On co-orbital quasi-periodic motion in the three-body problem

Within the framework of the planar three-body problem we establish the existence of quasi-periodic motions and KAM $4$-tori related to the co-orbital motion of two small moons about a large planet where the moons move in nearly circular orbits with almost equal radii. The approach is based on a combination of normal form and symplectic reduction theories and the application of a KAM theorem for high-order degenerate systems. To accomplish our results we need to expand the Hamiltonian of the three-body problem as a perturbation of two uncoupled Kepler problems. This approximation is valid in the region of phase space where co-orbital solutions occur.Peer ReviewedPostprint (author's final draft

Invariant manifolds of L_3 and horseshoe motion in the restricted three-body problem

In this paper, we consider horseshoe motion in the planar restricted three-body problem. On one hand, we deal with the families of horseshoe periodic orbits (which surround three equilibrium points called L3, L4 and L5), 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 horseshoe periodic orbits for any value of µ ∈ (0, 1/2] implies the understanding of the behaviour of the invariant manifolds of L3. So, a systematic analysis of such manifolds is carried out. As well the implications on the number of homoclinic connections to L3, and on the simple infinite and double infinite period homoclinic phenomena are also analysed. Finally, the relationship between the horseshoe homoclinic orbits and the horseshoe periodic orbits are considered in detail

Breakdown of homoclinic orbits to L3 in the RPC3BP (II). An asymptotic formula

PreprintThe Restricted 3-Body Problem models the motion of a body of negligible mass under the gravitational influence of two massive bodies called the primaries. If one assumes that the primaries perform circular motions and that all three bodies are coplanar, one has the Restricted Planar Circular 3-Body Problem (RPC3BP). In rotating coordinates, it can be modeled by a two degrees of freedom Hamiltonian, which has five critical points called the Lagrange points L1,.., L5. The Lagrange point L3 is a saddle-center critical point which is collinear with the primaries and beyond the largest of the two. In this paper, we obtain an asymptotic formula for the distance between the stable and unstable manifolds of L3 for small values of the mass ratio 0<µ«1. In particular we show that L3 cannot have (one round) homoclinic orbits. If the ratio between the masses of the primaries µ is small, the hyperbolic eigenvalues of L3 are weaker, by a factor of order µ--v, than the elliptic ones. This rapidly rotating dynamics makes the distance between manifolds exponentially small with respect to µ--v. Thus, classical perturbative methods (i.e the Melnikov-Poincaré method) can not be applied. The obtention of this asymptotic formula relies on the results obtained in the prequel paper on the complex singularities of the homoclinic of a certain averaged equation and on the associated inner equation. In this second paper, we relate the solutions of the inner equation to the analytic continuation of the parameterizations of the invariant manifolds of L3 via complex matching techniques. We complete the proof of the asymptotic formula for their distance showing that its dominant term is the one given by the analysis of the inner equation.Preprin

Breakdown of homoclinic orbits to L3 in the RPC3BP (II): an asymptotic formula

The Restricted 3-Body Problem models the motion of a body of negligible mass under the gravitational influence of two massive bodies called the primaries. If one assumes that the primaries perform circular motions and that all three bodies are coplanar, one has the Restricted Planar Circular 3-Body Problem (RPC3BP). In rotating coordinates, it can be modeled by a two degrees of freedom Hamiltonian, which has five critical points called the Lagrange points L1, . . . , L5. The Lagrange point L3 is a saddle-center critical point which is collinear with the primaries and beyond the largest of the two. In this paper, we obtain an asymptotic formula for the distance between the stable and unstable manifolds of L3 for small values of the mass ratio 0 < \mu \leqslant 1. In particular we show that L3 cannot have (one round) homoclinic orbits. If the ratio between the masses of the primaries \mu is small, the hyperbolic eigenvalues of L3 are weaker, by a factor of order \sqrt{\mu }, than the elliptic ones. This rapidly rotating dynamics makes the distance between manifolds exponentially small with respect to \sqrt{\mu } . Thus, classical perturbative methods (i.e. the Melnikov-Poincaré method) can not be applied.Peer ReviewedPostprint (published version

Coorbital periodic orbits in the three body problem

We consider the dynamics of coorbital motion of two small moons about a large planet which have nearly circular orbits with almost equal radii. These moons avoid collision because they switch orbits during each close encounter. We approach the problem as a perturbation of decoupled Kepler problems as in Poincar ́ e’s periodic orbits of the first kind. The perturbation is large but only in a small region in the phase space. We discuss the relationship required among the small quantities (radial separation, mass, and minimum angular separation). Persistence of the orbits is discussed.Peer Reviewe

Coorbital periodic orbits in the three body problem

We consider the dynamics of coorbital motion of two small moons about a large planet which have nearly circular orbits with almost equal radii. These moons avoid collision because they switch orbits during each close encounter. We approach the problem as a perturbation of decoupled Kepler problems as in Poincar ́ e’s periodic orbits of the first kind. The perturbation is large but only in a small region in the phase space. We discuss the relationship required among the small quantities (radial separation, mass, and minimum angular separation). Persistence of the orbits is discussed.Peer Reviewe