 Nested hydrogen-helium ionization zones

In a uniform sharp-front approximation, an inner ${\mathrm{He}}^{++}$ zone of radius $R_2$ lies inside a ${\mathrm{He}}^{+}$ shell ending at $R_1$, within the hydrogen ionization front $R_H$. For nuclear densities $n_H,n_{He}$, the electron densities are $n_H+2n_{He}$, $n_H+n_{He}$ and $n_H$ in the three zones. Thus the hydrogen recombination integral is $(4\pi /3){\beta }_H{n}_H[n_H{R}_H^3+n_{He}(R_1^3+R_2^3)]$. The helium integrals are $(4\pi /3){\beta }_{HeI}{n}_{He}(n_H+n_{He})(R_1^3-R_2^3)$ and $(4\pi /3){\beta }_{HeII}{n}_{He}(n_H+2n_{He})R_2^3$. These are balanced against species-specific effective photon budgets, not three independently reused cumulative stellar photon counts.