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 language | English (US) |
|---|---|
| Pages (from-to) | 249-264 |
| Number of pages | 16 |
| Journal | Icarus |
| Volume | 87 |
| Issue number | 2 |
| DOIs | |
| State | Published - 1990 |
| Externally published | Yes |
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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 journal › Article
}
TY - JOUR
T1 - Capture probabilities for secondary resonances
AU - Malhotra, Renu
PY - 1990
Y1 - 1990
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=0012466481&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=0012466481&partnerID=8YFLogxK
U2 - 10.1016/0019-1035(90)90133-T
DO - 10.1016/0019-1035(90)90133-T
M3 - Article
AN - SCOPUS:0012466481
VL - 87
SP - 249
EP - 264
JO - Icarus
JF - Icarus
SN - 0019-1035
IS - 2
ER -