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

An optically thick annulus radiates as a blackbody from both faces, soFor a steady Keplerian accretion disk with a zero-torque inner boundary condition,ConsequentlyandAway from the inner edge, $T\propto R^{-3/4}$.

Ignoring inclination and distance factors, the multitemperature blackbody disk spectrum isSet $x=h\bar{\nu }/(k_{B}T)$. Since $T\propto R^{-3/4}$, $RdR\propto {\bar{\nu }}^{-8/3}{x}^{5/3}dx$, whereas the Planck function contributes ${\bar{\nu }}^3/(e^x-1)$. In the stated intermediate-frequency range the radial endpoints become $0$ and $\infty$, leaving a frequency-independent convergent integral. Therefore

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.

Steady mass conservation gives $\dot{M}=-2\pi R\Sigma u_R$. Substituting this into the angular-momentum equation and integrating from the innermost stable circular orbit with zero torque givesThereforeUsing $\nu =\alpha c_{s}H$, $c_s\simeq H\Omega$, $l=R{u}_{\varphi }$, and $Rd\Omega /dR$ of order $-\Omega$ givesup to the order-unity Keplerian factor $3/2$.

At the sonic transition, $|u_R|\simeq c_s\simeq (H/R)u_{\varphi }$. Hencefor a geometrically thin disk with $\alpha \ll 1$. The specific angular momentum therefore differs only fractionally from $l_{\mathrm{ISCO}}$ before the gas enters the plunging region of a black-hole accretion disk. Its much shorter inflow time then prevents appreciable viscous transport, justifying angular-momentum conservation across the ISCO and the zero-torque boundary condition.