Celestial Mechanics and Dynamical Astronomy

, Volume 103, Issue 3, pp 209–225 | Cite as

Phase-space volume of regions of trapped motion: multiple ring components and arcs

Original article

Abstract

The phase-space volume of regions of regular or trapped motion, for bounded or scattering systems with two degrees of freedom respectively, displays universal properties. In particular, sudden reductions in the phase-space volume or gaps are observed at specific values of the parameter which tunes the dynamics; these locations are approximated by the stability resonances. The latter are defined by a resonant condition on the stability exponents of a central linearly stable periodic orbit. We show that, for more than two degrees of freedom, these resonances can be excited opening up gaps, which effectively separate and reduce the regions of trapped motion in phase space. Using the scattering approach to narrow rings and a billiard system as example, we demonstrate that this mechanism yields rings with two or more components. Arcs are also obtained, specifically when an additional (mean-motion) resonance condition is met. We obtain a complete representation of the phase-space volume occupied by the regions of trapped motion.

Keywords

Phase-space volume of trapped regions Scattering systems Narrow rings Multiple components Ring arcs 

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References

  1. Arnold V.I. (1964) Instability of dynamical systems with several degrees of freedom. Sov. Math. Dokl. 5: 581–585Google Scholar
  2. Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer-Verlag, New York (appendix 7) (1989)Google Scholar
  3. Arnold V.I., Kozlov V.V., Neishtadt A.I. (1988) Mathematical aspects of classical and celestial mechanics. In: Gamkrelidze R.V.(eds) Encyclopedia of Mathematical Sciences, vol. 3.. Springer-Verlag, New YorkGoogle Scholar
  4. Benet L (2001) Occurrence of planetary rings with shepherds. Celest. Mech. Dyn. Astron. 81: 123–128MATHCrossRefADSMathSciNetGoogle Scholar
  5. Benet L., Merlo O. (2004) Phase-space structure for narrow planetary rings. Regul. Chaotic Dyn. 9: 373–384 arXiv:nlin.CD/0410028MATHCrossRefMathSciNetGoogle Scholar
  6. Benet L., Merlo O. (2008) Multiple components in narrow planetary rings. Phys. Rev. Lett. 100: 014102 arXiv:nlin/0702039v2CrossRefADSGoogle Scholar
  7. Benet L., Seligman T.H. (2000) Generic occurrence of rings in rotating scattering systems. Phys. Lett. A. 273: 331–337 arXiv:nlin.CD/0001018MATHCrossRefADSMathSciNetGoogle Scholar
  8. Benet L., Broch J., Merlo O., Seligman T.H. (2005) Symmetry breaking: a heuristic approach to chaotic scattering in many dimensions. Phys. Rev. E 71: 036225CrossRefADSGoogle Scholar
  9. Birkhoff G.D. (1927) On the periodic motions of dynamical systems. Acta Math. 50: 359–379MATHCrossRefMathSciNetGoogle Scholar
  10. Contopoulos G., Dvorak R., Harsoula M., Freistetter F. (2005) Recurrence of order in chaos. Int. J. Bif. Chaos 15: 2865–2882MATHCrossRefMathSciNetGoogle Scholar
  11. Duarte P. (1994) Plenty of elliptic islands for the standard family of area-preserving maps. Ann. Inst. Henri Poincaré 11: 359–409MATHMathSciNetGoogle Scholar
  12. Dvorak, R., Freistetter, F.: 2005, Orbit dynamics, stability and chaos in planetary systems. In: Dvorak, R., Freistetter, F., Kurths, J. (eds.) Chaos and Stability in Planetary Systems, Lecture Notes in Physics, vol. 683, pp. 3–140. Springer, Berlin (2005)Google Scholar
  13. Dullin H.R. (1998) Linear stability in billiards with potential. Nonlinearity 11: 151–173MATHCrossRefMathSciNetGoogle Scholar
  14. Dumas C. et al (1999) Stability of Neptune’s ring arcs in question. Nature 400: 733–735CrossRefADSGoogle Scholar
  15. Esposito L.W. (2006) Planetary Rings. Cambridge University Press, CambridgeGoogle Scholar
  16. Gelfreich V. (2002) Near strongly resonant periodic orbits in a Hamiltonian system. Proc. Natl. Acad. Sci. 99: 13975–13979MATHCrossRefADSMathSciNetGoogle Scholar
  17. Goldreich P., Tremaine S., Borderies N. (1986) Towards a theory for Neptune’s arc rings. Astron. J. 92: 490–494CrossRefADSGoogle Scholar
  18. Hénon M. (1968) Sur les Orbites Interplanétaires qui Rencontrent Deux Fois la Terre. Bull. Astron. 3(3): 337–402Google Scholar
  19. Hitzl D.L., Hénon M. (1977) Critical generating orbits for second species periodic solutions of the restricted problem. Celest. Mech. 15: 421–452MATHCrossRefADSGoogle Scholar
  20. Jorba A., Villanueva J. (1997a) On the normal behaviour of partially elliptic lower dimensional tori of Hamiltonian Systems. Nonlinearity 10: 783–822MATHCrossRefMathSciNetGoogle Scholar
  21. Jorba A., Villanueva J. (1997b) On the persistence of lower dimensional invariant tori under quasi-periodic perturbations. J. Nonlinear Sci. 7: 427–473MATHCrossRefADSMathSciNetGoogle Scholar
  22. Jung C., Lipp C., Seligman T.H. (1999) The inverse scattering problem for chaotic systems. Ann. Phys. 275: 151–189MATHCrossRefMathSciNetGoogle Scholar
  23. Jung C., Mejía-Monasterio C., Merlo O., Seligman T.H. (2004) Self-pulsing effect in chaotic scattering. New J. Phys. 6: 48–00CrossRefADSGoogle Scholar
  24. Merlo O.: Through symmetry breaking to higher dimensional chaotic scattering. PhD Thesis, U. Basel, Switzerland (unpublished, 2004). Online version available at http://pages.unibas.ch/diss/2004/DissB_7024.htm
  25. Merlo O., Benet L. (2007) Strands and braids in narrow planetary rings: a scattering system approach. Celest. Mech. Dyn. Astron. 97: 49–72 arXiv:astro-ph/0609627MATHCrossRefADSMathSciNetGoogle Scholar
  26. Meyer N. et al (1995) Chaotic scattering off a rotating target. J. Phys. A: Math. Gen. 28: 2529–2544CrossRefADSGoogle Scholar
  27. Namouni F., Porco C. (2002) The confinement of Neptune’s ring arcs by the moon Galatea. Nature 417: 45–47CrossRefADSGoogle Scholar
  28. Nekhoroshev N.N. (1977) An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems. Russ. Math. Surv. 32: 5–66MATHCrossRefGoogle Scholar
  29. Porco C.C. (1991) An explanation for Neptune’s ring arcs. Science 253: 995–1001CrossRefADSGoogle Scholar
  30. Rückerl B., Jung C. (1994) Scaling properties of a scattering system with an incomplete horseshoe. J. Phys. A: Math. Gen. 27: 55–77MATHCrossRefGoogle Scholar
  31. Sicardy B. et al (1999) Images of Neptune’s ring arcs obtained by a ground-based telescope. Nature 400: 731–733CrossRefADSGoogle Scholar
  32. Simó, C.: Boundaries of Stability. Talk given at the Univ. de Barcelona (2006). See http://www.maia.ub.es/dsg/2006/index.html
  33. Simó, C.: Studies in Dynamics: From local to Global aspects. Lecture given at the opening of the 2007–2008 course in the Societat Catalana de Matematiques (Barcelona) (2007). See http://www.maia.ub.es/dsg/2007/index.shtm..
  34. Simó, C., Vieiro, A.: Global study of area preserving maps (in preparation)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Instituto de Ciencias FísicasUniversidad Nacional Autónoma de México (UNAM)CuernavacaMexico

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