= Gas-pressure alpha disk with inverse-density opacity
{title2=$\kappa\propto\rho^{-3/2}T^{5/4}\quad\Longrightarrow\quad\nu\propto\Sigma^{1/2}R^{9/4}$}
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.
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