# Mergers of neutron stars and black holes with cores of giant stars: a
population synthesis study

Aldana Grichener  Department of Physics, Technion, Haifa, 3200003, Israel;
<EMAIL_ADDRESS>
###### Abstract

We perform population synthesis of massive binaries to study the mergers of
neutron stars (NSs) and black holes (BHs) with the cores of their giant
secondaries during common envelope evolution (CEE). We use different values of
the efficiency parameter $\alpha_{\rm CE}$ in the framework of the energy
formalism for traditional CEE ($\alpha_{\rm CE}\leq 1$) and including
additional energy sources to unbind the envelope ($\alpha_{\rm CE}>1$). We
constrain the possible values of $\alpha_{\rm CE}$ by comparing the results of
our simulations with local rate densities of binary compact object mergers as
inferred from gravitational waves observations. We find two primary
evolutionary pathways of binary systems that result in NS-core mergers, while
only one of them can also lead to the merger of a BH with the core of the
giant star. We explore the zero age main sequence (ZAMS) statistical
properties of systems that result in NS/BH-core mergers and find that the two
evolutionary channels correspond to a bimodal distribution of orbital
separations. We estimate the percentage of the mergers’ event rates relative
to core collapse supernovae (CCSNe). We include the effect of mass accreted by
the NS/BH during CEE in a separate set of simulations and find it does not
affect the mergers’ event rates.

binaries: general – stars: neutron stars – stars: black holes – stars: massive
–methods: numerical

##  1 INTRODUCTION

Most massive stars are in close multiple systems consisting of two or more
stars in orbit around their common center of mass. The immense swelling of one
or both stars in a binary system at late evolutionary phases might lead to a
substantial decrease in the orbital separation of the system to the point
where mass transfer takes place. If the mass transfer becomes unstable then
the system might enter a common envelope evolution (CEE) phase (e.g.,
Paczynski 1976; Iben & Livio 1993; Taam & Sandquist 2000; Izzard et al. 2012;
Ivanova et al. 2013; Roepke & De Marco 2022) in which the envelopes of both
stars merge. In cases where at the onset of CEE the stars in the binary are a
neutron star (NS) or a black hole (BH) and a massive giant, the compact object
111 Throughout the manuscript ”compact object” refers only to a NS or a BH. is
immersed within the envelope of the giant star and spirals closer to its core.
This can lead to either the ejection of the envelope and a surviving compact
object-core pair or to the merger of the core with the NS/BH. Another proposed
alternative is the formation of a stable Thorne-Zytkow object (Thorne &
Zytkow, 1977). The fate of massive binaries is of significant interest due to
the various astrophysical phenomena it engenders.

Mergers of NSs/BHs with cores of giant stars are a topic of ongoing research
in a variety of contexts (e.g., Fryer & Woosley 1998; Zhang & Fryer 2001;
Barkov & Komissarov 2011; Chevalier 2012; Schrøder et al. 2020; Metzger 2022;
Guarini et al. 2022). In particular, such mergers can lead to transient events
known as common envelope jet supernovae (CEJSNe; Soker & Gilkis 2018). In a
CEJSN event a NS/BH is engulfed by a giant star and spirals-in inside its
envelope. The compact object accretes mass via an accretion disk and launches
two opposite jets that propagate through the giant’s envelope and expel mass.
Eventually, the NS/BH reaches the dense core of the giant star and launches
more energetic jets as they merge. The effects of the jets that the compact
object launches on the envelope and on the core of the giant star are broadly
studied in one dimensional and three dimensional simulations (e.g., Moreno
Méndez et al. 2017; Moriya 2018; Gilkis et al. 2019; López-Cámara et al. 2019;
Grichener et al. 2021; Ragoler et al. 2022; Hillel et al. 2022; Schreier et
al. 2022).

Unfortunately, the high opacity of the giant’s envelope does not allow for
direct observations of core-NS/BH mergers with existing facilities. Moreover,
even though the merger of a compact object with the core of a giant star emits
gravitational waves (e.g., Ginat et al. 2020), their signal is much lower than
in the case of binary compact object mergers, and is undetectable at the
moment. A future detection by next-generation gravitational waves detectors
can serve as an observational signature of a merger event.

Meanwhile, we can study the effect that NS/BH-core mergers would have on their
surroundings. Due to the high energies of the jets in their correlated CEJSN
transient, NS/BH-core mergers might account for several astrophysical
phenomena, such as heavy r-process nucleosynthesis that occurs inside jets
that a NS launches as it merges with the core of a giant star (Papish et al.
2015; Grichener & Soker 2019a; Grichener et al. 2022; Grichener & Soker 2022)
and high energy neutrinos emission from jets that a BH launches while
spiraling-in inside the envelope of a giant prior to the merger with its core
(Grichener & Soker, 2021). Moreover, some peculiar transient events such as
fast blue optical transients and other rare supernovae (SNe) might hint to
NS/BH-core mergers and the CEJSN mechanism as well (e.g., Thöne et al. 2011;
Soker & Gilkis 2018; Soker et al. 2019; Dong et al. 2021; Soker 2022a; Soker
2022b). The recently proposed hypernebula transient that is powered by jets
accompanying hyper-Eddington mass transfer from an evolved post-main sequence
(MS) star onto a NS/BH shortly before CEE (Sridhar & Metzger 2022; Sridhar et
al. 2022) might serve as a precursor of a NS/BH-core merger. Estimating the
mergers’ event rates is crucial to understand whether they can account for
said phenomena.

In this study we use the population synthesis code COMPAS (Team COMPAS: Riley
et al., 2022) to find the merger rates of NSs/BHs with cores of giant stars.
Many population synthesis studies of double compact object binaries formation
and merger were performed using COMPAS (e.g., Stevenson et al. 2017; Vigna-
Gómez et al. 2018; Neijssel et al. 2019; Vigna-Gómez et al. 2020; Broekgaarden
et al. 2021; Raveh et al. 2022). To our knowledge, the only study which
performed population synthesis of NS/BH-core mergers is Schrøder et al.
(2020). In the present work we include in the simulations the effects of
additional energy sources that are required to unbind the envelope and
estimate the merger rates in these scenarios. Moreover, we find different
evolutionary routes that lead to NS/BH-core mergers and study the structure of
the core during the merger event.

Our study is organized as follows. We begin by describing the initial
parameters and prescriptions we use in our population models (section 2). We
then present the main evolutionary channels towards NS/BH-core mergers, the
properties of binary systems that result in these mergers and their event
rates (section 3). We summarize our findings and discuss their relevance to
previous works in section 4.

##  2 Population Model

We use the population synthesis code Compact Object Mergers: Population,
Astrophysics and Statistics (COMPAS; Stevenson et al. 2017; Vigna-Gómez et al.
2018; Team COMPAS: Riley et al. 2022) to find the evolutionary pathways of
binary systems that result in the merger of a NS/BH and the core of a giant
star, explore their statistical properties and estimate the mergers’ event
rate.

COMPAS generates populations of isolated stellar binary systems and evolves
the stars in the binary using the analytical fits in Hurley et al. (2000)
based on the stellar models of Pols et al. (1998). We sample the initial
distribution of binary properties at the ZAMS (zero age main sequence)
according to the ”Fiducial model” of Vigna-Gómez et al. (2018) as described
below. Henceforth we refer to the initially heavier star as the primary star,
and the initially lighter star as the secondary star. We draw the mass of the
primary star from the Kroupa initial mass function (IMF) in the form
$dN/dM_{\rm 1,ZAMS}\propto M_{\rm 1,ZAMS}^{-2.3}$ (Kroupa, 2001) with masses
in the range $5\leq M_{\rm 1,ZAMS}/M_{\rm\odot}\leq 100$, and the mass of the
secondary star from a flat distribution in the mass ratio with $0.1\leq q_{\rm
ZAMS}\equiv M_{\rm 2,ZAMS}/M_{\rm 1,ZAMS}\leq 1$ (Sana et al., 2012). The
initial separations follow the flat-in-the-log distribution between $0.1\leq
a_{\rm ZAMS}/\rm AU\leq 1000$ (Sana et al., 2012). We take all of the orbits
to be circular at ZAMS (i.e., $e_{\rm ZAMS}=0$), and all the stars in our
sample have solar metalicity $Z=0.0142$ (e.g., Asplund et al. 2009).

In general, massive stars end their lives in SN explosions. COMPAS
differentiates between three SN scenarios: regular hydrogen-rich core collapse
supernovae (CCSNe), electron capture supernovae (ECSNe) and ultra stripped
supernovae (USSNe), according to the core and envelope masses prior to the
explosion. For NS remnants, we take a bimodal natal-kick velocity distribution
where CCSNe constitute the higher mode of $\sigma_{\rm high}=265\rm\;kms^{-1}$
and ECSNe together with USSNe contribute to the lower mode of $\sigma_{\rm
low}=30\rm\;kms^{-1}$. We draw the BH natal kicks from the same bimodal
distribution reduced according the fallback model of Fryer et al. (2012). We
set the maximum allowed NS mass to be $M_{\rm NS,mas}=2M_{\rm\odot}$, as
according to Özel et al. (2010) this value reproduces the observational mass
gap between NSs and BHs.

To determine the orbital separation between both stars in the binary system
after a CEE event, COMPAS uses the energy formalism in which the energy
difference between the orbital energies before and after the CEE phase is
compared with the binding energy of the envelope (van den Heuvel 1976; Webbink
1984; Livio & Soker 1988; Iben & Livio 1993; see Ivanova et al. 2013 for a
review). In the case of a NS/BH that is swallowed by a giant star

$\begin{split}E_{\rm bind}=\frac{\alpha_{\rm CE}GM_{\rm giant,pre-CE}M_{\rm
NS/BH}}{2a_{\rm pre-CE}}\\\ -\frac{\alpha_{\rm CE}GM_{\rm giant,post-CE}M_{\rm
NS/BH}}{2a_{\rm post-CE}}\end{split}$ (1)

where $\alpha_{\rm CE}$ is the common envelope efficiency parameter, $M_{\rm
NS/BH}$ is the mass of the compact object, $M_{\rm giant,pre-CE}$ and $M_{\rm
giant,post-CE}$ are the masses of the giant star before and after the CEE
phase respectively, and $a_{\rm pre-CE}$ and $a_{\rm post-CE}$ are the orbital
separations before and after CEE respectively. The binding energy is
calculated using the lambda formalism of de Kool (1990) as implemented by Xu &
Li (2010). To find whether the NS/BH merges with the giant star’s core, we
compare the orbital separation of the systems after the CEE phase with the
radius of the core, which is estimated using approximate analytical relations
between the mass and radius of the core in different evolutionary phases from
Hall & Tout (2014). If the orbital separation is smaller than the radius of
the core, we assume that a NS/BH-core merger has occurred.

Traditionally, the maximal value of the common envelope efficiency parameter
$\alpha_{\rm CE}$ is one, representing a case where the entire difference
between orbital energies goes to unbind the envelope of the giant star and is
precisely sufficient for this purpose. However, many hydrodynamical
simulations do not find full envelope ejection, in contradiction to
observations of post common envelope systems (e.g., Passy et al. 2012; Ricker
& Taam 2012; Kuruwita et al. 2016; Ohlmann et al. 2016; Iaconi et al. 2017;
Glanz & Perets 2021a; Glanz & Perets 2021b), implying that additional energy
sources are required to unbind the envelope. Several physical processes have
been suggested as possible mechanism for the ejection of the common envelope,
such as the core-companion system’s interaction with a circumbinary disk in
the final CEE stages (e.g., Kashi & Soker 2011; Kuruwita et al. 2016), jets
launched from the compact object while it is inside the envelope of the
secondary star (e.g., Sabach et al. 2017; Soker 2017), the recombination
energy of hydrogen and helium (e.g., Ivanova et al. 2015), envelope inflation
followed by long period pulsations (e.g., Clayton et al. 2017) and dust driven
winds (Glanz & Perets, 2018). In the energy formalism an additional energy
source can be represented by $\alpha_{\rm CE}>1$ regardless of its nature.

We perform simulations for $0.1\leq\alpha_{\rm CE}\leq 5$ to allow for both
traditional CEE and scenarios with additional energy sources. For each value
of $\alpha_{\rm CE}$, we evolve $10^{\rm 7}$ isolated binaries with the
initial parameter distribution described above.

##  3 Results

###  3.1 Constraining the values of $\alpha_{\rm CE}$

Due to the lack of ability to observe systems in the relatively short and low
luminosity CEE phase we cannot compare the results of our population synthesis
simulations to direct observations. However, since roughly $80-90\%$
progenitors of binary compact object mergers in our simulations go through the
same evolutionary stages as progenitors of NS/BH-core mergers until the CEE of
the compact objects with the giant star (Fig. 2), we find their merger rates
for different values of $\alpha_{\rm CE}$ and compare these rates with
observations to determine the possible values of the efficiency parameter.

We crudely estimate the local transient rate density of different events in
our COMPAS simulations using

$R_{\rm event}=f_{\rm event}\frac{n_{\rm SFR}f_{\rm pop}}{\left<M\right>},$
(2)

where $R_{\rm event}$ is the local transient rate density in $\rm
Gpc^{-3}~{}\rm{yr}^{-1}$, $f_{\rm event}=N_{\rm event}/N_{\rm total}$ is the
ratio between the number of systems that result in this type of event and the
total number of systems simulated in our COMPAS simulation, $n_{\rm SFR}\sim
10^{7}\rm M_{\rm\odot}Gpc^{-3}~{}\rm{yr}^{-1}$ is the local star formation
rate (Madau & Dickinson, 2014), $f_{\rm pop}$ is the fraction of the Kroupa
IMF (Kroupa, 2001) we simulate and $\left<M\right>$ is the average stellar
mass. Integrating over the Kroupa IMF between $M_{\rm min}=5M_{\rm\odot}$ and
$M_{\rm max}=100M_{\rm\odot}$ and dividing by the total mass we find that
$f_{\rm pop}\simeq 0.007$. The average stellar mass according to the Kroupa
IMF is $\left<M\right>\simeq 0.39M_{\rm\odot}$. Substituting all of the above
into equation (2) gives

$R_{\rm event}=1.79\times 10^{5}f_{\rm event}\rm\;Gpc^{-3}~{}\rm{yr}^{-1}.$
(3)

Figure 1: Local rate density of NS-NS mergers (blue dots), BH-BH mergers
(green dots) and NS-BH mergers (red dots) for different values of common
envelope efficiencies. The rectangular areas represent the error margin of
mergers observation taken from The LIGO Scientific Collaboration et al.
(2021).

In Fig 1 we present the local merger rate density of NS-NS binaries (upper
panel; blue dots), BH-BH binaries (middle panel; green dots) and NS-BH
binaries (lower panel; red dots) in our COMPAS simulations as computed by
using equation (3). The rectangular filled areas in each panel are the
possible observational ranges of these rates as inferred from gravitational
waves emission according to The LIGO Scientific Collaboration et al. (2021).
We conclude that the local merger rate densities are well within the
observational error margin for $\alpha_{\rm CE}\gtrsim 0.5$. This is in
accordance with previous studies which favour $\alpha_{\rm CE}\simeq 2$
compared to small $\alpha_{\rm CE}$ to explain the observed merger rates of
double compact object binaries (e.g., García et al. 2021; Zevin et al. 2021;
Broekgaarden & Berger 2021).

###  3.2 Evolutionary channels towards NS/BH-core mergers

Figure 2: A schematic illustration of the evolution of a massive binary
system. Two massive MS stars evolve towards a giant star-NS/BH CEE. The NS/BH
spirals-in inside the envelope of the giant star. The CEE can lead either to
the merger of the NS/BH with the core of the giant star potentially powering a
bright transient event, or to the formation of double compact objects binary
that might merge in the future by emission of gravitational waves. Left panel:
main evolution channel, where dynamically stable mass transfer from the post
MS primary star to its MS secondary companion takes place. Right Panel:
Secondary evolution channel in which a dynamically unstable mass transfer
leads to the formation of a common envelope that surrounds the secondary MS
star and the core of its post MS primary, reducing their orbital separation,
and allowing only for NS-core mergers.

The two main evolutionary channels of binary systems that result in the merger
of a NS/BH with the core of a giant star are presented in Fig. 2. Both
scenarios begin with two massive stars on the MS. The initially heavier star
$M_{\rm 1}$, to which we will refer as the primary star, evolves faster and
becomes a giant while the lighter secondary star $M_{\rm 2}$ keeps burning
hydrogen at its core.

In the main channel, denoted as channel I (left panel of Fig. 2), a stable
mass transfer episode from the giant primary to its MS secondary occurs at
this stage. The giant star eventually loses its envelope through mass transfer
and the remaining naked core keeps evolving until it explodes in an stripped-
envelope SN event in the same manner as Wolf-Rayet stars222We note that the
stripped core in our scenario is not necessarily a Wolf-Rayet star according
to the observational definition (Shenar et al., 2020). (e.g., Wheeler &
Levreault 1985; Woosley et al. 1995; Eldridge & Tout 2004; McClelland &
Eldridge 2016), leaving a NS/BH remnant, depending on the mass of the stripped
core. The natal kick might drive the compact object away from the SN location,
but a low enough kick allows for a bound NS/BH-MS binary system (e.g.,
Kochanek et al. 2019). The secondary star continues to evolve and when the
hydrogen in its core is depleted it enters the Hertzsprung Gap (HG) phase
where it expands until the ignition of helium in its core. From the HG phase
and onward, mass transfer occurs in the opposite direction, i.e. from the
secondary star to the primary. The expanding secondary becomes a giant and it
can engulf the NS/BH initiating a CEE phase where the compact object spirals-
in inside the envelope of the giant star. The compact object might be
swallowed by the giant star later in the evolution as well, e.g, during core
helium burning, as we show in table 2. Tidal forces can bring the NS/BH into
the envelope even if the binary separation is several times larger than the
radius of the giant star (e.g., Soker 1996). Currently the COMPAS code does
not include tidal evolution, implying that it underestimates the number of
systems that go through NS/BH-giant star CEE, and therefore the rates of NS/BH
core mergers and their resultant transients.

The giant star-NS/BH CEE has two possible outcomes. If the envelope is not
ejected before the NS/BH enters the core then the NS/BH merges with the core
of the giant star as shown on the left in the left panel of Fig. 2. However,
if the envelope is entirely unbound before the NS/BH reaches the core, then
the core will keep evolving and eventually ends its life in a stripped-
envelope SN explosion resulting in another NS/BH (right side in the left panel
of Fig. 2). If the binary remains bound after the second SN explosion, and the
two compact objects are close enough, they might merge on a timescale shorter
than the Hubble time emitting potentially detectable gravitational waves (the
majority of systems shown in Fig. 1).

Less massive and wider binaries evolve through a secondary channel (channel
II; right panel of Fig. 2), where dynamically unstable mass transfer between
the post MS primary star and its MS secondary leads to an early CEE phase that
brings the core of the giant and the MS star closer together. At the end of
CEE the envelope is ejected and the evolution continues as in channel I. We
note that this formation channel can only involve a NS compact object due to
the relatively low masses of the SN progenitor. A massive BH progenitor always
leads to stable mass transfer from the post-MS primary to its MS secondary
(second stage of channel I in Fig. 2), implying that all BH-core mergers
evolve through channel I. This evolution channel always ends with the merger
of the NS with the core of the giant star due to the small orbital separation
at the end of first CEE (see equation 1). In both formation channels presented
in Fig. 2) several short dynamically stable mass transfer episodes might occur
throughout the evolution besides the mass transfer events mentioned above.

Table 1: Percentage of NS-core mergers that evolved through channel I and channel II (Fig. 2) from all NS-core mergers for values of $\alpha_{\rm CE}$ that coincide with observations of double compact object mergers (Fig. 1).  $\alpha_{\rm CE}$ |  Channel I |  Channel II
---|---|---
0.5 |  73.3 % |  23.3 %
0.6 |  71.1 % |  25.7 %
0.75 |  73.1 % |  25.2 %
1 |  66.7 % |  32.2 %
1.25 |  56.7% |  42.7 %
1.5 |  54.9% |  45.0 %
1.75 |  73.2 % |  26.8 %
2 |  85.8 % |  14.2 %
2.5 |  99.6 % |  0.3%
3 |  99.8% |  0%
4 |  99.6% |  0%
5 |  99.7% |  0%

Table 1 presents the percentage of systems that evolve through channel I
(second column; left panel of Fig. 2) and through channel II (third column;
right panel of Fig. 2) for different values of $\alpha_{\rm CE}$ in our
simulations (first column). A remaining small percentage of the systems
evolved through various other channels, including binary evolution where there
is no mass transfer between the stars in the binary system when the primary
star is a giant. We note a non-monotonic behaviour with $\alpha_{\rm CE}$ as a
result of two separate CEE phases that are affected by this parameter. A
larger value of $\alpha_{\rm CE}$ implies a larger orbital separation in the
end of CEE. In channel II this means that on the one hand more systems survive
the first CEE phase (between the giant primary and the MS secondary) and
therefore can lead to NS-core mergers, but on the other hand in the second CEE
(NS primary-giant secondary) more NSs do not manage to enter the core of the
giant star.

###  3.3 Properties of binary systems that result in NS/BH-core mergers

As mentioned in section 2, we use COMPAS to explore the properties of NS/BH-
core merger progenitors. In Figs. 3-6 we show different distributions of these
progenitors at the ZAMS for representative values of $\alpha_{\rm CE}$.

Figure 3: Distributions of binary properties at the ZAMS of systems that
result in NS-core mergers (upper panels; orange) or in BH-core mergers (lower
panels; turquoise) for a common envelope efficiency parameter $\alpha_{\rm
CE}=0.5$. Left panels: initial mass of the NS/BH (orange dots/turquoise dots)
progenitors (primary star, $M_{\rm 1,ZAMS}$) vs the initial mass of the star
that swallows the NS/BH (secondary star, $M_{\rm 2,ZAMS}$). Middle panels:
initial mass ratio $q_{\rm ZAMS}\equiv M_{\rm 2,ZAMS}/M_{\rm 1,ZAMS}$ that
leads to NS/BH-core mergers. The orange bins represent the percentage of
binary systems that begin with a certain mass ratio and result in a NS-core
merger from all systems where the compact object merges with the core of the
giant star during the CEE phase. The turquoise bins depict this percentage for
BH-core mergers. Right panels: Similar to the middle panels for the binary
initial orbital separations.  Figure 4: Similar to Fig. 3 for a common
envelope efficiency parameter $\alpha_{\rm CE}=1$.  Figure 5: Similar to Fig.
3 for a common envelope efficiency parameter $\alpha_{\rm CE}=1.5$.  Figure 6:
Similar to Fig. 3 for a common envelope efficiency parameter $\alpha_{\rm
CE}=2$.

Several interesting trends emerge from these figures. While NS-core mergers
occur for $M_{\rm 1,ZAMS}\gtrsim 6.5M_{\rm\odot}$ up to the heaviest stars in
our mass distribution, BH-core mergers tend to involve much heavier stars in
the initial binary system, with $M_{\rm 1,ZAMS}\gtrsim 37.7M_{\rm\odot}$ (left
panels of Figs. 3-6). This can be explained by the heavier masses required to
produce a BH in a SN event under the assumptions of COMPAS. Binary systems
with higher mass ratios at the ZAMS are more likely to result in NS/BH-core
mergers. For both mergers of NSs and BHs with cores of giant stars most of the
progenitor binaries have initial mass ratios of $q_{\rm ZAMS}\equiv M_{\rm
2,ZAMS}/M_{\rm 1,ZAMS}\gtrsim 0.5$ (middle panels of Figs. 3-6), while NS-core
mergers can also originate from $0.25\lesssim q_{\rm ZAMS}\lesssim 0.5$. The
percentage of systems of the latter mass ratio strongly decreases with
$\alpha_{\rm CE}$.

We note a bimodal distribution in the initial orbital separation of systems
that result in a core-NS merger (right upper panels of Figs. 3-6) where each
peak corresponds to one of the evolution channels presented in Fig. 2. We find
that the higher/lower peak in the initial separation coincide with the
percentage of systems that evolve through channel I/II (table 1).

The larger population is in small separations and has a peak at $a_{\rm
ZAMS}\simeq\rm 0.65AU\simeq 130R_{\rm\odot}$ for most values of $\alpha_{\rm
CE}$ we simulated. It corresponds to channel I, which is characterized by
stable mass transfer from the post MS primary to the MS secondary. One might
expect that small initial separation would typically lead to unstable mass
transfer and CEE, and this is indeed a common outcome in our population
models. However, in most such systems the secondary star merges with the core
of the giant primary while still on the MS and hence cannot lead to a NS/BH-
core merger, i.e. the prevalence of stable mass transfer we find at small
separations is a selection effect. The stars in binary systems whose initial
orbital separation is sufficiently small remain close enough so that the
resulting NS can be swallowed by the secondary at its giant phase and
eventually merge with its core, while for systems that begin further apart the
NS and the giant star do not engage in CEE. BH-core mergers exhibit only the
first peak in the initial orbital separation, i.e., evolve only through
channel I333We made sure the single peak is not due to a small sample of BH-
core mergers by performing simulations of $10^{7}$ binaries drawing the mass
of the primary from Kroupa IMF with masses in the range $35\leq M_{\rm
1}/M_{\rm\odot}\leq 100$. This is equivalent to running $1.6\times 10^{8}$
systems in the range $5\leq M_{\rm 1}/M_{\rm\odot}\leq 100$. .

The second population of binaries that end their lives in a merger of a NS
with the core of a giant star during CEE is smaller and has a peak around
$a_{\rm ZAMS}\simeq\rm 4.65AU\simeq 1000R_{\rm\odot}$. This population
corresponds to channel II. Primaries with masses in the range
$6.5M_{\rm\odot}\lesssim M_{\rm 1,ZAMS}\lesssim 20M_{\rm\odot}$ evolve through
this channel. In this case a common envelope is formed around the core of the
post MS primary star and the MS secondary, and dynamical friction between the
core-MS system and the gas of the envelope reduces the orbital separation
significantly 444As there is still no analytical formulation for dynamical
friction, the reduction in orbital separation during CEE in COMPAS is handled
through the energy formalism.. At some point the common envelope is ejected
and a bound, more compact system of a MS star and the naked core remains
behind, and keeps evolving towards a NS-core merger as in channel I. Even
though stable mass transfer can occur between the post MS primary and the MS
secondary in this regime of initial orbital separations, the stars in such
binaries are not close enough to lead to a CEE between the NS and the giant at
a later stage, and therefore cannot lead to a NS-core merger event. For even
larger initial orbital separations stars in the binary systems would mainly
evolve separately.

###  3.4 Mergers’ event rate

Figure 7: Percentage of NS-core mergers (orange dots) and BH-core mergers
(turquoise dots) of all CCSNe for the values of common envelope efficiencies
that coincide with observations of binary compact object mergers (Fig. 1).

We can estimate the merger rates of NSs/BHs with the cores of the giant
secondaries using the results of our population synthesis models. We present
the percentage of these rates relative to CCSNe events in Fig. 7 for the
values of $\alpha_{\rm CE}$ we find in section 3.1. The orange and turquoise
dots are the percentages of NS-core and BH-core mergers, respectively, from
CCSNe in our population model.

We note that for both NS and BH mergers with the giant’s cores the event rate
decreases with $\alpha_{\rm CE}$, a trend we can also see by comparing the
left panels of Figs. 3-6. In the energy formalism (see equation 1), higher
values of $\alpha_{\rm CE}$ imply that more energy is transmitted from the
orbit (or from an additional energy source) to the envelope and it is ejected
earlier in the evolution. The underlying physical concept is that if a larger
amount of energy is deposited inside the envelope it expands to larger radii
before the outer layers are ejected, becoming less dense and reducing the
dynamical friction between the spiraling-in primary and the gas of the
secondary’s envelope. This results in larger orbital separations at the end of
CEE, implying that more NSs/BHs remain outside the core rather than merge with
it. BH-core mergers reduce to nearly nothing for $\alpha_{\rm CE}\gtrsim 4$
while the event rate of NS-core mergers drops about an order of magnitude
between the lowest and highest values of $\alpha_{\rm CE}$ we simulate.

Table 2: Percentage of NS-core mergers where the core of the giant secondary star is composed of helium (second column) or Carbon-Oxygen (third column) from all NS-core mergers for different values of the efficiency parameter $\alpha_{\rm CE}$ (first column). The fourth column lists the average mass of the secondary at the ZAMS for each value of $\alpha_{\rm CE}$ $\alpha_{\rm CE}$ |  He core |  CO core |  $\left<M_{\rm 2,ZAMS}\right>[M_{\rm\odot}]$
---|---|---|---
0.5 |  85.5 % |  14.5 % |  10.4
0.6 |  85.9 % |  14.1 % |  10.5
0.75 |  85.1 % |  14.8 % |  11.2
1 |  86.3 % |  13.7 % |  12.0
1.25 |  90.2 % |  9.8 % |  12.7
1.5 |  97.0 % |  3.0% |  13.9
1.75 |  98.6 % |  1.4 % |  16.4
2 |  99.6 % |  0.4 % |  18.3
2.5 |  99.9% |  0.1% |  21.4
3 |  100% |  0% |  22.9
4 |  100% |  0% |  25.0
5 |  100 % |  0 % |  25.6

In table 2 we present how the percentage of systems where a NS merges with the
helium (second column) or carbon-oxygen (third column) core of a giant star
from all NS-core mergers varies with the efficiency parameter $\alpha_{\rm
CE}$ (first column). For larger values of $\alpha_{\rm CE}$ the number of
mergers where the core is composed of helium is overall larger, i.e., the
mergers occur at earlier times during the giant-NS/BH CEE. As larger amounts
of energy go to unbind the envelope it is ejected at an earlier stage, and the
orbital separation at the end of the CEE phase is larger (equation 1). This
implies that mergers between the giant’s core and the NS for larger values of
$\alpha_{\rm CE}$ occur for more massive secondaries with higher envelope
binding energies (as we indeed see in the fourth column of table 2). Such
giants are larger to begin with and thus can swallow the NS as they expand at
earlier stages, before core helium burning. We find that binary systems whose
outcome is BH-core mergers require $\left<M_{\rm 2,ZAMS}\right>\gtrsim
37.7M_{\rm\odot}$ (section 3.3). Therefore, all BH-core mergers occur when the
core of the giant secondary is composed mainly of helium regardless of the
$\alpha_{\rm CE}$ value.

We examine the effects of mass accretion by the NS/BH while inside the
envelope of the giant star and find no statistically significant differences
in any of the quantities we have discussed (see appendix).

##  4 Summary and discussion

In this work we performed extensive population synthesis of massive binaries
in search of NSs and BHs mergers with cores of giant stars. We used COMPAS to
generate populations of massive binary systems (section 2) and followed the
evolution of binaries that result in these mergers (section 3.2). We used the
energy formalism to determine whether a merger occurred during the CEE of the
giant secondary star with the NS/BH. We simulated cases of traditional CEE in
which the orbital energy released due to the decay in the orbit is the main
cause for envelope ejection ($\alpha_{\rm CE}\leq 1$), and other cases where
we assume an additional energy source ($\alpha_{\rm CE}>1$). We constrained
the possible values of $\alpha_{\rm CE}$ by comparing observations of binary
compact object mergers to our simulated rates (section 3.1).

We found one main evolution route of NS-core and BH-core merger events
(channel I; left panel of Fig. 2) and an additional secondary evolution
pathway for NS-core mergers (channel II; right panel of Fig. 2). The right
panels of Figs. 3-6 exhibit a bimodal distribution in the initial orbital
separation of NS-core merger progenitors whose modes are consistent with the
two different evolution routes. For relatively close binaries with initial
orbital separation $a_{\rm ZAMS}\lesssim 1.3AU$ only cases where dynamically
stable mass transfer occurs when the primary star is in its post MS phase can
lead to NS (and also BH)-core mergers (channel I). However, at larger initial
orbital separations ($3AU\lesssim a_{\rm ZAMS}\lesssim 7AU$) a CEE is required
to reduce the separation of the core-MS binary and bring them close enough to
allow for a later CEE between the NS and the giant in which the NS might merge
with its core (channel II).

We compute the mergers’ event rates and find their percentage relative to
CCSNe for different values of the CEE efficiency parameter we simulated
(section 3.4). We find that for $\alpha_{\rm CE}=1$, which is the commonly
used value for CEEs that involve massive giants, there is about 1 NS-core
merger per 100 CCSN events and about 1 BH-core merger per 1000 CCSN events.
The number of mergers decreases with the CEE efficiency parameter $\alpha_{\rm
CE}$ as expected from the energy formalism and its underlying physics. For
$\alpha_{\rm CE}\gtrsim 4$ the event rate of BH-core mergers reduces to nearly
nothing within the accuracy of our simulations. We estimate the amount of
accreted mass by the NS and by the BH while inside the envelope of the giant
star in the CEJSN scenario (see appendix) and find it does not affect the
mergers’ event rate.

The energy formalism for CEE has several shortcomings. The large uncertainty
in the energy and mass transfer during the CEE phase might greatly affect the
rates of transients whose progenitor binaries go through CEE (e.g., Olejak et
al. 2021). Moreover, many hydrodynamical simulations find that the orbital
energy cannot be the sole energy source that contributes to envelope ejection
in massive binaries (see references in section 2). Using values of
$\alpha_{\rm CE}>1$ to represent additional energy sources, as we did in the
present study, disregards that these energy sources do not depend on the
orbital separation in the same way as the orbital energy. However, keeping in
mind this formalism is a phenomenological treatment, $\alpha_{\rm CE}>1$
yields that the spiraling-in star can end at a larger radius, and allows to
obtain reasonable results in population synthesis of massive stars. A possible
refinement of our analysis would take into account that $\alpha_{\rm CE}$ is
in general system-dependant. For instance, if jets are the additional energy
source that unbinds the envelope $\alpha_{\rm CE}$ is naturally larger for a
NS accretor than a MS star, and even larger for a BH. Another example is the
study of De Marco et al. 2011 which finds that $\alpha_{\rm CE}$ is inversely
correlated with the binary mass ratio. We note that other prescriptions of CEE
applicable to population synthesis, such as Hirai & Mandel (2022) and Di
Stefano et al. (2022) were recently suggested, but are not yet implemented in
the available codes. It would be interesting to compare the results presented
in this manuscript with results obtained using different CEE formalisms.

The results of this study can be used to test whether NS/BH-core mergers and
their resultant CEJSN transient events can account for several high energy
astrophysical phenomena (as proposed and studied in e.g., Papish et al. 2015;
Soker & Gilkis 2018; Soker et al. 2019; Grichener & Soker 2019a; Grichener &
Soker 2021; Soker 2021; Grichener et al. 2022; Soker 2022b) and what would be
their overall contribution to said events. Mergers with different core
structures, for instance, can lead to transient events with different
properties. R-process nucleosynthesis in CEJSN requires that the merger of the
giant’s core with the NS occurs when the core is CO rich. Grichener & Soker
(2019a) found that one CEJSN r-process events per $\simeq 1000$ CCSNe suffices
to explain the solar system r-process abundances by the CEJSN r-process
scenario, which is consistent with our results for $\alpha_{\rm CE}\simeq
1.25$ (Fig. 7 and table 2). For this scenario to account for a substantial
fraction of the r-process abundance (above $10\%$) $1.25\lesssim\alpha_{\rm
CE}\lesssim 1.75$ is required.

Grichener & Soker (2021) found that jets launched by a BH inside the envelope
of a giant star might emit neutrinos with energies of $10^{15}~{}\rm{eV}$ as
detected by IceCube (Aartsen et al., 2013). The event rate required to explain
the high energy neutrinos flux assuming the properties of the model described
in Grichener & Soker (2021), and based on Aartsen et al. (2021) is $\approx
3\%$ from CCSNe, and can be lower for jets with higher energies. We find that
the percentage of systems that result in CEE of a BH with the giant secondary
star (stage six in channel I; left panel of fig 2) is about $\simeq 0.6\%$
from CCSNe for all values of $\alpha_{\rm CE}$ we simulate. This implies that
the CEJSN scenario for high energy neutrinos might have a significant
contribution to the high energy neutrino flux.

We note that even though the motivation for this work was the CEJSN scenario
and its possible outcomes, the population model presented in this manuscript
is not exclusive for the jets powering mechanism, and the merger rates shown
in Fig. 7 can be used for any model that aims to explain NS/BH-core mergers in
both frameworks of traditional CEE ($\alpha_{\rm CE}\leq 1$), or models that
assume another energy source besides the orbital energy ($\alpha_{\rm CE}>1$).

## ACKNOWLEDGMENTS

I thank Noam Soker, Vladimir Kalnitsky, Amit Kashi, Avishai Gilkis, Jan J.
Eldridge, Hila Glanz and Dmitry Shishkin for helpful discussions and important
suggestions that helped in improving this manuscript. This research was
supported by a grant from the Israel Science Foundation (769/20). I
acknowledges support from Irwin and Joan Jacobs Fellowship. Simulations in
this paper made use of the COMPAS rapid binary population synthesis code
(version 02.31.06), which is freely available at
http://github.com/TeamCOMPAS/COMPAS.

## Data availability

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

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## The effect of mass accretion by the NS/BH during CEE with the giant star

The NS/BH merger with the core of the giant star can power a bright transient
event. In the CEJSN scenario (e.g., Soker et al. 2019) the NS/BH launches two
opposite jets, one on each side of the equatorial plane, while spiralling-in
inside the envelope and later through the core of the giant secondary star.
The collision of these jets with the giant’s envelope and the giant’s winds
converts kinetic energy to thermal energy and then radiation, leading to a
bright transient that might mimic a supernova explosion.

The energy that the two jets a NS launches deposit to the envelope of the
giant secondary star is

$E_{\rm 2j}=\eta\frac{GM_{\rm NS}M_{\rm acc,NS}}{R_{\rm NS}},$ (1)

where $M_{\rm NS}$ and $R_{\rm NS}$ are the mass and the radius of the NS,
respectively, and $\eta$ is an efficiency parameter. Assuming that the energy
of the jets is about the binding energy of the envelope $E_{\rm bind}$, as
required to unbind the envelope considering that hydrodynamical simulations
find that most of the envelope’s mass remains bound at the end of CEE (e.g.,
Passy et al. 2012), we can find the amount of mass accreted by the NS in the
CEE phase to be

$\begin{split}M_{\rm acc,NS}\simeq 0.001M_{\rm\odot}\left(\frac{E_{\rm
bind}}{5\times 10^{49}~{}\rm{erg}}\right)\left(\frac{R_{\rm
NS}}{10~{}\rm{km}}\right)\\\ \times\left(\frac{M_{\rm
NS}}{1.25M_{\rm\odot}}\right)^{-1}\left(\frac{\eta}{0.05}\right)^{-1}.\end{split}$
(2)

We calibrate $E_{\rm bind}$, $R_{\rm NS}$ and $M_{\rm NS}$ with typical values
from our COMPAS models and we take the efficiency parameter to be $\eta=0.05$
(e.g., López-Cámara et al. 2019).

The jets that a BH launches inside the envelope of the giant secondary star
are relativistic, and their energy is given by

$E_{\rm 2j}=\eta M_{\rm acc,BH}c^{2},$ (3)

implying that the amount of mass that the BH accretes while inside the
envelope of the giant secondary star is

$\begin{split}M_{\rm acc,BH}\simeq 0.002M_{\rm\odot}\left(\frac{E_{\rm
bind}}{3\times
10^{50}~{}\rm{erg}}\right)\left(\frac{\eta}{0.1}\right)^{-1}.\end{split}$ (4)

We take the typical value of $E_{\rm bind}$ from the COMPAS simulations and
the efficiency parameter $\eta=0.1$ from Frank et al. (2002).

Including the mass that is accreted by the NS/BH during CEE in our COMPAS
simulations yields practically no difference in the rates of NS/BH-core
mergers for all values of $\alpha_{\rm CE}$ we simulated. The rates remain
similar even for the higher bounds of the accreted masses inferred from
substituting the upper limits of the parameters from the simulations in
equations (2) and (4), which results in both cases in masses about an order of
magnitude higher than the calibrated values. Inside the core of the giant star
the energies of the jets are higher and hence the accretion has a more
significant role.