New results on orbital resonances

TL;DR

This paper uses non-perturbative numerical methods to reveal complex structures and behaviors in planetary resonances at various eccentricities, challenging previous divergence predictions and uncovering new resonance phenomena.

Contribution

It introduces a novel geometric perspective and numerical analysis that uncovers previously hidden resonance structures and clarifies the nature of resonance widths across eccentricities.

Findings

Resonance widths do not diverge at low or grazing eccentricities.

Discovery of low-eccentricity resonant bridges connecting neighboring resonances.

Identification of bifurcations and phase-shifted resonance zones at high eccentricities.

Abstract

Perturbative analyses of planetary resonances commonly predict singularities and/or divergences of resonance widths at very low and very high eccentricities. We have recently re-examined the nature of these divergences using non-perturbative numerical analyses, making use of Poincar\'e sections but from a different perspective relative to previous implementations of this method. This perspective reveals fine structure of resonances which otherwise remains hidden in conventional approaches, including analytical, semi-analytical and numerical-averaging approaches based on the critical resonant angle. At low eccentricity, first order resonances do not have diverging widths but have two asymmetric branches leading away from the nominal resonance location. A sequence of structures called ``low-eccentricity resonant bridges" connecting neighboring resonances is revealed. At planet-grazing eccentricity, the true resonance width is non-divergent. At higher eccentricities, the new results reveal hitherto unknown resonant structures and show that these parameter regions have a loss of some -- though not necessarily entire -- resonance libration zones to chaos. The chaos at high eccentricities was previously attributed to the overlap of neighboring resonances. The new results reveal the additional role of bifurcations and co-existence of phase-shifted resonance zones at higher eccentricities. By employing a geometric point of view, we relate the high eccentricity phase space structures and their transitions to the shapes of resonant orbits in the rotating frame. We outline some directions for future research to advance understanding of the dynamics of mean motion resonances.