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The origin of chaotic behaviour in the Miranda-Umbriel 3 : 1 resonances
Authors:Jacques Henrard  Massae Sato
Institution:(1) Département de Mathématique FUNDP 8, Rempart de la Vierge, B-5000 Namur, Belgium;(2) Instituto Astronomico e Geofisico, Universidade de São Paulo, Brazil
Abstract:We have investigated the pericentric resonances through which Miranda and Umbriel are believed to have passed when, due to tidal evolution, their orbital mean motions reached a 3 : 1 commensurability. Our investigation is based upon a perturbative treatment. The predictions of this theory are in good agreement with the results of numerical integrations concerning both the extend of the chaotic layers generated by the separatrices of the primary resonances and the location of the secondary resonances. The effect of tidal evolution is discussed on the bases of the adiatatic invariant theory and its extension to separatrix crossing. We recover qualitatively the mean features of the numerical experiments of Tittermore and Wisdom (1988–1989), Dermott et al (1988) and Malhotra and Dermott (1989).
Keywords:Resonance  chaotic motion  perturbation theory  tidal evolution  Miranda
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