How supermassive black holes shape central entropies in galaxy clusters

TL;DR

The paper addresses why central entropies in cool-core galaxy clusters are typically below the empirical threshold $K_e\sim 30\,\mathrm{keV\,cm^2}$. It develops a simple analytic model where Bondi-like accretion onto a central SMBH powers AGN jets whose heating balances cooling, with $\dot{M} \approx \frac{4 \pi G^2 M_\bullet^2}{(\gamma K_e)^{3/2}}$ and $\dot{E}=\epsilon \dot{M} c^2$, and relates cluster luminosity to mass via $L_\mathrm{bol} \approx 7\times10^{44} \left(\frac{M_{500c}}{10^{15}M_\odot}\right)^{3/2}$ erg s$^{-1}$. The model yields $K_e \approx 18.3\,\mathrm{keV\,cm^2} \left(\frac{\epsilon}{0.2}\right)^{2/3} \left(\frac{M_\bullet}{10^{10}M_\odot}\right)^{4/3} \left(\frac{M_{500c}}{10^{15}M_\odot}\right)^{-1}$ and a limiting SMBH mass of $M_\bullet \approx 1.4\times10^{10} M_\odot$ for a $10^{15}M_\odot$ cluster. Three-dimensional simulations of an isolated Perseus-like cluster show the central entropy tracks the SMBH mass, and when jet entropies are corrected in the X-ray emissivity, the results align with the analytic predictions, supporting a BH-mass–dependent entropy floor. The findings imply a tight link between SMBH growth and cluster thermodynamics, with implications for modeling cool-core populations in cosmological simulations and for upper limits on SMBH masses in massive halos.

Abstract

A significant fraction of galaxy clusters show central cooling times of less than 1 Gyr and associated central cluster entropies below $30\,\mathrm{keV}\,\mathrm{cm}^2$. We provide a straight forward explanation for these low central entropies in cool core systems and how this is related to accretion onto supermassive black holes (SMBHs). Assuming a time-averaged equilibrium between active galactic nucleus (AGN) jet heating of the radiatively cooling intracluster medium (ICM) as well as Bondi accretion, we derive an equilibrium entropy that scales with the SMBH and cluster mass as $K\propto M_\bullet^{4/3}M_{500\mathrm{c}}^{-1}$. At fixed cluster mass, overly massive SMBHs would raise the central entropy above the cool core threshold, thus implying a novel way of limiting SMBH masses in cool core clusters. We find a limiting mass of $1.4\times10^{10}\,\mathrm{M}_\odot$ in a cool core cluster of mass $10^{15}\,\mathrm{M}_\odot$. We carry out three-dimensional hydrodynamical simulations of an idealized Perseus-like cluster with AGN jets and find that they reproduce the predictions of our analytic model, once corrections for elevated jet entropies are applied in calculating X-ray emissivity-weighted cluster entropies. Our findings have significant implications for modelling galaxy clusters in cosmological simulations: a combination of overmassive SMBHs and high heating efficiencies preclude the formation of cool core clusters.

Figures (2)

• Figure 1: Entropy profiles of X-ray emitting gas of Perseus analogue self-regulated simulations with different SMBH masses (see legend). The coloured area indicates the 16th to 84th percentile of snapshots at times between $t=0.55\,\mathrm{Gyr}$ and $t=1\,\mathrm{Gyr}$. The dotted line denotes $30$ keV cm$^{2}$, an empirical dividing line between cool-core and non-cool-core clusters. The dashed line denotes the entropy profile of the Perseus cluster from Churazov2003, rescaled to $h=0.67$. The central entropy establishes an equilibrium value dependent on SMBH mass.
• Figure 2: Temperature vs. density phase diagram, colour coded by mass. The grey line indicates an electron entropy of $30\,\mathrm{keV}\,\mathrm{cm}^2$, the orange dashed line indicates the equilibrium ICM entropy predicted by our model (see text for details). Most of the gas mass is above our equilibrium entropy value, implying that efficient AGN jet feedback sets a SMBH mass-dependent entropy floor in the central ICM.