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Chaotic Layers in Resonance Problems

Published online by Cambridge University Press:  07 August 2017

Jacques Henrard ,
Michèle Moons  and
Alessandro Morbidelli
Jacques Henrard
Affiliation:
Département de mathématique FUNDP 8, Rempart de la Vierge, B-5000 Namur, Belgique
Michèle Moons
Affiliation:
Département de mathématique FUNDP 8, Rempart de la Vierge, B-5000 Namur, Belgique
Alessandro Morbidelli
Affiliation:
Département de mathématique FUNDP 8, Rempart de la Vierge, B-5000 Namur, Belgique

Abstract

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The recent numerical simulations of Tittemore and Wisdom (1988,1989,1990) and Dermott et al. (1988), Malhotra and Dermott (1990) concerning the tidal evolution through resonances of some pairs of Uranian satellites have revealed interesting dynamical phenomena related to the interactions between close-by resonances. These interactions produce chaotic layers and strong secondary resonances. The slow evolution of the satellite orbits in this dynamical lanscape is responsible for temporary capture into resonance, enhancement of eccentricity or inclination and subsequent escape from resonance. The present contribution aims at developing analytical tools for predicting the location and size of chaotic layers and secondary resonances. The problem of the 1:3 inclination resonance between Miranda and Umbriel is analysed.

Keywords

orbit-orbit resonancechaotic layerssecondary resonanceperturbation theory
Type
Part IV - Planetary Satellites
Information
Symposium - International Astronomical Union , Volume 152: Chaos, Resonance and Collective Dynamical Phenomena in the Solar System , 1992 , pp. 189 - 208
Copyright
Copyright © Kluwer 1992 

References

Arnold, V.I., (1963): “On a theorem of Liouville concerning integrable problems of dynamics”, Sib. mathem. zh. , 4, 2.Google Scholar
Delaunay, C., (1867): “Theorie du mouvement de la Lune”, Mem. Acad. Sci., Paris , 29.Google Scholar
Dermott, S.F.: 1984, “Origin and evolution of the Uranian and Neptunian satellites: some dynamical considerations”, in Uranus and Neptune (Bergstrahl, J Ed.), Nasa Conf. Pub. 2330, pp 377–404.Google Scholar
Dermott, S.F., Malhotra, R. and Murray, C.D.: 1988, “Dynamics of the Uranian and Saturnian satellite systems: A chaotic route to melting Miranda?” Icarus , 76, 295–334.CrossRefGoogle Scholar
Goldreich, P.: 1965, “An explanation of the frequent occurence of commensurable mean motions in the Solar system”, M.N.R.A.S. , 130, 159–181.CrossRefGoogle Scholar
Henrard, J.: 1982, “Capture into resonance: An extension of the use of the adiabatic invariants”, Celest. Mech. , 27, 3–22.CrossRefGoogle Scholar
Henrard, J.: 1990, “A semi-numerical perturbation method for separable Hamiltonian systems”, Celest. Mech. , 49, 43–68.CrossRefGoogle Scholar
Henrard, J. and Lemaître, A.: 1983, “A second fundamental model for resonance”, Celest. Mech. , 30, 197–218.CrossRefGoogle Scholar
Henrard, J., and Lemaître, A.: 1986, “A perturbation method for problems with two critical arguments”, Celest. Mech. , 39, 213–238.CrossRefGoogle Scholar
Henrard, J., and Moons, M.: 1992, “Capture probabilities for secondary resonances”, Submitted to Icarus.CrossRefGoogle Scholar
Malhotra, R. and Dermott, S.F.: 1990, “The role of secondary resonances in the orbital history of Miranda”, Icarus , 85,444–480.CrossRefGoogle Scholar
Moons, M., and Henrard, J.: 1992, “Surfaces of section in the Miranda-Umbriel 1:3 inclination problem”, Submitted to Celestial Mechanics.Google Scholar
Morbidelli, A. (1992): “On the successive eliminations of perturbation harmonics”, submitted to Celest. Mech. Google Scholar
Morbidelli, A., and Giorgilli, A., (1992): “Quantitative perturbation theory by successive elimination of harmonics”, in preparation.CrossRefGoogle Scholar
Peale, S.J.: 1986, “Orbital resonance, unusual configurations and exotic rotation states”, in Satellites (Burns, J. and Matthews, M. eds.), Univ. of Arizona Press, 159–223.CrossRefGoogle Scholar
Peale, S.J.: 1988, “Speculative histories of the Uranian satellite system”, Icarus , 74, 153–171.CrossRefGoogle Scholar
Tittemore, W.C. and Wisdom, J.: 1988, “Tidal evolution of the Uranian satellites. I. Passage of Ariel and Umbriel through the 5:3 mean-motion commensurability”, Icarus , 74, 172–230.CrossRefGoogle Scholar
Tittemore, W.C. and Wisdom, J.: 1989, “Tidal evolution of the Uranian satellites. II. An explanation of the anomalously high orbital inclination of Miranda”, Icarus , 78, 63–89.CrossRefGoogle Scholar
Tittemore, W.C. and Wisdom, J.: 1990, “Tidal evolution of the Uranian satellites. III. Evolution through the Miranda-Umbriel 3:1, Miranda-Ariel 5:3, and Arial-Umbriel 2:1 Mean-Motion Commensurabilities”, Icarus , 85, 394–443.CrossRefGoogle Scholar
Yoder, C.: 1973, “On the establishment and evolution of orbit-orbit resonances”, Ph.D. Dissertation , U. of California, Santa Barbara.Google Scholar
Yoder, C.F.: 1979, “Diagrammatic theory of transition of pendulum-like systems”, Celest. Mech. , 19, 3–29.CrossRefGoogle Scholar