Self-consistent modelling of tide-orbit-rotation couplings in the Galilean system

1. Motivation

Tides are a key driver of planetary system evolution: they govern the intensity of tidal heating in the moons' interiors, drive their rotations towards equilibrium states, and set the migration and circularisation rates of their orbits. A detailed characterisation of the satellites' current response to tidal forcing offers invaluable insight into their interior structure and properties, as well as into the present-day evolution of the system's orbital configuration. This is, in turn, essential for placing constraints on the long-term thermal-orbital evolution of the system and, for icy moons in particular, on the history of their internal oceans.

In the context of the upcoming Juice and Europa Clipper missions, radio-science tracking of both spacecraft during their close encounters with the Galilean moons (flybys and orbital phase) will be critical in this regard. Such measurements will constrain the moons' dynamics, and in particular the dynamical signatures of tidal effects, at an unprecedented level of detail and precision [1,2]. At the accuracy levels anticipated post-missions, however, the limiting factor for the inferred solution might no longer be the precision of the radio-science tracking, but instead the physical fidelity and self-consistency of the dynamical models that underpin the data analysis.

More specifically, tidal effects manifest themselves both in the spacecraft's trajectory and in the satellites' own dynamics. Reliably extracting these signatures from radio-science data therefore requires that tidal contributions be incorporated into the dynamical models of the spacecraft and moons in a fully consistent way. For synchronous satellites, the intricate coupling between tide, orbit, and rotation makes this particularly challenging: any mismodelling of these interactions leads to an erroneous dissipation signature and ultimately affects the recovered estimates [3]. Achieving such consistency in the Galilean system is further complicated by the Laplace resonance, which requires Io, Europa, and Ganymede to be modelled as a single, unified dynamical system to ensure that the strong gravitational couplings between them are properly captured. 

2. Approach

To address this challenge, we proposed a unified dynamical framework [4] in which each satellite's gravitational deformation is propagated through an ordinary differential equation derived from a prescribed rheology, along with the orbital and rotational dynamics (building on previous works focussing on exosystems [5,6]). Embedding the internal response directly within the integrated dynamics guarantees that all orbit-rotation-tide couplings are accounted for, at all forcing frequencies. Critically, the proposed model establishes a direct and physically motivated connection between interior properties (e.g., viscosity, rigidity) and tidal dissipation signatures in the moons’ dynamics, and ensures full consistency between the orbits, rotations and tidal deformation of all moons. We expanded our original two-dimensional proof-of-concept [4] into a complete N-body implementation that includes higher-order effects, such as moon-moon interactions, and three-dimensional effects in the moons’ orbits and rotations (non-zero inclination and obliquity).

3. Results and outlook

Applied to the Galilean system, the model successfully reproduces expected dynamical features: it maintains the Laplace resonance, yields realistic spin-orbit resonant rotations in a Cassini state, and recovers the orbit expansion and circularisation rates predicted by tidal theory [7]. Crucially, the propagated dynamics also captures coupling signatures that fall outside the reach of conventional formulations. Multi-frequency and indirect forcing and response of both tides and rotation emerge naturally from the concurrent integration of the coupled equations of motion, with additional forcings at the frequencies of the other Galilean moons appearing directly in each satellite's rotational and deformational response. This is particularly promising for Ganymede, whose gravitational deformation will be characterised with unprecedented detail, including sampling of the moon’s response at different frequencies [8]. More generally, these results demonstrate the potential of a fully coupled modelling approach for future analyses of Juice and Europa Clipper data, where the Galilean satellites’ orbits, rotations, and tidal interactions are self-consistently modelled as an integrated dynamical system.

References

[1] Magnanini, A. et al. Astronomy & Astrophysics 687 (2024): A132.

[2] Fayolle, S. Dissertation (2025)

[3] Magnanini, A., Zannoni, M., and Lainey, V. Astronomy & Astrophysics 707 (2026): A96.

[4] Fayolle, S., et al. Astronomy & Astrophysics 707 (2026): A224.

[5] Correia, A. et al. Astronomy & Astrophysics 571 (2014): A50.

[6] Boué, G., Correia, A., and Laskar, J. Celestial Mechanics and Dynamical Astronomy 126.1 (2016): 31-60.

[7] Goldreich, P., and Soter, S. Icarus 5.1-6 (1966): 375-389

[8] De Marchi, F., et al. Icarus 386 (2022): 115150