Studying the Properties of Spacetime with an Improved Dynamical Model of the Inner Solar System

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} div.type-section h2 { font-size: 20px; line-height: 26px; font-weight: 300; } div.type-section h3 { margin-left: 15px; margin-bottom: 0px; font-weight: 300; } .journal-tabs .tab-title.active a { } </style> <link rel="stylesheet" href="https://pub.mdpi-res.com/assets/css/slick.css?f38b2db10e01b157?1732615622"> <meta name="title" content="Studying the Properties of Spacetime with an Improved Dynamical Model of the Inner Solar System"> <meta name="description" content="Physical properties of the Sun (orientation of rotation axis, oblateness coefficient J2⊙, and change rate of the gravitational parameter μ˙⊙) are determined using a dynamical model describing the motion of the Sun, planets, the Moon, asteroids, and Trans-Neptunian objects (TNOs). Among the many kinds of observations used to determine the orbits and physical properties of the bodies, the most important for our study are precise interplanetary ranging data: Earth–Mercury ranges from MESSENGER spacecraft and Earth–Mars ranges from Odyssey and MRO. The findings allow us to improve the model of the Sun in modern planetary ephemerides. First, the dynamically determined direction of the Sun’s pole is ≈2° off the visible axis of rotation of the Sun’s surface, which is corroborated by present knowledge of the Sun’s interior. Second, the change rate of the Sun’s gravitational parameter is found to be smaller (in absolute value) than the nominal value derived from the estimate of mass loss through radiation and solar wind. Possible interpretations are discussed." > <link rel="image_src" href="https://pub.mdpi-res.com/img/journals/universe-logo.png?8600e93ff98dbf14" > <meta name="dc.title" content="Studying the Properties of Spacetime with an Improved Dynamical Model of the Inner Solar System"> <meta name="dc.creator" content="Dmitry Pavlov"> <meta name="dc.creator" content="Ivan Dolgakov"> <meta name="dc.type" content="Article"> <meta name="dc.source" content="Universe 2024, Vol. 10, Page 413"> <meta name="dc.date" content="2024-11-03"> <meta name ="dc.identifier" content="10.3390/universe10110413"> <meta name="dc.publisher" content="Multidisciplinary Digital Publishing Institute"> <meta name="dc.rights" content="http://creativecommons.org/licenses/by/3.0/"> <meta name="dc.format" content="application/pdf" > <meta name="dc.language" content="en" > <meta name="dc.description" content="Physical properties of the Sun (orientation of rotation axis, oblateness coefficient J2⊙, and change rate of the gravitational parameter μ˙⊙) are determined using a dynamical model describing the motion of the Sun, planets, the Moon, asteroids, and Trans-Neptunian objects (TNOs). Among the many kinds of observations used to determine the orbits and physical properties of the bodies, the most important for our study are precise interplanetary ranging data: Earth–Mercury ranges from MESSENGER spacecraft and Earth–Mars ranges from Odyssey and MRO. The findings allow us to improve the model of the Sun in modern planetary ephemerides. First, the dynamically determined direction of the Sun’s pole is ≈2° off the visible axis of rotation of the Sun’s surface, which is corroborated by present knowledge of the Sun’s interior. Second, the change rate of the Sun’s gravitational parameter is found to be smaller (in absolute value) than the nominal value derived from the estimate of mass loss through radiation and solar wind. Possible interpretations are discussed." > <meta name="dc.subject" content="sun" > <meta name="dc.subject" content="radio astronomy" > <meta name="dc.subject" content="gravitational constant" > <meta name="dc.subject" content="planetary ephemerides" > <meta name ="prism.issn" content="2218-1997"> <meta name ="prism.publicationName" content="Universe"> <meta name ="prism.publicationDate" content="2024-11-03"> <meta name ="prism.volume" content="10"> <meta name ="prism.number" content="11"> <meta name ="prism.section" content="Article" > <meta name ="prism.startingPage" content="413" > <meta name="citation_issn" content="2218-1997"> <meta name="citation_journal_title" content="Universe"> <meta name="citation_publisher" content="Multidisciplinary Digital Publishing Institute"> <meta name="citation_title" content="Studying the Properties of Spacetime with an Improved Dynamical Model of the Inner Solar System"> <meta name="citation_publication_date" content="2024/11"> <meta name="citation_online_date" content="2024/11/03"> <meta name="citation_volume" content="10"> <meta name="citation_issue" content="11"> <meta name="citation_firstpage" content="413"> <meta name="citation_author" content="Pavlov, Dmitry"> <meta name="citation_author" content="Dolgakov, Ivan"> <meta name="citation_doi" content="10.3390/universe10110413"> <meta name="citation_id" content="mdpi-universe10110413"> <meta name="citation_abstract_html_url" content="https://www.mdpi.com/2218-1997/10/11/413"> <meta name="citation_pdf_url" content="https://www.mdpi.com/2218-1997/10/11/413/pdf?version=1730710847"> <link rel="alternate" type="application/pdf" title="PDF Full-Text" href="https://www.mdpi.com/2218-1997/10/11/413/pdf?version=1730710847"> <meta name="fulltext_pdf" content="https://www.mdpi.com/2218-1997/10/11/413/pdf?version=1730710847"> <meta name="citation_fulltext_html_url" content="https://www.mdpi.com/2218-1997/10/11/413/htm"> <link rel="alternate" type="text/html" title="HTML Full-Text" href="https://www.mdpi.com/2218-1997/10/11/413/htm"> <meta name="fulltext_html" content="https://www.mdpi.com/2218-1997/10/11/413/htm"> <link rel="alternate" type="text/xml" title="XML Full-Text" href="https://www.mdpi.com/2218-1997/10/11/413/xml"> <meta name="fulltext_xml" content="https://www.mdpi.com/2218-1997/10/11/413/xml"> <meta name="citation_xml_url" content="https://www.mdpi.com/2218-1997/10/11/413/xml"> <meta name="twitter:card" content="summary" /> <meta name="twitter:site" content="@MDPIOpenAccess" /> <meta name="twitter:image" content="https://pub.mdpi-res.com/img/journals/universe-logo-social.png?8600e93ff98dbf14" /> <meta property="fb:app_id" content="131189377574"/> <meta property="og:site_name" content="MDPI"/> <meta property="og:type" content="article"/> <meta property="og:url" content="https://www.mdpi.com/2218-1997/10/11/413" /> <meta property="og:title" content="Studying the Properties of Spacetime with an Improved Dynamical Model of the Inner Solar System" /> <meta property="og:description" content="Physical properties of the Sun (orientation of rotation axis, oblateness coefficient J2⊙, and change rate of the gravitational parameter μ˙⊙) are determined using a dynamical model describing the motion of the Sun, planets, the Moon, asteroids, and Trans-Neptunian objects (TNOs). Among the many kinds of observations used to determine the orbits and physical properties of the bodies, the most important for our study are precise interplanetary ranging data: Earth–Mercury ranges from MESSENGER spacecraft and Earth–Mars ranges from Odyssey and MRO. The findings allow us to improve the model of the Sun in modern planetary ephemerides. First, the dynamically determined direction of the Sun’s pole is ≈2° off the visible axis of rotation of the Sun’s surface, which is corroborated by present knowledge of the Sun’s interior. Second, the change rate of the Sun’s gravitational parameter is found to be smaller (in absolute value) than the nominal value derived from the estimate of mass loss through radiation and solar wind. Possible interpretations are discussed." /> <meta property="og:image" content="https://pub.mdpi-res.com/universe/universe-10-00413/article_deploy/html/images/universe-10-00413-g001-550.jpg?1730710978" /> <link rel="alternate" type="application/rss+xml" title="MDPI Publishing - Latest articles" href="https://www.mdpi.com/rss"> <meta name="google-site-verification" content="PxTlsg7z2S00aHroktQd57fxygEjMiNHydKn3txhvwY"> <meta name="facebook-domain-verification" content="mcoq8dtq6sb2hf7z29j8w515jjoof7" /> <script id="Cookiebot" data-cfasync="false" src="https://consent.cookiebot.com/uc.js" data-cbid="51491ddd-fe7a-4425-ab39-69c78c55829f" type="text/javascript" async></script>  <script type="text/plain" data-cookieconsent="statistics"> (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start': new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0], j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src= 'https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f); 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data-options='is_hover:true, hover_timeout:5000'> Dmitry Pavlov</div><div id="profile-card-drop13352778" data-dropdown-content class="f-dropdown content profile-card-content" aria-hidden="true" tabindex="-1"><div class="profile-card__title"><div class="sciprofiles-link" style="display: inline-block"><div class="sciprofiles-link__link"><img class="sciprofiles-link__image" src="/bundles/mdpisciprofileslink/img/unknown-user.png" style="width: auto; height: 16px; border-radius: 50%;"><span class="sciprofiles-link__name">Dmitry Pavlov</span></div></div></div><div class="profile-card__buttons" style="margin-bottom: 10px;"><a href="https://sciprofiles.com/profile/3887721?utm_source=mdpi.com&utm_medium=website&utm_campaign=avatar_name" class="button button--color-inversed" target="_blank"> SciProfiles </a><a href="https://scilit.net/scholars?q=Dmitry%20Pavlov" class="button button--color-inversed" target="_blank"> Scilit </a><a 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tabindex="-1"><div class="profile-card__title"><div class="sciprofiles-link" style="display: inline-block"><div class="sciprofiles-link__link"><img class="sciprofiles-link__image" src="/bundles/mdpisciprofileslink/img/unknown-user.png" style="width: auto; height: 16px; border-radius: 50%;"><span class="sciprofiles-link__name">Ivan Dolgakov</span></div></div></div><div class="profile-card__buttons" style="margin-bottom: 10px;"><a href="https://sciprofiles.com/profile/author/YTBqdmdBWUZXZ3pVc2Zkbm1MaXdINDIwanhQd2VCdWdEUmR6aXRkU2RVbz0=?utm_source=mdpi.com&utm_medium=website&utm_campaign=avatar_name" class="button button--color-inversed" target="_blank"> SciProfiles </a><a href="https://scilit.net/scholars?q=Ivan%20Dolgakov" class="button button--color-inversed" target="_blank"> Scilit </a><a href="https://www.preprints.org/search?search1=Ivan%20Dolgakov&field1=authors" class="button button--color-inversed" target="_blank"> Preprints.org </a><a href="https://scholar.google.com/scholar?q=Ivan%20Dolgakov" class="button button--color-inversed" target="_blank" rels="noopener noreferrer"> Google Scholar </a></div></div><sup> 2</sup></span> </div> <div class="nrm"></div> <span style="display:block; height:6px;"></span> <div></div> <div style="margin: 5px 0 15px 0;" class="hypothesis_container"> <div class="art-affiliations"> <div class="affiliation "> <div class="affiliation-item"><sup>1</sup></div> <div class="affiliation-name ">Faculty of Computer Science and Technology, St. Petersburg Electrotechnical University, ul. Professora Popova 5, 197022 St. Petersburg, Russia</div> </div> <div class="affiliation "> <div class="affiliation-item"><sup>2</sup></div> <div class="affiliation-name ">Institute of Applied Astronomy RAS, Kutuzov Quay 10, 191187 St. Petersburg, Russia</div> </div> <div class="affiliation"> <div class="affiliation-item"><sup>*</sup></div> <div class="affiliation-name ">Author to whom correspondence should be addressed. </div> </div> </div> </div> <div class="bib-identity" style="margin-bottom: 10px;"> <em>Universe</em> <b>2024</b>, <em>10</em>(11), 413; <a href="https://doi.org/10.3390/universe10110413">https://doi.org/10.3390/universe10110413</a> </div> <div class="pubhistory" style="font-weight: bold; padding-bottom: 10px;"> <span style="display: inline-block">Submission received: 30 September 2024</span> / <span style="display: inline-block">Revised: 30 October 2024</span> / <span style="display: inline-block">Accepted: 1 November 2024</span> / <span style="display: inline-block">Published: 3 November 2024</span> </div> <div class="highlight-box1"> <div class="download"> <a class="button button--color-inversed button--drop-down" data-dropdown="drop-download-1513114" aria-controls="drop-supplementary-1513114" aria-expanded="false"> Download <i class="material-icons">keyboard_arrow_down</i> </a> <div id="drop-download-1513114" class="f-dropdown label__btn__dropdown label__btn__dropdown--button" data-dropdown-content aria-hidden="true" tabindex="-1"> <a class="UD_ArticlePDF" href="/2218-1997/10/11/413/pdf?version=1730710847" data-name="Studying the Properties of Spacetime with an Improved Dynamical Model of the Inner Solar System" data-journal="universe">Download PDF</a> <br/> <a id="js-pdf-with-cover-access-captcha" href="#" data-target="/2218-1997/10/11/413/pdf-with-cover" class="accessCaptcha">Download PDF with Cover</a> <br/> <a id="js-xml-access-captcha" href="#" data-target="/2218-1997/10/11/413/xml" class="accessCaptcha">Download XML</a> <br/> <a href="/2218-1997/10/11/413/epub" id="epub_link">Download Epub</a> <br/> </div> <div class="js-browse-figures" style="display: inline-block;"> <a href="#" class="button button--color-inversed margin-bottom-10 openpopupgallery UI_BrowseArticleFigures" data-target='article-popup' data-counterslink = "https://www.mdpi.com/2218-1997/10/11/413/browse" >Browse Figures</a> </div> <div id="article-popup" class="popupgallery" style="display: inline; line-height: 200%"> <a href="https://pub.mdpi-res.com/universe/universe-10-00413/article_deploy/html/images/universe-10-00413-g001.png?1730710976" title=" <strong>Figure 1</strong><br/> <p>Difference between model values of MESSENGER ranges (<math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mi>II</mi> </msub> <mo>−</mo> <msub> <mi>C</mi> <mi mathvariant="normal">I</mi> </msub> </mrow> </semantics></math>).</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00413/article_deploy/html/images/universe-10-00413-g002.png?1730710978" title=" <strong>Figure 2</strong><br/> <p>Difference between model values of MESSENGER ranges (<math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mi>III</mi> </msub> <mo>−</mo> <msub> <mi>C</mi> <mi>II</mi> </msub> </mrow> </semantics></math>).</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00413/article_deploy/html/images/universe-10-00413-g003.png?1730710980" title=" <strong>Figure 3</strong><br/> <p>Difference between model values of MESSENGER ranges (<math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mi>III</mi> </msub> <mo>−</mo> <msub> <mi>C</mi> <mi mathvariant="normal">I</mi> </msub> </mrow> </semantics></math>).</p> "> </a> <a href="https://pub.mdpi-res.com/universe/universe-10-00413/article_deploy/html/images/universe-10-00413-g004.png?1730710981" title=" <strong>Figure 4</strong><br/> <p>MESSENGER residuals <math display="inline"><semantics> <mrow> <mi>O</mi> <mo>−</mo> <msub> <mi>C</mi> <mi>III</mi> </msub> </mrow> </semantics></math>.</p> "> </a> </div> <a class="button button--color-inversed" href="/2218-1997/10/11/413/notes">Versions Notes</a> </div> </div> <div class="responsive-moving-container small hidden" data-id="article-counters" style="margin-top: 15px;"></div> <div class="html-dynamic"> <section> <div class="art-abstract art-abstract-new in-tab hypothesis_container"> <p> <div><section class="html-abstract" id="html-abstract"> <h2 id="html-abstract-title">Abstract</h2><b>:</b> <div class="html-p">Physical properties of the Sun (orientation of rotation axis, oblateness coefficient <math display="inline"><semantics> <msub> <mi>J</mi> <mrow> <mn>2</mn> <mo>⊙</mo> </mrow> </msub> </semantics></math>, and change rate of the gravitational parameter <math display="inline"><semantics> <msub> <mover accent="true"> <mi>μ</mi> <mo>˙</mo> </mover> <mo>⊙</mo> </msub> </semantics></math>) are determined using a dynamical model describing the motion of the Sun, planets, the Moon, asteroids, and Trans-Neptunian objects (TNOs). Among the many kinds of observations used to determine the orbits and physical properties of the bodies, the most important for our study are precise interplanetary ranging data: Earth–Mercury ranges from MESSENGER spacecraft and Earth–Mars ranges from Odyssey and MRO. The findings allow us to improve the model of the Sun in modern planetary ephemerides. First, the dynamically determined direction of the Sun’s pole is ≈2° off the visible axis of rotation of the Sun’s surface, which is corroborated by present knowledge of the Sun’s interior. Second, the change rate of the Sun’s gravitational parameter is found to be smaller (in absolute value) than the nominal value derived from the estimate of mass loss through radiation and solar wind. Possible interpretations are discussed.</div> </section> <div id="html-keywords"> <div class="html-gwd-group"><div id="html-keywords-title">Keywords: </div><a href="/search?q=sun">sun</a>; <a href="/search?q=radio+astronomy">radio astronomy</a>; <a href="/search?q=gravitational+constant">gravitational constant</a>; <a href="/search?q=planetary+ephemerides">planetary ephemerides</a></div> <div> </div> </div> </div> </p> </div> </section> </div> <div class="hypothesis_container"> <ul class="menu html-nav" data-prev-node="#html-quick-links-title"> </ul> <div class="html-body"> <section id='sec1-universe-10-00413' type='intro'><h2 data-nested='1'> 1. Introduction</h2><div class='html-p'>Since the beginning of radio astronomy and laser ranging, the precise determination of orbits has played an important role in geophysical studies of Solar system bodies and tests of General Relativity. One of such tests is the estimation of the possible change rate of gravitational constant <span class='html-italic'>G</span>. Such an estimation cannot be performed by directly measuring <span class='html-italic'>G</span> at different times because of the low accuracy of direct measurement via classical torsion balance: the current best estimate has the relative standard uncertainty of <math display='inline'><semantics> <mrow> <mn>2.2</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> [<a href="#B1-universe-10-00413" class="html-bibr">1</a>]; also, some modern measurements have a relative difference of about <math display='inline'><semantics> <mrow> <mn>5</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math>, which is hypothesized to come from currents in the Earth’s fluid core [<a href="#B2-universe-10-00413" class="html-bibr">2</a>].</div><div class='html-p'>Early attempts to estimate <math display='inline'><semantics> <mover accent="true"> <mi>G</mi> <mo>˙</mo> </mover> </semantics></math> relied on radar ranging of planets and spacecraft radio ranging (see e.g., refs. [<a href="#B3-universe-10-00413" class="html-bibr">3</a>,<a href="#B4-universe-10-00413" class="html-bibr">4</a>,<a href="#B5-universe-10-00413" class="html-bibr">5</a>]). As more observations become available, more complex and detailed models of the motion of the Solar system bodies and astrometric reductions are required to adequately represent the observations and infer scientific information. This work is a continuation of [<a href="#B6-universe-10-00413" class="html-bibr">6</a>], where constraints on <math display='inline'><semantics> <mover accent="true"> <mi>G</mi> <mo>˙</mo> </mover> </semantics></math> were obtained within the framework of the planetary ephemerides EPM. Since then, several improvements have been made: a new solar plasma delay model [<a href="#B7-universe-10-00413" class="html-bibr">7</a>] based on in situ electron density data; the use of a priori values of asteroid masses based on observed diameters [<a href="#B8-universe-10-00413" class="html-bibr">8</a>]; an updated rotation axis of the Sun (presented in this work). The improvements have allowed us to revisit the previously obtained results. Key observational data for this study are range points (hereinafter “ranges”) between DSN stations on Earth and the center of Mercury and Earth–Mars ranges obtained from MGS, Mars Odyssey, and MRO. Each range point is a pseudo-observation of the radio delay between the station and the center of the planet. It is formed from raw observations acquired during one tracking pass, whose systematic errors are strongly correlated.</div><div class='html-p'>The estimation of <math display='inline'><semantics> <mover accent="true"> <mi>G</mi> <mo>˙</mo> </mover> </semantics></math> is performed via the estimation of the change rate of the solar gravitational parameter <math display='inline'><semantics> <msub> <mi>μ</mi> <mo>⊙</mo> </msub> </semantics></math> (see <a href="#sec6dot3-universe-10-00413" class="html-sec">Section 6.3</a>), which is naturally negative because of the solar wind and energy lost by the Sun. The orbits of Mercury, Earth, and Mars are affected by <math display='inline'><semantics> <msub> <mover accent="true"> <mi>μ</mi> <mo>˙</mo> </mover> <mo>⊙</mo> </msub> </semantics></math> in a measurable way. However, since <math display='inline'><semantics> <msub> <mover accent="true"> <mi>μ</mi> <mo>˙</mo> </mover> <mo>⊙</mo> </msub> </semantics></math> manifests itself only via a slight change in the force of gravity, other factors affecting the force of gravity must be accounted for, most notably the value of <math display='inline'><semantics> <msub> <mi>μ</mi> <mo>⊙</mo> </msub> </semantics></math> itself and the solar oblateness factor (<math display='inline'><semantics> <msub> <mi>J</mi> <mrow> <mn>2</mn> <mo>⊙</mo> </mrow> </msub> </semantics></math>); they are both estimated as well. A precise estimation of <math display='inline'><semantics> <msub> <mi>μ</mi> <mo>⊙</mo> </msub> </semantics></math> poses a challenge because this parameter is correlated with the semi-major axes of planetary orbits, which, in turn, are correlated with transponder delays of spacecraft (see <a href="#sec2dot1-universe-10-00413" class="html-sec">Section 2.1</a>). Ranges to Jupiter and Saturn are of help, as they put additional constraints on the parameters of the orbit of the Earth (although introducing their own transponder delays). On a different note, the orbits of the inner planets and Jupiter are affected by hundreds of asteroids. The orbits of asteroids must be modeled and their masses (see <a href="#sec3dot1-universe-10-00413" class="html-sec">Section 3.1</a>) must be either estimated from ranges<a href="#fn001-universe-10-00413" class="html-fn">1</a>, or known with good accuracy from other techniques.</div><div class='html-p'>We also must model the Lense–Thirring effect (see <a href="#sec3dot3-universe-10-00413" class="html-sec">Section 3.3</a>) because it affects the orbit of Mercury enough to interfere with the determination of <math display='inline'><semantics> <msub> <mover accent="true"> <mi>μ</mi> <mo>˙</mo> </mover> <mo>⊙</mo> </msub> </semantics></math>. Finally, the orbit of the Earth is affected by the orbit of the Moon, which, in turn, is sensitive to <math display='inline'><semantics> <msub> <mi>μ</mi> <mo>⊙</mo> </msub> </semantics></math> and <math display='inline'><semantics> <msub> <mi>J</mi> <mrow> <mn>2</mn> <mo>⊙</mo> </mrow> </msub> </semantics></math>; in order to reduce systematic errors in the processing of interplanetary ranges, we model the motion of the Moon together with the Sun, planets, and asteroids, and fit the parameters of orbit and physical libration of the Moon to lunar laser ranging (LLR) observations.</div><div class='html-p'>In the following sections, we describe the data and models used and present results of numerical experiments obtained via the fitting of free parameters to the whole set of astrometrical data.</div></section><section id='sec2-universe-10-00413' type=''><h2 data-nested='1'> 2. Data</h2><section id='sec2dot1-universe-10-00413' type=''><h4 class='html-italic' data-nested='2'> 2.1. Observations</h4><div class='html-p'>MESSENGER data were obtained from the webpage of Solar System Dynamics (SSD) group at NASA JPL<a href="#fn002-universe-10-00413" class="html-fn">2</a>, as were the ranging data from MGS, Mars Odyssey, and MRO [<a href="#B9-universe-10-00413" class="html-bibr">9</a>], ranging and differenced range data from Viking [<a href="#B10-universe-10-00413" class="html-bibr">10</a>] and Pathfinder [<a href="#B11-universe-10-00413" class="html-bibr">11</a>], and older radar data. Mars Express ranges [<a href="#B12-universe-10-00413" class="html-bibr">12</a>] were downloaded from the Geoazur website<a href="#fn003-universe-10-00413" class="html-fn">3</a>. Radar ranging data from Crimea were available on the IAA RAS website<a href="#fn004-universe-10-00413" class="html-fn">4</a>.</div><div class='html-p'>While this work is focused on subtle effects visible in Earth–Mercury and Earth–Mars ranges, all massive bodies in the Solar system affect each other; Earth, Sun, Mercury, and Mars must be treated as part of the dynamical system involving all the planets, significant asteroids, and Trans-Neptunian objects; also, due to the orbit of the Earth being significantly dependent on the orbit of the Moon, it must be present too, on par with the Sun and the planets. Venus Express data were taken from the aforementioned Geoazur website; ranges to Juno and Cassini orbiters were obtained from the aforementioned SSD website. LLR data from observatories in Apache Point [<a href="#B13-universe-10-00413" class="html-bibr">13</a>,<a href="#B14-universe-10-00413" class="html-bibr">14</a>,<a href="#B15-universe-10-00413" class="html-bibr">15</a>] (NM, USA), Matera (Italy), Grasse [<a href="#B16-universe-10-00413" class="html-bibr">16</a>] (France), and Wettzell [<a href="#B17-universe-10-00413" class="html-bibr">17</a>] (Germany) were obtained from CDDIS<a href="#fn005-universe-10-00413" class="html-fn">5</a>, while historical observations from McDonald/MLRS1/MLRS2 [<a href="#B18-universe-10-00413" class="html-bibr">18</a>] and Haleakala [<a href="#B19-universe-10-00413" class="html-bibr">19</a>] observatories, as well as older data for Grasse (Ruby and YAG [<a href="#B20-universe-10-00413" class="html-bibr">20</a>] lasers) were downloaded from POLAC<a href="#fn006-universe-10-00413" class="html-fn">6</a>.</div><div class='html-p'>Optical observations for outer planets and their satellites were used to constrain the orbits of outer planets in the planetary solution. Also, a limited number of “3D normal points” of spacecraft flybys were used. Finally, VLBI observations of planetary orbiters were used to tie the Solar system to the celestial frame. Those observations are largely the same as were used in EPM2021 ephemeris and previous versions of EPM; for details, we refer the reader to [<a href="#B21-universe-10-00413" class="html-bibr">21</a>,<a href="#B22-universe-10-00413" class="html-bibr">22</a>,<a href="#B23-universe-10-00413" class="html-bibr">23</a>,<a href="#B24-universe-10-00413" class="html-bibr">24</a>], and references therein.</div><div class='html-p'>While the observations widely differ by time, instrument, team, observed body, kind of observed value (range or angular coordinates), and accuracy, they were all processed in a single dynamical model of the Solar system (see <a href="#sec3-universe-10-00413" class="html-sec">Section 3</a>). Each observation was assigned a weight according to its accuracy, and a model value that was calculated from the parameters of planetary orbits and other parameters. First-order derivatives were calculated of all observations with respect to parameters they were sensitive to, and then the parameters were fit to minimize the weighted least square difference between the model values and the actual observed values (see <a href="#sec4-universe-10-00413" class="html-sec">Section 4</a>).</div></section><section id='sec2dot2-universe-10-00413' type=''><h4 class='html-italic' data-nested='2'> 2.2. Other Data</h4><div class='html-p'>Orbital parameters of asteroids and TNOs were taken from JPL Horizons online service.<a href="#fn007-universe-10-00413" class="html-fn">7</a></div><div class='html-p'>A priori values based on diameters observed in the infrared [<a href="#B25-universe-10-00413" class="html-bibr">25</a>,<a href="#B26-universe-10-00413" class="html-bibr">26</a>] were used in the dynamical estimation of masses of asteroids and TNOs that could not be determined from precise techniques.</div><div class='html-p'>The EOP series required for astrometric reductions were obtained from the IERS website<a href="#fn008-universe-10-00413" class="html-fn">8</a>. The DSN stations’ coordinates were taken from the aforementioned SSD website.</div></section></section><section id='sec3-universe-10-00413' type=''><h2 data-nested='1'> 3. Dynamical Model</h2><section id='sec3dot1-universe-10-00413' type=''><h4 class='html-italic' data-nested='2'> 3.1. Bodies and Their Gravity</h4><div class='html-p'>The dynamical model used in this work was for the most part described in multiple papers dedicated to EPM ephemerides (see, e.g., ref. [<a href="#B24-universe-10-00413" class="html-bibr">24</a>] and references therein). All bodies present in the model were numerically integrated simultaneously. The numerical integration was performed via an Adams–Bashforth–Moulton predictor-corrector scheme of order 13 with the time step of 1.5 h. A modification of the scheme was carried out so as to handle the time-delayed terms in the dynamical equations of the rotation of the Moon [<a href="#B27-universe-10-00413" class="html-bibr">27</a>].</div><div class='html-p'>The list of bodies included the Sun, the Moon, the planets, selected asteroids and TNOs, as well as the discrete uniform 180-point annulus, approximating the gravitational action of the remaining asteroids, and the discrete uniform 160-point annulus, approximating the gravitational action of the remaining TNOs. The total number of individual asteroids was 249, which is less than 277 in [<a href="#B24-universe-10-00413" class="html-bibr">24</a>], because some more asteroids were removed whose masses were determined to be negative (always within uncertainty).</div><div class='html-p'>The masses of only 17 asteroids are known with good accuracy because they are either binary asteroids (e.g., Kalliope) or had spacecraft orbiting them (e.g., Ceres). The masses of some other asteroids are estimated by deflections of other asteroids’ orbits on approach (e.g., Iris), though estimates may differ across works. The masses of the remaining asteroids were determined dynamically in the planetary solution, with a priori values based on observed diameters and taxonomic classes [<a href="#B8-universe-10-00413" class="html-bibr">8</a>].</div><div class='html-p'>The gravitational parameter of the discrete TNO annulus was fixed to <math display='inline'><semantics> <mrow> <mn>8.847</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>12</mn> </mrow> </msup> <mspace width="0.166667em"/> <mspace width="0.166667em"/> <msup> <mi>au</mi> <mn>3</mn> </msup> <mo>/</mo> </mrow> </semantics></math><math display='inline'><semantics> <msup> <mi>day</mi> <mn>2</mn> </msup> </semantics></math> (which is close to <math display='inline'><semantics> <mrow> <mn>9.848</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>12</mn> </mrow> </msup> </mrow> </semantics></math> from [<a href="#B28-universe-10-00413" class="html-bibr">28</a>]). The gravitational parameter of the discrete asteroid annulus was determined dynamically.</div><div class='html-p'>Most bodies were modeled as point masses interacting either with Newtonian forces or (for the sixteen most important bodies) following the Einstein–Infeld–Hoffmann equations of motion. For the Earth and the Moon, a non-spherical gravitational potential up to degree and order 6 was modeled, together with the elasticity and liquid core of the Moon, and lunar and terrestrial tides [<a href="#B29-universe-10-00413" class="html-bibr">29</a>]. Effects that are of particular importance in this work are described in the further subsections.</div></section><section id='sec3dot2-universe-10-00413' type=''><h4 class='html-italic' data-nested='2'> 3.2. Acceleration from Solar Oblateness</h4><div class='html-p'><div class='html-disp-formula-info' id='FD1-universe-10-00413'> <div class='f'> <math display='block'><semantics> <mrow> <msubsup> <mi mathvariant="bold">a</mi> <mi>i</mi> <mrow> <mo>(</mo> <mi>obl</mi> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <mn>3</mn> <msub> <mi>J</mi> <mrow> <mn>2</mn> <mo>⊙</mo> </mrow> </msub> <msub> <mi>μ</mi> <mo>⊙</mo> </msub> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msup> <mi>R</mi> <mn>2</mn> </msup> <msubsup> <mi>r</mi> <mrow> <mi>i</mi> <mo>⊙</mo> </mrow> <mn>4</mn> </msubsup> </mfrac> </mstyle> <mfenced separators="" open="{" close="}"> <mfenced separators="" open="[" close="]"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> </mstyle> <msup> <mfenced separators="" open="(" close=")"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msub> <mi mathvariant="bold">r</mi> <mrow> <mi>i</mi> <mo>⊙</mo> </mrow> </msub> <msub> <mi>r</mi> <mrow> <mi>i</mi> <mo>⊙</mo> </mrow> </msub> </mfrac> </mstyle> <mo>·</mo> <mi mathvariant="bold">p</mi> </mfenced> <mn>2</mn> </msup> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mfenced> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msub> <mi mathvariant="bold">r</mi> <mrow> <mi>i</mi> <mo>⊙</mo> </mrow> </msub> <msub> <mi>r</mi> <mrow> <mi>i</mi> <mo>⊙</mo> </mrow> </msub> </mfrac> </mstyle> <mo>−</mo> <mfenced separators="" open="(" close=")"> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msub> <mi mathvariant="bold">r</mi> <mrow> <mi>i</mi> <mo>⊙</mo> </mrow> </msub> <msub> <mi>r</mi> <mrow> <mi>i</mi> <mo>⊙</mo> </mrow> </msub> </mfrac> </mstyle> <mo>·</mo> <mi mathvariant="bold">p</mi> </mfenced> <mi mathvariant="bold">p</mi> </mfenced> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(1)</label> </div> </div> where <math display='inline'><semantics> <mi mathvariant="bold">p</mi> </semantics></math> is a unit vector directed at the Sun’s north pole. In previously released versions of EPM, the direction was taken according to the rotational parameters of the Sun accepted by the IAU and derived from the visible motion of the sunspots: <div class='html-disp-formula-info' id='FD2-universe-10-00413'> <div class='f'> <math display='block'><semantics> <mrow> <mi mathvariant="bold">p</mi> <mo>=</mo> <msub> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">z</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>α</mi> <mn>0</mn> </msub> <mo>−</mo> <mn>270</mn> <mo>°</mo> <mo>)</mo> </mrow> <msub> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">x</mi> </msub> <mrow> <mo>(</mo> <mn>90</mn> <mo>°</mo> <mo>−</mo> <msub> <mi>δ</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mi mathvariant="bold">z</mi> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(2)</label> </div> </div> where <math display='inline'><semantics> <mrow> <mi mathvariant="bold">z</mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mn>0</mn> <mspace width="4pt"/> <mn>0</mn> <mspace width="4pt"/> <mn>1</mn> <mo>)</mo> </mrow> <mi>T</mi> </msup> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>α</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>286.13</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>δ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>63.87</mn> </mrow> </semantics></math> (see, e.g., ref. [<a href="#B30-universe-10-00413" class="html-bibr">30</a>]). The same approach was taken in DE [<a href="#B31-universe-10-00413" class="html-bibr">31</a>], INPOP [<a href="#B32-universe-10-00413" class="html-bibr">32</a>], and PETREL [<a href="#B33-universe-10-00413" class="html-bibr">33</a>] ephemerides. While the values of <math display='inline'><semantics> <msub> <mi>α</mi> <mn>0</mn> </msub> </semantics></math> and <math display='inline'><semantics> <msub> <mi>δ</mi> <mn>0</mn> </msub> </semantics></math> are probably accurate, we should not assume that the sunspots and the interior regions of the Sun rotate around one common axis. In this work, the angles <math display='inline'><semantics> <msubsup> <mi>α</mi> <mn>0</mn> <mo>∗</mo> </msubsup> </semantics></math> and <math display='inline'><semantics> <msubsup> <mi>δ</mi> <mn>0</mn> <mo>∗</mo> </msubsup> </semantics></math> (still defining a common rotation axis, but not necessarily coincident with the “visible” one) were determined as solution parameters. As for <math display='inline'><semantics> <msub> <mi>J</mi> <mrow> <mn>2</mn> <mo>⊙</mo> </mrow> </msub> </semantics></math>, while it has been determined by helioseismic inference of internal rotation in [<a href="#B34-universe-10-00413" class="html-bibr">34</a>] to be <math display='inline'><semantics> <mrow> <mrow> <mo>(</mo> <mn>2.21</mn> <mo>±</mo> <mn>0.01</mn> <mo>)</mo> </mrow> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>7</mn> </mrow> </msup> <mspace width="4pt"/> <mspace width="4pt"/> <mrow> <mo>(</mo> <mn>3</mn> <mi>σ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, it was nevertheless made a solution parameter in this work in order to verify this finding via dynamics.</div></section><section id='sec3dot3-universe-10-00413' type=''><h4 class='html-italic' data-nested='2'> 3.3. Lense–Thirring Effect</h4><div class='html-p'>The rotational frame-dragging effect in the gravitational field of a rotating massive body causes the orbits of objects around that body to precess. This effect, predicted by General Relativity and first derived in [<a href="#B35-universe-10-00413" class="html-bibr">35</a>], has been notably hard to observe to this day. One indisputably successful test has been the Gravity Probe B mission [<a href="#B36-universe-10-00413" class="html-bibr">36</a>,<a href="#B37-universe-10-00413" class="html-bibr">37</a>] (an Earth satellite), which confirmed the effect with an accuracy of 19%; however, it did not measure the precession of the spacecraft’s orbit but rather the precession of the spin axis of a freely moving gyroscope; such a test was first proposed in [<a href="#B38-universe-10-00413" class="html-bibr">38</a>,<a href="#B39-universe-10-00413" class="html-bibr">39</a>].</div><div class='html-p'>The Lense–Thirring acceleration can be formulated as follows [<a href="#B40-universe-10-00413" class="html-bibr">40</a>]:<div class='html-disp-formula-info' id='FD3-universe-10-00413'> <div class='f'> <math display='block'><semantics> <mrow> <msubsup> <mi mathvariant="bold">a</mi> <mi>i</mi> <mrow> <mo>(</mo> <mi>LT</mi> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>2</mn> <msup> <mi>c</mi> <mn>2</mn> </msup> </mfrac> </mstyle> <mi>G</mi> <msub> <mi>S</mi> <mo>⊙</mo> </msub> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <msubsup> <mi>r</mi> <mrow> <mi>i</mi> <mo>⊙</mo> </mrow> <mn>3</mn> </msubsup> </mfrac> </mstyle> <msub> <mi>M</mi> <mo>⊙</mo> </msub> <mfenced separators="" open="(" close=")"> <msub> <mover accent="true"> <mi mathvariant="bold">r</mi> <mo>˙</mo> </mover> <mrow> <mi mathvariant="bold">i</mi> <mo>⊙</mo> </mrow> </msub> <mo>×</mo> <mi mathvariant="bold">p</mi> <mo>+</mo> <mn>3</mn> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi mathvariant="bold">p</mi> <mo>·</mo> <msub> <mi mathvariant="bold">r</mi> <mrow> <mi mathvariant="bold">i</mi> <mo>⊙</mo> </mrow> </msub> </mrow> <msup> <mi>r</mi> <mn>2</mn> </msup> </mfrac> </mstyle> <msub> <mi mathvariant="bold">r</mi> <mrow> <mi mathvariant="bold">i</mi> <mo>⊙</mo> </mrow> </msub> <mo>×</mo> <msub> <mover accent="true"> <mi mathvariant="bold">r</mi> <mo>˙</mo> </mover> <mrow> <mi mathvariant="bold">i</mi> <mo>⊙</mo> </mrow> </msub> </mfenced> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(3)</label> </div> </div> where <math display='inline'><semantics> <msub> <mi>S</mi> <mo>⊙</mo> </msub> </semantics></math> is the angular momentum of the Sun. Since the launch of the second LAGEOS satellite, test were conducted to confirm the Lense–Thirring acceleration in the gravitational field of the Earth, see, e.g., ref. [<a href="#B41-universe-10-00413" class="html-bibr">41</a>], where an accuracy of 10% was reported, or a later work [<a href="#B42-universe-10-00413" class="html-bibr">42</a>] with a reported improved accuracy of 2%; however, there is an ongoing debate about the correctness of that methodology, see, e.g., refs. [<a href="#B43-universe-10-00413" class="html-bibr">43</a>,<a href="#B44-universe-10-00413" class="html-bibr">44</a>,<a href="#B45-universe-10-00413" class="html-bibr">45</a>,<a href="#B46-universe-10-00413" class="html-bibr">46</a>].</div><div class='html-p'>It was hoped that the Juno spacecraft, thanks to its polar orbit, would allow to test the Lense–Thirring effect via an estimation of the angular momentum of Jupiter [<a href="#B47-universe-10-00413" class="html-bibr">47</a>,<a href="#B48-universe-10-00413" class="html-bibr">48</a>], but it did not work out with the actual measurements [<a href="#B49-universe-10-00413" class="html-bibr">49</a>].</div><div class='html-p'>Possibilities for measurements of the Lense–Thirring effect from planetary dynamics, including the MESSENGER mission, were explored in [<a href="#B50-universe-10-00413" class="html-bibr">50</a>,<a href="#B51-universe-10-00413" class="html-bibr">51</a>]. It was later shown in [<a href="#B52-universe-10-00413" class="html-bibr">52</a>,<a href="#B53-universe-10-00413" class="html-bibr">53</a>,<a href="#B54-universe-10-00413" class="html-bibr">54</a>] that it is not possible at present to overcome the strong anticorrelation between <math display='inline'><semantics> <msub> <mi>S</mi> <mo>⊙</mo> </msub> </semantics></math> and <math display='inline'><semantics> <msub> <mi>J</mi> <mrow> <mn>2</mn> <mo>⊙</mo> </mrow> </msub> </semantics></math> in Earth–Mercury ranges. The numerically estimated anticorrelation was 92%. The present accuracy of the non-dynamical determination of <math display='inline'><semantics> <msub> <mi>J</mi> <mrow> <mn>2</mn> <mo>⊙</mo> </mrow> </msub> </semantics></math> (see <a href="#sec3dot2-universe-10-00413" class="html-sec">Section 3.2</a>) allows for an estimation of <math display='inline'><semantics> <msub> <mi>S</mi> <mo>⊙</mo> </msub> </semantics></math> from MESSENGER observations with an accuracy of 25% [<a href="#B52-universe-10-00413" class="html-bibr">52</a>]. The data from the extended phase of the BepiColombo mission is expected to improve this result [<a href="#B55-universe-10-00413" class="html-bibr">55</a>]. In our model, we fixed <math display='inline'><semantics> <msub> <mi>S</mi> <mo>⊙</mo> </msub> </semantics></math> to the value inferred from helioseismology in [<a href="#B56-universe-10-00413" class="html-bibr">56</a>]: <math display='inline'><semantics> <mrow> <mn>190</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>39</mn> </msup> <mspace width="4pt"/> <mi>kg</mi> <mo>/</mo> <msup> <mi mathvariant="normal">m</mi> <mn>2</mn> </msup> <mi mathvariant="normal">s</mi> </mrow> </semantics></math>.</div></section><section id='sec3dot4-universe-10-00413' type=''><h4 class='html-italic' data-nested='2'> 3.4. Sun Mass Loss Rate</h4><div class='html-p'>In this work, the gravitational parameter of the Sun was assigned a linear trend:<div class='html-disp-formula-info' id='FD4-universe-10-00413'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>μ</mi> <mo>⊙</mo> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mi>μ</mi> <mrow> <mo>⊙</mo> </mrow> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </msubsup> <mo>+</mo> <msub> <mover accent="true"> <mi>μ</mi> <mo>˙</mo> </mover> <mo>⊙</mo> </msub> <mo>·</mo> <mrow> <mo>(</mo> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(4)</label> </div> </div> where the epoch <math display='inline'><semantics> <msub> <mi>t</mi> <mn>0</mn> </msub> </semantics></math> was chosen to be JD 2454000.5 (22 September 2006) so that it was within the timespan of the precision ranging observations used in the solution: from the beginning of MGS data in 1999 till the end of available Odyssey and MRO data in 2017; MESSENGER ranges began in 2011 and ended in 2014.</div><div class='html-p'>The nominal value of <math display='inline'><semantics> <msub> <mover accent="true"> <mi>M</mi> <mo>˙</mo> </mover> <mo>⊙</mo> </msub> </semantics></math> is made up of radiation and solar wind terms (the normalized form is given):<div class='html-disp-formula-info' id='FD5-universe-10-00413'> <div class='f'> <math display='block'><semantics> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msub> <mover accent="true"> <mi>M</mi> <mo>˙</mo> </mover> <mo>⊙</mo> </msub> <msub> <mi>M</mi> <mo>⊙</mo> </msub> </mfrac> </mstyle> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msubsup> <mover accent="true"> <mi>M</mi> <mo>˙</mo> </mover> <mrow> <mo>⊙</mo> </mrow> <mi>rad</mi> </msubsup> <msub> <mi>M</mi> <mo>⊙</mo> </msub> </mfrac> </mstyle> <mo>+</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msubsup> <mover accent="true"> <mi>M</mi> <mo>˙</mo> </mover> <mrow> <mo>⊙</mo> </mrow> <mi>wind</mi> </msubsup> <msub> <mi>M</mi> <mo>⊙</mo> </msub> </mfrac> </mstyle> </mrow> </semantics></math> </div> <div class='l'> <label >(5)</label> </div> </div></div><div class='html-p'>The mass loss rate due to radiation was derived from the IAU nominal value of solar radiation: <math display='inline'><semantics> <mrow> <msubsup> <mi mathvariant="script">L</mi> <mrow> <mo>⊙</mo> </mrow> <mi mathvariant="normal">N</mi> </msubsup> <mo>=</mo> <mn>3.828</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>26</mn> </msup> <mspace width="4pt"/> <mi mathvariant="normal">W</mi> </mrow> </semantics></math> [<a href="#B57-universe-10-00413" class="html-bibr">57</a>], which gives <div class='html-disp-formula-info' id='FD6-universe-10-00413'> <div class='f'> <math display='block'><semantics> <mrow> <msubsup> <mover accent="true"> <mi>M</mi> <mo>˙</mo> </mover> <mrow> <mo>⊙</mo> </mrow> <mi>rad</mi> </msubsup> <mo>/</mo> <msub> <mi>M</mi> <mo>⊙</mo> </msub> <mo>=</mo> <mo>−</mo> <msubsup> <mi mathvariant="script">L</mi> <mrow> <mo>⊙</mo> </mrow> <mi mathvariant="normal">N</mi> </msubsup> <mo>/</mo> <mrow> <mo>(</mo> <msub> <mi>M</mi> <mo>⊙</mo> </msub> <msup> <mi>c</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mn>2.142</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>21</mn> </mrow> </msup> <mspace width="4pt"/> <msup> <mi mathvariant="normal">s</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>6.760</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> <mspace width="4pt"/> <msup> <mi>yr</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(6)</label> </div> </div></div><div class='html-p'>The actual value is known to be fluctuating; depending on the period, the estimates of amplitude of fluctuations are 0.2% for a 27-day rotation period and 0.1% for an 11-year solar cycle [<a href="#B58-universe-10-00413" class="html-bibr">58</a>]. On secular timescales, the estimate is about 0.11% [<a href="#B59-universe-10-00413" class="html-bibr">59</a>]. If we take the IAU-compliant value as the average, we can be sure that the error of this average is lower than the fluctuations, and obtain a safe margin of error <math display='inline'><semantics> <mrow> <mo>−</mo> <mn>6.77</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> <mo><</mo> <msubsup> <mover accent="true"> <mi>M</mi> <mo>˙</mo> </mover> <mrow> <mo>⊙</mo> </mrow> <mi>rad</mi> </msubsup> <mo>/</mo> <msub> <mi>M</mi> <mo>⊙</mo> </msub> <mo><</mo> <mo>−</mo> <mn>6.75</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> </mrow> </semantics></math>.</div><div class='html-p'>Mass loss due to the solar wind is harder to observe or estimate due to its significant variation in both latitude and time, and the non-perfection of solar wind models. Years of interest (1999–2017) were within solar cycles 23 (1996–2008) and 24 (2008–2019). Estimates in [<a href="#B60-universe-10-00413" class="html-bibr">60</a>] based on <span class='html-italic'>Wind</span>’s measurements concluded that the average value was “<math display='inline'><semantics> <mrow> <mn>2.1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> <mo> </mo> <msub> <mi>M</mi> <mo>⊙</mo> </msub> <mspace width="4pt"/> <msup> <mi>yr</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math> for cycle 23 and decreases by around 10% for cycle 24”. Estimates in [<a href="#B61-universe-10-00413" class="html-bibr">61</a>], based on OMNI data (combined from measurements from ACE and <span class='html-italic'>Wind</span> spacecraft) and Ulysses measurements, were in a similar range for the said timespan and were confirmed by a model. Estimate in [<a href="#B6-universe-10-00413" class="html-bibr">6</a>] for <math display='inline'><semantics> <mrow> <msubsup> <mover accent="true"> <mi>M</mi> <mo>˙</mo> </mover> <mrow> <mo>⊙</mo> </mrow> <mi>wind</mi> </msubsup> <mo>/</mo> <msub> <mi>M</mi> <mo>⊙</mo> </msub> </mrow> </semantics></math> was in the range from <math display='inline'><semantics> <mrow> <mo>−</mo> <mn>6.6</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> </mrow> </semantics></math> to <math display='inline'><semantics> <mrow> <mo>−</mo> <mn>3.0</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> </mrow> </semantics></math>. This estimate was based solely on the Ulysses data (SWOOPS) in its entire operating interval (1990–2008). Ulysses estimates shown in [<a href="#B61-universe-10-00413" class="html-bibr">61</a>] seemed to confirm the <math display='inline'><semantics> <mrow> <mo>−</mo> <mn>3.0</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> </mrow> </semantics></math> average value but only thanks to the high (above <math display='inline'><semantics> <mrow> <mn>3</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> </mrow> </semantics></math>) mass loss rate in the early 1990s, which was outside our timespan of interest; after 1999, an average mass loss rate of about <math display='inline'><semantics> <mrow> <mn>2</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> </mrow> </semantics></math> seemed very plausible. As a counterpoint, a model in [<a href="#B62-universe-10-00413" class="html-bibr">62</a>] gave the value of <math display='inline'><semantics> <mrow> <mn>4.1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> </mrow> </semantics></math> for a particular solar maximum (CR 2159, January 2015), which was not confirmed in [<a href="#B61-universe-10-00413" class="html-bibr">61</a>].</div><div class='html-p'>Solar mass is also lost to CMEs; however, CMEs account for no more than 10% of the mass loss as compared to solar wind [<a href="#B60-universe-10-00413" class="html-bibr">60</a>,<a href="#B63-universe-10-00413" class="html-bibr">63</a>]. Considering all of the above, and accounting for possible error due to variations in latitude (direct observations of solar wind outside ecliptic have not been made since 2008), we used the following safe bounds of mass loss rate due to solar wind and CMEs in 1999–2017: <math display='inline'><semantics> <mrow> <mo>−</mo> <mn>3.1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> <mo><</mo> <msubsup> <mover accent="true"> <mi>M</mi> <mo>˙</mo> </mover> <mrow> <mo>⊙</mo> </mrow> <mi>wind</mi> </msubsup> <mo>/</mo> <msub> <mi>M</mi> <mo>⊙</mo> </msub> <mo><</mo> <mo>−</mo> <mn>1.9</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> </mrow> </semantics></math>, with the nominal value of <math display='inline'><semantics> <mrow> <mo>−</mo> <mn>2.1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> </mrow> </semantics></math>.</div><div class='html-p'>Combining all of the above, we obtained the nominal value <div class='html-disp-formula-info' id='FD7-universe-10-00413'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mover accent="true"> <mi>M</mi> <mo>˙</mo> </mover> <mo>⊙</mo> </msub> <mo>/</mo> <msub> <mi>M</mi> <mo>⊙</mo> </msub> <mo>≈</mo> <mo>−</mo> <mn>8.9</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> <mspace width="4pt"/> <msup> <mi>yr</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math> </div> <div class='l'> <label >(7)</label> </div> </div> and the safe bounds <div class='html-disp-formula-info' id='FD8-universe-10-00413'> <div class='f'> <math display='block'><semantics> <mrow> <mo>−</mo> <mn>9.9</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> <mspace width="4pt"/> <msup> <mi>yr</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo><</mo> <msub> <mover accent="true"> <mi>M</mi> <mo>˙</mo> </mover> <mo>⊙</mo> </msub> <mo>/</mo> <msub> <mi>M</mi> <mo>⊙</mo> </msub> <mo><</mo> <mo>−</mo> <mn>8.6</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> <mspace width="4pt"/> <msup> <mi>yr</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math> </div> <div class='l'> <label >(8)</label> </div> </div></div></section></section><section id='sec4-universe-10-00413' type=''><h2 data-nested='1'> 4. Processing of Observations</h2><div class='html-p'>The full set of planetary parameters was fitted via the Gauss–Newton method to the full set of planetary observations (<a href="#sec2dot1-universe-10-00413" class="html-sec">Section 2.1</a>). The set of planetary parameters of EPM2021 was taken, including the following:</div><div class='html-p'><ul class='html-bullet'><li><div class='html-p'>Initial parameters of orbits of planets (at the epoch given in <a href="#sec3dot4-universe-10-00413" class="html-sec">Section 3.4</a>) in the reference plane of Earth’s mean equator at epoch J2000.</div></li><li><div class='html-p'>Masses of asteroids (see again <a href="#sec2dot1-universe-10-00413" class="html-sec">Section 2.1</a>);</div></li><li><div class='html-p'><math display='inline'><semantics> <msub> <mi>μ</mi> <mo>⊙</mo> </msub> </semantics></math> was determined via the scaling factor (correction to the value of astronomical unit) in EPM2021 but was determined directly in this work;</div></li><li><div class='html-p'><math display='inline'><semantics> <msub> <mi>μ</mi> <mi>annulus</mi> </msub> </semantics></math> of the 180-point discrete asteroid annulus;</div></li><li><div class='html-p'>The Earth/Moon mass ratio;</div></li><li><div class='html-p'><math display='inline'><semantics> <msub> <mi>J</mi> <mrow> <mn>2</mn> <mo>⊙</mo> </mrow> </msub> </semantics></math>;</div></li><li><div class='html-p'>Transponder delays for planetary orbiters and landers (while transponders are calibrated on Earth, this calibration is not precise enough and/or breaks in the conditions of outer space, so the additional delay must be determined, either positive or negative);</div></li><li><div class='html-p'>One factor of solar plasma delay [<a href="#B7-universe-10-00413" class="html-bibr">7</a>];</div></li><li><div class='html-p'>Two sets of biases for DSN stations [<a href="#B64-universe-10-00413" class="html-bibr">64</a>]: one for MGS and Odyssey spacecraft, the other for the MRO spacecraft.</div></li></ul></div><div class='html-p'>The listed parameters affect precision ranges to orbiters of inner planets. Other parameters were also determined in the planetary solution that were not directly important for this work: corrections to various systematic effects of earlier observations of planets and Martian landers, angles of orientation of the ephemeris frame to ICRF, etc.</div><div class='html-p'>An important detail is that MESSENGER ranges, unlike those of Martian spacecraft, are provided only in the form with the solar plasma delay subtracted from them according to a model developed by NASA JPL. Without knowing that model in detail, we cannot be sure that no systematic effects remain in the data that could in principle affect the results of this work. Ironically, as said in [<a href="#B65-universe-10-00413" class="html-bibr">65</a>], “mission’s published science objectives [<a href="#B66-universe-10-00413" class="html-bibr">66</a>] do not include determination of PPN parameters or <math display='inline'><semantics> <msub> <mi>J</mi> <mn>2</mn> </msub> </semantics></math> of the Sun. MESSENGER is equipped with only one radio link (X-band)”. That decision rendered impossible a reliable technique to determine solar plasma delay (Differenced Range Versus Integrated Doppler [<a href="#B67-universe-10-00413" class="html-bibr">67</a>]); and the JPL model of solar plasma delay has proven not to be the most accurate in some cases [<a href="#B7-universe-10-00413" class="html-bibr">7</a>].</div><div class='html-p'>Additional parameters were included in some of the solutions described in further sections: the <math display='inline'><semantics> <msubsup> <mi>α</mi> <mn>0</mn> <mo>∗</mo> </msubsup> </semantics></math> and <math display='inline'><semantics> <msubsup> <mi>δ</mi> <mn>0</mn> <mo>∗</mo> </msubsup> </semantics></math> angles of the Sun’s rotation axis and <math display='inline'><semantics> <msub> <mover accent="true"> <mi>μ</mi> <mo>˙</mo> </mover> <mo>⊙</mo> </msub> </semantics></math>.</div><div class='html-p'>Parameters of the Earth–Moon system and the geocentric orbit of the Moon were determined in a separate solution (lunar) according to the JPL DE model; for details, we refer to [<a href="#B23-universe-10-00413" class="html-bibr">23</a>,<a href="#B29-universe-10-00413" class="html-bibr">29</a>] and references therein. Planetary and lunar solutions in this work were obtained iteratively in an interleaving manner, so that the lunar solution accounted for the perturbations brought by the changes in planetary solution (most notably change in <math display='inline'><semantics> <msub> <mi>μ</mi> <mo>⊙</mo> </msub> </semantics></math>), and the planetary solution, in turn, benefited from a more correct orbit of the Moon (and hence a more correct orbit of the Earth).</div></section><section id='sec5-universe-10-00413' type=''><h2 data-nested='1'> 5. Experiment and Results</h2><div class='html-p'>Three solutions were obtained:</div><div class='html-p'><dl class='html-roman-upper'><dt id=''>I</dt><dd><div class='html-p'>With fixed <math display='inline'><semantics> <msub> <mi>α</mi> <mn>0</mn> </msub> </semantics></math> and <math display='inline'><semantics> <msub> <mi>δ</mi> <mn>0</mn> </msub> </semantics></math> and nominal <math display='inline'><semantics> <msub> <mover accent="true"> <mi>μ</mi> <mo>˙</mo> </mover> <mo>⊙</mo> </msub> </semantics></math>;</div></dd><dt id=''>II</dt><dd><div class='html-p'>With determined <math display='inline'><semantics> <msubsup> <mi>α</mi> <mn>0</mn> <mo>∗</mo> </msubsup> </semantics></math> and <math display='inline'><semantics> <msubsup> <mi>δ</mi> <mn>0</mn> <mo>∗</mo> </msubsup> </semantics></math> and nominal <math display='inline'><semantics> <msub> <mover accent="true"> <mi>μ</mi> <mo>˙</mo> </mover> <mo>⊙</mo> </msub> </semantics></math>;</div></dd><dt id=''>III</dt><dd><div class='html-p'>With determined <math display='inline'><semantics> <msubsup> <mi>α</mi> <mn>0</mn> <mo>∗</mo> </msubsup> </semantics></math>, <math display='inline'><semantics> <msubsup> <mi>δ</mi> <mn>0</mn> <mo>∗</mo> </msubsup> </semantics></math>, and <math display='inline'><semantics> <msub> <mover accent="true"> <mi>μ</mi> <mo>˙</mo> </mover> <mo>⊙</mo> </msub> </semantics></math>.</div></dd></dl></div><div class='html-p'>Selected parameters determined in the three solutions are listed in <a href="#universe-10-00413-t001" class="html-table">Table 1</a>, while selected one-way wrms are listed in <a href="#universe-10-00413-t002" class="html-table">Table 2</a>. Model differences in MESSENGER ranges between the three solutions are shown on <a href="#universe-10-00413-f001" class="html-fig">Figure 1</a>, <a href="#universe-10-00413-f002" class="html-fig">Figure 2</a> and <a href="#universe-10-00413-f003" class="html-fig">Figure 3</a>. MESSENGER range residuals for Solution III are shown in <a href="#universe-10-00413-f004" class="html-fig">Figure 4</a>.</div></section><section id='sec6-universe-10-00413' type='discussion'><h2 data-nested='1'> 6. Discussion</h2><section id='sec6dot1-universe-10-00413' type=''><h4 class='html-italic' data-nested='2'> 6.1. Comparison Between Solutions</h4><div class='html-p'>The determined <math display='inline'><semantics> <msub> <mi>J</mi> <mrow> <mn>2</mn> <mo>⊙</mo> </mrow> </msub> </semantics></math> values in all three solutions were in good agreement with the helioseismology value of <math display='inline'><semantics> <mrow> <mrow> <mo>(</mo> <mn>2.21</mn> <mo>±</mo> <mn>0.01</mn> <mo>)</mo> </mrow> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>7</mn> </mrow> </msup> </mrow> </semantics></math> [<a href="#B34-universe-10-00413" class="html-bibr">34</a>]. The introduction of <math display='inline'><semantics> <msubsup> <mi>α</mi> <mn>0</mn> <mo>∗</mo> </msubsup> </semantics></math> and <math display='inline'><semantics> <msubsup> <mi>δ</mi> <mn>0</mn> <mo>∗</mo> </msubsup> </semantics></math> angles as fitted parameters improved the MESSENGER wrms by 17 mm (or about 15 cm in the root-mean-square sense); the further introduction of <math display='inline'><semantics> <msub> <mover accent="true"> <mi>μ</mi> <mo>˙</mo> </mover> <mo>⊙</mo> </msub> </semantics></math> as fitted parameter improved the MESSENGER wrms by 5 mm (or 8 cm in the root-mean-square sense).</div></section><section id='sec6dot2-universe-10-00413' type=''><h4 class='html-italic' data-nested='2'> 6.2. Solar Rotation Axis</h4><div class='html-p'>The determined <math display='inline'><semantics> <msubsup> <mi>α</mi> <mn>0</mn> <mo>∗</mo> </msubsup> </semantics></math> was different from the nominal <math display='inline'><semantics> <msub> <mi>α</mi> <mn>0</mn> </msub> </semantics></math> by more than three estimated standard deviations, and <math display='inline'><semantics> <msubsup> <mi>δ</mi> <mn>0</mn> <mo>∗</mo> </msubsup> </semantics></math> was different from the nominal <math display='inline'><semantics> <msub> <mi>δ</mi> <mn>0</mn> </msub> </semantics></math> by more than three estimated standard deviations plus 1°. The dynamically determined rotation axis of the Sun was notably different from the observed one. While the nominal orientation <math display='inline'><semantics> <mrow> <msub> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">z</mi> </msub> <mrow> <mo>(</mo> <mn>16.13</mn> <mo>)</mo> </mrow> <msub> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">x</mi> </msub> <mrow> <mo>(</mo> <mn>26.13</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> in the equatorial system gave the solar equator inclined <math display='inline'><semantics> <mrow> <mn>7.25</mn> </mrow> </semantics></math> to the ecliptic, the determined orientation gave the inclination of <math display='inline'><semantics> <mrow> <mn>8.98</mn> </mrow> </semantics></math> (Solution II) or <math display='inline'><semantics> <mrow> <mn>8.91</mn> </mrow> </semantics></math> (Solution III). The azimuths of the two rotation axes (with respect to the ecliptic) differed by <math display='inline'><semantics> <mrow> <mn>8.94</mn> </mrow> </semantics></math> (Solution II vs. Solution I) or by <math display='inline'><semantics> <mrow> <mn>7.69</mn> </mrow> </semantics></math> (Solution III vs. Solution I). These numbers are within the <math display='inline'><semantics> <mrow> <mn>1.5</mn> <mi>σ</mi> </mrow> </semantics></math> confidence level of the “mean pulsation axis” estimates in [<a href="#B68-universe-10-00413" class="html-bibr">68</a>] obtained from the analysis of LOI measurements on SOHO. Also, our finding is consistent with the existence of a hypothetical inner region of the Sun (“core”) that rotates differently than the envelope (in [<a href="#B69-universe-10-00413" class="html-bibr">69</a>], this was assumed from the analysis of solar flares). Even so, we emphasize that the “rotation axis” determined within our model is not a real rotation axis of any part of the Sun, but merely an averaged approximation of how two different effects (solar oblateness and Lense–Thirring effect) jointly act in the case of a non-axially symmetric body. It is an open question whether it is possible to study the Sun’s interior by planetary dynamics via a more physically correct analysis, separating solar oblateness effect from the Lense–Thirring effect.</div></section><section id='sec6dot3-universe-10-00413' type=''><h4 class='html-italic' data-nested='2'> 6.3. Apparent Change of the Gravitational Constant</h4><div class='html-p'>Introducing the hypothetical change rate of the gravitational constant <span class='html-italic'>G</span>:<div class='html-disp-formula-info' id='FD9-universe-10-00413'> <div class='f'> <math display='block'><semantics> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msub> <mover accent="true"> <mi>μ</mi> <mo>˙</mo> </mover> <mo>⊙</mo> </msub> <msub> <mi>μ</mi> <mo>⊙</mo> </msub> </mfrac> </mstyle> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msub> <mover accent="true"> <mi>M</mi> <mo>˙</mo> </mover> <mo>⊙</mo> </msub> <msub> <mi>M</mi> <mo>⊙</mo> </msub> </mfrac> </mstyle> <mo>+</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mover accent="true"> <mi>G</mi> <mo>˙</mo> </mover> <mi>G</mi> </mfrac> </mstyle> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(9)</label> </div> </div> and taking the safe (<math display='inline'><semantics> <mrow> <mn>3</mn> <mi>σ</mi> </mrow> </semantics></math>) bounds for <math display='inline'><semantics> <msub> <mover accent="true"> <mi>μ</mi> <mo>˙</mo> </mover> <mo>⊙</mo> </msub> </semantics></math> obtained in Solution III:<div class='html-disp-formula-info' id='FD10-universe-10-00413'> <div class='f'> <math display='block'><semantics> <mrow> <mo>−</mo> <mn>6.5</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> <mspace width="4pt"/> <msup> <mi>yr</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo><</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msub> <mover accent="true"> <mi>μ</mi> <mo>˙</mo> </mover> <mo>⊙</mo> </msub> <msub> <mi>μ</mi> <mo>⊙</mo> </msub> </mfrac> </mstyle> <mo><</mo> <mo>−</mo> <mn>0.3</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> <mspace width="4pt"/> <msup> <mi>yr</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(10)</label> </div> </div> and taking into account the safe bounds for <math display='inline'><semantics> <msub> <mover accent="true"> <mi>M</mi> <mo>˙</mo> </mover> <mo>⊙</mo> </msub> </semantics></math> (<a href="#FD8-universe-10-00413" class="html-disp-formula">8</a>), we obtained the following constraints for <math display='inline'><semantics> <mover accent="true"> <mi>G</mi> <mo>˙</mo> </mover> </semantics></math>:<div class='html-disp-formula-info' id='FD11-universe-10-00413'> <div class='f'> <math display='block'><semantics> <mrow> <mn>2.0</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> <mspace width="4pt"/> <msup> <mi>yr</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo><</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mover accent="true"> <mi>G</mi> <mo>˙</mo> </mover> <mi>G</mi> </mfrac> </mstyle> <mo><</mo> <mn>9.6</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> <mspace width="4pt"/> <msup> <mi>yr</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math> </div> <div class='l'> <label >(11)</label> </div> </div></div><div class='html-p'>These bounds are within most of the earlier obtained bounds for <math display='inline'><semantics> <mover accent="true"> <mi>G</mi> <mo>˙</mo> </mover> </semantics></math> by non-dynamical methods:</div><div class='html-p'><ul class='html-bullet'><li><div class='html-p'><math display='inline'><semantics> <mrow> <mrow> <mo>(</mo> <mn>1.2</mn> <mo>±</mo> <mn>2.6</mn> <mo>)</mo> </mrow> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>12</mn> </mrow> </msup> </mrow> </semantics></math> in [<a href="#B70-universe-10-00413" class="html-bibr">70</a>] by the seismology of an ancient star;</div></li><li><div class='html-p'><math display='inline'><semantics> <mrow> <mrow> <mo>(</mo> <mn>3.6</mn> <mo>±</mo> <mn>4.5</mn> <mo>)</mo> </mrow> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>12</mn> </mrow> </msup> </mrow> </semantics></math> in [<a href="#B71-universe-10-00413" class="html-bibr">71</a>] by CMB observations;</div></li><li><div class='html-p'><math display='inline'><semantics> <mrow> <mrow> <mo>(</mo> <mn>1</mn> <mo>±</mo> <mn>9</mn> <mo>)</mo> </mrow> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>13</mn> </mrow> </msup> </mrow> </semantics></math> in [<a href="#B72-universe-10-00413" class="html-bibr">72</a>] by the timing of a single pulsar;</div></li><li><div class='html-p'><math display='inline'><semantics> <mrow> <mrow> <mo>(</mo> <mn>0</mn> <mo>±</mo> <mn>2</mn> <mo>)</mo> </mrow> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>13</mn> </mrow> </msup> </mrow> </semantics></math> in [<a href="#B73-universe-10-00413" class="html-bibr">73</a>] by helioseismology;</div></li><li><div class='html-p'><math display='inline'><semantics> <mrow> <mrow> <mo>(</mo> <mn>5.3</mn> <mo>±</mo> <mn>6</mn> <mo>)</mo> </mrow> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> </mrow> </semantics></math> in [<a href="#B74-universe-10-00413" class="html-bibr">74</a>] by observations of adsorption lines in quasar spectra.</div></li></ul></div><div class='html-p'>However, in later work [<a href="#B75-universe-10-00413" class="html-bibr">75</a>] the bounds obtained by quasar spectra were tightened to <math display='inline'><semantics> <mrow> <mrow> <mo>(</mo> <mn>0.92</mn> <mo>±</mo> <mn>2.83</mn> <mo>)</mo> </mrow> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>15</mn> </mrow> </msup> </mrow> </semantics></math>, which is not compatible with values found in this work.</div><div class='html-p'>Other estimates of <math display='inline'><semantics> <msub> <mover accent="true"> <mi>μ</mi> <mo>˙</mo> </mover> <mo>⊙</mo> </msub> </semantics></math> are known, derived from MESSENGER ranges. The first work of the kind [<a href="#B53-universe-10-00413" class="html-bibr">53</a>] involved processing of raw radio science data and the integration of the orbit of MESSENGER around Mercury, rather than using the range points provided by JPL, as in other works. It is not clear which model of solar plasma delay was used. Only Mercury’s orbit was fitted; the orbits of other planets were taken from a JPL DE ephemeris. The obtained estimate was <math display='inline'><semantics> <mrow> <msub> <mover accent="true"> <mi>μ</mi> <mo>˙</mo> </mover> <mo>⊙</mo> </msub> <mo>/</mo> <msub> <mi>μ</mi> <mo>⊙</mo> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mo>−</mo> <mn>6.13</mn> <mo>±</mo> <mn>4.41</mn> <mo>)</mo> </mrow> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> <mspace width="4pt"/> <mrow> <mo>(</mo> <mn>3</mn> <mi>σ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, which largely intersects with our estimate (<a href="#FD10-universe-10-00413" class="html-disp-formula">10</a>) but, unlike the latter, allows zero <math display='inline'><semantics> <mover accent="true"> <mi>G</mi> <mo>˙</mo> </mover> </semantics></math>.<a href="#fn009-universe-10-00413" class="html-fn">9</a></div><div class='html-p'>Another estimate based on MESSENGER is given in [<a href="#B32-universe-10-00413" class="html-bibr">32</a>]: <math display='inline'><semantics> <mrow> <msub> <mover accent="true"> <mi>μ</mi> <mo>˙</mo> </mover> <mo>⊙</mo> </msub> <mo>/</mo> <msub> <mi>μ</mi> <mo>⊙</mo> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mo>−</mo> <mn>3.7</mn> <mo>±</mo> <mn>3.2</mn> <mo>)</mo> </mrow> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> <mspace width="4pt"/> <mrow> <mo>(</mo> <mn>3</mn> <mi>σ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, which is close to our estimate (<a href="#FD10-universe-10-00413" class="html-disp-formula">10</a>). However, no conclusion about hypothetically positive <math display='inline'><semantics> <mover accent="true"> <mi>G</mi> <mo>˙</mo> </mover> </semantics></math> was made because the account of the Sun’s mass loss rate <math display='inline'><semantics> <mrow> <msub> <mover accent="true"> <mi>M</mi> <mo>˙</mo> </mover> <mo>⊙</mo> </msub> <mo>/</mo> <msub> <mi>M</mi> <mo>⊙</mo> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mo>−</mo> <mn>9.2</mn> <mo>±</mo> <mn>6.1</mn> <mo>)</mo> </mrow> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> <mspace width="4pt"/> <msup> <mi>yr</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math> (<math display='inline'><semantics> <mrow> <mn>3</mn> <mi>σ</mi> </mrow> </semantics></math>) was taken as a compromise between [<a href="#B76-universe-10-00413" class="html-bibr">76</a>,<a href="#B77-universe-10-00413" class="html-bibr">77</a>]. Neither paper is relevant to the timespan of MESSENGER observations; moreover, the margin of error was too wide because of the said compromise.</div><div class='html-p'>Another approach undertaken in [<a href="#B32-universe-10-00413" class="html-bibr">32</a>] was the WRSS filtering, which is more conservative than the least squares fitting used in this and other works. The obtained estimate was <math display='inline'><semantics> <mrow> <msub> <mover accent="true"> <mi>μ</mi> <mo>˙</mo> </mover> <mo>⊙</mo> </msub> <mo>/</mo> <msub> <mi>μ</mi> <mo>⊙</mo> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mo>−</mo> <mn>1.03</mn> <mo>±</mo> <mn>2.28</mn> <mo>)</mo> </mrow> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>13</mn> </mrow> </msup> </mrow> </semantics></math>, which rendered the estimates of <math display='inline'><semantics> <mrow> <mover accent="true"> <mi>G</mi> <mo>˙</mo> </mover> <mo>/</mo> <mi>G</mi> </mrow> </semantics></math> at the <math display='inline'><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> </semantics></math> level impossible.</div><div class='html-p'>All in all, we should not infer right away that the positive <math display='inline'><semantics> <mover accent="true"> <mi>G</mi> <mo>˙</mo> </mover> </semantics></math>, found in Solution III, indicates a secular change in <span class='html-italic'>G</span> in the whole Universe. We may instead suppose that it is an <span class='html-italic'>apparent</span> change in <math display='inline'><semantics> <mover accent="true"> <mi>G</mi> <mo>˙</mo> </mover> </semantics></math> that comes from either:</div><div class='html-p'><ul class='html-bullet'><li><div class='html-p'>An inaccurate modeling of the solar plasma delay of MESSENGER ranges. As said above, the ranges provided by NASA JPL and used in this work have the solar plasma delay subtracted from them; in [<a href="#B53-universe-10-00413" class="html-bibr">53</a>], a custom procedure of processing the raw radio science data was used, but the details of the solar plasma model are not clear. Possible systematic errors in the solar plasma delay model, which are in the scale of meters, surely can interfere with the processing of MESSENGER ranges whose residuals are in the scale of decimeters.</div></li><li><div class='html-p'>An inaccurate modeling of the asteroid belt. While the present model does a good job representing the “generic” part of the belt by 180 uniformly distributed point-masses [<a href="#B78-universe-10-00413" class="html-bibr">78</a>], there is no certainty that no systematic errors remain. Work is underway to make an improvement of the asteroid belt modeling in INPOP ephemerides [<a href="#B79-universe-10-00413" class="html-bibr">79</a>]. Also, it is important to note that the present procedure of individual asteroid masses’ determination [<a href="#B8-universe-10-00413" class="html-bibr">8</a>] does not disallow negative masses; they are excluded from the solution in an ad hoc manner, while in principle, a dedicated method (BVLS) should be applied.</div></li><li><div class='html-p'>An unmodeled change in <math display='inline'><semantics> <msub> <mi>J</mi> <mrow> <mn>2</mn> <mo>⊙</mo> </mrow> </msub> </semantics></math>. Solar gravitational moments are known to have temporal variations over a solar cycle, including <math display='inline'><semantics> <msub> <mi>J</mi> <mrow> <mn>2</mn> <mo>⊙</mo> </mrow> </msub> </semantics></math>. The helioseismological estimate made in [<a href="#B80-universe-10-00413" class="html-bibr">80</a>] was <math display='inline'><semantics> <mrow> <mn>1.0</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>10</mn> </mrow> </msup> </mrow> </semantics></math> in amplitude per 11 years. A numerical experiment (not included here) showed that a change in <math display='inline'><semantics> <msub> <mi>J</mi> <mrow> <mn>2</mn> <mo>⊙</mo> </mrow> </msub> </semantics></math> by <math display='inline'><semantics> <mrow> <mn>2.0</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>10</mn> </mrow> </msup> </mrow> </semantics></math> over 6.5 years (<math display='inline'><semantics> <mrow> <msub> <mover accent="true"> <mi>J</mi> <mo>˙</mo> </mover> <mrow> <mn>2</mn> <mo>⊙</mo> </mrow> </msub> <mo>/</mo> <msub> <mi>J</mi> <mrow> <mn>2</mn> <mo>⊙</mo> </mrow> </msub> <mo>≈</mo> <mn>0.014</mn> <mo>%</mo> <mspace width="4pt"/> <msup> <mi>yr</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math>) resulted in a notable signature in Earth–Mercury ranges, reaching 40 cm in amplitude, which is well comparable to the MESSENGER residuals. However, little is presently known about the phase of a possible <math display='inline'><semantics> <msub> <mi>J</mi> <mrow> <mn>2</mn> <mo>⊙</mo> </mrow> </msub> </semantics></math> oscillation during solar cycles 22 and 23, or possible higher-frequency oscillations. This issue requires further research.</div></li><li><div class='html-p'>Some local effect in the spacetime of the Solar system, rather than in the whole Universe or remote quasars. The hypotheses about such effects are outside the scope of this paper.</div></li></ul></div><div class='html-p'>Some other systematic effects that are less believed to affect the results are given in <a href="#sec6dot4-universe-10-00413" class="html-sec">Section 6.4</a>, <a href="#sec6dot5-universe-10-00413" class="html-sec">Section 6.5</a> and <a href="#sec6dot6-universe-10-00413" class="html-sec">Section 6.6</a>.</div></section><section id='sec6dot4-universe-10-00413' type=''><h4 class='html-italic' data-nested='2'> 6.4. Fall of Mass into the Sun</h4><div class='html-p'>Let us consider a positive <math display='inline'><semantics> <msubsup> <mover accent="true"> <mi>M</mi> <mo>˙</mo> </mover> <mrow> <mo>⊙</mo> </mrow> <mi>fall</mi> </msubsup> </semantics></math> due to comets, asteroids, and interplanetary dust falling into the Sun. Such effects, in principle, would move the estimated range of <math display='inline'><semantics> <mover accent="true"> <mi>G</mi> <mo>˙</mo> </mover> </semantics></math> to the left.</div><div class='html-p'>The comets that come in closest proximity to the Sun (sungrazers) have been observed by the SOHO space observatory since 1995, currently numbered in the thousands. Such comets rarely survive perihelion passage; however, the most frequent outcome is total evaporation of the comet before it reaches the surface, after which the small particles are pushed outwards by the solar pressure. A comet actually fallen into the Sun is a rare event; but even if 100 comets, each 1 km in radius, fell into the Sun in a year, the resulting mass acquisition relative rate would be less than <math display='inline'><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>18</mn> </mrow> </msup> </mrow> </semantics></math>, a negligible value compared to the mass loss by radiation and solar wind. The mass acquisition rate due to the fall of the interplanetary dust is also considered negligible, less than <math display='inline'><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>17</mn> </mrow> </msup> <mo> </mo> <msub> <mi>M</mi> <mo>⊙</mo> </msub> <msup> <mi>yr</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math>, see [<a href="#B6-universe-10-00413" class="html-bibr">6</a>].</div><div class='html-p'>Asteroids, unlike comets, have no record of falling into the Sun, while calculations show that such events can happen on a million-year timescale [<a href="#B81-universe-10-00413" class="html-bibr">81</a>]. For this effect to be noticeable in our studies (say <math display='inline'><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>15</mn> </mrow> </msup> <mo> </mo> <msub> <mi>M</mi> <mo>⊙</mo> </msub> <msup> <mi>yr</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math> of acquired mass), 90 asteroids, each 20 km in diameter, of the most dense type (5.3 g/cm<sup>3</sup>) should fall into the Sun per year, and not fall apart and be pushed out by solar pressure like most comets.</div><div class='html-p'>We could imagine that some small asteroids do fall into the Sun unnoticed; in that case, they should come from what is modeled as the 180-point annulus, a gravitational equivalent of all asteroids that are not in the model individually. The <math display='inline'><semantics> <mrow> <msub> <mi>μ</mi> <mi>annulus</mi> </msub> <mo>/</mo> <msub> <mi>μ</mi> <mo>⊙</mo> </msub> </mrow> </semantics></math> was determined dynamically to be ≈3.4 <math display='inline'><semantics> <mrow> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>11</mn> </mrow> </msup> </mrow> </semantics></math> in EPM2021 [<a href="#B82-universe-10-00413" class="html-bibr">82</a>] and ≈5.7 <math display='inline'><semantics> <mrow> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>11</mn> </mrow> </msup> <mo> </mo> <msub> <mi>M</mi> <mo>⊙</mo> </msub> </mrow> </semantics></math> in this work. If some <math display='inline'><semantics> <mrow> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>15</mn> </mrow> </msup> <mo> </mo> <msub> <mi>M</mi> <mo>⊙</mo> </msub> </mrow> </semantics></math> of asteroids (per year) were to leave the annulus to fall into the Sun in modern times, it would have meant that we happened to be close, on astronomical timescale, to witnessing the waning of the asteroid belt (with the exception of large asteroids). As such a circumstance is unlikely, we must conclude that <math display='inline'><semantics> <msubsup> <mover accent="true"> <mi>M</mi> <mo>˙</mo> </mover> <mrow> <mo>⊙</mo> </mrow> <mi>fall</mi> </msubsup> </semantics></math> is negligible in our study.</div><div class='html-p'>One could argue that the population of small asteroids is replenished by collisions between small and big asteroids. However, the estimates of the asteroid collision probabilities [<a href="#B83-universe-10-00413" class="html-bibr">83</a>] are nowhere near levels required for tens of collisions in a year.</div><div class='html-p'>For the sake of completeness, it must be noted that an asteroid does not have to fall into the Sun to increase the apparent <math display='inline'><semantics> <msub> <mi>M</mi> <mo>⊙</mo> </msub> </semantics></math>; asteroids on hypothetical orbits inside the orbit of Mercury (vulcanoids) would have produced similar gravitational effect. However, the population of vulcanoids is decreasing rather than increasing, if it exists at all [<a href="#B84-universe-10-00413" class="html-bibr">84</a>].</div></section><section id='sec6dot5-universe-10-00413' type=''><h4 class='html-italic' data-nested='2'> 6.5. Change of <math display='inline'><semantics> <msub> <mi>S</mi> <mo>⊙</mo> </msub> </semantics></math></h4><div class='html-p'>The Sun is slowly losing its angular momentum because of the torque exerted by its magnetic field interacting with solar wind. This effect has been studied since 1967 [<a href="#B85-universe-10-00413" class="html-bibr">85</a>]. The estimation of its amount requires estimating both the mass loss due to solar wind and the Alfvén radius; the latter in turn requires the estimation of the open magnetic flux. In [<a href="#B86-universe-10-00413" class="html-bibr">86</a>], the average angular momentum loss rate was found to be <math display='inline'><semantics> <mrow> <mn>2.2</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>30</mn> </msup> </mrow> </semantics></math> erg—both secular and since 2000, and in the latter case, the in situ measurements from the OMNI dataset were used to estimate the open magnetic flux. In [<a href="#B87-universe-10-00413" class="html-bibr">87</a>], based only on the <span class='html-italic'>Wind</span> spacecraft measurements, the average momentum loss was estimated to be <math display='inline'><semantics> <mrow> <mn>3.3</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>30</mn> </msup> </mrow> </semantics></math> erg.</div><div class='html-p'>According to [<a href="#B88-universe-10-00413" class="html-bibr">88</a>], the secular change in the semi-major axis <span class='html-italic'>a</span> of a planet due to Sun’s momentum loss rate <math display='inline'><semantics> <msub> <mover accent="true"> <mi>S</mi> <mo>˙</mo> </mover> <mo>⊙</mo> </msub> </semantics></math> is:<div class='html-disp-formula-info' id='FD12-universe-10-00413'> <div class='f'> <math display='block'><semantics> <mrow> <mrow> <mo>⟨</mo> <mover accent="true"> <mi>a</mi> <mo>˙</mo> </mover> <mo>⟩</mo> </mrow> <mo>=</mo> <mn>4</mn> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>G</mi> <msub> <mover accent="true"> <mi>S</mi> <mo>˙</mo> </mover> <mo>⊙</mo> </msub> </mrow> <mrow> <msup> <mi>c</mi> <mn>2</mn> </msup> <msqrt> <mrow> <msub> <mi>μ</mi> <mo>⊙</mo> </msub> <mi>a</mi> </mrow> </msqrt> </mrow> </mfrac> </mstyle> <mo form="prefix">cos</mo> <mi>ε</mi> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(12)</label> </div> </div> where <math display='inline'><semantics> <mi>ε</mi> </semantics></math> is the obliquity of the ecliptic. Using the value of <math display='inline'><semantics> <mrow> <mn>3.3</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>30</mn> </msup> </mrow> </semantics></math> erg for <math display='inline'><semantics> <msub> <mover accent="true"> <mi>S</mi> <mo>˙</mo> </mover> <mo>⊙</mo> </msub> </semantics></math>, we obtain the secular change in the semi-major axis of Mercury <math display='inline'><semantics> <mrow> <mrow> <mo>⟨</mo> <mover accent="true"> <mi>a</mi> <mo>˙</mo> </mover> <mo>⟩</mo> </mrow> <mo>≈</mo> <mo>−</mo> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>11</mn> </mrow> </msup> </mrow> </semantics></math> m/yr, which is negligible for our study. Secular variations in other orbital elements in [<a href="#B88-universe-10-00413" class="html-bibr">88</a>] are of similar order.</div></section><section id='sec6dot6-universe-10-00413' type=''><h4 class='html-italic' data-nested='2'> 6.6. Gravitational Wave Background</h4><div class='html-p'>Low-frequency gravitational waves were recently confirmed by pulsar timing [<a href="#B89-universe-10-00413" class="html-bibr">89</a>]; however, the reported strain amplitudes were on the order of <math display='inline'><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> </semantics></math>, which is too small to be detected by precision radio ranging (less than 1 mm change in Earth–Mercury distance).</div></section></section><section id='sec7-universe-10-00413' type=''><h2 data-nested='1'> 7. Determining <math display='inline'><semantics> <mover accent="true"> <mi mathvariant="bold-italic">G</mi> <mo>˙</mo> </mover> </semantics></math> in Lunar Solution</h2><div class='html-p'>An attempt was made to determine <math display='inline'><semantics> <mover accent="true"> <mi>G</mi> <mo>˙</mo> </mover> </semantics></math> in the lunar solution. Obviously a finite <math display='inline'><semantics> <mover accent="true"> <mi>G</mi> <mo>˙</mo> </mover> </semantics></math> would cause a trend to the gravitational parameter of the Earth–Moon system <math display='inline'><semantics> <msub> <mi>μ</mi> <mi>EM</mi> </msub> </semantics></math>, and LLR data, accumulated over more than half a century, are very sensitive to this parameter. Moreover, in the case of the Earth–Moon system, it is much easier to extract <math display='inline'><semantics> <mover accent="true"> <mi>G</mi> <mo>˙</mo> </mover> </semantics></math> from <math display='inline'><semantics> <mover accent="true"> <mi>μ</mi> <mo>˙</mo> </mover> </semantics></math>. The flux of meteoroids falling onto the Earth and the Moon is estimated to be less than <math display='inline'><semantics> <mrow> <mn>2.8</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>18</mn> </mrow> </msup> <mo> </mo> <msub> <mi>M</mi> <mo>⊕</mo> </msub> </mrow> </semantics></math> [<a href="#B90-universe-10-00413" class="html-bibr">90</a>]. As for the solar wind, even assuming that all its particles that reach the Earth magnetosphere are completely consumed, it would be less than <math display='inline'><semantics> <mrow> <mn>5</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>8</mn> </mrow> </msup> </mrow> </semantics></math> of the overall flux. Using the nominal value of <math display='inline'><semantics> <mrow> <msubsup> <mover accent="true"> <mi>M</mi> <mo>˙</mo> </mover> <mrow> <mo>⊙</mo> </mrow> <mrow> <mo>(</mo> <mi>wind</mi> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <mo>−</mo> <mn>2.1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> <mo> </mo> <msub> <mi>M</mi> <mo>⊙</mo> </msub> </mrow> </semantics></math>, we would obtain <math display='inline'><semantics> <mrow> <msubsup> <mover accent="true"> <mi>M</mi> <mo>˙</mo> </mover> <mrow> <mi>EM</mi> </mrow> <mrow> <mo>(</mo> <mi>wind</mi> <mo>)</mo> </mrow> </msubsup> <mo><</mo> <mn>1.05</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>21</mn> </mrow> </msup> <mo> </mo> <msub> <mi>M</mi> <mo>⊙</mo> </msub> <mo><</mo> <mn>3.5</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>16</mn> </mrow> </msup> <mo> </mo> <msub> <mi>M</mi> <mi>EM</mi> </msub> <mspace width="4pt"/> <msup> <mi>yr</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math>. In that case, the solar wind acquisition would dominate the meteoroids’ acquisition but would be still be negligible.</div><div class='html-p'>Unfortunately, as was already reported in [<a href="#B54-universe-10-00413" class="html-bibr">54</a>], <math display='inline'><semantics> <msub> <mover accent="true"> <mi>μ</mi> <mo>˙</mo> </mover> <mi>EM</mi> </msub> </semantics></math> cannot be reliably estimated in the lunar solution because it is hard to separate from two parameters of Earth tides, namely, the rotational delays of the Earth’s diurnal and semi-diurnal tides [<a href="#B91-universe-10-00413" class="html-bibr">91</a>]. The correlation of <math display='inline'><semantics> <msub> <mover accent="true"> <mi>μ</mi> <mo>˙</mo> </mover> <mi>EM</mi> </msub> </semantics></math> with <math display='inline'><semantics> <msub> <mi>τ</mi> <mrow> <mn>1</mn> <mi>R</mi> </mrow> </msub> </semantics></math> and <math display='inline'><semantics> <msub> <mi>τ</mi> <mrow> <mn>2</mn> <mi>R</mi> </mrow> </msub> </semantics></math> is 91.7% and 99.6%, respectively. Alternatively, the IERS tidal model [<a href="#B40-universe-10-00413" class="html-bibr">40</a>] does not need to be fit to LLR observations, but it does not provide as good a fit as the DE tidal model [<a href="#B23-universe-10-00413" class="html-bibr">23</a>,<a href="#B29-universe-10-00413" class="html-bibr">29</a>]. More research on Earth tides is needed before estimating <math display='inline'><semantics> <mover accent="true"> <mi>G</mi> <mo>˙</mo> </mover> </semantics></math> via LLR.</div></section><section id='sec8-universe-10-00413' type='conclusions'><h2 data-nested='1'> 8. Conclusions</h2><ul class='html-bullet'><li><div class='html-p'>The solar equator orientation angles, used in EPM and other ephemerides to this day to model acceleration from solar oblateness and the Lense–Thirring effect (in ephemerides where it is modeled), have no legitimate reason to be used for that purpose. The internal structure of the Sun is complex and there is no common rotation axis. An experiment showed that even a slight update to the model—keeping the single dynamical solar rotation axis but allowing it to differ from the “visible” one—significantly improved the modeling of MESSENGER ranges. It remains an open question whether more accurate modeling is possible.</div></li><li><div class='html-p'><math display='inline'><semantics> <mover accent="true"> <msub> <mi>μ</mi> <mo>⊙</mo> </msub> <mo>˙</mo> </mover> </semantics></math> was determined from the whole set of planetary observations, including MESSENGER, to <math display='inline'><semantics> <mrow> <mrow> <mo>(</mo> <mo>−</mo> <mn>3.4</mn> <mo>±</mo> <mn>3.1</mn> <mo>)</mo> </mrow> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> <mo> </mo> <msub> <mi>μ</mi> <mo>⊙</mo> </msub> <mo>/</mo> <mi>yr</mi> </mrow> </semantics></math> (<math display='inline'><semantics> <mrow> <mn>3</mn> <mi>σ</mi> </mrow> </semantics></math>). This, in turn, allowed us to estimate the time variation in the gravitational constant: <math display='inline'><semantics> <mrow> <mn>2.0</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> <mo>/</mo> <mi>yr</mi> <mo><</mo> <mover accent="true"> <mi>G</mi> <mo>˙</mo> </mover> <mo>/</mo> <mi>G</mi> <mo><</mo> <mn>9.6</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> <mo>/</mo> <mi>yr</mi> </mrow> </semantics></math>.</div></li><li><div class='html-p'>The positive value of <math display='inline'><semantics> <mover accent="true"> <mi>G</mi> <mo>˙</mo> </mover> </semantics></math> may not be real but apparent, owing to the imperfection of the models of solar plasma delay, asteroid belt, change in <math display='inline'><semantics> <msub> <mi>J</mi> <mrow> <mn>2</mn> <mo>⊙</mo> </mrow> </msub> </semantics></math>. Less likely but possibly, it may indicate some local effect in the spacetime of the Solar system in 2011–2014.</div></li><li><div class='html-p'>Care must be taken in research to properly estimate the mass loss of the Sun to the solar wind. The most modern models, and in situ observations made during the studied timespan, must be used. The model of solar plasma delay is also very important in research based on precise planetary radio ranges.</div></li><li><div class='html-p'>Also, the variation in <math display='inline'><semantics> <msub> <mi>J</mi> <mrow> <mn>2</mn> <mo>⊙</mo> </mrow> </msub> </semantics></math> is an important effect for the dynamics of Mercury, and a theory should be proposed in the future for the modeling of this effect.</div></li><li><div class='html-p'>Updated data can help to improve or disprove the <math display='inline'><semantics> <mover accent="true"> <mi>G</mi> <mo>˙</mo> </mover> </semantics></math> estimate obtained in this work: MESSENGER ranges without the solar plasma delay subtracted and MRO and Odyssey ranges past 2017.</div></li><li><div class='html-p'>Future Earth–Mercury ranges produced from the BepiColombo mission, presumably with centimeter accuracy [<a href="#B92-universe-10-00413" class="html-bibr">92</a>], will allow researchers to greatly improve the dynamical model of the Sun and the inner Solar system [<a href="#B32-universe-10-00413" class="html-bibr">32</a>,<a href="#B55-universe-10-00413" class="html-bibr">55</a>], including better estimates for <math display='inline'><semantics> <msub> <mover accent="true"> <mi>μ</mi> <mo>˙</mo> </mover> <mo>⊙</mo> </msub> </semantics></math> and <math display='inline'><semantics> <mover accent="true"> <mi>G</mi> <mo>˙</mo> </mover> </semantics></math>.</div></li><li><div class='html-p'>Research on a model of Earth tides is needed to unlock the big potential of LLR for estimating the change rate of the gravitational constant.</div></li></ul></section> </div> <div class="html-back"> <section class='html-notes'><h2 >Author Contributions</h2><div class='html-p'>Conceptualization, D.P.; methodology, D.P.; software, D.P. and I.D.; validation, D.P. and I.D.; writing, D.P. and I.D. All authors have read and agreed to the published version of the manuscript.</div></section><section class='html-notes'><h2>Funding</h2><div class='html-p'>The research of Dmitry Pavlov was funded by the Russian Science Foundation (project no. 24-22-00391).</div></section><section class='html-notes'><h2 >Data Availability Statement</h2><div class='html-p'>The observational data used in this work come from publicly available sources listed in <a href="#sec2-universe-10-00413" class="html-sec">Section 2</a>. Intermediate results of computations are available from authors upon reasonable request.</div></section><section id='html-ack' class='html-ack'><h2 >Acknowledgments</h2><div class='html-p'>Authors are grateful to Elena Pitjeva (IAA RAS) and Sergey Kurdubov (IAA RAS) for support provided and valuable advice given throughout this work; to Rodolfo Gustavo Cionco (CONICET-UTN) and Marshall Eubanks (Space Initiatives Inc.) for useful references; to Lorenzo Iorio (Ministry of Education and Merit of Italy) for support and encouragement. Planetary observations, including spacecraft ranges which are essential for this work, were collected from the NASA SSD webpage and are supported by Ryan Park, and at the Geoazur website supported by Agnès Fienga. Anonymous reviewers provided useful advice which allowed us to significantly improve the paper.</div></section><section class='html-notes'><h2 >Conflicts of Interest</h2><div class='html-p'>The authors declare no conflicts of interest.</div></section><section id='html-glossary'><h2 >Abbreviations</h2><div class='html-p'>The following abbreviations are used in this manuscript: <table class='html-array_table'><tbody ><tr ><td align='left' valign='middle' class='html-align-left' >ACE</td><td align='left' valign='middle' class='html-align-left' >Advanced Composition Explorer</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >BVLS</td><td align='left' valign='middle' class='html-align-left' >Bounded-value least squares</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >CDDIS</td><td align='left' valign='middle' class='html-align-left' >Crustal Dynamics Data Information System</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >CMB</td><td align='left' valign='middle' class='html-align-left' >Cosmic microwave background</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >CMEs</td><td align='left' valign='middle' class='html-align-left' >Coronal mass ejections</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >CONICET</td><td align='left' valign='middle' class='html-align-left' >Consejo Nacional de Investigaciones Científicas y Técnicas<br>(National Scientific and Technical Research Council)</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >DE</td><td align='left' valign='middle' class='html-align-left' >Development Ephmerides</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >DSN</td><td align='left' valign='middle' class='html-align-left' >Deep Space Network</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >EOP</td><td align='left' valign='middle' class='html-align-left' >Earth Orientation Parameters</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >EPM</td><td align='left' valign='middle' class='html-align-left' >Ephemerides of Planets and the Moon</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >IAA RAS</td><td align='left' valign='middle' class='html-align-left' >Institute of Applied Astronomy of the Russian Academy of Sciences</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >IAU</td><td align='left' valign='middle' class='html-align-left' >International Astronomical Union</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >IERS</td><td align='left' valign='middle' class='html-align-left' >International Earth Rotation and Reference Systems Service</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >LAGEOS</td><td align='left' valign='middle' class='html-align-left' >Laser Geodynamics Satellite or Laser Geometric Environmental Observation Survey</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >LLR</td><td align='left' valign='middle' class='html-align-left' >Lunar laser ranging</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >LOI</td><td align='left' valign='middle' class='html-align-left' >Luminosity Oscillations Imager</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >MESSENGER</td><td align='left' valign='middle' class='html-align-left' >Mercury Surface, Space Environment, Geochemistry, and Ranging</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >MLRS</td><td align='left' valign='middle' class='html-align-left' >McDonald Laser Ranging Station</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >MGS</td><td align='left' valign='middle' class='html-align-left' >Mars Global Surveyor</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >MRO</td><td align='left' valign='middle' class='html-align-left' >Mars Reconnaissance Orbiter</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >NASA JPL</td><td align='left' valign='middle' class='html-align-left' >Jet Propulsion Laboratory of the National Aeronautics and Space Administration</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >POLAC</td><td align='left' valign='middle' class='html-align-left' >Lunar Analysis Center of Paris Observatory</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >SOHO</td><td align='left' valign='middle' class='html-align-left' >Solar and Heliospheric Observatory</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >SLR</td><td align='left' valign='middle' class='html-align-left' >Satellite laser ranging</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >SSD</td><td align='left' valign='middle' class='html-align-left' >Solar system dynamics</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >SWOOPS</td><td align='left' valign='middle' class='html-align-left' >Solar Wind Observations Over the Poles of the Sun</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >TNOs</td><td align='left' valign='middle' class='html-align-left' >Trans-Neptunian objects</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >UTN</td><td align='left' valign='middle' class='html-align-left' >Universidad Tecnológica Nacional</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >VLBI</td><td align='left' valign='middle' class='html-align-left' >Very Long Baseline Interferometry</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >wrms</td><td align='left' valign='middle' class='html-align-left' >Weighted root-mean-square residual</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >WRSS</td><td align='left' valign='middle' class='html-align-left' >Weighted Residual Sum of Squares</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >YAG</td><td align='left' valign='middle' class='html-align-left' >Yttrium aluminum garnet</td></tr></tbody></table></div></section><section class='html-new-fn_group'><table><h2 >Notes</h2><tr id='fn001-universe-10-00413'><td><span class='html-label'>1</span></td><td><div class='html-p'>Only Earth–Mars spacecraft ranges are accurate enough and cover a long enough timespan to allow such an estimation.</div></td></tr><tr id='fn002-universe-10-00413'><td><span class='html-label'>2</span></td><td><div class='html-p'><a href='https://ssd.jpl.nasa.gov/?eph_data' target='_blank' rel="noopener noreferrer">https://ssd.jpl.nasa.gov/?eph_data</a> (accessed on 31 October 2024).</div></td></tr><tr id='fn003-universe-10-00413'><td><span class='html-label'>3</span></td><td><div class='html-p'><a href='http://www.geoazur.fr/astrogeo/?href=observations/base' target='_blank' rel="noopener noreferrer">http://www.geoazur.fr/astrogeo/?href=observations/base</a> (accessed on 31 October 2024).</div></td></tr><tr id='fn004-universe-10-00413'><td><span class='html-label'>4</span></td><td><div class='html-p'><a href='http://iaaras.ru/en/dept/ephemeris/observations' target='_blank' rel="noopener noreferrer">http://iaaras.ru/en/dept/ephemeris/observations</a> (accessed on 31 October 2024).</div></td></tr><tr id='fn005-universe-10-00413'><td><span class='html-label'>5</span></td><td><div class='html-p'><a href='http://sftp://gdc.cddis.gsfc.nasa.gov/slr/data/nptcrd' target='_blank' rel="noopener noreferrer">sftp://gdc.cddis.gsfc.nasa.gov/slr/data/nptcrd</a> (accessed on 31 October 2024).</div></td></tr><tr id='fn006-universe-10-00413'><td><span class='html-label'>6</span></td><td><div class='html-p'><a href='http://polac.obspm.fr/llrdatae.html' target='_blank' rel="noopener noreferrer">http://polac.obspm.fr/llrdatae.html</a> (accessed on 31 October 2024).</div></td></tr><tr id='fn007-universe-10-00413'><td><span class='html-label'>7</span></td><td><div class='html-p'><a href='https://ssd.jpl.nasa.gov/horizons/' target='_blank' rel="noopener noreferrer">https://ssd.jpl.nasa.gov/horizons/</a> (accessed on 31 October 2024).</div></td></tr><tr id='fn008-universe-10-00413'><td><span class='html-label'>8</span></td><td><div class='html-p'><a href='http://iers.org' target='_blank' rel="noopener noreferrer">http://iers.org</a> (accessed on 31 October 2024).</div></td></tr><tr id='fn009-universe-10-00413'><td><span class='html-label'>9</span></td><td><div class='html-p'>However, we must note that the account in [<a href="#B53-universe-10-00413" class="html-bibr">53</a>] of the Sun’s mass loss rate <math display='inline'><semantics> <mrow> <mo>−</mo> <mn>9</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> <mo><</mo> <msub> <mover accent="true"> <mi>M</mi> <mo>˙</mo> </mover> <mo>⊙</mo> </msub> <mo>/</mo> <msub> <mi>M</mi> <mo>⊙</mo> </msub> <mo><</mo> <mo>−</mo> <mn>11</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> </mrow> </semantics></math> was probably inaccurate because it was based on two papers published in 2011 [<a href="#B76-universe-10-00413" class="html-bibr">76</a>] and 2012 [<a href="#B77-universe-10-00413" class="html-bibr">77</a>], well before the end of the MESSENGER observations, and assumed that the activity of the solar wind was similar between consecutive solar cycles, which was not true for solar cycles 23 and 24. The more realistic estimate (<a href="#FD8-universe-10-00413" class="html-disp-formula">8</a>) would result in <math display='inline'><semantics> <mover accent="true"> <mi>G</mi> <mo>˙</mo> </mover> </semantics></math> being zero, assuming the <math display='inline'><semantics> <mrow> <mn>2</mn> <mi>σ</mi> </mrow> </semantics></math>-bounds of the determined <math display='inline'><semantics> <msub> <mover accent="true"> <mi>μ</mi> <mo>˙</mo> </mover> <mo>⊙</mo> </msub> </semantics></math>, not the <math display='inline'><semantics> <mrow> <mn>1</mn> <mi>σ</mi> </mrow> </semantics></math>-bounds as claimed in the paper.</div></td></tr></table></section><section id='html-references_list'><h2>References</h2><ol class='html-xx'><li id='B1-universe-10-00413' class='html-x' data-content='1.'>Mohr, P.; Newell, D.; Taylor, B.; Tiesinga, E. CODATA Recommended Values of the Fundamental Physical Constants: 2022. <span class='html-italic'>arXiv</span> <b>2024</b>. 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[<a href="https://scholar.google.com/scholar_lookup?title=Centimeter-Level+Ranging+Results+from+the+2022+BepiColombo+Solar+Conjunction&conference=Proceedings+of+the+AGU+Fall+Meeting&author=Paik,+M.&author=Asmar,+S.&author=Border,+J.&author=Shin,+D.K.&author=Martinez,+G.&author=Lippincott,+J.&author=Angkasa,+K.S.&author=Bedrossian,+A.&author=Iess,+L.&author=Cappuccio,+P.&publication_year=2022&pages=P42G-2483" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>]</li></ol></section><section id='FiguresandTables' type='display-objects'><div class="html-fig-wrap" id="universe-10-00413-f001"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/413/display" href="#fig_body_display_universe-10-00413-f001"> <img data-large="/universe/universe-10-00413/article_deploy/html/images/universe-10-00413-g001.png" data-original="/universe/universe-10-00413/article_deploy/html/images/universe-10-00413-g001.png" alt="Universe 10 00413 g001" data-lsrc="/universe/universe-10-00413/article_deploy/html/images/universe-10-00413-g001-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/413/display" href="#fig_body_display_universe-10-00413-f001"></a> </div> </div> <div class="html-fig_description"> <b>Figure 1.</b> Difference between model values of MESSENGER ranges (<math display='inline'><semantics> <mrow> <msub> <mi>C</mi> <mi>II</mi> </msub> <mo>−</mo> <msub> <mi>C</mi> <mi mathvariant="normal">I</mi> </msub> </mrow> </semantics></math>).  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00413-f001"> <div class="html-caption"> <b>Figure 1.</b> Difference between model values of MESSENGER ranges (<math display='inline'><semantics> <mrow> <msub> <mi>C</mi> <mi>II</mi> </msub> <mo>−</mo> <msub> <mi>C</mi> <mi mathvariant="normal">I</mi> </msub> </mrow> </semantics></math>).</div> <div class="html-img"><img data-large="/universe/universe-10-00413/article_deploy/html/images/universe-10-00413-g001.png" data-original="/universe/universe-10-00413/article_deploy/html/images/universe-10-00413-g001.png" alt="Universe 10 00413 g001" data-lsrc="/universe/universe-10-00413/article_deploy/html/images/universe-10-00413-g001.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00413-f002"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/413/display" href="#fig_body_display_universe-10-00413-f002"> <img data-large="/universe/universe-10-00413/article_deploy/html/images/universe-10-00413-g002.png" data-original="/universe/universe-10-00413/article_deploy/html/images/universe-10-00413-g002.png" alt="Universe 10 00413 g002" data-lsrc="/universe/universe-10-00413/article_deploy/html/images/universe-10-00413-g002-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/413/display" href="#fig_body_display_universe-10-00413-f002"></a> </div> </div> <div class="html-fig_description"> <b>Figure 2.</b> Difference between model values of MESSENGER ranges (<math display='inline'><semantics> <mrow> <msub> <mi>C</mi> <mi>III</mi> </msub> <mo>−</mo> <msub> <mi>C</mi> <mi>II</mi> </msub> </mrow> </semantics></math>).  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00413-f002"> <div class="html-caption"> <b>Figure 2.</b> Difference between model values of MESSENGER ranges (<math display='inline'><semantics> <mrow> <msub> <mi>C</mi> <mi>III</mi> </msub> <mo>−</mo> <msub> <mi>C</mi> <mi>II</mi> </msub> </mrow> </semantics></math>).</div> <div class="html-img"><img data-large="/universe/universe-10-00413/article_deploy/html/images/universe-10-00413-g002.png" data-original="/universe/universe-10-00413/article_deploy/html/images/universe-10-00413-g002.png" alt="Universe 10 00413 g002" data-lsrc="/universe/universe-10-00413/article_deploy/html/images/universe-10-00413-g002.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00413-f003"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/413/display" href="#fig_body_display_universe-10-00413-f003"> <img data-large="/universe/universe-10-00413/article_deploy/html/images/universe-10-00413-g003.png" data-original="/universe/universe-10-00413/article_deploy/html/images/universe-10-00413-g003.png" alt="Universe 10 00413 g003" data-lsrc="/universe/universe-10-00413/article_deploy/html/images/universe-10-00413-g003-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/413/display" href="#fig_body_display_universe-10-00413-f003"></a> </div> </div> <div class="html-fig_description"> <b>Figure 3.</b> Difference between model values of MESSENGER ranges (<math display='inline'><semantics> <mrow> <msub> <mi>C</mi> <mi>III</mi> </msub> <mo>−</mo> <msub> <mi>C</mi> <mi mathvariant="normal">I</mi> </msub> </mrow> </semantics></math>).  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00413-f003"> <div class="html-caption"> <b>Figure 3.</b> Difference between model values of MESSENGER ranges (<math display='inline'><semantics> <mrow> <msub> <mi>C</mi> <mi>III</mi> </msub> <mo>−</mo> <msub> <mi>C</mi> <mi mathvariant="normal">I</mi> </msub> </mrow> </semantics></math>).</div> <div class="html-img"><img data-large="/universe/universe-10-00413/article_deploy/html/images/universe-10-00413-g003.png" data-original="/universe/universe-10-00413/article_deploy/html/images/universe-10-00413-g003.png" alt="Universe 10 00413 g003" data-lsrc="/universe/universe-10-00413/article_deploy/html/images/universe-10-00413-g003.png" /></div> </div> <div class="html-fig-wrap" id="universe-10-00413-f004"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/413/display" href="#fig_body_display_universe-10-00413-f004"> <img data-large="/universe/universe-10-00413/article_deploy/html/images/universe-10-00413-g004.png" data-original="/universe/universe-10-00413/article_deploy/html/images/universe-10-00413-g004.png" alt="Universe 10 00413 g004" data-lsrc="/universe/universe-10-00413/article_deploy/html/images/universe-10-00413-g004-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/413/display" href="#fig_body_display_universe-10-00413-f004"></a> </div> </div> <div class="html-fig_description"> <b>Figure 4.</b> MESSENGER residuals <math display='inline'><semantics> <mrow> <mi>O</mi> <mo>−</mo> <msub> <mi>C</mi> <mi>III</mi> </msub> </mrow> </semantics></math>.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_universe-10-00413-f004"> <div class="html-caption"> <b>Figure 4.</b> MESSENGER residuals <math display='inline'><semantics> <mrow> <mi>O</mi> <mo>−</mo> <msub> <mi>C</mi> <mi>III</mi> </msub> </mrow> </semantics></math>.</div> <div class="html-img"><img data-large="/universe/universe-10-00413/article_deploy/html/images/universe-10-00413-g004.png" data-original="/universe/universe-10-00413/article_deploy/html/images/universe-10-00413-g004.png" alt="Universe 10 00413 g004" data-lsrc="/universe/universe-10-00413/article_deploy/html/images/universe-10-00413-g004.png" /></div> </div> <div class="html-table-wrap" id="universe-10-00413-t001"> <div class="html-table_wrap_td"> <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/413/display" href='#table_body_display_universe-10-00413-t001'> <img data-lsrc="https://pub.mdpi-res.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/413/display" href="#table_body_display_universe-10-00413-t001"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 1.</b> Comparison of parameters for the solutions I, II, and III. The uncertainty was <math display='inline'><semantics> <mrow> <mn>3</mn> <mi>σ</mi> </mrow> </semantics></math>. </div> </div> <div class="html-table_show mfp-hide " id="table_body_display_universe-10-00413-t001"> <div class="html-caption"><b>Table 1.</b> Comparison of parameters for the solutions I, II, and III. The uncertainty was <math display='inline'><semantics> <mrow> <mn>3</mn> <mi>σ</mi> </mrow> </semantics></math>.</div> <table > <thead ><tr ><th align='left' valign='middle' style='border-bottom:solid thin;border-top:solid thin' class='html-align-left' >Parameter</th><th align='right' valign='middle' style='border-bottom:solid thin;border-top:solid thin' class='html-align-right' >Solution I</th><th align='right' valign='middle' style='border-bottom:solid thin;border-top:solid thin' class='html-align-right' >Solution II</th><th align='right' valign='middle' style='border-bottom:solid thin;border-top:solid thin' class='html-align-right' >Solution III</th></tr></thead><tbody ><tr ><td align='left' valign='middle' class='html-align-left' ><math display='inline'><semantics> <mrow> <msubsup> <mi>μ</mi> <mrow> <mo>⊙</mo> </mrow> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </msubsup> <mo>−</mo> <mn>295912208283</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msup> <mi>au</mi> <mn>3</mn> </msup> <msup> <mi>day</mi> <mn>2</mn> </msup> </mfrac> </mstyle> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>15</mn> </mrow> </msup> </mrow> </semantics></math></td><td align='right' valign='middle' class='html-align-right' ><math display='inline'> <semantics> <mrow> <mn>0.78</mn> <mo>±</mo> <mn>0.27</mn> </mrow> </semantics> </math></td><td align='right' valign='middle' class='html-align-right' ><math display='inline'> <semantics> <mrow> <mn>0.37</mn> <mo>±</mo> <mn>0.27</mn> </mrow> </semantics> </math></td><td align='right' valign='middle' class='html-align-right' ><math display='inline'> <semantics> <mrow> <mn>0.51</mn> <mo>±</mo> <mn>0.27</mn> </mrow> </semantics> </math></td></tr><tr ><td align='left' valign='middle' class='html-align-left' ><math display='inline'><semantics> <msub> <mi>J</mi> <mrow> <mn>2</mn> <mo>⊙</mo> </mrow> </msub> </semantics></math>, <math display='inline'><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>7</mn> </mrow> </msup> </semantics></math></td><td align='right' valign='middle' class='html-align-right' ><math display='inline'> <semantics> <mrow> <mn>2.208</mn> <mo>±</mo> <mn>0.021</mn> </mrow> </semantics> </math></td><td align='right' valign='middle' class='html-align-right' ><math display='inline'> <semantics> <mrow> <mn>2.199</mn> <mo>±</mo> <mn>0.021</mn> </mrow> </semantics> </math></td><td align='right' valign='middle' class='html-align-right' ><math display='inline'> <semantics> <mrow> <mn>2.198</mn> <mo>±</mo> <mn>0.021</mn> </mrow> </semantics> </math></td></tr><tr ><td align='left' valign='middle' class='html-align-left' >Solar plasma delay factor</td><td align='right' valign='middle' class='html-align-right' ><math display='inline'> <semantics> <mrow> <mn>1.102</mn> <mo>±</mo> <mn>0.033</mn> </mrow> </semantics> </math></td><td align='right' valign='middle' class='html-align-right' ><math display='inline'> <semantics> <mrow> <mn>1.106</mn> <mo>±</mo> <mn>0.033</mn> </mrow> </semantics> </math></td><td align='right' valign='middle' class='html-align-right' ><math display='inline'> <semantics> <mrow> <mn>1.112</mn> <mo>±</mo> <mn>0.033</mn> </mrow> </semantics> </math></td></tr><tr ><td align='left' valign='middle' class='html-align-left' >MESSENGER transponder delay, m</td><td align='right' valign='middle' class='html-align-right' ><math display='inline'> <semantics> <mrow> <mo>−</mo> <mn>1.91</mn> <mo>±</mo> <mn>0.30</mn> </mrow> </semantics> </math></td><td align='right' valign='middle' class='html-align-right' ><math display='inline'> <semantics> <mrow> <mo>−</mo> <mn>1.82</mn> <mo>±</mo> <mn>0.30</mn> </mrow> </semantics> </math></td><td align='right' valign='middle' class='html-align-right' ><math display='inline'> <semantics> <mrow> <mo>−</mo> <mn>1.83</mn> <mo>±</mo> <mn>0.31</mn> </mrow> </semantics> </math></td></tr><tr ><td align='left' valign='middle' class='html-align-left' ><math display='inline'><semantics> <mrow> <msub> <mover accent="true"> <mi>μ</mi> <mo>˙</mo> </mover> <mo>⊙</mo> </msub> <mo>/</mo> <msub> <mi>μ</mi> <mo>⊙</mo> </msub> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msup> <mi>yr</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> </mrow> </semantics></math></td><td align='right' valign='middle' class='html-align-right' >fixed to <math display='inline'><semantics> <mrow> <mo>−</mo> <mn>8.9</mn> </mrow> </semantics></math></td><td align='right' valign='middle' class='html-align-right' >fixed to <math display='inline'><semantics> <mrow> <mo>−</mo> <mn>8.9</mn> </mrow> </semantics></math></td><td align='right' valign='middle' class='html-align-right' ><math display='inline'> <semantics> <mrow> <mo>−</mo> <mn>3.4</mn> <mo>±</mo> <mn>3.1</mn> </mrow> </semantics> </math></td></tr><tr ><td align='left' valign='middle' class='html-align-left' ><math display='inline'><semantics> <msubsup> <mi>α</mi> <mn>0</mn> <mo>∗</mo> </msubsup> </semantics></math>, °</td><td align='right' valign='middle' class='html-align-right' >fixed to <math display='inline'><semantics> <mrow> <mn>286.13</mn> </mrow> </semantics></math></td><td align='right' valign='middle' class='html-align-right' ><math display='inline'> <semantics> <mrow> <mn>287.72</mn> <mo>±</mo> <mn>1.32</mn> </mrow> </semantics> </math></td><td align='right' valign='middle' class='html-align-right' ><math display='inline'> <semantics> <mrow> <mn>287.86</mn> <mo>±</mo> <mn>1.32</mn> </mrow> </semantics> </math></td></tr><tr ><td align='left' valign='middle' style='border-bottom:solid thin' class='html-align-left' ><math display='inline'><semantics> <msubsup> <mi>δ</mi> <mn>0</mn> <mo>∗</mo> </msubsup> </semantics></math>, °</td><td align='right' valign='middle' style='border-bottom:solid thin' class='html-align-right' >fixed to <math display='inline'><semantics> <mrow> <mn>63.87</mn> </mrow> </semantics></math></td><td align='right' valign='middle' style='border-bottom:solid thin' class='html-align-right' ><math display='inline'> <semantics> <mrow> <mn>61.86</mn> <mo>±</mo> <mn>0.78</mn> </mrow> </semantics> </math></td><td align='right' valign='middle' style='border-bottom:solid thin' class='html-align-right' ><math display='inline'> <semantics> <mrow> <mn>61.98</mn> <mo>±</mo> <mn>0.78</mn> </mrow> </semantics> </math></td></tr></tbody> </table> </div> <div class="html-table-wrap" id="universe-10-00413-t002"> <div class="html-table_wrap_td"> <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/413/display" href='#table_body_display_universe-10-00413-t002'> <img data-lsrc="https://pub.mdpi-res.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2218-1997/10/11/413/display" href="#table_body_display_universe-10-00413-t002"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 2.</b> One-way wrms values for solutions I, II and III. </div> </div> <div class="html-table_show mfp-hide " id="table_body_display_universe-10-00413-t002"> <div class="html-caption"><b>Table 2.</b> One-way wrms values for solutions I, II and III.</div> <table > <thead ><tr ><th align='left' valign='middle' style='border-bottom:solid thin;border-top:solid thin' class='html-align-left' >Observation Group</th><th align='right' valign='middle' style='border-bottom:solid thin;border-top:solid thin' class='html-align-right' >No. of Ranges</th><th align='right' valign='middle' style='border-bottom:solid thin;border-top:solid thin' class='html-align-right' >Solution I</th><th align='right' valign='middle' style='border-bottom:solid thin;border-top:solid thin' class='html-align-right' >Solution II</th><th align='right' valign='middle' style='border-bottom:solid thin;border-top:solid thin' class='html-align-right' >Solution III</th></tr></thead><tbody ><tr ><td align='left' valign='middle' class='html-align-left' >MRO</td><td align='right' valign='middle' class='html-align-right' >1924</td><td align='right' valign='middle' class='html-align-right' >0.617 m</td><td align='right' valign='middle' class='html-align-right' >0.618 m</td><td align='right' valign='middle' class='html-align-right' >0.620 m</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >MGS</td><td align='right' valign='middle' class='html-align-right' >5590</td><td align='right' valign='middle' class='html-align-right' >0.921 m</td><td align='right' valign='middle' class='html-align-right' >0.921 m</td><td align='right' valign='middle' class='html-align-right' >0.922 m</td></tr><tr ><td align='left' valign='middle' class='html-align-left' >Odyssey</td><td align='right' valign='middle' class='html-align-right' >7988</td><td align='right' valign='middle' class='html-align-right' >0.565 m</td><td align='right' valign='middle' class='html-align-right' >0.565 m</td><td align='right' valign='middle' class='html-align-right' >0.565 m</td></tr><tr ><td align='left' valign='middle' style='border-bottom:solid thin' class='html-align-left' >MESSENGER</td><td align='right' valign='middle' style='border-bottom:solid thin' class='html-align-right' >1141</td><td align='right' valign='middle' style='border-bottom:solid thin' class='html-align-right' >0.669 m</td><td align='right' valign='middle' style='border-bottom:solid thin' class='html-align-right' >0.652 m</td><td align='right' valign='middle' style='border-bottom:solid thin' class='html-align-right' >0.647 m</td></tr></tbody> </table> </div> </section><section class='html-fn_group'><table><tr id=''><td></td><td><div class='html-p'><b>Disclaimer/Publisher’s Note:</b> The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) 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Studying the Properties of Spacetime with an Improved Dynamical Model of the Inner Solar System. <em>Universe</em> <b>2024</b>, <em>10</em>, 413. https://doi.org/10.3390/universe10110413 </p> <div style="display: block"> <b>AMA Style</b><br> <p> Pavlov D, Dolgakov I. Studying the Properties of Spacetime with an Improved Dynamical Model of the Inner Solar System. <em>Universe</em>. 2024; 10(11):413. https://doi.org/10.3390/universe10110413 </p> <b>Chicago/Turabian Style</b><br> <p> Pavlov, Dmitry, and Ivan Dolgakov. 2024. "Studying the Properties of Spacetime with an Improved Dynamical Model of the Inner Solar System" <em>Universe</em> 10, no. 11: 413. https://doi.org/10.3390/universe10110413 </p> <b>APA Style</b><br> <p> Pavlov, D., & Dolgakov, I. (2024). Studying the Properties of Spacetime with an Improved Dynamical Model of the Inner Solar System. <em>Universe</em>, <em>10</em>(11), 413. https://doi.org/10.3390/universe10110413 </p> </div> </div> <div class="info-box no-margin"> Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. 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Studying the Properties of Spacetime with an Improved Dynamical Model of the Inner Solar System