Spheroidal and ellipsoidal harmonic expansions of the gravitational potential of small Solar System bodies. Case study: Comet 67P/Churyumov-Gerasimenko

Gravitational features are a fundamental source of information to learn more about the interior structure and composition of planets, moons, asteroids, and comets. Gravitational field modeling typically approximates the target body with a sphere, leading to a representation in spherical harmonics. However, small celestial bodies are often irregular in shape and hence poorly approximated by a sphere. A much better suited geometrical fit is achieved by a triaxial ellipsoid. This is also mirrored in the fact that the associated harmonic expansion (ellipsoidal harmonics) shows a significantly better convergence behavior as opposed to spherical harmonics. Unfortunately, complex mathematics and numerical problems (arithmetic overflow) so far severely limited the applicability of ellipsoidal harmonics. In this paper, we present a method that allows expanding ellipsoidal harmonics to a considerably higher degree compared to existing techniques. We apply this novel approach to model the gravitational field of comet 67P, the final target of the Rosetta mission. The comparison of results based on the ellipsoidal parameterization with those based on the spheroidal and spherical approximations reveals that the latter is clearly inferior; the spheroidal solution, on the other hand, is virtually just as accurate as the ellipsoidal one. Finally, in order to generalize our findings, we assess the gravitational field modeling performance for some 400 small bodies in the Solar System. From this investigation we generally conclude that the spheroidal representation is an attractive alternative to the complex ellipsoidal parameterization, on the one hand, and the inadequate spherical representation, on the other hand.