Dynamics of two satellites in the 2/1 Mean–Motion resonance: application to the case of Enceladus and Dione

Abstract

The dynamics of a pair of satellites similar to Enceladus–Dione is investigated with a two-degrees-of-freedom model written in the domain of the planar general three-body problem. Using surfaces of section and spectral analysis methods, we study the phase space of the system in terms of several parameters, including the most recent data. A detailed study of the main possible regimes of motion is presented, and in particular we show that, besides the two separated resonances, the phase space is replete of secondary resonances.

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References

  1. 1.

    Beaugé, C., Callegari Jr., N., Ferraz-Mello, S., Michtchenko, T.A.: Resonances and stability of extra-solar planetary systems. In: Knežević Z., Milani A.(eds.). IAU Colloq. 197, 3–18 (2005)

  2. 2.

    Bevilacqua J.S. and Sessin W. (1987). First order perturbation in the Enceladus–Dione system. Rev. Mex. Astr. Astrof. 14: 627–630

    ADS  Google Scholar 

  3. 3.

    Bevilacqua, J.S., Sessin, W., Chen, J.: Periodic orbits in the Enceladus–Dione system’. In: Vieira Martins R. et al. (eds.) Orbital Dynamics of Natural and Artificial Objects, 13–21, Observatório Nacional, Rio de Janeiro, Brasil (1989).

  4. 4.

    Michtchenko T.A., Ferraz-Mello S. and Callegari N. (2004). Dynamics of two planets in the 2/1 mean–motion resonance. Celest. Mech. Dyn. Astr. 89: 201–234

    MATH  Article  ADS  MathSciNet  Google Scholar 

  5. 5.

    Ferraz-Mello S., Michtchenko T.A. and Callegari N. (2006). Dynamics of two planets in the 3/2 mean-motion resonance: application to the planetary system of the pulsar PSR B1252+12. Celest. Mech. Dyn. Astr. 94: 381–397

    MATH  Article  ADS  MathSciNet  Google Scholar 

  6. 6.

    Champenois S. and Vienne A. (1999). Chaos and secondary resonances in the Mimas-Tethys system. Celest. Mech. Dyn. Astr. 74: 111–146

    MATH  Article  ADS  MathSciNet  Google Scholar 

  7. 7.

    Ferraz-Mello S. (1985). First-order resonances in satellites orbits. In: Ferraz-Mello, S. and Sessin, W. (eds) Resonances in the Motion of the Planets, Satellites and Asteroids, pp 37–52. IAG/USP, Sao Paulo

    Google Scholar 

  8. 8.

    Ferraz-Mello S. and Dvorak R. (1987). Chaos and secular variations of planar orbits in 2:1 resonance with Dione. Astr. and Astrophys. 179: 304–310

    MATH  ADS  Google Scholar 

  9. 9.

    Ferraz-Mello, S., Michtchenko, T.A., Beaugé, C.: Regular motions in extra-solar planetary systems. In Steves, B.A., Maciejewicz, A.J., Hendry M. (eds.) Chaotic Worlds: From Order to Disorder in Gravitational N-Body Systems, pp. 255–288. Springer, Dordrecht. ArXiv: astro-ph/0402335 (2006)

  10. 10.

    Ferraz-Mello, S., Michtchenko, T.A., Beaugé, C., Callegari Jr. N.: Extra-solar planetary systems. In: Dvorak, R. et al. (eds.) Lecture Notes in Physics, 683, 219–271 (2005)

  11. 11.

    Goldreich P. (1965). An explanation of the frequent occurence of commensurable mean motion in the solar system. MNRAS 130: 159–181

    ADS  Google Scholar 

  12. 12.

    Henrard J. and Lemaitre A. (1983). A second fundamental model for resonance. Celest. Mech. 39: 197–218

    ADS  MathSciNet  Google Scholar 

  13. 13.

    Hori G. (1985). Mutual perturbations of 1:1 commensurable small bodies with the use of canonical relative coordinates. Part I. In: Ferraz-Mello, S. and Sessin, W. (eds) Resonances in the Motion of the Planets, Satellites and Asteroids., pp 33–66. IAG/USP, Sao Paulo

    Google Scholar 

  14. 14.

    Jacobson R.A. (2004). The orbits of the major Saturn Satellites and the gravity field of Saturn from spacecraft and Earth-based observations. Astron. J. 128: 492–501

    Article  ADS  Google Scholar 

  15. 15.

    Jacobson R.A., Antreasian P.G., Bordi J.J., Criddle K.E., Ionasescu R., Jones J.B., Mackenzie R.A., Pelletier F.J., Owen W.M., Roth D.C. and Stauch J.R. (2005). The gravity field of the Saturnian system and the orbits of the major Saturnian satellites. Bull. Am. Astron. Soc. 37: 729

    ADS  Google Scholar 

  16. 16.

    Jacobson R.A., Antreasian P.G., Bordi J.J., Criddle K.E., Ionasescu R., Jones J.B., Mackenzie R.A., Meek M.C., Parcher P., Pelletier F.J., Owen W.M., Roth D.C., Roundhill I.M. and Stauch J.R. (2006). The gravity field of the Saturnian system from satellite observations and spacecraft tracking data. Astron. J. 132: 2520–2526

    Article  ADS  Google Scholar 

  17. 17.

    Jancart S., Lemaitre A. and Istace A. (2002). Second fundamental model of resonance with asymmetric equilibria. Celest. Mech. Dyn. Astr. 84: 197–221

    MATH  Article  ADS  MathSciNet  Google Scholar 

  18. 18.

    Karch M. and Dvorak R. (1988). New results on the possible chaotic motion of Enceladus. Celest. Mech. Dyn. Astr. 43: 361–369

    ADS  MathSciNet  Google Scholar 

  19. 19.

    Laskar J. and Robutel P. (1995). Stability of the planetary three-body problem I: expansion of the planetary Hamiltonian. Celest. Mech. Dyn. Astr. 62: 193–217

    MATH  Article  ADS  MathSciNet  Google Scholar 

  20. 20.

    Lemaitre A. and Henrard J. (1990). On the origin of chaotic behavior in the 2/1 Kirkwood gap. Icarus 83: 391–409

    Article  ADS  Google Scholar 

  21. 21.

    Lichtenberg A.J. and Lieberman M.A. (1983). Regular and stochastic motion. Springer-Verlag, Heidelberg

    Google Scholar 

  22. 22.

    Malhotra R. and Dermott S.F. (1990). The role of secondary resonances in the orbital history of miranda. Icarus 85: 444–480

    Article  ADS  Google Scholar 

  23. 23.

    Message, P.J.: Orbits of Saturn’s satellites: some aspects of commensurabilities and periodic orbits. In: Steves, B.A., Roy, A.E. (eds.) The Dynamics of Small Bodies in the Solar System. Kluwer Academic Publishers, pp. 207–225 (1999)

  24. 24.

    Michtchenko T.A. and Ferraz-Mello S. (2001). Modeling the 5:2 mean–motion resonance in the Jupiter–Saturn planetary system. Icarus 149: 357–374

    Article  ADS  Google Scholar 

  25. 25.

    Michtchenko T.A., Beaugé C. and Ferraz-Mello S. (2006). Stationary orbits in resonant extrasolar planetary systems. Celest. Mech. Dyn. Astr. 94: 411–432

    MATH  Article  ADS  Google Scholar 

  26. 26.

    Peale, S.J: Orbital resonances, unsual configurations and exotic rotation states among planetary satellites. In: Burns, J.A. (ed.) Satellites pp. 159–223 (1986)

  27. 27.

    Peale S.J (1999). Origin and evolution of the natural atellites. Ann. Rev. Astron. Astrophys. 37: 533–602

    Article  ADS  Google Scholar 

  28. 28.

    Peale S.J (2003). Tidally induced volcanism. Celest. Mech. Dyn. Astr. 87: 129–155

    MATH  Article  ADS  Google Scholar 

  29. 29.

    Spencer J.R., Pearl J.C., Segura M., Flasar F.M., Mamoutkine A., Romani P., Buratti B.J., Hendrix A.R., Spilker L.J. and Lopes R.M.C. (2006). Cassini encounters enceladus: background and the discovery of a south polar hot spot. Science 311: 1401–1405

    Article  ADS  Google Scholar 

  30. 30.

    Salgado T.M.V. and Sessin W. (1985). The 2:1 commensurability in Enceladus–Dione system. In: Ferraz-Mello, S. and Sessin, W. (eds) Resonances in the Motion of the Planets, Satellites and Asteroids, pp 93–105. IAG/USP, Sao Paulo

    Google Scholar 

  31. 31.

    Sessin W. and Ferraz-Mello S. (1984). Motion of two planets with periods commensurable in the ratio 2:1. Solutions of the Hori auxiliaty system. Celest. Mech. 32: 307–332

    MATH  MathSciNet  Google Scholar 

  32. 32.

    Sinclair A.T. (1972). On the origin of the commensurabilities amongst the satellites of Saturn. MNRAS 160: 169–187

    ADS  Google Scholar 

  33. 33.

    Sinclair, A.T.: A re-consideration of the evolution hyphotesis of the origin of the resonances among Saturn’s satellites. In: Dynamical trapping and evolution in solar system; Proceedings of Seventy-four Colloquium, Gerakini, Greece, August 30-September 2, 1982 (A84-34976 16-89), Dordrecht, D. Reidel Publishing. Co., pp. 19–25 (1983)

  34. 34.

    Shinkin V.N. (2001). Approximate analytic solutions of the averaged three-body problem at first-order resonance with large oblateness of the central planet. Celest. Mech. Dyn. Astr. 79: 15–27

    MATH  Article  ADS  MathSciNet  Google Scholar 

  35. 35.

    Tittemore W. 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

    Article  ADS  Google Scholar 

  36. 36.

    Tittemore W. 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 Commensurability. Icarus 85: 394–443

    Article  ADS  Google Scholar 

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Callegari, N., Yokoyama, T. Dynamics of two satellites in the 2/1 Mean–Motion resonance: application to the case of Enceladus and Dione. Celestial Mech Dyn Astr 98, 5–30 (2007). https://doi.org/10.1007/s10569-007-9066-9

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Keywords

  • Enceladus
  • Dione
  • Mean–Motion resonance
  • Periodic orbits
  • Regular and chaotic motion
  • Secondary resonance
  • Saturn