Analytic transformation from osculating to mean elements under J2 perturbation

This work presents an analytical perturbation method to study the dynamics of an orbiting object subject to the term $J_2$ from the gravitational potential of the main celestial body. This is done using a power series expansion in the perturbation constant $J_2$ 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 solutions without the need to control the perturbed frequency of the system. The resultant approach allows to approximate the dynamics of the system in osculating elements for orbits at any eccentricity, and to obtain the approximate analytical transformation from osculating to mean elements in these orbits. This includes near circular, elliptic, parabolic and hyperbolic orbits at any inclination. Several examples of application are presented to show the accuracy of the perturbation approach and their related transformations.