Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 347 / 2 / b / Solution

The Shakura--Sunyaev thin disk model replaces poorly resolved turbulent angular-momentum transport by a stress proportional to pressure,Equivalently, an eddy viscosity has $\nu \sim v_{\mathrm{eddy}}{\ell }_{\mathrm{eddy}}$. Turbulent motions much faster than the sound speed $c_s$ would shock, while eddies much larger than the disk scale height $H$ would not fit within the disk. Writing $v_{\mathrm{eddy}}{\ell }_{\mathrm{eddy}}=\alpha c_{s}H$ therefore gives the alpha disk prescriptionwhere vertical hydrostatic balance gives $H\simeq c_s/{\Omega }_K$. The parameter $\alpha$ summarizes the correlation and efficiency of turbulent or magnetic stresses; it is not a molecular viscosity.