 Solution

Assume a uniform stellar disc of radius $R_{\star }$, an opaque planetary radius $R_p$, and a thin atmospheric annulus of thickness $z\ll R_p$. Neglect planetary light during transit and scattering back into the beam. The attenuation part of the radiative transfer equation gives $I_{\lambda }=I_{\lambda ,0}{e}^{-{\tau }_{\lambda }}$, where ${\tau }_{\lambda }$ is the slant optical depth along a stellar ray through the atmosphere. The fraction of light removed from that annulus is therefore $1-e^{-{\tau }_{\lambda }}$.

If the annulus is represented by one effective optical depth, its area divided by the stellar area isThe annulus model for transmission spectroscopy consequently gives the extra transit depthHere ${\delta }_{\lambda }$ is normalized to the unobscured stellar flux and excludes the opaque-disc depth $(R_p/R_{\star }{)}^2$. For a spectral feature measured relative to a continuum with slant optical depth ${\tau }_c$, the corresponding contrast is $A(e^{-{\tau }_c}-e^{-{\tau }_{\lambda }})$; the displayed formula takes the annular continuum to be transparent.

A real exoplanet transmission spectrum has an impact-parameter-dependent optical depth. Its more accurate expression isStellar limb darkening and horizontally varying clouds further modify the weighting. The single-annulus approximation is useful for estimating an atmospheric spectral-feature amplitude, rather than predicting every spectral line.