Effective stability around the Cassini state in the spin-orbit problem

  • Marco SansotteraEmail author
  • Christoph Lhotka
  • Anne Lemaître
Original Article


We investigate the long-time stability in the neighborhood of the Cassini state in the conservative spin-orbit problem. Starting with an expansion of the Hamiltonian in the canonical Andoyer-Delaunay variables, we construct a high-order Birkhoff normal form and give an estimate of the effective stability time in the Nekhoroshev sense. By extensively using algebraic manipulations on a computer, we explicitly apply our method to the rotation of Titan. We obtain physical bounds of Titan’s latitudinal and longitudinal librations, finding a stability time greatly exceeding the estimated age of the Universe. In addition, we study the dependence of the effective stability time on three relevant physical parameters: the orbital inclination, \(i\), the mean precession of the ascending node of Titan orbit, \(\dot{\varOmega }\), and the polar moment of inertia, \(C\).


Spin-orbit resonance Normal form methods Cassini state Titan  Long-time stability 



The work of C. L. was financially supported by the contract Prodex C90253 “ROMEO” from BELSPO, and partly by the Austrian FWF research grant P-J3206. The work of M. S. is supported by an FSR Incoming Post-doctoral Fellowship of the Académie universitaire Louvain, co-funded by the Marie Curie Actions of the European Commission.

Supplementary material

10569_2014_9547_MOESM1_ESM.pdf (54 kb)
Supplementary material 1 (pdf 54 KB)


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Marco Sansottera
    • 1
    Email author
  • Christoph Lhotka
    • 1
  • Anne Lemaître
    • 1
  1. 1.naXys, Department of MathematicsUniversity of NamurNamurBelgium

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