A study of temporary captures and collisions in the Circular Restricted Three-Body Problem with normalizations of the Levi-Civita Hamiltonian

The dynamics near the Lagrange equilibria $L_1$ and $L_2$ of the Circular Restricted Three-body Problem has gained attention in the last decades due to its relevance in some topics such as the temporary captures of comets and asteroids and the design of trajectories for space missions. In this paper we investigate the temporary captures using the tube manifolds of the horizontal Lyapunov orbits originating at $L_1$ and $L_2$ of the CR3BP at energy values which have not been considered so far. After showing that the radius of convergence of any Hamiltonian normalization at $L_1$ or $L_2$ computed with the Cartesian variables is limited in amplitude by $\|1-\mu-x_{L_1}\|$ ($\mu$ denoting the reduced mass of the problem), we investigate if regularizations allow us to overcome this limit. In particular, we consider the Hamiltonian describing the planar three-body problem in the Levi-Civita regularization and we compute its normalization for the Sun-Jupiter reduced mass for an interval of energy which overcomes the limit of Cartesian normalizations. As a result, for the largest values of the energy that we consider, we notice a transition in the structure of the tubes manifolds emanating from the Lyapunov orbit, which can contain orbits that collide with the secondary body before performing one full circulation around it. We discuss the relevance of this transition for temporary captures.