Normal Gravity Fields and Equipotential Ellipsoids of Small Objects in the Solar System: A Closed-form Solution in Ellipsoidal Harmonics up to the Second Degree

Abstract

We propose a definition for the normal gravity fields and normal figures of small objects in the solar system, such as asteroids, cometary nuclei, and planetary moons. Their gravity fields are represented as series of ellipsoidal harmonics, ensuring more robust field evaluation in the proximity of an arbitrary, convex shape than using spherical harmonics. The normal gravity field, approximate to the actual field, can be described by a finite series of three terms, that is, degree zero, and the zonal and sectoral harmonics of degree two. The normal gravity is that of an equipotential ellipsoid, defined as the normal ellipsoid of the body. The normal ellipsoid may be distinct from the actual figure. We present a rationale for specifying and a numerical method for determining the parameters of the normal ellipsoid. The definition presented here generalizes the convention of the normal spheroid of a large, hydrostatically equilibrated planet, such as Earth. Modeling the normal gravity and the normal ellipsoid is relevant to studying the formation of the “rubble pile” objects, which may have been accreted, or reorganized after disruption, under self-gravitation. While the proposed methodology applies to convex, approximately ellipsoidal objects, those bi-lobed objects can be treated as contact binaries comprising individual convex subunits. We study an exemplary case of the nearly ellipsoidal Martian moon, Phobos, subject to strong tidal influence in its present orbit around Mars. The results allude to the formation of Phobos via gravitational accretion at some further distance from Mars.