Realistic Estimations of the Effective Stability Region of the Trojan Asteroids

Abstract

The usual approach in analytical studies of the stability of the Trojan asteroids is to consider simple models for the system such as the two dimensional (2D) planar, and the three dimensional (3D) spatial restricted three body problem (RTBP) (Giorgilli et al., 1989; Simo, 1989; Celletti and Giorgilli, 1991; Celletti and Ferrara, 1996). As an example of a more complicated model for the problem we refer to the model developed by Gabern and Jorba (2001) where the effect of Saturn on the motion of the asteroid has been taken into account. The techniques used in these papers are based in normal forms or first integrals calculations. Roughly speaking one shows that the system admits a number of approximate integrals, whose time variation can be controlled to be small for an extremely long time. In this case we have effective stability, i.e. even when an orbit is not stable, the time needed for it to leave the neighborhood of the equilibrium is larger than the expected lifetime of the physical system studied. This is the basis to derive the classical Nekhoroshev’s estimates (Nekhoroshev, 1997). The first result that guarantees the effective stability of real asteroids was provided by Giorgilli and Skokos (1997) for the 2D RTBP. In the present paper we refer to some recent results for the 3D RTBP obtained by Skokos and Dokoumetzidis (2001).