Stability of librational motion in the spatial circular restricted three-body problem for high inclinations and mass ratios

TL;DR

This study investigates the stability of librational motion around Lagrangian points in the spatial three-body problem, revealing conditions for stability at high inclinations and mass ratios through numerical analysis.

Contribution

It provides new insights into the stability regions of librational motion at high inclinations and mass ratios, extending previous planar case results.

Findings

Stable librational motion exists below 60° inclination for μ<0.04.

Stable orbits are possible at higher μ and moderate inclinations over longer times.

Resonances and chaotic regions are identified using spectral analysis methods.

Abstract

In the spatial circular restricted three-body problem librational motion around the Lagrangian points $L_4$ and $L_5$ exists up to high inclinations of the massless object. We report the results of numerical investigations on the stability of this librational motion for systematic variations in the inclination of the Trojan and the mass-ratio $\mu$ of the two massive bodies. We show that stable motion prevails for inclinations below 60 degrees and mass-ratios $\mu < 0.04$ for typical integration times up to $10^6$ revolution periods. At even higher mass-ratios -- beyond the critical mass-ratio for the planar case -- stable orbits were found to exist for up to $10^7$ periods at moderate inclinations. We extracted the librational frequencies on a grid in the parameter space from the Fourier spectra and traced their variation. Several resonances between the short and long period librational frequency as well as the vertical frequency lie inside the investigated region. The application of the Lyapunov characteristic indicator and spectral number methods also reveals the chaotic regions. This simple model is equally applicable in the Solar system for low mass ratios for Trojans of the planets, as well as to Trojan-type exoplanets in binary star systems at high mass-ratios.