Shakura–Sunyaev alpha disk model

Remarks

Under these assumptions, $\Sigma$ obeys the diffusion equation $$\frac{\partial \Sigma}{\partial t} = \frac{3}{R} \frac{\partial}{\partial R} \left[ R^{1/2} \frac{\partial}{\partial R} \left( \nu \Sigma R^{1/2} \right) \right].$$ A steady disk with accretion rate $\dot M$ and zero torque at an inner radius $R_*$ satisfies $$\nu \Sigma = \frac{\dot M}{3\pi} \left( 1 - \sqrt{R_*/R} \right).$$ The heat released per unit area on each face is $D(R) = \frac{9}{8} \nu \Sigma \Omega^2 = \frac{3 G M \dot M}{8 \pi R^3} \left( 1 - \sqrt{R_*/R} \right)$. If each face radiates as a black body of temperature $T$, then $\sigma_{SB} T^4 = D(R)$, where $\sigma_{SB}$ is the Stefan–Boltzmann constant, so $T \propto R^{-3/4}$ for $R \gg R_*$.

Example: a disk with $c_s/(\Omega R) = 0.01$ and $\alpha = 0.1$ satisfies all three clauses; its viscosity is $\nu = 0.1 \, c_s^2/\Omega = 10^{-5} \, \Omega R^2$.

Non-example: a thick torus with $H$ comparable to $R$ fails clause 2, and its pressure support makes the rotation non-Keplerian, so it is not an alpha disk in this sense.

The bound $\alpha \le 1$ expresses that turbulent eddies are at most of size $H$ and at most sonic.