# Evolution of the post merger remnants from the coalescence of oxygen-neon
and carbon-oxygen white dwarf pairs

Chengyuan Wu Yunnan Observatories, Chinese Academy of Sciences, Kunming
650216, China<EMAIL_ADDRESS>Heran Xiong  Research School of
Astronomy and Astrophysics, The Australian National University, Canberra, ACT
2611, Australia  Jie Lin  Physics Department and Tsinghua Center for
Astrophysics, Tsinghua University, Beijing, 100084, People’s Republic of China
Yunlang Guo  Yunnan Observatories, Chinese Academy of Sciences, Kunming
650216, China  University of Chinese Academy of Sciences, Beijing 100049,
China  Xiaofeng Wang  Physics Department and Tsinghua Center for Astrophysics,
Tsinghua University, Beijing, 100084, People’s Republic of China  Zhanwen Han
Yunnan Observatories, Chinese Academy of Sciences, Kunming 650216, China  Bo
Wang  Yunnan Observatories, Chinese Academy of Sciences, Kunming 650216, China

###### Abstract

Although multidimensional simulations have investigated the processes of
double WD mergers, post-merger evolution only focused on the carbon-oxygen
(CO) WD or helium (He) WD merger remnants. In this work, we investigate for
the first time the evolution of the remnants stemmed from the merger of
oxygen-neon (ONe) WDs with CO WDs. Our simulation results indicate that the
merger remnants can evolve to hydrogen- and helium-deficient giants with
maximum radius of about 300${R}_{\odot}$. Our models show evidence that merger
remnants more massive than $1.95{M}_{\odot}$ can ignite Ne before significant
mass-loss ensues, and they thus would become electron-capture supernovae
(ECSNe). However, remnants with initial masses less than $1.90{M}_{\odot}$
will experience further core contraction and longer evolutionary time before
reaching at the conditions for Ne-burning. Therefore their fates are more
dependent on mass-loss rates due to stellar winds, and thus more uncertain.
Relatively high mass-loss rates would cause such remnants to end their lives
as ONe WDs. Our evolutionary models can naturally explain the observational
properties of the double WD merger remnant $-$ IRAS 00500+6713 (J005311). As
previously suggested in the literature, we propose and justify that J005311
may be the remnant from the coalescence of an ONe WD and an CO WD. We deduce
that the final outcome of J005311 would be a massive ONe WD rather than a
supernova explosion. Our investigations may be able to provide possible
constraints on the wind mass-loss properties of the giants which have CO-
dominant envelopes.

stars: evolution – binaries: close – stars: white dwarfs

##  1 Introduction

White dwarfs (WDs) are the most common evolutionary fate of single stars with
initial masses lower than $9-11{M}_{\odot}$ (e.g. Nomoto & Kondo 1991; Ritossa
et al. 1999; Eldridge & Tout 2004; Siess 2007; Poelarends 2007; Ibeling &
Heger 2013; Doherty et al. 2017; Siess 2010). It is estimated that there are
about $10^{10}$ WDs in our Galaxy out of which $2.5\times{10}^{8}$ reside in
binaries consisting of two WDs (e.g. Napiwotzki 2009; Nelemans et al. 2001;
Holberg 2009). About half of these binary WDs are close enough to begin mass
transfer within Hubble time.

The theoretical understanding of the evolution of WD merger remnants is based
on using of information from three-dimensional (3D) post-merger configurations
to build one-dimensional (1D) models, whose further evolution is computed in
detail through their longer-duration phases. This procedure has been applied a
number of times in the literature. For instance, Zhang et al. (2014)
investigated the merger of CO WD and He WD, and found that the merger remnant
can evolve to R CrB stars. Wu et al. (2022) investigated the post-merger
evolution of massive CO WD with He WD, and found that these systems contribute
to a portion of ultra-massive CO WDs. Brooks et al. (2017) investigated the
merger of ONe WD with He WD, and found that these systems are related to
electron capture supernovae (ECSNe). Schwab et al. (2016) investigated double
CO WD mergers (see also Schwab 2021), and found that the corresponding super-
Chandrasakha-mass remnants can undergo iron-core collapse supernova explosions
(Fe-CCSN). However, no post-merger evolutionary models has been built so far
to study ONe WD + CO WD. For the evolutionary outcomes of such systems,
although previous works suspected that they may related to ECSNe, SNe Iax or
NSs (e.g. Nomoto & Kondo 1991; Kitaura et al. 2006), but detail investigates
are desperately needed.

In this work, based on the results of three-dimensional (3D) simulations from
Dan et al. (2014), we construct the 1D structures of ONe+CO WD merger remnants
and investigate their evolution. We will show that the final outcomes of such
remnants are more complicated than previous deductions, which could be massive
ONe WD, ONeFe WD or NS depending on the mass of the merger remnant, the wind
mass-loss process and the effects of the treatment of convection. Besides, our
model can perfectly explain the main observational properties of object
J005311 (e.g. Gvaramadze et al. 2019; Oskinova et al. 2020; Lykou et al.
2022), which may provide a strong constrain on the wind mass-loss process of
such merger remnants. The article is organized as follows. In Sect. 2, we
introduce our initial models of merger remnants. The evolution process of the
remnants are shown in Sect. 3. The discussion of model uncertainties are given
in Sect. 4. We provide our analyze and final fate of J005311 in Sect. 5. The
summary are given in Sect. 6.

##  2 Initial models

Dan et al. (2014) performed a large set of WD merger simulations, including
225 pairs of double WD mergers. The mass range of ONe WDs and CO WDs in their
grid are from $1.10$ to $1.20{M}_{\odot}$ and from $0.65$ to $1.05{M}_{\odot}$
spaced at mass interval of $0.05{M}_{\odot}$, respectively. Almost all the
mergers in their simulations are able to form stable remnants, except the
double WD pairs with extremely low mass ratio (low mass He WDs merger with ONe
WDs), which do not show dynamical mass-transfer during the merger. For the
more massive merger remnant, Kashyap et al. (2018) simulated the process of
$1.20{M}_{\odot}$ ONe WD merger with $1.10{M}_{\odot}$ CO WD, and found that
the merger yields a failed detonation which may produce a very faint and
rapidly fading transient. In the present work, we only consider the post-
merger evolution of stable merger remnants (i.e. Dan et al. 2014).

According to Dan et al. (2014), a double WD merger will form a remnant
composed of a cool core, a hot envelope, a Keplerian disk and a tidal tail
from the inside out. Based on the results of 3D simulations, they provided 1D
remnant structure profiles, including key information of the merger remnants,
such as their maximum temperature (${T}_{\rm max}$), the density at the
location of the maximum temperature (${\rho}_{\rm max}$), their core mass
(${M}_{\rm core}$), their envelope mass (${M}_{\rm env}$), and their disk mass
(${M}_{\rm disk}$), among others. By considering the structures of merger
remnants from Dan et al. (2014), Schwab et al. (2016) developed the energy
injecting prescription to map the initial configurations of the remnants and
investigated the evolution of double CO WD mergers (see also Schwab 2021).

In this work, we use a prescription similar to the one described in Schwab et
al. (2016) to construct a series of ONe WD+CO WD merger remnants with ONe WD
mass in the range from 1.1 to $1.2{M}_{\odot}$ and total mass from 1.60 to
$2.25{M}_{\odot}$ spaced at mass interval of $0.05{M}_{\odot}$ as our fiducial
models by employing stellar evolutionary code MESA (version 12778; e.g. Paxton
et al., 2011, 2013, 2015, 2018, 2019). In addition, we also constructed double
CO WD merger remnants to compare with the observations (detailed information
can be found in Sect. 5). Since the mass of tidal tail only occupies a
significantly small portion of total mass of the remnants, we ignore the
influence of tidal tail and construct the merger remnants consisting only of a
core, an envelope and a disk. As proved by Schwab (2021), this approach has
almost negligible influence on the initial profile of the remnants.

The detail prescriptions in constructing initial models are presented as
follows. First, we create a series of pre-main sequence stars with masses
equal to the total masses of the merger remnants, and evolve these models with
nuclear reactions and elemental mixing turned off, until the central density
reached ${10}^{3}\,{\rm g}\,{\rm cm}^{-3}$. Secondly, we resume the evolution,
using the built-in capability of MESA to relax our model to a specified
composition. In Dan et al. (2014), the composition of ONe WDs include 60% of
${}^{\rm 16}{\rm O}$, 35% of ${}^{\rm 20}{\rm Ne}$, 5% of ${}^{\rm 24}{\rm
Mg}$, and CO WDs include 40% of ${}^{\rm 12}{\rm C}$ and 60% of ${}^{\rm
16}{\rm O}$. Thus, the merger remnants in our models consist of an ONe core
and an outer CO envelope. Once the composition relaxation is achieved, we
evolve these models until their central densities reach ${10}^{5}\,{\rm
g}\,{\rm cm}^{-3}$.

In order to construct the thermal structures of the merger remnants, we inject
energy into different mass zones to alter the temperature and density until
they reach target profiles, i.e. an isothermal core with a temperature of
${10}^{8}\,{\rm K}$, surrounded by a hot envelope and a disk. The target
profiles are as follows:

For $0\leq{M}_{\rm r}\leq{M}_{\rm{core}}$,

$T({M}_{\rm r})={T}_{\rm{core}}.$ (1)

For ${M}_{\rm{core}}\leq{M}_{\rm r}\leq{M}_{\rm{core}}+{M}_{\rm{env}}$,

$s({M}_{\rm
r})={s}_{\rm{core}}+[{s}_{\rm{env}}-s({M}_{\rm{core}})]\frac{{M}_{\rm{r}}-{M}_{\rm{core}}}{{M}_{\rm{peak}}}.$
(2)

And for ${M}_{\rm{core}}+{M}_{\rm{env}}\leq{M}_{\rm r}\leq{M}_{\rm{tot}}$,

$s({M}_{\rm r})={s}_{\rm{env}}.$ (3)

In the equations, ${M}_{\rm{core}}$, ${M}_{\rm{env}}$ and ${M}_{\rm{peak}}$
are, respectively, core masses, envelope masses and the mass coordinates of
the maximum temperature of the merger remnants, which are related to the total
masses of the remnants (${M}_{\rm{tot}}$) and the mass ratios of the double
WDs before mergers ($q$), i.e.,

${M}_{\rm{core}}={M}_{\rm{tot}}(0.7786-0.5114q),$ (4)
${M}_{\rm{env}}={M}_{\rm{tot}}(0.2779-0.464q+0.716{q}^{2}),$ (5)
${M}_{\rm{peak}}={M}_{\rm{tot}}(0.863-0.3335q).$ (6)

In equation (1), ${T}_{\rm{core}}={10}^{8}\,{\rm K}$ is the typical
temperature of the massive WD which used in our simulations. ${M}_{\rm peak}$
in equation (2) represents . ${s}_{\rm{core}}$ and ${s}_{\rm{env}}$ in
equations (2) and (3) are, respectively, the specific entropies of the core
and the envelope. We set ${s}_{\rm{core}}$ to be the specific entropy of the
core when ${T}_{\rm{core}}={10}^{8}\,{\rm K}$, and ${s}_{\rm{env}}$ equals to
the following equation:

${\rm
log}({s}_{\rm{env}}/{\rm{erg}}\,{\rm{g}^{-1}}\,{\rm{K}^{-1}})=8.7+0.3({M}_{\rm{tot}}/{M}_{\odot}-1.5).$
(7)

We use a very short time step (in seconds) to complete the energy injection
process to guarantee the corresponding process cannot change the target
profile (for more detail of the method, see Schwab et al. 2016 and Schwab
2021).

After the energy injection process, we obtained the initial structures of the
merger remnants. Fig. 1 shows the temperature-mass coordinate (${T}$-${M}_{\rm
r}$) and temperature-density (${T}$-${\rho}$) profiles of different merger
remnants after the energy injection processes. Our initial remnant models have
isothermal cores at ${10}^{8}\,{\rm K}$. ${M}_{\rm env}$ and ${T}_{\rm max}$
of the merger remnants increase with increasing mass ratio of the progenitor
double WDs and their total masses, which is consistent with 3D simulations.

Figure 1: Temperature-Mass coordinate profile of the merger remnants just
after the energy injection processes. Panel (a): temperature profiles of three
merger remnants of different total masses and the same ONe WD mass. Panel (b):
temperature profiles of three merger remnants of equal ONe WD mass and
different total masses. Different lines in panel (a) and (b) represent the
corresponding structures of double WD mergers with various mass ratios.

The comparison of some major parameters of our models with 3D simulations are
presented in Fig. 2. Although ${T}_{\rm max}$ and mass coordinate show
slightly different with 3D simulations, our models conform to the variation
trend of different merger remnants. Note, however, that the subsequent
evolution of the remnants may be sensitive to the structures of the initial
models. We analyse the influence of the initial conditions on the evolution of
merger remnants in sect. 4.1.

Figure 2: Main properties of the merger remnants (${T}_{\rm max}$ in panel a;
core mass in panel b; envelope mass in panel c; disk mass in panel d) with
different initial ONe WD masses and CO WD masses. Red, sea-green and blue
filled circles in each panel represent parameters of merger remnants with
initial ONe WD masses equal to $1.10{M}_{\odot}$, $1.15{M}_{\odot}$ and
$1.20{M}_{\odot}$ in our models, whereas grey diamonds in each panel represent
the corresponding parameters resulted from Dan et al. (2014).

##  3 Evolution of the remnant of ONe WD+CO WD mergers

###  3.1 Input physics

Since the outer layers of such merger remnants consist of carbon and oxygen,
we use OPAL Type 2 opacity table to calculate the evolution of the remnants
(e.g. Iglesias & Rogers 1996). When the remnants evolve to giant phase, the
surface temperature may decrease to about ${\rm log}(T/{\rm K})=3.65$. Since
the lower boundary of the OPAL tabulations is ${\rm log}(T/{\rm K})=3.75$, as
in Schwab et al. (2016), we adopt the Kasen opacities to deal with the low
temperature conditions, and blend the OPAL and Kasen opacities between ${\rm
log}({T}/{\rm K})=4.1$ and $4.2$. This supplement is suitable for H- and He-
deficient cool giant-like stars.

The nuclear reaction network used in the present work is
“${\tt{wd\\_aic.net}}$”, which includes “${\tt{co\\_burn.net}}$” in
“${\tt{MESA\,\,default}}$”, and we artificially added four isotopes (i.e.
${}^{\rm 20}{\rm O}$, ${}^{\rm 20}{\rm F}$, ${}^{\rm 24}{\rm Ne}$ and ${}^{\rm
20}{\rm Na}$) and the corresponding weak reactions in order to calculate the
electron-capture process. The weak interaction rates are from Schwab et al.
(2015). We apply “${\tt{MESA\,\,default}}$” screening factors to correct
nuclear reaction rates for plasma interactions (e.g. Chugunov et al. 2007).
The cooling rates from thermal neutrinos are derived from the fitting formulae
of Itoh et al. (1996).

As the remnant expands toward giant phase, the superadiabatic gradient arising
in radiation-dominated envelopes can force the adoption of prohibitively short
time step. To circumvent this issue, we apply the treatment of convective
energy transport “MLT++” (e.g. Paxton et al. 2013), to reduce the
superadiabaticity in the radiation-dominated convective regions. At soon as Ne
burning is ignited under the CO envelope, the inwardly propagating flame will
leave a thin burning shell with extremely high energy release rate. Previous
works found that the treatment of convection can affect the evolution of the
stars, for example, a small amount of convective boundary mixing can remove
the physical conditions required for the off-center burning flame in super-AGB
stars (e.g. Denissenkov et al. 2013; Jones et al. 2014; Farmer et al. 2015;
Jones et al. 2016a). In the present work, we consider overshooting process
with parameters of ${\rm f}=0.014$ and ${\rm f}_{\rm 0}=0.004$ to calculate
the inwardly propagating Ne flame (e.g. Denissenkov et al. 2013; Wu et al.
2020). We constrain the time resolution in our calculations by adopting
“${\tt{varcontral=3d-4}}$”. When the increasing central density of the remnant
reached $\approx$ ${10}^{9.5}\,{\rm g}\,{\rm{cm}^{-3}}$, we constrain the time
step by controlling the change of central density
(${\tt{delta\\_lgRho\\_cntr\\_limit=1d-3}}$) to guarantee the precision in
simulating the electron-capture process. In order to have a better treatment
on the propagation of off-center flame, we control the spatial resolution by
changing “${\tt{dlog\\_burn\\_c\\_dlogP\\_extra=0.1}}$” and
“${\tt{dlog\\_burn\\_ne\\_dlogP\\_extra=0.1}}$” to increase the mesh grid
around the flame.

Some physical mechanisms have not been included in our fiducial models, such
as wind mass-loss process, Urca process-cooling, stellar rotation, etc. We
will discuss the effects of the corresponding uncertainties in sect. 4.

###  3.2 Post-merger evolution

After constructing the initial models of the merger remnants, we restored the
nuclear reactions and mixing process to let the remnants evolve forward in
time. The detailed physical inputs of our fiducial models can be found in
Table. 2. The evolutionary features of two representative examples (a
$1.1{M}_{\odot}$ of ONe WD merges with a $0.8{M}_{\odot}$ of CO WD,
${M}_{\rm{tot}}=1.90{M}_{\odot}$; a $1.1{M}_{\odot}$ of ONe WD merges with a
$0.85{M}_{\odot}$ of CO WD, ${M}_{\rm{tot}}=1.95{M}_{\odot}$) are presented in
Fig. 3. Owing to the initially compact conﬁgurations, the remnants experiences
an extremely short thermal adjustment stage (8hrs) prior to the ignition of
carbon. Carbon burning starts below the convective envelope (the CO envelope
is partial convective because of the energy injection process), proceeds
through thermal pulses and forces the formation of an inner convective zone
due to the high energy release. Nuclear energy is then transformed into work
of expansion on the envelope, resulting in the increase of radius and opacity.
As the temperature gradient increases, the convective zone develops gradually
in the CO envelope. After the remnants approach the giant phase, carbon
burning proceed steadily on the bottom of the shell, resulting in the mass
increase of the ONe core. During the giant phase, the remnants can expand to
about $300{R}_{\odot}$ in tens of years after merger. This giant stage takes
the longest part of the evolution between the merger and the end of our
calculations.

Carbon burning becomes stationary after some thermal pulses. The steady carbon
burning leads to the mass increase of the degenerate core. The evolution of
such giant-like object complies with core mass-luminosity relation (e.g.
Jeffery 1988). The major difference between $1.90{M}_{\odot}$ and
$1.95{M}_{\odot}$ cases lays in the time when the Ne ignition occurs.

For the less massive remnants, carbon burning cannot triggers the Ne ignition
immediately after the merger, and the remnants’ cores need to experience
longer evolutionary times to increase in mass. In this case, the ignition of
Ne occurs after the electron capture process of ${}^{\rm 24}{\rm Mg}$ (i.e.
3015 years after merger) when the temperature and density of the Ne-burning
region reaches ${\rm log}({T}/{\rm K})=9.11$ and ${\rm log}({\rho}/{\rm
g}\,{\rm{cm}^{-3}})=7.23$, respectively. The mass coordinate of Ne ignition is
at ${M}_{\rm r}=1.368{M}_{\odot}$, just below the carbon burning shell
(${M}_{\rm r}=1.388{M}_{\odot}$, which corresponds to the mass of Ne core).

For the massive case, Ne ignition occurs $209$ years after the merger, at
${M}_{\rm r}=1.186{M}_{\odot}$ (relatively far away from the carbon burning
shell, located at ${M}_{\rm r}=1.360{M}_{\odot}$), when the temperature and
density of the Ne-burning region reaches ${\rm log}({T}/{\rm K})=9.23$ and
${\rm log}({\rho}/{\rm g}\,{\rm{cm}^{-3}})=7.37$, respectively. This means
that the massive remnant does not experience a significantly long core
increase phase, which leads to shorter evolutionary times of the remnants.

The major difference between these two cases can also be reviewed from the
elemental abundance distributions (panel c and d of Fig. 3). Owing to that the
position of Ne ignition is very close to the CO burning shell in
$1.90{M}_{\odot}$ case, the convective region caused by Ne burning penetrates
into the CO envelope, resulting in the dredge-up of ONe-rich material out to
the envelope. The abundance of Ne in CO envelope can reach about $30\%$.
Meanwhile, note that the thickness of Si mantle is equals to the width of
convective region caused by the Ne burning, therefore, the Si mantles in two
cases are different because of the different positions of Ne ignitions.

Figure 3: An example of the evolution of two merger remnants (a
$1.1{M}_{\odot}$ of ONe WD merges with a $0.8{M}_{\odot}$ of CO WD,
${M}_{\rm{tot}}=1.90{M}_{\odot}$; a $1.1{M}_{\odot}$ of ONe WD merges with a
$0.85{M}_{\odot}$ of CO WD, ${M}_{\rm{tot}}=1.95{M}_{\odot}$). Panel (a):
Hertzsprung–Russell diagram; panel (b): Central density-temperature evolution;
panel (c) and (d): elemental abundance distributions of $1.90{M}_{\odot}$ and
$1.95{M}_{\odot}$ merger remnants at the final stage of their evolution just
prior to the explosions. In panels (a) and (b), diamonds represents the
position of Ne ignition, whereas circles and triangles represent every 1000
years and 100 years after merger, respectively. Squares in panel (b)
represents the position of final evolutionary stage, nearby which the
corresponding evolutionary time are labeled. Black dotted lines in panel (b)
represents the carbon and oxygen burning lines, whereas brown and magenta
dotted lines represent starting lines of electron capture reactions of
${}^{\rm 24}{\rm Mg}$ and ${}^{\rm 20}{\rm Ne}$. Note that diamond on blue
line of panel (b) is out of range.

In addition to the above-mentioned evolutionary times, the final densities of
the two cases considered also show differences. To better analyse these
differences, we present the Hertzsprung–Russell diagram (HRD) and central
density-temperature profiles for a series of remnants in Fig. 4. Although the
evolutionary processes follow similar trends on HRD, they show differences in
${T}_{\rm c}$-${\rho}_{\rm c}$ profile. Due to the different ignition times of
Ne mentioned above, the corresponding two classes of remnants (${M}\leq
1.90{M}_{\odot}$ and ${M}\geq 1.95{M}_{\odot}$) can evolve to different
central densities when explosion occurs. Previous multidimensional dynamical
simulations of supernova explosion suggested that for the pure ONe core, Ne or
O explosion occur at higher central density (${\rm log}({\rho}_{\rm c}/{\rm
g}\,{\rm cm}^{-3})>9.97$) will result in the formation of NS, whereas for the
cases of which Ne or O explosion occur at lower central density (${\rm
log}({\rho}_{\rm c}/{\rm g}\,{\rm cm}^{-3})<9.97$), a ONeFe core would leave
behind after the explosion (e.g. Jones et al., 2016b, 2019). Our results
indicate that the final outcomes of double WD mergers strongly depend on the
initial remnant masses.

Figure 4: Evolutionary tracks of different remnants (${M}_{\rm ONe}^{\rm
i}=1.10{M}_{\odot}$) in HRD and in central temperature-central density
profile. Black dotted lines in panel (b) represent the carbon and oxygen
burning conditions, whereas brown and magenta dotted lines represent starting
lines of electron capture reactions of ${}^{\rm 24}{\rm Mg}$ and ${}^{\rm
20}{\rm Ne}$. Grey vertical line in panel (b) represents the critical density
below which the explosion of the core will lead to the formation of ONeFe WD,
whereas above which will lead to the formation of NS.

##  4 Model uncertainties

Previous studies indicate that some pivotal physical mechanisms such as wind
mass-loss process can significantly affect the evolution of double WD merger
remnants (e.g. Schwab 2019; Schwab 2021; Wu et al. 2022). We discuss the
influence of these mechanisms in the current section.

###  4.1 Initial structure

The initial structures of the merger remnants of our models result from the
fitting formulae in Dan et al. (2014). The comparison of some major parameters
of our models with those from 3D simulations are presented in Fig. 5. Although
${T}_{\rm max}$ and mass coordinate show slightly different with 3D
simulations, our models conform to the variation trend of different merger
remnants. However, the fitting formulae for different properties of merger
remnants in Dan et al. (2014) can only explain 75% to 90% of the variance. In
order to investigate the effects of initial model uncertainties, we
constructed models with different ${T}_{\rm max}$ by changing the entropy of
the envelope (${s}_{\rm env}$ in Eq. 4). Basically, a higher envelope entropy
can lead to a lower ${T}_{\rm max}$ because of the decreased in density of
envelope. On the contrary, a lower envelope entropy can result in a higher
${T}_{\rm max}$. Detail physical inputs are given in table. 3.

The evolution of $1.90{M}_{\odot}$ and $1.95{M}_{\odot}$ merger remnants with
different initial ${T}_{\rm max}$ are shown in Fig. 5. Decreasing the envelope
entropy (increasing ${T}_{\rm{max}}$) in our models shifts downwards the lower
mass threshold for the formation of an ONeFe WD. For the less massive remnant
(${M}=1.90{M}_{\odot}$), to increases the envelope entropy (a lower ${T}_{\rm
max}$) has no influence in the final outcome, whereas increasing in ${T}_{\rm
max}$ will lead to the similar evolutionary track as massive remnant. On the
other hand, the outcome of our $1.95{M}_{\odot}$ model is not affected by an
increase in ${T}_{\rm{max}}$, but delays Ne-ignition if ${T}_{\rm max}$ is
decreased. The dependence of the outcomes with ${T}_{\rm max}$ reflects the
influence of the initial remnant mass on the final fate of the considered
models.

Figure 5: Initial profiles and evolutionary tracks of $1.90{M}_{\odot}$ and
$1.95{M}_{\odot}$ merger remnants with different envelope entropy. Panel (a)
and (b): initial $T$-${M}_{\rm r}$ profiles. Red (blue) solid line represents
our fiducial model, whereas magenta (cyan) dashed-dotted-dotted line and
golden (navy) dotted line represent remnants with lower (${\rm ln}({s}_{\rm
env}/{\rm erg}\,{\rm{g}^{-1}}\,{\rm{K}^{-1}})=8.7$) and higher (${\rm
ln}({s}_{\rm env}/{\rm erg}\,{\rm{g}^{-1}}\,{\rm{K}^{-1}})=9.1$) envelope
entropy, respectively. Panel (c) and (d): evolutionary tracks of merger
remnants in ${\rm log}{T}_{\rm c}-{\rm log}{\rho}_{\rm c}$ diagram.

###  4.2 Mass-loss rate

The wind mass-loss prescription has not been considered in our fiducial
models. During their evolution, the merger remnants can evolve to giant phase
and produce dusty winds, which may have influence on their evolution. However,
the physics of the winds of such H-/He-deficient giants is not well
understood, and mass-loss rates cannot be constrained from the observation so
far. In the present work, we simply explore the effect of mass-loss by
considering Reimers’ (e.g. Reimers 1975) and Blocker’s (e.g. Bloecker 1995)
wind mass-loss prescription with “${\tt{MESA\,\,default}}$” values of wind
coefficients (${\eta}_{\rm R}=0.5$; ${\eta}_{\rm B}=0.05$). Detail physical
inputs are given in Table. 4.

Fig. 6 presents the evolutionary tracks of $1.90{M}_{\odot}$
($1.10$+$0.80{M}_{\odot}$) and $1.95{M}_{\odot}$ ($1.10$+$0.85{M}_{\odot}$)
ONe+CO WD merger remnants under Reimers’ and Blocker’s wind mass-loss
prescriptions. During the giant phase, the wind mass-loss rate with Reimers’
wind prescription is significantly less than that under Blocker’s wind
prescription, which only exceeds the order of magnitude of
${10}^{-4}\,{M}_{\odot}\,{\rm yr}^{-1}$ at the end of the evolution. This
means that Reimers’ wind cannot significantly change the total masses of the
remnants, and the final outcome of the remnants are same as those without wind
mass-loss. However, under the Blocker’s wind prescription, the remnants can
maintain high mass-loss rate during the whole giant phase, which effectively
decrease the total mass of the remnants. For the less massive remnants
(${M}\leq 1.90{M}_{\odot}$), all the CO envelope can be lost during the
evolution, resulting in the formation of ONe WDs. Although the strong wind
mass-loss process can obviously affect the massive remnants (${M}\geq
1.95{M}_{\odot}$), because of the short evolutionary time, the final mass of
the initially massive models can still be high enough to trigger the electron
capture reactions that lead to the formation of ONeFe WDs. Our results
indicate that wind prescriptions play a critical role in investigating the
evolution of double WD merger remnants. Whether NSs can be formed though
ONe+CO WD binary systems may strongly depend on the wind efficiency after the
merger.

Figure 6: Evolutionary of $1.90{M}_{\odot}$ and $1.95{M}_{\odot}$ ONe+CO WD
merger remnants under different wind mass-loss prescriptions. Panel (a): HRD.
The representative of each line are labeled. Squares on each line represent
the positions on HRD of the corresponding models when supernova explosion
occur.

Panel (b): evolution of mass-loss rate.

###  4.3 Initial helium abundance

According to the stellar evolutionary results, there exists a thin He shell
outside the CO core for the low mass CO WDs. The 0.5 to $0.6{M}_{\odot}$ CO
WDs in Dan et al. (2014) models have $0.1{M}_{\odot}$ of He envelope. During
the merger process, He is fully mixed with CO material and form the hot
envelope surrounding the post-merger ONe WD. The exist of He in the merger
remnant may influence the evolution of the remnants. To investigate the effect
of the existence of He in our initial models, we build an alternative
$1.60{M}_{\odot}$ ONe WD+CO WD merger remnant, by changing the composition
described in Sect. 2 to 40% of ${}^{\rm 12}{\rm C}$, 40% of ${}^{\rm 16}{\rm
O}$ and 20% of ${}^{\rm 4}{\rm He}$. Then we let this model evolve in time.
Detailed physical inputs are given in Table. 5.

In Fig. 7, we present the evolutionary tracks of $1.60{M}_{\odot}$ ONe+CO WD
(original composition) and the ONe+HeCO WD (alternative composition) merger
remnants on the HRD. Due to the non-negligible amount of He in the envelope,
the temperature required for the He-rich envelope burning is expected to be
lower than that without helium. Thus, the convective envelope develops very
fast in the He-rich envelope, it becomes fully convective in only 10 hours
after merger, resulting in the difference during the luminosity increase
phase. Furthermore, owing to the lower nuclear reaction rate in the He-rich
burning shell comparing with the pure CO counterpart, the ONe+HeCO WD merger
remnants need longer evolutionary time to increase the core mass until
explosive Ne/O ignition occurs. Although the existence of He in the envelope
influences the evolutionary time of the remnants, the final outcome are the
same.

Figure 7: Evolutionary tracks of $1.60{M}_{\odot}$ ONe+CO WD and ONe+HeCO WD
merger remnants on HRD. Black crosses on red curve represent every 5000 years
during evolution, whereas blue circles on blue curve represent every 1000
years during evolution. Black square represents final position on HRD for each
evolution track.

###  4.4 Urca-process cooling

Urca-process-cooling may effect the central temperature during the evolution.
We construct $1.90{M}_{\odot}$ and $1.95{M}_{\odot}$ merger remnants with 5%
of ${}^{\rm 23}{\rm Na}$ and 1% of ${}^{\rm 25}{\rm Mg}$ in the initial ONe WD
(e.g. Schwab et al. 2017), and simulate their evolution to investigate the
effect of Urca-process-cooling. Detail physical inputs are given in table. 6.

From Fig. 8, we can see that by considering Urca-process-cooling, the central
temperature decreased during the stage of electron capture reaction of
${}^{\rm 25}{\rm Mg}$ and ${}^{\rm 23}{\rm Na}$, which is similar to what
happened in mass-accretion WD systems (e.g. Schwab et al. 2017). The remnants
evolve faster when Urca-process-cooling is considered. This is because due to
the electron capture of ${}^{\rm 25}{\rm Mg}$ and ${}^{\rm 23}{\rm Na}$, the
central temperature of remnant decreased, resulting in the shrinking of the
ONe core which promote the occurrence of electron capture reaction of ${}^{\rm
24}{\rm Mg}$ and ${}^{\rm 20}{\rm Ne}$. The examples show that the final fates
of merger remnants do not changed when considering Urca-process cooling, which
means that the corresponding effect is a minor factor in such merger remnants.

Figure 8: Evolutionary tracks of $1.90{M}_{\odot}$ and $1.95{M}_{\odot}$
merger remnant in ${\rm log}{T}_{\rm c}-{\rm log}{\rho}_{\rm c}$ diagram.
Solid and dashed lines represent the corresponding tracks when considering and
without considering Urca-process-cooling during evolution, respectively.

###  4.5 Rotation

During the merger process, a portion of orbital angular momentum could be left
in the remnant. Rotation can decrease the density in the burning zone and
enhance the mixing process, which may have influence on the evolution of the
remnants.

In order to investigate the rotational effects, we inject torque into
different mass zones in $1.90{M}_{\odot}$ and $1.95{M}_{\odot}$ initial
models. 3D simulations predicted that the merger remnants is consist of a
corotating core with ${\Omega}_{\rm c}\approx 0.23-0.52\,{\rm rad}/{\rm s}$
(depended on total mass and mass ratio), and an differential rotating envelope
of which ${\Omega}_{\rm e}$ gradually decreases from the inside out. In the
present work, we consider two different rotational velocity. The fast
rotational models have uniform angular velocity of ONe core (${\Omega}_{\rm
c}=0.5\,{\rm rad}/{\rm s}$) and different rotational velocity in envelope
(${\Omega}_{\rm e}=0.5\,{\Omega}_{\rm crit}$), whereas the slow rotational
models have the same angular velocity distributions but with ${\Omega}_{\rm
c}=0.25\,{\rm rad}/{\rm s}$, and ${\Omega}_{\rm e}=0.25\,{\Omega}_{\rm crit}$,
respectively. Note that magnetic fields are quite effective in transporting
angular momentum in double WD merger remnants, therefore, we adopt the angular
momentum transport prescriptions described in Spruit (2002) (ST) and Fuller et
al. (2019) (FPJ) to simulate the evolution of the remnants, respectively.
Detail physical inputs are given in Table. 7.

From Fig. 9, we can see that the remnants computed with the FPJ prescription
can transport angular momentum from the core to the envelope more rapidly.
Thus, in this case, the envelope reaches the same angular velocity as the core
has within hours after merger. As the remnants expand towards the giant phase,
the angular velocity of the envelope obviously decreases. The opacity of the
outer shell changes due to the variation of temperature and rotational mixing
process, which leads to differences in the phase in which luminosity increase.
Although the final fate of the rotational models are same as those of the
fiducial models, the difference of core rotational velocity caused by the
different prescriptions of angular momentum transport implies that the
efficiency of angular momentum transport is one of the most important
mechanism in influencing the rotational velocity of massive WDs.

Figure 9: Evolutionary tracks when consider rotational effect. Panel (a):
initial rotation profile of $1.90{M}_{\odot}$ and $1.95{M}_{\odot}$ merger
remnants. Solid, dashed/dashed-dotted-dotted and dotted lines represent
angular velocity, critical angular velocity and the ratio of angular velocity
and critical angular velocity of $1.90$ and $1.95{M}_{\odot}$ merger remnants
with different initial rotational velocities, respectively. Panel (b):
evolution of core angular velocity (solid lines) and envelope angular velocity
(dotted lines) of four models under different angular momentum transport
prescriptions. Region inside the rectangle represents the most visible portion
of the angular momentum transfer process. Panel (c): evolutionary tracks of
four models comparing with non-rotational models on HRD. Similar to panel (b),
region inside the rectangle represents the most visible portion of the angular
momentum transfer process.

##  5 J005311

Owing to the small quantity of massive double WD systems (e.g. Liu & Wang
2020) and the extremely short evolutionary time of the corresponding merger
remnants (e.g. Schwab 2021) , it is very difficult to observe such remnants.
Fortunately, Gvaramadze et al. (2019) discovered a hot star in the
constellation Cassiopeia (RA = 00h 53 min 11.21s and dec. = +${67}^{\circ}$
30’ 2.1”, hereafter J005311), located at the center of a circular mid-infrared
nebula by using data from the Wide-field Infrared Survey Explorer (WISE). The
center star has V band magnitude of 15.5 mag and, according to GAIA DR2, is
located at a distance of 3.06 Kpc. The nebula appears as a circular shell and
a diffuse halo with linear radii of about 1.1pc and 1.6pc, respectively. The
optical follow-up spectroscopy of J005311 obtained from the Russian 6-m
telescope revealed an emission-line-dominated spectrum, similar to oxygen-rich
Wolf-Rayet (WO type) star. This emission-line-dominated spectrum was later
confirmed by Garnavich et al. (2020) who used the Large Binocular Telescope
(LBT) to obtained the spectra of J005311, and found significant variability in
the profile of strong O VI emission feature which was predicted to be caused
by rapidly shifting subpeaks generated by clumpiness in the stellar wind of
J005311. By employing the Potsdam Wolf-Rayet code, Gvaramadze et al. (2019)
analysed the optical spectrum of J005311, and found that the composition of
the material at the base of wind was dominated by oxygen and carbon with mass
fractions of 0.8$\pm$0.1 and 0.2$\pm$0.1, respectively. Based on the width and
strength of the O VI emission line, they estimated that the wind mass-loss
rate of the object was $\dot{M}=(3.5\pm
0.6)\times{10}^{-6}\,{M}_{\odot}\,{\rm{yr}^{-1}}$ with a terminal wind
velocity of ${v}_{\infty}=16000\pm 1000{\rm{km}}\,{\rm{s}^{-1}}$. On the basis
of these observations and the numerical results from Schwab et al. (2016),
Gvaramadze et al. (2019) argued that J005311 was the outcome of a super-
Chandrasekhar-mass double CO WD mergers. Furthermore, they suggested that it
would end its life as a Fe-CCSN.

Oskinova et al. (2020) reported on the first deep X-ray observation of J005311
and the first spectroscopic investigation of its nebula. They found that the
central star is more luminous in X-ray than single massive OB and WR stars.
Combining with optical data, they claimed that J005311 is a super-
Chandrasekhar mass ONe WD+CO WD merger remnant. Similar to Gvaramadze et al.
(2019), they suggested that J005311 would likely terminate its evolution as
type I supernova, in which the final core collapse to a neutron star might be
induced by electron captures process.

Ritter et al. (2021) later suggested that J005311 and its nebula are the
residual core and ejecta of the 1181 AD explosion. They argued that this
source is the result of an ONe+CO WD merger, and it is the only second type
Iax supernova living in our Galaxy, which provided strong observational
support for the double-degenerate merger scenario for type Iax supernovae.
Lykou et al. (2022) modelled the stellar remnant of J005311, and derived a
magnetic field with upper limit of 2.5MG, and the luminosity indicated a
remnant mass of $1.2\pm 0.2{M}_{\odot}$ with ejecta mass of $0.15\pm
0.05{M}_{\odot}$. Combining with the low mass ratio of Ne and O
(Ne/O$\,<0.15$), they suggested that the object is originated from the merger
of double CO WDs or CO+He WDs, argued against the ONe WD merger origin of this
object.

The origin of J005311 still remains uncertain. To understand the nature of
J005311, we recalculated ONe WD+CO WD merger models by considering wind mass-
loss and rotation process, since on one hand the observation supported that
the object has extremely strong mass-loss process, and on the other hand,
double WD merger remnants could keep a portion of orbital angular momentum. As
in the prescription described in Sect. 4.5, we inject torque into $1.6$ and
$1.8{M}_{\odot}$ merger remnants to change their initial rotational profiles
until their core (${\Omega}_{\rm c}$) and envelope (${\Omega}_{\rm e}$)
rotational velocity reached ${\Omega}_{\rm c}=0.5\,{\rm rad}/{\rm s}$ and
${\Omega}_{\rm e}=0.5\,{\Omega}_{\rm crit}$ (${\Omega}_{\rm crit}$ is the
Keplerian velocity for each mass zone). The corresponding rotational profile
can approximately describe the properties of the merger remnants with similar
total mass (e.g. Lorén-Aguilar et al. 2009). During the post-merger evolution,
we simply consider Blocker’s wind mass-loss prescription (e.g. Bloecker 1995)
with typical scaling factor of ${\eta}_{\rm B}=0.05$ to produce strong mass-
loss process. In comparison, we also calculated CO WD+CO WD merger models
under the same physical inputs.

Some key stellar parameters of J005311 and of our models are listed in Table 1
and Table 8, respectively. Fig. 10 shows that the evolutionary tracks on the
HRD of all these four models pass through a position close to J005311.
However, the double CO WD merger model has very low mass-loss rates, in the
range from ${10}^{-9}$ to a few of ${10}^{-8}\,{M}_{\odot}\,{\rm{yr}^{-1}}$,
when the corresponding remnant has a temperature similar to that of J005311.
The reason is that such merger remnant experiences off-center carbon burning
through flashes, which prevents stationary shell burning near the edge of the
degenerate core, resulting in a lower luminosity remnant. For the ONe WD+CO WD
model, because of the initially low mass and the short evolutionary time of
the remnant due to the strong wind mass-loss process, off-center neon burning
cannot be triggered after merger. Thus, the remnant still maintain shell
carbon burning, and has relatively high mass-loss rates comparable to the
values inferred from observations.

| Observation | ONe+CO WD | CO+CO WD
---|---|---|---
Parameter | J005311 | 1.08+0.52 | 1.10+0.70 | 0.85+0.75 | 0.95+0.85
T (K) | $211000_{-23000}^{+40000}$ | $18800-25100$ | $18800-25100$ | $18800-25100$ | $18800-25100$
L (${L}_{\odot})$ | $39810_{-10970}^{+20144}$ | $58606-59503$ | $76357-77520$ | $11000-14190$ | $16200-22527$
$\dot{M}$ (${M}_{\odot}\,{\rm{yr}}^{-1}$) | $3.5(\pm 0.6)\times{10}^{-6}$ | $3.2-6.06\times{10}^{-6}$ | $8.627-1.47\times{10}^{-6}$ | $3.2-7.65\times{10}^{-9}$ | $1.23-3.0\times{10}^{-8}$
M (${M}_{\odot}$) | $1.2\pm 0.2$ | $1.186$ | $1.252$ | $1.313$ | $1.463$
R (${R}_{\odot}$) | $0.15\pm 0.04$ | $0.128-0.23$ | $0.148-0.26$ | $0.06-0.108$ | $0.08-0.122$
age (yr) | unknown | $1586-1644$ | $1058-1082$ | $11371-18072$ | $12339-15863$
final fate | unknown | ONe WD | ONe WD | ONe WD | Fe-CCSN
Table 1:  Major parameters (temperature, luminosity, mass-loss rate, stellar
mass, stellar radius, stellar age and final fate) of J005311 from the
observations (second column) and from our theoretical models when the remnants
evolve to the same temperature as J005311 (columns 3-6).

Besides, results from double CO WD merger and ONe+CO mergers allow to infer
different model remnant masses when the associated tracks reach the position
of J005311 in the HRD. After experiencing extremely strong stellar wind, the
masses of ONe WD+CO WD merger remnants decrease and reach values in the range
from $1.186$ to $1.253{M}_{\odot}$. These sub-Chandrasekhar remnant mass
values are consistent with those derived from observation, i.e. $1.2\pm
0.2{M}_{\odot}$ (e.g. Lykou et al. 2022). For the double CO WD merger
counterparts, the initially massive remnant (${M}^{\rm i}=1.80{M}_{\odot}$) is
still super-Chandrasekhar mass ($1.473{M}_{\odot}$), which predict a different
evolutionary outcomes for J005311, i.e. it will undergo supernova explosion
through Fe-CCSN. The final fate of the less massive double CO merger remnant
(${M}^{\rm i}=1.60{M}_{\odot}$) is the same as that of the ONe+CO WD merger of
the same initial mass. However, the radius of the former is too small to
explain the observed properties of J005311. Note that for the super-
Chandrasekhar mass remnants, the deflagration of oxygen may produce ONeFe WD
with mass close to $1.2{M}_{\odot}$ (e.g. Jones et al. 2016b), which may also
show similar characteristics as J005311. However, such post-explosion remnant
should have extremely low carbon abundance (e.g. Jones et al. 2019), which may
not be consistent with J005311. Hence, we suggest that J005311 derives from an
ONe+CO WD merger and, in the future, it is expected to evolve to an ONe WD
with a mass close to $1.20{M}_{\odot}$ (see also Kashiyama et al. 2019).

Lykou et al. (2022) argued that J005311 originates from a double CO WD merger
rather than from a ONe WD+CO WD merger, since the ejecta of J005311 is
deficient in neon. Based on our simulations, the convective regions in the
envelope of neither the ONe WD+CO WD remnant nor the double CO WD remnants are
able to penetrate the carbon burning zone. This leads to expect neon deficient
ejecta at the corresponding evolutionary stage of J005311 (see panel c and d
of Fig. 10). Note that owing to the occurrence of mixing process during the
double WD merger, the base of CO envelope in the ONe+CO merger remnant could
show neon enhancement. However, according to Dan et al. (2014), the abundance
of neon at the base of the CO envelope cannot be higher than $20\%$, which may
still be consistent with a low mass ratio of Ne and O as the remnant evolves
to pre-WD.

Meanwhile, the nebula of J005311 is $1.1$ to $1.6$ pc far away from the
central star. Ritter et al. (2021) argued that the nebula is stemmed from the
supernova explosion of 1181 AD. If a NS is formed after the supernova
explosion, it is difficult to explain why the central remnant remain constant
luminosity within 100 years (e.g. Oskinova et al. 2020). Considering that the
wind velocity of pre-WD is in the order of magnitude of $1000{\rm km}/{\rm
s}$, it will spend $1120$ to $1600$ years to form the size of nebula of
J005311, which is consist with the evolutionary time of ONe+CO WD merger
remnant. Therefore, we suggest that J005311 may resulted from ONe WD+CO WD
merger, and its nebula may formed though the strong wind-loss process during
its evolution. In addition, we predict that this object will form a massive WD
rather than undergo supernova explosion thousand of years later.

Figure 10: Evolution of ONe+CO WD and CO+CO WD models when considering rotation (${\Omega}_{\rm c}=0.5\,{\rm rad}/{\rm s}$, ${\Omega}_{\rm e}=0.5{\Omega}_{\rm crit}$) and Blocker’s wind mass-loss prescription (${\eta}_{\rm B}=0.05$). Panel (a): HRD. Cyan and navy lines represent evolution of ONe+CO WD merger remnants with total masses equal to $1.60{M}_{\odot}$ and $1.80{M}_{\odot}$, respectively. In comparison, the evolution of double CO WD mergers with same total masses (mass ratio equal to 0.9) are shown as orange and red lines, respectively. Panel (b): evolution of wind mass-loss rate for the corresponding models. Brown point with error bar represents the position of J005311 on HRD, whereas the brown shaded area in panel (b) represents the estimated wind mass-loss rate of J005311. Thick solid portion of each lines in panel (a) and (b) represent the position of each models when they evolve to the same temperature as J005311. Panel (c) and (d): Elemental abundance distributions of ONe+CO WD and CO+CO WD models at the moment when they evolve to the position close to the object of J005311 on HRD, respectively. Model | ${M}_{\rm tot}/{M}_{\odot}$ | ${\rm ln}({s}_{\rm env}/{\rm erg}\,{\rm g}^{-1}\,{\rm K}^{-1})$ | ${T}_{\rm max}/{10}^{8}\,{\rm K}$ | Stopping condition | ${\rm t}/{\rm yr}$ | Expected fate
---|---|---|---|---|---|---
ONe+CO |  |  |  |  |  |
$1.10+0.65$ | $1.75$ | $8.875$ | $10.484$ | ${\rm log}({T}_{\rm c}/{\rm K})>9.34$ | $9687$ | NS
$1.10+0.70$ | $1.80$ | $8.890$ | $10.788$ | ${\rm log}({T}_{\rm c}/{\rm K})>9.34$ | $8173$ | NS
$1.15+0.65$ | $1.80$ | $8.890$ | $11.660$ | ${\rm log}({T}_{\rm c}/{\rm K})>9.34$ | $5681$ | NS
$1.10+0.75$ | $1.85$ | $8.905$ | $11.138$ |  | $6202$ | NS
$1.15+0.70$ | $1.85$ | $8.905$ | $11.929$ | ${\rm log}({T}_{\rm c}/{\rm K})>9.34$ | $4112$ | NS
$1.20+0.65$ | $1.85$ | $8.905$ | $13.114$ |  | $994$ | NS
$1.10+0.80$ | $1.90$ | $8.920$ | $11.579$ |  | $3125$ | NS
$1.15+0.75$ | $1.90$ | $8.920$ | $12.339$ | ${\rm log}({T}_{\rm c}/{\rm K})>9.34$ | $1729$ | NS
$1.20+0.70$ | $1.90$ | $8.920$ | $13.398$ |  | $178$ | ONeFe WD
$1.10+0.85$ | $1.95$ | $8.935$ | $12.154$ |  | $240$ | ONeFe WD
$1.15+0.80$ | $1.95$ | $8.935$ | $12.782$ | ${\rm log}({T}_{\rm c}/{\rm K})>9.34$ | $179$ | ONeFe WD
$1.20+0.75$ | $1.95$ | $8.935$ | $13.824$ |  | $107$ | ONeFe WD
$1.10+0.90$ | $2.00$ | $8.950$ | $12.649$ |  | $134$ | ONeFe WD
$1.15+0.85$ | $2.00$ | $8.950$ | $13.242$ | ${\rm log}({T}_{\rm c}/{\rm K})>9.34$ | $163$ | ONeFe WD
$1.20+0.80$ | $2.00$ | $8.950$ | $14.271$ |  | $47$ | ONeFe WD
$1.10+0.95$ | $2.05$ | $8.965$ | $13.330$ |  | $106$ | ONeFe WD
$1.15+0.90$ | $2.05$ | $8.965$ | $13.850$ | ${\rm log}({T}_{\rm c}/{\rm K})>9.34$ | $66$ | ONeFe WD
$1.20+0.85$ | $2.05$ | $8.965$ | $14.771$ |  | $39$ | ONeFe WD
$1.10+1.00$ | $2.10$ | $8.980$ | $14.053$ |  | $59$ | ONeFe WD
$1.15+0.95$ | $2.10$ | $8.980$ | $14.556$ | ${\rm log}({T}_{\rm c}/{\rm K})>9.34$ | $43$ | ONeFe WD
$1.20+0.90$ | $2.10$ | $8.980$ | $15.387$ |  | $25$ | ONeFe WD
$1.10+1.05$ | $2.15$ | $8.995$ | $14.959$ |  | $--$ | ONeFe WD
$1.15+1.00$ | $2.15$ | $8.995$ | $15.486$ | ${\rm log}({T}_{\rm c}/{\rm K})>9.34$ | $51$ | ONeFe WD
$1.20+0.95$ | $2.15$ | $8.995$ | $16.234$ |  | $20$ | ONeFe WD
$1.15+1.05$ | $2.20$ | $9.010$ | $16.441$ | ${\rm log}({T}_{\rm c}/{\rm K})>9.34$ | $232$ | ONeFe WD
$1.20+1.00$ | $2.20$ | $9.010$ | $10.720$ | ${\rm log}({T}_{\rm c}/{\rm K})>9.34$ | $14$ | ONeFe WD
$1.20+1.05$ | $2.25$ | $9.025$ | $18.350$ | ${\rm log}({T}_{\rm c}/{\rm K})>9.34$ | $8.4$ | ONeFe WD
Table 2:  Models involved in sect. 3.2. Columns 1-7 represent, (1) Model: masses of ONe WD and CO WD; (2) ${M}_{\rm tot}/{M}_{\odot}$: total mass of the merger remnant; (3) ${\rm ln}({s}_{\rm env}/{\rm erg}\,{\rm g}^{-1}\,{\rm K}^{-1})$: natural logarithm of envelope entropy of remnant; (4) ${T}_{\rm max}/{10}^{8}\,{\rm K}$: maximum temperature of remnant; (5) Stopping condition: stage at which calculation is stopped; (6) ${\rm t}/{\rm yr}$: evolution times from the beginning of our calculations to the stopping conditions (the model of $1.10+1.05$ meets some numerical problems during its evolution); (7) Expected fate: final evolutionary outcome of the remnant. NS represents that the remnant experiences ECSN to forms neutron star, whereas ONeFe WD represents that the remnant experiences ECSN to forms ONeFe WD. Model | ${M}_{\rm tot}/{M}_{\odot}$ | ${\rm ln}({s}_{\rm env}/{\rm erg}\,{\rm g}^{-1}\,{\rm K}^{-1})$ | ${T}_{\rm max}/{10}^{8}\,{\rm K}$ | Stopping condition | ${\rm t}/{\rm yr}$ | Expected fate
---|---|---|---|---|---|---
ONe+CO |  |  |  |  |  |
$1.10+0.80$ | $1.90$ | $8.70$ | $15.133$ | ${\rm log}({T}_{\rm c}/{\rm K})>9.34$ | $80$ | ONeFe WD
$1.10+0.80$ | $1.90$ | $9.10$ | $9.773$ | ${\rm log}({T}_{\rm c}/{\rm K})>9.34$ | $10963$ | NS
$1.10+0.85$ | $1.95$ | $8.70$ | $17.034$ | ${\rm log}({T}_{\rm c}/{\rm K})>9.34$ | $45.5$ | ONeFe WD
$1.10+0.85$ | $1.95$ | $9.10$ | $10.228$ | ${\rm log}({T}_{\rm c}/{\rm K})>9.34$ | $8788$ | NS
Table 3:  Models involved in sect. 4.1. Columns 1-7 represent, (1) Model: masses of ONe WD and CO WD; (2) ${M}_{\rm tot}/{M}_{\odot}$: total mass of the merger remnant; (3) ${\rm ln}({s}_{\rm env}/{\rm erg}\,{\rm g}^{-1}\,{\rm K}^{-1})$: natural logarithm of envelope entropy of remnant; (4) ${T}_{\rm max}/{10}^{8}\,{\rm K}$: maximum temperature of remnant; (5) Stopping condition: stage at which calculation is stopped; (6) ${\rm t}/{\rm yr}$: evolution times from the beginning of our calculations to the stopping conditions; (7) Expected fate: final evolutionary outcome of the remnant. Model | ${M}_{\rm tot}/{M}_{\odot}$ | Mass-loss prescription | Stopping condition | ${\rm t}/{\rm yr}$ | Expected fate
---|---|---|---|---|---
ONe+CO |  |  |  |  |
$1.10+0.80$ | $1.90$ | ${\eta}_{\rm R}=0.5$ | ${\rm log}({T}_{\rm c}/{\rm K})>9.34$ | $3121$ | NS
$1.10+0.80$ | $1.90$ | ${\eta}_{\rm B}=0.05$ | ${\rm log}({L}/{L}_{\odot})<1.0$ | $29960$ | ONe WD
$1.10+0.85$ | $1.95$ | ${\eta}_{\rm R}=0.5$ | ${\rm log}({T}_{\rm c}/{\rm K})>9.34$ | $242$ | ONeFe WD
$1.10+0.85$ | $1.95$ | ${\eta}_{\rm B}=0.05$ | ${\rm log}({T}_{\rm c}/{\rm K})>9.34$ | $241$ | ONeFe WD
Table 4:  Models involved in sect. 4.2. Columns 1-6 represent, (1) Model: masses of ONe WD and CO WD; (2) ${M}_{\rm tot}/{M}_{\odot}$: total mass of the merger remnant; (3) Mass-loss prescription: wind mass-loss prescription considered in the corresponding model. “${\eta}_{\rm R}=0.5$” represents models adopt Reimers’ wind mass-loss prescription with factor of ${\eta}_{\rm R}=0.5$, whereas “${\eta}_{\rm B}=0.05$” represents models adopt Blocker’s wind mass-loss prescription with factor of ${\eta}_{\rm B}=0.05$; (4) Stopping condition: stage at which calculation is stopped; (5) ${\rm t}/{\rm yr}$: evolution times from the beginning of our calculations to the stopping conditions; (6) Expected fate: final evolutionary outcome of the remnant. NS represents that the remnant experiences ECSN to forms neutron star, whereas ONeFe WD represents that the remnant experiences ECSN to forms ONeFe WD. Model | ${M}_{\rm tot}/{M}_{\odot}$ |  ${f}_{\rm He}$ in envelope | Stopping condition | ${\rm t}/{\rm yr}$ | Expected fate
---|---|---|---|---|---
ONe+CO |  |  |  |  |
$1.10+0.50$ | $1.60$ | $0\%$ | ${\rm log}({T}_{\rm c}/{\rm K})>9.34$ | $12507$ | NS
ONe+HeCO |  |  |  |  |
$1.10+0.50$ | $1.60$ | $20\%$ | ${\rm log}({T}_{\rm c}/{\rm K})>9.34$ | $27859$ | NS
Table 5:  Models involved in sect. 4.3. Columns 1-6 represent, (1) Model: masses of ONe WD and CO WD; (2) ${M}_{\rm tot}/{M}_{\odot}$: total mass of the merger remnant; (3) ${f}_{\rm He}$ in envelope: mass fraction of ${}^{\rm 4}{\rm He}$ in CO envelope; (4) Stopping condition: stage at which calculation is stopped; (5) ${\rm t}/{\rm yr}$: evolution times from the beginning of our calculations to the stopping conditions; (6) Expected fate: final evolutionary outcome of the remnant. NS represents that the remnant experiences ECSN to forms neutron star. Model | ${M}_{\rm tot}/{M}_{\odot}$ | ${f}_{{}^{\rm 16}{\rm O}}$ | ${f}_{{}^{\rm 20}{\rm Ne}}$ | ${f}_{{}^{\rm 24}{\rm Mg}}$ | ${f}_{{}^{\rm 23}{\rm Na}}$ | ${f}_{{}^{\rm 25}{\rm Mg}}$ | Stopping condition | ${\rm t}/{\rm yr}$ | Expected fate
---|---|---|---|---|---|---|---|---|---
ONe+CO |  |  |  |  |  |  |  |  |
$1.10+0.80$ | $1.90$ | $60\%$ | $35\%$ | $5\%$ | $0\%$ | $0\%$ | ${\rm log}({T}_{\rm c}/{\rm K})>9.34$ | $3125$ | NS
$1.10+0.80$ | $1.90$ | $54\%$ | $35\%$ | $5\%$ | $5\%$ | $1\%$ | ${\rm log}({T}_{\rm c}/{\rm K})>9.34$ | $1937$ | NS
$1.10+0.85$ | $1.95$ | $60\%$ | $35\%$ | $5\%$ | $0\%$ | $0\%$ | ${\rm log}({T}_{\rm c}/{\rm K})>9.34$ | $240$ | ONeFe WD
$1.10+0.85$ | $1.95$ | $54\%$ | $35\%$ | $5\%$ | $5\%$ | $1\%$ | ${\rm log}({T}_{\rm c}/{\rm K})>9.34$ | $197$ | ONeFe WD
Table 6:  Models involved in sect. 4.4. Columns 1-10 represent, (1) Model: masses of ONe WD and CO WD; (2) ${M}_{\rm tot}/{M}_{\odot}$: total mass of the merger remnant; (3) ${f}_{{}^{\rm 16}{\rm O}}$: mass fraction of ${}^{\rm 16}{\rm O}$ in ONe core; (4) ${f}_{{}^{\rm 20}{\rm Ne}}$: mass fraction of ${}^{\rm 20}{\rm Ne}$ in ONe core; (5) ${f}_{{}^{\rm 24}{\rm Mg}}$: mass fraction of ${}^{\rm 24}{\rm Mg}$ in ONe core; (6) ${f}_{{}^{\rm 23}{\rm Na}}$: mass fraction of ${}^{\rm 23}{\rm Na}$ in ONe core; (7) ${f}_{{}^{\rm 25}{\rm Mg}}$: mass fraction of ${}^{\rm 25}{\rm Mg}$ in ONe core; (8) Stopping condition: stage at which calculation is stopped; (9) ${\rm t}/{\rm yr}$: evolution times from the beginning of our calculations to the stopping conditions; (10) Expected fate: final evolutionary outcome of the remnant. NS represents that the remnant experiences ECSN to forms neutron star, whereas ONeFe WD represents that the remnant experiences ECSN to forms ONeFe WD. Model | ${M}_{\rm tot}/{M}_{\odot}$ | ${\Omega}_{\rm c}/{\rm rad}\,{\rm s}^{-1}$ | ${\Omega}_{\rm e}/{\Omega}_{\rm crit}$ | AM transport | Stopping condition | ${\rm t}/{\rm yr}$ | Expected fate
---|---|---|---|---|---|---|---
ONe+CO |  |  |  |  |  |  |
$1.10+0.80$ | $1.90$ | $0.50$ | $0.50$ | Set 1 | ${\rm log}({T}_{\rm c}/{\rm K})>9.34$ | $3020$ | NS
$1.10+0.80$ | $1.90$ | $0.50$ | $0.50$ | Set 2 | ${\rm log}({T}_{\rm c}/{\rm K})>9.34$ | $3050$ | NS
$1.10+0.80$ | $1.90$ | $0.25$ | $0.25$ | Set 1 | ${\rm log}({T}_{\rm c}/{\rm K})>9.34$ | $3127$ | NS
$1.10+0.80$ | $1.90$ | $0.25$ | $0.25$ | Set 2 | ${\rm log}({T}_{\rm c}/{\rm K})>9.34$ | $3135$ | NS
$1.10+0.85$ | $1.95$ | $0.50$ | $0.50$ | Set 1 | ${\rm log}({T}_{\rm c}/{\rm K})>9.34$ | $243$ | ONeFe WD
$1.10+0.85$ | $1.95$ | $0.50$ | $0.50$ | Set 2 | ${\rm log}({T}_{\rm c}/{\rm K})>9.34$ | $214$ | ONeFe WD
$1.10+0.85$ | $1.95$ | $0.25$ | $0.25$ | Set 1 | ${\rm log}({T}_{\rm c}/{\rm K})>9.34$ | $243$ | ONeFe WD
$1.10+0.85$ | $1.95$ | $0.25$ | $0.25$ | Set 2 | ${\rm log}({T}_{\rm c}/{\rm K})>9.34$ | $245$ | ONeFe WD
Table 7:  Models involved in sect. 4.5. Columns 1-8 represent, (1) Model: masses of ONe WD and CO WD; (2) ${M}_{\rm tot}/{M}_{\odot}$: total mass of the merger remnant; (3) ${\Omega}_{\rm c}/{\rm rad}\,{\rm s}^{-1}$: rotation velocity of core in ${\rm rad}/{\rm s}$; (4) ${\Omega}_{\rm e}/{\Omega}_{\rm crit}$: rotation velocity of each mass zone in envelope divided by Keplerian velocity of the corresponding mass zone; (5) AM transport: angular momentum transport prescription adopted in the corresponding models; Set 1 represents adopting the prescription described in Fuller et al. (2019), whereas Set 2 represents which described in Spruit (2002); (6) Stopping condition: stage at which calculation is stopped; (7) ${\rm t}/{\rm yr}$: evolution times from the beginning of our calculations to the stopping conditions; (8) Expected fate: final evolutionary outcome of the remnant. NS represents that the remnant experiences ECSN to forms neutron star, whereas ONeFe WD represents that the remnant experiences ECSN to forms ONeFe WD. Model | ${M}_{\rm tot}/{M}_{\odot}$ | ${\Omega}_{\rm c}/{\rm rad}\,{\rm s}^{-1}$ | ${\Omega}_{\rm e}/{\Omega}_{\rm crit}$ | Mass-loss prescription | Stopping condition | ${\rm t}/{\rm yr}$ | Expected fate
---|---|---|---|---|---|---|---
ONe+CO |  |  |  |  |  |  |
$1.08+0.52$ | $1.60$ | $0.50$ | $0.50$ | ${\eta}_{\rm B}=0.05$ | ${\rm log}({L}/{L}_{\odot})<1.0$ | $86306$ | ONe WD
$1.10+0.70$ | $1.80$ | $0.50$ | $0.50$ | ${\eta}_{\rm B}=0.05$ | ${\rm log}({L}/{L}_{\odot})<1.0$ | $56030$ | ONe WD
CO+CO |  |  |  |  |  |  |
$0.85+0.75$ | $1.60$ | $0.50$ | $0.50$ | ${\eta}_{\rm B}=0.05$ | ${\rm log}({L}/{L}_{\odot})<1.0$ | $52378$ | ONe WD
$0.95+0.85$ | $1.80$ | $0.50$ | $0.50$ | ${\eta}_{\rm B}=0.05$ | Ne ignition | $16276$ | Fe-CCSN
Table 8:  Models involved in sect. 5. Columns 1-8 represent, (1) Model: masses
of ONe WD and CO WD; (2) ${M}_{\rm tot}/{M}_{\odot}$: total mass of the merger
remnant; (3) ${\Omega}_{\rm c}/{\rm rad}\,{\rm s}^{-1}$: rotation velocity of
core in ${\rm rad}/{\rm s}$; (4) ${\Omega}_{\rm e}/{\Omega}_{\rm crit}$:
rotation velocity of each mass zone in envelope divided by Keplerian velocity
of the corresponding mass zone; (5) Mass-loss prescription: wind mass-loss
prescription considered in the corresponding model. “${\eta}_{\rm B}=0.05$”
represents models adopt Blocker’s wind mass-loss prescription with factor of
${\eta}_{\rm B}=0.05$; (6) Stopping condition: stage at which calculation is
stopped; (7) ${\rm t}/{\rm yr}$: evolution times from the beginning of our
calculations to the stopping conditions; (8) Expected fate: final evolutionary
outcome of the remnant. ONe WD represents that the remnant end its evolution
as ONe WD, whereas Fe-CCSN represents that the remnant experiences Fe-CCSN to
forms NS.

##  6 Summary

In this work, we investigated the evolution of ONe WD+CO WD merger remnants
for the first time. We found that the evolutionary outcomes of such merger
remnants are related to their initial masses, i.e. less massive remnants
(${M}\leq 1.90{M}_{\odot}$) would experience shell carbon burning phase to
increase the core mass before off-center neon burning, whereas more massive
remnants (${M}\geq 1.95{M}_{\odot}$) would trigger off-center neon burning
soon after the merger. By considering wind mass-loss prescriptions, the less
massive remnants show more sensitive to the wind mass-loss rate, resulting in
that their final fates may be ONe WDs or NSs. By contrast, the more massive
remnants may always experience supernova explosions to form ONeFe WDs. Using
the mass-loss prescription by Bloecker (1995), our models can explain the
observational properties of J005311, including temperature, luminosity, wind
mass-loss rate, the formation of a nebula, the evolutionary time, and the
abundances of the ejecta. From this, We can derive, first that our study may
be able to provide some constraints on the wind mass-loss rates of H-/He-
deficient giant-like objects similar to J005311. For example, due to the
relevance of dust in both environments, their wind mass-loss process could
present analogies with that occurring in AGB stars. Finally, we strongly
suggest that J005311 originates from the merger of an ONe WD with a CO WD, and
the final fate of this object would be to become a massive ONe WD rather than
supernova explosion.

We thank Matthias Kruckow, Xiangcun Meng, Zhengwei Liu, Jiangdan Li, Hailiang
Chen and Dongdong Liu for helpful discussion. This study is supported by the
National Key R&D Program of China (No. 2021YFA1600404), the National Natural
Science Foundation of China (NSFC grants 12003013, 12225304, 12288102 and
12033003), the Western Light Project of CAS (No. XBZG-ZDSYS-202117), the
science research grants from the China Manned Space Project (No CMS-
CSST-2021-A12), the Yunnan Fundamental Research Projects (No. 202001AS070029),
the Scholar Program of Beijing Academy of Science and Technology
(DZ:BS202002), and the Tencent Xplorer Prize. The source files for our MESA
models are publicly available on Zenodo at
https://doi.org/10.5281/zenodo.7456455.

## References

  * Bloecker (1995) Bloecker, T. 1995, A&A, 297, 727
  * Brooks et al. (2017) Brooks, J., Schwab, J., Bildsten, L., et al. 2017, ApJ, 850, 127, doi: 10.3847/1538-4357/aa9568
  * Chugunov et al. (2007) Chugunov, A. I., Dewitt, H. E., & Yakovlev, D. G. 2007, Phys. Rev. D, 76, 025028, doi: 10.1103/PhysRevD.76.025028
  * Dan et al. (2014) Dan, M., Rosswog, S., Brüggen, M., & Podsiadlowski, P. 2014, MNRAS, 438, 14, doi: 10.1093/mnras/stt1766
  * Denissenkov et al. (2013) Denissenkov, P. A., Herwig, F., Truran, J. W., & Paxton, B. 2013, ApJ, 772, 37, doi: 10.1088/0004-637X/772/1/37
  * Doherty et al. (2017) Doherty, C. L., Gil-Pons, P., Siess, L., & Lattanzio, J. C. 2017, PASA, 34, e056, doi: 10.1017/pasa.2017.52
  * Eldridge & Tout (2004) Eldridge, J. J., & Tout, C. A. 2004, MNRAS, 353, 87, doi: 10.1111/j.1365-2966.2004.08041.x
  * Farmer et al. (2015) Farmer, R., Fields, C. E., & Timmes, F. X. 2015, ApJ, 807, 184, doi: 10.1088/0004-637X/807/2/184
  * Fuller et al. (2019) Fuller, J., Piro, A. L., & Jermyn, A. S. 2019, MNRAS, 485, 3661, doi: 10.1093/mnras/stz514
  * Garnavich et al. (2020) Garnavich, P., Littlefield, C., Pogge, R., & Wood, C. 2020, Research Notes of the American Astronomical Society, 4, 167, doi: 10.3847/2515-5172/abbb8b
  * Gvaramadze et al. (2019) Gvaramadze, V. V., Gräfener, G., Langer, N., et al. 2019, Nature, 569, 684, doi: 10.1038/s41586-019-1216-1
  * Holberg (2009) Holberg, J. B. 2009, in Journal of Physics Conference Series, Vol. 172, Journal of Physics Conference Series, 012022, doi: 10.1088/1742-6596/172/1/012022
  * Ibeling & Heger (2013) Ibeling, D., & Heger, A. 2013, ApJ, 765, L43, doi: 10.1088/2041-8205/765/2/L43
  * Iglesias & Rogers (1996) Iglesias, C. A., & Rogers, F. J. 1996, ApJ, 464, 943, doi: 10.1086/177381
  * Itoh et al. (1996) Itoh, N., Hayashi, H., Nishikawa, A., & Kohyama, Y. 1996, ApJS, 102, 411, doi: 10.1086/192264
  * Jeffery (1988) Jeffery, C. S. 1988, MNRAS, 235, 1287, doi: 10.1093/mnras/235.4.1287
  * Jones et al. (2014) Jones, S., Hirschi, R., & Nomoto, K. 2014, ApJ, 797, 83, doi: 10.1088/0004-637X/797/2/83
  * Jones et al. (2016a) Jones, S., Ritter, C., Herwig, F., et al. 2016a, MNRAS, 455, 3848, doi: 10.1093/mnras/stv2488
  * Jones et al. (2016b) Jones, S., Röpke, F. K., Pakmor, R., et al. 2016b, A&A, 593, A72, doi: 10.1051/0004-6361/201628321
  * Jones et al. (2019) Jones, S., Röpke, F. K., Fryer, C., et al. 2019, A&A, 622, A74, doi: 10.1051/0004-6361/201834381
  * Kashiyama et al. (2019) Kashiyama, K., Fujisawa, K., & Shigeyama, T. 2019, ApJ, 887, 39, doi: 10.3847/1538-4357/ab4e97
  * Kashyap et al. (2018) Kashyap, R., Haque, T., Lorén-Aguilar, P., García-Berro, E., & Fisher, R. 2018, ApJ, 869, 140, doi: 10.3847/1538-4357/aaedb7
  * Kitaura et al. (2006) Kitaura, F. S., Janka, H. T., & Hillebrandt, W. 2006, A&A, 450, 345, doi: 10.1051/0004-6361:20054703
  * Liu & Wang (2020) Liu, D., & Wang, B. 2020, MNRAS, 494, 3422, doi: 10.1093/mnras/staa963
  * Lorén-Aguilar et al. (2009) Lorén-Aguilar, P., Isern, J., & García-Berro, E. 2009, A&A, 500, 1193, doi: 10.1051/0004-6361/200811060
  * Lykou et al. (2022) Lykou, F., Parker, Q. A., Ritter, A., et al. 2022, arXiv e-prints, arXiv:2208.03946.  https://arxiv.org/abs/2208.03946
  * Napiwotzki (2009) Napiwotzki, R. 2009, in Journal of Physics Conference Series, Vol. 172, Journal of Physics Conference Series, 012004, doi: 10.1088/1742-6596/172/1/012004
  * Nelemans et al. (2001) Nelemans, G., Yungelson, L. R., Portegies Zwart, S. F., & Verbunt, F. 2001, A&A, 365, 491, doi: 10.1051/0004-6361:20000147
  * Nomoto & Kondo (1991) Nomoto, K., & Kondo, Y. 1991, ApJ, 367, L19, doi: 10.1086/185922
  * Oskinova et al. (2020) Oskinova, L. M., Gvaramadze, V. V., Gräfener, G., Langer, N., & Todt, H. 2020, A&A, 644, L8, doi: 10.1051/0004-6361/202039232
  * Paxton et al. (2011) Paxton, B., Bildsten, L., Dotter, A., et al. 2011, ApJS, 192, 3, doi: 10.1088/0067-0049/192/1/3
  * Paxton et al. (2013) Paxton, B., Cantiello, M., Arras, P., et al. 2013, ApJS, 208, 4, doi: 10.1088/0067-0049/208/1/4
  * Paxton et al. (2015) Paxton, B., Marchant, P., Schwab, J., et al. 2015, ApJS, 220, 15, doi: 10.1088/0067-0049/220/1/15
  * Paxton et al. (2018) Paxton, B., Schwab, J., Bauer, E. B., et al. 2018, ApJS, 234, 34, doi: 10.3847/1538-4365/aaa5a8
  * Paxton et al. (2019) Paxton, B., Smolec, R., Schwab, J., et al. 2019, ApJS, 243, 10, doi: 10.3847/1538-4365/ab2241
  * Poelarends (2007) Poelarends, A. J. T. 2007, PhD thesis, University of Utrecht, Netherlands
  * Reimers (1975) Reimers, D. 1975, Memoires of the Societe Royale des Sciences de Liege, 8, 369
  * Ritossa et al. (1999) Ritossa, C., García-Berro, E., & Iben, Icko, J. 1999, ApJ, 515, 381, doi: 10.1086/307017
  * Ritter et al. (2021) Ritter, A., Parker, Q. A., Lykou, F., et al. 2021, ApJ, 918, L33, doi: 10.3847/2041-8213/ac2253
  * Schwab (2019) Schwab, J. 2019, ApJ, 885, 27, doi: 10.3847/1538-4357/ab425d
  * Schwab (2021) —. 2021, ApJ, 906, 53, doi: 10.3847/1538-4357/abc87e
  * Schwab et al. (2017) Schwab, J., Bildsten, L., & Quataert, E. 2017, MNRAS, 472, 3390, doi: 10.1093/mnras/stx2169
  * Schwab et al. (2015) Schwab, J., Quataert, E., & Bildsten, L. 2015, MNRAS, 453, 1910, doi: 10.1093/mnras/stv1804
  * Schwab et al. (2016) Schwab, J., Quataert, E., & Kasen, D. 2016, MNRAS, 463, 3461, doi: 10.1093/mnras/stw2249
  * Siess (2007) Siess, L. 2007, A&A, 476, 893, doi: 10.1051/0004-6361:20078132
  * Siess (2010) —. 2010, A&A, 512, A10, doi: 10.1051/0004-6361/200913556
  * Spruit (2002) Spruit, H. C. 2002, A&A, 381, 923, doi: 10.1051/0004-6361:20011465
  * Wu et al. (2020) Wu, C., Wang, B., Wang, X., Maeda, K., & Mazzali, P. 2020, MNRAS, 495, 1445, doi: 10.1093/mnras/staa1277
  * Wu et al. (2022) Wu, C., Xiong, H., & Wang, X. 2022, MNRAS, 512, 2972, doi: 10.1093/mnras/stac273
  * Zhang et al. (2014) Zhang, X., Jeffery, C. S., Chen, X., & Han, Z. 2014, MNRAS, 445, 660, doi: 10.1093/mnras/stu1741