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This work presents an analytical perturbation method to study the dynamics ofan orbiting object subject to the term $J_2$ from the gravitational potentialof the main celestial body. This is done using a power series expansion in theperturbation constant $J_2$ on all the variables of the system, and a timeregularization based on the argument of latitude of the orbit. This enables thegeneration of analytic solutions without the need to control the perturbedfrequency of the system. The resultant approach allows to approximate thedynamics of the system in osculating elements for orbits at any eccentricity,and to obtain the approximate analytical transformation from osculating to meanelements in these orbits. This includes near circular, elliptic, parabolic andhyperbolic orbits at any inclination. Several examples of application arepresented to show the accuracy of the perturbation approach and their relatedtransformations.

Analytic transformation from osculating to mean elements under J2 perturbation