= First-order spherical attenuation of a photoionization rate
{title2=$q'/q=-2/r-n_H(1-\zeta)\langle\sigma^2\rangle/\langle\sigma\rangle$}
At an unattenuated inner boundary, define photon-weighted <cross section> moments $\langle\sigma^k\rangle=Q^{-1}\int\sigma_\nu^k L_\nu/(h\nu)\,d\nu$. Across a layer $dr$, the inverse-square geometrical factor contributes $-2dr/r$, and expanding $e^{-n_H(1-\zeta)\sigma_\nu dr}$ contributes $-n_H(1-\zeta)dr\langle\sigma^2\rangle/\langle\sigma\rangle$. Both are first order; a plane-parallel treatment can suppress the former only as a separate geometrical approximation. The second moment occurs because the <photoionization rate> weights an already attenuated spectrum by another <cross section>.
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