The instability transition for the restricted 3-body problem

I. Theoretical approach

¹ Department of Physics, Science Hall, University of Texas at Arlington, Arlington, TX 76019-0059, USA e-mail: [wjeberle;cuntz;zmusielak]@uta.edu
² Institut für Theoretische Astrophysik, Universität Heidelberg, Albert Überle Str. 2, 69120 Heidelberg, Germany
³ Kiepenheuer-Institut für Sonnenphysik, Schöneckstr. 6, 79104 Freiburg, Germany

Received: 10 March 2008
Accepted: 24 June 2008

Abstract

Aims. We study the onset of orbital instability for a small object, identified as a planet, that is part of a stellar binary system with properties equivalent to the restricted three body problem.

Methods. Our study is based on both analytical and numerical means and makes use of a rotating (synodic) coordinate system keeping both binary stars at rest. This allows us to define a constant of motion (Jacobi's constant), which is used to describe the permissible region of motion for the planet. We illustrate the transition to instability by depicting sets of time-dependent simulations with star-planet systems of different mass and distance ratios.

Results. Our method utilizes the existence of an absolute stability limit. As the system parameters are varied, the permissible region of motion passes through the three collinear equilibrium points, which significantly changes the type of planetary orbit. Our simulations feature various illustrative examples of instability transitions.

Conclusions. Our study allows us to identify systems of absolute stability, where the stability limit does not depend on the specifics or duration of time-dependent simulations. We also find evidence of a quasi-stability region, superimposed on the region of instability, where the planetary orbits show quasi-periodic behavior. The analytically deduced onset of instability is found to be consistent with the behavior of the depicted time-dependent models, although the manifestation of long-term orbital stability will require more detailed studies.