Shakura-Sunyaev Accretion Disk Models

• Shakura-Sunyaev models are accretion disk frameworks that employ a constant or radius-dependent α parameter to parameterize angular momentum transport and enable tractable disk calculations.
• They encompass various branches including thin, radiation-pressure-dominated, and supercritical regimes, integrating insights from GRMHD simulations and thermal instability analyses.
• Extensions to other accretion environments illustrate the models' adaptability, as they incorporate finite thickness, magnetic support, and relativistic corrections to traditional disk theory.

Shakura-Sunyaev models are accretion-disk models organized around the assumption that angular-momentum transport can be represented by a dimensionless stress parameter $\alpha$, with the stress written relative to pressure. In the modern literature, the term covers the classical geometrically thin, optically thick $\alpha$-disk, the radiation-pressure-dominated and supercritical/super-Eddington branches, and a growing set of extensions in which $\alpha$ is radius-dependent, magnetic pressure contributes substantially to vertical support, or the emitting surface departs from the razor-thin approximation (Penna et al., 2012, Vinokurov et al., 2013). Their continuing importance lies in the fact that they compress poorly understood transport physics into a tractable closure while remaining flexible enough to be recalibrated against GRMHD, GRRMHD, and radiation-hydrodynamic calculations.

1. Canonical closure and thin-disk structure

The classical closure is the stress-pressure relation

$T_{\hat r \hat \phi}=\alpha p,$

or, in closely related notation,

$$W_{r\phi}=\alpha P_{\rm tot}, \qquad \mathbb{T}=\alpha p,$$

with $\alpha$ treated as a dimensionless ratio of stress to pressure. In the standard formulation, the disk is geometrically thin,

$H/R \ll 1,$

optically thick, and in nearly Keplerian rotation; vertical support and radiative cooling are treated in a vertically averaged framework, and the closure makes one-dimensional or semi-analytic disk calculations feasible despite the unresolved microphysics of turbulence (Abramowicz et al., 11 Mar 2026, Berdina et al., 2020).

In vertically integrated form, the conservation laws may be written as

$\dot{M} = \int^{+H}_{-H} 2\pi r \, \rho v \, dz,$

$$\dot{J} = \dot{M}\,j + \mathbb{T}, \qquad \dot{E} = \dot{M}\,e + \Omega \mathbb{T},$$

so that the local radiative flux follows as

$$F = \frac{\dot{M}}{4\pi r}\left( \frac{d\Omega}{dr} \right) \big[ j(r) - j_0 \big].$$

In the Newtonian thin-disk limit with $\alpha$0, this becomes

$\alpha$1

The standard thermal structure implies

$\alpha$2

which leads to the well-known size-wavelength scaling

$\alpha$3

That scaling underlies continuum-lag and microlensing analyses of thin disks in AGN, although several recent applications conclude that observed sizes can exceed the bare thin-disk prediction (Berdina et al., 2020).

2. Radiation-pressure-dominated thin disks and classical instability

Within the thin-disk branch, the most delicate regime is the luminous inner region in which radiation pressure dominates and opacity is mainly due to electron scattering. In that regime, the standard thermal-instability argument follows from the different pressure dependences of heating and cooling at fixed surface density: $\alpha$4 An upward perturbation in $\alpha$5 therefore makes heating rise faster than cooling, while a downward perturbation produces collapse. The same branch is also expected to be viscously unstable in the Lightman-Eardley sense (Mishra et al., 2022).

Recent global GRRMHD calculations have sharpened rather than erased that classical result. In simulations initialized from a Shakura-Sunyaev disk with

$\alpha$6

corresponding to

$\alpha$7

thin radiation-pressure-dominated disks remain thermally unstable when the magnetic field topology fails to build sustained midplane magnetic support. The dipole and multi-loop zero-net-flux configurations collapse vertically on the local thermal timescale and settle to lower-$\alpha$8, lower-luminosity states. This reproduces the classical expectation of thermal collapse, but with MRI-generated stresses rather than an imposed viscous source term (Mishra et al., 2022).

The same instability language also underlies time-dependent branch transitions. In a radiation-hydrodynamic model of Swift J1644+57, the adopted closure is explicitly the $\alpha$9-prescription in which the viscous stress is proportional to the total pressure, with $\alpha$0. There the source is interpreted as executing a limit cycle between a supercritically accreting slim disk and a subcritically accreting Shakura-Sunyaev disk. The slim-to-SSD transition begins in the inner disk and propagates outward on a timescale of about a day, while the reverse transition is predicted after

$\alpha$1

provided the supply rate remains above the threshold for a radiation-pressure-dominated disk (Kawashima et al., 2013).

3. Supercritical and slim-disk branches

The original framework also contains a supercritical extension. When the outer inflow exceeds the Eddington accretion rate,

$\alpha$2

the inner disk becomes thick inside the spherization radius,

$\alpha$3

with $\alpha$4 and a wind funnel of characteristic half-opening angle

$\alpha$5

In this regime the local accretion rate declines inward because radiation-driven mass loss removes material from the flow: $\alpha$6 The temperature profile flattens from the thin-disk law to

$\alpha$7

and the luminosity rises only logarithmically above Eddington,

$\alpha$8

Outside $\alpha$9, the thin outer disk contributes $T_{\hat r \hat \phi}=\alpha p,$0; inside it, the supercritical region contributes $T_{\hat r \hat \phi}=\alpha p,$1 (Vinokurov et al., 2013).

A direct observational consequence of the $T_{\hat r \hat \phi}=\alpha p,$2 law is

$T_{\hat r \hat \phi}=\alpha p,$3

or equivalently a flat

$T_{\hat r \hat \phi}=\alpha p,$4

segment. In the supercritical accretion-disk model for ULXs, that flat component is expected in the soft X-ray range around $T_{\hat r \hat \phi}=\alpha p,$5–$T_{\hat r \hat \phi}=\alpha p,$6, while the irradiated wind/funnel produces broad UV/optical emission. Applied to Holmberg II X-1, NGC 6946 ULX-1, NGC 1313 X-1, NGC 1313 X-2, and NGC 5408 X-1, the fitted parameters imply stellar-mass black holes, typically $T_{\hat r \hat \phi}=\alpha p,$7–$T_{\hat r \hat \phi}=\alpha p,$8, with

$T_{\hat r \hat \phi}=\alpha p,$9

rather than intermediate-mass black holes (Vinokurov et al., 2013).

The same supercritical logic has been invoked for quasars whose continuum lags exceed thin-disk expectations. For Q2237+0305, the measured rest-frame lags

$$W_{r\phi}=\alpha P_{\rm tot}, \qquad \mathbb{T}=\alpha p,$$0

are larger, and less chromatic, than the standard thin-disk prediction. A super-Eddington scattering envelope is therefore used, with effective radius and optical photosphere estimated from the Shakura-Sunyaev supercritical expressions

$$W_{r\phi}=\alpha P_{\rm tot}, \qquad \mathbb{T}=\alpha p,$$1

$$W_{r\phi}=\alpha P_{\rm tot}, \qquad \mathbb{T}=\alpha p,$$2

$$W_{r\phi}=\alpha P_{\rm tot}, \qquad \mathbb{T}=\alpha p,$$3

In that interpretation, the observed reverberation traces the larger envelope photosphere rather than bare thin-disk annuli (Berdina et al., 2020).

4. Radius-dependent $$W_{r\phi}=\alpha P_{\rm tot}, \qquad \mathbb{T}=\alpha p,$$4 and relativistic generalizations

The best-developed modernization of the classical closure keeps the stress prescription but replaces $$W_{r\phi}=\alpha P_{\rm tot}, \qquad \mathbb{T}=\alpha p,$$5 with $$W_{r\phi}=\alpha P_{\rm tot}, \qquad \mathbb{T}=\alpha p,$$6. GRMHD simulations show that $$W_{r\phi}=\alpha P_{\rm tot}, \qquad \mathbb{T}=\alpha p,$$7 varies systematically with radius, that Newtonian calculations tend to produce smaller values than relativistic ones, and that two distinct physical zones must be separated: an outer MRI-turbulent region and an inner laminar or plunging region dominated by stretched large-scale magnetic fields (Penna et al., 2012).

For the turbulent outer flow, the key quantity is the dimensionless shear parameter

$$W_{r\phi}=\alpha P_{\rm tot}, \qquad \mathbb{T}=\alpha p,$$8

with $$W_{r\phi}=\alpha P_{\rm tot}, \qquad \mathbb{T}=\alpha p,$$9. In Newtonian Keplerian disks, $\alpha$0 everywhere. For equatorial circular Kerr geodesics,

$\alpha$1

Using shearing-box guidance and GRMHD calibration, the turbulent stress is taken to scale as $\alpha$2 with $\alpha$3, leading, for thin disks, to the practical formula

$\alpha$4

Far from the hole, $\alpha$5 and one recovers $\alpha$6; inward, relativistic enhancement of the shear raises $\alpha$7. Near and inside the ISCO, a separate mean-field term is added,

$\alpha$8

while for $\alpha$9 only the mean-field term is retained. In slim-disk solutions, replacing constant $H/R \ll 1,$0 by $H/R \ll 1,$1 matters mainly between the ISCO and $H/R \ll 1,$2: the larger inward-rising $H/R \ll 1,$3 increases radial velocity by a factor $H/R \ll 1,$4–$H/R \ll 1,$5, lowers the midplane temperature by about $H/R \ll 1,$6, and, at high accretion rates, reduces the radiative flux because advection becomes stronger (Penna et al., 2012).

A second relativistic generalization argues that the radial shape of $H/R \ll 1,$7 may be approximately universal across independent GRMHD simulations. The reported common features are: $H/R \ll 1,$8 at the horizon, a maximum close to the circular photon orbit, and much smaller values at large radius than in the plunging region. In Schwarzschild form, the proposed fit behaves like

$H/R \ll 1,$9

or, in invariant language,

$\dot{M} = \int^{+H}_{-H} 2\pi r \, \rho v \, dz,$0

For the Lančová et al. puffy-disc simulation, the best fit is

$\dot{M} = \int^{+H}_{-H} 2\pi r \, \rho v \, dz,$1

This suggests that the outer asymptotics of $\dot{M} = \int^{+H}_{-H} 2\pi r \, \rho v \, dz,$2 remain tied to local MRI physics, while the rise, turnover, and horizon zero are controlled by global relativistic geometry. The authors emphasize, however, that the claimed universality is strongest in shape rather than normalization, and that the simulations compared do not use identical definitions of stress and pressure (Abramowicz et al., 11 Mar 2026).

5. Magnetic support, finite thickness, and departures from the classical branch

One of the central revisions of the classical picture is the recognition that a luminous thin disk need not be thermally unstable if magnetic support becomes dynamically important. In global GRRMHD simulations of radiation-pressure-dominated disks, the key stabilizing condition is

$\dot{M} = \int^{+H}_{-H} 2\pi r \, \rho v \, dz,$3

The vertical-field model reaches this threshold, remains stable for many thermal timescales, and achieves heating that matches or exceeds cooling. The quadrupole model is intermediate, while the dipole and multi-loop cases collapse. The stable magnetic disks retain approximately the intended

$\dot{M} = \int^{+H}_{-H} 2\pi r \, \rho v \, dz,$4

but they are “puffier,” with

$\dot{M} = \int^{+H}_{-H} 2\pi r \, \rho v \, dz,$5

at $\dot{M} = \int^{+H}_{-H} 2\pi r \, \rho v \, dz,$6, and surface densities larger than standard by factors of $\dot{M} = \int^{+H}_{-H} 2\pi r \, \rho v \, dz,$7–$\dot{M} = \int^{+H}_{-H} 2\pi r \, \rho v \, dz,$8. They are therefore not exact Shakura-Sunyaev disks with MRI replacing viscosity; they are modified thin disks supported primarily by magnetic pressure (Mishra et al., 2022).

An even stronger departure appears in cosmological radiation-MHD simulations of quasar fueling. There the accretion flow remains optically thick and radiatively cooled, but the pressure is dominated by primarily toroidal magnetic fields, with plasma

$\dot{M} = \int^{+H}_{-H} 2\pi r \, \rho v \, dz,$9

in the abstract and more generally

$$\dot{J} = \dot{M}\,j + \mathbb{T}, \qquad \dot{E} = \dot{M}\,e + \Omega \mathbb{T},$$0

in the body of the paper, even in the midplane. The disk has

$$\dot{J} = \dot{M}\,j + \mathbb{T}, \qquad \dot{E} = \dot{M}\,e + \Omega \mathbb{T},$$1

in the inner region, inflow rates of order

$$\dot{J} = \dot{M}\,j + \mathbb{T}, \qquad \dot{E} = \dot{M}\,e + \Omega \mathbb{T},$$2

and densities at $$\dot{J} = \dot{M}\,j + \mathbb{T}, \qquad \dot{E} = \dot{M}\,e + \Omega \mathbb{T},$$3 far below Shakura-Sunyaev expectations at fixed $$\dot{J} = \dot{M}\,j + \mathbb{T}, \qquad \dot{E} = \dot{M}\,e + \Omega \mathbb{T},$$4. Removing magnetic fields produces catastrophic fragmentation and leaves a razor-thin, much smaller, lower-$$\dot{J} = \dot{M}\,j + \mathbb{T}, \qquad \dot{E} = \dot{M}\,e + \Omega \mathbb{T},$$5 Shakura-Sunyaev-like disk. This suggests that, under realistic quasar-feeding boundary conditions, the classical weak-field branch may be replaced by a flux-fed magnetically dominated one (Hopkins et al., 2023).

Finite disk thickness also affects observables even when the underlying model remains Shakura-Sunyaev-like. In relativistic ray-tracing simulations of X-ray reflection, a radiation-pressure-dominated Shakura-Sunyaev surface is implemented through

$$\dot{J} = \dot{M}\,j + \mathbb{T}, \qquad \dot{E} = \dot{M}\,e + \Omega \mathbb{T},$$6

with $$\dot{J} = \dot{M}\,j + \mathbb{T}, \qquad \dot{E} = \dot{M}\,e + \Omega \mathbb{T},$$7. Compared with a flat infinitesimally thin reflector, finite thickness enhances illumination of the inner disk, strengthens the red wing of the Fe K$$\dot{J} = \dot{M}\,j + \mathbb{T}, \qquad \dot{E} = \dot{M}\,e + \Omega \mathbb{T},$$8 line, suppresses some less-redshifted outer-disk contribution, and, if ignored in fitting, tends to underestimate black-hole spin, corona height, and inclination angle. Warped disks are reported to be incompatible with the flat-disk approximation (Surgent et al., 23 Apr 2026).

6. Extensions to other accretion environments and model limitations

Shakura-Sunyaev structure has also been extended far beyond relativistic black-hole disks. In protoplanetary and young-stellar-object applications, the standard thin-disk equations are combined with the induction equation and ionization balance, while keeping the Shakura-Sunyaev hydrodynamic scaffold. The turbulent viscosity is written in dynamic form as

$$\dot{J} = \dot{M}\,j + \mathbb{T}, \qquad \dot{E} = \dot{M}\,e + \Omega \mathbb{T},$$9

and the disk obeys the usual steady thin-disk relations

$$F = \frac{\dot{M}}{4\pi r}\left( \frac{d\Omega}{dr} \right) \big[ j(r) - j_0 \big].$$0

augmented by ohmic diffusion, ambipolar diffusion, buoyancy, and ionization from cosmic rays, X-rays, and thermal processes. In this extension, the magnetic field is quasi-azimuthal near the inner boundary, quasi-poloidal in dusty “dead” zones, and quasi-radial in parts of the outer disk when grain growth or enhanced ionization improves coupling. The inner boundary of the dead zone is reported at

$$F = \frac{\dot{M}}{4\pi r}\left( \frac{d\Omega}{dr} \right) \big[ j(r) - j_0 \big].$$1

the outer boundary at

$$F = \frac{\dot{M}}{4\pi r}\left( \frac{d\Omega}{dr} \right) \big[ j(r) - j_0 \big].$$2

and the solid mass in the dead zone exceeds $$F = \frac{\dot{M}}{4\pi r}\left( \frac{d\Omega}{dr} \right) \big[ j(r) - j_0 \big].$$3 for stars with $$F = \frac{\dot{M}}{4\pi r}\left( \frac{d\Omega}{dr} \right) \big[ j(r) - j_0 \big].$$4 (Dudorov et al., 2014).

Across these variants, several limitations recur. The coefficients in GRMHD-calibrated $$F = \frac{\dot{M}}{4\pi r}\left( \frac{d\Omega}{dr} \right) \big[ j(r) - j_0 \big].$$5 laws, such as $$F = \frac{\dot{M}}{4\pi r}\left( \frac{d\Omega}{dr} \right) \big[ j(r) - j_0 \big].$$6 and $$F = \frac{\dot{M}}{4\pi r}\left( \frac{d\Omega}{dr} \right) \big[ j(r) - j_0 \big].$$7 in

$$F = \frac{\dot{M}}{4\pi r}\left( \frac{d\Omega}{dr} \right) \big[ j(r) - j_0 \big].$$8

are explicitly empirical and expected to change as better simulations become available. Thermal stability in radiation-pressure-dominated disks depends strongly on magnetic topology and simulation duration. Supercritical ULX spectral models omit Comptonization in the inner hot wind and neglect geometric beaming in the baseline fits. The proposed universal $$F = \frac{\dot{M}}{4\pi r}\left( \frac{d\Omega}{dr} \right) \big[ j(r) - j_0 \big].$$9 behavior is assembled from simulations that do not use identical stress or pressure definitions. Reflection calculations with finite-thickness Shakura-Sunyaev surfaces still assume Keplerian orbital motion even in very thick or super-Eddington-like cases (Penna et al., 2012, Mishra et al., 2022, Vinokurov et al., 2013, Abramowicz et al., 11 Mar 2026, Surgent et al., 23 Apr 2026).

Taken together, these developments define “Shakura-Sunyaev models” less as a single immutable solution than as a modeling lineage. The enduring element is the stress closure

$\alpha$00

or its close variants, coupled to thin-disk or slim-disk structure equations. What has changed is the status of the assumptions that once accompanied it: constant $\alpha$01, negligible magnetic support, a uniquely thin emitting surface, and a universal instability of radiation-pressure-dominated disks are all now treated as contingent rather than generic. In that sense, current work preserves the Shakura-Sunyaev framework in spirit while systematically replacing its least robust idealizations.