# A Channel to Form Fast-spinning Black Hole–Neutron Star Binary Mergers as
Multi-messenger Sources

Rui-Chong Hu Guangxi Key Laboratory for Relativistic Astrophysics, School of
Physical Science and Technology, Guangxi University, Nanning 530004, China
Jin-Ping Zhu Department of Astronomy, School of Physics, Peking University,
Beijing 100871, China  Ying Qin Department of Physics, Anhui Normal
University, Wuhu, Anhui, 241000, China  Guangxi Key Laboratory for
Relativistic Astrophysics, School of Physical Science and Technology, Guangxi
University, Nanning 530004, China  Bing Zhang Nevada Center for Astrophysics,
University of Nevada, Las Vegas, NV 89154, USA  Department of Physics and
Astronomy, University of Nevada, Las Vegas, NV 89154, USA  En-Wei Liang
Guangxi Key Laboratory for Relativistic Astrophysics, School of Physical
Science and Technology, Guangxi University, Nanning 530004, China  Yong Shao
Department of Astronomy, Nanjing University, Nanjing 210023, China

###### Abstract

After the successful detection of a gravitational-wave (GW) signal and its
associated electromagnetic (EM) counterparts from GW170817, neutron star–black
hole (NSBH) mergers have been highly expected to be the next type of multi-
messenger source. However, despite the detection of several of NSBH merger
candidates during the GW third observation run, no confirmed EM counterparts
from these sources have been identified. The most plausible explanation is
that these NSBH merger candidates were plunging events mainly because the
primary BHs had near-zero projected aligned-spins based on GW observations. In
view that NSs can be easily tidally disrupted by BHs with high projected
aligned-spins, we study an evolution channel to form NSBH binaries with fast-
spinning BHs, the properties of BH mass and spin, and their associated tidal
disruption probability. We find that if the NSs are born firstly, the
companion helium stars would be tidally spun up efficiently, and would thus
finally form fast-spinning BHs. If BHs do not receive significant natal kicks
at birth, these NSBH binaries that can merge within the Hubble time would have
BHs with the projected aligned-spins $\chi_{z}\gtrsim 0.8$ and, hence, can
certainly allow tidal disruption to happen. Even if significant BH kicks are
considered for a small fraction of NSBH binaries, the projected aligned-spins
of BHs are $\chi_{z}\gtrsim 0.2$. These systems can still be disrupted events
unless the NSs are very massive. Thus, NS-first-born NSBH mergers would be
promising multi-messenger sources. We discuss various potential EM
counterparts associated with these systems and their detectability in the
upcoming fourth observation run.

gravitational waves — binaries: close — stars: black holes — stars: neutron
stars

##  1 Introduction

Neutron star–black hole (NSBH) mergers are prime search targets for the
ground-based gravitational wave (GW) detectors, e.g., LIGO (LIGO Scientific
Collaboration et al., 2015), Virgo (Acernese et al., 2015) and KAGRA (Aso et
al., 2013). Recently, two high-confidence GW events (GW200105 and GW200115)
were announced by the LIGO-Virgo-KAGRA (LVK) Collaboration, which was, for the
first time, identified to come from mergers of NSBH binaries (Abbott et al.,
2021a; Nitz et al., 2021). Furthermore, two lower mass-gap sources (GW190814
and GW200210_092254) that could either be from a NSBH or a binary BH, and
several marginal NSBH candidates were discovered in GW during the third
observation run (O3) of LVK (Abbott et al., 2020b, 2021b; The LIGO Scientific
Collaboration et al., 2021a, b). The analysis of these NSBH candidates
displayed that their effective inspiral spins could be near-zero or very low,
suggesting that their primary BHs would plausibly have negligible projected
spins aligned to the direction of the orbital angular momentum (abbreviated as
projected aligned-spins, hereafter). Among these candidates, only the BH
component of GW200115 showed a non-zero BH spin with an apparent BH spin-orbit
misalignment angle (Abbott et al., 2021a), which may require a strong natal
kick for the BH or the NS (Fragione et al., 2021; Gompertz et al., 2021; Zhu,
2021). Conversely, Mandel & Smith (2021) argued that GW200115 could
potentially be a merger between a non-spinning BH and a typical mass of NS by
reanalysing its GW signal with the consideration of astrophysically motivated
priors.

NSBH mergers have long been proposed to be progenitors of some fast-evolving
electromagnetic (EM) counterparts, including short-duration gamma-ray bursts
(sGRBs; Paczynski, 1991; Narayan et al., 1992; Zhang, 2018) and kilonovae (Li
& Paczyński, 1998; Metzger et al., 2010). As main GW sources of ground-based
GW detectors, it was thus expected that the follow-up searches after GW
triggers could help find out the associated EM signals of NSBH mergers.
However, despite many efforts toward the follow-up observations, no confirmed
EM counterpart candidate was identified (e.g., Anand et al., 2021; Gompertz et
al., 2020b; Kasliwal et al., 2020; Coughlin et al., 2020a, b; Page et al.,
2020; Becerra et al., 2021). The tidal disruption probability of NSBH mergers
and the brightness of EM signals depend on NS mass, NS equation of state
(EoS), BH mass, and especially BH projected aligned-spin (e.g., Kyutoku et
al., 2015; Kawaguchi et al., 2016; Foucart et al., 2018; Barbieri et al.,
2019; Raaijmakers et al., 2021; Zhu et al., 2020, 2021a, 2021b, 2021c).
Disrupted events associated with brighter EM signals tend to occur only if a
NSBH binary has a low-mass NS component with a stiff EoS and a low-mass BH
component with a high projected aligned-spin. The most promising explanation
for the lack of EM counterparts is that these NSBH candidates were plunging
events without forming any bright EM signals, mainly due to their near-zero BH
projected aligned-spins (Zhu et al., 2021b; Fragione, 2021; D’Orazio et al.,
2021).

The most widely accepted formation channel for the majority of NSBH binaries
in the universe is the classical Common Envelope (CE) scenario (e.g., Giacobbo
& Mapelli, 2018; Belczynski et al., 2020; Drozda et al., 2020; Shao & Li,
2021; Broekgaarden et al., 2021). In this scenario, the immediate progenitor
of a NSBH binary just after the CE phase is a close binary system, which
consists of a compact object (NS or BH) and a helium star. On the one hand, if
the first-born compact object is a BH, its spin has been found to be
negligible (Qin et al., 2018; Belczynski et al., 2020). This was supported by
the finding of the near-zero distribution of BH spin in the O3 GW NSBH
candidates (Zhu et al., 2021a), which implies that the NSs in these NSBH
candidate systems may be directly plunged into the BHs. On the other hand, for
a fraction of NSBH progenitor systems in which the NS was first born, the
companion helium star can be spun up by the NS and finally form a BH with a
high projected aligned-spin. Since a fast-spinning BH can easily tidally
disrupt the NS and produce bright EM signals (i.e., sGRBs and kilonovae), NSBH
mergers formed via this formation channel can be potential multi-messenger
sources that allow us to discover their associated bright EM counterparts with
a high probability after GW triggers (Zhu et al., 2021c). Furthermore, as
studied recently by Román-Garza et al. (2021), core-collapse physics plays a
critical role in the observability of the EM signals produced by NSBH mergers.
In viewing of the lack of relevant researches on the BH spin properties of
such NS-first-born formation channel for NSBH mergers, we investigate this
formation channel of NSBHs with fast-spinning BHs in detail and study their
corresponding tidal disruption probability.

In this work, we investigate the detailed binary evolution process of forming
fast-spinning BHs in NSBH binaries by taking into account the accretion
feedback of the core-collapse processes, supernova (SN) kicks of newly-formed
BHs, and different NS EoSs. The main methods adopted in the stellar and binary
evolution models are shown in Section 2. We then present in Section 3 our
findings of NSBH mergers without and with natal kicks, respectively, along
with their associated probability for tidal disruption. The multi-messenger
observational signatures of this channel are discussed in Section 4. Finally,
main conclusions are summarized in Section 5 with some discussion.

##  2 Methods

We use the release 15140 of MESA stellar evolution code (Paxton et al., 2011,
2013, 2015, 2018, 2019) to perform all of the binary evolution calculations in
this work. A metallicity of $Z=Z_{\odot}$, where the solar metallicity is
$Z_{\odot}=0.0142$ (Asplund et al., 2009), is adopted. We create single helium
stars at Zero-Age Helium Main-Sequence (ZAHeMS) following the method in Qin et
al. (2018) and then relax the created helium stars to reach the thermal
equilibrium, where the He-burning luminosity just exceeds 99% of the total
luminosity. We model convection using the mixing-length theory (Böhm-Vitense,
1958) with a mixing-length parameter $\alpha=1.93$. The Ledoux criterion is
used to treat the boundaries of the convective zone, while we consider the
step overshooting as an extension given by $\alpha_{p}=0.1H_{p}$, where
$H_{p}$ is the pressure scale height at the Ledoux boundary limit.
Semiconvection (Langer et al., 1983) with an efficiency parameter $\alpha=1.0$
is also included in our model. The network of approx12.net is chosen for
nucleosynthesis.

Stellar winds are modelled with the standard “Dutch” scheme, calibrated by
multiplying with a scaling factor of 0.667 to match the recently updated
modeling of helium stars’ winds (Higgins et al., 2021). We model angular
momentum transport and rotational mixing diffusive processes (Heger & Langer,
2000), including the effects of Eddington–Sweet circulations, the
Goldreich–Schubert–Fricke instability, as well as secular and dynamical shear
mixing. We adopt diffusive element mixing from these processes with an
efficiency parameter of $f_{c}=1/30$ (Chaboyer & Zahn, 1992; Heger & Langer,
2000). Mass transfer is modeled following the Kolb scheme (Kolb & Ritter,
1990) and the implicit mass transfer method (Paxton et al., 2015) is adopted.

We model helium stars until the carbon depletion in the center. The baryonic
remnant mass is calculated following the “delayed” supernova prescription in
Fryer et al. (2012). As shown in Batta & Ramirez-Ruiz (2019), the newly formed
BH might be unable to accrete all of the available stellar material.
Therefore, in order to calculate the final mass and spin of the BH from the
direct collapse, we follow the framework given in Batta & Ramirez-Ruiz (2019),
which has already been implemented in a recent work (Bavera et al., 2020). The
neutrino loss as in Zevin et al. (2020) is taken into account. Furthermore,
the maximum NS mass is assumed to be $2.5\,M_{\odot}$ in this work.

BHs formed through direct collapse are considered to receive no mass loss and
thus no natal kick (Belczynski et al., 2008). Recently, an estimation of the
NSBH merger rate from the LIGO-Virgo O3a run showed that negligible kicks
imparted on the low-mass BHs is favored (Román-Garza et al., 2021).
Consequently, calculation with no natal kick included is considered as our
fiducial model. Additionally, we also take into account natal kicks onto BHs
formed through direct collapse. We follow the parametrised recipes of Mandel &
Müller (2020) to calculate the natal kicks of BHs. The binary properties just
after the natal kick is given based on the framework of Kalogera (1996); Wong
et al. (2012) and a recent work by Callister et al. (2021). For binaries
surviving after the kick, we estimate the merger time given from Peters (1964)
through GW emission, with the updated fitting formula for eccentricity in
Mandel (2021).

##  3 Results

Here we present a parameter space study of the initial binary properties for a
close binary system composed of a helium star and a NS. The parameters include
the initial helium star masses and the initial orbital periods. We cover the
helium star mass range $8-40\,M_{\odot}$, the NS star mass range
$1.2-2.5\,M_{\odot}$, as well as the orbital period from 0.2 to 2 days. We
also take into account the natal kicks imparted onto BHs in the core-collapse
models. We describe in detail our findings for the NSBH formation without and
with natal kicks, respectively, along with their associated tidal disruption
probability.

###  3.1 Fast-spinning BHs originated from tidal spin-up in NSBH binaries

Figure 1: Left panel: BH mass (the color bar) as a function of the helium star
initial mass and orbital period. The symbols represent different compact
objects formed through direct core-collapse of the helium stars, i.e., squares
for NS and dots for BH. Right panel: same as the left panel, but the color
represents the BH projected aligned-spins. The star symbols are marked with
NSBH binaries that can be merged via GW emission within Hubble time. The
numbers on the left side of the star symbols are the corresponding merger time
in units of billion years.

Figure 1 presents various outcomes of the detailed binary evolution for helium
stars at different initial masses orbiting around a NS of $1.4\,M_{\odot}$ in
close orbits. We choose 2 days as the upper limit of the orbital period,
beyond which tides are not important (Qin et al., 2018). We do not take into
account the parameter space where the initial orbital periods are shorter than
0.2 days, as the binary would experience either initial overflow or
dynamically unstable mass transfer. NSs with other masses have similar
results, which are not shown in the paper.

First, we note from the left panel of Figure 1 that helium stars with the
initial mass higher than $10\,M_{\odot}$ and initial orbital period larger
than $\sim 0.4$ days collapse to form BHs. Furthermore, helium stars in
tighter orbits tend to lose more mass due to rotationally enhanced mass loss
(Heger & Langer, 1998; Langer, 1998). This is why we can see that helium stars
in short orbits end up forming low-mass BHs. For low-mass ($<10\,M_{\odot}$)
helium stars, including a small fraction with initial orbital periods less
than $\sim 0.3$ days, NSs are formed instead. A further investigation of
binary NS systems formed in our grid is postponed in a future work.

In the right panel of Figure 1, we show the magnitudes of the BH projected
aligned-spin for different initial binary properties. As shown in Qin et al.
(2018), the spins of the resulting second-born BHs are exclusively dependent
on the tidal interaction and stellar winds of the helium stars. For initial
orbital periods considered in this gird, the BHs have the entire range of the
spin from zero to value allowed by maximally spinning. For more massive helium
stars, their stellar winds are stronger, which widen the binaries and thus
make NSBHs undetectable for GW emission within Hubble Time. Interestingly, we
note that for helium stars with different initial masses, the BH projected
aligned-spins have a similar trend in decreasing magnitudes. After the
formation of the second-born BH, the GW emission removes the orbital angular
momentum of the NSBH, shrinks the orbit and eventually leads to merger of the
two compact objects. The merger time is calculated as given in Peters (1964).
It is worth noting that all NSBHs with their merger time less than the Hubble
time have extremely fast spins. This finding is mainly attributed to the
strong tidal force in close binaries. However, BHs formed through direct core-
collapse may receive SN kicks, which can change the direction of the BH spins
and, hence, their projected aligned-spins. The impact of SN kicks is discussed
as follows.

###  3.2 Impact of SN kicks on BH projected aligned-spins and the probability
of merging NSBHs

Figure 2: Left panel: similar to the left panel in Figure 1, but with SN kicks
included. Right panel: the color represents the fraction of NSBH binaries
which can be merged within Hubble time with SN kicks considered. The numbers
on the left side of the star symbols are the medians of the corresponding
merger time in units of billion years that is less than Hubble time. Dark blue
dots: NSBH binaries that never merge within Hubble time.

The SN kicks imparted onto BHs are considered to cause a misalignment of the
BH spin to the orbital angular momentum. In addition, the post-SN orbital
separation and the eccentricity of the binary can also be modified. We briefly
describe these calculations in Appendix and more details can be found in
Kalogera (1996); Wong et al. (2012); Callister et al. (2021). Similar to
Mandel & Müller (2020), kicks are drawn from a Gaussian distribution and we
repeat $10^{5}$ times to obtain the average value of the kick and also its
associated angle $\theta$. The corresponding averaged BH projected aligned-
spins ($\bar{\chi}_{z}$) are shown in the left panel of Figure 2. We note that
$\bar{\chi}_{z}$ values are slightly less than those obtained assuming no
kicks (see the right panel of Figure 1). This is because the SN kicks we apply
to the BHs are typically small (see the red line for BH kicks in Figure 4 of
Mandel & Smith, 2021). Accordingly, we then obtain on average a small
misalignment for BH spins.

Let us see how kicks change the fraction of merging NSBHs. As shown in the
right panel of Figure 2, with the post-SN-explosion binary properties updated,
for each initial system we can then calculate the fraction of the post-SN-
explosion NSBH systems that can survive and merge within Hubble time. The BH
masses of these systems against their BH projected aligned-spins are presented
in Figure 3. As the kick is considered, massive BHs with $M_{\rm BH}\gtrsim
7.6\,M_{\odot}$ do not experience natal kicks (the last three columns from the
right side) as they are heavier than the carbon-oxygen core of their
progenitors at the pre-SN state. Some NSBH binaries with relatively low spins,
which can not merge in a wide orbit, can instead merge within Hubble time
caused by the SN kicks. However, the merger fraction deceases since the
systems have lower-mass BHs with lower projected aligned-spins. More
specifically, as shown in Figure 3, BHs with $M_{\rm BH}\lesssim 5\,M_{\odot}$
have a wide range of the projected aligned-spin $\chi_{z}$ from $\sim 0.2$ to
$\sim 1.0$, while more massive BHs are found to be extremely spinning, i.e.,
$\chi_{z}\gtrsim 0.8$.

The first-born BHs in NSBH mergers are expected to have near-zero spin
distributions (e.g., Qin et al., 2018; Belczynski et al., 2020). Broekgaarden
& Berger (2021); Kinugawa et al. (2022) suggested that GW200105 and GW200115
are expected to be formed via the classical BH-first-born isolated formation
channel. Their posterior distributions of BH projected aligned-spin versus BH
mass from Abbott et al. (2021a) are plotted in Figure 3. It is clear that the
distributions of the BH projected aligned-spin for GW200105 and GW200115 are
quite different from our calculated distributions for NS-first-born NSBH
mergers.

###  3.3 Tidal disruption probability

Figure 3: The median and $90\%$ confidence interval of BH projected aligned-
spin versus BH mass from the grid of calculations for those merger events
within Hubble time used to produce Figure 2 (red circles), and the parameter
space for NSBH merger systems to allow tidal disruption of the NS by the BH.
Two EoSs, i.e., AP4 (left panel) and DD2 (right panel), are considered. We
mark several values of the NS mass from $M_{\rm NS}=1\,M_{\odot}$ to $M_{\rm
NS}=2.2\,M_{\odot}$ for AP4 ($M_{\rm NS}=2.4\,M_{\odot}$ for DD2) as solid
lines in each panel. For a specific $M_{\rm NS}$, the NSBH mergers with BH
mass and BH projected aligned-spin located at the top left parameter space
(denoted by the direction of the arrows) can allow tidal disruptions to occur.
For GW200105 (green) and GW200115 (blue), the $90\%$ credible posterior
distributions (colored solid lines) of the parameters obtained from Abbott et
al. (2021a) are displayed, respectively. Corresponding median values of
$M_{\rm NS}$ for these two sources are marked as dashed lines. Both GW200105
and GW200115, which are believed to form via the BH-first-born formation
scenario (Broekgaarden & Berger, 2021; Kinugawa et al., 2022), are most likely
plunging events.

The tidal disruption probability of NSBH mergers is determined by NS mass, NS
EoS, BH mass, and BH projected aligned-spin. By considering two representative
EoSs, i.e., AP4 (Akmal & Pandharipande, 1997) and DD2 (Typel et al., 2010), in
which AP4 is one of the most likely EoSs while DD2 is one of the stiffest EoSs
constrained by GW170817 (Abbott et al., 2018, 2019), the parameter space where
the NS can be tidally disrupted using the formula from Foucart et al. (2018)
is shown in Figure 3. A NSBH binary system that has a lower-mass NS with a
stiffer EoS and a lower-mass BH with a higher projected aligned-spin can more
easily make tidal disruption. The Tolman-Oppenheimer-Volkov (TOV) mass for
EoSs of AP4 and DD2 is $M_{\rm TOV}=2.22\,M_{\odot}$ and $2.42\,M_{\odot}$,
respectively. On one hand, for almost all of the parameter space of the NS
mass, Figure 3 reveals that NSBH mergers formed via NS-first-born formation
scenario should be disrupted events if the remnant BHs are heavier than $\sim
5\,M_{\odot}$. Furthermore, for the BH mass located in the range of $M_{\rm
BH}\lesssim 5\,M_{\odot}$, NSBH mergers whose BH projected aligned-spins are
$\chi_{z}>0.8$ are expected to make tidal disruption as shown in Figure 3. On
the other hand, some mergers of NSBH binaries with relatively low BH projected
aligned-spins (i.e., $\chi_{z}\lesssim 0.8$) caused by the impact of SN kicks
on low-mass BH components could still be plunging events, if the mass of the
NS companions is $M_{\rm NS}\gtrsim 1.5\,M_{\odot}$ ($M_{\rm NS}\gtrsim
1.7\,M_{\odot}$) with the adoption of an EoS of AP4 (DD2). However, since the
fraction of NSBH binaries with low BH projected aligned-spins that can merge
within Hubble time after the SN kick is limited as discussed in Section 3.2,
plunging events may account for only a small part of NSBH mergers formed via
NS-first-born formation scenario. Therefore, regardless of the EoS we choose,
it is expected that most NSs can be tidally disrupted by the BHs in NS-first-
born NSBH merger systems and they would be plausible multi-messenger sources,
which may be discovered in the future.

##  4 Observational Signatures

###  4.1 Multi-messenger signals

In Section 3.3, we showed that most of NS-first-born NSBH mergers can easily
make tidal disruption. Disrupted NSBH mergers are believed to drive some
bright EM counterparts. After a NSBH merger, there might be a pair of
relativistic jets launched along the polar axis by the accreting remnant BH
via the Blandford–Znajek effect (Blandford & Znajek, 1977; Gompertz et al.,
2020a), which can power a sGRB and its broad-band afterglow emission as the
jet interacts with the interstellar medium. The radioactive decays of the
rapid neutron-capture-formed heavy nuclei can effectively heat the ejected
materials in the dynamically ejecta ejecta or the wind outflows launched from
the disk around the remnant BH, which would power the fast-evolving kilonova
emission (e.g., Kawaguchi et al., 2016; Barbieri et al., 2019; Zhu et al.,
2020; Darbha et al., 2021). The brightness of the emission may be further
enhanced by the energy injection from the central engine (e.g., Ma et al.,
2018; Qi et al., 2021).

In the fourth observing run (O4), the number of detected NSBH GWs is believed
to increase significantly (e.g., Abbott et al., 2020a; Zhu et al., 2021c). A
higher accuracy in both localization and distance measurement by the GW
observations is expected, which make the post-GW-trigger follow-up search
observations more efficiently. Furthermore, during the same GW period, several
more powerful telescopes, e.g., the Space Variable Objects Monitor (Wei et
al., 2016) in gamma-rays, the Einstein Probe (Yuan et al., 2016) in X-rays,
the Large Synoptic Survey Telescope (LSST Science Collaboration et al., 2009)
and the Wide-Field Infrared Transient Explorer (Frostig et al., 2021) in
optical-infrared bands, will start to operate and join the GW follow-up
observational campaigns. It is possible that multi-messenger signals from NSBH
mergers, especially for NS-first-born NSBH mergers, could be discovered in the
forthcoming O4.

###  4.2 A Possible Connection Between Long-duration and Short-duration GRBs

One interesting inference from this channel is that there could be a common
progenitor system for a small fraction of long-duration GRBs (lGRBs) and
sGRBs. As shown in Section 3, the resulting BHs in close binaries are found to
be highly spinning due to tidally spin-up. Such a fast-spinning BH can in
principle launch a relativistic jet and produce a lGRB through the Blandford-
Znajek mechanism (Blandford & Znajek, 1977) during the formation of the
second-born BH. After the formation of the BH, the NSBH binary would merge
long time later after losing orbital energy and angular momentum due to GW
emission. Eventually, a sGRB may be produced at the merger.

We show for each NSBH system the merger time, namely the time delay between
the lGRB and the following sGRB, on the left side of the symbol in the right
panel of Figure 1. The time delay varies from a few tens to hundred billion
years. When the SN kicks imparted onto the newly-formed BHs are considered,
the average merger time111We adopt the median of the merger time in $10^{5}$
times drawings as the average merger time. We exclude the systems that are
disrupted or whose merger times are beyond Hubble time. drops by roughly one
to two orders of magnitude (see the right panel of Figure 2. In any case, the
delay time is too long and it is impossible to directly test observationally
the existence of both a lGRB and a sGRB from the same progenitor system. One
possible approach to test this scenario is through host galaxy studies. It has
been well established that lGRBs and sGRBs have statistically very different
host properties (e.g. Fruchter et al., 2006; Fong et al., 2010; Li et al.,
2016). Cross comparing the host properties with the consideration of galaxy
evolution in the timescale of the delay between the SN explosion and merger
may offer some clues. This is beyond the scope of this paper.

##  5 Conclusions and Discussion

In this work, we first present a detailed binary evolution modeling of the
formation of NSBH systems, concentrating on a special channel in which NSs are
first formed and BHs are born with high natal spins. With this formation
scenario considered, we explore the tidal disruption probability of NSBH
mergers, taking into account the impact of accretion feedback of direct core-
collapse modeling on the mass and spin of the newly-formed BHs, SN kicks, as
well as different NS EoSs. We find that these NSBH mergers produce BHs with
extremely high natal spins. With no natal kicks for BHs, we note that NSBH
binaries that can merge within Hubble time would have BHs with the projected
aligned-spins $\chi_{z}\gtrsim 0.8$, and therefore, can definitely have tidal
disruption of the NSs at the merger. On the other hand, when natal kicks are
taken into account, BHs with $M_{\rm BH}\lesssim 5\,M_{\odot}$ and with
projected aligned-spins $\chi_{z}$ down to 0.2 (see the right panel of Figure
2) can still merge with a NS within Hubble time, but with a low probability.
These systems can still be disrupted events to produce bright EM signals
unless the NSs are very massive.

Very recently, Olsen et al. (2022) reported an marginal candidate,
GW190920_113516, whose secondary could be a heavy NS. With an effective
inspiral spin of $\chi_{\rm eff}=0.60^{+0.26}_{-0.07}$, this event could be a
potential NSBH merger with the NS first born. Román-Garza et al. (2021)
claimed that the fraction of the NS-first-born systems with different SN
engines is $\sim 10\%$ (see also Kinugawa et al., 2022). The event rate and
system parameter distribution for NS-first-born NSBH mergers are subject to
further studies in the future. With the upgrade of GW observatories and the
update of large survey telescopes, it is foreseen that one may detect high-
confidence GW signals with associated EM counterparts from NSBH mergers,
especially for NS-first-born NSBH mergers, in the near future GW-led multi-
messenger era.

YQ acknowledges the support from the Doctoral research start-up funding of
Anhui Normal University and from Key Laboratory for Relativistic Astrophysics
in Guangxi University. EWL is supported by the National Natural Science
Foundation of China (Grant Nos. 12133003, U1731239) and the Guangxi Science
Foundation (Grant No. AD17129006). This work is supported by the National
Natural Science Foundation of China (Grant Nos. 12192220, 12192221) and by the
Natural Science Foundation of Universities in Anhui Province (Grant No.
KJ2021A0106). Our implementation for natal kicks, imparted on the newly formed
BH, is based on the framework of Kalogera (1996); Wong et al. (2012) and a
recent work in Callister et al. (2021). Following the prescription in Mandel &
Müller (2020), the mean natal kick for BHs is

$\mu_{\mathrm{kick}}=v_{\mathrm{BH}}\frac{\max\left(M_{\mathrm{CO}}-M_{\mathrm{BH}},0\right)}{M_{\mathrm{BH}}},$
(1)

where $v_{\mathrm{BH}}$ is the BH scaling prefactor and $M_{\mathrm{CO}}$ is
the carbon-oxygen core (CO core) mass, respectively. We adopt
$v_{\mathrm{BH}}=200\,\mathrm{km\,s^{-1}}$ as suggested in Mandel & Müller
(2020). We also add an additional component drawn from a Gaussian distribution
with the standard deviation $0.3\mu_{\mathrm{kick}}$.

We update the semi-major axis and the eccentricity of the binary after the
supernova (SN) kick following Kalogera (1996)

$\begin{array}[]{r}a_{f}=G\left(M_{\rm NS}+M_{\rm
BH}\right)\bigg{[}\displaystyle\frac{2G\left(M_{\rm NS}+M_{\rm
BH}\right)}{a_{i}}-v_{\text{kick}}^{2}-v_{\text{orb}}^{2}-2v_{\text{kick},y}v_{\text{orb}}\bigg{]}^{-1}\end{array}$
(2)

and

$1-e^{2}=\frac{\left(v_{\text{kick},y}^{2}+v_{\text{kick},z}^{2}+v_{\text{orb}}^{2}+2v_{\text{kick},y}v_{\text{orb}}\right)a_{i}^{2}}{G\left(M_{\rm
NS}+M_{\rm BH}\right)a_{f}}$ (3)

where $G$ is Newton’s constant, $a_{i}$ and $a_{f}$ are the pre- and post-SN
semi-major axis, and $v_{\mathrm{kick},i}$ represents the $i$-th component of
the kick velocity $\mathbf{v}_{\text{kick}}$. Here $v_{\text{orb}}$, the pre-
SN orbital velocity of the BH progenitor relative to its companion (NS), is
given (Wong et al., 2012) by

$v_{\text{orb}}^{2}=G\left(M_{\rm NS}+M_{\rm pre-
SN}\right)\left(\frac{2}{r}-\frac{1}{a_{i}}\right),$ (4)

where $r$ is the orbital separation between the BH progenitor and its
companion at the pre-SN stage and $M_{\rm pre-SN}$ ($M_{\rm He}$ at the pre-SN
phase) is the mass of the BH progenitor just prior to the SN explosion. The
expression of $r$ is given in Equation (15) in Wong et al. (2012).

Following the same assumption (Kalogera, 1996) that the explosion is
instantaneous, $r$ remains unchanged. As a result, the angle $\theta$ between
the pre- and post-SN orbital planes is determined by

$\cos\theta=\frac{v_{\text{kick},y}+v_{\text{orb}}}{\left[\left(v_{\text{kick},y}+v_{\text{orb}}\right)^{2}+v_{\text{kick},z}^{2}\right]^{1/2}}.$
(5)

If the natal kick is too strong, the binary after the supernova explosion will
be disrupted. As shown in Callister et al. (2021), the binary is disrupted
when

$\beta<\frac{1}{2}+\frac{v_{\text{kick}}^{2}}{2v_{\text{orb}}^{2}}+\frac{\mathbf{v}_{\text{kick}}\cdot\mathbf{v}_{\text{orb}}}{v_{\text{orb}}^{2}},$
(6)

where $\beta$ is defined in Equation (7) in Kalogera (1996), the pre-SN-to-
post-SN ratio of the binary’s total mass. For binaries surviving after the SN
kicks, we can calculate the merger time via gravitational-wave emission
(Mandel, 2021), i.e.,

$T_{\rm merger}=\frac{5}{256}\frac{c^{5}a_{f}^{4}}{G^{3}(M_{\rm NS}+M_{\rm
BH})^{2}m_{r}}T(e),$ (7)

where $c$ is the speed of light and $m_{r}$ is the binary’s reduced mass, and
the function $T(e)$ is given by

$T(e)=\left(1+0.27e^{10}+0.33e^{20}+0.2e^{1000}\right)\left(1-e^{2}\right)^{7/2}.$
(8)

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