The two-point angular autocorrelation function is defined by the
joint probability
P of finding two sources in each of the elements of solid angle
and 
separated by angle 
:
where N is the mean surface density. Many derivations for estimators
of 
have been given before (see, for example, Peebles 1980).
For those unfamiliar with the field, we outline a basic derivation
given by Sicotte (1995).
It follows from the definition that for a catalog of n data points
covering a solid angle 
, the mean number
of sources at a distance 
from a randomly picked
data point is
where
is the mean solid angle about a randomly
chosen data point. The total number of
pairs with separations in the interval is then given
by 
. 
can be measured directly from the
catalog, and when combined with an estimate of 
, can give an estimate of . For an all-sky
catalog 
and 
. For
surveys with complicated geometries, it is more practical to calculate
using a field of randomly distributed
points
covering the same area as the survey. Assuming there are the same
number of random points as there are data points, the number
of pairs of random points with separations between
is given by
. One can
clearly measure this quantity and obtain a value for 
which is approximately equal to .
A simple estimator of is thus given by DD/RR-1. Strictly, one
needs to have more random points than data points so that the
estimate is not limited by statistical errors in the random points.
An improvement on this estimate of can
be obtained using the quantity DR(): the number of
pairs of points separated by angle where one point is taken from
the data field and one is from the random field. Using DR instead of RR
enables one to measure the mean solid
angle around data points [] as opposed to the mean solid angle around random
points [].
Sicotte (1995) gives a detailed comparison
of the various estimators of . We have chosen to quote
results using the
the standard (S) estimator [
] as well as the estimator
suggested by Landy and Szalay (1993, hereafter LS)
[
]. The LS estimator has been shown to have
smaller uncertainties on larger scales and Sicotte claims it will remove
the effects of large scale fluctuations in density for estimates of
at small . The method introduced by Hamilton (1993)
[
] gives results very similar
to the LS estimator.
To determine the uncertainties associated with each estimate, a
`bootstrap' analysis of
the errors was performed. This method of estimating uncertainties is described in
detail in Ling, Frenk and Barrow (1986) and in Fisher et al. (1994). A set of
`bootstrap' catalogs, each the same size as the data catalog, are
generated using the following procedure. A source is picked at random from the
data catalog and inserted into the first bootstrap catalog. Random sources
from the data catalog continue to be included in the bootstrap catalog
until it has the same number of sources as the data catalog.
Some of the sources in the
original data set will appear more than once in the bootstrap catalog,
some will not appear at all. A set of these bootstrap catalogs can then be
generated and the correlation function can be calculated
for each one. The result is that at each we produce a set of normally
distributed estimates of the correlation function. The variance around
the mean can then be used as an estimate
of the uncertainty in the measurement of w at each . This
process established that both the S estimator and the
LS estimator of had similar
uncertainties associated with
them out to about 
. The Poissonian estimate of
the uncertainties given by 
is less than the bootstrap estimate by a factor of 2 on
small scales (
) and by more than an order of magnitude on
larger scales (
).
As explained above, the random field measurements RR and DR
correct for the `edge effects'; that is, they
estimate the mean solid angle about points in the area of the survey for a
given . Therefore, the random field should contain all the biases
that the data
field contains. We consider the following:
of each other are considered a
single source. This constraint is reproduced in the random fields.
pixel coverage map where each pixel represents a
(
) area. Random points are
generated
in any region where there is data coverage. The flux density threshold
for detection of a source depends on the local rms noise, which varies
slighlty accross the
survey area. To check whether any systematic variations in sensitivity
could affect the correlation function estimate, we used the coverage map
to generate a random field which also `missed' sources in noisy areas even
though they had a flux density of S>1 mJy. Random sources were assigned
a random flux density according to Windhorst et al.'s (1985)
log N-log S curve. If the source flux density was below 5 times the
rms noise given in the coverage map
then it was discarded. The results of this are noted in 
where Dd is the number of radio sources
separated by
degrees from an Abell cluster center and Rd is the number of random
field points degrees from an Abell cluster center.
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