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4. Selection and completeness

 

4.1. Selection in position

  The ENACS galaxy samples were designed to constitute magnitude-limited subsets of the general galaxy population in the areas defined by the Optopus plates (see Table 5 for the centres of these circular, 31'-diameter, areas). In Sect. 3.1 (click here) we found that the ENACS samples contain a few galaxies (for which we measured an ENACS redshift, and therefore presumed real) which are not in the COSMOS catalogue. Apart from this fairly minor effect, the ENACS galaxy samples are indeed subsets of the COSMOS catalogue. A question which may be important for some types of analysis, is whether the ENACS galaxy samples, which were selected on magnitude, form unbiased subsets of the COSMOS catalogue as far as position is concerned. In other words: does the surface density of the ENACS galaxies more or less follow that of the COSMOS galaxies within the areas covered by the Optopus plates.

To investigate this question we have applied a 2-D Kolmogorov-Smirnov test (Fasano & Franceschini 1987) to the ENACS and COSMOS galaxy distributions in the solid angle of the ENACS survey (either a single Optopus area, or the union of several Optopus areas). In order to make the test meaningful we have applied it to subsamples of both the ENACS and COSMOS catalogues complete to magnitude limits, tex2html_wrap_inline1919 and tex2html_wrap_inline1921 that differ by the average value of < bj-R25 > for which we took 1.5. For each cluster, we applied the test for five pairs of (tex2html_wrap_inline1919, tex2html_wrap_inline1921), with tex2html_wrap_inline1919 = 16.5(0.5)18.5. Clearly, if tex2html_wrap_inline1919 becomes fainter than the actual magnitude limit of the ENACS data in a given cluster (see Sect. 4.2 (click here)), the ENACS galaxies represent a progressively smaller fraction of the COSMOS sample, and there will be a natural tendency for the two projected distributions to become different, if they were not so already at brighter limits.

We have analyzed for each cluster the KS-probabilities at the five different magnitude limits, and we conclude that for almost all clusters, the galaxy distributions in the ENACS and COSMOS catalogues are not different at a confidence level of more than 95%. There are only three clusters, viz. A3705, A3809 and A3825, for which it appears that the ENACS galaxy distribution differs from that of the COSMOS galaxies at more than 95% confidence level. In all three cases, inspection of contour maps of galaxy surface density visually supports this conclusion: the ENACS galaxies are relatively abundant at the edges of the concentrations present in the COSMOS surface density. In addition, there is one cluster, A3822, for which there is marginal evidence for a biased selection. However, in that case there is a secondary concentration in the COSMOS data which is not present in the distribution of the ENACS galaxies. For contour maps of projected galaxy density based on the COSMOS catalogue, we refer the reader to Adami et al. (1998).

4.2. Selection in magnitude for clusters with COSMOS data

  In Fig. 1 (click here) we have given the overall magnitude distribution of the galaxies for which the ENACS has yielded a redshift. This distribution shows that on average the redshift catalogues start to become incomplete below tex2html_wrap_inline1933 17, but that the fraction of ENACS galaxies with R25 between 17 and 19 is non-negligible. The decrease in the number of galaxies beyond tex2html_wrap_inline1933 17 is the result of two factors. First, the galaxy catalogues that we prepared have (fairly sharp) magnitude cut-offs at R25 between about 17.5 and 19.0. Second, our success in obtaining redshifts decreases quite strongly for R25 tex2html_wrap_inline1943 17.

In Fig. 4 (click here) we show as a function of R25, for the ENACS as a whole, the ratio of the number of galaxies for which the ENACS observations have yielded a redshift and the total number of galaxies that we observed in the ENACS. In other words: Fig. 4 (click here) shows our success-rate of obtaining a redshift as a function of magnitude. Figure 4 (click here) therefore quantifies our discussion in Sect. 5.5 of Paper I, where we already mentioned that our maximum success-rate was about 80%. The strong decrease for R25 tex2html_wrap_inline1943 17 is due to the smaller S/N-ratio of the absorption lines in the spectra of the fainter galaxies. The fact that we do not score 100% for the brightest galaxies must be due to the less-than-ideal match between the diameter of the Optopus fibres and the surface brightness distribution of some of the brightest galaxies, which can have a relatively low central surface brightness.

  figure452
Figure 4: The ratio of the number of galaxies for which the Optopus observations have yielded a redshift and the total number of galaxies observed in the ENACS, as a function of R25

For some types of discussion it may be necessary to know, as a function of magnitude, the fraction of COSMOS galaxies (i.e. cluster and field galaxies) for which we obtained an ENACS redshift. This fraction is shown graphically in Fig. 5 (click here), as a function of R25, for all 73 clusters for which this fraction could be meaningfully determined. For 4 clusters no distributions are given because either the number of ENACS galaxies is very small (A2502 and A3144), or the overlap between ENACS and COSMOS data is too limited (A0543 and A2915). Rather than show the fraction itself, we give the magnitude distributions of ENACS and COSMOS galaxies, both normalized to the number of ENACS galaxies in the most populated 0.5-mag bin. The two magnitude distributions refer to the same solid angle, i.e. the overlap "area" between the two surveys.

  figure460
Figure 5: The normalized magnitude distributions of COSMOS (line) and ENACS (circles) galaxies in the overlap areas between the COSMOS and ENACS catalogues, for 73 clusters (ACO number is given below frame; number of ENACS galaxies - N - shown in frame). Magnitudes are on the R25 scale (COSMOS bj magnitudes have been corrected for the average bj-R25 colour). Normalization is wrt the most populated 0.5-mag bin in the distribution for the ENACS galaxies

In general, the ENACS magnitude distribution coincides with, or falls below, that based on COSMOS, as expected. However, in some cases, the ENACS distribution exceeds that based on COSMOS, particularly at brighter magnitudes. One factor that may contribute to this is that some of the bright ENACS galaxies are not in the COSMOS catalogue (see Sect. 3.1 (click here)). Another reason for this small inconsistency which appears only occasionally, is that the two distributions are on different magnitude scales. The COSMOS distributions have been brought to the R25 scale by applying a correction for the average bj-R25 colour for the entire ENACS survey.

Figure 5 (click here) shows that, with one or two exceptions, the ENACS galaxy samples are essentially complete, magnitude-limited, subsets of the COSMOS samples up to an R25 of 16.5 tex2html_wrap_inline1861 0.5 (within the ENACS apertures!). The fainter galaxies can, of course, be used in discussions for which the completeness of the galaxy sample is not important.

4.3. Magnitude and redshift selection for the ENACS as a whole

  For several types of analysis it may be useful to have analytic expressions for the cut-offs towards faint magnitudes and higher redshifts for the ENACS as a whole. For the analysis of these cut-off functions we have restricted ourselves to the galaxies that are not in the main system, i.e. those in the field and in secondary systems. This avoids possible complications due to the rather uneven redshift distribution of galaxies in the rich systems. For reasons that are not important here, the analysis was done on a representative subset of 65 ENACS clusters, with a total of 681 "field" galaxies. The distribution of those 681 galaxies with respect to apparent (R25) magnitude and redshift is shown in the upper lefthand panel of Fig. 6 (click here).

We assume the following model when trying to reproduce this observed distribution:
displaymath1981
in which tex2html_wrap_inline1985 is the unbiased distribution for a constant luminosity function. In other words: we assume that there are two independent cut-off functions, S1(m) which describes the magnitude cut-off in the galaxy sample for which we attempted spectroscopy and S2(v) which describes the succes-rate of obtaining a redshift as a function of velocity.

In the upper righthand panel of Fig. 6 (click here) we show a predicted tex2html_wrap_inline1991 distribution, calculated without magnitude or redshift cut-off, for a sample of 681 galaxies using a Schechter luminosity function with M*R = -22.5 and tex2html_wrap_inline1997 = -1.25, and using tex2html_wrap_inline2001, to transform from absolute to apparent magnitude. Clearly, this prediction is very different from the observed distribution.

  figure491
Figure 6: The distribution of the galaxies that are not in the main system (i.e. in the "field") wrt to magnitude, R25, and redshift. The upper lefthand panel shows the observed distribution for 681 galaxies in 65 clusters. The upper righthand panel gives the result of a simulation for the same number of galaxies, without cut-offs in magnitude or redshift. The lower lefthand panel also results from a simulation, with the magnitude cut-off described in the text, but without redshift cut-off. The lower righthand panel shows the result of a simulation with the magnitude and redshift cut-off functions described in the text

The introduction of a magnitude cut-off of the form
eqnarray496
produces the distribution in the lower lefthand panel, which has more or less the correct total magnitude distribution. However, the total redshift distribution extends too much beyond tex2html_wrap_inline1719 40 000 km/s.

Finally, the application of a redshift cut-off of the form
eqnarray506
yields the distribution shown in the lower righthand panel. Note that we do not pretend that this is the best description one may give. However, we have chosen the functional forms of, and the parameters in the cut-off functions not just by looking at Fig. 6 (click here). Instead, we have tried to reproduce as closely as possible the observed redshift distributions in several rather narrow magnitude intervals. Short of introducing a possible dependence of e.g. S2(v) on m etc., we think the selection functions given here provide a sufficiently accurate description of the overall selection functions in the ENACS as a whole (but these may not be necessarily correct for individual clusters!).

That the selection functions given here, on the basis of the galaxies in the "field" are at least reasonable is also supported by the following evidence. For the galaxies in the clusters with v < 30000 km/s, only S1(m) is relevant. Using again a Schechter luminosity function, with M*R = -22.5 and tex2html_wrap_inline1997 = -1.25, we predict that the average number of observable galaxies in a cluster depends on the redshift of the cluster as z-1. This is consistent with what we find for the main systems in ENACS (see e.g. Paper I).


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