We present a statistical study of the occurrence and effects of the cooling cores in the clusters of galaxies in a flux-limited sample, HIFLUGCS, based on ROSAT and ASCA observations. About 49% of the clusters in this sample have a significant, classically-calculated cooling-flow, mass-deposition rate. The upper envelope of the derived mass-deposition rate is roughly proportional to the cluster mass, and the fraction of cooling core clusters is found to decrease with it. The cooling core clusters are found to have smaller core radii than non-cooling core clusters, while some non-cooling core clusters have high $beta$ values (> 0.8). In the relation of the X-ray luminosity vs. the temperature and the mass, the cooling core clusters show a significantly higher normalization. A systematic correlation analysis, also involving relations of the gas mass and the total infrared luminosity, indicates that this bias is shown to be mostly due to an enhanced X-ray luminosity for cooling core clusters, while the other parameters, like temperature, mass, and gas mass may be less affected by the occurrence of a cooling core. These results may be explained by at least some of the non-cooling core clusters being in dynamically young states compared with cooling core clusters, and they may turn into cooling core clusters in a later evolutionary stage.
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