# A Channel to Form Fast-spinning Black Hole-Neutron Star Binary Mergers as
Multimessenger Sources. II. Accretion-induced Spin-up

Zhen-Han-Tao Wang Guangxi Key Laboratory for Relativistic Astrophysics, School
of Physical Science and Technology, Guangxi University, Nanning 530004, China
Rui-Chong Hu 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  Ying Qin Department of Physics, Anhui Normal University,
Wuhu, Anhui, 241002, China  Jin-Ping Zhu School of Physics and Astronomy,
Monash University, Clayton Victoria 3800, Australia  OzGrav: The ARC Centre of
Excellence for Gravitational Wave Discovery, Australia  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  Shuang-Xi Yi School of Physics and Physical Engineering, Qufu
Normal University, Qufu, Shandong 273165, China  Qin-Wen Tang Department of
Physics, School of Physics and Materials Science, Nanchang University,
Nanchang 330031, China  Xin-Wen Shu Department of Physics, Anhui Normal
University, Wuhu, Anhui, 241002, China  Fen Lyu Institute of Astronomy and
Astrophysics, Anqing Normal University, Anqing 246133, People’s Republic of
China  Guangxi Key Laboratory for Relativistic Astrophysics, School of
Physical Science and Technology, Guangxi University, Nanning 530004, China
En-Wei Liang Guangxi Key Laboratory for Relativistic Astrophysics, School of
Physical Science and Technology, Guangxi University, Nanning 530004, China

###### Abstract

In this work, we investigate an alternative channel for the formation of fast-
spinning black hole-neutron star (BHNS) binaries, in which super-Eddington
accretion is expected to occur in accreting BHs during the stable mass
transfer phase within BH-stripped helium (BH–He-rich) star binary systems. We
evolve intensive MESA grids of close-orbit BH–He-rich star systems to
systematically explore the projected aligned spins of BHs in BHNS binaries, as
well as the impact of different accretion limits on the tidal disruption
probability and electromagnetic (EM) signature of BHNS mergers. Most of the
BHs in BHNS mergers cannot be effectively spun up through accretion, if the
accretion rate is limited to $\lesssim 10\,\dot{M}_{\rm Edd}$, where
$\dot{M}_{\rm Edd}$ is the standard Eddington accretion limit. In order to
reach high spins (e.g., $\chi_{\rm BH}\gtrsim 0.5$), the BHs are required to
be born less massive (e.g., $\lesssim 3.0\,M_{\odot}$) in binary systems with
initial periods of $\lesssim 0.2-0.3\,{\rm days}$ and accrete material at
$\sim 100\,\dot{M}_{\rm Edd}$. However, even under this high accretion limit,
$\gtrsim 6\,M_{\odot}$ BHs are typically challenging to significantly spin up
and generate detectable associated EM signals. Our population simulations
suggest that different accretion limits have a slight impact on the ratio of
tidal disruption events. However, as the accretion limit increases, the EM
counterparts from the cosmological BHNS population can become bright overall.

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

##  1 Introduction

Mergers of black hole–neutron star (BHNS) are the prime targeted
gravitational-wave (GW) sources for the Advanced LIGO (Aasi et al., 2015),
Advanced Virgo (Acernese et al., 2015), and KAGRA (Aso et al., 2013) GW
detectors. At the end of the third observing run (O3), the LIGO-Virgo-KAGRA
(LVK) Collaboration for the first time identified two high-confidence GWs from
BHNS mergers, i.e., GW200105 and GW200115 Abbott et al. (2021a); Nitz et al.
(2023). GW200105 was inferred to be a merger between a
$8.9^{+1.2}_{-1.5}\,M_{\odot}$ BH with a near-zero spin of $\chi_{\rm
BH}=0.08^{+0.22}_{-0.08}$ and a $1.9^{+0.3}_{-0.2}\,M_{\odot}$ NS. The source
of GW200115 has relatively lower component masses with the BH mass of
$5.7^{+1.8}_{-2.1}\,M_{\odot}$ and the NS mass of
$1.5^{+0.7}_{-0.3}\,M_{\odot}$. The BH component of GW200115 was reported to
have a non-negligible mis-aligned spin with $\chi_{\rm
BH}=0.33^{+0.48}_{-0.29}$ (see also Zhu, 2021). However, the posterior
distributions of component masses and BH spin of GW200115 could be highly
impacted by the prior inference. Applying an astrophysically-motivated spin
prior inference ($\chi_{\rm BH}\sim 0$) to GW200115 leads to much tighter
constraints on the component masses, which were measured to be
$7.0^{+0.4}_{-0.4}\,M_{\odot}$ and $1.25^{+0.09}_{-0.04}\,M_{\odot}$ (Mandel &
Smith, 2021), respectively. Furthermore, a few marginal events detected in the
O3, e.g., GW190426_1522155 and GW191219_163120, could potentially originate
from BHNS mergers if they have astrophysical origins (Abbott et al., 2021b;
The LIGO Scientific Collaboration et al., 2021; Abbott et al., 2023). As the
LVK Collaboration began its latest campaign of O4, two BHNS candidates,
S230518h and S230529a, have been reported recently (Ligo Scientific
Collaboration et al., 2023a, b). More and more BHNS mergers are expected to be
observed in the O4 and O5 observing runs, as well as the next-generation
ground-based detectors, e.g., the Einstein Telescope (Punturo et al., 2010)
and the Cosmic Explorer (Reitze et al., 2019).

BHNS mergers that undergo tidal disruption are believed to power
electromagnetic (EM) counterparts, such as kilonovae (e.g., Li & Paczyński,
1998; Metzger et al., 2010; Kasen et al., 2013, 2017; Kyutoku et al., 2013,
2015; Kawaguchi et al., 2016, 2020; Barbieri et al., 2019; Zhu et al., 2020,
2022a; Darbha et al., 2021; Gompertz et al., 2023) and gamma-ray bursts (e.g.,
Paczynski, 1986, 1991; Eichler et al., 1989; Narayan et al., 1992; Zhang,
2018; Gottlieb et al., 2023). The tidal disruption, however, is only expected
for a fraction of the BHNS population under certain system parameters. More
specifically, the systems that have a low-mass NS component with a stiff
equation of state (EoS) and a low-mass BH with a high projected aligned spin
are more easily to generate tidal disruption and generate bright EM
signals111Weak EM signals could also be potentially produced for plunging
events if the BH component is charged or the NS is strongly magnetized (e.g.,
Zhang, 2019; Dai, 2019; Chen & Dai, 2021; Sridhar et al., 2021; D’Orazio et
al., 2022; Yuan et al., 2023; Most & Philippov, 2023). (e.g., Shibata et al.,
2009; Kyutoku et al., 2015; Foucart et al., 2018; Barbieri et al., 2019;
Raaijmakers et al., 2021; Zhu et al., 2021a, b, 2022b; Clarke et al., 2023;
Sarin et al., 2023). Among the above parameters, the BH aligned spin could be
one of the most critical parameter that affects the tidal disruption and EM
detection for the population of BHNS merger in the universe. It is widely
accepted that the majority of BHNS binaries are formed through the classical
isolated common envelope (CE) channel (e.g., Phinney, 1991; Tutukov &
Yungelson, 1993; Hurley et al., 2002; Kalogera et al., 2007; Belczynski et
al., 2016; Ablimit & Maeda, 2018; Kruckow et al., 2018; Mapelli & Giacobbo,
2018; Neijssel et al., 2019; Bavera et al., 2020; Belczynski et al., 2020;
Drozda et al., 2022; Shao & Li, 2021; Broekgaarden et al., 2021; Román-Garza
et al., 2021; Hu et al., 2022; Mandel & Broekgaarden, 2022; Xing et al.,
2023). In this standard formation scenario, BHs typically formed at wide
orbits are expected to possess low spins. The low BH spins in GW200105 and
GW200115 revealed by the GW observations support that these BHNS binaries were
originated from this standard formation scenario (Broekgaarden & Berger, 2021;
Zhu et al., 2022b; Chattopadhyay et al., 2022; Biscoveanu et al., 2023; Jiang
et al., 2023). Therefore, GW200105 and GW200115 could be likely plunging
events and be hard to generate bright EM counterparts, which could explain why
EM signals were not found by the follow-up observations of these GW events
(e.g., Anand et al., 2021; Zhu et al., 2021a; Gompertz et al., 2022; D’Orazio
et al., 2022). The population synthesis simulations and analyses for
population properties of O3 NSBH events (e.g., Zhu et al., 2021b, 2022b;
Fragione, 2021; Drozda et al., 2022; Biscoveanu et al., 2023; Colombo et al.,
2023) also predicted that most of BHNS mergers in the universe could be
plunging events without making any bright EM counterparts since these binaries
unusually contain low-spin BH components, while detectable EM counterparts
tend to be observed in BHNS systems with high aligned spin BHs.

In order to possess high-spin BHs, the BHs or BH progenitors should be spun up
before merging. One feasible scenario is that massive He-rich stars in close
binaries after the CE phase can be efficiently spun up by strong tides to form
fast-spinning BHs (Qin et al., 2018; Bavera et al., 2020; Hu et al., 2022),
namely “tidal-induced spin-up”. Various groups have also independently
investigated the spin-up of BHs in close binary systems through tidal
interactions, i.e., Detmers et al. (2008); Kushnir et al. (2016); Olejak &
Belczynski (2021). In order to form fast-spinning BHs in BHNS binaries, the NS
could be born first due to a reversal in the mass ratio of progenitor stars
during the Case A mass transfer (MT) stage. The massive He-rich star of the BH
progenitor can be formed in close binaries later, which could be tidally spun
up by the primary NS to potentially form a fast-spinning BH. InPaper I (Hu et
al., 2022), we carried out detailed binary modeling to investigate this
peculiar channel, mainly focusing on the properties of resulting BHs and their
associated tidal disruption probabilities (see also Román-Garza et al., 2021;
Chattopadhyay et al., 2021). Alternatively, the immediate progenitor of the
BHNS binary could be a BH–He-rich star system formed after the CE phase or
through chemically homogeneous evolution (Marchant et al., 2016; Mandel & de
Mink, 2016; de Mink & Mandel, 2016; Song et al., 2016; du Buisson et al.,
2020; Riley et al., 2021; Zevin et al., 2021; Bavera et al., 2022a, b; Qin et
al., 2022c). The BH can be spun up via the accretion during the stable MT
phase (see recent studies, Qin et al., 2022a; Shao & Li, 2022; Zevin & Bavera,
2022; Steinle et al., 2023), called as “accretion-induced spin-up”. Xing et
al. (2023) recently adopted a new binary population synthesis code POSYDON
(Fragos et al., 2023) to investigate the formation of merging BHNS binaries at
solar metallicity, finding that the BH always forms first with a low spin
magnitude (e.g. $\chi_{\rm BH}\lesssim 0.2$), and can subsequently reach
moderately spin magnitudes (e.g. $\chi_{\rm BH}\lesssim 0.4$) through stable
MT. It was found that high spins of BHs can be achieved if super-Eddington
accretion is allowed (e.g., Zevin & Bavera, 2022; Shao & Li, 2022; Qin et al.,
2022a). Besides, to significantly spin up BHs in BH–He-rich star binaries,
super-Eddington accretion must be at play (see recent investigations in
Steinle et al., 2023), as the accretion time is very limited due to the short
lifetime of massive He-rich stars. As pointed out by Steinle et al. (2023),
super-Eddington accretion is not impossible in principle, since the Eddington
limit is dependent on the geometry of the accretion and various kinds of
instabilities. It has been suggested that the majority of ultraluminous X-ray
sources are stellar-mass BHs or NSs undergoing super-Eddington accretion
(e.g., Bachetti et al., 2013; Sutton et al., 2013; Walton et al., 2014;
Middleton et al., 2015; Kitaki et al., 2017). In particular, the Galactic
X-ray binary SS 433 has been considered to be a super-Eddington accretion
system (see the detailed review of theoretical and observational progress in
Fabrika, 2004; Okuda et al., 2009 and a recent update in Cherepashchuk et al.,
2020).

In this study, we utilize stellar structure and binary evolution code MESA
(Paxton et al., 2011, 2013, 2015, 2018, 2019; Jermyn et al., 2023) to
investigate how fast-spinning BHs in BHNS binaries as potential GW sources can
be formed through super-Eddington accretion and their associated tidal
disruption possibility. We first describe the setup of our binary evolution
models in Section 2. Our main results of detailed binary evolution
calculations and population simulations are presented in Section 3. Finally,
we summarize the conclusions with some discussion in Section 4.

##  2 Methods

###  2.1 Main Physics Adopted in MESA

We adopt the released version r15140 of the MESA stellar evolution code
(Paxton et al., 2011, 2013, 2015, 2018, 2019; Jermyn et al., 2023) to perform
the detailed binary evolutionary calculations. Our single He-rich star models
are built following the same method introduced in Qin et al. (2018); Bavera et
al. (2020); Hu et al. (2022, 2023); Fragos et al. (2023); Qin et al. (2023);
Zhu et al. (2023) with two steps: 1) create pure single He-rich stars at the
zero-age helium main sequence (ZaHeMS); 2) relax the ZaHeMS models at given
metallicity to reach the thermal equilibrium where the luminosity from core
helium-burning just exceeds 99% total luminosity. In our models, we evolve He-
rich stars at the solar-metallicity environment with a metallicity of
$Z=Z_{\odot}$, where $Z_{\odot}$ is set to be 0.0142 (Asplund et al., 2009).
We model convection using the mixing-length theory (Böhm-Vitense, 1958) with a
mixing-length of $\alpha_{\rm mlt}=1.93$. Semiconvection (Langer et al., 1983)
with an efficiency parameter $\alpha=1.0$ is adopted in our modeling. We treat
the boundaries of the convective zone using the Ledoux criterion and consider
the step overshooting as an extension given by $\alpha_{\rm p}=0.1H_{\rm p}$,
where $H_{\rm p}$ is the pressure scale height at the Ledoux boundary limit.
The network approx12.net is used for our nucleosynthesis calculations. We
treat rotational mixing and angular momentum transport as diffusive processes
(Heger & Langer, 2000), including the effects of the Goldreich–Schubert–Fricke
instability, Eddington–Sweet circulations, as well as secular and dynamical
shear mixing. We adopt diffusive element mixing from these processes with an
efficiency parameter of $f_{\rm c}=1/30$ (Chaboyer & Zahn, 1992; Heger &
Langer, 2000).

We model stellar winds of He-rich stars as in Paper I and also take into
account the rotationally-enhanced wind mass loss (Heger & Langer, 1998;
Langer, 1998) in the following:

$\centering\dot{M}(\omega)=\dot{M}(0)\left(\frac{1}{1-\omega/\omega_{\rm
crit}}\right)^{\xi},\@add@centering$ (1)

where $\omega$ and $\omega_{\rm crit}$ are the angular velocity and critical
angular velocity at the surface, respectively. The critical angular velocity
can be expressed as $\omega_{\rm crit}=\sqrt{(1-L/L_{\rm Edd})GM/R^{3}}$,
where $L$, $M$, $R$ represent star’s total luminosity, mass and radius,
$L_{\rm Edd}$ is the Eddington luminosity, and $G$ is the gravitational
constant. The default value of the exponent $\xi=0.43$ is adopted from Langer
(1998). For more details on wind mass loss enhanced by rotation, We refer
readers to Section 5.2 in Qin et al. (2018). No gravity darkening effect is
accounted for (see Maeder & Meynet, 2000, for discussions on the impact of
this process). The mass transfer stability is also affected by the star’s
rotation. For instance, when an accretor (i.e., BH in this work) rotates, the
accretion rate of the rotating accretor might be reduced by a factor of
$(1-\omega/\omega_{\rm crit})$ (e.g., Stancliffe & Eldridge, 2009; Ablimit,
2021; Ablimit et al., 2022; Oh et al., 2023).

Stars have extended atmospheres and thus MT is expected to occur through the
first Lagrangian point even when the star’s radius $R_{1}$ is inside its Roche
lobe $R_{\rm RL}$ (namely Ritter scheme, Ritter, 1988). Kolb & Ritter (1990)
further extended the Ritter scheme to cover the case $R_{1}>R_{\rm RL}$. In
this work, we modeled MT following the Kolb scheme (Kolb & Ritter, 1990) and
the implicit MT method (Paxton et al., 2015). In addition, conservative mass
transfer between a He-rich star and its companion is adopted for simplicity.
For the tidal interactions, we apply the dynamical tides to He-rich stars with
radiative envelopes (Zahn, 1977). The timescale of synchronization is
calculated using the prescription mentioned in Zahn (1977); Hut (1981); Hurley
et al. (2002), while the tidal torque coefficient $E_{2}$ is adopted from the
updated fitting formula as in Qin et al. (2018).

We evolve He-rich stars until the carbon depletion is reached in the center.
We assume a white dwarf (WD) is formed when the final carbon/oxygen (C/O) core
mass is not higher than 1.37 $M_{\odot}$. In order to calculate the baryonic
remnant mass, we adopt the “delayed” supernova (SN) prescription (Fryer et
al., 2012). We also take into account the neutrino loss mechanism as described
in the study by Zevin et al. (2020). We then transform the baryonic remnant
mass to the NS mass by Equation (13) and (14) described in Fryer et al.
(2012). In this work, the maximum NS mass is set to be 2.5 $M_{\odot}$.

###  2.2 Super-Eddington Accretion onto Stellar-mass BHs

A BH accretes material through a disk at a rate $\dot{M}_{\rm acc}$ with the
radiation efficiency $\eta$ determined by its innermost stable circular orbit,
which is further dependent on the BH spin. It has been suggested that the
natal spin of the BH could be typically negligible (e.g., Fragos & McClintock,
2015; Qin et al., 2018; Fuller & Ma, 2019; Belczynski et al., 2020). For an
initially non-rotating BH, its mass and BH dimensionless spin parameter
increase through accretion (Bardeen, 1970; Bardeen et al., 1972; Podsiadlowski
et al., 2003) according to

$\dot{M}_{\rm BH}=(1-\eta)\dot{M}_{\rm acc},$ (2)

and

$\chi_{\rm BH}=\sqrt{\frac{2}{3}}\frac{M_{\rm BH,init}}{M_{\rm
BH}}\left(4-\sqrt{18\left(\frac{M_{\rm BH,init}}{M_{\rm
BH}}\right)^{2}-2}\right),$ (3)

for $M_{\rm BH}<\sqrt{6}M_{\rm BH,init}$, where the radiation efficiency is
$\eta=1-\sqrt{1-\left({M_{\rm BH}}/{3M_{\rm BH,init}}\right)^{2}}$ with the
BH’s initial mass $M_{\rm BH,init}$ and current mass $M_{\rm BH}$.

Assuming an Eddington-limited accretion, i.e., the critical rate at which the
outward force from the radiation pressure balances the inward gravitational
pull, the maximum accretion rate is defined as

$\dot{M}_{\rm Edd}=\frac{4\pi GM_{\rm BH}}{\kappa c\eta},$ (4)

where $c$ is the speed of light, and $\kappa$ is the opacity mainly determined
by pure electron scattering, i.e., $\kappa=0.2(1+X)\,{\rm cm}^{2}\,{\rm
g}^{-1}$ with the hydrogen mass fraction $X$.

Figure 1: BH dimensionless spin $\chi_{\rm BH}$ as a function of its accreted
material with different initial masses (Solid line: 3 $M_{\odot}$; dashed
line: 6 $M_{\odot}$; dotted line: 9 $M_{\odot}$).

As previously mentioned in the Introduction, super-Eddington accretion for the
BH accretors is valid in principle. Here, we relax the standard Eddington-
limited accretion to be 10 $\dot{M}_{\rm Edd}$ and 100 $\dot{M}_{\rm Edd}$
beyond which the BH can no longer accrete any material. By combining Equations
(2) and (3) to model the BH spin as a function of the accreted material, we
show the evolution of the BH spin with its increased mass for different
initial masses in Figure 1. We note that it is possible that the BH could spin
down (see Eqs. 42–44 in Lei et al., 2017) when the accretion rate is high
(e.g., exceeding Eddington accretion rate) due to launching a Blandford-Znajek
jet (Blandford & Znajek, 1977). Therefore, the spin magnitudes calculated in
this work can be considered to be an upper limit.

As an example, we evolve a close binary of a $4\,M_{\odot}$ He-rich star and a
$6\,M_{\odot}$ BH companion with an initial orbital period $P_{\rm
init}=0.114\,{\rm days}$. Three different accretion limits (1, 10, and
$100\,\dot{M}_{\rm Edd}$) are assumed in this modeling. In all cases, the
binary system is found to experience MT from the shell-helium burning He-rich
star (so-called case BB MT222Case BA(BB) MT: MT initiates during the core-
helium (shell-helium) burning phase for He-rich stars; Case BC MT: MT
initiates after shell-helium burning phase for He-rich stars.) onto the BH.
For the standard accretion limit (1 $\dot{M}_{\rm Edd}$) assumption, the BH
has a spin magnitude $\chi_{\rm BH}$ below 0.01 due to a very small amount of
mass accretion (see black dashed line in Figure 2). At 10 times Eddington
limits (10 $\dot{M}_{\rm Edd}$), the BH can accrete mass close to 0.1
$M_{\odot}$ and its spin magnitude $\chi_{\rm BH}$ is still low (e.g.,
$\chi_{\rm BH}<0.05$, see cyan solid line). In contrast, given 100
$\dot{M}_{\rm Edd}$, the BH can be efficiently spun up and reach a moderate
value ($\chi_{\rm BH}\sim 0.4$) by accreting $>0.7\,M_{\odot}$ (see green
solid line).

Figure 2: As a case study of our binary modeling, we show the BH dimensionless
spin $\chi_{\rm BH}$ evolving with accreted mass for a binary system, which
contains a $6\,M_{\odot}$ BH and a $4\,M_{\odot}$ He-rich star with an initial
orbital period $P_{\rm init}=0.114\,{\rm days}$. Three different accretion
limits are considered: solid green line: $100\,\dot{M}_{\rm Edd}$, solid cyan
line: $10\,\dot{M}_{\rm Edd}$; dashed black line: $1\,\dot{M}_{\rm Edd}$.

###  2.3 SN Kick Impacted onto NSs in BHNS Binaries

In general, NSs are expected to receive natal kicks when they are born via
core-collapse supernovae because of asymmetric ejection of matter and possibly
neutrinos. Therefore, the effect of kicks on the NS itself and on the
properties of the binary system (including the angular momentum evolution) is
important for the formation of various NS binaries (i.e., Belczyński & Bulik,
1999; Mandel et al., 2021; Willcox et al., 2021; Ablimit & Soker, 2024). When
the second-born NS is formed, the SN gives a kick velocity to the NS,
affecting the orbit of the binary system or potentially disrupting the system.
Although the BH generally has a spin along the pre-SN orbital angular momentum
axis, the SN kick to the newborn NS could result in a misalignment angle
between the BH spin and system orbit (e.g., Hills, 1983; Brandt &
Podsiadlowski, 1995; Kalogera, 1996; Mandel & Müller, 2020; Zhu, 2021). SN
kick could also affect the tidal disruption probability and the amount of
ejecta mass for BHNS mergers, since they are determined by the BH spin
projected onto the orientation of orbital angular momentum (projected aligned
spin hereafter), rather than the intrinsic BH spin. We thus take into account
the SN kick imparted onto the final BHNS binary system according to the
framework of Kalogera (1996); Wong et al. (2012); Callister et al. (2021).

Following the prescription outlined in Mandel & Müller (2020), the mean natal
kick for NSs can be expressed as

$\mu_{\rm kick}=v_{\rm NS}\frac{M_{\rm CO}-M_{\rm NS}}{M_{\rm NS}},$ (5)

where $v_{\rm NS}$ is the scaling prefactor for NS, and $M_{\rm CO}$ the
carbon-oxygen core mass. We adopt a default value of $v_{\rm NS}=400\,{\rm
km}\,{\rm s}^{-1}$ suggested by Mandel & Müller (2020). An additional
component drawn from a Gaussian distribution with a standard deviation of
$0.3\mu_{\rm kick}$ is considered. Thus, the kick velocity is $v_{\rm
kick}\sim\mathcal{N}(\mu=\mu_{\rm kick},\sigma=0.3\mu_{\rm kick})$.

The orbit of the binary system undergoes changes due to the formation of the
NS and SN kick. We assume a random SN kick direction and, then, obtain the
velocity component along each axis (i.e., $v_{{\rm orb},x}$, $v_{{\rm
orb},y}$, and $v_{{\rm orb},z}$). Our calculations assume that the orbit in
the pre-SN stage is circular, which means that the orbital separation $r$ is
equivalent to the initial semimajor axis $a_{\rm i}$. The orbital separation
$r$ remains unchanged under the assumption of an instantaneous SN explosion.
We then update the semimajor axis $a_{\rm f}$ and eccentricity $e$ of the
binary system after the SN kick following the approach outlined by Kalogera
(1996), i.e.,

$\begin{split}a_{\rm f}=&G(M_{\rm NS}+M_{\rm BH})\left[\frac{2G(M_{\rm
NS}+M_{\rm BH})}{a_{\rm i}}-v_{\rm kick}^{2}-\right.\\\ &\left.v_{\rm
orb}^{2}-2v_{{\rm kick},y}v_{\rm orb}\right]^{-1}\end{split}$ (6)

and

$1-e^{2}=\frac{(v_{{\rm kick},y}^{2}+v_{{\rm kick},z}^{2}+v_{\rm
orb}^{2}+2v_{{\rm kick},y}v_{\rm orb})a_{\rm i}^{2}}{G(M_{\rm NS}+M_{\rm
BH})a_{\rm f}},$ (7)

where $v_{{\rm kick},i}$ represents the $i$-th component of the kick velocity
$v_{\rm kick}$. Here, we adopt $v_{\rm orb}$, the pre-SN orbital velocity of
the NS progenitor relative to its BH companion, given in Wong et al. (2012) as
follows:

$v_{\rm orb}^{2}=G(M_{\rm BH}+M_{\rm pre-SN})\left(\frac{2}{r}-\frac{1}{a_{\rm
i}}\right),$ (8)

where $M_{\rm pre-SN}$ is the pre-SN mass of the NS progenitor.

Following the same assumption suggested in Callister et al. (2021), the binary
system is disrupted if

$\beta<\frac{1}{2}+\frac{v_{\rm kick}^{2}}{2v_{\rm orb}^{2}}+\frac{v_{{\rm
kick,}y}v_{\rm orb}}{v_{\rm orb}^{2}},$ (9)

where $\beta=(M_{\rm NS}+M_{\rm BH})/(M_{\rm pre-SN}+M_{\rm BH})$ is defined
in Equation (7) of Kalogera (1996). For the binary systems that survive after
the SN kicks, we then calculate the merger time via GW emission (Peters, 1964)
by

$T_{\rm merge}=\frac{5}{256}\frac{c^{5}a_{\rm f}^{4}}{G^{3}(M_{\rm NS}+M_{\rm
BH})^{2}m_{\rm r}}T(e),$ (10)

where $m_{\rm r}=M_{\rm NS}M_{\rm BH}/(M_{\rm NS}+M_{\rm BH})$ is the reduced
mass of the binary and
$T(e)=(1+0.27e^{10}+0.33e^{20}+0.2e^{1000})(1-e^{2})^{7/2}$ is adopted
following an accurate analytical fitting as in Mandel (2021).

The angle $\theta$ between the pre-and post-SN orbital planes can be
calculated by

$\cos\theta=\frac{v_{{\rm orb},y}+v_{\rm orb}}{\sqrt{(v_{{\rm kick},y}+v_{\rm
orb})^{2}+v_{{\rm kick},z}^{2}}}.$ (11)

Thus, the dimensionless projected aligned spin can be expressed as $\chi_{{\rm
BH},z}=\chi_{\rm BH}\cos\theta$.

##  3 Results

###  3.1 Various Mass Transfer Phases in the Formation of BHNS Binaries

Figure 3: Accreted mass (see the color bar) as a function of the initial
orbital period and initial mass of He-rich stars. Square: Case BA; triangle:
Case BB; diamond: Case BC; cross: Initial Overflow; Star: DDI); circle: No MT.
The three columns correspond to different Eddington limits and the three rows
refer to different masses of the BH as a companion.

As demonstrated above, BHs can be efficiently spun up under the condition of
high accretion limits. Here we perform parameter space analysis of a close
binary system consisting of a BH and a He-rich star. In particular, we
investigate the impact of three key parameters including initial He-rich star
mass, initial BH mass, and initial orbital period. We cover the initial He-
rich star mass of $2.5-9.0\,M_{\odot}$ in steps of $0.5\,M_{\odot}$, BH as a
point mass of $3.0-15.0\,M_{\odot}$ in steps of $3.0\,M_{\odot}$, and initial
orbital period in a range of $0.04-40\,{\rm days}$ with $\Delta$log $(P_{\rm
init}$[day]) $\sim$ 0.16 dex.

Let us begin by describing the upper-left panel of Figure 3. In the standard
accretion assumption, the He-rich star’s companion is assumed to be a BH with
an initial mass of $3.0\,M_{\odot}$. Most binary systems are found to
experience MT when the initial orbital period is shorter than $\sim 1.0\,{\rm
day}$. The more massive a He-rich star is, the shorter an orbital period is
required for the binary system to undergo interactions in terms of all kinds
of MT. This is because massive He-rich stars are more compact when compared
with low-mass He-rich stars (see their Figure 1 in Qin et al., 2023). Systems
that have various interactions are divided into three categories (Case BA,
Case BB, and Case BC) based on MT occurring in different evolutionary phases
of He-rich stars. For binary systems with Case BA MT, the accreted mass onto a
BH can reach $\sim 0.3\,M_{\odot}$. For binary systems that experience Case BB
MT, the accreted mass onto BHs can be higher than $0.01\,M_{\odot}$ when
$M_{\rm ZamsHe}\leq 3.5\,M_{\odot}$. Furthermore, BHs in all binary systems
that have gone through Case BC MT can accrete mass $\sim$ $0.001\,M_{\odot}$.
We also note that a small fraction of systems with higher initial mass and
lower initial orbital periods (e.g., $P_{\rm init}\sim 0.1\,{\rm days}$ and
$7.5\,M_{\odot}\leq M_{\rm ZamsHe}\leq 9.0\,M_{\odot}$) are found to
experience unstable MT, leading to the dynamical delayed instability (DDI, see
Ivanova et al., 2003). Additionally, the systems with an initial orbital
period of $P_{\rm init}<0.1\,{\rm day}$ undergo initial overflow through the
first Lagrangian point, which is considered to be located in the forbidden
parameter space in this work.

When considering $10\,\dot{M}_{\rm Edd}$ (see the top-middle panel in the
first row), one can see that the initial conditions (initial mass and orbital
period) of the binary systems play no impact on the parameter space of the MT.
The stark difference, however, is that the system accretes more mass when
compared with the standard Eddington limit. Such difference becomes even more
obvious when considering $100\,\dot{M}_{\rm Edd}$ (see the top-right panel in
the first row). For instance, the accreted mass onto a BH via Case BB (BC) MT
can reach up to $\sim 1.0\,M_{\odot}$ ($0.1\,M_{\odot}$). It is worth noting
that the accreted mass through Case BA MT can even reach beyond
$1.0\,M_{\odot}$. For more massive primary BHs ($M_{\rm
BH,init}=6.0\,M_{\odot}$ in the second row and $M_{\rm
BH,init}=9.0\,M_{\odot}$ in the third row), we note that more systems instead
are found to undergo Case BA MT, producing much higher accreted mass even
under the condition of the standard Eddington limit.

Figure 4: As in Figure 3, but the color bar corresponds to the final spin of
the BH via accretion from its companion He-rich star.

As presented in Figure 2, the Eddington limit plays a critical role in the
accreted mass of the BH and thus its corresponding spin magnitude. We here
then carry out a more detailed investigation of the impact of the binary’s
different initial conditions on the spin magnitudes. It is found in the upper-
left panel of Figure 4 that Case BB/BC MT cannot significantly spin up the BH
(i.e., $\chi_{\rm BH}$ $\lesssim 0.1$) due to a limited time of accretion,
while Case BA MT instead can allow the BH spin to reach a moderate value of
$\chi_{\rm BH}\,\sim 0.3$. Similar results are obtained when a higher
Eddington accretion limit (e.g. 10 $\dot{M}_{\rm Edd}$, see the middle panel
in the first row) is assumed. Notably, the BH can be efficiently spun up under
100 $\dot{M}_{\rm Edd}$ assumption. The spin magnitude can reach a value of
$\chi_{\rm BH}$ close to $\sim 0.7$ ($\sim 0.5$) for Case BB (Case BC) MT,
even higher than 0.8 for Case BA MT. Nevertheless, the spin-up of a more
massive BH companion becomes much less efficient when compared with a low-mass
BH companion (see the second row for a 6 $M_{\odot}$ BH and the third row for
a 9 $M_{\odot}$ BH, respectively), which has been demonstrated at the
beginning of Section 3 (see Figure 1).

Figure 5: As in Figure 3, but three color bars from top to bottom rows refer
to the mass of NS formed based on the “delayed” SN prescription (Fryer et al.,
2012), the fraction of BHNS binaries that can be merged within Hubble time
with SN kicks considered, and $T_{\rm merger}$ of BHNS binaries due to GW
emissions, respectively. Square: WD; triangle: NS; dot: BH; cross: Initial
Overflow; Star: DDI. Left panel: 3.0 $M_{\odot}$ BH companion; middle panel:
6.0 $M_{\odot}$ BH companion; right panel: 9.0 $M_{\odot}$ BH companion. For
the merger fraction, we use black triangles to mark BHNS binaries whose merger
times are beyond Hubble time. Different Eddington accretion limits have little
impact on the mass of resulting NS, merger fraction, and $T_{\rm merger}$, and
hence other similar figures corresponding to higher Eddington limits are not
shown here.

Figure 5 presents various outcomes of the resulting compact binary objects
(BHWD, BHNS, and BHBH), and only BHNS binaries are presented in this work.
Since different accretion limits have little impact on the mass of newborn NS,
merger fraction, and $T_{\rm merger}$, we only describe the results under the
standard Eddington accretion assumption. In the left panel of the top row, we
present the mass of NS following the “delayed” SN prescription (Fryer et al.,
2012) after the He-rich star reaches its carbon depletion in the center.
Clearly, the initial orbital period has a slight impact on the final mass of
the massive He-rich star and thus the resulting NS mass. This is because He-
rich stars could be tidally locked by tides in close binaries (Qin et al.,
2018), which allows them to have different rotation rates depending on their
current orbital periods. Furthermore, rotation is found to enhance wind mass
loss (e.g., Langer, 1998), which potentially reduces the resulting mass for
massive stars (see their Figure 3 in Lyu et al., 2023). It is also found that
the BH companion with a slightly higher mass has negligible impact on the mass
of the resulting NS (see the middle and right panel).

The SN kick imparted onto a newborn NS can change the separation of the binary
systems, some of which might be disrupted. Similar to the same approach in
Paper I, we obtain the kick velocity and kick direction by drawing $10^{5}$
times from a Gaussian distribution and a random distribution, respectively. As
shown in the middle panels of Figure 5, the color represents the merger
fraction of each BHNS system that can be merged within Hubble time. In the
left panel of the middle row (i.e. the standard Eddington limit), we can see
there are two main findings. First of all, the merger fraction of the binary
system is higher when its initial orbital period is shorter. This is because
more BHNS binaries are disrupted or too wide to merge within Hubble time.
Besides, we note that the merger fraction decreases when increasing the
initial mass of the He-rich star, resulting in a higher kick velocity by
following the expression in Equation (5). After the NS is formed, GW emission
shrinks the orbit of the BHNS binaries by removing their orbital angular
momentum, leading to the merger of the two compact objects. The timescale for
the merger of a binary compact object due to GWs is calculated following
Equation (10). In order to take into account the impact of SN kicks, we adopt
the median of the merger time in $10^{5}$ times as $T_{\rm merger}$ in the
bottom panels of Figure 5. It is expected that $T_{\rm merger}$ is shorter for
BHNS binaries with an initially shorter orbit. Furthermore, $T_{\rm merger}$
is shorter when initial He-rich stars are more massive. This finding is mainly
due to the anticorrelation between the total mass and the merging timescale
(see Section 6 for more details in Qin et al., 2018). Additionally, it is
clearly shown that both the Eddington limit and the BH companion mass have a
negligible impact on the merger fraction and $T_{\rm merger}$.

###  3.2 Parameter Space of an NS Disrupted by a BH

Figure 6: As in Figure 5, but the color bar refer to $\cos\theta$ values given
in Equation (11). The grey region is marked for the parameter space in which
Case BB/BC MT occurs when forming BHNS binaries.

Whether an NS can be tidally disrupted by a BH is exclusively determined by
the amount of the remnant mass after the BHNS merger (hereafter abbreviated as
remnant mass), which can be given by a fitting formula from Foucart et al.
(2018), i.e.,

$\frac{M_{\rm total,fit}}{M^{\rm b}_{\rm NS}}=\left[{\rm
max}\left(\alpha\frac{1-2C_{\rm NS}}{\eta^{1/3}}-\beta\tilde{R}_{\rm
ISCO}\frac{C_{\rm NS}}{\eta}+\gamma\right),0\right]^{\delta},$ (12)

where $\alpha=0.406$, $\beta=0.139$, $\gamma=0.255$, $\delta=1.761$ are the
fitting parameters obtained following numerical relativity simulations,
$\tilde{R}_{\rm ISCO}=3+Z_{2}-{\rm sign}(\chi_{{\rm
BH},z})\sqrt{(3-Z_{1})(3+Z_{1}+2Z_{2})}$ is the normalized innermost stable
circular orbit radius (Bardeen et al., 1972) along the orbital angular
momentum, with $Z_{1}=1+(1-\chi_{{\rm BH},z}^{2})^{1/3}\left[(1+\chi_{{\rm
BH},z})^{1/3}+(1-\chi_{{\rm BH},z})^{1/3})\right]$, $Z_{2}=\sqrt{3\chi_{{\rm
BH},z}^{2}+Z_{1}^{2}}$, $\eta=Q/(1+Q)^{2}$, and $Q=M_{\rm BH}/M_{\rm NS}$ is
the mass ratio. $M_{\rm NS}^{\rm b}$ is the baryonic mass of the NS, which can
be expressed as an empirical formula with the NS mass for different EoS
introduced by Gao et al. (2020):

$M^{\rm b}_{\rm NS}=M_{\rm NS}+A_{1}M_{\rm NS}^{2}+A_{2}M_{\rm NS}^{3},$ (13)

where $M^{\rm b}_{\rm NS}$ and $M_{\rm NS}$ are in units of $M_{\odot}$ . We
use the commonly employed EoS DD2 (Typel et al., 2010), with a Tolman-
Oppenheimer-Volkoff mass of $2.42\,M_{\odot}$, which is close to the maximum
NS mass we have defined for our detailed binary evolution simulations (see
Section 2.1), as a representative EoS. In Equation (13), $A_{1}=0.046$, and
$A_{2}=0.010$ for DD2 given by Gao et al. (2020). We calculate the NS
compactness $C_{\rm NS}$ following Coughlin et al. (2017):

$C_{\rm NS}=1.1056\left(\frac{M^{\rm b}_{\rm NS}}{M_{\rm
NS}}-1\right)^{0.8277}.$ (14)

BHNS merger is considered to be a plunging event if $M_{\rm total,fit}=0$,
while the NS can be tidally disrupted by the BH companion to generate bright
EM signals when $M_{\rm tot,fit}>0$. We note that the remnant mass is not
equivalent to the ejecta mass. However, a higher remnant mass implies that the
EM signals from BHNS mergers can be bright. Hereafter, we use the remnant mass
to scale the brightness of the EM counterparts.

In Section 3.1, we explore the intrinsic spin of BH in BHNS binaries by
considering different accretion limits during the Case BB/BC MT phase.
Nevertheless, whether the tidal disruption of BHNS mergers can occur depends
on the projected aligned spin of the BH, as indicated by Equation (12), rather
than its intrinsic spin. The kick may result in a tilt angle $\theta$ between
the angular momentum of the BH spin and the orbital plane, which could affect
the tidal disruption of BHNS mergers. Following Equation (11), we present the
averaged $\cos\theta$ value for each BHNS system in Figure 6. It is shown that
the tilt angle tends to be larger when the He-rich star is initially more
massive and the initial orbit is wider, potentially producing a stronger kick
and a wider pre-SN orbital separation. This result is consistent with the
earlier finding (see their Figure 1 in Zhu, 2021). On the one hand, as shown
in Figure 6, we find that the kick has little effect on the tilt angle of
binary systems that experience Case BB/BC MT to spin up BH components. On the
other hand, the progenitor systems with $M_{\rm ZamsHe}\gtrsim 3.5\,M_{\odot}$
and $P_{\rm init}\gtrsim 2-3\,{\rm days}$ can finally remain BHNS binaries
consisting a negative-spin BH. However, due to the weak tidal effect to
inefficiently accelerate the BH progenitors 333Figure 2 in Qin et al. (2018)
shows that tides can be efficient for binary systems with $P_{\rm
init}\lesssim 2.0\,{\rm days}$. and the absence of an MT process to spin up
the BHs, these BHs thus typically have near-zero spins. Therefore, one can
directly use the intrinsic BH spin in Figure 4 to judge whether or not the
tidal disruption happens for a GW BHNS merger.

Figure 7: Parameter space (represented by the direction of the white arrows
for specific NS mass) of tidal disruption as a function of BH projected
aligned spin and its mass with different masses of NS (the color bar). NS mass
contours for $1.1,\,1.5,\,1.9,$ and $2.3\,M_{\odot}$ are marked with solid
white lines. For GW200105 (black solid line) and GW200115 (red solid line),
the 90% credible posterior distributions of the parameters obtained from
Abbott et al. (2021c) are displayed, respectively. Corresponding 90%
confidence interval of $M_{\rm NS}$ for these two sources are marked by dashed
lines (red dashed lines: GW200115, black dashed lines: GW200105).

Following Equation (13), Figure 7 illustrates the parameter space in which NSs
can be tidally disrupted by BHs. Tidal disruptions are more likely to occur
for binary systems comprising a lower-mass BH with higher projected aligned
spin and a lower-mass NS. Given the posterior distributions of the parameters
of GW200105 and GW200115 inferred by the LVK collaboration (Abbott et al.,
2021c), our detailed calculations could reproduce the two merging events. It
also shows that the two systems are most likely plunging events as shown in
Figure 7, consistent with earlier findings (i.e., Zhu et al., 2021a;
Broekgaarden & Berger, 2021; Fragione, 2021; Hu et al., 2022). On the one
hand, despite BHs having near-zero spins, BHNS mergers involving a
representative Galactic NS with a mass of $M_{\rm NS}=1.4\,M_{\odot}$ can
easily become disrupted events if the mass of the BH component is lower, i.e.,
$M_{\rm BH}\lesssim 6\,M_{\odot}$. On the other hand, as the BH mass
increases, the required minimum spin magnitude allowing for tidal disruptions
in BHNS mergers also increases accordingly. When considering BHs with a mass
of $M_{\rm BH}\gtrsim 9\,M_{\odot}$, BHNS mergers can always result in
plunging events unless the BH spin is $\chi_{\rm BH}\lesssim 0.5$, which are
hard to be reached even at high accretion limits (see Figure 4). Therefore, we
expect that accretion-induced spin-up would have an insignificant impact on
the tidal disruption probability for BHNS mergers. Similar findings can also
be obtained from Figure 8, where we evaluate tidal disruptions under various
initial conditions and different accretion limits.

Figure 8: As in Figure 5, but the color bar refers to the total remnant mass
outside the BH. The grey background masks the parameter space in which the
merger time of BHNS binaries due to the GW emission is not longer than Hubble
time.

###  3.3 Remnant mass of disrupted BHNS mergers

In Figure 8, we estimate the remnant mass for disrupted BHNS mergers with
consideration of various initial conditions and different accretion limits.
BHNS binaries, capable of merging within the Hubble time, are denoted by the
grey background. When the BH accretion rate is constrained to the standard
Eddington rate (i.e., $1\dot{M}_{\rm Edd}$), most primary BHs cannot be
efficiently spun up through accretion (see Figure 4). The initial orbital
period does not show any impact on the amount of the remnant mass. A more
massive He-rich star tends to form a higher-mass NS, allowing for the ejection
of less remnant mass when the NS merges with its BH companion. As a higher
accretion limit is allowed, lower-mass BHs in close-orbit BHNS binaries with
smaller $P_{\rm init}$ can be more significantly spun up through accretion
(refer to Figure 4), thus easily ejecting more materials when BHNS mergers
occur. This trend is more prominent for high accretion limits, especially
under a rate of $100\,\dot{M}_{\rm Edd}$. Therefore, it is expected that BHNS
binaries with a higher accretion rate limit can eject more remnant mass and
thus generate brighter EM signals upon merging. It is worth noting that
accretion-induced spin-up is less efficient when compared with BHs spun up
through tidal interaction on progenitors (see Paper I).

Based on Figure 8, we can further explore the effect of different accretion
limits on population properties of the tidal disruption and remnant mass. To
achieve this, we perform multi-dimensional interpolation using the calculated
remnant mass in Figure 8. In what follows, we describe how the initial
parameter distribution of BH–He-rich star binary systems just after the CE
phase is adopted. Referring to Figure 14 in Broekgaarden et al. (2021), the
distribution of initial BH mass in BH–He-rich star binaries is highly
dependent on the model assumptions. For simplicity, we adopt three
representative empirical BH mass spectra in the literature (see the first
column in Table 1), including I) Exponential decay given by Özel et al.
(2010):

$P(M_{\rm BH})=\begin{cases}{\rm exp}[(M_{\rm c}-M)/M_{\rm scale}]/M_{\rm
scale}&,M>M_{\rm c}\\\ 0&,M\leqslant M_{\rm c}\end{cases}$ (15)

where $M_{\rm c}=2.5\,M_{\odot}$, $M_{\rm scale}=1.57\,M_{\odot}$ (see also
Steiner et al., 2012). II) Uniform BH mass distribution in a range of 2.5 –
5.0 $M_{\odot}$ \+ Power law function from 5.0 $M_{\odot}$ to 15 $M_{\odot}$
with a slope index of –4.8, and here we adopt 5.0 $M_{\odot}$ as a connecting
point between the two functions (also see Power law distribution of BH mass in
Farr et al., 2011; Abbott et al., 2016); III) Normal distribution: $P(M_{\rm
BH}/M_{\odot})\sim\mathcal{N}(\mu=5.0,\sigma=2.5)$ (the same BH mass function
as in Özel et al., 2010, but with different values of mean and standard
deviation in order to match the observation of GWTC-3). In addition, we simply
assume a Power law distribution for the initial He-rich star mass (i.e.,
$dN/dM_{\rm ZamsHe}\varpropto M_{\rm ZamsHe}^{-3.3}$, similar to the initial
mass function as in Kroupa, 2001, but with an index of -3.3) within a range of
[2.5,10] $M_{\odot}$ in which NS is formed, and adopt a uniform distribution
for the initial periods of BH–He-rich binaries in logarithmic space, i.e.,
[0.04, 40] $\rm days$ (similar to the range as in Zhang et al., 2023, see the
Section 4.3).

BH mass spectrum |  $M_{\rm total,fit}$ [$10^{-2}$ $M_{\odot}$] ($1\,\dot{M}_{\rm Edd}$) |  $M_{\rm total,fit}$ [$10^{-2}$ $M_{\odot}$] ($10\,\dot{M}_{\rm Edd}$) |  $M_{\rm total,fit}$ [$10^{-2}$ $M_{\odot}$] ($100\,\dot{M}_{\rm Edd}$)
---|---|---|---
Model I: Exponential decay | $3.89^{+3.65}_{-3.86}$ | $4.08^{+4.27}_{-4.05}$ | $6.0^{+11.6}_{-5.9}$
Model II: Uniform + Power law | $3.74^{+3.61}_{-3.60}$ | $4.08^{+4.02}_{-3.94}$ | $6.5^{+10.3}_{-6.2}$
Model III: Normal | $0.85^{+5.99}_{-0.84}$ | $0.90^{+6.40}_{-0.88}$ | $2.6^{+11.1}_{-2.6}$
Table 1: Median values with 90% credible intervals of remnant mass $M_{\rm
total,fit}$ for BHNS mergers with different assumptions of BH mass spectrum.
Column 1: BH mass spectrum. The Column 2, 3, and 4 correspond to the remnant
mass for 1 $\dot{M}_{\rm Edd}$ 10 $\dot{M}_{\rm Edd}$ and 100 $\dot{M}_{\rm
Edd}$, respectively.

Based on different parameter distributions of the BH–He-rich star binaries, we
then perform $10^{5}$ sampling calculations. First of all, our findings show
that the probabilities of the tidal disruption of BHNS systems are moderately
dependent on the assumption of BH mass spectrum, i.e., Model I: 0.799, Model
II: 0.841, and Model III: 0.702 for the standard Eddington accretion limit.
Furthermore, given the same assumption of BH mass spectrum, the corresponding
probabilities of the tidal disruption slightly increase with higher Eddington
accretion limits, i.e., 10 $\dot{M}_{\rm Edd}$ (100 $\dot{M}_{\rm Edd}$):
0.808 (0.818), 0.842 (0.855), and 0.705 (0.736). Thus, a higher accretion-rate
MT does not significantly lead to more BHNS mergers undergoing tidal
disruption, consistent with the discussion above in Figures 7 and 8. Notably,
the fraction of disrupted events in BHNS mergers was recently inferred to be
$0.6-9\%$ in Drozda et al. (2022), $0-18.6\%$ in Xing et al. (2023), and $\sim
0-70\%$ in Broekgaarden et al. (2021), which are highly determined by model
assumptions. Our simulated fraction of disrupted events is slightly higher
than that in these studies. This difference could be attributed to the BH
spectra used in our work, which tend to generate more low-mass BHs with masses
of $\lesssim 7\,M_{\odot}$.

In Table 1, by assuming various Eddington accretion limits, we present the
median value with a 90% credible level of the remnant mass $M_{\rm total,fit}$
for disrupted BHNS population. We note that the BH mass spectrum plays a key
role in determining the remnant mass. However, across different BH spectrum
models, the trend of changes in the distribution of the remnant masses remains
consistent with varying accretion limits. The amount of the remnant mass when
adopting $1\dot{M}_{\rm Edd}$ shows a slight increase when compared to that
for $10\dot{M}_{\rm Edd}$. However, if the accretion rate is limited to
$100\,\dot{M}_{\rm Edd}$, the median and upper limit of the remnant mass
increase significantly. As an example, we present the probability density
function (PDF) of the remnant mass for Model I in Figure 9. Compared to
$1\dot{M}_{\rm Edd}$ and $10\dot{M}_{\rm Edd}$, where the remnant mass is
always below $\sim 0.1\,M_{\odot}$, the PDF obtained using $100\dot{M}_{\rm
Edd}$ can have a much broader distribution with a maximum value even extending
up to $\sim 0.25\,M_{\odot}$. Thus, we can conclude that the population
properties of disrupted BHNS mergers by considering different accretion limits
also reveal that higher Eddington accretion limits show an insignificant
effect on the tidal disruption probabilities. However, they can contribute to
the generation of more bright EM counterparts from the BHNS population.

Figure 9: Probability density function of remnant mass $M_{\rm total,fit}$ in
BHNS binaries for example Model I in Table 1. Solid, dashed, and dotted lines
represent the accretion limits of 1 $\dot{M}_{\rm Edd}$, 10 $\dot{M}_{\rm
Edd}$, and 100 $\dot{M}_{\rm Edd}$, respectively.

##  4 Conclusions and Discussion

As presented in Paper I, we study a channel (channel I) to form BHNS binaries
in which the tides in close orbits can efficiently spin up the progenitor of
BHs and thus produce fast-spinning BHs (also see earlier results in Qin et
al., 2018; Bavera et al., 2020). In this work, we explore an alternative
channel (channel II) where the BH forms first to produce fast-spinning BHs in
BHNS binaries as potential GW sources. It is expected the relative rate
density of BHNS binaries formed through channel II is higher when comparing
with BHNS binaries formed through channel I. Assuming the first-born BH with a
near-zero spin (e.g., $\chi_{\rm BH}\sim 0.01$, predicted by the revised
version of Tayler-Spruit dynamo as in Fuller & Ma, 2019), we first perform
detailed binary modeling for the evolution of the immediate progenitors of
BHNS binaries, in which BHs and massive He-rich stars are in close orbits.

For the initial parameter space, we cover the mass of the He-rich star from
2.5 $M_{\odot}$ to 9.0 $M_{\odot}$ and the BH companion with a mass range of
$3.0-15.0\,M_{\odot}$, as well as the orbital period in $0.04-40\,{\rm days}$.
In order to investigate how the BH can be significantly spun up through
accretion, we assume three different Eddington accretion limits, i.e., 1
$\dot{M}_{\rm Edd}$, 10 $\dot{M}_{\rm Edd}$, and 100 $\dot{M}_{\rm Edd}$. In
the standard Eddington accretion limit, the mass accreted onto BHs via Case BB
(Case BC) MT can reach $\sim 0.01\,M_{\odot}$ ($\sim 0.01\,M_{\odot}$). The
amount of accreted mass is rarely affected by the BH companion mass. However,
assuming 100 $\dot{M}_{\rm Edd}$, a 3 $M_{\odot}$ BH as the companion accretes
mass larger than 1.0 $M_{\odot}$ via Case BB MT. In this case, the BH can be
efficiently spun up (e.g., $\chi_{\rm BH}>0.6$). We note that the fraction of
binary systems that experience Case BB MT is $\sim 9\%$, some of which could
potentially lead to fast-spinning BHNS binaries with super-Eddington
accretion. When considering SN kicks imparted onto the newly formed NS, we
note that the initial orbital period $P_{\rm init}$ for producing mergeable
BHNS binaries (i.e., $T_{\rm merger}$ $<$ Hubble time) is required to be
shorter than $\sim 2\,{\rm days}$. Additionally, we find the SN kicks play a
negligible role in tilting the plane of the newly-formed BHNS binaries when
$M_{\rm ZamsHe}\lesssim 3.5\,M_{\odot}$ and $P_{\rm init}\lesssim 2-3\,{\rm
days}$, resulting in no impact on the amount of remnant mass of mergeable BHNS
systems within Hubble time.

Based on the detailed binary modeling of our intense MESA grids, we further
adopt multi-dimensional interpolation to estimate the remnant mass during the
tidal disruption of the BHNS binaries. Assuming various parameter
distributions of BH–He-rich star binaries (component masses and orbital
period) to study the population properties of BHNS mergers, we find that the
BH mass spectrum plays an essential role in determining the amount of remnant
mass of the tidal disruption in BHNS binaries. Based on our simulated
population properties of disrupted BHNS mergers by considering different
accretion limits, our results reveal that higher accretion limits show an
insignificant effect on the tidal disruption probabilities. However, they can
contribute to the generation of more bright EM counterparts from the BHNS
population.

The observability of EM counterparts of BHNS mergers is dependent mainly on
the magnitude of the BH-aligned spin. The physical process of forming fast-
spinning BHs in BHNS binaries remains uncertain. On the one hand, if NSs form
first, the BH progenitors can be efficiently spun up by strong tides,
resulting in fast-spinning BHs (e.g., Qin et al., 2018; Bavera et al., 2020;
Hu et al., 2022). On the other hand, the first-born BH could inherit a very
high spin from its progenitor assuming a weak angular momentum transport
between the stellar core and its envelope (Qin et al., 2019, 2022b). The
physics of the angular momentum transport within stars is still uncertain.
Angular momentum is transported mainly by meridional currents, the stellar
core-envelope coupling is less efficient (e.g., Georgy et al., 2012), which
allows the core to retain more angular momentum and thus forms fast-spinning
BHs.

It is worth noting that some other physical processes could have an impact on
the spin magnitudes of resulting BHs. We highlight that the magnetic braking
(i.e., Mestel, 1968) is not taken into account when modeling low-mass He-rich
stars. Therefore, He-rich stars could potentially experience spin-down arising
from the coupling of the stellar wind to a magnetic field.

As shown in Xing et al. (2023), BHs are found to be always formed first and
their spins are typically low (i.e., $\chi_{\rm BH}\lesssim 0.2$). However,
some BHs can be subsequently spun moderately up ($X_{\rm BH}\,\sim 0.4$)
through stable Case BB MT. We note that the inferred probabilities of tidal
disruption for BHNS binaries from different groups are very different
(Broekgaarden et al., 2021; Xing et al., 2023). These findings mainly arise
from the specific parameter distributions of BHNS binaries. For instance, the
BH mass spectrum, which depends on the model assumptions, is highly uncertain.
We expect that the LVK Collaboration will detect more BHNS binaries in
upcoming observing runs (O4 and O5), which will help uncover the properties of
BHNS mergers.

This work was supported by Anhui Provincial Natural Science Foundation (grant
No. 2308085MA29) and the Natural Science Foundation of Universities in Anhui
Province (grant No. KJ2021A0106). YQ acknowledges the support of the Key
Laboratory for Relativistic Astrophysics at Guangxi University. JPZ thanks the
COMPAS group at Monash University. QWT acknowledges support from the Natural
Science Foundation of Jiangxi Province of China (grant No. 20224ACB211001).
This work was partially supported by the National Natural Science Foundation
of China (grant Nos. 12065017, 12192220, 12192221, 12133003, U2038106)), The
inlists and input files to reproduce our simulations and associated data
products are available at https://doi.org/10.5281/zenodo.10477135 (catalog
https://doi.org/10.5281/zenodo.10477135). All figures are made with the free
Python module Matplotlib (Hunter, 2007).

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11institutetext: Indian Institute of Technology, Kharagpur, India
22institutetext: Nanyang Technological University, Singapore

# Automatic segmentation of skin lesions using deep learning

Joshua Peter Ebenezer 11 Jagath C. Rajapakse 22

###### Abstract

This paper summarizes the method used in our submission to Task 1 of the
International Skin Imaging Collaboration’s (ISIC) Skin Lesion Analysis Towards
Melanoma Detection challenge held in 2018. We used a fully automated method to
accurately segment lesion boundaries from dermoscopic images. A U-net deep
learning network is trained on publicly available data from ISIC. We introduce
the use of intensity, color, and texture enhancement operations as pre-
processing steps and morphological operations and contour identification as
post-processing steps.

###### Keywords:

contrast enhancement, deep learning, image segmentation, skin cancer, U-net

##  1 Introduction

The International Skin Imaging Collaboration111https://isic-archive.com
(ISIC), an international effort to improve melanoma diagnosis, holds a
recurring challenge every year for the developoment of image analysis tools
for segmentation of dermoscopic images. This paper summarises our submission
to Task 1 of the challenge, which is lesion boundary segmentation. The data
released for the ISIC 2018 challenge were used for our submission [1, 2]. No
private datasets or other publicly available datasets were used.

In our submission, we used a deep learning architecture and an image
processing pipeline for accurate segmentation of skin cancer lesions from
dermoscopic images. Images of lesions generally have poor contrast and
boundaries of the lesions are often hard to outline as lesion color is similar
to the color of the skin near the edges. The pre-processing is able to
significantly enhance the information in the images and suppress noise.

##  2 Methodology

###  2.1 Intensity contrast enhancement

Skin lesion images are color images. The contrast enhancement algorithm based
on the layered-difference representation of 2D histograms proposed by Lee at
al. [3] is used to enhance the contrast of the intensity of the image, where
intensity is defined as the arithmetic mean of the red (R), blue (G), and
green (G) channels.

###  2.2 Hue-preserving color enhancement

Color provides details of skin lesions and following Naik and Murthy [4], we
apply a hue-preserving color enhancement technique to enhance the contrast of
the color images. Hue is the attribute of color according to which an area of
an image appears similar to a perceived color. Color enhancement techniques
generally involve non-linear transformations between color spaces, which lead
to the gamut problem when pixel values go out of the acceptable bounds of the
RGB space during conversion, which causes undesirable changes in the hue. The
resulting dermoscopic images have enhanced color, their hue is preserved, and
they have clearer lesions.

###  2.3 Texture enhancement

We use the multiscale texture enhancement algorithm based on fractional
differential masks proposed by Pu et al. [5]. The second mask proposed is used
as it has the best precision, convergence, and texture enhancement properties,
according to the paper. 8 convolution kernels, one for each direction, are
created for rotation invariance as suggested by Pu et al., with fractional
order $v=0.5$, and applied to the intensity of the images. The magnitude of
the response is found using the Euclidean norm.

###  2.4 U-net architecture

A modified architecture of U-net [6] is used for this study. The primary
difference from the original architecture is that the output segmentation map
produced is of the same size as the input image, unlike in the original
architecture, and the number of filters is half of that in the original
architecture at each stage. The modified architecture is shown in Fig. 1 and
consists of a contracting path on the left and an expanding path on the right.
All input images are resized to 128x128 using bilinear interpolation. The
output of the network are sigmoid probabilities on a 128x128 map. After the
network is trained, the output is converted to a binary labeled image at a
threshold of 0.5 applied to logit probabilities.

Figure 1: Illustration of the modified U-net architecture.

###  2.5 Post-processing operations

The sigmoid probability map generated by the network does not take into
account the connectivity of segmented masks. Consequently, there are holes and
small blobs in the segmented image. Opening (erosion followed by dilation) and
closing (dilation followed by erosion) with a 5x5 square structuring element
are used to remove these. The largest connected contour in the image is found
and drawn as the predicted mask. After post-processing, the image is resized
to its original shape.

##  3 Experiments and results

The dataset consists of 2594 dermoscopic images and corresponding masks and
the images are of different sizes. 34 different random splits were performed
on the data released, and 30 U-net networks, each with different random
initializations as well, were trained on these splits. 3 randomly selected
groups of 25 networks each were used, whose outputs were averaged to create
three sets of submissions.

###  3.1 Pre-processing and augmentation

Aggressive on-line data augmentation was performed before contrast
enhancement. The images were allowed a random rotation range, flipping, and
zooming. The proposed method is to train the network with five channels: three
color enhanced channels, a contrast-enhanced intensity channel, and a texture-
enhanced intensity channel.

###  3.2 U-net training

The Adam optimizer [7] was used for training the U-net with a learning rate of
1e-4, regularization parameters $\beta_{1}=0.9$ and $\beta_{2}=0.999$, and no
decay in the learning rate over each update. The loss was defined as the
negative of the Dice similarity coefficient222For given sets $X$ and $Y$, Dice
coefficient = $\frac{2|X\cap Y|}{|X|+|Y|}$. The Jaccard index333Jaccard index
= $\frac{|X\cap Y|}{|X\cup Y|}$ was monitored as a metric. A mini-batch size
of 4 was used and the training was run for 200 epochs. Models were saved at
checkpoints of best Jaccard coefficient on random validation splits for each
network. The Keras 2 library444https://keras.io with a Tensorflow backend was
used for the training and evaluation and all experiments were run on a 16 GB
Nvidia Tesla P100 GPU.

###  3.3 Results on validation data

The three submissions received thresholded Jaccard indices of 0.756, 0.750,
and 0.746 on the validation set of 100 images. The thresholded index is
defined as 0 if the Jaccard index if less than 0.65, and equal to the Jaccard
index if more than 0.65.

##  4 Conclusion

We presented a deep learning image processing framework that achieved state-
of-the-art results. We used five channel inputs to train the U-net
architecture: the contrast-enhanced intensity channel, the three hue-preserved
color-enhanced RGB channels, and the texture-enhanced intensity channel; the
U-net architecture was modified to make both input and output image sizes
same; and a post-processing method that uses morphological operations and
contour identification was used to arrive at the output segmentation. The
framework is fast, simple, and efficient and can be extended to other
applications as well.

### Acknowledgement

This work is partially supported by AcRF Tier-1 grant RG149/17 by the Ministry
of Education, Singapore

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