First of all, we need to choose a definition of the "cooling flow" cluster that can be efficiently applied to the X-ray data of various statistical quality. The most common definition is based on the estimated central cooling time: cooling flow clusters have tcoo\ ^ tH (e.g., ). One could also use the mass deposition rate given by the standard cooling flow model ; cooling flow clusters have M > (10 — 100) Mq yr_1 . These definitions rely on spatially-resolved spectroscopic measurements which is a serious disadvantage for application at high redshifts. The data of sufficient quality to measure T (r) are available
Radius, kpc Radius, kpc
Fig. 1. X-ray surface brightness profiles typical for non-cooling flow (left; A401) and cooling flow (right; A85) clusters. Solid lines show the model X-ray brightness corresponding to the best-fit gas density model (see  for details).
only for a small number of high-z objects. We, therefore, look for a definition based solely on the X-imaging data.
At low redshifts, there is a clear connection between the presence of the cooling flow and the X-ray morphology. Clusters with tcoo\ > tH have X-ray brightness profiles with flat cores while those with tcooi ^ tH have characteristic central cusps in the X-ray brightness distribution (Fig. 1). The central cusp can be characterized by the power-law index of the gas density profile, a = d log pg/d log r. For uniformity, the radius at which a is measured should be chosen at a fixed fraction of the cluster virial radius. This radius should be sufficiently small so that the effects of cooling are strong. At very small radii, however, the density profiles even in clusters with strong cooling flows can flatten because of the outflows from the central AGN (see many papers in these proceedings). Empirically, a good choice is r = 0.04 R500,5 and so we define the "cuspiness parameter", a, as
d log r
Cuspiness can be measured by fitting a realistic 3-dimensional gas density model to the observed X-ray surface brightness (our procedure is described in ). Examples of the best-fit models are shown by solid lines in Fig. 1. Such modeling is feasible with moderate-exposure Chandra observations of high-redshift clusters. R500 can be estimated using low-scatter cluster mass proxies such as the average temperature (excluding the central cooling region). We use an even better proxy, the recently proposed YX parameter , which is remarkably insensitive to the cluster dynamical state and easily measured even in high-redshift clusters.
5 R500 is the radius at which the mean enclosed total mass overdensity is 500 relative to the critical density at the object redshift. R5oo ~ 0.5Rvir.
Fig. 2. The distribution of the cuspiness parameter in the low-z cluster sample. Arrows indicate the values for some well-known clusters. The boundary value of a = 0.5 approximately separates cooling flow and non-cooling flow clusters.
Our low-redshift cluster sample is a flux-limited subsample of 48 objects from the HIFLUGCS catalog , all with the archival Chandra observations. The distribution of the cuspiness parameter for these objects is shown in Fig. 2. Clearly, the value of a is closely connected to the more common cooling flow definitions. Clusters with a > 0.7 (e.g., A2065, A478, A2029, A2597, 2A 0035, A133) are known to host strong cooling flows. The objects with a < 0.5 (e.g., A2163, A399, A119, A2256, A754) are famous non cooling flow clusters. The clusters in the range 0.5 < a < 0.7 (e.g., A2589, A3571) host weak cooling flows. Therefore, the cooling flow clusters are those with a > 0.5. Approximately 2/3 of the low-redshift sample (31 of 48 objects) have cuspiness above this value, in line with the previous estimates of the cooling flow incidence rate (e.g. ).
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