Abstract
The recent numerical simulations of Tittemore and Wisdom (1988, 1989, 1990) and Dermottet 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 landscape 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 3:1 inclination resonance between Miranda and Umbriel is analysed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Dermott, S. F.: 1984, ‘Origin and Evolution of the Uranian and Neptunian Satellites: Some Dynamical Considerations’, inUranus and Neptune (J. Bergstrahl Ed.), NASA Conf. Pub. 2330, pp. 377–404.
Dermott, S. F. and Nicholson, P. D.: 1986, ‘Masses of the Satellites of Uranus’,Nature 319, 115–120.
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.
Elskens, Y. and Escande, O. F.: 1991, ‘Slowly Pulsating Separatrices Sweep Homoclinic Tangles Where Island Must Be Small: An Extension of Classical Adiabatic Theory’,Nonlinearity 4, 615–667.
Henrard, J.: 1982, ‘Capture into Resonance: An Extension of the Use of the Adiabatic Invariants’,Celest. Mech. 27, 3–22.
Henrard, J.: 1990, ‘A Semi-Numerical Perturbation Method for Separable Hamiltonian Systems’,Celest. Mech. 49, 43–68.
Henrard J. and Henrard, M.: 1991, ‘Slow Crossing of a Stochastic Layer’,Physica D 54, 135–146.
Henrard J. and Moons, M.: 1992, ‘Capture Probabilities for Secondary Resonances’,Icarus 95, 244–252.
Henrard, J. and Morbidelli, A.: 1993, ‘Slow Crossing of a Stochastic Layer’,Physica D 68, 187–200.
Henrard, J. and Sato, M.: 1990, ‘The Origin of Chaotic Behaviour in the Miranda Umbriel 3:1 Resonances’,Celest. Mech. 47, 391–417.
Laskar, J.: 1986, ‘A General Theory for the Uranian Satellites’,Astron. Astrophys. 166, 349–358.
Malhotra, R.: 1990, ‘Capture Probabilities for Secondary Resonances’,Icarus 87, 249–264.
Malhotra, R. and Dermott, S. F.: 1990, ‘The role of secondary resonances in the orbital history of Miranda’,Icarus 85, 444–480.
Morbidelli, A.: 1993, ‘On the Successive Elimination of Perturbation Harmonics’,Celest. Mech. 55, 101–130.
Peale, S. J.: 1988, ‘Speculative Histories of the Uranian Satellite System’,Icarus 74, 153–171.
Smith, B. A., Soderblom, L. A., Beebe, R., Bliss, D., Boyce, J. M., Brahic, A., Briggs, G. A., Brown, R. H., Collins, S. A., Cook II, A. F., Croft, S. K., Cuzzi, J. N., Danielson, G. e., Davies, M. E., Dowling, T. E., Godfrey, D., Hansen, C. J., Harris, C., Hunt, G. E., Ingersoll, A. P., Johnson, T. V., Krauss, R. J., Masursky, H., Morrisson, D., Owen, T., Plescia, J. B., Pollack, J. B., Porco, C. C., Rages, K., Sagan, C., Shoemaker, E. M., Sromovsky, L. A., Stoker, C., Strom, R. G., Suomi, V. E., Synnott, S. P., Terrile, R. J., Thomas, P., Thompson, W. R., and Veverka, J.: 1986, ‘Voyager 2 in the Uranian System: Imaging Science Results’,Science 233, 43–64.
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.
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.
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 Ariel-Umbriel 2:1 Mean-Motion Commensurabilities’,Icarus 85, 394–443.
Wisdom, J.: 1985, ‘A Perturbative Treatment of Motion near the 3/1 Commensurability’,Icarus 63, 272–289.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Moons, M., Henrard, J. Surfaces of section in the Miranda-Umbriel 3:1 inclination problem. Celestial Mech Dyn Astr 59, 129–148 (1994). https://doi.org/10.1007/BF00692129
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00692129