Celestial Mechanics and Dynamical Astronomy

, Volume 103, Issue 3, pp 267–279 | Cite as

Dynamics of planets in retrograde mean motion resonance

  • Julie Gayon
  • Eric Bois
  • Hans Scholl
Original Article


In a previous paper (Gayon and Bois 2008a), we have shown the general efficiency of retrograde resonances for stabilizing compact planetary systems. Such retrograde resonances can be found when two-planets of a three-body planetary system are both in mean motion resonance and revolve in opposite directions. For a particular two-planet system, we have also obtained a new orbital fit involving such a counter-revolving configuration and consistent with the observational data. In the present paper, we analytically investigate the three-body problem in this particular case of retrograde resonances. We therefore define a new set of canonical variables allowing to express correctly the resonance angles and obtain the Hamiltonian of a system harboring planets revolving in opposite directions. The acquiring of an analytical “rail” may notably contribute to a deeper understanding of our numerical investigation and provides the major structures related to the stability properties. A comparison between our analytical and numerical results is also carried out.


Hamiltonian systems Planetary systems Resonance Periodic orbits Retrograde planets Resonant average Hamiltonian 


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Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.CNRS, Observatoire de la Côte d’Azur, Laboratoire CassiopéeNice Sophia-Antipolis UniversityNice Cedex 4France

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