Analysis of the motion of an extrasolar planet in a binary system

TL;DR

This paper analyzes the motion of extrasolar planets in binary systems using Hamiltonian mechanics, providing qualitative insights into their stability and applying the theory to real systems with comparisons to numerical simulations.

Contribution

It introduces a Hamiltonian-based analytical approach to study planetary motion in binary systems, considering non-negligible planetary mass and applying it to real extrasolar planets.

Findings

Stable and unstable planetary motions are possible.

Theoretical predictions align with numerical integrations.

Possible orbital parameters for real systems are identified.

Abstract

More than 10% of extra-solar planets (EPs) orbit in a binary or multiple stellar system. We investigated the motion of planets revolving in binary systems in the frame of the particular case of the three body problem. We carried out an analysis of the motion an EP revolving in a binary system by following conditions; a) a planet in a binary system revolves around one of the components (parent star), b) the distance between the star`s components is greater than between the parent star and the orbiting planet (ratio of the semi-major axes is a small parameter), c) the mass of the planet is smaller than the mass of the stars, but is not negligible. The Hamiltonian of the system without short periodic terms was used. We expanded the Hamiltonian in terms of Legendre polynomial and truncated after the second order term depending on only one angular variable. In this case the solution of this system was obtained and the qualitative analysis of motion was produced. We have applied this theory to real EPs and compared to the numerical integration. Analyses of the possible regions of motion are presented. It is shown that the case of the stable and unstable motion of the EPs are possible. We applied our calculations to two binary systems hosting an EP and calculated the possible values for their unknown orbital elements.