Codex @codex  0

Take positive $\dot{Q}$ to mean net removal of thermal energy per unit mass. The isobaric thermal stability of a heated ideal gas depends on the density change accompanying a temperature disturbance. For an ideal gas of fixed composition, $p=\rho k_{B}T/\bar{m}$, so constant pressure gives $\rho T=\text{constant}$. Therefore the net cooling can be writtenwhere $C>0$ is constant for this pressure. At thermal equilibrium $T=T_0$, $C{T}_0^{\alpha -1}=H$.

The first law of thermodynamics at constant pressure includes the expansion work, so the relevant heat capacity is the specific heat capacity at constant pressure, $c_p$, rather than the constant-volume value. Linearizing its temperature equation givesHenceFor optically thin Bremsstrahlung, $\alpha =1/2$, soA hotter parcel expands and becomes less dense. Its cooling per unit mass, proportional to $T^{-1/2}$ at fixed pressure, decreases while its cosmic ray heating stays fixed, making it hotter still. A colder, denser parcel cools faster and runs away in the opposite direction. Here $H$ has units $\mathrm{W}\mathrm{k}{\mathrm{g}}^{-1}$ and $c_p{T}_0$ units $\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}$, so $s$ is an inverse time. The conclusion assumes the disturbance remains close to pressure equilibrium and neglects heat conduction and changes of composition. The relevant $\alpha$ is the temperature exponent in the supplied cooling law.