Disentangling degenerate solutions from primary transit and secondary eclipse spectroscopy of exoplanets

Abstract

Infrared transmission and emission spectroscopy of exoplanets, recorded from primary transit and secondary eclipse measurements, indicate the presence of the most abundant carbon and oxygen molecular species (H 2 O, CH 4 , CO and CO 2 ) in a few exoplanets. However, efforts to constrain the molecular abundances to within several orders of magnitude are thwarted by the broad range of degenerate solutions that fit the data. Here, we explore, with radiative transfer models and analytical approximations, the nature of the degenerate solution sets resulting from the sparse measurements of ‘hot Jupiter’ exoplanets. As demonstrated with simple analytical expressions, primary transit measurements probe roughly four atmospheric scale heights at each wavelength band. Derived mixing ratios from these data are highly sensitive to errors in the radius of the planet at a reference pressure. For example, an uncertainty of 1% in the radius of a 1000 K and H 2 -based exoplanet with Jupiter's radius and mass causes an uncertainty of a factor of approximately 100–10 000 in the derived gas mixing ratios. The degree of sensitivity depends on how the line strength increases with the optical depth (i.e. the curve of growth) and the atmospheric scale height. Temperature degeneracies in the solutions of the primary transit data, which manifest their effects through the scale height and absorption coefficients, are smaller. We argue that these challenges can be partially surmounted by a combination of selected wavelength sampling of optical and infrared measurements and, when possible, the joint analysis of transit and secondary eclipse data of exoplanets. However, additional work is needed to constrain other effects, such as those owing to planetary clouds and star spots. Given the current range of open questions in the field, both observations and theory, there is a need for detailed measurements with space-based large mirror platforms (e.g. James web space telescope) and smaller broad survey telescopes as well as ground-based efforts.

Refraction in planetary atmospheres: improved analytical expressions and comparison with a new ray-tracing algorithm

“…The total column abundance and deflection experienced by a ray is twice that quantity since the rays goes through these layers twice: once moving downward, and once moving upward through the atmosphere. Although the computational grid used in Bétrémieux & Kaltenegger (2013, 2014 was composed of 0.1 km-thick layers, we have found that the precision of the computations improves significantly when the thickness of the computational layers decreases (see section 4.1), and that the desired thickness depends on the atmospheric scale height. Hence, we now adjust this parameter as required by the atmosphere for which we do the computations.…”

“…This expression for the column abundance has been previously derived in terms of these modified Bessel functions (see e.g. Griffith 2014). However, this is only the leading term in our Taylor series, and corresponds to the non-refractive case.…”

“…This algorithm, which has been previously described by Bétrémieux & Kaltenegger (2013, 2014 and references therein, suffered from numerical instabilities in the computed ray deflection, which started at densities of about 5×10 −5 amagat, and became worse with lower densities, such that the algorithm was useless for densities below 3 × 10 −6 amagat. These numerical instabilities were not only due to improper treatment of integer to double precision conversion of variables, but also intrinsically due to the method.…”

“…1) due to the exponential increase of atmospheric refractivity (the refraction index minus one), with decreasing altitude. The trajectory of a ray is described by an invariant, b, which is equal to its impact parameter (Phinney & Anderson 1968;Bétrémieux & Kaltenegger 2013, 2014 b = r(1 + ν) sin θ = r0(1 + ν0) = Rtop sin θ0.…”

“…that travels in a straight light, through a homogeneous isothermal atmospheres (see e. g. Fortney 2005; Lecavelier des Etangs et al 2008;Benneke & Seager 2012;Howe & Burrows 2012;de Wit & Seager 2013;Griffith 2014). However, several papers (Hui & Seager 2002;Sidis & Sari 2010;García Muñoz et al 2012;Bétrémieux & Kaltenegger 2013, 2014Misra, Meadows & Crisp 2014; have emphasised the importance of the refractive bending of light ⋆ e-mail: betremieux@mpia.de † Now at the Institute for Pale Blue Dots, Cornell University by exoplanetary atmospheres on these lightcurves. Refraction is also important in various observational geometries in our Solar System.…”