Multitemperature blackbody disk

A multitemperature blackbody disk adds the radiation from annuli with different effective temperatures. For a steady, radiatively efficient accretion disk, the inner annuli are hotter and dominate higher photon frequencies, while cooler outer annuli contribute at lower frequencies. With $T(r)\propto r^{-3/4}$, integrating the Planck law produces the intermediate-frequency slope $F_{\nu }\propto {\nu }^{1/3}$. This slope applies between the outer thermal turnover and inner high-frequency cutoff, rather than across the entire spectrum.^[ref]

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.