# Hydrodynamical Simulations Favor a Pure Deflagration Origin of the Near-
Chandrasekhar Mass Supernova Remnant 3C 397

Vrutant Mehta1 , Jack Sullivan1,2, Robert Fisher1 , Yuken Ohshiro3, Hiroya
Yamaguchi 3, Khanak Bhargava4,5, and Sudarshan Neopane6
1Department of Physics, University of Massachusetts Dartmouth, 285, Old
Westport Road, North Dartmouth, 02740, USA
2 Department of Physics, University of Connecticut, 196A Auditorium Rd Unit
3046, Storrs, CT 06269, USA
3Department of Physics, Graduate School of Science, The University of Tokyo,
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
4Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY
11794-3800, USA
5Institute for Advanced Computational Science, Stony Brook University, Stony
Brook, NY 11794, USA
6Department of Physics and Astronomy, University of Tennessee Knoxville, 401
Nielsen Physics Building, 1408 Circle Drive, Knoxville TN 37996-1200, USA
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Suzaku X-ray observations of the Type Ia supernova remnant (SNR) 3C 397
discovered exceptionally high mass ratios of Mn/Fe, Ni/Fe, and Cr/Fe,
consistent with a near $M_{\rm Ch}$ progenitor white dwarf (WD). The Suzaku
observations have established 3C 397 as our best candidate for a near-$M_{\rm
Ch}$ SNR Ia, and opened the way to address additional outstanding questions
about the origin and explosion mechanism of these transients. In particular,
subsequent XMM-Newton observations revealed an unusually clumpy distribution
of iron group elemental (IGE) abundances within the ejecta of 3C 397. In this
paper, we undertake a suite of two dimensional hydrodynamical models, varying
both the explosion mechanism – either deflagration-to-detonation (DDT), or
pure deflagration – WD progenitors, and WD progenitor metallicity, and analyze
their detailed nucleosynthetic abundances and associated clumping. We find
that pure deflagrations naturally give rise to clumpy distributions of
neutronized species concentrated towards the outer limb of the remnant, and
confirm DDTs have smoothly structured ejecta with a central concentration of
neutronization. Our findings indicate that 3C 397 was most likely a pure
deflagration of a high central density WD. We discuss a range of implications
of these findings for the broader SN Ia progenitor problem.

###### keywords:

supernova remnants – supernovae:general – white dwarfs

††pubyear: 2023††pagerange: Hydrodynamical Simulations Favor a Pure
Deflagration Origin of the Near-Chandrasekhar Mass Supernova Remnant 3C 397–5

##  1 Introduction

Type Ia supernovae (SNe Ia) are among the most luminous and common optical
transients in the universe. SNe Ia have important ramifications across a wide
range of astrophysical domains, from galaxy formation (Gandhi et al., 2022)
and galactic nucleosynthesis (Kobayashi et al., 2020), to the Hubble tension
(Riess et al., 2016), and the nature of dark energy (Maoz et al., 2014). Yet,
we still do not know the stellar progenitor systems which give rise to SNe Ia.
We also do not understand precisely how a detonation arises within these
systems, and under what conditions detonations fail to initiate. These failed
detonation systems will lead to faint SNe Iax transient events with surviving
hypervelocity white dwarfs (WDs) (Jordan et al., 2012a; Kromer et al., 2013;
Foley et al., 2013; Raddi et al., 2018; El-Badry et al., 2023).

Galactic supernova remnants (SNRs) offer valuable insights into Type Ia
supernovae (SNe Ia) through spatially-resolved X-ray observations. These
observations complement optical and infrared studies of SNe Ia transients
(Vink, 2012). Among SNRs, 3C 397 stands out as a uniquely rich testing ground
for investigating the SN Ia progenitor problem. JAXA Suzaku X-ray observations
revealed high abundance ratios of Ni/Fe, Mn/Fe, and Cr/Fe (Yamaguchi et al.,
2015). Such high abundance ratios of iron-group elements (IGEs) point towards
a near-Chandrasekhar (near-$M_{\rm Ch}$) mass white dwarf (WD) progenitor for
3C 397, since the dense interiors of near-$M_{\rm Ch}$ WD SNe Ia undergo
efficient electron captures, producing neutronized isotopes in abundance,
including 62Ni, 55Fe (which subsequently decays to 55Mn), and 54Cr. Further
detailed nucleosynthesis from hydrodynamical simulations predicted that in
order to match the exceptionally high IGE abundance ratios observed in 3C 397,
the WD progenitor of 3C 397 either had to have either a high central density
($\rho_{c}\sim 5-6\times 10^{9}$ g cm-3), or represent a pure deflagration of
a canonical near-$M_{\rm Ch}$ WD ($\rho_{c}\sim 2\times 10^{9}$ g cm-3) often
adopted in the literature (Dave et al., 2017; Leung & Nomoto, 2018). These
predictions for a high-central density near-$M_{\rm Ch}$ WD progenitor for SNR
3C 397 gathered support in ESA XMM-Newton follow-up observations, which
uncovered the existence of clumps with high Cr/Fe abundances, indicative of
highly-neutronized burning (Ohshiro et al., 2021). However, the clumpiness of
the ejecta is in tension with the layered structure generally predicted by
deflagration-to-detonation transition (DDT) models (Gamezo et al., 2004). Both
the explosion mechanism as well as the structure of the WD progenitor of 3C
397 at ignition (which depends upon the initial WD mass, accretion rate, and
angular momentum in the single-degenerate near-$M_{\rm Ch}$ scenario) have
important implications for the SN Ia problem in general.

This successful dialog between X-ray observers and theorists lays the
foundation for the next stage of progress in understanding the origin of SNR
3C 397, and its connection to the wider SN Ia progenitor problem. While many
diverse channels have been predicted to contribute to the SN Ia rate, all
leading models for normal brightness SNe Ia share in common one physical
process: detonation. In this study, we investigate two distinct explosion
mechanisms in our simulations.

The first scenario we will investigate is the deflagration-to-detonation
transition (DDT). The DDT has been a well-explored explosion mechanism over
the past few decades in the general context of the near-$M_{\rm Ch}$ channel
of SNe Ia. In the DDT scenario, a flame bubble ignites runaway deflagration
burning inside the convective region of the near-$M_{\rm Ch}$ white dwarf, and
rises buoyantly until it transitions to a detonation. The nuclear energy
released during the detonation is sufficient to completely disrupt the white
dwarf, and typically produces a normal to bright type Ia event. The solar
abundance of the monoisotopic element 55Mn strongly suggests that near-$M_{\rm
Ch}$ SNe Ia explosions contribute at least partially to the total SNe Ia rate
(Seitenzahl et al., 2013b). Additionally, recent JWST observations of SN
2021aefx favor a DDT in a near-$M_{\rm Ch}$ WD progenitor model (DerKacy et
al., 2023).

In contrast, in the second scenario we investigate, a pure deflagration, the
flame bubble ignites close to the center of the WD and burns through the dense
core. After burning through the center of the WD, the bubble rises to the
surface without a transition into detonation. Only a fraction of the total
mass of the WD is burned; this model predicts that failed detonation
near-$M_{\rm Ch}$ events naturally produce significantly less 56Ni and ejected
mass than a DDT (Röpke et al., 2007; Long et al., 2014). The outcome is
therefore a subluminous and more rapidly declining transient than a normal
event (Jordan et al., 2012b; Kromer et al., 2013; Fink et al., 2014). These
predictions are broadly consistent with the class of subluminuous Iax events
(Foley et al., 2013). Moreover, because the initial ignition is necessarily
asymmetric, the ejecta of ash occurs preferentially along the initial ignition
axis. Of this burned mass, an even lesser fraction is ejected into the ISM,
while the rest of the ash mass remains gravitationally bound to the remnant
star, and falls back on to the surface. Because of the asymmetric ejection of
ash, the bound star gets kicked away from the ejecta with a speed of order
$f_{\rm burned}v_{\rm ej}$, where $f_{\rm burned}$ is the fraction of the star
that is burned, and $v_{\rm ej}\sim v_{\rm esc}\simeq 10^{4}$ km/s is a the
ejection velocity, approximately the escape speed from a near-$M_{\rm Ch}$ WD.
For burn fractions $f_{\rm burn}\sim 0.1$, typical of SNe Iax simulations of
CO WDs, the ejection velocity is $\sim 10^{3}$ km/s. This simple estimate is
consistent with speeds of a few hundred to approximately one thousand km s-1
observed in hypervelocity near-$M_{\rm Ch}$ SNe Iax primary WD remnants,
classified as LP 40-365 objects. A total of six such candidate hypervelocity
Iax remnants are now known, lending support to this scenario (Raddi et al.,
2018; El-Badry et al., 2023).

Previous theoretical studies (Dave et al., 2017; Leung & Nomoto, 2018) have
focused upon models for the global nucleosynthetic abundance ratios of 3C 397.
Spurred by the new ESA XMM-Newton abundance ratio measurements within
localized clumps in 3C 397, in this paper, we will calculate synthetic clump
abundance ratios. We will consider both DDT as well as pure deflagration
multidimensional hydrodynamical simulations whose global abundance ratios
match 3C 397, and compute their resulting nucleosynthetic spatially-mapped
yields. We will detail algorithms to isolate model clump analogs to the
observed clumps. These model clumps will form the basis for comparison against
3C 397, and will address key questions surrounding the SN Ia detonation
mechanism and progenitor problems: What was the explosion mechanism that
caused the SN Ia which gave rise to 3C 397: a DDT or a pure deflagration? How
does the explosion mechanism impact the mixing of the ejecta, as seen in the
XMM-Newton IGE abundances spatially mapped across the remnant? What are the
implications for the broader SN Ia progenitor problem?

In the near-$M_{\rm Ch}$ scenario, the initial deflagration shapes the ejecta,
retaining the imprint of the Rayleigh-Taylor and Kelvin-Helmholtz
instabilities (eg, Bell et al., 2004). The subsequent evolution of the SNR
also leads to the development of Rayleigh-Taylor instabilities at the
interface between the remnant and the circumstellar medium (Mandal et al.,
2023). Guided by the XMM-Newton observations of 3C 397, which clearly reveal
anisotropies in the neutronized ejecta, here we focus upon the role of the
initial deflagration in shaping the nucleosynthetic products of the explosion.

In §2, we lay out our methodology for the hydrodynamical simulations and
nucleosynthetic post-processing of the ejecta. In §3, we present our results,
including the algorithms employed for identifying model ejecta clumps to be
compared directly against the observations of (Ohshiro et al., 2021). In §4,
we discuss our findings in the context of other observations and theoretical
models. Finally, in §5, we conclude.

##  2 Methodology

Details of Models
---
Model Name | Explosion Mechanism | $\mathrm{\rho_{c}(g\ cm^{-3})}$ | Bubble Offset (km) | Bubble Radius (km) | C/O Ratio
DDT-HIGHDENS-HIGHOFF | DDT | $6.0\times 10^{9}$ | 200 | 8 | 30/70
DEF-HIGHDENS-NR-CENTER | Pure Deflagration | $6.0\times 10^{9}$ | 10 | 16 | 30/70
DEF-STDDENS-NR-CENTER | Pure Deflagration | $2.2\times 10^{9}$ | 8 | 16 | 50/50

Table 1: The table above show the configurations of the hydrodynamical models
that are presented in this paper.

We utilize the FLASH 4 (Fryxell et al., 2000b) hydrodynamical code with
adaptive mesh refinement (AMR), which facilitates high resolution in the
desired regions of the simulation domain. Our FLASH simulations capture the
nuclear burning within the flame which arises in the interior of near-$M_{\rm
Ch}$ white dwarfs in 2D axisymmetric cylindrical geometry. These near-$M_{\rm
Ch}$ models begin with a WD in hydrostatic equilibrium, parameterized by its
central density and composition (C/O ratio and stellar progenitor
metallicity). Because the initial near-$M_{\rm Ch}$ WD progenitor is
convectively unstable due to carbon burning in the core, we generate WD
initial conditions using an adiabatic WD model generator (Chamulak et al.,
2008). The WD models self-consistently include the simmering convective core
with an adiabatic temperature gradient, using the same Helmholtz equation of
state and carbon burning rate as is used in FLASH for self-consistency. Based
upon previous best fits to the global abundance ratios of Ni/Fe, Mn/Fe, and
Cr/Fe in 3C 397 (Dave et al., 2017), we generate two WD progenitors for use in
this study: one a high central density WD with central density
$\rho_{c}=6.0\times 10^{9}$ g cm-3, and the other a standard central density
WD $\rho_{c}=2.2\times 10^{9}$ g cm-3. The initial composition of the high-
central density ($\rm{6.0\times 10^{9}}$ g cm-3) WD model is 30/70 carbon-
oxygen with a central temperature $T_{c}=7.0\times 10^{8}$ K. The standard
central density ($\rm{2.2\times 10^{9}}$ g cm-3) WD model contains 50/50
carbon-oxygen to the initial composition with same Tc of $7.0\times 10^{8}$ K.
For our high central density WD model and standard central density WD model,
we adopted the background temperatures of $\rm{10^{8}}$ K and $\rm{3\times
10^{7}}$ K, respectively. The simulation begins with the mapping of the
near-$M_{\rm Ch}$ mass WD model at the center of the simulation domain with
initialization of hydrodynamic variables and passive Lagrangian tracer
particles to be used for detailed nucleosynthetic postprocessing (see below)
distributed by mass over the WD.

FLASH uses the Helmholtz equation of state to compute pressures and sound
speeds, including electrons with an arbitrary level of degeneracy and
relativity, radiation pressure, and non-degenerate nuclei. Local thermodynamic
equilibrium between the gas and the radiation is assumed throughout, so that a
single temperature applies to the electrons, positrons, ions, and radiation
field. FLASH captures the nuclear-burning flame using a three-stage
artifically-thickened flame burner, modeled as advection-diffusion reaction
equations along with an unsplit higher-order Godunov hydrodynamics solver
(Calder et al., 2007). The reactive hydrodynamics flame module contained
within FLASH has energetics and neutronization calibrated against self-heating
reaction networks, including neutronization, electron screening, and Coulomb
corrections self-consistently (Calder et al., 2007; Townsley et al., 2016).
Three separate scalar fields solving separate advection diffusion reaction
requations track carbon burning, burning to nuclear statistical quasi-
equilibrium (NSQE), and burning to nuclear statistical equilibrium (NSE).

Our choice of hydrodynamical models is guided by our previous study, in which
we matched the global abundance ratios of 3C 397 with two sets of models, one
a high central density WD undergoing a DDT, and the second a pure deflagration
of a standard central density WD (Dave et al., 2017). Table 1 shows the
initial conditions for the range of explosion models for the present study
with the mechanism, the WD central density, and the composition of the white
dwarf including initial ignition bubble parameters such as ignition point
offset from the center of the white dwarf, and initial bubble radius. This
choice of models brackets the best fits to the global yields found in Dave et
al. (2017).

In considering the pure deflagration models, the question arises as to how to
simultaneously select the ignition conditions while also varying the structure
of the progenitor. Our strategy for this set of models is to keep the ignition
parameters as relatively fixed, while varying the central density of the
progenitor WD. We do so by scaling the offset radii to a characteristic length
scale which arises when one compares the effects of buoyancy and deflagration
(Fisher & Jumper, 2015). We determine a critical boundary for this offset
radii using the equation 1.

$\mathrm{\displaystyle{\mathit{r}_{\mathrm{crit}}=19\ \mathrm{km}\
\left(\frac{\mathit{S}_{l}}{100\ \mathrm{km}\
\mathrm{s}^{-1}}\right)\left(\frac{0.09}{\mathit{A}}\right)^{1/2}\left(\frac{2.2\times
10^{9}\ \mathrm{g}\ \mathrm{cm}^{-3}}{\rho_{c}}\right)}}$ (1)

The critical offset radii is a function of laminar flame speed ($S_{l}$), the
Atwood number $A=(\rho_{f}-\rho_{a})/({\rho_{f}+\rho_{a}})$, where $\rho_{f}$
and $\rho_{a}$ are the fuel and ash densities, respectively, and the central
density $\mathrm{\rho_{c}}$. For ignition radii greater than this critical
radius, $r_{\rm crit}$, the flame bubble rapidly ascends to the surface due to
buoyancy. Conversely, when the offset distance falls below the critical
offset, the flame bubble initially expands and consumes the dense core of the
WD before buoyancy can take hold. The critical radii are $\sim$16 km for the
high central density model, and $\sim$14 km for the standard central density
model. We choose a dimensionless offset radius, $r_{0}/r_{\rm crit}=0.6$ for
the pure deflagration models, consistent with central ignitions. The
corresponding physical offset radii $r_{0}$ are 10 km for the high central
density WD and 8 km for the standard central density WD. Additionally, we also
selected an initial bubble radius of 16 km in both pure deflagration models,
and 8 km in the DDT model, all consistent with the parameters of Dave et al.
(2017). For our DDT model, we fix detonation initiation criterion at a density
of $\mathrm{2.6\times 10^{7}g\ cm^{-3}}$; a detonation is triggered once the
the flame front reaches this density.

The simulations are conducted with a 2D axisymmetric cylindrical geometry. The
$R=0$ boundary is axisymmetric by imposition, and diode boundary conditions
are employed for the remaining three sides. The domain spans 0 to $2^{17}$ km
= $131072$ km along the cylindrical $R$-axis and -131072 km to 131072 km along
$z$. The maximum linear resolution resolution is 4 km at the highest
refinement level 11, numbering the base level as 1. Our refinement criteria
are jointly based upon both the density gradient and the flame progress
variable. Regions of steep density gradients are resolved according to a
standard FLASH criterion based upon the discrete dimensionless density
gradient across cells (Lohner, 1987; Fryxell et al., 2000a). Regions of active
burning are always captured at the highest level of refinement (Townsley et
al., 2007). We fill the regions outside the WD with matter of low density with
$\mathrm{10^{-3}\ g\ cm^{-3}}$ and a temperature of $\mathrm{3\times 10^{7}\
K}$. This low density “fluff" has a negligible impact on the simulation other
than to fill in the vacuum upon the Eulerian mesh outside the star. Modeling
of 3C 397 suggests that the remnant is likely either in the later stages of
free expansion or the earlier stages of the Sedov phase (Leahy & Ranasinghe,
2016), based upon a distance range of 6.3 $\pm$ 0.1 - 9.7 $\pm$ 0.3 kpc, and
an age range of 1350 - 1750 yr, which will justify comparisons of the
morphology obtained through our models versus that observed.

Simulations incorporating a pure-deflagration explosion mechanism require an
additional step to separate the ejected material from the bound WD that is
kicked away from the ejecta. For this procedure, we employ a similar
methodology as Kashyap et al. (2018), using an energetic criterion to
delineate bound and unbound material. We calculate the specific binding energy
$e_{\rm b}$ for each particle at the last time step at the end of the
simulation, $e_{\rm b}=e_{\rm int}+\frac{1}{2}|v|^{2}+\Phi$, where $e_{\rm
int}$ is the specific internal energy, $\frac{1}{2}|v|^{2}$ the specific
kinetic energy, and $\Phi$ the gravitational potential. Negative binding
energy indicates the particle is gravitationally bound, and non-negative
binding energy implies the particle is in the ejecta. While this energetic
criterion provides a plausible estimate of the ejected mass, the general
criterion for gravitationally unbound material in multidimensional
hydrodynamics is not rigorously known, as hydrodynamical studies of common
envelope evolution have shown (Röpke & De Marco, 2023). In particular, we note
that this criterion based upon the internal energy for the bound mass
implicitly assumes that any additional mechanisms for mass loss, such as
through thermally-driven or magnetically-driven winds, can be neglected.

A further subtlety arises because the simulations do not possess a rigorously
defined homologous expansion phase during the simulation time. This issue is
particularly acute for pure deflagrations, which leave behind a bound remnant.
Broadly speaking, in pure deflagrations, the outermost region in velocity
space are ejected from the star, while the innermost ejecta will fall back
onto the WD remnant. Near the surface of zero total energy, material is at the
cusp of being unbound, and is sent out to large apocenter distances before
either falling back or becoming marginally unbound. In the results section, we
analyze the last few seconds of these simulations to assess the sensitivity of
the ejecta to the cutoff time.

The flame module suffices to capture the thermodynamics of the nuclear energy
release accurately. The nuclear energy release is crucial for the
hydrodynamical evolution, but for detailed nucleosynthetic abundances needed
to compare against SNR 3C 397, we must also utilize Lagrangian tracer
particles. 104 constant mass tracer particles are embedded within the grid-
based FLASH simulation and passively record the fluid density and temperature
as a function of time. This number of particles typically establishes the
nucleosynthetic yields within 10% for a near-$M_{\rm Ch}$ SNe Ia model
(Seitenzahl et al., 2010). In post-processing, we first restructure time
slices of particle data into Lagrangian thermodynamic temporal histories over
individual particles. This post-processing step also includes a cut on the
ejecta using the boundedness criterion identified above. In particular,
gravitationally bound particles defined by this criterion will fall back on to
the remnant, and are therefore not included in the ejecta yields included in
this paper.

These Lagrangian density-temperature trajectories are post-processed with
Torch, using a 489 isotope nuclear network (Timmes, 1999). In contrast, the
flame model employed inline in FLASH tracks the effective electron molar
fraction $Y_{e}$ along with the flame progress variables, but does not track
detailed abundances (Calder et al., 2007). The estimated 1350-1750 yr age of
3C 397 (Leahy & Ranasinghe, 2016) is sufficient for most of the unstable iron
group species to decay into stable species. However, the radioisotope 53Mn
subsequently decays into stable 53Cr by electron capture with a 3.7 Myr half
life. Similarly, the radioactive decay of 59Ni to 59Co occurs primarily
through electron capture with a $7.6\times 10^{4}$ kyr half life. In our
study, we include the parent nuclei 53Mn and 59Ni as effectively stable over
the lifetime of 3C 397. Throughout this paper, all quoted model atomic
abundances are derived from these modified decayed stable isotopes.

The ratios of the decayed stable atomic abundances obtained from our model
pipeline can then be directly compared against the ratios measured for 3C 397.
To understand the effect of metallicity, we also varied the initial
metallicity by including $\mathrm{{}^{22}Ne}$ as a proxy for the WD stellar
progenitor metallicity, as the result of reaction bottlenecks during the CNO
cycle in WD progenitors (Timmes et al., 2003). We considered same solar
metallicity ($Z_{\odot}$) from (Asplund et al., 2009) as previously used by
(Dave et al., 2017). Studies of the neutronization during the simmering phase
leading up to unstable deflagration ignition have generally found that the
effect of simmering is less significant (with neutron excesses of order
$10^{-4}$) (Piro & Bildsten, 2008; Martínez-Rodríguez et al., 2016). These
post-processed yields are aggregated into spatially mapped abundances. The
spatially-mapped abundances from the 2D simulations are crucial to
understanding the asymmetry evident in the recently resolved XMM-Newton data
of 3C 397.

##  3 Results

(a) DDT-HIGHDENS-HIGHOFF

(b) DEF-HIGHDENS-NR-CENTER

(c) DEF-STDDENS-NR-CENTER

Figure 1: Plots comparing the outcome of the simulations at end-time. Shown,
top-to-bottom, are models (a) DDT-HIGHDENS-HIGHOFF, (b) DEF-HIGHDENS-NR-
CENTER, and (c) DEF-STDDENS-NR-CENTER. Note that the pure deflagration models
have lower expansion velocities, and so are shown on smaller domains than the
DDT model. The right-side color bar represents the log of the mass density.
Additionally, each figure shows the log Fe abundance $X_{\rm Fe}$, the log Cr
abundance $X_{\rm Cr}$, and the log Cr/Fe abundance ratio $X_{\rm Cr}/X_{\rm
Fe}$ on unbound tracer particles. In order to highlight the clumps, the size
of the tracers is scaled proportional to the abundance and abundance ratios.
All models depicted have metallicity $Z=3.0Z_{\odot}$. The annotated black
dashed line contours in pure-deflagration models enclose the bound mass. The
annotated circles show the highest likelihood clumps that are identified for
all models; see Section 3.2.

(a) DDT-HIGHDENS-HIGHOFF

(b) DEF-HIGHDENS-NR-CENTER

(c) DEF-STDDENS-NR-CENTER

Figure 2: Plots comparing the distribution of electrons per baryons (Ye) among
our three models : (a) DDT-HIGHDENS-HIGHOFF, (b) DEF-HIGHDENS-NR-CENTER, and
(c) DEF-STDHDENS-NR-CENTER. Over-plotted particles represents Cr/Fe abundance
ratio on log scale. The annotated circles show groups of particles that
contains 7% of the total iron in the ejecta with the highest nucleosynthetic
likelihood to match the southern 3C 397 clump. All models depicted have
metallicity $Z=3.0Z_{\odot}$.

###  3.1 Spatial Distribution of the Neutronized Species

Figure 1 shows density slice plots of our three models. In particular, figure
1(a) showcases DDT-HIGHDENS-HIGHOFF model at the time t = 6s, figure 1(b)
displays DEF-HIGHDENS-NR-CENTER model at the time t = 8s, and figure 1(c)
illustrates the DEF-STDDENS-NR-CENTER model at the time t = 7s. The color bar
on the right side of the figures indicates density on the log scale. The
overplotted distribution of particles in the figures shows the abundance of
Fe, Cr, and the abundance ratio of Cr/Fe in the ejecta. The left column shows
Fe abundances, the center column Cr abundances, and the right column the
abundances ratios of Cr/Fe. For clarity, the particle size is scaled
proportional to the mass fraction or mass fraction ratio. The color bar below
each column shows the range of mass fractions on a log scale, with iron-rich
tracer particles in red, and chromium-rich tracers in blue.

These plots unveil the underlying connections between the ejecta morphology,
the explosion mechanisms, and the distribution of neutron-rich elements in
each model. Because the supersonic detonation propagates faster than the
hydrodynamical response time of the WD, the density profiles of the DDT-
HIGHDENS-HIGHOFF model reveal the hallmark of the DDT: a smoothly-stratified
structure of the burned material in the ejecta. This feature is highlighted
further in Figure 2, which overlays the tracer particles shaded by Cr/Fe as in
Figure 1, but now shown overplotted against the electron fraction $Y_{e}$,
defined as $Y_{e}=\sum Z_{i}X_{i}/A_{i}$, where $Z_{i}$ and $A_{i}$ are the
atomic and mass numbers of isotope $i$ with mass fraction $X_{i}$. The flame
module used in FLASH tracks $Y_{e}$ on the Eulerian mesh, and it is this field
which is plotted in Figure 2. In particular, the abundances of neutronized
regions (where $Y_{e}<0.5$) for model DDT-HIGHDENS-HIGHOFF are concentrated in
the central region with a lesser amount along the outskirts, the result of ash
ejected from the WD just prior to detonation. The clump identified as having
the highest mean Cr/Fe ratio is nearly centrally located for this model.
Furthermore, we also confirm the released kinetic energy in this DDT
simulation is $1.43\times 10^{51}$ erg, with the outer ejecta moving at
approximately 10,000 km s-1. We calculate the mass of 56Ni in the ejecta of
DDT-HIGHDENS-HIGHOFF to be 0.76 M⊙, which would roughly correspond to a bright
normal SN Ia in terms of its expected light curve.

In marked contrast, the failed detonation models DEF-HIGHDENS-NR-CENTER and
DEF-STDDENS-NR-CENTER, shown in (b) and (c) on Figures 1 and 2 exhibit highly
asymmetric ejecta, both in terms of their mass as well as their neutronized
isotope distributions. Critically, unlike the smooth structure of
neutronization within the DDT model, the distribution of neutronized isotopes
in the ejecta of DEF-HIGHDENS-NR-CENTER and DEF-STDDENS-NR-CENTER are highly
aspherical and clumpy. Fundamentally, this is because the pure deflagration
models never transition into detonation, causing the neutronized ash buoyed up
from the dense core to maintain the complex morphological structures developed
during the onset of the Rayleigh-Taylor and Kelvin-Helmholtz instabilities
during burning (Woosley et al., 2004; Bell et al., 2004; Hicks, 2015). These
structures are frozen in as the ejecta enter into homologous expansion, where
they contribute to the diversity of the class of SNe Iax (Lach et al., 2022).

The released kinetic energy in both DEF-HIGHDENS-NR-CENTER and DEF-STDDENS-NR-
CENTER models are $\sim 8\times 10^{49}$ erg and $\sim 1.7\times 10^{50}$, and
the ejecta velocity as large as a few thousand km s-1. Our analysis confirms
that large portions of the WD masses are still bound at the end of these two
simulations. In particular, the DEF-HIGHDENS-NR-CENTER model ejects about 0.19
$\mathrm{M_{\odot}}$ and DEF-STDDENS-NR-CENTER model ejects about 0.56
$\mathrm{M_{\odot}}$, leaving behind remnants of 1.2 $\mathrm{M_{\odot}}$ and
0.82 $\mathrm{M_{\odot}}$, respectively. Similar to previous investigations of
pure deflagrations, we find the 56Ni ejected by the HIGHDENS-NR-CENTER and
DEF-STDDENS-NR-CENTER models to be an order of magnitude or less than produced
in a normal SN Ia event. HIGHDENS-NR-CENTER and DEF-STDDENS-NR-CENTER
nucleosynthesize $\sim$ 0.032 $\mathrm{M_{\odot}}$ and $\sim$ 0.084
$\mathrm{M_{\odot}}$ of 56Ni ejecta, respectively.

###  3.2 Clump Identification and Analysis

A key goal of this study is to compare the clumps formed in simulations of SNR
3C 397 directly against the recent XMM-Newton observations of Ohshiro et al.
(2021). Accordingly, we proceed to constrain the model clumps rich in iron
group elements Ni, Mn, Cr, and Fe, and the newly-discovered Ti, all identified
in X-ray spectra of two clump regions of 3C 397 (“south" and “west").

The extraordinarily high abundances of Ti and iron group elements observed in
the southern clump of 3C 397 are consistent with burning in dense environments
(Ohshiro et al., 2021; Leung & Nomoto, 2018). Fundamentally, this is because
efficient electron captures occur during burning at high densities, reducing
the number of free electrons, and shifting nucleosynthesis yields towards
neutron-rich abundances ($Y_{e}<0.5$) (Lach et al., 2020; Mori et al., 2020).
As discussed in Dave et al. (2017), the resulting Cr/Fe abundance ratios are
particularly sensitive to the density of the burning. Because the southern
clump has a higher Cr/Fe abundance, it poses the tightest constraints on the
burning conditions, and we focus upon it solely here.

We identify clusters of particles based on nucleosynthetic abundances and
spatial distribution. To analyze model clumps, our algorithm proceeds via an
exhaustive search as follows. We loop over all $N_{\rm ej}$ ejected tracer
particles, picking each particle one at a time to represent the center of a
localized candidate clump. We subsequently add up the Fe from its nearest
neighbor particles, selected using the kd-tree method scipy.spatial.KDTree
(Friedman et al., 1977), until the cumulative mass of Fe in the candidate
clump reaches the same 7% of the overall iron abundance in the model ejecta as
that estimated in the southern clump of 3C 397 by Ohshiro et al. (2021).
Because we fix the total Fe mass in each candidate clump, the total mass or
equivalently the number of particles will vary both between clumps and between
models. This process of determining the $N_{\rm ej}$ candidate clumps is
repeated in turn for each of the $N_{\rm ej}$ ejecta particles in the dataset.
Because the pure deflagration models lead to lower ejecta masses, the total
mass of iron in the ejecta and also the cumulative iron mass in the particle
clumps also varies. We note that in the DDT-HIGHDENS-HIGHOFF models, the total
iron enclosed by each candidate clumps is $\sim$ 0.07 M⊙, while the clumps in
DEF-HIGHDENS-NR-CENTER models comprise $\sim$ 0.0034 M⊙ of iron. The DEF-
STDDENS-NR-CENTER models shows $\sim$ 0.0067 M⊙ of iron in the clumps.

In our next step, we compared the nucleosynthetic yields of candidate particle
clusters in our simulations to the southern clump observed in SNR 3C 397. We
define a likelihood function that compares the abundance ratios from our
models to the observations:

$\ln(\text{likelihood})=-\frac{1}{2}\sum_{i}\left(\frac{(X_{\text{obs},i}-X_{\text{model},i})^{2}}{\sigma_{i}^{2}}+\ln(2\pi\sigma_{i}^{2})\right)$
(2)

The index $i$ runs over the abundance ratios of Cr/Fe, Mn/Fe and Ni/Fe.
$X_{\text{obs},i}$ and $X_{\text{model},i}$ are the abundance ratios of the
observed clump and model clusters, respectively. ${\sigma_{i}}$ are the
observational abundance ratio errors.

We also examine the model statistical errors associated with Cr/Fe, Ni/Fe,
Mn/Fe, and Ti/Fe within the entire mass associated with the clump. We utilize
the jackknife resampling method astropy.stats.jackknife_stats (Quenouille,
1956) to elegantly capture the statistical uncertainty associated with our
clump analysis procedure. The jackknife method systematically resamples the
clump, removing one particle at a time, and estimates from these resamples the
confidence interval of individual isotopic abundances and mass ratios for the
cluster. The clumps each have of order $10^{2}$ particles, so the discrete
error associated with resampling produces uncertainties are typically on the a
few percent level. These model statistical errors are much less than the
observational error bars, which are of the order of tens of percent.
Consequently, the inclusion of only the experimental errors in the likelihood
function in Equation 2 is self-consistent.

Finally, we isolate the single most likely clump for each simulation model
with the highest likelihood values, out of all $N_{\rm ej}$ candidates, and
compare directly against the southern clump identified by Ohshiro et al.
(2021) in 3C 397. This final selection is then the most likely clump with an
equivalent Fe mass obtained by Ohshiro et al. (2021). We quoted the abundance
ratios from these most likely clumps in each models (including different
initial progenitor metallicity) in Table 2 along with the statistical
confidence intervals obtained from the jackknife statistical analysis. We
highlighted the most likely clump for each model by a white circle in Figures
1 and 2.

DDT-HIGHDENS-HIGHOFF
---
Metallicity (Z) | Ni/Fe | Mn/Fe | Cr/Fe | Ti/Fe
$Z=0$ | $0.34947^{+0.00316}_{-0.00319}$ | $0.04170^{+0.00067}_{-0.00068}$ | $0.01652^{+0.00059}_{-0.00060}$ | $(1.6\times 10^{-6})^{+7.9\times 10^{-8}}_{-8.0\times 10^{-8}}$
$Z=1.5Z_{\odot}$ | $0.33568^{+0.00295}_{-0.00298}$ | $0.04018^{+0.00073}_{-0.00074}$ | $0.01741^{+0.00060}_{-0.00056}$ | $(1.7\times 10^{-6})^{+9.3\times 10^{-8}}_{-9.4\times 10^{-8}}$
$Z=3.0Z_{\odot}$ | $0.31815^{+0.00260}_{-0.00262}$ | $0.03804^{+0.00079}_{-0.00079}$ | $0.01865^{+0.00061}_{-0.00062}$ | $(1.9\times 10^{-6})^{+1.1\times 10^{-7}}_{-1.1\times 10^{-7}}$
DEF-HIGHDENS-NR-CENTER
Metallicity (Z) | Ni/Fe | Mn/Fe | Cr/Fe | Ti/Fe
$Z=0$ | $0.30450^{+0.01286}_{-0.02077}$ | $0.03686^{+0.00121}_{-0.00195}$ | $0.13226^{+0.02053}_{-0.03316}$ | $0.02641^{+0.01153}_{-0.01862}$
$Z=1.5Z_{\odot}$ | $0.21491^{+0.01789}_{-0.03143}$ | $0.03298^{+0.00046}_{-0.00081}$ | $0.08887^{+0.01676}_{-0.02944}$ | $0.00846^{+0.00440}_{-0.00773}$
$Z=3.0Z_{\odot}$ | $0.20101^{+0.01729}_{-0.02992}$ | $0.03086^{+0.00045}_{-0.00078}$ | $0.08787^{+0.02246}_{-0.03888}$ | $0.01212^{+0.00497}_{-0.00860}$
DEF-STDDENS-NR-CENTER
Metallicity (Z) | Ni/Fe | Mn/Fe | Cr/Fe | Ti/Fe
$Z=0$ | $0.12422^{+0.00990}_{-0.01582}$ | $0.01315^{+0.00105}_{-0.00167}$ | $0.01278^{+0.00122}_{-0.00195}$ | $0.00056^{+5.6\times 10^{-5}}_{-9.0\times 10^{-5}}$
$Z=1.5Z_{\odot}$ | $0.14805^{+0.01480}_{-0.02397}$ | $0.01780^{+0.00049}_{-0.00080}$ | $0.01199^{+0.00105}_{-0.00170}$ | $0.00047^{+4.9\times 10^{-5}}_{-7.9\times 10^{-5}}$
$Z=3.0Z_{\odot}$ | $0.19729^{+0.01417}_{-0.02220}$ | $0.02764^{+6.7\times 10^{-5}}_{-1.0\times 10^{-4}}$ | $0.01247^{+0.00085}_{-0.00134}$ | $0.00043^{+4.5\times 10^{-5}}_{-7.0\times 10^{-5}}$
Observed clumps in the SNR 3C 397
Region | Ni/Fe | Mn/Fe | Cr/Fe | Ti/Fe
South | $0.18^{+0.07}_{-0.03}$ | $0.051^{+0.009}_{-0.009}$ | $0.106^{+0.011}_{-0.009}$ | $0.014^{+0.004}_{-0.004}$
West | $0.27^{+0.05}_{-0.03}$ | $0.050^{+0.006}_{-0.006}$ | $0.034^{+0.004}_{-0.004}$ | $<0.002$

Table 2: The tables above shows the localized clumps’ abundance ratios of
Ni/Fe, Mn/Fe, Cr/Fe and Ti/Fe from DDT-HIGHDENS-HIGHOFF, DEF-HIGHDENS-NR-
CENTER, and DEF-STDDENS-NR-CENTER models. The chosen clumps have the highest
nucleosynthetic likelihood and contains 7% of the total iron mass in the
ejecta. Three rows in each section indicates the different metallicities
$Z=0$, $Z=1.5Z_{\odot}$ and $Z=3.0Z_{\odot}$ of the initial WD progenitor
model. The solar metallicity ($Z_{\odot}$) is taken from (Asplund et al.,
2009). For comparison the mass fraction ratios of observed clump in the
southern and western regions of SNR are also reported.

Inspecting the most likely clumps from each of the hydrodynamic models, we
determined the most likely particle cluster in the DDT-HIGHDENS-HIGHOFF model
(Figure 1(a)) is situated at the innermost parts of the systematically layered
ejecta. However, the Cr/Fe abundance ratio of this particle cluster is less
than one-fifth of the reported value from the observed clump. This local clump
analysis establishes that a DDT even within a high central density WD does not
achieve the nucleosynthesis observed within the local neutronized clumps of 3C
397. Furthermore, the centralized location of the model clump in the inner
ejecta disagrees with the observed clump’s outer location in SNR 3C 397.

An identical comparison in DEF-STDDENS-NR-CENTER and the DEF-HIGHDENS-NR-
CENTER models (Figures 1(c) and 1(b)) shows the most likely particle cluster
in the pure deflagrations are centered towards the outskirts of the ejecta.
The Cr/Fe ratio of the most likely cluster from the DEF-STDDENS-NR-CENTER
model is nearly one-eighth of the value reported to clump’s observation data.
In contrast, the DEF-HIGHDENS-NR-CENTER model show multiple particle groups in
the ejecta that have Cr/Fe ratios comparable to, and in some cases even higher
than, the Cr/Fe ratios reported in the observations. The most likely DEF-
HIGHDENS-NR-CENTER model cluster has Cr/Fe abundance ratio more closely
aligned with the corresponding observational value.

Our analysis shows that overall, the most likely model with respect to both
morphology and nucleosynthesis is the high central density model with a pure-
deflagration explosion mechanism (DEF-HIGHDENS-NR-CENTER) and initial
metallicity of 3.0 Z⊙. For this model, the clump Mn/Fe and Cr/Fe ratios are
within 1 and 3 $\sigma$ of the observed values, respectively. The Ni/Fe ratios
are within 3 $\sigma$. The Ti/Fe is in tension with the observations at the 4
$\sigma$ level. In contrast, both the DDT-HIGHDENS-HIGHOFF and DEF-STDDENS-NR-
CENTER models are in large disagreement with observation, notably with Cr/Fe
disagreements at 9 $\sigma$ or greater. Overall, bearing in mind that we have
not fine tuned our hydrodynamic models to match the observations, the model
clump nucleosynthesis and morphology overwhelmingly favor the DEF-HIGHDENS-NR-
CENTER model.

###  3.3 Sensitivity of Ejecta Criterion

As mentioned in the methods section, we analyze the increment in the unbound
ejecta mass from the marginally bound ejecta over fixed intervals in time in
order to assess the sensitivity of the ejecta to the time cutoff of the
simulation. In DEF-HIGHDENS-NR-CENTER simulation, the ejecta mass increment
between 5s and 6s is $\sim$ 12%, between 6s and 7s is $\sim$ 7%, and between
7s and 8s is $\sim$ 5.3%. In DEF-STDDENS-NR-CENTER simulation, the increment
between 4s and 5s is $\sim$ 85.5%, between 5s and 6s is 36.6%, and between 6s
and 7s is $\sim$ 14.2%.

Nonetheless, once the expelled material becomes gravitationally unbound from
the star, the radial distribution of the mass does not alter significantly
under a free expansion. To quantify the degree to which to unbound ejecta
achieves free expansion, we follow an analysis procedure reported in Röpke
(2005). We divide the ejecta particles into radial shells of equal width and
find the mean velocity of particles in each shell. We then calculate the
deviations from the radial velocities from a linear homologous expansion. In
our DEF-HIGHDENS-NR-CENTER model, the maximum deviation is 17%, while the mass
weighted deviation remains less than 3%. The weighted deviation in the DEF-
STDDENS-NR-CENTER model also remains less than 3%. These results confirm that
that the ejecta criteria robustly identifies the ejecta material in homologous
expansion, even in models with bound remnants.

##  4 Discussion

As previous authors have established, buoyancy-driven plumes generally lead to
the ejection of neutronized material to the outer edge of the ejecta even in
the DDT scenario (e.g., Seitenzahl et al., 2013a). We note that our DDT
simulation also has a plume of neutronized ejecta along the +$z$ axis, where
the initial deflagration ignition point was located (Figure 2a). We are
agnostic with regards to where the selected model clump originates; in
principle this outermost region of DDT ejecta could have been selected if it
had been the best fit to the 3C 397 southern clump nucleosynthetic abundances.
However, for the DDT model, it is the central region and not the outer one
that produces a greater neutronization and a better match to the southern
clump data. Figure 2a demonstrates that the buoyancy-driven deflagration ash
is less neutronized than the material in the core, as it is burned at lower
densities on average. Thus the buoyancy-driven effect in the DDT model
considered here is not sufficient to explain the neutronization in the clumps
observed in 3C 397.

Thermal broadening in explosive transients is significantly larger than the
isotopic shifts for nuclei, limiting optical and X-ray observations of SNe and
their remnants to elemental rather than isotopic abundances. With the
exception of gamma-ray spectroscopy and monoisotopic elements like 55Mn,
astronomical observations generally cannot probe the nuclear isotopes crucial
for nuclear astrophysics. In a nutshell, with the exception of gamma-ray
spectroscopy and monisotopes like 55Mn, astronomical observations do not probe
the nuclear isotopes so critical for nuclear astrophysics. In contrast, the
isotopic abundances of meteorites can be analyzed in detail in the laboratory.
The clumpy structures prevalent in the pure deflagration models support the
idea that exceptionally high abundances of neutronized isotopes may persist
and be found in meteoretic clumps (Ohshiro et al., 2021). The highly
neutronized isotopes formed in 3C 397 point towards supersolar abundances of
IGE and IMGE isotopes such as 54Cr, 50Ti, 58Fe, 62Ni, and 48Ca. The
carbonaceous chondrite meteors Murchison and Allende have long been known to
have inclusions exhibiting correlated overabundance in both 48Ca and 50Ti,
with large variations in inferred neutron excess across grains (Zinner et al.,
1986). More recent measurements of the Orgueil CI meteorite have also revealed
high abundances of 54Cr and 50Ti (Nittler et al., 2018). These measurements
excessively abundant in neutronized isotopes are consistent with both
nucleosynthesis in electron capture SNe (ECSNe) as well as in high-density WD
SNe Ia progenitors (Nittler et al., 2018). As we have found, the neutronized
material inside a DDT is concentrated towards the center of the remnant, while
the dense WD SNe Iax progenitors naturally produce clumpy distributions of
neutronized material in the outer ejecta, lending support to the idea that
these clumps may survive and become incorporated into meteoretic grains.
Moreover, a direct comparison against the localized clumps such as those
presented here, versus the global nucleosynthetic yields will be a more
faithful representation of the actual composition of the neutronized grains.

In the pure deflagration scenario for the origin of SNe Iax, the asymmetric
ejection of the ash imparts a kick to the primary WD, causing it to be ejected
from the system as a hypervelocity WD (Jordan et al., 2012b). Six
hypervelocity candidates have now been discovered thanks to the Gaia survey
(El-Badry et al., 2023). Based on our finding that 3C 397 likely originated
from a pure deflagration SN Ia, the single degenerate scenario for
near-$M_{\rm Ch}$ WDs predicts that both the stripped non-degenerate donor as
well as the kicked accretor should have survived the explosion. The 8 - 9.7
kpc estimated distance (Leahy & Ranasinghe, 2016) to 3C 397 makes a direct
search for both survivors implausible through Gaia.

Qualitatively, the asymmetric structure of the ejecta introduced by the pure-
deflagration scenario is consistent with the extended Fe-K shell emission in
3C 397 (Ohshiro et al., 2021). On the other hand, the low explosion energy and
low mass ejecta inferred in the presented study seem to be in tension with the
bright luminosity of the Fe-K emission (Yamaguchi et al., 2014). In order to
clarify whether the characteristics of the pure-deflagration scenario can
explain the observed structure and the X-ray emission of 3C 397, a
multidimensional model that covers the evolutional phase from SN to SNR is
needed that includes the asymmetric ejecta distribution, the surrounding
environment, the fallback effect, and even impact interaction with the donor.
Observationally, the measurement of the detailed spatial distribution of
ejecta clumps and their mass ratios provides important information on the
explosion mechanism of SNe Ia, which is enabled by Chandra and XRISM
satellites.

While the overall agreement between our computed models and the observed clump
abundances in both the morphology and nucleosynthesis is best for the high
central density WD pure deflagration model, there are still some significant
error bars (up to 3 $\sigma$) on individual abundance ratios. There is
therefore some room in improvement in the hydrodynamical models in terms of
inferring the precise parameters of the explosion, both in terms of the WD
progenitor properties (in particular its central density), as well as the
uncertain ignition parameters. The need for further refinement of the models
is particularly evident in the high stellar progenitor WD metallicity (3
$Z_{\odot}$). A similar trend towards higher metallicity was seen in model
matches in Yamaguchi et al. (2015), which ultimately derived from the need for
higher central density WD model progenitors. We conjecture that a similar
effect is at play here, and that further optimization of the WD progenitor
structure will narrow down the central density of the progenitor to a slightly
higher value than was considered here for our high central density case.

The underlying explosion mechanism is one of the most central questions
concerning the origin of the class of SNe Iax events. Some studies have
suggested that the structure of SNe Iax are smoothly stratified, which would
point towards a detonation mechanism, at least in some brighter events like
2012Z with greater estimated 56Ni yields (Stritzinger et al., 2015; Barna et
al., 2017; Barna et al., 2018). Other investigators have found that a
comparison of artificially-mixed profiles of SNe Iax, ranging from smoothly
stratified to well-mixed, favors well-mixed ejecta, consistent with the
predictions of the failed detonation scenario (Magee et al., 2021). However,
as our model results show, the clumpy distribution observed by Ohshiro et al.
(2021) disfavors a smoothly stratified ejecta structure that would have
originated for a DDT-powered SN Ia transient for 3C 397. A pure deflagration
origin for 3C 397 opens up a new window onto this central question, with
spatially-mapped X-ray observations enabling a direct glimpse into the ejecta.

##  5 Conclusions

In this study, we conducted a detailed analysis of the nucleosynthetic yields
and morphology for a wide range of plausible hydrodynamical explosion
scenarios of a near-MCh WD progenitor for SNR 3C 397. Our hydrodynamical
models of the DDT release sufficient nuclear energy to unbind the whole WD,
though the model ejecta profiles are smoothly stratified, with the most
neutronized material located in the central region. Further, the DDT model
nucleosynthesis is significantly in disagreement with the observations of 3C
397, with over 9 $\sigma$ deviation in the the ratios of Cr/Fe. While pure
deflagration models burn less, and eject lesser still, their ejected mass
retains the imprint of the hydrodynamic instabilities from the deflagration
phase. Overlaid post-processed particles clearly show the clustering of
neutronized clumps in the pure deflagration ejecta. In particular, our clump
analysis of the high central density model reveals clumps rich in Cr/Fe ratios
and other neutronized species similar to the observed clump in the southern
region of SNR 3C 397. Based on our analysis, we find that SNR 3C 397 most
likely originated from the pure deflagration of a high central density of
near-MCh WD progenitor, with an optical SN Iax transient which gave rise to 3C
397.

SNe Iax are broadly diverse as a class of transients. In marked contrast to
the relatively homogeneous class of normal SNe Ia, optical observations
demonstrate a remarkable diversity in the spectra and light curves of the
class of SNe Iax transients (Foley et al., 2013). The diversity stems from a
number of factors, ranging from the viewing angle dependence of the asymmetric
ejecta, to the sensitivity to the inherent stochastic ignition in the core of
the near-$M_{\rm Ch}$ WD, to interaction with the non-degenerate donor (Bulla
et al., 2020). There may even be entirely distinct explosion channels beyond
the near-$M_{\rm Ch}$ scenario which contribute to the SN Iax population,
including electron capture SNe (Pumo et al., 2009) and the mergers of ONe WDs
with CO WDs (Kashyap et al., 2018). Our conclusions regarding a SN Iax origin
of 3C 397 will, of course, need to be confirmed and supported by additional
observational and theoretical investigations. “Reverse engineering” 3C 397
using simulation models may help us understand in detail the properties of the
optical transient which gave rise to the remnant in the context of the
near-$M_{\rm Ch}$ channel.

## Acknowledgements

R.T.F. acknowledges support from NASA ATP awards 80NSSC18K1013 and
80NSSC22K0630. This work used the Extreme Science and Engineering Discovery
Environment (XSEDE) Stampede 2 supercomputer at the University of Texas at
Austin’s Texas Advanced Computing Center through allocation TG-AST100038.
XSEDE is supported by National Science Foundation grant number ACI-1548562
(Towns et al., 2014). R.T.F. gratefully acknowledges support from the
Institute for Advanced Studies and the Heidelberg Institute for Theoretical
Studies.

This research has made use of FLASH 4.3 (Fryxell et al., 2000a; Dubey et al.,
2012), the FLASH SN Ia module (Townsley et al., 2016), Frank Timmes’s
Adiabatic WD generator code
(cococubed.com/code_pages/adiabatic_white_dwarf.shtml), yt (Turk et al.,
2011), the Python programming language (van Rossum & de Boer, 1991), Numpy
(van der Walt et al., 2011), Jupyter/IPython (Perez & Granger, 2007), and
Matplotlib (Hunter, 2007).

Data from the astronomical facilities Suzaku (Mitsuda et al., 2007) and XMM-
Newton (Jansen, F. et al., 2001) have been used.

## Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

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Species Name | Z = 0 | Z = 1.5 $Z_{\odot}$ | Z = 3.0 $Z_{\odot}$
---|---|---|---
${}^{4}\text{He}$ | $3.03\times 10^{-03}$ | $2.36\times 10^{-03}$ | $1.77\times 10^{-03}$
${}^{16}\text{O}$ | $4.28\times 10^{-04}$ | $5.07\times 10^{-04}$ | $5.10\times 10^{-04}$
${}^{24}\text{Mg}$ | $2.64\times 10^{-05}$ | $4.45\times 10^{-06}$ | $2.14\times 10^{-06}$
${}^{28}\text{Si}$ | $4.75\times 10^{-03}$ | $5.05\times 10^{-03}$ | $5.14\times 10^{-03}$
${}^{29}\text{Si}$ | $7.46\times 10^{-07}$ | $1.95\times 10^{-06}$ | $2.81\times 10^{-06}$
${}^{30}\text{Si}$ | $1.76\times 10^{-06}$ | $3.01\times 10^{-06}$ | $6.82\times 10^{-06}$
${}^{31}\text{P}$ | $9.83\times 10^{-07}$ | $1.89\times 10^{-06}$ | $3.30\times 10^{-06}$
${}^{32}\text{S}$ | $3.02\times 10^{-03}$ | $2.83\times 10^{-03}$ | $2.58\times 10^{-03}$
${}^{33}\text{S}$ | $1.34\times 10^{-07}$ | $2.33\times 10^{-06}$ | $3.74\times 10^{-06}$
${}^{34}\text{S}$ | $8.71\times 10^{-07}$ | $1.69\times 10^{-05}$ | $3.88\times 10^{-05}$
${}^{35}\text{Cl}$ | $5.24\times 10^{-07}$ | $1.78\times 10^{-06}$ | $2.68\times 10^{-06}$
${}^{36}\text{Ar}$ | $7.00\times 10^{-04}$ | $5.85\times 10^{-04}$ | $4.99\times 10^{-04}$
${}^{38}\text{Ar}$ | $1.58\times 10^{-07}$ | $1.57\times 10^{-05}$ | $3.44\times 10^{-05}$
${}^{39}\text{K}$ | $5.95\times 10^{-07}$ | $2.68\times 10^{-06}$ | $3.44\times 10^{-06}$
${}^{40}\text{Ca}$ | $7.07\times 10^{-04}$ | $5.46\times 10^{-04}$ | $4.53\times 10^{-04}$
${}^{42}\text{Ca}$ | $-$ | $6.52\times 10^{-07}$ | $1.33\times 10^{-06}$
${}^{44}\text{Ca}$ | $3.57\times 10^{-06}$ | $2.22\times 10^{-06}$ | $1.34\times 10^{-06}$
${}^{48}\text{Ti}$ | $2.44\times 10^{-05}$ | $1.85\times 10^{-05}$ | $1.45\times 10^{-05}$
${}^{49}\text{Ti}$ | $5.36\times 10^{-07}$ | $1.58\times 10^{-06}$ | $1.82\times 10^{-06}$
${}^{51}\text{V}$ | $1.83\times 10^{-05}$ | $2.17\times 10^{-05}$ | $2.37\times 10^{-05}$

Species Name | Z = 0 | Z = 1.5 $Z_{\odot}$ | Z = 3.0 $Z_{\odot}$
---|---|---|---
${}^{50}\text{Cr}$ | $1.19\times 10^{-04}$ | $1.30\times 10^{-04}$ | $1.45\times 10^{-04}$
${}^{52}\text{Cr}$ | $1.96\times 10^{-03}$ | $1.95\times 10^{-03}$ | $1.98\times 10^{-03}$
${}^{54}\text{Cr}$ | $9.48\times 10^{-07}$ | $1.12\times 10^{-06}$ | $1.35\times 10^{-06}$
${}^{53}\text{Mn}$ | $7.62\times 10^{-04}$ | $8.15\times 10^{-04}$ | $8.51\times 10^{-04}$
${}^{55}\text{Mn}$ | $1.45\times 10^{-02}$ | $1.54\times 10^{-02}$ | $1.62\times 10^{-02}$
${}^{54}\text{Fe}$ | $1.45\times 10^{-01}$ | $1.51\times 10^{-01}$ | $1.59\times 10^{-01}$
${}^{56}\text{Fe}$ | $5.47\times 10^{-01}$ | $5.20\times 10^{-01}$ | $4.94\times 10^{-01}$
${}^{57}\text{Fe}$ | $2.88\times 10^{-02}$ | $3.12\times 10^{-02}$ | $3.27\times 10^{-02}$
${}^{58}\text{Fe}$ | $2.91\times 10^{-05}$ | $3.38\times 10^{-05}$ | $3.98\times 10^{-05}$
${}^{58}\text{Ni}$ | $2.08\times 10^{-01}$ | $2.26\times 10^{-01}$ | $2.44\times 10^{-01}$
${}^{59}\text{Ni}$ | $2.82\times 10^{-03}$ | $2.94\times 10^{-03}$ | $2.99\times 10^{-03}$
${}^{60}\text{Ni}$ | $3.60\times 10^{-02}$ | $3.45\times 10^{-02}$ | $3.36\times 10^{-02}$
${}^{61}\text{Ni}$ | $3.39\times 10^{-04}$ | $3.45\times 10^{-04}$ | $2.97\times 10^{-04}$
${}^{62}\text{Ni}$ | $2.13\times 10^{-03}$ | $2.99\times 10^{-03}$ | $3.21\times 10^{-03}$
${}^{63}\text{Cu}$ | $1.63\times 10^{-05}$ | $4.33\times 10^{-06}$ | $4.61\times 10^{-06}$
${}^{65}\text{Cu}$ | $2.38\times 10^{-06}$ | $2.24\times 10^{-06}$ | $1.59\times 10^{-06}$
${}^{64}\text{Zn}$ | $1.05\times 10^{-04}$ | $3.05\times 10^{-05}$ | $1.72\times 10^{-05}$
${}^{66}\text{Zn}$ | $3.93\times 10^{-05}$ | $5.19\times 10^{-05}$ | $4.78\times 10^{-05}$
${}^{70}\text{Ge}$ | $1.12\times 10^{-06}$ | $1.23\times 10^{-06}$ | $9.08\times 10^{-07}$

Table 3:  The table contains information about the stable nucleosynthetic
yields for DDT-HIGHDENS-HIGHOFF model in mass fraction. The first column shows
the name of the species, the second column is the mass fraction for 0
metallicity, the third column is the mass fraction for 1.5$Z_{\odot}$, and the
fourth column shows the mass fraction for 3$Z_{\odot}$. The solar metallicity
($Z_{\odot}$) is taken from (Asplund et al., 2009). Please note that the
yields below $1\times 10^{-7}$ are not mentioned in this table.

Species Name | Z = 0 | Z = 1.5 $Z_{\odot}$ | Z = 3.0 $Z_{\odot}$
---|---|---|---
${}^{4}\text{He}$ | $6.78\times 10^{-06}$ | $3.33\times 10^{-07}$ | -
${}^{12}\text{C}$ | $1.48\times 10^{-01}$ | $1.45\times 10^{-01}$ | $1.42\times 10^{-01}$
${}^{14}\text{N}$ | $1.06\times 10^{-07}$ | - | -
${}^{16}\text{O}$ | $3.79\times 10^{-01}$ | $3.72\times 10^{-01}$ | $3.65\times 10^{-01}$
${}^{20}\text{Ne}$ | $5.67\times 10^{-03}$ | $4.74\times 10^{-03}$ | $4.17\times 10^{-03}$
${}^{21}\text{Ne}$ | $1.09\times 10^{-07}$ | $5.24\times 10^{-06}$ | $8.43\times 10^{-06}$
${}^{22}\text{Ne}$ | - | $1.03\times 10^{-02}$ | $2.07\times 10^{-02}$
${}^{23}\text{Na}$ | $1.55\times 10^{-05}$ | $1.09\times 10^{-04}$ | $1.70\times 10^{-04}$
${}^{24}\text{Mg}$ | $5.06\times 10^{-03}$ | $3.26\times 10^{-03}$ | $2.20\times 10^{-03}$
${}^{25}\text{Mg}$ | $3.97\times 10^{-07}$ | $1.47\times 10^{-04}$ | $2.92\times 10^{-04}$
${}^{26}\text{Mg}$ | $7.52\times 10^{-06}$ | $2.71\times 10^{-04}$ | $6.00\times 10^{-04}$
${}^{27}\text{Al}$ | $2.49\times 10^{-05}$ | $3.84\times 10^{-04}$ | $4.61\times 10^{-04}$
${}^{28}\text{Si}$ | $5.58\times 10^{-02}$ | $5.42\times 10^{-02}$ | $5.31\times 10^{-02}$
${}^{29}\text{Si}$ | $1.99\times 10^{-05}$ | $4.08\times 10^{-04}$ | $6.86\times 10^{-04}$
${}^{30}\text{Si}$ | $7.69\times 10^{-06}$ | $4.28\times 10^{-04}$ | $8.89\times 10^{-04}$
${}^{31}\text{P}$ | $1.96\times 10^{-05}$ | $1.52\times 10^{-04}$ | $2.67\times 10^{-04}$
${}^{32}\text{S}$ | $2.67\times 10^{-02}$ | $2.64\times 10^{-02}$ | $2.48\times 10^{-02}$
${}^{33}\text{S}$ | $1.39\times 10^{-05}$ | $8.18\times 10^{-05}$ | $1.09\times 10^{-04}$
${}^{34}\text{S}$ | $5.53\times 10^{-07}$ | $5.60\times 10^{-04}$ | $1.20\times 10^{-03}$
${}^{36}\text{S}$ | - | - | $2.57\times 10^{-07}$
${}^{35}\text{Cl}$ | $8.37\times 10^{-07}$ | $5.33\times 10^{-05}$ | $8.04\times 10^{-05}$
${}^{37}\text{Cl}$ | $2.19\times 10^{-06}$ | $1.37\times 10^{-05}$ | $1.88\times 10^{-05}$
${}^{36}\text{Ar}$ | $4.84\times 10^{-03}$ | $4.68\times 10^{-03}$ | $4.23\times 10^{-03}$
${}^{38}\text{Ar}$ | $5.23\times 10^{-07}$ | $5.54\times 10^{-04}$ | $1.18\times 10^{-03}$
${}^{39}\text{K}$ | $8.04\times 10^{-07}$ | $6.22\times 10^{-05}$ | $9.46\times 10^{-05}$
${}^{41}\text{K}$ | $4.48\times 10^{-07}$ | $3.78\times 10^{-06}$ | $5.12\times 10^{-06}$
${}^{40}\text{Ca}$ | $4.10\times 10^{-03}$ | $3.97\times 10^{-03}$ | $3.55\times 10^{-03}$
${}^{42}\text{Ca}$ - | $2.44\times 10^{-05}$ | $5.43\times 10^{-05}$ |
${}^{44}\text{Ca}$ | $2.52\times 10^{-06}$ | $2.50\times 10^{-06}$ | $2.31\times 10^{-06}$
${}^{48}\text{Ca}$ | $2.47\times 10^{-04}$ | $2.47\times 10^{-04}$ | $2.47\times 10^{-04}$
${}^{45}\text{Sc}$ | - | $1.52\times 10^{-07}$ | $1.98\times 10^{-07}$
${}^{46}\text{Ti}$ | - | $1.14\times 10^{-05}$ | $2.24\times 10^{-05}$
${}^{47}\text{Ti}$ | - | $2.54\times 10^{-07}$ | $5.86\times 10^{-07}$

Species Name | Z = 0 | Z = 1.5 $Z_{\odot}$ | Z = 3.0 $Z_{\odot}$
---|---|---|---
${}^{48}\text{Ti}$ | $6.97\times 10^{-05}$ | $5.98\times 10^{-05}$ | $5.21\times 10^{-05}$
${}^{49}\text{Ti}$ | $9.30\times 10^{-05}$ | $9.56\times 10^{-05}$ | $9.66\times 10^{-05}$
${}^{50}\text{Ti}$ | $4.05\times 10^{-03}$ | $4.05\times 10^{-03}$ | $4.06\times 10^{-03}$
${}^{51}\text{V}$ | $2.19\times 10^{-04}$ | $2.30\times 10^{-04}$ | $2.37\times 10^{-04}$
${}^{50}\text{Cr}$ | $5.55\times 10^{-05}$ | $1.10\times 10^{-04}$ | $1.75\times 10^{-04}$
${}^{52}\text{Cr}$ | $6.68\times 10^{-03}$ | $6.47\times 10^{-03}$ | $6.32\times 10^{-03}$
${}^{53}\text{Cr}$ | $8.34\times 10^{-04}$ | $9.19\times 10^{-04}$ | $9.65\times 10^{-04}$
${}^{54}\text{Cr}$ | $1.36\times 10^{-02}$ | $1.36\times 10^{-02}$ | $1.37\times 10^{-02}$
${}^{55}\text{Mn}$ | $7.13\times 10^{-03}$ | $7.89\times 10^{-03}$ | $8.41\times 10^{-03}$
${}^{54}\text{Fe}$ | $5.13\times 10^{-02}$ | $5.72\times 10^{-02}$ | $6.37\times 10^{-02}$
${}^{56}\text{Fe}$ | $1.69\times 10^{-01}$ | $1.62\times 10^{-01}$ | $1.55\times 10^{-01}$
${}^{57}\text{Fe}$ | $6.82\times 10^{-03}$ | $7.17\times 10^{-03}$ | $7.35\times 10^{-03}$
${}^{58}\text{Fe}$ | $2.80\times 10^{-02}$ | $2.80\times 10^{-02}$ | $2.81\times 10^{-02}$
${}^{59}\text{Co}$ | $1.25\times 10^{-03}$ | $1.25\times 10^{-03}$ | $1.26\times 10^{-03}$
${}^{58}\text{Ni}$ | $4.93\times 10^{-02}$ | $5.10\times 10^{-02}$ | $5.26\times 10^{-02}$
${}^{60}\text{Ni}$ | $1.43\times 10^{-02}$ | $1.43\times 10^{-02}$ | $1.44\times 10^{-02}$
${}^{61}\text{Ni}$ | $2.32\times 10^{-04}$ | $2.30\times 10^{-04}$ | $2.31\times 10^{-04}$
${}^{62}\text{Ni}$ | $9.28\times 10^{-03}$ | $9.26\times 10^{-03}$ | $9.28\times 10^{-03}$
${}^{64}\text{Ni}$ | $5.89\times 10^{-03}$ | $5.90\times 10^{-03}$ | $5.90\times 10^{-03}$
${}^{63}\text{Cu}$ | $6.59\times 10^{-05}$ | $6.60\times 10^{-05}$ | $6.60\times 10^{-05}$
${}^{65}\text{Cu}$ | $2.39\times 10^{-05}$ | $2.39\times 10^{-05}$ | $2.39\times 10^{-05}$
${}^{64}\text{Zn}$ | $6.51\times 10^{-07}$ | $5.85\times 10^{-07}$ | $5.91\times 10^{-07}$
${}^{66}\text{Zn}$ | $2.41\times 10^{-03}$ | $2.41\times 10^{-03}$ | $2.41\times 10^{-03}$
${}^{67}\text{Zn}$ | $2.10\times 10^{-05}$ | $2.10\times 10^{-05}$ | $2.10\times 10^{-05}$
${}^{68}\text{Zn}$ | $1.61\times 10^{-06}$ | $1.62\times 10^{-06}$ | $1.62\times 10^{-06}$
${}^{70}\text{Zn}$ | $4.96\times 10^{-06}$ | $4.97\times 10^{-06}$ | $4.97\times 10^{-06}$
${}^{69}\text{Ga}$ | $1.43\times 10^{-06}$ | $1.44\times 10^{-06}$ | $1.44\times 10^{-06}$
${}^{72}\text{Ge}$ | $1.05\times 10^{-06}$ | $1.05\times 10^{-06}$ | $1.05\times 10^{-06}$
${}^{76}\text{Ge}$ | $1.55\times 10^{-06}$ | $1.55\times 10^{-06}$ | $1.55\times 10^{-06}$
${}^{78}\text{Se}$ | $5.10\times 10^{-07}$ | $5.11\times 10^{-07}$ | $5.11\times 10^{-07}$
${}^{82}\text{Se}$ | $1.71\times 10^{-06}$ | $1.71\times 10^{-06}$ | $1.71\times 10^{-06}$
${}^{86}\text{Kr}$ | $1.87\times 10^{-06}$ | $1.87\times 10^{-06}$ | $1.87\times 10^{-06}$

Table 4: The table contains information about the stable nucleosynthetic
yields for DEF-HIGHDENS-NR-CENTER model in mass fraction. The first column
shows the name of the species, the second column is the mass fraction for 0
metallicities, the third column is the mass fraction for 1.5$Z_{\odot}$, and
the fourth column shows the mass fraction for 3$Z_{\odot}$. The solar
metallicity ($Z_{\odot}$) is taken from (Asplund et al., 2009). Please note
that the yields below $1\times 10^{-7}$ are not mentioned in this table.

Species Name | Z = 0 | Z = 1.5 $Z_{\odot}$ | Z = 3.0 $Z_{\odot}$
---|---|---|---
${}^{4}\text{He}$ | $8.52\times 10^{-05}$ | $2.45\times 10^{-05}$ | $2.11\times 10^{-06}$
${}^{12}\text{C}$ | $2.55\times 10^{-01}$ | $2.50\times 10^{-01}$ | $2.45\times 10^{-01}$
${}^{14}\text{N}$ | $2.65\times 10^{-07}$ | - | -
${}^{15}\text{N}$ | $1.12\times 10^{-07}$ | - | -
${}^{16}\text{O}$ | $3.21\times 10^{-01}$ | $3.17\times 10^{-01}$ | $3.13\times 10^{-01}$
${}^{20}\text{Ne}$ | $2.05\times 10^{-02}$ | $1.81\times 10^{-02}$ | $1.67\times 10^{-02}$
${}^{21}\text{Ne}$ | $1.78\times 10^{-07}$ | $1.40\times 10^{-05}$ | $2.28\times 10^{-05}$
${}^{22}\text{Ne}$ | - | $1.02\times 10^{-02}$ | $2.06\times 10^{-02}$
${}^{23}\text{Na}$ | $6.53\times 10^{-05}$ | $3.41\times 10^{-04}$ | $5.09\times 10^{-04}$
${}^{24}\text{Mg}$ | $2.80\times 10^{-02}$ | $2.09\times 10^{-02}$ | $1.58\times 10^{-02}$
${}^{25}\text{Mg}$ | $1.63\times 10^{-06}$ | $3.52\times 10^{-04}$ | $7.31\times 10^{-04}$
${}^{26}\text{Mg}$ | $3.65\times 10^{-05}$ | $7.03\times 10^{-04}$ | $1.43\times 10^{-03}$
${}^{27}\text{Al}$ | $1.44\times 10^{-04}$ | $1.61\times 10^{-03}$ | $2.08\times 10^{-03}$
${}^{28}\text{Si}$ | $1.25\times 10^{-01}$ | $1.24\times 10^{-01}$ | $1.19\times 10^{-01}$
${}^{29}\text{Si}$ | $6.93\times 10^{-05}$ | $1.23\times 10^{-03}$ | $2.24\times 10^{-03}$
${}^{30}\text{Si}$ | $2.52\times 10^{-05}$ | $1.59\times 10^{-03}$ | $3.28\times 10^{-03}$
${}^{31}\text{P}$ | $5.83\times 10^{-05}$ | $4.52\times 10^{-04}$ | $7.91\times 10^{-04}$
${}^{32}\text{S}$ | $3.83\times 10^{-02}$ | $3.75\times 10^{-02}$ | $3.67\times 10^{-02}$
${}^{33}\text{S}$ | $2.16\times 10^{-05}$ | $1.70\times 10^{-04}$ | $2.41\times 10^{-04}$
${}^{34}\text{S}$ | $1.55\times 10^{-06}$ | $1.44\times 10^{-03}$ | $3.03\times 10^{-03}$
${}^{36}\text{S}$ | $8.89\times 10^{-13}$ | - | $3.96\times 10^{-07}$
${}^{35}\text{Cl}$ | $1.60\times 10^{-06}$ | $7.60\times 10^{-05}$ | $1.28\times 10^{-04}$
${}^{37}\text{Cl}$ | $1.33\times 10^{-06}$ | $1.32\times 10^{-05}$ | $1.99\times 10^{-05}$
${}^{36}\text{Ar}$ | $5.49\times 10^{-03}$ | $5.04\times 10^{-03}$ | $4.80\times 10^{-03}$
${}^{38}\text{Ar}$ | $1.19\times 10^{-06}$ | $7.58\times 10^{-04}$ | $1.67\times 10^{-03}$
${}^{39}\text{K}$ | $9.85\times 10^{-07}$ | $3.96\times 10^{-05}$ | $6.62\times 10^{-05}$
${}^{41}\text{K}$ | $1.51\times 10^{-07}$ | $1.96\times 10^{-06}$ | $3.09\times 10^{-06}$

Species Name | Z = 0 | Z = 1.5 $Z_{\odot}$ | Z = 3.0 $Z_{\odot}$
---|---|---|---
${}^{40}\text{Ca}$ | $4.28\times 10^{-03}$ | $3.84\times 10^{-03}$ | $3.62\times 10^{-03}$
${}^{42}\text{Ca}$ | - | $1.58\times 10^{-05}$ | $3.89\times 10^{-05}$
${}^{44}\text{Ca}$ | $2.52\times 10^{-06}$ | $2.25\times 10^{-06}$ | $2.13\times 10^{-06}$
${}^{46}\text{Ti}$ | - | $6.45\times 10^{-06}$ | $1.42\times 10^{-05}$
${}^{47}\text{Ti}$ | - | $1.80\times 10^{-07}$ | $4.79\times 10^{-07}$
${}^{48}\text{Ti}$ | $8.69\times 10^{-05}$ | $7.31\times 10^{-05}$ | $6.46\times 10^{-05}$
${}^{49}\text{Ti}$ | $2.27\times 10^{-06}$ | $5.76\times 10^{-06}$ | $7.17\times 10^{-06}$
${}^{50}\text{Ti}$ | $5.84\times 10^{-06}$ | $5.87\times 10^{-06}$ | $5.93\times 10^{-06}$
${}^{51}\text{V}$ | $9.30\times 10^{-06}$ | $2.46\times 10^{-05}$ | $3.56\times 10^{-05}$
${}^{50}\text{Cr}$ | $1.60\times 10^{-05}$ | $1.05\times 10^{-04}$ | $1.94\times 10^{-04}$
${}^{52}\text{Cr}$ | $2.44\times 10^{-03}$ | $2.17\times 10^{-03}$ | $2.11\times 10^{-03}$
${}^{53}\text{Cr}$ | $1.63\times 10^{-04}$ | $2.87\times 10^{-04}$ | $3.78\times 10^{-04}$
${}^{54}\text{Cr}$ | $7.72\times 10^{-05}$ | $7.76\times 10^{-05}$ | $7.81\times 10^{-05}$
${}^{55}\text{Mn}$ | $2.22\times 10^{-03}$ | $3.32\times 10^{-03}$ | $4.22\times 10^{-03}$
${}^{54}\text{Fe}$ | $1.73\times 10^{-02}$ | $2.53\times 10^{-02}$ | $3.37\times 10^{-02}$
${}^{56}\text{Fe}$ | $1.51\times 10^{-01}$ | $1.41\times 10^{-01}$ | $1.32\times 10^{-01}$
${}^{57}\text{Fe}$ | $4.31\times 10^{-03}$ | $4.99\times 10^{-03}$ | $5.35\times 10^{-03}$
${}^{58}\text{Fe}$ | $3.50\times 10^{-04}$ | $3.51\times 10^{-04}$ | $3.53\times 10^{-04}$
${}^{59}\text{Co}$ | $1.52\times 10^{-04}$ | $1.40\times 10^{-04}$ | $1.30\times 10^{-04}$
${}^{58}\text{Ni}$ | $2.25\times 10^{-02}$ | $2.57\times 10^{-02}$ | $2.86\times 10^{-02}$
${}^{60}\text{Ni}$ | $1.54\times 10^{-03}$ | $1.36\times 10^{-03}$ | $1.32\times 10^{-03}$
${}^{61}\text{Ni}$ | $1.71\times 10^{-05}$ | $8.63\times 10^{-06}$ | $3.98\times 10^{-06}$
${}^{62}\text{Ni}$ | $2.97\times 10^{-04}$ | $2.39\times 10^{-04}$ | $1.86\times 10^{-04}$
${}^{64}\text{Ni}$ | $9.47\times 10^{-07}$ | $9.53\times 10^{-07}$ | $9.62\times 10^{-07}$
${}^{63}\text{Cu}$ | $2.49\times 10^{-07}$ | $2.08\times 10^{-07}$ | $1.71\times 10^{-07}$
${}^{64}\text{Zn}$ | $8.08\times 10^{-07}$ | $1.81\times 10^{-07}$ | -
${}^{66}\text{Zn}$ | $1.12\times 10^{-06}$ | $3.61\times 10^{-07}$ | -

Table 5:  The table contains information about the stable nucleosynthetic
yields for DEF-STDDENS-NR-CENTER model in mass fraction. The first column
shows the name of the species, the second column is the mass fraction for 0
metallicities, the third column is the mass fraction for 1.5$Z_{\odot}$, and
the fourth column shows the mass fraction for 3$Z_{\odot}$. The solar
metallicity ($Z_{\odot}$) is taken from (Asplund et al., 2009). Please note
that the yields below $1\times 10^{-7}$ are not mentioned in this table.