Translational–rotational couplings in the dynamics of Phobos

¹ Royal Observatory of Belgium, Ringlaan 3, 1180 Brussels, Belgium
² Delft university of Technology, Kluyverweg 1, Delft 2629HS, The Netherlands
³ European Space Research and Technology Centre, ESA, Keplerlaan 1, Noordwijk 2201AZ, The Netherlands

^★ Corresponding author.

Received: 1 May 2024
Accepted: 18 June 2025

Abstract

Context. Upcoming science missions to Phobos will potentially provide unprecedented observations of Phobos’s orbit in the form of orbiter and/or lander tracking data. This will likely require an updating of the dynamical models currently used to invert this data, with the coupling between the satellite’s orbit and rotation being of particular importance. State-of-the-art ephemerides estimations for tidally locked satellites rely on a decoupled approach where translational models are combined with a simplified analytical representation of the moon’s rotation (typically a single-frequency periodic variation superimposed to a synchronous rotation).

Aims. This paper investigates the coupled propagation of Phobos’s translational and rotational dynamics, and assesses the extent to which the most commonly used uncoupled model can emulate the results of the coupled integration, and what consequences the mis-modeling has on the products of data inversion.

Methods. We considered two models: a coupled model that propagates Phobos’s translational and rotational dynamics simultaneously, and an uncoupled model that assumes Phobos to be in a fully locked configuration with a once-per-orbit longitudinal libration. By simulating the dynamics for about ten years, first in a coupled and then in an uncoupled manner, we compared the results and used the coupled trajectory as simulated observations for an estimation of the different parameters using uncoupled translational dynamics.

Results. For identical initial states, differences between the coupled and uncoupled trajectories were found to accumulate to 40 m, most predominantly in Phobos’s direction of motion. Longitudinal librations were misrepresented by the uncoupled model particularly around the frequencies of the normal mode, where forcings are amplified up to 3.6 × 10⁻³ degrees. Long-term latitudinal librations also arise from forcings due to coupling-induced changes in orbital inclination. The use of uncoupled models in data inversion results in true errors in the estimated parameters. In this case, we performed estimations of different lengths up to 1000 days to estimate Phobos’s initial state, once-per-orbit libration amplitude, and harmonic coefficients Ċ_2,0 and Ċ_2,2. Errors in dynamical parameters were found to be on the order of 10⁻³ degrees for the physical libration amplitude and of 10⁻⁵ for the harmonic coefficients (relative errors of around 0.1%).

Conclusions. These true errors are one to three orders of magnitude above the formal errors expected from laser ranging measurements to a Phobos lander, which indicates that the typical single-frequency uncoupled model is not suitable for the proper inversion of such data. Refined rotation models will therefore be required, either by expanding the uncoupled model to multiple frequencies or by performing a fully coupled orbital-rotational propagation as proposed in this paper. We discuss the theoretical and practical limitations of an extended analytical parametrization in the specific case of tidally locked satellites, and advocate for the use of a fully coupled approach.