# Constraining Mass Transfer Models with Galactic Neutron Star$-$White Dwarf
Binaries as Gravitational Wave Sources

Jian-Guo He1,2, Yong Shao1,2, Xiao-Jie Xu1,2, Xiang-Dong Li1,2
1Department of Astronomy, Nanjing University, Nanjing 210023, People’s
Republic of China
2Key Laboratory of Modern Astronomy and Astrophysics, Nanjing University,
Ministry of Education, Nanjing 210023, People’s Republic of China E-mail:
<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Neutron star$-$white dwarf (NSWD) binaries are one of the most abundant
sources of gravitational waves (GW) in the Milky Way. These GW sources are the
evolutionary products of primordial binaries that experienced many processes
of binary interaction. We employ a binary population synthesis method to
investigate the properties of Galactic NSWD binaries detectable by the Laser
Interferometer Space Antenna (LISA). In this paper, only the NSWD systems with
a COWD or ONeWD component are included. We consider various models related to
mass transfer efficiencies during primordial binary evolution, supernova
explosion mechanisms at NS formation, common envelope ejection efficiencies,
and critical WD masses that determining the stability of mass transfer between
WDs and NSs. Based on our calculations, we estimate that tens to hundreds of
LISA NSWD binaries exist in the Milky Way. We find that the detection of LISA
NSWD binaries is able to provide profound insights into mass transfer
efficiencies during the evolution of primordial binaries and critical WD
masses during mass transfer from a WD to an NS.

###### keywords:

gravitational waves – binaries: general – stars: neutron – stars: white dwarf
– stars: evolution

††pubyear: 2023††pagerange: Constraining Mass Transfer Models with Galactic
Neutron Star$-$White Dwarf Binaries as Gravitational Wave Sources–A

##  1 Introduction

In recent years, ground-based gravitational wave (GW) detectors such as LIGO
and Virgo have identified nearly one hundred mergers of double compact objects
(Abbott et al., 2023), primarily composed of black holes (BHs), since the
groundbreaking detection of GW150914 (Abbott et al., 2016). Among these
mergers, two (GW170817 and GW190425) originate from double neutron star (NS)
systems. To date, no event involving the merger of an NS and a white dwarf
(WD) has been confirmed. The detection of GW signals from NSWD binaries is
able to help resolve some astrophysical problems, including the stability of
mass transfer between WDs and NSs (e.g., Verbunt & Rappaport, 1988; Bobrick et
al., 2017), the equation of state of NS matter (Tauris, 2018), the possible
origins of ultra-compact X-ray binaries (UCXBs, van Haaften et al., 2013; Wang
et al., 2021), repeating fast radio bursts (Gu et al., 2016), faint type Iax
supernovae (Bobrick et al., 2022), and peculiar gamma-ray bursts (e.g., Yang
et al., 2022; Kaltenborn et al., 2023). Future space-based GW detectors such
as LISA (Amaro-Seoane et al., 2017) and TianQin (Luo et al., 2016) are
promising to detect these GW signals in the mHz band.

It is expected that the Milky Way hosts hundreds of NSWD systems observable by
LISA (Amaro-Seoane et al., 2023). An early work by Nelemans et al. (2001b)
indicated that LISA may detect several hundreds of Galactic NSWD binaries.
Based on the observations of close binary radio millisecond pulsars, Tauris
(2018) inferred that at least a hundred NSWD binaries with helium WDs (HeWDs)
could be detected as LISA sources in the Milky Way. In addition, Chen et al.
(2020) estimated the existence of approximately 60$-$80 interacting LISA
NS$-$HeWD systems in the Galactic field. More recently, Korol et al. (2023)
suggested that around 80$-$350 detached Galactic NSWDs are detectable by LISA
over its 4-year duration, depending on models related to the kick velocities
of natal NSs and the treatments of common envelope (CE) evolution. Outside the
Milky Way, LISA is likely to detect about $1-6$ NSWD systems in M31 for a
10-year mission (He et al., 2023). In galaxies beyond the local group, GW
signals from merging NSWD binaries are challenging to observe due to their
large distances and limited chirp masses (Amaro-Seoane et al., 2023), unless
the operation of future next-generation detectors such as DO-OPT and DEC (Kang
et al., 2024).

Based on the canonical channels with isolated binary evolution, close NSWD
systems are expected to be the descendants of low-mass X-ray binaries that
experienced stable Roche lobe overflow (RLOF) or intermediate-mass X-ray
binaries that experienced CE evolution (Tauris & van den Heuvel, 2023). The
former channel always results in the formation of HeWDs, while the latter
tends to produce more massive WDs, i.e., carbon-oxygen WDs (COWDs) or oxygen-
neon WDs (ONeWDs). In some cases, it is possible that mass transfer during the
progenitor evolution of NSWD binaries can effect a reversal of the end states
of the two components, resulting in a WD that forms before an NS (e.g.,
Nelemans et al., 2001b). Observations of detached NSWD systems with orbital
periods of $<0.4$ days and HeWD masses of $<0.2M_{\odot}$ pose a challenge to
the RLOF channel since the formation of these binaries is very sensitive to
various factors such as initial NS masses, NS’s accretion efficiencies, and
magnetic braking mechanisms (Istrate et al., 2014; Chen et al., 2020). As the
orbits of these detached NSWD binaries significantly shrink due to GW
radiation, they are likely to become detectable GW sources. Subsequently,
these systems evolve to be UCXBs when the WD companion fills its Roche lobe
and transfers material to the NS. It has been shown that the duration
approximately one million years before and after the onset of UCXBs represents
a GW detection window (Tauris, 2018), indicating an overlap in the
evolutionary pathways of NSWD binaries as LISA sources and UCXBs. In recent
investigations on Galactic NSWD binaries as GW sources (Tauris, 2018; Chen et
al., 2020; Korol et al., 2023), UCXBs with HeWDs and detached systems with
HeWDs/COWDs/ONeWDs have been considered. It is possible that UCXBs with
COWDs/ONeWDs also contribute to the population of GW sources if a massive WD
can stably transfer matter to an NS.

The stability of mass transfer between WDs and NSs has been extensively
studied but remains uncertain. The traditional jet-only model (Verbunt &
Rappaport, 1988) suggests a critical WD mass of approximately $0.5M_{\odot}$
for stable mass transfer. Later, the isotropic re-emission mass-transfer
assumption gives a limit of around $0.4M_{\odot}$ for WD masses (Yungelson et
al., 2002; van Haaften et al., 2012), taking into account the inability to
eject a sufficient amount of transferred matter from the system, further
reducing the stability of mass transfer. By assuming that the NSWD systems
with WDs of masses $>0.38M_{\odot}$ lead to unstable mass transfer and merge,
van Haaften et al. (2013) pointed out that most UCXBs consist of an HeWD
component. Based on hydrodynamic simulations, Bobrick et al. (2017) revealed a
lower critical WD mass of $0.2M_{\odot}$ when considering the effect of disc
winds that developed at highly super-Eddington mass-transfer rates. In this
case, only NSWD systems containing an HeWD can evolve into UCXBs. Using the
same method, Church et al. (2017) obtained a critical value of approximately
$0.15-0.25M_{\odot}$ for WD masses, depending on the assumptions of initial
input parameters and specific physical models. However, in contrast to these
results, Chen et al. (2022) suggested that all NS$-$HeWD binaries with WD
masses of $0.17-0.45M_{\odot}$ are expected to undergo stable mass transfer
when using the detailed stellar evolution code MESA, which takes into account
the realistic structure of WDs. Chen et al. (2022) also demonstrated that the
stability of mass transfer from an HeWD to an NS is independent of NS’s mass
and its mass-accretion efficiency that characterizes the fraction of
transferred matter accreted by the NS. In a different approach, Yu et al.
(2021) developed a code to investigate the orbital evolution of mass-
transferring NSWD binaries and showed that the majority of these systems
experience stable mass transfer. They found that the maximum WD mass can reach
approximately $1.25-1.28M_{\odot}$ for stable mass transfer, beyond which
NSWDs directly merge when mass transfer begins.

In this paper, we provide a comprehensive study on the characteristic
distribution of Galactic NSWD binaries as GW sources. Using a binary
population synthesis (BPS, see a review by Han et al., 2020) method, we
consider the effect from various aspects such as the treatments of mass
transfer between binary components, the efficiencies of CE ejection, and the
mechanisms of supernova explosion. The structure of this paper is organized as
follows. In Section 2, we present the methodology employed, which includes the
adoption of different models. Subsequently, in Section 3, we present the
results derived from our calculations. In Section 4, we make some discussions
based on these results. Finally, we conclude in Section 5.

##  2 Method

We employ the BSE code developed by Hurley et al. (2002) to investigate the
population of Galactic GW sources of NSWD binaries across various models.
These models involve different supernova recipes, mass-transfer efficiencies
and its stability, as well as physical parameters related to CE evolution.
Additionally, we account for the star formation history of the Milky Way and
perform the integration of the spatial motion of NSWDs under the influence of
Galactic gravitational potential. Taking into account the spatial locations of
Galactic NSWD systems, we calculate the signal-to-noise ratio (S/N) for every
GW binary in the LISA band and obtain the population of detectable sources
accordingly.

###  2.1 Supernova Mechanisms

Regarding the types and the masses of stellar remnants following core-collapse
supernova (CCSN) explosions, we utilize the rapid mechanism (Fryer et al.,
2012), which correlates the remnant masses with the masses of the CO cores
prior to explosions. This mechanism can account for the existence of the mass
gap between NSs and BHs (Shao, 2022). Under this mechanism, we adopt a Maxwell
distribution with a standard deviation of
$\sigma=265\mathrm{~{}km}\mathrm{~{}s}^{-1}$ (Hobbs et al., 2005) for the kick
velocities of natal NSs. For NSs formed through electron capture supernovae
(ECSN) or accretion-induced collapse (AIC), we use smaller kick velocities
with $\sigma=30\mathrm{~{}km}\mathrm{~{}s}^{-1}$ (Vigna-Gómez et al., 2018;
Shao & Li, 2018).

In addition, we consider an alternative supernova explosion mechanism with the
stochastic recipe proposed by Mandel & Müller (2020). Unlike the rapid
mechanism, this recipe introduces randomness in compact remnant masses and
kick velocities. Both the rapid and the stochastic recipes have been
incorporated into the BSE code (Shao & Li, 2021).

###  2.2 Mass Transfer

In a binary, the stability of mass transfer is determined by considering the
adiabatic hydrostatic response of the donor star to mass loss, denoted as
$\zeta_{\mathrm{ad}}$,

$\zeta_{\mathrm{ad}}=\left.\frac{\partial\ln R_{2}}{\partial\ln
M_{2}}\right|_{\mathrm{ad}},$ (1)

as well as the response of the Roche lobe to mass loss, denoted as
$\zeta_{\mathrm{RL}}$,

$\zeta_{\mathrm{RL}}=\left.\frac{\partial\ln R_{\mathrm{L,2}}}{\partial\ln
M_{2}}\right|_{\mathrm{bin}},$ (2)

where $R_{2}$ and $R_{\mathrm{L,2}}$ are the radii of the donor star and its
Roche lobe, respectively, and $M_{2}$ is the mass of the donor star (see e.g.,
Soberman et al., 1997). When $\zeta_{\mathrm{RL}}<\zeta_{\mathrm{ad}}$, mass
transfer proceeds in a stable manner. Otherwise, dynamically unstable mass
transfer occurs and CE evolution is triggered.

####  2.2.1 Efficiency of mass transfer

During the evolution of the primordial binaries initially containing two zero-
age main-sequence stars, mass-transfer efficiency ($\eta_{\mathrm{MT}}$)
characterizes the fraction of the transferred matter that is accreted by the
secondary star, which plays a crucial role in determining whether the binaries
undergo stable mass transfer or evolve into a contact/CE phase (e.g., de Mink
et al., 2007). It has been demonstrated that a lower mass-transfer efficiency
tends to prevent significant expansion of the secondary star due to accretion,
allowing a larger parameter space of primordial binaries for stable mass
transfer. Based on the work of Shao & Li (2014), we employ three mass
accretion models to deal with the process of mass transfer during primordial
binary evolution. By default, we utilize the rotation-dependent mass accretion
model, referred to as MA1. In this model, the mass accretion rate of the
secondary star is assumed to be dependent on its rotational velocity, so the
mass-transfer process could be highly non-conservative with
$\eta_{\mathrm{MT}}\lesssim 20\%$. Alternatively, we consider two other
models: half mass accretion and thermal equilibrium-limited mass accretion,
referred to as MA2 and MA3, respectively, corresponding to
$\eta_{\mathrm{MT}}=50\%$ and $\eta_{\mathrm{MT}}\sim 100\%$. Each accretion
model is associated with a specific criterion to decide the stability of mass
transfer between binary components (Shao & Li, 2014). Previous investigations
have shown that the rotation-dependent model can better match the observations
of Galactic OBe-star binaries with a BH or a Wolf-Rayet star companion, while
the observations of Galactic Be-star binaries with an NS or a subdwarf
companion seem to favor the models with $\eta_{\mathrm{MT}}\gtrsim 0.5$ (Shao,
2022, and references therein).

When the accretor is an NS, we assume an accretion efficiency of 0.5 (Chen et
al., 2020). In addition, the accretion rate of the NS is constrained by the
Eddington limit. In our calculations, we adopt the isotropic re-emission
mechanism for non-conservative mass transfer. It is assumed that the material
lost from a binary system carries away the specific angular momentum of the
accretor.

####  2.2.2 Mass transfer from a nondegenerate star to an NS

In the case of non-conservative mass transfer with an isotropic re-emission
way, previous works (e.g., Soberman et al., 1997) have suggested a positive
correlation between $\zeta_{\mathrm{RL}}$ and $q=M_{\rm d}/M_{\rm NS}$ (mass
ratio of the donor to the NS). As a consequence, there are critical mass
ratios used to determine mass-transfer stability. Tauris et al. (2000) pointed
out that the NS binaries with $q\gtrsim 3-3.5$ always evolve into CE phases
while the systems with $q\lesssim 1.5-2$ are expected to undergo stable mass
transfer (see also Shao & Li, 2012; Misra et al., 2020). Based on detailed
evolutionary simulations of the BH binaries with nondegenerate donors, Shao &
Li (2021) obtained easy-to-use criteria for mass-transfer stability. In this
paper, we apply these criteria to the binaries with an NS accretor. It is
assumed that mass transfer is always stable if $q<2$ or always unstable if
$q>2.1+0.8M_{\rm NS}$. For the systems with mass ratios between them, mass
transfer stability is dependent on the properties of the donor stars.

For the binaries with a naked He star and an NS, we adopt the criteria
calculated by Tauris et al. (2015) to deal with mass-transfer stability.
According to their work, CE phases are expected to occur when the masses of
the He stars exceed $2.7M_{\odot}$ and the orbital periods are below $0.06$
days. So we assume that only the systems with He-star masses of
$>2.7M_{\odot}$ and orbital periods of $<0.06$ days evolve into CE phases.

####  2.2.3 Mass transfer from a WD to an NS

Considering that UCXBs with a WD donor and an NS accretor can last $\sim
1\,\rm Myr$ as GW sources, it is speculated that the stability of mass
transfer between WDs and NSs significantly impacts the population properties
of LISA NSWD binaries. Until now, however, the stability of mass transfer from
a WD to an NS still remains uncertain. By default, we adopt a threshold of
$0.2M_{\odot}$ for WD masses (Bobrick et al., 2017) to investigate the
properties of detached NSWD systems observable by LISA. This threshold does
not allow the NS binaries with COWDs/ONeWDs to evolve into long-standing
UCXBs. Also, we vary this threshold mass $M_{\mathrm{WD,crit}}$ from
$0.4M_{\odot}$ (van Haaften et al., 2012) to $1.25M_{\odot}$ (Yu et al., 2021)
and explore its influence on the number of interacting NSWD systems (UCXBs) in
the Milky Way.

For other types of binary systems not mentioned above, we use the default
criteria given by Hurley et al. (2002) to deal with the stability of mass
transfer.

####  2.2.4 CE Evolution

When CE evolution is triggered, we employ the energy conservation prescription
to determine the outcome of CE phases. The related formulae can be found in
Hurley et al. (2002) and Shao & Li (2014). In our study, we utilize the
binding energy parameter $\lambda$ fitted by Xu & Li (2010). By default, we
assume the efficiency of CE ejection to be unity, i.e.,
$\alpha_{\mathrm{CE}}=1.0$. This parameter determines the proportion of
orbital energy lost that used to eject donor’s envelope. To assess the impact
of $\alpha_{\mathrm{CE}}$ on the population of LISA NSWD systems, we consider
two additional efficiencies, by choosing $\alpha_{\mathrm{CE}}=0.3$ as
inferred from the parameter distribution of Galactic WDWD systems (Scherbak &
Fuller, 2023) and $\alpha_{\mathrm{CE}}=3.0$ as required by the formation of
the post-CE system IK Pegasi (Davis et al., 2010).

###  2.3 Primordial Binaries

In our study, we simulate the evolution of approximately $10^{6}$ primordial
binaries for each model. The initial parameters of primordial binaries are set
as follows. The primary masses range from $5$ to $100M_{\odot}$, and the
secondary masses range from $0.5$ to $100M_{\odot}$. All primordial binaries
are assumed to have circular orbits, with separations varying from $3$ to
$10000R_{\odot}$. The binary fraction among all stars is assumed to be unity.
We follow the method of Shao & Li (2021) to calculate the Galactic population
of LISA NSWD systems that evolved from primordial binaries.

###  2.4 Star Formation History and Orbital Integration

For the Milky Way, we assume a constant star formation rate of
$3M_{\odot}\mathrm{~{}yr}^{-1}$ and a constant metallicity of $Z_{\odot}=0.02$
throughout its entire lifespan of 10 $\mathrm{Gyr}$.

To account for the spatial motions of NSWD binaries, we utilize the galpy
package (with the MWPotential2014 model, Bovy, 2015) to numerically integrate
their tracks from the formation of NS until either the binaries merge or the
evolutionary time exceeds 10 $\mathrm{Gyr}$.

Regarding the initial locations of primordial binaries in the Milky Way, the
star number density distribution can be described as a function of radial
distance from the Galactic center $r$ and vertical distance from the Galactic
plane $z$, using the equation proposed by Bahcall & Soneira (1980) as

$\rho(r,z)=\rho_{\odot}\exp\left[-\frac{r-r_{\odot}}{h_{r}}-\frac{z}{h_{z}}\right],$
(3)

where $r_{\odot}=8.5\mathrm{~{}kpc}$ represents the radial distance of the Sun
from the Galactic center, and $\rho_{\odot}$ denotes the star number density
at the location of the Sun. Here, $h_{r}=3.5\mathrm{~{}kpc}$ and
$h_{z}=0.25\mathrm{~{}kpc}$ represent the scale length parallel and
perpendicular to the Galactic plane. In the Galactic reference frame, the
velocities of the mass centers of pre-SN systems are assumed to be consistent
with rotation curves, where the circular velocity of the Sun is 220
$\mathrm{km}\mathrm{~{}s}^{-1}$.

Following the approach of Hurley et al. (2002), the velocities of the new mass
centers of post-SN binaries are subject to changes caused by mass losses and
NS kicks during supernova explosions.

###  2.5 Signal-to-noise Ratio (S/N) of GW

We utilize the python package LEGWORK (Wagg et al., 2022) to calculate the S/N
of Galactic NSWD binaries and identify those with an S/N greater than 5 as
detectable LISA sources. Among optional spacecraft parameters, we select the
robson19 model (Robson et al., 2019) for the sub-mHz confusion noise
contributed by unresolved WDWD binaries in the Galaxy (Cornish & Robson, 2017;
Babak et al., 2021). By default, the observation time is set to 4 years, which
is the standard duration used in LISA mission simulations.

##  3 Result

It is challenging for rapid population synthesis to model the formation of
close NSWD systems with low-mass HeWDs that involving the RLOF channel and
requiring severe fine-tuning of input parameters (see e.g., Istrate et al.,
2014; Chen et al., 2020; Deng et al., 2021). Consequently, the NS$-$HeWD
systems as GW sources are absent in our results111Despite the absence of
NS$-$HeWD systems in our results, we can evaluate their impact on LISA
detection. Previous studies have demonstrated that LISA may detect more than
100 NS$-$HeWD binaries in the Milky Way (Tauris, 2018; Chen et al., 2020).
Magnetic braking mechanisms are thought to play a crucial role in forming
close NS$-$HeWD systems and alleviating the fine-tuning problem (Deng et al.,
2021; Chen et al., 2021). For LISA NSWD sources, one may differentiate HeWDs
from COWDs/ONeWDs with measured chirp masses in detached systems or observed
spectral lines in interacting systems. Possible detection of LISA NS$-$HeWD
systems can be used to constrain the mechanisms of magnetic braking.. And, we
consider LISA NSWD systems with NSs originating from CCSN and ECSN, while
those with NSs originating from AIC are disregarded here and discussed in
Section 4.3.

Our calculations reveal that the total number of Galactic NSWD systems in the
Milky Way is about $2\times 10^{6}$, which is consistent with the estimation
of Nelemans et al. (2001b). Expected numbers of detectable NSWD systems by
LISA vary across different models, as presented in Table 1. In this table, we
separately list the numbers of detached and interacting LISA sources, as well
as the merger rates of NSWD systems.

In Section 3.1, we discuss the evolutionary pathways and initial binary
parameter spaces to form LISA NSWD binaries. Subsequently, in Section 3.2, we
evaluate the influence of different models related to the options of
$\eta_{\mathrm{MT}}$, $\alpha_{\mathrm{CE}}$, and supernova recipes. We
analyze the distributions of various parameters of LISA NSWD binaries
including their orbital parameters, component masses, and Galactic locations.
In Section 3.3, we investigate the impact of $M_{\mathrm{WD,crit}}$ on
interacting LISA NSWDs. In Section 3.4, we estimate the merger rates of NSWD
binaries in the Milky Way and the local Universe.

Table 1: Expected numbers and merger rates ($R_{\mathrm{merger}}$) of LISA NSWD binaries in the Milky Way. Our models include different treatments of mass-transfer efficiencies during primordial binary evolution ($\eta_{\mathrm{MT}}$), critical WD masses for the stability of mass transfer between WDs and NSs ($M_{\mathrm{WD,cri}}$), CE ejection efficiencies ($\alpha_{\mathrm{CE}}$) and supernova recipes. MA1, MA2, and MA3 represent rotation-dependent mass accretion, half mass accretion ($\eta_{\mathrm{MT}}=50\%$), and near-conservative mass accretion ($\eta_{\mathrm{MT}}\sim 100\%$), respectively. The superscripts D and I refer to detached and interacting LISA NSWD binaries, respectively. $\mathcal{R}_{\mathrm{merger}}$ denotes estimated merger rate densities of NSWD systems in the local Universe. $\eta_{\mathrm{MT}}$ | $M_{\mathrm{WD,cri}}$ | $\alpha_{\mathrm{CE}}$ | SN recipes | $N^{\mathrm{D}}$ | $N^{\mathrm{I}}$ |  $N^{\mathrm{D}}$ ($N^{\mathrm{I}}$) |  $N^{\mathrm{D}}$ ($N^{\mathrm{I}}$) | $R_{\mathrm{merger}}$ | $\mathcal{R}_{\mathrm{merger}}$
---|---|---|---|---|---|---|---|---|---
| $(M_{\odot})$ |  |  |  |  | NS forms first | WD forms first | $(\mathrm{Myr}^{-1})$ | $(\mathrm{Gpc}^{-3}\mathrm{yr}^{-1})$
MA1 | 0.2 | 0.3 | rapid | 45 | $-$ | 34 | 11 | 17.4 | 174
MA1 | 0.2 | 1.0 | rapid | 105 | $-$ | 79 | 26 | 43.3 | 433
MA1 | 0.2 | 1.0 | stochastic | 162 | $-$ | 120 | 42 | 66.9 | 669
MA1 | 0.2 | 3.0 | rapid | 197 | $-$ | 149 | 48 | 83.3 | 833
MA2 | 0.2 | 0.3 | rapid | 17 | $-$ | 14 | 3 | 7.2 | 72
MA2 | 0.2 | 1.0 | rapid | 78 | $-$ | 55 | 23 | 31.2 | 312
MA2 | 0.2 | 1.0 | stochastic | 103 | $-$ | 72 | 31 | 42.3 | 423
MA2 | 0.2 | 3.0 | rapid | 153 | $-$ | 86 | 67 | 64.3 | 643
MA3 | 0.2 | 0.3 | rapid | 49 | $-$ | 11 | 38 | 18.1 | 181
MA3 | 0.2 | 1.0 | rapid | 145 | $-$ | 47 | 98 | 55.9 | 559
MA3 | 0.2 | 1.0 | stochastic | 211 | $-$ | 63 | 148 | 82.6 | 826
MA3 | 0.2 | 3.0 | rapid | 234 | $-$ | 64 | 170 | 88.9 | 889
MA1 | 1.25 | 1.0 | rapid | 105 | 194 | 79 (174) | 26 (20) | 5.1 | 51
MA2 | 1.25 | 1.0 | rapid | 78 | 116 | 55 (100) | 23 (16) | 8.7 | 87
MA3 | 1.25 | 1.0 | rapid | 145 | 120 | 47 (99) | 98 (21) | 31.8 | 318

###  3.1 Formation Scenarios and Binary Parameter Spaces

NSWD systems can be categorized based on the formation order of their binary
components, specifically whether the NS or the WD forms first. Therefore, we
classify the formation scenarios for LISA NSWD sources as follows:

Scenario 1: NS forms first.

Scenario 2: WD forms first.

Evolved from primordial binaries, Scenario 1 means that the primary stars
firstly evolve into NSs and then the secondary stars become WDs. There is a
common feature that the NSWD binaries formed via Scenario 1 have circular
orbits. The masses of the primary stars and the secondary stars fall within
the ranges of $6-20M_{\odot}$ and $2-10M_{\odot}$, respectively. The orbital
periods of the primordial binaries cover two separated ranges, i.e., a few
days to tens of days and hundreds to thousands of days. Generally, the
primordial binaries with narrow orbits undergo stable mass transfer during the
first mass-transfer stage, while those with wide orbits are expected to
experience CE evolution since the primary stars have developed a deep
convective envelope prior to mass transfer.

In Scenario 2, the primary stars typically have initial masses of $\gtrsim
5M_{\odot}$ and the secondary stars have similar masses. As the progenitors of
LISA NSWD sources, the corresponding primordial binaries have relatively
narrow orbits with periods of a few days to tens of days. During the
evolution, mass transfer occurs through stable RLOF. The primary stars firstly
turn into WDs. Since the secondary stars have accreted sufficient matter
during previous mass-transfer processes, they can evolve into He stars with
masses of $\gtrsim 2M_{\odot}$ after their hydrogen envelopes stripped via CE
phases, and eventually become NSs. Additionally, it is possible for Scenario 2
that a small fraction of the primordial binaries with long orbital periods
evolve to be double He-star systems before WD and NS formation (see also the
Pathway 3 described by Toonen et al., 2018). In contrast to Scenario 1, NSWD
binaries where WDs form first tend to exhibit large orbital eccentricities due
to the lack of an efficient mechanism of orbital circularization after NS
formation.

###  3.2 Detached NSWD Binaries

####  3.2.1 The impact of mass-transfer efficiency

Figure 1: Calculated number distributions of detached LISA NSWD systems in the
Milky Way, as a function of NS mass, WD mass, orbital period, and
eccentricity. The left, middle, and right panels correspond to the models MA1,
MA2 and MA3, respectively. Here, we adopt $\alpha_{\mathrm{CE}}=1$ and the
rapid mechanism of supernova explosions. In each panel, the blue contours
represent systems where NS forms first, while the orange contours represent
systems where WD forms first. Notably, all NSWD binaries where NS forms first
exhibit circular orbits, the corresponding blue contours do not appear to show
these systems in the plane of orbital period versus eccentricity.

Fig. 1 presents calculated number distributions of Galactic LISA sources of
detached NSWD binaries in the planes of NS mass versus WD mass (upper panels)
and orbital period versus eccentricity (lower panels). The left, middle, and
right panels correspond to the mass accretion models MA1, MA2, and MA3,
respectively. In this analysis, we adopt the default rapid supernova explosion
mechanism and set $\alpha_{\mathrm{CE}}=1$. The value of
$M_{\mathrm{WD,crit}}$ is fixed at $0.2M_{\odot}$, resulting in all LISA NSWD
systems being detached. The reason is that when NSs form first, WD masses are
typically above $0.4M_{\odot}$, whereas when WDs form first, WD masses are
generally larger than $0.8M_{\odot}$. This also indicates the absence of HeWDs
in all cases. Since the critical mass ratio of nondegenerate donors to NSs for
avoiding CE evolution is $\sim 2$ (see e.g., Misra et al., 2020), LISA NSWD
binaries formed via Scenario 1, as the evolutionary products of intermediate-
mass X-ray binaries, are expected to host COWDs or ONeWDs. On the other hand,
NSWD systems formed via Scenario 2 require the masses of both components of
the primordial binaries to exceed $5M_{\odot}$, leading to produce massive
WDs. The masses of NSs are distributed with two peaks at $\sim 1.1M_{\odot}$
and $\sim 1.3M_{\odot}$, corresponding to the NSs formed from CCSN and ECSN,
respectively. For CCSN NSs, we adopt the rapid mechanism (Fryer et al., 2012)
which predicts that stars with masses of $\sim 8-12M_{\odot}$ finally collapse
into $\sim 1.1M_{\odot}$ NSs. For ECSN NSs, we simply assume that they are
born with mass of $1.3M_{\odot}$ (see also He et al., 2023).

It is noteworthy that distinguishing the evolutionary origins of detached NSWD
systems from the RLOF and the CE channels is relatively straightforward. The
RLOF channel is expected to produce LISA NS$-$HeWD binaries with chirp masses
of $\lesssim 0.44M_{\odot}$, corresponding to the systems containing a
$\lesssim 2M_{\odot}$ NS and a $\sim 0.167M_{\odot}$ WD (Tauris, 2018). For
the CE channel, our calculations indicate a minimum chirp mass of $\sim
0.56M_{\odot}$ for LISA NSWD systems which contain a $\gtrsim 1.1M_{\odot}$ NS
and a $\gtrsim 0.4M_{\odot}$ WD. Consequently, we can confidently discern the
evolutionary channels of LISA NSWD systems. Next, we focus on the systems with
COWD/ONeWD components and perform quantitative analyses of binary parameters
under different models.

Among three mass accretion models, we see that MA3, which corresponds to
nearly conservative mass transfer, yields the highest number of LISA NSWD
systems where WDs form first. Additionally, the majority of these systems are
expected to host an ONeWD. As mass-transfer efficiency increases, the
component masses of primordial binaries in Scenario 2 shift towards lower
values, resulting in the formation of more systems where WDs form first due to
initial mass function. Specifically, the model MA3 predicts the existence of
approximately 50 detached NSWD systems with eccentricities exceeding 0.1. In
this case, the fraction of these eccentric binaries among all detached LISA
systems with COWDs/ONeWDs (the number is 145 from Table 1) can reach as high
as $\sim 0.34$. And, we estimate that $\sim 8$ systems are likely to have
relatively wide orbits with periods ranging from 0.1 to 1 day. In contrast,
the models MA1 and MA2 predict that almost no LISA NSWD binaries have large
eccentricities of $>0.5$, and only around 10 systems are expected to have
eccentricities exceeding 0.1, corresponding to their fraction of $\sim
0.1-0.14$ among all $78-105$ LISA binaries (see Table 1). Based on the
distributions of orbital parameters combined with the numbers of detached NSWD
systems detectable by LISA, it becomes feasible to provide constraints on
possible mass accretion models from future GW observations.

Figure 2: Pie charts illustrating the relative fractions of detached LISA NSWD
systems where NSs (blue) or WDs (red) form first. The left, middle, and right
panels represent the models MA1, MA2 and MA3, respectively. The panels from
top to bottom correspond to $\alpha_{\mathrm{CE}}=0.3$, 1.0, and 3.0,
respectively. Below each pie chart, we also give the corresponding numbers for
all detached binaries and the systems with eccentricities larger than 0.1.

####  3.2.2 The impact of $\alpha_{\mathrm{CE}}$

Fig. 2 shows the relative fractions of detached LISA NSWD binaries formed via
Scenario 1 or 2. The left, middle, and right panels represent the models MA1,
MA2 and MA3, respectively. The panels from top to bottom correspond to
$\alpha_{\mathrm{CE}}=0.3,$ 1.0, and 3.0, respectively. In each panel, we also
give the numbers of all detached LISA binaries from our calculations and the
systems with eccentricities larger than 0.1.

It is obvious that mass-transfer efficiencies during primordial binary
evolution are the dominant factor influencing the relative fractions of
systems where NSs or WDs form first. However, we note that the influence of
$\alpha_{\mathrm{CE}}$ cannot be disregarded. On the one hand, a lower value
of $\alpha_{\mathrm{CE}}$ make it more challenging to eject CE, resulting in a
significant reduction in the number of systems. Expected numbers of LISA NSWD
binaries ($N^{\rm D}$) decrease from 153$-$234, to 78$-$145, and to 17$-$49
when decreasing $\alpha_{\mathrm{CE}}$ from 3.0, to 1.0, and to 0.3,
respectively. On the other hand, the numbers of the systems with
eccentricities above 0.1 ($N^{\rm D}_{\rm e>0.1}$) are sensitive to the
options of $\alpha_{\mathrm{CE}}$ in the models MA1 and MA2. Overall, the
ratios of $N^{\rm D}_{\rm e>0.1}$ to $N^{\rm D}$ are always $\lesssim 0.2$ in
these two models. While for all our adopted $\alpha_{\mathrm{CE}}$, the model
MA3 predicts that $\sim 0.3-0.4$ of detectable NSWD binaries have
eccentricities larger than 0.1. This is because the primordial binaries in
Scenario 2 have the orbital periods of 8$-$27 days for the model MA3, while
they have the orbital periods of 2$-$20 days for the models MA1 and MA2. The
latter is more susceptible to the influence of $\alpha_{\mathrm{CE}}$ when CE
evolution occurs in the subsequent binaries with a giant star and a WD.

In the model MA3, a lower value of $\alpha_{\mathrm{CE}}=0.3$ leads to an
increase of the relative fraction of detached LISA NSWD binaries where WDs
form first, compared to the cases of $\alpha_{\mathrm{CE}}=1.0$ and
$\alpha_{\mathrm{CE}}=3.0$, which can potentially shift the chirp-mass
distribution of the whole population of NSWD GW sources to have a higher mass
peak. A similar conclusion was drawn by Korol et al. (2023), who assumed the
mass-accretion rate during primordial binary evolution to be limited by the
thermal timescale of the accreting secondary, resembling our model MA3.

####  3.2.3 The impact of supernova recipes

Figure 3: Probability density functions (PDFs) of detached LISA NSWD systems
in the Milky Way, as a function of chirp mass and vertical distance from the
Galactic plane. The left, middle, and right panels correspond to the models
MA1, MA2, and MA3, respectively. The top and bottom panels represent the rapid
and stochastic supernova mechanisms, respectively. Here we adopt
$\alpha_{\mathrm{CE}}=1.0$. In each panels, the blue dots denote systems where
NSs form first, while the orange dots denote systems where WDs form first. The
size of each dot reflects its offset distance from the Galactic center.

Fig. 3 shows the probability density functions of detached LISA NSWDs in the
Milky Way, as a function of chirp mass $M_{\rm chirp}$ and vertical distance
$z$ from the Galactic plane. Here, we adopt $\alpha_{\mathrm{CE}}=1.0$. The
blue and orange dots with big scatters correspond to the NSWD systems formed
via Scenarios 1 and 2, respectively. Different sizes of these dots represent
their offset distances from the Galactic center. The left, middle, and right
panels represent the models MA1, MA2, and MA3, respectively. The top and
bottom panels represent the rapid and stochastic supernova mechanisms,
respectively. There is a tendency that the vast majority of the binaries where
NSs form first are distributed with $z<1\,\rm kpc$, while the systems where
WDs form first are more likely to locate at relatively large $z$ with a tail
up to $\sim 2-5\,\rm kpc$. And, the latter systems are expected to have
significantly large offset distances from the Galactic center. The main reason
for these discrepancies is that the systems where WDs form first are more
susceptible to the kick velocities of natal NSs. Also, we should note that a
significant fraction of LISA NSWD binaries where WDs form first have eccentric
orbits.

For the supernova mechanisms changing from the rapid to the stochastic
recipes, we observe a slight shift in the overall distribution of chirp masses
of LISA NSWD sources towards higher values, especially in the model MA3. This
shift is because the stochastic mechanism can produce more massive NSs with
masses of $\sim 1.2M_{\odot}-1.6M_{\odot}$ than the rapid mechanism that forms
$\sim 1.1$ NSs. Furthermore, the stochastic mechanism allows part of NSs to
have small kick velocities, compared to the rapid mechanism, therefore
preventing the disruption of more binaries where WDs form first during
supernova explosions. As a result, we can see for the stochastic mechanism
that some LISA sources locate at relatively large $z$. However, this
difference from the spatial locations of LISA NSWD binaries cannot provide
strong constraints on our adopted supernova mechanisms. As pointed out by
Korol et al. (2023), the majority of LISA NSWDs locate within
$z=5\mathrm{~{}kpc}$ from the Galactic disk. Another avenue of exploration
lies in decoupling the component masses of observable binaries, which may
offer improved constraints on the remnant masses after supernova explosions.

###  3.3 Interacting NSWD Binaries

Figure 4: The influence of $M_{\mathrm{WD,crit}}$ on calculated numbers of
interacting LISA NSWD binaries (top panel), as well as number ratios of
interacting systems to detached systems (bottom panel). The blue, orange, and
green curves correspond to the models MA1, MA2, and MA3, respectively. Here
$\alpha_{\mathrm{CE}}=1.0$ and the rapid supernova mechanism are adopted.

For close detached NSWDs with orbital periods less than $\sim 0.4$ days, in
particular eccentric systems, the emission of GW is able to significantly
shrink their orbits. Within a Hubble time, RLOF occurs in these binaries,
leading to the formation of interacting systems. Depending on the stability of
mass transfer via RLOF, NSWD binaries may either merge or evolve into stable
UCXBs. Notably, a large value of $M_{\mathrm{WD,crit}}$ can increase the
number of interacting NSWD binaries.

Fig. 4 shows the expected number of interacting LISA NSWDs, as well as the
number ratio of interacting systems to detached systems, as a function of
$M_{\mathrm{WD,crit}}$. In this analysis, we choose $\alpha_{\mathrm{CE}}=1.0$
and the rapid supernova mechanism. Three mass accretion models (i.e. MA1, MA2,
and MA3) are taken into account for comparison. Across these models, a
pronounced increase in the number of observable sources occurs near
$0.8M_{\odot}$, as this is where the mass distribution of WDs has a peak (see
also Fig. 1). In each model, there is a plateau at the high-mass end of $\sim
1.1-1.2M_{\odot}$. This plateau arises because the orbital shrinkage of the
systems with $\gtrsim 1.1M_{\odot}$ ONeWDs caused by GW radiation dominates
and these systems always merge, although stable mass transfer is allowed to
happen. Overall, when $M_{\mathrm{WD,crit}}$ is below 0.6, the three models
exhibit a similar trend. For $M_{\mathrm{WD,crit}}$ exceeding 0.6, the model
MA1 is able to produce more interacting NSWD systems, since it allows the
formation of more close binaries with less-massive COWDs compared to the
models MA2 and MA3.

In total, the number ratios of interacting to detached NSWD binaries are less
than about 1.9/1.5/0.9, corresponding to the models MA1/MA2/MA3, respectively.
Importantly, these number ratios are sensitive to the options of
$M_{\mathrm{WD,crit}}$. In principle, we can provide some constraints on
$M_{\mathrm{WD,crit}}$ if a number of interacting and detached LISA NSWD
binaries with COWD/ONeWD components are identified in the future. However, it
is essential to consider the formation of interacting LISA sources via other
pathways such as the RLOF and the AIC channels. Distinguishing between these
channels becomes challenging. On the one hand, all detectable interacting
NSWDs exhibit eccentricities of $<0.001$ which are below the measurement
threshold of approximately 0.1 by LISA (Korol et al., 2023). On the other
hand, the RLOF channel can produce NS$-$HeWD systems with
$z>1\mathrm{~{}kpc}$, such as J0348+0432 (Antoniadis et al., 2013). It is
possible to distinguish the RLOF and the CE channels if one can observe
different spectra with components from originally HeWDs or COWDs/ONeWDs.
Furthermore, the AIC channel can lead to the formation of both NS$-$HeWD and
NS$-$COWD systems (see Section 4.3 for more details).

###  3.4 Merger rates

The merger rates ($R_{\mathrm{merger}}$) of Galactic NSWD binaries under
various models are presented in Table 1, varying in the range of about
$5-90\,\rm Myr^{-1}$. It is worth noting that the merger rates are relatively
low in the models with $\alpha_{\mathrm{CE}}=0.3$, which are consistent with
the small numbers of corresponding LISA binaries predicted in these models. In
the models MA1 and MA2, increasing $M_{\mathrm{WD,crit}}$ from $0.2M_{\odot}$
to $1.25M_{\odot}$ can greatly decrease the merger rates by a factor of $\sim
4-8$. However, the model MA3 shows an exception, as it allows the formation of
a substantial number of NS$-$ONeWD binaries, which are bound to merge despite
a large value of $M_{\mathrm{WD,crit}}$.

It is believed that NSWD mergers are relevant to some observable transients
outside the Milky Way (e.g., Bobrick et al., 2022; Kaltenborn et al., 2023).
One can calculate the merger rate density ($\mathcal{R}_{\mathrm{merger}}$) of
NSWD binaries in the local Universe, if simultaneously modeling the evolution
of star formation and metallicity as a function of redshift. Here we present a
rough estimation of $\mathcal{R}_{\mathrm{merger}}$, according to our obtained
$R_{\mathrm{merger}}$. We assume that the number density of Milky Way
equivalent galaxies within the local Universe is $0.01\rm\,Mpc^{-3}$ (e.g.,
Abadie et al., 2010). This produces a conversion factor between
$R_{\mathrm{merger}}=1\rm\,Myr^{-1}$ and
$\mathcal{R}_{\mathrm{merger}}=10\rm\,Gpc^{-3}\,yr^{-1}$. So we estimate the
merger rate density of NSWD binaries in the local Universe of $\sim
50-900\rm\,Gpc^{-3}\,yr^{-1}$, which is agreement with the theoretical
prediction of $390\rm\,Gpc^{-3}\,yr^{-1}$ made by Zhao et al. (2021).

##  4 Discussion

###  4.1 Identification of NSWDs

Although the detection of Galactic NSWD binaries as GW sources can provide
valuable information to constrain the origin of these sources and the physics
of binary interaction, not all NSWDs can be discerned among numerous GW
sources in the Milky Way.

The identification of LISA NSWD systems typically relies on the measurement of
chirp masses, which usually range from approximately $0.35$ to $1.2M_{\odot}$
(e.g., Korol et al., 2023). For comparison, the chirp mass distribution of
WDWD and NSNS systems exhibits a peak around $\sim 0.25-0.4\mathrm{M}_{\odot}$
(Korol et al., 2022) and $1.1M_{\odot}$ (Korol & Safarzadeh, 2021),
respectively. The determination of chirp masses can often be inferred through
the measurement of GW frequency derivative $\dot{f}_{\mathrm{GW}}$ for nearby
GW sources with exceptionally high S/N. Additionally, for other systems,
Tauris (2018) indicated that combining measurements of optical distance and GW
strain amplitude can effectively constrain $\dot{f}_{\mathrm{GW}}$.
Furthermore, eccentricities can serve as an alternative feature to distinguish
NSWD systems from WDWD systems in cases where chirp masses cannot be directly
measured. Korol et al. (2023) demonstrated that the minimum detectable
eccentricity, derived from full Bayesian parameter estimation (Moore et al.,
2023), can reach as low as $\sim 0.03$ when searching for realistic eccentric
NSWD systems. Moreover, as mentioned by Tauris (2018), a direct method to
identify NSWD systems is the search of both optical WDs and radio pulsars.

###  4.2 The degeneracy of multiple parameters

In our study, we include multiple parameters related to binary evolution,
which can potentially cross-influence the population statistics of LISA NSWD
systems and lead to possible degeneracy of these parameters.

The mass accretion models (MA1, MA2 and MA3) play a vital role in determining
the formation order of NS/WD, thereby affecting the eccentricity distribution
of detached LISA NSWD binaries. The formation of the systems with
eccentricities larger than 0.1 is sensitive to the option of these models.
Additionally, the mass accretion models can also influence the chirp mass
distributions, as the NSWD binaries where WDs from first tend to contain more
massive WDs than the systems where NSs form first. Combining the distributions
of orbital eccentricities and chirp masses for detached LISA NSWD binaries is
able to provide constraints on the mass accretion models.

Varying CE ejection efficiencies $\alpha_{\mathrm{CE}}$ can lead to different
relative fractions of detached LISA NSWD binaries where NSs or WDs form first.
The impact of $\alpha_{\mathrm{CE}}$ on the ratios of $N^{\rm D}_{\rm e>0.1}$
to $N^{\rm D}$ is considerably smaller than that of mass accretion models (see
Fig. 2). Furthermore, disentangling the impact of $\alpha_{\mathrm{CE}}$ from
that of supernova mechanisms is challenging, as both of them can significantly
influence the calculated numbers and the obtained parameter distributions of
detached LISA NSWD binaries (see Fig. 1 and Figs. 6$-$8 in Appendix A).

###  4.3 AIC Mechanism

Figure 5: The bar chart illustrates calculated numbers of LISA NSWD binaries
under various models. The blue bars represent the systems with CCSN/ECSN NSs,
while the orange bars represent the systems with AIC NSs. Since the criterion
of $M_{\mathrm{WD,crit}}=0.2M_{\odot}$ is used, all LISA binaries with
CCSN/ECSN NSs from our calculations are detached systems. Here, the LISA
binaries with AIC NSs include interacting systems.

Based on our calculations, we observe a significant contribution from the NSWD
binaries with AIC NSs to the whole population of Galactic LISA NSWD sources.
This contribution was not considered in previous analyses due to considerable
uncertainties of the AIC mechanism itself.

Beginning the evolution from a primordial binary, the primary star firstly
leaves an ONeWD. When the secondary star evolves to the giant branch and fills
its Roche lobe, a CE phase is triggered. This phase leads to the formation of
the ONeWD binary with a helium star or another WD companion. Subsequently, the
ONeWD collapse into an NS when its mass exceeds the critical mass of
$1.38M_{\odot}$ due to accretion.

In Fig. 5, we provide an estimate of the number of LISA NSWD binaries with AIC
NSs in the Milky Way. In contrast to the systems with CCSN/ECSN NSs, a lower
value of $\alpha_{\mathrm{CE}}$ tends to produce more LISA sources with AIC
NSs. Overall, our models predict that the Milky Way may host about $100-300$
LISA NSWD binaries with AIC NSs.

The AIC channel is not yet fully understood, including uncertainties from the
treatments of mass-transfer stability in binaries with WD accretors and mass-
retention efficiency of accreting WDs. For the stability of mass transfer
between giant-like donor stars and WDs, we adopt the default critical mass
ratio in the BSE code, as Equation (56) of Hurley et al. (2002). This
criterion of mass-transfer stability is derived by Hjellming & Webbink (1987),
assuming the transfer of mass and orbital angular momentum is conservative
during the evolution. However, the realistic value of $\zeta_{\mathrm{RL}}$
depends on specific mass-loss mechanisms when mass transfer is non-
conservative (Soberman et al., 1997). The default criterion involving a low
critical mass ratio implies that almost all of the systems with an ONeWD and a
giant donor experience a CE phase and evolve to be close binaries, thereby
significantly contributing to the population of LISA NSWD binaries during
subsequent evolution. However, Pavlovskii & Ivanova (2015) proposed that the
critical mass ratio for stable mass transfer in binaries with a giant donor
and a compact object ranges from 1.5 to 2.2. More recently, Ge et al. (2020)
showed that mass transfer in binaries with giant donor stars is more stable
than previously believed.

Furthermore, for stable mass transfer between two WDs, we adopt the critical
mass ratio of 0.628 (Hurley et al., 2002), which is similar to the value
proposed by Nelemans et al. (2001a). This critical mass ratio allows most
ONeWDs to effectively accrete mass from HeWDs or COWDs, eventually resulting
in the formation of NSs via AIC. Additionally, we assume that steady nuclear
burning occurs on the surface of accreting WDs if mass-transfer rate falls
within the range of $(1.03-2.71)\times
10^{-7}\mathrm{M}_{\odot}\mathrm{yr}^{-1}$. However, this range greatly
depends on the masses and the temperatures of the WDs involved, as well as the
components of accreted material. Therefore, there remains uncertainty
regarding whether the AIC mechanism significantly contributes the population
of LISA NSWD binaries.

##  5 Conclusions

In this study, we have performed binary population synthesis calculations to
investigate the origins of LISA NSWD binaries in the Milky Way and examine the
influences of different assumptions related to binary evolution on their
characteristic distribution. Our results reveal that approximately 17$-$234
detached NSWD binaries and less than 200 interacting systems can serve as
detectable LISA sources, excluding the NSWDs originating from the RLOF and the
AIC channels.

The model MA3, with near-conservative mass transfer during primordial binary
evolution, predicts most of detached LISA NSWD binaries are systems where WDs
form first (see Fig. 2). Among all detached LISA binaries from our
calculations, the fraction for the systems with eccentricities larger than 0.1
can reach as high as $\sim 0.3-0.4$. While for the models MA1 and MA2, we
obtain that $\lesssim 0.2$ of detached LISA NSWD systems have eccentricities
of larger than 0.1. These eccentric systems are more likely to locate at large
vertical distances from the Galactic plane. Furthermore, detached LISA NSWD
binaries are more likely to host a massive ONeWD in the model MA3, compared to
the models MA1 and MA2. By studying the distributions of the binary parameters
and the spatial locations of detached LISA NSWD sources, it is possible to
constrain mass accretion models (i.e., MA1, MA2, and MA3).

Also, we have demonstrated that CE ejection efficiencies
$\alpha_{\mathrm{CE}}$ can significantly influence the expected numbers of
LISA NSWD sources in the Milky Way. A smaller value of $\alpha_{\mathrm{CE}}$
tends to suppress the formation of the systems with eccentricities of $>0.1$,
particularly in the models MA1 and MA2. For supernova mechanisms, the adoption
of the stochastic recipe tends to generate more binaries with large chirp
masses, in contrast to the rapid recipe.

At last, we propose that the stability of mass transfer between WDs and NSs
could be constrained, according to the observations of interacting LISA NSWD
binaries also appearing as UCXBs.

## Acknowledgements

We thank the anonymous referee for helpful suggestions that improved this
paper. This work was supported by the National Key Research and Development
Program of China (Grant Nos. 2023YFA1607902, 2021YFA0718500), the Natural
Science Foundation of China (Nos. 12041301, 12121003, 12373034), the Strategic
Priority Research Program of the Chinese Academy of Sciences (Grant No.
XDB0550300), and the Project U1838201 supported by NSFC and CAS.

## Data Availability

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

## References

  * Abadie et al. (2010) Abadie J., et al., 2010, Classical and Quantum Gravity, 27, 173001
  * Abbott et al. (2016) Abbott B. P., et al., 2016, Physical Review X, 6, 041015
  * Abbott et al. (2023) Abbott R., et al., 2023, Physical Review X, 13, 011048
  * Amaro-Seoane et al. (2017) Amaro-Seoane P., et al., 2017, arXiv e-prints, p. arXiv:1702.00786
  * Amaro-Seoane et al. (2023) Amaro-Seoane P., et al., 2023, Living Reviews in Relativity, 26, 2
  * Antoniadis et al. (2013) Antoniadis J., et al., 2013, Science, 340, 448
  * Babak et al. (2021) Babak S., Hewitson M., Petiteau A., 2021, arXiv e-prints, p. arXiv:2108.01167
  * Bahcall & Soneira (1980) Bahcall J. N., Soneira R. M., 1980, ApJS, 44, 73
  * Bobrick et al. (2017) Bobrick A., Davies M. B., Church R. P., 2017, MNRAS, 467, 3556
  * Bobrick et al. (2022) Bobrick A., Zenati Y., Perets H. B., Davies M. B., Church R., 2022, MNRAS, 510, 3758
  * Bovy (2015) Bovy J., 2015, ApJS, 216, 29
  * Chen et al. (2020) Chen W.-C., Liu D.-D., Wang B., 2020, ApJ, 900, L8
  * Chen et al. (2021) Chen H.-L., Tauris T. M., Han Z., Chen X., 2021, MNRAS, 503, 3540
  * Chen et al. (2022) Chen H.-L., Tauris T. M., Chen X., Han Z., 2022, ApJ, 930, 134
  * Church et al. (2017) Church R. P., Strader J., Davies M. B., Bobrick A., 2017, ApJ, 851, L4
  * Cornish & Robson (2017) Cornish N., Robson T., 2017, in Journal of Physics Conference Series. p. 012024 (arXiv:1703.09858), doi:10.1088/1742-6596/840/1/012024
  * Davis et al. (2010) Davis P. J., Kolb U., Willems B., 2010, MNRAS, 403, 179
  * Deng et al. (2021) Deng Z.-L., Li X.-D., Gao Z.-F., Shao Y., 2021, ApJ, 909, 174
  * Fryer et al. (2012) Fryer C. L., Belczynski K., Wiktorowicz G., Dominik M., Kalogera V., Holz D. E., 2012, ApJ, 749, 91
  * Ge et al. (2020) Ge H., Webbink R. F., Chen X., Han Z., 2020, ApJ, 899, 132
  * Gu et al. (2016) Gu W.-M., Dong Y.-Z., Liu T., Ma R., Wang J., 2016, ApJ, 823, L28
  * Han et al. (2020) Han Z.-W., Ge H.-W., Chen X.-F., Chen H.-L., 2020, Research in Astronomy and Astrophysics, 20, 161
  * He et al. (2023) He J.-G., Shao Y., Gao S.-J., Li X.-D., 2023, ApJ, 953, 153
  * Hjellming & Webbink (1987) Hjellming M. S., Webbink R. F., 1987, ApJ, 318, 794
  * Hobbs et al. (2005) Hobbs G., Lorimer D. R., Lyne A. G., Kramer M., 2005, MNRAS, 360, 974
  * Hurley et al. (2002) Hurley J. R., Tout C. A., Pols O. R., 2002, MNRAS, 329, 897
  * Istrate et al. (2014) Istrate A. G., Tauris T. M., Langer N., 2014, A&A, 571, A45
  * Kaltenborn et al. (2023) Kaltenborn M. A. R., Fryer C. L., Wollaeger R. T., Belczynski K., Even W., Kouveliotou C., 2023, ApJ, 956, 71
  * Kang et al. (2024) Kang Y., et al., 2024, MNRAS, 528, 5309
  * Korol & Safarzadeh (2021) Korol V., Safarzadeh M., 2021, MNRAS, 502, 5576
  * Korol et al. (2022) Korol V., Hallakoun N., Toonen S., Karnesis N., 2022, MNRAS, 511, 5936
  * Korol et al. (2023) Korol V., Igoshev A. P., Toonen S., Karnesis N., Moore C. J., Finch E., Klein A., 2023, arXiv e-prints, p. arXiv:2310.06559
  * Luo et al. (2016) Luo J., et al., 2016, Classical and Quantum Gravity, 33, 035010
  * Mandel & Müller (2020) Mandel I., Müller B., 2020, MNRAS, 499, 3214
  * Misra et al. (2020) Misra D., Fragos T., Tauris T. M., Zapartas E., Aguilera-Dena D. R., 2020, A&A, 642, A174
  * Moore et al. (2023) Moore C. J., Finch E., Klein A., Korol V., Pham N., Robins D., 2023, arXiv e-prints, p. arXiv:2310.06568
  * Nelemans et al. (2001a) Nelemans G., Portegies Zwart S. F., Verbunt F., Yungelson L. R., 2001a, A&A, 368, 939
  * Nelemans et al. (2001b) Nelemans G., Yungelson L. R., Portegies Zwart S. F., 2001b, A&A, 375, 890
  * Pavlovskii & Ivanova (2015) Pavlovskii K., Ivanova N., 2015, MNRAS, 449, 4415
  * Robson et al. (2019) Robson T., Cornish N. J., Liu C., 2019, Classical and Quantum Gravity, 36, 105011
  * Scherbak & Fuller (2023) Scherbak P., Fuller J., 2023, MNRAS, 518, 3966
  * Shao (2022) Shao Y., 2022, Research in Astronomy and Astrophysics, 22, 122002
  * Shao & Li (2012) Shao Y., Li X.-D., 2012, ApJ, 756, 85
  * Shao & Li (2014) Shao Y., Li X.-D., 2014, ApJ, 796, 37
  * Shao & Li (2018) Shao Y., Li X.-D., 2018, ApJ, 867, 124
  * Shao & Li (2021) Shao Y., Li X.-D., 2021, ApJ, 920, 81
  * Soberman et al. (1997) Soberman G. E., Phinney E. S., van den Heuvel E. P. J., 1997, A&A, 327, 620
  * Tauris (2018) Tauris T. M., 2018, Phys. Rev. Lett., 121, 131105
  * Tauris & van den Heuvel (2023) Tauris T. M., van den Heuvel E. P. J., 2023, Physics of Binary Star Evolution. From Stars to X-ray Binaries and Gravitational Wave Sources, doi:10.48550/arXiv.2305.09388.
  * Tauris et al. (2000) Tauris T. M., van den Heuvel E. P. J., Savonije G. J., 2000, ApJ, 530, L93
  * Tauris et al. (2015) Tauris T. M., Langer N., Podsiadlowski P., 2015, MNRAS, 451, 2123
  * Toonen et al. (2018) Toonen S., Perets H. B., Igoshev A. P., Michaely E., Zenati Y., 2018, A&A, 619, A53
  * Verbunt & Rappaport (1988) Verbunt F., Rappaport S., 1988, ApJ, 332, 193
  * Vigna-Gómez et al. (2018) Vigna-Gómez A., et al., 2018, MNRAS, 481, 4009
  * Wagg et al. (2022) Wagg T., Breivik K., de Mink S. E., 2022, ApJS, 260, 52
  * Wang et al. (2021) Wang B., Chen W.-C., Liu D.-D., Chen H.-L., Wu C.-Y., Tang W.-S., Guo Y.-L., Han Z.-W., 2021, MNRAS, 506, 4654
  * Xu & Li (2010) Xu X.-J., Li X.-D., 2010, ApJ, 716, 114
  * Yang et al. (2022) Yang J., et al., 2022, Nature, 612, 232
  * Yu et al. (2021) Yu S., Lu Y., Jeffery C. S., 2021, MNRAS, 503, 2776
  * Yungelson et al. (2002) Yungelson L. R., Nelemans G., van den Heuvel E. P. J., 2002, A&A, 388, 546
  * Zhao et al. (2021) Zhao Z. Y., Zhang G. Q., Wang Y. Y., Tu Z.-L., Wang F. Y., 2021, ApJ, 907, 111
  * de Mink et al. (2007) de Mink S. E., Pols O. R., Hilditch R. W., 2007, A&A, 467, 1181
  * van Haaften et al. (2012) van Haaften L. M., Nelemans G., Voss R., Wood M. A., Kuijpers J., 2012, A&A, 537, A104
  * van Haaften et al. (2013) van Haaften L. M., Nelemans G., Voss R., Toonen S., Portegies Zwart S. F., Yungelson L. R., van der Sluys M. V., 2013, A&A, 552, A69

##  Appendix A Parameter distributions of detached LISA NSWD binaries

Figs. 6$-$8 show calculated number distributions of Galactic LISA sources of
detached NSWD systems as a function of binary parameters, similar to Fig. 1,
by assuming different CE efficiencies and supernova recipes.

Figure 6: Similar to Fig. 1, but assuming $\alpha_{\mathrm{CE}}=0.3$. Figure
7: Similar to Fig. 1, but assuming $\alpha_{\mathrm{CE}}=3.0$. Figure 8:
Similar to Fig. 1, but assuming the stochastic mechanism of supernova
explosions.