Stable Chaos in the Asteroid Belt

Abstract

Most asteroids are characterized by positive Lyapounov Characteristic Exponents (LCE), which means that their orbits are chaotic. Indeed, the striking fact is that for many of them the maximum LCE is very large, with a corresponding Lyapounov time shorter (often much shorter) than 100,000 years, as we report in this work. This finding is in apparent contradiction with the overall orbital stability observed in numerical integrations covering timespans orders of magnitude longer than the Lyapounov times. More importantly, it is in contradiction with the estimated astronomical age of these small planets. The phenomenon, of which other examples are already known, is referred to as stable chaos. We have studied in particular a region, that of the Veritas asteroid family, where other causes of chaotic motion are not relevant and stable chaos can be investigated as such in its various appearances. Stable chaos is found to be due to high order mean motion resonances with Jupiter in combination with secular perturbations on the perihelia of the asteroids. These perturbations cause a large number of critical arguments to get in and out of the resonance and can move the orbit from one high resonance to another, in a typical irregular behavior. Since chaos is driven by secular perturbations, it is no surprise that the Lyapounov times are very close to the timescales of these perturbations. The quasi-integrals of motion of asteroids with very short Lyapounov times are less stable and a slow diffusion is observed in eccentricity and inclination. There is essentially no diffusion in semimajor axis, although variations are large (e.g., between two high order resonances). These asteroids can therefore be regarded as cases of stable chaos.