Capture probabilities for secondary resonances

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Abstract

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 iM2 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 iM2 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.

Original languageEnglish (US)
Pages (from-to)249-264
Number of pages16
JournalIcarus
Volume87
Issue number2
DOIs
StatePublished - 1990
Externally publishedYes

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Miranda
Umbriel
orbitals
libration
pendulums
solar system
inclination
estimates
analysis

ASJC Scopus subject areas

  • Space and Planetary Science
  • Astronomy and Astrophysics

Cite this

Capture probabilities for secondary resonances. / Malhotra, Renu.

In: Icarus, Vol. 87, No. 2, 1990, p. 249-264.

Research output: Contribution to journalArticle

Malhotra, Renu. / Capture probabilities for secondary resonances. In: Icarus. 1990 ; Vol. 87, No. 2. pp. 249-264.
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