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 Incoming links: Past exam of the radiative processes and high energy astrophysics course of the University of Oxford / 2024 / 2 / b

Past exam of the radiative processes and high energy astrophysics course of the University of Oxford / 2024 / 2 / c / Solution 2026-10-25

Put

Δ = R - r ≪ R

. The near-surface mass density and mean free path from a and b are

ρ ≃ ρ_{c} \frac{Δ}{R}, ℓ ≃ \frac{L R}{Δ} .

A simple local escape estimate sets the remaining distance equal to the mean free path,

ℓ ≃ Δ

. It gives

Δ R_{esc, local} ≃ L R, \frac{Δ R _{esc, local}}{R} ≃ \frac{L}{R} .

For the Sun's parameters this is

Δ R_{esc, local} ≃ 1.94 km, Δ R_{esc, local} / R ≃ 2.79 \times 1 0^{- 6}

.

A more precise operational definition uses the outward radial optical depth. In this varying mass density,

τ (r) = \int_{r}^{R} \frac{d s}{ℓ ( s )} = \frac{1}{a L} \int_{a r}^{π} \frac{s i n x}{x} d x ≃ \frac{Δ ^{2}}{2 L R} .

The chance of an outward radial photon escaping without another scattering is

e^{- τ}

. Thus the conventional

τ = 1

last-scattering depth gives

Δ R_{esc, τ = 1} ≃ 2 L R ≃ 2.75 km, Δ R_{esc, τ = 1} / R ≃ 3.95 \times 1 0^{- 6} .

There is no sharp escape surface:

τ ≪ 1

gives nearly free propagation, and oblique rays have different optical depths. The local

ℓ = Δ

estimate actually has

τ ≃ 1/2

in this model, not

1

; the factor

2

is a convention-dependent order-one correction, not a discrepancy in the Thomson cross-section. Neither scattering criterion establishes a thermalization surface.

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