# Seismic Signatures of the 12C($\alpha,\gamma$)16O Reaction Rate in White
Dwarf Models with Overshooting

Morgan T. Chidester School of Earth and Space Exploration, Arizona State
University, Tempe, AZ 85287, USA  F.X. Timmes School of Earth and Space
Exploration, Arizona State University, Tempe, AZ 85287, USA  Ebraheem Farag
School of Earth and Space Exploration, Arizona State University, Tempe, AZ
85287, USA Morgan T. Chidester<EMAIL_ADDRESS>
###### Abstract

We consider the combined effects that overshooting and the
$\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ reaction rate have on variable white
dwarf stellar models. We find that carbon-oxygen white dwarf models continue
to yield pulsation signatures of the current experimental
$\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ reaction rate probability
distribution function when overshooting is included in the evolution. These
signatures hold because the resonating mantle region, encompassing $\simeq$
0.2 $\mathrm{M}_{\odot}$ in a typical $\simeq$ 0.6 $\mathrm{M}_{\odot}$ white
dwarf model, still undergoes radiative helium burning during the evolution to
a white dwarf. Our specific models show two potential low-order adiabatic
g-modes, $g_{2}$ and $g_{6}$, that signalize the
$\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ reaction rate probability
distribution function. Both g-mode signatures induce average relative period
shifts of $\Delta P/P=0.44\%$ and $\Delta P/P=1.33\%$ for $g_{2}$ and $g_{6}$
respectively. We find that $g_{6}$ is a trapped mode, and the $g_{2}$ period
signature is inversely proportional to the
$\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ reaction rate. The $g_{6}$ period
signature generally separates the slower and faster reaction rates, and has a
maximum relative period shift of $\Delta P/P=3.45\%$. We conclude that low-
order g-mode periods from carbon-oxygen white dwarfs may still serve as viable
probes for the $\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ reaction rate
probability distribution function when overshooting is included in the
evolution.

Asteroseismology(73); Nuclear astrophysics (1129); White dwarf stars (1799);
Stellar physics (1621)

††software:  MESA (Paxton et al., 2011, 2013, 2015, 2018, 2019; Jermyn et al.,
2023, https://docs.mesastar.org/), MESASDK 20190830 (Townsend, 2019a, b),
wd_builder https://github.com/jschwab/wd_builder, GYRE (Townsend & Teitler,
2013; Townsend et al., 2018, https://github.com/rhdtownsend/gyre), mkipp
https://github.com/orlox/mkipp, matplotlib (Hunter, 2007), NumPy (van der Walt
et al., 2011), amd ChatGPT (OpenAI, 2023).

##  1 Introduction

Helium burning is primarily the fusion of helium into carbon by the triple-
alpha (3$\alpha$) process. All stars born with more than $\simeq$ 0.5
$\mathrm{M}_{\odot}$ go through this stage of energy production as they evolve
beyond the main-sequence (e.g., Hansen et al., 2004). Helium burning also
plays a key role in transients such as Type I X-ray bursts (Weinberg et al.,
2006; Guichandut & Cumming, 2023), Type Ia supernovae (Shen et al., 2018;
Collins et al., 2022), and He-rich subdwarf O stars (Werner et al., 2022;
Miller Bertolami et al., 2022). Helium burning also impacts several classes of
distribution functions, such as the black hole mass distribution function
(Fryer & Kalogera, 2001; Sukhbold et al., 2018; Sajadian & Sahu, 2023)
including any mass gaps based on the pair-instability mechanism in the
evolution of massive stars (Fowler & Hoyle, 1964; Woosley et al., 2002; Farmer
et al., 2019, 2020; Renzo et al., 2020; Marchant & Moriya, 2020; Farag et al.,
2022).

He burning is triggered by the 3$\alpha$ process releasing 7.5 MeV in fusion
energy and producing 12C (Hoyle, 1954; Eriksen et al., 2020; Kibédi et al.,
2020; Cook et al., 2021). This is a unique process, setting stringent
conditions for helium ignition. The 3$\alpha$ process is followed by the
$\alpha$ capture reaction 12C($\alpha$, $\gamma$)16O, converting the 12C into
16O (deBoer et al., 2017; Mehta et al., 2022; Shen et al., 2023). These two
isotopes are the principal products of He burning. In addition, nearly all of
a star’s initial CNO abundances in the stellar interior are converted to 22Ne
at the onset of He burning (Timmes et al., 2003; Howell et al., 2009; Bravo et
al., 2010; Blondin et al., 2022; Meng et al., 2023). This marks the first time
in a star’s life where the core becomes neutron rich. We follow the convention
that 22Ne is the “metallicity” of a carbon-oxygen (CO) white dwarf (WD).

The interiors of CO WDs are, in principle, the best probe of the ashes of He
burning. A goal of WD seismology is to characterize the chemical profiles of
principal products of He burning (Metcalfe et al., 2001, 2002; Fontaine &
Brassard, 2002; Straniero et al., 2003; Metcalfe, 2003; Giammichele et al.,
2017; De Gerónimo et al., 2017; Giammichele et al., 2018; De Gerónimo et al.,
2019; Córsico et al., 2019; Pepper et al., 2022; Giammichele et al., 2022;
Córsico et al., 2022; Romero et al., 2023) and the chemical profile of the
trace 22Ne metallicity (Camisassa et al., 2016; Giammichele et al., 2018;
Chidester et al., 2021; Althaus & Córsico, 2022).

Furthermore, regions within a CO WD model that burn helium radiatively during
its prior evolution can offer potential constraints on the He burning nuclear
reaction rates. For example, Chidester et al. (2022, hereafter C22) found that
certain trapped adiabatic g-modes in WD models may provide a pulsation
signature that constrains the experimental
$\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ reaction rate probability
distribution function. These signature g-modes were shown to resonate with the
region of the CO WD model that underwent radiative He burning during its
previous evolution. The innermost boundary of this resonant cavity corresponds
to the molecular weight gradient at O$\rightarrow$C chemical transition, and
the outermost boundary to the molecular weight C$\rightarrow$He chemical
transition. The resonating region encompasses $\simeq$ 0.2
$\mathrm{M}_{\odot}$ of a typical $\simeq$ 0.6 $\mathrm{M}_{\odot}$ WD model.
C22 cautioned that the chemical structure and resulting pulsation spectrum is
sensitive to the width of the O$\rightarrow$C transition (Córsico et al.,
2002; Salaris & Cassisi, 2017; Pepper et al., 2022), the experimental
3$\alpha$ reaction rate probability distribution functions (deBoer et al.,
2017; Kibédi et al., 2020; Schatz et al., 2022), convective boundary mixing
processes during core He depletion (Salaris & Cassisi, 2017; Anders et al.,
2022), and the number of thermal pulses during the Asymptotic Giant Branch
(AGB) phase of evolution (De Gerónimo et al., 2017; Pepper et al., 2022).

Modeling convective boundary mixing processes at the convective-radiative
interface during core He burning in low- and intermediate-mass stellar models
is currently uncertain (Herwig, 2000; Salaris & Cassisi, 2017; Jermyn et al.,
2022; Anders et al., 2022; Blouin et al., 2023). Convective overshoot occurs
because the convective boundary is not the location where convective
velocities are zero, but the location where the buoyant acceleration of the
fluid is zero. An order–of-magnitude expression $\Delta x=u\Delta t$ provides
an estimate for how far convective motions overshoot (Anders et al., 2022).
Here $\Delta x$ is the overshoot distance, $u$ is the convective velocity, and
$\Delta t\simeq 1/N$ where $N$ is the Brunt-Väisälä frequency in the stable
region. There is disagreement on how to calculate $\Delta x$, but this
estimate broadly shows $\Delta x\ll H_{P}$ in stellar environments, where
$H_{P}$ is the pressure scale height. The exponential overshoot
parameterization (e.g., Herwig, 2000) is frequently implemented in 1D models
to describe this convective boundary mixing process, treating $\Delta x$ as a
free parameter. The values of $\Delta x$ needed to match the gravity modes
found in Slowly Pulsating B-type stars (Pedersen et al., 2021) suggest $\Delta
x/H_{P}\simeq 0.1$, which is larger than 3D hydrodynamical simulations of low
Mach number flows at stable interfaces indicate (Korre et al., 2019; Blouin et
al., 2023).

The injection of fresh He into the convective core enhances the rate of energy
production by the 12C$(\alpha,\gamma)^{16}$O reaction rate, increases the
central $\mathrm{{}^{16}O}$ mass fraction De Gerónimo et al. (e.g., 2017), and
modifies the lifetime through this phase of evolution. The resulting increase
in the radiative gradient can also lead to rapid growth in the convective He
core boundary (a “breathing pulse”). A consensus on breathing pulses being
physical or numerical has not yet been reached (Caputo et al., 1989; Cassisi
et al., 2003; Farmer et al., 2016; Constantino et al., 2017; Paxton et al.,
2019).

C22 found a pulsation signature of the
$\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ reaction rate probability
distribution function using evolutionary models that purposely excluded
overshooting. This article is novel in analyzing whether or not pulsation
signals of the $\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ reaction rate
probability distribution function still exist when overshooting at the inner
convective-radiative interface during core He burning (CHeB) is included in
the models’ evolution history. Here, the inner convective-radiative interface
is the transition from the convective core to the exterior radiative layer.
Section 2 describes our models, §3 analyzes our models, §4 discusses our
results, and we summarize our findings in §5. Appendix A lists the
microphysics used, and Appendix B discusses variations with the number of
isotopes in the reaction network and with the temporal resolution of our
models.

##  2 Stellar Evolutionary Models

We define the term “model” to mean an evolutionary sequence that begins at the
pre-main sequence, progresses through CHeB, and terminates as a cold WD. We
define the term “snapshot” to mean a specific instance in time or phase of
evolution within a model, and the term “set” to mean a suite of models or
snapshots that have identical input physics except for the value of the
$\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ reaction rate.

We use MESA version r15140 (Paxton et al., 2011, 2013, 2015, 2018, 2019;
Jermyn et al., 2023) to build 2.1 $\mathrm{M}_{\odot}$, $Z$ = 0.0151
metallicity, $Y$ = 0.266 He mass fraction, nonrotating models at the pre-main
sequence. We adopt the AGSS09 (Asplund et al., 2009) abundances and use a 23
isotope nuclear reaction network with 22Ne being the heaviest isotope111A
comparison to a 30 isotope network is given in Appendix B.. Our models employ
MESA’s Henyey mixing-length theory (MLT) of convection option, with an MLT
parameter of $\alpha$ = 1.5. This is consistent with the value used in C22. We
use the Ledoux criterion, and the predictive mixing scheme. Additional details
of the MESA microphysics are listed in Appendix A.

Figure 1: Top: Mass fraction profiles without overshooting (NOV) during the
CHeB phase. Left: Mass fraction profiles after core He-depletion, terminated
prior to the first thermal pulse at $\log(L/L_{\odot})>3.0$. Middle: Mass
Fraction profiles at the first wd_builder model step. These profiles have been
shaved of their excess H envelope prior to running on the WD cooling track.
These are the initial hot WD profiles. Right: Mass fraction profiles when the
models have cooled to $T_{\rm eff}$ = 10,000 K. The smoothness in the profiles
reflects the element diffusion processes included in the calculation. Bottom:
Mass fraction profiles with core overshooting (OV) during the CHeB phase, in
the same format as above. Green curves represent positive $\sigma_{i}$
$\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ reaction rates, grey curves represent
negative $\sigma_{i}$ $\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ reaction rates.
For both positive and negative $\sigma_{i}$, the shading grows fainter the
further $\sigma$ is from the standard rate ($\sigma=0$; black curve).

As in C22, we span the current experimental
$\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ reaction rate probability
distribution function (deBoer et al., 2017; Mehta et al., 2022) from
$\sigma=-3.0$ to $\sigma=+3.0$ in 0.5$\sigma$ steps, totaling to 13
$\sigma_{i}$ reaction rates; each model is prescribed one such $\sigma_{i}$
$\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ reaction rate value for its
evolution. We calculate one set of models without overshooting (NOV), and a
second set with overshooting (OV) at the inner radiative-convective interface
during the CHeB phase. Hence, each evolutionary model differs only in its
$\sigma_{i}$ $\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ reaction rate, and NOV
or OV mixing prescription. This yields 26 individual stellar evolutionary
models; 13 for the NOV set and 13 for the OV set. For
$i=(-3.0,-2.5,...,+2.5,+3.0)$, we use $\sigma_{i}$ and $\sigma=i$
interchangeably to reference a given $\sigma$ from the
$\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ reaction rate probability
distribution function.

After CHeB, the models evolve until $\log(L/L_{\odot})=3.0$, prior to the
first thermal pulse on the AGB. At this snapshot, we interrupt the evolution
of each model. All models at this snapshot thus have a C$\rightarrow$He
transition at nearly the same mass location. We use this snapshot to construct
H-dominated atmosphere (DA) WDs by removing the H envelopes until
$\log$(MH/M∗)$<-$3.5. The resulting composition profile structures are used to
build 0.56 $\mathrm{M}_{\odot}$ ab-initio WD models with wd_builder, as done
in C22. These WD models evolve until $T_{\rm eff}$ = 10,000 K. We discuss the
reasoning for constructing the WDs from the post-CHeB $\log(L/L_{\odot})=3.0$
snapshot in the following section.

We utilized version 6.0.1 of the GYRE code (Townsend & Teitler, 2013; Townsend
et al., 2018) to compute the adiabatic pulsations of our WD models throughout
their respective cooling tracks (from $\sim 50,000$ K to $10,000$ K). We
tracked the pulsations for the entire WD cooling track to observe the
evolution of the adiabatic modes. Further, this was the most convenient way to
auto-implement pulsation calculations for multiple models (i.e. we did not
have to post-process the pulsation calculations over a specifed $T_{\rm eff}$
range for each of the 26 models). We emphasize that the computed pulsations
are adiabatic, and that the observed instability strip for DAV WDs spans only
from $\sim 13,000$ K to $\sim 10,000$ K. The inlist parameters were set to
search for modes of harmonic degrees $\ell=1,2$ and radial orders $n\leq 25$,
where our models were assumed to be non-rotating, hence only $m=0$ azimuthal
orders were present. For the adiabatic mode analysis, we employed the fourth-
order Gauss-Legendre collocation difference equation scheme (Iserles, 1996;
Townsend & Teitler, 2013; Townsend et al., 2018).

Details of the MESA models and GYRE oscillation parameters are in the files to
reproduce our results at at doi:[https://doi.org/10.5281/zenodo.8126450
(catalog 10.5281/zenodo.8126450).

###  2.1 Core Overshooting prescription during the CHeB

During the CHeB phase, we use the following core overshooting parameters in
the MESA inlist for the OV set:

  * ! overshoot

  * min_overshoot_q = 1d-3

  * overshoot_scheme(1) = ‘exponential’

  * overshoot_zone_type(1) = ‘any’

  * overshoot_zone_loc(1) = ‘core’

  * overshoot_f(1) = 0.016

  * overshoot_f0(1) = 0.008

  * overshoot_mass_full_on(1) = 0.01

  * overshoot_mass_full_off(1) = 0.4

Details of the specific parameters are described in the MESA
documentation222https://docs.mesastar.org/en/latest/. We choose the
conventional Herwig (2000) value of `overshoot_f(1)=0.016`. This parameter
sets the fractional distance of $H_{p}$ to overshoot at the
$\nabla_{ad}=\nabla_{rad}$ interface, for the order of magnitude estimate
given in the introduction, $\Delta x=f_{0}\cdot H_{p}$.

The trapped mode seismic signatures found in C22 were resonating most with the
region that underwent radiative He burning, defined as R2. Their inner
boundary of R2 is near the molecular weight gradient at the O$\rightarrow$C
transition (the “O drop”) and their outer boundary is near the
C$\rightarrow$He transition. Mode trapping is sensitive to the location of
both of these boundaries because they define the width of the resonant cavity.

One approach to analyzing the sensitivity of the R2 trapped mode signatures is
to fix one boundary and vary the other boundary. We fix the R2 outer boundary
by excluding variations imposed from the thermal pulse history, hence the
interruption at the post-CHeB $\log(L/L_{\odot})=3.0$ snapshot for all models.
The phenomena that happens during the AGB phase is another source of model
uncertainty. Gautschy (2023) found that early post-AGB pulsations can cause
rapid growth of an instability that drives a super-wind which can shed much of
the outer layers in a few years. Further, their 2.0 $\mathrm{M}_{\odot}$,
Z=0.02 model shows a dynamic evolutionary track, especially during the AGB,
that is similar to the models in this article. Gautschy (2023) summarizes that
while the preliminary results show promise on future AGB and post-AGB
phenomenon, there are currently more questions than answers. We therefore
leave the thermal pulse history and the particular envelope ejection phenomena
on the AGB to future studies, and freeze the outermost R2 boundary before the
first thermal pulse occurs. In this vein, we isolate the sensitivity of the R2
region to its inner boundary, and specifically address how core overshooting
influences the pulsation signatures for the
$\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ reaction rate probability
distribution function.

We end this section by stating we are not advocating for a specific
evolutionary model or overshooting scheme. Rather, we are exploring one
approach to quantifying the coupled uncertainty between the
$\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ reaction rate probability
distribution function and a common overshooting model.

##  3 Results

###  3.1 Evolution of Composition Profiles

Figure 2:  Composition profiles after the NOV (top) and OV (bottom) models
cool to $T_{\rm eff}$ = 10,000 K. Region boundaries are indicated by vertical
blue lines, and the $\sigma_{i}$ colors are the same as in Figure 1.

Figure 3: Kippenhahn diagrams for the NOV (left) and OV (right) $\sigma=0.0$
models. The x-axis is the respective stellar model’s age, the left y-axis is
the mass coordinate, the right y-axis is the central mass fraction of the
isotopes. Bright green areas represent convection, blue shaded regions depict
nuclear burning (see colorbar), white areas represent radiation, and yellow-
gold areas represent overshooting (right figure). The light green area shows
the hydrogen envelope. The solid pink curve is the central $\mathrm{{}^{4}He}$
mass fraction, the solid dark blue curve is the central $\mathrm{{}^{16}O}$
mass fraction, and the dark yellow curve is the central $\mathrm{{}^{12}C}$
mass fraction. The dashed line shows core helium depletion. The evolution was
terminated when $\log(L/\mathrm{L}_{\odot})>3.0$ for all stellar models, and
the figures are plotted until that point. Annotated is the radiative R2
region’s edges and widths. An interactive figure is provided in the online
version. Its functionality compares the NOV and OV Kippenhahn diagrams for any
given $\sigma_{i}$ $\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ reaction rate. To
use the interactive figure, click on the ‘sigma’ slider at the top, and slide
through the indices 0 through 12. Sliding through each index provides the user
the side by side comparison of the Kippenhahn diagrams for the respective
$\sigma_{i}$ $\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ reaction rate used in
the evolution. Index 0 starts at $\sigma_{-3.0}$ and index 12 ends at
$\sigma_{3.0}$.

Figure 1 shows the mass fraction profiles for both sets at three evolutionary
snapshots. The top row shows the mass fraction profiles for the NOV set and
the bottom row shows the mass fraction profiles for the OV set. The left most
column shows the mass fraction profiles at the post-CHeB
$\log(L/L_{\odot})>3.0$ snapshot. At this point, our models have not lost much
mass and are all $\sim 2.1\ \mathrm{M}_{\odot}$. The middle column shows the
mass fraction profiles after removing the H envelopes until
$\log$(MH/M∗)$<-$3.5. This snapshot shows the initial hot WD profiles, after
completing one model step in wd_builder. The profiles shift slightly in mass
location, but the overall composition structure only differs from the left
panel in the thickness of the H envelope. The right column is the final
snapshot of the mass fraction profiles, when the models reach $T_{\rm
eff}=10,000$ K. Diffusion was included on the WD cooling track and leads to
the smoothness of the profiles in this column.

Figure 2 accentuates the differences between the NOV (top) and OV (bottom)
mass fraction profiles for the final WD structures (right column of Figure 1).
Here, we show the abundance in mass fraction with respect to fractional radius
$r/R$. We partition the WDs’ composition profiles into four regions: R1, R2,
R3, and R4. This is similar to that done in C22. The regions are defined to
estimate trapping (resonant) zones. Boundaries for mode trapping are typically
near composition transitions because they generally have large mean molecular
weight gradients. This may lead to partial reflections for a resonant mode(s),
“trapping” it within the local cavity (Brassard et al., 1991; Winget et al.,
1981). The Ledoux B profile (henceforth $B$) captures composition gradients
and can estimate trapping regions. We use $B$ as our primary guide to define
the region boundaries for a given model. The R1-R2 boundary is set at the
first local maximum in $B$ that occurs after reaching peak $\mathrm{{}^{16}O}$
in a given model’s chemical profile. The R2-R3 boundary is set at the second
local maximum in $B$. The R3-R4 boundary is set at the location where $\rm
X(^{1}H)>X(^{4}He)$.

In both NOV and OV sets, $\sigma_{i}$ impacts the magnitude of the 16O and 12C
profiles in R1. Core overshooting changes the structure of these profiles,
especially at $r/R\sim 0.37$ where the flatness of the profiles becomes
disrupted. This is due to additional He fuel ingested during CHeB, from
overshooting and/or convection. The fuel ingestion from overshooting and
convection is a coupled effect and specific to each $\sigma_{i}$ model. After
$r/R\sim 0.37$, there is some overlap in the profiles that perturbs the
proportional trend with $\sigma_{i}$.

For both sets, the first group of vertical blue lines marks the R1-R2
boundary, with each line representing a given $\sigma_{i}$. The NOV set shows
a steep composition gradient at the R1-R2 boundary, and the R1-R2 location is
nearly the same for all $\sigma_{i}$. There is greater variance in the R1-R2
location for the OV set. Further, core overshooting has softened the
$\mathrm{{}^{16}O}$ and $\mathrm{{}^{12}C}$ gradients, and the disruption of
the profiles’ regularity with $\sigma_{i}$ continues into the start of the R2
region. At $r/R\sim 0.6$, the proportionality of $\sigma_{i}$ to the
$\mathrm{{}^{12}C}$ and $\mathrm{{}^{16}O}$ profiles is restored.

By design from stopping at the first thermal pulse, the R3 and R4 regions are
almost identical between the NOV and OV sets. These regions are least affected
from mixing processes in the core (e.g. overshooting).

In Figures 1 and 2, the OV chemical profiles show a non-constant structure
from overshooting during CHeB in the O dominated central core (below
$\simeq$0.4 $\mathrm{M}_{\odot}$). While element diffusion is included during
the white dwarf cooling phase, these chemical profiles may be further
flattened by mixing processes not considered in this study such as time-
dependent convection (Jermyn et al., 2023), rotationally induced mixing,
semiconvection, thermohaline mixing, or first-order phase separation of the CO
mixture (Bauer, 2023).

###  3.2 Evolutionary differences after the main-sequence

How do the final WD profiles for the NOV and OV sets in Figure 2 relate to
their respective CHeB evolution histories? Figure 3 shows the Kippenhahn
diagrams for the $\sigma$ = 0.0 models for NOV (left) and OV (right). This
figure shows the CHeB phase until the $log(L/\mathrm{L}_{\odot})>3.0$
termination point, spanning $\simeq$ 0.93–1.10 Gyr. During this period the
total mass of our models is $\simeq$ 2.1 $\mathrm{M}_{\odot}$, but we show
only the innermost $\simeq$ 0.65 $\mathrm{M}_{\odot}$to capture the evolution
history that ultimately defines the CO WDs.

There are immediate differences between the NOV and OV CHeB evolution
histories for the $\sigma=0.0$ models. These differences are similar for any
given $\sigma_{i}$ models, and a link to an interactive figure is provided in
the online journal to see each rate’s OV vs. NOV comparison in greater detail.

For the NOV set, we see gradual growth of the convective core throughout the
CHeB phase; the noted central mass fraction isotopes smoothly deplete/grow to
reach their final mass fractions; the convective cores have no apparent
splitting during the CHeB phase. Further, there is a pure radiative zone
throughout the CHeB history. In comparison, the OV set shows convective cores
that ebb and flow in their extent, in a saw-tooth like manner; overshooting
extends past the inner convective core in a fairly consistent mass length; the
OV central mass fraction isotopes ebb and flow symmetrically with the mixing
phenomena at any given time.

We also see splittings of the convective core in the OV set. These splittings
were not observed in any of the NOV models during the CHeB phase. We presume
they are a result of overshoot inclusion. This introduces “pollution” to the
pureness of the radiative burning zone, which becomes the R2 region of the WD.
The pollution is seen by observing that some of the split-convection zone
surpasses the $\log(L/\mathrm{L}_{\odot})>3.0$ R2 inner edge boundary. This
boundary becomes the inner edge of R2 in the cool WDs. The amount of
convective pollution within the OV set is minor for $\sigma_{0.0}$, but varies
with $\sigma_{i}$. Figure 3 qualifies R2 as “Mostly Radiative” for the NOV set
due to localized, short-lived, subtle convective occurrences between $\simeq$
0.30–0.35 $\mathrm{M}_{\odot}$ near core He depletion energetics. Composition
profiles are less sensitive to mixing after CHeB is complete. Any convective
pollution from these brief convective periods in the NOV set are insignificant
compared to the convective pollution introduced in the OV set.

For both sets, nuclear burning primarily takes place within the convective
core. Both sets also show similar burning regions in the mantle outside the
core, in the radiative zone. Near the end of core He depletion, nuclear
burning in the core extends past the convective and overshooting core regions
in the OV set, and burns into the radiative zone. This is not seen in the NOV
set.

###  3.3 WD Adiabatic Pulsation Analysis

How do these evolutionary and WD structural differences impact the WD
$\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ reaction rate pulsation signatures?
We first stress the importance of the NOV models’ R2 pure radiative zone
during the CHeB. The trapped mode $\sigma_{i}$ signature found in C22
resonates the most with this region. We want to determine if this signature,
or any other $\sigma_{i}$ pulsation signature, exists when overshooting is
considered at the inner R2 boundary during CHeB. First we compare the NOV WD
pulsation signatures in this work to those in C22.

Figure 4: Top two: NOV set’s signature adiabatic pulsation modes shown at
$T_{\rm eff}=11,500$ (bright green), and $T_{\rm eff}=10,000$ (blue) K
respectively. The first panel shows the signal from the $g_{2}$ mode; the
second shows the signal from the $g_{6}$ trapped mode. Bottom: The kinetic
energy diagrams for all $\sigma_{i}$ at $T_{\rm eff}=11,500$ K (left) and
$T_{\rm eff}=10,000$ K (right) respectively. The green dots/lines represent
$E_{kin}$ for $\sigma_{i}>0$; grey for $\sigma_{i}<0$; black for $\sigma=0$.
The shading of color gets fainter the further away from $\sigma=0$.  Figure 5:
Pulsation periods as function of $T_{\rm eff}$ for the NOV (black) and OV
(grey) model sets.

###  3.4 NOV set vs. C22

In this section we briefly describe the main differences between the NOV and
C22 models. The models in C22 used a 30 isotope chemical network compared to
the 23 isotope network used here. See Appendix B for a comparison. Also, the
temporal resolution was greater in C22, especially through CHeB. The most
important difference in the NOV models is that we terminated the evolution
prior to the first thermal pulse; the models in C22 continued the evolution
through the thermal pulse phase of evolution. The overall composition
structure of the R1 and R2 regions in our NOV models are quite similar to
those in C22.

The NOV set of models in this work found two WD g-mode signals for
$\sigma_{i}$ rather than one. This is shown in the top two panels of Figure 4.
Both panels show snapshots of the percent period differences as a function of
$\sigma_{i}$, at $T_{\rm eff}=11,500$ K (bright green) and $T_{\rm
eff}=10,000$ K (blue) respectively. The y-axis label defines the period
differences as $(P_{\sigma_{0}}-P_{\sigma_{i}})/P_{\sigma_{0}}$. That is, they
are normalized to the pulsation periods of the $\sigma=0$ NOV model. The first
panel is the signal from $g_{2}$ and the second is the signal from $g_{6}$. In
C22, the g-mode signature was a trapped mode. Trapped modes are identified
from local minima in the kinetic energy diagram (Winget et al., 1981; Brassard
et al., 1991). The NOV kinetic energy diagrams for all $\sigma_{i}$ at these
snapshots are shown in the bottom left and right panels of Figure 4, following
Equation 2 in C22 (Unno et al., 1989; Córsico et al., 2002). The figure
caption explains the coloring for $\sigma_{i}$. At $T_{\rm eff}=11,500$ K
(bottom left panel), the first apparent trapped mode occurs at $g_{6}$ for all
$\sigma_{i}$, with the exception of $\sigma=0.5$, which has its first local
minimum of $E_{kin}$ at $g_{5}$. By $T_{\rm eff}=10,000$ K (bottom right
panel), all $\sigma_{i}$ have the first local minimum in $E_{kin}$ at $g_{6}$,
including $\sigma=0.5$. This is important as $g_{6}$ is one of our signature
modes for $\sigma_{i}$. These findings are in overall agreement with C22. The
trapped $g_{6}$ mode signature is not linear with $\sigma_{i}$, but overall
shows $\sigma_{i}<0$ to have longer periods than $\sigma=0.0$, and
$\sigma_{i}>0$ to have shorter periods than $\sigma=0.0$. The R2 contribution
to the $g_{6}$ period in our NOV models was $\sim 25$%. Other regions equally
contributed between $\sim 20-30$%, meaning that the trapped mode from our NOV
set is more equitably trapped among the four regions. Thus, its credibility
from R2 isn’t as strong as in C22. Nonetheless, it is not a negligible
contribution and can still serve as a viable probe for $\sigma_{i}$.

Our other g-mode signal, $g_{2}$, does not appear to be trapped by definition
(see other highlighted mode in bottom of Figure 4). However, the $g_{2}$
period differences are directly proportional to $\sigma_{i}$ (first panel of
Figure 4). This suggests that $g_{2}$ is likely distinguishing CO features in
the inner regions better than other g-modes. The additional $g_{2}$ signal was
either recovered or contrived as a consequence of excluding the thermal pulse
history in the evolution. This was the only procedural difference between our
models and those in C22. The direct impact of this procedural difference is
expressed by the nearly uniform $\mathrm{{}^{12}C}$ and $\mathrm{{}^{4}He}$
profiles after the C$\rightarrow$He transition (see Figure 1). C22 showed
variations in these profiles that stemmed from variations in the thermal pulse
histories. Eliminating such chemical variations near the R2-R3 interface can
placate the g-modes’ sensitivity to the R3 and R4 regions, especially for low-
order g-modes such as $g_{2}$. Figure 9 in C22 shows $g_{2}$ distinguishes
$\sigma_{i}$ in their thinner atmosphere sequence of models. Thinner
atmospheres may also lessen sensitivities to outer regions, allowing lower-
order g-modes like $g_{2}$ to probe deeper into the CO interior. We therefore
suspect $g_{2}$ is a viable probe for $\sigma_{i}$ if there are uniform
composition profiles at the R2-R3 boundary, and/or thinner WD atmosphere
models.

We conclude that our NOV pulsation signature results are overall consistent
with C22; we find certain low-order adiabatic WD g-modes which probe the
$\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ reaction rate probability
distribution function. With our two signature modes established, we now
discuss the impact that overshoot inclusion has on these pulsation signatures.

###  3.5 Detailed Analysis of Differences

We first show the pulsation periods as a function of surface temperature for
all $\sigma_{i}$ models in Figure 5. Black dots mark the NOV periods and grey
dots mark the OV periods. G-modes with radial orders $n=1-10$ are annotated,
all for $\ell$ = 1. Figure 5 shows that there are differences in the periods
between the NOV and OV sets, but there is no global systematic offset; the
differences between the OV and NOV periods for any given g-mode is random.
This is the case even when $\sigma_{i}$ is constant. We find that $g_{6}$
shows the largest spread in the periods of the models. Further, the kinetic
energy diagrams for all models show that $g_{6}$ was a trapped mode by $T_{\rm
eff}=10,000$ K for every model, regardless of the $\sigma_{i}$, NOV/OV
prescription. Since $g_{6}$ is one of the signals for $\sigma_{i}$ in the NOV
models, we point out this feature in Figure 5. We will touch on the cause of
the larger spread later, but now focus our attention on the detailed pulsation
properties of the signature $g_{2}$ and $g_{6}$ modes.

Figure 6: Top to bottom: Mass fractions of 12C and 16O; $B$ profile;
normalized weight function profile for the g6 mode; normalized weight function
profile for the g2mode. The left column shows the NOV results and the right
column shows the OV results. Both figures are for $\sigma=0.0$, $T_{\rm
eff}=10,000$ K. The R1-R4 region boundaries are indicated by dashed vertical
lines. An interactive figure is provided in the online version. Its
functionality compares the NOV and OV diagrams, as structured in this figure,
for any given $\sigma_{i}$ $\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ reaction
rate. To use the interactive figure, click on the ‘sigma’ slider at the top,
and slide through indices 0 through 12. Sliding through each index provides
the user the side by side comparison of the diagrams for the respective
$\sigma_{i}$ $\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ reaction rate used in
the evolution. Index 0 starts at $\sigma_{-3.0}$ and index 12 ends at
$\sigma_{3.0}$.

Figure 6 shows, from top to bottom, the mass fraction profiles, $B$, and the
$g_{6}$ and $g_{2}$ mode weight functions $\zeta$ for the final WDs at $T_{\rm
eff}=10,000$ K. The left and right columns are the NOV and OV results
respectively. Here, we show the comparison for $\sigma=0.0$, but an
interactive figure link is provided in the online article to compare these
properties for any $\sigma_{i}$. For all $\sigma_{i}$, NOV/OV comparisons, the
dotted vertical lines mark the region boundary locations in each panel. This
is useful to compare where the boundary locations are across multiple profile
properties. For instance, the R1-R2 boundary marks the C$\rightarrow$O
transition region, the first most prominent peak in $B$, and the first peak-
like features in $g_{6}\ \zeta$ and $g_{2}\ \zeta$ in the NOV case. Comparing
the OV column to the NOV column, we see the global impacts from overshooting.
Overall, prominent features in the NOV set are lessened in magnitude in the OV
set. The C$\rightarrow$O transition is more gradual, lessening the composition
gradient at the defined boundary. This remarkably impacts the shape of $B$.
The first prominent peak after max(O) is much less in magnitude for all
$\sigma_{i}$, and is not the only outstanding peak near the boundary. There
are now multiple, smaller peaks in $B$ and the $g_{6}\ \zeta$ near the R1-R2
boundary as opposed to one. There are slight deviations between NOV and OV in
these profiles for the R3 and R4 regions of the WD, but the R1$\rightarrow$R2
region in these profiles was affected most.

The $g_{6}\ \zeta$ and $g_{2}\ \zeta$ panels in Figure 6 note the weight
percentages per region in the WD. This tells each region’s contribution to the
overall mode period (frequency). An interesting result for all $\sigma_{i}$ is
that both the $g_{2}$ and $g_{6}$ modes decrease the amount of weight in R1
when overshoot is included, and increase the amount of weight in R2. There is
also a slight decrease in the weight of R3 for $g_{2}$ for all $\sigma_{i}$
when overshoot is included. These results are important. The R2 region is the
most reliable region in terms of extracting the $\sigma_{i}$ rate signature.
When overshoot is included, the R2 contribution to the overall pulsation modes
in $g_{2}$ and $g_{6}$ are accentuated, implying that these modes more
reliably distinguish $\sigma_{i}$ than the NOV set. A quantitative analysis of
each region’s weight percentage contribution per $\sigma_{i}$ is given for
both sets in Table 1 and Table 2 for $g_{2}$ and $g_{6}$ respectively.
Overall, Table 1 shows that R2 and R3 are the most heavily weighted regions
for $g_{2}$’s period. $G_{6}$ has more equitable weight dispersed across
regions, but the combined weight of R1 and R2 accounts for $\sim 50\%$ of the
$g_{6}$ period for any given model. As identified in Figure 2 and Figure 6, R1
and R2 are the most impacted regions in this study. A g-mode with about half
its weight from those regions may pick up the detailed differences more so
than modes weighted more in outer regions. This may explain why Figure 5 shows
a larger spread in the $g_{6}$ periods as this g-mode is likely picking up the
R1 and R2 contributions to its period better than other g-modes.

Table 1: $g_{2}$ Weight Function Percentages Per WD Region  | R1 | R2 | R3 | R4
---|---|---|---|---
$\sigma_{i}$ | NOV | OV | NOV | OV | NOV | OV | NOV | OV
-3.0 | 0.91 | 0.75 | 40.6 | 41.3 | 57.0 | 56.4 | 1.47 | 1.47
-2.5 | 1.14 | 0.99 | 40.2 | 44.2 | 57.2 | 52.9 | 1.43 | 1.94
-2.0 | 1.05 | 0.52 | 40.2 | 41.1 | 57.2 | 56.9 | 1.54 | 1.53
-1.5 | 1.18 | 0.53 | 39.5 | 41.7 | 57.9 | 56.2 | 1.50 | 1.50
-1.0 | 1.16 | 0.27 | 40.4 | 41.5 | 56.9 | 56.8 | 1.48 | 1.46
-0.5 | 1.15 | 0.18 | 38.8 | 42.1 | 58.6 | 56.3 | 1.43 | 1.49
0.0 | 1.25 | 0.38 | 40.6 | 42.0 | 56.6 | 56.1 | 1.52 | 1.47
0.5 | 1.44 | 0.49 | 40.8 | 41.9 | 56.2 | 56.2 | 1.52 | 1.47
1.0 | 1.28 | 0.31 | 40.4 | 41.4 | 56.9 | 56.7 | 1.49 | 1.58
1.5 | 1.32 | 0.28 | 39.9 | 41.4 | 57.2 | 56.8 | 1.50 | 1.51
2.0 | 1.35 | 0.19 | 39.4 | 40.8 | 57.8 | 57.5 | 1.50 | 1.49
2.5 | 1.25 | 0.42 | 38.3 | 41.6 | 58.9 | 56.6 | 1.47 | 1.45
3.0 | 1.39 | 2.06 | 40.2 | 39.6 | 56.9 | 56.8 | 1.59 | 1.52
Table 2: $g_{6}$ Weight Function Percentages Per WD Region  | R1 | R2 | R3 | R4
---|---|---|---|---
$\sigma_{i}$ | NOV | OV | NOV | OV | NOV | OV | NOV | OV
-3.0 | 25.5 | 20.1 | 25.6 | 32.4 | 21.1 | 19.8 | 27.8 | 27.8
-2.5 | 33.1 | 19.1 | 29.5 | 33.5 | 13.1 | 20.2 | 24.2 | 24.2
-2.0 | 32.3 | 16.6 | 30.8 | 36.3 | 13.9 | 19.7 | 23.0 | 23.0
-1.5 | 33.5 | 17.3 | 29.6 | 39.1 | 12.6 | 17.3 | 24.4 | 24.4
-1.0 | 33.8 | 13.4 | 30.0 | 43.1 | 12.9 | 17.4 | 23.3 | 23.3
-0.5 | 33.5 | 11.7 | 29.8 | 47.5 | 12.8 | 14.9 | 23.9 | 23.9
0.0 | 33.2 | 15.4 | 28.9 | 42.8 | 12.0 | 15.5 | 25.9 | 25.9
0.5 | 26.6 | 16.4 | 22.5 | 41.0 | 13.8 | 14.0 | 37.1 | 37.1
1.0 | 31.2 | 14.1 | 27.1 | 43.8 | 12.4 | 16.1 | 29.3 | 29.3
1.5 | 32.2 | 13.7 | 27.4 | 46.7 | 12.2 | 14.7 | 28.3 | 28.3
2.0 | 25.5 | 11.7 | 23.0 | 48.1 | 14.1 | 14.3 | 37.3 | 37.3
2.5 | 30.9 | 14.2 | 28.0 | 42.5 | 12.5 | 13.8 | 28.6 | 28.6
3.0 | 30.1 | 32.0 | 25.5 | 26.2 | 12.4 | 13.8 | 32.0 | 32.0

When an integer multiple $q$ of the local radial wavelength $\lambda_{r}$ for
a given g-mode nearly matches the width of a certain region(s) in a star, the
g-mode resonates with that region(s). Figure 7 shows
$q\cdot\lambda_{r}(R_{\odot})$ as a function of radius $R\ (R_{\odot})$ for
the $g_{2}$ and $g_{6}$ modes. The NOV set doesn’t show any particular close
matches for any region. But the closest matches to the R2 width were the
$\lambda_{r}$ curves of $g_{2},q=1$, and $g_{6},q=2$. Further, the $g_{2},q=2$
and $g_{6},q=3$ modes were best at resonating with R3. Larger $q$ values may
show stronger resonance with R4. The resonance with R2 is enhanced in the OV
set. The $g_{2},q=1$ and $g_{6},q=2$ $\lambda_{r}$ curves match much more
closely to the R2 width in the OV set. This implies that overshoot has
enhanced the g-mode resonance for our signature modes in the region that was
constructed mainly from radiative burning (Figure 3). We also see stronger
resonance within the R1 region with the $g_{2},q=1$ $\lambda_{r}$ curve.

Figure 7: Integer multiples of the local radial wavelengths
$q\cdot\lambda_{r}$ for $g_{2}$ and $g_{6}$ as a function of the star’s radius
$R$. Mode resonance occurs when $q\cdot\lambda_{r}$ closely matches the width
of a certain region(s) in the star. The left panel is the NOV set’s mode
resonance and the right panel is the OV set’s mode resonance. In both panels,
the black horizontal lines mark the respective region widths. Blue curves show
$q\cdot\lambda_{r}$ for $g_{2}$ and maroon curves show $q\cdot\lambda_{r}$ for
$g_{6}$. The $q$ values are stated in the legend.

Figure 8:  Adiabatic g2 (left) and $g_{6}$ (right) mode signatures for the NOV
and OV sets, at $T_{\rm eff}=11,500$ (bright green) and $T_{\rm eff}=10,000$ K
(blue) respectively

.

Will the differences between the NOV and OV sets in Figure 6 impact the WD
$\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ $\sigma_{i}$ pulsation signatures
shown in Figure 4? Figure 8 shows the resulting relative period percent
differences, as a function of $\sigma_{i}$ at $T_{\rm eff}=11,500$ K (bright
green) and $T_{\rm eff}=10,000$ K (blue). The period differences are negative
for $\sigma_{i}$ with longer periods than the $\sigma=0$ model, and are
positive for $\sigma_{i}$ with shorter periods than the $\sigma=0$ model for
the given NOV or OV set. The left of this figure shows the period differences
for $g_{2}$, and the right shows the period differences for $g_{6}$. The NOV
set is indicated by the dotted lines and the OV set is the solid lines.

Looking at $g_{2}$, the period differences between NOV and OV at $T_{\rm
eff}=11,500$ K are minimal; both sets show a trend of decreasing period with
increasing $\sigma_{i}$. At $T_{\rm eff}=10,000$ K, the OV set shows an
overall decrease in the percent differences, and a slightly greater variation
in the overall $\sigma_{i}$ vs. $g_{2}$ period difference shape. However, at
both temperatures, the same pattern of the $g_{2}$ period decreasing with
increasing $\sigma_{i}$ is sustained with overshoot inclusion. Further, the
magnitude of percent differences, ranging from $\simeq$ $-$1.5 to +1.0, is
within the detectable threshold (Chidester et al., 2021).

The OV set shows greater deviation from the NOV line of period percent
differences in $g_{6}$ more-so than $g_{2}$. This is most likely because
$g_{6}$ is more sensitive to changes from R1 than $g_{2}$. Nonetheless,
despite the $\sigma_{=-0.5}$ and $\sigma_{+1.0}$ outliers, the overall trend
remains: $\sigma_{i}<0$ generally have longer periods than $\sigma_{0}$ and
$\sigma_{i}>0$ generally have shorter periods than $\sigma_{0}$. Once again,
the magnitude of the relative period percent differences surpass the
observable threshold.

An interesting note is that for both $g_{2}$ ad $g_{6}$ signals, the percent
differences change more in the NOV set as the models cool from $T_{\rm
eff}=11,500$ to $T_{\rm eff}=10,000$ K than the OV set. The OV set showed
nearly the same period differences at both temperatures.

##  4 Discussion

C22 found pulsation signature(s) for the experimental
$\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ reaction rate probability
distribution function. They describe four sensitivities that may impact this
result: width of the O$\rightarrow$C transition, mixing during CHeB, thermal
pulse history on the AGB, and the $3\alpha$ reaction rate. This work
investigated the impact that overshoot inclusion had on the
$\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ reaction rate pulsation signature(s).
Doing so, we address the width of the O$\rightarrow$C transition and mixing
during CHeB. Further, by ignoring the thermal pulse history in our models, we
also address the sensitivity to the number of thermal pulses, albeit, the
trivial case when the number of thermal pulses is zero. In the following
paragraphs, we discuss how these three sensitivities impacted our results. We
further caution how our results could be impacted from further sensitivity
investigations.

Including overshooting overall increased the width of the O$\rightarrow$C
transition for all $\sigma_{i}$ cool WDs. This lessened the sharp peak in $B$
at the O$\rightarrow$C transition, and decreased the peak in $g_{6}\ \zeta$ at
the O$\rightarrow$C transition. While the transition peak was lessened and
dispersed into R2, widening the O$\rightarrow$C transition shows an
enhancement of both the weight contribution to the R2 region for $g_{2}$ and
$g_{6}$, and the R2 resonance with $\lambda_{r}$ for $g_{2}$ and $g_{6}$. The
widening of the O$\rightarrow$C transition was from the combined effects of
overshoot inclusion and the $\sigma_{i}$ prescription. We conclude that
widening the O$\rightarrow$C transition imposes differences in $B$, $\zeta$,
and the pulsation periods. Despite these changes, we still find the $g_{2}$
and $g_{6}$ relative period differences in the NOV and OV sets to distinguish
the $\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ reaction rate probability
distribution function. Namely, the pattern of decreasing period with
increasing $\sigma_{i}$ persisted in both NOV and OV sets. By itself, the
inclusion of overshooting does not destroy the seismic signatures of the
$\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ reaction rate in our WD models –
which was the primary question of this study.

We caution that increasing (decreasing) the width of the O$\rightarrow$C
transition in CO WD models could potentially yield different results. Our CO
WD models were informed from their evolution history, with the stated model
parameters. Thus, an increase (decrease) of the width of the O$\rightarrow$C
transition may come from choosing different mixing processes, prescriptions
and parameters, such as for convection and overshooting. A change in the width
of the O$\rightarrow$C transition may also come from mixing processes not
considered in this study such as time-dependent convection (Jermyn et al.,
2023), rotationally induced mixing, semiconvection, thermohaline mixing, or
first-order phase separations of the CO mixture (Bauer, 2023).

Ignoring the thermal pulse history gave an additional low-order adiabatic
g-mode signature for $\sigma_{i}$, namely the $g_{2}$ signal. This signal was
not found in C22, where the thermal pulse history was included. Future studies
on the thermal pulse phase of evolution with different temporal and spatial
resolutions are needed to determine the sustainability of the $g_{2}$ signal
as a probe for $\sigma_{i}$. Concurrently, future studies could also explore
the interaction, if any, between the thermal pulses and overshooting during
CHeB on the chemical profiles.

The CO cores of WDs are the result of the competition between $3\alpha$ and
$\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ during CHeB. An experimental
$3\alpha$ reaction rate probability distribution function, similar to the
existing one for $\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ (deBoer et al.,
2017; Mehta et al., 2022; Farag et al., 2022, C22), does not yet exist to our
knowledge, although a probability distribution function could be constructed
using the STARLIB reaction rate library (Sallaska et al., 2013; Fields et al.,
2016, 2018). Future studies involving both reaction rate probability
distribution functions could probe properties of DAV WD models in the
$3\alpha$ rate - $\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ rate plane. For
example, the $3\alpha$ reaction rate is likely to slowly modulate the central
16O mass fraction at any $\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ reaction
rate because $3\alpha$ controls the production of 12C. The
$\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ reaction rate will likely modulate
the central 16O mass fraction more strongly at any $3\alpha$ reaction rate. We
speculate that the radiative region R2 will exist in all such models. We also
suspect that all such models, whether terminated at the first thermal pulse or
evolved through the thermal pulse phase, will show a trapped mode, with
substantial trapping from R2, that best probes the
${}^{12}C(\alpha,\gamma)^{16}O$ burning reaction rates (i.e. $g_{6}$ in this
work, and see Figure 9 in C22). We caution that the relative period shifts we
find in this work from considering the
$\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ probability distribution and
overshooting may change when a 3$\alpha$ reaction rate probability
distribution function is also considered.

De Gerónimo et al. (2017) found that including overshooting impacted ensuing
WD pulsations by $\sim 2-5$ s. Their results were independent of their
$\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ reaction rate uncertainty evaluation.
We combined the effects of overshooting and the
$\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ reaction rate sensitivities in our
pulsation analysis, and likewise find period differences of similar
magnitudes. Our $\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ reaction rate
analysis spanned the current experimental probability distribution function,
which analyzed different rate values than those explored in De Gerónimo et al.
(2017). They concluded that the $\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$
uncertainty was less relevant than overshooting. In this study, we find that
the combined effects from overshooting and the
$\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ reaction rate probability
distribution function yields remarkable differences in the structure of the CO
WDs, and pulsation differences. Despite these differences, we still find
pulsation signatures for $\sigma_{i}$.

##  5 Summary

We conducted a search for signatures of the current experimental
$\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ reaction rate probability
distribution function in the pulsation periods of CO WD models with the
inclusion of overshooting. We found two signature adiabatic g-modes that show
period differences with the reaction rate probability distribution function
$\sigma_{i}$ trend regardless of whether or not overshoot is included. We find
a $g_{2}$ period difference signature is inversely proportional to
$\sigma_{i}$. Without overshoot, the $g_{2}$ relative period differences span
$\pm$ 0.9%. With overshoot, the $g_{2}$ relative period differences range from
-1.33% to 0.47%. The average magnitude of the relative period differences for
$g_{2}$ were 0.46% and 0.44% respectively. The $g_{6}$ period differences were
larger in magnitude, spanning from -3.44% to 1.78% for NOV and -2.02% to 1.58%
for OV. The average magnitude of the $g_{6}$ period differences were 1.21% and
0.95% respectively. The average magnitudes of the $g_{2}$ and $g_{6}$ period
differences were slightly decreased from the NOV set.

We found that the R2 weight contribution to these g-modes was enhanced with
overshoot inclusion. The R2 region remains the best identifying region for
tracing the $\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ reaction rate probability
distribution function. This is because even with overshoot inclusion, it is
predominantly constructed by radiative burning during CHeB. Regardless of
whether or not overshooting is considered, we find:

  1. 1.

two signature g-modes, $g_{2}$ and $g_{6}$ probe $\sigma_{i}$

  2. 2.

$g_{2}$ is inversely proportional to $\sigma_{i}$ and $g_{6}$ is a trapped
mode

  3. 3.

the $g_{2}$ and $g_{6}$ periods are generally shorter for positive
$\sigma_{i}$ and longer for negative $\sigma_{i}$

  4. 4.

both signatures have period deviations within the detectable regime

These findings suggest that an astrophysical constraint on the
$\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ reaction rate probability
distribution function remains, in principle, extractable from the period
spectrum of observed variable WDs.

##  6 Acknowledgements

We thank James Deboer for sharing the 12C$(\alpha,\gamma)^{16}$O probability
distribution function, Josiah Schwab for sharing wd_builder, and Pablo
Marchant for sharing mkipp. We acknowledge using ChatGPT (OpenAI, 2023) to
polish the language of one paragraph (Vishniac, 2023). This research is
supported by NASA under the Astrophysics Theory Program grant
NNH21ZDA001N-ATP, and in part by the National Science Foundation under Grant
No. NSF PHY-1748958. This research made extensive use of the SAO/NASA
Astrophysics Data System (ADS).

##  Appendix A Microphpysics in MESA

The MESA EOS is a blend of the OPAL (Rogers & Nayfonov, 2002), SCVH (Saumon et
al., 1995), FreeEOS (Irwin, 2004), HELM (Timmes & Swesty, 2000), PC (Potekhin
& Chabrier, 2010), and Skye (Jermyn et al., 2022) EOSes.

Radiative opacities are primarily from OPAL (Iglesias & Rogers, 1993, 1996),
with low-temperature data from Ferguson et al. (2005) and the high-
temperature, Compton-scattering dominated regime by Poutanen (2017). Electron
conduction opacities are from Cassisi et al. (2007) and Blouin et al. (2020).

Nuclear reaction rates are from JINA REACLIB (Cyburt et al., 2010), NACRE
(Angulo et al., 1999) and additional tabulated weak reaction rates Fuller et
al. (1985); Oda et al. (1994); Langanke & Martínez-Pinedo (2000). Screening is
included via the prescription of Chugunov et al. (2007). Thermal neutrino loss
rates are from Itoh et al. (1996).

##  Appendix B Model Optimization and Resolution

###  B.1 Reduced Chemical Network

The nature of our evolutionary models is computationally expensive. This paper
is concerned about overshooting and the
$\mathrm{{}^{12}C(\alpha,\gamma)^{16}O}$ reaction rate probability
distribution function, which primarily dictate the evolutionary processes and
consequences of the CHeB phase. The isotopes most impacted during CHeB are
$\mathrm{{}^{12}C}$, $\mathrm{{}^{16}O}$, and $\mathrm{{}^{4}He}$.
$\mathrm{{}^{14}N}$ and $\mathrm{{}^{20}Ne}$ are the next two most impacted
isotopes during CHeB. We thus optimize the efficiency of our models by
reducing the chemical network number of isotopes from 30 to 23. The eliminated
isotopes are 21Ne, 21,22,23Na, 23,24Mg, and 56Fe. A comparison of the
resulting inner mass fraction profiles for the 5 most abundant isotopes for
both networks is shown in Figure 9 for each chemical network. This figure
shows the profiles at the completion of CHeB. both network models used the
same temporal and spatial resolution during CHeB. The run-time was reduced
from a few days to a a few hours on 12 cores. All resolution studies were
conducted with $\sigma=0.0$ without overshoot (NOV).

Reducing the network impacted $\mathrm{{}^{22}Ne}$ most, with an offset of
$\sim 22\%$ more $\mathrm{{}^{22}Ne}$ in the 23 isotope network. We note that
C22 used a 30 network and our overall signature results persistent through
variations in heavier isotopes.

Figure 9: Mass fraction profiles for at the completion of CHeB for a 30
isotope (solid) and 23 isotope nuclear reaction network (dotted). Shown are
the 5 most abundant isotopes for both networks.

###  B.2 Temporal Resolution

Several timestep limiters in MESA help optimize convergence studies. In this
paper, we want to limit the timestep to achieve the temporal resolution that
yields a smooth evolution of the central $\mathrm{{}^{4}He}$,
$\mathrm{{}^{16}O}$, and $\mathrm{{}^{12}C}$ abundances during CHeB. We first
utilize the delta_XC_cntr_limit limiter. This limits the amount the central
$\mathrm{{}^{12}C}$ abundance can change in a given timestep. To help optimize
computational run-time, we begin limiting the change in central
$\mathrm{{}^{12}C}$ during CHeB which the central helium abundance
X$(\mathrm{{}^{4}He}_{c})<0.6$. This is done by adding the following lines of
code in the MESA run_star_extras.f90 file:

  * if ((s% center_h1$<$1d-6).and.(s% center_he4 $<$ 0.6).and.(s% delta_XC_cntr_limit$>$0.001))then

  * s% delta_XC_cntr_limit = 0.0005d0

  * end if

This temporal resolution was used for the 30 and 23 isotope network models. We
refer to it as resolution A. The remaining temporal resolution studies were
performed using the 23 isotope chemical network.

The next iteration of increased temporal resolution modified the
run_star_extras.f90 file to include the following:

  * if ((s% center_h1$<$1d-6).and.(s% center_he4 $<$ 0.5).and.(s% delta_XC_cntr_limit$>$0.001))then

  * s% delta_XC_cntr_limit = 0.0005d0

  * s% delta_lgT_cntr_limit = 5d-4

  * s% delta_lgT_cntr_hard_limit = 1d-3

  * s% delta_lgRho_cntr_limit = 1d-3

  * s% delta_lgRho_cntr_hard_limit = 5d-3

  * end if

This resolution is employed slightly earlier during CHeB, when
X$(\mathrm{{}^{4}He}_{c})<0.5$. We added limits to the change in central
temperature and density from resolution A. This is resolution B.

Our third resolution iteration used the following limiter controls in the
run_star_extras.f90 file:

  * if ((s% center_h1$<$1d-6).and.(s% center_he4 $<$ 0.999).and.(s% delta_XC_cntr_limit$>$0.001))then

  * s% delta_XC_cntr_limit = 0.00025

  * s% delta_XO_cntr_limit = 0.00025

  * s% delta_lgT_cntr_limit = 2.5d-4

  * s% delta_lgT_cntr_hard_limit = 0.5d-3

  * s% delta_lgRho_cntr_limit = 0.5d-3

  * s% delta_lgRho_cntr_hard_limit = 2.5d-3

  * end if

This is resolution C. We have set the limiters at the start of CHeB, and have
decreased the limiter values from those in resolution B.

A comparison for resolutions A, B, and C are shown in Figure 10. In each
column, the solid light curves represent resolution A, the dotted curves B,
and the dark solid curves C.

The left figure shows the evolution of central abundances of
$\mathrm{{}^{4}He}$, $\mathrm{{}^{12}C}$, and $\mathrm{{}^{16}O}$ during CHeB,
starting when X$(\mathrm{{}^{4}He}_{c})\lesssim 0.6$ until the completion of
CHeB. The central abundances for resolutions A and B are nearly identical.
Resolution C varies slightly, with the final central $\mathrm{{}^{16}O}$
abundance reaching a slightly larger amount than resolutions A and B. Further,
all three resolutions show a smooth evolution of these central abundances
throughout CHeB.

Figure 10: Left: Evolution of central mass fractions during CHeB until
$\log(L/\mathrm{L}_{\odot})$ = 3.0 at three resolutions. Middle: Mass fraction
profiles at $\log(L/\mathrm{L}_{\odot})$ = 3.0 for the three resolutions.
Right: Middle figure zoom-in to show differences.

The middle plot in Figure 10 shows the mass fraction profiles at the
completion of CHeB. We show the 5 most abundant isotope profiles for each
resolution. The $\mathrm{{}^{12}C}$ and $\mathrm{{}^{16}O}$ profiles for A are
noticeably different than the profiles for B and C, especially after the
O$\rightarrow$C transition. This is more apparent in the right plot of Figure
10, which zooms in on the $\mathrm{{}^{16}O}$ and $\mathrm{{}^{12}C}$ profiles
of the three resolutions. Resolution B follows A in the core, but then more
closely aligns with C after the O$\rightarrow$C transition. Since resolutions
B and C agree well, with only a slight difference in the central
$\mathrm{{}^{12}C}$ and $\mathrm{{}^{16}O}$ abundance, we set resolution C as
the standard temporal resolution for our 13 models.

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