# The updated BaSTI stellar evolution models and isochrones. III. White Dwarfs

Maurizio Salaris,1,2 Santi Cassisi,2,3 Adriano Pietrinferni2 and Sebastian
Hidalgo4,5
1Astrophysics Research Institute, Liverpool John Moores University, 146
Brownlow Hill, Liverpool, L3 5RF, UK
2INAF – Osservatorio Astronomico di Abruzzo, Via M. Maggini, s/n, I-64100,
Teramo, Italy
3INFN – Sezione di Pisa, Largo Pontecorvo 3, 56127 Pisa, Italy
4Instituto de Astrofisica de Canarias, Via Lactea s/n, La Laguna, Tenerife,
Spain
5Department of Astrophysics, University of La Laguna, Via Lactea s/n, La
Laguna, Tenerife, Spain E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present new cooling models for carbon-oxygen white dwarfs with both H- and
He-atmospheres, covering the whole relevant mass range, to extend our updated
BaSTI (a Bag of Stellar Tracks and Isochrones) stellar evolution archive. They
have been computed using core chemical stratifications obtained from new
progenitor calculations, adopting a semiempirical initial-final mass relation.
The physics inputs have been updated compared to our previous BaSTI
calculations: 22Ne diffusion in the core is now included, together with an
updated CO phase diagram, and updated electron conduction opacities. We have
calculated models with various different neon abundances in the core, suitable
to study white dwarfs in populations with metallicities ranging from super-
solar to metal poor, and have performed various tests/comparisons of the
chemical stratification and cooling times of our models. Two complete sets of
calculations are provided, for two different choices of the electron
conduction opacities, to reflect the current uncertainty in the evaluation of
the electron thermal conductivity in the transition regime between moderate
and strong degeneracy, crucial for the H and He envelopes. We have also made a
first, preliminary estimate of the effect –that turns out to be generally
small – of Fe sedimentation on the cooling times of white dwarf models,
following recent calculations of the phase diagrams of carbon-oxygen-iron
mixtures. We make publicly available the evolutionary tracks from both sets of
calculations, including cooling times and magnitudes in the Johnson-Cousins,
Sloan, Pan-STARSS, Galex, $Gaia$-DR2, $Gaia$-eDR3, $HST$-ACS, $HST$-WFC3, and
$JWST$ photometric systems.

###### keywords:

stars: evolution – stars: interiors – white dwarfs

††pubyear: 2015††pagerange: The updated BaSTI stellar evolution models and
isochrones. III. White Dwarfs–The updated BaSTI stellar evolution models and
isochrones. III. White Dwarfs

##  1 Introduction

According to theory, white dwarfs (WDs) are the most common final stage of the
evolution of single stars. Indeed, all single stars with initial mass up to
$\sim$9-10$M_{\odot}$ are predicted to end up as either He-core (for initial
masses below $\sim$0.5$M_{\odot}$), CO-core (initial masses between $\sim$0.5
and $\sim$6-7$M_{\odot}$), or ONe-core (initial masses between 6-7 and
9-10$M_{\odot}$) WDs, and due to the current age of the universe and the shape
of the stellar initial mass function, the overwhelming majority of WDs
produced so far by single star evolution are predicted to have a CO core.

In the last 25 years, thanks to steady advances in both observations and
theory, models of CO-core WDs have been used extensively in conjuction with
photometric, spectroscopic and asteroseismic data, to determine the ages of
field WDs (e.g., Winget et al., 1987; Oswalt et al., 1996; Torres & García-
Berro, 2016; Kilic et al., 2017; Tononi et al., 2019) for constraining the
star formation history of the Milky Way, the ages of WDs in open clusters
(e.g., Richer et al., 1998; von Hippel, 2005; Bedin et al., 2008; Bellini et
al., 2010; García-Berro et al., 2010; Bedin et al., 2010, 2015) and globular
clusters (e.g., Hansen et al., 2004, 2007; Winget et al., 2009; Bedin et al.,
2009; Goldsbury et al., 2012; Bedin et al., 2019), and even as probes to
investigate open questions in theoretical physics (e.g., Freese, 1984; Isern
et al., 1992; Garcia-Berro et al., 1995; Córsico et al., 2001; Benvenuto et
al., 2004; Bertone & Fairbairn, 2008; García-Berro et al., 2011; Isern et al.,
2018; Winget et al., 2004).

As part of an ongoing new release of the BaSTI (a Bag of Stellar Tracks and
Isochrones) stellar model library (Hidalgo et al., 2018; Pietrinferni et al.,
2021), and given the important role played by evolutionary WD models to
address a range of major astrophysics and physics questions, we present here
new sets of CO-core WD model computations, that update the Salaris et al.
(2010) calculations –hereafter S10– part of the previous BaSTI release, taking
advantage of recent improvements in a number of stellar physics inputs. In
these new calculations, that cover the whole relevant mass range, we have
included the diffusion of 22Ne in the liquid phase, which wasn’t accounted for
by S10, we have updated the electron conduction opacities, the phase diagram
for CO compositions, and also the CO stratification in the cores of the
models, as described in the next sections.

Following the recent analysis by Cassisi et al. (2021), we present
calculations for two different choices of the electron conduction opacities,
and we also provide a preliminary estimate of the effect on the model cooling
times of iron sedimentation in the CO cores, based on the very recent study by
Caplan et al. (2021) of the phase diagram of carbon, oxygen, and iron ternary
mixtures. We make publicly available at the official BaSTI
website111http://basti-iac.oa-abruzzo.inaf.it the evolutionary tracks (cooling
tracks) in the Hertzsprung-Russell diagram, including cooling times and
magnitudes in several photometric filters, obtained for all WD masses and
chemical stratifications presented here.

The plan of the paper is as follows. Section 2 describes the parameter space
covered by our WD calculations, the adopted physics inputs and initial
chemical abundance profiles, and the effect of the 22Ne diffusion on both the
chemical stratification and cooling times of the models. The following Sect. 3
presents numerical tests to quantify the effect of progenitor metallicity and
the adopted relation between progenitor mass and final WD mass, on chemical
profiles and cooling times. We also discuss how the cooling times are affected
by considering the idealized, sharp chemical transitions between the CO core
and the pure-He/H envelopes adopted in our models (and other models in the
literature), and by neglecting the small fraction of heavy elements and atomic
diffusion in the envelopes during the WD evolution. Section 4 compares the
cooling times obtained with the two choices for the conductive opacities,
while Sect. 5 presents a first preliminary estimate of the effect of iron
sedimentation of our model cooling times. Section 6 compares cooling times and
radii of our new calculations with models from the literature, and is followed
by a brief summary of our results which brings the paper to a close.

##  2 Models

We have calculated CO-core WD cooling models with pure-H (representative of
H-atmosphere WDs) and pure He (representative of He-atmosphere WDs) envelopes,
with masses $M_{\rm WD}$ equal to 0.54, 0.61, 0.68, 0.77, 0.87, 1.0 and 1.1
$M_{\odot}$, respectively. These masses are in common with our previous S10
calculations, apart from the highest value.

All computations include the important processes of diffusion of ${}^{22}{\rm
Ne}$ in the liquid phase of the CO-cores (see Bravo et al., 1992; Bildsten &
Hall, 2001; Deloye & Bildsten, 2002; García-Berro et al., 2008; Althaus et
al., 2010), latent heat release and phase separation upon crystallization
(see, e.g., Stevenson, 1980; Mochkovitch, 1983; Garcia-Berro et al., 1988),
which provide sizable contributions to the energy budget of the models.

As in our previous sets of calculations (Salaris et al., 2000, and S10), for
each mass an initial model has been converged at $\log(L/L_{\odot})\sim
1.0-1.5$, by considering a reference CO-core chemical stratification and a
reference thickness and chemical composition of the envelope layers, described
in Sect. 2.2.

For each mass and core chemical stratification we have computed models for two
different choices of the electron conduction opacities –from Cassisi et al.
(2007) and Blouin et al. (2020), respectively– as described in Sect. 2.1.
Figures 1 and 2 display, as an example, the radius of our grid of H-atmosphere
and He-atmosphere models calculated with Cassisi et al. (2007) opacities as a
function of the effective temperature $T_{\mathrm{e}ff}$, and the resulting
cooling tracks in selected colour magnitude diagrams (CMDs). We can easily
notice in three of the displayed CMDs of H-atmosphere models the turn to the
blue of the colours at faint magnitudes (more pronounced in infrared CMDs,
like the $HST$-WFC3 ($F115W,F200W$) diagram) caused by the ${\rm H_{2}}$
collision induced absorption in the atmospheres (see, e.g., Hansen, 1998;
Saumon & Jacobson, 1999). Mass-radius relations and CMDs for all models
calculated with Blouin et al. (2020) opacities are identical to those
displayed in these figures.

Figure 1: Radius of our grid of H-atmosphere (solid lines) and He-atmosphere
(dotted lines) models calculated with Cassisi et al. (2007) electron
conduction opacities, as a function of the models’ $T_{\mathrm{e}ff}$ (at
fixed $T_{\mathrm{e}ff}$ larger radii correspond to less massive models).
Figure 2: Selected CMDs of the models in Fig. 1 (solid lines for H-atmosphere
models, dotted lines for He-atmosphere models) in Johnson-Cousins (upper
panels), $HST$-WFC3 (bottom-left panel), and $Gaia$-eDR3 (bottom-right panel)
filters.

###  2.1 Physics inputs

The WD evolution code employed for our calculations is the same described in
Salaris et al. (2000) and S10. We briefly summarize here the main physics
inputs, and what has changed compared to S10.

The equation of state (EOS) for the H and He layers is from Saumon et al.
(1995), supplemented in the liquid phase by calculations following Segretain
et al. (1994) formalism; for the CO cores we adopted Segretain et al. (1994)
calculations in the solid and liquid phase, and Straniero (1988) EOS for the
gas phase. We have made some test calculations employing for the H and He
layers the recent EOS published by Chabrier et al. (2019), an update of the
Saumon et al. (1995) one. We have found negligible differences in the
structure and cooling times of the models, compared to computations with the
Saumon et al. (1995) EOS.

Phase separation upon crystallization of the CO mixture is included employing
the new phase diagram by Blouin & Daligault (2021), and the energy
contribution due to phase separation is calculated following Isern et al.
(2000).

The diffusion of ${}^{22}{\rm Ne}$ (hereafter Ne) in the liquid layers of the
CO cores is included following García-Berro et al. (2008) and Althaus et al.
(2010). Neon is considered to diffuse as a trace element in a one-component
plasma background – see Eqs. 19 and 20 in García-Berro et al. (2008)– made of
a fictitious element with mean atomic weight and charge calculated by
averaging over the local number fractions of C and O. The Ne diffusion
coefficients come from the most recent results by Hughto et al. (2010), and
the contribution of this process to the energy budget has been calculated
following García-Berro et al. (2008). We have not included phase separation of
Ne in the core in the solid phase, a process whose occurrence is under debate.
It is found to be efficient in WD cores by Blouin et al. (2021) calculations,
while Caplan et al. (2020) results show that Ne cannot separate for realistic
values of the Ne abundance in WDs.

Rates of neutrino energy losses are taken from Itoh et al. (1996), with the
exception of plasma neutrino emission, taken from the more accurate Haft et
al. (1994) calculations222The more recent determination of plasma neutrino
emission rates by Kantor & Gusakov (2007) provides results in agreement with
Haft et al. (1994), in the temperature-density regime of our calculations..
Radiative opacities for temperatures above 10,000 K are taken from Iglesias &
Rogers (1996), while below 10,000 K we employed Saumon & Jacobson (1999)
results for pure-H, and Alexander et al. (1997) for pure-He (and the mixed
H-He or C-O-He compositions in some tests described in Sect. 3) compositions.

Electron conduction opacities are from the calculations by Cassisi et al.
(2007) in one set of models, and from Blouin et al. (2020) for the H and He
envelopes (keeping Cassisi et al., 2007, opacities for the CO cores) in a
second set. Blouin et al. (2020) have recently published new, improved
calculations of electron conduction opacities for H and He compositions for
moderate degeneracy (i.e. when $\theta\sim 1$, where $\theta\equiv
T/T_{\mathrm{F}}$, and $T_{\mathrm{F}}$ is the Fermi temperature) which they
have bridged with Cassisi et al. (2007) calculations for strong degeneracy
($\theta\lesssim 0.1$). These new opacities are lower by a factor up to 2.5–3
near the boundary of the temperature-density domain where the new calculations
are valid. The treatment of the conductive opacities at the transition from
moderate to strong degeneracy is however still uncertain – as discussed in
detail by Cassisi et al. (2021), see also Blouin et al. (2020)– and is crucial
for the envelopes of WD models. Different ways to bridge Blouin et al. (2020)
calculations for moderate degeneracy with Cassisi et al. (2007) for strong
degeneracy can potentially ‘recover’ Cassisi et al. (2007) opacities in the
regime of H and He WD envelopes, and for this reason we have calculated models
using both sets of opacities for H and He compositions. We will show in Sect.
4 the main differences between these two sets of models.

Photospheric boundary conditions for $T_{\mathrm{e}ff}$ below 10,000 K have
been obtained from calculations of the non-grey model atmospheres described in
S10, while at higher $T_{\mathrm{e}ff}$ we use the integration of the
Eddington grey $T(\tau)$ relation. Superadiabatic convection in the envelope
is treated according to the Böhm-Vitense (1958) formalism of the mixing length
theory, with mixing length $\alpha_{\mathrm{M}LT}$=1.5 (see Ludwig et al.,
1994). Hydrogen burning through the $p-p$ cycle in the model envelopes is
included, using the same reaction rates as in Hidalgo et al. (2018). As
discussed later, its effect on our models is negligible.

The bolometric corrections necessary to place our WD cooling tracks in CMDs
are based on the results by Bergeron et al. (1995) and Holberg & Bergeron
(2006), the same employed by S10
333https://www.astro.umontreal.ca/~bergeron/CoolingModels/. For each track we
have calculated magnitudes in the Johnson-Cousins, Sloan, Pan-STARSS, Galex,
$Gaia$-DR2, $Gaia$-eDR3, $HST$-ACS, $HST$-WFC3, $JWST$ photometric systems.

To summarize, the main differences with respect to S10 regarding the physics
inputs of the models are the inclusion of Ne diffusion, the use of the Blouin
& Daligault (2021) CO phase diagram instead of Segretain & Chabrier (1993),
and the Cassisi et al. (2007) (and alternatively Blouin et al., 2020)
conductive opacities, instead of a combination of the older Hubbard & Lampe
(1969) tables, and Itoh et al. (1983), Itoh et al. (1984) analytic formulae
employed by S10.

###  2.2 Chemical stratification

To determine the initial chemical stratification of WD models of a given mass,
we need in principle to compute the full evolution of the progenitor, from the
main sequence through the challenging thermally-pulsing asymptotic giant
branch, and post asymptotic giant branch phases (see, e.g. Renedo et al.,
2010; Camisassa et al., 2016; Camisassa et al., 2017). It is however fair to
say that the modelling of these advanced evolutionary stages of low- and
intermediate-mass stars is subject to non-negligible uncertainties, mainly
related to the efficiency of mass loss, and internal mixing during the thermal
pulses. For example, independent evolutionary calculations show appreciable
differences in the relation between the initial mass of the progenitor and the
final WD mass (see, e.g., Marigo & Girardi, 2007; Weiss & Ferguson, 2009;
Cummings et al., 2018); given that the CO stratification in the WD progeny
depends on the initial mass of their progenitors, variations of the initial-
final mass relation (IFMR) imply variations of the predicted CO profile for a
fixed $M_{\rm WD}$. Also, the mass thickness of the H- and He-rich layers
sorrounding the CO core depend on how many thermal pulses have been
experienced by the progenitor, the timing of the last pulse, and the
efficiency of mass loss during the post asymptotic giant branch phase (see,
e.g., De Gerónimo et al., 2019; Althaus et al., 2005, 2015), that are still
difficult to predict accurately by means of stellar evolution calculations.

Following the philosophy of S10, we provide here calculations with a fixed
constant initial thickness of the H and He envelope layers for all WD masses
and all initial compositions of the progenitors (see also, e.g. Hansen, 1999;
Fontaine et al., 2001; Bédard et al., 2020, for a similar approach), and a
fixed, semiempirical IFMR to determine the core CO stratifications. This
choice provides a reference baseline to eventually help disentangle the effect
of the existing uncertainties related to core and envelope physical and
chemical properties, when comparisons with observations are made.

The H-atmosphere models have ‘thick’ H layers, as in S10; the envelope is made
of pure hydrogen layers enclosing a fraction $q({\rm H})$=$10^{-4}$ of the
total mass $M_{\rm WD}$, around pure-He layers of mass $q({\rm He})=10^{-2}$,
which surround the CO core (the same values chosen by Hansen, 1999; Fontaine
et al., 2001; Bédard et al., 2020). The He-atmosphere WD models have a pure
helium envelope with mass fraction $q({\rm He})=10^{-2}$ around the CO core.
The chosen orders of magnitude of $q({\rm H})$ and $q({\rm He})$ are informed
by results of evolutionary calculations of WD progenitors (see, e.g, Iben &
MacDonald, 1986; Renedo et al., 2010).

With these choices of $q({\rm H})$ and $q({\rm He})$ the surface convection,
which eventually develops during the cooling evolution, is not able cross the
H-He interface in the H-atmosphere models, or the He-CO interface in the He-
atmosphere models. Also, $p-p$ burning slowly changes with time the chemical
composition of the deeper parts of the H envelopes of the models, producing a
small (in mass) tail of increasing He abundance when moving inwards. However,
the cooling times of the models (and the total mass of H and He in the
envelope) are not changed appreciably compared to the case of no burning
allowed.

Whilst the H and He stratification has been set at the start of the cooling
sequence without modelling the previous evolution, the profiles of the C, O
and Ne abundances in the CO cores have been taken from the evolution of models
of the chosen progenitors. To this purpose, we have considered the
semiempirical IFMR by Cummings et al. (2018), and for each value of $M_{\rm
WD}$ in our model grid we have derived the corresponding value of the
progenitor mass $M_{\rm i}$ from Cummings et al. (2018) formula444We have
considered Cummings et al. (2018) results obtained using the Bressan et al.
(2012) models for the WD progenitor lifetimes, instead of the Choi et al.
(2016) ones. The differences in the predicted progenitor masses are however
negligible for $M_{\rm WD}$ lower than 1$M_{\odot}$, and reach at most
0.6$M_{\odot}$ for $M_{\rm WD}$=1.1$M_{\odot}$..

For any given $M_{\rm WD}$ we have then computed the evolution of the
progenitor with mass $M_{\rm i}$ starting from the pre-main sequence,
employing the same code and physics inputs of Hidalgo et al. (2018)
calculations, including overshooting from the convective cores during the main
sequence (but not from the convective envelopes, when present), semiconvection
during the core He-burning phase, mass loss using Reimers (1975) formula with
the free parameter $\eta$ set to 0.3, and an initial solar scaled chemical
composition with metal mass fraction $Z$=0.017 (about solar –see Hidalgo et
al., 2018, for details of the code). We are not interested in following the
details of the asymptotic giant branch evolution of these models, just in the
growth of the He-exhausted core; the initial CO stratification of the WD model
has been then taken as the CO abundance profile inside the He-free core of the
progenitor, when this has reached a mass equal to $M_{\rm WD}$.

Figure 3: Upper panel:  Stratification of the C, O, Ne, H and He abundances in
the inner layers of a 3.1$M_{\odot}$ asymptotic giant branch model with
initial chemical composition about solar. Abundances are given in mass
fractions (for the sake of clarity the Ne mass fraction is increased by a
factor of 10) and $m_{r}$ is the mass enclosed within a radial distance $r$
from the centre of the model. The stratification is taken around the time the
He-free core (whose boundary is marked by the thin dash-dotted vertical line)
has reached 0.77$M_{\odot}$, the final WD mass prescribed by the Cummings et
al. (2018) IFMR (see text for details). Lower panel:  Initial chemical
stratification adopted for the corresponding 0.77$M_{\odot}$ H-atmosphere WD
model ($M$ is the total mass of the model).

As an example, the upper panel of Fig. 3 shows the case of the models with
$M_{\rm WD}$=0.77$M_{\odot}$; it displays the inner chemical stratification of
the progenitor with mass $M_{\rm i}=3.1M_{\odot}$ around the time its He-free
core has reached 0.77$M_{\odot}$. As discussed in Salaris et al. (1997) the
chemical profile in the inner region of the He-exhausted core, with constant
abundances of C and O, is determined by the maximum extension of the central
He-burning convective region. Moving outwards from the centre, the peak in the
oxygen abundance (and the corresponding depression of the C abundance) is
produced when the He-burning shell crosses the semiconvective region partially
enriched in C and O, and ${}^{12}{\rm C}$ is converted into ${}^{16}{\rm O}$
through the ${}^{12}{\rm C}(\alpha,\gamma)^{16}{\rm O}$ reaction. Beyond this
region, the oxygen profile is built by the He-burning shell moving outwards.
Gravitational contraction increases temperature and density in the shell, and
given that the ratio between the ${}^{12}{\rm C}(\alpha,\gamma)^{16}{\rm O}$
and $3\alpha$ reaction rates is lower for larger temperatures, the oxygen
abundance steadily decreases in the external part of the CO core. The flat
abundance profile of Ne is caused by the processing of N in the He core during
He-burning through the reactions
${}^{14}\mathrm{N}\,(\alpha,\gamma){}^{18}\mathrm{F}\,(\beta^{+}){}^{18}\mathrm{O}\,(\alpha,\gamma){}^{22}\mathrm{Ne}\,$;
as a result, the final Ne mass fraction is about equal to the initial metal
mass fraction $Z$ of the progenitor models.

The lower panel of Fig. 3 shows the chosen initial chemical stratification of
the corresponding WD, after the chemical rehomogeneization of the inner core
caused by the molecular weight inversion due to the above-mentioned local peak
in the O abundance (and dip of the carbon mass fraction) discussed in Salaris
et al. (1997).

This kind of chemical profiles is admittedly ‘artificial’, because the CO-He-H
chemical transitions are not predicted to be sharp by fully evolutionary
calculations (see, e.g. Renedo et al., 2010; De Gerónimo et al., 2018, for
some examples); This can be seen also in the upper panel of the same figure,
which displays the beginning of a slowly rising He abundance from the boundary
of the He-free core, with C and O abundances different from zero in a region
where He is also present. In Sect. 3 we will show that imposing these sharp
transitions instead of more realistic ones does not have any major effect on
the cooling timescales of the models (but they have an impact on the model
asteroseismic properties, see, e.g., Giammichele et al., 2017).

Figure 4: Oxygen abundance (mass fraction) stratification adopted for our
models (see text for details).

In this way we have determined the initial chemical profiles for all selected
values of $M_{\rm WD}$, from calculations of progenitor models with $Z$=0.017.
Figure 4 displays the resulting oxygen stratifications (given in mass fraction
$X_{\rm O}$) across the CO cores, with the central value of $X_{\rm O}$
typically decreasing with increasing $M_{\rm WD}$. The corresponding carbon
mass fraction $X_{\rm C}$ is given by $X_{\rm C}$=1-$X_{\rm O}$-$Z$, where $Z$
is the progenitor metallicity, which is equal to a very good approximation to
the mass fraction of Ne ($X_{\rm Ne}$) across the core.

Given the importance of the amount of Ne in the core for the cooling
timescales of the models (see, e.g., García-Berro et al., 2008), for each WD
mass we have calculated models for five different initial Ne flat abundance
profiles, corresponding to progenitors’ metallicities equal to $Z=$0.006,
0.01, 0.017, 0.03, 0.04. When the mass fraction of Ne is below 0.006, we found
that the effect on both H-atmosphere and He-atmosphere models’ cooling
timescales is always negligible, hence we have also computed an additional set
with no Ne, representative of WDs from progenitors with $Z<$0.006. For all
these sets of calculations with varying Ne abundances, we have kept the same
CO profiles determined as described before (see Sect. 3).

Before closing this section, we wish to briefly comment on the results by
Giammichele et al. (2018), later refined by Charpinet et al. (2019), who have
determined the radial chemical stratification of the $\sim$0.56$M_{\odot}$
pulsating He-atmosphere WD KIC08626021, using asteroseismic techniques555See
also the preliminary results by Charpinet et al. (2021) for five more
objects.. In the helium-free layers they found the presence of a central
homogeneous core with a mass of $\sim$0.45 $M_{\odot}$ and $X_{\rm
O}\sim$0.84, and also an external, very thin (in mass) almost pure carbon
buffer (see Fig. 3 in Charpinet et al., 2019). As discussed extensively by De
Gerónimo et al. (2019), within our current understanding of the evolution of
WD progenitors, it is difficult to reproduce the chemical structure of this
star as derived from asteroseismology. For example, looking at the oxygen
stratification of our 0.54$M_{\odot}$ models (a mass very close to the value
estimated for KIC08626021) shown in Fig. 4, the central $X_{\rm O}$ is much
lower than the asteroseismic value, and the chemically homogeneous inner core
has a mass of $\sim$0.32$M_{\odot}$ instead of 0.45 $M_{\odot}$. We have
assessed the impact of this asteroseismic CO profile on our WD models by
computing the evolution of 0.54$M_{\odot}$ WDs with hydrogen and helium
atmospheres respectively (the thickness of the H and He envelopes is the same
adopted in the calculations described in this section) and no Ne –using
Cassisi et al. (2007) conductive opacities– employing the CO stratification
determined by Charpinet et al. (2019). We have found small differences in the
cooling times compared to our calculations with the chemical profile of Fig.
4, amounting at most to 4% (cooling times being shorter with the asteroseismic
chemical stratification) at luminosities log($L/L_{\odot}$)$\sim-$4.9.

###  2.3 The effect of Ne diffusion

As discussed in detail in the literature (see, e.g. Bildsten & Hall, 2001;
García-Berro et al., 2008), ${}^{22}{\rm Ne}$ nuclei have a larger mass-to-
charge ratio than ${}^{12}{\rm C}$ and ${}^{16}{\rm O}$ (the ratio between
atomic weight and charge is larger than 2 for ${}^{22}{\rm Ne}$); this results
in a net downward gravitational force on Ne, with a consequent slow diffusion
towards the centre in the liquid layers, and the release of gravitational
energy that contributes to the WD energy budget.

Figure 5: Initial (solid line) and final (dotted line) Ne abundance
stratification (abundance in mass fraction) of a 0.61$M_{\odot}$ H-atmosphere
model from a progenitor with $Z$=0.04 (see text for details).

Figure 5 displays the initial Ne abundance profile in the CO core of our
0.61$M_{\odot}$ H-atmosphere calculations with a $Z$=0.04 progenitor, and the
final Ne stratification when the core has fully crystallized. On the whole,
the final distribution of Ne is more concentrated towards the centre compared
to the initial one, and the details of the final abundance profile arise from
the interplay between the settling of Ne towards the centre, and the
inhibition of diffusion at the crystallization front, that moves outwards with
time.

Figure 6: Cooling time delay $\Delta t_{\rm cool}$ (in Gyr) as a function of
the luminosity, caused by the diffusion of Ne in 0.61$M_{\odot}$ (solid lines)
and 1.0$M_{\odot}$ (dashed lines) H-atmosphere WD models, from progenitors
with $Z$=0.006, 0.01, 0.017, 0.03, 0.04 (in order of increasing maximum
$\Delta t_{\rm cool}$) respectively.

As already mentioned, this displacement of Ne towards the centre causes a
release of gravitational energy, that slows down the cooling of the models.
This is shown by Figures 6 and 7, which display the extra time $\Delta t_{\rm
cool}$ that selected H-atmosphere and He-atmosphere models –calculated with
Cassisi et al. (2007) electron conduction opacities– take to cool down to a
given luminosity, due to the diffusion of Ne. In addition to the mass of the
model, the progenitor metallicity here plays a major role, because an increase
of the amount of Ne in the core increases $\Delta t_{\rm cool}$. These delays
can have the same order of magnitude of the effect of latent heat release plus
phase separation upon crystallization of the CO mixture (which amounts to
$\sim$1.85 Gyr for the 0.61$M_{\odot}$ H-atmosphere models), depending on the
initial Ne abundance.

Figure 7: As Fig. 6 but for the corresponding He-atmosphere models. The
displayed 0.61$M_{\odot}$ calculations are for progenitors with $Z$=0.03 and
0.04; the 1.0$M_{\odot}$ calculations are for progenitors with $Z$=0.01,
0.017, 0.03, 0.04 (in order of increasing maximum $\Delta t_{\rm cool}$). The
cooling delay for models from lower metallicity progenitors are negligible.

The values of $\Delta t_{\rm cool}$ due to Ne diffusion (and also
crystallization) are in general smaller for He-atmosphere models, due to the
lower opacities of the non-degenerate envelope, and for the lower mass models
they are already negligible for progenitors around solar metallicity.

##  3 Tests on the initial chemical profiles

In the previous section we have compared the cooling times of WD models from
progenitors with different initial metallicities. These models have been
computed with Ne abundances appropriate for the progenitor metallicity, but
with a fixed CO stratification at a given $M_{\rm WD}$, determined from
progenitors with metallicity about solar and mass derived from a single IFMR.
In this way we have quantified the effect of Ne diffusion on the cooling times
of our models, for different Ne abundances and $M_{\rm WD}$.

In this section we want first to assess whether changing just the Ne abundance
in the core is a reasonable approximation to determine reliable cooling times
for WDs produced by progenitors with varying metallicity. Whilst the Ne mass
fraction is always to a very good approximation equal to $Z$ and uniform
across the core for any $M_{\rm WD}$, irrespective of the assumed values of
$M_{\rm i}$, the situation for CO is more complex. In fact, the CO
stratification for a given $M_{\rm WD}$ depends in principle on the progenitor
$Z$ through a combination of the effect of $Z$ on the CO profiles at fixed
$M_{\rm i}$, and the effect of metallicity on the IFMR, that associates a
progenitor mass $M_{\rm i}$ to a given $M_{\rm WD}$.

Semiempirical determinations of the IFMR are based on objects of metallicity
typically around solar, apart from the lowest mass WDs included in Cummings et
al. (2018) IFMR adopted here, that come from observations of the globular
cluster NGC 6121. Therefore, we do not have empirical indications of how the
IFMR globally changes with changing progenitor metallicity, with the
consequent not well established impact on the CO stratification of the WD
models.

As already mentioned, theoretical predictions of the IFMR do not provide
consistent results. The only qualitative trend on which various theoretical
calculations of asymptotic giant branch evolution (see, e.g., Marigo &
Girardi, 2007; Weiss & Ferguson, 2009) generally agree is that, with
decreasing $Z$, the value of $M_{\rm i}$ for a fixed $M_{\rm WD}$ tends
typically to decrease, at least for $M_{\rm i}$ above $\sim$2-2.5$M_{\odot}$.
Assuming this qualitative trend is real, we have performed the following test
on H-atmosphere models, calculated with the Cassisi et al. (2007) conductive
opacities.

The upper panels of Fig. 8 display pairs of oxygen abundance profiles employed
to calculate 0.87$M_{\odot}$ and 1.1$M_{\odot}$ models, respectively. We have
considered first our reference abundances determined as described in the
previous section, which in this case come from progenitors with $M_{\rm
i}$=3.6$M_{\odot}$ (for $M_{\rm WD}$=0.87$M_{\odot}$), and $M_{\rm
i}$=6.4$M_{\odot}$ (for $M_{\rm WD}$=1.1$M_{\odot}$), both calculated with
$Z$=0.017. For these progenitor the mass of the He-free core has increased by,
respectively, $\sim$0.16$M_{\odot}$ and $\sim$0.18$M_{\odot}$ compared to the
values at the first thermal pulse.

The second set of oxygen stratifications have been obtained by considering
progenitors with $Z$=0.0002 (a low metallicity typical of metal-poor Galactic
globular clusters) and masses $M_{\rm i}$ such that the models reach $M_{\rm
WD}$=0.87$M_{\odot}$ and 1.1$M_{\odot}$ when the mass of their He-free cores
had grown by $\sim$0.16$M_{\odot}$ and $\sim$0.18$M_{\odot}$, respectively,
compared to the values at the first thermal pulse, like the case of our
reference IFMR. The corresponding values of $M_{\rm i}$ are equal to
2.5$M_{\odot}$ and 5$M_{\odot}$, lower than the values prescribed by our
reference IFMR, consistent with the qualitative trend expected from asymptotic
giant branch evolutionary models.

For each value of $M_{\rm WD}$, the two abundance profiles from progenitors
with different metallicities turn out to be very similar. In the assumption
that our assumed metal poor IFMR is the ‘real’ one, then we can expect that if
we use our reference CO stratification to calculate models for WDs in
$Z$=0.0002 populations, we will determine cooling times negligibly different
from the case of using the appropriate metal poor IFMR.

This is shown in the lower panels of the same figure, which compare the
cooling times of our reference 0.87$M_{\odot}$ and 1.1$M_{\odot}$ calculations
without Ne (because its effect on the cooling times is negligible at this
metallicity) with models calculated using the core stratification from the
appropriate $Z$=0.0002 progenitors (and no Ne). We find almost negligible
fractional differences of the cooling times, the largest values being at most
equal to 2%.

Given the results of this test –based on necessarily arbitrary assumptions–
that somehow mimics the qualitative behaviour of the IFMR with $Z$ as
predicted by theoretical asymptotic giant branch models, and the lack of
empirical assessments of how the IFMR varies with metallicity, we consider our
models calculated with fixed IFMR and CO profiles and varying Ne abundances
(see Sect. 2) to be an acceptable choice to study the cooling of WDs in
populations with different metallicities.

Figure 8: Upper panels: Pairs of oxygen abundance profiles (abundance in mass
fraction) employed for the calculation of 0.87 and 1.1$M_{\odot}$ H-atmosphere
WD models. The dashed lines display our adopted reference profiles, whilst the
solid lines correspond to the case of metal poor progenitors with $Z$=0.0002
(see text for details). Lower panels: Relative difference between the cooling
times obtained using the CO stratification of the metal poor progenitor, and
those obtained from the reference stratification, as a function of the
luminosity. Selected cooling ages from the calculations with the reference
profiles are marked.

###  3.1 Testing the idealized chemical interfaces

Another assumption –common in the literature– made in our WD model
calculations is that the CO-He-H chemical transitions are sharp, with an
abrupt switch from the CO (and Ne) core to a pure-He envelope, and from the He
envelope to a pure-H envelope. As mentioned in the previous section, this is
just an idealization. In reality, progenitor model calculations show a narrow
(in mass) transition region around the He-free core, where the He abundance
progressively increases outwards, and the C and O abundances drop to zero, due
to incomplete burning in the He-shell. Additionally, H and He are not
‘separated’ in the envelopes of the asymptotic giant branch progenitors, even
considering the effect of atomic diffusion during the previous evolution, and
pure-H and He envelopes do not account for the metals in the initial chemical
composition of the progenitor. The CNO of the original chemical composition
can be involved in hydrogen burning if the mass of the hydrogen layers is
large enough (see, e.g., Althaus et al., 2015, and below).

As discussed for example in Renedo et al. (2010), atomic diffusion during the
WD cooling makes helium and metals sink from the outer evelope, producing a
thin (in mass) pure-H layer that thickens as evolution proceeds. Chemical
transitions exhibit increasingly less sharp discontinuities, and the metals
progressively sink towards deeper and deeper regions of the envelope. Overall,
the chemical stratification of WD model envelopes is expected not only to be
initially without sharp chemical transitions, but also to change slowly during
the cooling. To test the effect of the simplified chemical transitions adopted
in our models on the predicted cooling ages, we have performed the following
numerical experiment.

Figure 9: Upper panel: Solid lines display the C, O, He and H stratification
(abundances in mass fraction) at the beginning of the cooling sequence of
0.61$M_{\odot}$ H-atmosphere models by Renedo et al. (2010). The dotted lines
display our adopted core stratification with CO profile determined as
described in Sect. 2.2, plus stratified pure-H and He envelopes which match
the total mass of H and He of Renedo et al. (2010) models. Lower panel:
Relative difference of cooling times as a function of luminosity with respect
to Renedo et al. (2010) calculations, of 0.61$M_{\odot}$ models calculated
using the stratification with pure-He and H envelopes shown in the upper panel
(dotted line – see text for details). Some reference cooling ages from Renedo
et al. (2010) calculations are marked.

We have considered the C, O, He, H chemical stratification of the
0.61$M_{\odot}$ H-atmosphere WD model by Renedo et al. (2010), calculated from
a progenitor with $M_{\rm i}$=2$M_{\odot}$ and metallicity $Z$=0.01, taken at
log($L/L_{\odot}$)$\sim$1.0, and displayed in the upper panel of Fig. 9. In
this plot the mass fractions of C and O below $10^{-4}$ found in the He- and
H-rich envelopes of Renedo et al. (2010) model are set to zero; the nitrogen
abundance, that is accounted for by Renedo et al. (2010) is also set to zero
everywhere. The total masses of hydrogen and helium in the envelope correspond
to $q({\rm H})$=$1.3\times 10^{-4}$ and $q({\rm He})$=$2.3\times 10^{-2}$,
respectively.

Starting from this chemical profiles, we have built the second chemical
stratification displayed in the upper panel of Fig. 9, which mimics the way we
have built the chemical make-up of our new BaSTI models presented in this
paper (see Sect. 2.2). The CO-He-H discontinuities are sharp, the envelope is
made of pure-H and pure-He layers with $q({\rm H})$=$1.3\times 10^{-4}$ and
$q({\rm He})$=$2.3\times 10^{-2}$, to preserve the total amounts of H and He
of Renedo et al. (2010) initial model, and the CO stratification has been
obtained by taking the CO abundance profile inside the He-free core of Renedo
et al. (2010) model. Given that we do not account for the small amount of C
and O in the abundance tails that overlap the helium envelope (i.e., in the
region between log($1-m_{r}/M$) between $\sim-$1.2 and $\sim-2.4$) we needed
to apply a very small rescaling of the mass coordinate in the CO core, to
conserve the total mass of the model. This profile is a proxy for Renedo et
al. (2010) initial profile, built essentially like the chemical stratification
of the set of new BaSTI WD models we are presenting in this paper.

Using this chemical stratification, we have calculated with our code the
evolution of a 0.61$M_{\odot}$ WD model using the same EOS, boundary
conditions, radiative and conductive opacities, neutrino energy loss rates,
and CO phase diagram employed by Renedo et al. (2010), who used the LPCODE for
their calculation. For this test –like for all our BaSTI WD models– we neglect
the evolution of the H and He profiles due to atomic diffusion, set all metal
abundances to zero in the non-degenerate envelope, and hence we neglect also
the occurrence –a small effect in Renedo et al. (2010) calculations– of CNO
burning in the hydrogen envelope.

The lower panel of Fig. 9 displays the relative difference of the cooling
times (zeroed at log($L/L_{\odot}$)$\sim$1.0) with respect to Renedo et al.
(2010) models666The result of this comparison is significant, given that in
Salaris et al. (2013) we have shown how WD models calculated with the LPCODE
and our BaSTI code using the same physics inputs and initial chemical
stratification, have cooling times within 2% or better, and radii consistent
at the 0.5% level.. Our calculation displays systematically shorter ages (up
to about 20% shorter) during the early phases of cooling, when the age of the
models is on the order of 10 Myr or less. This difference is most likely due
to the different thermal structures at log($L/L_{\odot}$)$\sim$1.0, the start
of our calculations777Renedo et al. (2010) followed the full evolution from
the main sequence to the white dwarf stage of the progenitor, whilst we have
converged an initial ‘artificial’ structure at log($L/L_{\odot}$)$\sim$1.0,
with the chosen total mass and chemical profile.. By
log($L/L_{\odot}$)$\sim-$1.8 and cooling ages around 100 Myr neutrino emission
–by now inefficient– has erased any difference of the initial thermal
structure of the models, and the cooling times become almost identical.
Differences reach at most about $\pm$6% when log($L/L_{\odot}$) goes below
$\sim-$4.0, corresponding to ages above $\sim$4.5 Gyr. The radius of our
models is systematically lower than Renedo et al. (2010) results, differences
amounting to just about 3% at log($L/L_{\odot}$)=0.5, decreasing to 1.5% and
log($L/L_{\odot}$)=$-$2.0, and below 1% when log($L/L_{\odot}$) is lower than
$\sim-3$.

To summarize, this test shows that the use of sharp chemical interfaces, the
neglect of the abundance of the metals in the H- and He-rich layers, and the
effect of atomic diffusion and CNO burning in the envelope of models with
total H and He mass fractions similar to the choice adopted in our BaSTI
calculations, have a minor effect on the models’ cooling times and radius,
amounting to at most a few percent.

##  4 The role of the electron conduction opacities

As motivated and discussed in Sect. 2.1, we have computed two sets of new
BaSTI WD calculations, employing the Cassisi et al. (2007) and Blouin et al.
(2020) conductive opacities, respectively. Blouin et al. (2020) and Cassisi et
al. (2021) have shown that WD models calculated with Blouin et al. (2020)
conductive opacities display longer cooling times at bright luminosities and
shorter cooling times at faint luminosities, compared to calculations with
Cassisi et al. (2007) opacities; Figs 10 and 11 confirm their results,
displaying the relative difference of the cooling ages of 0.61$M_{\odot}$ and
1.0$M_{\odot}$ H-atmosphere and He-atmosphere models calculated with both sets
of opacities.

Figure 10: Relative difference of cooling times as a function of luminosity
for our 0.61$M_{\odot}$ and 1.0$M_{\odot}$ H-atmosphere models with no Ne
(corresponding to metal poor populations), calculated with Blouin et al.
(2020) and Cassisi et al. (2007) opacities, respectively. A positive
difference means that models computed with Blouin et al. (2020) opacities have
longer cooling times. Some reference cooling ages from the calculations with
Cassisi et al. (2007) opacities are marked. Figure 11: Same as Fig. 10 but for
He-atmosphere models.

The lower Blouin et al. (2020) conductive opacities for the H and He layers
induce a faster cooling of the core, which in turns reduces the efficiency of
neutrino cooling at bright luminosities, and increases the cooling times at a
fixed value of log($L/L_{\odot}$). The absolute values of the cooling times at
these luminosities are however fairly short, well below 100 Myr. With
decreasing luminosities, the cooling times calculated with Blouin et al.
(2020) opacities become increasingly shorter than calculations with Cassisi et
al. (2007) opacities, because of the faster cooling of the structure, and
differences increase with increasing $M_{\rm WD}$. This trend is temporarily
broken by the earlier start of CO crystallization in the models with Blouin et
al. (2020) opacities, and the associated earlier onset of the release of
latent heat and extra energy due to phase separation.

For initial Ne abundances that have an impact on the cooling times of the
models, the trends shown in Figs. 10 and 11 are unchanged. Due to the lower
opacity of the denser layers of the envelope, the cooling delay caused by the
diffusion of Ne in the liquid phase is shorter in models with Blouin et al.
(2020) opacities, but the increase of the relative age differences at low
luminosities is however small, by at most a few percent for the higher masses
and higher Ne abundances.

##  5 Sedimentation of iron in the core

After C, O and Ne, the next more abundant element in the CO core is predicted
to be ${}^{56}{\rm Fe}$ (hereafter Fe), part of the initial metal content of
the WD progenitors. For the solar scaled metal mixture by Caffau et al. (2011)
used in our Hidalgo et al. (2018) progenitor calculations, the mass fraction
of Fe is equal to about 8.8% of $Z$.

Caplan et al. (2021) have very recently computed new phase diagrams for a COFe
ternary mixture and performed molecular dynamics calculations, showing the Fe
present in WD CO cores can precipitate to form an inner iron core during
crystallization. This process would release energy (like the diffusion of Ne,
or the phase separation of the CO mixture) slowing down the cooling of the
models. The sedimentation of Fe was already briefly explored by Segretain &
Chabrier (1993) and Segretain et al. (1994), and discussed in more details by
Xu & van Horn (1992), making use of calculations of earlier phase diagrams.
The details of this process are still uncertain, and various outcomes are
possible regarding the chemical stratification of the core during the
crystallization of the C, O and Fe mixture (see Caplan et al., 2021); here we
discuss the more standard scenario presented by Caplan et al. (2021), making a
very preliminary estimate of the effect of Fe sedimentation on the cooling
times of our models.

When $\Gamma_{\rm C}=6^{5/3}(2.275\,10^{5}/T)(\rho/2)^{1/3}$ (with density
given in g/$\rm cm^{3}$ and temperature in K – the Coulomb parameter of a
pure-C composition) reaches a critical value equal to 140 in a layer of the CO
core, its Fe precipitates on very short timescales towards the centre; we
assume here that the fraction of Fe that precipitates is 100% (it could be
less than this, according to Caplan et al., 2021) and that this iron settles
virtually instantaneously at the centre, replacing C and O (and we assume also
Ne, although the fate of Ne is not discussed in the paper by Caplan et al.,
2021, and another possibility is that this core containing the precipitated Fe
is made instead of a C, O, Fe alloy). As the cooling proceeds, more external
layers reach $\Gamma_{\rm C}$=140, and the mass of this pure-Fe core
increases, until the CO mixture starts to crystallize and Fe sedimentation
stops (Caplan et al., 2021, speculate that Fe sedimentation might continue
also after C and O start to crystallize). From this moment on, the CO
crystallization proceeds like in the case of neglecting the presence of Fe.

We have implemented this process to follow the evolution of our $M_{\rm
WD}$=0.54$M_{\odot}$ and $M_{\rm WD}$=1.0$M_{\odot}$ models, respectively,
coming from progenitors with $Z$=0.04, to maximize the amount of Fe in the
core. This metallicity corresponds to a Fe mass fraction $X_{\rm Fe}$=0.0035.
In the calculations we assumed that the diffusion of Ne continues also during
the sedimentation of Fe.

For the $M_{\rm WD}$=0.54$M_{\odot}$ case all iron within the inner 11% in
mass of the CO core is displaced to form a pure-Fe core with mass $M_{\rm
Fe}$=0.00021$M_{\odot}$, before the CO component starts to crystallize (this
happens when $\Gamma_{\rm C}\sim 160$ in the innermost CO layers outside the
Fe core). The layers depleted in Fe are within the flat part of the CO
abundance stratification, that spans the inner 55-60% in mass of the core (see
Fig. 3).

The C, O and Ne removed from the layers now belonging to the iron core replace
the heavier Fe in the layers affected by sedimentation, and this causes a
small inversion of the mean molecular weight with the overlying Fe-rich
layers. The whole region beyond the iron core that originally had an
homogeneous CO stratification gets mixed by convection to cancel this
inversion, and part of the energy gained by the sedimentation of Fe is
absorbed in these convective layers888Also during the CO crystallization part
of the energy gained is absorbed by the layers that are rehomogeneized by the
convective episodes caused by phase separation (see,e.g., Isern et al.,
2000).. As a result, the C and O abundances in these mixed layers are only
very slightly changed, by much less than 1%. The Ne abundance profile, which
is affected by diffusion, is rehomogeneised in these layers, but the effect on
the contribution of the further development of Ne diffusion to the model
energy budget is negligible.

The cooling age of the models increases by $\sim$150 Myr in the range between
log($L/L_{\odot}$) $\sim-$3.8 and $-$3.9 when we use the Blouin et al. (2020)
opacities (at cooling ages around 2.5 Gyr), and by about 190 Myr between
log($L/L_{\odot}$) $\sim-$3.9 and $-$4.0 in models calculated with Cassisi et
al. (2007) opacities (at cooling ages around 3.5 Gyr).

For the $M_{\rm WD}$=1.0 calculations the Fe core mass at the end of the
sedimentation is equal to $M_{\rm Fe}$=0.00057$M_{\odot}$, equivalent to about
16% of the total Fe in the CO core; as a result, the cooling age increases by
$\sim$120 Myr in the range between log($L/L_{\odot}$) $\sim-$2.6 and $-$2.8 in
models calculated with Blouin et al. (2020) opacities (at cooling ages around
790 Myr), and by about 200 Myr between log($L/L_{\odot}$) $\sim-$2.7 and
$-$2.9 in models calculated with Cassisi et al. (2007) opacities (at cooling
ages around 1 Gyr).

##  6 Comparison with other models

To put our new calculations in the broader context of existing WD models, we
make first a comparison with our previous S10 results. While the models’ radii
change by less than 1% compared to S10, the same is not true regarding the
cooling times. As an example, Figs. 12 and 13 compare the cooling times of our
0.61$M_{\odot}$ H-atmosphere and He-atmosphere calculations with their S10
counterparts, which do not include Ne diffusion.

Figure 12: Relative difference of cooling times (in Gyr) as a function of
luminosity with respect to the 0.61$M_{\odot}$ H-atmosphere calculations from
S10 of, respectively, our new 0.61$M_{\odot}$ H-atmosphere calculations with
Cassisi et al. (2007) opacities without Ne diffusion (solid line), with Ne
diffusion and $Z$=0.017 Ne abundance (dotted line), with Ne diffusion and
$Z$=0.04 Ne abundance (dashed line). Some reference ages from S10 calculations
are marked.

For log($L/L_{\odot}$) larger than about $\sim-$ 3.5, the $\pm 10$%
differences of cooling times with our models without Ne diffusion are due just
to the older conductive opacities adopted by S10. Below this luminosity we see
the effect also of the new CO phase diagram employed here. The Blouin &
Daligault (2021) diagram delays the onset of crystallization compared to the
Segretain & Chabrier (1993) diagram used by S10, and the amount of matter
redistributed by phase separation for a given $M_{\rm WD}$ is sizably smaller
when the Blouin & Daligault (2021) phase diagram is employed. This causes a
smaller amount of energy released by CO phase separation, causing a faster
cooling compared to S10 models999The differences with S10 results due to the
use of Blouin & Daligault (2021) phase diagram instead of the Segretain &
Chabrier (1993) one are very similar to the results discussed by Althaus et
al. (2012). These authors compared models calculated with Segretain & Chabrier
(1993) phase diagram and the Horowitz et al. (2010) one, which is very similar
to the more recent Blouin & Daligault (2021) diagram.. The CO stratification
is also not the same as in S10, although differences are small, and the impact
on the crystallization process is relatively minor. When we include Ne
diffusion the cooling times increase and eventually, in models calculated with
Cassisi et al. (2007) opacities and for super-solar Ne abundances, they become
longer than S10 results when log($L/L_{\odot}$) is below $\sim-$4\. In case of
the He-atmosphere models, for both choices of conductive opacities –even when
including Ne diffusion– the cooling times of our models at log($L/L_{\odot}$)
below $\sim-$4 are always shorter than S10 results.

Figure 13: As Fig. 12, but for our new models calculated with Blouin et al.
(2020) opacities.

In Fig. 14 we compare the cooling times of our 0.61$M_{\odot}$ H-atmosphere
models including Ne diffusion and a progenitor metallicity $Z$=0.017,
calculated with Cassisi et al. (2007) opacities, with results by Camisassa et
al. (2016) for a 0.6$M_{\odot}$ H-atmosphere WD computed with the same
conductive opacities, which include Ne diffusion and account for the full
evolution of a solar metallicity progenitor101010At
http://evolgroup.fcaglp.unlp.edu.ar/TRACKS/newtables.html. Camisassa et al.
(2016) models have a different thickness of the envelope layers compared to
our calculations, a different CO stratification and there are some differences
in the physics inputs, but we take the results of the comparison of cooling
times at face value.

Figure 14: Relative difference of the cooling times (zeroed at
log($L/L_{\odot}$)=1.1) as a function of luminosity between our
0.61$M_{\odot}$ H-atmosphere cooling track with Ne abundance from a $Z$=0.017
progenitor, and a 0.6$M_{\odot}$ H-atmosphere track with solar metallicity
progenitor from Camisassa et al. (2016). Some reference cooling ages from
Camisassa et al. (2016) calculations are marked.

At luminosities above log($L/L_{\odot}$) $\sim-1.5$ and ages below $\sim 100$
Myr our models display systematically lower cooling ages, most likely due to
differences in the thermal structure at the beginning of the cooling
evolution. Cooling times are then within $\pm$2-4% down to log($L/L_{\odot}$)
$\sim-$4, beyond which our models evolve at a slower speed and display cooling
times up to about 10% longer.

##  7 Summary

We have extended our updated BaSTI stellar evolution archive with the
inclusion of new models for H- and He-atmosphere CO-core WDs, computed using
the C,O, and Ne stratification obtained from new progenitor calculations,
adopting a semiempirical IFMR. We have also updated the model physics inputs
compared to the previous S10 BaSTI WD models; in particular, we now include Ne
diffusion in the core, an updated CO phase diagram, and updated electron
conduction opacities.

We have calculated the evolution of models with masses $M_{\rm WD}$ equal to
0.54, 0.61, 0.68, 0.77, 0.87, 1.0 and 1.1 $M_{\odot}$, made of a CO core
surrounded by pure-He layers of mass fraction $q({\rm H})=10^{-2}$, and (for
H-atmosphere models) outermost H layers enclosing a mass fraction $q({\rm
H})$=$10^{-4}$. For any given $M_{\rm WD}$ we have computed the evolution of
models with several Ne abundances, appropriate for initial metallicities of
the progenitors equal to $Z$=0.006, 0.01, 0.017, 0.03, and 0.04, corresponding
to [Fe/H]=$-$0.40, $-$0.20, 0.06, 0.30 and 0.45. We have also models
calculated with no Ne, representative of WDs from progenitors with [Fe/H]
lower than $-$0.40 ($Z$=0.006), because the corresponding small quantities of
Ne have a negligible effect on the cooling times of the models. In all these
calculations we have kept fixed the CO abundance profiles to those obtained
from progenitors with $Z$=0.017 ([Fe/H]=0.06), and the IFMR is kept also
unchanged in all calculations, for it is not yet established empirically
if/how the IFMR varies as a function of the progenitor metallicity. We
estimated that the effect of this assumption on the model cooling times is
potentially negligible, when we take into account the qualitative trend with
metallicity of the IFMRs predicted by theoretical asymptotic giant branch
models.

All our models comprise a CO (and Ne) core surrounded by pure-He and H (in
H-atmosphere models) envelopes. These sharp chemical transitions and envelopes
made of pure-He and H are actually a simplified description of the chemical
structure of a WD. We have performed test calculations which have quantified
to at most $\pm$6% the effect on the model cooling times of our idealization
of the chemical transitions, and the neglect of the metals in the H and He
envelopes.

We have also made a first, preliminary estimate of the effect of Fe
sedimentation on the cooling times of WD models, following the recent results
by Caplan et al. (2021) regarding the phase diagram of C, O and Fe mixtures.
The effect turns out to be generally minor, but there are still substantial
uncertainties on the details of this process.

Two complete sets of calculations have been performed, for two different
choices of the electron conduction opacities, to reflect the current
uncertainty in the evaluation of the electron thermal conductivity in the
crucial (for the H and He envelopes) transition regime between moderate and
strong degeneracy. Models calculated with different opacities indeed display
non-negligible differences in their cooling times, larger for higher values of
$M_{\rm WD}$. In a next paper we will employ both sets of calculations to
study WDs in old stellar populations, with the aim of setting some stringent
constraints on how to bridge calculations of conductive opacities for moderate
and strong degeneracy, as suggested by Cassisi et al. (2021).

We make publicly available the cooling tracks from both sets of calculations,
including cooling times and magnitudes in the Johnson-Cousins, Sloan, Pan-
STARSS, Galex, $Gaia$-DR2, $Gaia$-eDR3, $HST$-ACS, $HST$-WFC3, and $JWST$
photometric systems.

## Acknowledgements

We thank the referee for constructive comments that helped us improving the
manuscript. SC acknowledges support from Premiale INAF MITiC, from INFN
(Iniziativa specifica TAsP), and from PLATO ASI-INAF agreement
n.2015-019-R.1-2018. MS acknowledges support from STFC Consolidated Grant
ST/V00087X/1. SC warmly thanks the Instituto de Astrofisica de Canarias for
the hospitality and the ‘Programa de investigadores visitantes’de la Fundacion
Jesus Serra.

## Data Availability

All WD tracks in several photometric systems are available at the new official
BaSTI website http://basti-iac.oa-abruzzo.inaf.it.

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