The impact of spectral line wing cut-off: Recommended standard method with
application to MAESTRO opacity database–The impact of spectral line wing cut-
off: Recommended standard method with application to MAESTRO opacity database

# The impact of spectral line wing cut-off: Recommended standard method with
application to MAESTRO opacity database

Ehsan (Sam) Gharib-Nezhad  1,2 Natasha E. Batalha Lead Author,
<EMAIL_ADDRESS>authors have contributed equally to this study. 1
Katy Chubb  3 Richard Freedman  1,4 Iouli E. Gordon  5 Robert R. Gamache,6
Robert J. Hargreaves,5 Nikole K. Lewis,7 Jonathan Tennyson,8 Sergei N.
Yurchenko,8
1 Space Science and Astrobiology Division  NASA Ames Research Center  Moffett
Field  CA  USA;
2 Bay Area Environmental Research Institute  CA  USA;
3 School of Physics and Astronomy  University of St Andrews  St Andrews
United Kingdom;
4 Carl Sagan Center  SETI Institute  515 North Whisman Road  Mountain View  CA
94043.;
5 Atomic and Molecular Physics  Harvard-Smithsonian Center for Astrophysics
Cambridge  MA  USA;
6 Department of Environmental  Earth  and Atmospheric Sciences  University of
Massachusetts  Lowell  MA  USA;
7 Department of Astronomy  Cornell University  Ithaca  NY  USA;
8 Department of Physics and Astronomy  University College London  London
United Kingdom

(Accepted XXX. Received YYY; in original form ZZZ; 2023)

###### Abstract

When computing cross-sections from a line list, the result depends not only on
the line strength, but also the line shape, pressure-broadening parameters,
and line wing cut-off (i.e., the maximum distance calculated from each line
centre). Pressure-broadening can be described using the Lorentz lineshape, but
it is known to not represent the true absorption in the far wings. Both theory
and experiment have shown that far from the line centre, non-Lorentzian
behaviour controls the shape of the wings and the Lorentz lineshape fails to
accurately characterize the absorption, leading to an underestimation or
overestimation of the opacity continuum depending on the molecular species
involved. The line wing cut-off is an often overlooked parameter when
calculating absorption cross sections, but can have a significant effect on
the appearance of the spectrum since it dictates the extent of the line wing
that contributes to the calculation either side of every line centre.
Therefore, when used to analyse exoplanet and brown dwarf spectra, an
inaccurate choice for the line wing cut-off can result in errors in the
opacity continuum, which propagate into the modeled transit spectra, and
ultimately impact/bias the interpretation of observational spectra, and the
derived composition and thermal structure. Here, we examine the different
methods commonly utilized to calculate the wing cut-off and propose a standard
practice procedure (i.e., absolute value of 25 cm-1 for $P\leqslant$ 200 bar
and 100 cm-1 for $P>$ 200 bar) to generate molecular opacities which will be
used by the open-access MAESTRO (Molecules and Atoms in Exoplanet Science:
Tools and Resources for Opacities) database. The pressing need for new
measurements and theoretical studies of the far-wings is highlighted.

###### keywords:

Wing cut-off – Opacity – Line profile – Voigt profile – Lorentz profile –
Absorption Cross Section

##  1 Introduction

Observations of a diverse set of atmospheres, from brown dwarfs and hot gas
giants, to small rocky worlds, will be a legacy of JWST and the future next-
generation ground based telescopes. In order to robustly compare observations
to models, the community needs access to validated and up-to-date opacities.
In many cases, computing opacities for a single molecule, under a specified
pressure-temperature-gas mixture combination, is not trivial. For exoplanet
conditions specifically (i.e., high temperature and exotic, non-air gas
mixtures), data are often incomplete (as outlined in Fortney et al., 2019) and
could lead to noticeable errors in modeling transit spectra of exoplanets
(e.g., Gharib-Nezhad & Line, 2019; Niraula et al., 2022) and brown dwarf
atmospheres (e.g., Iyer et al., 2023). Beyond line list data use, other
modeling choices in opacity calculations affect the resultant calculation,
including line profile, intensity cutoffs, and wing cut-offs. This was the
motivation for the MAESTRO111science.data.nasa.gov/opacities/ (Molecules and
Atoms in Exoplanet Science: Tools and Resources for Opacities) database: an
open NASA-supported opacity service that can be accessed by the community via
a web interface. Of priority for the MAESTRO project is to develop community
standards in computing and publishing molecular and atomic opacity data. The
main principles of the MAESTRO interface are to highlight: 1) known updates
from line lists communities that need to be incorporated into the community
database, 2) opacity calculation cautions that specify when approximations or
estimations have been made, 3) limits of use for comparing models to
observations (e.g., completeness and line position accuracy), and 4) the full
breadth of data (including relevant citations) that are used for a single
opacity calculation. Currently, the focus of this work is on molecular, not
atomic opacities. Atomic opacities will be the focus of future work. For the
molecular opacity calculation cautions listed, we currently employ a set of
standard warnings. We have six standard warnings that pertain to the
broadening choices (e.g., “Laboratory or theoretical broadener information not
available at all” or “Broadener data for an air mixture was used in place of
this broadener gas”), and two standard warnings that pertain to the line list
choices (“Line list used is not suitable for this temperature regime” and “The
line position accuracy is suitable for JWST-quality spectra only
(R=100-3000)”)..

We aim to accept community contributions to the MAESTRO opacity database,
which will undergo an open peer review through our contributor portal on
Github222github.com/maestro-opacities/submit-data. In order to facilitate
community contributions we must first agree on a standard method for wing cut-
offs such that we can effectively execute community model comparisons.
Therefore, here, we document our standard methods that we will adopt in the
MAESTRO database. This manuscript specifically focuses on the impact of
different line wing cut-off methods and the “best” method implemented to
generate opacity datasets using Voigt profile. In a future MAESTRO update, we
aim to expand our calculation cautions to include those that pertain
specifically to wing-cutoffs choices such that users can assess how choices in
wing-cutoff might affect their opacity data.

The Voigt profile, which is a convolution of Lorentz and Doppler (Gaussian)
profiles, is a common profile when generating model spectra from
molecular/atomic line lists. While this profile is extensively utilized for
generating absorption cross sections (ACS), spectroscopic measurements have
shown that non-Lorentzian behaviour plays a key role in both the line centre
and the wings (For example, for the distortion from the line centre see Ngo et
al. (2013); Wcisło et al. (2016); Ngo et al. (2017); Burch et al. (1963);
Burch & Gryvnak (1971); Burch et al. (1968, 1967, 1956); Burch & Williams
(1962), and for the wings see (Hartmann et al., 2002)). Establishing a
consistent policy for determining the limit of the extent of the line profile
is a difficult and complex problem. The far wing is usually defined as the
region beyond a certain multiple of the line widths from the line centre. For
most species, knowledge of the far wing may not be well established either by
theory, experiment, or a combination of the two. This deviation from the
perfect Lorentzian behaviour begins at $\sim$5 - 30 cm-1 from the line centre,
and depends on the absorber, broadening agent, pressure, and temperature.

The inclusion of the non-Lorentz behaviour in the ACS data is challenging due
to the lack of complete spectroscopic parameters. The main technique used in
astrophysics to minimize the issue is to introduce a wing cut-off, $R_{\rm
cut}$, parameter. $R_{\rm cut}$ can be a fixed spectral distance (e.g., 25
cm-1 is typically used for calculations of terrestrial radiative transfer
(Mlawer et al., 2012)) applied to all lines in the ACS calculation equally, or
it can be dependant on a property of each line. For example, some studies have
used $R_{\rm cut}=200\gamma_{\mathrm{v}}$ to 500$\gamma_{\mathrm{v}}$
(Freedman et al., 2014; Grimm & Heng, 2015; Hedges & Madhusudhan, 2016; Chubb
et al., 2021; Grimm et al., 2021), where $\gamma_{\mathrm{v}}$ is the Voigt
half-width-at-half-maximum (HWHM). In addition, a maximum cutoff can be used
to prevent the line wings from becoming too large at higher pressures (Chubb
et al., 2021). Some studies have employed a fixed value for all or different
pressure ranges (MacDonald, 2019; Gharib-Nezhad et al., 2021a, b; Zhang et
al., 2020), or even use a pressure-dependent cutoff, such as $\min(25P;100)$
cm-1333To prevent the absorption from becoming unrealistically large, a
pressure-dependent cutoff with maximum value of 100 cm-1 is proposed., where
$P$ is the total gas pressure in atmospheres, and also apply normalization
correction factors to ensure that the total strength of the profile is
conserved (Sharp & Burrows, 2007). In another study, (Letchworth & Benner,
2007) have proposed different equations to control the truncation of the line
wing in under different P-T conditions including the elimination of the weak
lines and evaluating the resultant errors.

A few studies, however, calculated semi-empirical parameters or “correction
factors” to improve the Voigt result. Examples are limited to few
spectroscopic measurements and for room temperature including CH4-in-H2 for
$P_{\ce{H2}}=10-200$ atm (Hartmann et al., 2002), and CH4-in-CO2 for
$P_{\ce{H2}}=3-25$ bar (Tran et al., 2022). Correction factors are often
different for different bands of a molecule; for instance, the Spectral
Mapping Atmospheric Radiative Transfer model (Meadows & Crisp, 1996) derives
correction factors and their functional implementations for different bands of
CO2 based on existing experimental studies and spectra of Venus. The spectral
wings generated from all these different methods are not necessarily
consistent, and hence different ACS spectra are generated and used for
atmospheric modeling. This hinders cross comparisons between studies, even
when the source line lists used are identical. As a result of different
choices in the wing cut-off, this imposes difficulties when comparing the
resultant physical inferences from substellar spectra.

In addition to the inconsistency problem, inaccurate choice of $R_{\rm cut}$
results in many types of errors. First, a small value of $R_{\rm cut}$ results
in an underestimation of the opacity continuum, particularly at high pressures
($\gtrsim$200 bar) where the Voigt (or Lorentzian) HWHM can be a few tens of
wavenumbers. This means that the wing region should be extended. Second,
having a large value for $R_{\rm cut}$ leads to an overestimation of the
opacity continuum at low pressures ($\lesssim 10^{-2}$ bar) where the Voigt
HWHM is $\lesssim$ 0.01cm-1. Third, far wing effects are often included in
continuum models (Mlawer et al., 2012; Shine et al., 2016) to correct the
overall opacity continuum (see Section 3 below) and it is important to match
$R_{\rm cut}$ with the definition of these functions to avoid over/under
counting. Another related problem specific to low pressure calculations is
that the line profile is sometimes narrower than the size of the wavelength
grid intervals. This can lead to under-sampling, or no sampling at all, and
impact the accuracy of the ACS444To generate absorption cross sections, a
line-by-line code typically iterates over all transitions in the line list. A
wavelength grid vector is created before the loop to accumulate the opacity
values from the Voigt profile for each generated line profile within the loop.
Subsequently, these values are added to the wavelength grid vector. At very
low pressures and very low frequencies, the line widths become extremely
narrow, potentially some of these lines will not be added accurately to the
grid if they are very narrower than the chosen grid spacing. Therefore, the
key point here is that the wavelength grid spacing should be sufficiently
smaller than the spectral line widths to include the opacity contributions
from all the lines and minimize the loss of intensity in such cases. . Among
the solutions are moving the line centres to the nearest grid point (Sharp &
Burrows, 2007) or using the line profile binning over the grid point instead
of sampling (Hill et al., 2013). These inaccuracies become crucial in spectral
regions with a large number of spectral lines because they add up and make the
overall opacity continuum stronger or weaker.

Our main objective in this study is to propose a “standard practice” method to
control inaccuracies resulted by calculations of the non-Voigt behaviour for
the MAESTRO opacity database. MAESTRO spans a wide range of pressures and
temperatures (i.e., 10-6–3000 bar and 75–4000 K). The justification for such a
wide range of pressures and temperatures in the context of atmospheric
radiative transfer modeling is provided by §2. Spectroscopic drawbacks in
calculating the non-Voigt profile and its impacts on far wings are discussed
in §3. §4 deals with other challenges: the challenges with the theoretical
calculation of the thermochemical phase of the materials/gases in a very high-
pressure regime are discussed in §4.1, and the impact of the window regions in
§4.2. Two popular methods for choosing the $R_{\rm cut}$ to mitigate the
effect of non-Voigt behaviour of the spectral lines are quantitatively
analyzed and discussed for CH4-in-air system in §5.1. These two scenarios are
then tested using the experimental spectrum of CO2-in-N2 system in §5.2. The
accuracy of the super-lines technique and its impact on the opacity continuum
for the SO2-in-H2 system is assessed in §5.3. Our recommendations for a
current default line wing cut-off are presented in §6, followed by
instructions on how to control the line wing cut-off parameter in some example
well-used codes in §7. We finish with our conclusions in §8.

##  2 Astrophysical context for the temperature and pressure ranges

Missions to the gas giant planets in our own solar system (e.g., Cassini,
Galileo, Juno) have long sought to probe their deep atmospheres, which
critically shape the chemistry and circulation patterns in the visible portion
of their atmospheres. Since we cannot yet send spacecraft to visit planets
orbiting other stars, we must rely on a combination of remote observations and
atmospheric modeling to tie together the processes in their deep
atmospheres/interiors and observable upper atmospheres. Theoretical models of
giant exoplanet atmospheres critically depend on assumptions about atmospheric
opacity sources which control the radiative balance between external and
internal heating sources. Internal heat in giant exoplanets arising from both
their formation and external forcing critically shapes the deep adiabatic
region of their atmospheres (pressures greater than 10s to 100s of bars) which
feeds into the radiative processes in the upper atmosphere (e.g. Thorngren et
al., 2019). Figure 1 illustrates the thermal structure in an atmosphere of a
generic hot Jupiter across the pressure-temperature space (left), and
associated contribution functions over the relevant JWST wavelengths for
thermal emission (middle) (Lothringer et al., 2018) and transmission (Mollière
et al., 2019) observations.

Additionally, giant planets orbiting close to their host stars are expected to
have deep atmospheric radiative zones extending to pressures of 1000 bar or
more (e.g. Showman & Guillot, 2002; Fortney et al., 2005). In order to attempt
to capture the radiative, advective, and chemical processes that occur
throughout giant exoplanet atmospheres, we must make assumptions about the
atmospheric opacity at pressures well beyond the 1-10 bars typically
considered. Hence, our opacity data for the MAESTRO database are calculated
for a wide range of pressures, 10-6–3000 bar, to be applicable to modeling all
cases. Table 1 lists the temperature and pressure grid points in detail.

Figure 1: Typical hot Jupiter pressure-temperature profile (left) and associated contribution functions over relevant JWST wavelengths. Contribution functions dictate the pressure regions that the observable spectrum is sensitive to. The emission contribution function (middle) is defined in Lothringer et al. (2018). The transmission contribution (right) is defined in Mollière et al. (2019).  Table 1: Opacity computational details for MAESTRO database T[K] grid: (73 points) |  |
---|---|---
| 75 | 100 | 110 | 120 | 130 | 140 | 150 | 160
| 170 | 180 | 190 | 200 | 210 | 220 | 230 | 240
| 250 | 260 | 270 | 275 | 280 | 290 | 300 | 310
| 320 | 330 | 340 | 350 | 375 | 400 | 425 | 450
| 475 | 500 | 525 | 550 | 575 | 600 | 650 | 700
| 750 | 800 | 850 | 900 | 950 | 1000 | 1100 | 1200
| 1300 | 1400 | 1500 | 1600 | 1700 | 1800 | 1900 | 2000
| 2100 | 2200 | 2300 | 2400 | 2500 | 2600 | 2700 | 2800
| 2900 | 3000 | 3100 | 3200 | 3300 | 3400 | 3500 | 3750
| 4000 |  |  |  |  |  |  |
P[bar] grid: (20 points) |  |
|  10-6 |  3$\times$10-6 |  10-5 |  3$\times$10-5 |  10-4 |  3$\times$10-4 |  |
|  10-3 |  3$\times$10-3 |  10-1 |  3$\times$10-1 |  10-2 |  3$\times$10-2 |  |
| 1 | 3 | 10 | 30 | 100 | 300 |  |
| 1000 | 3000 |  |  |  |  |  |

##  3 A brief comment on the line shape theory and the challenges with
calculating the spectral Far wing

The dipole moment of an atom or molecule describes the separation of charge
within it. The line shape can be formally defined as the Laplace
transform555The Laplace transform is a mathematical technique used to convert
a function from the time domain to the frequency domain. By taking the Laplace
transform of the dipole autocorrelation function, the line shape can be
characterised as a function of frequency of the thermally-averaged dipole
autocorrelation function, which is a measure of the loss of coherence of the
radiation due to the environment (perturbing electrons, ions, molecules, …).
The relationship between frequency in the line shape and time in the
autocorrelation function shows that the time dependence of certain phenomena
will impact different parts of the line shape. Processes that happen within a
certain time $\Delta t$ will appear in the spectrum within a frequency range
of $\Delta f=1/\Delta t$. Because small $\Delta f$ samples large $\Delta t$,
the cores of lines are in the dynamic limit because the perturbing particles
move appreciably within a large $\Delta t$. In the wings of the lines, large
$\Delta f$ corresponds to short $\Delta t$. At short $\Delta t$ the perturbing
particles do not move appreciably and appear “static”. Thus, the line wings
can be determined by ‘quasi-static’ methods. For example, Ma & Tipping (1990a)
developed a theory for the calculation of the continuous absorption due to the
far wing contributions of allowed lines of H2O- H2O absorption based on a
generalization of Fano’s theory and later extended it using the quasistatic
approximation for the far wing limit and the binary collision approximation
(Ma & Tipping, 1991). Their development was limited to binary collisions
(first-order in density). Hussey et al. (1975) developed expressions
containing a generalized Fano operator that has terms containing the effects
of three- or four-body collisions for spectral line broadening in plasmas.
These terms would be necessary for high-pressure applications. The wings of
spectral lines can be either sub-Lorentzian or super-Lorentzian. It is the
interaction between the radiating and perturbing molecules in the quasistatic
limit that determines the sub- or super-Lorentzian behaviour.

The simulation of molecular spectra has become routine and automated since the
development of the spectroscopic databases in the early 1970s (McClatchey et
al., 1973): HITRAN (Gordon et al., 2022), GEISA (Delahaye et al., 2021), ATMOS
(Gunson et al., 1996), etc. Here we will use HITRAN as the standard. However,
many problems remain, and therefore these databases and their functionality
are periodically updated and extended. First, there is always the question of
the quality of the molecular parameters in these databases. Positions and
lower state energies (term values as the units are cm-1) are well-known for
many molecules, but the line intensities are relatively known for a number of
molecules in mostly low temperature ranges. The line shape parameters,
$\gamma$, and $\delta$ (i.e., the Lorentzian pressure-broadening and pressure-
shift parameters, respectively) are the least well-known. For limited number
of the molecules in HITRAN, there are measured half-widths and a small number
of shifts. However, for many cases, these pressure-broadening coefficients are
limited to few transitions, and hence they are estimated for all other lines.
The uncertainty in the estimated half-width ranges from $\sim$1-100$\%$, and
in some cases, where the experimental or theoretical data is lacking, the
values in HITRAN (Gordon et al., 2022) are estimated using different methods.
For instance, they can be estimated by scaling data from other collision
systems (Gamache & Laraia, 2009), using the mass dependence of half-widths
(Lamouroux et al., 2010), or using the ratios based on the few available
empirical values available in the literature (Tan et al., 2022), or by
averaging the values as a function of quantum numbers (for instance, for
unmeasured transitions of water vapor in HITEMP2010 (Rothman et al., 2010), or
simply a constant value for all transitions. The line shifts are even less
certain as they exhibit much stronger vibrational dependence.

One of the main problems associated with using the Voigt or Lorentz profiles
is that the model does not give the correct profile over the whole extent of a
spectral line. Burch et al. (Burch et al., 1963; Burch & Gryvnak, 1971; Burch
et al., 1968, 1967, 1956; Burch & Williams, 1962) made several studies in the
sixties and seventies showing that the Lorentzian line shape does not match
measured spectra. They note for CO2 “Beyond a few cm-1 from line centre all
lines absorb less that Lorentz-shaped lines with the same $\gamma$.” For water
vapor, the opposite is true, the absorption is greater than that modeled by
Lorentzian as one moves away from the line centre. At around the same time,
the radiative transfer model FASCODE (Smith et al., 1978; Clough et al., 1981)
was being developed (now it is the LBLRTM (Clough et al., 2005) code). To deal
with this problem and to reduce the computation time, line wings were cut off
at $\pm$25 cm-1 from the line centre. For broad (low J) lines, of H2O and CO2
at room temperature and 1 atmosphere, this amounts to going out to $\sim$333
half-widths, but note as Burch et al. have suggested there are noticeable
differences in the absorption at $\sim$25 half-widths from the line centre.
Note also that for narrow lines (generally high J), going out $\pm$25 cm-1
corresponds to many more half-widths. To correct for this cut-off additional
absorption is accounted for by what has been called the continuum, designated
as the correction factor or $\chi$ factor. Such a correction factor was used
for simulating CO2 emission to compare to observations of the night side of
Venus (Pollack et al., 1993). The continuum is different for each absorbing
molecule. For water vapour, it was based on the work of Clough et al. and was
called CKD (Clough et al., 1989; Ma & Tipping, 1992) This model has been
updated many times based on new measurements and simulations. The current
version is called MT$\\_$CKD (Mlawer et al., 2012). The use of the continuum
recognizes that the absorption arises from several mechanisms and that
corrections need to be made by cutting off the lines at 25 cm-1. First, there
is the far-wing collision-induced absorption as modeled by Ma and Tipping (Ma
& Tipping, 1990b, c, 1991, 1994, 1992; Ma et al., 1999), and there is also
absorption by metastable species (Simonova et al., 2022) (dimers, trimers,
etc. although most of this can be attributed to the dimers). For example, the
H2O absorption can be broken into collision-induced absorption, dimer
spectrum, and a correction for the line cut-off. It will be similar to other
molecules.

##  4 Challenges

###  4.1 Computing opacities for high pressures: Line mixing

Spectroscopic studies of exoplanets largely probe regions whose pressure are
less than a few bars and, in many cases, a lot lower than this. For these
regions, the Voigt profile is known to give a reasonable approximation to the
pressure and temperature-dependent line shape in the region of the line centre
although subtle collision effects are known to lead to minor distortions of
this (Tennyson et al., 2014). At high pressures, there are a number of
complications in simulating the spectrum. Table 1 lists the T and P grid
points for the MAESTRO database. The line shape parameters are determined
based on the number density of the perturbing molecules with the pressure of 1
atm. At high pressures, non-ideal behavior of the gases occurs and the line
shape parameters should be taken at the corrected pressure corresponding to
the number density of the real gas. The correction can be found in the
appendix of Sung et al. (2019) who discuss O2; information on compressibility
for key molecules can by found in the on-line NIST Chemistry Webbook
666https://webbook.nist.gov/chemistry/. However, the largest distortion away
from a Voigt profile is caused by line mixing. Physically line mixing is
occurs due to collisions that connect the two transitions (Lévy et al., 1992;
Hartmann et al., 2008; Pieroni et al., 2001). To see this, consider two
transitions that mix, f $\leftarrow$ i and f′ $\leftarrow$ i′. Line mixing
provides an alternative path to go from $i$ to $f$. A molecule in state i
undergoes a collisional transition to i′, which then absorbs a photon and goes
to f′; f′ then undergoes a collisional transition to f, thus completing the
path from i to f. This path amounts to a redistribution of population, which
does not affect the line position, but the line intensity and line shape
parameters can be greatly affected. To be efficient, the frequencies of the
two transitions should be very close to each other and the probability of the
collisional connections, which are given by reduced matrix elements (Lamouroux
et al., 2014; Gamache et al., 2019), should be large. As pressure increases
collision induced absorption (CIA), which depends on the square of the
pressure, becomes increasingly dominant.

Firstly, Sung et al. (2019) discussed it only for one single molecule, O2. One
good source of compressibility would be the NIST Chembook on-line. Secondly,
what is underlined is not quite true. Collision-Induced Absorption (cia) is
density-square dependent while line mixing effect is proportional to density.
Therefore, toward high pressures, the cia effect is expected to be significant
( 10 bars), substantial ( 100 bars), and eventually dominant ( 1000 bars) over
the ACS based on the Voigt profile. The effect of line mixing with increasing
pressure is gradually blending discrete lines into a single, strongly
broadened absorption feature (see Fig. IV.2 of Hartmann et al. (2008) for a
nice illustration of this process). At terrestrial temperatures and pressures,
line mixing is only significant in a few special cases where lines lie
particularly close together such as Q-branch transitions of CO2. Thus, for
example, the current HITRAN2020 release (Gordon et al., 2022) implements full
and first-order line-mixing for air- and self-broadened CO2 lines (Hashemi et
al., 2020) and first-order line mixing of N2O and CO lines (Hashemi et al.,
2021); see also a recent experimental analysis of line mixing models for CO2
at high temperature and pressure (Cole et al., 2023).

However, models of gas giant exoplanets need to consider radiative transport
effects in regions where the pressure is much higher than 1 bar; the MAESTRO
database provides cross sections extending up to 3000 bar. In this high-
pressure regime, the ideal gas law is not longer obeyed and line mixing not
only becomes important but increasingly becomes the dominant pressure-
broadening mechanism. The standard procedure for modeling line mixing is via
the use of a frequency-dependent relaxation matrix (Lévy et al., 1992;
Hartmann et al., 2008). Empirical tests of the methodology using the spectrum
of NO molecule at pressures of 30 - 100 bar and temperatures over 500 K (Fu et
al., 1996; Almodovar et al., 2021) suggest that this methodology is reasonably
successful at modeling the effects of line mixing. We note that in this
pressure – temperature regime, the individual transitions merge into single
quasi-continuous features more characteristic of a condensed phase than the
usually heavily-structured line spectra which characterize gas phase spectra.
In this context, we note that in the high-pressure regime of interest for the
interior of gas giant (exo)-planets the species generally exist in an
environment well above their triple point meaning that there is no longer a
distinction between gas and liquid phases. It is therefore not altogether
surprising that the resulting spectra take on some of the characteristics of a
condensed phase spectrum.

Very recently, Ren et al. (2023) performed an experimental study of high-
pressure absorption coefficients for CO2, CO, and H2O for pressures up to 80
bar. They found that an empirical pseudo-Lorentz line shape model, which
considers line-mixing effects is easy to implement and relatively accurate at
modeling high-pressure gas spectra for the conditions they tested. This
suggestion certainly merits further investigation.

We are not aware of any exoplanetary models which allow for the effects of
line mixing. This is not surprising as the explicit coupling of the very many
transitions that occur in a hot spectrum via a relaxation matrix which links
these transitions together represents a significant extra complication in the
radiative transport problem. It is likely that this problem is best addressed
by generating effective temperature and pressure dependent cross section sets
which include line-mixing effects which can simply be used in the appropriate
models. This idea is left to future work. The onset of high-precision JWST
spectra as well as high-precision medium-resolution ground-based spectra of
Brown Dwarfs and giant planets will enable constraints on pressure-temperature
profiles to pressures of $\sim$1-10 bar (e.g. Hood et al., 2023, and as seen
in Figure 1). Therefore, we will soon be able to shed light on whether line
mixing seems to be driving temperature structure by comparing radiative-
convective-derived models to retrieved temperature profiles.

###  4.2 Spectral Windows regions

The question of how to limit the extent of the line wings as a function of
pressure and temperature is complex. At higher pressures, in order to capture
the majority of the opacity in any given line transition it may be necessary
to extend the line wings far beyond the line core and thus far beyond the
validity of the Lorentzian profile. The lack of an extension will result in
underestimating the total opacity, which will predominantly effect regions of
low opacity — so-called “spectral windows”. Because these regions of low
opacity contribute significantly to the character of the emerging radiation,
they have to be modeled correctly. The areas of the spectrum that are already
optically thick will not be affected to any great degree by decisions to
modify the line wing extent. However because the far wings in many cases are
known to be very sub-Lorentzian, the correct approach to the modeling of the
spectra is complicated. One approach is to establish a cutoff [in cm-1] from
the line centre for the profile and then attempt to compensate for the missing
line wings by a pseudo-continuum contribution that would have to be computed
separately. Although this approach is already used in many applications
related to the Earth’s atmosphere for water opacity (as discussed in §3), how
this would be handled in the astronomical case is less clear. A recent study
by Anisman et al. (2022) found that transmission spectra of exoplanet
atmospheres which are close to Earth’s atmospheric temperature can be
significantly impacted if data for the water continuum is not included in
models. The significance of the continuum is less clear at higher
temperatures, but is generally assumed to follow a negative temperature
dependence (Shine et al., 2016). Exoplanetary and brown dwarf atmospheric
radiative-transfer models cover a wide range of temperatures and pressures
that are never encountered on the Earth and include many species whose spectra
may not be fully documented.

##  5 Case studies

###  5.1 Benchmark the Effect of Different wing cut-off Methods on CH4-in-air
System

Given the challenges on the theory side as well as the lack of sufficient
experimental spectroscopic parameters, our goal is to propose the standard
practice for calculating the wing cut-off for the astronomy community. But
first, we need a quantitative overview of the range of Voigt HWHM (i.e.,
$\Gamma_{\rm V}$) values. Fig. 2 provides this information across a wide range
of pressures and wavenumbers for room temperature 777Note that the Voigt HWHM
($\Gamma_{V}$) in Fig. 2 is calculated with the following equation:
$\Gamma_{\rm V}=0.5346\enspace\Gamma_{\rm L}+\sqrt{0.2166\enspace\Gamma_{\rm
L}^{2}+\Gamma_{\rm G}^{2}}$ where $\Gamma_{\rm G}$ is the Doppler HWHM in cm-1
(Olivero & Longbothum, 1977). . For the sake of simplicity in comparison, a
constant Lorentz coefficient of $\gamma_{L}$ = 0.07 cm-1/bar is used here but
in reality, $\gamma_{L}$ is dependent on the molecular symmetry and quantum
numbers of the transition. In this figure shows that $\Gamma_{\rm V}$ HWHM has
a wide range of $10^{-1}$–150 cm-1 across the pressure-wavenumber space. Cases
with very low pressures ($\lesssim 10^{-2}$ bar) and low wavenumbers $\lesssim
10^{3}$ cm-1 (or $\gtrsim$ 10 ), have $\Gamma_{\rm V}$ lower than 10-2 cm-1.
On the other hand, very high pressures ($\gtrsim$101 bar) lead to a large
$\Gamma_{\rm V}$ HWHM, i.e., > 20 cm-1, and up to $\sim$150 cm-1.

Figure 2:  Estimated Voigt HWHM across a wide range of spectral range and
pressure for CH4 broadened by air with a constant Lorentz coefficient of 0.07
cm-1/bar and for room temperature. The calculated Voigt HWHM varies smoothly
with pressure and position, the apparent steps are an artifact of the number
of points calculated.

Fig. 3 represents the generated ACS datasets for the CH4-in-air system for a
wide range of pressures and temperatures, 10-6 – 103 bar and 75 – 4000 K, and
for four wavenumber (WN) regions: 90 – 110, 900 – 1100, 4900 – 5100, and 9900
– 10100 cm-1 (or 100, 10, and 2, 1 , respectively) with $R_{\rm cut}$
calculated using in Eqs (1) and (2). To quantify the impact of the wing cut-
off, Fig. 4 represents the differences in MAPE (the mean absolute percentage
error) metric between the ACS datasets for hundreds of spectra.

$R_{\rm cut,Abs}=\left\\{\begin{matrix}30~{}\textrm{cm}^{-1}~{}\ \ \textrm{for
P}~{}\leqslant 200~{}\textrm{bar}\\\ 150~{}\textrm{cm}^{-1}~{}\textrm{for
P}>200~{}\textrm{bar}\end{matrix}\right.$ (1)

$R_{\rm cut,HWHM}=500\times\gamma_{\rm Voigt}$ (2)

If we take the $R_{\rm cut,Abs}$ as the actual value, then MAPE formula is
expressed as follows:

$\textrm{MAPE}=\left(\frac{100}{n_{\rm gp}}\right)\sum_{i=1}^{\nu_{\rm
gp}}\left|\frac{\sigma_{R_{\rm cut,Abs}(\nu,T,P)}-\sigma_{R_{\rm
cut,HWHM}(\nu,T,P)}}{\sigma_{R_{\rm cut,Abs}(\nu,T,P)}}\right|$ (3)

where $n_{\rm gp}$ is the number of grid points in the pressure-temperature
space and $\nu_{\rm gp}$ is the wavelength of the transition. Few examples of
the generated ACS spectra from the two scenarios of the wing cut-off are
represented in Fig. 3 for T= 300 K and pressures 10-3, 10-1, 10 bar and
wavelength 90 – 110, 490 – 510, 990 – 1010, and 9990 – 10010 cm-1. We see the
largest discrepancies in the generated ACS are from cases with low pressures,
P < 0.1, and low wavenumbers.

Following that, Fig. 4 illustrates the differences between these two scenarios
in the pressure–temperature space and across a wide spectral range. The
balance between the Doppler widths and Lorentz widths plays a key role in the
MAPE amount between these two $R_{\rm cut}$ scenarios. This difference is
$>$50$\%$ for very low wavenumbers (i.e., WN$\leq$1000cm-1 or at far-infrared
$\lambda$>10). MAPE error becomes less than 10$\%$ for high wavenumbers. The
lesson learned from this figure is that the generated opacities at low
wavenumbers, low pressures, and low temperatures are highly sensitive to the
spectral wings and wing cut-off approaches. This is in agreement with the
findings of Niraula et al. (2023) where the effect (or lack of effect) of
pressure broadening and far-wing on the retrievals from the WASP 39b spectra
(which probed the low-pressure/high-temperature layer of the atmosphere) was
investigated. Please note that this figure serves as an illustration of the
magnitude of the MAPE error for this specific absorber. For absorbers with
different molecular masses (and consequently different Doppler widths), the
errors would differ accordingly.

The current focus of exoplanet and brown dwarf atmospheric spectroscopy with
JWST is on the spectral range at wavenumbers $>500$ cm-1. For example, the
very first JWST Early Release Science observations focused on exoplanets and
brown dwarfs featured a 1-20 $\micron$ R$\sim$1000-3700 spectrum of VHS
1256-1257 b (Miles et al., 2022), and a 1-5$\micron$ R$<300$ spectrum of
WASP-39 b (JWST Transiting Exoplanet Community Early Release Science Team et
al., 2023). Additionally, radiative-convective climate modeling demands
accurate opacities throughout the expected spectral energy distribution of the
planet’s emitted flux. Figure 3 in Mukherjee et al. (2023) shows typical
wavelength bins used for radiative-convective models of planets with
T${}_{\mathrm{eff}}=300-1000$ K. These wavelength bins extend to 30 cm-1 (333
$\micron$). Therefore, from an observational and heat transport perspective,
it is essential to establish uniform methods of computing wing cut-offs.

Figure 3:  Calculated cross sections of CH4 at 300 K broadened by air at three
sample pressures (0.001, 0.1, 10.0 bar) in four spectral regions, using fixed
line wings ($R_{\rm cut,Abs}$, (red spectra) and line wings dependant on the
pressure-broadening ($R_{\rm cut,HWHM}$, grey spectra) represented in Eqs.
1-2. Figure 4:  Mean Absolute Percentage Error (MAPE) from the two wing cut-
off approaches for different spectral ranges: From left to right: $\sim$100 ,
$\sim$10 , $\sim$2 , and $\sim$1 for CH4 in air. This figure shows that the
generated opacities at low wavenumbers, low pressures, and low temperatures
are highly sensitive to the spectral wings and wing cut-off approaches. The
lack of broadening coefficients and appropriate line profile for very high
pressures is another issue. See section §5.1 for further details.

###  5.2 Benchmark the Effect of Different wing cut-off Methods on CO2-in-N2
System

The strong $\nu_{3}$ vibrational mode of CO2 near 2350 cm-1 (4.25 $\mu$m)
provides an opportunity to test the impact of the line wing cut-off over a
complete absorption band. This band is readily observed in terrestrial
atmospheres and has recently been detected in JWST observations of WASP-39b
(JWST Transiting Exoplanet Community Early Release Science Team et al., 2023).

Figure 5 compares the effect of the line wing cut-off for the $\nu_{3}$ region
of CO2. The left panel displays an experimental ACS of CO2 at 25∘C broadened
by 1.0 atm of N2 recorded by the Pacific Northwest National Laboratory, PNNL
(Sharpe et al., 2004). The experimental measurement (black line) is shown in
the region of the strong $\nu_{3}$ vibrational mode near 2350 cm-1 on a
logarithmic scale. The approximate sensitivity of the experimental measurement
is $\sim$10-23 cm2/molecule. Overplotted are four spectra calculated with
Voigt line shapes using HAPI (Kochanov et al., 2016) and the CO2 line list
from HITRAN2020 (Gordon et al., 2022). The only difference between the four
calculated spectra is the treatment of the line wing cut-offs. These include
line wings calculated from each line centre in multiples of the Voigt HWHM
(250$\times$ and 500$\times$) as well as fixed widths (25 or 100 cm-1). The
effect of the different line wing treatments is clearly seen in the region >
2380 cm-1.

The right panel of Figure 5 shows a zoomed-in region near the centre of the
band for three different pressures (the example at 1.0 atm is the same as the
left panel). At lower pressures, the limitation of calculating a line wing
that is dependent on the line broadening is shown to produce unexpected steps
in the ACS between the strongest lines. In this example, these steps are a
consequence of calculating the line wings up to 250$\times$HWHM. This can be
seen for the line indicated by an asterisk, where at 0.01 atm the broadening
is 0.002 cm-1, which results in line wings of approximately $\sim$0.5 cm-1
either side of the line centre.

It should be noted that the comparisons in Figure 5 use the air-broadening
half-widths from HITRAN. For a more accurate comparison to the PNNL spectra,
it would be necessary to use N2-broadening parameters along with line-mixing.
However, the effects observed by the different choices in line wing cut-off
would still be observed.

Figure 5: A demonstration of the effect of the line wing cut-off for the
$\nu_{3}$ region of CO2. a) An experimental absorption cross-section of CO2 at
25∘C broadened by 1.0 atm of N2 recorded by the Pacific Northwest National
Laboratory, PNNL (Sharpe et al., 2004), compared to four calculated spectra
with different line wing cut-off limits. b) A zoomed in comparison at 25∘C for
three pressures. Note, only the plots at 1.0 atm include the PNNL spectrum.
See Section 5.2 for further details.

###  5.3 Assess the Impact of the Super-Lines Method on wing cut-off:
SO2-in-H2/He system

Here we assess the effect of using the super-lines method to compute
absorption cross-sections as a comparison to the effect of line-wing cut-off.
Some line lists are very large and can contain billions of transitions between
many millions of energy states. When computing cross-sections from such large
line lists, studies such as Rey et al. (2016) and Yurchenko et al. (2017) have
employed so-called super-lines. The general principle is to first compute line
intensities $I_{ij}$ at a given temperature (i.e. stick spectra) and then to
sum together the intensities of all lines within a specified spectral bin to
create a so-called super-line. Thus super-lines are effectively temperature-
dependent spectral histograms computed on a spectral grid. In order to ensure
error transmission is kept to a minimum, a non-uniform grid of R = 1,000,000
is typically used for the first step of the super-lines computations with a
typical size of a few million grid points (super-lines) for each of the 20–30
temperatures used. A Voigt broadening profile is then applied to each of these
super-lines. Since the computation of the Voigt profile is the most expensive
part of a simulation, this two-step procedure provides a huge reduction in the
computation time.

The main source of error of the super-line procedure is that any dependence of
the pressure broadening line profiles on the quantum numbers and molecular
symmetry is ignored. This is because each super-line is a combination of
multiple transitions connecting different states falling into the given
spectroscopic bin.

Here we consider the high-resolution cross-sections computed for ExoMolOP
(Chubb et al., 2021), prior to sampling down to sampled cross-sections or
k-tables888The general principle of k-tables is to order spectral lines within
a given spectral bin, producing a smooth cumulative distribution function to
represent opacity. This smooth distribution can be more efficiently sampled,
allowing k-tables to reach comparable accuracy as sampled cross-sections for a
much smaller R = $\frac{\lambda}{\delta\lambda}$.. We compare the cross-
sections of SO2, using the ExoCross (Yurchenko et al., 2018) code with the
ExoMol ExoAmes line list of Underwood et al. (2016), as an illustrative
example, with and without using the super-lines method. The super-lines
methods necessitate that average-$J$ rather than $J$-dependent broadening
parameters are used. The SO2 line list of Underwood et al. (2016) contains 1.4
billion transitions between 3.3 million ro-vibrational energy levels and
covers up to 8000 cm-1 (down to 1.25 $\mu$m). We use average broadening
parameters (i.e. not $J$-dependent) for the super-lines method, compared to
$J$-dependent broadening parameters extracted from the HITRAN database (Gordon
et al., 2022) for the broadening of SO2 by H2 and He. A fixed line wing cut-
off of 25 cm-1 was used for both the super-lines and the standard non-super-
lines methods. A variable grid spacing is used to try and ensure at least 4
sampling points for each line, with a maximum separation of 0.01 cm-1.

To compute the mean absolute percentage error (MAPE), we use the metric of Eq.
(3). Here we take the non-super-lines approach as the actual value, where we
also use a fixed line wing cut-off of 25 cm-1. We focus on the region 1300 -
1400 cm-1 (7.1 - 7.7 $\mu$m). The MAPE value between the non-super-lines and
the super-lines method for a selection of pressures and temperatures using SO2
as an example is summarized in Table 2. It can be seen that the error
increases with lower pressures and temperatures, where the lines are narrower
than at higher pressures and temperatures. An example of cross-sections of SO2
computed using the two methods at 500 K and 1 $\times$ 10-5 bar can be seen in
Figure 6.

Figure 6:  An example of cross-sections of SO2 computed using ExoCross (Yurchenko et al., 2018), with the super-lines method compared to without, at 500 K and 1 $\times$ 10-5 bar. See section §5.3 for further details.  Table 2: Mean absolute percentage error (MAPE) of the super-lines vs non-super-lines method for computing cross-sections of SO2 for a selection of pressures and temperatures.  Temperature (K) | Pressure (bar) | MAPE ($\%$)
---|---|---
1500 | 1 | 7.8$\times$10-4
1500 | 1$\times$10-2 | 0.09
1500 | 1$\times$10-5 | 0.18
1000 | 1 | 5.90$\times$10-3
1000 | 1$\times$10-2 | 0.21
1000 | 1$\times$10-5 | 0.83
500 | 1 | 0.03
500 | 1$\times$10-2 | 0.03
500 | 1$\times$10-5 | 6.34

The cross-sections computed using the super-lines technique (which
necessitates the use of average-$J$ broadening parameters) have negligible
differences when compared to those computed without, while they have a huge
saving in efficiency. This is particularly true for large line lists with many
millions of levels and billions of transitions. The differences appear to be
much smaller than the typical differences found between using different wing
cut-off methods. The latter is illustrated by Figure 4. We assess that cross-
sections computed with the use of super-lines are appropriate for low- to mid-
resolution applications, such as modeling JWST observations. As long as each
of the super-lines is adequately sampled then the overall opacity will largely
be conserved.

##  6 Our recommendation for choosing the Wing cut-off

The scope of our paper does not encompass resolving current challenges such as
line mixing, non-Lorentzian behavior, or inaccuracies in various Voigt
algorithms. Our primary objective is to establish a standardized approach for
opacity calculations within the exoplanetary and astrophysical communities,
specifically focusing on wing cutoffs. For the systems where the empirical
parameterizations of far-wing exist, they should be used, although one has to
be aware that there is still room for improvement. Further theoretical studies
and spectroscopic measurements are still required to create comprehensive
insights into the physics of the far wings across the pressure regime and
within the different mixtures of atmospheric gases that haven’t been studied
yet. However, given the different scenarios, we examined in Figs. 3–5, we
propose the following method, assuming a Voigt profile, to calculate the wing
cut-off for these pressure ranges:

$R_{\rm cut,Abs}=\left\\{\begin{matrix}25~{}\textrm{cm}^{-1}~{}\ \ \textrm{for
P}~{}\leqslant 200~{}\textrm{bar}\\\ 100~{}\textrm{cm}^{-1}~{}\textrm{for
P}>200~{}\textrm{bar}\end{matrix}\right.$ (4)

This method is chosen for the following reasons :

  1. 1.

For intermediate pressures and high wavenumber (> 5000 cm-1), the two common
methods in Eqs. 1 and 2 are in a very good agreement. However, for very low
pressures (< 10-1) and for some spectral ranges, they result in some
disagreement as shown in Figs. 3–5. At low pressures, the Voigt HWHM is
$<0.01$cm-1 (see Fig. 2) and the values calculated using 2 are physically
small, therefore using fixed values is beneficial.

  2. 2.

For very high pressure i.e., > 200 bars, the Voigt HWHM is more than 20 cm-1
and so to model the core of the line profile and a part of the wing, an
absolute value of 100 cm-1 is proposed. It is worthwhile to note that this
value might seem very small for P > 1000 bar, but the lack of sufficient
thermodynamics and spectroscopic studies in this high-pressure regime forced
us to conservatively choose this number of 100 cm-1 (see §4.1).

  3. 3.

A line wing of 25 cm-1 is typically used for calculations of terrestrial
radiative transfer (Mlawer et al., 2012). Hence, our proposed method is very
well aligned with the planetary community and could be utilized for their
modeling studies as well.

##  7 Instruction on how to control $R_{cut}$

The pre-generated ACS data in the current version of MAESTRO were computed
using HAPI and ExoCross codes. Hence, you can see a short description on how
to control the $R_{\rm cut}$ in these tools.

###  7.1 HAPI

The HITRAN Application Programming Interface (HAPI) (Kochanov et al., 2016) is
a set of Python libraries that allow users to download the HITRAN data and
carry out different calculations, including absorption cross-sections,
emission, transmission, etc. HAPI has a number of line shape functions, and
complex probability functions implemented. The users can select among these
functions or provide their own. One can also specify the wing cut-off.
Although the default value is 50 HWHM, for the applications to planetary
atmospheres under pressures encountered in the terrestrial atmosphere, it is
recommended to use 25 cm-1. Going outside of the terrestrial conditions, one
can implement user-defined functions on a case-by-case basis. We will
implement recommendations from this paper in future HAPI distributions. In any
case, the flexibility of the open-source code gives all power to the user to
implement different line shape functions.

###  7.2 ExoCross

ExoCross (Yurchenko et al., 2018) is a Fortran 2003 open-source program to
compute spectra of molecules developed and maintained by the ExoMol project
(Tennyson et al., 2020). ExoCross provides several line profiles as well as
algorithms to compute them. Apart from the main-stream algorithm by Humlíček
(1979), two fast Voigt methods are also provided in ExoCross: an approximated
vectorized Voigt and a binned Voigt profile with analytical integrals.
Different line profiles can be easily implemented in ExoCross.

The default wing cut-off value is 25 cm-1 which however can be changed by the
user. A python version of ExoCross, PyExoCross has also recently been
developed (Zhang et al., 2023) and treats the wing cut-offs similarly.

##  8 Conclusions

The main goal of this study is to provide a standard method for controlling
the spectral line wing cut-off. Inconsistencies, potential
inaccuracies/biases, and the lack of necessary theoretical studies or
laboratory measurements are reviewed in this work to highlight the
difficulties involved in implementing line shapes and wing cut-off relevant to
modeling extrasolar atmospheres.

The generated absorption cross-sections can be very sensitive to the choice of
wing cut-off, particularly in certain pressure and temperature ranges across
the wavelength space. With the incredible advancements in space exploration
over the last decade and a large number of detected exoplanets and brown
dwarfs, it is crucial to utilize accurate and standardized cross-sections
across different research groups. Our main objective in this study is to
promote consistency in atmospheric modeling results by recommending a
“standard” and “default” method for the wing cut-off. The recommendations and
discussions provided here aim to provide insights into the applications of the
MAESTRO database, but they are also valid and applicable across the fields of
astronomy and spectroscopy.

Given the spectral parameters, the simulation of a spectrum has a number of
other complications, most are associated with the line shape. Some molecules
exhibit line-coupling effects (or line-mixing effects), which can greatly
change the simulated spectra. There are theoretical models for each mechanism.
However, the main challenge to simulate a spectrum is the choice of a line-
shape model. Most of the line-shape data on HITRAN is for the Lorentz profile;
in applications, this is coupled with Doppler broadening to create the Voigt
profile. The suggested model is the Hartmann-Tran Profile (Tran et al., 2013,
2014; Tennyson et al., 2014), but little data are available, especially for
large-scale simulations. There are parameters available for a small number of
transitions for other more sophisticated line-shape models.

The lack of pressure broadening data for pressure and temperature ranges
relevant to exo-atmospheres is another issue. Most laboratory spectroscopic
measurements are limited to Terrestrial and Jovian atmospheric temperature
ranges ($\leq$350 K) and up to a few tens of bars. At high pressures, the
assumption of the Voigt profile (binary collisions between absorbers and
broadeners) is not valid, and more sophisticated line shapes are required to
account for non-binary collisions and the intensive interactions that exist
between these species.

Therefore, for now, we recommend the default or standard practice, in
particular with regards to the MAESTRO (Molecules and Atoms in Exoplanet
Science: Tools and Resources for Opacities) database, to be the use of a Voigt
profile with a line-wing cut-off of 25 cm-1 for pressures of 200 bar or less,
and 100 cm-1 for pressures above 200 bar. Our main reasons for this choice are
outlined in Section 6. We hope future theoretical and laboratory work will
facilitate more precise line profiles for a wide range of species, which can,
in turn, be easily implemented in codes to compute cross-sections and thus
included in future releases of MAESTRO. Having a current “standard” or
“default” recommendation will at least allow for a consistency and thus fair
comparison between different studies. Such a comparison is particularly
important for the open-access MAESTRO database, as it will allow for an easier
peer review process999github.com/maestro-opacities/submit-data for allowing
community contributions of opacity data.

## Acknowledgements

N.B. & E.G.N. both acknowledge support from the NASA Astrophysics Division.
E.G.N., N. B., & B. G. acknowledge support from the NASA XRP funding
NNH21ZDA001N-XRP. N.E.B, N.K.L, I.E.G., R.J.H. et al. MAESTRO Grant (NASA
Grant Number 80NSSC19K1036). K.L.C acknowledges funding from STFC under
project number ST/V000861/1. S.N.Y. and J.T. acknowledge supported by the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme through Advance Grant number 883830
(ExoMolHD). We also would like to thank the anonymous reviewers for their
insightful comments.

## Software

HAPI Kochanov et al. (2016)
ExoCross (Yurchenko et al., 2018) which is available from
https://github.com/ExoMol/ExoCross.

## Author Contributions

All authors have contributed equally to this study.

## Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

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$\alpha(x)({\mathrm{Path}}_{\sim}(X))=\tilde{\mu}_{P}(\mathrm{Path}(X)\cap\uparrow\\!{\varepsilon_{x}})=\tilde{\mu}_{P}(\uparrow\\!{\varepsilon_{x}})=\mu_{P}(\varepsilon_{x})=1$.
∎

######  Theorem V.6 (Theorem 29).

Two states are probabilistic delay bisimilar if and only if they are
$\mathcal{G}\circ\mathrm{Path}_{\sim}$-behaviourally equivalent.

###### Proof.

$\Rightarrow$ For this direction, it was left to show that $\nu_{f(x)}$ in (3)
is well-defined. We first recall a technical result from [21, Lemma 24]. For
any branching bisimulation $R$, any $R$-saturated set $M\subseteq X$, and any
saturated block $W\subseteq A_{\tau}^{*}$, the following equation holds

$P(x,W,M)=P(x^{\prime},W,M),\quad\text{for any $x,x^{\prime}\in X$ such that
$xRx^{\prime}$}.$ (6)

A subset $M\subseteq X$ is $R$-saturated if for all $x\in M$, the whole
equivalence class of $x$ is contained in $M$, i.e., $\forall_{x,x^{\prime}}\
(x\in M\land xRx^{\prime})\implies x^{\prime}\in M$. A subset $W\subseteq
A_{\tau}^{*}$ is saturated if it can be decomposed into a collection of blocks
$B_{i}$, for some index set $I$, i.e., $W=\bigcup_{i\in I}B_{i}$, where each
block $B_{i}=\tau^{*}a_{1}\tau^{*}\cdots\tau^{*}a_{n}\tau^{*}$, for a word
$a_{1}a_{2}\cdots a_{n}\in A_{\tau}^{*}$.

Let $U={\pi}^{-1}({{\mathrm{Path}}_{\sim}(f)}^{-1}V)$. First we claim that
$\mathrm{trace}(U)=\\{\mathrm{trace}(p)\mid{[fp]}_{\sim}\in V\\}$ is saturated
by establishing the equation:

$\mathrm{trace}(U)=\bigcup_{{[fp]}_{\sim}\in
V}g(\mathrm{trace}({p}^{{\dagger}})),$

where the operation $A_{\tau}^{*}\rTo^{g}\mathcal{P}(A_{\tau}^{*})$ is defined
inductively as follows: $g(\varepsilon)=\tau^{*}$ and $g(\sigma
a)=g(\sigma)\tau^{*}a\tau^{*}$. The direction
$\mathrm{trace}(U)\subseteq\bigcup_{{[fp]}_{\sim}\in
U}g(\mathrm{trace}({p}^{{\dagger}}))$ because $\sigma\in g(\sigma)$, for any
$\sigma\in A_{\tau}^{*}$. For the other direction, let $\sigma\in
g(\mathrm{trace}({p}^{{\dagger}}))$, for some ${[fp]}_{\sim}\in V$. Let
$\mathrm{trace}({p}^{{\dagger}})=a_{1}a_{2}\cdots a_{n}$ (for some $n\geq 1$).
Then, we find
$\sigma=\tau^{m_{1}}a_{1}\tau^{m_{2}}\cdots\tau^{m_{n}}a_{n}\tau^{m_{n+1}}$.

Define a path $q$ whose domain is $\sigma$ as follows:

$q(\sigma^{\prime})=\begin{cases}{p}^{{\dagger}}(\varepsilon)&\text{if}\
\sigma^{\prime}\preceq\tau^{m_{i}},\\\ {p}^{{\dagger}}(a_{1}\cdots
a_{i})&\text{if}\
\tau^{m_{1}}a_{1}\cdots\tau^{m_{i}}a_{i}\prec\sigma^{\prime}\preceq\tau^{m_{1}}a_{1}\cdots\tau^{m_{i}}a_{i}\tau^{m_{i}+1}\land
i\leq n,\\\ {p}^{{\dagger}}(a_{1}\cdots a_{i})&\text{if}\
\sigma^{\prime}=\tau^{m_{1}}a_{1}\cdots\tau^{m_{i}}a_{i}\land i\leq
n.\end{cases}$

Clearly, ${p}^{{\dagger}}={q}^{{\dagger}}$, i.e., $p\sim q$. Thus,
${[fp]}_{\sim}={[fq]}_{\sim}\in V$; hence, $q\in\mathrm{trace}(U)$.

Next, we verify that $\mathrm{last}(U)$ is an $R$-saturated set. To see this
let, $x\in\mathrm{last}(U)$ and $xRx^{\prime}$. Then, there is a path $p\in U$
such that $\mathrm{last}(p)=x$ and ${[fp]}_{\sim}\in V$. Suppose
$\mathrm{dom}(p)=\varepsilon$. Then, we find that
$f\circ\varepsilon_{x}=f\circ\varepsilon_{x^{\prime}}$ (because
$xRx^{\prime}$). I.e., $\varepsilon_{x^{\prime}}\in U$. Now suppose $p$ has a
non-empty domain, i.e., there is some path $p^{\prime}\prec p$ such that
$\max\mathrm{dom}(p^{\prime})=\sigma$, $\max\mathrm{dom}(p)=\sigma a$, for
some $\sigma\in A_{\tau}^{*},a\in A_{\tau}$. Then, we define a path $q$ whose
domain is identical to $p$ as follows:
$q(\sigma^{\prime})=p^{\prime}(\sigma^{\prime})$ if
$\sigma^{\prime}\in\mathrm{dom}(p^{\prime})$ and $q(\sigma a)=x^{\prime}$.
Again, we find that $f\circ p=f\circ q$ because $xRx^{\prime}$. Thus, $q\in
U$, i.e., $x^{\prime}\in U$.

Now we can prove that $\nu_{f(x)}$ in (3) is well-defined. Suppose
$xRx^{\prime}$. Then it suffices to show that
$\tilde{\mu}(U\cap\varepsilon_{x})=\tilde{\mu}(U\cap\varepsilon_{x}^{\prime})$,
where $U={\pi}^{-1}({{\mathrm{Path}}_{\sim}(f)}^{-1}V)$ and
$V\in\mathcal{O}_{{\mathrm{Path}}_{\sim}(X/R)}$. This follows directly from
the fact that
$\tilde{\mu}(U\cap\varepsilon_{x})=\mu((U\cap\varepsilon_{x})^{\star})=P(x,\mathrm{trace}(U),\mathrm{last}(U))$
and Equation 6.

($\Leftarrow$) Let $(X,A_{\tau},P)$ be a fully probabilistic system and
$(X,\alpha)$ be the corresponding
$\mathcal{G}\circ\mathrm{Path}_{\sim}$-coalgebra. Moreover, let $(Y,\beta)$ be
a $\mathcal{G}\circ\mathrm{Path}_{\sim}$-coalgebra and $X\rTo^{f}Y$ be a
$\mathcal{G}\circ\mathrm{Path}_{\sim}$ coalgebra homomorphism. We will show
that the equivalence relation $xRx^{\prime}\iff f(x)=f(x^{\prime})$ is a
probabilistic branching bisimulation in two stages. First, we claim that

$\beta(f(x))\Big{(}\bigcup_{q\in f(x)\ext@arrow
0395\arrowfill@\relbar\relbar\twoheadrightarrow{}{\hat{a}}{}\\{y\\}}\uparrow\\!{{[q]}_{\sim}}\Big{)}=\alpha(x)\Big{(}\bigcup_{p\in
x\ext@arrow
0395\arrowfill@\relbar\relbar\twoheadrightarrow{}{\hat{a}}{}{f}^{-1}\\{y\\}}\uparrow\\!{{[p]}_{\sim}}\Big{)}\qquad\text{for
any $x\in X,y\in Y$ }.$ (7)

Here, we abbreviate $f(x)\ext@arrow
0395\arrowfill@\relbar\relbar\twoheadrightarrow{}{\tau^{*}\hat{a}}{}\\{y\\}$
by $f(x)\ext@arrow
0395\arrowfill@\relbar\relbar\twoheadrightarrow{}{\hat{a}}{}\\{y\\}$. Notice
that the sets $\bigcup_{q\in f(x)\ext@arrow
0395\arrowfill@\relbar\relbar\twoheadrightarrow{}{\hat{a}}{}\\{y\\}}\uparrow\\!{{[q]}_{\sim}}$
and $\bigcup_{p\in x\ext@arrow
0395\arrowfill@\relbar\relbar\twoheadrightarrow{}{\hat{a}}{}{f}^{-1}\\{y\\}}\uparrow\\!{{[p]}_{\sim}}$
are open sets (hence Borel sets). Second, we claim that

$P(x,\tau^{*}\hat{a},{f}^{-1}\\{y\\})=\alpha(x)\Big{(}\bigcup_{p\in
x\ext@arrow
0395\arrowfill@\relbar\relbar\twoheadrightarrow{}{\hat{a}}{}{f}^{-1}\\{y\\}}\uparrow\\!{{[p]}_{\sim}}\Big{)}\qquad\text{for
any $x\in X,y\in Y$}.$ (8)

Using the above two properties we can now prove the transfer property of
bisimulation as follows:

$\displaystyle P(x,\tau^{*}\hat{a},{f}^{-1}\\{y\\})=$ $\displaystyle\
\alpha(x)\Big{(}\bigcup_{p\in x\ext@arrow
0395\arrowfill@\relbar\relbar\twoheadrightarrow{}{\hat{a}}{}{f}^{-1}\\{y\\}}\uparrow\\!{{[p]}_{\sim}}\Big{)}$
$\displaystyle(\text{Using Equation\leavevmode\nobreak\ \eqref{eq:2}})$
$\displaystyle=$ $\displaystyle\ \beta(f(x))\Big{(}\bigcup_{q\in
f(x)\ext@arrow
0395\arrowfill@\relbar\relbar\twoheadrightarrow{}{\hat{a}}{}\\{y\\}}\uparrow\\!{{[q]}_{\sim}}\Big{)}$
$\displaystyle(\text{Using Equation\leavevmode\nobreak\ \eqref{eq:1}})$
$\displaystyle=$ $\displaystyle\ \beta(f(x^{\prime}))\Big{(}\bigcup_{q\in
f(x^{\prime})\ext@arrow
0395\arrowfill@\relbar\relbar\twoheadrightarrow{}{\hat{a}}{}\\{y\\}}\uparrow\\!{{[q]}_{\sim}}\Big{)}$
$\displaystyle(\because f(x)=f(x^{\prime}))$ $\displaystyle=$ $\displaystyle\
\alpha(x^{\prime})\Big{(}\bigcup_{p\in x^{\prime}\ext@arrow
0395\arrowfill@\relbar\relbar\twoheadrightarrow{}{\hat{a}}{}{f}^{-1}\\{y\\}}\uparrow\\!{{[p]}_{\sim}}\Big{)}$
$\displaystyle(\text{Using Equation\leavevmode\nobreak\ \eqref{eq:1}})$
$\displaystyle=$ $\displaystyle\
P(x^{\prime},\tau^{*}\hat{a},{f}^{-1}\\{y\\})$ $\displaystyle(\text{Using
Equation\leavevmode\nobreak\ \eqref{eq:2}}).$

Next, we show the satisfaction of Property (7). For this we derive

$\displaystyle\beta(f(x))\Big{(}\bigcup_{q\in f(x)\ext@arrow
0395\arrowfill@\relbar\relbar\twoheadrightarrow{}{\hat{a}}{}\\{y\\}}\uparrow\\!{{[q]}_{\sim}}\Big{)}$
$\displaystyle=\tilde{\mu}\bigg{(}{\mathrm{Path}(f)}^{-1}\Big{(}{\pi_{Y}}^{-1}(\bigcup_{q\in
f(x)\ext@arrow
0395\arrowfill@\relbar\relbar\twoheadrightarrow{}{\hat{a}}{}\\{y\\}}\uparrow\\!{{[q]}_{\sim}})\Big{)}\cap\uparrow\\!{\varepsilon_{x}}\bigg{)}$
$\displaystyle=\tilde{\mu}\bigg{(}{\mathrm{Path}(f)}^{-1}\Big{(}\bigcup_{q\in
f(x)\ext@arrow
0395\arrowfill@\relbar\relbar\twoheadrightarrow{}{\hat{a}}{}\\{y\\}}{\pi_{Y}}^{-1}(\uparrow\\!{{[q]}_{\sim}})\Big{)}\cap\uparrow\\!{\varepsilon_{x}}\bigg{)}$
(Lemma 10)
$\displaystyle=\tilde{\mu}\bigg{(}{\mathrm{Path}(f)}^{-1}\Big{(}\bigcup_{q^{\prime}\in\\{q^{\prime}\mid\exists_{q}\
q^{\prime}\sim q\land q\in f(x)\ext@arrow
0395\arrowfill@\relbar\relbar\twoheadrightarrow{}{\hat{a}}{}\\{y\\}\\}}\uparrow\\!{q^{\prime}}\Big{)}\cap\uparrow\\!{\varepsilon_{x}}\bigg{)}$
$\displaystyle=\tilde{\mu}\bigg{(}\Big{(}\bigcup_{q^{\prime}\in\\{q^{\prime}\mid\exists_{q}\
q^{\prime}\sim q\land q\in f(x)\ext@arrow
0395\arrowfill@\relbar\relbar\twoheadrightarrow{}{\hat{a}}{}\\{y\\}\\}}{\mathrm{Path}(f)}^{-1}(\uparrow\\!{q^{\prime}})\Big{)}\cap\uparrow\\!{\varepsilon_{x}}\bigg{)}$
$\displaystyle=\tilde{\mu}\bigg{(}\Big{(}\bigcup_{p\in f(x)\ext@arrow
0395\arrowfill@\relbar\relbar\twoheadrightarrow{}{\hat{a}}{}_{\\!\\!X}\\{y\\}}\uparrow\\!{p}\Big{)}\cap\uparrow\\!{\varepsilon_{x}}\bigg{)}$
(where, $f(x)\ext@arrow
0395\arrowfill@\relbar\relbar\twoheadrightarrow{}{\hat{a}}{}_{\\!\\!X}\\{y\\}=\\{p\mid\exists_{q}\
fp\sim q\land q\in f(x)\ext@arrow
0395\arrowfill@\relbar\relbar\twoheadrightarrow{}{\hat{a}}{}\\{y\\}\\}$)
$\displaystyle=\tilde{\mu}\bigg{(}\bigcup_{p\in f(x)\ext@arrow
0395\arrowfill@\relbar\relbar\twoheadrightarrow{}{\hat{a}}{}_{\\!\\!X}\\{y\\}}\Big{(}\uparrow\\!{p}\cap\uparrow\\!{\varepsilon_{x}}\Big{)}\bigg{)}$
$\displaystyle=\tilde{\mu}\bigg{(}\bigcup_{p\in f(x)\ext@arrow
0395\arrowfill@\relbar\relbar\twoheadrightarrow{}{\hat{a}}{}_{\\!\\!X}\\{y\\}}\uparrow\\!{p}\bigg{)}$
$\displaystyle=\tilde{\mu}(\bigcup_{p\in x\ext@arrow
0395\arrowfill@\relbar\relbar\twoheadrightarrow{}{\hat{a}}{}{f}^{-1}\\{y\\}}\uparrow\\!{p}).$

To show that why the last two steps in the above derivation are equivalent, we
prove that $f(x)\ext@arrow
0395\arrowfill@\relbar\relbar\twoheadrightarrow{}{\hat{a}}{}_{\\!\\!X}\\{y\\}=x\ext@arrow
0395\arrowfill@\relbar\relbar\twoheadrightarrow{}{\hat{a}}{}{f}^{-1}\\{y\\}$.
Let $p\in f(x)\ext@arrow
0395\arrowfill@\relbar\relbar\twoheadrightarrow{}{\hat{a}}{}_{\\!\\!X}\\{y\\}$.
Then we find $\varepsilon_{x}\preceq p$, $fp\sim q$, and $q\in f(x)\ext@arrow
0395\arrowfill@\relbar\relbar\twoheadrightarrow{}{\hat{a}}{}\\{y\\}$, for some
$q\in\mathrm{Path}(Y)$. From Lemma 6(1) and 6(3) we find that $p\in
x\ext@arrow
0395\arrowfill@\relbar\relbar\twoheadrightarrow{}{\hat{a}}{}{f}^{-1}\\{y\\}$.
For the other direction, take $q=fp$ for any $p\in x\ext@arrow
0395\arrowfill@\relbar\relbar\twoheadrightarrow{}{\hat{a}}{}{f}^{-1}\\{y\\}$.

Lastly, we show the satisfaction of (8). Let $x\in X,y\in Y$. Then we derive

$\displaystyle P(x,\tau^{*}\hat{a},{f}^{-1}\\{y\\})=$ $\displaystyle\
\mu_{P}((x\ext@arrow
0395\arrowfill@\relbar\relbar\twoheadrightarrow{}{\tau^{*}\hat{a}}{}{f}^{-1}\\{y\\})^{\star})\quad(\text{Proposition\leavevmode\nobreak\
\ref{prop:separated}.\ref{prop:separated:future} and
$\uparrow\\!{U}=\bigcup_{p\in U}\uparrow\\!{p}$})$ $\displaystyle=$
$\displaystyle\ \mu_{P}\bigg{(}\Big{(}\bigcup_{p\in x\ext@arrow
0395\arrowfill@\relbar\relbar\twoheadrightarrow{}{\hat{a}}{}{f}^{-1}\\{y\\}}\uparrow\\!{p}\Big{)}^{\star}\bigg{)}$
$\displaystyle=$ $\displaystyle\ \tilde{\mu}_{P}\bigg{(}\bigcup_{p\in
x\ext@arrow
0395\arrowfill@\relbar\relbar\twoheadrightarrow{}{\hat{a}}{}{f}^{-1}\\{y\\}}\uparrow\\!{p}\bigg{)}$
$\displaystyle=$ $\displaystyle\ \alpha(x)(\bigcup_{p\in x\ext@arrow
0395\arrowfill@\relbar\relbar\twoheadrightarrow{}{\hat{a}}{}{f}^{-1}\\{y\\}}\uparrow\\!{{[p]}_{\sim}}).\qed$