Relativistic variations in Mars rotation
Authors: Baland Rose-Marie , Hees Aurélien , Yseboodt Marie , Bourgoin Adrien , Le Maistre Sébastien 
Published: September 2022

&lt;p&gt;&lt;strong&gt;Introduction&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;The orientation of Mars with respect to its orbit can be described with a set of Euler angles (longitude &amp;#968;, obliquity &amp;#949;, and rotation &amp;#966;). The rotation model for Mars, as described in the Barycentric Celestial Reference System (BCRS), includes two relativistic contributions: (1) a contribution to the precession and nutations in longitude which mainly comes from the Geodetic effect, (2) a contribution to the variations in rotation angle which mainly comes from the reference frame transformation between the local Martian frame and the BCRS frame and their respective time. We present a new estimation of the two relativistic contributions to Mars orientation.&amp;#160;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Results&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;We compute the geodetic and Lense-Thirring effects. The geodetic effect (e.g. Fukushima, 1991, Baland et al. 2020) is due to Mars motion along its orbit and only affects the longitude angle with precession and nutations terms:&lt;/p&gt;
&lt;p&gt;&amp;#968;&lt;sub&gt;geodetic&lt;/sub&gt;(t) = 6.754 mas/yr t + 0.565 mas sin l + 0.039 mas sin 2l + 0.004 mas sin 3l,&lt;/p&gt;
&lt;p&gt;l being the mean anomaly of Mars. The Lense-Thirring effect is due to the rotation of the Sun, and affects the three orientation angles (e.g. Barker and O&amp;#8217;Connell, 1970, or Poisson and Will, 2014). We find that the Lense-Thirring effect is 4 to 5 orders of magnitude smaller than the geodetic effect.&lt;/p&gt;
&lt;p&gt;Another relativistic correction [&amp;#966;]&lt;sub&gt;GR&lt;/sub&gt;(t) must be included in the model for the rotation angle (e.g. Eq. (18) of Konopliv et al. (2006), after Yoder and Standish 1997, with annual, semi-annual, and ter-annual terms, and amplitudes in mas) :&lt;/p&gt;
&lt;p&gt;[&amp;#966;]&lt;sub&gt;GR&lt;/sub&gt;(t) = - 175.80 sin l - 8.20 sin 2l - 0.60 sin 3l.&lt;/p&gt;
&lt;p&gt;This correction is obtained as the product of the mean rotation rate and of the difference between the local Martian time and the Barycenteric Dynamical Time. This time difference is related to relativistic time transformation between reference frames and is impacted by the velocity of Mars with respect to the solar system barycenter and by the gravitational potential experienced by Mars in its motion. We update the estimation of Yoder and Standish (1997), using three independent approaches: (1) a toy model where the planets evolve on Keplerian orbit, (2) a fit to a numerical integration using the DE440 planetary ephemerides (Park et al. 2021), (3) a computation using the analytical planetary theory VSOP87 (Bretagnon &amp; Francou 1988). We obtain a solution of the form:&lt;/p&gt;
&lt;p&gt;[&amp;#966;]&lt;sub&gt;GR&lt;/sub&gt;(t)&amp;#160; = &amp;#966;'&lt;sub&gt;GR &lt;/sub&gt;t + &amp;#931;&lt;sub&gt;j&lt;/sub&gt; &amp;#966;&lt;sub&gt;j&lt;/sub&gt; sin (f&lt;sub&gt;j&lt;/sub&gt; t + &amp;#958;&lt;sub&gt;j&lt;/sub&gt;),&lt;/p&gt;
&lt;p&gt;with &amp;#966;&lt;sub&gt;j&lt;/sub&gt; the amplitudes, f&lt;sub&gt;j&lt;/sub&gt; the angular frequencies, and &amp;#958;&lt;sub&gt;j&lt;/sub&gt; the phases at t=0 of a periodic series. This solution includes seasonal terms with amplitudes which differ from those of Yoder and Standish, (1997), e.g. by about 10 mas for the annual term, which is 10 times larger than the current formal uncertainty (Le Maistre et al. 2022). It also includes new terms, in particular one term at the synodic period between Jupiter and Mars (2.24 y), with about the same amplitude as the ter-annual term, and a secular term with the rate &amp;#966;'&lt;sub&gt;GR&lt;/sub&gt; which adds to the local sidereal rotation rate to produce the sidereal BCRS rotation rate (&amp;#966;'=&amp;#966;'&lt;sub&gt;local&lt;/sub&gt;+&amp;#966;'&lt;sub&gt;GR&lt;/sub&gt;).&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Discussion&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;The geodetic precession is about 3 times larger than the current formal uncertainty on the determination of the precession rate. It must be taken into account to avoid an error of about 0.1 % in the determination of the polar moment of inertia from the measured precession rate (e.g. Le Maistre et al. 2022).&lt;/p&gt;
&lt;p&gt;We provided a new set of relativistic corrections, more accurate than previous solutions, and that must be applied to the measured rotation parameters to avoid errors in their interpretation in terms of local physics (exchange of angular momentum between the atmosphere and the solid planet).&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Acknowledgements&lt;/strong&gt;&lt;/p&gt;
&lt;p align=&quot;left&quot;&gt;This work is done in the frame of the analysis of 2 Martian radioscience experiments (RISE on InSight, and LaRa on ExoMars), and is financially supported by the Belgian PRODEX program managed by the European Space Agency in collaboration with the Belgian Federal Science Policy Office.&lt;/p&gt;
&lt;p align=&quot;left&quot;&gt;&lt;strong&gt;References&lt;/strong&gt;&lt;/p&gt;
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&lt;p align=&quot;left&quot;&gt;Fukushima, T. 1991, A&amp;A, 244, L11&lt;/p&gt;
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