Effectively Calculating Gaseous Absorption in Radiative Transfer Models of Exoplanetary and Brown Dwarf Atmospheres
MMNRAS , 1– ?? (2019) Preprint 12 March 2019 Compiled using MNRAS L A TEX style file v3.0
Effectively Calculating Gaseous Absorption in RadiativeTransfer Models of Exoplanetary and Brown DwarfAtmospheres
Ryan Garland, (cid:63) Patrick G. J. Irwin, Department of Physics, University of Oxford, Oxford, United Kingdom
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
Sophisticated atmospheric retrieval algorithms, such as Nested Sampling, explorelarge parameter spaces by iterating over millions of radiative transfer (RT) calculations.Probability distribution functions for retrieved parameters are highly sensitive toassumptions made within the RT forward model. One key difference between RTmodels is the computation of the gaseous absorption throughout the atmosphere. Wecompare two methods of calculating gaseous absorption, cross-sections and correlated- k , by examining their resulting spectra of a number of typical H -He dominatedexoplanetary and brown dwarf atmospheres. We also consider the effects of includingH -He pressure-broadening in some of these examples. We use NEMESIS to computeforward models. Our k -tables are verified by comparison to ExoMol cross-sectionsprovided online and a line-by-line calculation. For test cases with typical resolutions( ∆ ν = cm − ), we show that the cross-section method overestimates the amount ofabsorption present in the atmosphere and should be used with caution. For mixed-gasatmospheres the morphology of the spectra changes, producing ‘ghost’ features. Thetwo methods produce differences in flux of up to a few orders of magnitude. Theaddition of pressure broadening of lines adds up to an additional order of magnitudechange in flux. These effects are more pronounced for brown dwarfs and secondaryeclipse geometries. We note that correlated- k can produce similar results to very high-resolution cross-sections, but is much less computationally expensive. We concludethat inaccurate use of cross-sections and omission of pressure broadening can be keysources of error in the modelling of brown dwarf and exoplanet atmospheres. Key words:
Planetary Systems – planets and satellites: atmospheres – planets andsatellites: composition – planets and satellites: gaseous planets
The Solar System contains a wonderful variety of planetaryatmospheres. Each planet has its own unique formation andevolutionary history resulting in brilliant rings, icy moons,and complex atmospheric compositions. In order to betterunderstand the physical processes taking place within thesecelestial bodies, we extract data across almost the entire elec-tromagnetic spectrum, primarily in the visible and infraredas these are the wavelengths where light is best reflectedand thermally emitted. Radiative transfer models allow us tosimulate the physical processes on these bodies, producingspectra that can be compared to our data sets. Usually theseradiative transfer models are coupled to an inverse method, (cid:63)
E-mail: [email protected] which iteratively explores parameter space to retrieve themost likely set of parameters describing the atmosphere ofthe (exo)planet or brown dwarf (e.g., Irwin et al. (2008);Waldmann et al. (2015); Line et al. (2015)).This method of determining the atmospheric structureand composition is reliant on the accuracy of our radiativetransfer models. For Bayesian retrieval algorithms such asNested Sampling (e.g. MultiNest Feroz et al. (2013)) andMarkov Chain Monte Carlo (MCMC, e.g. EMCEE Foreman-Mackey et al. (2013)), the radiative transfer forward modelwill be executed millions of times while it explores a largeparameter space. Because of the nature of these highly di-mensional retrieval calculations, small differences in forwardmodels could potentially produce non-trivial changes in theretrieved probability distributions. For example, if one modelproduces an extra, erroneous absorption band for, say, NH , © a r X i v : . [ a s t r o - ph . E P ] M a r Garland & Irwin the retrieval would try to compensate for this error duringthe fitting procedure. While the ‘true’ value of NH shouldfit the spectrum (assuming the rest of the model is perfect),the extra absorption band would mean that the total amountof retrieved NH will be smaller than the ‘true’ value, as itfinds a medium-ground answer which best fits all (true +erroneous) absorption bands. If we do not assume the rest ofthe model is perfect, then the problem is even graver - theretrieval will try to fit the spectrum by adjusting the modelin a non-trivial way, such as adjusting the surface gravityor the temperature profile. In this way, one forward modelerror propagates through to all other retrieved posteriorprobability distribution functions (PDFs).A key difference between a number of radiative transfermodels is how the gaseous absorption is calculated - this isbecause it is a computationally expensive step. A line-by-linecalculation, where the absorption coefficient is calculatedfor the exact temperature and pressure for each individualline, is the most accurate and correct as it resolves eachindividual line exactly. However, this method is far too slowto be used in sophisticated retrieval procedures because ofthe enormous number of lines involved, especially at hightemperatures ( > k (e.g., Lacis & Oinas 1991). We also briefly consider theopacity sampling method (see e.g. Hubeny & Mihalas 2014),though not as extensively as the previous two as this wouldrequire a major overhaul of our code. We note that, as long asthe pressure and temperature resolution in the cross-sectionlook-up table is high enough, there is no distinction betweena high-resolution ( ∆ ν = . cm − ) cross-section and a line-by-line calculation. Individual gas cross-sections have beenused to solve the radiative transfer equation (MacDonald &Madhusudhan 2017). The cross-sections are absorption coef-ficients calculated from a line-by-line calculation on a grid ofpressures and temperatures for a given gas, then integratedpreserving area to a (typical) resolution of ∆ ν = cm − . Otherauthors calculate a high-resolution cross-section and sampleit at a resolution of ∆ ν = cm − (Line et al. 2015; Sharp& Burrows 2007). Hedges & Madhusudhan (2016) foundthat these individual gas cross-sections could have mediandifferences of <
1% for low-resolution ( R ∼ R (cid:46) R ∼ × ), introducedby various aspects of pressure broadening. This is before thenon-trivial task of quantifying these differences over a wholeatmosphere.Another approach to calculating the spectra of browndwarfs and exoplanets is using premixed k -coefficients(Saumon & Marley 2008) or on-the-fly mixing for thecorrelated- k method (Irwin et al. 2008; Barstow et al. 2014a;Lee et al. 2014). Note that for the rest of the paper, ‘cross-sections’ refers to cross-sections which are not premixed, i.e.individual cross-sections for individual gases. k -tables areproduced by performing line-by-line calculations of the ab-sorption coefficient then rewriting the absorption coefficientstrength distribution in terms of a cumulative frequencydistribution over bins of specified wavenumber/wavelength width by ranking and sampling the distribution according toabsorption coefficient strength. The inverse of this distribu-tion is known as the k -distribution (Lacis & Oinas 1991). The k -distribution is a smooth, monotonically increasing functionand so can be sampled with only 10-20 points, compared with ∼ to for the cross-section or line-by-line methods.Within a single atmospheric layer, we may simply combine the k -distributions of different gases by assuming that the linesare randomly overlapping. Note that there are many waysof combining k -coefficients for different gases. Among browndwarf and gas giant atmosphere models, ATMO (Tremblinet al. 2015; Drummond et al. 2016) and PETIT (Molli`ere et al.2015) also use random overlap, HELIOS (Malik et al. 2017)assumes perfect correlation, and Amundsen et al. (2016) useequivalent extinction. For further information see Amundsen,David S. et al. (2017). Then, by assuming that the wavenum-bers at which the (total) cross section takes a certain valueare vertically correlated these k -distributions may be used tocalculate the transmission, thermal emission or scattering ofan atmosphere using the correlated- k method. From previousstudies (Irwin et al. 2008), we have found the correlated- k ap-proximation to be accurate to better than 5%. This is why weprimarily use this as our benchmark absorption method forthe paper. For more information on our correlated- k method,refer to (Irwin et al. 2008) and references therein.In this paper, we argue that it is inaccurate to calculatethe gaseous absorption in both single- and mixed-gas atmo-spheres by combining individual gas cross-sections with large-and moderately-sized bins ( ∆ ν = , cm − ), or inefficientwith high-resolution bins ( ∆ ν = . cm − ). Throughoutthe paper we compare our calculations to the correlated- k method, and verify our methods with a line-by-line cal-culation. We also consider the effects of including H -Hepressure-broadening in the spectral calculations.Section one describes the methods and ingredients usedin the correlated- k tables (‘ k -tables’) and cross-sections suchas the pressure-broadening parameters and line lists used.Section two verifies our methods with simple single-layeratmospheres and a line-by-line calculation. More realisticatmospheres are then used to contrast the spectra of thetwo methods (correlated- k and cross-sections) and the effectof introducing H -He pressure broadening. The examplecases include a simple Hot Jupiter, a typical late-T dwarf,and HD189733b in primary transit and secondary eclipsegeometries.Section three summarizes our findings. A wide range of line list databases exists to provide the rel-evant molecular information for calculating spectra in theatmospheres of exoplanets and brown dwarfs. The key fac-tors in deciding which line lists are most appropriate are thewavelength and temperature ranges for which they are valid.For example, a widely used line list database is the High Res-olution Transmission (HITRAN) database (Rothman et al.2013), which is collated from multiple experimental and the-oretical sources. This database is, however, used mainly forrepresenting the spectrum of the Earth, and is therefore only
MNRAS , 1– ?? (2019) ffectively Calculating Gaseous Absorption in Radiative Transfer Models of Exoplanetary and Brown Dwarf Atmospheres reliable for temperatures up to ∼ ab initio calculations ratherthan measured in a laboratory and as such line positions andintensities may contain larger errors than those found exper-imentally in, e.g. HITRAN, for overlapping validity ranges.We note that HITEMP, a sister database to HITRAN, containline lists appropriate for higher temperatures, and similarlycontains a mixture of experimental and ab initio data.In Table 1, we present a summary of the line lists thatwe selected according to the criteria above, where Q ( T ) liststhe chosen source of the partition function data, necessary tocalculate the line intensities at the temperature of interest.We also use H -H and H -He collision-induced absorptionfrom HITRAN 2012 (Richard et al. 2012) for any calculationcontaining H -He. Another measure of appropriateness for each line list is thevalidity of its pressure-broadening coefficients. Unfortunately,line list databases which are suitable for the pressure broad-ening found in H -He-dominated atmospheres are scarce.HITRAN’s Earth-centric lists exhibit broadening parameterssuited to air-broadening (N and O ). ExoMol now pro-vide H and He pressure broadening parameters for most oftheir gases. For all molecules ExoMol provides cross-sectionswith only Doppler (i.e. thermal) broadening, but no pressurebroadening.In order to correctly estimate the pressure broadeninginduced on spectral lines in brown dwarf and exoplanetatmospheres, we performed a literature search and found thatAmundsen et al. (2014) had openly discussed their sourcesfor H -He pressure broadening. This work was performedprior to ExoMol providing pressure broadening parametersfor H and He, hence we have not used those values. Instead,most of the information found in Table 2 that we use forour pressure broadening parameters is from Amundsen et al.(2014), with a few additional sources added. Note that wehave not chosen to implement the same procedure for Naand K, as they both produce massive broadening wingsin the optical and are not experimentally well sampled forH and He broadening; instead we arbitrarily set the air-broadened widths to 0.075cm -1 atm − based on experiencewith terrestrial radiative transfer studies. We also note thatusing Voigt lineshapes, as we have done for all gases, isespecially dubious for the Na and K resonance lines Burrowset al. (2001). We intend on updating the Na and K k -tables inthe future for more realistic lineshapes; however this will notdrastically change the outcomes of this paper. For methodstaken by other groups, see e.g. Tremblin et al. (2015); Baudinoet al. (2015).In many cases these sources only provided broaden- H γ He J low H -He Figure 1.
Pressure broadening coefficients for CO. Pink is air-broadening, blue circles are data points, green lines are fourth-orderpolynomial fits. Top panel: H broadening coefficients. Middlepanel: He broadening coefficients. Bottom: H -He broadeningcoefficients with 85:15 ratio. ing parameters for the lower rotational quantum number, J low , up to 8-20 for the maximum J low value dependingon the gas, while our line lists contain data up to J low = 300. We implemented the broadening parameters intoour line lists shown in Table 2 by first converting theminto a single ‘foreign broadening parameter’ γ assuming anatmospheric ratio of 85:15 for H :He, using the weightedsum γ = γ H VMR H + γ He VMR He , where VMR representsthe volume mixing ratio. We then fitted this foreign broad-ening coefficient with a fourth order polynomial given by γ ( J low ) = (cid:205) i = i = α i J ilow , where α i represents each order’s con-stant, up until the available data, then using the last availablebroadening coefficient for any J low higher than the maximumavailable. Note that Amundsen et al. (2014) used a linearapproach up to the maximum J low , and then a constantvalue as we have.This is of course not ideal, but does not introduce anycomplex error propagation which might be the case witha more sophisticated modelling that extrapolates to higher J low . An empirical approach was considered, but while manygases show a gradual flattening of broadening coefficient withincreasing J low (see e.g., Buldyreva et al. 2011), there is noclear or simple relationship that allows us to model all ofthe gases after this maximum. In general, the constant valueappears to be a good first-order approximation. An examplefor the molecule CO is presented in Figure 1, where wecompare our new H -He foreign broadening to that providedfor air by HITRAN.The pressure-broadened line half-width (cm -1 ), is cal-culated from γ = γ (cid:16) PP (cid:17) (cid:16) T T (cid:17) n where the half-width at half-maximum of the Lorentzian profile γ is determined at astandard temperature T and pressure P (i.e. 296K and 1atm), n is an empirically derived pressure-broadening temper-ature exponent found in Table 2, and T and P are the desiredtemperature and pressure of the line respectively. The tem-perature exponent is constant over all quantum rotationalnumbers. MNRAS , 1– ?? (2019) Garland & Irwin
In this subsection we investigate two methods of solving theradiative transfer equation regarding optical depth, and showhow they affect the spectra produced by forward models ofbrown dwarfs and exoplanets. As our retrievals are entirelydependent on our forward models being ‘correct’, this is inan important point to address.We now describe the exact details of the k -tables we usein our radiative transfer calculations.For each gas in Table 1, we must calculate k -tables tobe used in the correlated- k method of solving the radia-tive transfer equations. We developed a method to removeinsignificant lines in the line databases at specified tempera-tures, as this will shorten the length of computation time ofthe k -tables. The method first calculates the line intensitiesat a particular temperature, and orders the line intensitiesfrom smallest to largest. It then creates a cumulative sumfrom the smallest to largest line intensities, and removes thesmallest n% of contributions to the total line intensities. Weuse n = × − , a very small percentage, so that we donot underestimate any continuum effects. We do this for ourentire temperature range, i.e. 100 - 2950K, over 20 equallyspaced (150K) temperature points as different lines becomeimportant at different temperatures. At the lower tempera-tures, a majority of the lines are stripped away (leaving ∼ × lines), while at higher temperatures the line lists areessentially identical (up to ∼ × ).For the k -tables, we first calculate the underlying ab-sorption spectrum monochromatically (at least 1/6 Voigt linewidth) over the entire spectral region (0.3-30 µ m). We cal-culate the absorption and hence cumulative k -distributionsat output wavelengths having separation ∆ λ =0.001 µ m, withsquare bins of width double that of the separation (i.e. aresolution of λ =0.002 µ m). We use Gauss-Lobatto quadratureto sample the distributions. To include the contribution ofwings of lines centred outside the spectral region of interest,the total wavelength range considered is defined as a rangebetween ν min − ν cut to ν max + ν cut . The total spectral inter-val is then subdivided into 1.5cm − bins where line data isstored. Absorption at a particular output wavelength is thencalculated by considering lines stored in the adjacent binsand the bin in the middle. Wing contributions from the linescentred outside these bins are calculated at the middle andend bins using a quadratic polynomial (because the wingshape follows a Lorentzian), and added on to the absorptionat the output wavelength. For each line calculation, a linewing outside the cutoff (25cm − from centre) is ignored.We calculate a grid of spectral opacities with the 20aforementioned temperature points, and 20 pressure levelsequally spaced in logspace from ∼ × − to 100 atm, using20 g-ordinates for Gauss-Lobatto quadrature. The lineshapesare given by a Voigt profile, where the line-wing cut-off ( ν cut )is at 25cm − for all gases but alkali, where the cut-off is at6000cm − .The pressure-temperature grid is linearly interpolatedin temperature-log-pressure space to find the desired spec-tral opacity. The k -tables may also be resampled into lowerresolutions to increase calculation speed, if necessary. For amore detailed description of how these calculations are done,the reader is referred to Irwin et al. (2008). These k -tablescalculated here will be adopted in future works. For the cross-section calculations in the literature, Exo-Mol (Hill et al. 2013) calculate a high-resolution ( < − )cross-section and the integral of the cross-section is preservedfor all requested resolutions. MacDonald & Madhusudhan(2017) calculate their cross-sections at 0.01cm − (from Hedges& Madhusudhan (2016)), and bin them down (presumablypreserving area) to 1cm − resolution before using them. Lineet al. (2016) use a 1cm − resolution opacity sampling methodon pre-computed cross sections which have a variable resolu-tion wavenumber grid that samples the lines at 1/4 of theirVoigt half widths from Freedman et al. (2008, 2014).One of the major issues with low- and medium-resolutioncross-sections is that they cannot combine gases effectivelydue to their insufficient resolution. The usual approach is tosum the individual contributions of the various gases andweight them by their volume mixing ratio (Sharp & Burrows2007; Waldmann et al. 2015). The multiplication propertyof transmission is only valid when these calculations aredone monochromatically. To illustrate this issue for low- andmedium-resolution cross-sections, consider the simple two-gas problem in a single layer where each gas has an identicaltransmission spectrum over a given interval which is largerthan the bin size for the cross-section, ∆ ν , shown in Figure 2.In this extreme example, the multiplication using these twomethods produces mean transmissions that are a factor of twodifferent. Correlated- k preserves this multiplication propertyof transmissions (Goody & Yung 1989), i.e. the calculationsare the same as for monochromaticity. Using cross-sectionsinvokes using a mean transmission over the interval and hencethe multiplication can produce fallacious results.The opacity sampling method requires a sufficient num-ber of samples within a specified bin to correctly estimate thearea of the cross-section, though this number is ill-definedand depends on spectral resolution. Line et al. (2016) showthat the 1cm − resolution opacity sampling is sufficient fortheir purposes (1.0-2.5 µ m); however, this may not be a goodresolution for either higher spectral resolutions or longerwavelengths. We briefly explore the effects of using opacitysampling as a means of computing the gaseous absorption inthe next section, but do not include it in our main resultssection as it would require a large overhaul of our code.There are two key issues with using the Doppler-broadened cross-sections provided by the ExoMol project(Hill et al. 2013). One is that they fundamentally should notbe used to solve the radiative transfer equation in mixed-gasatmospheres by combining single gas cross-sections as theyare either ineffective (low- and medium-resolution) or com-putationally expensive (high-resolution). The second is thatpressure-broadening is a key property in calculating atmo-spheric spectra that should not be ignored. These points willbe addressed in the next section.At low pressures, where all of the pressure-broadeninghas ceased and the lineshape is dominated by Doppler broad-ening, the mean opacity (i.e. ¯ k = (cid:205) NGi = k i ∆ g i ) of our k -tablesshould be equivalent to the ExoMol cross-sections. We showthis to be true for CH ∆ ν = cm − . The spectra are binned to ∆ λ = µ mresolution. This is also the case for the other relevant gases.By flattening our k -distribution, i.e. replacing k i for eachg-ordinate with ¯ k so that the k -distribution k ( g ) is now a flat(constant) distribution, our k -tables at these low pressures MNRAS , 1– ?? (2019) ffectively Calculating Gaseous Absorption in Radiative Transfer Models of Exoplanetary and Brown Dwarf Atmospheres Spectral Interval, ∆ ν T r an s m i ss i on , T Correlated- k X-Sec, ¯ T X-Sec, ¯ T Figure 2.
An illustrative example of the effects on transmissionwhen multiplying monochromatic and non-monochromatic wave-length regions. The Correlated- k transmission (pink) squared isidentical to the non-squared, as the first half of the interval is1 squared, and the second half is 0 squared. The cross-sectiontransmission (green) is averaged over the bin, such that whensquared it becomes a yet smaller value (blue). λ [ m] -24 -23 -22 -21 -20 -19 k ( c m m o l e c u l e − ) CH X-SecCH Corr- k Figure 3.
The CH k -table opacity at low-pressure (blue) iscompared to the Doppler-broadened cross-sections at a resolutionof ∆ ν = cm − from the ExoMol project (pink) for 1600K. Thespectra are binned to ∆ λ = µ m resolution. are mathematically identical to using cross-sections, and sowill fall foul to the aforementioned gas-mixing issues.Instead of using the ExoMol cross-sections directly wherewe are limited to the temperatures ranges supplied online,we carry out our comparison using only the lowest pressurepoints of our flattened k -tables as a proxy. In the next sectionwe show first that this is a good approximation to using cross-sections of resolution ∆ ν = cm − . First we would like to show the inability to effectively mixgases with cross-sections. We create a simple one-layer modelat low pressure (and therefore only Doppler-broadened), witha temperature of 1000K, and containing 50% H O and 50%CH . We calculate the mean transmission for the correlated- k using: ¯ T = NG (cid:213) i = NG (cid:213) j = e −( k i m a + k j m b ) ∆ g i ∆ g j (1)and the cross-sections: ¯ T = e −( k a m a + k b m b ) (2)where i and j represent the index of the weights and a and b are labels for the first and second gas respectively, k is the absorption coefficient (cm molecule − ), and m isthe absorber amount (molecule cm − ). Absorption (1 - ¯ T ) iscalculated with four different methods: 1) using our k -tablesdescribed earlier, the online ExoMol cross-sections directly attwo different resolutions - 2) 1cm − and 3) 25cm − ); 4) ourcross-section proxy k -tables described in the previous section;and 5) the opacity-sampling method at 1cm − resolutionusing the 0.01cm − ExoMol cross-sections. Methods 1 and 4are combined using equation 1, while methods 2,3 and 5 arecombined using equation 2. The spectra are then binned to ∆ λ = µ m resolution, and shown in Figure 4. We choosethese two resolutions for the cross-sections because the prioris the usual resolution used by other modellers, and thelatter because it is closer to the typical resolution that weare presenting in Figure 8 and subsequent figures in thevisible ( ∼ µ m). Beyond ∼ µ m, our k -tables are in facthigher resolution than ∆ ν = cm − . This means that weexpect the cross-sections (and opacity sampling method)to become relatively less accurate at longer wavelengths.This is also generally true because the Doppler width isproportional to the wavenumber, i.e. at long wavelengthslines become increasingly narrow and consequently require ahigher resolution to be resolved.In Figures 4 and 5 we see that the opacity samplingmethod produces absorption that is approximately of thesame order as the correlated- k and line-by-line methods.However it also exhibits a slight underestimation of theabsorption and a large increase in noise as we go to longer,less-sampled wavelengths. The effect is especially pronouncedin certain bands such as the 2.5-3 µ m band in Figure 5. Whilethese effects are interesting, it would require a major overhaulof our code to further investigate, and hence do not include itin further analysis. Figure 4 also shows that the overlappingbands can cause up to ∼ k and cross-sectionmethods. We believe these ‘ghost features’ are a consequenceof one-cross section being more accurate than the other, sothat in a mixture, when the inaccuracy dominates the totalabsorption, a ghost feature is formed. This is because theaverage transmissions computed from equation 2 does notcapture the non-linear relationship between transmission andopacity. As expected, beyond 10 µ m the absorption variesenormously due to an increase in the number of binned linesper resolution element in the cross-sections. Also it shows MNRAS , 1– ?? (2019) Garland & Irwin λ [ m] -5 -4 -3 -2 -1 A b s o r p t i on OS 1cm − XSEC 1cm − Corr- k XSECXSEC 25cm − Corr- k Figure 4.
Absorption calculated for a layer consisting of 50%CH , 50% H O at 1000K and path amount of × moleculecm − . Pink is opacity-sampling at 1cm − resolution using the0.01cm − resolution ExoMol cross-sections. Gold and blue are thecross-section absorption spectra at 1cm − and 25cm − resolutionrespectively, black line is the correlated- k method with a resolutionof 0.002 µ m the green line is the cross-section derived from ourcorrelated- k method. λ [ m] -3 -2 -1 A b s o r p t i on OS 1cm − XSEC 1cm − Corr- k XSECXSEC 25cm − Corr- k LBL
Figure 5.
Same as Figure 4 but zoomed in to 2-3 µ m. The purpledots are a line-by-line calculation calculated at ∆ ν = . cm − . that our k -table proxy cross-sections show approximately thesame (if not smaller) effects than the two different ExoMolcross-sections, hence we can conclude that using these cross-sections are a reasonable proxy to use for the rest of thispaper. As shown in Figure 5, the line-by-line calculationbetween 2-3 µ m agrees very well with our correlated- k methodand shows that there is a real concern for combining cross-sections at a resolution of ∆ ν = cm − , especially at longerwavelengths. By comparing our correlated- k method to a high-resolution cross-section (0.01cm − ) in Figure 6, it is evidentthat they now share a similar spectral morphology; however,the increase in resolution of the cross-section now causes alarge increase in computation time. Figure 6 also shows amedium-resolution case (0.1cm − ) which shows inaccuracies [ m] A b s o r p t i on XSEC 0.1cm XSEC 0.01cm Corr k Figure 6.
Similar to Figure 4, except pink is the cross-sectionabsorption spectra at 0.1cm − resolution, gold is cross-section at0.01cm − resolution, and black is the same correlated- k calculation. beginning to occur in a non-negligible way. In a vast majorityof real-life data resolutions, using the correlated- k method isfaster or more accurate that using these high- and medium-resolution cross-sections because the the correlated- k methodcan be precalculated at exactly the right resolution of thedata, whereas the cross-section method must always be at amuch higher resolution than that of the data. In this example we take a simplified but more realistic at-mosphere found in exoplanetary science, and compare itsresulting spectra using the two different gaseous absorptionmethods. We do this using an isothermal (2000K) Hot Jupiter-esque primary transit as our example ( . M Sat , . R Jup around a Solar-size star), which we refer to as PT1.Here we limit the k -tables to have only Doppler broad-ening by using the lowest pressure point in our grid( × − atm). This means we are directly comparing k -tableswith cross-sections, i.e. the effect of flattening over the g -distribution. From this section onwards, we also includecollision-induced absorption from H -H and H -He in ourrealistic atmosphere calculations where appropriate.In Figure 7, we have two different atmospheres, each witha set of spectra calculated by the correlated- k method andthe cross-section method. We have a 100% CH atmosphere,and a 100% H O atmosphere. Here, no mixing of the gasesis required, and therefore the two different methods producealmost identical results, with the differences being morepronounced for H O. These results are relatively similarbecause the propagating error in transmission multiplicationthroughout the pressure-varying path due to bin-averagingis small compared to those when averaging with overlappinggaseous bands. Note that the changes are not due to pressure-broadening effects, as the k -tables only contain Dopplerbroadening.We introduce a third atmosphere, composed of 50%CH and 50% H O. In Figure 8, we compare the spectra
MNRAS , 1– ?? (2019) ffectively Calculating Gaseous Absorption in Radiative Transfer Models of Exoplanetary and Brown Dwarf Atmospheres λ [ m] ( R p / R ∗ ) [ % ] CH Corr- k CH X-SecH O Corr- k H O X-Sec
Figure 7.
Spectra for two PT1 (defined in manuscript) exoplanetsconsisting of a single gas (100% CH in pink and black, 100%H O in gold and green) calculated using two methods. Legendlabels X-Sec and Corr- k represent cross-sections and correlated- k (Doppler-broadened only). λ [ m] ( R p / R ∗ ) [ % ]
50% CH ,50% H O50% CH ,50% H O XSEC
Figure 8.
Spectra for one PT1 exoplanet consisting of 50%CH , 50% H O. Pink line is the correlated- k method (Doppler-broadened only), blue is the cross-section method. produced by the Doppler-broadened correlated- k and thecross-section methods. We find that not only do the cross-sections overestimate the transit depth, but indeed theychange the morphology of the spectrum itself, similar to theresults for in Figure 4. In the previous two subsections we verified our correlated- k method via line-by-line calculations, and showed the impactof various resolutions of cross-sections on the morphology ofthe spectra and absorption profiles of simple exoplanet atmo-spheres. In this subsection and the next we intend to showtwo things for multiple types of realistic H -He-dominated at-mospheres: 1) the effects of using cross-sections to incorrectly [ m] F [ W m m ] Corr k PresCorr k No PresXSec No PresXSec Pres [ m] T r an s m i ss i on CH NH H ONaK
Figure 9.
Top plot: Spectra calculated from best fit parametersfor GL570D using various methods of calculation. The spectra arecalculated for cross-sections with (translucent green) and withoutpressure-broadening (translucent black), and correlated- k with(red) and without pressure broadening (blue). Bottom plot: thetransmission calculated for the three major absorbing species inthe atmosphere in a single layer of atmosphere at 700K, whereeach gas has been weighted by its volume mixing ratio for a givenpath amount. mix gases, and 2) the effects of including pressure-broadeningon the resulting spectra.To illustrate the effects of using cross-sections, correlated- k with no pressure-broadening (smallest pressure level avail-able at our k -tables, ∼ k withpressure-broadening to calculate the spectra of brown dwarfatmospheres, we use a typical late-T dwarf as an example,as it contains significant amounts of H O, CH , and NH ( . × − , . × − , . × − respectively for their vol-ume mixing ratios, which are representative values that areconstant in height). The spectrum also contains Na and Kwith VMRs of . × − and . × − . The mass and radiusare 41.5M Jup and 1.38R
Jup respectively, and the tempera-ture profile is a 673.5K grid model from Saumon & Marley(2008) with the appropriate surface gravity. The temperatureprofile, along with those from the next two sections, is shownin Figure 10. The exact numbers for these parameters donot significantly change the outcome of the results, and arethe best fit parameters of a previous unpublished retrievalperformed on the object GL570D.Figure 9 contains two plots. In the upper plot, we havespectra calculated using the cross-section method with andwithout pressure broadening, and the correlated- k method MNRAS , 1– ?? (2019) Garland & Irwin
Temperature [K] -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 P r e ss u r e [ a t m ] T dwarfEclipsePrimary
Figure 10.
Temperature profiles for the brown dwarf (‘T dwarf’in pink), the primary transit of HD189733b (‘Primary’ in blue)and secondary eclipse of HD189733b (‘Eclipse’ in green). with and without pressure broadening. In the bottom plot,we have calculated the transmission in a single layer at 700Kand 1 atm, weighted by the volume mixing ratios with agiven path amount ( × molecules/cm − ) for NH , CH ,H O, Na and K, so that it can be easily seen which spectralfeatures belong to which species.In all of the large bands we find a discrepancy betweenthe cross-section method and the correlated- k method, usu-ally of multiple orders of magnitude. For example, the 2.8 µ mwater band produces luminosities that vary by up to ∼ µ m CH band. In molecular bands, the addition of pressure broaden-ing can produce an additional order of magnitude change.These are huge effects that would certainly be reflected inparameter estimation during retrievals. In Figure 11 we present a fiducial model from Barstow et al.(2014b) of the primary transit spectrum of HD189733b, cal-culated using the three methods as described before for thebrown dwarfs. The atmosphere primarily consists of the usualH , He, H O, Na and K gases, along with a haze to coverthe observed Rayleigh slope.Similar to the brown dwarf cases, we find that the totalopacity and morphology of the spectra differ greatly betweenthe methods. The transit depth can vary by up to 1%, asseen at the 2.8 µ m water band. Contrary to the brown dwarfs,we note that the pressure broadening effects are subduedfor primary transit. This is because we are probing muchhigher in the atmosphere than in the brown dwarf case, whereDoppler broadening is the main agent of broadening. TheNa and K lines are changed vastly, although we must notethat the visible region is especially subject to large changesin ∆ ν (a constant ∆ λ , as we have, produces larger ∆ ν in thevisible), and thus the effects are more pronounced than forthe infrared.For secondary eclipse, the VMRs used are slightly dif-ferent than for primary transit. H O, CO , CO and CH all [ m] ( R p / R * ) [ % ] Corr k PresCorr k No PresXSec No PresXSec
Figure 11.
A fiducial HD189733b atmosphere for primary transit.The spectra are calculated for cross-sections with (translucentgreen) and without pressure-broadening (translucent black), andcorrelated- k with (red) and without pressure broadening (blue). [ m] F p / F s t a r Corr k PresCorr k No PresXSec No PresXSec
Figure 12.
A fiducial HD189733b atmosphere for secondaryeclipse. The spectra are calculated for cross-sections with (translu-cent green) and without pressure-broadening (translucent black),and correlated- k with (red) and without pressure broadening(blue). have VMRs of × − . H , He, Na, and K have VMRs of0.9, 0.1, × − , and × − respectively. In Figure 12 wesee a familiar increase in opacity and change in morphologyfor the cross-section method. The pressure-broadening is alsoslightly more significant compared to primary transit, as weare probing lower in the atmosphere. The effects here aresmaller than for the brown dwarfs, where a change in themethod can increase the flux by half an order of magnitudein the mid- and far- infrared, or change it by an order ofmagnitude in the near-IR. The effects of pressure-broadeningin secondary eclipse is smaller than on brown dwarfs (becausethe contribution functions peak at a lower pressure in theatmosphere) but still appreciable enough to become apparentin the spectra. MNRAS , 1– ?? (2019) ffectively Calculating Gaseous Absorption in Radiative Transfer Models of Exoplanetary and Brown Dwarf Atmospheres A major discrepancy between different radiative transfer mod-els is how gaseous absorption is calculated. We investigatedthe effects of using two different gaseous absorption methods(correlated- k and cross-sections) to calculate spectra in avariety of atmospheres of varying complexities. We also inves-tigated the effects of including H -He pressure broadening inthe more complex atmospheres. These investigations are im-portant because radiative transfer models are often coupledto inverse methods, which can iterate over millions of forwardmodels in a large parameter space; a single forward modelerror, such as a ‘ghost feature’, would propagate through toall retrieved PDFs influencing them in a non-trivial way, i.e.an error in the absorption spectrum of NH could influencethe retrieved temperature profile.We first showed that for test cases with resolutionsof ∆ ν = cm − the cross-section method overestimates theamount of absorption present in the atmosphere and shouldtherefore be used with caution. The morphology of the spec-tra also changes and produces ‘ghost’ features for mixed-gasatmospheres. When considering our flattened k -table cross-sections, the effect can produce multiple orders of magnitudechange in the flux received from brown dwarfs in certain wave-length regions. The effect is similar but smaller for primarytransit, and is closer in order to a ∼
1% change in transitdepth. The flux ratio of secondary eclipse exoplanets canfind an order of magnitude change in the near-IR, with theeffect becoming lesser for longer wavelengths. Correlated- k can produce similar results to line-by-line and very high-resolution cross-sections, but is much less computationallyexpensive. The inclusion of H -He pressure broadening sim-ilarly changes the total flux found in the spectra of browndwarfs and secondary transit exoplanets by up to an order ofmagnitude, while only making slight changes to the spectraof transiting exoplanets.If we take into account the sizable discrepancies foundby Hedges & Madhusudhan (2016) between the differentaspects of pressure broadening from medium- and higherresolutions, the issues discussed in this paper might be evenmore serious than suggested. However, at higher resolutionsthe differences we present here will become less significant,and so the main error source will switch from those discussedhere to those discussed by Hedges & Madhusudhan (2016).We conclude that inaccurate use of cross-sections andomission of pressure broadening can be key sources of errorin the modelling of brown dwarf and exoplanet atmospheres.These sources of error in the forward model may producestrong biases in the probability distribution functions ofretrieved parameters. ACKNOWLEDGEMENTS
R.G. thanks and acknowledges the support of the Scienceand Technology Facilities Council. P.G.J.I. also receivesfunding from the Science and Technology Facilities Council(ST/K00106X/1).
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Summary of Line Lists Selected. Data sources: HITEMP: , CDSD: ftp://ftp.iao.ru/pub/CDSD-4000 , ExoMol: , VALD: http://vald.astro.uu.se .Molecule Reference Q(T) AvailableCO Rothman et al. (2010) Rothman et al. (2010) HITEMPCO Tashkun & Perevalov (2011) Rothman et al. (2013) CDSDH O Barber et al. (2006) Barber et al. (2006) ExoMolNH Yurchenko et al. (2011) Yurchenko et al. (2011) ExoMolCH Yurchenko & Tennyson (2014) Yurchenko & Tennyson (2014) ExoMolTiO R.S. Freedman (priv. com.) Sauval & Tatum (1984) N/AVO R.S. Freedman (priv. com.) Sauval & Tatum (1984) N/ANa Kupka et al. (2000) Sauval & Tatum (1984) VALDK Kupka et al. (2000) Sauval & Tatum (1984) VALD M N R A S , ?? ( ) G a r l a n d & I r w i n Table 2.
Summary of Line Broadening DataMolecule Broadener Line width reference n (Average value) Temp. exponent referenceH O H Gamache et al. (1996) 0.44 Gamache et al. (1996)He Solodov & Starikov (2009); Steyert et al. (2004) 0.44 Gamache et al. (1996)CH H Pine (1992); Margolis (1993) 0.44 Margolis (1993)He Pine (1992) 0.28 Varanasi & Chudamani (1990)CO H Padmanabhan et al. (2014) 0.60 Sharp & Burrows (2007)He Thibault et al. (1992) 0.60 Thibault et al. (2000)CO H Regalia-Jarlot et al. (2005) 0.60 Moal & Severin (1986)He BelBruno et al. (1982); Mantz et al. (2005) 0.55 Mantz et al. (2005)NH H Hadded et al. (2001); Pine et al. (1993) 0.64 Nouri et al. (2004)He Hadded et al. (2001); Pine et al. (1993) 0.40 Sharp & Burrows (2007)TiO H Sharp & Burrows (2007) 0.60 Sharp & Burrows (2007)He Sharp & Burrows (2007) 0.40 Sharp & Burrows (2007)VO H Sharp & Burrows (2007) 0.60 Sharp & Burrows (2007)He Sharp & Burrows (2007) 0.40 Sharp & Burrows (2007) M N R A S , ?? (2019