AGN Feedback

An active galactic nucleus is a fusion reactor's nightmare made real: growing a supermassive black hole liberates $E_{\rm acc}=\eta M_{\rm BH}c^2$ , and with radiative efficiency $\eta\approx0.1$ a $10^8\,M_\odot$ hole pours out $\sim2\times10^{61}$ erg — a few hundred times the $\sim10^{59}$ erg that binds the gas of its $\sim10^{11}\,M_\odot$ host. The energy is never in doubt. The whole subject of AGN feedback is the far subtler question of coupling: what fraction of that torrent actually thermalizes in the surrounding gas, and through which carrier it arrives. There are three carriers, each with its own physics. Radiation exerts pressure on dust and, through resonance lines, launches ultraviolet winds; broad-absorption-line quasars show outflows at $0.01$ – $0.3c$ . Those nuclear winds shock against the interstellar medium and sweep it up in a bubble that is either momentum-conserving or energy-conserving — a distinction that changes the galaxy-scale outcome by orders of magnitude. And relativistic jets, powered by black-hole spin and disk magnetic fields, drill into hot halos and inflate the X-ray cavities seen in cool-core clusters. The three are siblings of one accretion flow, and their balance shifts with Eddington ratio. Phenomenologically this splits into two modes. The radiative (quasar) mode runs near the Eddington limit during rapid growth, quenching star formation in massive galaxies and closing the M–sigma relation; the kinetic (radio, or maintenance) mode runs at low accretion rates through advection-dominated flows feeding jets that keep hot atmospheres from cooling — a duty-cycled drip of heat, not a violent expulsion. Together they are the keystone of galaxy formation: without feedback, massive halos overcool and the bright end of the galaxy mass function is overpredicted more than tenfold. AGN feedback is how the smallest scale in astrophysics reaches up and disciplines the largest.

Key takeaways

AGN feedback is the coupling of an accreting black hole's output — radiation, wide-angle winds, and collimated jets — into the surrounding interstellar and intracluster gas, regulating both black-hole and galaxy growth.
The energy is never the bottleneck: growing the hole releases $\eta M_{\rm BH}c^2$ , hundreds of times the gas binding energy, so a coupling of only $\sim0.5\%$ suffices to unbind or heat the reservoir.
Two modes operate: a radiative (quasar) mode of high-Eddington outflows that expel galaxy-scale gas, and a kinetic (radio) mode of low-luminosity jets that inflate cavities and offset cooling.
Self-regulation closes the M–sigma relation: the hole grows until its wind can expel its own fuel, freezing $M_{\rm BH}$ at a value set by the bulge potential — $M_{\rm BH}\propto\sigma^4$ (momentum-driven) to $\sigma^5$ (energy-driven).
In clusters the enthalpy $4pV$ of jet-blown X-ray cavities, divided by the buoyant rise time, delivers a cavity power that tracks the X-ray cooling luminosity — the standard cure for the cooling-flow problem.
Key idea: because the energy released in assembling a black hole vastly exceeds a galaxy's gaseous binding energy, even a percent-level coupling of AGN output to the surrounding gas can regulate — and ultimately self-limit — both black-hole and galaxy growth, closing the M–sigma relation and quenching star formation.

Where this lives today

AGN feedback is now a load-bearing wall of every cosmological simulation. EAGLE, IllustrisTNG, and SIMBA all inject black-hole energy — thermally, kinetically, or both — to truncate the bright end of the galaxy stellar-mass function; switch it off and they overproduce $>10^{11}\,M_\odot$ galaxies more than tenfold. Observationally the field is being remade by instruments that resolve the outflows and cavities directly. X-ray spectroscopy from XMM-Newton and the XRISM microcalorimeter measures the $\sim0.1c$ ultra-fast outflows blueshift by blueshift, weighing their kinetic power against the $\sim5\%L$ prediction; ALMA maps molecular outflows on kiloparsec scales; Chandra extends the $P_{\rm cav}$ – $L_{\rm cool}$ cavity relation down into individual galaxies; and the Event Horizon Telescope's image of the M87 jet base ties kinetic-mode power to the spinning hole that launches it. The open frontier is the coupling efficiency itself: no first-principles theory yet predicts what fraction of $L$ thermalizes, so it remains a calibrated knob — the single largest systematic uncertainty between accretion physics and the galaxy population it shapes.

The physical picture

Picture the black hole as a valve on its own fuel line. Gas falls in, lights up the nucleus, and the radiation and winds that result push back on the very reservoir that feeds them. When the push is strong enough to expel or heat the gas, accretion starves and the AGN fades — a self-regulating thermostat that ties the hole's final mass to the depth of the galaxy's potential well. The first figure makes the energetics concrete: the accretion energy dwarfs the binding energy, so the outcome is decided by the coupling fraction, not the supply.

Figure

The self-regulating engine: an accreting black hole releases E_acc = η M_BH c² ≈ 2×10⁶¹ erg for a 10⁸ M_⊙ hole, hundreds of times the E_bind ≈ 10⁵⁹ erg that binds a 10¹¹ M_⊙ gas reservoir, so only ~0.5% coupling of the wind momentum L/c against gravity is needed to drive the shell out

The difference between momentum-driven and energy-driven winds is the difference between a snowplow and a pressure cooker. In the momentum-driven case the shocked wind radiates away its thermal energy almost instantly, so only its momentum $L/c$ pushes the swept-up shell; in the energy-driven case the shock cannot cool, so a hot bubble of pressure $\sim\tfrac12\dot M_w v_w^2$ expands adiabatically and does far more work. Same central engine, wildly different galaxy-scale consequences — and the split is set by a cooling radius of order a kiloparsec. The second figure shows the payoff of self-regulation: the observed M–sigma relation, and the $\sigma^4$ line the momentum-driven argument predicts, passing through the anchor $M_{\rm BH}\approx3\times10^8\,M_\odot$ at $\sigma=200\,\mathrm{km\,s^{-1}}$ .

Figure

The self-regulation payoff: the momentum-driven prediction M_BH = f_g σ_T σ⁴ /(π G² m_p) plotted as a slope-4 line in log-log, pinned at M_BH ≈ 3.7×10⁸ M_⊙ for σ = 200 km/s with f_g = 0.16, threading real black holes from the Milky Way (σ ≈ 105) to M87 (σ ≈ 375)

The core equations

The premise of the whole field is an inequality, not an equation. Compare the energy released in building the black hole to the gravitational binding energy of the gas that must be removed; the first overwhelms the second, so feedback is a coupling problem.

E_{\rm acc}=\eta\,M_{\rm BH}\,c^2 \;\gg\; E_{\rm bind}\sim \frac{G M_{\rm gal}^2}{R_{\rm gal}}\sim M_{\rm gas}\,\sigma^2 \qquad (1)

where: $E_{\rm acc}$ — total energy released growing the hole; $\eta\approx0.1$ — thin-disk radiative efficiency; $M_{\rm BH}$ — black-hole mass; $E_{\rm bind}$ — binding energy of the host's gas; $M_{\rm gas},\sigma$ — gas mass and stellar velocity dispersion. For $M_{\rm BH}=10^8\,M_\odot$ and $M_{\rm gas}=10^{11}\,M_\odot$ at $\sigma=200\,\mathrm{km\,s^{-1}}$ the ratio is $\approx225$ : only sub-percent coupling is required. The luminosity cannot climb without bound. When radiation pressure on ionized gas balances gravity the accretion self-limits at the Eddington luminosity, the natural ceiling for radiative-mode feedback.

L_{\rm Edd}=\frac{4\pi G M_{\rm BH} m_p c}{\sigma_T}\approx 1.26\times10^{38}\left(\frac{M_{\rm BH}}{M_\odot}\right)\ \mathrm{erg\,s^{-1}} \qquad (2)

where: $m_p$ — proton mass; $\sigma_T=6.65\times10^{-25}\,\mathrm{cm^2}$ — Thomson cross section; $c,G$ — speed of light and gravitational constant. A $10^8\,M_\odot$ hole radiating at $L_{\rm Edd}$ shines at $1.26\times10^{46}\,\mathrm{erg\,s^{-1}}$ ; the wind and jet powers are gauged against this ceiling. The carriers deposit momentum, not just energy. Radiation carries a momentum flux $L/c$ ; a wind that scatters those photons carries a comparable flux scaled by its optical depth.

\dot p_{\rm rad}=\frac{L}{c},\qquad \dot M_w v_w \simeq \tau\,\frac{L}{c} \qquad (3)

where: $\dot p_{\rm rad}$ — radiation momentum flux; $\dot M_w,v_w$ — wind mass-loss rate and speed; $\tau$ — scattering optical depth, $\tau\approx1$ for a single-scattering nuclear wind. Setting $\tau\approx1$ makes the wind's momentum flux equal to the radiation's — the defining condition of the momentum-driven regime. Single scattering at the Eddington limit fixes the wind speed and its kinetic power in one stroke, and the answer is mildly relativistic — exactly the ultra-fast outflows X-ray spectroscopy reveals.

v_w \approx \frac{L_{\rm Edd}}{\dot M_w c}\sim \eta\,c\approx 0.1\,c,\qquad \frac{1}{2}\dot M_w v_w^2\approx\frac{v_w}{2c}\,L\approx 0.05\,L \qquad (4)

where: $v_w$ — nuclear wind speed; $\dot M_w\sim\dot M_{\rm Edd}=L_{\rm Edd}/(\eta c^2)$ — Eddington mass flux. Because $v_w\sim\eta c$ , the wind carries roughly $5\%$ of the bolometric luminosity as kinetic energy — well above the sub-percent that needs to couple, so energy-driven winds are potent. The heart of self-regulation is the critical black-hole mass at which the wind momentum can finally lift the gas out of an isothermal halo. Balancing the two forces gives the momentum-driven M–sigma relation.

M_{\rm BH}=\frac{f_g\,\sigma_T}{\pi G^2 m_p}\,\sigma^4 \;\Rightarrow\; M_{\rm BH}\propto\sigma^4 \qquad (5)

where: $\sigma$ — bulge stellar velocity dispersion; $f_g\approx\Omega_b/\Omega_m\approx0.16$ — gas fraction. With $f_g=0.16$ and $\sigma=200\,\mathrm{km\,s^{-1}}$ this evaluates to $3.7\times10^8\,M_\odot$ , matching the observed normalization without any tuning — the single most important quantitative success of the theory. Why is feedback needed at all in massive halos? Because the gas can radiate away its thermal energy faster than the Hubble time and pile up at the center — the overcooling problem.

t_{\rm cool}=\frac{\tfrac{3}{2}n k_B T}{n_e n_H \Lambda(T)}\;\lesssim\; t_{\rm Hubble} \qquad (6)

where: $n,n_e,n_H$ — total, electron, and hydrogen number densities; $T$ — gas temperature; $\Lambda(T)$ — radiative cooling function; $k_B$ — Boltzmann constant. Where $t_{\rm cool}\ll t_{\rm Hubble}$ — the cores of massive halos — unchecked cooling would build far too many overweight galaxies. The maintenance-mode cure is a heating rate that, averaged over the jet duty cycle, matches the cooling. It is measured directly from the enthalpy of jet-blown X-ray cavities.

P_{\rm cav}=\frac{4pV}{t_{\rm buoy}}\;\approx\; L_{\rm cool} \qquad (7)

where: $p,V$ — pressure and volume of an X-ray cavity; $4pV$ — its enthalpy ( $pV$ of work plus $3pV$ of internal energy for a relativistic gas); $t_{\rm buoy}$ — buoyant rise time; $L_{\rm cool}$ — X-ray cooling luminosity inside the cooling radius. Across cool-core clusters $P_{\rm cav}$ tracks $L_{\rm cool}$ over two decades in power — the observational backbone of maintenance-mode feedback.

Derivation AThe momentum-driven M–sigma relation

The goal is to derive the scaling $M_{\rm BH}\propto\sigma^4$ , together with its absolute normalization, from a single physical picture: an AGN radiating at its Eddington limit tries to push a shell of gas out of an isothermal galaxy, and the critical black-hole mass is the one at which the outward momentum flux exactly balances the inward pull of gravity. Model the host as a singular isothermal sphere of dispersion $\sigma$ carrying a gas fraction $f_g$ , take single-scattering radiation pressure as the driving force, and demand force balance on the swept-up shell. Everything below the critical mass stalls; everything above it expels the gas and starves the hole.

Step 1. For a singular isothermal sphere the enclosed mass rises linearly with radius, $M(r)=2\sigma^2 r/G$ , and the swept-up gas mass is $M_g=f_g M(r)$ . Why: the isothermal profile is the equilibrium of a self-gravitating gas at constant $\sigma$ , and it is the standard idealization of a galaxy halo — its linear $M(r)$ is what makes the forces come out radius-independent.

Step 2. Compute the gravitational force on the shell: $F_{\rm grav}=G M(r)M_g/r^2 = G\,(2\sigma^2 r/G)(f_g\,2\sigma^2 r/G)/r^2 = 4 f_g\sigma^4/G$ . Why: the two factors of $r$ from $M(r)$ and $M_g$ cancel the $r^{-2}$ of gravity, so the restraining force is constant with radius — a shell that can be pushed at one radius can be pushed at all of them, which is what makes expulsion an all-or-nothing threshold.

Step 3. Take the maximum outward push as the single-scattering radiation momentum flux at the Eddington limit, $F_{\rm drive}=L_{\rm Edd}/c = 4\pi G M_{\rm BH} m_p/\sigma_T$ . Why: at $\tau\approx1$ each photon transfers its momentum once, so the wind flux equals $L/c$ ; using $L_{\rm Edd}$ ties the driving directly to $M_{\rm BH}$ and asks the sharpest question — can the hole, at its own luminosity ceiling, clear its host?

Step 4. Set $F_{\rm drive}=F_{\rm grav}$ : $4\pi G M_{\rm BH} m_p/\sigma_T = 4 f_g\sigma^4/G$ . Why: this is the self-regulation condition — below the balancing mass the shell falls back and accretion continues; at balance the gas is driven off and the fuel supply is cut, freezing the mass.

Step 5. Solve for the critical mass: $M_{\rm BH}=f_g\,\sigma_T\,\sigma^4/(\pi G^2 m_p)$ . Inserting $f_g=0.16$ and $\sigma=200\,\mathrm{km\,s^{-1}}$ gives $M_{\rm BH}\approx3.7\times10^8\,M_\odot$ . Why: the result is not fitted — the coefficient is built from fundamental constants and one cosmological gas fraction, and it lands on the observed M–sigma normalization, which is the theory's decisive quantitative win. *Direct substitution gives $3.7\times10^8\,M_\odot$ at $\sigma=200\,\mathrm{km\,s^{-1}}$ , within a factor of a few of the empirical fit.*

Go deeper · Derivation B · 1 minThe coupling efficiency needed to unbind the gas

Derivation BThe coupling efficiency needed to unbind the gas

The goal is to show quantitatively that the physics of AGN feedback is limited by coupling rather than by energy supply, by computing the minimum fraction $\epsilon_{\rm min}$ of the accretion energy that must be delivered to the gas to unbind it. Compare the binding energy of the gas — the energy needed to lift it out at roughly the escape speed — to the total energy the black hole radiates over its growth. The ratio is a pure number, and it comes out at the half-percent level, which is why simulations treat the coupling efficiency as their single most consequential free parameter.

Step 1. Estimate the binding energy of the gas as $E_{\rm bind}\simeq\tfrac12 M_{\rm gas}v_{\rm esc}^2\sim M_{\rm gas}\sigma^2$ , using $v_{\rm esc}^2\sim2\sigma^2$ for a self-gravitating system. For $M_{\rm gas}=10^{11}\,M_\odot$ and $\sigma=200\,\mathrm{km\,s^{-1}}$ this is $\approx8\times10^{58}\,\mathrm{erg}$ . Why: unbinding means giving each gas parcel of order its own kinetic escape energy, and $\sigma$ is the directly observable proxy for the potential depth.

Step 2. Write the available accretion energy $E_{\rm acc}=\eta M_{\rm BH}c^2$ . For $\eta=0.1$ and $M_{\rm BH}=10^8\,M_\odot$ this is $1.8\times10^{61}\,\mathrm{erg}$ . Why: this is the total electromagnetic energy the hole liberates over its lifetime — the reservoir from which any coupling fraction is drawn.

Step 3. Form the ratio $\epsilon_{\rm min}=E_{\rm bind}/E_{\rm acc}\approx 8\times10^{58}/1.8\times10^{61}\approx0.45\%$ . Using $\sigma^2/c^2\approx4.4\times10^{-7}$ and $M_{\rm gas}/M_{\rm BH}\sim10^3$ recovers the same figure analytically. Why: the smallness of $\sigma^2/c^2$ is the crux — the potential well is shallow in units of $c^2$ , so only a sliver of the relativistic accretion energy is ever needed; the rest can escape as radiation without touching the gas.

Step 4. Read the physical conclusion: since only $\sim0.5\%$ of $E_{\rm acc}$ must couple, feedback strength is governed by how the energy is delivered — momentum vs energy conservation, wind vs jet — not by whether enough exists. Why: this is why a factor-of-two change in the assumed coupling efficiency moves a simulation's galaxy stellar-mass function by an order of magnitude, and why the field's central uncertainty is a coupling number.

Go deeper · Derivation C · 2 minWhy the wind switches from momentum- to energy-driven

Derivation CWhy the wind switches from momentum- to energy-driven

The goal is to explain the transition that steepens the M–sigma slope from $\sigma^4$ to $\sigma^5$ : close to the nucleus the shocked wind cools so efficiently that only its momentum survives to push the gas, but beyond a cooling radius the shock cannot radiate and its full kinetic energy inflates an adiabatic bubble that does far more work. Track the inverse-Compton cooling time of the shocked wind electrons against the flow time, find the radius where they cross, and identify the two regimes on either side. The switch is what reconciles the modest momentum-driven prediction with the large energetics that observed outflows actually carry.

Step 1. The shocked wind is heated to $\sim10^{10}\,\mathrm{K}$ and cools chiefly by inverse-Compton scattering off the AGN radiation field, with $t_{\rm IC}\propto1/U_{\rm rad}\propto r^2/L$ . Why: the radiation energy density dilutes as $r^{-2}$ , so the cooling time lengthens with distance — cooling is fast in the intense inner field and slow far out.

Step 2. Set the cooling time equal to the flow time to define the cooling radius, $t_{\rm IC}(R_C)=t_{\rm flow}(R_C)$ , which lands at $R_C\sim0.5$ – $1\,\mathrm{kpc}$ . Why: inside $R_C$ the shock dumps its thermal energy back into radiation and only $L/c$ of momentum is transmitted (momentum-driven); outside $R_C$ the energy is retained.

Step 3. Beyond $R_C$ the full wind kinetic power drives the bubble, $\tfrac12\dot M_w v_w^2 \to \tfrac{\mathrm d}{\mathrm dt}(\tfrac32 pV)$ , and the shocked gas expands adiabatically. Why: with no radiative losses, the conserved quantity is energy, not momentum, and a hot bubble pushing a shell does $pV$ work as it grows — a much larger effect than a bare momentum kick.

Step 4. Energy conservation for the expanding bubble in an isothermal halo yields $M_{\rm BH}\propto\sigma^5$ with a coefficient far below the momentum-driven mass — so once a wind reaches $R_C$ it clears the galaxy at a black-hole mass smaller than the momentum-driven estimate. Why: energy-driven feedback is intrinsically more effective, which is why observed outflow kinetic powers exceed the pure momentum-driven expectation and why the effective M–sigma slope sits between 4 and 5. *The $R_C\sim1\,\mathrm{kpc}$ transition explains observed outflow energetics that exceed the momentum-driven bound.*

Limiting cases

The character of feedback is set by three switches — the Eddington ratio, whether the shocked wind can cool, and whether radiation or jets carry the power:

Case	Result	Meaning
High Eddington ratio ( $\dot M\gtrsim\dot M_{\rm Edd}$ )	$L\approx L_{\rm Edd}=\dfrac{4\pi G M_{\rm BH} m_p c}{\sigma_T}$	Radiative (quasar) mode: radiation pressure and line-driven winds dominate, sweeping galaxy-scale gas and quenching star formation.
Low Eddington ratio ( $\dot M\ll\dot M_{\rm Edd}$ )	$L_{\rm rad}\propto\dot M^2,\ \ P_{\rm jet}\gtrsim L_{\rm rad}$	Kinetic (radio) mode via an ADAF: radiation collapses but spin-powered jets persist, inflating cavities and providing maintenance heat.
Momentum-conserving wind (fast cooling, $r<R_C$ )	$M_{\rm BH}=\dfrac{f_g\,\sigma_T}{\pi G^2 m_p}\,\sigma^4$	Inner shocked shell radiates its heat; only $L/c$ pushes the shell, giving the $\sigma^4$ M–sigma scaling.
Energy-conserving wind (no cooling, $r>R_C$ )	$M_{\rm BH}\propto\sigma^5$	Hot bubble retains thermal energy and does adiabatic work — steeper, far more effective feedback beyond the cooling radius.

Common trapDo not confuse the AGN emitting enough energy with the gas receiving enough energy.

The luminosity of a quasar is staggering, and it is tempting to conclude that feedback is therefore trivially guaranteed. It is not. The energy budget only tells you that the supply is ample — hundreds of times the gas binding energy. The physically decisive quantities are the coupling efficiency (what fraction of $L$ thermalizes in the gas) and the thermodynamic regime of the shocked wind: momentum-conserving winds lose most of their energy to radiation and push only with $L/c$ , giving $M_{\rm BH}\propto\sigma^4$ , while energy-conserving winds retain their heat and give the steeper, stronger $\sigma^5$ scaling. Quote a luminosity and you have said nothing about the outcome; the M–sigma slope, the mass-loading of the outflow, and the quenching timescale all live in the coupling and the cooling, not in $L$ itself.

Worked examples

1 · The energy budget and the required coupling (fully worked)

A black hole grows to $M_{\rm BH}=10^8\,M_\odot$ at radiative efficiency $\eta=0.1$ inside a host whose gas reservoir is $M_{\rm gas}=10^{11}\,M_\odot$ with velocity dispersion $\sigma=200\,\mathrm{km\,s^{-1}}$ . Find the accretion energy, the gas binding energy, and the minimum coupling efficiency needed to unbind the gas.

Step 1. Accretion energy. With $M_{\rm BH}=10^8\,M_\odot=1.99\times10^{41}\,\mathrm g$ and $c^2=9.0\times10^{20}\,\mathrm{cm^2\,s^{-2}}$ : $E_{\rm acc}=\eta M_{\rm BH}c^2 = 0.1\times1.99\times10^{41}\times9.0\times10^{20}=\mathbf{1.8\times10^{61}\ erg}$ .

Step 2. Binding energy. With $M_{\rm gas}=10^{11}\,M_\odot=1.99\times10^{44}\,\mathrm g$ and $\sigma=2\times10^{7}\,\mathrm{cm\,s^{-1}}$ , so $\sigma^2=4\times10^{14}\,\mathrm{cm^2\,s^{-2}}$ : $E_{\rm bind}\sim M_{\rm gas}\sigma^2 = 1.99\times10^{44}\times4\times10^{14}=\mathbf{8\times10^{58}\ erg}$ .

Step 3. Ratio and coupling. $E_{\rm acc}/E_{\rm bind}=1.8\times10^{61}/8\times10^{58}\approx\mathbf{225}$ , so the minimum coupling is $\epsilon_{\rm min}=E_{\rm bind}/E_{\rm acc}\approx\mathbf{0.45\%}$ .

Step 4. Interpretation. Fewer than one part in two hundred of the accretion energy has to reach the gas. The engine is never energy-starved; whether the gas is actually expelled depends on how that energy is delivered. E_acc ≈ 1.8×10⁶¹ erg, E_bind ≈ 8×10⁵⁸ erg, ε_min ≈ 0.5%.

2 · The M–sigma normalization and the wind it takes (fully worked)

For $\sigma=200\,\mathrm{km\,s^{-1}}$ and gas fraction $f_g=0.16$ , evaluate the momentum-driven critical black-hole mass, and find the speed and kinetic power of the nuclear wind that enforces it.

Step 1. Assemble the constants: $\sigma_T=6.65\times10^{-25}\,\mathrm{cm^2}$ , $G=6.67\times10^{-8}$ cgs, $m_p=1.67\times10^{-24}\,\mathrm g$ , $\sigma^4=(2\times10^7)^4=1.6\times10^{29}\,\mathrm{cm^4\,s^{-4}}$ . Numerator $f_g\sigma_T\sigma^4=0.16\times6.65\times10^{-25}\times1.6\times10^{29}=1.70\times10^{4}$ .

Step 2. Denominator $\pi G^2 m_p = 3.1416\times(6.67\times10^{-8})^2\times1.67\times10^{-24}=2.34\times10^{-38}$ . Divide: $M_{\rm BH}=1.70\times10^{4}/2.34\times10^{-38}=7.3\times10^{41}\,\mathrm g=\mathbf{3.7\times10^{8}\ M_\odot}$ — squarely on the observed relation.

Step 3. The wind that does this is single-scattering and Eddington-limited, so $v_w\sim\eta c\approx0.1c=\mathbf{3\times10^{7}\ m\,s^{-1}}$ , matching the $0.03$ – $0.3c$ ultra-fast outflows seen in X-rays.

Step 4. Its kinetic luminosity is $\tfrac12\dot M_w v_w^2\approx(v_w/2c)L\approx\mathbf{0.05\,L}$ — about $5\%$ of bolometric, ten times the sub-percent that must couple, so the wind is comfortably energetic enough. M_BH = 3.7×10⁸ M_⊙, v_w ≈ 0.1c, wind carries ≈ 5% of L.

3 · Does a cluster cavity offset its cooling? (fully worked)

A radio jet inflates an X-ray cavity of radius $R=10\,\mathrm{kpc}$ in intracluster gas of pressure $p=3\times10^{-10}\,\mathrm{erg\,cm^{-3}}$ ; the bubble rises buoyantly on a timescale $t_{\rm buoy}=3\times10^{7}\,\mathrm{yr}$ . Compare its power to a cool-core cooling luminosity of $L_{\rm cool}\sim10^{44}\,\mathrm{erg\,s^{-1}}$ .

Step 1. Cavity volume: $R=10\,\mathrm{kpc}=3.09\times10^{22}\,\mathrm{cm}$ , so $V=\tfrac43\pi R^3=1.23\times10^{68}\,\mathrm{cm^3}$ .

Step 2. Enthalpy: $4pV=4\times3\times10^{-10}\times1.23\times10^{68}=\mathbf{1.5\times10^{59}\ erg}$ — the energy the jet stored in the bubble.

Step 3. Cavity power: $t_{\rm buoy}=3\times10^{7}\,\mathrm{yr}=9.5\times10^{14}\,\mathrm s$ , so $P_{\rm cav}=4pV/t_{\rm buoy}=1.5\times10^{59}/9.5\times10^{14}=\mathbf{1.6\times10^{44}\ erg\,s^{-1}}$ .

Step 4. Verdict: $P_{\rm cav}\approx1.6\times10^{44}\gtrsim L_{\rm cool}\sim10^{44}\,\mathrm{erg\,s^{-1}}$ . A single duty cycle of one bubble roughly offsets the entire cooling luminosity — precisely the balance measured across cool-core clusters. P_cav ≈ 1.6×10⁴⁴ erg/s ≈ L_cool — maintenance-mode feedback confirmed.

Physics of the portrait

AGN feedback is a theory with a lineage: Silk and Rees imagined the self-regulating quasar in 1998, and Andrew King turned the picture into the sharp analytic scalings — $\sigma^4$ and $\sigma^5$ — that this article is built around.

Origins · 1 minAndrew King and how this was discovered

Andrew King

momentum- and energy-driven black-hole winds, M ∝ σ⁴ and σ⁵ · self-regulated AGN feedback and the M–sigma relation

The idea that a growing black hole might set its own mass was floated by Joseph Silk and Martin Rees in 1998, just after black-hole masses were found to correlate with their host bulges: perhaps, they argued, the quasar wind expels the gas once the hole is massive enough, freezing the relation. It was Andrew King, a British astrophysicist long associated with the University of Leicester and known for his work on accretion in X-ray binaries and AGN, who in 2003 made the argument quantitative. Modelling the host as an isothermal sphere and demanding that the Eddington-limited wind momentum $L/c$ balance gravity, King derived $M_{\rm BH}=f_g\sigma_T\sigma^4/(\pi G^2 m_p)$ — a normalization built from fundamental constants alone that landed, astonishingly, on the observed M–sigma value. In 2005 he sharpened it, showing the shocked wind is momentum-conserving only inside an inverse-Compton cooling radius of about a kiloparsec; beyond it the flow becomes energy-conserving and the scaling steepens to $\sigma^5$ . King also argued that the ultra-fast outflows — the $\sim0.1c$ X-ray winds discovered by Pounds, Reeves, and collaborators — are exactly the single-scattering, Eddington-driven winds his theory required, closing the loop between nuclear physics and the galaxy-scale relation. The observational counterpart, the X-ray cavities and suppressed cooling flows of clusters, was assembled largely by Andrew Fabian and collaborators; together the two strands made AGN feedback indispensable to modern galaxy formation.

History

1969

— Lynden-Bell identifies accretion onto supermassive black holes as the quasar engine

1998

— Silk and Rees propose self-regulating AGN feedback to close the M–sigma relation

2000

— the M–sigma relation is measured (Ferrarese-Merritt; Gebhardt and collaborators)

2003

— King derives the momentum-driven M ∝ σ⁴ limit from an isothermal-sphere argument

2012

— Fabian reviews the X-ray cavity and cooling-flow evidence for AGN feedback

Going further

Texts. Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., Ch. 28 (Active Galaxies) · Mo, van den Bosch & White, Galaxy Formation and Evolution, Ch. 14 (feedback and galaxy formation) · Binney & Tremaine, Galactic Dynamics, 2nd ed. (galaxy energetics and gas dynamics) · Dodelson, Modern Cosmology (baryonic physics of structure formation) · King & Pounds, ARA&A 53, 115 (2015), Powerful Outflows and Feedback from AGN · Fabian, ARA&A 50, 455 (2012), Observational Evidence of AGN Feedback. The Eddington luminosity is $L_{\rm Edd}=1.26\times10^{38}(M/M_\odot)\,\mathrm{erg\,s^{-1}}$ . The energy-budget figures ( $E_{\rm acc}\approx1.8\times10^{61}$ erg, $E_{\rm bind}\approx8\times10^{58}$ erg, ratio $\approx225$ , $\epsilon_{\rm min}\approx0.5\%$ ) follow directly from $\eta M_{\rm BH}c^2$ and $M_{\rm gas}\sigma^2$ . The momentum-driven M–sigma relation $M_{\rm BH}=f_g\sigma_T\sigma^4/(\pi G^2 m_p)$ , with $f_g=0.16$ and $\sigma=200\,\mathrm{km\,s^{-1}}$ , gives $3.7\times10^8\,M_\odot$ . The momentum-to-energy transition and $\sigma^5$ scaling arise beyond the inverse-Compton cooling radius. The cavity balance $P_{\rm cav}=4pV/t_{\rm buoy}\approx L_{\rm cool}$ gives a worked cavity power of $1.6\times10^{44}\,\mathrm{erg\,s^{-1}}$ for the parameters in Worked Example 3.

Prerequisites. Eddington Luminosity and Accretion Limits · Supermassive Black Holes · Thin Accretion Disks and the Shakura–Sunyaev Model · Advection-Dominated Accretion Flows · Relativistic Jets and Launching Mechanisms. Leads to. The M–sigma Relation and Black-Hole–Galaxy Coevolution · Quasars and Quasar Evolution · Black-Hole Spin and the Ergosphere.

Ask the tutor: Ask me to walk through the isothermal-sphere force balance that turns "the wind can lift the gas" into the exact $M_{\rm BH}\propto\sigma^4$ normalization — or to contrast it with the energy-driven $\sigma^5$ case step by step.

If a quasar is so luminous, isn't it obvious it can blow its galaxy apart?

No — that intuition skips the crucial step. The luminosity guarantees only that the energy supply is enormous, hundreds of times the gas binding energy. What matters is how much of that energy actually couples to the gas, and by what mechanism. Radiation streaming freely out of a transparent nucleus deposits almost nothing; a wind that shocks and cools radiates its energy away and delivers only momentum. The observed regulation requires that a small but nonzero fraction — around half a percent effectively — thermalizes in the gas, and the whole difficulty of the subject is predicting that fraction from first principles, which no one can yet do.

What is the difference between quasar-mode and radio-mode feedback?

They are the two faces of the same engine at different accretion rates. Quasar (radiative) mode operates near the Eddington limit during rapid black-hole growth, typically after gas-rich mergers; it drives fast, wide-angle radiation- and line-driven winds that sweep cold gas out of the galaxy and abruptly quench star formation. Radio (kinetic, or maintenance) mode operates at low accretion rates, where the flow becomes an advection-dominated accretion flow with feeble radiation but powerful, spin-launched jets. Those jets inflate X-ray cavities in hot halos and gently reheat the gas over gigayears, preventing it from cooling and forming new stars. One expels; the other maintains.

Which really operates — momentum-driven or energy-driven winds?

Both, in different places. The distinction is set by whether the shocked nuclear wind can radiate away its thermal energy before it does work. Close to the black hole, inside an inverse-Compton cooling radius of roughly a kiloparsec, the shock cools almost instantly, so only the wind's momentum $L/c$ survives to push the gas — the momentum-driven regime that gives $M_{\rm BH}\propto\sigma^4$ . Beyond that radius the shocked gas cannot cool, retains its full kinetic energy, and inflates an adiabatic bubble that does far more work, steepening the scaling toward $\sigma^5$ . Real outflows appear to carry more energy than pure momentum-driving allows, which is evidence that the energy-driven phase is reached.

How do we know jets actually balance cooling in clusters?

Through a direct energy accounting made possible by X-ray imaging. Chandra resolves the cavities that radio jets carve into the hot intracluster gas; each bubble has a measurable pressure and volume, so its enthalpy is $4pV$ . Dividing by the buoyant rise time gives a mechanical power, and when this is plotted against the X-ray cooling luminosity inside the cooling radius for a large sample of cool-core clusters, the two track each other over roughly two decades in power. That correlation is the observational backbone of maintenance-mode feedback: it shows the jets deposit, on average, just about the energy needed to offset radiative cooling.

Does AGN feedback cause the M–sigma relation or merely correlate with it?

The theoretical case is causal, and it is compelling because the normalization is predicted, not fitted. The self-regulation argument says the black hole grows until its Eddington-limited wind can expel the gas from the bulge, and that condition fixes $M_{\rm BH}=f_g\sigma_T\sigma^4/(\pi G^2 m_p)$ — a coefficient built entirely from fundamental constants and the cosmic gas fraction. Plugging in $\sigma=200\,\mathrm{km\,s^{-1}}$ yields $\sim3.7\times10^8\,M_\odot$ , on the observed relation with no free parameters. Alternative accounts exist — hierarchical merging can also produce a tight relation by central-limit averaging — so the community treats feedback as the leading but not exclusive explanation.

Why can't supernova feedback do the job in massive galaxies?

Because supernovae run out of leverage exactly where it is needed most. Each supernova injects a fixed energy, so stellar feedback scales with the star-formation rate and works well in shallow potential wells — dwarf and low-mass galaxies, where modest energy input can drive winds. But in massive halos the escape velocity is high and the gas binding energy grows faster than the supernova budget can match; the hot gas simply cannot be expelled by stellar processes alone. AGN feedback fills the gap because its energy source, $\eta M_{\rm BH}c^2$ , is tied to the black hole rather than the stars and vastly exceeds the supernova output in exactly the massive systems where overcooling would otherwise run away.