A new perturbative solution to the motion around triangular Lagrangian points in the elliptic restricted three-body problem

TL;DR

This paper introduces an analytic perturbative method for solving the motion around triangular Lagrangian points in the elliptic restricted three-body problem, valid for small eccentricities and mass parameters.

Contribution

The paper presents a new analytic solution approach by transforming the equations into Hill's equations and applying Floquet theory, extending solutions to systems with small eccentricity and mass parameter.

Findings

Valid for eccentricity $0 < e \,\leq\, 0.05$ and mass parameter $0 < \mu \leq 0.01$

Provides explicit transformation details for the method

Analytic solutions derived for the motion around Lagrangian points

Abstract

The equations of motion of planar elliptic restricted three body problem are transformed to four decoupled Hill's equations. By using the Floquet theorem analytic solution to the oscillator equations with time dependent periodic coefficients are presented. We show that the new analytic approach is valid for system parameters $0 < e \leq 0.05$ and $0 < \mu \leq 0.01$ where $e$ denotes the eccentricity of primaries while $\mu$ is the mass parameter, respectively. We also clarify the transformation details that provide the applicability of the method.