Mutual gravitational potential of two solid bodies revisited

Application to Mars and Phobos orbit perturbations in the context of mission MMX

CNES – Centre spatial de Toulouse, 18 avenue Edouard Belin, 31401 Toulouse Cedex 9, France

^★ Corresponding author; julien.laurent-varin@cnes.fr

Received: 20 November 2024
Accepted: 24 January 2025

Abstract

Context. The discovery in the Solar System of more and more binary asteroids, or even triple asteroids with odd shapes, has triggered studies and new space missions to these bodies related to their dynamical behavior. The Martian moons, due to their characteristics, are the focus of similar interest.

Aims. In this context, the aims of this paper are to revisit the mutual gravitational potential of two celestial bodies, and evaluate the dynamical effects on their orbit of terms, which are often neglected. The rotational dynamics of the bodies is not addressed here.

Methods. The method uses complex gravity spherical harmonic coefficients of each body and takes into account their exact orientation in space. These coefficients are usually estimated from the spherical harmonic representation of the shape and assuming constant density. The analytical derivation of the potential is achieved by using the rotation and translation formulas for the spherical harmonic functions and coefficients. The different components of the potential function are categorized into central direct, indirect, and coupling terms.

Results. Application to the Didymos–Dimorphos pair shows the importance of indirect and coupling terms, for which accelerations reach, respectively, 5% and 1% of the direct terms. This approach was then applied to the Mars-Phobos system, for which the effects on the Phobos orbit due to its gravity field coupling with Mars field were numerically evaluated; effects can reach several tens of meters, as is shown after orbit adjustment on currently adopted Phobos reference ephemeris. The analysis of the radio science data from the coming MMX mission (Martian Moons eXploration) may ultimately require such refinement.