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

Abstract

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.In light of this, analytical formalisms to compute exoplanetary spectra are extremely useful. They allow one to understand how key atmospheric parameters shape exoplanetary spectra, can constitute the backbone of forward models, and increase computational speed tremendously if they are general and accurate enough. They can also test, under the same conditions for which they are valid, the accuracy of more general numerical methods. One important formalism for transmission spectroscopy, against which radiative

The Precision of Mass Measurements Required for Robust Atmospheric Characterization of Transiting Exoplanets

“…This leads to spectral features that are highly muted, relative to WASP-17b and HAT-P-1b. Cloud-coverage is a welldocumented effect that can sometimes impede precise retrievals of atmospheric parameters (Bétrémieux 2016;Bétrémieux & Swain 2017;Line & Parmentier 2016;MacDonald & Madhusudhan 2017). Therefore, WASP-12b exhibits wider retrieved posteriors for temperature, metallicity, reference radius, and mass, as shown in Figure 2, relative to the other two hot Jupiters.…”

“…Des Etangs et al 2008;Heng & Kitzmann 2017) to retrieval analyses (e.g. Benneke & Seager 2012;Benneke & Seager 2013;Line & Parmentier 2016;Bétrémieux 2016;Bétrémieux & Swain 2017;MacDonald & Madhusudhan 2017;Welbanks & Madhusudhan 2019). For example, Benneke & Seager (2013) investigated degeneracies between mean molecular weight and cloud top pressure, and Heng & Kitzmann (2017) & Welbanks & Madhusudhan (2019 focused on the degeneracy between radius, reference pressure, and composition.…”

ACCESS: A Visual to Near-infrared Spectrum of the Hot Jupiter WASP-43b with Evidence of H2O, but No Evidence of Na or K

“…The details of the retrieval code are given in our previous study of WASP-19b (Espinoza et al 2019), and we only briefly summarize the methodology here. Following the semi-analytical formalism from Bétrémieux & Swain (2017) and Heng & Kitzmann (2017), we assume an isothermal and isobaric atmosphere, with an optically thick base region with radius (R p /R s ) 0 and reference pressure P 0 , which we interpret as the cloud-top pressure. Above this region is an optically thin planetary atmosphere with average temperature T that can have either (i) a set of atomic and molecular species and/or (ii) a scattering haze defined by σ haze (λ) = aσ 0 (λ/λ 0 ) γ haze (MacDonald & Madhusudhan 2017), where σ 0 = 5.31 × 10 −27 cm 2 is the Rayleigh scattering cross-section of H 2 at the reference wavelength λ 0 = 350 nm, and a and γ haze are free parameters.…”

Refraction in exoplanet atmospheres

“…Refraction has attracted much attention as it could increase the effective radius of a planet and set a 'surface' above the deepest layers of the atmosphere, which has critical consequences for transmission spectroscopy (García Muñoz et al 2012;Bétrémieux & Kaltenegger 2013, 2014Bétrémieux 2016;Bétrémieux & Swain 2017). This effect is more pronounced for in-transit transmission spectra and it was emphasised by that deeper layers can be probed out of transit in such cases.…”

“…Refraction in exoplanet atmospheres have received increased attention over the past few years and several authors have emphasised that refraction can set the effective planet radius (García Muñoz et al 2012;Bétrémieux & Kaltenegger 2013, 2014. The implications of the effective surface depth due to refraction on transmission spectroscopy have also been studied (Bétrémieux 2016;Bétrémieux & Swain 2017). Several authors have studied how transmission spectra depend on transit phase and the possibility of using refraction to probe the atmospheric composition at different altitudes (Sidis & Sari 2010;García Muñoz et al 2012;.…”