Branches and bifurcations of ejection–collision orbits in the planar circular restricted three body problem

Abstract The goal of this paper it to prove existence theorems for one parameter families (branches) of ejection-collision orbits in the planar circular restricted three body problem (CRTBP), and to study some of bifurcations of these branches. The CRTBP considers the dynamics of an infinitesimal particle moving in the gravitational field of two massive primary bodies. The motion of the primaries assumed to be circular, and we study ejection–collision orbits where the infinitesimal body is ejected from one primary and collides with the other (as opposed to more local ejections–collisions where the infinitesimal body collides with a single primary body in both forward and backward time). We consider branches of ejection–collision orbits which are (i) parameterized by the Jacobi integral (energy like quantity conserved by the CRTBP) with the masses of the primaries fixed, and (ii) parameterized by the mass ratio of the primary bodies with energy fixed. The method of proof is constructive and computer assisted, hence can be applied in non-perturbative settings and (potentially) to other conservative systems of differential equations. The main requirement is that the system should admit a change of coordinates which regularizes the collision singularities. In the planar CRTBP, the necessary regularization is provided by the classical Levi–Civita transformation.