Celestial Mechanics and Dynamical Astronomy

, Volume 89, Issue 3, pp 267–283 | Cite as

The Spin-Orbit Resonant Rotation of Mercury: A Two Degree of Freedom Hamiltonian Model

  • Sandrine D'hoedt
  • Anne Lemaitre


The paper develops a hamiltonian formulation describing the coupled orbital and spin motions of a rigid Mercury rotation about its axis of maximum moment of inertia in the frame of a 3:2 spin orbit resonance; the (ecliptic) obliquity is not constant, the gravitational potential of mercury is developed up to the second degree terms (the only ones for which an approximate numerical value can be given) and is reduced to a two degree of freedom model in the absence of planetary perturbations. Four equilibria can be calculated, corresponding to four different values of the (ecliptic) obliquity. The present situation of Mercury corresponds to one of them, which is proved to be stable. We introduce action-angle variables in the neighborhood of this stable equilibrium, by several successive canonical transformations, so to get two constant frequencies, the first one for the free spin-orbit libration, the other one for the 1:1 resonant precession of both nodes (orbital and rotational) on the ecliptic plane. The numerical values obtained by this simplified model are in perfect agreement with those obtained by Rambaux and Bois [Astron. Astrophys. 413, 381–393].

Cassini laws Hamiltonian formalism Mercury planetary rotation resonance spin-orbit 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Anderson, J. D., Colombo, G., Esposito, P. B., Lau, E. L. and Trager, G. B.: 1987, 'The mass, gravity field and ephemeris of Mercury', Icarus 71, 337–349.CrossRefADSGoogle Scholar
  2. Balogh, A. and Giampieri, G.: 2002, 'Mercury: the planet and its orbit', Reports Prog. Phys. 65, 529–560.CrossRefADSGoogle Scholar
  3. Beletskii, V. V.: 1972, 'Resonance rotation of celestial bodies and Cassini's laws', Celestial Mech. 6, 359–378.MathSciNetCrossRefADSGoogle Scholar
  4. Bouquillon, S.: 2001, 'Mercury libration: first stage', Journées 2001, Syste`mes de Référence Spatio-Temporels (<nt>Ed.</nt> N. Capitaine), 135–140.Google Scholar
  5. Brouwer, D. and Clemence, G. M.: 1961, Methods of Celestial Mechanics, Academic Press New York.Google Scholar
  6. Burns, T. J.: 1979, 'On the rotation of Mercury', Celestial Mech. 19, 297–313.MATHMathSciNetCrossRefADSGoogle Scholar
  7. Carpentier, G. and Roosbeek, F.: 2003, 'Analytical development of rigid Mercury nutation series', Celestial Mech. Dyn. Astr. 86, 223–236.MATHCrossRefADSGoogle Scholar
  8. Colombo, G.: 1965, Nature 208, 575.CrossRefADSGoogle Scholar
  9. Chapront, J., Chapront-Touzé, M. and Francou, G.: 1999, 'Complements to Moons' lunar librationtheory', Celestial Mech. Dyn. Astr. 73, 317–328.MATHCrossRefADSGoogle Scholar
  10. Deprit, A.: 1967, 'Free rotation of a rigid body studied in the phase plane', Am. J. Phs. 35 (5), 424–428.CrossRefGoogle Scholar
  11. Eckhardt, D. H.: 1981, 'Theory of the libration of the Moon', Moon Planets 25, 3–49.MATHCrossRefADSGoogle Scholar
  12. ESA-SCI: 2000, 'BepiColombo, An Interdisciplinary Cornerstone Mission to the Planet Mercury', System and Technology Study Report.Google Scholar
  13. Henrard, J and Schwanen, G.: 2004, 'Rotation of synchronous satellites: Application to the Gazillan, satellites, Celestial Mech. Dyn. Astr. 89, 181–200.MATHMathSciNetCrossRefADSGoogle Scholar
  14. Kinoshita, H.: 1972, 'First-order perturbations of the two finite body problem', Publ. Astron. Soc. Jpn. 24, 423–457.MathSciNetADSGoogle Scholar
  15. Migus, A.: 1980, 'Analytical lunar libration tables', Moon Planets 23, 391–427.MATHCrossRefADSGoogle Scholar
  16. Moons, M.: 1982, 'Analytical theory of the libration of the Moon', Moon Planets 27, 257–284.MATHCrossRefADSGoogle Scholar
  17. Moons, M.: 1984, 'Planetary perturbations on the libration of the Moon', Celestial Mech. 34, 263–273.MATHCrossRefADSGoogle Scholar
  18. Peale, S. J.: 1969, 'Generalized Cassini's laws', Astron. J. 74(3), 483–489.CrossRefADSGoogle Scholar
  19. Peale, S. J.: 1972, 'Determination of parameters related to the interior of Mercury,' Icarus 17, 168–173.CrossRefADSGoogle Scholar
  20. Peale, S. J.: 1974, 'Possible histories of the obliquity of Mercury', Astron. J. 79(6), 722–744.CrossRefADSGoogle Scholar
  21. Rambaux, N. and Bois, E.: 2004, 'Theory of the Mercury's spin-orbit motion and analysis of its main librations', Astron. Astrophys., 413, 381–393.CrossRefADSGoogle Scholar
  22. Williams, J. G., Boggs, D. H., Yoder, Ch. F., Ratcliff, J. T. and Dickey, J. O.: 2001, 'Lunar Rotational dissipation in solid body and molten core.' J. Geophys. Res. 106(E11), 27933–27968.CrossRefADSGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Sandrine D'hoedt
    • 1
  • Anne Lemaitre
    • 2
  1. 1.Département de mathématiqueFUNDPNamurBelgium
  2. 2.Département de mathématiqueFUNDPBelgium

Personalised recommendations