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Use a characteristic midplane temperature $T$, density $\rho$ and disk scale height $H$. For an ideal gas of fixed composition, dominant gas pressure gives $c_s^2\propto T$, where $c_s$ is the sound speed. Vertical hydrostatic equilibrium and the alpha disk prescription giveNumerical factors in the vertical averaging do not affect the exponents. The stipulated opacity becomesFor an optically thick, radiatively efficient disk, radiative diffusion gives a cooling rate per unit area proportional to $T^4/(\kappa \Sigma )$. The viscous heating rate is proportional to $\nu \Sigma {\Omega }^2$. ThusEquating heating and cooling gives $T\propto \alpha {\Sigma }^{1/2}{\Omega }^{-1/2}$, soAt fixed $\alpha$ and central mass, $\Omega \propto R^{-3/2}$. Therefore the gas-pressure alpha disk with inverse-density opacity satisfies

For thermal stability of an accretion disk, perturb $T$ at fixed $R$ and $\Sigma$, since the thermal adjustment is faster than radial mass redistribution. The logarithmic heating and cooling slopes are respectively one and two. At equilibrium, a fractional temperature increase therefore makes the excess heating $Q^{+}-Q^{-}$ decrease, and the disk cools back. Thus the local thermal equilibrium is stable within the assumed gas-pressure, radiative-diffusion closure.

For viscous stability of an accretion disk, use the thermally equilibrated relation between viscosity and surface density. It givesLocally freezing the background radius in the Keplerian viscous diffusion equation, a short radial Fourier mode has growth rate $-3k_R^2{\partial }_{\Sigma }(\nu \Sigma )<0$. Density enhancements diffuse rather than grow, so the disk is locally viscously stable as well.