TP13
Session assets
Introduction Tidal observations have provided crucial insight into the interior of various Solar System bodies; including Mercury (1), Venus (2), the Moon (3, 4), Mars (5, 6), Jupiter (7), Io (8) and Titan (9, 10). Previous efforts to characterize planetary interior structures using tides have focused on inferring one-dimensional (i.e., radial) structure. However, lateral variations in properties can induce patterns of tidal deformation that are distinct from those produced by radial structure, thus allowing for a 3D characterization of the interior through future detailed observations of time-varying gravity or shape.
In a spherically symmetric body, the tidal response mirrors the wavelength of the tidal forcing. However, lateral heterogeneities cause mode coupling (i.e., interaction between forcing and response at mutually distinct harmonics). This wavelength-dependent variation in tidal deformation forms the basis of tidal tomography — a technique that seeks to constrain internal 3D structure by measuring the body's tidal response across different wavelengths (11–14).
Here, we present the first application of tidal tomography beyond Earth, explore its potential use in ongoing missions, and advocate for its broader consideration in future planetary exploration efforts.
First Uses of Tidal Tomography Beyond Earth Due to the high precision required to detect signals associated with 3D structure, tidal tomography had been applied only to the Earth (15). But, recently, a reanalysis of GRAIL Lunar gravity data has revealed a nearside/farside dichotomy in the Moon’s mantle from its unexpectedly large-amplitude degree three response, suggesting a ~100K thermal anomaly (16). While this constitutes the first application of this technique to an extraterrestrial body, more examples might soon follow. Juice’s 3GM instrument will be precise enough to unveil variations of Ganymede’s outer ice shell of a few percent its mean shell thickness (17). The effects of 3D tides might also be recorded on the surface of planetary bodies. The distribution of Io’s volcanoes might be indicative of tidal dissipation in a body with an asthenosphere with a laterally varying melt distribution (18).
Future Applications of Tidal Tomography The first use of tidal tomography beyond Earth demonstrates the potential to map the three-dimensional interior structure of planetary bodies remotely. We argue that future missions should consider using tidal tomography to characterize 3D structures in planetary interiors. Enceladus is perhaps the clearest example. Static gravity and shape data hints at thickness variations in Enceladus’ crust (19, 20), which are expected to drive an exceptionally large-amplitude mode coupled tidal response for the body (12, 21, 22). Lateral variations in interior properties are the fingerprint of processes relevant to habitability (e.g., ocean circulation, the distribution of tidal heating). With the tiny Saturnian moon being one of the top priority planetary science targets for both ESA and NASA, tidal tomography should be considered in future Enceladan missions.
Figure 1: Schematic representation of the tidal response of the Moon to various tidal components for a 1D and a 3D Moon
References
1. S. Padovan, J. L. Margot, S. A. Hauck, W. B. Moore, S. C. Solomon, J Geophys Res Planets. 119, 850–866 (2014).
2. A. S. Konopliv, C. F. Yoder, Geophys Res Lett. 23, 1857–1860 (1996).
3. I. Matsuyama et al., Geophys Res Lett. 43, 8365–8375 (2016).
4. J. G. Williams et al., J Geophys Res Planets. 119, 1546–1578 (2014).
5. C. F. Yoder, A. S. Konopliv, D. N. Yuan, E. M. Standish, W. M. Folkner, Science (1979). 300, 299–303 (2003).
6. L. Pou et al., J Geophys Res Planets, in press, doi:https://doi.org/10.1029/2022JE007291.
7. D. Durante et al., Geophys Res Lett, in press, doi:https://doi.org/10.1029/2019GL086572.
8. R. S. Park et al., Nature. 638, 69–73 (2025).
9. D. Durante, D. J. Hemingway, P. Racioppa, L. Iess, D. J. Stevenson, Icarus. 326, 123–132 (2019).
10. S. Goossens, B. van Noort, A. Mateo, E. Mazarico, W. van der Wal, Nat Astron. 8, 846–855 (2024).
11. C. Qin, S. Zhong, J. Wahr, Geophys J Int. 207, 89–110 (2016).
12. M. Rovira-Navarro, I. Matsuyama, A. Berne, Planet Sci J. 5, 129 (2024).
1.3 S. Zhong, C. Qin, G. A, J. Wahr, Geophys Res Lett. 39 (2012).
14. L. Métivier, M. Greff-Lefftz, M. Diament, Geophys J Int. 167, 1060–1076 (2006).
15. H. C. P. Lau et al., Nature. 551, 321–326 (2017).
16. R. S. Park and A. Berne and A. S. Konopliv and J. T. Keane and I. Matsuyama and F. Nimmo and M. Rovira-Navarro and M. P. Panning and M. Simons and D. J. Stevenson and R. C. Weber, Nature (accepted)(2025).
17. M. Rovira-Navarro et al., Geophys Res Lett (2025).
18. A. Veenstra, M. Rovira-Navarro, W. van der Wal, T. Steinke, A. Davies, Res Sq (2025), doi:10.21203/rs.3.rs-5522583/v1.
19. D. J. Hemingway, T. Mittal, Icarus. 332, 111–131 (2019).
20. M. Beuthe, A. Rivoldini, A. Trinh, Geophys Res Lett. 43, 10, 10–88, 96 (2016).
21. A. Berne, M. Simons, J. T. Keane, R. S. Park, Geophys Res Lett. 50 (2023), doi:10.1029/2023GL106656.
22. M. Beuthe, Icarus. 302, 145–174 (2018).
How to cite: Rovira-Navarro, M., Berne, A., Park, R. S., Veenstra, A., Calliess, D., Dirkx, D., Fayolle, S., Matsutayama, I., Nimmo, F., Simons, M., and van der Wal, W.: Using Tidal Tomography to Unveil the 3D Structure of Moons, EPSC-DPS Joint Meeting 2025, Helsinki, Finland, 7–13 Sep 2025, EPSC-DPS2025-49, https://doi.org/10.5194/epsc-dps2025-49, 2025.
Five Venus missions are under development to study the planet in the next decade, with ESA's EnVision and NASA's VERITAS both including a geophysical investigation among their objectives[1]. Their radar and gravity experiments will observe Venus's orientation, enabling spin dynamics analyses to infer geophysical and atmospheric properties.
To support these future observations, we developed a polar motion model for Venus. Polar motion describes the motion of a planet's spin axis relative to its surface. It is distinct from precession-nutation, which describes the motion of the spin axis relative to the fixed celestial sphere. Both motions provide complementary constraints for interior models.
Our polar motion model accounts for solar torque, deformations of a viscoelastic mantle, and atmospheric dynamics. We revisited the expression for the free motion period: solar torque is the dominant phenomenon affecting Venus's Chandler period, accelerating the wobble by a factor of 2.75. Using available interior models[2], we predict a Chandler period in the range [12900 ; 18900] years (Fig.1 below)[3]. During EnVision's four-year primary mission, this manifests as a linear drift of about 90 meters (±20 m) of the spin pole on Venus's surface, comparable to the expected precession of 240 meters.
Therefore, when anticipating the upcoming orbiter missions, polar motion should be incorporated into Venus rotation models, alongside the precession and spin rate variations whose recovery has already been simulated[4,5]. A combined recovery of both the precession and Chandler periods could reveal the state and estimate the size of Venus’s core (Fig.2 below)[3].
Figure 1 - Chandler period of Venus computed for different interior models. Colored bars correspond to the density profiles from Shah et al. [2], featuring either a fully liquid core (green), a solid inner core with a liquid outer core (blue), or a fully solid core (orange). For each core state, the 99% contour of a kernel density estimate is shown. Black bars correspond to the density profiles from Dumoulin et al. [6], all featuring fully liquid cores.
Figure 2 - Chandler and precession periods of Venus computed for the density profiles of Shah et al. [2], featuring either a fully liquid core (green), a solid inner core with a liquid outer core (blue), or a fully solid core (orange). For each core state, the 99% contour of a kernel density estimate is shown. Black level lines indicate the core size of the models in kilometers, to within 50 km.
[1] Widemann et al. (2023), Space Science Reviews, doi:10.1007/s11214-023-00992-w
[2] Shah et al. (2022), The Astrophysical Journal, doi:10.3847/1538-4357/ac410d
[3] Phan and Rambaux (2025, under review)
[4] Rosenblatt et al. (2021), Remote Sensing, doi:10.3390/rs13091624
[5] Cascioli et al. (2021), Planetary Science Journal, doi:10.3847/psj/ac26c0
[6] Dumoulin et al. (2017), JGR Planets, doi:10.1002/2016je005249
How to cite: Phan, P.-L. and Rambaux, N.: Towards a comprehensive rotation model for Venus, EPSC-DPS Joint Meeting 2025, Helsinki, Finland, 7–13 Sep 2025, EPSC-DPS2025-1617, https://doi.org/10.5194/epsc-dps2025-1617, 2025.
Mercury is locked in a 3:2 spin-orbit resonance. Moreover, the combination of its non-spherical shape and eccentric orbit lead to reversing torques that produce small periodic changes in its spin rate, a motion referred to as forced (physical) libration. The annual (88-day) libration of Mercury has been determined independently based on observations from Earth-based radar (Margot et al., 2012), camera and/or laser altimeter (Stark et al., 2015; Bertone et al., 2021), and radio tracking (Mazarico et al., 2014; Genova et al., 2019; Konopliv et al., 2020). Unfortunately, the precision of such annual libration measurements does not allow the detection of a large solid inner core (Van Hoolst et al., 2012). Additional long-period forced librations due to the non-Keplerian characteristics of Mercury’s orbits could be amplified by resonances if their periods are close to that of a libration eigenperiod (Yseboodt et al., 2013). Constraints on the amplifications of these additional librations would inform us about the free libration periods, which can be related to the interior structure of Mercury. Currently, these long-period librations remain largely unknown.
We devise an alternative approach based on the self-registration of laser altimetric profiles (Xiao et al., 2022; Xiao et al., 2025a) to maximize their self-consistency, while precisely tracking temporal variations of the rotation angle of the prime meridian. This allows us to simultaneously measure the annual libration and the long-period librations. We also take advantage of the fact that mismodelings in the rotation angle and external error sources would shift the laser profiles in different ways. The orbit, pointing, and timing errors are typically slow-varying and thus lead to near-constant shifts. In contrast, offsets in the rotation angles would rotate the profiles around Mercury’s spin axis. We apply our approach to MESSENGER’s Mercury Laser Altimeter (MLA) data to constrain Mercury’s librations (Xiao et al., 2025b).
Figure 1. Rotation angle variation, with respect to Mercury’s resonant rotation, using IAU2015 model values as priors. Each dot represents the measurement from an individual MLA profile. Results using other a priori values feature similar trends.
Simulations using realistic synthetic profiles verify that there is negligible bias associated with the proposed approach. In terms of the actual profiles, we have experimented with various a priori rotation and orientation values, i.e., Stark2015, IAU2015 (Archinal et al., 2018), Genova2019, and Bertone2021. An example of the obtained temporal variation of the rotation angle is shown in Figure 1. In addition, Figure 2 shows the amplitudes of our measured mean rotation rate and annual libration. The obtained annual libration converges to 37.53±0.56 as, which is significantly smaller than the existing estimates and indicative for a slightly smaller outer core and a larger inner core. The amplified long-period libration is clearly visible in Figure 1 with its amplitude constrained to be 18.50±1.12 as. The captured long-period libration features a period of around 6 years, and is either induced by Venus (5.66 y), Jupiter (5.93 y), or the Earth (6.57 y).
For interpretation, we have performed libration modelling taking into account the gravitational coupling between layers (Dumberry et al., 2013) and different core compositions to investigate the insights these measurements provide into Mercury's interior (Rivoldini et al., this meeting; Yseboodt et al., this meeting). Preliminary models favor an inner core size larger than 1,000 km. These findings will critically enhance our understanding of Mercury’s thermal evolution and magnetic dynamo. Further improvements can be expected from data by the upcoming BepiColombo Laser Altimeter (BELA) onboard ESA/JAXA’s BepiColombo mission to Mercury (Thomas et al., 2021).
Figure 2. Rotation rates and annual libration amplitudes estimated from various a priori values, in comparison with existing estimates. The 3-σ errors for our weighted mean rotation rate and annual libration are 0.53×10-6 °/day and 0.56 as, respectively.
References
Archinal et al., 2018. Celest. Mech. Dyn. Astron., 130. https://doi.org/10.1007/s10569-017-9805-5.
Bertone et al., 2021. JGR: Planets, 126(4). https://doi.org/10.1029/2020JE006683.
Dumberry et al., 2013. Icarus, 225(1), 62-74. https://doi.org/10.1016/j.icarus.2013.03.001.
Genova et al., 2019. GRL, 46(7). https://doi.org/10.1029/2018GL081135.
Konopliv et al., 2020. Icarus, 335, 113386. https://doi.org/10.1016/j.icarus.2019.07.020.
Margot et al., 2012. JGR: Planets, 117(E12). https://doi.org/10.1029/2012JE00416.
Mazarico et al., 2014. JGR: Planets, 119(12). https://doi.org/10.1002/2014JE004675.
Stark et al., 2015. GRL, 42(19). https://doi.org/10.1002/2015GL065152.
Thomas et al., 2021. SSR, 217, 1-62. https://doi.org/10.1007/s11214-021-00794-y.
Van Hoolst et al., 2012. EPSL, 333. https://doi.org/10.1016/j.epsl.2012.04.014.
Xiao et al., 2022. PSS, 214, 105446. https://doi.org/10.1016/j.pss.2022.105446.
Xiao et al., 2025a. GRL, 52(7), e2024GL112266. https://doi.org/10.1029/2024GL112266.
Xiao et al., 2025b. Lunar and Planetary Science Conference. https://www.hou.usra.edu/meetings/lpsc2025/pdf/1980.pdf. E-poster available at https://lpsc2025.ipostersessions.com/Default.aspx?s=6F-4E-28-F7-3A-45-58-51-CB-5C-5D-57-69-39-F5-9B.
Yseboodt et al., 2013. Icarus, 226(1). https://doi.org/10.1016/j.icarus.2013.05.011.
How to cite: Xiao, H., Stark, A., Rivoldini, A., Yseboodt, M., Bertone, S., Baland, R.-M., Stenzel, O., Briaud, A., Hussmann, H., Lara, L., and Gutiérrez, P.: Mercury’s librations from self-registration of MESSENGER laser profiles, EPSC-DPS Joint Meeting 2025, Helsinki, Finland, 7–13 Sep 2025, EPSC-DPS2025-325, https://doi.org/10.5194/epsc-dps2025-325, 2025.
Like our Moon, Mercury is assumed to be driven by dissipative torques into an equilibrium Cassini state 1, where its spin axis, its orbit normal, and the Laplace pole remain coplanar during the precessional cycle. The orientation of Mercury’s spin axis has been determined independently based on observations from Earth-based radar (Margot et al., 2007, 2012), camera and/or laser altimeter (Stark et al., 2015; Bertone et al., 2021), and radio tracking (Mazarico et al., 2014; Verma and Margot, 2016; Genova et al., 2019; Konopliv et al., 2020). The obliquity, the separation angle between the spin axis and the orbital normal, is constrained to be approximately 2 arcmin and deviation of the spin axis from coplanarity is limited to a phase lag of around 10 arcsec.
If Mercury is indeed forced into an equilibrium Cassini state and the whole planet precesses as a rigid body (Dumberry, 2021), its obliquity can be used to infer its normalized polar moment of inertia which depends on Mercury’s radial mass distribution (e.g., Peale, 1981; Baland et al., 2017). In addition, a precise estimate of the angular offset of Mercury’s spin axis to the Cassini state can place a constraint on its tidal quality factor Q and thus on the bulk viscosity of its mantle (Baland et al., 2017; MacPherson and Dumberry, 2022).
In this study, we look into Mercury’s Cassini state by applying the co-registration techniques to the MESSENGER Mercury Laser Altimeter (MLA) profiles. First, we reprocess the profiles (Xiao et al., 2021) and self-register them to maximize their self-consistency by shifting the profiles in both lateral and radial directions (Xiao et al., 2022; Xiao et al., 2025). After that, we build a reference terrain model out of these self-registered profiles. Finally, we merge all original profiles and co-register them as a whole to the reference terrain model by solving for corrections to the mismodeling in the orientation angles. We have carried out some preliminary experiments at various zonal bands and with different a priori orientation angles (Figure 1). In these cases, the prime meridian angle is fixed and set to match our measured annual and long-period libratons (Xiao et al., this meeting). We can see that our estimates converge well and place Mercury within 2 arcsec (0.0006 degree) of the Cassini state although some yet-to-be-explained dependence on latitude can be observed.
We will further refine the proposed approach and carry out simulations using realistic synthetic profiles to validate our approach and quantify the uncertainty. Once the assessment of Mercury’s Cassini state is finalized, we will combine it with our estimated annual and long-term librations of Mercury (Xiao et al., this meeting), and its tidal Love number h2 (Xiao et al., 2025) to shed comprehensive insights into the planet’s interior structure and compositions (e.g., Dumberry and Rivoldini, 2015; Briaud et al., this meeting; Rivoldini et al., this meeting; Yseboodt et al., this meeting). This information can critically constrain the thermal evolution of Mercury and its magnetic dynamo.
Figure 1. Right ascension (RA) and declination (DEC) angles of Mercury’s spin axis at J2000.0 in the International Celestial Reference Frame (ICRF). Circles are the existing surface-based measurements, and squares correspond to the gravity-based solutions. The thick black reference line represents the classical Cassini state reconstructed from ephemerides data (Baland et al., 2017). Diamonds, triangles, and stars represent our measurements at [81°N, 84°N], [75°N, 84°N], and [60°N, 84°N], respectively. Colors denote the a priori orientation angles used that correspond to the existing estimates.
References
Baland et al., 2017. Icarus, 291, 136-159. https://doi.org/10.1016/j.icarus.2017.03.020.
Bertone et al., 2021. JGR: Planets, 126(4). https://doi.org/10.1029/2020JE006683.
Dumberry, 2021. JGR: Planets, 126(2), e2020JE006621. https://doi.org/10.1029/2020JE006621.
Genova et al., 2019. GRL, 46(7). https://doi.org/10.1029/2018GL081135.
Konopliv et al., 2020. Icarus, 335, 113386. https://doi.org/10.1016/j.icarus.2019.07.020.
MacPherson and Dumberry, 2022. JGR: Planets, 127(4), e2022JE007184. https://doi.org/10.1029/2022JE007184.
Margot et al., 2007. Science, 316(5825), 710-714. https://doi.org/10.1126/science.1140514.
Margot et al., 2012. JGR: Planets, 117(E12). https://doi.org/10.1029/2012JE00416.
Mazarico et al., 2014. JGR: Planets, 119(12). https://doi.org/10.1002/2014JE004675.
Peale, 1981. Icarus, 48(1), 143-145. https://doi.org/10.1016/0019-1035(81)90160-3.
Dumberry and Rivoldini, 2015. Icarus, 248, 254-268. https://doi.org/10.1016/j.icarus.2014.10.038.
Stark et al., 2015. GRL, 42(19). https://doi.org/10.1002/2015GL065152.
Verma and Margot, 2016. JGR: Planets, 121(9), 1627-1640. https://doi.org/10.1002/2016JE005037.
Xiao et al., 2021. JoG, 95(2), 22. https://doi.org/10.1007/s00190-020-01467-4.
Xiao et al., 2022. PSS, 214, 105446. https://doi.org/10.1016/j.pss.2022.105446.
Xiao et al., 2025. GRL, 52(7), e2024GL112266. https://doi.org/10.1029/2024GL112266.
How to cite: Xiao, H., Stark, A., Baland, R.-M., Dumberry, M., Rivoldini, A., Van Hoolst, T., Yseboodt, M., Briaud, A., Lara, L., and Gutiérrez, P.: Mercury’s Cassini state from co-registration of MESSENGER laser profiles, EPSC-DPS Joint Meeting 2025, Helsinki, Finland, 7–13 Sep 2025, EPSC-DPS2025-641, https://doi.org/10.5194/epsc-dps2025-641, 2025.
The TU Delft Astrodynamics Toolbox (Tudat) [1] is a free open-source software (FOSS) suite geared towards research and education in computational astrodynamics (https://docs.tudat.space/). The primary focus of its functionality is numerical propagation and estimation of the dynamics of objects in space, with applications to a broad range of past, current, and future planetary missions and data sets.
The uniqueness of Tudat and its powerful estimation module stems from its versatility and modularity, combined with its fully open-source nature. It handles diverse data sets (e.g. radio range and Doppler, optical astrometry, VLBI, radar), while offering high-fidelity models for both spacecraft and Solar System bodies dynamics alike, covering a wide range of different dynamical regimes. These features make the Tudat software suite equally suitable for estimating the dynamics (and related physical parameters) of spacecraft, small bodies (asteroids, minor moons), natural satellites, and planets.
Building upon a strong legacy in simulation (see https://docs.tudat.space/en/latest/_src_about/research_output.html for a list of publications), Tudat’s capabilities have recently been expanded to real data analyses, with a development team comprised of both professional research software engineers and planetary scientists/astrodynamicists. Here we showcase these capabilities, demonstrating the readiness level of our observation and dynamical models for state-of-the-art analyses, and providing an outlook on our future plans.
Tudat makes it possible for the user to both perform and disseminate their analyses with full transparency regarding their underlying assumptions, settings, and models. Ongoing studies by the research team using these new features of Tudat will be published along with the full analysis code and results, allowing anyone in the community to entirely and easily reproduce all results. This not only allows for cross-examination of the results and underlying models by the full community, but will also facilitate building upon this analysis, allowing new ideas and methods to be tested and applied with a minimum of effort by any interested party, and not only the original authors. Such an approach is expected to improve the scientific return of current and future dynamical studies of various Solar System bodies.
At present, Tudat provides capabilities to process several categories of Earth- and space-based observational data: (i) deep-space range and (closed-loop) Doppler tracking data of planetary missions (e.g. [2]) collected by the Deep Space Network (DSN) and ESA’s ESTRACK, supporting multiple standard data formats such as IFMS, ATDF, ODF and TNF; (ii) deep-space (open-loop) Doppler and VLBI tracking data of planetary missions collected by the Planetary and Radio Interferometry and Doppler Experiment (PRIDE) with radio(astronomy) telescopes [3]; (iii) optical astrometry and radar tracking archived by the Minor Planet Center (MPC)[4] and the Natural Satellite Data Center (NSDC)[5].
Tudat’s deep-space radiometric processing capabilities have been successfully tested for several missions including GRAIL (Fig. 1) and Cassini (Fig. 2). By comparing post-fit residuals and estimated orbits against those obtained from reference solutions, we validate that our dynamical and observation models reproduce the measured data within the expected noise characteristic of the data.
Fig. 1: Post-fit residuals and orbit differences for GRAIL from S-band Doppler data.
Fig. 2: Pre-fit residuals for Cassini from X-band Doppler data at Titan flyby T033.[8]
In another demonstration involving different system dynamics and data types/quality, our software has been successfully applied to orbit estimation of small bodies from astrometric observations. Fig. 3 shows the post-fit residuals for the MPC observations of Eros. The difference between our estimated orbit and the JPL Horizons orbit (used as a reference) is consistent with the expected level of uncertainty.
Fig 3: Orbit differences w.r.t. JPL Horizons (top) and post-fit residuals (bottom) for Eros from a six-year arc of MPC optical astrometry
Finally, the scope and accuracy of our dynamical models have been demonstrated to reproduce the motion of both spacecraft and natural bodies with the fidelity level required for high precision orbit estimations (see Fig. 4 for an example of Galilean satellites). This ensures that our software has the capability to fully exploit the quality of the available data sets and provide physically realistic orbital solutions and physical parameter estimates.
Fig 4: Orbit differences of Galilean moons over 1 year between Tudat and NOE ephemerides (as generated during validation of analyses in [7])
Tudat’s real data processing capabilities are now at the core of several research projects, primarily focusing on the orbits, interiors, and/or dynamical evolution of Solar System bodies.
Current analyses include the investigation of long-term time variations of Mars’ gravity field as a means to probe the physics of its deep interior, by combining tracking data from various Martian orbiters [7]. A re-analysis of Cassini tracking data acquired during Titan flybys is also ongoing (to be ultimately combined with long-term astrometry observations), with a particular focus on solving existing discrepancies in the estimation of tidal parameters [8]. Data analysis of Rosetta Doppler tracking coupled with high-fidelity coma density modelling will commence soon with the goal of improving understanding of the interior and dynamics of comet 67P [9]. Tudat will also be used to process the open-loop Doppler and VLBI data of the PRIDE experiment on the JUICE mission.
[1] Dirkx, D., et al. (2022). The open-source astrodynamics Tudatpy software–overview for planetary mission design and science analysis. EPSC2022, (EPSC2022-253).
[2] Gurvits, L. I., et al. (2023). Space Science Reviews, 219(8), 79.
[3] Asmar, S. W. (2022). Radio science techniques for deep space exploration. John Wiley & Sons.
[4] Minor Planet Center. (n.d.). Minor Planet Circulars (MPCs). Cambridge, MA: Smithsonian Astrophysical Observatory. https://www.minorplanetcenter.net/
[5] Arlot, J. E., & Emelyanov, N. V. (2009). Astronomy & Astrophysics, 503(2), 631-638.
[6] Fayolle, M.,et al. (2023). Astronomy & Astrophysics, 677, A42.\
[7] Alkahal, R. and Root, B.C. (2024) Satellite gravity-rate observations to uncover Martian plume-lithosphere dynamics. EPSC2024, (EPSC2024-952).
[8] Hener, J. et al. (2025), Re-visiting determination of Saturn-Titan tidal interaction parameters using the open-source Tudat software.these proceedings
[9] Reichel, M. et al. (2025), Assessing the impact of perturbations on the motion of Rosetta spacecraft near Comet 67P/Churyumov-Gerasimenko, these proceedings
How to cite: Dirkx, D., Fayolle, S., Avillez, M., Dijkstra, T., Filice, V., Gisolfi, L., Hener, J., Hinüber, L., Sanchez Rodriguez, A., Alkahal, R., Cowan, K., Gehly, S., Hu, X., Jeanjean, M., Langbroek, M., Lopez Rivera, A., Minervino Amodio, A., Plumaris, M., Reichel, M., and Van Hulle, S.: Tudat: an open-source, high-fidelity orbit and parameter estimation software – Overview of planetary mission data analysis capabilities, EPSC-DPS Joint Meeting 2025, Helsinki, Finland, 7–13 Sep 2025, EPSC-DPS2025-673, https://doi.org/10.5194/epsc-dps2025-673, 2025.
[Introduction]
Between 2006 and 2016, Cassini collected high-precision Doppler measurements during ten of its Titan fly-bys. These data allowed the determination of Titan’s gravity field up to degree and order 5 [1, 2] and its tidal Love number k2 [2,3]. In this work we revisit the determination of this parameter, seeing as two existing determinations of k2 [2, 3] are statistically inconsistent, implying different characteristics of Titan’s interior structure, specifically the ocean density [3]. The conflicting results may stem from modelling differences, estimation settings or under-estimation of the error bars associated with the determined values. Using the open-source orbit and parameter estimation software Tudat [4] and a rigorous open-science approach, we aim to provide an independent, fully transparent re-analysis of the Titan flybys’ radio-science data, and thus help resolve the conflict from prior k2 estimates and its implications for the interior models. Here we present the results of the observational and dynamical models test, which demonstrate the suitability of the Tudat software to carry out this analysis.
[Modelling Results]
To isolate potential mis-modelling in the computation of the radiometric observables, we adopt a reference trajectory of the Cassini spacecraft from a spice-disseminated navigation solution, simulate the expected X- and Ka-band Doppler observables and compare them against the real observables. The inherent noise level of the 2-way Doppler observables is between 0.02 and 0.07 mm/s [3]. In the example of tracking pass T033, we demonstrate the computation of the observables to their noise level. When embedding the observables computation in an estimation, the small biases in the residual statistics are expected to be absorbed by ground station bias and clock synchronisation parameters. We conclude that the observables model is at the expected level.
Figure 1: Residuals from computed Doppler observables during Titan flyby T033, based on the Cassini reference trajectory.
The detection of the acceleration exerted on the spacecraft due to the tidal response of Titan's gravity field demands a dynamical model of Titan that is sufficiently accurate over the timescale of the inversion. In the case of multi-arc estimation of the Titan ephemeris, as employed in prior analyses [2, 3], this timescale is in the order of two days. We aim to retrieve k2 alongside a single arc solution for the Titan ephemeris, which is beneficial for the accuracy and consistency of the determined value [2]. This requires the Titan dynamical model to be sufficiently accurate over multiple years [5], including subtle effects such as Saturn’s tidal dissipation at Titan’s forcing frequency and Saturn’s rotation pole motion. Eventually, a consistent single-arc ephemeris solution will also be critical to extract the tidal dissipation signature from Titan’s dynamics. Figure 2 demonstrates the accuracy of our model, which we derive from comparison with the numerical solution of Titan’s orbit from the Paris Observatory [6]. Titan position residuals over ± 24 hours (Fig. 2a) are contained within ± 1 m. The dynamical model is likely even more accurate, as the dominating in-plane signature is believed to originate from interpolation functions of the ephemeris kernel. Residuals over ± 5 years are contained within ± 100 m in-plane and grow up to 1500 m out-of-plane.
(a) Residuals over ± 24 h. Initial Titan state fitted to the reference ephemeris.
(b) Residuals over ± 5 years. Initial Titan state and Saturn's tidal quality factor fitted to the reference.
Figure 2: Titan position residuals in the orbit-defined RSW frame over the timescale of a single flyby pass (a) and all passes (b).
In addition to accurate dynamical models of the Titan dynamics, the trajectory of the Cassini spacecraft needs adequate modelling. This includes non-conservative forces like solar radiation pressure, thermal emissions from RTGs, and Titan’s atmosphere during low-altitude passes [2, 3]. Another challenge is reconstructing the motion of the low gain antenna phase center used during the high-latitude T110 flyby. These modelling efforts are still ongoing.
[Preliminary Conclusions and Outlook]
The models for Doppler observables were shown to be sufficiently mature to reproduce the Cassini observables to the accuracy of their inherent noise level. In several instances residual signatures were found. Their origin is still under investigation, but suspected to stem from the Cassini reference trajectory instead of the observable computation itself. We demonstrated a Titan dynamical model accuracy of < 1 m (Fig. 2a) on the timescale of individual passes, which is considered satisfactory for a multi-arc estimation. The current state of residuals over the 10 year interval (Fig. 2b) is promising with regards to enabling a global inversion with a single, consistent Titan ephemeris. Further work will investigate and mitigate the out-of-plane residual signature. The implementation of selected dynamical models for the Cassini spacecraft is ongoing.
Further tests of the Titan dynamical model over a timespan of 180 years show in-plane residuals within ± 2500 m and 50000 m out-of-plane. After mitigation of remaining out-of-plane deviations, we aim for an inversion of Titan’s dynamical model over periods much greater than the Cassini mission. Such an inversion could be constrained with additional astrometric observables from e.g. Voyager 1&2 and ground-based campaigns. We aim to recover parameters relating to the long-term evolution of Titan’s orbit (specifically the tidal quality factor of Saturn at Titan’s forcing frequency), with implications for the long-term history of the entire Saturnian system [6].
[1] Iess, L., et al. (2012). The tides of Titan. Science, 337(6093)
[2] Durante, D., et al. (2019). Titan's gravity field and interior structure after Cassini. Icarus, 326
[3] Goosens, S., et al. (2024). A low-density ocean inside Titan inferred from Cassini data. Nature Astronomy, 8(7)
[4] Dirkx, D., et al. (2022). The open-source astrodynamics Tudatpy software–overview for planetary mission design and science analysis. EPSC2022
[5] Fayolle, M., et al. (2022). Decoupled and coupled moons’ ephemerides estimation strategies application to the JUICE mission. Planetary and Space Science, 219
[6] Lainey, V., et al. (2020). Resonance locking in giant planets indicated by the rapid orbital expansion of Titan. Nature Astronomy, 4(11)
How to cite: Hener, J., Dirkx, D., Fayolle, S., Gisolfi, L., Filice, V., Lainey, V., and Visser, P.: Re-visiting determination of Saturn-Titan tidal interaction parameters using the open-source Tudat software., EPSC-DPS Joint Meeting 2025, Helsinki, Finland, 7–13 Sep 2025, EPSC-DPS2025-1567, https://doi.org/10.5194/epsc-dps2025-1567, 2025.
Tides are key drivers of planetary system long-term evolution. Both planet and satellite tides indeed yield a secular change in the satellites’ semi-major axis and eccentricity. They also induce tidal heating in the satellites’ interiors, thus controlling their thermal evolution.
Consistently accounting for the effect of tides on the gravitational potential of natural satellites is essential to study their dynamics and that of spacecraft flybying and/or orbiting them. Various strategies exist to include tides in dynamical models. But, for natural satellites in 1:1 spin-orbit resonances, the effects of satellite tides (i.e., tides raised on the satellite by the host planet) on the satellite’s own orbit and rotation are particularly challenging to model. This becomes apparent when looking at Kaula’s potential expansion (all quantities with refer to the perturbing body):
All eccentricity functions Glmp(e*) are O(e*) terms, except for G220(e*). The latter therefore represents the dominating component of the tidal potential, and occurs at the forcing frequency:
with n the satellite’s mean motion and θ the rotation angle of the body undergoing tides. This mode typically dominates the tidal response, and for instance accounts for the leading term in the secular evolution of the satellite’s orbit due to planet tides (tides raised on the satellite on the planet).
However, for satellite tides, the commensurability between the satellite’s orbital and rotational periods causes this forcing mode to vanish. The tidal response becomes then dominated by the next leading terms, whose forcings occur at the following frequencies:
The dominant forcing modes thus exactly coincide with the main once-per-orbit frequency driving the satellite’s translational and rotational dynamics. This requires highly consistent modelling of the interplay between the satellite’s orbit, rotation, and tidal deformation. Any inconsistency would indeed lead to unphysical dynamics, either breaking the spin-orbit resonance and/or predicting wrong secular rates for the da/dt or de/dt. For instance, neglecting the role of a small misalignment between the moon’s long-axis and the planet-satellite line at periapsis implies either one of the following [1]:
- The resonance is maintained, but the energy dissipated through tides is overestimated, leading to the following semi-major axis changing rate [2]
instead of [3]
with k2 and Q the satellite’s tidal Love number and quality factor, respectively.
- The energy dissipation is accurately modelled, but the satellite is expected to be in a pseudo-synchronous state, which we know to be unphysical [4].
Both inconsistencies above stem from the use of decoupled models which cannot reproduce the delicate balance between i) how the satellite’s orbit and rotation define its instantaneous response to tidal forcing; and ii) how this response in turn affects the satellite’s translational and rotational dynamics.
When focusing on the effect of satellite tides on the satellite’s own orbit, as in e.g., natural satellite ephemerides studies, a common approach is to account for the averaged contribution (over one orbit) of tidal dissipation to the system’s dynamics [5]. This is done by introducing an additional perturbing force on the satellite (see Fayolle, 2025 for derivation from Eq. 1):
This approach, however, only reproduces the average – rather than instantaneous - orbital effects of satellite tides. Moreover, as it does not explicitly model the satellite’s gravitational deformation. While sufficient to extract the signature of tidal dissipation from the satellite’s own dynamics (given quality of existing data), this is ill-suited to model tidal effects in the dynamics of a nearby spacecraft. For future missions (JUICE, Europa Clipper), the signature of satellite tides on both their own dynamics and that of the spacecraft will be visible in the data. Concurrently modelling both effects will thus be crucial to obtain a consistent quantification of tidal parameters, which current models do not allow.
To resolve this, we therefore propose a coupled propagation of the satellite’s orbital motion, rotational dynamics, and tidal deformation, based on [6,7]. In this approach, the tidal deformation is formulated as an ordinary differential equation of the satellite’s gravity coefficients. Numerically integrating these alongside the satellite’s translational and rotational states yields time-dependent variations of the gravity coefficients, which are by definition consistent with the orbit, rotation, and assumed rheology of the satellite. The concurrent propagation of this extended dynamical model automatically accounts for all couplings at play. Such an approach not only provides a unified way to model the orbit-rotation-tide interactions, but is also equally applicable to include satellite tides in the dynamics of the satellite itself, or in those of a nearby spacecraft.
We show that our model - implemented in the open-source Tudat software [8] - correctly reproduces the full coupled orbital-rotational-tidal dynamics, using both the Earth-Moon and Mars-Phobos systems as test cases, and assuming a simple Maxwell rheology. We obtain the expected secular rates for the satellite’s semi-major axis and eccentricity, while conserving rotational energy and maintaining the spin-orbit resonance. In the case of Phobos, the non-negligible enhancement of tidal dissipation effects due to librations also matches theoretical expectations [9]. We numerically verified that the resulting time history of our gravity coefficients matches that predicted by using Eq. 1 up to at least 3rd order in eccentricity (after expanding Eq. 1 to account for librations, following [10]).
In the context of upcoming missions such as JUICE and Europa Clipper, such a unified model will be invaluable to obtain consistent estimates of tidal parameters from their combined effects on natural satellites’ and spacecraft’s dynamics. Future work will include adopting more complex and realistic rheology models, and investigating the possibility to directly estimate interior properties (e.g., modulus and rigidity instead of Love numbers, etc.).
References
[1] Fayolle (2025) PhD dissertation
[2] Boué & Efroimsky (2019) CMDA 131
[3] Goldreich & Soter (1966) Icarus 5.1-6
[4] Makarov & Efroimsky (2013) The Astrophysical Journal 764.1
[5] Lainey et al. (2009) Nature 459.7249
[6] Correia et al. (2014) A&A 57:A50
[7] Boue et al. (2016) CMDA 12631-60
[8] Dirkx et al. (2022) EPSC2022-253
[9] Efroimsky (2018) Icarus 306
[10] Frouard & Efroimsky (2017) CMDA 129.1-2
How to cite: Fayolle, S., Dirkx, D., Rovira Navarro, M., Mol, J., and Roubiou, S.: A consistent numerical integration of orbit, tides, and rotation for synchronous satellites, EPSC-DPS Joint Meeting 2025, Helsinki, Finland, 7–13 Sep 2025, EPSC-DPS2025-949, https://doi.org/10.5194/epsc-dps2025-949, 2025.