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The Cross-correlations

The actual cross-correlations were done as follows: north of $\delta\gt^\circ$ WGACAT was correlated with the GB6 catalog with a correlation radius of one arc minute. The resulting sample, which included 1,119 sources, was then correlated with the NORTH20CM, this time with a correlation radius of 3 arc minutes, as the positional uncertainties of the NORTH20CM catalog are considerably worse than those of the GB6 catalog (160 arcsec at the 90% level compared to 10-15 arcsec at the $1 \sigma$ level). This produced a list of 570 sources, 262 of which are unclassified (see below for details on the classification of WGACAT sources). The 6-20 cm spectral index was then calculated and 89 sources turned out to have $\alpha_{\rm r} \leq 0.7$ and $\vert b\vert \gt 10^\circ$.The total number of candidates in the south is still growing. The correlation of WGACAT with the PMN catalog, with a correlation radius of one arcmin, produced a list of 541 objects, 310 of which are unclassified. Of these, 223 are at $\vert b\vert \gt 10^\circ$ and $\delta<0^\circ$ (the PMN sample reaches $\delta \sim 10^\circ$). So far, 103 have turned out to have $\alpha_{\rm r} \leq 0.7$, but this number is bound to increase with the analysis of our October 1997 ATCA observations and the completion of the NVSS (almost complete as of October 1997).

To evaluate our completeness it is important to derive WGACAT positional error circles. This was done as follows. WGACAT was cross-correlated with the Hipparcos Input Catalog (see e.g., Torra & Turon 1985) and the offsets between the two databases were obtained for 6 bins of the distance from the WGACAT field center (0 - 10 arcmin, 10 - 20 arcmin etc.). One $\sigma$ positional errors were then estimated by sorting the offsets in ascending order and taking the value which included 68% of the objects in the bin. These values, reported in Table 3, range from 13 arcsec for the inner 10 arcmin of the PSPC field to 53 arcsec for the 50 - 60 arcmin ring.

Since the positional accuracy of radio catalogs decreases with flux, we investigated the possibility that a one arcminute cross-correlation radius might not be large enough at lower radio fluxes and/or large PSPC center offsets. We therefore cross-correlated WGACAT with the GB6 and PMN radio catalogs with a 1.5 arcmin radius. Total positional errors were derived by summing in quadrature the X-ray and radio uncertainties, the latter obtained from the GB6 catalog and from the PMN radio fluxes via the formulae given in the PMN papers. The significance of the match was quantified by the ratio between X-ray/radio offset and combined positional error. (Note that 1.5 arcmin is roughly equal to twice the combined X-ray and radio uncertainty of a source with PSPC offset $\sim 30$ arcmin and radio flux $\sim 50$ mJy.)

The results are as follows: for the WGACAT/PMN correlation, the number of X-ray/radio matches using a correlation radius of 1.5 arcmin increases by 40%. Dividing the WGACAT sample in an inner (PSPC offset $\le 30$ arcmin) and outer (PSPC offset > 30 arcmin) region, there is a 38% increase in the inner region and a 45% increase in the outer region. In the inner region most of the increase is due to ``spurious'' associations, which we define for the purpose of this experiment as those matches with ratio between offset and positional error > 2. Of the 144 new sources, 104 are spurious, with a net increase of ``good'' sources $\sim 12\%$. In the outer region, of the 81 new sources, only 13 are spurious, so the number of good sources increases by $\sim 38\%$. This simply reflects the fact that WGACAT sources with larger offsets have larger positional uncertainties and ``real'' matches can have X-ray/radio offsets larger than one arcmin. For the WGACAT/GB6 correlation, the results were slightly different: the increase is only 25%, practically independent of WGACAT offset. The net increase of ``good'' matches is only $\sim 4\%$ in the inner region and $\sim 21\%$ in the outer one. That is, as the GB6 positions are better than the PMN ones, increasing the correlation radius has a bigger effect on the WGACAT/PMN ``real'' matches than it has on the WGACAT/GB6 matches.

As a result of this experiment, we then added to the WGACAT/PMN candidate list the 29 unclassified sources obtained from the 1.5 arcmin correlation and having PSPC offsets $\le 30$ arcmin, ratio between X-ray/radio offset and positional error $\le 2$, $\vert b\vert \gt 10^\circ$ and $\delta<0^\circ$. (Note that these WGACAT/PMN additional candidates all have relatively small radio fluxes within a factor 2 of the PMN completeness limit.) By comparison, only 5 of the additional GB6 sources with PSPC offset $\le 30$ arcmin derived from the correlation with the larger radius had entries in the NORTH20CM catalog but all of them had $\alpha_{\rm r} \gt 0.7$. This is easily explained: all these sources had relatively large positional errors and therefore relatively small radio fluxes (F(6 cm) $\sim 20 -30$ mJy). Therefore, only steep-spectrum radio sources could have a 20 cm flux > 100 mJy, the limit of the NORTH20CM catalog.

In summary, based on the positional accuracy of both WGACAT and the radio catalogs used to construct our candidate list, we expect our final sample to be complete in terms of radio/X-ray correlations to the radio flux limit for center offsets $\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}30$ arcmin. At present, these make up about 60% of our sources. At larger offsets, our completeness limit will be at somewhat higher radio fluxes.

We have already mentioned the possibility that some of the X-ray/radio associations are simply chance coincidences. From the number counts of flat-spectrum radio sources (Condon 1984), we estimate that only about 20 $\pm$ 4 objects in our sample are spurious X-ray/radio sources. Most of these will be singled out because of their large value of the ratio between X-ray/radio offset and positional error.