 Isochoric thermal stability of a collisionally excited line

At fixed density below critical density, optically thin two-level cooling scales as $\mathcal{C}\propto T^{-1/2}{e}^{-\theta /T}$, with $\theta =\Delta E/k_B$. Under temperature-independent heating, $d\ln \mathcal{C}/d\ln T=-1/2+\theta /T$ gives restoring thermal balance for $T<2\theta$ and instability for $T>2\theta$. This boundary assumes fixed ionic abundance and does not describe a constant-pressure perturbation.