Capture probabilities for secondary resonances

Secondary resonances are commensurabilities between the libration and circulation frequencies of two neighboring mean motion resonances. We present a general analysis of secondary resonances within a perturbed pendulum model. There exists a self-similarity between secondary resonances and primary resonances which is made apparent in this analysis. We derive formulas for the probability of capture into secondary resonances by making use of J. Henrard's (1982, Celest. Mech. 27, 3-22) theory for separatrix crossing transitions. The importance of secondary resonances in long-time evolutions has been discovered in several recent studies of orbital dynamics problems in the solar system. We apply the theory developed here to the tidal evolution of Miranda and Umbriel in the 1:3 i_M² resonance and find significant probabilities of capture into secondary resonances. Our analytical estimates are consistent with previous numerical results, and support the scenario that tidal capture into the 1:3 i_M² resonance followed by disruption of the resonance due to capture into a secondary resonance provides an explanation for the anomalously large orbital inclination (4.3°) of Miranda.