Weak Stability Boundary and Invariant Manifolds

The concept of weak stability boundary has been successfully used in the design of several fuel efficient space missions. In this paper we give a rigorous definition of the weak stability boundary in the context of the planar circular restricted three-body problem, and we provide a geometric argument for the fact that, for some energy range, the points in the weak stability boundary of the small primary are the points with zero radial velocity that lie on the stable manifolds of the Lyapunov orbits about the libration points L1 and L2, provided that these manifolds satisfy some topological conditions. The geometric method is based on the property of the invariant manifolds of Lyapunov orbits being separatrices of the energy manifold. We support our geometric argument with numerical experiments.