# New full evolutionary sequences of H and He atmosphere massive white dwarf
stars using MESA

G.R. Lauffer,1 A. D. Romero,1 S.O. Kepler1
1Instituto de Física, Universidade Federal do Rio Grande do Sul, Av. Bento
Goncalves 9500, Porto Alegre 91501-970, RS, Brazil E-mail:
<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We explore the evolution of hydrogen-rich and hydrogen-deficient white dwarf
stars with masses between $1.012$ and $1.307$ M⊙, and initial metallicity of
$Z=0.02$. These sequences are the result of main sequence stars with masses
between $8.8$ and $11.8$ M⊙. The simulations were performed with MESA,
starting at the zero-age main sequence, through thermally pulsing and mass-
loss phases, ending at the white dwarfs cooling sequence. We present reliable
chemical profiles for the whole mass range considered, covering the different
expected central compositions, i.e. C/O, O/Ne and Ne/O/Mg, and its dependence
with the stellar mass. In addition, we present detailed chemical profiles of
hybrid C/O-O/Ne core white dwarfs, found in the mass range between $1.024$ and
$1.15$ M⊙. We present the initial-to-final mass relation, mass-radius
relation, and cooling times considering the effects of atmosphere and core
composition.

###### keywords:

stars: white dwarfs, evolution – methods: numerical

††pubyear: 2018††pagerange: New full evolutionary sequences of H and He
atmosphere massive white dwarf stars using MESA–A

##  1 Introduction

White dwarf stars are the most common final stage of stellar evolution. They
are the end product of main sequence stars with masses up to $5-12$ M⊙
(García-Berro et al., 1997a; Poelarends et al., 2008; Siess, 2010; Langer,
2012; Woosley & Heger, 2015; Doherty et al., 2015), depending on metallicity,
which correspond to $95-97\%$ of all stars in the Galaxy. White dwarf stars
are excellent galactic chronometers as they are slowly cooling, releasing the
heat stored in the core. The age of stellar populations can be obtained by
comparisons of theoretical computations with their distribution in the color-
magnitude diagram (Hansen et al., 2007; Campos et al., 2016) or from the white
dwarf luminosity function (Winget et al., 1987; Bedin et al., 2010; García-
Berro et al., 2014). The connection between the properties of progenitor stars
and white dwarfs is printed in the initial-to-final mass relation (IFMR) which
leads to constraints on the upper mass limit that separates white dwarfs
progenitors of Type II supernovae (Siess, 2007; Catalán et al., 2008). Also
binary systems with white dwarfs are believed to produce Type Ia supernovae as
result of accretion that exceeds the upper mass limit (e.g. Wang et al. 2017).
Finally, a comparison between models and observations of white dwarfs can
enhance the comprehension of the physical properties of high density matter
(e.g. Prada Moroni & Straniero 2002; Isern et al. 2010).

The white dwarf stars can be divided into two main classes: DA and non-DA
types, based on the main component of their atmosphere. The DA spectral class
shows an hydrogen rich atmosphere and represent $\simeq 84\%$ of all white
dwarfs (e.g. Kepler et al. 2007) while non-DA’s have hydrogen deficient
atmospheres. The non-DA white dwarfs can be classified by the dominant element
in the spectra and effective temperature: DO class with strong lines of HeII
and effective temperatures of $T_{\rm eff}\approx 45,000-200,000$ K, DB class
with strong HeI lines and $T_{\rm eff}\approx 11,000-30,000$ K, and for
effective temperatures bellow $T_{\rm eff}\approx 11,000$ K there are DC, DQ
and DZ types with traces of carbon and metals in their spectra. The helium-
rich white dwarfs are believed to be progeny of PG 1159 stars (McGraw et al.,
1979) formed by a Born-Again episode (Schoenberner, 1979; Iben et al., 1983)
in which a helium flash occurs during the cooling sequence (i.e. very late
thermal pulse) forcing the star to evolve back to the asymptotic giant branch
(AGB). As a result, this very late thermal pulse burns out all hydrogen left
in the atmosphere of the star.

White dwarfs with masses in the range $0.4-1.05$ M⊙ are believed to harbor a
carbon-oxygen (C/O) core, after central hydrogen and helium burning on earlier
stages of evolution. On the other hand, the progenitors of white dwarfs with
masses larger than $\approx 1.05$ M⊙ (García-Berro et al., 1997a; Siess, 2007)
should reach temperature high enough to achieve stable carbon burning, forming
an oxygen-neon (O/Ne) or neon-oxygen-magnesium (Ne/O/Mg) core, depending if
carbon burning ignites off-center or at the center of the star. When carbon
burning starts off-center, ignition is followed by a inward propagation of the
burning front (hereafter carbon flame). This carbon flame may or may not reach
the center. If the flame does not reach the center a hybrid C/O+O/Ne white
dwarf will form (Denissenkov et al., 2013; Farmer et al., 2015), whilst a
flame which reaches the center will form a Ne/O/Mg white dwarf. These results
are dependent on the evolutionary code and input physics considered. Siess
(2007) calculated sequences for several metallicities, with and without
overshooting, and found that for $Z=0.02$ the minimum core mass for O/Ne/
cores is $1.04$ M⊙. Doherty et al. (2015), based on sequences calculated until
the thermally pulsing asymptotic giant branch phase (TP-AGB), found that for
$Z=0.02$ the minimum core mass for a O/Ne core white dwarf star is $1.154$ M⊙.
The determination of the mass range for each type of core composition is
fundamental to improve our understanding of the populations of massive white
dwarf stars.

The number of spectroscopically confirmed white dwarf stars has increase to
more than $\approx 30\,000$ with surveys like the Sloan Digital Sky survey
(Ahn et al., 2014). The mass distribution for DA white dwarf stars presented
by Kleinman et al. (2013) is centered at $0.649$ M⊙ but presents another peak
at $0.811$ M⊙ (Kepler et al., 2015). In addition, an analysis of $112$
isolated massive white dwarfs performed by Należyty & Madej (2004) showed
that, for known white dwarfs with masses $\geq 0.8$ M⊙, the mass distribution
has a primary peak at $\approx 0.8$ M⊙ and a secondary peak around $1.04$ M⊙.
Around $20\%$ of all DA white dwarfs are more massive than $0.8$ M⊙ (Liebert
et al., 2005; Kepler et al., 2007) and about $9\%$ have mass around $1.12$ M⊙
(Kepler et al., 2007). Recent works presents a high mass tail on the mass
distribution with an excess for masses around $M_{\rm WD}\approx 1.0$ M⊙
(Falcon et al., 2010; Tremblay et al., 2013; Rebassa-Mansergas et al., 2015;
Kepler et al., 2016).

In the literature, several evolutionary models for low and intermediate mass
white dwarf stars can be found. For instance Romero et al. (2012, 2013) and
Althaus et al. (2009) for C/O core DA and non-DA white dwarfs, respectively,
and Althaus et al. (2013); Istrate et al. (2014, 2016); Istrate et al. (2017);
Sun & Arras (2017) for low and extremely low mass white dwarfs, are just some
recent works. On the other hand, there is a lack of full evolutionary models
for massive white dwarfs. Some recent work computed the evolution of super
asymptotic branch stars until the end of the thermally pulsing phase (Siess,
2010; Gil-Pons et al., 2013; Jones et al., 2013; Doherty et al., 2015), but
they do not compute the following post-AGB stage nor the white dwarf
evolution. Other authors computed white dwarf models by generating
static/politropic models at the top of the cooling sequence and/or by
rescaling the mass and/or artificially inserting a previously calculated
chemical profile with fixed O/Ne ratios and fixed hydrogen and helium layers
on the atmosphere (Benvenuto & Althaus, 1999; Salaris et al., 2000; Althaus et
al., 2005b, 2007; Salaris et al., 2013).

The present work is intended to fill the gap due to the lack of full
evolutionary sequences for massive white dwarfs, computed from the Zero Age
Main Sequence (ZAMS) through the red giant branch (RGB) and AGB to the white
dwarf cooling sequence. To this end we used the MESA evolutionary code (Paxton
et al., 2011; Paxton et al., 2013, 2015, 2018). We computed full evolutionary
sequences with initial masses in the range of $8.8-11.8$ $M_{\rm\odot}$
resulting in white dwarfs models with masses in the range of $1.01-1.31$ M⊙,
adding both hydrogen and helium-rich envelopes. This paper is organized as
follows. On section 2 we present the MESA evolutionary code and the input
physics considered in our computations. The results from our full evolutionary
computations are presented in section 3. Section 4 discusses the mass-radius
relation of our sequences. Finally we present our final remarks on section 5.

##  2 Numerical Simulations

The numerical simulations where performed using the MESA code (Paxton et al.,
2011; Paxton et al., 2013, 2015, 2018) version r8845. We started the
computations on the ZAMS, considering an initial metallicity $Z=0.02$. We
computed the hydrogen burning and helium burning stages, and the RGB and AGB
mass loss phases. The sequences ended as C/O or O/Ne/Mg white dwarfs cooling
until $\log L/L_{\rm\odot}\sim-4$, which represents $T_{\rm eff}\approx
7\,000-10\,000$ K depending on final mass.

The MESA evolutionary code has been extensively used to perform calculations
of extremely low-mass white dwarfs (Istrate et al., 2016; Istrate et al.,
2017; Sun & Arras, 2017), hybrid C/O/Ne white dwarfs and Type Ia SN
progenitors (Jones et al., 2013; Denissenkov et al., 2013; Chen et al., 2014;
Farmer et al., 2015; Brooks et al., 2017a), accreting white dwarf binaries
with C/O core (Brooks et al., 2016; Wang et al., 2017) and O/Ne core (Schwab
et al., 2017; Brooks et al., 2017b), white dwarf isochrones (Dotter, 2016) and
sets of sequences covering the white dwarf mass range (Choi et al., 2016;
Pignatari et al., 2016; Ritter et al., 2017).

We detail the input physics considered prior and during white dwarf evolution
in the following subsections.

###  2.1 Pre-white dwarf evolution

On the evolutionary phases prior to the cooling sequence, we considered the
nuclear network co_burn_plus.net which has 16 elements from hydrogen to
silicon, 1H, 3,4He, 12,13C, 13-15N, 16-18O, 19F, 20,22Ne, 24Mg, and 28Si. This
network has 67 nuclear reactions, including pp-chain, CNO cycle, triple-alpha
and some auxiliary reactions. We used JINA reaclib (Cyburt et al., 2010) with
electron screening factors for thermonuclear reactions from Graboske et al.
(1973) and DeWitt et al. (1973) for weak and intermediate screening, and
Alastuey & Jancovici (1978) for the strong screening regime. Plasma parameters
are those from Itoh et al. (1979). The energy-loss rates including the
derivatives from thermal neutrinos are from Itoh et al. (1996). The equation
of state (EOS) is based on the OPAL EOS tables (Rogers & Nayfonov, 2002)
including the SCVH tables (Saumon et al., 1995) for lower temperatures and
densities and the HELM (Timmes & Swesty, 2000) and PC (Potekhin & Chabrier,
2010) EOS for higher temperatures and densities.

Convection was consider using Ledoux criterion, which takes into account the
composition gradient, along with the Cox implementation of MLT (Cox & Giuli,
1968) with a mixing length free parameter $\alpha_{\rm MLT}=2$. The diffusion
and gravitational settling in MESA are calculated by solving the equations
from Burgers (1969) with coefficients of Thoul et al. (1994). In the overshoot
region MESA treats convective mixing as a time-dependent diffusion process,
with a diffusion coefficient given by,

$D_{\rm OV}=D_{\rm conv,0}\exp{-\frac{2z}{f\lambda_{\rm P,0}}}$ (1)

where $D_{\rm conv,0}$ is the MLT derived diffusion coefficient at the
convective boundary, $z$ is the distance from the convective boundary,
$\lambda_{\rm P,0}$ is the local pressure scale height and $f$ is the
adjustable parameter (Herwig, 2000), which can have different values at the
upper and lower convective boundaries for H-burning, He-burning, metal-burning
(i.e all burning regions that are not H or He-burning) and non-burning
convection zones. Due to convergence issues, overshooting was only considered
for the lower boundary of metal-burning convective region with parameters set
to overshoot_f_below_burn_z_shell $=0.1$ and overshoot_f0_below_burn_z_shell
$=0.01$. The mixing in regions unstable to Schwarzschild but stable to Ledoux
criteria is treated by semiconvection (Langer et al., 1983) using the
dimensionless efficiency parameter $\alpha_{\rm sc}=0.01$. Thermohaline mixing
(Ulrich, 1972; Kippenhahn et al., 1980) is also considered with efficiency
parameter set to $\alpha_{\rm th}=2$. For details in the implementation of
semiconvection and thermohaline mixing the reader is referred to section 4 of
Paxton et al. (2013).

Mass loss was considered during the giant phases, i.e. RGB and AGB phases. We
used the mass loss formula from Reimers (1975) with $\eta=0.1$ during RGB,
followed by the Bloecker (1995) scheme with $\eta=10$ on the AGB. The opacity
tables are those from OPAL type 2 (Iglesias & Rogers, 1996) for enhanced C/O
variations in composition. We used a gray atmosphere for the entire evolution.
Considering white dwarf stars has rotation periods of $\sim 1$ day (Kepler &
Romero, 2017), in most cases, the effects of rotation are negligible. Even
thought there are a few known fast rotators, we do no consider rotation in our
computations.

###  2.2 White dwarf evolution

Chemical diffusion and gravitational settling was modeled following the
formulation from Burgers (1969). Sun & Arras (2017) suggested that the routine
diffusion_use_cgs_solver is better suited for electron degeneracy, however
they considered only very low mass white dwarf models with stellar mass below
$M_{\rm WD}<0.17$ M⊙. Employing this formulation for diffusion in massive
models leads to numerical instabilities and to non-physical chemical profiles.

In our models, convection was shut down due to numerical instabilities during
the cooling sequence. Computations with or without convection show no
differences in the final ages of our cooling sequences. The same happens when
the Sun & Arras (2017) formulation is considered. Thus, our treatment of
diffusion and convection do not affect the cooling times in the mass range
considered in this work.

The energy-loss rates from thermal neutrinos and its derivatives are from Itoh
et al. (1996).

White dwarf stars are expected to undergo crystallization as a result of
strong Coulomb interactions in their dense interiors (van Horn, 1968). The
transition occurs when the energy of the Coulomb interaction between
neighboring ions is much larger than their thermal energy. The ratio between
these two contributions can be expressed as a coupling parameter
$\Gamma=\bar{Z}^{2}e^{2}/a_{i}k_{B}T$ where $\bar{Z}$ is the average ion
charge, $a_{i}$ is the mean ion spacing and the other symbols have their usual
meaning. Crystallization for pure oxygen plasma begins when $\Gamma\approx
175$ (Van Horn, 1969). The onset of crystallization also depends on the
adopted phase diagram. Near the crystallization limit ($\Gamma\sim 175$) MESA
uses the Potekhin & Chabrier (2010) equation of state (PC EOS) which accounts
for the thermodynamics of crystals. By default, MESA changes from HELM EOS to
PC EOS when the Coulomb coupling parameter $\Gamma>80$ and considers a mixture
of solid and liquid for $\Gamma_{\rm i}=150$ and a full crystal when
$\Gamma_{\rm full}=175$. Those values where obtained by Potekhin & Chabrier
(2010). Results from the asteroseismological analysis performed by Romero et
al. (2013) indicates that the azeotropic type phase diagram from Horowitz et
al. (2010) (see also Schneider et al., 2012; Hughto et al., 2012) better
represents the crystallization on white dwarfs cores. Hence, we modified the
values for the coupling constant $\Gamma$ to $\Gamma_{\rm i}=215$ and
$\Gamma_{\rm full}=220$ in our computations. Thus, the release of latent heat
occur for $\Gamma=215-220$. This modification had to be done on the file
pc_eos.f located inside MESA eos module. Until MESA version r8845 used in this
work, there was no option to control those parameters.

The Debye cooling regime, which affects the cooling of white dwarfs at lower
luminosities, was not considered in our simulations. However, this effect will
not be important in our results, since it occurs for temperatures below $\sim
10\,000$ K and luminosities lower than $\log L/L_{\rm\odot}<-4.06$ (Althaus et
al., 2007) for the mass range considered in this work (see section 3.6).

##  3 Evolution from the ZAMS to the cooling curve

We calculated 16 full evolutionary sequences with initial metallicity $Z=0.02$
and initial mass at the ZAMS between 8.80 and 11.80 M⊙. As a result we
obtained white dwarf models with stellar masses ranging from $1.012$ to
$1.307$ M⊙. From our computations, we find that a sequence with an initial
mass of $11.9$ M⊙ do not result in a white dwarf but possibly ends up as a
neutron star. Thus, a white dwarf with the Chandrasekhar limiting mass should
have a progenitor with an initial mass between $11.8$ and $11.9$ M⊙. Further
calculations are needed to determine the model which lead to the Chandrasekhar
mass limit. Note that within this mass range there are two possible core
compositions, i.e, C/O and O/Ne/Mg. Our computations complement at high
stellar masses the works of Romero et al. (2012, 2013) who computed C/O core
white dwarf sequences with masses below $M_{\rm WD}=1.050$ M⊙. Our H and He
atmosphere sequences have evolved equally on the stages prior to the cooling
sequence, therefore there is no distinction on the evolution previous to the
white dwarf stage.

During the AGB, the computations present numerical difficulties related to the
mass loss episodes. At this stage, we use the mass loss prescription of
Bloecker (1995) with a factor $\eta$=10, that leads to the complete removal of
the hydrogen envelope, preventing the occurrence of thermal pulses, in all
sequences. A less efficient mass loss rate, achieve by decreasing the value of
$\eta$, does not allow us to compute the stages following the AGB stage but
still leads to the complete removal of hydrogen. The same occurs for other
MESA built in mass loss schemes. García-Berro et al. (1997a) discussed that
models with rapid radiative wind during AGB phase are expected to lose all its
hydrogen layer and most of its helium layer leading to a PG 1159 star, but
real massive white dwarfs are known with H and He envelopes. In order to
produce hydrogen atmosphere white dwarf sequences, we artificially added
hydrogen at the surface of the models. This procedure was performed at high
effective temperature, near the beginning of the cooling sequences, so the
transitory effects caused by the artificial procedure are rapidly washed out.
The amount of hydrogen added to each sequence was computed by a linear
extrapolation to higher stellar mass pf the results of Romero et al. (2012,
2013). The amount of hydrogen as a function of the white dwarf stellar mass is
shown in Figure 1. Circles correspond to the results from Romero et al. (2012,
2013), while the triangles correspond to the values extrapolated for higher
stellar masses.

Figure 1: Range of star mass and hydrogen envelope mass. Circles are from
Romero et al. (2012, 2013) and triangles are the extrapolated data.

As an example, Figure 2 shows two chemical profile in terms of outer mass
fraction at $T_{\rm eff}\approx 80\,000$ K on the cooling curve for a sequence
with stellar mass $M_{\rm WD}=1.019$ M⊙ and a C/O core. The bottom panel shows
the chemical profile for the helium atmosphere case and the top panel presents
the chemical profile for the hydrogen atmosphere sequence, after $M_{\rm
H}=10^{-5.7}M_{\rm WD}$ of hydrogen was artificially added to the model. Note
that the only difference between the two chemical profiles shown in Figure 2
is the chemical abundance of the outer layers, while the core regions remain
the same.

Figure 2: Example of chemical profile in terms of the outer mass fraction at
$T_{\rm eff}\approx 80,000$ K on the cooling sequence for H atmosphere (top)
and He atmosphere (bottom) white dwarf with 1.019 M⊙. Hydrogen had to be added
on all sequences due to losses on mass loss phase. The quantities added on
each evolution are shown in table 1. This figure show that only the atmosphere
of the star was changed. The nuclei composition are identical to both H and He
atmosphere sequences.

The sequences with final masses $M_{\rm WD}\geq 1.024$ M⊙ experience loops at
high luminosity and high effective temperature region in the post-AGB stage
(see Figure 3). The loops are due to carbon shell-burning followed by helium
shell-burning which, for sequences with $M_{\rm WD}\geq 1.132$ M⊙, burns all
the helium content in the outer layers. These helium shell-burning happens due
to the huge temperature gradient on the shells surrounding the core. Thus,
enough shells must be added in the numerical models in these regions to
account for the large temperature gradients and composition gradients in the
shell burning regions.

For sequences with $M_{\rm WD}\geq 1.132$ M⊙ helium was added on top of the
model at the beginning of the cooling sequence using the same procedure used
to add hydrogen. The amount of helium to be added was extrapolated from our
sequences which had not burnt all helium. The amount of hydrogen and helium
for each sequence is presented in columns 3 and 4 of Table 1, along with the
initial mass at the ZAMS and the white dwarf mass in column 1 and 2,
respectively. Helium atmosphere sequences were produced by not adding hydrogen
on top of the models.

Table 1: Values of initial mass at ZAMS ($M_{\rm ZAMS}$), White dwarfs masses ($M_{\rm WD}$) and hydrogen content for H-rich sequences and helium contents for both H and He atmosphere sequences. Hydrogen abundances where extrapolated from Romero et al. (2012, 2013). The Helium abundances were extrapolated for sequences with $M_{\rm WD}\geq 1.132$. $M_{\rm ZAMS}$ $[M_{\rm\odot}]$ |  $M_{\rm WD}$ $[M_{\rm\odot}]$ | $-\log M_{\rm H}/M_{\rm WD}$ | $-\log M_{\rm He}/M_{\rm WD}$
---|---|---|---
8.80 | 1.012 | 5.722 | 2.451
8.90 | 1.019 | 5.732 | 2.466
9.00 | 1.024 | 5.740 | 3.477
9.20 | 1.036 | 5.785 | 3.349
9.50 | 1.064 | 6.076 | 2.958
9.80 | 1.088 | 6.365 | 4.006
10.00 | 1.110 | 6.446 | 3.517
10.10 | 1.125 | 6.484 | 2.447
10.20 | 1.131 | 6.498 | 2.489
10.30 | 1.132 | 6.500 | 2.495
10.45 | 1.147 | 6.536 | 2.597
10.50 | 1.151 | 6.547 | 2.628
10.80 | 1.216 | 6.711 | 3.089
11.00 | 1.220 | 6.720 | 3.116
11.40 | 1.267 | 6.838 | 3.449
11.80 | 1.307 | 6.939 | 3.735

The Hertzprung-Russell diagram from ZAMS to white dwarf cooling sequence is
shown in Figure 3 for three sequences with initial masses 8.9, 10 and 11 M⊙
and final masses of 1.019, 1.11 and 1.22 M⊙ in the cooling curve,
respectively. Each sequence in this figure has a different core composition as
a white dwarf star. The solid, dashed, and dotted lines represent a C/O core
white dwarf with $M_{\rm WD}=1.019$ M⊙, an O/Ne core white dwarf with $M_{\rm
WD}=1.11$ M⊙ and a Ne/O/Mg white dwarf with $M_{\rm WD}=1.22$ M⊙,
respectively.

The $M_{\rm WD}=1.019$ M⊙ sequence with C/O core experienced a core hydrogen
burning stage lasting for $24.51$ Myr, and then during the next $2.80$ Myr
helium has burn until depletion at the center. For more massive sequences the
time spent on main sequence and central helium burning decreases. For the
sequences with final mass $M_{\rm WD}=1.11$ M⊙, with a O/Ne core, the total
main sequence time is $19.75$ Myr plus $2.28$ Myr to burn all helium. Lastly,
the sequence with final mass $M_{\rm WD}=1.22$ M⊙ burnt hydrogen for $16.83$
Myr until depletion at the center and burnt helium for $1.81$ Myr until
extinction at the center. The ages for hydrogen and helium depletion at center
and progenitor age, i.e., the age at the beginning of the white dwarf cooling
sequence, are listed in columns 3, 4 and 5 of Table 3, respectively (see
section 3.6).

Figure 3: Evolutionary tracks for three sequences in the HR diagram, with
initial masses 8.9 (solid line), 10 (dashed line) and 11 M⊙ (dotted line) and
final masses 1.019, 1.11 and 1.22 M⊙, respectively. Each sequence has a
different core composition in the white dwarf cooling curve stage, being C/O,
O/Ne and Ne/O/Mg for the sequences with white dwarf masses 1.019, 1.11 and
1.22 M⊙, respectively.

###  3.1 Carbon flame

As can be seen from Figure 3, the sequences characterized with white dwarf
masses of $1.11M_{\rm\odot}$ and $1.22M_{\rm\odot}$ presents loops on the
upper left region of the HR diagram, prior to entering the white dwarf cooling
sequence. Those loops are due to carbon burning which ignites off-center and
moves inwards to the center of the star. In particular, towards the end of the
loop, just before the extinction of the carbon burning, helium ignites on a
shell near the surface. All sequences with $M_{\rm WD}\geq 1.024$ M⊙
experiences an off-center carbon ignition prior to the white dwarf stage, and
for sequences with $M_{\rm WD}\geq 1.132$ M⊙ the subsequent helium burning
consumes all the helium content in the outer layers. If the carbon flame do
not reach the center of the star, a hybrid C/O-O/Ne white dwarf will form
(Denissenkov et al., 2013), while if the flame propagates to the center a
O/Ne/Mg white dwarf will be produced. The occurrence of the late carbon
flashes can be an explanation for the luminosity variations observed in the
star in the Stingray Nebula, CD-59 6479 or SAO 24567 (Henize, 1976; Reindl et
al., 2014). The luminosity variation timescale at the beginning of the carbon
burning in our sequence with $M_{\rm WD}=1.024$ M⊙ are similar with the one
observed, however, the total change in luminosity in CD-59 6479 is larger than
the one showed in our models by a factor of $\sim 100$ (Schaefer & Edwards,
2015) and the stellar masses are very different.

All sequences which experienced a carbon flame presented subsequent flames
which started further away from the center and were quenched before the
starting point of the previous flame. Figure 4 shows the location in mass
where the first carbon ignition occurs (full line) and the location of the
extinction of the carbon flame which got closest to center (dashed line), as a
function of the white dwarf mass which experienced a carbon flame. Note that
the sequences with stellar mass $M_{\rm WD}<1.024$ M⊙ do not experience carbon
ignition. The dotted vertical lines separate the mass range for different core
compositions (see subsection 3.3). As can be seen from this figure, the
position of the first carbon ignition gets closer to the center of the model
for increasing white dwarf mass. As the core mass increases, the central
degeneracy is lower, allowing the inner maximum temperature to be near the
center, causing a carbon ignition closer to center on more massive sequences.

Specifically, the sequence with $M_{\rm ZAMS}=10$ M⊙ ($M_{\rm WD}=1.11$ M⊙)
experience an off-center carbon ignition starting at $M_{\rm r}/M_{\rm WD}\sim
0.3$ which moves inwards until $M_{\rm r}/M_{\rm WD}\sim 0.1$, while for the
sequence with $M_{\rm ZAMS}=11.0$ M⊙ ($M_{\rm WD}=1.220$ M⊙) carbon ignition
starts at $M_{\rm r}/M_{\rm WD}\sim 0.094$ and, in this case, reaches the
center of the star. Carbon ignition in all sequences started on regions with
density $\rho\sim 1.5\times 10^{6}$ g/cm3, agreeing with the results for
super-AGB sequences from Farmer et al. (2015). For a detailed discussion about
carbon burning we refer to the works of Siess (2006, 2007) and of Farmer et
al. (2015) and references therein.

From our computations, some sequences with C/O core in the white dwarf stages
experienced a carbon ignition, but the carbon flame only reaches $M_{\rm
r}/M_{\rm WD}\sim 0.2$ for those models. When the carbon flame reaches the
regions closer than $M_{\rm r}/M_{\rm WD}<0.2$, a hybrid C/O-O/Ne white dwarf
with O/Ne core is formed. For sequences with $M_{\rm WD}>1.15$ M⊙ the carbon
flame reaches the center of the star producing a Ne/O/Mg core white dwarf. The
details of core composition and chemical profile will be analyzed on the
following subsections.

Figure 4: Mass location of the first ignition of carbon and mass location of
closest to center flame extinction as function of white dwarf masses which
experienced a carbon flame. Vertical dotted lines delimits the mass range for
core composition. See text for details.

###  3.2 The Initial-to-Final Mass Relation

Figure 5 presents the initial-to-final mass relation obtained from our
computations compared to the results of Siess (2010) and Doherty et al.
(2015), within the stellar mass range considered in this work. We consider
models with metallicity $Z=0.02$. The final mass obtained in this work is
systematically lower than previous results, due to the different input physics
considered in each case. Siess (2010) considers the Vassiliadis & Wood (1993)
mass loss scheme, a mixing length parameter of $\alpha_{\rm MLT}=1.75$ and no
core overshooting. On the other hand, Doherty et al. (2015) consider mass loss
rate from Reimers (1975) on the RGB, followed by Bloecker (1995) at high
luminosities and change to Vassiliadis & Wood (1993) on thermally pulsing
phase, with similar convection treatment than Siess (2010). Also, both authors
evolved their sequences only until the TP-AGB phase, and then consider the
mass of the He-free core as the final mass at the white dwarf stage. As
discussed in Siess (2010) the final mass is dependent on core growth and mass
loss scheme. In our computations, we calculated full evolutionary sequences by
employing the mass loss formula from Reimers (1975) on RGB and Bloecker (1995)
on the AGB with a larger efficiency parameter than the mass loss rates
considered by previous authors, thus evolving the sequences to post-AGB phase
and leading to a lower final mass in the cooling curve for the same initial
mass. Thus, from our computations, the upper limit of progenitor mass to
produce a white dwarf star is $M_{ZAMS}=11.8$ M⊙ for $Z=0.02$, $21.6\%$
greater than the value found by Doherty et al. (2015) and $12.4\%$ greater
compared to Siess (2010) for the same initial metallicity. Considering the
dispersion in the initial-to final Mass relation from the different authors in
figure 5, we can estimate for a given initial mass in the ZAMS a range of
final white dwarf masses. For instance, a sequence with initial mass of $9$ M⊙
can produce a white dwarf with masses ranging from $1.024$ to $1.225$ M⊙,
depending on the input physics applied, specially the mass loss formulation.
The present day observational data cannot distinguish the best model.

Figure 5: Initial-to-final mass relation obtained from our computations
(triangle) as compared to the results from Siess (2010) (circle) and Doherty
et al. (2015) (square). In all cases the initial metallicity is $Z=0.02$. We
only show the massive models of Doherty et al. (2015).

###  3.3 Central abundances in the cooling sequence

Within the stellar mass range studied in this work, we expect to have three
main core compositions, i.e., C/O and O/Ne (García-Berro et al., 1997b;
Doherty et al., 2015) or Ne/O/Mg, changing with increasing stellar mass.
Figure 6 shows central abundances of carbon, oxygen, neon and magnesium in a
logarithmic scale, as a function of the white dwarf mass. The values are taken
at $T_{\rm eff}\approx 40\,000$ K, a effective temperature low enough to give
time for diffusion to act on the central composition and still high enough to
be above crystallization. As shown on Figure 6, the C/O core is dominant for
$M_{\rm WD}\leq 1.088$ M⊙ due to carbon being more abundant than neon. Between
$1.088<M_{\rm WD}\leq 1.147$ M⊙ we have an O/Ne core as the amount of neon
increases but is still lower than oxygen, whilst for masses $M_{\rm WD}>1.147$
M⊙ neon is the most abundant element on the central composition, followed by
oxygen and then by magnesium, defining the core composition as Ne/O/Mg core.
Note that sequences with $M_{\rm WD}>1.15$ M⊙ experienced a carbon flame which
reaches the center of the stars as shown on Figure 4 and discussed on
subsection 3.1. Also, the sequence with $M_{\rm WD}=1.024$ M⊙ is the sequences
with the lowest stellar mass to experience carbon burning, but it does not
reach the center, producing an hybrid C/O-O/Ne white dwarf. This explains the
step in the central 20Ne abundance observed in Figure 6 for that stellar mass.
Doherty et al. (2015) computed sequences for super-AGB stars and found that,
for $Z=0.02$, sequences with core masses $M_{\rm c}<1.075$ M⊙ become a C/O
core white dwarf, sequences with core masses $1.075\leq M_{\rm c}<1.154$ M⊙
presented a C/O/Ne core and for core masses $M_{\rm c}\geq 1.154$ M⊙ the
sequences are O/Ne white dwarfs. Woosley & Heger (2015) also calculated
sequences on the limit of SN progenitors and obtained models with O/Ne core
for a range of core mass $1.088\leq M_{\rm c}\leq 1.345$ M⊙. Chemical profiles
from Doherty et al. (2015) and Woosley & Heger (2015) are not publicly
available so we can not compare the internal abundances of the last models
computed. It is important to note that Doherty et al. (2015) and Woosley &
Heger (2015) did not compute the cooling curves, opposite to the present work.
They also uses different evolutionary codes with different nuclear reaction
and input physics.

Figure 6: Central abundances versus final mass of the white dwarfs stars
before crystallization starts ($T_{\rm eff}\approx 40\,000$ K). The black
circles represent 12C, triangles are 16O, squares are 20Ne and empty circles
are 24Mg. We can identify three regions, one for C/O core for masses below or
equal to $1.088$ M⊙ where neon is lower than carbon, another region between
$1.088$ and $1.147$ M⊙ in which neon is greater than carbon and lower than
oxygen but magnesium is lower than carbon defining a O/Ne core, and a third
region for masses greater than $1.147$ M⊙ with a O/Ne/Mg core due to neon
being greater than oxygen and magnesium greater than carbon.

###  3.4 Chemical profiles in the white dwarf cooling sequence

In Figure 7 we show three chemical profiles for models along the white dwarf
cooling sequence for an O/Ne core H–atmosphere white dwarf with stellar mass
$M_{\rm WD}=1.11$ M⊙. The abundance profiles are in terms of outer mass
fraction and correspond to three effective temperatures, $T_{\rm eff}\approx
80\,000,40\,000$ and $10\,000$ K, on top, middle and bottom panels,
respectively. The effects of diffusion and gravitational settling on the
entire chemical profile as the star cools are evident from this figure.
Initially, the oxygen central abundance is $X_{\rm O}=0.515$ while for neon it
is $X_{\rm Ne}=0.195$ (top panel). Rehomogenization processes changes the
central abundances to a flat profile with $X_{\rm O}=0.406$ and $X_{\rm
Ne}=0.458$ when the star reaches effective temperatures $\sim 10,000$ K
(bottom panel). Also there is a carbon-oxygen-helium triple-layer structure at
$2.7<-\log(1-M_{\rm r}/M)<5.5$ for high effective temperatures, which diminish
due to diffusion processes. Note that, for the sequence with white dwarf mass
$1.11$ M⊙ the carbon flame experienced in the pre-white dwarf stages, did not
reach the center of the model, giving rise to a hybrid C/O-O/Ne core white
dwarfs (Denissenkov et al., 2013; Doherty et al., 2015; Farmer et al., 2015).
The hybrid models presented in Denissenkov et al. (2013) are characterized by
a C/O core surrounded by an O/Ne region at the super AGB stage. This chemical
structure differs from the one characterizing our models (see Figure 7) ,
since ours are taken from the white dwarf cooling curve, where
rehomogenization process are taken into account. To our knowledge this is the
first time a chemical profile for the entire star is presented for a hybrid
C/O-O/Ne white dwarf. The shaded region on the bottom panel depicts the
crystallized core, which is defined as the regions where $\Gamma\geq 220$ (see
section 3.5). For this sequence, crystallization starts at $T_{\rm eff}\approx
20\,000$ K. Finally, the chemical structure of the star can be separated into
the outer layers, composed by hydrogen followed by an extended helium buffer,
and the inner regions showing an triple-layer of carbon-oxygen-helium from
$2<-\log(1-M_{\rm r}/M)<4$ with a carbon peak ($-\log M_{\rm r}/M\sim 3.1$) on
top of the Ne/O core. This chemical structure of the core is the result of
burning events occurring on previous stages to the white dwarf cooling
sequence.

Figure 7: Chemical profile in terms of outer mass fraction for selected
elements for three different stages of the cooling sequence: $T_{\rm
eff}\approx 80\,000$ K on top, $T_{\rm eff}\approx 40\,000$ K on middle and
$T_{\rm eff}\approx 10\,000$ K on bottom panels. The chemical profiles are for
a H atmosphere white dwarf with $M_{\rm WD}=1.11M_{\rm\odot}$. The shaded
region represents the crystallized zone of the star.

The composition of the core and its size has a strong impact on the chemical
profile of the entire star. Figure 8 shows the chemical profiles in terms of
outer mass fraction for three hydrogen atmosphere white dwarfs models with
$T_{\rm eff}\approx 10\,000$ K, with stellar mass $M_{\rm WD}=1.019$ M⊙ (top
panel), $M_{\rm WD}=1.11$ M⊙ (middle panel) and $M_{\rm WD}=1.22$ M⊙ (bottom
panel). The shaded region represents the crystallized region of the model.
From this figure, we note that the crystallization front is closer to the
surface of the model for higher stellar mass, located at $-\log(1-M_{\rm
r}/M)\approx 0.9,1.2$ and $2$ for $M_{\rm WD}=1.019$, $M_{\rm WD}=1.11$ and
$M_{\rm WD}=1.22$ M⊙ models, respectively. The sequence characterized by a
stellar mass $M_{\rm WD}=1.019$ M⊙ has a C/O core with a core mass of $M_{\rm
c}=1.016$ M⊙ extending to $-\log(1-M_{\rm r}/M_{*})=2.6$ and present a triple-
layer of carbon-oxygen-helium at $1.8<-\log(1-M_{\rm r}/M_{*})<2.6$. On top of
the core, the model has a helium buffer of $M_{\rm He}=10^{-2.47}M_{\rm WD}$
and a hydrogen envelope with $M_{\rm H}=10^{-5.73}M_{\rm WD}$. All these
structures have strong impact on the propagation of stellar oscillations, when
present (Winget et al., 1997). Althaus et al. (2010) and Romero et al. (2012,
2013) also obtained a triple-layer structure in their C/O core white dwarf
models.

The sequence with $M_{\rm WD}=1.11$ M⊙ has an O/Ne core, extending to
$-\log(1-M_{\rm r}/M)=4$, which represents in mass $M_{\rm c}=1.1099$ M⊙. As
discussed before, this O/Ne core model shows a triple-layer oxygen-carbon-
helium with a peak of carbon at the outer boundary of the core, a helium
buffer of $M_{\rm He}=10^{-3.52}M_{\rm WD}$ and a hydrogen envelope of $M_{\rm
H}=10^{-6.45}M_{\rm WD}$. The $M_{\rm WD}=1.22$ M⊙ sequence has a Ne/O/Mg core
with $M_{\rm c}=1.219$ M⊙ which extends to $-\log(1-M_{\rm r}/M)=3.5$. It also
presents a triple-layer of oxygen-carbon-helium with a sharp oxygen peak,
instead of a carbon peak in the O/Ne core case. The Ne/O/Mg core model has an
helium buffer of $M_{\rm He}=10^{-3.12}M_{\rm WD}$ and a hydrogen envelope of
$M_{\rm H}=10^{-6.72}M_{\rm WD}$. The difference in core composition are due
to final mass as shown on Figure 6. Asteroseismology will be able to
distinguish these structures.

In Appendix A we present all chemical profiles for both H and He atmosphere
sequences at effective temperatures $T_{\rm eff}\approx 10\,000$ K.

Figure 8: Chemical profile in terms of outer mass fraction for selected
elements and for three different H atmosphere white dwarf with $T_{\rm
eff}\approx 10\,000$ K. From top to bottom we depict the chemical profile for
a $M_{\rm WD}=1.019,1.11,1.22$ M⊙. The shaded region represents the
crystallized zone of the star

###  3.5 Crystallization and the Coulomb coupling parameter

As stated before, the MESA code considers as a standard input value that a
mixture of solid and liquid phases occurs for a coupling parameter
$\Gamma_{\rm i}=150$ while a full crystal structure occurs when $\Gamma_{\rm
full}=175$. However, Romero et al. (2013) calculated white dwarfs sequences
considering two types of phase diagram and concluded that the azeotropic type
from Horowitz et al. (2010) better represents the crystallization on the
nuclei of white dwarfs. Hence, we modify the values of the $\Gamma$ parameter
to $\Gamma_{\rm i}=215$ for a liquid/solid coexisting phases and $\Gamma_{\rm
full}=220$ for a full crystal phase, in line with the results obtained using
the Horowitz et al. (2010) phase diagram. Paxton et al. (2018) calculated
cooling sequences for a $0.6$ M⊙ white dwarf star, using MESA, varying
$\Gamma_{\rm i}=174\ \text{to}\ 220$ and $\Gamma_{\rm full}=176\ \text{to}\
240$ and found a difference in the cooling times of the order of $\sim 0.2$
Gyr. For the stellar mass range considered in this work, to assume
$\Gamma_{\rm i}=215$ and $\Gamma_{\rm full}=220$, makes the effective
temperature at which crystallization starts $\approx 2000$ K lower, and the
age $\approx 0.23$ Gyr larger than the ones using the standard input values.
The effective temperature and age on the onset of crystallization for three
sequences considering the two values of $\Gamma_{\rm full}$, are listed in
table 2.

We compare the values of the effective temperature at the onset of
crystallization, as a function of the white dwarf mass, with the results
obtained by Romero et al. (2013) in Figure 9. Note that the results obtained
in this work (triangles) are in good agreement with those from Romero et al.
(2013) (circles), in the stellar mass range where both works overlap.

Table 2: Age and effective temperature on the onset of crystallization for different Coulomb coupling parameter. The age of crystallization is defined as time starting at the cooling sequence and ending at the onset of crystallization and is given in Gyr. Effective temperature in Kelvin. The effective temperature for $\Gamma_{\rm full}=220$ on the last row agrees with table 2 of Romero et al. (2013). | $\Gamma_{\rm full}=175$ | $\Gamma_{\rm full}=220$
---|---|---
$M_{\rm WD}$ [M⊙] |  $T_{\rm eff}$ [K] | Age [Gyr] |  $T_{\rm eff}$ [K] | Age [Gyr]
1.012 | 15095 | 0.647 | 13152 | 0.871
1.019 | 15376 | 0.628 | 13396 | 0.848
1.024 | 15552 | 0.678 | 13584 | 0.925
Figure 9: Comparison of effective temperature on onset of crystallization for
a range of white dwarfs mass. Effective temperature on units of $10^{4}$ K.
Circles are the data extracted from Romero et al. (2013) and triangles are
results from this work. There is a agreement between both works for values of
mass $M_{\rm WD}\sim 1$ M⊙ and the effective temperature raises as mass also
increases.

###  3.6 White dwarf cooling times

The cooling time is defined as the time spent by a star in the white dwarf
cooling sequence ($t_{\rm cool}$). We define the beginning of the cooling
sequences as the point of maximum effective temperature in the post-AGB stage,
before the star enters the cooling sequence. The progenitor age ($t_{\rm i}$)
is defined as the time from the ZAMS to the beginning of the cooling sequence.
Then the total age of a white dwarf star can be computed as $t_{\rm
total}=t_{i}+t_{\rm cool}$. Table 3 summarizes the characteristic time-scales
for all the sequences computed in this work. Specifically, for each sequence
we listed the initial mass in the ZAMS (column 1) and the final mass at the
cooling curve (column 2), along with the age for hydrogen and helium
exhaustion at the core in columns 3 and 4, respectively. The progenitor age
$t_{i}$ is listed in column 5. The age for hydrogen and helium depletion and
the progenitor age are equal for both hydrogen and helium atmosphere white
dwarfs sequences, since both have evolved equally on the stages prior to the
cooling sequence. We also listed the effective temperature and the age in the
cooling curve at the onset of crystallization for the hydrogen (columns 6 and
7) and helium atmosphere (columns 9 and 10) sequences. Finally, the cooling
times corresponding to an effective temperature of $10\,000$ K are listed in
columns 8 and 12 for sequences with hydrogen and helium atmospheres,
respectively.

Table 3: Mass at ZAMS, white dwarf mass, age when hydrogen is depleted at center ($t_{\rm H}$), age when helium is totally consumed at the center ($t_{\rm He}$), progenitor ages ($t_{\rm i}$), effective temperature on crystallization $(T_{\rm eff_{c}})$ and age at crystallization ($t_{\rm cryst}$, accounted from the beginning of the cooling sequence) and cooling time ($t_{\rm cool}=$ age at $T_{\rm eff}=1\times 10^{4}$ K minus $t_{i}$) for H and He atmosphere white dwarfs. Ages ($t_{\rm H}$, $t_{\rm He}$, $t_{\rm i}$) in units of Myr and crystallization and cooling times on units of Gyr. |  |  |  |  | H-atmosphere | He-atmosphere
---|---|---|---|---|---|---
$M_{\rm ZAMS}$ | $M_{\rm WD}$ | $t_{\rm H}$ | $t_{\rm He}$ | $t_{i}$ | $T_{\rm eff_{c}}$ | $t_{\rm cryst}$ | $t_{\rm cool}$ | $T_{\rm eff_{c}}$ | $t_{\rm cryst}$ | $t_{\rm cool}$
[M⊙] | [M⊙] | [Myr] | [Myr] | [Myr] | [K] | [Gyr] | [Gyr] | [K] | [Gyr] | [Gyr]
8.80 | 1.012 | 24.99 | 27.87 | 28.53 | 13152 | 0.870 | 3.65 | 13738 | 0.839 | 2.99
8.90 | 1.019 | 24.50 | 27.30 | 27.94 | 13396 | 0.847 | 3.69 | 13837 | 0.839 | 2.99
9.00 | 1.024 | 23.88 | 26.61 | 27.26 | 13584 | 0.924 | 4.09 | 13634 | 0.933 | 3.48
9.20 | 1.036 | 22.99 | 25.73 | 26.24 | 14281 | 0.810 | 3.75 | 14714 | 0.756 | 3.36
9.50 | 1.064 | 21.63 | 24.17 | 24.63 | 14643 | 0.754 | 3.13 | 14098 | 0.924 | 2.85
9.80 | 1.088 | 20.49 | 22.82 | 23.24 | 16585 | 0.636 | 3.39 | 15858 | 0.742 | 2.90
10.00 | 1.11 | 19.75 | 22.03 | 22.45 | 19356 | 0.415 | 3.04 | 17945 | 0.538 | 2.62
10.10 | 1.125 | 19.42 | 21.64 | 22.03 | 20941 | 0.338 | 2.65 | 19552 | 0.405 | 2.34
10.20 | 1.131 | 19.06 | 21.23 | 21.62 | 21963 | 0.310 | 2.72 | 19815 | 0.387 | 2.48
10.30 | 1.132 | 18.75 | 20.87 | 21.26 | 22276 | 0.296 | 2.71 | 20383 | 0.376 | 2.57
10.45 | 1.147 | 18.35 | 20.37 | 20.74 | 23673 | 0.253 | 2.52 | 23006 | 0.285 | 2.21
10.50 | 1.151 | 18.19 | 20.17 | 20.54 | 24001 | 0.250 | 2.51 | 23339 | 0.278 | 2.21
10.80 | 1.216 | 17.38 | 19.31 | 19.55 | 29651 | 0.183 | 2.28 | 29982 | 0.181 | 1.91
11.00 | 1.22 | 16.83 | 18.63 | 18.88 | 30177 | 0.177 | 2.25 | 30244 | 0.179 | 1.90
11.40 | 1.267 | 15.86 | 17.51 | 17.73 | 36584 | 0.133 | 1.93 | 36808 | 0.132 | 1.71
11.80 | 1.307 | 15.02 | 16.55 | 16.75 | 44953 | 0.098 | 1.54 | 45137 | 0.099 | 1.31
Figure 10: Cooling times for selected H and He atmosphere cooling sequences.
We present 3 sets of H and He atmosphere sequences: a C/O core set with mass
1.019 M⊙ as solid line. A O/Ne set with mass 1.11 as dashed line and a Ne/O/Mg
set with mass 1.22 M⊙ as dotted line. Gray lines represent H atmosphere and
black lines are He atmosphere white dwarfs sequences. The final chemical
profile for the H atmosphere are shown in Figure 8.

Figure 10 shows the cooling times for H and He atmosphere sequences with the
same stellar mass, 1.019 (solid line), 1.11 (dashed line) and 1.22 $M_{\odot}$
(dotted line), whose chemical profiles are presented in Figure 8. Gray lines
represent hydrogen atmosphere and black lines correspond to helium atmosphere
sequences. For luminosities $\log L/L_{\rm\odot}\gtrsim-2.8$, He-atmosphere
sequences cool somewhat slower then the H-atmosphere counterparts. On the
other hand, for $\log L/L_{\rm\odot}\lesssim-2.8$ He-atmosphere sequences
cools considerably faster leading to differences in cooling age of $\sim 1$
Gyr for luminosities of $\log L/L_{\rm\odot}\sim-4$. These behavior can be
explain by means of the central temperature of the star. Figure 11 depicts the
central temperature $T_{\rm c}$, in a logarithmic scale, in terms of the
surface luminosity for the sequences presented in Figure 10.

On luminosities higher than $\log L/L_{\rm\odot}\approx-3$ the central
temperature for the He atmosphere sequences is slightly larger than in the
case of a H-atmosphere sequence with the same mass. However, for lower
luminosities there is a large difference in $T_{\rm c}$ between H and He
atmosphere sequences. The release of internal energy due to crystallization
occurs at higher luminosities for He atmosphere sequences, delaying the
cooling for those luminosities. Since helium is more transparent than
hydrogen, He-atmosphere sequences lose energy faster than H atmosphere
sequences at lower luminosities. The results presented in Figure 10 agrees
with the results from Camisassa et al. (2017) (see their Figures 7 and 11),
who calculated evolutionary sequences for lower mass white dwarfs ($M_{\rm
WD}\leq 1$ M⊙) with hydrogen-deficient atmosphere and compared against
hydrogen rich white dwarfs from Camisassa et al. (2016).

The core composition also impacts the cooling times of white dwarfs stars.
García-Berro et al. (1997a) compared cooling times for sequences with same
mass and two core compositions, C/O and O/Ne , and obtained a difference in
cooling time of the order of $\sim 2$ Gyr. This is due to the difference in
the heat capacity of the core, which is lower for a O/Ne mixture than for an
C/O mixture, reducing the cooling time for a core with a higher mean molecular
weight.

The Debye regime of fast cooling for white dwarfs is important for very low
luminosities. In particular, Althaus et al. (2007) found that the Debye
cooling regime for H-atmosphere white dwarfs starts to be important for
luminosities below $\log L/L_{\rm\odot}\sim-4$ for masses $M_{\rm WD}\geq
1.28$ M⊙. Since our computations end at those luminosities, the cooling times
presented in Figure 10 are not affected by Debye cooling.

In the region of $0>\log L/L_{\rm\odot}>-1.5$ the effects of neutrino losses
are present and dominant, as can be see from Figure 10. The neutrino emission
decreases at higher luminosities for the more massive sequences when compared
to lower masses sequences, however, there are no difference between the H and
He atmospheres sequences at those luminosities. The rate of neutrino emission
becomes negligible at $\log L/L_{\rm\odot}\sim-1$ for the sequences with
masses $1.019$ M⊙ and $1.11$ M⊙, and at $\log L/L_{\rm\odot}\sim-0.5$ for the
sequence with $1.22$ M⊙, in agreement with the results from Althaus et al.
(2007). However, the evolution for luminosities lower than $\log
L/L_{\rm\odot}\sim-2$ in our computations differs from the ones computed by
Althaus et al. (2007). In our computations we consider a crystallization
treatment that mimics the results from the phase diagram presented by Horowitz
et al. (2010), while Althaus et al. (2007) considers the phase diagram from
Segretain et al. (1994), which increases the crystallization temperature by
$\sim 2\,000$ K (see Romero et al. (2013) for details). Also Althaus et al.
(2007) considers a fixed O/Ne core chemical profile for all stellar masses,
while in the cooling sequences calculated in this work, the chemical profiles
come from evolutionary computations and are consistent with the stellar mass
and covers C/O, O/Ne and Ne/O/Mg cores. The difference in the cooling times
between our computations and those from Althaus et al. (2007), are $\sim 0.6$
Gyr at $\log L/L_{\rm\odot}\sim-4$ for a stellar mass of $\sim
1.21M_{\rm\odot}$.

Figure 11: Logarithm of central temperature in terms of star luminosity. Three
sets of H and He atmosphere are shown, the same as Figure 10. Faint lines
represent H atmosphere and strong lines are He atmosphere white dwarfs.

##  4 Mass-Radius Relation

Figure 12: Mass-Radius relation for H and He atmosphere white dwarfs models,
represented as solid and dashed lines, respectively. From top to bottom, the
lines represent effective temperatures of $150\,000$ K, $100\,000$ K,
$50\,000$ K, $25\,000$ K and $10\,000$ K. See text for details.

Figure 12 shows the mass-radius relation of our sequences with H and He
atmosphere, as solid and dashed lines, respectively. We consider effective
temperatures of $150\,000$ K, $100\,000$ K, $50\,000$ K, $25\,000$ K and
$10\,000$ K, shown from top to bottom. The rate of contraction is higher for
high effective temperatures compared to low effective temperatures. Due to
their higher surface gravity, more massive sequence show a smaller range in
the value of the radius for the different effective temperatures considered
here. For a sequence with H atmosphere characterized by a stellar mass of
$1.267$ M⊙ the total radius changes from $4.83\times 10^{-3}$ R⊙ to
$4.34\times 10^{-3}$ R⊙ when the star cools from $150\,000$ K to $10\,000$ K.
On the other hand, for a sequence with a mass of $1.064$ M⊙, the radius
decreases from $8.31\times 10^{-3}$ R⊙ to $6.95\times 10^{-3}$ R⊙ for the same
range of effective temperatures. This means a change in radius of $\sim 10\%$
for the more massive and $\sim 16\%$ for the latter. Those values change when
sequences with He atmosphere are considered. For a He atmosphere sequence with
$1.267$ M⊙ the radius change from $4.74\times 10^{-3}$ R⊙ to $4.32\times
10^{-3}$ R⊙ ($\sim 9\%$) and for the $1.064$ M⊙ sequence there is a change
from $8.02\times 10^{-3}$ R⊙ to $6.90\times 10^{-3}$ ($\sim 14\%$) when
considering the same effective temperature range. Note that, although the
contraction of H atmosphere sequences are slightly higher, their final radii
are higher compared to the He atmosphere counterparts, showing that a small
amount of hydrogen of the order of $M_{\rm H}\approx 10^{-6}$ MWD can change
the radius by $5.4\times 10^{-5}$ R⊙ ($\sim 1\%$) for the $M_{\rm WD}=1.064$
M⊙ sequence.

The surface gravity $\log g$ as a function of effective temperature $T_{\rm
eff}$ for H and He atmosphere sequences are presented in Figures 13 and 14,
respectively. The solid lines correspond to C/O cores sequences while dashed
and dotted lines correspond to O/Ne and Ne/O/Mg core sequences, respectively.
Stellar mass decreases from top to bottom. Also plotted, as empty circles, are
a selection of white dwarfs with spectroscopic masses $M_{\rm WD}>1$ M⊙ from
the SDSS DR12 catalog presented by (Kepler et al., 2016). Surface gravities
and effective temperatures for H-atmosphere white dwarfs are corrected to 3D
convection (Tremblay et al., 2013) and have mean values of $\left<\log
g\right>=8.987\pm 0.21$ and $\left<T_{\rm eff}\right>=11\,591\pm 501$. From
Figure 13 we find a good agreement between observations and our theoretical
cooling sequences with masses $M_{\rm WD}\leq 1.151$ M⊙. Comparing Figures 13
and 14 we can see that the sequences are slightly shifted to smaller
gravities, on average $\Delta\log g=0.0135$, for lower effective temperatures
due to contraction, which reduced the radius by $\Delta R=3.246\times 10^{-5}$
$R_{\rm\odot}$ on average. The increase in $\log g$ at $\log T_{\rm eff}\sim
4.8-4.9$ presented in figures 13 and 14 for the Ne/O/Mg sequences with $M_{\rm
WD}>1.2$ M⊙ is due to the decrease in the neutrino luminosity, which causes a
reduction in the radius of the model. This behavior agrees with the results
from Althaus et al. (2005b) who were the first to report this phenomenon.

Figure 13: $\log g$ \- $T_{\rm eff}$ relation for H atmosphere white dwarfs.
$T_{\rm eff}$ from $\approx 250\,000$ K to $\approx 10\,000$ K. Empty circles
correspond to H atmosphere white dwarf stars from the SDSS catalog presented
in (Kepler et al., 2016). See text for details. Figure 14: $\log g$ \- $T_{\rm
eff}$ relation for He atmosphere white dwarfs. $T_{\rm eff}$ from $\approx
250\,000$ K to $\approx 10\,000$ K. Empty circles correspond to He atmosphere
white dwarf stars from the SDSS catalog presented in (Kepler et al., 2016).
See text for details.

For comparison proposes, we compute the stellar mass of a sample of 252 stars
from the SDSS DR12 catalog of Kepler et al. (2016) using the cooling sequence
computed in this work and those obtained using the LPCODE evolutionary code
(Althaus et al., 2005a; Romero et al., 2013). The results are shown in Figure
15 where we depict the stellar mass obtained from the MESA tracks against the
mass computed using the LPCODE tracks. As can be seen, there is practically no
difference between the two determinations. The deviations from the 1:1
correspondence for stellar masses around $1.05M_{\rm\odot}$ are due to the
transition from a C/O to a O/Ne core composition, which occurs at
$1.05M_{\rm\odot}$ for the LPCODE, while in our simulations it occurs at
$1.088M_{\rm\odot}$.

Figure 15: Comparison between the stellar mass for a sample of 252
spectroscopically identified DA white dwarf stars from the SDSS DR12 catalog,
computed using the MESA sequences (x-axis) and LPCODE sequences (y-axis). The
dashed line indicates the 1:1 correspondence.

##  5 Conclusion

In the present work we have calculated full evolutionary sequences with
$Z=0.02$ for massive H and He atmosphere white dwarf stars using the
evolutionary code MESA version r8845. Our sequences started from the zero-age
main sequence with masses from 8.8 to 11.8 M⊙, evolving through central
hydrogen and helium burning, thermally pulsing and mass-loss phases ending as
white dwarfs on the cooling sequence. The resulting white dwarfs sequences
show stellar masses in a range of $M_{\rm WD}=1.012-1.307$ M⊙. We presented
full chemical profiles, core mass range, cooling times with crystallization
effects, mass-radius relation and initial-to-final mass relation. We also
calculated masses from our evolutionary tracks for a selection of 252 massive
stars from the SDSS DR12 catalog (Kepler et al., 2016) and compared with
derived masses from other evolutionary tracks presented on the literature.

Unlike models that only evolve to the AGB, our white dwarf sequences considers
the entire evolutionary history of the corresponding progenitors, leading to
realistic chemical profiles which are also consistent with stellar mass. Note
that the amount of hydrogen and helium affects both the age and mass
determinations of DA and non-DA white dwarfs. The final mass range of our
sequences cover the entire range of C/O, O/Ne and Ne/O/Mg core composition and
also accounts for hybrids C/O-O/Ne white dwarfs. Reliable chemical composition
of core and the chemical transition regions presented on Figure 8 are of
relevance for asteroseimological studies of pulsating white dwarf. The set of
tracks presented here is the first in the literature to cover those core
composition for massive white dwarfs, considering the evolutionary history of
the progenitors, and presenting the entire chemical profiles for the
sequences.

Our main results are the following:

  1. 1.

From our simulations we obtained white dwarf sequences characterized with C/O,
O/Ne and Ne/O/Mg cores being: C/O cores for $M_{\rm WD}\leq 1.088$ M⊙, O/Ne
cores for $1.088<M_{\rm WD}\leq 1.147$ M⊙, Ne/O/Mg cores for $M_{\rm
WD}>1.147$ M⊙. Note that these values correspond to sequences with initial
metallicity $Z=0.02$. We expect them to vary for different initial
metallicities (Doherty et al., 2015) . We compute a sequence with $M_{\rm
WD}=1.13$ M⊙ and $Z=0.015$ and found that the chemical structure does not
change considerably from the sequence with $Z=0.02$ and a similar stellar mass
in the cooling curve.

  2. 2.

All sequences with $M_{\rm WD}\geq 1.024$ M⊙ experienced inward propagating
carbon flames with subsequent helium burning of the outer shells, in the post-
AGB stage. The carbon flame for sequences with masses $1.024\leq M_{\rm
WD}\leq 1.147$ M⊙ does not reach the central regions, forming a hybrid
C/O-O/Ne white dwarf. For sequences with $M_{\rm WD}>1.147$ M⊙ the carbon
flame reaches the center of the model producing a Ne/O/Mg core. Our Hybrid
models presents a O/Ne core surrounded by a triple-layer of oxygen-carbon-
helium, differing from the work of Denissenkov et al. (2013) where they
reported a C/O core surrounded by a O/Ne zone for hybrid C/O/Ne Super AGB
stars.

  3. 3.

Our choice of the Coulomb coupling parameter for crystallization $\Gamma_{\rm
full}=220$, based on the latest asteroseismology results, instead of the MESA
default $\Gamma=175$, can increase the cooling times by $\approx 0.26$ Gyr and
decrease the effective temperature at the onset of crystallization by $\approx
2000K$.

  4. 4.

The amount of hydrogen left at the envelope of a white dwarf has an important
impact in determining the stellar mass and the cooling times. Even low
quantities of hydrogen, as the ones expected for massive white dwarfs, can
impact the radii and cooling times. In particular, our H-atmosphere sequences
shows cooling times $\sim 0.23-0.66$ Gyr larger at $T_{\rm eff}\approx
10\,000$ K, depending on stellar mass, when compared to He atmosphere sequence
with the same mass.

  5. 5.

The derived masses for a selection of 252 massive stars from the SDSS DR12
catalog (Kepler et al., 2016) using our simulations are in fine agreement with
masses computed using the evolutionary tracks of LPCODE Althaus et al.
(2005a); Romero et al. (2012, 2013), except around masses $M_{\rm
WD}=1.05-1.088$ M⊙.

## Acknowledgements

We acknowledge the valuable report of the anonymous referee. GRL acknowledges
the support by CAPES-Brazil. ADR and SOK thank support from CNPq and PRONEX-
FAPERGS/CNPq (Brazil). This research has made use of NASA’s Astrophysics Data
System.

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##  Appendix A Chemical Profiles at the end of cooling sequence

In this section is presented the chemical profiles of all our models for H and
He atmosphere white dwarfs. Figure 16 shows the chemical profiles for H
atmosphere on the column on the left and He atmosphere profiles on column on
the right. Each row presents a H atmosphere and a He atmosphere models for a
same mass. All chemical profiles presented in this section are for $T_{\rm
eff}\approx 10,000$ K.

Figure 16: Chemical profiles in terms of outer mass fractions for H atmosphere
(left) and He atmosphere (right) models at $T_{\rm eff}\approx 10\,000$ K in
the cooling sequence. The shaded regions means crystallized portion of core.
Each row presents a H atmosphere (left) and a He atmosphere (right) for a same
mass Figure 17: continued Figure 18: continued