Celestial Mechanics and Dynamical Astronomy

, Volume 117, Issue 4, pp 405–421 | Cite as

Retrograde resonance in the planar three-body problem

  • M. H. M. Morais
  • F. Namouni
Original Article


We continue the investigation of the dynamics of retrograde resonances initiated in Morais and Giuppone (Mon Notices R Astron Soc 424:52–64, doi: 10.1111/j.1365-2966.2012.21151.x, 2012). After deriving a procedure to deduce the retrograde resonance terms from the standard expansion of the three-dimensional disturbing function, we concentrate on the planar problem and construct surfaces of section that explore phase-space in the vicinity of the main retrograde resonances (2/\(-\)1, 1/\(-\)1 and 1/\(-\)2). In the case of the 1/\(-\)1 resonance for which the standard expansion is not adequate to describe the dynamics, we develop a semi-analytic model based on numerical averaging of the unexpanded disturbing function, and show that the predicted libration modes are in agreement with the behavior seen in the surfaces of section.


Resonance Three-body problem Surface of section Co-orbital resonance 



We thank both reviewers for helpful suggestions that improved the article’s clarity. We acknowledge financial support from FCT-Portugal (PEst-C/CTM/LA0025/2011). The surfaces of section computations were performed on the Blafis cluster at the University of Aveiro.

Supplementary material

10569_2013_9519_MOESM1_ESM.pdf (1.7 mb)
Supplementary material 1 (pdf 1732 KB)
10569_2013_9519_MOESM2_ESM.pdf (8.4 mb)
Supplementary material 2 (pdf 8596 KB)


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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of Physics & I3NUniversity of AveiroAveiroPortugal
  2. 2.Université de Nice, CNRSNiceFrance

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