<figure><img src="image_1.png" /></figure>

# ÆSOPUS 2.1: Low-temperature Opacities Extended to High Pressure

Paola Marigo¹,<fnref n="5" /> ID, Francesco Addari² ID, Diego Bossini¹ ID, Alessandro Bressan² ID, Guglielmo Costa³,⁴ ID, Léo Girardi⁴ ID, Giada Pastorelli¹ ID, Michele Trabucchi¹,⁴ ID, and Guglielmo Volpato¹,⁴ ID

¹ Department of Physics and Astronomy G. Galilei, University of Padova, Vicolo dell'Osservatorio 3, I-35122, Padova, Italy; diego.bossini@unipd.it

² Scuola Internazionale Superiore di Studi Avanzati, Via Bonomea, 265, I-34136, Trieste, Italy

³ Univ Lyon, Univ Lyon1, Ens de Lyon, CNRS, Centre de Recherche Astrophysique de Lyon UMR5574, F-69230 Saint-Genis-Laval, France

⁴ INAF-Osservatorio Astronomico di Padova, Vicolo dell'Osservatorio 5, I-35122, Padova, Italy

Received 2024 May 15; revised 2024 August 29; accepted 2024 September 14; published 2024 November 12

## Abstract

We address the critical need for accurate Rosseland mean gas opacities in high-pressure environments, spanning temperatures from 100 K to 32,000 K. Current opacity tables from Wichita State University and ÆSOPUS 2.0 are limited to $log(R) \le 1$, where $R = \rho T_6^{-3}$ in units of g cm$^{-3}$(10$^6$K)$^{-3}$. This is insufficient for modeling very low-mass stars, brown dwarfs, and planets with atmospheres exhibiting higher densities and pressures ($log(R) > 1$). Leveraging extensive databases such as ExoMol, ExoMolOP, MoLLIST, and HITEMP, we focus on expanding the ÆSOPUS opacity calculations to cover a broad range of pressure and density conditions ($-8 \le \log(R) \le +6$). We incorporate the thermal Doppler mechanism and microturbulence velocity. Pressure-broadening effects on molecular transitions, leading to Lorentzian or Voigt profiles, are explored in the context of atmospheric profiles for exoplanets, brown dwarfs, and low-mass stars. We also delve into the impact of electron degeneracy and nonideal effects, such as ionization potential depression under high-density conditions, emphasizing its notable influence on Rosseland mean opacities at temperatures exceeding 10,000 K. As a result, this study expands the ÆSOPUS public web interface for customized gas chemical mixtures, promoting flexibility in opacity calculations based on specific research needs. Additionally, precomputed opacity tables, inclusive of condensates, are provided. We present a preliminary application to evolutionary models for very low-mass stars.

Unified Astronomy Thesaurus concepts: Stellar atmospheric opacity (1585); Astrochemistry (75); Low mass stars (2050); Brown dwarfs (185); Exoplanets (498); Collisional broadening (2083)

## 1. Introduction

Thanks to extensive databases such as ExoMol (J. Tennyson & S. N. Yurchenko 2012; J. Tennyson et al. 2016), ExoMolOP (K. L. Chubb et al. 2021), MoLLIST (P. F. Bernath 2020) and HITEMP (L. S. Rothman et al. 2010), we can now rely on the availability of a substantial amount of molecular line list data, which is a significant asset for modeling the atmospheres of hot exoplanets, as well as cool stellar and substellar atmospheres.

Rosseland mean opacities are critical components in modeling stars and substellar objects. To our knowledge, the most widely distributed suppliers of Rosseland mean opacities for $T \lesssim 10,000$ K are the Wichita State University group (D. R. Alexander 1975; D. R. Alexander & J. W. Ferguson 1994; J. W. Ferguson et al. 2005), and the ÆSOPUS team (P. Marigo & B. Aringer 2009; P. Marigo et al. 2022). Two standard variables are used to build Rosseland mean opacity tables:

$$T \text{ and } R = \rho T_6^{-3}, \qquad (1)$$

where T is the temperature (in Kelvin) and the R parameter (in g cm$^{-3}$(10$^6$ K)$^{-3}$) includes both the temperature ($T_6 = T/(10^6$ K)), and the gas mass density $\rho$ (in g cm$^{-3}$). The fact

that smooth opacity interpolations are feasible is the primary motivation behind the selection of these two parameters.

So far, low-temperature opacities have been computed encompassing the range $-8 \le \log(R) \le +1$ (J. W. Ferguson et al. 2005; P. Marigo et al. 2022). This range, however, is insufficiently broad to cover the structural characteristics of very low-mass stars, brown dwarfs, and planets (see Section 2). In fact, these objects have atmospheres with high density and pressure, so $\log(R)$ can easily exceed 1. Very low-mass stars ($0.08 \lesssim M/M_\odot \lesssim 0.6$) and substellar objects ($10^{-3} \lesssim M/M_\odot \lesssim 13$ for exoplanets, $13 \lesssim M/M_\odot \lesssim 90$ for brown dwarfs, with $M_\odot \equiv 1$ Jupiter mass) have atmospheres that typically cover temperature ranges of $100 \lesssim T/K \lesssim 3500$ and gas pressure ranges of $10^{-4} \lesssim P_{\text{gas}}/\text{bar} \lesssim 10^3$ (A. Burrows et al. 2001; D. S. Spiegel et al. 2011; P. A. Wilson et al. 2016; G. D. Mulders et al. 2021). This demonstrates the need of extending the Rosseland mean opacity tables at higher densities and pressures (with $\log(R) > 1$).

The thermal Doppler mechanism is used by both the Wichita State University and ÆSOPUS teams for molecular line broadening. Microturbulence velocity is also included in ÆSOPUS. As we will see in Section 3.2, this approximation is mostly valid over the standard range $-8 \le \log(R) \le +1$, but it becomes inadequate when higher R values, hence larger densities and pressures are considered. When moving into the high-density regime, pressure effects broaden molecular transitions, resulting in either Lorentzian or Voigt profiles (A. Burrows et al. 2001; C. M. Sharp & A. Burrows 2007). Figure 1 depicts the pressure-temperature atmospheric profiles of two exoplanets, one brown dwarf, and very low-mass stars. Pressure broadening of spectral molecular lines affects all of

<figure><img src="image_2.png" /></figure>

<footnote n="5">⁵ This manuscript represents the final work of Prof. Paola Marigo, who passed away before its publication. As her group, we would like to express our deepest gratitude for her guidance, and our commitment to carry forward her legacy.</footnote>
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Atmospheric pressure-temperature profiles for exoplanets, brown dwarfs, and very low-mass stars. The two hot Jupiter exoplanets, WASP-17b with a mass of 0.51 *M*<sub>J</sub>, and WASP-19b with a mass of 1.14 *M*<sub>J</sub>, were analyzed by D. K. Sing et al. (2016) through Hubble Space Telescope/Spitzer observations. E. Calamari et al. (2022) used an atmospheric retrieval analysis to derive the profile for the brown dwarf Gliese 229B with an estimated mass of 50 *M*<sub>J</sub>. We also add two theoretical profiles for brown dwarfs with a surface gravity of log(g) = 4.5 (I. Baraffe et al. 2002), and a very low-mass star model from PARSEC (A. Bressan et al. 2012). The dashed black line denotes the limit below which pressure broadening for molecular spectral lines should be included. Refer to Section 3.2 for more details.</figcaption></figure>

these objects' atmospheres (below the dashed black line). Few
works in the past literature, to our knowledge, computed
Rosseland mean gas opacities at high densities, namely
R. L. Kurucz (1993), R. S. Freedman et al. (2008), M. G. Mal-
ygin et al. (2014), R. S. Freedman et al. (2014), adopting scaled
solar abundances according to N. Grevesse & A. J. Sauval
(1998) or K. Lodders (2003). For zero-metallicity gas, we also
refer to the work of P. Lenzuni et al. (1991).

Under high-density conditions, additional important phenomena, namely electron degeneracy and Coulomb interactions between charged particles, must be accounted for in the equation of state (J. Cox & R. Giuli 1968; J. P. Hansen & P. Vieillefosse 1976; A. Y. Potekhin et al. 2009). Furthermore, all charges present in a radiating gas, electrons, and ions, contribute to reducing the energy required to free an electron in the fundamental state. This process is designated as ionization potential depression (IPD; G. Ecker & W. Kröll 1963; J. C. Stewart & K. D. J. Pyatt 1966). We will show that the IPD effect will have a notable impact on Rosseland mean opacities, especially for T > 10,000 K. As a result, appropriate physics must be included to describe the gas at high pressure and its opacity interaction with radiation.

This study aims to increase the accessibility of Rosseland mean gas opacities in high-pressure environments across a broad temperature range from 100 K to 32,000 K. Expanding the opacities to cover a wide range of pressure and density conditions ($-8 \le \log(R) \le +6$) is essential for accurate modeling of substellar objects and very low-mass stars. We provide our results in a public web interface that allows users to customize the chemical mixture according to their specific research requirements. This flexibility is crucial because different research projects may focus on different chemical compositions and environmental conditions. In addition, we produce precomputed opacity tables with the inclusion of condensates, similar to P. Marigo et al. (2023).

This paper is organized as follows. Section 2 recaps the basic ingredients of the equation of state in ÆSOPUS and GGchem codes, as well as the method for calculating the Rosseland mean opacity. We additionally present some physical structures of very low-mass stars to demonstrate the need for the pressure and density ranges of the opacity tables to be expanded to higher values. Section 3 describes how we implement the IPD effect in ÆSOPUS partition functions and abundance differential equations, as well as how we treat line pressure broadening for atomic and molecular transitions. In Section 4 we analyze and discuss the findings, focusing on the impact of the IPD effect and pressure broadening on Rosseland mean opacities and comparing our results to others found in the literature. Section 5 presents a few examples of Rosseland mean opacities with solid grains included, while Section 6 briefly discusses the impact of these opacities in low-mass stellar models.

**2. Equation of State and Rosseland Mean Opacities**

The ÆSOPUS code solves the equation of state encompass-
ing more than 800 species, including about 300 atoms and ions
and 500 molecules, in the gas phase under conditions of
thermodynamic and instantaneous chemical equilibrium (see
Appendix). In order to expand the opacity computations in the
low-temperature regime where solid grains condense, we
(P. Marigo et al. 2023) recently coupled ÆSOPUS
(P. Marigo & B. Aringer 2009; P. Marigo et al. 2022) and
GGchem codes (P. Woitke et al. 2018). In practice, we use
ÆSOPUS for temperatures between 30,000 K and 3000 K, and
then we switch to GGchem for temperatures between 3000 K
and 400 K. We recall that GGchem computes the abundances
of approximately 568 gas molecules, 55 liquid species, and
nearly 200 types of solid particles.

There is an extensive description of how to calculate the
Rosseland mean opacity in P. Marigo et al. (2022) and
P. Marigo & B. Aringer (2009), so it will not be repeated here.
To summarize, we compute the total monochromatic opacity
cross section per unit mass (in cm² g⁻¹) for any chosen (ρ, T)
pair by incorporating all the contributions from true absorption
and scattering, for both gas and solid particles.

As previously stated in Section 1, the standard range of
Rosseland mean opacities, -8 ≲ log(R) ≲ 1, is not enough to
describe the physical properties of very low-mass stars, brown
dwarfs, and planets, which reach much higher densities and
pressures. To demonstrate this fact, we display a few stellar
structures in Figure 2 that correspond to brown dwarfs and very
low-mass stars with initial masses Mᵢ between 0.05 M☉ and
0.7 M☉.

Looking at the left panel of Figure 2, it is evident that a
sizable portion of the structures of stars with Mᵢ ≲ 0.7 M☉
exceeds the upper limit of log(R) = 1, reaching values up to
log(R) ≃ 4. The brown dwarf model with Mᵢ = 0.05 M☉
extends up to log(R) ≃ 6. As we will discuss in Section 6,
opacity extrapolation outside of the validity range of tables may
result in incorrect structural properties.

From stellar evolution theory, we know that main-sequence
stars with 0.08 ≲ Mᵢ/M☉ ≲ 0.35 are completely convective and
adiabatic, regardless of the convection model used. However,
convection becomes less effective for transporting energy
closer to the surface, particularly in layers where hydrogen and
helium are partially ionized, and the true temperature gradient,
∇ = d log(T)/d log(P), becomes superadiabatic, that is
<figure><img src="image_4.png" /><figcaption>**Figure 2.** Structures of very low-mass stars and brown dwarfs computed with the PARSEC code (A. Bressan et al. 2012), from the center to the atmosphere, in a stage close to the zero-age main sequence. The initial mass changes in increments of 0.05 $M_\odot$, from 0.05 $M_\odot$ to 0.7 $M_\odot$. The initial chemical composition is scaled solar according to E. Caffau et al. (2011), with a metallicity $Z = 0.014$, and helium abundance $Y = 0.273$. The horizontal line marks the maximum value, $log(R) = 1$, currently available in the low-temperature Rosseland mean opacities (J. W. Ferguson et al. 2005; P. Marigo et al. 2022). Left panel: structures in the $log(T) - log(R)$ plane, color coded according to the initial stellar mass. Right panel: structures in the $log(T) - log(R)$ plane, with convective regions color coded according to the degree of superadiabaticity, $\nabla - \nabla_{ad}$. The gray sections correspond to convective regions treated as adiabatic, while the black sections refer to the radiative regions. See text for more details.</figcaption></figure>

$\nabla - \nabla_{ad} > 0$. In these regions, microphysics plays a crucial role, as factors such as the equation of state and opacities become essential for determining the solution of the stellar structure. The right panel of Figure 2 highlights the regions where the temperature gradient becomes superadiabatic. The mixing-length theory (E. Böhm-Vitense 1958) is used to estimate $\nabla - \nabla_{ad}$. It is clear that in very low-mass stars, low-temperature opacities extend in stellar layers where $log(R) > 1$ and convection is superadiabatic, and thus accurate opacity estimation is critical.

What are the chemical species that have to be considered in the calculation of Rosseland mean opacities at high values of $R$? To answer this question, Figure 3 depicts the concentrations of several molecules in the gas phase as a function of temperature for $log(R) = 6$. At high density, the most abundant molecules are molecular hydrogen ($H_2$) and water vapor ($H_2O$), carbon monoxide (CO), and silicon monoxide (SiO), followed by methane ($CH_4$), ammonia ($NH_3$), and silane ($SiH_4$) toward lower temperatures.

Figure 4 shows a number of condensed species, including metal oxides such as SiO, ZrO, VO, silicates, Cr, and Fe. It is worth noting that corundum ($Al_2O_3$) does not condense in appreciable amounts at high densities (hence, it is not shown in Figure 4). The condensation sequence at low temperatures is closed by iron sulfide (FeS), water ice ($H_2O$), and ammonia ice ($NH_3$).

### 3. Opacities at High Density

When entering a high-density regime, opacities become increasingly complex and must be handled with great care due to a variety of physical processes, including electron degeneracy, which increases gas pressure; nonideal effects due to Coulomb interactions among charged particles, which can lower atom and molecule ionization potentials; and line pressure broadening of molecular bands and atomic transitions. In Sections 3.1 and 3.2, we will detail the necessary improvements we made in ÆSOPUS to deal with these high-density conditions.

#### 3.1. Ionization Potential Depression

The ÆSOPUS opacity calculations are extended up to $log(R) = 6$ in a high-density regime. At the highest temperature, $log(T/K) = 4.5$, this corresponds to a mass density of $\rho \approx 32 \text{ g cm}^{-3}$. As shown in Figure 5 (left panel), electrons are partially degenerate at this density. We recall that ÆSOPUS incorporates an accurate treatment of electron degeneracy, based on generalized Fermi–Dirac integrals (J. Cox & R. Giuli 1968). At increasing densities, some nonideal effects appear, such as the lowering of the ionization potentials of atoms and molecules.

Let us briefly discuss this aspect. We know that the ionization potential $U$ of an ion embedded in a plasma is reduced due to the interaction of all charged particles (ions and electrons) with that ion. The IPD effect is accounted for using the method developed by G. Ecker & W. Kröll (1963), who formulated a generalized Saha equation as a function of the chemical potential of the plasma.

The IPD is calculated with

$$ \Delta U(z) = \frac{z e^2}{\lambda_D}, \qquad (2) $$

where $z$ is the charge of the ion after the ionization occurrence, $e$ is the electron's charge, and $\lambda_D$ is a generalized Debye length, which is computed through

$$ \lambda_D = \sqrt{\frac{kT}{4\pi(n_e + n_{\text{ion}})}}, \qquad (3) $$

where $k$ is the Boltzmann constant, $n_e$ and $n_{\text{ion}}$ are the number densities of free electron and ions in the plasma.

The implementation of the IPD effect in ÆSOPUS is carried out as follows. The ÆSOPUS code uses the Newton–Raphson technique to solve the equation of state assuming instantaneous chemical equilibrium, employing a set of dissociation–recombination and/or ionization equilibrium constants for each of the 800 particles. Concerning the Saha equation for ionization, at each iteration we correct the ionization potential
<figure><img src="image_5.png" /><figcaption>**Figure 3.** Gaseous molecular concentrations, in number density, for some selected species as a function of temperature, without condensation. We assume log(R) = 6, with a chemical composition solar-scaled according to E. Magg et al. (2022), with metallicity Z = 0.01 and hydrogen abundance X = 0.7.</figcaption></figure>

using the IPD effect inside the equilibrium constant, which depends on U(z) − ΔU(z), T, n_e, and n_ion until convergence is reached.

This is illustrated in the right panel of Figure 5. It compares the hydrogen ionization degree x(H)<fnref n="6" /> with and without the IPD effect as a function of temperature for two R parameter values. Particularly for log(R) = 6, with the classical Saha equation, x (H) remains extremely low, whereas, with the IPD plasma effect, x(H) increases significantly at the highest temperatures.

The locus of points in the log(T) − log(ρ) diagram where the ionization degree for hydrogen x(H) is equal to 0.5 and 0.2 is plotted in Figure 5 (left panel). We observe that, when the IPD effect is considered, the same ionization degree is obtained at higher densities than in the ideal case. This is especially noticeable at the highest temperatures for x(H) = 0.2, while at lower densities, the differences become less and less significant.

The reduction in the ionization potential of atoms and molecules has a sizable impact on the Rosseland mean opacities at high density, as discussed in Section 4.1.

## 3.2. Line Pressure Broadening for Atomic and Molecular Transitions

Various processes in planet, substellar, and stellar atmospheres naturally broaden spectral lines. Doppler and pressure broadening are the most common types of line broadening. Doppler broadening is caused by the thermal velocities of each atom and molecule and is normally described by a Gaussian line profile (S. N. Yurchenko et al. 2018a). The width of the Doppler line core is directly related to the temperature. As the temperature increases, the thermal motion of particles becomes more significant, causing broader line profiles.

For instance, J. W. Ferguson et al. (2005) use a pure thermal Doppler profile in their opacity computations. In ÆSOPUS we do similarly but also account for microturbulence velocity by producing a normalized broadening profile, φ(ν), according to the equation:

$$ \phi(\nu) = \frac{1}{\Delta_{\nu} \sqrt{\pi}} e^{-(\nu - \nu_0)/\Delta_{\nu}}^2, \qquad (4) $$

where ν₀ is the line center position in frequency, and Δ_ν is the line width, obtained with

$$ \Delta_{\nu} = \frac{\nu_0}{c} \sqrt{\frac{2k_B T}{m} + \xi^2}. \qquad (5) $$

<footnote n="6">⁶ The ionization degree of hydrogen is defined as the ratio x(H) = n(H⁺)/[n(H) + n(H⁺)], where n(H) and n(H⁺) are the number densities of neutral and ionized hydrogen.</footnote>
<figure><img src="image_6.png" /><figcaption>**Figure 4.** Distribution of several condensed species as a function of temperature at log(R) = 6, for solar abundances in phase equilibrium, computed with *GGchem*. The abundances are evaluated with respect to the number density of hydrogen nuclei. The assumed log(R) value and chemical composition are the same as in Figure 4.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Figure 5.** Left panel: the equation of state for a gas of free particles in the log(T) – log(ρ) plane. The dashed blue lines are approximate boundaries between regions where radiation pressure, classical ideal gas pressure, and nonrelativistic electron degeneracy dominate. We assume E. Magg et al. (2022)'s solar composition with X = 0.7 and Z = 0.01. The gray area denotes the upper extension of our opacity tables. The red and black lines mark the locus of points where the ionization degree of hydrogen is 0.5 and 0.2, as indicated. Right panel: ionization degree of hydrogen as a function of temperature for log(R) = 6 (black line) and log(R) = 3 (magenta line). In both panels, the plasma IPD effect is represented by solid lines, while dashed lines are the result of the classical Saha ionization equation. See text for more details.</figcaption></figure>

In this Equation, *c* denotes the speed of light, *k*<sub>B</sub> the Boltzmann constant, *m* the molecule's mass, and ξ the microturbulent velocity, which is set to 2.5 km s<sup>-1</sup> (see P. Marigo & B. Aringer 2009; P. Marigo et al. 2022, for more details).

Pressure broadening, which varies depending on the perturbing species (such as H, He, H<sub>2</sub>) in addition to pressure, results in a Lorentzian or Voigt profile. While a Lorentzian profile is typically pressure dependent, a Voigt profile is a
convolution of the Doppler and Lorentzian profiles. It accounts for both thermal motion and collisional broadening, making it suitable for modeling line shapes in a broader range of conditions.

In addition to temperature and pressure, the line width in Lorentzian and Voigt profiles is also influenced by various other quantities, the so-called broadening parameters, that can be challenging to determine accurately. Broadening parameters, if available, are present in the ExoMol database (E. J. Barton et al. 2017; S. N. Yurchenko et al. 2017c). Based on ExoMol broadening data, K. L. Chubb et al. (2021) recently created a publicly accessible database (ExoMolOP<sup>7</sup>) of opacities for over 80 molecules of astrophysical interest computed at various pressures (10<sup>-5</sup>-10<sup>2</sup> bar) and temperatures (the range depends on the line list). Atomic data for the alkali-neutral metals, Na and K, is additionally provided, based on the NIST database (A. Kramida et al. 2022) and the most recent profiles for the resonance lines (N. F. Allard et al. 2016, 2019). The data can be recovered in a variety of formats that are compatible with different exoplanet atmosphere retrieval codes.

For this work we use the cross section data for the retrieval code TauREx (I. P. Waldmann et al. 2015a, 2015b; A. F. Al-Refaie et al. 2021), in HDF5 format, with a spectral resolution of $R = \lambda/\Delta\lambda = 15,000$, wavenumber coverage of 200–33,333 cm<sup>-1</sup>. The TauREx table format is compatible with another retrieval code petitRADTRANS (P. Mollière et al. 2019), so we could benefit from its publicly available software. Utilizing the ExoK<sup>8</sup> code (J. Leconte 2021), we can create cross section tables appropriate for ÆSOPUS that are interpolated in pressure for each temperature value. The grids of temperature and pressure typically have 18 and 15 nodes, respectively, distributed throughout the corresponding ranges (the number of temperature nodes, as well as the wavenumber grid, may vary depending on the line list). Table 1 contains a complete list of molecules, alkali-neutral metals, and other atoms for which we have pressure-broadening line profiles. For atoms with temperatures above 4000 K, we use Opacity Project cross sections (M. J. Seaton et al. 1994), which are expressed as a function of temperature and electron density. Line broadening includes effects produced by thermal Doppler, radiation damping, and pressure.

The molecular data consist of 65 species, which is slightly less than the 80 species included in P. Marigo et al. (2022, see their Table 2). The molecules that currently lack pressure broadening are: O<sub>3</sub>, ClO, HI, CS<sub>2</sub>, OCS, NaO, N<sub>2</sub>, KOH, H<sub>2</sub>, HCl, ZrO, C<sub>3</sub>, CH<sub>3</sub>Cl, SO. We will incorporate new data into ÆSOPUS as soon as it becomes available. To avoid opacity gaps, we include monochromatic cross sections for these molecules using thermal Doppler plus microturbulent velocity profiles for any (T, R) combinations. We interpolate the TauREx tables as a function of wavenumber, temperature, and pressure to compute monochromatic cross sections for molecules, and alkali atoms Na and K. The same procedure is applied for all molecules at T values above the T<sub>max</sub> limits listed in Table 1.

It is important to know where in the pressure–temperature (P–T) space each of the two broadening mechanisms contributes most significantly. C. Hedges & N. Madhusudhan (2016) compared Doppler and Lorentzian broadening profiles

over the P–T diagram in terms of half-width at half-maximum to gain a picture of where each profile impacts substantially. Based on their findings, we depict the two broadening regimes in Figure 6. According to K. L. Chubb et al.'s (2021) analysis, we also add a lower pressure limit (10<sup>6</sup> bar, black line) above which molecular lines are treated with a Voigt profile. The data can be found in the ExoMolOP database. As expected, thermal Doppler broadening contributes significantly to the final profile core at low pressures, whereas pressure (Lorentzian) broadening is more effective at high pressures. Both broadening mechanisms are likely to contribute considerably to the core of the line profiles closer to the border between these two regimes. In this work, we use the Voigt profile for molecular lines, which is a convolution of Lorentzian and Doppler broadening mechanisms. This convolution takes into account both pressure broadening (Lorentzian) and temperature-induced broadening (Doppler), making it a versatile tool for accurately modeling spectral lines in a variety of physical conditions. Below the black horizontal line in Figure 6 we assume that pressure effects become insignificant, and we use the thermal Doppler plus microturbulence velocity broadening.

High pressure can have a sizable impact on the monochromatic absorption cross sections $\sigma$. To illustrate the effect, Figure 7 compares $\sigma$ with applied Doppler plus microturbulence broadening to pressure broadening for two molecules, water vapor (H<sub>2</sub>O) and methane (CH<sub>4</sub>). It is evident that pressure broadening reduces the excursion of $\sigma$ to higher and lower values. In the case of water vapor, this is particularly clear at 2000 K. When compared to the thermal Doppler Gaussian profiles, the Lorentzian line profiles produce a $\sigma$ that is most concentrated at higher values at a pressure of 1 bar. At 100 bar of pressure and 1000 K of temperature, methane experiences a similar effect, with a cross section that does not exhibit large fluctuations when compared to the Doppler line profiles. The tendency of $\sigma$ toward higher values at increasing pressure will have a noticeable impact on Rosseland mean gas opacities, which will tend to increase.

# 4. Results

## 4.1. Ionization Potential Depression Effects

To illustrate the impact of the IPD on the opacities, we compare in Figure 8 the different contributions to the opacity for a scaled solar mixture, assuming $log(R) = 6$. Moving up in temperature, we notice the significant contribution of collision-induced absorption (CIA) at low temperatures, which is primarily caused mainly by H<sub>2</sub>–H<sub>2</sub> collisions. The inclusion of the IPD for ions does not produce discernible effects in $log(\kappa_R)$, with differences <0.02 dex up to $log(T) = 3.75$. Beyond this temperature, the contribution of different opacity sources can vary significantly as the temperature and density increase (we assume $log(R) = 6$). When IPD is considered, the most striking facts are: a significant decrease in H<sup>-</sup> opacity as the number density of neutral hydrogen decreases (see right panel of Figure 5), a shift to lower temperatures of the bound–bound hydrogen line opacity, and a remarkable increase in H<sub>2</sub><sup>+</sup> opacity given its higher abundance. We also notice that at high density, metals contribute significantly to opacity in the temperature range 3.5 ≲ log(T/K) ≲ 4.0. There is a noticeable opacity bump at 3.6 ≲ log(T/K) ≲ 3.7, and we verify that the major absorption contributions come from Fe, Al, Na, and Ca. Such a bump will be visible in the Rosseland mean opacity as well (see Section 4.2).

<footnote n="7"><sup>7</sup> Data is available at https://www.exomol.com.</footnote>

<footnote n="8"><sup>8</sup> https://perso.astrophy.u-bordeaux.fr/~jleconte/exo_k-doc/index.html</footnote>
<table><caption>**Table 1**
Spectral Line Data for Molecular Absorption with Pressure Broadening Taken from the ExoMoLOP Database</caption><thead><tr><th>Group</th><th>Species</th><th>Line List</th><th>References</th><th>λ<sub>1</sub><br>(μm)</th><th>λ<sub>u</sub><br>(μm)</th><th>T<sub>max</sub><br>(K)</th></tr></thead><tbody><tr><td rowspan="5">Metal Oxides</td><td>AlO</td><td>ExoMol ATP</td><td>C. A. Bowesman et al. (2021)</td><td>0.29</td><td>100</td><td>8000</td></tr><tr><td>CaO</td><td>ExoMol VBATHY</td><td>S. N. Yurchenko et al. (2016)</td><td>0.5</td><td>100</td><td>5000</td></tr><tr><td>MgO</td><td>ExoMol LiTY</td><td>H. Y. Li et al. (2019)</td><td>0.3</td><td>100</td><td>5000</td></tr><tr><td>SiO</td><td>ExoMol EBJT</td><td>S. N. Yurchenko et al. (2022)</td><td>1.65</td><td>100</td><td>9000</td></tr><tr><td>TiO</td><td>ExoMol TOTO</td><td>L. K. McKemmish et al. (2019)</td><td>0.33</td><td>100</td><td>5000</td></tr><tr><td rowspan="4">Other Oxides</td><td>VO</td><td>ExoMol VOMYT</td><td>L. K. McKemmish et al. (2016)</td><td>0.29</td><td>100</td><td>5000</td></tr><tr><td>CO</td><td>Li 2015</td><td>G. Li et al. (2015)</td><td>0.43</td><td>100</td><td>5000</td></tr><tr><td>NO</td><td>HITEMP-2019</td><td>Q. Qu et al. (2021)</td><td>0.37</td><td>100</td><td>4000</td></tr><tr><td>O<sub>2</sub></td><td>HITRAN</td><td>K. L. Chubb et al. (2021)</td><td>1.43</td><td>100</td><td>296</td></tr><tr><td rowspan="5">Triatomics</td><td>PO</td><td>ExoMol POPS</td><td>L. Prajapat et al. (2017)</td><td>0.83</td><td>100</td><td>5000</td></tr><tr><td>CO<sub>2</sub></td><td>ExoMol UCL-4000</td><td>S. N. Yurchenko et al. (2020a)</td><td>0.5</td><td>100</td><td>4000</td></tr><tr><td>H<sub>2</sub>O</td><td>POKAZATEL</td><td>O. L. Polyansky et al. (2018)</td><td>0.24</td><td>100</td><td>4000</td></tr><tr><td>H<sub>2</sub>S</td><td>ExoMol AYT2</td><td>E. Roueff et al. (2019)</td><td>0.91</td><td>100</td><td>2000</td></tr><tr><td>HCN</td><td>ExoMol Harris</td><td>G. J. Harris et al. (2006)</td><td>0.56</td><td>100</td><td>4000</td></tr><tr><td rowspan="5">Metal hydrides</td><td>O<sub>3</sub></td><td>HITRAN</td><td>I. E. Gordon et al. (2022)</td><td>1.43</td><td>100</td><td>...</td></tr><tr><td>SiH<sub>2</sub></td><td>ExoMol CATS</td><td>S. N. Yurchenko et al. (2017b)</td><td>1.00</td><td>100</td><td>2000</td></tr><tr><td>SiO<sub>2</sub></td><td>ExoMol OYT3</td><td>A. Owens et al. (2020)</td><td>1.67</td><td>100</td><td>3000</td></tr><tr><td>SO<sub>2</sub></td><td>ExoMol ExoAmes</td><td>K. L. Chubb et al. (2021)</td><td>1.25</td><td>100</td><td>2000</td></tr><tr><td>AlH</td><td>ExoMol AlHambra</td><td>S. N. Yurchenko et al. (2018d)</td><td>0.37</td><td>100</td><td>5000</td></tr><tr><td rowspan="10">Other hydrides</td><td>BeH</td><td>ExoMol</td><td>D. Darby-Lewis et al. (2018)</td><td>0.24</td><td>100</td><td>2000</td></tr><tr><td>CaH</td><td>MoLLIST</td><td>A. Owens et al. (2022)</td><td>0.45</td><td>100</td><td>5000</td></tr><tr><td>CrH</td><td>MoLLIST</td><td>P. F. Bernath (2020)</td><td>0.69</td><td>100</td><td>5000</td></tr><tr><td>FeH</td><td>MoLLIST</td><td>M. C. W. Dulick et al. (2003)</td><td>0.67</td><td>100</td><td>5000</td></tr><tr><td>LiH</td><td>CLT</td><td>C. M. Coppola et al. (2011)</td><td>0.5</td><td>100</td><td>2000</td></tr><tr><td>MgH</td><td>MoLLIST</td><td>A. Owens et al. (2022)</td><td>0.34</td><td>100</td><td>2000</td></tr><tr><td>NaH</td><td>ExoMol Rivlin</td><td>T. Rivlin et al. (2015)</td><td>0.27</td><td>100</td><td>7000</td></tr><tr><td>ScH</td><td>LYT</td><td>K. L. Chubb et al. (2021)</td><td>0.63</td><td>100</td><td>2000</td></tr><tr><td>TiH</td><td>MoLLIST</td><td>A. Burrows et al. (2005)</td><td>0.42</td><td>100</td><td>5000</td></tr><tr><td>CH</td><td>MoLLIST</td><td>T. Masseron et al. (2014)</td><td>0.26</td><td>100</td><td>5000</td></tr><tr><td rowspan="5">Other diatomics</td><td>HBr</td><td>HITRAN</td><td>G. Li et al. (2013)</td><td>0.62</td><td>100</td><td>5000</td></tr><tr><td>HCl</td><td>HITRAN</td><td>G. Li et al. (2013)</td><td>0.49</td><td>100</td><td>5000</td></tr><tr><td>HF</td><td>HITRAN</td><td>G. Li et al. (2013)</td><td>0.31</td><td>100</td><td>5000</td></tr><tr><td>HI</td><td>HITRAN</td><td>G. Li et al. (2013)</td><td>0.71</td><td>100</td><td>5000</td></tr><tr><td>NH</td><td>MoLLIST</td><td>A. M. Fernando et al. (2018)</td><td>0.59</td><td>100</td><td>5000</td></tr><tr><td rowspan="5">AlCl</td><td>OH</td><td>MoLLIST</td><td>J. N. Hodges & P. F. Bernath (2017)</td><td>0.23</td><td>100</td><td>5000</td></tr><tr><td>PH</td><td>ExoMol LaTY</td><td>J. Langleben et al. (2019)</td><td>0.41</td><td>100</td><td>4000</td></tr><tr><td>SiH</td><td>ExoMol SiGHTLY</td><td>S. N. Yurchenko et al. (2017b)</td><td>0.32</td><td>100</td><td>5000</td></tr><tr><td>SH</td><td>ExoMol GYT</td><td>M. N. Gorman et al. (2019)</td><td>0.26</td><td>100</td><td>5000</td></tr><tr><td>AlF</td><td>MoLLIST</td><td>P. F. Bernath (2020)</td><td>4.26</td><td>100</td><td>5000</td></tr><tr><td rowspan="5">C<sub>2</sub></td><td>MoLLIST</td><td>P. F. Bernath (2020)</td><td>2.58</td><td>100</td><td>5000</td></tr><tr><td>ExoMol 8states</td><td>S. N. Yurchenko et al. (2018c)</td><td>0.21</td><td>100</td><td>5000</td></tr><tr><td>CaF</td><td>MoLLIST</td><td>S. Hou & P. F. Bernath (2018)</td><td>1.79</td><td>100</td><td>5000</td></tr><tr><td>CN</td><td>MoLLIST</td><td>A.-M. Syme & L. K. McKemmish (2021)</td><td>0.23</td><td>100</td><td>5000</td></tr><tr><td>CP</td><td>MoLLIST</td><td>Z. Qin et al. (2021)</td><td>0.67</td><td>100</td><td>5000</td></tr><tr><td rowspan="5">Si</td><td>CS</td><td>ExoMol JnK</td><td>G. Paulose et al. (2015)</td><td>0.91</td><td>100</td><td>3000</td></tr><tr><td>H<sub>2</sub></td><td>RACPPK</td><td>E. Roueff et al. (2019)</td><td>0.28</td><td>100</td><td>5000</td></tr><tr><td>KCl</td><td>ExoMol Barton</td><td>E. J. Barton et al. (2014)</td><td>3.45</td><td>100</td><td>3000</td></tr><tr><td>KF</td><td>MoLLIST</td><td>D. J. Frohman et al. (2016)</td><td>2.49</td><td>100</td><td>5000</td></tr><tr><td>LiCl</td><td>MoLLIST</td><td>D. M. Bittner & P. F. Bernath (2018)</td><td>2.07</td><td>100</td><td>5000</td></tr><tr><td rowspan="5">NaCl</td><td>LiF</td><td>MoLLIST</td><td>D. M. Bittner & P. F. Bernath (2018)</td><td>5.52</td><td>100</td><td>5000</td></tr><tr><td>MgF</td><td>MoLLIST</td><td>S. Hou & P. F. Bernath (2017)</td><td>1.83</td><td>100</td><td>5000</td></tr><tr><td>NaF</td><td>ExoMol Barton</td><td>E. J. Barton et al. (2014)</td><td>4.00</td><td>100</td><td>3000</td></tr><tr><td>NS</td><td>MoLLIST</td><td>D. J. Frohman et al. (2016)</td><td>2.01</td><td>100</td><td>5000</td></tr><tr><td>PN</td><td>ExoMol SNaSH</td><td>S. N. Yurchenko et al. (2018b)</td><td>0.26</td><td>100</td><td>5000</td></tr><tr><td rowspan="3">PS</td><td>ExoMol YYLT</td><td>L. Yorke et al. (2014)</td><td>1.54</td><td>100</td><td>5000</td></tr><tr><td>ExoMol POPS</td><td>L. Prajapat et al. (2017)</td><td>0.27</td><td>100</td><td>5000</td></tr><tr><td>SiS</td><td>ExoMol UCTY</td><td>A. Upadhyay et al. (2018)</td><td>2.70</td><td>100</td><td>5000</td></tr></tbody></table>
<table><caption>**Table 1**
(Continued)</caption><thead><tr><th>Group</th><th>Species</th><th>Line List</th><th>References</th><th>λ<sub>l</sub><br>(μm)</th><th>λ<sub>u</sub><br>(μm)</th><th>T<sub>max</sub><br>(K)</th></tr></thead><tbody><tr><td rowspan="8">Larger molecules</td><td>C<sub>2</sub>H<sub>2</sub></td><td>ExoMol aCeTY</td><td>K. L. Chubb et al. (2020)</td><td>1.00</td><td>100</td><td>2200</td></tr><tr><td>C<sub>2</sub>H<sub>4</sub></td><td>ExoMol MaYTY</td><td>B. P. Mant et al. (2018)</td><td>1.41</td><td>100</td><td>700</td></tr><tr><td>CH<sub>3</sub></td><td>ExoMol AYYJ</td><td>A. Y. Adam et al. (2019)</td><td>1.00</td><td>100</td><td>1500</td></tr><tr><td>CH<sub>3</sub>Cl</td><td>ExoMol OYT</td><td>A. Owens et al. (2018)</td><td>1.56</td><td>100</td><td>1200</td></tr><tr><td>CH<sub>4</sub></td><td>ExoMol 34to10</td><td>S. N. Yurchenko et al. (2017a)</td><td>0.56</td><td>100</td><td>2000</td></tr><tr><td>NH<sub>3</sub></td><td>ExoMol CoYuTe</td><td>A. R. Al Derzi et al. (2015)</td><td>0.5</td><td>100</td><td>1500</td></tr><tr><td>PH<sub>3</sub></td><td>ExoMol SAITY</td><td>C. Sousa-Silva et al. (2014)</td><td>1.00</td><td>100</td><td>1500</td></tr><tr><td>SiH<sub>4</sub></td><td>ExoMol OY2T</td><td>A. Owens et al. (2017)</td><td>2.00</td><td>100</td><td>1200</td></tr><tr><td rowspan="5">Molecular ions</td><td>H<sub>3</sub><sup>+</sup></td><td>ExoMol MiZATeP</td><td>I. I. Mizus et al. (2017)</td><td>0.4</td><td>100</td><td>5000</td></tr><tr><td>H<sub>3</sub>O<sup>+</sup></td><td>ExoMol eXeL</td><td>S. N. Yurchenko et al. (2020b)</td><td>1.00</td><td>100</td><td>1500</td></tr><tr><td>HeH<sup>+</sup></td><td>ADJSAAM</td><td>P. H. R. Amaral et al. (2019)</td><td>0.67</td><td>100</td><td>4000</td></tr><tr><td>LiH<sup>+</sup></td><td>CLT</td><td>C. M. Coppola et al. (2011)</td><td>10.87</td><td>100</td><td>2000</td></tr><tr><td>OH<sup>+</sup></td><td>MoLLIST</td><td>J. N. Hodges & P. F. Bernath (2017)</td><td>0.33</td><td>100</td><td>5000</td></tr><tr><td rowspan="2">Alkali-neutral atoms</td><td>K</td><td>NIST</td><td>J. Coursey et al. (2015)</td><td>0.29</td><td>100</td><td>5000</td></tr><tr><td>Na</td><td>NIST</td><td>J. Coursey et al. (2015)</td><td>...</td><td>...</td><td>5000</td></tr><tr><td rowspan="5">Other atoms</td><td>C N O</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Ne Na Mg</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Al Si S</td><td>Opacity Project</td><td>M. J. Seaton et al. (1994)</td><td>...</td><td>...</td><td>10<sup>8</sup></td></tr><tr><td>Ar Ca Cr</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Mn Fe Ni</td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>

Note. For each species, $\lambda_l$ and $\lambda_u$ denote the minimum and maximum wavelength of the corresponding line list; $T_{max}$ is the highest temperature available.

<figure><img src="image_8.png" /><figcaption>**Figure 6.** Comparison of the widths of Doppler line cores vs. Lorentzian profiles in the pressure-temperature diagram according to the analysis of C. Hedges & N. Madhusudhan (2016). The boundary between the two regimes is denoted by the white line. Above the horizontal black line, molecular transitions are treated with a Voigt profile extracted from the ExoMolOP database (K. L. Chubb et al. 2021).</figcaption></figure>

Finally, Figure 9 shows a map of the differences in $\log \kappa_R$ when the IPD effect is taken into account or neglected. The major consequence of IPD is increasing H and $H_2^+$ ionization, which results in higher Rosseland mean opacity for $\log(T) > 4.1$ and $\log(R) > 1$ (red area).

## 4.2. Pressure-broadening Effects on Mean Gas Opacity

As thoroughly discussed in several studies (e.g., R. S. Freedman et al. 2008; C. Helling & W. Lucas 2009; M. G. Malygin et al. 2014) mean gas opacities without a dust continuum contribution

have several astrophysical applications. They are important, for example, in a dust-depleted low-metallicity medium or when the equilibrium temperature exceeds the local dust sublimation temperature. As a result, gas opacities can be relevant in describing the inner regions of accretion disks (J. Muzerolle et al. 2004), calculating the energy balance of Type Ia supernovae (L. Dessart et al. 2014), estimating the cooling of nonaccreting hot white dwarfs (R. D. Rohrmann et al. 2012), quantifying stellar feedback processes in the interstellar medium (F. I. Pelupessy & P. P. Papadopoulos 2009), and simulating star and planet formation (C. Helling & W. Lucas 2009).

To assess the impact of pressure broadening on Rosseland mean gas opacities, we performed two independent runs of ÆSOPUS, one adopting the thermal Doppler plus microturbulent velocity molecular line profiles, and the other assuming the Lorentzian molecular line profiles that depend on both temperature and pressure.

The results are illustrated in Figure 10. We note that the dynamical range of $\kappa_R$ is extremely broad, spanning ~14 orders of magnitude, making eye comparison somewhat difficult. Nonetheless, we reckon it is useful to show the opacity trends in the two cases. To quantify the differences in $\kappa_R$, we create a map that spans the whole area of the table, as shown in the right panel of Figure 11. In Section 3.1, the IPD effect has already been discussed. To help the discussion, in the left panel, we also draw a map of the gas pressure, indicating the contour level (white line) above which we begin to consider pressure broadening of molecular transitions.

### 4.2.1. The Opacity Significance of Neutral Alkalis Na and K

The contribution of alkali atoms in atmospheric opacity of cool substellar objects was initially established by studying the
<figure><img src="image_9.png" /><figcaption>**Figure 7.** Comparison of two broadening profiles used for monochromatic absorption cross sections: the Lorentzian profile produced by pressure and perturbing species (blue line) and the Gaussian thermal Doppler plus microturbulent velocity profile (magenta line; see Equation (5)). Left panel: water vapor; right panel: methane. See Table 11 for details. Values for pressure and temperature are labeled.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Figure 8.** Properties of the Rosseland mean opacity toward the highest temperature of our tables, at high density for $log(R) = 6$. The chemical composition is defined by $X = 0.7$, $Z = 0.01$, with a scaled solar chemical mixture following E. Magg et al. (2022). Each curve corresponds to $log(κ_R) - log(κ_R^{off})$, where $κ_R$ is the full opacity, including all opacity sources considered here, and $κ_R^{off}$ denotes the reduced opacity obtained by excluding the specific absorbing species. This displays the temperature window to which a given opacity source contributes the most. The IPD effect is ignored in the computations of the left panel, whereas it is incorporated in the results of the right panel. In both cases, we apply thermal Doppler plus microturbulent velocity profiles for molecular transitions.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Figure 9.** Map of the differences in Rosseland mean opacities when the IPD effect is included or ignored at high density. Contour levels are distributed every 0.025 dex in $\Delta log(κ_R)$.</figcaption></figure>

far-red spectra of T dwarfs (A. Burrows et al. 2000). Atomic pressure-broadened lines, especially those of Na and K, are major opacity sources over certain spectral ranges, temperatures, and densities (R. S. Freedman et al. 2008). With their large absorption cross sections at near-infrared and far-red wavelengths, sodium and potassium fill what would otherwise be a spectral region of relatively low opacity.

The left panel of Figure 12 compares the total molecular opacity (magenta line) as a function of wavenumber for a gas temperature of 1585 K and a gas pressure of 211 bar, with a calculation that does not account for alkali opacity (blue line). Above about a wavenumber of 10,000 cm$^{-1}$, the alkali opacity plays a significant role in determining the total summed opacity. The resonance Na doublet at $\approx 17,000$ cm$^{-1}$ stands out as a prominent source of absorption. The right panel of Figure 12 compares the Rosseland mean opacity with and without the contribution of alkali metals at various densities. It is evident that the alkali opacity fills in the opacity minimum from about 1000 to 3200 K at higher densities. Na and K lines play a much smaller role at lower densities, so the differences are minor.
<figure><img src="image_12.png" /><figcaption>**Figure 10.** Rosseland mean gas opacities computed in the temperature range $100 \le T/K \le 30,000$ and encompassing the $R$ interval $-8 \le \log(R) \le 6$ in steps of 0.5 dex. The chemical composition is defined by $X = 0.7$, $Z = 0.01$, with a scaled solar chemical mixture following E. Magg et al. (2022). Left panel: opacities assuming Doppler plus microturbulent velocity molecular line profiles and ignoring the IPD effect. Right panel: opacities assuming pressure broadening for molecular line profiles and accounting for the IPD effect.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>**Figure 11.** Left panel: map of the gas pressure as a function of $T$ and $R$. Contour levels are distributed every 1 dex in $\log(P_{\text{gas}})$. The white contour draws the locus of points where the logarithmic difference in Rosseland mean opacity treated with a thermal Doppler profile and pressure broadening equals 0.001 dex in the regime where molecular transitions are important, for $T \le 4000$ K. Right panel: map of the differences in Rosseland mean opacities between computations that include the IPD effect and pressure broadening for molecular transitions and those that ignore the IPD and assume Doppler broadening regardless of pressure. Contour levels are distributed every 0.05 dex in $\Delta \log(\kappa_{\text{R}})$. The chemical composition is the same as in Figure 10.</figcaption></figure>

### 4.2.2. Comparison with Other Authors

To test our results, we consider the Rosseland mean gas opacities computed by M. G. Malygin et al. (2014) and R. S. Freedman et al. (2014). In M. G. Malygin et al. (2014) work Kurucz's CD-ROMs were used to extract the line and continuum opacity data (R. L. Kurucz 1993), and the mean opacities were calculated using the publicly available DFSYNTHE code (F. Castelli 2005). The results are presented in the left panel of Figure 13 for a temperature range of 700 K–10,000 K. The agreement between the two opacity sets is good for $\log(T) > 3.6$. At lower temperatures, some deviations start to appear, most likely due to different line lists for atoms and molecules, as well as different line pressure broadening treatments. The most prominent discrepancy shows up at $\log(T) < 3.3$. While our opacity results remain either relatively flat or even increase for the highest $R$ values, M. G. Malygin et al. (2014) opacities do, in fact, significantly decrease at lower temperatures. For example, our opacity curve for $\log(R) = -7$ only includes thermal Doppler profiles for the molecular transitions, as pressure is irrelevant in

this case. Despite this, the decrease at lower temperatures is much less pronounced than in M. G. Malygin et al. (2014). One plausible explanation is that we rely on molecular line lists that extend down to 100 K in temperature, whereas in Kurucz opacity, the contributions of molecules cease for $T < 1995$ K, and therefore extrapolations below that limit may be inaccurate.

In comparison to R. S. Freedman et al. (2014) results shown in the right panel of Figure 13, there are significant differences in Rosseland mean opacities. We found no explicit information about pressure broadening for molecular transitions, and molecular absorption is limited to 12 species. Conversely, we consider 80 absorbing molecules in our study. Both facts could explain the disparity in outcomes.

## 5. Rosseland Mean Opacities with Solid Grains

We present a few examples of Rosseland mean opacities with solid grains included. The dust prescriptions are the same as in P. Marigo et al. (2023). We are aware that dust clouds in brown dwarfs and planets are crucial for understanding their atmospheric
<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /><figcaption>**Figure 12.** Left panel: molecular gas opacity as a function of wavenumber at $T = 1585$ K, gas pressure $P_{\text{gas}} = 211$ bar, and density $\rho = 0.004$ g cm$^{-3}$. The reference solar chemical composition is taken from E. Magg et al. (2022), with metallicity $Z = 0.02$ and hydrogen abundance $X = 0.7$. The blue curve represents the monochromatic opacity without the contribution of alkali atoms Na and K, whereas the magenta curve includes the two atoms’ opacity contributions. Right panel: the Rosseland mean opacity as a function of temperature at three densities (g cm$^{-3}$). Magenta lines include the opacity from alkali atoms Na and K, whereas dashed blue lines do not. The chemical composition is scaled solar according to E. Magg et al. (2022), with metallicity $Z = 0.01$ and hydrogen abundance $X = 0.7$.</figcaption></figure>

<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /><figcaption>**Figure 13.** Comparison of Rosseland mean gas opacities computed in this work and by other studies. Left panel: Comparison with M. G. Malygin et al. (2014). The reference solar chemical composition is taken from N. Grevesse & A. J. Sauval (1998), with metallicity $Z = 0.01696$ and hydrogen abundance $X = 0.7347$. Right panel: Comparison with R. S. Freedman et al. (2014). The reference solar chemical composition is taken from K. Lodders (2003), with metallicity $Z = 0.0133$ and hydrogen abundance $X = 0.7491$. Values for $\log(R)$ are labeled.</figcaption></figure>

properties, formation mechanisms, and overall behavior, providing valuable insights into the broader field of planetary science. Dust clouds can significantly impact their atmospheric composition. They often consist of various particles, including silicates, iron, and other compounds, contributing to the chemical makeup of the atmosphere. In this context, important contributions were provided by T. Tsuji et al. (1996), A. Burrows et al. (2000), A. S. Ackerman & M. S. Marley (2001), T. Tsuji (2002), C. M. Sharp & A. Burrows (2007), C. Helling et al. (2008), S. Witte et al. (2009), F. Allard et al. (2012), D. Juncher et al. (2017), P. Woitke et al. (2020). Because our primary interest is in very low-mass stars, in this work we do not take into account the formation of dust clouds, and we postpone the effort to a future study.

Figure 14 compares major solid species at low and high densities that contribute most to the opacity. There are significant differences between the two regimes.

First, we notice that the corundum opacity bump that appears for $\log(R) = -3$ at temperatures ranging from 1500 to 1200 K is missing for $\log(R) = 6$. Al$_2$O$_3$ does not condense at high-pressure conditions. Even at $\log(R) = -3$, where corundum contributes to opacity, molecular band absorption by water

molecules continues to play a significant role in opacity. This extends down to around 400 K. In contrast, for $\log(R) = 6$, the opacity contribution of water vapor is significantly reduced. Another major distinction is that at high density, solid iron is the dominant opacity source from $\approx 1870$ to $650$ K, whereas at lower temperatures, silicates begin to prevail. Furthermore, we see that amorphous carbon does not condense, whereas Troilite (FeS) has a discernible contribution, from about 700 to 380 K.

Figure 15 shows the behavior of Rosseland mean opacities over the range $-8 \le \log(R) \le 6$. As previously discussed, we see that the corundum opacity bump (at $3.08 \le \log(T) \le 3.17$) is present for $\log(R) < 2$, while for $\log(R) > 2$ the species does not condense and solid iron makes the most important opacity contribution for $650 \le T/K \le 1870$ ($2.81 \le \log(T) \le 3.27$).

## 6. Impact on Stellar Models

Figure 2 suggests that the updated Rosseland mean opacities could have a significant impact on stellar models of masses below ~0.7 $M_{\odot}$, particularly influencing the temperature gradient in their superadiabatic regions. This conjecture holds true, as illustrated in Figure 16, depicting the consequences of
<figure><img src="image_18.png" /><figcaption>**Figure 14.** Properties of the Rosseland mean opacity for temperatures where solid grains dominate, at low density ($\log(R) = -3$, left panel) and high density ($\log(R) = 6$, right panel). The chemical composition is defined by $X = 0.735$, $Z = 0.0165$, with scaled solar elemental abundances following E. Magg et al. (2022). Each curve represents the contribution of major solid species to the total Rosseland mean opacity, and it is calculated as $\log(\kappa_R) - \log(\kappa_R^{\text{off}})$, where $\kappa_R$ is the full opacity including all opacity sources considered here, and $\kappa_R^{\text{off}}$ is the reduced opacity computed by excluding the specific absorbing species.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>**Figure 15.** Rosseland mean opacities with the inclusion of condensed dust grains. The chemical composition is the same as in Figure 14. The curves are distributed every 1 dex in $\log(R)$, in the interval $-8 \le \log(R) \le 6$.</figcaption></figure>

employing the revised opacities on the fundamental properties of PARSEC models within the mass range of 0.1 to 0.85 $M_{\odot}$. These results are compared with models utilizing the prior version of ÆSOPUS opacities (specifically, ÆSOPUS v2.0 from P. Marigo et al. 2022). In this latter case, the opacities were not readily available for $\log(R)$ values exceeding 1, prompting PARSEC to use the opacity values taken at the border of the available tables. This assumption was grounded in the expectation that, in the extensive convective regions of such stars, the temperature gradient would approach adiabatic conditions, rendering it largely insensitive to variations in the opacity tables.

The models in Figure 16 are computed with PARSEC code version 2.0 (G. Costa et al. 2019; C. T. Nguyen et al. 2022) and adopting $T-\tau$ relations interpolated from the PHOENIX stellar atmosphere models (F. Allard et al. 2012), implemented and discussed in Y. Chen et al. (2014). Here, suffice it to recall that we use the solar composition from E. Caffau et al. (2011)<fnref n="9" /> and

the mixing-length theory with a parameter $\alpha = 1.74$ derived from the calibration of the Solar model (A. Bressan et al. 2012). We present models either using (labeled with S) or not using the shift in the $T-\tau$ relation advocated by Y. Chen et al. (2014). Further details will be discussed in a subsequent paper dedicated to very low-mass star models.

Since low-mass models evolve minimally after settling into their main sequences, only their properties at the age of 5 Gyr are presented in Figure 16. From the Hertzsprung-Russell (HR) diagram, it is evident that the use of new opacities consistently produces cooler and fainter models. Furthermore, the impact of the new opacities diminishes as we approach stellar models with a mass of 0.6 $M_{\odot}$, as expected for stellar structures that predominantly evolve in the range of $\log(R) < 1$ (refer to Figure 2).

The right panel of Figure 16 displays the same models in the mass-radius plane. In this instance, the empirical data derived from double-lined eclipsing binary catalogs is superimposed. As observed, our PARSEC models computed with ÆSOPUS v2.0 opacities generally align with the lower limit of the empirical mass-radius relation. This once again highlights the ongoing manifestation of the mass-radius discrepancy, extensively documented in various studies (see, for example, Y. Chen et al. 2014; G. Torres et al. 2014; G. Somers et al. 2020). It is evident that the use of the new ÆSOPUS v2.1 opacities somewhat mitigates this discrepancy by slightly inflating all models in the 0.1-0.6 $M_{\odot}$ range. Therefore, we advocate the use of proper opacity tables extended to high densities and pressures to describe very low-mass models.

Figure 17 shows the relative differences between the observed and model radii, for the same models and data as in Figure 16, so as to allow a better visualization of the small improvements reached by using ÆSOPUS v2.1 instead of v2.0 opacities. An additional panel presented additional models, labeled ÆSOPUS 2.1 S, in which the $T-\tau$ relation is shifted by just half the amount calibrated in Y. Chen et al. (2014)—that is, these models adopt a shift of $\Delta \log(T/T_{\text{eff}}) = 0.03$ dex for $\log(T_{\text{eff}}/\text{K}) < 3.5$, and gradually reduce this shift to 0 for $\log(T_{\text{eff}}/\text{K})$ between 3.5 and 3.765. These latter models practically cancel out the discrepancy in the mass-radius relation at low masses.

<footnote n="9">⁹ In this case, the E. Caffau et al. (2011) composition is used in PARSEC and in the opacity tables but not in the PHOENIX atmosphere models, which are based on the M. Asplund et al. (2009) solar composition. Obtaining full consistency between all these components is beyond our reach at the moment.</footnote>
<figure><img src="image_20.png" /><figcaption>**Figure 16.** The impact of our new Rosseland mean opacities on low-mass stellar models computed with the PARSEC v2.0 code. The left panel shows the H-R diagram for models in the mass range from 0.1 to 0.85 $M_{\odot}$ and computed at intervals of 0.05 $M_{\odot}$ (from bottom right to top left) and plotted at ages of 5 Gyr. The blue symbols indicate models computed with the previous version of the opacity tables from P. Marigo et al. (2022), while the red and green symbols are for models computed with present opacity tables. The green symbols correspond to models with a shift in their $T-\tau$ relation (see text for details). The right panel shows the same models in the mass-radius plane, now compared with the empirical data of low-mass stars in double-lined eclipsing binary catalogs from D. Ségransan et al. (2003), B. O. Demory et al. (2009), G. Torres et al. (2010), J. A. Carter et al. (2011), L. R. Doyle et al. (2011), A. L. Kraus et al. (2011), S. G. Parsons et al. (2012a), S. G. Parsons et al. (2012b), J. Southworth (2015).</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>**Figure 17.** The relative difference between the observed and model radii for the three sets of models and the data presented in the right panel of Figure 16 as a function of stellar mass. It can be seen that while our new opacities reduce the systematic discrepancies between the predicted and observed radii for stars in the $M < 0.6 M_{\odot}$ range, a shift in the $T-\tau$ relation (or other alternatives as discussed in the text) seems to still be necessary to eliminate it completely.</figcaption></figure>

Therefore, while our new ÆSOPUS v2.1 opacities reduce the systematic discrepancies between the predicted and observed radii of very low-mass stars, additional model assumptions—

like the shift in the $T-\tau$ relation by Y. Chen et al. (2014) or the stellar spots by G. Somers et al. (2020)—seem to still be necessary to eliminate them completely.
## 7. Concluding Remarks

We compute the equation of state and provide Rosseland mean opacity tables for temperatures ranging from 32,000 to 100 K. These tables are expected to be useful for a series of applications, which we leave to the readers to explore. Tables for different chemical compositions can be retrieved with the updated *ÆSOPUS* web interface<fnref n="10" /> where one will also find links to a set of precomputed tables for 10 popular chemical mixtures (i.e., E. Anders & N. Grevesse 1989; N. Grevesse & A. Noels 1993; N. Grevesse & A. J. Sauval 1998; H. Holwe-ger 2001; K. Lodders 2003; N. Grevesse et al. 2007; M. Asplund et al. 2009; E. Caffau et al. 2011; M. Asplund et al. 2021, and E. Magg et al. 2022), and in each case spanning wide ranges in Z and X values. We have so far verified that these new opacity tables lead to important changes in the modeling of very low-mass stars at near-solar metallicities, causing shifts in their mass-effective temperature and mass-luminosity rela-tions. More modest shifts are present in the mass-radius relation, and they go in the right direction to alleviate the discrepancies in the radii of very low-mass stars that have been widely reported in the literature.

## Acknowledgments

We acknowledge support from Padova University through the research project PRD 2021. P.M. and A.B. acknowledge the Italian Ministerial grant PRIN2022, “Radiative opacities for astrophysical applications,” No. 2022NEXMP8. G.C. acknowledges support from the Agence Nationale de la Recherche grant POPSYCLE No. ANR-19-CE31-0022. L.G. acknowledges partial support from an INAF Theory grant in 2022.

*Software: ÆSOPUS* (P. Marigo & B. Aringer 2009; P. Marigo et al. 2022), *EXOCROSS* (S. N. Yurchenko et al. 2018a), *DIANA* (P. Woitke et al. 2016), *GGchem* (P. Woitke et al. 2018).

# Appendix

## Chemical Species Considered in ÆSOPUS

For completeness, Table 2 presents the complete list of chemical species considered at the moment in the equation of state of ÆSOPUS 2.1.

<footnote n="10">¹⁰ The long-lasting link for the ÆSOPUS 2.1 web interface is http://stev.oapd.inaf.it/cgi-bin/aesopus_2.1. We recall that the previous versions of ÆSOPUS are still available in their original locations at http://stev.oapd.inaf.it/cgi-bin/aesopus_1.0 and http://stev.oapd.inaf.it/cgi-bin/aesopus_2.0.</footnote>
<table><caption>**Table 2**

Chemical Species Considered in the Equation of State of *ÆSOPUS*</caption><thead><tr><th></th><th>H<sup>+</sup></th><th>He</th><th>He<sup>+</sup></th><th>He<sup>++</sup></th><th>Li</th><th>Li<sup>+</sup></th><th>Li<sup>++</sup></th><th>Be</th><th>Be<sup>+</sup></th><th>Be<sup>++</sup></th><th>B</th><th>B<sup>+</sup></th></tr></thead><tbody><tr><td>H</td><td>H<sup>+</sup></td><td>He</td><td>He<sup>+</sup></td><td>He<sup>++</sup></td><td>Li</td><td>Li<sup>+</sup></td><td>Li<sup>++</sup></td><td>Be</td><td>Be<sup>+</sup></td><td>Be<sup>++</sup></td><td>B</td><td>B<sup>+</sup></td></tr><tr><td>B<sup>++</sup></td><td>C</td><td>C<sup>+</sup></td><td>C<sup>++</sup></td><td>C<sup>+++</sup></td><td>C<sup>+++</sup></td><td>N</td><td>N<sup>+</sup></td><td>N<sup>+++</sup></td><td>N<sup>+++</sup></td><td>N<sup>+++</sup></td><td>O</td><td>O<sup>+</sup></td></tr><tr><td>O<sup>++</sup></td><td>O<sup>+++</sup></td><td>O<sup>+++</sup></td><td>O<sup>++++</sup></td><td>O<sup>++++</sup></td><td>F</td><td>F<sup>+</sup></td><td>F<sup>++</sup></td><td>F<sup>+++</sup></td><td>F<sup>+++</sup></td><td>Ne</td><td>Ne<sup>+</sup></td><td>Ne<sup>++</sup></td></tr><tr><td>Ne<sup>++++</sup></td><td>Ne<sup>++++</sup></td><td>Na</td><td>Na<sup>+</sup></td><td>Na<sup>++</sup></td><td>Na<sup>+++</sup></td><td>Na<sup>+++</sup></td><td>Mg</td><td>Mg<sup>+</sup></td><td>Mg<sup>++</sup></td><td>Mg<sup>+++</sup></td><td>Mg<sup>++++</sup></td><td>Al</td></tr><tr><td>Al<sup>+</sup></td><td>Al<sup>++</sup></td><td>Al<sup>+++</sup></td><td>Al<sup>++++</sup></td><td>Si</td><td>Si<sup>+</sup></td><td>Si<sup>++</sup></td><td>Si<sup>+++</sup></td><td>Si<sup>+++</sup></td><td>P</td><td>P<sup>+</sup></td><td>P<sup>++</sup></td><td>P<sup>+++</sup></td></tr><tr><td>P<sup>++++</sup></td><td>S</td><td>S<sup>+</sup></td><td>S<sup>++</sup></td><td>S<sup>+++</sup></td><td>S<sup>+++</sup></td><td>Cl</td><td>Cl<sup>+</sup></td><td>Cl<sup>++</sup></td><td>Cl<sup>+++</sup></td><td>Cl<sup>+++</sup></td><td>Ar</td><td>Ar<sup>+</sup></td></tr><tr><td>Ar<sup>++</sup></td><td>Ar<sup>+++</sup></td><td>Ar<sup>++++</sup></td><td>K</td><td>K<sup>+</sup></td><td>K<sup>++</sup></td><td>K<sup>+++</sup></td><td>K<sup>++</sup></td><td>K<sup>+++</sup></td><td>Ca</td><td>Ca<sup>+</sup></td><td>Ca<sup>++</sup></td><td>Ca<sup>+++</sup></td></tr><tr><td>Sc</td><td>Sc<sup>+</sup></td><td>Sc<sup>++</sup></td><td>Sc<sup>+++</sup></td><td>Sc<sup>++++</sup></td><td>Ti</td><td>Ti<sup>+</sup></td><td>Ti<sup>++</sup></td><td>Ti<sup>+++</sup></td><td>Ti<sup>+++</sup></td><td>V</td><td>V<sup>+</sup></td><td>V<sup>++</sup></td></tr><tr><td>V<sup>+++</sup></td><td>V<sup>++++</sup></td><td>Cr</td><td>Cr<sup>+</sup></td><td>Cr<sup>++</sup></td><td>Cr<sup>+++</sup></td><td>Cr<sup>++++</sup></td><td>Mn</td><td>Mn<sup>+</sup></td><td>Mn<sup>++</sup></td><td>Mn<sup>+++</sup></td><td>Mn<sup>++++</sup></td><td>Fe</td></tr><tr><td>Fe<sup>+</sup></td><td>Fe<sup>++</sup></td><td>Fe<sup>+++</sup></td><td>Fe<sup>++++</sup></td><td>Co</td><td>Co<sup>+</sup></td><td>Co<sup>++</sup></td><td>Co<sup>+++</sup></td><td>Co<sup>++++</sup></td><td>Ni</td><td>Ni<sup>+</sup></td><td>Ni<sup>++</sup></td><td>Ni<sup>+++</sup></td></tr><tr><td>Ni<sup>++++</sup></td><td>Cu</td><td>Cu<sup>+</sup></td><td>Cu<sup>++</sup></td><td>Zn</td><td>Zn<sup>+</sup></td><td>Zn<sup>++</sup></td><td>Ga</td><td>Ga<sup>+</sup></td><td>Ga<sup>++</sup></td><td>Ge</td><td>Ge<sup>+</sup></td><td>Ge<sup>++</sup></td></tr><tr><td>As</td><td>As<sup>+</sup></td><td>As<sup>++</sup></td><td>Se</td><td>Se<sup>+</sup></td><td>Se<sup>++</sup></td><td>Br</td><td>Br<sup>+</sup></td><td>Br<sup>++</sup></td><td>Kr</td><td>Kr<sup>+</sup></td><td>Kr<sup>++</sup></td><td>Rb</td></tr><tr><td>Rb<sup>+</sup></td><td>Rb<sup>++</sup></td><td>Sr</td><td>Sr<sup>+</sup></td><td>Sr<sup>++</sup></td><td>Y</td><td>Y<sup>+</sup></td><td>Y<sup>++</sup></td><td>Zr</td><td>Zr<sup>+</sup></td><td>Zr<sup>++</sup></td><td>Nb</td><td>Nb<sup>+</sup></td></tr><tr><td>Nb<sup>++</sup></td><td>Mo</td><td>Mo<sup>+</sup></td><td>Mo<sup>++</sup></td><td>Tc</td><td>Tc<sup>+</sup></td><td>Tc<sup>++</sup></td><td>Ru</td><td>Ru<sup>+</sup></td><td>Ru<sup>++</sup></td><td>Rh</td><td>Rh<sup>+</sup></td><td>Rh<sup>++</sup></td></tr><tr><td>Pd</td><td>Pd<sup>+</sup></td><td>Pd<sup>++</sup></td><td>Ag</td><td>Ag<sup>+</sup></td><td>Ag<sup>++</sup></td><td>Cd</td><td>Cd<sup>+</sup></td><td>Cd<sup>++</sup></td><td>In</td><td>In<sup>+</sup></td><td>In<sup>++</sup></td><td>Sn</td></tr><tr><td>Sn<sup>+</sup></td><td>Sn<sup>++</sup></td><td>Sb</td><td>Sb<sup>+</sup></td><td>Sb<sup>++</sup></td><td>Te</td><td>Te<sup>+</sup></td><td>Te<sup>++</sup></td><td>I</td><td>I<sup>+</sup></td><td>I<sup>++</sup></td><td>Xe</td><td>Xe<sup>+</sup></td></tr><tr><td>Xe<sup>++</sup></td><td>Cs</td><td>Cs<sup>+</sup></td><td>Cs<sup>++</sup></td><td>Ba</td><td>Ba<sup>+</sup></td><td>Ba<sup>++</sup></td><td>La</td><td>La<sup>+</sup></td><td>La<sup>++</sup></td><td>Ce</td><td>Ce<sup>+</sup></td><td>Ce<sup>++</sup></td></tr><tr><td>Pr</td><td>Pr<sup>+</sup></td><td>Nd</td><td>Nd<sup>+</sup></td><td>Nd<sup>+</sup></td><td>Nd<sup>++</sup></td><td>Pm</td><td>Pm<sup>+</sup></td><td>Pm<sup>++</sup></td><td>Sm</td><td>Sm<sup>+</sup></td><td>Sm<sup>++</sup></td><td>Eu</td></tr><tr><td>Eu<sup>+</sup></td><td>Eu<sup>++</sup></td><td>Gd</td><td>Gd<sup>+</sup></td><td>Gd<sup>++</sup></td><td>Tb</td><td>Tb<sup>+</sup></td><td>Tb<sup>++</sup></td><td>Dy</td><td>Dy<sup>+</sup></td><td>Dy<sup>++</sup></td><td>Ho</td><td>Ho<sup>+</sup></td></tr><tr><td>Ho<sup>++</sup></td><td>Er</td><td>Er<sup>+</sup></td><td>Er<sup>++</sup></td><td>Tm</td><td>Tm<sup>+</sup></td><td>Tm<sup>++</sup></td><td>Yb</td><td>Yb<sup>+</sup></td><td>Yb<sup>++</sup></td><td>Lu</td><td>Lu<sup>+</sup></td><td>Lu<sup>++</sup></td></tr><tr><td>Hf</td><td>Hf<sup>+</sup></td><td>Hf<sup>++</sup></td><td>Ta</td><td>Ta<sup>+</sup></td><td>Ta<sup>++</sup></td><td>W</td><td>W<sup>+</sup></td><td>W<sup>++</sup></td><td>Re</td><td>Re<sup>+</sup></td><td>Re<sup>++</sup></td><td>O<sub>2</sub></td></tr><tr><td>Os<sup>+</sup></td><td>Os<sup>++</sup></td><td>Ir</td><td>Ir<sup>+</sup></td><td>Ir<sup>++</sup></td><td>Pt</td><td>Pt<sup>+</sup></td><td>Pt<sup>++</sup></td><td>Au</td><td>Au<sup>+</sup></td><td>Au<sup>++</sup></td><td>Hg</td><td>Hg<sup>+</sup></td></tr><tr><td>Hg<sup>++</sup></td><td>Tl</td><td>Tl<sup>+</sup></td><td>Tl<sup>++</sup></td><td>Pb</td><td>Pb<sup>+</sup></td><td>Pb<sup>++</sup></td><td>Bi</td><td>Bi<sup>+</sup></td><td>Bi<sup>++</sup></td><td>Po</td><td>Po<sup>+</sup></td><td>Po<sup>++</sup></td></tr><tr><td>At</td><td>At<sup>+</sup></td><td>At<sup>++</sup></td><td>Rn</td><td>Rn<sup>+</sup></td><td>Rn<sup>++</sup></td><td>Fr</td><td>Fr<sup>+</sup></td><td>Fr<sup>++</sup></td><td>Ra</td><td>Ra<sup>+</sup></td><td>Ra<sup>++</sup></td><td>Ac</td></tr><tr><td>Ac<sup>+</sup></td><td>Ac<sup>++</sup></td><td>Th</td><td>Th<sup>+</sup></td><td>Th<sup>++</sup></td><td>Pa</td><td>Pa<sup>+</sup></td><td>Pa<sup>++</sup></td><td>U</td><td>U<sup>+</sup></td><td>U<sup>++</sup></td><td>H<sup>-</sup></td><td>H<sub>2</sub></td></tr><tr><td>H<sub>2</sub><sup>+</sup></td><td>HF</td><td>HF<sup>+</sup></td><td>HCl</td><td>HCl<sup>+</sup></td><td>HBr</td><td>HBr<sup>+</sup></td><td>HI</td><td>HeH<sup>+</sup></td><td>He<sub>2</sub></td><td>He<sub>2</sub><sup>+</sup></td><td>HeNe<sup>+</sup></td><td>LiH</td></tr><tr><td>LiH<sup>+</sup></td><td>Li<sub>2</sub></td><td>LiO</td><td>LiF</td><td>LiNa</td><td>LiCl</td><td>LiK</td><td>LiBr</td><td>LiI</td><td>BeH</td><td>BeH<sup>+</sup></td><td>BeO</td><td>BeF</td></tr><tr><td>BeS</td><td>BeCl</td><td>BH</td><td>BH<sup>+</sup></td><td>B<sub>2</sub></td><td>BO</td><td>BF</td><td>BS</td><td>BCl</td><td>BBr</td><td>CH</td><td>CH<sup>+</sup></td><td>C<sub>2</sub></td></tr><tr><td>C<sub>2</sub><sup>+</sup></td><td>CN</td><td>CN<sup>+</sup></td><td>CO</td><td>CO<sup>+</sup></td><td>CF</td><td>CP</td><td>CS</td><td>CS<sup>+</sup></td><td>CCI</td><td>CSe</td><td>CBr</td><td>NH<sub>2</sub></td></tr><tr><td>NH<sub>2</sub><sup>+</sup></td><td>N<sub>2</sub></td><td>N<sub>2</sub><sup>+</sup></td><td>NO</td><td>NO<sup>+</sup></td><td>NF</td><td>NS</td><td>NS<sup>+</sup></td><td>NS<sub>2</sub></td><td>NBr</td><td>OH</td><td>OH<sup>+</sup></td><td>O<sub>2</sub></td></tr><tr><td>O<sub>2</sub><sup>+</sup></td><td>F<sup>-</sup></td><td>FO</td><td>F<sub>2</sub></td><td>F<sub>2</sub><sup>+</sup></td><td>NeH<sup>+</sup></td><td>Ne<sub>2</sub></td><td>Ne<sub>2</sub><sup>+</sup></td><td>NaH</td><td>NaO</td><td>NaF</td><td>Na<sub>2</sub></td><td>Na<sub>2</sub><sup>+</sup></td></tr><tr><td>NaCl</td><td>NaBr</td><td>NaI</td><td>MgH</td><td>MgH<sup>+</sup></td><td>MgO</td><td>MgF</td><td>Mg<sub>2</sub></td><td>MgS</td><td>MgCl</td><td>MgBr</td><td>AlH</td><td>AlO</td></tr><tr><td>AlF</td><td>Al<sub>2</sub></td><td>AlS</td><td>AlCl</td><td>AlSe</td><td>AlBr</td><td>All</td><td>SiH</td><td>SiH<sup>+</sup></td><td>SiC</td><td>SiO</td><td>SiO<sup>+</sup></td><td>SiF</td></tr><tr><td>Si<sub>2</sub></td><td>SiS</td><td>SiSe</td><td>SiTe</td><td>SiI</td><td>PH</td><td>PH<sup>+</sup></td><td>PN</td><td>PO</td><td>PO<sup>+</sup></td><td>P<sub>2</sub></td><td>P<sub>2</sub><sup>+</sup></td><td>PS</td></tr><tr><td>SH</td><td>SH<sup>+</sup></td><td>SO</td><td>SO<sup>+</sup></td><td>SF</td><td>S<sub>2</sub></td><td>S<sub>2</sub><sup>+</sup></td><td>Cl<sup>-</sup></td><td>ClO</td><td>ClF</td><td>Cl<sub>2</sub></td><td>Cl<sub>2</sub><sup>+</sup></td><td>Ar<sub>2</sub></td></tr><tr><td>KH</td><td>KO</td><td>KF</td><td>KCl</td><td>K<sub>2</sub></td><td>KBr</td><td>KI</td><td>CaH</td><td>CaO</td><td>CaF</td><td>CaS</td><td>CaCl</td><td>CaCa</td></tr><tr><td>ScH</td><td>ScO</td><td>ScF</td><td>ScS</td><td>ScCl</td><td>TiH</td><td>TiN</td><td>TiO</td><td>TiS</td><td>VO</td><td>CrH</td><td>CrO</td><td>CrS</td></tr><tr><td>MnH</td><td>MnO</td><td>MnF</td><td>MnS</td><td>MnCl</td><td>FeH</td><td>FeO</td><td>CoH</td><td>CoCl</td><td>NiH</td><td>NiO</td><td>NiCl</td><td>NiBr</td></tr><tr><td>CuH</td><td>CuO</td><td>CuF</td><td>CuS</td><td>CuCl</td><td>Cu<sub>2</sub></td><td>CuSe</td><td>CuBr</td><td>CuTe</td><td>CuI</td><td>ZnH</td><td>ZnH<sup>+</sup></td><td>ZnCl</td></tr><tr><td>GaH</td><td>GaO</td><td>GaF</td><td>GaCl</td><td>GaBr</td><td>GaI</td><td>GeH</td><td>GeO</td><td>GeF</td><td>GeS</td><td>GeCl</td><td>GeSe</td><td>GeTe</td></tr><tr><td>AsH</td><td>AsO</td><td>AsF</td><td>AsAs</td><td>SeH</td><td>SeO</td><td>SeF</td><td>SeS</td><td>Se<sub>2</sub></td><td>BrO</td><td>BrF</td><td>BrCl</td><td>Br<sub>2</sub></td></tr><tr><td>Br<sub>2</sub><sup>+</sup></td><td>KrF<sup>+</sup></td><td>RbF</td><td>RbCl</td><td>RbBr</td><td>RbI</td><td>SrH</td><td>SrO</td><td>SrF</td><td>SrS</td><td>SrCl</td><td>YO</td><td>YF</td></tr><tr><td>YS</td><td>ZrN</td><td>ZrO</td><td>NbO</td><td>RuC</td><td>RuO</td><td>RhC</td><td>AgH</td><td>AgO</td><td>AgF</td><td>AgAl</td><td>AgCl</td><td>AgBr</td></tr><tr><td>AgI</td><td>CdH</td><td>CdH<sup>+</sup></td><td>CdF</td><td>CdCl</td><td>InH</td><td>InO</td><td>InF</td><td>InCl</td><td>TeS</td><td>TeSe</td><td>Te<sub>2</sub></td><td>IO</td></tr><tr><td>SnF</td><td>SnS</td><td>SnSe</td><td>SnTe</td><td>SbO</td><td>SbF</td><td>SbP</td><td>SbSb</td><td>TeO</td><td>TeO</td><td>TeS</td><td>Te<sub>2</sub></td><td>IO</td></tr><tr><td>IF</td><td>ICI</td><td>IBr</td><td>II</td><td>Xe<sub>2</sub></td><td>CsH</td><td>CsF</td><td>CsCl</td><td>CsBr</td><td>CsI</td><td>Cs<sub>2</sub></td><td>BaH</td><td>BaO</td></tr><tr><td>BaF</td><td>BaS</td><td>BaCl</td><td>LaO</td><td>LaS</td><td>CeO</td><td>PrO</td><td>TbO</td><td>HoF</td><td>YbH</td><td>YbF</td><td>LuO</td><td>LuF</td></tr><tr><td>HfO</td><td>TaO</td><td>TaO<sup>+</sup></td><td>WO</td><td>IrC</td><td>IrO</td><td>PtH</td><td>PtC</td><td>PtO</td><td>AuH</td><td>Au<sub>2</sub></td><td>AuSi</td><td>AuCl</td></tr><tr><td>Au<sub>2</sub></td><td>HgH</td><td>HgH<sup>+</sup></td><td>HgF</td><td>HgCl</td><td>HgAr<sup>+</sup></td><td>TlH</td><td>TlF</td><td>TlCl</td><td>TlBr</td><td>TlI</td><td>PbH</td><td>PbO</td></tr><tr><td>PbF</td><td>PbS</td><td>PbCl</td><td>PbSe</td><td>PbTe</td><td>BiH</td><td>BiO</td><td>BiS</td><td>BiCl</td><td>Bi<sub>2</sub></td><td>ThO</td><td>H<sub>2</sub><sup>-</sup></td><td>H<sub>3</sub><sup>+</sup></td></tr><tr><td>H<sub>2</sub>O</td><td>H<sub>2</sub>S</td><td>HBO</td><td>HBS</td><td>HCN</td><td>HCO</td><td>HCO<sup>+</sup></td><td>LiOH</td><td>BeH<sub>2</sub></td><td>Be<sub>2</sub>O</td><td>BeC<sub>2</sub></td><td>BeOH</td><td>BeF<sub>2</sub></td></tr><tr><td>BeClF</td><td>BeCl<sub>2</sub></td><td>BH<sub>2</sub></td><td>BO<sub>2</sub></td><td>CH<sub>2</sub></td><td>CHF</td><td>CHP</td><td>CHCl</td><td>C<sub>2</sub>O</td><td>C<sub>2</sub>H</td><td>C<sub>2</sub>O</td><td>C<sub>2</sub>O</td><td>CN<sup>-</sup></td></tr><tr><td>CNC</td><td>CO<sub>2</sub></td><td>COS</td><td>CS<sub>2</sub></td><td>NH<sub>2</sub></td><td>NO<sup>-</sup></td><td>OH<sup>-</sup></td><td>OBF</td><td>O<sub>3</sub></td><td>OAlH</td><td>OAlF</td><td>OAlCl</td><td>OTiF</td></tr><tr><td>OTiCl</td><td>F<sub>2</sub></td><td>NaCN</td><td>NaOH</td><td>MgOH</td><td>MgOH<sub>2</sub></td><td>MgClF</td><td>MgCl<sub>2</sub></td><td>MgBr<sub>2</sub></td><td>AlOH</td><td>AlOH<sup>+</sup></td><td>AlF<sub>2</sub></td><td>Al<sub>2</sub>O</td></tr><tr><td>AlClF</td><td>AlCl<sub>2</sub></td><td>SiH</td><td>SiC<sub>2</sub></td><td>SiO<sub>2</sub></td><td>Si<sub>2</sub>C</td><td>Si<sub>2</sub>N</td><td>Si<sub>3</sub></td><td>PH<sub>2</sub></td><td>PO<sub>2</sub></td><td>SH<sup>-</sup></td><td>SO<sub>2</sub></td><td>ClCN</td></tr><tr><td>KCN</td><td>KOH</td><td>CaOH</td><td>CaOH<sup>+</sup></td><td>CaF<sub>2</sub></td><td>CaCl<sub>2</sub></td><td>CaBr<sub>2</sub></td><td>CaI<sub>2</sub></td><td>TiO<sub>2</sub></td><td>TiF<sub>2</sub></td><td>TiCl<sub>2</sub></td><td>TiBr<sub>2</sub></td><td>TiI<sub>2</sub></td></tr><tr><td>VO<sub>2</sub></td><td>CrO<sub>2</sub></td><td>FeCl<sub>2</sub></td><td>FeBr<sub>2</sub></td><td>BrCN</td><td>SrOH</td><td>SrOH<sup>+</sup></td><td>SrF<sub>2</sub></td><td>SrCl<sub>2</sub></td><td>SrBr<sub>2</sub></td><td>ZrO<sub>2</sub></td><td>ZrF<sub>2</sub></td><td>ZrCl<sub>2</sub></td></tr><tr><td>ZrBr<sub>2</sub></td><td>ZrI<sub>2</sub></td><td>NbO<sub>2</sub></td><td>MoO<sub>2</sub></td><td>CsOH</td><td>BaOH</td><td>BaOH<sup>+</sup></td><td>BaF<sub>2</sub></td><td>BaCl<sub>2</sub></td><td>BaBr<sub>2</sub></td><td>TaO<sub>2</sub></td><td>WO<sub>2</sub></td><td>WCl<sub>2</sub></td></tr><tr><td>H<sub>3</sub>O<sup>+</sup></td><td>HBO<sub>2</sub></td><td>LiBO<sub>2</sub></td><td>BeBO<sub>2</sub></td><td>BH<sub>3</sub></td><td>BO<sub>2</sub></td><td>CH<sub>3</sub></td><td>C<sub>2</sub>H<sub>2</sub></td><td>C<sub>2</sub>HCl</td><td>C<sub>4</sub></td><td>(CN)<sub>2</sub></td><td>(CN)<sub>2</sub></td><td>CO<sub>2</sub><sup>-</sup></td></tr><tr><td>NH<sub>3</sub></td><td>OAlOH</td><td>OAlF<sub>2</sub></td><td>NaBO<sub>2</sub></td><td>(NaCl)<sub>2</sub></td><td>(NaBr)<sub>2</sub></td><td>AlBO<sub>2</sub></td><td>AlOH<sup>-</sup></td><td>AlO<sub>2</sub><sup>-</sup></td><td>(AlO)<sub>2</sub></td><td>AlF<sub>2</sub><sup>-</sup></td><td>AlCl<sub>2</sub><sup>-</sup></td><td>PH<sub>3</sub></td></tr><tr><td>P<sub>4</sub></td><td>KBO<sub>2</sub></td><td>(KCl)<sub>2</sub></td><td>(KBr)<sub>2</sub></td><td>TiOCl<sub>2</sub></td><td>TiF<sub>3</sub></td><td>TiCl<sub>3</sub></td><td>ZrCl<sub>3</sub></td><td>MoO<sub>3</sub></td><td>WO<sub>3</sub></td><td>Be(OH)<sub>2</sub></td><td>B(OH)<sub>2</sub></td><td>CH<sub>4</sub></td></tr><tr><td>CH<sub>3</sub>Cl</td><td>C<sub>5</sub></td><td>OAlF<sub>2</sub><sup>-</sup></td><td>Mg(OH)<sub>2</sub></td><td>SiH<sub>4</sub></td><td>SiH<sub>3</sub>F</td><td>SiH<sub>3</sub>Cl</td><td>SiH<sub>3</sub>Br</td><td>SiH<sub>3</sub>I</td><td>SiH<sub>2</sub>F<sub>2</sub></td><td>Ca(OH)<sub>2</sub></td><td>Fe(OH)<sub>2</sub></td><td>Sr(OH)<sub>2</sub></td></tr><tr><td>ZrF<sub>4</sub></td><td>ZrCl<sub>4</sub></td><td>Ba(OH)<sub>2</sub></td><td>WO<sub>2</sub>Cl<sub>2</sub></td><td>(LiOH)<sub>2</sub></td><td>(BeO)<sub>3</sub></td><td>C<sub>2</sub>H<sub>4</sub></td><td>(NaCN)<sub>2</sub></td><td>(NaOH)<sub>2</sub></td><td>AlF<sub>4</sub><sup>-</sup></td><td>(KOH)<sub>2</sub></td><td>H<sub>3</sub>BO<sub>3</sub></td><td>H<sub>2</sub>MoO<sub>4</sub></td></tr><tr><td>O<sub>2</sub>W(OH)<sub>2</sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>
ORCID iDs

Paola Marigo https://orcid.org/0000-0002-9137-0773

Francesco Addari https://orcid.org/0000-0002-3867-9966

Diego Bossini https://orcid.org/0000-0002-9480-8400

Alessandro Bressan https://orcid.org/0000-0002-7922-8440

Guglielmo Costa https://orcid.org/0000-0002-6213-6988

Léo Girardi https://orcid.org/0000-0002-6301-3269

Giada Pastorelli https://orcid.org/0000-0002-9300-7409

Michele Trabucchi https://orcid.org/0000-0002-1429-2388

Guglielmo Volpato https://orcid.org/0000-0002-8691-4940

References

Ackerman, A. S., & Marley, M. S. 2001, ApJ, 556, 872

Adam, A. Y., Yachmenev, A., Yurchenko, S. N., & Jensen, P. 2019, JPCA, 123, 4755

Al Derzi, A. R., Furtenbacher, T., Tennyson, J., Yurchenko, S. N., & Csaszar, A. G. 2015, JQSRT, 161, 117

Alexander, D. R. 1975, ApJS, 29, 363

Alexander, D. R., & Ferguson, J. W. 1994, ApJ, 437, 879

Allard, F., Homeier, D., & Freytag, B. 2012, RSPTA, 370, 2765

Allard, N. F., Spiegelman, F., & Kielkopf, J. F. 2016, A&A, 589, A21

Allard, N. F., Spiegelman, F., Leininger, T., & Molliere, P. 2019, A&A, 628, A120

Al-Refaie, A. F., Changeat, Q., Waldmann, I. P., & Tinetti, G. 2021, ApJ, 917, 37

Amaral, P. H. R., Diniz, L. G., Jones, K. A., et al. 2019, ApJ, 878, 95

Anders, E., & Grevesse, N. 1989, GeCoA, 53, 197

Asplund, M., Amarsi, A. M., & Grevesse, N. 2021, A&A, 653, A141

Asplund, M., Grevesse, N., Sauval, A. J., & Scott, P. 2009, ARA&A, 47, 481

Baraffe, I., Chabrier, G., Allard, F., & Hauschildt, P. H. 2002, A&A, 382, 563

Barton, E. J., Chiu, C., Golpayegani, S., et al. 2014, MNRAS, 442, 1821

Barton, E. J., Hill, C., Yurchenko, S. N., et al. 2017, JQSRT, 187, 453

Bernath, P. F. 2020, JQSRT, 240, 106687

Bittner, D. M., & Bernath, P. F. 2018, ApJS, 235, 8

Böhm-Vitense, E. 1958, ZAp, 46, 108

Bowesman, C. A., Shuai, M., Yurchenko, S. N., & Tennyson, J. 2021, MNRAS, 508, 3181

Bressan, A., Marigo, P., Girardi, L., et al. 2012, MNRAS, 427, 127

Burrows, A., Dulick, M., C. W. Bauschlicher, J., et al. 2005, ApJ, 624, 988

Burrows, A., Hubbard, W. B., Lunine, J. I., & Liebert, J. 2001, RvMP, 73, 719

Burrows, A., Marley, M. S., & Sharp, C. M. 2000, ApJ, 531, 438

Caffau, E., Ludwig, H. G., Steffen, M., Freytag, B., & Bonifacio, P. 2011, SoPh, 268, 255

Calamari, E., Faherty, J. K., Burningham, B., et al. 2022, ApJ, 940, 164

Carter, J. A., Fabrycky, D. C., Ragozzine, D., et al. 2011, Sci, 331, 562

Castelli, F. 2005, MSAIS, 8, 34

Chen, Y., Girardi, L., Bressan, A., et al. 2014, MNRAS, 444, 2525

Chubb, K. L., Rocchetto, M., Yurchenko, S. N., et al. 2021, A&A, 646, A21

Chubb, K. L., Tennyson, J., & Yurchenko, S. N. 2020, MNRAS, 493, 1531

Coppola, C. M., Lodi, L., & Tennyson, J. 2011, MNRAS, 415, 487

Costa, G., Girardi, L., Bressan, A., et al. 2019, MNRAS, 485, 4641

Courtney, J., Schwab, D., Tsai, J., & Dragoset, R., 2015 Atomic Weights and Isotopic Compositions v4.1, NIST, http://physics.nist.gov/Comp

Cox, J., & Giuli, R. 1968, Principles of Stellar Structure (London: Gordon & Breach)

Darby-Lewis, D., Tennyson, J., Lawson, K. D., et al. 2018, JPhB, 51, 185701

Demory, B. O., Ségransan, D., Forveille, T., et al. 2009, A&A, 505, 205

Dessart, L., Hillier, D. J., Blondin, S., & Khokhlov, A. 2014, MNRAS, 441, 3249

Doyle, L. R., Carter, J. A., Fabrycky, D. C., et al. 2011, Sci, 333, 1602

Dulick, M., Bauschlicher, J., Burrows, A., et al. 2003, ApJ, 594, 651

Ecker, G., & Kröll, W. 1963, PhFl, 6, 62

Ferguson, J. W., Alexander, D. R., Allard, F., et al. 2005, ApJ, 623, 585

Fernando, A. M., Bernath, P. F., Hodges, J. N., & Masseron, T. 2018, JQSRT, 217, 29

Freedman, R. S., Lustig-Yaeger, J., Fortney, J. J., et al. 2014, ApJS, 214, 25

Freedman, R. S., Marley, M. S., & Lodders, K. 2008, ApJS, 174, 504

Frohman, D. J., Bernath, P. F., & Brooke, J. S. 2016, JQSRT, 169, 104

Gordon, I. E., Rothman, L. S., Hargreaves, R. J., et al. 2022, JQSRT, 277, 107949

Gorman, M. N., Yurchenko, S. N., & Tennyson, J. 2019, MNRAS, 490, 1652

Grevesse, N., Asplund, M., & Sauval, A. J. 2007, SSRv, 130, 105

Grevesse, N., & Noels, A. 1993, PhST, 47, 133

Grevesse, N., & Sauval, A. J. 1998, SSRv, 85, 161

Hansen, J. P., & Vieillefosse, P. 1976, PhRvL, 37, 391

Harris, G. J., Tennyson, J., Kaminsky, B. M., Pavlenko, Y. V., & Jones, H. R. A. 2006, MNRAS, 367, 400

Hedges, C., & Madhusudhan, N. 2016, MNRAS, 458, 1427

Helling, C., Ackerman, A., Allard, F., et al. 2008, MNRAS, 391, 1854

Helling, C., & Lucas, W. 2009, MNRAS, 398, 985

Hodges, J. N., & Bernath, P. F. 2017, ApJ, 840, 81

Holweger, H. 2001, in AIP Conf. Ser. 598, Solar and Galactic Composition, ed. R. F. Wimmer-Schweingruber (Melville, NY: AIP), 23

Hou, S., & Bernath, P. F. 2017, JQSRT, 203, 511

Hou, S., & Bernath, P. F. 2018, JQSRT, 210, 44

Juncher, D., Jørgensen, U. G., & Helling, C. 2017, A&A, 608, A70

Kramida, D., Ralchenko, Yu., Reader, J. & NIST ASD Team, 2022 Atomic Spectra Database v5.10, NIST, https://physics.nist.gov/asd

Kraus, A. L., Tucker, R. A., Thompson, M. I., Craine, E. R., & Hillenbrand, L. A. 2011, ApJ, 728, 48

Kurucz, R. L. 1993, Kurucz CD-ROM (Cambridge, MA: Smithsonian Astrophysical Observatory)

Langleben, J., Tennyson, J., Yurchenko, S. N., & Bernath, P. 2019, MNRAS, 488, 2332

Leconte, J. 2021, A&A, 645, A20

Lenzuni, P., Chernoff, D. F., & Salpeter, E. E. 1991, ApJS, 76, 759

Li, G., Gordon, I. E., Hajigeorgiou, P. G., Coxon, J. A., & Rothman, L. S. 2013, JQSRT, 130, 284

Li, G., Gordon, I. E., Rothman, L. S., et al. 2015, ApJS, 216, 15

Li, H. Y., Tennyson, J., & Yurchenko, S. N. 2019, MNRAS, 486, 2351

Lodders, K. 2003, ApJ, 591, 1220

Magg, E., Bergemann, M., Serenelli, A., et al. 2022, A&A, 661, A140

Malygin, M. G., Kuiper, R., Klahr, H., Dullemond, C. P., & Henning, T. 2014, A&A, 568, A91

Mant, B. P., Yachmenev, A., Tennyson, J., & Yurchenko, S. N. 2018, MNRAS, 478, 3220

Marigo, P., & Aringer, B. 2009, A&A, 508, 1539

Marigo, P., Aringer, B., Girardi, L., & Bressan, A. 2022, ApJ, 940, 129

Marigo, P., Woitke, P., Tognelli, E., et al. 2024, ApJ, 960, 18

Masseron, T., Plez, B., Van Eck, S., et al. 2014, A&A, 571, A47

McKemmish, L. K., Masseron, T., Hoeijmakers, H. J., et al. 2019, MNRAS, 488, 2836

McKemmish, L. K., Yurchenko, S. N., & Tennyson, J. 2016, MNRAS, 463, 771

Mizus, I. I., Alijah, A., Zobov, N. F., et al. 2017, MNRAS, 468, 1717

Mollière, P., Wardenier, J. P., van Boekel, R., et al. 2019, A&A, 627, A67

Mulders, G. D., Pascucci, I., Ciesla, F. J., & Fernandes, R. B. 2021, ApJ, 920, 66

Muzerolle, J., D'Alessio, P., Calvet, N., & Hartmann, L. 2004, ApJ, 617, 406

Nguyen, C. T., Costa, G., Girardi, L., et al. 2022, A&A, 665, A126

Owens, A., Conway, E. K., Tennyson, J., & Yurchenko, S. N. 2020, MNRAS, 495, 1927

Owens, A., Dooley, S., McLaughlin, L., et al. 2022, MNRAS, 511, 5448

Owens, A., Yachmenev, A., Thiel, W., Tennyson, J., & Yurchenko, S. N. 2017, MNRAS, 471, 5025

Owens, A., Yachmenev, A., Thiel, W., et al. 2018, MNRAS, 479, 3002

Parsons, S. G., Marsh, T. R., Gänsicke, B. T., et al. 2012a, MNRAS, 419, 304

Parsons, S. G., Marsh, T. R., Gänsicke, B. T., et al. 2012b, MNRAS, 420, 3281

Paulose, G., Barton, E. J., Yurchenko, S. N., & Tennyson, J. 2015, MNRAS, 454, 1931

Pelupessy, F. I., & Papadopoulos, P. P. 2009, ApJ, 707, 954

Polyansky, O. L., Kyuberis, A. A., Zobov, N. F., et al. 2018, MNRAS, 480, 2597

Potekhin, A. Y., Chabrier, G., Chugunov, A. I., Dewitt, H. E., & Rogers, F. J. 2009, PhRvE, 80, 047401

Prajapat, L., Jagoda, P., Lodi, L., et al. 2017, MNRAS, 472, 3648

Qin, Z., Bai, T., & Liu, L. 2021, JQSRT, 258, 107352

Qu, Q., Yurchenko, S. N., & Tennyson, J. 2021, MNRAS, 504, 5768

Rivlin, T., Lodi, L., Yurchenko, S. N., Tennyson, J., & Le Roy, R. J. 2015, MNRAS, 451, 634

Rohrmann, R. D., Althaus, L. G., García-Berro, E., Córsico, A. H., & Miller Bertolami, M. M. 2012, A&A, 546, A119

Rothman, L. S., Gordon, I. E., Barber, R. J., et al. 2010, JQSRT, 111, 2139

Roueff, E., Abgrall, H., Czachorowski, P., et al. 2019, A&A, 630, A58

Seaton, M. J., Yan, Y., Mihalas, D., & Pradhan, A. K. 1994, MNRAS, 266, 805

Ségransan, D., Kervella, P., Forveille, T., & Queloz, D. 2003, A&A, 397, L5

Sharp, C. M., & Burrows, A. 2007, ApJS, 168, 140
Sing, D. K., Fortney, J. J., Nikolov, N., et al. 2016, Natur, 529, 59

Somers, G., Cao, L., & Pinsonneault, M. H. 2020, ApJ, 891, 29

Sousa-Silva, C., Al-Refaie, A. F., Tennyson, J., & Yurchenko, S. N. 2014, MNRAS, 446, 2337

Southworth, J. 2015, in ASP Conf. Ser. 496, Living Together: Planets, Host Stars and Binaries, ed. S. M. Rucinski, G. Torres, & M. Zejda (San Francisco, CA: ASP), 164

Spiegel, D. S., Burrows, A., & Milsom, J. A. 2011, ApJ, 727, 57

Stewart, J. C., & Pyatt, K. D. J. 1966, ApJ, 144, 1203

Syme, A.-M., & McKemmish, L. K. 2021, MNRAS, 505, 4383

Tennyson, J., & Yurchenko, S. N. 2012, MNRAS, 425, 21

Tennyson, J., Yurchenko, S. N., Al-Refaie, A. F., et al. 2016, JMoSp, 327, 73

Torres, G., Andersen, J., & Giménez, A. 2010, A&ARv, 18, 67

Torres, G., Sandberg Lacy, C. H., Pavlovski, K., et al. 2014, ApJ, 797, 31

Tsuji, T. 2002, ApJ, 575, 264

Tsuji, T., Ohnaka, K., & Aoki, W. 1996, A&A, 305, L1

Upadhyay, A., Conway, E. K., Tennyson, J., & Yurchenko, S. N. 2018, MNRAS, 477, 1520

Waldmann, I. P., Rocchetto, M., Tinetti, G., et al. 2015a, ApJ, 813, 13

Waldmann, I. P., Tinetti, G., Rocchetto, M., et al. 2015b, ApJ, 802, 107

Wilson, P. A., Hébrard, G., Santos, N. C., et al. 2016, A&A, 588, A144

Witte, S., Helling, C., & Hauschildt, P. H. 2009, A&A, 506, 1367

Woitke, P., Helling, C., & Gunn, O. 2020, A&A, 634, A23

Woitke, P., Helling, C., Hunter, G. H., et al. 2018, A&A, 614, A1

Woitke, P., Min, M., Pinte, C., et al. 2016, A&A, 586, A103

Yorke, L., Yurchenko, S. N., Lodi, L., & Tennyson, J. 2014, MNRAS, 445, 1383

Yurchenko, S. N., Al-Refaie, A. F., & Tennyson, J. 2018a, A&A, 614, A131
Yurchenko, S. N., Amundsen, D. S., Tennyson, J., & Waldmann, I. P. 2017a, A&A, 605, A95

Yurchenko, S. N., Blissett, A., Asari, U., et al. 2016, MNRAS, 456, 4524

Yurchenko, S. N., Bond, W., Gorman, M. N., et al. 2018b, MNRAS, 478, 270

Yurchenko, S. N., Mellor, T. M., Freedman, R. S., & Tennyson, J. 2020a, MNRAS, 496, 5282

Yurchenko, S. N., Sinden, F., Lodi, L., et al. 2017b, MNRAS, 473, 5324

Yurchenko, S. N., Szabó, I., Pyatenko, E., & Tennyson, J. 2018c, MNRAS, 480, 3397

Yurchenko, S. N., Tennyson, J., & Barton, E. J. 2017c, JPhCS, 810, 012010

Yurchenko, S. N., Tennyson, J., Miller, S., et al. 2020b, MNRAS, 497, 2340

Yurchenko, S. N., Tennyson, J., Syme, A.-M., et al. 2022, MNRAS, 510, 903

Yurchenko, S. N., Williams, H., Leyland, P. C., Lodi, L., & Tennyson, J. 2018d, MNRAS, 479, 1401