Constant-opacity gas-pressure disk viscosity law ($\bar{\nu }\propto {\alpha }^{4/3}{\Sigma }^{2/3}{\Omega }^{-2/3}$)

The density-weighted mean kinematic viscosity of the universal vertical profile is $\bar{\nu }=\alpha \Omega h^2{\int }_{-{\zeta }_s}^{{\zeta }_s}Pd\zeta$. Substitution of the height scale gives $\bar{\nu }=C{\alpha }^{4/3}(\kappa /\sigma {)}^{1/3}{\mathcal{R}}^{4/3}{\Sigma }^{2/3}{\Omega }^{-2/3}$. For Keplerian rotation, ${\Omega }^{-2/3}=(GM{)}^{-1/3}r$. The positive numerical constant depends only on the dimensionless vertical profile and the convention for $\alpha$.

// Target: astrophysics.bigb