Critical ionizing photon rate of a cored power-law nebula ($Q_{\mathrm{crit}}$)

For the cored power-law density profile that is constant inside $r_c$ and falls as $r^{-\beta }$ outside, with $\beta >3/2$, the maximum integrated Case B recombination rate isA hydrogen-ionizing photon production rate greater than this has no finite radius at photoionization equilibrium. Equality requires an infinite ionized radius. Other core shapes change the coefficient but retain the convergence condition $\beta >3/2$.