Application of chaos indicators in the study of dynamics of S-type extrasolar planets in stellar binaries

TL;DR

This study applies chaos indicators like Hill stability, Lyapunov exponents, and MEGNO maps to analyze the orbital stability of S-type extrasolar planets in binary star systems, providing a comprehensive dynamical assessment.

Contribution

It introduces a combined approach using multiple chaos indicators to evaluate the stability of planets in binary systems, enhancing the understanding of their long-term orbital behavior.

Findings

Planets show stability and quasi-periodicity for at least ten million years.

MLE and MEGNO maps effectively identify stable and unstable regions.

The Hill stability criterion aligns well with other chaos indicators.

Abstract

The orbits of two individual planets in two known binary star systems, \gamma Cephei and HD 196885 are numerically integrated using various numerical techniques to assess the chaotic or quasi-periodic nature of the dynamical system considered. The Hill stability (HS) function which measures the orbital perturbation of a planet around the primary star due to the secondary star is calculated for each system. The maximum Lyapunov exponent (MLE) time series are generated to measure the divergence/convergence rate of stable manifolds, which are used to differentiate between chaotic and non-chaotic orbits. Then, we calculate dynamical Mean Exponential Growth factor of Nearby Orbits (MEGNO) maps from solving the variational equations along with the equations of motion. These maps allow us to accurately differentiate between stable and unstable dynamical systems. The results obtained from the analysis of HS, MLE, and MEGNO maps are analysed for their dynamical variations and resemblance. The HS test for the planets shows stability and quasi-periodicity for at least ten million years. The MLE and the MEGNO maps have also indicated the local quasi-periodicity and global stability in a relatively short integration period. The orbital stability of the systems is tested using each indicator for various values of planet inclinations (i_{pl} \le 25^\circ) and binary eccentricities. The reliability of HS criterion is also discussed based on its stability results compared with the MLE and MEGNO maps.