Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 347 / 2 / b / Solution

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.