# The impact of the uncertainties in the 12C($\alpha$, $\gamma$)16O reaction
rate on the evolution of low– to intermediate–mass stars

Ben T. Pepper,1 A. G. Istrate2, A. D. Romero1 and S. O. Kepler1
1Physics Institute, Universidade Federal do Rio Grande do Sul, 91501-900
Porto-Alegre, RS, Brazil
2Department of Astrophysics/IMAPP, Radboud University, P O Box 9010, NL-6500
GL Nijmegen, The Netherlands
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

One of the largest uncertainties in stellar evolutionary computations is the
accuracy of the considered reaction rates. The 12C($\alpha$, $\gamma$)16O
reaction is particularly important for the study of low- and intermediate-mass
stars as it determines the final C/O ratio in the core which influences the
white dwarf cooling evolution. Thus, there is a need for a study of how the
computations of white dwarfs and their progenitors that are made to date may
be affected by the uncertainties of the 12C($\alpha$, $\gamma$)16O reaction
rates. In this work we compute fully evolutionary sequences using the MESA
code with initial masses in the range of $0.90\leq M_{i}/M_{\odot}\leq 3.05$.
We consider different adopted reaction rates, obtained from the literature, as
well as the extreme limits within their uncertainties. As expected, we find
that previous to the core helium burning stage, there are no changes to the
evolution of the stars. However, the subsequent stages are all affected by the
uncertainties of the considered reaction rate. In particular, we find
differences to the convective core mass during the core helium burning stage
which may affect pulsation properties of subdwarfs, the number of thermal
pulses during the asymptotic giant branch and trends between final oxygen
abundance in the core and the progenitor masses of the remnant white dwarfs.

###### keywords:

nuclear reactions – stars: abundances – stars: evolution

††pubyear: 2021††pagerange: The impact of the uncertainties in the
12C($\alpha$, $\gamma$)16O reaction rate on the evolution of low– to
intermediate–mass stars–B

##  1 Introduction

Single stellar evolution is fuelled by nuclear reactions that occur within the
stellar interior (Bethe, 1939; Hoyle, 1946, 1954; Burbidge et al., 1957).
These reactions not only release energy which allows the star to support
itself against gravitational collapse and remain in hydrostatic equilibrium,
but also change the composition of the star: this is known as nucleosynthesis
(Eddington, 1920; Hoyle, 1954; Burbidge et al., 1957). The study of these
nuclear reactions is where nuclear physics and astronomy come hand-in-hand; an
understanding of what happens at the fundamental level provides a better
knowledge of how stars evolve and influence their environment. Particularly,
improved estimations of the often uncertain reaction rate data, including
formula fitted to such data, will improve the accuracy of stellar evolution
codes and the understanding of stellar evolution (Caughlan & Fowler, 1988;
Angulo et al., 1999; Katsuma, 2012; Xu et al., 2013; An et al., 2016). Such
estimations are hereafter referred to as ’reaction rates’.

The 12C($\alpha$, $\gamma$)16O reaction during the central helium burning
stage is considered to be the most important mechanism for defining the white
dwarf (WD) core composition (Salaris & Cassisi, 2005; D’Antona & Mazzitelli,
1990; De Gerónimo et al., 2017; Deboer et al., 2019). However, the reaction
rate for this reaction has an extremely large uncertainty (Fowler et al.,
1967; Caughlan & Fowler, 1988; Kunz et al., 2002; An et al., 2016; Deboer et
al., 2017, 2019). The main entrance channel for the ${}^{12}\text{C}+\alpha$
mechanism ($E_{\alpha_{0}}=7.16\,\text{MeV}$) does not have a resonance
channel close to this threshold, the closest occurring at
$E_{x}=9.59\,\text{MeV}$. Instead, the low energy cross-section is largely
influenced by the $1^{-1}$ ($E_{x}=7.12\,\text{MeV}$) and $2^{+}$
($E_{x}=6.92\,\text{MeV}$) subthreshold states (see Figure 2 of Deboer et al.,
2017, for details). The primary influence of these two nearby subthreshold
states and the addition of possible resonant transitions in the wings of the
broad channel at $E_{x}=9.59\,\text{MeV}$ makes the nuclear cross-section
extremely difficult to estimate (see Fowler et al., 1967; Kunz et al., 2002;
An et al., 2016; Deboer et al., 2017, 2019; Aliotta et al., 2021).

During the core helium burning (CHB) stage, carbon is produced from the fusion
of three helium nuclei via the triple-$\alpha$ process (Salpeter, 1952;
Kippenhahn & Weigert, 1990; Salaris & Cassisi, 2005; Prialnik, 2009). As the
abundance of helium in the core depletes, the probability of carbon
interacting with helium to produce oxygen [via 12C($\alpha$, $\gamma$)16O] is
larger than that of the triple-$\alpha$ process at late times during the core
helium burning stage (Salaris & Cassisi, 2005). Thus, the 12C($\alpha$,
$\gamma$)16O reaction is of great importance and is vital to model the carbon-
oxygen (C/O) abundance in the inner chemical profiles for all stellar masses,
but particularly low– and intermediate–mass stars (Woosley & Weaver, 1995;
Weaver & Woosley, 1993; Wallerstein et al., 1997).

The C/O abundance, therefore the 12C($\alpha$, $\gamma$)16O reaction, is
important in many areas of stellar evolution. Such as, influencing the
pulsation properties of ZZ Ceti stars (De Gerónimo et al., 2015, 2017).
Differences between the considered 12C($\alpha$, $\gamma$)16O reaction rate
will also affect the duration of the core helium burning stage (Deboer et al.,
2017). In addition, the 12C($\alpha$, $\gamma$)16O reaction impacts supernova
explosions as the outcome is related to the composition of the final WD (e.g.
Iben & Tutukov, 1984; Wu et al., 2020) and third dredge-up episodes (TDUs)
during the asymptotic giant branch (AGB) stage (Frost & Lattanzio, 1996;
Karakas et al., 2002; Marigo, 2002; Karakas & Lattanzio, 2003; Marigo, 2007;
Cristallo et al., 2009; Weiss & Ferguson, 2009; Ventura & Marigo, 2009;
Kalirai et al., 2014; Matteucci, 2021). Furthermore, thermonuclear explosions
of C/O WDs impacts the ignition of Type 1a supernovae, an important event in
constraining cosmological parameters (Perlmutter et al., 1999; Riess et al.,
1998). The enrichment of the outer layer of the AGB stars from dredge-up and
the mass-loss affects the chemical evolution of galaxies (Matteucci, 2012;
Boothroyd & Sackmann, 1988; Kobayashi et al., 2020; Ventura et al., 2020;
Cristallo et al., 2015; Matteucci, 2021). Additionally, the 12C($\alpha$,
$\gamma$)16O reaction governs whether a star will form a neutron star or black
hole (Brown et al., 2001; Heger et al., 2002; Tur et al., 2007; West et al.,
2013; Sukhbold & Adams, 2020). Gravitational wave detections from black hole
mergers can also be used to constrain the 12C($\alpha$, $\gamma$)16O reaction
rate by determining the mass of the black hole and the fraction of carbon and
oxygen that remains (see Farmer et al., 2020, for details).

De Gerónimo et al. (2015) and De Gerónimo et al. (2017) consider 3 different
reaction rates: an adopted rate from Angulo et al. (1999) and the high and low
rates from Kunz et al. (2002). They consider these alternate rates for the CHB
until the thermally pulsing asymptotic giant branch (TP-AGB) phase with a sole
focus on how the pulsational properties are affected in ZZ Ceti stars, rather
than all stages as we attempt in this work.

In this work, we use stellar evolutionary models as tools to study the impact
of the 12C($\alpha$, $\gamma$)16O reaction rate uncertainties on the stellar
structure and evolution of low– and intermediate–mass stars. The paper is
organised as follows. Section 2 describes the input physics and numerical tool
used to compute the evolutionary sequences, as well as a deeper discussion of
the considered 12C($\alpha$, $\gamma$)16O reaction rates used in this work. In
section 3 we present and discuss our results. We summarise our work in section
4, concluding our findings and indicating future areas where the impact of
this work may affect.

##  2 Numerical Tools

###  2.1 MESA Input Physics

In this work we employ the Modules for Experiments in Stellar Astrophysics
(MESA) code version-r15140 (see Paxton et al. (2011); Paxton et al. (2013,
2015, 2018), for details). We compute the full evolutionary sequence from the
zero age main sequence (ZAMS) through both core hydrogen and helium burning
stages, leading to the AGB and the white dwarf stage (WD). The computation
stops when the stellar model reaches a luminosity of $\log(L/L_{\odot})=-3$ on
the WD cooling track. This stopping condition is applied such that the
sequences have experienced their evolution through the DAV instability strip
(Fontaine & Brassard, 2008; Winget & Kepler, 2008; Althaus et al., 2010). This
allows for asteroseismology of ZZ Ceti stars to be performed in the future.
The final WD masses obtained in this work range from $0.513M_{\odot}\leq
M_{f}/M_{\odot}\leq 0.691\,M_{\odot}$. The initial mass range considered in
this work is selected such that all sequences evolve into a carbon–oxygen WD
(examples of works which consider/include a similar mass range are Renedo et
al., 2010; Romero et al., 2015; De Gerónimo et al., 2017; Marigo et al.,
2020).

We compute a total of 246 sequences, with an initial metallicity of
$Z_{i}=0.01$ and 41 initial masses in the range of $0.90\leq
M_{i}/M_{\odot}\leq 3.05$. For each initial mass, we compute the full
evolution considering 6 different formulae for the 12C($\alpha$, $\gamma$)16O
reaction rate. The 6 reaction rates are adapted from Angulo et al. (1999) and
An et al. (2016). Each source comprises 3 reaction rates: the adopted rate,
the low and high limiting values, given by the reported uncertainties of the
respective rate (see Section 2.2). The rates taken from Angulo et al. (1999)
are part of the NACRE compilation and have been used extensively in other
computations (Renedo et al., 2010; Romero et al., 2015; De Gerónimo et al.,
2017). The reaction rates from An et al. (2016) are less recognised, but boast
a lower uncertainty on their reported adopted reaction rate. More detail on
these rates and their significance can be found in Section 2.2.

We use the reaction network ’basic.net’, which comprises 33 individual
reactions including the full p-p chain, CNO cycle, 3$\alpha$ up until 24Mg,
which contains the 12C($\alpha$, $\gamma$)16O reaction. This network also
includes 8 individual isotopes: 1H, 3He, 4He, 12C, 14N, 16O, 20Ne, 24Mg in
addition to elementary and $\alpha$ particles.

In our computations, we consider the default radiative opacity tables within
MESA. These are from Ferguson et al. (2005) (for $2.7\leq\textrm{log}T\leq
4.5$) and from the OPAL project (for $3.75\leq\textrm{log}T\leq 8.7$)
(Iglesias & Rogers, 1993, 1996). Furthermore, we consider OPAL Type 2 tables
as they allow for varying amounts of C and O, which are needed for helium
burning and beyond (Iglesias & Rogers, 1996; Paxton et al., 2011).

We adopt the standard mixing length free parameter as $\alpha=2.0118$. This
value is adopted from the work of Guzik et al. (2016) who found this value to
be a good approximation for sequences that consider the solar metallicity when
using the opacity tables from the OPAL project. To derive this value, Guzik et
al. (2016) compared calculated non-adiabatic solar oscillation frequencies and
solar interior sound speeds to observed frequencies and helioseismic
inferences. However, it should be noted that Guzik et al. (2016) consider an
initial metallicity of $Z_{i}=0.015$, rather than the value we consider in
this work ($Z_{i}=0.01$). Such a difference would alter the value of the
$\alpha$ parameter if a similar analysis was performed with this initial
metallicity consideration. Convective mixing is treated as a time-dependent
diffusion process, with the diffusion coefficient given as,

$D_{\mathrm{EM}}=D_{0}\exp(-2z/fH_{P})$ (1)

where $H_{P}$ is the pressure scale-height at the convective boundary, $D_{0}$
is the diffusion coefficient of the unstable regions that are near the
convective boundary, and $z$ is the geometric distance from the convective
boundary. $f$ is an adjustable free parameter that controls the efficiency of
mixing by setting the size of the overshooting region (Herwig et al., 1997;
Herwig, 2000). We take the value of $f=0.016$ for all regions of the model for
this work, following the same consideration of overshooting as Herwig (2000);
Weiss & Ferguson (2009); De Gerónimo et al. (2017). This treatment of the
convective boundaries was also adopted by other authors for single stellar
evolution computations (Weiss & Ferguson, 2009; Romero et al., 2015; De
Gerónimo et al., 2017).

The presence of dredge-up episodes during the core helium burning stage is
relevant for the final composition of WDs (Prada Moroni & Straniero, 2002;
Straniero et al., 2003; Renedo et al., 2010). During the thermally pulsing AGB
phase, although overshooting was considered at the boundary of the convective
H-rich envelope during the TP-AGB, the third dredge-up episodes did not occur.
Therefore, the evolution of the hydrogen–exhausted core (which is hereafter
simply referred to as "the helium core mass") and the final mass of the
sequences for those which should experience some third dredge-up episodes will
be affected (see Section 3.2). We define the "helium core mass" as the region
from the centre until the local abundance of hydrogen is greater than
$10^{-6}$. Additional models were computed to assess the impact of the third
dredge-up on the core mass growth during the thermal pulses (see Section 3.2
and Appendix B for details).

For regions stable against convection according to the Ledoux criterion, but
there is an inversion of mean molecular weight, we employ thermohaline mixing.
In MESA this is treated as a diffusion process, as above, with a diffusion
coefficient produced by the stability analysis of Ulrich (1972) and Kippenhahn
et al. (1980). For the efficiency parameter of thermohaline mixing, we
consider $\alpha_{th}=1.0$ (see Equation 14 of Paxton et al., 2013, for
details)). Thermohaline mixing was considered in order to smooth a
discontinuity in the carbon and oxygen chemical profiles at the edge of the
C/O core, during the early-AGB.

Towards the end of the core helium burning stage, when the central He
abundance is lower than $\sim$10%, breathing pulse–like instabilities may
appear. However, these events are attributed to adopted algorithms rather than
to the physics of convection (see Straniero et al., 2003; Romero et al., 2015;
Constantino et al., 2016, 2017, for details). To suppress the breathing
pulses, when the central abundance of He drops below 0.13, we neglect
convection until the central abundance of helium decreases below $10^{-6}$,
similar to the prescription used by Renedo et al. (2010) and Romero et al.
(2015). Without this prescription, the final carbon-to-oxygen (C/O) ratios can
vary rapidly (up to $\pm 0.1$) with small increments of initial mass
($0.05\,M_{\odot}$).

During the main sequence (MS), red–giant branch (RGB) and core helium burning
stages, the mass-loss due to stellar winds follows the rate based on the
Reimers formula (see Reimers, 1975). The asymptotic giant branch and
subsequent evolution follow a rate based on the Bloecker formula instead (see
Bloecker, 1995). We set our scale factors to be $\eta_{R}=0.5$ and
$\eta_{B}=0.2$ for the Reimers and Bloecker formulae, respectively. These
values are chosen as they reproduce a WD with a similar final mass to that
found by Renedo et al. (2010) for $M_{i}=1.00M_{\odot}$ with $Z_{i}=0.01$.

A grey atmosphere is employed for the entire evolution of all sequences, which
utilises the grey Eddington $\tau$ relation. We consider the equations of
state ELM EOS and DT2 EOS, which are derived from the HELM EOS (Timmes &
Swesty, 2000) and the SCVH tables (Saumon et al., 1995), respectively.

Once the star leaves the AGB, we employ an element diffusion process from the
work of Burgers (1969). We refer to element diffusion as the physical
mechanism for mixing chemicals that includes gravitational settling, thermal
diffusion and chemical diffusion. Gravitational settling leads to denser
element diffusing towards the core, while lighter elements float towards the
surface. Thermal diffusion acts in the same direction as gravitational
settling, although to a lesser extent, bringing highly charged and more
massive species to the central regions of the star. Chemical diffusion,
however, works against this general direction (see Iben & MacDonald, 1985;
Thoul et al., 1994, for details). In addition to the aforementioned processes,
MESA includes radiative accelerations (Hu et al., 2011) into their element
diffusion prescription. These radiative forces are negligible in hot regions,
as well as being computationally demanding. Hence, we do not consider the
effects of radiative levitation. Our element diffusion process is applied to
the following isotopes: 1H, 3He, 4He, 12C, 14N, 16O, 20Ne, 24Mg.

###  2.2 The 12C($\alpha$, $\gamma$)16O Reaction

Here we discuss a brief, yet relevant, history of 12C($\alpha$, $\gamma$)16O
reaction rate evaluations. We lead this into further detail for the
12C($\alpha$, $\gamma$)16O reaction rate prescriptions from Angulo et al.
(1999) and An et al. (2016), discussing their differences to the previous
determinations from the literature.

Fowler et al. (1967) organised the first symposium of reaction rate cross-
sections that included the 12C($\alpha$, $\gamma$)16O reaction. At the time,
many resonant factors were neglected and were updated by Caughlan & Fowler
(1988). However, it is believed that some resonances were still neglected and
the treatment of the S-factor in this work produced values that are too small
and require a scale factor of $\sim$2 to produce a realistic S-Factor (Angulo
et al., 1999; Kunz et al., 2002; Heil et al., 2008; An et al., 2016; Deboer et
al., 2017, 2019).

Built upon the works of Fowler et al. (1967); Caughlan & Fowler (1988) and
those associated works in between, Angulo et al. (1999) provided a strong
basis for the 12C($\alpha$, $\gamma$)16O reaction within the NACRE
compilation. Angulo et al. (1999) provided the reaction rates for 86 different
reactions, including 12C($\alpha$, $\gamma$)16O. For the S-factor
calculations, Angulo et al. (1999) considered the values for non-resonant
energies. For narrow resonances, however, they fit the resulting cross-section
using a Briet-Wigner model. When the effects of different resonant energies
overlap, they use a multi-resonance fit, shown in equation 29 of Angulo et al.
(1999). Angulo et al. (1999) state that their analysis is numerical for the
majority, although they do provide an analytical approach for each reaction,
for completeness. They find that their numerical approach yields a higher
accuracy for their calculated reaction rates. The quoted S-Factor value from
Angulo et al. (1999) for a stellar energy of 300 keV is $S(\text{300\,
\text{keV}})=199\pm 64\,\text{keV\,b}$, resulting in a reaction rate ($RR$) of
$RR(\text{300\, \text{keV}})=(9.11^{+3.69}_{-3.67})\cdot
10^{-15}\text{cm\textsuperscript{3}\, mole\textsuperscript{-1}
s\textsuperscript{-1}}$. A stellar energy of $E=\text{300\, \text{keV}}$ is
often chosen as the energy at which to compare the S-factors across different
works, as it is associated with the ignition of core helium burning. In this
work, we consider the adopted rate of Angulo et al. (1999) (NACRE_A) and the
highest and lowest reaction rate within the uncertainties (NACRE_H and
NACRE_L, respectively). Hereafter, we refer to the collective 12C($\alpha$,
$\gamma$)16O reaction rates from Angulo et al. (1999) as ’NACRE’.

An et al. (2016) point out that the resonance parameters used by Kunz et al.
(2002), which were taken from Tilley et al. (1993), neglect the ground state
transitions from the works of Brochard, F. et al. (1975); Ophel et al. (1976).
This results in a larger value for the expected reaction rate at helium
burning temperatures. Instead, An et al. (2016) use the reduced R-matrix and
S-factor derived by An et al. (2015) to estimate the reaction rate, which
accounted for all transitions.

In their computations, An et al. (2015) and An et al. (2016) found a
significant reduction to the uncertainty of their S-factors when compared to
that of Angulo et al. (1999), $S(\text{300\, keV})=162.7\pm 7.3\,\text{keV\,
b}$. The reaction rate for the same energy resulted
$RR(\text{300\,keV})=(7.83\pm 0.35)\times
10^{-15}\text{cm\textsuperscript{3}\, mole\textsuperscript{-1}
s\textsuperscript{-1}}$. We consider the adopted rate from An et al. (2016)
(An_A) and the highest and lowest reaction rate within the uncertainties (An_H
and An_L, respectively). However, the S-factor calculation of An et al.
(2015), seems to neglect external contributions for ground state energy
levels, making this approximation not valid for high precision analysis
(Deboer et al., 2017). Therefore, we treat the uncertainties of the
12C($\alpha$, $\gamma$)16O reaction rate from An et al. (2016) as arbitrary
differences to determine the effect of the urgent need for more precise
12C($\alpha$, $\gamma$)16O reaction rate uncertainties, as claimed by Kunz et
al. (2002); Tur et al. (2010). Some works further claim that the uncertainty
must be less than 10% to be on par with non-nuclear physical uncertainties
(see Woosley et al., 2003; Deboer et al., 2017, for details).

Figure 1 shows a comparison between the adopted reaction rates from An et al.
(2016), NACRE and all of their associated uncertainties. In this figure, we
depict for each rate, the ratio between the rate and the value for NACRE_A, as
a function of temperature. For an analysis including other works, see Figure 4
of An et al. (2016). As can be seen from Figure 1, for energies characteristic
of stellar energies, the An_A, An_H and An_L reaction rates are lower than for
NACRE_A for most temperatures within the blue shaded region, characteristic of
core helium burning temperatures. We therefore expect to have a larger C/O
ratio in the core after the central helium burning stage for the sequences
which consider the rate from An et al. (2016) when compared to those sequences
which consider NACRE_A. It can also be seen in Figure 1 that the range between
NACRE_H and NACRE_L includes all the other prescriptions within the region of
helium burning temperatures, which will lead to the largest differences in the
C/O ratio after the core helium burning stage. At higher temperatures (greater
than those considered to be helium burning temperatures) the reaction rate
from An et al. (2016) is larger than that from NACRE. These temperatures are
not reached in the sequences computed within this work.

Figure 1: Ratios of each reaction rate considered when compared to the adopted
NACRE rate for the 12C($\alpha$, $\gamma$)16O reaction, as a function of
temperature, where $T_{9}=T/10^{9}$. The beige shaded region defines the
temperatures where helium burning occurs. During the core helium burning stage
is also where the 12C($\alpha$, $\gamma$)16O reaction is most prominent. The
light-orange dotted and red dashed lines represent the NACRE_L and NACRE_H
considerations, respectively. The solid blue line defines the adopted rate
from An et al. (2016) with the An_L and An_H rates being depicted as light-
blue dotted and dark-blue dashed lines, respectively.

##  3 Results and Discussions

In this section we describe in detail the effects that the uncertainties of
the 12C($\alpha$, $\gamma$)16O reaction rate have on the inner structure and
evolution for low- and intermediate-mass single stars. As expected, during the
pre-main sequence, main sequence (MS) and red-giant branch (RGB), we find no
differences to the evolution since the 12C($\alpha$, $\gamma$)16O reaction
only becomes important, and increasingly more dominant, during the CHB as the
central helium abundance decreases (Salaris & Cassisi, 2005; Spruit, 2015;
Deboer et al., 2019). Thus, we report no difference between the different
12C($\alpha$, $\gamma$)16O reaction rates at the time of, or shortly after,
the helium-flash or a non-degenerate helium ignition. We only show the results
from the CHB, AGB and WD stages where we expect some differences to occur due
to the uncertainties and separate literature sources of the 12C($\alpha$,
$\gamma$)16O reaction rate. We consider each evolutionary stage separately in
chronological order.

###  3.1 The Core Helium Burning Phase

Figure 2 shows the carbon–to–oxygen (C/O) ratio for each star at the end of
CHB, as a function of initial mass. As expected due to the large uncertainties
of the reaction rate from NACRE, the smallest and largest C/O ratios come from
the NACRE_H and NACRE_L rates, respectively. Note that when all reaction rates
from An et al. (2016) are considered, the values for the C/O ratios are
between the values corresponding to NACRE_A and NACRE_L.

We find that the C/O ratio at the end of the CHB decreases for all considered
reaction rates around an initial mass of $M_{i}=1.90\,M_{\odot}$. This mass
corresponds to the minimum mass for which helium burning starts in non-
degenerate conditions, and will be referred to as the transition mass. The C/O
ratio increases again for higher initial masses (between $2.20\leq
M_{i}/M_{\odot}\leq 2.45$). We find that the initial mass where the increase
of the C/O ratio occurs is dependent on the considered 12C($\alpha$,
$\gamma$)16O reaction rate, such that higher reaction rates have a wider
initial mass range for the decreased C/O ratio and lower reaction rates have a
narrower initial mass range. For example, the NACRE_H has the widest range
($1.90\leq M_{i}/M_{\odot}\leq 2.45$) whereas the NACRE_L has the narrowest
range ($1.90\leq M_{i}/M_{\odot}\leq 2.20$). Furthermore, we find no
difference to the initial mass range between the adopted rate from An et al.
(2016) and the An_H and An_L rates. We also add that the decrease in the C/O
ratio is more pronounced for less efficient reaction rates, see Figure 2, for
details.

Figure 2: Central C/O ratio at the end of the CHB as a function of initial
mass. The red points represent the reaction rates considered by NACRE and the
blue points are those considered by An et al. (2016). Additionally, squares
represent the respective adopted rates while darker-coloured triangles and
lighter-coloured upside-down triangles represent the high and low limit
uncertainties, respectively.

Figure 3 shows the time spent in the CHB as a function of initial mass for the
High and Low reaction rate formulas for NACRE (left panel) and An et al.
(2016) (right panel). We consider the difference in the CHB age from the
values obtained using the respective adopted reaction rate for each panel.
Considering the NACRE rates (left panel of Figure 3), we find that the CHB
lifetime can be up to 12 Myr shorter (longer) from the adopted rate if we
consider NACRE_L (NACRE_H) reaction rate, which is roughly a 7% difference. On
the other hand the differences between the An et al. (2016) rates are much
lower (right panel of Figure 3), up to 4 Myr translating to a difference of
4%. Such changes to the CHB lifetimes due to limits of the uncertainties on
the 12C($\alpha$, $\gamma$)16O reaction rate are not negligible, particularly
for the rate taken from NACRE. Constantino et al. (2016) found that the
difference in the the ratio of HB–to–AGB stars in a sample of 48 globular
clusters could be explained by the differences in the CHB duration due to the
uncertainties in the 12C($\alpha$, $\gamma$)16O reaction rate.

Figure 3: Differences to the duration of the CHB stage due to associated
reaction rate uncertainties as a function of initial mass. The differences are
calculated between each limit of the reaction rate due to their uncertainties
and the adopted rate of each case. The left panel shows the differences of the
uncertainties of the rate calculated by NACRE and the right panel shows the
same from the rate of An et al. (2016). Darker-coloured triangles and lighter-
coloured upside-down triangles represent the high and low limit uncertainties,
respectively.

The top panel of Figure 4 shows the CHB history of the convective mass. The
convective mass is defined as the mass-coordinate of the core convective
boundary, such that convection occurs between this mass-coordinate and the
centre. Additionally, the bottom panel of Figure 4 shows the luminosities of
the 3$\alpha$ process and the 12C($\alpha$, $\gamma$)16O reaction (the latter
will be referred to as C$\alpha$ luminosity), for the NACRE reaction rates. As
expected, the C$\alpha$ luminosity increases when the more efficient reaction
rates are considered. Furthermore, the contribution from the 3$\alpha$ process
decreases for higher reaction rates due to the helium reservoir being depleted
faster by the more efficient 12C($\alpha$, $\gamma$)16O reaction.

Mixing episodes due to the convective core during the CHB extends from the C/O
core to the He-rich layers above, so we define the convective mass as the mass
of the convective core. Figure 4 also shows that higher reaction rates produce
more mixing episodes which are characterised by sudden increases of the
convective mass. These enhanced convective episodes bring fresh helium from
the helium region above the C/O core which not only increases the duration of
the CHB but also increases the abundance of oxygen in the core (Ghasemi et
al., 2017; Guo & Li, 2018).

Convective mixing episodes induce a chemical discontinuity between the fully
mixed core and the radiative layer, increasing the opacity beyond the
convective boundary. In a class of CHB pulsating stars, sdB stars (see Heber,
2009, for an in depth discussion), g-modes propagate from the surface all the
way until the boundary of the convective core (Ghasemi et al., 2017). Since we
find significant differences to the size of the convective core and number of
mixing episodes between the NACRE adopted reaction rate and its uncertainties
for the 12C($\alpha$, $\gamma$)16O reaction rate, the precision of
astereoseismology for these objects is limited and must be considered in the
calculations of the pulsation period spectrum. However for the adopted rate
taken from An et al. (2016), the high and low limits (An_H and An_L,
respectively) do not produce a significant change to the convective core mass
and the total number of mixing episodes and would therefore produce a more
precise study of the g-mode pulsations (see Figure 13 in Appendix A, for an
example of the same case that considers the reaction rates from An et al.
(2016)). The implications for asteroseismology from the treatment to mixing
during the CHB has been studied by Constantino et al. (2015) who found that
changes to the composition and He-burning reaction rates do not significantly
change the period spacing of pulsations for pulsators during the CHB stage.
However, the period values could be more sensitive to the changes in the
chemical profile.

Figure 4: History of the convective mass (top panel), 3$\alpha$ luminosity and
the luminosity of the 12C($\alpha$, $\gamma$)16O reaction during the CHB
(bottom panel). The history is given in terms of the CHB duration. This plot
in particular considers all NACRE prescriptions for the 12C($\alpha$,
$\gamma$)16O reaction rate for an initial mass of $M_{i}=2.45\,M_{\odot}$.
Blue lines represent NACRE_H, orange-brown depicts NACRE_A and dark-brown
shows NACRE_L. Furthermore, the solid line represents the convective mass,
dotted lines show the luminosity of the 12C($\alpha$, $\gamma$)16O reaction
and dot-dash lines portray the 3$\alpha$ luminosity.

The total energy produced by the 12C($\alpha$, $\gamma$)16O reaction during
the CHB is presented in Figure 5. The values shown in Figure 5 are moving
averages. We compute the total energy by integrating the C$\alpha$ luminosity
with respect to time for the CHB duration. Figure 5 shows the ratio between
the different reaction rates and the NACRE_A (top panel) and An_A (bottom
panel) reaction rates, as a function of initial mass. If we consider the
reaction rates from An et al. (2016), the differences are generally smaller
than 10%, the largest difference occurs for the sequence with an initial mass
of $2.85\,M_{\odot}$ that considers An_H. In most cases, the differences are
no larger than 5% (70.7% of the sequences for An_H and 82.9% of the sequences
for An_L).

We find larger differences between the limits of 12C($\alpha$, $\gamma$)16O
NACRE rates when compared to the NACRE_A formula, as shown in the top panel of
Figure 5. In this case we also compare the adopted reaction rate from An et
al. (2016). If we consider how NACRE_H differs from NACRE_A, we find that the
energy production for the majority of the sequences are greater than 10% than
that of the NACRE_A case, with a few exceeding a difference of 20%. For
NACRE_L, the carbon energy produced differs more than 30% from the NACRE_A
rate. The extra energy produced from the high rates when compared to the
adopted rates increases the temperature gradient further allowing convection
to continue, causing the extra mixing episodes shown in Figure 4 (Kippenhahn &
Weigert, 1990; Prialnik, 2009).

Considering the adopted rate from An et al. (2016), the absolute value of the
differences in carbon energy produced due to An_H and An_L appears to be
independent of either selection. This is not the case for the NACRE rates. A
limiting factor for the amount of energy produced is the abundance of
available helium. This is more of a limit for the NACRE_H case due to lack of
available helium inhibiting further reactions to occur. The NACRE_L will
always produce less carbon energy and so is not limited by the helium
abundance or lack thereof. The smaller uncertainties of the rates taken from
An et al. (2016) are not large enough to produce such an effect.

Figure 5: Ratios of the total energy produced by the 12C($\alpha$,
$\gamma$)16O reaction as a function of initial mass. Values are presented in
the form of moving averages. The energy produced is calculated by integrating
the C$\alpha$ luminosity shown in Figure 4 and is integrated with respect to
time. The ratios in the top panel are in terms of the NACRE_A rate and the
ratios in the bottom panel are made in terms of An_A. The red points represent
the reaction rates considered by NACRE and the blue points are those
considered by An et al. (2016). Additionally, squares represent the respective
adopted rates while darker-coloured triangles and lighter-coloured upside-down
triangles represent the high and low limit uncertainties, respectively.

The CHB stage is where the 12C($\alpha$, $\gamma$)16O reaction is the most
active. In particular, we find that the largest differences due to the
considered 12C($\alpha$, $\gamma$)16O reaction rate appear in the final C/O
ratio, CHB duration, energy generation rate and the number of experienced
mixing episodes. The primary reason that we find such changes to these
properties is due to the changes in energy generation that affects the
convection efficiency in this phase. Furthermore, we find that the differences
between the An_H and An_L rates from the An_A rate are generally
insignificant, unlike those of the NACRE uncertainties which are intrinsically
larger. A final point to add is that, in future works, the use of overshooting
parameters specifically designed for the CHB would be interesting. Works such
as Spruit (2015) claim to keep the convective boundaries stable inhibiting the
need for manual breathing pulse suppression, as performed in this work, whilst
keeping "stable" convection active throughout the evolution (Spruit, 2015;
Constantino et al., 2017).

###  3.2 The Asymptotic Giant Branch Phase

During the AGB the energy production is given by two shell sources, the
hydrogen-shell at the base of the hydrogen-rich envelope and the He-shell on
top of the C/O core. Hydrogen burning occurs through the CNO cycle, while He-
burning is through the 3$\alpha$ process. Towards the end of the AGB, the He-
burning shell will become thin enough to trigger unstable burning, and the
thermal pulses (TPs) begin (e.g. Kippenhahn & Weigert, 1990; Iben, 1991).
During the interpulse period between the TPs, the outer convection zone may be
deep enough to bring the products of He-shell burning to the surface, this is
known as the third dredge-up (TDU) (Wallerstein et al., 1997; Busso et al.,
1999; Herwig, 2005; Karakas & Lattanzio, 2014).

Well known consequences of TDUs are a reduction of the helium core mass and
changes to the surface composition, leading to the formation of C-stars (Frost
& Lattanzio, 1996; Busso et al., 1999; Karakas et al., 2002; Weiss & Ferguson,
2009; Romero et al., 2015; Marigo et al., 2020). The extent of the reduction
of the helium core mass from TDU episodes is parameterised by the dredge-up
efficiency parameter, $\lambda_{d}$111The dredge-up efficiency parameter is
defined as the fraction of helium core mass lost during the TDU episode over
the helium core mass growth since the last TDU (see Karakas et al., 2002;
Marigo et al., 2013, for details). The 12C($\alpha$, $\gamma$)16O reaction
during this stage is essentially inactive. There may be some fusion reactions
between 12C and alpha particles at the edge of the C/O core but they are,
however, insignificant. Thus, any difference between the sequences during the
AGB is due to the effect that the 12C($\alpha$, $\gamma$)16O reaction rate has
during the CHB.

Figure 6 shows the helium core mass at the first TP of each sequence as a
function of initial mass. A minimum value occurs for an initial mass
$M_{i}=1.90\,M_{\odot}$, which is the transition point as described in Section
3.1. The same result was found in the work of Kalirai et al. (2014), whose
initial models come from those produced in Bressan et al. (2012). However,
their transition point occurs for $M_{i}=2.00\,M_{\odot}$ due the larger
initial metallicity affecting the mass for which core helium burning ignites
in degenerate conditions (Bertelli et al., 1986; Romero et al., 2015). We find
that there is no significant difference to the helium core mass at the first
TP as a result of different 12C($\alpha$, $\gamma$)16O reaction rates for
masses lower than the transition point. Above this mass, the maximum
difference between the NACRE rates is $\sim 0.01M_{\odot}$, with NACRE_L
producing lower helium core masses and NACRE_H producing larger helium core
masses. This is due to the difference in energy outputs between the adopted
rate, NACRE_A, and the NACRE_H/NACRE_L rates. Higher reaction rates during the
CHB increase the temperature throughout the star which favours the CNO-cycle
(Boeltzig et al., 2016), allowing the helium core mass to develop further than
sequences which consider lower reaction rates. There are no significant
differences in the helium core mass at the first TP between the adopted rate
from An et al. (2016) and An_H/An_L for any of the considered initial masses.

Figure 6: Helium core mass at the start of the first TP as a function of
initial mass. All of the considered reaction rates and their uncertainties are
shown within this figure. We find a minimum to the helium core mass for the
same initial mass which corresponds to the transition mass where core helium
burning begins on a non-degenerate core rather an electron degenerate core
($M_{i}=1.90\,M_{\odot}$). Within the uncertainties, we find differences up to
0.01 M⊙ for masses larger than $M_{i}=1.90\,M_{\odot}$. The red points
represent the reaction rates considered by NACRE and the blue points are those
from An et al. (2016). Additionally, squares represent the respective adopted
rates while darker-coloured triangles and lighter-coloured upside-down
triangles represent the high and low limit uncertainties, respectively.

Figure 7 shows the growth of the helium core mass during the TP-AGB as a
function of initial mass for each considered reaction rate. We find that the
dramatic increase of core growth (for helium core mass growth $\geq 10\%$
(Kalirai et al., 2014)) occurs in the range $1.70\leq M_{i}/M_{\odot}\leq
2.60$, with a maximum increase of 19% occurring at $M_{i}\approx
2.00\,M_{\odot}$. This result is in agreement with that of Bird & Pinsonneault
(2011) and is similar to that of Kalirai et al. (2014), who find a helium core
growth up to 30%. This discrepancy between their work and ours is due to not
only a different initial metallicity, but also their consideration of a less
efficient mass-loss scheme for stages previous to the AGB (Reimers law with
$\eta_{R}=0.2$ (Bressan et al., 2012)). Thus, the models used by Kalirai et
al. (2014) have a larger mass of hydrogen fuel to produce a larger final mass
(see Table 1 for our values of this variable and Bird & Pinsonneault (2011)
for an in-depth discussion of the hydrogen fuel variable). Furthermore,
possible differences to the energy produced in the H-rich envelope during the
TP-AGB may affect the rate of the helium core growth (see Forestini &
Charbonnel, 1997; Marigo et al., 2013; Kalirai et al., 2014, for details).

Considering only the difference in helium core mass growth for NACRE_A rate
and it’s NACRE_H/NACRE_L limits, we find that NACRE_L has a larger core growth
and NACRE_H has smaller core growth. The increased core growth during the AGB
for the NACRE_L sequences is due to the smaller helium core mass at the first
TP (see Figure 6) and as such more fuel to keep He-shell burning sustained,
particularly for initial masses above the transition point where the core
growth differences are greater (see Table 1). Additionally, during the TP-AGB,
we find differences in the energy generation from the CNO cycle between the
NACRE_H/NACRE_L limits in comparison with the NACRE_A. The energy generation
can be up to 25% lower (higher) when the NACRE_H (NACRE_L) reaction rate is
considered.

Figure 7: Percentage growth of the helium core mass during the AGB as a
function of initial mass. Growth is calculated as the difference between the
final mass of the core and the helium core mass described in Figure 6. We find
that the largest growth occurs for initial masses $\approx 2.00\,M_{\odot}$,
peaking at 19%. Above initial masses of $M_{i}=2.90\,M_{\odot}$, it appears
that the growth begins to plateau around 8-9%. The red points represent the
reaction rates considered by NACRE and the blue points are those from An et
al. (2016). Additionally, squares represent the respective adopted rates while
darker-coloured triangles and lighter-coloured upside-down triangles represent
the high and low limit uncertainties, respectively.

Figure 8 shows the number of thermal pulses as a function of initial mass for
each considered reaction rate. Moreover, it shows that lower reaction rates
experience more TPs than higher reaction rates. This is related to the larger
amount of available hydrogen to aid the outward growth of the helium core
through a greater number of unstable He-shell burning episodes - TPs. We do
not find any M-star to C-star transitions (see Marigo et al., 2020, for
example) as convective overshooting about the boundary between the helium core
and the He–exhausted core was disregarded during the TP-AGB, inhibiting the
TDU (Herwig, 2000; Romero et al., 2015). However, overshooting still occurred
at the boundary of the H–rich core. We define the "He–exhausted core" as the
region from the centre until the local abundance of helium is greater than
$10^{-6}$.

Thermal pulses are strongly dependent on the mass–loss rate, helium core mass
and initial metallicity (Karakas et al., 2002; Cristallo et al., 2009; Weiss &
Ferguson, 2009; Renedo et al., 2010; Romero et al., 2015; De Gerónimo et al.,
2017). We find that the number of thermal pulses in our computations is lower
than that from the works of Weiss & Ferguson (2009); Renedo et al. (2010) and
Romero et al. (2015) for a given initial mass, a similar treatment of
convection and a similar helium core mass at the beginning of the TP-AGB
phase. Difference in the number of TPs could be related to the different
mass–loss schemes during the RGB stage. In this work we consider the mass–loss
prescription from Bloecker (1995) while the works of Weiss & Ferguson (2009);
Renedo et al. (2010) and Romero et al. (2015) consider a mass–loss scheme that
produces a "super wind" stage towards the last TPs, making it more efficient
in these last TPs but less so in the early TP-AGB (see Vassiliadis & Wood,
1993; van Loon et al., 2005, for details). However, the trend in the number of
experienced TPs as a function of initial mass obtained in our work agrees with
other works (see Weiss & Ferguson, 2009; Renedo et al., 2010; Romero et al.,
2015).

To assess the effect of the TDU during the TP-AGB, we computed additional
sequences, allowing convective overshooting to occur at all fully– or
semi–convective boundaries, with $f=0.016$ (see Appendix B, for details on
it’s effect). For sequences that consider the NACRE_A prescription, TDU
episodes occur for initial masses larger than $M_{i}\geq 2.40\;M_{\odot}$,
with the dredge-up efficiency parameter ($\lambda_{d}$) showing values of
$\lambda_{d}=0.033-0.124$ that increases with increasing initial mass. The
abundance of carbon and oxygen at the surface does increase during each TDU in
these additional models, but the C/O is still lower than 1 meaning that our
models show an oxygen dominated surface. A higher value of the overshooting
parameter may be necessary to produce C–stars (see Herwig et al., 1997;
Karakas et al., 2002; Weiss & Ferguson, 2009; Romero et al., 2015; Marigo et
al., 2020, for examples of C-star transitions). For sequences where convective
overshooting was considered across all boundaries during the AGB we find a
decrease in the final helium core mass up to 0.63%. This value is much lower
than the 15% decrease found by Karakas et al. (2002); Romero et al. (2015).

The sequences that have initial masses $M_{i}<2.40\;M_{\odot}$ do not show any
third dredge–up episodes, as such we do not expect any difference to the
growth of the helium core or the final mass. For those sequences with initial
masses $M_{i}\geq 2.40\;M_{\odot}$, a more detailed study of the convective
boundaries during the TP-AGB is required for more thorough analysis of why we
find such weak dredge–up efficiency parameters.

In the case of NACRE_H and NACRE_L, we find that TDU episodes occur for the
same initial mass range as that of the NACRE_A sequences ($2.40\leq
M_{i}/M_{\odot}\leq 3.05$). Additionally, the dredge-up efficiency parameters
are also similar to those of the NACRE_A sequences, with
$\lambda_{d}=0.040-0.123$. From the results gathered in this work, we find
that the uncertainties of current 12C($\alpha$, $\gamma$)16O reaction rates
are not significant in modelling the TDU.

The 12C($\alpha$, $\gamma$)16O reaction during the AGB is negligible during
the TP-AGB. Instead, the main energy source occurs through the 3$\alpha$
reaction series and the CNO-cycle within the H-rich envelope (Herwig, 2005;
Karakas & Lattanzio, 2014). Thus, we do not find any significant change to the
peak TP luminosity nor the depth of each TDU, since the changes in core mass
at the beginning of the TP-AGB are negligible as a result of the uncertainties
of the 12C($\alpha$, $\gamma$)16O reaction rate, as shown in Figure 6 (see
Wallerstein et al., 1997; Wagenhuber & Groenewegen, 1998; Busso et al., 1999;
Herwig, 2005; Karakas & Lattanzio, 2014, for details). However, the
uncertainties of the overshooting efficiency raises a greater uncertainty in
the surface composition during the AGB, as such we leave a detailed discussion
for a future work that considers the overshooting efficiency in more detail
(Abia et al., 2002; Herwig, 2005; Cristallo et al., 2009; Ventura & Marigo,
2009; Karakas & Lattanzio, 2014).

Figure 8: Number of TPs experienced as a function of initial mass. Each reaction rate consideration and their uncertainties are shown. We find that the number of TPs peaks at initial masses $\approx 2.00\,M_{\odot}$, in-line with the largest core growth, as in Figure 7. We also show that lower reaction rates for the 12C($\alpha$, $\gamma$)16O reaction produce more TPs. The red points represent the reaction rates considered by NACRE and the blue points are those from An et al. (2016). Additionally, squares represent the respective adopted rates while darker-coloured triangles and lighter-coloured upside-down triangles represent the high and low limit uncertainties, respectively. $M_{i}$/$M_{\odot}$ | $\Delta M_{\text{growth}}/M_{\odot}$ | $M_{\text{fuel}}/M_{\odot}$
---|---|---
NACRE_H | NACRE_A | NACRE_L | An_A | NACRE_H | NACRE_A | NACRE_L | An_A
1.00 | 0.009 | 0.010 | 0.009 | 0.009 | 0.007 | 0.008 | 0.007 | 0.008
1.50 | 0.030 | 0.027 | 0.031 | 0.031 | 0.024 | 0.022 | 0.026 | 0.025
1.60 | 0.037 | 0.038 | 0.039 | 0.038 | 0.030 | 0.031 | 0.031 | 0.031
2.00 | 0.085 | 0.091 | 0.089 | 0.091 | 0.069 | 0.073 | 0.072 | 0.073
2.90 | 0.043 | 0.044 | 0.050 | 0.048 | 0.035 | 0.035 | 0.040 | 0.039
Table 1: Values showing the TP-AGB helium core mass growth and fuel mass. We
report the values from the following reaction rate considerations: NACRE_H,
NACRE_A, NACRE_L and An_A. We do not report the values from the uncertainties
of the rate taken from An et al. (2016) since they are negligible when
compared to their adopted rate.

###  3.3 The White Dwarf Final Cooling Track

Figure 9 shows the initial-to-final mass relation (IFMR) for all sequences
produced in this work. We find that there is no significant difference in the
final mass of any given initial mass due to the 12C($\alpha$, $\gamma$)16O
reaction rate. Considering the largest difference in the reaction rates,
between NACRE_H and NACRE_L, the largest difference in the final mass for a
given initial mass is less than $0.01\,M_{\odot}$ ($<2\%$).

In the interest of the pursuit for a global IFMR, we compare our IFMR to those
of other works of a similar metallicity. We consider the IFMRs from the works
of Weidemann (2000); Salaris et al. (2009) and Renedo et al. (2010). We find a
similar trend with the work of Weidemann (2000), both of which consider the
same mass-loss scheme from Bloecker (1995) for the AGB phase. The IFMRs from
the works of Salaris et al. (2009) and Renedo et al. (2010) consider the
mass–loss scheme from Vassiliadis & Wood (1993) for the AGB and show a much
steeper gradient in their IFMRs. However, the core masses between this work
and the works of Weidemann (2000); Salaris et al. (2009) and Renedo et al.
(2010) are similar at the first TP. Thus, it is reasonable to assume that the
difference is due to their considered mass-loss scheme for the IFMR
determination.

By considering the third-order polynomial nature of the IFMR computed in this
work, we fit a function to the NACRE_A final masses to produce a general
relation from the results of this work. This allows for a comparison to other
IFMRs as well as other masses to be easily estimated, if desired. The
following IFMR reproduces the IFMR of NACRE_A well, such that the R-square
value is $R^{2}=0.9995$:

$M_{f}=0.02047M_{i}^{3}-0.1051M_{i}^{2}+0.2323M_{i}+0.3783M_{\odot}$ (2)

where $M_{f}$ is the final mass and $M_{i}$ is the initial mass. The non-
linear relationship described by Equation 2 is caused by the mass-loss rate
adopted on the AGB. The Bloecker (1995) scheme in particular has a large
dependency on luminosity. It would be interesting to see how our IFMR holds
for observational data as well as it’s dependency on metallicity - an
important dependence as discussed in Romero et al. (2015).

Figure 9: Initial-to-final mass relation of all sequences calculated as part
of this work. Also shown are other IFMRs from the works of Weidemann (2000);
Salaris et al. (2009); Renedo et al. (2010) (yellow stars, purple dashed line
and black squares, respectively) for a comparison of their trends. The red
points represent the reaction rates considered by NACRE, and the blue points
are those from An et al. (2016). Additionally, squares represent the
respective adopted rates while darker-coloured triangles and lighter-coloured
upside-down triangles represent the high and low limit uncertainties,
respectively. We find that the slope of the IFMR has a strong dependency on
the considered mass-loss scheme considered during the AGB, with the scheme
from Vassiliadis & Wood (1993) producing a steeper gradient and that from
Bloecker (1995) showing a shallower gradient.

In Figure 10 we show, in panel a), the final ages of a WD that has cooled to
an effective temperature of $T_{\textrm{eff}}=10\,000$K (log scale) as a
function of initial mass for all the sequences computed in this work. The
differences in the final ages due to the High/Low limits of each considered
12C($\alpha$, $\gamma$)16O reaction rate are in general negligible, with
variations of the order $\sim 0.01$ Gyr for both the NACRE and An et al.
(2016) 12C($\alpha$, $\gamma$)16O reaction rate. The variations in the
reported final ages due to the uncertainties of the 12C($\alpha$, $\gamma$)16O
reaction rate are a magnitude lower than the populations studied in the works
of Hansen et al. (2013); Forbes et al. (2015); Campos et al. (2016). As such,
the impact that the 12C($\alpha$, $\gamma$)16O reaction rate has on final ages
of WD models is currently negligible as compared to the greater uncertainty of
ageing stellar populations.

Panels c) and d) of Figure 10 show the moving average for the time spent on
the cooling track for the NACRE and An et al. (2016) 12C($\alpha$,
$\gamma$)16O reaction rates, respectively. We define this quantity as the time
taken for a star on the final cooling track to cool from it’s maximum
effective temperature until an effective temperature of
$T_{\text{eff}}=10\,000$K. During the final cooling track, the differences in
the duration due to the reaction rates between the Adopted and High/Low limits
generally differ up to $0.030$ Gyr for those of NACRE and up to $0.015$ Gyr
for An et al. (2016). The general trend is in agreement with past discussions
of the effect of the 12C($\alpha$, $\gamma$)16O reaction rate and cooling time
during this stage of evolution, such that more oxygen-rich cores will produce
a lower cooling time. This is due to the gravitational energy release during
stratification occurring at earlier times for more oxygen-rich cores. As a
consequence, the WD is left with a lower thermal content to feed the surface
luminosity at later times. The larger the luminosity at which the
stratification occurs, the shorter the resulting cooling times will be
(D’Antona & Mazzitelli, 1990; Prada Moroni & Straniero, 2002; Salaris et al.,
2010). Furthermore, for the High/Low limits of the NACRE rate, we find that
NACRE_L produces a greater absolute difference than that of NACRE_H. This is
due to the availability of helium during the CHB as discussed in Section 3.1.

Figure 10: Panel a) shows the final age (log scale) of the star on the final
cooling track with an effective temperature $T_{\text{eff}}=10\,000$K. Panel
b) shows the time spent of the cooling track, defined as the time taken for a
WD on the final cooling track to cool from its maximum effective temperature
to an effective temperature of $T_{\text{eff}}=10\,000$K. Panel c) and d) show
the moving average for the difference of cooling times between the High/Low
limits and the Adopted rate for the NACRE and An et al. (2016) 12C($\alpha$,
$\gamma$)16O reaction rate, respectively. All panels are represented as
functions of initial mass. The NACRE reaction rates are shown as different
shades of red and those from An et al. (2016) are depicted by shades of blue.
Furthermore, squares represent the respective adopted rates while darker-
coloured triangles and lighter-coloured upside-down triangles represent the
high and low limit uncertainties, respectively. In general, we find that the
uncertainties of the 12C($\alpha$, $\gamma$)16O reaction rate have an
negligible effect on the final ages of the stars at this point, whereas the
cooling time can differ up to 8%.

After the settling and diffusion processes described in Section 2, the final
oxygen abundances within the core of the sequences are presented in Figure 11,
as a function of initial mass. We find similar trends to the oxygen mass
fraction in this stage to those found at the end of the CHB. Although there
are slight increases to the oxygen mass fraction due to the aforementioned
diffusion processes (Unglaub & Bues, 2000). Additionally, diffusion affects
the C/O ratio throughout the star up to the surface and not just in the core
(see Herwig, 2000; Straniero et al., 2003, for details).

The onset of crystallisation begins when the core cools to a certain
temperature, $T_{c}$ (Segretain et al., 1994; Horowitz et al., 2010). This
temperature is dependent on the internal composition of the star. Through
observations of the globular cluster NGC 6397, Winget et al. (2009) report
that the crystallisation of the WD core is similar to that of a pure carbon
core. According to the phase diagram produced in Horowitz et al. (2010) and
their limits for the maximum crystallisation temperature, this would require a
limit to the oxygen mass fraction of $X_{\text{O}}\leq 0.64$. This requires
that the maximum S-factor at 300 keV has an upper limit of
$S(300\,\text{keV})\leq 170\,\text{keV b}$. Considering the relationship
between oxygen mass fraction and initial mass presented in Figure 11, we find
that NACRE_H and NACRE_A produce central oxygen abundances that are too large
for a crystallisation process similar to that found by Horowitz et al. (2010).
Meanwhile, the rates An et al. (2016) agree not only with the oxygen mass
fraction limit presented by Horowitz et al. (2010), but also their derived
S-factor for an energy of 300 keV. Thus, we find that sequences dedicated to
studying crystallisation using the method presented by Horowitz et al. (2010)
should consider a lower reaction rate than that from NACRE for the
12C($\alpha$, $\gamma$)16O reaction to keep their analysis consistent with the
input physics that they use.

Figure 11: Central oxygen mass fraction for the final WD as a function of
initial mass. We show each calculated sequence. The trends for each considered
reaction rate are similar to those found in Figure 2. There has been a slight
increase in the central oxygen abundance since the CHB due to diffusion
processes in the star. Additionally, squares represent the respective adopted
rates while darker-coloured triangles and lighter-coloured upside-down
triangles represent the high and low limit uncertainties, respectively.

Figure 12 shows the abundance profiles of white dwarf models with a stellar
mass of $M_{*}=0.548M_{\odot}$, $T_{\textrm{eff}}=20\,000$K and an initial
mass of $M_{i}=1.30\,M_{\odot}$. Sequences that consider a reaction rate from
NACRE are shown in the top panel and those from An et al. (2016) are
represented in the bottom panel. All sequences finish with similar structure
to those shown in Figure 12. The profiles depict a DA white dwarf
configuration, with a hydrogen-rich envelope, a helium buffer and a C/O core.
Where the abundance of carbon reaches it’s maximum, we hereafter refer to this
as the carbon peak.

We show that the interior of the star has a consistent trend where the carbon
peak is higher for lower reaction rates - an outcome of a less efficient
reaction rate which leaves behind a larger abundance of carbon. Furthermore,
the position of the carbon peak changes with the reaction rates, moving away
from the centre as the reaction rate increases. We find in general that
differences between An_A and the An_H/An_L reaction rates do not affect this
region drastically (bottom panel), unlike that of the NACRE 12C($\alpha$,
$\gamma$)16O reaction rate considerations (top panel).

The abundance profile and composition gradients in these central regions that
lie within the range of $1<-\textrm{log}_{10}(1-M_{r}/M_{*})<2$ affect the
peaks in the Brunt-Väisälä frequency, which disturbs the period spectrum
structure (see Córsico & Althaus, 2006; Romero et al., 2012b, for more
details). This is an outcome of the pulsation modes that are trapped in this
region through the mode-trapping mechanism. We confirm that uncertainties of
the 12C($\alpha$, $\gamma$)16O reaction rate may affect the pulsation period
spectrum. Another region where the Brunt-Väisälä frequency is affected is in
the He/H transition region. In particular, the position of the He/H transition
will impact the period spectrum (Romero et al., 2012a, 2013).

Figure 12: In both panels we show the abundance profiles of sequences
considering an initial mass of $M_{i}=1.30\,M_{\odot}$. The top panel
represents the adopted rate and it’s uncertainties for the NACRE rate, and the
same for the An et al. (2016) rates in the bottom panel. The line-styles for
each rate are shown in the legend in the bottom panel and the colours for each
element is shown in the legend in the top panel. Colour version is available
online.

##  4 Conclusions

In this work we analyse the impact that the limits of the 12C($\alpha$,
$\gamma$)16O reaction rate has on the inner structure and evolutionary
properties of low- and intermediate-mass stars. We consider the 12C($\alpha$,
$\gamma$)16O reaction rates from NACRE (Angulo et al., 1999) and An et al.
(2016). We have computed stellar sequences from the ZAMS until the remnant
white dwarf reaches a luminosity of $\text{log}(L/L_{\odot})=-3$. We applied
similar starting parameters for different ensembles of reaction rates where we
consider the adopted rate along with the upper and lower limits within the
uncertainties of each source. We summarise our main results below.

  1. 1.

The C/O ratio of the core in the final model of each sequence is affected by
the 12C($\alpha$, $\gamma$)16O reaction rate as expected, with lower C/O
ratios for larger reaction rates. We find that the decreased C/O ratio for
initial masses greater than the transition mass increase again at higher
masses. The mass at which this increase occurs is dependent on the considered
12C($\alpha$, $\gamma$)16O reaction rate, such that it occurs for higher
masses if higher reaction rates are considered. This is due to an increased
number of mixing episodes, a cause of larger energy outputs increasing
convective efficiency which brings fresh helium to the core during the CHB.
Note that significant differences between the adopted rate and high/low limits
occur only for those rates taken from NACRE which has a much larger
uncertainty than those from An et al. (2016).

  2. 2.

CHB lifetime is dependent on the considered reaction rate, a higher reaction
rate produces a greater lifetime. We deem this to be a consequence in the
number of mixing episodes extending the core helium burning lifetime, although
further research would be beneficial to confirm this. Between the adopted rate
and high/low limits, we find a difference up to 12 Myr for the NACRE rates and
up to 4 Myr for those from An et al. (2016).

  3. 3.

The helium core mass at the beginning of the first TP is independent of the
considered 12C($\alpha$, $\gamma$)16O reaction rate up to and including the
transition mass. Above this mass, we find a maximum difference of $\approx
0.01M_{\odot}$ between NACRE_H and NACRE_L, with lower reaction rates
producing a lower helium core mass. Additionally, our minimum helium core mass
at this point occurs at our transition mass.

  4. 4.

Growth of the helium core mass between the first TP and the final mass reaches
a maximum of 19%, with growths greater than 10% occurring in the mass range
$1.70\leq M_{i}/M_{\odot}\leq 2.60$ which is in agreement with Bird &
Pinsonneault (2011) and Kalirai et al. (2014). The largest growths occur for
the lower reaction rates due to more available hydrogen which remained after
the CHB. There are no significant differences between the rates taken from An
et al. (2016) due to the limits being smaller in relation to their adopted
rate than those from NACRE.

  5. 5.

The number of TPs during the TP-AGB is dependent on the considered
12C($\alpha$, $\gamma$)16O reaction rate. We find that lower reaction rates
increase the number of TPs due to a larger hydrogen fuel aiding the outward
growth of the helium core mass by fuelling the unstable He-shell with a
greater supply of fresh helium.

  6. 6.

TDU episodes occur for sequences in the initial mass range of $2.40\leq
M_{i}/M_{\odot}\leq 3.05$ with dredge-up efficiency parameters
$\lambda_{d}=0.033-0.124$. This mass range is independent of the considered
12C($\alpha$, $\gamma$)16O reaction rate. Additionally, the values of
$\lambda_{d}$ between the considered 12C($\alpha$, $\gamma$)16O reaction rate
uncertainties are not significant. Furthermore, the depth of each TDU is
independent of the 12C($\alpha$, $\gamma$)16O reaction rate.

  7. 7.

The IFMR produced in this work has a similar trend to that of Weidemann
(2000), who also consider a similar mass-loss prescription during the AGB. The
IFMRs of Renedo et al. (2010) and Salaris et al. (2009) show a much steeper
gradient and they consider the Vassiliadis & Wood (1993) mass-loss
prescription during the AGB.

  8. 8.

We find that the final ages of the sequences are in general independent of the
considered reaction rate. However, during the final cooling track, we find
differences up to 10% between the adopted rates and high/low limits. This is
true for both those rates taken from NACRE and An et al. (2016). This
difference in the cooling time agrees with the works of Prada Moroni &
Straniero (2002); Salaris et al. (2010); Isern et al. (2013).

  9. 9.

The final C/O ratio in the core shows a similar trend to that at the end of
the CHB. The oxygen abundance increases slightly due to the diffusion
processes. The final oxygen mass fraction for NACRE_A and NACRE_H sequences
are greater than the values derived by Horowitz et al. (2010) for
crystallisation of a C/O core. The reaction rates from An et al. (2016) agree
closely with the derived values of Horowitz et al. (2010). As such, future
works should consider a lower reaction rate than that of NACRE when
considering the crystallisation process of Horowitz et al. (2010).

  10. 10.

The inner structure of the star is affected by the uncertainties within the
considered reaction rates, particularly those from NACRE. The position and
height of the carbon peak is significantly affected by the difference between
the adopted rate and high/low limits of the reaction rate for the NACRE
considerations. This may affect the modes in which pulsations can occur during
the ZZ Ceti instability strip (Córsico & Althaus, 2006; Romero et al., 2012a).

Although we analyse the possible evolutionary stages where more accurate
12C($\alpha$, $\gamma$)16O reaction rates are needed, a deeper analysis of
some effects are still required. For instance, a quantification of how the
pulsation modes of sdB’s and ZZ Ceti stars are affected, for example.
Furthermore, we conclude that a lower reaction than that of NACRE_A is
favourable for the Horowitz et al. (2010) considerations of crystallisation,
however, this must be further analysed as well. By limiting the uncertainties
of 12C($\alpha$, $\gamma$)16O reaction rates to 10% of the adopted rate, as in
An et al. (2016), reports a much better consistency of stellar parameters.

## Acknowledgements

BTP, ADR and SOK acknowledge support by CNPq and PRONEX-FAPERGS/CNPq. This
study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de
Nível Superior - Brasil (CAPES) - Finance Code 001. This research has made use
of NASA’s Astrophysics Data System. AGI acknowledges support from the
Netherlands Organisation for Scientific Research (NWO). We also thank
developers of the MESA software, which was used extensively in this work.
Finlly, we thank the anonymous referee for their input to make it a more
complete work.

## Data Availability

The data is available upon request to the corresponding author.

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##  Appendix A Convection during CHB for An rates

Figure 13 shows the CHB history of the convective mass and the luminosities of
the 3$\alpha$ process and the 12C($\alpha$, $\gamma$)16O reaction, for the
reaction rates taken from An et al. (2016). This figure is analogous to that
of Figure 4 which shows the same for the NACRE rates. We provide this figure
to prove that we do not find any significant difference between the number of
mixing episodes, luminosity from the 3$\alpha$ process and the 12C($\alpha$,
$\gamma$)16O reaction. Thus, the high/low limits for the 12C($\alpha$,
$\gamma$)16O reaction rate from An et al. (2016) does not affect the CHB in
terms of energy production, mixing episodes or CHB duration. This was not
found for the NACRE case, which is discussed in Section 3.1.

Figure 13: History of the convective mass (top panel), 3$\alpha$ luminosity
and the luminosity of the 12C($\alpha$, $\gamma$)16O reaction during the CHB
(bottom panel). The history is given in terms of the CHB duration. This plot
in particular considers all An et al. (2016) prescriptions for the
12C($\alpha$, $\gamma$)16O reaction rate for an initial mass of
$M_{i}=2.45\,M_{\odot}$. Blue lines represent An_H, orange-brown depicts An_A
and dark-brown shows An_L. Furthermore, the solid line represents the
convective mass, dotted lines show the luminosity of the 12C($\alpha$,
$\gamma$)16O reaction and dot-dash lines portray the 3$\alpha$ luminosity.

##  Appendix B Additional AGB Models

Figure 14 shows the Kippenhahn diagram for the case of $M_{i}=3.05\,M_{\odot}$
during the TP-AGB in the original NACRE_A models. We represent the mass co-
ordinate on the first y–axis and the surface C/O ratio on the second y–axis.
Both values are plotted against the age of the sequence. These models did not
consider convective overshooting around the border of the He–exhausted core.
Green slashed areas show convective regions, red back slashed areas represent
semi-convective regions and the purple regions are where overshooting occurs.
The purple dotted line shows the history of the He–exhausted core mass and the
blue dotted line represents the history of the helium core mass. The colour
bar measures the energy generation rate from nuclear reactions. The solid
orange line represents the C/O ratio at the surface. It can be seen that the
overshooting occurs close to the envelope boundary and there is no
overshooting about the semi-convective region of the He–exhausted core. As a
result of this, we do not observe TDU episodes in the original models. We can
be sure that there are no TDU episodes because of the lack of change in helium
core mass and that the surface C/O ratio remains constant, which would change
if TDUs were experienced (Frost & Lattanzio, 1996; Herwig et al., 1999;
Karakas et al., 2002; Weiss & Ferguson, 2009; Romero et al., 2015; De Gerónimo
et al., 2017; Marigo et al., 2020).

Figure 15 shows the same as Figure 14 but allows for convective overshooting
at each boundary. We find that, with the new prescription, convection and
overshooting extends throughout the helium buffer. For this reason material
can be "dredged-up" from the core to the surface. This results in the helium
core mass and He–exhausted core masses changing with each convective episode -
an outcome of TDU episodes (Frost & Lattanzio, 1996; Herwig et al., 1999;
Karakas et al., 2002; Weiss & Ferguson, 2009; Romero et al., 2015; De Gerónimo
et al., 2017; Marigo et al., 2020). Furthermore, we find an increase in the
surface C/O ratio with each TDU as material travels from the stellar interior
to the surface. The surface C/O ratio, however, remains less than 1. This
indicates a larger overshooting parameter is required for M–star to C–star
transitions.

Figure 14: Kippenhahn diagram for during the TP-AGB for the case of
$M_{i}=3.05\,M_{\odot}$ of the original models. We represent the mass co-
ordinate on the first y–axis and the surface C/O ratio on the second y–axis.
Both values are plotted against the age of the sequence. This model did not
consider convective overshooting at boundary of the He–exhausted core which
inhibited the TDU. The colour bar measures the energy generation rate from
nuclear reactions. The blue dotted line represents the helium core mass while
the purple dotted line represents the He–exhausted core. Green slashed regions
show convection and the red back slashed regions represent where regions of
the star are semi-convective. Finally, purple areas are where overshooting
occurs. Figure 15: Kippenhahn diagram for during the TP-AGB for the case of
$M_{i}=3.05\,M_{\odot}$ of the new models. We represent the mass co-ordinate
on the first y–axis and the surface C/O ratio on the second y–axis. Both
values are plotted against the age of the sequence. The colour bar measures
the energy generation rate from nuclear reactions. This model considered
convective overshooting at all convective boundaries, allowing fro TDUs to
occur. The blue dotted line represents the helium core mass while the purple
dotted line represents the He–exhausted core. Green slashed regions show
convection and the red back slashed regions represent where regions of the
star are semi-convective. Finally, purple areas are where overshooting occurs.