A consistent numerical integration of orbit, tides, and rotation for synchronous satellites

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 G_lmp(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