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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 $4\pi a^2{\sigma }_{\mathrm{SB}}{T}^4$. 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,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.