4 Reanalysis of the X-ray data of the RASS sources with the GCA technique

Since we found from our previous studies that the flux of extended sources is underestimated by the standard RASS source detection algorithm (Ebeling et al. 1996; De Grandi et al. 1997), a reanalysis of the source fluxes is necessary before we introduce a count rate cut or flux limit in the source list, that will subsequently serve as the basis for the construction of the X-ray flux-limited cluster sample. This is especially important in the present study since the majority of the cluster sources feature a significant extent and many appear not perfectly spherically symmetric. To this end we have developed a new source characterization technique, the growth curve analysis (GCA) method, which is described in detail in Böhringer et al. (2000). The strategy for the development of the new algorithm is to obtain reliable fluxes for extended sources and to extract as much useful information from the raw data as possible with a simple and robust technique. The simplicity of the technique is particularly important in devising a model for the source detection from theoretically constructed catalogue data in order to simulate possible selection effects in the sample. We have given preference to use the GCA method for this analysis over the methods described by Ebeling et al. (1996) and De Grandi et al. (1997) since it makes more extensive use of the photon count information and provides reliable count rates also for lower fluxes. Also the influence of assumptions made about the applied cluster model is minimized, since (as shown in Böhringer et al. 2000) almost 90% of the total flux is sampled directly with GCA and only for the final minor correction a model assumption is used. A more detailed comparison between the different methods to measure the count rate will be given elsewhere (see also Böhringer et al. 2000 for a comparison of GCA results with BCS data from Ebeling 1998). Here we give only a brief outline on the GCA.

For each source GCA returns (among other information) the following most important parameters which will be used in the source selection work: $\circ$ observed source count rate (background subtracted) $\circ$ Poisson error (photon statistics) for the count rate $\circ$ locally redetermined center of the source $\circ$ mean exposure for the source region $\circ$ significance of the source detection $\circ$ estimate of the radius out to which the source emission is significantly detected $\circ$ extrapolated source count rate (obtained by model fitting to the source emission distribution) $\circ$ hardness ratio characterizing the source spectrum and its photon statistical error $\circ$ fitted source core radius $\circ$ Kolmogorov-Smirnov test probability that the source shape is consistent with a point source The basic parameters are derived for the photon distribution in the three energy bands "hard'' (0.5 to 2.0 keV, channels 52-201), "broad'' (0.1 to 2.4 keV, channels 11-240), and "soft'' (0.1 to 0.4 keV, channel 11-40). The band definitions are the same as those used in the standard analysis (Voges et al. 1999). Here we are only using the hard band results, since the clusters are detected in this band with the highest signal to noise ratio. An exception is the hardness ratio which requires the results from the hard and soft bands.

Figure 3: Example of the set-up of the source characterization method used in the GCA technique. The image shows the hard band photon distribution from an area of the RASS in a 1.5 degree box around the X-ray source. The outer two circles enclose the area of the background determination. This background area is divided into 12 sectors. The two sectors marked by a cross are discarded from the background determination. They are flagged by a $2.3\sigma$ clipping technique indicating a possible contamination or strong fluctuation (see Böhringer et al. 2000 for details). The inner ring marks the outer radius out to which significant X-ray emission from the source is detected

The source count rate is determined from the cumulative, radial source count rate profile ("growth curve'') after background subtraction. The construction of the growth curve is preceded by a redetermination of the source center and by the background measurement. As a typical example, the growth curve for the source shown in Fig. 3 is displayed in Fig. 4. In addition to the count rate as a function of integration radius, the uncertainty limits determined by photon statistical error (including the error for the background subtraction) also are calculated and displayed in Fig. 4 as dashed lines.

The count rate is determined in two alternative ways. In the first determination an outer radius of significant X-ray emission, $r_{\rm out}$, is determined from the point where the increase in the $1 \sigma$ error is larger than the increase of the source signal. The integrated count rate is then taken at this radius. In the second method a horizontal level is fitted to the outer region of the growth curve (at $r \ge r_{\rm out}$), and this plateau is adopted as the source flux. We use the second approach as the standard method but use also the first method as a check, and a way to estimate systematic uncertainties in the count rate determination in addition to the pure photon statistical errors. We also determine a fitted total count rate by means of a $\beta$-model as described below. For sources where close neighbours disturb the count rate measurement we run a separate deblending analysis.

Figure 4: Integrated count rate profile for the source shown in Fig. 3. The integrated count rate profile is background subtracted. The two dashed curves give the photon statistical error ($1 \sigma$) of the count rate which includes the uncertainty of the signal and background determination. The vertical dashed line shows the outer source radius as explained in the text. The lower dotted line shows the $\chi ^2$ fit of a point source to the data while the upper dotted line shows the best King model fit (including the convolution with the RASS PSF)

Figure 5: Flow diagram illustrating the major data reduction steps conducted in the construction of the REFLEX sample. Also shown are two side branches of the data analysis used to test the sample completeness based on a separate search for X-ray emission in RASS II for the clusters of Abell et al. (1989) and an inspection of all extended X-ray sources in the REFLEX area above the REFLEX flux-limit

The two most important source quality parameters determined within GCA are the spectral hardness ratio and the source extent. The hardness ratio, HR, is defined as $HR = {H - S \over H + S}$ where H is the hard band and S the soft band source count rate (both determined for the same outer radius limit).

The source extent is investigated in two ways. In the first analysis a $\beta$-model profile (Cavaliere & Fusco-Femiano 1976) convolved with the averaged survey PSF (G. Hasinger, private communication) is fitted to the differential count rate profile (using a fixed value of $\beta$ of 2/3) yielding a core radius, $r_{\rm c}$, and a fitted total count rate. Secondly, a Kolmogorov-Smirnov test is used to estimate the probability that the source is consistent with a point source. The source is flagged to be extended when the KS probability is less than 0.01. Tests with X-ray sources which have been identified with stars or AGN show a false classification rate as extended sources of about 5% (these results will be discussed in detail in a subsequent paper).

All 54076 RASS II sources in the REFLEX study region were subjected to the GCA reanalysis. All sources with a count rate $\ge 0.08$ ctss^-1were retained for the primary sample.

For the first sample cut in count rate we have been very conservative. In addition to selecting all sources with a count rate $\ge 0.08$ ctss^-1 as measured at $r_{\rm out}$ we have also retained all sources featuring a fitted total source count rate above this value in the $\beta$-model fit and a significance for the source detection $\ge 3\sigma$. While this leads to the inclusion of a significant fraction of sources below the count rate cut (due to less successful $\beta$-model fits) it also ensures that sources with pathological count rate profiles featuring an underestimate of $r_{\rm out}$ are not lost before all sources can individually be inspected in the GCA diagnostic plots. A comparison of the GCA determined count rate (first method) and the fitted count rate is shown in Fig. 6. There is a good correlation of the two count rate values above a measured count rate of about 0.1 ctss^-1. At low values of the GCA count rate, the fitted count rates show a large scatter. This is mostly due to the poor photon statistics providing not enough constraints on the source shape for a good enough $\beta$-model fit. A closer inspection of the results shows that at low count rates the fitted results give overestimates in more than one fourth of the sources, leading to an oversampling of about 20%. Simulations have shown that the reason for this is an overestimate of the core radius for sources with small photon numbers. This is one reason why it is preferable not to use model fits or the SRT method described in De Grandi et al. (1997) for sources with low photon statistics. The oversampling is of no harm to the final sample construction, since the final REFLEX sample is obtained by another cut in flux well above this limit.

In total the first count rate cut leads to a sample of 6593 sources. This sample contains still a large number of original multiple detections of extended sources by the RASS II standard analysis. The new analysis method offers an efficient way of removing most of these multiple detections. In the redetermination of the source center in the GCA analysis, the technique usually finds a common center for the multiple detection of clusters within small numerical differences in the position (generally < 1 arcmin). Since at a separation of 2 arcmin also point sources are already overlapping, we have used a maximal separation of 2 arcmin to identify these multiple detections as a single structure in the further identification process. Removing the redundant detections the source list shrinks to 4410 sources. This is the sample that was subjected to the first X-ray optical correlation as described in the next section. Further screening revealed another 204 redundant detections in very extended clusters where the method has settled in different local maxima (with a separation larger than 2 arcmin), but which are easily recognized visually as continuous patches in the photon distribution. Figure 5 summarizes the number of sources obtained in the subsequent steps of the X-ray source sample construction as well as the further data reduction steps described in the following.

Figure 6: Comparison of the count rates determined for the RASS sources in the REFLEX study area with both techniques, the "measured'' count rate out to the radius of significant X-ray emission and the count rate obtained by fitting a King profile to the source count rate profile shape. The diagonal line gives the location of the points for which both measures are equal. The vertical and horizontal line give the count rate cut values for the two techniques, respectively. The data for which the significance of the detected signal is found to be greater than 3 are marked by heavy dots, while the data below this significance threshold are plotted by light dots. In the graph for clarity only the first 5000 sources of the total sample of 54076 sources are shown

To illustrate how well the X-ray properties can still be characterized near the flux limit of this sample we provide some statistics on the source photon number and detection significance for the sources in the sample. This is interesting in the light of the discussion in Sect. 2, where we outline the strategy for the survey depth and where we argue that the depth of a survey is limited, if we require a certain accuracy for the derived X-ray properties requiring a minimum source photon number. In Fig. 7 we show the number of photons detected for the sample of 4206 RASS II sources above the count rate cut. Most sources at the count rate limit have more than 10 photons allowing for a flux determination with a formal accuracy of at least 30%. Note that besides the "main sequence'' of data points there is also a fraction of sources well below these typical data. These data points come from the low exposure areas in RASS II. To avoid unwanted selection effects in the sample it is useful to introduce a source count limit, and to correct for this cut in the sample selection function as discussed later.

Figure 8 shows in a similar way the typical signal-to-noise ratio ( $S/N = N_{\rm s} / \sqrt{N_{\rm s} + N_{\rm b}}$, where $N_{\rm s}$ are the source counts and $N_{\rm b}$ are the background counts in the source region) for the flux measurement of a source as a function of the X-ray flux. Again there are some sources with a low significance for the source flux determination, which come from the low exposure areas.

Figure 7: Distribution of the number of source photons (background subtracted) obtained as a function of the X-ray flux

Figure 8: Distribution of the signal-to-noise of the flux determination as a function of the X-ray flux. For the definition of the significance parameter see the text