# An Expanded Set of Los Alamos OPLIB Tables in MESA:
Type-1 Rosseland-mean Opacities and Solar Models

Ebraheem Farag School of Earth and Space Exploration, Arizona State
University, Tempe, AZ 85287, USA  Center for Theoretical Astrophysics, Los
Alamos National Laboratory, Los Alamos, NM 87545  Department of Astronomy,
Yale University, New Haven, CT 06511  Christopher J. Fontes Center for
Theoretical Astrophysics, Los Alamos National Laboratory, Los Alamos, NM 87545
F.X. Timmes School of Earth and Space Exploration, Arizona State University,
Tempe, AZ 85287, USA  Earl P. Bellinger Department of Astronomy, Yale
University, New Haven, CT 06511  Max-Planck-Institut für Astrophysik, Karl-
Schwarzschild-Straße 1, D-85741 Garching, Germany  Stellar Astrophysics
Centre, Aarhus University, Aarhus, Denmark  Joyce A. Guzik Center for
Theoretical Astrophysics, Los Alamos National Laboratory, Los Alamos, NM 87545
Evan B. Bauer Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden
Street, Cambridge, MA 02138, USA  Suzannah R. Wood Center for Theoretical
Astrophysics, Los Alamos National Laboratory, Los Alamos, NM 87545  Katie
Mussack Center for Theoretical Astrophysics, Los Alamos National Laboratory,
Los Alamos, NM 87545  Peter Hakel Los Alamos National Laboratory, Los Alamos,
NM 87545  James Colgan Los Alamos National Laboratory, Los Alamos, NM 87545
David P. Kilcrease Los Alamos National Laboratory, Los Alamos, NM 87545
Manolo E. Sherrill Los Alamos National Laboratory, Los Alamos, NM 87545
Tryston C. Raecke Los Alamos National Laboratory, Los Alamos, NM 87545  Morgan
T. Chidester School of Earth and Space Exploration, Arizona State University,
Tempe, AZ 85287, USA Ebraheem Farag<EMAIL_ADDRESS>
###### Abstract

We present a set of 1194 Type-1 Rosseland-mean opacity tables for four
different metallicity mixtures. These new Los Alamos OPLIB atomic radiative
opacity tables are an order of magnitude larger in number than any previous
opacity table release, and span regimes where previous opacity tables have not
existed. For example, the new set of opacity tables expands the metallicity
range to $Z$ = 10-6 to $Z$ = 0.2 which allows improved accuracy of opacities
at low and high metallicity, increases the table density in the metallicity
range $Z$ = 10-4 to $Z$ = 0.1 to enhance the accuracy of opacities drawn from
interpolations across neighboring metallicities, and adds entries for hydrogen
mass fractions between $X$ = 0 and $X$ = 0.1 including $X$ =
$10^{-2},10^{-3},10^{-4},10^{-5},10^{-6}$ that can improve stellar models of
hydrogen deficient stars. We implement these new OPLIB radiative opacity
tables in MESA, and find that calibrated solar models agree broadly with
previously published helioseismic and solar neutrino results. We find
differences between using the new 1194 OPLIB opacity tables and the 126 OPAL
opacity tables range from $\approx$ 20–80% across individual chemical
mixtures, up to $\approx$ 8% and $\approx$ 15% at the bottom and top of the
solar convection zone respectively, and $\approx$ 7% in the solar core. We
also find differences between standard solar models using different opacity
table sources that are on par with altering the initial abundance mixture. We
conclude that this new, open-access set of OPLIB opacity tables does not solve
the solar modeling problem, and suggest the investigation of physical
mechanisms other than the atomic radiative opacity.

Stellar opacity(1585), Stellar interiors(1606), Stellar physics(1621), Stellar
evolution(1599)

††software:  MESA (Paxton et al., 2011, 2013, 2015, 2018, 2019; Jermyn et al.,
2023, https://docs.mesastar.org), MESASDK 20190830 (Townsend, 2019a, b),
matplotlib (Hunter, 2007), Scipy (Virtanen et al., 2020), and NumPy (van der
Walt et al., 2011).

##  1 Introduction

The radiative opacity $\kappa$, a cross-section per unit mass usually
expressed in cm2 g-1, describes the absorption and scattering of photons in a
medium. It is a crucial component of stellar physics that, in the interior,
relates the diffusive transport of the photon energy flux $F_{\gamma}$ to
spatial gradients in the energy density of the radiation field: $F_{\gamma}$
$\propto$ (1/$\kappa$) d($T^{4}$)/d$r$, where $T$ is the temperature and $r$
is the radial distance (Mihalas & Mihalas, 1984).

The radiative opacity depends on the energy of the photons and three other
quantities shared with the stellar equation of state – the temperature $T$,
density $\rho$, and composition vector. As a result, $\kappa$ and its partial
derivatives with respect to thermodynamic quantities can impact a wide range
of stellar phenomena.

For example, the $\kappa$-mechanism drives changes in the luminosity of many
types of pulsating variable stars (Eddington, 1926; Cox, 1980; Aerts et al.,
2010). In regions where the opacity increases with temperature (e.g., where
hydrogen and helium are partly ionized), the atmosphere becomes unstable
against pulsations (Hansen et al., 2004; Kippenhahn et al., 2012; Das et al.,
2020; Kurtz, 2022). Hydrogen ionization drives the pulsations of Mira
variables (Fox & Wood, 1982; Fadeyev, 2022) and red supergiants (Heger et al.,
1997; Yoon & Cantiello, 2010), the high-overtone, low-degree, non-radial
pressure modes of rapidly oscillating Ap stars (Kurtz, 1982; Shibahashi &
Takata, 1993; Holdsworth et al., 2021; Bigot & Dziembowski, 2002), and ZZ Ceti
variables (Landolt, 1968; Córsico et al., 2019). Helium ionization drives
pulsations in RR Lyrae variables (Smith, 2004; Ngeow et al., 2022) and
$\delta$ Scuti variables (Balona, 2018; Bowman & Kurtz, 2018; Guzik, 2021;
Murphy et al., 2023), DBV white dwarf variables (Córsico et al., 2019; Saumon
et al., 2022), and is furthermore responsible for acoustic glitches in solar-
like oscillators (Gough, 1990; Basu et al., 2004; Mazumdar et al., 2014; Verma
et al., 2017, 2019; Saunders et al., 2023). Other opacity increases from the
iron group elements (Cr, Fe, Ni and Cu) at temperatures of $\simeq$
2$\times$105 K and densities of $\simeq$10-7 g cm-3 are likely the cause of
pulsations in B-type stars (Townsend, 2005; Aerts et al., 2010; Guzik et al.,
2018; Shi et al., 2023), and $\beta$-Cephei variables (e.g., $\beta$ Centauri,
$\beta$ Centauri, $\gamma$ Pegasi and $\nu$ Eridani) where $\kappa$ may
account for differences between the observed and calculated pulsation periods
(Daszyńska-Daszkiewicz & Walczak, 2009, 2010; Cugier, 2012; Walczak et al.,
2015).

Another example is the solar structure (Basu, 2016; Buldgen et al., 2019;
Christensen-Dalsgaard, 2021) where the ionization of C, N, O, Ne, and Fe group
elements near the base of the solar convection zone induces bound-free
transitions that can be a source of the discrepancy between inferences from
helioseismology measurements and solar photosphere composition determinations
(Turck-Chièze et al., 2004; Bahcall et al., 2006; Guzik, 2008; Basu & Antia,
2008; Guzik & Mussack, 2010; Neuforge-Verheecke et al., 2001a, b; Krief et
al., 2016; Pradhan, 2023).

Differences between theoretical opacities and observation-adjusted opacities
are discussed in Eddington (1926); Daszyńska-Daszkiewicz et al. (2017); Guzik
et al. (2018); Daszyńska-Daszkiewicz et al. (2023). Considerations of negative
hydrogen ion transitions (Wildt, 1939; Chandrasekhar, 1944; Chandrasekhar &
Breen, 1946; Ohmura & Ohmura, 1960; Ohmura, 1964; Doughty & Fraser, 1966),
bound-bound transitions (Mayer, 1947; Meyerott & Moszkowski, 1951; Moszkowski
& Meyerott, 1951), bound-free transitions (photoionization), free-free
transitions (inverse bremsstrahlung) and electron scattering (Vitense, 1951;
Schwarzschild, 1958; Vardya, 1964; Jin, 1982; Meyer-Hofmeister, 1982) have
been followed by an extensive literature reporting atomic radiative opacity
calculations.

Examples include Los Alamos/OPLIB (Cox & Stewart, 1962; Cox, 1965; Cox &
Stewart, 1970a, b; Cox & Tabor, 1976; Hübner et al., 1977; Weiss et al., 1990;
Magee et al., 1995; Colgan et al., 2016), the Opacity Project (Seaton, 1987;
Seaton et al., 1994; Seaton & Badnell, 2004; Seaton, 2005; Badnell et al.,
2005; Pradhan et al., 2023; Nahar et al., 2023; Pradhan, 2023; Zhao et al.,
2023), Livermore OPAL (Rogers & Iglesias, 1992; Iglesias & Rogers, 1993,
1996), analytic fitting (Christy, 1966; Iben, 1975; Stellingwerf, 1975a, b),
the Scotland model (Carson et al., 1968; Carson, 1976), the OPAS model
(Blancard et al., 2012; Le Pennec et al., 2015; Mondet et al., 2015), the
hybrid model SCO-RCG (Iglesias & Sonnad, 2012; Pain & Gilleron, 2015; Pain et
al., 2017; Pain & Gilleron, 2019), for actinides (Fontes et al., 2023; Flörs
et al., 2023), and for He-dominated compositions, He- free–free (Somerville,
1965; John, 1994), He${}^{+}_{2}$ bound–free and free-free (Ignjatović et al.,
2009), He Rayleigh scattering (Iglesias et al., 2002; Rohrmann, 2018), and
triple-He collisions (Kowalski, 2014; Blouin et al., 2019; Saumon et al.,
2022).

In this article we add a novel contribution to this canon by providing open
access to an expanded set of 1194 Type-1 Rosseland-mean opacity tables for
four different heavy element mixtures, with improvements to the composition
range, table coverage, and table resolution. Section 2 describes the methods
used to construct the opacity tables and highlights a few representative
results. Section 3 implements the new opacity tables in MESA and applies them
to the helioseimology and neutrino fluxes of standard solar models, and
Section 4 offers concluding remarks. Appendices A to C detail the
implementation and verification of the new Los Alamos opacity tables in MESA.

Important symbols are defined in Table 1. In this article “log” refers to the
base-10 logarithm; where the natural logarithm is intended, we use “ln”.

Table 1: Important symbols.

Name | Description | Appears
---|---|---
$\alpha$ |  4He particle | 3.3
$a$ | Radiation constant = 4$\sigma/c$ | 2
$A$ | Atomic number | 1
$c$ | Speed of light in the medium | 2
$c_{s}$ | Acoustic sound speed | 3.3
$G$ | Gravitational constant | 2
$E$ | Energy | 2
$\gamma$ | Photon | 3.3
$\Gamma_{3}$ | $\equiv d{\rm ln}T/d{\rm ln}\rho|_{S}+1$ | 2
$h$ | Planck constant | 2
$k_{\rm B}$ | Boltzmann constant | 2
$\kappa$ | Opacity | 1
$\kappa_{\nu}$ | Monochromatic opacity | 2
$\kappa_{P}$ | Planck mean opacity | 2
$\kappa_{R}$ | Rosseland mean opacity | 2
$\lambda$ | Mean free path | 2
$L$ | Luminosity | 2
$M$ | Stellar mass | 2
$\nu$ | Frequency | 2
$\nu_{e}$ | Electron neutrino | 3.3
$\nabla_{\text{rad}}$ | Radiative temperature gradient | 2
$r$ | Radial coordinate | 1
$R$ | Opacity coordinate | 2
$R_{o}$ | Stellar radius | 2
$\rho$ | Mass density | 1
$\Phi$ | Neutrino Flux | 3.3
$P$ | Pressure | 2
$\sigma$ | Stefan–Boltzmann constant | 2
$S$ | Entropy per gram | 2
$t$ | Time | 2
$T$ | Temperature | 1
$X$ | Hydrogen mass fraction | 2
$Y$ | Helium mass fraction | 2
$Z$ | Metal mass fraction | 2
Z | Atomic charge | 2

††footnotetext: Note: Some symbols may be further subscripted, for example, by
an $\mathrm{a}$ (indicating an absorption quantity), by an $\mathrm{s}$
(indicating a scattering quantity), or by a $\mathrm{t}$ (indicating a total
quantity).

##  2 Stellar Opacity

The total monochromatic radiative opacity $\kappa_{\nu,{\rm t}}$ is the sum of
the absorption opacity $\kappa_{\nu,{\rm a}}$ and scattering opacity
$\kappa_{\nu,{\rm s}}$ at a specific $T$, $\rho$ and composition. The mean (or
gray) opacity represents, in a single number, the tendency of a material to
absorb and scatter radiation of all frequencies. Two common mean opacities are
the Planck mean (or emission mean) and Rosseland mean opacities. Other
examples include a flux-weighted mean and an absorption mean. These various
means arise in order to obtain correct values for a particular frequency-
integrated physical quantity, such as the radiation flux or energy (Cox &
Giuli, 1968; Mihalas, 1978; Cowan, 1981; Mihalas & Mihalas, 1984; Hansen et
al., 2004; Kippenhahn et al., 2012; Huebner & Barfield, 2014; Fontes et al.,
2015, 2023).

The Planck mean opacity $\kappa_{P}$ yields the correct value for the
integrated thermal emission for an optically thin plasma

$\kappa_{P}=\frac{\int_{0}^{\infty}\ \kappa_{\nu,{\rm a}}\ B_{\nu}(T)\
d\nu}{\int_{0}^{\infty}B_{\nu}(T)\ d\nu}\ ,$ (1)

where $B_{\nu}(T)$ = $2h\nu^{3}$/$c^{2}$ $\cdot$ 1/($\exp(h\nu/k_{\rm B}T)-1$)
is the Planck function, $k_{\rm B}$ is the Boltzmann constant, $h$ is the
Planck constant, $c$ is the medium’s speed of light, and $\nu$ is the photon
frequency. Note $\kappa_{P}$ is calculated only from $\kappa_{\nu,{\rm a}}$.
This weighting function peaks at $h\nu$ $\simeq$ 2.8 $k_{\rm B}T$, indicating
where $\kappa_{\nu,{\rm a}}$ is most strongly sampled.

The Rosseland mean opacity $\kappa_{R}$ yields the correct value for the
integrated energy flux of an optically thick plasma

$\frac{1}{\kappa_{R}}=\frac{\int_{0}^{\infty}\frac{1}{\kappa_{\nu,t}}\
\frac{dB_{\nu}(T)}{dT}\ d\nu}{\int_{0}^{\infty}\frac{dB_{\nu}(T)}{dT}\ d\nu}\
.$ (2)

The use of a harmonic average means the individual contributions (bound-bound,
bound-free, free-free, scattering) cannot be averaged first and then added to
obtain the proper mean value. This weighting function peaks at $h\nu$ $\simeq$
3.8 $k_{\rm B}T$, indicating where $\kappa_{\nu,{\rm t}}$ is strongly sampled.

The radiation transport equation in the gray-diffusion approximation (e.g.,
Mihalas, 1978)

$\frac{\partial E_{r}}{\partial
t}=\bm{\nabla}\cdot\left(\frac{c/3}{\rho\kappa_{R}}\bm{\nabla}E_{r}\right)+c\rho\kappa_{P}(aT_{e}^{4}-E_{r})$
(3)

admits two mean free paths; $\lambda_{R}$ = 1/$\rho$$\kappa_{R}$ (used in the
diffusion coefficient term) and $\lambda_{P}$ = 1/$\rho$$\kappa_{P}$ (used in
the radiation-electron coupling term). The two mean free paths can differ by
several orders of magnitude due to the different averaging prescriptions. Here
$E_{r}$ is the radiation energy, $a$ is the radiation constant, and $T_{e}$ is
the electron temperature. In stellar evolution models, $\kappa_{R}$ is used
for the radiative temperature gradient

$\nabla_{\text{rad}}=\left(\frac{d\ln T}{d\ln
P}\right)_{\text{rad}}=\frac{3\kappa_{R}LP}{16\pi acGM_{r}T^{4}}\ ,$ (4)

where $L$ is the luminosity, $P$ is the pressure, $G$ is the gravitational
constant, and $M_{r}$ is the mass contained within radius $r$.
$\nabla_{\text{rad}}$ in the mixing-length theory of convection (MLT)
determines convective stability, the radiative flux in convective regions, and
the actual temperature gradient in radiative regions.

Radiative opacities depend on the composition, which can change rapidly in
mass or time due to nuclear reactions, diffusion or chemical mixing. A typical
approach is to adopt a fixed solar abundance mixture of metals with mass
fraction $Z$, hydrogen mass fraction $X$, and helium mass fraction $Y$.
Individual Rosseland mean opacity tables are calculated for various
combinations of $X$ and $Z$ where $Y$ = 1 $-$ $X$ $-$ $Z$, and the chemical
elements that compose $Z$ have a fixed distribution. Tabulated Rosseland mean
opacity values computed with a fixed metal distribution are an acceptable
alternative to calculating self-consistent Rosseland mean opacities that
reflect the exact metal distribution of a stellar model as it evolves. The
adoption of self-consistent Rosseland mean opacity tables results in a
$\lesssim$ 2% change to the total opacity in a stellar model (Hui-Bon-Hoa,
2021).

We use MESA’s kap module to implement the new atomic radiative opacity tables
and compare them to previous atomic opacity releases. The kap module builds
opacity tables by combining the radiative opacities and electron conduction
opacities

$\frac{1}{\kappa}=\frac{1}{\kappa_{R}}+\frac{1}{\kappa_{\rm cond}}$ (5)

where $\kappa_{\rm cond}$ = 16$\sigma$T3/$\rho$$K$ converts an electron
conductivity $K$ to an opacity. Electron conduction opacities are tabulated
for atomic charges 1 $\leq$ Z $\leq$ 60 between 3 $\leq$ log $T$ $\leq$ 10 and
-6 $\leq$ log $\rho$ $\leq$ 11.5 (Cassisi et al., 2007; Blouin et al., 2020).

The cores of stellar models mostly evolve along lines of constant specific
radiation entropy $S_{\rm rad}$ $\sim$ $T_{\rm c}^{3}$/$\rho_{\rm c}$ $\sim$
$M$2 for a fixed non-degenerate stellar mass $M$. Atomic and molecular
radiative opacities in MESA are tabulated using the conventional OPAL log $R$
– log $T$ format, where $R$ = $\rho$/$T_{6}^{3}$ which scales as
$\sim$1/$S_{\rm rad}$. The advantage of using $T_{6}$ = $T$/106 K and $R$ is
that the range of interest for stellar physics can be covered by a rectangular
array in these variables (Bahcall & Ulrich, 1988).

Figure 1: Location of each Type-1 opacity table in the $X$–$Z$ plane (left
panel) and the $\log T$ –$\log R$ plane (right panel). Orange circles mark the
location of the 126 OPAL Type-1 tables (Rogers & Iglesias, 1992). Blue circles
mark the location of the new 1194 Type-1 opacity tables.

Molecular transitions (e.g., H2, H2O, TiO, CO) are the primary source of the
radiative opacity for log $T$ $\leq$ 4.5. In this regime the kap module
provides radiative opacities from Ferguson et al. (2005). The kap module also
supplies a privately communicated set of Freedman et al. (2008) low-
temperature opacities, and supports ÆSOPUS opacities (Marigo & Aringer, 2009)
for arbitrary chemical mixtures (see Jermyn et al., 2023, for details). MESA
currently defaults to blending the higher temperature and lower temperature
opacity tables between 3.80 $\leq$ $\log T$ $\leq$ 3.88 (see Paxton et al.,
2013). Compton scattering dominates the radiative opacities for log $T$ $\geq$
8.7, where the kap module uses Poutanen (2017). For intermediate temperatures,
atomic radiative opacities have been supplied by OPAL (Iglesias & Rogers,
1993, 1996) and OP (Seaton, 2005; Badnell et al., 2005). Type-1 OPAL tables,
the subject of this article, have a fixed heavy element metal distribution
(e.g., H burning). Type-2 OPAL tables provide radiative opacities for C and O
rich mixtures (e.g., He burning, Iglesias & Rogers, 1993).

###  2.1 OPLIB Database

The Los Alamos OPLIB opacity database has been publicly available for more
than forty years, and is currently accessible at the website
http://aphysics2.lanl.gov/opacity/lanl. The database contains monochromatic
opacities for the first thirty elements of the periodic table, using a photon-
energy grid of 14,900 points in the variable $u$ = $h\nu/k_{\rm B}T$, with
values ranging from 10-4 $\leq$ $u$ $\leq$ 30,000. The website can produce
monochromatic, multigroup and gray opacities for either pure elements or
arbitrary mixtures. The most recent database release (Colgan et al., 2016),
which is referred to simply as “OPLIB” in the comparisons provided in this
article, was generated with the ATOMIC code, while the previous release, which
is referred to as “OPLIB-L”, was generated with the LEDCOP code (Magee et al.,
1995).

Since this article focuses on the latest OPLIB release, which is intended to
supersede the OPLIB-L release, we provide a brief summary of the relevant
computational methods. ATOMIC is a multi-purpose plasma modeling code (Magee
et al., 2004; Hakel et al., 2006; Fontes et al., 2015) that can be run in
local thermodynamic equilibrium (LTE) or non-LTE mode to calculate the atomic-
state populations. In this article, we focus on LTE populations. These
populations are combined with atomic data, e.g. oscillator strengths and
photoionization cross-sections, to obtain the monochromatic opacities, which
are constructed from the standard four contributions: bound–bound, bound–free,
free–free, and scattering.

When calculating the populations, ATOMIC uses the ChemEOS model to account for
the effects of the plasma environment on the equation of state (Hakel &
Kilcrease, 2004; Hakel et al., 2006; Kilcrease et al., 2015). Whereas OPAL is
based on the physical picture that treats plasma as a collection of nuclei and
electrons Coulomb-interacting in the grand canonical ensemble, ChemEOS uses
the chemical picture in which the free electrons and the various ion stages
are identified as individual species in the canonical ensemble; the associated
Helmholtz free energy is then minimized in ChemEOS to yield the species’
populations. From that perspective ChemEOS is similar to OP, with which it
also shares the use of occupation probabilities to model the gradual
dissolution of ion-stages’ bound states by the plasma environment. ChemEOS
differs from OP in the precise form of the occupation probabilities due to its
adopted plasma microfield model, and in the details of excluded-volume
considerations for the ion-stages’ finite sizes. Furthermore, in order to
extend the validity of the chemical picture to high densities ChemEOS adopts a
model for the species’ Coulomb interactions that smoothly bridges the
transition from the Debye-Hückel limit at low densities to the Thomas-Fermi
limit in the strongly-coupled high-density regime.

For the latest OPLIB release, ATOMIC used semi-relativistic Hartree-Fock
atomic data that were generated with the Los Alamos suite of atomic physics
codes (Fontes et al., 2015). Improvements over the previous (LEDCOP) OPLIB
release include the addition of 24 isotherms to reduce interpolation errors,
extending the calculations to higher densities and the inclusion of
significantly more lines via a histogram method (Abdallah et al., 2007).
Additional details about the latest OPLIB release, as well as comparisons with
other opacity databases, are provided in Colgan et al. (2016). These
improvements have made possible the generation of higher fidelity opacity
tables for pre-tabulated ($X$,$Y$,$Z$) mixtures of interest, as compared to
the pre-tabulated OPLIB-L tables publicly available at
http://aphysics2.lanl.gov/opacity/lanl. We detail these improvements in the
next section.

Before leaving this section, we provide some general information about
uncertainties in the calculated opacities. Uncertainties in Rosseland mean
opacity are caused by uncertainties in the calculation of the fundamental
atomic cross sections, plasma effects caused by perturbing ions, and
computational limitations. For example, measurements of fundamental cross
sections are usually carried out on neutral atoms, rather than on charged
ions, due to the difficulty in preparing a sample in a specific ion stage and
because of the myriad possibilities of excited levels. On the other hand,
cross sections of neutral atoms are more difficult to calculate accurately
because of the many-body, electron-electron interactions. Thus, comparison of
calculations with measured cross sections for neutral species can provide an
upper bound on cross section uncertainties. For Sun-like conditions, estimates
of the uncertainty in the opacity are $\simeq$5% when electron scattering
dominates at high $T$ and low $\rho$ (Huebner & Barfield, 2014). As the $\rho$
increases and free-free processes become more important, the uncertainty is
less than $\simeq$10%. As $T$ decreases and bound-free processes become
important, the uncertainty increases to $\simeq$20%. As $T$ decreases further,
bound-bound processes can contribute, and the uncertainty rises to $\simeq$30%
(Huebner & Barfield, 2014). From atomic-theory considerations, the
uncertainties in the calculated cross sections, particularly those that
involve bound electrons, progressively decrease as the ionic charge increases
toward the limiting case of one-electron (hydrogenic) ions, provided that
plasma effects do not become too important. Such conditions exist for a
variety of astrophysical applications, such as the solar modeling discussed in
Section 3.

###  2.2 New OPLIB Opacity Tables

Figure 1 shows the OPLIB tables in the $X$–$Z$ plane (left panel). Previous
opacity releases by OP and OPAL contain 126 tables spanning $Z$ = 10-4 to $Z$
= 0.1, and $Z$ = 0.0. We have improved on these grids in three ways. First, we
expand the range to $Z$ = 10-6 to $Z$ = 0.2, allowing for improved accuracy of
opacities at low and high $Z$. Second, we substantially increase the table
density in the range $Z$ = 10-4 to $Z$ = 0.1 enhancing the accuracy of
opacities drawn from interpolations across neighboring metallicities. Third,
we add opacity tables between $X$ = 0 and $X$ = 0.1. The addition of tables at
$X$ $=10^{-2},10^{-3},10^{-4},10^{-5},10^{-6}$ can, for example, improve
stellar models with thin H-depleted mixtures, such as in the atmospheres of
hot subdwarfs or other stripped stars. These new OPLIB tables encompass 1194
individual tables, an order of magnitude larger in number than any previous
opacity table release, and span regimes where previous opacity tables have not
existed.

Figure 1 also shows the OPLIB tables in the $\log T$–$\log R$ plane (right
panel). OP and OPAL radiative opacity tables are tabulated for 3.75 $\leq$
log($T$/K) $\leq$ 8.7, and $-$8 $\leq$ log(R) $\leq$ 1, with 70 tabulated
points in log($T$/K) and 19 tabulated points in log($R$). The new OPLIB
radiative opacity tables are tabulated over a larger space of 3.764 $\leq$
log($T$/K) $\leq$ 9.065, and $-$8 $\leq$ log($R$) $\leq$ 1.5 with 74 tabulated
points in log($T$/K) and 39 tabulated points in log($R$). Table values are
written to four decimal place precision, an improvement over other works which
write values to three decimal place precision. Each opacity table is
calculated with a mixture of 25 elements: H, He C, N, O, F, Ne, Na, Mg, Al,
Si, P, S, Cl, Ar, K, Ca, Sc, Ti, V, Cr, Mn, Fe, Co, Ni.

###  2.3 OPLIB Opacity Tables in MESA

The 126 OPAL and OP opacity tables are pre-processed via smoothing and spline-
fitting routines provided in Seaton (1993) to ensure smooth opacity
derivatives. When run through these routines, OP and OPAL tables are smoothed
and interpolated from 70 log($T$/K) $\times$ 19 log($R$) point tables spanning
3.75 $\leq$ log($T$/K) $\leq$ 8.70 and $-$8.0 $\leq$ log($R$) $\leq$ 1.0 into
evenly spaced 138 log($T$/K) $\times$ 37 log($R$) opacity tables using bi-
cubic splines. We take a similar approach by applying bi-cubic splines without
a smoothing filter to the OPLIB tables, which interpolate our 74 log($T$/K)
points up to $213$ points. Further discussion of the interpolation method
along with comparisons between the raw and interpolated tables are provided in
Appendix A.

MESA’s kap module computes the radiative opacity given $\rho$, $T$, $X$, $Z$
from a cell. For a fixed composition opacity table in the $X$–$Z$ plane, MESA
interpolates between $\rho$ and $T$ values with an on-the-fly bi-cubic spline.
The spline returns $\kappa_{R}(\rho,T)$, and its partial derivatives,
$\partial{\kappa_{R}(\rho,T)}/\partial{T}$ and
$\partial{\kappa_{R}(\rho,T)}/\partial{\rho}$.

MESA currently offers two choices for interpolating between opacity tables in
$X$–$Z$: linear or monotonic Hermite cubic spline functions (see Paxton et
al., 2011). The default is linear interpolation, as exemplified in MESA’s test
suite (Wolf et al., 2023). In this article we activate cubic interpolation in
MESA’s inlist controls to return $\kappa_{R}(\rho,T,X,Z)$:

  * cubic_interpolation_in_X = .true.

  * cubic_interpolation_in_Z = .true.

Appendix B explores the impact of adopting cubic versus linear interpolation
of opacity data tables across $X$–$Z$ in MESA. We find that linear
interpolation systematically under predicts the opacity as compared to cubic
interpolation, and recommend cubic interpolation be activated when using MESA.
We leave further exploration and improvements to MESA’s opacity interpolation
methods to future work.

Figure 2: Opacities and partial derivatives with respect to temperature and
density for a $X$ = 0.7, $Z$ = 0.02 Grevesse & Sauval (1998) abundances
generated from OPLIB radiative opacities and MESA’s kap module. Left column:
These quantities as a function of $T$ for different $\rho$ (colorbar). Dashed
black lines show scaling relations for H- and Kramers opacities. Black dots
mark the locations where a pure H composition is 1/2 ionized. Colored regions
show thermal ionization stages of key elements for nondegenerate material.
Right column: These quantities as contours in the $\rho-T$ plane. Regions
where different opacity sources dominate are labeled, as are the log R = $-$8
and log R = 1.5 table limits of the OPLIB radiative opacities. Both columns:
Black curves show the profile of a standard solar model (see Section 3). Red
circles on the solar profile mark the inner and outer boundaries of the
convective region and are connected with a dashed blue curve.

Figure 2 shows $\kappa$ and its partial derivatives
${{\partial{\kappa}}/{\partial{T}}}$ and
${{\partial{\kappa}}/{\partial{\rho}}}$ generated by the OPLIB radiative
opacities and MESA’s kap module for a $X$ = 0.7, $Z$ = 0.02 Grevesse & Sauval
(1998) abundance mixture. The left column shows these quantities as a function
of $T$ for different $\rho$. The dependence of $\kappa$ with $T$ in the upper
left plot can be approximated with three temperature ranges. At low
temperatures, log($T$/K) $\lesssim$ 4, the H- opacity dominates and scales as
$\kappa$ $\sim$ $\rho^{1/2}T^{9}$ (Hansen et al., 2004) as shown by the
labeled dotted black line in the upper left plot of Figure 2. At higher
temperatures, 4 $\lesssim$ log($T$/K) $\lesssim$ 8, the free-free (inverse
thermal bremsstrahlung) and bound-free (radiative recombination) opacity
dominates and scales as $\kappa$ $\sim$ $\rho T^{7/2}$ (Kramers, 1923; Gaunt,
1930) also shown by a labeled dotted black line. At still higher temperatures,
log($T$/K) $\gtrsim$ 8, all the opacity curves converge to $\kappa\sim$
constant, the flat plateau at the foot of the “$\kappa$ mountain” (Kippenhahn
& Weigert, 1990). When the atoms are completely ionized, then there are only
two sources of opacity, free-free and Compton scattering. For low densities,
Compton scattering dominates $\kappa_{R}$ and does not impact $\kappa_{P}$.
For high densities, free-free transitions dominate $\kappa_{R}$ and
$\kappa_{P}$.

Two additional features of the $\kappa$ curves in the upper left plot of
Figure 2 are noteworthy. The first is that the location of peak $\kappa$
shifts towards higher $T$ as $\rho$ increases. The H- opacity depends on the
abundance of neutral hydrogen. A pure atomic H composition is 1/2 ionized by
the Saha equation when (Hansen et al., 2004)

$\rho=8.02\times 10^{-9}\ T^{3/2}\ \exp(-1.578\times 10^{5}/T)\ .$ (6)

This locus of points is shown by the black circles and reflects the broad
trend in the location of peak $\kappa$, where the dominant opacity smoothly
transitions from H- to bound-free. The second feature is the Fe group opacity
bump (or $Z$ bump) at log($T$/K) $\simeq$ 5.35, centered in the colored region
denoting the thermal ionization regime of Fe, most prominent at low densities
(blue curves).

The standard solar model shown in upper left plot of Figure 2, which is
detailed in Section 3, begins at the photosphere on the left with log($T$/K) =
3.762 and log($\rho$/g cm-3) = $-$6.66. Progressing inwards, towards higher
$T$, the opacity sharply increases due to ionization of elements with low-
lying first ionization stages (e.g., Na, Mg, Al, K, Ca, and H). These stages
provide electrons for H- formation, and $\kappa$ rises by several orders of
magnitude until it reaches a maximum at approximately the half-ionization
curve for pure H, when most H is ionized and thus is not available for H-
formation (Kippenhahn & Weigert, 1990; Hansen et al., 2004). Progressing
further inwards, bound-free transitions become the main source of opacity and
still further inwards free-free transitions dominate. The solar core remains
in the domain of free-free transitions (Kippenhahn & Weigert, 1990).

The upper right plot of Figure 2 shows contours of $\kappa$ in the $\rho-T$
plane over a larger range of $T$ and $\rho$ than in the corresponding upper
left plot. The red $\kappa$ mountain is prominent. The overlaid standard solar
model profile shows that high-energy photons generated in the solar core
follow one trajectory in traversing a face of the $\kappa$ mountain before
eventually being released as lower-energy photons at the photosphere. Another
face of the $\kappa$ mountain, towards larger $\rho$, is bounded by the region
where $\kappa_{\rm cond}$ dominates the opacity. Note the log($R$) =1.5
opacity tables define the peak of the $\kappa$ mountain.

The partial derivatives of the opacity with respect to temperature
${{\partial{\kappa}}/{\partial{T}}}$ and with respect to density
${{\partial{\kappa}}/{\partial{\rho}}}$ are useful for assessing excitation of
a $\kappa$-mechanism (Saio, 1993)

$\frac{{\rm d}}{{\rm d}r}\left(\frac{\partial\kappa}{\partial
T}+\frac{\partial\kappa}{\partial\rho}\frac{1}{\Gamma_{3}-1}\right)>0\ ,$ (7)

where $\Gamma_{3}$ $\equiv$ $d{\rm ln}T/d{\rm ln}\rho|_{S}+1$ is the third
dimensionless adiabatic exponent from the equation of state. Opacity
derivatives are also useful for constructing the Jacobian matrix of
derivatives for Newton-like iterations to solve the stellar evolution
hydrodynamic equations.

Figure 3: Opacity as a function of $T$ for three values of $\log(\rho)$ (black
labels) and five compositions with Grevesse & Sauval (1998) abundances
(colored curves), generated from OPLIB radiative opacities and MESA’s kap
module. Dashed black lines show scaling relations for H- and Kramers
opacities. Black circles mark the half-ionization points for the pure atomic H
composition (dark blue curves, Equation 6) and the pure atomic He composition
(dark orange curves, Equation 8). Colored regions show thermal ionization
stages of key elements for nondegenerate material.
Figure 4: Relative differences between the 1194 OPLIB and 126 OPLIB opacity
table grids using Grevesse & Sauval (1998) abundances, generated from MESA’s
kap module, for six mixtures. The left column shows mixtures with $X$ = 0.6
and varying $Z$, and the right column shows mixtures with $Z$ = 0.015 and
varying $X$. The OPLIB log($R$) = $-$8, 1.5 table boundaries are marked with a
solid black line. The approximate location of the Z-dependent transition to an
electron conduction dominated opacity is marked with dot-dash blue curve.
Figure 5: Relative differences between the 1194 OPLIB and 126 OPAL opacity
table grids using Grevesse & Sauval (1998) abundances, generated from MESA’s
kap module, for six mixtures. The left column shows mixtures with $X$ = 0.7
and varying $Z$, and the right column shows mixtures with $Z$ = 0.02 and
varying $X$. The OP/OPAL log($R$)=1 table boundary is marked with a dashed
black line, the OPLIB log($R$) = 1.5 table boundary is marked with a solid
black line, and the approximate location of the Z-dependent transition to an
electron conduction dominated opacity is marked with dot-dash blue curve. The
thick black curve for the $X$=0.7, $Z$=0.02 mixture marks the location of a
standard solar model; the black dots mark the approximate beginning and end of
the solar convection zone (overlaid blue dashed curve). Interpolation across
the $X$–$Z$ plane uses cubic splines.

The middle row of Figure 2 shows ${{\partial{\kappa}}/{\partial{T}}}$ and the
bottom row shows ${{\partial{\kappa}}/{\partial{\rho}}}$ for this chemical
mixture. The left column shows these quantities as a function of $T$ for
different $\rho$, and the right column shows contours of these partial
derivatives in the $\rho-T$ plane. Peaks and changes of slope in these four
plots correlate with ionization stages due to the injection of additional
electrons into the plasma. Examples include the onset of light metal
ionization at low $T$, and the ionization stages of H and He at larger $T$.
The sign and magnitude of the partial derivative features are relevant for
variable stars driven by the ionization stages of different elements (e.g., He
ionization and RR Lyrae variables).

Figure 3 compares the $\kappa$ generated from OPLIB radiative opacities and
MESA’s kap module at log($\rho$/g cm-3) = $-$2, $-$6, and $-$10 for five
compositions: the same $X$ and $Z$ as in Figure 2, pure H, a solar $Z$ with
the remainder H, pure He, and a solar $Z$ with the remainder He. We first
analyze the three H-rich mixtures and then the two He-rich mixtures.

All three H-rich mixtures show a peak $\kappa$ that shifts towards higher $T$
as $\rho$ increases, reflecting the half-ionization curve of H discussed for
Figure 2. A difference occurs in $\kappa$ between these three H-rich mixtures
at log($T$/K) $\lesssim$ 3.8, reaching a three orders of magnitude difference
for log($\rho$/g cm-3) = $-$2 at log($T$/K) = 3.5. Here, the lack of electrons
from metals with relatively low ionization potentials in the pure H
composition (blue curve) delays the onset of a dominant H- opacity until H
ionizes. Another difference occurs at log($T$/K) $\gtrsim$ 5.0 for all three
densities, where the lack of electrons from Fe group atoms in the pure H
composition (blue curve) suppresses the bound-free and free-free opacities
relative to the two mixtures with a solar compliment of Fe.

Helium contributes four main opacity sources: He- free–free (Somerville, 1965;
John, 1994), He${}^{+}_{2}$ bound–free and free-free (Ignjatović et al.,
2009), He Rayleigh scattering (Iglesias et al., 2002; Rohrmann, 2018), and
triple-He collision (Kowalski, 2014; Blouin et al., 2019). The two He-rich
mixtures in Figure 3 (orange curves) have opacities that are usually about an
order of magnitude smaller for a given $T$ and $\rho$ than the opacities of
the H-rich mixtures before electron scattering dominates. Below log($T$/K)
$\lesssim$ 4.2, the presence of free electrons from the ionization of trace
metals with relatively low ionization potentials introduces He- free–free
electron scattering and He${}^{+}_{2}$ bound–free absorption that increases
$\kappa$ by several orders of magnitude in the He composition with solar $Z$
(Saumon et al., 2022). Above log($T$/K) $\gtrsim$ 5.0 for all three densities,
the lack of electrons from Fe group atoms in the pure He composition (dark
orange curve) reduces the bound-free and free-free opacities relative to the
mixture with a solar Fe abundance (light orange curve). The $\kappa$ curves
for log($\rho$/g cm-3) = $-$2, $-$6 show undulations due to He0 in excited
states, H0 in its ground state and He+ in excited states, and He+ in its
ground state (Seaton et al., 1994). Like the three H-rich mixtures, both He-
rich mixtures also show a peak $\kappa$ that shifts towards higher $T$ as
$\rho$ increases. An approximate expression for the half-ionization curve for
a pure atomic He mixture is

$\rho=4\cdot 8.02\times 10^{-9}\ T^{3/2}\ \exp(-2.853\times 10^{5}/T)\ .$ (8)

The half-ionization points for pure atomic H (Equation 6) and pure atomic He
(Equation 8) mixtures are shown by the black circles in Figure 3. These
approximations broadly reflect the trend, but miss the $\kappa$ peaks due to
missing physics in the approximations, for example the abundance of H- in a
hydrogen mixture.

###  2.4 Verification of the OPLIB opacities

Here we illustrate some differences between the 1194 OPLIB opacity tables, a
degraded set of 126 OPLIB opacity tables, and the 126 OPAL opacity tables.
Figure 4 shows the relative opacity differences between the OPLIB 1194 table
grid (denoted $\kappa_{OPLIB_{1194}}$) and the OPLIB 126 table grid (denoted
$\kappa_{OPLIB_{126}}$) for six mixtures in the $\rho-T$ plane. The six
mixtures are chosen so that they lie between or outside the 126 OPLIB table
grid in the $X$–$Z$ plane. For constant $X=0.6$ across $Z=0.0005,0.015,0.2$,
we find interpolation over the 1194 table grid results in $\approx$ 2–2.5%
change in opacity over the 126 table grid for $Z\leq 0.1$. Since the 126 OPLIB
opacity tables only reach $Z$ = 0.1, MESA returns the opacity value at $Z$ =
0.1 for $Z$ $>$ 0.1. For $Z$ = 0.2 in the top left panel of Figure 4, the 1194
OPLIB tables provide improvements up to $\approx$ 50% in MESA. The right
column of Figure 4 shows the opacity differences at $Z=0.015$ across
$X=0.05,0.15,0.925$. We find interpolation over the 1194 OPLIB tables show a
$\approx$ 2% change in the opacity at larger H mass fractions and up to 75%
change in the opacity at smaller H mass fractions in the temperature range 3.8
$\lesssim$ $\log$T $\lesssim$ 4.8 due to the ionization of H and He. These
differences are a direct result of the enhanced table density between 0 $\leq
X$ $\leq$ 0.1.

Figure 5 shows the opacity differences between the OPLIB 1194 tables and the
OPAL 126 tables for six mixtures in the $\rho-T$ plane. In regions between the
atomic radiative opacities and the conductive opacities with no table
coverage, there can be large differences since MESA uses the values from the
log($R$) table edge and then extends/mixes these radiative opacities with the
conductive opacities. Overall, we find differences between the 1194 OPLIB
tables and the 126 OPAL tables extend from $\simeq$ 20% at $X$=0.7, $Z$ = 0.02
to upwards of 40–80% for other mixtures, especially at low $T$ and moderate
$\rho$. In the case a solar model profile, shown in the left–middle panel at
approximately solar metallicity, $X=0.7$ and $Z=0.02$, OPLIB opacity tables
consistently predict a higher opacity at the base of the convective envelope
and lower opacity in the Solar core at otherwise identical conditions.

##  3 Standard Solar Models

In this section we detail the construction of standard solar models and their
input physics. We quantify the resulting differences in their internal
structure and compare with observational helioseismic and neutrino flux
constraints. Standard solar models have been previously calculated using OPLIB
opacities in Colgan et al. (2016); Guzik et al. (2016); Raecke (2022) and
OPLIB-L opacities by Neuforge-Verheecke et al. (2001b). We construct standard
solar models with four differing photospheric estimates of the solar heavy
element abundance: $Z/X$ $=0.0181$ (AGSS09, Asplund et al., 2009), $Z/X$
$=0.0229$ (GS98, Grevesse & Sauval, 1998), $Z/X$ $=0.0187$ (AAG21, Asplund et
al., 2021), and $Z/X$ $=0.0225$ (MB22, Magg et al., 2022). We compare solar
models computed with OPLIB opacities to the default OP and OPAL opacity tables
provided by MESA. We broadly find that standard solar models produced with
OPLIB opacities have systematically lower core opacities and temperatures, and
higher core densities resulting in lower neutrino fluxes and markedly
different helioseismology than OP/OPAL models. We also find that standard
solar models computed with higher metallicity mixtures (GS98 or MB22) are in
better agreement with helioseismic and neutrino constraints than low
metallicity mixtures (AGSS09 or AAG21) regardless of the adopted opacity
table.

###  3.1 Input Physics

We use MESA version r22.11.1 to construct our stellar models (Paxton et al.,
2011, 2013, 2015, 2018, 2019; Jermyn et al., 2023). Each star is modeled as a
single, non-rotating, non-mass losing, solar metallicity object. We use the
built-in MESA nuclear reaction network mesa_28. Relatively large nuclear
networks are required to fully capture the energy generation rate (Farmer et
al., 2016), and thus for example the neutrino luminosity (Farag et al., 2023).
The current defaults for nuclear reaction rates are described in Appendix A.2
of Paxton et al. (2019).

Rates are taken from a combination of NACRE (Angulo et al., 1999) and the
Joint Institute for Nuclear Astrophysics REACLIB library (default version,
dated 2017-10-20) (Cyburt et al., 2010), supplemented with NACRE II reaction
rates provided in Xu et al. (2013). We adopt the weak-decay rate for 7Be from
Simonucci et al. (2013). Note there is a long literature on the sensitivity of
solar model neutrino fluxes to uncertainties in the nuclear physics (Bahcall &
Ulmer, 1996; Degl’Innoccenti et al., 1997; Bahcall & Pinsonneault, 2004;
Bahcall et al., 2006; Haxton & Serenelli, 2008; Adelberger et al., 2011;
Haxton et al., 2013; Vissani, 2019; Villante & Serenelli, 2021; Bellinger &
Christensen-Dalsgaard, 2022). The MESA screening corrections are from Chugunov
et al. (2007), which includes a physical parameterization for the intermediate
screening regime and reduces to the familiar weak (Dewitt et al., 1973;
Graboske et al., 1973) and strong (Alastuey & Jancovici, 1978; Itoh et al.,
1979) limits at small and large values of the plasma coupling parameter. All
the weak reaction rates are based (in order of precedence) on the tabulations
of Langanke & Martínez-Pinedo (2000), Oda et al. (1994), and Fuller et al.
(1985).

We use MESA’s default equation of state (Jermyn et al., 2023), where a
standard solar model’s $\rho-T$ profile lies in the domain of FreeEOS
calculated with a Grevesse & Sauval (1998) composition. The atmosphere is
modeled using the solar-Hopf relation analytic fit by Ball (2021) to the Hopf
function for the solar simulation by Trampedach et al. (2014).

We adopt the MESA default time-dependent convection model which reduces to the
Cox MLT model (Cox & Giuli, 1968) on long timescales typical of a solar model
(Jermyn et al., 2023). We adopt the Ledoux criterion for convective stability,
and we adopt a semiconvection efficiency coefficient of $\alpha$ = 0.1. We use
the convective premixing routine (Paxton et al., 2019) to determine the
location of convective boundaries, and we do not include the effects of
convective overshooting, rotational deformation, or the effects of rotational
mixing. We include the effects of element diffusion for every isotope included
in our nuclear network by solving the unmodified Burgers equations in cgs
units and including the heat flow vector terms (Paxton et al., 2018). Each
model is self-consistently evolved with the hydrodynamics such that a radial
velocity variable is present throughout the evolution.

Table 1: Solar Calibration Parameters and properties

Component |  $X$0 |  $Y$0 |  $Z$0 | $\alpha_{mlt}$ | ($Z/X$)surf | $L_{\nu,\odot}/L_{\gamma,\odot}$ |  $R$cz,b/$R_{\odot}$ |  $Y$surf
---|---|---|---|---|---|---|---|---
Currently in MESA |  |  |  |  |  |  |  |
OPAL_126 GS98 | 0.70964 | 0.27181 | 0.01855 | 1.799 | 0.0229 | 0.024181 | 0.718 | 0.24423
OPAL_126 AGSS09 | 0.71990 | 0.265125 | 0.01498 | 1.778 | 0.0181 | 0.023909 | 0.726 | 0.23697
OP_126 GS98 | 0.71114 | 0.27028 | 0.01857 | 1.810 | 0.0229 | 0.024133 | 0.718 | 0.24298
OP_126 AGSS09 | 0.72009 | 0.26490 | 0.01500 | 1.783 | 0.0181 | 0.023882 | 0.726 | 0.23674
From this Work |  |  |  |  |  |  |  |
OPLIB-L_126_50T GS98 | 0.71156 | 0.26975 | 0.01869 | 1.9054 | 0.0229 | 0.024150 | 0.7235 | 0.24186
OPLIB_126_50T GS98 | 0.71379 | 0.26762 | 0.01859 | 1.9494 | 0.0229 | 0.023977 | 0.7167 | 0.24108
OPLIB_126 GS98 | 0.71202 | 0.26944 | 0.01854 | 1.9442 | 0.0229 | 0.024010 | 0.7166 | 0.24275
OPLIB_126 AGSS09 | 0.72564 | 0.25931 | 0.01504 | 1.9403 | 0.0181 | 0.023679 | 0.7230 | 0.23258
OPLIB_126 AAG21 | 0.72654 | 0.25795 | 0.01551 | 1.9140 | 0.0181 | 0.023658 | 0.7214 | 0.23172
OPLIB_126 MB22 | 0.71480 | 0.26694 | 0.01826 | 1.9339 | 0.0225 | 0.023986 | 0.7151 | 0.24076
OPLIB_1194 GS98 | 0.71191 | 0.26955 | 0.01854 | 1.9437 | 0.0229 | 0.024012 | 0.7166 | 0.24284
OPLIB_1194 AGSS09 | 0.72607 | 0.25886 | 0.01507 | 1.9350 | 0.0181 | 0.023670 | 0.7248 | 0.23193
OPLIB_1194 AAG21 | 0.72684 | 0.25762 | 0.01554 | 1.9095 | 0.0181 | 0.023651 | 0.7225 | 0.23122
OPLIB_1194 MB22 | 0.71469 | 0.26705 | 0.01826 | 1.9331 | 0.0225 | 0.023988 | 0.7159 | 0.24082
Vinyoles et al. (2017) |  |  |  |  |  |  |  |
OP_126 GS98 | 0.7095 | 0.2718 | 0.0187 | 2.18 | 0.0229 | - | 0.7116 | 0.2426
OP_126 AGSS09met | 0.7238 | 0.2613 | 0.0149 | 2.11 | 0.0187 | - | 0.7223 | 0.2317
Magg et al. (2022) |  |  |  |  |  |  |  |
OP_126 GS98 | 0.7095 | 0.2718 | 0.0187 | - | 0.0229 | - | 0.7122 | 0.2425
OP_126 AGSS09met | 0.7237 | 0.2614 | 0.0149 | - | 0.0178 | - | 0.7231 | 0.2316
OP_126 AAG21 | 0.7207 | 0.2638 | 0.0155 | - | 0.0187 | - | 0.7197 | 0.2343
OP_126 MB22 | 0.7090 | 0.2734 | 0.0176 | - | 0.0225 | - | 0.7123 | 0.2439
Observationa | - | - | - | - | - | - | 0.713 $\pm$ 0.001 | 0.2485 $\pm$ 0.0035

aafootnotetext: The helioseismic derived radius at the bottom of the
convective zone, $R$cz,b, and surface He mass fraction, $Y$surf, are from Basu
& Antia (1997) and Basu & Antia (2004).

Figure 6:  Fractional sound speed differences $\delta c_{s}$ $=$ $(c_{\rm
obs}$ \- $c_{s}(r))/c_{s}(r)$ between values predicted by a calibrated MESA
standard solar model $c(r)$ and $c_{\rm obs}$ values inferred from
helioseismic data (Basu et al., 2009). The 1$\sigma$ observational
uncertainties are shown as the blue bands at ordinates of zero. Top, middle,
and bottom panels are for GS98, AGSS09, and MB22+AAG21 mixtures respectively.
Black circles mark locations where $\delta c_{s}$ is evaluated. Gray bands
show the convective regions, labeled CZ.

###  3.2 Standard Solar Model Calibrations

We perform solar model calibrations to generate standard solar models and
compare our results with present day helioseismic and neutrino observation
data. We iterate on differences between the final model at $t_{\sun}$ = 4.568
Gyr (Bouvier & Wadhwa, 2010) and the solar radius, $R_{\sun}=6.957\times
10^{10}$ cm, solar luminosity, $L_{\gamma,\sun}=3.828\times 10^{33}$ erg s-1
(Prša et al., 2016), and surface heavy element abundance $Z/X$.

We use the built-in MESA simplex module to iteratively vary the mixing-length
parameter, $\alpha$, and the initial composition $X$, $Y$, and $Z$. This
calibration is performed for four estimates of the heavy element abundance at
the surface of the Sun: $Z$/X $=0.0181$ (AGSS09, Asplund et al., 2009), $Z/X$
$=0.0229$ (GS98, Grevesse & Sauval, 1998), $Z/X$ $=0.0187$ (AAG21, Asplund et
al., 2021), and $Z/X$ $=0.0225$ (MB22, Magg et al., 2022).

Calibrated parameters are listed in Table 1. The solar models are calculated
using OPAL (Iglesias & Rogers, 1996), Opacity Project (OP) (Badnell et al.,
2005), OPLIB, and OPLIB-L (see Section 2.1) Rosseland-mean radiative
opacities.

Each of the 16 stellar models calculated in this work has approximately 5000
mass zones. Each model takes approximately 1200 timesteps. Each model takes
approximately 2-4 hours to run on a 12–16 core machine. Each solar model is
run roughly 150-200 times to calibrate, taking roughly 2–4 weeks to complete.
The MESA files to reproduce our models, along with the python scipts to
reproduce all of our plots are available at
http://doi.org/10.5281/zenodo.10798600 (catalog
http://doi.org/10.5281/zenodo.10798600).

###  3.3 Helioseismic Observables

Helioseismic inversions of solar models in combination with solar oscillation
data provide observational estimates of quantities such as the radius of the
solar convection zone base (Christensen-Dalsgaard et al., 1991; Kosovichev,
1993; Basu & Antia, 1997), the convection zone helium mass fraction (Vorontsov
et al., 1991; Dziembowski et al., 1991; Antia & Basu, 1994a; Basu & Antia,
1995; Richard et al., 1998), radial profiles of sound speed and density
(Christensen-Dalsgaard et al., 1985; Antia & Basu, 1994b) two-dimensional
profiles of the rotational velocity (Thompson et al., 1996; Brown & Morrow,
1987), and other quantities such the Ledoux discriminant profile (Buldgen et
al., 2020). In this section we compare inferred helioseismic quantities from
Basu & Antia (1997, 2004); Basu et al. (2009) with those predicted by standard
solar models with (GS98, AGSS09, AAG21, and MB22) abundances and (OPLIB, OPAL,
and OP) opacity tables.

Figure 7:  Opacity (top), temperature (middle), and density (bottom)
differences between standard solar models normalized to a standard solar model
computed with OP opacities (Badnell et al., 2005). The left column is for the
Asplund et al. (2009) mixture and the right column is for the Grevesse &
Sauval (1998) mixture. New results are labeled in bold font. We use a
smoothing floor to reduce the noise generated by taking finite differences
between small changes in radii.

Figure 8:  Fractional sound speed and density differences, $\delta c_{s}$ $=$
$(c_{\rm obs}$ \- $c_{s}(r))/c_{s}(r)$ and $\delta\rho$ $=$ $(\rho_{\rm obs}$
\- $\rho(r))/\rho(r)$, between the values predicted by a calibrated MESA
standard solar model, $c_{s}(r)$ and $\rho(r)$, and the $c_{\rm obs}$ and
$\rho_{\rm obs}$ values inferred from helioseismic data (Basu et al., 2009).
The 1$\sigma$ observational uncertainties are shown as the blue bands at
ordinates of zero. Black circles mark locations where $\delta c_{s}$ and
$\delta\rho$ are evaluated. Gray bands shows the convective regions, labeled
CZ. New results are labeled in bold font.

Calculated helioseismic quantities are shown in Table 1 for each set of
opacity tables considered in this work and from Vinyoles et al. (2017) and
Magg et al. (2022). Since we have evolved these solar models with otherwise
identical input physics, disagreements between models of identical chemical
mixtures should arise from differences in the treatment of atomic opacities.
In Appendix C we highlight the necessity of adopting cubic interpolation
across $X$–$Z$ in MESA to reproduce similar helioseismic results found in the
other works.

The first column of Table 1 uses the notation “Source_N”, where “Source” is
OPLIB, OPAL, or OP and “N” is either 126 or 1194 signifying the number of
individual tables. With two exceptions, all the models shown in Table 1 use an
individual $\kappa$ table resolution described in Section 2.2. The two
exceptions are entry OPLIB_126_50T which refers to a calibrated solar model
computed with 126 individual OPLIB tables at a table resolution of 50
temperature points, and entry OPLIB-L_126_50T which uses the older OPLIB-L
tables at a table resolution also of 50 temperature points.

Table 1 also lists the initial abundances $X$0, $Y$0 and $Z$0 along with the
calibrated dimensionless mixing length parameter $\alpha_{mlt}$, the surface
metal to H abundance ratio ($Z/X$)surf, the neutrino luminosity
$L_{\nu,\odot}/L_{\gamma,\odot}$, the radius of the convection zone base
$R$cz,b/$R_{\odot}$, and the surface helium mass fraction $Y$surf. We find
both OP GS98 and AGSS09 MESA solar models agree well with the results found in
both Vinyoles et al. (2017) and Magg et al. (2022), both of which use the
GARSTEC (GARching STEllar Code, Weiss & Schlattl, 2008) with the most up to
date physical prescriptions from the “B16” solar models. These MESAmodels
adopts the photospheric AGSS09 abundance mixtures. Vinyoles et al. (2017) and
Magg et al. (2022) adopt the AGSS09 photospheric abundance mixture
supplemented by meteoric abundances for refractory elements (labeled
“AGSS09met” in Table 1). Some of the difference between the MESA OP AGSS09 and
the Vinyoles et al. (2017) OP AGSS09met models can be attributed to these
composition differences. Overall, OPLIB standard solar models produce a
smaller $R$cz,b/$R_{\odot}$ and smaller $Y$surf as compared to OP or OPAL
solar models for similar compositions.

Figure 6 shows the fractional difference in sound speed, $\delta c_{s}$, for
each calibrated solar model in Table 1 with respect to the helioseismic sound
speeds inferred in Basu et al. (2009). The OPLIB-L_126_50T model in the top
panel (blue curve) agrees more with OP and OPAL models (gray and purple curves
respectively) in the stellar core for $r/R_{o}$ $\lesssim$ 0.4, but has a
significantly larger $\delta c_{s}$ as compared to the OPLIB, OP, or OPAL
solar models for $r/R_{o}$ $\gtrsim$ 0.4. The OPLIB_126_50T model (red curve)
shows similar sound speeds to OPLIB_126 (green curve) showing that the
improved individual table resolution in the OPLIB_126 model does not have a
measurable large impact on $\delta c_{s}$. The three OPLIB models have higher
sound speeds compared to the OP, OPAL, and OPLIB-L models for $r/R_{o}$
$\lesssim$ 0.2, and lower sound speeds for 0.2 $\lesssim$ $r/R_{o}$ $\lesssim$
0.4. In the envelope for $r/R_{o}$ $\gtrsim$ 0.4 the three OPLIB models are
closer to the OP and OPAL models, yet show a difference in $\delta c_{s}$.

The AGSS09 mixture in the middle panel show improvement in $\delta c_{s}$ with
OPLIB opacities (dark and light blue curves). The AAG21 and MB22 mixtures with
OPLIB opacities in the bottom panel show overall better agreement with the
inferred solar sound speed than the GS98 and AGSS09 mixtures.

Near the peaks in $\delta c_{s}$, below the convection zone base, the 1194
OPLIB models have a larger $\delta c_{s}$ than the 126 OPLIB models for the
AGSS09 and AAG21 mixtures (second and third panels) due to a lower opacity
from the more accurate interpolations offered by the 1194 table set. The
difference is more pronounced for the AGSS09 and AAG21 mixtures due to their
lower metallicity, with $Z_{0}$ $\sim$ 0.015 as opposed to $Z_{0}$ $\sim$
0.018-0.019 in the GS98 and MB22 mixtures where using 1194 or 126 tables has
little consequence. A metallicity of $Z_{0}$ $\sim$ 0.015 lies between the $Z$
$\sim$ 0.01 and $Z$ $\sim$ 0.02 tabulations of the default 126 table set. The
new 1194 table set provides tables at $Z$ $\sim$ 0.014, 0.015, 0.016.

Figure 7 compares the $\kappa$, $T$, and $\rho$ profiles of standard solar
models evolved with GS98 and AGSS09 mixtures. The differences are normalized
to standard solar models computed with OP opacities (Badnell et al., 2005), as
they are commonly adopted standard in solar modeling studies. Models evolved
with OPLIB opacities (blue curves) have $\sim$ 6–7% lower core opacity than OP
(dashed horizontal line) over $r/R_{o}$ $\lesssim$ 0.15, while the model with
OPAL opacity (purple curve) is within $\sim$ 1% of OP across the entire
radiative core. Between 0.15 $\lesssim$ $r/R_{o}$ $\lesssim$ 0.4 the OPLIB
normalized opacity difference decreases to a local minimum near $r/R_{o}$
$\sim$ 0.4, and increases to $\sim$ $3$% at $r/R_{o}$ $\sim$ 0.65 below the
convection zone base.

Differences in $\kappa$ arising from the number of OPLIB tables adopted are
visible in Figure 7 at $r/R_{o}$ $\gtrsim$ 0.7 for the GS98 and $r/R_{o}$
$\gtrsim$ 0.5 for AGSS09 mixtures. Overall, we find differences in the opacity
table source (OP, OPAL, OPLIB) result in solar model opacity differences of
$\approx 8\%$ and $\approx 15\%$ at the bottom and top of the solar convection
zone, and up to $\approx 7\%$ in the solar core. It should be noted that only
a small portion of the total luminous flux in a convective envelope is carried
by radiation, hence its structure is primarily determined by the EOS not the
opacity. However opacity differences are important near the bottom and top of
the solar convection zone, because the convective boundary location is where
adiabatic and radiative temperature gradients are comparable to one another.
The opacity is important in the shallow super-adiabatic layers, as it sets the
entropy gradient for the convection zone as a whole, as well as the radiative
atmosphere. In more massive, hotter stars than the Sun, these opacity
differences can become very important because their envelopes can become
radiation dominated and are not always fully convective.

Standard solar models with the OPAL and OP tables have $T$ and $\rho$ profiles
that are within 0.1% of one another, while models with the OPLIB opacities
that have 0.8–1% lower $T$ and 2.5–3% higher $\rho$ in the core. Outside of
0.2 $\leq$ $r/R_{o}$ $\leq$ 0.4 OPLIB solar models show $\gtrsim$ 1–2% lower
$\rho$ than OP/OPAL solar models.

Figure 8 shows the fractional difference in sound speed and density between
calibrated solar models and inferred helioseismic values (Basu et al., 2009).
Models using the 1194 OPLIB table set are compared to models using the OP and
OPAL tables across different initial compositions. Overall, the MB22 model
shows the smallest $\delta c_{s}$ and $\delta\rho$ differences. This suggests
a high-$Z$ solution to the solar model problem (Guzik, 2008; Guzik & Mussack,
2010; Serenelli et al., 2009; Salmon et al., 2021). Recent helioseismic
determination of the solar mass-fractions by Buldgen et al. (2023b, a) favors
low-$Z$ mixtures such as AAG21. However, these two results are not necessarily
in tension as the model sets are constructed and calibrated differently.

Figure 9: Neutrino fluxes and uncertainties compared to solar neutrino
observations from the Borexino Collaboration (Bellini et al., 2011) as
presented in Haxton et al. (2013), Villante et al. (2014), and Kumaran et al.
(2021). Standard solar model results using previous opacity sets are shown in
orange, results using the new 1194 OPLIB opacity tables are shown in blue, and
results from Vinyoles et al. (2017) are shown in black. See Table 2 for the
data behind this figure.

###  3.4 Solar Neutrino Fluxes

Neutrinos are produced during H-burning on the main-sequence from the proton-
proton chain reactions p(p,e+$\nu_{e}$)2H, p(e-p,$\nu_{e}$)2H,
3He(p,e+$\nu_{e}$)4He, 7Be(e-,$\nu_{e}$)7Li, 8B(,e+$\nu_{e}$)8Be, and the CNO
cycle reactions 13N(,e+$\nu_{e}$)13C, 13N(e-,$\nu_{e}$)13C,
15O(,e+$\nu_{e}$)15N, 15O(e-,$\nu_{e}$)15N, 17F(,e+$\nu_{e}$)17O,
17F(e-,$\nu_{e}$)17O, 18F(,e+$\nu_{e}$)18O, where electron capture reactions
on CNO nuclei are included (Stonehill et al., 2004). The neutrino flux in the
solar interior is strongly dependent on the core $T$ (see Bahcall & Ulmer,
1996). Standard solar models that accurately predict the $T$ profile of the
solar core should also generate comparable neutrino fluxes to solar neutrino
data (Farag et al., 2020).

Neutrino fluxes $\Phi$ from each MESA calibrated solar model and Vinyoles et
al. (2017) models are compared to the observations in Figure 9 and Table 2.
$\Phi$(pp), and $\Phi$(pep) are well within 1 $\sigma$ of their inferred
observational value. The fractional variation in $\delta\Phi$(pp),
$\delta\Phi$(pep) $\propto$ $-$0.9, $-$1.4 $\delta T$ (Villante & Serenelli,
2021)is a nearly linear relationship with the fractional variation in the
temperature $\delta T$. Therefore the variation of nearly $\sim$ $9$% in the
observational value for $\Phi$(pp) and $\Phi$(pep) allow for a similar level
of deviation in these nuclear reaction rates. However, these rates are
constrained at the nearly $\sim$ 1% level (Adelberger et al., 2011),
suggesting the current neutrino flux statistics for $\Phi$(pp) and $\Phi$(pep)
do not presently constrain the $T$ of the solar core. Our results show similar
$\Phi$(pp) regardless off the adopted opacity table and slightly higher
neutrino fluxes from $\Phi$(pep) in the OPLIB based models then the OP and
OPAL based models. All solar models, regardless of the adopted opacity table,
reliably predict the value of $\Phi$(hep) to be roughly an order of magnitude
lower than the currently measured upper limit.

The $\Phi$(7Be) and $\Phi$(8B) fluxes place the tightest constraints on the
solar core $T$ with $\delta\Phi$(7Be), $\delta\Phi$(8B) $\propto$ $-$11, $-$24
$\delta T$ (Villante & Serenelli, 2021). The lower core $T$ produced by the
OPLIB based solar models result in significantly lower $\Phi$(7Be) and
$\Phi$(8B) versus the OP or OPAL models regardless of initial $Z$. Higher $Z$
solar models systematically produce larger 7Be and 8B neutrino fluxes. The
metallicity is most important in determining the CNO flux $\Phi$(CNO) which
appears to agree only with the latter metallicity GS98/MB22 models regardless
of the chosen opacity table. OPLIB based solar models show a slight decrease
in $\Phi$(CNO) resulting from their lower core $T$.

Despite many differences in model physics between our MESA models and the B16
standard solar models from Vinyoles et al. (2017), our neutrino flux results
are in good agreement with theirs. Overall, OPLIB opacities produce
sufficiently different neutrino flux and helioseismic quantities to warrant
the re-investigation of solar models with additional input physics such as
rotation, accretion, and other mixing mechanisms which could alter the
chemical stratification of these solar models (Guzik & Mussack, 2010; Wood et
al., 2018; Zhang et al., 2019; Kunitomo & Guillot, 2021; Kunitomo et al.,
2022). It is conceivable that the true solar abundances in the core are higher
than in the envelope.

Table 2: Solar Neutrino Fluxes

Component |  $\Phi$pp |  $\Phi$pep |  $\Phi$hep |  $\Phi$Be7 |  $\Phi$B8 |  $\Phi$CNO
---|---|---|---|---|---|---
Observed | $6.1\pm(0.5^{+0.3}_{-0.5})$ | $1.39\pm(0.19^{+0.08}_{-0.13})$ |  $\leq 180$ (90 % CL) | $4.99\pm(0.11^{+0.06}_{-0.08})$ |  $5.68\pm(^{+0.39}_{-0.41}$${}^{+0.03}_{-0.03})$ | $7.0^{+3.0}_{-2.0}$
Currently in MESA |  |  |  |  |  |
OPAL_126 GS98 | 5.964 | 1.412 | 7.615 | 4.856 | 5.374 | 5.395
OPAL_126 AGSS09 | 6.003 | 1.430 | 7.794 | 4.573 | 4.754 | 4.060
OP_126 GS98 | 5.970 | 1.414 | 7.641 | 4.796 | 5.221 | 5.291
OP_126 AGSS09 | 6.007 | 1.429 | 7.803 | 4.540 | 4.662 | 4.012
This Work |  |  |  |  |  |
OPLIB_168 GS98 | 5.987 | 1.447 | 7.719 | 4.640 | 4.692 | 4.892
OPLIB_126 AGSS09 | 6.035 | 1.472 | 7.955 | 4.274 | 3.974 | 3.597
OPLIB_126 AAG21 | 6.038 | 1.477 | 7.976 | 4.227 | 3.872 | 3.751
OPLIB_126 MB22 | 5.991 | 1.452 | 7.762 | 4.562 | 4.551 | 5.288
OPLIB_1194 GS98 | 5.987 | 1.447 | 7.717 | 4.643 | 4.697 | 4.896
OPLIB_1194 AGSS09 | 6.036 | 1.472 | 7.961 | 4.260 | 3.952 | 3.590
OPLIB_1194 AAG21 | 6.039 | 1.476 | 7.980 | 4.217 | 3.857 | 3.748
OPLIB_1194 MB22 | 5.991 | 1.452 | 7.760 | 4.565 | 4.555 | 5.293
Vinyoles et al. (2017) |  |  |  |  |  |
OP_126 GS98 | 5.98 | 1.44 | 7.98 | 4.93 | 5.46 | 4.88
OP_126 AGSS09met | 6.03 | 1.46 | 8.25 | 4.50 | 4.50 | 3.52

aafootnotetext: Observations from the Borexino Collaboration (Bellini et al.,
2011) as presented in Haxton et al. (2013); Villante et al. (2014); Kumaran et
al. (2021). Flux scales for $\Phi$ (in cm-2 s-1) are: $10^{10}$ (pp); $10^{8}$
(pep); $10^{3}$ (hep); $10^{9}$ (Be); $10^{6}$ (B); $10^{8}$ (CNO).

##  4 Conclusion

We presented highlights of a new set of 1194 Type-1 Rosseland-mean opacity
tables for four different metallicity mixtures. These new Los Alamos OPLIB
radiative opacity tables are an order of magnitude larger in number than any
previous opacity table release, and span regimes where previous opacity tables
have not existed. For example, the new set of opacity tables expands the
metallicity range to $Z$ = 10-6 to $Z$ = 0.2 which allows improved accuracy of
opacities at low and high metallicity. In addition, the table density in the
metallicity range $Z$ = 10-4 to $Z$ = 0.1 are enhanced to improve the accuracy
of opacities drawn from interpolations across neighboring metallicities.
Finally, there are new opacity tables for hydrogen mass fractions between $X$
= 0 and $X$ = 0.1 including $X$ = $10^{-2},10^{-3},10^{-4},10^{-5},10^{-6}$
that can improve stellar models of hydrogen deficient stars. This larger set
of opacity tables result in a $\approx 2-2.5\%$ improvement across the $X$–$Z$
plane. At $Z>0.1$ the improvement can reach up to $50\%$ in MESA. The largest
improvement, up to $\approx 70$, is observed between $3.8\leq$ log$T$ $\leq
4.8$ and $0<X<0.1$: the hydrogen poor regime which can be encountered in
stellar models with effective gravitational settling. Differences between
using the new 1194 OPLIB opacity tables and the 126 OPAL opacity tables range
from $\approx 20-80\%$ across individual chemical mixtures.

We implemented and verified these new OPLIB radiative opacity tables in MESA.
We found that calibrated solar models produced with these new OPLIB tables
agree broadly with previously published helioseismic and neutrino results
found in the literature. In hotter and denser regimes, our calculated OPLIB
opacity tables yield higher opacities than OP/OPAL. In colder models such as
solar models, OPLIB opacity tables predict lower core opacities, lower core
$T$, and higher core $\rho$ than OP/OPAL. Overall, we find differences in the
opacity table source (OP, OPAL, OPLIB) result in solar model opacity
differences of $\approx 8\%$ and $\approx 15\%$ at the bottom and top of the
solar convection zone, and up to $\approx 7\%$ in the solar core. We find that
standard solar models produced with the OPLIB opacities produce lower neutrino
fluxes and markedly different helioseismology than OP/OPAL models. We also
find that standard solar models computed with higher metallicity mixtures
(GS98 or MB22) are in better agreement with helioseismic and neutrino
constraints than low metallicity mixtures (AGSS09 or AAG21) regardless of the
adopted opacity table. We find that the new OPLIB opacity tables do not solve
the solar modeling problem when used in standard solar models. This suggests
that physical mechanisms other than the atomic radiative opacity should be
further investigated in order to solve the solar modeling problem (Guzik,
2008; Guzik & Mussack, 2010; Serenelli et al., 2009; Salmon et al., 2021;
Eggenberger et al., 2022; Buldgen et al., 2023a).

We also tested the opacity interpolation schemes adopted in MESA for
interpolation across the $X$–$Z$ plane. We find that linear interpolation
systematically under predicts the opacity by $3-60\%$ across metallicities and
up to $10\%$ in solar models, as compared to cubic interpolation. We find MESA
solar models must adopt cubic interpolation in $X$–$Z$ to produce helioseismic
results consistent with other published works. We leave further exploration
and improvements to MESA’s opacity interpolation methods to future work.

The low temperature opacity tables used in this article will be included in
the forthcoming public release of MESA, and can be directly downloaded from
http://www.wichita.edu/academics/fairmount_las/physics/Research/opacity.php.
The new Los Alamos OPLIB radiative opacity tables presented in this article
will be available at http://aphysics2.lanl.gov/opacity/lanl and also included
in a forthcoming public release of MESA. Users can also generate opacity
tables for their own desired mixtures at
http://aphysics2.lanl.gov/opacity/lanl. We encourage future stellar physics
research to experiment with this expanded set of Type-1 Rosseland-mean opacity
tables.

## Acknowledgements

We thank Luis Trivino at Los Alamos National Laboratory for his pivotal help
in providing computing resources for this project. We also thank Jason
Ferguson and David Alexander for providing the low temperature opacity tables
for the MB22 and AAG21 mixtures. This research was partially supported by the
National Science Foundation under grant 2154339 and by the U.S. Department of
Energy through the Los Alamos National Laboratory. Los Alamos National
Laboratory is operated by Triad National Security, LLC, for the National
Nuclear Security Administration of U.S. Department of Energy (Contract No.
89233218CNA000001). This research made extensive use of the SAO/NASA
Astrophysics Data System (ADS).

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Appendices A, B and C detail the implementation and verification of the new
OPLIB tables in MESA. Appendix A discusses the use of bicubic splines on the
raw OPLIB tables, Appendix B compares linear and cubic interpolation of
opacities across $X$–$Z$ in MESA, and Appendix C compares the helioseismic
differences between MESA solar models calculated with linear and cubic
interpolation of the opacities across the $X$–$Z$ plane.

##  Appendix A Bicubic splines of OPLIB tables in MESA

Figure 10: Top panel: opacity versus temperature for different log($R$) from a
GS98 OPLIB opacity table at $X$=0.7, $Z$=0.02 for different log($R$). The
bicubic spline interpolation are overlaid as a dashed curve. Bottom panel:
normalized difference between the bicubic spline interpolations and the raw
OPLIB tables at various values of log($R$).

To ensure smooth opacity derivatives, OPAL and OP opacity tables have
historically been run through smoothing and spline-fitting routines, see
Seaton (1993). We generate bicubic spline versions of the raw OPLIB tables
using a python Scipy routine. We interpolate the original 74 log($T$/K)
$\times$ 39 log($R$) point OPLIB tables spanning 3.764 $\leq$ log($T$/K)
$\leq$ 9.065 and $-$8.0 $\leq$ log($R$) $\leq$ 1.5 into evenly spaced 213
log($T$/K) $\times$ 39 log($R$) opacity tables spanning 3.75 $\leq$ log($T$/K)
$\leq$ 9.05 with $\Delta T$ $=$ $0.025$, and identical spacing in log($R$).
Figure 10 shows the raw OPLIB and bicubic spline fits for a $X$ $=$ $0.7$, $Z$
$=$ $0.02$ mixture along constant log($R$) and their respective normalized
differences. Overall, the bicubic spline smoothing of the OPLIB tables results
in $\lesssim$ 0.5% changes to the overall opacity, except for log($T$/K)
$\lesssim$ 4 where the OPLIB tables are less smooth.

##  Appendix B  MESA linear versus cubic interpolation in $X$–$Z$

Figure 11: Relative differences between the 126 OPLIB opacity tables with
cubic versus linear interpolation using Grevesse & Sauval (1998) abundances,
generated from MESA’s kap module, for six mixtures. The left column shows
mixtures with $X$ = 0.6 and varying $Z$, and the right column shows mixtures
with $Z$ = 0.015 and varying $X$. The OPLIB log($R$) = -8, 1.5 table
boundaries are marked with a solid black line. The approximate location of the
Z-dependent transition to an electron conduction dominated opacity is marked
with dot-dash blue curve.

The default interpolation scheme across the $X$–$Z$ plane in MESA uses linear
interpolation:

  * cubic_interpolation_in_X = .false.

  * cubic_interpolation_in_Z = .false.

Figure 11 shows the normalized opacity differences between the cubic
interpolated OPLIB 126 table set (denoted $\kappa_{cubic}$) and the linear
interpolated OPLIB 126 table set (denoted $\kappa_{linear}$) for six mixtures
in the $\rho-T$ plane. The six mixtures are chosen to lie between the
available tables in the $X$–$Z$ plane. Overall we find linear interpolation
systematically under predicts the opacity by up to 60% depending on the
metallicity.

##  Appendix C MESA solar models with linear versus cubic interpolation in
$X$–$Z$

We briefly illustrate the helioseismic differences arising from using linear
and cubic interpolation in the $X$–$Z$ plane for standard solar models
calibrated with the OP/OPAL opacity tables and GS98/AGSS09 abundances included
in MESA. Figure 12 shows the relative sound speed and density profiles of
solar models calculated with linear and cubic interpolation. The agreement
with helioseismic data is systemically worse for solar models that use linear
interpolation in $X$–$Z$. The sound speed and density profiles from models
that use cubic interpolation are in better agreement with solar model results
found in the literature (e.g., Vinyoles et al., 2017; Magg et al., 2022) than
those run with linear interpolation.

Figure 12:  Fractional sound speed and density differences, $\delta c_{s}$ $=$
$(c_{\rm obs}$ \- $c_{s}(r))/c_{s}(r)$ and $\delta\rho$ $=$ $(\rho_{\rm obs}$
\- $\rho(r))/\rho(r)$, between the values predicted by a calibrated MESA
standard solar model, $c_{s}(r)$ and $\rho(r)$, and the $c_{\rm obs}$ and
$\rho_{\rm obs}$ values inferred from helioseismic data (Basu et al., 2009).
The 1$\sigma$ observational uncertainties are shown as the blue bands at
ordinates of zero. Black circles mark locations where $\delta c_{s}$ and
$\delta\rho$ are evaluated. Gray bands shows the convective regions, labeled
CZ.