Retrieval Study of Brown Dwarfs across the L-T Sequence

We solve the radiative transfer equation for a one-dimensional, plane-parallel atmosphere using the method of short characteristics (Olson & Kunasz 1987), as previously implemented within the Helios-r2 code (Kitzmann et al. 2020). The method of short characteristics allows for a stable and efficient solution of the transfer equation in the absence of scattering. As stated in Kitzmann et al. (2020), we use the first-order version of the short characteristic method presented in Olson & Kunasz (1987). Due to the neglect of scattering, we only need to calculate the spectral intensity in the upward direction. It is calculated for a total of two different, discrete polar angles, which is equivalent to a four-stream radiative transfer method. The result is then numerically integrated over these angles with a Gaussian quadrature to yield the outgoing flux ${F}_{\nu }^{+}$.

Due to the low resolution of the SpeX instrument, we calculate the theoretical high-resolution spectra with a constant step size of 1 cm⁻¹ in wavenumber space (Line et al. 2015; Kitzmann et al. 2020). The resulting spectrum is convolved with an appropriate instrument line profile before it is binned down to the resolution of the measured spectra.

An additional scaling factor f is used in the radius–distance relation to scale the outgoing flux ${F}_{\nu }^{+}$ of the brown dwarf to the one measured by the observer (F_ν ):

$$\begin{eqnarray}&&{F}_{\nu }={F}_{\nu }^{+}f{\left(\displaystyle \frac{{R}_{p}}{d}\right)}^{2},\end{eqnarray} \tag{ 1 }$$

where d is the distance between the observer and the brown dwarf and R_p the (prior) radius. For R_p we choose a fixed value of R_p = 1R_J throughout this study. In practice, f serves as a "catchall" scaling factor that absorbs uncertainties in the radius and distance, as well as inaccuracies in the atmospheric models and flux calibration of the measured spectra. Since all of these quantities are essentially degenerate, we choose to combine them all in a single factor.

Our distances listed in Table 1 are not all based on measured parallaxes, but partly on estimated values based on spectroscopic data, such as our T8 (2MASS J04151954–0935066). For our T8 brown dwarf, we have used the distance d based on spectroscopy (Lodieu et al. 2012). Other studies have published distances based on parallax measurements (e.g., Faherty et al. 2012; d = 5.736 ± 0.362 pc). These small differences in d are absorbed into the retrieved value of f and thus do not affect the final outcome.

If f is assumed to only contain the deviations from the assumed prior radius of 1R_J, it can be converted into the actual radius of the brown dwarf in units of Jupiter radii via

$$\begin{eqnarray}&&R=\sqrt{f}.\end{eqnarray} \tag{ 2 }$$

In practice, however, f usually also involves other uncertainties and missing or inaccurate model physics, such that the derived radius R should not always be considered as the true radius of the brown dwarf (Kitzmann et al. 2020).

For this study, we divide the atmosphere into a total of 70 levels (i.e., 69 layers). The levels are distributed equidistantly in log pressure-space. The specification of a temperature–pressure profile in atmospheric retrievals has a long history. Approaches include the specification of a temperature value in each model atmospheric layer (Irwin et al. 2008); the use of a nine-parameter, ad hoc fitting function (Madhusudhan & Seager 2009); and self-consistent but simplified profiles (Guillot 2010; Parmentier & Guillot 2014; Heng et al. 2012, 2014). In particular, the self-consistent temperature–pressure profiles invoke strong assumptions that can produce an artificially isothermal atmosphere at low pressures (Heng et al. 2014).

As discussed in Kitzmann et al. (2020), Helios-r2 implements a description of the temperature profile based on a finite-element approach. This ensures a continuous temperature–pressure profile, described by a relatively small number of free parameters. Unless stated otherwise, we use six first-order elements for the temperature profile, which results in a total of seven free parameters. Due to inherent, continuous nature of this finite-element approach, we can evaluate the temperature for any given pressure in the forward model.

Since this study is focused on brown dwarfs for which temperature inversions are not anticipated, we force the profiles to be monotonically decreasing with pressure (Kitzmann et al. 2020). As shown in Bourrier et al. (2020), Helios-r2 is also able to retrieve temperature inversions if the aforementioned assumption of a monotonically decreasing profile is not employed.

In its original version Helios-r2 has has the option of adding a gray-cloud layer to the atmosphere. This approximation usually assumes that the cloud particles are large compared to the wavelength range of the measured spectrum.

For this study we extend Helios-r2 to also optionally use non-gray-cloud layers. Here we only aim at parameterizing the particles' extinction coefficients. No attempt is made to actually model the formation of these clouds.

We follow the work of Kitzmann & Heng (2018) to describe the extinction efficiency of the cloud particles as a function of wavelength. Their approach assumes cloud particles with single radii and then approximates the extinction efficiencies Q_ext resulting from Mie theory calculations with a simple, analytic equation:

$$\begin{eqnarray}&&{Q}_{\mathrm{ext}}(\lambda )=\displaystyle \frac{{Q}_{1}}{{Q}_{0}{x}_{\lambda }^{-{a}_{0}}+{x}_{\lambda }^{0.2}},\end{eqnarray} \tag{ 3 }$$

where Q₁ is a normalization constant, Q₀ determines the x-value at which Q_ext is peaking, x_λ = 2π a/λ is the dimensionless size parameter, a is the particle radius, and a₀ is the power-law index in the small particle limit, where Mie theory converges to the limit of Rayleigh scattering.

The equation is not supposed to describe the exact behavior of the extinction efficiencies, but should rather serve as a first-order approximation. While the full Mie absorption and scattering efficiencies of single particles usually exhibit low- and high-frequency oscillations, the analytic fit provides a smooth description. It is worth noting that Q₀ can be a proxy for the cloud particle composition (see Table 2 of Kitzmann & Heng 2018).

The optical depth τ of the cloud layer is then given by

$$\begin{eqnarray}&&\tau (\lambda )={\sigma }_{\mathrm{ext}}(\lambda ){n}_{c}{\rm{\Delta }}z={Q}_{\mathrm{ext}}(\lambda )\pi {a}^{2}{n}_{c}{\rm{\Delta }}z\,,\end{eqnarray} \tag{ 4 }$$

with the extinction cross section σ_ext=Q_ext π a², the cloud particle number density n_c , and the vertical extent of the cloud layer Δz.

Since it is difficult to estimate a good prior for n_c , we replace it with an optical depth at a reference wavelength of λ_ref = 1 μm. The optical depth is then given as

$$\begin{eqnarray}&&\tau (\lambda )={\tau }_{\mathrm{ref}}\displaystyle \frac{{Q}_{\mathrm{ext}}(\lambda )}{{Q}_{\mathrm{ext}}({\lambda }_{\mathrm{ref}})}={\tau }_{\mathrm{ref}}\displaystyle \frac{{Q}_{0}{x}_{{\lambda }_{\mathrm{ref}}}^{-{a}_{0}}+{x}_{{\lambda }_{\mathrm{ref}}}^{0.2}}{{Q}_{0}{x}_{\lambda }^{-{a}_{0}}+{x}_{\lambda }^{0.2}}.\end{eqnarray} \tag{ 5 }$$

The position of the cloud layer in the atmosphere and its vertical extent Δz are described by two more free parameters: the cloud's top pressure p _t and its bottom pressure p _b . Instead of using p _b directly as a free retrieval parameter, we instead use a factor b_c ≥ 1, such that p _b = b_c p _t . The factor b_c is limited to a maximum value of 10, such that a cloud layer can at most span over one order of magnitude in pressure.

For the non-gray-cloud description we thus have six free parameters in total, whereas the gray cloud requires three. A summary of all cloud retrieval parameters and their prior distributions is given in Table 3.

We incorporate all of the components described in the previous subsections into a Bayesian inference framework (see Figure 3). Our choice of method to explore the multidimensional parameter space is nested sampling (Skilling 2006) in its MultiNest implementation (Feroz & Hobson 2008; Feroz et al. 2009), which was previously implemented in Helios-r2 by Kitzmann et al. (2020). Nested sampling was introduced to the exoplanet atmospheric retrieval literature by Benneke & Seager (2013). We assume that the prior distributions of our free parameters are uniform, log-uniform, or Gaussian (see Section 3.5 and Table 3).

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A key ingredient of nested sampling is the specification of the likelihood function ${ \mathcal L }$, which is the mathematical relationship between the model D_j,m, the observational data D_j , and uncertainties s_j associated with the data for all measured data points j. It is common practice to assume a Gaussian likelihood function (Kitzmann et al. 2020)

$$\begin{eqnarray}&&\mathrm{ln}{ \mathcal L }=-\displaystyle \frac{1}{2}\sum _{j=1}^{J}\displaystyle \frac{{\left({D}_{j}-{D}_{j,{\rm{m}}}\right)}^{2}}{{s}_{j}^{2}}-\displaystyle \frac{1}{2}\mathrm{ln}(2\pi {s}_{j}^{2}).\end{eqnarray} \tag{ 6 }$$

Implicitly, this assumes not only that the data uncertainties s_j are Gaussian distributed but also that they are uncorrelated for all j.

Following Line et al. (2015) and Kitzmann et al. (2020), we account for the possibility that stated uncertainties of the measured fluxes have been underestimated by implementing the procedure of Hogg et al. (2010). Let the standard deviation of the measured fluxes at each jth data point be σ_j . The effective standard deviation of the jth data point is then given by

$$\begin{eqnarray}&&{s}_{j}^{2}={\sigma }_{j}^{2}+{e}^{2\,\mathrm{ln}\delta },\end{eqnarray} \tag{ 7 }$$

where the parameter $\mathrm{ln}\delta$ is part of the fit. Specifying $\mathrm{ln}\delta$ as a fitting parameter ensures that δ > 0 (Foreman-Mackey et al. 2013). For example, $\mathrm{ln}\delta =-4$ corresponds to δ ≈ 2%.

The key advantage of nested sampling is that it allows for the calculation of the marginalized likelihood or Bayesian evidence, which may be used to implement a formal form of Occam's razor known as Bayesian model comparison (Trotta 2008).

A pair of models of differing complexity (characterized by different numbers of parameters or prior distributions) are compared by taking the ratio of their Bayesian evidences, which is known as the Bayes factor (Trotta 2008). There is an established correspondence between the Bayes factor and the number of standard deviations that one of the models is disfavored by the data, which we reproduce in Table 2. It is worth pointing out, though, that Bayesian model comparison may fail to exclude unphysical scenarios (e.g., Fisher & Heng 2019).

Table 2. Correspondence of the Bayes Factor B_ij to the Number of Standard Deviations σ

Bij	$\mathrm{ln}{B}_{{ij}}$	σ	Category
2.5	0.9	2.0
2.9	1.0	2.1	"weak" at best
8.0	2.1	2.6
12	2.5	2.7	"moderate" at best
21	3.0	3.0
53	4.0	3.3
150	5.0	3.6	"strong" at best
43,000	11	5.0

Note. Reproduced from Trotta (2008).

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The Bayesian framework requires us to define prior distributions for all free parameters of the forward model. All parameters and their priors are shown in Table 3.

Table 3. Summary of Retrieval Parameters and Prior Distributions for the Free-chemistry Approach Used in the Cloud-free, Gray-cloud, and Non-gray-cloud Models

Parameter		Prior
	Type		Value
$\mathrm{log}g$	uniform		3.5 to 6.0 cm s−2
d	Gaussian		measured i
f	uniform		0.1 to 5.0
T1	uniform		1000 to 5000 K
bi	uniform		0.1 to 0.95
$\mathrm{ln}\delta$	uniform		−10 to 1.0
xi	log-uniform		10−12 to 0.1
Gray Clouds
p t	log-uniform		10−2 to 50 bar
b c	log-uniform		1 to 10
τ	log-uniform		10−5 to 20
Nongray Clouds
p t	log-uniform		10−2 to 50 bar
b c	log-uniform		1 to 10
τref	log-uniform		10−5 to 20
Q0	log-uniform		1 to 100
a0	uniform		3 to 7
a	log-uniform		0.1 to 50 μm

Note.

ⁱ Note that all the measured distances d can be found in Table 1.

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For the general description of the brown dwarf atmosphere we require the surface gravity ${\rm{log}}\,g$, the distance d, and the calibration factor f. ⁷ We use the measured distances and the corresponding errors with a Gaussian prior (see Table 1). This procedure propagates the error in the measured distances through all other retrieval parameters. The temperature profile is described by seven free parameters in total (see Section 3.2).

For the abundances of the chemical species we make the usual assumptions that they are isoprofiles throughout the atmosphere. Each considered chemical species, therefore, requires one free parameter for its mixing ratio x_i .

Specifically, we retrieve mixing ratios of the following species: H₂O, CH₄, NH₃, CO₂, CO, H₂S, CrH, FeH, CaH, TiH, and K. The mixing ratio of Na is determined from the one of potassium by using the solar element abundance ratio of K and Na (see also Kitzmann et al. 2020). Since the SpeX spectra are only sensitive to the far line wings of the strong alkali resonance lines, the abundances of K and Na are essentially degenerate. It is, therefore, only possible to directly constrain one of them (Line et al. 2015). The abundances of H₂ and He are derived from the remaining background atmosphere, assuming a solar H/He element abundance ratio.

Additionally, for the cloudy models we require the cloud parameters as discussed in Section 3.3. For the nongray clouds we use six free parameters, while the gray cloud needs three parameters in total.

Besides the free retrieval parameters that are used directly within the nested sampling, Helios-r2 also provides posterior distributions for a set of derived quantities. One is the effective temperature of the brown dwarf. This quantity is obtained by integrating the high-resolution spectra of all posterior samples over wavelengths and then converting the resulting total flux to an effective temperature via the Stefan–Boltzmann law.

Two other derived quantities are the C/O ratio and the overall metallicity [M/H]. The C/O ratios are calculated by counting the amount of carbon and oxygen atoms using the retrieved mixing ratios of all carbon and oxygen carriers (e.g., Equation (19) of Line et al. 2013):

$$\begin{eqnarray}&&{\rm{C}}/{\rm{O}}=\displaystyle \frac{{\mathrm{CH}}_{4}+\mathrm{CO}+{\mathrm{CO}}_{2}}{\mathrm{CO}+2{\mathrm{CO}}_{2}+{{\rm{H}}}_{2}{\rm{O}}}.\end{eqnarray} \tag{ 8 }$$

The metallicity [M/H], on the other hand, is approximated by summing up the constant mixing ratios for each species weighted by the number of metal atoms and divided by the abundance of hydrogen. The result is then compared to the sum of solar metals relative to hydrogen.

Major absorbers (H₂O, CH₄, NH₃, CO₂, CO, H₂S, CrH, FeH, CaH, and TiH, as well as the alkali metals Na and K) are considered in this study to cover the wavelength range of the SpeX instrument, which is from 0.85 to about 2.45 μm.

Most opacities are calculated by the open-source HELIOS-K opacity calculator (Grimm & Heng 2015; Grimm et al. 2021). The line list data for these molecules are taken from the ExoMol database (Barber et al. 2006; Yurchenko et al. 2011; Yurchenko & Tennyson 2014; Azzam et al. 2016) and the HITEMP database (Rothman et al. 2010). The collision-induced absorption coefficients for H₂H₂ and H₂He are based on Abel et al. (2011) and Abel et al. (2012), respectively. We refer the reader to Tennyson & Yurchenko (2017) for a review of the spectroscopic databases.

For the alkali metals K and Na, we use the descriptions of their resonance line wings published by Allard et al. (2016) and Allard et al. (2019), respectively. The computations of the Na and K opacities are described in Kitzmann et al. (2020).

Before we discuss the outcome of our retrieval calculations, we first introduce the following terminology and the reasoning behind it:

• 1.  
  Oreshenko et al. (2020) previously demonstrated that disregarding data in each spectrum blueward of 1.2 μm circumvents unresolved issues with the shapes of the alkali metal resonance line wings but retains enough information in the spectrum to constrain the surface gravity spectroscopically. Following this approach, spectra with and without this cut are referred to as "restricted" and "full," respectively.
• 2.  
  If the full set of chemical species is used in the retrieval (see Section 3.6), the label "all" is employed. In a follow-up retrieval, species that only have upper limits on their abundances are removed; we refer to this as the "reduced" set of species. We explicitly check that inferred quantities from the "all" versus "reduced" retrievals are consistent with each other.
• 3.  
  We consider both cloud-free and cloudy atmospheres. For the cloudy cases, we either use gray or nongray clouds as described in Section 3.3.

In total, there are 12 permutations of models for each spectrum. However, we find that the Bayesian evidence consistently favors models with a reduced set of chemical species. This is to be expected since the Bayesian evidence effectively penalizes models with more free parameters over those with fewer if both models provide an adequate fit to the measured spectrum. Therefore, we have only listed six models for each object in Tables 5 and 6.

Table 6. Summary of Retrieval Outcomes for the Standard T Dwarfs

Model	Parameter	T0	T1	T2	T3	T4	T5	T6	T7	T8
FC	log g	${4.36}_{-0.32}^{+0.89}$	${4.46}_{-0.07}^{+0.07}$	${3.72}_{-0.08}^{+0.08}$	${5.36}_{-0.12}^{+0.12}$	${3.72}_{-0.10}^{+0.10}$	${4.83}_{-0.10}^{+0.11}$	${4.61}_{-0.25}^{+0.25}$	${4.92}_{-0.13}^{+0.16}$	${3.67}_{-0.10}^{+0.12}$
FG	log g	${4.27}_{-0.24}^{+0.32}$	${4.48}_{-0.08}^{+0.09}$	${3.93}_{-0.24}^{+0.22}$	${5.38}_{-0.13}^{+0.14}$	${3.75}_{-0.11}^{+0.18}$	${4.83}_{-0.10}^{+0.11}$	${4.60}_{-0.24}^{+0.24}$	${4.91}_{-0.12}^{+0.15}$	${3.67}_{-0.10}^{+0.12}$
FN	log g	${4.26}_{-0.25}^{+0.45}$	${4.47}_{-0.07}^{+0.07}$	${3.76}_{-0.11}^{+0.24}$	${5.38}_{-0.12}^{+0.13}$	${3.74}_{-0.09}^{+0.09}$	${4.82}_{-0.09}^{+0.11}$	${4.60}_{-0.23}^{+0.24}$	${4.59}_{-0.15}^{+0.17}$	${3.67}_{-0.10}^{+0.11}$
RC	log g	${4.06}_{-0.19}^{+0.25}$	${4.50}_{-0.10}^{+0.15}$	${3.71}_{-0.11}^{+0.12}$	${5.37}_{-0.11}^{+0.11}$	${4.36}_{-0.23}^{+0.26}$	${4.73}_{-0.13}^{+0.14}$	${4.53}_{-0.52}^{+0.32}$	${3.91}_{-0.17}^{+0.20}$	${3.66}_{-0.10}^{+0.16}$
RG	log g	${4.05}_{-0.18}^{+0.20}$	${4.45}_{-0.08}^{+0.09}$	${3.72}_{-0.11}^{+0.12}$	${5.41}_{-0.13}^{+0.21}$	${4.37}_{-0.22}^{+0.24}$	${4.73}_{-0.13}^{+0.13}$	${4.53}_{-0.46}^{+0.30}$	${3.82}_{-0.14}^{+0.16}$	${3.67}_{-0.10}^{+0.15}$
RN	log g	${4.04}_{-0.16}^{+0.18}$	${4.48}_{-0.08}^{+0.09}$	${3.71}_{-0.10}^{+0.11}$	${5.35}_{-0.10}^{+0.08}$	${4.37}_{-0.20}^{+0.23}$	${4.73}_{-0.12}^{+0.12}$	${4.56}_{-0.42}^{+0.27}$	${3.91}_{-0.16}^{+0.18}$	${4.52}_{-0.30}^{+0.32}$
FC	R	${0.68}_{-0.10}^{+0.10}$	${0.48}_{-0.03}^{+0.03}$	${0.94}_{-0.04}^{+0.05}$	${0.66}_{-0.06}^{+0.06}$	${0.79}_{-0.10}^{+0.10}$	${0.71}_{-0.04}^{+0.04}$	${0.85}_{-0.10}^{+0.11}$	${0.69}_{-0.04}^{+0.04}$	${0.63}_{-0.12}^{+0.13}$
FG	R	${0.69}_{-0.10}^{+0.09}$	${0.48}_{-0.03}^{+0.03}$	${0.93}_{-0.04}^{+0.04}$	${0.66}_{-0.06}^{+0.06}$	${0.79}_{-0.10}^{+0.09}$	${0.71}_{-0.04}^{+0.04}$	${0.86}_{-0.09}^{+0.10}$	${0.69}_{-0.04}^{+0.04}$	${0.63}_{-0.12}^{+0.12}$
FN	R	${0.68}_{-0.09}^{+0.09}$	${0.48}_{-0.03}^{+0.03}$	${0.93}_{-0.04}^{+0.04}$	${0.66}_{-0.05}^{+0.06}$	${0.79}_{-0.09}^{+0.09}$	${0.71}_{-0.04}^{+0.04}$	${0.85}_{-0.09}^{+0.09}$	${0.68}_{-0.03}^{+0.03}$	${0.63}_{-0.12}^{+0.12}$
RC	R	${0.72}_{-0.11}^{+0.10}$	${0.49}_{-0.04}^{+0.04}$	${0.99}_{-0.05}^{+0.06}$	${0.61}_{-0.06}^{+0.06}$	${0.69}_{-0.09}^{+0.09}$	${0.69}_{-0.05}^{+0.05}$	${0.84}_{-0.13}^{+0.15}$	${0.81}_{-0.06}^{+0.06}$	${0.52}_{-0.10}^{+0.11}$
RG	R	${0.72}_{-0.10}^{+0.10}$	${0.50}_{-0.04}^{+0.04}$	${0.99}_{-0.06}^{+0.06}$	${0.60}_{-0.06}^{+0.06}$	${0.68}_{-0.09}^{+0.09}$	${0.69}_{-0.05}^{+0.05}$	${0.84}_{-0.12}^{+0.14}$	${0.86}_{-0.06}^{+0.06}$	${0.53}_{-0.10}^{+0.11}$
RN	R	${0.72}_{-0.09}^{+0.09}$	${0.50}_{-0.03}^{+0.04}$	${0.99}_{-0.05}^{+0.05}$	${0.61}_{-0.05}^{+0.06}$	${0.68}_{-0.08}^{+0.08}$	${0.69}_{-0.05}^{+0.05}$	${0.81}_{-0.11}^{+0.12}$	${0.81}_{-0.05}^{+0.06}$	${0.40}_{-0.05}^{+0.07}$
FC	Teff	${1333.79}_{-31.91}^{+32.71}$	${1464.51}_{-26.14}^{+26.97}$	${1169.07}_{-21.85}^{+22.71}$	${1295.39}_{-36.28}^{+37.30}$	${1124.45}_{-24.47}^{+25.38}$	${1049.15}_{-26.50}^{+30.15}$	${853.72}_{-47.68}^{+52.44}$	${847.62}_{-20.41}^{+21.15}$	${717.17}_{-30.67}^{+32.00}$
FG	Teff	${1325.20}_{-28.88}^{+29.94}$	${1461.66}_{-24.90}^{+26.13}$	${1176.50}_{-22.57}^{+21.59}$	${1297.46}_{-37.22}^{+37.79}$	${1124.28}_{-23.83}^{+24.38}$	${1047.87}_{-25.79}^{+29.33}$	${852.41}_{-45.69}^{+50.57}$	${845.48}_{-19.99}^{+20.98}$	${715.34}_{-30.18}^{+31.38}$
FN	Teff	${1328.49}_{-29.06}^{+29.28}$	${1465.75}_{-24.78}^{+24.85}$	${1171.62}_{-22.00}^{+21.58}$	${1295.48}_{-35.03}^{+36.89}$	${1120.24}_{-22.77}^{+24.30}$	${1048.70}_{-24.88}^{+27.85}$	${856.36}_{-43.37}^{+48.99}$	${852.59}_{-18.69}^{+20.49}$	${716.90}_{-29.83}^{+30.65}$
RC	Teff	${1220.51}_{-34.74}^{+34.67}$	${1349.59}_{-36.57}^{+39.18}$	${1061.97}_{-25.82}^{+26.21}$	${1227.07}_{-47.49}^{+48.33}$	${1116.11}_{-37.32}^{+38.24}$	${968.86}_{-34.22}^{+34.15}$	${787.84}_{-61.32}^{+71.06}$	${712.02}_{-22.57}^{+26.18}$	${722.01}_{-40.77}^{+42.74}$
RG	Teff	${1218.15}_{-33.68}^{+32.98}$	${1334.25}_{-33.94}^{+32.75}$	${1062.43}_{-26.39}^{+28.11}$	${1237.30}_{-48.99}^{+52.22}$	${1115.92}_{-35.27}^{+36.39}$	${967.32}_{-32.82}^{+33.55}$	${787.84}_{-59.38}^{+65.91}$	${691.80}_{-20.36}^{+26.09}$	${719.76}_{-39.73}^{+41.50}$
RN	Teff	${1217.48}_{-32.82}^{+31.47}$	${1340.13}_{-32.94}^{+31.06}$	${1059.99}_{-24.68}^{+25.65}$	${1221.94}_{-41.75}^{+43.60}$	${1116.21}_{-32.88}^{+34.58}$	${964.77}_{-30.74}^{+33.75}$	${798.66}_{-54.31}^{+63.54}$	${710.72}_{-21.75}^{+23.84}$	${844.14}_{-53.89}^{+53.45}$
FC	log H2O	$-{3.97}_{-0.11}^{+0.35}$	$-{3.81}_{-0.05}^{+0.05}$	$-{3.91}_{-0.04}^{+0.04}$	$-{3.64}_{-0.05}^{+0.05}$	$-{3.76}_{-0.05}^{+0.05}$	$-{3.44}_{-0.04}^{+0.04}$	$-{3.60}_{-0.10}^{+0.10}$	$-{3.02}_{-0.06}^{+0.07}$	$-{3.59}_{-0.05}^{+0.06}$
FG	log H2O	$-{4.00}_{-0.08}^{+0.11}$	$-{3.81}_{-0.05}^{+0.05}$	$-{3.85}_{-0.07}^{+0.07}$	$-{3.63}_{-0.05}^{+0.06}$	$-{3.75}_{-0.05}^{+0.06}$	$-{3.44}_{-0.04}^{+0.04}$	$-{3.60}_{-0.09}^{+0.10}$	$-{3.02}_{-0.06}^{+0.06}$	$-{3.59}_{-0.05}^{+0.05}$
FN	log H2O	$-{4.00}_{-0.09}^{+0.15}$	$-{3.81}_{-0.04}^{+0.05}$	$-{3.89}_{-0.05}^{+0.06}$	$-{3.63}_{-0.05}^{+0.05}$	$-{3.74}_{-0.05}^{+0.05}$	$-{3.44}_{-0.04}^{+0.04}$	$-{3.60}_{-0.09}^{+0.10}$	$-{3.07}_{-0.05}^{+0.05}$	$-{3.59}_{-0.05}^{+0.05}$
RC	log H2O	$-{4.08}_{-0.08}^{+0.09}$	$-{3.82}_{-0.06}^{+0.08}$	$-{3.93}_{-0.04}^{+0.05}$	$-{3.61}_{-0.05}^{+0.06}$	$-{3.50}_{-0.12}^{+0.15}$	$-{3.45}_{-0.05}^{+0.05}$	$-{3.63}_{-0.15}^{+0.14}$	$-{3.24}_{-0.05}^{+0.06}$	$-{3.58}_{-0.06}^{+0.07}$
RG	log H2O	$-{4.08}_{-0.07}^{+0.07}$	$-{3.84}_{-0.05}^{+0.06}$	$-{3.93}_{-0.04}^{+0.05}$	$-{3.59}_{-0.06}^{+0.10}$	$-{3.50}_{-0.12}^{+0.14}$	$-{3.45}_{-0.05}^{+0.05}$	$-{3.63}_{-0.14}^{+0.13}$	$-{3.26}_{-0.04}^{+0.05}$	$-{3.58}_{-0.06}^{+0.06}$
RN	log H2O	$-{4.09}_{-0.07}^{+0.07}$	$-{3.83}_{-0.05}^{+0.06}$	$-{3.93}_{-0.04}^{+0.04}$	$-{3.61}_{-0.05}^{+0.05}$	$-{3.49}_{-0.11}^{+0.13}$	$-{3.45}_{-0.05}^{+0.05}$	$-{3.61}_{-0.13}^{+0.12}$	$-{3.24}_{-0.05}^{+0.06}$	$-{3.29}_{-0.13}^{+0.14}$
FC	log CH4	$-{5.09}_{-0.17}^{+0.61}$	$-{4.86}_{-0.07}^{+0.06}$	$-{4.92}_{-0.07}^{+0.06}$	$-{4.09}_{-0.07}^{+0.07}$	$-{4.37}_{-0.06}^{+0.06}$	$-{3.73}_{-0.06}^{+0.06}$	$-{3.75}_{-0.15}^{+0.14}$	$-{3.26}_{-0.08}^{+0.09}$	$-{3.94}_{-0.06}^{+0.07}$
FG	log CH4	$-{5.13}_{-0.13}^{+0.19}$	$-{4.84}_{-0.07}^{+0.08}$	$-{4.81}_{-0.12}^{+0.14}$	$-{4.08}_{-0.07}^{+0.08}$	$-{4.35}_{-0.07}^{+0.09}$	$-{3.73}_{-0.06}^{+0.06}$	$-{3.76}_{-0.14}^{+0.13}$	$-{3.26}_{-0.07}^{+0.08}$	$-{3.94}_{-0.05}^{+0.06}$
FN	log CH4	$-{5.13}_{-0.13}^{+0.25}$	$-{4.85}_{-0.07}^{+0.07}$	$-{4.89}_{-0.08}^{+0.12}$	$-{4.08}_{-0.07}^{+0.07}$	$-{4.35}_{-0.06}^{+0.06}$	$-{3.73}_{-0.05}^{+0.06}$	$-{3.75}_{-0.14}^{+0.13}$	$-{3.39}_{-0.08}^{+0.08}$	$-{3.94}_{-0.05}^{+0.06}$
RC	log CH4	$-{5.23}_{-0.12}^{+0.14}$	$-{4.84}_{-0.10}^{+0.13}$	$-{4.94}_{-0.07}^{+0.07}$	$-{4.06}_{-0.07}^{+0.08}$	$-{3.95}_{-0.16}^{+0.19}$	$-{3.76}_{-0.07}^{+0.07}$	$-{3.80}_{-0.27}^{+0.19}$	$-{3.73}_{-0.08}^{+0.11}$	$-{3.94}_{-0.06}^{+0.08}$
RG	log CH4	$-{5.24}_{-0.11}^{+0.11}$	$-{4.89}_{-0.08}^{+0.09}$	$-{4.94}_{-0.07}^{+0.07}$	$-{4.03}_{-0.09}^{+0.14}$	$-{3.95}_{-0.16}^{+0.18}$	$-{3.76}_{-0.07}^{+0.07}$	$-{3.79}_{-0.25}^{+0.18}$	$-{3.79}_{-0.06}^{+0.08}$	$-{3.94}_{-0.06}^{+0.08}$
RN	log CH4	$-{5.24}_{-0.11}^{+0.11}$	$-{4.86}_{-0.08}^{+0.09}$	$-{4.94}_{-0.07}^{+0.07}$	$-{4.07}_{-0.06}^{+0.06}$	$-{3.94}_{-0.15}^{+0.17}$	$-{3.76}_{-0.07}^{+0.07}$	$-{3.77}_{-0.23}^{+0.16}$	$-{3.73}_{-0.08}^{+0.10}$	$-{3.47}_{-0.18}^{+0.18}$
FC	log NH3	$-{5.63}_{-1.11}^{+0.70}$	⋯	⋯	⋯	⋯	$-{5.39}_{-0.40}^{+0.18}$	$-{4.97}_{-0.21}^{+0.18}$	⋯	⋯
FG	log NH3	$-{5.76}_{-1.32}^{+0.35}$	⋯	⋯	⋯	⋯	$-{5.39}_{-0.37}^{+0.18}$	$-{4.97}_{-0.20}^{+0.18}$	⋯	⋯
FN	log NH3	$-{5.77}_{-1.79}^{+0.47}$	⋯	⋯	⋯	⋯	$-{5.39}_{-0.33}^{+0.17}$	$-{4.97}_{-0.20}^{+0.18}$	⋯	⋯
RC	log NH3	$-{6.09}_{-3.08}^{+0.46}$	⋯	⋯	⋯	⋯	$-{5.65}_{-3.17}^{+0.31}$	$-{5.02}_{-0.34}^{+0.24}$	⋯	⋯
RG	log NH3	$-{6.13}_{-3.12}^{+0.43}$	⋯	⋯	⋯	⋯	$-{5.65}_{-3.21}^{+0.31}$	$-{5.01}_{-0.32}^{+0.23}$	⋯	⋯
RN	log NH3	$-{6.22}_{-3.22}^{+0.47}$	⋯	⋯	⋯	⋯	$-{5.62}_{-2.82}^{+0.28}$	$-{5.00}_{-0.31}^{+0.22}$	⋯	⋯
FC	log CO2	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
FG	log CO2	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
FN	log CO2	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
RC	log CO2	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
RG	log CO2	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
RN	log CO2	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
FC	log CO	⋯	⋯	$-{3.69}_{-0.29}^{+0.25}$	$-{2.85}_{-0.21}^{+0.19}$	⋯	⋯	⋯	⋯	⋯
FG	log CO	⋯	⋯	$-{3.66}_{-0.23}^{+0.19}$	$-{2.83}_{-0.21}^{+0.18}$	⋯	⋯	⋯	⋯	⋯
FN	log CO	⋯	⋯	$-{3.69}_{-0.24}^{+0.21}$	$-{2.84}_{-0.20}^{+0.18}$	⋯	⋯	⋯	⋯	⋯
RC	log CO	⋯	⋯	$-{3.80}_{-0.26}^{+0.24}$	$-{2.87}_{-0.24}^{+0.20}$	⋯	⋯	⋯	⋯	⋯
RG	log CO	⋯	⋯	$-{3.80}_{-0.26}^{+0.23}$	$-{2.82}_{-0.23}^{+0.21}$	⋯	⋯	⋯	⋯	⋯
RN	log CO	⋯	⋯	$-{3.81}_{-0.25}^{+0.22}$	$-{2.84}_{-0.21}^{+0.19}$	⋯	⋯	⋯	⋯	⋯
FC	log H2S	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
FG	log H2S	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
FN	log H2S	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
RC	log H2S	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
RG	log H2S	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
RN	log H2S	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
FC	log K	$-{7.29}_{-0.16}^{+0.17}$	$-{6.46}_{-0.10}^{+0.10}$	$-{6.78}_{-0.08}^{+0.08}$	$-{7.04}_{-0.11}^{+0.11}$	$-{6.51}_{-0.08}^{+0.08}$	$-{6.87}_{-0.05}^{+0.06}$	$-{7.17}_{-0.12}^{+0.13}$	$-{6.68}_{-0.08}^{+0.08}$	$-{7.07}_{-0.11}^{+0.11}$
FG	log K	$-{7.32}_{-0.14}^{+0.15}$	$-{6.46}_{-0.10}^{+0.10}$	$-{6.79}_{-0.07}^{+0.07}$	$-{7.02}_{-0.11}^{+0.12}$	$-{6.52}_{-0.08}^{+0.08}$	$-{6.87}_{-0.05}^{+0.06}$	$-{7.16}_{-0.11}^{+0.12}$	$-{6.68}_{-0.08}^{+0.08}$	$-{7.07}_{-0.11}^{+0.11}$
FN	log K	$-{7.31}_{-0.15}^{+0.15}$	$-{6.44}_{-0.10}^{+0.09}$	$-{6.79}_{-0.07}^{+0.07}$	$-{7.04}_{-0.11}^{+0.11}$	$-{6.49}_{-0.08}^{+0.07}$	$-{6.87}_{-0.05}^{+0.05}$	$-{7.15}_{-0.11}^{+0.12}$	$-{6.86}_{-0.12}^{+0.10}$	$-{7.05}_{-0.11}^{+0.11}$
RC	log K	⋯	$-{6.47}_{-0.92}^{+0.43}$	$-{6.83}_{-1.93}^{+0.49}$	⋯	⋯	⋯	⋯	⋯	⋯
RG	log K	⋯	$-{7.27}_{-2.17}^{+0.60}$	$-{6.87}_{-1.92}^{+0.51}$	⋯	⋯	⋯	⋯	$-{5.99}_{-3.27}^{+0.42}$	⋯
RN	log K	⋯	$-{6.69}_{-1.18}^{+0.46}$	$-{6.90}_{-1.89}^{+0.51}$	⋯	⋯	⋯	⋯	⋯	⋯
FC	log CrH	$-{9.01}_{-0.18}^{+0.32}$	⋯	$-{9.47}_{-0.21}^{+0.17}$	⋯	⋯	⋯	⋯	⋯	⋯
FG	log CrH	$-{9.06}_{-0.15}^{+0.14}$	⋯	$-{9.40}_{-0.20}^{+0.16}$	⋯	⋯	⋯	⋯	⋯	⋯
FN	log CrH	$-{9.06}_{-0.16}^{+0.18}$	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
RC	log CrH	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
RG	log CrH	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
RN	log CrH	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	$-{7.16}_{-0.50}^{+0.30}$
FC	log FeH	⋯	⋯	⋯	⋯	$-{9.44}_{-0.20}^{+0.17}$	$-{9.60}_{-0.12}^{+0.11}$	$-{10.25}_{-0.76}^{+0.33}$	$-{10.12}_{-0.96}^{+0.38}$	⋯
FG	log FeH	⋯	⋯	⋯	⋯	$-{9.42}_{-0.19}^{+0.16}$	$-{9.60}_{-0.12}^{+0.11}$	$-{10.24}_{-0.72}^{+0.32}$	$-{10.11}_{-0.92}^{+0.38}$	⋯
FN	log FeH	⋯	⋯	⋯	⋯	$-{9.42}_{-0.18}^{+0.16}$	$-{9.62}_{-0.14}^{+0.11}$	$-{10.25}_{-0.69}^{+0.31}$	⋯	⋯
RC	log FeH	⋯	⋯	⋯	⋯	$-{7.27}_{-0.20}^{+0.20}$	$-{7.96}_{-0.18}^{+0.15}$	⋯	$-{7.79}_{-0.25}^{+0.19}$	⋯
RG	log FeH	⋯	⋯	⋯	⋯	$-{7.27}_{-0.19}^{+0.18}$	$-{7.96}_{-0.17}^{+0.15}$	⋯	$-{8.69}_{-2.13}^{+0.90}$	⋯
RN	log FeH	⋯	⋯	⋯	⋯	$-{7.26}_{-0.17}^{+0.17}$	$-{7.96}_{-0.15}^{+0.13}$	⋯	$-{7.79}_{-0.24}^{+0.17}$	⋯
FC	log CaH	⋯	⋯	⋯	⋯	${3.52}_{-0.12}^{+0.12}$	⋯	⋯	⋯	⋯
FG	log CaH	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
FN	log CaH	⋯	⋯	⋯	⋯	$-{3.59}_{-4.94}^{+0.91}$	⋯	⋯	⋯	⋯
RC	log CaH	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
RG	log CaH	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
RN	log CaH	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
FC	log TiH	⋯	⋯	⋯	⋯	⋯	$-{10.50}_{-0.55}^{+0.28}$	$-{10.24}_{-0.52}^{+0.32}$	$-{10.09}_{-0.74}^{+0.37}$	⋯
FG	log TiH	⋯	⋯	⋯	⋯	⋯	$-{10.50}_{-0.52}^{+0.28}$	$-{10.24}_{-0.51}^{+0.31}$	$-{10.08}_{-0.69}^{+0.35}$	⋯
FN	log TiH	⋯	⋯	⋯	⋯	⋯	$-{10.54}_{-0.57}^{+0.30}$	$-{10.24}_{-0.48}^{+0.31}$	⋯	⋯
RC	log TiH	⋯	$-{8.60}_{-1.83}^{+0.45}$	⋯	⋯	⋯	⋯	⋯	⋯	⋯
RG	log TiH	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
RN	log TiH	⋯	$-{8.93}_{-1.78}^{+0.58}$	⋯	⋯	⋯	⋯	⋯	⋯	⋯
FC	p t	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
FG	p t	${0.54}_{-1.60}^{+0.57}$	${0.58}_{-1.60}^{+0.54}$	${0.55}_{-0.67}^{+0.17}$	${0.50}_{-1.76}^{+0.96}$	${0.47}_{-1.51}^{+0.66}$	${0.12}_{-1.44}^{+1.17}$	${0.08}_{-1.39}^{+1.16}$	${0.14}_{-1.47}^{+1.13}$	${0.20}_{-1.49}^{+1.03}$
FN	p t	${0.33}_{-1.46}^{+0.80}$	${0.42}_{-1.52}^{+0.77}$	${0.53}_{-1.33}^{+0.44}$	${0.29}_{-1.54}^{+1.16}$	${0.28}_{-1.44}^{+0.90}$	${0.36}_{-1.58}^{+0.89}$	${0.09}_{-1.39}^{+1.14}$	${0.40}_{-0.16}^{+0.12}$	${0.19}_{-1.47}^{+1.03}$
RC	p t	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
RG	p t	${0.22}_{-1.38}^{+0.92}$	${0.41}_{-1.52}^{+0.73}$	${0.32}_{-1.46}^{+0.82}$	${0.76}_{-1.86}^{+0.68}$	${0.22}_{-1.43}^{+1.00}$	${0.09}_{-1.37}^{+1.15}$	${0.07}_{-1.36}^{+1.11}$	${0.25}_{-1.45}^{+0.94}$	${0.27}_{-1.51}^{+0.93}$
RN	p t	${0.22}_{-1.33}^{+0.92}$	${0.25}_{-1.40}^{+0.92}$	${0.29}_{-1.40}^{+0.86}$	${0.09}_{-1.37}^{+1.16}$	${0.20}_{-1.39}^{+0.99}$	${0.11}_{-1.36}^{+1.12}$	${0.09}_{-1.35}^{+1.10}$	${0.23}_{-1.42}^{+0.96}$	${0.23}_{-1.43}^{+0.90}$
FC	p b	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
FG	p b	${0.50}_{-0.32}^{+0.32}$	${0.50}_{-0.32}^{+0.32}$	${0.49}_{-0.31}^{+0.32}$	${0.50}_{-0.33}^{+0.33}$	${0.49}_{-0.32}^{+0.33}$	${0.51}_{-0.34}^{+0.33}$	${0.50}_{-0.33}^{+0.33}$	${0.50}_{-0.34}^{+0.33}$	${0.50}_{-0.33}^{+0.33}$
FN	p b	${0.50}_{-0.32}^{+0.31}$	${0.51}_{-0.32}^{+0.31}$	${0.49}_{-0.31}^{+0.31}$	${0.50}_{-0.33}^{+0.32}$	${0.50}_{-0.32}^{+0.32}$	${0.52}_{-0.34}^{+0.32}$	${0.50}_{-0.33}^{+0.33}$	${0.54}_{-0.31}^{+0.29}$	${0.50}_{-0.32}^{+0.32}$
RC	p b	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
RG	p b	${0.50}_{-0.31}^{+0.32}$	${0.51}_{-0.32}^{+0.31}$	${0.50}_{-0.32}^{+0.31}$	${0.50}_{-0.33}^{+0.33}$	${0.50}_{-0.32}^{+0.32}$	${0.50}_{-0.32}^{+0.32}$	${0.50}_{-0.32}^{+0.33}$	${0.50}_{-0.32}^{+0.32}$	${0.51}_{-0.33}^{+0.32}$
RN	p b	${0.51}_{-0.31}^{+0.30}$	${0.50}_{-0.31}^{+0.31}$	${0.51}_{-0.31}^{+0.31}$	${0.51}_{-0.33}^{+0.32}$	${0.51}_{-0.31}^{+0.31}$	${0.51}_{-0.32}^{+0.31}$	${0.50}_{-0.31}^{+0.32}$	${0.50}_{-0.32}^{+0.32}$	${0.50}_{-0.31}^{+0.32}$
FC	τ	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
FG	τ	$-{2.19}_{-1.81}^{+2.29}$	$-{2.19}_{-1.86}^{+2.50}$	${0.4}_{-3.43}^{+0.55}$	$-{2.55}_{-1.63}^{+2.78}$	$-{2.17}_{-1.86}^{+2.69}$	$-{2.99}_{-1.34}^{+1.37}$	$-{2.76}_{-1.49}^{+1.52}$	$-{2.86}_{-1.43}^{+1.49}$	$-{2.68}_{-1.52}^{+1.68}$
FN	τ	$-{2.63}_{-1.53}^{+2.03}$	$-{2.66}_{-1.50}^{+2.18}$	$-{1.62}_{-2.19}^{+2.41}$	$-{2.77}_{-1.46}^{+1.97}$	$-{2.68}_{-1.48}^{+1.85}$	$-{2.77}_{-1.49}^{+1.88}$	$-{2.76}_{-1.48}^{+1.52}$	${0.04}_{-0.15}^{+0.16}$	$-{2.66}_{-1.53}^{+1.68}$
RC	τ	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
RG	τ	$-{2.64}_{-1.50}^{+1.66}$	$-{2.56}_{-1.56}^{+2.31}$	$-{2.51}_{-1.58}^{+2.06}$	$-{2.21}_{-1.85}^{+2.90}$	$-{2.78}_{-1.41}^{+1.60}$	$-{2.92}_{-1.35}^{+1.41}$	$-{2.66}_{-1.52}^{+1.57}$	$-{2.67}_{-1.50}^{+1.71}$	$-{2.52}_{-1.60}^{+1.83}$
RN	τ	$-{2.70}_{-1.43}^{+1.57}$	$-{2.79}_{-1.38}^{+1.66}$	$-{2.60}_{-1.50}^{+1.76}$	$-{2.92}_{-1.36}^{+1.41}$	$-{2.76}_{-1.43}^{+1.58}$	$-{2.91}_{-1.34}^{+1.42}$	$-{2.64}_{-1.51}^{+1.52}$	$-{2.66}_{-1.50}^{+1.67}$	$-{2.48}_{-1.59}^{+1.82}$
FC	Q0	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
FG	Q0	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
FN	Q0	${9.83}_{-7.48}^{+32.86}$	${10.06}_{-7.70}^{+33.82}$	${9.56}_{-7.25}^{+31.66}$	${9.91}_{-7.67}^{+35.19}$	${9.89}_{-7.57}^{+32.32}$	${10.78}_{-8.51}^{+36.02}$	${10.24}_{-8.02}^{+34.98}$	${18.68}_{-14.39}^{+35.32}$	${10.18}_{-7.87}^{+34.11}$
RC	Q0	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
RG	Q0	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
RN	Q0	${9.85}_{-7.43}^{+30.42}$	${10.32}_{-7.86}^{+31.79}$	${9.96}_{-7.57}^{+30.88}$	${10.25}_{-7.99}^{+35.31}$	${10.08}_{-7.68}^{+32.27}$	${10.14}_{-7.78}^{+32.93}$	${10.13}_{-7.81}^{+33.77}$	${10.22}_{-7.88}^{+33.94}$	${9.78}_{-7.45}^{+31.84}$
FC	a0	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
FG	a0	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
FN	a0	${0.70}_{-0.13}^{+0.10}$	${0.70}_{-0.13}^{+0.10}$	${0.70}_{-0.12}^{+0.10}$	${0.70}_{-0.13}^{+0.10}$	${0.70}_{-0.13}^{+0.10}$	${0.70}_{-0.13}^{+0.10}$	${0.70}_{-0.13}^{+0.10}$	${0.69}_{-0.09}^{+0.08}$	${0.70}_{-0.13}^{+0.10}$
RC	a0	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
RG	a0	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
RN	a0	${0.70}_{-0.12}^{+0.10}$	${0.70}_{-0.12}^{+0.10}$	${0.70}_{-0.12}^{+0.10}$	${0.70}_{-0.13}^{+0.10}$	${0.70}_{-0.12}^{+0.10}$	${0.70}_{-0.13}^{+0.10}$	${0.70}_{-0.13}^{+0.10}$	${0.70}_{-0.13}^{+0.10}$	${0.70}_{-0.13}^{+0.10}$
FC	a	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
FG	a	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
FN	a	${0.35}_{-0.85}^{+0.86}$	${0.35}_{-0.84}^{+0.84}$	${0.41}_{-0.81}^{+0.80}$	${0.33}_{-0.88}^{+0.88}$	${0.36}_{-0.86}^{+0.86}$	${0.15}_{-0.89}^{+1.03}$	${0.30}_{-0.88}^{+0.91}$	$-{0.81}_{-0.11}^{+0.14}$	${0.34}_{-0.89}^{+0.88}$
RC	a	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
RG	a	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
RN	a	${0.34}_{-0.81}^{+0.83}$	${0.35}_{-0.84}^{+0.84}$	${0.37}_{-0.83}^{+0.83}$	${0.30}_{-0.88}^{+0.89}$	${0.33}_{-0.86}^{+0.86}$	${0.30}_{-0.86}^{+0.88}$	${0.32}_{-0.87}^{+0.88}$	${0.32}_{-0.87}^{+0.88}$	${0.32}_{-0.88}^{+0.87}$

Note. Only models with the reduced set of molecules are tabulated. Variations of the models shown are as follows: full spectra cloud-free (FC), full spectra gray (FG), full spectra nongray (FN), restricted spectra cloud-free (RC), restricted spectra gray (RG), and restricted spectra nongray (RN).

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