A Generalization of the Lagrangian Points: Studies of Resonance for Highly Eccentric Orbits

Abstract

We develop a framework based on energy kicks for the evolution of high-eccentricity long-period orbits with Jacobi constant close to 3 in the restricted circular planar three-body problem where the secondary and primary masses have mass ratio mu << 1. We use this framework to explore mean-motion resonances between the test particle and the secondary mass. This approach leads to (i) a redefinition of resonance orders to reflect the importance of interactions at periapse; (ii) a pendulum-like equation describing the librations of resonance orbits; (iii) an analogy between these new fixed points and the Lagrangian points as well as between librations around the fixed points and the well known tadpole and horseshoe orbits; (iv) a condition a ~ mu^(-2/5) for the onset of chaos at large semimajor axis a; (v) the existence of a range mu < ~5 x 10^(-6) in secondary mass for which a test particle initially close to the secondary cannot escape from the system, at least in the planar problem; (vi) a simple explanation for the presence of asymmetric librations in exterior 1:N resonances.