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A local thermal imbalance can amplify a gas perturbation even without an initial displacement of an interface. Let $\mathcal{L}(\rho ,T)$ denote net cooling per unit mass, including heating with a negative sign, and suppose $\mathcal{L}({\rho }_0,T_0)=0$. When the sound-crossing time is short compared with the thermal evolution time, pressure equilibrates and an ideal gas perturbation is approximately isobaric. The linearized thermal equation isThus a negative constant-pressure derivative produces isobaric thermal instability: a cooler parcel experiences a net loss that cools it further, while a warmer parcel heats further. For example, isobaric thermal stability of a heated ideal gas with $\mathcal{L}=A\rho T^{\alpha }-H$ gives $\sigma =(1-\alpha )H/(c_p{T}_0)$ and instability for $\alpha <1$. Different heating laws and fixed-density perturbations give different criteria; thermal conduction damps sufficiently short-scale temperature contrasts. Such feedback helps a gas separate into denser cool and rarer warm regions when stable thermal branches exist.

The Rayleigh-Taylor instability instead releases gravitational potential energy when a heavier fluid lies above a lighter fluid in a downward gravitational field. For two inviscid incompressible half-spaces with no surface tension or magnetic field, a horizontal perturbation of wavenumber $k$ hasThe Atwood number measures the density contrast. A descending heavy-fluid finger and an ascending light-fluid bubble both lower the total gravitational potential energy, increasing the original displacement. Interchanging the layers gives an oscillatory stable interface. An accelerating or decelerating astrophysical shell has an analogous effective acceleration, allowing mixing at suitable ejecta interfaces in a supernova remnant. Finite interface width, viscosity, surface tension and magnetic tension modify the short-wavelength behavior; the ideal sharp-interface formula does not select a finite fastest-growing wavelength.

Let $\mathcal{L}$ be net cooling per unit volume, positive when energy is lost. At thermal equilibrium $\mathcal{L}=0$. For a slow perturbation that maintains pressure balance, the linearized thermal equation isThe heat capacity factor is positive. A positive derivative restores equilibrium: hotter gas cools more strongly, while cooler gas is heated. This is isobaric thermal stability of a heated ideal gas.

For fully ionized hydrogen, both protons and electrons contribute to the ideal gas law, so $P=2n{k}_{B}T$. Treat the stated constant heat input $\Gamma$ as a fixed heating rate per unit volume. Substituting $n=P/(2k_{B}T)$ into the cooling givesAt equilibrium $T_0$, its derivative isWith a temperature-independent recombination coefficient, the derivative is $-\Gamma /T_0$: the equilibrium is thermally unstable to isobaric perturbations. More generally, any nonincreasing ${\alpha }_B(T)$ also gives instability; stability requires its logarithmic slope to exceed one. Heating at fixed volume and pressure-balanced perturbations are distinct assumptions: the density falls as a parcel is heated, reducing the quadratic-density cooling.

No temperature law for ${\alpha }_B$ is supplied, so the displayed slope criterion states the general result without inventing one. A constant incident ultraviolet flux need not imply constant local volumetric heating in a self-consistent ionization-balance calculation. If the heating instead varies with density, temperature or neutral fraction, its derivative must be included; the conclusion here uses the explicit constant-$\Gamma$ model.

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.