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Classifying orbits in the restricted three-body problem

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Abstract

The case of the planar circular restricted three-body problem is used as a test field in order to determine the character of the orbits of a small body which moves under the gravitational influence of the two heavy primary bodies. We conduct a thorough numerical analysis on the phase space mixing by classifying initial conditions of orbits and distinguishing between three types of motion: (1) bounded, (2) escape and (3) collisional. The presented outcomes reveal the high complexity of this dynamical system. Furthermore, our numerical analysis shows a remarkable presence of fractal basin boundaries along all the escape regimes. Interpreting the collisional motion as leaking in the phase space, we related our results to both chaotic scattering and the theory of leaking Hamiltonian systems. We also determined the escape and collisional basins and computed the corresponding escape/collisional times. We hope our contribution to be useful for a further understanding of the escape and collisional mechanism of orbits in the restricted three-body problem.

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Notes

  1. It should be emphasized that the momenta \(\dot{x}\) and \(\dot{y}\) are not the canonical momenta \(p_x\) and \(p_y\), respectively.

  2. An infinite number of regions of (stable) quasi-periodic (or small scale chaotic) motion is expected from classical chaos theory.

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Acknowledgments

I would like to express my warmest thanks to J. Nagler and Ch. Jung for all the illuminating and inspiring discussions during this research. My thanks also go to the anonymous referees for the careful reading of the manuscript and for all the apt suggestions and comments which allowed us to improve both the quality and the clarity of the paper.

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  1. Department of Physics, School of Science, Aristotle University of Thessaloniki, 541 24, Thessaloniki, Greece

    Euaggelos E. Zotos

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Zotos, E.E. Classifying orbits in the restricted three-body problem. Nonlinear Dyn 82, 1233–1250 (2015). https://doi.org/10.1007/s11071-015-2229-4

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