Radiative cooling functions and thermal instability

Define a positive optically thin volumetric radiative loss as $C=n_en_H\Lambda(T)$, where n H counts hydrogen nuclei and Lambda has SI units watts cubic metres. C is watts per cubic metre, integrated over escaping photon frequencies and solid angle. It is not the earlier spectral emissivity j. Tabulations using electron times ion density, hydrogen density squared, or energy loss per mass have different normalizations. Always identify the density factors before using a numerical coefficient, especially when combining composition or ionization models.

A temperature-only table commonly assumes specified abundances and ionization equilibrium. Collisional ionization equilibrium is not the same as equilibrium under an external photoionizing field, and neither describes every rapidly cooling parcel. Ion fractions can lag the temperature when recombination is slow. Line photons may also be trapped instead of removing energy locally. Metal abundance, radiation, dust, density-dependent collisional deexcitation and optical depth can therefore alter the loss. A curve supplied for one environment is not a universal property of matter at that temperature alone.

At fixed density, the translational thermal energy uses three halves k T per free particle. Fully ionized pure hydrogen has one electron and one proton per nucleus, so its total particle density is twice n H. At fixed pressure an ideal monatomic parcel also does expansion or compression work; its enthalpy gives a five-halves factor per particle instead. Ionization energy, changing particle counts, molecular degrees of freedom and relativistic particles require an appropriate energy equation. A familiar numerical cooling-time formula can fail when those assumptions change.

With volumetric heating H, define the net loss as L equal to C minus H. A cooling-only timescale can remain finite at thermal equilibrium even though the net loss is zero. Conversely, dividing thermal energy by a nearly vanishing net loss does not determine the response to a small perturbation. Stability requires the derivative of the net energy exchange along the permitted perturbation, together with the parcel's heat capacity and mechanical constraint. Pressure equilibration by sound waves, conduction and the perturbation wavelength help decide which constraint is physically appropriate.

For the toy power law, a constant-density perturbation is stable when a is positive: a hotter parcel cools more while its per-volume heating stays fixed. At constant pressure, however, raising temperature lowers density. Cooling carries two density factors whereas the chosen heating carries only one. Differentiating at equilibrium gives a factor a minus one, so isobaric stability requires a greater than one. A cooling function that rises with temperature can consequently still be unstable to pressure-balanced perturbations under this particular heating prescription.

This is a local linear result around an equilibrium with a declared heating law, not a claim that every cooling cloud fragments whenever a slope is small. Marginal exponents require additional terms or higher-order analysis. Thermal conduction can damp sufficiently short wavelength temperature variations, and magnetic geometry can make conduction anisotropic. Gravity, turbulent mixing, acoustic dynamics and a changing ionization state can couple additional modes. Applying a named instability criterion therefore requires identifying the energy equation and response times, rather than reading a slope from an unrelated equilibrium table.