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

Abstract

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 $L_1, \ldots, L_5$. The Lagrange point $L_3$ 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 $L_3$ for small values of the mass ratio $0<\mu \ll 1$. In particular we show that $L_3$ cannot have (one round) homoclinic orbits. If the ratio between the masses of the primaries $\mu$ is small, the hyperbolic eigenvalues of $L_3$ are weaker, by a factor of order $\sqrt{\mu}$, than the elliptic ones. This rapidly rotating dynamics makes the distance between manifolds exponentially small with respect to $\sqrt{\mu}$. Thus, classical perturbative methods (i.e. the Melnikov-Poincaré method) can not be applied.