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 is
$$
\delta_\lambda=A(1-e^{-\tau_\lambda}),
\qquad
A=\frac{(R_p+z)^2-R_p^2}{R_\star^2}\simeq\frac{2R_pz}{R_\star^2}.
$$
The general <exoplanet transmission spectrum> instead uses an integral over impact parameter $b$,
$$
\delta_\lambda=\frac2{R_\star^2}\int_{R_p}^{\infty}b[1-e^{-\tau_\lambda(b)}]\,db.
$$
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.


 Annulus model for transmission spectroscopy

ID: annulus-model-for-transmission-spectroscopy