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Symmetric periodic orbits in the Moulton–Copenhagen problem

  • Manuel PalaciosEmail author
  • Mercedes Arribas
  • Alberto Abad
  • Antonio Elipe
Original Article
  • 116 Downloads

Abstract

We consider the planar restricted four-body problem proposed by Moulton. One infinitesimal mass moves under the attraction of three mass points in collinear Euler’s configuration, namely \(P_0 (m_0 = \mu \,m)\) is placed at the origin, and other two identical points \(P_1 (m)\) and \(P_2 (m)\) are placed at the same distance from the origin. The problem is an extension of the well-known Copenhagen problem, in which \(P_0\) does not exist, and therefore, the name is chosen for the considered problem. We perform a study on the evolution of families of symmetric periodic orbits (characteristic curves) as the mass parameter \(\mu \) evolves. Compared with the Copenhagen problem, we find new families of periodic orbits and how the classical ones change. We also analyse the number and evolution of spiral points, which represent the heteroclinic orbits connecting equilibrium points.

Keywords

Moulton–Copenhagen problem Four-body problem Symmetric periodic orbits Characteristic curves Asymptotic orbits 

Notes

Acknowledgements

Authors are thankful to the two anonymous reviewers, whose comments and suggestions have been very useful in improving the manuscript. This research was supported by: the Ministerio de economía y competitividad (Spain), Project ESP2017-87113-R (AEI/FEDER, UE) and the DGA (Government of Aragón), Project E24_17R. APEDIF (Aplicaciones de Ecuaciones DIFerenciales).

Compliance with ethical standards

Conflict of interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Grupo de Mecánica Espacial and IUMAUniversity of ZaragozaZaragozaSpain
  2. 2.ZaragozaSpain

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