The instability transition for the restricted 3-body problem

II. The hodograph eccentricity criterion

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

Received: 14 May 2009
Accepted: 20 January 2010

Abstract

Aims. We present a new method that allows identifying the onset of orbital instability, as well as quasi-periodicity and multi-periodicity, for planets in binary systems. This method is given for the special case of the circular restricted 3-body problem (CR3BP).

Methods. Our method relies on an approach given by differential geometry that analyzes the curvature of the planetary orbit in the synodic coordinate system. The centerpiece of the method consists in inspecting the effective (instantaneous) eccentricity of the orbit based on the hodograph in rotated coordinates and in calculating the mean and median values of the eccentricity distribution.

Results. Orbital stability and instability can be mapped by numerically inspecting the hodograph and/or the effective eccentricity of the orbit in the synodic coordinate system. The behavior of the system depends solely on the mass ratio μ of the binary components and the initial distance ratio of the planet relative to the stellar separation distance noting that the stellar components move on circular orbits. Our study indicates that orbital instability occurs when the median of the effective eccentricity distribution exceeds unity. This instability criterion can be compared to other criteria, including those based on Jacobi's integral and the zero-velocity contour of the planetary orbit.

Conclusions. The method can be used during detailed numerical simulations and in contrast to other methods such as methods based on the Lyapunov exponent does not require a piece-wise secondary integration of the planetary orbit. Although the method has been deduced for the CR3BP, it is likely that it can be expanded to more general cases.