Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 315 / 3 / b / Solution

Let the atmosphere occupy a thin annulus from $R_p$ to $R_a=R_p+\Delta R$. With mass extinction coefficient $k_{\nu }$, constant density $\rho$, and the assumed common chord length $l$, its optical depth isThe opaque solid planet removes area $\pi R_p^2$, while the annulus removes the fraction $1-e^{-{\tau }_{\nu }}$ of the incident specific intensity. Neglecting limb darkening, the exoplanet transmission spectrum is thereforeFor a thin annulus,If the geometrical thickness is estimated as $\Delta R=N_{H}H$, then the atmospheric scale height is $H=k_B{T}_p/(\mu m_{H}G{M}_p/R_p^2)$. A wavelength-independent $k_{\nu }$ makes this idealized spectrum flat; real molecular opacities create its features.