Accretion Disk

Why disks form, not spheres

Spherical free-fall onto a compact object is rare in the real universe. Almost any parcel of gas falling toward a star, neutron star, or black hole arrives with at least a little angular momentum — picked up from a binary partner, a galactic potential, or just turbulence in a molecular cloud. Even a tiny rotation has consequences. As gas falls inward, conservation of angular momentum demands that its rotation speed up like the inverse of distance. At some radius, the centrifugal barrier matches gravity and the gas can fall no further radially; instead it settles onto a rotationally supported disk in the equatorial plane. Out-of-plane motion is rapidly damped by collisions, while the rotation persists.

The disk that results is a kind of clearing house. Matter cannot simply spiral in on its own — it must surrender its angular momentum to something. In real disks, that something is a turbulent stress that effectively acts like viscosity: shear between adjacent rings transports angular momentum outward. As a parcel loses L, it drops to a smaller orbit, gives up gravitational potential energy, and radiates the excess. This is the fundamental cycle: viscosity moves angular momentum out, mass drifts in, binding energy comes out as light.

The Shakura-Sunyaev α-disk

The benchmark theoretical framework is the 1973 paper by Nikolai Shakura and Rashid Sunyaev. They wrote down the equations for a steady, geometrically thin, optically thick disk in which the unknown viscosity ν is parameterised as

ν = α c_s H

where c_s is the local sound speed, H is the vertical scale height, and α is a dimensionless number presumed to be of order unity or less. Empirically, α-values that match observation cluster between 0.01 and 0.3.

The model gives clean predictions for the radial profiles of surface density Σ, mid-plane temperature T, and effective temperature T_eff. The key result for an observer is that

T_eff(r) = [3GM Ṁ / (8π σ r³)]^(1/4) × [1 − (r_in/r)^(1/2)]^(1/4)
       ∝ r^(−3/4)   (far from the inner edge)

Each annulus radiates approximately as a blackbody at its local T_eff. The total spectrum is then a sum over annuli — a "multi-colour disk blackbody" that turns over near the temperature of the hottest, innermost ring. That single curve, with its characteristic flat-topped F_ν ∝ ν^(1/3) middle and its high-frequency Wien cutoff, is the diagnostic fingerprint of a thin disk.

What actually drives the viscosity

The α-prescription was deliberately a black box. The molecular viscosity of a hot ionised plasma is many orders of magnitude too small to explain observed accretion rates: it would predict α ~ 10^(-12), whereas inferred values are 10^(-2) to 10^(-1). For two decades the source of the missing stress was a notorious open problem.

The breakthrough came from Steven Balbus and John Hawley in 1991, who realised that the magnetorotational instability (MRI) — known to plasma physicists since Velikhov (1959) and Chandrasekhar (1960) but never connected to disks — is generic in a magnetised, differentially rotating fluid. A weak magnetic field is destabilised by the shear, drives turbulence that extracts angular momentum from inner annuli and deposits it on outer ones, and self-sustains via dynamo action. MRI turbulence reproduces α ~ 0.01 to 0.1 in shearing-box simulations and is now the canonical picture for ionised disks. Cool, partly neutral protoplanetary disks have "dead zones" where the MRI is suppressed; in those regions other transport mechanisms (gravitational instabilities, magnetic disk winds) take over.

How much energy is liberated

For a particle moving from infinity to a circular orbit at radius r around a non-rotating mass M, the binding energy released is

E_bind = (1/2) GM/r  (per unit mass, Newtonian)

Half the gravitational potential energy goes into orbital kinetic energy; the other half is radiated as the particle works its way inward. By the time it reaches the ISCO at 6 GM/c² of a Schwarzschild black hole, a fully relativistic calculation gives the total efficiency η = 1 − √(8/9) ≈ 0.057 of rest mass — about 6%. If you count the rest-mass energy of the matter that ultimately falls through the horizon as input, the fraction that escapes as light is roughly 5.7%, conventionally rounded to "about 10%". For a maximally spinning Kerr black hole (a = M, prograde orbit), the ISCO retreats to 1.235 GM/c² and the efficiency rises to 0.423 — 42 % of mc². This is more than 50 times the efficiency of hydrogen fusion (0.7%) and explains why active galactic nuclei can outshine their host galaxies from a region the size of the solar system.

Disk temperatures and the mass scaling

For a thin disk radiating at the Eddington limit, the peak T_eff scales as M^(-1/4). Plugging numbers into the Shakura-Sunyaev formula at r = 10 GM/c²:

Object	Mass	T_peak	Spectral peak	Class
White dwarf binary	~1 M☉ (WD)	~5 × 10⁴ K	UV / soft X-ray	Cataclysmic variable
Neutron star binary	1.4 M☉	~10⁷ K	X-ray (1 keV)	LMXB / HMXB
Stellar-mass black hole	10 M☉	~10⁷ K	Soft X-ray	X-ray binary
Intermediate-mass BH	10³ M☉	~10⁶ K	EUV	ULX candidate
Seyfert nucleus	10⁷ M☉	~10⁵ K	UV (Big Blue Bump)	AGN
Quasar	10⁹ M☉	~3 × 10⁴ K	Near UV / blue	AGN
Protostar (T Tauri)	0.5 M☉	~10³ K	Near IR	Protoplanetary disk

This single scaling explains an enormous amount of observational phenomenology. AGN UV continuum, X-ray binary soft excess, the IR excess of T Tauri stars — all are different annuli of the same multi-temperature disk seen across an enormous mass range.

Worked example: how much mass feeds a quasar?

Consider a 10⁸ M☉ supermassive black hole radiating at the Eddington limit. The Eddington luminosity is

L_Edd = 1.26 × 10³⁸ (M/M☉) erg/s
      = 1.26 × 10⁴⁶ erg/s   for M = 10⁸ M☉

Adopt a radiative efficiency η = 0.1 (a non-spinning hole, conservative). The relation between accretion rate and luminosity is

L = η Ṁ c²    →    Ṁ = L / (η c²)

Plugging in L = 1.26 × 10⁴⁶ erg/s, η = 0.1, c² = 9 × 10²⁰ cm²/s²:

Ṁ = 1.26 × 10⁴⁶ / (0.1 × 9 × 10²⁰)
   = 1.4 × 10²⁵ g/s
   ≈ 1.4 × 10²⁵ × (3.156 × 10⁷ s/yr) g/yr
   ≈ 4.4 × 10³² g/yr
   ≈ 2.2 M☉/yr        (M☉ = 2 × 10³³ g)

So a luminous quasar consumes about two solar masses of gas every year. Over a typical 10⁷-year duty cycle that builds up to roughly 2 × 10⁷ M☉ — comparable to the entire SMBH mass. This is the dimensional argument behind the Soltan relation linking quasar luminosity density to local SMBH mass density: the integrated AGN light tells you how much of today's black-hole mass came from luminous accretion.

If instead we assume η = 0.42 (maximally spinning Kerr), the required Ṁ is only 0.5 M☉/yr — illustrating why spin-up of SMBHs by aligned accretion is an important observational driver of inferred fuelling rates.

Regimes: thin, thick, slim, ADAF

The Shakura-Sunyaev solution applies in a window of accretion rate roughly 0.01 ≲ Ṁ/Ṁ_Edd ≲ 0.3. Outside that window, other regimes take over, each with its own observational signatures.

Regime	Ṁ / Ṁ_Edd	Geometry	Cooling	Spectrum
Standard thin disk	0.01 – 0.3	H/r ≪ 1	Radiative, efficient	Multi-colour blackbody
Slim disk	~0.3 – 1+	H/r ~ 1	Photons trapped, advected	Saturated blackbody
Super-Eddington (funnel)	> 1	Funnel + outflow	Radiation-driven wind	Wind + soft X-ray
ADAF / RIAF	< 10⁻³	H/r ~ 1, hot	Advected, inefficient	Hard X-ray, low-power jet
Quiescent low-state	~ 10⁻⁵	Truncated thin + corona	Inefficient + Comptonised	Power-law hard X-ray
Self-gravitating disk	varies	Q ~ 1 outer disk	Heating by GI	Spiral arms, fragmentation

The two extremes — slim disks and ADAFs — both arise when radiative cooling cannot keep pace with viscous heating. In the slim disk, photons are trapped in the inflow because diffusion is slower than advection; the disk puffs up and luminosity saturates near a few L_Edd. In the ADAF, the gas is so tenuous that it is collisionless on relevant timescales; ions cannot transfer their thermal energy to electrons, the gas heats to ~10⁹ K, and most of the energy is advected through the horizon rather than radiated. Sgr A* and the centres of nearly every massive elliptical are ADAFs.

Variants and extensions

• Novikov-Thorne disk. The general-relativistic version of Shakura-Sunyaev, introduced in 1973 by Igor Novikov and Kip Thorne. Adds GR corrections to gravity, redshift, and the inner boundary condition; reproduces the Schwarzschild and Kerr ISCO efficiencies.
• Slim disk (Abramowicz et al. 1988). Relaxes the "thin" approximation. At Ṁ near or above L_Edd, the disk becomes geometrically thick, photon trapping makes cooling inefficient, and luminosity saturates. The model that fuels the modern picture of ULXs.
• ADAF / RIAF (Narayan-Yi 1994; Yuan-Narayan). Advection-dominated accretion flow / radiatively inefficient accretion flow. At very low Ṁ, ions and electrons decouple and most of the heat is advected through the horizon. The basis for modelling Sgr A* and M87*.
• Truncated disk + hot corona. Composite geometry invoked for X-ray binaries in their hard state: a thin disk extends from the outer edge to ~10–100 r_g, where it transitions into a hot, optically thin Comptonising corona that produces the power-law tail.
• Magnetically arrested disk (MAD). When magnetic flux accumulating near the hole becomes strong enough to choke off accretion in a steady-state way, the disk transitions to the MAD regime. Observed in EHT polarimetry of M87* and may correlate with relativistic jet launching.

Where accretion disks show up

• Active galactic nuclei. 10⁶–10¹⁰ M☉ supermassive black holes accreting at 10⁻²–1 L_Edd. Disk temperatures 10⁴–10⁵ K produce the UV "Big Blue Bump"; coronal Comptonisation gives the 2–100 keV X-ray power law. Quasars (the most luminous AGN) reach L ~ 10⁴⁷ erg/s.
• X-ray binaries. Stellar-mass NS or BH (1.4–30 M☉) accreting from a stellar companion. Disks peak at ~1 keV, give the "soft state" spectrum; in the "hard state" the inner disk is replaced by a hot Comptonising corona. Cygnus X-1 and GX 339-4 are canonical examples.
• Cataclysmic variables. White dwarfs (~1 M☉) accreting from low-mass companions via Roche-lobe overflow. Disk temperatures 10⁴–10⁵ K, peak in UV. Dwarf novae like SS Cygni cycle through thermal-viscous instability outbursts every ~50 days, an excellent local laboratory for disk physics.
• Protoplanetary disks. 0.001–0.1 M☉ of gas and dust around a forming T Tauri or Herbig Ae/Be star. Cool (~10–1500 K), partly neutral. Peak in IR and submillimetre; ALMA has resolved gaps and rings (HL Tau, TW Hya) interpreted as forming planets clearing their feeding zones.
• Tidal disruption events. When a star strays inside the tidal radius of a SMBH, it is torn apart and roughly half its mass returns on highly eccentric orbits to circularise into a transient accretion disk. The result is a months-to-years X-ray/UV flare, e.g. ASASSN-14li, PS1-10jh.

Time variability — the disk is not steady

Real disks are observed to vary on every timescale from milliseconds (X-ray binary high-frequency QPOs) to decades (AGN long-term changing-look events). The relevant disk timescales at radius r are:

t_dyn  = 1 / Ω_K       ≈ 10⁻⁴ s × (r/r_g)^(3/2)  for 10 M☉
t_th   = t_dyn / α     thermal timescale
t_visc = t_dyn / (α (H/r)²)   viscous (radial drift)

For a 10 M☉ stellar BH at r = 10 r_g and α = 0.1, t_dyn ~ 3 ms, t_th ~ 30 ms, t_visc ~ several seconds (assuming H/r ~ 0.1). For a 10⁸ M☉ SMBH the same orbits scale up to hours and centuries respectively. Observed AGN UV variability on years-to-decades timescales is much faster than the canonical thin-disk t_visc, an open puzzle that has motivated revised disk models (e.g. magnetic stress-driven inflow).

Common pitfalls

• Confusing the ISCO with the event horizon. ISCO is the inner edge of the disk; the event horizon (Schwarzschild radius 2 GM/c²) is much smaller. Photons emitted inside the ISCO can still escape; matter inside it plunges in roughly free-fall.
• Treating α as a constant of nature. α is a parameterisation, not a fundamental quantity. It varies with disk state, magnetic topology, ionisation fraction. MHD simulations show "α" rises from 0.01 in stratified MRI boxes to 0.1+ in MAD configurations.
• Forgetting the inner-edge boundary correction. The radial profile of T_eff includes a factor [1 − (r_in/r)^(1/2)]^(1/4) that vanishes at the inner edge — without it, T appears to diverge as r → r_in. The factor reflects the fact that no torque is exerted at the inner boundary.
• Misapplying L = η Ṁ c² above Eddington. The thin-disk efficiency η is a property of the geometry. When Ṁ exceeds Ṁ_Edd, photon trapping reduces the effective efficiency and the slim-disk regime takes over; using the η of a thin disk overestimates the luminosity dramatically.
• Ignoring the corona. The optically thin hot corona accounts for 10–80% of the X-ray luminosity in real systems and is not part of the standard disk solution. Modelling broadband X-ray spectra without it drives the inferred disk parameters into nonsense.