Equinoctial elements for Vinti theory: Generalizations to an oblate spheroidal geometry

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Vinti theory constructs orbits on an oblate spheroidal geometry, naturally encoding the gravitational potential of an oblate spheroid in the coordinates. Classical techniques use spherical geometry. Recent work applied Vinti theory to the relative motion problem by way of a linear dynamical model, which is nonsingular in the oblate spheroidal element space. But as with classical spherical elements, the linear mapping between classical spheroidal elements and inertial rectangular coordinates becomes singular for small eccentricities and/or inclinations in the sense of linear dependence of columns in the Jacobian. To mitigate these practical issues, the standard (spherical) equinoctial elements are chosen to inform in a natural way their generalization to a new nonsingular element set: the oblate spheroidal equinoctial orbital elements. The spherical equinoctial elements can be considered a special case of the spheroidal equinoctial elements in the same way that spherical coordinates can be considered a special case of oblate spheroidal coordinates. The new element set is defined and algorithms for converting between spheroidal equinoctial elements and inertial coordinates are derived. Similarities and differences between spheroidal and spherical equinoctial elements are emphasized for clarity, both in terms of the form of equations and geometrical interpretation. The transformations are valid away from the nearly rectilinear orbit regime and are exact except near the poles. When near the poles, the transformations match the accuracy of the approximate analytical solution, which has been developed to the third order in J2 in the literature. As a result, the singularity on the poles is completely eliminated for the first time.