Dynamical Stability of the Laplace Resonance

TL;DR

This paper investigates the stability of the Laplace resonance in multi-planet systems, identifying equilibrium states and analyzing their stability, including potential new equilibria at high eccentricities, with implications for planetary system dynamics.

Contribution

It provides a detailed analysis of the de Sitter equilibria in multi-resonant planetary systems, including the explicit location and stability of these states and potential new equilibria.

Findings

Standard de Sitter equilibrium is Lyapunov stable.

Additional equilibria may exist at large forced eccentricities.

Equilibria are traced up to crossing exact commensurability.

Abstract

We analyse the stability of the de Sitter equilibria in multi-resonant planetary systems. The de Sitter equilibrium is the dynamical state of the Laplace resonance in which all resonant arguments are librating. The sequence of equilibria exists all along the possible states balancing resonance offsets and forced eccentricities. Possible additional new-de Sitter equilibria may exist when at least one of the forced eccentricities is large (the paradigmatic case is Gliese-876). In the present work, these families of equilibria are traced up to crossing exact commensurability, where approximate first-order solutions diverge. Explicit exact location of the equilibria are determined allowing us to verify the Lyapunov stability of the standard de Sitter equilibrium and of the stable branches of the additional ones.