= 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.
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