# Near-Chandrasekhar-Mass Type Ia Supernovae from the Double-Degenerate
Channel

Sudarshan Neopane These authors contributed equally to this work.  Department
of Physics, University of Massachusetts Dartmouth, 285 Old Westport Road,
North Dartmouth, MA 02747, USA  Khanak Bhargava These authors contributed
equally to this work.  Department of Physics, University of Massachusetts
Dartmouth, 285 Old Westport Road, North Dartmouth, MA 02747, USA  Robert
Fisher<EMAIL_ADDRESS>Department of Physics, University of Massachusetts
Dartmouth, 285 Old Westport Road, North Dartmouth, MA 02747, USA  Mckenzie
Ferrari Department of Physics, University of Massachusetts Dartmouth, 285 Old
Westport Road, North Dartmouth, MA 02747, USA  Shin’ichirou Yoshida Department
of Earth Science and Astronomy, Graduate School of Arts and Sciences, The
University of Tokyo,Komaba 3-8-1, Meguro-ku, Tokyo 153-8902, Japan  Silvia
Toonen Anton Pannekoek Institute, University of Amsterdam, Science Park 904,
1098 XH Amsterdam, Netherlands  Eduardo Bravo E.T.S. Arquitectura del
Valle$\grave{s}$, Universitat Polite$\grave{c}$nica de Catalunya, Carrer Pere
Serra 1-15, 08173 Sant Cugat del Valle$\grave{s}$, Spain Robert Fisher

(Received October 18, 2021; Accepted November 17, 2021)

###### Abstract

Recent observational evidence has demonstrated that white dwarf (WD) mergers
are a highly efficient mechanism for mass accretion onto WDs in the galaxy. In
this paper, we show that WD mergers naturally produce highly-magnetized,
uniformly-rotating WDs, including a substantial population within a narrow
mass range close to the Chandrasekhar mass ($M_{\rm Ch}$). These near-$M_{\rm
Ch}$ WD mergers subsequently undergo rapid spin up and compression on a $\sim
10^{2}$ yr timescale, either leading to central ignition and a normal SN Ia
via the DDT mechanism, or alternatively to a failed detonation and SN Iax
through pure deflagration. The resulting SNe Ia and SNe Iax will have spectra,
light curves, polarimetry, and nucleosynthetic yields similar to those
predicted to arise through the canonical near-$M_{\rm Ch}$ single degenerate
(SD) channel, but with a $t^{-1}$ delay time distribution characteristic of
the double-degenerate (DD) channel. Furthermore, in contrast to the SD
channel, WD merger near-$M_{\rm Ch}$ SNe Ia and SNe Iax will not produce
observable companion signatures. We discuss a range of implications of these
findings, from SNe Ia explosion mechanisms, to galactic nucleosynthesis of
iron peak elements including manganese.

editorials, notices — miscellaneous — catalogs — surveys

††journal: The Astrophysical Journal††software:  SeBa (Toonen et al., 2012),
FLASH 4.3 (Fryxell et al., 2000; Dubey et al., 2012), FLASH SN Ia module
Townsley et al. (2016) (http://pages.astronomy.ua.edu/townsley/code), yt (Turk
et al., 2011), Python programming language (van Rossum & de Boer, 1991), Numpy
(van der Walt et al., 2011), IPython (Perez & Granger, 2007), Matplotlib
(Hunter, 2007)

##  1 Introduction

Type Ia supernovae (SNe Ia) are the radioactively-powered thermonuclear
explosions of white dwarfs (WDs), and play a crucial role as standardizable
candles for cosmology (Pankey, 1962; Phillips, 1993). The stellar progenitors
of SNe Ia have remained a subject of intense investigation for decades. For
many years, the single-degenerate (SD) channel, in which a main sequence or
red giant star donates material to a CO white dwarf until it reaches a near-
Chandrasekhar mass $M_{\rm Ch}$ and subsequently detonates as a SN Ia (Whelan
& Iben, 1973), was considered canonical. More recent evidence has begun to
favor the double-degenerate (DD) SNe Ia channel, resulting from the merger of
two white dwarfs (Webbink, 1984). However, numerous questions still surround
both channels, leaving the nature of the SNe Ia progenitors a largely unsolved
problem (Maoz et al., 2013).

A key outstanding puzzle surrounding SNe Ia stems from the nucleosynthesis of
iron group elements (IGEs). The fundamental physics of electron degeneracy
requires that the central density of WDs increases with increasing WD mass
(Chandrasekhar, 1935). Consequently, near-$M_{\rm Ch}$ WD progenitors undergo
more efficient electron capture in their cores than sub-$M_{\rm Ch}$
progenitors. This efficient electron capture in near-$M_{\rm Ch}$ WD
progenitors in turn results in enhanced abundances of neutron-rich isotopes,
most notably the radiosotope 55Fe, which decays to the monoisotopic 55Mn
(Röpke et al., 2012). Seitenzahl et al. (2013a) compellingly argue that a
substantial fraction of near-$M_{\rm Ch}$ WD events, as much as half of the
total SNe Ia rate, is required to produce a solar abundance of 55Mn. Detailed
galactic nucleosynthesis models find the fraction of near-$M_{\rm Ch}$ SNe Ia
to be as high as 75% in the solar neighborhood, though lower in dwarf galaxies
(Kobayashi et al., 2019).

Another puzzle surrounding near-$M_{\rm Ch}$ events is connected to
subluminous SNe Iax. SNe Iax are relatively commonplace transients, accounting
for 10- 30% of the total SNe Ia rate (Foley et al., 2013). The leading
theoretical models for SNe Iax are failed detonations of near-$M_{\rm Ch}$ WDs
(Jordan et al., 2012; Fink et al., 2014). Yet, the rates for the production of
near-$M_{\rm Ch}$ WDs predicted from binary population synthesis (BPS) models
through the canonical SD channel is typically an order of magnitude less than
the total SNe Ia rate, and depending on model assumptions, orders of magnitude
less (Maoz et al., 2014). Thus there is also a tension between the requisite
SNe Iax rate and that predicted from BPS models, if SNe Iax originate as
failed detonations of near-$M_{\rm Ch}$ WDs through the classical SD channel.

Furthermore, numerous observations place tight constraints on the prevalence
of near-$M_{\rm Ch}$ SNe Ia events produced through the canonical single-
degenerate channel. Observations of the nebular phase spectra of a large
sample of 111 SNe Ia including normal, 91T-like, and 91bg-like (but excluding
Ia-CSM and Iax) events found no evidence of H${\alpha}$ from the stripped
companions expected from the SD channel (Tucker et al., 2019), contrary to
theoretical expectations (Botyánszki et al., 2018). Similarly tight limits on
single-degenerate SNe Ia are obtained from companion (Li et al., 2011; Shappee
et al., 2016), and ex-companion (González Hernández et al., 2012; Schaefer &
Pagnotta, 2012a) searches, the delay-time distribution (DTD) (Strolger et al.,
2020), as well as radio (Chomiuk et al., 2016) and X-ray (Margutti et al.,
2012, 2014) constraints.

In this paper, we present a possible solution to the challenges posed by
near-$M_{\rm Ch}$ SNe Ia and SNe Iax. We propose that near-$M_{\rm Ch}$ SNe Ia
and SNe Iax are produced primarily not through the SD channel, as is widely
believed, but instead through the DD channel. Our line of reasoning builds
upon recent observational advances, showing that the galactic WD merger rate
exceeds the SN Ia rate (Maoz et al., 2018; Cheng et al., 2019), and that at
least half of the most massive WDs in the solar neighbourhood are merger
products (Kilic et al., 2021).

The production of super-$M_{\rm Ch}$ WD mergers through the DD channel exceeds
the predicted SD channel rate, provided that only a small fraction of
super-$M_{\rm Ch}$ mergers promptly detonate upon merger, as both recent
hydrodynamical simulations (Raskin et al., 2012, 2014; Kashyap et al., 2017)
and observations (Shen et al., 2018) suggest. Additional theoretical and
observational investigations have demonstrated that the product of the merger
is a high-field magnetic white dwarf (HFMWD) . We will show that a substantial
population of mergers result in rigidly-rotating WDs in a narrow mass range
close to $M_{\rm Ch}$ lose angular momentum through both a propeller-driven
phase as well as magnetic dipole radiation. These mergers have been previously
shown to undergo central ignition (Becerra et al., 2018, 2019). We will
consider both the possibility that these central ignitions lead to SNe Ia as
well as failed detonation SNe Iax, and explore a SN Ia model in detail.

The structure of this paper is as follows. In §2, we demonstrate how WD
mergers are naturally expected to give rise to near-$M_{\rm Ch}$ SNe Ia and
SNe Iax progenitors. In §3, we describe the methodology used to numerically
simulate a near-$M_{\rm Ch}$ DD SNe Ia event, including the calculation of its
detailed nucleosynthetic abundances and synthetic spectra. In §4, we present
the simulation results. In §5, we discuss the observational and theoretical
implications of WD merger near-$M_{\rm Ch}$ SNe Ia and SNe Iax. Finally, in
§6, we conclude.

##  2 Near-Chandrasekhar Mass SNe Ia From DD WD Mergers

###  2.1 Post-Merger HFMWD Production

Decades of theoretical work have elucidated the physical processes at work
during WD mergers. WDs in tight binaries formed through previous common
envelope phases of evolution (Nelemans & Tout, 2005; Di Stefano, 2010) are
driven together in a final inspiral by gravitational wave radiation. As the
secondary WD approaches the tidal radius, it is tidally disrupted, producing a
hot, flared accretion disk surrounding the primary WD (Benz et al., 1990;
Lorén-Aguilar et al., 2009). The differential rotation within the disk is
unstable to the magnetorotational instability (Balbus & Hawley, 1991), so that
even a dynamically weak seed field is rapidly amplified over several disk
rotational periods (Ji et al., 2013). Consequently, magnetohydrodynamics plays
a crucial role both in regulating the accretion and outflow rates, and in
determining the final thermodynamic structure and spin of the WD merger.

Figure 1: Schematic diagram showing the processes involved in the formation of
near-$M_{\rm Ch}$ SNe Ia from a DD WD merger. (a) The inspiral phase of the WD
merger lasts $\sim$ 1 - 10 Gyr. (b) The inspiral phase is followed by the
tidal disruption of the secondary on a timescale of $\sim 10^{3}$s. (c) The
differentially-rotating disk gives rise to the magnetorotational instability
(MRI), the accretion of which onto the primary produces a rigidly-rotating,
high field magnetic white dwarf (HFMWD). (d) As the primary accretes from the
disk, it gains angular momentum and spins up. On a timescale of $\sim
10^{6}$s, the system enters the propeller phase. During the propeller phase,
the primary ejects the remaining disk from the system and spins down. (e) The
misaligned magnetic and rotational axes of the the merger cause it to spin up
on a timescale of $\sim 10^{2}$yr and compress the core, resulting in central
ignition, and either a SN Ia or SN Iax.

###  2.2 Accretion onto Magnetically-Braked HFMWD

During the merger itself, material accreted from the tidally-disrupted
accretion disk will initially possess a higher specific angular momentum than
the primary WD. As a result, the accretion initially generates a differential
shear within the WD merger. However, the differential shear within the nascent
HFMWD results in the propagation of torsional Alfvén waves, which magnetically
brake the differential rotation of the HFMWD merger rapidly on a timescale of
minutes (Shapiro, 2000; Piro, 2008),

$t_{\rm A}={R\over v_{A}}\approx 7\times 10^{2}\left(B_{0}\over 10^{10}\ {\rm
G}\right)^{-1}\left(R_{\rm WD}\over 2\times 10^{3}\ {\rm
km}\right)^{-1/2}\left(M\over 1.4\ M_{\odot}\right)^{1/2}{\rm s}$ (1)

Here $M$ and $R_{\rm WD}$ are the radius and mass of the HFMWD merger, and
$B_{0}$ is the surface magnetic field strength. $v_{A}$ is the Alfvén velocity
$B/\sqrt{4\pi\rho}$. The Alfvénic timescale $t_{\rm A}$ is shorter than the
timescale to accrete the tidally-disrupted disk,

$t_{\rm acc}=\alpha^{-1}t_{\rm dyn}\approx 3\times
10^{3}\left(0.01\over\alpha\right)\left(R_{\rm disk}\over 5\times 10^{4}\ {\rm
km}\right)^{3/2}\left(M\over 1.4\ M_{\odot}\right)^{-1/2}{\rm s}$ (2)

Where $t_{\rm dyn}$ is the dynamical timescale corresponding to the outer disk
radius $R_{\rm disk}$. Consequently, the HFMWD is rapidly braked, and is in a
state of uniform rotation even as it continues to accrete mass from the disk.
This conclusion is confirmed by multidimensional MHD simulations of post-
merger accretion (Ji et al., 2013), where the resulting HFMWD is found to be
uniformly rotating.

Magnetohydroynamics is also responsible for converting differential shear into
magnetic energy through the magnetorotational (MRI) instability, and in
driving jets and outflows. As such, magnetohydrodynamics plays a leading role
in establishing the thermodynamic profile of the disk, and consequently key
observable properties of the merger, including its final radius and surface
temperature. The energy budget of the disk is particularly important because
the disk is non-degenerate, and net heating injected into it can result in
expanded merger sizes. In particular, if the deposition of differential shear
is converted into heat over a viscous timescale through the Shakura-Sunyaev
$\alpha$ prescription, the merger expands to $\sim 10^{10}-10^{11}$ cm (Shen
et al., 2012; Schwab et al., 2012). In contrast, magnetohydrodynamics converts
shear energy into magnetic energy, and propels a significant amount of energy
into jets and winds, producing WD-sized mergers with $\sim 10^{9}$ cm
envelopes (Ji et al., 2013; Becerra et al., 2018).

###  2.3 Propeller Regime

If the central WD is rotating sufficiently rapidly, then accretion from the
disk onto the WD cannot overcome its centrifugal barrier, and the system
enters into the propeller regime. Theory and simulations demonstrate that
accretion efficiency in the propeller regime is greatly reduced. Instead, in
the propeller regime, disk material is efficiently ejected into unbound
trajectories through magnetically-driven disk winds powered by the rotation of
the central star and torquing it down (Illarionov & Sunyaev, 1975; Ghosh &
Lamb, 1979; Lovelace et al., 1999; Romanova et al., 2005).

We now demonstrate that every double-degenerate white dwarf merger necessarily
enters into the propeller regime at some juncture in the accretion process,
provided solely that it does not first detonate promptly as a SN Ia. Ghosh &
Lamb (1979) defined the fastness parameter $\omega_{*}$ of the central WD as

$\omega_{*}={\Omega_{\rm WD}\over\Omega_{K}(R_{\rm in})}=\left(R_{\rm in}\over
R_{\rm co}\right)^{3/2}$ (3)

Here $R_{\rm co}$ is the corotation radius $R_{\rm CO}=(GM_{1}/\omega_{\rm
WD}^{2})^{1/3}$. When $\omega_{*}>1$, the disk is in the propeller regime. One
can estimate the inner disk radius $R_{\rm in}$ as a fraction $\xi$ of the
Alfvén radius, where the inflow velocity of spherical accretion equals the
Alfvén velocity, such that $R_{\rm in}=\xi R_{\rm A}$,

$R_{\rm A}=\left({\mu_{*}^{2}\over\sqrt{2GM}\dot{M}}\right)^{2/7}$ (4)

Here $\mu_{*}$ is the stellar magnetic dipole moment, and $\dot{M}$ is the
mass accretion rate. The propeller regime condition is then simply expressed
as $\Omega_{\rm WD}>\Omega_{K}(R_{\rm in})$, or

$\Omega_{\rm WD}>\left(GM_{\rm WD}\over\xi^{3}R_{A}^{3}\right)^{1/2}$ (5)

Solving for $\dot{M}$, the propeller regime condition is

$\dot{M}<\dot{M}_{\rm crit}={\xi^{7/2}B_{s}^{2}R_{\rm WD}^{6}\over\sqrt{2}\
G^{5/3}M_{\rm WD}^{5/3}}\Omega_{\rm WD}^{7/3}={\xi^{7/2}R_{\rm
WD}\over\sqrt{2}\ \Omega_{\rm max}}\left({\Omega_{\rm WD}\over\Omega_{\rm
max}}\right)^{7/3}B_{s}^{2}={\xi^{7/2}\over 2^{9/2}\pi^{2}}{\Phi_{s}^{2}\over
R_{\rm WD}^{2}v_{\rm max}}\left({\Omega_{\rm WD}\over\Omega_{\rm
max}}\right)^{7/3},$ (6)

where $v_{\rm max}=\Omega_{\rm max}R_{\rm WD}$ is the breakup velocity of the
WD, $\Omega_{\rm max}=\sqrt{GM_{\rm WD}/R_{\rm WD}^{3}}$ is the breakup
angular velocity of the WD, and $\Phi_{s}=4\pi R_{\rm WD}^{2}B_{s}$ is the
mean surface magnetic flux.

The rotation rate $\Omega_{\rm WD}$ of a rigidly-rotating WD cannot exceed the
maximal value, $\Omega_{\rm max}=\sqrt{GM_{\rm WD}/R_{\rm WD}^{3}}$.
Consequently, there is a minimum accretion rate $\dot{M}_{\rm crit}$ crossed
when the mass of the accretion disk falls beneath a minimum value, $M_{\rm
disk}\sim\dot{M}_{\rm crit}t_{\rm acc}$, below which the system must enter
into the propeller regime. The propeller regime has important ramifications
for double-degenerate systems. In particular, as a result of the propeller
regime, the entire mass of the donor cannot be efficiently accreted onto the
WD merger, and there is a minimum amount of mass, of order $\dot{M}_{\rm
crit}t_{\rm acc}$, which must be ejected from the system. We note that the
propelled mass is in addition to both the tidal tail (Benz et al., 1990;
Raskin & Kasen, 2013) and magnetically-driven jets and outflows (Ji et al.,
2013) arising during the earlier phases of the merger; combined this earlier
ejected mass amounts to $\sim$ few $\times 10^{-3}M_{\odot}$.

It is important to note that the critical mass accretion rate $\dot{M}_{\rm
crit}$ is in turn set by the surface magnetic field of the merger. The
propelled mass is clearly a magnetohydrodynamical phenomenon which is absent
from a purely hydrodynamical treatment of the merger.

An upper bound estimate of the mass ejected in the propeller phase can be
obtained by considering $\dot{M}_{\rm crit}$ and the spin-down timescale,
defined as $t_{\rm sd}\approx\Omega_{\rm WD}/|\dot{\Omega}|$. The spin-down of
the WD merger in the propeller phase is due to the magnetic torque acting at
the interface of magnetosphere and the inner radius of accretion disk;
simulations of the MRI show that the Reynolds stress is negligible in
comparison to the Maxwell stress (Davis et al., 2010). The spin-down of the WD
can then be calculated by integrating the magnetic torque over the inner edge
of the disk, assuming that the field at the inner edge of the disk is
dominated by a dipolar HFMWD field threading the inner disk during the
propeller phase, and the approximate equality of the radial and toroidal
components of the magnetic field, $B_{r}\approx B_{\phi}$ (Romanova et al.,
2003),

$I\frac{d\Omega}{dt}=-4\pi R_{\rm in}^{2}f\left(R_{\rm in}\frac{B_{R_{\rm
in}}^{2}}{8\pi}\right)$ (7)

Then, the spin-down timescale is given by

$t_{\rm sd}\approx\frac{\Omega_{\rm
WD}}{|\dot{\Omega}|}=2^{4/7}\beta\xi^{3}f^{-1}G^{1/14}B_{\rm
s}^{-2/7}\dot{M}^{-6/7}M_{\rm WD}^{15/14}R_{\rm WD}^{-5/14}\left(\Omega_{\rm
WD}\over\Omega_{\rm max}\right)$ (8)

Here $\beta=I/(MR^{2})$, where $I$ is the moment of inertia of the WD and $f$
is the fraction of the full spherical solid angle over which matter is being
propelled.

An upper bound estimate for the mass ejected from the system $M_{\rm ej}$ can
then be obtained by the product of the critical mass accretion rate
$\dot{M}_{\rm crit}$ and spin-down timescale $t_{\rm sd}$,

$M_{\rm ej}<\dot{M}_{\rm crit}t_{\rm sd}=2^{1/2}\beta\xi^{7/2}f^{-1}M_{\rm
WD}\left(\frac{\Omega_{\rm WD}}{\Omega_{\rm max}}\right)^{4/3}\approx
0.2M_{\odot}\ \left(\frac{\beta}{0.1}\right)\
\left(\frac{\xi}{1.0}\right)^{7/2}\
\left(\frac{f}{1.0}\right)^{-1}\left(\frac{M_{\rm WD}}{1.4\ M_{\odot}}\right)\
\ \left(\frac{\Omega_{\rm WD}}{\Omega_{\rm max}}\right)^{4/3}$ (9)

Physically, this upper-bound estimate on the propelled mass $M_{\rm ej}$ also
follows immediately from the conversion of rotational energy of the WD into
propelled mass being ejected at the escape speed at the Alfvén radius.
Consequently, because the rotational energy of the WD is the ultimate source
of energy of the propeller, this upper bound is independent of the magnetic
field strength. It is also important to note that the upper bound to the
ejected mass $M_{\rm ej}$ is also proportional to the rotational rate of the
WD to the $4/3$ power; mergers rotating more slowly than breakup will propel
less mass.

###  2.4 Central Ignition

An off-centered ignition burns through the WD envelope to its center on a
timescale of order $t_{\rm burn}\sim 10^{4}$ yr (Nomoto & Iben, 1985; Saio &
Nomoto, 1985; Schwab et al., 2016). The inwardly-propagating carbon burning
flame initiates a convective zone (Schwab et al., 2016), but continues to
stably progress inwards; Lecoanet et al. (2016) found the possibility that
mixing induced by convective overshoot was unlikely to disrupt the flame.
Consequently, in the absence of a strong magnetic field or substantial mass
loss, an off-centered ignition invariably results in the formation of a
neutron star (Schwab et al., 2016).

However, the magnetic field generated through the action of the turbulent MRI
and accreted onto the rigidly-rotating HFMWDs will in general be misaligned
from its rotational axis. Consequently, the HFMWD will undergo angular
momentum loss through the magnetic dipole radiation torque on a characteristic
timescale $t_{B}$,

$t_{B}\simeq{Ic^{3}\over B^{2}R^{6}\Omega^{2}}\approx 10^{2}\ {\rm
yr}\left(\frac{\beta}{0.1}\right)^{1}\left(\frac{M}{1.4M_{\odot}}\right)\left(\frac{B}{10^{10}\rm
G}\right)^{-2}\left(\Omega_{\rm max}\over\Omega\right)^{2}\left(\Omega_{\rm
max}\over{5\ {\rm s}^{-1}}\right)^{2/3}.$ (10)

Here $I$ is the moment of inertia of the WD, $I=\beta MR^{2}$, where
$\beta\simeq 0.1$ (Yoon & Langer, 2005), and $\Omega$ is the final angular
velocity.

The subsequent spin evolution of the WD hinges critically upon its mass. A
body of literature, beginning with Finn & Shapiro (1990) and Shapiro et al.
(1990) showed that relativistically-degenerate stars actually spin up and
compress their cores with angular momentum loss. This seemingly non-intuitive
effect is driven fundamentally by the compression induced by angular momentum
loss, since the rigidly-rotating near-$M_{\rm Ch}$ WD sequence greatly reduces
its moment of inertia as it loses angular momentum. Geroyannis & Papasotiriou
(2000) expanded on these findings, generating evolutionary sequences of
rapidly-rotating magnetic WDs, and demonstrating that near-$M_{\rm Ch}$ WDs
with masses $\geq 1.32M_{\odot}$ continue to spin up, and would compress their
cores to extremely high densities $\rho_{c}=4\times 10^{10}$ g cm-3, in the
absence of nuclear reactions.

Becerra et al. (2018) showed that WD mergers with accretion rates set by the
viscous timescale initially undergo rapid spin up due to the torque exerted by
the accreting material. When the Alfvén radius becomes larger than the WD
radius, the remnant system enters the propeller phase, spinning down and
compressing the WD and causing the central density to rapidly rise.
Subsequently, Becerra et al. (2018) demonstrated that the timescale of the
evolution of the WD is governed by the magnetic dipole radiation torque
timescale $t_{B}$. In a subsequent paper, Becerra et al. (2019) examined the
influence of a strong magnetic field misaligned with the rotational axis of an
isolated WD, appropriate for this final phase of evolution of the merger,
following the propeller phase. Becerra et al. (2019) demonstrated that the
combined effects of neutrino cooling and angular momentum loss due to dipole
emission lead to the central nuclear ignition of the merger for magnetic field
strengths in excess of $10^{7}$ G. Their detailed models predict timescales of
the same order as our simpler estimate for $t_{B}$ (Becerra et al., 2019).

As a consequence of the relativistically-degenerate cores of near-$M_{\rm Ch}$
WDs, uniformally-rotating WDs will undergo spin up and compression until they
reach carbon ignition in their cores. The magnetic dipole radiation timescale
$t_{B}$ given by equation 10 predicts that for a magnetic field of $10^{6}$ G,
a typical field strength for field WDs (Ferrario et al., 2015), $t_{B}$
exceeds the Hubble time. Thus, we emphasize that the production of a HFMWD
crucially enables this process of dipole-driven central ignition to be
astrophysically relevant.

Figure 2: The delay time distribution of double WD mergers, along with the
distribution of near-$\rm M_{ch}$ CO mergers, obtained from the $\alpha\alpha$
and $\alpha\gamma$ BPS models. The near-$\rm M_{ch}$ WDs have a total mass
$M\geq 1.38M_{\odot}$ and $M\leq 1.48M_{\odot}$.

###  2.5 WD Merger Rate and Delay Time Distribution

Next, to examine the rate of near-$M_{\rm Ch}$ DD mergers and their associated
delay time distribution, we employ two sets of binary population synthesis
(BPS) models from Toonen et al. (2012, 2017) using the SeBa code (Portegies
Zwart & Verbunt, 1996; Toonen et al., 2012). Both models assume a single burst
of star formation, with initial periods determined by a flat distribution in
the log of the initial semi-major axis. The models differ in the treatment of
the common-envelope (CE) phase during the binary merger (Toonen et al., 2017).
Model $\alpha\alpha$ assumes conservation of orbital energy for every common
envelope phase (Webbink, 1984). In contrast, model $\alpha\gamma$ adopts
angular momentum conservation (Nelemans et al., 2000), unless a common
envelope phase is initiated by a tidal instability, or if the binary contains
a WD. Toonen et al. (2012, 2017) provide further details on the BPS models.

Figure 2 shows the delay time distribution (DTD) of all the WD mergers and the
subset of near-$M_{\rm Ch}$ CO WD mergers for both common envelope
prescriptions, along with a $\rm t^{-1}$ power law, consistent with the
observed delay time distribution (Maoz et al., 2014). To create this DTD plot,
we first created 20 equally spaced delay time bins. Then, we divided the
number of mergers in each bin by the total BPS simulation mass to obtain the
specific number of WD mergers per solar mass. Finally, the specific number of
WD mergers is divided by the width of each delay time bin, yielding the
specific rate of WD mergers per solar mass per year.

We calculate the fraction of WD mergers producing a double CO WD binary under
two assumptions regarding mass accretion and ejection during the merger. In
the first model, we assume mass accretion is completely conservative during
the merger, without any mass lost through any of the possible processes
discussed previously in §2 : tidal tails, magnetized jets and outflows, and
propeller-driven outflows. In the second model variation, we assume non-
conservative mass accretion, with the maximal propelled mass according to eqn.
9. These two model variations span the possible range of mass transfer during
the merger. In particular, in order to produce a near-$M_{\rm Ch}$ mass
merger, mass ejection requires even more massive, and consequently rarer, WDs.
As a result, fully conservative and maximally non-conservative mass accretion
mergers yield upper and lower bounds, respectively, to the fraction of
near-$M_{\rm Ch}$ WDs produced. The fraction of near-$M_{\rm Ch}$ CO WD
mergers is then estimated by computing the fraction of total post-merger
system masses $M$ spanning the range of bounds for rigidly-rotating near
$M_{\rm Ch}$ WDs ($1.38M_{\odot}\leq M\leq 1.48M_{\odot}$) from the BPS
simulation datasets. The $1.38M_{\odot}$ lower bound mass bound is adopted
from the minimum mass of a rigidly-rotating WD which undergoes spin up upon
angular momentum loss in full GR, and the upper bound of $1.48M_{\odot}$ the
maximum stable mass of a C-O rigidly-rotating WD (Boshkayev et al., 2013).

The maximally non-conservative mass transfer assumption yields 2.7% and 2.0%
of near-$M_{\rm Ch}$ CO WD systems for $\alpha\alpha$ and $\alpha\gamma$
models, respectively. Similarly, under the fully-conservative mass transfer
assumption, these fractions increase to 3.5% and 3.1% for $\alpha\alpha$ and
$\alpha\gamma$ models, respectively. Combined, we estimate that near-$M_{\rm
Ch}$ CO WD mergers may account for 2% - 3.5% of all WD mergers, with the
estimates spanning the range of maximally non-conservative to fully-
conservative mass transfer during the merger, and accounting for both the
$\alpha\alpha$ and $\alpha\gamma$ common envelope prescriptions. A radial
velocity survey by Maoz et al. (2018) shows that the double WD (DWD) merger
rate is approximately 6 times the specific SN Ia rate in our galaxy. Thus,
near-$M_{\rm Ch}$ DWD mergers may account for up to 12% - 21% of all SNe Ia
events, if all near-$M_{\rm Ch}$ CO DWD mergers result in a SN Ia.

However, because the fundamental physical process of detonation initiation in
a turbulent, unconfined medium is an extremely challenging one to address
(Poludnenko et al., 2011; Fenn & Plewa, 2017; Fisher et al., 2019; Poludnenko
et al., 2019), the explosive outcome of a near-$M_{\rm Ch}$ WD remains
uncertain from a first-principles standpoint. A central ignition in the core
of the near-$M_{\rm Ch}$ WD leads to a deflagration bubble (Nomoto et al.,
1984) at a single offset location, according to ab initio numerical
simulations (Zingale et al., 2011; Nonaka et al., 2012a; Malone et al., 2014).
This flame bubble will buoyantly rise and become unstable to the reactive
Rayleigh-Taylor and Kelvin-Helmholtz instabilities, driving turbulence that
may cause the deflagration to transition to a detonation (Khokhlov, 1991), and
thereby lead to a SN Ia. Alternatively, the bubble may continue to buoyantly
rise until it erupts from the surface of the WD. Under this second
possibility, a portion of the buoyant ash will be expelled from the
gravitationally-bound WD, which is kicked in the opposite direction and
enriched from the remaining ash, which falls back onto the surface of the WD.
Observational (Jha, 2017) and theoretical evidence (Jordan et al., 2012; Fink
et al., 2014) points towards the possibility that at least some of these
failed detonation events will be visible as SNe Iax (Foley et al., 2013;
Stritzinger et al., 2015), and leave behind surviving kicked WDs enriched by
ash rich in IGEs resulting from high-density burning, such as LP 40-365 (Raddi
et al., 2019).

Therefore, if alternatively a near-$M_{\rm Ch}$ DWD merger ignites but fails
to detonate, it will instead produce a SN Iax (Jordan et al., 2012; Fink et
al., 2014). This near-$M_{\rm Ch}$ DWD merger rate is close to the 10 - 30% of
the total SNe Ia rate estimated for SNe Iax by Foley et al. (2013), suggesting
that near-$M_{\rm Ch}$ DWD mergers may largely account for the SNe Iax
population if the majority of mergers ignite as pure deflagrations but fail to
detonate.

##  3 Hydrodynamical Simulation and Postprocessing Methodology

We next consider the observable signatures of SNe Ia DD near-$M_{\rm Ch}$
events. In order to capture the detailed nucleosynthetic yields and synthetic
spectral signatures of SNe Ia events originating DD near-$M_{\rm Ch}$ mergers,
we turn to hydrodynamical simulations, combined with a nucleosynthetic and
radiative transfer post-processing pipeline.

In the following analysis, we will consider the possibility that the DD
near-$M_{\rm Ch}$ WD progenitor leads to successful SNe Ia via the DDT
mechanism. We adopt an equilibrium WD progenitor model with a cold degenerate
core and a hot, thick isentropic envelope to model the end state of the merger
following spin down and ignition (Yoshida, 2019). In particular, as described
above, he strong magnetic fields cause nascent HFMWDs to be in a state of
uniform rotation. The addition of rigid rotation has only a minimal impact on
the density structure of the WD (Ostriker & Bodenheimer, 1968; Yoshida, 2019),
so we utilize a non-rotating WD model for the purpose of computing
nucleosynthetic yields and synthetic spectra. We further neglect the influence
of the magnetic field upon the WD structure, as the pressure is dominated by
the degenerate electron pressure for HFMWDs with surface fields of $10^{10}$ G
and less (Ji et al., 2013), and simulate the evolution in pure hydrodynamics.

The central density of a near-$M_{\rm Ch}$ SN Ia event plays a primary role in
determining the nucleosynthesis of key iron group elements including 56Ni as
well as neutronized isotopes such as 55Fe, the parent isotope of 55Mn. For
illustrative purposes, we adopt only a single set of representative parameters
for the merger for this paper. We select a model with a central density
$2\times 10^{9}$ g cm-3, which has been widely adopted for near-$M_{\rm Ch}$
SNe Ia (Woosley, 1997; Nomoto et al., 1997). The total mass of the model is
$1.378M_{\odot}$, with a core mass of $0.795M_{\odot}$ and an envelope mass of
$0.593M_{\odot}$. If the merger process conserved mass, the core and envelope
masses would equate to the primary and secondary WD masses, respectively. In
reality, mass accretion will be non-conservative due to
magnetohydrodynamically-driven jets and winds and tidal tail ejection,
implying the secondary mass will be slightly more massive than the
$0.593M_{\odot}$ of the envelope in order to account for mass ejected from the
system.

As noted in section 2.2, the temperature of the envelope is the result of the
complex magnetohydrodynamical processes involved during the merger, including
the development of MRI turbulence and a magnetized corona, as well as jets and
outflows. In our model, this complexity is subsumed into the entropy of the
envelope, or equivalently the temperature at the base of the envelope. The
primary WD is modeled as isothermal with a temperature of $10^{8}$ K, while
the tidally-disrupted secondary forms a hot, thick isentropic envelope around
the primary with a base temperature $10^{9}$ K. The composition of the
hydrodynamical model is taken to be 50/50 C/O, though we also consider the
effect of solar metallicity in nucleosynthetic post-processing.

As noted above, while there has been significant advances made in our
understanding of turbulent detonation initiation, much work still needs to be
done to incorporate these physically-realistic detonation mechanisms into
first-principles multidimensional simulations of SNe Ia. However, the ab
initio simulations of the deflagration phase point towards the ignition of a
single buoyancy-driven bubble with an mean offset of 40 km (Nonaka et al.,
2012b; Zingale et al., 2011; Malone et al., 2014). While the process of
ignition is inherently a stochastic one, the ab initio simulations indicate
the majority of ignitions lead to buoyancy-driven deflagration phases with
weak pre-expansion (Fisher & Jumper, 2015; Byrohl et al., 2019). Furthermore,
recent advances regarding turbulently-driven detonation initiation point
towards a higher DDT transition density, $\sim 10^{8}$ g cm-3, earlier than
the distributed burning density of $\sim 10^{7}$ g cm-3 previously assumed
(Poludnenko et al., 2019).

Accordingly, in our DDT model, we adopt a simplified explosion scenario.
Incorporating these recent SN IA combustion advances, we neglect the weak pre-
expansion of the deflagration phase, and centrally detonate the merger model
with a detonation radius of $23$ km. The hydrostatic progenitor model is
mapped to 2D axisymmetry in the Eulerian adaptive mesh refinement (AMR) code
FLASH (Fryxell et al., 2000; Dubey et al., 2009, 2014).111Though the model
presented here is essentially spherically symmetric, the 2D axisymmetric setup
enables broader sets of initial conditions for comparison against other
published runs. We ran the hydrodynamical simulation on a 2D azimuthally-
symmetric cylindrical $r-z$ domain, with $z$ axis extending from
$-1.31072\times 10^{6}$ km to $1.31072\times 10^{6}$ km, and $r$ axis
extending from $0$ to $1.31072\times 10^{6}$ km. The region surrounding the WD
in the domain is filled with a low density matter with negligible inertia and
thermal energy, having an initial density of $10^{-3}$ g cm-3 and a
temperature of $3\times 10^{7}$ K. FLASH utilizes (Timmes & Swesty, 2000). An
advection-diffusion-reaction equation is employed for the , along with a
simplified treatment of (Townsley et al., 2007, 2009). The Poisson equation
for self-gravity is solved using a multipole moment method (Couch et al.,
2013) with isolated boundary conditions, including terms up to $l=6$ in the
expansion. In the DDT model, the system is evolved until it reaches free
expansion. To follow the nuclear burning regions with maximal refinement, and
to derefine regions with low density, we use several refinement criteria based
on Townsley et al. (2007) and Townsley et al. (2009). In our run, refinement
and derefinement of the grid is based on a dimensionless gradient parameter of
density and burnt fraction, a scalar field which tracks the , and ranges from
0 for pure fuel to 1 for pure ash (Townsley et al., 2016). The maximum spatial
resolution for the hydrodynamical simulations presented here is $4$ km.

We also incorporate passively-advected Lagrangian tracer particles (Dubey et
al., 2012) to record the thermodynamic state along the particle trajectories
throughout the simulation. The tracer particles are initialized proportional
to the mass in the simulation domain. In the runs presented here, we used
$10^{4}$ particles, which provides good precision for near-$M_{\rm Ch}$ WDs in
2D (Seitenzahl et al., 2010). Thermodynamic histories of these particles are
restructured into trajectory files which are then post-processed in Torch
nuclear network (Timmes, 1999), with 489 species, to obtain the detailed
nucleosynthetic yields. A convergence test for sample trajectories showed that
the maximum final abundance error for integration time step tolerances using
abundance normed errors between $10^{-6}$ and $10^{-8}$ was $\sim 0.3\%$.
Consequently, in the results presented here, an integration timestep tolerance
of $10^{-6}$ was used for all tracer particles.

To generate synthetic spectra, we map the nucleosynthetic yields onto a 2D
axisymmetric velocity mesh. This homologously-expanding mesh is then followed
with the radiative transport code SuperNu (Wollaeger et al., 2013; Wollaeger &
van Rossum, 2014), which employs the implicit Monte Carlo (IMC) and discrete
diffusion Monte Carlo (DDMC) methods. We bin photons at every time step into
512 logarithmically-spaced wavelength bins within 103 – 104.5 Å. To classify
the synthetic spectra, we use the supernova identification code SNID (Blondin,
2011), which cross correlates the input model spectra with templates of
previously-observed events, and identifies the best-matched supernova type as
well as its epoch.

##  4 Simulation Results

###  4.1 DD Near-Chandrasekhar Mass DDT

The hydrodynamical simulation of the detonation through free expansion yields
a total nuclear energy of $2.13\times 10^{51}\rm erg$. Overall, the energetics
and nucleosynthetic yields are comparable to the most energetic deflagration-
to-detonation near-$M_{\rm Ch}$ previously published (Seitenzahl et al.,
2013b; Ohlmann et al., 2014; Leung & Nomoto, 2018), though with the marked
difference of less ${}^{28}\rm Si$, owing to the less vigorous pre-expansion
resulting from our deflagration and turbulently-driven deflagration-to-
detonation model assumptions. Table 1 contains stable mean nucleosynthetic
yields for isotopes with abundances greater than $10^{-6}M_{\odot}$. Complete
mean nucleosynthetic yields, which include both stable and radioisotopes
obtained at the onset of free expansion, $t=2.5$ s, with abundances greater
than $10^{-6}M_{\odot}$ are given in Table 2.

Isotope | $Z=0$ | $Z=Z_{\odot}$
---|---|---
${}^{4}\rm He$ | $9.21\times 10^{-3}$ | $8.09\times 10^{-3}$
${}^{12}\rm C$ | $2.85\times 10^{-6}$ | $3.53\times 10^{-6}$
${}^{16}\rm O$ | $9.33\times 10^{-4}$ | $1.04\times 10^{-3}$
${}^{24}\rm Mg$ | $7.78\times 10^{-5}$ | $1.55\times 10^{-5}$
${}^{28}\rm Si$ | $7.76\times 10^{-3}$ | $7.95\times 10^{-3}$
${}^{29}\rm Si$ | $1.97\times 10^{-6}$ | $3.84\times 10^{-6}$
${}^{30}\rm Si$ | $8.23\times 10^{-6}$ | $4.68\times 10^{-6}$
${}^{31}\rm P$ | $1.81\times 10^{-6}$ | $3.47\times 10^{-6}$
${}^{32}\rm S$ | $4.93\times 10^{-3}$ | $4.80\times 10^{-3}$
${}^{33}\rm S$ | $2.59\times 10^{-7}$ | $3.62\times 10^{-6}$
${}^{34}\rm S$ | $3.93\times 10^{-6}$ | $2.64\times 10^{-5}$
${}^{35}\rm Cl$ | $1.43\times 10^{-6}$ | $2.96\times 10^{-6}$
${}^{36}\rm Ar$ | $1.22\times 10^{-3}$ | $1.12\times 10^{-3}$
${}^{38}\rm Ar$ | $5.69\times 10^{-7}$ | $1.60\times 10^{-5}$
${}^{39}\rm K$ | $1.70\times 10^{-6}$ | $4.23\times 10^{-6}$
${}^{40}\rm Ca$ | $1.32\times 10^{-3}$ | $1.18\times 10^{-3}$
${}^{44}\rm Ca$ | $1.10\times 10^{-5}$ | $8.67\times 10^{-6}$
${}^{48}\rm Ti$ | $6.29\times 10^{-5}$ | $5.55\times 10^{-5}$
${}^{49}\rm Ti$ | $5.09\times 10^{-7}$ | $2.67\times 10^{-6}$
${}^{51}\rm V$ | $1.12\times 10^{-5}$ | $1.59\times 10^{-5}$
${}^{50}\rm Cr$ | $3.88\times 10^{-5}$ | $4.74\times 10^{-5}$
${}^{52}\rm Cr$ | $1.26\times 10^{-3}$ | $1.19\times 10^{-3}$
${}^{53}\rm Cr$ | $2.31\times 10^{-4}$ | $3.07\times 10^{-4}$
${}^{55}\rm Mn$ | $9.19\times 10^{-3}$ | $1.02\times 10^{-2}$
${}^{54}\rm Fe$ | $7.79\times 10^{-2}$ | $8.52\times 10^{-2}$
${}^{56}\rm Fe$ | $9.98\times 10^{-1}$ | $9.60\times 10^{-1}$
${}^{57}\rm Fe$ | $4.24\times 10^{-2}$ | $4.72\times 10^{-2}$
${}^{59}\rm Co$ | $1.45\times 10^{-3}$ | $1.54\times 10^{-3}$
${}^{58}\rm Ni$ | $1.98\times 10^{-1}$ | $2.26\times 10^{-1}$
${}^{60}\rm Ni$ | $1.78\times 10^{-2}$ | $1.45\times 10^{-2}$
${}^{61}\rm Ni$ | $7.18\times 10^{-4}$ | $7.90\times 10^{-4}$
${}^{62}\rm Ni$ | $4.61\times 10^{-3}$ | $6.42\times 10^{-3}$
${}^{63}\rm Cu$ | $4.75\times 10^{-5}$ | $6.58\times 10^{-6}$
${}^{65}\rm Cu$ | $5.22\times 10^{-6}$ | $5.69\times 10^{-6}$
${}^{64}\rm Zn$ | $2.77\times 10^{-4}$ | $8.66\times 10^{-5}$
${}^{66}\rm Zn$ | $8.67\times 10^{-5}$ | $1.17\times 10^{-4}$
${}^{68}\rm Zn$ | $2.04\times 10^{-6}$ | $1.35\times 10^{-7}$
${}^{70}\rm Ge$ | $2.31\times 10^{-6}$ | $2.31\times 10^{-6}$

Table 1: DDT stable mean nucleosynthetic yields in solar masses. Only isotopes
in excess of $10^{-6}\ M_{\odot}$ in either model are shown.

Note. — Abundances for all isotopes are provided in machine-readable format.

Isotope | $Z=0$ | $Z=Z_{\odot}$
---|---|---
${}^{4}\rm He$ | $9.21\times 10^{-3}$ | $8.09\times 10^{-3}$
${}^{12}\rm C$ | $2.85\times 10^{-6}$ | $3.53\times 10^{-6}$
${}^{16}\rm O$ | $9.33\times 10^{-4}$ | $1.04\times 10^{-3}$
${}^{24}\rm Mg$ | $7.78\times 10^{-5}$ | $1.55\times 10^{-5}$
${}^{28}\rm Si$ | $7.76\times 10^{-3}$ | $7.95\times 10^{-3}$
${}^{29}\rm Si$ | $7.47\times 10^{-7}$ | $3.84\times 10^{-6}$
${}^{30}\rm Si$ | $1.55\times 10^{-7}$ | $4.65\times 10^{-6}$
${}^{30}\rm P$ | $5.85\times 10^{-6}$ | $3.38\times 10^{-8}$
${}^{30}\rm S$ | $2.22\times 10^{-6}$ | $7.15\times 10^{-12}$
${}^{31}\rm S$ | $9.57\times 10^{-7}$ | $5.44\times 10^{-8}$
${}^{32}\rm S$ | $4.93\times 10^{-3}$ | $4.80\times 10^{-3}$
${}^{34}\rm S$ | $1.62\times 10^{-6}$ | $2.64\times 10^{-5}$
${}^{34}\rm Cl$ | $1.65\times 10^{-6}$ | $5.59\times 10^{-9}$
${}^{35}\rm Cl$ | $1.11\times 10^{-6}$ | $2.94\times 10^{-6}$
${}^{36}\rm Ar$ | $1.22\times 10^{-3}$ | $1.12\times 10^{-3}$
${}^{38}\rm Ar$ | $1.75\times 10^{-7}$ | $1.60\times 10^{-5}$
${}^{39}\rm K$ | $1.61\times 10^{-6}$ | $4.17\times 10^{-6}$
${}^{40}\rm Ca$ | $1.32\times 10^{-3}$ | $1.18\times 10^{-3}$
${}^{44}\rm Ti$ | $1.10\times 10^{-5}$ | $8.67\times 10^{-6}$
${}^{48}\rm Cr$ | $6.29\times 10^{-5}$ | $5.55\times 10^{-5}$
${}^{49}\rm Cr$ | $5.08\times 10^{-7}$ | $2.67\times 10^{-6}$
${}^{50}\rm Cr$ | $3.88\times 10^{-5}$ | $4.74\times 10^{-5}$
${}^{51}\rm Mn$ | $1.11\times 10^{-5}$ | $1.58\times 10^{-5}$
${}^{53}\rm Mn$ | $3.25\times 10^{-6}$ | $4.29\times 10^{-6}$
${}^{52}\rm Fe$ | $1.26\times 10^{-3}$ | $1.19\times 10^{-3}$
${}^{53}\rm Fe$ | $2.28\times 10^{-4}$ | $3.03\times 10^{-4}$
${}^{54}\rm Fe$ | $7.79\times 10^{-2}$ | $8.52\times 10^{-2}$
${}^{55}\rm Fe$ | $1.55\times 10^{-4}$ | $1.83\times 10^{-4}$
${}^{56}\rm Fe$ | $5.38\times 10^{-5}$ | $7.85\times 10^{-5}$
${}^{55}\rm Co$ | $9.04\times 10^{-3}$ | $1.00\times 10^{-2}$
${}^{56}\rm Co$ | $1.48\times 10^{-4}$ | $1.60\times 10^{-4}$
${}^{57}\rm Co$ | $8.77\times 10^{-5}$ | $1.04\times 10^{-4}$
${}^{56}\rm Ni$ | $9.98\times 10^{-1}$ | $9.60\times 10^{-1}$
${}^{57}\rm Ni$ | $4.24\times 10^{-2}$ | $4.71\times 10^{-2}$
${}^{58}\rm Ni$ | $1.98\times 10^{-1}$ | $2.26\times 10^{-1}$
${}^{59}\rm Ni$ | $1.98\times 10^{-4}$ | $2.21\times 10^{-4}$
${}^{60}\rm Ni$ | $1.35\times 10^{-4}$ | $1.63\times 10^{-4}$
${}^{58}\rm Cu$ | $2.80\times 10^{-4}$ | $2.22\times 10^{-6}$
${}^{60}\rm Cu$ | $8.22\times 10^{-5}$ | $6.92\times 10^{-5}$
${}^{61}\rm Cu$ | $1.18\times 10^{-5}$ | $1.33\times 10^{-5}$
${}^{60}\rm Zn$ | $1.76\times 10^{-2}$ | $1.43\times 10^{-2}$
${}^{61}\rm Zn$ | $7.07\times 10^{-4}$ | $7.77\times 10^{-4}$
${}^{62}\rm Zn$ | $4.61\times 10^{-3}$ | $6.42\times 10^{-3}$
${}^{63}\rm Zn$ | $4.38\times 10^{-6}$ | $3.29\times 10^{-6}$
${}^{63}\rm Ga$ | $4.26\times 10^{-5}$ | $3.28\times 10^{-6}$
${}^{64}\rm Ga$ | $4.65\times 10^{-6}$ | $1.93\times 10^{-6}$
${}^{64}\rm Ge$ | $2.72\times 10^{-4}$ | $8.46\times 10^{-5}$
${}^{65}\rm Ge$ | $4.69\times 10^{-6}$ | $5.09\times 10^{-6}$
${}^{66}\rm Ge$ | $8.66\times 10^{-5}$ | $1.17\times 10^{-4}$
${}^{68}\rm Se$ | $1.92\times 10^{-6}$ | $8.50\times 10^{-8}$
${}^{70}\rm Se$ | $2.31\times 10^{-6}$ | $2.84\times 10^{-6}$

Table 2: DDT frozen-out mean nucleosynthetic yields in solar masses. Only
isotopes in excess of $10^{-6}\ M_{\odot}$ in either model are shown.

Note. — Abundances for all isotopes are provided in machine-readable format.

Plots of the frozen-out electron abundance fraction $Y_{e}$ =
$\sum_{i}X_{i}Z_{i}/A_{i}$, where $Z_{i}$ is the atomic number, $A_{i}$ is the
atomic mass number, and $X_{i}$ is the mass abundance of the isotope $i$, are
depicted near the onset of homologous expansion in Figure 3. Consistent with
the findings of previous simulations of near-$M_{\rm Ch}$ SNe Ia DDT models
(including Meyer et al. 1996; Nomoto et al. 1997; Woosley 1997; Brachwitz et
al. 2000; Krueger et al. 2012), we find that the greater electron capture
rates in the cores of denser WD progenitors lead to greater abundances of
stable IGEs in the n-NSE regime.

Figure 3: Slice plot of $Y_{e}$ at $t=$ 240 ms, when the nucleosynthetic
yields have frozen out, showing the centrally-concentrated region of
neutronized isotopes produced from electron captures. Note that this plot
represents the hydrodynamic model output, which is at zero WD progenitor
metallicity. The $10^{6}$ g cm-3 density isocontour is represented by a black
line.

Next, in conducting our synthetic spectroscopic classification, we compare not
to a single event, but rather to a template library, just as new optical
transients are classified observationally. Accordingly, we utilize the
supernova identification code SNID, which defines the quality parameter rlap
to quantify the strength of the correlation between the input synthetic
spectra and previously observed events (Blondin, 2011). Tonry & Davis (1979)
defined the parameter $r$ in terms of the ratio of the height $h$ to the root
mean square of the antisymmetric term of the normalized cross-correlation
function between the input and the template spectra, $\sigma_{a}$, so that
$r=2^{-1/2}h\sigma_{a}^{-1}$. In addition, SNID also uses the overlap of the
spectra in log wavelength space to quantify the reliability of the correlation
peak. To match with the wavelength range of the input spectrum at the
correlation peak redshift, the template spectra are trimmed. Then the overlap
parameter, named lap, in log wavelength space is given by $0\leq$ lap
$\leq\ln(\lambda_{1}/\lambda_{0})$, where $[\lambda_{0},\lambda_{1}]$ is the
rest-frame wavelength range of the input spectrum. Values of rlap, defined as
the product of $r$ and lap, over 5 are considered to be good matches. We
constrain our synthetic spectra to the rest frame at redshift $z=0$.

SNID classifies the best-matches to the synthetic spectra at peak-brightness
for both WD progenitor metallicities $Z=0$ and $Z=Z_{\odot}$ as SN 1999ee, a
type Ia-norm, at an epoch of -9.1 d (Hamuy et al., 2002). These matches have
rlap scores 7.5 and 8.6, respectively. Figure 4 shows the synthetic spectrum
for $Z=Z_{\odot}$, plotted against its best-match spectrum. We emphasize that
the match is obtained directly from the near-$M_{\rm Ch}$ progenitor without
any fine-tuned parameters. The overall agreement between the two spectra is
quite good, with the noted exception of a blueshift in the synthetic spectral
Si II absorption feature near 6000 Å, in comparison to SN 1999ee, as well as
in the pre-peak epoch identification. The inclusion of non-LTE in the
radiative transfer is expected to shift the Si II feature towards the red,
which may partially account for this disagreement (Shen et al., 2021). The
blueward shift of the Si II feature caused by the LTE approximation may also
partially account for the SNID epoch identification prior to peak brightness,
as the Si II feature generally shifts redward at later epochs (Pereira et al.,
2013).

We also note that SN 1999ee is a member of a class of SNe Ia with high-
velocity Si II and Ca II features (HVFs). The origin of these features in SN
1999ee and other similar SNe Ia HVF events has been discussed in the
literature, with possibilities ranging from density and abundance enhancements
in the progenitor WD to interactions with circumstellar material (Mazzali et
al., 2005). Our models support the viewpoint that the high velocity Si II HVF
in particular may be connected to nuclear burning on massive white dwarfs
(Kato et al., 2018).

To compare against observations of supernova remnants, we consider stable
isotopic abundances. Because supernova remnants typically have ages of
hundreds to thousands of years, in the decay chain ${}^{53}_{26}\rm Fe$
$\xrightarrow{\text{9m}}$ ${}^{53}_{25}\rm Mn$ $\xrightarrow{\text{4Myr}}$
${}^{53}_{24}\rm Cr$, we restrict the ${}^{53}_{26}\rm Fe$ decay to
${}^{53}_{25}\rm Mn$ only (Unterweger et al., 1992). The decayed IGE abundance
ratios relative to solar abundances, with a WD progenitor with solar
metallicity $Z_{\odot}=0.0139$ (Asplund et al., 2021), and C/O ratio of unity
yield [Ni/Fe] = 0.62, [Cr/Fe] = -1.11 and [Mn/Fe] = 0.02, all relative to
solar. The [Ni/Fe] ratio is higher than Tycho or Kepler, but comparable to SNR
3C 397. We emphasize that this comparison is based upon our specific choice of
parameters, and is therefore illustrative rather than exhaustive of the range
of possibilities of DD near-$M_{\rm Ch}$ SNe Ia.

Figure 4: Synthetic peak brightness model spectrum (black) at WD progenitor
metallicity $Z=Z_{\odot}$ compared with the best matching spectral template,
SN 1999ee, a normal Ia (red).

##  5 Observational Implications

The classical single-degenerate near-$M_{\rm Ch}$ DDT mechanism requires fine-
tuning of the ignition in order to obtain the vigorous pre-expansion necessary
for a normal SN Ia. In contrast, DD near-$M_{\rm Ch}$ SNe Ia naturally begin
with an extended heated envelope of tidally-disrupted material from the
secondary. The braking timescale is less than the cooling timescale for the WD
merger for a HFMWD, allowing the WD progenitor to retain this extended
envelope at the onset of detonation. As a direct result, a DDT within a DD
near-$M_{\rm Ch}$ naturally predicts a normal SN Ia with high velocity Si II
features without any fine-tuning of parameters, even though the silicon
abundance is significantly sub-solar.

A similar magnetic dipole mechanism for nuclear ignition as discussed here is
also envisioned for the core degenerate scenario (Ilkov & Soker, 2012). The
two scenarios differ in the physical explanation for the delay-time
distribution. In the core degenerate scenario, the delay-time distribution is
explained by the magnetic braking timescale, whereas in our scenario the DTD
is explained instead by gravitational wave inspiral, as in the standard
double-degenerate scenario. Because the core degenerate scenario predicts a
long post-merger delay, and our proposed scenario a relatively prompt delay,
the two scenarios offer different possibilities for SN ejecta interactions,
which can be used in observational tests.

SNe Ia and their remnants are known to have a high degree of spherical
symmetry. Spectropolarimetry of normal SNe Ia generally reveal a low degree of
polarization, typically 0.5% in the Si II absorption line feature at peak
brightness, which is a sensitive probe of the outskirts of the SN (Wang &
Wheeler, 2008; Cikota et al., 2019). This degree of polarization in SNe Ia is
particularly striking in comparison with core-collapse SNe, particularly
stripped envelope CCSNe, which can exhibit much greater polarizations up to
several percent, which translate into intrinsic SN asymmetries of tens of
percent (Wang et al., 2001; Wang & Wheeler, 2008). Additionally, the line and
thermal X-ray emission of SNe Ia remnants also possess a high degree of
spherical symmetry (Lopez et al., 2009, 2011). Taken together, these
observational findings place stringent limits on SNe Ia channels which predict
large asymmetries, either in the progenitor or the explosion mechanism, or
through the SN blast wave interacting with a non-degenerate companion or
circumstellar medium. For example, the violent double degenerate merger
predicts Si II polarization of $\sim 1-3\%$, in excess of most observations of
normal SNe Ia (Cikota et al., 2019). Additionally, the interaction of the SN
blast wave with the non-degenerate stellar companion in the classical single-
degenerate near-$M_{\rm Ch}$ scenario leaves a large “shadow” which is present
in the young remnant for hundreds of years (Gray et al., 2016).

In contrast, the DD near-$M_{\rm Ch}$ channel predicts the WD merger
progenitor will be uniformly and slowly rotating at the time of detonation,
and so will be very highly spherical. Furthermore, as emphasized in our
discussion of deflagration initiation and the turbulent deflagration-to-
detonation mechanism, recent work points towards a mild pre-expansion and
relatively prompt detonation. Consequently, DD near-$M_{\rm Ch}$ SNe Ia will
have very low Si II polarization near peak brightness.222The failed detonation
of DD near-$M_{\rm Ch}$ progenitors leading to SNe Iax will have a much
greater degree of assymmetry originating from the explosion mechanism itself:
the pure deflagration within its interior. While the first DDT simulations of
DD near-$M_{\rm Ch}$ SNe Ia presented in this paper neglect the initial
deflagration phase, we expect that more realistic simulations including the
initial deflagration phase through the DDT will produce Si II polarization
levels similar to prior near-$M_{\rm Ch}$ SNe Ia DDT models, typically well
under 1%, in line with observational constraints (Cikota et al., 2019).
Further, DD near-$M_{\rm Ch}$ SNe Ia remnants will also naturally be nearly
spherically symmetric, without the shadow predicted to arise from the
companion in the classical single-degenerate near-$M_{\rm Ch}$ scenario.

A large body of observational evidence has also placed stringent constraints
on SNe Ia nebular H $\alpha$ (Leonard, 2007; Tucker et al., 2020), X-ray
(Margutti et al., 2014), and radio emission (Chomiuk et al., 2016), and the
existence of stellar binary companions (Li et al., 2011) and ex-companions
(Schaefer & Pagnotta, 2012b) pre-explosion and post-explosion, respectively,
in the vast majority of normal SNe Ia systems. A natural prediction of the
near-$M_{\rm Ch}$ DD scenario is that there is no surviving companion,
implying that there can also be no nebular H $\alpha$ emission. The
near-$M_{\rm Ch}$ DD scenario further predicts an absence of strong prompt
X-ray and radio emission, although there are prospects for delayed interaction
with the propelled and ejected material.

Recent work on the ratio on the abundances of Ni II / Fe II, accessible
through optical and NIR lines, has elucidated that this line ratio is
sensitive to the mass of the progenitor, owing largely to the production of
stable Ni isotopes in the cores of near-$M_{Ch}$ events, particularly 58Ni
(Flörs et al., 2018, 2020). The use of stable Ni is a therefore a promising
discriminant between near-$M_{\rm Ch}$ and sub-$M_{\rm Ch}$ events (though
also subject to caveats – see (Blondin et al., 2021)). The nucleosynthetic
yields presented here predict a relatively high abundance of 58Ni / 56Fe = 0.2
for a DD near-$M_{\rm Ch}$ event, in comparison to other previously-computed
near-$M_{\rm Ch}$ models (Blondin et al., 2021). While the 58Ni / 56Fe
abundance ratio will decrease in a more realistic explosion scenario including
the deflagration phase, the stable Ni test may help shed light not only upon
the mass of the SN Ia progenitor, but also the physics of its formation and
the detonation mechanism.

While substantial evidence points towards the likelihood that partially burnt
near-$M_{\rm Ch}$ WDs lead to SNe Iax (eg, Jha, 2017), there is a relative
scarcity of direct observational evidence for helium through the helium-
accreting single degenerate channel. Jacobson-Galán et al. (2019) find early-
time He I CSM lines in two of a sample of 44 SNe Iax; with a luminosity limit
cutoff, they conclude He I CSM features are present in 33% of the remaining
sample. They further find no evidence for stripped helium in the late-time
nebular spectra of 11 SNe Iax, placing an upper bound limit on the amount of
helium to be less than $10^{-2}M_{\odot}$ in these events, with even tighter
bounds of $10^{-3}M_{\odot}$ for stripped hydrogen. In the context of the
near-$M_{\rm Ch}$ DD scenario, the absence of large amounts of helium in SNe
Iax spectra is a natural consequence of the complete merger of the double
degenerate system prior to the SN Iax event. Most of the thin
$10^{-3}-10^{-1}M_{\odot}$ (Lawlor & MacDonald, 2006) helium surface layers
atop the two white dwarfs will be susceptible to nuclear burning during the
final merger. However, the rapid accretion just prior to and during the tidal
disruption of the donor may cause a small amount of helium to be ejected
through the outer Lagrange point L2 (Livio et al., 1979; MacLeod et al.,
2018). Such ejected helium may account for the early-time He I CSM seen in
some SNe Iax.

Another strong constraint on SNe Ia channels is the $t^{-1}$ observed delay
time distribution (DTD) (Maoz et al., 2010; Maoz & Badenes, 2010; Graur et
al., 2011; Maoz et al., 2012). DD models generically predict a $t^{-1}$ DTD
from gravitational wave inspiral. In contrast, theoretical models of the SD
typically produce a sharper decay owing the absence of the requisite
intermediate mass SD donors over 10 Gyr timescales. While multi-component SD
DTDs invoking both main sequence and red giant donor channels have found
agreement with a $t^{-1}$ DTD (Hachisu et al., 2008), these same models
encounter the aforementioned tensions with the observed absence of companion
signatures in the majority of SNe Ia events. In contrast, DD near-$M_{\rm Ch}$
SNe Ia originate from the same gravitational wave inspiral mechanism which
underlies the overall DD population and do not possess companion signatures.
Consequently, as we have demonstrated in section 2.5, the DTD of DD
near-$M_{\rm Ch}$ SNe Ia retains the $t^{-1}$ DTD of the wider DD population.
Furthermore, the characteristic $t^{-1}$ DTD shape is not significantly biased
by a mass selection effect, even taking into account the maximum possible mass
lost through the propeller phase.

Recent models of double-detonations on sub-$M_{\rm Ch}$ WDs suggest that
carbon-enriched helium surface layers may produce sufficient 55Mn to reduce
but not entirely eliminate the need for near-$M_{\rm Ch}$ SNe Ia events
(Gronow et al., 2021). Consequently, the classical SD scenario and its
corresponding predictions for companion and ex-companion interactions is in
tension with the requisite need to produce 55Mn through near-$M_{\rm Ch}$. The
nucleosynthetic yields for the DD near-$M_{\rm Ch}$ event presented here shows
an approximately solar abundance of Mn/Fe. Late-time light curves (Seitenzahl
et al., 2009) offer a window into the 55 and 57 isobar decay chains
terminating in 55Mn and 57Fe, respectively, and may help shed light on both
the progenitor as well as the explosion scenario.

There are two primary means of observationally determining the galactic WD
merger rate. The first method infers the WD merger rate from observations of
WD field binaries (Maoz et al., 2018). The second examines single WDs, and
discriminates those produced by mergers from those resulting from isolated
evolution (Cheng et al., 2019). The field WD merger rate obtained in the most
recent observations is systematically higher than that inferred from merged
single WDs, which may be at least in part be due to unaccounted systematics
between these two techniques. Explosive transients, including the near-$M_{\rm
Ch}$ DD mergers described here, may account for a key systematic bias between
these two sets of observations.

Observational surveys of the abundance ratios of [Ni/Fe] and [Mn/Fe] have
yielded a rich picture of the nucleosynthetic histories of different galactic
environments. In particular, the dwarf spheroidal Sculptor, which had a single
burst of star formation after its formation, has decreasing ratios of [Ni/Fe]
and [Mn/Fe] as a function of metallicity [Fe/H], a proxy for age (Kirby et
al., 2019; de los Reyes et al., 2020). In contrast, other galactic
environments with ongoing star formation histories, including Leo I, Fornax,
and the Milky Way, exhibit increasing ratios of [Ni/Fe] and [Mn/Fe] with
[Fe/H] (Kirby et al., 2019; de los Reyes et al., 2020). Because near-$M_{\rm
Ch}$ events, including the DD channel presented here, generally have higher
[Ni/Fe] and [Mn/Fe] in comparison to sub-$M_{\rm Ch}$ events, the
nucleosynthetic histories of stellar environments such as Sculptor undergoing
a single star formation burst point towards an increasing predominance of
sub-$M_{\rm Ch}$ relative to near-$M_{\rm Ch}$ events with time. Other stellar
environments with ongoing star formation histories point instead towards an
increasing ratio of near-$M_{\rm Ch}$ to sub-$M_{\rm Ch}$ events with time. A
possible explanation for these diverse SNe Ia histories stems from
considerations of both the sub-$M_{\rm Ch}$ as well as the DD near-$M_{\rm
Ch}$ DTDs. The DTD of sub-$M_{\rm Ch}$ double detonations from In contrast,
the characteristic $t^{-1}$ shape of the DD near-$M_{\rm Ch}$ DTD the
qualitative difference of the DD near-$M_{\rm Ch}$ DTD with the classical SD
near-$M_{\rm Ch}$ DTD, which cuts off sharply after a few Gyr (Kobayashi et
al., 2019), may offer a potentially observable test of these different
scenarios for the production of near-$M_{\rm Ch}$ SNe Ia.

During the final writing of this paper, the discovery of a rapidly-rotating,
strongly-magnetized near-$M_{\rm Ch}$ WD, ZTF J190132.9+145808.7, was reported
(Caiazzo et al., 2021). The surface magnetic field was estimated to be 600 -
800 MG, with an inferred internal O/Ne composition approaching a central
density of $\sim$ several $\times 10^{9}$ g cm-3 and central temperature $\sim
7\times 10^{8}$ K. This rapidly-rotating, strongly-magnetized near-$M_{\rm
Ch}$ WD is similar to the DD near-$M_{\rm Ch}$ progenitors discussed in this
paper. The inferred internal composition and thermodynamic structure, as well
as the inferred age of this WD will, however, hinge crucially on
magnetohydrodynamical processes which give rise to such a strongly-magnetized
WD, and further modeling is required

##  6 Conclusion

In this paper, we have presented a model for the production of near-$M_{\rm
Ch}$ WDs from the DD channel resulting from the merger and inspiral of two C/O
WDs. The magnetic field plays a crucial role in this mechanism, first during
the production of a high field magnetic WD during the merger, and secondly in
compressing the core of the WD to ignition conditions during spin down. We
have also demonstrated the onset of a propeller phase during the merger, with
potentially observable interaction signatures.

To summarize the key observable signatures of SNe Ia and SNe Iax produced by
DD near-$M_{\rm Ch}$:

  * •

A $t^{-1}$ DTD both DD near-$M_{\rm Ch}$ SNe and SNe Iax.

  * •

The absence of companion and ex-companion signatures in both DD near-$M_{\rm
Ch}$ SNe and SNe Iax.

  * •

Weakly-polarized Si II lines in DD near-$M_{\rm Ch}$ SNe Ia.

  * •

In near-$M_{\rm Ch}$ DD SNe Ia, the production of a bright normal to
potentially overluminous (91T-like) event.

  * •

The absence of strong H and He signatures in both DD near-$M_{\rm Ch}$ SNe and
SNe Iax. Trace He abundaces may be possible by ejection through the outer
Lagrange point L2, and may account for He I CSM in some events.

  * •

Solar or super-solar abundances of [Mn/Fe] and [Ni/Fe].

  * •

Further theoretical modeling and simulation is required to obtain a clearer
picture of the role which magnetohydrodynamics plays in WD mergers, and help
address which circumstances lead to explosive transients, and which lead to
stable massive magnetized mergers. While magnetohydrodynamics plays a key role
during the WD merger, the process has been long neglected in the majority of
models. Massive magnetized WDs like ZTF J190132.9+145808.7 are providing us
with a strong clue that the stellar merger, magnetohydrodynamics, and the SN
Ia mechanism are all inextricably linked.

New observational instruments, including JWST and , will help discriminate
both between sub-$M_{\rm Ch}$ and super-$M_{\rm Ch}$ events and the explosion
scenario using a wide range of observational tests, including the
radioisotopic decay of SNe Ia late-time light curves, and the presence of
stable Ni in the nebular phase of SNe Ia. Detailed surveys of galactic SN Ia
SNRs, particularly of the near-$M_{\rm Ch}$ SNR 3C 397, will shed a
complementary light on the stellar progenitor and the explosion mechanism.

The authors acknowledge insightful conversations with Jorge Rueda, Laura
Becerra, Evan Kirby, Mithi Alexa (Mia) de los Reyes, Lifan Wang, Noam Soker,
and Hagai Perets. R.T.F. and S.N. acknowledge support from NASA ATP award
80NSSC18K1013, and K.B. support from NASA XMM-Newton award 80NSSC19K0601. 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). .

## References

  * Asplund et al. (2021) Asplund, M., Amarsi, A. M., & Grevesse, N. 2021, The chemical make-up of the Sun: A 2020 vision.  https://arxiv.org/abs/2105.01661
  * Balbus & Hawley (1991) Balbus, S. A., & Hawley, J. F. 1991, ApJ, 376, 214, doi: 10.1086/170270
  * Becerra et al. (2019) Becerra, L., Boshkayev, K., Rueda, J. A., & Ruffini, R. 2019, MNRAS, 487, 812, doi: 10.1093/mnras/stz1394
  * Becerra et al. (2018) Becerra, L., Rueda, J. A., Lorén-Aguilar, P., & García-Berro, E. 2018, ApJ, 857, 134, doi: 10.3847/1538-4357/aabc12
  * Benz et al. (1990) Benz, W., Bowers, R. L., Cameron, A. G. W., & Press, W. H. . 1990, ApJ, 348, 647, doi: 10.1086/168273
  * Blondin (2011) Blondin, Stéphane; Tonry, J. L. 2011, Astrophysics Source Code Library
  * Blondin et al. (2021) Blondin, S., Bravo, E., Timmes, F., Dessart, L., & Hillier, D. J. 2021, arXiv e-prints, arXiv:2109.13840.  https://arxiv.org/abs/2109.13840
  * Boshkayev et al. (2013) Boshkayev, K., Rueda, J. A., Ruffini, R., & Siutsou, I. 2013, ApJ, 762, 117, doi: 10.1088/0004-637X/762/2/117
  * Botyánszki et al. (2018) Botyánszki, J., Kasen, D., & Plewa, T. 2018, ApJ, 852, L6, doi: 10.3847/2041-8213/aaa07b
  * Brachwitz et al. (2000) Brachwitz, F., Dean, D. J., Hix, W. R., et al. 2000, ApJ, 536, 934, doi: 10.1086/308968
  * Byrohl et al. (2019) Byrohl, C., Fisher, R., & Townsley, D. 2019, ApJ, 878, 67, doi: 10.3847/1538-4357/ab1f73
  * Caiazzo et al. (2021) Caiazzo, I., Burdge, K. B., Fuller, J., et al. 2021, Nature, 595, 39, doi: 10.1038/s41586-021-03615-y
  * Chandrasekhar (1935) Chandrasekhar, S. 1935, MNRAS, 95, 207, doi: 10.1093/mnras/95.3.207
  * Cheng et al. (2019) Cheng, S., Cummings, J. D., Ménard, B., & Toonen, S. 2019, arXiv e-prints, arXiv:1910.09558.  https://arxiv.org/abs/1910.09558
  * Chomiuk et al. (2016) Chomiuk, L., Soderberg, A. M., Chevalier, R. A., et al. 2016, ApJ, 821, 119, doi: 10.3847/0004-637X/821/2/119
  * Cikota et al. (2019) Cikota, A., Patat, F., Wang, L., et al. 2019, MNRAS, 490, 578, doi: 10.1093/mnras/stz2322
  * Couch et al. (2013) Couch, S. M., Graziani, C., & Flocke, N. 2013, ApJ, 778, 181.  http://dx.doi.org/10.1088/0004-637X/778/2/181
  * Davis et al. (2010) Davis, S. W., Stone, J. M., & Pessah, M. E. 2010, The Astrophysical Journal, 713, 52, doi: 10.1088/0004-637x/713/1/52
  * de los Reyes et al. (2020) de los Reyes, M. A. C., Kirby, E. N., Seitenzahl, I. R., & Shen, K. J. 2020, ApJ, 891, 85, doi: 10.3847/1538-4357/ab736f
  * Di Stefano (2010) Di Stefano, R. 2010, ApJ, 712, 728, doi: 10.1088/0004-637X/712/1/728
  * Dubey et al. (2012) Dubey, A., Daley, C., ZuHone, J., et al. 2012, ApJS, 201, 27, doi: 10.1088/0067-0049/201/2/27
  * Dubey et al. (2009) Dubey, A., Reid, L. B., Weide, K., et al. 2009, arXiv e-prints, arXiv:0903.4875.  https://arxiv.org/abs/0903.4875
  * Dubey et al. (2014) Dubey, A., Antypas, K., Calder, A. C., et al. 2014, The International Journal of High Performance Computing Applications, 28, 225, doi: 10.1177/1094342013505656
  * Fenn & Plewa (2017) Fenn, D., & Plewa, T. 2017, MNRAS, 468, 1361, doi: 10.1093/mnras/stx524
  * Ferrario et al. (2015) Ferrario, L., de Martino, D., & Gänsicke, B. T. 2015, Space Sci. Rev., 191, 111, doi: 10.1007/s11214-015-0152-0
  * Fink et al. (2014) Fink, M., Kromer, M., Seitenzahl, I. R., et al. 2014, MNRAS, 438, 1762, doi: 10.1093/mnras/stt2315
  * Finn & Shapiro (1990) Finn, L. S., & Shapiro, S. L. 1990, ApJ, 359, 444, doi: 10.1086/169077
  * Fisher & Jumper (2015) Fisher, R., & Jumper, K. 2015, ApJ, 805, 150, doi: 10.1088/0004-637X/805/2/150
  * Fisher et al. (2019) Fisher, R., Mozumdar, P., & Casabona, G. 2019, ApJ, 876, 64, doi: 10.3847/1538-4357/ab15d8
  * Flörs et al. (2018) Flörs, A., Spyromilio, J., Maguire, K., et al. 2018, A&A, 620, A200, doi: 10.1051/0004-6361/201833512
  * Flörs et al. (2020) Flörs, A., Spyromilio, J., Taubenberger, S., et al. 2020, MNRAS, 491, 2902, doi: 10.1093/mnras/stz3013
  * Foley et al. (2013) Foley, R. J., Challis, P. J., Chornock, R., et al. 2013, ApJ, 767, 57, doi: 10.1088/0004-637X/767/1/57
  * Fryxell et al. (2000) Fryxell, B., Olson, K., Ricker, P., et al. 2000, The Astrophysical Journal Supplement Series, 131, 273–334, doi: 10.1086/317361
  * Geroyannis & Papasotiriou (2000) Geroyannis, V. S., & Papasotiriou, P. J. 2000, ApJ, 534, 359, doi: 10.1086/308728
  * Ghosh & Lamb (1979) Ghosh, P., & Lamb, F. K. 1979, ApJ, 232, 259, doi: 10.1086/157285
  * González Hernández et al. (2012) González Hernández, J. I., Ruiz-Lapuente, P., Tabernero, H. M., et al. 2012, Nature, 489, 533, doi: 10.1038/nature11447
  * Graur et al. (2011) Graur, O., Poznanski, D., Maoz, D., et al. 2011, MNRAS, 417, 916, doi: 10.1111/j.1365-2966.2011.19287.x
  * Gray et al. (2016) Gray, W. J., Raskin, C., & Owen, J. M. 2016, ApJ, 833, 62, doi: 10.3847/1538-4357/833/1/62
  * Gronow et al. (2021) Gronow, S., Cote, B., Lach, F., et al. 2021, arXiv e-prints, arXiv:2103.14050.  https://arxiv.org/abs/2103.14050
  * Gvaramadze et al. (2019) Gvaramadze, V. V., Gräfener, G., Langer, N., et al. 2019, Nature, 569, 684, doi: 10.1038/s41586-019-1216-1
  * Hachisu (1986) Hachisu, I. 1986, ApJS, 61, 479, doi: 10.1086/191121
  * Hachisu et al. (2008) Hachisu, I., Kato, M., & Nomoto, K. 2008, ApJ, 683, L127, doi: 10.1086/591646
  * Hamuy et al. (2002) Hamuy, M., Maza, J., Pinto, P. A., et al. 2002, AJ, 124, 417, doi: 10.1086/340968
  * Hunter (2007) Hunter, J. D. 2007, Computing in Science Engineering, 9, 90, doi: 10.1109/MCSE.2007.55
  * Ilkov & Soker (2012) Ilkov, M., & Soker, N. 2012, MNRAS, 419, 1695, doi: 10.1111/j.1365-2966.2011.19833.x
  * Illarionov & Sunyaev (1975) Illarionov, A. F., & Sunyaev, R. A. 1975, A&A, 39, 185
  * Jacobson-Galán et al. (2019) Jacobson-Galán, W. V., Foley, R. J., Schwab, J., et al. 2019, MNRAS, 487, 2538, doi: 10.1093/mnras/stz1305
  * Jha (2017) Jha, S. W. 2017, Type Iax Supernovae (Springer), 375, doi: 10.1007/978-3-319-21846-5_42
  * Ji et al. (2013) Ji, S., Fisher, R. T., García-Berro, E., et al. 2013, ApJ, 773, 136, doi: 10.1088/0004-637X/773/2/136
  * Jordan et al. (2012) Jordan, George C., I., Perets, H. B., Fisher, R. T., & van Rossum, D. R. 2012, ApJ, 761, L23, doi: 10.1088/2041-8205/761/2/L23
  * Kashyap et al. (2017) Kashyap, R., Fisher, R., García-Berro, E., et al. 2017, ApJ, 840, 16, doi: 10.3847/1538-4357/aa6afb
  * Kato et al. (2018) Kato, M., Saio, H., & Hachisu, I. 2018, ApJ, 863, 125, doi: 10.3847/1538-4357/aad327
  * Khokhlov (1991) Khokhlov, A. M. 1991, A&A, 245, 114
  * Kilic et al. (2021) Kilic, M., Bergeron, P., Blouin, S., & Bedard, A. 2021, The Most Massive White Dwarfs in the Solar Neighborhood.  https://arxiv.org/abs/2103.06906
  * Kirby et al. (2019) Kirby, E. N., Xie, J. L., Guo, R., et al. 2019, ApJ, 881, 45, doi: 10.3847/1538-4357/ab2c02
  * Kobayashi et al. (2019) Kobayashi, C., Leung, S.-C., & Nomoto, K. 2019, arXiv e-prints, arXiv:1906.09980.  https://arxiv.org/abs/1906.09980
  * Krueger et al. (2012) Krueger, B. K., Jackson, A. P., Calder, A. C., et al. 2012, ApJ, 757, 175, doi: 10.1088/0004-637X/757/2/175
  * Lawlor & MacDonald (2006) Lawlor, T. M., & MacDonald, J. 2006, MNRAS, 371, 263, doi: 10.1111/j.1365-2966.2006.10641.x
  * Lecoanet et al. (2016) Lecoanet, D., Schwab, J., Quataert, E., et al. 2016, ApJ, 832, 71, doi: 10.3847/0004-637X/832/1/71
  * Lee & Deane (2009) Lee, D., & Deane, A. E. 2009, Journal of Computational Physics, 228, 952–975, doi: 10.1016/j.jcp.2008.08.026
  * Leonard (2007) Leonard, D. C. 2007, ApJ, 670, 1275, doi: 10.1086/522367
  * Leung & Nomoto (2018) Leung, S.-C., & Nomoto, K. 2018, The Astrophysical Journal, 861, 143, doi: 10.3847/1538-4357/aac2df
  * Li et al. (2011) Li, W., Bloom, J. S., Podsiadlowski, P., et al. 2011, Nature, 480, 348, doi: 10.1038/nature10646
  * Livio et al. (1979) Livio, M., Salzman, J., & Shaviv, G. 1979, MNRAS, 188, 1, doi: 10.1093/mnras/188.1.1
  * Lopez et al. (2009) Lopez, L. A., Ramirez-Ruiz, E., Badenes, C., et al. 2009, ApJ, 706, L106, doi: 10.1088/0004-637X/706/1/L106
  * Lopez et al. (2011) Lopez, L. A., Ramirez-Ruiz, E., Huppenkothen, D., Badenes, C., & Pooley, D. A. 2011, ApJ, 732, 114, doi: 10.1088/0004-637X/732/2/114
  * Lorén-Aguilar et al. (2009) Lorén-Aguilar, P., Isern, J., & García-Berro, E. 2009, A&A, 500, 1193, doi: 10.1051/0004-6361/200811060
  * Lovelace et al. (1999) Lovelace, R. V. E., Romanova, M. M., & Bisnovatyi-Kogan, G. S. 1999, ApJ, 514, 368, doi: 10.1086/306945
  * MacLeod et al. (2018) MacLeod, M., Ostriker, E. C., & Stone, J. M. 2018, ApJ, 863, 5, doi: 10.3847/1538-4357/aacf08
  * Malone et al. (2014) Malone, C. M., Nonaka, A., Woosley, S. E., et al. 2014, ApJ, 782, 11, doi: 10.1088/0004-637X/782/1/11
  * Maoz & Badenes (2010) Maoz, D., & Badenes, C. 2010, MNRAS, 407, 1314, doi: 10.1111/j.1365-2966.2010.16988.x
  * Maoz et al. (2018) Maoz, D., Hallakoun, N., & Badenes, C. 2018, MNRAS, 476, 2584, doi: 10.1093/mnras/sty339
  * Maoz et al. (2012) Maoz, D., Mannucci, F., & Brandt, T. D. 2012, MNRAS, 426, 3282, doi: 10.1111/j.1365-2966.2012.21871.x
  * Maoz et al. (2013) Maoz, D., Mannucci, F., & Nelemans, G. 2013, ArXiv e-prints
  * Maoz et al. (2014) —. 2014, ARA&A, 52, 107, doi: 10.1146/annurev-astro-082812-141031
  * Maoz et al. (2010) Maoz, D., Sharon, K., & Gal-Yam, A. 2010, ApJ, 722, 1879, doi: 10.1088/0004-637X/722/2/1879
  * Margutti et al. (2014) Margutti, R., Parrent, J., Kamble, A., et al. 2014, ApJ, 790, 52, doi: 10.1088/0004-637X/790/1/52
  * Margutti et al. (2012) Margutti, R., Soderberg, A. M., Chomiuk, L., et al. 2012, ApJ, 751, 134, doi: 10.1088/0004-637X/751/2/134
  * Mazzali et al. (2005) Mazzali, P. A., Benetti, S., Stehle, M., et al. 2005, MNRAS, 357, 200, doi: 10.1111/j.1365-2966.2005.08640.x
  * Meyer et al. (1996) Meyer, B. S., Krishnan, T. D., & Clayton, D. D. 1996, ApJ, 462, 825, doi: 10.1086/177197
  * Nelemans & Tout (2005) Nelemans, G., & Tout, C. A. 2005, MNRAS, 356, 753, doi: 10.1111/j.1365-2966.2004.08496.x
  * Nelemans et al. (2000) Nelemans, G., Verbunt, F., Yungelson, L. R., & Portegies Zwart, S. F. 2000, A&A, 360, 1011
  * Nomoto & Iben (1985) Nomoto, K., & Iben, I., J. 1985, ApJ, 297, 531, doi: 10.1086/163547
  * Nomoto et al. (1997) Nomoto, K., Iwamoto, K., Nakasato, N., et al. 1997, Nuclear Physics A, 621, 467, doi: 10.1016/S0375-9474(97)00291-1
  * Nomoto et al. (1984) Nomoto, K., Thielemann, F.-K., & Yokoi, K. 1984, ApJ, 286, 644, doi: 10.1086/162639
  * Nonaka et al. (2012a) Nonaka, A., Aspden, A. J., Zingale, M., et al. 2012a, ApJ, 745, 73, doi: 10.1088/0004-637X/745/1/73
  * Nonaka et al. (2012b) —. 2012b, ApJ, 745, 73, doi: 10.1088/0004-637X/745/1/73
  * Ohlmann et al. (2014) Ohlmann, S. T., Kromer, M., Fink, M., et al. 2014, A&A, 572, A57, doi: 10.1051/0004-6361/201423924
  * Ostriker & Bodenheimer (1968) Ostriker, J. P., & Bodenheimer, P. 1968, ApJ, 151, 1089, doi: 10.1086/149507
  * Pankey (1962) Pankey, Titus, J. 1962, PhD thesis, HOWARD UNIVERSITY.
  * Pereira et al. (2013) Pereira, R., Thomas, R. C., Aldering, G., et al. 2013, A&A, 554, A27, doi: 10.1051/0004-6361/201221008
  * Perez & Granger (2007) Perez, F., & Granger, B. E. 2007, Computing in Science Engineering, 9, 21, doi: 10.1109/MCSE.2007.53
  * Phillips (1993) Phillips, M. M. 1993, ApJ, 413, L105, doi: 10.1086/186970
  * Piro (2008) Piro, A. L. 2008, ApJ, 679, 616, doi: 10.1086/529363
  * Poludnenko et al. (2019) Poludnenko, A. Y., Chambers, J., Ahmed, K., Gamezo, V. N., & Taylor, B. D. 2019, Science, 366, aau7365, doi: 10.1126/science.aau7365
  * Poludnenko et al. (2011) Poludnenko, A. Y., Gardiner, T. A., & Oran, E. S. 2011, Physical Review Letters, 107, 054501, doi: 10.1103/PhysRevLett.107.054501
  * Portegies Zwart & Verbunt (1996) Portegies Zwart, S. F., & Verbunt, F. 1996, A&A, 309, 179
  * Raddi et al. (2019) Raddi, R., Hollands, M. A., Koester, D., et al. 2019, MNRAS, 489, 1489, doi: 10.1093/mnras/stz1618
  * Raskin & Kasen (2013) Raskin, C., & Kasen, D. 2013, ApJ, 772, 1, doi: 10.1088/0004-637X/772/1/1
  * Raskin et al. (2014) Raskin, C., Kasen, D., Moll, R., Schwab, J., & Woosley, S. 2014, ApJ, 788, 75.  http://dx.doi.org/10.1088/0004-637X/788/1/75
  * Raskin et al. (2012) Raskin, C., Scannapieco, E., Fryer, C., Rockefeller, G., & Timmes, F. X. 2012, ApJ, 746, 62, doi: 10.1088/0004-637X/746/1/62
  * Romanova et al. (2003) Romanova, M. M., Toropina, O. D., Toropin, Y. M., & Lovelace, R. V. E. 2003, The Astrophysical Journal, 588, 400, doi: 10.1086/373990
  * Romanova et al. (2005) Romanova, M. M., Ustyugova, G. V., Koldoba, A. V., & Lovelace, R. V. E. 2005, ApJ, 635, L165, doi: 10.1086/499560
  * Röpke et al. (2012) Röpke, F. K., Kromer, M., Seitenzahl, I. R., et al. 2012, ApJ, 750, L19, doi: 10.1088/2041-8205/750/1/L19
  * Ruiter et al. (2014) Ruiter, A. J., Belczynski, K., Sim, S. A., Seitenzahl, I. R., & Kwiatkowski, D. 2014, MNRAS, 440, L101, doi: 10.1093/mnrasl/slu030
  * Saio & Nomoto (1985) Saio, H., & Nomoto, K. 1985, A&A, 150, L21
  * Schaefer & Pagnotta (2012a) Schaefer, B. E., & Pagnotta, A. 2012a, Nature, 481, 164, doi: 10.1038/nature10692
  * Schaefer & Pagnotta (2012b) —. 2012b, Nature, 481, 164, doi: 10.1038/nature10692
  * Schwab et al. (2016) Schwab, J., Quataert, E., & Kasen, D. 2016, MNRAS, 463, 3461, doi: 10.1093/mnras/stw2249
  * Schwab et al. (2012) Schwab, J., Shen, K. J., Quataert, E., Dan, M., & Rosswog, S. 2012, MNRAS, 427, 190, doi: 10.1111/j.1365-2966.2012.21993.x
  * Seitenzahl et al. (2013a) Seitenzahl, I. R., Cescutti, G., Röpke, F. K., Ruiter, A. J., & Pakmor, R. 2013a, A&A, 559, L5, doi: 10.1051/0004-6361/201322599
  * Seitenzahl et al. (2010) Seitenzahl, I. R., Röpke, F. K., Fink, M., & Pakmor, R. 2010, MNRAS, 407, 2297, doi: 10.1111/j.1365-2966.2010.17106.x
  * Seitenzahl et al. (2009) Seitenzahl, I. R., Taubenberger, S., & Sim, S. A. 2009, MNRAS, 400, 531, doi: 10.1111/j.1365-2966.2009.15478.x
  * Seitenzahl et al. (2013b) Seitenzahl, I. R., Ciaraldi-Schoolmann, F., Röpke, F. K., et al. 2013b, MNRAS, 429, 1156, doi: 10.1093/mnras/sts402
  * Shapiro (2000) Shapiro, S. L. 2000, ApJ, 544, 397, doi: 10.1086/317209
  * Shapiro & Teukolsky (1986) Shapiro, S. L., & Teukolsky, S. A. 1986, Black Holes, White Dwarfs and Neutron Stars: The Physics of Compact Objects
  * Shapiro et al. (1990) Shapiro, S. L., Teukolsky, S. A., & Nakamura, T. 1990, ApJ, 357, L17, doi: 10.1086/185756
  * Shappee et al. (2016) Shappee, B. J., Piro, A. L., Stanek, K. Z., et al. 2016, ArXiv e-prints.  https://arxiv.org/abs/1610.07601
  * Shen et al. (2012) Shen, K. J., Bildsten, L., Kasen, D., & Quataert, E. 2012, ApJ, 748, 35, doi: 10.1088/0004-637X/748/1/35
  * Shen et al. (2021) Shen, K. J., Blondin, S., Kasen, D., et al. 2021, ApJ, 909, L18, doi: 10.3847/2041-8213/abe69b
  * Shen et al. (2018) Shen, K. J., Boubert, D., Gänsicke, B. T., et al. 2018, ApJ, 865, 15, doi: 10.3847/1538-4357/aad55b
  * Stritzinger et al. (2015) Stritzinger, M. D., Valenti, S., Hoeflich, P., et al. 2015, A&A, 573, A2, doi: 10.1051/0004-6361/201424168
  * Strolger et al. (2020) Strolger, L.-G., Rodney, S. A., Pacifici, C., Narayan, G., & Graur, O. 2020, arXiv e-prints, arXiv:2001.05967.  https://arxiv.org/abs/2001.05967
  * Timmes (1999) Timmes, F. X. 1999, The Astrophysical Journal Supplement Series, 124, 241–263, doi: 10.1086/313257
  * Timmes & Swesty (2000) Timmes, F. X., & Swesty, F. D. 2000, The Astrophysical Journal Supplement Series, 126, 501–516, doi: 10.1086/313304
  * Tonry & Davis (1979) Tonry, J., & Davis, M. 1979, AJ, 84, 1511, doi: 10.1086/112569
  * Toonen et al. (2017) Toonen, S., Hollands, M., Gänsicke, B. T., & Boekholt, T. 2017, A&A, 602, A16, doi: 10.1051/0004-6361/201629978
  * Toonen et al. (2012) Toonen, S., Nelemans, G., & Portegies Zwart, S. 2012, A&A, 546, A70, doi: 10.1051/0004-6361/201218966
  * Towns et al. (2014) Towns, J., Cockerill, T., Dahan, M., et al. 2014, Computing in Science Engineering, 16, 62, doi: 10.1109/MCSE.2014.80
  * Townsley et al. (2007) Townsley, D. M., Calder, A. C., Asida, S. M., et al. 2007, ApJ, 668, 1118, doi: 10.1086/521013
  * Townsley et al. (2009) Townsley, D. M., Jackson, A. P., Calder, A. C., et al. 2009, ApJ, 701, 1582, doi: 10.1088/0004-637X/701/2/1582
  * Townsley et al. (2016) Townsley, D. M., Miles, B. J., Timmes, F. X., Calder, A. C., & Brown, E. F. 2016, ApJS, 225, 3, doi: 10.3847/0067-0049/225/1/3
  * Tucker et al. (2019) Tucker, M. A., Shappee, B. J., Vallely, P. J., et al. 2019, MNRAS, 3025, doi: 10.1093/mnras/stz3390
  * Tucker et al. (2020) —. 2020, MNRAS, 493, 1044, doi: 10.1093/mnras/stz3390
  * Turk et al. (2011) Turk, M. J., Smith, B. D., Oishi, J. S., et al. 2011, ApJS, 192, 9.  http://dx.doi.org/10.1088/0067-0049/192/1/9
  * Unterweger et al. (1992) Unterweger, M. P., Hoppes, D. D., & Schima, F. J. 1992, Nuclear Instruments and Methods in Physics Research A, 312, 349, doi: 10.1016/0168-9002(92)90180-C
  * van der Walt et al. (2011) van der Walt, S., Colbert, S. C., & Varoquaux, G. 2011, Computing in Science Engineering, 13, 22, doi: 10.1109/MCSE.2011.37
  * van Rossum & de Boer (1991) van Rossum, G., & de Boer, J. 1991, CWI Quarterly, 4, 283
  * Wang et al. (2001) Wang, L., Howell, D. A., Höflich, P., & Wheeler, J. C. 2001, ApJ, 550, 1030, doi: 10.1086/319822
  * Wang & Wheeler (2008) Wang, L., & Wheeler, J. C. 2008, ARA&A, 46, 433, doi: 10.1146/annurev.astro.46.060407.145139
  * Webbink (1984) Webbink, R. F. 1984, ApJ, 277, 355, doi: 10.1086/161701
  * Whelan & Iben (1973) Whelan, J., & Iben, Jr., I. 1973, ApJ, 186, 1007.  http://dx.doi.org/10.1086/152565
  * Wollaeger & van Rossum (2014) Wollaeger, R. T., & van Rossum, D. R. 2014, ApJS, 214, 21, doi: 10.1088/0067-0049/214/2/28
  * Wollaeger et al. (2013) Wollaeger, R. T., van Rossum, D. R., Graziani, C., et al. 2013, ApJS, 209, 21, doi: 10.1088/0067-0049/209/2/36
  * Woosley (1997) Woosley, S. E. 1997, ApJ, 476, 801
  * Yoon & Langer (2005) Yoon, S.-C., & Langer, N. 2005, A&A, 435, 967, doi: 10.1051/0004-6361:20042542
  * Yoshida (2019) Yoshida, S. 2019, Monthly Notices of the Royal Astronomical Society, 486, 2982, doi: 10.1093/mnras/stz1030
  * Zhu et al. (2013) Zhu, C., Chang, P., van Kerkwijk, M. H., & Wadsley, J. 2013, ApJ, 767, 164.  http://dx.doi.org/10.1088/0004-637X/767/2/164
  * Zingale et al. (2011) Zingale, M., Nonaka, A., Almgren, A. S., et al. 2011, ApJ, 740, 8, doi: 10.1088/0004-637X/740/1/8


* [GV23] B. Grabmayr and A. Visser.  Self-reference upfront: a study of self-referential Gödel numberings.  The Review of Symbolic Logic, 16:385–424, 2023.
  * [Han65] W. Hanf.  Model-theoretic methods in the study of elementary logic.  In J.W. Addison, L. Henkin, and A. Tarski, editors, The Theory of Models, Proceedings of the 1963 International Symposium at Berkeley, pages 132–145. North Holland, Amsterdam, 1965.
  * [Jan53] A. Janiczak.  Undecidability of some simple formalized theories.  Fundamenta Mathematicæ, 40:131–139, 1953.
  * [JdJ98] G. Japaridze and D. de Jongh.  The logic of provability.  In S. Buss, editor, Handbook of proof theory, pages 475–546. North-Holland Publishing Co., Amsterdam, 1998.
  * [JS83] J.P. Jones and J.C. Shepherdson.  Variants of Robinson’s essentially undecidable theory R.  Archiv für Mathematische Logik und Grundlagenforschung, 23:61–64, 1983.
  * [Kri23] S.A. Kripke.  Gödel’s theorem and direct self-reference.  The Review of Symbolic Logic, 16:650–654, 2023.
  * [Kur22] T. Kurahashi.  Incompleteness and undecidability of theories consistent with $\mathsf{R}$.  arXiv.2211.15455, 2022.
  * [Pak19] F. Pakhomov.  A weak set theory that proves its own consistency.  arXiv:1907.00877, 2019.
  * [Per97] M. G. Peretyat’kin.  Finitely axiomatizable theories.  Consultants Bureau, New York, 1997.
  * [PMV22] F. Pakhomov, J. Murwanashyaka, and A. Visser.  There are no minimal essentially undecidable theories.  arXiv.2207.0817, 2022.
  * [Put57] H. Putnam.  Decidability and essential undecidability.  The Journal of Symbolic Logic, 22(1):39–54, 1957.
  * [Rog67] H. Rogers.  Theory of recursive functions and effective computability.  McGraw-Hill Book Company, New York, 1967.
  * [Sho58] J. R. Shoenfield.  Degrees of formal systems 1.  The Journal of Symbolic Logic, 23(4):389–392, 1958.
  * [Soa16] R.I. Soare.  Turing computability: Theory and applications.  Springer, Berlin, 2016.
  * [Szm55] W. Szmielew.  Elementary properties of Abelian groups.  Fundamenta Mathematicæ, 41(2):203–271, 1955.
  * [TMR53] A. Tarski, A. Mostowski, and R.M. Robinson.  Undecidable theories.  North–Holland, Amsterdam, 1953.
  * [Vau62] R.L. Vaught.  On a theorem of Cobham concerning undecidable theories.  In E. Nagel, P. Suppes, and A. Tarski, editors, Logic, Methodology and Philosophy of Science. Proceedings of the 1960 International Congress, pages 14–25. Stanford University Press, Stanford, 1962.
  * [Vis11] A. Visser.  Can we make the Second Incompleteness Theorem coordinate free?  Journal of Logic and Computation, 21(4):543–560, 2011.
  * [Vis12] A. Visser.  Vaught’s theorem on axiomatizability by a scheme.  Bulletin of Symbolic Logic, 18(3):382–402, 2012.
  * [Vis14a] A. Visser.  The interpretability of inconsistency, Feferman’s theorem and related results.  Logic Group Preprint Series 318, Faculty of Humanities, Philosophy, Utrecht University, Janskerkhof 13, 3512 BL Utrecht, http://www.phil.uu.nl/preprints/lgps/, 2014.
  * [Vis14b] A. Visser.  Why the theory R is special.  In Neil Tennant, editor, Foundational Adventures. Essays in honour of Harvey Friedman, pages 7–23. College Publications, UK, 2014.  Originally published online by Templeton Press in 2012. See http://foundationaladventures.com/.
  * [Vis17] A. Visser.  On Q.  Soft Computing, 21(1):39–56, 2017.
  * [Vis22] A. Visser.  Friedman-reflexivity.  Annals of Pure and Applied Logic, 173(9):103160, 2022.

##  Appendix A Parametrically Local Interpretability

In this appendix, we present a reduction notion, to wit _c-lax
interpretability_ , that improves upon both lax interpretability and
parametrically local interpretability w.r.t. the treatment of essential
hereditary undecidability.

###  A.1. Parameters and Finite Axiomatisability

Our preferred treatment of interpretations with parameters uses a parameter
domain $\pi$. Let $V$ be the interpreting theory and let $U$ be the
interpreted theory and let $\tau$ be the translation. We demand that (a)
$V\vdash\exists\vv p\,\pi(\vv p)$ and (b) if $U\vdash\varphi$, then
$V\vdash\forall\vv q\,(\pi(\vv q)\to\varphi^{\tau,\vv q})$.

In case $U$ is finitely axiomatised, we can take a short-cut and drop the
parameter domain. Let $\alpha$ be the conjunction of the axioms of $U$ plus
the axioms of the theory of identity. We ask: (a′) $V\vdash\exists\vv
p\;\alpha^{\tau,\vv p}$.

We note that (a) and (b) immediately imply (a′). Conversely, if we have (a′),
we can define a parameter domain $\pi^{\ast}(\vv
p):\leftrightarrow\alpha^{\tau,\vv p}$ and use the equivalence between axioms-
and theorems-interpretability to obtain (a) and (b) for $\pi^{\ast}$.

We note that, if we start with a parameter-domain $\pi$ that satisfies (a) and
(b), then $\pi^{\ast}$ is an extension of $\pi$.

###  A.2. Parametric Local Interpretability

We say that _$V$ parametrically locally interprets $U$_, or $V\rhd_{\sf pl}U$,
iff, there is a parametric translation $\tau$ of the $U$-language to the
$V$-language, such that, whenever $U\vdash\varphi$, then $V\vdash\exists\vv
p\;\varphi^{\tau,\vv p}$. It is easy to see that $\rhd_{\sf pl}$ is
transitive, using the known properties of parametric interpretability.

######  Theorem A.1.

Let $U$ be a theory in a language $\mathcal{L}$ extended with constants $\vv
c$. Let $U^{\ast}$ be the theory axiomatised by the $\mathcal{L}$-consequences
of $U$. Then, $U\bowtie_{\sf pl}U^{\ast}$.

###### Proof.

We interpret $U$ in $U^{\ast}$ via $\tau$ which is the identical translation
on the vocabulary of $\mathcal{L}$ and which interprets constant $c_{i}$ as
parameter $p_{i}$. ❑

###  A.3. c-Essential Undecidability

Consider a theory $U$. We write $U^{(\vv c)}$ for the theory $U$ with its
signature expanded with constants $\vv c$. Tarski in [TMR53] calls this _an
inessential extension_.

We write $U\subseteq_{\sf c}V$ for: for some $\vv c$, we have $U^{(\vv
c)}\subseteq_{\sf e}V$. We use _c-essentially_ in the obvious way.

######  Theorem A.2.

  1. a.

$U$ is c-essentially undecidable iff $U$ is essentially undecidable.

  2. b.

$U$ is c-essentially hereditarily undecidable iff $U$ is essentially
hereditarily undecidable.

###### Proof.

_We prove(a)._ The left-to-right direction is trivial. We treat right-to-left.
Suppose $U$ is essentially undecidable. Consider a consistent
$U^{\prime}\supseteq_{\sf c}U$. Suppose $U^{\prime}$ is decidable. Let
$U^{\ast}$ be the theory given by the $U^{\prime}$-theorems in the
$U$-language. Then, $U^{\ast}\supseteq_{\sf e}U$ and $U^{\ast}$ is consistent
and decidable. _Quod impossibile_.

_We prove(b)._ The left-to-right direction is trivial. We treat right-to-left.
Suppose $U$ is essentially hereditarily undecidable. Suppose
$U^{\prime}\supseteq_{\sf c}U$ and $W\subseteq_{\sf e}U^{\prime}$. Let
${U^{\prime}}^{\ast}$ be the set of consequences of $U^{\prime}$ in the
$U$-language and let $W^{\ast}$ be the set of consequences of $W$ in the
$U$-language. Then $U\subseteq_{\sf e}{U^{\prime}}^{\ast}$ and
$W^{\ast}\subseteq_{\sf e}{U^{\prime}}^{\ast}$. If $W$ would be decidable,
then so would be $W^{\ast}$. _Quod impossibile_. ❑

The next theorem combines Theorems 4 and 5 of [TMR53, Part I].

######  Theorem A.3.

Suppose $U$ is decidable and $\varphi$ is a sentence in the $U^{(\vv
c)}$-language. Then $U^{(\vv c)}+\varphi$ is decidable.

###### Proof.

We have $U^{(\vv c)}+\varphi\vdash\psi$ iff $U\vdash\forall\vv w\,(\varphi[\vv
c:\vv w]\to\psi[\vv c:\vv w])$. ❑

###  A.4. c-Lax Interpretability

We define _$V$ c-laxly interprets $U$_, or
$V\mathrel{\mbox{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>c}U$, iff, for all consistent $V^{\prime}\supseteq_{\sf c}V$, there is a
consistent $V^{\prime\prime}\supseteq_{\sf c}V^{\prime}$, such that
$V^{\prime\prime}\rhd U$.

######  Theorem A.4.

$V\mathrel{\mbox{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>c}U$ iff, for all $V^{\prime}\supseteq_{\sf e}V$, there is a
$V^{\prime\prime}\supseteq_{\sf c}V$, such that $V^{\prime\prime}\rhd U$.

###### Proof.

Left-to-right is trivial. We treat right-to-left. We assume the right-hand-
side of our equivalence. Suppose $V^{\prime}\supseteq_{\sf c}V$ and
$V^{\prime}$ is consistent. Let $\vv c$ be the relevant new constants. We
define ${V^{\prime}}^{\ast}$ as the theory given by the consequences of
$V^{\prime}$ in the $V$-language. Let $Z\supseteq_{\sf c}{V^{\prime}}^{\ast}$
be a consistent theory, such that $Z\rhd U$. Clearly, we may choose the new
constants of $Z$, say $\vv d,$ distinct from $\vv c$. We take
$V^{\prime\prime}:={V^{\prime}}^{(\vv d)}\cup Z^{(\vv c)}$. Clearly,
$V^{\prime\prime}\supseteq_{\sf c}V^{\prime}$ and $V^{\prime\prime}\rhd U$.

We claim that $V^{\prime\prime}$ is consistent. If not, for some $\psi(\vv
c,\vv d)$, we would have ${V^{\prime}}^{(\vv d)}\vdash\psi(\vv c,\vv d)$ and
$Z^{(\vv c)}\vdash\neg\,\psi(\vv c,\vv d)$. It follows that
${V^{\prime}}^{\ast}\vdash\exists\vv v\,\forall\vv w\,\psi(\vv v,\vv w)$, So,
$Z\vdash\exists\vv v\,\forall\vv w\,\psi(\vv v,\vv w)$. On the other hand, it
follows that $Z\vdash\forall\vv v\,\neg\,\psi(\vv v,\vv d)$. This contradicts
the fact that $Z$ is consistent. ❑

We prove that essential hereditary undecidability is preserved over c-lax
interpretability.

######  Theorem A.5.

Suppose
$V\mathrel{\mbox{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>c}U$ and $U$ is essentially hereditarily undecidable. Then $V$ is essentially
hereditarily undecidable.

###### Proof.

Suppose
$V\mathrel{\mbox{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>c}U$ and $U$ is essentially hereditarily undecidable. Suppose $W$ is a
decidable theory in the same language as $V$ that is consistent with $V$. We
have $V\subseteq_{\sf e}V\cup W=:V^{\prime}$. So, there is a consistent
$V^{\prime\prime}\supseteq_{\sf c}V^{\prime}$ such that $V^{\prime\prime}\rhd
U$. Say the witnessing constants are $\vec{c}$ and the witnessing translation
is $\tau$. Let $Z:=W^{(\vv c)}+\mathfrak{id}^{\tau}$. Then, by Theorem A.3,
$Z$ decidable. Since $Z\subseteq_{\sf e}V^{\prime\prime}$, so
$T:=\\{\varphi\mid Z\vdash\varphi^{\tau}\\}$ is consistent with $U$. Since $Z$
is decidable $T$ is decidable. _Quod non_. ❑

######  Theorem A.6.

Suppose $V\rhd_{\sf pl}U$. Then,
$V\mathrel{\mbox{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>c}U$.

###### Proof.

Suppose $V\rhd_{\sf pl}U$. Let $\tau$ be the witnessing translation. Consider
any consistent $V^{\prime}\supseteq_{\sf e}V$. We find that $V^{\prime}$
proves $\exists\vv p\;\varphi^{\tau,\vv p}$, for all $\varphi$ such that
$U\vdash\varphi$. We take a sequence of fresh constants $\vv c$ of the length
of $\vv p$. Then, by compactness,
$V^{\prime\prime}:=V^{\prime}+\\{\varphi^{\tau,\vv c}\mid U\vdash\varphi\\}$
is consistent and, clearly, $V^{\prime\prime}\supseteq_{\sf c}V^{\prime}$ and
$V^{\prime\prime}\rhd U$. ❑

##  Appendix B An Alternative Proof of Theorem 5.5

We show that if $U$ is $\Sigma^{0}_{1}$-representative, then it is effectively
inseparable. Let $\varPhi$ witness that $U$ is
$\Sigma^{0}_{1}$-representative. Suppose $\mathcal{X}_{0}$ and
$\mathcal{X}_{1}$ are disjoint RE sets. Via a bijective Gödel numbering, we
can view these as sets of $U$-sentences. Suppose that
$U_{\mathfrak{p}}\subseteq\mathcal{X}_{0}$ and
$U_{\mathfrak{r}}\subseteq\mathcal{X}_{1}$.

Suppose $\exists u\,\delta_{i}(u,x)$ represents $\varPhi(x)\in\mathcal{X}_{i}$
where $\delta_{i}$ is a pure $\Delta_{0}$-formula.

We need a special form of the Gödel fixed point construction. We substitute a
formula $\rho$ in $\sigma_{0}(v)<\sigma_{1}(v)$ in the following way. First we
substitute the numeral of the Gödel number of $\rho$ in each of the
$\sigma_{i}$. Then, we rewrite each of the resulting formulas to pure
1-$\Sigma^{0}_{1}$-form and, subsequently, we apply witness comparison.
Suppose Sub is this function. Let ${\sf sub}(x,y,z)$ be a pure
1-$\Sigma^{0}_{1}$-representation of the graph of Sub. Say ${\sf sub}(x,y,z)$
is $\exists w\,\nu(w,x,y,z)$, where $\nu$ is pure $\Delta_{0}$ and the witness
$w$ will always majorise $x$, $y$ and $z$. We consider the formula
$\varphi(v)$:

$(\exists a\,\exists w\mathbin{<}a\,\exists
z\mathbin{<}a\,(\nu(w,v,v,z)\wedge\exists
u\mathbin{<}a\,\delta_{1}(u,z)))\;<\\\ (\exists b\,\exists
w\mathbin{<}a\,\exists z\mathbin{<}b\,(\nu(w,v,v,z)\wedge\exists
u\mathbin{<}b\,\delta_{0}(u,z))).$

Let $\rho:={\sf Sub}(\varphi,\varphi)$. We note that $\rho$ is of the literal
form $\rho_{1}<\rho_{0}$, where the $\rho_{i}$ are pure
1-$\Sigma^{0}_{1}$-sentences.

Suppose $\varPhi(\rho)\in\mathcal{X}_{i}$. Then $\rho_{i}$. So either $\rho$
or $\rho^{\bot}$. If we have $\rho$, then $\varPhi(\rho)\in\mathcal{X}_{1}$.
Moreover, by $\Sigma 1$, we find $U\vdash\varPhi(\rho)$, and, hence,
$\varPhi(\rho)\in\mathcal{X}_{0}$. _Quod impossibile_. If we have
$\rho^{\bot}$, then $\varPhi(\rho)\in\mathcal{X}_{0}$. Moreover, by $\Sigma
3$, we have $U\vdash\neg\,\varPhi(\rho)$ and, hence,
$\varPhi(\rho)\in\mathcal{X}_{1}$. _Quod impossibile iterum_. _Ergo_ , the
formula $\varPhi(\rho)$ is in none of the $\mathcal{X}_{i}$.

######  Remark B.1.

Inspection of the proof shows that in order to derive Rosser’s theorem we do
not need the formalisation of the Fixed Point Lemma inside the given theory. ❍

######  Remark B.2.

The above proof can be somewhat simplified by employing a self-referential
Gödel numbering. See, e.g., [Kri23] or [GV23]. However, even with this
strategy, there will be some detail on how to handle substitution of numerals.
❍