= Ionization fraction at an empty nebular cavity boundary
{title2=$\zeta=2/(1+\sqrt{1+4\Theta})$}
For pure hydrogen illuminated by an isotropic central source with <hydrogen-ionizing photon production rate> $Q$, an empty cavity of physical radius $r_{\rm in}$ gives <photoionization rate> $q=Q\langle\sigma\rangle/(4\pi r_{\rm in}^2)$ per neutral atom. Local <photoionization equilibrium> is $(1-\zeta)q=\alpha_Bn_H\zeta^2$, so $\Theta=\alpha_Bn_H/q$ and the physical <quadratic equation> root is the displayed rationalized expression. If an unextinguished total recombination-line <observed radiative flux> $F_\ell$ has photon yield $y_\ell$, then $Q=4\pi D^2F_\ell\lambda_\ell/(y_\ell hc)$ and $\Theta=y_\ell\alpha_Bn_Hhc(r_{\rm in}/D)^2/(\langle\sigma\rangle F_\ell\lambda_\ell)$. The ratio $r_{\rm in}/D$, rather than a product of lengths, makes $\Theta$ dimensionless.
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