It has often been observed that orbits near an

invariant surface become "trapped" into almost

regular orbits for long (but finite) times (Perry

and Wiggins, 1994; Kaneko and Konishi, 1994;

also Wisdom, 1983). Jensen (1984, a review)

and Lai et al. (1992, recent work) apply this to

the surviving fraction of un-ionized atoms in a

model of the ionization of surface-state elec-

trons in an oscillating electromagnetic field.

Such "stickiness" will have a profound effect on

the statistical properties of a system, including

diffusion rates or escape rates. A population of

trajectories will contain some members lying

closer than others to a KAM surface, leading to

a distribution of escape times, some of which

will be long.

Perry and Wiggins (1994) show that the

survival probability for an N-degree of freedom

Hamiltonian depends strongly on the dynamics

near the KAM tori. Solutions beginning there

simply take a long time to move away, despite

the fact that the motion is chaotic. They rigor-

ously derive a lower bound on the time it takes

to move away from a torus:

(3)

T m Cr e

(

K b / r

)

a

where T is the lower bound on the time it takes

to double the distance from the surface, r is the

distance from the surface, C and K are positive

constants that depend on the analytic properties

of the Hamiltonian,

> 0 is a measure of the

irrationality of the flow,

= 1/(N+2), and N is

the number of degrees of freedom. The more

irrational the flow (i.e., the higher the reso-

nance order), the more "sticky" is the surface.

Lower bounds like this were first obtained by

Nekhoroshev (1977) for a class of near-

integrable Hamiltonian systems and are there-

fore known as Nekhoroshev type bounds.

They were worked out for the time to move

away from elliptic equilibrium points by Gior-

gilli (1988) and Giorgilli et al. (1989).

The extension to the outer-belt asteroids

follows naturally, and we do not therefore

require the existence of some mysterious, new

class of objects or type of motion in order to

explain the observed behavior. It is, in fact,

quite consistent with the above-mentioned pre-

vious studies. The association with KAM tori

also explains our observation that the outer

asteroid orbits considered here all lie near a

high-order resonance with Jupiter, the main

perturber.

*Chaotic Motion in the Outer Asteroid Belt*

page 10

**Figure 3.**

Log of observed event time, T

e

, vs. the log of

the Lyapunov time, T

L

, for the 440 survey orbits calcu-

lated with m

2

=10 M

J

. The solid line is an unweighted

least squares fit, with slope *b*=1.74 ± 0.03 and intercept

*a*=1.30 ± 0.03. Dotted lines mark the 2

G

boundaries of a

Gaussian fit to the residuals.