Part III: Galaxies & Cosmology | Chapter 2

Active Galactic Nuclei

Supermassive black holes as cosmic engines: the unified model, accretion disk physics, the Eddington luminosity, and relativistic jet formation

Overview

Active galactic nuclei (AGN) are the luminous cores of galaxies powered by accretion onto supermassive black holes (SMBHs) with masses $10^6\text{--}10^{10}\,M_\odot$. They are among the most luminous persistent objects in the Universe, with bolometric luminosities reaching $10^{48}$ erg/s (for the brightest quasars), outshining their entire host galaxy by factors of 100 or more. AGN phenomena include Seyfert galaxies, quasars, radio galaxies, blazars, and LINERs — a zoo of observational classes that the unified model explains through viewing angle effects.

In this chapter we derive the Eddington luminosity, develop the standard thin accretion disk model (Shakura-Sunyaev), explain the unified AGN model, derive the radiative efficiency of accretion, and analyze the physics of relativistic jet formation.

1. The Eddington Luminosity

The Eddington luminosity is the maximum luminosity at which a source can shine while maintaining hydrostatic equilibrium against radiation pressure.

1.1 Derivation from Force Balance

Consider a fully ionized hydrogen gas element at distance $r$ from a luminous source of luminosity $L$. The radiation force per electron (via Thomson scattering) acts outward:

$$F_{\text{rad}} = \frac{\sigma_T L}{4\pi r^2 c}$$

The gravitational force per electron-proton pair (treating the electron as coupled to the proton) acts inward:

$$F_{\text{grav}} = \frac{GM m_p}{r^2}$$

Setting $F_{\text{rad}} = F_{\text{grav}}$ and solving for $L$:

$$\boxed{L_{\text{Edd}} = \frac{4\pi G M m_p c}{\sigma_T} \approx 1.26 \times 10^{38}\left(\frac{M}{M_\odot}\right)\;\text{erg/s} \approx 3.3 \times 10^4\left(\frac{M}{M_\odot}\right)\;L_\odot}$$

For a $10^8\,M_\odot$ SMBH, $L_{\text{Edd}} \approx 1.3 \times 10^{46}$ erg/s, comparable to the luminosity of a bright quasar. The Eddington ratio $\lambda_{\text{Edd}} = L/L_{\text{Edd}}$characterizes the accretion state: $\lambda_{\text{Edd}} \sim 0.01\text{--}1$for typical AGN, with super-Eddington accretion possible in some systems.

1.2 The Salpeter Time

If a black hole accretes at the Eddington rate with radiative efficiency $\eta$, its mass grows exponentially:

$$M(t) = M_0\,\exp\left(\frac{t}{t_{\text{Sal}}}\right), \qquad t_{\text{Sal}} = \frac{\eta \sigma_T c}{4\pi G m_p} \approx 4.5 \times 10^8\left(\frac{\eta}{0.1}\right)\;\text{yr}$$

The Salpeter time $t_{\text{Sal}} \approx 45$ Myr (for $\eta = 0.1$) sets the e-folding timescale for black hole growth. Growing a $10^9\,M_\odot$SMBH from a $100\,M_\odot$ seed by $z = 7$requires $\sim 16$ e-foldings, or $\sim 720$ Myr of continuous Eddington-rate accretion — a significant challenge for early Universe black hole growth.

2. Accretion Disk Physics

The enormous luminosity of AGN is powered by the gravitational potential energy released as matter spirals inward through an accretion disk.

2.1 Radiative Efficiency

The maximum energy extractable from accretion is set by the binding energy at the innermost stable circular orbit (ISCO). For a Schwarzschild black hole, the ISCO is at $r_{\text{ISCO}} = 6GM/c^2 = 3r_s$, where $r_s = 2GM/c^2$is the Schwarzschild radius. The specific binding energy at the ISCO gives:

$$\boxed{\eta = 1 - \sqrt{1 - \frac{2}{3}} \approx 0.057 \;\;\text{(Schwarzschild)}}$$

For a maximally spinning Kerr black hole ($a = 1$), the ISCO shrinks to $r_{\text{ISCO}} = GM/c^2$ and:

$$\eta \approx 0.42 \;\;\text{(maximal Kerr)}$$

This makes accretion onto black holes the most efficient energy conversion process in nature, far exceeding nuclear fusion ($\eta \approx 0.007$).

2.2 The Shakura-Sunyaev Disk

The standard thin disk model (Shakura & Sunyaev 1973) assumes a geometrically thin, optically thick disk in which viscous torques transport angular momentum outward. The effective temperature profile is:

$$\boxed{T(r) = \left(\frac{3GM\dot{M}}{8\pi\sigma_{\text{SB}} r^3}\right)^{1/4}\left(1 - \sqrt{\frac{r_{\text{ISCO}}}{r}}\right)^{1/4}}$$

The peak temperature occurs at $r \approx 1.36\,r_{\text{ISCO}}$ and scales as:

$$T_{\text{max}} \approx 6.3 \times 10^5\left(\frac{M}{10^8\,M_\odot}\right)^{-1/4}\left(\frac{\dot{M}}{\dot{M}_{\text{Edd}}}\right)^{1/4}\;\text{K}$$

For stellar-mass black holes, $T_{\text{max}} \sim 10^7$ K (X-ray). For SMBHs, $T_{\text{max}} \sim 10^5$ K (UV), explaining why quasars are powerful ultraviolet sources and produce the "big blue bump" in their spectral energy distribution.

3. The Unified AGN Model

The remarkable diversity of AGN observational classes can be explained primarily by viewing angle relative to an obscuring torus of gas and dust surrounding the central engine.

3.1 The Central Engine Components

The unified model (Antonucci 1993; Urry & Padovani 1995) identifies several key structural components:

Central SMBH and accretion disk: The primary energy source, producing thermal UV/optical emission.

Broad-line region (BLR): Clouds at$r \sim 0.01\text{--}0.1$ pc moving at$v \sim 3000\text{--}10{,}000$ km/s, producing Doppler-broadened emission lines.

Dusty torus: A geometrically thick structure at $r \sim 0.1\text{--}10$ pc that obscures the central engine and BLR when viewed edge-on.

Narrow-line region (NLR): Clouds at$r \sim 100\text{--}1000$ pc with $v \sim 300\text{--}1000$ km/s, visible from all angles.

3.2 Viewing Angle Classification

The classification depends on orientation:

Face-on (Type 1): Direct view of the accretion disk and BLR. Appears as a Seyfert 1 galaxy or quasar, with broad emission lines and a strong UV/X-ray continuum.

Edge-on (Type 2): The torus blocks the BLR and continuum source. Appears as a Seyfert 2 galaxy, with only narrow emission lines visible. Scattered broad lines may be seen in polarized light.

Jet-on (blazars): When a relativistic jet points toward the observer, the Doppler-boosted jet emission dominates, producing rapid variability, high polarization, and apparent superluminal motion.

4. Black Hole Mass Estimation

Measuring SMBH masses is crucial for understanding AGN physics and the co-evolution of black holes and galaxies.

4.1 Reverberation Mapping

Reverberation mapping exploits the light-travel time delay between continuum variations (from the accretion disk) and the response of broad emission lines (from the BLR). The time lag $\tau$ gives the BLR size $R_{\text{BLR}} = c\tau$. Combined with the emission line velocity dispersion $\sigma_{\text{line}}$ or FWHM, the virial mass is:

$$\boxed{M_{\text{BH}} = f\,\frac{R_{\text{BLR}}\,v^2}{G} = f\,\frac{c\tau\,\sigma_{\text{line}}^2}{G}}$$

where $f \approx 4\text{--}5$ is a geometric factor calibrated against independent mass measurements. The BLR size-luminosity relation $R_{\text{BLR}} \propto L^{0.5}$ allows single-epoch mass estimates for large AGN samples.

4.2 The M-sigma Relation

A remarkably tight correlation exists between the SMBH mass and the stellar velocity dispersion of the host galaxy bulge:

$$\boxed{M_{\text{BH}} \approx 1.9 \times 10^8\left(\frac{\sigma}{200\;\text{km/s}}\right)^{4.4}\;M_\odot}$$

This implies a fundamental connection between SMBH growth and galaxy evolution, mediated by AGN feedback that regulates star formation in the host galaxy.

5. Relativistic Jet Formation

About 10% of AGN produce powerful collimated jets of relativistic plasma extending from parsec to megaparsec scales.

5.1 The Blandford-Znajek Mechanism

The leading model for jet launching is the Blandford-Znajek (1977) mechanism, which extracts rotational energy from a spinning black hole through magnetic field lines threading the ergosphere. The jet power is:

$$\boxed{P_{\text{BZ}} \approx \frac{1}{32}\frac{\Phi_B^2 \Omega_H^2}{c} \propto a^2 M^2 B^2}$$

where $\Phi_B$ is the magnetic flux threading the horizon, $\Omega_H = ac/(2r_+)$is the angular velocity of the horizon, and $a = J/(Mc)$ is the dimensionless spin parameter. The maximum extractable energy is 29% of the black hole rest mass energy for $a = 1$, far exceeding the accretion luminosity.

5.2 Doppler Boosting

For a jet moving at angle $\theta$ to the line of sight with Lorentz factor $\Gamma$, the Doppler factor is:

$$\delta = \frac{1}{\Gamma(1 - \beta\cos\theta)}$$

The observed flux is boosted by $\delta^{3+\alpha}$ (for a continuous jet) where $\alpha$ is the spectral index. For $\Gamma = 10$and $\theta = 5°$, $\delta \approx 10$, boosting the observed flux by a factor of $\sim 10^4$.

Applications

AGN Feedback and Galaxy Quenching

AGN feedback is now recognized as a crucial ingredient in galaxy evolution. In the "quasar mode" (radiative feedback), radiation pressure and winds from the AGN can expel gas from the galaxy, shutting down star formation. In the "radio mode" (kinetic feedback), relativistic jets heat the intracluster medium, preventing cooling flows in galaxy clusters. AGN feedback is essential for reproducing the observed galaxy luminosity function and the bimodality of galaxy colors (red sequence vs blue cloud).

The Event Horizon Telescope

The Event Horizon Telescope (EHT) achieved the first direct images of SMBH shadows: M87* in 2019 ($M \approx 6.5 \times 10^9\,M_\odot$) and Sgr A* in 2022 ($M \approx 4 \times 10^6\,M_\odot$). These images resolve structures at the event horizon scale and confirm predictions of general relativity, including the photon ring and the shadow size.

Tidal Disruption Events

When a star passes within the tidal radius $r_t = R_*(M_{\text{BH}}/M_*)^{1/3}$of a supermassive black hole, it is torn apart by tidal forces. About half the stellar debris falls back onto the black hole, producing a luminous flare that rises on a timescale of weeks and decays as $t^{-5/3}$ (the fallback rate). TDEs provide a way to detect otherwise quiescent SMBHs in normal galaxies and probe the demographics of black holes in the mass range $10^5\text{--}10^8\,M_\odot$. The Zwicky Transient Facility and LSST are expected to discover thousands of TDEs.

AGN Variability and Time-Domain Science

AGN are variable at all wavelengths and on all timescales from minutes to decades. The variability amplitude is described by the structure function or power spectral density, which typically follows a damped random walk (DRW) model with characteristic timescale$\tau \sim 100\text{--}1000$ days correlated with black hole mass. Continuum reverberation mapping uses the correlated variability between different wavelength bands to map the accretion disk temperature profile, testing the $T \propto r^{-3/4}$prediction of the Shakura-Sunyaev model. Changing-look AGN — objects that transition between Type 1 and Type 2 on timescales of years — challenge the static unified model and suggest that the accretion rate itself varies dramatically.

AGN as Cosmic Beacons

Quasar absorption line spectroscopy probes the intergalactic and circumgalactic medium along the line of sight. Damped Lyman-alpha systems (DLAs, with$N_{\text{HI}} > 2 \times 10^{20}$ cm$^{-2}$) trace the neutral gas reservoirs of galaxies at all redshifts. The metallicity evolution of DLAs maps the chemical enrichment history of the Universe. Metal absorption lines (C IV, Mg II, O VI) trace the circumgalactic medium and galactic winds out to hundreds of kpc from galaxies. The proximity effect — the deficit of Lyman-alpha absorption near quasars — measures the UV background intensity and quasar ionizing luminosity.

Quasar Demographics and the Luminosity Function

The quasar luminosity function (QLF) describes the comoving number density of AGN as a function of luminosity and redshift. It follows a double power law with a characteristic break luminosity $L_*$ that evolves with redshift. The total AGN emissivity peaks at $z \sim 2$ and declines by a factor of $\sim 100$ to the present day. Integrating the QLF over cosmic time yields the total accreted mass density $\rho_{\text{BH}} \sim 4 \times 10^5\,M_\odot$ Mpc$^{-3}$, consistent with the local SMBH mass function measured from the M-sigma relation — the Soltan argument. This consistency confirms that SMBHs grew primarily by radiatively efficient accretion during luminous AGN phases.

Historical Notes

Carl Seyfert identified the first AGN in 1943 as galaxies with unusually bright, broad-lined nuclei. The discovery of quasars began with the identification of radio source 3C 273 by Maarten Schmidt in 1963, who recognized the redshifted Balmer lines implying a cosmological distance and enormous luminosity. Lynden-Bell (1969) proposed that quasars are powered by accretion onto SMBHs. The Shakura-Sunyaev disk model (1973) provided the quantitative framework for accretion physics. The unified model was crystallized by Antonucci and Miller (1985), who discovered broad emission lines in the polarized spectrum of the Seyfert 2 galaxy NGC 1068, proving the presence of a hidden BLR. The M-sigma relation was discovered independently by Ferrarese & Merritt and Gebhardt et al. in 2000, revolutionizing our understanding of the black hole–galaxy connection.

The era of SMBH imaging began with the Event Horizon Telescope's first image of the shadow of M87* in April 2019, followed by the image of Sgr A* in May 2022. These observations resolved structures at scales of a few Schwarzschild radii, directly confirming the existence of event horizons and testing the Kerr metric predictions of general relativity. Meanwhile, JWST has revealed luminous AGN at redshifts$z > 10$, just a few hundred million years after the Big Bang, posing severe challenges for models of early SMBH formation and growth.

Ultra-Fast Outflows and AGN Winds

High-resolution X-ray spectroscopy has revealed ultra-fast outflows (UFOs) in many AGN, with velocities reaching $v \sim 0.1\text{--}0.4\,c$. These winds, detected through blue-shifted Fe XXV and Fe XXVI absorption lines, carry kinetic luminosities of $\sim 5\text{--}20\%$ of the bolometric luminosity — sufficient to drive AGN feedback and regulate star formation in the host galaxy. The mechanical energy injection rate is:

$\dot{E}_{\text{kin}} = \frac{1}{2}\dot{M}_w v_w^2 \approx 10^{45}(\dot{M}_w/M_\odot\,\text{yr}^{-1})(v_w/0.1c)^2$ erg/s

When these winds shock against the interstellar medium of the host galaxy, they can sweep up gas and drive large-scale molecular outflows observed at millimeter wavelengths (with ALMA), removing the fuel for star formation. This provides the physical mechanism connecting SMBH growth to galaxy quenching as demanded by the M-sigma relation.

The Cosmic Evolution of AGN

The quasar luminosity function peaks at $z \sim 2\text{--}3$ (the epoch of "cosmic noon"), coinciding with the peak of cosmic star formation. The number density of luminous quasars has declined by a factor of $\sim 1000$from $z = 2$ to the present, a phenomenon called "AGN downsizing": the most luminous AGN were most common at earlier epochs. This anti-hierarchical behavior (in contrast to the hierarchical assembly of dark matter halos) reflects the declining gas supply available for accretion as galaxies consume and expel their gas reservoirs. JWST has detected luminous AGN at $z > 10$, raising questions about the formation of their seed black holes within the first few hundred million years after the Big Bang.

Accretion Modes and State Transitions

The structure of the accretion flow depends critically on the Eddington ratio $\lambda_{\text{Edd}} = L/L_{\text{Edd}}$:

High accretion rate ($\lambda_{\text{Edd}} > 0.01$): The standard Shakura-Sunyaev thin disk applies, producing thermal UV/optical emission and the classic Type 1 AGN spectrum. At super-Eddington rates ($\lambda_{\text{Edd}} > 1$), the disk puffs up due to radiation pressure, forming a "slim disk" with advective energy transport and photon trapping.

Low accretion rate ($\lambda_{\text{Edd}} < 0.01$): The flow transitions to a radiatively inefficient accretion flow (RIAF), where the gas is geometrically thick and optically thin. The ions and electrons are thermally decoupled, with ion temperatures reaching $\sim 10^{12}$ K. RIAFs produce hard X-ray emission and are associated with radio-loud AGN and jet production. Most nearby SMBHs, including Sgr A* in the Milky Way center, are in this low-luminosity state.

The disk-jet connection: Observations suggest that jets are preferentially launched from SMBHs accreting in the RIAF mode, possibly because the geometrically thick flow more effectively threads magnetic flux through the black hole horizon. This connection explains the observation that radio-loud AGN tend to reside in massive elliptical galaxies with low star formation rates and correspondingly low gas supply rates.

Computational Exploration

The following simulation computes the accretion disk temperature profile, models the spectral energy distribution, illustrates the M-sigma relation, and demonstrates Doppler boosting of relativistic jets.

Accretion Disk Spectra, M-sigma Relation, and Jet Doppler Boosting

Python

script.py219 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

fig, axes = plt.subplots(2, 2, figsize=(14, 11))
fig.patch.set_facecolor('#0a0a0a')
for ax in axes.flat:
    ax.set_facecolor('#0f0f0f')
    ax.tick_params(colors='white')
    ax.xaxis.label.set_color('white')
    ax.yaxis.label.set_color('white')
    ax.title.set_color('white')
    for spine in ax.spines.values():
        spine.set_color('#333')

# Constants
G = 6.674e-11
c = 3e8
M_sun = 1.989e30
sigma_SB = 5.67e-8
sigma_T = 6.65e-29
m_p = 1.673e-27
h = 6.626e-34
k_B = 1.381e-23
L_sun = 3.828e26

# --- Plot 1: Accretion disk temperature and spectrum ---
ax1 = axes[0, 0]

masses_bh = [10, 1e6, 1e8, 1e10]  # solar masses
colors_m = ['#ff4444', '#ffa500', '#ffdd00', '#44bbff']
labels_m = ['10 M_sun (XRB)', '10^6 M_sun', '10^8 M_sun', '10^10 M_sun']

for M_val, col, lab in zip(masses_bh, colors_m, labels_m):
    M = M_val * M_sun
    r_s = 2 * G * M / c**2
    r_isco = 3 * r_s  # Schwarzschild

# Eddington accretion rate
    L_edd = 4 * np.pi * G * M * m_p * c / sigma_T
    eta = 0.1
    Mdot = L_edd / (eta * c**2)

r_range = np.logspace(np.log10(r_isco), np.log10(1e4 * r_isco), 200)

T_disk = np.zeros_like(r_range)
    for i, r in enumerate(r_range):
        factor = 1 - np.sqrt(r_isco / r)
        if factor > 0:
            T_disk[i] = (3 * G * M * Mdot / (8 * np.pi * sigma_SB * r**3) * factor)**0.25

# Plot T vs r/r_s
    r_normalized = r_range / r_s
    valid = T_disk > 0
    ax1.loglog(r_normalized[valid], T_disk[valid], color=col, linewidth=2.5, label=lab)

ax1.set_xlabel('r / r_s', fontsize=12)
ax1.set_ylabel('Temperature (K)', fontsize=12)
ax1.set_title('Accretion Disk Temperature Profiles', fontsize=14, fontweight='bold')
ax1.legend(fontsize=9, facecolor='#1a1a1a', edgecolor='#444', labelcolor='white')
ax1.set_xlim(1, 1e4)
ax1.set_ylim(1e2, 1e8)
ax1.grid(True, alpha=0.15)

# Mark spectral regions
ax1.axhspan(1e7, 1e8, alpha=0.05, color='#ff4444')
ax1.text(3, 3e7, 'X-ray', color='#ff6644', fontsize=9)
ax1.axhspan(1e4, 1e6, alpha=0.05, color='#44bbff')
ax1.text(3, 1e5, 'UV/Optical', color='#44aaff', fontsize=9)

# --- Plot 2: M-sigma relation ---
ax2 = axes[0, 1]

# Generate synthetic M-sigma data
rng = np.random.default_rng(42)
N_gal = 80
log_sigma = rng.uniform(1.7, 2.6, N_gal)  # km/s
# M-sigma: log(M/Msun) = 8.13 + 4.02*log(sigma/200)
log_M_true = 8.13 + 4.38 * (log_sigma - np.log10(200))
scatter = rng.normal(0, 0.3, N_gal)
log_M_obs = log_M_true + scatter

sigma_vals = 10**log_sigma
M_vals = 10**log_M_obs

# Color by type
elliptical = sigma_vals > 180
spiral = ~elliptical

ax2.scatter(sigma_vals[elliptical], M_vals[elliptical], s=40, color='#ff4444', alpha=0.7,
            edgecolor='white', linewidth=0.3, label='Elliptical', zorder=5)
ax2.scatter(sigma_vals[spiral], M_vals[spiral], s=30, color='#44bbff', alpha=0.7,
            edgecolor='white', linewidth=0.3, label='Spiral', zorder=5)

# Best fit line
sigma_fit = np.logspace(1.7, 2.6, 100)
M_fit = 10**(8.13 + 4.38 * (np.log10(sigma_fit) - np.log10(200)))
ax2.plot(sigma_fit, M_fit, color='#ffa500', linewidth=3, label=r'$M propto sigma^{4.4}$')

# Mark notable BHs
notable = [('Sgr A*', 100, 4e6), ('M87', 375, 6.5e9), ('NGC 4889', 400, 2e10)]
for name, sig, mass in notable:
    ax2.plot(sig, mass, '*', color='#ffdd00', markersize=15, zorder=10)
    ax2.text(sig * 1.1, mass * 0.5, name, color='#ffdd00', fontsize=9)

ax2.set_xscale('log')
ax2.set_yscale('log')
ax2.set_xlabel('Velocity Dispersion sigma (km/s)', fontsize=12)
ax2.set_ylabel('Black Hole Mass (M_sun)', fontsize=12)
ax2.set_title('The M-sigma Relation', fontsize=14, fontweight='bold')
ax2.legend(fontsize=9, facecolor='#1a1a1a', edgecolor='#444', labelcolor='white')
ax2.set_xlim(50, 500)
ax2.set_ylim(1e5, 5e10)
ax2.grid(True, alpha=0.15)

# --- Plot 3: Doppler boosting ---
ax3 = axes[1, 0]

theta_range = np.linspace(0.1, 180, 500)  # degrees
theta_rad = theta_range * np.pi / 180

Gamma_values = [2, 5, 10, 20]
Gamma_colors = ['#44bbff', '#ffa500', '#ffdd00', '#ff4444']

for Gamma, col in zip(Gamma_values, Gamma_colors):
    beta = np.sqrt(1 - 1/Gamma**2)
    delta = 1 / (Gamma * (1 - beta * np.cos(theta_rad)))
    # Flux boost for continuous jet: delta^(3+alpha), alpha~0.7
    alpha_spec = 0.7
    boost = delta**(3 + alpha_spec)

ax3.semilogy(theta_range, boost, color=col, linewidth=2.5, label=f'Gamma = {Gamma}')

ax3.axhline(y=1.0, color='#888', linestyle='--', linewidth=1, alpha=0.5)
ax3.set_xlabel('Viewing angle theta (degrees)', fontsize=12)
ax3.set_ylabel('Flux boost factor', fontsize=12)
ax3.set_title('Relativistic Doppler Boosting', fontsize=14, fontweight='bold')
ax3.legend(fontsize=10, facecolor='#1a1a1a', edgecolor='#444', labelcolor='white')
ax3.set_xlim(0, 90)
ax3.set_ylim(1e-3, 1e7)
ax3.grid(True, alpha=0.15)

# Annotate blazar zone
ax3.axvspan(0, 10, alpha=0.1, color='#ff4444')
ax3.text(5, 1e6, 'Blazar\nzone', color='#ff6644', fontsize=10, ha='center')

# --- Plot 4: Eddington luminosity and BH growth ---
ax4 = axes[1, 1]

# BH mass growth curves
t_range = np.linspace(0, 1, 500)  # Gyr

M_seeds = [100, 1e4, 1e6]  # solar masses
seed_colors = ['#44bbff', '#ffa500', '#ff4444']
seed_labels = ['100 M_sun seed', '10^4 M_sun seed', '10^6 M_sun seed']

eta_acc = 0.1
t_sal = eta_acc * sigma_T * c / (4 * np.pi * G * m_p)  # seconds
t_sal_Gyr = t_sal / (3.156e16)

for M_seed, col, lab in zip(M_seeds, seed_colors, seed_labels):
    # Growth at Eddington rate
    M_growth = M_seed * np.exp(t_range / t_sal_Gyr)
    ax4.semilogy(t_range, M_growth, color=col, linewidth=2.5, label=lab)

# Sub-Eddington (lambda = 0.3)
    M_sub = M_seed * np.exp(0.3 * t_range / t_sal_Gyr)
    ax4.semilogy(t_range, M_sub, '--', color=col, linewidth=1.5, alpha=0.5)

# Mark observed high-z quasars
ax4.axhline(y=1e9, color='#888', linestyle=':', linewidth=1.5, alpha=0.5)
ax4.text(0.05, 1.5e9, 'Observed z>7 quasars', color='#aaa', fontsize=10)

# Available time from z=30 to z=7
ax4.axvline(x=0.72, color='#ff4444', linestyle=':', linewidth=1.5, alpha=0.5)
ax4.text(0.73, 1e3, 'z=7\n(t~0.72 Gyr)', color='#ff6644', fontsize=9)

ax4.set_xlabel('Time (Gyr)', fontsize=12)
ax4.set_ylabel('Black Hole Mass (M_sun)', fontsize=12)
ax4.set_title('SMBH Mass Growth (Eddington rate)', fontsize=14, fontweight='bold')
ax4.legend(fontsize=9, facecolor='#1a1a1a', edgecolor='#444', labelcolor='white')
ax4.set_xlim(0, 1)
ax4.set_ylim(10, 1e12)
ax4.grid(True, alpha=0.15)

# Print summary
print("=" * 65)
print("ACTIVE GALACTIC NUCLEI: KEY RESULTS")
print("=" * 65)

print("\nEddington luminosities:")
for M_val in [1e6, 1e8, 1e9, 1e10]:
    L_edd = 1.26e38 * M_val
    print(f"  M = {M_val:.0e} M_sun: L_Edd = {L_edd:.2e} erg/s = {L_edd/L_sun:.2e} L_sun")

print(f"\nSalpeter time (eta=0.1): {t_sal_Gyr*1e3:.0f} Myr")

print("\nRadiative efficiencies:")
print(f"  Schwarzschild (a=0): eta = 0.057")
print(f"  Maximal Kerr (a=1): eta = 0.42")
print(f"  Nuclear fusion:      eta = 0.007")

print("\nPeak disk temperatures (Eddington rate):")
for M_val in [10, 1e6, 1e8, 1e10]:
    M = M_val * M_sun
    T_max = 6.3e5 * (M_val / 1e8)**(-0.25)
    band = "X-ray" if T_max > 1e6 else ("UV" if T_max > 1e4 else "Optical")
    print(f"  M = {M_val:.0e} M_sun: T_max = {T_max:.2e} K ({band})")

print("\nM-sigma relation:")
for sigma in [100, 200, 300, 400]:
    M_bh = 10**(8.13 + 4.38 * np.log10(sigma/200))
    print(f"  sigma = {sigma} km/s: M_BH = {M_bh:.2e} M_sun")

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a0a')
print("\nPlot saved to output.png")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Chapter Summary

AGN are powered by accretion onto SMBHs, with the Eddington luminosity $L_{\text{Edd}} \approx 1.3 \times 10^{46}(M/10^8\,M_\odot)$ erg/s setting the maximum luminosity. The Shakura-Sunyaev disk model predicts temperature profiles peaking in the UV for SMBHs, with radiative efficiencies of$6\text{--}42\%$ depending on black hole spin.

The unified model explains the diversity of AGN classes through viewing angle: Type 1 (face-on, broad lines visible), Type 2 (edge-on, obscured), and blazars (jet-on, Doppler boosted).

The M-sigma relation $M_{\text{BH}} \propto \sigma^{4.4}$ reveals a deep connection between SMBH growth and galaxy evolution, mediated by AGN feedback. Relativistic jets, powered by the Blandford-Znajek mechanism, extract black hole spin energy and regulate star formation on galaxy-cluster scales.

← Galaxy Formation Dark Matter Halos →