Doubly-symmetric horseshoe orbits in the general planar three-body problem

Author： Bengochea Abimael Galán Jorge Pérez-Chavela Ernesto

ISSN： 0004-640X

Source： Astrophysics and Space Science, Vol.348, Iss.2, 2013-12, pp. : 403-415

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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.