---
title: Active Galactic Nuclei
draft: false
module: Galaxies and Dark Matter
moduleNumber: 10
lessonNumber: 4
order: 1004
summary: >
  A small fraction of galaxies pour out enormous luminosity from a region smaller
  than the solar system. Accretion onto a supermassive black hole, limited by the
  Eddington balance of radiation pressure and gravity, powers the Seyferts, quasars,
  radio galaxies, and blazars — one engine seen from different angles through an
  obscuring torus. Relativistic jets produce apparent superluminal motion, reverberation
  mapping and stellar dynamics weigh the central mass, and the M–sigma relation ties
  that mass to the host bulge.
topics: [Galaxies and Dark Matter]
sources:
  - book: Carroll & Ostlie
    ref: "Ch. 28 — Active Galaxies; §28.1 Observations; §28.2 A Unified Model"
  - book: Maoz
    ref: "Ch. 7 — Active Galactic Nuclei and Quasars"
---

An **active galactic nucleus** (AGN) outshines the entire surrounding galaxy from a
central region light-days across. The luminosity, up to $10^{40}\ \mathrm{W}$ for the
brightest quasars, the rapid variability that bounds the emitting size, and the broad
emission lines all point to accretion onto a supermassive black hole. This lesson
derives the Eddington luminosity that caps accretion power and the radiative
efficiency that sets the fuel requirement, catalogues the AGN types and unifies them
through orientation, treats the relativistic jets and their apparent superluminal
motion, and connects the black-hole mass — measured by reverberation mapping and
stellar dynamics — to the host galaxy through the $M$–$\sigma$ relation.

## The central engine and the Eddington limit

The compactness argument comes first. AGN vary on timescales as short as hours, and a
source cannot vary coherently faster than light crosses it, so the emitting region is
smaller than $R \lesssim c\,\Delta t \sim$ light-hours to light-days — smaller than the
solar system, yet outshining $10^{11}$ stars. Only accretion onto a black hole
liberates enough energy in so small a volume.

Radiation escaping the accreting gas pushes outward on the infalling plasma through
Thomson scattering off electrons, while gravity pulls it in. Balancing the two on a
fully ionized hydrogen plasma gives the **Eddington luminosity**, the maximum steady
luminosity for accretion powered by gravity. The radiative force per electron is
$\sigma_T L/(4\pi r^2 c)$, and the gravitational force on the electron–proton pair is
$G M m_p/r^2$. Setting them equal (the $r^{-2}$ cancels):

$$
L_{\mathrm{Edd}} = \frac{4\pi G M m_p c}{\sigma_T}
= 1.26\times10^{31}\left(\frac{M}{M_\odot}\right)\ \mathrm{W}.
$$

An AGN radiating $10^{40}\ \mathrm{W}$ therefore requires
$M \gtrsim 10^{8}\,M_\odot$: the black hole must be supermassive. Above $L_{\mathrm{Edd}}$
radiation pressure would blow the accretion flow apart, so the Eddington luminosity is
a natural cap and a lower bound on the black-hole mass.

The luminosity comes from the gravitational binding energy released as matter spirals
inward. Matter accreting at rate $\dot M$ radiates

$$
L = \eta\,\dot M\, c^2,
$$

with **radiative efficiency** $\eta$ set by the binding energy at the innermost stable
circular orbit: $\eta \approx 0.057$ for a non-rotating (Schwarzschild) hole and up to
$\eta \approx 0.42$ for a maximally rotating (Kerr) hole, with $\eta \approx 0.1$
typical. Accretion is an order of magnitude more efficient than hydrogen fusion
($\eta_{\mathrm{fusion}} \approx 0.007$), which is why so much power emerges from so
little mass. Producing $10^{40}\ \mathrm{W}$ at $\eta = 0.1$ consumes
$\dot M \approx 2\,M_\odot\,\mathrm{yr^{-1}}$.

$$
% caption: Energy budget of the central engine: gas spirals through an accretion disk
% releasing binding energy as radiation with efficiency near ten percent, while a
% fraction is channeled into relativistic jets along the spin axis.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% black hole
\fill[black!75] (4.2,0) circle (0.3);
\node[black!70, anchor=north] at (4.2,-0.4) {black hole};
% accretion disk (edge-on)
\fill[acc!16] (4.2,0) ellipse (2.6 and 0.5);
\draw[acc, very thick] (4.2,0) ellipse (2.6 and 0.5);
\draw[acc, very thick] (1.6,0) arc (180:360:2.6 and 0.5);
\node[acc, anchor=north] at (6.4,0.35) {accretion disk};
% infall arrows
\draw[->, black] (7.2,0.2) -- (6.6,0.1);
\draw[->, black] (1.2,-0.2) -- (1.8,-0.1);
% jets along axis
\draw[very thick, ->] (4.2,0.4) -- (4.2,2.7);
\draw[very thick, ->] (4.2,-0.4) -- (4.2,-2.7);
\node[black, anchor=west] at (4.35,2.2) {relativistic jet};
% radiation
\foreach \a in {20,45,135,160} \draw[->, black] (4.2,0) -- ($(4.2,0)+(\a:1.6)$);
\node[black, anchor=south west] at (5.4,1.0) {radiation};
\end{tikzpicture}
$$

## The AGN zoo and the unified model

AGN come in observationally distinct classes:

- **Seyfert galaxies.** Spiral hosts with bright point-like nuclei. Type 1 show both
  broad and narrow emission lines; type 2 show only narrow lines.
- **Quasars (QSOs).** The most luminous AGN, so bright the host is hard to see, with
  broad lines and (in radio-loud cases) jets. Common at high redshift, marking the
  peak epoch of black-hole growth.
- **Radio galaxies.** Elliptical hosts with powerful double-lobed radio emission fed
  by jets, again split into broad-line and narrow-line varieties.
- **Blazars (BL Lac objects and OVV quasars).** Rapidly variable, highly polarized,
  with a nearly featureless continuum and apparent superluminal motion.

The **unified model** attributes this diversity largely to **orientation** of a single
axisymmetric structure. Surrounding the accretion disk is a dusty, molecular **torus**;
close to the hole, fast-moving gas emits the **broad-line region** (BLR) lines
(velocity widths of thousands of $\mathrm{km\,s^{-1}}$); farther out, slower gas emits
the **narrow-line region** (NLR) lines. Viewing angle then determines the class:

- **Edge-on**, the torus hides the disk and BLR, leaving only the narrow lines — a
  type 2 Seyfert or narrow-line radio galaxy.
- **Intermediate**, the BLR is visible over the torus — a type 1 Seyfert or quasar.
- **Down the jet**, relativistic beaming dominates and the continuum swamps the lines
  — a blazar.

Radio-loud versus radio-quiet is a second axis, correlated with host type and possibly
black-hole spin, not with orientation.

$$
% caption: The unified model: an accretion disk and dusty torus around the black hole
% with a broad-line region inside and a narrow-line region outside; the observed AGN
% class depends on whether the line of sight clears the torus or looks down the jet.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% black hole and disk
\fill[black!75] (0,0) circle (0.22);
\draw[very thick] (0,0) ellipse (1.2 and 0.28);
% torus (two lobes)
\fill[black] (1.9,0) ellipse (0.75 and 0.9);
\draw[black, thick] (1.9,0) ellipse (0.75 and 0.9);
\fill[black] (-1.9,0) ellipse (0.75 and 0.9);
\draw[black, thick] (-1.9,0) ellipse (0.75 and 0.9);
\node[black, anchor=north] at (0,-1.2) {dusty torus};
% BLR / NLR labels
\node[black, anchor=south] at (0,0.35) {BLR};
\node[black, anchor=south] at (3.4,0.4) {NLR};
\fill[black] (3.2,0.2) circle (1pt);
\fill[black] (3.5,-0.2) circle (1pt);
\fill[black] (-3.2,0.2) circle (1pt);
% jet
\draw[acc, very thick, ->] (0,0.28) -- (0,2.4);
\draw[acc, very thick, ->] (0,-0.28) -- (0,-2.4);
% sightlines
\draw[->, black] (5.2,2.2) -- (0.6,0.35);
\node[black, anchor=west] at (5.3,2.2) {type 1 (over torus)};
\draw[->, black] (5.2,0) -- (2.7,0);
\node[black, anchor=west] at (5.3,0) {type 2 (through torus)};
\draw[->, black] (1.2,3.0) -- (0.15,2.3);
\node[acc, anchor=west] at (1.3,3.0) {blazar (down jet)};
\end{tikzpicture}
$$

## Relativistic jets and superluminal motion

Radio jets show blobs that appear to move across the sky **faster than light**. This is
a projection effect for material moving at nearly $c$ almost toward the observer. A
blob moving at speed $v = \beta c$ at angle $\theta$ to the line of sight has an
apparent transverse velocity

$$
v_{\mathrm{app}} = \frac{v\sin\theta}{1 - \beta\cos\theta}.
$$

The denominator shrinks because the blob nearly keeps pace with the light it emits, so
successive signals arrive bunched in time and the motion looks fast. Maximizing over
$\theta$ (setting $\cos\theta = \beta$) gives

$$
v_{\mathrm{app}}^{\max} = \gamma\beta c,
\qquad \gamma = (1 - \beta^2)^{-1/2},
$$

which exceeds $c$ whenever $\gamma\beta > 1$, i.e. $\beta > 1/\sqrt2$. No signal travels
faster than light; the apparent superluminal speed is geometry plus light-travel-time
compression, and it directly diagnoses bulk Lorentz factors $\gamma \sim 5$–$30$ in the
jets.

> **Worked example.** A blob with $\gamma = 10$ ($\beta = 0.995$) at the optimal angle
> $\cos\theta = \beta$, i.e. $\theta \approx 5.7^\circ$, shows
> $v_{\mathrm{app}} = \gamma\beta c \approx 9.95\,c$: an apparent motion of ten times
> the speed of light from strictly subluminal gas pointed nearly at us.

$$
% caption: Superluminal geometry: a blob moving at nearly the speed of light at a
% small angle to the line of sight nearly overtakes its own emitted light, compressing
% the arrival times so its projected motion across the sky appears to exceed c.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% observer at right
\fill[black] (9.0,0.4) circle (1.6pt);
\node[black, anchor=west] at (9.1,0.4) {to observer};
% source
\fill[black] (0.6,0.4) circle (2pt);
\node[black, anchor=east] at (0.5,0.4) {core};
% blob path at small angle to sightline
\draw[acc, very thick, ->] (0.6,0.4) -- (6.6,1.9);
\node[acc, anchor=south] at (3.6,1.35) {blob at nearly c};
% line of sight
\draw[black, densely dashed] (0.6,0.4) -- (9.0,0.4);
% angle marker
\draw[black] (1.6,0.4) arc (0:14:1.0);
\node[black, anchor=west] at (1.7,0.62) {small angle};
% projected transverse motion
\draw[<->, black] (6.6,1.9) -- (6.6,0.4);
\node[black, anchor=west] at (6.7,1.2) {projected motion};
\end{tikzpicture}
$$

## Weighing the black hole

Two methods measure the central mass. **Reverberation mapping** uses the continuum's
variability. The accretion-disk continuum flickers, and the broad emission lines
respond after a light-travel delay $\tau$ as the ionizing flash reaches the BLR at
radius $R_{\mathrm{BLR}} = c\,\tau$. Combining that radius with the line width
$\Delta V$ (the virial velocity of BLR gas) gives

$$
M_\bullet = f\,\frac{R_{\mathrm{BLR}}\,(\Delta V)^2}{G},
$$

where $f \sim 1$ is a geometric factor for the unknown BLR structure. The method reaches
distant AGN because it needs only timing and spectroscopy, not spatial resolution. For
the nearest galaxies, **resolved stellar and gas dynamics** — the rise of the velocity
dispersion toward the center — weigh the hole directly, as the S-star orbits do for
[Sgr A$^\ast$](/astrophysics-cosmology/galaxies/the-milky-way).

## The M–sigma relation and co-evolution

Black-hole masses across galaxies are not random. The mass correlates tightly with the
**velocity dispersion** $\sigma$ of the host's bulge,

$$
M_\bullet \approx 2\times10^{8}\,M_\odot\left(\frac{\sigma}{200\ \mathrm{km\,s^{-1}}}\right)^{4},
$$

the **$M$–$\sigma$ relation**, with a slope near $4$–$5$ and scatter small enough to look fundamental.
The black hole knows about the bulge even though its sphere of gravitational influence
is a tiny fraction of the galaxy. The link points to **co-evolution**: AGN feedback —
the energy and momentum the accreting hole deposits in the surrounding gas through
radiation and jets — regulates star formation and gas retention in the bulge, tying
the two growths together. A hole that grows too large drives out the gas that feeds it
and forms stars, self-limiting at the observed $M_\bullet$–$\sigma$ ratio. The relation
is a central clue that supermassive black holes and their host galaxies build up in
concert.

$$
% caption: The M-sigma relation: black-hole mass rises as roughly the fourth power of
% the host bulge velocity dispersion, a tight correlation across galaxy types pointing
% to feedback-regulated co-evolution.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (8.8,0) node[right, black!70] {log dispersion};
\draw[->, black] (0,0) -- (0,4.6) node[above, black!70] {log black-hole mass};
% slope-4 relation
\draw[acc, very thick] (0.9,0.5) -- (7.8,4.2);
% data with scatter
\foreach \x/\y in {1.4/0.9, 2.2/1.15, 2.8/1.8, 3.6/1.95, 4.2/2.6, 5.0/2.75, 5.8/3.4, 6.6/3.7} {
  \fill[black] (\x,\y) circle (1.5pt);
}
\node[acc, anchor=north west] at (4.2,2.5) {M proportional to sigma to the fourth};
\end{tikzpicture}
$$

## Summary

AGN are powered by accretion onto supermassive black holes: the compactness set by
variability plus the Eddington luminosity
$L_{\mathrm{Edd}} = 1.26\times10^{31}(M/M_\odot)\ \mathrm{W}$ demand
$M \gtrsim 10^{8}\,M_\odot$, and the luminosity $L = \eta\dot M c^2$ with
$\eta \approx 0.1$ makes accretion an order of magnitude more efficient than fusion.
The unified model explains Seyferts, quasars, radio galaxies, and blazars as one engine
— disk, torus, broad- and narrow-line regions, jet — seen at different orientations.
Relativistic jets produce apparent superluminal motion $v_{\mathrm{app}} = v\sin\theta/(1-\beta\cos\theta)$,
peaking at $\gamma\beta c$, diagnosing $\gamma \sim 5$–$30$. Reverberation mapping
($M_\bullet = f R_{\mathrm{BLR}}\Delta V^2/G$) and stellar dynamics weigh the hole, and
the $M$–$\sigma$ relation $M_\bullet \propto \sigma^4$ ties it to the host bulge through
feedback-driven co-evolution.[^co-agn][^maoz-agn]

[^co-agn]: Carroll & Ostlie, §28.1–28.2 — active-galaxy observations, the Eddington limit, superluminal motion, and the unified model of AGN.
[^maoz-agn]: Maoz, Ch. 7 — active galactic nuclei: accretion power, the AGN classes, reverberation mapping, and the black-hole–bulge relation.