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Astronomical Applications Department, U.S. Naval Observatory - lyap2 (Page 9)

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Astronomical Applications Department, U.S. Naval Observatory - lyap2
, and the rest
lie well within 2
. This distribution is consis-
tent with the notion that in the outer-belt aster-
oids we are seeing the remnants of a larger
original distribution. The bulk of that popula-
tion has been cleared out by Jupiter, leaving the
long-T
e
tail members to survive to the present.
The observed value of
fluctuates with
small changes of the initial conditions. For
example, Figure 2 shows the Lyapunov times-
cale as a function of initial semimajor axis for
(522) Helga. The value of T
L
is greater than
32,000 yr (corresponding to T
e
~ 22 Myr),
except in a narrow interval around the 12:7
resonance, a
0
[0.69770, 0.69815], where T
L
averages about 5800 yr (T
e
~ 0.95 Myr). The
open triangles denote a lower limit, since the
Lyapunov exponent in these cases did not level
off to a definite value by the end of the integra-
tion (cf. curves A and B of Figure 1). Also
shown in Figure 2 are the corresponding mean
values of semimajor axis, eccentricity, and incli-
nation, averaged over the length of the integra-
tions (100,000 T
J
). The 12:7 mean motion
resonance is at a=0.697922 a
J
, marked by the
symbol in the second panel of Figure 2. We
noted variations in T
L
of roughly 12 percent
within the resonance. The chaos exhibited by
the motion of (522) Helga is apparently associ-
ated with the 12:7 resonance, and the semima-
jor axis width of this resonance is clearly
delineated by the behavior of T
L
, the mean
eccentricity, and the mean inclination. The last
column in Table II shows the nearest mean
motion resonance encompassed by the given
orbit over the length of the 10
5
T
J
integrations.
Those with short Lyapunov times are, like
(522) Helga, possibly associated with the cor-
responding resonance for the particular initial
conditions we used. The range in semimajor
axis for (522) Helga was more than a factor of
7 larger than the width of the resonance as evi-
denced in Figure 2. Asteroid (1390) Abastu-
mani, falling between 15:8 and 13:7, was the
only one whose semimajor axis did not at any
time in our numerical integration cross a reso-
nance. (This asteroid exhibited behavior con-
sistent with quasiperiodic motion, cf. Table II.)
We are not certain to what extent, if any, the
asteroids are affected by the corresponding
resonances, but the association is suggestive
(see below).
An Interpretation of Short Lyapunov
Times
We propose the following picture for the
dynamics leading to an "event." First we
review briefly the relevant dynamics of a two
degree of freedom system, then we conjecture
that analogous behavior is occurring in the
much more complicated, many degree of free-
dom, Hamiltonian system represented by the
outer-belt asteroids.
It is well-known that, in a two degree of
freedom Hamiltonian system, invariant surfaces
(KAM tori) divide the phase space. Under the
influence of a sufficiently strong perturbation,
the "outermost" invariant surfaces are
destroyed, giving rise to a global sea of chaos
surrounding an inner stable region, where
invariant surfaces still exist. At the core of the
stable region is a stable period one orbit. Fur-
ther out, past the outermost intact surface and
embedded in the chaotic sea, are secondary
invariant surfaces surrounding elliptic period n
Chaotic Motion in the Outer Asteroid Belt
page 8

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