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Horseshoe orbits in the restricted four-body problem

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Abstract

The circular restricted four-body problem studies the dynamics of a massless particle under the gravitational force produced by three point masses that follow circular orbits with constant angular velocity, the configuration of these circular orbits forms an equilateral triangle for all time; this problem can be considered as an extension of the celebrated restricted three-body problem. In this work we investigate the orbits which emanate from some equilibrium points. In particular, we focus on the horseshoe shaped orbits (rotating frame), which are well known in the restricted three-body problem. We study some families of symmetric horseshoe orbits and show their relation with a family of the so called Lyapunov orbits.

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Notes

  1. http://www.minorplanetcenter.net/iau/lists/Trojans.html.

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Acknowledgements

The beginning of this research of the first author was developed during a postdoctoral stay at the Instituto Tecnológico Autónomo de México where the author thanks the hospitality. The second author is pleased to acknowledge the financial support from SNI and PROMEP-SEP.

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Authors and Affiliations

  1. Facultad de Ciencias Físico Matemáticas, Universidad Autónoma de Coahuila, Unidad Campo redondo, Edificio A, Saltillo, Coahuila, 25020, Mexico

    Jaime Burgos-Garcia

  2. Departamento Académico de Matemáticas, Instituto Tecnológico Autónomo de México, 01080, México, Ciudad de México, Mexico

    Abimael Bengochea

Authors
  1. Jaime Burgos-Garcia
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  2. Abimael Bengochea
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Corresponding author

Correspondence to Jaime Burgos-Garcia.

Appendix: Initial conditions for critical orbits

Appendix: Initial conditions for critical orbits

Table 1 Initial conditions for some horizontally critical orbits. The initial condition for \(\dot{y}_{0}\) can be obtained from the Jacobi first integral
Full size table
Table 2 Initial conditions for some vertically critical orbits. The initial condition for \(\dot{y}_{0}\) can be obtained from the Jacobi first integral
Full size table

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Burgos-Garcia, J., Bengochea, A. Horseshoe orbits in the restricted four-body problem. Astrophys Space Sci 362, 212 (2017). https://doi.org/10.1007/s10509-017-3193-x

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