# Analytic model for the long-term evolution of circular Earth satellite orbits including lunar node regression

## TL;DR

This paper develops an analytic model to predict the long-term evolution of circular Earth satellite orbits, incorporating lunar node regression and gravitational perturbations, improving understanding of orbital dynamics over extended periods.

## Contribution

It introduces an approximate analytic model that accounts for lunar orbital inclination and node regression, extending previous solutions for orbit evolution under Earth's $J_2$ and lunar effects.

## Key findings

- Model agrees well with numerical results outside resonant cases.
- Provides explicit analytic expressions for orbit evolution.
- Highlights limitations near resonance conditions.

## Abstract

This paper aims to obtain an analytic approximation to the evolution of circular orbits governed by the Earth's $J_2$ and the luni-solar gravitational perturbations. Assuming that the lunar orbital plane coincides with the ecliptic plane, Allan and Cook (1964) derived an analytic solution to the orbital plane evolution of circular orbits. Using their result as an intermediate solution, we establish an approximate analytic model with lunar orbital inclination and its node regression be taken into account. Finally, an approximate analytic expression is derived, which is accurate compared to the numerical results except for the resonant cases when the period of the reference orbit approximately equals the integer multiples (especially 1 or 2 times) of lunar node regression period.