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.