A restricted four-body model for the dynamics near the Lagrangian points of the Sun-Jupiter system

We focus on the dynamics of a small particle near the Lagrangian points of the Sun-Jupiter system. To try to account for the effect of Saturn, we develop a specific model based on the computation of a true solution of the planar three-body problem for Sun, Jupiter and Saturn, close to the real motion of these three bodies. Then, we can write the equations of motion of a fourth infinitesimal particle moving under the attraction of these three masses. Using suitable coordinates, the model is written as a time-dependent perturbation of the well-known spatial Restricted Three-Body Problem. Then, we study the dynamics of this model near the triangular points. The tools are based on computing, up to high order, suitable normal forms and first integrals. From these expansions, it is not difficult to derive approximations to invariant tori (of dimensions 2, 3 and 4) as well as bounds on the speed of diffusion on suitable domains. We have also included some comparisons with the motion of a few Trojan asteroids in the real Solar system.