Shannon Entropy Applied to the Planar Restricted Three-Body Problem

TL;DR

This paper demonstrates that Shannon entropy can efficiently analyze the stability of the planar restricted three-body problem, providing reliable results with less computational effort compared to traditional methods.

Contribution

It introduces the use of Shannon entropy for dynamical analysis in planetary systems, enabling faster and ensemble-free stability assessments near mean-motion resonances.

Findings

Entropy method yields comparable results to variance analysis

Entropy provides reliable stability results without ensembles

Good agreement with N-body simulations on instability times

Abstract

We present a numerical study of the application of the Shannon entropy technique to the planar restricted three-body problem in the vicinity of first-order interior mean-motion resonances with the perturber. We estimate the diffusion coefficient for a series of initial conditions and compare the results with calculations obtained from the time evolution of the variance in the semimajor-axis and eccentricity plane. Adopting adequate normalization factors, both methods yield comparable results, although much shorter integration times are required for entropy calculations. A second advantage of the use of entropy is that it is possible to obtain reliable results even without the use of ensembles or analysis restricted to surfaces of section or representative planes. This allows for a much more numerically efficient tool that may be incorporated into a working N-body code and applied to numerous dynamical problems in planetary dynamics. Finally, we estimate instability times for a series of initial conditions in the 2/1 and 3/2 mean-motion resonances and compare them with times of escape obtained from directed N-body simulations. We find very good agreement in all cases, not only with respect to average values but also in their dispersion for near-by trajectories