Shakura--Sunyaev thin disk

The Shakura--Sunyaev thin disk is a geometrically thin, optically thick, radiatively efficient accretion-disk model whose turbulent stress is parametrized by the alpha prescription.

A multitemperature blackbody disk is an optically thick disk approximated as concentric blackbody annuli of different effective temperatures. If $T\propto R^{-3/4}$, its intermediate-frequency spectrum satisfies $S_{\nu }\propto {\nu }^{1/3}$.

For a blackbody disk with $T\propto r^{-3/4}$, the substitution $x=h\nu /(k_{B}T)$ in $F_{\nu }\propto {\nu }^3\int r(e^x-1{)}^{-1}dr$ produces $F_{\nu }\propto {\nu }^{1/3}\int x^{5/3}(e^x-1{)}^{-1}dx$. When the inner and outer temperatures make the limits approximately zero and infinity, that convergent integral is independent of frequency. The law applies between the outer-edge turnover and the hot inner-disk cutoff.

For a finite multitemperature blackbody disk, the low-frequency Rayleigh-Jeans law gives $F_{\nu }\propto {\nu }^2\int rT(r)dr$. If $T\propto r^{-3/4}$ away from the inner edge, outer annuli dominate because the weight per logarithmic radial interval grows as $r^{5/4}$. A measure-zero zero-temperature inner edge does not invalidate the leading law when $rT(r)$ is integrable.

The standard thin-disk solution separates approximately into an inner radiation-pressure and electron-scattering zone, a middle gas-pressure and electron-scattering zone, and an outer gas-pressure and free-free-opacity zone.

The inner Shakura--Sunyaev thin disk is supported mainly by radiation pressure and has nearly constant electron-scattering opacity. Equating radiative diffusion with local dissipation and imposing vertical hydrostatic equilibrium gives the characteristic half-thicknessThe radiation-supported height approaches a plateau away from the inner boundary. The formal zero at that boundary is outside the reliable radiation-dominated approximation. At an Eddington accretion rate defined using efficiency $0.1$, this height can make the inner flow only marginally thin.

In a steady thin alpha disk, $|u_R|/c_s$ is of order $\alpha H/R\ll 1$, whereas $u_{\varphi }/c_s$ is of order $R/H\gg 1$. Matter therefore completes many nearly circular orbits during its slow viscous drift inward.

For a steady Keplerian disk with a zero-torque inner edge $R_{\mathrm{in}}$, the flux summed over both faces is

A Keplerian accretion disk with a zero-torque inner boundary condition and local blackbody cooling has $T_{\mathrm{eff}}^4=3GM\dot{M}(1-\sqrt{r_{\mathrm{in}}/r})/(8\pi \sigma r^3)$. The Stefan–Boltzmann law uses the flux through one face; the standard thin-disk dissipation flux includes both faces. The scale $T_{\mathrm{in}}=[3GM\dot{M}/(8\pi \sigma r_{\mathrm{in}}^3){]}^{1/4}$ is a normalization, not the actual temperature at the stress-free inner edge.

For the effective-temperature profile of a zero-torque disk, differentiation of $y^{-3}(1-y^{-1/2})$ with $y=r/r_{\mathrm{in}}$ gives $y_{\max }=49/36$. Thus $T_{\max }=(6/7{)}^{3/2}{7}^{-1/4}{T}_{\mathrm{in}}\simeq 0.488T_{\mathrm{in}}$. This distinguishes a genuine maximum effective temperature from the normalization used in a power-law approximation.

Integrating the standard thin-disk dissipation flux givesHalf of the Newtonian potential-energy decrease remains as orbital kinetic energy until the inner edge.