1 Introduction

The spatial and angular correlation functions, $\xi(r)$ and $w(\theta )$respectively, have been extensively used to quantify the clustering of galaxies. The former has the important property of being the Fourier conjugate of the power spectrum, P(k), of the galaxy distribution (e.g. Peacock & Nicholson 1991). The latter is the two dimensional projection of $\xi(r)$ on the sky and is related to it by the Limber equation (Limber 1953; Phillipps et al. 1978). Although $w(\theta )$conveys less information on galaxy clustering than $\xi(r)$ due to its integral nature, nevertheless it is a powerful probe of the large scale distribution of galaxies, especially at faint magnitudes where redshift information is difficult to obtain.

Wide-area surveys at optical wavelengths have provided information on the large-scale structure at redshifts $z\approx0.1$ (Loveday et al. 1995; Maddox et al. 1990b). Additionally, deep optical and near-infrared samples over smaller areas (Villumsen et al. 1997; Hudon et al. 1996; Roche et al. 1998a,b; Carlberg et al. 1997) give useful information on the large-scale distribution of galaxies at higher redshifts ($z\ \le$ 1.0). There is a general consensus that at optical wavelengths, $w(\theta )$ is a power law of the form $w(\theta)=A_{w}\,\theta^{-(\gamma-1)}$, with $\gamma\approx1.8$ (Maddox et al. 1990b). The amplitude, A_w, decreases with increasing depth of the survey, while there is also evidence that the slope, $\gamma$, flattens at faint magnitudes (Infante & Pritchet 1995). The spatial correlation function has the form $\xi(r)=(r/r_0)^{-\gamma}$, where the correlation length, r₀, is found to be r₀=5.4h^-1Mpc (Davis & Peebles 1983). However, it has been demonstrated that early-type galaxies are more clustered (higher values of r₀) than late types and that lower luminosity galaxies are a factor of $\approx2$ less clustered than their brighter counterparts of the same Hubble type (Loveday et al. 1995). There is also evidence that r₀ decreases with increasing depth of the survey (Infante & Pritchet 1995), due to the presence of a population of weakly clustered faint galaxies.

Radio surveys, unlike the optical ones, are not affected by galactic dust extinction and mainly consist of high redshift objects ( $\overline{z}\approx1$). Therefore, they sample much larger volumes than optically selected samples, albeit with sparser coverage, providing the opportunity to study the clustering of matter at much larger physical scales. However, the broad luminosity function of extragalactic radio sources (Condon 1984; Dunlop & Peacock 1990) implies that a flux density-limited sample spans a wide range of redshifts. The projection on the sky of all detected objects results in a distribution close to random, smearing out any information on the large-scale structure. This is supported by the fact that no signal has been detected in the angular (2-D) correlation analysis of bright radio galaxies (Webster 1976; Masson 1979). However, Shaver & Pierre (1989) found evidence of anisotropic distribution of radio sources towards the supergalactic plane extending out to at least $z\approx0.02$. More recently, a clustering length of r₀=11h^-1Mpc was estimated at 1.4GHz from the spatial (3-D) correlation analysis of radio sources having flux densities >0.5Jy and redshifts in the range 0.01<z<0.1 (Peacock & Nicholson 1991). Cress et al. (1996) have suggested that at lower flux densities, projection effects may become less significant and hence, a 2-D correlation analysis can be applied successfully to investigate the clustering of radio galaxies. Indeed, a non-zero amplitude has been found for the angular correlation function of radio sources with $S_{4.85\ \rm GHz}>50$mJy in both the Green Bank (northern hemisphere) and the Parkes-MIT-NRAO (southern hemisphere) 4.85GHz sky surveys (Loan et al. 1997; Kooiman et al. 1995). More recently, using the FIRST radio survey (1.4GHz; Becker et al. 1995), with a uniform sensitivity of 1mJy over an area of 1500deg², a non-zero and clearly significant amplitude for the angular correlation function is estimated (Cress et al. 1996). Adopting the radio luminosity functions (RLF) determined independently by Condon (1984) and Dunlop & Peacock (1990), Cress et al. (1997) inferred a clustering length of r₀=6-8h^-1Mpc for S_1.4> 1mJy radio sources.

The recent large-area radio surveys described above, have provided information on the two-dimensional projected distribution of relatively bright ( $S_{1.4}\ge 1$mJy) radio sources, dominated by bright ellipticals and AGNs. However, there is still limited information on the clustering properties of the faint (sub-mJy) radio population. At flux densities below few mJy there is evidence for the appearance of a new population of faint radio sources, likely to comprise a large fraction of starbursts (Benn et al. 1993; Georgakakis et al. 1999). In particular, the radio luminosity function models developed by Condon (1984) predict that the surface density of starbursts increases from $10\%$ of the radio population at $S_{1.4}\approx1$mJy to over $30\%$ at $S_{1.4}\approx0.4$mJy. Moreover, studies of the spatial distribution of starbursts, selected at optical and infrared wavelengths, show that these objects are expected to have different clustering properties (Davis & Geller 1976; Giovanelli et al. 1986; Saunders et al. 1992; Loveday et al. 1995) compared to those of early-type galaxies, with which bright radio sources are often associated. Therefore, study of the angular correlation function at sub-mJy flux density levels has the potential to reveal differences in the clustering properties of the faint radio population.

Recently, Benn & Wall (1995) demonstrated that the isotropy (or anisotropy) in the radio source counts in different fields of similar geometry and sensitivity can be used to set limits on the scale of the largest cellular structures in the Universe. However, they argued that at sub-mJy and $\mu$Jy flux densities, such a study is hampered by the small number of comparable surveys and by the small number of sources detected in each field. This is also supported by Windhorst et al. (1990), who reviewed the field-to-field variations in the radio source counts of small area radio surveys ( $\le1\mathrm{\,deg^{2}}$). They concluded that the differences, although exceeding the random distribution expectation, are due to statistical fluctuations, rather than of cosmic nature. Nevertheless, a non-uniform angular correlation function was estimated by Oort (1987a) for the radio sources detected in the deep ( S_1.4>0.1mJy) Lynx fields (Oort 1987b). More recently, Richards (1999) also reported the detection of clustering signal for radio sources brighter than $40\,\mu$Jy, detected within a 40arcmin diameter radio survey (1.4GHz) centred on the HDF.

The Phoenix radio (1.4GHz) survey, covering an area of 3.14deg², larger than any other survey at a similar flux density limit ( S_1.4=0.4mJy) and reaching surface densities $\approx 6.6\ 10^{5}\mathrm{\,sources\,sr^{-1}}$, provides a unique opportunity to study the clustering properties of the faint radio population. In this paper the angular correlation function of the radio sources detected in the Phoenix field is estimated. Section 2 gives a brief description of the observations. The method for calculating $w(\theta )$ is outlined in Sect. 3. In Sect. 4 we estimate the correlation function of the radio sample. Section 5 presents the simulations carried out to investigate the significance of our results, while in Sect. 6 the correlation function amplitudes are compared with 3-D clustering models. The results from the radio correlation analysis are discussed in Sect. 7. Finally, we summarise our conclusions in Sect. 8.