Qualitative and analytical results of the bifurcation thresholds to halo orbits

Abstract

We study the dynamics in the neighborhood of the collinear Lagrangian points in the spatial, circular, restricted three--body problem. We consider the case in which one of the primaries is a radiating body and the other is oblate (although the latter is a minor effect). Beside having an intrinsic mathematical interest, this model is particularly suited for the description of a mission of a spacecraft (e.g., a solar sail) to an asteroid. The aim of our study is to investigate the occurrence of bifurcations to halo orbits, which take place as the energy level is varied. The estimate of the bifurcation thresholds is performed by analytical and numerical methods: we find a remarkable agreement between the two approaches. As a side result, we also evaluate the influence of the different parameters, most notably the solar radiation pressure coefficient, on the dynamical behavior of the model. To perform the analytical and numerical computations, we start by implementing a center manifold reduction. Next, we estimate the bifurcation values using qualitative techniques (e.g. Poincar\'e surfaces, frequency analysis, FLIs). Concerning the analytical approach, following \cite{CPS} we implement a resonant normal form, we transform to suitable action-angle variables and we introduce a detuning parameter measuring the displacement from the synchronous resonance. The bifurcation thresholds are then determined as series expansions in the detuning. Three concrete examples are considered and we find in all cases a very good agreement between the analytical and numerical results.