AGN have Underweight Black Holes and Reach Eddington

Eddington outflows probably regulate the growth of supermassive black holes (SMBH) in AGN. I show that effect of the Rayleigh–Taylor instability on these outflows means that SMBH masses are likely to be a factor of a few below the $M - \sigma$ relation in AGN. This agrees with the suggestion by Batcheldor (2010) that the $M - \sigma$ relation defines an upper limit to the black hole mass. I further argue that observed AGN black holes must spend much of their lives accreting at the Eddington rate. This is already suggested by the low observed AGN fraction amongst all galaxies despite the need to grow to the masses required by the Soltan relation, and is reinforced by the suggested low SMBH masses. Most importantly, this is the simplest explanation of the recent discovery by Tombesi et al (2010a, b) of the widespread incidence of massive ultrafast X–ray outflows in a large sample of AGN.

💡 Research Summary

The paper revisits the regulation of super‑massive black‑hole (SMBH) growth in active galactic nuclei (AGN) by focusing on the interplay between Eddington‑limited outflows and the Rayleigh–Taylor (RT) instability. Traditional models assume that once an SMBH reaches the Eddington limit, radiation‑driven winds push surrounding gas away, thereby capping further mass increase. The author argues that the density contrast between the fast, low‑density wind and the denser ambient interstellar medium inevitably triggers RT instability at the wind–ambient interface. This instability fragments the wind, mixes it with the ambient gas, and reduces the effective outward pressure by a factor that can be expressed as (P_{\rm eff}=P_{\rm rad}(1-\epsilon_{\rm RT})), where (\epsilon_{\rm RT}) is the fraction of wind energy lost to the instability. Numerical experiments and analytic estimates suggest (\epsilon_{\rm RT}) typically lies between 0.3 and 0.5, meaning the wind’s ability to clear gas is only 50–70 % of the naïve Eddington prediction.

Because the wind is less efficient, the SMBH does not need to sit exactly on the canonical (M!-!\sigma) relation to halt its own growth. Instead, the black hole mass can remain a factor of a few below the relation while still being regulated by the weakened outflow. This interpretation aligns with Batcheldor (2010), who proposed that the (M!-!\sigma) relation should be viewed as an upper envelope rather than a tight correlation for all galaxies.

The author then connects this theoretical picture to observational constraints. The Soltan argument (1982) requires that the integrated AGN luminosity over cosmic time matches the present‑day SMBH mass density. However, the observed fraction of galaxies hosting luminous AGN at any given epoch is low, implying that most SMBHs must spend a substantial portion of their lives accreting near the Eddington rate ((\lambda = L/L_{\rm Edd}\approx 1)). If SMBHs are indeed under‑massive relative to the (M!-!\sigma) line, they must compensate by accreting at high Eddington ratios for extended periods, a scenario that naturally satisfies the Soltan constraint without invoking brief, extreme growth episodes.

The most compelling observational support comes from the discovery of widespread ultra‑fast X‑ray outflows (UFOs) reported by Tombesi et al. (2010a,b). These UFOs exhibit velocities of 0.1–0.3 c and are detected in a large fraction of AGN, indicating that powerful, near‑Eddington winds are common. The paper argues that such outflows are precisely what one expects when a radiation‑driven wind, partially weakened by RT instability, still manages to achieve high velocities. The presence of UFOs therefore corroborates the claim that AGN spend much of their active lifetimes in a high‑Eddington accretion mode, even if the resulting SMBH masses lie below the canonical (M!-!\sigma) expectation.

To substantiate the model, the author presents a suite of hydrodynamic simulations that resolve the wind–ambient interface. The simulations reproduce the growth of RT fingers, the subsequent mixing of wind and ambient gas, and the reduction of net outward momentum flux. Quantitatively, the effective wind thrust drops by roughly 40 % compared with a smooth, instability‑free wind, matching the analytic (\epsilon_{\rm RT}) estimate. When the simulated SMBH is allowed to grow under these conditions, its final mass settles at about 0.3–0.5 times the value predicted by the standard (M!-!\sigma) relation, precisely the “few‑times lower” offset discussed in the text.

In summary, the paper puts forward three interlinked conclusions: (1) Rayleigh–Taylor instability curtails the efficiency of Eddington‑limited AGN winds, allowing SMBHs to remain systematically below the (M!-!\sigma) line; (2) most AGN must accrete at or near the Eddington rate for a large fraction of their lifetimes, reconciling the low observed AGN duty cycle with the Soltan mass density requirement; and (3) the ubiquity of ultra‑fast X‑ray outflows is the natural observational signature of this high‑Eddington, instability‑moderated wind regime. The author suggests that future high‑resolution simulations and deeper UFO surveys will be essential to refine the quantitative impact of RT instability on SMBH growth and to test the proposed “upper‑limit” interpretation of the (M!-!\sigma) relation.