Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 315 / 3 / b

Solution

Let the atmosphere occupy a thin annulus from

R_{p}

R_{a} = R_{p} + Δ R

. With mass extinction coefficient

k_{ν}

, constant density

ρ

, and the assumed common chord length

l

τ_{ν} = k_{ν} ρl .

The opaque solid planet removes area

π R_{p}^{2}

, while the annulus removes the fraction

1 - e^{- τ_{ν}}

D_{ν} \equiv 1 - \frac{F _{ν}^{in}}{F _{ν}^{out}} = \frac{R _{p}^{2} + ( R _{a}^{2} - R _{p}^{2} ) ( 1 - e ^{- k_{ν} ρl} )}{R _{s}^{2}} .

For a thin annulus,

D_{ν} ≃ (\frac{R _{p}}{R _{s}})^{2} + \frac{2 R _{p} Δ R}{R _{s}^{2}} (1 - e^{- k_{ν} ρl}) .

If the geometrical thickness is estimated as

Δ R = N_{H} H

H = k_{B} T_{p} / (μ m_{H} G M_{p} / R_{p}^{2})

. A wavelength-independent

k_{ν}

makes this idealized spectrum flat; real molecular opacities create its features.

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