= Solution
For an optically thin, low-density plasma, the <astrophysical cooling function> packages collisional radiation losses and temperature-dependent ion fractions into the coefficient multiplying $n_H^2$. Because the volume loss rate has units $\mathrm{erg\,cm^{-3}\,s^{-1}}$, the coefficient has units \b[$\mathrm{erg\,cm^3\,s^{-1}}$]. The energy-density unit printed in the numerical hint cannot be the unit of this coefficient. Interpret the quoted logarithmic values in the dimensionally consistent cooling-coefficient unit.
Assume <collisional ionization equilibrium>, a primordial hydrogen-helium mixture and no external photoheating. In the specified range, the <primordial atomic cooling curve> has the following features. Just above $10^4\,\mathrm K$, thermal <Electrons> begin to excite neutral <hydrogen> efficiently; subsequent line emission, especially <Lyman-alpha emission>, causes a steep rise. The excitation rate contains a threshold factor of order $e^{-10.2\,\mathrm{eV}/(k_BT)}$. <Hydrogen> line cooling is strong near a few times $10^4\,\mathrm K$; the supplied value at $1.5\times10^4\,\mathrm K$ provides a useful low-temperature label.
As <hydrogen> becomes ionized, neutral-hydrogen line cooling declines. <Helium> excitation and <ionization> produce a further shoulder or peak around $10^5\,\mathrm K$. <Collisional excitation>, <collisional ionization> and <radiative recombination> all contribute: excitation photons remove <Electron> kinetic energy, <ionization> consumes it, and recombination produces free-bound radiation. Once <hydrogen> and <helium> are almost fully stripped, their bound-state cooling disappears and the curve falls into a relatively inefficient interval. At high temperature, <thermal bremsstrahlung> dominates, with an approximate \b[$\Lambda\propto T^{1/2}$] tail and a weak Gaunt-factor correction.
The sketch uses the two supplied numerical labels and a qualitative hydrogen-helium interpolation; it is not a tabulated atomic-rate calculation.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/iii/paper-61-primordial-cooling.png]
{title=Qualitative primordial atomic cooling curve with hydrogen and helium line features, a bremsstrahlung tail, and an illustrative metal-enriched comparison}
{height=440}
<Metal-line cooling> raises the cooling coefficient markedly over much of $10^5$–$10^7\,\mathrm K$, because heavier elements supply many ions and excitation transitions after <hydrogen> and <helium> have lost their bound <Electrons>. It also broadens and reshapes the line-cooling peaks. Fine-structure lines can permit cooling below the <hydrogen> atomic threshold; <molecular hydrogen> can likewise cool metal-free gas below that threshold, but lies outside the requested temperature range. At sufficiently high temperatures <thermal free-free emission> again dominates the continuum. The enhanced curve in the figure is a schematic comparison, not a numerical claim about a specified metallicity. Thus \b[metals generally shorten the cooling time and extend the temperature range of efficient cooling]; <ionization> state, abundance and radiation field determine the actual curve.
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