# Electron-capture supernovae in NS+He star systems and the double neutron
star systems

Yun-Lang Guo,1,2,3 Bo Wang,3,4,5 Wen-Cong Chen,6 Xiang-Dong Li,1,2 Hong-Wei
Ge,3,4,5 Long Jiang6 and Zhan-Wen Han3,4,5
1School of Astronomy and Space Science, Nanjing University, Nanjing 210023,
China
2Key Laboratory of Modern Astronomy and Astrophysics, Nanjing University,
Ministry of Education, Nanjing 210023, China
3University of Chinese Academy of Sciences, Beijing 100049, China
4Yunnan Observatories, Chinese Academy of Sciences, Kunming 650216, China
5International Centre of Supernovae, Yunnan Key Laboratory, Kunming 650216,
China
6School of Science, Qingdao University of Technology, Qingdao 266525, China
E-mail:yunlang@nju.edu.cnE-mail:wangbo<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Electron-capture supernovae (EC-SNe) provide an alternative channel for
producing neutron stars (NSs). They play an important role in the formation of
double NS (DNS) systems and the chemical evolution of galaxies, and contribute
to the NS mass distribution in observations. It is generally believed that EC-
SNe originate from $e$-captures on $\rm{}^{24}Mg$ and $\rm{}^{20}Ne$ in the
massive degenerate oxygen-neon (ONe) cores with masses close to the
Chandrasekhar limit ($M_{\rm Ch}$). However, the origin of EC-SNe is still
uncertain. In this paper, we systematically studied the EC-SNe in NS+He star
systems by considering the explosive oxygen burning that may occur in the
near-$M_{\rm Ch}$ ONe core. We provided the initial parameter spaces for
producing EC-SNe in the initial orbital period $-$ initial He star mass
(log$P_{\rm orb}^{\rm i}-M_{\rm He}^{\rm i}$) diagram, and found that both
$M_{\rm He}^{\rm i}$ and minimum $P_{\rm orb}^{\rm i}$ for EC-SNe increase
with metallicity. Then, by considering NS kicks added to the newborn NS, we
investigated the properties of the formed DNS systems after the He star
companions collapse into NSs, such as the orbital periods, eccentricities and
spin periods of recycle pulsars ($P_{\rm spin}$), etc. The results show that
most of the observed DNS systems can be produced by NS kicks of $\lesssim
50\rm\,km\,s^{-1}$. In addition, we found that NSs could accrete more material
if the residual H envelope on the He star companions is considered, which can
form the mildly recycled pulsars ($P_{\rm spin}\sim 20$ ms) in DNS systems.

###### keywords:

binaries: close – stars: evolution – pulsars: general – supernovae: general.

††pubyear: 2022††pagerange: Electron-capture supernovae in NS+He star systems
and the double neutron star systems–15

##  1 Introduction

Electron-capture supernovae (EC-SNe) are induced by the $e$-capture on
${}^{24}\rm Mg$ and ${}^{20}\rm Ne$ in the strongly degenerate oxygen-neon
(ONe) cores with masses close to the Chandrasekhar limit ($M_{\rm Ch}$; Nomoto
1987). It is generally suggested that the cores will eventually collapse into
neutron stars (NSs) after undergoing EC-SNe (e.g. Nomoto, 1984; Siess, 2007;
Jones et al., 2013; Doherty et al., 2015; Tauris et al., 2015). Because of the
relatively steep density gradient in the outer cores, EC-SNe are expected to
experience a prompt or fast explosion (see Podsiadlowski et al., 2004). Thus
they may have relatively low explosion energies ($\rm\sim 10^{50}\,erg$,
Kitaura et al., 2006), small amounts of ${}^{56}\rm Ni$ ($\rm\sim
0.002-0.015\,M_{\odot}$, Kitaura et al., 2006; Wanajo et al., 2009), and low-
velocity kicks ($\rm\lesssim 100\,\rm km\,s^{-1}$, Gessner & Janka, 2018;
Stockinger et al., 2020). In view of this, EC-SNe can explain a class of high-
mass X-ray binaries with long orbital periods and low eccentricities, as well
as the retention for NSs in globular clusters (e.g. Pfahl et al., 2002;
Podsiadlowski et al., 2004). Meanwhile, EC-SNe can be used to explain some low
luminosity SNe, e.g. SNe 1997D (Turatto et al., 1998), SN 2005cs (Pastorello
et al., 2006, 2009), SN 2008S (Botticella et al., 2009) and SN 2018zd
(Hiramatsu et al., 2021). They are also potential sources for the $r$-process,
which can synthesize heavy elements and contribute to the chemical evolution
of galaxies (Ning et al., 2007). In addition, Schwab et al. (2010) studied the
masses of 14 well-measured NSs, and found that the mass distribution of NSs
may be bimodal, in which the lower mass peak ($\sim 1.25\,\rm M_{\odot}$) may
arise from EC-SNe.

Double neutron star (DNS) systems are important potential gravitational wave
sources, which can be formed in NS+He star systems after the He star
companions undergo EC-SNe or iron core collapse supernovae (Fe CC-SNe; e.g.
Dewi et al., 2002; Dewi & Pols, 2003; Ivanova et al., 2003; Andrews et al.,
2015; Abbott et al., 2017; Tauris et al., 2015; Tauris et al., 2017; Kruckow
et al., 2018; Jiang et al., 2021). Since the discovery of the first DNS system
decades ago (PSR $\rm B1913+16$; Hulse & Taylor, 1975), about 24 sources have
been detected so far111ATNF Pulsar Catalogue,
http://www.atnf.csiro.au/research/pulsar/psrcat (version 1.71, November 2023;
Manchester et al., 2005). The orbital periods of the observed DNS systems span
a wide range from $0.1$ d to $45.0$ d, the orbital eccentricities range from
$0.064$ to $0.828$, and the component masses mainly range from
$1.1-1.5\rm\,M_{\odot}$. The observed features of DNSs are important for
exploring their origin (e.g. Tauris et al., 2017; Shao & Li, 2018; Andrews &
Mandel, 2019). It seems that most DNS systems are produced by small NS kick
velocities, indicating that the EC-SN channel may provide an important
contribution to the formation of DNS binaries. However, there are still some
uncertainties for the origin of EC-SNe, although they play a vital role in the
formation of DNS binaries and the NS mass distribution in observations (e.g.
Tauris et al., 2015; Jones et al., 2016; Chanlaridis et al., 2022).

Many works have been devoted to the study of the initial mass range for
producing EC-SNe, which can be used in population synthesis studies (e.g.
Nomoto, 1984; Poelarends, 2007; Doherty et al., 2015; Tauris et al., 2015; Han
et al., 2020). It is generally believed that the initial masses of single
main-sequence stars for producing EC-SNe are in the range of $\sim
8-10\rm\,M_{\odot}$ for solar metallicity, and this mass range is metallicity
dependent (see e.g. Doherty et al., 2017). In the classic picture, the lower
limit of metal core mass for producing EC-SNe is $\sim 1.37\,\rm M_{\odot}$
(e.g. Nomoto, 1984; Takahashi et al., 2013), and the upper limit of metal core
mass is $\sim 1.43\,\rm M_{\odot}$ (Tauris et al., 2015). For the metal cores
with masses less than $\sim 1.37\,\rm M_{\odot}$, stars will evolve into ONe
white dwarfs (WDs). For the metal cores with masses larger than $\sim
1.43\,\rm M_{\odot}$, the stars will collapse into NSs through Fe CC-SNe. For
the metal cores with masses of $1.37-1.43\,\rm M_{\odot}$, the oxygen
deflagration is initiated in the center due to the heating of $e$-capture on
${}^{20}\rm Ne$. Subsequently, as the central temperature rises steeply,
nuclear statistical equilibrium (NSE) will be achieved when temperature
exceeds $5\times 10^{9}$ K. The $e$-capture reactions in NSE region accelerate
core contraction, resulting in the formation of NSs eventually if the
$e$-capture rate exceeds the nuclear burning rate.

However, Guo et al. (2023, hereafter Paper I) recently found that explosive
oxygen burning may be triggered after He stars develop ONe cores with masses
close to $M_{\rm Ch}$, and then the He stars may explode as Type Ia supernovae
(SNe Ia) instead of collapsing into NSs (see also Antoniadis et al., 2020;
Chanlaridis et al., 2022). It is interesting to see how these new results
affect the evolution of binary NSs resulting from EC-SNe. Accordingly, we will
explore the formation of the EC-SNe in NS+He star binaries, and the properties
of DNSs originating from the EC-SN channel. The basic method and assumptions
for the binary evolution simulations are introduced in Section 2. The
simulation results for EC-SNe in NS+He star binaries are presented in Section
3. We show the investigation of the properties of DNSs produced by EC-SNe in
Section 4. Finally, we give discussions in Section 5 and a summary in Section
6.

##  2 Numerical methods and assumptions

We performed detailed binary evolution calculations of NS+He star systems by
using the stellar evolution code Modules for Experiments in Stellar
Astrophysics (MESA, version 10398; see Paxton et al. 2018), in which the
initial NS mass ($M_{\rm NS}^{\rm i}$) is set to be $1.35\rm\,M_{\odot}$.
Instead of solving the stellar structure equations of NSs, we assume that NSs
to be point masses. In the present work, we constructed zero-age main-sequence
(ZAMS) He star models based on suite case create_zams. We mainly explored the
parameter space for producing EC-SNe, where the initial He star mass ($M_{\rm
He}^{\rm i}$) ranges from $\sim 2.4\rm\,M_{\odot}$ to $3.0\rm\,M_{\odot}$ with
a resolution of $0.01\rm\,M_{\odot}$, and the initial orbital period ($P_{\rm
orb}^{\rm i}$) ranges from $\sim 0.07$ d to $10$ d with $\Delta$log $P_{\rm
orb}^{\rm i}=0.5$.

During the binary evolution, the NS starts to accrete material from He star
companion once the companion fills its Roche lobe. The scheme proposed by Kolb
& Ritter (1990) is adpoted to compute the mass-transfer rate (see also the
Appendix in Ge et al., 2010), and the fraction of mass lost from the vicinity
of the NS is set to be $0.5$. Accordingly, the mass increase rate of NSs is
$\dot{M}_{\rm NS}$ $=$ min($0.5\times\dot{M}_{\rm tran},\dot{M}_{\rm Edd}$),
in which $\dot{M}_{\rm Edd}=3\times 10^{-8}\rm\,M_{\odot}\,\rm yr^{-1}$ is the
Eddington accretion rate and $\dot{M}_{\rm tran}$ is the mass-transfer rate
(e.g. Podsiadlowski et al., 2002; Chen et al., 2020; Wang et al., 2021). In
this work, the simulated mass-transfer rates are often $3-4$ orders of
magnitude larger than $\dot{M}_{\rm Edd}$ and therefore limited by
$\dot{M}_{\rm Edd}$ (e.g. Tauris et al. 2015; Paper I, ). We assume that the
excess material ($\dot{M}_{\rm tran}-\dot{M}_{\rm NS}$) is ejected from the
vicinity of the NS, carrying away the specific orbital angular momentum of the
accreting NS (e.g. Dewi et al., 2002; Chen et al., 2011; Wang et al., 2021;
Guo et al., 2022).

During the evolution of He star companions, we adopted the ‘Dutch’
prescription with a scaling factor of $1.0$ as the stellar wind mass-loss
mechanism (e.g. Glebbeek et al., 2009; Antoniadis et al., 2020). We used Type
2 Rosseland mean opacity tables provided by Iglesias & Rogers (1996), which
can be applied to the enhanced carbon-oxygen caused by He-burning. The nuclear
reaction network is coupled by $43$ isotopes from $\rm{}^{1}H-^{58}Ni$,
including NeNa and MgAl cycles, Urca processes of ${}^{23}\rm
Na\rightleftharpoons$ ${}^{23}\rm Ne$, ${}^{23}\rm Ne\rightleftharpoons$
${}^{23}\rm F$, ${}^{25}\rm Mg\rightleftharpoons$ ${}^{25}\rm Na$ and
${}^{25}\rm Na\rightleftharpoons$ ${}^{25}\rm Ne$, and the $e$-capture chains,
i.e. ${}^{24}\rm Mg$$(e^{-},\nu_{e})^{24}\rm Na$$(e^{-},\nu_{e})^{24}\rm Ne$
and ${}^{20}\rm Ne$$(e^{-},\nu_{e})^{20}\rm F$$(e^{-},\nu_{e})^{20}\rm O$. We
used the weak interaction rates from Suzuki et al. (2016). In addition, we
adopted the HELM and PC equations-of-state (e.g. Timmes & Swesty, 2000;
Potekhin & Chabrier, 2010; Schwab et al., 2017).

In our simulations, we employed the Ledoux criterion and set the mixing-length
parameter to be $2.0$. We set the overshooting parameter ($f_{\rm ov}$) to be
$0.014$ (Jones et al., 2013; Antoniadis et al., 2020), and considered semi-
convection (Langer et al., 1983) and thermohaline mixing (Kippenhahn et al.,
1980)222The MESA inlists for this work are publicly available at
https://zenodo.org/record/7655310#.Y_LaNS-KGx9.. We stop the code if the
$e$-capture on $\rm{}^{20}Ne$ or the explosive oxygen burning occurs. For the
models that can undergo Fe CC-SNe, the code is stopped after the formation of
Si core.

##  3 Results

Figure 1:  Mass-loss rate versus remaining time ($t_{\rm f}-t$) for the
$2.67\,\rm M_{\odot}$ He star companion, in which $t$ and $t_{\rm f}$ are the
stellar age and the total evolutionary time, respectively. The purple, red and
yellow lines represent the stellar wind mass-loss rate, the mass-transfer rate
and the total mass-loss rate, respectively. The dashed line indicates the
Eddington accretion rate of the NS, i.e. $\dot{M}_{\rm Edd}=3\times
10^{-8}\,\rm M_{\odot}\rm yr^{-1}$ for accretion of helium. Figure 2: H-R
diagram of a $2.67\,\rm M_{\odot}$ He star companion from He-ZAMS to
$e$-capture on $\rm{}^{20}Ne$, in which the brown dots denote the moment when
the He star evolution, the Case BB RLOF, the Ne-O flashes and the $e$-capture
on $\rm{}^{20}Ne$ begin.

###  3.1 Typical binary evolutionary example for EC-SNe

In Figs 1$-$4, we present an example of the evolution of a NS+He star binary
with $M_{\rm He}^{\rm i}=2.67\rm\,M_{\odot}$ and $P_{\rm orb}^{\rm i}=1.0$ d,
in which the final fate of the He star companion is an EC-SN. Fig. 1 shows the
evolution of mass-loss rate of He star companion as a function of remaining
time ($t_{\rm f}-t$) until $e$-capture on $\rm{}^{20}Ne$ take place, where $t$
and $t_{\rm f}$ are the stellar age and the total evolutionary time,
respectively. The total simulated stellar age form He-ZAMS to $e$-capture is
$\sim 2.37$ Myr. After the central helium exhaustion, the He star gradually
expands, thereby leading to the initiation of Roche-lobe overflow (so-called
Case BB RLOF) at $t\sim 2.35$ Myr. Fig. 2 shows the evolution of the companion
in the H-R diagram. The points marked by the brown dots along the evolutionary
track correspond to He star ZAMS, the onset of Case BB RLOF and other key
processes. Fig. 3 illustrates the whole evolution of central temperature
($T_{\rm c}$) and central density ($\rho_{\rm c}$) for the He star companion.

Figure 3: Central temperature versus central density for the
$2.67\rm\,M_{\odot}$ He star companion. Grey shading denotes the region where
the $e$-capture on $\rm{}^{20}Ne$ occurs. The dotted lines represent helium,
carbon and oxygen burning ignition curves, respectively. The dashed line
denotes the separation of degenerate and non-degenerate regions
($\epsilon_{\rm F}/kT\simeq 4$).

Following central helium and carbon burning stage, the He star companion
develops a $\sim 1.385\,\rm M_{\odot}$ ONe core which is surrounded by a $\sim
0.1\,\rm M_{\odot}$ He-layer. Fig. 4a shows the chemical structure of the mass
donor after the carbon burning stage. At this moment, the ONe core is mainly
composed of oxygen, neon, sodium and magnesium. Subsequently, neon is ignited
off-center at mass coordinates ($M_{\rm c}$) of $\sim 0.8\,\rm M_{\odot}$ due
to the temperature inversion caused by the neutrino emission in the center of
core (i.e. Ne-shell flash). From Fig. 3, it can be seen that the center
reaches higher density and temperature after each Ne-shell flash quenches
(Jones et al., 2013). Following a few Ne-shell flashes, the central density
increases to $\rm log_{10}(\rho_{c}/g\,cm^{-3})\approx 9.1$, resulting in the
initiation of Urca processes. The central temperature decreases because of the
Urca reactions. Meanwhile, the Urca reactions accelerate the contraction of
the metal core, thereby leading to higher central density.

As the central density gradually increases, $e$-capture on ${}^{24}\rm Mg$ and
${}^{20}\rm Ne$ take place at $\rm log_{10}(\rho_{c}/g\,cm^{-3})\approx 9.6$
and $\rm log_{10}(\rho_{c}/g\,cm^{-3})\approx 9.9$, respectively (see Fig. 3).
Fig. 4b shows the chemical structure of the mass donor when the $e$-capture on
$\rm{}^{20}Ne$ occurs. We can see that the isotope ${}^{20}\rm O$ is produced
in the center owing to the reactions of ${}^{20}\rm
Ne$$(e^{-},\nu_{e})^{20}\rm F$$(e^{-},\nu_{e})^{20}\rm O$. Meanwhile, a thick
Si-rich mantle is formed because of the Ne-shell flashes. At this moment, the
final mass of the He envelope is $\sim 8.2\times 10^{-4}\rm\,M_{\odot}$, thus
the He lines are not visible in the spectra after the core-collapse event
(Hachinger et al., 2012).

Figure 4: Panel (a): the chemical structure of the donor after carbon burning
stage. At this moment, the donor mass is $\sim 1.50\rm\,M_{\odot}$, and the
donor consists a $\sim 1.385\rm\,M_{\odot}$ ONe core and a $\sim
0.115\rm\,M_{\odot}$ He envelope. Panel (b): the chemical structure of the
donor when the $e$-capture reactions on ${}^{20}\rm Ne$ occur. At this moment,
the donor consists of an ONe core and a thick Si shell. Table 1: Evolutionary
properties of the NS+He star binaries ($Z=0.02$) at boundaries of the
parameter space for EC-SNe, where the selected models correspond to the lowest
simulated masses (for the given initial orbital period) to produce either Fe
CC-SNe or EC-SNe. $M_{\rm He}^{\rm i}$ and log$P_{\rm orb}^{\rm i}$ are the
initial mass of the He star companion and the initial orbital period; $M_{\rm
core,f}$ and $M_{\rm env,f}$ are the final metal core mass and the remaining
helium envelope mass of the donor prior to SN; $\Delta M_{\rm NS}$ is the
accreted mass of NS; $P_{\rm spin}^{\rm min}$ is the minimum spin period of
NS; $P_{\rm orb}^{\rm pre}$ is the final orbital period prior to SN; $t_{\rm
f}$ is the total evolution time of He star companion; the last column is the
evolutionary outcome of the He star companion.

Set |  | $M_{\rm He}^{\rm i}$ | log$P_{\rm orb}^{\rm i}$ | $M_{\rm core,f}$ | $M_{\rm env,f}$ | $\Delta M_{\rm NS}$ | $P_{\rm spin}^{\rm min}$ | $P_{\rm orb}^{\rm pre}$ | $t_{\rm f}$ | Final fate
---|---|---|---|---|---|---|---|---|---|---
|  | ($\rm M_{\odot}$) | (d) | ($\rm M_{\odot}$) | ($\rm M_{\odot}$) | $(\rm M_{\odot})$ | (ms) | (d) | (Myr) |
1 |  | $2.95$ | $-1.15$ | $1.431$ | 4.5e-3 | 1.8e-3 | $38.7$ | $0.069$ | $2.02$ | Fe CC-SN
2 |  | $2.84$ | $-1.15$ | $1.381$ | 3.0e-4 | 2.0e-3 | $36.0$ | $0.075$ | $2.14$ | EC-SN
3 |  | $2.88$ | $-1.00$ | $1.431$ | 3.0e-3 | 1.6e-3 | $43.3$ | $0.107$ | $2.10$ | Fe CC-SN
4 |  | $2.79$ | $-1.00$ | $1.389$ | 5.0e-4 | 1.7e-3 | $40.6$ | $0.113$ | $2.20$ | EC-SN
7 |  | $2.78$ | $-0.50$ | $1.430$ | 4.7e-3 | 9.0e-4 | $65.7$ | $0.361$ | $2.22$ | Fe CC-SN
6 |  | $2.70$ | $-0.50$ | $1.396$ | 7.3e-4 | 8.7e-4 | $67.1$ | $0.378$ | $2.32$ | EC-SN
7 |  | $2.77$ | $0$ | $1.436$ | 3.0e-3 | 7.1e-4 | $78.2$ | $1.153$ | $2.23$ | Fe CC-SN
8 |  | $2.67$ | $0$ | $1.385$ | 6.3e-4 | 8.2e-4 | $70.2$ | $1.226$ | $2.37$ | EC-SN
9 |  | $2.76$ | $0.5$ | $1.435$ | 5.4e-3 | 5.7e-4 | $92.2$ | $3.676$ | $2.25$ | Fe CC-SN
10 |  | $2.65$ | $0.5$ | $1.383$ | 4.5e-4 | 6.7e-4 | $81.6$ | $3.915$ | $2.40$ | EC-SN
11 |  | $2.76$ | $1$ | $1.433$ | 6.4e-3 | 3.3e-4 | $138.3$ | $11.65$ | $2.25$ | Fe CC-SN
12 |  | $2.66$ | $1$ | $1.384$ | 2.5e-4 | 4.8e-4 | $104.8$ | $12.39$ | $2.38$ | EC-SN
13a |  | $2.73$ | $-1.15$ | $1.433$ | 1.6e-3 | 2.0e-3 | $36.8$ | $0.057$ | $2.23$ | Fe CC-SN
14a |  | $2.64$ | $-1.15$ | $1.385$ | 4.2e-4 | 2.1e-3 | $34.2$ | $0.062$ | $2.35$ | EC-SN
15a |  | $2.54$ | $1$ | $1.443$ | 1.7e-2 | 2.0e-4 | $208.5$ | $9.502$ | $2.56$ | Fe CC-SN
16a |  | $2.49$ | $1$ | $1.393$ | 7.0e-4 | 3.0e-4 | $157.1$ | $10.10$ | $2.66$ | EC-SN

  * a Models with a stellar wind efficiency of $0.1$.

###  3.2 Parameter space for EC-SNe

Figure 5: Evolutionary track of the central temperature and the central
density for NS+naked He star binaries with different initial He star masses
(i.e. $M_{\rm He}^{\rm i}=2.65,2.67$ and $3.0\rm\,M_{\odot}$), in which the
initial orbital period of all models is $1.0$ d.

By calculating a large number of NS+He star systems with different $M_{\rm
He}^{\rm i}$ and $P_{\rm orb}^{\rm i}$, we obtained the initial contours for
producing EC-SNe. Table 1 lists the information about the evolution of NS+He
star binaries ($Z=0.02$) at the boundaries of the parameter space for EC-SNe,
in which the selected models correspond to the lowest simulated masses (for
the given initial orbital period) to produce either Fe CC-SNe or EC-SNe. The
metal core mass for EC-SNe ranges from $\sim 1.385\rm\,M_{\odot}$ to
$1.43\rm\,M_{\odot}$. If the final core mass is lower than $\sim
1.385\rm\,M_{\odot}$, then the explosive oxygen burning can be triggered owing
to the convective Urca process, resulting in that the He star companions
explode as SNe Ia eventually (see Paper I, ). If the final core mass is larger
than $\sim 1.43\rm\,M_{\odot}$, then the final fates of He star companions are
Fe CC-SNe.

Figure 6: Final chemical structure for the He star companion with initial mass
of $3.0\rm\,M_{\odot}$, in which the final core mass is $\sim
1.55\rm\,M_{\odot}$ and its final fate is an Fe CC-SN.

Fig. 5 shows $T_{\rm c}$ versus $\rho_{\rm c}$ for three representative NS+He
star binaries which terminate their evolution as an SN Ia, an EC-SN and an Fe
CC-SN, respectively. For the model with $M^{\rm i}_{\rm He}=2.65\,\rm
M_{\odot}$, the He star companion develops a $\sim 1.378\,\rm M_{\odot}$ ONe
core, and the explosive oxygen burning takes place owing to the convective
Urca process, resulting in the formation of an SN Ia (see Paper I, ). For the
model with $M^{\rm i}_{\rm He}=2.67\,\rm M_{\odot}$, the He star companion
develops a $\sim 1.385\,\rm M_{\odot}$ ONe core, and the He star will undergo
EC-SN after $e$-capture on $\rm{}^{20}Ne$ occurs. For the model with $M^{\rm
i}_{\rm He}=3.0\,\rm M_{\odot}$, the He star companion develops a $\sim
1.55\,\rm M_{\odot}$ ONe core, and the Ne-shell flash takes place at $M_{\rm
c}\approx 0.3\,\rm M_{\odot}$, which is closer to the center than the model
with a $2.67\,\rm M_{\odot}$ He star companion. Fig. 6 shows the final
chemical structure of the He star companion. The Ne-O burning flame can
propagate to the center, leading to the formation of a silicon core, and
thereby an iron core-collapse.

Figure 7: Initial parameter spaces in the $\rm log\,$$P^{\rm i}_{\rm
orb}-M^{\rm i}_{\rm He}$ plane for NS+naked He star binaries that can produce
EC-SNe. The dashed lines represent the boundaries between EC-SNe and Fe CC-
SNe, and the solid lines represent the boundaries between SNe Ia and EC-SNe.

Fig. 7 represents the initial parameter spaces for EC-SNe in the log$P_{\rm
orb}^{\rm i}-M_{\rm He}^{\rm i}$ diagram with two metallicities (i.e.
$Z=0.0001,0.02$), from which the metallicity has a strong influence on the
parameter spaces. The left boundaries of the parameter spaces are set by the
condition that RLOF takes place when the He star companions are on He-ZAMS
stage. If the initial orbital period is $\gtrsim 10$ d, then the He star
companions have a similar final fate. The dashed lines represent the
boundaries between EC-SNe and Fe CC-SNe, and the solid lines represent the
boundaries between SNe Ia and EC-SNe. We found that for a lower metallicity,
the parameter spaces shifted to low-mass He star companion. This is because
lower metallicities result in weaker stellar wind mass-loss and larger metal
core for a given initial He star mass (e.g. Chanlaridis et al., 2022;
Aguilera-Dena et al., 2022). Meanwhile, lower metallicities lead to smaller
radii of He ZAMS stars, thus the left boundaries of the parameter space shifts
slightly to shorter initial period (see also Wang & Han, 2010).

##  4 Double neutron star systems

The simulations show that the He star companions in NS binaries may explode as
SNe Ia if the degenerate ONe core masses range from $\sim 1.335-1.385\,\rm
M_{\odot}$ (see Paper I, ). Meanwhile, the NSs spin up during the mass-
accretion phase, resulting in the formation of isolated mildly recycled
pulsars after the He star companions experience SN Ia explosions. If the ONe
core masses are larger than $\sim 1.385\,\rm M_{\odot}$, then the He star
companions collapse into NSs through EC-SNe or Fe CC-SNe, which may result in
the formation of DNS systems (e.g. Tauris et al., 2017; Kruckow et al., 2018).
Accordingly, we will study the DNS systems originating from the EC-SN channel,
and compare our calculation results with observations.

###  4.1 The effects of SN explosions

It has been suggested that the eccentricity ($e$) and the orbital period of NS
binaries will be changed after the companion star undergoes a SN explosion,
due to the sudden mass loss and the kick velocity ($V_{\rm k}$) added to the
newly-born NS. The orientation of the NS kick velocity is controlled by two
angles, i.e. the angle between $V_{\rm k}$ and the pre-SN orbital velocity
($0\leq\theta\leq 180^{\circ}$), and the positional angle of $V_{\rm k}$ with
respect to the orbital plane ($-180^{\circ}<\phi\leq 180^{\circ}$). Thus, the
ratio of pre-SN orbital separation ($a_{0}$) to the post-SN orbital separation
($a$) can be given by (e.g. Hills, 1983; Dewi & Pols, 2003; Shao & Li, 2016)

$\frac{a_{0}}{a}=2-\frac{M_{0}}{M_{0}-\Delta M}(1+\nu^{2}+2\nu\,\rm
cos\,\theta),$ (1)

where $M_{0}$ and $\Delta M$ are the pre-SN total mass, and the ejected mass
from the exploding star, respectively. In addition, we defined $\nu=V_{\rm
k}/V_{0}$, in which $V_{0}$ is relative velocity between the two stars. The
eccentricity after SN explosions can be written as

$1-e^{2}=\frac{a_{0}}{a}\frac{M_{0}}{M_{0}-\Delta M}[1+2\nu\,\rm
cos\,\theta+\nu^{2}(\rm cos^{2}\theta+sin^{2}\theta\,sin^{2}\phi)].$ (2)

We assume that the NS mass is $1.35\,\rm M_{\odot}$. In our simulations, the
final companion star mass prior to SN ranges from $\sim 1.385\,\rm M_{\odot}$
to $1.43\,\rm M_{\odot}$, thus we adopt an average value of $1.4\,\rm
M_{\odot}$. Accordingly, the pre-SN total mass is $2.75\,\rm M_{\odot}$. The
ejected mass for EC-SNe is still highly uncertain. Previous studies generally
suggested that the collapse of these objects can produce small ejected mass
from $0.01-0.2\,\rm M_{\odot}$ (e.g. Timmes et al., 1996; Kitaura et al.,
2006; Fryer et al., 2012). In this work, the value of $\Delta M$ is set to be
$0.08\,\rm M_{\odot}$ (Shao & Li, 2018). The small value of $\Delta M$ means
that the properties of post-SN binaries are dominated by the natal NS kicks.
In addition, it has been suggested that EC-SNe may produce low NS kicks owing
to the fast explosions and relatively low explosion energies, while the value
of NS kicks is still under debate (e.g. Podsiadlowski et al., 2004; Kitaura et
al., 2006; Verbunt et al., 2017; Janka, 2017; Gessner & Janka, 2018;
Stockinger et al., 2020). Tauris et al. (2015) suggested that the amount of
ejecta is extremely small for EC-SNe and low-mass Fe CC-SNe, resulting in
small NS kicks less than $\sim 10\,\rm km\,s^{-1}$. In addition, simulations
of EC-SNe show that the explosion energy is about $10^{50}$ erg, which leads
to the kick velocities lower than $\sim 50\,\rm km\,s^{-1}$ (e.g. Kitaura et
al., 2006; Dessart et al., 2006; Janka, 2017). Accordingly, we assume that
$V_{\rm k}\leq 50\,\rm km\,s^{-1}$ for EC-SNe.

###  4.2 DNS systems in the $P_{\rm orb}-e$ diagram

Figure 8: Distribution of the simulated post-SN systems in the $P_{\rm orb}-e$
diagram, in which we performed $2000$ cycles with randomly oriented NS kicks
for each pre-SN system. We adopted four pre-SN orbital periods (i.e. $0.1$,
$1$, $10$ and $40$ d) and two NS kicks (i.e. $V_{\rm k}=20$ and $40\,\rm
km\,s^{-1}$).

Fig. 8 shows the effect of NS kicks and pre-SN orbital period on the
distribution of post-SN systems in the $P_{\rm orb}-e$ diagram, where we
performed $2000$ cycles with randomly oriented NS kicks for each pre-SN
system. Obviously, larger $V_{\rm k}$ and longer pre-SN orbital period easily
lead to higher post-SN eccentricity and longer post-SN $P_{\rm orb}$. We note
that if the systems have short pre-SN $P_{\rm orb}$ ($\lesssim 1$ d) and
relatively low $V_{\rm k}$, then the post-SN $P_{\rm orb}$ does not change
significantly. Additionally, the binaries may be disrupted if they have wide-
orbits. For example, if a pre-SN binary has parameters with ($V_{\rm k},P_{\rm
orb})=(40\,\rm km\,s^{-1},40\,d)$, then the disruption probability for this
system is $32\%$. Moreover, we found that the probability of the post-SN
system located in the minimum and maximum period is relatively high,
corresponding to $\theta$ around $180^{\circ}$ and $0$, respectively.

Figure 9: Distribution of the simulated post-SN systems in the $P_{\rm orb}-e$
diagram (the purple regions), in which we set $500$ pre-SN $P_{\rm orb}$ that
vary logarithmically from $0.07$ d to $50$ d and the NS kicks are in the range
of $10-50\,\rm km\,s^{-1}$. We calculated $1000$ cycles with randomly oriented
NS kicks for each pre-SN system. Lines I, II and III are the regions where
post-SN systems are located with relatively high probability, corresponding to
$\theta\sim 180^{\circ}$, $\theta\sim 0$ and $\phi\sim 0,\pm 180^{\circ}$,
respectively. The red stars are the Galactic DNS systems in observations. The
observed data of DNS binaries are taken from the ATNF Pulsar Catalogue,
http://www.atnf.csiro.au/research/pulsar/psrcat (version 1.71, November 2023;
Manchester et al., 2005). Table 2: Properties of Galactic DNS systems. $M_{\rm
total}$, $M_{\rm psr}$, $M_{\rm comp}$ are the binary total mass, the mass of
radio pulsar and the mass of companion; $P_{\rm orb}$, $e$, $P_{\rm spin}$ and
$\dot{P}_{\rm spin}$ are the binary orbital period, the eccentricity of DNS
system, the spin period of pulsar and its derivative; $V_{\rm k}$ and $P_{\rm
orb}^{\rm pre}$ are the NS kick velocity and pre-SN orbital period estimated
in this work.

|  | $M_{\rm total}$ | $M_{\rm psr}$ | $M_{\rm comp}$ | $P_{\rm orb}$ | $e$ | $P_{\rm spin}$ | $\dot{P}_{\rm spin}$ | $V_{\rm k}$ | $P_{\rm orb}^{\rm pre}$
---|---|---|---|---|---|---|---|---|---|---
Pulsar name |  | ($\rm M_{\odot}$) | ($\rm M_{\odot}$) | ($\rm M_{\odot}$) | (d) |  | (ms) | ($10^{-18}$) | ($\rm km\,s^{-1}$) | (d)
$\rm J0453+1559^{1}$ |  | $2.734$ | $1.559$ | $1.174$ | $4.072$ | $0.113$ | $45.8$ | $0.186$ | $\lesssim 80$ | $\sim 3-5$
$\rm J0509+3801^{2}$ |  | $2.805$ | $1.348$ | $1.468$ | $0.380$ | $0.586$ | $76.5$ | $80.54$ | $\gtrsim 150$ | $\lesssim 0.9$
$\rm J0737-3039^{3}$ |  | $2.587$ | $1.338$ | $1.249$ | $0.102$ | $0.088$ | $22.7$ | $1.76$ | $\sim 20-200$ | $\sim 0.08-0.1$
$\rm J1411+2551^{4}$ |  | $2.538$ | $\textless 1.64$ | $\textgreater 0.92$ | $2.616$ | $0.170$ | $62.5$ | $0.096$ | $\lesssim 100$ | $\sim 2-4$
$\rm J1518+4904^{5}$ |  | $2.718$ | $1.41$ | $1.31$ | $8.634$ | $0.249$ | $40.9$ | $0.027$ | $\sim 20-80$ | $\sim 4-15$
$\rm J1537+1155^{6}$ |  | $2.678$ | $1.333$ | $1.346$ | $0.421$ | $0.274$ | $37.9$ | $2.42$ | $\sim 30-200$ | $\sim 0.2-0.7$
$\rm J1753-2240^{7}$ |  | … | … | … | $13.64$ | $0.304$ | $95.1$ | $0.970$ | $\sim 20-80$ | $\sim 7-20$
$\rm J1755-2550^{8}$ |  | … | … | … | $9.696$ | $0.089$ | $315$ | $2434$ | $\lesssim 30$ | $\sim 8-12$
$\rm J1756-2251^{9}$ |  | $2.569$ | $1.341$ | $1.230$ | $0.320$ | $0.181$ | $28.5$ | $1.018$ | $\sim 30-150$ | $\sim 0.2-0.5$
$\rm J1757-1854^{10}$ |  | $2.733$ | $1.338$ | $1.395$ | $0.184$ | $0.606$ | $21.5$ | $2.630$ | $\gtrsim 150$ | $\lesssim 0.3$
$\rm J1759+5036^{11}$ |  | $2.62$ | $\textless 1.92$ | $\textgreater 0.70$ | $2.043$ | $0.308$ | $176$ | $0.243$ | $\sim 30-150$ | $\sim 1-4$
$\rm J1811-1736^{12}$ |  | $2.57$ | $\textless 1.75$ | $\textgreater 0.91$ | $18.78$ | $0.828$ | $104$ | $0.901$ | $\sim 80-300$ | $\sim 1-50$
$\rm J1829+2456^{13}$ |  | $2.59$ | $\textless 1.38$ | $\textgreater 1.22$ | $1.176$ | $0.139$ | $41.0$ | $0.053$ | $\sim 20-100$ | $\sim 0.8-1.5$
$\rm J1906+0746^{14}$ |  | $2.613$ | $1.291$ | $1.322$ | $0.166$ | $0.085$ | $144.1$ | $20300$ | $\sim 20-150$ | $\sim 0.13-0.2$
$\rm J1913+1102^{15}$ |  | $2.89$ | $1.65$ | $1.24$ | $0.206$ | $0.090$ | $27.3$ | $0.161$ | $\sim 20-150$ | $\sim 0.15-0.3$
$\rm J1915+1606^{16}$ |  | … | … | … | $0.323$ | $0.617$ | $59.03$ | $8.618$ | $\gtrsim 150$ | $\lesssim 1.0$
$\rm J1930-1852^{17}$ |  | … | … | … | $45.06$ | $0.400$ | $185.5$ | $19.00$ | $\sim 20-80$ | $\sim 15-100$
$\rm J1946+2052^{18}$ |  | $2.50$ | $\textless 1.35$ | $\textgreater 1.17$ | $0.078$ | $0.064$ | $17.0$ | $0.92$ | $\lesssim 150$ | $\sim 0.07-0.09$

  * References. 1: Martinez et al. (2015); 2: Lynch et al. (2018); 3: Kramer et al. (2006); 4: Martinez et al. (2017); 5: Janssen et al. (2008); 6: Fonseca et al. (2014); 7: Keith et al. (2009); 8: Ng et al. (2015); 9: Ferdman et al. (2014); 10: Cameron et al. (2018); 11: Agazie et al. (2021); 12: Corongiu et al. (2007); 13: Champion et al. (2005); 14: van Leeuwen et al. (2015); 15: Ferdman & PALFA Collaboration (2018); 16: Weisberg et al. (2010); 17: Swiggum et al. (2015); 18: Stovall et al. (2018).

Fig. 9 represents the distribution of the simulated post-SN systems in the
$P_{\rm orb}-e$ diagram, as well as the Galactic DNS systems in observations
(red stars)333This probability distribution is not necessarily reflective of
reality since we assume a uniform distribution of pre-SN log$P_{\rm orb}$.. We
set $500$ pre-SN $P_{\rm orb}$ that vary logarithmically from $0.07$ d to $50$
d, and each panel corresponds to different $V_{\rm k}$, i.e. $10,20,30$ and
$50\,\rm km\,s^{-1}$. We note that the post-SN systems are more likely to be
located in the regions where the kick direction has a small impact on the
$e-a$ relationship. Accordingly, we can conclude that lines I, II and III in
Fig. 9 are the regions where post-SN systems are located with relatively high
probability, corresponding to $\theta\sim 180^{\circ}$, $\theta\sim 0$ and
$\phi\sim 0,\pm 180^{\circ}$, respectively. It can be seen that the NS kicks
$\lesssim 50\,\rm km\,s^{-1}$ could reproduce the most DNS systems, especially
for the binaries with relatively low eccentricities. For example, PSR $\rm
J1755-2550$ (Ng et al., 2015) is a DNS candidate with a long orbital ($P_{\rm
orb}=9.7$ d) and a low eccentricity ($e=0.09$). Ng et al. (2018) suggested
that the minimum mass for the companion star of PSR $\rm J1755-2550$ is
$0.9\,\rm M_{\odot}$, but it is not completely determined whether the
companion star is a NS or a WD. By assuming that PSR $\rm J1755-2550$ is a DNS
system, Tauris et al. (2017) suggested that this system may be formed by a NS
kick lower than $50\,\rm km\,s^{-1}$. Our simulations also show that a NS kick
of $\lesssim 30\,\rm km\,s^{-1}$ (most likely $\sim 10\,\rm km\,s^{-1}$) may
explain the formation of PSR $\rm J1755-2550$.

PSR $\rm J1930-1852$ is the widest DNS system with a orbital period of $45$ d
(Swiggum et al., 2015). From Fig. 9, it is seen that its NS kick velocity was
likely in the range of $\sim 20-50\rm\,km\,s^{-1}$. PSR J$1811-1736$ has a
long orbital period ($P_{\rm orb}=18.8$ d) and the highest eccentricity
($e=0.83$) among the known Galactic DNS systems (Corongiu et al., 2007).
Although the results show that PSR J$1811-1736$ can be produced by kicks of
$\sim 50\rm\,km\,s^{-1}$, this source may come from different formation
channel because of the huge gap at long periods and moderate eccentricities.
In addition, kicks lower than $\sim 50\rm\,km\,s^{-1}$ cannot account for
three DNSs with high eccentricity, i.e. PSRs $\rm J0509+3801$, $\rm
J1757-1854$ and $\rm J1915+1606$. The distribution of the simulated post-SN
systems for higher NS kick velocities are shown in Fig. 14. We can see that
these three high eccentricity DNSs may originate from high NS kicks larger
than $\sim 150\,\rm km\,s^{-1}$ (see also Tauris et al., 2017).

###  4.3 Properties of pre-SN systems

Table 2 shows the properties of Galactic DNS systems, as well as their NS kick
velocities and pre-SN orbital periods estimated in this work. We found that
the DNSs with short orbital periods and high eccentricities (e.g. PSRs
J$0509+3801$, J$1757-1854$ and J$1915+1606$) may be produced by high kick
velocities (see also Tauris et al., 2017). However, it has been suggested that
such systems may originate from other formation channel (see Andrews & Mandel,
2019, and discussion in Section 5.2). In addition, half of the DNSs have pre-
SN orbital periods less than $\sim 1$ d and most DNSs may originate from the
relatively low NS kicks. This may be caused by the selection effects, that is,
the probability of surviving the pre-SN system is higher if it has a short
orbital period or a low kick velocity. Tauris et al. (2017) calculated the
probability of surviving the second SN in a binary with different pre-SN
orbital periods and NS kicks. Their results indicate that for relatively tight
systems with $P_{\rm orb}^{\rm pre}\lesssim 1\rm\,d$, all systems will survive
if the NS kicks are lower than $100\rm\,km\,s^{-1}$. Meanwhile, the orbital
periods of such tight systems does not change significantly after the SN
explosions, which is helpful for studying their pre-SN properties.

###  4.4 DNS systems in the $P_{\rm orb}-P_{\rm spin}$ diagram

####  4.4.1 The $P_{\rm orb}-P_{\rm spin}$ correlation

NSs in close X-ray binaries could be spun up during the mass-accretion process
(e.g. Alpar et al., 1982; Bhattacharya & van den Heuvel, 1991), and the
minimum spin period of the recycled pulsars can be expressed as (Tauris et
al., 2012)444The initial spin angular momentum can be negligible owing to the
spin-down process during the detached stage with a timescale of $\sim 10^{6}$
yr. For example, the NS with an initial spin period of $\sim 10$ ms and a
magnetic field of $10^{12}$ G spins down on a timescale of $\sim 10^{4}$ yr
due to magnetic dipole radiation (see also equation 2 in Kasen & Bildsten,
2010). In addition, the accretion will not play as significant a role if
pulsars are already spinning at a high rate because pulsars have equilibrium
periods.:

$P_{\rm spin}^{\rm min}\approx 0.34\times(\Delta M_{\rm NS}/\rm
M_{\odot})^{-3/4}\,ms,$ (3)

in which $\Delta M_{\rm NS}$ is the accreted mass of NSs. For Case BB RLOF,
the mass-transfer rate is much higher than $\dot{M}_{\rm Edd}$ (see Fig. 1),
resulting in that most of the transferred material is ejected from the
binaries and only a small amount of material are accreted by NS. From Table 1,
we can see that the accreted masses of NSs range from $2.0\times 10^{-4}\,\rm
M_{\odot}$ to $2.0\times 10^{-3}\,\rm M_{\odot}$, corresponding to initial
orbital periods of $0.07-10$ d. This results in the formation of recycled
pulsars with $P_{\rm spin}^{\rm min}\sim 36-210$ ms after the He star
companions undergo EC-SNe. Table 1 lists $P_{\rm spin}^{\rm min}$ and final
orbital periods of NSs prior to EC-SNe, and their relation can be written as:

$P_{\rm spin}^{\rm min}\approx 68\pm 6\rm\,ms\times(\textit{P}_{\rm orb}^{\rm
pre}\rm/d)^{0.22\pm 0.05}.$ (4)

In addition, longer $P_{\rm orb}^{\rm i}$ and larger $M_{\rm He}^{\rm i}$
result in lower $\Delta M_{\rm NS}$, thereby leading to longer spin periods of
the recycled pulsars (see also Tauris et al., 2015).

Figure 10: The $P_{\rm orb}-P_{\rm spin}$ diagram for the Galactic DNS systems
(red stars) in observations. The black line (Equation 4) is a fit to our
simulated results for the ($P_{\rm spin}^{\rm min},P_{\rm orb}^{\rm pre}$)
correlation, and the brown shaded regions represent the effect of the NS kick
velocities on the post-SN $P_{\rm orb}$.

Fig. 10 represents the Galactic DNS systems (red stars) in the $P_{\rm
orb}-P_{\rm spin}$ diagram. The black line indicates the relation between the
simulated $P_{\rm spin}^{\rm min}$ and $P_{\rm orb}^{\rm pre}$ prior to SN
(see Equation 4). From this figure, we can see that the DNSs with longer
$P_{\rm orb}$ are preferentially accompanied by the recycled pulsars with
higher $P_{\rm spin}$, resulting from the shorter duration of the mass-
accretion phase (e.g. Tauris et al. 2015; Paper I, ). The brown shaded regions
indicate the effect of NS kicks on the post-SN $P_{\rm orb}$. As shown in this
figure, longer $P_{\rm orb}^{\rm pre}$ and higher NS kicks will result in a
wider range of the post-SN orbital periods. However, we note that this results
hardly explain the DNS systems with relatively rapidly spinning recycled
pulsars ($P_{\rm spin}\lesssim 30$ ms). For example, the DNS system PSR
J$1946+2052$ has a spin period of $17$ ms, which is lower than the minimum
value in our simulations ($\sim 38$ ms, see Table 1). It has been suggested
that such mildly recycled pulsars may be caused by the super-Eddington
accretion, that is, Case BB RLOF allows first-born NS to accrete material at
mass-accretion rates of $2-3$ times $\dot{M}_{\rm Edd}$ (e.g. Lazarus et al.,
2014; Tauris et al., 2017). In this work, we also provide a possible
explanation for the relatively low spin period of pulsars by considering the
residual H envelope on the surface of the He star companion, see Section
4.4.2.

####  4.4.2 The effect of residual H envelope on $P_{\rm spin}$

It is generally believed that the NS+He star system that can form DNS has gone
through the common envelope (CE) phase, in which the dynamic friction
generated by the motion of NS inside the giant star’s envelope leads to the
loss of orbital angular momentum and the ejection of the H-rich envelope (e.g.
Paczynski, 1976; Dewi & Pols, 2003; Ivanova et al., 2013; Tauris et al.,
2017). However, recent simulations suggest that a small amount of H shell
still remains on the surface of He star companion after CE ejection (e.g.
Fragos et al., 2019; Ge et al., 2022), which may affect the evolution of NS+He
star systems. Accordingly, we calculated the evolution of three binaries
taking into account the residual H envelope.

Figure 11: An example of the chemical structure of the He-ZAMS model with H
envelope, where the total mass, He core mass and H envelope mass are
$2.73\rm\,M_{\odot}$, $2.128\rm\,M_{\odot}$ and $0.598\rm\,M_{\odot}$,
respectively.

We evolved a $11\rm\,M_{\odot}$ MS star to the stage near the tip of red
supergiant phase, where central helium burning has not yet occurred. We then
removed the envelope until its radius less than the roche lobe radius, e.g.
for binary with $M_{\rm He}^{\rm i}\sim 2.7\rm\,M_{\odot}$ and $P_{\rm
orb}^{\rm i}=0.1$ d, the initial radius of companion star should be less than
$\sim 0.63R_{\odot}$. Meanwhile, the mass of the remaining H envelope should
be in the range of $\sim 0.1-0.6\rm\,M_{\odot}$ (see, e.g. Fragos et al.,
2019), consistent with the extended test of a $13\rm\,M_{\odot}$ primordial
star based on Ge et al. (2022). After that, we put the He-ZAMS model with H
envelope into the binary module to calculate the binary evolution. Fig. 11
shows an example of the chemical structure of the He star companion with H
envelope. The He core mass and the H envelope mass are $\sim
2.13\rm\,M_{\odot}$ and $\sim 0.60\rm\,M_{\odot}$, in which the H abundance in
the He core is assumed to be less than $0.01$. Note that the outer layer is a
mixture of H and He rather than a pure H shell. Table 3 lists evolutionary
properties of three NS+He star binaries by considering the H envelope, as well
as a binary system with a naked He star companion for comparison. We can see
that the accreted mass of NS increases several times if the residual H
envelope is assumed to remain on the surface of the He star companion.
Meanwhile, the residual H envelope can also affect the final fate of the He
star companion.

Table 3:  Characteristics of the binaries considering the residual H envelope on the surface of the He star companion, in which we set the stellar wind efficiency to be $0.1$. $M_{\rm He}^{\rm i}$ is the mass of the He-ZAMS model with H envelope; $P_{\rm orb}^{\rm i}$ is the orbital period after the artificial stripping event; $M_{\rm He-core}$, $M_{\rm H-env}$ and log$R^{\rm i}$ are the initial He core mass, H envelope mass and the radius of the He star companion, respectively.  Set |  | $M_{\rm He}^{\rm i}$ | log$P_{\rm orb}^{\rm i}$ | $M_{\rm He-core}$ | $M_{\rm H-env}$ | log$R^{\rm i}$ | $M_{\rm core,f}$ | $M_{\rm env,f}$ | $\Delta M_{\rm NS}$ | $P_{\rm spin}^{\rm min}$ | $P_{\rm orb}^{\rm f}$ | $t_{\rm f}$ | Final fate
---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  | ($\rm M_{\odot}$) | (d) | ($\rm M_{\odot}$) | ($\rm M_{\odot}$) | ($\rm R_{\odot}$) | $(\rm M_{\odot})$ | ($\rm M_{\odot}$) | ($\rm M_{\odot}$) | (ms) | (d) | (Myr) |
17 |  | $2.66$ | $-1.00$ | $2.500$ | $0.160$ | $-0.20$ | $1.386$ | 1.2e-2 | 4.1e-3 | $21$ | $0.088$ | $2.25$ | EC-SN
18 |  | $2.73$ | $0$ | $2.128$ | $0.598$ | $0.42$ | $1.388$ | 1.0e-1 | 1.5e-3 | $45$ | $0.833$ | $2.66$ | EC-SN
19 |  | $2.72$ | $1.00$ | $2.125$ | $0.595$ | $0.36$ | $1.470$ | 3.8e-1 | 1.0e-3 | $60$ | $7.701$ | $2.46$ | Fe CC-SN
20 |  | $2.73$ | $0$ | $2.73$ | $0$ | $-0.37$ | $1.513$ | 2.5e-1 | 3.6e-4 | $130$ | $0.780$ | $2.30$ | Fe CC-SN
Figure 12: The effect of the H envelope on the mass-transfer rate, where the
initial donor mass is $2.73\rm\,M_{\odot}$ (see sets 18 and 20 in Table 3).
The black line and the red-dashed line denote the model considering the
residual H envelope and the model with naked He star, respectively. Figure 13:
The $P_{\rm orb}-P_{\rm spin}$ diagram for the Galactic DNS systems (red
stars) and our simulations. The black line and the gray dashed line represent
the fitting results for the models with and without residual H envelope,
respectively. The data of black dots come from sets $17-19$ in Table 3. The
green and red shaded regions represent the effect of the NS kick velocity
($V_{\rm k}=50\rm\,km\,s^{-1}$).

Fig. 12 shows the effect of the residual H envelope on the mass-transfer rate,
in which the initial donor mass in both models is $2.73\rm\,M_{\odot}$ (see
sets 18 and 20 in Table 3). The He star with a H envelope has a larger radius
than the naked He star, causing it to fill its Roche lobe more easily. It can
be seen that the H envelope results in a longer duration of the mass-transfer
phase (Bauer & Kupfer, 2021), thereby a larger accreted mass and a lower spin
period of NS. Fig. 13 shows the theoretical ($P_{\rm orb}$, $P_{\rm spin}$)
correlation calculated with and without the residual H envelope. The
simulation results can better explain the observational characteristics of the
DNS systems if we consider the H envelope on the surface of the He star
companions. It is worth noting that the structure of the He star companion and
the mass of the H envelope depend on the process of CE ejection. The impact of
the residual H envelope on the evolution of pulsars needs to be systematically
studied in future.

##  5 Discussion

###  5.1 Uncertainties in the modelling

In our default models, we adopted the ‘Dutch’ stellar wind with a efficiency
of $1.0$ to calculate the wind mass-loss rate. However, different stellar wind
efficiencies may change the evolutionary outcomes of the He stars. To further
explore the influence of stellar wind efficiencies on the evolution of NS+He
star systems, we computed four models with a stellar wind efficiency of $0.1$
(see Table 1). The most significant effect is that lower wind efficiencies
result in lower initial mass of the He star companion for EC-SNe. In addition,
we note that the remaining helium envelope mass ($M_{\rm He,f}^{\rm env}$)
increases with the decrease of wind efficiency for the binaries with
relatively long initial orbital periods, e.g. $P_{\rm orb}^{\rm i}=10$ d.
However, the value of $M_{\rm He,f}^{\rm env}$ is still less than $0.06\,\rm
M_{\odot}$ even though a low stellar wind efficiency is adopted, indicating
that the EC-SN is likely to be classified as a Type Ic SN (Hachinger et al.,
2012). In addition, convective overshooting is also an important physical
process, which can result in higher effective temperature/luminosity, longer
HeMS lifetime and a larger convective core (e.g. Yan et al., 2016; Chanlaridis
et al., 2022). In the present work, we set the overshooting parameter to be
$0.014$ (e.g. Jones et al., 2013; Chanlaridis et al., 2022). We expect that
the initial parameter space for EC-SNe will move up if a lower overshooting
parameter is taken.

We assume that the ejected mass of the exploding star as $0.08\,\rm
M_{\odot}$. However, by using the gravitational binding energy provided by
Lattimer & Yahil (1989), Tauris et al. (2015) calculated the gravitational
masses of the NS remnants, and found that about $0.2\,\rm M_{\odot}$ of the
material could be ejected when the He star companions undergo EC-SNe.
Accordingly, we calculated the distribution of post-SN systems in the $P_{\rm
orb}-e$ diagram by assuming the ejected mass as $0.2\,\rm M_{\odot}$ (see Fig.
15). We found that higher ejected mass leads to a wider range of eccentricity
and the post-SN $P_{\rm orb}$, although the influence is not significant. In
addition, Tauris et al. (2017) suggested that post-SN simulations are also
less dependent on the mass of the first-born NS.

###  5.2 Comparison to previous studies

Tauris et al. (2015) studied the parameter spaces for CO WDs, ONe WDs, EC-SNe
and Fe CC-SNe in NS+He star systems. They assumed that EC-SNe will be produced
if final metal core masses range from $1.37\,\rm M_{\odot}$ to $1.43\,\rm
M_{\odot}$. However, our simulations show that the He star companions may
explode as SNe Ia if the metal core masses range from $\sim 1.335-1.385\,\rm
M_{\odot}$ ( Paper I, ). This could result in a $25\%$ reduction in the
parameter space for producing EC-SNe. It has been suggested that the EC-SN
channel mainly contributes to the formation of $\sim 1.24-1.29\,\rm M_{\odot}$
second-born NSs in DNSs (e.g. Hüdepohl et al., 2010; Kruckow et al., 2018).
Kruckow et al. (2018) studied the formation rate of DNSs by performing the
binary population syntheses. From their Fig. 8, it can be seen that the final
masses of second-born NSs range from $\sim 1.13-1.7\,\rm M_{\odot}$, and Fe
CC-SN is the mainly channel for producing DNSs. Thus, the reduced EC-SN
parameter space has no significant impact on the formation rate of DNSs.

By comparing with the results of Tauris et al. (2015), we note that the
remaining helium envelope mass calculated in their study is larger than that
of this work, and their simulations show that EC-SNe in some NS binaries may
appear as ultra-stripped Type Ib SNe. This is because Tauris et al. (2015)
stopped the code when the ONe cores are formed, while we continued the
calculation to a later evolution stage, i.e. the $e$-capture on $\rm{}^{20}Ne$
(see Fig. 4). We note that the donors still lose material at a rate of $\sim
10^{-4}-10^{-3}\rm\,M_{\odot}\,\rm yr^{-1}$ from ONe core formation to the
$e$-capture on $\rm{}^{20}Ne$ (see Fig. 1), which reduces the donor mass by
$\sim 0.1\rm\,M_{\odot}$. Our simulations show that the final He envelope
masses are lower than $0.06\rm\,M_{\odot}$ for NS+He star binaries with
$P_{\rm orb}^{\rm i}\lesssim 10$ d, and such EC-SNe may be observed as Type Ic
SNe.

Shao & Li (2018) investigated the effect of the NS kicks on the distribution
of DNS systems with population synthesis studies. They suggested that the
second-born NSs in most DNS systems have kick velocities less than $\rm
80\,km\,s^{-1}$. Tauris et al. (2017) explored the pre-SN characters of DNS
systems, and obtained the distribution of parameters of each source. Their
results indicate that the kick velocity and pre-SN orbital period cover a wide
range, and the peak values of kick velocities for most of the observed DNS
systems tend to be less than $\sim 50\rm\,km\,s^{-1}$ (see their Figs
$24-38$). Previous studies show that EC-SNe have explosion energies about
$\sim 10^{50}$ erg and low ejected mass between $0.01-0.2\rm\,M_{\odot}$ (e.g.
Kitaura et al., 2006; Dessart et al., 2006; Zha et al., 2022), which may lead
to the small NS kicks $\lesssim 50\rm\,km\,s^{-1}$ or even $\lesssim
10\rm\,km\,s^{-1}$ (e.g. Janka, 2017; Tauris et al., 2017). Meanwhile, our
results show that the NS kick velocities of $\lesssim 50\rm\,km\,s^{-1}$ can
produce most observed DNS systems, indicating that EC-SNe may play an
important role in the formation of DNSs.

Janka (2017) studied how the NS kick velocity depends on the explosion energy,
ejecta mass, NS mass, and other relevant parameters. They found that massive
metal cores tend to result in larger ejected masses and higher explosion
energies, and therefore higher NS kick velocities. Meanwhile, Tauris et al.
(2017) made statistics related to the masses of the second-born NSs and the
kick velocities. They found that the second-born NSs with masses $\gtrsim
1.33\,\rm M_{\odot}$ are more likely to have high kick velocities, and
suggested that smaller NS masses are preferentially accompanied by smaller NS
kicks. Table 2 lists the NS kick velocities estimated from this work and the
masses of the second-born NSs in observations. We also note that the DNS
systems with high second-born NS masses may originate from high NS kicks (e.g.
PSRs J$0509+3801$ and J$1757-1854$).

Andrews & Mandel (2019) divided the observed DNSs into three distinct subtypes
according to the orbital characteristics: (1) DNSs with tight orbits and low
eccentricities, which can merge within the Hubble time; (2) DNSs with wide
orbits that cannot merge within the Hubble time; (3) DNSs with tight orbits
and high eccentricities. Our theoretical results show that both subtypes (1)
and (2) can be reproduced by the NS kicks $\lesssim 50\,\rm km\,s^{-1}$. To
explain the clustering of DNSs in subtype (3), i.e. PSRs $\rm J0509+3801$,
$\rm J1757-1854$ and $\rm J1915+1606$, Andrews & Mandel (2019) suggested that
if the DNSs are the product of isolated binary evolution, then the second-born
NSs must have a small natal kicks $\lesssim 25\,\rm km\,s^{-1}$ and the pre-SN
He star masses narrowly distributed around $3.2\,\rm M_{\odot}$. In addition,
they also proposed that the DNSs in subtype (3) may originate from dynamical
formation channel, that is, the DNSs firstly formed in globular cluster, and
was then dynamically kicked into the Milky Way field (see Phinney &
Sigurdsson, 1991).

PSR J$1952+2630$ is a mildly recycled $20.7$ ms pulsar orbiting together with
a massive WD companion ($\gtrsim 0.93\rm\,M_{\odot}$), and its orbital period
is $9.4$ h (Lazarus et al., 2014). Such tight orbit indicates that this system
has undergone CE evolution and form a NS+He star system, and then the NS spins
up through the Case BB RLOF. Lazarus et al. (2014) suggested that the
relatively fast spin of this pulsar may be caused by the super-Eddington
accretion with mass-accretion rate between $\sim 100-300$ per cent of
$\dot{M}_{\rm Edd}$. According to our simulations, we concluded that such
mildly recycled pulsar+WD systems can also be produced if the residual H
envelope on the surface of He star companion is considered.

###  5.3 Thermonuclear EC-SNe

Previous studies usually assumed that the $e$-capture on $\rm{}^{20}Ne$ takes
place if ONe cores have masses $\gtrsim 1.37\,\rm M_{\odot}$, resulting in
that the ONe cores collapse into NS through EC-SNe (e.g. Nomoto, 1984; Tauris
et al., 2015; Doherty et al., 2015). However, by performing the
multidimensional hydrodynamic simulations for oxygen deflagration in ONe cores
with three different central ignition densities, Jones et al. (2016) found
that the ONe cores may not collapse into NSs if oxygen deflagration is
triggered at ignition density $\rm log_{10}(\rho_{\rm c}/g\,cm^{-3})\lesssim
10$, while $\sim 0.1-1\rm\,M_{\odot}$ of material will be ejected and leave
bound ONeFe WD remnants, called the thermonuclear EC-SNe (see also Isern et
al., 1991; Kirsebom et al., 2019; Jones et al., 2019). On the other hand, the
ONe cores will collapse into NSs at relatively high ignition density $\rm
log_{10}(\rho_{\rm c}/g\,cm^{-3})\gtrsim 10$. Fig. 3 shows that the explosive
oxygen burning occurs at central density of $\rm log_{10}(\rho_{\rm
c}/g\,cm^{-3})\sim 9.8$, indicating that this NS+He star binaries may evolve
into eccentric pulsar+WD systems after SN explosions.

PSR $\rm J0453+1559$ is a DNS system candidate with $P_{\rm orb}=4.07$ d and
$e=0.1125$ (Martinez et al., 2015). The mass of the recycled pulsar and its
companion are $1.559(5)\,\rm M_{\odot}$ and $1.174(4)\,\rm M_{\odot}$,
respectively. However, it is unclear whether the companion is a NS or a WD.
The mass of its companion is significantly lower than that of the second-born
NSs in other DNS systems, indicating that the companion is more likely not a
NS. On the other hand, the eccentricity of $0.1125$ is high for normal NS+WD
systems. In view of this, Tauris & Janka (2019) used the thermonuclear EC-SN
channel to explain the formation of this system, and they found that PSR $\rm
J0453+1559$ may be produced by a kick velocity from $\sim 70\rm\,km\,s^{-1}$
to $100\rm\,km\,s^{-1}$. More numerical simulations and observations about
thermonuclear EC-SNe are needed to verify this scenario.

##  6 Summary

Using the stellar evolution code MESA, we systematically studied EC-SNe in
NS+He star binaries with $P^{\rm i}_{\rm orb}<10$ d. In our simulations, the
$e$-captures on $\rm{}^{20}Ne$ are triggered if the He star companions develop
highly degenerate ONe cores with masses from $\sim 1.385\rm\,M_{\odot}$ to
$1.43\rm\,M_{\odot}$. SNe Ia are produced if the core masses $\lesssim
1.385\rm\,M_{\odot}$, owing to the explosive oxygen burning caused by the
convective Urca process. The final fates of the He stars will be Fe CC-SNe if
the core masses $\gtrsim 1.43\rm\,M_{\odot}$. We calculated a series of NS+He
star binaries and obtained initial parameter spaces for producing EC-SNe in
the log$P_{\rm orb}^{\rm i}-M_{\rm He}^{\rm i}$ diagram, and found that higher
metallicity can result in higher $M_{\rm He}^{\rm i}$ and minimum $P_{\rm
orb}^{\rm i}$ for EC-SNe.

By considering the kicks added to the second-born NS, we then explored the
properties of the formed DNS systems after the He star companions collapse
into NSs through EC-SNe, such as the post-SN orbital period, eccentricity and
the spin period of recycle pulsar, etc. Based on the comparison between
numerical simulations and observations, we found that the majority of the
observed DNS systems can be explained by the NS kicks of $\lesssim
50\rm\,km\,s^{-1}$, indicating that the EC-SN channel may play an important
role in the formation of DNS systems. We also estimated the properties of the
pre-SN systems of the observed DNS systems, and found that half of the
observed DNS systems seem to have tight pre-SN orbit ($\lesssim 1$ d),
resulting from the higher probability of surviving post-SN systems. In
addition, by assuming that the residual H envelope remains on the surface of
the He star companion, we found that the first-born NSs could accrete enough
material to form the mildly recycled pulsars ($P_{\rm spin}\sim 20$ ms) in DNS
systems.

## Acknowledgements

We thank the anonymous referee for valuable comments that help to improve the
paper. This study is supported by the the National Natural Science Foundation
of China (Nos 12041301, 12121003, 12225304, 12090040/12090043, 12273014,
12288102, 12125303, 12173081 and 11733009), National Key R&D Program of China
(Nos 2021YFA0718500, 2021YFA1600404 and 2021YFA1600403), the Western Light
Project of CAS (No. XBZG-ZDSYS-202117), the science research grant from the
China Manned Space Project (No. CMS-CSST-2021-A12), the Yunnan Fundamental
Research Project (Nos 202101AV070001 and 202201BC070003), the Frontier
Scientific Research Program of Deep Space Exploration Laboratory (No.
2022-QYKYJH-ZYTS-016), the key research program of frontier sciences, CAS, No.
ZDBS-LY-7005, and the Shandong Fundamental Research Project (No. ZR2021MA013).
The authors also acknowledge the “PHOENIX Supercomputing Platform” jointly
operated by the Binary Population Synthesis Group and the Stellar Astrophysics
Group at Yunnan Observatories, Chinese Academy of Sciences.

## Data availability

Results will be shared on reasonable request to corresponding author.

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##  Appendix A The effect of higher kicks and ejected mass

Figure 14: Similar to Fig. 9, but for high NS kick velocities, i.e. $V_{\rm
k}=100,200,300$ and $\rm 400\,km\,s^{-1}$. Figure 15: Similar to Fig. 9, but
for $\Delta M=0.2\rm\,M_{\odot}$.