= Solution
Take $R_p$ to be the <radius> of an opaque planetary disc and $H$ the geometric thickness of the model atmosphere, not necessarily one <atmospheric scale height>. Assume a uniform stellar <specific intensity>, a fully projected non-grazing transit, negligible planetary emission in the measured band, and no scattering or refraction returning light to the beam. The cylindrical approximation assigns the same slant <optical depth> $\tau_\lambda$ to every ray through the annulus.
The opaque disc blocks area $\pi R_p^2$. The annulus has projected area $\pi[(R_p+H)^2-R_p^2]$, and the <radiative transfer equation> transmits fraction $e^{-\tau_\lambda}$ through it. Its blocked fraction is therefore $1-e^{-\tau_\lambda}$. Dividing the missing light by the unobscured stellar-disc light gives the <exoplanet transmission spectrum>
$$
\boxed{D_\lambda=\frac{R_p^2+[(R_p+H)^2-R_p^2](1-e^{-\tau_\lambda})}{R_s^2}.}
$$
For $H\ll R_p$, the <annulus model for transmission spectroscopy> becomes
$$
D_\lambda\simeq\left(\frac{R_p}{R_s}\right)^2
+\frac{2R_pH}{R_s^2}(1-e^{-\tau_\lambda}).
$$
The <optically thin> excess is $2R_pH\tau_\lambda/R_s^2$; the <optically thick> limit is the area ratio $(R_p+H)^2/R_s^2$ before the thin-annulus approximation. For a realistic atmosphere, the slant <optical depth> varies with the ray's impact parameter $b$, giving instead
$$
D_\lambda=\frac{R_p^2+2\int_{R_p}^{R_p+H}b[1-e^{-\tau_\lambda(b)}]\,db}{R_s^2}.
$$
<Limb darkening> replaces the simple area weighting by the local stellar <specific intensity>. The constant-depth model is therefore an explicit geometric approximation, not the slant-depth law of a spherical hydrostatic atmosphere.
Back to article page
