Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 48 / 2 / b

At $X_e=1/2$, the left side of the Saha equation is two. With its rounded coefficient, the recombination temperature therefore satisfiesNeglecting the logarithmic prefactor gives $y\sim 36$ andThis is the requested rough scale. Retaining the prefactor and the unrounded logarithm gives $y\simeq 42.0$ and $T_{\mathrm{rec}}\simeq 0.32$ eV; the displayed $0.4$ eV is not a high-accuracy numerical root of the Saha equation.

The small baryon-to-photon ratio means that radiation and the entropy of the free charged particles strongly favor ionization. A low mean photon energy does not eliminate the energetic tail, and ionized particles have a large phase-space advantage. The exponential binding factor must become very large before neutral atoms dominate this dilute gas. Accordingly $\mathcal{B}/T$ is of order forty rather than order one, so recombination occurs well below the hydrogen binding energy.