11institutetext: Department of Astronomy, University of Geneva, Chemin Pegasi
51, 1290 Versoix, Switzerland
11email<EMAIL_ADDRESS>22institutetext: IRAP, CNRS UMR 5277 &
Université de Toulouse, 14 avenue Edouard Belin, 31400 Toulouse, France
33institutetext: Institut d’Astrophysique de Paris, UMR7095 CNRS, Sorbonne
Université, 98bis Bd. Arago, 75104 Paris, France 44institutetext: University
of Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg, France
55institutetext: LUPM, Université de Montpellier, CNRS, Place Eugène
Bataillon, 34095 Montpellier, France

# A coherent view of Li depletion and angular momentum transport to explain
the Li plateau – from Population II to Population I stars

Sviatoslav Borisov 11 Corinne Charbonnel  Nikos Prantzos 1122 Thibaut Dumont
33 Ana Palacios 4455

(Received September 15, 1996; accepted March 16, 1997)

Unraveling the cosmological Li problem – the discrepancy between Big Bang
nucleosynthesis predictions and observed values in the Spite plateau –
requires a comprehensive exploration of stellar evolution. In this study, we
utilized the code STAREVOL to compute the stellar evolution models with atomic
diffusion, rotation-induced processes, parametric turbulence, and additional
viscosity. We calibrated the models to fit the abundance of Li in Population
II stars selected from the GALAH DR3 spectroscopic survey and literature
compilation based on their chemical composition. The calibration reveals the
significance of parametric turbulence in counteracting atomic diffusion
effects. These models predict the constancy of the Spite plateau as a function
of $T_{\mathrm{eff}}$ and [Fe/H] which agrees with the observational trend
found after a detailed selection of dwarf non-peculiar stars. Other dwarfs
that lie below the Spite plateau are either CEMP or have other types of
chemical peculiarities, reinforcing the notion of their environmental origin.
The Li abundance near the Spite plateau of the most Fe-deficient star,
J0023+0307, which is not CEMP, provides additional evidence for the stellar
depletion solution of the Li cosmological problem. Also, our models predict a
transition from Li constancy at low metallicities to dispersion at high
metallicities which is seen in observations. In addition, we extend our
analysis to include a comparison with observational data from the globular
cluster NGC 6752, showcasing excellent agreement between model predictions and
Li and Mg trends in post-turnoff stars. This opens avenues for refining the
estimates of initial Li abundance in metal-rich globular clusters which would
help to constrain Li evolution in the Milky Way.

###### Key Words.:

stars: abundances – stars: evolution – stars: Population II – Galaxy:
abundances – Galaxy: evolution – Galaxy: halo

##  1 Introduction

The heaviest and most abundant lithium isotope (7Li, hereafter Li) is known to
be the only one in the periodic table that has three different astrophysical
sources: thermonuclear reactions in the expanding universe during the first
minutes after the Big Bang (so-called Big Bang nucleosynthesis, or BBN),
spallation nuclear reactions between Galactic cosmic rays (GCR) and
interstellar matter, and thermonuclear reactions in stars. The first of the
three sources of 7Li is theoretically predicted by BBN models (Wagoner et al.,
1967) and observationally confirmed by the discovery of the so-called “Spite
plateau” (Spite & Spite, 1982a). The second one is theoretically predicted by
Reeves et al. (1970) and observationally inferred by the fact that the minor
isotope 6Li can only be produced in GCR (Meneguzzi et al., 1971). The
contribution of those two sources together has been quantitatively evaluated
to less than half of the solar abundance of 7Li (Prantzos, 2012); this makes
the stellar source the most important Li producer in the Universe. However,
although several stellar sites of 7Li production have been identified over the
years (novae, Red Giants, Asymptotic Giant Branch stars, core-collapse
supernovae and even merging white dwarfs) and despite decades of theoretical
and observational efforts, the major contributor (assuming there is one) has
not been identified yet. The reasons are both theoretical and observational.

Indeed, the yields of the various candidate sources are still highly
uncertain. This concerns stars on the red giant branch (RGB, Sackmann &
Boothroyd, 1999), the asymptotic giant branch (AGB, Sackmann & Boothroyd,
1992; Iwamoto, 2009; van Raai et al., 2012; Karakas & Lugaro, 2016), classical
novae (Arnould & Norgaard, 1975; Starrfield et al., 1978, 2020; José &
Hernanz, 1998; José et al., 2006; Denissenkov et al., 2014), core-collapse
supernovae (Woosley et al., 1990; Woosley & Weaver, 1995; Nakamura et al.,
2010; Kusakabe et al., 2019), or merging white dwarfs (Longland et al., 2012).
This limits the predictive power of Galactic chemical evolution models
(Travaglio et al., 2001; Prantzos, 2012; Matteucci et al., 2021; Roca-Fàbrega
et al., 2021; Romano et al., 2021; Kemp et al., 2022; Grohs & Fuller, 2023).
Interest in novae has been revived in recent years, through observations and
quantitative evaluation of their Li yields (see Molaro et al., 2023, and
references therein). However, the uncertainties affecting the evolution of the
nova rate, as well as the overproduction of minor CNO isotopes by nova models
if novae are the major Li producers, still prevent a definitive conclusion
(José & Hernanz, 1998; Prantzos et al., 2017).

On the other hand, the theoretical contribution of BBN (so-called primordial
Li) is constrained to a very high degree of accuracy in the standard
cosmological model from the analysis of the Cosmic Microwave Background (CMB,
Planck Collaboration et al. 2020, 2021) and nuclear and particle physics (e.g.
Steigman, 2007; Cyburt et al., 2016; Pitrou et al., 2018; Iliadis & Coc, 2020;
Hayakawa et al., 2021; Pizzone et al., 2023). However, the primordial Li/H
value predicted by BBN+CMB exceeds by a factor of $\sim$three the so-called Li
plateau on which the highest photometric Li abundances observed in old, metal-
poor Galactic turnoff stars, which are the only observable relics of the early
Universe for this isotope, are lying (Spite & Spite, 1982b; Rebolo et al.,
1988; Thorburn, 1994; Ryan et al., 2001b; Charbonnel & Primas, 2005; Sbordone
et al., 2010; Hosford et al., 2010; Gao et al., 2020). The Li plateau extends
over a wide range in [Fe/H] (up to -1.5 dex, with a possible meltdown in the
very Fe-poor regime below $\sim-3.0$ dex; e.g., Matsuno et al. (2017) and
references therein) and a specific range in effective temperature (above $\sim
5800-6000$ K depending on the observational studies; e.g., Norris et al.
(2023) (hereafter N23) and references therein). A similar Li plateau has been
discovered in dwarf spheroidal and ultra-faint galaxies, as well as in
accreted satellites such as Gaia-Enceladus and three members of the S2-Stream
(Mucciarelli et al., 2014; Hill et al., 2019; Molaro et al., 2020; Aguado et
al., 2021; Matteucci et al., 2021). It has also been found in $\omega$
Centauri (Monaco et al., 2010). This pattern thus does not depend on
environmental effects, even though the lack of stars with Li plateau values in
the extremely Fe-poor regime ([Fe/H] $\leq-4$ dex) might require explanations.
It has been suggested that the astration of a large fraction of the
interstellar medium (ISM) of galactic haloes by a first generation of stars
would have effectively destroyed Li before the formation of the halo dwarfs we
are observing today (Piau et al., 2006), but this would lead to an important
overproduction of metals (Prantzos, 2010). Alternatively, a strong primordial
magnetic field could lead to the chemical separation of Li+ ions during the
hierarchical structure formation and reduce its abundance within the collapsed
structures (Kusakabe & Kawasaki, 2019). Physics beyond the Standard
$\Lambda$CDM model has also been invoked to lower the theoretical primordial
Li abundance and reconcile it with the level of the Spite plateau (Jedamzik,
2004; Iocco et al., 2009; Fields, 2011; Iocco et al., 2009; Fields, 2011; Coc
& Vangioni, 2017; Deal & Martins, 2021). In this case, however, the challenge
is to keep the predictions for the primordial abundances of deuterium and
helium-4 unaffected, since the present agreement with those deduced from the
absorption spectra of high-redshift quasars and observations of HII regions in
metal-poor galaxies, respectively, is excellent (Izotov et al., 2014; Valerdi
et al., 2021; Cooke et al., 2016; Zavarygin et al., 2018; Hsyu et al., 2020;
Aver et al., 2021), making the concordance between CMB and BBN one of the
greatest triumphs of physics.

While Beyond Standard Model physics remains open (Grohs & Fuller, 2023, and
references therein), many studies have explored the so-called stellar
depletion solution to reconcile the Li value predicted by standard BBN and its
abundance along the plateau, considering that this fragile element burns at
relatively low temperature ($\sim 2.5\times 10^{6}$ K) in stellar interiors.
This is supported by decades of spectroscopic surveys that have depicted the
Li abundances in low-mass stars belonging to different Galactic subpopulations
(halo, open and globular clusters, thin and thick discs). It is now firmly
established that the photospheric abundance of Li decreases as low-mass stars
(including the Sun) age on the pre-main and main sequences, in proportions
that depend on several factors among which the mass and the metallicity of the
star, and its rotation rate (e.g. Soderblom et al., 1993; Sestito & Randich,
2005; Bensby & Lind, 2018; Deliyannis et al., 2019; Jeffries et al., 2021). It
is also well accepted that microscopic (i.e., atomic diffusion) and
macroscopic transport processes easily modify the chemical element
distribution within stellar interiors as well as the surface composition
compared to that with which low-mass stars were born (e.g. Michaud et al.,
2015, and references therein). Clear signatures of their interactions and
global effects were evidenced through the observed trends with evolutionary
stage of the atmospheric abundances of various heavy elements in Population II
turnoff and subgiant stars in several globular clusters (Korn et al., 2006,
2007a; Nordlander et al., 2012a; Gruyters et al., 2013, 2016; Gavel et al.,
2021a). In their seminal paper, Richard et al. (2002a) concluded that “the
Spite plateau for Li in low-metallicity field stars remains the strongest
argument for the presence of a (hydrodynamic) process competing with atomic
diffusion”. These macroscopic processes can also lead to the photospheric Li
depletion in Population I as well as in Population II low-mass stars (Pop I
and PopII hereafter; e.g. Michaud et al., 1984; Vauclair, 1988; Proffitt &
Michaud, 1991; Chaboyer & Demarque, 1994; Vauclair & Charbonnel, 1995, 1998;
Pinsonneault et al., 2000; Salaris & Weiss, 2001; Richard et al., 2002b, 2005;
Korn et al., 2006; Piau, 2008; Vick et al., 2013; Deal et al., 2020). However,
while atomic diffusion can be computed from first principles as it results
from internal gradients of pressure, temperature, and composition, the
description of macroscopic flows of different natures (turbulence, rotational
currents, magnetic instabilities, convection, mass loss, accretion) and of
their interactions remains extremely challenging in 1D stellar evolution
models. The fact that some of these mechanisms transport both chemical
elements and angular momentum in stellar interiors adds another layer of
complexity, leading in some cases to the use of simple parameterizations that
can then constrain the underlying physics (e.g. Talon et al., 2006; Decressin
et al., 2009a; Mathis, 2013; Dumont et al., 2021b), with additional
constraints from asteroseismology in the case of Population I stars (e.g.
Richard et al., 1996; Piau et al., 2003; Guzik & Mussack, 2010; Mathis et al.,
2021; Eggenberger et al., 2022a; Buldgen et al., 2023).

The actual challenge in reproducing the Li plateau lies in its constancy over
a wide range in [Fe/H] and a specific range in effective temperature for Pop
II stars, whereas Li varies greatly among Pop I stars. This is what we explore
in this work, applying in stellar models of low-metallicity stars along the Li
plateau the same assumptions for the transport of chemicals and angular
momentum as Dumont et al. (2021a, b, hereafter D21a,b) who studied lithium and
rotation in solar-type field stars and F- and G-type stars in Galactic open
clusters. This paper is structured as follows: in §2, we discuss the input
physics and transport processes included in our models. §3 is dedicated to the
calibration of the models to fit the Li plateau value. We discuss in §4 data
selection and properties of the observed field Pop II stars and how they match
the model prediction. In this section, we also discuss the reasons for the
dispersion of Li abundance at high metallicities. In addition, we also compare
our models with the behaviour of Li and Mg in post-turnoff stars of the
globular cluster NGC 6752. We summarize our results in §5.

##  2 Stellar evolution models

###  2.1 Basic input physics of the grid of models

For stellar evolution modeling, we use the code STAREVOL (Siess et al., 2000;
Palacios et al., 2006; Decressin et al., 2009b; Lagarde et al., 2012; Amard et
al., 2019) with the same basic input physics and prescriptions for various
transport processes of chemicals and angular momentum (see §2.2) as in D21b.
The models are computed with the OPAL opacity tables (Iglesias & Rogers, 1996)
and Wichita opacity database (Ferguson, J. W., private communication) for the
high- and low-temperature regimes ($T>8000$ K and $T<8000$ K respectively),
nuclear reaction rates from the NACRE2 database (Xu et al., 2013a, b), and the
analytical equation of state of Eggleton et al. (1973) and Pols et al. (1995)
described in Siess et al. (2000). The surface boundary conditions account for
Krishna Swamy (1966) correction to the grey atmosphere approximation with the
numerical surface and the connection to the atmosphere set at optical depths
equal to $5\times 10^{-3}$ and 2 respectively (for details see Amard et al.,
2019). The convective boundaries are obtained with the Schwarzschild
criterion, with the value for the mixing length parameter
($\alpha_{MLT}=2.2236$) that was calibrated on the solar model with the same
input physics including atomic diffusion and rotation by D21b. We adopt the
corresponding initial mass fraction Y for He in the Sun (0.2718). For models
of different metallicities (Z, mass fraction of metals heavier than He), we
assume the $\Delta Y/\Delta Z$ value derived from the above-mentioned solar
calibration with $Y_{0}=0.2484$ (Coc & Vangioni, 2017) and assume the solar
chemical mixture from Asplund et al. (2009) with Ne modified as suggested by
Young (2018).

With this input physics, we computed two grids of models that are illustrated
in Fig. 1:

  * •

a grid of Pop II models with [Fe/H] between $-4.0$ and $-0.5$ dex and
[$\alpha$/Fe] enhancement by 0.3 dex, with an extension down to -5.8 dex with
tailor-made models for the most Fe-poor star of our sample, J0023$+$0307 (§
4.1.1 and 4.1.5). For this grid, we assume an initial A(Li) equal to the
primordial BBN prediction A(Li)=2.75 dex (Pitrou et al., 2018; Coc & Vangioni,
2017). While Pop II stars have [Fe/H] lower than -1.5 or -1 dex (depending on
studies), we intentionally add models up to [Fe/H]=-0.5 dex for comparison
purposes with the Pop I models. The mass range covered by this grid
corresponds to stars presently (i.e., at an age of $\sim$12 Gyr) lying in the
$T_{\mathrm{eff}}$ range of the Spite plateau (most stars have
$T_{\mathrm{eff}}$ between $\sim$5800 K and 6500 K) as illustrated on the
lower panel of Fig. 1 (we show only models with [Fe/H] down to -4, as the
iso-$T_{\mathrm{eff}}$ lines stay vertical below that value).

  * •

a grid of Pop I models with [Fe/H] between $-0.25$ and $<+0.5$ dex and
[$\alpha$/Fe]=0 with initial masses chosen so that the models had the same
effective temperature on the zero-age main sequence (ZAMS) as the Pop II stars
of the plateau. This is illustrated in the upper panel of Fig. 1 with color-
coded $T_{\mathrm{eff}}^{\mathrm{ZAMS}}$. For these models, we assume a linear
increase of initial Li with Fe-content up to the meteoritic Li value of 3.31
dex (Anders & Grevesse, 1989) at the metallicity of the Sun.

All the models are computed from the pre-main sequence including the transport
processes described below.

Figure 1: The model grid computed for Pop I and Pop II stars down to [Fe/H]=-4
dex (upper and bottom panels respectively). The circles (including the solar
symbol) indicate the masses and [Fe/H] values of the computed models, whilst
the background shows the predicted values of
$T_{\mathrm{eff}}^{\mathrm{ZAMS}}$ which have been interpolated with a third-
degree polynomial.

###  2.2 Transport processes of angular momentum and chemicals

####  2.2.1 Global picture

Consistently adopting the nomenclature established by D21a,b, we specifically
focus on their ${}_{\nu}R1^{T_{0}}_{A}$ model111In the adopted nomenclature
${}_{\nu}R1^{T_{0}}_{A}$, $\nu$ stands for the ad hoc vertical viscosity
$\nu_{add}$, $T_{0}$ is the logarithm of the temperature at which the
parametric turbulence is anchored, and $A$ stands for the prescription for
penetrative convection. that they calibrated to fit the observed abundance of
Li of stars around solar metallicity and the radial rotation profile of the
Sun. It includes the transport of angular momentum driven by meridional
circulation, shear-induced turbulence, and parametric vertical viscosity, as
well as the transport of chemicals by atomic diffusion, rotation-induced
processes, penetrative convection, and parametric turbulence, with the
prescriptions we recall below. The corresponding references are summarized in
Table 3 together with the values of the key parameters that were adjusted by
D21a,b for field solar-type stars, G- and F-type stars, with the exception of
stars in the Li-dip222It was shown by D21a that a different prescription for
the vertical shear was required to reproduce the Li-dip that is observed in a
group of F-type stars centered around $\approx$6600 K, as observed e.g. in the
Hyades and Praesepe open clusters (Cummings et al., 2017)., in open clusters
with [Fe/H] between -0.4 and +0.15 dex.

The evolution of the abundances in the models is described by the following
equation

$\rho\frac{\partial X_{i}}{\partial t}=\frac{1}{r^{2}}\frac{\partial}{\partial
r}\left(r^{2}\rho\left(D\frac{\partial X_{i}}{\partial
r}-X_{i}\mathrm{v}_{i}\right)\right)+m_{i}\left[\sum_{j}r_{ji}-\sum_{k}r_{ik}\right],$
(1)

where $\rho$ and $r$ are the density and the radius, $X_{i}$ is the mass
fraction of the nuclei $i$ of mass $m_{i}$ and of atomic diffusion velocity
vi, $r_{ji}$ and $r_{ik}$ are the rates of the nuclear reactions producing and
destructing nuclei $i$ from nuclei $j$ and $k$ respectively. Finally, $D$ is
the sum of all the considered diffusion coefficients that describe rotation-
induced mixing (meridional circulation and shear-induced turbulence),
penetrative convection, and parametric turbulence as described below.

Table 1: Transport processes included in our models.

Process [Quantity] | Reference | Parameters adjusted |
---|---|---|---
|  | for Pop I (D21ab) | for Pop II
Atomic diffusion [C] | Thoul et al. (1994); Paquette et al. (1986) | – |
Parametric turbulence $\mathrm{D_{T}}$ [C] | Richer et al. (2000); Richard et al. (2005) | log$T_{0}$=6.42 | log$T_{0}$=6.28
Penetrative convection $\mathrm{D_{A}}$ [C] | Augustson & Mathis (2019) | $d_{\mathrm{ov}}=0.0325$ |
Horizontal shear $\mathrm{D_{h}}-\mathrm{\nu_{h}}$ [C-AM] | Mathis et al. (2018) | – |
Vertical shear $\mathrm{D_{v}}-\mathrm{\nu_{v}}$ [C-AM] | Zahn (1992) | – |
Parametric viscosity $\mathrm{\nu_{add}}$ [AM] |  Eggenberger et al. (2012); D21a |  $\nu_{add}$=3.5$\times$104 cm2 s-1 |
Magnetic torque [AM] | Matt et al. (2015); Amard et al. (2019) | K=7.5$\times$1030 |

333C for chemicals and AM for angular momentum.

####  2.2.2 Transport of angular momentum and rotation-induced mixing

The treatment of stellar rotation in our models follows the formalism
developed by Zahn (1992), Maeder & Zahn (1998), and Mathis & Zahn (2004) under
the so-called shellular rotation hypothesis. The transport of angular momentum
obeys the following advection-diffusion equation:

$\rho\frac{d}{dt}(r^{2}\Omega)=\frac{1}{5r^{2}}\frac{\partial}{\partial
r}\left(\rho r^{4}\Omega U_{2}\right)+\frac{1}{r^{2}}\frac{\partial}{\partial
r}\left((\nu_{\mathrm{v}}+\nu_{add})r^{4}\frac{\partial\Omega}{\partial
r}\right),$ (2)

where $\rho$, $r$, $\Omega$, $U_{2}$, and $\nu_{\mathrm{v}}$ are the density,
radius, angular velocity, meridional circulation velocity on an isobar, and
vertical shellular component of the turbulent viscosity, respectively. For the
chemicals, the transport by meridional circulation is described with the
diffusion coefficient $\mathrm{D_{eff}}=\left|rU_{2}(r)\right|^{2}/(30D_{h})$
entering Eq. 1 (Chaboyer & Zahn, 1992), with Dh the horizontal diffusion
coefficient. As in the R1 models of D21a,b, we use the prescriptions of Mathis
et al. (2018) and Zahn (1992) for horizontal and vertical diffusivities Dh and
Dv respectively and assume a proportionality factor of 1 with the horizontal
and vertical viscosities. For the parametric vertical viscosity $\nu_{add}$
that was originally introduced by Eggenberger et al. (2012) in order to
flatten the internal rotation profile as evidenced by helio- and
asteroseismology of low-mass stars, we use the same value of $3.5\times
10^{4}$ cm2s-1 as calibrated on the solar rotation profile by D21a,b and
discuss the impact of this parameter in §3.1.

The extraction of angular momentum at the stellar surface due to magnetized
winds is accounted for following Matt et al. (2015) formalism as described in
Amard et al. (2019) with the parameters $m$=0.22 and $p$=2.1, which refer
respectively to an exponent related to the magnetic field geometry and the
exponent relating rotation and activity. For the parameter $K$ that is linked
to magnetized wind braking, we use the value of $7.5\times 10^{30}$ erg that
was calibrated by D21a to reproduce the solar surface rotation at the age of
the Sun. We assume the same initial rotation rates on the PMS as observed in
open clusters (Gallet & Bouvier, 2015, see also Amard et al. 2016 and D21a,b).
Median, slow, and fast rotation refer to initial rotation periods of 4.5, 9.0,
and 1.6 days respectively. We set the timescale of the disc coupling at 5 Myrs
in the first two cases, and 2.5 Myrs in the last one. We check the importance
of these assumptions in §3.2.

####  2.2.3 Penetrative convection

Our models include penetrative convection treated as an overshoot with the
formalism of Augustson & Mathis (2019) that affects only the chemicals. The
corresponding diffusion coefficient that enters Eq. 1 is

$D_{A}(r)=D_{0}\left[1-\exp{\left(-\exp{\left(\frac{r-r_{bcz}}{d_{\mathrm{ov}}\cdot(\mathrm{v/v_{0}})^{3/2}}+\frac{\mu}{\lambda}\right)}\right)}\right],$
(3)

where $D_{0}=\alpha_{MLT}\mathrm{v}_{conv}H_{p}/3$, vconv is the mean velocity
of convective elements, $H_{p}=\mathrm{d}r/\mathrm{d}\ln P$ is the pressure
scale height, $d_{\mathrm{ov}}$ is the parameter that controls the overshoot
depth, and $(\mathrm{v}/\mathrm{v}_{0})$ is the ratio of the velocity of the
convective elements when taking rotation into account to the non-rotating
inviscid value. We used the coefficients $\mu=5\times 10^{-3}$ and
$\lambda=6\times 10^{-3}$ prescribed by Baraffe et al. (2017), and
$d_{\mathrm{ov}}$=0.0325 as calibrated by D21a,b to reproduce the surface Li
abundance of young open clusters.

####  2.2.4 Atomic diffusion and parametric turbulence

Atomic diffusion is treated with the formalism of Thoul et al. (1994) to solve
the Burgers equations and compute the individual atomic diffusion velocities
of all the isotopes that are taken into account in STAREVOL. The collision
integrals are computed following Paquette et al. (1986). We account for the
partial ionization of chemical elements for temperatures lower than $5\times
10^{6}$ K (Schlattl 2002). Our models do not include radiative accelerations,
which are always smaller than gravity for Li in Population II stars along the
plateau (e.g. Vick et al., 2013).

Turbulence in the radiative zone is parametrized following the expression
initially proposed by Richer et al. (2000) to compete with atomic diffusion in
AmFm stars. The corresponding diffusion coefficient that enters Eq. 1 follows
a simple algebraic dependence on density:

$D_{\mathrm{T}}(r)=400D_{\mathrm{He}}(T_{0})\left[\frac{\rho(r)}{\rho(T_{0})}\right]^{-3},$
(4)

where $D_{\mathrm{He}}(T_{0})$ is the He atomic diffusion coefficient at the
layer within the star where the temperature is $T_{0}$ (see below), and that
is computed with the analytical approximation $D_{\mathrm{He}}=3.3\cdot
10^{-15}T^{2.5}/[4\leavevmode\nobreak\ \rho\ln(1+1.125\cdot
10^{-15}T^{3}/\rho)]$ from Richer et al. (2000)444This approximation for
$D_{\mathrm{He}}$ is done only when computing $D_{\mathrm{T}}(r)$, but not for
the stellar structure computation.. Since Li burning is very sensitive to
temperature, the free parameter $T_{0}$ can be adjusted to lead to more or
less photospheric Li depletion. The $\rho^{-3}$ dependence resulted from
fitting the abundance anomalies in AmFm stars by Richer et al. (2000). We
discuss the calibration of $T_{0}$ and compare it to the value derived by
D21a,b for Pop I stars in §3.2.

##  3 Model calibration

In this section, we discuss the validity of the parameters that were
calibrated for Pop I stars (see Table 3) in the models of Pop II plateau stars
for which we do not have information on their initial and internal rotation
rates. We assume A(Li)=2.3 dex555A(Li)=$\log_{10}$(NLi/NH)+12 where NLi and NH
are the number density of Lithium and Hydrogen respectively. for the plateau
value (Norris et al., 2023) and 12 Gyr for the mean age of the plateau stars.
We first focus on a “reference” star, which we chose assuming that the typical
mass of the Spite stars is M=0.74M⊙ with the chemical composition of
[Fe/H]=-2.0 and [$\alpha$/Fe]=0.3. This star has an effective temperature of
6234 K at 12 Gyr (according to our model). We then test the robustness of the
calibration across the [Fe/H] and $T_{\mathrm{eff}}$ range of the Spite
plateau.

###  3.1 Additional parametric vertical viscosity

As extensively discussed in the literature, with the currently available
prescriptions, shear turbulence and meridional circulation alone do not
account for enough angular momentum transport to reproduce the internal
rotation profile of the Sun and of some well-studied Pop I main sequence stars
(see references in Sect.1). As of today, there is no asteroseismic data to
depict the internal rotation of Pop II main sequence stars and to calibrate
the value of the parametric additional viscosity $\nu_{add}$ for them. To the
best of our knowledge, no theoretical estimate exists of the relative
efficiency in Pop II stars of the magnetic Tayler instability, which is one of
the main candidates for efficient angular momentum transport in the Sun and
solar-type stars (Eggenberger et al., 2022b, and references therein). On the
other hand, Talon & Charbonnel (2004) showed that for Pop II stars lying along
the Li plateau, the other main candidate, namely internal gravity waves,
should efficiently lead to a quasi-solid rotation state, similar to that of
the Sun (Talon et al., 2002; Charbonnel & Talon, 2005).

We assume that the theoretical behaviour of internal gravity waves legitimates
the use of the same value of $\nu_{add}$ as in the R1 models of D21a,b (i.e.,
$3.5\times 10^{4}$cm2 s-1, constant in space and time). We show in Fig. 2 the
internal profiles of angular velocity at different evolution stages for two
models of the reference star computed with $\nu_{add}$=0 and $3.5\times
10^{4}$cm2 s-1. As in the case of Pop I low-mass stars, the additional
vertical viscosity strongly flattens the internal rotation profile along the
main sequence.

Figure 2: Angular velocity profiles within the models with M=0.74M⊙,
[Fe/H]=-2.0, [$\alpha$/Fe]=0.3 computed with and without additional viscosity
($\nu_{add}=3.5\times 10^{4}$ cm2s-1 and 0, red and orange lines respectively)
at different evolution stages. Both models include rotation at the median
rate. Figure 3: Evolution of the surface Li abundance and 4He in mass fraction
(solid and dashed lines respectively) as a function of age in models with
M=0.74M⊙, [Fe/H]=-2.0, [$\alpha$/Fe]=0.3 with no additional viscosity and
parametric turbulence(red line), with additional viscosity but without
parametric turbulence (orange line), and with both taken into account (with
log $T_{0}$=6.28, blue line). The light blue band around the solid blue line
indicates the A(Li) evolution for models with $\nu_{add}=(3.5\pm 1.5)\times
10^{4}$ cm2s-1 and parametric turbulence (log $T_{0}$=6.28). The green lines
are for a model with $\nu_{add}=3.5\times 10^{4}$ cm2s-1 and parametric
turbulence as in Pop I stars (log $T_{0}$=6.42). All the models include
rotation at the median rate.

We show in Fig. 3 the impact of the $\nu_{add}$ parameter on the evolution of
the Li and He surface abundances for the same reference star models. In these
two specific cases, we turned off the parametric turbulence for chemicals
($D_{\mathrm{T}}(r)=0$, Eq. 4). Li depletion at the surface is due to the
combined effect of atomic diffusion, meridional circulation, penetrative
convection, and shear-induced turbulence, which eventually transports this
fragile element from the convective envelope to slightly deeper radiation
layers where it can burn. He abundance variations, on the other hand, result
from the relative efficiency of the hydrodynamic processes that slow down the
effects of atomic diffusion. As expected, Li is more depleted when
$\nu_{add}$=0 because of the steep rotation gradients that build inside the
star (Fig. 2), leading to more efficient shear-induced mixing, hence more Li
destruction, but also less efficient atomic diffusion on He which decreases
more slowly (see description of Fig. 4 below). When efficient angular momentum
transport is simulated by the additional viscosity, the radial profile of
angular velocity is much flatter, hence shear-induced mixing is less
efficient. Li is less efficiently driven to the burning region, leading to
slower surface Li depletion. On the other hand, He decreases more rapidly at
the surface of the star because atomic diffusion is less counteracted by the
hydrodynamical processes. In both models, however, the Li abundance at the age
of $\sim$12 Gyr is higher than the assumed plateau value of 2.3 dex. As in the
case of the ${}_{\nu}R1^{T_{0}}_{A}$ models of Pop I stars (D21a,b), an
additional process for the transport of chemicals is required to fit the Li
data, that we treat below as parametric turbulence.

###  3.2 Calibration of parametric turbulence and choice of the initial
rotation period

Figure 4: Radial profiles of diffusion coefficients (left axis) of meridional
circulation ($\mathrm{D_{eff}}$), vertical shear ($\mathrm{D_{v}}$),
parametric turbulence ($\mathrm{D_{T6.28}}$), penetrative convection
($\mathrm{D_{A}}$) and of the Li and He mass fractions (right axis) for the
models with M=0.74M⊙ ([Fe/H]=-2.0 and [$\alpha$/Fe]=0.3) with median, slow,
and fast rotation as indicated, shown on the PMS (shortly after the core
became radiative, age$\approx$1.3 Myr), at the ZAMS, and on the MS when the
central hydrogen mass fraction $X_{C}$=0.5$X_{0}$ and $0.1X_{0}$, where
$X_{0}$ is the initial H mass fraction (from left to right). The upper row
corresponds to the model without additional parametric viscosity
($\nu_{add}$=0) and the other rows are for models computed with
$\nu_{add}$=3.5 $\times 10^{4}$cm2s-1 and three initial rotation rates. The
pinkish area corresponds to the convective envelope. For the PMS structure, we
show the profiles down to the center to demonstrate the initial (almost flat)
behaviour of Li in the models.

As can be seen in Fig. 3, Li depletion in the reference star is too strong
when assuming the $T_{0}$ value for parametric turbulence $D_{\mathrm{T}}$
(Eq. 4) that was calibrated in Pop I solar-type dwarfs stars by D21b (log
$T_{0}$=6.42). We have adjusted the value of $T_{0}$ to reach the plateau
value (A(Li)=2.3 dex; Norris et al. 2023, see §4) at the age of 12 Gyr for the
reference star computed with $\nu_{add}=3.5\times 10^{4}$ cm2s-1 and median
rotation velocity. Although for calibration purposes we adopt the value of 12
Gyr as the average age of Pop II stars, we allow an age variation of about 1
Gyr for most of them. The value that best fits our assumptions is $\log
T_{0}$=6.28 (see discussion in §3.3). The time evolution of Li and He at the
surface of the corresponding model is shown in Fig. 3. Parametric turbulence
slows down the effect of atomic diffusion, as indicated by the behaviour of He
which is less depleted at the surface than in the previous models. On the
other hand, it transports more efficiently Li down to its nuclear burning
temperature below the convective envelope, leading to more Li depletion than
when the transport by only meridional circulation and shear turbulence were
accounted for.

Figure 5: Upper panel: Li evolution in the ${}_{\nu_{add}}R1^{T_{0}}_{A}$
models with M=0.74M⊙ ([Fe/H]=-2.0 and [$\alpha$/Fe]=0.3) and three different
rotation rates (slow, median, and fast). Bottom panel: Difference between Li
abundance as a function of age with respect to the model with median rotation.

The different diffusion coefficients for chemicals are shown in Fig. 4 for the
reference star models computed with different assumptions. The first three
rows from the bottom are for $\nu_{add}=3.5\times 10^{4}$ cm2s-1, parametric
turbulence $D_{\mathrm{T}}$ with $\log T_{0}$=6.28, and different initial
rotation rates taken in the same range as for Pop I stars. When angular
momentum extraction by the wind is efficient, i.e., on the PMS (left column,
the age of each model is about 1.3 Myr) and at the beginning of the MS, the
diffusion coefficients related to meridional circulation and vertical shear
($D_{\mathrm{eff}}$ and $D_{\mathrm{v}}$ respectively) are of the same order
of magnitude as the diffusion coefficient $D_{\mathrm{T}}$. Later along the
MS, however, $D_{\mathrm{eff}}$ and $D_{\mathrm{v}}$ decrease together with
the magnetic torque, and the transport of chemicals is driven by
$D_{\mathrm{T}}$. In the middle of the MS ($X_{C}$=0.5$X_{0}$, corresponding
to $\sim$7 Gyr for these models), parametric turbulence is $\sim$4 magnitudes
stronger than rotation-induced mixing and becomes even stronger at later
stages. While Li depletion on the PMS is slightly affected by the initial
rotation rate via the depth of convective penetration through the ratio
(v/v0), the final Li depletion is almost independent of the initial rotation
rate in the range considered for Pop I stars (Fig. 5). At the age of 12 Gyr,
the A(Li) difference between the models with slow and fast rotation is only
0.01 dex. Due to a lack of data on the initial rotation of young Pop II stars
or evidence for different behaviour, we decided to keep the median rotation
rate for the grid of models.

The first row from the top in Fig. 4 corresponds to a reference star model
with $\nu_{add}=0$, parametric turbulence $D_{\mathrm{T}}$ with $\log
T_{0}$=6.28, and median rotation rate. Comparison with the second row from the
top illustrates the effect on $\nu_{add}$ on meridional circulation and
vertical shear as discussed in §3.1. In particular, the vertical shear is
stronger due to the steeper differential rotation after angular momentum has
been extracted by the stellar wind. Finally, we show in Fig. 3 with the light
blue band the Li predictions for models of the same star computed with
different values of $\nu_{add}$ (2$\times 10^{4}$ and 5$\times 10^{4}$) and
with the same value of the parametric turbulence calibrated to reach the
plateau (compare the light blue band to the blue line). Since parametric
turbulence drives the Li depletion in that case, the actual. Without internal
rotation information for Pop II stars, we thus face a slight degeneracy
between the determination of the additional viscosity for the transport of
angular momentum and that of the parametric turbulence for chemicals. For this
reason, we decided not to explore possible variations of additional viscosity
in space and time as proposed e.g. by Spada et al. (2016).

For our grid of models, we thus assume the value of $\nu_{add}=3.5\times
10^{4}$ cm2.s-1 for all our models of ${}_{\nu_{add}}R1^{T_{0}}_{A}$ Pop II
stars as in the R1 models of D21a,b for Pop I stars, and median rotation with
the same initial period as for Pop I stars. The only parameter that we need to
change to reach the Li plateau value is the anchoring temperature $T_{0}$ for
parametric turbulence, with log $T_{0}$=6.28 instead of 6.42 in Pop I stars
(see further discussion in §3.3). This is consistent with Richard et al.
(2005) who determined log$T_{0}\sim 6.25-6.28$ to best fit the constancy of
the Li plateau against the effects of gravitational settling. In their study,
the log $T_{0}$ range came from the consideration of different temperature
scales used in the observational samples they considered, which lead to
different Li abundance determinations (see also Charbonnel & Primas, 2005),
and from the mean age assumed for the sample stars. The robustness across the
Li plateau (in both [Fe/H] and $T_{\mathrm{eff}}$) of the calibration we did
with the M=0.74M⊙, [Fe/H]=-2.0 models is discussed below.

Figure 6: Atomic diffusion coefficient of He just below the base of the
convective envelope for Pop II and Pop I models (open and filled circles
respectively) at the ZAMS and in the middle of the MS when $X_{C}=0.5X_{0}$
(upper and bottom panels respectively). The [Fe/H] value of the models is
color-coded. With black five-pointed stars, we show the predicted values for
the model with [Fe/H]=-5.8 computed for the most Fe-poor star J0023+0307 (§
4.1.1 and 4.1.5).

###  3.3 Robustness of the calibration across the Spite plateau and key
differences between Pop II and Pop I stars

Figure 7: Atomic diffusion coefficients of He and density gradient just below
the base of the convective envelope (left and right, respectively) for the
model grid of Pop I and Pop II stars (upper and bottom panels respectively).
The circles (including the solar symbol) show the actual values for the grid
points, whilst the colour-coded background shows the predicted values of the
same parameters which have been interpolated with a third-degree polynomial.

In their seminal paper, Richard et al. (2005, see also ) discussed the
properties of Pop II stars that lead to the constancy of the Li plateau, both
in [Fe/H] and $T_{\mathrm{eff}}$. They showed that for low-metallicity MS
stars across a large range in [Fe/H], the two main quantities that drive Li
settling through atomic diffusion and Li nuclear burning, namely the mass of
the convective envelope and the temperature at its base, depend only on the
stellar effective temperature (see Fig.2 of Richard et al. 2002b and Fig.1 of
Talon & Charbonnel 2004). With our (a priori) more sophisticated models that
include rotation-induced processes but where parametric turbulence is still
required to counteract atomic diffusion and transport Li to the burning
region, we humbly confirm their findings and extend the comparison to Pop I
stars. We discuss in particular the behaviour of the key physical properties
that potentially explain the difference in Li behaviour in the low and high
metallicity regime using our entire grid of ${}_{\nu}R1^{T_{0}}_{A}$ Pop II
and Pop I models.

Besides the mass and the temperature of the convective envelope, the important
quantities for the combined effect of atomic diffusion and turbulence on the
surface Li depletion are the density below the base of the convective envelope
and the density gradient in the radiation layers down to the Li burning
region. The first one determines the atomic diffusion timescale if turbulence
is not present, which can be represented by the atomic diffusion coefficient
just below the convective envelope. $D_{\mathrm{He}}$(BCE) is shown in Fig. 6
for Pop II and Pop I stars at the ZAMS (see also Fig. 7) and in the middle of
the main sequence. For [Fe/H] between -4.0 and -1.5 dex, Pop II stars are very
compact, with a very similar density below the base of the convective
envelope, leading to similarly small atomic diffusion coefficients at a given
effective temperature and stellar mass. The models predict similar compactness
down to [Fe/H]=-5.8 dex for which we have adopted the C and Na abundances of
the most Fe-poor sample star (yet, this star is not a CEMP-object, see § 4.1.1
and 4.1.5). This explains why Li depletion for Pop II stars of similar
effective temperature is potentially independent of [Fe/H]. At the same time,
density gradients $d\rho/dr$ at the BCE also show independence on metallicity
up to [Fe/H]$\approx$-1.5 dex. However, they decrease together with the atomic
diffusion timescale when the stellar mass increases (Fig. 7). These facts,
combined with the prescription for parametric turbulence (Eq. 4), and more
specifically the fact that it changes as $\rho^{-3}$, imply the existence of a
relatively constant value of $T_{0}$ at which the equilibrium is maintained
between atomic diffusion and parametric turbulence for Pop II stars across the
$T_{\mathrm{eff}}$ and the [Fe/H] ranges of the plateau.

When [Fe/H] increases above $\simeq$ -1.5 dex, however, the density and the
atomic diffusion coefficient below the base of the convective envelope
strongly vary with both [Fe/H] and $T_{\mathrm{eff}}$ (and mass, Fig. 6 and
7). This explains the need for a different (higher, see D21a,b) anchoring
temperature $T_{0}$ for parametric turbulence above a certain limit in
metallicity around [Fe/H]$\sim$-0.5 dex) to counteract the effect of the
atomic diffusion and lead to the observed Li depletion at a given effective
temperature.

The evolution of the surface Li abundances as a function of $T_{\mathrm{eff}}$
and [Fe/H] is shown in Fig. 8 for our grid of Pop II
${}_{\nu_{add}}R1^{T_{0}}_{A}$ models. As can be seen from Fig. 3, the
reference model star with [Fe/H]=-2.0 and M=0.74M⊙ reaches the value of
A(Li)=2.3 dex at almost exactly 12 Gyr. At this age, the model has
$T_{\mathrm{eff}}$=6234 K. While some stellar models with effective
temperatures $T_{\mathrm{eff}}$$>$6000 K and an associated lithium abundance
A(Li) of 2.3 dex reach this specific value at varying ages, the majority of
them consistently do so within the age range of 11 to 13 Gyr. This fits within
our aforementioned assumptions about age variations of Pop II stars, and,
thus, we conclude that our models can effectively describe the Spite plateau.

Figure 8: Evolution of the surface Li abundance as a function of
$T_{\mathrm{eff}}$ for stars of different initial masses and different initial
Fe content. Short horizontal ticks and numbers along the tracks indicate age
in Gyr. The dashed horizontal line and surrounding pinkish area show the mean
value of the Spite plateau (according to Norris et al. (2023)) and 0.05 dex
scatter around it. The reference model is denoted with a four-pointed star.

##  4 Observational tests to model predictions

In §3, we calibrated our models assuming the plateau value to be A(Li)=2.3 dex
according to Norris et al. (2023). We now look in more detail at their sample
of carefully selected field Galactic halo stars and complement it with data
for field stars from GALAH DR3 (Buder et al., 2021). We characterize the
sample stars, eliminate peculiar objects, and compare to our model predictions
for Li across the plateau ([Fe/H]$\leq-1.5$ dex and $T_{\mathrm{eff}}$ $\geq
5800$ K) and in more Fe-rich dwarf stars (in what follows, we use the term
“dwarf” for main sequence and main sequence turnoff stars). Finally, we use
data in the globular cluster NGC 6752 to test our model predictions for Li and
metals after the MS turnoff.

###  4.1 Field dwarf stars

####  4.1.1 N23 sample

N23 re-investigated the shape of the Spite plateau using literature data for
$\sim$200 stars with [Fe/H] between -6 and -1.5 dex, $T_{\mathrm{eff}}$
between 6700 and 5500 K, and log g $\gtrsim$ 3.5 (see references in N23).
After a careful evaluation of the uncertainties in the Li abundances resulting
from the different methods used to determine stellar $T_{\mathrm{eff}}$ in the
original papers (see also Charbonnel & Primas, 2005), they have homogeneously
re-determined the 1D,LTE Li and [Fe/H] abundances of the literature sample
stars by anchoring $T_{\mathrm{eff}}$ to the Infrared Flux Method (IRFM) of
Meléndez et al. (2010). Here we use their “Corrected-Literature” values as
recommended by the authors, and we adopt the value of 5’800 K they find for
the minimum $T_{\mathrm{eff}}$ of the Li plateau. Also, we computed 3D,NLTE
corrections with the code Breidablik (Wang et al., 2021). For this, we used
$T_{\mathrm{eff}}$, [Fe/H], and A(Li) values from N23 and log $g$ computed
from stellar luminosity $L$ and mass $M$ (see §4.1.3 and §4.1.4 for details
about the computation of luminosity and mass) as $\log g=\log
g_{\odot}+\log(M/M_{\odot})+4\log\left(T_{\mathrm{eff}}/T_{\mathrm{eff,\odot}}\right)-\log(L/L_{\odot})$,
where we adopted $T_{\mathrm{eff,\odot}}$=5772 K and $\log g_{\odot}$=4.438.
Although the corrections are very low (the mean absolute value is
$\langle|\Delta\mathrm{NLTE}|\rangle\approx 0.016$ dex), hereafter, we will
use 3D,NLTE abundance of Li for N23.

As underlined by N23, their literature sample contains stars with
peculiarities potentially caused by binarity-induced processes (e.g., mass
transfer from a companion) or environmental effects (e.g. very early Galactic
chemical evolution under the influence of Pop III stars, see e.g.Piau et al.
2006). Those should be discarded when studying the primordial Li abundance and
especially when testing the stellar Li depletion solution. Following their
discussion, we excluded stars classified as blue stragglers (BS) according to
catalogs by Ryan et al. (2001a, c) and Carney et al. (2005). We also
eliminated CEMP(-no) stars from the sample by cross-checking with Lucey et al.
(2023) who follow Beers & Christlieb (2005) and set the lower [C/Fe] limit for
CEMP stars to +0.7 dex. Finally, we excluded binaries and chemically peculiar
stars based on the classification provided by the SIMBAD database (Wenger et
al., 2000). Last, we applied for the remaining sample the method for dwarf
selection described in §4.1.3. As a result, the final sample we keep from N23
contains 57 “normal” Plateau dwarfs.

Fig. 9 shows the position of all the stars from N23 in the HR diagram for
different metallicity bins up to [Fe/H]=-1.5$\pm 0.25$ dex, with the above-
mentioned peculiarities indicated by the symbol styles, and the “normal”
dwarfs colour-coded with their Li content. Their distribution in the A(Li)
versus [Fe/H] diagram is shown in Fig. 10. In the [Fe/H] domain below $-$3 dex
where the so-called “meltdown” of the Li plateau was reported (Sbordone et
al., 2010; Bonifacio et al., 2012, 2015, 2018), the corresponding large spread
of Li abundances below the plateau is due to peculiar stars that we have
excluded based on the criteria described above. In particular and as
underlined by N23, the incidence of C-rich stars in their original literature
sample is very high in this Li-[Fe/H] domain. In the most iron-poor range
([Fe/H]$\leq$-3.75), all stars except one are extremely C-rich. These Li
plateau “outliers” are thus expected to have been Li-deficient at birth (see
e.g. Meynet et al. 2010; Norris & Yong 2019; N23). The most Fe-poor dwarf that
belongs to this bin, J0023$+$0307, is, however, not a CEMP, and it has a Li
value slightly below the plateau. The tracks we show in the corresponding
panel are for tailor-made models that we computed with [Fe/H]=-5.8 dex and the
observed [Na/Fe] and [C/Fe] values (1.9 and 3.5 dex respectively, hence not a
CEMP star; Frebel et al. 2019). This star presents abundances of
$\alpha$-elements (Mg, Si, Ca, Ti) at very different levels (from
[Ca/Fe]$\sim$0 dex to [Mg/Fe]$\sim$2.8 dex). For this reason, we computed
models with three different values of $\alpha$-enhancement –
[$\alpha$/Fe]=0.5, 1.0, and 2.0 dex. However, the difference between models
with different [$\alpha$/Fe] is negligible, and we show only models with
[$\alpha$/Fe]=1.0 dex. In our following discussion about the stellar depletion
solution, we keep only the 57 “normal” (colour-coded) Plateau dwarf stars
(with $T_{\mathrm{eff}}$ $\geq 5800$ K, and [Fe/H]$\leq$-1.5 dex).

Figure 9: Position in the HRD of all the N23 sample stars in different
metallicity bins (we use their “Corrected-Literature” value for
$T_{\mathrm{eff}}$). Grey symbols correspond to stars on the subgiant branch
(SGB) and to the peculiar objects (legend in the insert). Li abundance
(3D,NLTE) is colour-coded for the 57 “normal” (colour-coded) Plateau dwarf
stars (indicated as MS in the insert) that we use for the comparison with the
theoretical Li predictions. Solid lines show the MS part of the computed
evolutionary tracks computed for the middle [Fe/H] values of each bin except
for the upper left bin that shows models at [Fe/H]=-5.8 dex that we computed
assuming the C and Na content of the most Fe-poor dwarf, J0023$+$0307, for
which we also show the error bars on its luminosity. Figure 10: Li abundance
(3D,NLTE) as a function of [Fe/H] for the N23 sample (“Corrected-Literature”
values). We keep the same symbol notation as in Fig. 9. For the “normal”
Plateau dwarf stars (red circles), the vertical arrows indicate Li upper
limits. For the most Fe-poor dwarf, (shown with the red five-pointed star),
J0023+0307, we also show the Li value determined by Aguado et al. (2019, green
five-pointed star). The horizontal arrows indicate upper limits for [Fe/H].
The dashed line indicates the Spite plateau value of A(Li)=2.3 dex used for
the calibration of our models.

####  4.1.2 GALAH sample

We selected Plateau field dwarf stars from the GALAH DR3 sample with
-2.84$\leq$[Fe/H]$\leq$-1.25 dex and $T_{\mathrm{eff}}$ $\geq 5800$ K. The
lower limit of the metallicity range corresponds to the most metal-poor star
in GALAH DR3 after applying all the selection criteria (see below). We applied
the quality flags on stellar parameters ($T_{\mathrm{eff}}$ and log g), Li
abundance and [Fe/H]: flag_sp=0, flag_li_fe=0, flag_fe_h=0. The flag_sp
criterion is very likely to eliminate peculiar stars as those we excluded in
N23 sample. Also, we selected only objects with S/N per pixel $\geq$30 as
recommended by Buder et al. (2021) and age uncertainty
$\sigma_{age}$/age$\leq$10% with age (and $\sigma_{age}$) from the value-added
catalog of GALAH DR3. We kept only stars with low values of interstellar
extinction ($A_{G}\leq 0.2$). As was discussed in Charbonnel et al. (2021),
this parameter might be in a strong degeneracy with $T_{\mathrm{eff}}$, which
affects the precision of the latter and luminosity, thus causing an incorrect
determination of age. Finally, to select dwarfs, we initially selected stars
with log $g\geq 3.8$ and then performed the dwarf selection algorithm to
eliminate possible contamination by subgiants (see §4.1.3). The final GALAH
sample of dwarf stars along the [Fe/H] and $T_{\mathrm{eff}}$ domain of the Li
plateau consists of 72 stars. It is worth mentioning that, unlike the N23
sample, GALAH DR3 provides 1D,NLTE abundances of Li.

GALAH DR3 contains a large number of dwarf stars at higher metallicities,
including the metallicities of Pop I stars even at super-solar metallicities,
which is necessary for the comparison of our model predictions with
observational data in the high-metallicity regime (see §4.1.6). For this
purpose, we completed the sample with stars at [Fe/H]$>$-1.25 dex and log
$g\geq 4.0$ that meet the above mentioned quality flags. The additional “high-
metallicity” subsample covers the $T_{\mathrm{eff}}$ range of the plateau from
5800 K to 6500 K.

####  4.1.3 Final step of the selection of the dwarfs

For both the N23 and GALAH samples, we performed the following procedure to
carefully select dwarf stars (MS and MS turnoff). First, we computed the
stellar luminosities of the sample stars using the Gaia DR3 data. For this, we
took into account bolometric correction $BC_{G}$ computed with the tool by
Creevey et al. (2023) and interstellar extinction $A_{G}$:
$L/L_{\odot}=10^{0.4\cdot(M_{\odot}^{bol}-(G+5-5\log(d)+BC_{G}-A_{G}))}$,
where $M_{\odot}^{bol}=4.74^{m}$ (Prša et al., 2016), $G$ is the apparent
magnitude, and $d$ is the geometric distance (Bailer-Jones et al., 2021).
Then, with the use of our evolutionary tracks, we defined the main sequence
(MS) area in the Hertzsprung-Russell Diagram (HRD) for each metallicity of the
grid (see Fig. 11). This area is represented by a polygon with Zero-Age Main
Sequence (ZAMS) as vertices on one side and either the Terminal Age Main
Sequence (TAMS) for stars more massive than $\sim$0.8M⊙ or points with the age
of 13.8 Gyrs (age of the Universe according to Planck Collaboration et al.
2016) for the less massive ones. In STAREVOL, the TAMS is defined as a moment
when central hydrogen content XC drops to $10^{-7}$. The last step is to
determine whether an individual star is inside or outside the MS area.
However, for consistency, for each star, we performed linear interpolation of
the position of all the vertices on the corresponding value of [Fe/H] and
[$\alpha$/Fe] of the star. We considered that a star is a dwarf if the values
of its luminosity and $T_{\mathrm{eff}}$ fall into the polygon at least within
the 1-$\sigma$ confidence interval. However, if the polygon and the star’s
luminosity and $T_{\mathrm{eff}}$ confidence intervals do not overlap, we
marked this star as non-dwarf.

Figure 11: Process of selection of dwarfs. Lines show evolutionary tracks for
masses between 0.65 and 0.82M⊙ and metallicities of [Fe/H]=-2.0 (left group
tracks) and [Fe/H]=-1.5 (right group of tracks). For better visibility, the
tracks for [Fe/H]=-2.0 and Fe/H]=-1.5 were shifted by 500 K to the left and
the right correspondingly. Black parts of the lines and pinkish areas indicate
the main sequence, while grey lines show the subgiant stage. The green area is
an interpolation of the main sequences from the presented tracks to the
metallicity of [Fe/H]=-1.65. The points show mock stars that have different
positions with respect to the interpolated main sequence area: one star is
completely outside of the area even with its uncertainties, another one is
outside but hits the area with its $T_{\mathrm{eff}}$ uncertainty, and the
last one is inside the area (last two are shown in green). Figure 12: Milky
Way map with the Pop II dwarf stars shown as red and orange points (the N23
and GALAH subsamples respectively). The inset shows the globular cluster NGC
6752.

The [Fe/H] and $T_{\mathrm{eff}}$ distributions of both the N23 and GALAH
Plateau samples are presented in the upper panels of Fig. 13. Their positions,
as well as the position of the globular cluster NGC 6752 (see §4.2), in the
Milky Way map are shown in Fig. 12. In this figure, it is seen that the vast
majority of the selected stars are located well above the mid-plane of the
Galactic disc, i.e. in the halo ($\sim$94% of the samples stars have galactic
latitude $|b|>$15∘).

Figure 13: [Fe/H], $T_{\mathrm{eff}}$, age, and mass distributions of the
“normal” dwarf stars in the [Fe/H] and $T_{\mathrm{eff}}$ ranges of the
plateau from the GALAH and N23 samples (orange and red histograms
respectively).

####  4.1.4 Determination of masses and ages of the sample stars

We computed the masses and ages of all the field dwarf stars we selected from
N23+GALAH (except for J0023$+$0307) using SPInS (Lebreton & Reese, 2020) with
our present grid of models, following the steps described in Borisov et al.
(2022). For luminosities, we used the values that we previously computed and
described in §4.1.3. For the stars from N23, we adopted values of 100 K, 0.1
dex, and 0.2 dex for uncertainties of $T_{\mathrm{eff}}$, [Fe/H], and log $g$
respectively. For the GALAH sample, we used the uncertainties from the
original publication for individual stars.

Figure 14: Upper panel: Evolution of metallicity [M/H]=$\log(Z/Z_{\odot})$
(blue line, left axis) and Li content A(Li) (red line, right axis) in the
M=0.81M⊙ star with [Fe/H]=-1.5 and [$\alpha$/Fe]=0.3. Bottom panel: Kippenhahn
diagram for the same model star. The green areas indicate the convective
regions. Light blue, yellow, and orange areas show the radiative zone at the
pre-main sequence, main sequence, and subgiant stages correspondingly.

We accounted for metallicity [M/H]=$\log(Z/Z_{\odot})$ variations during the
evolution due to atomic diffusion and other transport processes. Although this
effect does not significantly change the surface metallicity of a star in the
domain we explore ($\Delta$[M/H] rarely exceeds 0.05 dex), it is known to
decrease the theoretical ages and increase the estimated masses compared to
models without atomic diffusion (e.g. VandenBerg et al., 2002). Fig. 14 shows
the evolution of the surface [M/H] and A(Li) and of the structure of a star
with M=0.81M⊙, [Fe/H]=-1.5, and [$\alpha$/Fe]=0.3. It is clearly seen that
once the star leaves the MS and becomes a subgiant, its convective envelope
rapidly deepens (so-called first dredge-up) and brings back to the surface the
metals that have “sunk” in the radiative layers along the MS. This is,
however, not the case for Li as it has not accumulated below the convective
envelope but has burned at relatively low burning temperature where it was
transported by the turbulence (see §3.2).

The bottom panels of Fig. 13 show the age and mass distributions of the Pop II
dwarf stars selected from the samples of N23 and GALAH in the [Fe/H] and
$T_{\mathrm{eff}}$ regime of the Li plateau. The age of the selected sample
stars is between $\sim$9 and 13 Gyr, with a peak around 12 Gyr (as the typical
age we assumed in §3.2). Their [Fe/H] goes from around -4 dex, with only
J0023$+$0307 with [Fe/H] below -4 dex (see §4.1.1 and 4.1.5). For this star,
we did not determine its mass and age but a rough visual estimation based on
its position in HRD (see Fig. 9)suggests its mass lies approximately between
0.72 and 0.76M⊙.

####  4.1.5 Comparison between models and observational data in the
$T_{\mathrm{eff}}$ \- [Fe/H] regime of the plateau

We first focus on the [Fe/H] and $T_{\mathrm{eff}}$ ranges of the Li plateau.
We compare in Fig. 15 the observed Li abundances of normal Plateau dwarf stars
as a function of their effective temperature to the corresponding theoretical
Li isochrones in different [Fe/H] bins. Since some stars of the N23 sample
have ages between 9 and 11 Gyr (see §4.1.4 for details on age determination),
we show isochrones at two different ages, 12$\pm$1 Gyr and 10$\pm$1 Gyr for
this sample. For the GALAH sample which contains mostly stars with ages above
$\sim$ 11 Gyr, we only show the former one. Each metallicity bin extends by
0.25 dex to each side of the metallicity values of the grid. However, we made
the lowest metallicity bins wider (both for the N23 and GALAH Pop II samples)
to include the star J0023$+$0307 with [Fe/H]=-5.8 dex (N23) and two stars that
have [Fe/H] slightly below -2.75 (GALAH). Also, as well as in Fig. 9, the
lowest metallicity bin shows models [Fe/H]=-5.8 with C- and Na-enhancement.

We see very good agreement between the models and the observations within
1$\sigma$ for most stars of both N23 and GALAH samples including the ones in
the low-metallicity bins with -5.8$<$[Fe/H]$<$-2.75 dex. There are, however, a
couple of low metallicity dwarfs that sit below the theoretical plateau. Two
of them have only lithium upper limit values. The third outlier is the most
Fe-poor star of the N23 sample, J0023$+$0307\. It shows no iron line but an
upper limit for its [Fe/H] was derived from CaII K ($<$-5.8 and $<$-6.6 from
Frebel et al. 2019 and Aguado et al. 2019 respectively). It is marginally
enriched in C for such an extremely Fe-poor star ([C/Fe]$>$3.9 dex),
potentially excluding contamination from a binary companion. As discussed in
the original papers, its abundance patterns including its overabundances in Mg
and Si point towards a second-generation star formed out of material enriched
by a single Population III massive star exploding as a fallback supernova.
While Frebel et al. (2019) and N23 derived values of 1.7 and 1.86 respectively
for its A(Li), Aguado et al. (2019) derived a value of 2.02$\pm$0.08, much
closer to the Plateau and our model predictions.

In Fig. 16 we show, for different ages, the theoretical upper and lower Li
envelopes as a function of [Fe/H]. These lines connect respectively the
highest and lowest Li abundances at a given [Fe/H] and age predicted by the
models in the $T_{\mathrm{eff}}$ range of the plateau. As expected, the
theoretical dispersion at a given age between the upper and lower Li envelopes
is very small in the [Fe/H] range of the Spite plateau. Additionally, the
envelopes become slightly positively sloped when age increases. Following N23,
we computed the theoretical slope within the [Fe/H] interval of -3 to -1.5
dex, resulting in values of 0.04 and 0.05 at 10 and 13 Gyr respectively. At
the same time, the observational slope among the normal sample stars (in the
same metallicity range) is $0.08\pm 0.02$ and $0.05\pm 0.02$ for the N23 and
combined N23+GALAH samples. This is in good agreement with Asplund et al. 2006
who reports the value of $0.10\pm 0.01$. The fact that the theoretical slope
is even slightly lower than the observational one underscores the constancy of
the $\log T_{0}$ value in the metallicity range of the Plateau. Otherwise,
increasing turbulence would lead to a decrease in slope or even potential
inversion. This imposes a strong constraint on the underlying mechanism, which
is imitated in our models by parametric turbulence. Finally, Li depletion as
modeled here does not predict a Li meltdown regime at very low metallicity,
which further supports the non-stellar origin of this feature that seems to
appear observationally only when CEMP stars are considered (see § 4.1.1).
Contrary to the conclusion of Aguado et al. (2019), we thus claim that the Li
abundance in the most Fe-poor and not C-enhanced star, J0023+0307, which
extends the Li plateau over more than five orders of magnitude in [Fe/H],
strongly supports the stellar depletion solution to the cosmological Li
problem. The determination of Li abundances in very metal-poor stars without
(or with limited) C-enhancement is thus strongly encouraged.

####  4.1.6 Comparison between models and observational data – Li dispersion
in more Fe-rich stars

As discussed above (see also §3.3), the stellar Li-depletion solution to the
Li plateau constancy is sustained by the fact that very Fe-poor low-mass stars
have very similar compact structures on the main sequence. Stellar compactness
decreases however around [Fe/H]$\sim$-1.5 dex, inducing larger variations in
the temperature and density gradients in the stellar radiative zones as the
stellar mass and metallicity vary. In addition, while at low metallicities we
see only old stars, stellar populations in the more Fe-rich domain of the
Milky Way have high dispersion in ages. These two facts are expected to result
in a high dispersion of Li abundance in more Fe-rich dwarfs because the Li
depletion rate depends on [Fe/H], mass, and age. Stellar lifetime is also
highly dependent on mass and, to a much lesser extent, on metallicity which
leads to the fact that the upper mass limit decreases when we look at older
stars.

The corresponding theoretical behaviour of the upper and lower Li envelopes
(hence of the expected dispersion) at different ages is shown and compared to
the observations in Fig. 16. The data correspond to the normal dwarf stars in
the $T_{\mathrm{eff}}$ range from 5800 K to 6500 K from N23 and the extended
sample of GALAH DR3 with ages colour-coded (the points are plotted in the age
ascending order to show the oldest stars that exist in different parts of this
diagram). The dashed line shows the initial Li value assumed in our models as
a function of [Fe/H] (see §2.1). Although the linear interpolation we used
between the BBN and meteoritic Li values is not fully justified and might
cause discrepancies between the observed Li abundances and model predictions,
it follows the observational trend in the youngest and most Fe-rich stars, and
it cannot affect the theoretical Li dispersion pattern qualitatively. At the
metallicities higher than that of the Li plateau, the models predict
increasing Li dispersion and fit very well the observed upper and lower Li
envelopes at different ages. However, there are still many stars with [Fe/H]
above $\sim-0.5$ dex and ages higher than $\sim$ 6 Gyrs that present Li values
below the theoretical predictions. This could be due to the fact that our
models are computed only for $\log T_{0}$=6.28. Indeed and as discussed in
§3.2, stars with [Fe/H] above $\sim-1.5$ dex require an increase of this
parameter, which would lead to an increase in Li depletion rate and, hence, a
decrease of Li abundance. For this purpose, we also show upper values at the
solar metallicity predicted by models for dwarfs with $\log T_{0}$=6.42 and
$T_{\mathrm{eff}}^{\mathrm{ZAMS}}$$<$6500 K from D21b for which turbulence was
adjusted to reproduce solar-type stars around solar metallicity. Since these
models are computed with a mass step of 0.1M⊙, for higher precision, we
interpolated the Li values from the models at ages 2, 6, and 8 Gyr.
Interpolated values approximately correspond to masses of 1.25, 1.15, and
1.05M⊙. These results shall help future studies of the physical origin of the
process that we simulate as parametric turbulence.

Figure 15: Comparison between Li observations and theoretical Li isochrones as
a function of $T_{\mathrm{eff}}$ in different [Fe/H] bins. Blue and red
isochrone lines and data points correspond to two age ranges, 10$\pm$1 and
12$\pm$1 Gyr respectively. Circles and arrows correspond to Li abundance
determinations and upper limits respectively. Error bars show mean
$T_{\mathrm{eff}}$ and A(Li) uncertainty of stars in a given bin. Top and
bottom: N23 and GALAH sample stars respectively. In the upper left panel, we
show the position of the star most Fe-poor dwarf of our sample, J0023+037,
with the Li and $T_{\mathrm{eff}}$ values from N23 (red five-pointed star) and
from Aguado et al. (2019, green five-pointed star), with the theoretical Li-
isochrones for the corresponding [Fe/H]=-5-8 dex models

.

Figure 16: Li abundance (3D,NLTE) as a function of [Fe/H] for the N23 and
GALAH DR3 sample stars (squares and circles respectively) with age color-
coded. Solid lines show the theoretical Li upper and lower envelopes (upper
and bottom panels respectively) at different ages (see the legend), i.e., the
maximum and minimum Li values at a given [Fe/H] and age expected from the
models with 5800 K$\lesssim$$T_{\mathrm{eff}}^{\mathrm{ZAMS}}$$\lesssim$6500
K. We do not show it here, but the highest/ lowest A(Li) values for the models
at [Fe/H]=-5.8 dex are 2.34/2.29 and 2.23/2.18 at the ages of 10 and 13 Gyr
respectively. The dashed line shows the initial Li abundance assumed in the
models. In the upper panel, the four-pointed stars show predictions from the
models with log$T_{0}$=6.42 and solar metallicity from D21b with age similarly
colour-coded.

###  4.2 Globular cluster stars - Lithium and metals

The abundance variations of metals around the MS turnoff in globular clusters
have been successfully used to pinpoint the combined effects of atomic
diffusion and turbulence in Pop II low-mass stars (e.g. Korn et al., 2007b;
Lind et al., 2008; Nordlander et al., 2012b; Gruyters et al., 2013, 2014;
Gavel et al., 2021b; Nordlander et al., 2024). We have carried out this
additional test on our models using Mg and Li data for the globular cluster
NGC 6752 from Gruyters et al. (2013) and Gruyters et al. (2014) (hereafter G13
and G14 correspondingly), where the former one provides measurements for
individual post-turnoff stars and the latter one averaged values for four
groups of stars at different evolutionary stages: turnoff, SGB, base of the
RGB, and RGB. Although the G14 sample is more complete, the G13 sample covers
a wider $T_{\mathrm{eff}}$ range. The G14 sample contains 146 and 124 stars
with measurements of Mg and Li abundance correspondingly. To avoid initial
abundance variations among the multiple stellar populations hosted by the
cluster (for a review on multiple stellar populations in globular clusters,
see Gratton et al., 2019), we consider only those stars from G14 that were
identified as being part of the “primordial” (i.e., first) population. These
stars should have the same original composition as field halo stars of similar
[Fe/H]. We are left with 25 and 34 stars with measurements of Li and Mg
respectively. We adopt the same age for the cluster as G14 (13.5 Gyr). The
authors derive [Fe/H]=-1.6 and [$\alpha$/Fe]=0.3 for the cluster. Other
estimations suggest its [Fe/H] value is around -1.5 (e.g. [Fe/H]=-1.49, Souza
et al. 2020; -1.48, Gratton et al. 2005; -1.55, Harris 1996). Considering
these slight differences and in order to save computational time, we used the
already computed models for [Fe/H]=-1.5 and [$\alpha$/Fe]=0.3. We completed
them with a few additional models around the cluster turnoff mass of
0.8M⊙666To this end, we computed models with initial masses of 0.795, 0.798,
0.802, 0.805, 0.808, and 0.81M⊙. to trace the SGB and RGB with a higher level
of discreteness.

We compare the model predictions to the Li and Mg observations as a function
of $T_{\mathrm{eff}}$ (as a proxy for the evolution stage) in Fig. 17. For
both elements, the magenta lines connect the theoretical abundances in models
of different initial masses at 13.5 Gyr (i.e., isochrones for Li and Mg).
Although we calibrated models to reproduce Li abundance in turnoff MS stars,
we obtain a very good agreement between the observational data and model for
Li beyond the MS until the end of the first dredge-up. As discussed in §4.1.4,
the abundances of heavier elements (including Mg) that have accumulated below
the convective envelope along the MS due to atomic diffusion recover their
initial values as the first dredge-up occurs (see also Fig. 14 for the global
metallicity [M/H]). The theoretical predictions for Mg were homogeneously
shifted by -0.14 dex to account for the differences between the initial Mg
content assumed in our models and the observations. The shift value was
computed to minimize the $\chi^{2}$ discrepancy between the model and
observation. Although the models were not calibrated to fit the Mg abundance
increase in post-turnoff stars, they still reproduce the observed trend at a
fairly high level of confidence. Our results are in agreement with those of
the models of G14 with atomic diffusion and parametric turbulence derived a
slightly lower value of log T0=6.20 than ours for parametric turbulence in
order to recover the main observational trends for metals in NGC 6752. The
corresponding initial Li abundances they retrieved with their optimal model
are thus slightly lower (A(Li)=2.58$\pm$0.10) than the BBN+CMB value we assume
in our models with log T0=6.28. We consider, however, that the difference is
not compiling, for two reasons. First, the different input microphysics in G14
and our models can lead to slight stellar structure differences, with
differences in the optimal log T0 simply hiding our ignorance about the origin
of the parametric turbulence. Second, the significance of the trends for
metals is relatively weak in NGC 6752.

Figure 17: Upper panel: trend of Li abundance in stars of the primordial
population of NGC 6752. Black points correspond to the data points from G14
and blue diamonds show the averaged values for different evolutionary stages
of stars from G13 (turnoff, SGB, base of the RGB, RGB from left to right). The
model prediction at 13.5 Gyr is shown with the magenta line with points
labeled according to their mass. Bottom panel: The same, but for the abundance
of Mg.

##  5 Summary and conclusions

The discrepancy between the Li value predicted by BBN+CMB and the one observed
along the Spite plateau has long been a matter of debate. In this work, we
investigated the stellar depletion solution using the same assumptions as in
the models of Dumont et al. (2021b, a) that explain the Li behaviour and the
surface and internal rotation of Population I low-mass stars. They include
rotation-induced processes, parametric vertical viscosity, atomic diffusion,
penetrative convection, and parametric turbulence. Given the lack of
asteroseismic data for Pop II Plateau dwarf stars, we adopted the same value
for the vertical viscosity as required in the ${}_{\nu}R1^{T_{0}}_{A}$ models
of D21a/b to explain the internal rotation profile of solar-type stars, and we
show that variations around this value do not significantly impact the Li
content at the age of Pop II stars (see §3.1). We adopted the same initial
rotation periods as Amard et al. (2019) and D21a,b for Pop I low-mass stars
and show that Li depletion at the age of the Plateau dwarfs is marginally
impacted by the initial rotation rate. Finally, the only difference to the
${}_{\nu}R1^{T_{0}}_{A}$ models of D21a/b is the value of $\log T_{0}$ where
the efficiency of parametric turbulence is anchored, which we adjusted to
counteract atomic diffusion and fit the surface abundance of Li observed along
the Plateau. Using the BBN+CMB Li value as the initial Li abundance for Pop II
stars, we found that the value of $\log T_{0}$=6.28 fits the best the today-
observed value of A(Li)=2.3. This value is in good agreement with $\log
T_{0}$=6.25 found by Richard et al. (2005) whose models include only atomic
diffusion and parametric turbulence that turn out to be the dominant processes
for surface Li depletion along the plateau compared to rotation-induced
meridional circulation and horizontal shear as modeled in the
${}_{\nu}R1^{T_{0}}_{A}$ configuration.

We have demonstrated that the constancy of the Li Spite plateau over a large
range of metallicities results naturally from the similar compactness of the
plateau dwarf stars, whose structure does not significantly change in a wide
metallicity range. Consequently, these stars are expected to undergo the
combined action of parametric turbulence and atomic diffusion at a similar
level of efficiency.

At the same time, we have explained the constancy of the Spite plateau in
quite a wide range of $T_{\mathrm{eff}}$. Although, at a given metallicity,
atomic diffusion efficiency increases with $T_{\mathrm{eff}}$, its effect is
balanced by the changing of the density gradient below the base of the
convective envelope. Eventually, these two effects create a domain in the
[Fe/H]-$T_{\mathrm{eff}}$ space where stars have very similar rates of Li
depletion, which results in a plateau with very little Li dispersion. When
using the same $T_{0}$ value over a [Fe/H] range covering five orders of
magnitude below -1.5 dex, the ${}_{\nu}R1^{T_{0}}_{A}$ models predict a very
modest positive slope for the Li plateau, but no Li meltdown at [Fe/H] below
-2.5 dex. This is in excellent agreement with the trend found after a detailed
analysis of the peculiarities of dwarf stars with Li data in the very low
metallicity regime. There, the dwarf stars that lie below the plateau are all
CEMP or other chemically peculiar stars, supporting the environmental origin
(influence of Pop III stars in the original composition, and/or binarity-
induced processes) of the so-called meltdown. The Li abundance close to the
Plateau of the most Fe-poor star, J0023+0307, which is not C-enriched,
provides strong additional support to the stellar depletion solution of the
cosmological problem.

We compared trends of Li and Mg in post-turnoff stars of the globular cluster
NGC 6752 with the model predictions. These stars experience the first dredge-
up caused by the rapid deepening of the convective envelope which results in
the increase of most metals and the decrease of Li. We found a perfect
agreement between observed Li abundances and models that have the most recent
measurement of the primordial Li abundance of A(Li)=2.75 dex (Pitrou et al.,
2018; Coc & Vangioni, 2017) as a starting point. Also, the modeled trend for
Mg abundance fits well with the observational data. This further supports the
stellar depletion solution to the cosmological Li problem. Possibly, Mg and Li
data in post-turnoff stars in metal-rich globular clusters could help estimate
how the Li abundance increased in the interstellar matter from the BBN to the
meteoretic value in the -1$<$[Fe/H]$<$-0.5 dex range. Up to date, it is in
this range that there are the greatest uncertainties in the behaviour of Li.
Understanding this may help shed light on the history of enrichment in the
Galaxy.

The same stellar structure considerations consistently explain the change of
Li depletion and dispersion regime for [Fe/H] around -1.5 dex, i.e., at the
transition in metallicity between Pop II to Pop I stars. The stellar
compactness decreases strongly above this metallicity, and the density at the
base of the convective envelope (hence the atomic diffusion efficiency) varies
strongly with $T_{\mathrm{eff}}$ and metallicity. This explains the need for a
higher anchoring temperature for parametric turbulence for Pop I dwarfs than
for Pop II stars along the plateau. This also leads naturally to higher Li
dispersion in Pop I stars at a given age (e.g., $\sim$0.6-0.7 dex at the solar
metallicity at 2 Gyr), and more generally since Pop I low-mass stars that
cover a large range in age. This plateau-to-scatter transition pattern is in
good agreement with the observational data, which further supports a
consistent stellar depletion solution to the Li behaviour in low-mass stars
from different Galactic populations and to the Li cosmological problem.

###### Acknowledgements.

SB and CC acknowledge financial support from the Swiss National Science
Foundation (SNF; Project 200021$-$212160). TD is supported by the European
Union, ChETEC-INFRA, project no. 101008324. We thank John Norris for kindly
providing data on stellar parameters and abundances. This research has made
use of NASA’s Astrophysics Data System Bibliographic Services. We extensively
used the SIMBAD database (Wenger et al., 2000), VizieR catalog access tool
(Ochsenbein et al., 2000), and TOPCAT (Taylor, 2005).

## References

  * Aguado et al. (2019) Aguado, D. S., González Hernández, J. I., Allende Prieto, C., & Rebolo, R. 2019, ApJ, 874, L21
  * Aguado et al. (2021) Aguado, D. S., Myeong, G. C., Belokurov, V., et al. 2021, MNRAS, 500, 889
  * Amard et al. (2016) Amard, L., Palacios, A., Charbonnel, C., Gallet, F., & Bouvier, J. 2016, A&A, 587, A105
  * Amard et al. (2019) Amard, L., Palacios, A., Charbonnel, C., et al. 2019, A&A, 631, A77
  * Anders & Grevesse (1989) Anders, E. & Grevesse, N. 1989, Geochim. Cosmochim. Acta., 53, 197
  * Arnould & Norgaard (1975) Arnould, M. & Norgaard, H. 1975, A&A, 42, 55
  * Asplund et al. (2009) Asplund, M., Grevesse, N., Sauval, A. J., & Scott, P. 2009, ARA&A, 47, 481
  * Asplund et al. (2006) Asplund, M., Lambert, D. L., Nissen, P. E., Primas, F., & Smith, V. V. 2006, ApJ, 644, 229
  * Augustson & Mathis (2019) Augustson, K. C. & Mathis, S. 2019, ApJ, 874, 83
  * Aver et al. (2021) Aver, E., Berg, D. A., Olive, K. A., et al. 2021, J. Cosmology Astropart. Phys., 2021, 027
  * Bailer-Jones et al. (2021) Bailer-Jones, C. A. L., Rybizki, J., Fouesneau, M., Demleitner, M., & Andrae, R. 2021, AJ, 161, 147
  * Baraffe et al. (2017) Baraffe, I., Pratt, J., Goffrey, T., et al. 2017, ApJ, 845, L6
  * Beers & Christlieb (2005) Beers, T. C. & Christlieb, N. 2005, ARA&A, 43, 531
  * Bensby & Lind (2018) Bensby, T. & Lind, K. 2018, A&A, 615, A151
  * Bonifacio et al. (2015) Bonifacio, P., Caffau, E., Spite, M., et al. 2015, A&A, 579, A28
  * Bonifacio et al. (2018) Bonifacio, P., Caffau, E., Spite, M., et al. 2018, A&A, 612, A65
  * Bonifacio et al. (2012) Bonifacio, P., Sbordone, L., Caffau, E., et al. 2012, A&A, 542, A87
  * Borisov et al. (2022) Borisov, S., Prantzos, N., & Charbonnel, C. 2022, A&A, 668, A181
  * Buder et al. (2021) Buder, S., Sharma, S., Kos, J., et al. 2021, MNRAS, 506, 150
  * Buldgen et al. (2023) Buldgen, G., Eggenberger, P., Noels, A., et al. 2023, A&A, 669, L9
  * Carney et al. (2005) Carney, B. W., Latham, D. W., & Laird, J. B. 2005, AJ, 129, 466
  * Chaboyer & Demarque (1994) Chaboyer, B. & Demarque, P. 1994, ApJ, 433, 510
  * Chaboyer & Zahn (1992) Chaboyer, B. & Zahn, J. P. 1992, A&A, 253, 173
  * Charbonnel et al. (2021) Charbonnel, C., Borisov, S., de Laverny, P., & Prantzos, N. 2021, A&A, 649, L10
  * Charbonnel & Primas (2005) Charbonnel, C. & Primas, F. 2005, A&A, 442, 961
  * Charbonnel & Talon (2005) Charbonnel, C. & Talon, S. 2005, Science, 309, 2189
  * Coc & Vangioni (2017) Coc, A. & Vangioni, E. 2017, International Journal of Modern Physics E, 26, 1741002
  * Cooke et al. (2016) Cooke, R. J., Pettini, M., Nollett, K. M., & Jorgenson, R. 2016, ApJ, 830, 148
  * Creevey et al. (2023) Creevey, O. L., Sordo, R., Pailler, F., et al. 2023, A&A, 674, A26
  * Cummings et al. (2017) Cummings, J. D., Deliyannis, C. P., Maderak, R. M., & Steinhauer, A. 2017, AJ, 153, 128
  * Cyburt et al. (2016) Cyburt, R. H., Fields, B. D., Olive, K. A., & Yeh, T.-H. 2016, Reviews of Modern Physics, 88, 015004
  * Deal et al. (2020) Deal, M., Goupil, M. J., Marques, J. P., Reese, D. R., & Lebreton, Y. 2020, A&A, 633, A23
  * Deal & Martins (2021) Deal, M. & Martins, C. J. A. P. 2021, A&A, 653, A48
  * Decressin et al. (2009a) Decressin, T., Mathis, S., Palacios, A., et al. 2009a, A&A, 495, 271
  * Decressin et al. (2009b) Decressin, T., Mathis, S., Palacios, A., et al. 2009b, A&A, 495, 271
  * Deliyannis et al. (2019) Deliyannis, C. P., Anthony-Twarog, B. J., Lee-Brown, D. B., & Twarog, B. A. 2019, AJ, 158, 163
  * Denissenkov et al. (2014) Denissenkov, P. A., Truran, J. W., Pignatari, M., et al. 2014, MNRAS, 442, 2058
  * Dumont et al. (2021a) Dumont, T., Charbonnel, C., Palacios, A., & Borisov, S. 2021a, A&A, 654, A46
  * Dumont et al. (2021b) Dumont, T., Palacios, A., Charbonnel, C., et al. 2021b, A&A, 646, A48
  * Eggenberger et al. (2022a) Eggenberger, P., Buldgen, G., Salmon, S. J. A. J., et al. 2022a, Nature Astronomy, 6, 788
  * Eggenberger et al. (2012) Eggenberger, P., Montalbán, J., & Miglio, A. 2012, A&A, 544, L4
  * Eggenberger et al. (2022b) Eggenberger, P., Moyano, F. D., & den Hartogh, J. W. 2022b, A&A, 664, L16
  * Eggleton et al. (1973) Eggleton, P. P., Faulkner, J., & Flannery, B. P. 1973, A&A, 23, 325
  * Fields (2011) Fields, B. D. 2011, Annual Review of Nuclear and Particle Science, 61, 47
  * Frebel et al. (2019) Frebel, A., Ji, A. P., Ezzeddine, R., et al. 2019, ApJ, 871, 146
  * Gallet & Bouvier (2015) Gallet, F. & Bouvier, J. 2015, A&A, 577, A98
  * Gao et al. (2020) Gao, X., Lind, K., Amarsi, A. M., et al. 2020, MNRAS, 497, L30
  * Gavel et al. (2021a) Gavel, A., Gruyters, P., Heiter, U., et al. 2021a, A&A, 652, A75
  * Gavel et al. (2021b) Gavel, A., Gruyters, P., Heiter, U., et al. 2021b, A&A, 652, A75
  * Gratton et al. (2019) Gratton, R., Bragaglia, A., Carretta, E., et al. 2019, A&A Rev., 27, 8
  * Gratton et al. (2005) Gratton, R. G., Bragaglia, A., Carretta, E., et al. 2005, A&A, 440, 901
  * Grohs & Fuller (2023) Grohs, E. & Fuller, G. M. 2023, arXiv e-prints, arXiv:2301.12299
  * Gruyters et al. (2013) Gruyters, P., Korn, A. J., Richard, O., et al. 2013, A&A, 555, A31
  * Gruyters et al. (2016) Gruyters, P., Lind, K., Richard, O., et al. 2016, A&A, 589, A61
  * Gruyters et al. (2014) Gruyters, P., Nordlander, T., & Korn, A. J. 2014, A&A, 567, A72
  * Guzik & Mussack (2010) Guzik, J. A. & Mussack, K. 2010, ApJ, 713, 1108
  * Harris (1996) Harris, W. E. 1996, AJ, 112, 1487
  * Hayakawa et al. (2021) Hayakawa, S., La Cognata, M., Lamia, L., et al. 2021, ApJ, 915, L13
  * Hill et al. (2019) Hill, V., Skúladóttir, Á., Tolstoy, E., et al. 2019, A&A, 626, A15
  * Hosford et al. (2010) Hosford, A., García Pérez, A. E., Collet, R., et al. 2010, A&A, 511, A47
  * Hsyu et al. (2020) Hsyu, T., Cooke, R. J., Prochaska, J. X., & Bolte, M. 2020, ApJ, 896, 77
  * Iglesias & Rogers (1996) Iglesias, C. A. & Rogers, F. J. 1996, ApJ, 464, 943
  * Iliadis & Coc (2020) Iliadis, C. & Coc, A. 2020, ApJ, 901, 127
  * Iocco et al. (2009) Iocco, F., Mangano, G., Miele, G., Pisanti, O., & Serpico, P. D. 2009, Phys. Rep, 472, 1
  * Iwamoto (2009) Iwamoto, N. 2009, PASA, 26, 145
  * Izotov et al. (2014) Izotov, Y. I., Thuan, T. X., & Guseva, N. G. 2014, MNRAS, 445, 778
  * Jedamzik (2004) Jedamzik, K. 2004, Phys. Rev. D, 70, 063524
  * Jeffries et al. (2021) Jeffries, R. D., Jackson, R. J., Sun, Q., & Deliyannis, C. P. 2021, MNRAS, 500, 1158
  * José & Hernanz (1998) José, J. & Hernanz, M. 1998, ApJ, 494, 680
  * José et al. (2006) José, J., Hernanz, M., & Iliadis, C. 2006, Nucl. Phys. A, 777, 550
  * Karakas & Lugaro (2016) Karakas, A. I. & Lugaro, M. 2016, ApJ, 825, 26
  * Kemp et al. (2022) Kemp, A. J., Karakas, A. I., Casey, A. R., et al. 2022, ApJ, 933, L30
  * Korn et al. (2006) Korn, A. J., Grundahl, F., Richard, O., et al. 2006, Nature, 442, 657
  * Korn et al. (2007a) Korn, A. J., Grundahl, F., Richard, O., et al. 2007a, ApJ, 671, 402
  * Korn et al. (2007b) Korn, A. J., Grundahl, F., Richard, O., et al. 2007b, ApJ, 671, 402
  * Krishna Swamy (1966) Krishna Swamy, K. S. 1966, ApJ, 145, 174
  * Kusakabe et al. (2019) Kusakabe, M., Cheoun, M.-K., Kim, K. S., et al. 2019, ApJ, 872, 164
  * Kusakabe & Kawasaki (2019) Kusakabe, M. & Kawasaki, M. 2019, ApJ, 876, L30
  * Lagarde et al. (2012) Lagarde, N., Decressin, T., Charbonnel, C., et al. 2012, A&A, 543, A108
  * Lebreton & Reese (2020) Lebreton, Y. & Reese, D. R. 2020, A&A, 642, A88
  * Lind et al. (2008) Lind, K., Korn, A. J., Barklem, P. S., & Grundahl, F. 2008, A&A, 490, 777
  * Longland et al. (2012) Longland, R., Lorén-Aguilar, P., José, J., García-Berro, E., & Althaus, L. G. 2012, A&A, 542, A117
  * Lucey et al. (2023) Lucey, M., Al Kharusi, N., Hawkins, K., et al. 2023, MNRAS, 523, 4049
  * Maeder & Zahn (1998) Maeder, A. & Zahn, J.-P. 1998, A&A, 334, 1000
  * Mathis (2013) Mathis, S. 2013, in Lecture Notes in Physics, Berlin Springer Verlag, ed. M. Goupil, K. Belkacem, C. Neiner, F. Lignières, & J. J. Green, Vol. 865, 23
  * Mathis et al. (2021) Mathis, S., Bugnet, L., Prat, V., et al. 2021, A&A, 647, A122
  * Mathis et al. (2018) Mathis, S., Prat, V., Amard, L., et al. 2018, A&A, 620, A22
  * Mathis & Zahn (2004) Mathis, S. & Zahn, J. P. 2004, A&A, 425, 229
  * Matsuno et al. (2017) Matsuno, T., Aoki, W., Beers, T. C., Lee, Y. S., & Honda, S. 2017, AJ, 154, 52
  * Matt et al. (2015) Matt, S. P., Brun, A. S., Baraffe, I., Bouvier, J., & Chabrier, G. 2015, ApJ, 799, L23
  * Matteucci et al. (2021) Matteucci, F., Molero, M., Aguado, D. S., & Romano, D. 2021, MNRAS, 505, 200
  * Meléndez et al. (2010) Meléndez, J., Casagrande, L., Ramírez, I., Asplund, M., & Schuster, W. J. 2010, A&A, 515, L3
  * Meneguzzi et al. (1971) Meneguzzi, M., Audouze, J., & Reeves, H. 1971, A&A, 15, 337
  * Meynet et al. (2010) Meynet, G., Hirschi, R., Ekstrom, S., et al. 2010, A&A, 521, A30
  * Michaud et al. (2015) Michaud, G., Alecian, G., & Richer, J. 2015, Atomic Diffusion in Stars
  * Michaud et al. (1984) Michaud, G., Fontaine, G., & Beaudet, G. 1984, ApJ, 282, 206
  * Molaro et al. (2020) Molaro, P., Cescutti, G., & Fu, X. 2020, MNRAS, 496, 2902
  * Molaro et al. (2023) Molaro, P., Izzo, L., Selvelli, P., et al. 2023, MNRAS, 518, 2614
  * Monaco et al. (2010) Monaco, L., Bonifacio, P., Sbordone, L., Villanova, S., & Pancino, E. 2010, A&A, 519, L3
  * Mucciarelli et al. (2014) Mucciarelli, A., Salaris, M., Bonifacio, P., Monaco, L., & Villanova, S. 2014, MNRAS, 444, 1812
  * Nakamura et al. (2010) Nakamura, K., Yoshida, T., Shigeyama, T., & Kajino, T. 2010, in Light Elements in the Universe, ed. C. Charbonnel, M. Tosi, F. Primas, & C. Chiappini, Vol. 268, 463–468
  * Nordlander et al. (2024) Nordlander, T., Gruyters, P., Richard, O., & Korn, A. J. 2024, MNRAS, 527, 12120
  * Nordlander et al. (2012a) Nordlander, T., Korn, A. J., Richard, O., & Lind, K. 2012a, ApJ, 753, 48
  * Nordlander et al. (2012b) Nordlander, T., Korn, A. J., Richard, O., & Lind, K. 2012b, ApJ, 753, 48
  * Norris & Yong (2019) Norris, J. E. & Yong, D. 2019, ApJ, 879, 37
  * Norris et al. (2023) Norris, J. E., Yong, D., Frebel, A., & Ryan, S. G. 2023, MNRAS, 522, 1358
  * Ochsenbein et al. (2000) Ochsenbein, F., Bauer, P., & Marcout, J. 2000, A&AS, 143, 23
  * Palacios et al. (2006) Palacios, A., Charbonnel, C., Talon, S., & Siess, L. 2006, A&A, 453, 261
  * Paquette et al. (1986) Paquette, C., Pelletier, C., Fontaine, G., & Michaud, G. 1986, ApJS, 61, 177
  * Piau (2008) Piau, L. 2008, ApJ, 689, 1279
  * Piau et al. (2006) Piau, L., Beers, T. C., Balsara, D. S., et al. 2006, ApJ, 653, 300
  * Piau et al. (2003) Piau, L., Randich, S., & Palla, F. 2003, A&A, 408, 1037
  * Pinsonneault et al. (2000) Pinsonneault, M. H., Charbonnel, C., & Deliyannis, C. P. 2000, arXiv e-prints, astro
  * Pitrou et al. (2018) Pitrou, C., Coc, A., Uzan, J.-P., & Vangioni, E. 2018, Phys. Rep, 754, 1
  * Pizzone et al. (2023) Pizzone, R. G., Spampinato, C., Sergi, M. L., Spartá, R., & Zanatta-Martinez, T. 2023, in Journal of Physics Conference Series, Vol. 2619, Journal of Physics Conference Series, 012012
  * Planck Collaboration et al. (2016) Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2016, A&A, 594, A13
  * Planck Collaboration et al. (2020) Planck Collaboration, Aghanim, N., Akrami, Y., et al. 2020, A&A, 641, A6
  * Planck Collaboration et al. (2021) Planck Collaboration, Aghanim, N., Akrami, Y., et al. 2021, A&A, 652, C4
  * Pols et al. (1995) Pols, O. R., Tout, C. A., Eggleton, P. P., & Han, Z. 1995, MNRAS, 274, 964
  * Prantzos (2010) Prantzos, N. 2010, in Proceedings of International Symposium on Nuclear Astrophysics - Nuclei in the Cosmos - IX (2006) — PoS(NIC-IX), Vol. 028, 254
  * Prantzos (2012) Prantzos, N. 2012, A&A, 542, A67
  * Prantzos et al. (2017) Prantzos, N., de Laverny, P., Guiglion, G., Recio-Blanco, A., & Worley, C. C. 2017, A&A, 606, A132
  * Proffitt & Michaud (1991) Proffitt, C. R. & Michaud, G. 1991, ApJ, 371, 584
  * Prša et al. (2016) Prša, A., Harmanec, P., Torres, G., et al. 2016, AJ, 152, 41
  * Rebolo et al. (1988) Rebolo, R., Molaro, P., & Beckman, J. E. 1988, A&A, 192, 192
  * Reeves et al. (1970) Reeves, H., Fowler, W. A., & Hoyle, F. 1970, Nature, 226, 727
  * Richard et al. (2002a) Richard, O., Michaud, G., & Richer, J. 2002a, ApJ, 580, 1100
  * Richard et al. (2005) Richard, O., Michaud, G., & Richer, J. 2005, ApJ, 619, 538
  * Richard et al. (2002b) Richard, O., Michaud, G., Richer, J., et al. 2002b, ApJ, 568, 979
  * Richard et al. (1996) Richard, O., Vauclair, S., Charbonnel, C., & Dziembowski, W. A. 1996, A&A, 312, 1000
  * Richer et al. (2000) Richer, J., Michaud, G., & Turcotte, S. 2000, ApJ, 529, 338
  * Roca-Fàbrega et al. (2021) Roca-Fàbrega, S., Llorente de Andrés, F., Chavero, C., Cifuentes, C., & de la Reza, R. 2021, A&A, 656, A64
  * Romano et al. (2021) Romano, D., Magrini, L., Randich, S., et al. 2021, A&A, 653, A72
  * Ryan et al. (2001a) Ryan, S. G., Beers, T. C., Kajino, T., & Rosolankova, K. 2001a, ApJ, 547, 231
  * Ryan et al. (2001b) Ryan, S. G., Kajino, T., Beers, T. C., et al. 2001b, ApJ, 549, 55
  * Ryan et al. (2001c) Ryan, S. G., Kajino, T., Beers, T. C., et al. 2001c, ApJ, 549, 55
  * Sackmann & Boothroyd (1992) Sackmann, I. J. & Boothroyd, A. I. 1992, ApJ, 392, L71
  * Sackmann & Boothroyd (1999) Sackmann, I. J. & Boothroyd, A. I. 1999, ApJ, 510, 217
  * Salaris & Weiss (2001) Salaris, M. & Weiss, A. 2001, A&A, 376, 955
  * Sbordone et al. (2010) Sbordone, L., Bonifacio, P., Caffau, E., et al. 2010, A&A, 522, A26
  * Sestito & Randich (2005) Sestito, P. & Randich, S. 2005, A&A, 442, 615
  * Siess et al. (2000) Siess, L., Dufour, E., & Forestini, M. 2000, A&A, 358, 593
  * Soderblom et al. (1993) Soderblom, D. R., Jones, B. F., Balachandran, S., et al. 1993, AJ, 106, 1059
  * Souza et al. (2020) Souza, S. O., Kerber, L. O., Barbuy, B., et al. 2020, ApJ, 890, 38
  * Spada et al. (2016) Spada, F., Gellert, M., Arlt, R., & Deheuvels, S. 2016, A&A, 589, A23
  * Spite & Spite (1982a) Spite, F. & Spite, M. 1982a, A&A, 115, 357
  * Spite & Spite (1982b) Spite, M. & Spite, F. 1982b, Nature, 297, 483
  * Starrfield et al. (2020) Starrfield, S., Bose, M., Iliadis, C., et al. 2020, ApJ, 895, 70
  * Starrfield et al. (1978) Starrfield, S., Truran, J. W., Sparks, W. M., & Arnould, M. 1978, ApJ, 222, 600
  * Steigman (2007) Steigman, G. 2007, Annual Review of Nuclear and Particle Science, 57, 463
  * Talon & Charbonnel (2004) Talon, S. & Charbonnel, C. 2004, A&A, 418, 1051
  * Talon et al. (2002) Talon, S., Kumar, P., & Zahn, J.-P. 2002, ApJ, 574, L175
  * Talon et al. (2006) Talon, S., Richard, O., & Michaud, G. 2006, ApJ, 645, 634
  * Taylor (2005) Taylor, M. B. 2005, in Astronomical Society of the Pacific Conference Series, Vol. 347, Astronomical Data Analysis Software and Systems XIV, ed. P. Shopbell, M. Britton, & R. Ebert, 29
  * Thorburn (1994) Thorburn, J. A. 1994, ApJ, 421, 318
  * Thoul et al. (1994) Thoul, A. A., Bahcall, J. N., & Loeb, A. 1994, ApJ, 421, 828
  * Travaglio et al. (2001) Travaglio, C., Randich, S., Galli, D., et al. 2001, ApJ, 559, 909
  * Valerdi et al. (2021) Valerdi, M., Peimbert, A., & Peimbert, M. 2021, MNRAS, 505, 3624
  * van Raai et al. (2012) van Raai, M. A., Lugaro, M., Karakas, A. I., García-Hernández, D. A., & Yong, D. 2012, A&A, 540, A44
  * VandenBerg et al. (2002) VandenBerg, D. A., Richard, O., Michaud, G., & Richer, J. 2002, ApJ, 571, 487
  * Vauclair (1988) Vauclair, S. 1988, ApJ, 335, 971
  * Vauclair & Charbonnel (1995) Vauclair, S. & Charbonnel, C. 1995, A&A, 295, 715
  * Vauclair & Charbonnel (1998) Vauclair, S. & Charbonnel, C. 1998, ApJ, 502, 372
  * Vick et al. (2013) Vick, M., Michaud, G., Richer, J., & Richard, O. 2013, A&A, 552, A131
  * Wagoner et al. (1967) Wagoner, R. V., Fowler, W. A., & Hoyle, F. 1967, ApJ, 148, 3
  * Wang et al. (2021) Wang, E. X., Nordlander, T., Asplund, M., et al. 2021, MNRAS, 500, 2159
  * Wenger et al. (2000) Wenger, M., Ochsenbein, F., Egret, D., et al. 2000, A&AS, 143, 9
  * Woosley et al. (1990) Woosley, S. E., Hartmann, D. H., Hoffman, R. D., & Haxton, W. C. 1990, ApJ, 356, 272
  * Woosley & Weaver (1995) Woosley, S. E. & Weaver, T. A. 1995, ApJS, 101, 181
  * Xu et al. (2013a) Xu, Y., Goriely, S., Jorissen, A., Chen, G. L., & Arnould, M. 2013a, A&A, 549, A106
  * Xu et al. (2013b) Xu, Y., Takahashi, K., Goriely, S., et al. 2013b, Nucl. Phys. A, 918, 61
  * Young (2018) Young, P. R. 2018, ApJ, 855, 15
  * Zahn (1992) Zahn, J. P. 1992, A&A, 265, 115
  * Zavarygin et al. (2018) Zavarygin, E. O., Webb, J. K., Dumont, V., & Riemer-Sørensen, S. 2018, MNRAS, 477, 5536