

Astronomical Applications Department, U.S. Naval Observatory  lyap2 (Page 13)Wodocs >> Science : Astronomy >> Astronomical Applications Department, U.S. Naval Observatory lyap2 Page 13
fit. We also calculated the skewness S of the distribution (all data included), finding S/
S
=
2.6, where
is the theoretical
r
S
=
15/N
standard deviation for the skewness of an ideal Gaussian distribution. It is difficult to interpret the significance of a nonzero skewness, since it is highly sensitive to the tail, which in turn suf fers from small number statistics. A value 2.6 standard deviations from the expected is possi bly significant.
The nonzero skewness of our distribution
arises almost entirely from the presence of the four orbits making up the bump near 1.4 in Fig ure 4. Removing these orbits drops the skew ness to approximately one standard deviation from a perfect Gaussian. Are these four orbits anomalous, or do they just represent statistical fluctuations? One can ask this question: given the expected number of orbits in a particular bin, what is the probability of finding the observed number? One may also adjust bin width to gauge sensitivity to bin boundary placement. For the bump orbits, that probabil ity ranges, depending on bin boundaries, from six to eighteen percent not unreasonably small, and relatively insensitive to bin width. Nevertheless, we reintegrated these orbits with microscopically different initial conditions and found that they shifted out of the tail, removing the bump. As a further check, we then reinte grated seven other orbits chosen at random from the distribution and observed the same kind of movement within the distribution. Thus, we conclude that the noise in T
L
is such
G
boundaries of the Gaussian fit. The standard deviation of this distribution is almost a factor of three smaller than the
calculated from the
LFM data involving Jupiter at its actual mass. Thus, the distribution width is a function of mass ratio. It is also apparent from the figures in LFM that
is an increasing function of
orbital inclination of the test particle.
Summary
We have examined the 25 "nonresonant"
outerbelt asteroids in light of the T
L
T
e
relation
and found that their predicted event times, though significantly less than the age of the solar system T
SS
, are statistically consistent
with their being present today. The key to this conclusion is that chaotic orbits are, to a close approximation, normally distributed about the mean T
L
T
e
relation for the particular mass
ratio and dynamical configuration in question. We think that the 25 objects are the expected distribution tail of an originally much larger population, which has been thinned out by Jupi ter. The adjustable parameters of the T
L
T
e
relation are relatively insensitive to mass ratio and dynamical configuration. However, the distribution width crucial for interpreting the significance of predicted event times is a function of the mass ratio. We have noted the existence of one such misinterpretation in the recent literature, and we urge caution to the dynamical community to prevent such mistakes in the future.
The two facets of the problem discussed in
Chaotic Motion in the Outer Asteroid Belt
page 12
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