= Jeans-length ionization correction for the Lyman-alpha forest
{c}
{title2=$N_H\propto N_{\rm HI}^{1/3}\Gamma^{1/3}[f_gT/\alpha(T)]^{1/3}$}
Assume optically thin <photoionization equilibrium> and an absorber depth of order its local <Jeans length>, $L\propto(f_gT/n_H)^{1/2}$, where $f_g$ is the local gas fraction. Then $N_{\rm HI}\simeq\alpha(T)n_H^2L/\Gamma\propto[\alpha(T)/\Gamma](f_gT)^{1/2}n_H^{3/2}$. Solving for $n_H$ and substituting in $N_H=n_HL$ gives the displayed correction. Thus the ionized mass integral weights the <neutral-hydrogen column-density distribution> by $N_{\rm HI}^{1/3}$ rather than $N_{\rm HI}$. This model does not apply without modification to optically thick systems or arbitrary unconfined gas.
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