Figure 20 shows the distribution of the X-ray luminosities and redshifts for the 449 clusters with redshift information. Details on the way the fluxes and luminosities of the clusters are calculated can be obtained from Böhringer et al. (2000, 2001a). The parabolic boundary in the plot reflects the flux limit of the sample. The sample is covering a luminosity range from about 1 10⁴² ergs^-1to 6 10⁴⁵ ergs^-1. The objects with luminosities below 10⁴³ ergs^-1 are Hickson type groups and even smaller units down to elliptical galaxies with extended X-ray halos. In the latter objects the extended X-ray emission is still tracing a massive dark matter halo which is in principle not different from a scaled down cluster. Therefore we have included them in the cluster sample with the caveat that we are not certain at present how well the population of these objects below a luminosity of 10⁴³ ergs^-1 is sampled in this project. This is because some of them feature a very small membership number which may not always guarantee that they are detected by the galaxy count search.

At high redshifts, beyond z = 0.3, only exceptionally luminous objects are observed, with X-ray luminosities of several 10⁴⁵ ergs^-1. Even in this simple distribution plot we can recognize inhomogeneities in the cluster distribution which can be attributed in a more detailed analysis to the large-scale structure of the Universe (Collins et al. 2000; Schuecker et al. 2000). The paucity of the data at very low redshifts in Fig. 20 is an effect of the small sampling volume. The apparent deficiency of clusters with $L_x \ge 10^{45}$ erg s^-1in the redshift interval z = 0-0.15 is certainly an effect of large-scale structure. Only about 3 such X-ray luminous clusters are expected in this region. While we do not expect the sample to be complete above a redshift of z = 0.3, the expected number of objects at these high redshifts is indeed very small in a no-evolution model. We explore this further in a forthcoming paper.

Figure 20 also shows which of the clusters in the luminosity redshift distribution are clusters already catalogued by Abell et al. (1989) and which are mostly new. Since the difference of the two different populations is not so easily recognized in this figure we have plotted the non-Abell clusters separately in Fig. 21. One notes that the non-ACO clusters are distributed over the whole range of parameters covered by the total REFLEX sample. As we had expected, many non-ACO clusters are found among the nearby low luminosity, poor clusters which fail Abell's richness threshold and among the most distant clusters, which are not covered well in the optical plates. To our surprise there is also a large fraction of non-ACO clusters found in the intermediate redshift range with X-ray luminosities implying more typical Abell type cluster masses. These latter clusters indicate an incompleteness effect in the Abell catalogue. A similar result was found for the northern BCS sample as shown in Ebeling et al. (1998).

$\begin{figure} \par\includegraphics[width=8.6cm,clip]{aa10210f22.ps}\end{figure}$

Figure 20: Distribution of the REFLEX sample clusters in redshift and X-ray luminosity. The clusters catalogued by Abell et al. (1989) and the non-ACO clusters are marked differently. The luminosities are calculated for a Hubble constant of 50 km s^-1 Mpc^-1

$\begin{figure} \par\includegraphics[width=8.6cm,clip]{aa10210f23.ps}\end{figure}$

Figure 21: Distribution of the non-ACO clusters in the REFLEX sample in redshift and X-ray luminosity. These clusters cover practically the whole distribution range of all REFLEX clusters. The clusters catalogued by Abell et al. (1989) are also shown as very light points

$\begin{figure} \par\includegraphics[width=13.6cm,clip]{aa10210f24.ps} % \end{figure}$

Figure 22: Sensitivity map of RASS II in the area of the REFLEX survey. Five levels of increasing grey scale have been used for the coding the sensitivity levels given in units of the number of photons detected at the flux limit: > 60 , 30-60 , 20-30, 15-20, and < 15, respectively. The coordinate system is equatorial for the epoch J2000

Since we do not have a homogeneous exposure coverage of the REFLEX survey area as described in Sect. 3 we have to apply a corresponding correction to any statistical study of the REFLEX sample. The best way to take the effect of the varying exposure and the effect of the interstellar absorption into account is to calculate for each sky position the number of photons needed to reach a certain flux limit. This includes both the exposure and the sensitivity modification by interstellar extinction. In total the sensitivity variation due to extinction is less than a factor of 1.25 in the REFLEX survey area (see also Böhringer et al. 2000 for details and numerical values). The so defined sensitivity distribution across the REFLEX study region is shown in Fig. 22. Since for the relatively short exposures in the RASS the source detection process is practically always source photon limited and not background limited (except for the most diffuse, low-surface brightness structures) the success rate of detection depends mostly on the number of photons. The use of the ROSAT hard band to characterize the cluster emission further reduces the background which is a great advantage for this analysis. Thus fixing a minimum number of photons per source we can calculate the effective survey depth in terms of the flux limit at any position on the sky. The integral of this survey depth versus sky coverage is shown in Fig. 23 for the three cases of a minimum detection of 10, 20, and 30 photons. Also shown is the nominal flux limit of 3 10^-12 ergs^-1 cm^-2. We note that for a detection requirement of 10 photons the sky coverage is 97% at a flux limit of 3 10^-12 ergs^-1 cm^-2. For the much more conservative requirement of at least 30 photons per source the sky coverage for the nominal flux limit of the survey is about 78%. For the remaining part of the survey area the flux limit is slightly reduced. Since the sensitivity map is available for the whole study area (Fig. 22) we can for any choice of the minimum number of photons calculate the correction for the missing sky coverage as a function of flux also for the three-dimensional analyses e.g. the determination of the correlation function and the power spectrum of the cluster density distribution (see Collins et al. 2000; Schuecker et al. 2000).

$\begin{figure} \par\includegraphics[width=8.1cm,clip]{aa10210f25.ps} % \end{figure}$

Figure 23: Effective sky coverage of the REFLEX sample. The thick line gives the effective sky area for the nominal flux limit of 3 10^-12 erg s^-1 cm^-2and a minimum number of 30 photons per source as used e.g. for the correction of the $\log N{-}\log S$ -curve shown in Fig. 24. For further details see text

In Fig. 24 we give the integral surface number counts of clusters for the REFLEX sample as a function of X-ray flux ( $\log N{-}\log S$ -curve). For this determination we have chosen the conservative requirement of a minimum of 30 counts. The figure also shows the result of a maximum likelihood fit of a power law function to the data for the corrected fluxes. The likelihood analysis takes the uncertainties of the flux measurement (analogous to the description of Murdoch et al. 1973) and the variations of the effective sky coverage for a count limit of 30 photons (as given in Fig. 23) into account. The resulting power law index is constraint to the range -1.39 with a 1 $\sigma$ error of $\pm 0.07$ . The normalization in Fig. 24 is fixed to be consistent with the total number of clusters found. This result is in good agreement within the errors with other determinations of the cluster number counts as the results by Ebeling et al. (1998); De Grandi et al. (1999) and Rosati et al. (1998). Note that the flux values used correspond to the observed fluxes. The currently best estimate for the total flux implies an average correction by a factor of about 1.1. The fact that the observed $\log N{-}\log S$ -distribution follows the straight line so closely down to the lowest fluxes shows clearly that there is no significant incompleteness effect close to the flux limit.

$\begin{figure} \par\includegraphics[width=8cm,clip]{aa10210f26.ps}\end{figure}$

Figure 24: Log N-Log S-distribution of the REFLEX sample clusters. The solid line shows the $\log N{-}\log S$ function for the nominal fluxes (determined for an assumed temperature of 5 keV and z = 0) which is used for the REFLEX flux cut while the dashed line shows the same function for the corrected fluxes as described in Sect. 2. The straight line shows the result of a maximum likelihood fit of a power law function to the data yielding a slope value of $-1.39 (\pm 0.07)$

Given the $\log N{-}\log S$ -distribution corrected for the varying flux limit as shown in Fig. 24, we can now also calculate the number of clusters we expect to be detected with a certain number of counts. This distribution is shown in Fig. 25. Here we are first of all interested in checking the completeness of the sample concerning detections at low photon numbers (< 30 photons). Since the $\log N{-}\log S$ -distribution was constructed based on clusters with more than 30 counts only, it provides an independent check on the relative completeness of the sample for low compared to high photon numbers. We note that the number of clusters to be detected with low photon numbers is quite small and also that there is no striking deficit of clusters at low counts. Below a detection with 10 counts 3.8 clusters are expected and 1 is detected. In the interval between a detection of 10 to 20 counts there is no deficit and for the interval between 10 and 30 counts the expectation is about 37 clusters compared to 26 found, a $2\sigma$ deviation. Therefore we expect very little difference for the statistical analyses using different cuts in count rate, as long as the corresponding sky coverage is taken into account. In fact in the construction of the luminosity function we find only a difference of less than 2 percent (in the fitting parameters for an analysis using a 10 photon count and a 30 photon count limit, respectively (Böhringer et al. 2001a). The proper corrections for the effective sky area will become increasingly important, however, when the sample is extended to lower flux limits.

$\begin{figure} \par\includegraphics[width=8cm,clip]{aa10210f27.ps}\end{figure}$

Figure 25: Distribution of the number of source counts per cluster for the REFLEX sample. The numbers are given as the number of objects per bin of ten photons width. The solid line gives the expected numbers as calculated from the $\log N{-}\log S$ distribution while the stars give the actual number counts with their Poissonian errors

9 Properties of the REFLEX cluster sample