Celestial Mechanics and Dynamical Astronomy

, Volume 96, Issue 3–4, pp 253–258 | Cite as

Note on Mercury’s rotation: the four equilibria of the Hamiltonian model

  • Sandrine D’HoedtEmail author
  • Anne Lemaitre
  • Nicolas Rambaux
Original Paper


Mercury is observed in a stable Cassini’s state, close to a 3:2 spin-orbit resonance, and a 1:1 node resonance. This present situation is not the only possible mathematical stable state, as it is shown here through a simple model limited to the second-order in harmonics and where Mercury is considered as a rigid body. In this framework, using a Hamiltonian formalism, four different sets of resonant angles are computed from the differential Hamiltonian equations, and each of them corresponds to four values of the obliquity; thanks to the calculation of the corresponding eigenvalues, their linear stability is analyzed. In this simplified model, two equilibria (one of which corresponding to the present state of Mercury) are stable, one is unstable, and the fourth one is degenerate. This degenerate status disappears with the introduction of the orbit (node and pericenter) precessions. The influence of these precession rates on the proper frequencies of the rotation is also analyzed and quantified, for different planetary models.


Mercury Equilibria Stability Precession 


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  1. Anderson J.D., Colombo G., Esposito P.B., Lau E.L., Trager G.B. (1987) The mass, gravity field and ephemeris of Mercury. Icarus 71, 337–349CrossRefADSGoogle Scholar
  2. Beletskii V.V. (1972) Resonance rotation of celestial bodies and Cassini’s laws. Celest. Mech. Dyn. Astron. 6, 359–378MathSciNetGoogle Scholar
  3. D’Hoedt S., Lemaitre A. (2004a) The spin-orbit resonant rotation of Mercury: a two degree of freedom Hamiltonian model. Celest. Mech. Dyn. Astron. 89, 267–283MathSciNetCrossRefADSGoogle Scholar
  4. D’Hoedt S., Lemaitre A. (2004b) The spin-orbit resonance of Mercury: a Hamiltonian approach. In: Kurtz D.W. (ed) Proceedings of the International Astronomical Union 196, Vol. 7–11 June 2004. Preston, UK, pp. 263–270Google Scholar
  5. Lemaitre A., D’Hoedt S., Rambaux N. (2006) The 3:2 spin-orbit resonant motion of Mercury. In: Proceedings of CELMEC IV. Celest. Mech. Dyn. Astron. 95, 213–224Google Scholar
  6. Peale S.J. (1974) Possible histories of the obliquity of Mercury. Astron. J. 79(6): 722–744CrossRefADSGoogle Scholar
  7. Rambaux N., Bois E. (2004) Theory of the Mercury’s spin-orbit motion and analysis of its main librations. Astron. Astrophys. 413, 381–393CrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2006

Authors and Affiliations

  • Sandrine D’Hoedt
    • 1
  • Anne Lemaitre
    • 1
  • Nicolas Rambaux
    • 1
  1. 1.Département de mathématiqueFUNDPNamurBelgique

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