The Stability of the Planetary Three-Body Problem Influence of the Secular Resonances

Abstract

At first sight, K.A.M. theory seems to be very useful for obtaining results of global stability in planetary problems. In fact, it describes the topology of nondegenerate quasi-integrable elliptical problems and provides result of stability for all time. In 1963 Arnold (Arnold, 1963) demonstrates a theorem of conservation of invariant tori in degenerated problems, that he apply to the planar planetary three-body problem to prove its stability. This result of stability was extended in 1992 to the spatial case (Robutel, 1992). But the proof of these results is only valid for very small values of the planetary masses, eccentricities and inclinations, which are largely inferior to these met in our solar system. In order to overcome this difficulty, we are going to use Laskar’s numerical method of frequency analysis (Laskar, 1992a) which enable us to have a global vision of the secular phase space for two planets in the neighbouring conditions of those of Jupiter and Saturn. The study of such a problem, which is more realistic than those based upon the K.A.M. theory, will lead us to give results of stability available in finite but very long time.