The life of central radio galaxies in clusters: AGN-ICM studies of eRASS1 clusters in the ASKAP fields

The mechanical feedback from the central AGNs can be crucial for balancing the radiative cooling of the intracluster medium at the cluster centre. We aim to understand the relationship between the power of AGN feedback and the cooling of gas in the centres of galaxy clusters by correlating the radio properties of the brightest cluster galaxies (BCGs) with the X-ray properties of their host clusters. We used catalogues from the first SRG/eROSITA All-Sky Survey (eRASS1) along with ASKAP radio data. In total, we identified 134 radio sources associated with BCGs of the 151 eRASS1 clusters located in the PS1, PS2, and SWAG-X ASKAP fields. Non-detections were treated as upper limits. We correlated BCG radio luminosity, largest linear size (LLS), and BCG offset with the integrated X-ray luminosity of their host clusters. To characterise cool cores (CCs) and non-cool cores (NCCs), we used the concentration parameter $c_{R_{500}}$ and combined it with the BCG offset to assess cluster dynamical state. We analysed the correlation between radio mechanical power and X-ray luminosity within the CC subsample. We observe a potential positive trend between LLS and BCG offset, suggesting an environmental effect on radio-source morphology. We find a weak trend where more luminous central radio galaxies are found in clusters with higher X-ray luminosity. Within the CC subsample, there is a positive but highly scattered relationship between the mechanical luminosity of AGN jets and the X-ray cooling luminosity. This finding is supported by bootstrap resampling and flux-flux analyses. The correlation indicates that AGN feedback is ineffective in high-luminosity (high-mass) clusters. At a cooling luminosity of $L_{\mathrm{X},~r<R_\mathrm{cool}}\approx 5.50\times10^{43}$ erg/s, on average, AGN feedback appears to contribute only about 13%-22% of the energy needed to offset the radiative losses in the ICM.