Analytical Treatment of Planetary Resonances

TL;DR

This paper develops an approximate, integrable theory for first-order planetary resonances, providing insights into their dynamical behavior, resonance overlap, chaos onset, and orbital configurations in extrasolar planetary systems.

Contribution

It introduces a novel analytical framework for understanding planetary resonances, including criteria for secondary resonances and chaos, advancing theoretical comprehension of planetary system dynamics.

Findings

Resonance overlap leads to rapid chaos in planetary systems.

Secondary resonances can be analytically characterized.

Divergent encounters in the adiabatic regime cause apsidal anti-alignment.

Abstract

An ever-growing observational aggregate of extrasolar planets has revealed that systems of planets that reside in or near mean-motion resonances are relatively common. While the origin of such systems is attributed to protoplanetary disk-driven migration, a qualitative description of the dynamical evolution of resonant planets remains largely elusive. Aided by the pioneering works of the last century, we formulate an approximate, integrable theory for first-order resonant motion. We utilize the developed theory to construct an intuitive, geometrical representation of resonances within the context of the unrestricted three-body problem. Moreover, we derive a simple analytical criterion for the appearance of secondary resonances between resonant and secular motion. Subsequently, we demonstrate the onset of rapid chaotic motion as a result of overlap among neighboring first-order mean-motion resonances, as well as the appearance of slow chaos as a result of secular modulation of the planetary orbits. Finally, we take advantage of the integrable theory to analytically show that, in the adiabatic regime, divergent encounters with first-order mean-motion resonances always lead to persistent apsidal anti-alignment.