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 mu much less than 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 similar to mu(-2/5) for the onset of chaos at large semimajor axis a; 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; 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.