Gas-pressure alpha disk with inverse-density opacity ($\kappa \propto {\rho }^{-3/2}{T}^{5/4}⟹\nu \propto {\Sigma }^{1/2}{R}^{9/4}$)

For an optically thick alpha disk dominated by gas pressure, vertical hydrostatic equilibrium gives $H\propto T^{1/2}/\Omega$ and $\rho \propto \Sigma \Omega T^{-1/2}$. The specified opacity therefore scales as $\kappa \propto {\Sigma }^{-3/2}{\Omega }^{-3/2}{T}^2$. Viscous heating scales as $Q^{+}\propto \alpha \Sigma T\Omega$, while radiative diffusion gives $Q^{-}\propto T^4/(\kappa \Sigma )\propto {\Sigma }^{1/2}{\Omega }^{3/2}{T}^2$. Their balance gives $T\propto \alpha {\Sigma }^{1/2}{\Omega }^{-1/2}$, hence $\nu \propto {\alpha }^2{\Sigma }^{1/2}{\Omega }^{-3/2}$. Fixed $\alpha$ and Keplerian rotation yield the displayed law.

The local thermal stability of an accretion disk follows because, at fixed surface density and radius, cooling has logarithmic temperature slope two while heating has slope one. The viscous stability of an accretion disk follows from ${\partial }_{\Sigma }(\nu \Sigma )>0$ on the thermal-equilibrium branch. Both conclusions apply within this pressure, opacity and transport closure.