Celestial Mechanics and Dynamical Astronomy

, Volume 59, Issue 2, pp 129–148 | Cite as

Surfaces of section in the Miranda-Umbriel 3:1 inclination problem

  • Michèle Moons
  • Jacques Henrard


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.

Key words

orbit-orbit resonance chaotic layers secondary resonances perturbation theory 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 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.Google Scholar
  2. Dermott, S. F. and Nicholson, P. D.: 1986, ‘Masses of the Satellites of Uranus’,Nature 319, 115–120.Google Scholar
  3. 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.Google Scholar
  4. 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.Google Scholar
  5. Henrard, J.: 1982, ‘Capture into Resonance: An Extension of the Use of the Adiabatic Invariants’,Celest. Mech. 27, 3–22.Google Scholar
  6. Henrard, J.: 1990, ‘A Semi-Numerical Perturbation Method for Separable Hamiltonian Systems’,Celest. Mech. 49, 43–68.Google Scholar
  7. Henrard J. and Henrard, M.: 1991, ‘Slow Crossing of a Stochastic Layer’,Physica D 54, 135–146.Google Scholar
  8. Henrard J. and Moons, M.: 1992, ‘Capture Probabilities for Secondary Resonances’,Icarus 95, 244–252.Google Scholar
  9. Henrard, J. and Morbidelli, A.: 1993, ‘Slow Crossing of a Stochastic Layer’,Physica D 68, 187–200.Google Scholar
  10. Henrard, J. and Sato, M.: 1990, ‘The Origin of Chaotic Behaviour in the Miranda Umbriel 3:1 Resonances’,Celest. Mech. 47, 391–417.Google Scholar
  11. Laskar, J.: 1986, ‘A General Theory for the Uranian Satellites’,Astron. Astrophys. 166, 349–358.Google Scholar
  12. Malhotra, R.: 1990, ‘Capture Probabilities for Secondary Resonances’,Icarus 87, 249–264.Google Scholar
  13. Malhotra, R. and Dermott, S. F.: 1990, ‘The role of secondary resonances in the orbital history of Miranda’,Icarus 85, 444–480.Google Scholar
  14. Morbidelli, A.: 1993, ‘On the Successive Elimination of Perturbation Harmonics’,Celest. Mech. 55, 101–130.Google Scholar
  15. Peale, S. J.: 1988, ‘Speculative Histories of the Uranian Satellite System’,Icarus 74, 153–171.Google Scholar
  16. 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.Google Scholar
  17. 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.Google Scholar
  18. 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.Google Scholar
  19. 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.Google Scholar
  20. Wisdom, J.: 1985, ‘A Perturbative Treatment of Motion near the 3/1 Commensurability’,Icarus 63, 272–289.Google Scholar

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • Michèle Moons
    • 1
  • Jacques Henrard
    • 1
  1. 1.Département de Mathématique FUNDP 8NamurBelgique

Personalised recommendations