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Celestial Mechanics and Dynamical Astronomy

, Volume 125, Issue 2, pp 223–246 | Cite as

On the rotation of co-orbital bodies in eccentric orbits

  • A. Leleu
  • P. Robutel
  • A. C. M. Correia
Original Article

Abstract

We investigate the resonant rotation of co-orbital bodies in eccentric and planar orbits. We develop a simple analytical model to study the impact of the eccentricity and orbital perturbations on the spin dynamics. This model is relevant in the entire domain of horseshoe and tadpole orbit, for moderate eccentricities. We show that there are three different families of spin–orbit resonances, one depending on the eccentricity, one depending on the orbital libration frequency, and another depending on the pericenter’s dynamics. We can estimate the width and the location of the different resonant islands in the phase space, predicting which are the more likely to capture the spin of the rotating body. In some regions of the phase space the resonant islands may overlap, giving rise to chaotic rotation.

Keywords

Co-orbitals Spin–orbit resonances Lagrange Planetary problem Three-body problem Tadpoles Horseshoe configuration 

References

  1. Charlier, C.V.L.: Über den Planeten 1906 TG. Astron. Nachr. 171, 213 (1906)CrossRefADSGoogle Scholar
  2. Chirikov, B.V.: A universal instability of many-dimensional oscillator systems. Phys. Rep. 52, 263–379 (1979)MathSciNetCrossRefADSGoogle Scholar
  3. Colombo, G.: Rotational period of the planet mercury. Nature 208, 575 (1965)CrossRefADSGoogle Scholar
  4. Correia, A.C.M.: Secular evolution of a satellite by tidal effect: application to triton. Astrophys. J. Lett. 704, L1–L4 (2009)CrossRefADSGoogle Scholar
  5. Correia, A.C.M., Laskar, J.: Mercury’s capture into the 3/2 spin–orbit resonance including the effect of core–mantle friction. Icarus 201, 1–11 (2009)CrossRefADSGoogle Scholar
  6. Correia, A.C.M., Robutel, P.: Spin–orbit coupling and chaotic rotation for coorbital bodies in quasi-circular orbits. Astrophys. J. 779, 20 (2013)CrossRefADSGoogle Scholar
  7. Correia, A.C.M., Rodríguez, A.: On the equilibrium figure of close-in planets and satellites. Astrophys. J. 767, 128 (2013)CrossRefADSGoogle Scholar
  8. Danby, J.M.A.: Stability of the triangular points in the elliptic restricted problem of three bodies. Astron. Astrophys. 69, 165 (1964)MathSciNetADSGoogle Scholar
  9. Dermott, S.F., Murray, C.D.: The dynamics of tadpole and horseshoe orbits. I—theory. Icarus 48, 1–11 (1981)CrossRefADSGoogle Scholar
  10. Gascheau, G.: Examen d’une classe d’quations diffrentielles et application un cas particulier du problme des trois corps. Comptes Rendus 16, 393 (1843)Google Scholar
  11. Giuppone, C.A., Beaugé, C., Michtchenko, T.A., Ferraz-Mello, S.: Dynamics of two planets in co-orbital motion. Mon. Not. R Astron. Soc. 407, 390–398 (2010)CrossRefADSGoogle Scholar
  12. Goldreich, P., Peale, S.: Spin–orbit coupling in the solar system. Astron. J. 71, 425 (1966)CrossRefADSGoogle Scholar
  13. Hansen, P.A.: Entwickelung der products einer potenz des radius vectors mit dem sinus oder cosinus eines vielfachen der wahren anomalie in reihen. Abhandl. d. K. S. Ges. d, Wissensch, IV. 182281 (1855)Google Scholar
  14. Lagrange: Œuvres complètes, vol. VI, p. 272. Gouthier-Villars, Paris (1772/1869)Google Scholar
  15. Laskar, J.: Frequency analysis of a dynamical system. Celest. Mech. Dyn. Astron. 56, 191–196 (1993)MathSciNetCrossRefMATHADSGoogle Scholar
  16. Leleu, A., Robutel, P., Correia, A.C.M.: Detectability of quasi-circular co-orbital planets. Application to the radial velocity technique. Astron. Astrophys. 581, A128 (2015)CrossRefADSGoogle Scholar
  17. MacDonald, G.J.F.: Tidal friction. Rev. Geophys. Space Phys. 2, 467–541 (1964)CrossRefADSGoogle Scholar
  18. Mikkola, S., Innanen, K., Wiegert, P., Connors, M., Brasser, R.: Stability limits for the quasi-satellite orbit. Mon. Not. R. Astron. Soc. 369, 15–24 (2006)CrossRefADSGoogle Scholar
  19. Morais, M.H.M.: A secular theory for trojan-type motion. Astron. Astrophys. 350, 318–326 (1999)ADSGoogle Scholar
  20. Morais, M.H.M., Namouni, F.: Asteroids in retrograde resonance with Jupiter and Saturn. Mon. Not. R. Astron. Soc. 436, L30–L34 (2013)CrossRefADSGoogle Scholar
  21. Morbidelli, A.: Modern celestial mechanics: aspects of solar system dynamics. Taylor & Francis, London (2002)Google Scholar
  22. Murray, C.D., Dermott, S.F.: Solar system dynamics. Cambridge University Press, Cambridge (1999)Google Scholar
  23. Namouni, F.: Secular interactions of coorbiting objects. Icarus 137, 293–314 (1999)CrossRefADSGoogle Scholar
  24. Nauenberg, M.: Stability and eccentricity for two planets in a 1:1 resonance, and their possible occurrence in extrasolar planetary systems. Astron. J. 124, 2332–2338 (2002)CrossRefADSGoogle Scholar
  25. Robutel, P., Laskar, J.: Frequency map and global dynamics in the solar system I: short period dynamics of massless particles. Icarus 152, 4–28 (2001)CrossRefADSGoogle Scholar
  26. Robutel, P., Niederman, L., Pousse, A.: Rigorous treatment of the averaging process for co-orbital motions in the planetary problem. ArXiv e-prints (2015)Google Scholar
  27. Robutel, P., Pousse, A.: On the co-orbital motion of two planets in quasi-circular orbits. Celest. Mech. Dyn. Astron. 117, 17–40 (2013)MathSciNetCrossRefMATHADSGoogle Scholar
  28. Robutel, P., Rambaux, N., El Moutamid, M.: Influence of the coorbital resonance on the rotation of the trojan satellites of saturn. Celest. Mech. Dyn. Astron. 113(1), 1–22 (2012)MathSciNetCrossRefMATHADSGoogle Scholar
  29. Wisdom, J., Peale, S.J., Mignard, F.: The chaotic rotation of Hyperion. Icarus 58, 137–152 (1984a)CrossRefADSGoogle Scholar
  30. Wisdom, J., Peale, S.J., Mignard, F.: The chaotic rotation of Hyperion. Icarus 58, 137–152 (1984b)CrossRefADSGoogle Scholar
  31. Wolf, M.: Photographische Aufnahmen von kleinen Planeten. Astron. Nachr. 170, 353 (1906)CrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.IMCCE, Observatoire de Paris, PSL Research University, UPMC Univ. Paris 06, Univ. Lille 1CNRSParisFrance
  2. 2.Departemento de Fìsica, I3NUniversidade de AveiroAveiroPortugal

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