Introduction of tidal models in lunar ephemerides
Authors: Baguet Daniel , Rambaux Nicolas , Fienga Agnès , Mémin Anthony , Briaud Arthur , Hussmann Hauke , Stark Alexander , Hu Xuanyu , Viswanathan Vishnu , Gastineau Mickaël 
Published: September 2022

&lt;p&gt;&lt;strong&gt;1) Introduction&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;In the context of the Artemis program, the interest of lunar study has been renewed. The Lunar Laser Ranging (LLR) experiment allows since fifty years to determine the Earth-Moon distance at a few centimeters accuracy and the Moon&amp;#8217;s librations at a one milliarcsecond accuracy ([4], [3]) Such accuracy allows a refined description of the Moon&amp;#8217;s rotation.&lt;/p&gt;
&lt;p&gt;The tidal response of the Moon depends on its density and rheology and it affects its rotation. Therefore it provides important information on the lunar internal structure (e.g. [6], [7], [5], [1]). The variation of the harmonic degree 2 gravitational potential due to the tidal response can be quantified by the tidal Love number&amp;#160;&lt;em&gt;k&lt;/em&gt;&lt;em&gt;&lt;sub&gt;2&lt;/sub&gt;&lt;/em&gt;. The latter depends on the forcing frequency, which are mainly exerted by the Earth and the Sun. The dissipation is quantified by the dissipation factor&amp;#160;&lt;em&gt;Q&lt;/em&gt;&amp;#160;which is related to the value of the time delay.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;2) Time-delay lunar tides&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;Furthermore, because of the dissipation due to the viscosity of the lunar interior, the tidal deformation is not instantaneous. The current version of the planetary and lunar ephemeris INPOP only account for unique&amp;#160;&lt;em&gt;k&lt;/em&gt;&lt;em&gt;&lt;sub&gt;2&lt;/sub&gt;&lt;/em&gt;&amp;#160;and time delay [4]. Time delay can be described by a complex Love number&amp;#160;&lt;em&gt;k&lt;/em&gt;&lt;em&gt;&lt;sub&gt;2&lt;/sub&gt;&lt;/em&gt;, which in this case, depends also on the forcing frequency [7]. The dissipation factor&amp;#160;&lt;em&gt;Q&lt;/em&gt;&amp;#160;is related to the imaginary part of the Love number.&lt;/p&gt;
&lt;p&gt;Here we explore the introduction of tidal models in numerical ephemerides. We aim to include the&amp;#160;&lt;em&gt;k&lt;/em&gt;&lt;em&gt;&lt;sub&gt;2&lt;/sub&gt;&lt;/em&gt;&amp;#160;and&amp;#160;&lt;em&gt;Q&lt;/em&gt;&amp;#160;frequency dependency in INPOP.&lt;/p&gt;
&lt;p&gt;The formulation in Fourier series of the distortion coefficients (also called variation of the Stokes coefficients) by Williams and Boggs (2015) ([7]) allows to describe the tidal gravitational variation in accounting for the frequency dependency of the complex&amp;#160;&lt;em&gt;k&lt;/em&gt;&lt;em&gt;&lt;sub&gt;2&lt;/sub&gt;&lt;/em&gt;. The distortion coefficients vary according to the forcing frequencies formulated as linear combinations of the Delaunay arguments, the latter being described as a polynomial expansion with respect to the time.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;3) Representation of the orbit of the Moon&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;However, the representation in series of the distortion coefficients needs to be consistent with the ephemerides that we use. For that, we can use a semi-analytical representation of the ephemerides.&lt;/p&gt;
&lt;p&gt;Numerical ephemerides can described the orbit and the rotation of the Moon with a good accuracy according to the observational error. In complement, the semi-analytical approach is useful to extract information and to disentangle the different physical contributions contains in the numerical approaches. The &amp;#201;ph&amp;#233;m&amp;#233;ride Lunaire Parisienne (ELP) provides the most accurate semi-analytical model of the orbital motion of the Moon, in the form of Fourier and Poisson series [2].&lt;/p&gt;
&lt;p&gt;Figure 1 shows that the difference of the Earth-Moon distance between the solution of INPOP19a and the solution of ELP fitted to INPOP19a reach a maximum value of the order of 4 m. Then, we deduce the amplitude of the distortion coefficients from the fitted semi-analytical representation.&lt;/p&gt;
&lt;p&gt;&lt;img src=&quot;&quot; alt=&quot;&quot; width=&quot;334&quot; height=&quot;237&quot; /&gt;&lt;/p&gt;
&lt;p&gt;Figure 1. Difference of the Earth-Moon distance between the solution of IN- POP19a and the solution of ELP fitted to INPOP19a. The origin of time is J2000.&lt;/p&gt;
&lt;p&gt;&amp;#160;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;References&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;[1] &amp;#160;A. Briaud, A. Fienga, D. Melini, N. Rambaux, A. M &amp;#769;emin, G. Spada, C. Sal- iby, H. Hussmann, and A. Stark. Constraints on the Moon&amp;#8217;s Deep Interior from Tidal Deformation. In LPI Contributions, volume 2678 of LPI Contri- butions, page 1349, March 2022.&lt;/p&gt;
&lt;p&gt;[2] &amp;#160;J. Chapront and G. Francou. The lunar theory ELP revisited. Introduction of new planetary perturbations. , 404:735&amp;#8211;742, June 2003.&lt;/p&gt;
&lt;p&gt;[3] &amp;#160;Ryan S. Park, William M. Folkner, James G. Williams, and Dale H. Boggs. The JPL Planetary and Lunar Ephemerides DE440 and DE441. , 161(3):105, March 2021.&lt;/p&gt;
&lt;p&gt;[4] &amp;#160;V. Viswanathan, A. Fienga, O. Minazzoli, L. Bernus, J. Laskar, and M. Gastineau. The new lunar ephemeris INPOP17a and its application to fundamental physics. , 476(2):1877&amp;#8211;1888, May 2018.&lt;/p&gt;
&lt;p&gt;[5] &amp;#160;V. Viswanathan, N. Rambaux, A. Fienga, J. Laskar, and M. Gastineau. Observational Constraint on the Radius and Oblateness of the Lunar Core- Mantle Boundary. , 46(13):7295&amp;#8211;7303, July 2019.&lt;/p&gt;
&lt;p&gt;[6] &amp;#160;James Williams, Dale Boggs, Charles Yoder, James Ratcliff, and J. Dickey. Lunar rotational dissipation in solid body and molten core. Journal of Geo- physical Research, 106:27933&amp;#8211;27968, 11 2001.&lt;/p&gt;
&lt;p&gt;[7] &amp;#160;James G. Williams and Dale. H. Boggs. Tides on the Moon: Theory and determination of dissipation. Journal of Geophysical Research (Planets), 120(4):689&amp;#8211;724, April 2015.&lt;/p&gt;