Advertisement

Tidal evolution of hierarchical and inclined systems

  • Alexandre C. M. CorreiaEmail author
  • Jacques Laskar
  • François Farago
  • Gwenaël Boué
Original Article

Abstract

We investigate the dynamical evolution of hierarchical three-body systems under the effect of tides, when the ratio of the orbital semi-major axes is small and the mutual inclination is relatively large (greater than 20°). Using the quadrupolar non-restricted approximation for the gravitational interactions and the viscous linear model for tides, we derive the averaged equations of motion in a vectorial formalism which is suitable to model the long-term evolution of a large variety of exoplanetary systems in very eccentric and inclined orbits. In particular, it can be used to derive constraints for stellar spin-orbit misalignment, capture in Cassini states, tidal-Kozai migration, or damping of the mutual inclination. Because our model is valid for the non-restricted problem, it can be used to study systems of identical mass or for the outer restricted problem, such as the evolution of a planet around a binary of stars. Here, we apply our model to various situations in the HD 11964, HD 80606, and HD 98800 systems.

Keywords

Restricted problems Extended body Dissipative forces Planetary systems Rotation HD 11964 HD 80606 HD 98800 

References

  1. Boden A.F., Sargent A.I., Akeson R.L., Carpenter J.M., Torres G., Latham D.W., Soderblom D.R., Nelan E., Franz O.G., Wasserman L.H.: Dynamical masses for low-mass pre-main-sequence stars: a preliminary physical orbit for HD 98800 B. Astrophys. J. 635, 442–451 (2005)ADSCrossRefGoogle Scholar
  2. Boué G., Laskar J.: Precession of a planet with a satellite. Icarus 185, 312–330 (2006)ADSCrossRefGoogle Scholar
  3. Boué G., Laskar J.: Spin axis evolution of two interacting bodies. Icarus 201, 750–767 (2009)ADSCrossRefGoogle Scholar
  4. Butler R.P., Wright J.T., Marcy G.W., Fischer D.A., Vogt S.S., Tinney C.G., Jones H.R.A., Carter B.D., Johnson J.A., McCarthym C., Penny A.J.: Catalog of nearby exoplanets. Astrophys. J. 646, 505–522 (2006)ADSCrossRefGoogle Scholar
  5. Chatterjee S., Ford E.B., Matsumura S., Rasio F.A.: Dynamical outcomes of planet-planet scattering. Astrophys. J. 686, 580–602 (2008)ADSCrossRefGoogle Scholar
  6. Colombo G.: Cassini’s second and third laws. Astron. J. 71, 891–896 (1966)ADSCrossRefGoogle Scholar
  7. Correia A.C.M.: Secular evolution of a satellite by tidal effect: application to triton. Astrophys. J. 704, L1–L4 (2009)ADSCrossRefGoogle Scholar
  8. Correia A.C.M., Laskar J.: Mercury’s capture into the 3/2 spin-orbit resonance as a result of its chaotic dynamics. Nature 429, 848–850 (2004)ADSCrossRefGoogle Scholar
  9. 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)ADSCrossRefGoogle Scholar
  10. Correia A.C.M., Laskar J.: Long-term evolution of the spin of Mercury. I. Effect of the obliquity and core-mantle friction. Icarus 205, 338–355 (2010a)ADSCrossRefGoogle Scholar
  11. Correia, A.C.M., Laskar, J.: Tidal evolution of exoplanets. In: Exoplanets. University of Arizona Press, Arizona, pp 534–575 (2010b)Google Scholar
  12. Correia A.C.M., Laskar J., Néron de Surgy O.: Long-term evolution of the spin of Venus I. Theory. Icarus 163, 1–23 (2003)ADSCrossRefGoogle Scholar
  13. D’Angelo C., van Kerkwijk M.H., Rucinski S.M.: Contact binaries with additional components. II. A spectroscopic search for faint tertiaries. Astron. J. 132, 650–662 (2006)ADSCrossRefGoogle Scholar
  14. Efroimsky M., Williams J.G.: Tidal torques: a critical review of some techniques. Celest. Mech. Dyn. Astron. 104, 257–289 (2009)MathSciNetADSCrossRefzbMATHGoogle Scholar
  15. Eggenberger A., Udry S., Mayor M.: Statistical properties of exoplanets. III. Planet properties and stellar multiplicity. Astron. Astrophys. 417, 353–360 (2004)Google Scholar
  16. Eggleton P.P., Kiseleva-Eggleton L.: Orbital evolution in binary and triple stars, with an application to SS Lacertae. Astrophys. J. 562, 1012–1030 (2001)ADSCrossRefGoogle Scholar
  17. Fabrycky D., Tremaine S.: Shrinking Binary and Planetary Orbits by Kozai Cycles with Tidal Friction. Astrophys. J. 669, 1298–1315 (2007)ADSCrossRefGoogle Scholar
  18. Farago F., Laskar J.: High-inclination orbits in the secular quadrupolar three-body problem. Mon. Not. R. Astron. Soc. 401, 1189–1198 (2010)ADSCrossRefGoogle Scholar
  19. Fischer D.A., Marcy G.W., Butler R.P., Vogt S.S., Frink S., Apps K.: Planetary companions to HD 12661, HD 92788, and HD 38529 and variations in Keplerian residuals of extrasolar planets. Astrophys. J. 551, 1107–1118 (2001)ADSCrossRefGoogle Scholar
  20. Ford E.B., Kozinsky B., Rasio F.A.: Secular evolution of hierarchical triple star systems. Astrophys. J. 535, 385–401 (2000)ADSCrossRefGoogle Scholar
  21. Furlan E., Sargent B., Calvet N., Forrest W.J., D’Alessio P., Hartmann L., Watson D.M., Green J.D., Najita J., Chen C.H.: HD 98800: a 10 Myr old transition disk. Astrophys. J. 664, 1176–1184 (2007)ADSCrossRefGoogle Scholar
  22. Kastner J.H., Zuckerman B., Weintraub D.A., Forveille T.: X-ray and molecular emission from the nearest region of recent star formation. Science 277, 67–71 (1997)ADSCrossRefGoogle Scholar
  23. Kaula W.M.: Tidal dissipation by solid friction and the resulting orbital evolution. Rev. Geophys. 2, 661–685 (1964)ADSCrossRefGoogle Scholar
  24. Koerner D.W., Jensen E.L.N., Cruz K.L., Guild T.B., Gultekin K.: A single circumbinary disk in the HD 98800 quadruple system. Astrophys. J. 533, L37–L40 (2000)ADSCrossRefGoogle Scholar
  25. Kozai Y.: Secular perturbations of asteroids with high inclination and eccentricity. Astron. J. 67, 591–598 (1962)MathSciNetADSCrossRefGoogle Scholar
  26. Lambeck, K.: Geophysical geodesy: the slow deformations of the earth Lambeck. Oxford [England]: Clarendon Press ; New York: Oxford University Press (1988)Google Scholar
  27. Laskar J.: Large scale chaos and the spacing of the inner planets. Astron. Astrophys. 317, L75–L78 (1997)ADSGoogle Scholar
  28. Laskar J., Boué G.: Explicit expansion of the three-body disturbing function for arbitrary eccentricities and inclinations. Astron. Astrophys. 522, A60 (2010)ADSCrossRefGoogle Scholar
  29. Lee M.H., Peale S.J.: Secular evolution of hierarchical planetary systems. Astrophys. J. 592, 1201–1216 (2003)ADSCrossRefGoogle Scholar
  30. Lidov M.L.: Evolution of the planets artificial satellites orbits under effect of the outer bodies gravity perturbations. Iskus sputniky Zemly (in Russian) 8, 5–45 (1961)Google Scholar
  31. Lidov M.L.: The evolution of orbits of artificial satellites of planets under the action of gravitational perturbations of external bodies. Plan. Space Sci. 9, 719–759 (1962)ADSCrossRefGoogle Scholar
  32. Lidov M.L., Ziglin S.L.: Non-restricted double-averaged three body problem in Hill’s case. Celest. Mech. 13, 471–489 (1976)MathSciNetADSCrossRefzbMATHGoogle Scholar
  33. Marchal C.: The Three-Body Problem. Elsevier, Amsterdam (1990)zbMATHGoogle Scholar
  34. Migaszewski C., Goździewski K.: Secular dynamics of a coplanar, non-resonant planetary system under the general relativity and quadrupole moment perturbations. Mon. Not. R. Astron. Soc. 392, 2–18 (2009)ADSCrossRefGoogle Scholar
  35. Mignard F.: The evolution of the lunar orbit revisited. I. Moon Planets 20, 301–315 (1979)ADSCrossRefzbMATHGoogle Scholar
  36. Naef D., Latham D.W., Mayor M., Mazeh T., Beuzit J.L., Drukier G.A., Perrier-Bellet C., Queloz D., Sivan J.P., Torres G., Udry S., Zucker S.: HD 80606 b, a planet on an extremely elongated orbit. Astron. Astrophys. 375, L27–L30 (2001)ADSCrossRefGoogle Scholar
  37. Nagasawa M., Ida S., Bessho T.: Formation of hot planets by a combination of planet scattering, tidal circularization, and the kozai mechanism. Astrophys. J. 678, 498–508 (2008)ADSCrossRefGoogle Scholar
  38. Néron de Surgy O., Laskar J.: On the long term evolution of the spin of the earth. Astron. Astrophys. 318, 975–989 (1997)ADSGoogle Scholar
  39. Palacián J.F., Yanguas P., Fernández S., Nicotra M.A.: Searching for periodic orbits of the spatial elliptic restricted three-body problem by double averaging. Physica D Nonlinear Phenomena 213, 15–24 (2006)MathSciNetADSzbMATHCrossRefGoogle Scholar
  40. Peale S.J.: Generalized Cassini’s laws. Astron. J. 74, 483–489 (1969)ADSCrossRefGoogle Scholar
  41. Pont F., Hébrard G., Irwin J.M., Bouchy F., Moutou C., Ehrenreich D., Guillot T., Aigrain S., Bonfils X., Berta Z., Boisse I., Burke C., Charbonneau D., Delfosse X., Desort M., Eggenberger A., Forveille T., Lagrange A., Lovis C., Nutzman P., Pepe F., Perrier C., Queloz D., Santos N.C., Ségransan D., Udry S., Vidal-Madjar A.: Spin-orbit misalignment in the HD 80606 planetary system. Astron. Astrophys. 502, 695–703 (2009)ADSCrossRefGoogle Scholar
  42. Pont, F., Husnoo, N., Mazeh, T., Fabrycky, D.: Determining eccentricities of transiting planets: a divide in the mass-period plane. Mon. Not. R. Astron. Soc. 414, 1278–1284 (2011)Google Scholar
  43. Saffe, C., Gómez, M., Chavero, C.: On the ages of exoplanet host stars. Astron. Astrophys. 443, 609–626 (2005)Google Scholar
  44. Schutz B.F.: A First Course in General Relativity. Cambridge University Press, Cambridge (1985)Google Scholar
  45. Singer S.F.: The origin of the moon and geophysical consequences. Geophys. J. R. Astron. Soc. 15, 205–226 (1968)CrossRefGoogle Scholar
  46. Smart W.M.: Celestial Mechanics. Longmans, Green, London, New York (1953)zbMATHGoogle Scholar
  47. Tokovinin, A., Thomas, S., Sterzik, M., Udry, S.: Tertiary companions to close spectroscopic binaries. Astron. Astrophys. 450, 681–693 (2006)Google Scholar
  48. Tokovinin A.A.: The visual orbit of HD 98800. Astron. Lett. 25, 669–671 (1999)ADSGoogle Scholar
  49. Torres G., Stefanik R.P., Latham D.W., Mazeh T.: Study of spectroscopic binaries with TODCOR. IV. The multiplicity of the young nearby star HD 98800. Astrophys. J. 452, 870–878 (1995)ADSCrossRefGoogle Scholar
  50. Tremaine S., Touma J., Namouni F.: Satellite dynamics on the Laplace surface. Astron. J. 137, 3706–3717 (2009)ADSCrossRefGoogle Scholar
  51. Triaud A.H.M.J., Collier Cameron A., Queloz D., Anderson D.R., Gillon M., Hebb L., Hellier C., Loeillet B., Maxted P.F.L., Mayor M., Pepe F., Pollacco D., Ségransan D., Smalley B., Udry S., West R.G., Wheatley P.J.: Spin-orbit angle measurements for six southern transiting planets. New insights into the dynamical origins of hot Jupiters. Astron. Astrophys. 524, A25 (2010)ADSCrossRefGoogle Scholar
  52. Veras D., Ford E.B.: Secular orbital dynamics of hierarchical two-planet systems. Astrophys. J. 715, 803–822 (2010)ADSCrossRefGoogle Scholar
  53. Verrier P.E., Evans N.W.: High-inclination planets and asteroids in multistellar systems. Mon. Not. R. Astron. Soc. 394, 1721–1726 (2009)ADSCrossRefGoogle Scholar
  54. Ward W.R., Hamilton D.P.: Tilting saturn. I. Analytic model. Astron. J. 128, 2501–2509 (2004)ADSCrossRefGoogle Scholar
  55. Wright J.T., Upadhyay S., Marcy G.W., Fischer D.A., Ford E.B., Johnson J.A.: Ten new and updated multiplanet systems and a survey of exoplanetary systems. Astrophys. J. 693, 1084–1099 (2009)ADSCrossRefGoogle Scholar
  56. Wu, Y., Goldreich, P.: Tidal evolution of the planetary system around HD 83443. Astrophys. J. 564, 1024–1027 (2002)ADSCrossRefGoogle Scholar
  57. Wu, Y., Murray, N.: Planet migration and binary companions: The case of HD 80606b. Astrophys. J. 589, 605–614 (2003)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • Alexandre C. M. Correia
    • 1
    Email author
  • Jacques Laskar
    • 2
  • François Farago
    • 2
  • Gwenaël Boué
    • 3
  1. 1.Department of PhysicsI3N, University of Aveiro, Campus Universitário de SantiagoAveiroPortugal
  2. 2.Astronomie et Systèmes DynamiquesIMCCE-CNRS UMR 8028, Observatoire de ParisParisFrance
  3. 3.Centro de AstrofísicaUniversity of PortoPortoPortugal

Personalised recommendations