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In the Toomre stability criterionthe disk's gravitational potential supplies the destabilizing attraction, gas pressure represented by $c_s$ suppresses short wavelengths, and the Keplerian shear represented by the angular frequency $\Omega$ prevents coherent collapse on long scales through epicyclic motion. Axisymmetric perturbations are stable for $Q>1$ and gravitationally unstable for $Q<1$.

The state $Q\sim 1$ is a self-regulating fixed point. If cooling lowers $c_s$ until $Q<1$, gravitational instability produces shocks and turbulence that heat the disk and raise its effective velocity dispersion. If $Q$ becomes appreciably larger than one, the instability and its heating switch off, while turbulent dissipation and radiative cooling reduce $c_s$. A sustained gravito-turbulent state therefore remains close to marginal stability.

The standard Shakura--Sunyaev thin disk has three qualitative radial zones. Its hot inner part is dominated by radiation pressure and electron-scattering opacity; farther out, gas pressure overtakes radiation pressure while electron scattering can remain the main opacity; in the cool outer zone, gas pressure remains dominant and free-free opacity becomes important. The transition radii vary with $M$, $\dot{M}$, and $\alpha$.

For the inner zone, hold the surface density $\Sigma$ fixed during a local thermal perturbation. Vertical hydrostatic equilibrium givesso $H\propto T_c^4/\Sigma$. The alpha disk prescription then yieldswhereas optically thick radiative diffusion with nearly constant electron-scattering opacity givesAt equilibrium $Q^{+}=Q^{-}=Q_0$. For net cooling $\dot{Q}=Q^{-}-Q^{+}$,Thus the total-pressure alpha prescription predicts thermal instability of a radiation-pressure-dominated alpha disk: a temperature increase makes heating outrun cooling. The absence of ubiquitous, large-amplitude thermal limit cycles in luminous AGN light curves indicates that this local model omits stabilizing effects, plausibly magnetic pressure and stress, vertical advection, winds, or a stress law that does not simply track total pressure.