Radiative cooling

A body loses internal energy by emitting thermal radiation faster than it absorbs radiation or receives other heating. In a gravitating gas this can remove pressure support and allow further contraction.

Hot electrons can transfer energy to cosmic microwave background photons through Compton scattering. For a fixed electron fraction $x_e$ and electron temperature well above the radiation temperature, the optically thin gas cooling time scales approximately as $(1+z{)}^{-4}/x_e$. Its volume loss scales with electron density, not with $n_H^2$, because the radiation density is independent of local gas density. The process tends toward the radiation temperature; it heats rather than cools electrons colder than that radiation.

Dust grains emit thermal radiation after absorbing energy from radiation or collisions. Collisions can transfer thermal energy from dense gas to grains, whose radiation then cools the gas. This channel requires dust and does not operate in strictly pristine hydrogen-helium material without prior enrichment.

Thermal electrons can transfer energy to lower-energy radiation photons through Compton scattering. In the nonrelativistic Thomson regime, the net cooling power per electron is $4{\sigma }_{T}c{U}_{\gamma }{k}_B(T_e-T_{\gamma })/(m_e{c}^2)$ for a blackbody photon bath. A sufficiently hot electron population therefore loses energy; a colder population is heated instead. The cosmic microwave background radiation energy density grows as $(1+z{)}^4$, making this channel more important for high-redshift diffuse ionized gas.

An astrophysical cooling function packages radiative losses of a plasma into a temperature-dependent coefficient. In the hydrogen-density-squared convention, the volume energy loss is $n_H^2\Lambda (T)$, so the coefficient has units $\mathrm{erg}\mathrm{c}{\mathrm{m}}^3{\mathrm{s}}^{-1}$. Its dependence on ionization state, chemical abundances and radiation field must be specified; a low-density collisional-equilibrium curve is not universal.

In an optically thin plasma, one convention defines the radiative cooling power per volume by $\mathcal{C}=n_e{n}_H{\Lambda }_{eH}(T)$, where $n_H$ counts hydrogen nuclei. A convention $\mathcal{C}=n_H^2{\Lambda }_{HH}$ instead has ${\Lambda }_{HH}=(n_e/n_H){\Lambda }_{eH}$. Thus the tabulated functions differ even for the same gas. With total particle density $n$, a monatomic-gas radiative gas cooling time is $3n{k}_{B}T/(2\mathcal{C})$ when heating is negligible.^[ref]

Radiation from rotational and vibrational transitions of molecules removes gas thermal energy. In primordial gas below the atomic threshold, molecular hydrogen is a principal coolant; hydrogen deuteride can matter in suitable chemical conditions. Molecular formation, photodissociation, optical depth and density-dependent level populations control the rate. This allows some small systems below the atomic virial temperature threshold to form stars.

Photoionization changes the ion population, often removing bound-state coolants at temperatures where collisional equilibrium would retain them. It can therefore suppress primordial line-cooling peaks. It also supplies photoionization heating. The net loss is cooling minus heating and depends on radiation spectrum, intensity, shielding and gas density. A universal density-independent function of temperature alone is not generally sufficient for irradiated gas.

For a nonrelativistic ideal gas with $n_{\mathrm{tot}}$ thermal particles per volume, divide its thermal energy $3n_{\mathrm{tot}}{k}_{B}T/2$ by the optically thin volume radiation loss. In the $n_H^2$ cooling convention this becomes $3\chi k_{B}T/(2n_H\Lambda )$, where $\chi =n_{\mathrm{tot}}/n_H$. Density scaling and ionization-dependent particle count must be kept consistent.

Equating the optically thin cooling time to the uniform-sphere free-fall time gives a critical gas density proportional to $T^2/{\Lambda }^2$. With hydrogen mass fraction $X$, particle ratio $\chi$ and gravitating gas fraction $f_g$, it is $[32G{m}_p/(3\pi X{f}_g)][3\chi k_{B}T/(2\Lambda ){]}^2$. At fixed temperature, gas above this density cools faster than free fall. A smaller gravitating gas fraction moves the boundary upward because gravity is faster at a fixed gas density.

The primordial atomic cooling curve describes hydrogen-helium gas without heavy elements. Near the hydrogen excitation threshold it rises sharply, then shows hydrogen and helium excitation/ionization features. Bound-state losses weaken when both elements become highly ionized. At sufficiently high temperatures thermal bremsstrahlung gives a slowly rising tail. Composition, density convention and ionization assumptions fix the numerical normalization.

Atomic line cooling removes gas kinetic energy by collisional excitation followed by photon escape. It is strongest where atoms or ions have bound electrons and thermally accessible transitions. If photons are trapped, or if the relevant atoms are fully ionized, the simple optically thin line-cooling rate no longer gives the same loss.

Heavy elements provide many ionic transitions at temperatures where hydrogen and helium have lost their most effective line emitters. Metal-line cooling can therefore increase the plasma cooling function substantially and broaden its temperature features. Low-energy fine-structure transitions also permit cooling below the hydrogen atomic threshold when the relevant ions and electrons are present.