The Resonance Overlap and Hill Stability Criteria Revisited

TL;DR

This paper revisits the resonance overlap and Hill stability criteria in the planar circular restricted three-body problem, providing new estimates, detailed dynamical maps, and insights into the complex boundary between stable and unstable orbits.

Contribution

It introduces refined estimates of the resonance overlap and Hill stability limits, highlighting the complex structure of the stability boundary and proposing a dual critical semimajor axis criterion.

Findings

The boundary between stable and unstable orbits is complex and generated by superimposed resonances.

Two critical semimajor axes define stability: one Hill-stability limit and one near a resonance overlap condition.

The Hill-stability limit depends on eccentricity, while the resonance overlap limit is nearly eccentricity independent.

Abstract

We review the orbital stability of the planar circular restricted three-body problem, in the case of massless particles initially located between both massive bodies. We present new estimates of the resonance overlap criterion and the Hill stability limit, and compare their predictions with detailed dynamical maps constructed with N-body simulations. We show that the boundary between (Hill) stable and unstable orbits is not smooth but characterized by a rich structure generated by the superposition of different mean-motion resonances which does not allow for a simple global expression for stability. We propose that, for a given perturbing mass $m_1$ and initial eccentricity $e$, there are actually two critical values of the semimajor axis. All values $a < a_{\rm Hill}$ are Hill-stable, while all values $a > a_{\rm unstable}$ are unstable in the Hill sense. The first limit is given by the Hill-stability criterion and is a function of the eccentricity. The second limit is virtually insensitive to the initial eccentricity, and closely resembles a new resonance overlap condition (for circular orbits) developed in terms of the intersection between first and second-order mean-motion resonances.