Celestial Mechanics and Dynamical Astronomy

, Volume 116, Issue 4, pp 389–416 | Cite as

Dynamics of rotation of super-Earths

  • Nelson CallegariJr.
  • Ádrian Rodríguez
Original Article


We numerically investigate the dynamics of rotation of several close-in terrestrial exoplanet candidates. In our model, the rotation of the planet is disturbed by the torque of the central star due to the asymmetric equilibrium figure of the planet. We model the shape of the planet by a Jeans spheroid. We use surfaces of section and spectral analysis to explore numerically the rotation phase space of the systems adopting different sets of parameters and initial conditions close to the main spin–orbit resonant states. One of the parameters, the orbital eccentricity, is critically discussed here within the domain of validity of orbital circularization timescales given by tidal models. We show that, depending on some parameters of the system like the radius and mass of the planet, eccentricity etc., the rotation can be strongly perturbed and a chaotic layer around the synchronous state may occupy a significant region of the phase space. 55 Cnc e is an example.


Spin–orbit resonance Terrestrial exoplanets 55 Cnc planets CoRoT-7 planets KOI-55 planets GJ 876d 



FAPESP (2006/58000-2 (NCJ); 2009/16900-5 (ARC).)


  1. Anglada-Escudé, G., Lpez-Morales, M.: How eccentric orbital solutions can hide planetary systems in 2:1 resonant orbits. Astrophys. J. 709, 168–178 (2010)ADSCrossRefGoogle Scholar
  2. Baraffe, I., Chabrier, G., Barman, T.: The physical properties of extra-solar planets. Rep. Progr. Phys. 73(1), 016901 (2010)ADSCrossRefGoogle Scholar
  3. Barnes, J.W., Fortney, J.J.: Measuring the oblateness and rotation of transiting extrasolar giant planets. Astrophys. J. 588, 545–556 (2003)ADSCrossRefGoogle Scholar
  4. Batalha, N.M., Borucki, W.J., Bryson, S.T., Buchhave, L.A., Caldwell, D.A., Christensen-Dalsgaard, J., et al.: Kepler’s first rocky planet: Kepler-10b. Astrophys. J. 729, 27 (pp 24) (2011)Google Scholar
  5. Batygin, K., Bodenheimer, P., Laughlin, G.: Determination of the interior structure of transiting planets in multiple-planet systems. Astrophys. J. Lett. 704, L49–L53 (2009)ADSCrossRefGoogle Scholar
  6. Beutler, G.: Methods of Celestial Mechanics, vol. I. Springer, Berlin (2005)Google Scholar
  7. Bills, B.G., Nimmo, F., Karatekin, O., van Hoolst, T., Rambaux, N., Levrard, B., et al.: Rotational dynamics of Europa. In: Pappalardo, R.T., McKinnon, W.B., Khurana, K.K. (eds.) with the assistance of René Dotson with 85 collaborating authors. Europa. University of Arizona Press, Tucson. The University of Arizona space science series, p. 119. ISBN: 9780816528448 (2009)Google Scholar
  8. Callegari Jr., N., Yokoyama, T.: Numerical exploration of resonant dynamics in the system of Saturnian inner satellites. Planet. Space Sci. 58, 1906–1921 (2010)Google Scholar
  9. Carter, J.A., Winn, J.N.: Empirical constraints on the oblateness of an exoplanet. Astrophys. J. 709, 1219–1229 (2010)ADSCrossRefGoogle Scholar
  10. Castan, T., Menou, K.: Atmospheres of hot super-Earths. Astrophys. J. Lett. 743(2), article id. L36 (2011)Google Scholar
  11. Celletti, A., Voyatzis, G.: Regions of stability in rotational dynamics. Celest. Mech. Dyn. Astron. 107, 101–113 (2010)MathSciNetADSzbMATHCrossRefGoogle Scholar
  12. Chandrasekhar, S.: Ellipsoidal figures of equilibrium, chap. VIII. Yale Univ. Press, New Haven (1969).Google Scholar
  13. Charbonneau, D., Berta, Z.K., Irwin, J., Burke, C.J., Philip, N., Buchhave, L.A., et al.: A super-Earth transiting a nearby low-mass star. Nature 462, 891–894 (2009)Google Scholar
  14. Charpinet, S., Fontaine, G., Brassard, P., Green, E.M., van Grootel, V., Randall, S.K., et al.: A compact system of small planets around a former red-giant star. Nature 480, 496–499 (2011)Google Scholar
  15. Correia, A.C.M., Levrard, B., Laskar, J.: On the equilibrium rotation of Earth-like extra-solar planets. Astron. Astrophys. 488, L63–L66 (2008)ADSzbMATHCrossRefGoogle Scholar
  16. Correia, A.C.M.: Secular evolution of a satellite by tidal effect: application to triton. Astrophys. J. Lett. 704, L1–L4 (2009)ADSCrossRefGoogle Scholar
  17. Correia, A.C.M., Couetdic, J., Laskar, J., Bonfils, X., Mayor, M., Bertaux, J.-L., et al.: The HARPS search for southern extra-solar planets. XIX. Characterization and dynamics of the GJ 876 planetary system. Astron. Astrophys. 511, id. A21 (2010)Google Scholar
  18. Correia, A.C.M., Bou, G., Laskar, J.: Pumping the eccentricity of exoplanets by tidal effect. Astrophys. J. Lett. 744(2), article id. L23 (2012)Google Scholar
  19. Correia, A.C.M., Rodríguez, A.: On the equilibrium figure of close-in planets and satellites. Astrophys. J. 767, 128 (2013)Google Scholar
  20. Danby, J.M.A.: Fundamentals of Celestial Mechanics, 2nd edn. Willmann-Bell, Richmond (1988)Google Scholar
  21. Dobrovolskis, A.R.: Spin states and climates of eccentric exoplanets. Icarus 192, 1–23 (2007)ADSCrossRefGoogle Scholar
  22. Dobrovolskis, A.R.: Insolation patterns on synchronous exoplanets with obliquity. Icarus 2004, 1–10 (2009)ADSCrossRefGoogle Scholar
  23. Dobbs-Dixon, I., Lin, D.N.C., Mardling, R.A.: Spin–orbit evolution of short-period planets. Astrophys. J. 610, 464–476 (2004)ADSCrossRefGoogle Scholar
  24. Everhart, E.: An efficient integrator that uses Gauss–Radau spacings. In: IAU Coloquium, vol. 83, pp. 185–202 (1985)Google Scholar
  25. Ferraz-Mello, S., Rodríguez, A., Hussmann, H.: Tidal friction in close-in satellites and exoplanets: the Darwin theory re-visited. Celest. Mech. Dyn. Astron. 101, 171–201 (2008) (Errata: Celest. Mech. Dyn. Astwn. 104, 319–320 (2009))Google Scholar
  26. Ferraz-Mello, S., Tadeu dos Santos, M., Beauge, C., Michtchenko, T.A., Rodríguez, A.: On planetary mass determination in the case of super-Earths orbiting active stars. The case of the CoRoT-7 system. A &A, 531A (2011)Google Scholar
  27. Giampieri, G.: A note on the tidally induced potential of a satellite in eccentric orbit. Icarus 167, 228–230 (2004)ADSCrossRefGoogle Scholar
  28. Gillon, M., Demory, B.-O., Benneke, B., Valencia, D., Deming, D., Seager, S., et al.: Astron. Astrophys. 539, id.A28 eprint (2012)Google Scholar
  29. Goldreich, P., Peale, S.: Spin–orbit coupling in the solar system. Astron. J. 71, 425–437 (1966)ADSCrossRefGoogle Scholar
  30. Henrard, J.: Spin–orbit esonance and the adiabatic invarint. In: Ferraz-Mello, S., Sessin, W. (eds.) Resonances in the Motion of the Planets, Satellites and Asteroids, pp. 19–26. IAG/USP, Sao Paulo (1985)Google Scholar
  31. Hébrard, G., Ehrenreich, D., Bouchy, F., Delfosse, X., Moutou, C., Arnold, L., et al.: The retrograde orbit of the HAT-P-6b exoplanet. Astron. Astrophys. 527, id. L11 (2011)Google Scholar
  32. Iess, L., Rappaport, N.J., Jacobson, R.A., et al.: Gravity field, shape, and moment of inertia of Titan. Science 327, 1367 (2010)ADSCrossRefGoogle Scholar
  33. Jackson, B., Greenberg, R., Barnes, R.: Tidal evolution of close-in extrasolar planets. Astrophys. J. 678, 1396–1406 (2008a)ADSCrossRefGoogle Scholar
  34. Jackson, B., Barnes, R., Greenberg, R.: Tidal heating of terrestrial extrasolar planets and implications for their habitability. Mon. Not. R. Astron. Soc. 391, 237–245 (2008b)ADSCrossRefGoogle Scholar
  35. Jackson, B., Greenberg, R., Barnes, R.: Tidal heating of extrasolar planets. Astrophys. J. 681, 1631–1638 (2008c)ADSCrossRefGoogle Scholar
  36. Kitiashvili, I.N., Alexander, G.: Rotational evolution of exoplanets under the action of gravitational and magnetic perturbations. Celest. Mech. Dyn. Astron. 100, 121–140 (2008)Google Scholar
  37. Kramm, U., Nettelmann, N., Redmer, R., Stevenson, D.J.: On the degeneracy of the tidal Love number k2 in multi-layer planetary models: application to Saturn and GJ436b. Astron. Astrophys. 528, id.A18 (2011)Google Scholar
  38. Lammer, H., Khodachenko, M.L., Herbert, I.M., Lichtenegger, I.M., Kulinov, Y.N.: Impact of stellar activity on the evolution of planetary atmospheres and hability. In: Dvorak, R. (ed.) Extrasolar Planets: Formation, Detection and Dynamics, p. 127. Wiley-VCH (2008)Google Scholar
  39. Lammer, H., Bredehft, J.H., Coustenis, A., Khodachenko, M.L., Kaltenegger, L., Grasset, O., et al.: What makes a planet habitable? Astron. Astrophys. Rev. 17, 181–249 (2009)Google Scholar
  40. Léger, A., Rouan, D., Schneider, J., Barge, P., Fridlund, M., Samuel, B., et al.: Transiting exoplanets from the CoRoT space mission. VIII. CoRoT-7 b: the first super-Earth with measured radius. Astron. Astrophys. 506, 287–302 (2009)Google Scholar
  41. Léger, A., Grasset, O., Fegley, B., Codron, F., Albarede, A.F., Barge, P., et al.: The extreme physical properties of the CoRoT-7b super-Earth. Icarus 213, 1–11 (2011)Google Scholar
  42. Levrard, B., Correia, A.C.M., Chabrier, G., Baraffe, I., Selsis, F., Laskar, J.: Tidal dissipation within hot Jupiters: a new appraisal. Astron. Astrophys. 462, L5–L8 (2007)ADSCrossRefGoogle Scholar
  43. Madhusudhan, N., Lee, K.K.M., Mousis, O.: A possible carbon-rich interior in super-Earth 55 Cancri e. Astrophys. J. Lett. 759, L40 (2012)Google Scholar
  44. Mardling, R.A.: Long-term tidal evolution of short-period planets with companions. Mon. Notices R. Astron. Soc. 382, 1768–1790 (2007)ADSCrossRefGoogle Scholar
  45. Matsumura, S., Takeda, G., Rasio, F.A.: On the origins of eccentric close-in planets. Astrophy. J. 686(1), L29–L32 (2008)ADSCrossRefGoogle Scholar
  46. Michtchenko, T.A., Ferraz-Mello, S.: Resonant structure of the outer solar system in the neighborhood of the planets. Astron. J. 122, 474–481 (2001)ADSCrossRefGoogle Scholar
  47. Mignard, F.: The evolution of the lunar orbit revisited—I. Moon Planet. 20, 301–315 (1979)ADSzbMATHCrossRefGoogle Scholar
  48. Muirhead, P.S., Johnson, J.A., Apps, K., Carter, J.A., Morton, T.D., Fabrycky, D.C., et al.: Characterizing the cool kois. iii. KOI 961: a small star with large proper motion and three small planets. Astrophys. J. 747(2), article id. 144 (2012)Google Scholar
  49. Murray, C.D., Dermott, S.F.: Solar System Dynamics. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  50. Rappaport, N., Bertotti, B., Giampieri, G., Anderson, J.D.: Doppler measurements of the quadrupole moments of Titan. Icarus 126, 313–323 (1997)ADSCrossRefGoogle Scholar
  51. Ragozzine, D., Wolf, A.S.: Probing the Interiors of very hot Jupiters using transit light curves. Astrophys. J. 698, 1778–1794 (2009)ADSCrossRefGoogle Scholar
  52. Rivera, E.J., Laughlin, G., Butler, R.P., Vogt, S.S., Haghighipou, N., Meschiari, S.: The Lick-Carnegie exoplanet survey: a Uranus-mass fourth planet for GJ 876 in an extrasolar laplace configuration. Astrophys. J. 719, 890–899 (2010)ADSCrossRefGoogle Scholar
  53. Rodríguez, A., Ferraz-Mello, S.: Tidal decay and circularization of the orbits of short-period planets. EAS Publ. Ser. 42, 411–418 (2010)CrossRefGoogle Scholar
  54. Rodríguez, A.: Evolução Orbital de Planetas Quentes Atribuída ao Efeito de Maré. Ph.D. thesis, Universidade de Sao Paulo, Brazil (2010)Google Scholar
  55. Rodríguez, A., Ferraz-Mello, S., Michtchenko, T.A., Beaugé, C., Miloni, O.: Tidal decay and orbital circularization in close-in two-planet systems. Mon. Notices R. Astron. Soc. 415, 2349–2358 (2011)ADSCrossRefGoogle Scholar
  56. Rodríguez, A.C., Callegari Jr., N., Michtchenko, T., Hussmann, H.: Spin-orbit evolution of hot super-Earths. Mon. Notices R. Astron. Soc. 427, 2239–2250 (2012)Google Scholar
  57. Seager, S., Hui, L.: Constraining the rotation rate of transiting extrasolar planets by oblateness measurements. Astrophys. J. 574, 1004–1010 (2002)ADSCrossRefGoogle Scholar
  58. Showman, A.P., Polvani, L.M.: Equatorial superrotation on hot Jupiters. Astrophys. J. 738(1), article id. 71 (2011)Google Scholar
  59. Schubert, G., Anderson, J.D., Spohn, T., McKinnon, W.B.: Interior composition, structure and dynamics of the Galilean satellites. In: Bagenal, F., Dowling, T.E., McKinnon, W.B. (eds.) Jupiter. The Planet, Satellites and Magnetosphere. Cambridge planetary science, vol. 1, pp. 281–306. Cambridge University Press, Cambridge, UK (2004)Google Scholar
  60. Spiegel, D.S., Raymond, S.N., Dressing, C.D., Scharf, C.A., Mitchell, J.L.: Generalized Milankovitch cycles and long-term climatic habitability. Astrophys. J. 721, 1308–1318 (2010)ADSCrossRefGoogle Scholar
  61. Tadeu dos Santos, M., Silva, G.G., Ferraz-Mello, S., Michtchenko, T.A.: A new analysis of the GJ581 extrasolar planetary system. Celest. Mech. Dyn. Astron. 113, 49–62 (2012)ADSCrossRefGoogle Scholar
  62. Torres, G., Fressin, F., Batalha, N.M., Borucki, W.J., Brown, T.M., Bryson, S.T., et al.: Modeling Kepler transit light curves as false positives: rejection of blend scenarios for Kepler-9, and validation of Kepler-9 d, a super-Earth-size planet in a multiple system. Astrophys. J. 727(1), article id. 24 (2011)Google Scholar
  63. van Hoolst, T., Rambaux, N., Karatekin, O., Dehant, V., Rivoldini, A.: The librations, shape, and icy shell of Europa. Icarus 195(1), 386–399 (2008)ADSCrossRefGoogle Scholar
  64. Valencia, D., Sasselov, D.D., O’Connell, R.J.: Detailed models of super-Earths: how well can we infer bulk properties? Astrophys. J. 665, 1413–1420 (2007a)ADSCrossRefGoogle Scholar
  65. Valencia, D., Sasselov, D.D., O’Connell, R.J.: Radius and structure models of the first super-Earth planet. Astrophys. J. 656, 545–551 (2007b)ADSCrossRefGoogle Scholar
  66. Valencia, D.: Characterising super-Earths. In: EPJ Web of Conferences, vol. 11, p. 03001 (2011)Google Scholar
  67. Zuluaga, Z.I., Cuartas-Restrepo, P.A.: The role of rotation on the evolution of dynamo generated magnetic fields in super Earths. Icarus 217, 88–102 (2012)ADSCrossRefGoogle Scholar
  68. Wisdom, J., Peale, S.J., Mignard, F.: The chaotic rotation of Hyperion. Icarus 58, 137–152 (1984)ADSCrossRefGoogle Scholar
  69. Wisdom, J.: Rotational dynamics of irregularly shaped natural satellites. Astron. J. 94, 1350–1360 (1987)ADSCrossRefGoogle Scholar
  70. Wisdom, J.: Spin–orbit secondary resonance dynamics of Enceladus. Astron. J. 128, 484–491 (2004)ADSCrossRefGoogle Scholar
  71. Wisdom, J.: Tidal dissipation at arbitrary eccentricity and obliquity. Icarus 193, 637–640 (2008)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Instituto de Geociências e Ciências ExatasUnesp, Univ Estadual PaulistaRio ClaroBrazil
  2. 2.Departamento de Astronomia, Instituto de Astronomia, Geofísica e Ciências AtmosféricasUniversidade de São PauloSão PauloBrazil

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