The Instability Transition for the Restricted 3-Body Problem. III. The Lyapunov Exponent Criterion

Abstract

We establish a criterion for the stability of planetary orbits in stellar binary systems by using Lyapunov exponents and power spectra for the special case of the circular restricted 3-body problem (CR3BP). The centerpiece of our method is the concept of Lyapunov exponents, which are incorporated into the analysis of orbital stability by integrating the Jacobian of the CR3BP and orthogonalizing the tangent vectors via a well-established algorithm originally developed by Wolf et al. The criterion for orbital stability based on the Lyapunov exponents is independently verified by using power spectra. The obtained results are compared to results presented in the two previous papers of this series. It is shown that the maximum Lyapunov exponent can be used as an indicator for chaotic behaviour of planetary orbits, which is consistent with previous applications of this method, particularly studies for the Solar System. The chaotic behaviour corresponds to either orbital stability or instability, and it depends solely on the mass ratio of the binary components and the initial distance ratio of the planet relative to the stellar separation distance. Our theoretical results allow us to link the study of planetary orbital stability to chaos theory noting that there is a large array of literature on the properties and significance of Lyapunov exponents. Although our results are given for the special case of the CR3BP, we expect that it may be possible to augment the proposed Lyapunov exponent criterion to studies of planets in generalized stellar binary systems, which is strongly motivated by existing observational results as well as results expected from ongoing and future planet search missions.

10 pages, 8 figures, 3 tables; accepted by Astronomy and Astrophysics

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• Two Suns in The Sky: Stellar Multiplicity in Exoplanet Systems
• DARWIN - A Mission to Detect, and Search for Life on, Extrasolar Planets
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• Stability of prograde and retrograde planets in circular binary systems
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• Habitability of Earth-type Planets and Moons in the Kepler-16 System
• Orbital Stability of Circumstellar Planets in Binary Systems
• Chaotic dynamics of the planet in HD 196885 AB
• The Stability of the Suggested Planet in the nu Octantis System: A Numerical and Statistical Study
• Application of chaos indicators in the study of dynamics of S-type extrasolar planets in stellar binaries
• Study of Resonances for the Restricted 3-Body Problem
• Partial suppression of chaos in relativistic three-body problems
• The PAIRS project: a global formation model for planets in binaries. II. Gravitational perturbation effects from secondary stars