Active Galactic Nuclei

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 $GM{m}_p/r^2$. Setting them equal (the $r^{-2}$ cancels):

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

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 $\dot{M}\approx 2M_{\odot }\mathrm{y}{\mathrm{r}}^{-1}$.

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 $\mathrm{km}{\mathrm{s}}^{-1}$); 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.

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

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

which exceeds $c$ whenever $\gamma \beta >1$, i.e. $\beta >1/\sqrt{2}$. 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.

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

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${}^{\ast }$.

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,

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.

Summary

AGN are powered by accretion onto supermassive black holes: the compactness set by variability plus the Eddington luminosity $L_{\mathrm{Edd}}=1.26\times 10^{31}(M/M_{\odot })\mathrm{W}$ 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.¹²