Stability of outer planetary orbits around binary stars: A comparison of Hill's and Laplace's stability criteria

Abstract

A comparison is made between the stability criteria of Hill and that of Laplace to determine the stability of outer planetary orbits encircling binary stars. The restricted, analytically determined results of Hill’s method by Szebehely and co-workers and the general, numerically integrated results of Laplace’s method by Graziani and Black are compared for varying values of the mass parameter μ = m2/(m1 + m2). For 0 ≤ μ ≤ 0.15, the closest orbit (lower limit of radius) an outer planet in a binary system can have and still remain stable is determined by Hill’s stability criterion. For μ > 0.15, the critical radius is determined by Laplace’s stability criterion. It appears that the Graziani-Black stability criterion describes the critical orbit within a few percent for all values of μ.