First-order spherical attenuation of a photoionization rate ($q^{\prime }/q=-2/r-n_H(1-\zeta )⟨{\sigma }^2⟩/⟨\sigma ⟩$)

At an unattenuated inner boundary, define photon-weighted cross section moments $⟨{\sigma }^k⟩=Q^{-1}\int {\sigma }_{\nu }^k{L}_{\nu }/(h\nu )d\nu$. Across a layer $dr$, the inverse-square geometrical factor contributes $-2dr/r$, and expanding $e^{-n_H(1-\zeta ){\sigma }_{\nu }dr}$ contributes $-n_H(1-\zeta )dr⟨{\sigma }^2⟩/⟨\sigma ⟩$. 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.