Invariant manifolds ofL₃and horseshoe motion in the restricted three-body problem

Abstract

In this paper, we consider horseshoe motion in the planar restricted threebody problem. On one hand, we deal with the families of horseshoe periodic orbits (HPOs) (which surround three equilibrium points called L 3 , L 4 and L 5 ), when the mass parameter µ is positive and small; we describe the structure of such families from the two-body problem (µ = 0). On the other hand, the region of existence of HPOs for any value of µ ∈ (0, 1/2] implies the understanding of the behaviour of the invariant manifolds of L 3 . So, a systematic analysis of such manifolds is carried out. As well the implications on the number of homoclinic connections to L 3 and on the simple infinite and double infinite period homoclinic phenomena are analysed. Finally, the relationship between the horseshoe homoclinic orbits and the HPO is considered in detail.

Dynamical aspects of multi-round horseshoe-shaped homoclinic orbits in the RTBP

“…This means that for µ > 0.01173615, the dynamics around the small primary and the equilibrium points L 1 and L 2 play a role (and the corresponding Lyapunov orbits and their invariant manifolds). In particular, there exist values of µ for which one of the invariant manifolds collides with the small primary (see Barrabés and Ollé, 2006) and values of µ for which there are not homoclinic but heteroclinic connections between L 3 and a Lyapunov periodic orbit around L 1 or L 2 (see Fig. 1).…”

“…The computation of horseshoe periodic orbits (HPO) in the spatial RTBP has been done for several authors, Schanzle (see for example 1967), or Taylor (1981) were some families of horseshoe periodic orbits are shown for the Sun-Jupiter mass ratio. More recently, in Barrabés and Mikkola (2005), the computation and description of the organization of families was done, and in Barrabés and Ollé (2006), the existence of symmetrical HPO in the planar RTBP from the dynamical behavior of the invariant manifolds of L 3 was studied. Furthermore, there is numerical evidence (Farrés, 2005;Gómez et al, 2001;Simó, 2006) on the fact that the stable and unstable manifolds of the objects (Lyapunov periodic orbits and 2D tori) of the center manifold of L 3 in the 3D RTBP confine regions of effective stability around the triangular points L 4 and L 5 .…”

“…In Koltsova et al (2005), the authors analyzed the homoclinic orbits to invariant tori near a homoclinic orbit to a center×center×saddle equilibrium point (in a Hamiltonian with three degrees of freedom). On the other hand, the existence of an infinite set of periodic orbits accumulating to a given homoclinic orbit (the so called blue sky catastrophe phenomenon after Devaney, see Devaney (1977) and also explains the evolution of some families of horseshoe periodic orbits, when varying the mass parameter µ and the Jacobi constant C (see Barrabés and Ollé, 2006). The paper is structured as follows.…”

Spacecraft trajectories to the L3 point of the Sun–Earth three-body problem

“…The numbers n 1 and n 2 of revolutions on the two elliptical, resonant orbits are integer and are chosen by means of a grid search. An important characteristic of this type of solutions is that the s/c encounters the Earth in a swingby with adequate dynamical parameters and reaches L 3 with a small insertion maneuver 1 . Of course, the price is a much extended TOF with respect to the standard solution.…”

THE SYMMETRIC HORSESHOE PERIODIC FAMILIES AND THE LYAPUNOV PLANAR FAMILY AROUNDL3

“…Later, Taylor (1981) proved that this conjecture was wrong: the smooth horseshoe orbits belong to different families. Schanzle (1967) proposed a useful way to compute the symmetric horseshoe periodic orbits, and later Barrabés et al (Barrabés & Mikkola 2005;Barrabés & Ollé 2006) made a more systematic study of the structures of the symmetric horseshoe periodic families using a similar method to that proposed by Schanzle. In any case, they did not give the natural terminations of these families.…”