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Origin and continuation of 3/2, 5/2, 3/1, 4/1 and 5/1 resonant periodic orbits in the circular and elliptic restricted three-body problem

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Abstract

We consider a planetary system consisting of two primaries, namely a star and a giant planet, and a massless secondary, say a terrestrial planet or an asteroid, which moves under their gravitational attraction. We study the dynamics of this system in the framework of the circular and elliptic restricted three-body problem, when the motion of the giant planet describes circular and elliptic orbits, respectively. Originating from the circular family, families of symmetric periodic orbits in the 3/2, 5/2, 3/1, 4/1 and 5/1 mean-motion resonances are continued in the circular and the elliptic problems. New bifurcation points from the circular to the elliptic problem are found for each of the above resonances, and thus, new families continued from these points are herein presented. Stable segments of periodic orbits were found at high eccentricity values of the already known families considered as whole unstable previously. Moreover, new isolated (not continued from bifurcation points) families are computed in the elliptic restricted problem. The majority of the new families mainly consists of stable periodic orbits at high eccentricities. The families of the 5/1 resonance are investigated for the first time in the restricted three-body problems. We highlight the effect of stable periodic orbits on the formation of stable regions in their vicinity and unveil the boundaries of such domains in phase space by computing maps of dynamical stability. The long-term stable evolution of the terrestrial planets or asteroids is dependent on the existence of regular domains in their dynamical neighbourhood in phase space, which could host them for long-time spans. This study, besides other celestial architectures that can be efficiently modelled by the circular and elliptic restricted problems, is particularly appropriate for the discovery of terrestrial companions among the single-giant planet systems discovered so far.

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Notes

  1. The reader may refer to exoplanet.eu (Schneider et al. 2011) and exoplanets.org (Han et al. 2014).

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Acknowledgements

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. Computational resources have been provided by the Consortium des Équipements de Calcul Intensif (CÉCI), funded by the Fonds de la Recherche Scientifique de Belgique (F.R.S.-FNRS) under Grant No. 2.5020.11.

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  1. NaXys, Department of Mathematics, University of Namur, 8 Rempart de la Vierge, 5000, Namur, Belgium

    Kyriaki I. Antoniadou & Anne-Sophie Libert

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  1. Kyriaki I. Antoniadou
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  2. Anne-Sophie Libert
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Correspondence to Kyriaki I. Antoniadou.

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This article is part of the topical collection on Recent advances in the study of the dynamics of N-body problem.

Guest Editors: Giovanni Federico Gronchi, Ugo Locatelli, Giuseppe Pucacco and Alessandra Celletti.

Appendix A: Maps of dynamical stability for the exact MMRs

Appendix A: Maps of dynamical stability for the exact MMRs

See Fig. 16.

Fig. 16
figure 16

DS-maps computed for the exact MMR for the four symmetric configurations with a \(a_1=2/1^{-2/3}\), b \(a_1=3/2^{-2/3}\), c \(a_1=5/2^{-2/3}\), d \(a_1=3/1^{-2/3}\), e \(a_1=4/1^{-2/3}\) and f \(a_1=5/1^{-2/3}\). Overplotted are the families of periodic orbits in each MMR. The resolution of the initial conditions per configuration is \(200\times 200\)

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Antoniadou, K.I., Libert, AS. Origin and continuation of 3/2, 5/2, 3/1, 4/1 and 5/1 resonant periodic orbits in the circular and elliptic restricted three-body problem. Celest Mech Dyn Astr 130, 41 (2018). https://doi.org/10.1007/s10569-018-9834-8

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