Analytical Models of Exoplanetary Atmospheres. IV. Improved Two-stream Radiative Transfer for the Treatment of Aerosols

We present a novel generalization of the two-stream method of radiative transfer, which allows for the accurate treatment of radiative transfer in the presence of strong infrared scattering by aerosols. We prove that this generalization involves only a simple modification of the coupling coefficients and transmission functions in the hemispheric two-stream method. This modification originates from allowing the ratio of the first Eddington coefficients to depart from unity. At the heart of the method is the fact that this ratio may be computed once and for all over the entire range of values of the single-scattering albedo and scattering asymmetry factor. We benchmark our improved two-stream method by calculating the fraction of flux reflected by a single atmospheric layer (the reflectivity) and comparing these calculations to those performed using a 32-stream discrete-ordinates method. We further compare our improved two-stream method to the two-stream source function (16 streams) and delta-Eddington methods, demonstrating that it is often more accurate at the order-of-magnitude level. Finally, we illustrate its accuracy using a toy model of the early Martian atmosphere hosting a cloud layer composed of carbon-dioxide ice particles. The simplicity of implementation and accuracy of our improved two-stream method renders it suitable for implementation in three-dimensional general circulation models. In other words, our improved two-stream method has the ease of implementation of a standard two-stream method, but the accuracy of a 32-stream method.Comment: Accepted by ApJS. 7 pages, 5 figure

Exoplanet Characterization by Multi-Observatory Transit Photometry with TESS and CHEOPS

Space-based photometric surveys have discovered large numbers of planets transiting other stars, but these observe in a single band-pass and yield only the planet radius, orbital period, and transit duration. Information on the masses, compositions, and any atmospheres of these planets requires additional observations from the ground or space. The Transiting Exoplanet Survey Satellite (TESS) will yield thousands of planets around bright stars suitable for such follow-up. In the absence of spectroscopy or spectrophotometry from space, observations through the different pass-bands of multiple space telescopes provide some spectral information useful for identifying false positive signals, distinguishing between reflected light and thermal emission from hot Jupiters, and detecting Rayleigh scattering by planetary atmospheres. We calculated the expected difference in transit depths measured by the TESS and Characterizing Exoplanets Satellites (CHEOPS) missions, which will be more sensitive to redder and bluer optical wavelengths, respectively. The difference due to companion or background stars is small (<3% for main sequence companions) and likely to be negligible and undetectable. For only a few "hot" Jupiters, can combined photometry disambiguate between the reflected and thermal signals from planets. However, Rayleigh scattering by hazy atmospheres with particles sizes near 0.04 $\mu$m and at pressure altitudes above ~1 mbar can be detected for ~100 transiting planets, assuming every planet has such an atmosphere. Hazes with this characteristic particle size do not obscure observations at longer (near-infrared) wavelengths; CHEOPS follow-up of TESS-detected planets could thus identify candidates suitable for further study with the James Webb Space Telescope.Comment: MNRAS, in pres

Optical properties of potential condensates in exoplanetary atmospheres

The prevalence of clouds in currently observable exoplanetary atmospheres motivates the compilation and calculation of their optical properties. First, we present a new open-source Mie scattering code known as LX-MIE, which is able to consider large size parameters ($\sim 10^7$) using a single computational treatment. We validate LX-MIE against the classical MIEV0 code as well as previous studies. Second, we embark on an expanded survey of the published literature for both the real and imaginary components of the refractive indices of 32 condensate species. As much as possible, we rely on experimental measurements of the refractive indices and resort to obtaining the real from the imaginary component (or vice versa), via the Kramers-Kronig relation, only in the absence of data. We use these refractive indices as input for LX-MIE to compute the absorption, scattering and extinction efficiencies of all 32 condensate species. Finally, we use a three-parameter function to provide convenient fits to the shape of the extinction efficiency curve. We show that the errors associated with these simple fits in the Wide Field Camera 3 (WFC3), J, H and K wavebands are $\sim 10\%$. These fits allow for the extinction cross section or opacity of the condensate species to be easily included in retrieval analyses of transmission spectra. We discuss prospects for future experimental work. The compilation of the optical constants and LX-MIE are publicly available as part of the open-source Exoclime Simulation Platform (http://www.exoclime.org).Comment: accepted version; 15 pages, 5 figures, 3 table

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]Comment: Accepted by MNRAS. 11 pages, 7 figures, 2 table

Clouds in the atmospheres of extrasolar planets. IV. On the scattering greenhouse effect of CO2 ice particles: Numerical radiative transfer studies

Owing to their wavelengths dependent absorption and scattering properties, clouds have a strong impact on the climate of planetary atmospheres. Especially, the potential greenhouse effect of CO2 ice clouds in the atmospheres of terrestrial extrasolar planets is of particular interest because it might influence the position and thus the extension of the outer boundary of the classic habitable zone around main sequence stars. We study the radiative effects of CO2 ice particles obtained by different numerical treatments to solve the radiative transfer equation. The comparison between the results of a high-order discrete ordinate method and simpler two-stream approaches reveals large deviations in terms of a potential scattering efficiency of the greenhouse effect. The two-stream methods overestimate the transmitted and reflected radiation, thereby yielding a higher scattering greenhouse effect. For the particular case of a cool M-type dwarf the CO2 ice particles show no strong effective scattering greenhouse effect by using the high-order discrete ordinate method, whereas a positive net greenhouse effect was found in case of the two-stream radiative transfer schemes. As a result, previous studies on the effects of CO2 ice clouds using two-stream approximations overrated the atmospheric warming caused by the scattering greenhouse effect. Consequently, the scattering greenhouse effect of CO2 ice particles seems to be less effective than previously estimated. In general, higher order radiative transfer methods are necessary to describe the effects of CO2 ice clouds accurately as indicated by our numerical radiative transfer studies.Comment: accepted for publication in A&

Linking the evolution of terrestrial interiors and an early outgassed atmosphere to astrophysical observations

A terrestrial planet is molten during formation and may remain so if subject to intense insolation or tidal forces. Observations continue to favour the detection and characterisation of hot planets, potentially with large outgassed atmospheres. We aim to determine the radius of hot Earth-like planets with large outgassed atmospheres and explore differences between molten and solid silicate planets and their influence on the mass-radius relationship and transmission and emission spectra. An interior-atmosphere model, combined with static structure calculations, tracks the evolving radius of a rocky mantle that is outgassing CO$_2$ and H$_2$O. Synthetic emission and transmission spectra are generated for CO$_2$ and H$_2$O dominated atmospheres. Atmospheres dominated by CO$_2$ suppress the outgassing of H$_2$O to a greater extent than previously realised, as previous studies have applied an erroneous relationship between volatile mass and partial pressure. We therefore predict more H$_2$O can be retained by the interior during the later stages of magma ocean crystallisation. Furthermore, formation of a lid at the surface can tie outgassing of H$_2$O to the efficiency of heat transport through the lid, rather than the atmosphere's radiative timescale. Contraction of the mantle as it solidifies gives $\sim5\%$ radius decrease, which can partly be offset by addition of a relatively light species to the atmosphere. We conclude that a molten silicate mantle can increase the radius of a terrestrial planet by around $5\%$ compared to its solid counterpart, or equivalently account for a $13\%$ decrease in bulk density. An outgassing atmosphere can perturb the total radius according to its speciation. Atmospheres of terrestrial planets around M-stars that are dominated by CO$_2$ or H$_2$O can be distinguished by observing facilities with extended wavelength coverage (e.g., JWST).Comment: 19 pages, published in A&A, abstract shortene

Clouds in the atmospheres of extrasolar planets. II. Thermal emission spectra of Earth-like planets influenced by low and high-level clouds

We study the impact of multi-layered clouds (low-level water and high-level ice clouds) on the thermal emission spectra of Earth-like planets orbiting different types of stars. Clouds have an important influence on such planetary emission spectra due to their wavelength dependent absorption and scattering properties. We also investigate the influence of clouds on the ability to derive information about planetary surface temperatures from low-resolution spectra.Comment: accepted for publication in A&

On the climatic impact of CO2 ice particles in atmospheres of terrestrial exoplanets

Dieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG geförderten) Allianz- bzw. Nationallizenz frei zugänglich.This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively.Clouds play a significant role for the energy budget in planetary atmospheres. They can scatter incident stellar radiation back to space, effectively cooling the surface of terrestrial planets. On the other hand, they may contribute to the atmospheric greenhouse effect by trapping outgoing thermal radiation. For exoplanets near the outer boundary of the habitable zone, condensation of CO2 can occur due to the low atmospheric temperatures. These CO2 ice clouds may play an important role for the surface temperature and, therefore, for the question of habitability of those planets. However, the optical properties of CO2 ice crystals differ significantly from those of water droplets or water ice particles. Except for a small number of strong absorption bands, they are almost transparent with respect to absorption. Instead, they are highly effective scatterers at long and short wavelengths. Therefore, the climatic effect of a CO2 ice cloud will depend on how much incident stellar radiation is scattered to space in comparison to the amount of thermal radiation scattered back towards the planetary surface. This contribution aims at the potential greenhouse effect of CO2 ice particles. Their scattering and absorption properties are calculated for assumed particle size distributions with different effective radii and particle densities. An accurate radiative transfer model is used to determine the atmospheric radiation field affected by such CO2 particles. These results are compared to less detailed radiative transfer schemes employed in previous studies