Part IV

Accretion Disks

Infalling matter with angular momentum cannot plunge directly into a black hole; it forms a disk that dissipates angular momentum outward through viscosity and radiates binding energy as thermal + non-thermal photons. Shakura-Sunyaev 1973 thin-disk theory underpins quasar observations; advection-dominated accretion flows (ADAFs) cover the low-luminosity regime.

Shakura-Sunyaev Thin Disk

For a thin Keplerian disk with α-viscosity parametrisation, the local dissipation at radius r produces a temperature:

\[ T(r) \;=\; \left(\frac{3 G M \dot M}{8\pi r^3 \sigma}\right)^{1/4}\left(1 - \sqrt{r_{in}/r}\right)^{1/4} \]

The r^-3/4 power law dominates at large r; the inner-edge correction kills T at r → r_in. Spectral energy distribution is a sum of Planck blackbodies weighted by disk area — a “multicolour blackbody” peaked at keV for stellar BHs and UV for supermassive BHs.

Eddington Luminosity

Radiation pressure from accretion balances gravitational pull at the Eddington limit:

\[ L_{Edd} \;=\; \frac{4\pi G M m_p c}{\sigma_T} \;\approx\; 1.26\times 10^{31}\,\left(\frac{M}{M_\odot}\right)\ \text{W} \]

Super-Eddington accretion produces radiation-driven outflows (UFOs, broad-line-region winds) and is common in tidal disruption events.

Simulation: L_Edd & Disk Temperature

Python

script.py42 lines

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

# Shakura-Sunyaev thin disk: T(r) ~ r^-3/4
# Edddington luminosity limit
M = np.logspace(0, 10, 200)     # solar masses
L_edd = 1.26e31 * M              # Watts

# Disk temperature profile
r_rs = np.logspace(0, 3, 200)
T_SS = 1e7 * r_rs**(-0.75)

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5), facecolor='#0a0a1a')
for ax in (ax1, ax2):
    ax.set_facecolor('#111827'); ax.tick_params(colors='#cbd5e1')
    for s in ax.spines.values(): s.set_color('#334155')
    ax.grid(True, color='#334155', alpha=0.3, which='both')

ax1.loglog(M, L_edd, color='#fbbf24', lw=2.6)
ax1.plot(10, 1e32, 'o', color='#f87171', markersize=10, label='Cyg X-1 ~10 Msun')
ax1.plot(4e6, 5e37, 'o', color='#34d399', markersize=10, label='Sgr A* ~4e6 Msun')
ax1.plot(6e9, 8e40, 'o', color='#a78bfa', markersize=10, label='M87* ~6e9 Msun')
ax1.set_xlabel('BH mass (Msun)', color='#cbd5e1')
ax1.set_ylabel('Eddington luminosity (W)', color='#cbd5e1')
ax1.set_title('L_Edd scales linearly with BH mass',
              color='#fde68a', fontweight='bold')
ax1.legend(facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')

ax2.loglog(r_rs, T_SS, color='#f472b6', lw=2.6)
ax2.axhline(1e4, color='#60a5fa', ls='--', label='UV 10 eV')
ax2.axhline(1e7, color='#fbbf24', ls='--', label='Soft X-ray keV')
ax2.set_xlabel('r / r_s', color='#cbd5e1')
ax2.set_ylabel('Temperature (K)', color='#cbd5e1')
ax2.set_title('Shakura-Sunyaev T(r) ~ r^(-3/4)',
              color='#fde68a', fontweight='bold')
ax2.legend(facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')

plt.tight_layout()
plt.savefig('output.png', dpi=120, bbox_inches='tight', facecolor='#0a0a1a')
print('Shakura-Sunyaev 1973: thin disk with alpha-viscosity')
print('Inner disk T ~ 10^7 K (stellar BH) to ~10^5 K (supermassive)')

Click Run to execute the Python code

Code will be executed with Python 3 on the server

ADAF & RIAF Regimes

At low accretion rates (<0.01 L_Edd), the gas becomes radiatively inefficient: advected thermal energy dominates over radiated energy. This is the regime of Sgr A* and M87* (EHT targets). Radiatively- inefficient accretion flows (RIAFs) are two-temperature plasmas with T_electron ≈ 10⁹ K and T_ion ≈ 10¹² K, producing synchrotron radiation that the EHT images.

Key References

• Shakura, N. I. & Sunyaev, R. A. (1973). “Black holes in binary systems.” Astron. Astrophys., 24, 337–355.

• Narayan, R. & Yi, I. (1994). “Advection-dominated accretion.” Astrophys. J., 428, L13–L16.

• Yuan, F. & Narayan, R. (2014). “Hot accretion flows around black holes.” Annu. Rev. Astron. Astrophys., 52, 529–588.

• Frank, J., King, A. & Raine, D. (2002). Accretion Power in Astrophysics, 3rd ed.

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