On the Stability of Resonant Asteroid Orbits

Abstract

The stability of the asteroid orbits has been studied by the method of surface of section. Families of simple symmetric periodic orbits of the asteroid and their stability have been computed and this served as a guide for the selection of the energy levels for the surface of section. In this way all possible cases for the structure of phase space have been obtained. It was found that the region in phase space around the resonant orbits at the resonances 1/3, 3/5, 5/7,…. is unstable, but small stability regions of doubly symmetric periodic orbits near the above resonances are also present. At the resonances 1/2, 2/3, 3/4,…. it was found that there exist two separate regions in phase space at about the same resonance 1/2, 2/3, 3/4,…., respectively, one being stable and the other unstable. At certain energy levels only the stable region appears. The above results are consistent with the observed distribution of the asteroids.