Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 349 / 4 / Solution

A star-forming galaxy contains short-lived, massive O and B stars whose hot photospheres dominate the ultraviolet and blue continuum and ionize surrounding gas. Once star formation ceases, these stars disappear quickly and an older, cooler stellar population produces a redder spectrum with stronger stellar absorption features and a prominent 4000-angstrom break.

In star-forming regions, direct stellar continuum is accompanied by nebular free-bound and free-free continuum, hydrogen and helium recombination lines, collisionally excited metal lines, and infrared emission from dust that absorbed shorter-wavelength photons. Supernova remnants and cosmic rays add synchrotron radio emission, while hot shocked gas can emit X-rays.

The Strömgren sphere model assumes a steady ionizing source in uniform, static, pure hydrogen of number density $n_H$, with a sharp ionization front enclosing fully ionized gas. If $N_i=(4\pi /3)R^3{n}_H$ is the number of ions, photon conservation givesThe equilibrium Strömgren radius and recombination time areConsequently the radius obeysWriting $x=(R/R_S{)}^3$ turns this into $dx/d(t/t_{\mathrm{rec}})=1-x$. For an initially neutral medium, $x(0)=0$, and the Ionization-front growth of a Strömgren sphere isThe front initially expands rapidly because few ions are recombining and asymptotically approaches $R_S$ as recombinations balance ionizations. This photon-counting solution precedes any pressure-driven hydrodynamic expansion of the H II region.