 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 isThe 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.