We have extracted from the original catalogue four flux limited samples with flux density cutoffs at 0.4, 0.5, 0.6 and 0.7mJy. Furthermore, to investigate changes in the clustering properties of radio sources with flux density, two independent sub-samples were considered, having the same number of objects and flux densities in the range 0.4<S_1.4<0.9mJy and S_1.4>0.9mJy. Table 1 lists the number of sources in each sub-sample. The correlation function $w(\theta )$ is calculated for each sub-sample for angular separations ranging from 0.03deg to 1.3deg, within 9 equally separated logarithmic bins. The results are shown in Figs. 3 and 4.

$\begin{figure} \psfig{figure=ag8556f3.eps,width=6in,height=5in,angle=0} \end{figure}$

Figure 3: Angular correlation function $w(\theta )$ of the Phoenix radio survey for subsamples with flux density cutoffs at 0.4, 0.5, 0.6 and 0.7mJy. The error bars shown are Poisson estimates. The solid and dashed lines represent the fits to the data assuming $\delta =0.8$ and $\delta =1.1$ respectively. Points in the range $3^{\prime } <\theta <20^{\prime }$ are used in the fitting algorithm

The amplitude of the correlation function is estimated by fitting the following function to the observations

$\begin{displaymath} w(\theta)=A_{w} \times \theta^{-\delta}-\omega_{\Omega}(\delta). \end{displaymath}$

(9)

where the integral constraint is evaluated for the two different values of $\delta$ adopted in previous studies; $\delta =0.8$ (Peacock & Nicholson 1991; Loan et al. 1997) and $\delta =1.1$ (Cress et al. 1996). The calculated amplitudes and the Poissonian uncertainties are listed in Table 2. Assuming Poisson errors, these amplitudes are non-zero at the 2 $\sigma$ confidence level for all the sub-samples, except from those with S_1.4>0.4mJy and 0.4<S_1.4<0.9mJy. However, since the formal bootstrap errors are $\approx3$ times larger than the Poisson expectation, the amplitudes derived here are treated as upper limits.

Additionally, there are two effects that might further increase the correlation amplitude errors: (i) the inter-dependence of the $w(\theta )$ measurements at different angles $\theta$ and (ii) the dependence of the uncertainty estimates on higher order correlations (skewness & kurtosis) that are ignored in this study. However, Mo et al. (1992) demonstrated that the underestimation of the correlation amplitude uncertainty, $\delta A_{w}$ , by these two effects is compensated by the overestimation of the $w(\theta )$ errors by the bootstrap resampling method.

Furthermore, because of the small solid angle of the Phoenix field, cosmic variance may affect the estimated correlation amplitudes. The effect of cosmic variance is studied in detail in the next section.

$\begin{figure}{\psfig{figure=ag8556f4.eps,width=6in,height=3in,angle=270} } \end{figure}$

Figure 4: Angular correlation function for sources with flux densities S_1.4>0.9mJy and 0.4<S_1.4<0.9mJy. The error bars shown are Poisson estimates. The solid and dashed lines represent the fit to the data assuming $\delta =0.8$ and $\delta =1.1$ respectively. Points in the range $3^{\prime } <\theta <20^{\prime }$ are used in the fitting algorithm

**Table 2:** Angular correlation function amplitudes and the associated Poisson errors
Flux Density (mJy)	$A_{w}^{(\delta=1.1)}$	$A_{w}^{(\delta=0.8)}$
>0.4	0.001 $\pm$ 0.001	0.003 $\pm$ 0.003
>0.5	0.003 $\pm$ 0.001	0.006 $\pm$ 0.003
>0.6	0.004 $\pm$ 0.002	0.008 $\pm$ 0.004
>0.7	0.005 $\pm$ 0.002	0.010 $\pm$ 0.004
>0.9	0.004 $\pm$ 0.002	0.009 $\pm$ 0.005
[0.4, 0.9]	0.003 $\pm$ 0.002	0.007 $\pm$ 0.005

4 Correlation function of the Phoenix radio survey