CHAOTIC MOTION IN THE OUTER ASTEROID BELT AND ITS RELATION TO THE AGEOF THE SOLAR-SYSTEM

Recently, we analyzed a relation, found for chaotic orbits, between th e Lyapunov time T-L (the inverse of the maximum Lyapunov exponent) and the ''event'' time T-e (the time at which an orbit becomes clearly un stable). In this paper we treat two new problems. First, we apply this T-L-T-e relation to numerical integrations of 25 outer-belt asteroids and show that, when viewed in the proper context of a Gaussian distri bution of event time residuals, none of the 25 objects exhibit an anom alously short Lyapunov time. The current age of the solar system is ap proximately three standard deviations or less from the anticipated eve nt times of all of these asteroids. We argue that the Lyapunov times o f the 25 remaining bodies are each consistent with the age of the sola r system, and that we are therefore seeing the remnants of a larger or iginal distribution. The bulk of that population has been ejected by J upiter, leaving the ''tail members'' as present-day survivors. This in terpretation is consistent with current understanding of the behavior of trajectories near KAM tori in Hamiltonian systems. In particular, t here is no need to invoke a new type of motion or class of dynamical o bjects to explain the short Lyapunov time scales found for solar syste m objects. Second, we discuss integrations of 440 fictitious outer-bel t asteroids and show that the slope and offset parameters of the T-L-T -e relation do not change with an increase in Jupiter's mass by a fact or of 10, and that the distribution of residuals in log T-e is Gaussia n. This allows us to sensibly and quantitatively interpret the signifi cance of the Lyapunov time scale. However, the width of the residuals distribution is a function of mass ratio. Since knowledge of the distr ibution width is needed in order to interpret the significance of pred icted event times, a calibration must be performed at the mass ratio o f interest.