# The solution of the Generalized Kepler's equation

## TL;DR

This paper develops a first-order closed-form theory for satellite motion considering perturbations, and discusses numerical solutions for a generalized Kepler's equation essential for orbit analysis.

## Contribution

It introduces a novel first-order theory in closed form for perturbed satellite motion and explores numerical methods for solving the generalized Kepler's equation.

## Key findings

- Effective numerical techniques for solving the generalized Kepler's equation.
- Three initial guesses improve convergence of solutions.
- First-order theory simplifies perturbation analysis.

## Abstract

In the context of general perturbation theories, the main problem of the artificial satellite analyses the motion of an orbiter around an Earth-like planet, only perturbed by its equatorial bulge or J2 effect. By means of a Lie transform and the Krylov-Bogoliubov-Mitropolsky method, a first-order theory in closed form of the eccentricity is produced. During the evaluation of the theory it is necessary to solve a generalization of the classical Kepler's equation. In this work, the application of a numerical technique and three initial guesses to the Generalized Kepler's equation are discussed.