Chaotic motion in the outer asteroid belt and its relation to the age of the solar system

Abstract

Recently, we analyzed a relation, found for chaotic orbits, between the 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 unstable). 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 distribution of event time residuals, none of the 25 objects exhibits an anomalously short Lyapunov time. The current age of the solar system is approximately three standard deviations or less from the anticipated event times of all of these asteroids. We argue that the Lyapunov times of the 25 remaining bodies are each consistent with the age of the solar system, and that we are therefore seeing the remnants of a larger original distribution. The bulk of that population has been ejected by Jupiter, leaving the "tail members" as present-day survivors. This interpretation is consistent with current understanding of the behavior of trajectories near KAM tori in Hamiltonian systems. In particular, there is no need to invoke a new type of motion or class of dynamical objects to explain the short Lyapunov timescales found for solar system objects. Second, we discuss integrations of 440 fictitious outer-belt asteroids and show that the slope and offset parameters of the TL-T e relation do not change with an increase in Jupiter's mass by a factor of 10, and that the distribution of residuals in log T e is Gaussian. This allows us to sensibly and quantitatively interpret the significance of the Lyapunov timescale. However, the width of the residuals distribution is a function of mass ratio. Since knowledge of the distribution width is needed in order to interpret the significance of predicted event times, a calibration must be performed at the mass ratio of interest.

Short Lyapunov time: a method for identifying confined chaos

“…The time required for that to happen is called the "event time" (T E ). Several studies involving the instability of asteroids disturbed by Jupiter and/or other planets (Soper et al 1990; Lecar et al 1992;Franklin et al 1993;Levison & Duncan 1993;Murison et al 1994;Whipple 1995) have claimed that there is a simple relation between the Lyapunov time, T L , and the event time, T E (to become a Jupiter's crosser, for example) given by…”

Regimes of stability and scaling relations for the removal time in the asteroid belt: a simple kinetic model and numerical tests

“…Obviously, we have a substantially better fit. be regarded as "laws" in the sense of Murison, Lecar & Franklin (1994). Furthermore, due to their statistical nature, they cannot be used for any particular object, only for populations.…”

Stability of the HD 12661 Planetary System

“…Because the motion is close to a separatrix (or takes place within it), it can be influenced by the stickiness phenomenon. This effect was studied, e.g., by Murison et al (1994), and recently, by Tsiganis et al (2000Tsiganis et al ( , 2002, for the models of asteroid motion in the Solar system. The former authors found that the long-term evolution of chaotic orbits initiated in the vicinity of some high-order mean motion resonances with Jupiter is possible if there exist no resonant periodic orbits in the framework of the elliptic-three body problem.…”