The Capture of Particles by Chaotic Resonances During Orbital Migration

Because a chaotic zone can reduce the long timescale capture probabilities and cause catastrophic events such as close encounters with a planet or star during temporary capture, the dynamics of migrating planets is likely to be strongly dependent on the widths of the resonance chaotic zones. Previous theoretical work on the resonant capture of particles into mean-motion resonances by orbital migration has been restricted to the study of integrable models. By exploring toy drifting Hamiltonian models we illustrate how the structure in phase space of a resonance can be used to generalize this integrable capture theory to include the richer phenomenology of chaotic resonances. We show that particles are temporarily captured into the chaotic zone of a resonance with fixed shape for a time that is given by the width of the chaotic zone divided by the resonance drift rate. Particles can be permanently captured into a drifting chaotic resonance only if they are captured into a growing integrable region. Therefore resonances containing wide chaotic zones have lower permanent capture probabilities than those lacking chaotic zones. We expect large deviations from the predictions of integrable capture theories when the chaotic zone is large and the migration rate is sufficiently long that many Lyapunov times pass while particles are temporarily captured in the resonance. The 2:1 mean-motion resonance with Neptune in the Kuiper Belt contains a chaotic zone, even when integrated on short timescales such as a million years. Capture probabilities are lower than estimated previously from drifting integrable models. This may offer an explanation for low eccentricity Kuiper Belt objects between 45-47 AU which should have been previously captured in the 2:1 resonance by Neptune's migration.