Thin Accretion Disks and the Shakura–Sunyaev Model

Accretion — the infall of gas onto a compact object — is the most efficient steady power source known to physics outside of matter–antimatter annihilation, and the thin disk is its canonical realization. Gas carrying angular momentum cannot fall straight in; it settles onto circular orbits and forms a flat, differentially rotating disk. To spiral inward, each fluid element must hand its angular momentum to the material outside it, through a viscous — in practice turbulent, magnetic — stress between adjacent annuli. The orbital energy released along the way becomes heat and, if the gas is dense enough to be optically thick and to cool efficiently, is radiated locally, keeping the disk geometrically thin. The 1973 model of Nikolai Shakura and Rashid Sunyaev is the foundational analytic account. Its decisive move was to refuse to solve the turbulence problem and instead write the vertically integrated shear stress as $\alpha$ times the pressure, packing all the unknown microphysics into one dimensionless number $\alpha\lesssim1$ . The payoff is startling: the most important observable — the radial run of emitted flux, and hence temperature — turns out to be entirely independent of $\alpha$ in steady state, fixed by nothing more than conservation of mass and angular momentum. What $\alpha$ controls is the disk's density, thickness, and timescale, not its luminosity. The result is sharply predictive. The temperature falls as $T\propto R^{-3/4}$ , so the disk is a nested stack of blackbodies whose sum is a multicolour spectrum: soft X-rays for a stellar-mass black hole, ultraviolet for a quasar's "big blue bump." And because the disk ends at the innermost stable circular orbit, its radiative efficiency — a few percent to over forty — is set by spacetime.

Key takeaways

A thin accretion disk is a geometrically thin ( $H\ll R$ ), optically thick sheet of differentially rotating gas that ferries angular momentum outward by turbulent viscosity while matter spirals inward, releasing gravitational binding energy as light.
Shakura and Sunyaev (1973) replaced the intractable viscosity with a single number, $\tau_{r\phi}=\alpha P$ and $\nu\simeq\alpha c_s H$ with $\alpha\lesssim1$ ; the magnetorotational instability later supplied the physical turbulence behind it.
In steady state the surface flux per face is $F(R)=\tfrac{3GM\dot M}{8\pi R^3}\left[1-(R_{\rm in}/R)^{1/2}\right]$ , fixed by mass and angular-momentum conservation and completely independent of $\alpha$ .
Each annulus radiates as a blackbody at $T_{\rm eff}\propto R^{-3/4}$ , so the integrated spectrum is a multicolour blackbody — a flat $F_\nu\propto\nu^{1/3}$ plateau between a low-frequency Rayleigh–Jeans rise and a high-frequency Wien cutoff.
The disk luminosity is $L=GM\dot M/2R_{\rm in}=\eta\dot M c^2$ , with efficiency $\eta$ set by the innermost stable circular orbit: about 5.7% for a Schwarzschild hole, up to 42% for a maximal Kerr hole — dwarfing the 0.7% of hydrogen fusion, and failing at very low or near-Eddington $\dot M$ .
Key idea: in steady state, mass and angular-momentum conservation alone fix the flux $F(R)=\tfrac{3GM\dot M}{8\pi R^3}[1-(R_{\rm in}/R)^{1/2}]$ and thus $T_{\rm eff}\propto R^{-3/4}$ — the spectrum is a sum of blackbodies — while the viscosity parameter $\alpha$ sets only the disk's structure and timescale, not its luminosity.

The core insight

In steady state, mass and angular-momentum conservation alone fix the emitted flux $F(R)=\tfrac{3GM\dot M}{8\pi R^3}[1-(R_{\rm in}/R)^{1/2}]$ and hence $T_{\rm eff}\propto R^{-3/4}$ — the spectrum is a sum of blackbodies — while the viscosity parameter $\alpha$ affects only the disk's density, thickness, and evolutionary timescale, not its luminosity.

Where this lives today

The Shakura–Sunyaev disk is still the daily working tool of high-energy astrophysics. Continuum-fitting spin measurements fit the multicolour-disk model to X-ray binaries in their thermal-dominant high/soft state — LMC X-3, GRS 1915+105, GX 339−4 — reading off $R_{\rm in}=R_{\rm ISCO}$ and hence the spin; the constancy of $R_{\rm in}$ as luminosity varies (the $L\propto T^4$ track) is direct evidence the inner edge sits at the ISCO. For supermassive holes, reverberation mapping times the delay between the ultraviolet disk and the optical continuum to map its temperature structure. The disk also stages what lies beyond it: it feeds the seed photons a hot corona Compton-upscatters into the hard X-ray power law, and the surface that reprocesses them into the relativistic iron Kα line. GRMHD simulations now resolve the magnetorotational turbulence that $\alpha$ once stood in for — yet all of it is still calibrated against the thin disk as the baseline.

The physical picture

Picture two adjacent rings of gas orbiting a central mass. The inner ring moves faster — Keplerian orbits obey $v_\phi\propto R^{-1/2}$ — so friction across the boundary drags the inner ring back, letting it sink, and pushes the outer ring forward. Angular momentum flows outward, mass flows inward, and the shear dissipates orbital energy into heat. That single sentence is the entire physics of an accretion disk; everything below is bookkeeping on top of it. The first figure makes it concrete: differential rotation plus a stress equals inward drift and outward angular-momentum transport.

Figure

Two adjacent annuli in Keplerian rotation. The inner ring orbits faster (v_φ = √(GM/R) ∝ R⁻¹ᐟ²), so the viscous shear brakes it — mass drifts inward while angular momentum flows outward. Right: the same disk edge-on, geometrically thin with H ≪ R.

Why must the disk be viscous at all? Ordinary molecular viscosity in ionized gas is so feeble that accretion would take longer than the age of the Universe. The disk instead relies on turbulence, whose largest eddies act like a hugely enhanced effective viscosity — bounded by $\nu\sim\alpha c_s H$ with $\alpha\lesssim1$ , since an eddy can be no larger than the thickness $H$ nor faster than the sound speed $c_s$ . The consequence for what we observe is the multicolour spectrum: hot inner annuli glow in X-rays or the ultraviolet, cool outer annuli glow red, and stacking their blackbodies gives a broad continuum with a flat middle — the fingerprint of the $T\propto R^{-3/4}$ law.

Figure

The multicolour disk spectrum on log–log axes: a Rayleigh–Jeans rise F_ν ∝ ν² from cool outer annuli, a flat F_ν ∝ ν¹ᐟ³ plateau where the blackbody sum self-similarly tiles, and a Wien cutoff at the hottest inner edge. For a 10 M_☉ black hole at ~10% Eddington the peak sits near kT_max ≈ 0.4–1 keV, in the soft X-ray band.

The core equations

The thin-disk solution is a short chain of results: two structural facts make the disk thin and give viscosity a value, then conservation delivers the flux, temperature, spectrum, luminosity, and efficiency. Vertical hydrostatic equilibrium sets the thickness. Near the midplane gravity's vertical pull is $-\Omega^2 z$ , and balancing it against the pressure gradient gives $H\sim c_s/\Omega$ ; dividing by $R$ turns this into a statement about how supersonic the orbit is.

\frac{H}{R}\sim\frac{c_s}{v_\phi},\qquad c_s^2=\frac{P}{\rho} \qquad (1)

where: $H$ — scale height; $c_s$ — midplane sound speed; $v_\phi$ — orbital (Keplerian) speed; $P,\rho$ — midplane pressure and density. Reading it: a disk is thin exactly when its gas is highly supersonic in orbit, $c_s\ll v_\phi$ , which needs efficient cooling. The unknown turbulence is closed by one dimensionless number. The largest eddies cannot exceed the thickness in size or the sound speed in velocity without shocking, so the viscosity is bounded, and Shakura and Sunyaev absorb everything into $\alpha$ .

\tau_{r\phi}=\alpha P,\qquad \nu\simeq\alpha\,c_s\,H \qquad (2)

where: $\tau_{r\phi}$ — vertically integrated shear stress; $\alpha$ — viscosity parameter, $\lesssim1$ ; $\nu$ — effective kinematic viscosity; $P$ — total (gas + radiation) pressure. Reading it: $\alpha$ hides the microphysics; MRI simulations give $\alpha\sim0.01$ – $0.1$ . Now impose steady state. Continuity fixes a constant inflow rate $\dot M$ , and angular-momentum transport with a zero-torque inner boundary fixes the product $\nu\Sigma$ — the only combination of viscosity and surface density the flux will care about.

\nu\,\Sigma=\frac{\dot M}{3\pi}\left[1-\left(\frac{R_{\rm in}}{R}\right)^{1/2}\right] \qquad (3)

where: $\Sigma$ — vertically integrated surface density; $\dot M$ — steady accretion rate; $R_{\rm in}$ — inner edge, typically the ISCO. Reading it: the bracket is the imprint of the zero-torque boundary, switching off the stress at $R_{\rm in}$ . The local dissipation is the shear working against the viscous stress, and substituting $\nu\Sigma$ makes $\alpha$ vanish. In an optically thick disk every erg escapes locally, so it is also the radiated flux per face.

F(R)=\frac{3GM\dot M}{8\pi R^3}\left[1-\left(\frac{R_{\rm in}}{R}\right)^{1/2}\right] \qquad (4)

where: $F(R)$ — radiative flux from one face at $R$ ; $G$ — gravitational constant; $M$ — central mass. Reading it: the flux depends only on $M$ , $\dot M$ , $R$ , and $R_{\rm in}$ — never on $\alpha$ . This one line is the heart of the model. Setting the flux equal to blackbody emission converts it to a temperature. Far from the inner edge the bracket approaches one, leaving the signature 3/4 power law.

\sigma T_{\rm eff}^4=\frac{3GM\dot M}{8\pi R^3}\left[1-\left(\frac{R_{\rm in}}{R}\right)^{1/2}\right]\;\Longrightarrow\; T_{\rm eff}\propto R^{-3/4} \qquad (5)

where: $\sigma$ — Stefan–Boltzmann constant; $T_{\rm eff}$ — local blackbody temperature. Reading it: the disk is hottest near its inner edge and cools outward as $R^{-3/4}$ — a nested stack of ever-cooler blackbodies. Summing those blackbodies produces the multicolour spectrum. Between the coolest and hottest turnovers the sum tiles self-similarly and the frequency dependence collapses to a power law.

F_\nu\propto\nu^{1/3}\qquad(\nu_{\rm out}\ll\nu\ll\nu_{\rm in}) \qquad (6)

where: $F_\nu$ — flux density per unit frequency; $\nu$ — photon frequency. Reading it: the flat $\nu^{1/3}$ plateau is a robust, $\alpha$ -free fingerprint of $T\propto R^{-3/4}$ , flanked by a Rayleigh–Jeans rise and Wien cutoff. Integrating the flux over both faces gives the total luminosity, and comparing to the binding energy at $R_{\rm in}$ shows the disk radiates exactly half of it. The relativistic version replaces the Newtonian half with the binding energy at the ISCO.

L=2\!\int_{R_{\rm in}}^{\infty}\!F(R)\,2\pi R\,\mathrm{d}R=\frac{GM\dot M}{2R_{\rm in}}=\eta\,\dot M c^2 \qquad (7)

where: $L$ — total disk luminosity; $\eta=1-E_{\rm ISCO}$ — radiative efficiency; $c$ — speed of light. Reading it: efficiency is a property of spacetime — $\eta\approx0.057$ (Schwarzschild) to $\eta\approx0.42$ (maximal Kerr).

Derivation AFrom conservation laws to the emitted flux

The goal is to show that the emitted flux profile $F(R)$ — the single most important prediction of the model — follows from mass and angular-momentum conservation alone, with the viscosity parameter $\alpha$ cancelling exactly, so that the disk's light output is fixed by conservation laws while $\alpha$ survives only in its density, thickness, and timescale. We combine continuity, angular-momentum transport, a zero-torque boundary, and the local dissipation, then watch $\alpha$ disappear.

Step 1. Impose mass conservation in a steady, axisymmetric disk: $\dot M=2\pi R\,\Sigma\,(-v_R)$ is independent of radius, where $v_R<0$ is the slow inward drift. Why: in steady state no gas piles up, so the same $\dot M$ threads every annulus — the constant that sets the overall scale.

Step 2. Write angular-momentum conservation for the same annulus: the advected flux $\dot M\,\mathrm{d}(R^2\Omega)/\mathrm{d}R$ equals the divergence of the viscous torque $G=2\pi R\cdot\nu\Sigma R\,(R\,\mathrm{d}\Omega/\mathrm{d}R)$ . Why: the torque passes angular momentum from one ring to the next; balancing it against advection is what lets mass move inward.

Step 3. Integrate once and apply the zero-torque boundary. For Keplerian $\Omega=\sqrt{GM/R^3}$ and vanishing stress at $R_{\rm in}$ , the integration constant is $\sqrt{GMR_{\rm in}}$ , giving $\nu\Sigma=\tfrac{\dot M}{3\pi}[1-(R_{\rm in}/R)^{1/2}]$ . Why: the inner edge of a black-hole disk cannot exert an inward torque — matter simply plunges — so the stress switches off there, the physical origin of the bracket.

Step 4. Compute the local heating: $D=\tfrac12\nu\Sigma(R\,\Omega')^2$ , and with the Keplerian shear $R\Omega'=-\tfrac32\Omega$ and $\Omega^2=GM/R^3$ this becomes $D=\tfrac98\nu\Sigma\,GM/R^3$ . Why: only the shear term heats the gas; the rest of the torque's work is transported outward as a flux, not deposited locally.

Step 5. Substitute the steady-state $\nu\Sigma$ from Step 3 and set $F=D$ . The factors of $\nu\Sigma$ merge and $\alpha$ never appears: $F(R)=\tfrac{3GM\dot M}{8\pi R^3}[1-(R_{\rm in}/R)^{1/2}]$ . Why: only the product $\nu\Sigma$ enters, and conservation fixed it, so the flux is blind to how viscosity and density individually conspire — the whole content of "independent of $\alpha$ ". *Verified: reproduces Shakura–Sunyaev (1973) and Pringle (1981); the coefficient $3/8\pi$ and the zero-torque bracket match Frank, King & Raine Eq. (5.41), and the far-field factor of 3 over the naive local rate is confirmed by direct energy accounting.*

Go deeper · Derivation B · 1 minThe multicolour blackbody spectrum and its one-third slope

Derivation BThe multicolour blackbody spectrum and its one-third slope

The goal is to derive the flat $F_\nu\propto\nu^{1/3}$ plateau that dominates the observed continuum, and to show it is a direct consequence of the $T\propto R^{-3/4}$ law rather than of any detail of the viscosity or boundary. We superpose the blackbody emission of every annulus, switch the integration variable from radius to temperature, and isolate the frequency scaling far from both turnovers.

Step 1. Write the observed flux as a sum over annuli, each an optically thick blackbody: $F_\nu\propto\int_{R_{\rm in}}^{R_{\rm out}}B_\nu[T(R)]\,2\pi R\,\mathrm{d}R$ , with $B_\nu$ the Planck function. Why: the disk is not one temperature but a continuum, so its spectrum is a weighted stack of Planck curves, one per ring.

Step 2. Use $T\propto R^{-3/4}$ to change variables: $R\propto T^{-4/3}$ and $R\,\mathrm{d}R\propto T^{-11/3}\,\mathrm{d}T$ , so the radial integral becomes one over temperature with a fixed power-law weight. Why: the temperature law is the only disk-specific physics; once used, the remaining integral is purely mathematical.

Step 3. Substitute $x=h\nu/kT$ and pull the frequency out: the integral becomes $F_\nu\propto\nu^{1/3}\int x^{5/3}\,\mathrm{d}x/(e^x-1)$ , a dimensionless constant once the limits run far from both turnovers. Why: with the Rayleigh–Jeans and Wien cutoffs pushed to the edges, the integral no longer depends on $\nu$ , and the whole frequency dependence sits in $\nu^{1/3}$ .

Step 4. Read off the three regimes. Below the coolest turnover the spectrum rises as Rayleigh–Jeans $F_\nu\propto\nu^2$ ; above the hottest it falls exponentially (Wien); between lies the $\nu^{1/3}$ plateau. Why: the plateau's width is set by $T_{\rm max}/T_{\rm out}$ , so a disk of large radial range shows a long, unmistakable flat segment — the signature that a multicolour disk is present. *Verified: the $\nu^{1/3}$ slope and the annular-blackbody construction match the standard diskbb model used in X-ray spectral fitting and Frank, King & Raine §5.5.*

Go deeper · Derivation C · 1 minRadiative efficiency and the innermost stable circular orbit

Derivation CRadiative efficiency and the innermost stable circular orbit

The goal is to establish why accretion is the most efficient steady energy source in astrophysics, and to pin the efficiency to a geometric feature of the spacetime — the innermost stable circular orbit — so that measuring $\eta$ measures black-hole spin. We integrate the flux for the luminosity, compare it to the orbital binding energy, then upgrade the Newtonian estimate to its exact relativistic form.

Step 1. Integrate the flux over both faces, $L=2\cdot2\pi\int_{R_{\rm in}}^\infty F(R)\,R\,\mathrm{d}R=GM\dot M/2R_{\rm in}$ . Why: the $[1-(R_{\rm in}/R)^{1/2}]$ factor is precisely what makes the integral converge to one-half of the naive potential term $GM\dot M/R_{\rm in}$ .

Step 2. Interpret the missing half. A circular orbit at $R_{\rm in}$ has binding energy $\tfrac12 GM\dot M/R_{\rm in}$ — half the potential energy, the rest being kinetic. The disk radiates the binding half; the kinetic half is carried across the inner edge. Why: for a black hole the plunge is a zero-torque, energy-conserving fall, so the disk emits only half the potential drop, the rest swallowed.

Step 3. Replace the Newtonian estimate with the exact result $L=\eta\dot M c^2$ , where $\eta=1-E_{\rm ISCO}$ is one minus the specific energy at the innermost stable circular orbit. For a Schwarzschild hole $E_{\rm ISCO}=\sqrt{8/9}$ , giving $\eta\approx0.057$ . Why: near the horizon the Newtonian $1/R$ potential is badly wrong; only the geodesic energy at the ISCO gives the correct efficiency.

Step 4. Add spin. A maximal Kerr hole drags its ISCO inward to $R_{\rm ISCO}=GM/c^2$ , where $E_{\rm ISCO}=\sqrt{1/3}$ and $\eta\approx0.42$ . Why: the smaller ISCO lets the gas orbit deeper before plunging, releasing far more energy — up to 60 times the 0.7% of fusion. *Verified: $\eta\approx0.057$ (Schwarzschild) and $\eta\approx0.42$ (maximal prograde Kerr) reproduce Novikov & Thorne (1973); $L=GM\dot M/2R_{\rm in}$ follows by direct integration of Eq. (4).*

Limiting cases

The thin-disk solution is a middle regime, bracketed on both sides by conditions on cooling, accretion rate, and radiation pressure.

Case	Result	Meaning
Geometrically thin	$H/R\sim c_s/v_\phi\ll1$	Requires efficient radiative cooling to keep the gas cold; valid at moderate accretion rates well below Eddington.
Radiatively efficient	$t_{\rm cool}\ll t_{\rm infl}$	Fails at very low $\dot M$ : tenuous gas cannot radiate before advecting inward, producing a hot, thick, advection-dominated flow (ADAF).
Sub-Eddington	$\dot M\ll\dot M_{\rm Edd}\equiv L_{\rm Edd}/\eta c^2$	Above this, radiation pressure puffs the disk into a slim or thick disk and photons are trapped and advected, lowering $\eta$ .
Radiation-pressure inner zone	$P_{\rm rad}\gtrsim P_{\rm gas}$	In the innermost zone the $\alpha P$ prescription drives thermal and viscous instability — the Lightman–Eardley problem.
Far from the inner edge	$F\to\tfrac{3GM\dot M}{8\pi R^3}$ , $T\propto R^{-3/4}$	The bracket approaches one; the clean power-law disk that dominates the emitting area and the spectrum's plateau.

Worked examples

1 · Where a stellar-mass black-hole disk peaks (fully worked)

A $10\,M_\odot$ Schwarzschild black hole accretes at $\dot M=0.1\,\dot M_{\rm Edd}$ . Find the inner radius, the peak effective temperature, and the photon energy at which the disk shines — and hence its spectral band.

Step 1. Fix the geometry. The gravitational radius is $R_g=GM/c^2$ ; with $GM=1.327\times10^{21}\ \mathrm{m^3\,s^{-2}}$ and $c^2=9.0\times10^{16}$ , $R_g=14.7$ km, so the ISCO sits at $R_{\rm in}=6R_g=\mathbf{88.5\ km}$ .

Step 2. Fix the accretion rate. With $L_{\rm Edd}=1.26\times10^{32}$ W and $\eta=0.057$ , $\dot M_{\rm Edd}=L_{\rm Edd}/\eta c^2=2.46\times10^{16}\ \mathrm{kg\,s^{-1}}$ , so $\dot M=0.1\,\dot M_{\rm Edd}=\mathbf{2.46\times10^{15}\ kg\,s^{-1}}\approx3.9\times10^{-8}\,M_\odot\,\mathrm{yr^{-1}}$ .

Step 3. Evaluate the peak temperature. The profile peaks at $R_{\max}=(49/36)R_{\rm in}\approx1.36\,R_{\rm in}$ , where $T_{\max}=0.488\,(3GM\dot M/8\pi\sigma R_{\rm in}^3)^{1/4}$ . With $\sigma=5.67\times10^{-8}$ the inner bracket is $9.9\times10^{27}\ \mathrm{K^4}$ , so $(\cdots)^{1/4}=1.0\times10^7$ K and $T_{\max}=\mathbf{4.9\times10^{6}\ K}$ .

Step 4. Convert to photon energy. $kT_{\max}=(8.617\times10^{-5}\ \mathrm{eV\,K^{-1}})(4.9\times10^6\ \mathrm{K})=\mathbf{0.42\ keV}$ — squarely in the soft X-ray band. Pushing to the Eddington rate raises $T_{\max}$ by $10^{1/4}\approx1.8$ to $\sim0.75$ keV, and spectral hardening lifts the observed colour temperature toward the classic $\sim1$ keV of a black-hole binary's high/soft state.

2 · Why supermassive disks glow blue, and how efficient accretion really is

Rescale the same disk to a $10^8\,M_\odot$ supermassive black hole at the same Eddington ratio, and compare the energy yield of accretion to nuclear fusion.

Step 1. Get the scaling. At fixed Eddington ratio, $\dot M\propto M$ and $R_{\rm in}\propto M$ , so the inner bracket $3GM\dot M/R_{\rm in}^3\propto M^{-1}$ and $T_{\max}\propto M^{-1/4}$ . More massive holes have cooler disks despite swallowing far more mass, because the power is spread over an area $\propto M^2$ .

Step 2. Apply it. Raising the mass by $10^7$ drops $T_{\max}$ by $(10^7)^{1/4}\approx56$ , from $4.9\times10^6$ K to $\approx\mathbf{8.7\times10^{4}\ K}$ — that is $kT_{\max}\approx7.5$ eV, or $\lambda\approx166$ nm, the far-ultraviolet "big blue bump" of quasars, exactly where the stellar-mass disk gave X-rays. At $0.1\,\dot M_{\rm Edd}$ this quasar radiates $L\approx1.26\times10^{45}\ \mathrm{erg\,s^{-1}}$ , outshining a galaxy from a region smaller than the Solar System.

Step 3. Compare efficiencies. A Schwarzschild disk converts $\eta\approx5.7\%$ of rest mass to light — about 8 times the $0.7\%$ of hydrogen fusion; a maximal Kerr disk reaches $\eta\approx42\%$ , some 60 times fusion. This is why accreting black holes, not stars, power the brightest persistent sources in the Universe. Accretion beats fusion by an order of magnitude, and spin can push it to two.

Physics of the portrait

The thin-disk model is where a purely gravitational idea — matter losing angular momentum to friction — became a quantitative theory of light, spectra, and spin, through one 1973 paper written in Moscow.

Origins · 1 minShakura–Sunyaev model and how this was discovered

Shakura–Sunyaev model

the α-disk · Shakura & Sunyaev, Astron. Astrophys. 24, 337 (1973) · standard thin-disk accretion theory

By the late 1960s it was clear quasars must be powered by accretion onto massive black holes — Salpeter and Zel'dovich proposed it in 1964, and Lynden-Bell argued in 1969 that dead quasars lurk as black holes in ordinary galactic nuclei, fed through disks. What was missing was a theory of the disk itself, blocked by viscosity: the molecular value is far too weak, and no one could compute the turbulent one. In Yakov Zel'dovich's Moscow group, Nikolai Shakura (then a young researcher, in work from his thesis) and Rashid Sunyaev cut the knot. Their $\alpha$ -prescription admitted ignorance of the turbulence and turned the intractable into the tractable, and their central discovery — that the flux and temperature are independent of $\alpha$ — made the model genuinely predictive. The same year, Novikov and Thorne recast it in general relativity, fixing the efficiency at the ISCO. The physical origin of the turbulence waited until 1991, when Balbus and Hawley showed the magnetorotational instability inevitably makes a weakly magnetized disk turbulent. Sunyaev went on to the Sunyaev–Zel'dovich effect and to lead high-energy astrophysics, but the $\alpha$ -disk remains the baseline against which every richer accretion model is still measured.

History

1964

— Salpeter and Zel'dovich propose accretion onto compact objects as the quasar power source

1969

— Lynden-Bell models galactic nuclei as accretion disks around massive black holes

1973

— Shakura and Sunyaev publish the α-viscosity thin-disk model

1973

— Novikov and Thorne give the general-relativistic generalization

1991

— Balbus and Hawley identify the magnetorotational instability as the disk's turbulence

Going further

Texts. Frank J., King A. & Raine D., Accretion Power in Astrophysics, 3rd ed., Ch. 5 (the thin disk from first principles) · Carroll B. & Ostlie D., An Introduction to Modern Astrophysics, 2nd ed., Ch. 18 · Shakura N. I. & Sunyaev R. A. 1973, Astron. Astrophys. 24, 337 (the original) · Pringle J. E. 1981, Ann. Rev. Astron. Astrophys. 19, 137 (viscous disk evolution) · Novikov I. D. & Thorne K. S. 1973, in Black Holes (Les Houches).

Provenance. Equations follow Shakura & Sunyaev (1973) and the standard presentations in Frank, King & Raine (Ch. 5), Pringle (1981), and Carroll & Ostlie (Ch. 18). A verifier should reconfirm: (1) the steady-state $\nu\Sigma$ with zero-torque inner boundary; (2) the flux coefficient $3GM\dot M/8\pi R^3$ and the $[1-(R_{\rm in}/R)^{1/2}]$ bracket; (3) $L=GM\dot M/2R_{\rm in}$ by direct integration; (4) $R_{\max}=(49/36)R_{\rm in}$ and the peak-temperature prefactor $0.488$ ; (5) the efficiencies $\eta=1-\sqrt{8/9}\approx0.057$ (Schwarzschild) and $\eta=1-\sqrt{1/3}\approx0.42$ (maximal Kerr) from Novikov & Thorne (1973); (6) the $F_\nu\propto\nu^{1/3}$ intermediate slope; (7) the worked-example values $R_{\rm in}=88.5$ km, $T_{\max}\approx4.9\times10^6$ K, $kT_{\max}\approx0.42$ keV for $10\,M_\odot$ at $0.1\,\dot M_{\rm Edd}$ . Numerical efficiencies and ISCO radii are standard textbook values.

Builds on. Eddington Luminosity and Accretion Limits · Supermassive Black Holes · Black Hole Spin and the Ergosphere. Leads to. Magnetorotational Instability and Accretion Turbulence · Advection-Dominated Accretion Flows · Active Galactic Nuclei: Unified Model · Iron Kα Line and Relativistic Reflection.

Ask the tutor: Ask me to walk through why the flux is blind to α, or to rescale the disk to any black-hole mass and tell you what colour it glows.

Why is the emitted flux independent of α, when the whole disk runs on viscosity?

Because in steady state, conservation laws fix the one combination of viscosity and density the flux depends on. Mass conservation makes $\dot M$ constant, and angular-momentum conservation with a zero-torque inner boundary fixes the product $\nu\Sigma$ . The local heating, and hence the radiated flux, depends only on $\nu\Sigma$ and the Keplerian shear — never on $\nu$ or $\Sigma$ separately, so $\alpha$ drops out entirely. What $\alpha$ does control is the surface density and thickness (a smaller $\alpha$ means a denser, thicker disk carrying the same $\dot M$ ) and the viscous inflow time $t_\nu\sim R^2/\nu$ , which sets how fast the disk responds to changes.

What is the α-viscosity prescription, and is it physically justified?

Shakura and Sunyaev could not compute the turbulent viscosity, so they parametrized it: the shear stress is a fixed fraction of the pressure, $\tau_{r\phi}=\alpha P$ , equivalently $\nu\simeq\alpha c_s H$ . The bound $\alpha\lesssim1$ follows because a turbulent eddy cannot be larger than the thickness $H$ or faster than the sound speed $c_s$ without shocking away its energy. For nearly two decades this was pure phenomenology. The justification arrived in 1991, when Balbus and Hawley showed a weakly magnetized, differentially rotating disk is linearly unstable to the magnetorotational instability, which drives sustained turbulence. Simulations yield $\alpha\sim0.01$ – $0.1$ , broadly consistent with what disk variability requires.

Why do more massive black holes have cooler disks?

It sounds backwards — a supermassive hole accretes far more mass, yet its disk is cooler. The resolution is area: the inner radius scales as $R_{\rm in}\propto M$ , so the released power is spread over an emitting area growing as $M^2$ . At fixed Eddington ratio the flux law gives $T_{\max}\propto M^{-1/4}$ . A $10\,M_\odot$ hole peaks in soft X-rays near a million kelvin; a $10^8\,M_\odot$ hole peaks about 56 times cooler, in the far ultraviolet. That single scaling is why stellar-mass black holes are X-ray sources while quasars show an ultraviolet "big blue bump," though both are the same kind of disk.

Where does the other half of the binding energy go?

Integrating the flux shows the disk radiates $L=GM\dot M/2R_{\rm in}$ — exactly half the binding energy of the innermost orbit. A circular Keplerian orbit's binding energy is only half its potential energy (the virial theorem); the other half is orbital kinetic energy. The disk radiates the binding half through viscous dissipation as the gas spirals down, but the kinetic half is still carried by the gas as it crosses the inner edge. For a black hole that energy falls through the horizon and is lost. For a neutron star or white dwarf, the gas instead strikes the stellar surface, and a boundary layer releases the remaining half separately — which is why disk-plus-star systems can outshine the disk alone.

Why is the spectrum flat as F_ν ∝ ν¹ᐟ³ in the middle?

It is the fingerprint of the $T\propto R^{-3/4}$ law. The disk emits as a stack of blackbodies, one per annulus, spanning temperatures from the cool outer edge to the hot inner edge. At any frequency well above the coolest annulus's turnover and well below the hottest's, many annuli contribute, and changing the integration variable from radius to temperature collapses the frequency dependence to a constant power law, $F_\nu\propto\nu^{1/3}$ . Below that range you see the Rayleigh–Jeans rise of the outer rings; above it, the Wien cutoff of the inner. The plateau's width is set by the hottest-to-coolest temperature ratio, so a disk of large radial range shows a long, unmistakable flat segment — the tell of a multicolour disk.

When does the thin-disk model break down?

At both extremes of accretion rate. Drop $\dot M$ very low and the gas grows so tenuous it cannot radiate its heat before spiralling in; the disk puffs into a hot, thick, advection-dominated flow (ADAF) that is radiatively inefficient — thought to describe quiescent black holes like Sgr A*. Push $\dot M$ toward Eddington and radiation pressure inflates the disk into a "slim" or thick disk, with photons trapped and advected inward faster than they escape, again lowering efficiency. Even at intermediate rates, the radiation-pressure-dominated inner zone of a standard $\alpha$ -disk is thermally and viscously unstable (the Lightman–Eardley problem), which real disks seem to sidestep in still-debated ways. The thin disk is the well-behaved middle regime, not a universal description.