Frequencies and resonances around $L_4$ in the elliptic restricted three-body problem

TL;DR

This paper analyzes the stability and frequency resonances around the Lagrangian point L4 in the elliptic restricted three-body problem, revealing how resonances influence escape times of particles.

Contribution

It provides a detailed analysis of the frequencies and resonances around L4, including the extension of 1:1 resonances across unstable domains and their impact on particle escape times.

Findings

Stable and unstable domains are mapped in the parameter plane.

Fitting functions for frequencies depending on parameters are derived.

Resonances, especially 1:1, extend beyond single curves into unstable regions.

Abstract

The stability of the Lagrangian point $L_4$ is investigated in the elliptic restricted three-body problem by using Floquet's theory. Stable and unstable domains are determined in the parameter plane of the mass parameter and the eccentricity by computing the characteristic exponents. Frequencies of motion around $L_4$ have been determined both in the stable and unstable domains and fitting functions for the frequencies are derived depending on the mass parameter and the eccentricity. Resonances between the frequencies are studied in the whole parameter plane. It is shown that the 1:1 resonances are not restricted only to single curves but extend to the whole unstable domain. In the unstable domains longer escape times of the test particle from the neighbourhood of $L_4$ are related to certain resonances, but changing the parameters the same resonances may lead to faster escape.