Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 55 / 1 / ii

If the gas can cool appreciably below the virial temperature, radiative cooling removes thermal energy and pressure support. In a dark-matter halo the gas then contracts, dissipating more energy as it falls. Efficient condensation requires the radiative gas cooling time to be short enough compared with the relevant dynamical or assembly time; merely having an available low-temperature transition does not guarantee that the gas reaches it quickly.

The collisionless dark matter cannot lose comparable energy through radiation and remains extended. Gas with appreciable conserved angular momentum stops radial collapse when rotation supports it, often forming a disk; lower-angular-momentum gas reaches a more compact central region. Cold dense gas can fragment into self-gravitating clouds and form stars if its gravitational instability overcomes remaining support. Stellar feedback subsequently reheats or expels gas and regulates the conversion. Efficient cooling enables central baryonic condensation and star formation; angular momentum and feedback determine the resulting galaxy.

Below about $10^4\mathrm{K}$, neutral hydrogen electronic excitation becomes inefficient because the lowest relevant excitation energies greatly exceed the typical particle thermal energy. Primordial gas therefore needs molecular hydrogen cooling through rotational and vibrational transitions; HD can cool still colder gas where it is sufficiently abundant. Without molecules or metals, cooling can stall near the atomic threshold.

In enriched gas, metal-line cooling from low-energy fine-structure transitions, notably singly ionized carbon and neutral oxygen, remains effective below that threshold. At higher densities, molecular rotational lines such as CO and energy transfer from gas to dust followed by dust infrared emission are important. Cold-gas cooling is primarily molecular, fine-structure, or dust-mediated, according to composition and density. Molecule formation, dissociating radiation and the Cosmic microwave background temperature floor constrain how far cooling proceeds.

In the intermediate-temperature interval, atomic line cooling is generally efficient. Collisional excitation of hydrogen and helium followed by photon emission removes thermal energy; collisional ionization and subsequent recombination also contribute. As the gas becomes more ionized, different transitions enter and leave the cooling budget.

For enriched gas, metal-line cooling is often dominant over substantial parts of this interval, because heavy ions provide many ultraviolet and optical transitions. The cooling curve consequently has pronounced peaks rather than one smooth universal power law. Hydrogen/helium atomic processes and metal lines provide the main cooling channels here. Their relative strengths depend on metallicity, ionization state, density and the incident radiation field; these temperature bands describe typical gas, not composition-independent boundaries.

In sufficiently hot ionized gas, electrons radiate when accelerated in ion Coulomb fields: thermal bremsstrahlung is the main continuum cooling process. In the optically thin nonrelativistic limit its emission rate scales approximately asFor hot metal-poor gas this is the principal high-temperature channel. Metal ions still give important metal-line cooling near $10^6\mathrm{K}$, and in enriched gas can remain important up to several million kelvin; one should not infer that crossing $10^6\mathrm{K}$ instantly eliminates all lines. At sufficiently high temperatures most ions are stripped and free-free emission dominates. Inverse Compton cooling can also matter for ionized, diffuse gas in a strong radiation field, especially the high-redshift Cosmic microwave background. The high-temperature asymptote is bremsstrahlung cooling, with metal-line and Compton qualifications where appropriate.