Global gravitational field modelling for spheroidal planetary bodies: non-singular solutions

The standard theoretical framework for the gravitational field determination often relies on spherical approximation. However, Earth’s shape is much closer to a rotational ellipsoid flattened at the poles, as proved by the legendary expeditions of the French Academy of Sciences to South America and Lapland already in the 18th century. Contemporary investigations of solar system planetary bodies have revealed that many resemble prolate or oblate ellipsoids, whereas a high amount of them is flattened more significantly than the Earth. Four such spheroidal bodies have recently been subject to immense research interest: 1) Mars being explored by satellite and lander missions as it represents a potential target for future colonisation, 2) the asteroid Bennu explored by the sample-return satellite mission OSIRIS-REx, 3) the dwarf planet Ceres, and 4) the asteroid Vesta, both explored by the satellite mission Dawn. Moreover, several comets and asteroids with spheroidal (ellipsoidal) shapes have been subjected to intense small-body research. Consequently, there is an urgent need to formulate a modern theoretical framework for the gravitational field determination.

In this contribution, we formulate a mathematical theory for modelling of gravitational fields generated by ellipsoidal bodies. In addition, we present both theory and software considering non-singular solution for derived equations using recurrences instead of classical approach, which depends on reduced spheroidal latitude. Using recurrences, we can eliminate singularities on poles and computational errors in their proximity caused by latitude dependence.