Annulus model for transmission spectroscopy

Take a planet with opaque radius $R_p$ and an atmosphere of thickness $z\ll R_p$, projected against a uniform stellar disc of radius $R_{\star }$. If the atmospheric annulus has one representative slant optical depth ${\tau }_{\lambda }$, its additional transit depth isThe general exoplanet transmission spectrum instead uses an integral over impact parameter $b$,The annulus model replaces this varying slant opacity by an effective constant. The atmospheric signal scales with optical depth when optically thin and saturates at the annular area when opaque.

An exoplanet cloud deck divides the projected atmospheric annulus into cloudy and clear radial zones. Their contributions to the exoplanet transmission spectrum add as projected areas multiplied by $1-e^{-{\tau }_{\lambda }}$ for each zone's slant optical depth.