= Solution
\b[The dust-cooled equilibrium is thermally stable under the stated constant irradiation, with fixed dust properties and dust-to-<gas> ratio.] At a fixed position, absorbed power per grain is independent of its <temperature>, while emission is $4\pi a^2\sigma_{\rm SB}T^4$. A positive <temperature> perturbation increases cooling, causing a restoring energy loss; a negative perturbation reduces cooling, allowing reheating.
For heat capacity $C>0$ of a grain together with its thermally coupled <gas>,
$$
C\,\delta\dot T=-16\pi a^2\sigma_{\rm SB}T_0^3\delta T,
$$
so the perturbation decays. The <thermal stability of externally heated blackbody dust> relies on remaining below sublimation with the grain population intact; destruction, variable illumination or finite dust-<gas> equilibration can change the response. An isobaric parcel with a fixed grain-to-<gas> ratio has the same restoring sign when heating and cooling are compared per particle.
Back to article page
