Symmetric horseshoe periodic orbits in the general planar three-body problem

Abstract

We present some families of horseshoe periodic orbits in the general planar three-body problem for the case of two equal masses. The considered system is a symmetric version of the one formed by Saturn, Janus and Epimetheus. We use a mass ratio equal to 35 × 10 −5 , corresponding to 10 5 times the Saturn-Janus mass parameter of the restricted case; for this mass ratio the satellites have a significantly bigger influence on the planet than in the classical Saturn, Janus and Epimetheus system. To obtain periodic orbits, we search those horseshoe orbits passing through two reversible configurations. A particular kind of periodic orbits where the minor bodies follow the same path is discussed.

Exchange orbits in the planar 1+4 body problem

“…Thus, we follow a different way. We use as Γ an initial condition of a periodic horseshoe orbit computed for the threebody problem [9] (any initial condition given in such reference works-it was corroborated that these initial conditions give rise to exchange-a orbits in the five-body problem). In the three-body problem, take the initial condition of that horseshoe orbit and integrate it numerically until the isosceles reversible configuration is reached.…”

“…First, obtain an initial condition Γ with the additional requirement of appropriate encounters by the corresponding orbit, that is of exchange-a type (horseshoeshaped in the rotating frame). To obtain Γ we use as a guide an initial condition of some known horseshoe orbit belonging to the three-body problem [9]. Second, determine S restricted to those orbits with appropriate encounters.…”

“…In [9], in order to obtain the corresponding Γ , the system of the three bodies was approximated by two decoupled Kepler problems. It was done in a region where the interaction between the minor bodies is negligible, far away of the encounters.…”

“…Determine Γ = (y 30 , vx 30 , vy 30 ). We have computed Γ using an initial condition of a periodic horseshoe orbit of the threebody problem [9] (see the explanation of this at the first paragraph of Section 5.1). With this we have implicitly a set of initial conditions S. 2.…”

“…In [8][9][10][11] we have studied the horseshoe motion in the general planar three-body problem (we avoid the massless restriction on one of the bodies and the circular orbit prescribed for the other two, as it is assumed in the restricted circular planar three-body problem). In the periodic aspect, we have focused on the most symmetric orbits, those related with the reversing symmetries of the equations of motion.…”