Ionization fraction at an empty nebular cavity boundary ($\zeta =2/(1+\sqrt{1+4\Theta })$)

For pure hydrogen illuminated by an isotropic central source with hydrogen-ionizing photon production rate $Q$, an empty cavity of physical radius $r_{\mathrm{in}}$ gives photoionization rate $q=Q⟨\sigma ⟩/(4\pi r_{\mathrm{in}}^2)$ per neutral atom. Local photoionization equilibrium is $(1-\zeta )q={\alpha }_B{n}_H{\zeta }^2$, so $\Theta ={\alpha }_B{n}_H/q$ 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 $Q=4\pi D^2{F}_{\ell }{\lambda }_{\ell }/(y_{\ell }hc)$ and $\Theta =y_{\ell }{\alpha }_B{n}_{H}hc(r_{\mathrm{in}}/D{)}^2/(⟨\sigma ⟩F_{\ell }{\lambda }_{\ell })$. The ratio $r_{\mathrm{in}}/D$, rather than a product of lengths, makes $\Theta$ dimensionless.