# Stellar Core-Merger-Induced Collapse: New Formation Pathways for Black
Holes, Thorne - $\dot{\rm Z}$ytkow objects, Magnetars and Superluminous
Supernovae

Iminhaji Ablimit1,4,, Philipp Podsiadlowski2, Ryosuke Hirai3, James Wicker5
1CAS Key Laboratory for Optical Astronomy, National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100101, China.
2Department of Physics, University of Oxford, Keble Rd, Oxford OX1 3RH, UK
3Monash Centre for Astrophysics, School of Physics and Astronomy, Monash
University, Clayton, Victoria 3800, Australia
4Department of Astronomy, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyo-
ku, Kyoto 606-8502, Japan
5National Astronomical Observatories, Chinese Academy of Sciences, Beijing
100101, China E-mail<EMAIL_ADDRESS>(I.A.)

###### Abstract

Most neutron stars (NSs) and black holes (BHs) are believed to be the final
remnants in the evolution of massive stars. In this study, we propose a new
formation channel for the formation of BHs and peculiar NSs (specifically,
magnetars and Thorne-$\dot{\rm Z}$ytkow objects [T$\dot{\rm Z}$Os]), which we
refer to as the core merger-induced collapse (CMIC) model. This model involves
the merger during a common-envelope phase of an oxygen/neon/magnesium
composition white dwarf and the core of a hydrogen-rich or helium-rich non-
degenerate star, leading to the creation of peculiar new types of objects. The
results of binary population synthesis simulations show that the CMIC channel
could make important contributions to the populations of (millisecond)
pulsars, T$\dot{\rm Z}$Os, magnetars and BHs. The possibility of superluminous
supernovae powered by T$\dot{\rm Z}$Os, magnetars and BHs formed through the
CMIC model is also being investigated. Magnetars with immediate matter
surroundings formed after the CMIC might be good sources for fast radio
bursts.

###### keywords:

stars: evolution - (stars:) binaries (including multiple): close - (stars:)
white dwarfs - stars: neutron - stars: black holes - (stars:) supernovae:
general.

††pagerange: Stellar Core-Merger-Induced Collapse: New Formation Pathways for
Black Holes, Thorne - $\dot{\rm Z}$ytkow objects, Magnetars and Superluminous
Supernovae–Stellar Core-Merger-Induced Collapse: New Formation Pathways for
Black Holes, Thorne - $\dot{\rm Z}$ytkow objects, Magnetars and Superluminous
Supernovae

##  1 Introduction

The remnants of core-collapse supernovae (CCSNe) at the end of the evolution
of single hydrogen-rich massive stars ($\gtrsim 8\,M_{\odot}$) can be neutron
stars111NSs also may be born through the electron-capture SNe (ECSNe from
$\sim 8-10M_{\odot}$ main-sequence stars) (e.g., Miyaji et al. 1980). (NSs) or
stellar-mass black holes (BHs) (e.g., Baade & Zwicky 1934; Woosley et al.
2002; Heger et al. 2003). There are also other formation pathways to form NSs
and BHs, such as accretion-induced collapse (AIC) or merger-induced collapse
(MIC) in interacting binaries: in the AIC channel, a white dwarf (WD) with an
oxygen/neon/magnesium (ONeMg) composition or a NS accretes matter from a
companion, the donor star, and grow in mass to a critical limiting mass (e.g.,
Michel 1987; Ivanova & Taam 2004; Ablimit & Li 2015) when they collapse to an
even more compact state and become a NS or BH, respectively (e.g., Nomoto &
Kondo 1991; Timmes et al. 1996). The merger of two WDs or the merger of two
NSs may form NSs or BHs in the MIC pathway; these are also very promising
sources for producing gravitational waves (GW) (e.g., Ivanova et al. 2008;
Bauswein et al. 2013; Shapiro 2017; Lyutikov & Toonen 2019; Ruiter et al.
2019; Chattopadhyay et al. 2020; Abbott et al. 2020). The AIC route to form
NSs is an important channel to produce pulsars and magnetars (e.g., Hurley et
al. 2010; Tauris et al. 2013; Ablimit & Li 2015; Jones et al. 2016; Ablimit
2019, 2022), including young pulsars in old populations (van den Heuvel 1981,
1987), and are promising contributors to the NS population in globular
clusters because of their expected low kick velocities (e.g., Podsiadlowski et
al. 2004; Ivanova et al. 2007). BHs formed through AIC or MIC may become
members of low-mass or intermediate-mass X-ray binaries or even become single
BHs (e.g., Belczynski & Taam 2004; Bauswein et al. 2013; Bernuzzi et al.
2020). The upper limit for the mass of NSs is a key parameter in these
formation channels. However, the upper mass limit of NSs, above which they
collapse to BHs, is still controversial due to poor observational constraints
(e.g., Margalit & Metzger 2017).

In a binary, two stars can merge due to orbital angular momentum loss. Stellar
mergers in binaries are important for a variety of astronomical phenomena,
such as peculiar SNe, NSs and BHs. Indeed, a significant fraction of single
massive stars may be the product of the merger of binary stars, and binary
interactions, including mergers, are important for producing various types of
SNe and unusual objects (e.g., Podsiadlowski et al. 1992; Langer 2012; de Mink
et al. 2014; Andrews et al. 2018; Schneider et al. 2019; Horiuchi et al. 2020;
Mandel et al. 2021). Wiktorowicz et al. (2019) studied possible merger routes
to produce BHs by performing binary population synthesis (BPS) simulations. In
the current study, we consider core mergers in WD binaries during the common
envelope (CE) phase as an alternative pathway to form unusual objects
including BHs.

Two stars evolve into a CE if mass transfer by Roche-lobe overflow in a binary
is dynamically unstable (Paczy$\acute{\rm n}$ski 1976). Although CE evolution
is still very poorly understood and not all binaries undergo the CE (e.g.,
Kemp et al. 2021), it has been seen as a crucial phase in binary evolution,
irrespective of whether the stars merge in the CE or survive from the CE (see
Ivanova et al. (2013) for a detailed review of the uncertainties and the
importance of this phase). It has been suggested that future observations of
planetary nebulae and luminous red novae may provide useful constraints to
improve our understanding of many of the open questions of the CE phase
(Blagorodnova et al. 2017; MacLeod et al. 2018; Jones 2020). Kruckow et al.
(2021) presented a catalog of candidate post-CE binaries, and they showed that
their catalog and its future extensions will provide insight into the
evolution of close binaries through the CE phase. As long as a few decades
ago, it has been suggested that the merger inside a CE could potentially
trigger a peculiar supernova event (Sparks & Stecher 1974). The merger of a CO
(carbon/oxygen) WD and the core of an asymptotic giant branch (AGB) star
during the CE phase has been proposed as a possible progenitor channel for
Type Ia SNe (e.g., Soker 2011; Soker et al. 2013). Mergers between WDs and
cores of hydrogen-rich normal stars during the CE phase have been pointed out
for the possible origins of intermediate-luminosity optical transients (Red
Transient; Red Nova) and/or peculiar SNe (Sabach & Soker 2014). More recently,
Ablimit (2021) investigated stellar core mergers between CO WDs and cores of
hydrogen-rich intermediate-mass, cores of massive stars, or cores of helium-
rich non-degenerate stars during the CE phase as origins for peculiar type Ia
SNe and type II SNe with the framework of the core-merger detonation (CMD)
model. The Galactic rates calculated with the CMD model show that this
scenario has significant contributions to SNe Ia (i.e. overluminous/super-
Chandrasekhar-mass SNe Ia and SNe Iax) and SN 2006gy-like type II
superluminous SNe (SLSNe; see Ablimit (2021) for the details of CMD model and
discussions).

The merger of a NS and the core of massive companion star inside the CE can
produce a Thorne-$\dot{\rm Z}$ytkow object (T$\dot{\rm Z}$O; Thorne &
$\dot{\rm Z}$ytkow 1977; Taam et al. 1978). T$\dot{\rm Z}$Os are an exotic
class of stellar objects comprised of a neutron core surrounded by a extended
envelope; from the outside they have the appearance of a red supergiant
(Thorne & $\dot{\rm Z}$ytkow 1977; Levesque et al. 2014; DeMarchi et al.
2021). They might eventually explode as (SLSNe)222Superluminous SNe (SLSNe)
have a luminosity around $10^{44}\,\rm erg\,\rm s^{-1}$ and radiate a total
energy of $\sim 10^{51}\,\rm erg$ over a period of several months (see Moriya
et al. (2018a) and Gal-Yam (2019) for reviews). Type I SLSNe do not show
hydrogen lines in their spectra, while Type II SLSNe do. (Moriya 2018; also
see Chevalier (2012)333In Chevalier’s paper, the SLSN occurs in the spiral-in
process itself, not in a T$\dot{\rm Z}$O.). Magnetars and accreting BHs have
also been studied as potential central engines for powering SLSNe (Kasen &
Bildsten 2010; Woosley 2010; Inserra et al. 2013; Dexter & Kasen 2013). In
this study we consider a possibly important (see our results and discussions
for important contributions of the channel to peculiar transients and
objects), alternative channel for producing SLSNe, involving T$\dot{\rm Z}$Os,
magnetars and BHs.

Here, we systematically explore the presumed core-collapse SN-like event
following the merger of an ONeMg WD and the core of a non-degenerate star
inside a CE. We call this channel, involving the merger of a ONeMg WD with the
He core of a massive star in a CE, the core merger-induced collapse (CMIC)
scenario in order to distinguish it from the more ‘normal’ AIC because the
timescale of CMIC is much shorter than the timescale of AIC in accreting ONeMg
WDs through stable mass transfer. We propose that the CMIC scenario could be
an alternative formation channel for single (millisecond) pulsars, magnetars
and BHs with an accretion disk, and that SLSNe could be powered by these
compact objects formed through CMIC. We also study the mergers between ONeMg
WDs and cores of stripped, non-degenerate helium stars and low-/intermediate-
mass stars as a potential channel to produce single (millisecond) pulsars. The
rates of ONeMg WD – NS binaries surviving the CE phase are also being
investigated as they are potential GW sources. We also explore how the final
outcomes of these binary populations depend on the modeling of the more poorly
understood physical phases and key parameters in binary evolution studies,
specifically involving mass transfer (e.g., its stability depending on the
mass ratio and the mass-transfer efficiency), the modeling of the common-
envelope phase (e.g, the CE efficiency and the binding energy parameter),
metallicity, etc. In §2, we describe our method to treat the physical binary
processes in the BPS simulations and discuss the assumptions for the CMIC. The
main results are presented in §3. The possible contribution and possible
observational implications are discussed in §4, and a summary is given in §5.

##  2 Methods

###  2.1 Binary Model Set

In order to derive the population of WD$+$non-degenerate star binaries, we
perform Monte-Carlo simulations by using an updated $BSE$ population synthesis
code based on Hurley et al. (2002; see Ablimit et al. (2016) and Ablimit &
Maeda (2018) for the updated version), starting with $10^{7}$ primordial
binaries (consisting of two zero-age main sequence stars) with chosen
distributions of primary mass, secondary mass and orbital separation. Initial
binary parameters of primary mass, secondary mass and orbital separation can
be set up with the $\rm n_{x}$ grid points of parameter $\chi$ logarithmically
spaced,

$\delta{\rm ln\chi}=\frac{1}{\rm n_{x}-1}({\rm ln{\chi_{\rm max}}}-{\rm
ln{\chi_{\rm min}}}),$ (1)

For each set of initial parameters, we evolve the binary system to an age of
the Hubble time, or until it is destroyed. Each phase of the evolution, such
as tidal evolution, angular momentum changes due to mass variations etc., is
followed in detail according to the algorithms described in Hurley et
al.(2002). The initial input parameters of the primordial binaries are set the
same as in Ablimit et al.(2016). The initial mass function of Kroupa et al.
(1993) is adopted for the primary star ($M_{1}$),

$f(M_{1})=\left\\{\begin{array}[]{ll}0&\textrm{${M_{1}/M_{\odot}}<0.1$}\\\
0.29056{(M_{1}/M_{\odot})}^{-1.3}&\textrm{$0.1\leq{M_{1}/M_{\odot}}<0.5$}\\\
0.1557{(M_{1}/M_{\odot})}^{-2.2}&\textrm{$0.5\leq{M_{1}/M_{\odot}}<1.0$}\\\
0.1557{(M_{1}/M_{\odot})}^{-2.7}&\textrm{$1.0\leq{M_{1}/M_{\odot}}\leq
150$},\end{array}\right.$ (2)

The distribution of the mass of the secondary ($M_{2}$; where the secondary is
the star with the lower initial mass) is determined by the distribution of the
initial mass ratio,

$n(q)=\left\\{\begin{array}[]{ll}0&\textrm{$q>1$}\\\ \mu
q^{\nu}&\textrm{$0\leq q<1$},\end{array}\right.$ (3)

where $q=M_{2}/M_{1}$ and $\mu$ is the normalization factor for the assumed
power-law distribution with index $\nu$. We consider a flat distribution
($\nu=0$ and $n(q)=$constant) for the initial mass-ratio distribution. The
chosen distribution for the initial orbital separation, $a_{\rm i}$, is

$n(a_{\rm i})=\left\\{\begin{array}[]{ll}0&\textrm{$a_{\rm i}/R_{\odot}<3$ or
$a_{\rm i}/R_{\odot}>10^{6}$},\\\ 0.078636{(a_{\rm
i}/R_{\odot})}^{-1}&\textrm{$3\leq a_{\rm i}/R_{\odot}\leq 10^{6}$}\
.\end{array}\right.$ (4)

We also adopt a thermal distribution for the initial eccentricity between 0
and 1. For the SN remnant calculation, we adopted the rapid remnant-mass model
of Fryer et al. (2012) in the subroutine $hrdiag$ of the updated $BSE$ code,
and note that a typo in Fryer et al. (2012) has been corrected in our code
(corrected as $a_{1}=0.25-1.275/{(M_{1}-M_{\rm proto})}$, and the proto-
compact object mass is set as $M_{\rm proto}=1.0M_{\odot}$). For the natal
supernova kick velocity, imparted to the newborn NS at its birth, we adopt
Maxwellian distributions with a velocity dispersion of $\sigma_{\rm
k}=265\,{\rm km\,s^{-1}}$ (Hobbs et al. 2005) for CCSNe and $\sigma_{\rm
k}=40\,{\rm km\,s^{-1}}$ for ECSNe/AIC (e.g., Hurley et al. 2010). For the
wind mass loss, the prescription of Vink et al. (2001) is used for O and B
stars in different stages (hot stars), and $1.5\times
10^{-4}\,\dot{M}_{\odot}\,\rm yr^{-1}$ for luminous blue variables (Vink & de
Koter 2002) in the subroutine $mlwind$ of the code (see the wind model 2 of
Ablimit & Maeda (2018)). Note that these prescriptions (for massive stars),
i.e., the remnant-mass determination, wind mass loss and kick velocity do not
influence the main results of this study as the formation of the ONeMg WD
systems of interest here never suffer from CCSN explosions nor do they involve
very massive stars. One of the important issues is how ONeMg WD form. Stellar
evolution remnants composed mainly of O and Ne after the ejection of the
envelope of a thermally pulsing AGB stars are considered as ONeMg WDs in the
code (e.g., Hurley et al. 2000). Note that the mass range of stars that form
ONeMg WDs strongly depends on the modeling of a variety of, sometimes
uncertain, processes in stellar evolution codes, such as the initial
composition, the adopted overshooting, nuclear reaction rates and the role of
rotation (e.g., Doherty et al. 2017).

As discussed in section 1, CE evolution is a key phase in the evolution of
many binary systems. Whether it occurs depends most importantly on the mass
ratio. If the mass ratio is larger than a critical mass ratio ($q_{\rm c}$),
mass transfer between the two binary components is dynamically unstable and a
CE forms. When the donor star is on the main sequence (MS) or crosses the
Hertzsprung gap (HG), our adopted default value is $q_{\rm c}=q_{\rm
const}=4.0$ (Hurley et al. 2002). As an alternative we also use a prescription
of $q_{\rm c}=q_{\rm cs}$ in the subroutine $evolv2$ of the code, where
$q_{\rm cs}=M_{1}/M_{2}$ is determined with the mass-transfer model by taking
the spin of the accretor into account (for more details see Ablimit et al.
(2016) and Ablimit & Maeda (2018), and see Table 2). This rotation-dependent
mass accretion model leads to the accretion efficiency, because the accretion
rate of a rotating accretor is reduced by a factor of
($1-\Omega/{{\Omega}_{\rm cr}}$), where ${\Omega}$ is the angular velocity of
the star and ${\Omega}_{\rm cr}$ is its critical rotation value (e.g.,
Stancliffe & Eldridge (2009)). An accreting star can spin very rapidly even
when only a small amount of mass has been transferred from the donor (Packet
1981); this drastically reduces the mass-transfer efficiency (to values
$<0.2$) and can even stop it completely when the accretor rotates at
${\Omega}_{\rm cr}$ (but also see Paczy$\acute{\rm n}$ski (1991)). The mass-
transfer efficiency determines how much of the transferred matter is accreted
by the accretor and how much escapes from the binary. It is assumed that the
matter is escaped/ejected from the binary when it is no longer affected by the
binary. We assume that material that is ejected from the system carries the
specific angular momentum of the accretor (e.g., Hurley et al. 2002). In this
prescription, the maximum initial mass ratio of the primary to the secondary
in the primordial binaries can be as high as $\sim$5-6, and a larger number of
primordial binaries can avoid a contact phase and hence experience stable mass
transfer until the primary’s envelope is completely exhausted (see Abdusalam
et al. (2020) for similar discussions and results). When the original primary
is on the first giant branch (FGB), AGB or is a helium (He) star, the
prescriptions in Hurley et al. (2002) are applied.

The standard CE model, based on energy conservation (Paczy$\acute{\rm n}$ski
1976; Ivanova 2013), is utilized in our simulations,

$E_{\rm{bind}}={\alpha_{\rm CE}}\Delta E_{\rm orb}\ ,$ (5)

where $E_{\rm{bind}}$ and $\Delta E_{\rm orb}$ are the binding energy of the
envelope and the change in the orbital energy during the CE phase,
respectively. In the specific model of Webbink (1984), which we adopt here,
the binding energy of the envelope is parameterized as,

$E_{\rm{bind}}=-\frac{GM_{1}M_{\rm{en}}}{\lambda{R}_{1}},$ (6)

where $M_{1}$, $M_{\rm{en}}$ and ${R}_{1}$ are the total mass, envelope mass
and radius of the primary star, respectively. We use two values for the CE
efficiency as $\alpha_{\rm CE}=1.0$ and $\alpha_{\rm CE}=0.1$ in the
subroutine $comenv$ of the code. The value of $\lambda$ varies when a star
evolves and strongly depends on the star’s initial mass, its evolutionary
state and other possible energy sources, such as, e.g., recombination energy
(see, e.g., Han et al. 1994; Tauris & Dewi 2001; Podsiadlowski et al. 2003;
Wang et al. 2016a,b). The results of Wang et al. (2016a,b) are supported by
Klencki et al. (2021), they are also consistent with results of the very
recent CE study of Vigna-G$\acute{\rm o}$mez et al. (2021), and these
consistency show the fact that results of Wang et al. (2016a,b) are the
correct and reliable ones. We take either a constant value $\lambda=0.5$ or
the detailed prescription $\lambda=\lambda_{\rm w}$ ($\lambda_{\rm w}$ is
taken from $\lambda_{\rm b}$ of Wang et al. (2016b)) for the binding energy
parameter. Whether the two stars merge completely or survive from the CE to
continue their evolution critically depends on these two parameters. We set
the initial metallicity of stars to be $Z=0.02$ and $Z=0.001$. Table 2
summarizes our main simulation models. For other parameters we use the default
values in Hurley et al. (2000, 2002).

###  2.2 The Core Merger-Induced Collapse during the CE phase

In the traditional AIC scenario, the ONeMg WD grows towards the Chandrasekhar
mass by accreting H- and/or He-rich material transferred from its non-
degenerate companion through stable mass transfer. In addition to the AIC
channel, two WDs in a compact binary (with a combined mass $\geq$ the
Chandrasekhar mass where at least one of them is an ONeMg WD) may merge and
collapse to a NS in the so-called merger-induced collapse (MIC). Because the
energy released by nuclear fusion in a O+Ne deflagration that is expected to
take place during the merging process is not sufficient to cause an explosion
of the tightly bound core (Miyaji et al. 1980), further electron captures
eventually lead to the gravitational collapse of the core and the formation of
a NS (e.g., Nomoto & Kondo 1991). AIC/MIC is the favored alternative way to
produce pulsars in globular clusters that have characteristic ages
significantly less than the age of the clusters (Lyne et al. 1996; Boyles et
al. 2011), and also to form millisecond pulsars (MSPs) in addition to the
standard recycling scenario involving NSs formed in a CCSN (Chanmugam &
Brecher 1987; Michel 1987; Kulkarni & Narayan 1988; Bailyn & Grindlay 1990).

In this work we consider the case where, after a ONeMg WD CE system formed,
the two cores, i.e. the WD and the core of the donor star, spiral towards each
other; if the CE cannot be expelled, they eventually coalesce and mix
completely when the cores are both degenerate. If one core is considerably
more compact (usually a WD) than the other (e.g. a non-degenerate He core),
the non-degenerate component will eventually be tidally disrupted. The WD will
sink to the center of the CE system without mixing, while it rapidly accretes
material from the disrupted component. In both cases, the final outcome will
be the collapse of the WD to form a more compact object. We present several
evolutionary routes involving a CE phase, assuming that the CMIC happens
inside the CE if the combined mass of an ONeMg WD and the core mass of the
companion star is larger than the Chandrasekhar limit mass (1.44$M_{\odot}$).

How the final merging of the compact core and the companion object happen is
not very well understood, and it is not entirely clear whether and how the
core is being spun up, but there are various possibilities for spin-up. For
example, the matter stream of the spiraling-in companion may impact with the
more compact core and spin it up in a "slow merger" phase (e.g., Podsiadlowski
et al. 2010). Another possibility is that the spiral-in occurs on a dynamical
timescale and the companion is immediately dynamically disrupted; this would
naturally lead to some type of accretion disk around the more compact
component from which it could be spun-up by disk accretion.

If the combined core masses are larger than 1.44$M_{\odot}$ and the total mass
(the two cores and the envelope) is less than 15 $M_{\odot}$ during the CE,
the core merger through the CMIC could form NSs/pulsars (of which a fraction
could be MSPs) or magnetars (see Figure 1 as one example; this will be further
discussed in subsequent sections). The newborn NS formed by CMIC is likely to
be surrounded by a substantial amount of material because not all the CE
material may be ejected from the newborn NS in the collapse event444For
comparison, only 10% of the envelope typically becomes unbound when MS stars
merge (e.g., Owocki et al. 2019), and the unbound ejecta is probably around 1%
for degenerate mergers (e.g., see Fig. 13 in Dan et al. (2014)).. Here, the
newborn NS formed from CMIC in the WD + massive star binary could be an
unusual type of object, because this NS would be surrounded by a very massive
envelope, potentially much more massive than the central compact object
itself. Thus, this would produce a configuration similar to that of a
T$\dot{\rm Z}$O (e.g., DeMarchi et al. 2021); we therefore postulate that
T$\dot{\rm Z}$Os could be formed in the CMIC scenario.

As the envelope loses its thermal energy by radiation from the surface of the
merger product, an accretion disk may form around the compact core, in
particular as neutrino cooling could be very effective near the NS (e.g.,
Fryer et al. 1996; Taylor et al. 2011). Fryer et al. (1996) predicted that a
range of rapid-infall neutron star accretors present in certain low-mass X-ray
binaries, common-envelope systems, cases of supernova fallback, and T$\dot{\rm
Z}$Os would lead to explosions by neutrino heating similar to the neutrino-
driven explosions in standard CCSNe.

A certain fraction of known WDs in binaries has been found with bipolar
magnetic fields (Kahabka 1995; Sokoloski & Bildsten 1999; Osborne et al. 2001;
Ferrario et al. 2015; Pala et al. 2020), and some observed WDs are highly
magnetized (e.g., Schmidt et al. 1999). The magnetic field of a WD has been
pointed out as one important physical parameter that influences the evolution
of a WD in a binary (e.g., Wheeler 2012; Ablimit et al. 2014; Farihi et al.
2018). Recently, detailed and systematic binary evolution studies show that
the magnetic field of the WD can affect the accretion phase (Ablimit & Maeda
2019a, b), and Ablimit (2019) showed that it can play a crucial role in the
AIC formation route for peculiar MSPs. A magnetic WD can be spun up by gaining
angular momentum during the merger process inside the CE as discussed above.
If the WD in the CMIC channel is strongly magnetized, the newborn MSP/pulsar
would inherit the large magnetic field through magnetic field amplification
during the CMIC process of the highly magnetized WD (e.g., Duncan & Thompson
1992). Indeed, several previous studies proposed that the large magnetic
fields of magnetic WDs can be formed or amplified during CE evolution (e.g.,
Tout et al. 2008; Wickramasinghe et al. 2014; Ohlmann et al. 2016). Based on
these observational and theoretical studies, we here hypothesize that a
magnetar could be the result of this CMIC formation channel if it formed in a
WD CE system with a highly magnetized WD.

The first magnetar was identified about three decades ago; typically magnetars
are isolated and relatively rapidly rotating neutron stars (NSs) with very
strong magnetic fields (up to $\sim 10^{15}$ G; Duncan & Thompson 1992). The
majority of the magnetars known to date (29 in the compilation of Olausen &
Kaspi (2014)) have been discovered just over the past two decades (with a
range of spin periods between 1 and 11 s) through their distinctive high-
energy phenomenology: bursts of X-ray/gamma-ray emission and/or enhancements
of their persistent X-ray luminosity, dubbed ’outbursts’ (see Kaspi &
Beloborodov 2017; Esposito et al. 2021). A magnetar surrounded by ejected
material in this channel may power X-ray emission and possibly ’outbursts’. It
is also possible that a BH may be formed in a CMIC during the CE phase when
the combined core masses are larger than 1.44$M_{\odot}$ and the total mass
(cores and envelope) is larger than 15 $M_{\odot}$; the reason why we set 15
$M_{\odot}$ as a cut to form pulsars/magnetars or BHs will be discussed
further in subsections below (especially in subsection 3.2). The possibility
that the central newborn objects, such as T$\dot{\rm Z}$Os/magnetars/BHs, may
power transient events like SLSNe will also be considered in later sections.

##  3 Results

We first examine the evolution of two MS stars leading to a CE event involving
an ONeMg WD and the core of a non-degenerate star. In a Monte-Carlo simulation
of $10^{7}$ binaries with the standard model parameters (model 1), there are
3535 CE events which involve ONeMg WDs and non-degenerate stars, and 1907 of
them end up with mergers between the ONeMg WDs and the cores of the non-
degenerate stars inside the CE. We focus on the core mergers during the CE
phase as progenitors of some transients and objects and considering different
binary stellar physics assumptions to estimate the importance of the different
channels.

###  3.1 Pulsars, Thorne - $\dot{\rm Z}$ytkow object, Magnetar and
Superluminous SNe through CMIC inside the CE

In a typical example, illustrated in Figure 1, the primary star evolves
through the MS, HG, helium-rich MS (HeMS) and helium-rich giant (HeG) phases
and forms an ONeMg WD555Note that the mass of zero-age MSs to form ONeMg WDs
may be different (comparing to single stellar evolution) due to possible early
mass loss or accretion caused by the binary interaction (e.g., Hurley et al.
2000, 2002).. Because of the two stable RLOF mass-transfer phases from the
primary, the secondary becomes a relatively massive star. The WD is engulfed
by the mass gained from the massive secondary when the secondary fills its
Roche lobe during its core helium burning (CHeB) stage. The system fails to
eject the CE, and the WD and the core of the massive secondary merge during
the CE phase. This evolutionary route of the merger between an ONeMg WD and
the core of a H-rich normal star finally produces a NS. The large amount of
angular momentum transferred onto the compact object during the merger process
may make the newborn NS a fast rotator like a pulsar or a millisecond pulsar.
If the WD has a strong magnetic field before the merger, the newborn NS most
likely would become a magnetar.

The typical ranges of the initial binary parameters for the ONeMg WD + H-rich
star CE event channel are: initial primary masses, $M_{1,\rm i}\sim
6.3-11.0M_{\odot}$, secondary initial masses, $M_{2,\rm i}\sim
1.0-10.2M_{\odot}$; initial orbital periods, $P_{\rm orb,i}\sim 0.01-122.0$
days. Figure 2 shows the mass distributions in model 1 for the merger case
which cause CMIC. Compared to the initial binary parameters of all ONeMg WD +
H-rich star systems with a CE, the initial conditions (mass ranges) of
binaries that lead to the core merger during the CE are quite similar (Figure
2). Figure 2 shows that mergers with more massive donors generally have a
larger CE mass.

The rates of all ONeMg + non-degenerate star systems associated with a CE and
core merger events for the different models considered are summarized in Table
3. Table 3 shows that the rates of all ONeMg WD + non-degenerate H-rich star
CE events vary significantly with the $\alpha$ parameter, metallicity and mass
transfer assumptions. The pessimistic value of $\alpha=0.1$ in model 2 causes
more mergers between non-degenerate stars during the CE events, therefore the
number of ONeMg WD binaries with CE phases are reduced from 0.33% to 0.15%.
The model with the pessimistic value $\alpha=0.1$ is adopted for illustrating
the effect of $\alpha$, though it should be noted that with such a pessimistic
value, no close double NS systems as GW sources can be formed (see Table 3).
For all merger cases (0.005% - 0.20% mergers of $10^{7}$ binaries), more
binaries can have stable mass transfer and avoid the CE phase with the mass-
transfer (critical mass ratio) model adopted in model 5 (see Section 2), thus
reducing CE events in general. Because more energy (i.e. internal energy) is
considered in the $\lambda$ prescriptions of models 4 and 5 for the CE
expulsion, there are more successful CE ejections in the binaries, which
reduces the number of CE merger systems. Zapartas et al. (2017) studied
mergers of ONeMg WDs and HG stars as a contributor to ’late’ core-collapse
SNe, and their rate ($\sim 2.1\times 10^{-4}{M_{\odot}}^{-1}$) is in good
agreement with our rate derived with mergers of ONeMg WDs and normal stars
($\sim 1.9\times 10^{-4}{M_{\odot}}^{-1}$; see R1 or R3 in line 2 of Table 3).
In this work, we investigated more new pathways as origins of various
transients and peculiar objects comparing to literatures.

Now first consider cases with condition (1), i.e., systems with a combined
core mass $M_{\rm com}\geq 1.44M_{\odot}$ and total mass $M_{\rm
all}<8.0M_{\odot}$ for mergers during the CE phase. This includes mergers
between ONeMg WDs and He cores of evolved hydrogen-rich stars, and also
mergers between ONeMg WDs and degenerate cores during the late stages of AGB
stars (e.g., Canals et al. 2018). The evolutionary delay times in this channel
range from $\sim 0.1$ to 12.5 Gyr. The rates for these systems range from
$0.24\times 10^{-5}{M_{\odot}}^{-1}$ to $7.38\times 10^{-5}{M_{\odot}}^{-1}$
for the different models in Table 3. The newborn NSs with condition (1) could
be MSPs with relatively smaller envelope mass (see Section 2 for the
discussion of formation pathway, and Figure 2). Derived rates with CMIC are
consistent with rates of AIC in previous studies (e.g., Hurley et al. 2010;
Ablimit 2022).

Mergers with condition (2) (i.e. $M_{\rm com}\geq 1.44M_{\odot}$ and $8.0\leq
M_{\rm all}<15.0M_{\odot}$) tend to form NSs surrounded by a large amount of
material because of the more massive CE (see the discussion in Figure 2). As
discussed in Section 2, newborn NSs surrounded by massive envelopes naturally
produce T$\dot{\rm Z}$Os. The typical T$\dot{\rm Z}$O formation rate from the
CMIC channel is $\sim 6\times 10^{-5}\,{\rm yr}^{-1}$ if we adopt a star-
formation rate of 2 $M_{\odot}\,{\rm yr}^{-1}$ (e.g., Misiriotis et al. 2006;
Robitaille & Whitney 2010) and binary fraction of 0.7; our rate is lower than
the rate of $\sim 2\times 10^{-4}\,{\rm yr}^{-1}$ estimated by Podsiadlowski
et al. (1995). The evolutionary delay times in this channel are between $\sim
14.18$ and 86.22 Myr. If we consider systems with a total mass $M_{\rm
all}\geq 10.0M_{\odot}$ for producing T$\dot{\rm Z}$Os, the Galactic rate from
the CMIC channel would be one order of magnitude lower than the estimate of
Podsiadlowski et al. (1995), and these T$\dot{\rm Z}$Os may evolve into other
objects on a relatively short timescale. T$\dot{\rm Z}$Os born here would have
very massive envelope (often larger than 8$M_{\odot}$). Moriya (2018) has
studied the strong accretion that occurs onto the central neutron core when
the nuclear reactions that support such massive T$\dot{\rm Z}$O terminate. As
the accretion is neutrino driven, it can be highly super-Eddington and trigger
a supernova event when the NS contracts and ultimately collapses. Such
explosions would be observed as energetic Type II supernovae or even as
superluminous supernovae of Type II. A strong large-scale outflow or a jet can
be launched from the super-Eddington accretion disc formed during the collapse
(Moriya 2018). The fallback of material towards the neutron star after a
successful explosion is large in this case, and a black hole is formed very
quickly (i.e. a few seconds; Fryer et al. 1996).

With conditions (1) and (2), it is possible that the newborn NSs/pulsars/MSPs
formed through CMIC could be magnetars if the collapsing WDs are strongly
magnetized (as discussed in section 2). The fraction of highly magnetized WDs
among the WD population has been estimated from observations to be around 15%
($\sim$15%-18%; Ferrario et al. 2015; see also Kepler et al. 2013, 2015).
Adopting 15%, we can estimate a possible birthrate for magnetars via CMIC. It
lies between $0.03\times 10^{-5}{M_{\odot}}^{-1}$ and $2.46\times
10^{-5}{M_{\odot}}^{-1}$ for the different chosen models (see Table 3). Note
that these rates are simply obtained by combining the results for conditions
(1) and (2) and multiplying them with 0.15. These rates have a good agreement
with the Galactic rate of magnetars formed by AIC from magnetized WD binaries
(Ablimit 2022). A rotating magnetar radiating according to the classic dipole
formula could power a very luminous supernova (e.g., Woosley 2010); thus
magnetars formed via CMIC may power Type II SLSNe. We will compare the
predicted rates with observationally inferred rate of peculiar SNe in Section
4.

The combined total mass distributions (Figure 3) for mergers demonstrate that
metallicity, the details of mass transfer and the binding energy parameter in
the CE phase play an important role. The central temperatures and densities
are higher, and more hydrogen burns stably for the lower-metallicity and more
massive MS stars; this leads to larger core masses and different evolutionary
lifetimes for the merger systems. Thus, more ONeMg WD binaries can be formed
at lower metallicity and for a higher total mass. There are few or no WD CE
merger systems with massive star companions (especially with a combined mass
$>15{M_{\odot}}$) in models 4 and 5, in comparison to the other models,
because more massive star systems can survive or avoid the CE phase for the
adopted $\lambda$ and mass-transfer prescriptions in model 4 and 5 (Table 1
and Figure 3) for the same reasons as discussed above.

Now consider ONeMg WD CE events in which the companion star is a non-
degenerate He star. The ONeMg WD + He star CE event channel is realized for
the following ranges in the initial binary parameters: initial primary mass,
$M_{1,\rm i}\sim 6.34-9.1M_{\odot}$, secondary initial mass $M_{2,\rm i}\sim
2.1-7.9M_{\odot}$ and initial orbital period $P_{\rm orb,i}\sim 0.03-40.0$ d.
The mass range of the ONeMg WDs in this channel lies between $\sim 1.07$ and
$\sim 1.42M_{\odot}$ while the He stars’ mass is mainly distributed between
$\sim 0.75$ and $\sim 1.5M_{\odot}$ (up to $\sim 2.25M_{\odot}$; see Figure
4). The rates ($0.34-10.78\times 10^{-5}{M_{\odot}}^{-1}$) in Table 2 show
that at most $\sim$10% of all CE events of CO WDs $+$ He stars end up with
mergers for the condition $M_{\rm com}\geq 1.44M_{\odot}$. The evolutionary
delay times lie between $\sim$0.05 and 1.0 Gyr. The different patches in
Figure 4 are caused by different evolutionary routes to form WD $+$ He star
binaries. Figure 4 shows that the mass distributions for the cores and
envelopes of He stars for the standard model 1 have a relatively narrow range
between $\sim 0.3$ and $\sim 1.0M_{\odot}$. The lower rates and narrow mass
ranges suggest that ONeMg WDs that merge with the core of a He star companion
may not play as important a role as mergers with different types of
companions. However, this might be the origin for some rare transients and
objects where no hydrogen lines are detected. Our result suggests that this
channel could produce millisecond pulsars as well as magnetars. Moriya et al.
(2018b) studied the fallback accretion central engine model for hydrogen-poor
SLSNe; they showed that $\geq 2M_{\odot}$ needs to be accreted for powering
SLSNe by fallback accretion. The derived envelope masses in Figure 4 are
smaller than the minimum value suggested by Moriya et al. (2018b; Nicholl et
al. 2017); therefore it is not clear whether the central objects formed in
this channel may be able to produce fallback accretion powered type I SLSNe.

###  3.2 Possible BH Formation through CMIC inside the CE

Stellar evolution models suggest that stellar mass BHs are the final products
of massive stars $\gtrsim 20-25\,M_{\odot}$ (e.g., Woosley & Weaver 1995).
Recent numerical simulations (e.g., O’Connor & Ott 2011; Ugliano et al. 2012;
Sukhbold & Woosley 2014; Pejcha & Thompson 2015; Ertl et al. 2016; Sukhbold et
al. 2016; Schneider et al. 2021) indicate that the outcome of neutrino-driven
explosions is largely controlled by the final core structure of massive stars,
and a clear mass limit for a progenitor star to end its evolution as a BH is
hard to establish. The likelihood of a successful explosion or of BH formation
is related to the compactness of the stellar core (e.g., O’Connor & Ott 2011;
Sukhbold et al. 2016; Raithel et al. 2018). Observational and theoretical
results suggest different progenitor mass ranges for failed SNe and BH
formation (Smartt 2009, 2015; Kochanek 2014; Schneider et al. 2021). Sukhbold
et al. (2016) suggested that single zero-age MS stars with masses as low as
15$M_{\odot}$ could produce BHs, while Schneider et al. (2021) showed that,
for stripped stars in binaries, a much higher initial mass is likely to be
required. On the other hand, in binary systems there are other ways, e.g.,
stellar mergers (Wiktorowicz et al. 2019) that could produce BHs. Here, we
propose the CMIC scenario as an additional channel for producing BHs which is
the final outcome of the merger of an ONeMg WD and the core of its companion
during the CE phase. Here we assume that a BH is formed when the total mass of
the system is larger than 15$M_{\odot}$ if the merger occurs in a CE. The
newborn BH formed through CMIC may be surrounded by an accretion disc formed
by the ejected material as shown in the typical example illustrated in Figure
5. Note that the core of the merger system with the whole mass of
$15M_{\odot}$ in our model is already an ONeMg WD, thus the core density is
significantly higher than that of a normal MS star with the same mass studied
in Sukhbold et al. (2016). In this merger process, there may be a quick
transition from a NS to a BH (e.g., Moriya et al. 2016) due to the heavy core
and the large amount of fallback material after the collapse of the heavy
ONeMG WD (in Figure 5, we only show the final possible outcome).

The results of our calculations in Figure 2 show that the initial mass for the
primaries can be as high as $\sim$11 $M_{\odot}$. For the mergers, the
secondary mass, secondary core mass, secondary envelope mass can be as high as
$\sim 21M_{\odot}$, $\sim 6.5M_{\odot}$ and $\sim 17M_{\odot}$, respectively.
The ONeMg WDs tend to be more massive ($>1.2M_{\odot}$). The mass
distributions are mainly affected by the metallicity, details of mass transfer
and CE evolution (i.e., Figure 3; see the earlier discussion). The rate for BH
formation in the CMIC channel is as high as $3.48\times
10^{-5}{M_{\odot}}^{-1}$ (Table 2). This implies that this channel makes a
contribution of at least a few percent to the BH population. The massive CE
would not be ejected far away during the CMIC and fast NS to BH transition
process, and at least some fraction of the ejected material in a core collapse
supernova explosion may remain bound to the compact remnant. Thus, the BH
formed in this way may be surrounded by an accretion disk due to the turn-
around and fallback as shown in the example in Figure 5. The accretion power
released when material falls back onto the compact remnant at late times could
power unusual SNe such as superluminous or otherwise peculiar supernovae
(e.g., Dexter & Kasen 2013). Therefore, the BH formed in this proposed
formation channel could potentially power type II SLSNe.

##  4 Discussion

$Rates/Contributions$: Adopting a star formation rate of 2 $M_{\odot}\,{\rm
yr}^{-1}$ and binary fraction of 0.7, we can estimate the Galactic rates of
peculiar objects and transients formed via the CMIC scenario as follow (the
two values given below refer to the standard model and our model 4,
respectively): (1) MSPs/Pulsars have rates of $8.81\times 10^{-5}\,{\rm
yr}^{-1}$ and $6.57\times 10^{-5}\,{\rm yr}^{-1}$. These pulsars can be the
explanation for the retention of NSs in globular clusters. (2) The rate of
T$\dot{\rm Z}$Os, including the possibility of Type II CCSNe/SLSNe and BHs
powered by AIC of the cores, are $1.36\times 10^{-4}\,{\rm yr}^{-1}$ and
$5.76\times 10^{-5}{\rm yr}^{-1}$. (3) Magnetars and SLSNe powered by these
magnetars have rates of $3.36\times 10^{-5}\,{\rm yr}^{-1}$ and $1.95\times
10^{-5}{\rm yr}^{-1}$. (4) BHs and possible SLSNe have rates of $3.23\times
10^{-5}\,{\rm yr}^{-1}$ and $0.11\times 10^{-5}\,{\rm yr}^{-1}$. Contributions
of three objects (T$\dot{\rm Z}$O, magnetar and BH) to produce SLSNe lie
between $7.82\times 10^{-5}{\rm yr}^{-1}$ and $\sim 2\times 10^{-4}{\rm
yr}^{-1}$. If we combine the rates for BHs formed right after CMIC and formed
through AIC of core NSs in T$\dot{\rm Z}$Os, the Galactic rates of BHs range
from $5.87\times 10^{-5}{\rm yr}^{-1}$ to $1.68\times 10^{-4}{\rm yr}^{-1}$;
this means that these BH formation pathways make a non-negligible contribution
to the overall BH population.

Up to April 2014, according to the ATNF Pulsar Catalogue (Manchester et al.
2005), 2016 pulsars with measured values of spin period and its time
derivative had been discovered. Among these, 1847 were single pulsars, a few
percent millisecond pulsars, 55 of them in SN remnants, and 29 single
magnetars. Based on the CMIC model results, about 1000 – 6500 pulsars can be
formed, a large fraction of which are likely to be millisecond pulsars, and
about 30-100 could be magnetars (if we take the observed fraction of
magnetized WDs into consideration for CMIC; Note that here it should be
combined rates of all three channels of CMIC). Thus, those observed numbers
given above can be reproduced by the CMIC channel proposed in this work.
However, it is worth noting that the consistency between our rates and
observational results only shows the importance of the formation scenarios in
this work, and it does not mean that other possible formation ways could be
excluded. Most interestingly, previous studies show the importance of the
immediate surroundings of a magnetar for fast radio bursts (FRBs) production
in relativistic shock wave models (e.g., Khangulyan et al. 2021). Magnetars
formed through CMIC in this work can be immediately surrounded by envelope
matters, thus the newborn magnetars in this work may produce FRBs. Compared to
the NS population, the CMIC channel makes a small contribution to the BH
population. Because the rate of type II SLSNe is not observationally well-
determined yet, we roughly take the rate of type II SLSNe as $\sim 10^{-5}{\rm
yr}^{-1}$ according to Quimby et al. (2013). Considering all cases of NS/BH
central engine models in our study, the predicted rates range from $\sim
10^{-5}{\rm yr}^{-1}$ up to $\sim 2\times 10^{-4}{\rm yr}^{-1}$). Table 4
shows general rates of SLSNe which might be originated from mergers in
different types of WD binaries. For rates of various transients and different
objects formed through the stellar-core mergers from WD binaries, see Ablimit
(2021) and this paper. From the table, it can be seen that different
assumptions (especially different models for the CE evolution and mass
transfer) could change the rate by more than an order of magnitude, and it
again shows the important role of CE evolution, mass transfer and metallicity
(see above sections for more discussions). Derived rates in the merger
scenarios are compatible with the observational estimate, although both of
these estimates are inherently uncertain (see below for other limitations).

$Related\,further\,evolution$: Above we investigated only mergers inside a CE.
But what about the systems that survive from the CE as WD binaries? WD
binaries are the one kind of X-ray binaries, and these X-ray binaries have the
connection to GW sources (e.g., Abdusalam et al. 2020). One particularly
interesting final outcome of a ONeMg WD with a massive star companion evolved
from the CE is a WD-NS binary or possibly a WD-BH system with a sufficiently
close orbit to merge within a Hubble time. These WD-NS and WD-BH binaries are
excellent GW sources and may create peculiar transients (Portegies Zwart &
Yungelson 1999; Paschalidis et al. 2004; Sabach & Soker 2014; Margalit &
Metzger 2016; Toonen et al. 2018; Abdusalam et al. 2020; Bobrick et al. 2021),
potentially observable with current and future transient surveys such as LISA.
Because the formation of close WD-BH system is very model-dependent, there is
no such system with the rapid remnant-mass model of Fryer et al. (2012), and
ONeMg WD- NS systems are the final outcomes survived from the CE evolution of
ONeMg WD-massive star binaries in this study. After CE ejection, the two stars
move closer towards each other due to various angular-momentum loss mechanisms
(i.e. GW radiation), and the resulting close ONeMg WD-NS systems will merge
within the Hubble time if they are sufficiently close after the CE phase.
Table 3 shows the rates of these ONeMg WD-NS binaries for the different
models, where the rate could be as high as $4.95\times
10^{-5}{M_{\odot}}^{-1}$. The rather non-conservative mass-transfer model
adopted in this work makes more primordial binaries have stable mass transfer,
reducing the number of CE events. With the potentially more reliable treatment
of $\lambda$ in model 4, more binaries can successfully survive from the CE
phase if they evolve into a CE phase in the first place. Considering the
formation of WD-NS systems and the contribution of mergers to these peculiar
objects, our model 4 is the most favorable one compared to the other models.

$Limitations/Uncertainties$: It is worth to raise that the critical total mass
of the WD core merged system to form a BH or a NS is very uncertain (it might
be lower or higher than the adopted value (15$M_{\odot}$) in this work), we
just adopted a possible value according to the previous work as discussed
above, and this needs further investigations. The central NS objects (e.g.,
pulsars and T$\dot{\rm Z}$Os etc.) formed through CMIC could eventually become
BHs if they accretes enough mass from fallback/surrounded materials to exceed
the maximum NS mass. One of the other biggest uncertainties in this work is
the detailed nature of newborn NSs formed through CMIC; we have discussed
various possibilities, but this clearly needs further investigation. Besides,
there are many physical processes and parameters in the binary evolution
pathways which still remain uncertain, specifically the CE phase, the
existence of accretion discs around the collapsed object, and the physical
response at very high mass-transfer rates. It is relatively easy to simulate
all these parameters in BPS simulations, i.e. the initial conditions,
prescriptions for stable/unstable mass transfer, treatments of common envelope
(CE) evolution etc. The numbers of binaries, special objects and merger
systems are affected strongly by the treatment of all these physical processes
and parameters, most of all by the parameters for the CE phase and the nature
of the mass transfer. In our BPS simulations we used a range of
acceptable/common ways by considering new evolutionary models. Future more
detailed theoretical and observational studies should provide more information
and constraints to improve our understanding.

##  5 Conclusion

We have proposed several possible new evolutionary pathways to form
(millisecond) pulsars, Thorne-$\dot{\rm Z}$ytkow objects, magnetars and BHs,
through core merger-induced collapse during the CE phase of ONeMg WD binaries
with hydrogen-rich intermediate-mass/massive stars and stripped helium-rich
stars. We have systematically investigated the CMIC model from ONeMg binaries
as origin for those objects, and discussed possibilities of powering two types
of SLSNe and FRBs etc. By comparing our simulations with observationally
inferred values of known peculiar NS objects, BH population and estimates of
the SLSN rate, we conclude that the CMIC scenario can make a significant
contribution to those peculiar transients and objects. We expect that future
observations with more information including robust estimates of the
number/rate of these systems and transients will be useful for further testing
the scenario proposed in this work.

## Acknowledgements

I.A. thank T. Moriya and S. Popov for useful comments. This work is supported
by NSF.

## Data Availability:

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

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Table 1:  Meanings of abbreviations used in the text and figures. Abbreviation | Meaning
---|---
AGB | Asymptotic giant branch (star)
AIC | Accretion-induced collapse
BH | Black hole
BPS | Binary population synthesis
CCSN(e) | Core-collapse supernova(e)
CE | Common envelope
CHeB | Core helium burning (star)
CMIC | Core merger-induced collapse
CO | Carbon/oxygen
ECSN(e) | Electron-capture supernova(e)
GW | Gravitational wave
HeMS | Star on the equivalent of the main sequence
| for hydrogen-poor helium-burning stars
HeG | Hydrogen-poor helium-burning giant
HG | Hertzsprung-gap (star)
k1 | Type of the primary star in a binary
k2 | Type of the secondary star in a binary
MIC | merger-induced collapse
MS | Main sequence
MSP | Millisecond pulsar
NS | Neutron star
ONeMg | Oxygen/neon/magnesium
SN(e) | Supernova(e)
SLSNe | Superluminous Supernovae
T$\dot{\rm Z}$O | Thorne-$\dot{\rm Z}$ytkow object
WD | White dwarf

Table 2: Different models in our calculation  Model | $\alpha_{\rm CE}$ | $\lambda$ | $q_{\rm c}$ | $Z$
---|---|---|---|---
mod1 | 1.0 | 0.5 | $q_{\rm const}$ | 0.02
mod2 | 0.1 | 0.5 | $q_{\rm const}$ | 0.02
mod3 | 1.0 | 0.5 | $q_{\rm const}$ | 0.001
mod4 | 1.0 | $\lambda_{\rm w}$ | $q_{\rm const}$ | 0.02
mod5 | 1.0 | $\lambda_{\rm w}$ | $q_{\rm cs}$ | 0.02

Table 3: Calculated rates ($R$) of ONeMg WD/non-degenerate star CE events for
five different models (in $10^{-5}$ ${M_{\odot}}^{-1}$)

CE events of ONeMg WDs/companion stars | $R1$ | $R2$ | $R3$ | $R4$ | $R5$
---|---|---|---|---|---
ONeMg WDs/normal stars (all CE) | 33.04 | 15.43 | 41.34 | 33.94 | 15.94
ONeMg WDs/normal stars (all merger) | 18.51 | 12.03 | 19.90 | 9.62 | 0.52
ONeMg WDs/normal stars (merger con1) | 6.29 | 4.96 | 7.38 | 4.69 | 0.24
ONeMg WDs/normal stars (merger con2) | 9.75 | 4.57 | 9.04 | 4.12 | 0.00
ONeMg WDs/normal stars (merger con3) | 2.31 | 2.32 | 3.48 | 0.08 | 0.00
ONeMg WDs/He stars (all CE) | 2.31 | 0.34 | 3.78 | 10.78 | 10.62
ONeMg WDs/He stars (Merger with $M_{\rm com}\geq 1.44M_{\odot}$) | 0.56 | 0.26 | 0.36 | 1.06 | 0.47
ONeMg WDs - neutron star systems (experienced CE) | 0.11 | 0.0 | 0.06 | 4.27 | 4.95

Note: Rates in the table are calculated with $R=\frac{\rm Event\,numbers}{\rm
Total\,mass(\sim
10^{7}M_{\odot}\,in\,this\,work)\,yielded\,in\,a\,simulation\,run}$. Normal
stars mean sub-giant or giant branch stars with hydrogen-rich envelopes; He-
rich stars are (sub)giant helium stars, and all mergers with He stars have
combined core masses of $M_{\rm com}\geq 1.44$. Merger con1 means that the
combined mass of the core and the secondary is $M_{\rm com}\geq 1.44M_{\odot}$
and the whole mass $M_{\rm all}<8.0M_{\odot}$, merger con2 means that $M_{\rm
com}\geq 1.44M_{\odot}$ and $8.0\leq M_{\rm all}<15.0M_{\odot}$, and merger
con3 that $M_{\rm com}\geq 1.44M_{\odot}$ and $M_{\rm all}\geq 15.0M_{\odot}$
during the CE phase. Event numbers can easily be calculated by multiplying the
numbers in the table by $R\times 10^{7}$, and the birthrate can be determined
by choosing a star formation rate in $M_{\odot}\,{\rm yr}^{1}$ ($SFR$) and
binary fraction ($f_{\rm bin}$) and multiplying the numbers in the table by
$SFR\times f_{\rm bin}\times R$.

Table 4: Galactic birthrates ($R_{\rm B}$) of stellar core-mergers which may
produce SLSNe: a comparison between mergers of CO WD/massive star CE events
($R_{\rm B,S-CO}$) and mergers of ONeMg WD/normal star CE events with two
massive cases ($R_{\rm B,S-ONe}$) for five different models (in $10^{-5}$
${\rm yr}^{-1}$)

Model |  $R_{\rm B,S-CO}$ ($M_{\rm WD}\geq 0.9M_{\odot}$) |  $R_{\rm B,S-ONe}$ ($8.0\leq M_{\rm all}<15.0M_{\odot}$) |  $R_{\rm B,S-ONe}$ ($M_{\rm all}\geq 15.0M_{\odot}$)
---|---|---|---
mod1 | 10.10 | 19.50 | 4.62
mod2 | 24.36 | 9.14 | 4.64
mod3 | 17.88 | 18.08 | 6.96
mod4 | 7.08 | 8.24 | 0.16
mod5 | 1.70 | 0.0 | 0.0

Note: This table only shows the Galactic rates (adopting SFR as 2
$M_{\odot}\,{\rm yr}^{-1}$) of SLSNe originated from mergers of between CO WDs
($M_{\rm WD}\geq 0.9M_{\odot}$) and cores of massive stars ($M_{2}\geq
8.0M_{\odot}$) inside the CE, and from massive merger products ($M_{\rm
all}\geq 8.0M_{\odot}$) of ONeMg WDs and non-degenerate hydrogen-rich stars CE
events. This table does not include other possibilities like SLSNe triggered
by mergers of between CO WDs ($M_{\rm WD}\geq 1.0M_{\odot}$) and cores of
massive stars or mergers between WDs and cores of less massive ones
($<8.0M_{\odot}$) or mergers of between WDs and cores of stripped He stars.
For other transients like CMIC or SNe Ia or other possibilities for SLSNe,
other tables including discussions in this work and Ablimit (2021) can be
used, and different SFR can be adopted with rates of those tables in this work
and Ablimit (2021) to calculate the Galactic rates for a real comparison with
future works.

Figure 1: Example of the evolution of a system to the channel for the merger
between a ONeMg WD and He core of a massive star based on model 1. The primary
(mass is M1) experiences two phases of mass transfer before forming the ONeMg
WD. Afterwards, the secondary (mass is M2) fills its Roche lobe at the core
helium burning stage of the evolution (CHeB), and its envelope engulfs the
ONeMg WD and its core due to the unstable mass transfer. The abbreviations of
the stellar types (k1 is the type of the primary star, and k2 is type of the
secondary star in a binary) are defined in the text. The evolution time is in
Myr, and separation (sep) between two stars is in $R_{\odot}$.

Figure 2: The normalized/weighted number density distributions from simulated
results of model 1 (standard model). (a) and (b) show the initial mass
relation of primary MS ($M_{\rm 1,i}$) and secondary MS stars ($M_{\rm 2,i}$)
in primordial binaries which form all ONeMg WD + normal star binaries with CE
(upper left), and ONeMg WD + normal star systems which experience the CE and
leading to core mergers (upper right), respectively. Mass relations at the
beginning of the CE between the ONeMg WD and its companion star for ONeMg WD +
normal star systems, and secondaries’ core and envelope mass relations for the
merger case under the condition of $M_{\rm WD}+M_{\rm 2,C}\geq 1.44$ are given
in (c) and (d), respectively.

Figure 3: The total mass distributions of merger systems between the ONeMg WDs
and cores of the normal stars under the condition of $M_{\rm WD}+M_{\rm
2,C}\geq 1.44$ for five different models.

Figure 4: The normalized/weighted number density distributions from simulated
results of model 1 (standard model) for the merger case of the ONeMg WD and
core of the He stars under the condition of $M_{\rm WD}+M_{\rm 2,C}\geq 1.44$
at the onset of CE. (a) demonstrates the initial mass relation of primary MS
($M_{\rm 1,i}$) and secondary MS stars ($M_{\rm 2,i}$) in primordial binaries
which form all ONeMg WD + He star binaries with CE. (b) shows the mass
relation of the ONeMg WD ($M_{\rm WD}$) and secondary He stars ($M_{\rm He}$).
Mass relations between the ONeMg WD and core of the He star for the merger
case are given in (c).

Figure 5: Example of the merger between a ONeMg WD and He core of a massive
star inside the CE based on model 1. The evolution is very similar to that of
Fig. 1, and the main difference compared to the first example is that the
combined core and total mass are higher, the CE event happens when the
secondary becomes the HG star, and the evolution time is longer due to the
different initial conditions. The column headings and units are same as that
of Fig. 1.