AGN heating and ICM cooling in the

Active galactic nuclei (AGN) at the center of galaxy clusters with gas cooling times that are much shorter than the Hubble time have emerged as heating agents powerful enough to prevent further cooling of the intracluster medium (ICM). We carried out an intensive study of the AGN heating-ICM cooling network by comparing various cluster parameters to the integrated radio luminosity of the central AGN, $L_{\mathrm {R}}$, defined as the total synchrotron power between 10 MHz and 15 GHz. This study is based on the $\textit{HIFLUGCS}$ sample comprising the 64 X-ray brightest galaxy clusters. We adopted the central cooling time, $t_{\mathrm{cool}}$, as the diagnostic to ascertain cooling properties of the $\textit{HIFLUGCS}$ sample and classify clusters with $t_{\mathrm{cool}}< 1$ Gyr as strong cool-core (SCC) clusters, with 1 Gyr $< t_{\mathrm{cool}}<7.7$ Gyr as weak cool-core (WCC) clusters and with $t_{\mathrm{cool}}> 7.7$ Gyr as non-cool-core (NCC) clusters. We find 48 out of 64 clusters (75%) contain cluster center radio sources (CCRS) cospatial with or within 50$~h_{71}^{-1}~$ kpc of the X-ray peak emission. Furthermore, we find that the probability of finding a CCRS increases from 45% to 67% to 100% for NCC, WCC, and SCC clusters, respectively. We use a total of ~140 independent radio flux-density measurements, with data at more than two frequencies for more than 54% of the sources extending below 500 MHz, enabling the determination of accurate estimates of $L_{\mathrm {R}}$. We find that $L_{\mathrm {R}}$ in SCC clusters depends strongly on the cluster scale such that more massive clusters harbor more powerful radio AGN. The same trend is observed between $L_{\mathrm {R}}$ and the classical mass deposition rate, $\dot{M}_{\mathrm{classical}}$ in SCC and partly also in WCC clusters, and can be quantified as $L_{\mathrm {R}}\propto \dot{M}_{\mathrm{classical}}^{1.69\pm0.25}$. We also perform correlations of the luminosity for the brightest cluster galaxy, $L_{\mathrm{BCG}}$, close to the X-ray peak in all 64 clusters with $L_{\mathrm {R}}$ and cluster parameters, such as the virial mass, $M_{{500}}$, and the bolometric X-ray luminosity, $L_{\mathrm {X}}$. To this end, we use the 2MASS K-band magnitudes and invoke the near-infrared bulge luminosity-black hole mass relation to convert $L_{\mathrm{BCG}}$ to supermassive black hole mass, $M_{\mathrm{BH}}$. We find a weak correlation between $M_{\mathrm{BH}}$ and $L_{\mathrm {R}}$ for SCC clusters, $L_{\mathrm {R}}\sim M_{\mathrm{BH}}^{4.10 \pm 0.42}$, although with a few outliers. We find an excellent correlation of $L_{\mathrm{BCG}}$ with $M_{{500}}$ and $L_{\mathrm {X}}$ for the entire sample, the SCC clusters showing a tighter trend in both the cases. We discuss the plausible reasons behind these scaling relations in the context of cooling flows and AGN feedback. Our results strongly suggest an AGN-feedback machinery in SCC clusters, which regulates the cooling in the central regions. Since the dispersion in these correlations, such as that between $L_{\mathrm {R}}$ and $\dot{M}_{\mathrm{classical}}$ or $L_{\mathrm {R}}$ and $M_{\mathrm{BH}}$, increases in going from SCC to WCC clusters, we conclude there must be secondary processes that work either in conjunction with the AGN heating or independently to counteract the radiative losses in WCC clusters.