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Astronomical Applications Department, U.S. Naval Observatory - lyap2 (Page 2)

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Astronomical Applications Department, U.S. Naval Observatory - lyap2
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 survi -
vors. 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 sys-
tem objects.
Second, we discuss integrations of 440 fictitious outer-belt 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
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 per -
formed at the mass ratio of interest.
Murison, Lecar, and Franklin

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