Analytic frozen orbits under the zonal harmonics perturbation from an Earth-like planet

TL;DR

This paper derives analytical expressions for frozen orbits around Earth-like planets considering zonal harmonic perturbations, enabling precise initial condition determination and bifurcation analysis through purely analytical methods.

Contribution

It introduces a second-order analytical solution for frozen orbits under zonal harmonics, including J2 to J7, and provides a methodology for studying bifurcations analytically.

Findings

Analytical expressions for initial conditions of frozen orbits are derived.

The methodology allows bifurcation analysis near critical inclination.

Error performance of the analytical solutions is demonstrated.

Abstract

This work focuses on providing closed form analytical expressions to define frozen orbits under the effects of the zonal harmonics of an Earth-like planet. Particularly, the perturbation effects from the terms J2, J3, J4, J5, J6, and J7 are considered in this work. This is done using a power series expansion in the small parameter that allow not only to provide an approximate solution to the system, but also to determine the analytical expressions that define the initial osculating conditions that generate frozen orbits. As a result of that, the proposed methodology allows to study the bifurcation of frozen orbits close to the critical inclination by purely analytical methods. Additionally, the derivation of the secular variation of the orbital elements as well as the transformation from osculating to mean elements is provided based on the second order analytical solution proposed in this work. Examples of application are also provided to show the error performance of the results included in this document.