5.15.m7.1.1.3" xref="S3.F5.15.m7.1.1.3.cmml"><mn id="S3.F5.15.m7.1.1.3.2" xref="S3.F5.15.m7.1.1.3.2.cmml">10</mn><mrow id="S3.F5.15.m7.1.1.3.3" xref="S3.F5.15.m7.1.1.3.3.cmml"><mo id="S3.F5.15.m7.1.1.3.3b" xref="S3.F5.15.m7.1.1.3.3.cmml">−</mo><mn id="S3.F5.15.m7.1.1.3.3.2" xref="S3.F5.15.m7.1.1.3.3.2.cmml">16</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.F5.15.m7.1c"><apply id="S3.F5.15.m7.1.1.cmml" xref="S3.F5.15.m7.1.1"><csymbol cd="latexml" id="S3.F5.15.m7.1.1.1.cmml" xref="S3.F5.15.m7.1.1.1">less-than-or-similar-to</csymbol><apply id="S3.F5.15.m7.1.1.2.cmml" xref="S3.F5.15.m7.1.1.2"><csymbol cd="ambiguous" id="S3.F5.15.m7.1.1.2.1.cmml" xref="S3.F5.15.m7.1.1.2">subscript</csymbol><ci id="S3.F5.15.m7.1.1.2.2.cmml" xref="S3.F5.15.m7.1.1.2.2">𝜎</ci><ci id="S3.F5.15.m7.1.1.2.3.cmml" xref="S3.F5.15.m7.1.1.2.3">𝑦</ci></apply><apply id="S3.F5.15.m7.1.1.3.cmml" xref="S3.F5.15.m7.1.1.3"><csymbol cd="ambiguous" id="S3.F5.15.m7.1.1.3.1.cmml" xref="S3.F5.15.m7.1.1.3">superscript</csymbol><cn id="S3.F5.15.m7.1.1.3.2.cmml" type="integer" xref="S3.F5.15.m7.1.1.3.2">10</cn><apply id="S3.F5.15.m7.1.1.3.3.cmml" xref="S3.F5.15.m7.1.1.3.3"><minus id="S3.F5.15.m7.1.1.3.3.1.cmml" xref="S3.F5.15.m7.1.1.3.3"></minus><cn id="S3.F5.15.m7.1.1.3.3.2.cmml" type="integer" xref="S3.F5.15.m7.1.1.3.3.2">16</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.15.m7.1d">\sigma_{y}\lesssim 10^{-16}</annotation><annotation encoding="application/x-llamapun" id="S3.F5.15.m7.1e">italic_σ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ≲ 10 start_POSTSUPERSCRIPT - 16 end_POSTSUPERSCRIPT</annotation></semantics></math>. All sensitivity curves assume a geometric suppression of the accelerations by a factor of <math alttext="0.5" class="ltx_Math" display="inline" id="S3.F5.16.m8.1"><semantics id="S3.F5.16.m8.1b"><mn id="S3.F5.16.m8.1.1" xref="S3.F5.16.m8.1.1.cmml">0.5</mn><annotation-xml encoding="MathML-Content" id="S3.F5.16.m8.1c"><cn id="S3.F5.16.m8.1.1.cmml" type="float" xref="S3.F5.16.m8.1.1">0.5</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.16.m8.1d">0.5</annotation><annotation encoding="application/x-llamapun" id="S3.F5.16.m8.1e">0.5</annotation></semantics></math>.</figcaption> </figure> </section> <section class="ltx_subsection" id="S3.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.2 </span>Single-arm with atomic clock forecasts</h3> <div class="ltx_para" id="S3.SS2.p1"> <p class="ltx_p" id="S3.SS2.p1.5">A simpler mission would use a single arm, but this would also require the inclusion of an atomic clock to reach interesting sensitivities. Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S3.F5" title="Figure 5 ‣ 3.1 Two-arm time-delay interferometry forecasts ‣ 3 Strain power sensitivity ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">5</span></a> considers this one-arm setup for <math alttext="r=L=10" class="ltx_Math" display="inline" id="S3.SS2.p1.1.m1.1"><semantics id="S3.SS2.p1.1.m1.1a"><mrow id="S3.SS2.p1.1.m1.1.1" xref="S3.SS2.p1.1.m1.1.1.cmml"><mi id="S3.SS2.p1.1.m1.1.1.2" xref="S3.SS2.p1.1.m1.1.1.2.cmml">r</mi><mo id="S3.SS2.p1.1.m1.1.1.3" xref="S3.SS2.p1.1.m1.1.1.3.cmml">=</mo><mi id="S3.SS2.p1.1.m1.1.1.4" xref="S3.SS2.p1.1.m1.1.1.4.cmml">L</mi><mo id="S3.SS2.p1.1.m1.1.1.5" xref="S3.SS2.p1.1.m1.1.1.5.cmml">=</mo><mn id="S3.SS2.p1.1.m1.1.1.6" xref="S3.SS2.p1.1.m1.1.1.6.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.1.m1.1b"><apply id="S3.SS2.p1.1.m1.1.1.cmml" xref="S3.SS2.p1.1.m1.1.1"><and id="S3.SS2.p1.1.m1.1.1a.cmml" xref="S3.SS2.p1.1.m1.1.1"></and><apply id="S3.SS2.p1.1.m1.1.1b.cmml" xref="S3.SS2.p1.1.m1.1.1"><eq id="S3.SS2.p1.1.m1.1.1.3.cmml" xref="S3.SS2.p1.1.m1.1.1.3"></eq><ci id="S3.SS2.p1.1.m1.1.1.2.cmml" xref="S3.SS2.p1.1.m1.1.1.2">𝑟</ci><ci id="S3.SS2.p1.1.m1.1.1.4.cmml" xref="S3.SS2.p1.1.m1.1.1.4">𝐿</ci></apply><apply id="S3.SS2.p1.1.m1.1.1c.cmml" xref="S3.SS2.p1.1.m1.1.1"><eq id="S3.SS2.p1.1.m1.1.1.5.cmml" xref="S3.SS2.p1.1.m1.1.1.5"></eq><share href="https://arxiv.org/html/2411.15072v1#S3.SS2.p1.1.m1.1.1.4.cmml" id="S3.SS2.p1.1.m1.1.1d.cmml" xref="S3.SS2.p1.1.m1.1.1"></share><cn id="S3.SS2.p1.1.m1.1.1.6.cmml" type="integer" xref="S3.SS2.p1.1.m1.1.1.6">10</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.1.m1.1c">r=L=10</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.1.m1.1d">italic_r = italic_L = 10</annotation></semantics></math> and <math alttext="30" class="ltx_Math" display="inline" id="S3.SS2.p1.2.m2.1"><semantics id="S3.SS2.p1.2.m2.1a"><mn id="S3.SS2.p1.2.m2.1.1" xref="S3.SS2.p1.2.m2.1.1.cmml">30</mn><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.2.m2.1b"><cn id="S3.SS2.p1.2.m2.1.1.cmml" type="integer" xref="S3.SS2.p1.2.m2.1.1">30</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.2.m2.1c">30</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.2.m2.1d">30</annotation></semantics></math> AU, but note that the clock noise power is independent of the baseline length for <math alttext="fL\ll 1" class="ltx_Math" display="inline" id="S3.SS2.p1.3.m3.1"><semantics id="S3.SS2.p1.3.m3.1a"><mrow id="S3.SS2.p1.3.m3.1.1" xref="S3.SS2.p1.3.m3.1.1.cmml"><mrow id="S3.SS2.p1.3.m3.1.1.2" xref="S3.SS2.p1.3.m3.1.1.2.cmml"><mi id="S3.SS2.p1.3.m3.1.1.2.2" xref="S3.SS2.p1.3.m3.1.1.2.2.cmml">f</mi><mo id="S3.SS2.p1.3.m3.1.1.2.1" xref="S3.SS2.p1.3.m3.1.1.2.1.cmml">⁢</mo><mi id="S3.SS2.p1.3.m3.1.1.2.3" xref="S3.SS2.p1.3.m3.1.1.2.3.cmml">L</mi></mrow><mo id="S3.SS2.p1.3.m3.1.1.1" xref="S3.SS2.p1.3.m3.1.1.1.cmml">≪</mo><mn id="S3.SS2.p1.3.m3.1.1.3" xref="S3.SS2.p1.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.3.m3.1b"><apply id="S3.SS2.p1.3.m3.1.1.cmml" xref="S3.SS2.p1.3.m3.1.1"><csymbol cd="latexml" id="S3.SS2.p1.3.m3.1.1.1.cmml" xref="S3.SS2.p1.3.m3.1.1.1">much-less-than</csymbol><apply id="S3.SS2.p1.3.m3.1.1.2.cmml" xref="S3.SS2.p1.3.m3.1.1.2"><times id="S3.SS2.p1.3.m3.1.1.2.1.cmml" xref="S3.SS2.p1.3.m3.1.1.2.1"></times><ci id="S3.SS2.p1.3.m3.1.1.2.2.cmml" xref="S3.SS2.p1.3.m3.1.1.2.2">𝑓</ci><ci id="S3.SS2.p1.3.m3.1.1.2.3.cmml" xref="S3.SS2.p1.3.m3.1.1.2.3">𝐿</ci></apply><cn id="S3.SS2.p1.3.m3.1.1.3.cmml" type="integer" xref="S3.SS2.p1.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.3.m3.1c">fL\ll 1</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.3.m3.1d">italic_f italic_L ≪ 1</annotation></semantics></math>. For this calculation, we generalize equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S3.E18" title="In 3 Strain power sensitivity ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">18</span></a>) to also include the clock noise given by equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.E3" title="In Single-arm and an atomic clock ‣ 2.1 Clock noise and its mitigation ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">3</span></a>), requiring the replacement <math alttext="2{\cal A}^{2}S_{n}\rightarrow 2{\cal A}^{2}S_{n}+S_{\rm os,tot}" class="ltx_Math" display="inline" id="S3.SS2.p1.4.m4.2"><semantics id="S3.SS2.p1.4.m4.2a"><mrow id="S3.SS2.p1.4.m4.2.3" xref="S3.SS2.p1.4.m4.2.3.cmml"><mrow id="S3.SS2.p1.4.m4.2.3.2" 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xref="S3.SS2.p1.4.m4.1.1.1.1">os</ci><ci id="S3.SS2.p1.4.m4.2.2.2.2.cmml" xref="S3.SS2.p1.4.m4.2.2.2.2">tot</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.4.m4.2c">2{\cal A}^{2}S_{n}\rightarrow 2{\cal A}^{2}S_{n}+S_{\rm os,tot}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.4.m4.2d">2 caligraphic_A start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT → 2 caligraphic_A start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT roman_os , roman_tot end_POSTSUBSCRIPT</annotation></semantics></math>. The thick solid curves are for the <math alttext="\lambda=1~{}" class="ltx_Math" display="inline" id="S3.SS2.p1.5.m5.1"><semantics id="S3.SS2.p1.5.m5.1a"><mrow id="S3.SS2.p1.5.m5.1.1" xref="S3.SS2.p1.5.m5.1.1.cmml"><mi id="S3.SS2.p1.5.m5.1.1.2" xref="S3.SS2.p1.5.m5.1.1.2.cmml">λ</mi><mo id="S3.SS2.p1.5.m5.1.1.1" xref="S3.SS2.p1.5.m5.1.1.1.cmml">=</mo><mn id="S3.SS2.p1.5.m5.1.1.3" xref="S3.SS2.p1.5.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.5.m5.1b"><apply id="S3.SS2.p1.5.m5.1.1.cmml" xref="S3.SS2.p1.5.m5.1.1"><eq id="S3.SS2.p1.5.m5.1.1.1.cmml" xref="S3.SS2.p1.5.m5.1.1.1"></eq><ci id="S3.SS2.p1.5.m5.1.1.2.cmml" xref="S3.SS2.p1.5.m5.1.1.2">𝜆</ci><cn id="S3.SS2.p1.5.m5.1.1.3.cmml" type="integer" xref="S3.SS2.p1.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.5.m5.1c">\lambda=1~{}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.5.m5.1d">italic_λ = 1</annotation></semantics></math>cm radio case, but we note the laser and radio sensitivities are the same to the extent that the sensitivity is limited by clock noise rather than radiometer noise.</p> </div> <div class="ltx_para" id="S3.SS2.p2"> <p class="ltx_p" id="S3.SS2.p2.6">The clock noise is comparable to radiometer noise in the curves that assume the most precise clock with <math alttext="\sigma_{y}(\tau=1\;{\rm s})=10^{-16}" class="ltx_Math" display="inline" id="S3.SS2.p2.1.m1.1"><semantics id="S3.SS2.p2.1.m1.1a"><mrow id="S3.SS2.p2.1.m1.1.1" xref="S3.SS2.p2.1.m1.1.1.cmml"><mrow id="S3.SS2.p2.1.m1.1.1.1" xref="S3.SS2.p2.1.m1.1.1.1.cmml"><msub id="S3.SS2.p2.1.m1.1.1.1.3" xref="S3.SS2.p2.1.m1.1.1.1.3.cmml"><mi id="S3.SS2.p2.1.m1.1.1.1.3.2" xref="S3.SS2.p2.1.m1.1.1.1.3.2.cmml">σ</mi><mi id="S3.SS2.p2.1.m1.1.1.1.3.3" xref="S3.SS2.p2.1.m1.1.1.1.3.3.cmml">y</mi></msub><mo id="S3.SS2.p2.1.m1.1.1.1.2" xref="S3.SS2.p2.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS2.p2.1.m1.1.1.1.1.1" xref="S3.SS2.p2.1.m1.1.1.1.1.1.1.cmml"><mo id="S3.SS2.p2.1.m1.1.1.1.1.1.2" stretchy="false" xref="S3.SS2.p2.1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS2.p2.1.m1.1.1.1.1.1.1" xref="S3.SS2.p2.1.m1.1.1.1.1.1.1.cmml"><mi id="S3.SS2.p2.1.m1.1.1.1.1.1.1.2" xref="S3.SS2.p2.1.m1.1.1.1.1.1.1.2.cmml">τ</mi><mo id="S3.SS2.p2.1.m1.1.1.1.1.1.1.1" xref="S3.SS2.p2.1.m1.1.1.1.1.1.1.1.cmml">=</mo><mrow id="S3.SS2.p2.1.m1.1.1.1.1.1.1.3" xref="S3.SS2.p2.1.m1.1.1.1.1.1.1.3.cmml"><mn id="S3.SS2.p2.1.m1.1.1.1.1.1.1.3.2" xref="S3.SS2.p2.1.m1.1.1.1.1.1.1.3.2.cmml">1</mn><mo id="S3.SS2.p2.1.m1.1.1.1.1.1.1.3.1" lspace="0.280em" xref="S3.SS2.p2.1.m1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.p2.1.m1.1.1.1.1.1.1.3.3" mathvariant="normal" xref="S3.SS2.p2.1.m1.1.1.1.1.1.1.3.3.cmml">s</mi></mrow></mrow><mo id="S3.SS2.p2.1.m1.1.1.1.1.1.3" stretchy="false" xref="S3.SS2.p2.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS2.p2.1.m1.1.1.2" xref="S3.SS2.p2.1.m1.1.1.2.cmml">=</mo><msup id="S3.SS2.p2.1.m1.1.1.3" xref="S3.SS2.p2.1.m1.1.1.3.cmml"><mn id="S3.SS2.p2.1.m1.1.1.3.2" xref="S3.SS2.p2.1.m1.1.1.3.2.cmml">10</mn><mrow id="S3.SS2.p2.1.m1.1.1.3.3" xref="S3.SS2.p2.1.m1.1.1.3.3.cmml"><mo id="S3.SS2.p2.1.m1.1.1.3.3a" xref="S3.SS2.p2.1.m1.1.1.3.3.cmml">−</mo><mn id="S3.SS2.p2.1.m1.1.1.3.3.2" xref="S3.SS2.p2.1.m1.1.1.3.3.2.cmml">16</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.1.m1.1b"><apply id="S3.SS2.p2.1.m1.1.1.cmml" xref="S3.SS2.p2.1.m1.1.1"><eq id="S3.SS2.p2.1.m1.1.1.2.cmml" xref="S3.SS2.p2.1.m1.1.1.2"></eq><apply id="S3.SS2.p2.1.m1.1.1.1.cmml" xref="S3.SS2.p2.1.m1.1.1.1"><times id="S3.SS2.p2.1.m1.1.1.1.2.cmml" xref="S3.SS2.p2.1.m1.1.1.1.2"></times><apply id="S3.SS2.p2.1.m1.1.1.1.3.cmml" xref="S3.SS2.p2.1.m1.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.p2.1.m1.1.1.1.3.1.cmml" xref="S3.SS2.p2.1.m1.1.1.1.3">subscript</csymbol><ci id="S3.SS2.p2.1.m1.1.1.1.3.2.cmml" xref="S3.SS2.p2.1.m1.1.1.1.3.2">𝜎</ci><ci id="S3.SS2.p2.1.m1.1.1.1.3.3.cmml" xref="S3.SS2.p2.1.m1.1.1.1.3.3">𝑦</ci></apply><apply id="S3.SS2.p2.1.m1.1.1.1.1.1.1.cmml" xref="S3.SS2.p2.1.m1.1.1.1.1.1"><eq id="S3.SS2.p2.1.m1.1.1.1.1.1.1.1.cmml" xref="S3.SS2.p2.1.m1.1.1.1.1.1.1.1"></eq><ci id="S3.SS2.p2.1.m1.1.1.1.1.1.1.2.cmml" xref="S3.SS2.p2.1.m1.1.1.1.1.1.1.2">𝜏</ci><apply id="S3.SS2.p2.1.m1.1.1.1.1.1.1.3.cmml" xref="S3.SS2.p2.1.m1.1.1.1.1.1.1.3"><times id="S3.SS2.p2.1.m1.1.1.1.1.1.1.3.1.cmml" xref="S3.SS2.p2.1.m1.1.1.1.1.1.1.3.1"></times><cn id="S3.SS2.p2.1.m1.1.1.1.1.1.1.3.2.cmml" type="integer" xref="S3.SS2.p2.1.m1.1.1.1.1.1.1.3.2">1</cn><ci id="S3.SS2.p2.1.m1.1.1.1.1.1.1.3.3.cmml" xref="S3.SS2.p2.1.m1.1.1.1.1.1.1.3.3">s</ci></apply></apply></apply><apply id="S3.SS2.p2.1.m1.1.1.3.cmml" xref="S3.SS2.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.p2.1.m1.1.1.3.1.cmml" xref="S3.SS2.p2.1.m1.1.1.3">superscript</csymbol><cn id="S3.SS2.p2.1.m1.1.1.3.2.cmml" type="integer" xref="S3.SS2.p2.1.m1.1.1.3.2">10</cn><apply id="S3.SS2.p2.1.m1.1.1.3.3.cmml" xref="S3.SS2.p2.1.m1.1.1.3.3"><minus id="S3.SS2.p2.1.m1.1.1.3.3.1.cmml" xref="S3.SS2.p2.1.m1.1.1.3.3"></minus><cn id="S3.SS2.p2.1.m1.1.1.3.3.2.cmml" type="integer" xref="S3.SS2.p2.1.m1.1.1.3.3.2">16</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.1.m1.1c">\sigma_{y}(\tau=1\;{\rm s})=10^{-16}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.1.m1.1d">italic_σ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ( italic_τ = 1 roman_s ) = 10 start_POSTSUPERSCRIPT - 16 end_POSTSUPERSCRIPT</annotation></semantics></math>, as can be seen by noting their sensitivity is comparable to the corresponding time-delay interferometry curve. (The transfer function <math alttext="R(f)" class="ltx_Math" display="inline" id="S3.SS2.p2.2.m2.1"><semantics id="S3.SS2.p2.2.m2.1a"><mrow id="S3.SS2.p2.2.m2.1.2" xref="S3.SS2.p2.2.m2.1.2.cmml"><mi id="S3.SS2.p2.2.m2.1.2.2" xref="S3.SS2.p2.2.m2.1.2.2.cmml">R</mi><mo id="S3.SS2.p2.2.m2.1.2.1" xref="S3.SS2.p2.2.m2.1.2.1.cmml">⁢</mo><mrow id="S3.SS2.p2.2.m2.1.2.3.2" xref="S3.SS2.p2.2.m2.1.2.cmml"><mo id="S3.SS2.p2.2.m2.1.2.3.2.1" stretchy="false" xref="S3.SS2.p2.2.m2.1.2.cmml">(</mo><mi id="S3.SS2.p2.2.m2.1.1" xref="S3.SS2.p2.2.m2.1.1.cmml">f</mi><mo id="S3.SS2.p2.2.m2.1.2.3.2.2" stretchy="false" xref="S3.SS2.p2.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.2.m2.1b"><apply id="S3.SS2.p2.2.m2.1.2.cmml" xref="S3.SS2.p2.2.m2.1.2"><times id="S3.SS2.p2.2.m2.1.2.1.cmml" xref="S3.SS2.p2.2.m2.1.2.1"></times><ci id="S3.SS2.p2.2.m2.1.2.2.cmml" xref="S3.SS2.p2.2.m2.1.2.2">𝑅</ci><ci id="S3.SS2.p2.2.m2.1.1.cmml" xref="S3.SS2.p2.2.m2.1.1">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.2.m2.1c">R(f)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.2.m2.1d">italic_R ( italic_f )</annotation></semantics></math> is modestly different compared to the two-arm concept.<span class="ltx_note ltx_role_footnote" id="footnote11"><sup class="ltx_note_mark">11</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">11</sup><span class="ltx_tag ltx_tag_note">11</span>To compute <math alttext="R(f)" class="ltx_Math" display="inline" id="footnote11.m1.1"><semantics id="footnote11.m1.1b"><mrow id="footnote11.m1.1.2" xref="footnote11.m1.1.2.cmml"><mi id="footnote11.m1.1.2.2" xref="footnote11.m1.1.2.2.cmml">R</mi><mo id="footnote11.m1.1.2.1" xref="footnote11.m1.1.2.1.cmml">⁢</mo><mrow id="footnote11.m1.1.2.3.2" xref="footnote11.m1.1.2.cmml"><mo id="footnote11.m1.1.2.3.2.1" stretchy="false" xref="footnote11.m1.1.2.cmml">(</mo><mi id="footnote11.m1.1.1" xref="footnote11.m1.1.1.cmml">f</mi><mo id="footnote11.m1.1.2.3.2.2" stretchy="false" xref="footnote11.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote11.m1.1c"><apply id="footnote11.m1.1.2.cmml" xref="footnote11.m1.1.2"><times id="footnote11.m1.1.2.1.cmml" xref="footnote11.m1.1.2.1"></times><ci id="footnote11.m1.1.2.2.cmml" xref="footnote11.m1.1.2.2">𝑅</ci><ci id="footnote11.m1.1.1.cmml" xref="footnote11.m1.1.1">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote11.m1.1d">R(f)</annotation><annotation encoding="application/x-llamapun" id="footnote11.m1.1e">italic_R ( italic_f )</annotation></semantics></math> for one arm, we drop the <math alttext="T_{3}" class="ltx_Math" display="inline" id="footnote11.m2.1"><semantics id="footnote11.m2.1b"><msub id="footnote11.m2.1.1" xref="footnote11.m2.1.1.cmml"><mi id="footnote11.m2.1.1.2" xref="footnote11.m2.1.1.2.cmml">T</mi><mn id="footnote11.m2.1.1.3" xref="footnote11.m2.1.1.3.cmml">3</mn></msub><annotation-xml encoding="MathML-Content" id="footnote11.m2.1c"><apply id="footnote11.m2.1.1.cmml" xref="footnote11.m2.1.1"><csymbol cd="ambiguous" id="footnote11.m2.1.1.1.cmml" xref="footnote11.m2.1.1">subscript</csymbol><ci id="footnote11.m2.1.1.2.cmml" xref="footnote11.m2.1.1.2">𝑇</ci><cn id="footnote11.m2.1.1.3.cmml" type="integer" xref="footnote11.m2.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote11.m2.1d">T_{3}</annotation><annotation encoding="application/x-llamapun" id="footnote11.m2.1e">italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math> term in equation (23) in <cite class="ltx_cite ltx_citemacro_citet">Larson et al. (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib43" title="">2000</a>)</cite> that accounts for the cross-correlation of the gravitational wave signal between two arms, but include all other terms in their calculation of <math alttext="R(f)" class="ltx_Math" display="inline" id="footnote11.m3.1"><semantics id="footnote11.m3.1b"><mrow id="footnote11.m3.1.2" xref="footnote11.m3.1.2.cmml"><mi id="footnote11.m3.1.2.2" xref="footnote11.m3.1.2.2.cmml">R</mi><mo id="footnote11.m3.1.2.1" xref="footnote11.m3.1.2.1.cmml">⁢</mo><mrow id="footnote11.m3.1.2.3.2" xref="footnote11.m3.1.2.cmml"><mo id="footnote11.m3.1.2.3.2.1" stretchy="false" xref="footnote11.m3.1.2.cmml">(</mo><mi id="footnote11.m3.1.1" xref="footnote11.m3.1.1.cmml">f</mi><mo id="footnote11.m3.1.2.3.2.2" stretchy="false" xref="footnote11.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote11.m3.1c"><apply id="footnote11.m3.1.2.cmml" xref="footnote11.m3.1.2"><times id="footnote11.m3.1.2.1.cmml" xref="footnote11.m3.1.2.1"></times><ci id="footnote11.m3.1.2.2.cmml" xref="footnote11.m3.1.2.2">𝑅</ci><ci id="footnote11.m3.1.1.cmml" xref="footnote11.m3.1.1">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote11.m3.1d">R(f)</annotation><annotation encoding="application/x-llamapun" id="footnote11.m3.1e">italic_R ( italic_f )</annotation></semantics></math>. This results in an <math alttext="R(f)" class="ltx_Math" display="inline" id="footnote11.m4.1"><semantics id="footnote11.m4.1b"><mrow id="footnote11.m4.1.2" xref="footnote11.m4.1.2.cmml"><mi id="footnote11.m4.1.2.2" xref="footnote11.m4.1.2.2.cmml">R</mi><mo id="footnote11.m4.1.2.1" xref="footnote11.m4.1.2.1.cmml">⁢</mo><mrow id="footnote11.m4.1.2.3.2" xref="footnote11.m4.1.2.cmml"><mo id="footnote11.m4.1.2.3.2.1" stretchy="false" xref="footnote11.m4.1.2.cmml">(</mo><mi id="footnote11.m4.1.1" xref="footnote11.m4.1.1.cmml">f</mi><mo id="footnote11.m4.1.2.3.2.2" stretchy="false" xref="footnote11.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote11.m4.1c"><apply id="footnote11.m4.1.2.cmml" xref="footnote11.m4.1.2"><times id="footnote11.m4.1.2.1.cmml" xref="footnote11.m4.1.2.1"></times><ci id="footnote11.m4.1.2.2.cmml" xref="footnote11.m4.1.2.2">𝑅</ci><ci id="footnote11.m4.1.1.cmml" xref="footnote11.m4.1.1">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote11.m4.1d">R(f)</annotation><annotation encoding="application/x-llamapun" id="footnote11.m4.1e">italic_R ( italic_f )</annotation></semantics></math> that is a factor of two larger than might be most natural to define for a single arm, but this extra fact of two cancels the factor of two in the front of equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S3.E18" title="In 3 Strain power sensitivity ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">18</span></a>) that owes to noise in both arms, allowing this equation to still apply for our one-arm concepts.</span></span></span>) At larger <math alttext="f" class="ltx_Math" display="inline" id="S3.SS2.p2.3.m3.1"><semantics id="S3.SS2.p2.3.m3.1a"><mi id="S3.SS2.p2.3.m3.1.1" xref="S3.SS2.p2.3.m3.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.3.m3.1b"><ci id="S3.SS2.p2.3.m3.1.1.cmml" xref="S3.SS2.p2.3.m3.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.3.m3.1c">f</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.3.m3.1d">italic_f</annotation></semantics></math>, distinct downward spikes in the noise are present at <math alttext="2Lf=n" class="ltx_Math" display="inline" id="S3.SS2.p2.4.m4.1"><semantics id="S3.SS2.p2.4.m4.1a"><mrow id="S3.SS2.p2.4.m4.1.1" xref="S3.SS2.p2.4.m4.1.1.cmml"><mrow id="S3.SS2.p2.4.m4.1.1.2" xref="S3.SS2.p2.4.m4.1.1.2.cmml"><mn id="S3.SS2.p2.4.m4.1.1.2.2" xref="S3.SS2.p2.4.m4.1.1.2.2.cmml">2</mn><mo id="S3.SS2.p2.4.m4.1.1.2.1" xref="S3.SS2.p2.4.m4.1.1.2.1.cmml">⁢</mo><mi id="S3.SS2.p2.4.m4.1.1.2.3" xref="S3.SS2.p2.4.m4.1.1.2.3.cmml">L</mi><mo id="S3.SS2.p2.4.m4.1.1.2.1a" xref="S3.SS2.p2.4.m4.1.1.2.1.cmml">⁢</mo><mi id="S3.SS2.p2.4.m4.1.1.2.4" xref="S3.SS2.p2.4.m4.1.1.2.4.cmml">f</mi></mrow><mo id="S3.SS2.p2.4.m4.1.1.1" xref="S3.SS2.p2.4.m4.1.1.1.cmml">=</mo><mi id="S3.SS2.p2.4.m4.1.1.3" xref="S3.SS2.p2.4.m4.1.1.3.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.4.m4.1b"><apply id="S3.SS2.p2.4.m4.1.1.cmml" xref="S3.SS2.p2.4.m4.1.1"><eq id="S3.SS2.p2.4.m4.1.1.1.cmml" xref="S3.SS2.p2.4.m4.1.1.1"></eq><apply id="S3.SS2.p2.4.m4.1.1.2.cmml" xref="S3.SS2.p2.4.m4.1.1.2"><times id="S3.SS2.p2.4.m4.1.1.2.1.cmml" xref="S3.SS2.p2.4.m4.1.1.2.1"></times><cn id="S3.SS2.p2.4.m4.1.1.2.2.cmml" type="integer" xref="S3.SS2.p2.4.m4.1.1.2.2">2</cn><ci id="S3.SS2.p2.4.m4.1.1.2.3.cmml" xref="S3.SS2.p2.4.m4.1.1.2.3">𝐿</ci><ci id="S3.SS2.p2.4.m4.1.1.2.4.cmml" xref="S3.SS2.p2.4.m4.1.1.2.4">𝑓</ci></apply><ci id="S3.SS2.p2.4.m4.1.1.3.cmml" xref="S3.SS2.p2.4.m4.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.4.m4.1c">2Lf=n</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.4.m4.1d">2 italic_L italic_f = italic_n</annotation></semantics></math> for integer <math alttext="n" class="ltx_Math" display="inline" id="S3.SS2.p2.5.m5.1"><semantics id="S3.SS2.p2.5.m5.1a"><mi id="S3.SS2.p2.5.m5.1.1" xref="S3.SS2.p2.5.m5.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.5.m5.1b"><ci id="S3.SS2.p2.5.m5.1.1.cmml" xref="S3.SS2.p2.5.m5.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.5.m5.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.5.m5.1d">italic_n</annotation></semantics></math>, due to the clock noise canceling in the noise power. One can see that the amplitude of these downward spikes decreases with decreasing <math alttext="\sigma_{y}" class="ltx_Math" display="inline" id="S3.SS2.p2.6.m6.1"><semantics id="S3.SS2.p2.6.m6.1a"><msub id="S3.SS2.p2.6.m6.1.1" xref="S3.SS2.p2.6.m6.1.1.cmml"><mi id="S3.SS2.p2.6.m6.1.1.2" xref="S3.SS2.p2.6.m6.1.1.2.cmml">σ</mi><mi id="S3.SS2.p2.6.m6.1.1.3" xref="S3.SS2.p2.6.m6.1.1.3.cmml">y</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.6.m6.1b"><apply id="S3.SS2.p2.6.m6.1.1.cmml" xref="S3.SS2.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S3.SS2.p2.6.m6.1.1.1.cmml" xref="S3.SS2.p2.6.m6.1.1">subscript</csymbol><ci id="S3.SS2.p2.6.m6.1.1.2.cmml" xref="S3.SS2.p2.6.m6.1.1.2">𝜎</ci><ci id="S3.SS2.p2.6.m6.1.1.3.cmml" xref="S3.SS2.p2.6.m6.1.1.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.6.m6.1c">\sigma_{y}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.6.m6.1d">italic_σ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT</annotation></semantics></math> as clock noise becomes less of a limiting factor.</p> </div> <div class="ltx_para" id="S3.SS2.p3"> <p class="ltx_p" id="S3.SS2.p3.12">For the case of radio transmissions, phase measurements referenced to a local clock in each arm would achieve nearly the sensitivity that would be achieved by time-delay interferometry if the clock has <math alttext="\sigma_{y}\lesssim 10^{-16}" class="ltx_Math" display="inline" id="S3.SS2.p3.1.m1.1"><semantics id="S3.SS2.p3.1.m1.1a"><mrow id="S3.SS2.p3.1.m1.1.1" xref="S3.SS2.p3.1.m1.1.1.cmml"><msub id="S3.SS2.p3.1.m1.1.1.2" xref="S3.SS2.p3.1.m1.1.1.2.cmml"><mi id="S3.SS2.p3.1.m1.1.1.2.2" xref="S3.SS2.p3.1.m1.1.1.2.2.cmml">σ</mi><mi id="S3.SS2.p3.1.m1.1.1.2.3" xref="S3.SS2.p3.1.m1.1.1.2.3.cmml">y</mi></msub><mo id="S3.SS2.p3.1.m1.1.1.1" xref="S3.SS2.p3.1.m1.1.1.1.cmml">≲</mo><msup id="S3.SS2.p3.1.m1.1.1.3" xref="S3.SS2.p3.1.m1.1.1.3.cmml"><mn id="S3.SS2.p3.1.m1.1.1.3.2" xref="S3.SS2.p3.1.m1.1.1.3.2.cmml">10</mn><mrow id="S3.SS2.p3.1.m1.1.1.3.3" xref="S3.SS2.p3.1.m1.1.1.3.3.cmml"><mo id="S3.SS2.p3.1.m1.1.1.3.3a" xref="S3.SS2.p3.1.m1.1.1.3.3.cmml">−</mo><mn id="S3.SS2.p3.1.m1.1.1.3.3.2" xref="S3.SS2.p3.1.m1.1.1.3.3.2.cmml">16</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.1.m1.1b"><apply id="S3.SS2.p3.1.m1.1.1.cmml" xref="S3.SS2.p3.1.m1.1.1"><csymbol cd="latexml" id="S3.SS2.p3.1.m1.1.1.1.cmml" xref="S3.SS2.p3.1.m1.1.1.1">less-than-or-similar-to</csymbol><apply id="S3.SS2.p3.1.m1.1.1.2.cmml" xref="S3.SS2.p3.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.p3.1.m1.1.1.2.1.cmml" xref="S3.SS2.p3.1.m1.1.1.2">subscript</csymbol><ci id="S3.SS2.p3.1.m1.1.1.2.2.cmml" xref="S3.SS2.p3.1.m1.1.1.2.2">𝜎</ci><ci id="S3.SS2.p3.1.m1.1.1.2.3.cmml" xref="S3.SS2.p3.1.m1.1.1.2.3">𝑦</ci></apply><apply id="S3.SS2.p3.1.m1.1.1.3.cmml" xref="S3.SS2.p3.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.p3.1.m1.1.1.3.1.cmml" xref="S3.SS2.p3.1.m1.1.1.3">superscript</csymbol><cn id="S3.SS2.p3.1.m1.1.1.3.2.cmml" type="integer" xref="S3.SS2.p3.1.m1.1.1.3.2">10</cn><apply id="S3.SS2.p3.1.m1.1.1.3.3.cmml" xref="S3.SS2.p3.1.m1.1.1.3.3"><minus id="S3.SS2.p3.1.m1.1.1.3.3.1.cmml" xref="S3.SS2.p3.1.m1.1.1.3.3"></minus><cn id="S3.SS2.p3.1.m1.1.1.3.3.2.cmml" type="integer" xref="S3.SS2.p3.1.m1.1.1.3.3.2">16</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.1.m1.1c">\sigma_{y}\lesssim 10^{-16}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.1.m1.1d">italic_σ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ≲ 10 start_POSTSUPERSCRIPT - 16 end_POSTSUPERSCRIPT</annotation></semantics></math>. Allan deviations for <math alttext="\tau=1\," class="ltx_Math" display="inline" id="S3.SS2.p3.2.m2.1"><semantics id="S3.SS2.p3.2.m2.1a"><mrow id="S3.SS2.p3.2.m2.1.1" xref="S3.SS2.p3.2.m2.1.1.cmml"><mi id="S3.SS2.p3.2.m2.1.1.2" xref="S3.SS2.p3.2.m2.1.1.2.cmml">τ</mi><mo id="S3.SS2.p3.2.m2.1.1.1" xref="S3.SS2.p3.2.m2.1.1.1.cmml">=</mo><mn id="S3.SS2.p3.2.m2.1.1.3" xref="S3.SS2.p3.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.2.m2.1b"><apply id="S3.SS2.p3.2.m2.1.1.cmml" xref="S3.SS2.p3.2.m2.1.1"><eq id="S3.SS2.p3.2.m2.1.1.1.cmml" xref="S3.SS2.p3.2.m2.1.1.1"></eq><ci id="S3.SS2.p3.2.m2.1.1.2.cmml" xref="S3.SS2.p3.2.m2.1.1.2">𝜏</ci><cn id="S3.SS2.p3.2.m2.1.1.3.cmml" type="integer" xref="S3.SS2.p3.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.2.m2.1c">\tau=1\,</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.2.m2.1d">italic_τ = 1</annotation></semantics></math>s of <math alttext="\sigma_{y}=10^{-13}" class="ltx_Math" display="inline" id="S3.SS2.p3.3.m3.1"><semantics id="S3.SS2.p3.3.m3.1a"><mrow id="S3.SS2.p3.3.m3.1.1" xref="S3.SS2.p3.3.m3.1.1.cmml"><msub id="S3.SS2.p3.3.m3.1.1.2" xref="S3.SS2.p3.3.m3.1.1.2.cmml"><mi id="S3.SS2.p3.3.m3.1.1.2.2" xref="S3.SS2.p3.3.m3.1.1.2.2.cmml">σ</mi><mi id="S3.SS2.p3.3.m3.1.1.2.3" xref="S3.SS2.p3.3.m3.1.1.2.3.cmml">y</mi></msub><mo id="S3.SS2.p3.3.m3.1.1.1" xref="S3.SS2.p3.3.m3.1.1.1.cmml">=</mo><msup id="S3.SS2.p3.3.m3.1.1.3" xref="S3.SS2.p3.3.m3.1.1.3.cmml"><mn id="S3.SS2.p3.3.m3.1.1.3.2" xref="S3.SS2.p3.3.m3.1.1.3.2.cmml">10</mn><mrow id="S3.SS2.p3.3.m3.1.1.3.3" xref="S3.SS2.p3.3.m3.1.1.3.3.cmml"><mo id="S3.SS2.p3.3.m3.1.1.3.3a" xref="S3.SS2.p3.3.m3.1.1.3.3.cmml">−</mo><mn id="S3.SS2.p3.3.m3.1.1.3.3.2" xref="S3.SS2.p3.3.m3.1.1.3.3.2.cmml">13</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.3.m3.1b"><apply id="S3.SS2.p3.3.m3.1.1.cmml" xref="S3.SS2.p3.3.m3.1.1"><eq id="S3.SS2.p3.3.m3.1.1.1.cmml" xref="S3.SS2.p3.3.m3.1.1.1"></eq><apply id="S3.SS2.p3.3.m3.1.1.2.cmml" xref="S3.SS2.p3.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.p3.3.m3.1.1.2.1.cmml" xref="S3.SS2.p3.3.m3.1.1.2">subscript</csymbol><ci id="S3.SS2.p3.3.m3.1.1.2.2.cmml" xref="S3.SS2.p3.3.m3.1.1.2.2">𝜎</ci><ci id="S3.SS2.p3.3.m3.1.1.2.3.cmml" xref="S3.SS2.p3.3.m3.1.1.2.3">𝑦</ci></apply><apply id="S3.SS2.p3.3.m3.1.1.3.cmml" xref="S3.SS2.p3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.p3.3.m3.1.1.3.1.cmml" xref="S3.SS2.p3.3.m3.1.1.3">superscript</csymbol><cn id="S3.SS2.p3.3.m3.1.1.3.2.cmml" type="integer" xref="S3.SS2.p3.3.m3.1.1.3.2">10</cn><apply id="S3.SS2.p3.3.m3.1.1.3.3.cmml" xref="S3.SS2.p3.3.m3.1.1.3.3"><minus id="S3.SS2.p3.3.m3.1.1.3.3.1.cmml" xref="S3.SS2.p3.3.m3.1.1.3.3"></minus><cn id="S3.SS2.p3.3.m3.1.1.3.3.2.cmml" type="integer" xref="S3.SS2.p3.3.m3.1.1.3.3.2">13</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.3.m3.1c">\sigma_{y}=10^{-13}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.3.m3.1d">italic_σ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT = 10 start_POSTSUPERSCRIPT - 13 end_POSTSUPERSCRIPT</annotation></semantics></math> are similar to the Deep Space Atomic Clock, whereas <math alttext="\sigma_{y}=10^{-16}" class="ltx_Math" display="inline" id="S3.SS2.p3.4.m4.1"><semantics id="S3.SS2.p3.4.m4.1a"><mrow id="S3.SS2.p3.4.m4.1.1" xref="S3.SS2.p3.4.m4.1.1.cmml"><msub id="S3.SS2.p3.4.m4.1.1.2" xref="S3.SS2.p3.4.m4.1.1.2.cmml"><mi id="S3.SS2.p3.4.m4.1.1.2.2" xref="S3.SS2.p3.4.m4.1.1.2.2.cmml">σ</mi><mi id="S3.SS2.p3.4.m4.1.1.2.3" xref="S3.SS2.p3.4.m4.1.1.2.3.cmml">y</mi></msub><mo id="S3.SS2.p3.4.m4.1.1.1" xref="S3.SS2.p3.4.m4.1.1.1.cmml">=</mo><msup id="S3.SS2.p3.4.m4.1.1.3" xref="S3.SS2.p3.4.m4.1.1.3.cmml"><mn id="S3.SS2.p3.4.m4.1.1.3.2" xref="S3.SS2.p3.4.m4.1.1.3.2.cmml">10</mn><mrow id="S3.SS2.p3.4.m4.1.1.3.3" xref="S3.SS2.p3.4.m4.1.1.3.3.cmml"><mo id="S3.SS2.p3.4.m4.1.1.3.3a" xref="S3.SS2.p3.4.m4.1.1.3.3.cmml">−</mo><mn id="S3.SS2.p3.4.m4.1.1.3.3.2" xref="S3.SS2.p3.4.m4.1.1.3.3.2.cmml">16</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.4.m4.1b"><apply id="S3.SS2.p3.4.m4.1.1.cmml" xref="S3.SS2.p3.4.m4.1.1"><eq id="S3.SS2.p3.4.m4.1.1.1.cmml" xref="S3.SS2.p3.4.m4.1.1.1"></eq><apply id="S3.SS2.p3.4.m4.1.1.2.cmml" xref="S3.SS2.p3.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.p3.4.m4.1.1.2.1.cmml" xref="S3.SS2.p3.4.m4.1.1.2">subscript</csymbol><ci id="S3.SS2.p3.4.m4.1.1.2.2.cmml" xref="S3.SS2.p3.4.m4.1.1.2.2">𝜎</ci><ci id="S3.SS2.p3.4.m4.1.1.2.3.cmml" xref="S3.SS2.p3.4.m4.1.1.2.3">𝑦</ci></apply><apply id="S3.SS2.p3.4.m4.1.1.3.cmml" xref="S3.SS2.p3.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.p3.4.m4.1.1.3.1.cmml" xref="S3.SS2.p3.4.m4.1.1.3">superscript</csymbol><cn id="S3.SS2.p3.4.m4.1.1.3.2.cmml" type="integer" xref="S3.SS2.p3.4.m4.1.1.3.2">10</cn><apply id="S3.SS2.p3.4.m4.1.1.3.3.cmml" xref="S3.SS2.p3.4.m4.1.1.3.3"><minus id="S3.SS2.p3.4.m4.1.1.3.3.1.cmml" xref="S3.SS2.p3.4.m4.1.1.3.3"></minus><cn id="S3.SS2.p3.4.m4.1.1.3.3.2.cmml" type="integer" xref="S3.SS2.p3.4.m4.1.1.3.3.2">16</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.4.m4.1c">\sigma_{y}=10^{-16}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.4.m4.1d">italic_σ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT = 10 start_POSTSUPERSCRIPT - 16 end_POSTSUPERSCRIPT</annotation></semantics></math> has been bested by three orders of magnitude by the most precise Earth-based clocks <cite class="ltx_cite ltx_citemacro_citep">(Oelker et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib57" title="">2019</a>)</cite>. 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1 end_POSTSUPERSCRIPT</annotation></semantics></math> as applies in the case white frequency modulation noise, a scaling demonstrated to hold for <math alttext="\tau<10^{5}\;" class="ltx_Math" display="inline" id="S3.SS2.p3.6.m6.1"><semantics id="S3.SS2.p3.6.m6.1a"><mrow id="S3.SS2.p3.6.m6.1.1" xref="S3.SS2.p3.6.m6.1.1.cmml"><mi id="S3.SS2.p3.6.m6.1.1.2" xref="S3.SS2.p3.6.m6.1.1.2.cmml">τ</mi><mo id="S3.SS2.p3.6.m6.1.1.1" xref="S3.SS2.p3.6.m6.1.1.1.cmml"><</mo><msup id="S3.SS2.p3.6.m6.1.1.3" xref="S3.SS2.p3.6.m6.1.1.3.cmml"><mn id="S3.SS2.p3.6.m6.1.1.3.2" xref="S3.SS2.p3.6.m6.1.1.3.2.cmml">10</mn><mn id="S3.SS2.p3.6.m6.1.1.3.3" xref="S3.SS2.p3.6.m6.1.1.3.3.cmml">5</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.6.m6.1b"><apply id="S3.SS2.p3.6.m6.1.1.cmml" xref="S3.SS2.p3.6.m6.1.1"><lt id="S3.SS2.p3.6.m6.1.1.1.cmml" xref="S3.SS2.p3.6.m6.1.1.1"></lt><ci id="S3.SS2.p3.6.m6.1.1.2.cmml" xref="S3.SS2.p3.6.m6.1.1.2">𝜏</ci><apply id="S3.SS2.p3.6.m6.1.1.3.cmml" xref="S3.SS2.p3.6.m6.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.p3.6.m6.1.1.3.1.cmml" xref="S3.SS2.p3.6.m6.1.1.3">superscript</csymbol><cn id="S3.SS2.p3.6.m6.1.1.3.2.cmml" type="integer" xref="S3.SS2.p3.6.m6.1.1.3.2">10</cn><cn id="S3.SS2.p3.6.m6.1.1.3.3.cmml" type="integer" xref="S3.SS2.p3.6.m6.1.1.3.3">5</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.6.m6.1c">\tau<10^{5}\;</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.6.m6.1d">italic_τ < 10 start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT</annotation></semantics></math>s in the case of the Deep Space Atomic Clock (but sometimes only to <math alttext="\tau\sim 10^{4}\;" class="ltx_Math" display="inline" id="S3.SS2.p3.7.m7.1"><semantics id="S3.SS2.p3.7.m7.1a"><mrow id="S3.SS2.p3.7.m7.1.1" xref="S3.SS2.p3.7.m7.1.1.cmml"><mi id="S3.SS2.p3.7.m7.1.1.2" xref="S3.SS2.p3.7.m7.1.1.2.cmml">τ</mi><mo id="S3.SS2.p3.7.m7.1.1.1" xref="S3.SS2.p3.7.m7.1.1.1.cmml">∼</mo><msup id="S3.SS2.p3.7.m7.1.1.3" xref="S3.SS2.p3.7.m7.1.1.3.cmml"><mn id="S3.SS2.p3.7.m7.1.1.3.2" xref="S3.SS2.p3.7.m7.1.1.3.2.cmml">10</mn><mn id="S3.SS2.p3.7.m7.1.1.3.3" xref="S3.SS2.p3.7.m7.1.1.3.3.cmml">4</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.7.m7.1b"><apply id="S3.SS2.p3.7.m7.1.1.cmml" xref="S3.SS2.p3.7.m7.1.1"><csymbol cd="latexml" id="S3.SS2.p3.7.m7.1.1.1.cmml" xref="S3.SS2.p3.7.m7.1.1.1">similar-to</csymbol><ci id="S3.SS2.p3.7.m7.1.1.2.cmml" xref="S3.SS2.p3.7.m7.1.1.2">𝜏</ci><apply id="S3.SS2.p3.7.m7.1.1.3.cmml" xref="S3.SS2.p3.7.m7.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.p3.7.m7.1.1.3.1.cmml" xref="S3.SS2.p3.7.m7.1.1.3">superscript</csymbol><cn id="S3.SS2.p3.7.m7.1.1.3.2.cmml" type="integer" xref="S3.SS2.p3.7.m7.1.1.3.2">10</cn><cn id="S3.SS2.p3.7.m7.1.1.3.3.cmml" type="integer" xref="S3.SS2.p3.7.m7.1.1.3.3">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.7.m7.1c">\tau\sim 10^{4}\;</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.7.m7.1d">italic_τ ∼ 10 start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT</annotation></semantics></math>s for the most precise atomic clocks). When white frequency modulation noise no longer applies, clocks often enter the ‘flicker frequency modulation’ noise regime where <math alttext="\sigma_{y}(\tau)" class="ltx_Math" display="inline" id="S3.SS2.p3.8.m8.1"><semantics id="S3.SS2.p3.8.m8.1a"><mrow id="S3.SS2.p3.8.m8.1.2" xref="S3.SS2.p3.8.m8.1.2.cmml"><msub id="S3.SS2.p3.8.m8.1.2.2" xref="S3.SS2.p3.8.m8.1.2.2.cmml"><mi id="S3.SS2.p3.8.m8.1.2.2.2" xref="S3.SS2.p3.8.m8.1.2.2.2.cmml">σ</mi><mi id="S3.SS2.p3.8.m8.1.2.2.3" xref="S3.SS2.p3.8.m8.1.2.2.3.cmml">y</mi></msub><mo id="S3.SS2.p3.8.m8.1.2.1" xref="S3.SS2.p3.8.m8.1.2.1.cmml">⁢</mo><mrow id="S3.SS2.p3.8.m8.1.2.3.2" xref="S3.SS2.p3.8.m8.1.2.cmml"><mo id="S3.SS2.p3.8.m8.1.2.3.2.1" stretchy="false" xref="S3.SS2.p3.8.m8.1.2.cmml">(</mo><mi id="S3.SS2.p3.8.m8.1.1" xref="S3.SS2.p3.8.m8.1.1.cmml">τ</mi><mo id="S3.SS2.p3.8.m8.1.2.3.2.2" stretchy="false" xref="S3.SS2.p3.8.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.8.m8.1b"><apply id="S3.SS2.p3.8.m8.1.2.cmml" xref="S3.SS2.p3.8.m8.1.2"><times id="S3.SS2.p3.8.m8.1.2.1.cmml" xref="S3.SS2.p3.8.m8.1.2.1"></times><apply id="S3.SS2.p3.8.m8.1.2.2.cmml" xref="S3.SS2.p3.8.m8.1.2.2"><csymbol cd="ambiguous" id="S3.SS2.p3.8.m8.1.2.2.1.cmml" xref="S3.SS2.p3.8.m8.1.2.2">subscript</csymbol><ci id="S3.SS2.p3.8.m8.1.2.2.2.cmml" xref="S3.SS2.p3.8.m8.1.2.2.2">𝜎</ci><ci id="S3.SS2.p3.8.m8.1.2.2.3.cmml" xref="S3.SS2.p3.8.m8.1.2.2.3">𝑦</ci></apply><ci id="S3.SS2.p3.8.m8.1.1.cmml" xref="S3.SS2.p3.8.m8.1.1">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.8.m8.1c">\sigma_{y}(\tau)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.8.m8.1d">italic_σ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ( italic_τ )</annotation></semantics></math> is constant with <math alttext="\tau" class="ltx_Math" display="inline" id="S3.SS2.p3.9.m9.1"><semantics id="S3.SS2.p3.9.m9.1a"><mi id="S3.SS2.p3.9.m9.1.1" xref="S3.SS2.p3.9.m9.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.9.m9.1b"><ci id="S3.SS2.p3.9.m9.1.1.cmml" xref="S3.SS2.p3.9.m9.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.9.m9.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.9.m9.1d">italic_τ</annotation></semantics></math> and the clock noise power flattens in <math alttext="f" class="ltx_Math" display="inline" id="S3.SS2.p3.10.m10.1"><semantics id="S3.SS2.p3.10.m10.1a"><mi id="S3.SS2.p3.10.m10.1.1" xref="S3.SS2.p3.10.m10.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.10.m10.1b"><ci id="S3.SS2.p3.10.m10.1.1.cmml" xref="S3.SS2.p3.10.m10.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.10.m10.1c">f</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.10.m10.1d">italic_f</annotation></semantics></math> by one index. The pink dashed curves in Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S3.F5" title="Figure 5 ‣ 3.1 Two-arm time-delay interferometry forecasts ‣ 3 Strain power sensitivity ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">5</span></a> show that transitioning this flicker clock noise scaling at <math alttext="f<10^{-4}" class="ltx_Math" display="inline" id="S3.SS2.p3.11.m11.1"><semantics id="S3.SS2.p3.11.m11.1a"><mrow id="S3.SS2.p3.11.m11.1.1" xref="S3.SS2.p3.11.m11.1.1.cmml"><mi id="S3.SS2.p3.11.m11.1.1.2" xref="S3.SS2.p3.11.m11.1.1.2.cmml">f</mi><mo id="S3.SS2.p3.11.m11.1.1.1" xref="S3.SS2.p3.11.m11.1.1.1.cmml"><</mo><msup id="S3.SS2.p3.11.m11.1.1.3" xref="S3.SS2.p3.11.m11.1.1.3.cmml"><mn id="S3.SS2.p3.11.m11.1.1.3.2" xref="S3.SS2.p3.11.m11.1.1.3.2.cmml">10</mn><mrow id="S3.SS2.p3.11.m11.1.1.3.3" xref="S3.SS2.p3.11.m11.1.1.3.3.cmml"><mo id="S3.SS2.p3.11.m11.1.1.3.3a" xref="S3.SS2.p3.11.m11.1.1.3.3.cmml">−</mo><mn id="S3.SS2.p3.11.m11.1.1.3.3.2" xref="S3.SS2.p3.11.m11.1.1.3.3.2.cmml">4</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.11.m11.1b"><apply id="S3.SS2.p3.11.m11.1.1.cmml" xref="S3.SS2.p3.11.m11.1.1"><lt id="S3.SS2.p3.11.m11.1.1.1.cmml" xref="S3.SS2.p3.11.m11.1.1.1"></lt><ci id="S3.SS2.p3.11.m11.1.1.2.cmml" xref="S3.SS2.p3.11.m11.1.1.2">𝑓</ci><apply id="S3.SS2.p3.11.m11.1.1.3.cmml" xref="S3.SS2.p3.11.m11.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.p3.11.m11.1.1.3.1.cmml" xref="S3.SS2.p3.11.m11.1.1.3">superscript</csymbol><cn id="S3.SS2.p3.11.m11.1.1.3.2.cmml" type="integer" xref="S3.SS2.p3.11.m11.1.1.3.2">10</cn><apply id="S3.SS2.p3.11.m11.1.1.3.3.cmml" xref="S3.SS2.p3.11.m11.1.1.3.3"><minus id="S3.SS2.p3.11.m11.1.1.3.3.1.cmml" xref="S3.SS2.p3.11.m11.1.1.3.3"></minus><cn id="S3.SS2.p3.11.m11.1.1.3.3.2.cmml" type="integer" xref="S3.SS2.p3.11.m11.1.1.3.3.2">4</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.11.m11.1c">f<10^{-4}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.11.m11.1d">italic_f < 10 start_POSTSUPERSCRIPT - 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15 end_POSTSUPERSCRIPT</annotation></semantics></math> has a relatively modest affect on the sensitivity.</p> </div> </section> <section class="ltx_subsection" id="S3.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.3 </span>Doppler tracking forecasts</h3> <div class="ltx_para" id="S3.SS3.p1"> <p class="ltx_p" id="S3.SS3.p1.5">A potentially simpler design uses Earth for one element in the single-arm setup. This approach has an extensive history and is called Doppler tracking (for a review, see <cite class="ltx_cite ltx_citemacro_citet">Armstrong <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib9" title="">2006b</a></cite>). Beyond requiring just a single spacecraft, such a setup has several other advantages: 1) the Earth station’s non-gravitational accelerations are potentially negligible, 2) substantial resources are available at the Earth station, allowing kilowatt up-links, larger collecting areas, and cryogenic cooling, 3) the Earth station can be equipped with an ultra-precise atomic clock. However, the mechanical distortions of the instruments from gravity and terrestrial temperature cycles are larger on Earth, and an Earth station has to further contend with propagation delays due to the Earth’s atmosphere. For reference, our estimates for a purely space-based time-delay interferometry setup with <math alttext="r=L=30\," class="ltx_Math" display="inline" id="S3.SS3.p1.1.m1.1"><semantics id="S3.SS3.p1.1.m1.1a"><mrow id="S3.SS3.p1.1.m1.1.1" xref="S3.SS3.p1.1.m1.1.1.cmml"><mi id="S3.SS3.p1.1.m1.1.1.2" xref="S3.SS3.p1.1.m1.1.1.2.cmml">r</mi><mo id="S3.SS3.p1.1.m1.1.1.3" xref="S3.SS3.p1.1.m1.1.1.3.cmml">=</mo><mi id="S3.SS3.p1.1.m1.1.1.4" xref="S3.SS3.p1.1.m1.1.1.4.cmml">L</mi><mo id="S3.SS3.p1.1.m1.1.1.5" xref="S3.SS3.p1.1.m1.1.1.5.cmml">=</mo><mn id="S3.SS3.p1.1.m1.1.1.6" xref="S3.SS3.p1.1.m1.1.1.6.cmml">30</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p1.1.m1.1b"><apply id="S3.SS3.p1.1.m1.1.1.cmml" xref="S3.SS3.p1.1.m1.1.1"><and id="S3.SS3.p1.1.m1.1.1a.cmml" xref="S3.SS3.p1.1.m1.1.1"></and><apply id="S3.SS3.p1.1.m1.1.1b.cmml" xref="S3.SS3.p1.1.m1.1.1"><eq id="S3.SS3.p1.1.m1.1.1.3.cmml" xref="S3.SS3.p1.1.m1.1.1.3"></eq><ci id="S3.SS3.p1.1.m1.1.1.2.cmml" xref="S3.SS3.p1.1.m1.1.1.2">𝑟</ci><ci id="S3.SS3.p1.1.m1.1.1.4.cmml" xref="S3.SS3.p1.1.m1.1.1.4">𝐿</ci></apply><apply id="S3.SS3.p1.1.m1.1.1c.cmml" xref="S3.SS3.p1.1.m1.1.1"><eq id="S3.SS3.p1.1.m1.1.1.5.cmml" xref="S3.SS3.p1.1.m1.1.1.5"></eq><share href="https://arxiv.org/html/2411.15072v1#S3.SS3.p1.1.m1.1.1.4.cmml" id="S3.SS3.p1.1.m1.1.1d.cmml" xref="S3.SS3.p1.1.m1.1.1"></share><cn id="S3.SS3.p1.1.m1.1.1.6.cmml" type="integer" xref="S3.SS3.p1.1.m1.1.1.6">30</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p1.1.m1.1c">r=L=30\,</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p1.1.m1.1d">italic_r = italic_L = 30</annotation></semantics></math> AU have sensitivities of <math alttext="\sqrt{fS_{h}}\sim 10^{-16}" class="ltx_Math" display="inline" id="S3.SS3.p1.2.m2.1"><semantics id="S3.SS3.p1.2.m2.1a"><mrow id="S3.SS3.p1.2.m2.1.1" xref="S3.SS3.p1.2.m2.1.1.cmml"><msqrt id="S3.SS3.p1.2.m2.1.1.2" xref="S3.SS3.p1.2.m2.1.1.2.cmml"><mrow id="S3.SS3.p1.2.m2.1.1.2.2" xref="S3.SS3.p1.2.m2.1.1.2.2.cmml"><mi id="S3.SS3.p1.2.m2.1.1.2.2.2" xref="S3.SS3.p1.2.m2.1.1.2.2.2.cmml">f</mi><mo id="S3.SS3.p1.2.m2.1.1.2.2.1" xref="S3.SS3.p1.2.m2.1.1.2.2.1.cmml">⁢</mo><msub id="S3.SS3.p1.2.m2.1.1.2.2.3" xref="S3.SS3.p1.2.m2.1.1.2.2.3.cmml"><mi id="S3.SS3.p1.2.m2.1.1.2.2.3.2" xref="S3.SS3.p1.2.m2.1.1.2.2.3.2.cmml">S</mi><mi id="S3.SS3.p1.2.m2.1.1.2.2.3.3" xref="S3.SS3.p1.2.m2.1.1.2.2.3.3.cmml">h</mi></msub></mrow></msqrt><mo id="S3.SS3.p1.2.m2.1.1.1" xref="S3.SS3.p1.2.m2.1.1.1.cmml">∼</mo><msup id="S3.SS3.p1.2.m2.1.1.3" xref="S3.SS3.p1.2.m2.1.1.3.cmml"><mn id="S3.SS3.p1.2.m2.1.1.3.2" xref="S3.SS3.p1.2.m2.1.1.3.2.cmml">10</mn><mrow id="S3.SS3.p1.2.m2.1.1.3.3" xref="S3.SS3.p1.2.m2.1.1.3.3.cmml"><mo id="S3.SS3.p1.2.m2.1.1.3.3a" xref="S3.SS3.p1.2.m2.1.1.3.3.cmml">−</mo><mn id="S3.SS3.p1.2.m2.1.1.3.3.2" xref="S3.SS3.p1.2.m2.1.1.3.3.2.cmml">16</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p1.2.m2.1b"><apply id="S3.SS3.p1.2.m2.1.1.cmml" xref="S3.SS3.p1.2.m2.1.1"><csymbol cd="latexml" id="S3.SS3.p1.2.m2.1.1.1.cmml" xref="S3.SS3.p1.2.m2.1.1.1">similar-to</csymbol><apply id="S3.SS3.p1.2.m2.1.1.2.cmml" xref="S3.SS3.p1.2.m2.1.1.2"><root id="S3.SS3.p1.2.m2.1.1.2a.cmml" xref="S3.SS3.p1.2.m2.1.1.2"></root><apply id="S3.SS3.p1.2.m2.1.1.2.2.cmml" xref="S3.SS3.p1.2.m2.1.1.2.2"><times id="S3.SS3.p1.2.m2.1.1.2.2.1.cmml" xref="S3.SS3.p1.2.m2.1.1.2.2.1"></times><ci id="S3.SS3.p1.2.m2.1.1.2.2.2.cmml" xref="S3.SS3.p1.2.m2.1.1.2.2.2">𝑓</ci><apply id="S3.SS3.p1.2.m2.1.1.2.2.3.cmml" xref="S3.SS3.p1.2.m2.1.1.2.2.3"><csymbol cd="ambiguous" id="S3.SS3.p1.2.m2.1.1.2.2.3.1.cmml" xref="S3.SS3.p1.2.m2.1.1.2.2.3">subscript</csymbol><ci id="S3.SS3.p1.2.m2.1.1.2.2.3.2.cmml" xref="S3.SS3.p1.2.m2.1.1.2.2.3.2">𝑆</ci><ci id="S3.SS3.p1.2.m2.1.1.2.2.3.3.cmml" xref="S3.SS3.p1.2.m2.1.1.2.2.3.3">ℎ</ci></apply></apply></apply><apply id="S3.SS3.p1.2.m2.1.1.3.cmml" xref="S3.SS3.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.p1.2.m2.1.1.3.1.cmml" xref="S3.SS3.p1.2.m2.1.1.3">superscript</csymbol><cn id="S3.SS3.p1.2.m2.1.1.3.2.cmml" type="integer" xref="S3.SS3.p1.2.m2.1.1.3.2">10</cn><apply id="S3.SS3.p1.2.m2.1.1.3.3.cmml" xref="S3.SS3.p1.2.m2.1.1.3.3"><minus id="S3.SS3.p1.2.m2.1.1.3.3.1.cmml" xref="S3.SS3.p1.2.m2.1.1.3.3"></minus><cn id="S3.SS3.p1.2.m2.1.1.3.3.2.cmml" type="integer" xref="S3.SS3.p1.2.m2.1.1.3.3.2">16</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p1.2.m2.1c">\sqrt{fS_{h}}\sim 10^{-16}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p1.2.m2.1d">square-root start_ARG italic_f italic_S start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT end_ARG ∼ 10 start_POSTSUPERSCRIPT - 16 end_POSTSUPERSCRIPT</annotation></semantics></math> for <math alttext="f=10^{-6}" class="ltx_Math" display="inline" id="S3.SS3.p1.3.m3.1"><semantics id="S3.SS3.p1.3.m3.1a"><mrow id="S3.SS3.p1.3.m3.1.1" xref="S3.SS3.p1.3.m3.1.1.cmml"><mi id="S3.SS3.p1.3.m3.1.1.2" xref="S3.SS3.p1.3.m3.1.1.2.cmml">f</mi><mo id="S3.SS3.p1.3.m3.1.1.1" xref="S3.SS3.p1.3.m3.1.1.1.cmml">=</mo><msup id="S3.SS3.p1.3.m3.1.1.3" xref="S3.SS3.p1.3.m3.1.1.3.cmml"><mn id="S3.SS3.p1.3.m3.1.1.3.2" xref="S3.SS3.p1.3.m3.1.1.3.2.cmml">10</mn><mrow id="S3.SS3.p1.3.m3.1.1.3.3" xref="S3.SS3.p1.3.m3.1.1.3.3.cmml"><mo id="S3.SS3.p1.3.m3.1.1.3.3a" xref="S3.SS3.p1.3.m3.1.1.3.3.cmml">−</mo><mn id="S3.SS3.p1.3.m3.1.1.3.3.2" xref="S3.SS3.p1.3.m3.1.1.3.3.2.cmml">6</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p1.3.m3.1b"><apply id="S3.SS3.p1.3.m3.1.1.cmml" xref="S3.SS3.p1.3.m3.1.1"><eq id="S3.SS3.p1.3.m3.1.1.1.cmml" xref="S3.SS3.p1.3.m3.1.1.1"></eq><ci id="S3.SS3.p1.3.m3.1.1.2.cmml" xref="S3.SS3.p1.3.m3.1.1.2">𝑓</ci><apply id="S3.SS3.p1.3.m3.1.1.3.cmml" xref="S3.SS3.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.p1.3.m3.1.1.3.1.cmml" xref="S3.SS3.p1.3.m3.1.1.3">superscript</csymbol><cn id="S3.SS3.p1.3.m3.1.1.3.2.cmml" type="integer" xref="S3.SS3.p1.3.m3.1.1.3.2">10</cn><apply id="S3.SS3.p1.3.m3.1.1.3.3.cmml" xref="S3.SS3.p1.3.m3.1.1.3.3"><minus id="S3.SS3.p1.3.m3.1.1.3.3.1.cmml" xref="S3.SS3.p1.3.m3.1.1.3.3"></minus><cn id="S3.SS3.p1.3.m3.1.1.3.3.2.cmml" type="integer" xref="S3.SS3.p1.3.m3.1.1.3.3.2">6</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p1.3.m3.1c">f=10^{-6}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p1.3.m3.1d">italic_f = 10 start_POSTSUPERSCRIPT - 6 end_POSTSUPERSCRIPT</annotation></semantics></math>Hz, which corresponds to detecting variations of <math alttext="0.3\," class="ltx_Math" display="inline" id="S3.SS3.p1.4.m4.1"><semantics id="S3.SS3.p1.4.m4.1a"><mn id="S3.SS3.p1.4.m4.1.1" xref="S3.SS3.p1.4.m4.1.1.cmml">0.3</mn><annotation-xml encoding="MathML-Content" id="S3.SS3.p1.4.m4.1b"><cn id="S3.SS3.p1.4.m4.1.1.cmml" type="float" xref="S3.SS3.p1.4.m4.1.1">0.3</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p1.4.m4.1c">0.3\,</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p1.4.m4.1d">0.3</annotation></semantics></math>cm. It is conceivable that comparable distance control could be achieved from Earth since lunar laser ranging has achieved <math alttext="<0.1~{}" class="ltx_Math" display="inline" id="S3.SS3.p1.5.m5.1"><semantics id="S3.SS3.p1.5.m5.1a"><mrow id="S3.SS3.p1.5.m5.1.1" xref="S3.SS3.p1.5.m5.1.1.cmml"><mi id="S3.SS3.p1.5.m5.1.1.2" xref="S3.SS3.p1.5.m5.1.1.2.cmml"></mi><mo id="S3.SS3.p1.5.m5.1.1.1" xref="S3.SS3.p1.5.m5.1.1.1.cmml"><</mo><mn id="S3.SS3.p1.5.m5.1.1.3" xref="S3.SS3.p1.5.m5.1.1.3.cmml">0.1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p1.5.m5.1b"><apply id="S3.SS3.p1.5.m5.1.1.cmml" xref="S3.SS3.p1.5.m5.1.1"><lt id="S3.SS3.p1.5.m5.1.1.1.cmml" xref="S3.SS3.p1.5.m5.1.1.1"></lt><csymbol cd="latexml" id="S3.SS3.p1.5.m5.1.1.2.cmml" xref="S3.SS3.p1.5.m5.1.1.2">absent</csymbol><cn id="S3.SS3.p1.5.m5.1.1.3.cmml" type="float" xref="S3.SS3.p1.5.m5.1.1.3">0.1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p1.5.m5.1c"><0.1~{}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p1.5.m5.1d">< 0.1</annotation></semantics></math>cm measurements of the Earth-Moon distance <cite class="ltx_cite ltx_citemacro_citep">(Colmenares et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib20" title="">2023</a>)</cite>.</p> </div> <div class="ltx_para" id="S3.SS3.p2"> <p class="ltx_p" id="S3.SS3.p2.1">Previous Doppler tracking, which has been most successfully executed with the Cassini spacecraft, has been limited by mechanical variations in the analog path owing to e.g. thermal expansion or gravitational stresses. For Cassini, this resulted in centimeter-scale noise over relevant frequencies <cite class="ltx_cite ltx_citemacro_citep">(Armstrong, <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib9" title="">2006b</a>)</cite>. However, mechanical distortions could potentially be further reduced by using a smaller terrestrial radio dish constructed to be more rigid <cite class="ltx_cite ltx_citemacro_citep">(Armstrong, <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib9" title="">2006b</a>)</cite>, by using an optical laser setup, or by employing a precise atomic clock on the spacecraft. More details on the latter two are provided below.</p> </div> <div class="ltx_para" id="S3.SS3.p3"> <p class="ltx_p" id="S3.SS3.p3.1">If mechanical noise can be significantly reduced, atmospheric delays are likely to become the limiting factor. The atmospheric delay that is most worrisome is the tropospheric “wet” delay from water vapor in the atmosphere because it is frequency independent – it cannot be removed by observing multiple frequencies. Although typical wet delays vary by several centimeters on day timescales towards zenith, much of this delay can be removed by precise atmospheric monitoring, such as with water vapor radiometers. Indeed, such monitoring occurred for the Cassini Doppler tracking system, for which <cite class="ltx_cite ltx_citemacro_citet">Woo & Guo (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib73" title="">2004</a>)</cite> estimated its error by differencing the predictions of two identical systems. They predicted that the delay could be corrected to <math alttext="\sim 0.01~{}" class="ltx_Math" display="inline" id="S3.SS3.p3.1.m1.1"><semantics id="S3.SS3.p3.1.m1.1a"><mrow id="S3.SS3.p3.1.m1.1.1" xref="S3.SS3.p3.1.m1.1.1.cmml"><mi id="S3.SS3.p3.1.m1.1.1.2" xref="S3.SS3.p3.1.m1.1.1.2.cmml"></mi><mo id="S3.SS3.p3.1.m1.1.1.1" xref="S3.SS3.p3.1.m1.1.1.1.cmml">∼</mo><mn id="S3.SS3.p3.1.m1.1.1.3" xref="S3.SS3.p3.1.m1.1.1.3.cmml">0.01</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.1.m1.1b"><apply id="S3.SS3.p3.1.m1.1.1.cmml" xref="S3.SS3.p3.1.m1.1.1"><csymbol cd="latexml" id="S3.SS3.p3.1.m1.1.1.1.cmml" xref="S3.SS3.p3.1.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S3.SS3.p3.1.m1.1.1.2.cmml" xref="S3.SS3.p3.1.m1.1.1.2">absent</csymbol><cn id="S3.SS3.p3.1.m1.1.1.3.cmml" type="float" xref="S3.SS3.p3.1.m1.1.1.3">0.01</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.1.m1.1c">\sim 0.01~{}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.1.m1.1d">∼ 0.01</annotation></semantics></math> cm over times of hours to months, although their differencing methodology should be taken as a lower bound on the true error as it does not account for modeling uncertainties.</p> </div> <figure class="ltx_figure" id="S3.F6"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="332" id="S3.F6.g1" src="x6.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 6: </span>Gravitational wave strain sensitivities for the <span class="ltx_text ltx_font_bold" id="S3.F6.22.1">third design architecture: the Doppler tracking concept</span>, sending <math alttext="\lambda=1\," class="ltx_Math" display="inline" id="S3.F6.11.m1.1"><semantics id="S3.F6.11.m1.1b"><mrow id="S3.F6.11.m1.1.1" xref="S3.F6.11.m1.1.1.cmml"><mi id="S3.F6.11.m1.1.1.2" xref="S3.F6.11.m1.1.1.2.cmml">λ</mi><mo id="S3.F6.11.m1.1.1.1" xref="S3.F6.11.m1.1.1.1.cmml">=</mo><mn id="S3.F6.11.m1.1.1.3" xref="S3.F6.11.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.F6.11.m1.1c"><apply id="S3.F6.11.m1.1.1.cmml" xref="S3.F6.11.m1.1.1"><eq id="S3.F6.11.m1.1.1.1.cmml" xref="S3.F6.11.m1.1.1.1"></eq><ci id="S3.F6.11.m1.1.1.2.cmml" xref="S3.F6.11.m1.1.1.2">𝜆</ci><cn id="S3.F6.11.m1.1.1.3.cmml" type="integer" xref="S3.F6.11.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F6.11.m1.1d">\lambda=1\,</annotation><annotation encoding="application/x-llamapun" id="S3.F6.11.m1.1e">italic_λ = 1</annotation></semantics></math>cm radio signals between an Earth station and a <math alttext="r=10~{}" class="ltx_Math" display="inline" id="S3.F6.12.m2.1"><semantics id="S3.F6.12.m2.1b"><mrow id="S3.F6.12.m2.1.1" xref="S3.F6.12.m2.1.1.cmml"><mi id="S3.F6.12.m2.1.1.2" xref="S3.F6.12.m2.1.1.2.cmml">r</mi><mo id="S3.F6.12.m2.1.1.1" xref="S3.F6.12.m2.1.1.1.cmml">=</mo><mn id="S3.F6.12.m2.1.1.3" xref="S3.F6.12.m2.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.F6.12.m2.1c"><apply id="S3.F6.12.m2.1.1.cmml" xref="S3.F6.12.m2.1.1"><eq id="S3.F6.12.m2.1.1.1.cmml" xref="S3.F6.12.m2.1.1.1"></eq><ci id="S3.F6.12.m2.1.1.2.cmml" xref="S3.F6.12.m2.1.1.2">𝑟</ci><cn id="S3.F6.12.m2.1.1.3.cmml" type="integer" xref="S3.F6.12.m2.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F6.12.m2.1d">r=10~{}</annotation><annotation encoding="application/x-llamapun" id="S3.F6.12.m2.1e">italic_r = 10</annotation></semantics></math>AU (left panel) and <math alttext="r=30~{}" class="ltx_Math" display="inline" id="S3.F6.13.m3.1"><semantics id="S3.F6.13.m3.1b"><mrow id="S3.F6.13.m3.1.1" xref="S3.F6.13.m3.1.1.cmml"><mi id="S3.F6.13.m3.1.1.2" xref="S3.F6.13.m3.1.1.2.cmml">r</mi><mo id="S3.F6.13.m3.1.1.1" xref="S3.F6.13.m3.1.1.1.cmml">=</mo><mn id="S3.F6.13.m3.1.1.3" xref="S3.F6.13.m3.1.1.3.cmml">30</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.F6.13.m3.1c"><apply id="S3.F6.13.m3.1.1.cmml" xref="S3.F6.13.m3.1.1"><eq id="S3.F6.13.m3.1.1.1.cmml" xref="S3.F6.13.m3.1.1.1"></eq><ci id="S3.F6.13.m3.1.1.2.cmml" xref="S3.F6.13.m3.1.1.2">𝑟</ci><cn id="S3.F6.13.m3.1.1.3.cmml" type="integer" xref="S3.F6.13.m3.1.1.3">30</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F6.13.m3.1d">r=30~{}</annotation><annotation encoding="application/x-llamapun" id="S3.F6.13.m3.1e">italic_r = 30</annotation></semantics></math>AU (right panel) spacecraft. The solid blue curve is the strain sensitivity that an ideal instrument could achieve if its sensitivity is set by spacecraft accelerations and radiometer noise with the specified <math alttext="C/N_{0}" class="ltx_Math" display="inline" id="S3.F6.14.m4.1"><semantics id="S3.F6.14.m4.1b"><mrow id="S3.F6.14.m4.1.1" xref="S3.F6.14.m4.1.1.cmml"><mi id="S3.F6.14.m4.1.1.2" xref="S3.F6.14.m4.1.1.2.cmml">C</mi><mo id="S3.F6.14.m4.1.1.1" xref="S3.F6.14.m4.1.1.1.cmml">/</mo><msub id="S3.F6.14.m4.1.1.3" xref="S3.F6.14.m4.1.1.3.cmml"><mi id="S3.F6.14.m4.1.1.3.2" xref="S3.F6.14.m4.1.1.3.2.cmml">N</mi><mn id="S3.F6.14.m4.1.1.3.3" xref="S3.F6.14.m4.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.F6.14.m4.1c"><apply id="S3.F6.14.m4.1.1.cmml" xref="S3.F6.14.m4.1.1"><divide id="S3.F6.14.m4.1.1.1.cmml" xref="S3.F6.14.m4.1.1.1"></divide><ci id="S3.F6.14.m4.1.1.2.cmml" xref="S3.F6.14.m4.1.1.2">𝐶</ci><apply id="S3.F6.14.m4.1.1.3.cmml" xref="S3.F6.14.m4.1.1.3"><csymbol cd="ambiguous" id="S3.F6.14.m4.1.1.3.1.cmml" xref="S3.F6.14.m4.1.1.3">subscript</csymbol><ci id="S3.F6.14.m4.1.1.3.2.cmml" xref="S3.F6.14.m4.1.1.3.2">𝑁</ci><cn id="S3.F6.14.m4.1.1.3.3.cmml" type="integer" xref="S3.F6.14.m4.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F6.14.m4.1d">C/N_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.F6.14.m4.1e">italic_C / italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>. Mechanical noise and Earth’s atmosphere likely limit the sensitivity over this ideal curve. The magenta dashed curves illustrate the residual noise power if the atmosphere and mechanical noise can be subtracted to a residual power that has equal variance per <math alttext="\log f" class="ltx_Math" display="inline" id="S3.F6.15.m5.1"><semantics id="S3.F6.15.m5.1b"><mrow id="S3.F6.15.m5.1.1" xref="S3.F6.15.m5.1.1.cmml"><mi id="S3.F6.15.m5.1.1.1" xref="S3.F6.15.m5.1.1.1.cmml">log</mi><mo id="S3.F6.15.m5.1.1b" lspace="0.167em" xref="S3.F6.15.m5.1.1.cmml">⁡</mo><mi id="S3.F6.15.m5.1.1.2" xref="S3.F6.15.m5.1.1.2.cmml">f</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.F6.15.m5.1c"><apply id="S3.F6.15.m5.1.1.cmml" xref="S3.F6.15.m5.1.1"><log id="S3.F6.15.m5.1.1.1.cmml" xref="S3.F6.15.m5.1.1.1"></log><ci id="S3.F6.15.m5.1.1.2.cmml" xref="S3.F6.15.m5.1.1.2">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F6.15.m5.1d">\log f</annotation><annotation encoding="application/x-llamapun" id="S3.F6.15.m5.1e">roman_log italic_f</annotation></semantics></math> and equal to <math alttext="0.1\," class="ltx_Math" display="inline" id="S3.F6.16.m6.1"><semantics id="S3.F6.16.m6.1b"><mn id="S3.F6.16.m6.1.1" xref="S3.F6.16.m6.1.1.cmml">0.1</mn><annotation-xml encoding="MathML-Content" id="S3.F6.16.m6.1c"><cn id="S3.F6.16.m6.1.1.cmml" type="float" xref="S3.F6.16.m6.1.1">0.1</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.F6.16.m6.1d">0.1\,</annotation><annotation encoding="application/x-llamapun" id="S3.F6.16.m6.1e">0.1</annotation></semantics></math>cm and <math alttext="0.01\," class="ltx_Math" display="inline" id="S3.F6.17.m7.1"><semantics id="S3.F6.17.m7.1b"><mn id="S3.F6.17.m7.1.1" xref="S3.F6.17.m7.1.1.cmml">0.01</mn><annotation-xml encoding="MathML-Content" id="S3.F6.17.m7.1c"><cn id="S3.F6.17.m7.1.1.cmml" type="float" xref="S3.F6.17.m7.1.1">0.01</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.F6.17.m7.1d">0.01\,</annotation><annotation encoding="application/x-llamapun" id="S3.F6.17.m7.1e">0.01</annotation></semantics></math>cm, numbers motivated in the text. The orange Doppler-tracking sensitivity curve in the right panel is the same as the blue one except that it includes clock noise with <math alttext="\sigma_{y}(1s)=10^{-14}" class="ltx_Math" display="inline" id="S3.F6.18.m8.1"><semantics id="S3.F6.18.m8.1b"><mrow id="S3.F6.18.m8.1.1" xref="S3.F6.18.m8.1.1.cmml"><mrow id="S3.F6.18.m8.1.1.1" xref="S3.F6.18.m8.1.1.1.cmml"><msub id="S3.F6.18.m8.1.1.1.3" xref="S3.F6.18.m8.1.1.1.3.cmml"><mi id="S3.F6.18.m8.1.1.1.3.2" xref="S3.F6.18.m8.1.1.1.3.2.cmml">σ</mi><mi id="S3.F6.18.m8.1.1.1.3.3" xref="S3.F6.18.m8.1.1.1.3.3.cmml">y</mi></msub><mo id="S3.F6.18.m8.1.1.1.2" xref="S3.F6.18.m8.1.1.1.2.cmml">⁢</mo><mrow id="S3.F6.18.m8.1.1.1.1.1" xref="S3.F6.18.m8.1.1.1.1.1.1.cmml"><mo id="S3.F6.18.m8.1.1.1.1.1.2" stretchy="false" xref="S3.F6.18.m8.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.F6.18.m8.1.1.1.1.1.1" xref="S3.F6.18.m8.1.1.1.1.1.1.cmml"><mn id="S3.F6.18.m8.1.1.1.1.1.1.2" xref="S3.F6.18.m8.1.1.1.1.1.1.2.cmml">1</mn><mo id="S3.F6.18.m8.1.1.1.1.1.1.1" xref="S3.F6.18.m8.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S3.F6.18.m8.1.1.1.1.1.1.3" xref="S3.F6.18.m8.1.1.1.1.1.1.3.cmml">s</mi></mrow><mo id="S3.F6.18.m8.1.1.1.1.1.3" stretchy="false" xref="S3.F6.18.m8.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.F6.18.m8.1.1.2" xref="S3.F6.18.m8.1.1.2.cmml">=</mo><msup id="S3.F6.18.m8.1.1.3" xref="S3.F6.18.m8.1.1.3.cmml"><mn id="S3.F6.18.m8.1.1.3.2" xref="S3.F6.18.m8.1.1.3.2.cmml">10</mn><mrow id="S3.F6.18.m8.1.1.3.3" xref="S3.F6.18.m8.1.1.3.3.cmml"><mo id="S3.F6.18.m8.1.1.3.3b" xref="S3.F6.18.m8.1.1.3.3.cmml">−</mo><mn id="S3.F6.18.m8.1.1.3.3.2" xref="S3.F6.18.m8.1.1.3.3.2.cmml">14</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.F6.18.m8.1c"><apply id="S3.F6.18.m8.1.1.cmml" xref="S3.F6.18.m8.1.1"><eq id="S3.F6.18.m8.1.1.2.cmml" xref="S3.F6.18.m8.1.1.2"></eq><apply id="S3.F6.18.m8.1.1.1.cmml" xref="S3.F6.18.m8.1.1.1"><times id="S3.F6.18.m8.1.1.1.2.cmml" xref="S3.F6.18.m8.1.1.1.2"></times><apply id="S3.F6.18.m8.1.1.1.3.cmml" xref="S3.F6.18.m8.1.1.1.3"><csymbol cd="ambiguous" id="S3.F6.18.m8.1.1.1.3.1.cmml" xref="S3.F6.18.m8.1.1.1.3">subscript</csymbol><ci id="S3.F6.18.m8.1.1.1.3.2.cmml" xref="S3.F6.18.m8.1.1.1.3.2">𝜎</ci><ci id="S3.F6.18.m8.1.1.1.3.3.cmml" xref="S3.F6.18.m8.1.1.1.3.3">𝑦</ci></apply><apply id="S3.F6.18.m8.1.1.1.1.1.1.cmml" xref="S3.F6.18.m8.1.1.1.1.1"><times id="S3.F6.18.m8.1.1.1.1.1.1.1.cmml" xref="S3.F6.18.m8.1.1.1.1.1.1.1"></times><cn id="S3.F6.18.m8.1.1.1.1.1.1.2.cmml" type="integer" xref="S3.F6.18.m8.1.1.1.1.1.1.2">1</cn><ci id="S3.F6.18.m8.1.1.1.1.1.1.3.cmml" xref="S3.F6.18.m8.1.1.1.1.1.1.3">𝑠</ci></apply></apply><apply id="S3.F6.18.m8.1.1.3.cmml" xref="S3.F6.18.m8.1.1.3"><csymbol cd="ambiguous" id="S3.F6.18.m8.1.1.3.1.cmml" xref="S3.F6.18.m8.1.1.3">superscript</csymbol><cn id="S3.F6.18.m8.1.1.3.2.cmml" type="integer" xref="S3.F6.18.m8.1.1.3.2">10</cn><apply id="S3.F6.18.m8.1.1.3.3.cmml" xref="S3.F6.18.m8.1.1.3.3"><minus id="S3.F6.18.m8.1.1.3.3.1.cmml" xref="S3.F6.18.m8.1.1.3.3"></minus><cn id="S3.F6.18.m8.1.1.3.3.2.cmml" type="integer" xref="S3.F6.18.m8.1.1.3.3.2">14</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F6.18.m8.1d">\sigma_{y}(1s)=10^{-14}</annotation><annotation encoding="application/x-llamapun" id="S3.F6.18.m8.1e">italic_σ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ( 1 italic_s ) = 10 start_POSTSUPERSCRIPT - 14 end_POSTSUPERSCRIPT</annotation></semantics></math>. If the spacecraft is equipped with such a clock, an observable exists that can remove atmospheric and mechanical delays. The green dot dashed curve shows the <math alttext="f<10^{-5}" class="ltx_Math" display="inline" id="S3.F6.19.m9.1"><semantics id="S3.F6.19.m9.1b"><mrow id="S3.F6.19.m9.1.1" xref="S3.F6.19.m9.1.1.cmml"><mi id="S3.F6.19.m9.1.1.2" xref="S3.F6.19.m9.1.1.2.cmml">f</mi><mo id="S3.F6.19.m9.1.1.1" xref="S3.F6.19.m9.1.1.1.cmml"><</mo><msup id="S3.F6.19.m9.1.1.3" xref="S3.F6.19.m9.1.1.3.cmml"><mn id="S3.F6.19.m9.1.1.3.2" xref="S3.F6.19.m9.1.1.3.2.cmml">10</mn><mrow id="S3.F6.19.m9.1.1.3.3" xref="S3.F6.19.m9.1.1.3.3.cmml"><mo id="S3.F6.19.m9.1.1.3.3b" xref="S3.F6.19.m9.1.1.3.3.cmml">−</mo><mn id="S3.F6.19.m9.1.1.3.3.2" xref="S3.F6.19.m9.1.1.3.3.2.cmml">5</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.F6.19.m9.1c"><apply id="S3.F6.19.m9.1.1.cmml" xref="S3.F6.19.m9.1.1"><lt id="S3.F6.19.m9.1.1.1.cmml" xref="S3.F6.19.m9.1.1.1"></lt><ci id="S3.F6.19.m9.1.1.2.cmml" xref="S3.F6.19.m9.1.1.2">𝑓</ci><apply id="S3.F6.19.m9.1.1.3.cmml" xref="S3.F6.19.m9.1.1.3"><csymbol cd="ambiguous" id="S3.F6.19.m9.1.1.3.1.cmml" xref="S3.F6.19.m9.1.1.3">superscript</csymbol><cn id="S3.F6.19.m9.1.1.3.2.cmml" type="integer" xref="S3.F6.19.m9.1.1.3.2">10</cn><apply id="S3.F6.19.m9.1.1.3.3.cmml" xref="S3.F6.19.m9.1.1.3.3"><minus id="S3.F6.19.m9.1.1.3.3.1.cmml" xref="S3.F6.19.m9.1.1.3.3"></minus><cn id="S3.F6.19.m9.1.1.3.3.2.cmml" type="integer" xref="S3.F6.19.m9.1.1.3.3.2">5</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F6.19.m9.1d">f<10^{-5}</annotation><annotation encoding="application/x-llamapun" id="S3.F6.19.m9.1e">italic_f < 10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT</annotation></semantics></math>Hz ‘priority’ sensitivity goal for a future mission to Uranus of <cite class="ltx_cite ltx_citemacro_citet">Zwick et al. (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib75" title="">2024</a>)</cite>. The dotted blue points in the left panel are the best constraints using Doppler tracking, achieved with the Cassini mission with <math alttext="L\approx 10~{}" class="ltx_Math" display="inline" id="S3.F6.20.m10.1"><semantics id="S3.F6.20.m10.1b"><mrow id="S3.F6.20.m10.1.1" xref="S3.F6.20.m10.1.1.cmml"><mi id="S3.F6.20.m10.1.1.2" xref="S3.F6.20.m10.1.1.2.cmml">L</mi><mo id="S3.F6.20.m10.1.1.1" xref="S3.F6.20.m10.1.1.1.cmml">≈</mo><mn id="S3.F6.20.m10.1.1.3" xref="S3.F6.20.m10.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.F6.20.m10.1c"><apply id="S3.F6.20.m10.1.1.cmml" xref="S3.F6.20.m10.1.1"><approx id="S3.F6.20.m10.1.1.1.cmml" xref="S3.F6.20.m10.1.1.1"></approx><ci id="S3.F6.20.m10.1.1.2.cmml" xref="S3.F6.20.m10.1.1.2">𝐿</ci><cn id="S3.F6.20.m10.1.1.3.cmml" type="integer" xref="S3.F6.20.m10.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F6.20.m10.1d">L\approx 10~{}</annotation><annotation encoding="application/x-llamapun" id="S3.F6.20.m10.1e">italic_L ≈ 10</annotation></semantics></math>AU <cite class="ltx_cite ltx_citemacro_citep">(Armstrong, <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib9" title="">2006b</a>)</cite>. </figcaption> </figure> <div class="ltx_para" id="S3.SS3.p4"> <p class="ltx_p" id="S3.SS3.p4.6">Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S3.F6" title="Figure 6 ‣ 3.3 Doppler tracking forecasts ‣ 3 Strain power sensitivity ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">6</span></a> investigates the potential sensitivity of Doppler tracking of an <math alttext="L=10~{}" class="ltx_Math" display="inline" id="S3.SS3.p4.1.m1.1"><semantics id="S3.SS3.p4.1.m1.1a"><mrow id="S3.SS3.p4.1.m1.1.1" xref="S3.SS3.p4.1.m1.1.1.cmml"><mi id="S3.SS3.p4.1.m1.1.1.2" xref="S3.SS3.p4.1.m1.1.1.2.cmml">L</mi><mo id="S3.SS3.p4.1.m1.1.1.1" xref="S3.SS3.p4.1.m1.1.1.1.cmml">=</mo><mn id="S3.SS3.p4.1.m1.1.1.3" xref="S3.SS3.p4.1.m1.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p4.1.m1.1b"><apply id="S3.SS3.p4.1.m1.1.1.cmml" xref="S3.SS3.p4.1.m1.1.1"><eq id="S3.SS3.p4.1.m1.1.1.1.cmml" xref="S3.SS3.p4.1.m1.1.1.1"></eq><ci id="S3.SS3.p4.1.m1.1.1.2.cmml" xref="S3.SS3.p4.1.m1.1.1.2">𝐿</ci><cn id="S3.SS3.p4.1.m1.1.1.3.cmml" type="integer" xref="S3.SS3.p4.1.m1.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p4.1.m1.1c">L=10~{}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p4.1.m1.1d">italic_L = 10</annotation></semantics></math>AU (left panel) and <math alttext="L=30~{}" class="ltx_Math" display="inline" id="S3.SS3.p4.2.m2.1"><semantics id="S3.SS3.p4.2.m2.1a"><mrow id="S3.SS3.p4.2.m2.1.1" xref="S3.SS3.p4.2.m2.1.1.cmml"><mi id="S3.SS3.p4.2.m2.1.1.2" xref="S3.SS3.p4.2.m2.1.1.2.cmml">L</mi><mo id="S3.SS3.p4.2.m2.1.1.1" xref="S3.SS3.p4.2.m2.1.1.1.cmml">=</mo><mn id="S3.SS3.p4.2.m2.1.1.3" xref="S3.SS3.p4.2.m2.1.1.3.cmml">30</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p4.2.m2.1b"><apply id="S3.SS3.p4.2.m2.1.1.cmml" xref="S3.SS3.p4.2.m2.1.1"><eq id="S3.SS3.p4.2.m2.1.1.1.cmml" xref="S3.SS3.p4.2.m2.1.1.1"></eq><ci id="S3.SS3.p4.2.m2.1.1.2.cmml" xref="S3.SS3.p4.2.m2.1.1.2">𝐿</ci><cn id="S3.SS3.p4.2.m2.1.1.3.cmml" type="integer" xref="S3.SS3.p4.2.m2.1.1.3">30</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p4.2.m2.1c">L=30~{}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p4.2.m2.1d">italic_L = 30</annotation></semantics></math>AU (right panel) spacecraft. The solid blue curve is the strain sensitivity that a Doppler tracking arm could achieve if its sensitivity is set by spacecraft accelerations and radiometer noise with <math alttext="C/N_{0}=60" class="ltx_Math" display="inline" id="S3.SS3.p4.3.m3.1"><semantics id="S3.SS3.p4.3.m3.1a"><mrow id="S3.SS3.p4.3.m3.1.1" xref="S3.SS3.p4.3.m3.1.1.cmml"><mrow id="S3.SS3.p4.3.m3.1.1.2" xref="S3.SS3.p4.3.m3.1.1.2.cmml"><mi id="S3.SS3.p4.3.m3.1.1.2.2" xref="S3.SS3.p4.3.m3.1.1.2.2.cmml">C</mi><mo id="S3.SS3.p4.3.m3.1.1.2.1" xref="S3.SS3.p4.3.m3.1.1.2.1.cmml">/</mo><msub id="S3.SS3.p4.3.m3.1.1.2.3" xref="S3.SS3.p4.3.m3.1.1.2.3.cmml"><mi id="S3.SS3.p4.3.m3.1.1.2.3.2" xref="S3.SS3.p4.3.m3.1.1.2.3.2.cmml">N</mi><mn id="S3.SS3.p4.3.m3.1.1.2.3.3" xref="S3.SS3.p4.3.m3.1.1.2.3.3.cmml">0</mn></msub></mrow><mo id="S3.SS3.p4.3.m3.1.1.1" xref="S3.SS3.p4.3.m3.1.1.1.cmml">=</mo><mn id="S3.SS3.p4.3.m3.1.1.3" xref="S3.SS3.p4.3.m3.1.1.3.cmml">60</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p4.3.m3.1b"><apply id="S3.SS3.p4.3.m3.1.1.cmml" xref="S3.SS3.p4.3.m3.1.1"><eq id="S3.SS3.p4.3.m3.1.1.1.cmml" xref="S3.SS3.p4.3.m3.1.1.1"></eq><apply id="S3.SS3.p4.3.m3.1.1.2.cmml" xref="S3.SS3.p4.3.m3.1.1.2"><divide id="S3.SS3.p4.3.m3.1.1.2.1.cmml" xref="S3.SS3.p4.3.m3.1.1.2.1"></divide><ci id="S3.SS3.p4.3.m3.1.1.2.2.cmml" xref="S3.SS3.p4.3.m3.1.1.2.2">𝐶</ci><apply id="S3.SS3.p4.3.m3.1.1.2.3.cmml" xref="S3.SS3.p4.3.m3.1.1.2.3"><csymbol cd="ambiguous" id="S3.SS3.p4.3.m3.1.1.2.3.1.cmml" xref="S3.SS3.p4.3.m3.1.1.2.3">subscript</csymbol><ci id="S3.SS3.p4.3.m3.1.1.2.3.2.cmml" xref="S3.SS3.p4.3.m3.1.1.2.3.2">𝑁</ci><cn id="S3.SS3.p4.3.m3.1.1.2.3.3.cmml" type="integer" xref="S3.SS3.p4.3.m3.1.1.2.3.3">0</cn></apply></apply><cn id="S3.SS3.p4.3.m3.1.1.3.cmml" type="integer" xref="S3.SS3.p4.3.m3.1.1.3">60</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p4.3.m3.1c">C/N_{0}=60</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p4.3.m3.1d">italic_C / italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 60</annotation></semantics></math> dB-Hz (left panel) and <math alttext="40" class="ltx_Math" display="inline" id="S3.SS3.p4.4.m4.1"><semantics id="S3.SS3.p4.4.m4.1a"><mn id="S3.SS3.p4.4.m4.1.1" xref="S3.SS3.p4.4.m4.1.1.cmml">40</mn><annotation-xml encoding="MathML-Content" id="S3.SS3.p4.4.m4.1b"><cn id="S3.SS3.p4.4.m4.1.1.cmml" type="integer" xref="S3.SS3.p4.4.m4.1.1">40</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p4.4.m4.1c">40</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p4.4.m4.1d">40</annotation></semantics></math> dB-Hz (right panel) and broadcasting in the Ka band at <math alttext="\lambda=1~{}" class="ltx_Math" display="inline" id="S3.SS3.p4.5.m5.1"><semantics id="S3.SS3.p4.5.m5.1a"><mrow id="S3.SS3.p4.5.m5.1.1" xref="S3.SS3.p4.5.m5.1.1.cmml"><mi id="S3.SS3.p4.5.m5.1.1.2" xref="S3.SS3.p4.5.m5.1.1.2.cmml">λ</mi><mo id="S3.SS3.p4.5.m5.1.1.1" xref="S3.SS3.p4.5.m5.1.1.1.cmml">=</mo><mn id="S3.SS3.p4.5.m5.1.1.3" xref="S3.SS3.p4.5.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p4.5.m5.1b"><apply id="S3.SS3.p4.5.m5.1.1.cmml" xref="S3.SS3.p4.5.m5.1.1"><eq id="S3.SS3.p4.5.m5.1.1.1.cmml" xref="S3.SS3.p4.5.m5.1.1.1"></eq><ci id="S3.SS3.p4.5.m5.1.1.2.cmml" xref="S3.SS3.p4.5.m5.1.1.2">𝜆</ci><cn id="S3.SS3.p4.5.m5.1.1.3.cmml" type="integer" xref="S3.SS3.p4.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p4.5.m5.1c">\lambda=1~{}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p4.5.m5.1d">italic_λ = 1</annotation></semantics></math>cm, with these larger dB-Hz values than in previous plots reflecting that more collecting area can potentially be available for the Earth station (and the downlink is the limiting step in terms of phase noise). Unlike in previous figures, where we assumed some geometric cancellation of the largely radial spacecraft acceleration, the radial nature of Doppler tracking to outer Solar System spacecraft means that nearly all of the acceleration projects onto the Earth-spacecraft chord. Mechanical noise and the atmosphere are likely to limit the sensitivity over this ideal curve. Indeed, the dotted blue points are the best constraints using Doppler tracking, achieved with the Cassini mission with <math alttext="L\approx 10~{}" class="ltx_Math" display="inline" id="S3.SS3.p4.6.m6.1"><semantics id="S3.SS3.p4.6.m6.1a"><mrow id="S3.SS3.p4.6.m6.1.1" xref="S3.SS3.p4.6.m6.1.1.cmml"><mi id="S3.SS3.p4.6.m6.1.1.2" xref="S3.SS3.p4.6.m6.1.1.2.cmml">L</mi><mo id="S3.SS3.p4.6.m6.1.1.1" xref="S3.SS3.p4.6.m6.1.1.1.cmml">≈</mo><mn id="S3.SS3.p4.6.m6.1.1.3" xref="S3.SS3.p4.6.m6.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p4.6.m6.1b"><apply id="S3.SS3.p4.6.m6.1.1.cmml" xref="S3.SS3.p4.6.m6.1.1"><approx id="S3.SS3.p4.6.m6.1.1.1.cmml" xref="S3.SS3.p4.6.m6.1.1.1"></approx><ci id="S3.SS3.p4.6.m6.1.1.2.cmml" xref="S3.SS3.p4.6.m6.1.1.2">𝐿</ci><cn id="S3.SS3.p4.6.m6.1.1.3.cmml" type="integer" xref="S3.SS3.p4.6.m6.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p4.6.m6.1c">L\approx 10~{}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p4.6.m6.1d">italic_L ≈ 10</annotation></semantics></math>AU, with the sensitivity likely limited by mechanical noise <cite class="ltx_cite ltx_citemacro_citep">(Armstrong, <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib9" title="">2006b</a>)</cite>. Improvements in the Cassini noise by as much as several orders of magnitude would be required to reach the blue solid curve.</p> </div> <div class="ltx_para" id="S3.SS3.p5"> <p class="ltx_p" id="S3.SS3.p5.5">Atmospheric and mechanical delays could also be reduced by using a precise clock on the spacecraft and then doing one-way and two-way ranging (‘One-way’ uses the clock on the spacecraft as the phase reference that is differenced with a signal sent from Earth. ‘Two-way’ phases the spacecraft to the terrestrial phase and so does not require a precise clock on the spacecraft and, therefore, is the mode used by previous Doppler tracking experiments.) Analogous to time-delay interferometry, an observable can be constructed from the phases of the one-way and two-way ranging signals that cancels out any delays that occur at the time of emission and reception by the Earth station and does not remove all of the gravitational wave signal <cite class="ltx_cite ltx_citemacro_citep">(Armstrong, <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib10" title="">2021</a>)</cite>, namely by differencing the phases in this manner: <math alttext="\big{(}\text{two-way}(t)\big{)}" class="ltx_Math" display="inline" id="S3.SS3.p5.1.m1.2"><semantics id="S3.SS3.p5.1.m1.2a"><mrow id="S3.SS3.p5.1.m1.2.2.1" xref="S3.SS3.p5.1.m1.2.2.1.1.cmml"><mo id="S3.SS3.p5.1.m1.2.2.1.2" maxsize="120%" minsize="120%" xref="S3.SS3.p5.1.m1.2.2.1.1.cmml">(</mo><mrow id="S3.SS3.p5.1.m1.2.2.1.1" xref="S3.SS3.p5.1.m1.2.2.1.1.cmml"><mtext id="S3.SS3.p5.1.m1.2.2.1.1.2" xref="S3.SS3.p5.1.m1.2.2.1.1.2a.cmml">two-way</mtext><mo id="S3.SS3.p5.1.m1.2.2.1.1.1" xref="S3.SS3.p5.1.m1.2.2.1.1.1.cmml">⁢</mo><mrow id="S3.SS3.p5.1.m1.2.2.1.1.3.2" xref="S3.SS3.p5.1.m1.2.2.1.1.cmml"><mo id="S3.SS3.p5.1.m1.2.2.1.1.3.2.1" stretchy="false" xref="S3.SS3.p5.1.m1.2.2.1.1.cmml">(</mo><mi id="S3.SS3.p5.1.m1.1.1" xref="S3.SS3.p5.1.m1.1.1.cmml">t</mi><mo id="S3.SS3.p5.1.m1.2.2.1.1.3.2.2" stretchy="false" xref="S3.SS3.p5.1.m1.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS3.p5.1.m1.2.2.1.3" maxsize="120%" minsize="120%" xref="S3.SS3.p5.1.m1.2.2.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p5.1.m1.2b"><apply id="S3.SS3.p5.1.m1.2.2.1.1.cmml" xref="S3.SS3.p5.1.m1.2.2.1"><times id="S3.SS3.p5.1.m1.2.2.1.1.1.cmml" xref="S3.SS3.p5.1.m1.2.2.1.1.1"></times><ci id="S3.SS3.p5.1.m1.2.2.1.1.2a.cmml" xref="S3.SS3.p5.1.m1.2.2.1.1.2"><mtext id="S3.SS3.p5.1.m1.2.2.1.1.2.cmml" xref="S3.SS3.p5.1.m1.2.2.1.1.2">two-way</mtext></ci><ci id="S3.SS3.p5.1.m1.1.1.cmml" xref="S3.SS3.p5.1.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p5.1.m1.2c">\big{(}\text{two-way}(t)\big{)}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p5.1.m1.2d">( two-way ( italic_t ) )</annotation></semantics></math> - <math alttext="\big{(}\text{one-way}(t)\big{)}" class="ltx_Math" display="inline" id="S3.SS3.p5.2.m2.2"><semantics id="S3.SS3.p5.2.m2.2a"><mrow id="S3.SS3.p5.2.m2.2.2.1" xref="S3.SS3.p5.2.m2.2.2.1.1.cmml"><mo id="S3.SS3.p5.2.m2.2.2.1.2" maxsize="120%" minsize="120%" xref="S3.SS3.p5.2.m2.2.2.1.1.cmml">(</mo><mrow id="S3.SS3.p5.2.m2.2.2.1.1" xref="S3.SS3.p5.2.m2.2.2.1.1.cmml"><mtext id="S3.SS3.p5.2.m2.2.2.1.1.2" xref="S3.SS3.p5.2.m2.2.2.1.1.2a.cmml">one-way</mtext><mo id="S3.SS3.p5.2.m2.2.2.1.1.1" xref="S3.SS3.p5.2.m2.2.2.1.1.1.cmml">⁢</mo><mrow id="S3.SS3.p5.2.m2.2.2.1.1.3.2" xref="S3.SS3.p5.2.m2.2.2.1.1.cmml"><mo id="S3.SS3.p5.2.m2.2.2.1.1.3.2.1" stretchy="false" xref="S3.SS3.p5.2.m2.2.2.1.1.cmml">(</mo><mi id="S3.SS3.p5.2.m2.1.1" xref="S3.SS3.p5.2.m2.1.1.cmml">t</mi><mo id="S3.SS3.p5.2.m2.2.2.1.1.3.2.2" stretchy="false" xref="S3.SS3.p5.2.m2.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS3.p5.2.m2.2.2.1.3" maxsize="120%" minsize="120%" xref="S3.SS3.p5.2.m2.2.2.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p5.2.m2.2b"><apply id="S3.SS3.p5.2.m2.2.2.1.1.cmml" xref="S3.SS3.p5.2.m2.2.2.1"><times id="S3.SS3.p5.2.m2.2.2.1.1.1.cmml" xref="S3.SS3.p5.2.m2.2.2.1.1.1"></times><ci id="S3.SS3.p5.2.m2.2.2.1.1.2a.cmml" xref="S3.SS3.p5.2.m2.2.2.1.1.2"><mtext id="S3.SS3.p5.2.m2.2.2.1.1.2.cmml" xref="S3.SS3.p5.2.m2.2.2.1.1.2">one-way</mtext></ci><ci id="S3.SS3.p5.2.m2.1.1.cmml" xref="S3.SS3.p5.2.m2.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p5.2.m2.2c">\big{(}\text{one-way}(t)\big{)}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p5.2.m2.2d">( one-way ( italic_t ) )</annotation></semantics></math> - <math alttext="\big{(}\text{one-way}(t+L/c)\big{)}" class="ltx_Math" display="inline" id="S3.SS3.p5.3.m3.1"><semantics id="S3.SS3.p5.3.m3.1a"><mrow id="S3.SS3.p5.3.m3.1.1.1" xref="S3.SS3.p5.3.m3.1.1.1.1.cmml"><mo id="S3.SS3.p5.3.m3.1.1.1.2" maxsize="120%" minsize="120%" xref="S3.SS3.p5.3.m3.1.1.1.1.cmml">(</mo><mrow id="S3.SS3.p5.3.m3.1.1.1.1" xref="S3.SS3.p5.3.m3.1.1.1.1.cmml"><mtext id="S3.SS3.p5.3.m3.1.1.1.1.3" xref="S3.SS3.p5.3.m3.1.1.1.1.3a.cmml">one-way</mtext><mo id="S3.SS3.p5.3.m3.1.1.1.1.2" xref="S3.SS3.p5.3.m3.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS3.p5.3.m3.1.1.1.1.1.1" xref="S3.SS3.p5.3.m3.1.1.1.1.1.1.1.cmml"><mo id="S3.SS3.p5.3.m3.1.1.1.1.1.1.2" stretchy="false" xref="S3.SS3.p5.3.m3.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS3.p5.3.m3.1.1.1.1.1.1.1" xref="S3.SS3.p5.3.m3.1.1.1.1.1.1.1.cmml"><mi id="S3.SS3.p5.3.m3.1.1.1.1.1.1.1.2" xref="S3.SS3.p5.3.m3.1.1.1.1.1.1.1.2.cmml">t</mi><mo id="S3.SS3.p5.3.m3.1.1.1.1.1.1.1.1" xref="S3.SS3.p5.3.m3.1.1.1.1.1.1.1.1.cmml">+</mo><mrow id="S3.SS3.p5.3.m3.1.1.1.1.1.1.1.3" xref="S3.SS3.p5.3.m3.1.1.1.1.1.1.1.3.cmml"><mi id="S3.SS3.p5.3.m3.1.1.1.1.1.1.1.3.2" xref="S3.SS3.p5.3.m3.1.1.1.1.1.1.1.3.2.cmml">L</mi><mo id="S3.SS3.p5.3.m3.1.1.1.1.1.1.1.3.1" xref="S3.SS3.p5.3.m3.1.1.1.1.1.1.1.3.1.cmml">/</mo><mi id="S3.SS3.p5.3.m3.1.1.1.1.1.1.1.3.3" xref="S3.SS3.p5.3.m3.1.1.1.1.1.1.1.3.3.cmml">c</mi></mrow></mrow><mo id="S3.SS3.p5.3.m3.1.1.1.1.1.1.3" stretchy="false" xref="S3.SS3.p5.3.m3.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS3.p5.3.m3.1.1.1.3" maxsize="120%" minsize="120%" xref="S3.SS3.p5.3.m3.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p5.3.m3.1b"><apply id="S3.SS3.p5.3.m3.1.1.1.1.cmml" xref="S3.SS3.p5.3.m3.1.1.1"><times id="S3.SS3.p5.3.m3.1.1.1.1.2.cmml" xref="S3.SS3.p5.3.m3.1.1.1.1.2"></times><ci id="S3.SS3.p5.3.m3.1.1.1.1.3a.cmml" xref="S3.SS3.p5.3.m3.1.1.1.1.3"><mtext id="S3.SS3.p5.3.m3.1.1.1.1.3.cmml" xref="S3.SS3.p5.3.m3.1.1.1.1.3">one-way</mtext></ci><apply id="S3.SS3.p5.3.m3.1.1.1.1.1.1.1.cmml" xref="S3.SS3.p5.3.m3.1.1.1.1.1.1"><plus id="S3.SS3.p5.3.m3.1.1.1.1.1.1.1.1.cmml" xref="S3.SS3.p5.3.m3.1.1.1.1.1.1.1.1"></plus><ci id="S3.SS3.p5.3.m3.1.1.1.1.1.1.1.2.cmml" xref="S3.SS3.p5.3.m3.1.1.1.1.1.1.1.2">𝑡</ci><apply id="S3.SS3.p5.3.m3.1.1.1.1.1.1.1.3.cmml" xref="S3.SS3.p5.3.m3.1.1.1.1.1.1.1.3"><divide id="S3.SS3.p5.3.m3.1.1.1.1.1.1.1.3.1.cmml" xref="S3.SS3.p5.3.m3.1.1.1.1.1.1.1.3.1"></divide><ci id="S3.SS3.p5.3.m3.1.1.1.1.1.1.1.3.2.cmml" xref="S3.SS3.p5.3.m3.1.1.1.1.1.1.1.3.2">𝐿</ci><ci id="S3.SS3.p5.3.m3.1.1.1.1.1.1.1.3.3.cmml" xref="S3.SS3.p5.3.m3.1.1.1.1.1.1.1.3.3">𝑐</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p5.3.m3.1c">\big{(}\text{one-way}(t+L/c)\big{)}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p5.3.m3.1d">( one-way ( italic_t + italic_L / italic_c ) )</annotation></semantics></math>. The orange curve in Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S3.F6" title="Figure 6 ‣ 3.3 Doppler tracking forecasts ‣ 3 Strain power sensitivity ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">6</span></a> is the same as the blue curve but also includes clock noise with <math alttext="\sigma_{y}(1s)=10^{-14}" class="ltx_Math" display="inline" id="S3.SS3.p5.4.m4.1"><semantics id="S3.SS3.p5.4.m4.1a"><mrow id="S3.SS3.p5.4.m4.1.1" xref="S3.SS3.p5.4.m4.1.1.cmml"><mrow id="S3.SS3.p5.4.m4.1.1.1" xref="S3.SS3.p5.4.m4.1.1.1.cmml"><msub id="S3.SS3.p5.4.m4.1.1.1.3" xref="S3.SS3.p5.4.m4.1.1.1.3.cmml"><mi id="S3.SS3.p5.4.m4.1.1.1.3.2" xref="S3.SS3.p5.4.m4.1.1.1.3.2.cmml">σ</mi><mi id="S3.SS3.p5.4.m4.1.1.1.3.3" xref="S3.SS3.p5.4.m4.1.1.1.3.3.cmml">y</mi></msub><mo id="S3.SS3.p5.4.m4.1.1.1.2" xref="S3.SS3.p5.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS3.p5.4.m4.1.1.1.1.1" xref="S3.SS3.p5.4.m4.1.1.1.1.1.1.cmml"><mo id="S3.SS3.p5.4.m4.1.1.1.1.1.2" stretchy="false" xref="S3.SS3.p5.4.m4.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS3.p5.4.m4.1.1.1.1.1.1" xref="S3.SS3.p5.4.m4.1.1.1.1.1.1.cmml"><mn id="S3.SS3.p5.4.m4.1.1.1.1.1.1.2" xref="S3.SS3.p5.4.m4.1.1.1.1.1.1.2.cmml">1</mn><mo id="S3.SS3.p5.4.m4.1.1.1.1.1.1.1" xref="S3.SS3.p5.4.m4.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S3.SS3.p5.4.m4.1.1.1.1.1.1.3" xref="S3.SS3.p5.4.m4.1.1.1.1.1.1.3.cmml">s</mi></mrow><mo id="S3.SS3.p5.4.m4.1.1.1.1.1.3" stretchy="false" xref="S3.SS3.p5.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS3.p5.4.m4.1.1.2" xref="S3.SS3.p5.4.m4.1.1.2.cmml">=</mo><msup id="S3.SS3.p5.4.m4.1.1.3" xref="S3.SS3.p5.4.m4.1.1.3.cmml"><mn id="S3.SS3.p5.4.m4.1.1.3.2" xref="S3.SS3.p5.4.m4.1.1.3.2.cmml">10</mn><mrow id="S3.SS3.p5.4.m4.1.1.3.3" xref="S3.SS3.p5.4.m4.1.1.3.3.cmml"><mo id="S3.SS3.p5.4.m4.1.1.3.3a" xref="S3.SS3.p5.4.m4.1.1.3.3.cmml">−</mo><mn id="S3.SS3.p5.4.m4.1.1.3.3.2" xref="S3.SS3.p5.4.m4.1.1.3.3.2.cmml">14</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p5.4.m4.1b"><apply id="S3.SS3.p5.4.m4.1.1.cmml" xref="S3.SS3.p5.4.m4.1.1"><eq id="S3.SS3.p5.4.m4.1.1.2.cmml" xref="S3.SS3.p5.4.m4.1.1.2"></eq><apply id="S3.SS3.p5.4.m4.1.1.1.cmml" xref="S3.SS3.p5.4.m4.1.1.1"><times id="S3.SS3.p5.4.m4.1.1.1.2.cmml" xref="S3.SS3.p5.4.m4.1.1.1.2"></times><apply id="S3.SS3.p5.4.m4.1.1.1.3.cmml" xref="S3.SS3.p5.4.m4.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.p5.4.m4.1.1.1.3.1.cmml" xref="S3.SS3.p5.4.m4.1.1.1.3">subscript</csymbol><ci id="S3.SS3.p5.4.m4.1.1.1.3.2.cmml" xref="S3.SS3.p5.4.m4.1.1.1.3.2">𝜎</ci><ci id="S3.SS3.p5.4.m4.1.1.1.3.3.cmml" xref="S3.SS3.p5.4.m4.1.1.1.3.3">𝑦</ci></apply><apply id="S3.SS3.p5.4.m4.1.1.1.1.1.1.cmml" xref="S3.SS3.p5.4.m4.1.1.1.1.1"><times id="S3.SS3.p5.4.m4.1.1.1.1.1.1.1.cmml" xref="S3.SS3.p5.4.m4.1.1.1.1.1.1.1"></times><cn id="S3.SS3.p5.4.m4.1.1.1.1.1.1.2.cmml" type="integer" xref="S3.SS3.p5.4.m4.1.1.1.1.1.1.2">1</cn><ci id="S3.SS3.p5.4.m4.1.1.1.1.1.1.3.cmml" xref="S3.SS3.p5.4.m4.1.1.1.1.1.1.3">𝑠</ci></apply></apply><apply id="S3.SS3.p5.4.m4.1.1.3.cmml" xref="S3.SS3.p5.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.p5.4.m4.1.1.3.1.cmml" xref="S3.SS3.p5.4.m4.1.1.3">superscript</csymbol><cn id="S3.SS3.p5.4.m4.1.1.3.2.cmml" type="integer" xref="S3.SS3.p5.4.m4.1.1.3.2">10</cn><apply id="S3.SS3.p5.4.m4.1.1.3.3.cmml" xref="S3.SS3.p5.4.m4.1.1.3.3"><minus id="S3.SS3.p5.4.m4.1.1.3.3.1.cmml" xref="S3.SS3.p5.4.m4.1.1.3.3"></minus><cn id="S3.SS3.p5.4.m4.1.1.3.3.2.cmml" type="integer" xref="S3.SS3.p5.4.m4.1.1.3.3.2">14</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p5.4.m4.1c">\sigma_{y}(1s)=10^{-14}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p5.4.m4.1d">italic_σ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ( 1 italic_s ) = 10 start_POSTSUPERSCRIPT - 14 end_POSTSUPERSCRIPT</annotation></semantics></math> – ten times lower <math alttext="\sigma_{y}(1s)" class="ltx_Math" display="inline" id="S3.SS3.p5.5.m5.1"><semantics id="S3.SS3.p5.5.m5.1a"><mrow id="S3.SS3.p5.5.m5.1.1" xref="S3.SS3.p5.5.m5.1.1.cmml"><msub id="S3.SS3.p5.5.m5.1.1.3" xref="S3.SS3.p5.5.m5.1.1.3.cmml"><mi id="S3.SS3.p5.5.m5.1.1.3.2" xref="S3.SS3.p5.5.m5.1.1.3.2.cmml">σ</mi><mi id="S3.SS3.p5.5.m5.1.1.3.3" xref="S3.SS3.p5.5.m5.1.1.3.3.cmml">y</mi></msub><mo id="S3.SS3.p5.5.m5.1.1.2" xref="S3.SS3.p5.5.m5.1.1.2.cmml">⁢</mo><mrow id="S3.SS3.p5.5.m5.1.1.1.1" xref="S3.SS3.p5.5.m5.1.1.1.1.1.cmml"><mo id="S3.SS3.p5.5.m5.1.1.1.1.2" stretchy="false" xref="S3.SS3.p5.5.m5.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS3.p5.5.m5.1.1.1.1.1" xref="S3.SS3.p5.5.m5.1.1.1.1.1.cmml"><mn id="S3.SS3.p5.5.m5.1.1.1.1.1.2" xref="S3.SS3.p5.5.m5.1.1.1.1.1.2.cmml">1</mn><mo id="S3.SS3.p5.5.m5.1.1.1.1.1.1" xref="S3.SS3.p5.5.m5.1.1.1.1.1.1.cmml">⁢</mo><mi id="S3.SS3.p5.5.m5.1.1.1.1.1.3" xref="S3.SS3.p5.5.m5.1.1.1.1.1.3.cmml">s</mi></mrow><mo id="S3.SS3.p5.5.m5.1.1.1.1.3" stretchy="false" xref="S3.SS3.p5.5.m5.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p5.5.m5.1b"><apply id="S3.SS3.p5.5.m5.1.1.cmml" xref="S3.SS3.p5.5.m5.1.1"><times id="S3.SS3.p5.5.m5.1.1.2.cmml" xref="S3.SS3.p5.5.m5.1.1.2"></times><apply id="S3.SS3.p5.5.m5.1.1.3.cmml" xref="S3.SS3.p5.5.m5.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.p5.5.m5.1.1.3.1.cmml" xref="S3.SS3.p5.5.m5.1.1.3">subscript</csymbol><ci id="S3.SS3.p5.5.m5.1.1.3.2.cmml" xref="S3.SS3.p5.5.m5.1.1.3.2">𝜎</ci><ci id="S3.SS3.p5.5.m5.1.1.3.3.cmml" xref="S3.SS3.p5.5.m5.1.1.3.3">𝑦</ci></apply><apply id="S3.SS3.p5.5.m5.1.1.1.1.1.cmml" xref="S3.SS3.p5.5.m5.1.1.1.1"><times id="S3.SS3.p5.5.m5.1.1.1.1.1.1.cmml" xref="S3.SS3.p5.5.m5.1.1.1.1.1.1"></times><cn id="S3.SS3.p5.5.m5.1.1.1.1.1.2.cmml" type="integer" xref="S3.SS3.p5.5.m5.1.1.1.1.1.2">1</cn><ci id="S3.SS3.p5.5.m5.1.1.1.1.1.3.cmml" xref="S3.SS3.p5.5.m5.1.1.1.1.1.3">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p5.5.m5.1c">\sigma_{y}(1s)</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p5.5.m5.1d">italic_σ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ( 1 italic_s )</annotation></semantics></math> than the Deep Space Atomic clock.<span class="ltx_note ltx_role_footnote" id="footnote12"><sup class="ltx_note_mark">12</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">12</sup><span class="ltx_tag ltx_tag_note">12</span>For mechanical and other instrumental delays, the cancellation in the estimator of <cite class="ltx_cite ltx_citemacro_citet">Armstrong (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib10" title="">2021</a>)</cite> occurs to the extent that the one-way and two-way ranging share the same analog path (even during transmit and receive). In our calculations, we have not used the correct transfer function of this observable, which we anticipate will result in a <math alttext="{\cal O}(1)" class="ltx_Math" display="inline" id="footnote12.m1.1"><semantics id="footnote12.m1.1b"><mrow id="footnote12.m1.1.2" xref="footnote12.m1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="footnote12.m1.1.2.2" xref="footnote12.m1.1.2.2.cmml">𝒪</mi><mo id="footnote12.m1.1.2.1" xref="footnote12.m1.1.2.1.cmml">⁢</mo><mrow id="footnote12.m1.1.2.3.2" xref="footnote12.m1.1.2.cmml"><mo id="footnote12.m1.1.2.3.2.1" stretchy="false" xref="footnote12.m1.1.2.cmml">(</mo><mn id="footnote12.m1.1.1" xref="footnote12.m1.1.1.cmml">1</mn><mo id="footnote12.m1.1.2.3.2.2" stretchy="false" xref="footnote12.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote12.m1.1c"><apply id="footnote12.m1.1.2.cmml" xref="footnote12.m1.1.2"><times id="footnote12.m1.1.2.1.cmml" xref="footnote12.m1.1.2.1"></times><ci id="footnote12.m1.1.2.2.cmml" xref="footnote12.m1.1.2.2">𝒪</ci><cn id="footnote12.m1.1.1.cmml" type="integer" xref="footnote12.m1.1.1">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote12.m1.1d">{\cal O}(1)</annotation><annotation encoding="application/x-llamapun" id="footnote12.m1.1e">caligraphic_O ( 1 )</annotation></semantics></math> reduction in the sensitivity relative to the single-arm transfer function used for our calculation due to the cancellation of some of the gravitational strain terms in this combination. </span></span></span> (This type of observable also eliminates acceleration noise at one satellite. It could motivate a configuration with a more-resourced satellite in the high-acceleration near-Earth environment broadcasting to one in the outer Solar System or, alternatively, the satellite near Earth having a precise accelerometer and the acceleration noise nulled for the outer solar system satellite.)</p> </div> <div class="ltx_para" id="S3.SS3.p6"> <p class="ltx_p" id="S3.SS3.p6.9">The magenta dashed curves shown at <math alttext="f<10^{-4}" class="ltx_Math" display="inline" id="S3.SS3.p6.1.m1.1"><semantics id="S3.SS3.p6.1.m1.1a"><mrow id="S3.SS3.p6.1.m1.1.1" xref="S3.SS3.p6.1.m1.1.1.cmml"><mi id="S3.SS3.p6.1.m1.1.1.2" xref="S3.SS3.p6.1.m1.1.1.2.cmml">f</mi><mo id="S3.SS3.p6.1.m1.1.1.1" xref="S3.SS3.p6.1.m1.1.1.1.cmml"><</mo><msup id="S3.SS3.p6.1.m1.1.1.3" xref="S3.SS3.p6.1.m1.1.1.3.cmml"><mn id="S3.SS3.p6.1.m1.1.1.3.2" xref="S3.SS3.p6.1.m1.1.1.3.2.cmml">10</mn><mrow id="S3.SS3.p6.1.m1.1.1.3.3" xref="S3.SS3.p6.1.m1.1.1.3.3.cmml"><mo id="S3.SS3.p6.1.m1.1.1.3.3a" xref="S3.SS3.p6.1.m1.1.1.3.3.cmml">−</mo><mn id="S3.SS3.p6.1.m1.1.1.3.3.2" xref="S3.SS3.p6.1.m1.1.1.3.3.2.cmml">4</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p6.1.m1.1b"><apply id="S3.SS3.p6.1.m1.1.1.cmml" xref="S3.SS3.p6.1.m1.1.1"><lt id="S3.SS3.p6.1.m1.1.1.1.cmml" xref="S3.SS3.p6.1.m1.1.1.1"></lt><ci id="S3.SS3.p6.1.m1.1.1.2.cmml" xref="S3.SS3.p6.1.m1.1.1.2">𝑓</ci><apply id="S3.SS3.p6.1.m1.1.1.3.cmml" xref="S3.SS3.p6.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.p6.1.m1.1.1.3.1.cmml" xref="S3.SS3.p6.1.m1.1.1.3">superscript</csymbol><cn id="S3.SS3.p6.1.m1.1.1.3.2.cmml" type="integer" xref="S3.SS3.p6.1.m1.1.1.3.2">10</cn><apply id="S3.SS3.p6.1.m1.1.1.3.3.cmml" xref="S3.SS3.p6.1.m1.1.1.3.3"><minus id="S3.SS3.p6.1.m1.1.1.3.3.1.cmml" xref="S3.SS3.p6.1.m1.1.1.3.3"></minus><cn id="S3.SS3.p6.1.m1.1.1.3.3.2.cmml" type="integer" xref="S3.SS3.p6.1.m1.1.1.3.3.2">4</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p6.1.m1.1c">f<10^{-4}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p6.1.m1.1d">italic_f < 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT</annotation></semantics></math> show the residual noise power if the atmosphere and mechanical noise can be subtracted to a residual power that has an equal variance per <math alttext="\log f" class="ltx_Math" display="inline" id="S3.SS3.p6.2.m2.1"><semantics id="S3.SS3.p6.2.m2.1a"><mrow id="S3.SS3.p6.2.m2.1.1" xref="S3.SS3.p6.2.m2.1.1.cmml"><mi id="S3.SS3.p6.2.m2.1.1.1" xref="S3.SS3.p6.2.m2.1.1.1.cmml">log</mi><mo id="S3.SS3.p6.2.m2.1.1a" lspace="0.167em" xref="S3.SS3.p6.2.m2.1.1.cmml">⁡</mo><mi id="S3.SS3.p6.2.m2.1.1.2" xref="S3.SS3.p6.2.m2.1.1.2.cmml">f</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p6.2.m2.1b"><apply id="S3.SS3.p6.2.m2.1.1.cmml" xref="S3.SS3.p6.2.m2.1.1"><log id="S3.SS3.p6.2.m2.1.1.1.cmml" xref="S3.SS3.p6.2.m2.1.1.1"></log><ci id="S3.SS3.p6.2.m2.1.1.2.cmml" xref="S3.SS3.p6.2.m2.1.1.2">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p6.2.m2.1c">\log f</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p6.2.m2.1d">roman_log italic_f</annotation></semantics></math> (i.e. <math alttext="1/f" class="ltx_Math" display="inline" id="S3.SS3.p6.3.m3.1"><semantics id="S3.SS3.p6.3.m3.1a"><mrow id="S3.SS3.p6.3.m3.1.1" xref="S3.SS3.p6.3.m3.1.1.cmml"><mn id="S3.SS3.p6.3.m3.1.1.2" xref="S3.SS3.p6.3.m3.1.1.2.cmml">1</mn><mo id="S3.SS3.p6.3.m3.1.1.1" xref="S3.SS3.p6.3.m3.1.1.1.cmml">/</mo><mi id="S3.SS3.p6.3.m3.1.1.3" xref="S3.SS3.p6.3.m3.1.1.3.cmml">f</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p6.3.m3.1b"><apply id="S3.SS3.p6.3.m3.1.1.cmml" xref="S3.SS3.p6.3.m3.1.1"><divide id="S3.SS3.p6.3.m3.1.1.1.cmml" xref="S3.SS3.p6.3.m3.1.1.1"></divide><cn id="S3.SS3.p6.3.m3.1.1.2.cmml" type="integer" xref="S3.SS3.p6.3.m3.1.1.2">1</cn><ci id="S3.SS3.p6.3.m3.1.1.3.cmml" xref="S3.SS3.p6.3.m3.1.1.3">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p6.3.m3.1c">1/f</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p6.3.m3.1d">1 / italic_f</annotation></semantics></math> noise) and equal to <math alttext="0.1\," class="ltx_Math" display="inline" id="S3.SS3.p6.4.m4.1"><semantics id="S3.SS3.p6.4.m4.1a"><mn id="S3.SS3.p6.4.m4.1.1" xref="S3.SS3.p6.4.m4.1.1.cmml">0.1</mn><annotation-xml encoding="MathML-Content" id="S3.SS3.p6.4.m4.1b"><cn id="S3.SS3.p6.4.m4.1.1.cmml" type="float" xref="S3.SS3.p6.4.m4.1.1">0.1</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p6.4.m4.1c">0.1\,</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p6.4.m4.1d">0.1</annotation></semantics></math>cm and <math alttext="0.01\," class="ltx_Math" display="inline" id="S3.SS3.p6.5.m5.1"><semantics id="S3.SS3.p6.5.m5.1a"><mn id="S3.SS3.p6.5.m5.1.1" xref="S3.SS3.p6.5.m5.1.1.cmml">0.01</mn><annotation-xml encoding="MathML-Content" id="S3.SS3.p6.5.m5.1b"><cn id="S3.SS3.p6.5.m5.1.1.cmml" type="float" xref="S3.SS3.p6.5.m5.1.1">0.01</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p6.5.m5.1c">0.01\,</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p6.5.m5.1d">0.01</annotation></semantics></math>cm, numbers motivated above for the degree to which wet tropospheric delays could be cleaned. Although tropospheric delays have a redder spectrum than <math alttext="1/f" class="ltx_Math" display="inline" id="S3.SS3.p6.6.m6.1"><semantics id="S3.SS3.p6.6.m6.1a"><mrow id="S3.SS3.p6.6.m6.1.1" xref="S3.SS3.p6.6.m6.1.1.cmml"><mn id="S3.SS3.p6.6.m6.1.1.2" xref="S3.SS3.p6.6.m6.1.1.2.cmml">1</mn><mo id="S3.SS3.p6.6.m6.1.1.1" xref="S3.SS3.p6.6.m6.1.1.1.cmml">/</mo><mi id="S3.SS3.p6.6.m6.1.1.3" xref="S3.SS3.p6.6.m6.1.1.3.cmml">f</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p6.6.m6.1b"><apply id="S3.SS3.p6.6.m6.1.1.cmml" xref="S3.SS3.p6.6.m6.1.1"><divide id="S3.SS3.p6.6.m6.1.1.1.cmml" xref="S3.SS3.p6.6.m6.1.1.1"></divide><cn id="S3.SS3.p6.6.m6.1.1.2.cmml" type="integer" xref="S3.SS3.p6.6.m6.1.1.2">1</cn><ci id="S3.SS3.p6.6.m6.1.1.3.cmml" xref="S3.SS3.p6.6.m6.1.1.3">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p6.6.m6.1c">1/f</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p6.6.m6.1d">1 / italic_f</annotation></semantics></math>, the spectrum <cite class="ltx_cite ltx_citemacro_citet">Woo & Guo (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib73" title="">2004</a>)</cite> found after atmospheric correction was much closer to the <math alttext="1/f" class="ltx_Math" display="inline" id="S3.SS3.p6.7.m7.1"><semantics id="S3.SS3.p6.7.m7.1a"><mrow id="S3.SS3.p6.7.m7.1.1" xref="S3.SS3.p6.7.m7.1.1.cmml"><mn id="S3.SS3.p6.7.m7.1.1.2" xref="S3.SS3.p6.7.m7.1.1.2.cmml">1</mn><mo id="S3.SS3.p6.7.m7.1.1.1" xref="S3.SS3.p6.7.m7.1.1.1.cmml">/</mo><mi id="S3.SS3.p6.7.m7.1.1.3" xref="S3.SS3.p6.7.m7.1.1.3.cmml">f</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p6.7.m7.1b"><apply id="S3.SS3.p6.7.m7.1.1.cmml" xref="S3.SS3.p6.7.m7.1.1"><divide id="S3.SS3.p6.7.m7.1.1.1.cmml" xref="S3.SS3.p6.7.m7.1.1.1"></divide><cn id="S3.SS3.p6.7.m7.1.1.2.cmml" type="integer" xref="S3.SS3.p6.7.m7.1.1.2">1</cn><ci id="S3.SS3.p6.7.m7.1.1.3.cmml" xref="S3.SS3.p6.7.m7.1.1.3">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p6.7.m7.1c">1/f</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p6.7.m7.1d">1 / italic_f</annotation></semantics></math> form. These curves show that for <math alttext="\lesssim 10^{-5.5}" class="ltx_Math" display="inline" id="S3.SS3.p6.8.m8.1"><semantics id="S3.SS3.p6.8.m8.1a"><mrow id="S3.SS3.p6.8.m8.1.1" xref="S3.SS3.p6.8.m8.1.1.cmml"><mi id="S3.SS3.p6.8.m8.1.1.2" xref="S3.SS3.p6.8.m8.1.1.2.cmml"></mi><mo id="S3.SS3.p6.8.m8.1.1.1" xref="S3.SS3.p6.8.m8.1.1.1.cmml">≲</mo><msup id="S3.SS3.p6.8.m8.1.1.3" xref="S3.SS3.p6.8.m8.1.1.3.cmml"><mn id="S3.SS3.p6.8.m8.1.1.3.2" xref="S3.SS3.p6.8.m8.1.1.3.2.cmml">10</mn><mrow id="S3.SS3.p6.8.m8.1.1.3.3" xref="S3.SS3.p6.8.m8.1.1.3.3.cmml"><mo id="S3.SS3.p6.8.m8.1.1.3.3a" xref="S3.SS3.p6.8.m8.1.1.3.3.cmml">−</mo><mn id="S3.SS3.p6.8.m8.1.1.3.3.2" xref="S3.SS3.p6.8.m8.1.1.3.3.2.cmml">5.5</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p6.8.m8.1b"><apply id="S3.SS3.p6.8.m8.1.1.cmml" xref="S3.SS3.p6.8.m8.1.1"><csymbol cd="latexml" id="S3.SS3.p6.8.m8.1.1.1.cmml" xref="S3.SS3.p6.8.m8.1.1.1">less-than-or-similar-to</csymbol><csymbol cd="latexml" id="S3.SS3.p6.8.m8.1.1.2.cmml" xref="S3.SS3.p6.8.m8.1.1.2">absent</csymbol><apply id="S3.SS3.p6.8.m8.1.1.3.cmml" xref="S3.SS3.p6.8.m8.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.p6.8.m8.1.1.3.1.cmml" xref="S3.SS3.p6.8.m8.1.1.3">superscript</csymbol><cn id="S3.SS3.p6.8.m8.1.1.3.2.cmml" type="integer" xref="S3.SS3.p6.8.m8.1.1.3.2">10</cn><apply id="S3.SS3.p6.8.m8.1.1.3.3.cmml" xref="S3.SS3.p6.8.m8.1.1.3.3"><minus id="S3.SS3.p6.8.m8.1.1.3.3.1.cmml" xref="S3.SS3.p6.8.m8.1.1.3.3"></minus><cn id="S3.SS3.p6.8.m8.1.1.3.3.2.cmml" type="float" xref="S3.SS3.p6.8.m8.1.1.3.3.2">5.5</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p6.8.m8.1c">\lesssim 10^{-5.5}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p6.8.m8.1d">≲ 10 start_POSTSUPERSCRIPT - 5.5 end_POSTSUPERSCRIPT</annotation></semantics></math>Hz, achieving <math alttext="0.01-0.1\," class="ltx_Math" display="inline" id="S3.SS3.p6.9.m9.1"><semantics id="S3.SS3.p6.9.m9.1a"><mrow id="S3.SS3.p6.9.m9.1.1" xref="S3.SS3.p6.9.m9.1.1.cmml"><mn id="S3.SS3.p6.9.m9.1.1.2" xref="S3.SS3.p6.9.m9.1.1.2.cmml">0.01</mn><mo id="S3.SS3.p6.9.m9.1.1.1" xref="S3.SS3.p6.9.m9.1.1.1.cmml">−</mo><mn id="S3.SS3.p6.9.m9.1.1.3" xref="S3.SS3.p6.9.m9.1.1.3.cmml">0.1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p6.9.m9.1b"><apply id="S3.SS3.p6.9.m9.1.1.cmml" xref="S3.SS3.p6.9.m9.1.1"><minus id="S3.SS3.p6.9.m9.1.1.1.cmml" xref="S3.SS3.p6.9.m9.1.1.1"></minus><cn id="S3.SS3.p6.9.m9.1.1.2.cmml" type="float" xref="S3.SS3.p6.9.m9.1.1.2">0.01</cn><cn id="S3.SS3.p6.9.m9.1.1.3.cmml" type="float" xref="S3.SS3.p6.9.m9.1.1.3">0.1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p6.9.m9.1c">0.01-0.1\,</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p6.9.m9.1d">0.01 - 0.1</annotation></semantics></math>cm error in mechanical and tropospheric delays would be sufficient to be limited by accelerations in the outer Solar System.<span class="ltx_note ltx_role_footnote" id="footnote13"><sup class="ltx_note_mark">13</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">13</sup><span class="ltx_tag ltx_tag_note">13</span>Our estimates for noise of a Doppler tracking do not include the error from the Earth’s ephemerides. Earth’s position needs to be tracked and is uncertain at the <math alttext="\sim 100\;" class="ltx_Math" display="inline" id="footnote13.m1.1"><semantics id="footnote13.m1.1b"><mrow id="footnote13.m1.1.1" xref="footnote13.m1.1.1.cmml"><mi id="footnote13.m1.1.1.2" xref="footnote13.m1.1.1.2.cmml"></mi><mo id="footnote13.m1.1.1.1" xref="footnote13.m1.1.1.1.cmml">∼</mo><mn id="footnote13.m1.1.1.3" xref="footnote13.m1.1.1.3.cmml">100</mn></mrow><annotation-xml encoding="MathML-Content" id="footnote13.m1.1c"><apply id="footnote13.m1.1.1.cmml" xref="footnote13.m1.1.1"><csymbol cd="latexml" id="footnote13.m1.1.1.1.cmml" xref="footnote13.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="footnote13.m1.1.1.2.cmml" xref="footnote13.m1.1.1.2">absent</csymbol><cn id="footnote13.m1.1.1.3.cmml" type="integer" xref="footnote13.m1.1.1.3">100</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote13.m1.1d">\sim 100\;</annotation><annotation encoding="application/x-llamapun" id="footnote13.m1.1e">∼ 100</annotation></semantics></math>m level, with these errors on year- and longer-timescales <cite class="ltx_cite ltx_citemacro_citep">(Vallisneri et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib67" title="">2020</a>)</cite>. Ranging can be used to reduce this ephemerides error along the vector to the spacecraft, but this will come at the cost of being less sensitive to gravitational waves on the timescales for which the uncertainties ephemerides need to be fit for. Such fitting will bleed into shorter timescales than Earth’s orbital period and possibly set the error for the lowest frequencies considered in our plots of <math alttext="f\sim 10^{-7}" class="ltx_Math" display="inline" id="footnote13.m2.1"><semantics id="footnote13.m2.1b"><mrow id="footnote13.m2.1.1" xref="footnote13.m2.1.1.cmml"><mi id="footnote13.m2.1.1.2" xref="footnote13.m2.1.1.2.cmml">f</mi><mo id="footnote13.m2.1.1.1" xref="footnote13.m2.1.1.1.cmml">∼</mo><msup id="footnote13.m2.1.1.3" xref="footnote13.m2.1.1.3.cmml"><mn id="footnote13.m2.1.1.3.2" xref="footnote13.m2.1.1.3.2.cmml">10</mn><mrow id="footnote13.m2.1.1.3.3" xref="footnote13.m2.1.1.3.3.cmml"><mo id="footnote13.m2.1.1.3.3b" xref="footnote13.m2.1.1.3.3.cmml">−</mo><mn id="footnote13.m2.1.1.3.3.2" xref="footnote13.m2.1.1.3.3.2.cmml">7</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="footnote13.m2.1c"><apply id="footnote13.m2.1.1.cmml" xref="footnote13.m2.1.1"><csymbol cd="latexml" id="footnote13.m2.1.1.1.cmml" xref="footnote13.m2.1.1.1">similar-to</csymbol><ci id="footnote13.m2.1.1.2.cmml" xref="footnote13.m2.1.1.2">𝑓</ci><apply id="footnote13.m2.1.1.3.cmml" xref="footnote13.m2.1.1.3"><csymbol cd="ambiguous" id="footnote13.m2.1.1.3.1.cmml" xref="footnote13.m2.1.1.3">superscript</csymbol><cn id="footnote13.m2.1.1.3.2.cmml" type="integer" xref="footnote13.m2.1.1.3.2">10</cn><apply id="footnote13.m2.1.1.3.3.cmml" xref="footnote13.m2.1.1.3.3"><minus id="footnote13.m2.1.1.3.3.1.cmml" xref="footnote13.m2.1.1.3.3"></minus><cn id="footnote13.m2.1.1.3.3.2.cmml" type="integer" xref="footnote13.m2.1.1.3.3.2">7</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote13.m2.1d">f\sim 10^{-7}</annotation><annotation encoding="application/x-llamapun" id="footnote13.m2.1e">italic_f ∼ 10 start_POSTSUPERSCRIPT - 7 end_POSTSUPERSCRIPT</annotation></semantics></math>Hz (as <math alttext="\sim 100\;" class="ltx_Math" display="inline" id="footnote13.m3.1"><semantics id="footnote13.m3.1b"><mrow id="footnote13.m3.1.1" xref="footnote13.m3.1.1.cmml"><mi id="footnote13.m3.1.1.2" xref="footnote13.m3.1.1.2.cmml"></mi><mo id="footnote13.m3.1.1.1" xref="footnote13.m3.1.1.1.cmml">∼</mo><mn id="footnote13.m3.1.1.3" xref="footnote13.m3.1.1.3.cmml">100</mn></mrow><annotation-xml encoding="MathML-Content" id="footnote13.m3.1c"><apply id="footnote13.m3.1.1.cmml" xref="footnote13.m3.1.1"><csymbol cd="latexml" id="footnote13.m3.1.1.1.cmml" xref="footnote13.m3.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="footnote13.m3.1.1.2.cmml" xref="footnote13.m3.1.1.2">absent</csymbol><cn id="footnote13.m3.1.1.3.cmml" type="integer" xref="footnote13.m3.1.1.3">100</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote13.m3.1d">\sim 100\;</annotation><annotation encoding="application/x-llamapun" id="footnote13.m3.1e">∼ 100</annotation></semantics></math>cm residuals will lead to larger errors than accelerations at such frequencies). A dedicated study is needed to understand the frequencies over which ephemerides errors limit the strain sensitivity.</span></span></span></p> </div> <div class="ltx_para" id="S3.SS3.p7"> <p class="ltx_p" id="S3.SS3.p7.3">There has been some recent interest in Doppler tracking for the <math alttext="\mu" class="ltx_Math" display="inline" id="S3.SS3.p7.1.m1.1"><semantics id="S3.SS3.p7.1.m1.1a"><mi id="S3.SS3.p7.1.m1.1.1" xref="S3.SS3.p7.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p7.1.m1.1b"><ci id="S3.SS3.p7.1.m1.1.1.cmml" xref="S3.SS3.p7.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p7.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p7.1.m1.1d">italic_μ</annotation></semantics></math>Hz band in the context of a flagship Uranus mission <cite class="ltx_cite ltx_citemacro_citep">(Zwick et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib75" title="">2024</a>)</cite>. The green dot-dashed curve in Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S3.F6" title="Figure 6 ‣ 3.3 Doppler tracking forecasts ‣ 3 Strain power sensitivity ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">6</span></a> shows the <math alttext="f<10^{-5}" class="ltx_Math" display="inline" id="S3.SS3.p7.2.m2.1"><semantics id="S3.SS3.p7.2.m2.1a"><mrow id="S3.SS3.p7.2.m2.1.1" xref="S3.SS3.p7.2.m2.1.1.cmml"><mi id="S3.SS3.p7.2.m2.1.1.2" xref="S3.SS3.p7.2.m2.1.1.2.cmml">f</mi><mo id="S3.SS3.p7.2.m2.1.1.1" xref="S3.SS3.p7.2.m2.1.1.1.cmml"><</mo><msup id="S3.SS3.p7.2.m2.1.1.3" xref="S3.SS3.p7.2.m2.1.1.3.cmml"><mn id="S3.SS3.p7.2.m2.1.1.3.2" xref="S3.SS3.p7.2.m2.1.1.3.2.cmml">10</mn><mrow id="S3.SS3.p7.2.m2.1.1.3.3" xref="S3.SS3.p7.2.m2.1.1.3.3.cmml"><mo id="S3.SS3.p7.2.m2.1.1.3.3a" xref="S3.SS3.p7.2.m2.1.1.3.3.cmml">−</mo><mn id="S3.SS3.p7.2.m2.1.1.3.3.2" xref="S3.SS3.p7.2.m2.1.1.3.3.2.cmml">5</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p7.2.m2.1b"><apply id="S3.SS3.p7.2.m2.1.1.cmml" xref="S3.SS3.p7.2.m2.1.1"><lt id="S3.SS3.p7.2.m2.1.1.1.cmml" xref="S3.SS3.p7.2.m2.1.1.1"></lt><ci id="S3.SS3.p7.2.m2.1.1.2.cmml" xref="S3.SS3.p7.2.m2.1.1.2">𝑓</ci><apply id="S3.SS3.p7.2.m2.1.1.3.cmml" xref="S3.SS3.p7.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.p7.2.m2.1.1.3.1.cmml" xref="S3.SS3.p7.2.m2.1.1.3">superscript</csymbol><cn id="S3.SS3.p7.2.m2.1.1.3.2.cmml" type="integer" xref="S3.SS3.p7.2.m2.1.1.3.2">10</cn><apply id="S3.SS3.p7.2.m2.1.1.3.3.cmml" xref="S3.SS3.p7.2.m2.1.1.3.3"><minus id="S3.SS3.p7.2.m2.1.1.3.3.1.cmml" xref="S3.SS3.p7.2.m2.1.1.3.3"></minus><cn id="S3.SS3.p7.2.m2.1.1.3.3.2.cmml" type="integer" xref="S3.SS3.p7.2.m2.1.1.3.3.2">5</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p7.2.m2.1c">f<10^{-5}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p7.2.m2.1d">italic_f < 10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT</annotation></semantics></math>Hz ‘priority’ sensitivity goal for this “Uranus Orbiter and Probe” (UOP) mission. This goal assumes an order of magnitude improvement of the theoretical Cassini noise, which was an order of magnitude better than what was actually achieved, and extrapolates this noise curve to lower frequencies. Our estimates show that at <math alttext="f<10^{-6}" class="ltx_Math" display="inline" id="S3.SS3.p7.3.m3.1"><semantics id="S3.SS3.p7.3.m3.1a"><mrow id="S3.SS3.p7.3.m3.1.1" xref="S3.SS3.p7.3.m3.1.1.cmml"><mi id="S3.SS3.p7.3.m3.1.1.2" xref="S3.SS3.p7.3.m3.1.1.2.cmml">f</mi><mo id="S3.SS3.p7.3.m3.1.1.1" xref="S3.SS3.p7.3.m3.1.1.1.cmml"><</mo><msup id="S3.SS3.p7.3.m3.1.1.3" xref="S3.SS3.p7.3.m3.1.1.3.cmml"><mn id="S3.SS3.p7.3.m3.1.1.3.2" xref="S3.SS3.p7.3.m3.1.1.3.2.cmml">10</mn><mrow id="S3.SS3.p7.3.m3.1.1.3.3" xref="S3.SS3.p7.3.m3.1.1.3.3.cmml"><mo id="S3.SS3.p7.3.m3.1.1.3.3a" xref="S3.SS3.p7.3.m3.1.1.3.3.cmml">−</mo><mn id="S3.SS3.p7.3.m3.1.1.3.3.2" xref="S3.SS3.p7.3.m3.1.1.3.3.2.cmml">6</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p7.3.m3.1b"><apply id="S3.SS3.p7.3.m3.1.1.cmml" xref="S3.SS3.p7.3.m3.1.1"><lt id="S3.SS3.p7.3.m3.1.1.1.cmml" xref="S3.SS3.p7.3.m3.1.1.1"></lt><ci id="S3.SS3.p7.3.m3.1.1.2.cmml" xref="S3.SS3.p7.3.m3.1.1.2">𝑓</ci><apply id="S3.SS3.p7.3.m3.1.1.3.cmml" xref="S3.SS3.p7.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.p7.3.m3.1.1.3.1.cmml" xref="S3.SS3.p7.3.m3.1.1.3">superscript</csymbol><cn id="S3.SS3.p7.3.m3.1.1.3.2.cmml" type="integer" xref="S3.SS3.p7.3.m3.1.1.3.2">10</cn><apply id="S3.SS3.p7.3.m3.1.1.3.3.cmml" xref="S3.SS3.p7.3.m3.1.1.3.3"><minus id="S3.SS3.p7.3.m3.1.1.3.3.1.cmml" xref="S3.SS3.p7.3.m3.1.1.3.3"></minus><cn id="S3.SS3.p7.3.m3.1.1.3.3.2.cmml" type="integer" xref="S3.SS3.p7.3.m3.1.1.3.3.2">6</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p7.3.m3.1c">f<10^{-6}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p7.3.m3.1d">italic_f < 10 start_POSTSUPERSCRIPT - 6 end_POSTSUPERSCRIPT</annotation></semantics></math>Hz, the accelerations on the spacecraft from the solar irradiance and solar wind must be corrected (likely by equipping the spacecraft with solar irradiance and plasma monitors) to achieve this sensitivity goal.</p> </div> <div class="ltx_para" id="S3.SS3.p8"> <p class="ltx_p" id="S3.SS3.p8.7">As discussed in <cite class="ltx_cite ltx_citemacro_citet">Zwick et al. (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib75" title="">2024</a>)</cite> for the UOP, Doppler tracking using lasers might also be possible, but such an approach is unlikely to improve the sensitivity with respect to our radio estimates. Earth’s turbulent atmosphere and the low <math alttext="C/N_{0}" class="ltx_Math" display="inline" id="S3.SS3.p8.1.m1.1"><semantics id="S3.SS3.p8.1.m1.1a"><mrow id="S3.SS3.p8.1.m1.1.1" xref="S3.SS3.p8.1.m1.1.1.cmml"><mi id="S3.SS3.p8.1.m1.1.1.2" xref="S3.SS3.p8.1.m1.1.1.2.cmml">C</mi><mo id="S3.SS3.p8.1.m1.1.1.1" xref="S3.SS3.p8.1.m1.1.1.1.cmml">/</mo><msub id="S3.SS3.p8.1.m1.1.1.3" xref="S3.SS3.p8.1.m1.1.1.3.cmml"><mi id="S3.SS3.p8.1.m1.1.1.3.2" xref="S3.SS3.p8.1.m1.1.1.3.2.cmml">N</mi><mn id="S3.SS3.p8.1.m1.1.1.3.3" xref="S3.SS3.p8.1.m1.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p8.1.m1.1b"><apply id="S3.SS3.p8.1.m1.1.1.cmml" xref="S3.SS3.p8.1.m1.1.1"><divide id="S3.SS3.p8.1.m1.1.1.1.cmml" xref="S3.SS3.p8.1.m1.1.1.1"></divide><ci id="S3.SS3.p8.1.m1.1.1.2.cmml" xref="S3.SS3.p8.1.m1.1.1.2">𝐶</ci><apply id="S3.SS3.p8.1.m1.1.1.3.cmml" xref="S3.SS3.p8.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.p8.1.m1.1.1.3.1.cmml" xref="S3.SS3.p8.1.m1.1.1.3">subscript</csymbol><ci id="S3.SS3.p8.1.m1.1.1.3.2.cmml" xref="S3.SS3.p8.1.m1.1.1.3.2">𝑁</ci><cn id="S3.SS3.p8.1.m1.1.1.3.3.cmml" type="integer" xref="S3.SS3.p8.1.m1.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p8.1.m1.1c">C/N_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p8.1.m1.1d">italic_C / italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> downlink make phase locking in the optical impractical (as this requires both adaptive optics and a <math alttext="C/N_{0}" class="ltx_Math" display="inline" id="S3.SS3.p8.2.m2.1"><semantics id="S3.SS3.p8.2.m2.1a"><mrow id="S3.SS3.p8.2.m2.1.1" xref="S3.SS3.p8.2.m2.1.1.cmml"><mi id="S3.SS3.p8.2.m2.1.1.2" xref="S3.SS3.p8.2.m2.1.1.2.cmml">C</mi><mo id="S3.SS3.p8.2.m2.1.1.1" xref="S3.SS3.p8.2.m2.1.1.1.cmml">/</mo><msub id="S3.SS3.p8.2.m2.1.1.3" xref="S3.SS3.p8.2.m2.1.1.3.cmml"><mi id="S3.SS3.p8.2.m2.1.1.3.2" xref="S3.SS3.p8.2.m2.1.1.3.2.cmml">N</mi><mn id="S3.SS3.p8.2.m2.1.1.3.3" xref="S3.SS3.p8.2.m2.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p8.2.m2.1b"><apply id="S3.SS3.p8.2.m2.1.1.cmml" xref="S3.SS3.p8.2.m2.1.1"><divide id="S3.SS3.p8.2.m2.1.1.1.cmml" xref="S3.SS3.p8.2.m2.1.1.1"></divide><ci id="S3.SS3.p8.2.m2.1.1.2.cmml" xref="S3.SS3.p8.2.m2.1.1.2">𝐶</ci><apply id="S3.SS3.p8.2.m2.1.1.3.cmml" xref="S3.SS3.p8.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.p8.2.m2.1.1.3.1.cmml" xref="S3.SS3.p8.2.m2.1.1.3">subscript</csymbol><ci id="S3.SS3.p8.2.m2.1.1.3.2.cmml" xref="S3.SS3.p8.2.m2.1.1.3.2">𝑁</ci><cn id="S3.SS3.p8.2.m2.1.1.3.3.cmml" type="integer" xref="S3.SS3.p8.2.m2.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p8.2.m2.1c">C/N_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p8.2.m2.1d">italic_C / italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> sufficient to achieve phase lock in the millisecond before the atmosphere changes). Thus, a laser Doppler tracking system would use the beat of laser pulses rather than the carrier phase. In contrast to timing the carrier phase – which allows a matched filter-like approach that results in the timing sensitivity scaling as <math alttext="L^{-1}" class="ltx_Math" display="inline" id="S3.SS3.p8.3.m3.1"><semantics id="S3.SS3.p8.3.m3.1a"><msup id="S3.SS3.p8.3.m3.1.1" xref="S3.SS3.p8.3.m3.1.1.cmml"><mi id="S3.SS3.p8.3.m3.1.1.2" xref="S3.SS3.p8.3.m3.1.1.2.cmml">L</mi><mrow id="S3.SS3.p8.3.m3.1.1.3" xref="S3.SS3.p8.3.m3.1.1.3.cmml"><mo id="S3.SS3.p8.3.m3.1.1.3a" xref="S3.SS3.p8.3.m3.1.1.3.cmml">−</mo><mn id="S3.SS3.p8.3.m3.1.1.3.2" xref="S3.SS3.p8.3.m3.1.1.3.2.cmml">1</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.SS3.p8.3.m3.1b"><apply id="S3.SS3.p8.3.m3.1.1.cmml" xref="S3.SS3.p8.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS3.p8.3.m3.1.1.1.cmml" xref="S3.SS3.p8.3.m3.1.1">superscript</csymbol><ci id="S3.SS3.p8.3.m3.1.1.2.cmml" xref="S3.SS3.p8.3.m3.1.1.2">𝐿</ci><apply id="S3.SS3.p8.3.m3.1.1.3.cmml" xref="S3.SS3.p8.3.m3.1.1.3"><minus id="S3.SS3.p8.3.m3.1.1.3.1.cmml" xref="S3.SS3.p8.3.m3.1.1.3"></minus><cn id="S3.SS3.p8.3.m3.1.1.3.2.cmml" type="integer" xref="S3.SS3.p8.3.m3.1.1.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p8.3.m3.1c">L^{-1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p8.3.m3.1d">italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> –, the timing sensitivity for pulsed lasers scales as <math alttext="L^{-2}" class="ltx_Math" display="inline" id="S3.SS3.p8.4.m4.1"><semantics id="S3.SS3.p8.4.m4.1a"><msup id="S3.SS3.p8.4.m4.1.1" xref="S3.SS3.p8.4.m4.1.1.cmml"><mi id="S3.SS3.p8.4.m4.1.1.2" xref="S3.SS3.p8.4.m4.1.1.2.cmml">L</mi><mrow id="S3.SS3.p8.4.m4.1.1.3" xref="S3.SS3.p8.4.m4.1.1.3.cmml"><mo id="S3.SS3.p8.4.m4.1.1.3a" xref="S3.SS3.p8.4.m4.1.1.3.cmml">−</mo><mn id="S3.SS3.p8.4.m4.1.1.3.2" xref="S3.SS3.p8.4.m4.1.1.3.2.cmml">2</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.SS3.p8.4.m4.1b"><apply id="S3.SS3.p8.4.m4.1.1.cmml" xref="S3.SS3.p8.4.m4.1.1"><csymbol cd="ambiguous" id="S3.SS3.p8.4.m4.1.1.1.cmml" xref="S3.SS3.p8.4.m4.1.1">superscript</csymbol><ci id="S3.SS3.p8.4.m4.1.1.2.cmml" xref="S3.SS3.p8.4.m4.1.1.2">𝐿</ci><apply id="S3.SS3.p8.4.m4.1.1.3.cmml" xref="S3.SS3.p8.4.m4.1.1.3"><minus id="S3.SS3.p8.4.m4.1.1.3.1.cmml" xref="S3.SS3.p8.4.m4.1.1.3"></minus><cn id="S3.SS3.p8.4.m4.1.1.3.2.cmml" type="integer" xref="S3.SS3.p8.4.m4.1.1.3.2">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p8.4.m4.1c">L^{-2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p8.4.m4.1d">italic_L start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT</annotation></semantics></math>, which is problematic for the envisioned Solar System-scale baselines. Equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.E13" title="In 2.2 Radiometer noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">13</span></a>) in comparison to equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.E12" title="In 2.2 Radiometer noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">12</span></a>) shows that at <math alttext="L=30~{}" class="ltx_Math" display="inline" id="S3.SS3.p8.5.m5.1"><semantics id="S3.SS3.p8.5.m5.1a"><mrow id="S3.SS3.p8.5.m5.1.1" xref="S3.SS3.p8.5.m5.1.1.cmml"><mi id="S3.SS3.p8.5.m5.1.1.2" xref="S3.SS3.p8.5.m5.1.1.2.cmml">L</mi><mo id="S3.SS3.p8.5.m5.1.1.1" xref="S3.SS3.p8.5.m5.1.1.1.cmml">=</mo><mn id="S3.SS3.p8.5.m5.1.1.3" xref="S3.SS3.p8.5.m5.1.1.3.cmml">30</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p8.5.m5.1b"><apply id="S3.SS3.p8.5.m5.1.1.cmml" xref="S3.SS3.p8.5.m5.1.1"><eq id="S3.SS3.p8.5.m5.1.1.1.cmml" xref="S3.SS3.p8.5.m5.1.1.1"></eq><ci id="S3.SS3.p8.5.m5.1.1.2.cmml" xref="S3.SS3.p8.5.m5.1.1.2">𝐿</ci><cn id="S3.SS3.p8.5.m5.1.1.3.cmml" type="integer" xref="S3.SS3.p8.5.m5.1.1.3">30</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p8.5.m5.1c">L=30~{}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p8.5.m5.1d">italic_L = 30</annotation></semantics></math>AU the timing noise for pulsed lasers will only be comparable with radio transmissions if a <math alttext="D_{2}\sim 10\," class="ltx_Math" display="inline" id="S3.SS3.p8.6.m6.1"><semantics id="S3.SS3.p8.6.m6.1a"><mrow id="S3.SS3.p8.6.m6.1.1" xref="S3.SS3.p8.6.m6.1.1.cmml"><msub id="S3.SS3.p8.6.m6.1.1.2" xref="S3.SS3.p8.6.m6.1.1.2.cmml"><mi id="S3.SS3.p8.6.m6.1.1.2.2" xref="S3.SS3.p8.6.m6.1.1.2.2.cmml">D</mi><mn id="S3.SS3.p8.6.m6.1.1.2.3" xref="S3.SS3.p8.6.m6.1.1.2.3.cmml">2</mn></msub><mo id="S3.SS3.p8.6.m6.1.1.1" xref="S3.SS3.p8.6.m6.1.1.1.cmml">∼</mo><mn id="S3.SS3.p8.6.m6.1.1.3" xref="S3.SS3.p8.6.m6.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p8.6.m6.1b"><apply id="S3.SS3.p8.6.m6.1.1.cmml" xref="S3.SS3.p8.6.m6.1.1"><csymbol cd="latexml" id="S3.SS3.p8.6.m6.1.1.1.cmml" xref="S3.SS3.p8.6.m6.1.1.1">similar-to</csymbol><apply id="S3.SS3.p8.6.m6.1.1.2.cmml" xref="S3.SS3.p8.6.m6.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.p8.6.m6.1.1.2.1.cmml" xref="S3.SS3.p8.6.m6.1.1.2">subscript</csymbol><ci id="S3.SS3.p8.6.m6.1.1.2.2.cmml" xref="S3.SS3.p8.6.m6.1.1.2.2">𝐷</ci><cn id="S3.SS3.p8.6.m6.1.1.2.3.cmml" type="integer" xref="S3.SS3.p8.6.m6.1.1.2.3">2</cn></apply><cn id="S3.SS3.p8.6.m6.1.1.3.cmml" type="integer" xref="S3.SS3.p8.6.m6.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p8.6.m6.1c">D_{2}\sim 10\,</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p8.6.m6.1d">italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∼ 10</annotation></semantics></math>m telescope on Earth is deployed along with a <math alttext="D_{1}\sim 1\," class="ltx_Math" display="inline" id="S3.SS3.p8.7.m7.1"><semantics id="S3.SS3.p8.7.m7.1a"><mrow id="S3.SS3.p8.7.m7.1.1" xref="S3.SS3.p8.7.m7.1.1.cmml"><msub id="S3.SS3.p8.7.m7.1.1.2" xref="S3.SS3.p8.7.m7.1.1.2.cmml"><mi id="S3.SS3.p8.7.m7.1.1.2.2" xref="S3.SS3.p8.7.m7.1.1.2.2.cmml">D</mi><mn id="S3.SS3.p8.7.m7.1.1.2.3" xref="S3.SS3.p8.7.m7.1.1.2.3.cmml">1</mn></msub><mo id="S3.SS3.p8.7.m7.1.1.1" xref="S3.SS3.p8.7.m7.1.1.1.cmml">∼</mo><mn id="S3.SS3.p8.7.m7.1.1.3" xref="S3.SS3.p8.7.m7.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p8.7.m7.1b"><apply id="S3.SS3.p8.7.m7.1.1.cmml" xref="S3.SS3.p8.7.m7.1.1"><csymbol cd="latexml" id="S3.SS3.p8.7.m7.1.1.1.cmml" xref="S3.SS3.p8.7.m7.1.1.1">similar-to</csymbol><apply id="S3.SS3.p8.7.m7.1.1.2.cmml" xref="S3.SS3.p8.7.m7.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.p8.7.m7.1.1.2.1.cmml" xref="S3.SS3.p8.7.m7.1.1.2">subscript</csymbol><ci id="S3.SS3.p8.7.m7.1.1.2.2.cmml" xref="S3.SS3.p8.7.m7.1.1.2.2">𝐷</ci><cn id="S3.SS3.p8.7.m7.1.1.2.3.cmml" type="integer" xref="S3.SS3.p8.7.m7.1.1.2.3">1</cn></apply><cn id="S3.SS3.p8.7.m7.1.1.3.cmml" type="integer" xref="S3.SS3.p8.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p8.7.m7.1c">D_{1}\sim 1\,</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p8.7.m7.1d">italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∼ 1</annotation></semantics></math>m on the spacecraft. Additionally, optical transmissions experience a time-varying delay from the atmosphere that is similar in magnitude to the wet tropospheric delay for radio waves: Near zenith, this delay is a couple of centimeters and can be corrected using atmospheric instrumentation similar to those used for the wet tropospheric delay. Attempts to correct for this error in the context of lunar laser ranging has resulted in millimeter-scale errors <cite class="ltx_cite ltx_citemacro_citep">(Marini & C. W. Murray, <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib47" title="">1973</a>)</cite>.</p> </div> </section> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">4 </span>Instrumental considerations</h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.1">When fixing the strain noise power due to shot noise at long wavelengths, the timing error, displacement error, and square of the angular pointing error are relaxed in proportion to the baseline distance. Thus, the extremely long arms of the presented concepts may reduce many design tolerances. Of course, long arms do come with the challenge that the transmitted signal strengths are much weaker compared to, e.g., LISA; Section <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.SS2" title="2.2 Radiometer noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">2.2</span></a> argued that the strengths are still sufficient to acquire stable phase locks. We now discuss other challenges of the laser and radio concepts, as well as challenges with placing spacecraft in the outer Solar System.</p> </div> <section class="ltx_subsection" id="S4.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.1 </span>Laser design</h3> <div class="ltx_para" id="S4.SS1.p1"> <p class="ltx_p" id="S4.SS1.p1.6">Perhaps the biggest challenge in our laser-based concept is tuning the laser frequencies to cancel the relative velocities of the spacecraft. The spacecraft will likely have much larger relative velocities compared to LISA, which are kept to <math alttext="\lesssim 10\," class="ltx_Math" display="inline" id="S4.SS1.p1.1.m1.1"><semantics id="S4.SS1.p1.1.m1.1a"><mrow id="S4.SS1.p1.1.m1.1.1" xref="S4.SS1.p1.1.m1.1.1.cmml"><mi id="S4.SS1.p1.1.m1.1.1.2" xref="S4.SS1.p1.1.m1.1.1.2.cmml"></mi><mo id="S4.SS1.p1.1.m1.1.1.1" xref="S4.SS1.p1.1.m1.1.1.1.cmml">≲</mo><mn id="S4.SS1.p1.1.m1.1.1.3" xref="S4.SS1.p1.1.m1.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.1.m1.1b"><apply id="S4.SS1.p1.1.m1.1.1.cmml" xref="S4.SS1.p1.1.m1.1.1"><csymbol cd="latexml" id="S4.SS1.p1.1.m1.1.1.1.cmml" xref="S4.SS1.p1.1.m1.1.1.1">less-than-or-similar-to</csymbol><csymbol cd="latexml" id="S4.SS1.p1.1.m1.1.1.2.cmml" xref="S4.SS1.p1.1.m1.1.1.2">absent</csymbol><cn id="S4.SS1.p1.1.m1.1.1.3.cmml" type="integer" xref="S4.SS1.p1.1.m1.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.1.m1.1c">\lesssim 10\,</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.1.m1.1d">≲ 10</annotation></semantics></math>m s<sup class="ltx_sup" id="S4.SS1.p1.6.1"><span class="ltx_text ltx_font_italic" id="S4.SS1.p1.6.1.1">-1</span></sup>. (The way to avoid large relative velocities is if the outer Solar System spacecraft were in a circular orbit, each separated by <math alttext="120^{\circ}" class="ltx_Math" display="inline" id="S4.SS1.p1.3.m3.1"><semantics id="S4.SS1.p1.3.m3.1a"><msup id="S4.SS1.p1.3.m3.1.1" xref="S4.SS1.p1.3.m3.1.1.cmml"><mn id="S4.SS1.p1.3.m3.1.1.2" xref="S4.SS1.p1.3.m3.1.1.2.cmml">120</mn><mo id="S4.SS1.p1.3.m3.1.1.3" xref="S4.SS1.p1.3.m3.1.1.3.cmml">∘</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.3.m3.1b"><apply id="S4.SS1.p1.3.m3.1.1.cmml" xref="S4.SS1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.3.m3.1.1.1.cmml" xref="S4.SS1.p1.3.m3.1.1">superscript</csymbol><cn id="S4.SS1.p1.3.m3.1.1.2.cmml" type="integer" xref="S4.SS1.p1.3.m3.1.1.2">120</cn><compose id="S4.SS1.p1.3.m3.1.1.3.cmml" xref="S4.SS1.p1.3.m3.1.1.3"></compose></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.3.m3.1c">120^{\circ}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.3.m3.1d">120 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT</annotation></semantics></math> in orbital phase; <cite class="ltx_cite ltx_citemacro_citet">Folkner <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib30" title="">2011</a>; Sesana et al. <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib62" title="">2021</a></cite>. Achieving such a configuration in the outer Solar System would be challenging.) Relative velocities are problematic in that they spoil the cancellation of clock errors from time-delay interferometry, as the observed phase difference (eqn. <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.E6" title="In Two-arm and time-delay interferometry ‣ 2.1 Clock noise and its mitigation ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">6</span></a>) now has a strongly time-dependent phase given by <math alttext="\phi_{\rm het}=2\pi f_{\rm het}t" class="ltx_Math" display="inline" id="S4.SS1.p1.4.m4.1"><semantics id="S4.SS1.p1.4.m4.1a"><mrow id="S4.SS1.p1.4.m4.1.1" xref="S4.SS1.p1.4.m4.1.1.cmml"><msub id="S4.SS1.p1.4.m4.1.1.2" xref="S4.SS1.p1.4.m4.1.1.2.cmml"><mi id="S4.SS1.p1.4.m4.1.1.2.2" xref="S4.SS1.p1.4.m4.1.1.2.2.cmml">ϕ</mi><mi id="S4.SS1.p1.4.m4.1.1.2.3" xref="S4.SS1.p1.4.m4.1.1.2.3.cmml">het</mi></msub><mo id="S4.SS1.p1.4.m4.1.1.1" xref="S4.SS1.p1.4.m4.1.1.1.cmml">=</mo><mrow id="S4.SS1.p1.4.m4.1.1.3" xref="S4.SS1.p1.4.m4.1.1.3.cmml"><mn id="S4.SS1.p1.4.m4.1.1.3.2" xref="S4.SS1.p1.4.m4.1.1.3.2.cmml">2</mn><mo id="S4.SS1.p1.4.m4.1.1.3.1" xref="S4.SS1.p1.4.m4.1.1.3.1.cmml">⁢</mo><mi id="S4.SS1.p1.4.m4.1.1.3.3" xref="S4.SS1.p1.4.m4.1.1.3.3.cmml">π</mi><mo id="S4.SS1.p1.4.m4.1.1.3.1a" xref="S4.SS1.p1.4.m4.1.1.3.1.cmml">⁢</mo><msub id="S4.SS1.p1.4.m4.1.1.3.4" xref="S4.SS1.p1.4.m4.1.1.3.4.cmml"><mi id="S4.SS1.p1.4.m4.1.1.3.4.2" xref="S4.SS1.p1.4.m4.1.1.3.4.2.cmml">f</mi><mi id="S4.SS1.p1.4.m4.1.1.3.4.3" xref="S4.SS1.p1.4.m4.1.1.3.4.3.cmml">het</mi></msub><mo id="S4.SS1.p1.4.m4.1.1.3.1b" xref="S4.SS1.p1.4.m4.1.1.3.1.cmml">⁢</mo><mi id="S4.SS1.p1.4.m4.1.1.3.5" xref="S4.SS1.p1.4.m4.1.1.3.5.cmml">t</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.4.m4.1b"><apply id="S4.SS1.p1.4.m4.1.1.cmml" xref="S4.SS1.p1.4.m4.1.1"><eq id="S4.SS1.p1.4.m4.1.1.1.cmml" xref="S4.SS1.p1.4.m4.1.1.1"></eq><apply id="S4.SS1.p1.4.m4.1.1.2.cmml" xref="S4.SS1.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p1.4.m4.1.1.2.1.cmml" xref="S4.SS1.p1.4.m4.1.1.2">subscript</csymbol><ci id="S4.SS1.p1.4.m4.1.1.2.2.cmml" xref="S4.SS1.p1.4.m4.1.1.2.2">italic-ϕ</ci><ci id="S4.SS1.p1.4.m4.1.1.2.3.cmml" xref="S4.SS1.p1.4.m4.1.1.2.3">het</ci></apply><apply id="S4.SS1.p1.4.m4.1.1.3.cmml" xref="S4.SS1.p1.4.m4.1.1.3"><times id="S4.SS1.p1.4.m4.1.1.3.1.cmml" xref="S4.SS1.p1.4.m4.1.1.3.1"></times><cn id="S4.SS1.p1.4.m4.1.1.3.2.cmml" type="integer" xref="S4.SS1.p1.4.m4.1.1.3.2">2</cn><ci id="S4.SS1.p1.4.m4.1.1.3.3.cmml" xref="S4.SS1.p1.4.m4.1.1.3.3">𝜋</ci><apply id="S4.SS1.p1.4.m4.1.1.3.4.cmml" xref="S4.SS1.p1.4.m4.1.1.3.4"><csymbol cd="ambiguous" id="S4.SS1.p1.4.m4.1.1.3.4.1.cmml" xref="S4.SS1.p1.4.m4.1.1.3.4">subscript</csymbol><ci id="S4.SS1.p1.4.m4.1.1.3.4.2.cmml" xref="S4.SS1.p1.4.m4.1.1.3.4.2">𝑓</ci><ci id="S4.SS1.p1.4.m4.1.1.3.4.3.cmml" xref="S4.SS1.p1.4.m4.1.1.3.4.3">het</ci></apply><ci id="S4.SS1.p1.4.m4.1.1.3.5.cmml" xref="S4.SS1.p1.4.m4.1.1.3.5">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.4.m4.1c">\phi_{\rm het}=2\pi f_{\rm het}t</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.4.m4.1d">italic_ϕ start_POSTSUBSCRIPT roman_het end_POSTSUBSCRIPT = 2 italic_π italic_f start_POSTSUBSCRIPT roman_het end_POSTSUBSCRIPT italic_t</annotation></semantics></math>, where <math alttext="f_{\rm het}=v/\lambda" class="ltx_Math" display="inline" id="S4.SS1.p1.5.m5.1"><semantics id="S4.SS1.p1.5.m5.1a"><mrow id="S4.SS1.p1.5.m5.1.1" xref="S4.SS1.p1.5.m5.1.1.cmml"><msub id="S4.SS1.p1.5.m5.1.1.2" xref="S4.SS1.p1.5.m5.1.1.2.cmml"><mi id="S4.SS1.p1.5.m5.1.1.2.2" xref="S4.SS1.p1.5.m5.1.1.2.2.cmml">f</mi><mi id="S4.SS1.p1.5.m5.1.1.2.3" xref="S4.SS1.p1.5.m5.1.1.2.3.cmml">het</mi></msub><mo id="S4.SS1.p1.5.m5.1.1.1" xref="S4.SS1.p1.5.m5.1.1.1.cmml">=</mo><mrow id="S4.SS1.p1.5.m5.1.1.3" xref="S4.SS1.p1.5.m5.1.1.3.cmml"><mi id="S4.SS1.p1.5.m5.1.1.3.2" xref="S4.SS1.p1.5.m5.1.1.3.2.cmml">v</mi><mo id="S4.SS1.p1.5.m5.1.1.3.1" xref="S4.SS1.p1.5.m5.1.1.3.1.cmml">/</mo><mi id="S4.SS1.p1.5.m5.1.1.3.3" xref="S4.SS1.p1.5.m5.1.1.3.3.cmml">λ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.5.m5.1b"><apply id="S4.SS1.p1.5.m5.1.1.cmml" xref="S4.SS1.p1.5.m5.1.1"><eq id="S4.SS1.p1.5.m5.1.1.1.cmml" xref="S4.SS1.p1.5.m5.1.1.1"></eq><apply id="S4.SS1.p1.5.m5.1.1.2.cmml" xref="S4.SS1.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p1.5.m5.1.1.2.1.cmml" xref="S4.SS1.p1.5.m5.1.1.2">subscript</csymbol><ci id="S4.SS1.p1.5.m5.1.1.2.2.cmml" xref="S4.SS1.p1.5.m5.1.1.2.2">𝑓</ci><ci id="S4.SS1.p1.5.m5.1.1.2.3.cmml" xref="S4.SS1.p1.5.m5.1.1.2.3">het</ci></apply><apply id="S4.SS1.p1.5.m5.1.1.3.cmml" xref="S4.SS1.p1.5.m5.1.1.3"><divide id="S4.SS1.p1.5.m5.1.1.3.1.cmml" xref="S4.SS1.p1.5.m5.1.1.3.1"></divide><ci id="S4.SS1.p1.5.m5.1.1.3.2.cmml" xref="S4.SS1.p1.5.m5.1.1.3.2">𝑣</ci><ci id="S4.SS1.p1.5.m5.1.1.3.3.cmml" xref="S4.SS1.p1.5.m5.1.1.3.3">𝜆</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.5.m5.1c">f_{\rm het}=v/\lambda</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.5.m5.1d">italic_f start_POSTSUBSCRIPT roman_het end_POSTSUBSCRIPT = italic_v / italic_λ</annotation></semantics></math>. This time-dependent phase means that, when it is recorded, the clock error <math alttext="t_{\rm os}" class="ltx_Math" display="inline" id="S4.SS1.p1.6.m6.1"><semantics id="S4.SS1.p1.6.m6.1a"><msub id="S4.SS1.p1.6.m6.1.1" xref="S4.SS1.p1.6.m6.1.1.cmml"><mi id="S4.SS1.p1.6.m6.1.1.2" xref="S4.SS1.p1.6.m6.1.1.2.cmml">t</mi><mi id="S4.SS1.p1.6.m6.1.1.3" xref="S4.SS1.p1.6.m6.1.1.3.cmml">os</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.6.m6.1b"><apply id="S4.SS1.p1.6.m6.1.1.cmml" xref="S4.SS1.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.6.m6.1.1.1.cmml" xref="S4.SS1.p1.6.m6.1.1">subscript</csymbol><ci id="S4.SS1.p1.6.m6.1.1.2.cmml" xref="S4.SS1.p1.6.m6.1.1.2">𝑡</ci><ci id="S4.SS1.p1.6.m6.1.1.3.cmml" xref="S4.SS1.p1.6.m6.1.1.3">os</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.6.m6.1c">t_{\rm os}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.6.m6.1d">italic_t start_POSTSUBSCRIPT roman_os end_POSTSUBSCRIPT</annotation></semantics></math> – or any timing error – again enters and results in a phase error of</p> <table class="ltx_equation ltx_eqn_table" id="S4.E19"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\delta\phi_{\rm het}=2\pi f_{\rm het}\delta t_{\rm os}." class="ltx_Math" display="block" id="S4.E19.m1.1"><semantics id="S4.E19.m1.1a"><mrow id="S4.E19.m1.1.1.1" xref="S4.E19.m1.1.1.1.1.cmml"><mrow id="S4.E19.m1.1.1.1.1" xref="S4.E19.m1.1.1.1.1.cmml"><mrow id="S4.E19.m1.1.1.1.1.2" xref="S4.E19.m1.1.1.1.1.2.cmml"><mi id="S4.E19.m1.1.1.1.1.2.2" xref="S4.E19.m1.1.1.1.1.2.2.cmml">δ</mi><mo id="S4.E19.m1.1.1.1.1.2.1" xref="S4.E19.m1.1.1.1.1.2.1.cmml">⁢</mo><msub id="S4.E19.m1.1.1.1.1.2.3" xref="S4.E19.m1.1.1.1.1.2.3.cmml"><mi id="S4.E19.m1.1.1.1.1.2.3.2" xref="S4.E19.m1.1.1.1.1.2.3.2.cmml">ϕ</mi><mi id="S4.E19.m1.1.1.1.1.2.3.3" xref="S4.E19.m1.1.1.1.1.2.3.3.cmml">het</mi></msub></mrow><mo id="S4.E19.m1.1.1.1.1.1" xref="S4.E19.m1.1.1.1.1.1.cmml">=</mo><mrow id="S4.E19.m1.1.1.1.1.3" xref="S4.E19.m1.1.1.1.1.3.cmml"><mn id="S4.E19.m1.1.1.1.1.3.2" xref="S4.E19.m1.1.1.1.1.3.2.cmml">2</mn><mo id="S4.E19.m1.1.1.1.1.3.1" xref="S4.E19.m1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S4.E19.m1.1.1.1.1.3.3" xref="S4.E19.m1.1.1.1.1.3.3.cmml">π</mi><mo id="S4.E19.m1.1.1.1.1.3.1a" xref="S4.E19.m1.1.1.1.1.3.1.cmml">⁢</mo><msub id="S4.E19.m1.1.1.1.1.3.4" xref="S4.E19.m1.1.1.1.1.3.4.cmml"><mi id="S4.E19.m1.1.1.1.1.3.4.2" xref="S4.E19.m1.1.1.1.1.3.4.2.cmml">f</mi><mi id="S4.E19.m1.1.1.1.1.3.4.3" xref="S4.E19.m1.1.1.1.1.3.4.3.cmml">het</mi></msub><mo id="S4.E19.m1.1.1.1.1.3.1b" xref="S4.E19.m1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S4.E19.m1.1.1.1.1.3.5" xref="S4.E19.m1.1.1.1.1.3.5.cmml">δ</mi><mo id="S4.E19.m1.1.1.1.1.3.1c" xref="S4.E19.m1.1.1.1.1.3.1.cmml">⁢</mo><msub id="S4.E19.m1.1.1.1.1.3.6" xref="S4.E19.m1.1.1.1.1.3.6.cmml"><mi id="S4.E19.m1.1.1.1.1.3.6.2" xref="S4.E19.m1.1.1.1.1.3.6.2.cmml">t</mi><mi id="S4.E19.m1.1.1.1.1.3.6.3" xref="S4.E19.m1.1.1.1.1.3.6.3.cmml">os</mi></msub></mrow></mrow><mo id="S4.E19.m1.1.1.1.2" lspace="0em" xref="S4.E19.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E19.m1.1b"><apply id="S4.E19.m1.1.1.1.1.cmml" xref="S4.E19.m1.1.1.1"><eq id="S4.E19.m1.1.1.1.1.1.cmml" xref="S4.E19.m1.1.1.1.1.1"></eq><apply id="S4.E19.m1.1.1.1.1.2.cmml" xref="S4.E19.m1.1.1.1.1.2"><times id="S4.E19.m1.1.1.1.1.2.1.cmml" xref="S4.E19.m1.1.1.1.1.2.1"></times><ci id="S4.E19.m1.1.1.1.1.2.2.cmml" xref="S4.E19.m1.1.1.1.1.2.2">𝛿</ci><apply id="S4.E19.m1.1.1.1.1.2.3.cmml" xref="S4.E19.m1.1.1.1.1.2.3"><csymbol cd="ambiguous" id="S4.E19.m1.1.1.1.1.2.3.1.cmml" xref="S4.E19.m1.1.1.1.1.2.3">subscript</csymbol><ci id="S4.E19.m1.1.1.1.1.2.3.2.cmml" xref="S4.E19.m1.1.1.1.1.2.3.2">italic-ϕ</ci><ci id="S4.E19.m1.1.1.1.1.2.3.3.cmml" xref="S4.E19.m1.1.1.1.1.2.3.3">het</ci></apply></apply><apply id="S4.E19.m1.1.1.1.1.3.cmml" xref="S4.E19.m1.1.1.1.1.3"><times id="S4.E19.m1.1.1.1.1.3.1.cmml" xref="S4.E19.m1.1.1.1.1.3.1"></times><cn id="S4.E19.m1.1.1.1.1.3.2.cmml" type="integer" xref="S4.E19.m1.1.1.1.1.3.2">2</cn><ci id="S4.E19.m1.1.1.1.1.3.3.cmml" xref="S4.E19.m1.1.1.1.1.3.3">𝜋</ci><apply id="S4.E19.m1.1.1.1.1.3.4.cmml" xref="S4.E19.m1.1.1.1.1.3.4"><csymbol cd="ambiguous" id="S4.E19.m1.1.1.1.1.3.4.1.cmml" xref="S4.E19.m1.1.1.1.1.3.4">subscript</csymbol><ci id="S4.E19.m1.1.1.1.1.3.4.2.cmml" xref="S4.E19.m1.1.1.1.1.3.4.2">𝑓</ci><ci id="S4.E19.m1.1.1.1.1.3.4.3.cmml" xref="S4.E19.m1.1.1.1.1.3.4.3">het</ci></apply><ci id="S4.E19.m1.1.1.1.1.3.5.cmml" xref="S4.E19.m1.1.1.1.1.3.5">𝛿</ci><apply id="S4.E19.m1.1.1.1.1.3.6.cmml" xref="S4.E19.m1.1.1.1.1.3.6"><csymbol cd="ambiguous" id="S4.E19.m1.1.1.1.1.3.6.1.cmml" xref="S4.E19.m1.1.1.1.1.3.6">subscript</csymbol><ci id="S4.E19.m1.1.1.1.1.3.6.2.cmml" xref="S4.E19.m1.1.1.1.1.3.6.2">𝑡</ci><ci id="S4.E19.m1.1.1.1.1.3.6.3.cmml" xref="S4.E19.m1.1.1.1.1.3.6.3">os</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E19.m1.1c">\delta\phi_{\rm het}=2\pi f_{\rm het}\delta t_{\rm os}.</annotation><annotation encoding="application/x-llamapun" id="S4.E19.m1.1d">italic_δ italic_ϕ start_POSTSUBSCRIPT roman_het end_POSTSUBSCRIPT = 2 italic_π italic_f start_POSTSUBSCRIPT roman_het end_POSTSUBSCRIPT italic_δ italic_t start_POSTSUBSCRIPT roman_os end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(19)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p1.10">For LISA, <math alttext="\delta\phi_{\rm het}" class="ltx_Math" display="inline" id="S4.SS1.p1.7.m1.1"><semantics id="S4.SS1.p1.7.m1.1a"><mrow id="S4.SS1.p1.7.m1.1.1" xref="S4.SS1.p1.7.m1.1.1.cmml"><mi id="S4.SS1.p1.7.m1.1.1.2" xref="S4.SS1.p1.7.m1.1.1.2.cmml">δ</mi><mo id="S4.SS1.p1.7.m1.1.1.1" xref="S4.SS1.p1.7.m1.1.1.1.cmml">⁢</mo><msub id="S4.SS1.p1.7.m1.1.1.3" xref="S4.SS1.p1.7.m1.1.1.3.cmml"><mi id="S4.SS1.p1.7.m1.1.1.3.2" xref="S4.SS1.p1.7.m1.1.1.3.2.cmml">ϕ</mi><mi id="S4.SS1.p1.7.m1.1.1.3.3" xref="S4.SS1.p1.7.m1.1.1.3.3.cmml">het</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.7.m1.1b"><apply id="S4.SS1.p1.7.m1.1.1.cmml" xref="S4.SS1.p1.7.m1.1.1"><times id="S4.SS1.p1.7.m1.1.1.1.cmml" xref="S4.SS1.p1.7.m1.1.1.1"></times><ci id="S4.SS1.p1.7.m1.1.1.2.cmml" xref="S4.SS1.p1.7.m1.1.1.2">𝛿</ci><apply id="S4.SS1.p1.7.m1.1.1.3.cmml" xref="S4.SS1.p1.7.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p1.7.m1.1.1.3.1.cmml" xref="S4.SS1.p1.7.m1.1.1.3">subscript</csymbol><ci id="S4.SS1.p1.7.m1.1.1.3.2.cmml" xref="S4.SS1.p1.7.m1.1.1.3.2">italic-ϕ</ci><ci id="S4.SS1.p1.7.m1.1.1.3.3.cmml" xref="S4.SS1.p1.7.m1.1.1.3.3">het</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.7.m1.1c">\delta\phi_{\rm het}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.7.m1.1d">italic_δ italic_ϕ start_POSTSUBSCRIPT roman_het end_POSTSUBSCRIPT</annotation></semantics></math> is large enough to substantially reduce instrument performance. The LISA design includes an elaborate scheme for correcting this error by superimposing weak pilot tones on top of the primary laser signal. The phase of these pilot tones contains the <math alttext="\delta t_{\rm os}" class="ltx_Math" display="inline" id="S4.SS1.p1.8.m2.1"><semantics id="S4.SS1.p1.8.m2.1a"><mrow id="S4.SS1.p1.8.m2.1.1" xref="S4.SS1.p1.8.m2.1.1.cmml"><mi id="S4.SS1.p1.8.m2.1.1.2" xref="S4.SS1.p1.8.m2.1.1.2.cmml">δ</mi><mo id="S4.SS1.p1.8.m2.1.1.1" xref="S4.SS1.p1.8.m2.1.1.1.cmml">⁢</mo><msub id="S4.SS1.p1.8.m2.1.1.3" xref="S4.SS1.p1.8.m2.1.1.3.cmml"><mi id="S4.SS1.p1.8.m2.1.1.3.2" xref="S4.SS1.p1.8.m2.1.1.3.2.cmml">t</mi><mi id="S4.SS1.p1.8.m2.1.1.3.3" xref="S4.SS1.p1.8.m2.1.1.3.3.cmml">os</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.8.m2.1b"><apply id="S4.SS1.p1.8.m2.1.1.cmml" xref="S4.SS1.p1.8.m2.1.1"><times id="S4.SS1.p1.8.m2.1.1.1.cmml" xref="S4.SS1.p1.8.m2.1.1.1"></times><ci id="S4.SS1.p1.8.m2.1.1.2.cmml" xref="S4.SS1.p1.8.m2.1.1.2">𝛿</ci><apply id="S4.SS1.p1.8.m2.1.1.3.cmml" xref="S4.SS1.p1.8.m2.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p1.8.m2.1.1.3.1.cmml" xref="S4.SS1.p1.8.m2.1.1.3">subscript</csymbol><ci id="S4.SS1.p1.8.m2.1.1.3.2.cmml" xref="S4.SS1.p1.8.m2.1.1.3.2">𝑡</ci><ci id="S4.SS1.p1.8.m2.1.1.3.3.cmml" xref="S4.SS1.p1.8.m2.1.1.3.3">os</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.8.m2.1c">\delta t_{\rm os}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.8.m2.1d">italic_δ italic_t start_POSTSUBSCRIPT roman_os end_POSTSUBSCRIPT</annotation></semantics></math> information that allows <math alttext="\delta\phi_{\rm het}" class="ltx_Math" display="inline" id="S4.SS1.p1.9.m3.1"><semantics id="S4.SS1.p1.9.m3.1a"><mrow id="S4.SS1.p1.9.m3.1.1" xref="S4.SS1.p1.9.m3.1.1.cmml"><mi id="S4.SS1.p1.9.m3.1.1.2" xref="S4.SS1.p1.9.m3.1.1.2.cmml">δ</mi><mo id="S4.SS1.p1.9.m3.1.1.1" xref="S4.SS1.p1.9.m3.1.1.1.cmml">⁢</mo><msub id="S4.SS1.p1.9.m3.1.1.3" xref="S4.SS1.p1.9.m3.1.1.3.cmml"><mi id="S4.SS1.p1.9.m3.1.1.3.2" xref="S4.SS1.p1.9.m3.1.1.3.2.cmml">ϕ</mi><mi id="S4.SS1.p1.9.m3.1.1.3.3" xref="S4.SS1.p1.9.m3.1.1.3.3.cmml">het</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.9.m3.1b"><apply id="S4.SS1.p1.9.m3.1.1.cmml" xref="S4.SS1.p1.9.m3.1.1"><times id="S4.SS1.p1.9.m3.1.1.1.cmml" xref="S4.SS1.p1.9.m3.1.1.1"></times><ci id="S4.SS1.p1.9.m3.1.1.2.cmml" xref="S4.SS1.p1.9.m3.1.1.2">𝛿</ci><apply id="S4.SS1.p1.9.m3.1.1.3.cmml" xref="S4.SS1.p1.9.m3.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p1.9.m3.1.1.3.1.cmml" xref="S4.SS1.p1.9.m3.1.1.3">subscript</csymbol><ci id="S4.SS1.p1.9.m3.1.1.3.2.cmml" xref="S4.SS1.p1.9.m3.1.1.3.2">italic-ϕ</ci><ci id="S4.SS1.p1.9.m3.1.1.3.3.cmml" xref="S4.SS1.p1.9.m3.1.1.3.3">het</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.9.m3.1c">\delta\phi_{\rm het}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.9.m3.1d">italic_δ italic_ϕ start_POSTSUBSCRIPT roman_het end_POSTSUBSCRIPT</annotation></semantics></math> to be corrected and essentially eliminated from the phase noise. The low <math alttext="C/N_{0}" class="ltx_Math" display="inline" id="S4.SS1.p1.10.m4.1"><semantics id="S4.SS1.p1.10.m4.1a"><mrow id="S4.SS1.p1.10.m4.1.1" xref="S4.SS1.p1.10.m4.1.1.cmml"><mi id="S4.SS1.p1.10.m4.1.1.2" xref="S4.SS1.p1.10.m4.1.1.2.cmml">C</mi><mo id="S4.SS1.p1.10.m4.1.1.1" xref="S4.SS1.p1.10.m4.1.1.1.cmml">/</mo><msub id="S4.SS1.p1.10.m4.1.1.3" xref="S4.SS1.p1.10.m4.1.1.3.cmml"><mi id="S4.SS1.p1.10.m4.1.1.3.2" xref="S4.SS1.p1.10.m4.1.1.3.2.cmml">N</mi><mn id="S4.SS1.p1.10.m4.1.1.3.3" xref="S4.SS1.p1.10.m4.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.10.m4.1b"><apply id="S4.SS1.p1.10.m4.1.1.cmml" xref="S4.SS1.p1.10.m4.1.1"><divide id="S4.SS1.p1.10.m4.1.1.1.cmml" xref="S4.SS1.p1.10.m4.1.1.1"></divide><ci id="S4.SS1.p1.10.m4.1.1.2.cmml" xref="S4.SS1.p1.10.m4.1.1.2">𝐶</ci><apply id="S4.SS1.p1.10.m4.1.1.3.cmml" xref="S4.SS1.p1.10.m4.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p1.10.m4.1.1.3.1.cmml" xref="S4.SS1.p1.10.m4.1.1.3">subscript</csymbol><ci id="S4.SS1.p1.10.m4.1.1.3.2.cmml" xref="S4.SS1.p1.10.m4.1.1.3.2">𝑁</ci><cn id="S4.SS1.p1.10.m4.1.1.3.3.cmml" type="integer" xref="S4.SS1.p1.10.m4.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.10.m4.1c">C/N_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.10.m4.1d">italic_C / italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> of the inter-spacecraft broadcasts in our concept likely makes this strategy of superimposing weaker pilot tones not viable.</p> </div> <div class="ltx_para" id="S4.SS1.p2"> <p class="ltx_p" id="S4.SS1.p2.2">Fortunately, our concept’s much larger phase error tolerance may allow <math alttext="\delta\phi_{\rm het}" class="ltx_Math" display="inline" id="S4.SS1.p2.1.m1.1"><semantics id="S4.SS1.p2.1.m1.1a"><mrow id="S4.SS1.p2.1.m1.1.1" xref="S4.SS1.p2.1.m1.1.1.cmml"><mi id="S4.SS1.p2.1.m1.1.1.2" xref="S4.SS1.p2.1.m1.1.1.2.cmml">δ</mi><mo id="S4.SS1.p2.1.m1.1.1.1" xref="S4.SS1.p2.1.m1.1.1.1.cmml">⁢</mo><msub id="S4.SS1.p2.1.m1.1.1.3" xref="S4.SS1.p2.1.m1.1.1.3.cmml"><mi id="S4.SS1.p2.1.m1.1.1.3.2" xref="S4.SS1.p2.1.m1.1.1.3.2.cmml">ϕ</mi><mi id="S4.SS1.p2.1.m1.1.1.3.3" xref="S4.SS1.p2.1.m1.1.1.3.3.cmml">het</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.1.m1.1b"><apply id="S4.SS1.p2.1.m1.1.1.cmml" xref="S4.SS1.p2.1.m1.1.1"><times id="S4.SS1.p2.1.m1.1.1.1.cmml" xref="S4.SS1.p2.1.m1.1.1.1"></times><ci id="S4.SS1.p2.1.m1.1.1.2.cmml" xref="S4.SS1.p2.1.m1.1.1.2">𝛿</ci><apply id="S4.SS1.p2.1.m1.1.1.3.cmml" xref="S4.SS1.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p2.1.m1.1.1.3.1.cmml" xref="S4.SS1.p2.1.m1.1.1.3">subscript</csymbol><ci id="S4.SS1.p2.1.m1.1.1.3.2.cmml" xref="S4.SS1.p2.1.m1.1.1.3.2">italic-ϕ</ci><ci id="S4.SS1.p2.1.m1.1.1.3.3.cmml" xref="S4.SS1.p2.1.m1.1.1.3.3">het</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.1.m1.1c">\delta\phi_{\rm het}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.1.m1.1d">italic_δ italic_ϕ start_POSTSUBSCRIPT roman_het end_POSTSUBSCRIPT</annotation></semantics></math> to be sufficiently small as to not dominate the error if the spacecraft adjoining both arms is equipped with an atomic clock. We can write the standard deviation of <math alttext="\widetilde{\delta\phi}_{\rm het}" class="ltx_Math" display="inline" id="S4.SS1.p2.2.m2.1"><semantics id="S4.SS1.p2.2.m2.1a"><msub id="S4.SS1.p2.2.m2.1.1" xref="S4.SS1.p2.2.m2.1.1.cmml"><mover accent="true" id="S4.SS1.p2.2.m2.1.1.2" xref="S4.SS1.p2.2.m2.1.1.2.cmml"><mrow id="S4.SS1.p2.2.m2.1.1.2.2" xref="S4.SS1.p2.2.m2.1.1.2.2.cmml"><mi id="S4.SS1.p2.2.m2.1.1.2.2.2" xref="S4.SS1.p2.2.m2.1.1.2.2.2.cmml">δ</mi><mo id="S4.SS1.p2.2.m2.1.1.2.2.1" xref="S4.SS1.p2.2.m2.1.1.2.2.1.cmml">⁢</mo><mi id="S4.SS1.p2.2.m2.1.1.2.2.3" xref="S4.SS1.p2.2.m2.1.1.2.2.3.cmml">ϕ</mi></mrow><mo id="S4.SS1.p2.2.m2.1.1.2.1" xref="S4.SS1.p2.2.m2.1.1.2.1.cmml">~</mo></mover><mi id="S4.SS1.p2.2.m2.1.1.3" xref="S4.SS1.p2.2.m2.1.1.3.cmml">het</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.2.m2.1b"><apply id="S4.SS1.p2.2.m2.1.1.cmml" xref="S4.SS1.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.2.m2.1.1.1.cmml" xref="S4.SS1.p2.2.m2.1.1">subscript</csymbol><apply id="S4.SS1.p2.2.m2.1.1.2.cmml" xref="S4.SS1.p2.2.m2.1.1.2"><ci id="S4.SS1.p2.2.m2.1.1.2.1.cmml" xref="S4.SS1.p2.2.m2.1.1.2.1">~</ci><apply id="S4.SS1.p2.2.m2.1.1.2.2.cmml" xref="S4.SS1.p2.2.m2.1.1.2.2"><times id="S4.SS1.p2.2.m2.1.1.2.2.1.cmml" xref="S4.SS1.p2.2.m2.1.1.2.2.1"></times><ci id="S4.SS1.p2.2.m2.1.1.2.2.2.cmml" xref="S4.SS1.p2.2.m2.1.1.2.2.2">𝛿</ci><ci id="S4.SS1.p2.2.m2.1.1.2.2.3.cmml" xref="S4.SS1.p2.2.m2.1.1.2.2.3">italic-ϕ</ci></apply></apply><ci id="S4.SS1.p2.2.m2.1.1.3.cmml" xref="S4.SS1.p2.2.m2.1.1.3">het</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.2.m2.1c">\widetilde{\delta\phi}_{\rm het}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.2.m2.1d">over~ start_ARG italic_δ italic_ϕ end_ARG start_POSTSUBSCRIPT roman_het end_POSTSUBSCRIPT</annotation></semantics></math> in terms of the Allan deviation of its clock (cf. eqn. <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.E2" title="In Single-arm and an atomic clock ‣ 2.1 Clock noise and its mitigation ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">2</span></a>):</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="A3.EGx5"> <tbody id="S4.E20"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\widetilde{\phi}_{\rm het,rms}=\frac{\sqrt{2\tau}(2\pi f_{\rm het% })\sigma_{y}(\tau)}{(2\pi f)}=0.1~{}~{}\text{rad~{}Hz}^{-1/2}\left(\frac{% \sigma_{y}(1~{}\rm s)}{10^{-13}}\right)\left(\frac{f_{\rm het}}{100~{}\rm MHz}% \right)\left(\frac{f}{10^{-5}~{}\rm Hz}\right)^{-1}." class="ltx_Math" display="inline" id="S4.E20.m1.9"><semantics id="S4.E20.m1.9a"><mrow id="S4.E20.m1.9.9.1" 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id="S4.E20.m1.8.8.3.3.cmml" xref="S4.E20.m1.8.8.3.3">Hz</ci></apply></apply><apply id="S4.E20.m1.9.9.1.1.5.6.3.cmml" xref="S4.E20.m1.9.9.1.1.5.6.3"><minus id="S4.E20.m1.9.9.1.1.5.6.3.1.cmml" xref="S4.E20.m1.9.9.1.1.5.6.3"></minus><cn id="S4.E20.m1.9.9.1.1.5.6.3.2.cmml" type="integer" xref="S4.E20.m1.9.9.1.1.5.6.3.2">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E20.m1.9c">\displaystyle\widetilde{\phi}_{\rm het,rms}=\frac{\sqrt{2\tau}(2\pi f_{\rm het% })\sigma_{y}(\tau)}{(2\pi f)}=0.1~{}~{}\text{rad~{}Hz}^{-1/2}\left(\frac{% \sigma_{y}(1~{}\rm s)}{10^{-13}}\right)\left(\frac{f_{\rm het}}{100~{}\rm MHz}% \right)\left(\frac{f}{10^{-5}~{}\rm Hz}\right)^{-1}.</annotation><annotation encoding="application/x-llamapun" id="S4.E20.m1.9d">over~ start_ARG italic_ϕ end_ARG start_POSTSUBSCRIPT roman_het , roman_rms end_POSTSUBSCRIPT = divide start_ARG square-root start_ARG 2 italic_τ end_ARG ( 2 italic_π italic_f start_POSTSUBSCRIPT roman_het end_POSTSUBSCRIPT ) italic_σ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ( italic_τ ) end_ARG start_ARG ( 2 italic_π italic_f ) end_ARG = 0.1 rad Hz start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT ( divide start_ARG italic_σ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ( 1 roman_s ) end_ARG start_ARG 10 start_POSTSUPERSCRIPT - 13 end_POSTSUPERSCRIPT end_ARG ) ( divide start_ARG italic_f start_POSTSUBSCRIPT roman_het end_POSTSUBSCRIPT end_ARG start_ARG 100 roman_MHz end_ARG ) ( divide start_ARG italic_f end_ARG start_ARG 10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT roman_Hz end_ARG ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(20)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p2.7">For reference, the shot noise errors on the phase are <math alttext="0.03~{}\text{rad~{}Hz}^{-1/2}" class="ltx_Math" display="inline" id="S4.SS1.p2.3.m1.1"><semantics id="S4.SS1.p2.3.m1.1a"><mrow id="S4.SS1.p2.3.m1.1.1" xref="S4.SS1.p2.3.m1.1.1.cmml"><mn id="S4.SS1.p2.3.m1.1.1.2" xref="S4.SS1.p2.3.m1.1.1.2.cmml">0.03</mn><mo id="S4.SS1.p2.3.m1.1.1.1" lspace="0.330em" xref="S4.SS1.p2.3.m1.1.1.1.cmml">⁢</mo><msup id="S4.SS1.p2.3.m1.1.1.3" xref="S4.SS1.p2.3.m1.1.1.3.cmml"><mtext id="S4.SS1.p2.3.m1.1.1.3.2" xref="S4.SS1.p2.3.m1.1.1.3.2a.cmml">rad Hz</mtext><mrow id="S4.SS1.p2.3.m1.1.1.3.3" xref="S4.SS1.p2.3.m1.1.1.3.3.cmml"><mo id="S4.SS1.p2.3.m1.1.1.3.3a" xref="S4.SS1.p2.3.m1.1.1.3.3.cmml">−</mo><mrow id="S4.SS1.p2.3.m1.1.1.3.3.2" xref="S4.SS1.p2.3.m1.1.1.3.3.2.cmml"><mn id="S4.SS1.p2.3.m1.1.1.3.3.2.2" xref="S4.SS1.p2.3.m1.1.1.3.3.2.2.cmml">1</mn><mo id="S4.SS1.p2.3.m1.1.1.3.3.2.1" xref="S4.SS1.p2.3.m1.1.1.3.3.2.1.cmml">/</mo><mn id="S4.SS1.p2.3.m1.1.1.3.3.2.3" xref="S4.SS1.p2.3.m1.1.1.3.3.2.3.cmml">2</mn></mrow></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.3.m1.1b"><apply id="S4.SS1.p2.3.m1.1.1.cmml" xref="S4.SS1.p2.3.m1.1.1"><times id="S4.SS1.p2.3.m1.1.1.1.cmml" xref="S4.SS1.p2.3.m1.1.1.1"></times><cn id="S4.SS1.p2.3.m1.1.1.2.cmml" type="float" xref="S4.SS1.p2.3.m1.1.1.2">0.03</cn><apply id="S4.SS1.p2.3.m1.1.1.3.cmml" xref="S4.SS1.p2.3.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p2.3.m1.1.1.3.1.cmml" xref="S4.SS1.p2.3.m1.1.1.3">superscript</csymbol><ci id="S4.SS1.p2.3.m1.1.1.3.2a.cmml" xref="S4.SS1.p2.3.m1.1.1.3.2"><mtext id="S4.SS1.p2.3.m1.1.1.3.2.cmml" xref="S4.SS1.p2.3.m1.1.1.3.2">rad Hz</mtext></ci><apply id="S4.SS1.p2.3.m1.1.1.3.3.cmml" xref="S4.SS1.p2.3.m1.1.1.3.3"><minus id="S4.SS1.p2.3.m1.1.1.3.3.1.cmml" xref="S4.SS1.p2.3.m1.1.1.3.3"></minus><apply id="S4.SS1.p2.3.m1.1.1.3.3.2.cmml" xref="S4.SS1.p2.3.m1.1.1.3.3.2"><divide id="S4.SS1.p2.3.m1.1.1.3.3.2.1.cmml" xref="S4.SS1.p2.3.m1.1.1.3.3.2.1"></divide><cn id="S4.SS1.p2.3.m1.1.1.3.3.2.2.cmml" type="integer" xref="S4.SS1.p2.3.m1.1.1.3.3.2.2">1</cn><cn id="S4.SS1.p2.3.m1.1.1.3.3.2.3.cmml" type="integer" xref="S4.SS1.p2.3.m1.1.1.3.3.2.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.3.m1.1c">0.03~{}\text{rad~{}Hz}^{-1/2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.3.m1.1d">0.03 rad Hz start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT</annotation></semantics></math> for <math alttext="[C/N_{0}]_{\rm dB-Hz}=30\," class="ltx_Math" display="inline" id="S4.SS1.p2.4.m2.1"><semantics id="S4.SS1.p2.4.m2.1a"><mrow id="S4.SS1.p2.4.m2.1.1" xref="S4.SS1.p2.4.m2.1.1.cmml"><msub id="S4.SS1.p2.4.m2.1.1.1" xref="S4.SS1.p2.4.m2.1.1.1.cmml"><mrow id="S4.SS1.p2.4.m2.1.1.1.1.1" xref="S4.SS1.p2.4.m2.1.1.1.1.2.cmml"><mo id="S4.SS1.p2.4.m2.1.1.1.1.1.2" stretchy="false" xref="S4.SS1.p2.4.m2.1.1.1.1.2.1.cmml">[</mo><mrow 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xref="S4.SS1.p2.4.m2.1.1.2.cmml">=</mo><mn id="S4.SS1.p2.4.m2.1.1.3" xref="S4.SS1.p2.4.m2.1.1.3.cmml">30</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.4.m2.1b"><apply id="S4.SS1.p2.4.m2.1.1.cmml" xref="S4.SS1.p2.4.m2.1.1"><eq id="S4.SS1.p2.4.m2.1.1.2.cmml" xref="S4.SS1.p2.4.m2.1.1.2"></eq><apply id="S4.SS1.p2.4.m2.1.1.1.cmml" xref="S4.SS1.p2.4.m2.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.4.m2.1.1.1.2.cmml" xref="S4.SS1.p2.4.m2.1.1.1">subscript</csymbol><apply id="S4.SS1.p2.4.m2.1.1.1.1.2.cmml" xref="S4.SS1.p2.4.m2.1.1.1.1.1"><csymbol cd="latexml" id="S4.SS1.p2.4.m2.1.1.1.1.2.1.cmml" xref="S4.SS1.p2.4.m2.1.1.1.1.1.2">delimited-[]</csymbol><apply id="S4.SS1.p2.4.m2.1.1.1.1.1.1.cmml" xref="S4.SS1.p2.4.m2.1.1.1.1.1.1"><divide id="S4.SS1.p2.4.m2.1.1.1.1.1.1.1.cmml" xref="S4.SS1.p2.4.m2.1.1.1.1.1.1.1"></divide><ci id="S4.SS1.p2.4.m2.1.1.1.1.1.1.2.cmml" xref="S4.SS1.p2.4.m2.1.1.1.1.1.1.2">𝐶</ci><apply id="S4.SS1.p2.4.m2.1.1.1.1.1.1.3.cmml" xref="S4.SS1.p2.4.m2.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p2.4.m2.1.1.1.1.1.1.3.1.cmml" xref="S4.SS1.p2.4.m2.1.1.1.1.1.1.3">subscript</csymbol><ci id="S4.SS1.p2.4.m2.1.1.1.1.1.1.3.2.cmml" xref="S4.SS1.p2.4.m2.1.1.1.1.1.1.3.2">𝑁</ci><cn id="S4.SS1.p2.4.m2.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S4.SS1.p2.4.m2.1.1.1.1.1.1.3.3">0</cn></apply></apply></apply><apply id="S4.SS1.p2.4.m2.1.1.1.3.cmml" xref="S4.SS1.p2.4.m2.1.1.1.3"><minus id="S4.SS1.p2.4.m2.1.1.1.3.1.cmml" xref="S4.SS1.p2.4.m2.1.1.1.3.1"></minus><ci id="S4.SS1.p2.4.m2.1.1.1.3.2.cmml" xref="S4.SS1.p2.4.m2.1.1.1.3.2">dB</ci><ci id="S4.SS1.p2.4.m2.1.1.1.3.3.cmml" xref="S4.SS1.p2.4.m2.1.1.1.3.3">Hz</ci></apply></apply><cn id="S4.SS1.p2.4.m2.1.1.3.cmml" type="integer" xref="S4.SS1.p2.4.m2.1.1.3">30</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.4.m2.1c">[C/N_{0}]_{\rm dB-Hz}=30\,</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.4.m2.1d">[ italic_C / italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT roman_dB - roman_Hz end_POSTSUBSCRIPT = 30</annotation></semantics></math>dB-Hz (eqn. <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.E10" title="In 2.2 Radiometer noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">10</span></a>). We conclude that the heterodyne phase error is manageable with an atomic clock like the Deep Space Atomic Clock for which <math alttext="\sigma_{y}(1~{}\rm s)\sim 10^{-13}" class="ltx_Math" display="inline" id="S4.SS1.p2.5.m3.1"><semantics id="S4.SS1.p2.5.m3.1a"><mrow id="S4.SS1.p2.5.m3.1.1" xref="S4.SS1.p2.5.m3.1.1.cmml"><mrow id="S4.SS1.p2.5.m3.1.1.1" xref="S4.SS1.p2.5.m3.1.1.1.cmml"><msub id="S4.SS1.p2.5.m3.1.1.1.3" xref="S4.SS1.p2.5.m3.1.1.1.3.cmml"><mi id="S4.SS1.p2.5.m3.1.1.1.3.2" xref="S4.SS1.p2.5.m3.1.1.1.3.2.cmml">σ</mi><mi id="S4.SS1.p2.5.m3.1.1.1.3.3" xref="S4.SS1.p2.5.m3.1.1.1.3.3.cmml">y</mi></msub><mo id="S4.SS1.p2.5.m3.1.1.1.2" xref="S4.SS1.p2.5.m3.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS1.p2.5.m3.1.1.1.1.1" xref="S4.SS1.p2.5.m3.1.1.1.1.1.1.cmml"><mo id="S4.SS1.p2.5.m3.1.1.1.1.1.2" stretchy="false" xref="S4.SS1.p2.5.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS1.p2.5.m3.1.1.1.1.1.1" xref="S4.SS1.p2.5.m3.1.1.1.1.1.1.cmml"><mn id="S4.SS1.p2.5.m3.1.1.1.1.1.1.2" xref="S4.SS1.p2.5.m3.1.1.1.1.1.1.2.cmml">1</mn><mo id="S4.SS1.p2.5.m3.1.1.1.1.1.1.1" lspace="0.330em" xref="S4.SS1.p2.5.m3.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S4.SS1.p2.5.m3.1.1.1.1.1.1.3" mathvariant="normal" xref="S4.SS1.p2.5.m3.1.1.1.1.1.1.3.cmml">s</mi></mrow><mo id="S4.SS1.p2.5.m3.1.1.1.1.1.3" stretchy="false" xref="S4.SS1.p2.5.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p2.5.m3.1.1.2" xref="S4.SS1.p2.5.m3.1.1.2.cmml">∼</mo><msup id="S4.SS1.p2.5.m3.1.1.3" xref="S4.SS1.p2.5.m3.1.1.3.cmml"><mn id="S4.SS1.p2.5.m3.1.1.3.2" xref="S4.SS1.p2.5.m3.1.1.3.2.cmml">10</mn><mrow id="S4.SS1.p2.5.m3.1.1.3.3" xref="S4.SS1.p2.5.m3.1.1.3.3.cmml"><mo id="S4.SS1.p2.5.m3.1.1.3.3a" xref="S4.SS1.p2.5.m3.1.1.3.3.cmml">−</mo><mn id="S4.SS1.p2.5.m3.1.1.3.3.2" xref="S4.SS1.p2.5.m3.1.1.3.3.2.cmml">13</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.5.m3.1b"><apply id="S4.SS1.p2.5.m3.1.1.cmml" xref="S4.SS1.p2.5.m3.1.1"><csymbol cd="latexml" id="S4.SS1.p2.5.m3.1.1.2.cmml" xref="S4.SS1.p2.5.m3.1.1.2">similar-to</csymbol><apply id="S4.SS1.p2.5.m3.1.1.1.cmml" xref="S4.SS1.p2.5.m3.1.1.1"><times id="S4.SS1.p2.5.m3.1.1.1.2.cmml" xref="S4.SS1.p2.5.m3.1.1.1.2"></times><apply id="S4.SS1.p2.5.m3.1.1.1.3.cmml" xref="S4.SS1.p2.5.m3.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p2.5.m3.1.1.1.3.1.cmml" xref="S4.SS1.p2.5.m3.1.1.1.3">subscript</csymbol><ci id="S4.SS1.p2.5.m3.1.1.1.3.2.cmml" xref="S4.SS1.p2.5.m3.1.1.1.3.2">𝜎</ci><ci id="S4.SS1.p2.5.m3.1.1.1.3.3.cmml" xref="S4.SS1.p2.5.m3.1.1.1.3.3">𝑦</ci></apply><apply id="S4.SS1.p2.5.m3.1.1.1.1.1.1.cmml" xref="S4.SS1.p2.5.m3.1.1.1.1.1"><times id="S4.SS1.p2.5.m3.1.1.1.1.1.1.1.cmml" xref="S4.SS1.p2.5.m3.1.1.1.1.1.1.1"></times><cn id="S4.SS1.p2.5.m3.1.1.1.1.1.1.2.cmml" type="integer" xref="S4.SS1.p2.5.m3.1.1.1.1.1.1.2">1</cn><ci id="S4.SS1.p2.5.m3.1.1.1.1.1.1.3.cmml" xref="S4.SS1.p2.5.m3.1.1.1.1.1.1.3">s</ci></apply></apply><apply id="S4.SS1.p2.5.m3.1.1.3.cmml" xref="S4.SS1.p2.5.m3.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p2.5.m3.1.1.3.1.cmml" xref="S4.SS1.p2.5.m3.1.1.3">superscript</csymbol><cn id="S4.SS1.p2.5.m3.1.1.3.2.cmml" type="integer" xref="S4.SS1.p2.5.m3.1.1.3.2">10</cn><apply id="S4.SS1.p2.5.m3.1.1.3.3.cmml" xref="S4.SS1.p2.5.m3.1.1.3.3"><minus id="S4.SS1.p2.5.m3.1.1.3.3.1.cmml" xref="S4.SS1.p2.5.m3.1.1.3.3"></minus><cn id="S4.SS1.p2.5.m3.1.1.3.3.2.cmml" type="integer" xref="S4.SS1.p2.5.m3.1.1.3.3.2">13</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.5.m3.1c">\sigma_{y}(1~{}\rm s)\sim 10^{-13}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.5.m3.1d">italic_σ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ( 1 roman_s ) ∼ 10 start_POSTSUPERSCRIPT - 13 end_POSTSUPERSCRIPT</annotation></semantics></math> if <math alttext="f_{\rm het}\sim 10^{2}" class="ltx_Math" display="inline" id="S4.SS1.p2.6.m4.1"><semantics id="S4.SS1.p2.6.m4.1a"><mrow id="S4.SS1.p2.6.m4.1.1" xref="S4.SS1.p2.6.m4.1.1.cmml"><msub id="S4.SS1.p2.6.m4.1.1.2" xref="S4.SS1.p2.6.m4.1.1.2.cmml"><mi id="S4.SS1.p2.6.m4.1.1.2.2" xref="S4.SS1.p2.6.m4.1.1.2.2.cmml">f</mi><mi id="S4.SS1.p2.6.m4.1.1.2.3" xref="S4.SS1.p2.6.m4.1.1.2.3.cmml">het</mi></msub><mo id="S4.SS1.p2.6.m4.1.1.1" xref="S4.SS1.p2.6.m4.1.1.1.cmml">∼</mo><msup id="S4.SS1.p2.6.m4.1.1.3" xref="S4.SS1.p2.6.m4.1.1.3.cmml"><mn id="S4.SS1.p2.6.m4.1.1.3.2" xref="S4.SS1.p2.6.m4.1.1.3.2.cmml">10</mn><mn id="S4.SS1.p2.6.m4.1.1.3.3" xref="S4.SS1.p2.6.m4.1.1.3.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.6.m4.1b"><apply id="S4.SS1.p2.6.m4.1.1.cmml" xref="S4.SS1.p2.6.m4.1.1"><csymbol cd="latexml" id="S4.SS1.p2.6.m4.1.1.1.cmml" xref="S4.SS1.p2.6.m4.1.1.1">similar-to</csymbol><apply id="S4.SS1.p2.6.m4.1.1.2.cmml" xref="S4.SS1.p2.6.m4.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p2.6.m4.1.1.2.1.cmml" xref="S4.SS1.p2.6.m4.1.1.2">subscript</csymbol><ci id="S4.SS1.p2.6.m4.1.1.2.2.cmml" xref="S4.SS1.p2.6.m4.1.1.2.2">𝑓</ci><ci id="S4.SS1.p2.6.m4.1.1.2.3.cmml" xref="S4.SS1.p2.6.m4.1.1.2.3">het</ci></apply><apply id="S4.SS1.p2.6.m4.1.1.3.cmml" xref="S4.SS1.p2.6.m4.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p2.6.m4.1.1.3.1.cmml" xref="S4.SS1.p2.6.m4.1.1.3">superscript</csymbol><cn id="S4.SS1.p2.6.m4.1.1.3.2.cmml" type="integer" xref="S4.SS1.p2.6.m4.1.1.3.2">10</cn><cn id="S4.SS1.p2.6.m4.1.1.3.3.cmml" type="integer" xref="S4.SS1.p2.6.m4.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.6.m4.1c">f_{\rm het}\sim 10^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.6.m4.1d">italic_f start_POSTSUBSCRIPT roman_het end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> MHz, an order of magnitude higher than the maximum heterodyne frequency allowed for LISA. Relaxing these specifications further may also be possible since the sensitivity floor set by shot noise is already well below the stochastic gravitational wave background from white dwarf binaries: with an averaging time of a minute in the phase-lock loop, it might be possible to accommodate <math alttext="\widetilde{\phi}_{\rm het,rms}\sim 1~{}\text{rad~{}Hz}^{-1/2}" class="ltx_Math" display="inline" id="S4.SS1.p2.7.m5.2"><semantics id="S4.SS1.p2.7.m5.2a"><mrow id="S4.SS1.p2.7.m5.2.3" xref="S4.SS1.p2.7.m5.2.3.cmml"><msub id="S4.SS1.p2.7.m5.2.3.2" xref="S4.SS1.p2.7.m5.2.3.2.cmml"><mover accent="true" id="S4.SS1.p2.7.m5.2.3.2.2" xref="S4.SS1.p2.7.m5.2.3.2.2.cmml"><mi id="S4.SS1.p2.7.m5.2.3.2.2.2" xref="S4.SS1.p2.7.m5.2.3.2.2.2.cmml">ϕ</mi><mo id="S4.SS1.p2.7.m5.2.3.2.2.1" xref="S4.SS1.p2.7.m5.2.3.2.2.1.cmml">~</mo></mover><mrow id="S4.SS1.p2.7.m5.2.2.2.4" xref="S4.SS1.p2.7.m5.2.2.2.3.cmml"><mi id="S4.SS1.p2.7.m5.1.1.1.1" xref="S4.SS1.p2.7.m5.1.1.1.1.cmml">het</mi><mo id="S4.SS1.p2.7.m5.2.2.2.4.1" xref="S4.SS1.p2.7.m5.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p2.7.m5.2.2.2.2" xref="S4.SS1.p2.7.m5.2.2.2.2.cmml">rms</mi></mrow></msub><mo id="S4.SS1.p2.7.m5.2.3.1" xref="S4.SS1.p2.7.m5.2.3.1.cmml">∼</mo><mrow id="S4.SS1.p2.7.m5.2.3.3" xref="S4.SS1.p2.7.m5.2.3.3.cmml"><mn id="S4.SS1.p2.7.m5.2.3.3.2" xref="S4.SS1.p2.7.m5.2.3.3.2.cmml">1</mn><mo id="S4.SS1.p2.7.m5.2.3.3.1" lspace="0.330em" xref="S4.SS1.p2.7.m5.2.3.3.1.cmml">⁢</mo><msup id="S4.SS1.p2.7.m5.2.3.3.3" xref="S4.SS1.p2.7.m5.2.3.3.3.cmml"><mtext id="S4.SS1.p2.7.m5.2.3.3.3.2" xref="S4.SS1.p2.7.m5.2.3.3.3.2a.cmml">rad Hz</mtext><mrow id="S4.SS1.p2.7.m5.2.3.3.3.3" xref="S4.SS1.p2.7.m5.2.3.3.3.3.cmml"><mo id="S4.SS1.p2.7.m5.2.3.3.3.3a" xref="S4.SS1.p2.7.m5.2.3.3.3.3.cmml">−</mo><mrow id="S4.SS1.p2.7.m5.2.3.3.3.3.2" xref="S4.SS1.p2.7.m5.2.3.3.3.3.2.cmml"><mn id="S4.SS1.p2.7.m5.2.3.3.3.3.2.2" xref="S4.SS1.p2.7.m5.2.3.3.3.3.2.2.cmml">1</mn><mo id="S4.SS1.p2.7.m5.2.3.3.3.3.2.1" xref="S4.SS1.p2.7.m5.2.3.3.3.3.2.1.cmml">/</mo><mn id="S4.SS1.p2.7.m5.2.3.3.3.3.2.3" xref="S4.SS1.p2.7.m5.2.3.3.3.3.2.3.cmml">2</mn></mrow></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.7.m5.2b"><apply id="S4.SS1.p2.7.m5.2.3.cmml" xref="S4.SS1.p2.7.m5.2.3"><csymbol cd="latexml" id="S4.SS1.p2.7.m5.2.3.1.cmml" xref="S4.SS1.p2.7.m5.2.3.1">similar-to</csymbol><apply id="S4.SS1.p2.7.m5.2.3.2.cmml" xref="S4.SS1.p2.7.m5.2.3.2"><csymbol cd="ambiguous" id="S4.SS1.p2.7.m5.2.3.2.1.cmml" xref="S4.SS1.p2.7.m5.2.3.2">subscript</csymbol><apply id="S4.SS1.p2.7.m5.2.3.2.2.cmml" xref="S4.SS1.p2.7.m5.2.3.2.2"><ci id="S4.SS1.p2.7.m5.2.3.2.2.1.cmml" xref="S4.SS1.p2.7.m5.2.3.2.2.1">~</ci><ci id="S4.SS1.p2.7.m5.2.3.2.2.2.cmml" xref="S4.SS1.p2.7.m5.2.3.2.2.2">italic-ϕ</ci></apply><list id="S4.SS1.p2.7.m5.2.2.2.3.cmml" xref="S4.SS1.p2.7.m5.2.2.2.4"><ci id="S4.SS1.p2.7.m5.1.1.1.1.cmml" xref="S4.SS1.p2.7.m5.1.1.1.1">het</ci><ci id="S4.SS1.p2.7.m5.2.2.2.2.cmml" xref="S4.SS1.p2.7.m5.2.2.2.2">rms</ci></list></apply><apply id="S4.SS1.p2.7.m5.2.3.3.cmml" xref="S4.SS1.p2.7.m5.2.3.3"><times id="S4.SS1.p2.7.m5.2.3.3.1.cmml" xref="S4.SS1.p2.7.m5.2.3.3.1"></times><cn id="S4.SS1.p2.7.m5.2.3.3.2.cmml" type="integer" xref="S4.SS1.p2.7.m5.2.3.3.2">1</cn><apply id="S4.SS1.p2.7.m5.2.3.3.3.cmml" xref="S4.SS1.p2.7.m5.2.3.3.3"><csymbol cd="ambiguous" id="S4.SS1.p2.7.m5.2.3.3.3.1.cmml" xref="S4.SS1.p2.7.m5.2.3.3.3">superscript</csymbol><ci id="S4.SS1.p2.7.m5.2.3.3.3.2a.cmml" xref="S4.SS1.p2.7.m5.2.3.3.3.2"><mtext id="S4.SS1.p2.7.m5.2.3.3.3.2.cmml" xref="S4.SS1.p2.7.m5.2.3.3.3.2">rad Hz</mtext></ci><apply id="S4.SS1.p2.7.m5.2.3.3.3.3.cmml" xref="S4.SS1.p2.7.m5.2.3.3.3.3"><minus id="S4.SS1.p2.7.m5.2.3.3.3.3.1.cmml" xref="S4.SS1.p2.7.m5.2.3.3.3.3"></minus><apply id="S4.SS1.p2.7.m5.2.3.3.3.3.2.cmml" xref="S4.SS1.p2.7.m5.2.3.3.3.3.2"><divide id="S4.SS1.p2.7.m5.2.3.3.3.3.2.1.cmml" xref="S4.SS1.p2.7.m5.2.3.3.3.3.2.1"></divide><cn id="S4.SS1.p2.7.m5.2.3.3.3.3.2.2.cmml" type="integer" xref="S4.SS1.p2.7.m5.2.3.3.3.3.2.2">1</cn><cn id="S4.SS1.p2.7.m5.2.3.3.3.3.2.3.cmml" type="integer" xref="S4.SS1.p2.7.m5.2.3.3.3.3.2.3">2</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.7.m5.2c">\widetilde{\phi}_{\rm het,rms}\sim 1~{}\text{rad~{}Hz}^{-1/2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.7.m5.2d">over~ start_ARG italic_ϕ end_ARG start_POSTSUBSCRIPT roman_het , roman_rms end_POSTSUBSCRIPT ∼ 1 rad Hz start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT</annotation></semantics></math> and still achieve a stable phase lock.</p> </div> <div class="ltx_para" id="S4.SS1.p3"> <p class="ltx_p" id="S4.SS1.p3.6">The simplest orbits would send the spacecraft on radial trajectories with respect to the Sun. In this case, the relative velocities between the spacecraft could be tens of kilometers per second. A <math alttext="10\," class="ltx_Math" display="inline" id="S4.SS1.p3.1.m1.1"><semantics id="S4.SS1.p3.1.m1.1a"><mn id="S4.SS1.p3.1.m1.1.1" xref="S4.SS1.p3.1.m1.1.1.cmml">10</mn><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.1.m1.1b"><cn id="S4.SS1.p3.1.m1.1.1.cmml" type="integer" xref="S4.SS1.p3.1.m1.1.1">10</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.1.m1.1c">10\,</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.1.m1.1d">10</annotation></semantics></math>km s<sup class="ltx_sup" id="S4.SS1.p3.6.1"><span class="ltx_text ltx_font_italic" id="S4.SS1.p3.6.1.1">-1</span></sup> offset velocity would have to be compensated for by a <math alttext="\sim 0.3" class="ltx_Math" display="inline" id="S4.SS1.p3.3.m3.1"><semantics id="S4.SS1.p3.3.m3.1a"><mrow id="S4.SS1.p3.3.m3.1.1" xref="S4.SS1.p3.3.m3.1.1.cmml"><mi id="S4.SS1.p3.3.m3.1.1.2" xref="S4.SS1.p3.3.m3.1.1.2.cmml"></mi><mo id="S4.SS1.p3.3.m3.1.1.1" xref="S4.SS1.p3.3.m3.1.1.1.cmml">∼</mo><mn id="S4.SS1.p3.3.m3.1.1.3" xref="S4.SS1.p3.3.m3.1.1.3.cmml">0.3</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.3.m3.1b"><apply id="S4.SS1.p3.3.m3.1.1.cmml" xref="S4.SS1.p3.3.m3.1.1"><csymbol cd="latexml" id="S4.SS1.p3.3.m3.1.1.1.cmml" xref="S4.SS1.p3.3.m3.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S4.SS1.p3.3.m3.1.1.2.cmml" xref="S4.SS1.p3.3.m3.1.1.2">absent</csymbol><cn id="S4.SS1.p3.3.m3.1.1.3.cmml" type="float" xref="S4.SS1.p3.3.m3.1.1.3">0.3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.3.m3.1c">\sim 0.3</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.3.m3.1d">∼ 0.3</annotation></semantics></math>Å shift of the output wavelength of a <math alttext="\lambda=10^{4}" class="ltx_Math" display="inline" id="S4.SS1.p3.4.m4.1"><semantics id="S4.SS1.p3.4.m4.1a"><mrow id="S4.SS1.p3.4.m4.1.1" xref="S4.SS1.p3.4.m4.1.1.cmml"><mi id="S4.SS1.p3.4.m4.1.1.2" xref="S4.SS1.p3.4.m4.1.1.2.cmml">λ</mi><mo id="S4.SS1.p3.4.m4.1.1.1" xref="S4.SS1.p3.4.m4.1.1.1.cmml">=</mo><msup id="S4.SS1.p3.4.m4.1.1.3" xref="S4.SS1.p3.4.m4.1.1.3.cmml"><mn id="S4.SS1.p3.4.m4.1.1.3.2" xref="S4.SS1.p3.4.m4.1.1.3.2.cmml">10</mn><mn id="S4.SS1.p3.4.m4.1.1.3.3" xref="S4.SS1.p3.4.m4.1.1.3.3.cmml">4</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.4.m4.1b"><apply id="S4.SS1.p3.4.m4.1.1.cmml" xref="S4.SS1.p3.4.m4.1.1"><eq id="S4.SS1.p3.4.m4.1.1.1.cmml" xref="S4.SS1.p3.4.m4.1.1.1"></eq><ci id="S4.SS1.p3.4.m4.1.1.2.cmml" xref="S4.SS1.p3.4.m4.1.1.2">𝜆</ci><apply id="S4.SS1.p3.4.m4.1.1.3.cmml" xref="S4.SS1.p3.4.m4.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p3.4.m4.1.1.3.1.cmml" xref="S4.SS1.p3.4.m4.1.1.3">superscript</csymbol><cn id="S4.SS1.p3.4.m4.1.1.3.2.cmml" type="integer" xref="S4.SS1.p3.4.m4.1.1.3.2">10</cn><cn id="S4.SS1.p3.4.m4.1.1.3.3.cmml" type="integer" xref="S4.SS1.p3.4.m4.1.1.3.3">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.4.m4.1c">\lambda=10^{4}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.4.m4.1d">italic_λ = 10 start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT</annotation></semantics></math>Å laser in order for it to interfere within the heterodyne specification. Tuneable lasers that can adjust their wavelength even to <math alttext=">1" class="ltx_Math" display="inline" id="S4.SS1.p3.5.m5.1"><semantics id="S4.SS1.p3.5.m5.1a"><mrow id="S4.SS1.p3.5.m5.1.1" xref="S4.SS1.p3.5.m5.1.1.cmml"><mi id="S4.SS1.p3.5.m5.1.1.2" xref="S4.SS1.p3.5.m5.1.1.2.cmml"></mi><mo id="S4.SS1.p3.5.m5.1.1.1" xref="S4.SS1.p3.5.m5.1.1.1.cmml">></mo><mn id="S4.SS1.p3.5.m5.1.1.3" xref="S4.SS1.p3.5.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.5.m5.1b"><apply id="S4.SS1.p3.5.m5.1.1.cmml" xref="S4.SS1.p3.5.m5.1.1"><gt id="S4.SS1.p3.5.m5.1.1.1.cmml" xref="S4.SS1.p3.5.m5.1.1.1"></gt><csymbol cd="latexml" id="S4.SS1.p3.5.m5.1.1.2.cmml" xref="S4.SS1.p3.5.m5.1.1.2">absent</csymbol><cn id="S4.SS1.p3.5.m5.1.1.3.cmml" type="integer" xref="S4.SS1.p3.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.5.m5.1c">>1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.5.m5.1d">> 1</annotation></semantics></math>Å exist. Additionally, space interferometers like LISA must adjust the laser frequency on month timescales to compensate for this drift and stay under the maximum <math alttext="f_{\rm het}" class="ltx_Math" display="inline" id="S4.SS1.p3.6.m6.1"><semantics id="S4.SS1.p3.6.m6.1a"><msub id="S4.SS1.p3.6.m6.1.1" xref="S4.SS1.p3.6.m6.1.1.cmml"><mi id="S4.SS1.p3.6.m6.1.1.2" xref="S4.SS1.p3.6.m6.1.1.2.cmml">f</mi><mi id="S4.SS1.p3.6.m6.1.1.3" xref="S4.SS1.p3.6.m6.1.1.3.cmml">het</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.6.m6.1b"><apply id="S4.SS1.p3.6.m6.1.1.cmml" xref="S4.SS1.p3.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS1.p3.6.m6.1.1.1.cmml" xref="S4.SS1.p3.6.m6.1.1">subscript</csymbol><ci id="S4.SS1.p3.6.m6.1.1.2.cmml" xref="S4.SS1.p3.6.m6.1.1.2">𝑓</ci><ci id="S4.SS1.p3.6.m6.1.1.3.cmml" xref="S4.SS1.p3.6.m6.1.1.3">het</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.6.m6.1c">f_{\rm het}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.6.m6.1d">italic_f start_POSTSUBSCRIPT roman_het end_POSTSUBSCRIPT</annotation></semantics></math> that the system can tolerate <cite class="ltx_cite ltx_citemacro_citep">(Barke, <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib12" title="">2015</a>)</cite>. For spacecraft on radial trajectories, the rate of change of the frequency from Doppler shifting between a spacecraft and some fixed reference is</p> <table class="ltx_equation ltx_eqn_table" id="S4.E21"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\frac{df}{dt}=\frac{GM_{\odot}}{r^{2}\lambda}=2\times 10^{9}\left(\frac{r}{30% \rm AU}\right)^{-2}\left(\frac{\lambda}{1000\ \text{\AA}}\right)^{-1}~{}~{}~{}% ~{}~{}\text{GHz ~{}yr${}^{-1}$}." class="ltx_Math" display="block" id="S4.E21.m1.4"><semantics id="S4.E21.m1.4a"><mrow id="S4.E21.m1.4.4.1"><mrow id="S4.E21.m1.4.4.1.1.1" xref="S4.E21.m1.4.4.1.1.2.cmml"><mrow id="S4.E21.m1.4.4.1.1.1.1" xref="S4.E21.m1.4.4.1.1.1.1.cmml"><mfrac id="S4.E21.m1.4.4.1.1.1.1.2" xref="S4.E21.m1.4.4.1.1.1.1.2.cmml"><mrow id="S4.E21.m1.4.4.1.1.1.1.2.2" xref="S4.E21.m1.4.4.1.1.1.1.2.2.cmml"><mi id="S4.E21.m1.4.4.1.1.1.1.2.2.2" 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AU}\right)^{-2}\left(\frac{\lambda}{1000\ \text{\AA}}\right)^{-1}~{}~{}~{}% ~{}~{}\text{GHz ~{}yr${}^{-1}$}.</annotation><annotation encoding="application/x-llamapun" id="S4.E21.m1.4d">divide start_ARG italic_d italic_f end_ARG start_ARG italic_d italic_t end_ARG = divide start_ARG italic_G italic_M start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT end_ARG start_ARG italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_λ end_ARG = 2 × 10 start_POSTSUPERSCRIPT 9 end_POSTSUPERSCRIPT ( divide start_ARG italic_r end_ARG start_ARG 30 roman_A roman_U end_ARG ) start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT ( divide start_ARG italic_λ end_ARG start_ARG 1000 Å end_ARG ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT GHz yr .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(21)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p3.8">Equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S4.E21" title="In 4.1 Laser design ‣ 4 Instrumental considerations ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">21</span></a>) suggests that for radial trajectories and spacecraft at <math alttext="r=10-30~{}" class="ltx_Math" display="inline" id="S4.SS1.p3.7.m1.1"><semantics id="S4.SS1.p3.7.m1.1a"><mrow id="S4.SS1.p3.7.m1.1.1" xref="S4.SS1.p3.7.m1.1.1.cmml"><mi id="S4.SS1.p3.7.m1.1.1.2" xref="S4.SS1.p3.7.m1.1.1.2.cmml">r</mi><mo id="S4.SS1.p3.7.m1.1.1.1" xref="S4.SS1.p3.7.m1.1.1.1.cmml">=</mo><mrow id="S4.SS1.p3.7.m1.1.1.3" xref="S4.SS1.p3.7.m1.1.1.3.cmml"><mn id="S4.SS1.p3.7.m1.1.1.3.2" xref="S4.SS1.p3.7.m1.1.1.3.2.cmml">10</mn><mo id="S4.SS1.p3.7.m1.1.1.3.1" xref="S4.SS1.p3.7.m1.1.1.3.1.cmml">−</mo><mn id="S4.SS1.p3.7.m1.1.1.3.3" xref="S4.SS1.p3.7.m1.1.1.3.3.cmml">30</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.7.m1.1b"><apply id="S4.SS1.p3.7.m1.1.1.cmml" xref="S4.SS1.p3.7.m1.1.1"><eq id="S4.SS1.p3.7.m1.1.1.1.cmml" xref="S4.SS1.p3.7.m1.1.1.1"></eq><ci id="S4.SS1.p3.7.m1.1.1.2.cmml" xref="S4.SS1.p3.7.m1.1.1.2">𝑟</ci><apply id="S4.SS1.p3.7.m1.1.1.3.cmml" xref="S4.SS1.p3.7.m1.1.1.3"><minus id="S4.SS1.p3.7.m1.1.1.3.1.cmml" xref="S4.SS1.p3.7.m1.1.1.3.1"></minus><cn id="S4.SS1.p3.7.m1.1.1.3.2.cmml" type="integer" xref="S4.SS1.p3.7.m1.1.1.3.2">10</cn><cn id="S4.SS1.p3.7.m1.1.1.3.3.cmml" type="integer" xref="S4.SS1.p3.7.m1.1.1.3.3">30</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.7.m1.1c">r=10-30~{}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.7.m1.1d">italic_r = 10 - 30</annotation></semantics></math>AU this tuning would also have to be done weekly or monthly to maintain a maximum heterodyne frequency of <math alttext="\sim 100~{}" class="ltx_Math" display="inline" id="S4.SS1.p3.8.m2.1"><semantics id="S4.SS1.p3.8.m2.1a"><mrow id="S4.SS1.p3.8.m2.1.1" xref="S4.SS1.p3.8.m2.1.1.cmml"><mi id="S4.SS1.p3.8.m2.1.1.2" xref="S4.SS1.p3.8.m2.1.1.2.cmml"></mi><mo id="S4.SS1.p3.8.m2.1.1.1" xref="S4.SS1.p3.8.m2.1.1.1.cmml">∼</mo><mn id="S4.SS1.p3.8.m2.1.1.3" xref="S4.SS1.p3.8.m2.1.1.3.cmml">100</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.8.m2.1b"><apply id="S4.SS1.p3.8.m2.1.1.cmml" xref="S4.SS1.p3.8.m2.1.1"><csymbol cd="latexml" id="S4.SS1.p3.8.m2.1.1.1.cmml" xref="S4.SS1.p3.8.m2.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S4.SS1.p3.8.m2.1.1.2.cmml" xref="S4.SS1.p3.8.m2.1.1.2">absent</csymbol><cn id="S4.SS1.p3.8.m2.1.1.3.cmml" type="integer" xref="S4.SS1.p3.8.m2.1.1.3">100</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.8.m2.1c">\sim 100~{}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.8.m2.1d">∼ 100</annotation></semantics></math>MHz.</p> </div> <div class="ltx_para" id="S4.SS1.p4"> <p class="ltx_p" id="S4.SS1.p4.4">A second challenge is controlling the relative intensity noise of the laser at the heterodyne frequency. As the relative intensity noise scales in the same way with <math alttext="C/N_{0}" class="ltx_Math" display="inline" id="S4.SS1.p4.1.m1.1"><semantics id="S4.SS1.p4.1.m1.1a"><mrow id="S4.SS1.p4.1.m1.1.1" xref="S4.SS1.p4.1.m1.1.1.cmml"><mi id="S4.SS1.p4.1.m1.1.1.2" xref="S4.SS1.p4.1.m1.1.1.2.cmml">C</mi><mo id="S4.SS1.p4.1.m1.1.1.1" xref="S4.SS1.p4.1.m1.1.1.1.cmml">/</mo><msub id="S4.SS1.p4.1.m1.1.1.3" xref="S4.SS1.p4.1.m1.1.1.3.cmml"><mi id="S4.SS1.p4.1.m1.1.1.3.2" xref="S4.SS1.p4.1.m1.1.1.3.2.cmml">N</mi><mn id="S4.SS1.p4.1.m1.1.1.3.3" xref="S4.SS1.p4.1.m1.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.1.m1.1b"><apply id="S4.SS1.p4.1.m1.1.1.cmml" xref="S4.SS1.p4.1.m1.1.1"><divide id="S4.SS1.p4.1.m1.1.1.1.cmml" xref="S4.SS1.p4.1.m1.1.1.1"></divide><ci id="S4.SS1.p4.1.m1.1.1.2.cmml" xref="S4.SS1.p4.1.m1.1.1.2">𝐶</ci><apply id="S4.SS1.p4.1.m1.1.1.3.cmml" xref="S4.SS1.p4.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p4.1.m1.1.1.3.1.cmml" xref="S4.SS1.p4.1.m1.1.1.3">subscript</csymbol><ci id="S4.SS1.p4.1.m1.1.1.3.2.cmml" xref="S4.SS1.p4.1.m1.1.1.3.2">𝑁</ci><cn id="S4.SS1.p4.1.m1.1.1.3.3.cmml" type="integer" xref="S4.SS1.p4.1.m1.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.1.m1.1c">C/N_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.1.m1.1d">italic_C / italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> as the shot noise, the relative intensity noise of our concept has the same requirement as LISA of <math alttext="\widetilde{I}_{\rm rms}\approx 10^{-4}~{}" class="ltx_Math" display="inline" id="S4.SS1.p4.2.m2.1"><semantics id="S4.SS1.p4.2.m2.1a"><mrow id="S4.SS1.p4.2.m2.1.1" xref="S4.SS1.p4.2.m2.1.1.cmml"><msub id="S4.SS1.p4.2.m2.1.1.2" xref="S4.SS1.p4.2.m2.1.1.2.cmml"><mover accent="true" id="S4.SS1.p4.2.m2.1.1.2.2" xref="S4.SS1.p4.2.m2.1.1.2.2.cmml"><mi id="S4.SS1.p4.2.m2.1.1.2.2.2" xref="S4.SS1.p4.2.m2.1.1.2.2.2.cmml">I</mi><mo id="S4.SS1.p4.2.m2.1.1.2.2.1" xref="S4.SS1.p4.2.m2.1.1.2.2.1.cmml">~</mo></mover><mi id="S4.SS1.p4.2.m2.1.1.2.3" xref="S4.SS1.p4.2.m2.1.1.2.3.cmml">rms</mi></msub><mo id="S4.SS1.p4.2.m2.1.1.1" xref="S4.SS1.p4.2.m2.1.1.1.cmml">≈</mo><msup id="S4.SS1.p4.2.m2.1.1.3" xref="S4.SS1.p4.2.m2.1.1.3.cmml"><mn id="S4.SS1.p4.2.m2.1.1.3.2" xref="S4.SS1.p4.2.m2.1.1.3.2.cmml">10</mn><mrow id="S4.SS1.p4.2.m2.1.1.3.3" xref="S4.SS1.p4.2.m2.1.1.3.3.cmml"><mo id="S4.SS1.p4.2.m2.1.1.3.3a" xref="S4.SS1.p4.2.m2.1.1.3.3.cmml">−</mo><mn id="S4.SS1.p4.2.m2.1.1.3.3.2" xref="S4.SS1.p4.2.m2.1.1.3.3.2.cmml">4</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.2.m2.1b"><apply id="S4.SS1.p4.2.m2.1.1.cmml" xref="S4.SS1.p4.2.m2.1.1"><approx id="S4.SS1.p4.2.m2.1.1.1.cmml" xref="S4.SS1.p4.2.m2.1.1.1"></approx><apply id="S4.SS1.p4.2.m2.1.1.2.cmml" xref="S4.SS1.p4.2.m2.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p4.2.m2.1.1.2.1.cmml" xref="S4.SS1.p4.2.m2.1.1.2">subscript</csymbol><apply id="S4.SS1.p4.2.m2.1.1.2.2.cmml" xref="S4.SS1.p4.2.m2.1.1.2.2"><ci id="S4.SS1.p4.2.m2.1.1.2.2.1.cmml" xref="S4.SS1.p4.2.m2.1.1.2.2.1">~</ci><ci id="S4.SS1.p4.2.m2.1.1.2.2.2.cmml" xref="S4.SS1.p4.2.m2.1.1.2.2.2">𝐼</ci></apply><ci id="S4.SS1.p4.2.m2.1.1.2.3.cmml" xref="S4.SS1.p4.2.m2.1.1.2.3">rms</ci></apply><apply id="S4.SS1.p4.2.m2.1.1.3.cmml" xref="S4.SS1.p4.2.m2.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p4.2.m2.1.1.3.1.cmml" xref="S4.SS1.p4.2.m2.1.1.3">superscript</csymbol><cn id="S4.SS1.p4.2.m2.1.1.3.2.cmml" type="integer" xref="S4.SS1.p4.2.m2.1.1.3.2">10</cn><apply id="S4.SS1.p4.2.m2.1.1.3.3.cmml" xref="S4.SS1.p4.2.m2.1.1.3.3"><minus id="S4.SS1.p4.2.m2.1.1.3.3.1.cmml" xref="S4.SS1.p4.2.m2.1.1.3.3"></minus><cn id="S4.SS1.p4.2.m2.1.1.3.3.2.cmml" type="integer" xref="S4.SS1.p4.2.m2.1.1.3.3.2">4</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.2.m2.1c">\widetilde{I}_{\rm rms}\approx 10^{-4}~{}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.2.m2.1d">over~ start_ARG italic_I end_ARG start_POSTSUBSCRIPT roman_rms end_POSTSUBSCRIPT ≈ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT</annotation></semantics></math>Hz<sup class="ltx_sup" id="S4.SS1.p4.4.1"><span class="ltx_text ltx_font_italic" id="S4.SS1.p4.4.1.1">-1/2</span></sup> in order not to exceed the shot noise <cite class="ltx_cite ltx_citemacro_citep">(Barke et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib14" title="">2015</a>)</cite>. As with <math alttext="\widetilde{\phi}_{\rm het,rms}" class="ltx_Math" display="inline" id="S4.SS1.p4.4.m4.2"><semantics id="S4.SS1.p4.4.m4.2a"><msub id="S4.SS1.p4.4.m4.2.3" xref="S4.SS1.p4.4.m4.2.3.cmml"><mover accent="true" id="S4.SS1.p4.4.m4.2.3.2" xref="S4.SS1.p4.4.m4.2.3.2.cmml"><mi id="S4.SS1.p4.4.m4.2.3.2.2" xref="S4.SS1.p4.4.m4.2.3.2.2.cmml">ϕ</mi><mo id="S4.SS1.p4.4.m4.2.3.2.1" xref="S4.SS1.p4.4.m4.2.3.2.1.cmml">~</mo></mover><mrow id="S4.SS1.p4.4.m4.2.2.2.4" xref="S4.SS1.p4.4.m4.2.2.2.3.cmml"><mi id="S4.SS1.p4.4.m4.1.1.1.1" xref="S4.SS1.p4.4.m4.1.1.1.1.cmml">het</mi><mo id="S4.SS1.p4.4.m4.2.2.2.4.1" xref="S4.SS1.p4.4.m4.2.2.2.3.cmml">,</mo><mi id="S4.SS1.p4.4.m4.2.2.2.2" xref="S4.SS1.p4.4.m4.2.2.2.2.cmml">rms</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.4.m4.2b"><apply id="S4.SS1.p4.4.m4.2.3.cmml" xref="S4.SS1.p4.4.m4.2.3"><csymbol cd="ambiguous" id="S4.SS1.p4.4.m4.2.3.1.cmml" xref="S4.SS1.p4.4.m4.2.3">subscript</csymbol><apply id="S4.SS1.p4.4.m4.2.3.2.cmml" xref="S4.SS1.p4.4.m4.2.3.2"><ci id="S4.SS1.p4.4.m4.2.3.2.1.cmml" xref="S4.SS1.p4.4.m4.2.3.2.1">~</ci><ci id="S4.SS1.p4.4.m4.2.3.2.2.cmml" xref="S4.SS1.p4.4.m4.2.3.2.2">italic-ϕ</ci></apply><list id="S4.SS1.p4.4.m4.2.2.2.3.cmml" xref="S4.SS1.p4.4.m4.2.2.2.4"><ci id="S4.SS1.p4.4.m4.1.1.1.1.cmml" xref="S4.SS1.p4.4.m4.1.1.1.1">het</ci><ci id="S4.SS1.p4.4.m4.2.2.2.2.cmml" xref="S4.SS1.p4.4.m4.2.2.2.2">rms</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.4.m4.2c">\widetilde{\phi}_{\rm het,rms}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.4.m4.2d">over~ start_ARG italic_ϕ end_ARG start_POSTSUBSCRIPT roman_het , roman_rms end_POSTSUBSCRIPT</annotation></semantics></math>, relaxing the specification on the lasers’ relative intensity noise to be larger than the shot noise may also be a viable approach given that affected frequencies are likely to be limited by stochastic backgrounds.</p> </div> </section> <section class="ltx_subsection" id="S4.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.2 </span>Radio design</h3> <div class="ltx_para" id="S4.SS2.p1"> <p class="ltx_p" id="S4.SS2.p1.1">There are many aspects that are easier for an experiment that relies on radio transmissions. For example, the displacement requirements are relaxed relative to laser transmissions by the factor of <math alttext="\sim 10^{4}" class="ltx_Math" display="inline" id="S4.SS2.p1.1.m1.1"><semantics id="S4.SS2.p1.1.m1.1a"><mrow id="S4.SS2.p1.1.m1.1.1" xref="S4.SS2.p1.1.m1.1.1.cmml"><mi id="S4.SS2.p1.1.m1.1.1.2" xref="S4.SS2.p1.1.m1.1.1.2.cmml"></mi><mo id="S4.SS2.p1.1.m1.1.1.1" xref="S4.SS2.p1.1.m1.1.1.1.cmml">∼</mo><msup id="S4.SS2.p1.1.m1.1.1.3" xref="S4.SS2.p1.1.m1.1.1.3.cmml"><mn id="S4.SS2.p1.1.m1.1.1.3.2" xref="S4.SS2.p1.1.m1.1.1.3.2.cmml">10</mn><mn id="S4.SS2.p1.1.m1.1.1.3.3" xref="S4.SS2.p1.1.m1.1.1.3.3.cmml">4</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.1.m1.1b"><apply id="S4.SS2.p1.1.m1.1.1.cmml" xref="S4.SS2.p1.1.m1.1.1"><csymbol cd="latexml" id="S4.SS2.p1.1.m1.1.1.1.cmml" xref="S4.SS2.p1.1.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S4.SS2.p1.1.m1.1.1.2.cmml" xref="S4.SS2.p1.1.m1.1.1.2">absent</csymbol><apply id="S4.SS2.p1.1.m1.1.1.3.cmml" xref="S4.SS2.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p1.1.m1.1.1.3.1.cmml" xref="S4.SS2.p1.1.m1.1.1.3">superscript</csymbol><cn id="S4.SS2.p1.1.m1.1.1.3.2.cmml" type="integer" xref="S4.SS2.p1.1.m1.1.1.3.2">10</cn><cn id="S4.SS2.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S4.SS2.p1.1.m1.1.1.3.3">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.1.m1.1c">\sim 10^{4}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.1.m1.1d">∼ 10 start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT</annotation></semantics></math> ratio of the wavelengths. Additionally, relative velocities lead to much smaller frequency differences and, hence, much smaller timing errors in the phase (eqn. <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S4.E19" title="In 4.1 Laser design ‣ 4 Instrumental considerations ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">19</span></a>). Because the phase of the signal itself would be fed directly into the phase-lock loop for radio transmissions, rather than the phase of the received laser being beat against a reference laser, relative intensity noise is not a concern. The pointing requirements are also relaxed relative to lasers, as lasers generate much more planar wavefronts than radio dishes (which leads to larger phase errors from mispointing).</p> </div> <div class="ltx_para" id="S4.SS2.p2"> <p class="ltx_p" id="S4.SS2.p2.4">The radio design requires large dishes and an analog instrumental path via wires, such that thermal path length variations are likely to be larger than the optical design. For metals, the thermal expansion factor for <math alttext="1\;" class="ltx_Math" display="inline" id="S4.SS2.p2.1.m1.1"><semantics id="S4.SS2.p2.1.m1.1a"><mn id="S4.SS2.p2.1.m1.1.1" xref="S4.SS2.p2.1.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.1.m1.1b"><cn id="S4.SS2.p2.1.m1.1.1.cmml" type="integer" xref="S4.SS2.p2.1.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.1.m1.1c">1\;</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.1.m1.1d">1</annotation></semantics></math>m of pathlength is typically <math alttext="\sim 10^{-6}-10^{-5}" class="ltx_Math" display="inline" id="S4.SS2.p2.2.m2.1"><semantics id="S4.SS2.p2.2.m2.1a"><mrow id="S4.SS2.p2.2.m2.1.1" xref="S4.SS2.p2.2.m2.1.1.cmml"><mi id="S4.SS2.p2.2.m2.1.1.2" xref="S4.SS2.p2.2.m2.1.1.2.cmml"></mi><mo id="S4.SS2.p2.2.m2.1.1.1" xref="S4.SS2.p2.2.m2.1.1.1.cmml">∼</mo><mrow id="S4.SS2.p2.2.m2.1.1.3" xref="S4.SS2.p2.2.m2.1.1.3.cmml"><msup id="S4.SS2.p2.2.m2.1.1.3.2" xref="S4.SS2.p2.2.m2.1.1.3.2.cmml"><mn id="S4.SS2.p2.2.m2.1.1.3.2.2" xref="S4.SS2.p2.2.m2.1.1.3.2.2.cmml">10</mn><mrow id="S4.SS2.p2.2.m2.1.1.3.2.3" xref="S4.SS2.p2.2.m2.1.1.3.2.3.cmml"><mo id="S4.SS2.p2.2.m2.1.1.3.2.3a" xref="S4.SS2.p2.2.m2.1.1.3.2.3.cmml">−</mo><mn id="S4.SS2.p2.2.m2.1.1.3.2.3.2" xref="S4.SS2.p2.2.m2.1.1.3.2.3.2.cmml">6</mn></mrow></msup><mo id="S4.SS2.p2.2.m2.1.1.3.1" xref="S4.SS2.p2.2.m2.1.1.3.1.cmml">−</mo><msup id="S4.SS2.p2.2.m2.1.1.3.3" xref="S4.SS2.p2.2.m2.1.1.3.3.cmml"><mn id="S4.SS2.p2.2.m2.1.1.3.3.2" xref="S4.SS2.p2.2.m2.1.1.3.3.2.cmml">10</mn><mrow id="S4.SS2.p2.2.m2.1.1.3.3.3" xref="S4.SS2.p2.2.m2.1.1.3.3.3.cmml"><mo id="S4.SS2.p2.2.m2.1.1.3.3.3a" xref="S4.SS2.p2.2.m2.1.1.3.3.3.cmml">−</mo><mn id="S4.SS2.p2.2.m2.1.1.3.3.3.2" xref="S4.SS2.p2.2.m2.1.1.3.3.3.2.cmml">5</mn></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.2.m2.1b"><apply id="S4.SS2.p2.2.m2.1.1.cmml" xref="S4.SS2.p2.2.m2.1.1"><csymbol cd="latexml" id="S4.SS2.p2.2.m2.1.1.1.cmml" xref="S4.SS2.p2.2.m2.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S4.SS2.p2.2.m2.1.1.2.cmml" xref="S4.SS2.p2.2.m2.1.1.2">absent</csymbol><apply id="S4.SS2.p2.2.m2.1.1.3.cmml" xref="S4.SS2.p2.2.m2.1.1.3"><minus id="S4.SS2.p2.2.m2.1.1.3.1.cmml" xref="S4.SS2.p2.2.m2.1.1.3.1"></minus><apply id="S4.SS2.p2.2.m2.1.1.3.2.cmml" xref="S4.SS2.p2.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS2.p2.2.m2.1.1.3.2.1.cmml" xref="S4.SS2.p2.2.m2.1.1.3.2">superscript</csymbol><cn id="S4.SS2.p2.2.m2.1.1.3.2.2.cmml" type="integer" xref="S4.SS2.p2.2.m2.1.1.3.2.2">10</cn><apply id="S4.SS2.p2.2.m2.1.1.3.2.3.cmml" xref="S4.SS2.p2.2.m2.1.1.3.2.3"><minus id="S4.SS2.p2.2.m2.1.1.3.2.3.1.cmml" xref="S4.SS2.p2.2.m2.1.1.3.2.3"></minus><cn id="S4.SS2.p2.2.m2.1.1.3.2.3.2.cmml" type="integer" xref="S4.SS2.p2.2.m2.1.1.3.2.3.2">6</cn></apply></apply><apply id="S4.SS2.p2.2.m2.1.1.3.3.cmml" xref="S4.SS2.p2.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS2.p2.2.m2.1.1.3.3.1.cmml" xref="S4.SS2.p2.2.m2.1.1.3.3">superscript</csymbol><cn id="S4.SS2.p2.2.m2.1.1.3.3.2.cmml" type="integer" xref="S4.SS2.p2.2.m2.1.1.3.3.2">10</cn><apply id="S4.SS2.p2.2.m2.1.1.3.3.3.cmml" xref="S4.SS2.p2.2.m2.1.1.3.3.3"><minus id="S4.SS2.p2.2.m2.1.1.3.3.3.1.cmml" xref="S4.SS2.p2.2.m2.1.1.3.3.3"></minus><cn id="S4.SS2.p2.2.m2.1.1.3.3.3.2.cmml" type="integer" xref="S4.SS2.p2.2.m2.1.1.3.3.3.2">5</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.2.m2.1c">\sim 10^{-6}-10^{-5}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.2.m2.1d">∼ 10 start_POSTSUPERSCRIPT - 6 end_POSTSUPERSCRIPT - 10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT</annotation></semantics></math>K<sup class="ltx_sup" id="S4.SS2.p2.4.1"><span class="ltx_text ltx_font_italic" id="S4.SS2.p2.4.1.1">-1</span></sup>, with the smaller values for temperatures an order of magnitude below room temperature. To keep thermal expansion to <math alttext="10^{-4}" class="ltx_Math" display="inline" id="S4.SS2.p2.4.m4.1"><semantics id="S4.SS2.p2.4.m4.1a"><msup id="S4.SS2.p2.4.m4.1.1" xref="S4.SS2.p2.4.m4.1.1.cmml"><mn id="S4.SS2.p2.4.m4.1.1.2" xref="S4.SS2.p2.4.m4.1.1.2.cmml">10</mn><mrow id="S4.SS2.p2.4.m4.1.1.3" xref="S4.SS2.p2.4.m4.1.1.3.cmml"><mo id="S4.SS2.p2.4.m4.1.1.3a" xref="S4.SS2.p2.4.m4.1.1.3.cmml">−</mo><mn id="S4.SS2.p2.4.m4.1.1.3.2" xref="S4.SS2.p2.4.m4.1.1.3.2.cmml">4</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.4.m4.1b"><apply id="S4.SS2.p2.4.m4.1.1.cmml" xref="S4.SS2.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.4.m4.1.1.1.cmml" xref="S4.SS2.p2.4.m4.1.1">superscript</csymbol><cn id="S4.SS2.p2.4.m4.1.1.2.cmml" type="integer" xref="S4.SS2.p2.4.m4.1.1.2">10</cn><apply id="S4.SS2.p2.4.m4.1.1.3.cmml" xref="S4.SS2.p2.4.m4.1.1.3"><minus id="S4.SS2.p2.4.m4.1.1.3.1.cmml" xref="S4.SS2.p2.4.m4.1.1.3"></minus><cn id="S4.SS2.p2.4.m4.1.1.3.2.cmml" type="integer" xref="S4.SS2.p2.4.m4.1.1.3.2">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.4.m4.1c">10^{-4}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.4.m4.1d">10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT</annotation></semantics></math>cm, as required for these pathlength changes to be a tenth or so of the radiometer noise on displacements (eqn. <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.E11" title="In 2.2 Radiometer noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">11</span></a>), the temperature should be controlled to just tens of Kelvin.</p> </div> </section> <section class="ltx_subsection" id="S4.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.3 </span>Outer Solar System considerations</h3> <div class="ltx_para" id="S4.SS3.p1"> <p class="ltx_p" id="S4.SS3.p1.1">One challenge to our proposal is that outer Solar System missions are significantly restricted when it comes to downlink data rates, to their mass, and to the power budget. Here we discuss each.</p> </div> <div class="ltx_para" id="S4.SS3.p2"> <dl class="ltx_description" id="S4.I1"> <dt class="ltx_item" id="S4.I1.ix1"><span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_font_bold" id="S4.I1.ix1.1.1.1">downlink data rates</span></span></dt> <dd class="ltx_item"> <div class="ltx_para" id="S4.I1.ix1.p1"> <p class="ltx_p" id="S4.I1.ix1.p1.1">Appendix <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#A3" title="Appendix C Data rates ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">C</span></a> shows that, due to the low gravitational wave frequencies targeted, even <math alttext="10\," class="ltx_Math" display="inline" id="S4.I1.ix1.p1.1.m1.1"><semantics id="S4.I1.ix1.p1.1.m1.1a"><mn id="S4.I1.ix1.p1.1.m1.1.1" xref="S4.I1.ix1.p1.1.m1.1.1.cmml">10</mn><annotation-xml encoding="MathML-Content" id="S4.I1.ix1.p1.1.m1.1b"><cn id="S4.I1.ix1.p1.1.m1.1.1.cmml" type="integer" xref="S4.I1.ix1.p1.1.m1.1.1">10</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.ix1.p1.1.m1.1c">10\,</annotation><annotation encoding="application/x-llamapun" id="S4.I1.ix1.p1.1.m1.1d">10</annotation></semantics></math>kbps hour-long downlinks every several months could be sufficient. Such downlink rates have been achieved to outer Solar System spacecraft, including New Horizons.</p> </div> </dd> <dt class="ltx_item" id="S4.I1.ix2"><span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_font_bold" id="S4.I1.ix2.1.1.1">mass and orbits</span></span></dt> <dd class="ltx_item"> <div class="ltx_para" id="S4.I1.ix2.p1"> <p class="ltx_p" id="S4.I1.ix2.p1.1">The New Horizons spacecraft took nearly a decade to reach Pluto at 34 AU despite weighting just <math alttext="500\;" class="ltx_Math" display="inline" id="S4.I1.ix2.p1.1.m1.1"><semantics id="S4.I1.ix2.p1.1.m1.1a"><mn id="S4.I1.ix2.p1.1.m1.1.1" xref="S4.I1.ix2.p1.1.m1.1.1.cmml">500</mn><annotation-xml encoding="MathML-Content" id="S4.I1.ix2.p1.1.m1.1b"><cn id="S4.I1.ix2.p1.1.m1.1.1.cmml" type="integer" xref="S4.I1.ix2.p1.1.m1.1.1">500</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.ix2.p1.1.m1.1c">500\;</annotation><annotation encoding="application/x-llamapun" id="S4.I1.ix2.p1.1.m1.1d">500</annotation></semantics></math>kg <cite class="ltx_cite ltx_citemacro_citep">(Fountain et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib31" title="">2008</a>)</cite>. However, with new Block 1B and Block 2 rockets from the Space Launch System, plus advances in third/fourth stage boosters, it will soon be possible to launch several times more massive spacecraft at the same C3 velocity as New Horizons <cite class="ltx_cite ltx_citemacro_citep">(NASA, <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib52" title="">2020</a>)</cite> and it is likely that similar specifications will be met by SpaceX and Blue Origin rockets. These new rockets would allow a spacecraft with the mass of New Horizons to reach Pluto in half the time. As achieving a particular orbit would require substantial fuel to slow down, the most feasible architecture would likely involve spacecraft that are continually drifting further from the Sun, possibly on unbound trajectories from the Solar System.<span class="ltx_note ltx_role_footnote" id="footnote14"><sup class="ltx_note_mark">14</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">14</sup><span class="ltx_tag ltx_tag_note">14</span>Spacecraft trajectories that do not require achieving precise orbits may be an advantage over other gravitational wave interferometric concepts aimed at the microHertz band, which require precise placement at <math alttext="120^{\circ}" class="ltx_Math" display="inline" id="footnote14.m1.1"><semantics id="footnote14.m1.1b"><msup id="footnote14.m1.1.1" xref="footnote14.m1.1.1.cmml"><mn id="footnote14.m1.1.1.2" xref="footnote14.m1.1.1.2.cmml">120</mn><mo id="footnote14.m1.1.1.3" xref="footnote14.m1.1.1.3.cmml">∘</mo></msup><annotation-xml encoding="MathML-Content" id="footnote14.m1.1c"><apply id="footnote14.m1.1.1.cmml" xref="footnote14.m1.1.1"><csymbol cd="ambiguous" id="footnote14.m1.1.1.1.cmml" xref="footnote14.m1.1.1">superscript</csymbol><cn id="footnote14.m1.1.1.2.cmml" type="integer" xref="footnote14.m1.1.1.2">120</cn><compose id="footnote14.m1.1.1.3.cmml" xref="footnote14.m1.1.1.3"></compose></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote14.m1.1d">120^{\circ}</annotation><annotation encoding="application/x-llamapun" id="footnote14.m1.1e">120 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT</annotation></semantics></math> in orbital phase at <math alttext="1-3~{}" class="ltx_Math" display="inline" id="footnote14.m2.1"><semantics id="footnote14.m2.1b"><mrow id="footnote14.m2.1.1" xref="footnote14.m2.1.1.cmml"><mn id="footnote14.m2.1.1.2" xref="footnote14.m2.1.1.2.cmml">1</mn><mo id="footnote14.m2.1.1.1" xref="footnote14.m2.1.1.1.cmml">−</mo><mn id="footnote14.m2.1.1.3" xref="footnote14.m2.1.1.3.cmml">3</mn></mrow><annotation-xml encoding="MathML-Content" id="footnote14.m2.1c"><apply id="footnote14.m2.1.1.cmml" xref="footnote14.m2.1.1"><minus id="footnote14.m2.1.1.1.cmml" xref="footnote14.m2.1.1.1"></minus><cn id="footnote14.m2.1.1.2.cmml" type="integer" xref="footnote14.m2.1.1.2">1</cn><cn id="footnote14.m2.1.1.3.cmml" type="integer" xref="footnote14.m2.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote14.m2.1d">1-3~{}</annotation><annotation encoding="application/x-llamapun" id="footnote14.m2.1e">1 - 3</annotation></semantics></math>AU <cite class="ltx_cite ltx_citemacro_citep">(Folkner, <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib30" title="">2011</a>; Martens et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib48" title="">2023</a>; Sesana et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib62" title="">2021</a>; Martens et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib48" title="">2023</a>)</cite>.</span></span></span></p> </div> </dd> <dt class="ltx_item" id="S4.I1.ix3"><span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_font_bold" id="S4.I1.ix3.1.1.1">power</span></span></dt> <dd class="ltx_item"> <div class="ltx_para" id="S4.I1.ix3.p1"> <p class="ltx_p" id="S4.I1.ix3.p1.2">Outer Solar System spacecraft must rely on radioisotope power. If the power output is similar to the radioisotope thermoelectic generator on the New Horizons spacecraft, the total power budget would be <math alttext="250~{}" class="ltx_Math" display="inline" id="S4.I1.ix3.p1.1.m1.1"><semantics id="S4.I1.ix3.p1.1.m1.1a"><mn id="S4.I1.ix3.p1.1.m1.1.1" xref="S4.I1.ix3.p1.1.m1.1.1.cmml">250</mn><annotation-xml encoding="MathML-Content" id="S4.I1.ix3.p1.1.m1.1b"><cn id="S4.I1.ix3.p1.1.m1.1.1.cmml" type="integer" xref="S4.I1.ix3.p1.1.m1.1.1">250</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.ix3.p1.1.m1.1c">250~{}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.ix3.p1.1.m1.1d">250</annotation></semantics></math>Watt, and only a fraction of this could be dedicated to the spacecraft’s science system. Meeting such a restrictive power budget seems potentially feasible given our result that <math alttext="\sim 1-10~{}" class="ltx_Math" display="inline" id="S4.I1.ix3.p1.2.m2.1"><semantics id="S4.I1.ix3.p1.2.m2.1a"><mrow id="S4.I1.ix3.p1.2.m2.1.1" xref="S4.I1.ix3.p1.2.m2.1.1.cmml"><mi id="S4.I1.ix3.p1.2.m2.1.1.2" xref="S4.I1.ix3.p1.2.m2.1.1.2.cmml"></mi><mo id="S4.I1.ix3.p1.2.m2.1.1.1" xref="S4.I1.ix3.p1.2.m2.1.1.1.cmml">∼</mo><mrow id="S4.I1.ix3.p1.2.m2.1.1.3" xref="S4.I1.ix3.p1.2.m2.1.1.3.cmml"><mn id="S4.I1.ix3.p1.2.m2.1.1.3.2" xref="S4.I1.ix3.p1.2.m2.1.1.3.2.cmml">1</mn><mo id="S4.I1.ix3.p1.2.m2.1.1.3.1" xref="S4.I1.ix3.p1.2.m2.1.1.3.1.cmml">−</mo><mn id="S4.I1.ix3.p1.2.m2.1.1.3.3" xref="S4.I1.ix3.p1.2.m2.1.1.3.3.cmml">10</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.ix3.p1.2.m2.1b"><apply id="S4.I1.ix3.p1.2.m2.1.1.cmml" xref="S4.I1.ix3.p1.2.m2.1.1"><csymbol cd="latexml" id="S4.I1.ix3.p1.2.m2.1.1.1.cmml" xref="S4.I1.ix3.p1.2.m2.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S4.I1.ix3.p1.2.m2.1.1.2.cmml" xref="S4.I1.ix3.p1.2.m2.1.1.2">absent</csymbol><apply id="S4.I1.ix3.p1.2.m2.1.1.3.cmml" xref="S4.I1.ix3.p1.2.m2.1.1.3"><minus id="S4.I1.ix3.p1.2.m2.1.1.3.1.cmml" xref="S4.I1.ix3.p1.2.m2.1.1.3.1"></minus><cn id="S4.I1.ix3.p1.2.m2.1.1.3.2.cmml" type="integer" xref="S4.I1.ix3.p1.2.m2.1.1.3.2">1</cn><cn id="S4.I1.ix3.p1.2.m2.1.1.3.3.cmml" type="integer" xref="S4.I1.ix3.p1.2.m2.1.1.3.3">10</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.ix3.p1.2.m2.1c">\sim 1-10~{}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.ix3.p1.2.m2.1d">∼ 1 - 10</annotation></semantics></math>Watt inter-spacecraft transmissions are sufficient. We further estimate similar power requirements for other systems, such as for compute and for the reaction wheels. Indeed, such budgeting is aided by removing the requirement of ultra-precise drag-free control – one of the most power-intensive systems on LISA.</p> </div> </dd> </dl> </div> </section> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">5 </span>Conclusions</h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.3">This paper discussed the feasibility of detecting microhertz gravitational waves using outer Solar System spacecraft. This waveband probes the merger of supermassive black holes as well as a host of other gravitational wave phenomena. Taking advantage of the low acceleration environment beyond <math alttext="10" class="ltx_Math" display="inline" id="S5.p1.1.m1.1"><semantics id="S5.p1.1.m1.1a"><mn id="S5.p1.1.m1.1.1" xref="S5.p1.1.m1.1.1.cmml">10</mn><annotation-xml encoding="MathML-Content" id="S5.p1.1.m1.1b"><cn id="S5.p1.1.m1.1.1.cmml" type="integer" xref="S5.p1.1.m1.1.1">10</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.1.m1.1c">10</annotation><annotation encoding="application/x-llamapun" id="S5.p1.1.m1.1d">10</annotation></semantics></math> AU, such a system could avoid the substantial technological development required for sufficient drag-free control at lower frequencies of <math alttext="f<2\times 10^{-5}" class="ltx_Math" display="inline" id="S5.p1.2.m2.1"><semantics id="S5.p1.2.m2.1a"><mrow id="S5.p1.2.m2.1.1" xref="S5.p1.2.m2.1.1.cmml"><mi id="S5.p1.2.m2.1.1.2" xref="S5.p1.2.m2.1.1.2.cmml">f</mi><mo id="S5.p1.2.m2.1.1.1" xref="S5.p1.2.m2.1.1.1.cmml"><</mo><mrow id="S5.p1.2.m2.1.1.3" xref="S5.p1.2.m2.1.1.3.cmml"><mn id="S5.p1.2.m2.1.1.3.2" xref="S5.p1.2.m2.1.1.3.2.cmml">2</mn><mo id="S5.p1.2.m2.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S5.p1.2.m2.1.1.3.1.cmml">×</mo><msup id="S5.p1.2.m2.1.1.3.3" xref="S5.p1.2.m2.1.1.3.3.cmml"><mn id="S5.p1.2.m2.1.1.3.3.2" xref="S5.p1.2.m2.1.1.3.3.2.cmml">10</mn><mrow id="S5.p1.2.m2.1.1.3.3.3" xref="S5.p1.2.m2.1.1.3.3.3.cmml"><mo id="S5.p1.2.m2.1.1.3.3.3a" xref="S5.p1.2.m2.1.1.3.3.3.cmml">−</mo><mn id="S5.p1.2.m2.1.1.3.3.3.2" xref="S5.p1.2.m2.1.1.3.3.3.2.cmml">5</mn></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.2.m2.1b"><apply id="S5.p1.2.m2.1.1.cmml" xref="S5.p1.2.m2.1.1"><lt id="S5.p1.2.m2.1.1.1.cmml" xref="S5.p1.2.m2.1.1.1"></lt><ci id="S5.p1.2.m2.1.1.2.cmml" xref="S5.p1.2.m2.1.1.2">𝑓</ci><apply id="S5.p1.2.m2.1.1.3.cmml" xref="S5.p1.2.m2.1.1.3"><times id="S5.p1.2.m2.1.1.3.1.cmml" xref="S5.p1.2.m2.1.1.3.1"></times><cn id="S5.p1.2.m2.1.1.3.2.cmml" type="integer" xref="S5.p1.2.m2.1.1.3.2">2</cn><apply id="S5.p1.2.m2.1.1.3.3.cmml" xref="S5.p1.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S5.p1.2.m2.1.1.3.3.1.cmml" xref="S5.p1.2.m2.1.1.3.3">superscript</csymbol><cn id="S5.p1.2.m2.1.1.3.3.2.cmml" type="integer" xref="S5.p1.2.m2.1.1.3.3.2">10</cn><apply id="S5.p1.2.m2.1.1.3.3.3.cmml" xref="S5.p1.2.m2.1.1.3.3.3"><minus id="S5.p1.2.m2.1.1.3.3.3.1.cmml" xref="S5.p1.2.m2.1.1.3.3.3"></minus><cn id="S5.p1.2.m2.1.1.3.3.3.2.cmml" type="integer" xref="S5.p1.2.m2.1.1.3.3.3.2">5</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.2.m2.1c">f<2\times 10^{-5}</annotation><annotation encoding="application/x-llamapun" id="S5.p1.2.m2.1d">italic_f < 2 × 10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT</annotation></semantics></math>Hz than the LISA mission targets. For solar distances as well as inter-spacecraft separations of <math alttext="\geq 10" class="ltx_Math" display="inline" id="S5.p1.3.m3.1"><semantics id="S5.p1.3.m3.1a"><mrow id="S5.p1.3.m3.1.1" xref="S5.p1.3.m3.1.1.cmml"><mi id="S5.p1.3.m3.1.1.2" xref="S5.p1.3.m3.1.1.2.cmml"></mi><mo id="S5.p1.3.m3.1.1.1" xref="S5.p1.3.m3.1.1.1.cmml">≥</mo><mn id="S5.p1.3.m3.1.1.3" xref="S5.p1.3.m3.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.3.m3.1b"><apply id="S5.p1.3.m3.1.1.cmml" xref="S5.p1.3.m3.1.1"><geq id="S5.p1.3.m3.1.1.1.cmml" xref="S5.p1.3.m3.1.1.1"></geq><csymbol cd="latexml" id="S5.p1.3.m3.1.1.2.cmml" xref="S5.p1.3.m3.1.1.2">absent</csymbol><cn id="S5.p1.3.m3.1.1.3.cmml" type="integer" xref="S5.p1.3.m3.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.3.m3.1c">\geq 10</annotation><annotation encoding="application/x-llamapun" id="S5.p1.3.m3.1d">≥ 10</annotation></semantics></math> AU, we showed that the various interplanetary acceleration sources are small enough that even reaching a sensitivity where the noise is set by the stochastic gravitational wave background from massive black holes and white dwarf binaries appears achievable. We showed that such an acceleration-limited system would be easily able to detect the mergers of supermassive black holes at all likely redshifts.</p> </div> <div class="ltx_para" id="S5.p2"> <p class="ltx_p" id="S5.p2.4">We investigated systems that lock onto and time the phase from both laser and radio transmissions between the spacecraft. For both, we argued that even for <math alttext="30\;" class="ltx_Math" display="inline" id="S5.p2.1.m1.1"><semantics id="S5.p2.1.m1.1a"><mn id="S5.p2.1.m1.1.1" xref="S5.p2.1.m1.1.1.cmml">30</mn><annotation-xml encoding="MathML-Content" id="S5.p2.1.m1.1b"><cn id="S5.p2.1.m1.1.1.cmml" type="integer" xref="S5.p2.1.m1.1.1">30</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.1.m1.1c">30\;</annotation><annotation encoding="application/x-llamapun" id="S5.p2.1.m1.1d">30</annotation></semantics></math>AU separations, transmission powers of <math alttext="\sim 1-10~{}" class="ltx_Math" display="inline" id="S5.p2.2.m2.1"><semantics id="S5.p2.2.m2.1a"><mrow id="S5.p2.2.m2.1.1" xref="S5.p2.2.m2.1.1.cmml"><mi id="S5.p2.2.m2.1.1.2" xref="S5.p2.2.m2.1.1.2.cmml"></mi><mo id="S5.p2.2.m2.1.1.1" xref="S5.p2.2.m2.1.1.1.cmml">∼</mo><mrow id="S5.p2.2.m2.1.1.3" xref="S5.p2.2.m2.1.1.3.cmml"><mn id="S5.p2.2.m2.1.1.3.2" xref="S5.p2.2.m2.1.1.3.2.cmml">1</mn><mo id="S5.p2.2.m2.1.1.3.1" xref="S5.p2.2.m2.1.1.3.1.cmml">−</mo><mn id="S5.p2.2.m2.1.1.3.3" xref="S5.p2.2.m2.1.1.3.3.cmml">10</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.2.m2.1b"><apply id="S5.p2.2.m2.1.1.cmml" xref="S5.p2.2.m2.1.1"><csymbol cd="latexml" id="S5.p2.2.m2.1.1.1.cmml" xref="S5.p2.2.m2.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S5.p2.2.m2.1.1.2.cmml" xref="S5.p2.2.m2.1.1.2">absent</csymbol><apply id="S5.p2.2.m2.1.1.3.cmml" xref="S5.p2.2.m2.1.1.3"><minus id="S5.p2.2.m2.1.1.3.1.cmml" xref="S5.p2.2.m2.1.1.3.1"></minus><cn id="S5.p2.2.m2.1.1.3.2.cmml" type="integer" xref="S5.p2.2.m2.1.1.3.2">1</cn><cn id="S5.p2.2.m2.1.1.3.3.cmml" type="integer" xref="S5.p2.2.m2.1.1.3.3">10</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.2.m2.1c">\sim 1-10~{}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.2.m2.1d">∼ 1 - 10</annotation></semantics></math>Watt and reasonable mirror/dish sizes, stable phase locks should be achievable. Additionally, despite the much longer wavelengths of radio compared to laser transmissions, we showed that the sensitivity is likely to still be set by acceleration noise at <math alttext="\lesssim 3\times 10^{-4}~{}" class="ltx_Math" display="inline" id="S5.p2.3.m3.1"><semantics id="S5.p2.3.m3.1a"><mrow id="S5.p2.3.m3.1.1" xref="S5.p2.3.m3.1.1.cmml"><mi id="S5.p2.3.m3.1.1.2" xref="S5.p2.3.m3.1.1.2.cmml"></mi><mo id="S5.p2.3.m3.1.1.1" xref="S5.p2.3.m3.1.1.1.cmml">≲</mo><mrow id="S5.p2.3.m3.1.1.3" xref="S5.p2.3.m3.1.1.3.cmml"><mn id="S5.p2.3.m3.1.1.3.2" xref="S5.p2.3.m3.1.1.3.2.cmml">3</mn><mo id="S5.p2.3.m3.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S5.p2.3.m3.1.1.3.1.cmml">×</mo><msup id="S5.p2.3.m3.1.1.3.3" xref="S5.p2.3.m3.1.1.3.3.cmml"><mn id="S5.p2.3.m3.1.1.3.3.2" xref="S5.p2.3.m3.1.1.3.3.2.cmml">10</mn><mrow id="S5.p2.3.m3.1.1.3.3.3" xref="S5.p2.3.m3.1.1.3.3.3.cmml"><mo id="S5.p2.3.m3.1.1.3.3.3a" xref="S5.p2.3.m3.1.1.3.3.3.cmml">−</mo><mn id="S5.p2.3.m3.1.1.3.3.3.2" xref="S5.p2.3.m3.1.1.3.3.3.2.cmml">4</mn></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.3.m3.1b"><apply id="S5.p2.3.m3.1.1.cmml" xref="S5.p2.3.m3.1.1"><csymbol cd="latexml" id="S5.p2.3.m3.1.1.1.cmml" xref="S5.p2.3.m3.1.1.1">less-than-or-similar-to</csymbol><csymbol cd="latexml" id="S5.p2.3.m3.1.1.2.cmml" xref="S5.p2.3.m3.1.1.2">absent</csymbol><apply id="S5.p2.3.m3.1.1.3.cmml" xref="S5.p2.3.m3.1.1.3"><times id="S5.p2.3.m3.1.1.3.1.cmml" xref="S5.p2.3.m3.1.1.3.1"></times><cn id="S5.p2.3.m3.1.1.3.2.cmml" type="integer" xref="S5.p2.3.m3.1.1.3.2">3</cn><apply id="S5.p2.3.m3.1.1.3.3.cmml" xref="S5.p2.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S5.p2.3.m3.1.1.3.3.1.cmml" xref="S5.p2.3.m3.1.1.3.3">superscript</csymbol><cn id="S5.p2.3.m3.1.1.3.3.2.cmml" type="integer" xref="S5.p2.3.m3.1.1.3.3.2">10</cn><apply id="S5.p2.3.m3.1.1.3.3.3.cmml" xref="S5.p2.3.m3.1.1.3.3.3"><minus id="S5.p2.3.m3.1.1.3.3.3.1.cmml" xref="S5.p2.3.m3.1.1.3.3.3"></minus><cn id="S5.p2.3.m3.1.1.3.3.3.2.cmml" type="integer" xref="S5.p2.3.m3.1.1.3.3.3.2">4</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.3.m3.1c">\lesssim 3\times 10^{-4}~{}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.3.m3.1d">≲ 3 × 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT</annotation></semantics></math>Hz and, hence, independent of the wavelength of the transmissions. A system that uses the radio significantly reduces many design tolerances, such as those regarding pointing, transmission intensity variations, and spacecraft relative velocities. For radio implementations, interplanetary plasma contributes phase noise that can be effectively eliminated by transmitting at two wavelengths, with only a modest (factor of <math alttext="\sim 1.5" class="ltx_Math" display="inline" id="S5.p2.4.m4.1"><semantics id="S5.p2.4.m4.1a"><mrow id="S5.p2.4.m4.1.1" xref="S5.p2.4.m4.1.1.cmml"><mi id="S5.p2.4.m4.1.1.2" xref="S5.p2.4.m4.1.1.2.cmml"></mi><mo id="S5.p2.4.m4.1.1.1" xref="S5.p2.4.m4.1.1.1.cmml">∼</mo><mn id="S5.p2.4.m4.1.1.3" xref="S5.p2.4.m4.1.1.3.cmml">1.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.4.m4.1b"><apply id="S5.p2.4.m4.1.1.cmml" xref="S5.p2.4.m4.1.1"><csymbol cd="latexml" id="S5.p2.4.m4.1.1.1.cmml" xref="S5.p2.4.m4.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S5.p2.4.m4.1.1.2.cmml" xref="S5.p2.4.m4.1.1.2">absent</csymbol><cn id="S5.p2.4.m4.1.1.3.cmml" type="float" xref="S5.p2.4.m4.1.1.3">1.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.4.m4.1c">\sim 1.5</annotation><annotation encoding="application/x-llamapun" id="S5.p2.4.m4.1d">∼ 1.5</annotation></semantics></math>) reduction in strain sensitivity.</p> </div> <div class="ltx_para" id="S5.p3"> <p class="ltx_p" id="S5.p3.5">This paper considered three possible architectures. The first was a two-arm (three-spacecraft) configuration that allows time-delay interferometry. This configuration is also the most sensitive without substantial improvement in space-certified atomic clocks. We considered configurations where the arms are at solar distances of <math alttext="10" class="ltx_Math" display="inline" id="S5.p3.1.m1.1"><semantics id="S5.p3.1.m1.1a"><mn id="S5.p3.1.m1.1.1" xref="S5.p3.1.m1.1.1.cmml">10</mn><annotation-xml encoding="MathML-Content" id="S5.p3.1.m1.1b"><cn id="S5.p3.1.m1.1.1.cmml" type="integer" xref="S5.p3.1.m1.1.1">10</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.1.m1.1c">10</annotation><annotation encoding="application/x-llamapun" id="S5.p3.1.m1.1d">10</annotation></semantics></math> and <math alttext="30~{}" class="ltx_Math" display="inline" id="S5.p3.2.m2.1"><semantics id="S5.p3.2.m2.1a"><mn id="S5.p3.2.m2.1.1" xref="S5.p3.2.m2.1.1.cmml">30</mn><annotation-xml encoding="MathML-Content" id="S5.p3.2.m2.1b"><cn id="S5.p3.2.m2.1.1.cmml" type="integer" xref="S5.p3.2.m2.1.1">30</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.2.m2.1c">30~{}</annotation><annotation encoding="application/x-llamapun" id="S5.p3.2.m2.1d">30</annotation></semantics></math>AU, and where the arm lengths were the same as the solar distance. Both configurations were able to detect merging <math alttext="10^{7}-10^{10}M_{\odot}" class="ltx_Math" display="inline" id="S5.p3.3.m3.1"><semantics id="S5.p3.3.m3.1a"><mrow id="S5.p3.3.m3.1.1" xref="S5.p3.3.m3.1.1.cmml"><msup id="S5.p3.3.m3.1.1.2" xref="S5.p3.3.m3.1.1.2.cmml"><mn id="S5.p3.3.m3.1.1.2.2" xref="S5.p3.3.m3.1.1.2.2.cmml">10</mn><mn id="S5.p3.3.m3.1.1.2.3" xref="S5.p3.3.m3.1.1.2.3.cmml">7</mn></msup><mo id="S5.p3.3.m3.1.1.1" xref="S5.p3.3.m3.1.1.1.cmml">−</mo><mrow id="S5.p3.3.m3.1.1.3" xref="S5.p3.3.m3.1.1.3.cmml"><msup id="S5.p3.3.m3.1.1.3.2" xref="S5.p3.3.m3.1.1.3.2.cmml"><mn id="S5.p3.3.m3.1.1.3.2.2" xref="S5.p3.3.m3.1.1.3.2.2.cmml">10</mn><mn id="S5.p3.3.m3.1.1.3.2.3" xref="S5.p3.3.m3.1.1.3.2.3.cmml">10</mn></msup><mo id="S5.p3.3.m3.1.1.3.1" xref="S5.p3.3.m3.1.1.3.1.cmml">⁢</mo><msub id="S5.p3.3.m3.1.1.3.3" xref="S5.p3.3.m3.1.1.3.3.cmml"><mi id="S5.p3.3.m3.1.1.3.3.2" xref="S5.p3.3.m3.1.1.3.3.2.cmml">M</mi><mo id="S5.p3.3.m3.1.1.3.3.3" xref="S5.p3.3.m3.1.1.3.3.3.cmml">⊙</mo></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p3.3.m3.1b"><apply id="S5.p3.3.m3.1.1.cmml" xref="S5.p3.3.m3.1.1"><minus id="S5.p3.3.m3.1.1.1.cmml" xref="S5.p3.3.m3.1.1.1"></minus><apply id="S5.p3.3.m3.1.1.2.cmml" xref="S5.p3.3.m3.1.1.2"><csymbol cd="ambiguous" id="S5.p3.3.m3.1.1.2.1.cmml" xref="S5.p3.3.m3.1.1.2">superscript</csymbol><cn id="S5.p3.3.m3.1.1.2.2.cmml" type="integer" xref="S5.p3.3.m3.1.1.2.2">10</cn><cn id="S5.p3.3.m3.1.1.2.3.cmml" type="integer" xref="S5.p3.3.m3.1.1.2.3">7</cn></apply><apply id="S5.p3.3.m3.1.1.3.cmml" xref="S5.p3.3.m3.1.1.3"><times id="S5.p3.3.m3.1.1.3.1.cmml" xref="S5.p3.3.m3.1.1.3.1"></times><apply id="S5.p3.3.m3.1.1.3.2.cmml" xref="S5.p3.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="S5.p3.3.m3.1.1.3.2.1.cmml" xref="S5.p3.3.m3.1.1.3.2">superscript</csymbol><cn id="S5.p3.3.m3.1.1.3.2.2.cmml" type="integer" xref="S5.p3.3.m3.1.1.3.2.2">10</cn><cn id="S5.p3.3.m3.1.1.3.2.3.cmml" type="integer" xref="S5.p3.3.m3.1.1.3.2.3">10</cn></apply><apply id="S5.p3.3.m3.1.1.3.3.cmml" xref="S5.p3.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S5.p3.3.m3.1.1.3.3.1.cmml" xref="S5.p3.3.m3.1.1.3.3">subscript</csymbol><ci id="S5.p3.3.m3.1.1.3.3.2.cmml" xref="S5.p3.3.m3.1.1.3.3.2">𝑀</ci><csymbol cd="latexml" id="S5.p3.3.m3.1.1.3.3.3.cmml" xref="S5.p3.3.m3.1.1.3.3.3">direct-product</csymbol></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.3.m3.1c">10^{7}-10^{10}M_{\odot}</annotation><annotation encoding="application/x-llamapun" id="S5.p3.3.m3.1d">10 start_POSTSUPERSCRIPT 7 end_POSTSUPERSCRIPT - 10 start_POSTSUPERSCRIPT 10 end_POSTSUPERSCRIPT italic_M start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT</annotation></semantics></math> black holes out to substantial redshifts, and the <math alttext="30~{}" class="ltx_Math" display="inline" id="S5.p3.4.m4.1"><semantics id="S5.p3.4.m4.1a"><mn id="S5.p3.4.m4.1.1" xref="S5.p3.4.m4.1.1.cmml">30</mn><annotation-xml encoding="MathML-Content" id="S5.p3.4.m4.1b"><cn id="S5.p3.4.m4.1.1.cmml" type="integer" xref="S5.p3.4.m4.1.1">30</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.4.m4.1c">30~{}</annotation><annotation encoding="application/x-llamapun" id="S5.p3.4.m4.1d">30</annotation></semantics></math>AU case was sensitive to middle-of-the-road predictions for the stochastic gravitational background at <math alttext="\sim 10^{-5}" class="ltx_Math" display="inline" id="S5.p3.5.m5.1"><semantics id="S5.p3.5.m5.1a"><mrow id="S5.p3.5.m5.1.1" xref="S5.p3.5.m5.1.1.cmml"><mi id="S5.p3.5.m5.1.1.2" xref="S5.p3.5.m5.1.1.2.cmml"></mi><mo id="S5.p3.5.m5.1.1.1" xref="S5.p3.5.m5.1.1.1.cmml">∼</mo><msup id="S5.p3.5.m5.1.1.3" xref="S5.p3.5.m5.1.1.3.cmml"><mn id="S5.p3.5.m5.1.1.3.2" xref="S5.p3.5.m5.1.1.3.2.cmml">10</mn><mrow id="S5.p3.5.m5.1.1.3.3" xref="S5.p3.5.m5.1.1.3.3.cmml"><mo id="S5.p3.5.m5.1.1.3.3a" xref="S5.p3.5.m5.1.1.3.3.cmml">−</mo><mn id="S5.p3.5.m5.1.1.3.3.2" xref="S5.p3.5.m5.1.1.3.3.2.cmml">5</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.p3.5.m5.1b"><apply id="S5.p3.5.m5.1.1.cmml" xref="S5.p3.5.m5.1.1"><csymbol cd="latexml" id="S5.p3.5.m5.1.1.1.cmml" xref="S5.p3.5.m5.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S5.p3.5.m5.1.1.2.cmml" xref="S5.p3.5.m5.1.1.2">absent</csymbol><apply id="S5.p3.5.m5.1.1.3.cmml" xref="S5.p3.5.m5.1.1.3"><csymbol cd="ambiguous" id="S5.p3.5.m5.1.1.3.1.cmml" xref="S5.p3.5.m5.1.1.3">superscript</csymbol><cn id="S5.p3.5.m5.1.1.3.2.cmml" type="integer" xref="S5.p3.5.m5.1.1.3.2">10</cn><apply id="S5.p3.5.m5.1.1.3.3.cmml" xref="S5.p3.5.m5.1.1.3.3"><minus id="S5.p3.5.m5.1.1.3.3.1.cmml" xref="S5.p3.5.m5.1.1.3.3"></minus><cn id="S5.p3.5.m5.1.1.3.3.2.cmml" type="integer" xref="S5.p3.5.m5.1.1.3.3.2">5</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.5.m5.1c">\sim 10^{-5}</annotation><annotation encoding="application/x-llamapun" id="S5.p3.5.m5.1d">∼ 10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT</annotation></semantics></math>Hz. The sensitivity can be further improved by correcting for accelerations by monitoring solar irradiance variations and the solar wind.</p> </div> <div class="ltx_para" id="S5.p4"> <p class="ltx_p" id="S5.p4.3">The second architecture we considered involved just a single-arm. As a single arm cannot do interferometry to essentially eliminate clock noise, a single-arm design must incorporate a precise atomic clock on at least one spacecraft. We showed that the single-arm architecture equipped with a clock similar to the Deep Space Atomic Clock – a clock scoped for future interplanetary missions – could be sensitive to the characteristic strains of <math alttext="10^{8}M_{\odot}" class="ltx_Math" display="inline" id="S5.p4.1.m1.1"><semantics id="S5.p4.1.m1.1a"><mrow id="S5.p4.1.m1.1.1" xref="S5.p4.1.m1.1.1.cmml"><msup id="S5.p4.1.m1.1.1.2" xref="S5.p4.1.m1.1.1.2.cmml"><mn id="S5.p4.1.m1.1.1.2.2" xref="S5.p4.1.m1.1.1.2.2.cmml">10</mn><mn id="S5.p4.1.m1.1.1.2.3" xref="S5.p4.1.m1.1.1.2.3.cmml">8</mn></msup><mo id="S5.p4.1.m1.1.1.1" xref="S5.p4.1.m1.1.1.1.cmml">⁢</mo><msub id="S5.p4.1.m1.1.1.3" xref="S5.p4.1.m1.1.1.3.cmml"><mi id="S5.p4.1.m1.1.1.3.2" xref="S5.p4.1.m1.1.1.3.2.cmml">M</mi><mo id="S5.p4.1.m1.1.1.3.3" xref="S5.p4.1.m1.1.1.3.3.cmml">⊙</mo></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.p4.1.m1.1b"><apply id="S5.p4.1.m1.1.1.cmml" xref="S5.p4.1.m1.1.1"><times id="S5.p4.1.m1.1.1.1.cmml" xref="S5.p4.1.m1.1.1.1"></times><apply id="S5.p4.1.m1.1.1.2.cmml" xref="S5.p4.1.m1.1.1.2"><csymbol cd="ambiguous" id="S5.p4.1.m1.1.1.2.1.cmml" xref="S5.p4.1.m1.1.1.2">superscript</csymbol><cn id="S5.p4.1.m1.1.1.2.2.cmml" type="integer" xref="S5.p4.1.m1.1.1.2.2">10</cn><cn id="S5.p4.1.m1.1.1.2.3.cmml" type="integer" xref="S5.p4.1.m1.1.1.2.3">8</cn></apply><apply id="S5.p4.1.m1.1.1.3.cmml" xref="S5.p4.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.p4.1.m1.1.1.3.1.cmml" xref="S5.p4.1.m1.1.1.3">subscript</csymbol><ci id="S5.p4.1.m1.1.1.3.2.cmml" xref="S5.p4.1.m1.1.1.3.2">𝑀</ci><csymbol cd="latexml" id="S5.p4.1.m1.1.1.3.3.cmml" xref="S5.p4.1.m1.1.1.3.3">direct-product</csymbol></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p4.1.m1.1c">10^{8}M_{\odot}</annotation><annotation encoding="application/x-llamapun" id="S5.p4.1.m1.1d">10 start_POSTSUPERSCRIPT 8 end_POSTSUPERSCRIPT italic_M start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="10^{9}M_{\odot}" class="ltx_Math" display="inline" id="S5.p4.2.m2.1"><semantics id="S5.p4.2.m2.1a"><mrow id="S5.p4.2.m2.1.1" xref="S5.p4.2.m2.1.1.cmml"><msup id="S5.p4.2.m2.1.1.2" xref="S5.p4.2.m2.1.1.2.cmml"><mn id="S5.p4.2.m2.1.1.2.2" xref="S5.p4.2.m2.1.1.2.2.cmml">10</mn><mn id="S5.p4.2.m2.1.1.2.3" xref="S5.p4.2.m2.1.1.2.3.cmml">9</mn></msup><mo id="S5.p4.2.m2.1.1.1" xref="S5.p4.2.m2.1.1.1.cmml">⁢</mo><msub id="S5.p4.2.m2.1.1.3" xref="S5.p4.2.m2.1.1.3.cmml"><mi id="S5.p4.2.m2.1.1.3.2" xref="S5.p4.2.m2.1.1.3.2.cmml">M</mi><mo id="S5.p4.2.m2.1.1.3.3" xref="S5.p4.2.m2.1.1.3.3.cmml">⊙</mo></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.p4.2.m2.1b"><apply id="S5.p4.2.m2.1.1.cmml" xref="S5.p4.2.m2.1.1"><times id="S5.p4.2.m2.1.1.1.cmml" xref="S5.p4.2.m2.1.1.1"></times><apply id="S5.p4.2.m2.1.1.2.cmml" xref="S5.p4.2.m2.1.1.2"><csymbol cd="ambiguous" id="S5.p4.2.m2.1.1.2.1.cmml" xref="S5.p4.2.m2.1.1.2">superscript</csymbol><cn id="S5.p4.2.m2.1.1.2.2.cmml" type="integer" xref="S5.p4.2.m2.1.1.2.2">10</cn><cn id="S5.p4.2.m2.1.1.2.3.cmml" type="integer" xref="S5.p4.2.m2.1.1.2.3">9</cn></apply><apply id="S5.p4.2.m2.1.1.3.cmml" xref="S5.p4.2.m2.1.1.3"><csymbol cd="ambiguous" id="S5.p4.2.m2.1.1.3.1.cmml" xref="S5.p4.2.m2.1.1.3">subscript</csymbol><ci id="S5.p4.2.m2.1.1.3.2.cmml" xref="S5.p4.2.m2.1.1.3.2">𝑀</ci><csymbol cd="latexml" id="S5.p4.2.m2.1.1.3.3.cmml" xref="S5.p4.2.m2.1.1.3.3">direct-product</csymbol></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p4.2.m2.1c">10^{9}M_{\odot}</annotation><annotation encoding="application/x-llamapun" id="S5.p4.2.m2.1d">10 start_POSTSUPERSCRIPT 9 end_POSTSUPERSCRIPT italic_M start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT</annotation></semantics></math> supermassive black hole mergers, one of the most exciting signals anticipated in the microhertz waveband. Three orders of magnitude improvements in the timing precision over the Deep Space Atomic Clock, still far from the precision of the most precise terrestrial clocks, could achieve sensitivity similar to the interferometric configuration when comparing at the same <math alttext="L" class="ltx_Math" display="inline" id="S5.p4.3.m3.1"><semantics id="S5.p4.3.m3.1a"><mi id="S5.p4.3.m3.1.1" xref="S5.p4.3.m3.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S5.p4.3.m3.1b"><ci id="S5.p4.3.m3.1.1.cmml" xref="S5.p4.3.m3.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p4.3.m3.1c">L</annotation><annotation encoding="application/x-llamapun" id="S5.p4.3.m3.1d">italic_L</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.p5"> <p class="ltx_p" id="S5.p5.1">The single-arm design becomes Doppler tracking of outer Solar System spacecraft when one of the nodes is located on Earth. Doppler tracking using outer Solar System spacecraft has a rich history <cite class="ltx_cite ltx_citemacro_citep">(Armstrong, <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib8" title="">2006a</a>)</cite>. Doppler tracking is the final architecture that we considered. We discussed the atmospheric and ground-station delay requirements for Doppler tracking to reach interesting sensitivity benchmarks. We showed that for gravitational waves with <math alttext="f\lesssim 1\mu" class="ltx_Math" display="inline" id="S5.p5.1.m1.1"><semantics id="S5.p5.1.m1.1a"><mrow id="S5.p5.1.m1.1.1" xref="S5.p5.1.m1.1.1.cmml"><mi id="S5.p5.1.m1.1.1.2" xref="S5.p5.1.m1.1.1.2.cmml">f</mi><mo id="S5.p5.1.m1.1.1.1" xref="S5.p5.1.m1.1.1.1.cmml">≲</mo><mrow id="S5.p5.1.m1.1.1.3" xref="S5.p5.1.m1.1.1.3.cmml"><mn id="S5.p5.1.m1.1.1.3.2" xref="S5.p5.1.m1.1.1.3.2.cmml">1</mn><mo id="S5.p5.1.m1.1.1.3.1" xref="S5.p5.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S5.p5.1.m1.1.1.3.3" xref="S5.p5.1.m1.1.1.3.3.cmml">μ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p5.1.m1.1b"><apply id="S5.p5.1.m1.1.1.cmml" xref="S5.p5.1.m1.1.1"><csymbol cd="latexml" id="S5.p5.1.m1.1.1.1.cmml" xref="S5.p5.1.m1.1.1.1">less-than-or-similar-to</csymbol><ci id="S5.p5.1.m1.1.1.2.cmml" xref="S5.p5.1.m1.1.1.2">𝑓</ci><apply id="S5.p5.1.m1.1.1.3.cmml" xref="S5.p5.1.m1.1.1.3"><times id="S5.p5.1.m1.1.1.3.1.cmml" xref="S5.p5.1.m1.1.1.3.1"></times><cn id="S5.p5.1.m1.1.1.3.2.cmml" type="integer" xref="S5.p5.1.m1.1.1.3.2">1</cn><ci id="S5.p5.1.m1.1.1.3.3.cmml" xref="S5.p5.1.m1.1.1.3.3">𝜇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p5.1.m1.1c">f\lesssim 1\mu</annotation><annotation encoding="application/x-llamapun" id="S5.p5.1.m1.1d">italic_f ≲ 1 italic_μ</annotation></semantics></math>Hz, spacecraft accelerations must be corrected with on-board instrumentation for Doppler tracking to achieve the sensitivity goals of <cite class="ltx_cite ltx_citemacro_citet">Zwick et al. (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib75" title="">2024</a>)</cite>, envisioned in the context of a future Uranus probe. We also investigated the sensitivity of Doppler tracking if the spacecraft could be equipped with an atomic clock, allowing for a time-delay observable that nulls out atmospheric and some mechanical delays.</p> </div> <div class="ltx_para" id="S5.p6"> <p class="ltx_p" id="S5.p6.2">Placing spacecraft in the outer solar system puts severe limits on the downlink rates, mass requirements, and power considerations. We showed that the achievable downlink rates should be sufficient because of the low frequencies of the targeted gravitational waves. We also argued that several tens of Watts of power could conceivably power the science systems on the spacecraft, within the realm of what can be supplied radioisotope thermoelectric generator. The mass of each spacecraft would likely have to be under <math alttext="10^{3}" class="ltx_Math" display="inline" id="S5.p6.1.m1.1"><semantics id="S5.p6.1.m1.1a"><msup id="S5.p6.1.m1.1.1" xref="S5.p6.1.m1.1.1.cmml"><mn id="S5.p6.1.m1.1.1.2" xref="S5.p6.1.m1.1.1.2.cmml">10</mn><mn id="S5.p6.1.m1.1.1.3" xref="S5.p6.1.m1.1.1.3.cmml">3</mn></msup><annotation-xml encoding="MathML-Content" id="S5.p6.1.m1.1b"><apply id="S5.p6.1.m1.1.1.cmml" xref="S5.p6.1.m1.1.1"><csymbol cd="ambiguous" id="S5.p6.1.m1.1.1.1.cmml" xref="S5.p6.1.m1.1.1">superscript</csymbol><cn id="S5.p6.1.m1.1.1.2.cmml" type="integer" xref="S5.p6.1.m1.1.1.2">10</cn><cn id="S5.p6.1.m1.1.1.3.cmml" type="integer" xref="S5.p6.1.m1.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p6.1.m1.1c">10^{3}</annotation><annotation encoding="application/x-llamapun" id="S5.p6.1.m1.1d">10 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT</annotation></semantics></math>kg in order to be launched to tens of astronomical units in <math alttext="5-10~{}" class="ltx_Math" display="inline" id="S5.p6.2.m2.1"><semantics id="S5.p6.2.m2.1a"><mrow id="S5.p6.2.m2.1.1" xref="S5.p6.2.m2.1.1.cmml"><mn id="S5.p6.2.m2.1.1.2" xref="S5.p6.2.m2.1.1.2.cmml">5</mn><mo id="S5.p6.2.m2.1.1.1" xref="S5.p6.2.m2.1.1.1.cmml">−</mo><mn id="S5.p6.2.m2.1.1.3" xref="S5.p6.2.m2.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.p6.2.m2.1b"><apply id="S5.p6.2.m2.1.1.cmml" xref="S5.p6.2.m2.1.1"><minus id="S5.p6.2.m2.1.1.1.cmml" xref="S5.p6.2.m2.1.1.1"></minus><cn id="S5.p6.2.m2.1.1.2.cmml" type="integer" xref="S5.p6.2.m2.1.1.2">5</cn><cn id="S5.p6.2.m2.1.1.3.cmml" type="integer" xref="S5.p6.2.m2.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p6.2.m2.1c">5-10~{}</annotation><annotation encoding="application/x-llamapun" id="S5.p6.2.m2.1d">5 - 10</annotation></semantics></math>yr. The spacecraft requirement that we identified as potentially concerning (and in need of further study) is the severe restrictions on off-gassing (§ <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.SS3.SSS6" title="2.3.6 Off-gassing and other spacecraft emissions ‣ 2.3 Acceleration noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">2.3.6</span></a>).</p> </div> <div class="ltx_para" id="S5.p7"> <p class="ltx_p" id="S5.p7.1">Although the discussion in this paper focused on concepts without acceleration control, some of our results could also apply to an outer Solar System concept that includes an ultra-precise accelerometer. The extremely long arms of our hypothetical concepts mean that the accelerometer would not need to be as precise as in designs with shorter arms to reach the same sensitivity. Furthermore, the stable thermal and acceleration environment of the outer Solar System may facilitate acceleration control over the long periods of our targeted gravitational waves. <br class="ltx_break"/></p> </div> </section> <section class="ltx_section" id="Sx1"> <h2 class="ltx_title ltx_title_section">Acknowledgments</h2> <div class="ltx_para" id="Sx1.p1"> <p class="ltx_p" id="Sx1.p1.1">The authors thank Eric Agol, Miguel Morales, and Joseph Lazio for useful discussions. We thank Jacob Slutsky for feedback on an earlier version of the manuscript. This work is supported by NASA/NIAC Phase I award 80NSSC24K0644. C.M. would like to acknowledge support by NASA under award number 80GSFC24M0006.</p> </div> <div class="ltx_para" id="Sx1.p2"> <p class="ltx_p" id="Sx1.p2.1"><span class="ltx_text ltx_font_italic" id="Sx1.p2.1.1">Materials Availability.</span> The code for the calculations presented in this work can be found at <a class="ltx_ref ltx_url ltx_font_typewriter" href="https://github.com/astromcquinn/GWwithDragFree.git" title="">https://github.com/astromcquinn/GWwithDragFree.git</a>.</p> </div> <div class="ltx_para" id="Sx1.p3"> <p class="ltx_p" id="Sx1.p3.1"><span class="ltx_text ltx_font_italic" id="Sx1.p3.1.1">Software.</span> This work made use of the following software packages: <span class="ltx_text ltx_font_typewriter" id="Sx1.p3.1.2">Jupyter </span> <cite class="ltx_cite ltx_citemacro_citep">(Perez & Granger, <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib58" title="">2007</a>; Kluyver et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib40" title="">2016</a>)</cite>, <span class="ltx_text ltx_font_typewriter" id="Sx1.p3.1.3">matplotlib </span> <cite class="ltx_cite ltx_citemacro_citep">(Hunter, <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib37" title="">2007</a>)</cite>, <span class="ltx_text ltx_font_typewriter" id="Sx1.p3.1.4">numpy </span> <cite class="ltx_cite ltx_citemacro_citep">(Harris et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib35" title="">2020</a>)</cite>, and <span class="ltx_text ltx_font_typewriter" id="Sx1.p3.1.5">python </span> <cite class="ltx_cite ltx_citemacro_citep">(Van Rossum & Drake, <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib68" title="">2009</a>)</cite>. Software citation information aggregated using <a class="ltx_ref ltx_href ltx_font_typewriter" href="https://www.tomwagg.com/software-citation-station/" title="">The Software Citation Station</a> <cite class="ltx_cite ltx_citemacro_citep">(Wagg & Broekgaarden, <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib69" title="">2024</a>)</cite>.</p> </div> </section> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title ltx_title_bibliography">References</h2> <ul class="ltx_biblist"> <li class="ltx_bibitem" id="bib.bib1"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Abbott et al. (2016)</span> <span class="ltx_bibblock"> Abbott B. P., et al., 2016, <a class="ltx_ref ltx_href" href="http://dx.doi.org/10.1103/PhysRevLett.116.061102" title="">Phys. Rev. Lett.</a>, 116, 061102 </span> </li> <li class="ltx_bibitem" id="bib.bib2"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Abbott et al. (2017)</span> <span class="ltx_bibblock"> Abbott B. 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(2024)</span> <span class="ltx_bibblock"> Zwick L., et al., 2024, <a class="ltx_ref ltx_href" href="http://dx.doi.org/10.48550/arXiv.2406.02306" title="">arXiv e-prints</a>, <a class="ltx_ref ltx_href" href="https://ui.adsabs.harvard.edu/abs/2024arXiv240602306Z" title="">p. arXiv:2406.02306</a> </span> </li> </ul> </section> <section class="ltx_appendix" id="A1"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix A </span>Dispersion</h2> <figure class="ltx_figure" id="A1.F7"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="295" id="A1.F7.1.g1" src="x7.png" width="498"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 7: </span>Illustration of the setup of the calculation presented in Appendixes <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#A1" title="Appendix A Dispersion ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">A</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#A2" title="Appendix B Refraction and diffraction ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">B</span></a> for the dispersive delays owing to the interplanetary plasma. Parallel paths <math alttext="{\cal P}_{1}" class="ltx_Math" display="inline" id="A1.F7.9.m1.1"><semantics id="A1.F7.9.m1.1b"><msub id="A1.F7.9.m1.1.1" xref="A1.F7.9.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.F7.9.m1.1.1.2" xref="A1.F7.9.m1.1.1.2.cmml">𝒫</mi><mn id="A1.F7.9.m1.1.1.3" xref="A1.F7.9.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="A1.F7.9.m1.1c"><apply id="A1.F7.9.m1.1.1.cmml" xref="A1.F7.9.m1.1.1"><csymbol cd="ambiguous" id="A1.F7.9.m1.1.1.1.cmml" xref="A1.F7.9.m1.1.1">subscript</csymbol><ci id="A1.F7.9.m1.1.1.2.cmml" xref="A1.F7.9.m1.1.1.2">𝒫</ci><cn id="A1.F7.9.m1.1.1.3.cmml" type="integer" xref="A1.F7.9.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.F7.9.m1.1d">{\cal P}_{1}</annotation><annotation encoding="application/x-llamapun" id="A1.F7.9.m1.1e">caligraphic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="{\cal P}_{2}" class="ltx_Math" display="inline" id="A1.F7.10.m2.1"><semantics id="A1.F7.10.m2.1b"><msub id="A1.F7.10.m2.1.1" xref="A1.F7.10.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.F7.10.m2.1.1.2" xref="A1.F7.10.m2.1.1.2.cmml">𝒫</mi><mn id="A1.F7.10.m2.1.1.3" xref="A1.F7.10.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="A1.F7.10.m2.1c"><apply id="A1.F7.10.m2.1.1.cmml" xref="A1.F7.10.m2.1.1"><csymbol cd="ambiguous" id="A1.F7.10.m2.1.1.1.cmml" xref="A1.F7.10.m2.1.1">subscript</csymbol><ci id="A1.F7.10.m2.1.1.2.cmml" xref="A1.F7.10.m2.1.1.2">𝒫</ci><cn id="A1.F7.10.m2.1.1.3.cmml" type="integer" xref="A1.F7.10.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.F7.10.m2.1d">{\cal P}_{2}</annotation><annotation encoding="application/x-llamapun" id="A1.F7.10.m2.1e">caligraphic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> represent the arms linking two spacecraft at different times. 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One such nearly perpendicular mode of electron density fluctuations is illustrated.</figcaption> </figure> <div class="ltx_para" id="A1.p1"> <p class="ltx_p" id="A1.p1.2">An issue with radio observations that is not present for our laser setup is that dispersion in the interplanetary plasma will contribute phase noise. 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class="ltx_sup" id="A1.p1.2.1"><span class="ltx_text ltx_font_italic" id="A1.p1.2.1.1">-3</span></sup> (e.g., <cite class="ltx_cite ltx_citemacro_citet">Bellamy et al. <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib16" title="">2005</a></cite>), this leads to an error on the strain of</p> <table class="ltx_equation ltx_eqn_table" id="A1.E1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\delta h\sim c\kappa n_{e}=2.6\times 10^{-15}\left(\frac{r}{30~{}{\rm AU}}% \right)^{-2}\left(\frac{\lambda}{1~{}\rm cm}\right)^{2}," class="ltx_Math" display="block" id="A1.E1.m1.3"><semantics id="A1.E1.m1.3a"><mrow id="A1.E1.m1.3.3.1" xref="A1.E1.m1.3.3.1.1.cmml"><mrow id="A1.E1.m1.3.3.1.1" xref="A1.E1.m1.3.3.1.1.cmml"><mrow id="A1.E1.m1.3.3.1.1.2" xref="A1.E1.m1.3.3.1.1.2.cmml"><mi id="A1.E1.m1.3.3.1.1.2.2" xref="A1.E1.m1.3.3.1.1.2.2.cmml">δ</mi><mo 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start_ARG italic_r end_ARG start_ARG 30 roman_AU end_ARG ) start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT ( divide start_ARG italic_λ end_ARG start_ARG 1 roman_cm end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(A1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A1.p1.6">where <math alttext="\kappa=\lambda^{2}e^{2}/(2\pi m_{e}c^{3})" class="ltx_Math" display="inline" id="A1.p1.3.m1.1"><semantics id="A1.p1.3.m1.1a"><mrow id="A1.p1.3.m1.1.1" xref="A1.p1.3.m1.1.1.cmml"><mi id="A1.p1.3.m1.1.1.3" xref="A1.p1.3.m1.1.1.3.cmml">κ</mi><mo id="A1.p1.3.m1.1.1.2" xref="A1.p1.3.m1.1.1.2.cmml">=</mo><mrow id="A1.p1.3.m1.1.1.1" xref="A1.p1.3.m1.1.1.1.cmml"><mrow id="A1.p1.3.m1.1.1.1.3" xref="A1.p1.3.m1.1.1.1.3.cmml"><msup id="A1.p1.3.m1.1.1.1.3.2" 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xref="A1.p1.3.m1.1.1.1.1.1.1.3.cmml">π</mi><mo id="A1.p1.3.m1.1.1.1.1.1.1.1a" xref="A1.p1.3.m1.1.1.1.1.1.1.1.cmml">⁢</mo><msub id="A1.p1.3.m1.1.1.1.1.1.1.4" xref="A1.p1.3.m1.1.1.1.1.1.1.4.cmml"><mi id="A1.p1.3.m1.1.1.1.1.1.1.4.2" xref="A1.p1.3.m1.1.1.1.1.1.1.4.2.cmml">m</mi><mi id="A1.p1.3.m1.1.1.1.1.1.1.4.3" xref="A1.p1.3.m1.1.1.1.1.1.1.4.3.cmml">e</mi></msub><mo id="A1.p1.3.m1.1.1.1.1.1.1.1b" xref="A1.p1.3.m1.1.1.1.1.1.1.1.cmml">⁢</mo><msup id="A1.p1.3.m1.1.1.1.1.1.1.5" xref="A1.p1.3.m1.1.1.1.1.1.1.5.cmml"><mi id="A1.p1.3.m1.1.1.1.1.1.1.5.2" xref="A1.p1.3.m1.1.1.1.1.1.1.5.2.cmml">c</mi><mn id="A1.p1.3.m1.1.1.1.1.1.1.5.3" xref="A1.p1.3.m1.1.1.1.1.1.1.5.3.cmml">3</mn></msup></mrow><mo id="A1.p1.3.m1.1.1.1.1.1.3" stretchy="false" xref="A1.p1.3.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.p1.3.m1.1b"><apply id="A1.p1.3.m1.1.1.cmml" xref="A1.p1.3.m1.1.1"><eq id="A1.p1.3.m1.1.1.2.cmml" xref="A1.p1.3.m1.1.1.2"></eq><ci 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id="A1.p1.3.m1.1.1.1.1.1.1.5.3.cmml" type="integer" xref="A1.p1.3.m1.1.1.1.1.1.1.5.3">3</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.3.m1.1c">\kappa=\lambda^{2}e^{2}/(2\pi m_{e}c^{3})</annotation><annotation encoding="application/x-llamapun" id="A1.p1.3.m1.1d">italic_κ = italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / ( 2 italic_π italic_m start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_c start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT )</annotation></semantics></math> converts an electron column density to a phase delay. Of course, the uniform solar wind signal does not look like gravitational waves. We can calculate the inhomogeneous part in terms of the power spectrum of the solar wind electrons, <math alttext="P_{e}" class="ltx_Math" display="inline" id="A1.p1.4.m2.1"><semantics id="A1.p1.4.m2.1a"><msub id="A1.p1.4.m2.1.1" xref="A1.p1.4.m2.1.1.cmml"><mi id="A1.p1.4.m2.1.1.2" xref="A1.p1.4.m2.1.1.2.cmml">P</mi><mi id="A1.p1.4.m2.1.1.3" xref="A1.p1.4.m2.1.1.3.cmml">e</mi></msub><annotation-xml encoding="MathML-Content" id="A1.p1.4.m2.1b"><apply id="A1.p1.4.m2.1.1.cmml" xref="A1.p1.4.m2.1.1"><csymbol cd="ambiguous" id="A1.p1.4.m2.1.1.1.cmml" xref="A1.p1.4.m2.1.1">subscript</csymbol><ci id="A1.p1.4.m2.1.1.2.cmml" xref="A1.p1.4.m2.1.1.2">𝑃</ci><ci id="A1.p1.4.m2.1.1.3.cmml" xref="A1.p1.4.m2.1.1.3">𝑒</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.4.m2.1c">P_{e}</annotation><annotation encoding="application/x-llamapun" id="A1.p1.4.m2.1d">italic_P start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT</annotation></semantics></math>, by first calculating the temporal correlation function of the plasma delay between the two dishes. Additionally, because the solar wind acts to radially translate the inhomogeneities, the correlation function of delays between parallel paths can be related to the temporal correlation function that we desire. We assume that the paths are at a fixed solar distance <math alttext="r" class="ltx_Math" display="inline" id="A1.p1.5.m3.1"><semantics id="A1.p1.5.m3.1a"><mi id="A1.p1.5.m3.1.1" xref="A1.p1.5.m3.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="A1.p1.5.m3.1b"><ci id="A1.p1.5.m3.1.1.cmml" xref="A1.p1.5.m3.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.5.m3.1c">r</annotation><annotation encoding="application/x-llamapun" id="A1.p1.5.m3.1d">italic_r</annotation></semantics></math> such that the arms are nearly perpendicular to the direction of the Sun. In detail, this cannot be exactly true, but is sufficient for our estimate. We additionally assume that the perturbations along the path that the light travels do not change over the round-trip light-travel time for an arm, which means that our calculation will somewhat overestimate the effect for frequencies that satisfy <math alttext="f\gtrsim c/(2L)" class="ltx_Math" display="inline" id="A1.p1.6.m4.1"><semantics id="A1.p1.6.m4.1a"><mrow id="A1.p1.6.m4.1.1" xref="A1.p1.6.m4.1.1.cmml"><mi id="A1.p1.6.m4.1.1.3" xref="A1.p1.6.m4.1.1.3.cmml">f</mi><mo id="A1.p1.6.m4.1.1.2" xref="A1.p1.6.m4.1.1.2.cmml">≳</mo><mrow id="A1.p1.6.m4.1.1.1" xref="A1.p1.6.m4.1.1.1.cmml"><mi id="A1.p1.6.m4.1.1.1.3" xref="A1.p1.6.m4.1.1.1.3.cmml">c</mi><mo id="A1.p1.6.m4.1.1.1.2" xref="A1.p1.6.m4.1.1.1.2.cmml">/</mo><mrow id="A1.p1.6.m4.1.1.1.1.1" xref="A1.p1.6.m4.1.1.1.1.1.1.cmml"><mo id="A1.p1.6.m4.1.1.1.1.1.2" stretchy="false" xref="A1.p1.6.m4.1.1.1.1.1.1.cmml">(</mo><mrow id="A1.p1.6.m4.1.1.1.1.1.1" xref="A1.p1.6.m4.1.1.1.1.1.1.cmml"><mn id="A1.p1.6.m4.1.1.1.1.1.1.2" xref="A1.p1.6.m4.1.1.1.1.1.1.2.cmml">2</mn><mo id="A1.p1.6.m4.1.1.1.1.1.1.1" xref="A1.p1.6.m4.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="A1.p1.6.m4.1.1.1.1.1.1.3" xref="A1.p1.6.m4.1.1.1.1.1.1.3.cmml">L</mi></mrow><mo id="A1.p1.6.m4.1.1.1.1.1.3" stretchy="false" xref="A1.p1.6.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.p1.6.m4.1b"><apply id="A1.p1.6.m4.1.1.cmml" xref="A1.p1.6.m4.1.1"><csymbol cd="latexml" id="A1.p1.6.m4.1.1.2.cmml" xref="A1.p1.6.m4.1.1.2">greater-than-or-equivalent-to</csymbol><ci id="A1.p1.6.m4.1.1.3.cmml" xref="A1.p1.6.m4.1.1.3">𝑓</ci><apply id="A1.p1.6.m4.1.1.1.cmml" xref="A1.p1.6.m4.1.1.1"><divide id="A1.p1.6.m4.1.1.1.2.cmml" xref="A1.p1.6.m4.1.1.1.2"></divide><ci id="A1.p1.6.m4.1.1.1.3.cmml" xref="A1.p1.6.m4.1.1.1.3">𝑐</ci><apply id="A1.p1.6.m4.1.1.1.1.1.1.cmml" xref="A1.p1.6.m4.1.1.1.1.1"><times id="A1.p1.6.m4.1.1.1.1.1.1.1.cmml" xref="A1.p1.6.m4.1.1.1.1.1.1.1"></times><cn id="A1.p1.6.m4.1.1.1.1.1.1.2.cmml" type="integer" xref="A1.p1.6.m4.1.1.1.1.1.1.2">2</cn><ci id="A1.p1.6.m4.1.1.1.1.1.1.3.cmml" xref="A1.p1.6.m4.1.1.1.1.1.1.3">𝐿</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.6.m4.1c">f\gtrsim c/(2L)</annotation><annotation encoding="application/x-llamapun" id="A1.p1.6.m4.1d">italic_f ≳ italic_c / ( 2 italic_L )</annotation></semantics></math>. Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#A1.F7" title="Figure 7 ‣ Appendix A Dispersion ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">7</span></a> illustrates the setup.</p> </div> <div class="ltx_para" id="A1.p2"> <p class="ltx_p" id="A1.p2.11">The correlation of the phase delays between two parallel paths is</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A3.EGx6"> <tbody id="A1.E2"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\left\langle\Delta\tau_{d,1}\Delta\tau_{d,2}\right\rangle" class="ltx_Math" display="inline" id="A1.E2.m1.5"><semantics id="A1.E2.m1.5a"><mrow id="A1.E2.m1.5.5.1" xref="A1.E2.m1.5.5.2.cmml"><mo id="A1.E2.m1.5.5.1.2" xref="A1.E2.m1.5.5.2.1.cmml">⟨</mo><mrow id="A1.E2.m1.5.5.1.1" xref="A1.E2.m1.5.5.1.1.cmml"><mi id="A1.E2.m1.5.5.1.1.2" mathvariant="normal" xref="A1.E2.m1.5.5.1.1.2.cmml">Δ</mi><mo id="A1.E2.m1.5.5.1.1.1" xref="A1.E2.m1.5.5.1.1.1.cmml">⁢</mo><msub id="A1.E2.m1.5.5.1.1.3" xref="A1.E2.m1.5.5.1.1.3.cmml"><mi id="A1.E2.m1.5.5.1.1.3.2" xref="A1.E2.m1.5.5.1.1.3.2.cmml">τ</mi><mrow id="A1.E2.m1.2.2.2.4" xref="A1.E2.m1.2.2.2.3.cmml"><mi id="A1.E2.m1.1.1.1.1" xref="A1.E2.m1.1.1.1.1.cmml">d</mi><mo id="A1.E2.m1.2.2.2.4.1" xref="A1.E2.m1.2.2.2.3.cmml">,</mo><mn id="A1.E2.m1.2.2.2.2" xref="A1.E2.m1.2.2.2.2.cmml">1</mn></mrow></msub><mo id="A1.E2.m1.5.5.1.1.1a" xref="A1.E2.m1.5.5.1.1.1.cmml">⁢</mo><mi id="A1.E2.m1.5.5.1.1.4" mathvariant="normal" xref="A1.E2.m1.5.5.1.1.4.cmml">Δ</mi><mo id="A1.E2.m1.5.5.1.1.1b" xref="A1.E2.m1.5.5.1.1.1.cmml">⁢</mo><msub id="A1.E2.m1.5.5.1.1.5" xref="A1.E2.m1.5.5.1.1.5.cmml"><mi id="A1.E2.m1.5.5.1.1.5.2" xref="A1.E2.m1.5.5.1.1.5.2.cmml">τ</mi><mrow id="A1.E2.m1.4.4.2.4" xref="A1.E2.m1.4.4.2.3.cmml"><mi id="A1.E2.m1.3.3.1.1" xref="A1.E2.m1.3.3.1.1.cmml">d</mi><mo id="A1.E2.m1.4.4.2.4.1" xref="A1.E2.m1.4.4.2.3.cmml">,</mo><mn id="A1.E2.m1.4.4.2.2" xref="A1.E2.m1.4.4.2.2.cmml">2</mn></mrow></msub></mrow><mo id="A1.E2.m1.5.5.1.3" xref="A1.E2.m1.5.5.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.E2.m1.5b"><apply id="A1.E2.m1.5.5.2.cmml" xref="A1.E2.m1.5.5.1"><csymbol cd="latexml" id="A1.E2.m1.5.5.2.1.cmml" xref="A1.E2.m1.5.5.1.2">delimited-⟨⟩</csymbol><apply id="A1.E2.m1.5.5.1.1.cmml" xref="A1.E2.m1.5.5.1.1"><times id="A1.E2.m1.5.5.1.1.1.cmml" xref="A1.E2.m1.5.5.1.1.1"></times><ci id="A1.E2.m1.5.5.1.1.2.cmml" xref="A1.E2.m1.5.5.1.1.2">Δ</ci><apply id="A1.E2.m1.5.5.1.1.3.cmml" xref="A1.E2.m1.5.5.1.1.3"><csymbol cd="ambiguous" id="A1.E2.m1.5.5.1.1.3.1.cmml" xref="A1.E2.m1.5.5.1.1.3">subscript</csymbol><ci id="A1.E2.m1.5.5.1.1.3.2.cmml" xref="A1.E2.m1.5.5.1.1.3.2">𝜏</ci><list id="A1.E2.m1.2.2.2.3.cmml" xref="A1.E2.m1.2.2.2.4"><ci id="A1.E2.m1.1.1.1.1.cmml" 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⟩</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\kappa^{2}\left\langle\int_{{\cal P}_{1}}dx_{1}\Delta n_{e}(% \boldsymbol{x}_{1})\int_{{\cal P}_{2}}dx_{2}\Delta n_{e}(\boldsymbol{x}_{2})% \right\rangle," class="ltx_Math" display="inline" id="A1.E2.m2.1"><semantics id="A1.E2.m2.1a"><mrow id="A1.E2.m2.1.1.1" xref="A1.E2.m2.1.1.1.1.cmml"><mrow id="A1.E2.m2.1.1.1.1" xref="A1.E2.m2.1.1.1.1.cmml"><mi id="A1.E2.m2.1.1.1.1.3" xref="A1.E2.m2.1.1.1.1.3.cmml"></mi><mo id="A1.E2.m2.1.1.1.1.2" xref="A1.E2.m2.1.1.1.1.2.cmml">=</mo><mrow id="A1.E2.m2.1.1.1.1.1" xref="A1.E2.m2.1.1.1.1.1.cmml"><msup id="A1.E2.m2.1.1.1.1.1.3" xref="A1.E2.m2.1.1.1.1.1.3.cmml"><mi id="A1.E2.m2.1.1.1.1.1.3.2" xref="A1.E2.m2.1.1.1.1.1.3.2.cmml">κ</mi><mn id="A1.E2.m2.1.1.1.1.1.3.3" xref="A1.E2.m2.1.1.1.1.1.3.3.cmml">2</mn></msup><mo id="A1.E2.m2.1.1.1.1.1.2" xref="A1.E2.m2.1.1.1.1.1.2.cmml">⁢</mo><mrow id="A1.E2.m2.1.1.1.1.1.1.1" 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n_{e}(\boldsymbol{x}_{2})% \right\rangle,</annotation><annotation encoding="application/x-llamapun" id="A1.E2.m2.1d">= italic_κ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⟨ ∫ start_POSTSUBSCRIPT caligraphic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_d italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_Δ italic_n start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ( bold_italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ∫ start_POSTSUBSCRIPT caligraphic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_d italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT roman_Δ italic_n start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ( bold_italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ⟩ ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(A2)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A1.p2.2">where <math alttext="\Delta n_{e}" class="ltx_Math" display="inline" id="A1.p2.1.m1.1"><semantics id="A1.p2.1.m1.1a"><mrow id="A1.p2.1.m1.1.1" xref="A1.p2.1.m1.1.1.cmml"><mi id="A1.p2.1.m1.1.1.2" mathvariant="normal" xref="A1.p2.1.m1.1.1.2.cmml">Δ</mi><mo id="A1.p2.1.m1.1.1.1" xref="A1.p2.1.m1.1.1.1.cmml">⁢</mo><msub id="A1.p2.1.m1.1.1.3" xref="A1.p2.1.m1.1.1.3.cmml"><mi id="A1.p2.1.m1.1.1.3.2" xref="A1.p2.1.m1.1.1.3.2.cmml">n</mi><mi id="A1.p2.1.m1.1.1.3.3" xref="A1.p2.1.m1.1.1.3.3.cmml">e</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A1.p2.1.m1.1b"><apply id="A1.p2.1.m1.1.1.cmml" xref="A1.p2.1.m1.1.1"><times id="A1.p2.1.m1.1.1.1.cmml" xref="A1.p2.1.m1.1.1.1"></times><ci id="A1.p2.1.m1.1.1.2.cmml" xref="A1.p2.1.m1.1.1.2">Δ</ci><apply id="A1.p2.1.m1.1.1.3.cmml" xref="A1.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="A1.p2.1.m1.1.1.3.1.cmml" xref="A1.p2.1.m1.1.1.3">subscript</csymbol><ci id="A1.p2.1.m1.1.1.3.2.cmml" xref="A1.p2.1.m1.1.1.3.2">𝑛</ci><ci id="A1.p2.1.m1.1.1.3.3.cmml" xref="A1.p2.1.m1.1.1.3.3">𝑒</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.1.m1.1c">\Delta n_{e}</annotation><annotation encoding="application/x-llamapun" id="A1.p2.1.m1.1d">roman_Δ italic_n start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT</annotation></semantics></math> is the 3D field of electron density fluctuations. Writing <math alttext="\Delta n_{e}" class="ltx_Math" display="inline" id="A1.p2.2.m2.1"><semantics id="A1.p2.2.m2.1a"><mrow id="A1.p2.2.m2.1.1" xref="A1.p2.2.m2.1.1.cmml"><mi id="A1.p2.2.m2.1.1.2" mathvariant="normal" xref="A1.p2.2.m2.1.1.2.cmml">Δ</mi><mo id="A1.p2.2.m2.1.1.1" xref="A1.p2.2.m2.1.1.1.cmml">⁢</mo><msub id="A1.p2.2.m2.1.1.3" xref="A1.p2.2.m2.1.1.3.cmml"><mi id="A1.p2.2.m2.1.1.3.2" xref="A1.p2.2.m2.1.1.3.2.cmml">n</mi><mi id="A1.p2.2.m2.1.1.3.3" xref="A1.p2.2.m2.1.1.3.3.cmml">e</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A1.p2.2.m2.1b"><apply id="A1.p2.2.m2.1.1.cmml" xref="A1.p2.2.m2.1.1"><times id="A1.p2.2.m2.1.1.1.cmml" xref="A1.p2.2.m2.1.1.1"></times><ci id="A1.p2.2.m2.1.1.2.cmml" xref="A1.p2.2.m2.1.1.2">Δ</ci><apply id="A1.p2.2.m2.1.1.3.cmml" xref="A1.p2.2.m2.1.1.3"><csymbol cd="ambiguous" id="A1.p2.2.m2.1.1.3.1.cmml" xref="A1.p2.2.m2.1.1.3">subscript</csymbol><ci id="A1.p2.2.m2.1.1.3.2.cmml" xref="A1.p2.2.m2.1.1.3.2">𝑛</ci><ci id="A1.p2.2.m2.1.1.3.3.cmml" xref="A1.p2.2.m2.1.1.3.3">𝑒</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.2.m2.1c">\Delta n_{e}</annotation><annotation encoding="application/x-llamapun" id="A1.p2.2.m2.1d">roman_Δ italic_n start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT</annotation></semantics></math> in terms of its Fourier transform, the expectation value becomes</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A3.EGx7"> <tbody id="A1.E3"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\left\langle\Delta\tau_{d,1}\Delta\tau_{d,2}\right\rangle=" class="ltx_Math" display="inline" id="A1.E3.m1.5"><semantics id="A1.E3.m1.5a"><mrow id="A1.E3.m1.5.5" xref="A1.E3.m1.5.5.cmml"><mrow id="A1.E3.m1.5.5.1.1" xref="A1.E3.m1.5.5.1.2.cmml"><mo id="A1.E3.m1.5.5.1.1.2" xref="A1.E3.m1.5.5.1.2.1.cmml">⟨</mo><mrow id="A1.E3.m1.5.5.1.1.1" xref="A1.E3.m1.5.5.1.1.1.cmml"><mi id="A1.E3.m1.5.5.1.1.1.2" mathvariant="normal" xref="A1.E3.m1.5.5.1.1.1.2.cmml">Δ</mi><mo id="A1.E3.m1.5.5.1.1.1.1" xref="A1.E3.m1.5.5.1.1.1.1.cmml">⁢</mo><msub id="A1.E3.m1.5.5.1.1.1.3" xref="A1.E3.m1.5.5.1.1.1.3.cmml"><mi id="A1.E3.m1.5.5.1.1.1.3.2" xref="A1.E3.m1.5.5.1.1.1.3.2.cmml">τ</mi><mrow id="A1.E3.m1.2.2.2.4" xref="A1.E3.m1.2.2.2.3.cmml"><mi id="A1.E3.m1.1.1.1.1" xref="A1.E3.m1.1.1.1.1.cmml">d</mi><mo id="A1.E3.m1.2.2.2.4.1" xref="A1.E3.m1.2.2.2.3.cmml">,</mo><mn id="A1.E3.m1.2.2.2.2" xref="A1.E3.m1.2.2.2.2.cmml">1</mn></mrow></msub><mo id="A1.E3.m1.5.5.1.1.1.1a" xref="A1.E3.m1.5.5.1.1.1.1.cmml">⁢</mo><mi id="A1.E3.m1.5.5.1.1.1.4" mathvariant="normal" xref="A1.E3.m1.5.5.1.1.1.4.cmml">Δ</mi><mo id="A1.E3.m1.5.5.1.1.1.1b" xref="A1.E3.m1.5.5.1.1.1.1.cmml">⁢</mo><msub id="A1.E3.m1.5.5.1.1.1.5" xref="A1.E3.m1.5.5.1.1.1.5.cmml"><mi id="A1.E3.m1.5.5.1.1.1.5.2" xref="A1.E3.m1.5.5.1.1.1.5.2.cmml">τ</mi><mrow id="A1.E3.m1.4.4.2.4" xref="A1.E3.m1.4.4.2.3.cmml"><mi id="A1.E3.m1.3.3.1.1" xref="A1.E3.m1.3.3.1.1.cmml">d</mi><mo id="A1.E3.m1.4.4.2.4.1" xref="A1.E3.m1.4.4.2.3.cmml">,</mo><mn id="A1.E3.m1.4.4.2.2" xref="A1.E3.m1.4.4.2.2.cmml">2</mn></mrow></msub></mrow><mo id="A1.E3.m1.5.5.1.1.3" xref="A1.E3.m1.5.5.1.2.1.cmml">⟩</mo></mrow><mo id="A1.E3.m1.5.5.2" xref="A1.E3.m1.5.5.2.cmml">=</mo><mi id="A1.E3.m1.5.5.3" xref="A1.E3.m1.5.5.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="A1.E3.m1.5b"><apply id="A1.E3.m1.5.5.cmml" xref="A1.E3.m1.5.5"><eq id="A1.E3.m1.5.5.2.cmml" xref="A1.E3.m1.5.5.2"></eq><apply id="A1.E3.m1.5.5.1.2.cmml" xref="A1.E3.m1.5.5.1.1"><csymbol cd="latexml" id="A1.E3.m1.5.5.1.2.1.cmml" xref="A1.E3.m1.5.5.1.1.2">delimited-⟨⟩</csymbol><apply id="A1.E3.m1.5.5.1.1.1.cmml" xref="A1.E3.m1.5.5.1.1.1"><times id="A1.E3.m1.5.5.1.1.1.1.cmml" xref="A1.E3.m1.5.5.1.1.1.1"></times><ci id="A1.E3.m1.5.5.1.1.1.2.cmml" xref="A1.E3.m1.5.5.1.1.1.2">Δ</ci><apply id="A1.E3.m1.5.5.1.1.1.3.cmml" xref="A1.E3.m1.5.5.1.1.1.3"><csymbol cd="ambiguous" id="A1.E3.m1.5.5.1.1.1.3.1.cmml" xref="A1.E3.m1.5.5.1.1.1.3">subscript</csymbol><ci id="A1.E3.m1.5.5.1.1.1.3.2.cmml" xref="A1.E3.m1.5.5.1.1.1.3.2">𝜏</ci><list id="A1.E3.m1.2.2.2.3.cmml" xref="A1.E3.m1.2.2.2.4"><ci id="A1.E3.m1.1.1.1.1.cmml" xref="A1.E3.m1.1.1.1.1">𝑑</ci><cn id="A1.E3.m1.2.2.2.2.cmml" type="integer" xref="A1.E3.m1.2.2.2.2">1</cn></list></apply><ci id="A1.E3.m1.5.5.1.1.1.4.cmml" xref="A1.E3.m1.5.5.1.1.1.4">Δ</ci><apply id="A1.E3.m1.5.5.1.1.1.5.cmml" xref="A1.E3.m1.5.5.1.1.1.5"><csymbol cd="ambiguous" id="A1.E3.m1.5.5.1.1.1.5.1.cmml" xref="A1.E3.m1.5.5.1.1.1.5">subscript</csymbol><ci id="A1.E3.m1.5.5.1.1.1.5.2.cmml" xref="A1.E3.m1.5.5.1.1.1.5.2">𝜏</ci><list id="A1.E3.m1.4.4.2.3.cmml" xref="A1.E3.m1.4.4.2.4"><ci id="A1.E3.m1.3.3.1.1.cmml" xref="A1.E3.m1.3.3.1.1">𝑑</ci><cn id="A1.E3.m1.4.4.2.2.cmml" type="integer" xref="A1.E3.m1.4.4.2.2">2</cn></list></apply></apply></apply><csymbol cd="latexml" id="A1.E3.m1.5.5.3.cmml" xref="A1.E3.m1.5.5.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E3.m1.5c">\displaystyle\left\langle\Delta\tau_{d,1}\Delta\tau_{d,2}\right\rangle=</annotation><annotation encoding="application/x-llamapun" id="A1.E3.m1.5d">⟨ roman_Δ italic_τ start_POSTSUBSCRIPT italic_d , 1 end_POSTSUBSCRIPT roman_Δ italic_τ start_POSTSUBSCRIPT italic_d , 2 end_POSTSUBSCRIPT ⟩ =</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\kappa^{2}\int_{{\cal P}_{1}}dx_{1}\int_{{\cal P}_{2}}dx_{2}\iint% \frac{d^{3}{\boldsymbol{k}}_{1}d^{3}{\boldsymbol{k}}_{2}}{(2\pi)^{6}}\left% \langle\widetilde{\Delta n_{e}}({\boldsymbol{k}}_{1})\widetilde{\Delta n_{e}}(% {\boldsymbol{k}}_{2})^{*}\right\rangle e^{i({\boldsymbol{k}}_{1}\cdot% \boldsymbol{x}_{1}-{\boldsymbol{k}}_{2}\cdot\boldsymbol{x}_{2})}," 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xref="A1.E3.m2.2.2.1.1.1.1.3.3.3">2</cn></apply></apply></apply></apply></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E3.m2.3c">\displaystyle\kappa^{2}\int_{{\cal P}_{1}}dx_{1}\int_{{\cal P}_{2}}dx_{2}\iint% \frac{d^{3}{\boldsymbol{k}}_{1}d^{3}{\boldsymbol{k}}_{2}}{(2\pi)^{6}}\left% \langle\widetilde{\Delta n_{e}}({\boldsymbol{k}}_{1})\widetilde{\Delta n_{e}}(% {\boldsymbol{k}}_{2})^{*}\right\rangle e^{i({\boldsymbol{k}}_{1}\cdot% \boldsymbol{x}_{1}-{\boldsymbol{k}}_{2}\cdot\boldsymbol{x}_{2})},</annotation><annotation encoding="application/x-llamapun" id="A1.E3.m2.3d">italic_κ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∫ start_POSTSUBSCRIPT caligraphic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_d italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∫ start_POSTSUBSCRIPT caligraphic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_d italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∬ divide start_ARG italic_d start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT bold_italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_d start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT bold_italic_k start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG start_ARG ( 2 italic_π ) start_POSTSUPERSCRIPT 6 end_POSTSUPERSCRIPT end_ARG ⟨ over~ start_ARG roman_Δ italic_n start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT end_ARG ( bold_italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) over~ start_ARG roman_Δ italic_n start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT end_ARG ( bold_italic_k start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ⟩ italic_e start_POSTSUPERSCRIPT italic_i ( bold_italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⋅ bold_italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - bold_italic_k start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⋅ bold_italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(A3)</span></td> </tr></tbody> <tbody id="A1.Ex1"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="A1.Ex1.m1.1"><semantics id="A1.Ex1.m1.1a"><mo id="A1.Ex1.m1.1.1" xref="A1.Ex1.m1.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="A1.Ex1.m1.1b"><eq id="A1.Ex1.m1.1.1.cmml" xref="A1.Ex1.m1.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex1.m1.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="A1.Ex1.m1.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\kappa^{2}\int_{{\cal P}_{1}}dx_{1}\int_{{\cal P}_{2}}dx_{2}\int% \frac{d^{3}{\boldsymbol{k}}}{(2\pi)^{3}}P_{e}(k)e^{i{\boldsymbol{k}}\cdot(% \boldsymbol{x}_{1}-\boldsymbol{x}_{2})}," class="ltx_Math" display="inline" id="A1.Ex1.m2.4"><semantics id="A1.Ex1.m2.4a"><mrow id="A1.Ex1.m2.4.4.1" xref="A1.Ex1.m2.4.4.1.1.cmml"><mrow id="A1.Ex1.m2.4.4.1.1" xref="A1.Ex1.m2.4.4.1.1.cmml"><msup id="A1.Ex1.m2.4.4.1.1.2" xref="A1.Ex1.m2.4.4.1.1.2.cmml"><mi id="A1.Ex1.m2.4.4.1.1.2.2" xref="A1.Ex1.m2.4.4.1.1.2.2.cmml">κ</mi><mn id="A1.Ex1.m2.4.4.1.1.2.3" xref="A1.Ex1.m2.4.4.1.1.2.3.cmml">2</mn></msup><mo id="A1.Ex1.m2.4.4.1.1.1" xref="A1.Ex1.m2.4.4.1.1.1.cmml">⁢</mo><mrow id="A1.Ex1.m2.4.4.1.1.3" xref="A1.Ex1.m2.4.4.1.1.3.cmml"><mstyle displaystyle="true" id="A1.Ex1.m2.4.4.1.1.3.1" xref="A1.Ex1.m2.4.4.1.1.3.1.cmml"><msub id="A1.Ex1.m2.4.4.1.1.3.1a" xref="A1.Ex1.m2.4.4.1.1.3.1.cmml"><mo id="A1.Ex1.m2.4.4.1.1.3.1.2" xref="A1.Ex1.m2.4.4.1.1.3.1.2.cmml">∫</mo><msub id="A1.Ex1.m2.4.4.1.1.3.1.3" 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start_POSTSUBSCRIPT caligraphic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_d italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∫ divide start_ARG italic_d start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT bold_italic_k end_ARG start_ARG ( 2 italic_π ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG italic_P start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ( italic_k ) italic_e start_POSTSUPERSCRIPT italic_i bold_italic_k ⋅ ( bold_italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - bold_italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A1.p2.9">where we have defined the electron density power spectrum as <math alttext="\left\langle\widetilde{\Delta n_{e}}({\boldsymbol{k}}_{1})\widetilde{\Delta n_% {e}}({\boldsymbol{k}}_{2})^{*}\right\rangle=(2\pi)^{3}P_{e}(k_{1})\delta^{D}({% 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id="A1.p2.3.m1.4.4.4.3.1.1.3.3.cmml" type="integer" xref="A1.p2.3.m1.4.4.4.3.1.1.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.3.m1.4c">\left\langle\widetilde{\Delta n_{e}}({\boldsymbol{k}}_{1})\widetilde{\Delta n_% {e}}({\boldsymbol{k}}_{2})^{*}\right\rangle=(2\pi)^{3}P_{e}(k_{1})\delta^{D}({% \boldsymbol{k}}_{1}-{\boldsymbol{k}}_{2})</annotation><annotation encoding="application/x-llamapun" id="A1.p2.3.m1.4d">⟨ over~ start_ARG roman_Δ italic_n start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT end_ARG ( bold_italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) over~ start_ARG roman_Δ italic_n start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT end_ARG ( bold_italic_k start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ⟩ = ( 2 italic_π ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_P start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ( italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) italic_δ start_POSTSUPERSCRIPT italic_D end_POSTSUPERSCRIPT ( bold_italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - bold_italic_k start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math>. In the ‘Limber approximation limit’ that applies when the integral’s support comes from <math alttext="k\gg 2\pi/L" class="ltx_Math" display="inline" id="A1.p2.4.m2.1"><semantics id="A1.p2.4.m2.1a"><mrow id="A1.p2.4.m2.1.1" xref="A1.p2.4.m2.1.1.cmml"><mi id="A1.p2.4.m2.1.1.2" xref="A1.p2.4.m2.1.1.2.cmml">k</mi><mo id="A1.p2.4.m2.1.1.1" xref="A1.p2.4.m2.1.1.1.cmml">≫</mo><mrow id="A1.p2.4.m2.1.1.3" xref="A1.p2.4.m2.1.1.3.cmml"><mrow id="A1.p2.4.m2.1.1.3.2" xref="A1.p2.4.m2.1.1.3.2.cmml"><mn id="A1.p2.4.m2.1.1.3.2.2" xref="A1.p2.4.m2.1.1.3.2.2.cmml">2</mn><mo id="A1.p2.4.m2.1.1.3.2.1" xref="A1.p2.4.m2.1.1.3.2.1.cmml">⁢</mo><mi id="A1.p2.4.m2.1.1.3.2.3" xref="A1.p2.4.m2.1.1.3.2.3.cmml">π</mi></mrow><mo id="A1.p2.4.m2.1.1.3.1" xref="A1.p2.4.m2.1.1.3.1.cmml">/</mo><mi id="A1.p2.4.m2.1.1.3.3" xref="A1.p2.4.m2.1.1.3.3.cmml">L</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.p2.4.m2.1b"><apply id="A1.p2.4.m2.1.1.cmml" xref="A1.p2.4.m2.1.1"><csymbol cd="latexml" id="A1.p2.4.m2.1.1.1.cmml" xref="A1.p2.4.m2.1.1.1">much-greater-than</csymbol><ci id="A1.p2.4.m2.1.1.2.cmml" xref="A1.p2.4.m2.1.1.2">𝑘</ci><apply id="A1.p2.4.m2.1.1.3.cmml" xref="A1.p2.4.m2.1.1.3"><divide id="A1.p2.4.m2.1.1.3.1.cmml" xref="A1.p2.4.m2.1.1.3.1"></divide><apply id="A1.p2.4.m2.1.1.3.2.cmml" xref="A1.p2.4.m2.1.1.3.2"><times id="A1.p2.4.m2.1.1.3.2.1.cmml" xref="A1.p2.4.m2.1.1.3.2.1"></times><cn id="A1.p2.4.m2.1.1.3.2.2.cmml" type="integer" xref="A1.p2.4.m2.1.1.3.2.2">2</cn><ci id="A1.p2.4.m2.1.1.3.2.3.cmml" xref="A1.p2.4.m2.1.1.3.2.3">𝜋</ci></apply><ci id="A1.p2.4.m2.1.1.3.3.cmml" xref="A1.p2.4.m2.1.1.3.3">𝐿</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.4.m2.1c">k\gg 2\pi/L</annotation><annotation encoding="application/x-llamapun" id="A1.p2.4.m2.1d">italic_k ≫ 2 italic_π / italic_L</annotation></semantics></math>, the integral over the line-of-sight wavevector along <math alttext="{\cal P}_{1}" class="ltx_Math" display="inline" id="A1.p2.5.m3.1"><semantics id="A1.p2.5.m3.1a"><msub id="A1.p2.5.m3.1.1" xref="A1.p2.5.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.p2.5.m3.1.1.2" xref="A1.p2.5.m3.1.1.2.cmml">𝒫</mi><mn id="A1.p2.5.m3.1.1.3" xref="A1.p2.5.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="A1.p2.5.m3.1b"><apply id="A1.p2.5.m3.1.1.cmml" xref="A1.p2.5.m3.1.1"><csymbol cd="ambiguous" id="A1.p2.5.m3.1.1.1.cmml" xref="A1.p2.5.m3.1.1">subscript</csymbol><ci id="A1.p2.5.m3.1.1.2.cmml" xref="A1.p2.5.m3.1.1.2">𝒫</ci><cn id="A1.p2.5.m3.1.1.3.cmml" type="integer" xref="A1.p2.5.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.5.m3.1c">{\cal P}_{1}</annotation><annotation encoding="application/x-llamapun" id="A1.p2.5.m3.1d">caligraphic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> can be approximated as a <math alttext="\delta" class="ltx_Math" display="inline" id="A1.p2.6.m4.1"><semantics id="A1.p2.6.m4.1a"><mi id="A1.p2.6.m4.1.1" xref="A1.p2.6.m4.1.1.cmml">δ</mi><annotation-xml encoding="MathML-Content" id="A1.p2.6.m4.1b"><ci id="A1.p2.6.m4.1.1.cmml" xref="A1.p2.6.m4.1.1">𝛿</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.6.m4.1c">\delta</annotation><annotation encoding="application/x-llamapun" id="A1.p2.6.m4.1d">italic_δ</annotation></semantics></math>-function. The <math alttext="\delta" class="ltx_Math" display="inline" id="A1.p2.7.m5.1"><semantics id="A1.p2.7.m5.1a"><mi id="A1.p2.7.m5.1.1" xref="A1.p2.7.m5.1.1.cmml">δ</mi><annotation-xml encoding="MathML-Content" id="A1.p2.7.m5.1b"><ci id="A1.p2.7.m5.1.1.cmml" xref="A1.p2.7.m5.1.1">𝛿</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.7.m5.1c">\delta</annotation><annotation encoding="application/x-llamapun" id="A1.p2.7.m5.1d">italic_δ</annotation></semantics></math>-function can then be used to eliminate the integral over the wavevector along the path. The remaining integral along <math alttext="{\cal P}_{2}" class="ltx_Math" display="inline" id="A1.p2.8.m6.1"><semantics id="A1.p2.8.m6.1a"><msub id="A1.p2.8.m6.1.1" xref="A1.p2.8.m6.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.p2.8.m6.1.1.2" xref="A1.p2.8.m6.1.1.2.cmml">𝒫</mi><mn id="A1.p2.8.m6.1.1.3" xref="A1.p2.8.m6.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="A1.p2.8.m6.1b"><apply id="A1.p2.8.m6.1.1.cmml" xref="A1.p2.8.m6.1.1"><csymbol cd="ambiguous" id="A1.p2.8.m6.1.1.1.cmml" xref="A1.p2.8.m6.1.1">subscript</csymbol><ci id="A1.p2.8.m6.1.1.2.cmml" xref="A1.p2.8.m6.1.1.2">𝒫</ci><cn id="A1.p2.8.m6.1.1.3.cmml" type="integer" xref="A1.p2.8.m6.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.8.m6.1c">{\cal P}_{2}</annotation><annotation encoding="application/x-llamapun" id="A1.p2.8.m6.1d">caligraphic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> evaluates to its length, <math alttext="L" class="ltx_Math" display="inline" id="A1.p2.9.m7.1"><semantics id="A1.p2.9.m7.1a"><mi id="A1.p2.9.m7.1.1" xref="A1.p2.9.m7.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="A1.p2.9.m7.1b"><ci id="A1.p2.9.m7.1.1.cmml" xref="A1.p2.9.m7.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.9.m7.1c">L</annotation><annotation encoding="application/x-llamapun" id="A1.p2.9.m7.1d">italic_L</annotation></semantics></math>, yielding</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A3.EGx8"> <tbody id="A1.E4"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\left\langle\Delta\tau_{d,1}\Delta\tau_{d,2}\right\rangle\approx% \kappa^{2}L\int\frac{d^{2}{\boldsymbol{k}}_{\perp}}{(2\pi)^{2}}P_{e}(k)e^{i{% \boldsymbol{k}}_{\perp}\cdot(\boldsymbol{x}_{\perp,1}-\boldsymbol{x}_{\perp,2}% )}," class="ltx_Math" display="inline" id="A1.E4.m1.12"><semantics id="A1.E4.m1.12a"><mrow id="A1.E4.m1.12.12.1" xref="A1.E4.m1.12.12.1.1.cmml"><mrow id="A1.E4.m1.12.12.1.1" xref="A1.E4.m1.12.12.1.1.cmml"><mrow id="A1.E4.m1.12.12.1.1.1.1" xref="A1.E4.m1.12.12.1.1.1.2.cmml"><mo id="A1.E4.m1.12.12.1.1.1.1.2" xref="A1.E4.m1.12.12.1.1.1.2.1.cmml">⟨</mo><mrow id="A1.E4.m1.12.12.1.1.1.1.1" xref="A1.E4.m1.12.12.1.1.1.1.1.cmml"><mi id="A1.E4.m1.12.12.1.1.1.1.1.2" mathvariant="normal" xref="A1.E4.m1.12.12.1.1.1.1.1.2.cmml">Δ</mi><mo id="A1.E4.m1.12.12.1.1.1.1.1.1" xref="A1.E4.m1.12.12.1.1.1.1.1.1.cmml">⁢</mo><msub id="A1.E4.m1.12.12.1.1.1.1.1.3" xref="A1.E4.m1.12.12.1.1.1.1.1.3.cmml"><mi id="A1.E4.m1.12.12.1.1.1.1.1.3.2" xref="A1.E4.m1.12.12.1.1.1.1.1.3.2.cmml">τ</mi><mrow id="A1.E4.m1.2.2.2.4" xref="A1.E4.m1.2.2.2.3.cmml"><mi id="A1.E4.m1.1.1.1.1" xref="A1.E4.m1.1.1.1.1.cmml">d</mi><mo id="A1.E4.m1.2.2.2.4.1" xref="A1.E4.m1.2.2.2.3.cmml">,</mo><mn id="A1.E4.m1.2.2.2.2" xref="A1.E4.m1.2.2.2.2.cmml">1</mn></mrow></msub><mo id="A1.E4.m1.12.12.1.1.1.1.1.1a" xref="A1.E4.m1.12.12.1.1.1.1.1.1.cmml">⁢</mo><mi id="A1.E4.m1.12.12.1.1.1.1.1.4" mathvariant="normal" xref="A1.E4.m1.12.12.1.1.1.1.1.4.cmml">Δ</mi><mo id="A1.E4.m1.12.12.1.1.1.1.1.1b" xref="A1.E4.m1.12.12.1.1.1.1.1.1.cmml">⁢</mo><msub id="A1.E4.m1.12.12.1.1.1.1.1.5" xref="A1.E4.m1.12.12.1.1.1.1.1.5.cmml"><mi id="A1.E4.m1.12.12.1.1.1.1.1.5.2" xref="A1.E4.m1.12.12.1.1.1.1.1.5.2.cmml">τ</mi><mrow id="A1.E4.m1.4.4.2.4" xref="A1.E4.m1.4.4.2.3.cmml"><mi id="A1.E4.m1.3.3.1.1" xref="A1.E4.m1.3.3.1.1.cmml">d</mi><mo id="A1.E4.m1.4.4.2.4.1" xref="A1.E4.m1.4.4.2.3.cmml">,</mo><mn id="A1.E4.m1.4.4.2.2" xref="A1.E4.m1.4.4.2.2.cmml">2</mn></mrow></msub></mrow><mo id="A1.E4.m1.12.12.1.1.1.1.3" xref="A1.E4.m1.12.12.1.1.1.2.1.cmml">⟩</mo></mrow><mo id="A1.E4.m1.12.12.1.1.2" xref="A1.E4.m1.12.12.1.1.2.cmml">≈</mo><mrow id="A1.E4.m1.12.12.1.1.3" xref="A1.E4.m1.12.12.1.1.3.cmml"><msup id="A1.E4.m1.12.12.1.1.3.2" xref="A1.E4.m1.12.12.1.1.3.2.cmml"><mi id="A1.E4.m1.12.12.1.1.3.2.2" xref="A1.E4.m1.12.12.1.1.3.2.2.cmml">κ</mi><mn id="A1.E4.m1.12.12.1.1.3.2.3" xref="A1.E4.m1.12.12.1.1.3.2.3.cmml">2</mn></msup><mo id="A1.E4.m1.12.12.1.1.3.1" xref="A1.E4.m1.12.12.1.1.3.1.cmml">⁢</mo><mi id="A1.E4.m1.12.12.1.1.3.3" xref="A1.E4.m1.12.12.1.1.3.3.cmml">L</mi><mo id="A1.E4.m1.12.12.1.1.3.1a" xref="A1.E4.m1.12.12.1.1.3.1.cmml">⁢</mo><mstyle displaystyle="true" id="A1.E4.m1.12.12.1.1.3.4" xref="A1.E4.m1.12.12.1.1.3.4.cmml"><mrow id="A1.E4.m1.12.12.1.1.3.4a" xref="A1.E4.m1.12.12.1.1.3.4.cmml"><mo id="A1.E4.m1.12.12.1.1.3.4.1" xref="A1.E4.m1.12.12.1.1.3.4.1.cmml">∫</mo><mrow id="A1.E4.m1.12.12.1.1.3.4.2" xref="A1.E4.m1.12.12.1.1.3.4.2.cmml"><mfrac id="A1.E4.m1.5.5" xref="A1.E4.m1.5.5.cmml"><mrow id="A1.E4.m1.5.5.3" xref="A1.E4.m1.5.5.3.cmml"><msup id="A1.E4.m1.5.5.3.2" xref="A1.E4.m1.5.5.3.2.cmml"><mi id="A1.E4.m1.5.5.3.2.2" xref="A1.E4.m1.5.5.3.2.2.cmml">d</mi><mn id="A1.E4.m1.5.5.3.2.3" xref="A1.E4.m1.5.5.3.2.3.cmml">2</mn></msup><mo id="A1.E4.m1.5.5.3.1" xref="A1.E4.m1.5.5.3.1.cmml">⁢</mo><msub id="A1.E4.m1.5.5.3.3" xref="A1.E4.m1.5.5.3.3.cmml"><mi id="A1.E4.m1.5.5.3.3.2" xref="A1.E4.m1.5.5.3.3.2.cmml">𝒌</mi><mo id="A1.E4.m1.5.5.3.3.3" xref="A1.E4.m1.5.5.3.3.3.cmml">⟂</mo></msub></mrow><msup id="A1.E4.m1.5.5.1" xref="A1.E4.m1.5.5.1.cmml"><mrow id="A1.E4.m1.5.5.1.1.1" xref="A1.E4.m1.5.5.1.1.1.1.cmml"><mo id="A1.E4.m1.5.5.1.1.1.2" stretchy="false" xref="A1.E4.m1.5.5.1.1.1.1.cmml">(</mo><mrow id="A1.E4.m1.5.5.1.1.1.1" xref="A1.E4.m1.5.5.1.1.1.1.cmml"><mn id="A1.E4.m1.5.5.1.1.1.1.2" xref="A1.E4.m1.5.5.1.1.1.1.2.cmml">2</mn><mo id="A1.E4.m1.5.5.1.1.1.1.1" xref="A1.E4.m1.5.5.1.1.1.1.1.cmml">⁢</mo><mi id="A1.E4.m1.5.5.1.1.1.1.3" xref="A1.E4.m1.5.5.1.1.1.1.3.cmml">π</mi></mrow><mo id="A1.E4.m1.5.5.1.1.1.3" stretchy="false" xref="A1.E4.m1.5.5.1.1.1.1.cmml">)</mo></mrow><mn id="A1.E4.m1.5.5.1.3" xref="A1.E4.m1.5.5.1.3.cmml">2</mn></msup></mfrac><mo 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xref="A1.E4.m1.12.12.1.1.3.4.2.4.2.cmml">e</mi><mrow id="A1.E4.m1.10.10.5" xref="A1.E4.m1.10.10.5.cmml"><mrow id="A1.E4.m1.10.10.5.7" xref="A1.E4.m1.10.10.5.7.cmml"><mi id="A1.E4.m1.10.10.5.7.2" xref="A1.E4.m1.10.10.5.7.2.cmml">i</mi><mo id="A1.E4.m1.10.10.5.7.1" xref="A1.E4.m1.10.10.5.7.1.cmml">⁢</mo><msub id="A1.E4.m1.10.10.5.7.3" xref="A1.E4.m1.10.10.5.7.3.cmml"><mi id="A1.E4.m1.10.10.5.7.3.2" xref="A1.E4.m1.10.10.5.7.3.2.cmml">𝒌</mi><mo id="A1.E4.m1.10.10.5.7.3.3" xref="A1.E4.m1.10.10.5.7.3.3.cmml">⟂</mo></msub></mrow><mo id="A1.E4.m1.10.10.5.6" lspace="0.222em" rspace="0.222em" xref="A1.E4.m1.10.10.5.6.cmml">⋅</mo><mrow id="A1.E4.m1.10.10.5.5.1" xref="A1.E4.m1.10.10.5.5.1.1.cmml"><mo id="A1.E4.m1.10.10.5.5.1.2" stretchy="false" xref="A1.E4.m1.10.10.5.5.1.1.cmml">(</mo><mrow id="A1.E4.m1.10.10.5.5.1.1" xref="A1.E4.m1.10.10.5.5.1.1.cmml"><msub id="A1.E4.m1.10.10.5.5.1.1.2" xref="A1.E4.m1.10.10.5.5.1.1.2.cmml"><mi id="A1.E4.m1.10.10.5.5.1.1.2.2" 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xref="A1.E4.m1.9.9.4.4.2.2">2</cn></list></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E4.m1.12c">\displaystyle\left\langle\Delta\tau_{d,1}\Delta\tau_{d,2}\right\rangle\approx% \kappa^{2}L\int\frac{d^{2}{\boldsymbol{k}}_{\perp}}{(2\pi)^{2}}P_{e}(k)e^{i{% \boldsymbol{k}}_{\perp}\cdot(\boldsymbol{x}_{\perp,1}-\boldsymbol{x}_{\perp,2}% )},</annotation><annotation encoding="application/x-llamapun" id="A1.E4.m1.12d">⟨ roman_Δ italic_τ start_POSTSUBSCRIPT italic_d , 1 end_POSTSUBSCRIPT roman_Δ italic_τ start_POSTSUBSCRIPT italic_d , 2 end_POSTSUBSCRIPT ⟩ ≈ italic_κ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_L ∫ divide start_ARG italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_italic_k start_POSTSUBSCRIPT ⟂ end_POSTSUBSCRIPT end_ARG start_ARG ( 2 italic_π ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG italic_P start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ( italic_k ) italic_e start_POSTSUPERSCRIPT italic_i bold_italic_k start_POSTSUBSCRIPT ⟂ end_POSTSUBSCRIPT ⋅ ( bold_italic_x start_POSTSUBSCRIPT ⟂ , 1 end_POSTSUBSCRIPT - bold_italic_x start_POSTSUBSCRIPT ⟂ , 2 end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(A4)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A1.p2.10">where <math alttext="\boldsymbol{x}_{\perp,1}-\boldsymbol{x}_{\perp,2}" class="ltx_Math" display="inline" id="A1.p2.10.m1.4"><semantics id="A1.p2.10.m1.4a"><mrow id="A1.p2.10.m1.4.5" xref="A1.p2.10.m1.4.5.cmml"><msub id="A1.p2.10.m1.4.5.2" xref="A1.p2.10.m1.4.5.2.cmml"><mi id="A1.p2.10.m1.4.5.2.2" xref="A1.p2.10.m1.4.5.2.2.cmml">𝒙</mi><mrow id="A1.p2.10.m1.2.2.2.4" xref="A1.p2.10.m1.2.2.2.3.cmml"><mo id="A1.p2.10.m1.1.1.1.1" rspace="0em" 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⟩</annotation></semantics></math>, using that time is related to position by <math alttext="\boldsymbol{x}_{\perp,1}-\boldsymbol{x}_{\perp,2}=t/v_{\rm sw}{\boldsymbol{% \widehat{n}}}" class="ltx_Math" display="inline" id="A1.p3.2.m2.4"><semantics id="A1.p3.2.m2.4a"><mrow id="A1.p3.2.m2.4.5" xref="A1.p3.2.m2.4.5.cmml"><mrow id="A1.p3.2.m2.4.5.2" xref="A1.p3.2.m2.4.5.2.cmml"><msub id="A1.p3.2.m2.4.5.2.2" xref="A1.p3.2.m2.4.5.2.2.cmml"><mi id="A1.p3.2.m2.4.5.2.2.2" xref="A1.p3.2.m2.4.5.2.2.2.cmml">𝒙</mi><mrow id="A1.p3.2.m2.2.2.2.4" xref="A1.p3.2.m2.2.2.2.3.cmml"><mo id="A1.p3.2.m2.1.1.1.1" rspace="0em" xref="A1.p3.2.m2.1.1.1.1.cmml">⟂</mo><mo id="A1.p3.2.m2.2.2.2.4.1" xref="A1.p3.2.m2.2.2.2.3.cmml">,</mo><mn id="A1.p3.2.m2.2.2.2.2" xref="A1.p3.2.m2.2.2.2.2.cmml">1</mn></mrow></msub><mo id="A1.p3.2.m2.4.5.2.1" xref="A1.p3.2.m2.4.5.2.1.cmml">−</mo><msub id="A1.p3.2.m2.4.5.2.3" xref="A1.p3.2.m2.4.5.2.3.cmml"><mi id="A1.p3.2.m2.4.5.2.3.2" xref="A1.p3.2.m2.4.5.2.3.2.cmml">𝒙</mi><mrow 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id="A1.p3.2.m2.4.5.3.2.3.2.cmml" xref="A1.p3.2.m2.4.5.3.2.3.2">𝑣</ci><ci id="A1.p3.2.m2.4.5.3.2.3.3.cmml" xref="A1.p3.2.m2.4.5.3.2.3.3">sw</ci></apply></apply><apply id="A1.p3.2.m2.4.5.3.3.cmml" xref="A1.p3.2.m2.4.5.3.3"><ci id="A1.p3.2.m2.4.5.3.3.1.cmml" xref="A1.p3.2.m2.4.5.3.3.1">bold-^</ci><ci id="A1.p3.2.m2.4.5.3.3.2.cmml" xref="A1.p3.2.m2.4.5.3.3.2">𝒏</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p3.2.m2.4c">\boldsymbol{x}_{\perp,1}-\boldsymbol{x}_{\perp,2}=t/v_{\rm sw}{\boldsymbol{% \widehat{n}}}</annotation><annotation encoding="application/x-llamapun" id="A1.p3.2.m2.4d">bold_italic_x start_POSTSUBSCRIPT ⟂ , 1 end_POSTSUBSCRIPT - bold_italic_x start_POSTSUBSCRIPT ⟂ , 2 end_POSTSUBSCRIPT = italic_t / italic_v start_POSTSUBSCRIPT roman_sw end_POSTSUBSCRIPT overbold_^ start_ARG bold_italic_n end_ARG</annotation></semantics></math>, where <math alttext="{\boldsymbol{\widehat{n}}}" class="ltx_Math" display="inline" id="A1.p3.3.m3.1"><semantics id="A1.p3.3.m3.1a"><mover accent="true" id="A1.p3.3.m3.1.1" xref="A1.p3.3.m3.1.1.cmml"><mi id="A1.p3.3.m3.1.1.2" xref="A1.p3.3.m3.1.1.2.cmml">𝒏</mi><mo class="ltx_mathvariant_bold" id="A1.p3.3.m3.1.1.1" mathvariant="bold" xref="A1.p3.3.m3.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="A1.p3.3.m3.1b"><apply id="A1.p3.3.m3.1.1.cmml" xref="A1.p3.3.m3.1.1"><ci id="A1.p3.3.m3.1.1.1.cmml" xref="A1.p3.3.m3.1.1.1">bold-^</ci><ci id="A1.p3.3.m3.1.1.2.cmml" xref="A1.p3.3.m3.1.1.2">𝒏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p3.3.m3.1c">{\boldsymbol{\widehat{n}}}</annotation><annotation encoding="application/x-llamapun" id="A1.p3.3.m3.1d">overbold_^ start_ARG bold_italic_n end_ARG</annotation></semantics></math> is the direction of the solar wind as well as the direction perpendicular to our two parallel paths. Thus,</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="A3.EGx9"> <tbody id="A1.E5"> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle P_{\Delta\tau_{d}}(f)" class="ltx_Math" display="inline" id="A1.Ex2.m1.1"><semantics id="A1.Ex2.m1.1a"><mrow id="A1.Ex2.m1.1.2" xref="A1.Ex2.m1.1.2.cmml"><msub id="A1.Ex2.m1.1.2.2" xref="A1.Ex2.m1.1.2.2.cmml"><mi id="A1.Ex2.m1.1.2.2.2" xref="A1.Ex2.m1.1.2.2.2.cmml">P</mi><mrow id="A1.Ex2.m1.1.2.2.3" xref="A1.Ex2.m1.1.2.2.3.cmml"><mi id="A1.Ex2.m1.1.2.2.3.2" mathvariant="normal" xref="A1.Ex2.m1.1.2.2.3.2.cmml">Δ</mi><mo id="A1.Ex2.m1.1.2.2.3.1" xref="A1.Ex2.m1.1.2.2.3.1.cmml">⁢</mo><msub id="A1.Ex2.m1.1.2.2.3.3" xref="A1.Ex2.m1.1.2.2.3.3.cmml"><mi id="A1.Ex2.m1.1.2.2.3.3.2" xref="A1.Ex2.m1.1.2.2.3.3.2.cmml">τ</mi><mi id="A1.Ex2.m1.1.2.2.3.3.3" xref="A1.Ex2.m1.1.2.2.3.3.3.cmml">d</mi></msub></mrow></msub><mo id="A1.Ex2.m1.1.2.1" xref="A1.Ex2.m1.1.2.1.cmml">⁢</mo><mrow id="A1.Ex2.m1.1.2.3.2" xref="A1.Ex2.m1.1.2.cmml"><mo id="A1.Ex2.m1.1.2.3.2.1" stretchy="false" xref="A1.Ex2.m1.1.2.cmml">(</mo><mi id="A1.Ex2.m1.1.1" xref="A1.Ex2.m1.1.1.cmml">f</mi><mo id="A1.Ex2.m1.1.2.3.2.2" stretchy="false" xref="A1.Ex2.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Ex2.m1.1b"><apply id="A1.Ex2.m1.1.2.cmml" xref="A1.Ex2.m1.1.2"><times id="A1.Ex2.m1.1.2.1.cmml" xref="A1.Ex2.m1.1.2.1"></times><apply id="A1.Ex2.m1.1.2.2.cmml" xref="A1.Ex2.m1.1.2.2"><csymbol cd="ambiguous" id="A1.Ex2.m1.1.2.2.1.cmml" xref="A1.Ex2.m1.1.2.2">subscript</csymbol><ci id="A1.Ex2.m1.1.2.2.2.cmml" xref="A1.Ex2.m1.1.2.2.2">𝑃</ci><apply id="A1.Ex2.m1.1.2.2.3.cmml" xref="A1.Ex2.m1.1.2.2.3"><times id="A1.Ex2.m1.1.2.2.3.1.cmml" xref="A1.Ex2.m1.1.2.2.3.1"></times><ci id="A1.Ex2.m1.1.2.2.3.2.cmml" xref="A1.Ex2.m1.1.2.2.3.2">Δ</ci><apply id="A1.Ex2.m1.1.2.2.3.3.cmml" xref="A1.Ex2.m1.1.2.2.3.3"><csymbol cd="ambiguous" id="A1.Ex2.m1.1.2.2.3.3.1.cmml" xref="A1.Ex2.m1.1.2.2.3.3">subscript</csymbol><ci id="A1.Ex2.m1.1.2.2.3.3.2.cmml" xref="A1.Ex2.m1.1.2.2.3.3.2">𝜏</ci><ci id="A1.Ex2.m1.1.2.2.3.3.3.cmml" xref="A1.Ex2.m1.1.2.2.3.3.3">𝑑</ci></apply></apply></apply><ci id="A1.Ex2.m1.1.1.cmml" xref="A1.Ex2.m1.1.1">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex2.m1.1c">\displaystyle P_{\Delta\tau_{d}}(f)</annotation><annotation encoding="application/x-llamapun" id="A1.Ex2.m1.1d">italic_P start_POSTSUBSCRIPT roman_Δ italic_τ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_f )</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="A1.Ex2.m2.1"><semantics id="A1.Ex2.m2.1a"><mo id="A1.Ex2.m2.1.1" xref="A1.Ex2.m2.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" 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italic_d italic_t italic_e start_POSTSUPERSCRIPT - italic_i italic_ω italic_t end_POSTSUPERSCRIPT ⟨ roman_Δ italic_τ start_POSTSUBSCRIPT italic_d , 1 end_POSTSUBSCRIPT roman_Δ italic_τ start_POSTSUBSCRIPT italic_d , 2 end_POSTSUBSCRIPT ⟩ = 2 italic_κ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_L ∫ divide start_ARG italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_italic_k start_POSTSUBSCRIPT ⟂ end_POSTSUBSCRIPT end_ARG start_ARG ( 2 italic_π ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG italic_P start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ( italic_k ) ( 2 italic_π ) italic_δ start_POSTSUPERSCRIPT italic_D end_POSTSUPERSCRIPT ( italic_ω - italic_v start_POSTSUBSCRIPT roman_w , ⊙ end_POSTSUBSCRIPT bold_italic_k start_POSTSUBSCRIPT ⟂ end_POSTSUBSCRIPT ⋅ overbold_^ start_ARG bold_italic_n end_ARG ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="2"><span class="ltx_tag ltx_tag_equation ltx_align_right">(A5)</span></td> </tr> <tr class="ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="A1.E5.m1.1"><semantics id="A1.E5.m1.1a"><mo id="A1.E5.m1.1.1" xref="A1.E5.m1.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="A1.E5.m1.1b"><eq id="A1.E5.m1.1.1.cmml" xref="A1.E5.m1.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="A1.E5.m1.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="A1.E5.m1.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\frac{2\kappa^{2}L}{v_{\rm w,\odot}}\int_{-\infty}^{\infty}\frac{% dk_{x}}{(2\pi)}P_{e}\left(\sqrt{k_{x}^{2}+\frac{\omega^{2}}{v_{\rm sw}^{2}}}% \right)," class="ltx_Math" 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start_POSTSUBSCRIPT - ∞ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT divide start_ARG italic_d italic_k start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_ARG start_ARG ( 2 italic_π ) end_ARG italic_P start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ( square-root start_ARG italic_k start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + divide start_ARG italic_ω start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_v start_POSTSUBSCRIPT roman_sw end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG end_ARG ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </tbody> <tbody id="A1.E6"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle\approx" class="ltx_Math" display="inline" id="A1.E6.m1.1"><semantics id="A1.E6.m1.1a"><mo id="A1.E6.m1.1.1" xref="A1.E6.m1.1.1.cmml">≈</mo><annotation-xml encoding="MathML-Content" id="A1.E6.m1.1b"><approx id="A1.E6.m1.1.1.cmml" xref="A1.E6.m1.1.1"></approx></annotation-xml><annotation encoding="application/x-tex" id="A1.E6.m1.1c">\displaystyle\approx</annotation><annotation encoding="application/x-llamapun" id="A1.E6.m1.1d">≈</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle 1.5\,\kappa^{2}\,L\frac{v_{\rm sw}}{\omega}P_{e}^{\rm 1D}(f,r)," class="ltx_Math" display="inline" id="A1.E6.m2.3"><semantics id="A1.E6.m2.3a"><mrow id="A1.E6.m2.3.3.1" xref="A1.E6.m2.3.3.1.1.cmml"><mrow id="A1.E6.m2.3.3.1.1" xref="A1.E6.m2.3.3.1.1.cmml"><mn id="A1.E6.m2.3.3.1.1.2" xref="A1.E6.m2.3.3.1.1.2.cmml">1.5</mn><mo id="A1.E6.m2.3.3.1.1.1" lspace="0.170em" xref="A1.E6.m2.3.3.1.1.1.cmml">⁢</mo><msup id="A1.E6.m2.3.3.1.1.3" xref="A1.E6.m2.3.3.1.1.3.cmml"><mi id="A1.E6.m2.3.3.1.1.3.2" xref="A1.E6.m2.3.3.1.1.3.2.cmml">κ</mi><mn id="A1.E6.m2.3.3.1.1.3.3" xref="A1.E6.m2.3.3.1.1.3.3.cmml">2</mn></msup><mo id="A1.E6.m2.3.3.1.1.1a" xref="A1.E6.m2.3.3.1.1.1.cmml">⁢</mo><mi id="A1.E6.m2.3.3.1.1.4" xref="A1.E6.m2.3.3.1.1.4.cmml">L</mi><mo id="A1.E6.m2.3.3.1.1.1b" xref="A1.E6.m2.3.3.1.1.1.cmml">⁢</mo><mstyle displaystyle="true" id="A1.E6.m2.3.3.1.1.5" xref="A1.E6.m2.3.3.1.1.5.cmml"><mfrac id="A1.E6.m2.3.3.1.1.5a" xref="A1.E6.m2.3.3.1.1.5.cmml"><msub id="A1.E6.m2.3.3.1.1.5.2" xref="A1.E6.m2.3.3.1.1.5.2.cmml"><mi id="A1.E6.m2.3.3.1.1.5.2.2" xref="A1.E6.m2.3.3.1.1.5.2.2.cmml">v</mi><mi id="A1.E6.m2.3.3.1.1.5.2.3" xref="A1.E6.m2.3.3.1.1.5.2.3.cmml">sw</mi></msub><mi id="A1.E6.m2.3.3.1.1.5.3" xref="A1.E6.m2.3.3.1.1.5.3.cmml">ω</mi></mfrac></mstyle><mo id="A1.E6.m2.3.3.1.1.1c" xref="A1.E6.m2.3.3.1.1.1.cmml">⁢</mo><msubsup id="A1.E6.m2.3.3.1.1.6" xref="A1.E6.m2.3.3.1.1.6.cmml"><mi id="A1.E6.m2.3.3.1.1.6.2.2" xref="A1.E6.m2.3.3.1.1.6.2.2.cmml">P</mi><mi id="A1.E6.m2.3.3.1.1.6.2.3" xref="A1.E6.m2.3.3.1.1.6.2.3.cmml">e</mi><mrow id="A1.E6.m2.3.3.1.1.6.3" xref="A1.E6.m2.3.3.1.1.6.3.cmml"><mn id="A1.E6.m2.3.3.1.1.6.3.2" xref="A1.E6.m2.3.3.1.1.6.3.2.cmml">1</mn><mo id="A1.E6.m2.3.3.1.1.6.3.1" xref="A1.E6.m2.3.3.1.1.6.3.1.cmml">⁢</mo><mi id="A1.E6.m2.3.3.1.1.6.3.3" mathvariant="normal" xref="A1.E6.m2.3.3.1.1.6.3.3.cmml">D</mi></mrow></msubsup><mo id="A1.E6.m2.3.3.1.1.1d" xref="A1.E6.m2.3.3.1.1.1.cmml">⁢</mo><mrow id="A1.E6.m2.3.3.1.1.7.2" xref="A1.E6.m2.3.3.1.1.7.1.cmml"><mo id="A1.E6.m2.3.3.1.1.7.2.1" stretchy="false" xref="A1.E6.m2.3.3.1.1.7.1.cmml">(</mo><mi id="A1.E6.m2.1.1" xref="A1.E6.m2.1.1.cmml">f</mi><mo id="A1.E6.m2.3.3.1.1.7.2.2" xref="A1.E6.m2.3.3.1.1.7.1.cmml">,</mo><mi id="A1.E6.m2.2.2" xref="A1.E6.m2.2.2.cmml">r</mi><mo id="A1.E6.m2.3.3.1.1.7.2.3" stretchy="false" xref="A1.E6.m2.3.3.1.1.7.1.cmml">)</mo></mrow></mrow><mo id="A1.E6.m2.3.3.1.2" xref="A1.E6.m2.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.E6.m2.3b"><apply id="A1.E6.m2.3.3.1.1.cmml" xref="A1.E6.m2.3.3.1"><times id="A1.E6.m2.3.3.1.1.1.cmml" xref="A1.E6.m2.3.3.1.1.1"></times><cn id="A1.E6.m2.3.3.1.1.2.cmml" type="float" xref="A1.E6.m2.3.3.1.1.2">1.5</cn><apply id="A1.E6.m2.3.3.1.1.3.cmml" xref="A1.E6.m2.3.3.1.1.3"><csymbol cd="ambiguous" id="A1.E6.m2.3.3.1.1.3.1.cmml" xref="A1.E6.m2.3.3.1.1.3">superscript</csymbol><ci id="A1.E6.m2.3.3.1.1.3.2.cmml" xref="A1.E6.m2.3.3.1.1.3.2">𝜅</ci><cn id="A1.E6.m2.3.3.1.1.3.3.cmml" type="integer" xref="A1.E6.m2.3.3.1.1.3.3">2</cn></apply><ci id="A1.E6.m2.3.3.1.1.4.cmml" xref="A1.E6.m2.3.3.1.1.4">𝐿</ci><apply id="A1.E6.m2.3.3.1.1.5.cmml" xref="A1.E6.m2.3.3.1.1.5"><divide id="A1.E6.m2.3.3.1.1.5.1.cmml" xref="A1.E6.m2.3.3.1.1.5"></divide><apply id="A1.E6.m2.3.3.1.1.5.2.cmml" xref="A1.E6.m2.3.3.1.1.5.2"><csymbol cd="ambiguous" id="A1.E6.m2.3.3.1.1.5.2.1.cmml" xref="A1.E6.m2.3.3.1.1.5.2">subscript</csymbol><ci id="A1.E6.m2.3.3.1.1.5.2.2.cmml" xref="A1.E6.m2.3.3.1.1.5.2.2">𝑣</ci><ci id="A1.E6.m2.3.3.1.1.5.2.3.cmml" xref="A1.E6.m2.3.3.1.1.5.2.3">sw</ci></apply><ci id="A1.E6.m2.3.3.1.1.5.3.cmml" xref="A1.E6.m2.3.3.1.1.5.3">𝜔</ci></apply><apply id="A1.E6.m2.3.3.1.1.6.cmml" xref="A1.E6.m2.3.3.1.1.6"><csymbol cd="ambiguous" id="A1.E6.m2.3.3.1.1.6.1.cmml" xref="A1.E6.m2.3.3.1.1.6">superscript</csymbol><apply id="A1.E6.m2.3.3.1.1.6.2.cmml" xref="A1.E6.m2.3.3.1.1.6"><csymbol cd="ambiguous" id="A1.E6.m2.3.3.1.1.6.2.1.cmml" xref="A1.E6.m2.3.3.1.1.6">subscript</csymbol><ci id="A1.E6.m2.3.3.1.1.6.2.2.cmml" xref="A1.E6.m2.3.3.1.1.6.2.2">𝑃</ci><ci id="A1.E6.m2.3.3.1.1.6.2.3.cmml" xref="A1.E6.m2.3.3.1.1.6.2.3">𝑒</ci></apply><apply id="A1.E6.m2.3.3.1.1.6.3.cmml" xref="A1.E6.m2.3.3.1.1.6.3"><times id="A1.E6.m2.3.3.1.1.6.3.1.cmml" xref="A1.E6.m2.3.3.1.1.6.3.1"></times><cn id="A1.E6.m2.3.3.1.1.6.3.2.cmml" type="integer" xref="A1.E6.m2.3.3.1.1.6.3.2">1</cn><ci id="A1.E6.m2.3.3.1.1.6.3.3.cmml" xref="A1.E6.m2.3.3.1.1.6.3.3">D</ci></apply></apply><interval closure="open" id="A1.E6.m2.3.3.1.1.7.1.cmml" xref="A1.E6.m2.3.3.1.1.7.2"><ci id="A1.E6.m2.1.1.cmml" xref="A1.E6.m2.1.1">𝑓</ci><ci id="A1.E6.m2.2.2.cmml" xref="A1.E6.m2.2.2">𝑟</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E6.m2.3c">\displaystyle 1.5\,\kappa^{2}\,L\frac{v_{\rm sw}}{\omega}P_{e}^{\rm 1D}(f,r),</annotation><annotation encoding="application/x-llamapun" id="A1.E6.m2.3d">1.5 italic_κ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_L divide start_ARG italic_v start_POSTSUBSCRIPT roman_sw end_POSTSUBSCRIPT end_ARG start_ARG italic_ω end_ARG italic_P start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 roman_D end_POSTSUPERSCRIPT ( italic_f , italic_r ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(A6)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A1.p3.5">where <math alttext="f\equiv\omega/2\pi" class="ltx_Math" display="inline" id="A1.p3.4.m1.1"><semantics id="A1.p3.4.m1.1a"><mrow id="A1.p3.4.m1.1.1" xref="A1.p3.4.m1.1.1.cmml"><mi id="A1.p3.4.m1.1.1.2" xref="A1.p3.4.m1.1.1.2.cmml">f</mi><mo id="A1.p3.4.m1.1.1.1" xref="A1.p3.4.m1.1.1.1.cmml">≡</mo><mrow id="A1.p3.4.m1.1.1.3" xref="A1.p3.4.m1.1.1.3.cmml"><mrow id="A1.p3.4.m1.1.1.3.2" xref="A1.p3.4.m1.1.1.3.2.cmml"><mi id="A1.p3.4.m1.1.1.3.2.2" xref="A1.p3.4.m1.1.1.3.2.2.cmml">ω</mi><mo id="A1.p3.4.m1.1.1.3.2.1" xref="A1.p3.4.m1.1.1.3.2.1.cmml">/</mo><mn id="A1.p3.4.m1.1.1.3.2.3" xref="A1.p3.4.m1.1.1.3.2.3.cmml">2</mn></mrow><mo id="A1.p3.4.m1.1.1.3.1" xref="A1.p3.4.m1.1.1.3.1.cmml">⁢</mo><mi id="A1.p3.4.m1.1.1.3.3" xref="A1.p3.4.m1.1.1.3.3.cmml">π</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.p3.4.m1.1b"><apply id="A1.p3.4.m1.1.1.cmml" xref="A1.p3.4.m1.1.1"><equivalent id="A1.p3.4.m1.1.1.1.cmml" xref="A1.p3.4.m1.1.1.1"></equivalent><ci id="A1.p3.4.m1.1.1.2.cmml" xref="A1.p3.4.m1.1.1.2">𝑓</ci><apply id="A1.p3.4.m1.1.1.3.cmml" xref="A1.p3.4.m1.1.1.3"><times id="A1.p3.4.m1.1.1.3.1.cmml" xref="A1.p3.4.m1.1.1.3.1"></times><apply id="A1.p3.4.m1.1.1.3.2.cmml" xref="A1.p3.4.m1.1.1.3.2"><divide id="A1.p3.4.m1.1.1.3.2.1.cmml" xref="A1.p3.4.m1.1.1.3.2.1"></divide><ci id="A1.p3.4.m1.1.1.3.2.2.cmml" xref="A1.p3.4.m1.1.1.3.2.2">𝜔</ci><cn id="A1.p3.4.m1.1.1.3.2.3.cmml" type="integer" xref="A1.p3.4.m1.1.1.3.2.3">2</cn></apply><ci id="A1.p3.4.m1.1.1.3.3.cmml" xref="A1.p3.4.m1.1.1.3.3">𝜋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p3.4.m1.1c">f\equiv\omega/2\pi</annotation><annotation encoding="application/x-llamapun" id="A1.p3.4.m1.1d">italic_f ≡ italic_ω / 2 italic_π</annotation></semantics></math> and <math alttext="k_{x}" class="ltx_Math" display="inline" id="A1.p3.5.m2.1"><semantics id="A1.p3.5.m2.1a"><msub id="A1.p3.5.m2.1.1" xref="A1.p3.5.m2.1.1.cmml"><mi id="A1.p3.5.m2.1.1.2" xref="A1.p3.5.m2.1.1.2.cmml">k</mi><mi id="A1.p3.5.m2.1.1.3" xref="A1.p3.5.m2.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="A1.p3.5.m2.1b"><apply id="A1.p3.5.m2.1.1.cmml" xref="A1.p3.5.m2.1.1"><csymbol cd="ambiguous" id="A1.p3.5.m2.1.1.1.cmml" xref="A1.p3.5.m2.1.1">subscript</csymbol><ci id="A1.p3.5.m2.1.1.2.cmml" xref="A1.p3.5.m2.1.1.2">𝑘</ci><ci id="A1.p3.5.m2.1.1.3.cmml" xref="A1.p3.5.m2.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p3.5.m2.1c">k_{x}</annotation><annotation encoding="application/x-llamapun" id="A1.p3.5.m2.1d">italic_k start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> is the component of the wavevector perpendicular to the solar wind. Equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#A1.E6" title="In Appendix A Dispersion ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">A6</span></a>) used that the 1D density power that is measured by, e.g., Voyager 2 is related to the 3D in the above expression by</p> <table class="ltx_equation ltx_eqn_table" id="A1.E7"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="P_{e}^{\rm 1D}(f)=\frac{2}{v_{\rm sw,\odot}}\int_{0}^{\infty}\frac{dk}{(2\pi)}% kP_{e}\left(\sqrt{k^{2}+\frac{\omega^{2}}{v_{\rm sw}^{2}}}\right)," class="ltx_Math" display="block" id="A1.E7.m1.6"><semantics id="A1.E7.m1.6a"><mrow id="A1.E7.m1.6.6.1" xref="A1.E7.m1.6.6.1.1.cmml"><mrow id="A1.E7.m1.6.6.1.1" xref="A1.E7.m1.6.6.1.1.cmml"><mrow id="A1.E7.m1.6.6.1.1.2" 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end_ARG start_ARG italic_v start_POSTSUBSCRIPT roman_sw , ⊙ end_POSTSUBSCRIPT end_ARG ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT divide start_ARG italic_d italic_k end_ARG start_ARG ( 2 italic_π ) end_ARG italic_k italic_P start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ( square-root start_ARG italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + divide start_ARG italic_ω start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_v start_POSTSUBSCRIPT roman_sw end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG end_ARG ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(A7)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A1.p3.8">where the factor of <math alttext="2" class="ltx_Math" display="inline" id="A1.p3.6.m1.1"><semantics id="A1.p3.6.m1.1a"><mn id="A1.p3.6.m1.1.1" xref="A1.p3.6.m1.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="A1.p3.6.m1.1b"><cn id="A1.p3.6.m1.1.1.cmml" type="integer" xref="A1.p3.6.m1.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="A1.p3.6.m1.1c">2</annotation><annotation encoding="application/x-llamapun" id="A1.p3.6.m1.1d">2</annotation></semantics></math> is because this is the half bandwidth power. Since the integral that yields <math alttext="P_{e}^{\rm 1D}(f)" class="ltx_Math" display="inline" id="A1.p3.7.m2.1"><semantics id="A1.p3.7.m2.1a"><mrow id="A1.p3.7.m2.1.2" xref="A1.p3.7.m2.1.2.cmml"><msubsup id="A1.p3.7.m2.1.2.2" xref="A1.p3.7.m2.1.2.2.cmml"><mi id="A1.p3.7.m2.1.2.2.2.2" xref="A1.p3.7.m2.1.2.2.2.2.cmml">P</mi><mi id="A1.p3.7.m2.1.2.2.2.3" xref="A1.p3.7.m2.1.2.2.2.3.cmml">e</mi><mrow id="A1.p3.7.m2.1.2.2.3" xref="A1.p3.7.m2.1.2.2.3.cmml"><mn id="A1.p3.7.m2.1.2.2.3.2" xref="A1.p3.7.m2.1.2.2.3.2.cmml">1</mn><mo id="A1.p3.7.m2.1.2.2.3.1" xref="A1.p3.7.m2.1.2.2.3.1.cmml">⁢</mo><mi id="A1.p3.7.m2.1.2.2.3.3" mathvariant="normal" xref="A1.p3.7.m2.1.2.2.3.3.cmml">D</mi></mrow></msubsup><mo id="A1.p3.7.m2.1.2.1" xref="A1.p3.7.m2.1.2.1.cmml">⁢</mo><mrow id="A1.p3.7.m2.1.2.3.2" xref="A1.p3.7.m2.1.2.cmml"><mo id="A1.p3.7.m2.1.2.3.2.1" stretchy="false" xref="A1.p3.7.m2.1.2.cmml">(</mo><mi id="A1.p3.7.m2.1.1" xref="A1.p3.7.m2.1.1.cmml">f</mi><mo id="A1.p3.7.m2.1.2.3.2.2" stretchy="false" xref="A1.p3.7.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.p3.7.m2.1b"><apply id="A1.p3.7.m2.1.2.cmml" xref="A1.p3.7.m2.1.2"><times id="A1.p3.7.m2.1.2.1.cmml" xref="A1.p3.7.m2.1.2.1"></times><apply id="A1.p3.7.m2.1.2.2.cmml" xref="A1.p3.7.m2.1.2.2"><csymbol cd="ambiguous" id="A1.p3.7.m2.1.2.2.1.cmml" xref="A1.p3.7.m2.1.2.2">superscript</csymbol><apply id="A1.p3.7.m2.1.2.2.2.cmml" xref="A1.p3.7.m2.1.2.2"><csymbol cd="ambiguous" id="A1.p3.7.m2.1.2.2.2.1.cmml" xref="A1.p3.7.m2.1.2.2">subscript</csymbol><ci id="A1.p3.7.m2.1.2.2.2.2.cmml" xref="A1.p3.7.m2.1.2.2.2.2">𝑃</ci><ci id="A1.p3.7.m2.1.2.2.2.3.cmml" xref="A1.p3.7.m2.1.2.2.2.3">𝑒</ci></apply><apply id="A1.p3.7.m2.1.2.2.3.cmml" xref="A1.p3.7.m2.1.2.2.3"><times id="A1.p3.7.m2.1.2.2.3.1.cmml" xref="A1.p3.7.m2.1.2.2.3.1"></times><cn id="A1.p3.7.m2.1.2.2.3.2.cmml" type="integer" xref="A1.p3.7.m2.1.2.2.3.2">1</cn><ci id="A1.p3.7.m2.1.2.2.3.3.cmml" xref="A1.p3.7.m2.1.2.2.3.3">D</ci></apply></apply><ci id="A1.p3.7.m2.1.1.cmml" xref="A1.p3.7.m2.1.1">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p3.7.m2.1c">P_{e}^{\rm 1D}(f)</annotation><annotation encoding="application/x-llamapun" id="A1.p3.7.m2.1d">italic_P start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 roman_D end_POSTSUPERSCRIPT ( italic_f )</annotation></semantics></math> is somewhat different than that in equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#A1.E5" title="In Appendix A Dispersion ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">A5</span></a>), to determine the numerical coefficient in equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#A1.E6" title="In Appendix A Dispersion ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">A6</span></a>) we assumed the scaling <math alttext="P_{e}^{\rm 1D}\propto f^{-1.5}" class="ltx_Math" display="inline" id="A1.p3.8.m3.1"><semantics id="A1.p3.8.m3.1a"><mrow id="A1.p3.8.m3.1.1" xref="A1.p3.8.m3.1.1.cmml"><msubsup id="A1.p3.8.m3.1.1.2" xref="A1.p3.8.m3.1.1.2.cmml"><mi id="A1.p3.8.m3.1.1.2.2.2" xref="A1.p3.8.m3.1.1.2.2.2.cmml">P</mi><mi id="A1.p3.8.m3.1.1.2.2.3" xref="A1.p3.8.m3.1.1.2.2.3.cmml">e</mi><mrow id="A1.p3.8.m3.1.1.2.3" xref="A1.p3.8.m3.1.1.2.3.cmml"><mn id="A1.p3.8.m3.1.1.2.3.2" xref="A1.p3.8.m3.1.1.2.3.2.cmml">1</mn><mo id="A1.p3.8.m3.1.1.2.3.1" xref="A1.p3.8.m3.1.1.2.3.1.cmml">⁢</mo><mi id="A1.p3.8.m3.1.1.2.3.3" mathvariant="normal" xref="A1.p3.8.m3.1.1.2.3.3.cmml">D</mi></mrow></msubsup><mo id="A1.p3.8.m3.1.1.1" xref="A1.p3.8.m3.1.1.1.cmml">∝</mo><msup id="A1.p3.8.m3.1.1.3" xref="A1.p3.8.m3.1.1.3.cmml"><mi id="A1.p3.8.m3.1.1.3.2" xref="A1.p3.8.m3.1.1.3.2.cmml">f</mi><mrow id="A1.p3.8.m3.1.1.3.3" 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xref="A1.p3.8.m3.1.1.2.3.1"></times><cn id="A1.p3.8.m3.1.1.2.3.2.cmml" type="integer" xref="A1.p3.8.m3.1.1.2.3.2">1</cn><ci id="A1.p3.8.m3.1.1.2.3.3.cmml" xref="A1.p3.8.m3.1.1.2.3.3">D</ci></apply></apply><apply id="A1.p3.8.m3.1.1.3.cmml" xref="A1.p3.8.m3.1.1.3"><csymbol cd="ambiguous" id="A1.p3.8.m3.1.1.3.1.cmml" xref="A1.p3.8.m3.1.1.3">superscript</csymbol><ci id="A1.p3.8.m3.1.1.3.2.cmml" xref="A1.p3.8.m3.1.1.3.2">𝑓</ci><apply id="A1.p3.8.m3.1.1.3.3.cmml" xref="A1.p3.8.m3.1.1.3.3"><minus id="A1.p3.8.m3.1.1.3.3.1.cmml" xref="A1.p3.8.m3.1.1.3.3"></minus><cn id="A1.p3.8.m3.1.1.3.3.2.cmml" type="float" xref="A1.p3.8.m3.1.1.3.3.2">1.5</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p3.8.m3.1c">P_{e}^{\rm 1D}\propto f^{-1.5}</annotation><annotation encoding="application/x-llamapun" id="A1.p3.8.m3.1d">italic_P start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 roman_D end_POSTSUPERSCRIPT ∝ italic_f start_POSTSUPERSCRIPT - 1.5 end_POSTSUPERSCRIPT</annotation></semantics></math> as motivated in § <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.SS3.SSS2" title="2.3.2 Solar wind drag ‣ 2.3 Acceleration noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">2.3.2</span></a>. However, we find this coefficient depends weakly on what power-law index is assumed over relevant indices.<span class="ltx_note ltx_role_footnote" id="footnote15"><sup class="ltx_note_mark">15</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">15</sup><span class="ltx_tag ltx_tag_note">15</span>The coefficient changes from <math alttext="1.5" class="ltx_Math" display="inline" id="footnote15.m1.1"><semantics id="footnote15.m1.1b"><mn id="footnote15.m1.1.1" xref="footnote15.m1.1.1.cmml">1.5</mn><annotation-xml encoding="MathML-Content" id="footnote15.m1.1c"><cn id="footnote15.m1.1.1.cmml" type="float" xref="footnote15.m1.1.1">1.5</cn></annotation-xml><annotation encoding="application/x-tex" id="footnote15.m1.1d">1.5</annotation><annotation encoding="application/x-llamapun" id="footnote15.m1.1e">1.5</annotation></semantics></math> to <math alttext="1.6" class="ltx_Math" display="inline" id="footnote15.m2.1"><semantics id="footnote15.m2.1b"><mn id="footnote15.m2.1.1" xref="footnote15.m2.1.1.cmml">1.6</mn><annotation-xml encoding="MathML-Content" id="footnote15.m2.1c"><cn id="footnote15.m2.1.1.cmml" type="float" xref="footnote15.m2.1.1">1.6</cn></annotation-xml><annotation encoding="application/x-tex" id="footnote15.m2.1d">1.6</annotation><annotation encoding="application/x-llamapun" id="footnote15.m2.1e">1.6</annotation></semantics></math> if we use <math alttext="P_{e}^{\rm 1D}\propto f^{-2}" class="ltx_Math" display="inline" id="footnote15.m3.1"><semantics id="footnote15.m3.1b"><mrow id="footnote15.m3.1.1" xref="footnote15.m3.1.1.cmml"><msubsup id="footnote15.m3.1.1.2" xref="footnote15.m3.1.1.2.cmml"><mi id="footnote15.m3.1.1.2.2.2" xref="footnote15.m3.1.1.2.2.2.cmml">P</mi><mi id="footnote15.m3.1.1.2.2.3" xref="footnote15.m3.1.1.2.2.3.cmml">e</mi><mrow id="footnote15.m3.1.1.2.3" xref="footnote15.m3.1.1.2.3.cmml"><mn id="footnote15.m3.1.1.2.3.2" xref="footnote15.m3.1.1.2.3.2.cmml">1</mn><mo id="footnote15.m3.1.1.2.3.1" xref="footnote15.m3.1.1.2.3.1.cmml">⁢</mo><mi id="footnote15.m3.1.1.2.3.3" mathvariant="normal" xref="footnote15.m3.1.1.2.3.3.cmml">D</mi></mrow></msubsup><mo id="footnote15.m3.1.1.1" xref="footnote15.m3.1.1.1.cmml">∝</mo><msup id="footnote15.m3.1.1.3" xref="footnote15.m3.1.1.3.cmml"><mi id="footnote15.m3.1.1.3.2" xref="footnote15.m3.1.1.3.2.cmml">f</mi><mrow id="footnote15.m3.1.1.3.3" xref="footnote15.m3.1.1.3.3.cmml"><mo id="footnote15.m3.1.1.3.3b" xref="footnote15.m3.1.1.3.3.cmml">−</mo><mn id="footnote15.m3.1.1.3.3.2" xref="footnote15.m3.1.1.3.3.2.cmml">2</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="footnote15.m3.1c"><apply id="footnote15.m3.1.1.cmml" xref="footnote15.m3.1.1"><csymbol cd="latexml" id="footnote15.m3.1.1.1.cmml" xref="footnote15.m3.1.1.1">proportional-to</csymbol><apply id="footnote15.m3.1.1.2.cmml" xref="footnote15.m3.1.1.2"><csymbol cd="ambiguous" id="footnote15.m3.1.1.2.1.cmml" xref="footnote15.m3.1.1.2">superscript</csymbol><apply id="footnote15.m3.1.1.2.2.cmml" xref="footnote15.m3.1.1.2"><csymbol cd="ambiguous" id="footnote15.m3.1.1.2.2.1.cmml" xref="footnote15.m3.1.1.2">subscript</csymbol><ci id="footnote15.m3.1.1.2.2.2.cmml" xref="footnote15.m3.1.1.2.2.2">𝑃</ci><ci id="footnote15.m3.1.1.2.2.3.cmml" xref="footnote15.m3.1.1.2.2.3">𝑒</ci></apply><apply id="footnote15.m3.1.1.2.3.cmml" xref="footnote15.m3.1.1.2.3"><times id="footnote15.m3.1.1.2.3.1.cmml" xref="footnote15.m3.1.1.2.3.1"></times><cn id="footnote15.m3.1.1.2.3.2.cmml" type="integer" xref="footnote15.m3.1.1.2.3.2">1</cn><ci id="footnote15.m3.1.1.2.3.3.cmml" xref="footnote15.m3.1.1.2.3.3">D</ci></apply></apply><apply id="footnote15.m3.1.1.3.cmml" xref="footnote15.m3.1.1.3"><csymbol cd="ambiguous" id="footnote15.m3.1.1.3.1.cmml" xref="footnote15.m3.1.1.3">superscript</csymbol><ci id="footnote15.m3.1.1.3.2.cmml" xref="footnote15.m3.1.1.3.2">𝑓</ci><apply id="footnote15.m3.1.1.3.3.cmml" xref="footnote15.m3.1.1.3.3"><minus id="footnote15.m3.1.1.3.3.1.cmml" xref="footnote15.m3.1.1.3.3"></minus><cn id="footnote15.m3.1.1.3.3.2.cmml" type="integer" xref="footnote15.m3.1.1.3.3.2">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote15.m3.1d">P_{e}^{\rm 1D}\propto f^{-2}</annotation><annotation encoding="application/x-llamapun" id="footnote15.m3.1e">italic_P start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 roman_D end_POSTSUPERSCRIPT ∝ italic_f start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT</annotation></semantics></math>, which may be more appropriate at <math alttext="f\gtrsim 10^{-5}" class="ltx_Math" display="inline" id="footnote15.m4.1"><semantics id="footnote15.m4.1b"><mrow id="footnote15.m4.1.1" xref="footnote15.m4.1.1.cmml"><mi id="footnote15.m4.1.1.2" xref="footnote15.m4.1.1.2.cmml">f</mi><mo id="footnote15.m4.1.1.1" xref="footnote15.m4.1.1.1.cmml">≳</mo><msup id="footnote15.m4.1.1.3" xref="footnote15.m4.1.1.3.cmml"><mn id="footnote15.m4.1.1.3.2" xref="footnote15.m4.1.1.3.2.cmml">10</mn><mrow id="footnote15.m4.1.1.3.3" xref="footnote15.m4.1.1.3.3.cmml"><mo id="footnote15.m4.1.1.3.3b" xref="footnote15.m4.1.1.3.3.cmml">−</mo><mn id="footnote15.m4.1.1.3.3.2" xref="footnote15.m4.1.1.3.3.2.cmml">5</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="footnote15.m4.1c"><apply id="footnote15.m4.1.1.cmml" xref="footnote15.m4.1.1"><csymbol cd="latexml" id="footnote15.m4.1.1.1.cmml" xref="footnote15.m4.1.1.1">greater-than-or-equivalent-to</csymbol><ci id="footnote15.m4.1.1.2.cmml" xref="footnote15.m4.1.1.2">𝑓</ci><apply id="footnote15.m4.1.1.3.cmml" xref="footnote15.m4.1.1.3"><csymbol cd="ambiguous" id="footnote15.m4.1.1.3.1.cmml" xref="footnote15.m4.1.1.3">superscript</csymbol><cn id="footnote15.m4.1.1.3.2.cmml" type="integer" xref="footnote15.m4.1.1.3.2">10</cn><apply id="footnote15.m4.1.1.3.3.cmml" xref="footnote15.m4.1.1.3.3"><minus id="footnote15.m4.1.1.3.3.1.cmml" xref="footnote15.m4.1.1.3.3"></minus><cn id="footnote15.m4.1.1.3.3.2.cmml" type="integer" xref="footnote15.m4.1.1.3.3.2">5</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote15.m4.1d">f\gtrsim 10^{-5}</annotation><annotation encoding="application/x-llamapun" id="footnote15.m4.1e">italic_f ≳ 10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT</annotation></semantics></math>Hz.</span></span></span></p> </div> <div class="ltx_para" id="A1.p4"> <p class="ltx_p" id="A1.p4.3">We can convert this to fluctuations in phase as</p> <table class="ltx_equation ltx_eqn_table" id="A1.E8"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\widetilde{\phi}_{\rm disp,rms}=\frac{c}{2\pi\lambda}\sqrt{4P_{\Delta\tau_{d}}% (\omega)}," class="ltx_Math" display="block" id="A1.E8.m1.4"><semantics id="A1.E8.m1.4a"><mrow id="A1.E8.m1.4.4.1" xref="A1.E8.m1.4.4.1.1.cmml"><mrow 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xref="A1.E8.m1.4.4.1.1.3.2.3.2.cmml">2</mn><mo id="A1.E8.m1.4.4.1.1.3.2.3.1" xref="A1.E8.m1.4.4.1.1.3.2.3.1.cmml">⁢</mo><mi id="A1.E8.m1.4.4.1.1.3.2.3.3" xref="A1.E8.m1.4.4.1.1.3.2.3.3.cmml">π</mi><mo id="A1.E8.m1.4.4.1.1.3.2.3.1a" xref="A1.E8.m1.4.4.1.1.3.2.3.1.cmml">⁢</mo><mi id="A1.E8.m1.4.4.1.1.3.2.3.4" xref="A1.E8.m1.4.4.1.1.3.2.3.4.cmml">λ</mi></mrow></mfrac><mo id="A1.E8.m1.4.4.1.1.3.1" xref="A1.E8.m1.4.4.1.1.3.1.cmml">⁢</mo><msqrt id="A1.E8.m1.3.3" xref="A1.E8.m1.3.3.cmml"><mrow id="A1.E8.m1.3.3.1" xref="A1.E8.m1.3.3.1.cmml"><mn id="A1.E8.m1.3.3.1.3" xref="A1.E8.m1.3.3.1.3.cmml">4</mn><mo id="A1.E8.m1.3.3.1.2" xref="A1.E8.m1.3.3.1.2.cmml">⁢</mo><msub id="A1.E8.m1.3.3.1.4" xref="A1.E8.m1.3.3.1.4.cmml"><mi id="A1.E8.m1.3.3.1.4.2" xref="A1.E8.m1.3.3.1.4.2.cmml">P</mi><mrow id="A1.E8.m1.3.3.1.4.3" xref="A1.E8.m1.3.3.1.4.3.cmml"><mi id="A1.E8.m1.3.3.1.4.3.2" mathvariant="normal" xref="A1.E8.m1.3.3.1.4.3.2.cmml">Δ</mi><mo id="A1.E8.m1.3.3.1.4.3.1" 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id="A1.E8.m1.4.4.1.1.2.1.cmml" xref="A1.E8.m1.4.4.1.1.2">subscript</csymbol><apply id="A1.E8.m1.4.4.1.1.2.2.cmml" xref="A1.E8.m1.4.4.1.1.2.2"><ci id="A1.E8.m1.4.4.1.1.2.2.1.cmml" xref="A1.E8.m1.4.4.1.1.2.2.1">~</ci><ci id="A1.E8.m1.4.4.1.1.2.2.2.cmml" xref="A1.E8.m1.4.4.1.1.2.2.2">italic-ϕ</ci></apply><list id="A1.E8.m1.2.2.2.3.cmml" xref="A1.E8.m1.2.2.2.4"><ci id="A1.E8.m1.1.1.1.1.cmml" xref="A1.E8.m1.1.1.1.1">disp</ci><ci id="A1.E8.m1.2.2.2.2.cmml" xref="A1.E8.m1.2.2.2.2">rms</ci></list></apply><apply id="A1.E8.m1.4.4.1.1.3.cmml" xref="A1.E8.m1.4.4.1.1.3"><times id="A1.E8.m1.4.4.1.1.3.1.cmml" xref="A1.E8.m1.4.4.1.1.3.1"></times><apply id="A1.E8.m1.4.4.1.1.3.2.cmml" xref="A1.E8.m1.4.4.1.1.3.2"><divide id="A1.E8.m1.4.4.1.1.3.2.1.cmml" xref="A1.E8.m1.4.4.1.1.3.2"></divide><ci id="A1.E8.m1.4.4.1.1.3.2.2.cmml" xref="A1.E8.m1.4.4.1.1.3.2.2">𝑐</ci><apply id="A1.E8.m1.4.4.1.1.3.2.3.cmml" xref="A1.E8.m1.4.4.1.1.3.2.3"><times id="A1.E8.m1.4.4.1.1.3.2.3.1.cmml" 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xref="A1.E8.m1.3.3.1.4.3.2">Δ</ci><apply id="A1.E8.m1.3.3.1.4.3.3.cmml" xref="A1.E8.m1.3.3.1.4.3.3"><csymbol cd="ambiguous" id="A1.E8.m1.3.3.1.4.3.3.1.cmml" xref="A1.E8.m1.3.3.1.4.3.3">subscript</csymbol><ci id="A1.E8.m1.3.3.1.4.3.3.2.cmml" xref="A1.E8.m1.3.3.1.4.3.3.2">𝜏</ci><ci id="A1.E8.m1.3.3.1.4.3.3.3.cmml" xref="A1.E8.m1.3.3.1.4.3.3.3">𝑑</ci></apply></apply></apply><ci id="A1.E8.m1.3.3.1.1.cmml" xref="A1.E8.m1.3.3.1.1">𝜔</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E8.m1.4c">\widetilde{\phi}_{\rm disp,rms}=\frac{c}{2\pi\lambda}\sqrt{4P_{\Delta\tau_{d}}% (\omega)},</annotation><annotation encoding="application/x-llamapun" id="A1.E8.m1.4d">over~ start_ARG italic_ϕ end_ARG start_POSTSUBSCRIPT roman_disp , roman_rms end_POSTSUBSCRIPT = divide start_ARG italic_c end_ARG start_ARG 2 italic_π italic_λ end_ARG square-root start_ARG 4 italic_P start_POSTSUBSCRIPT roman_Δ italic_τ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_ω ) end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(A8)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A1.p4.2">where the four is because the light travels out and back for each arm, probing essentially the same electron field for <math alttext="f\lesssim c/(2L)" class="ltx_Math" display="inline" id="A1.p4.1.m1.1"><semantics id="A1.p4.1.m1.1a"><mrow id="A1.p4.1.m1.1.1" xref="A1.p4.1.m1.1.1.cmml"><mi id="A1.p4.1.m1.1.1.3" xref="A1.p4.1.m1.1.1.3.cmml">f</mi><mo id="A1.p4.1.m1.1.1.2" xref="A1.p4.1.m1.1.1.2.cmml">≲</mo><mrow id="A1.p4.1.m1.1.1.1" xref="A1.p4.1.m1.1.1.1.cmml"><mi id="A1.p4.1.m1.1.1.1.3" xref="A1.p4.1.m1.1.1.1.3.cmml">c</mi><mo id="A1.p4.1.m1.1.1.1.2" xref="A1.p4.1.m1.1.1.1.2.cmml">/</mo><mrow id="A1.p4.1.m1.1.1.1.1.1" xref="A1.p4.1.m1.1.1.1.1.1.1.cmml"><mo id="A1.p4.1.m1.1.1.1.1.1.2" stretchy="false" xref="A1.p4.1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="A1.p4.1.m1.1.1.1.1.1.1" xref="A1.p4.1.m1.1.1.1.1.1.1.cmml"><mn id="A1.p4.1.m1.1.1.1.1.1.1.2" xref="A1.p4.1.m1.1.1.1.1.1.1.2.cmml">2</mn><mo id="A1.p4.1.m1.1.1.1.1.1.1.1" xref="A1.p4.1.m1.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="A1.p4.1.m1.1.1.1.1.1.1.3" xref="A1.p4.1.m1.1.1.1.1.1.1.3.cmml">L</mi></mrow><mo id="A1.p4.1.m1.1.1.1.1.1.3" stretchy="false" xref="A1.p4.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.p4.1.m1.1b"><apply id="A1.p4.1.m1.1.1.cmml" xref="A1.p4.1.m1.1.1"><csymbol cd="latexml" id="A1.p4.1.m1.1.1.2.cmml" xref="A1.p4.1.m1.1.1.2">less-than-or-similar-to</csymbol><ci id="A1.p4.1.m1.1.1.3.cmml" xref="A1.p4.1.m1.1.1.3">𝑓</ci><apply id="A1.p4.1.m1.1.1.1.cmml" xref="A1.p4.1.m1.1.1.1"><divide id="A1.p4.1.m1.1.1.1.2.cmml" xref="A1.p4.1.m1.1.1.1.2"></divide><ci id="A1.p4.1.m1.1.1.1.3.cmml" xref="A1.p4.1.m1.1.1.1.3">𝑐</ci><apply id="A1.p4.1.m1.1.1.1.1.1.1.cmml" xref="A1.p4.1.m1.1.1.1.1.1"><times id="A1.p4.1.m1.1.1.1.1.1.1.1.cmml" xref="A1.p4.1.m1.1.1.1.1.1.1.1"></times><cn id="A1.p4.1.m1.1.1.1.1.1.1.2.cmml" type="integer" xref="A1.p4.1.m1.1.1.1.1.1.1.2">2</cn><ci id="A1.p4.1.m1.1.1.1.1.1.1.3.cmml" xref="A1.p4.1.m1.1.1.1.1.1.1.3">𝐿</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p4.1.m1.1c">f\lesssim c/(2L)</annotation><annotation encoding="application/x-llamapun" id="A1.p4.1.m1.1d">italic_f ≲ italic_c / ( 2 italic_L )</annotation></semantics></math>. At higher frequencies, the return path will see different electrons such that we should replace <math alttext="4\rightarrow 2" class="ltx_Math" display="inline" id="A1.p4.2.m2.1"><semantics id="A1.p4.2.m2.1a"><mrow id="A1.p4.2.m2.1.1" xref="A1.p4.2.m2.1.1.cmml"><mn id="A1.p4.2.m2.1.1.2" xref="A1.p4.2.m2.1.1.2.cmml">4</mn><mo id="A1.p4.2.m2.1.1.1" stretchy="false" xref="A1.p4.2.m2.1.1.1.cmml">→</mo><mn id="A1.p4.2.m2.1.1.3" xref="A1.p4.2.m2.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.p4.2.m2.1b"><apply id="A1.p4.2.m2.1.1.cmml" xref="A1.p4.2.m2.1.1"><ci id="A1.p4.2.m2.1.1.1.cmml" xref="A1.p4.2.m2.1.1.1">→</ci><cn id="A1.p4.2.m2.1.1.2.cmml" type="integer" xref="A1.p4.2.m2.1.1.2">4</cn><cn id="A1.p4.2.m2.1.1.3.cmml" type="integer" xref="A1.p4.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p4.2.m2.1c">4\rightarrow 2</annotation><annotation encoding="application/x-llamapun" id="A1.p4.2.m2.1d">4 → 2</annotation></semantics></math> – a correction which we ignore.</p> </div> <div class="ltx_para" id="A1.p5"> <p class="ltx_p" id="A1.p5.4">Converting the phase error to the error on the gravitational wave strain yields</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="A3.EGx10"> <tbody id="A1.E9"> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\widetilde{h}_{\rm disp,rms}" class="ltx_Math" display="inline" id="A1.Ex3.m1.2"><semantics id="A1.Ex3.m1.2a"><msub id="A1.Ex3.m1.2.3" xref="A1.Ex3.m1.2.3.cmml"><mover accent="true" id="A1.Ex3.m1.2.3.2" xref="A1.Ex3.m1.2.3.2.cmml"><mi id="A1.Ex3.m1.2.3.2.2" xref="A1.Ex3.m1.2.3.2.2.cmml">h</mi><mo id="A1.Ex3.m1.2.3.2.1" xref="A1.Ex3.m1.2.3.2.1.cmml">~</mo></mover><mrow id="A1.Ex3.m1.2.2.2.4" xref="A1.Ex3.m1.2.2.2.3.cmml"><mi id="A1.Ex3.m1.1.1.1.1" xref="A1.Ex3.m1.1.1.1.1.cmml">disp</mi><mo id="A1.Ex3.m1.2.2.2.4.1" xref="A1.Ex3.m1.2.2.2.3.cmml">,</mo><mi id="A1.Ex3.m1.2.2.2.2" xref="A1.Ex3.m1.2.2.2.2.cmml">rms</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A1.Ex3.m1.2b"><apply id="A1.Ex3.m1.2.3.cmml" xref="A1.Ex3.m1.2.3"><csymbol cd="ambiguous" id="A1.Ex3.m1.2.3.1.cmml" xref="A1.Ex3.m1.2.3">subscript</csymbol><apply id="A1.Ex3.m1.2.3.2.cmml" xref="A1.Ex3.m1.2.3.2"><ci id="A1.Ex3.m1.2.3.2.1.cmml" xref="A1.Ex3.m1.2.3.2.1">~</ci><ci id="A1.Ex3.m1.2.3.2.2.cmml" xref="A1.Ex3.m1.2.3.2.2">ℎ</ci></apply><list id="A1.Ex3.m1.2.2.2.3.cmml" xref="A1.Ex3.m1.2.2.2.4"><ci id="A1.Ex3.m1.1.1.1.1.cmml" xref="A1.Ex3.m1.1.1.1.1">disp</ci><ci id="A1.Ex3.m1.2.2.2.2.cmml" xref="A1.Ex3.m1.2.2.2.2">rms</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex3.m1.2c">\displaystyle\widetilde{h}_{\rm disp,rms}</annotation><annotation encoding="application/x-llamapun" id="A1.Ex3.m1.2d">over~ start_ARG italic_h end_ARG start_POSTSUBSCRIPT roman_disp , roman_rms end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="A1.Ex3.m2.1"><semantics id="A1.Ex3.m2.1a"><mo id="A1.Ex3.m2.1.1" xref="A1.Ex3.m2.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="A1.Ex3.m2.1b"><eq id="A1.Ex3.m2.1.1.cmml" xref="A1.Ex3.m2.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex3.m2.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="A1.Ex3.m2.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle 4\times 10^{-15}\,\text{Hz}^{-1/2}~{}R(f)^{-1/2}\left(\frac{L}{3% 0\,{\rm AU}}\right)^{-1/2}\left(\frac{r}{30\,{\rm AU}}\right)^{-1.5}" class="ltx_Math" display="inline" id="A1.Ex3.m3.3"><semantics id="A1.Ex3.m3.3a"><mrow id="A1.Ex3.m3.3.4" xref="A1.Ex3.m3.3.4.cmml"><mrow 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encoding="application/x-llamapun" id="A1.Ex3.m3.3d">4 × 10 start_POSTSUPERSCRIPT - 15 end_POSTSUPERSCRIPT Hz start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT italic_R ( italic_f ) start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT ( divide start_ARG italic_L end_ARG start_ARG 30 roman_AU end_ARG ) start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT ( divide start_ARG italic_r end_ARG start_ARG 30 roman_AU end_ARG ) start_POSTSUPERSCRIPT - 1.5 end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="2"><span class="ltx_tag ltx_tag_equation ltx_align_right">(A9)</span></td> </tr> <tr class="ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\times\left(\frac{f}{10^{-5}{\rm 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encoding="application/x-tex" id="A1.E9.m1.6c">\displaystyle\times\left(\frac{f}{10^{-5}{\rm Hz}}\right)^{-(\beta+1)/2}\left(% \frac{P_{e,0}^{\rm 1D}}{{10^{11.1}{\rm m}^{-6}{\rm Hz}^{-1}}}\right)^{1/2}% \left(\frac{\lambda}{1~{}{\rm cm}}\right)^{2},</annotation><annotation encoding="application/x-llamapun" id="A1.E9.m1.6d">× ( divide start_ARG italic_f end_ARG start_ARG 10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT roman_Hz end_ARG ) start_POSTSUPERSCRIPT - ( italic_β + 1 ) / 2 end_POSTSUPERSCRIPT ( divide start_ARG italic_P start_POSTSUBSCRIPT italic_e , 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 roman_D end_POSTSUPERSCRIPT end_ARG start_ARG 10 start_POSTSUPERSCRIPT 11.1 end_POSTSUPERSCRIPT roman_m start_POSTSUPERSCRIPT - 6 end_POSTSUPERSCRIPT roman_Hz start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT end_ARG ) start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT ( divide start_ARG italic_λ end_ARG start_ARG 1 roman_cm end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </tbody> </table> <p class="ltx_p" id="A1.p5.3">where we are using the form of the 1D solar wind density power given by equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.E14" title="In 2.3.2 Solar wind drag ‣ 2.3 Acceleration noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">14</span></a>), noting that <math alttext="\beta\approx 2" class="ltx_Math" display="inline" id="A1.p5.1.m1.1"><semantics id="A1.p5.1.m1.1a"><mrow id="A1.p5.1.m1.1.1" xref="A1.p5.1.m1.1.1.cmml"><mi id="A1.p5.1.m1.1.1.2" xref="A1.p5.1.m1.1.1.2.cmml">β</mi><mo id="A1.p5.1.m1.1.1.1" xref="A1.p5.1.m1.1.1.1.cmml">≈</mo><mn id="A1.p5.1.m1.1.1.3" xref="A1.p5.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.p5.1.m1.1b"><apply id="A1.p5.1.m1.1.1.cmml" xref="A1.p5.1.m1.1.1"><approx id="A1.p5.1.m1.1.1.1.cmml" xref="A1.p5.1.m1.1.1.1"></approx><ci id="A1.p5.1.m1.1.1.2.cmml" xref="A1.p5.1.m1.1.1.2">𝛽</ci><cn id="A1.p5.1.m1.1.1.3.cmml" type="integer" xref="A1.p5.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p5.1.m1.1c">\beta\approx 2</annotation><annotation encoding="application/x-llamapun" id="A1.p5.1.m1.1d">italic_β ≈ 2</annotation></semantics></math> for <math alttext="f<10^{-5}~{}" class="ltx_Math" display="inline" id="A1.p5.2.m2.1"><semantics id="A1.p5.2.m2.1a"><mrow id="A1.p5.2.m2.1.1" xref="A1.p5.2.m2.1.1.cmml"><mi id="A1.p5.2.m2.1.1.2" xref="A1.p5.2.m2.1.1.2.cmml">f</mi><mo id="A1.p5.2.m2.1.1.1" xref="A1.p5.2.m2.1.1.1.cmml"><</mo><msup id="A1.p5.2.m2.1.1.3" xref="A1.p5.2.m2.1.1.3.cmml"><mn id="A1.p5.2.m2.1.1.3.2" xref="A1.p5.2.m2.1.1.3.2.cmml">10</mn><mrow id="A1.p5.2.m2.1.1.3.3" xref="A1.p5.2.m2.1.1.3.3.cmml"><mo id="A1.p5.2.m2.1.1.3.3a" xref="A1.p5.2.m2.1.1.3.3.cmml">−</mo><mn id="A1.p5.2.m2.1.1.3.3.2" xref="A1.p5.2.m2.1.1.3.3.2.cmml">5</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="A1.p5.2.m2.1b"><apply id="A1.p5.2.m2.1.1.cmml" xref="A1.p5.2.m2.1.1"><lt id="A1.p5.2.m2.1.1.1.cmml" xref="A1.p5.2.m2.1.1.1"></lt><ci id="A1.p5.2.m2.1.1.2.cmml" xref="A1.p5.2.m2.1.1.2">𝑓</ci><apply id="A1.p5.2.m2.1.1.3.cmml" xref="A1.p5.2.m2.1.1.3"><csymbol cd="ambiguous" id="A1.p5.2.m2.1.1.3.1.cmml" xref="A1.p5.2.m2.1.1.3">superscript</csymbol><cn id="A1.p5.2.m2.1.1.3.2.cmml" type="integer" xref="A1.p5.2.m2.1.1.3.2">10</cn><apply id="A1.p5.2.m2.1.1.3.3.cmml" xref="A1.p5.2.m2.1.1.3.3"><minus id="A1.p5.2.m2.1.1.3.3.1.cmml" xref="A1.p5.2.m2.1.1.3.3"></minus><cn id="A1.p5.2.m2.1.1.3.3.2.cmml" type="integer" xref="A1.p5.2.m2.1.1.3.3.2">5</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p5.2.m2.1c">f<10^{-5}~{}</annotation><annotation encoding="application/x-llamapun" id="A1.p5.2.m2.1d">italic_f < 10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT</annotation></semantics></math>Hz (§ <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.SS3.SSS2" title="2.3.2 Solar wind drag ‣ 2.3 Acceleration noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">2.3.2</span></a>). We include the transfer function <math alttext="R(f)" class="ltx_Math" display="inline" id="A1.p5.3.m3.1"><semantics id="A1.p5.3.m3.1a"><mrow id="A1.p5.3.m3.1.2" xref="A1.p5.3.m3.1.2.cmml"><mi id="A1.p5.3.m3.1.2.2" xref="A1.p5.3.m3.1.2.2.cmml">R</mi><mo id="A1.p5.3.m3.1.2.1" xref="A1.p5.3.m3.1.2.1.cmml">⁢</mo><mrow id="A1.p5.3.m3.1.2.3.2" xref="A1.p5.3.m3.1.2.cmml"><mo id="A1.p5.3.m3.1.2.3.2.1" stretchy="false" xref="A1.p5.3.m3.1.2.cmml">(</mo><mi id="A1.p5.3.m3.1.1" xref="A1.p5.3.m3.1.1.cmml">f</mi><mo id="A1.p5.3.m3.1.2.3.2.2" stretchy="false" xref="A1.p5.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.p5.3.m3.1b"><apply id="A1.p5.3.m3.1.2.cmml" xref="A1.p5.3.m3.1.2"><times id="A1.p5.3.m3.1.2.1.cmml" xref="A1.p5.3.m3.1.2.1"></times><ci id="A1.p5.3.m3.1.2.2.cmml" xref="A1.p5.3.m3.1.2.2">𝑅</ci><ci id="A1.p5.3.m3.1.1.cmml" xref="A1.p5.3.m3.1.1">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p5.3.m3.1c">R(f)</annotation><annotation encoding="application/x-llamapun" id="A1.p5.3.m3.1d">italic_R ( italic_f )</annotation></semantics></math> to convert from the long wavelength limit (cf. eqn. <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S3.E18" title="In 3 Strain power sensitivity ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">18</span></a>).</p> </div> <div class="ltx_para" id="A1.p6"> <p class="ltx_p" id="A1.p6.3">Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#A1.F8" title="Figure 8 ‣ Appendix A Dispersion ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">8</span></a> shows the effect of dispersion for an interferometer operating at <math alttext="\lambda=1\," class="ltx_Math" display="inline" id="A1.p6.1.m1.1"><semantics id="A1.p6.1.m1.1a"><mrow id="A1.p6.1.m1.1.1" xref="A1.p6.1.m1.1.1.cmml"><mi id="A1.p6.1.m1.1.1.2" xref="A1.p6.1.m1.1.1.2.cmml">λ</mi><mo id="A1.p6.1.m1.1.1.1" xref="A1.p6.1.m1.1.1.1.cmml">=</mo><mn id="A1.p6.1.m1.1.1.3" xref="A1.p6.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.p6.1.m1.1b"><apply id="A1.p6.1.m1.1.1.cmml" xref="A1.p6.1.m1.1.1"><eq id="A1.p6.1.m1.1.1.1.cmml" xref="A1.p6.1.m1.1.1.1"></eq><ci id="A1.p6.1.m1.1.1.2.cmml" xref="A1.p6.1.m1.1.1.2">𝜆</ci><cn id="A1.p6.1.m1.1.1.3.cmml" type="integer" xref="A1.p6.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p6.1.m1.1c">\lambda=1\,</annotation><annotation encoding="application/x-llamapun" id="A1.p6.1.m1.1d">italic_λ = 1</annotation></semantics></math>cm. The dashed curves shows the strain noise power we estimate from plasma dispersion. The other curves are our predictions for the noise once correcting for plasma dispersion by transmitting at two frequencies (see below). The left panel shows <math alttext="r=L=10\," class="ltx_Math" display="inline" id="A1.p6.2.m2.1"><semantics id="A1.p6.2.m2.1a"><mrow id="A1.p6.2.m2.1.1" xref="A1.p6.2.m2.1.1.cmml"><mi id="A1.p6.2.m2.1.1.2" xref="A1.p6.2.m2.1.1.2.cmml">r</mi><mo id="A1.p6.2.m2.1.1.3" xref="A1.p6.2.m2.1.1.3.cmml">=</mo><mi id="A1.p6.2.m2.1.1.4" xref="A1.p6.2.m2.1.1.4.cmml">L</mi><mo id="A1.p6.2.m2.1.1.5" xref="A1.p6.2.m2.1.1.5.cmml">=</mo><mn id="A1.p6.2.m2.1.1.6" xref="A1.p6.2.m2.1.1.6.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.p6.2.m2.1b"><apply id="A1.p6.2.m2.1.1.cmml" xref="A1.p6.2.m2.1.1"><and id="A1.p6.2.m2.1.1a.cmml" xref="A1.p6.2.m2.1.1"></and><apply id="A1.p6.2.m2.1.1b.cmml" xref="A1.p6.2.m2.1.1"><eq id="A1.p6.2.m2.1.1.3.cmml" xref="A1.p6.2.m2.1.1.3"></eq><ci id="A1.p6.2.m2.1.1.2.cmml" xref="A1.p6.2.m2.1.1.2">𝑟</ci><ci id="A1.p6.2.m2.1.1.4.cmml" xref="A1.p6.2.m2.1.1.4">𝐿</ci></apply><apply id="A1.p6.2.m2.1.1c.cmml" xref="A1.p6.2.m2.1.1"><eq id="A1.p6.2.m2.1.1.5.cmml" xref="A1.p6.2.m2.1.1.5"></eq><share href="https://arxiv.org/html/2411.15072v1#A1.p6.2.m2.1.1.4.cmml" id="A1.p6.2.m2.1.1d.cmml" xref="A1.p6.2.m2.1.1"></share><cn id="A1.p6.2.m2.1.1.6.cmml" type="integer" xref="A1.p6.2.m2.1.1.6">10</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p6.2.m2.1c">r=L=10\,</annotation><annotation encoding="application/x-llamapun" id="A1.p6.2.m2.1d">italic_r = italic_L = 10</annotation></semantics></math>AU, and the right panel <math alttext="r=L=30\," class="ltx_Math" display="inline" id="A1.p6.3.m3.1"><semantics id="A1.p6.3.m3.1a"><mrow id="A1.p6.3.m3.1.1" xref="A1.p6.3.m3.1.1.cmml"><mi id="A1.p6.3.m3.1.1.2" xref="A1.p6.3.m3.1.1.2.cmml">r</mi><mo id="A1.p6.3.m3.1.1.3" xref="A1.p6.3.m3.1.1.3.cmml">=</mo><mi id="A1.p6.3.m3.1.1.4" xref="A1.p6.3.m3.1.1.4.cmml">L</mi><mo id="A1.p6.3.m3.1.1.5" xref="A1.p6.3.m3.1.1.5.cmml">=</mo><mn id="A1.p6.3.m3.1.1.6" xref="A1.p6.3.m3.1.1.6.cmml">30</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.p6.3.m3.1b"><apply id="A1.p6.3.m3.1.1.cmml" xref="A1.p6.3.m3.1.1"><and id="A1.p6.3.m3.1.1a.cmml" xref="A1.p6.3.m3.1.1"></and><apply id="A1.p6.3.m3.1.1b.cmml" xref="A1.p6.3.m3.1.1"><eq id="A1.p6.3.m3.1.1.3.cmml" xref="A1.p6.3.m3.1.1.3"></eq><ci id="A1.p6.3.m3.1.1.2.cmml" xref="A1.p6.3.m3.1.1.2">𝑟</ci><ci id="A1.p6.3.m3.1.1.4.cmml" xref="A1.p6.3.m3.1.1.4">𝐿</ci></apply><apply id="A1.p6.3.m3.1.1c.cmml" xref="A1.p6.3.m3.1.1"><eq id="A1.p6.3.m3.1.1.5.cmml" xref="A1.p6.3.m3.1.1.5"></eq><share href="https://arxiv.org/html/2411.15072v1#A1.p6.3.m3.1.1.4.cmml" id="A1.p6.3.m3.1.1d.cmml" xref="A1.p6.3.m3.1.1"></share><cn id="A1.p6.3.m3.1.1.6.cmml" type="integer" xref="A1.p6.3.m3.1.1.6">30</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p6.3.m3.1c">r=L=30\,</annotation><annotation encoding="application/x-llamapun" id="A1.p6.3.m3.1d">italic_r = italic_L = 30</annotation></semantics></math>AU assuming the spacecraft separation is nearly orthogonal to the radial direction so that the above calculations apply. Rather than assuming a power-law scaling as in equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#A1.E9" title="In Appendix A Dispersion ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">A9</span></a>), the estimates in this figure use equations (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#A1.E6" title="In Appendix A Dispersion ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">A6</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#A1.E8" title="In Appendix A Dispersion ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">A8</span></a>) as well as the 1D electron density power spectrum of the solar wind measured in <cite class="ltx_cite ltx_citemacro_citet">Bellamy et al. (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib16" title="">2005</a>)</cite> using Voyager 2 data.<span class="ltx_note ltx_role_footnote" id="footnote16"><sup class="ltx_note_mark">16</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">16</sup><span class="ltx_tag ltx_tag_note">16</span>Since our derivation of equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#A1.E6" title="In Appendix A Dispersion ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">A6</span></a>) assumed a power law, it is not rigorous to use the measured 1D electron density power spectrum: The different wavenumber weighting of the integrand over <math alttext="P_{e}" class="ltx_Math" display="inline" id="footnote16.m1.1"><semantics id="footnote16.m1.1b"><msub id="footnote16.m1.1.1" xref="footnote16.m1.1.1.cmml"><mi id="footnote16.m1.1.1.2" xref="footnote16.m1.1.1.2.cmml">P</mi><mi id="footnote16.m1.1.1.3" xref="footnote16.m1.1.1.3.cmml">e</mi></msub><annotation-xml encoding="MathML-Content" id="footnote16.m1.1c"><apply id="footnote16.m1.1.1.cmml" xref="footnote16.m1.1.1"><csymbol cd="ambiguous" id="footnote16.m1.1.1.1.cmml" xref="footnote16.m1.1.1">subscript</csymbol><ci id="footnote16.m1.1.1.2.cmml" xref="footnote16.m1.1.1.2">𝑃</ci><ci id="footnote16.m1.1.1.3.cmml" xref="footnote16.m1.1.1.3">𝑒</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote16.m1.1d">P_{e}</annotation><annotation encoding="application/x-llamapun" id="footnote16.m1.1e">italic_P start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT</annotation></semantics></math> that yields the measured 1D electron density spectrum and the phase noise means that the features in the measured <math alttext="P^{\rm 1D}_{e}(f)" class="ltx_Math" display="inline" id="footnote16.m2.1"><semantics id="footnote16.m2.1b"><mrow id="footnote16.m2.1.2" xref="footnote16.m2.1.2.cmml"><msubsup id="footnote16.m2.1.2.2" xref="footnote16.m2.1.2.2.cmml"><mi id="footnote16.m2.1.2.2.2.2" xref="footnote16.m2.1.2.2.2.2.cmml">P</mi><mi id="footnote16.m2.1.2.2.3" xref="footnote16.m2.1.2.2.3.cmml">e</mi><mrow id="footnote16.m2.1.2.2.2.3" xref="footnote16.m2.1.2.2.2.3.cmml"><mn id="footnote16.m2.1.2.2.2.3.2" xref="footnote16.m2.1.2.2.2.3.2.cmml">1</mn><mo id="footnote16.m2.1.2.2.2.3.1" xref="footnote16.m2.1.2.2.2.3.1.cmml">⁢</mo><mi id="footnote16.m2.1.2.2.2.3.3" mathvariant="normal" xref="footnote16.m2.1.2.2.2.3.3.cmml">D</mi></mrow></msubsup><mo id="footnote16.m2.1.2.1" xref="footnote16.m2.1.2.1.cmml">⁢</mo><mrow id="footnote16.m2.1.2.3.2" xref="footnote16.m2.1.2.cmml"><mo id="footnote16.m2.1.2.3.2.1" stretchy="false" xref="footnote16.m2.1.2.cmml">(</mo><mi id="footnote16.m2.1.1" xref="footnote16.m2.1.1.cmml">f</mi><mo id="footnote16.m2.1.2.3.2.2" stretchy="false" xref="footnote16.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml 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alttext="\lambda=1\," class="ltx_Math" display="inline" id="A1.F8.6.m1.1"><semantics id="A1.F8.6.m1.1b"><mrow id="A1.F8.6.m1.1.1" xref="A1.F8.6.m1.1.1.cmml"><mi id="A1.F8.6.m1.1.1.2" xref="A1.F8.6.m1.1.1.2.cmml">λ</mi><mo id="A1.F8.6.m1.1.1.1" xref="A1.F8.6.m1.1.1.1.cmml">=</mo><mn id="A1.F8.6.m1.1.1.3" xref="A1.F8.6.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.F8.6.m1.1c"><apply id="A1.F8.6.m1.1.1.cmml" xref="A1.F8.6.m1.1.1"><eq id="A1.F8.6.m1.1.1.1.cmml" xref="A1.F8.6.m1.1.1.1"></eq><ci id="A1.F8.6.m1.1.1.2.cmml" xref="A1.F8.6.m1.1.1.2">𝜆</ci><cn id="A1.F8.6.m1.1.1.3.cmml" type="integer" xref="A1.F8.6.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.F8.6.m1.1d">\lambda=1\,</annotation><annotation encoding="application/x-llamapun" id="A1.F8.6.m1.1e">italic_λ = 1</annotation></semantics></math>cm. The dashed curve shows the strain noise power we estimate from plasma dispersion. The other curves show the total noise once correcting for plasma dispersion with <math alttext="{\cal A}=1.5" class="ltx_Math" display="inline" id="A1.F8.7.m2.1"><semantics id="A1.F8.7.m2.1b"><mrow id="A1.F8.7.m2.1.1" xref="A1.F8.7.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.F8.7.m2.1.1.2" xref="A1.F8.7.m2.1.1.2.cmml">𝒜</mi><mo id="A1.F8.7.m2.1.1.1" xref="A1.F8.7.m2.1.1.1.cmml">=</mo><mn id="A1.F8.7.m2.1.1.3" xref="A1.F8.7.m2.1.1.3.cmml">1.5</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.F8.7.m2.1c"><apply id="A1.F8.7.m2.1.1.cmml" xref="A1.F8.7.m2.1.1"><eq id="A1.F8.7.m2.1.1.1.cmml" xref="A1.F8.7.m2.1.1.1"></eq><ci id="A1.F8.7.m2.1.1.2.cmml" xref="A1.F8.7.m2.1.1.2">𝒜</ci><cn id="A1.F8.7.m2.1.1.3.cmml" type="float" xref="A1.F8.7.m2.1.1.3">1.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.F8.7.m2.1d">{\cal A}=1.5</annotation><annotation encoding="application/x-llamapun" id="A1.F8.7.m2.1e">caligraphic_A = 1.5</annotation></semantics></math> in the time-delay interferometer setup. The left panel shows the case <math alttext="r=L=10\;" class="ltx_Math" display="inline" id="A1.F8.8.m3.1"><semantics id="A1.F8.8.m3.1b"><mrow id="A1.F8.8.m3.1.1" xref="A1.F8.8.m3.1.1.cmml"><mi id="A1.F8.8.m3.1.1.2" xref="A1.F8.8.m3.1.1.2.cmml">r</mi><mo id="A1.F8.8.m3.1.1.3" xref="A1.F8.8.m3.1.1.3.cmml">=</mo><mi id="A1.F8.8.m3.1.1.4" xref="A1.F8.8.m3.1.1.4.cmml">L</mi><mo id="A1.F8.8.m3.1.1.5" xref="A1.F8.8.m3.1.1.5.cmml">=</mo><mn id="A1.F8.8.m3.1.1.6" xref="A1.F8.8.m3.1.1.6.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.F8.8.m3.1c"><apply id="A1.F8.8.m3.1.1.cmml" xref="A1.F8.8.m3.1.1"><and id="A1.F8.8.m3.1.1a.cmml" xref="A1.F8.8.m3.1.1"></and><apply id="A1.F8.8.m3.1.1b.cmml" xref="A1.F8.8.m3.1.1"><eq id="A1.F8.8.m3.1.1.3.cmml" xref="A1.F8.8.m3.1.1.3"></eq><ci id="A1.F8.8.m3.1.1.2.cmml" xref="A1.F8.8.m3.1.1.2">𝑟</ci><ci id="A1.F8.8.m3.1.1.4.cmml" xref="A1.F8.8.m3.1.1.4">𝐿</ci></apply><apply id="A1.F8.8.m3.1.1c.cmml" xref="A1.F8.8.m3.1.1"><eq id="A1.F8.8.m3.1.1.5.cmml" xref="A1.F8.8.m3.1.1.5"></eq><share href="https://arxiv.org/html/2411.15072v1#A1.F8.8.m3.1.1.4.cmml" id="A1.F8.8.m3.1.1d.cmml" xref="A1.F8.8.m3.1.1"></share><cn id="A1.F8.8.m3.1.1.6.cmml" type="integer" xref="A1.F8.8.m3.1.1.6">10</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.F8.8.m3.1d">r=L=10\;</annotation><annotation encoding="application/x-llamapun" id="A1.F8.8.m3.1e">italic_r = italic_L = 10</annotation></semantics></math>AU, and the right panel <math alttext="r=L=30\;" class="ltx_Math" display="inline" id="A1.F8.9.m4.1"><semantics id="A1.F8.9.m4.1b"><mrow id="A1.F8.9.m4.1.1" xref="A1.F8.9.m4.1.1.cmml"><mi id="A1.F8.9.m4.1.1.2" xref="A1.F8.9.m4.1.1.2.cmml">r</mi><mo id="A1.F8.9.m4.1.1.3" xref="A1.F8.9.m4.1.1.3.cmml">=</mo><mi id="A1.F8.9.m4.1.1.4" xref="A1.F8.9.m4.1.1.4.cmml">L</mi><mo id="A1.F8.9.m4.1.1.5" xref="A1.F8.9.m4.1.1.5.cmml">=</mo><mn id="A1.F8.9.m4.1.1.6" xref="A1.F8.9.m4.1.1.6.cmml">30</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.F8.9.m4.1c"><apply id="A1.F8.9.m4.1.1.cmml" xref="A1.F8.9.m4.1.1"><and id="A1.F8.9.m4.1.1a.cmml" xref="A1.F8.9.m4.1.1"></and><apply id="A1.F8.9.m4.1.1b.cmml" xref="A1.F8.9.m4.1.1"><eq id="A1.F8.9.m4.1.1.3.cmml" xref="A1.F8.9.m4.1.1.3"></eq><ci id="A1.F8.9.m4.1.1.2.cmml" xref="A1.F8.9.m4.1.1.2">𝑟</ci><ci id="A1.F8.9.m4.1.1.4.cmml" xref="A1.F8.9.m4.1.1.4">𝐿</ci></apply><apply id="A1.F8.9.m4.1.1c.cmml" xref="A1.F8.9.m4.1.1"><eq id="A1.F8.9.m4.1.1.5.cmml" xref="A1.F8.9.m4.1.1.5"></eq><share href="https://arxiv.org/html/2411.15072v1#A1.F8.9.m4.1.1.4.cmml" id="A1.F8.9.m4.1.1d.cmml" xref="A1.F8.9.m4.1.1"></share><cn id="A1.F8.9.m4.1.1.6.cmml" type="integer" xref="A1.F8.9.m4.1.1.6">30</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.F8.9.m4.1d">r=L=30\;</annotation><annotation encoding="application/x-llamapun" id="A1.F8.9.m4.1e">italic_r = italic_L = 30</annotation></semantics></math>AU, and the spacecraft separation is assumed to be nearly orthogonal to the radial direction. The estimates in this figure use equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#A1.E8" title="In Appendix A Dispersion ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">A8</span></a>) and the electron density power spectrum of the solar wind measured by Voyager 2 and reported in <cite class="ltx_cite ltx_citemacro_citet">Bellamy et al. (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib16" title="">2005</a>)</cite>.</figcaption> </figure> <div class="ltx_para" id="A1.p7"> <p class="ltx_p" id="A1.p7.2">While the phase noise due to the plasma can limit the sensitivity, it can be essentially eliminated by broadcasting at two wavelengths, <math alttext="\lambda_{1}" class="ltx_Math" display="inline" id="A1.p7.1.m1.1"><semantics id="A1.p7.1.m1.1a"><msub id="A1.p7.1.m1.1.1" xref="A1.p7.1.m1.1.1.cmml"><mi id="A1.p7.1.m1.1.1.2" xref="A1.p7.1.m1.1.1.2.cmml">λ</mi><mn id="A1.p7.1.m1.1.1.3" xref="A1.p7.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="A1.p7.1.m1.1b"><apply id="A1.p7.1.m1.1.1.cmml" xref="A1.p7.1.m1.1.1"><csymbol cd="ambiguous" id="A1.p7.1.m1.1.1.1.cmml" xref="A1.p7.1.m1.1.1">subscript</csymbol><ci id="A1.p7.1.m1.1.1.2.cmml" xref="A1.p7.1.m1.1.1.2">𝜆</ci><cn id="A1.p7.1.m1.1.1.3.cmml" type="integer" xref="A1.p7.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation 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xref="A1.E10.m1.2.2.1.1.1.1.1.1.3.3.2">italic-ϕ</ci><cn id="A1.E10.m1.2.2.1.1.1.1.1.1.3.3.3.cmml" type="integer" xref="A1.E10.m1.2.2.1.1.1.1.1.1.3.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E10.m1.2c">\widehat{\Delta x}=\frac{\lambda_{1}\lambda_{2}}{2\pi(\lambda_{1}^{2}-\lambda_% {2}^{2})}\left(\lambda_{1}\phi_{2}-\lambda_{2}\phi_{1}\right).</annotation><annotation encoding="application/x-llamapun" id="A1.E10.m1.2d">over^ start_ARG roman_Δ italic_x end_ARG = divide start_ARG italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG start_ARG 2 italic_π ( italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_λ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_ARG ( italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_ϕ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - italic_λ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(A10)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A1.p7.21">Accelerations and gravitational waves lead to the same <math alttext="\Delta x" class="ltx_Math" display="inline" id="A1.p7.3.m1.1"><semantics id="A1.p7.3.m1.1a"><mrow id="A1.p7.3.m1.1.1" xref="A1.p7.3.m1.1.1.cmml"><mi id="A1.p7.3.m1.1.1.2" mathvariant="normal" xref="A1.p7.3.m1.1.1.2.cmml">Δ</mi><mo id="A1.p7.3.m1.1.1.1" xref="A1.p7.3.m1.1.1.1.cmml">⁢</mo><mi id="A1.p7.3.m1.1.1.3" xref="A1.p7.3.m1.1.1.3.cmml">x</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.p7.3.m1.1b"><apply id="A1.p7.3.m1.1.1.cmml" xref="A1.p7.3.m1.1.1"><times id="A1.p7.3.m1.1.1.1.cmml" xref="A1.p7.3.m1.1.1.1"></times><ci id="A1.p7.3.m1.1.1.2.cmml" xref="A1.p7.3.m1.1.1.2">Δ</ci><ci id="A1.p7.3.m1.1.1.3.cmml" xref="A1.p7.3.m1.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p7.3.m1.1c">\Delta x</annotation><annotation encoding="application/x-llamapun" id="A1.p7.3.m1.1d">roman_Δ italic_x</annotation></semantics></math> as the previous single-phase estimator. Therefore, this estimator’s gravitational wave strain sensitivity is not affected when accelerations set the noise. However, this estimator does come with the cost of an increased error when radiometer noise is important. When radiometer noise dominates, if we assume <math alttext="\delta\phi_{j}\propto\lambda_{j}" class="ltx_Math" display="inline" id="A1.p7.4.m2.1"><semantics id="A1.p7.4.m2.1a"><mrow id="A1.p7.4.m2.1.1" xref="A1.p7.4.m2.1.1.cmml"><mrow id="A1.p7.4.m2.1.1.2" xref="A1.p7.4.m2.1.1.2.cmml"><mi id="A1.p7.4.m2.1.1.2.2" xref="A1.p7.4.m2.1.1.2.2.cmml">δ</mi><mo id="A1.p7.4.m2.1.1.2.1" xref="A1.p7.4.m2.1.1.2.1.cmml">⁢</mo><msub id="A1.p7.4.m2.1.1.2.3" xref="A1.p7.4.m2.1.1.2.3.cmml"><mi id="A1.p7.4.m2.1.1.2.3.2" xref="A1.p7.4.m2.1.1.2.3.2.cmml">ϕ</mi><mi id="A1.p7.4.m2.1.1.2.3.3" xref="A1.p7.4.m2.1.1.2.3.3.cmml">j</mi></msub></mrow><mo id="A1.p7.4.m2.1.1.1" xref="A1.p7.4.m2.1.1.1.cmml">∝</mo><msub id="A1.p7.4.m2.1.1.3" xref="A1.p7.4.m2.1.1.3.cmml"><mi id="A1.p7.4.m2.1.1.3.2" xref="A1.p7.4.m2.1.1.3.2.cmml">λ</mi><mi id="A1.p7.4.m2.1.1.3.3" xref="A1.p7.4.m2.1.1.3.3.cmml">j</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A1.p7.4.m2.1b"><apply id="A1.p7.4.m2.1.1.cmml" xref="A1.p7.4.m2.1.1"><csymbol cd="latexml" id="A1.p7.4.m2.1.1.1.cmml" xref="A1.p7.4.m2.1.1.1">proportional-to</csymbol><apply id="A1.p7.4.m2.1.1.2.cmml" xref="A1.p7.4.m2.1.1.2"><times id="A1.p7.4.m2.1.1.2.1.cmml" xref="A1.p7.4.m2.1.1.2.1"></times><ci id="A1.p7.4.m2.1.1.2.2.cmml" xref="A1.p7.4.m2.1.1.2.2">𝛿</ci><apply id="A1.p7.4.m2.1.1.2.3.cmml" xref="A1.p7.4.m2.1.1.2.3"><csymbol cd="ambiguous" id="A1.p7.4.m2.1.1.2.3.1.cmml" xref="A1.p7.4.m2.1.1.2.3">subscript</csymbol><ci id="A1.p7.4.m2.1.1.2.3.2.cmml" xref="A1.p7.4.m2.1.1.2.3.2">italic-ϕ</ci><ci id="A1.p7.4.m2.1.1.2.3.3.cmml" xref="A1.p7.4.m2.1.1.2.3.3">𝑗</ci></apply></apply><apply id="A1.p7.4.m2.1.1.3.cmml" xref="A1.p7.4.m2.1.1.3"><csymbol cd="ambiguous" id="A1.p7.4.m2.1.1.3.1.cmml" xref="A1.p7.4.m2.1.1.3">subscript</csymbol><ci id="A1.p7.4.m2.1.1.3.2.cmml" xref="A1.p7.4.m2.1.1.3.2">𝜆</ci><ci id="A1.p7.4.m2.1.1.3.3.cmml" xref="A1.p7.4.m2.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p7.4.m2.1c">\delta\phi_{j}\propto\lambda_{j}</annotation><annotation encoding="application/x-llamapun" id="A1.p7.4.m2.1d">italic_δ italic_ϕ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∝ italic_λ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> as applies if <math alttext="A_{\rm eff}" class="ltx_Math" display="inline" id="A1.p7.5.m3.1"><semantics id="A1.p7.5.m3.1a"><msub id="A1.p7.5.m3.1.1" xref="A1.p7.5.m3.1.1.cmml"><mi id="A1.p7.5.m3.1.1.2" xref="A1.p7.5.m3.1.1.2.cmml">A</mi><mi id="A1.p7.5.m3.1.1.3" xref="A1.p7.5.m3.1.1.3.cmml">eff</mi></msub><annotation-xml encoding="MathML-Content" id="A1.p7.5.m3.1b"><apply id="A1.p7.5.m3.1.1.cmml" xref="A1.p7.5.m3.1.1"><csymbol cd="ambiguous" id="A1.p7.5.m3.1.1.1.cmml" xref="A1.p7.5.m3.1.1">subscript</csymbol><ci id="A1.p7.5.m3.1.1.2.cmml" xref="A1.p7.5.m3.1.1.2">𝐴</ci><ci id="A1.p7.5.m3.1.1.3.cmml" xref="A1.p7.5.m3.1.1.3">eff</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p7.5.m3.1c">A_{\rm eff}</annotation><annotation encoding="application/x-llamapun" id="A1.p7.5.m3.1d">italic_A start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="T_{\rm sys}" class="ltx_Math" display="inline" id="A1.p7.6.m4.1"><semantics id="A1.p7.6.m4.1a"><msub id="A1.p7.6.m4.1.1" xref="A1.p7.6.m4.1.1.cmml"><mi id="A1.p7.6.m4.1.1.2" xref="A1.p7.6.m4.1.1.2.cmml">T</mi><mi id="A1.p7.6.m4.1.1.3" xref="A1.p7.6.m4.1.1.3.cmml">sys</mi></msub><annotation-xml encoding="MathML-Content" id="A1.p7.6.m4.1b"><apply id="A1.p7.6.m4.1.1.cmml" xref="A1.p7.6.m4.1.1"><csymbol cd="ambiguous" id="A1.p7.6.m4.1.1.1.cmml" xref="A1.p7.6.m4.1.1">subscript</csymbol><ci id="A1.p7.6.m4.1.1.2.cmml" xref="A1.p7.6.m4.1.1.2">𝑇</ci><ci id="A1.p7.6.m4.1.1.3.cmml" xref="A1.p7.6.m4.1.1.3">sys</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p7.6.m4.1c">T_{\rm sys}</annotation><annotation encoding="application/x-llamapun" id="A1.p7.6.m4.1d">italic_T start_POSTSUBSCRIPT roman_sys end_POSTSUBSCRIPT</annotation></semantics></math> are the same for both <math alttext="\lambda_{1}" class="ltx_Math" display="inline" id="A1.p7.7.m5.1"><semantics id="A1.p7.7.m5.1a"><msub id="A1.p7.7.m5.1.1" xref="A1.p7.7.m5.1.1.cmml"><mi id="A1.p7.7.m5.1.1.2" xref="A1.p7.7.m5.1.1.2.cmml">λ</mi><mn id="A1.p7.7.m5.1.1.3" xref="A1.p7.7.m5.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="A1.p7.7.m5.1b"><apply id="A1.p7.7.m5.1.1.cmml" xref="A1.p7.7.m5.1.1"><csymbol cd="ambiguous" id="A1.p7.7.m5.1.1.1.cmml" xref="A1.p7.7.m5.1.1">subscript</csymbol><ci id="A1.p7.7.m5.1.1.2.cmml" xref="A1.p7.7.m5.1.1.2">𝜆</ci><cn id="A1.p7.7.m5.1.1.3.cmml" type="integer" xref="A1.p7.7.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p7.7.m5.1c">\lambda_{1}</annotation><annotation encoding="application/x-llamapun" id="A1.p7.7.m5.1d">italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\lambda_{2}" class="ltx_Math" display="inline" id="A1.p7.8.m6.1"><semantics id="A1.p7.8.m6.1a"><msub id="A1.p7.8.m6.1.1" xref="A1.p7.8.m6.1.1.cmml"><mi id="A1.p7.8.m6.1.1.2" xref="A1.p7.8.m6.1.1.2.cmml">λ</mi><mn id="A1.p7.8.m6.1.1.3" xref="A1.p7.8.m6.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="A1.p7.8.m6.1b"><apply id="A1.p7.8.m6.1.1.cmml" xref="A1.p7.8.m6.1.1"><csymbol cd="ambiguous" id="A1.p7.8.m6.1.1.1.cmml" xref="A1.p7.8.m6.1.1">subscript</csymbol><ci id="A1.p7.8.m6.1.1.2.cmml" xref="A1.p7.8.m6.1.1.2">𝜆</ci><cn id="A1.p7.8.m6.1.1.3.cmml" type="integer" xref="A1.p7.8.m6.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p7.8.m6.1c">\lambda_{2}</annotation><annotation encoding="application/x-llamapun" id="A1.p7.8.m6.1d">italic_λ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, then phase errors are mapped to total errors as <math 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xref="A1.p7.11.m9.1.1.1.1.3.3">1</cn></apply><apply id="A1.p7.11.m9.1.1.1.1.1.2.cmml" xref="A1.p7.11.m9.1.1.1.1.1.1"><csymbol cd="latexml" id="A1.p7.11.m9.1.1.1.1.1.2.1.cmml" xref="A1.p7.11.m9.1.1.1.1.1.1.2">delimited-[]</csymbol><apply id="A1.p7.11.m9.1.1.1.1.1.1.1.cmml" xref="A1.p7.11.m9.1.1.1.1.1.1.1"><times id="A1.p7.11.m9.1.1.1.1.1.1.1.1.cmml" xref="A1.p7.11.m9.1.1.1.1.1.1.1.1"></times><cn id="A1.p7.11.m9.1.1.1.1.1.1.1.2.cmml" type="integer" xref="A1.p7.11.m9.1.1.1.1.1.1.1.2">2</cn><ci id="A1.p7.11.m9.1.1.1.1.1.1.1.3.cmml" xref="A1.p7.11.m9.1.1.1.1.1.1.1.3">𝜋</ci></apply></apply></apply><apply id="A1.p7.11.m9.1.1.1.3.cmml" xref="A1.p7.11.m9.1.1.1.3"><csymbol cd="ambiguous" id="A1.p7.11.m9.1.1.1.3.1.cmml" xref="A1.p7.11.m9.1.1.1.3">subscript</csymbol><ci id="A1.p7.11.m9.1.1.1.3.2.cmml" xref="A1.p7.11.m9.1.1.1.3.2">𝜎</ci><apply id="A1.p7.11.m9.1.1.1.3.3.cmml" xref="A1.p7.11.m9.1.1.1.3.3"><csymbol cd="ambiguous" id="A1.p7.11.m9.1.1.1.3.3.1.cmml" 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Thus, it results in an increase in the estimator error by the factor <math alttext="\sqrt{2}\lambda_{2}^{2}/|\lambda_{2}^{2}-\lambda_{1}^{2}|" class="ltx_Math" display="inline" id="A1.p7.12.m10.1"><semantics id="A1.p7.12.m10.1a"><mrow id="A1.p7.12.m10.1.1" xref="A1.p7.12.m10.1.1.cmml"><mrow id="A1.p7.12.m10.1.1.3" xref="A1.p7.12.m10.1.1.3.cmml"><msqrt id="A1.p7.12.m10.1.1.3.2" xref="A1.p7.12.m10.1.1.3.2.cmml"><mn id="A1.p7.12.m10.1.1.3.2.2" xref="A1.p7.12.m10.1.1.3.2.2.cmml">2</mn></msqrt><mo id="A1.p7.12.m10.1.1.3.1" xref="A1.p7.12.m10.1.1.3.1.cmml">⁢</mo><msubsup id="A1.p7.12.m10.1.1.3.3" xref="A1.p7.12.m10.1.1.3.3.cmml"><mi id="A1.p7.12.m10.1.1.3.3.2.2" xref="A1.p7.12.m10.1.1.3.3.2.2.cmml">λ</mi><mn id="A1.p7.12.m10.1.1.3.3.2.3" xref="A1.p7.12.m10.1.1.3.3.2.3.cmml">2</mn><mn id="A1.p7.12.m10.1.1.3.3.3" xref="A1.p7.12.m10.1.1.3.3.3.cmml">2</mn></msubsup></mrow><mo id="A1.p7.12.m10.1.1.2" xref="A1.p7.12.m10.1.1.2.cmml">/</mo><mrow id="A1.p7.12.m10.1.1.1.1" 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If <math alttext="\lambda_{1}=1\," class="ltx_Math" display="inline" id="A1.p7.13.m11.1"><semantics id="A1.p7.13.m11.1a"><mrow id="A1.p7.13.m11.1.1" xref="A1.p7.13.m11.1.1.cmml"><msub id="A1.p7.13.m11.1.1.2" xref="A1.p7.13.m11.1.1.2.cmml"><mi id="A1.p7.13.m11.1.1.2.2" xref="A1.p7.13.m11.1.1.2.2.cmml">λ</mi><mn id="A1.p7.13.m11.1.1.2.3" xref="A1.p7.13.m11.1.1.2.3.cmml">1</mn></msub><mo id="A1.p7.13.m11.1.1.1" xref="A1.p7.13.m11.1.1.1.cmml">=</mo><mn id="A1.p7.13.m11.1.1.3" xref="A1.p7.13.m11.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.p7.13.m11.1b"><apply id="A1.p7.13.m11.1.1.cmml" xref="A1.p7.13.m11.1.1"><eq id="A1.p7.13.m11.1.1.1.cmml" xref="A1.p7.13.m11.1.1.1"></eq><apply id="A1.p7.13.m11.1.1.2.cmml" xref="A1.p7.13.m11.1.1.2"><csymbol cd="ambiguous" id="A1.p7.13.m11.1.1.2.1.cmml" xref="A1.p7.13.m11.1.1.2">subscript</csymbol><ci id="A1.p7.13.m11.1.1.2.2.cmml" xref="A1.p7.13.m11.1.1.2.2">𝜆</ci><cn id="A1.p7.13.m11.1.1.2.3.cmml" type="integer" xref="A1.p7.13.m11.1.1.2.3">1</cn></apply><cn id="A1.p7.13.m11.1.1.3.cmml" type="integer" xref="A1.p7.13.m11.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p7.13.m11.1c">\lambda_{1}=1\,</annotation><annotation encoding="application/x-llamapun" id="A1.p7.13.m11.1d">italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1</annotation></semantics></math>cm and <math alttext="\lambda_{2}=2\," class="ltx_Math" display="inline" id="A1.p7.14.m12.1"><semantics id="A1.p7.14.m12.1a"><mrow id="A1.p7.14.m12.1.1" xref="A1.p7.14.m12.1.1.cmml"><msub id="A1.p7.14.m12.1.1.2" xref="A1.p7.14.m12.1.1.2.cmml"><mi id="A1.p7.14.m12.1.1.2.2" xref="A1.p7.14.m12.1.1.2.2.cmml">λ</mi><mn id="A1.p7.14.m12.1.1.2.3" xref="A1.p7.14.m12.1.1.2.3.cmml">2</mn></msub><mo id="A1.p7.14.m12.1.1.1" xref="A1.p7.14.m12.1.1.1.cmml">=</mo><mn id="A1.p7.14.m12.1.1.3" xref="A1.p7.14.m12.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.p7.14.m12.1b"><apply id="A1.p7.14.m12.1.1.cmml" xref="A1.p7.14.m12.1.1"><eq id="A1.p7.14.m12.1.1.1.cmml" xref="A1.p7.14.m12.1.1.1"></eq><apply id="A1.p7.14.m12.1.1.2.cmml" xref="A1.p7.14.m12.1.1.2"><csymbol cd="ambiguous" id="A1.p7.14.m12.1.1.2.1.cmml" xref="A1.p7.14.m12.1.1.2">subscript</csymbol><ci id="A1.p7.14.m12.1.1.2.2.cmml" xref="A1.p7.14.m12.1.1.2.2">𝜆</ci><cn id="A1.p7.14.m12.1.1.2.3.cmml" type="integer" xref="A1.p7.14.m12.1.1.2.3">2</cn></apply><cn id="A1.p7.14.m12.1.1.3.cmml" type="integer" xref="A1.p7.14.m12.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p7.14.m12.1c">\lambda_{2}=2\,</annotation><annotation encoding="application/x-llamapun" id="A1.p7.14.m12.1d">italic_λ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 2</annotation></semantics></math>cm (<math alttext="\lambda_{2}=4" class="ltx_Math" display="inline" id="A1.p7.15.m13.1"><semantics id="A1.p7.15.m13.1a"><mrow id="A1.p7.15.m13.1.1" xref="A1.p7.15.m13.1.1.cmml"><msub id="A1.p7.15.m13.1.1.2" xref="A1.p7.15.m13.1.1.2.cmml"><mi id="A1.p7.15.m13.1.1.2.2" xref="A1.p7.15.m13.1.1.2.2.cmml">λ</mi><mn id="A1.p7.15.m13.1.1.2.3" xref="A1.p7.15.m13.1.1.2.3.cmml">2</mn></msub><mo id="A1.p7.15.m13.1.1.1" xref="A1.p7.15.m13.1.1.1.cmml">=</mo><mn id="A1.p7.15.m13.1.1.3" xref="A1.p7.15.m13.1.1.3.cmml">4</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.p7.15.m13.1b"><apply id="A1.p7.15.m13.1.1.cmml" xref="A1.p7.15.m13.1.1"><eq id="A1.p7.15.m13.1.1.1.cmml" xref="A1.p7.15.m13.1.1.1"></eq><apply id="A1.p7.15.m13.1.1.2.cmml" xref="A1.p7.15.m13.1.1.2"><csymbol cd="ambiguous" id="A1.p7.15.m13.1.1.2.1.cmml" xref="A1.p7.15.m13.1.1.2">subscript</csymbol><ci id="A1.p7.15.m13.1.1.2.2.cmml" xref="A1.p7.15.m13.1.1.2.2">𝜆</ci><cn id="A1.p7.15.m13.1.1.2.3.cmml" type="integer" xref="A1.p7.15.m13.1.1.2.3">2</cn></apply><cn id="A1.p7.15.m13.1.1.3.cmml" type="integer" xref="A1.p7.15.m13.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p7.15.m13.1c">\lambda_{2}=4</annotation><annotation encoding="application/x-llamapun" id="A1.p7.15.m13.1d">italic_λ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 4</annotation></semantics></math> cm), this results in an increase in the error of <math alttext="1.8" class="ltx_Math" display="inline" id="A1.p7.16.m14.1"><semantics id="A1.p7.16.m14.1a"><mn id="A1.p7.16.m14.1.1" xref="A1.p7.16.m14.1.1.cmml">1.8</mn><annotation-xml encoding="MathML-Content" id="A1.p7.16.m14.1b"><cn id="A1.p7.16.m14.1.1.cmml" type="float" xref="A1.p7.16.m14.1.1">1.8</cn></annotation-xml><annotation encoding="application/x-tex" id="A1.p7.16.m14.1c">1.8</annotation><annotation encoding="application/x-llamapun" id="A1.p7.16.m14.1d">1.8</annotation></semantics></math> (<math alttext="1.5" class="ltx_Math" display="inline" id="A1.p7.17.m15.1"><semantics id="A1.p7.17.m15.1a"><mn id="A1.p7.17.m15.1.1" xref="A1.p7.17.m15.1.1.cmml">1.5</mn><annotation-xml encoding="MathML-Content" id="A1.p7.17.m15.1b"><cn id="A1.p7.17.m15.1.1.cmml" type="float" xref="A1.p7.17.m15.1.1">1.5</cn></annotation-xml><annotation encoding="application/x-tex" id="A1.p7.17.m15.1c">1.5</annotation><annotation encoding="application/x-llamapun" id="A1.p7.17.m15.1d">1.5</annotation></semantics></math>) relative to what the error would be if we used the single phase estimator at <math alttext="\lambda_{1}" class="ltx_Math" display="inline" id="A1.p7.18.m16.1"><semantics id="A1.p7.18.m16.1a"><msub id="A1.p7.18.m16.1.1" xref="A1.p7.18.m16.1.1.cmml"><mi id="A1.p7.18.m16.1.1.2" xref="A1.p7.18.m16.1.1.2.cmml">λ</mi><mn id="A1.p7.18.m16.1.1.3" xref="A1.p7.18.m16.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="A1.p7.18.m16.1b"><apply id="A1.p7.18.m16.1.1.cmml" xref="A1.p7.18.m16.1.1"><csymbol cd="ambiguous" id="A1.p7.18.m16.1.1.1.cmml" xref="A1.p7.18.m16.1.1">subscript</csymbol><ci id="A1.p7.18.m16.1.1.2.cmml" xref="A1.p7.18.m16.1.1.2">𝜆</ci><cn id="A1.p7.18.m16.1.1.3.cmml" type="integer" xref="A1.p7.18.m16.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p7.18.m16.1c">\lambda_{1}</annotation><annotation encoding="application/x-llamapun" id="A1.p7.18.m16.1d">italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. We parameterize the increase in noise power from de-dispersion relative to the (higher <math alttext="S/N" class="ltx_Math" display="inline" id="A1.p7.19.m17.1"><semantics id="A1.p7.19.m17.1a"><mrow id="A1.p7.19.m17.1.1" xref="A1.p7.19.m17.1.1.cmml"><mi id="A1.p7.19.m17.1.1.2" xref="A1.p7.19.m17.1.1.2.cmml">S</mi><mo id="A1.p7.19.m17.1.1.1" xref="A1.p7.19.m17.1.1.1.cmml">/</mo><mi id="A1.p7.19.m17.1.1.3" xref="A1.p7.19.m17.1.1.3.cmml">N</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.p7.19.m17.1b"><apply id="A1.p7.19.m17.1.1.cmml" xref="A1.p7.19.m17.1.1"><divide id="A1.p7.19.m17.1.1.1.cmml" xref="A1.p7.19.m17.1.1.1"></divide><ci id="A1.p7.19.m17.1.1.2.cmml" xref="A1.p7.19.m17.1.1.2">𝑆</ci><ci id="A1.p7.19.m17.1.1.3.cmml" xref="A1.p7.19.m17.1.1.3">𝑁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p7.19.m17.1c">S/N</annotation><annotation encoding="application/x-llamapun" id="A1.p7.19.m17.1d">italic_S / italic_N</annotation></semantics></math>) shortest wavelength band by the factor <math alttext="{\cal A}" class="ltx_Math" display="inline" id="A1.p7.20.m18.1"><semantics id="A1.p7.20.m18.1a"><mi class="ltx_font_mathcaligraphic" id="A1.p7.20.m18.1.1" xref="A1.p7.20.m18.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="A1.p7.20.m18.1b"><ci id="A1.p7.20.m18.1.1.cmml" xref="A1.p7.20.m18.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.p7.20.m18.1c">{\cal A}</annotation><annotation encoding="application/x-llamapun" id="A1.p7.20.m18.1d">caligraphic_A</annotation></semantics></math> and include this in our error calculations, taking a fiducial value of <math alttext="{\cal A}=1.5" class="ltx_Math" display="inline" id="A1.p7.21.m19.1"><semantics id="A1.p7.21.m19.1a"><mrow id="A1.p7.21.m19.1.1" xref="A1.p7.21.m19.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.p7.21.m19.1.1.2" xref="A1.p7.21.m19.1.1.2.cmml">𝒜</mi><mo id="A1.p7.21.m19.1.1.1" xref="A1.p7.21.m19.1.1.1.cmml">=</mo><mn id="A1.p7.21.m19.1.1.3" xref="A1.p7.21.m19.1.1.3.cmml">1.5</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.p7.21.m19.1b"><apply id="A1.p7.21.m19.1.1.cmml" xref="A1.p7.21.m19.1.1"><eq id="A1.p7.21.m19.1.1.1.cmml" xref="A1.p7.21.m19.1.1.1"></eq><ci id="A1.p7.21.m19.1.1.2.cmml" xref="A1.p7.21.m19.1.1.2">𝒜</ci><cn id="A1.p7.21.m19.1.1.3.cmml" type="float" xref="A1.p7.21.m19.1.1.3">1.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p7.21.m19.1c">{\cal A}=1.5</annotation><annotation encoding="application/x-llamapun" id="A1.p7.21.m19.1d">caligraphic_A = 1.5</annotation></semantics></math> (cf. eqn. <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S3.E18" title="In 3 Strain power sensitivity ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">18</span></a>).</p> </div> </section> <section class="ltx_appendix" id="A2"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix B </span>Refraction and diffraction</h2> <div class="ltx_para" id="A2.p1"> <p class="ltx_p" id="A2.p1.5">Since Appendix <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#A1" title="Appendix A Dispersion ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">A</span></a> showed that dispersion (which scales as <math alttext="\phi_{\rm disp}\propto\lambda" class="ltx_Math" display="inline" id="A2.p1.1.m1.1"><semantics id="A2.p1.1.m1.1a"><mrow id="A2.p1.1.m1.1.1" xref="A2.p1.1.m1.1.1.cmml"><msub id="A2.p1.1.m1.1.1.2" xref="A2.p1.1.m1.1.1.2.cmml"><mi id="A2.p1.1.m1.1.1.2.2" xref="A2.p1.1.m1.1.1.2.2.cmml">ϕ</mi><mi id="A2.p1.1.m1.1.1.2.3" xref="A2.p1.1.m1.1.1.2.3.cmml">disp</mi></msub><mo id="A2.p1.1.m1.1.1.1" xref="A2.p1.1.m1.1.1.1.cmml">∝</mo><mi id="A2.p1.1.m1.1.1.3" xref="A2.p1.1.m1.1.1.3.cmml">λ</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.p1.1.m1.1b"><apply id="A2.p1.1.m1.1.1.cmml" xref="A2.p1.1.m1.1.1"><csymbol cd="latexml" id="A2.p1.1.m1.1.1.1.cmml" xref="A2.p1.1.m1.1.1.1">proportional-to</csymbol><apply id="A2.p1.1.m1.1.1.2.cmml" xref="A2.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="A2.p1.1.m1.1.1.2.1.cmml" xref="A2.p1.1.m1.1.1.2">subscript</csymbol><ci id="A2.p1.1.m1.1.1.2.2.cmml" xref="A2.p1.1.m1.1.1.2.2">italic-ϕ</ci><ci id="A2.p1.1.m1.1.1.2.3.cmml" xref="A2.p1.1.m1.1.1.2.3">disp</ci></apply><ci id="A2.p1.1.m1.1.1.3.cmml" xref="A2.p1.1.m1.1.1.3">𝜆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p1.1.m1.1c">\phi_{\rm disp}\propto\lambda</annotation><annotation encoding="application/x-llamapun" id="A2.p1.1.m1.1d">italic_ϕ start_POSTSUBSCRIPT roman_disp end_POSTSUBSCRIPT ∝ italic_λ</annotation></semantics></math>) can be perfectly removed, the next question is how good of an assumption is it that <math alttext="\phi_{\rm disp}\propto\lambda" class="ltx_Math" display="inline" id="A2.p1.2.m2.1"><semantics id="A2.p1.2.m2.1a"><mrow id="A2.p1.2.m2.1.1" xref="A2.p1.2.m2.1.1.cmml"><msub id="A2.p1.2.m2.1.1.2" xref="A2.p1.2.m2.1.1.2.cmml"><mi id="A2.p1.2.m2.1.1.2.2" xref="A2.p1.2.m2.1.1.2.2.cmml">ϕ</mi><mi id="A2.p1.2.m2.1.1.2.3" xref="A2.p1.2.m2.1.1.2.3.cmml">disp</mi></msub><mo id="A2.p1.2.m2.1.1.1" xref="A2.p1.2.m2.1.1.1.cmml">∝</mo><mi id="A2.p1.2.m2.1.1.3" xref="A2.p1.2.m2.1.1.3.cmml">λ</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.p1.2.m2.1b"><apply id="A2.p1.2.m2.1.1.cmml" xref="A2.p1.2.m2.1.1"><csymbol cd="latexml" id="A2.p1.2.m2.1.1.1.cmml" xref="A2.p1.2.m2.1.1.1">proportional-to</csymbol><apply id="A2.p1.2.m2.1.1.2.cmml" xref="A2.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="A2.p1.2.m2.1.1.2.1.cmml" xref="A2.p1.2.m2.1.1.2">subscript</csymbol><ci id="A2.p1.2.m2.1.1.2.2.cmml" xref="A2.p1.2.m2.1.1.2.2">italic-ϕ</ci><ci id="A2.p1.2.m2.1.1.2.3.cmml" xref="A2.p1.2.m2.1.1.2.3">disp</ci></apply><ci id="A2.p1.2.m2.1.1.3.cmml" xref="A2.p1.2.m2.1.1.3">𝜆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p1.2.m2.1c">\phi_{\rm disp}\propto\lambda</annotation><annotation encoding="application/x-llamapun" id="A2.p1.2.m2.1d">italic_ϕ start_POSTSUBSCRIPT roman_disp end_POSTSUBSCRIPT ∝ italic_λ</annotation></semantics></math>? For a homogeneous plasma, the correction to this long-wavelength scaling of the dispersion relation is suppressed by the ratio of the plasma frequency to the frequency squared. Since the plasma frequency for interplanetary gas is very small (<math alttext="\sim 10" class="ltx_Math" display="inline" id="A2.p1.3.m3.1"><semantics id="A2.p1.3.m3.1a"><mrow id="A2.p1.3.m3.1.1" xref="A2.p1.3.m3.1.1.cmml"><mi id="A2.p1.3.m3.1.1.2" xref="A2.p1.3.m3.1.1.2.cmml"></mi><mo id="A2.p1.3.m3.1.1.1" xref="A2.p1.3.m3.1.1.1.cmml">∼</mo><mn id="A2.p1.3.m3.1.1.3" xref="A2.p1.3.m3.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.p1.3.m3.1b"><apply id="A2.p1.3.m3.1.1.cmml" xref="A2.p1.3.m3.1.1"><csymbol cd="latexml" id="A2.p1.3.m3.1.1.1.cmml" xref="A2.p1.3.m3.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="A2.p1.3.m3.1.1.2.cmml" xref="A2.p1.3.m3.1.1.2">absent</csymbol><cn id="A2.p1.3.m3.1.1.3.cmml" type="integer" xref="A2.p1.3.m3.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p1.3.m3.1c">\sim 10</annotation><annotation encoding="application/x-llamapun" id="A2.p1.3.m3.1d">∼ 10</annotation></semantics></math> kHz) relative to the radio-wave frequencies of interest (<math alttext="\sim 10" class="ltx_Math" display="inline" id="A2.p1.4.m4.1"><semantics id="A2.p1.4.m4.1a"><mrow id="A2.p1.4.m4.1.1" xref="A2.p1.4.m4.1.1.cmml"><mi id="A2.p1.4.m4.1.1.2" xref="A2.p1.4.m4.1.1.2.cmml"></mi><mo id="A2.p1.4.m4.1.1.1" xref="A2.p1.4.m4.1.1.1.cmml">∼</mo><mn id="A2.p1.4.m4.1.1.3" xref="A2.p1.4.m4.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.p1.4.m4.1b"><apply id="A2.p1.4.m4.1.1.cmml" xref="A2.p1.4.m4.1.1"><csymbol cd="latexml" id="A2.p1.4.m4.1.1.1.cmml" xref="A2.p1.4.m4.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="A2.p1.4.m4.1.1.2.cmml" xref="A2.p1.4.m4.1.1.2">absent</csymbol><cn id="A2.p1.4.m4.1.1.3.cmml" type="integer" xref="A2.p1.4.m4.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p1.4.m4.1c">\sim 10</annotation><annotation encoding="application/x-llamapun" id="A2.p1.4.m4.1d">∼ 10</annotation></semantics></math> GHz), this correction to the <math alttext="\phi_{\rm disp}\propto\lambda" class="ltx_Math" display="inline" id="A2.p1.5.m5.1"><semantics id="A2.p1.5.m5.1a"><mrow id="A2.p1.5.m5.1.1" xref="A2.p1.5.m5.1.1.cmml"><msub id="A2.p1.5.m5.1.1.2" xref="A2.p1.5.m5.1.1.2.cmml"><mi id="A2.p1.5.m5.1.1.2.2" xref="A2.p1.5.m5.1.1.2.2.cmml">ϕ</mi><mi id="A2.p1.5.m5.1.1.2.3" xref="A2.p1.5.m5.1.1.2.3.cmml">disp</mi></msub><mo id="A2.p1.5.m5.1.1.1" xref="A2.p1.5.m5.1.1.1.cmml">∝</mo><mi id="A2.p1.5.m5.1.1.3" xref="A2.p1.5.m5.1.1.3.cmml">λ</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.p1.5.m5.1b"><apply id="A2.p1.5.m5.1.1.cmml" xref="A2.p1.5.m5.1.1"><csymbol cd="latexml" id="A2.p1.5.m5.1.1.1.cmml" xref="A2.p1.5.m5.1.1.1">proportional-to</csymbol><apply id="A2.p1.5.m5.1.1.2.cmml" xref="A2.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="A2.p1.5.m5.1.1.2.1.cmml" xref="A2.p1.5.m5.1.1.2">subscript</csymbol><ci id="A2.p1.5.m5.1.1.2.2.cmml" xref="A2.p1.5.m5.1.1.2.2">italic-ϕ</ci><ci id="A2.p1.5.m5.1.1.2.3.cmml" xref="A2.p1.5.m5.1.1.2.3">disp</ci></apply><ci id="A2.p1.5.m5.1.1.3.cmml" xref="A2.p1.5.m5.1.1.3">𝜆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p1.5.m5.1c">\phi_{\rm disp}\propto\lambda</annotation><annotation encoding="application/x-llamapun" id="A2.p1.5.m5.1d">italic_ϕ start_POSTSUBSCRIPT roman_disp end_POSTSUBSCRIPT ∝ italic_λ</annotation></semantics></math> scaling is extremely small.</p> </div> <div class="ltx_para" id="A2.p2"> <p class="ltx_p" id="A2.p2.7">A potentially larger effect is that the radio waves at different frequencies travel different paths owing to diffraction or refraction caused by the density inhomogeneities in the solar wind. Diffraction – or multipath propagation – results from phase fluctuations below the Fresnel scale <math alttext="\sim\sqrt{\lambda L/(4\pi)}" class="ltx_Math" display="inline" id="A2.p2.1.m1.1"><semantics id="A2.p2.1.m1.1a"><mrow id="A2.p2.1.m1.1.2" xref="A2.p2.1.m1.1.2.cmml"><mi id="A2.p2.1.m1.1.2.2" xref="A2.p2.1.m1.1.2.2.cmml"></mi><mo id="A2.p2.1.m1.1.2.1" xref="A2.p2.1.m1.1.2.1.cmml">∼</mo><msqrt id="A2.p2.1.m1.1.1" xref="A2.p2.1.m1.1.1.cmml"><mrow id="A2.p2.1.m1.1.1.1" xref="A2.p2.1.m1.1.1.1.cmml"><mrow id="A2.p2.1.m1.1.1.1.3" xref="A2.p2.1.m1.1.1.1.3.cmml"><mi id="A2.p2.1.m1.1.1.1.3.2" xref="A2.p2.1.m1.1.1.1.3.2.cmml">λ</mi><mo id="A2.p2.1.m1.1.1.1.3.1" xref="A2.p2.1.m1.1.1.1.3.1.cmml">⁢</mo><mi id="A2.p2.1.m1.1.1.1.3.3" xref="A2.p2.1.m1.1.1.1.3.3.cmml">L</mi></mrow><mo id="A2.p2.1.m1.1.1.1.2" xref="A2.p2.1.m1.1.1.1.2.cmml">/</mo><mrow id="A2.p2.1.m1.1.1.1.1.1" xref="A2.p2.1.m1.1.1.1.1.1.1.cmml"><mo id="A2.p2.1.m1.1.1.1.1.1.2" stretchy="false" xref="A2.p2.1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="A2.p2.1.m1.1.1.1.1.1.1" xref="A2.p2.1.m1.1.1.1.1.1.1.cmml"><mn id="A2.p2.1.m1.1.1.1.1.1.1.2" xref="A2.p2.1.m1.1.1.1.1.1.1.2.cmml">4</mn><mo id="A2.p2.1.m1.1.1.1.1.1.1.1" xref="A2.p2.1.m1.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="A2.p2.1.m1.1.1.1.1.1.1.3" xref="A2.p2.1.m1.1.1.1.1.1.1.3.cmml">π</mi></mrow><mo id="A2.p2.1.m1.1.1.1.1.1.3" stretchy="false" xref="A2.p2.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></msqrt></mrow><annotation-xml encoding="MathML-Content" id="A2.p2.1.m1.1b"><apply id="A2.p2.1.m1.1.2.cmml" xref="A2.p2.1.m1.1.2"><csymbol cd="latexml" id="A2.p2.1.m1.1.2.1.cmml" xref="A2.p2.1.m1.1.2.1">similar-to</csymbol><csymbol cd="latexml" id="A2.p2.1.m1.1.2.2.cmml" xref="A2.p2.1.m1.1.2.2">absent</csymbol><apply id="A2.p2.1.m1.1.1.cmml" xref="A2.p2.1.m1.1.1"><root id="A2.p2.1.m1.1.1a.cmml" xref="A2.p2.1.m1.1.1"></root><apply id="A2.p2.1.m1.1.1.1.cmml" xref="A2.p2.1.m1.1.1.1"><divide id="A2.p2.1.m1.1.1.1.2.cmml" xref="A2.p2.1.m1.1.1.1.2"></divide><apply id="A2.p2.1.m1.1.1.1.3.cmml" xref="A2.p2.1.m1.1.1.1.3"><times id="A2.p2.1.m1.1.1.1.3.1.cmml" xref="A2.p2.1.m1.1.1.1.3.1"></times><ci id="A2.p2.1.m1.1.1.1.3.2.cmml" xref="A2.p2.1.m1.1.1.1.3.2">𝜆</ci><ci id="A2.p2.1.m1.1.1.1.3.3.cmml" xref="A2.p2.1.m1.1.1.1.3.3">𝐿</ci></apply><apply id="A2.p2.1.m1.1.1.1.1.1.1.cmml" xref="A2.p2.1.m1.1.1.1.1.1"><times id="A2.p2.1.m1.1.1.1.1.1.1.1.cmml" xref="A2.p2.1.m1.1.1.1.1.1.1.1"></times><cn id="A2.p2.1.m1.1.1.1.1.1.1.2.cmml" type="integer" xref="A2.p2.1.m1.1.1.1.1.1.1.2">4</cn><ci id="A2.p2.1.m1.1.1.1.1.1.1.3.cmml" xref="A2.p2.1.m1.1.1.1.1.1.1.3">𝜋</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p2.1.m1.1c">\sim\sqrt{\lambda L/(4\pi)}</annotation><annotation encoding="application/x-llamapun" id="A2.p2.1.m1.1d">∼ square-root start_ARG italic_λ italic_L / ( 4 italic_π ) end_ARG</annotation></semantics></math>, which is the scale perpendicular to the sightline over which most light paths travel that contribute constructively to the image. For the case at hand, the Fresnel scale corresponds to perturbations in the solar wind that pass the spacecraft with frequencies of <math alttext="\omega_{F}\sim v_{\rm sw}/\sqrt{\lambda L/(4\pi)}\sim 10\,(L/30{\rm\,AU}\times% \lambda/1\,{\rm cm})^{1/2}~{}" class="ltx_Math" display="inline" id="A2.p2.2.m2.2"><semantics id="A2.p2.2.m2.2a"><mrow id="A2.p2.2.m2.2.2" xref="A2.p2.2.m2.2.2.cmml"><msub id="A2.p2.2.m2.2.2.3" xref="A2.p2.2.m2.2.2.3.cmml"><mi id="A2.p2.2.m2.2.2.3.2" xref="A2.p2.2.m2.2.2.3.2.cmml">ω</mi><mi id="A2.p2.2.m2.2.2.3.3" xref="A2.p2.2.m2.2.2.3.3.cmml">F</mi></msub><mo id="A2.p2.2.m2.2.2.4" xref="A2.p2.2.m2.2.2.4.cmml">∼</mo><mrow id="A2.p2.2.m2.2.2.5" xref="A2.p2.2.m2.2.2.5.cmml"><msub id="A2.p2.2.m2.2.2.5.2" xref="A2.p2.2.m2.2.2.5.2.cmml"><mi id="A2.p2.2.m2.2.2.5.2.2" xref="A2.p2.2.m2.2.2.5.2.2.cmml">v</mi><mi id="A2.p2.2.m2.2.2.5.2.3" xref="A2.p2.2.m2.2.2.5.2.3.cmml">sw</mi></msub><mo id="A2.p2.2.m2.2.2.5.1" xref="A2.p2.2.m2.2.2.5.1.cmml">/</mo><msqrt 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xref="A2.p2.2.m2.2.2.1.1.3.3">2</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p2.2.m2.2c">\omega_{F}\sim v_{\rm sw}/\sqrt{\lambda L/(4\pi)}\sim 10\,(L/30{\rm\,AU}\times% \lambda/1\,{\rm cm})^{1/2}~{}</annotation><annotation encoding="application/x-llamapun" id="A2.p2.2.m2.2d">italic_ω start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ∼ italic_v start_POSTSUBSCRIPT roman_sw end_POSTSUBSCRIPT / square-root start_ARG italic_λ italic_L / ( 4 italic_π ) end_ARG ∼ 10 ( italic_L / 30 roman_AU × italic_λ / 1 roman_cm ) start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT</annotation></semantics></math>Hz. We can use our estimate for the RMS phase fluctuations (eqn. <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#A1.E8" title="In Appendix A Dispersion ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">A8</span></a>), evaluating at the fiducial parameters where the parentheses in equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#A1.E9" title="In Appendix A Dispersion ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">A9</span></a>) evaluate to unity, the phase fluctuations over for <math alttext="\omega=10\," class="ltx_Math" display="inline" id="A2.p2.3.m3.1"><semantics id="A2.p2.3.m3.1a"><mrow id="A2.p2.3.m3.1.1" xref="A2.p2.3.m3.1.1.cmml"><mi id="A2.p2.3.m3.1.1.2" xref="A2.p2.3.m3.1.1.2.cmml">ω</mi><mo id="A2.p2.3.m3.1.1.1" xref="A2.p2.3.m3.1.1.1.cmml">=</mo><mn id="A2.p2.3.m3.1.1.3" xref="A2.p2.3.m3.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.p2.3.m3.1b"><apply id="A2.p2.3.m3.1.1.cmml" xref="A2.p2.3.m3.1.1"><eq id="A2.p2.3.m3.1.1.1.cmml" xref="A2.p2.3.m3.1.1.1"></eq><ci id="A2.p2.3.m3.1.1.2.cmml" xref="A2.p2.3.m3.1.1.2">𝜔</ci><cn id="A2.p2.3.m3.1.1.3.cmml" type="integer" xref="A2.p2.3.m3.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p2.3.m3.1c">\omega=10\,</annotation><annotation encoding="application/x-llamapun" id="A2.p2.3.m3.1d">italic_ω = 10</annotation></semantics></math>Hz are <math alttext="f^{1/2}\widetilde{\phi}_{\rm disp,rms}\sim 10^{-7}" class="ltx_Math" display="inline" id="A2.p2.4.m4.2"><semantics id="A2.p2.4.m4.2a"><mrow id="A2.p2.4.m4.2.3" xref="A2.p2.4.m4.2.3.cmml"><mrow id="A2.p2.4.m4.2.3.2" xref="A2.p2.4.m4.2.3.2.cmml"><msup id="A2.p2.4.m4.2.3.2.2" xref="A2.p2.4.m4.2.3.2.2.cmml"><mi id="A2.p2.4.m4.2.3.2.2.2" xref="A2.p2.4.m4.2.3.2.2.2.cmml">f</mi><mrow id="A2.p2.4.m4.2.3.2.2.3" xref="A2.p2.4.m4.2.3.2.2.3.cmml"><mn id="A2.p2.4.m4.2.3.2.2.3.2" xref="A2.p2.4.m4.2.3.2.2.3.2.cmml">1</mn><mo id="A2.p2.4.m4.2.3.2.2.3.1" xref="A2.p2.4.m4.2.3.2.2.3.1.cmml">/</mo><mn id="A2.p2.4.m4.2.3.2.2.3.3" xref="A2.p2.4.m4.2.3.2.2.3.3.cmml">2</mn></mrow></msup><mo id="A2.p2.4.m4.2.3.2.1" xref="A2.p2.4.m4.2.3.2.1.cmml">⁢</mo><msub id="A2.p2.4.m4.2.3.2.3" xref="A2.p2.4.m4.2.3.2.3.cmml"><mover accent="true" id="A2.p2.4.m4.2.3.2.3.2" xref="A2.p2.4.m4.2.3.2.3.2.cmml"><mi id="A2.p2.4.m4.2.3.2.3.2.2" xref="A2.p2.4.m4.2.3.2.3.2.2.cmml">ϕ</mi><mo id="A2.p2.4.m4.2.3.2.3.2.1" xref="A2.p2.4.m4.2.3.2.3.2.1.cmml">~</mo></mover><mrow id="A2.p2.4.m4.2.2.2.4" xref="A2.p2.4.m4.2.2.2.3.cmml"><mi id="A2.p2.4.m4.1.1.1.1" xref="A2.p2.4.m4.1.1.1.1.cmml">disp</mi><mo id="A2.p2.4.m4.2.2.2.4.1" xref="A2.p2.4.m4.2.2.2.3.cmml">,</mo><mi id="A2.p2.4.m4.2.2.2.2" xref="A2.p2.4.m4.2.2.2.2.cmml">rms</mi></mrow></msub></mrow><mo id="A2.p2.4.m4.2.3.1" 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id="A2.p2.4.m4.2.3.3.1.cmml" xref="A2.p2.4.m4.2.3.3">superscript</csymbol><cn id="A2.p2.4.m4.2.3.3.2.cmml" type="integer" xref="A2.p2.4.m4.2.3.3.2">10</cn><apply id="A2.p2.4.m4.2.3.3.3.cmml" xref="A2.p2.4.m4.2.3.3.3"><minus id="A2.p2.4.m4.2.3.3.3.1.cmml" xref="A2.p2.4.m4.2.3.3.3"></minus><cn id="A2.p2.4.m4.2.3.3.3.2.cmml" type="integer" xref="A2.p2.4.m4.2.3.3.3.2">7</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p2.4.m4.2c">f^{1/2}\widetilde{\phi}_{\rm disp,rms}\sim 10^{-7}</annotation><annotation encoding="application/x-llamapun" id="A2.p2.4.m4.2d">italic_f start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT over~ start_ARG italic_ϕ end_ARG start_POSTSUBSCRIPT roman_disp , roman_rms end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 7 end_POSTSUPERSCRIPT</annotation></semantics></math> and the leading-order contribution of diffraction to the phase scales quadratically in <math alttext="f^{1/2}\widetilde{\phi}_{\rm disp,rms}" class="ltx_Math" display="inline" id="A2.p2.5.m5.2"><semantics id="A2.p2.5.m5.2a"><mrow id="A2.p2.5.m5.2.3" xref="A2.p2.5.m5.2.3.cmml"><msup id="A2.p2.5.m5.2.3.2" xref="A2.p2.5.m5.2.3.2.cmml"><mi id="A2.p2.5.m5.2.3.2.2" xref="A2.p2.5.m5.2.3.2.2.cmml">f</mi><mrow id="A2.p2.5.m5.2.3.2.3" xref="A2.p2.5.m5.2.3.2.3.cmml"><mn id="A2.p2.5.m5.2.3.2.3.2" xref="A2.p2.5.m5.2.3.2.3.2.cmml">1</mn><mo id="A2.p2.5.m5.2.3.2.3.1" xref="A2.p2.5.m5.2.3.2.3.1.cmml">/</mo><mn id="A2.p2.5.m5.2.3.2.3.3" xref="A2.p2.5.m5.2.3.2.3.3.cmml">2</mn></mrow></msup><mo id="A2.p2.5.m5.2.3.1" xref="A2.p2.5.m5.2.3.1.cmml">⁢</mo><msub id="A2.p2.5.m5.2.3.3" xref="A2.p2.5.m5.2.3.3.cmml"><mover accent="true" id="A2.p2.5.m5.2.3.3.2" xref="A2.p2.5.m5.2.3.3.2.cmml"><mi id="A2.p2.5.m5.2.3.3.2.2" xref="A2.p2.5.m5.2.3.3.2.2.cmml">ϕ</mi><mo id="A2.p2.5.m5.2.3.3.2.1" xref="A2.p2.5.m5.2.3.3.2.1.cmml">~</mo></mover><mrow id="A2.p2.5.m5.2.2.2.4" xref="A2.p2.5.m5.2.2.2.3.cmml"><mi id="A2.p2.5.m5.1.1.1.1" xref="A2.p2.5.m5.1.1.1.1.cmml">disp</mi><mo 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xref="A2.p2.5.m5.2.3.3">subscript</csymbol><apply id="A2.p2.5.m5.2.3.3.2.cmml" xref="A2.p2.5.m5.2.3.3.2"><ci id="A2.p2.5.m5.2.3.3.2.1.cmml" xref="A2.p2.5.m5.2.3.3.2.1">~</ci><ci id="A2.p2.5.m5.2.3.3.2.2.cmml" xref="A2.p2.5.m5.2.3.3.2.2">italic-ϕ</ci></apply><list id="A2.p2.5.m5.2.2.2.3.cmml" xref="A2.p2.5.m5.2.2.2.4"><ci id="A2.p2.5.m5.1.1.1.1.cmml" xref="A2.p2.5.m5.1.1.1.1">disp</ci><ci id="A2.p2.5.m5.2.2.2.2.cmml" xref="A2.p2.5.m5.2.2.2.2">rms</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p2.5.m5.2c">f^{1/2}\widetilde{\phi}_{\rm disp,rms}</annotation><annotation encoding="application/x-llamapun" id="A2.p2.5.m5.2d">italic_f start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT over~ start_ARG italic_ϕ end_ARG start_POSTSUBSCRIPT roman_disp , roman_rms end_POSTSUBSCRIPT</annotation></semantics></math> (as the linear order cancels since both positive and negative sub-Fresnel phase fluctuations contribute to the total). This suggests that the refractive phase fluctuations should be an absolutely negligible <math alttext="\sim(f^{1/2}\widetilde{\phi}_{\rm disp,rms})^{2}\sim 10^{-14}" class="ltx_Math" display="inline" id="A2.p2.6.m6.3"><semantics id="A2.p2.6.m6.3a"><mrow id="A2.p2.6.m6.3.3" xref="A2.p2.6.m6.3.3.cmml"><mi id="A2.p2.6.m6.3.3.3" xref="A2.p2.6.m6.3.3.3.cmml"></mi><mo id="A2.p2.6.m6.3.3.4" xref="A2.p2.6.m6.3.3.4.cmml">∼</mo><msup id="A2.p2.6.m6.3.3.1" xref="A2.p2.6.m6.3.3.1.cmml"><mrow id="A2.p2.6.m6.3.3.1.1.1" xref="A2.p2.6.m6.3.3.1.1.1.1.cmml"><mo id="A2.p2.6.m6.3.3.1.1.1.2" stretchy="false" xref="A2.p2.6.m6.3.3.1.1.1.1.cmml">(</mo><mrow id="A2.p2.6.m6.3.3.1.1.1.1" xref="A2.p2.6.m6.3.3.1.1.1.1.cmml"><msup id="A2.p2.6.m6.3.3.1.1.1.1.2" xref="A2.p2.6.m6.3.3.1.1.1.1.2.cmml"><mi id="A2.p2.6.m6.3.3.1.1.1.1.2.2" xref="A2.p2.6.m6.3.3.1.1.1.1.2.2.cmml">f</mi><mrow id="A2.p2.6.m6.3.3.1.1.1.1.2.3" xref="A2.p2.6.m6.3.3.1.1.1.1.2.3.cmml"><mn id="A2.p2.6.m6.3.3.1.1.1.1.2.3.2" 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xref="A2.p2.6.m6.3.3.6"><csymbol cd="ambiguous" id="A2.p2.6.m6.3.3.6.1.cmml" xref="A2.p2.6.m6.3.3.6">superscript</csymbol><cn id="A2.p2.6.m6.3.3.6.2.cmml" type="integer" xref="A2.p2.6.m6.3.3.6.2">10</cn><apply id="A2.p2.6.m6.3.3.6.3.cmml" xref="A2.p2.6.m6.3.3.6.3"><minus id="A2.p2.6.m6.3.3.6.3.1.cmml" xref="A2.p2.6.m6.3.3.6.3"></minus><cn id="A2.p2.6.m6.3.3.6.3.2.cmml" type="integer" xref="A2.p2.6.m6.3.3.6.3.2">14</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p2.6.m6.3c">\sim(f^{1/2}\widetilde{\phi}_{\rm disp,rms})^{2}\sim 10^{-14}</annotation><annotation encoding="application/x-llamapun" id="A2.p2.6.m6.3d">∼ ( italic_f start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT over~ start_ARG italic_ϕ end_ARG start_POSTSUBSCRIPT roman_disp , roman_rms end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 14 end_POSTSUPERSCRIPT</annotation></semantics></math>. Furthermore, the sub-Fresnel fluctuations that drive the diffractive phase perturbations will decorrelate on timescales of <math alttext="\omega_{F}^{-1}" class="ltx_Math" display="inline" id="A2.p2.7.m7.1"><semantics id="A2.p2.7.m7.1a"><msubsup id="A2.p2.7.m7.1.1" xref="A2.p2.7.m7.1.1.cmml"><mi id="A2.p2.7.m7.1.1.2.2" xref="A2.p2.7.m7.1.1.2.2.cmml">ω</mi><mi id="A2.p2.7.m7.1.1.2.3" xref="A2.p2.7.m7.1.1.2.3.cmml">F</mi><mrow id="A2.p2.7.m7.1.1.3" xref="A2.p2.7.m7.1.1.3.cmml"><mo id="A2.p2.7.m7.1.1.3a" xref="A2.p2.7.m7.1.1.3.cmml">−</mo><mn id="A2.p2.7.m7.1.1.3.2" xref="A2.p2.7.m7.1.1.3.2.cmml">1</mn></mrow></msubsup><annotation-xml encoding="MathML-Content" id="A2.p2.7.m7.1b"><apply id="A2.p2.7.m7.1.1.cmml" xref="A2.p2.7.m7.1.1"><csymbol cd="ambiguous" id="A2.p2.7.m7.1.1.1.cmml" xref="A2.p2.7.m7.1.1">superscript</csymbol><apply id="A2.p2.7.m7.1.1.2.cmml" xref="A2.p2.7.m7.1.1"><csymbol cd="ambiguous" id="A2.p2.7.m7.1.1.2.1.cmml" xref="A2.p2.7.m7.1.1">subscript</csymbol><ci id="A2.p2.7.m7.1.1.2.2.cmml" xref="A2.p2.7.m7.1.1.2.2">𝜔</ci><ci id="A2.p2.7.m7.1.1.2.3.cmml" xref="A2.p2.7.m7.1.1.2.3">𝐹</ci></apply><apply id="A2.p2.7.m7.1.1.3.cmml" xref="A2.p2.7.m7.1.1.3"><minus id="A2.p2.7.m7.1.1.3.1.cmml" xref="A2.p2.7.m7.1.1.3"></minus><cn id="A2.p2.7.m7.1.1.3.2.cmml" type="integer" xref="A2.p2.7.m7.1.1.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p2.7.m7.1c">\omega_{F}^{-1}</annotation><annotation encoding="application/x-llamapun" id="A2.p2.7.m7.1d">italic_ω start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>, leading to further suppression over this estimate. Thus, phase fluctuations from diffraction are likely to be extremely small.</p> </div> <div class="ltx_para" id="A2.p3"> <p class="ltx_p" id="A2.p3.1">Refraction is the displacement of images from a super-Fresnel density gradient. Refraction will contribute a larger effect since the phase perturbations in the solar wind are larger for <math alttext="\omega\ll\omega_{F}" class="ltx_Math" display="inline" id="A2.p3.1.m1.1"><semantics id="A2.p3.1.m1.1a"><mrow id="A2.p3.1.m1.1.1" xref="A2.p3.1.m1.1.1.cmml"><mi id="A2.p3.1.m1.1.1.2" xref="A2.p3.1.m1.1.1.2.cmml">ω</mi><mo id="A2.p3.1.m1.1.1.1" xref="A2.p3.1.m1.1.1.1.cmml">≪</mo><msub id="A2.p3.1.m1.1.1.3" xref="A2.p3.1.m1.1.1.3.cmml"><mi id="A2.p3.1.m1.1.1.3.2" xref="A2.p3.1.m1.1.1.3.2.cmml">ω</mi><mi id="A2.p3.1.m1.1.1.3.3" xref="A2.p3.1.m1.1.1.3.3.cmml">F</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A2.p3.1.m1.1b"><apply id="A2.p3.1.m1.1.1.cmml" xref="A2.p3.1.m1.1.1"><csymbol cd="latexml" id="A2.p3.1.m1.1.1.1.cmml" xref="A2.p3.1.m1.1.1.1">much-less-than</csymbol><ci id="A2.p3.1.m1.1.1.2.cmml" xref="A2.p3.1.m1.1.1.2">𝜔</ci><apply id="A2.p3.1.m1.1.1.3.cmml" xref="A2.p3.1.m1.1.1.3"><csymbol cd="ambiguous" id="A2.p3.1.m1.1.1.3.1.cmml" xref="A2.p3.1.m1.1.1.3">subscript</csymbol><ci id="A2.p3.1.m1.1.1.3.2.cmml" xref="A2.p3.1.m1.1.1.3.2">𝜔</ci><ci id="A2.p3.1.m1.1.1.3.3.cmml" xref="A2.p3.1.m1.1.1.3.3">𝐹</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p3.1.m1.1c">\omega\ll\omega_{F}</annotation><annotation encoding="application/x-llamapun" id="A2.p3.1.m1.1d">italic_ω ≪ italic_ω start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT</annotation></semantics></math>, and the longer wavenumbers of the participating modes mean there will be less decorrelation compared to diffraction. However, in the following, we show that refraction also sources a negligible amount of phase noise.</p> </div> <div class="ltx_para" id="A2.p4"> <p class="ltx_p" id="A2.p4.1">To compute the effect of refraction on the phase, let us assume that all phase is acquired halfway between the two spacecraft in an arm, an assumption that will put an upper bound on refractive-phase fluctuations. The electric field’s phase received at one receiver at wavelength <math alttext="\lambda" class="ltx_Math" display="inline" id="A2.p4.1.m1.1"><semantics id="A2.p4.1.m1.1a"><mi id="A2.p4.1.m1.1.1" xref="A2.p4.1.m1.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="A2.p4.1.m1.1b"><ci id="A2.p4.1.m1.1.1.cmml" xref="A2.p4.1.m1.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.p4.1.m1.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="A2.p4.1.m1.1d">italic_λ</annotation></semantics></math> is</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="A3.EGx11"> <tbody id="A2.E1"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\phi_{\lambda}" class="ltx_Math" display="inline" id="A2.E1.m1.1"><semantics id="A2.E1.m1.1a"><msub id="A2.E1.m1.1.1" xref="A2.E1.m1.1.1.cmml"><mi id="A2.E1.m1.1.1.2" xref="A2.E1.m1.1.1.2.cmml">ϕ</mi><mi id="A2.E1.m1.1.1.3" xref="A2.E1.m1.1.1.3.cmml">λ</mi></msub><annotation-xml encoding="MathML-Content" id="A2.E1.m1.1b"><apply id="A2.E1.m1.1.1.cmml" xref="A2.E1.m1.1.1"><csymbol cd="ambiguous" id="A2.E1.m1.1.1.1.cmml" xref="A2.E1.m1.1.1">subscript</csymbol><ci id="A2.E1.m1.1.1.2.cmml" xref="A2.E1.m1.1.1.2">italic-ϕ</ci><ci id="A2.E1.m1.1.1.3.cmml" xref="A2.E1.m1.1.1.3">𝜆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E1.m1.1c">\displaystyle\phi_{\lambda}</annotation><annotation encoding="application/x-llamapun" id="A2.E1.m1.1d">italic_ϕ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="A2.E1.m2.1"><semantics id="A2.E1.m2.1a"><mo id="A2.E1.m2.1.1" xref="A2.E1.m2.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="A2.E1.m2.1b"><eq 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id="A2.E1.m3.4.4.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="A2.E1.m3.4.4.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2">𝜆</ci><ci id="A2.E1.m3.4.4.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="A2.E1.m3.4.4.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.3">𝑗</ci></apply><ci id="A2.E1.m3.1.1.cmml" xref="A2.E1.m3.1.1">𝒙</ci></interval></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E1.m3.4c">\displaystyle\arg\left(\frac{1}{2\pi ir_{F}^{2}}\int d^{2}\boldsymbol{x}\exp% \left[i\frac{\boldsymbol{x}^{2}}{2r_{F}^{2}}+i\phi_{d}(\lambda_{j},\boldsymbol% {x})\right]\right).</annotation><annotation encoding="application/x-llamapun" id="A2.E1.m3.4d">roman_arg ( divide start_ARG 1 end_ARG start_ARG 2 italic_π italic_i italic_r start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ∫ italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_italic_x roman_exp [ italic_i divide start_ARG bold_italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 italic_r start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG + italic_i italic_ϕ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_λ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , bold_italic_x ) ] ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(B1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A2.p4.2">The integral is over the plane halfway between the two spacecraft and its support comes mostly from around the Fresnel length defined as <math alttext="r_{F}^{2}\equiv\lambda L/(4\pi)" class="ltx_Math" display="inline" id="A2.p4.2.m1.1"><semantics id="A2.p4.2.m1.1a"><mrow id="A2.p4.2.m1.1.1" xref="A2.p4.2.m1.1.1.cmml"><msubsup id="A2.p4.2.m1.1.1.3" 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type="integer" xref="A2.p4.2.m1.1.1.3.3">2</cn></apply><apply id="A2.p4.2.m1.1.1.1.cmml" xref="A2.p4.2.m1.1.1.1"><divide id="A2.p4.2.m1.1.1.1.2.cmml" xref="A2.p4.2.m1.1.1.1.2"></divide><apply id="A2.p4.2.m1.1.1.1.3.cmml" xref="A2.p4.2.m1.1.1.1.3"><times id="A2.p4.2.m1.1.1.1.3.1.cmml" xref="A2.p4.2.m1.1.1.1.3.1"></times><ci id="A2.p4.2.m1.1.1.1.3.2.cmml" xref="A2.p4.2.m1.1.1.1.3.2">𝜆</ci><ci id="A2.p4.2.m1.1.1.1.3.3.cmml" xref="A2.p4.2.m1.1.1.1.3.3">𝐿</ci></apply><apply id="A2.p4.2.m1.1.1.1.1.1.1.cmml" xref="A2.p4.2.m1.1.1.1.1.1"><times id="A2.p4.2.m1.1.1.1.1.1.1.1.cmml" xref="A2.p4.2.m1.1.1.1.1.1.1.1"></times><cn id="A2.p4.2.m1.1.1.1.1.1.1.2.cmml" type="integer" xref="A2.p4.2.m1.1.1.1.1.1.1.2">4</cn><ci id="A2.p4.2.m1.1.1.1.1.1.1.3.cmml" xref="A2.p4.2.m1.1.1.1.1.1.1.3">𝜋</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p4.2.m1.1c">r_{F}^{2}\equiv\lambda L/(4\pi)</annotation><annotation encoding="application/x-llamapun" id="A2.p4.2.m1.1d">italic_r start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≡ italic_λ italic_L / ( 4 italic_π )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="A2.p5"> <p class="ltx_p" id="A2.p5.3">Since refraction owes to the smooth contribution from longer wavelengths than <math alttext="r_{F}" class="ltx_Math" display="inline" id="A2.p5.1.m1.1"><semantics id="A2.p5.1.m1.1a"><msub id="A2.p5.1.m1.1.1" xref="A2.p5.1.m1.1.1.cmml"><mi id="A2.p5.1.m1.1.1.2" xref="A2.p5.1.m1.1.1.2.cmml">r</mi><mi id="A2.p5.1.m1.1.1.3" xref="A2.p5.1.m1.1.1.3.cmml">F</mi></msub><annotation-xml encoding="MathML-Content" id="A2.p5.1.m1.1b"><apply id="A2.p5.1.m1.1.1.cmml" xref="A2.p5.1.m1.1.1"><csymbol cd="ambiguous" id="A2.p5.1.m1.1.1.1.cmml" xref="A2.p5.1.m1.1.1">subscript</csymbol><ci id="A2.p5.1.m1.1.1.2.cmml" xref="A2.p5.1.m1.1.1.2">𝑟</ci><ci id="A2.p5.1.m1.1.1.3.cmml" xref="A2.p5.1.m1.1.1.3">𝐹</ci></apply></annotation-xml><annotation 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start_POSTSUBSCRIPT italic_d , italic_l end_POSTSUBSCRIPT + ∇ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ϕ start_POSTSUBSCRIPT italic_d , italic_l end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + ∇ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∇ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_ϕ start_POSTSUBSCRIPT italic_d , italic_l end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>, where all the coefficients are evaluated at <math alttext="\boldsymbol{x}=0" class="ltx_Math" display="inline" id="A2.p5.3.m3.1"><semantics id="A2.p5.3.m3.1a"><mrow id="A2.p5.3.m3.1.1" xref="A2.p5.3.m3.1.1.cmml"><mi id="A2.p5.3.m3.1.1.2" xref="A2.p5.3.m3.1.1.2.cmml">𝒙</mi><mo id="A2.p5.3.m3.1.1.1" xref="A2.p5.3.m3.1.1.1.cmml">=</mo><mn id="A2.p5.3.m3.1.1.3" xref="A2.p5.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.p5.3.m3.1b"><apply id="A2.p5.3.m3.1.1.cmml" xref="A2.p5.3.m3.1.1"><eq id="A2.p5.3.m3.1.1.1.cmml" xref="A2.p5.3.m3.1.1.1"></eq><ci id="A2.p5.3.m3.1.1.2.cmml" xref="A2.p5.3.m3.1.1.2">𝒙</ci><cn id="A2.p5.3.m3.1.1.3.cmml" type="integer" xref="A2.p5.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p5.3.m3.1c">\boldsymbol{x}=0</annotation><annotation encoding="application/x-llamapun" id="A2.p5.3.m3.1d">bold_italic_x = 0</annotation></semantics></math>. Equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#A2.E1" title="In Appendix B Refraction and diffraction ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">B1</span></a>) then evaluates to</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="A3.EGx12"> <tbody id="A2.E2"> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\phi_{\lambda}" class="ltx_Math" display="inline" id="A2.Ex1.m1.1"><semantics id="A2.Ex1.m1.1a"><msub id="A2.Ex1.m1.1.1" xref="A2.Ex1.m1.1.1.cmml"><mi id="A2.Ex1.m1.1.1.2" xref="A2.Ex1.m1.1.1.2.cmml">ϕ</mi><mi id="A2.Ex1.m1.1.1.3" xref="A2.Ex1.m1.1.1.3.cmml">λ</mi></msub><annotation-xml encoding="MathML-Content" id="A2.Ex1.m1.1b"><apply id="A2.Ex1.m1.1.1.cmml" xref="A2.Ex1.m1.1.1"><csymbol cd="ambiguous" id="A2.Ex1.m1.1.1.1.cmml" xref="A2.Ex1.m1.1.1">subscript</csymbol><ci id="A2.Ex1.m1.1.1.2.cmml" xref="A2.Ex1.m1.1.1.2">italic-ϕ</ci><ci id="A2.Ex1.m1.1.1.3.cmml" xref="A2.Ex1.m1.1.1.3">𝜆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex1.m1.1c">\displaystyle\phi_{\lambda}</annotation><annotation encoding="application/x-llamapun" id="A2.Ex1.m1.1d">italic_ϕ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle\subseteq" class="ltx_Math" display="inline" id="A2.Ex1.m2.1"><semantics id="A2.Ex1.m2.1a"><mo id="A2.Ex1.m2.1.1" xref="A2.Ex1.m2.1.1.cmml">⊆</mo><annotation-xml encoding="MathML-Content" id="A2.Ex1.m2.1b"><subset id="A2.Ex1.m2.1.1.cmml" xref="A2.Ex1.m2.1.1"></subset></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex1.m2.1c">\displaystyle\subseteq</annotation><annotation encoding="application/x-llamapun" id="A2.Ex1.m2.1d">⊆</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-\frac{r_{F}(\lambda)^{2}}{2}\left(\frac{\nabla_{i}\phi_{d,l}(% \boldsymbol{x})}{1+r_{F}(\lambda)^{2}\nabla_{i}^{2}\phi_{d,l}(\boldsymbol{x})}% \right)^{2}," class="ltx_Math" display="inline" id="A2.Ex1.m3.9"><semantics id="A2.Ex1.m3.9a"><mrow id="A2.Ex1.m3.9.9.1" xref="A2.Ex1.m3.9.9.1.1.cmml"><mrow id="A2.Ex1.m3.9.9.1.1" xref="A2.Ex1.m3.9.9.1.1.cmml"><mo id="A2.Ex1.m3.9.9.1.1a" xref="A2.Ex1.m3.9.9.1.1.cmml">−</mo><mrow id="A2.Ex1.m3.9.9.1.1.2" xref="A2.Ex1.m3.9.9.1.1.2.cmml"><mstyle displaystyle="true" id="A2.Ex1.m3.1.1" xref="A2.Ex1.m3.1.1.cmml"><mfrac id="A2.Ex1.m3.1.1a" xref="A2.Ex1.m3.1.1.cmml"><mrow id="A2.Ex1.m3.1.1.1" xref="A2.Ex1.m3.1.1.1.cmml"><msub id="A2.Ex1.m3.1.1.1.3" xref="A2.Ex1.m3.1.1.1.3.cmml"><mi id="A2.Ex1.m3.1.1.1.3.2" xref="A2.Ex1.m3.1.1.1.3.2.cmml">r</mi><mi id="A2.Ex1.m3.1.1.1.3.3" xref="A2.Ex1.m3.1.1.1.3.3.cmml">F</mi></msub><mo 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\right)^{2},</annotation><annotation encoding="application/x-llamapun" id="A2.Ex1.m3.9d">- divide start_ARG italic_r start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_λ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG ( divide start_ARG ∇ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ϕ start_POSTSUBSCRIPT italic_d , italic_l end_POSTSUBSCRIPT ( bold_italic_x ) end_ARG start_ARG 1 + italic_r start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_λ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∇ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ϕ start_POSTSUBSCRIPT italic_d , italic_l end_POSTSUBSCRIPT ( bold_italic_x ) end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="2"><span class="ltx_tag ltx_tag_equation 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id="A2.E2.m2.12c">\displaystyle-\frac{r_{F}(\lambda)^{2}}{2}\Big{|}\boldsymbol{\nabla}\phi_{d,l}% (\boldsymbol{x})\Big{|}^{2}+\frac{r_{F}(\lambda)^{4}}{2}\Big{(}\nabla_{i}\phi_% {d,l}(\boldsymbol{x})\Big{)}^{2}\nabla_{i}^{2}\phi_{d,l}(\boldsymbol{x})+...\ ,</annotation><annotation encoding="application/x-llamapun" id="A2.E2.m2.12d">- divide start_ARG italic_r start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_λ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG | bold_∇ italic_ϕ start_POSTSUBSCRIPT italic_d , italic_l end_POSTSUBSCRIPT ( bold_italic_x ) | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + divide start_ARG italic_r start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_λ ) start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG ( ∇ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ϕ start_POSTSUBSCRIPT italic_d , italic_l end_POSTSUBSCRIPT ( bold_italic_x ) ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∇ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ϕ start_POSTSUBSCRIPT italic_d , italic_l end_POSTSUBSCRIPT ( bold_italic_x ) + … ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </tbody> </table> <p class="ltx_p" id="A2.p5.17">where we have reconstituted the <math alttext="\boldsymbol{x}" class="ltx_Math" display="inline" id="A2.p5.4.m1.1"><semantics id="A2.p5.4.m1.1a"><mi id="A2.p5.4.m1.1.1" xref="A2.p5.4.m1.1.1.cmml">𝒙</mi><annotation-xml encoding="MathML-Content" id="A2.p5.4.m1.1b"><ci id="A2.p5.4.m1.1.1.cmml" xref="A2.p5.4.m1.1.1">𝒙</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.p5.4.m1.1c">\boldsymbol{x}</annotation><annotation encoding="application/x-llamapun" id="A2.p5.4.m1.1d">bold_italic_x</annotation></semantics></math> argument for the coefficients in the expansion, as well as omitted the wavelength argument in <math alttext="\phi_{d}(\lambda_{j},\boldsymbol{x})" class="ltx_Math" display="inline" id="A2.p5.5.m2.2"><semantics id="A2.p5.5.m2.2a"><mrow id="A2.p5.5.m2.2.2" xref="A2.p5.5.m2.2.2.cmml"><msub id="A2.p5.5.m2.2.2.3" xref="A2.p5.5.m2.2.2.3.cmml"><mi id="A2.p5.5.m2.2.2.3.2" xref="A2.p5.5.m2.2.2.3.2.cmml">ϕ</mi><mi id="A2.p5.5.m2.2.2.3.3" xref="A2.p5.5.m2.2.2.3.3.cmml">d</mi></msub><mo id="A2.p5.5.m2.2.2.2" xref="A2.p5.5.m2.2.2.2.cmml">⁢</mo><mrow id="A2.p5.5.m2.2.2.1.1" xref="A2.p5.5.m2.2.2.1.2.cmml"><mo id="A2.p5.5.m2.2.2.1.1.2" stretchy="false" xref="A2.p5.5.m2.2.2.1.2.cmml">(</mo><msub id="A2.p5.5.m2.2.2.1.1.1" xref="A2.p5.5.m2.2.2.1.1.1.cmml"><mi id="A2.p5.5.m2.2.2.1.1.1.2" xref="A2.p5.5.m2.2.2.1.1.1.2.cmml">λ</mi><mi id="A2.p5.5.m2.2.2.1.1.1.3" xref="A2.p5.5.m2.2.2.1.1.1.3.cmml">j</mi></msub><mo id="A2.p5.5.m2.2.2.1.1.3" xref="A2.p5.5.m2.2.2.1.2.cmml">,</mo><mi id="A2.p5.5.m2.1.1" xref="A2.p5.5.m2.1.1.cmml">𝒙</mi><mo id="A2.p5.5.m2.2.2.1.1.4" stretchy="false" xref="A2.p5.5.m2.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml 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encoding="application/x-tex" id="A2.p5.5.m2.2c">\phi_{d}(\lambda_{j},\boldsymbol{x})</annotation><annotation encoding="application/x-llamapun" id="A2.p5.5.m2.2d">italic_ϕ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_λ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , bold_italic_x )</annotation></semantics></math> to simplify notation, and we have dropped an overall constant as this term is what was considered in Appendix <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#A1" title="Appendix A Dispersion ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">A</span></a> and is perfectly removed by the de-dispersion estimator given by equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#A1.E10" title="In Appendix A Dispersion ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">A10</span></a>). We have rotated to the basis where <math alttext="\nabla_{i}\nabla_{j}\phi_{d,l}(0)" class="ltx_Math" display="inline" id="A2.p5.6.m3.3"><semantics id="A2.p5.6.m3.3a"><mrow id="A2.p5.6.m3.3.4" xref="A2.p5.6.m3.3.4.cmml"><mrow id="A2.p5.6.m3.3.4.2" xref="A2.p5.6.m3.3.4.2.cmml"><mrow id="A2.p5.6.m3.3.4.2.1" xref="A2.p5.6.m3.3.4.2.1.cmml"><msub id="A2.p5.6.m3.3.4.2.1.1" xref="A2.p5.6.m3.3.4.2.1.1.cmml"><mo id="A2.p5.6.m3.3.4.2.1.1.2" xref="A2.p5.6.m3.3.4.2.1.1.2.cmml">∇</mo><mi id="A2.p5.6.m3.3.4.2.1.1.3" xref="A2.p5.6.m3.3.4.2.1.1.3.cmml">i</mi></msub><msub id="A2.p5.6.m3.3.4.2.1.2" xref="A2.p5.6.m3.3.4.2.1.2.cmml"><mo id="A2.p5.6.m3.3.4.2.1.2.2" lspace="0.167em" xref="A2.p5.6.m3.3.4.2.1.2.2.cmml">∇</mo><mi id="A2.p5.6.m3.3.4.2.1.2.3" xref="A2.p5.6.m3.3.4.2.1.2.3.cmml">j</mi></msub></mrow><mo id="A2.p5.6.m3.3.4.2a" lspace="0.167em" xref="A2.p5.6.m3.3.4.2.cmml">⁡</mo><msub id="A2.p5.6.m3.3.4.2.2" xref="A2.p5.6.m3.3.4.2.2.cmml"><mi id="A2.p5.6.m3.3.4.2.2.2" xref="A2.p5.6.m3.3.4.2.2.2.cmml">ϕ</mi><mrow id="A2.p5.6.m3.2.2.2.4" xref="A2.p5.6.m3.2.2.2.3.cmml"><mi id="A2.p5.6.m3.1.1.1.1" xref="A2.p5.6.m3.1.1.1.1.cmml">d</mi><mo id="A2.p5.6.m3.2.2.2.4.1" xref="A2.p5.6.m3.2.2.2.3.cmml">,</mo><mi id="A2.p5.6.m3.2.2.2.2" xref="A2.p5.6.m3.2.2.2.2.cmml">l</mi></mrow></msub></mrow><mo id="A2.p5.6.m3.3.4.1" xref="A2.p5.6.m3.3.4.1.cmml">⁢</mo><mrow id="A2.p5.6.m3.3.4.3.2" xref="A2.p5.6.m3.3.4.cmml"><mo id="A2.p5.6.m3.3.4.3.2.1" stretchy="false" xref="A2.p5.6.m3.3.4.cmml">(</mo><mn id="A2.p5.6.m3.3.3" xref="A2.p5.6.m3.3.3.cmml">0</mn><mo id="A2.p5.6.m3.3.4.3.2.2" stretchy="false" xref="A2.p5.6.m3.3.4.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.p5.6.m3.3b"><apply id="A2.p5.6.m3.3.4.cmml" xref="A2.p5.6.m3.3.4"><times id="A2.p5.6.m3.3.4.1.cmml" xref="A2.p5.6.m3.3.4.1"></times><apply id="A2.p5.6.m3.3.4.2.cmml" xref="A2.p5.6.m3.3.4.2"><apply id="A2.p5.6.m3.3.4.2.1.cmml" xref="A2.p5.6.m3.3.4.2.1"><apply id="A2.p5.6.m3.3.4.2.1.1.cmml" xref="A2.p5.6.m3.3.4.2.1.1"><csymbol cd="ambiguous" id="A2.p5.6.m3.3.4.2.1.1.1.cmml" xref="A2.p5.6.m3.3.4.2.1.1">subscript</csymbol><ci id="A2.p5.6.m3.3.4.2.1.1.2.cmml" xref="A2.p5.6.m3.3.4.2.1.1.2">∇</ci><ci id="A2.p5.6.m3.3.4.2.1.1.3.cmml" xref="A2.p5.6.m3.3.4.2.1.1.3">𝑖</ci></apply><apply id="A2.p5.6.m3.3.4.2.1.2.cmml" xref="A2.p5.6.m3.3.4.2.1.2"><csymbol cd="ambiguous" id="A2.p5.6.m3.3.4.2.1.2.1.cmml" xref="A2.p5.6.m3.3.4.2.1.2">subscript</csymbol><ci id="A2.p5.6.m3.3.4.2.1.2.2.cmml" xref="A2.p5.6.m3.3.4.2.1.2.2">∇</ci><ci id="A2.p5.6.m3.3.4.2.1.2.3.cmml" xref="A2.p5.6.m3.3.4.2.1.2.3">𝑗</ci></apply></apply><apply id="A2.p5.6.m3.3.4.2.2.cmml" xref="A2.p5.6.m3.3.4.2.2"><csymbol cd="ambiguous" id="A2.p5.6.m3.3.4.2.2.1.cmml" xref="A2.p5.6.m3.3.4.2.2">subscript</csymbol><ci id="A2.p5.6.m3.3.4.2.2.2.cmml" xref="A2.p5.6.m3.3.4.2.2.2">italic-ϕ</ci><list id="A2.p5.6.m3.2.2.2.3.cmml" xref="A2.p5.6.m3.2.2.2.4"><ci id="A2.p5.6.m3.1.1.1.1.cmml" xref="A2.p5.6.m3.1.1.1.1">𝑑</ci><ci id="A2.p5.6.m3.2.2.2.2.cmml" xref="A2.p5.6.m3.2.2.2.2">𝑙</ci></list></apply></apply><cn id="A2.p5.6.m3.3.3.cmml" type="integer" xref="A2.p5.6.m3.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p5.6.m3.3c">\nabla_{i}\nabla_{j}\phi_{d,l}(0)</annotation><annotation encoding="application/x-llamapun" id="A2.p5.6.m3.3d">∇ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∇ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_ϕ start_POSTSUBSCRIPT italic_d , italic_l end_POSTSUBSCRIPT ( 0 )</annotation></semantics></math> is diagonal, and <math alttext="i" class="ltx_Math" display="inline" id="A2.p5.7.m4.1"><semantics id="A2.p5.7.m4.1a"><mi id="A2.p5.7.m4.1.1" xref="A2.p5.7.m4.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="A2.p5.7.m4.1b"><ci id="A2.p5.7.m4.1.1.cmml" xref="A2.p5.7.m4.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.p5.7.m4.1c">i</annotation><annotation encoding="application/x-llamapun" id="A2.p5.7.m4.1d">italic_i</annotation></semantics></math> indices are implicitly summed. Equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#A2.E2" title="In Appendix B Refraction and diffraction ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">B2</span></a>) shows that the leading refractive term is the square of the phase gradient times the Fresnel scale. Since each <math alttext="r_{F}\nabla" class="ltx_Math" display="inline" id="A2.p5.8.m5.1"><semantics id="A2.p5.8.m5.1a"><mrow id="A2.p5.8.m5.1.1" xref="A2.p5.8.m5.1.1.cmml"><msub id="A2.p5.8.m5.1.1.2" xref="A2.p5.8.m5.1.1.2.cmml"><mi id="A2.p5.8.m5.1.1.2.2" xref="A2.p5.8.m5.1.1.2.2.cmml">r</mi><mi id="A2.p5.8.m5.1.1.2.3" xref="A2.p5.8.m5.1.1.2.3.cmml">F</mi></msub><mo id="A2.p5.8.m5.1.1.1" lspace="0.167em" xref="A2.p5.8.m5.1.1.1.cmml">⁢</mo><mo id="A2.p5.8.m5.1.1.3" xref="A2.p5.8.m5.1.1.3.cmml">∇</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.p5.8.m5.1b"><apply id="A2.p5.8.m5.1.1.cmml" xref="A2.p5.8.m5.1.1"><times id="A2.p5.8.m5.1.1.1.cmml" xref="A2.p5.8.m5.1.1.1"></times><apply id="A2.p5.8.m5.1.1.2.cmml" xref="A2.p5.8.m5.1.1.2"><csymbol cd="ambiguous" id="A2.p5.8.m5.1.1.2.1.cmml" xref="A2.p5.8.m5.1.1.2">subscript</csymbol><ci id="A2.p5.8.m5.1.1.2.2.cmml" xref="A2.p5.8.m5.1.1.2.2">𝑟</ci><ci id="A2.p5.8.m5.1.1.2.3.cmml" xref="A2.p5.8.m5.1.1.2.3">𝐹</ci></apply><ci id="A2.p5.8.m5.1.1.3.cmml" xref="A2.p5.8.m5.1.1.3">∇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p5.8.m5.1c">r_{F}\nabla</annotation><annotation encoding="application/x-llamapun" id="A2.p5.8.m5.1d">italic_r start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ∇</annotation></semantics></math> brings in a <math alttext="f/f_{F}" class="ltx_Math" display="inline" id="A2.p5.9.m6.1"><semantics id="A2.p5.9.m6.1a"><mrow id="A2.p5.9.m6.1.1" xref="A2.p5.9.m6.1.1.cmml"><mi id="A2.p5.9.m6.1.1.2" xref="A2.p5.9.m6.1.1.2.cmml">f</mi><mo id="A2.p5.9.m6.1.1.1" xref="A2.p5.9.m6.1.1.1.cmml">/</mo><msub id="A2.p5.9.m6.1.1.3" xref="A2.p5.9.m6.1.1.3.cmml"><mi id="A2.p5.9.m6.1.1.3.2" xref="A2.p5.9.m6.1.1.3.2.cmml">f</mi><mi id="A2.p5.9.m6.1.1.3.3" xref="A2.p5.9.m6.1.1.3.3.cmml">F</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A2.p5.9.m6.1b"><apply id="A2.p5.9.m6.1.1.cmml" xref="A2.p5.9.m6.1.1"><divide id="A2.p5.9.m6.1.1.1.cmml" xref="A2.p5.9.m6.1.1.1"></divide><ci id="A2.p5.9.m6.1.1.2.cmml" xref="A2.p5.9.m6.1.1.2">𝑓</ci><apply id="A2.p5.9.m6.1.1.3.cmml" xref="A2.p5.9.m6.1.1.3"><csymbol cd="ambiguous" id="A2.p5.9.m6.1.1.3.1.cmml" xref="A2.p5.9.m6.1.1.3">subscript</csymbol><ci id="A2.p5.9.m6.1.1.3.2.cmml" xref="A2.p5.9.m6.1.1.3.2">𝑓</ci><ci id="A2.p5.9.m6.1.1.3.3.cmml" xref="A2.p5.9.m6.1.1.3.3">𝐹</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p5.9.m6.1c">f/f_{F}</annotation><annotation encoding="application/x-llamapun" id="A2.p5.9.m6.1d">italic_f / italic_f start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT</annotation></semantics></math> where <math alttext="f_{F}\equiv\omega_{F}/(2\pi)" class="ltx_Math" display="inline" id="A2.p5.10.m7.1"><semantics id="A2.p5.10.m7.1a"><mrow id="A2.p5.10.m7.1.1" xref="A2.p5.10.m7.1.1.cmml"><msub id="A2.p5.10.m7.1.1.3" xref="A2.p5.10.m7.1.1.3.cmml"><mi id="A2.p5.10.m7.1.1.3.2" xref="A2.p5.10.m7.1.1.3.2.cmml">f</mi><mi id="A2.p5.10.m7.1.1.3.3" xref="A2.p5.10.m7.1.1.3.3.cmml">F</mi></msub><mo id="A2.p5.10.m7.1.1.2" xref="A2.p5.10.m7.1.1.2.cmml">≡</mo><mrow id="A2.p5.10.m7.1.1.1" xref="A2.p5.10.m7.1.1.1.cmml"><msub id="A2.p5.10.m7.1.1.1.3" xref="A2.p5.10.m7.1.1.1.3.cmml"><mi id="A2.p5.10.m7.1.1.1.3.2" xref="A2.p5.10.m7.1.1.1.3.2.cmml">ω</mi><mi id="A2.p5.10.m7.1.1.1.3.3" xref="A2.p5.10.m7.1.1.1.3.3.cmml">F</mi></msub><mo id="A2.p5.10.m7.1.1.1.2" xref="A2.p5.10.m7.1.1.1.2.cmml">/</mo><mrow id="A2.p5.10.m7.1.1.1.1.1" xref="A2.p5.10.m7.1.1.1.1.1.1.cmml"><mo id="A2.p5.10.m7.1.1.1.1.1.2" stretchy="false" xref="A2.p5.10.m7.1.1.1.1.1.1.cmml">(</mo><mrow id="A2.p5.10.m7.1.1.1.1.1.1" xref="A2.p5.10.m7.1.1.1.1.1.1.cmml"><mn id="A2.p5.10.m7.1.1.1.1.1.1.2" xref="A2.p5.10.m7.1.1.1.1.1.1.2.cmml">2</mn><mo id="A2.p5.10.m7.1.1.1.1.1.1.1" xref="A2.p5.10.m7.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="A2.p5.10.m7.1.1.1.1.1.1.3" xref="A2.p5.10.m7.1.1.1.1.1.1.3.cmml">π</mi></mrow><mo id="A2.p5.10.m7.1.1.1.1.1.3" stretchy="false" xref="A2.p5.10.m7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.p5.10.m7.1b"><apply id="A2.p5.10.m7.1.1.cmml" xref="A2.p5.10.m7.1.1"><equivalent id="A2.p5.10.m7.1.1.2.cmml" xref="A2.p5.10.m7.1.1.2"></equivalent><apply id="A2.p5.10.m7.1.1.3.cmml" xref="A2.p5.10.m7.1.1.3"><csymbol cd="ambiguous" id="A2.p5.10.m7.1.1.3.1.cmml" xref="A2.p5.10.m7.1.1.3">subscript</csymbol><ci id="A2.p5.10.m7.1.1.3.2.cmml" xref="A2.p5.10.m7.1.1.3.2">𝑓</ci><ci id="A2.p5.10.m7.1.1.3.3.cmml" xref="A2.p5.10.m7.1.1.3.3">𝐹</ci></apply><apply id="A2.p5.10.m7.1.1.1.cmml" xref="A2.p5.10.m7.1.1.1"><divide id="A2.p5.10.m7.1.1.1.2.cmml" xref="A2.p5.10.m7.1.1.1.2"></divide><apply id="A2.p5.10.m7.1.1.1.3.cmml" xref="A2.p5.10.m7.1.1.1.3"><csymbol cd="ambiguous" id="A2.p5.10.m7.1.1.1.3.1.cmml" xref="A2.p5.10.m7.1.1.1.3">subscript</csymbol><ci id="A2.p5.10.m7.1.1.1.3.2.cmml" xref="A2.p5.10.m7.1.1.1.3.2">𝜔</ci><ci id="A2.p5.10.m7.1.1.1.3.3.cmml" xref="A2.p5.10.m7.1.1.1.3.3">𝐹</ci></apply><apply id="A2.p5.10.m7.1.1.1.1.1.1.cmml" xref="A2.p5.10.m7.1.1.1.1.1"><times id="A2.p5.10.m7.1.1.1.1.1.1.1.cmml" xref="A2.p5.10.m7.1.1.1.1.1.1.1"></times><cn id="A2.p5.10.m7.1.1.1.1.1.1.2.cmml" type="integer" xref="A2.p5.10.m7.1.1.1.1.1.1.2">2</cn><ci id="A2.p5.10.m7.1.1.1.1.1.1.3.cmml" xref="A2.p5.10.m7.1.1.1.1.1.1.3">𝜋</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p5.10.m7.1c">f_{F}\equiv\omega_{F}/(2\pi)</annotation><annotation encoding="application/x-llamapun" id="A2.p5.10.m7.1d">italic_f start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ≡ italic_ω start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT / ( 2 italic_π )</annotation></semantics></math>, which is <math alttext="\sim 10^{-6}" class="ltx_Math" display="inline" id="A2.p5.11.m8.1"><semantics id="A2.p5.11.m8.1a"><mrow id="A2.p5.11.m8.1.1" xref="A2.p5.11.m8.1.1.cmml"><mi id="A2.p5.11.m8.1.1.2" xref="A2.p5.11.m8.1.1.2.cmml"></mi><mo id="A2.p5.11.m8.1.1.1" xref="A2.p5.11.m8.1.1.1.cmml">∼</mo><msup id="A2.p5.11.m8.1.1.3" xref="A2.p5.11.m8.1.1.3.cmml"><mn id="A2.p5.11.m8.1.1.3.2" xref="A2.p5.11.m8.1.1.3.2.cmml">10</mn><mrow id="A2.p5.11.m8.1.1.3.3" xref="A2.p5.11.m8.1.1.3.3.cmml"><mo id="A2.p5.11.m8.1.1.3.3a" xref="A2.p5.11.m8.1.1.3.3.cmml">−</mo><mn id="A2.p5.11.m8.1.1.3.3.2" xref="A2.p5.11.m8.1.1.3.3.2.cmml">6</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="A2.p5.11.m8.1b"><apply id="A2.p5.11.m8.1.1.cmml" xref="A2.p5.11.m8.1.1"><csymbol cd="latexml" id="A2.p5.11.m8.1.1.1.cmml" xref="A2.p5.11.m8.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="A2.p5.11.m8.1.1.2.cmml" xref="A2.p5.11.m8.1.1.2">absent</csymbol><apply id="A2.p5.11.m8.1.1.3.cmml" xref="A2.p5.11.m8.1.1.3"><csymbol cd="ambiguous" id="A2.p5.11.m8.1.1.3.1.cmml" xref="A2.p5.11.m8.1.1.3">superscript</csymbol><cn id="A2.p5.11.m8.1.1.3.2.cmml" type="integer" xref="A2.p5.11.m8.1.1.3.2">10</cn><apply id="A2.p5.11.m8.1.1.3.3.cmml" xref="A2.p5.11.m8.1.1.3.3"><minus id="A2.p5.11.m8.1.1.3.3.1.cmml" xref="A2.p5.11.m8.1.1.3.3"></minus><cn id="A2.p5.11.m8.1.1.3.3.2.cmml" type="integer" xref="A2.p5.11.m8.1.1.3.3.2">6</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p5.11.m8.1c">\sim 10^{-6}</annotation><annotation encoding="application/x-llamapun" id="A2.p5.11.m8.1d">∼ 10 start_POSTSUPERSCRIPT - 6 end_POSTSUPERSCRIPT</annotation></semantics></math> for <math alttext="f\sim 10^{-5}\;" class="ltx_Math" display="inline" id="A2.p5.12.m9.1"><semantics id="A2.p5.12.m9.1a"><mrow id="A2.p5.12.m9.1.1" xref="A2.p5.12.m9.1.1.cmml"><mi id="A2.p5.12.m9.1.1.2" xref="A2.p5.12.m9.1.1.2.cmml">f</mi><mo id="A2.p5.12.m9.1.1.1" xref="A2.p5.12.m9.1.1.1.cmml">∼</mo><msup id="A2.p5.12.m9.1.1.3" xref="A2.p5.12.m9.1.1.3.cmml"><mn id="A2.p5.12.m9.1.1.3.2" xref="A2.p5.12.m9.1.1.3.2.cmml">10</mn><mrow id="A2.p5.12.m9.1.1.3.3" xref="A2.p5.12.m9.1.1.3.3.cmml"><mo id="A2.p5.12.m9.1.1.3.3a" xref="A2.p5.12.m9.1.1.3.3.cmml">−</mo><mn id="A2.p5.12.m9.1.1.3.3.2" xref="A2.p5.12.m9.1.1.3.3.2.cmml">5</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="A2.p5.12.m9.1b"><apply id="A2.p5.12.m9.1.1.cmml" xref="A2.p5.12.m9.1.1"><csymbol cd="latexml" id="A2.p5.12.m9.1.1.1.cmml" xref="A2.p5.12.m9.1.1.1">similar-to</csymbol><ci id="A2.p5.12.m9.1.1.2.cmml" xref="A2.p5.12.m9.1.1.2">𝑓</ci><apply id="A2.p5.12.m9.1.1.3.cmml" xref="A2.p5.12.m9.1.1.3"><csymbol cd="ambiguous" id="A2.p5.12.m9.1.1.3.1.cmml" xref="A2.p5.12.m9.1.1.3">superscript</csymbol><cn id="A2.p5.12.m9.1.1.3.2.cmml" type="integer" xref="A2.p5.12.m9.1.1.3.2">10</cn><apply id="A2.p5.12.m9.1.1.3.3.cmml" xref="A2.p5.12.m9.1.1.3.3"><minus id="A2.p5.12.m9.1.1.3.3.1.cmml" xref="A2.p5.12.m9.1.1.3.3"></minus><cn id="A2.p5.12.m9.1.1.3.3.2.cmml" type="integer" xref="A2.p5.12.m9.1.1.3.3.2">5</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p5.12.m9.1c">f\sim 10^{-5}\;</annotation><annotation encoding="application/x-llamapun" id="A2.p5.12.m9.1d">italic_f ∼ 10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT</annotation></semantics></math>Hz, the refractive effect should be suppressed at this frequency and <math alttext="\lambda=1~{}" class="ltx_Math" display="inline" id="A2.p5.13.m10.1"><semantics id="A2.p5.13.m10.1a"><mrow id="A2.p5.13.m10.1.1" xref="A2.p5.13.m10.1.1.cmml"><mi id="A2.p5.13.m10.1.1.2" xref="A2.p5.13.m10.1.1.2.cmml">λ</mi><mo id="A2.p5.13.m10.1.1.1" xref="A2.p5.13.m10.1.1.1.cmml">=</mo><mn id="A2.p5.13.m10.1.1.3" xref="A2.p5.13.m10.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.p5.13.m10.1b"><apply id="A2.p5.13.m10.1.1.cmml" xref="A2.p5.13.m10.1.1"><eq id="A2.p5.13.m10.1.1.1.cmml" xref="A2.p5.13.m10.1.1.1"></eq><ci id="A2.p5.13.m10.1.1.2.cmml" xref="A2.p5.13.m10.1.1.2">𝜆</ci><cn id="A2.p5.13.m10.1.1.3.cmml" type="integer" xref="A2.p5.13.m10.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p5.13.m10.1c">\lambda=1~{}</annotation><annotation encoding="application/x-llamapun" id="A2.p5.13.m10.1d">italic_λ = 1</annotation></semantics></math>cm by a factor of <math alttext="\sim 10^{-13}" class="ltx_Math" display="inline" id="A2.p5.14.m11.1"><semantics id="A2.p5.14.m11.1a"><mrow id="A2.p5.14.m11.1.1" xref="A2.p5.14.m11.1.1.cmml"><mi id="A2.p5.14.m11.1.1.2" xref="A2.p5.14.m11.1.1.2.cmml"></mi><mo id="A2.p5.14.m11.1.1.1" xref="A2.p5.14.m11.1.1.1.cmml">∼</mo><msup id="A2.p5.14.m11.1.1.3" xref="A2.p5.14.m11.1.1.3.cmml"><mn id="A2.p5.14.m11.1.1.3.2" xref="A2.p5.14.m11.1.1.3.2.cmml">10</mn><mrow id="A2.p5.14.m11.1.1.3.3" xref="A2.p5.14.m11.1.1.3.3.cmml"><mo id="A2.p5.14.m11.1.1.3.3a" xref="A2.p5.14.m11.1.1.3.3.cmml">−</mo><mn id="A2.p5.14.m11.1.1.3.3.2" xref="A2.p5.14.m11.1.1.3.3.2.cmml">13</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="A2.p5.14.m11.1b"><apply id="A2.p5.14.m11.1.1.cmml" xref="A2.p5.14.m11.1.1"><csymbol cd="latexml" id="A2.p5.14.m11.1.1.1.cmml" xref="A2.p5.14.m11.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="A2.p5.14.m11.1.1.2.cmml" xref="A2.p5.14.m11.1.1.2">absent</csymbol><apply id="A2.p5.14.m11.1.1.3.cmml" xref="A2.p5.14.m11.1.1.3"><csymbol cd="ambiguous" id="A2.p5.14.m11.1.1.3.1.cmml" xref="A2.p5.14.m11.1.1.3">superscript</csymbol><cn id="A2.p5.14.m11.1.1.3.2.cmml" type="integer" xref="A2.p5.14.m11.1.1.3.2">10</cn><apply id="A2.p5.14.m11.1.1.3.3.cmml" xref="A2.p5.14.m11.1.1.3.3"><minus id="A2.p5.14.m11.1.1.3.3.1.cmml" xref="A2.p5.14.m11.1.1.3.3"></minus><cn id="A2.p5.14.m11.1.1.3.3.2.cmml" type="integer" xref="A2.p5.14.m11.1.1.3.3.2">13</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p5.14.m11.1c">\sim 10^{-13}</annotation><annotation encoding="application/x-llamapun" id="A2.p5.14.m11.1d">∼ 10 start_POSTSUPERSCRIPT - 13 end_POSTSUPERSCRIPT</annotation></semantics></math> relative to the mean dispersive term, which is <math alttext="\propto\phi_{d,l}" class="ltx_Math" display="inline" id="A2.p5.15.m12.2"><semantics id="A2.p5.15.m12.2a"><mrow id="A2.p5.15.m12.2.3" xref="A2.p5.15.m12.2.3.cmml"><mi id="A2.p5.15.m12.2.3.2" xref="A2.p5.15.m12.2.3.2.cmml"></mi><mo id="A2.p5.15.m12.2.3.1" xref="A2.p5.15.m12.2.3.1.cmml">∝</mo><msub id="A2.p5.15.m12.2.3.3" xref="A2.p5.15.m12.2.3.3.cmml"><mi id="A2.p5.15.m12.2.3.3.2" xref="A2.p5.15.m12.2.3.3.2.cmml">ϕ</mi><mrow id="A2.p5.15.m12.2.2.2.4" xref="A2.p5.15.m12.2.2.2.3.cmml"><mi id="A2.p5.15.m12.1.1.1.1" xref="A2.p5.15.m12.1.1.1.1.cmml">d</mi><mo id="A2.p5.15.m12.2.2.2.4.1" xref="A2.p5.15.m12.2.2.2.3.cmml">,</mo><mi id="A2.p5.15.m12.2.2.2.2" xref="A2.p5.15.m12.2.2.2.2.cmml">l</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="A2.p5.15.m12.2b"><apply id="A2.p5.15.m12.2.3.cmml" xref="A2.p5.15.m12.2.3"><csymbol cd="latexml" id="A2.p5.15.m12.2.3.1.cmml" xref="A2.p5.15.m12.2.3.1">proportional-to</csymbol><csymbol cd="latexml" id="A2.p5.15.m12.2.3.2.cmml" xref="A2.p5.15.m12.2.3.2">absent</csymbol><apply id="A2.p5.15.m12.2.3.3.cmml" xref="A2.p5.15.m12.2.3.3"><csymbol cd="ambiguous" id="A2.p5.15.m12.2.3.3.1.cmml" xref="A2.p5.15.m12.2.3.3">subscript</csymbol><ci id="A2.p5.15.m12.2.3.3.2.cmml" xref="A2.p5.15.m12.2.3.3.2">italic-ϕ</ci><list id="A2.p5.15.m12.2.2.2.3.cmml" xref="A2.p5.15.m12.2.2.2.4"><ci id="A2.p5.15.m12.1.1.1.1.cmml" xref="A2.p5.15.m12.1.1.1.1">𝑑</ci><ci id="A2.p5.15.m12.2.2.2.2.cmml" xref="A2.p5.15.m12.2.2.2.2">𝑙</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p5.15.m12.2c">\propto\phi_{d,l}</annotation><annotation encoding="application/x-llamapun" id="A2.p5.15.m12.2d">∝ italic_ϕ start_POSTSUBSCRIPT italic_d , italic_l end_POSTSUBSCRIPT</annotation></semantics></math> and whose effect was estimated in the previous section. This order-of-magnitude estimate uses that the phase fluctuation at <math alttext="f\sim 10^{-5}\;" class="ltx_Math" display="inline" id="A2.p5.16.m13.1"><semantics id="A2.p5.16.m13.1a"><mrow id="A2.p5.16.m13.1.1" xref="A2.p5.16.m13.1.1.cmml"><mi id="A2.p5.16.m13.1.1.2" xref="A2.p5.16.m13.1.1.2.cmml">f</mi><mo id="A2.p5.16.m13.1.1.1" xref="A2.p5.16.m13.1.1.1.cmml">∼</mo><msup id="A2.p5.16.m13.1.1.3" xref="A2.p5.16.m13.1.1.3.cmml"><mn id="A2.p5.16.m13.1.1.3.2" xref="A2.p5.16.m13.1.1.3.2.cmml">10</mn><mrow id="A2.p5.16.m13.1.1.3.3" xref="A2.p5.16.m13.1.1.3.3.cmml"><mo id="A2.p5.16.m13.1.1.3.3a" xref="A2.p5.16.m13.1.1.3.3.cmml">−</mo><mn id="A2.p5.16.m13.1.1.3.3.2" xref="A2.p5.16.m13.1.1.3.3.2.cmml">5</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="A2.p5.16.m13.1b"><apply id="A2.p5.16.m13.1.1.cmml" xref="A2.p5.16.m13.1.1"><csymbol cd="latexml" id="A2.p5.16.m13.1.1.1.cmml" xref="A2.p5.16.m13.1.1.1">similar-to</csymbol><ci id="A2.p5.16.m13.1.1.2.cmml" xref="A2.p5.16.m13.1.1.2">𝑓</ci><apply id="A2.p5.16.m13.1.1.3.cmml" xref="A2.p5.16.m13.1.1.3"><csymbol cd="ambiguous" id="A2.p5.16.m13.1.1.3.1.cmml" xref="A2.p5.16.m13.1.1.3">superscript</csymbol><cn id="A2.p5.16.m13.1.1.3.2.cmml" type="integer" xref="A2.p5.16.m13.1.1.3.2">10</cn><apply id="A2.p5.16.m13.1.1.3.3.cmml" xref="A2.p5.16.m13.1.1.3.3"><minus id="A2.p5.16.m13.1.1.3.3.1.cmml" xref="A2.p5.16.m13.1.1.3.3"></minus><cn id="A2.p5.16.m13.1.1.3.3.2.cmml" type="integer" xref="A2.p5.16.m13.1.1.3.3.2">5</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p5.16.m13.1c">f\sim 10^{-5}\;</annotation><annotation encoding="application/x-llamapun" id="A2.p5.16.m13.1d">italic_f ∼ 10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT</annotation></semantics></math>Hz is <math alttext="\sim 0.1" class="ltx_Math" display="inline" id="A2.p5.17.m14.1"><semantics id="A2.p5.17.m14.1a"><mrow id="A2.p5.17.m14.1.1" xref="A2.p5.17.m14.1.1.cmml"><mi id="A2.p5.17.m14.1.1.2" xref="A2.p5.17.m14.1.1.2.cmml"></mi><mo id="A2.p5.17.m14.1.1.1" xref="A2.p5.17.m14.1.1.1.cmml">∼</mo><mn id="A2.p5.17.m14.1.1.3" xref="A2.p5.17.m14.1.1.3.cmml">0.1</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.p5.17.m14.1b"><apply id="A2.p5.17.m14.1.1.cmml" xref="A2.p5.17.m14.1.1"><csymbol cd="latexml" id="A2.p5.17.m14.1.1.1.cmml" xref="A2.p5.17.m14.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="A2.p5.17.m14.1.1.2.cmml" xref="A2.p5.17.m14.1.1.2">absent</csymbol><cn id="A2.p5.17.m14.1.1.3.cmml" type="float" xref="A2.p5.17.m14.1.1.3">0.1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p5.17.m14.1c">\sim 0.1</annotation><annotation encoding="application/x-llamapun" id="A2.p5.17.m14.1d">∼ 0.1</annotation></semantics></math>. This simple estimate is consistent with the following more detailed calculation.</p> </div> <div class="ltx_para" id="A2.p6"> <p class="ltx_p" id="A2.p6.11">To calculate the phase noise from refraction, let us consider the lowest-order non-constant term in equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#A2.E2" title="In Appendix B Refraction and diffraction ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">B2</span></a>). Assuming the same setup as in Appendix <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#A1" title="Appendix A Dispersion ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">A</span></a> and illustrated in Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#A1.F7" title="Figure 7 ‣ Appendix A Dispersion ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">7</span></a>, where temporal correlations can be related to spatial correlations of the advecting solar wind, we can calculate the corrections from refraction owing to phase correlations as</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="A3.EGx13"> <tbody id="A2.E3"> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\Big{\langle}\phi_{\lambda}(t)\phi_{\lambda}(t+\tau)\Big{\rangle}\ " class="ltx_Math" display="inline" id="A2.Ex2.m1.2"><semantics id="A2.Ex2.m1.2a"><mrow id="A2.Ex2.m1.2.2.1" xref="A2.Ex2.m1.2.2.2.cmml"><mo id="A2.Ex2.m1.2.2.1.2" maxsize="160%" minsize="160%" xref="A2.Ex2.m1.2.2.2.1.cmml">⟨</mo><mrow id="A2.Ex2.m1.2.2.1.1" xref="A2.Ex2.m1.2.2.1.1.cmml"><msub id="A2.Ex2.m1.2.2.1.1.3" xref="A2.Ex2.m1.2.2.1.1.3.cmml"><mi id="A2.Ex2.m1.2.2.1.1.3.2" xref="A2.Ex2.m1.2.2.1.1.3.2.cmml">ϕ</mi><mi id="A2.Ex2.m1.2.2.1.1.3.3" xref="A2.Ex2.m1.2.2.1.1.3.3.cmml">λ</mi></msub><mo id="A2.Ex2.m1.2.2.1.1.2" xref="A2.Ex2.m1.2.2.1.1.2.cmml">⁢</mo><mrow id="A2.Ex2.m1.2.2.1.1.4.2" xref="A2.Ex2.m1.2.2.1.1.cmml"><mo id="A2.Ex2.m1.2.2.1.1.4.2.1" stretchy="false" xref="A2.Ex2.m1.2.2.1.1.cmml">(</mo><mi id="A2.Ex2.m1.1.1" xref="A2.Ex2.m1.1.1.cmml">t</mi><mo id="A2.Ex2.m1.2.2.1.1.4.2.2" stretchy="false" xref="A2.Ex2.m1.2.2.1.1.cmml">)</mo></mrow><mo id="A2.Ex2.m1.2.2.1.1.2a" xref="A2.Ex2.m1.2.2.1.1.2.cmml">⁢</mo><msub id="A2.Ex2.m1.2.2.1.1.5" xref="A2.Ex2.m1.2.2.1.1.5.cmml"><mi id="A2.Ex2.m1.2.2.1.1.5.2" xref="A2.Ex2.m1.2.2.1.1.5.2.cmml">ϕ</mi><mi id="A2.Ex2.m1.2.2.1.1.5.3" xref="A2.Ex2.m1.2.2.1.1.5.3.cmml">λ</mi></msub><mo id="A2.Ex2.m1.2.2.1.1.2b" xref="A2.Ex2.m1.2.2.1.1.2.cmml">⁢</mo><mrow id="A2.Ex2.m1.2.2.1.1.1.1" xref="A2.Ex2.m1.2.2.1.1.1.1.1.cmml"><mo id="A2.Ex2.m1.2.2.1.1.1.1.2" stretchy="false" xref="A2.Ex2.m1.2.2.1.1.1.1.1.cmml">(</mo><mrow id="A2.Ex2.m1.2.2.1.1.1.1.1" xref="A2.Ex2.m1.2.2.1.1.1.1.1.cmml"><mi id="A2.Ex2.m1.2.2.1.1.1.1.1.2" xref="A2.Ex2.m1.2.2.1.1.1.1.1.2.cmml">t</mi><mo id="A2.Ex2.m1.2.2.1.1.1.1.1.1" xref="A2.Ex2.m1.2.2.1.1.1.1.1.1.cmml">+</mo><mi id="A2.Ex2.m1.2.2.1.1.1.1.1.3" xref="A2.Ex2.m1.2.2.1.1.1.1.1.3.cmml">τ</mi></mrow><mo id="A2.Ex2.m1.2.2.1.1.1.1.3" stretchy="false" xref="A2.Ex2.m1.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A2.Ex2.m1.2.2.1.3" maxsize="160%" minsize="160%" xref="A2.Ex2.m1.2.2.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.Ex2.m1.2b"><apply id="A2.Ex2.m1.2.2.2.cmml" xref="A2.Ex2.m1.2.2.1"><csymbol cd="latexml" id="A2.Ex2.m1.2.2.2.1.cmml" xref="A2.Ex2.m1.2.2.1.2">delimited-⟨⟩</csymbol><apply id="A2.Ex2.m1.2.2.1.1.cmml" xref="A2.Ex2.m1.2.2.1.1"><times id="A2.Ex2.m1.2.2.1.1.2.cmml" xref="A2.Ex2.m1.2.2.1.1.2"></times><apply id="A2.Ex2.m1.2.2.1.1.3.cmml" xref="A2.Ex2.m1.2.2.1.1.3"><csymbol cd="ambiguous" id="A2.Ex2.m1.2.2.1.1.3.1.cmml" xref="A2.Ex2.m1.2.2.1.1.3">subscript</csymbol><ci id="A2.Ex2.m1.2.2.1.1.3.2.cmml" xref="A2.Ex2.m1.2.2.1.1.3.2">italic-ϕ</ci><ci id="A2.Ex2.m1.2.2.1.1.3.3.cmml" xref="A2.Ex2.m1.2.2.1.1.3.3">𝜆</ci></apply><ci id="A2.Ex2.m1.1.1.cmml" xref="A2.Ex2.m1.1.1">𝑡</ci><apply id="A2.Ex2.m1.2.2.1.1.5.cmml" xref="A2.Ex2.m1.2.2.1.1.5"><csymbol cd="ambiguous" id="A2.Ex2.m1.2.2.1.1.5.1.cmml" xref="A2.Ex2.m1.2.2.1.1.5">subscript</csymbol><ci id="A2.Ex2.m1.2.2.1.1.5.2.cmml" xref="A2.Ex2.m1.2.2.1.1.5.2">italic-ϕ</ci><ci id="A2.Ex2.m1.2.2.1.1.5.3.cmml" xref="A2.Ex2.m1.2.2.1.1.5.3">𝜆</ci></apply><apply id="A2.Ex2.m1.2.2.1.1.1.1.1.cmml" xref="A2.Ex2.m1.2.2.1.1.1.1"><plus id="A2.Ex2.m1.2.2.1.1.1.1.1.1.cmml" xref="A2.Ex2.m1.2.2.1.1.1.1.1.1"></plus><ci id="A2.Ex2.m1.2.2.1.1.1.1.1.2.cmml" xref="A2.Ex2.m1.2.2.1.1.1.1.1.2">𝑡</ci><ci id="A2.Ex2.m1.2.2.1.1.1.1.1.3.cmml" xref="A2.Ex2.m1.2.2.1.1.1.1.1.3">𝜏</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex2.m1.2c">\displaystyle\Big{\langle}\phi_{\lambda}(t)\phi_{\lambda}(t+\tau)\Big{\rangle}\ </annotation><annotation encoding="application/x-llamapun" id="A2.Ex2.m1.2d">⟨ italic_ϕ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_t ) italic_ϕ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_t + italic_τ ) ⟩</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle\approx" class="ltx_Math" display="inline" 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end_POSTSUBSCRIPT italic_τ overbold_^ start_ARG bold_italic_n end_ARG ) | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⟩</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="2"><span class="ltx_tag ltx_tag_equation ltx_align_right">(B3)</span></td> </tr> <tr class="ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle\subseteq" class="ltx_Math" display="inline" id="A2.E3.m1.1"><semantics id="A2.E3.m1.1a"><mo id="A2.E3.m1.1.1" xref="A2.E3.m1.1.1.cmml">⊆</mo><annotation-xml encoding="MathML-Content" id="A2.E3.m1.1b"><subset id="A2.E3.m1.1.1.cmml" xref="A2.E3.m1.1.1"></subset></annotation-xml><annotation encoding="application/x-tex" id="A2.E3.m1.1c">\displaystyle\subseteq</annotation><annotation 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xref="A2.E3.m2.3.3.1.1.6.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E3.m2.3c">\displaystyle\ 2r_{F}(\lambda)^{4}\xi_{\nabla\phi}(\tau)^{2},</annotation><annotation encoding="application/x-llamapun" id="A2.E3.m2.3d">2 italic_r start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_λ ) start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT italic_ξ start_POSTSUBSCRIPT ∇ italic_ϕ end_POSTSUBSCRIPT ( italic_τ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </tbody> </table> <p class="ltx_p" id="A2.p6.12">where in addition to dropping an irrelevant constant, the last line assumed statistical isotropy and Gaussian statistics to express the four-point correlation function in terms of the two-point correlation function, where </p> <table class="ltx_equation ltx_eqn_table" id="A2.Ex3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td 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id="A2.p6.13">The half-bandwidth power spectrum of phase fluctuations from refraction can be calculated from the Fourier transform of equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#A2.E3" title="In Appendix B Refraction and diffraction ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">B3</span></a>):</p> <table class="ltx_equation ltx_eqn_table" id="A2.E4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="P_{\rm\phi,ref}(f=2\pi\omega)=2r_{F}(\lambda)^{4}\int dte^{-i\omega t}\xi_{% \nabla\phi}(t)^{2}=2r_{F}(\lambda)^{4}\int\frac{d\omega^{\prime}}{2\pi}P_{% \nabla\phi}(\omega^{\prime})P_{\nabla\phi}(\omega-\omega^{\prime})," class="ltx_Math" display="block" id="A2.E4.m1.6"><semantics id="A2.E4.m1.6a"><mrow id="A2.E4.m1.6.6.1" 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xref="A2.E4.m1.6.6.1.1.3.2.2.2.1.1.3.2">𝜔</ci><ci id="A2.E4.m1.6.6.1.1.3.2.2.2.1.1.3.3.cmml" xref="A2.E4.m1.6.6.1.1.3.2.2.2.1.1.3.3">′</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E4.m1.6c">P_{\rm\phi,ref}(f=2\pi\omega)=2r_{F}(\lambda)^{4}\int dte^{-i\omega t}\xi_{% \nabla\phi}(t)^{2}=2r_{F}(\lambda)^{4}\int\frac{d\omega^{\prime}}{2\pi}P_{% \nabla\phi}(\omega^{\prime})P_{\nabla\phi}(\omega-\omega^{\prime}),</annotation><annotation encoding="application/x-llamapun" id="A2.E4.m1.6d">italic_P start_POSTSUBSCRIPT italic_ϕ , roman_ref end_POSTSUBSCRIPT ( italic_f = 2 italic_π italic_ω ) = 2 italic_r start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_λ ) start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ∫ italic_d italic_t italic_e start_POSTSUPERSCRIPT - italic_i italic_ω italic_t end_POSTSUPERSCRIPT italic_ξ start_POSTSUBSCRIPT ∇ italic_ϕ end_POSTSUBSCRIPT ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 2 italic_r start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_λ ) start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ∫ divide start_ARG italic_d italic_ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG 2 italic_π end_ARG italic_P start_POSTSUBSCRIPT ∇ italic_ϕ end_POSTSUBSCRIPT ( italic_ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) italic_P start_POSTSUBSCRIPT ∇ italic_ϕ end_POSTSUBSCRIPT ( italic_ω - italic_ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(B4)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A2.p6.1">where the full-bandwidth power spectrum of <math alttext="|\boldsymbol{\nabla}\phi|" class="ltx_Math" display="inline" id="A2.p6.1.m1.1"><semantics id="A2.p6.1.m1.1a"><mrow id="A2.p6.1.m1.1.1.1" 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id="A2.p6.1.m1.1c">|\boldsymbol{\nabla}\phi|</annotation><annotation encoding="application/x-llamapun" id="A2.p6.1.m1.1d">| bold_∇ italic_ϕ |</annotation></semantics></math> is defined as</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="A3.EGx14"> <tbody id="A2.E5"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle P_{\nabla\phi}(\omega)" class="ltx_Math" display="inline" id="A2.E5.m1.1"><semantics id="A2.E5.m1.1a"><mrow id="A2.E5.m1.1.2" xref="A2.E5.m1.1.2.cmml"><msub id="A2.E5.m1.1.2.2" xref="A2.E5.m1.1.2.2.cmml"><mi id="A2.E5.m1.1.2.2.2" xref="A2.E5.m1.1.2.2.2.cmml">P</mi><mrow id="A2.E5.m1.1.2.2.3" xref="A2.E5.m1.1.2.2.3.cmml"><mo id="A2.E5.m1.1.2.2.3.1" rspace="0.167em" xref="A2.E5.m1.1.2.2.3.1.cmml">∇</mo><mi id="A2.E5.m1.1.2.2.3.2" xref="A2.E5.m1.1.2.2.3.2.cmml">ϕ</mi></mrow></msub><mo id="A2.E5.m1.1.2.1" xref="A2.E5.m1.1.2.1.cmml">⁢</mo><mrow id="A2.E5.m1.1.2.3.2" xref="A2.E5.m1.1.2.cmml"><mo id="A2.E5.m1.1.2.3.2.1" stretchy="false" xref="A2.E5.m1.1.2.cmml">(</mo><mi id="A2.E5.m1.1.1" xref="A2.E5.m1.1.1.cmml">ω</mi><mo id="A2.E5.m1.1.2.3.2.2" stretchy="false" xref="A2.E5.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.E5.m1.1b"><apply id="A2.E5.m1.1.2.cmml" xref="A2.E5.m1.1.2"><times id="A2.E5.m1.1.2.1.cmml" xref="A2.E5.m1.1.2.1"></times><apply id="A2.E5.m1.1.2.2.cmml" xref="A2.E5.m1.1.2.2"><csymbol cd="ambiguous" id="A2.E5.m1.1.2.2.1.cmml" xref="A2.E5.m1.1.2.2">subscript</csymbol><ci id="A2.E5.m1.1.2.2.2.cmml" xref="A2.E5.m1.1.2.2.2">𝑃</ci><apply id="A2.E5.m1.1.2.2.3.cmml" xref="A2.E5.m1.1.2.2.3"><ci id="A2.E5.m1.1.2.2.3.1.cmml" xref="A2.E5.m1.1.2.2.3.1">∇</ci><ci id="A2.E5.m1.1.2.2.3.2.cmml" xref="A2.E5.m1.1.2.2.3.2">italic-ϕ</ci></apply></apply><ci id="A2.E5.m1.1.1.cmml" xref="A2.E5.m1.1.1">𝜔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E5.m1.1c">\displaystyle P_{\nabla\phi}(\omega)</annotation><annotation encoding="application/x-llamapun" id="A2.E5.m1.1d">italic_P start_POSTSUBSCRIPT ∇ italic_ϕ end_POSTSUBSCRIPT ( italic_ω )</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle\equiv" class="ltx_Math" display="inline" id="A2.E5.m2.1"><semantics id="A2.E5.m2.1a"><mo id="A2.E5.m2.1.1" xref="A2.E5.m2.1.1.cmml">≡</mo><annotation-xml encoding="MathML-Content" id="A2.E5.m2.1b"><equivalent id="A2.E5.m2.1.1.cmml" xref="A2.E5.m2.1.1"></equivalent></annotation-xml><annotation encoding="application/x-tex" id="A2.E5.m2.1c">\displaystyle\equiv</annotation><annotation encoding="application/x-llamapun" id="A2.E5.m2.1d">≡</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\int dte^{-i\omega t}\xi_{\nabla\phi}(t)=\frac{(2\pi c)^{2}}{% \lambda^{2}}\int dte^{-i\omega t}\left\langle\nabla_{i}\Delta\tau_{d,1}\nabla_% {i}\Delta\tau_{d,2}\right\rangle(v_{\rm sw}t)," class="ltx_Math" display="inline" id="A2.E5.m3.7"><semantics id="A2.E5.m3.7a"><mrow id="A2.E5.m3.7.7.1" xref="A2.E5.m3.7.7.1.1.cmml"><mrow id="A2.E5.m3.7.7.1.1" xref="A2.E5.m3.7.7.1.1.cmml"><mstyle displaystyle="true" id="A2.E5.m3.7.7.1.1.4" xref="A2.E5.m3.7.7.1.1.4.cmml"><mrow id="A2.E5.m3.7.7.1.1.4a" xref="A2.E5.m3.7.7.1.1.4.cmml"><mo id="A2.E5.m3.7.7.1.1.4.1" xref="A2.E5.m3.7.7.1.1.4.1.cmml">∫</mo><mrow id="A2.E5.m3.7.7.1.1.4.2" xref="A2.E5.m3.7.7.1.1.4.2.cmml"><mrow id="A2.E5.m3.7.7.1.1.4.2.2" xref="A2.E5.m3.7.7.1.1.4.2.2.cmml"><mo id="A2.E5.m3.7.7.1.1.4.2.2.1" lspace="0em" rspace="0em" xref="A2.E5.m3.7.7.1.1.4.2.2.1.cmml">𝑑</mo><mi id="A2.E5.m3.7.7.1.1.4.2.2.2" xref="A2.E5.m3.7.7.1.1.4.2.2.2.cmml">t</mi></mrow><mo id="A2.E5.m3.7.7.1.1.4.2.1" xref="A2.E5.m3.7.7.1.1.4.2.1.cmml">⁢</mo><msup id="A2.E5.m3.7.7.1.1.4.2.3" xref="A2.E5.m3.7.7.1.1.4.2.3.cmml"><mi id="A2.E5.m3.7.7.1.1.4.2.3.2" 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xref="A2.E5.m3.7.7.1.1.2.2.2.2.1.1.2.3">sw</ci></apply><ci id="A2.E5.m3.7.7.1.1.2.2.2.2.1.1.3.cmml" xref="A2.E5.m3.7.7.1.1.2.2.2.2.1.1.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E5.m3.7c">\displaystyle\int dte^{-i\omega t}\xi_{\nabla\phi}(t)=\frac{(2\pi c)^{2}}{% \lambda^{2}}\int dte^{-i\omega t}\left\langle\nabla_{i}\Delta\tau_{d,1}\nabla_% {i}\Delta\tau_{d,2}\right\rangle(v_{\rm sw}t),</annotation><annotation encoding="application/x-llamapun" id="A2.E5.m3.7d">∫ italic_d italic_t italic_e start_POSTSUPERSCRIPT - italic_i italic_ω italic_t end_POSTSUPERSCRIPT italic_ξ start_POSTSUBSCRIPT ∇ italic_ϕ end_POSTSUBSCRIPT ( italic_t ) = divide start_ARG ( 2 italic_π italic_c ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ∫ italic_d italic_t italic_e start_POSTSUPERSCRIPT - italic_i italic_ω italic_t end_POSTSUPERSCRIPT ⟨ ∇ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_Δ italic_τ start_POSTSUBSCRIPT italic_d , 1 end_POSTSUBSCRIPT ∇ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_Δ italic_τ start_POSTSUBSCRIPT italic_d , 2 end_POSTSUBSCRIPT ⟩ ( italic_v start_POSTSUBSCRIPT roman_sw end_POSTSUBSCRIPT italic_t ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(B5)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A2.p6.4">and we have converted to gradients of the time delay rather than phase using that <math alttext="\tau_{d}=2\pi c/\lambda\,\phi_{d}" class="ltx_Math" display="inline" id="A2.p6.2.m1.1"><semantics id="A2.p6.2.m1.1a"><mrow id="A2.p6.2.m1.1.1" xref="A2.p6.2.m1.1.1.cmml"><msub id="A2.p6.2.m1.1.1.2" xref="A2.p6.2.m1.1.1.2.cmml"><mi id="A2.p6.2.m1.1.1.2.2" xref="A2.p6.2.m1.1.1.2.2.cmml">τ</mi><mi id="A2.p6.2.m1.1.1.2.3" xref="A2.p6.2.m1.1.1.2.3.cmml">d</mi></msub><mo id="A2.p6.2.m1.1.1.1" xref="A2.p6.2.m1.1.1.1.cmml">=</mo><mrow id="A2.p6.2.m1.1.1.3" xref="A2.p6.2.m1.1.1.3.cmml"><mrow id="A2.p6.2.m1.1.1.3.2" xref="A2.p6.2.m1.1.1.3.2.cmml"><mrow id="A2.p6.2.m1.1.1.3.2.2" xref="A2.p6.2.m1.1.1.3.2.2.cmml"><mn id="A2.p6.2.m1.1.1.3.2.2.2" xref="A2.p6.2.m1.1.1.3.2.2.2.cmml">2</mn><mo id="A2.p6.2.m1.1.1.3.2.2.1" xref="A2.p6.2.m1.1.1.3.2.2.1.cmml">⁢</mo><mi id="A2.p6.2.m1.1.1.3.2.2.3" xref="A2.p6.2.m1.1.1.3.2.2.3.cmml">π</mi><mo id="A2.p6.2.m1.1.1.3.2.2.1a" xref="A2.p6.2.m1.1.1.3.2.2.1.cmml">⁢</mo><mi id="A2.p6.2.m1.1.1.3.2.2.4" xref="A2.p6.2.m1.1.1.3.2.2.4.cmml">c</mi></mrow><mo id="A2.p6.2.m1.1.1.3.2.1" xref="A2.p6.2.m1.1.1.3.2.1.cmml">/</mo><mi id="A2.p6.2.m1.1.1.3.2.3" xref="A2.p6.2.m1.1.1.3.2.3.cmml">λ</mi></mrow><mo id="A2.p6.2.m1.1.1.3.1" lspace="0.170em" xref="A2.p6.2.m1.1.1.3.1.cmml">⁢</mo><msub id="A2.p6.2.m1.1.1.3.3" xref="A2.p6.2.m1.1.1.3.3.cmml"><mi id="A2.p6.2.m1.1.1.3.3.2" xref="A2.p6.2.m1.1.1.3.3.2.cmml">ϕ</mi><mi id="A2.p6.2.m1.1.1.3.3.3" xref="A2.p6.2.m1.1.1.3.3.3.cmml">d</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.p6.2.m1.1b"><apply id="A2.p6.2.m1.1.1.cmml" xref="A2.p6.2.m1.1.1"><eq id="A2.p6.2.m1.1.1.1.cmml" xref="A2.p6.2.m1.1.1.1"></eq><apply id="A2.p6.2.m1.1.1.2.cmml" xref="A2.p6.2.m1.1.1.2"><csymbol cd="ambiguous" id="A2.p6.2.m1.1.1.2.1.cmml" xref="A2.p6.2.m1.1.1.2">subscript</csymbol><ci id="A2.p6.2.m1.1.1.2.2.cmml" xref="A2.p6.2.m1.1.1.2.2">𝜏</ci><ci id="A2.p6.2.m1.1.1.2.3.cmml" xref="A2.p6.2.m1.1.1.2.3">𝑑</ci></apply><apply id="A2.p6.2.m1.1.1.3.cmml" xref="A2.p6.2.m1.1.1.3"><times id="A2.p6.2.m1.1.1.3.1.cmml" xref="A2.p6.2.m1.1.1.3.1"></times><apply id="A2.p6.2.m1.1.1.3.2.cmml" xref="A2.p6.2.m1.1.1.3.2"><divide id="A2.p6.2.m1.1.1.3.2.1.cmml" xref="A2.p6.2.m1.1.1.3.2.1"></divide><apply id="A2.p6.2.m1.1.1.3.2.2.cmml" xref="A2.p6.2.m1.1.1.3.2.2"><times id="A2.p6.2.m1.1.1.3.2.2.1.cmml" xref="A2.p6.2.m1.1.1.3.2.2.1"></times><cn id="A2.p6.2.m1.1.1.3.2.2.2.cmml" type="integer" xref="A2.p6.2.m1.1.1.3.2.2.2">2</cn><ci id="A2.p6.2.m1.1.1.3.2.2.3.cmml" xref="A2.p6.2.m1.1.1.3.2.2.3">𝜋</ci><ci id="A2.p6.2.m1.1.1.3.2.2.4.cmml" xref="A2.p6.2.m1.1.1.3.2.2.4">𝑐</ci></apply><ci id="A2.p6.2.m1.1.1.3.2.3.cmml" xref="A2.p6.2.m1.1.1.3.2.3">𝜆</ci></apply><apply id="A2.p6.2.m1.1.1.3.3.cmml" xref="A2.p6.2.m1.1.1.3.3"><csymbol cd="ambiguous" id="A2.p6.2.m1.1.1.3.3.1.cmml" xref="A2.p6.2.m1.1.1.3.3">subscript</csymbol><ci id="A2.p6.2.m1.1.1.3.3.2.cmml" xref="A2.p6.2.m1.1.1.3.3.2">italic-ϕ</ci><ci id="A2.p6.2.m1.1.1.3.3.3.cmml" xref="A2.p6.2.m1.1.1.3.3.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p6.2.m1.1c">\tau_{d}=2\pi c/\lambda\,\phi_{d}</annotation><annotation encoding="application/x-llamapun" id="A2.p6.2.m1.1d">italic_τ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT = 2 italic_π italic_c / italic_λ italic_ϕ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT</annotation></semantics></math>. Using the integral expression given by equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#A1.E4" title="In Appendix A Dispersion ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">A4</span></a>) for <math alttext="\left\langle\Delta\tau_{d,1}\Delta\tau_{d,2}\right\rangle" class="ltx_Math" display="inline" id="A2.p6.3.m2.5"><semantics id="A2.p6.3.m2.5a"><mrow id="A2.p6.3.m2.5.5.1" xref="A2.p6.3.m2.5.5.2.cmml"><mo id="A2.p6.3.m2.5.5.1.2" xref="A2.p6.3.m2.5.5.2.1.cmml">⟨</mo><mrow id="A2.p6.3.m2.5.5.1.1" xref="A2.p6.3.m2.5.5.1.1.cmml"><mi id="A2.p6.3.m2.5.5.1.1.2" mathvariant="normal" xref="A2.p6.3.m2.5.5.1.1.2.cmml">Δ</mi><mo id="A2.p6.3.m2.5.5.1.1.1" xref="A2.p6.3.m2.5.5.1.1.1.cmml">⁢</mo><msub id="A2.p6.3.m2.5.5.1.1.3" xref="A2.p6.3.m2.5.5.1.1.3.cmml"><mi id="A2.p6.3.m2.5.5.1.1.3.2" xref="A2.p6.3.m2.5.5.1.1.3.2.cmml">τ</mi><mrow id="A2.p6.3.m2.2.2.2.4" xref="A2.p6.3.m2.2.2.2.3.cmml"><mi id="A2.p6.3.m2.1.1.1.1" xref="A2.p6.3.m2.1.1.1.1.cmml">d</mi><mo id="A2.p6.3.m2.2.2.2.4.1" xref="A2.p6.3.m2.2.2.2.3.cmml">,</mo><mn id="A2.p6.3.m2.2.2.2.2" xref="A2.p6.3.m2.2.2.2.2.cmml">1</mn></mrow></msub><mo id="A2.p6.3.m2.5.5.1.1.1a" xref="A2.p6.3.m2.5.5.1.1.1.cmml">⁢</mo><mi id="A2.p6.3.m2.5.5.1.1.4" mathvariant="normal" xref="A2.p6.3.m2.5.5.1.1.4.cmml">Δ</mi><mo id="A2.p6.3.m2.5.5.1.1.1b" xref="A2.p6.3.m2.5.5.1.1.1.cmml">⁢</mo><msub id="A2.p6.3.m2.5.5.1.1.5" xref="A2.p6.3.m2.5.5.1.1.5.cmml"><mi id="A2.p6.3.m2.5.5.1.1.5.2" xref="A2.p6.3.m2.5.5.1.1.5.2.cmml">τ</mi><mrow id="A2.p6.3.m2.4.4.2.4" xref="A2.p6.3.m2.4.4.2.3.cmml"><mi id="A2.p6.3.m2.3.3.1.1" xref="A2.p6.3.m2.3.3.1.1.cmml">d</mi><mo id="A2.p6.3.m2.4.4.2.4.1" xref="A2.p6.3.m2.4.4.2.3.cmml">,</mo><mn id="A2.p6.3.m2.4.4.2.2" xref="A2.p6.3.m2.4.4.2.2.cmml">2</mn></mrow></msub></mrow><mo id="A2.p6.3.m2.5.5.1.3" xref="A2.p6.3.m2.5.5.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.p6.3.m2.5b"><apply id="A2.p6.3.m2.5.5.2.cmml" xref="A2.p6.3.m2.5.5.1"><csymbol cd="latexml" id="A2.p6.3.m2.5.5.2.1.cmml" xref="A2.p6.3.m2.5.5.1.2">delimited-⟨⟩</csymbol><apply id="A2.p6.3.m2.5.5.1.1.cmml" xref="A2.p6.3.m2.5.5.1.1"><times id="A2.p6.3.m2.5.5.1.1.1.cmml" xref="A2.p6.3.m2.5.5.1.1.1"></times><ci id="A2.p6.3.m2.5.5.1.1.2.cmml" xref="A2.p6.3.m2.5.5.1.1.2">Δ</ci><apply id="A2.p6.3.m2.5.5.1.1.3.cmml" xref="A2.p6.3.m2.5.5.1.1.3"><csymbol cd="ambiguous" id="A2.p6.3.m2.5.5.1.1.3.1.cmml" xref="A2.p6.3.m2.5.5.1.1.3">subscript</csymbol><ci id="A2.p6.3.m2.5.5.1.1.3.2.cmml" xref="A2.p6.3.m2.5.5.1.1.3.2">𝜏</ci><list id="A2.p6.3.m2.2.2.2.3.cmml" xref="A2.p6.3.m2.2.2.2.4"><ci id="A2.p6.3.m2.1.1.1.1.cmml" xref="A2.p6.3.m2.1.1.1.1">𝑑</ci><cn id="A2.p6.3.m2.2.2.2.2.cmml" type="integer" xref="A2.p6.3.m2.2.2.2.2">1</cn></list></apply><ci id="A2.p6.3.m2.5.5.1.1.4.cmml" xref="A2.p6.3.m2.5.5.1.1.4">Δ</ci><apply id="A2.p6.3.m2.5.5.1.1.5.cmml" xref="A2.p6.3.m2.5.5.1.1.5"><csymbol cd="ambiguous" id="A2.p6.3.m2.5.5.1.1.5.1.cmml" xref="A2.p6.3.m2.5.5.1.1.5">subscript</csymbol><ci id="A2.p6.3.m2.5.5.1.1.5.2.cmml" xref="A2.p6.3.m2.5.5.1.1.5.2">𝜏</ci><list id="A2.p6.3.m2.4.4.2.3.cmml" xref="A2.p6.3.m2.4.4.2.4"><ci id="A2.p6.3.m2.3.3.1.1.cmml" xref="A2.p6.3.m2.3.3.1.1">𝑑</ci><cn id="A2.p6.3.m2.4.4.2.2.cmml" type="integer" xref="A2.p6.3.m2.4.4.2.2">2</cn></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p6.3.m2.5c">\left\langle\Delta\tau_{d,1}\Delta\tau_{d,2}\right\rangle</annotation><annotation encoding="application/x-llamapun" id="A2.p6.3.m2.5d">⟨ roman_Δ italic_τ start_POSTSUBSCRIPT italic_d , 1 end_POSTSUBSCRIPT roman_Δ italic_τ start_POSTSUBSCRIPT italic_d , 2 end_POSTSUBSCRIPT ⟩</annotation></semantics></math>, and that the derivatives bring down factors of <math alttext="{\boldsymbol{k}}_{\perp}" class="ltx_Math" display="inline" id="A2.p6.4.m3.1"><semantics id="A2.p6.4.m3.1a"><msub id="A2.p6.4.m3.1.1" xref="A2.p6.4.m3.1.1.cmml"><mi id="A2.p6.4.m3.1.1.2" xref="A2.p6.4.m3.1.1.2.cmml">𝒌</mi><mo id="A2.p6.4.m3.1.1.3" xref="A2.p6.4.m3.1.1.3.cmml">⟂</mo></msub><annotation-xml encoding="MathML-Content" id="A2.p6.4.m3.1b"><apply id="A2.p6.4.m3.1.1.cmml" xref="A2.p6.4.m3.1.1"><csymbol cd="ambiguous" id="A2.p6.4.m3.1.1.1.cmml" xref="A2.p6.4.m3.1.1">subscript</csymbol><ci id="A2.p6.4.m3.1.1.2.cmml" xref="A2.p6.4.m3.1.1.2">𝒌</ci><csymbol cd="latexml" id="A2.p6.4.m3.1.1.3.cmml" xref="A2.p6.4.m3.1.1.3">perpendicular-to</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p6.4.m3.1c">{\boldsymbol{k}}_{\perp}</annotation><annotation encoding="application/x-llamapun" id="A2.p6.4.m3.1d">bold_italic_k start_POSTSUBSCRIPT ⟂ end_POSTSUBSCRIPT</annotation></semantics></math> under the integrand, we can write this as</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="A3.EGx15"> <tbody id="A2.E6"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle P_{\nabla\phi}(\omega)" class="ltx_Math" display="inline" id="A2.E6.m1.1"><semantics id="A2.E6.m1.1a"><mrow id="A2.E6.m1.1.2" xref="A2.E6.m1.1.2.cmml"><msub id="A2.E6.m1.1.2.2" xref="A2.E6.m1.1.2.2.cmml"><mi id="A2.E6.m1.1.2.2.2" xref="A2.E6.m1.1.2.2.2.cmml">P</mi><mrow id="A2.E6.m1.1.2.2.3" xref="A2.E6.m1.1.2.2.3.cmml"><mo id="A2.E6.m1.1.2.2.3.1" rspace="0.167em" xref="A2.E6.m1.1.2.2.3.1.cmml">∇</mo><mi id="A2.E6.m1.1.2.2.3.2" xref="A2.E6.m1.1.2.2.3.2.cmml">ϕ</mi></mrow></msub><mo id="A2.E6.m1.1.2.1" xref="A2.E6.m1.1.2.1.cmml">⁢</mo><mrow id="A2.E6.m1.1.2.3.2" xref="A2.E6.m1.1.2.cmml"><mo id="A2.E6.m1.1.2.3.2.1" stretchy="false" xref="A2.E6.m1.1.2.cmml">(</mo><mi id="A2.E6.m1.1.1" xref="A2.E6.m1.1.1.cmml">ω</mi><mo id="A2.E6.m1.1.2.3.2.2" stretchy="false" xref="A2.E6.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.E6.m1.1b"><apply id="A2.E6.m1.1.2.cmml" xref="A2.E6.m1.1.2"><times id="A2.E6.m1.1.2.1.cmml" xref="A2.E6.m1.1.2.1"></times><apply id="A2.E6.m1.1.2.2.cmml" xref="A2.E6.m1.1.2.2"><csymbol cd="ambiguous" id="A2.E6.m1.1.2.2.1.cmml" xref="A2.E6.m1.1.2.2">subscript</csymbol><ci id="A2.E6.m1.1.2.2.2.cmml" xref="A2.E6.m1.1.2.2.2">𝑃</ci><apply id="A2.E6.m1.1.2.2.3.cmml" xref="A2.E6.m1.1.2.2.3"><ci id="A2.E6.m1.1.2.2.3.1.cmml" xref="A2.E6.m1.1.2.2.3.1">∇</ci><ci id="A2.E6.m1.1.2.2.3.2.cmml" xref="A2.E6.m1.1.2.2.3.2">italic-ϕ</ci></apply></apply><ci id="A2.E6.m1.1.1.cmml" xref="A2.E6.m1.1.1">𝜔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E6.m1.1c">\displaystyle P_{\nabla\phi}(\omega)</annotation><annotation encoding="application/x-llamapun" id="A2.E6.m1.1d">italic_P start_POSTSUBSCRIPT ∇ italic_ϕ end_POSTSUBSCRIPT ( italic_ω )</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="A2.E6.m2.1"><semantics id="A2.E6.m2.1a"><mo id="A2.E6.m2.1.1" xref="A2.E6.m2.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="A2.E6.m2.1b"><eq id="A2.E6.m2.1.1.cmml" xref="A2.E6.m2.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="A2.E6.m2.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="A2.E6.m2.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\int dte^{-i\omega t}\kappa^{\prime 2}L\int\frac{d^{2}{% \boldsymbol{k}}_{\perp}}{(2\pi)^{2}}P_{e}(k)\Big{|}{\boldsymbol{k}}_{\perp}% \Big{|}^{2}e^{i{\boldsymbol{k}}_{\perp}\cdot{\boldsymbol{\widehat{n}}}v_{\rm sw% }t}," class="ltx_Math" display="inline" id="A2.E6.m3.5"><semantics id="A2.E6.m3.5a"><mrow id="A2.E6.m3.5.5.1" xref="A2.E6.m3.5.5.1.1.cmml"><mstyle displaystyle="true" id="A2.E6.m3.5.5.1.1" 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end_POSTSUPERSCRIPT italic_L ∫ divide start_ARG italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_italic_k start_POSTSUBSCRIPT ⟂ end_POSTSUBSCRIPT end_ARG start_ARG ( 2 italic_π ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG italic_P start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ( italic_k ) | bold_italic_k start_POSTSUBSCRIPT ⟂ end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT italic_i bold_italic_k start_POSTSUBSCRIPT ⟂ end_POSTSUBSCRIPT ⋅ overbold_^ start_ARG bold_italic_n end_ARG italic_v start_POSTSUBSCRIPT roman_sw end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(B6)</span></td> </tr></tbody> <tbody id="A2.E7"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="A2.E7.m1.1"><semantics id="A2.E7.m1.1a"><mo id="A2.E7.m1.1.1" xref="A2.E7.m1.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="A2.E7.m1.1b"><eq id="A2.E7.m1.1.1.cmml" xref="A2.E7.m1.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="A2.E7.m1.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="A2.E7.m1.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\frac{\kappa^{\prime 2}L}{v_{\rm sw}}\int_{-\infty}^{\infty}\frac% {dk_{x}}{(2\pi)}P_{e}\left(\sqrt{k_{x}^{2}+\frac{\omega^{2}}{v_{\rm sw}^{2}}}% \right)\left(k_{x}^{2}+\frac{\omega^{2}}{v_{\rm sw}^{2}}\right)," class="ltx_Math" display="inline" id="A2.E7.m2.5"><semantics id="A2.E7.m2.5a"><mrow id="A2.E7.m2.5.5.1" 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xref="A2.E7.m2.5.5.1.1.1.1.1.1.1.3.3.2.3">sw</ci></apply><cn id="A2.E7.m2.5.5.1.1.1.1.1.1.1.3.3.3.cmml" type="integer" xref="A2.E7.m2.5.5.1.1.1.1.1.1.1.3.3.3">2</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E7.m2.5c">\displaystyle\frac{\kappa^{\prime 2}L}{v_{\rm sw}}\int_{-\infty}^{\infty}\frac% {dk_{x}}{(2\pi)}P_{e}\left(\sqrt{k_{x}^{2}+\frac{\omega^{2}}{v_{\rm sw}^{2}}}% \right)\left(k_{x}^{2}+\frac{\omega^{2}}{v_{\rm sw}^{2}}\right),</annotation><annotation encoding="application/x-llamapun" id="A2.E7.m2.5d">divide start_ARG italic_κ start_POSTSUPERSCRIPT ′ 2 end_POSTSUPERSCRIPT italic_L end_ARG start_ARG italic_v start_POSTSUBSCRIPT roman_sw end_POSTSUBSCRIPT end_ARG ∫ start_POSTSUBSCRIPT - ∞ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT divide start_ARG italic_d italic_k start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_ARG start_ARG ( 2 italic_π ) end_ARG italic_P start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ( square-root start_ARG italic_k start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + divide start_ARG italic_ω start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_v start_POSTSUBSCRIPT roman_sw end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG end_ARG ) ( italic_k start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + divide start_ARG italic_ω start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_v start_POSTSUBSCRIPT roman_sw end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(B7)</span></td> </tr></tbody> <tbody id="A2.E8"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle\approx" class="ltx_Math" display="inline" id="A2.E8.m1.1"><semantics id="A2.E8.m1.1a"><mo id="A2.E8.m1.1.1" xref="A2.E8.m1.1.1.cmml">≈</mo><annotation-xml encoding="MathML-Content" id="A2.E8.m1.1b"><approx id="A2.E8.m1.1.1.cmml" xref="A2.E8.m1.1.1"></approx></annotation-xml><annotation encoding="application/x-tex" id="A2.E8.m1.1c">\displaystyle\approx</annotation><annotation encoding="application/x-llamapun" id="A2.E8.m1.1d">≈</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\frac{4.4\kappa^{\prime 2}L|\omega|}{v_{\rm sw}}P_{e}^{\rm 1D}(f,% r)," class="ltx_Math" display="inline" id="A2.E8.m2.6"><semantics id="A2.E8.m2.6a"><mrow id="A2.E8.m2.6.6.1" xref="A2.E8.m2.6.6.1.1.cmml"><mrow id="A2.E8.m2.6.6.1.1" xref="A2.E8.m2.6.6.1.1.cmml"><mstyle displaystyle="true" id="A2.E8.m2.3.3" xref="A2.E8.m2.3.3.cmml"><mfrac id="A2.E8.m2.3.3a" xref="A2.E8.m2.3.3.cmml"><mrow id="A2.E8.m2.3.3.3" xref="A2.E8.m2.3.3.3.cmml"><mn id="A2.E8.m2.3.3.3.5" xref="A2.E8.m2.3.3.3.5.cmml">4.4</mn><mo id="A2.E8.m2.3.3.3.4" xref="A2.E8.m2.3.3.3.4.cmml">⁢</mo><msup id="A2.E8.m2.3.3.3.6" xref="A2.E8.m2.3.3.3.6.cmml"><mi id="A2.E8.m2.3.3.3.6.2" xref="A2.E8.m2.3.3.3.6.2.cmml">κ</mi><mrow id="A2.E8.m2.2.2.2.2.2.2" xref="A2.E8.m2.2.2.2.2.2.3.cmml"><mo id="A2.E8.m2.2.2.2.2.2.2.1" mathsize="142%" xref="A2.E8.m2.2.2.2.2.2.2.1.cmml">′</mo><mo id="A2.E8.m2.2.2.2.2.2.2.2" lspace="0em" xref="A2.E8.m2.2.2.2.2.2.3.cmml">⁣</mo><mn id="A2.E8.m2.1.1.1.1.1.1" xref="A2.E8.m2.1.1.1.1.1.1.cmml">2</mn></mrow></msup><mo id="A2.E8.m2.3.3.3.4a" xref="A2.E8.m2.3.3.3.4.cmml">⁢</mo><mi id="A2.E8.m2.3.3.3.7" xref="A2.E8.m2.3.3.3.7.cmml">L</mi><mo id="A2.E8.m2.3.3.3.4b" xref="A2.E8.m2.3.3.3.4.cmml">⁢</mo><mrow id="A2.E8.m2.3.3.3.8.2" xref="A2.E8.m2.3.3.3.8.1.cmml"><mo id="A2.E8.m2.3.3.3.8.2.1" stretchy="false" xref="A2.E8.m2.3.3.3.8.1.1.cmml">|</mo><mi id="A2.E8.m2.3.3.3.3" xref="A2.E8.m2.3.3.3.3.cmml">ω</mi><mo id="A2.E8.m2.3.3.3.8.2.2" stretchy="false" xref="A2.E8.m2.3.3.3.8.1.1.cmml">|</mo></mrow></mrow><msub id="A2.E8.m2.3.3.5" xref="A2.E8.m2.3.3.5.cmml"><mi id="A2.E8.m2.3.3.5.2" xref="A2.E8.m2.3.3.5.2.cmml">v</mi><mi id="A2.E8.m2.3.3.5.3" xref="A2.E8.m2.3.3.5.3.cmml">sw</mi></msub></mfrac></mstyle><mo id="A2.E8.m2.6.6.1.1.1" xref="A2.E8.m2.6.6.1.1.1.cmml">⁢</mo><msubsup id="A2.E8.m2.6.6.1.1.2" xref="A2.E8.m2.6.6.1.1.2.cmml"><mi id="A2.E8.m2.6.6.1.1.2.2.2" xref="A2.E8.m2.6.6.1.1.2.2.2.cmml">P</mi><mi id="A2.E8.m2.6.6.1.1.2.2.3" xref="A2.E8.m2.6.6.1.1.2.2.3.cmml">e</mi><mrow id="A2.E8.m2.6.6.1.1.2.3" xref="A2.E8.m2.6.6.1.1.2.3.cmml"><mn id="A2.E8.m2.6.6.1.1.2.3.2" xref="A2.E8.m2.6.6.1.1.2.3.2.cmml">1</mn><mo id="A2.E8.m2.6.6.1.1.2.3.1" xref="A2.E8.m2.6.6.1.1.2.3.1.cmml">⁢</mo><mi id="A2.E8.m2.6.6.1.1.2.3.3" mathvariant="normal" 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xref="A2.E8.m2.3.3.5">subscript</csymbol><ci id="A2.E8.m2.3.3.5.2.cmml" xref="A2.E8.m2.3.3.5.2">𝑣</ci><ci id="A2.E8.m2.3.3.5.3.cmml" xref="A2.E8.m2.3.3.5.3">sw</ci></apply></apply><apply id="A2.E8.m2.6.6.1.1.2.cmml" xref="A2.E8.m2.6.6.1.1.2"><csymbol cd="ambiguous" id="A2.E8.m2.6.6.1.1.2.1.cmml" xref="A2.E8.m2.6.6.1.1.2">superscript</csymbol><apply id="A2.E8.m2.6.6.1.1.2.2.cmml" xref="A2.E8.m2.6.6.1.1.2"><csymbol cd="ambiguous" id="A2.E8.m2.6.6.1.1.2.2.1.cmml" xref="A2.E8.m2.6.6.1.1.2">subscript</csymbol><ci id="A2.E8.m2.6.6.1.1.2.2.2.cmml" xref="A2.E8.m2.6.6.1.1.2.2.2">𝑃</ci><ci id="A2.E8.m2.6.6.1.1.2.2.3.cmml" xref="A2.E8.m2.6.6.1.1.2.2.3">𝑒</ci></apply><apply id="A2.E8.m2.6.6.1.1.2.3.cmml" xref="A2.E8.m2.6.6.1.1.2.3"><times id="A2.E8.m2.6.6.1.1.2.3.1.cmml" xref="A2.E8.m2.6.6.1.1.2.3.1"></times><cn id="A2.E8.m2.6.6.1.1.2.3.2.cmml" type="integer" xref="A2.E8.m2.6.6.1.1.2.3.2">1</cn><ci id="A2.E8.m2.6.6.1.1.2.3.3.cmml" xref="A2.E8.m2.6.6.1.1.2.3.3">D</ci></apply></apply><interval closure="open" id="A2.E8.m2.6.6.1.1.3.1.cmml" xref="A2.E8.m2.6.6.1.1.3.2"><ci id="A2.E8.m2.4.4.cmml" xref="A2.E8.m2.4.4">𝑓</ci><ci id="A2.E8.m2.5.5.cmml" xref="A2.E8.m2.5.5">𝑟</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E8.m2.6c">\displaystyle\frac{4.4\kappa^{\prime 2}L|\omega|}{v_{\rm sw}}P_{e}^{\rm 1D}(f,% r),</annotation><annotation encoding="application/x-llamapun" id="A2.E8.m2.6d">divide start_ARG 4.4 italic_κ start_POSTSUPERSCRIPT ′ 2 end_POSTSUPERSCRIPT italic_L | italic_ω | end_ARG start_ARG italic_v start_POSTSUBSCRIPT roman_sw end_POSTSUBSCRIPT end_ARG italic_P start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 roman_D end_POSTSUPERSCRIPT ( italic_f , italic_r ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(B8)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A2.p6.9">where <math alttext="\kappa^{\prime}=2\pi c\kappa/\lambda" class="ltx_Math" display="inline" id="A2.p6.5.m1.1"><semantics id="A2.p6.5.m1.1a"><mrow id="A2.p6.5.m1.1.1" xref="A2.p6.5.m1.1.1.cmml"><msup id="A2.p6.5.m1.1.1.2" xref="A2.p6.5.m1.1.1.2.cmml"><mi id="A2.p6.5.m1.1.1.2.2" xref="A2.p6.5.m1.1.1.2.2.cmml">κ</mi><mo id="A2.p6.5.m1.1.1.2.3" xref="A2.p6.5.m1.1.1.2.3.cmml">′</mo></msup><mo id="A2.p6.5.m1.1.1.1" xref="A2.p6.5.m1.1.1.1.cmml">=</mo><mrow id="A2.p6.5.m1.1.1.3" xref="A2.p6.5.m1.1.1.3.cmml"><mrow id="A2.p6.5.m1.1.1.3.2" xref="A2.p6.5.m1.1.1.3.2.cmml"><mn id="A2.p6.5.m1.1.1.3.2.2" xref="A2.p6.5.m1.1.1.3.2.2.cmml">2</mn><mo id="A2.p6.5.m1.1.1.3.2.1" xref="A2.p6.5.m1.1.1.3.2.1.cmml">⁢</mo><mi id="A2.p6.5.m1.1.1.3.2.3" xref="A2.p6.5.m1.1.1.3.2.3.cmml">π</mi><mo id="A2.p6.5.m1.1.1.3.2.1a" xref="A2.p6.5.m1.1.1.3.2.1.cmml">⁢</mo><mi id="A2.p6.5.m1.1.1.3.2.4" xref="A2.p6.5.m1.1.1.3.2.4.cmml">c</mi><mo id="A2.p6.5.m1.1.1.3.2.1b" xref="A2.p6.5.m1.1.1.3.2.1.cmml">⁢</mo><mi id="A2.p6.5.m1.1.1.3.2.5" xref="A2.p6.5.m1.1.1.3.2.5.cmml">κ</mi></mrow><mo id="A2.p6.5.m1.1.1.3.1" xref="A2.p6.5.m1.1.1.3.1.cmml">/</mo><mi id="A2.p6.5.m1.1.1.3.3" xref="A2.p6.5.m1.1.1.3.3.cmml">λ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.p6.5.m1.1b"><apply id="A2.p6.5.m1.1.1.cmml" xref="A2.p6.5.m1.1.1"><eq id="A2.p6.5.m1.1.1.1.cmml" xref="A2.p6.5.m1.1.1.1"></eq><apply id="A2.p6.5.m1.1.1.2.cmml" xref="A2.p6.5.m1.1.1.2"><csymbol cd="ambiguous" id="A2.p6.5.m1.1.1.2.1.cmml" xref="A2.p6.5.m1.1.1.2">superscript</csymbol><ci id="A2.p6.5.m1.1.1.2.2.cmml" xref="A2.p6.5.m1.1.1.2.2">𝜅</ci><ci id="A2.p6.5.m1.1.1.2.3.cmml" xref="A2.p6.5.m1.1.1.2.3">′</ci></apply><apply id="A2.p6.5.m1.1.1.3.cmml" xref="A2.p6.5.m1.1.1.3"><divide id="A2.p6.5.m1.1.1.3.1.cmml" xref="A2.p6.5.m1.1.1.3.1"></divide><apply id="A2.p6.5.m1.1.1.3.2.cmml" xref="A2.p6.5.m1.1.1.3.2"><times id="A2.p6.5.m1.1.1.3.2.1.cmml" xref="A2.p6.5.m1.1.1.3.2.1"></times><cn id="A2.p6.5.m1.1.1.3.2.2.cmml" type="integer" xref="A2.p6.5.m1.1.1.3.2.2">2</cn><ci id="A2.p6.5.m1.1.1.3.2.3.cmml" xref="A2.p6.5.m1.1.1.3.2.3">𝜋</ci><ci id="A2.p6.5.m1.1.1.3.2.4.cmml" xref="A2.p6.5.m1.1.1.3.2.4">𝑐</ci><ci id="A2.p6.5.m1.1.1.3.2.5.cmml" xref="A2.p6.5.m1.1.1.3.2.5">𝜅</ci></apply><ci id="A2.p6.5.m1.1.1.3.3.cmml" xref="A2.p6.5.m1.1.1.3.3">𝜆</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p6.5.m1.1c">\kappa^{\prime}=2\pi c\kappa/\lambda</annotation><annotation encoding="application/x-llamapun" id="A2.p6.5.m1.1d">italic_κ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = 2 italic_π italic_c italic_κ / italic_λ</annotation></semantics></math> and, similarly to how we reached equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#A1.E6" title="In Appendix A Dispersion ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">A6</span></a>), we assumed a power-law for the electron density power spectrum. (The coefficient changes to 3.1 if <math alttext="\beta=2" class="ltx_Math" display="inline" id="A2.p6.6.m2.1"><semantics id="A2.p6.6.m2.1a"><mrow id="A2.p6.6.m2.1.1" xref="A2.p6.6.m2.1.1.cmml"><mi id="A2.p6.6.m2.1.1.2" xref="A2.p6.6.m2.1.1.2.cmml">β</mi><mo id="A2.p6.6.m2.1.1.1" xref="A2.p6.6.m2.1.1.1.cmml">=</mo><mn id="A2.p6.6.m2.1.1.3" xref="A2.p6.6.m2.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.p6.6.m2.1b"><apply id="A2.p6.6.m2.1.1.cmml" xref="A2.p6.6.m2.1.1"><eq id="A2.p6.6.m2.1.1.1.cmml" xref="A2.p6.6.m2.1.1.1"></eq><ci id="A2.p6.6.m2.1.1.2.cmml" xref="A2.p6.6.m2.1.1.2">𝛽</ci><cn id="A2.p6.6.m2.1.1.3.cmml" type="integer" xref="A2.p6.6.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p6.6.m2.1c">\beta=2</annotation><annotation encoding="application/x-llamapun" id="A2.p6.6.m2.1d">italic_β = 2</annotation></semantics></math> is taken for the spectral slope of the 1D power spectrum rather than <math alttext="\beta=1.5" class="ltx_Math" display="inline" id="A2.p6.7.m3.1"><semantics id="A2.p6.7.m3.1a"><mrow id="A2.p6.7.m3.1.1" xref="A2.p6.7.m3.1.1.cmml"><mi id="A2.p6.7.m3.1.1.2" xref="A2.p6.7.m3.1.1.2.cmml">β</mi><mo id="A2.p6.7.m3.1.1.1" xref="A2.p6.7.m3.1.1.1.cmml">=</mo><mn id="A2.p6.7.m3.1.1.3" xref="A2.p6.7.m3.1.1.3.cmml">1.5</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.p6.7.m3.1b"><apply id="A2.p6.7.m3.1.1.cmml" xref="A2.p6.7.m3.1.1"><eq id="A2.p6.7.m3.1.1.1.cmml" xref="A2.p6.7.m3.1.1.1"></eq><ci id="A2.p6.7.m3.1.1.2.cmml" xref="A2.p6.7.m3.1.1.2">𝛽</ci><cn id="A2.p6.7.m3.1.1.3.cmml" type="float" xref="A2.p6.7.m3.1.1.3">1.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p6.7.m3.1c">\beta=1.5</annotation><annotation encoding="application/x-llamapun" id="A2.p6.7.m3.1d">italic_β = 1.5</annotation></semantics></math>, as was assumed to evaluate this expression.) The convolution that yields the phase noise power <math alttext="P_{\rm\phi,ref}" class="ltx_Math" display="inline" id="A2.p6.8.m4.2"><semantics id="A2.p6.8.m4.2a"><msub id="A2.p6.8.m4.2.3" xref="A2.p6.8.m4.2.3.cmml"><mi id="A2.p6.8.m4.2.3.2" xref="A2.p6.8.m4.2.3.2.cmml">P</mi><mrow id="A2.p6.8.m4.2.2.2.4" xref="A2.p6.8.m4.2.2.2.3.cmml"><mi id="A2.p6.8.m4.1.1.1.1" xref="A2.p6.8.m4.1.1.1.1.cmml">ϕ</mi><mo id="A2.p6.8.m4.2.2.2.4.1" xref="A2.p6.8.m4.2.2.2.3.cmml">,</mo><mi id="A2.p6.8.m4.2.2.2.2" xref="A2.p6.8.m4.2.2.2.2.cmml">ref</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A2.p6.8.m4.2b"><apply id="A2.p6.8.m4.2.3.cmml" xref="A2.p6.8.m4.2.3"><csymbol cd="ambiguous" id="A2.p6.8.m4.2.3.1.cmml" xref="A2.p6.8.m4.2.3">subscript</csymbol><ci id="A2.p6.8.m4.2.3.2.cmml" xref="A2.p6.8.m4.2.3.2">𝑃</ci><list id="A2.p6.8.m4.2.2.2.3.cmml" xref="A2.p6.8.m4.2.2.2.4"><ci id="A2.p6.8.m4.1.1.1.1.cmml" xref="A2.p6.8.m4.1.1.1.1">italic-ϕ</ci><ci id="A2.p6.8.m4.2.2.2.2.cmml" xref="A2.p6.8.m4.2.2.2.2">ref</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p6.8.m4.2c">P_{\rm\phi,ref}</annotation><annotation encoding="application/x-llamapun" id="A2.p6.8.m4.2d">italic_P start_POSTSUBSCRIPT italic_ϕ , roman_ref end_POSTSUBSCRIPT</annotation></semantics></math> (eqn. <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#A2.E4" title="In Appendix B Refraction and diffraction ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">B4</span></a>) is only convergent for <math alttext="3/2<\beta<2" class="ltx_Math" display="inline" id="A2.p6.9.m5.1"><semantics id="A2.p6.9.m5.1a"><mrow id="A2.p6.9.m5.1.1" xref="A2.p6.9.m5.1.1.cmml"><mrow id="A2.p6.9.m5.1.1.2" xref="A2.p6.9.m5.1.1.2.cmml"><mn id="A2.p6.9.m5.1.1.2.2" xref="A2.p6.9.m5.1.1.2.2.cmml">3</mn><mo id="A2.p6.9.m5.1.1.2.1" xref="A2.p6.9.m5.1.1.2.1.cmml">/</mo><mn id="A2.p6.9.m5.1.1.2.3" xref="A2.p6.9.m5.1.1.2.3.cmml">2</mn></mrow><mo id="A2.p6.9.m5.1.1.3" xref="A2.p6.9.m5.1.1.3.cmml"><</mo><mi id="A2.p6.9.m5.1.1.4" xref="A2.p6.9.m5.1.1.4.cmml">β</mi><mo id="A2.p6.9.m5.1.1.5" xref="A2.p6.9.m5.1.1.5.cmml"><</mo><mn id="A2.p6.9.m5.1.1.6" xref="A2.p6.9.m5.1.1.6.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.p6.9.m5.1b"><apply id="A2.p6.9.m5.1.1.cmml" xref="A2.p6.9.m5.1.1"><and id="A2.p6.9.m5.1.1a.cmml" xref="A2.p6.9.m5.1.1"></and><apply id="A2.p6.9.m5.1.1b.cmml" xref="A2.p6.9.m5.1.1"><lt id="A2.p6.9.m5.1.1.3.cmml" xref="A2.p6.9.m5.1.1.3"></lt><apply id="A2.p6.9.m5.1.1.2.cmml" xref="A2.p6.9.m5.1.1.2"><divide id="A2.p6.9.m5.1.1.2.1.cmml" xref="A2.p6.9.m5.1.1.2.1"></divide><cn id="A2.p6.9.m5.1.1.2.2.cmml" type="integer" xref="A2.p6.9.m5.1.1.2.2">3</cn><cn id="A2.p6.9.m5.1.1.2.3.cmml" type="integer" xref="A2.p6.9.m5.1.1.2.3">2</cn></apply><ci id="A2.p6.9.m5.1.1.4.cmml" xref="A2.p6.9.m5.1.1.4">𝛽</ci></apply><apply id="A2.p6.9.m5.1.1c.cmml" xref="A2.p6.9.m5.1.1"><lt id="A2.p6.9.m5.1.1.5.cmml" xref="A2.p6.9.m5.1.1.5"></lt><share href="https://arxiv.org/html/2411.15072v1#A2.p6.9.m5.1.1.4.cmml" id="A2.p6.9.m5.1.1d.cmml" xref="A2.p6.9.m5.1.1"></share><cn id="A2.p6.9.m5.1.1.6.cmml" type="integer" xref="A2.p6.9.m5.1.1.6">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p6.9.m5.1c">3/2<\beta<2</annotation><annotation encoding="application/x-llamapun" id="A2.p6.9.m5.1d">3 / 2 < italic_β < 2</annotation></semantics></math>, which conveniently is the rough range of indices favored by the Voyager 2 data, and has the scaling</p> <table class="ltx_equation ltx_eqn_table" id="A2.E9"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="P_{\rm\phi,ref}(f)\sim 10^{2}\frac{r_{F}(\lambda)^{4}\kappa^{\prime 4}L^{2}f}{% v_{\rm sw}^{2}}\Big{[}fP_{e}^{1D}(f,r)\Big{]}^{2}=10^{2}\frac{\pi^{2}\kappa^{4% }L^{4}f}{\lambda^{2}v_{\rm sw}^{2}}\Big{[}fP_{e}^{\rm 1D}(f,r)\Big{]}^{2}," class="ltx_Math" display="block" id="A2.E9.m1.11"><semantics id="A2.E9.m1.11a"><mrow id="A2.E9.m1.11.11.1" xref="A2.E9.m1.11.11.1.1.cmml"><mrow id="A2.E9.m1.11.11.1.1" xref="A2.E9.m1.11.11.1.1.cmml"><mrow id="A2.E9.m1.11.11.1.1.4" xref="A2.E9.m1.11.11.1.1.4.cmml"><msub id="A2.E9.m1.11.11.1.1.4.2" xref="A2.E9.m1.11.11.1.1.4.2.cmml"><mi id="A2.E9.m1.11.11.1.1.4.2.2" xref="A2.E9.m1.11.11.1.1.4.2.2.cmml">P</mi><mrow id="A2.E9.m1.2.2.2.4" xref="A2.E9.m1.2.2.2.3.cmml"><mi id="A2.E9.m1.1.1.1.1" xref="A2.E9.m1.1.1.1.1.cmml">ϕ</mi><mo id="A2.E9.m1.2.2.2.4.1" xref="A2.E9.m1.2.2.2.3.cmml">,</mo><mi id="A2.E9.m1.2.2.2.2" xref="A2.E9.m1.2.2.2.2.cmml">ref</mi></mrow></msub><mo id="A2.E9.m1.11.11.1.1.4.1" xref="A2.E9.m1.11.11.1.1.4.1.cmml">⁢</mo><mrow id="A2.E9.m1.11.11.1.1.4.3.2" xref="A2.E9.m1.11.11.1.1.4.cmml"><mo id="A2.E9.m1.11.11.1.1.4.3.2.1" 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id="A2.E9.m1.10.10.cmml" xref="A2.E9.m1.10.10">𝑟</ci></interval></apply></apply><cn id="A2.E9.m1.11.11.1.1.2.1.3.cmml" type="integer" xref="A2.E9.m1.11.11.1.1.2.1.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E9.m1.11c">P_{\rm\phi,ref}(f)\sim 10^{2}\frac{r_{F}(\lambda)^{4}\kappa^{\prime 4}L^{2}f}{% v_{\rm sw}^{2}}\Big{[}fP_{e}^{1D}(f,r)\Big{]}^{2}=10^{2}\frac{\pi^{2}\kappa^{4% }L^{4}f}{\lambda^{2}v_{\rm sw}^{2}}\Big{[}fP_{e}^{\rm 1D}(f,r)\Big{]}^{2},</annotation><annotation encoding="application/x-llamapun" id="A2.E9.m1.11d">italic_P start_POSTSUBSCRIPT italic_ϕ , roman_ref end_POSTSUBSCRIPT ( italic_f ) ∼ 10 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_r start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_λ ) start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT italic_κ start_POSTSUPERSCRIPT ′ 4 end_POSTSUPERSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_f end_ARG start_ARG italic_v start_POSTSUBSCRIPT roman_sw end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG [ italic_f italic_P start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 italic_D end_POSTSUPERSCRIPT ( italic_f , italic_r ) ] start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 10 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_π start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_κ start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT italic_L start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT italic_f end_ARG start_ARG italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_v start_POSTSUBSCRIPT roman_sw end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG [ italic_f italic_P start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 roman_D end_POSTSUPERSCRIPT ( italic_f , italic_r ) ] start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(B9)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A2.p6.10">where we have stuck in a <math alttext="10^{2}" class="ltx_Math" display="inline" id="A2.p6.10.m1.1"><semantics id="A2.p6.10.m1.1a"><msup id="A2.p6.10.m1.1.1" xref="A2.p6.10.m1.1.1.cmml"><mn id="A2.p6.10.m1.1.1.2" xref="A2.p6.10.m1.1.1.2.cmml">10</mn><mn id="A2.p6.10.m1.1.1.3" xref="A2.p6.10.m1.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="A2.p6.10.m1.1b"><apply id="A2.p6.10.m1.1.1.cmml" xref="A2.p6.10.m1.1.1"><csymbol cd="ambiguous" id="A2.p6.10.m1.1.1.1.cmml" xref="A2.p6.10.m1.1.1">superscript</csymbol><cn id="A2.p6.10.m1.1.1.2.cmml" type="integer" xref="A2.p6.10.m1.1.1.2">10</cn><cn id="A2.p6.10.m1.1.1.3.cmml" type="integer" xref="A2.p6.10.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p6.10.m1.1c">10^{2}</annotation><annotation encoding="application/x-llamapun" id="A2.p6.10.m1.1d">10 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> prefactor to match our order-of magnitude estimates from evaluating this integrand.</p> </div> <div class="ltx_para" id="A2.p7"> <p class="ltx_p" id="A2.p7.1">Finally, our expression for the RMS strain error using this estimate for <math alttext="P_{\rm\phi,ref}" class="ltx_Math" display="inline" id="A2.p7.1.m1.2"><semantics id="A2.p7.1.m1.2a"><msub id="A2.p7.1.m1.2.3" xref="A2.p7.1.m1.2.3.cmml"><mi id="A2.p7.1.m1.2.3.2" xref="A2.p7.1.m1.2.3.2.cmml">P</mi><mrow id="A2.p7.1.m1.2.2.2.4" xref="A2.p7.1.m1.2.2.2.3.cmml"><mi id="A2.p7.1.m1.1.1.1.1" xref="A2.p7.1.m1.1.1.1.1.cmml">ϕ</mi><mo id="A2.p7.1.m1.2.2.2.4.1" xref="A2.p7.1.m1.2.2.2.3.cmml">,</mo><mi id="A2.p7.1.m1.2.2.2.2" xref="A2.p7.1.m1.2.2.2.2.cmml">ref</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A2.p7.1.m1.2b"><apply id="A2.p7.1.m1.2.3.cmml" xref="A2.p7.1.m1.2.3"><csymbol cd="ambiguous" id="A2.p7.1.m1.2.3.1.cmml" xref="A2.p7.1.m1.2.3">subscript</csymbol><ci id="A2.p7.1.m1.2.3.2.cmml" xref="A2.p7.1.m1.2.3.2">𝑃</ci><list id="A2.p7.1.m1.2.2.2.3.cmml" xref="A2.p7.1.m1.2.2.2.4"><ci id="A2.p7.1.m1.1.1.1.1.cmml" xref="A2.p7.1.m1.1.1.1.1">italic-ϕ</ci><ci id="A2.p7.1.m1.2.2.2.2.cmml" xref="A2.p7.1.m1.2.2.2.2">ref</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p7.1.m1.2c">P_{\rm\phi,ref}</annotation><annotation encoding="application/x-llamapun" id="A2.p7.1.m1.2d">italic_P start_POSTSUBSCRIPT italic_ϕ , roman_ref end_POSTSUBSCRIPT</annotation></semantics></math> and our de-dispersion estimator given in equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#A1.E10" title="In Appendix A Dispersion ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">A10</span></a>) becomes</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="A3.EGx16"> <tbody id="A2.E10"> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\widetilde{h}_{\rm ref,rms}\ " class="ltx_Math" display="inline" id="A2.Ex4.m1.2"><semantics id="A2.Ex4.m1.2a"><msub id="A2.Ex4.m1.2.3" xref="A2.Ex4.m1.2.3.cmml"><mover accent="true" id="A2.Ex4.m1.2.3.2" xref="A2.Ex4.m1.2.3.2.cmml"><mi id="A2.Ex4.m1.2.3.2.2" xref="A2.Ex4.m1.2.3.2.2.cmml">h</mi><mo id="A2.Ex4.m1.2.3.2.1" xref="A2.Ex4.m1.2.3.2.1.cmml">~</mo></mover><mrow id="A2.Ex4.m1.2.2.2.4" xref="A2.Ex4.m1.2.2.2.3.cmml"><mi id="A2.Ex4.m1.1.1.1.1" xref="A2.Ex4.m1.1.1.1.1.cmml">ref</mi><mo id="A2.Ex4.m1.2.2.2.4.1" xref="A2.Ex4.m1.2.2.2.3.cmml">,</mo><mi id="A2.Ex4.m1.2.2.2.2" xref="A2.Ex4.m1.2.2.2.2.cmml">rms</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A2.Ex4.m1.2b"><apply id="A2.Ex4.m1.2.3.cmml" xref="A2.Ex4.m1.2.3"><csymbol cd="ambiguous" id="A2.Ex4.m1.2.3.1.cmml" xref="A2.Ex4.m1.2.3">subscript</csymbol><apply id="A2.Ex4.m1.2.3.2.cmml" xref="A2.Ex4.m1.2.3.2"><ci id="A2.Ex4.m1.2.3.2.1.cmml" xref="A2.Ex4.m1.2.3.2.1">~</ci><ci id="A2.Ex4.m1.2.3.2.2.cmml" xref="A2.Ex4.m1.2.3.2.2">ℎ</ci></apply><list id="A2.Ex4.m1.2.2.2.3.cmml" xref="A2.Ex4.m1.2.2.2.4"><ci id="A2.Ex4.m1.1.1.1.1.cmml" xref="A2.Ex4.m1.1.1.1.1">ref</ci><ci id="A2.Ex4.m1.2.2.2.2.cmml" xref="A2.Ex4.m1.2.2.2.2">rms</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex4.m1.2c">\displaystyle\widetilde{h}_{\rm ref,rms}\ </annotation><annotation encoding="application/x-llamapun" id="A2.Ex4.m1.2d">over~ start_ARG italic_h end_ARG start_POSTSUBSCRIPT roman_ref , roman_rms end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="A2.Ex4.m2.1"><semantics id="A2.Ex4.m2.1a"><mo id="A2.Ex4.m2.1.1" xref="A2.Ex4.m2.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="A2.Ex4.m2.1b"><eq id="A2.Ex4.m2.1.1.cmml" xref="A2.Ex4.m2.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex4.m2.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="A2.Ex4.m2.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\ f(\lambda_{1},\lambda_{2})\frac{\lambda_{1}\sqrt{4P_{\rm\phi,% ref}(f)|_{\lambda_{1}}}}{2\pi L~{}\sqrt{R(f)}}," class="ltx_Math" display="inline" id="A2.Ex4.m3.7"><semantics id="A2.Ex4.m3.7a"><mrow id="A2.Ex4.m3.7.7.1" xref="A2.Ex4.m3.7.7.1.1.cmml"><mrow id="A2.Ex4.m3.7.7.1.1" xref="A2.Ex4.m3.7.7.1.1.cmml"><mi id="A2.Ex4.m3.7.7.1.1.4" xref="A2.Ex4.m3.7.7.1.1.4.cmml">f</mi><mo id="A2.Ex4.m3.7.7.1.1.3" xref="A2.Ex4.m3.7.7.1.1.3.cmml">⁢</mo><mrow id="A2.Ex4.m3.7.7.1.1.2.2" xref="A2.Ex4.m3.7.7.1.1.2.3.cmml"><mo 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italic_π italic_L square-root start_ARG italic_R ( italic_f ) end_ARG end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="3"><span class="ltx_tag ltx_tag_equation ltx_align_right">(B10)</span></td> </tr> <tr class="ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle\ \sim" class="ltx_Math" display="inline" id="A2.Ex5.m1.1"><semantics id="A2.Ex5.m1.1a"><mo id="A2.Ex5.m1.1.1" xref="A2.Ex5.m1.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="A2.Ex5.m1.1b"><csymbol cd="latexml" id="A2.Ex5.m1.1.1.cmml" xref="A2.Ex5.m1.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex5.m1.1c">\displaystyle\ \sim</annotation><annotation encoding="application/x-llamapun" id="A2.Ex5.m1.1d">∼</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\ 10^{-25}~{}~{}{\rm Hz}^{-1/2}~{}R(f)^{-1/2}\left(\frac{f(% \lambda_{1},\lambda_{2})}{4}\right)\left(\frac{\lambda_{1}}{1\,\rm cm}\right)^% {4}\left(\frac{f}{10^{-5}{\rm Hz}}\right)^{(3-2\beta)/2}," class="ltx_Math" display="inline" id="A2.Ex5.m2.7"><semantics id="A2.Ex5.m2.7a"><mrow id="A2.Ex5.m2.7.7.1" xref="A2.Ex5.m2.7.7.1.1.cmml"><mrow id="A2.Ex5.m2.7.7.1.1" xref="A2.Ex5.m2.7.7.1.1.cmml"><msup id="A2.Ex5.m2.7.7.1.1.2" xref="A2.Ex5.m2.7.7.1.1.2.cmml"><mn id="A2.Ex5.m2.7.7.1.1.2.2" xref="A2.Ex5.m2.7.7.1.1.2.2.cmml">10</mn><mrow id="A2.Ex5.m2.7.7.1.1.2.3" xref="A2.Ex5.m2.7.7.1.1.2.3.cmml"><mo id="A2.Ex5.m2.7.7.1.1.2.3a" xref="A2.Ex5.m2.7.7.1.1.2.3.cmml">−</mo><mn id="A2.Ex5.m2.7.7.1.1.2.3.2" xref="A2.Ex5.m2.7.7.1.1.2.3.2.cmml">25</mn></mrow></msup><mo id="A2.Ex5.m2.7.7.1.1.1" lspace="0.660em" xref="A2.Ex5.m2.7.7.1.1.1.cmml">⁢</mo><msup 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encoding="application/x-llamapun" id="A2.Ex5.m2.7d">10 start_POSTSUPERSCRIPT - 25 end_POSTSUPERSCRIPT roman_Hz start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT italic_R ( italic_f ) start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT ( divide start_ARG italic_f ( italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_λ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_ARG start_ARG 4 end_ARG ) ( divide start_ARG italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 1 roman_cm end_ARG ) start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ( divide start_ARG italic_f end_ARG start_ARG 10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT roman_Hz end_ARG ) start_POSTSUPERSCRIPT ( 3 - 2 italic_β ) / 2 end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr class="ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_eqn_cell"></td> 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id="A2.E10.m1.6.6.3.1.cmml" xref="A2.E10.m1.6.6.3.1"></times><cn id="A2.E10.m1.6.6.3.2.cmml" type="integer" xref="A2.E10.m1.6.6.3.2">30</cn><ci id="A2.E10.m1.6.6.3.3.cmml" xref="A2.E10.m1.6.6.3.3">AU</ci></apply></apply><cn id="A2.E10.m1.7.7.1.1.3.4.3.cmml" type="integer" xref="A2.E10.m1.7.7.1.1.3.4.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E10.m1.7c">\displaystyle\ \times\left(\frac{P_{e,0}^{\rm 1D}}{10^{11.1}~{}\text{m${}^{-6}% $ Hz${}^{-1}$}}\right)\left(\frac{r}{30\,\rm AU}\right)^{-3}\left(\frac{L}{30% \,\rm AU}\right)^{2}.</annotation><annotation encoding="application/x-llamapun" id="A2.E10.m1.7d">× ( divide start_ARG italic_P start_POSTSUBSCRIPT italic_e , 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 roman_D end_POSTSUPERSCRIPT end_ARG start_ARG 10 start_POSTSUPERSCRIPT 11.1 end_POSTSUPERSCRIPT m Hz end_ARG ) ( divide start_ARG italic_r end_ARG start_ARG 30 roman_AU end_ARG ) start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT ( divide start_ARG italic_L end_ARG start_ARG 30 roman_AU end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </tbody> </table> <p class="ltx_p" id="A2.p7.4">where <math alttext="f(\lambda_{1},\lambda_{2})=\Big{|}\frac{\lambda_{2}/\lambda_{1}^{3}}{(\lambda_% {1}^{2}-\lambda_{2}^{2})}\left(\lambda_{1}\lambda_{2}^{3}-\lambda_{2}\lambda_{% 1}^{3}\right)\Big{|}" class="ltx_Math" display="inline" id="A2.p7.2.m1.4"><semantics id="A2.p7.2.m1.4a"><mrow id="A2.p7.2.m1.4.4" xref="A2.p7.2.m1.4.4.cmml"><mrow id="A2.p7.2.m1.3.3.2" xref="A2.p7.2.m1.3.3.2.cmml"><mi id="A2.p7.2.m1.3.3.2.4" xref="A2.p7.2.m1.3.3.2.4.cmml">f</mi><mo id="A2.p7.2.m1.3.3.2.3" xref="A2.p7.2.m1.3.3.2.3.cmml">⁢</mo><mrow id="A2.p7.2.m1.3.3.2.2.2" xref="A2.p7.2.m1.3.3.2.2.3.cmml"><mo id="A2.p7.2.m1.3.3.2.2.2.3" stretchy="false" xref="A2.p7.2.m1.3.3.2.2.3.cmml">(</mo><msub id="A2.p7.2.m1.2.2.1.1.1.1" 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id="A2.p7.2.m1.4.4.3.1.1.1.1.1.3.3.3.cmml" type="integer" xref="A2.p7.2.m1.4.4.3.1.1.1.1.1.3.3.3">3</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p7.2.m1.4c">f(\lambda_{1},\lambda_{2})=\Big{|}\frac{\lambda_{2}/\lambda_{1}^{3}}{(\lambda_% {1}^{2}-\lambda_{2}^{2})}\left(\lambda_{1}\lambda_{2}^{3}-\lambda_{2}\lambda_{% 1}^{3}\right)\Big{|}</annotation><annotation encoding="application/x-llamapun" id="A2.p7.2.m1.4d">italic_f ( italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_λ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = | divide start_ARG italic_λ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT / italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG start_ARG ( italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_λ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_ARG ( italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT - italic_λ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) |</annotation></semantics></math>, which evaluates to <math alttext="f(\lambda_{1},\lambda_{2})=4" class="ltx_Math" display="inline" id="A2.p7.3.m2.2"><semantics id="A2.p7.3.m2.2a"><mrow id="A2.p7.3.m2.2.2" xref="A2.p7.3.m2.2.2.cmml"><mrow id="A2.p7.3.m2.2.2.2" xref="A2.p7.3.m2.2.2.2.cmml"><mi id="A2.p7.3.m2.2.2.2.4" xref="A2.p7.3.m2.2.2.2.4.cmml">f</mi><mo id="A2.p7.3.m2.2.2.2.3" xref="A2.p7.3.m2.2.2.2.3.cmml">⁢</mo><mrow id="A2.p7.3.m2.2.2.2.2.2" xref="A2.p7.3.m2.2.2.2.2.3.cmml"><mo id="A2.p7.3.m2.2.2.2.2.2.3" stretchy="false" xref="A2.p7.3.m2.2.2.2.2.3.cmml">(</mo><msub id="A2.p7.3.m2.1.1.1.1.1.1" xref="A2.p7.3.m2.1.1.1.1.1.1.cmml"><mi id="A2.p7.3.m2.1.1.1.1.1.1.2" xref="A2.p7.3.m2.1.1.1.1.1.1.2.cmml">λ</mi><mn id="A2.p7.3.m2.1.1.1.1.1.1.3" xref="A2.p7.3.m2.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="A2.p7.3.m2.2.2.2.2.2.4" xref="A2.p7.3.m2.2.2.2.2.3.cmml">,</mo><msub id="A2.p7.3.m2.2.2.2.2.2.2" xref="A2.p7.3.m2.2.2.2.2.2.2.cmml"><mi id="A2.p7.3.m2.2.2.2.2.2.2.2" xref="A2.p7.3.m2.2.2.2.2.2.2.2.cmml">λ</mi><mn id="A2.p7.3.m2.2.2.2.2.2.2.3" xref="A2.p7.3.m2.2.2.2.2.2.2.3.cmml">2</mn></msub><mo id="A2.p7.3.m2.2.2.2.2.2.5" stretchy="false" xref="A2.p7.3.m2.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="A2.p7.3.m2.2.2.3" xref="A2.p7.3.m2.2.2.3.cmml">=</mo><mn id="A2.p7.3.m2.2.2.4" xref="A2.p7.3.m2.2.2.4.cmml">4</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.p7.3.m2.2b"><apply id="A2.p7.3.m2.2.2.cmml" xref="A2.p7.3.m2.2.2"><eq id="A2.p7.3.m2.2.2.3.cmml" xref="A2.p7.3.m2.2.2.3"></eq><apply id="A2.p7.3.m2.2.2.2.cmml" xref="A2.p7.3.m2.2.2.2"><times id="A2.p7.3.m2.2.2.2.3.cmml" xref="A2.p7.3.m2.2.2.2.3"></times><ci id="A2.p7.3.m2.2.2.2.4.cmml" xref="A2.p7.3.m2.2.2.2.4">𝑓</ci><interval closure="open" id="A2.p7.3.m2.2.2.2.2.3.cmml" xref="A2.p7.3.m2.2.2.2.2.2"><apply id="A2.p7.3.m2.1.1.1.1.1.1.cmml" xref="A2.p7.3.m2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.p7.3.m2.1.1.1.1.1.1.1.cmml" xref="A2.p7.3.m2.1.1.1.1.1.1">subscript</csymbol><ci id="A2.p7.3.m2.1.1.1.1.1.1.2.cmml" xref="A2.p7.3.m2.1.1.1.1.1.1.2">𝜆</ci><cn id="A2.p7.3.m2.1.1.1.1.1.1.3.cmml" type="integer" xref="A2.p7.3.m2.1.1.1.1.1.1.3">1</cn></apply><apply id="A2.p7.3.m2.2.2.2.2.2.2.cmml" xref="A2.p7.3.m2.2.2.2.2.2.2"><csymbol cd="ambiguous" id="A2.p7.3.m2.2.2.2.2.2.2.1.cmml" xref="A2.p7.3.m2.2.2.2.2.2.2">subscript</csymbol><ci id="A2.p7.3.m2.2.2.2.2.2.2.2.cmml" xref="A2.p7.3.m2.2.2.2.2.2.2.2">𝜆</ci><cn id="A2.p7.3.m2.2.2.2.2.2.2.3.cmml" type="integer" xref="A2.p7.3.m2.2.2.2.2.2.2.3">2</cn></apply></interval></apply><cn id="A2.p7.3.m2.2.2.4.cmml" type="integer" xref="A2.p7.3.m2.2.2.4">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p7.3.m2.2c">f(\lambda_{1},\lambda_{2})=4</annotation><annotation encoding="application/x-llamapun" id="A2.p7.3.m2.2d">italic_f ( italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_λ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = 4</annotation></semantics></math> if <math alttext="\lambda_{2}=2\lambda_{1}" class="ltx_Math" display="inline" id="A2.p7.4.m3.1"><semantics id="A2.p7.4.m3.1a"><mrow id="A2.p7.4.m3.1.1" xref="A2.p7.4.m3.1.1.cmml"><msub id="A2.p7.4.m3.1.1.2" xref="A2.p7.4.m3.1.1.2.cmml"><mi id="A2.p7.4.m3.1.1.2.2" xref="A2.p7.4.m3.1.1.2.2.cmml">λ</mi><mn id="A2.p7.4.m3.1.1.2.3" xref="A2.p7.4.m3.1.1.2.3.cmml">2</mn></msub><mo id="A2.p7.4.m3.1.1.1" xref="A2.p7.4.m3.1.1.1.cmml">=</mo><mrow id="A2.p7.4.m3.1.1.3" xref="A2.p7.4.m3.1.1.3.cmml"><mn id="A2.p7.4.m3.1.1.3.2" xref="A2.p7.4.m3.1.1.3.2.cmml">2</mn><mo id="A2.p7.4.m3.1.1.3.1" xref="A2.p7.4.m3.1.1.3.1.cmml">⁢</mo><msub id="A2.p7.4.m3.1.1.3.3" xref="A2.p7.4.m3.1.1.3.3.cmml"><mi id="A2.p7.4.m3.1.1.3.3.2" xref="A2.p7.4.m3.1.1.3.3.2.cmml">λ</mi><mn id="A2.p7.4.m3.1.1.3.3.3" xref="A2.p7.4.m3.1.1.3.3.3.cmml">1</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.p7.4.m3.1b"><apply id="A2.p7.4.m3.1.1.cmml" xref="A2.p7.4.m3.1.1"><eq id="A2.p7.4.m3.1.1.1.cmml" xref="A2.p7.4.m3.1.1.1"></eq><apply id="A2.p7.4.m3.1.1.2.cmml" xref="A2.p7.4.m3.1.1.2"><csymbol cd="ambiguous" id="A2.p7.4.m3.1.1.2.1.cmml" xref="A2.p7.4.m3.1.1.2">subscript</csymbol><ci id="A2.p7.4.m3.1.1.2.2.cmml" xref="A2.p7.4.m3.1.1.2.2">𝜆</ci><cn id="A2.p7.4.m3.1.1.2.3.cmml" type="integer" xref="A2.p7.4.m3.1.1.2.3">2</cn></apply><apply id="A2.p7.4.m3.1.1.3.cmml" xref="A2.p7.4.m3.1.1.3"><times id="A2.p7.4.m3.1.1.3.1.cmml" xref="A2.p7.4.m3.1.1.3.1"></times><cn id="A2.p7.4.m3.1.1.3.2.cmml" type="integer" xref="A2.p7.4.m3.1.1.3.2">2</cn><apply id="A2.p7.4.m3.1.1.3.3.cmml" xref="A2.p7.4.m3.1.1.3.3"><csymbol cd="ambiguous" id="A2.p7.4.m3.1.1.3.3.1.cmml" xref="A2.p7.4.m3.1.1.3.3">subscript</csymbol><ci id="A2.p7.4.m3.1.1.3.3.2.cmml" xref="A2.p7.4.m3.1.1.3.3.2">𝜆</ci><cn id="A2.p7.4.m3.1.1.3.3.3.cmml" type="integer" xref="A2.p7.4.m3.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p7.4.m3.1c">\lambda_{2}=2\lambda_{1}</annotation><annotation encoding="application/x-llamapun" id="A2.p7.4.m3.1d">italic_λ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 2 italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and we have used the parameterization of the power given by equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#S2.E14" title="In 2.3.2 Solar wind drag ‣ 2.3 Acceleration noise ‣ 2 Strain noise sources ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">14</span></a>). Equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#A2.E10" title="In Appendix B Refraction and diffraction ‣ Outer Solar System spacecraft without drag-free control to probe the 𝜇Hz gravitational wave frontier"><span class="ltx_text ltx_ref_tag">B10</span></a>) shows that the effect of refraction is very small relative to other sources of noise.</p> </div> </section> <section class="ltx_appendix" id="A3"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix C </span>Data rates</h2> <div class="ltx_para" id="A3.p1"> <p class="ltx_p" id="A3.p1.3">Due to the tens of astronomical units distances of the spacecraft in our concept, the concept developed here is likely constrained to a downlink data rate of <math alttext="\sim 10" class="ltx_Math" display="inline" id="A3.p1.1.m1.1"><semantics id="A3.p1.1.m1.1a"><mrow id="A3.p1.1.m1.1.1" xref="A3.p1.1.m1.1.1.cmml"><mi id="A3.p1.1.m1.1.1.2" xref="A3.p1.1.m1.1.1.2.cmml"></mi><mo id="A3.p1.1.m1.1.1.1" xref="A3.p1.1.m1.1.1.1.cmml">∼</mo><mn id="A3.p1.1.m1.1.1.3" xref="A3.p1.1.m1.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.p1.1.m1.1b"><apply id="A3.p1.1.m1.1.1.cmml" xref="A3.p1.1.m1.1.1"><csymbol cd="latexml" id="A3.p1.1.m1.1.1.1.cmml" xref="A3.p1.1.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="A3.p1.1.m1.1.1.2.cmml" xref="A3.p1.1.m1.1.1.2">absent</csymbol><cn id="A3.p1.1.m1.1.1.3.cmml" type="integer" xref="A3.p1.1.m1.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.1.m1.1c">\sim 10</annotation><annotation encoding="application/x-llamapun" id="A3.p1.1.m1.1d">∼ 10</annotation></semantics></math> kbps as has been achieved by previous outer Solar System missions such as the New Horizons spacecraft <cite class="ltx_cite ltx_citemacro_citep">(Fountain et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib31" title="">2008</a>)</cite>. To estimate how much data would need to be telemetered back to Earth, let us compare the proposed mission with the LISA mission. LISA aims to send a downlink with a data rate of <math alttext="230\;" class="ltx_Math" display="inline" id="A3.p1.2.m2.1"><semantics id="A3.p1.2.m2.1a"><mn id="A3.p1.2.m2.1.1" xref="A3.p1.2.m2.1.1.cmml">230</mn><annotation-xml encoding="MathML-Content" id="A3.p1.2.m2.1b"><cn id="A3.p1.2.m2.1.1.cmml" type="integer" xref="A3.p1.2.m2.1.1">230</cn></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.2.m2.1c">230\;</annotation><annotation encoding="application/x-llamapun" id="A3.p1.2.m2.1d">230</annotation></semantics></math>kbps for eight hours every day <cite class="ltx_cite ltx_citemacro_citep">(Colpi et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib21" title="">2024</a>)</cite>. Since LISA’s goal is to constrain gravitational waves to frequencies as high as <math alttext="1" class="ltx_Math" display="inline" id="A3.p1.3.m3.1"><semantics id="A3.p1.3.m3.1a"><mn id="A3.p1.3.m3.1.1" xref="A3.p1.3.m3.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="A3.p1.3.m3.1b"><cn id="A3.p1.3.m3.1.1.cmml" type="integer" xref="A3.p1.3.m3.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.3.m3.1c">1</annotation><annotation encoding="application/x-llamapun" id="A3.p1.3.m3.1d">1</annotation></semantics></math> Hz, it is designed to sample at 4 Hz <cite class="ltx_cite ltx_citemacro_citep">(Bayle & Hartwig, <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib15" title="">2023</a>; Colpi et al., <a class="ltx_ref" href="https://arxiv.org/html/2411.15072v1#bib.bib21" title="">2024</a>)</cite>. Because LISA has three interferometric arms, the downlinked data include data streams from the six interferometric observables as well as from the positions of six test masses.</p> </div> <div class="ltx_para" id="A3.p2"> <p class="ltx_p" id="A3.p2.4">The proposed mission likely could accommodate a total data transmission that is a factor of <math alttext="\sim 10^{5}" class="ltx_Math" display="inline" id="A3.p2.1.m1.1"><semantics id="A3.p2.1.m1.1a"><mrow id="A3.p2.1.m1.1.1" xref="A3.p2.1.m1.1.1.cmml"><mi id="A3.p2.1.m1.1.1.2" xref="A3.p2.1.m1.1.1.2.cmml"></mi><mo id="A3.p2.1.m1.1.1.1" xref="A3.p2.1.m1.1.1.1.cmml">∼</mo><msup id="A3.p2.1.m1.1.1.3" xref="A3.p2.1.m1.1.1.3.cmml"><mn id="A3.p2.1.m1.1.1.3.2" xref="A3.p2.1.m1.1.1.3.2.cmml">10</mn><mn id="A3.p2.1.m1.1.1.3.3" xref="A3.p2.1.m1.1.1.3.3.cmml">5</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="A3.p2.1.m1.1b"><apply id="A3.p2.1.m1.1.1.cmml" xref="A3.p2.1.m1.1.1"><csymbol cd="latexml" id="A3.p2.1.m1.1.1.1.cmml" xref="A3.p2.1.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="A3.p2.1.m1.1.1.2.cmml" xref="A3.p2.1.m1.1.1.2">absent</csymbol><apply id="A3.p2.1.m1.1.1.3.cmml" xref="A3.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="A3.p2.1.m1.1.1.3.1.cmml" xref="A3.p2.1.m1.1.1.3">superscript</csymbol><cn id="A3.p2.1.m1.1.1.3.2.cmml" type="integer" xref="A3.p2.1.m1.1.1.3.2">10</cn><cn id="A3.p2.1.m1.1.1.3.3.cmml" type="integer" xref="A3.p2.1.m1.1.1.3.3">5</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p2.1.m1.1c">\sim 10^{5}</annotation><annotation encoding="application/x-llamapun" id="A3.p2.1.m1.1d">∼ 10 start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT</annotation></semantics></math> smaller than LISA because (1) it targets <math alttext="f<10^{-4}" class="ltx_Math" display="inline" id="A3.p2.2.m2.1"><semantics id="A3.p2.2.m2.1a"><mrow id="A3.p2.2.m2.1.1" xref="A3.p2.2.m2.1.1.cmml"><mi id="A3.p2.2.m2.1.1.2" xref="A3.p2.2.m2.1.1.2.cmml">f</mi><mo id="A3.p2.2.m2.1.1.1" xref="A3.p2.2.m2.1.1.1.cmml"><</mo><msup id="A3.p2.2.m2.1.1.3" xref="A3.p2.2.m2.1.1.3.cmml"><mn id="A3.p2.2.m2.1.1.3.2" xref="A3.p2.2.m2.1.1.3.2.cmml">10</mn><mrow id="A3.p2.2.m2.1.1.3.3" xref="A3.p2.2.m2.1.1.3.3.cmml"><mo id="A3.p2.2.m2.1.1.3.3a" xref="A3.p2.2.m2.1.1.3.3.cmml">−</mo><mn id="A3.p2.2.m2.1.1.3.3.2" xref="A3.p2.2.m2.1.1.3.3.2.cmml">4</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="A3.p2.2.m2.1b"><apply id="A3.p2.2.m2.1.1.cmml" xref="A3.p2.2.m2.1.1"><lt id="A3.p2.2.m2.1.1.1.cmml" xref="A3.p2.2.m2.1.1.1"></lt><ci id="A3.p2.2.m2.1.1.2.cmml" xref="A3.p2.2.m2.1.1.2">𝑓</ci><apply id="A3.p2.2.m2.1.1.3.cmml" xref="A3.p2.2.m2.1.1.3"><csymbol cd="ambiguous" id="A3.p2.2.m2.1.1.3.1.cmml" xref="A3.p2.2.m2.1.1.3">superscript</csymbol><cn id="A3.p2.2.m2.1.1.3.2.cmml" type="integer" xref="A3.p2.2.m2.1.1.3.2">10</cn><apply id="A3.p2.2.m2.1.1.3.3.cmml" xref="A3.p2.2.m2.1.1.3.3"><minus id="A3.p2.2.m2.1.1.3.3.1.cmml" xref="A3.p2.2.m2.1.1.3.3"></minus><cn id="A3.p2.2.m2.1.1.3.3.2.cmml" type="integer" xref="A3.p2.2.m2.1.1.3.3.2">4</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p2.2.m2.1c">f<10^{-4}</annotation><annotation encoding="application/x-llamapun" id="A3.p2.2.m2.1d">italic_f < 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT</annotation></semantics></math>Hz and so can sample on <math alttext="10^{4}\times" class="ltx_math_unparsed" display="inline" id="A3.p2.3.m3.1"><semantics id="A3.p2.3.m3.1a"><mrow id="A3.p2.3.m3.1b"><msup id="A3.p2.3.m3.1.1"><mn id="A3.p2.3.m3.1.1.2">10</mn><mn id="A3.p2.3.m3.1.1.3">4</mn></msup><mo id="A3.p2.3.m3.1.2" lspace="0.222em">×</mo></mrow><annotation encoding="application/x-tex" id="A3.p2.3.m3.1c">10^{4}\times</annotation><annotation encoding="application/x-llamapun" id="A3.p2.3.m3.1d">10 start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ×</annotation></semantics></math> longer timescales, (2) it has one or two arms and no test masses such that there are fewer data streams to downlink, and (3) it would have <math alttext="\sim 10^{4}" class="ltx_Math" display="inline" id="A3.p2.4.m4.1"><semantics id="A3.p2.4.m4.1a"><mrow id="A3.p2.4.m4.1.1" xref="A3.p2.4.m4.1.1.cmml"><mi id="A3.p2.4.m4.1.1.2" xref="A3.p2.4.m4.1.1.2.cmml"></mi><mo id="A3.p2.4.m4.1.1.1" xref="A3.p2.4.m4.1.1.1.cmml">∼</mo><msup id="A3.p2.4.m4.1.1.3" xref="A3.p2.4.m4.1.1.3.cmml"><mn id="A3.p2.4.m4.1.1.3.2" xref="A3.p2.4.m4.1.1.3.2.cmml">10</mn><mn id="A3.p2.4.m4.1.1.3.3" xref="A3.p2.4.m4.1.1.3.3.cmml">4</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="A3.p2.4.m4.1b"><apply id="A3.p2.4.m4.1.1.cmml" xref="A3.p2.4.m4.1.1"><csymbol cd="latexml" id="A3.p2.4.m4.1.1.1.cmml" xref="A3.p2.4.m4.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="A3.p2.4.m4.1.1.2.cmml" xref="A3.p2.4.m4.1.1.2">absent</csymbol><apply id="A3.p2.4.m4.1.1.3.cmml" xref="A3.p2.4.m4.1.1.3"><csymbol cd="ambiguous" id="A3.p2.4.m4.1.1.3.1.cmml" xref="A3.p2.4.m4.1.1.3">superscript</csymbol><cn id="A3.p2.4.m4.1.1.3.2.cmml" type="integer" xref="A3.p2.4.m4.1.1.3.2">10</cn><cn id="A3.p2.4.m4.1.1.3.3.cmml" type="integer" xref="A3.p2.4.m4.1.1.3.3">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p2.4.m4.1c">\sim 10^{4}</annotation><annotation encoding="application/x-llamapun" id="A3.p2.4.m4.1d">∼ 10 start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT</annotation></semantics></math> times less precision in the phase measurements relative to LISA and, hence, require fewer bits per sample. This much lower data rate means that a one-hour downlink at 10 kbps could potentially download an entire year of science data. Even downlinking two orders of magnitude more data than this simple estimate seems possible.</p> </div> </section> </article> </div> <footer class="ltx_page_footer"> <div class="ltx_page_logo">Generated on Fri Nov 22 16:58:29 2024 by <a class="ltx_LaTeXML_logo" href="http://dlmf.nist.gov/LaTeXML/"><span style="letter-spacing:-0.2em; margin-right:0.1em;">L<span class="ltx_font_smallcaps" style="position:relative; bottom:2.2pt;">a</span>T<span class="ltx_font_smallcaps" style="font-size:120%;position:relative; bottom:-0.2ex;">e</span></span><span style="font-size:90%; position:relative; bottom:-0.2ex;">XML</span><img alt="Mascot Sammy" src="data:image/png;base64,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"/></a> </div></footer> </div> </body> </html>