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
page 2
