The continuum exoplanet transit photometry depth is approximately the opaque projected area ratio, assuming a corrected or negligible limb darkening contribution:

D_{0} = (\frac{R _{p}}{R _{s}})^{2} = 0.01, R_{p} = 0.1 R_{s} .

For a Sun-sized host, $R_{p} ≃ 6.96 \times 1 0^{7} m ≃ R_{J} ≃ 11 R_{\oplus}$ . The natural size analogue is Jupiter, not Earth.

The expected bulk constituent is molecular hydrogen, with helium next most abundant for a retained primary planetary atmosphere. The water band detects a strong trace absorber; it does not imply that water is the dominant gas. This interpretation assumes a conventional H/He gas giant; the measured radius by itself is not a measurement of mass or composition.

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Solution

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Interpret the quoted band amplitude as an absolute transit-depth increment

Δ D = 0.0003

, or

300

parts per million. For a thin atmospheric annulus of extra height

Δ z

, the atmospheric spectral-feature amplitude is

Δ D = \frac{( R _{p} + Δ z ) ^{2} - R _{p}^{2}}{R _{s}^{2}} ≃ \frac{2 R _{p} Δ z}{R _{s}^{2}}, Δ z ≃ \frac{Δ D}{2 D _{0}} R_{p} = 0.015 R_{p} ≃ 1040 km .

(1)

Let the water band span

N_{H}

atmospheric scale heights, so

Δ z = N_{H} H

. For a nearly isothermal atmosphere with constant gravity and mean molecular weight

μ

H = k_{B} T / (μ m_{u} g)

. The temperature from a transmission-band amplitude is therefore

T ≃ \frac{μ m _{u} g}{k _{B}} \frac{Δ D R _{s}^{2}}{2 R _{p} N _{H}}, \frac{T}{T _{0}} = \frac{0.015 R _{p}}{N _{H} H _{0}} \frac{μ}{μ _{0}} \frac{g}{g _{0}} .

(2)

Here

T_{0}, H_{0}, μ_{0}, g_{0}

refer to the chosen Jupiter reference layer. Assuming

g = g_{J} ≃ 25 m s^{- 2}

, an H/He exoplanet atmosphere with

μ ≃ 2.3

, and a clear strong band with

N_{H} ≃ 5

, gives

H ≃ 210 km

and $T ≃ 1.4 \times 1 0^{3} K$ . Equivalently, a Jovian reference with

T_{0} ≃ 165 K

has

H_{0} ≃ 24 km

, giving

H / H_{0} ≃ 8.7

The assumptions are hydrostatic, approximately isothermal limb gas, constant gravity, negligible cloud truncation, H/He composition, and an effective opacity contrast yielding about five scale heights. In an ideal clear limb,