Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 315 / 1 / a

Solution

Treat the morning and evening halves of the day-night terminator as independent, isothermal, hydrostatic atmospheres with the same reference radius

R_{0}

P_{0}

, composition, gravity

g

, and extinction coefficient

κ_{λ}

H_{m} = \frac{k _{B} T _{m}}{μ m _{H} g}, H_{e} = \frac{k _{B} T _{e}}{μ m _{H} g} .

For

H_{i} ≪ R_{0}

, the slant optical depth of half

i

z

is approximately

τ_{λ, i} (z) = \frac{κ _{λ} P _{0}}{g} \frac{2 π R _{0}}{H _{i}} e^{- z / H_{i}} .

The standard isothermal effective altitude is therefore

z_{λ, i} = H_{i} [γ_{E} + l o g (\frac{κ _{λ} P _{0}}{g} \frac{2 π R _{0}}{H _{i}})],

up to a wavelength-independent choice of reference radius. The two semicircular limbs add in projected area, so the exoplanet transmission spectrum is

D_{λ} = \frac{( R _{0} + z _{λ, m} ) ^{2} + ( R _{0} + z _{λ, e} ) ^{2}}{2 R _{*}^{2}} .

Equivalently,

R_{tr}^{2} = [(R_{0} + z_{m})^{2} + (R_{0} + z_{e})^{2}] /2

. Differentiating with respect to

lo g κ_{λ}

gives

\frac{d R _{tr}}{d l o g κ _{λ}} = \frac{( R _{0} + z _{m} ) H _{m} + ( R _{0} + z _{e} ) H _{e}}{2 R _{tr}} ≃ \frac{H _{m} + H _{e}}{2} .

Thus a homogeneous retrieval measures, to leading order,

H_{av} = \frac{k _{B} ( T _{m} + T _{e} )}{2 μ m _{H} g},

provided the opacity and composition do not themselves differ between the two limbs.

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