Analytic Transformation Between Osculating and Mean Elements in the J2 Problem

Abstract

This work presents an analytical perturbation method to study the dynamics of an orbiting object subject to the term from the gravitational potential of the main celestial body. In particular, this paper focuses on the generation of the analytical transformations between osculating and mean elements under this perturbation. This is done using a power series expansion in the perturbation constant on all the variables of the system, and a time regularization based on the argument of latitude of the orbit. This enables the generation of analytic approximate solutions without the need to control the perturbed frequency of the system. The resultant approximations provide the osculating behavior of the problem as well as the transformations between osculating and mean elements for orbits at any eccentricity, including near-circular, elliptic, parabolic, and hyperbolic orbits. Several examples of application are presented to show the accuracy of the perturbation approach and their related transformations.