On the vertical families of two-dimensional tori near the triangular points of the bicircular problem

Abstract

This paper focuses on some aspects of the motion of a small particle moving near the Lagrangian points of the Earth–Moon system. The model for the motion of the particle is the so-called bicircular problem (BCP), that includes the effect of Earth and Moon as in the spatial restricted three body problem (RTBP), plus the effect of the Sun as a periodic time-dependent perturbation of the RTBP. Due to this periodic forcing coming from the Sun, the Lagrangian points are no longer equilibrium solutions for the BCP. On the other hand, the BCP has three periodic orbits (with the same period as the forcing) that can be seen as the dynamical equivalent of the Lagrangian points. In this work, we first discuss some numerical methods for the accurate computation of quasi-periodic solutions, and then we apply them to the BCP to obtain families of 2-D tori in an extended neighbourhood of the Lagrangian points. These families start on the three periodic orbits mentioned above and they are continued in the vertical (z and z) direction up to a high distance. These (Cantor) families can be seen as the continuation, into the BCP, of the Lyapunov family of periodic orbits of the Lagrangian points that goes in the (z, z) direction. These results are used in a forthcoming work [9] to find regions where trajectories remain confined for a very long time. It is remarkable that these regions seem to persist in the real system.