## Abstract

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.

Original language | American English |
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Pages (from-to) | 1418-1429 |

Number of pages | 12 |

Journal | Astronomical Journal |

Volume | 128 |

Issue number | 3 1785 |

DOIs | |

State | Published - Sep 2004 |

Externally published | Yes |

## Keywords

- Celestial mechanics
- Minor planets, asteroids
- Solar system: general