= Saha neutral fraction from a measured neutral density
{c}
{title2=$f_{\rm HI}=(1+\sqrt{S/n_{\rm HI}})^{-1}$}
For charge-neutral pure <hydrogen> in <thermal equilibrium>, write the <Saha equation> as $n_en_p/n_{\rm HI}=S(T)$. A measured neutral <number density> fixes $n_e=n_p=\sqrt{S(T)n_{\rm HI}}$, so the neutral fraction is $f_{\rm HI}=n_{\rm HI}/(n_p+n_{\rm HI})=(1+\sqrt{S/n_{\rm HI}})^{-1}$. At high ionization it scales as $\sqrt{n_{\rm HI}/S}$, rather than $n_{\rm HI}/S$. The latter mistake confuses measured neutral <number density> with total <number density>. <Photoionization equilibrium> with an external nonthermal field generally requires a rate equation instead of the <Saha equation>.
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