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4 Results

The correlation function fit parameters for various subsamples are given in Table 1.

Table 1: Correlation Analysis Results

Sample          Source Type     Source Number   Amplitude (1E-3)  Power (gamma)
------          -----------     ------------    ----------------  -------------
all sources     mixed             109,873         3.7 +/- 0.3     -1.06 +/- 0.03
double+multi    FR I & II          17,773         3.8 +/- 0.5*    -1.39 +/- 0.4
single          UFR, SB & C        92,057         5.2 +/- 0.4     -0.84 +/- 0.05
single, S>3mJy  UFR, C & SB        33,800         1.9 +/- 0.4     -1.10 +/- 0.07
single, S<2mJy  SB, UFR & C        45,312         8.2 +/- 1.0     -0.84 +/- 0.05
all X Abell     mixed X clusters  109,873 X 382  13.5 +/- 4.8     -1.07 +/- 0.2
Fitted parameters for various subsamples of data. Values are obtained from the (DD/DR-1) method of estimation. Fitting is done out to 2\r{} except for the one labled with *, which is fitted to 0.2\r{}. C refers to `compact' sources, SB refers to starburst galaxies and FR refers to the Fanaroff-Riley classification. U stands for unresolved.

4.1 The Whole Sample

A catalog of 109,873 sources was generated by collapsing all sources that are within tex2html_wrap_inline457 of each other to a single source, since the majority of such cases represent multiple components of a single host object (e.g., double radio lobes). This sample predominantly contains radio-loud, giant elliptical galaxies, quasars, and starbursting galaxies (Windhorst et al. 1985). (Becker et al. (1996) have shown that stars make up less than 0.1% of the sources.) The correlation function of such a mixed sample is difficult to interpret but it serves as a starting point for investigating specific subsamples. The correlation function for all sources in the catalog with tex2html_wrap_inline505 and tex2html_wrap_inline641 was determined using both the LS estimator [tex2html_wrap_inline591] and the S estimator [tex2html_wrap_inline589]. The RR and DR used are the average of 10 random field generations. The results for the S estimate are displayed in Figure 1. The error bars shown are random errors determined using the bootstrap algorithm described above.

Figure 1: The squares show the autocorrelation function (tex2html_wrap_inline589) calculated for the whole sample (109,873 sources, where sources separated by less than tex2html_wrap_inline457 have been collapsed to a single source). The triangles show the correlation function obtained when sources are not collapsed. Error bars are obtained using the bootstrapping technique described in Section 3.

To determine the parameters of a power-law fit of the form tex2html_wrap_inline461, a straight line was fitted to the log-log plots (for collapsed sources) out to tex2html_wrap_inline459 using standard tex2html_wrap_inline655 minimization. This yields tex2html_wrap_inline657 and tex2html_wrap_inline659 for the S estimator and tex2html_wrap_inline661 and tex2html_wrap_inline663 for the LS estimator. For the LS estimate to be a better estimate than the S estimate, one requires many more random points in a field than data points. With about tex2html_wrap_inline665 data points, the number of random points is severely limited by computer time. The S estimate is thus probably more reliable.

Also shown in Figure 1 are correlation function estimates obtained when sources separated by less then tex2html_wrap_inline457 are not collapsed. As is expected, one sees a large increase in the correlation function on small scales resulting from single sources being counted as two or more sources. On the scales of our fit, however there is little difference between the two, indicating that the small percentage of unassociated sources which have been merged do not affect the correlation function significantly.

There are two problems with fitting a straight line to determine power law parameters. The first is that normal errors in the original data do not translate to normal errors in the log-log plot as required for tex2html_wrap_inline655 fitting. To check the effects of this, we used the Levenberg-Marquardt technique to fit a function of the form tex2html_wrap_inline461 to the original (linear-linear) data. It was found that the slopes and amplitudes agreed to well within tex2html_wrap_inline673 with the straight-line fits; we thus consider the straight-line fits to be adequate. The second problem is that the correlation function estimates at different tex2html_wrap_inline529 are not independent. In calculating the correlation function for a sample of optical galaxies, Bernstein (1994) used principal component analysis to obtain linearly independent combinations of their measurements which could then be used in a tex2html_wrap_inline655 fit. It did not appear to affect their estimates of the parameters significantly (although the effect on error estimates can be more substantial).

As a test of our procedures, random fields were generated and analyzed as though they were data fields. It was found that the tex2html_wrap_inline535 were consistent with zero.

Some systematic differences in the uniformity of the survey are evident in the correlation functions. The result for the sample that includes the sources flagged as sidelobes (not plotted) shows a sudden increase in tex2html_wrap_inline535 at tex2html_wrap_inline685. When flagged sources are removed (as in the figures shown here), the bump decreases in size but is still noticeable, particularly in the result for double and multi-component sources. Six arcminutes is the distance of the first sidelobe in uncleaned VLA B-configuration images so the `bump' is a clear indicator that not all sidelobes have been removed. The situation is worse for double and multi-component sources because the sidelobes from such extended sources are harder to remove in the standard cleaning process. To investigate further the effect of sidelobes on the correlation function, varying fractions of spurious sources were added at tex2html_wrap_inline685 from real data points. Results indicate that the distortion in the correlation function is fairly well localised to tex2html_wrap_inline685 and that about tex2html_wrap_inline691 of all the (collapsed) sources remaining in the catalog are sidelobes. Efforts to improve the efficiency of our sidelobe flagging algorithms are in progress.

The first point in the correlation function for the whole sample appears high. This could be attributed to the presence of double-lobed galaxies that have separations larger than tex2html_wrap_inline457, the adopted radius within which we call all detected components a single radio source. There is also a significant dip in tex2html_wrap_inline535 at tex2html_wrap_inline697. This could be related to the cleaning procedure or to the systematic non-uniformity in sensitivity that is repeated from one coadded map to another. To investigate this further we included the sensitivity fluctuations given in the coverage map in the random field (as described in tex2html_wrap_inline699). This did result in a smoother estimate at tex2html_wrap_inline701; in addition, the first point in the correlation function was also slightly lower. Fitting a straight line to the log-log data which included the sensitivity fluctuations returned parameters which were within tex2html_wrap_inline673 of those which did not include sensitivity fluctuations (given above).

Cross-correlating the whole sample with Abell clusters results in a power law fit with an amplitude tex2html_wrap_inline705 times that of the autocorrelation (see Figure 2). In the future, we will investigate the possibility of combining this information with spatial information for clusters to infer spatial information for radio sources.

Figure 2: The cross-correlation function of the whole sample of radio sources with Abell clusters using the Dd/Rd-1 estimator. The solid line shows the best fit to the cross-correlation results while the dotted line shows the best fit to the auto-correlation of the radio sources (S) estimator. Error bars show Poissonian errors (tex2html_wrap_inline709)

Double and Multi-component sources

A catalog of double and multi-component sources (resolved extended sources) was generated by considering any sources separated by less than tex2html_wrap_inline457 to be part of a single multi-component source. Results for this subsample of 17,773 sources using the S estimator are shown for tex2html_wrap_inline467 in Fig. 3. Force-fitting a line with the same slope as the whole sample yields an amplitude of tex2html_wrap_inline715, a factor of 2.7 times larger than that for the sample as a whole. Fitting a straight line on these small scales yields tex2html_wrap_inline717 and tex2html_wrap_inline719. On larger scales many of the error bars extend below zero so we fit the linear-linear data using the Levenberg-Marquardt technique. This yields tex2html_wrap_inline721 and tex2html_wrap_inline723 out to tex2html_wrap_inline725, but this does not fit the points on small scales very well.

Figure 3: Open squares show the autocorrelation function for the double and multi-component systems using the S estimator with the solid line showing the best fit to the points. The filled squares and dashed line are the same as those shown in Figure 1.

Traditionally, those radio sources which can be resolved into components have been classified into one of two groups (Fanaroff and Riley 1974): FR II sources are the more luminous (tex2html_wrap_inline727) `classical doubles', while FR I sources have distorted lobe structures and generally have lower luminosities. FR II sources are known to prefer lower density environments than FR I sources at low z, but Hill and Lilly (1991) have shown that at ztex2html_wrap_inline4970.5 FR II sources also exist in higher density regions. In an attempt to estimate the correlation for FR II and FR I sources separately, the catalog of multiple sources was split into a catalog containing double sources only, and a catalog containing sources with more than two components. The small number of points resulted in a large amount of scatter, particularly on larger scales, but on small scales (tex2html_wrap_inline467), it appeared that the multi-component sources (predominantly FR I's) were, as expected, more clustered than the double sources (FR II's) by about a factor of 2-3.

Using the preliminary catalog of sources with tex2html_wrap_inline733 that was available at the beginning of 1995, a catalog of double sources was generated using stricter selection criteria than those used here. In addition to the requirement that there be two (and only two) sources separated by less than tex2html_wrap_inline457, the fluxes of the sources were required to be within a factor of five of each other. The correlation function determined using the more limited sample agrees well with the correlation function determined for the sample with the simpler selection criteria.

The cross correlation of Abell clusters with double and multi-component sources was determined. The correlation amplitude is a factor of tex2html_wrap_inline737 larger than that obtained for the cross-correlation with the whole sample and tex2html_wrap_inline739 larger than that obtained for the autocorrelation of this subsample.

Flux Cuts and the Presence of Star-Bursting Galaxies

We have also analyzed a sample that excluded all double and multi-component sources. Fitting a straight line, one obtains tex2html_wrap_inline743 and tex2html_wrap_inline745 using the S estimate and tex2html_wrap_inline747 and tex2html_wrap_inline749 using the LS estimate. These slopes are shallower than those obtained for the sample as a whole, and are more consistent with values determined from optical surveys. The difference is statistically not very significant but it could be related to the fact that this subsample contains a larger fraction of starbursting galaxies as compared to the sample as a whole. Bright, low-redshift starbusting galaxies are an important component of extragalactic IRAS sources which have been shown to have spatial clustering properties similar to those found for optical galaxies (Davis et al. 1988). The relative contributions of starburst galaxies and AGN to the cumulative radio source counts is given in BWH as a function of flux density (based on Windhorst et al. (1985) and Condon (1984)). Below 1.0 mJy the ratio of the number of starbursting galaxies to the number of AGN (including all giant ellipticals) is approaching unity. Above 3 mJy this ratio becomes orders of magnitude smaller. To investigate the contribution of starbursting galaxies to the correlation function, the `singles' catalog was divided into catalogs of sources with flux densities below 2 mJy and above 3 mJy, respectively. The results are shown in Fig. 4. Best fit parameters to the S<2 mJy sources are tex2html_wrap_inline753 and tex2html_wrap_inline745 using the S estimator and tex2html_wrap_inline757 and tex2html_wrap_inline759 using the LS estimator. A similar result is obtained for a sample with 2 mJy<S<3 mJy, indicating that incompleteness is not a problem here. The shallow slope and large amplitude is consistent with there being a larger contribution from `nearby' starbursting galaxies with clustering properties more similar to optical galaxies. Best fit parameters to the S>3 mJy sample are tex2html_wrap_inline765 and tex2html_wrap_inline767 using the S estimator and tex2html_wrap_inline769 and tex2html_wrap_inline771 using the LS estimator. Above 3 mJy the slope is more similar to that obtained for all sources, but the amplitude is significantly lower. Assuming that clustering does not decrease with time, the clustering amplitude of a sample must decrease as the average distance to the sources increases. The results are thus consistent with the S> 3mJy sources being dominated by more distant, unresolved FR I's and FR II's. The lower amplitude could also be related to the presence of a large fraction of quasars--although radio loud quasars are thought to have a large clustering amplitude (Bahcall & Chokski 1991), their average distance is larger than that of normal radio galaxies which will push the clustering amplitude down.

Figure 4: The autocorrelation function of two subsamples: the circles show the result for all single-component sources with flux densities < 2 mJy, the stars show the result for single component sources with flux densities >3 mJy.

The whole sample was also divided into various flux density bins. We determined the correlation function for 3 samples in which all sources below 2 mJy, 3 mJy and 10 mJy, were, in turn, excluded. The results were all similar to that determined for the whole survey. This is also true for samples containing sources in 2-10 mJy and 10-35 mJy flux density bins (although the scatter increases as the number of sources decreases). In contrast, all but the deepest of optical surveys display a decrease in the amplitude of the angular correlation as the limiting magnitude is increased (e.g., Maddox et al. 1990). A similar result is not found for radio sources because a lower flux density threshold does not correspond to a deeper survey as it does for optically selected sources. The intrinsic luminosities of radio sources vary over many orders of magnitude, resulting in contributions from sources with a wide range of redshifts regardless of the flux density threshold . In addition, the appearance of starbursting galaxies decreases the average redshift as thresholds reach 1 mJy. This effect is also seen in the deepest optical surveys. The correlation function for the 1-2 mJy flux cut has a slope tex2html_wrap_inline779, consistent with a sample containing a significant fraction of starbursting galaxies with clustering properties more similar to optical galaxies.

next up previous
Next: 5 Discussion Up: The Angular Two-Point Correlation Previous: 3 Determining the Correlation

Richard L. White,
FIRST Home Page
1996 Dec 20