Abstract
We present some results about the continuation of doubly-symmetric horseshoe orbits in the general planar three-body problem. This is done by means of solving a boundary value problem with one free parameter which is the quotient of the masses of two bodies μ 3=m 3/m 1, keeping constant μ 2=m 2/m 1 (m 1 represents the mass of a big planet whereas m 2 and m 3 of minor bodies). For the numerical continuation of the horseshoe orbits we have considered m 2/m 1=3.5×10−4, and the variation of μ 3 from 3.5×10−4 to 9.7×10−5 or vice versa, depending on the orbit selected as “seed”. We discuss some issues related to the periodicity and symmetry of the orbits. We study the stability of some of them taking the limit μ 3→0. The numerical continuation was done using the software AUTO.
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Acknowledgements
The first author is pleased to acknowledge the financial support from CONACYT-México which allows him a postdoctoral stay in the Departamento de Matemática Aplicada II of the Universidad de Sevilla, and the facilities given by the Universidad de Sevilla. The second author wishes to acknowledge financial support from the Spanish government through grants MTM2009-07849 and MTM2012-31821. The first and third authors have been partially supported by CONACYT, Grant 128790.
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Bengochea, A., Galán, J. & Pérez-Chavela, E. Doubly-symmetric horseshoe orbits in the general planar three-body problem. Astrophys Space Sci 348, 403–415 (2013). https://doi.org/10.1007/s10509-013-1590-3
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DOI: https://doi.org/10.1007/s10509-013-1590-3