Part VII, Chapter 5

Accretion Disks

Shakura-Sunyaev α-disk model, Eddington luminosity, and the magneto-rotational instability

5.1 Accretion Luminosity

Accretion — the gravitational capture of matter — is the most efficient energy source in the universe. When mass Ṁ falls onto an object of mass M and radius R, the gravitational potential energy released per unit time is:

$$\boxed{L_{acc} = \frac{GM\dot{M}}{R} = \eta \dot{M} c^2}$$

where the efficiency η = GM/(Rc²) = r_g/(2R) with r_g = 2GM/c² (Schwarzschild radius).

Efficiency Comparison

• Chemical burning: η ∼ 10⁻¹⁰
• Nuclear fusion (pp chain): η = 0.007
• Accretion onto neutron star: η = GM/(Rc²) ≈ 0.1
• Accretion onto black hole (ISCO): η ≈ 0.06 (Schwarzschild) to 0.42 (extreme Kerr)

5.1.1 Eddington Luminosity

There is a maximum luminosity set by the balance between radiation pressure and gravity.

Derivation of the Eddington Luminosity

Step 1. Consider a fully ionized hydrogen plasma at distance r from a luminous source. The outward radiation force on a single electron (which drags its proton along via Coulomb coupling) is:

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

where σ_T = 6.65 × 10⁻²⁹ m² is the Thomson cross-section.

Step 2. The inward gravitational force on the electron-proton pair (mass ≈ m_p):

$$F_{grav} = \frac{GMm_p}{r^2}$$

Step 3. Setting F_rad = F_grav:

$$\frac{\sigma_T L_{Edd}}{4\pi r^2 c} = \frac{GM m_p}{r^2}$$

The r² cancels (the limit is independent of distance!):

$$\boxed{L_{Edd} = \frac{4\pi G M m_p c}{\sigma_T} \approx 1.3 \times 10^{31} \frac{M}{M_\odot} \text{ W} \approx 3.3 \times 10^4 \frac{M}{M_\odot} L_\odot}$$

The corresponding Eddington accretion rate is:

$$\dot{M}_{Edd} = \frac{L_{Edd}}{\eta c^2} = \frac{4\pi G M m_p}{\eta \sigma_T c} \approx 2.2 \times 10^{-8} \frac{M}{M_\odot} \text{ M}_\odot\text{/yr}$$

for η = 0.1. Accreting faster than Ṁ_Edd drives a radiation-pressure-driven outflow.

5.2 Thin Disk Theory

5.2.1 Angular Momentum Problem

Matter in circular Keplerian orbit at radius r has specific angular momentum:

$$\ell = r v_\phi = r\sqrt{\frac{GM}{r}} = \sqrt{GMr}$$

For matter to accrete inward (decrease r), it must lose angular momentum. Some mechanism must transport angular momentum outward. This is the fundamental problem of accretion disk physics.

5.2.2 Viscous Angular Momentum Transport

Consider a thin disk with surface density Σ = ∫ρ dz. The angular momentum equation is:

$$\frac{\partial}{\partial t}(\Sigma r^2 \Omega) + \frac{1}{r}\frac{\partial}{\partial r}(\Sigma r^3 v_r \Omega) = \frac{1}{r}\frac{\partial}{\partial r}(r^2 \nu \Sigma r \frac{d\Omega}{dr})$$

Combined with mass conservation, this gives the disk diffusion equation:

$$\frac{\partial \Sigma}{\partial t} = \frac{3}{r}\frac{\partial}{\partial r}\left[r^{1/2}\frac{\partial}{\partial r}\left(\nu \Sigma r^{1/2}\right)\right]$$

This is a diffusion equation: the disk spreads on the viscous timescale:

$$t_{visc} = \frac{r^2}{\nu}$$

5.2.3 Steady-State Accretion Rate

For a steady disk (∂Σ/∂t = 0) with Keplerian Ω = √(GM/r³), the mass accretion rate is:

$$\boxed{\dot{M} = 3\pi \nu \Sigma \left[1 - \sqrt{\frac{r_{in}}{r}}\right]}$$

Far from the inner boundary (r ≫ r_in): Ṁ ≈ 3πνΣ. This relates the accretion rate directly to the viscosity and surface density.

5.3 Shakura-Sunyaev α-Disk

Shakura & Sunyaev (1973) parametrized the unknown viscosity as:

$$\boxed{\nu = \alpha c_s H}$$

where α ≤ 1 is a dimensionless parameter, c_s is the sound speed, and H is the disk scale height. This assumes the viscous stress is a fraction α of the thermal pressure: t_rφ = αp.

5.3.1 Disk Temperature Profile

Derivation of Disk Temperature

Step 1. The viscous dissipation rate per unit area (both sides) is:

$$D(r) = \frac{3GM\dot{M}}{4\pi r^3}\left[1 - \sqrt{\frac{r_{in}}{r}}\right]$$

Step 2. For an optically thick disk in local thermal equilibrium, this energy is radiated as blackbody emission from both surfaces:

$$D(r) = 2\sigma_{SB} T_{eff}^4(r)$$

Step 3. Solving for T_eff(r) far from the inner boundary:

$$T_{eff}(r) = \left(\frac{3GM\dot{M}}{8\pi \sigma_{SB} r^3}\right)^{1/4} \propto r^{-3/4}$$

$$\boxed{T_{eff}(r) = \left(\frac{3GM\dot{M}}{8\pi \sigma_{SB} r^3}\right)^{1/4} \propto r^{-3/4}}$$

The maximum temperature occurs near the inner edge. For a black hole of mass M accreting at Eddington rate with r_in = 3r_g:

$$T_{max} \approx 6\times10^7 \left(\frac{M}{M_\odot}\right)^{-1/4} \text{ K}$$

Multi-Temperature Spectrum

The total spectrum is a sum of blackbodies at different temperatures:

$$L_\nu = \int_{r_{in}}^{r_{out}} B_\nu(T(r)) \cdot 4\pi r\, dr$$

This produces the characteristic multi-color disk spectrum: F_ν ∝ ν^1/3at intermediate frequencies (between peak temperatures of inner and outer disk), with exponential cutoffs at both ends.

5.3.2 Disk Scale Height

Vertical hydrostatic equilibrium in the disk:

$$\frac{1}{\rho}\frac{dp}{dz} = -\Omega_K^2 z \quad \Rightarrow \quad H = \frac{c_s}{\Omega_K} = c_s\sqrt{\frac{r^3}{GM}}$$

The aspect ratio H/r = c_s/v_K. For a thin disk, H ≪ r requires c_s ≪ v_K(the disk is cold compared to the virial temperature).

5.3.3 Total Disk Luminosity

Integrating the dissipation over the entire disk:

$$L_{disk} = \int_{r_{in}}^{\infty} D(r) 2\pi r\, dr = \frac{GM\dot{M}}{2r_{in}}$$

This is exactly half the total gravitational energy released. The other half goes into kinetic energy of the orbiting gas at r_in, which is either radiated in a boundary layer (neutron star) or swallowed by the event horizon (black hole).

5.4 Magneto-Rotational Instability (MRI)

The α-viscosity was purely phenomenological. Balbus & Hawley (1991) showed that a weak magnetic field in a differentially rotating disk is linearly unstable — the magneto-rotational instability (MRI) provides the turbulent angular momentum transport.

5.4.1 Physical Mechanism

Consider two fluid elements connected by a magnetic field line in a Keplerian disk. The inner element orbits faster (Ω ∝ r^−3/2). The magnetic tension acts like a spring, transferring angular momentum from the inner to the outer element. The inner element loses angular momentum, moves further inward, orbiting even faster — positive feedback!

5.4.2 Derivation of the MRI Dispersion Relation

Derivation

Step 1. Consider an incompressible MHD fluid with a vertical magnetic field B₀ = B₀ẑ in a disk with angular velocity Ω(r). Linearize the equations of motion for axisymmetric perturbations ∝ exp(ik_zz − iωt).

Step 2. The radial and azimuthal momentum equations in the rotating frame (including Coriolis and magnetic tension):

$$-i\omega \delta v_r - 2\Omega \delta v_\phi = -\frac{k_z^2 v_A^2}{-i\omega}\delta v_r$$

$$-i\omega \delta v_\phi + \frac{\kappa^2}{2\Omega}\delta v_r = -\frac{k_z^2 v_A^2}{-i\omega}\delta v_\phi$$

where v_A = B₀/√(μ₀ρ) and κ² = (2Ω/r)d(r²Ω)/dr is the epicyclic frequency squared. For Keplerian rotation, κ² = Ω².

Step 3. Defining ω_A² = k_z²v_A², the dispersion relation from the determinant condition is:

$$(\omega^2 - \omega_A^2)^2 + \kappa^2(\omega^2 - \omega_A^2) - 4\Omega^2\omega_A^2 = 0$$

Step 4. This is a quadratic in (ω² − ω_A²). Solving and finding the instability criterion (ω² < 0):

$$\omega_A^2 < -\frac{d\Omega^2}{d\ln r} = 3\Omega^2 \quad \text{(for Keplerian)}$$

$$\boxed{\text{MRI criterion: } \frac{d\Omega^2}{d\ln r} < 0 \quad \text{i.e., angular velocity decreases outward}}$$

This is satisfied for all astrophysical disks (Keplerian: Ω ∝ r^−3/2, so dΩ²/d ln r = −3Ω² < 0). The maximum growth rate is:

$$\gamma_{max} = \frac{3}{4}\Omega \quad \text{at} \quad k_z v_A = \frac{\sqrt{15}}{4}\Omega$$

The growth rate is of order Ω — the instability operates on the dynamical timescale, which is extremely fast. The condition v_A ≪ c_s (weak field) ensures the most unstable wavelength fits within the disk (λ ∼ v_A/Ω < H).

MRI-Driven Turbulence

Nonlinear MRI simulations show that turbulence develops with effective α ∼ 0.01−0.1, consistent with observational constraints. The Maxwell stress −B_rB_φ/μ₀ dominates the angular momentum transport, with the Reynolds stress ρδv_rδv_φcontributing ∼20-30% of the total.

5.5 Astrophysical Applications

5.5.1 Active Galactic Nuclei (AGN)

AGN are powered by accretion onto supermassive black holes (M ∼ 10⁶−10¹⁰ M_☉). The luminosity of bright quasars L ∼ 10³⁹−10⁴⁰ W requires:

$$\dot{M} = \frac{L}{\eta c^2} \sim \frac{10^{39}}{0.1 \times (3\times10^8)^2} \sim 1\,M_\odot\text{/yr}$$

The innermost stable circular orbit (ISCO) sets the inner disk radius. For a Schwarzschild black hole:

$$r_{ISCO} = 3r_g = \frac{6GM}{c^2} \approx 9\times10^9 \frac{M}{10^8 M_\odot} \text{ m}$$

5.5.2 X-Ray Binaries

Stellar-mass black holes (M ∼ 10 M_☉) accreting from companion stars produce X-ray emission. The peak disk temperature from the T ∝ r^−3/4 profile at r_ISCO:

$$k_B T_{max} \approx 1\text{ keV}\left(\frac{M}{10\,M_\odot}\right)^{-1/4}\left(\frac{\dot{M}}{\dot{M}_{Edd}}\right)^{1/4}$$

5.5.3 Relativistic Jets

Many accreting black holes launch relativistic jets — collimated outflows with Lorentz factors Γ ∼ 10. The jet power is related to the accretion power and the black hole spin. The Blandford-Znajek mechanism extracts rotational energy:

$$P_{BZ} \approx \frac{\kappa}{4\pi c}\Omega_H^2 \Phi_{BH}^2 \propto a^2 \dot{M} c^2$$

where Ω_H is the horizon angular velocity, Φ_BH is the magnetic flux threading the horizon, and a = J/(Mc) is the spin parameter.

Protoplanetary Disks

The same physics governs disks around young stars where planets form. Key differences: T ∼ 100−1000 K (dust sublimation), Ṁ ∼ 10⁻⁸ M_☉/yr, and the disk is partly neutral (MRI requires sufficient ionization). Dead zones where MRI is suppressed may play a role in planet formation.

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