Past exam of the formation of Structure in the Universe course of the University of Cambridge / 2015 / Paper 1 / 2 / c / Solution

At distance $r$, one grain intercepts power through its projected area and emits through its full surface:The blackbody dust sublimation radius follows by setting $T_d=T_{\mathrm{sub}}$:Here the supplied luminosity is $3.8\times 10^{37}\mathrm{W}$. The grain radius cancels because both cross-sections scale as $a^2$. The temperature exterior to this boundary scales as $T_d\propto r^{-1/2}$. This estimate assumes isotropic illumination, no shielding, and unit absorption/emission efficiency as specified; real grain emissivities can change it. The numerical evaluation uses the external conversion values ${\sigma }_{\mathrm{SB}}\simeq 5.6704\times 10^{-8}\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-4}$ and $1\mathrm{pc}\simeq 3.0857\times 10^{16}\mathrm{m}$.^[ref]