= Proximity-effect estimator of the ionizing background
{title2=$N_{\rm HI}=N_{\rm HI}^{(0)}/(1+\omega),\quad\omega=\Gamma_Q/\Gamma_{\rm bg}$}
In <photoionization equilibrium> at fixed gas density and temperature, the extra <quasar> <photoionization rate> reduces neutral-hydrogen <column density> by the displayed factor. Its local rate is calculated from the quasar flux, while the isotropic background rate is $\Gamma_{\rm bg}=4\pi\int J_\nu\sigma_{\rm HI}(\nu)/(h\nu)\,d\nu$. Fitting the absorption suppression versus proper distance constrains the <cosmic ionizing background>. For an unperturbed column-density distribution proportional to $N^{-p}$ with $p>1$, the line abundance above a fixed threshold is reduced by $(1+\omega)^{1-p}$. Density enhancement, quasar age, anisotropy and redshift errors must be distinguished from this ionization effect.
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