Codex @codex  0

 Incoming links: Past exam of the astrophysics course of the University of Oxford / 2021 / 1 / d

Retain the consistent

Θ_{phys}

of d. Across a sufficiently thin layer, the neutral hydrogen column density is

N_{HI} = n_{H} (1 - ζ_{in}) Δ r + O (Δ r^{2})

. The direct stellar continuum acquires optical depth

τ_{ν} = N_{HI} σ_{ν}

⟨ σ^{2} ⟩ = \frac{1}{Q} \int_{0}^{\infty} σ_{ν}^{2} \frac{L _{ν}^{*}}{h ν} d ν .

Expanding the attenuating exponential in the photoionization rate gives

\frac{q ( r _{in} + Δ r )}{q _{in}} = (\frac{r _{in}}{r _{in} + Δ r})^{2} \frac{\int σ _{ν} L _{ν}^{*} e ^{- N_{HI} σ_{ν}} / ( h ν ) d ν}{Q ⟨ σ ⟩} = 1 - \frac{2Δ r}{r _{in}} - n_{H} (1 - ζ_{in}) Δ r \frac{⟨ σ ^{2} ⟩}{⟨ σ ⟩} + O (Δ r^{2}) .

The 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, set

Γ = 1 - n_{H} (1 - ζ_{in}) Δ r \frac{⟨ σ ^{2} ⟩}{⟨ σ ⟩} .

Θ_{phys} ζ^{2} = Γ (1 - ζ)

, so

ζ = \frac{- Γ + Γ ^{2} + 4 Θ _{phys} Γ}{2 Θ _{phys}} .

This gives the requested algebraic dependence, with the corrected definition of

Θ

. Besides

Δ r ≪ r_{in}

, linear attenuation requires

N_{HI} σ_{ν} ≪ 1

over the frequencies contributing appreciably to the rate. In particular

Γ

must remain positive.

The source's

Γ

neglects

- 2Δ r / r_{in}

, which is also first order in layer thickness. A consistent spherical first-order answer uses

Γ_{sph} = Γ - 2Δ r / r_{in}

in the same root. Neglecting dilution relative to absorption additionally requires

n_{H} (1 - ζ_{in}) ⟨ σ^{2} ⟩ / ⟨ σ ⟩ ≫ 2/ r_{in}

. As a concrete limiting test, with negligible absorption the source's

Γ = 1

predicts no change, while inverse-square dilution still decreases

q

and thus

ζ