Following Jones & Forman (1999) and Mohr et al. (1995), the morphological analyses presented in Schuecker et al. (2001c) is another attempt to measure the substructure occurrence rate for a large sample of X-ray galaxy clusters in a systematic way. Other projects have significantly smaller number of clusters (below 30) so that it is questionable whether they provide a statistically representative census of cluster substructure (see, e.g., Neumann 1997; Rizza et al. 1998; Rhee & Rogers 1998; Kolokotronis et al. 2001).

'The main advantage of RASS-3 compared to RASS-2 (second processing of the RASS) is that its less stringent constraints on the attitude solutions yield a larger number of accepted X-ray photon events resulting in a higher signal-to-noise without a significant increase ofthe measurement errors of the individual photons. The advantages of RASS second processing versus the first processing are discussed in Voges et al. (1999).

In Schuecker et al. (2001c) the two largest and almost complete X-ray cluster surveys available to date are used. The working sample counts 452 clusters from the ROSAT-ESO Flux-Limited X-ray (REFLEX) cluster survey (Bohringer et al. 2001b), and 201 clusters from the Brightest Cluster Sample (BCS, Ebeling et al. 1998). The occurrence rates of substructure and elongation are determined within a metric aperture of 1 Mpc using data from the RASS-3. The rates are compared with those obtained for clusters with radio halo/relic and cooling flow signatures (see § 4.5).

Figure 5.12. Left: Fraction, /, of REFLEX+BCS clusters obtained from the RASS-3 as a function of the minimum number of X-ray photons, Nm-,n, with significances less or equal the S = 0.005 (lower curves with formal Poisson error bars), 0.01 (middle curves) and 0.02 (upper curves) level. Right: Observed significance curves for the REFLEX+BCS clusters with redshifts z < 0.08 (dashed lines) compared to curves obtained with an empirical model described in the text (continuous lines) covering the same redshift range. The input substructure fractions, /inp, of the template sample used to fit the observations Eire shown in the upper left of each panel. The central continuous lines represent the best fits of the substructure fraction to the observed curves. All curves are given for the 99% confidence limit.

The observed substructure and elongation occurrence rates, /, of the REFLEX+BCS clusters obtained with /?, LEE, and FEL statistics are plotted in Fig. 5.12 (left). The observed fraction of substructured and elongated clusters increases with the minimum number of X-ray photons, iVmin- Similar curves determined for subsamples with upper redshift limits between z = 0.04 and 0.40 show ^-dependent effects on the 10 % level and must be taken into account.

The observed fractions range between lower limits of 10% to 20% and plateau values of 60% to 78%. As expected, the number of X-ray photons per cluster is an important factor which clearly biases observed SORs. Similar biases are expected when optical galaxies instead of X-ray photons are used to trace substructure. However, the smoothing introduced in X-rays by the pointspread function of the X-ray telescope and detector leads to stronger redshift-dependent effects in X-rays compared to analyses of optical cluster galaxies.

Less biased SORs can be determined from f{Nm-m) curves shown in Figs. 5.12 (left) by the comparison with template samples with known fractions of substructured clusters, /inp- In order to estimate the SOR this input fraction is varied iteratively until good fits of the REFLEX+ BCS curves are obtained (see Schuecker et al. 2001c for more details). The best fits (continuous lines in Fig. 5.12, right) give SORs ranging from 46% to 59%. The flatter dotted reference lines obtained for FEL suggest that elongation is less affected at small Armin. This is of great interest when alignment effects of cluster major axes are studied. The final estimate of the 'true' SOR as obtained from the formal mean of the three results is (52 ± 7)%.

How does this SOR estimate compares to results obtained with similar projects in X-rays? As mentioned above, Jones & Forman (1999) find a substructure occurrence rate of 41% by visual inspecting 208 Einstein IPC images. Mohr et al. (1995) analysed 65 Einstein IPC images using the emission-weighted centroid variation for substructure detection. Kolmogorov-Smirnov tests suggest that the sample is representative. They found a SOR of 61% for the same confidence level (99%) as used for the REFLEX+BCS sample.

It is thus seen that the three largest presently available systematic X-ray cluster works give SORs of about 50 %. However, the conservative (formal) 3cr standard deviation of 30 % between the three estimates already indicates that there is still considerable scatter between different samples and methods. The conservative interval of substructure occurrence rates

20 < / < 80 percent (99% confidence range), (1)

for nearby clusters with might give a realistic picture of the current situation of statistical work on X-ray SORs.

The next step towards a physical understanding of the observed SOR should be the determination of the mass scales of the subclumps and the dynamical time scales involved. Note that the individual contributions of major mergers and accretion to (1) are not given by the measurements. Obtaining quantitative estimates appears to be quite difficult, even if the analysis would have been done with better data and refined substructure tests. However, large sample sizes offer the possibility to calibrate the substructure events at least in a statistical way by the application of the same substructure tests to both observed and simulated cluster X-ray images distributed in flux and redshift in the same way. This would establish the link between substructure as defined by the various measures and the dynamical state of a cluster. Some interesting statistical results obtained from the combination of observational work and numerical experiments can be found, for instance, in Mohr et al. (1995).

Depending on the accuracy of this comparison one should also try to investigate redshift-dependent effects where no information is available. High-resolution N-body simulations of Gottlober et al. (2001) suggest an increase of major merger rates by a factor of about 2 between redshift z = 0 and 0.25.

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