Study of Resonances for the Restricted 3-Body Problem

TL;DR

This study systematically identifies and classifies mean-motion resonances in the coplanar circular restricted three-body problem across various mass ratios, revealing dominant resonances and their stability characteristics relevant to planetary systems.

Contribution

It introduces a comprehensive method combining Lyapunov exponents, FFT, and orbital analysis to classify and understand resonances in the CR3BP, extending previous approaches.

Findings

2:1 resonance is most frequent

3:1 resonance is second most common

High resonance ratios correlate with orbital stability

Abstract

Our aim is to identify and classify mean-motion resonances (MMRs) for the coplanar circular restricted three-body problem (CR3BP) for mass ratios between 0.10 and 0.50. Our methods include the maximum Lyapunov exponent, which is used as an indicator for the location of the resonances, the Fast Fourier Transform (FFT) used for determining what kind of resonances are present, and the inspection of the orbital elements to classify the periodicity. We show that the 2:1 resonance occurs the most frequently. Among other resonances, the 3:1 resonance is the second most common, and furthermore both 3:2 and 5:3 resonances occur more often than the 4:1 resonance. Moreover, the resonances in the coplanar CR3BP are classified based on the behaviour of the orbits. We show that orbital stability is ensured for high values of resonance (i.e., high ratios) where only a single resonance is present. The resonances attained are consistent with the previously established resonances for the Solar System, i.e., specifically, in regards to the asteroid belt. Previous work employed digital filtering and Lyapunov characteristic exponents to determine stochasticity of the eccentricity, which is found to be consistent with our usage of Lyapunov exponents as an alternate approach based on varying the mass ratio instead of the eccentricity. Our results are expected to be of principal interest to future studies, including augmentations to observed or proposed resonances, of extra-solar planets in binary stellar systems.