The theory of transmission spectra revisited: a semi-analytical method for interpreting WFC3 data and an unresolved challenge

The computation of transmission spectra is a central ingredient in the study of exoplanetary atmospheres. First, we revisit the theory of transmission spectra, unifying ideas from several workers in the literature. Transmission spectra lack an absolute normalization due to the a priori unknown value of a reference transit radius, which is tied to an unknown reference pressure. We show that there is a degeneracy between the uncertainty in the transit radius, the assumed value of the reference pressure (typically set to 10 bar) and the inferred value of the water abundance when interpreting a WFC3 transmission spectrum. Second, we show that the transmission spectra of isothermal atmospheres are nearly isobaric. We validate the isothermal, isobaric analytical formula for the transmission spectrum against full numerical calculations and show that the typical errors are ~0.1% (~10 ppm) within the WFC3 range of wavelengths for temperatures of 1500 K (or higher). Third, we generalize the previous expression for the transit radius to include a small temperature gradient. Finally, we analyze the measured WFC3 transmission spectrum of WASP-12b and demonstrate that we obtain consistent results with the retrieval approach of Kreidberg et al. (2015) if the reference transit radius and reference pressure are fixed to assumed values. The unknown functional relationship between the reference transit radius and reference pressure implies that it is the product of the water abundance and reference pressure that is being retrieved from the data, and not just the water abundance alone. This degeneracy leads to a limitation on how accurately we may extract molecular abundances from transmission spectra using WFC3 data alone. Finally, we compare our study to that of Griffith (2014) and discuss why the degeneracy was missed in previous retrieval studies. [abridged]