# Accretion induced merger leading to core collapse supernovae in old stellar
populations

Jessica Braudo  Department of Physics, Technion – Israel Institute of
Technology, Haifa 3200003, Israel  Ealeal Bear  Department of Physics,
Technion – Israel Institute of Technology, Haifa 3200003, Israel  Noam Soker
Department of Physics, Technion – Israel Institute of Technology, Haifa
3200003, Israel  Guangdong Technion Israel Institute of Technology, Guangdong
Province, Shantou 515069, China<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We examine a triple-star evolution that might lead to core collapse supernovae
(CCSNe) in stellar populations that are too old to allow for single or binary
evolution to form CCSNe, i.e., where the most massive stars that evolve off
the main sequence have masses of $\simeq 4-5M_{\odot}$. In the scenario we
examine the most massive star in the triple system, of mass $\simeq
4-5M_{\odot}$, transfers mass to an inner binary system at an orbital
separation of $\simeq 100-1000R_{\odot}$. The initial orbital separation of
the inner binary is $\simeq 10-50R_{\odot}$. The inner binary accretes most of
the mass that the primary star loses and the two stars expand and their mutual
orbit contracts until merger. The merger product is a main sequence star of
mass $\simeq 8-10M_{\odot}$ that later experiences a CCSN explosion and leaves
a NS remnant, bound or unbound to the white dwarf (WD) remnant of the primary
star. We estimate the event rate of this WD-NS reverse evolution scenario to
be a fraction of $\approx 5\times 10^{-5}$ of all CCSNe. We expect that in the
coming decade sky surveys will detect 1-5 such events.

Supernovae: general — transients: supernovae — binaries (including multiple):
close

##  1 Introduction

Singly-evolving stars explode as core collapse supernovae (CCSNe), including
electron-capture supernovae (SNe), if their zero-age main sequence (ZAMS) mass
is $M_{\rm ZAMS}\gtrsim 8.5-9M_{\odot}$. This mass limit depends on some
uncertain stellar structure parameters as well as on initial metallicity
(e.g., Heger et al. 2003; Poelarends et al. 2008; Ibeling & Heger 2013;
Doherty et al. 2017; Gil-Pons et al. 2018).

Binary evolution, through merger and mass transfer, might allow ZAMS stars
with lower mass to end as CCSNe and leave neutron star (NS) remnants, e.g.,
electron-capture SNe by mass transfer (e.g., Siess & Lebreuilly 2018) and by
binary merger (e.g., Zapartas et al. 2017). Zapartas et al. (2017) conduct a
population synthesis study and find that $\simeq 10-20\%$ of CCSNe occur by
the merging of two stars in the mass range of $\simeq 4-8M_{\odot}$. This
merger yields a massive enough star to end as a CCSN.

If a star with an initial mass of $M_{\rm ZAMS}\lesssim 8M_{\odot}$ explodes
as a CCSN, it must be in a multiple stellar system, e.g., a binary or a triple
stellar system. The main routes involve the merger of two stars or mass
transfer, both processes bring the CCSN progenitor to a mass of $\gtrsim
8.5M_{\odot}$ (or even $>8M_{\odot}$ might allow CCSN). In that case some of
the evolutionary routes form a white dwarf (WD) before the CCSN explosion.
This binary evolutionary channel is termed a WD-NS reverse evolution (e.g.,
Sabach & Soker 2014). The initial more massive star in the binary system, the
primary star, has a mass of $5M_{\odot}\lesssim M_{\rm 1,ZAMS}\lesssim
8-8.5M_{\odot}$. It evolves first, transfers mass to its companion (the
secondary star) and forms the WD. The mass-transfer brings the secondary star
to have a post-accretion mass of $M_{\rm 2,B,f}\gtrsim 8.5M_{\odot}$, and so
it might end its life as a CCSN, leaving behind a NS remnant. If the mass
transfer takes place after the secondary star has developed a helium core,
i.e., it has already left the main sequence, then the secondary star is not
likely to explode as a CCSN even if its mass after mass transfer is $M_{\rm
2,B,f}\gtrsim 8.5M_{\odot}$ (e.g., Bear & Soker 2021a). We refer to the
initial second massive star as the ‘secondary star’, whether it loses or gains
mass. Similarly, we refer to the initial more massive star as the ‘primary
star’. We refer to the initially least massive star in a triple system as the
‘tertiary star’.

Accretion induced collapse, where a WD accretes mass and collapses to a NS,
and similar processes involve the formation of a WD before a NS (e.g. Ruiter
et al. 2019; Tauris & Janka 2019; Liu & Wang 2020; Wang & Liu 2020). As we
mentioned above, another WD-NS reverse evolution is of a binary system that
experiences a mass-transfer process that brings the secondary star to evolve
toward a CCSN and leaves behind a WD-NS binary system, bound or unbound (e.g.,
Tutukov & Yungelson 1993; Portegies Zwart & Verbunt 1996; Tauris & Sennels
2000; Brown et al. 2001; Nelemans et al. 2001; Kim et al. 2003; Kalogera et
al. 2005; van Haaften et al. 2013; Toonen et al. 2018; Breivik et al. 2020).
The WD-NS reverse evolution might account for the binary radio pulsars PSR
J1141-6545 and PSR B2303+46 that have a NS-WD binary system each (e.g.,
Portegies Zwart & Yungelson 1999; van Kerkwijk & Kulkarni 1999; Tauris &
Sennels 2000; Brown et al. 2001; Davies et al. 2002; Church et al. 2006).
Sabach & Soker (2014) and Ablimit (2021) discuss other outcomes of the WD-NS
reverse evolution, including the merger of the WD with the core of the NS
progenitor, and some types of bright transient events (intermediate luminosity
optical transients; ILOTs).

There are rare and peculiar exploding stars that require the WD-NS reverse
evolution to take place in triple stellar systems. Such is the formation of
CCSN inside a planetary nebula (Bear & Soker, 2021a). Bear & Soker (2021a)
find that in binary systems that experience the WD-NS reverse evolution the
minimum time from the formation of planetary nebula and the CCSN explosion is
about million years. By the explosion time the planetary nebula has long
dispersed into the interstellar medium. Starting with two stars of very close
initial masses does not help because in this case the primary transfers mass
to the secondary star after the secondary star has left the main sequence and
developed a helium core. This late mass transfer implies that even that its
mass exceeds $\simeq 8.5M_{\odot}$ the secondary star does not explode, but
rather forms a planetary nebula and leaves a WD remnant. Bear & Soker (2021a)
conclude that a CCSN explosion might occur inside a planetary nebula only if a
third star forms the PN. This is a rare case because the third star should be
less massive than the secondary star but by no more than ${\rm few}\times
0.01M_{\odot}$. Bear & Soker (2021a) estimate that about one CCSN inside a
planetary nebula occurs out of $10^{4}$ CCSNe.

In the present study we examine a WD-NS reverse evolution where the most
massive star in the system has an initial (ZAMS) mass of $M_{\rm 1,ZAMS}\simeq
4-5M_{\odot}$. In a binary system mass transfer will not bring the secondary
star to be sufficiently massive to explode as a CCSN. A merger of two stars
with combined mass of $\gtrsim 8.5M_{\odot}$ might lead to CCSN, but for a
CCSN a very fine tuned mass range is needed if at all it can take place, and
no WD formation will take place. We consider therefore a triple stellar system
where the primary star transfers mass to an inner (tight) binary system, and
forms a WD. The mass transfer causes the two stars of the inner binary to
merge while they are on the main sequence such that the mass of the inner
binary system together with the mass it accreted from the primary star add up
to a star of mass $M_{\rm 23acc}\gtrsim 8.5M_{\odot}$, that later explodes as
a CCSN. We base our scenario on the process where the mass transfer causes the
inner binary to merge (for a recent thorough study of this process see, e.g.,
Di Stefano 2020).

de Vries et al. (2014) already studied the mass transfer from a tertiary star
to an inner tight binary system, but not for systems that can experience the
WD-NS reverse evolution. They claim that triple-star systems where one star
suffers Roche lobe overflow (RLOF) to a tight (inner) binary system amount to
$\lesssim 1\%$ of triple-star systems. However, in a newer paper Leigh et al.
(2020) find that $>10\%$ of all triple star systems might have circumbinary
disk around the inner binary. de Vries et al. (2014) also find that under
their assumptions for the cases they studied the accreted mass does not form
an accretion disk around neither of the stars in the inner binary system, nor
around the binary (a circumbinary disk). The inner binary system ejects most
of the mass, and accretes only about fifth of the mass that the primary star
transfers. They used an adiabatic equation of state and did not allow for
radiative cooling. In the large orbital separations we use here we expect the
gas that leaves the primary star to be optically thin and cool by radiation.
This cooling is likely to allow a higher accretion rate. The orbit of the
inner binary does shrink. In the two cases that de Vries et al. (2014)
simulated the initial ratio of the inner semi-major axis (of the inner binary
system) to the triple semi-major axis were $a_{\rm in}/a_{\rm out}=0.046$ and
$a_{\rm in}/a_{\rm out}=0.108$. They simulated observed systems.

In a later study Portegies Zwart & Leigh (2019) examine the mass transfer from
a low mass star to an inner binary system through an accretion disk. This
leads to the formation of two blue stragglers. They find that the inner binary
system accretes at least $80\%$ of the mass that the primary star transfers
(they use the term tertiary star for their mass-losing star). We will
therefore also assume that the inner binary system accretes a large fraction
of the mass that the primary star transfers.

Portegies Zwart & Leigh (2019) mention that the accreting binary system might
evolve towards a mass ratio 1 in some cases (the lower mass star in the inner
binary system accretes more mass.), but not in all cases. de Vries et al.
(2014) find that the more massive star in the inner binary accretes more mass.
We will assume that the two stars of the inner binary accrete relative to
their respective masses. In section 2 we describe the proposed scenario.

We describe the evolution to mass transfer in section 3 and the merger of the
tight binary system in section 4. In section 5 we crudely estimate the rate of
such very rare events. We summarize our study in section 6

##  2 The proposed scenario

The main process of the scenario is the merger of two stars with an additional
mass from the primary star, such that the mass of the merger remnant is
$M_{\rm 23acc}\simeq 8-10M_{\odot}$, i.e., sufficiently massive to explode as
a CCSN. This supposes to take place in an old enough stellar population where
binary merger alone cannot bring a merger remnant to be massive enough to
explode as a CCSN. We schematically present the proposed scenario in Fig. 1.

Figure 1: A schematic diagram of our proposed WD-NS reverse triple star
evolution to a CCSN in an old stellar population where binary mass transfer
cannot lead to a CCSN. Thick arrows depict phases we do simulate, while thin
arrows are phases we do not simulate in this study. Abbreviation: AGB:
asymptotic giant branch; CCSN: core-collapse supernova; MS: main sequence; NS:
neutron star; RGB: red giant branch. WD: white dwarf.

We will refer to the initially most massive star as the primary along the
entire evolution. Its initial mass is $M_{\rm 1,ZAMS}\simeq 4-5M_{\odot}$ and
it is on a wide orbit of $a_{\rm out,0}\approx 0.5-5{~{}\rm AU}$ with an inner
tight binary system with initial masses of $M_{\rm 2,ZAMS}$ and $M_{\rm
3,ZAMS}$. The primary star evolves to become a giant, and either on its red
giant branch (RGB; for $a_{\rm out,0}\lesssim 1{~{}\rm AU}$, depending on
eccentricity) or on its asymptotic giant branch (AGB; for $a_{\rm
out,0}\gtrsim 1{~{}\rm AU}$) it transfers mass to the inner binary system via
RLOF such that the inner binary system accretes most of the mass that the
primary loses.

To the condition that the primary mass is not large enough to allow CCSN by
mass transfer, there are two additional conditions on the initial masses of
the inner binary components. ($i$) The primary star should expand to become a
giant and transfer mass to the tight binary system before the secondary
develops a helium core, i.e., before it leaves the main sequence. Otherwise it
will not explode later as a CCSN (e.g., Bear & Soker 2021a). This condition
reads $M_{\rm 2,ZAMS}\lesssim M_{\rm 1,ZAMS}-0.2M_{\odot}$. ($ii$) The mass of
the merger remnant of the inner binary system with the mass it accretes from
the primary star, $M_{\rm acc}$, should form a main sequence star that is
massive enough to explode as a CCSN, i.e., $M_{23acc}=M_{\rm 2,ZAMS}+M_{\rm
3,ZAMS}+M_{\rm acc}\simeq 8-10M_{\odot}$. More massive stars also explode, but
in the scenario we study here the mass will not be much larger that $M_{\rm
23acc}\simeq 8-9M_{\odot}$. This condition reads $2.5M_{\odot}\lesssim M_{\rm
3,ZAMS}\leq M_{\rm 2,ZAMS}$.

The conditions on the outer and inner orbits are as follows. The outer orbit,
of the primary star and the inner (tight) binary system, should allow a RLOF
mass transfer. A common envelope evolution might also take place, but the
scenario does not allow a too large mass ejection as the tight binary system
should accrete most of the mass that the primary system loses. A common
envelope evolution is likely to eject too much mass. This condition reads
therefore $0.5\lesssim a_{\rm out,0}\lesssim 5{~{}\rm AU}$, strongly depending
on the eccentricity. We will not explore this parameter space in the present
study because our goal is to demonstrate the possibility of the scenario. We
will assume a circular orbit and examine one initial value of $a_{\rm out,0}$.

The condition on the initial inner orbit $a_{\rm in,0}$, that of the inner
binary system, is that the two stars merge during (or shortly after) mass
transfer. We study this constrain on $a_{\rm in,0}$ in section 4.

The final outcome of this triple star WD-NS reverse evolution is a CCSN that
leaves a NS in a bound or unbound orbit with a WD. As we emphasize, this
scenario might take place in relatively old stellar populations where even
binary merger at late times cannot lead to CCSNe.

##  3 Evolution to mass transfer

To follow the evolution from the main sequence to the end of the phase of mass
transfer from the primary to the inner binary system we use version 10398 of
the stellar evolutionary code mesa-binary (Paxton et al., 2011, 2013, 2015,
2018, 2019).

As an example of a triple stellar system that might lead to a CCSN (including
the possibility of electron capture SN) we start with three ZAMS stars of
$M_{\rm 1,ZAMS}=4M_{\odot}$ (the primary) on a large orbit, and an inner
(tight) binary system of masses $M_{\rm 2,ZAMS}=3.5M_{\odot}$ and $M_{\rm
3,ZAMS}=3M_{\odot}$. We consider only circular orbits. We take the orbital
separation of the primary star from the center of mass of the inner binary
system to be $a_{\rm out,0}=500R_{\odot}$. As we show later, the initial inner
orbital separation should be $10\lesssim a_{\rm in,0}\lesssim 50R_{\odot}$. We
also assume that the triple system is coplanar, i.e., the inner and outer
orbital planes are the same, and the orbits are circular. As both
eccentricities and the inclination of the two orbital planes are not large
this system is stable to the Kozai-Lidov mechanism (e.g., Katz, Dong, &
Malhotra 2011; Li et al. 2014). Moreover, tidal forces further stabilize the
system against the Kozai-Lidov mechanism (e.g., Liu et al. 2015). The system
obeys also the dynamical stability criterion for coplanar prograde systems
according to equation (90) of Mardling & Aarseth (2001), which for the
parameters of the systems implies that $a_{\rm in,0}\lesssim 0.3a_{\rm
out,0}$. Namely, the inner binary will not experience merger before mass
transfer occurs.

In the first part of the simulation we do not follow the evolution of the
inner binary system, but rather refer to it as a point mass of $M_{\rm
point}=M_{\rm 2,ZAMS}+M_{\rm 3,ZAMS}=6.5M_{\odot}$ (for a recent population
synthesis study of mass transfer to an inner binary see Hamers, Glanz, &
Neunteufel 2021). This point mass can accrete mass. We continue the evolution
to the end of the mass transfer phase from the primary to the inner binary
system.

As in our earlier paper Bear & Soker (2021a) we base our mesa-binary mode on
the inlists (i.e., code prescription) of Gibson & Stencel (2018). This uses
the mass-transfer scheme of Kolb & Ritter (1990) with mass-transfer efficiency
scheme from Soberman et al. (1997) (as Gibson & Stencel 2018 apply). We take
the fraction of the mass-loss from the fast wind in the vicinity of the donor
star $M_{1}$ to be $\alpha=0.1$ and the mass-loss fraction from the vicinity
of the accretor point mass (that represents the inner binary system) to be
$\beta=0.1$. We take $\delta=0.1$ to be the mass-loss fraction from the
circumbinary co-planar toroid. These three identical factors imply that the
inner binary accreted $90\%$ of the mass that the primary loses. The radius of
this circumbinary toroid is $\gamma^{2}a$, where $a$ is the binary semi-major
axis and $\gamma=1.3$ (Gibson & Stencel, 2018). The initial metalicity is
$Z=0.01$ and the initial equatorial surface rotation velocity of $M_{1}$ is
$v_{\rm ZAMS,1}=2~{}{\rm km}~{}{\rm s}^{-1}$.

In Fig. 2 we present the mass accretion rate onto the tight binary system as
function of time. The mass transfer takes place while the primary star is on
the AGB as it has a radius of $164R_{\odot}$, has developed a carbon core of
$M({\rm C,c})=0.51M_{\odot}$, and the boundary of the helium core is at
$M({\rm He,c})=0.79M_{\odot}$.

Figure 2: Mass accretion rate by the inner binary system (blue line) and the
total accreted mass (red line) as function of time from ZAMS. The total mass
the inner binary accretes under our assumptions is $\Delta M_{\rm
acc}=2.82M_{\odot}$. The graph presents a time period of $2\times
10^{5}{~{}\rm yr}$.

At the end of the mass transfer phase the outer orbital separation is $a_{\rm
out}=2430R_{\odot}$ and the WD mass is $M_{\rm WD}=0.8M_{\odot}$. Such a large
orbital separation implies that the merger product will evolve as a single
star for most, or even all, of its evolution. In section 4 we describe a
simulation we conduct that shows that the WD in the case we simulate does not
influence much the evolution of the merger remnant and does not prevent its
final explosion.

##  4 Merger of the inner binary system

We follow the response of the two stars of the inner (tight) binary system to
the mass they accrete. Mass accretion by the binary system is likely to cause
orbital contraction (e.g., Comerford et al. 2019). We assume that the mass
that each star accretes is proportional to its mass. The secondary and the
tertiary stars accreted each a total mass of $\Delta M_{\rm
2,acc}=1.52M_{\odot}$ and $\Delta M_{\rm 3,acc}=1.3M_{\odot}$, respectively.
In Figs. 3 and 4 we present the evolution of the radius and luminosity of each
of these two stars during the main mass transfer phases.

Figure 3: The evolution of the radius (upper panel) and luminosity (lower
panel) of the $M_{\rm ZAMS,2}=3.5M_{\odot}$ component of the inner binary
during the main mass accretion phase. The total accreted mass by this star
under our assumptions is $\Delta M_{\rm acc,2}=1.52M_{\odot}$. The time
between two consecutive small ticks is $1000{~{}\rm yr}$.

Figure 4: Similar to Fig. 3 but for the $M_{\rm ZAMS,3}=3M_{\odot}$ component
of the inner binary.

At the peak of mass transfer rate the mass accretion time scale
$M/\dot{M}_{\rm acc}\simeq 10^{4}{~{}\rm yr}$ is much shorter than the thermal
time scale of each of the two stars $\tau_{\rm KH}>10^{5}{~{}\rm yr}$. This
explains the very large expansion of the stars.

The orbital separation of the inner binary system changes as a result of mass
accretion in a way that depends on the angular momentum of the accreted mass.
For a zero angular momentum of the accreted mass, $J_{\rm acc}=0$, angular
momentum conservation gives the final orbital separation as

$a_{\rm in}(J_{\rm acc}=0)=a_{\rm in,0}\left(\frac{M_{\rm 2,f}+M_{\rm
3,f}}{M_{\rm 2,b}+M_{\rm 3,b}}\right)\left(\frac{M_{\rm 2,b}M_{\rm
3,b}}{M_{\rm 2,f}M_{\rm 3,f}}\right)^{2},$ (1)

where subscript $b$ stands for just before mass accretion starts and $f$
stands for after mass transfer ends. We take the stellar masses just before
mass transfer starts to be very similar to their ZAMS masses as winds on the
main sequence do not carry much mass. Since we distribute the accreted mass by
the initial mass of the stars we can write equation (1) as

$a_{\rm in}(J_{\rm acc}=0)=a_{\rm in,0}\left(\frac{M_{\rm 2,b}+M_{\rm
3,b}}{M_{\rm 2,b}+M_{\rm 3,b}+\Delta M_{\rm acc}}\right)^{3}.$ (2)

The accreted mass has some angular momentum. If the angular momentum of the
inner binary system and the triple system have the same sense, so will the
accreted mass and the final orbital separation will be larger than what
equation (2) gives. However, the angular momenta of the two stars might also
be misaligned, or even have opposite sense such that the final orbital
separation is smaller than what equation (2) gives. In checking whether the
mass transfer by itself can induce merger we will use equation (2) as it is.
Overall, we take as our merger condition by mass accretion that at the end of
mass transfer

$R_{\rm 2}+R_{\rm 3}\geq a_{\rm in}(J_{\rm acc}=0).$ (3)

The evolution of the stars (Figs. 3 and 4) show that the large expansion takes
place during the high mass transfer rate, before much mass has been accreted.
To apply condition (3) after the inner binary accreted most of the mass, i.e.,
$\Delta M_{\rm acc}=2.8M_{\odot}$, would require using $R_{\rm 2}+R_{\rm
3}\simeq 8R_{\odot}$. With this left hand side of equation (3), the merger
condition reads $a_{\rm in,0}\lesssim 24R_{\odot}$. However, the two stars
expand to huge dimensions. This does not mean that we can place them at much
large separations as the inflated envelopes are of very low density as we show
in Fig. 5.

Figure 5: The density profiles (blue lines) and the accreted mass profile (red
lines) of the two stars of the binary system at their maximum expansion.

At maximum expansion $(R_{2}+R_{3})_{\rm max}=130R_{\odot}$. However, due to
the very low masses that reside in the very outer zones of the two inflated
envelopes, we crudely take the maximum orbital separation for merger to be
$\approx 50R_{\odot}$. To prevent merger before mass transfer (so the binary
merger does not evolve fast and off the main sequence) we require the initial
orbital separation to be $a_{\rm in,0}\gtrsim 10R_{\odot}$. This allowed range
of initial orbital separation, $10\lesssim a_{\rm in,0}\lesssim 50R_{\odot}$,
corresponds to an allowed period interval of $1~{}{\rm day}\lesssim P_{\rm
in,0}\lesssim 16{\rm day}$.

We cannot simulate the full merger process. Instead, we evolved a star with an
initial mass of $M_{\rm 2,ZAMS}=3.5M_{\odot}$ to the time when it merges with
the $M_{\rm 3,ZAMS}=3M_{\odot}$ after they have accreted a mass of $\Delta
M_{\rm acc}=2.82M_{\odot}$. We added a mass of $3M_{\odot}+2.82M_{\odot}$ of
solar composition to the evolved $M_{\rm ZAMS}=3.5M_{\odot}$ star to form a
new star of mass $M_{\rm 23acc}=9.32M_{\odot}$. We evolved this simulated
merger product with a WD companion of mass $M_{\rm WD}=0.8M_{\odot}$. (a point
mass) starting with an orbital separation of $a=a_{\rm out}=2430R_{\odot}$
using mesa-binary. Because of numerical difficulties we could not follow the
evolution to the point of electron capture that leads to explosion. We had to
terminate the evolution when the ONeMg core reached a mass of $M_{\rm
core}=1.2M_{\odot}$. The fate of such stars, i.e., in the mass range we focus
on here $M_{\rm ZAMS}\approx 9M_{\odot}$, is determined by the competition
between core growth and stellar mass loss (e.g., discussion by Poelarends et
al. 2017). We find that at the termination time of the evolution of the
remnant product the ratio of the stellar mass loss rate to core-mass growth
rate is about 10. The envelope mass at the time we terminated the evolution is
$M_{\rm env}=7.4M_{\odot}$. By the time the core will reach a mass of
$1.4M_{\odot}$ the mass loss would remove only about $2M_{\odot}$ from the
envelope. There are uncertainties in the final fate of the star because the
competition between core growth and mass loss depends on some uncertain
processes, such as the amount of overshooting introduced into the model and
the stellar mass loss rate. However, since for the present parameters our
stellar model will lose less than third of the envelope mass before it reaches
the minimum core mass for explosion $\simeq 1.4M_{\odot}$, there is a large
margin for the core to reach this minimum mass. We also find that the orbital
separation of the WD increases according to the mass loss rate. Namely, tidal
interaction is not strong enough to shrink the orbit of the WD. This implies
that the WD will not influence the fate of the merger product to explode.

From the above discussion we conclude that with a large margin, the core will
reach the mass of electron capture long before the star loses its envelope.
The star will explode as a type II CCSN (electron capture) as the hydrogen-
rich envelope mass will be non-negligible. Furthermore, Poelarends et al.
(2008) find that stars that reach a maximum luminosity of $L>10^{5}L_{\odot}$
in the above mass range explode as electron capture SNe. Our model of the
merger remnant when we terminate its evolution has a luminosity of $1.2\times
10^{5}L_{\odot}$. We safely conclude that the merger remnant of our scenario
will explode as an electron capture SN at $t=1.77\times 10^{8}{~{}\rm yr}$
after star formation.

The main conclusion from this section is that there is a non-negligible volume
of the parameter space that allows the triple star WD-NS reverse evolution
that we study here to take place.

##  5 Rate

We crudely estimate the rate of CCSNe from triple star evolution in
populations that are too old for binary merger or binary mass transfer to form
CCSNe. Namely, populations where stars of $4M_{\odot}<M_{\rm
1,ZAMS}<5M_{\odot}$ are now evolving off-the main sequence. We find the
following factors.

(1) For an initial mass function of $dN\propto M^{-2.35}$ (e.g. Bastian et al.
2010) the fraction of primary stars with $4M_{\odot}<M_{\rm
1,ZAMS}<5M_{\odot}$ to the number of CCSNe, which we take as all stars with
mass $>8M_{\odot}$, is $f_{M1}\simeq 0.7$.

(2) The probability of such stars to be in triple or higher order stellar
systems if $f_{\rm T}\simeq 0.3$ (e.g., Moe & Di Stefano 2017).

(3) From the mass ratio distributions that Moe & Di Stefano (2017) give for
these primary masses, we find that the probability of the secondary star to be
of mass $M_{\rm 2,ZAMS}\simeq 0.8-0.95M_{\rm 1,ZAMS}$ is $f_{2}\simeq 0.15$,
with a similar ratio for the tertiary star $f_{3}\simeq 0.15$.

(4) From Moe & Di Stefano (2017) we find that the fraction of systems with an
outer orbital semi-major axis of $a_{\rm out,0}\simeq 100-1000R_{\odot}$,
i.e., an orbital period range of $P_{\rm out,0}\simeq 30-1000{~{}\rm days}$,
is $f_{\rm out}\simeq 0.15$.

(5) In section 4 we found that the initial orbital period range for the inner
binary is $1~{}{\rm day}\lesssim P_{\rm in,0}\lesssim 16{\rm day}$. From Moe &
Di Stefano (2017) we find the probability for this range to be $f_{\rm
in}\simeq 0.07$.

Over all, the fraction of CCSNe in old stellar population where only triple
star mass transfer can lead to CCSNe to the total CCSNe rate is

$F_{\rm Old~{}Triple}\simeq f_{M1}f_{\rm T}f_{2}f_{3}f_{\rm out}f_{\rm
in}\approx 5\times 10^{-5}.$ (4)

As expected, old-triple CCSNe are very rare. However, we expect some to be
observed. With new sky transient surveys with an expected CCSN detection rate
of $\approx 10^{4}{~{}\rm yr}^{-1}$, we can expect that within the next decade
one to few CCSNe will be observed in old stellar populations where a binary
mass transfer cannot account for these CCSNe, but triple star evolution can.

##  6 Summary

Sky surveys, e.g., the All-Sky Automated Survey for Supernovae (ASAS-SN;
Kochanek et al. 2017), the Zwicky Transient Facility (ZTF; Bellm et al. 2019),
the Southern Hemisphere Variability Survey (LSQ; Baltay et al. 2013), and the
Large Synoptic Survey Telescope (LSST; Ivezić et al. 2019), will detect
$\approx 10^{4}$ CCSNe per year. This number implies that during a time period
of several years these surveys will observe even very rare types of CCSNe and
CCSN impostors. This motivates us to conduct a series of studies of a variety
of rare CCSNe and impostors (e.g., Bear & Soker 2021b; Schreier et al. 2021).
The present study is another study in that series of papers, now focusing on
CCSNe in stellar populations that are too old for binary mass transfer to
result in a CCSN. We rather consider a triple-star evolution that might lead
to CCSNe in stellar populations where the most massive stars that evolve off
the main sequence have masses of $M_{\rm 1,ZAMS}\simeq 4-5M_{\odot}$ (Fig. 1).
As the evolution forms a WD before it forms the NS remnant of the CCSN, it is
a WD-NS reverse evolution.

In the present study we do not search the parameter space for this scenario
because the uncertainties are too large, e.g., how much mass the inner binary
accretes from the mass the primary loses. We demonstrated the scenario for one
set of parameters by examining the mass transfer rate as function of time
(section 3; Fig. 2), and then by following the evolution of each of the two
stars of the inner binary system as they accrete mass (section 4; Figs. 3-5).
For the parameters we use here explosion takes place at $t=1.77\times
10^{8}{~{}\rm yr}$ after star formation. We then estimated the range of
initial orbital separations for which the inner binary might merge as a result
of the mass transfer (equation 3).

In section 5 we crudely estimated the event rate of the scenario we studied
here to be a fraction $F_{\rm Old~{}Triple}\approx 5\times 10^{-5}$ of CCSNe.
Although very rare, as we mentioned above, we expect that in the coming decade
sky surveys will detect 1-5 such events.

The results of this study add to the call to consider rare triple-star
evolutionary routes when analysing observations of peculiar and very rare
CCSNe and CCSN impostors.

## Acknowledgments

We thank Evgeni Grishin, Erez Michaely, and an anonymous referee for helpful
comments.

This research was supported by a grant from the Israel Science Foundation
(769/20).

Data availability

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

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