A generalization of the Lagrangian points: Studies of resonance for highly eccentric orbits

We develop a framework based on energy kicks for the evolution of high-eccentricity long-period orbits in the circular planar restricted three-body problem with Jacobi constant close to 3 and with secondary-to-primary mass ratio μ ≪ 1. We use this framework to explore mean motion resonances between the test particle and the massive bodies. This approach leads to a redefinition of resonance orders for the high-eccentricity regime, in which a p: (p + q) resonance is called "pth order" instead of the usual "qth order" to reflect the importance of interactions at periapse. This approach also produces a pendulum-like equation describing the librations of resonance orbits about fixed points that correspond to periodic trajectories in the rotating frame. A striking analogy exists 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; we call the new fixed points the "generalized Lagrangian points." Finally, our approach gives a condition a ∼ μ^-2/5 for the onset of chaos at large semimajor axis a; a range μ < ∼5 × 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; and a simple explanation for the presence of asymmetric librations in exterior 1:N resonances and the absence of these librations in other exterior resonances.