Stability limits in double stars

A study of inclined planetary orbits

Institute for Astronomy, University of Vienna, Türkenschanzstrasse 17, 1180 Vienna, Austria

Corresponding author: E. Pilat-Lohinger, lohinger@astro.univie.ac.at

Received: 25 July 2002
Accepted: 12 November 2002

Abstract

We treat the problem of the stability of planetary orbits in double stars with the aid of numerical studies in the model of the elliptic restricted three–body problem. Many investigations exist for the limits of the stability of the S-type planetary motion (i.e. the motion around one component of the binary). In this paper we present a numerical investigation of the so-called P-type orbits which are defined as orbits of the planet around both primary masses. The growing interest of such stability studies is due to the discovery of the numerous extra-solar planets, where five of them move in double star systems. Two different methods are used; in a first study the stability was investigated systematically for different initial conditions of the planet, which is regarded to be massless. We vary the initial starting distance from the barycenter (always with velocities corresponding to circular orbits at the beginning), the angle to the connecting line of the binary and the starting position of the primaries (peri-astron and apo-astron); the primaries' masses are considered to be equal. The numerical stability criterion is set to the condition that during the whole integration time (50 000 periods of the primaries) the planet should not suffer from a close encounter to one of the primaries. The binary's eccentricity (e) is increased form 0 to 0.5 with , and the initial distance () to the barycenter is taken of the range from 1.8 to (for ) or to (for ) with . For the first time, we study inclined P-type orbits ( with a step of ) for which we analyse the border line of the stable zone. Additionally we also record the escape times for all orbits. In the second part of our study we apply the Fast Lyapunov Indicator (FLI) to distinguish between the differnt types of motion, taking the same initial conditions, only the grid for the initial distance to the barycenter was finer (). Since the FLI is known to be a fast chaos indicator we reduced the integration time to 10⁴ periods of the primaries. A comparison of the results of the two studies show that they are in good agreement. It turned out that the inclination does not influence the stability significantly. The stability limit itself varies between (for and ) and (for and ).