An analytical formalism accounting for clouds and other ‘surfaces’ for exoplanet transmission spectroscopy

Although the formalism of Lecavelier des Etangs et al. (2008) is extremely useful to understand what shapes transmission spectra of exoplanets, it does not include the effects of a sharp change in flux with altitude generally associated with surfaces and optically thick clouds. Recent advances in understanding the effects of refraction in exoplanet transmission spectra have, however, demonstrated that even clear thick atmospheres have such a sharp change in flux due to a refractive boundary. We derive a more widely applicable analytical formalism by including first-order effects from all these "surfaces" to compute an exoplanet's effective radius, effective atmospheric thickness, and spectral modulation for an atmosphere with a constant scale height. We show that the effective radius cannot be located below these "surfaces" and that our formalism matches Lecavelier des Etangs et al. (2008)'s in the case of a clear atmosphere. Our formalism explains why clouds and refraction reduce the contrast of spectral features, and why refraction decreases the Rayleigh scattering slope as wavelength increases, but also shows that these are common effects of all "surfaces". We introduce the concept of a "surface" cross section, the minimum mean cross section that can be observed, as an index to characterize the location of "surfaces" and provide a simple method to estimate their effects on the spectral modulation of homogeneous atmospheres. We finally devise a numerical recipe that extends our formalism to atmospheres with a non-constant scale height and arbitrary sources of opacity, a potentially necessary step to interpret observations.