III. RESULTS

*Outer-Belt Asteroids and Long-Term*

Chaotic Motion

We present our results for the 25 outer-belt

J

(11.86 yr). The

Lyapunov time T

L

is shown in column 3, while

the predicted escape time T

e

, in units of 10

6

T

J

,

is in column 4. T

e

using *b*=1.74 and *a*=1.30. These values were

determined from the semimajor axis survey

results of this study, presented below. They

agree with the values found in LFM for aster-

oid orbits interior to Jupiter. The 25 orbits fall

into three categories, as denoted in column 5.

An "A" signifies a clearly chaotic orbit, where

the Lyapunov exponent as a function of time

appears to have leveled off to a nonzero value.

A typical plot of

in such a case is shown in

appears to be

leveling off but the integration was not long

enough to determine a value. The values of T

L

calculated for these orbits are therefore lower

limits. A typical plot for this case is shown in

Figure 1, curve B. A "C" in column 5 means

that the orbit was quasiperiodic (or very weakly

chaotic at best), where the Lyapunov exponent

is asymptotically zero (or nearly zero). Curve

C of Figure 1 shows

vs. t for a typical orbit of

this type. The actual T

L

for these orbits is very

large. We could have performed longer inte-

grations and, possibly, thereby removed some

of the > signs from columns containing T

L

and

T

e

i.e., converted class C into class B, or

class B into class A. This would be a noble

deed, but not an especially valuable use of time

because our real interest and concern centers

on just those minor planets with short Lyapu-

nov times.

An important characterization of the T

L

-T

e

L

and therefore relate it to physical

systems. In data obtained from LFM, we found

that the distribution of residuals in log T

e

for

the Jupiter-Sun system and asteroid orbits

inside Jupiter's orbit was consistent with a

Gaussian shape with standard deviation

=

0.98. Thus, for a given population of objects,

some nonzero percentage would be expected

to lie in the tail of the distribution. Take for

example the asteroid in Table II with the

*Chaotic Motion in the Outer Asteroid Belt*

page 5

**Figure 1.**

Typical behavior of the Lyapunov exponent

as a function of time (in Jovian periods), illustrating three

types. The curve labeled A is strongly chaotic; an

approximate value for

is quickly found. Curve B is

"possibly-chaotic," and further integratio n is required to

determine the asymptotic value of

. Curve C represents a

typical quasiperiodic (or at best very weakly chaotic) orbit.