Session 1: Dynamical Astronomy and Celestial Mechanics

Title: Stable Chaos and Kirkwood Gaps
Author(s):  H. Varvoglis, K. Tsiganis, J. Hadjidemetriou (Oral)

It is well known since the 19th century that the Kirkwood gaps, observed in the distribution of orbital periods of the asteroids, correspond to simple commensurabilities with the orbital period of Jupiter. Furthermore it is now widely accepted that the main gap-depletion mechanism is related to the chaotic motion of asteroids, generated in the vicinity of these resonances, which resulted in an increase of the asteroids' eccentricities and eventually forced them to suffer close encounters with the major planets. However, what was not well understood so far, is why there are gaps only in the observed commensurabilities and not in others, of comparable order, where chaotic motion is also dominant, as it is evident in numerical simulations of even the simplest dynamical models. We suggest that the answer to this question comes from the fact that Jupiter does not follow a circular trajectory around the Sun, as assumed in evaluating the commensurabilities. We show that a resonance, calculated in the circular restricted three-body problem, does not necessarily entail the existence of a corresponding periodic trajectory in the elliptic one. As a consequence gaps could not be formed in the vicinity of most of the resonances of the circular problem, due to the above fact, unless secular mechanisms (in more sophisticated dynamical models) act as well. In a resonance where a gap cannot be formed, there exist semi-confined chaotic trajectories with small Lyapunov times, for which a secular growth of the eccentricity s almost undetectable. Therefore, although the trajectory of the asteroid is chaotic, its orbital elements do not change appreciably for hundreds of million years. This phenomenon is known as stable chaos.