= Recombination integral for a power-law nebula
For fully ionized pure hydrogen of density $n(r)=n_0(r/r_0)^{-\beta}$ and constant <Case B recombination> coefficient $\alpha$, the total recombination rate is
$$
\mathcal R(R;r_{\rm in})=4\pi\alpha n_0^2r_0^{2\beta}\int_{r_{\rm in}}^R r^{2-2\beta}\,dr.
$$
The central integral converges only for $\beta<3/2$; an envelope extending to infinity has finite recombination capacity only for $\beta>3/2$. At $\beta=3/2$, each logarithmic radius interval contributes equally. A <cored power-law density profile> removes the central singularity.
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