Blackbody dust sublimation radius ($r_{\mathrm{sub}}\propto L^{1/2}{T}_{\mathrm{sub}}^{-2}$)

A spherical black grain of radius $a$ absorbs luminosity $L$ through projected area $\pi a^2$ and emits over $4\pi a^2$. Radiative balance at sublimation temperature gives $r_{\mathrm{sub}}=[L/(16\pi {\sigma }_{\mathrm{SB}}{T}_{\mathrm{sub}}^4){]}^{1/2}$. Grain size cancels only under the stated unit-efficiency approximation; shielding and wavelength-dependent emissivity change the result.

At fixed irradiation, heating per grain is independent of temperature while cooling is proportional to $T^4$. With positive heat capacity and intact grains, a positive temperature perturbation increases net cooling, and a negative one increases net heating. Fixed dust properties and sufficient dust-gas thermal coupling are necessary assumptions.