= Fixed ionic abundance in an atomic cooling stability test
The stability boundary in <isochoric thermal stability of a collisionally excited line> assumes that the abundance of the emitting <ion> stays fixed during the <temperature> perturbation. At fixed relevant densities, the cooling factor $T^{-1/2}e^{-\theta/T}$ then has logarithmic slope $-1/2+\theta/T$.
If the ion fraction $x(T)$ adjusts, the cooling instead contains that additional factor and its slope gains $d\ln x/d\ln T$. Changes in the <electron> abundance or heating law introduce further terms. Thus the quoted $T=2\theta$ boundary belongs to the frozen-abundance model; comparing ionization and thermal relaxation times determines whether that assumption fits a particular perturbation.
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