Analytical Solution to the Vinti Problem in Oblate Spheroidal Equinoctial Orbital Elements

Abstract

Equinoctial orbital elements were recently generalized from spherical geometry to the oblate spheroidal geometry of Vinti theory. For the symmetric Vinti potential, which accounts exactly for oblateness, these nonsingular elements are defined for all nondegenerate orbital regimes and resolve the usual problems found in the classical elements associated with angle ambiguities. In the present work, the generalized equinoctial elements are used to solve Vinti’s initial value problem, leading to a fully nonsingular analytical solution for bounded orbits. The result is akin to deriving the equinoctial form of Kepler’s equation for the two-body problem and then solving it, formally completing the introduction of the new equinoctial element set. Derivations of the equinoctial integrals of the motion are included, as well as techniques to eliminate all polar orbit singularities and solve a generalized Kepler’s equation. Multiple examples are presented. Code for predicting the Vinti orbit in oblate spheroidal equinoctial elements is provided online as supplementary material.