Central Black Hole Mass and Lifting of a Galaxy's Atmosphere

G. Mark Voit

Black holes at the centers of massive galaxies have masses ( $M_{\rm BH}$ ) that are highly correlated with circumgalactic gas temperature ( $kT$ ), as shown by the black points (from Gaspari et al. 2019), indicating that AGN feedback tunes the masses of those black holes to the energy output ( $E_{\rm CGM}$ ) required to significantly lift atmospheric gas out of the halo’s potential well (blue dashed line).

The figure above is from a recent Physics Reports review article by Donahue & Voit (2020) [1]. Its black points come from a study by Gaspari et al. (2019) [2] showing that directly measured black hole masses (M_BH) are highly correlated with circumgalactic gas temperature (T_CGM), following

\frac {M_{\rm BH}} {M_\odot} \approx 10^{9.4} \, \left( \frac{kT_{\rm CGM}} {1 \, {\rm keV}} \right)^{2.7}

with an intrinsic scatter of only 0.21 in log M_BH. This correlation is the strongest one that Gaspari et al. found, with even less scatter than the relationship between M_BH and central stellar velocity dispersion ( $\sigma_v$ ) among the galaxies in their sample.1

Interestingly, the observed M_BH–T_CGM relation dovetails with the hypothesis that black holes grow until they have released enough energy to counteract accretion, causing cumulative black-hole feedback output to scale with the energy (E_CGM) required to lift a massive galaxy’s atmosphere (e.g.,[3][4][5]). That amount of energy can be estimated from the relationship observed between halo mass (M₅₀₀) and X-ray gas temperature ( $kT_X$ )

\frac {M_{500}} {M_\odot} \approx 10^{13.5} \, \left( \frac{kT_{\rm X}} {1 \, {\rm keV}} \right)^{1.7}

among halos with masses in the $10^{13} \, M_\odot$ to $10^{14} \, M_\odot$ range [6]. The specific binding energy of a nearly hydrostatic atmosphere scales with T_X , and so the energy required to significantly lift a halo’s baryons scales as

E_{\rm CGM} \, \propto \, M_{500} T_{\rm X} \, \propto \, T_{\rm X}^{2.7}

for halos in this mass range.

The similarity in power-law slope between the M_BH–T_CGM relation and E_CGM–T_X relation strongly suggests that M_BH is indeed directly proportional to E_CGM. One subtlety is that T_CGM and T_X are usually similar but are not identical.2 However, equating them yields a simple numerical constant that connects M_BH to E_CGM. The dashed blue line in the figure shows the M_BH–T_CGM relation obtained by assuming $E_{\rm CGM} = 10^{-2.3} M_{\rm BH} c^2$ .

Presumably, the similarity of that M_BH–T_CGM line to the observations indicates that ~0.5% of the central black hole’s rest-mass energy has gone into lifting the surrounding CGM. Observations of the gaseous contents of galaxy groups show that something has lifted their atmospheres (e.g., [7]), and no other energy source provides enough energy. For example, the solid red line marking the upper boundary of the pink region shows that the total energy released from the central galaxy’s supernovae (E_SN,central) falls far below E_CGM in halos more massive than $10^{13} \, M_\odot$ . Also, the dotted red line representing the total supernova energy from all of a halo’s stars (E_SN,halo) likewise falls short of E_CGM in halos more massive than $\sim 10^{13.3} \, M_\odot$ .

One final feature to note about the plot is the orange dashed line (labeled L_X/H₀) representing an estimate of the total radiative energy losses from these atmospheres. It falls far below the blue dashed line indicating E_CGM, implying that black hole feedback on group scales puts far more energy into atmospheric lifting than into replenishing radiative losses.