Instability times in the HD 181433 exoplanetary system

Abstract

The present work consists on a study of the dynamical stability of a 3-body system, taking advantage of the Shannon entropy approachto estimate the diffusivity (D S ) in a Delaunay's action-like phase space. We outline the main features of a numerical computation ofD S from the solutions of the equations of motion, and therefrom how to estimate a macroscopic instability time-scale τ inst (roughlyspeaking, the lifetime of the system) associated to a given set of initial conditions. Through such estimates we characterize the system?sspace of initial conditions in terms of its orbital stability, applying numerical integrations on the construction of dynamical maps. Wecompare these measures of chaotic diffusion with other indicators, firstly in a qualitative fashion, and then more quantitatively bymeans of long direct integrations. We address an analysis of a particular, near-resonant system, namely the HD 181433, and show thatthe entropy may provide a complementary analysis in regard to other dynamical indicators. This work is part of a series of studiesdevoted to both present the Shannon entropy approach and expose its possibilities, as a numerical tool, to provide information onchaotic diffusion and dynamical stability of multidimensional dynamical systems.