Isobaric thermal stability of a heated ideal gas ($s=(1-\alpha )H/(c_p{T}_0)$)

Let the net cooling per unit mass of an ideal gas be $\mathcal{L}=A\rho T^{\alpha }-H$, with positive constants $A,H$. At fixed pressure and composition, $\rho T$ is constant, so $\mathcal{L}=C{T}^{\alpha -1}-H$. At a thermal equilibrium $T_0$, $C{T}_0^{\alpha -1}=H$. The specific heat capacity at constant pressure gives $c_{p}d(\delta T)/dt=-(\partial \mathcal{L}/\partial T{)}_p\delta T$. Thus $\delta T\propto e^{st}$ with the displayed growth rate. The equilibrium is stable for $\alpha >1$, unstable for $\alpha <1$, and marginal at linear order for $\alpha =1$. In particular, optically thin Bremsstrahlung with $\alpha =1/2$ is unstable under this heating law and the isobaric constraint. Fixed-density perturbations have a different criterion.

At thermal equilibrium, let $\mathcal{L}(\rho ,T)$ be net cooling per unit mass, including a specified heating law. A gas perturbation that stays at constant pressure obeys $c_{p}d\delta T/dt=-(\partial \mathcal{L}/\partial T{)}_p\delta T$. It grows when this derivative is negative. The isobaric approximation requires pressure communication faster than thermal evolution; a volumetric cooling function instead introduces a background mass density in the denominator. Thermal conduction alters the short-wavelength criterion.

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.