Past exam of the astrophysics course of the University of Oxford / 2021 / 1 / e

Retain the consistent ${\Theta }_{\mathrm{phys}}$ of d. Across a sufficiently thin layer, the neutral hydrogen column density is $N_{\mathrm{HI}}=n_H(1-{\zeta }_{\mathrm{in}})\Delta r+O(\Delta r^2)$. The direct stellar continuum acquires optical depth ${\tau }_{\nu }=N_{\mathrm{HI}}{\sigma }_{\nu }$. Define the photon-weighted second momentExpanding the attenuating exponential in the photoionization rate givesThe second cross section moment appears because the radiation both undergoes absorption and is weighted by the photoionization cross section in computing the rate.

In a local plane-parallel approximation that suppresses geometrical dilution, setThen local photoionization equilibrium becomes ${\Theta }_{\mathrm{phys}}{\zeta }^2=\Gamma (1-\zeta )$, soThis gives the requested algebraic dependence, with the corrected definition of $\Theta$. Besides $\Delta r\ll r_{\mathrm{in}}$, linear attenuation requires $N_{\mathrm{HI}}{\sigma }_{\nu }\ll 1$ over the frequencies contributing appreciably to the rate. In particular $\Gamma$ must remain positive.

The source's $\Gamma$ neglects $-2\Delta r/r_{\mathrm{in}}$, which is also first order in layer thickness. A consistent spherical first-order answer uses ${\Gamma }_{\mathrm{sph}}=\Gamma -2\Delta r/r_{\mathrm{in}}$ in the same root. Neglecting dilution relative to absorption additionally requires $n_H(1-{\zeta }_{\mathrm{in}})⟨{\sigma }^2⟩/⟨\sigma ⟩\gg 2/r_{\mathrm{in}}$. As a concrete limiting test, with negligible absorption the source's $\Gamma =1$ predicts no change, while inverse-square dilution still decreases $q$ and thus $\zeta$.