The dynamics of the de Sitter resonance

TL;DR

This paper analyzes the stability and dynamics of the de Sitter resonance among the Galilean satellites, using Hamiltonian normal form to understand equilibrium configurations and their relation to Laplace resonances.

Contribution

It introduces a Hamiltonian normal form approach to study the de Sitter resonance and clarifies its relation to Laplace resonant states, providing insights into stability and sensitivity.

Findings

Normal form accurately locates equilibrium positions

Phase-plane analysis reveals Laplace-like configurations

Sensitivity measures of the equilibrium to perturbations

Abstract

We study the dynamics of the de Sitter resonance, namely the stable equilibrium configuration of the first three Galilean satellites. We clarify the relation between this family of configurations and the more general Laplace resonant states. In order to describe the dynamics around the de Sitter stable equilibrium, a one-degree of freedom Hamiltonian normal form is constructed and exploited to identify initial conditions leading to the two families. The normal form Hamiltonian is used to check the accuracy in the location of the equilibrium positions. Besides, it gives a measure of how sensitive it is with respect to the different perturbations acting on the system. By looking at the phase-plane of the normal form, we can identify a \sl Laplace-like \rm configuration, which highlights many substantial aspects of the observed one.