A set of orbital elements to fully represent the zonal harmonics around an oblate celestial body

TL;DR

This paper introduces a novel set of orbital elements that comprehensively represent zonal harmonic effects around oblate celestial bodies, enabling complete linear and polynomial systems without singularities for all orbit types.

Contribution

The paper presents a new orbital element set that fully captures zonal harmonic perturbations and is applicable to all orbit types, improving modeling accuracy and mathematical completeness.

Findings

Complete linear system for unperturbed problem

Polynomial system for zonal harmonic perturbations

First order approximate solution for J2 perturbation

Abstract

This work introduces a new set of orbital elements to fully represent the zonal harmonics problem around an oblate celestial body. This new set of orbital elements allows to obtain a complete linear system for the unperturbed problem and, in addition, a complete polynomial system when considering the perturbation produced by the zonal harmonics from the gravitational force of an oblate celestial body. These orbital elements present no singularities and are able to represent any kind of orbit, including elliptic, parabolic and hyperbolic orbits. In addition, an application to this formulation of the Poincar\'e-Lindstedt perturbation method is included to obtain an approximate first order solution of the problem for the case of the J2 perturbation.