# Accretion discs/Analytic models of accretion discs/Thin discs

## 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 Shakura-Sunyaev 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 $$\tau < 1$$ non-thermal 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 non-singular 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 Shakura-Sunyaev 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 semi-analytic accretion disc models

 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): Shakura-Sunyaev (gas pressure) + Shakura-Sunyaev (radiation pressure) + Slim. Branch II (green): Shakura-Sunyaev (gas pressure) + SLE Branch III (yellow): SLE + Adaf. Branch IV (pink): Polish doughnut. Chen et al. Unified description of accretion flows around black holes, Ap.J., 443, L61 (195) Figure 1: The four branches of analytic models of accretion discs.