Mass Transfer in Eccentric Black Hole
–
Neutron Star Mergers
Yossef Zenati
1
,
2
,
9
, Mor Rozner
3
,
4
,
5
,
6
, Julian H. Krolik
1
, and Elias R. Most
7
,
8
1
Physics and Astronomy Department, Johns Hopkins University, Baltimore, MD 21218, USA;
yzenati1@jhu.edu
2
Space Telescope Science Institute, Baltimore, MD 21218, USA
3
Gonville & Caius College, Trinity Street, Cambridge CB2 1TA, UK
4
Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK
5
Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA
6
Technion-Israel Institute of Technology, Physics Department, Haifa 3200002, Israel
7
TAPIR, Mailcode 350-17, California Institute of Technology, Pasadena, CA 91125, USA
8
Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USA
Received 2024 October 03; revised 2024 November 26; accepted 2024 December 06; published 2025 January 3
Abstract
Black hole
–
neutron star binaries are of interest in many ways: they are intrinsically transient, radiate gravitational
waves detectable by LIGO, and may produce
γ
-ray bursts. Although it has long been assumed that their late-stage
orbital evolution is driven entirely by gravitational wave emission, we show here that in certain circumstances,
mass transfer from the neutron star onto the black hole can both alter the binary's orbital evolution and signi
fi
cantly
reduce the neutron star's mass: when the fraction of its mass transferred per orbit is
10
−
2
, the neutron star's mass
diminishes by order unity, leading to mergers in which the neutron star mass is exceptionally small. The mass
transfer creates a gas disk around the black hole
before
merger that can be comparable in mass to the debris
remaining after merger, i.e.,
~
0.1
M
e
. These processes are most important when the initial neutron star
–
black hole
mass ratio
q
is in the range
≈
0.2
–
0.8, the orbital semimajor axis is 40
a
0
/
r
g
300
(
r
g
≡
GM
BH
/
c
2
)
, and the
eccentricity is large at
e
0
0.8. Systems of this sort may be generated through the dynamical evolution of a triple
system, as well as by other means.
Uni
fi
ed Astronomy Thesaurus concepts:
Neutron stars
(
1108
)
;
Astrophysical black holes
(
98
)
;
General relativity
(
641
)
;
Roche lobe over
fl
ow
(
2155
)
1. Introduction
The
fi
rst direct detection of gravitational waves
(
B. P. Abbott
et al.
2016
)
opened a new era in astrophysics, especially in
terms of compact objects. The scienti
fi
c power of this approach
was further ampli
fi
ed when the
fi
rst binary neutron star merger
whose gravitational wave signal was detected
(
B. P. Abbott
et al.
2017a
)
was also seen in observations across a large part of
the electromagnetic
(
EM
)
spectrum, from radio frequencies to
γ
-rays
(
B. P. Abbott et al.
2017b
,
2017c
; R. Abbott et al.
2023a
,
2023b
)
. Since then, LIGO has observed many more
double neutron star and black hole
–
neutron star mergers
(
E. R. Most et al.
2020
; M. Zevin et al.
2020
; R. Abbott et al.
2021
; D. A. Godzieba et al.
2021
; P. Drozda et al.
2022
;
T. Martineau et al.
2024
)
but no additional EM counterparts.
The
fi
rst multimessenger gravitational wave event,
GW170817, prompted intense study of these events, beginning
with the formation of the original stars that ultimately evolved
to the merging neutron stars and black holes. Whether the
system was born with one or more companions
(
a triple or a
binary
)
or the binary was formed well after the stars by other
dynamical processes, there is now a very large literature on the
evolutionary tracks that might be followed by progenitor
systems of LIGO events
(
R. Abbott et al.
2020
,
2021
;
E. R. Most et al.
2020
; M. Zevin et al.
2020
; The LIGO
Scienti
fi
c Collaboration et al.
2024
)
. In this paper, we will
focus on a speci
fi
c subset of mergers involving neutron stars:
those in which the partner is a black hole
(
henceforth
abbreviated to BH
/
NS
)
.
Moreover, we will, for the most part, further restrict our
attention to the
fi
nal stage of the binary: the period in which its
separation diminishes from
~
10
2
r
g
(
for black hole gravitational
radius
r
g
≡
GM
BH
/
c
2
, with
M
BH
the mass of the black hole
)
to
a distance so close that the two objects merge. Hitherto, it has
been widely assumed that when the separation is this small,
orbital evolution is dominated by gravitational radiation
(
M. Shibata & K. Ury
ū
2006
; Z. B. Etienne et al.
2009
;
F. Pannarale et al.
2011
; K. Kyutoku et al.
2013
; F. Foucart
et al.
2014
,
2018
; K. Kyutoku et al.
2018
; M. Shibata &
K. Hotokezaka
2019
; D. Radice et al.
2020
; M. Shibata et al.
2021
; T. Martineau et al.
2024
)
. This assumption has several
immediate consequences. First, everything about the event is
determined by the masses of the two objects, the black hole
spin parameter, and the spin's orientation relative to the orbital
axis. Second, any binary whose lifetime is at least a few times
the timescale on which gravitational radiation shrinks the orbit
by a factor
~
O
(
1
)
becomes very nearly circular. Third, the
masses of the neutron star and black hole are unlikely to change
until the merger begins.
Despite the general support given to the third consequence,
there have been a few studies of classical
“
Roche lobe
over
fl
ow
”
of matter from the neutron star to the black hole
(
J. P. A. Clark & D. M. Eardley
1977
; S. I. Blinnikov et al.
1984
,
1990
; A. V. Yudin et al.
2020
; N. I. Kramarev et al.
2024
)
. Here, we will show that if the orbit is
eccentric
during
the later stages of inspiral, mass loss from the neutron star
during pericenter passage can alter the course of orbital
evolution and substantially diminish the neutron star's mass.
The Astrophysical Journal,
978:126
(
8pp
)
, 2025 January 10
https:
//
doi.org
/
10.3847
/
1538-4357
/
ad9b87
© 2025. The Author
(
s
)
. Published by the American Astronomical Society.
9
ISEF International Fellowship.
Original content from this work may be used under the terms
of the
Creative Commons Attribution 4.0 licence
. Any further
distribution of this work must maintain attribution to the author
(
s
)
and the title
of the work, journal citation and DOI.
1
This thought has been previously explored
(
M. B. Davies et al.
2005
)
; we return to it now using a much-improved treatment of
angular momentum and energy
fl
ow in the course of mass
transfer
(
A. S. Hamers & F. Dosopoulou
2019
)
. With this new
formalism, we also examine a broader range of
M
BH
and
M
NS
,
including events that do not necessarily reach the lower bound
of neutron star mass. A complementary study focusing on
single, very deep pericenter passages was carried out by
W. E. East et al.
(
2012
)
.
Eccentric BH
/
NS binaries with semimajor axes as small as
~
10
2
r
g
might be created through several channels. If the BH
/
NS binary is accompanied by a third component that is either a
black hole or a neutron star, Kozai
–
Lidov oscillations can give
the inner binary episodes of very high eccentricity
(
see, e.g.,
F. Antonini & H. B. Perets
2012
; S. Naoz
2016
; A. S. Hamers
et al.
2021
; C. Shariat et al.
2024
)
. When this mechanism is
active, the mass transfer could occur during a single high-
eccentricity Kozai
–
Lidov excursion lasting enough orbits to
accomplish a sizable total mass transfer or as a cumulative
process spanning many such episodes.
In a dense star cluster, a BS
/
NH binary orbiting an
intermediate-mass black hole could also exhibit Kozai
–
Lidov
oscillations
(
B.-M. Hoang et al.
2018
)
. Dense star clusters
provide another path to high eccentricity in the form of binary
–
single or binary
–
binary interactions
(
Z. Xuan et al.
2024
)
or
through cluster tides
(
C. Hamilton & R. R. Ra
fi
kov
2019
)
.
Investigating these possibilities also carries other interesting
rami
fi
cations. We will discuss in detail
(
Sections
2.4
and
3
)
the
range of black hole masses for which events of this sort might
happen. Here, we use an order of magnitude estimate to de
fi
ne
the issue. For mass transfer to occur, the distance at which the
black hole's tidal gravity becomes competitive with the neutron
star's self-gravity must be larger than the pericenter distance
r
p
(
J. M. Lattimer & D. N. Schramm
1976
)
. On the other hand,
r
p
must also be large enough that the black hole does not pass
through the neutron star. These two conditions taken together
demand that during the entire period of mass transfer,
r
p
must
be not much more than
;
8
r
g
(
M
e
/
M
BH
)
if we take the neutron
star's radius to be
;
12 km, independent of
M
NS
. In other words,
this process cannot occur if
M
BH
is more than several
M
e
.
Thus, searching for BH
/
NS systems with mass transfer
amounts to a search for black holes in the
fi
rst mass gap
(
R. Abbott et al.
2020
; The LIGO Scienti
fi
c Collaboration et al.
2024
)
. In addition, this mass-exchange process may lead to
mergers in which the neutron star has shallow and potentially
subsolar mass.
2. Formalism
2.1. Orbital Evolution by Gravitational Radiation
For pure gravitational radiation-driven orbital evolution, we
take the classic results found by P. C. Peters
(
1964
)
in the
lowest-order post-Newtonian approximation:
/
/
()( )
()
m
=-
-
+ +
--
da
dt
car
e
e
e
64
5
11
73
24
37
96
1
g
3272
2 4
⎛
⎝
⎞
⎠
and
/
/
()( )
()
m
=-
-
+
--
de
dt
c
r
ear
e
e
304
15
11
121
304
.2
g
g
4252
2
⎜⎟
⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
Here,
μ
is the symmetric mass ratio,
a
is the orbital semimajor
axis, and
e
is the orbital eccentricity. When it is more
convenient, we also use
/
/
()
()
== -
+
++
de
da
de dt
da dt
e
a
e
e
ee
19
12
1
1
1
.3
2
121
304
2
73
24
2
37
96
4
2.2. Mass Transfer
2.2.1. Onset
We begin with the criterion for when mass transfer occurs. In
the Introduction, we argued qualitatively that the primary
condition to satisfy is for the black hole's tidal gravity to be at
least competitive with the star's self-gravity. We follow
A. S. Hamers & F. Dosopoulou
(
2019
)
in making this
condition quantitative by de
fi
ning the quantity
()
()
x
º-
Fe
1,
4
q
where the function
F
q
is the ratio of the radius of the star's
effective Roche lobe to the orbit's semimajor axis. It is given by
/
//
()
()
=
++
F
a
R
q
qq
0.49
0.6
ln 1
,5
q
NS
23
23
13
where
R
NS
is the neutron star's radius, and
q
(
1
)
is the binary's
mass ratio
(
P. P. Eggleton
1983
)
. We speak of the
“
effective
Roche lobe
”
because true Roche lobes exist only for circular-
orbit binaries in which the stars corotate with the orbit. When
ξ
>
1, the effective Roche lobe of the neutron star is larger than
the star even at pericenter, so no mass transfer can take place;
mass transfer begins when
ξ
drops below unity. Note that
F
q
is
a rather weak function of
q
: it increases from
≈
0.2 for
q
=
0.1
to
≈
0.35 for
q
=
1.
To determine
R
NS
, we use Tolman
–
Oppenheimer
–
Volkoff
solutions assuming the SFHo equation of state
(
A. W. Steiner
et al.
2013
)
. As for all plausible equations of state, these
solutions yield a mass
–
radius relation in which the radius is
nearly independent of mass from
M
NS
;
0.3 to
;
2.2
M
e
;in
this case,
R
NS
≈
12 km.
2.2.2. Magnitude
When the orbit is eccentric, most mass transfer happens
when the mass-losing star is near pericenter, and rather little
takes place when the star is near apocenter. For the purpose of
studying long-term orbital evolution, it is therefore best to work
with the fractional mass loss per orbit. This quantity should
grow with greater
“
overhang
”
of the star beyond its effective
Roche lobe when it is at pericenter, a quantity given roughly by
1
−
ξ
. Although main-sequence stars have rather steep density
pro
fi
les, so that their outer layers contain only a small fraction
of their total mass, neutron stars have much shallower density
pro
fi
les out to very nearly the stellar radius
(
see Figure
1
)
.As
this
fi
gure shows, even for an
ξ
as large as
≈
0.9, the mass
fraction beyond the Roche lobe is
10%. Consequently,
neutron stars on eccentric orbits are likely to lose a
substantially larger fraction of their mass
(
Δ
M
/
M
)
in a single
orbit than a main-sequence star would. Note that
Δ
M
/
M
refers
to the fraction of the instantaneous mass of the neutron star.
2
The Astrophysical Journal,
978:126
(
8pp
)
, 2025 January 10
Zenati et al.
2.3. Orbital Evolution for a Given
Δ
M
/
M
M. B. Davies et al.
(
2005
)
assume that the mass transferred from
the neutron star to the black hole carries half the speci
fi
c angular
momentum of the neutron star
(
they do not state how much of its
orbital energy is given to the black hole
)
.Bycontrast,A.S.Hamers
& F. Dosopoulou
(
2019
)
calculate the angular momentum and
energy transferred, accounting for numerous details and their
variation with orbital phase: the location in the star from which the
mass is taken and where it goes into orbit around the black hole;
the velocities of the mass when it leaves the star and enters an orbit
around the black hole; and the change in position of the system
center of mass. Orbit-averaging these quantities, assuming that the
instantaneous mass-transfer rate is proportional to the cube of the
instantaneous fractional overhang, leads to the following orbital
evolution equations in the absence of gravitational radiation:
/
/
/
/
()
() ( )
)(
)
(
()
()
p
=-
D
́- +
-
da
dt
cr
MM
ar f eF
a
qf eF F
R
a
geF
q
R
a
heF
,
1,
,
,6
g
g
M
q
a
qq
a
q
A
aq
32
NS
⎡
⎣
⎛
⎝
⎞
⎠
⎤
⎦
and
/
/
/
/
()
() ( )
[(
) (
)
()
()
()
p
=-
D
́-
+-
de
dt
cr
MM
ar f eF
qf eF
F
R
a
geF q
R
a
heF
,
1,
,,.7
g
g
M
q
e
q
q
e
q
A
eq
32
NS
⎤
⎦
The functions
()
feF
,
M
q
,
f
a
,
e
(
e
,
F
q
)
,
g
a
,
e
(
e
,
F
q
)
,and
h
a
,
e
(
e
,
F
q
)
are all given explicitly in Appendix B of A. S. Hamers &
F. Dosopoulou
(
2019
)
; we do not reproduce them here because
the expressions are very lengthy. We assume that their accretion
delay parameter
τ
=
0and
R
A
is the radius of the accretion disk
the transferred mass forms around the black hole. We choose
R
A
=
0.3
a
because the Newtonian tidal truncation radius as
computed by B. Paczynski
(
1977
)
can be well
fi
tby
;
0.3
aq
−
1
/
3
.
To describe the evolution including both mass transfer and
gravitational wave emission, we simply add the contributions
to
da
/
dt
and
de
/
dt
. Note, however, that because the mass-
transfer part assumes Newtonian dynamics, the orbits them-
selves do not re
fl
ect any relativistic effects.
The preceding equations describe the orbital evolution, but
because many of the functions depend on the mass ratio
q
, and
mass transfer changes
q
, its evolution must also be tracked:
/
/
/
()
()
()
p
=-
D
+
dq
dt
c
r
MM
ar
qq
2
.8
gg
32
2
Here, we assume that the transfer conserves mass.
2.4. Implications of the Evolution Equations
Several qualitative conclusions can be drawn immediately from
the forms of these equations. First, as already broached in the
Introduction, the
ξ
1 criterion for mass transfer places a constraint
on the semimajor axis and eccentricity, requiring the pericenter
a
(
1
−
e
)
to be smaller than the neutron star's Roche lobe:
a
(
1
−
e
)
R
NS
/
F
q
. For a more-or-less
fi
xed neutron star
radius of
≈
12 km, this constraint becomes
()
-
a
e
1
//
()(
)
--
FMMr
27 0.3
qg
1
BH
1
. Second, the concept of mass
transfer via Roche lobe over
fl
ow requires that the pericenter
/
()
(
)
-> =
-
a
eR MMr
18
g
NS
BH
1
; otherwise, the black hole
would pass through the neutron star. This second constraint is
strengthened by the fact that Newtonian dynamics are a good
description of orbits only when the separation is
10
–
15
r
g
. Thus,
the parameter space in which mass transfer from an eccentric orbit
can occur is rather limited, and particularly so if
M
BH
is greater than
a few solar masses.
A different qualitative conclusion is that the orbital evolution
of such a system changes sharply when the mass lost per orbit
is great enough that mass transfer dominates gravitational wave
radiation as a driver of orbital evolution. Contrasting the orbital
evolution rates in Equations
(
6
)
and
(
7
)
with the gravitational
radiation evolution rates in Equations
(
1
)
and
(
2
)
yields a
criterion for this regime change,
/
/
()
()
m
D
́
--
M
M
ar f
210 50
,
9
g
M
352
where we have dropped the order-unity factor describing the
eccentricity dependence. Thus, the fractional mass loss per
orbit must be signi
fi
cant, but it does not need to be a large
fraction in order for mass transfer to in
fl
uence orbit evolution.
However, as the semimajor axis shrinks, the threshold at which
mass transfer drives orbital evolution rises.
Lastly, for mass transfer to continue,
a
(
1
−
e
)
R
NS
/
F
q
.As
the binary's semimajor axis shrinks, its eccentricity does also;
note that both gravitational radiation and mass transfer tend to
diminish
a
and
e
. The question therefore arises,
“
Does
a
diminish more rapidly than 1
−
e
increases?
”
This question
cannot be answered by estimates; it demands a proper
calculation.
2.5. Validity of the Post-Newtonian Approximation
The gravity in the region we are considering
—
distances
~
10
–
100
r
g
from a black hole, near and inside a neutron star
—
is
strong enough for relativistic corrections to be noticeable.
However, we treat the orbits as being purely Newtonian with
the exception of allowance for the binary's evolution by
radiation of gravitational waves. Here, we explain why we
believe our approximations are suitable for this
fi
rst exploration
of mass transfer from neutron stars to black holes.
There exist a number of formalisms by which relativistic binary
orbital evolution can be followed i
n more-or-less analytic fashion,
including the effective-one-bod
y approximation and its extension
Figure 1.
Fraction of a neutron star's mass outside radius
r
. Here,
M
NS
=
1.35
M
e
and the SFHo equation of state
(
A. W. Steiner et al.
2013
)
is assumed; other masses and equations of state produce very similar curves.
3
The Astrophysical Journal,
978:126
(
8pp
)
, 2025 January 10
Zenati et al.
to the inspiral-merger-ringd
own multipolar waveform model
(
D. Chiaramello & A. Nagar
2020
;M.Khaliletal.
2021
;
A. Ramos-Buades et al.
2022a
,
2022b
)
. The primary aim of these
methods is to provide a way to generate gravitational waveforms
beginning while the binary system i
s still tracing an almost-closed
elliptical orbit, and then continui
ng seamlessly through the plunge
and ringdown stages. Here, we are concerned only with the
fi
rst
stage: the almost-closed ellip
tical orbit. In this regime, the
P. C. Peters
(
1964
)
quadrupole approximation remains reliable,
even though it needs modi
fi
cation in the plunge.
Bearing this in mind, our evolutions all stop near the point at
which the plunge begins. This point can be determined by
comparing the binary's orbital energy to the maximum in the
Kerr spacetime's effective potential. If the black hole in the
binary has a spin parameter that is either prograde with respect
to the orbit, and is
0.8, or is retrograde, these evolutions stop
~
50
–
100 orbits beyond the beginning of the plunge; if the spin
parameter is prograde and larger, they stop a similar
time before the plunge. Because the timescale for change in
the orbital elements at this stage is several hundred orbits
(
see
Figure
3
)
, these offsets from the optimal stopping time make
little difference in terms of orbital evolution. Thus, in this
respect, our approximations should be fairly sound.
Other relativistic effects might al
so affect the orbit, both apsidal
and nodal precession. However, neither of these precessional
motions change the size of the or
bit; they merely rotate it. They
are therefore unlikely to alter mass-transfer effects.
Tidal stresses can also depart from the Newtonian form in
this regime. We have not calculated how important the changes
are in this context; they should be the focus of the next effort
examining this mechanism.
3. Results
To answer the question just posed, as well as a great many
more, we have integrated the or
bital evolution equations for a
number of combinations of parameters within the bounds
1
M
e
M
NS
2.1
M
e
,2.78
M
e
M
BH
5
M
e
, initial semimajor
axis
a
0
=
50, 60, or 100
r
g
, and initial eccentricity
e
0
chosen so
that the initial state has
ξ
very slightly less th
an 1. Consequently,
the initial values of
a
and
q
determine the initial value of
e
.In
practice, 0.84
e
0
0.87 for
a
0
=
50
r
g
,0.88
e
0
0.90 for
a
0
=
60
r
g
, and 0.90
e
0
0.94 for
a
0
=
100
r
g
(
see Table
1
)
.All
evolutions were stopped when the pericenter distance became
smaller than
≈
6
–
8
r
g
, where numerical relativity studies indicate
that for the systems we consi
der, the NS will always tidally
disrupt, leading to the formation of a remnant debris disk
(
F. Foucart
2012
; F. Foucart et al.
2018
)
. Note that the
requirement of beginning with
ξ
=
1 is, in fact, not restrictive
because any system evolving in a smooth fashion from an initial
state without mass transfer must begin mass transfer when
ξ
=
1.
3.1. Continuity of Mass Transfer and Orbital Evolution
Figure
2
illustrates how the mass-transfer discriminant
ξ
evolves in several cases differing only in
Δ
M
/
M
. When
Δ
M
/
M
10
−
3
or
ξ
>
1, mass transfer has essentially no effect
Table 1
Final Masses of Neutron Star
a
0
e
0
e
f
M
BH
́
-
D
1
0
M
M
3
M
NS
0
M
f
N
S
(
r
g
)(
M
e
)(
M
e
)(
M
e
)
50
0.842
0.153
3.8
20
1.0
0.633
0.846
0.192
1.1
0.724
0.851
0.108
1.2
0.818
0.853
0.204
1.3
0.909
0.856
0.219
1.4
1.003
0.859
0.213
1.5
1.181
0.861
0.171
1.6
1.181
0.863
0.201
1.7
1.273
0.865
0.175
1.8
1.366
0.867
0.198
1.9
1.453
0.869
0.227
2.0
1.533
0.871
0.215
2.1
1.619
0.842
0.176
3.8
50
1.0
0.353
0.842
0.118
3.8
5
1.0
0.885
0.842
0.222
2.78
5
1.0
0.833
60
0.882
0.138
3.8
20
1.0
0.691
0.885
0.164
1.1
0.784
0.887
0.111
1.2
0.885
0.891
0.202
1.3
0.982
0.893
0.212
1.4
1.071
0.895
0.208
1.5
1.168
0.896
0.173
1.6
1.261
0.898
0.212
1.7
1.337
0.900
0.178
1.8
1.409
0.902
0.186
1.9
1.402
0.903
0.209
2.0
1.621
0.904
0.207
2.1
1.613
0.881
0.124
3.8
50
1.0
0.402
0.881
0.117
3.8
5
1.0
0.890
0.846
0.200
2.78
5
1.0
0.852
100
0.923
0.098
3.8
20
1.0
0.635
0.925
0.211
1.1
0.725
0.927
0.205
1.2
0.824
0.928
0.182
1.3
0.915
0.930
0.126
1.4
0.965
0.931
0.196
1.5
1.102
0.932
0.105
1.6
1.193
0.934
0.188
1.7
1.279
0.935
0.147
1.8
1.376
0.935
0.192
1.9
1.461
0.936
0.206
2.0
1.557
0.937
0.221
2.1
1.639
0.923
0.209
3.8
5
1.0
0.334
0.923
0.101
3.8
50
1.0
0.889
0.902
0.093
2.78
50
1.0
0.827
Note.
Columns are initial semimajor axis, initial eccentricity, premerger
eccentricity, initial black hole mass, mass transfer, initial neutron star mass, and
fi
nal neutron star mass.
Figure 2.
Evolution of the mass-transfer discriminant,
ξ
, as a result of both
gravitational radiation and mass transfer
(
i.e., Equations
(
1
)
,
(
6
)
,
(
2
)
,
(
7
)
, and
(
8
))
, plotted as a function of the number of orbits,
N
orb
, for each curve. The
dashed curve shows the evolution as dictated by gravitational radiation without
mass transfer. The solid curves show the evolution of
ξ
when both mechanisms
contribute. They are distinguished by
Δ
M
/
M
:5
×
10
−
4
(
yellow
)
,5
×
10
−
3
(
blue
)
,5
×
10
−
2
(
orange
)
, and 10
−
1
(
purple
)
. The initial mass of the NS is
1.2
M
e
, and the initial mass of the BH is 3.3
M
e
. The initial orbital parameters
are
a
0
=
100
r
g
and
e
0
=
0.917 for the
Δ
M
/
M
≠
0 cases, but there is a slightly
larger value of
a
0
for
Δ
M
/
M
=
0.
4
The Astrophysical Journal,
978:126
(
8pp
)
, 2025 January 10
Zenati et al.
on the orbital evolution, just as predicted by our qualitative
estimate. In this limit,
ξ
declines gradually over time with a
characteristic shape illustrated equally well by the dashed curve
in Figure
2
(
no mass transfer at all
)
and the yellow curve
(
mass
transfer with
Δ
M
/
M
=
5
×
10
−
4
, a factor of a few below the
critical value
)
. When
Δ
M
/
M
=
0, over a span of more than 800
orbits,
a
shrinks from
;
100 to
;
40
r
g
, and
e
falls from
;
0.92
to
;
0.8. Importantly, even when
ξ
>
1, so that no mass transfer
can take place, evolution by gravitational wave emission alone
leads to a slow decline in
ξ
so that
ξ
ultimately falls below
unity, permitting mass transfer.
Because mass transfer acts in the same sense as gravitational
radiation
—
diminishing both
a
and
e
—
any mass transfer
accelerates the orbital evolution, and mass transfer at a rate
great enough to dominate gravitational radiation causes orbital
evolution faster than the rate gravitational radiation can drive.
When
Δ
M
/
M
is only a factor of several above the threshold
(
e.g., 5
×
10
−
3
)
, the time from the onset of mass transfer to the
beginning of merger with the black hole is
500 orbits; only
;
300 orbits are needed when
Δ
M
/
M
is a factor
~
20
×
larger.
Moreover, once
ξ
<
1 and mass transfer begins, if
Δ
M
/
M
10
−
3
,
ξ
decreases monotonically, so that mass transfer continues
all the way to merger. Although only three cases are shown here,
all the other cases we examined behaved in the same way. In other
words, once the binary begins mass transfer, the neutron star
continues to lose mass until either the merger takes place or some
external effect changes the orbit.
To gain a more speci
fi
c sense of how the orbit evolves, three
views of three sample histories are shown in Figure
3
. The top
and bottom panels
(
a
as a function of
N
orb
and
e
as a function of
N
orb
)
tell closely related stories. In both, the contrast between
the
a
0
=
100 and the
a
0
=
60
r
g
curves, despite their identical
values of
Δ
M
/
M
and initial
ξ
, shows that the semimajor axis at
which mass transfer begins remains imprinted on the system's
orbital evolution throughout its progress toward merger. By
contrast, the close similarity of the evolutions for
a
0
=
50 and
a
0
=
60
r
g
, despite having
Δ
M
/
M
values differing by a factor
of 2, demonstrates that even though
da
/
dN
orb
and
de
/
dN
orb
are
explicitly
∝
Δ
M
/
M
, these dependences can be largely canceled
by the
implicit
dependence of
a
(
N
orb
)
and
e
(
N
orb
)
on
Δ
M
/
M
through
q
(
N
orb
)(
see Equation
(
8
))
; terms proportional to both
q
and 1
−
q
appear in the evolution equations for both
a
and
e
,
and the functions
f
a
,
e
,
g
a
,
e
, and
h
a
,
e
have further implicit
dependences on
q
. Such a cancellation is not, however,
necessarily a general effect.
Although the mass-transfer-driven dependence of
a
and
e
on
time exhibits interesting parameter dependences, the relation
between
a
and
e
is universal
(
bottom panel of Figure
3
)
. All
three cases shown in the upper two panels lie on almost exactly
the same curve, differing only in the value of
a
at which they
enter it. Initially, their track is decently approximated by
the relation
a
∝
R
NS
/
(
1
−
e
)
because this is equivalent to
ξ
=
const.
, which is not exactly true for the early stages of
orbital evolution but is also not grossly wrong.
However, as illustrated in Figure
2
,
ξ
=
const
. does
eventually break down, and this happens sooner when
Δ
M
/
M
is larger. From this point onward,
a
declines more rapidly with
respect to
e
than the
ξ
;
const
. approximation would predict. In
fact, the relation between
a
and
e
over the entirety of the
binary's evolution when it is driven by mass transfer is well
described by the function
/
()
( )
ba
a
er
e
exp
g
with
α
constrained to lie in the range
[
0.87, 1.02
]
and
β
;
a
0
/
R
NS
.
In other words, as
e
decreases,
a
shrinks exponentially, and the
asymptotic value of
a
for
e
→
0is
;
a
0
/
R
NS
. This follows
because
( )()
()
()
( )()
()
()()
-+ -
-+ -
qf eF F geF q h eF
qfeF F geF q heF
1,
,
,
1,
,
,10
a
qq
R
a
a
q
R
a
aq
e
qq
R
a
e
q
R
a
eq
A
A
NS
NS
with remarkable precision despite the variations in
q
,
e
, and
a
during the binary evolution and the fact that the individual
function pairs
(
e.g.,
f
a
and
f
e
)
have similar values but are not as
close as the speci
fi
c combinations of this equation. Conse-
quently, the ratio of Equations
(
6
)
and
(
7
)
simpli
fi
es to
da
/
de
;
a
.
All of the orbital evolution properties in the mass-transfer-
dominated regime are quite different from how the evolution
proceeds when only gravitational radiation matters. Pure
gravitational wave emission causes
e
to decrease more sharply
as a function of
a
than when mass transfer dominates. As a
result, the orbit becomes almost circular when
a
has shrunk by
a factor of
~
2
–
3, in sharp contrast with the mass-transfer case,
in which the eccentricity remains
0.2 even when
a
has shrunk
by a factor
~
10
–
20. Ultimately, in the limit as
e
→
0,
Figure 3.
Evolution of semimajor axis
(
top panel
)
and eccentricity
(
middle panel
)
as functions of the number of o
rbits. The lower panel shows
a
as a function of
e
;
note that
e
decreases from left to right. Three d
ifferent initial semimajor axes are
shown: 50
r
g
(
purple dashed curves
)
,60
r
g
(
red curves
)
,and100
r
g
(
blue curves
)
,all
having
M
NS
=
1.2
M
e
,
M
BH
=
3.3
M
e
. For the two larger ini
tial semimajor axes,
Δ
M
/
M
=
5
×
10
−
3
;for
a
0
=
50
r
g
,
Δ
M
/
M
=
10
−
2
. The dotted curve in the
bottom panel shows
a
(
e
)
for evolution driven solely by gravitational radiation; the
dashed curve shows evolution at
fi
xed
ξ
∝
a
(
1
−
e
)
/
q
0
. The yellow curves
correspond to
the relation
()
(
)
ba
ae
e
exp
for two values of the three
parameters
(
α
,
β
,
Δ
M
/
M
)
:
(
1.02, 6.35, 1
×
10
−
2
)
and
(
0.87, 4.41, 5
×
10
−
3
)
.Both
α
values are close to unity.
5
The Astrophysical Journal,
978:126
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8pp
)
, 2025 January 10
Zenati et al.
gravitational radiation-controlled evolution leads to
a
∝
e
12
/
19
(
see Equation
(
3
))
.
3.2. Total Amount and Pace of Mass Exchange
We have already determined that whether gravitational
radiation or mass transfer dominates orbital evolution is largely
governed by
Δ
M
/
M
; when
Δ
M
/
M
10
−
3
, mass transfer
plays a role at least comparable to that of gravitational wave
emission. Not surprisingly, the fractional mass loss per orbit is
also the critical parameter for determining whether the total
mass transferred from the neutron star to the black hole is a
sizable fraction of
M
NS
.
This fact is demonstrated in Figure
4
. Whether the initial
mass ratio
q
=
0.24
(
the upper panel of this
fi
gure
)
or
q
=
0.64
(
the lower panel
)
, the mass lost by the neutron star is at least
;
10% of its original mass when
Δ
M
/
M
10
−
2
. The criterion
for substantial total mass transfer is therefore a factor of 10
more stringent than the one determining which mechanism
controls orbital evolution.
The same
fi
gure also points out the fact that signi
fi
cant mass
transfer starts when the semimajor axis shrinks to
50
r
g
.As
shown in Figure
3
, this is also the evolutionary stage at which
the eccentricity drops below
~
0.8
±
. However, the pace of mass
transfer accelerates once it starts, both in terms of
dM
/
da
(
as
shown in Figure
4
)
and even more so in terms of
dM
/
dt
because
/////
/
()()()()
=μ
-
d
Mdt dMdadadt dMdaar
g
32
.
Nonetheless, although the precise distribution of mass loss
with semimajor axis depends on the parameters, the bulk of the
mass loss occurs when
a
>
10
r
g
, and, particularly for larger
Δ
M
/
M
, when
a
>
20
r
g
(
see also Figure
5
)
.
3.3. Final Neutron Star Mass
Substantial mass loss from a neutron star leads, naturally
enough, to a substantially smaller neutron star mass at the time
of the actual merger. How much smaller
M
NS
can be is shown
in Figure
5
.
The upper panel of this
fi
gure underlines what we have
already seen: that the mass loss increases with
Δ
M
/
M
and is
substantial when
Δ
M
/
M
10
−
2
. However, another depend-
ence is also illustrated by these curves: for
fi
xed
M
NS,0
,
smaller
M
BH
promotes
greater
mass transfer. The contrast in total mass
loss between the cases of
M
BH
=
3.3 and
M
BH
=
5
M
e
is a
factor
~
2.
The fruits of a larger sample of parameter values are shown
in the lower panel of Figure
5
. A complete tally of all the cases
we studied can be found in Table
1
. As these points show,
when
Δ
M
/
M
10
−
2
, the neutron star's mass can be reduced
by tens of percent or more by the time it reaches merger.
4. Potential Consequences
Just as Roche lobe over
fl
ow in ordinary stellar binaries leads
to the creation of an accretion disk around the mass-receiving
star, the mass lost from the neutron star should form such a disk
around the black hole; the only requirement is that the mass
transferred should have an angular momentum relative to the
black hole large enough to place the matter outside the black
hole's ISCO. The mass contained in this disk, potentially as
much as a few tenths of a solar mass, could be comparable to
the disk that might be formed
after
the merger, which for
masses in the range treated here might have a mass
~
0.1
M
e
(
F. Foucart et al.
2018
)
.
The dynamical history of such a disk is the result of several
competing processes. In circular binaries, the tidal truncation
radius of a disk around the more massive partner is
;
0.3
aq
−
1
/
3
; however, when the mass transfer occurs, these
binaries are still moderately eccentric at
e
≈
0.2
–
0.3, throwing
that estimate into some doubt. Pericenter passage could, for a
brief time, signi
fi
cantly disturb the disk, both by the stronger
tidal gravity and by the impact of additional matter peeled off
the neutron star. Moreover, in the semimajor axis range of
greatest mass transfer
(
several tens of
r
g
)
, even the circular-
Figure 4.
Evolution of the mass ratio as a function of the semimajor axis for
fi
ve different values of
Δ
M
/
M
. Two pairs of black hole and neutron star
masses are portrayed:
M
BH
=
5.0
M
e
,
M
NS
=
1.2
M
e
(
upper panel
)
; and
M
BH
=
2.78
M
e
,
M
NS
=
1.8
M
e
(
lower panel
)
.
Figure 5.
Upper panel: for a single initial neutron star mass
(
M
NS,0
=
1.4
M
e
)
,
M
NS
as a function of semimajor axis for four different values of
Δ
M
/
M
and
two black hole masses
(
M
BH
=
3.3, 5.0
M
e
)
. In order of increasing
Δ
M
/
M
, the
curves have colors blue, yellow-green, and purple. The smaller black hole mass
is shown with solid curves, the larger with dotted
–
dashed curves. Lower panel:
initial neutron star mass
M
NS,0
vs.
fi
nal mass
M
NS,f
for
Δ
M
/
M
=
10
−
3
(
stars
)
,
10
−
2
(
circles
)
, and 3
×
10
−
2
(
diamonds
)
and for
M
BH
=
3.3
(
purple
)
and
M
BH
=
5.0
M
e
(
blue
)
. When
Δ
M
/
M
=
10
−
3
, the total mass loss is so small
that the points for both black hole masses are superposed at
M
NS,f
=
M
NS,0
.
6
The Astrophysical Journal,
978:126
(
8pp
)
, 2025 January 10
Zenati et al.
orbit truncation scale is quite small: for example, when
a
=
20
r
g
, the
(
Newtonian
)
circular-orbit truncation radius is
only
~
8
r
g
. This is close to the ISCO unless the black hole spins
fairly rapidly
(
spin oblique to the orbital axis could, of
course, lead to further complications
)
; this structure echoes
that of disks formed after a binary neutron star merger
(
A. Camilletti et al.
2024
; Y. Zenati et al.
2024
)
. Interestingly,
the characteristic duration for the mass-transfer process
(
~
300
binary orbits
)
is equivalent to a time
1 s, which is close
to the expected lifetime of postmerger disks
(
M. Shibata &
K. Taniguchi
2006
; F. Foucart
2012
; V. Paschalidis et al.
2015
)
. Thus, in respect to their mass, radial scale, and lifetime,
these premerger disks resemble postmerger debris disks.
Material removed from a neutron star should already be
threaded with a fairly strong magnetic
fi
eld; if the accretion
time for matter in the disk is at least
~
10 orbits within the disk,
the magnetorotational instability could amplify it further while
also stirring MHD turbulence. Because the dynamics of mass
transfer likely seed the disk with a predominantly toroidal
fi
eld,
poloidal components, the sort necessary to support a jet, may
grow slowly
(
E. R. Most et al.
2021
)
. As the result of mass
transfer beginning
~
1 s before merger, such jets may appear
earlier relative to the gravitational wave signal than jets
supported by ordinary debris disks. Unfortunately, it is dif
fi
cult
to make a clear statement of how much earlier they might be
launched because uncertainties in the jet-launching timescale
can be of comparable magnitude
(
O. Gottlieb
2023
; K. Hayashi
et al.
2024
)
.
The periodic perturbations at pericenter passage can affect
the neutron star as well as the mass-transfer stream and the
debris disk. If the crust can reform as the neutron star goes
through apocenter, each orbit's onset of mass transfer will
likely trigger a shattering of the neutron star crust, which may
potentially lead to EM transients
(
A. J. Penner et al.
2012
;
D. Tsang et al.
2012
; E. R. Most et al.
2024
)
. Additionally,
strong deformations of the star away from axisymmetry can be
induced near black hole passages
(
W. E. East et al.
2012
)
,
which may further complicate the mass-loss picture during the
fi
nal orbits. Such deformations may also excite
f
-modes at
every pericenter
fl
yby
(
C. Chirenti et al.
2017
; S. Rosofsky
et al.
2019
)
.
The last complication is that, after
~
300 binary orbits, the
remnant neutron star merges with the black hole. What happens
to the matter delivered in advance can only be ascertained by a
calculation including all these effects; such an effort is far
beyond the present paper's scope. The options include every-
thing from quick capture into the
(
enlarged
)
black hole to
mixture with additional matter drawn from the neutron star
during the merger proper.
There are several possibly observable signals from such a disk.
Because its physical properties resemble those found in
postmerger disks, some of the f
amiliar postmerger phenomenol-
ogy may be replicated: creation of
γ
-ray bursts, for example.
Suf
fi
cient heat production within the disk, whether due to
dissipation of MHD turbulence
or to nuclear reactions, might
drive the sort of wind thought to result in kilonova afterglows
(
R. Fernández & B. D. Metzger
2013
;F.Foucartetal.
2017
;
D. M. Siegel & B. D. Metzger
2017
)
. Just as for conventional
kilonovae, the neutron-rich co
mposition of the disk is likely to
result in
r
-process nucleosynthesis producing many lanthanide
nuclei
(
R. Fernández & B. D. Metzger
2013
;A.Peregoetal.
2014
; S. Wanajo et al.
2014
; J. Lippuner & L. F. Roberts
2015
)
,
whose optical opacity causes the emergent spectrum to be rather
red
(
D. Kasen et al.
2013
; M. Tanaka & K. Hotokezaka
2013
)
.
However, in the event that only a small amount
(
1% of the
neutron star's mass
)
is transferred, the portion of the neutron star
from which it was taken is its outermost layers
(
Y. Zenati et al.
2023
)
, where the lepton fraction
Y
e
is larger. When this is the case,
the kilonova might stay blue, although the disk can also
neutronize if its density becomes large enough
(
A. M. Beloboro-
dov
2003
; S. De & D. M. Siegel
2021
)
.
Acknowledgments
NASA partially supported this work through grants
NNH17ZDA001N and 80NSSC24K0100. Y.Z. and J.K. were
partially supported by NSF grant AST-2009260; in addition, J.K.
received support from NSF grant PHY-2110339. Y.Z. thanks
Hagai Perets and Jeremy Schnittman for helpful discussions.
E.R.M. acknowledges partial support by the National Science
Foundation under grant Nos. PHY-2309210 and AST-2307394.
Software:
astropy
(
Astropy Collaboration et al.
2018
)
,
Matplotlib
(
J. D. Hunter
2007
)
, and NumPy
(
C. R. Harris
et al.
2020
)
.
ORCID iDs
Yossef Zenati
https:
/
/
orcid.org
/
0000-0002-0632-8897
Mor Rozner
https:
/
/
orcid.org
/
0000-0002-2728-0132
Julian H. Krolik
https:
/
/
orcid.org
/
0000-0002-2995-7717
Elias R. Most
https:
/
/
orcid.org
/
0000-0002-0491-1210
References
Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2016,
PhRvL
,
116, 061102
Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2017a,
PhRvL
,
119, 161101
Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2017b,
ApJL
,
848, L12
Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2017c,
Natur
,
551, 85
Abbott, R., Abbott, T. D., Abraham, S., et al. 2020,
ApJL
,
896, L44
Abbott, R., Abbott, T. D., Abraham, S., et al. 2021,
ApJL
,
915, L5
Abbott, R., Abbott, T. D., Acernese, F., et al. 2023a,
PhRvX
,
13, 041039
Abbott, R., Abbott, T. D., Acernese, F., et al. 2023b,
PhRvX
,
13, 011048
Antonini, F., & Perets, H. B. 2012,
ApJ
,
757, 27
Astropy Collaboration, Price-Whelan, A. M., & Sip
ő
cz, B. M. 2018,
AJ
,
156, 123
Beloborodov, A. M. 2003,
ApJ
,
588, 931
Blinnikov, S. I., Imshennik, V. S., Nadezhin, D. K., et al. 1990, SvA,
34, 595
Blinnikov, S. I., Novikov, I. D., Perevodchikova, T. V., & Polnarev, A. G.
1984, SvAL,
10, 177
Camilletti, A., Perego, A., Guercilena, F. M., Bernuzzi, S., & Radice, D. 2024,
PhRvD
,
109, 063023
Chiaramello, D., & Nagar, A. 2020,
PhRvD
,
101, 101501
Chirenti, C., Gold, R., & Miller, M. C. 2017,
ApJ
,
837, 67
Clark, J. P. A., & Eardley, D. M. 1977,
ApJ
,
215, 311
Davies, M. B., Levan, A. J., & King, A. R. 2005,
MNRAS
,
356, 54
De, S., & Siegel, D. M. 2021,
ApJ
,
921, 94
Drozda, P., Belczynski, K., O'Shaughnessy, R., Bulik, T., & Fryer, C. L. 2022,
A&A
,
667, A126
East, W. E., Pretorius, F., & Stephens, B. C. 2012,
PhRvD
,
85, 124009
Eggleton, P. P. 1983,
ApJ
,
268, 368
Etienne, Z. B., Liu, Y. T., Shapiro, S. L., & Baumgarte, T. W. 2009,
PhRvD
,
79, 044024
Fernández, R., Foucart, F., Kasen, D., et al. 2017,
CQGra
,
34, 154001
Fernández, R., & Metzger, B. D. 2013,
MNRAS
,
435, 502
Foucart, F. 2012,
PhRvD
,
86, 124007
Foucart, F., Hinderer, T., & Nissanke, S. 2018,
PhRvD
,
98, 081501
Foucart, F., Deaton, M. B., Duez, M. D., et al. 2014,
PhRvD
,
90, 024026
Godzieba, D. A., Radice, D., & Bernuzzi, S. 2021,
ApJ
,
908, 122
Gottlieb, O., et al. 2023,
ApJ
,
954, L21
Hamers, A. S., & Dosopoulou, F. 2019,
ApJ
,
872, 119
Hamers, A. S., Rantala, A., Neunteufel, P., Preece, H., & Vynatheya, P. 2021,
MNRAS
,
502, 4479
Hamilton, C., & Ra
fi
kov, R. R. 2019,
ApJ
,
881, L13
7
The Astrophysical Journal,
978:126
(
8pp
)
, 2025 January 10
Zenati et al.
Harris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020,
Natur
,
585, 357
Hayashi, K., Kiuchi, K., Kyutoku, K., Sekiguchi, Y., & Shibata, M. 2024,
arXiv:
2410.10958
Hoang, B.-M., Naoz, S., Kocsis, B., Rasio, F. A., & Dosopoulou, F. 2018,
ApJ
,
856, 140
Hunter, J. D. 2007,
CSE
,
9, 90
Kasen, D., Badnell, N. R., & Barnes, J. 2013,
ApJ
,
774, 25
Khalil, M., Buonanno, A., Steinhoff, J., & Vines, J. 2021,
PhRvD
,
104, 024046
Kramarev, N. I., Kuranov, A. G., Yudin, A. V., & Postnov, K. A. 2024,
AstL
,
50, 302
Kyutoku, K., Ioka, K., & Shibata, M. 2013,
PhRvD
,
88, 041503
Kyutoku, K., Kiuchi, K., Sekiguchi, Y., Shibata, M., & Taniguchi, K. 2018,
PhRvD
,
97, 023009
Lattimer, J. M., & Schramm, D. N. 1976,
ApJ
,
210, 549
Lippuner, J., & Roberts, L. F. 2015,
ApJ
,
815, 82
Martineau, T., Foucart, F., Scheel, M., et al. 2024, arXiv:
2405.06819
Most, E. R., Kim, Y., Chatziioannou, K., & Legred, I. 2024,
ApJL
,
973, L37
Most, E. R., Papenfort, L. J., Tootle, S. D., & Rezzolla, L. 2021,
MNRAS
,
506, 3511
Most, E. R., Papenfort, L. J., Weih, L. R., & Rezzolla, L. 2020,
MNRAS
,
499, L82
Naoz, S. 2016,
ARA&A
,
54, 441
Paczynski, B. 1977,
ApJ
,
216, 822
Pannarale, F., Tonita, A., & Rezzolla, L. 2011,
ApJ
,
727, 95
Paschalidis, V., Ruiz, M., & Shapiro, S. L. 2015,
ApJL
,
806, L14
Penner, A. J., Andersson, N., Jones, D. I., Samuelsson, L., & Hawke, I. 2012,
ApJL
,
749, L36
Perego, A., Rosswog, S., Cabezón, R. M., et al. 2014,
MNRAS
,
443, 3134
Peters, P. C. 1964,
PhRv
,
136, 1224
Radice, D., Bernuzzi, S., & Perego, A. 2020,
ARNPS
,
70, 95
Ramos-Buades, A., Buonanno, A., Khalil, M., & Ossokine, S. 2022a,
PhRvD
,
105, 044035
Ramos-Buades, A., van de Meent, M., Pfeiffer, H. P., et al. 2022b,
PhRvD
,
106, 124040
Rosofsky, S., Gold, R., Chirenti, C., Huerta, E. A., & Miller, M. C. 2019,
PhRvD
,
99, 084024
Shariat, C., Naoz, S., El-Badry, K., et al. 2024, arXiv:
2407.06257
Shibata, M., Fujibayashi, S., & Sekiguchi, Y. 2021,
PhRvD
,
104, 063026
Shibata, M., & Hotokezaka, K. 2019,
ARNPS
,
69, 41
Shibata, M., & Taniguchi, K. 2006,
PhRvD
,
73, 064027
Shibata, M., & Ury
ū
, K. 2006,
PhRvD
,
74, 121503
Siegel, D. M., & Metzger, B. D. 2017,
PhRvL
,
119, 231102
Steiner, A. W., Hempel, M., & Fischer, T. 2013,
ApJ
,
774, 17
Tanaka, M., & Hotokezaka, K. 2013,
ApJ
,
775, 113
The LIGO Scienti
fi
c Collaborationthe Virgo Collaborationthe KAGRA
Collaboration 2024,
ApJL
,
970, L34
Tsang, D., Read, J. S., Hinderer, T., Piro, A. L., & Bondarescu, R. 2012,
PhRvL
,
108, 011102
Wanajo, S., Sekiguchi, Y., Nishimura, N., et al. 2014,
ApJL
,
789, L39
Xuan, Z., Naoz, S., Li, A. K. Y., et al. 2024, arXiv:
2409.15413
Yudin, A. V., Razinkova, T. L., & Blinnikov, S. I. 2020,
AstL
,
45, 847
Zenati, Y., Krolik, J., Werneck, L., et al. 2024,
ApJ
,
971, 50
Zenati, Y., Krolik, J. H., Werneck, L. R., et al. 2023,
ApJ
,
958, 161
Zevin, M., Spera, M., Berry, C. P. L., & Kalogera, V. 2020,
ApJL
,
899, L1
8
The Astrophysical Journal,
978:126
(
8pp
)
, 2025 January 10
Zenati et al.