Effective stability of the Trojan asteroids

We study the spatial circular restricted problem of three bodies in the lig ht of Nekhoroshev theory of stability over large time intervals. We conside r in particular the Sun-Jupiter model and the Trojan asteroids in the neigh borhood of the Lagrangian point Lq. We find a region of effective stability around the point Lq such that if the initial point of an orbit is inside t his region the orbit is confined in a slightly larger neighborhood of the e quilibrium (in phase space) for a very long time interval. By combining ana lytical methods and numerical approximations we are able to prove that stab ility over the age of the universe is guaranteed on a realistic legion, big enough to include one real asteroid. By comparing this result with the one obtained for the planar problem we see that the regions of stability in th e two cases are of the same magnitude.