= Critical ionizing photon rate of a cored power-law nebula
{title2=$Q_{\rm crit}$}
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 is
$$
Q_{\rm crit}=4\pi\alpha n_0^2r_c^3\left(\frac13+\frac1{2\beta-3}\right).
$$
A <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 $\beta>3/2$.
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