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

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