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Introduction Most spacecraft missions to small bodies provide the gravity field of their targets. As the extended gravity field is an expression of the mass distribution within the body, it can provide information about its interior structure, and that in turn can constrain its origin and history. We present a method to detect density anomalies within a small body from the global gravitational potential. As is well known, such an inverse problem is ill-posed [1], and requires regularization. Our main assumption, which mitigates the non-uniqueness of the problem, is that the interior of the body is composed of a finite number of domains, in which the density is uniform. The boundary of each domain is therefore a surface, that we describe as the level-set of a function. This approach, known as the level-set method, has been extensively used in local inversions of Earth’s gravity measurements [2]. It simplifies the manipulation of complex shapes, which in our case delimit the density anomalies. In order to reduce the dimensionality of the problem, we employ parametric expressions for the level-set functions. The aim is then to determine the parameters of each level-set function, which describe the location and size of the anomalies, along with the density within each anomaly and the bulk density of the body. Methods We model the gravity field of the body through a mascons approach [3]. The observables we consider are the coefficients of the spherical harmonic expansion of the potential. By implementing a continuous approximation for the unit step function, the partials derivatives of the measurements with respect to the level-set function parameters can be computed via the chain rule [4]. Then, the function parameters and the densities can be estimated via non-linear least-squares inversion of the gravity data. However, the problem is generally non-convex, which means that in order to converge to a global optimum the a priori values for the unknowns should be close to this solution. Since we assume no initial knowledge of the distribution to retrieve, a preliminary step is needed for the definition of an a priori model. To this end, we generate a Voronoi partition of the interior from randomly sampled points within the body. The domains are now fixed, and only the densities within each Voronoi cell are estimated from the data. This inversion problem is linear, but the solution will depend on the partition of the body. However, averaging solutions from many of such random partitions can provide a good approximation of the true model [5]. This averaged solution is then used to initialize the values of the level-set algorithm. Discussion and outlook We have tested the ability of our level-set implementation to retrieve interior models from simulated gravity data. Figure 1 shows an example of the inversion results for a Bennu-shaped body. In this case, the level-set functions are chosen so that the anomalies they describe are cuboids. The model in Figure 1a is the ground truth, used to simulate gravity coefficients up to degree 11. Figure 1b shows the density retrieved from the noise-free data, and Figure 1c the results when the data are perturbed by 1% Gaussian noise. The method can retrieve the location and density of the anomalies with good approximation, even in the perturbed case. For anomalies where the density and the size cannot be estimated simultaneously, such as with nearly-spherical shapes, the method can usually still correctly determine the total mass and the position of the center-of-mass. The size could then be constrained by independent observations from other instruments or by theoretical models. In a more realistic scenario, the resolution of the gravity would be lower, and the noise profile would be increasing with the degree of gravity. This could aggravate the non-uniqueness problem, since having fewer and less precise measurements could lead to different models fitting the data equally well. Current work is focused on improving the robustness of our approach and on better defining its limitations in its application to real gravity measurements. Figure 1. Interior density models, shown via sections along 3 planes perpendicular to the body axes and containing the origin: a) Ground truth; b) Solution from noise-free data; c) Solution from data with 1% noise Acknowledgements This work was financially supported by the French community of Belgium within the framework of a FRIA grant, and by the Belgian PRODEX program, managed by the European Space Agency in collaboration with the Belgian Federal Science Policy Office. References [1] V. Isakov. Inverse Source Problems. American Mathematical Soc., 1990. [2] J. Giraud, M. Lindsay, and M. Jessell. Generalization of level-set inversion to an arbitrary number of geologic units in a regularized least-squares framework. GEOPHYSICS, 86(4):R623–R637, July 2021. [3] S. Le Maistre, A. Rivoldini, and P. Rosenblatt. Signature of Phobos’ interior structure in its gravity field and libration. Icarus, 321:272–290, March 2019. [4] A. Aghasi, M. Kilmer, and E. L. Miller. Parametric Level Set Methods for Inverse Problems. SIAM Journal on Imaging Sciences, 4(2):618–650, January 2011. [5] L.-I. Sorsa, M. Takala, P. Bambach, et al. Tomographic inversion of gravity gradient field for a synthetic Itokawa model. Icarus, 336:113425, January 2020.

In order to describe the orientation of the spin axis of Mars, the IAU (International Astronomical Union) Working Group on Cartographic Coordinates and Rotational Elements (WGCCRE) uses the right ascension and declination angles, the equatorial coordinates orienting the planet with respect to the Earth equator. This solution is useful to make cartographic products.Differently from IAU, the radioscience community commonly uses Euler angles, with obliquity and node’s longitude defined with respect to the planet mean orbit at epoch. In both sets of angles, a third angle, the rotation angle, is used to position the prime meridian. Its trend is the diurnal rotation. The most usual way to transform Euler angles into IAU angles is to numerically evaluate the IAU angles over a given time interval with the help of spherical geometry, then to perform a frequency analysis on the so-obtained time series (e.g. Jacobson, 2010, Kuchynka et al., 2014 and Jacobson et al., 2018 for Mars). Unfortunately, such a method does not take into account the physical meaning of the planet’s rotational dynamics, which relies on well-known periodicities governed by the celestial mechanics. In the present study, we provide analytical expressions to precisely transform a set of angles in the other in the case of Mars. The targeted precision of the transformation is smaller than 0.1 mas for each angle on an interval of about 30 years before and after J2000. Each angle is modeled by the sum of a quadratic polynomial, a periodic series (nutation or rotation variations) and a Poisson series (a periodic series multiplied by the time). The analytical precise transformation shows that even when the Euler angles do not include quadratic terms (as commonly assumed by the radioscience community), the IAU angles do. As a result, since no quadratic term is considered in the IAU-like solutions as inferred by Jacobson (2010), Kuchynka et al. (2014), and Jacobson et al. (2018), an artificial very long period signal with well-chosen values for the amplitude, phase and frequency is needed to mimic the quadratic behavior on an interval of a few tens of years around J2000. Adding a long period modulation instead of a quadratic term largely and artificially alters the angle values at J2000 as well as their rates. For example, the correction was as high as 9° for the initial value of the angle or 100 mas/year for the rate. This work is done in the frame of the analysis of two 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.

Introduction The orientation of Mars with respect to its orbit can be described with a set of Euler angles (longitude ψ, obliquity ε, and rotation φ). 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. Results 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: ψgeodetic(t) = 6.754 mas/yr t + 0.565 mas sin l + 0.039 mas sin 2l + 0.004 mas sin 3l, 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’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. Another relativistic correction [φ]GR(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) : [φ]GR(t) = - 175.80 sin l - 8.20 sin 2l - 0.60 sin 3l. 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 & Francou 1988). We obtain a solution of the form: [φ]GR(t) = φ'GR t + Σj φj sin (fj t + ξj), with φj the amplitudes, fj the angular frequencies, and ξj 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 φ'GR which adds to the local sidereal rotation rate to produce the sidereal BCRS rotation rate (φ'=φ'local+φ'GR). Discussion 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). 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). Acknowledgements 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. References Baland, R.-M., Yseboodt, M., Le Maistre, S., et al. 2020, Celestial Mechanics and Dynamical Astronomy, 132, 47 Barker, B. M. & O’Connell, R. F. 1970, Phys. Rev. D, 2, 1428 Bretagnon, P. & Francou, G. 1988, A&A, 202, 309 Fukushima, T. 1991, A&A, 244, L11 Konopliv, A. S., Yoder, C. F., Standish, E. M., Yuan, D.-N., & Sjogren, W. L. 2006, Icarus, 182, 23 Le Maistre, S., Rivoldini, A., Caldiero, A., et al. 2022, in review Park, R. S., Folkner, W. M., Williams, J. G., & Boggs, D. H. 2021, AJ, 161, 105 Poisson, E. & Will, C. M. 2014, Gravit Yoder, C. F. & Standish, E. M. 1997, J. Geophys. Res., 102, 4065

The ideal conditions for RS measurements require an optimization of the communications and tracking systems, the spacecraft trajectories and mission operations for RS investigation. However, these ideal conditions are rarely achieved because of the several trades off that should be made with many other mission and science requirements. This is particularly true for missions in the Neptune system where the identified scientific goals are broad and varied. The exploration of Triton has been always identified as a key science goal for such a mission. In fact, Neptune’s largest moon Triton is presumably one of the ocean worlds in the outer solar system. With its own atmosphere, active geology, plumes of nitrogen gas and dust entrained deposited up to 150 km downwind, Triton intrigues planetary scientists, who proclaimed it as key ‘ocean world’ for future exploration [1]. According to space agencies’ long term plans (e.g. the US decadal surveys in planetary science, the ESA Cosmic Vision 2 and Voyage 2050 programs), Triton will certainly be visited by the end of the 21st century as also recommended by the radio-science (RS) community which declared Triton’s liquid layer investigation a “key RS goal” [2]. As a result of the general enthusiasm for Triton and similar icy moons of the outer solar system, several mission concepts have recently been proposed, all within the 2029–2035 time frame (e.g. [3-8]). According to the mission designs proposed in literature, most of the time Neptune is identified as the main target, but fly-bys of Triton will provide in-situ measurements whether there will be just one as Voyager 2 did and the TRIDENT team proposed [9], or multiple fly-bys by leveraging particular Neptune centric orbit (periodic and quasi-synchronous orbits) as Cassini did in the Saturn system or the as the “Triton tour” proposed by the Neptune Odyssey mission concept [10]. We focus on gravity science, i.e. the study of gravitational field with the purpose of inferring planetary properties such as bulk mass, mass/density distribution, internal density gradients, and interior structures and composition. In fact, radio tracking observations of changes in the velocity vectors of a spacecraft as it flies near Triton can be used to define the gravitational field. We perform a sensitivity study in order to asses the expected precision of the gravity recovery of a combination of different onboard instrumentation, trajectory geometries and radio science experiment configurations. In particular, we perform a covariance analysis in order to determine how the precision in the estimated parameter depend upon the various characteristics of an RS campaign, including: Orbit geometry Measurement accuracy/data quality (Doppler noise) Nongravitational forces Arc length Mission duration As an example of the proposed discussion, in Fig.1 we investigated the impact of different orbit geometries: we perform a covariance analysis on several fly-by geometries (see Fig.2), simulating Doppler tracking data performed with an onboard coherent X-band transponder and an High Gain Antenna (accuracy of 0.1mm/s at 60s integration time) over an 8h long arc. We modeled the gravity field the as a spherical harmonic expansion characterized by the Cl,m, Sl,m coefficients [11]. Since the spherical harmonics of Triton have never been measured, we used the Kaula power law in order to have an estimation of the power spectrum of the gravity field. Given the Doppler geometry configuration and relative position between Triton and the Earth in January 2030, we see that the C2,1 and S2,1 will be better determined if the spacecraft has an highly inclined orbit, whereas the estimation of C2,2 and S2,2 has a low uncertainty with slightly inclined orbits. Similar sensitivity analyses will be discussed over a wide range of mission scenarios, that is with different values for the aforementioned RS settings. The end goal of this study is to provide not only the optimal combination of these settings, but a complete parametric study that will help the design of a future mission to Triton. Acknowledgments This work was financially supported by the French Community of Belgium within the framework of the financing of a FRIA grant. References [1] Hendrix, A.R. et al., jan 2019. doi:10.1089/ast.2018.1955. [2] Asmar, S. et al., mar 2021. doi:10.3847/25c2cfeb.9d29ef85. [3] Arridge, C. et al. ,104:122–140, dec 2014. doi:10.1016/j.pss.2014.08.009. [4] Hofstadter, M. et al., Planetary and Space Science, 177:104680, nov 2019. doi:10.1016/j.pss.2019.06.004. [5] Masters, A. et al., Planetary and Space Science, 104:108–121, dec 2014. doi:10.1016/j.pss.2014.05.008. [6] Simon, A.A., Stern, S.A. and Hofstadter, July 2018. [7] Simon, A.A. et al., feb 2020. doi:10.1007/s11214-020-0639-1. [8] Turrini, D. et al., Planetary and Space Science, 104:93–107, dec 2014. doi:10.1016/j.pss.2014.09.005. [9] Prockter, L.M. et al., In Lunar and Planetary Science Conference, Lunar and Planetary Science Conference, page 3188, Mar. 2019. [10] Rymer, A.M. et al., sep 2021. doi:10.3847/psj/abf654. [11] Kaula, W.M. Theory of satellite geodesy. Applications of satellites to geodesy. 1966.

The Asteroid Impact and Deflection Assessment (AIDA) is an international collaboration supported by ESA and NASA to assess the feasibility of the kinetic impactor technique to deflect an asteroid, combining data obtained from NASA’s DART and ESA’s Hera missions. Together the missions represent the first humankind’s investigations of a planetary defense technique. In 2022, DART will impact Dimorphos, the secondary of the binary near-Earth asteroid (65803) Didymos. After 4 years, Hera will follow-up with a detailed post-impact survey of Didymos, to fully characterize and validate this planetary defense technique. In addition, Hera will deploy two CubeSats around Didymos once the Early Characterization Phase has completed, to augment the observations of the mother spacecraft. Juventas, the first Cubesat, will complete a low-frequency radar survey of the secondary, to unveil its interior. Milani, the second Cubesat, will perform a global mapping of Didymos and Dimorphos, with a focus on their compositional difference and their surface properties. One of the main objectives of Hera is to determine the binary system’s mass, gravity field, and dynamical state using radio tracking data in combination with imaging data. The gravity science experiment includes classical ground-based radiometric measurements between Hera and ground stations on Earth by means of a standard two-way X-band link, onboard images of Didymos, and spacecraft-to-spacecraft inter-satellite (ISL) radiometric tracking between Hera and the Cubesats. The satellite-to-satellite link is a crucial add-on to the gravity estimation of low-gravity bodies by exploiting the Cubesats’ proximity to the binary, as the range-rate measurements carried out by the inter-satellite link contain information on the dynamics of the system, i.e., masses and gravity field of Didymos primary and secondary.We will describe the updated mission scenario for the Hera radio science experiment to be jointly carried out by the three mission elements, i.e., Hera, Juventas and Milani. To conclude, our updated analysis and latest results, as well as the achievable accuracy for the estimation of the mass and gravity field of Didymos and Dimorphos, are presented.

We present here a method for the retrieval of the internal density distribution of a small body, via a least squares inversion of its gravity field and rotational state. Multiple solutions are averaged in order to limit the effect of the a priori density distribution. Simulations show successful density map estimations even with little to no initial information on the interior structure of the body. 1. Introduction The gravity field of a body is a property among the most readily obtainable by a spacecraft mission. Be it by employing radio-tracking, optical, or gravimetric data, several missions have provided and will provide gravity expansions of degree 2 and higher for small bodies. In turn, the extended gravity field is an effect of the distribution of the mass inside the body. However, the inverse problem of estimating the density distribution from the gravity field is not a trivial one, and generally with non-unique solutions [1]. Here, we propose a method to obtain a unique solution for the internal mass distribution of a small body by restricting our estimation to zones of constant density, which are limited in number by the accuracy of the gravity model [2]. Where available, additional observables such as the libration amplitudes could help in the estimation of the interior structure. Averaging of the solutions obtained for different subdivisions of zones provides then the estimated density map [3]. 2. Methods The code builds from available software which allows to compute the gravity harmonics of a body approximated by a collection of cubes, each with a given density [4]. Here, the cubes are grouped in subregions of constant density, as many as there are measured coefficients. Each subregion is assigned an initial density, from which the software computes gravity coefficients matching in degree and order those observed. The difference between the coefficients thus computed and the observed ones is then minimized in a least-squares sense [5]. The solution of the regularized least-squares inversion is the density for each of the initial zones, along with its uncertainty. However, this solution is heavily dependent on the initial subdivision of the body into zones. Hence, a wide range of possible initial zones is explored, and the corresponding density solutions averaged together. From such an average solution, a new set of subregions is generated, with the help of a blob detection algorithm [6]. Finally, the gravity coefficients are inverted using this new zone subdivision. This step is necessary because, while providing a good distribution of the density zones, the density values of the averaged map are biased by solutions coming from inaccurate initial zones subdivisions. The data used in the inversion are the spherical harmonics coefficients of the gravity field expansion. The advantage of these observables is that they can be measured without the need for specific payload. Although the method can be easily adapted to process surface gravity measurements, the larger errors on the surface gravity coming from the cubes discretization make it less appealing for these type of data. 3. Discussion and outlook Figure 1 shows the results obtained when applying the method to a fictitious density distribution for asteroid Bennu. The nominal density distribution, shown in Figure 1a, was used to produce a synthetic gravity field expansion of degree 11. An optimistic noise profile was assumed for these simulated observables, with a 1% error on each gravity coefficient. The spherical harmonics coefficients were then inverted via the iterative process described in Section 2, resulting in the density distribution shown in Figure 1b. With a gravity field of this resolution, the inversion method is clearly able to distinguish the 4 density anomalies inside the body. Still, the comparison between the nominal and the estimated density models reveals the current limitations of the method in accurately reconstructing the size of the anomalies or their density — which is equivalent, given the constraint on the total mass. Once a better agreement between the retrieved and the nominal models will be achieved with the current (high) resolution of the gravity harmonics, the synthetic gravity fields will be truncated at lower degrees, making them more realistic as outputs of spacecraft missions to small bodies. Future simulation studies will also make use of more realistic noise profiles, with the errors on the observed coefficient increasing as a function of the degree. (a) (b) Figure 1: (a) density model used to simulate a degree-11 gravity field; (b) density distribution retrieved from said gravity field, without any initial assumptions about the true distribution Acknowledgements This work was financially supported by the French community of Belgium within the framework of a FRIA grant, and by the Belgian PRODEX program, managed by the European Space Agency in collaboration with the Belgian Federal Science Policy Office. References [1] P. Tricarico. Global gravity inversion of bodies with arbitrary shape. Geophysical Journal International, 195(1):260–275, 2013.[2] Y. Takahashi and D. J. Scheeres. Morphology driven density distribution estimation for small bodies. Icarus, 233:179–193, 2014.[3] L.-I. Sorsa, M. Takala, P. Bambach, et al. Tomographic inversion of gravity gradient field for a synthetic Itokawa model. Icarus, 336:113425, 2020.[4] S. Le Maistre, A. Rivoldini, and P. Rosenblatt. Signature of Phobos’ interior structure in its gravity field and libration. Icarus, 321:272–290, 2019.[5] D. J. Scheeres, B. Khushalani, and R. A. Werner. Estimating asteroid density distributions from shape and gravity information. Planetary and Space Science, 48(10):965–971, 2000.[6] S. v. d. Walt, J. L. Schönberger, J. Nunez-Iglesias, et al. scikit-image: image processing in Python. PeerJ, 2:e453, 2014.

The nutations of Mars are about to be estimated with unprecedented accuracy (a few milliarcseconds) with the radioscience experiments RISE (Rotation and Interior Structure Experiment, Folkner et al. 2018) and LaRa (Lander Radioscience, Dehant et al. 2020) of the InSight and ExoMars 2020 missions, allowing to detect the contributions due to the liquid core and tidal deformations and to constrain the interior of Mars.To properly identify the non-rigid contribution, an accurate precession and nutation model for a rigidly behaving Mars is needed. We develop such a model, based on the Torque approach, and include the forcings by the Sun, Phobos, Deimos, and the other planets of the Solar System, as well as geodetic precession and nutations. Both semi-analytical developments (for the Solar and planetary torques) and analytical solutions (for Phobos and Deimos torques and the geodetic precession and nutations) are considered.We identify 43 nutation terms with an amplitude above the chosen truncation criterion of 0.025 milliarcseconds in prograde and/or retrograde nutations. Uncertainties related to modelling choices are negligible in comparison to the uncertainty coming from the observational uncertainty on the current determination of the precession rate of Mars (7608.3+/-pm2.1 mas/yr, Konopliv et al. 2016). Our model predicts a dynamical flattening HD=(C-A)/C=0.00538017+/-0.00000148 and a normalized polar moment of inertia C/MR2=0.36367+/-0.00010 for Mars.References:Folkner et al., 2018. doi: 10.1007/s11214-018-0530-5. Dehant et al., 2020. doi: 10.1016/j.pss.2019.104776.Konopliv et al., 2016. doi: 10.1016/j.icarus.2016.02.052.

The Lander Radioscience (LaRa) experiment on the ESA-Roscosmos ExoMars 2020 mission is designed to obtain coherent two-way Doppler measurements from the radio link between a lander on Mars and the Earth over at least one Martian year. The Doppler measurements will be used to determine the orientation and rotation of Mars in space (precession, nutations, and length-of-day variations). LaRa, on another location on the Martian surface with respect to the InSight mission could allow to observe the polar motion of Mars, in addition to further increase the accuracy on precession, nutation, and length-of-day measurements. The ultimate objective of LaRa is to obtain information on the Martian interior and about the sublimation/condensation cycle of atmospheric CO2. Concerning the nutations, a knowledge of the rigid body nutation can be computed and shall be used to constrain the interior properties of Mars.

The NASA InSight mission is operating from the surface of Mars for more than a year. RISE (for Rotation and Interior Structure Experiment) is one of the scientific payloads of InSight. This radio-science experiment consisting in an X-band transponder and two horn-antennas enabling two-way coherent radio-link between Mars and the Earth [Folkner et al., 2018]. The main goal of RISE is to measure the slight modulations of the nutational motion of the spin axis of Mars induced by the liquid core of the planet in order to constrain its interior structure and core properties. To increase our chance to achieve this challenging goal, we must calibrate the RISE Doppler data by accounting to 2nd order effects like the Mars atmospheric noise. This study shows the predicted contribution of the Martian ionosphere to the RISE data collected so far. To do so, we use a new empirical model of the Mars’ ionosphere called MoMo [Bergeot et al. 2019]. This model is based on the large database of Total Electron Content (TEC) derived from the subsurface mode of the Mars Express MARSIS radar. The model provides the vertical TEC as a function of solar zenith angle, solar activity, solar longitude and the location. Using MoMo, we produce vTEC maps for Mars that are then used to estimate the slant TEC in the Earth line of sight, enabling to infer the phase delay and Doppler shift affecting the RISE X-band measurements. These computed effects are shown to be of the order of 10-3mm.s-1 in Doppler observables, with a larger effect around sunrise and sunset. This is about one order of magnitude below the typical measurement noise of RISE, but it is comparable to the contribution of the liquid core in the Doppler (~10-3-10-2mm.s-1).The MoMo model is suitable for any Mars radio-science data calibration, and in particular the forthcoming ExoMars 2020 LaRa measurements [Dehant et al. 2019]. The predictions made with MoMo will be of great use either for the data corrections or to define the timing of observations in order to avoid operating when the TEC rapidly varies (i.e. close to sunrise and sunset). The model output is further discussed here in terms of climatologic behavior of the Mars’ ionosphere. For comparison, we also investigate the contribution of the Earth ionosphere using Global Ionospheric Maps (GIMs) based on GNSS data.

This paper is a case study providing some insights on what improvement could be achieved on the Mars Orientation and rotation Parameters (MOP) determination using radio tracking data from a moving rover. Thanks to high-performance mobility systems onboard new generation of rovers like ExoMars 2020, the position of the rover can be precisely known with respect to its previous position. This characteristic, together with the long life of the rovers and their steerable high-gain-antenna communication system, is shown here to provide an unexpected opportunity to improve the MOP determination. This paper presents the results of numerical simulations involving radio-science experiments between the moving rover and the Earth ground stations as well as between the rover and an orbiting spacecraft. The benefits of combining both links (direct-to-Earth and rover-orbiter) for the MOP determination is also assessed. The impacts of the spacecraft position accuracy as well as the frequency band used to communicate with it are quantified. It is shown that, after one Martian year of operation, the polar motion could be determined with 5 milliarcsecond (mas) of precision (formal error) from the rover-orbiter Doppler link, while it cannot be determined with usual equatorial lander-to-Earth radio link. This would allow for the first time the direct detection of the Chandler wobble amplitude in the polar motion of Mars, which is an important quantity to constrain the planet interior and atmospheric models. Although the moving rover Doppler data alone barely improve the current precision on the other MOP (like the length-of-day and nutation), a combination of those together with historical and future lander data would definitely help to fill gaps in the MOP signal and to decorrelate between the estimated parameters, thereby reducing the uncertainties in their determination.

The nutations of Mars are about to be estimated to a few milliarcseconds accuracy with the radioscience experiments onboard InSight and ExoMars 2022. The contribution to the nutations due to the liquid core and tidal deformations will be detected, allowing to constrain the interior of Mars. To avoid introducing systematic errors in the determination of the core properties, an accurate precession and nutation model for a rigidly behaving Mars is needed. Here, we develop such a model with adequate accuracy based on the Torque approach and compare it to previous models. We include in the model the forcings by the Sun, Phobos, Deimos, and the other planets of the solar system. We also include the geodetic precession and nutations. We use semi-analytical developments for the solar and planetary torques, and analytical solutions for the effect of Phobos and Deimos and for the geodetic precession and nutations. With a truncation criterion of 0.025 milliarcseconds in prograde and/or retrograde amplitude, we identify 43 nutation terms. The uncertainty on our solution mainly derives from the observational uncertainty on the current determination of the precession rate of Mars. Uncertainties related to our modeling choices are negligible in comparison. Given the current determination of the precession rate ( $$7608.3\pm 2.1$$ mas/yr, Konopliv et al. in Icarus 274:253–260, 2016. https://doi.org/10.1016/j.icarus.2016.02.052 ), our model predicts a dynamical flattening $$H_{{D}}=0.00538017\pm 0.00000148$$ and a normalized polar moment of inertia $$C/\mathrm{MR}^2=0.36367\pm 0.00010$$ for Mars.

Chandler Wobble of Mars From Orbiting Spacecraft and interior of Mars.

The Interior Exploration using Seismic Investigations, Geodesy, and Heat Transport (InSight) mission has provided an opportunity to improve our knowledge of Mars’ interior via the Rotation and Interior Structure Experiment (RISE). RISE provides information on the rotation of Mars by measuring the Doppler shift of radio transmissions from InSight to NASA’s Deep Space Network (DSN) on Earth. Through the combination of one year of Doppler data from InSight with previous data from the Viking-1 lander, Mars Pathfinder, and Opportunity missions, data spanning 43 years from 1976 to 2019 are used to estimate the Mars precession rate as −7605 ​± ​3 milliarcseconds per year. This result is consistent with the precession rate estimated from Doppler tracking of previous landers and orbiters alone.

The Mars orientation parameters (MOPs) are important information for constraining Martian interior properties, ice caps sublimation, and condensation process. Considering the future Mars landing mission, we performed numerical simulations of MOP determination using direct-to-Earth radio signals. The results suggest that after 200 d of tracking using high-quality levels of both two-way Doppler and two-way range measurements, the MOP uncertainties will be significantly reduced. Then, after about 800 mission days the accuracy of precession parameters will be improved by 5–10 times compared with the present. Also, the accuracy of nutation parameters will reach 10–30 mas, and the accuracy of length-of-day and Chandler wobble parameters will reach 5–10 mas, which meets the need to investigate the internal structure of Mars and the dynamics of the atmosphere. By analyzing the solution results of different latitude landers, two-way range data can provide a good complement to two-way Doppler data. Therefore, it is necessary to perform a two-way range measurement for a high-latitude lander. In addition, the study has also found the limit of the ephemeris error of Mars for MOP determination. When using the two-way Doppler data, the ephemeris error should not be more than 75 m; and for the two-way range measurement, the error should be less than 0.15 m.

Aims: Several scientific landers and rovers have reached the Martian surface since the 1970s. Communication between the asset (i.e., lander or rover) and Mars orbiters or Earth antennas uses radio signals in UHF to X-band frequencies passing through the Mars’ ionosphere. It is consequently necessary to take into account electron density variation in the Mars’ ionosphere to correct the refraction of the signal transmitted. Methods: We developed a new empirical model of the Mars’ ionosphere called MoMo. It is based on the large database of Total Electron Content (TEC) derived from the subsurface mode of the Mars Express MARSIS radar. The model provides vertical TEC as a function of solar zenith angle, solar activity, solar longitude and location. For validation, the model is compared with Mars Express radio occultation data as well as with the numerical model IPIM (IRAP Plasmasphere-Ionosphere Model). Results: We discussed the output of the model in terms of climatology behaviour of the Mars’ ionosphere. The output of MoMo is then uses to quantify the impact of the Martian ionosphere for radio-science experiments. From our results, the effect is of the order of 10−3 mm s−1 in Doppler observables especially around sunrise and sunset. Consequently, this new model could be used to support the data analysis of any radio-science experiment and especially for present InSight RISE and futur ExoMars LARA instruments aiming at better understand the deep-interior of Mars.

LaRa (Lander Radioscience) is an experiment on the ExoMars 2020 mission that uses the Doppler shift on the radio link due to the motion of the ExoMars platform tied to the surface of Mars with respect to the Earth ground stations (e.g. the deep space network stations of NASA), in order to precisely measure the relative velocity of the lander on Mars with respect to the Earth. The LaRa measurements shall improve the understanding of the structure and processes in the deep interior of Mars by obtaining the rotation and orientation of Mars with a better precision compared to the previous missions. In this paper, we provide the analysis done until now for the best realization of these objectives. We explain the geophysical observation that will be reached with LaRa (Length-of-day variations, precession, nutation, and possibly polar motion). We develop the experiment set up, which includes the ground stations on Earth (so-called ground segment). We describe the instrument, i.e. the transponder and its three antennas. We further detail the link budget and the expected noise level that will be reached. Finally, we detail the expected results, which encompasses the explanation of how we shall determine Mars' orientation parameters, and the way we shall deduce Mars' interior structure and Mars' atmosphere from them. Lastly, we explain briefly how we will be able to determine the Surface platform position., Comment: 43 pages, 39 figures, accepted in Planetary and Space Science

The Rotation and Interior Structure Experiment (RISE) on-board the InSight mission will use the lander’s X-band (8 GHz) radio system in combination with tracking stations of the NASA Deep Space Network (DSN) to determine the rotation of Mars. RISE will measure the nutation of the Martian spin axis, detecting for the first time the effect of the liquid core of Mars and providing in turn new constraints on the core radius and density. RISE will also measure changes in the rotation rate of Mars on seasonal time-scales thereby constraining the atmospheric angular momentum budget. Finally, RISE will provide a superb tie between the cartographic and inertial reference frames. This paper describes the RISE scientific objectives and measurements, and provides the expected results of the experiment.

The Opportunity Mars Exploration Rover remained stationary between January and May 2012 in order to conserve solar energy for running its survival heaters during martian winter. While stationary, extra Doppler tracking was performed in order to allow an improved estimate of the martian precession rate. In this study, we determine Mars rotation by combining the new Opportunity tracking data with historic tracking data from the Viking and Pathfinder landers and tracking data from Mars orbiters (Mars Global Surveyor, Mars Odyssey and Mars Reconnaissance Orbiter). The estimated rotation parameters are stable in cross-validation tests and compare well with previously published values. In particular, the Mars precession rate is estimated to be −7606.1 ± 3.5 mas/yr. A representation of Mars rotation as a series expansion based on the determined rotation parameters is provided.

Mars' Orientation and rotation Parameters (called here MOP): precession, nutation, polar motion and length of day (LOD) variations, are related to the interior of the planet as well as to the dynamics of its atmosphere. The MOP can be determined using the Doppler shift on radio signal data due to the motion of a probe landed on Mars relative to tracking stations on Earth. In this paper we perform numerical simulations for assessing the precision on the determination of the MOP using Direct-To-Earth (DTE) X-band Doppler measurements for a nearly equatorial lander. We then discuss how a better knowledge of the MOP could improve our understanding on the interior structure. The sensitivity of such a DTE radio link to these rotation parameters is first investigated. This shows that the latitude of the landing site must be higher than 20° to detect the Chandler Wobble component of the polar motion in DTE Doppler data and must be at least 40° to get tight constraints on it. It is found that the precision in the determination of the seasonal LOD variations will be significantly improved after about 350 days of operation, reaching the 5% level after 550 days, thereby better constraining the CO 2 mass budget in the Martian atmosphere and ice caps. The current precision in the precession rate (25 milliarcsecond (mas) per year) will be matched after 150 days of mission. An uncertainty of less than 5 mas/year will be reached after 700 days, improving the precision on the polar moment of inertia by a factor of five. The precision on the determination of the amplitudes of nutation is estimated at a few mas after one Martian year of mission (about 12 mas on the prograde/retrograde semi and terannual amplitudes) allowing for the detection of the contribution expected from the liquid core. Considering Mars with a liquid core in accordance with recent geodesic measurements, the Free-Core-Nutation (FCN) period is estimated with a precision of less than 10 days after 550 days of mission in case of an actual FCN period equal to −240 days and with a precision of 1 day after 300 days of observation for an FCN period at −230 days, which is close to the forcing terannual period of 228.9 days (resonance). It is shown that only 100 days of mission duration are needed to detect the liquid core contribution to the nutation near the resonance from the estimates of the amplitudes of nutation. Finally, the precisions on core physical parameters are inferred from the FCN period estimate. We show that a precision of 30% for the core moment of inertia and 5% for the dynamical flattening can be obtained after one Martian year of mission duration.

The paper presents the concept, the objectives, the approach used, and the expected performances and accuracies of a radioscience experiment based on a radio link between the Earth and the surface of Mars. This experiment involves radioscience equipment installed on a lander at the surface of Mars. The experiment with the generic name lander radioscience (LaRa) consists of an X-band transponder that has been designed to obtain, over at least one Martian year, two-way Doppler measurements from the radio link between the ExoMars lander and the Earth (ExoMars is an ESA mission to Mars due to launch in 2013). These Doppler measurements will be used to obtain Mars’ orientation in space and rotation (precession and nutations, and length-of-day variations). More specifically, the relative position of the lander on the surface of Mars with respect to the Earth ground stations allows reconstructing Mars’ time varying orientation and rotation in space. Precession will be determined with an accuracy better by a factor of 4 (better than the 0.1% level) with respect to the present-day accuracy after only a few months at the Martian surface. This precession determination will, in turn, improve the determination of the moment of inertia of the whole planet (mantle plus core) and the radius of the core: for a specific interior composition or even for a range of possible compositions, the core radius is expected to be determined with a precision decreasing to a few tens of kilometers. A fairly precise measurement of variations in the orientation of Mars’ spin axis will enable, in addition to the determination of the moment of inertia of the core, an even better determination of the size of the core via the core resonance in the nutation amplitudes. When the core is liquid, the free core nutation (FCN) resonance induces a change in the nutation amplitudes, with respect to their values for a solid planet, at the percent level in the large semi-annual prograde nutation amplitude and even more (a few percent, a few tens of percent or more, depending on the FCN period) for the retrograde ter-annual nutation amplitude. The resonance amplification depends on the size, moment of inertia, and flattening of the core. For a large core, the amplification can be very large, ensuring the detection of the FCN, and determination of the core moment of inertia. The measurement of variations in Mars’ rotation also determines variations of the angular momentum due to seasonal mass transfer between the atmosphere and ice caps. Observations even for a short period of 180 days at the surface of Mars will decrease the uncertainty by a factor of two with respect to the present knowledge of these quantities (at the 10% level). The ultimate objectives of the proposed experiment are to obtain information on Mars’ interior and on the sublimation/condensation of CO2 in Mars’ atmosphere. Improved knowledge of the interior will help us to better understand the formation and evolution of Mars. Improved knowledge of the CO2 sublimation/condensation cycle will enable better understanding of the circulation and dynamics of Mars’ atmosphere.

Numerical simulations have been performed to assess the precision that can be obtained on Phobos libration angles by using two types of data acquired by a probe landed on the Martian moon: (1) Direct-To-Earth (DTE) Doppler data and (2) Star-Tracker (ST) data. Compared to independent estimates, combination of DTE and ST data provides the more precise estimates of Phobos libration angles at the 10 − 3 − 10 − 5 degree level. Short period libration amplitudes are functions of the relative moments of inertia. Thus their determination would provide constraints on mass distributions inside Phobos. We show that the longitudinal libration amplitude at the orbital period, while being clearly distinct from the resonance period, is the most interesting signal for that purpose, because it leads to the best precision on ( B − A ) / C ( 10 − 5 after only 10 weeks of operation). Nevertheless, we showed that inferring the individual moments of inertia A, B and C with a precision sufficient ( 1 % ) to distinguish the rotational behavior of an homogenous from that of an heterogenous body requires the determination of one of the degree-two gravity field coefficient at the percent level. In addition, we stress that ST and DTE data combination allows de-correlating librational motion from orbital motion since ST measurements are sensitive to the moon's rotation and do not depend on its ephemeris, whereas DTE Doppler data are sensitive to both motions as well as to Phobos' surface displacement due to the tides raised by Mars. Furthermore, the ephemeris of Phobos has to be known at the centimeter level to allow measuring the tidal surface displacement with enough precision to conclude on the nature of Phobos' interior (rubble pile versus monolithic) as one step toward constraining its origin and evolution.

International audience; In November 2018, for the first time a dedicated geophysical station, the InSight lander, will be deployed on the surface of Mars. Along with the two main geophysical packages, the Seismic Experiment for Interior Structure (SEIS) and the Heat-Flow and Physical Properties Package (HP3), the InSight lander holds a highly sensitive pressure sensor (PS) and the Temperature and Winds for InSight (TWINS) instrument, both of which (along with the InSight FluxGate (IFG) Magnetometer) form the Auxiliary Sensor Payload Suite (APSS). Associated with the RADiometer (RAD) instrument which will measure the surface brightness temperature, and the Instrument Deployment Camera (IDC) which will be used to quantify atmospheric opacity, this will make InSight capable to act as a meteorological station at the surface of Mars. While probing the internal structure of Mars is the primary scientific goal of the mission, atmospheric science remains a key science objective for InSight. InSight has the potential to provide a more continuous and higher-frequency record of pressure, air temperature and winds at the surface of Mars than previous in situ missions. In the paper, key results from multiscale meteorological modeling, from Global Climate Models to Large-Eddy Simulations, are described as a reference for future studies based on the InSight measurements during operations. We summarize the capabilities of InSight for atmospheric observations, from profiling during Entry, Descent and Landing to surface measurements (pressure, temperature, winds, angular momentum), and the plans for how InSight’s sensors will be used during operations, as well as possible synergies with orbital observations. In a dedicated section, we describe the seismic impact of atmospheric phenomena (from the point of view of both “noise” to be decorrelated from the seismic signal and “signal” to provide information on atmospheric processes). We discuss in this framework Planetary Boundary Layer turbulence, with a focus on convective vortices and dust devils, gravity waves (with idealized modeling), and large-scale circulations. Our paper also presents possible new, exploratory, studies with the InSight instrumentation: surface layer scaling and exploration of the Monin-Obukhov model, aeolian surface changes and saltation / lifing studies, and monitoring of secular pressure changes. The InSight mission will be instrumental in broadening the knowledge of the Martian atmosphere, with a unique set of measurements from the surface of Mars.