Central stable/unstable manifolds and the destruction of KAM tori in the planar Hill problem

Abstract

The classical Hill’s problem is a simplified version of the restricted three body problem (RTBP) where the distance of the two massive bodies (say, primary for the largest one and secondary for the smallest) is made infinity through the use of Hill’s variables and a limiting procedure so that a neighborhood of the secondary can be studied in detail. In this way it is the zeroth-order approximation in powers of μ 1/3. The Levi-Civita regularization takes the Hamiltonian into the form of two uncoupled harmonic oscillators perturbed by the Coriolis force and the Sun action, polynomials of degree 4 and 6, respectively. The goal of this paper is multiple. It presents a detailed description of the main features, including a global description of the dynamics, when the zero velocity curve (zvc) confines the motion. Then it focuses on the collinear equilibrium points and its nearby periodic orbits. Several homoclinic and heteroclinic connections are displayed. Persistence of confined motion when the zvc opens is one of the major concerns. The geometrical behavior of the center-stable/unstable manifolds of the libration points L 1 and L 2 is studied. Suitable Poincaré sections make apparent the relation between these manifolds and the destruction of the invariant KAM tori surrounding the secondary. These results extend immediately to the RTBP. Some practical applications to astronomy and space missions are mentioned. The methodology presented here can be useful on a more general framework for many readers in other areas and not only in celestial mechanics.