Inclined asymmetric librations in exterior resonances

  • G. Voyatzis
  • K. Tsiganis
  • K. I. Antoniadou
Original Article


Librational motion in Celestial Mechanics is generally associated with the existence of stable resonant configurations and signified by the existence of stable periodic solutions and oscillation of critical (resonant) angles. When such an oscillation takes place around a value different than 0 or \(\pi \), the libration is called asymmetric. In the context of the planar circular restricted three-body problem, asymmetric librations have been identified for the exterior mean motion resonances (MMRs) 1:2, 1:3, etc., as well as for co-orbital motion (1:1). In exterior MMRs the massless body is the outer one. In this paper, we study asymmetric librations in the three-dimensional space. We employ the computational approach of Markellos (Mon Not R Astron Soc 184:273–281,, 1978) and compute families of asymmetric periodic orbits and their stability. Stable asymmetric periodic orbits are surrounded in phase space by domains of initial conditions which correspond to stable evolution and librating resonant angles. Our computations were focused on the spatial circular restricted three-body model of the Sun–Neptune–TNO system (TNO = trans-Neptunian object). We compare our results with numerical integrations of observed TNOs, which reveal that some of them perform 1:2 resonant, inclined asymmetric librations. For the stable 1:2 TNO librators, we find that their libration seems to be related to the vertically stable planar asymmetric orbits of our model, rather than the three-dimensional ones found in the present study.


Circular restricted TBP Exterior resonances Spatial asymmetric periodic orbits Trans-Neptunian object dynamics 



The work of KIA was supported by the Fonds de la Recherche Scientifique-FNRS under Grant No. T.0029.13 (“ExtraOrDynHa” research project) and the University of Namur.

Compliance with ethical standards

Conflict of interest

The authors G. Voyatzis and K. Tsiganis declare that they have no conflict of interest.


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Authors and Affiliations

  1. 1.Department of PhysicsAristotle University of ThessalonikiThessalonikiGreece
  2. 2.NaXys, Department of MathematicsUniversity of NamurNamurBelgium

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