Approximating Orbits in a Rotating Gravity Field with Oblateness and Ellipticity Perturbations

TL;DR

This paper develops analytical approximations for the orbital motion of near-circular orbits in a rotating gravitational field with oblateness and ellipticity perturbations, using the Jacobi integral for first-order solutions.

Contribution

It introduces a method to approximate orbital states in rotating, perturbed gravity fields, focusing on near-circular orbits and addressing the effects of oblateness and ellipticity.

Findings

Approximations are valid for angular rate ratio $ ext{Gamma} > 1$.

Accuracy decreases as $ ext{Gamma} ightarrow 1$.

Methodology extends to eccentric orbits with noted challenges.

Abstract

This paper explores the problem of analytically approximating the orbital state for a subset of orbits in a rotating potential with oblateness and ellipticity perturbations. This is done by isolating approximate differential equations for the orbit radius and other elements. The conservation of the Jacobi integral is used to make the problem solvable to first-order in the perturbations. The solutions are characterized as small deviations from an unperturbed circular orbit. The approximations are developed for near-circular orbits with initial mean motion $n_{0}$ around a body with rotation rate $c$. The approximations are shown to be valid for values of angular rate ratio $\Gamma = c/n_{0} > 1$, with accuracy decreasing as $\Gamma \rightarrow 1$, and singularities at and near $\Gamma = 1$. Extensions of the methodology to eccentric orbits are discussed, with commentary on the challenges of obtaining generally valid solutions for both near-circular and eccentric orbits.