Isobaric recombination-cooling stability with constant volumetric heating (${\mathcal{L}}_T=(\Gamma /T_0)(d\ln {\alpha }_B/d\ln T-1)$)

Take fully ionized hydrogen with electron and proton number densities both $n$, fixed local heat input $\Gamma$ per unit volume, and volumetric cooling $3n^2{\alpha }_B(T)k_{B}T/2$. The ideal gas law is $P=2n{k}_{B}T$. At fixed pressure, net cooling becomes $\mathcal{L}=3P^2{\alpha }_B(T)/(8k_{B}T)-\Gamma$. At an equilibrium $T_0$, differentiation gives the displayed derivative.

The linearized thermal equation is ${\rho }_0{c}_p\dot{\delta T}=-{\mathcal{L}}_T\delta T$, so equilibrium is stable exactly when the logarithmic slope of ${\alpha }_B$ exceeds one. A constant or decreasing coefficient gives instability. A different heating prescription must be differentiated too; constant incident radiation does not generally imply a density-independent local heating rate.