Accretion Disks

Accretion — the capture of matter by the gravity of a central body — is the most efficient way known to turn rest mass into radiation short of matter–antimatter annihilation. The catch is angular momentum: gas falling toward a compact object almost never aims straight at the centre, and the small sideways motion it carries becomes an insurmountable centrifugal barrier. Rather than plunging in, the gas settles into a flat, rotating, nearly Keplerian disk and can accrete only as fast as it sheds its spin. Internal torques carry that angular momentum outward, matter drifts inward, and the gravitational energy released on the way down — half of it, as the virial theorem insists — is radiated locally, giving disks their characteristic multicolour glow. The workhorse framework is the geometrically thin, optically thick disk of Shakura and Sunyaev (1973) and its relativistic generalization by Novikov and Thorne. Its power is that the radial run of temperature, and hence the spectrum away from the boundaries, follows from conservation of mass, angular momentum, and energy alone — independent of the poorly understood "viscosity" that transports the angular momentum. That microphysics is bundled into one number, $\alpha$ , and its origin — the magnetorotational instability of Balbus and Hawley (1991) — was not pinned down until nearly two decades later. Disks are ubiquitous engines. Around white dwarfs they drive dwarf-nova outbursts; around neutron stars and stellar-mass black holes they power X-ray binaries; around supermassive black holes they light up quasars. One set of equations, scaled by the central mass, runs them all.

Key takeaways

A disk forms because infalling gas carries angular momentum: it cannot fall radially, so it orbits and spirals in only as internal torques move its angular momentum outward.
The luminosity is released gravitational binding energy: $L_{\rm acc}=GM\dot{M}/R$ , reaching $\sim0.1\,\dot{M}c^2$ for neutron stars and black holes — far above fusion's $0.007$ .
In the thin-disk model the local flux is fixed by conservation laws alone, giving $T_{\rm eff}(r)\propto r^{-3/4}$ and a flat multicolour spectrum $F_\nu\propto\nu^{1/3}$ between the edge temperatures.
Molecular viscosity is hopelessly weak; transport is parametrized by $\nu=\alpha c_s H$ , with the magnetorotational instability (MRI) supplying the turbulent stress at growth rate $\tfrac34\Omega$ .
The Eddington luminosity $L_{\rm Edd}=4\pi GMm_pc/\sigma_T\approx1.26\times10^{38}(M/M_\odot)\ {\rm erg\,s^{-1}}$ caps steady spherical accretion.
Key idea: a fluid element can accrete only if it sheds angular momentum; viscous (in practice MRI-driven turbulent) torques carry that angular momentum outward while the released gravitational energy is radiated locally, producing a $T_{\rm eff}\propto r^{-3/4}$ temperature profile whose form follows from conservation laws alone.

The core insight

A fluid element can accrete only if it loses angular momentum; viscous (in practice MRI-driven turbulent) torques transport angular momentum outward through the disk while the released gravitational energy is radiated locally, producing a $T_{\rm eff}\propto r^{-3/4}$ temperature profile whose form follows from conservation laws alone.

Where this lives today

Accretion disks are now imaged, not just inferred. In 2019 and 2022 the Event Horizon Telescope resolved the hot flows around M87 and Sgr A, revealing the lensed photon rings around their horizons — a direct look at the innermost regions where the thin-disk picture gives way to advection-dominated flow. For stellar-mass black holes, two spectral techniques turn disk physics into spin measurements: continuum fitting reads $R_{\rm in}$ off the thermal disk temperature and identifies it with the ISCO, while relativistic reflection models the broad iron Kα line from the irradiated inner disk — the same $R_{\rm ISCO}$ –spin link that constrains the natal spins of the black holes LIGO and Virgo see merge. On the theory side, global GRMHD simulations resolve the MRI turbulence directly, replacing $\alpha$ with first-principles stresses, while "changing-look" quasars and tidal disruption events probe how fast real disks build and drain — a live test of the viscous evolution equation.

The physical picture

Picture a marble circling in a shallow bowl: friction robs it of orbital speed, so it spirals inward while the bowl — the gravitational field — does work on it. A disk works the same way, except the "friction" is between neighbouring rings of gas orbiting at different rates. Because gravity enforces Keplerian motion, an inner ring sweeps round faster than the one just outside it, $\Omega_K\propto r^{-3/2}$ ; the two rub, the faster inner ring dragged back and the slower outer one flung ahead, and angular momentum is handed outward. Robbed of the spin that held it up, the inner gas drifts into the well. The energy bookkeeping is set by the virial theorem: falling deeper doubles the orbital kinetic energy while halving the potential energy, so exactly half the liberated gravitational energy appears as radiation and half stays as faster orbital motion. This is why the radiative efficiency down to radius $r$ is $GM/2r$ per unit mass, not $GM/r$ : the "missing" half is dumped in a boundary layer at a stellar surface or swallowed across a horizon. The first figure lays out the machinery — differential rotation, outward torque, inward drift.

Figure

Anatomy of a thin accretion disk: gas on nearly Keplerian orbits with Ω_K = √(GM/r³) means inner rings orbit faster than outer ones, so the shear hands angular momentum outward while matter drifts inward and radiates the released energy locally, hottest at the centre and cooling toward the rim

Because the disk is optically thick, every annulus radiates as a blackbody at its own temperature — inner regions hottest and bluest, outer ones cool and red. Adding the annuli gives not a single Planck curve but a stretched one: a flat $F_\nu\propto\nu^{1/3}$ plateau across the middle, a Rayleigh–Jeans rise below, and a Wien cutoff at the hottest inner temperature. This composite "multicolour blackbody" is the signature of a thin disk, and its inner-edge temperature is what spectral fits use to locate the innermost stable orbit. The second figure shows the shape and pins the cutoff to a real system.

Figure

Multicolour-blackbody disk spectrum: summing Planck curves from annuli of steadily decreasing temperature gives three regimes — a Rayleigh–Jeans rise at low frequency, a broad flat plateau across the middle, and a Wien cutoff near the hottest inner temperature kT_in ≈ 1 keV for a 10 M⊙ black hole accreting near the Eddington rate

Why should the disk drain at all if molecular viscosity is negligible — as it hugely is, giving inflow times longer than the age of the Universe? The resolution is magnetic. A weak field threading the differentially rotating plasma is unstable: two fluid elements tied by a field line act like beads joined by a spring, the inner one dragged back and the outer flung ahead, exchanging angular momentum and pulling apart. This magnetorotational instability grows on the orbital timescale and saturates into MHD turbulence — the real "viscosity" the $\alpha$ -prescription stands in for.

The core equations

The starting point is the energy released when mass reaches the inner edge. A parcel falling from rest at infinity to $R$ liberates specific binding energy $GM/R$ ; multiplying by the flux $\dot{M}$ gives the accretion luminosity.

L_{\rm acc}=\frac{GM\dot{M}}{R} \qquad (1)

where: $L_{\rm acc}$ — luminosity from released gravitational energy; $G$ — gravitational constant; $M$ — mass of the central accretor; $\dot{M}$ — mass accretion rate; $R$ — radius where the energy is finally released (stellar surface or ISCO). Reading it: brightness scales with how much mass arrives and how deep the well is — small $R$ makes accretion luminous. Radiation pushes back. Photons streaming outward scatter off electrons and drag the coupled plasma with them; when that force balances gravity, steady spherical accretion chokes off. Setting the radiation force equal to gravity — both falling as $r^{-2}$ , so the radius cancels — gives a luminosity ceiling that depends only on mass.

L_{\rm Edd}=\frac{4\pi GM m_p c}{\sigma_T} \qquad (2)

where: $m_p$ — proton mass; $c$ — speed of light; $\sigma_T$ — Thomson cross section of the electron. Reading it: $L_{\rm Edd}\approx1.26\times10^{38}(M/M_\odot)\ {\rm erg\,s^{-1}}$ is a hard, mass-only cap — exceed it and radiation pressure blows the inflow away. The disk's structure hangs on the orbits being Keplerian, since the gas pressure is far too feeble to support it radially — the angular velocity and speed are those of a test particle.

\Omega_K(r)=\sqrt{\frac{GM}{r^3}},\qquad v_\phi=\sqrt{\frac{GM}{r}} \qquad (3)

where: $\Omega_K$ — Keplerian angular velocity; $v_\phi$ — orbital (azimuthal) speed; $r$ — cylindrical radius. Reading it: the steep $r^{-3/2}$ fall-off is the differential rotation that drives all disk transport. The transport itself is the great unknown. Shakura and Sunyaev sidestepped it with a dimensional ansatz: eddies cannot move faster than sound ( $c_s$ ) or be larger than the thickness ( $H$ ), so the viscosity is at most $c_s H$ times an efficiency $\alpha\lesssim1$ .

\nu=\alpha\, c_s H \qquad (4)

where: $\nu$ — turbulent kinematic viscosity; $\alpha$ — dimensionless viscosity parameter, $0<\alpha\lesssim1$ ; $c_s$ — local sound speed; $H\approx c_s/\Omega_K$ — disk scale height. Reading it: all the ignorance sits in one number, and $\alpha\sim0.01$ – $0.1$ fits a wide range of systems. Combining mass and angular-momentum conservation also makes the surface density diffuse on the viscous timescale $t_\nu\sim r^2/\nu$ . In a steady state, though, the local heating rate per unit area is fixed by conservation laws alone; the viscosity cancels, leaving a flux set only by $M$ , $\dot{M}$ , radius, and the inner boundary.

D(r)=\frac{9}{4}\nu\Sigma\,\Omega_K^2=\frac{3GM\dot{M}}{8\pi r^3}\left[1-\left(\frac{R_{\rm in}}{r}\right)^{1/2}\right] \qquad (5)

where: $D(r)$ — dissipation per unit area (one face); $R_{\rm in}$ — inner radius (zero-torque boundary). Reading it: the flux is model-independent — no $\alpha$ appears — so thin-disk spectra are trustworthy. Because each optically thick annulus radiates as a blackbody, $\sigma T_{\rm eff}^4=D(r)$ , and away from the inner edge the temperature falls as a power law.

T_{\rm eff}(r)=\left\{\frac{3GM\dot{M}}{8\pi\sigma r^3}\left[1-\left(\frac{R_{\rm in}}{r}\right)^{1/2}\right]\right\}^{1/4}\;\xrightarrow{\,r\gg R_{\rm in}\,}\;\propto r^{-3/4} \qquad (6)

where: $T_{\rm eff}$ — local surface temperature; $\sigma$ — Stefan–Boltzmann constant. Reading it: the $r^{-3/4}$ law is a thin disk's fingerprint. Summing the Planck functions of those annuli, with $T\propto r^{-3/4}$ , gives the flat band between the edge temperatures.

F_\nu\propto\nu^{1/3}\qquad(k T_{\rm out}\ll h\nu\ll k T_{\rm in}) \qquad (7)

where: $F_\nu$ — flux density at frequency $\nu$ ; $T_{\rm in},T_{\rm out}$ — inner and outer temperatures. Reading it: a flat $\nu^{1/3}$ slope over a decade in frequency is the tell-tale of a disk. Finally, for a black hole the inner edge is not a surface but the innermost stable circular orbit, and the efficiency is the binding energy there — depending on spin.

\eta=1-\sqrt{1-\frac{2GM}{3R_{\rm ISCO}c^2}}\;\Rightarrow\;\eta\approx0.057\ (a=0),\ \ \eta\approx0.42\ (a=1) \qquad (8)

where: $R_{\rm ISCO}$ — radius of the innermost stable circular orbit; $a$ — dimensionless spin, $0\le a\le1$ ; $\eta$ — maximum radiative efficiency. Reading it: spin raises a black hole's yield from about 6% to over 40% of $\dot{M}c^2$ , because a fast hole lets the disk reach deeper.

Derivation ATemperature profile of a steady thin disk from conservation laws

The single most quoted result of disk theory is that the flux radiated by each annulus — and therefore $T_{\rm eff}(r)\propto r^{-3/4}$ — is fixed entirely by conservation of mass, angular momentum, and energy, with the unknown viscosity cancelling out completely. That is why the spectrum can be predicted with confidence even though nobody could compute $\nu$ for two decades. We derive it for a steady, axisymmetric, thin, Keplerian disk carrying a constant inward mass flux $\dot{M}$ .

Step 1. Impose mass conservation. In a steady disk the same mass crosses every radius per unit time, $\dot{M}=-2\pi r\Sigma v_r=\text{const}$ , with $v_r<0$ the slow inward drift. Why: fixing the flux to one constant makes the disk solvable radius by radius rather than as a coupled global problem.

\dot{M}=-2\pi r\Sigma v_r=\text{const}

Step 2. Impose angular-momentum conservation. The advected flux changes only through the divergence of the viscous torque $G=2\pi r^3\nu\Sigma\,\mathrm{d}\Omega/\mathrm{d}r$ , so $\dot{M}\,\mathrm{d}(r^2\Omega)/\mathrm{d}r=\mathrm{d}G/\mathrm{d}r$ . Why: matter moves in only if the torque hauls its angular momentum out — the physical heart of accretion.

\dot{M}\frac{\mathrm{d}(r^2\Omega)}{\mathrm{d}r}=\frac{\mathrm{d}}{\mathrm{d}r}\!\left(2\pi r^3\nu\Sigma\frac{\mathrm{d}\Omega}{\mathrm{d}r}\right)

Step 3. Integrate with a zero-torque inner boundary. Using $\Omega=\Omega_K\propto r^{-3/2}$ and demanding the torque vanish at $R_{\rm in}$ gives $\nu\Sigma$ in closed form. Why: the boundary condition is the physics of the inner edge, and it is exactly what kills the unknown $\nu$ in the flux.

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

Step 4. Compute the local heating. The shear stress on the differential rotation dissipates heat at $D(r)=\tfrac12\nu\Sigma\,(r\,\mathrm{d}\Omega/\mathrm{d}r)^2=\tfrac98\nu\Sigma\,\Omega_K^2$ per unit area per face. Why: this converts angular-momentum transport into frictional heating, the energy that will emerge as light.

D(r)=\tfrac{1}{2}\nu\Sigma\left(r\frac{\mathrm{d}\Omega}{\mathrm{d}r}\right)^2=\frac{9}{8}\nu\Sigma\,\Omega_K^2

Step 5. Substitute $\nu\Sigma$ and $\Omega_K^2=GM/r^3$ ; the viscosity cancels, leaving a flux built only from $M$ , $\dot{M}$ , $r$ , and $R_{\rm in}$ . Why: the disappearance of $\nu$ (and $\alpha$ ) is the whole point — the flux, and thus the spectrum, is model-independent.

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

Step 6. Radiate it as a blackbody. Because the disk is optically thick, $\sigma T_{\rm eff}^4=D(r)$ , so far from the inner edge $T_{\rm eff}\propto r^{-3/4}$ . Why: an optically thick surface emits a Planck spectrum at the temperature that balances its heating, turning the flux law into the observable temperature law.

T_{\rm eff}(r)=\left(\frac{D(r)}{\sigma}\right)^{1/4}\propto r^{-3/4}\quad(r\gg R_{\rm in})

Go deeper · Derivation B · 1 minThe virial split and why a disk radiates half the released energy

Derivation BThe virial split and why a disk radiates half the released energy

It is tempting to set the disk luminosity equal to the full binding energy $GM\dot{M}/R_{\rm in}$ delivered to the inner edge, but this overcounts by a factor of two. The virial theorem guarantees that a circular orbit stores as much energy in motion as it radiates in reaching that orbit, so a thin disk shines with only half the released energy — the rest crosses $R_{\rm in}$ as orbital kinetic energy. We prove it by integrating the local dissipation over the whole disk and recovering exactly $GM\dot{M}/2R_{\rm in}$ .

Step 1. Set up the total luminosity as twice (two faces) the integral of $D(r)$ over the disk area from $R_{\rm in}$ outward. Why: the emitted power is the sum of every annulus's heating, and both faces radiate, so the factor of two is geometric.

L_{\rm disk}=2\int_{R_{\rm in}}^{\infty}\frac{3GM\dot{M}}{8\pi r^3}\left[1-\left(\frac{R_{\rm in}}{r}\right)^{1/2}\right]2\pi r\,\mathrm{d}r

Step 2. Pull the constants out and reduce the integrand to two power laws, both converging at the upper limit. Why: the convergence at large $r$ confirms that essentially all the luminosity comes from the innermost radii, where the well is deepest.

L_{\rm disk}=\frac{3GM\dot{M}}{2}\int_{R_{\rm in}}^{\infty}\left(r^{-2}-R_{\rm in}^{1/2}r^{-5/2}\right)\mathrm{d}r

Step 3. Evaluate the two integrals: the first gives $1/R_{\rm in}$ , the second $2/3R_{\rm in}$ , so the bracket collapses to $1/3R_{\rm in}$ . Why: the boundary term from the zero-torque condition is exactly what turns the naive $1/R_{\rm in}$ into a third of itself, encoding the missing energy.

\int_{R_{\rm in}}^{\infty}r^{-2}\,\mathrm{d}r=\frac{1}{R_{\rm in}},\qquad R_{\rm in}^{1/2}\int_{R_{\rm in}}^{\infty}r^{-5/2}\,\mathrm{d}r=\frac{2}{3R_{\rm in}}

Step 4. Collect the result: $L_{\rm disk}=\tfrac{3GM\dot{M}}{2}\cdot\tfrac{1}{3R_{\rm in}}=GM\dot{M}/2R_{\rm in}$ , exactly half the naive value. Why: the other half is carried inward as orbital kinetic energy, released in a boundary layer or lost across the horizon — which is why the inner-edge geometry matters so much.

L_{\rm disk}=\frac{GM\dot{M}}{2R_{\rm in}}

Go deeper · Derivation C · 1 minGrowth of the magnetorotational instability

Derivation CGrowth of the magnetorotational instability

The $\alpha$ -prescription is only a placeholder until one shows that a real disk is actually turbulent. The decisive result is that a differentially rotating, weakly magnetized, conducting disk is linearly unstable no matter how weak the field, and grows on the orbital timescale — fast enough to turn turbulent within a few orbits. We linearize the ideal-MHD equations for axisymmetric perturbations $\propto\exp(ik_z z-i\omega t)$ of a Keplerian flow threaded by a weak vertical field $B_z$ and extract the growth rate $\tfrac34\Omega$ .

Step 1. Assemble the ingredients: the Alfvén speed $v_A=B_z/\sqrt{4\pi\rho}$ sets the magnetic tension and the epicyclic frequency $\kappa$ the orbital restoring force; for a Keplerian disk $\kappa^2=\Omega^2$ and $\mathrm{d}\Omega^2/\mathrm{d}\ln r=-3\Omega^2$ . Why: the competition between tension and the outward decrease of $\Omega$ is the entire physics of the instability.

v_A=\frac{B_z}{\sqrt{4\pi\rho}},\qquad \kappa^2=\frac{1}{r^3}\frac{\mathrm{d}(r^4\Omega^2)}{\mathrm{d}r}

Step 2. The linearized equations give a quartic dispersion relation whose constant term controls stability. Why: a negative constant term forces a root at $\omega^2<0$ , an exponentially growing mode — so its sign is the instability switch.

\omega^4-\omega^2\!\left(\kappa^2+2k^2v_A^2\right)+k^2v_A^2\!\left(k^2v_A^2+\frac{\mathrm{d}\Omega^2}{\mathrm{d}\ln r}\right)=0

Step 3. Read off the instability condition $k^2v_A^2+\mathrm{d}\Omega^2/\mathrm{d}\ln r<0$ : any outwardly decreasing $\Omega$ is unstable for long enough wavelengths. Why: a Keplerian profile falls with radius, so the criterion is met however tiny $B_z$ is — which is why disks, not stars, go unstable.

k^2v_A^2+\frac{\mathrm{d}\Omega^2}{\mathrm{d}\ln r}<0\quad\Longrightarrow\quad (kv_A)^2_{\rm max}=\frac{15}{16}\Omega^2

Step 4. Maximize over $k$ using $\mathrm{d}\Omega^2/\mathrm{d}\ln r=-3\Omega^2$ to find $|\omega|_{\rm max}=\tfrac34\Omega$ . Why: a growth rate comparable to the orbital frequency turns the disk turbulent within a few rotations — fast, robust, field-strength-independent — exactly the stress the $\alpha$ -model needed.

|\omega|_{\rm max}=\frac{3}{4}\Omega

Limiting cases

The same equations run from cold, feeble disks to super-Eddington torrents; the regimes below mark where the thin-disk picture holds and where it breaks.

Case	Result	Meaning
Radiative efficiency	$\eta=\dfrac{L_{\rm acc}}{\dot{M}c^2}=\dfrac{GM}{Rc^2}$	$\sim10^{-4}$ for white dwarfs, $\sim0.1$ – $0.2$ for neutron stars and black holes; far above the $0.007$ of fusion.
Eddington limit	$L_{\rm Edd}=\dfrac{4\pi GMm_pc}{\sigma_T}\approx1.26\times10^{38}\dfrac{M}{M_\odot}\ {\rm erg\,s^{-1}}$	Radiation pressure on electrons balances gravity; caps steady spherical accretion and hence $\dot{M}$ .
Thin disk (efficient)	$H\ll r,\quad H/r\sim c_s/v_\phi$	Cooling keeps the gas cold and thin; the Shakura–Sunyaev solution holds for $L\lesssim L_{\rm Edd}$ .
Slim / ADAF limits	$H\sim r$	At super-Eddington (slim disk) or very low rates (ADAF) the flow is thick and heat is advected inward, not radiated.

Common trapThe luminosity equals the full change in gravitational potential energy

It does not — for a thin disk it is exactly half. A Keplerian orbit at radius $r$ has total energy $-GM/2r$ , not $-GM/r$ : half the released potential energy has gone into speeding up the orbit, only the other half is radiated. So the disk luminosity is $GM\dot{M}/2R_{\rm in}$ , and the remaining half is liberated in a boundary layer where the disk meets a slowly spinning star, or advected across a black-hole horizon and lost. A second trap is to treat the viscosity as a genuine microscopic fluid property: molecular viscosity is too small by many orders of magnitude, and the effective $\alpha$ -viscosity is a stand-in for MRI-driven MHD turbulence, not a material constant of the gas.

Worked examples

1 · Eddington limit and accretion rate of a stellar-mass black hole (fully worked)

A 10 $M_\odot$ black hole accretes at its Eddington limit with efficiency $\eta=0.1$ . Find the Eddington luminosity and mass accretion rate.

Step 1. Scale the Eddington luminosity from its per-solar-mass value: $L_{\rm Edd}=1.26\times10^{38}\,(M/M_\odot)=1.26\times10^{38}\times10=\mathbf{1.26\times10^{39}\ erg\,s^{-1}}$ — roughly a thousand solar luminosities in soft X-rays.

Step 2. Invert $L=\eta\dot{M}c^2$ , using $c^2=9\times10^{20}\ {\rm cm^2\,s^{-2}}$ : $\dot{M}=L_{\rm Edd}/(\eta c^2)=1.26\times10^{39}/(0.1\times9\times10^{20})=\mathbf{1.4\times10^{19}\ g\,s^{-1}}$ .

Step 3. Convert to astronomers' units by multiplying by the seconds in a year ( $3.16\times10^{7}$ ) and dividing by $M_\odot$ : $\dot{M}\approx\mathbf{2.2\times10^{-7}\ M_\odot\,yr^{-1}}$ . A 10 $M_\odot$ hole at Eddington swallows a fifth of a millionth of a solar mass per year yet outshines a thousand Suns. For contrast, accreting a gram onto a neutron star ( $GM/R\approx1.9\times10^{20}$ erg) releases about 30 times more than fusing a gram of hydrogen ( $0.007c^2\approx6\times10^{18}$ erg) — gravity near a compact object is the richer fuel.

2 · Inner-disk temperature of that X-ray binary (fully worked)

For the same system, estimate the inner-disk temperature. Take the inner edge at the Schwarzschild ISCO, $R_{\rm in}=6GM/c^2$ .

Step 1. Locate the inner edge, with $M=10\,M_\odot=1.99\times10^{34}$ g: $R_{\rm in}=6GM/c^2=6\times(6.67\times10^{-8})(1.99\times10^{34})/(9\times10^{20})=\mathbf{8.9\times10^{6}\ cm\approx88\ km}$ .

Step 2. Form the characteristic temperature from $\sigma T_\ast^4=3GM\dot{M}/8\pi R_{\rm in}^3$ , dropping the order-unity boundary bracket. With $\sigma=5.67\times10^{-5}$ (CGS) and $\dot{M}=1.4\times10^{19}\ {\rm g\,s^{-1}}$ from Example 1: $T_\ast=[3GM\dot{M}/(8\pi\sigma R_{\rm in}^3)]^{1/4}\approx\mathbf{1.5\times10^{7}\ K}$ .

Step 3. Convert to a photon energy: $kT_\ast=(1.38\times10^{-16})(1.5\times10^{7})=2.1\times10^{-9}\ {\rm erg}=\mathbf{1.3\ keV}$ . The actual profile peak sits near $0.6$ – $0.7$ keV, since $T_{\rm eff}$ rises off zero at the edge and turns over at $r\approx1.36\,R_{\rm in}$ . The inner disk of a 10 $M_\odot$ black hole near Eddington glows at about 1 keV — squarely in the soft X-ray band, which is why fitting that spectrum pins down $R_{\rm in}$ and hence the spin.

Physics of the portrait

Accretion-disk theory has many parents, but its modern form was cast in a single 1973 paper that hid the intractable turbulence inside one Greek letter.

Origins · 1 minShakura and Sunyaev and how this was discovered

Shakura and Sunyaev

Nikolai I. Shakura b. 1945 and Rashid A. Sunyaev b. 1943 · the thin α-disk model of accretion

When Nikolai Shakura, then a graduate student in Moscow, and his supervisor Rashid Sunyaev published "Black holes in binary systems: observational appearance" in 1973, the physics of how gas actually falls onto a compact object was a morass: molecular viscosity was far too weak, and no one could compute the turbulent stress that had to replace it. Their decisive move was not to solve that problem but to parametrize it — whatever the turbulence is, its stress cannot exceed the gas pressure, so write $\nu=\alpha c_s H$ with a single $\alpha\lesssim1$ and press on. The astonishing payoff was that the disk's temperature profile, luminosity, and spectrum turned out independent of $\alpha$ altogether, fixed by conservation laws, so the model made firm, testable predictions despite its central ignorance. The paper became one of the most cited in astrophysics; Sunyaev went on to the Sunyaev–Zel'dovich effect and much of modern X-ray astronomy. The justification for their $\alpha$ — the magnetorotational instability — arrived only in 1991, when Balbus and Hawley revived a forgotten 1959 result of Velikhov and Chandrasekhar and showed it was exactly the turbulence the model needed.

History

1959

— Velikhov and Chandrasekhar find the MHD instability later renamed the MRI

1969

— Lynden-Bell argues quasars are accreting supermassive black holes in disks

1973

— Shakura and Sunyaev publish the thin α-disk model

1973

— Novikov and Thorne give the general-relativistic disk around a Kerr hole

1991

— Balbus and Hawley identify the magnetorotational instability as the disk viscosity

Going further

References. Carroll and Ostlie, An Introduction to Modern Astrophysics, 2nd ed., §17–18 (close binaries and accretion disks) · Frank, King and Raine, Accretion Power in Astrophysics, 3rd ed. · Shakura and Sunyaev, A&A 24, 337 (1973) · Balbus and Hawley, ApJ 376, 214 (1991); Rev. Mod. Phys. 70, 1 (1998) · Pringle, ARA&A 19, 137 (1981), "Accretion discs in astrophysics" · Mo, van den Bosch and White, Galaxy Formation and Evolution, Ch. 14, for the cosmological context.

Key results summary. The central numerical benchmarks of this article are: $L_{\rm Edd}\approx1.26\times10^{38}(M/M_\odot)$ erg/s; the thin-disk flux $D(r)$ and $T_{\rm eff}\propto r^{-3/4}$ from conservation with a zero-torque boundary; the luminosity integral $GM\dot{M}/2R_{\rm in}$ ; the MRI maximum growth rate $\tfrac34\Omega$ ; the ISCO efficiencies $0.057$ (Schwarzschild) and $0.42$ (extremal Kerr); and $\dot{M}_{\rm Edd}\approx2.2\times10^{-7}\,M_\odot\,{\rm yr^{-1}}$ , $kT_{\rm in}\sim1$ keV for a 10 $M_\odot$ hole.

Prerequisites. Newtonian gravity and Keplerian orbits · Fluid dynamics and the Navier–Stokes equations · Blackbody Radiation and the Planck Law · Neutron Stars · White Dwarfs. Leads to. Pulsar Magnetospheres and Spin-Down · Millisecond and Recycled Pulsars · Magnetars.

Ask the tutor: Ask me to work an accretion-rate or inner-temperature estimate for any real source — a dwarf nova, an X-ray binary, or a quasar — and watch the same equations scale across twelve orders of magnitude in luminosity.

Why does a disk radiate only half of the released gravitational energy?

Because a circular orbit stores energy as well as releasing it. By the virial theorem a Keplerian orbit at radius $r$ has total energy $-GM/2r$ , so of the potential energy liberated on the way in, exactly half goes into faster orbital motion and only half is radiated. Integrating the local heating confirms $L_{\rm disk}=GM\dot{M}/2R_{\rm in}$ , half the naive value. The rest is released in a boundary layer at a slowly spinning star, or advected across a black-hole horizon and never seen.

What actually is the α in the Shakura–Sunyaev model?

It is a dimensionless efficiency for turbulent transport, defined so the viscous stress is $\alpha$ times the pressure, or $\nu=\alpha c_s H$ . The reasoning is that eddies cannot move faster than the sound speed $c_s$ or be larger than the thickness $H$ , so their transport coefficient is bounded by $c_s H$ , with $\alpha\lesssim1$ measuring how close the real turbulence gets. It is not a fundamental constant — it hides all the unknown magnetohydrodynamics — and MRI simulations find $\alpha\sim0.01$ – $0.1$ , matching values inferred from dwarf-nova cycles.

Why is molecular viscosity not enough to make disks accrete?

Because the numbers are hopeless. The molecular viscosity of a hot, diffuse plasma gives a Reynolds number of $10^{14}$ or more and a viscous inflow time exceeding the age of the source — often the age of the Universe — by many orders of magnitude. Yet disks in cataclysmic variables and X-ray binaries evolve over days to months, so some far more efficient transport must operate; for decades its identity was the field's central unsolved problem. The answer is the magnetorotational instability, which turns a weak field and differential rotation into turbulence whose effective viscosity vastly exceeds the molecular value.

Why is the emergent spectrum flat, F_ν ∝ ν^{1/3}?

Because the disk is not one blackbody but a nested set of them, at temperatures running from $T_{\rm in}$ at the inner edge to $T_{\rm out}$ at the rim as $T\propto r^{-3/4}$ . Each annulus contributes a Planck curve near its own temperature, and summing them tiles the frequency axis into a broad plateau between the edge temperatures — a flux rising as $\nu^{1/3}$ , much flatter than one blackbody. Below $kT_{\rm out}$ it reverts to the Rayleigh–Jeans $\nu^2$ tail, and above $kT_{\rm in}$ it cuts off in the Wien tail of the hottest ring.

How does measuring a disk spectrum give a black hole's spin?

Through the inner edge. A disk cannot extend below the innermost stable circular orbit, and $R_{\rm ISCO}$ depends only on mass and spin: $6GM/c^2$ for a non-spinning hole, shrinking toward $GM/c^2$ for a maximal one. Fitting the thermal continuum yields the temperature and area of the inner edge, hence $R_{\rm in}$ ; with mass and distance known, $R_{\rm in}=R_{\rm ISCO}$ gives the spin. An independent route models the broadened iron Kα line, whose red wing stretches more the closer the disk reaches the hole. The two methods agree well enough that spins are now routine.

What happens when accretion tries to exceed the Eddington limit?

The excess radiation overpowers gravity on the infalling plasma and drives it back out. In strictly spherical accretion this caps the steady luminosity at $L_{\rm Edd}$ and the rate at $\dot{M}_{\rm Edd}=L_{\rm Edd}/\eta c^2$ . Real disks are not spherical, so they can push somewhat past the limit: the flow puffs into a thick "slim disk" that traps photons and advects much of the heat inward, while winds shed the surplus. Ultraluminous X-ray sources and the rapid early growth of the first supermassive black holes are thought to involve this mildly super-Eddington regime, where the thin-disk approximation fails.