*Semimajor Axis Survey and Residuals*

Distribution

The results of a semimajor axis survey with

m

2

= 10 M

J

line is the unweighted least squares fit to 440

orbits, with slope *b*=1.74 ± 0.03 and intercept

*a*=1.30 ± 0.03 (all quoted errors are 1

formal

uncertainties). The standard deviation of the

residuals is

= 0.41. Agreement with the LFM

values for orbits interior to Jupiter (*b*=1.73 ±

0.19, *a*=1.53 ± 0.34) is well within the formal

uncertainties. This agreement illustrates the

apparent robustness of the relation. It appears

that *a* and *b* are insensitive to the mass ratio for

this dynamical configuration.

The distribution of the data points in log T

e

from the least squares fit (i.e., the residuals) is

approximately Gaussian. Figure 4 is a histo-

gram of the distance in log T
e

from the solid

line in Figure 3. The smooth curve in Figure 4

is the best-fit Gaussian to the histogram data.

It was found that the best fit is achieved by

excluding the "bump" in the right-hand tail

(represented by four orbits near 1.4). With

standard deviation and mean as free parame-

ters, we find
µ

=-0.039 and

G

=0.39, which is

reassuringly close to the RMS deviation,

=

0.41. A more quantitative view of the fit of the

histogram data to a Gaussian is shown in Fig-

ure 5. Here we show the difference between

the observed and expected fraction of data

points falling into the histogram bins. The

error bars are ±1

and represent the error in

the fraction, which is proportional to

(and

n

not the fractional error, which goes as

).

1/ n

The abscissa is the residual in units of

G

.

There are no significant deviations, including

the points in the tail that were excluded in the

*Chaotic Motion in the Outer Asteroid Belt*

page 11

**Figure 4.**

Histogram of the difference of log T

e

(ob-

served event time) from the linear fit to the semimajor

axis survey data (solid line) of Figure 3. Smooth curve

is the best-fit Gaussian, with

G

=0.39.

**Figure 5.**

Difference between observed and expected

fraction of data points falling into histogram bins of

Figure 4, for a Gaussian distribution. Error bars are

±1

.