Melnikov method for non-conservative perturbations of the three-body problem

TL;DR

This paper applies a Melnikov method to analyze how non-conservative, time-dependent perturbations affect the dynamics of spacecraft near Lagrange points in the Earth-Moon system, providing a way to predict trajectory changes.

Contribution

It extends Melnikov theory to non-conservative perturbations in the three-body problem without requiring special orbit types, and computes the first-order perturbed scattering map.

Findings

Derived first-order approximation of the perturbed scattering map.

Applicable to spacecraft trajectory planning with thrust or solar radiation effects.

Provides a method to predict trajectory changes under non-conservative forces.

Abstract

We consider the planar circular restricted three-body problem (PCRTBP), as a model for the motion of a spacecraft relative to the Earth-Moon system. We focus on the Lagrange equilibrium points $L_1$ and $L_2$. There are families of Lyapunov periodic orbits around either $L_1$ or $L_2$, forming Lyapunov manifolds. There also exist homoclinic orbits to the Lyapunov manifolds around either $L_1$ or $L_2$, as well as heteroclinic orbits between the Lyapunov manifold around $L_1$ and the one around $L_2$. The motion along the homoclinic/heteroclinic orbits can be described via the scattering map, which gives the future asymptotic of a homoclinic orbit as a function of the past asymptotic. In contrast with the more customary Melnikov theory, we do not need to assume that the asymptotic orbits have a special nature (periodic, quasi-periodic, etc.). We add a non-conservative, time-dependent perturbation, as a model for a thrust applied to the spacecraft for some duration of time, or for some other effect, such as solar radiation pressure. We compute the first order approximation of the perturbed scattering map, in terms of fast convergent integrals of the perturbation along homoclinic/heteroclinic orbits of the unperturbed system. As a possible application, this result can be used to determine the trajectory of the spacecraft upon using the thrust.