Instability times in the HD 181433 exoplanetary system

Abstract

The present work consists of a study of the dynamical stability of a three-body system that takes advantage of the Shannon entropy approach to estimate the diffusivity (DS) in a Delaunay’s action-like phase space. We outline the main features of a numerical computation ofDSfrom the solutions of the equations of motion and, thereupon, we consider how to estimate a macroscopic instability timescale,τinst, (roughly speaking, the lifetime of the system) associated with a given set of initial conditions. Through such estimates, we are able to characterize the system’s space of initial conditions in terms of its orbital stability by applying numerical integrations to the construction of dynamical maps. We compare these measures of chaotic diffusion with other indicators, first in a qualitative fashion and then more quantitatively, by means of long direct integrations. We address an analysis of a particular, near-resonant system, namely HD 181433, and we show that the entropy may provide a complementary analysis with regard to other dynamical indicators. This work is part of a series of studies devoted to presenting the Shannon entropy approach and its possibilities as a numerical tool providing information on chaotic diffusion and the dynamical stability of multidimensional dynamical systems.