 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 spectrumFor $H\ll R_p$, the annulus model for transmission spectroscopy becomesThe optically thin excess is $2R_{p}H{\tau }_{\lambda }/R_s^2$; 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 insteadLimb 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.