# Double common envelope jets supernovae (CEJSNe) by triple-star systems

Noam Soker  Department of Physics, Technion – Israel Institute of Technology,
Haifa 3200003, Israel  Guangdong Technion Israel Institute of Technology,
Guangdong Province, Shantou 515069, China

###### Abstract

I propose a new type of common envelope jets supernova (CEJSN) events where
instead of a single neutron star (NS; or a black hole; BH) a tight binary
system of a NS and a main sequence star enters a common envelope evolution
(CEE) with a red supergiant. The NS and the main sequence star of the tight
binary system merge inside the red supergiant envelope and enter a CEE of
their own. The NS accretes some mass through an accretion disk and launches
jets that explodes the main sequence star. I estimate that the two jets that
the NS launches at this phase carry an energy of $\approx 10^{52}{~{}\rm
erg}$, about the same order of magnitude as the energy that the jets will
carry when the NS or its BH remnant will enter the core in a later phase. For
that, I term the entire event a double CEJSN. The outcome of the double CEJSN
is a very long, months to years, and very energetic event, a total energy of
$\approx 10^{52}-10^{53}{~{}\rm erg}$, that will be observationally classified
as a peculiar super-energetic event. I crudely estimate that new transient
surveys should detect about one CEJSN event from a triple star system per
year.

(stars:) binaries (including multiple): close; (stars:) supernovae: general;
transients: supernovae; stars: jets

##  1 Introduction

In a common envelope jets supernova (CEJSN) event a neutron star (NS) or a
black hole (BH) accretes mass of a red supergiant (RSG) star from within its
envelope and launches jets as it spirals-in in a common envelope evolution
(CEE; e.g., Gilkis et al. 2019; Soker et al. 2019; Grichener & Soker 2019a;
Schrøder et al. 2020; Grichener & Soker 2021). The non-uniform density in the
RSG envelope implies sufficiently large specific angular momentum of the mass
that the NS/BH accretes to form an accretion disk around the NS/BH (e.g.,
Armitage & Livio 2000; Papish et al. 2015; Soker & Gilkis 2018; for recent
hydrodynamical studies of disk formation see, e.g., López-Cámara et al. 2019,
2020, and for a discussion on the formation of the accretion disk with more
references see Hillel et al. 2021). The density gradient in the envelope
causes the NS/BH to accrete more mass from the denser side. The accreted gas
from the denser side and the accreted gas from the other side of the NS/BH
have opposite signs of the angular momentum around the NS/BH. Since more mass
is accreted from the denser side, the angular momenta of the two sides do not
cancel each other and the accreted mass has a net angular momentum. Because of
the very small radius of the NS/BH the specific angular momentum of the
accreted matter is sufficient to form an accretion disk around the NS/BH.

The spiralling-in NS/BH ends in one of two possibilities. It might eject the
entire envelope before it reaches the core, such that the dynamical friction
ceases and the spiralling-in processes stops (e.g., Soker et al. 2019). At a
later time the core explodes as a CCSN and the remnant is a system of two NSs,
or NS + BH, or two BHs, bound or unbound (e.g., Soker et al. 2019). More
relevant to the present study is the case where the NS/BH reaches the core and
continues with the CEE inside the core. The last powering phase is when the
NS/BH destroys the core and the destroyed core gas forms a massive accretion
disk that launches energetic jets (e.g., Grichener & Soker 2019a). The
observed CEJSN event might be similar to a CCSN, but it is likely to be more
energetic with explosions energy of up to $\simeq{\rm several}\times
10^{52}{~{}\rm erg}$, might last for a longer time of up to few years,
typically contains a more massive circumstellar matter (CSM) of up to several
solar masses, and in general it would be a peculiar supernova (SN; e.g., Soker
et al. 2019; Schrøder et al. 2020). The peculiarity might be in the long
lasting and energetic light curve, and/or in several peaks in the light curve,
and/or in abnormal composition due to nucleosynthesis inside the jets that the
NS/BH launches while inside the core (e.g., Grichener & Soker 2019a), and/or a
highly non-spherical explosion (e.g., Soker et al. 2019). As well, the NS/BH
merger with the core might be a source of gravitational waves (e.g., Ginat et
al. 2020). Soker & Gilkis (2018) attributed the enigmatic SN iPTF14hls (Arcavi
et al., 2017), and now also the similar SN 2020faa (Yang et al., 2021) to a
CEJSN event. Soker et al. (2019) developed the polar CEJSN scenario where the
jets that the NS launches as it spirals-in inside the RSG envelope clear two
opposite polar regions, allowing the later jets that the NS launches as it
spirals-in inside the core to expand freely. They suggested this scenario to
explain the fast-rising blue optical transient AT2018cow (e.g., Prentice et
al. 2018; Margutti et al. 2019).

A second key process that allows the CEJSN, the first one being the formation
of an accretion disk around the NS/BH, is the efficient neutrino-cooling of
the accreted mass when the accretion rate onto a NS is $\dot{M}_{\rm
acc}\gtrsim 10^{-3}M_{\odot}{~{}\rm yr}^{-1}$ (Houck & Chevalier, 1991;
Chevalier, 1993, 2012). In the case of a BH the accreted gas can in addition
carry a large fraction of the energy into the BH (e.g., Popham et al. 1999).
Namely, for accreting NS/BHs in a CEE the Eddington accretion limit is not
relevant and the NS/BH might accrete mass at a very high rate, i.e., up to a
tenth of a solar mass per second. The high accretion rate implies that the
accretion disk might launch energetic jets that in turn host high-energy
processes, such as r-process nucleosynthesis (Papish et al., 2015; Grichener &
Soker, 2019a, b) and the formation of high-energy ($\approx 10^{15}{~{}\rm
eV}$) neutrinos when the accretor is a BH (Grichener & Soker, 2021). As well,
the jets that the NS/BH launches unbind large amounts of envelope mass and can
supply extra energy source to make the CEE efficiency parameter to be
$\alpha_{\rm CE}>1$, as some scenarios require (e.g. Fragos et al. 2019; Zevin
et al. 2021; García et al. 2021).

The increasing number of sky surveys is observing more and more peculiar SNe
(e.g., Gal-Yam 2012, 2019; Graham et al. 2019; Holoien et al. 2019). The goal
of the present study is to further explore evolutionary routes that might lead
to peculiar super-energetic supernovae (SNe) that are not pair instability
SNe. Being super-energetic, i.e., with explosion energy of $E_{\rm
exp}>3\times 10^{51}{~{}\rm erg}$, these cannot be driven by the commonly
studied delayed neutrino mechanism. Rather, these SNe must be driven by jets.
The large fraction of massive stars in triple (or more) star systems (e.g.,
Sana et al. 2014; Moe & Di Stefano 2017) motivate this study of CEJSNe in
triple-star systems. One such evolutionary route, the double CEJSN, is the
subject of the present study.

I describe the double CEJSN scenario in section 2, and quantify some of its
parameters in section 3. In section 4 I estimate the rate of double CEJSN
event relative to CCSN events. I summarise in section 5.

##  2 The double CEJSN scenario

In this section I discuss the different evolutionary phases of the double
CEJSN event that I propose, which I schematic present in Fig. 1. In the double
CEJSN scenario there are four types of CEE. (1) The tight binary system enters
the RSG envelope. (2) The NS enters the envelope of its MS companion, and this
systems continues to spiral-in inside the RSG envelope. The system experiences
a CEE within a CEE. (3) The NS/BH remnant spirals-in inside the RSG envelope.
(4) The NS/BH remnant enters the core of the RSG for the final CEE (this
occurs if the NS/BH remnant does not remove the entire envelope before it
reaches the core).

Figure 1: A schematic diagram of the proposed double CEJSN scenario. The boxes
on the left list the duration of each phase and for the last four phases the
energy that the two jets that the NS launches carry and the mass that the NS
accretes, respectively. In the fourth phase the NS might collapse to a BH, and
much more likely so in the last phase. P1-P6 stand for the six phases.
Abbreviation: BH: black hole; CCSN: core-collapse supernova; CEE: common
envelope evolution; MS: main sequence; NS: neutron star; RSG: Red supergiant.

Phase 1: The initial triple-star system. The initial state is of a triple-star
system of three main sequence (MS) stars, at least two of them massive (CCSN
progenitors). The most massive of them, of mass $M_{\rm H,0}$, is in the tight
(inner) binary system of this triple system, and the second most massive star,
of mass $M_{\rm W,0}$, is on a wide orbit with the tight binary system. The
third star of initial mass $M_{\rm L,0}$ might be less or more massive that
$8M_{\odot}$, but for most cases I study here where the most massive star
forms a NS, it is less massive than $8M_{\odot}$. I estimate the requirement
on the initial semi-major axes to be $a_{\rm tb}\lesssim 0.5{~{}\rm AU}$ for
the tight binary system, and $a_{\rm wide}\approx 3-10AU$ for the wide one
(larger eccentricities allow higher values of $a_{\rm wide}$). This system
seems to be stable (e.g., Aarseth & Mardling 2001). This phase is marked by P1
in Fig. 1. I estimate these values from the following considerations. I
require the wide semi-major axis to be sufficiently small for the wide star to
engulf the tight binary system during its RSG phase. I estimate this
constraint to be $a_{\rm wide}\lesssim 10AU$. I also require the tight binary
system to form a NS-MS binary with a short orbital period so that it later
enters a NS-MS CEE. For that, the tight binary system cannot be too wide such
that it avoids both a CEE and a strong tidal interaction. I estimate this
constraint to be $a_{\rm tb}\lesssim 0.5{~{}\rm AU}$. For the system to be
stable, the wide star cannot be too close, namely, $a_{\rm wide}\gtrsim{\rm
several}\times a_{\rm tb}$ (depending on the eccentricities and on the mutual
inclination of the two orbital planes). This sets the lower limit on the side
semi-major axis.

Phase 2: The formation of the NS (BH). The post-MS evolution brings the most
massive star, of initial mass $M_{\rm H,0}\gtrsim 10M_{\odot}$, to explode as
a CCSN after an evolutionary time of $\Delta t_{1}\simeq 10^{7}{~{}\rm yr}$,
and to form either a NS or a BH. Before explosion it transfers mass to the two
other stars so that it explodes as a stripped-envelope CCSN (Type Ib or Ic)
and ejects little mass, so that the tight binary system stays bound. For
example, the lowest-mass star enters the envelope of the most massive star as
it becomes a RSG, accretes some of the envelope mass, $\approx 0.1M_{\rm
H,0}-0.5M_{\rm H,0}$, and ejects the rest of the envelope during the CEE. The
wide-orbit star might accrete a large fraction of this ejected mass, crudely
up to $10\%-50\%$ (or even more) of the ejected mass. Namely, the wide star
might crudely accrete $\approx 1M_{\odot}-0.4M_{\rm H,0}$. After this phase
the mass of the wide-orbit star is $M_{\rm W}>10M_{\odot}$. As we see below
(section 3.1), more favourable to the present scenario would be $M_{\rm
W}\gtrsim 20M_{\odot}$ as a more massive wider companion is more likely to
bring the NS into its core as it become a RSG star. At the end of the first
CCSN the triple-star system is of a tight binary system of a MS star and a
NS/BH, and a wide-orbit massive star (second line in Fig. 1 marked by P2). In
the present study I concentrate on a NS, but a similar scenario with more
massive stars might hold for the formation of a BH in this first CCSN. The
tight binary system is similar to the progenitors of low and intermediate
X-ray binaries (e.g., Podsiadlowski et al. 2002; Tauris & van den Heuvel
2006), but there is no need for a mass transfer from the MS star to the NS
star.

To have the double CEJSN evolution this phase must end with the wide companion
more massive than the MS star of the tight binary system, $M_{\rm W}>M_{\rm
L}$, so that the wide companion becomes a giant next. For this condition to
hold, the initial mass of the wide companion should be substantially larger
than that of the lowest star mass star because the lowest mass star might
accrete more mass from the most massive star as it evolves first to form the
NS. In some case where this phase ends with $M_{\rm W}<M_{\rm L}$, the double
CEJSN will not take place.

Phase 3: The beginning of the CEJSN event with a triple-CEE. The next phase,
at about $\Delta t_{2}\simeq 10^{6}{~{}\rm yr}$ to ${\rm few}\times
10^{6}{~{}\rm yr}$ after the NS/BH formation, starts when the wide-orbit star
expands to become a RSG star and engulfs the tight binary system (third line
in Fig. 1 marked by P3). At early phases in the double-CEJSN event the NS
accretes mass from the RSG envelope and launches jets. The powering is as in
regular CEJSN (e.g., Soker et al. 2019). If the orbital plane of the tight
binary is inclined to the triple-orbital plane the jets are inclined as well,
with consequences on the outflow morphology that forms a CSM that lacks any
symmetry (Schreier et al., 2019). The accretion rate is also different. The
more massive tight binary mass, several$\times M_{\odot}$ compared with
$M_{\rm NS}\simeq 1.4M_{\odot}$ in the regular CEJSN, attracts more mass to
the binary vicinity (for the simulations of the accretion process by a binary
star from an ambient medium see, e.g., Comerford et al. 2019). However, the
more massive MS star might accrete most of this mass.

Let me elaborate on this. de Vries et al. (2014) find that when a wide
tertiary star overflows its Roche lobe and transfers mass to a tight binary
system, the tight binary system ejects most of the mass, and accretes only
about fifth of the mass that the tertiary star transfers. The process that I
study here involves a NS-MS tight binary that accretes mass inside the
envelope rather than a Roche lobe overflow. It is not clear what the dynamical
effect of the binary system is on the accretion rate. In any case, I expect
that the main reduction in the mass accretion rate results from the removal of
envelope mass from the tight binary vicinity by the jets it launches. On the
other hand, to have the jets some non-negligible accretion must take place (a
negative feedback cycle). Future three-dimensional hydrodynamical simulations
of this triple-star CEE will teach us on the accretion rate of each of the
stars in the tight binary system.

Phase 4: The CEJSN inside a CEJSN phase. When a tight binary system enters a
CEE with a giant star there are several classes of outcomes (Sabach & Soker
2015; Comerford & Izzard 2020; Glanz & Perets 2021). The tight binary might
survive as it spirals-in, the two stars of the tight binary system might merge
inside the envelope (e.g., Hillel et al. 2017), the two stars might enter the
core, or the tight binary system might break up. The last class leads to
several sub-classes according to the fate of the two stars of the tight binary
system, e.g., whether one of the stars leaves the system or whether one or two
of the stars merge with the core. Relevant to the present study is the
following merger evolution (fourth line in Fig. 1 marked by P4). Due to the
expansion of the MS star as it accretes mass (because it has a radiative
envelope) and/or the gravitational friction inside the RSG envelope, the NS
and the MS star merge to start a CEE. Now the NS accretes mass from the MS
star, namely, from a much denser environment than that of the RSG envelope
(section 3). This results in the launching of very energetic jets that lead to
a peak in the light curve. This process ends with the formation of an
accretion disk around the NS from the destroyed MS star. The lifetime of the
accretion disk is about the viscosity time of the disk which itself is about
tens times the dynamical time of the MS star. This entire phase lasts for a
time period of (very crudely) $\Delta t_{4}\approx 1~{}{\rm day}-1~{}{\rm
week}$. Because the total energy of this phase is similar to that of a CEJSN
where a NS/BH enters the core of a RSG (section 3), I also term this phase a
CEJSN. This merger is different than that of a NS/BH merger with a star
outside a RSG envelope, as the massive envelope and CSM ensure a long-duration
peak in the light curve that is similar to a CCSN. A NS/BH merger with a MS
star in an optically thin medium will lead to a bluer and shorter event.

Phase 5: Binary-remnant evolution. Now the system is of a NS (or a BH) and a
RSG in a CEE, as in regular CEJSNe (fifth line in Fig. 1 marked by P5). The
NS/BH might eject the entire envelope and leave behind a stable core-NS/BH
binary system, where later the core explodes as a CCSN and the system ends as
two NSs (or a NS with a BH), bound or unbound.

Phase 6: NS-core merger. The other possibility to end the CEE is that the
NS/BH enters the core and accretes mass at a very high rate, leading to a very
bright CEJSN peak. In the later case the jets might serve as a r-process site
(Papish et al., 2015; Grichener & Soker, 2019a, b).

Observations will classify a double-CEJSN event as peculiar super-energetic
SN. It is peculiar due to the very long duration, months to few years, very
massive ejecta, up to tens of $M_{\odot}$, very high energy of $\approx
10^{52}-10^{53}{~{}\rm erg}$, and likely (but not necessarily) with two main
peaks in the light curve.

##  3 Timescales and powering

In this section I emphasise the powering role of jets. In addition there is
the gravitational energy that the system releases in the CEE processes. A
large fraction of the orbital energy goes to unbind the envelopes, e.g., the
recent 3D numerical simulation of a NS-RSG CEE by Law-Smith et al. (2021) and
Schrøder et al. (2020). As for the energy that accretion liberates via jets,
which also depends on the response of the NS to accretion (Holgado et al.,
2021), it can be about comparable to or much larger than the orbital energy
that the in-spiralling system release. I schematically (not to scale) present
the expected light curve in Fig. 2.

Figure 2:  A schematic drawing (not to scaled) of the expected light curve of
the double CEJSN event. The first four phases are the last four phases of Fig.
1. If the NS does not enter the core, the core explodes as a CCSN. The light
curve beyond the phase of a NS inside the RSG envelope depends in that case
(not shown) on the time from the ejection of the envelope to the explosion,
i.e., the properties of the CSM at the explosion time. The total energy,
kinetic and radiation, that the jets carry in their interaction with the
envelope and CSM that leads to each of the two large peaks in the light curve
is $\approx 10^{52}{~{}\rm erg}$.

The CEE proceeds in several phases (e.g., Ivanova & Nandez 2016 for a recent
discussion). Relevant to the present study are the plunge-in rapid phase and
the much slower self-regulating spiral-in phase. In the plunge-in phase the
compact star reaches deep into the envelope of the large star during about the
Keplerian orbital time on the surface of the giant star (e.g., Terman et al.
1995; Taam & Ricker 2010; Passy et al. 2012; Ricker & Taam 2012; Ivanova &
Nandez 2016; Ohlmann et al. 2016; Shiber et al. 2019; Chamandy et al. 2020;
Kramer et al. 2020; Reichardt et al. 2020; Sand et al. 2020; Glanz & Perets
2021). The self-regulating phase lasts for a longer time. The same behavior of
CEE in binary systems applies for triple star systems, when observing the
distance between the center of mass of the tight binary system and the core of
the RSG (Glanz & Perets, 2021).

In what follows, the subscripts correspond to the phases P3-P6 in Fig. 1 that
I also described in section 2. For example, phase 4 is the CEE of the NS
inside the MS star.

###  3.1 Tight binary system inside the RSG envelope

During the plunge-in time the NS accretes mass and launches jets (P3 in Fig.
1). The accretion rate of a single NS in a RSG envelope is $\dot{M}_{\rm
acc}\approx 0.1-0.2\dot{M}_{\rm BHL}$, where $\dot{M}_{\rm BHL}$ is the Bondi-
Hoyle-Lyttleton mass accretion rate (e.g., López-Cámara et al. 2020). Due to
negative feedback from the jets (as they remove mass from the envelope), the
mass accretion rate in the loosely bound envelope is expected to be smaller
even. The total energy that the jet carry might be several times the binding
energy of the envelope (Soker et al. 2019; a large fraction of the energy of
the jets goes to accelerate the ejected envelope gas to velocities larger than
the escape velocity). For a RSG, the binding energy of the envelope above
radius $r$ depends on the RSG radius $R_{\rm RSG}$ and its mass. Typical
values for the binding energy of the envelope residing above $r\gtrsim
10R_{\odot}$ are $E_{\rm bind}(10R_{\odot})\approx 10^{47}-10^{49}{~{}\rm
erg}$, where the lower values are for RSG masses of $M_{\rm RSG}\approx
10M_{\odot}$ and radii of $R_{\rm RSG}\simeq 10^{3}R_{\odot}$ and the larger
values are for RSG of $M_{\rm RSG}\approx 30M_{\odot}$ and $R_{\rm RSG}\simeq
200R_{\odot}$ (e.g., Lohev et al. 2019).

If the two jets together carry a fraction $\eta\simeq 0.1$ of the accreted
mass at a velocity of $v_{j}\simeq 10^{5}{~{}\rm km}{~{}\rm s}^{-1}$, and
these jets carry a total energy of $q$ times the envelope binding energy (see
Soker et al. 2019), the total mass that the NS accretes in this phase is

$\displaystyle\begin{aligned} M_{\rm acc,3}\approx
10^{-3}\left(\frac{q}{10}\right)\left(\frac{\eta}{0.1}\right)^{-1}\left(\frac{E_{\rm
bind}}{10^{48}{~{}\rm erg}}\right)M_{\odot}.\end{aligned}$ (1)

I assume that during the plunge-in phase or shortly after it the tight binary
suffers a strong enough perturbation, by the expansion of the MS star as it
accretes mass and by gravitational friction, that they merge. Namely, the NS-
MS merger occurs at a time $\Delta t_{3}\approx\Delta t_{\rm plunge}$. This is
about the Keplerian orbital time on the surface of the RSG of radius $R_{\rm
RSG}$

$\displaystyle\begin{aligned} \Delta t_{3}\approx 0.1-0.2\left(\frac{R_{\rm
RSG}}{200R_{\odot}}\right)^{3/2}\left(\frac{M_{\rm
total}}{30M_{\odot}}\right)^{-1/2}{~{}\rm yr},\end{aligned}$ (2)

where $M_{\rm total}$ is the total mass of the triple system at the onset of
the CEE.

From equations (1) and (2) the typical accretion rate onto the NS in this
phase is $\dot{M}_{\rm acc,3}\approx 0.01M_{\odot}{~{}\rm yr}^{-1}$. This is
high enough to allow neutrino cooling (Houck & Chevalier, 1991). In the
relevant zones of $r\approx 10-100R_{\odot}$ the envelope density is $\approx
10^{-6}-10^{-4}{~{}\rm g}{~{}\rm cm}^{-3}$ and the Bondi-Hoyle-Lyttleton mass
accretion rate of a single NS is very crudely $\dot{M}_{\rm BHL}\approx
0.01-0.1M_{\odot}{~{}\rm yr}^{-1}$. López-Cámara et al. (2019) and López-
Cámara et al. (2020) find the NS to accrete at a rate of $\dot{M}_{\rm
acc}\approx 0.1-0.2\dot{M}_{\rm BHL}$, and so the NS can indeed accrete at the
required rate (here there is the complication of the presence of the MS close
companion).

Overall, the phase of tight binary inside the RSG in the double CEJSN scenario
lasts for months to few years, and the energy that the two jets carry is

$E_{\rm 2j,3}\approx 10^{48}-10^{50}{~{}\rm erg},$ (3)

with an expected large variations between different systems. During this phase
the NS accretes too little mass to become a BH. Note that, as mentioned above,
the binding energy of the envelope residing above $\simeq 10R_{\odot}$ is
$\approx 10^{47}{~{}\rm erg}$ for lower mass RSG progenitors of CCSNe, and up
to $\approx 10^{50}{~{}\rm erg}$ for the envelope residing above a radius of
$\simeq 0.1R_{\odot}$ of more massive CCSN progenitors (e.g., Lohev et al.
2019). Therefore, the orbital energy that the triple system releases during
this phase might be up to about few times the envelope binding energy (e.g.,
as in the NS-RSG simulation of Terman et al. 1995), and so it is typically
smaller than the energy the jets carry. In some case it will be smaller but
not by much, and in some other cases it might even be comparable and somewhat
larger than the energy the jets carry.

###  3.2 A NS-MS merger inside the RSG envelope

This phase is a new ingredient I introduce into the family of the CEJSN
scenario. Unlike giant stars, the MS star does not have the structure of a
tenuous envelope and a very dense core. The CEE brings the NS close to the
center of the MS, and due to its huge gravity the NS tidally destroys the MS
star. For a NS inside a MS star this takes place when the stellar mass inner
to the NS orbit is about equal to or less than the NS mass (because then the
tidal disruption radius becomes larger than the orbit), as in a micro-tidal
disruption event (micro-TDE) where a NS/BH disrupts a star (Perets et al.,
2016). Because of the large angular momentum of the binary system the
destroyed MS gas forms an accretion disk around the NS. The first part of the
spiralling-in lasts for about the Keplerian time on the surface of the MS star
$t_{\rm{Kep}}^{\rm MS}$, about few hours. The accretion of the rest of the
destroyed MS gas lasts for about the viscous timescale of the accretion disk.
In the $\alpha$ disk model where the viscosity is $\nu=\alpha_{\rm d}C_{s}H$,
where $C_{s}$ is the sound speed and $H(R)$ the disk height at radius $R$ from
the NS, the viscous timescale is given by

$t_{\rm{visc}}^{\rm MS}=\frac{R^{2}}{\nu}=\frac{R^{2}}{\alpha C_{s}H}\simeq
t_{\rm{Kep}}^{\rm MS}\left(\frac{R}{H}\right)^{2}\frac{1}{2\pi\alpha_{\rm
d}}.$ (4)

The large uncertainty here is in the ratio $R/H$. In a thin accretion disk the
typical value is $(R/H)\simeq 10$. However, here the destroyed MS starts as a
sphere, and it takes time for the disk to collapse to the equatorial plane.
Another unknown parameter is the viscosity coefficient $\alpha_{\rm d}$. It is
likely that over several dynamical timescales the outer edge of the disk
expands, increasing the dynamical time of the outer parts of the disk, and the
disk becomes thinner, increasing the value of $R/H(R)$. Both these effects
increase the viscous timescale. Overall, this accretion phase decays over a
timescale of tens to about a hundred times the Keplerian orbital time of the
MS star, or about a day to a week

$\Delta t_{4}\approx t_{\rm{visc}}^{\rm MS}\approx
1\left(\frac{t_{\rm{Kep}}^{\rm MS}}{2{\rm
h}}\right)\left(\frac{R}{3H(R)}\right)^{2}\frac{1}{2\pi\alpha_{\rm d}}{\rm
day}.$ (5)

The fraction of the MS mass that the NS accretes depends on how efficient the
polar jets are in removing the mass. The NS is likely to accrete up to about
half the mass, or $M_{\rm acc,4}\approx 0.3-3M_{\odot}$. This might bring the
NS to collapse to a BH. Over all, the accretion rate and the total energy that
the two jets carry are

$\displaystyle\begin{aligned} &\dot{M}_{\rm acc,4}\approx
10-1000M_{\odot}{~{}\rm yr}^{-1}.\\\ &E_{\rm 2j,4}\approx
10^{52}\left(\frac{\eta}{0.1}\right)\left(\frac{M_{\rm
acc}}{1M_{\odot}}\right){~{}\rm erg},\end{aligned}$ (6)

respectively.

The jets will eject most of the RSG envelope outer to the orbit of the NS or
the BH remnant, but some equatorial gas might stay bound. The outflow will be
very fast, but it will collide with the CSM that the system ejected in the
previous phase. The collision of the ejecta with the CSM will transfer a large
fraction of the kinetic energy to radiation (e.g., Schrøder et al. 2020),
forming a super-luminous event, even if the NS accretes only $0.1M_{\odot}$.
The peak in the light curve will last for months to a year due to the long
photon diffusion time in the massive CSM (Fig. 2).

Some of the MS gas might end as bound gas, i.e., it becomes part of the common
envelope (RSG envelope).

If the NS-MS merger event ejects most of the envelope, such that the NS or BH
remnant does not spiral-in to the core, the event ends here, until much later
(more than hundreds of years) the core explodes as a CCSN.

###  3.3 CEE after the destruction the MS star

Now the system proceeds as a CEJSN of a binary star, a NS/BH and an RSG. For
example, if the remnant is a black hole the relativistic jets it launches can
be a strong source of $\approx 10^{15}{~{}\rm eV}$ neutrinos (Grichener &
Soker, 2021).

In the double CEE event the NS-MS merger process and its jets do not remove
the entire envelope, and the RSG envelope that is still bound is massive
enough to cause the NS/BH remnant to spiral-in to the core. As said, part of
the MS mass might mix with the original RSG envelope. The NS/BH remnant starts
this phase very close to the core, orbital separation much smaller than the
initial radius of the RSG star. However, the envelope might be synchronised
with the orbital motion and the spiralling-in phase might be many times the
Keplerian orbital period. It is very hard to estimate the duration of this
phase, the accretion rate, and the total power. But due to the lower envelope
mass, the jets that the NS/BH launch in this phase do not add much to the
energy of the event. The main energy source of this phase is the orbital
energy $E_{\rm orb}$ of the core-NS/BH binary systems that spirals-in to
merge.

The duration of this phase and the energy of the jets are highly uncertain. To
give some plausible values I assume the following. (1) I assume that when the
mass of the reminding envelope is large so is the dynamical friction and the
spiralling-in time to the core, i.e., of the order of the orbital period where
the previous phase (NS-MS CEE) took place $a_{4}$, about two weeks for
$a_{4}\simeq 50R_{\odot}$. If the reminding envelope is of low mass, the
evolution might be much longer and take years. For example, if the envelope
shrinks to be less that the orbital separation $a_{4}$, then the evolution is
due to tidal interaction that can take tens of orbital periods. (2) I assume
that the jets operate in a negative feedback cycle. This implies that if the
jets are too strong they remove large parts of the envelope and accretion
decreases. In addition, in the spiralling-in towards the core the NS releases
a large amount of energy that by itself can unbind most of the envelope and
leads to the decreasing of the accretion rate. I assume therefore, that at
this phase the released orbital energy plays a larger role that the jets,
i.e., releases more energy that the jets carry. Very crudely I summarise the
duration of this phase and the jets power as

$\displaystyle\begin{aligned} &\Delta t_{5}\approx{\rm weeks-years}\\\ &E_{\rm
2j,5}\ll E_{\rm 2j,4}\qquad{\rm and}\\\ &E_{\rm 2j,5}\ll E_{\rm orb}\simeq
7\times 10^{50}\left(\frac{M_{\rm core}}{5M_{\odot}}\right)\left(\frac{M_{\rm
NS}}{1.4M_{\odot}}\right)\\\
&\qquad\qquad\qquad\times\left(\frac{a_{5}}{0.02R_{\odot}}\right)^{-1}{~{}\rm
erg},\end{aligned}$ (7)

where $M_{\rm core}$ and $a_{5}$ are the core mass and final orbital
separation at this phase before the final spiralling-in into the core starts.

###  3.4 NS/BH inside the core

When the NS/BH enters the core we are in the regular process of a CEJSN. It is
hard to estimate the accretion rate of this process due to the complicated
gravitational interaction and the role of jets. Unlike the case of NS star
merging with a WD, here the core is more massive than the NS. For that, the
Bondi-Hoyle-Lyttleton mass accretion rate might be a crude estimate at the
early times of the NS-core merger (but not for a BH that is more massive).
Grichener & Soker (2019a) find this rate to be $\dot{M}_{\rm BHL}\approx
0.01-0.1M_{\odot}{~{}\rm s}^{-1}$. A BH will tidally destroy the core in at
early times, a process similar to a micro-TDE event where a NS/BH merges with
a star (Perets et al., 2016). I estimate the mass accretion as I did in
equation (5), but now the Keplerian orbital time is that of a core, e.g.,
$t_{\rm{Kep}}^{\rm core}\simeq 100{~{}\rm s}$. According to Grichener & Soker
(2019a) the highest value of $\dot{M}_{\rm BHL}$ is when the NS is at $a\simeq
0.01-0.03R_{\odot}$ from the center of the core and the inner core mass is
$\approx 1M_{\odot}$. I take this last phase when the inner mass is about that
value, and so the Keplerian time of this inner core is (for a NS of mass
$1.4M_{\odot}$) $t_{\rm{Kep}}^{\rm in}\simeq 5-30{~{}\rm s}$. The viscous
timescale, which I also take to be the duration of the highest accretion rate
phase from the core, is then

$\Delta t_{6}\approx t_{\rm{visc}}\simeq 100\left(\frac{t_{\rm{Kep}}^{\rm
in}}{10{~{}\rm
s}}\right)\left(\frac{R}{3H(R)}\right)^{2}\frac{1}{2\pi\alpha_{\rm d}}{~{}\rm
s}.$ (8)

The NS/BH accretes $\approx 1M_{\odot}$ (Grichener & Soker, 2019a), which is
most of the core (as the jets eject only the polar regions). Before that
phase, for about hundreds to few thousands seconds the NS/BH might accrete
mass from the outer parts of the envelope. As well, it might accrete fallback
gas at later times. However, the highest accretion rate lasts for about 100
seconds and the accretion rate and total energy that the two jets carry are

$\displaystyle\begin{aligned} &\dot{M}_{\rm acc,6}\approx 0.01M_{\odot}{~{}\rm
s}^{-1},\\\ &E_{\rm 2j,6}\approx
10^{52}\left(\frac{\eta}{0.1}\right)\left(\frac{M_{\rm
acc}}{1M_{\odot}}\right){~{}\rm erg},\end{aligned}$ (9)

respectively.

The large energy of $E_{\rm 2j,6}\approx 10^{52}{~{}\rm erg}$ forms a second
energetic peak in the light curve of the long-lasting event. Due to the high
opacity of the newly ejected core gas and the previously ejected MS gas and
RSG envelope gas, the optical depth of the entire system is large, and the
second peak might last for several years. If the jets break through the polar
directions, on the other hand, then the second peak might be much shorter than
a year, similarly to the polar-CEJSN scenario that Soker et al. (2019)
proposed for the fast-rising blue optical transient AT2018cow.

The accretion rate $\dot{M}_{\rm acc,6}$ is the average value over about 100
seconds. It is about 0.1 times the highest value of $\dot{M}_{\rm BHL}$ that
Grichener & Soker (2019a) find, which is expected as the largest values is
only in a specific radius inside the core. I find that the two estimates of
the mass accretion rate inside the core are compatible. Moreover, these values
are compatible with the results of Zhang & Fryer (2001) when I scale to a
higher core-density than what they use (as they take a helium core and not a
CO core as I assume here). The average accretion rate of $\approx
0.01M_{\odot}$ is above the threshold of $0.002M_{\odot}{~{}\rm s}^{-1}$ for
r-process nucleosynthesis according to Siegel et al. (2019). This process
might be a source of r-process nucleosynthesis (Grichener & Soker, 2019a).

The above discussion was for a RSG core after core helium burning. In case of
a helium core, the density is lower and the mass accretion rate is lower
(e.g., Siegel et al. 2019), and it is below the threshold for the r-process
nucleosynthesis. The total energy that the jet carry, thought, might be the
same.

##  4 Event rate

Schrøder et al. (2020) performed population synthesis calculations and found
that for RSG stars of mass $\gtrsim 10M_{\odot}$ the frequency of NS/BH
entering a RSG envelope is about $1.5\times 10^{-4}M^{-1}_{\odot}$, i.e., rate
per solar mass of stars formed. This rate is $2.6\%$ of the CCSN rate
(Chevalier 2012 estimate this rate to be about $1\%$). For the CCSN rate in
the local Universe (e.g., Strolger et al. 2015) this rate of NS/BH entring RSG
envelope is $\simeq 2\times 10^{-6}{~{}\rm yr}^{-1}{~{}\rm Mpc}^{-1}$. From
these, Schrøder et al. (2020) find that in about $22\%$ the NS/BH enters the
RSG core, i.e., a CEJSN event. In the present nomenclature the rest, about
$78\%$ are CEJSN impostors that form NS/BH binaries that later might merge as
a gravitational waves source. Schrøder et al. (2020) find that in about half
of all CEJSNe the compact object is a NS and in the other half it is a BH. In
total, the CEJSN rate they find is $0.6\%$ of the rate of CCSNe. Grichener &
Soker (2019a) use the r-process CEJSN scenario to argue that the rate of NS
entering a CO core of a RSG should be about $0.1\%$ of the CCSN rate. Other
cases where a NS enters the RSG core before helium burning do not lead to
r-process. These different estimates are compatible. I take the CEJSN rate to
be $f_{\rm CEJ}\simeq 0.006$ times the CCSN rate, and CEJSN cases with post-
helium burning RSG core to be about $\simeq 0.3-0.5$ of those cases, namely a
ratio of $f_{\rm CEJ,He}\simeq 0.002$ to CCSN rate.

Let me now estimate the number of cases where instead of being a single NS/BH,
the NS/BH is in a tight binary system with a MS star. I assume that this
fraction is about equal to the fraction of binary systems where a NS/BH has a
MS companion. I base this assumption on that the evolution of the most massive
star in the triple did not eject its close MS companion, and so most likely it
did not eject the wider star. As well, I assume that the fraction of NS-MS
close binary systems does not depend on the presence of a wide companion.
Pfahl et al. (2003) estimate the birthrate of low- and intermediate (MS mass
of $<4M_{\odot}$) X-ray binaries to be $\approx 10^{-4}-10^{-2}$ of the CCSN
birthrate. Shao & Li (2015) estimate the birth rate of all X-ray binaries to
be a fraction of $\approx 0.01$ of CCSN birthrate, most of them with a MS star
of $M>5M_{\odot}$. Overall, I take that, very crudely, $f_{\rm tb}\approx
0.003$ of all NS form the tight binary systems that I require in this study
(the range can be $f_{\rm tb}\approx 10^{-3}-0.01$). The fraction of NS in
low-mass X-ray binaries can be as low as $f_{\rm tb}({\rm low})\approx 0.001$,
but the fraction of high mass X-ray binaries can be two orders of magnitude
higher $f_{\rm tb}({\rm high})\approx 0.1$ (Tauris & van den Heuvel, 2021).
Shao & Li (2020) claim that X-ray binaries with a BH companion favour low mass
BHs. The formation rate is less certain, and I take it to be the same as that
of X-ray binaries with a NS companion.

My crude estimated ratio of double CEJSN event rate with a NS or BH companion
to CCSNe rate is

$f\approx f_{\rm tb}f_{\rm CEJ}\approx 6\times 10^{-6}-6\times 10^{-5}.$ (10)

If one to take one value, I crudely estimate $f\approx 2\times 10^{-5}$. About
half of these are with a NS star, what I term here double CEJSN, i.e., $f_{\rm
DCEJ}\approx 10^{-5}$. Half of these cases are with a BH, what I term a micro-
TDE CEJSN, i.e., $f_{\rm BH,CEJ}\approx 10^{-5}$. The rate of double CEJSN
events with a NS companion and a CO core of the RSG is $f_{\rm
DCEJ}{\rm(CO)}\approx 3\times 10^{-6}$.

With new and future transient surveys, e.g., All-Sky Automated Survey for
Supernovae (ASAS-SN; Kochanek et al. 2017), The Zwicky Transient Facility
(ZTF; Bellm et al. 2019), and the Southern Hemisphere Variability Survey (LSQ;
Baltay et al. 2013), which might detect about $10^{4}$ CCSNe per year, I very
crudely expect the detection of one event per year that is a double CEJSN or a
micro-TDE. The reason that the ratio of $\approx 10^{-4}$ to CCSN detection
rate is larger than the value given by equation (10) is that the double CEJSN
events are much brighter than typical CCSN events, and therefore the volume
coverage is much larger.

##  5 Summary

I presented the basic phases, including timescales, energy, and NS mass
accretion, of the double CEJSN in Fig. 1. In equations (1)-(9) I gave my crude
estimates of the timescales and jet-powers of the four CEE phases when the NS
(or BH) launch jets. In proposing this scenario I made the following main
assumptions. (1) I assumed that when a tight NS-MS binary system enters a RSG
envelope the interaction with the RSG envelope can bring the NS-MS binary
system to merge. One process that might prevent the merger is if the jets that
the NS launches remove too much gas from the NS-MS binary vicinity and the
interaction of the NS-MS binary with the envelope is not sufficiently strong
to bring the NS and MS star to merge. Namely, the NS-MS binary system performs
the grazing envelope evolution rather than the CEE (e.g., Shiber & Soker
2018). (2) I assumed that the NS-MS merger process ends with a large fraction
of the MS mass forming an accretion disk around the NS, and that the NS
accretes a large fraction of the disk mass. Here, again, the early jets might
efficiently remove most of the MS stellar mass such that the total accreted
mass is very low, i.e., $M_{\rm acc,4}\lesssim 10^{-2}M_{\odot}$.

For these uncertainties, all these estimates require better determinations by
future studies, mainly of complicated and highly demanding 3D hydrodynamical
simulations.

In section 4 I crudely estimated the rate of the double CEJSN events to the
rate of CCSNe. About 1 double CEJSN event takes place per $10^{5}$ CCSNe.
Since CEJSNe are much brighter than typical CCSNe, their detection can cover a
larger volume and the detection rate, including also the micro-TDE CEJSN
events where there is a BH instead of a NS might be $1:10^{4}$. This implies
that new transient surveys might detect in total one triple-star CEJSN event
per year.

The double CEJSN event is one of several different types of triple-star CEJSN
events (cases). I end by calling the attention of future discoverer of
peculiar super-energetic SNe to the CEJSN scenario, in binaries and in
triples.

## Acknowledgments

I thank Aldana Grichener for helpful discussions and comments. I thank an
anonymous referee for detailed comments and helpful suggestions. This research
was supported by the Amnon Pazy Research Foundation.

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

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