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.

A thermal-stability argument that holds dust properties fixed can fail near a sublimation front. A rise in temperature can destroy grains, changing the absorption and emission efficiencies, the dust mass and the optical depth through which radiation reaches neighboring material. The heating and cooling laws then change along with the temperature.

Write the net heating schematically as $Q(T,a,{\rho }_d)$, where $a$ is grain size and ${\rho }_d$ is dust density. The response along an evolving state includes $Q_{a}da/dT$ and $Q_{{\rho }_d}d{\rho }_d/dT$, not only the fixed-grain derivative $Q_T$. These additional terms can reinforce or oppose the perturbation. The sign depends on geometry, radiation spectrum and dust–gas coupling, so the fixed-property result in thermal stability of externally heated blackbody dust does not by itself establish the stability of a sublimation front.