Breakdown of homoclinic orbits to L3 in the RPC3BP (II). An asymptotic formula

The Restricted 3-Body Problem models the motion of a body of negligible mass under the gravitational influence of two massive bodies called the primaries. If one assumes that the primaries perform circular motions and that all three bodies are coplanar, one has the Restricted Planar Circular 3-Body Problem (RPC3BP). In rotating coordinates, it can be modeled by a two degrees of freedom Hamiltonian, which has five critical points called the Lagrange points L1,…,L5. The Lagrange point L3 is a saddle-center critical point which is collinear with the primaries and beyond the largest of the two. In this paper, we obtain an asymptotic formula for the distance between the stable and unstable manifolds of L3 for small values of the mass ratio 0<μ≪1. In particular we show that L3 cannot have (one round) homoclinic orbits. If the ratio between the masses of the primaries μ is small, the hyperbolic eigenvalues of L3 are weaker, by a factor of order μ, than the elliptic ones. This rapidly rotating dynamics makes the distance between manifolds exponentially small with respect to μ. Thus, classical perturbative methods (i.e. the Melnikov-Poincaré method) can not be applied. The obtention of this asymptotic formula relies on the results obtained in the prequel paper [10] on the complex singularities of the homoclinic of a certain averaged equation and on the associated inner equation. In this second paper, we relate the solutions of the inner equation to the analytic continuation of the parameterizations of the invariant manifolds of L3 via complex matching techniques. We complete the proof of the asymptotic formula for their distance showing that its dominant term is the one given by the analysis of the inner equation.