I. INTRODUCTION

Strikingly large values of the maximum

Lyapunov exponent are associated with cha-

otic motion in the solar system (e.g. Sussman

and Wisdom 1988, 1992; Laskar 1989; Torbett

1989; Mikkola and Innanen 1992). In this

paper, we continue our efforts to interpret the

Lyapunov time, T

L

=1/

, in the solar system in

light of a relation between T

L

and the "event"

time T

e

(Soper et al. 1990; Lecar et al. 1992a;

(1)

log

T

e

T

0

=

a

+

b log

T

L

T

0

where T

0

is an appropriate normalizing period.

The event time is the timescale on which the

qualitative character of the motion changes.

Events are indicated by, for example, a close

approach to a planet, the crossing of a plane-

tary orbit, or the escape of a satellite in gen-

eral, an ejection or a collision. We use the

notation T

e

, rather than T

c

, the planetary orbit

crossing time (Lecar et al., 1992a), to reflect

this more generalized meaning of an event. For

this study, T

0

=T

J

, the orbital period of Jupiter.

Notice that, unlike the slope *b*, the value of the

offset parameter *a* scales with T

0

. For orbits

interior to Jupiter, we find in this paper that

*a*=1.30 ± 0.03 and *b*=1.74 ± 0.03, in good

agreement with the preliminary results of LFM.

The T

L

-T

e

relation is the only known method

for *prediction* of the long-term instability

timescale of solar-system bodies.

Here we focus our attention primarily on

two issues. First, we have calculated T

L

for all

25 known outer-belt asteroids not associated

with a major resonance (3.43 < a < 3.76 AU).

Just beyond the upper limit of the range consid-

ered here are the Hilda group asteroids. See

Franklin et al. (1993) for further discussion and

an application of the T

L

-T

e

relation to these

interesting objects. The Lyapunov times of the

25 orbits considered here range from 3200 yr

to greater than 96,000 yr. We interpret these

values in terms of the T

L

-T

e

relation and argue

that the few remaining bodies with short T

L

are

the expected remnants of an initially much

larger population. An alternative view (Milani

and Nobili, 1992), that existing bodies with

short T

L

are members of a curious class of

objects described by the misleading label "sta-

ble chaos," seems unlikely. The existence of

this purported class was inferred on the basis of

numerical integrations of a single body, (522)

Helga, and is, we claim, a misinterpretation of

the significance of the observed T

L

. Our inter-

pretation is in fact consistent with well-

established behavior of trajectories near

invariant surfaces of Hamiltonian systems.

The T

L

-T

e

relation suffers from being

poorly established for times greater than 10

6-7

Jovian periods. Levison and Duncan (1992)

have completed some integrations for up to 4

Gyr in the region of the proposed Kuiper belt,

and their results appear to follow the T

L

-T

e

relation. (They also report, for integrations

between Jupiter and Saturn and between Ura-

nus and Neptune, approximate values of *a *

1.4, *b *

* *1.9, in good agreement with our val-

ues.) However, encouraging as these results

are, there is still an insufficient number of T

e

>

10

6

T

0

orbits calculated to date. Since the

needed long-term integrations are difficult to

obtain, we also investigate here the conse-

*Chaotic Motion in the Outer Asteroid Belt*

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