Analytic inversion of closed form solutions of the satellite's $J_2$ problem

TL;DR

This paper derives and inverts closed-form solutions for a satellite's equatorial motion considering Earth's oblateness, using elliptic functions and Fourier series, providing new analytical tools for the $J_2$ problem.

Contribution

It introduces a novel inversion method for time laws in the $J_2$ problem using Fourier series, enhancing analytical understanding of satellite motion.

Findings

Derived explicit solutions involving elliptic integrals and Jacobi functions.

Developed a Fourier series expansion for inverse time functions to facilitate inversion.

Validated the approach by recovering classical Keplerian results.

Abstract

This report provides some closed form solutions -- and their inversion -- to a satellite's bounded motion on the equatorial plane of a spheroidal attractor (planet) considering the $J_{2}$ spherical zonal harmonic. The equatorial track of satellite motion -- assuming the co-latitude $\varphi$ fixed at $\pi/2$ -- is investigated: the relevant time laws and trajectories are evaluated as combinations of elliptic integrals of first, second, third kind and Jacobi elliptic functions. The new feature of this report is: from the inverse $t = t(c)$ to get the period $T$ of some functions $c(t)$ of mechanical interest and then to construct the relevant $c(t)$ expansion in Fourier series, in such a way performing the inversion. Such approach -- which led to new formulations for time laws of a $J_{2}$ problem -- is benchmarked by applying it to the basic case of keplerian motion, finding again the classic results through our different analytic path. Keywords: $J_2$ problem, bounded satellite motion, Fourier series, elliptic integrals, Jacobi elliptic functions.