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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}}%