Astronomy and Astrophysics – Astronomy
Scientific paper
Dec 1994
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1994aj....108.2323m&link_type=abstract
Astronomical Journal (ISSN 0004-6256), vol. 108, no. 6, p. 2323-2329
Astronomy and Astrophysics
Astronomy
32
Asteroid Belts, Astronomical Models, Celestial Mechanics, Chronology, Liapunov Functions, Solar System Evolution, Computerized Simulation, Jupiter (Planet), Mass Distribution, Mass Ratios, Normal Density Functions, Numerical Integration
Scientific paper
Recently, we analyzed a relation, found for chaotic orbits, between the Lyapunov time TL (the inverse of the maximum Lyapunov exponent) and the 'event' time Te (the time at which an orbit becomes clearly unstable). In this paper we treat two new problems. First, we apply this TL - Te 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 exhibit 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 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 time scales 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 - Te relation do not change with an increase in Jupiter's mass by a factor of 10, and that the distribution of residuals in log Te is Gaussian. This allows us to sensibly and quantitatively interpret the significance of the Lyapunov time scale. 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.
Franklin Fred A.
Lecar Myron
Murison Marc Allen
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