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

¹ European Space Agency (ESA), European Space Research and Technology Centre (ESTEC), Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands
² Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands

^★ Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 26 September 2025
Accepted: 19 December 2025

Abstract

Context. Consistently modelling the effects of the tides raised on a satellite on the dynamics of the satellite itself and on those of a nearby spacecraft (either in orbit or performing a flyby) requires accounting for the instantaneous tidal deformation of the satellite’s gravitational potential. For synchronous satellites, the spin-orbit resonance causes perfect commensurability between the orbital and rotational periods, and the main tidal forcing frequency. This imposes stringent consistency requirements on the modelling of the delicate interplay between the satellite’s orbital motion, rotational dynamics, and tidal deformation. These three aspects of satellite dynamics are typically handled separately (at least partially) in classical modelling approaches, which are therefore highly inconsistency-prone for the specific spin-orbit resonance case.

Aims. Modelling inconsistencies can lead to the under- or overestimation of tidal parameters when dissipation signatures are extracted simultaneously from both spacecraft and moon dynamics, a combined approach that is increasingly critical for current and upcoming mission analyses. As a promising alternative, we propose a coupled integration of the satellite’s orbit, rotation, and tidal deformation.

Methods. Integrating the satellite’s deformation requires introducing an additional set of differential equations for its degree-two gravity coefficients to complement the translational and rotational equations of motion. A concurrent integration ensures that the satellite’s instantaneous tidal response is fully consistent with its orbit and rotation, while automatically accounting for all dynamical couplings at play. In this paper, we present a two-dimensional implementation of this coupled propagation framework and investigate its ability to produce realistic dynamics with expected tidal dissipation signatures. As a proof-of-concept, we validated the physical self-consistency of the results using the Earth–Moon and Mars–Phobos systems as conceptual test cases.

Results. Our coupled propagation naturally maintains the spin-orbit resonance while producing the expected orbital migration and circularisation rates (including libration-induced tidal dissipation enhancement in the case of Phobos), a delicate balance that is hard to achieve with decoupled modelling strategies. The time history of the satellite’s gravitational tidal response (obtained as a direct output of our integration) is also in agreement with analytical predictions derived from the tidal potential theory.

Conclusions. Our coupled approach thus provides a unified and consistent way to model orbit-rotation-tide interactions. Crucially, it is equally applicable to representing tidal effects on the satellite itself and on a nearby spacecraft. This is critical for planetary missions such as Juice and Europa Clipper, where tidal dissipation signatures can (and will) be extracted from both the spacecraft’s and moons’ dynamics.