Numerical Results on a Simple Model for the Confinement of Saturn’s F Ring

  • Luis BenetEmail author
  • Àngel Jorba
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 54)


In this paper we discuss a simple model for the confinement of Saturn’s F ring and present some preliminary numerical results. The model involves the gravitational interaction of independent test particles with Saturn, including its second zonal harmonic, the shepherd moons Prometheus and Pandora, and Titan, the largest of Saturn’s satellites. We perform accurate long-time integrations (3.2 × 106 revolutions of Prometheus) to check if the particle has escaped or remains trapped in the region between the shepherds. A particle escapes if its orbit crosses the region between the shepherds, or if it displays a physical collisions (lies with Hill’s region) with them. We find a wide region of initial conditions of the test particle that remain confined. We carry out a frequency analysis and use the ratio of the standard deviation over the average main frequencies as stability index. This indicator separates clearly the set of trapped initial conditions of the test particles, displaying some localised structures for the most stable ones. Retaining only those particles which are more stable according to our indicator, we obtain a narrow elliptic ring displaying sharp edges which agrees with the nominal location of Saturn’s F ring.


Test Particle Stability Index Time Dependent Perturbation Kepler Motion Ring Particle 
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This work was initiated during LB’s sabbatical year at the Universitat de Barcelona, which was partially supported by DGAPA (UNAM) and Min. Educación (SAB2010-0123, Spain). We acknowledge financial support provided by the projects IN–110110 (DGAPA–UNAM), 79988 and 144684 (CONACyT), 334309729-9729-4-9 (Min. Ciencia e Innovación), MTM2009-09723 (Min. Ciencia e Innovación) and 2009 SGR 67 (Generalitat de Catalunya). It is a pleasure to thank Carles Simó for his encouragement, valuable comments, questions and discussions.


  1. 1.
    Benet, L., Seligman, T.H.: Generic occurrence of rings in rotating scattering systems. Phys. Lett. A 273, 331–337 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Charnoz, S., Dones, L., Esposito, L.W., Estrada, P.R., Hedman, M.M.: Origin and evolution of Saturn’s ring system. In: Dougherty, M.K., Esposito, L.W., Krimigis, S.M. (eds.) Saturn from Cassini-Huygens, pp. 537–575. Springer, Dordrecht (2009)CrossRefGoogle Scholar
  3. 3.
    Esposito, L.W.: Planetary Rings. Cambridge University Press, Cambridge (2006)Google Scholar
  4. 4.
    French, R.G., McGhee, C.A., Dones, L., Lissauer, J.J.: Saturn’s wayward shepherds: the peregrinations of Prometheus and Pandora. Icarus 162, 143–160 (2003)CrossRefGoogle Scholar
  5. 5.
    Goldreich, P., Tremaine, S.D.: Towards a theory for the Uranian rings. Nature 277, 97–99 (1979)CrossRefGoogle Scholar
  6. 6.
    Gómez, G., Mondelo, J.M., Simó, C.: A collocation method for the numerical Fourier analysis of quasiperiodic functions I: numerical tests and examples. DCDS Ser. B 14, 41–74 (2010)CrossRefzbMATHGoogle Scholar
  7. 7.
    Jacobson, R.A., et al.: The gravity field of the Saturnian system from satellite observations and spacecraft tracking data. Astron. J. 132, 2520 (2006)CrossRefGoogle Scholar
  8. 8.
    Jorba, À., Simó, C.: On quasiperiodic perturbations of elliptic equilibrium points. SIAM J. Math. Anal. 27, 1704–1737 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Jorba, À., Villanueva, J.: On the persistence of lower dimensional invariant tori under quasi-periodic perturbations. J. Nonlinear Sci. 7, 427–473 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jorba, À., Zou, M.: A software package for the numerical integration of ODE by means of high-order Taylor methods. Exp. Math. 14, 99–117 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Laskar, J.: The chaotic behavior of the solar system: a numerical estimate of the chaotic zones. Icarus 88, 266–291 (1992)CrossRefGoogle Scholar
  12. 12.
    Laskar, J.: Introduction to frequency map analysis. In: Simó, C. (ed) Hamiltonian Systems with Three or More Degrees of Freedom. NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 533, pp. 134–150. Kluwer, Dordrecht (1999)CrossRefGoogle Scholar
  13. 13.
    Merlo, O., Benet, L.: Strands and braids in narrow planetary rings: a scattering system approach. Celest. Mech. Dyn. Astron. 97, 49–72 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Murray, C.D., Beurle, K., Cooper, N.J., Evans, M.W., Williams G.A., Charnoz, S.: The determination of the structure of Saturn’s F ring by neraby moonlets. Nature 453, 739–744 (2008)CrossRefGoogle Scholar
  15. 15.
    Ohtsuki, K.: Capture probability of colliding planetesimals – dynamical constraints on accretion of planets, satellites, and ring particles. Icarus 106, 228–246 (1993)CrossRefGoogle Scholar
  16. 16.
    Poulet, F., Sicardy, B., Nicholson, P.D., Karkoschka, E., Caldwell, J.: Saturn’s ring-plane crossings of August and November 1995: a model for the new F-ring objects. Icarus 144, 135–148 (2000)CrossRefGoogle Scholar
  17. 17.
    Seidelmann, P.K., et al.: Report of the IAU/IAG working group on cartographic coordinates and rotational elements: 2006. Celest. Mech. Dyn. Astron. 98, 155–180 (2007)CrossRefzbMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Instituto de Ciencias FísicasUniversidad Nacional Autónoma de México (UNAM)CuernavacaMexico
  2. 2.Departament de Matemàtica Aplicada i AnàlisiUniversitat de BarcelonaBarcelonaSpain

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