Celestial Mechanics and Dynamical Astronomy

, Volume 126, Issue 4, pp 369–403 | Cite as

Long-term dynamics beyond Neptune: secular models to study the regular motions

  • Melaine SaillenfestEmail author
  • Marc Fouchard
  • Giacomo Tommei
  • Giovanni B. Valsecchi
Original Article


Two semi-analytical one-degree-of-freedom secular models are presented for the motion of small bodies beyond Neptune. A special attention is given to trajectories entirely exterior to the planetary orbits. The first one is the well-known non-resonant model of Kozai (Astron J 67:591, 1962) adapted to the transneptunian region. Contrary to previous papers, the dynamics is fully characterized with respect to the fixed parameters. A maximum perihelion excursion possible of 16.4 AU is determined. The second model handles the occurrence of a mean-motion resonance with one of the planets. In that case, the one-degree-of-freedom integrable approximation is obtained by postulating the adiabatic invariance, and is much more general and accurate than previous secular models found in the literature. It brings out in a plain way the possibility of perihelion oscillations with a very high amplitude. Such a model could thus be used in future studies to deeper explore that kind of motion. For complex resonant orbits (especially of type 1 : k), a segmented secular description is necessary since the trajectories are only “integrable by parts”. The two models are applied to the Solar System but the notations are kept general so that it could be used for any quasi-circular and coplanar planetary system.


Secular model Lidov–Kozai mechanism Mean-motion resonance Transneptunian object (TNO) High-perihelion TNOs Resonant secular model 



We thank Giovanni F. Gronchi and Andrea Milani for their precious advice: sometimes, even a few words can be of great help. We thank also two anonymous referees who helped us to improve the paper.


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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Melaine Saillenfest
    • 1
    • 2
    Email author
  • Marc Fouchard
    • 1
    • 3
  • Giacomo Tommei
    • 2
  • Giovanni B. Valsecchi
    • 4
    • 5
  1. 1.IMCCEObservatoire de ParisParisFrance
  2. 2.Dipartimento di MatematicaUniversità di Pisa, Largo Bruno Pontecorvo 5PisaItaly
  3. 3.LAL-IMCCEUniversité de Lille, 1 Impasse de l’ObservatoireLilleFrance
  4. 4.IAPS-INAFRomaItaly
  5. 5.IFAC-CNRSesto FiorentinoItaly

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