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Celestial Mechanics and Dynamical Astronomy

, Volume 91, Issue 1–2, pp 151–171 | Cite as

Resonantly Forced Eccentric Ringlets: Relationships Between Surface Density, Resonance Location, Eccentricity And Eccentricity-Gradient

  • M. D Melita
  • J. C. B Papaloizou
Article

Abstract

We use a simple model of the dynamics of a narrow-eccentric ring, to put some constraints on some of the observable properties of the real systems. In this work we concentrate on the case of the ‘Titan ringlet of Saturn’.

Our approach is fluid-like, since our description is based on normal modes of oscillation rather than in individual particle orbits. Thus, the rigid precession of the ring is described as a global m = 1 mode, which originates from a standing wave superposed on an axisymmetric background. An integral balance condition for the maintenance of the m=1 standing-wave can be set up, in which the differential precession induced by the oblateness of the central planet must cancel the contributions of self-gravity, the resonant satellite forcing and collisional effects. We expect that in nearly circular narrow rings dominated by self-gravity, the eccentricity varies linearly across the ring. Thus, we take a first order expansion and we derive two integral relationships from the rigid-precession condition. These relate the surface density of the ring with the eccentricity at the centre, the eccentricity gradient and the location of the secular resonance.

These relationships are applied to the Titan ringlet of Saturn, which has a secular resonance with the satellite Titan in which the ring precession period is close to Titan’s orbital period. In this case, we estimate the mean surface density and the location of the secular resonance.

Keywords

eccentric ringlets planetary rings 

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References

  1. Borderies, N., Goldreich, P., Tremaine, S. 1983‘The dynamics of elliptical rings’Astron. J8815601568CrossRefGoogle Scholar
  2. Chiang, E. I., Goldreich, P. 2000‘Apse Alignment of Narrow Eccentric Planetary Rings’Astrophys. J54010841090CrossRefGoogle Scholar
  3. Dermott, S. F., Murray, C. D. 1980‘The origin of the eccentricity gradient and the apse alignment of the epsilon-ring of Uranus’Icarus43338349CrossRefGoogle Scholar
  4. French, R. G., Nicholson, P. D., Porco, C. and Marrouf, E. A.: 1984, ‘Dynamics and structure of the uranian rings’, In, Planetary Rings, Richard Greenberg and Andre Brahic (eds.), University of Arizona press, Tucson, Arizona, pp. 513–561. Google Scholar
  5. Goldreich, P., Porco, C. C. 1987‘Shepherding of the uranian rings. II. Dynamics’Astron. J93730737CrossRefGoogle Scholar
  6. Goldreich, P., Tremaine, S. 1979‘Precession of the epsilon ring of Uranus’Astron. J8416381641CrossRefGoogle Scholar
  7. Goldreich, P., Tremaine, S. 1981‘The origin of the eccentricities of the rings of Uranus’Astrophys. J24310621075CrossRefGoogle Scholar
  8. Graps, A. L., Showalter, M. R., Lissauer, J. J., Kary, D.M. 1995‘Optical depths profiles and streamlines of the uranian (epsilon) ring’Astron. J10922622273CrossRefGoogle Scholar
  9. Lebovitz, N. R. 1967Astrophys J.150203212CrossRefGoogle Scholar
  10. Longaretti, P. Y., Rappaport, N. 1995‘Viscous overstabilities in dense narrow planetary rings’Icarus116376396CrossRefGoogle Scholar
  11. Murray, C. D., Dermott, S. 1999Solar System DynamicsCambridge University pressCambridge, United KingdomGoogle Scholar
  12. Mosqueira, I., Estrada, P. R. 2002‘Apse alignment of the uranian rings’Icarus158545556CrossRefGoogle Scholar
  13. Moulton, F. R.: 1935, An Introduction to Celestial Mechanics. Ed: The Macmillan company, London. Google Scholar
  14. Papaloizou, J. C. B. and Melita, M. D.: 2005, Icarus, in press.Google Scholar
  15. Porco, C., Nicholson, P. D., Borderies, N., Danielson, G. E., Goldreich, P., Holberg, J. B., Lane, A. L. 1984‘The eccentric Saturnian rings at 1.29RS and 1.45RSIcarus60116CrossRefGoogle Scholar
  16. Shu, F. H., Yuan, C., Lissauer, J. J. 1985‘Nonlinear spiral density waves: An inviscid theory’Astrophys. J291356376CrossRefGoogle Scholar
  17. Tyler, G. L., Eshleman, V. R., Hinson, D. P., Marouf, E. A., Simpson, R. A., Sweetnam, D. N., Anderson, J. D., Campbell, J. K., Levy, G. S., Lindal, G. F. 1986‘Voyager 2 radio science observations of the uranian system atmosphere, rings, and satellites’Science2337984Google Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Astronomy Unit, Queen MaryUniversity of LondonLondon

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