The parameters
The three parameters of thin disc models are the mass of the central accreting object \(M\ ,\) the accretion rate \({\dot M}\ ,\) and the viscosity parameter \(\alpha\ .\)
The equations
Newtonian hydrodynamical models of stationary and axially symmetric, thin accretion discs are described by equations similar to (or equivalent to) the 12 equations given in the table below. A "model" should give each of the 12 unknown quantities, for example the matter density \(\rho\ ,\) as a function of the radius \(R\ ,\) and the three model parameters \(\rho = \rho(R, M, {\dot M}, \alpha)\ .\)
The standard thin accretion disc equations 

Unknown functions 
(01) 
\(\rho\,V\frac{dV}{dR} = \rho (\Omega^2  \Omega_K^2 )\,R  \frac{dP}{dR}\) 
radial balance of forces 

\(\rho\) 
density of matter 
(02) 
\({\dot M}\left[ \frac{dU}{dR} + P\,\frac{d}{dR}\left(\frac{1}{\rho}\right)\right]
= 4\pi R^2 H(\tau_{R \phi}) \frac{d\Omega}{dR} + 4\pi R\,F\) 
energy conservation 

\(V\) 
radial velocity 
(03) 
\({\dot M} = 4\pi\,R\,H\rho\,V\) 
mass conservation 

\(\Omega\) 
angular velocity of matter 
(04) 
\({\dot M}\left(j  j_{0}\right) = 4\pi\,R^2\,H(\tau_{R\phi})  X_{0}\) 
angular momentum conservation 

\(P\) 
pressure 
(05) 
\(F = \frac{acT^{4}}{[\kappa\,\rho\,H]}\) 
vertical radiative transfer 

\(U\) 
internal energy (thermodynamics) 
(06) 
\(\frac{C_s}{\Omega_K\,R} = \frac{H}{R}\) 
vertical momentum balance 

\(H\) 
vertical thickness 
(07) 
\((\tau_{R\phi}) = \rho[\alpha\,C_s\,H]R\frac{d\Omega}{dR}\) 
"viscous" torque \(\alpha\)prescription 

\((\tau_{R\phi})\) 
viscous torque 
(08) 
\(P = P(\rho, T)\) 
equation of state 

\(F\) 
flux of radiation 
(09) 
\(U = U(\rho, T)\) 
equation of state (internal energy) 

\(T\) 
temperature 
(10) 
\(C_s = C_s(\rho, T)\) 
equation of state (sound speed) 

\(j\) 
angular momentum 
(11) 
\(\kappa = \kappa (\rho, T)\) 
opacity 

\(\kappa\) 
opacity coefficient 
(12) 
\(j = \Omega\,R^2\) 
angular momentum and angular velocity 

\(C_s\) 
sound speed 
The gravitation field of the central object enters the above Newtonian equations only through the Keplerian angular velocity \(\Omega_K(R)\ .\) In the general version of (01)(12) the gravity (i.e. the spacetime curvature) enters also through components of the metric tensor \(g_{\mu\nu} = g_{\mu\nu}(R)\) the equatorial plane. The Kerr geometry version of (01)(12) was written first by Lasota (1994), and later elaborated by Abramowicz, Chen, Granath and Lasota (1996). However, equations (01)(12) in their form above are often used to model the black hole accretion discs. This is possible because of a brilliant discovery by Paczynski of the Newtonian model for the black hole gravity.
Particular models make several additional simplifying assumptions. For example, several models assume that \(\Omega(R) = \Omega_K(R)\ ,\) with \(\Omega_K(R)\) being the Keplerian angular velocity, which is known since the gravitational field of the central object is known (for a spherical body with the mass \(M\) Newton's theory yields \(\Omega_K(R) = (GM/R^3)^{1/2}\)). Note, that in this case the derivative \(d\Omega/dR\) that appears in equations (02) and (07) becomes a known function of \(R\ .\) Equation (07) postulates the form of the "viscous" stress \((\tau_{R\phi})\) in therms of an ad hoc ansatz that introduces the dimensionless \(\alpha\)viscosity. Note that the quantity that appears in square bracket is called in hydrodynamics the "kinematic viscosity". The original ShakuraSunyaev ansatz postulated \((\tau_{R\phi}) = \alpha\,P\ .\) Equation (05) gives the flux of radiation in (a very rough) diffusion approximation. Note that the quantity in square brackets in this equation is the optical depth, \(\tau = [\kappa\,\rho\,H]\) in the vertical direction. The equation is valid only if \(\tau \gg 1\ ,\) and if
\(\tau < 1\) nonthermal radiative processes should be considered, and equation (05) replaced by \(F = F(\rho, T)\ .\)
In equation (04), \(j_{0}, X_{0}\) are the angular momentum and the viscous torque at some undefined radius \(R_{0}\ .\) In the black hole accretion discs models, it is customary to take \(R_{0} = R_G = (black ~horizon ~hole ~radius)\ ,\) because the viscus torque at the horizon vanishes. Then, \(j_{0}\) is the (a priori unknown) angular momentum of matter at the horizon. With respect to first order derivatives, quations (01)(12) form a linear system that may be solved for each derivative. For \(dV/dR\) this gives,
\[ \frac{dV}{dR} = \frac{N(R, \rho, V, \Omega, ...)}{V^2  C^2_s}.\]
Any black hole accretion flow must be transonic, i.e. somewhere it must pass the sonic radius \(R_s\ ,\) where \(V(R_s) = C_s(R_s)\ .\) In order that \(dV/dR\) and all other derivatives are nonsingular there, it must be,
\[N(R_s, \rho, V, \Omega, ...) = 0.\]
The above sonic point regularity condition makes the system (01)(12) over constrained, i.e. an eigenvalue problem, with the eigenvalue being the angular momentum at the horizon, \(j_{0}\ .\)
Analytic models describe black hole accretion discs down to a certain "inner edge" \(R_{in}\) which locates close to the central accreting object. The inner edge is a theoretical concept introduced for convenience, because at \(R \approx R_{in} \approx R_s\) the accretion flow changes its character. In the case of the black hole accretion, the change goes from almost circular orbits to almost radial free fall. It is therefore convenient to separately model the two regions\[R > R_{in}\] where matter moves on circular orbits, and \(R < R_{in}\) where matter free falls. Of course, in reality the situation is more complicated, as the change of the flow character occurs smoothly in an extended region on both sides of \(R_{in}\ .\)
For black hole accretion, \(R_{MB} \le R_{in} \le R_{ISCO}\ .\) For very efficient ShakuraSunyaev discs, \(R_{in} \approx R_{ISCO}\ ,\) while for RIFs \(R_{in} \approx R_{MB}\ .\) For stellar accretion, \(R_{in}\) is located near the surface of the star and the flow there is described by a boundary layer model.
For more details, see Beckwith, Hawley and Krolik (2008) and references quoted there.
A catalog of the analytic and semianalytic accretion disc models
Model's name

Short characteristic

Links to online references

ShakuraSunyaev
=standard=
=thin disc=

Axially symmetric, stationary, local analytic model. Explicit formulae
give all physical characteristics in terms of M, Mdot,
alpha and R. Geometrically thin in the vertical
direction (H/R < 1), has a disclike shape).
Accretion rate very subEddington. Opacity very high. The gas goes down
on tight spirals, approximated by circular, free (Keplerian, geodesic)
orbits. For black hole and (very compact) neutron star the
inner edge at ISCO. High luminosity, high efficiency of radiative
cooling. Electromagnetic spectra not much different from that of
a sum of black bodies. Alpha viscosity prescription. Diffusion
approximation for radiative transfer. Dynamically stable. When the
gas is cold and radiation pressure negligible also thermally and
viscously stable, otherwise unstable in both respects.
Applications: YSOs, CVs, LMXRB, AGNs. The best known
and studied theoretical model.

Standard reference:
Shakura, Sunyaev (1974)
, one of the most often quoted papers in modern astrophysics
(quotation counts).
Similar ideas:
Pringle, Rees (1972);
LyndenBell, Pringle (1974)
Fully relativistic version (Novikov, Thorne 1974); for the detailed description see:
Page, Thorne (1974).
Recent application to spectral fits:
Shafee et al. (2006);
Middleton et al. (2006).
Recommended review:
Pringle (1981).

ShakuraSunyaev
=modifications=

Thin discs: strong opacity variation with temperature 
???? 
Thin discs: warps 
???? 
Thin discs: selfgravity 
???? 
Adafs

SubEddington accretion, very small opacity.
Adafs are cooled by advection (heat captured by moving matter)
rather than by radiation. They are very radiatively inefficient,
geometrically extended, similar in shape to a sphere (or a "corona")
rather than a disc, and very hot (close to the virial temperature).
Because of their low efficiency, adafs are much less luminous than
the ShakuraSunyaev thin discs. Adafs emit a powerlaw, nonthermal
radiation, often with a strong Compton component. For black hole
and (very compact) neutron star the inner edge at a radius
smaller than ISCO. Dynamically, thermally and viscously stable.
Applications: mostly LMXRB, AGNs, with good fits to observed spectra.
Numerical 1.5D (vertically integrated) stationary transonic models (in Kerr):
Abramowicz et al. (1996);
Narayan et al. (1997);
Popham, Gammie (1998)
Numerical 2D nonstationary models (in PaczynskiWiita):
Igumenshev et al. (1996).

Most influential paper, describing a Newtonian, selfsimilar, stationary, axially symmetric analytic, model:
Narayan, Yi (1994).
Immediate followup by the same group:
Narayan, Yi (1995);
Narayan, Mahadevan (1995);
Narayan et al. (1996).
The idea mentioned first time:
Ichimaru (1987),
see also:
Rees et al. (1982);
Abramowicz et al. (1995).
Recommended review:
Narayan, McClintock (2008).

Slim

Nearly Eddington accretion. Large opacity. Cooled by radiation and
strong advection. Radiatively less efficient than ShakuraSunyaev.
H/R slightly less than one. For black hole and (very compact) neutron
star the inner edge at a radius smaller than ISCO. Dynamically, thermally
and viscously stable.
Applications: mostly LMXRB, AGNs, with good fits to observed spectra.
??? ??? ???

??? ??? ???

KluzniakKita

Fully two dimensional analytic solution (stationary, axially symmetric)
obtained through a mathematically exact expansion in the small parameter
H/R of the equations of viscous hydrodynamics. Significant backflows in the
midplane of the disk have been found.

Kluzniak, Kita (2000),
numerical follow up:
Umurhan et al. (2006).


Lines correspond to fixed M, r, and \(\alpha\ .\) An example of each of the four branches is shown
in a corresponding color: pink, blue, green, and yellow. The congruence of all branches has a critical point, corresponding to
\(\alpha = \alpha_{crit}\ .\) In different places of the parameter space, the cooling is
dominated by black body radiation, bremsstrahlung, Compton losses, pair production, or by
advection, as indicated by arrows. Figure adapted from
Björnsson et al. (1996).
Branch I (blue): ShakuraSunyaev (gas pressure) + ShakuraSunyaev (radiation pressure) + Slim.
Branch II (green): ShakuraSunyaev (gas pressure) + SLE
Branch III (yellow): SLE + Adaf.
Branch IV (pink): Polish doughnut.

Figure 1: The four branches of analytic models of accretion discs.

