Past exam of the formation of Structure in the Universe course of the University of Cambridge / 2014 / Paper 1 / 3 / a / Solution

For a monatomic ideal gas, write the thermal energy density as $u=3n{k}_{B}T/2$. At low optical depth, radiative loss uses an astrophysical cooling function, often written $\mathcal{C}=n_e{n}_H\Lambda (T,Z)$, while $\mathcal{H}$ contains heating. At fixed density the net radiative gas cooling time is approximately $u/(\mathcal{C}-\mathcal{H})$ when the denominator is positive; a cooling flow closer to constant pressure must include enthalpy and compressional work. Comparing this time with the free-fall time of a uniform sphere determines whether pressure can be lost during gravitational contraction.

Hot ionized gas cools by bremsstrahlung and recombination; collisional excitation followed by photon escape supplies atomic and metal-line cooling at lower temperatures. molecular line cooling through rotational or vibrational lines and dust thermal emission become important in cold dense gas. Abundances, ionization state, density-dependent collisional de-excitation, and photon trapping all change the cooling function. Optically thick gas requires radiative transfer, not a simple local density-squared loss law.

Heating comes from photoionization, grain photoelectric heating driven by ultraviolet light, cosmic rays, shocks, turbulent dissipation and compression. photoionization heating deposits the photon energy above the ionization threshold as electron kinetic energy. A thermal equilibrium satisfies $\mathcal{H}=\mathcal{C}$; its stability depends on how the net loss changes under the relevant pressure or density constraint, not merely on equality. Different stable balances permit a multiphase interstellar medium. Spectral-line ratios constrain temperature and ionization, but interpreting them requires density diagnostics, interstellar extinction corrections and a radiation-field model.