The Voigt and complex error function: Huml\'i\v{c}ek's rational approximation generalized

Accurate yet efficient computation of the Voigt and complex error function is a challenge since decades in astrophysics and other areas of physics. Rational approximations have attracted considerable attention and are used in many codes, often in combination with other techniques. The 12-term code "cpf12" of Huml\'i\v{c}ek (1979) achieves an accuracy of five to six significant digits throughout the entire complex plane. Here we generalize this algorithm to a larger (even) number of terms. The $n=16$ approximation has a relative accuracy better than $10^{-5}$ for almost the entire complex plane except for very small imaginary values of the argument even without the correction term required for the cpf12 algorithm. With 20 terms the accuracy is better than $10^{-6}$. In addition to the accuracy assessment we discuss methods for optimization and propose a combination of the 16-term approximation with the asymptotic approximation of Huml\'i\v{c}ek (1982) for high efficiency.Comment: 9 pages, 5 figure

Computational Aspects of Speed-Dependent Voigt Profiles

The increasing quality of atmospheric spectroscopy observations has indicated the limitations of the Voigt profile routinely used for line-by-line modeling, and physical processes beyond pressure and Doppler broadening have to be considered. The speed-dependent Voigt (SDV) profile can be readily computed as the difference of the real part of two complex error functions (i.e. Voigt functions). Using a highly accurate code as a reference, various implementations of the SDV function based on Humlíček's rational approximations are examined for typical speed dependences of pressure broadening and the range of wavenumber distances and Lorentz to Doppler width ratios encountered in infrared applications. Neither of these implementations appears to be optimal, and a new algorithm based on a combination of the Humlíček (1982) and Weideman (1994) rational approximations is suggested

Impact of Molecular Spectroscopy on Carbon Monoxide Abundances from SCIAMACHY

High-quality observations have indicated the need for improved molecular spectroscopy for accurate atmospheric characterization. Line data provided by the new SEOM-IAS (Scientific Exploitation of Operational Missions - Improved Atmospheric Spectroscopy) database in the shortwave infrared (SWIR) region were used to retrieve CO total vertical columns from a selected set of nadir SCIAMACHY (SCanning Imaging Absorption SpectroMeter for Atmospheric CHartographY) observations. In order to assess the quality of the retrieval results, differences in the spectral fitting residuals with respect to the HITRAN 2016 (High-resolution TRANsmission molecular absorption) and GEISA 2015 (Gestion et Etude des Informations Spectroscopiques Atmosphériques) line lists were quantified and column-averaged dry-air CO mole fractions were compared to NDACC (Network for the Detection of Atmospheric Composition Change) and TCCON (Total Carbon Column Observing Network) ground-based measurements. In general, it was found that using SEOM-IAS line data with corresponding line models improve the spectral quality of the retrieval (smaller residuals) and increase the fitted CO columns, thereby reducing the bias to both ground-based networks

Evolution and Spectral Response of a Steam Atmosphere for Early Earth with a coupled climate-interior model

The evolution of Earth's early atmosphere and the emergence of habitable conditions on our planet are intricately coupled with the development and duration of the magma ocean phase during the early Hadean period (4 to 4.5 Ga). In this paper, we deal with the evolution of the steam atmosphere during the magma ocean period. We obtain the outgoing longwave radiation using a line-by-line radiative transfer code GARLIC. Our study suggests that an atmosphere consisting of pure H$_{2}$O, built as a result of outgassing extends the magma ocean lifetime to several million years. The thermal emission as a function of solidification timescale of magma ocean is shown. We study the effect of thermal dissociation of H$_{2}$O at higher temperatures by applying atmospheric chemical equilibrium which results in the formation of H$_{2}$ and O$_{2}$ during the early phase of the magma ocean. A 1-6\% reduction in the OLR is seen. We also obtain the effective height of the atmosphere by calculating the transmission spectra for the whole duration of the magma ocean. An atmosphere of depth ~100 km is seen for pure water atmospheres. The effect of thermal dissociation on the effective height of the atmosphere is also shown. Due to the difference in the absorption behavior at different altitudes, the spectral features of H$_{2}$ and O$_{2}$ are seen at different altitudes of the atmosphere. Therefore, these species along with H$_{2}$O have a significant contribution to the transmission spectra and could be useful for placing observational constraints upon magma ocean exoplanets.Comment: 22 pages, 17 Figures, accepted for publication in ApJ on March

Transmission Spectroscopy with the ACE-FTS Infrared Spectral Atlas of Earth: A Model Validation and Feasibility Study

Infrared solar occultation measurements are well established for remote sensing of Earth's atmosphere, and the corresponding primary transit spectroscopy has turned out to be valuable for characterization of extrasolar planets. Our objective is an assessment of the detectability of molecular signatures in Earth's transit spectra. To this end, we take a limb sequence of representative cloud-free transmission spectra recorded by the space-borne ACE-FTS Earth observation mission (Hughes et al., ACE infrared spectral atlases of the Earth's atmosphere, JQSRT 2014) and combine these spectra to the effective height of the atmosphere. These data are compared to spectra modeled with an atmospheric radiative transfer line-by-line infrared code to study the impact of individual molecules, spectral resolution, the choice of auxiliary data, and numerical approximations. Moreover, the study serves as a validation of our infrared radiative transfer code. The largest impact is due to water, carbon dioxide, ozone, methane, nitrous oxide, nitrogen, nitric acid, oxygen, and some chlorofluorocarbons (CFC11 and CFC12). The effect of further molecules considered in the modeling is either marginal or absent. The best matching model has a mean residuum of 0.4 km and a maximum difference of 2 km to the measured effective height. For a quantitative estimate of visibility and detectability we consider the maximum change of the residual spectrum, the relative change of the residual norm, the additional transit depth, and signal-to-noise ratios for a JWST setup. In conclusion, our study provides a list of molecules that are relevant for modeling transmission spectra of Earth-like exoplanets and discusses the feasibility of retrieval.Comment: 25 pages, 15 figures, 3 table

Assessment of a Physics-based Retrieval of Exoplanet Atmospheric Temperatures from Infrared Emission Spectra

Atmospheric temperatures are to be estimated from thermal emission spectra of Earth-like exoplanets orbiting M-stars as observed by current and future planned missions. To this end, a line-by-line radiative transfer code is used to generate synthetic thermal infrared (TIR) observations. The range of 'observed' intensities provides a rough hint of the atmospheric temperature range without any a priori knowledge. The equivalent brightness temperature (related to intensities by Planck's function) at certain wavenumbers can be used to estimate the atmospheric temperature at corresponding altitudes. To exploit the full information provided by the measurement we generalize Chahine's original approach and infer atmospheric temperatures from all spectral data using the wavenumber-to-altitude mapping defined by the weighting functions. Chahine relaxation allows an iterative refinement of this 'first guess'. Analysis of the 4.3{\mu}m and 15{\mu}m carbon dioxide TIR bands enables an estimate of atmospheric temperatures for rocky exoplanets even for low signal to noise ratios of 10 and medium resolution. Inference of Trappist-1e temperatures is, however, more challenging especially for CO2 dominated atmospheres: the 'standard' 4.3{\mu}m and 15{\mu}m regions are optically thick and an extension of the spectral range towards atmospheric window regions is important. If atmospheric composition (essentially CO2 concentration) is known temperatures can be estimated remarkably well, quality measures such as the residual norm provide hints on incorrect abundances. In conclusion, temperature in the mid atmosphere of Earth-like planets orbiting cooler stars can be quickly estimated from thermal IR emission spectra with moderate resolution.Comment: 16 pages, 19 figures, 1 tabl

Computational Aspects of Speed-Dependent Voigt and Rautian Profiles

For accurate line-by-line modeling of molecular cross sections several physical processes "beyond Voigt" have to be considered. For the speed-dependent Voigt and Rautian profiles (SDV, SDR) and the Hartmann-Tran profile the difference of two complex error functions (essentially Voigt functions) has to be evaluated where the function arguments z± are given by the sum and difference of two square roots. These two terms describing z± can be huge and the default implementation of the difference can lead to large cancellation errors. First we demonstrate that these problems can be avoided by a simple reformulation of z-. Furthermore we show that a single rational approximation of the complex error function valid in the whole complex plane (e.g. by Humlicek, 1979 or Weideman, 1994) allows an evaluation of the SDV and SDR with four significant digits or better. Our benchmarks indicate that the SDV and SDR function evaluations are about a factor 2.2 slower compared to the Voigt function, but for evaluation of molecular cross sections this time lag does not significantly prolong the overall program execution because speed-dependent parameters are available only for a fraction of strong lines

Spectral features of Earth-like planets and their detectability at different orbital distances around F, G, and K-type stars

We investigate the spectral appearance of Earth-like exoplanets in the HZ of different main sequence stars at different orbital distances. We furthermore discuss for which of these scenarios biomarker absorption bands may be detected during primary or secondary transit with near-future telescopes and instruments.We analyze the spectra taking into account different filter bandpasses of two photometric instruments planned to be mounted to the JWST. We analyze in which filters and for which scenarios molecular absorption bands are detectable when using the space-borne JWST or the ground-based telescope E-ELT. Absorption bands of CO2, H2O, CH4 and O3 are clearly visible in high-resolution spectra as well as in the filters of photometric instruments. However, only during primary eclipse bands of CO2, H2O and O3 are detectable for all scenarios when using photometric instruments and an E-ELT telescope setup. CH4 is only detectable at the outer HZ of the K star since here the atmospheric modeling results in very high abundances. Since the detectable CO2 and H2O bands overlap, separate bands need to be observed to prove their existence in the atmosphere. In order to detect H2O in a separate band, a S/N>7 needs to be achieved for E-ELT observations, e.g. by co-adding at least 10 transit observations. Using a spaceborne telescope like the JWST enables the detection of CO2 at 4.3mu, which is not possible for ground-based observations due to the Earth's atmospheric absorption. Hence combining observations of spaceborne and groundbased telescopes might allow to detect the presence of the biomarker molecule O3 and the related compounds H2O and CO2 in a planetary atmosphere. Other absorption bands using the JWST can only be detected for much higher S/Ns, which is not achievable by just co-adding transit observations since this would be far beyond the planned mission time of JWST.(abridged)Comment: 15 pages, 8 figure

Remote Sensing of Stratospheric Trace Gases by TELIS

TELIS (TErahertz and submillimeter LImb Sounder) is a balloon-borne cryogenic heterodyne spectrometer with two far infrared and submillimeter channels (1.8 THz and 480--650 GHz developed by DLR and SRON, respectively). The instrument was designed to investigate atmospheric chemistry and dynamics with a focus on the stratosphere. TELIS participated in three scientific campaigns in Kiruna, Sweden between 2009 and 2011. The recent campaign took place in 2014 over Ontario, Canada. During previous campaigns, TELIS shared a balloon gondola with MIPAS-B and mini-DOAS. The primary scientific goal of these campaigns has been to monitor the time-dependent chemistry of chlorine and bromine, and to achieve the closure of chemical families inside the polar vortex. In this work, we present retrieved profiles of ozone (O3), hydrogen chlorine (HCl), carbon monoxide (CO), and hydroxyl radical (OH) obtained by the 1.8 THz channel from the polar winter flights during 2009--2011. Furthermore, the corresponding retrieval algorithm is described. The quality of the retrieval products is analyzed in a quantitative manner including: error characterization, internal comparisons of the two different channels, and external comparisons with coincident spaceborne observations. The errors due to the instrument parameters and pressure dominate in the upper troposphere and lower stratosphere, while the errors at higher altitudes are mainly due to the spectroscopic parameters and the radiometric calibration. The comparisons with other limb sounders help us to assess the measurement capabilities of TELIS, thereby establishing the instrument as a valuable tool to study the chemical interactions in the stratosphere