Analytic Osculating Frozen Orbits Under J2 Perturbation

Abstract

This work provides a set of closed-form analytical expressions to define osculating frozen orbits under the perturbation effects of the oblateness of the main celestial body. To this end, an analytical perturbation method based on osculating elements is proposed to characterize, define, and study the three existing families of frozen orbits in closed form: the two families of frozen orbits close to the critical inclination and the family of frozen orbits appearing at low eccentricity values. As such, this work aims to complement other analytical approaches based on mean elements by providing an alternative methodology based on the more natural osculating elements that is able to generate closed-form expressions for all known frozen conditions in the main satellite problem. Additionally, this work includes the first- and second-order approximate solutions of the proposed perturbation method, including their applications to the analytical definition of frozen orbits, repeating ground-track orbits, and sun-synchronous orbits under this perturbation. Examples of applications are also provided to show the expected error performance of the proposed approach.