20XX Vol. X No. XX, 000–000

11institutetext:  School of Astronomy and Space Sciences, University of
Chinese Academy of Sciences, Beijing, People’s Republic of China;
<EMAIL_ADDRESS>
22institutetext: Key Laboratory of Optical Astronomy, National Astronomical
Observatories, Chinese Academy of Sciences, Beijing, People’s Republic of
China
33institutetext: Department of Astronomy, Nanjing University, Nanjing 210023,
People’s Republic of China
44institutetext: Key Laboratory of Modern Astronomy and Astrophysics, Nanjing
University, Ministry of Education, Nanjing 210023, People’s Republic of China
55institutetext: WHU-NAOC Joint Center for Astronomy, Wuhan University, Wuhan,
People’s Republic of China
Received 20XX Month Day; accepted 20XX Month Day

# Are there magnetars in high-mass X-ray binaries?∗ 00footnotetext: $*$
Supported by the National Natural Science Foundation of China.

Kun Xu 112233 Xiang-Dong Li 3344 Zhe Cui 33 Qiao-Chu Li 3344 Yong Shao 3344
Xilong Liang 1122 Jifeng Liu 112255

###### Abstract

Magnetars form a special population of neutron stars with strong magnetic
fields and long spin periods. About 30 magnetars and magnetar candidates known
currently are probably isolated. But the possibility that magnetars are in
binaries hasn’t been excluded. In this work, we perform spin evolution of
neutron stars with different magnetic fields in wind-fed high-mass X-ray
binaries and compare the spin period distribution with observations, aiming to
find magnetars in binaries. Our simulation shows that some of the neutron
stars, which have long spin periods or in wide-separation systems, need strong
magnetic fields to explain their spin evolution. This implies that there are
probably magnetars in high-mass X-ray binaries. Moreover, this can further
provide a theoretical basis for some unclear astronomical phenomena, such as
the possible origin of periodic fast radio bursts from magnetars in binary
systems.

###### keywords:

stars: magnetars — stars: neutron — stars: rotation — stars: supergiants —
accretion, accretion discs — X-rays: binaries

##  1 Introduction

Magnetars are thought to be a special population of neutron stars (NSs) with
magnetic fields $B\gtrsim 10^{14}$ G (Duncan & Thompson 1992; Kaspi &
Beloborodov 2017) in general. They pulsate in X-ray band as persistent
emission sources with bursts and outbursts, which are powered by their strong
magnetic fields (An & Archibald 2019; Beniamini et al. 2019; Kaspi &
Beloborodov 2017). While low-field magnetars were found in observations (Rea
et al. 2010; Zhou et al. 2014), and the dipole magnetic field of a few $\times
10^{12}$ G indicates that they are possibly old magnetars (Turolla et al.
2011; Dall’Osso et al. 2012; Tong & Xu 2012). About 30 magnetars and
candidates known currently are listed in the McGill magnetar catalog 111The
url of the catalog is
http://www.physics.mcgill.ca/$\sim$pulsar/magnetar/main.html (Olausen & Kaspi
2014) with characteristic magnetic fields from $\sim 6\times 10^{12}$ G to
$\sim 2\times 10^{15}$ G. In addition, some central compact objects (CCOs) may
be magnetar candidates, e.g., the one located close to the center of RCW 103
(1E 161348$-$5055) is thought to be a magnetar with $B\gtrsim 5\times 10^{15}$
G (De Luca et al. 2006; Ho & Andersson 2017; Tong et al. 2016; Xu & Li 2019).
All magnetars and candidates are probably isolated NSs because no
optical/near-infrared counterpart of them has been identified (King & Lasota
2019). But in the theory of binary evolution, one can’t exclude the
possibility that magnetars are in binaries.

So here comes the question, are there magnetars in binary systems? It seems to
be answered when the first ultraluminous X-ray pulsar (ULXP) M82 X-2 was found
(Bachetti et al. 2014). The peaked luminosity of M82 X-2 is $\sim 10^{40}$ erg
s-1, so its magnetic field has been estimated to be $\sim 10^{14}$ G (Eksi et
al. 2015; Tsygankov et al. 2016). While other estimation implies that its
magnetic field is $\sim 10^{12}-10^{13}$ G (Dall’Osso et al. 2015; Xu & Li
2017) or even lower ($\lesssim 10^{9}$ G, Kluzniak & Lasota (2015)). Then more
and more ULXPs are discovered but none of them was confirmed to be a magnetar
(King & Lasota 2019). However, an interesting idea is that M82 X-2 is an
accreting low magnetic field magnetar (Tong 2015; Tong & Wang 2019) with
$B\sim 10^{12}$ G (Chen 2017). This is in accord with the current view that
the magnetars’ ultrahigh fields decay on a timescale $<10^{5}-10^{6}$ yr by
Hall draft and Ohm diffusion (Turolla et al. 2015). After decay, the magnetic
fields of old magnetars (or low magnetic field magnetars, or post-magnetars)
are in the similar magnitude as that of other pulsars’. Therefore, the problem
now is how to pick old magnetars out from the zone of NSs. The long period
($\sim 2.6$ h) X-ray pulsar in the high-mass X-ray binary (HMXB) 4U 0114+65
provides some inspirations, which could be an old magnetar (Li & van den
Heuvel 1999; Sanjurjo-Ferrrín et al. 2017) with age of 2.4$-$5 Myr (Igoshev &
Popov 2018). Li & van den Heuvel (1999) thought that it was born as a magnetar
and spun down to $\sim 10$ s by magnetic dipole radiation on a timescale of
$10^{4}-10^{5}$ yr, then its magnetosphere started to interact with the wind
material from the companion and the NS spun down to $\sim 10^{4}$ s within
$10^{5}$ yr. This can lead one to find some clues to the problem in wind-fed
HMXBs.

Some mysterious electromagnetic radiations can be interpreted by magnetars in
binaries, such as fast radio bursts (FRBs). FRBs are millisecond-duration and
extremely bright radio transients (for reviews, see Cordes & Chatterjee 2019;
Petroff et al. 2019; Zhang 2020; Xiao et al. 2021). Since 2007, when the first
FRB was discovered, there have been a large number of models proposed (e.g.,
Lyubarsky 2014; Geng & Huang 2015; Dai et al. 2016; Zhang 2017; Yang & Zhang
2018, 2021; Platts et al. 2019). However, the radiation mechanism of FRBs is
still unclear. Remarkably, FRB 200428 was detected in association with an
X-ray burst from the Galactic magnetar SGR 1935+2146 (e.g., Bochenek et al.
2020; CHIME/FRB Collaboration et al. 2020; Mereghetti et al. 2020; Li et al.
2021b; Tavani et al. 2021). This indicate that magnetars can be a striking
scenario for the origin of FRBs (e.g., Zhang 2020). More interestingly, there
are some repeating FRBs showing long periodic activities, e.g. $\sim$ 16.35
days for FRB 180916.J0158+65 (Chime/Frb Collaboration et al. 2020) and a
possible $\sim 160$-day for FRB 121102 (Rajwade et al. 2020; Cruces et al.
2021). Many scenarios have been proposed to explain the periodicities (e.g.,
Smallwood et al. 2019; Beniamini et al. 2020; Dai & Zhong 2020; Gu et al.
2020; Levin et al. 2020; Lyutikov et al. 2020; Deng et al. 2021; Geng et al.
2021; Kuerban et al. 2021; Li & Zanazzi 2021; Wada et al. 2021; Xu et al.
2021), in which a leading one invokes magnetars in high-mass binaries (e.g.,
Ioka & Zhang 2020; Li et al. 2021a). Therefore, it also becomes a very
interesting subject in the FRB field whether magnetars can exist in binary
systems (Zhang & Gao 2020).

Popov & Turolla (2012) have modeled the formation channel for the NS in the
Be/X-ray binary SXP 1062, which has a long spin period ($P_{\rm s}=1062$ s)
and short age, indicating that its initial magnetic field may be larger than
$10^{14}$ G and decayed to $10^{13}$ G. Zhang et al. (2004) have calculated
the spin evolution of NSs in OB/X-ray binaries before steady wind accretion.
Shakura et al. (2012) proposed that wind matter around an accreting NS should
form a quasi-spherical shell above the magnetosphere rather than directly
accreted if the cooling time of the wind plasma is longer than its free-fall
time. The results of population synthesis (Li et al. 2016) with the subsonic
settling accretion (Shakura et al. 2012) are better consistent with
observations. Karino (2020) found that the wind velocity makes a great
difference on the spin evolution of NSs in wind-fed HMXBs.

In this paper, we perform the spin evolution of neutron stars in wind-fed
high-mass X-ray binaries and compare the spin periods distribution with the
observations, aiming to see if there is any difference between old magnetars
and other NSs. In Section 2, we describe the wind accretion regimes and
Section 3 presents the numerical results as well as the comparison with
observations. Finally, our discussion and summary are shown in Section 4 and
5, respectively.

##  2 Model

###  2.1 Wind Accretion Regimes

We consider a wind-fed accretion binary system consisting of a NS of mass
$M_{\rm NS}=1.4M_{\odot}$ and a massive main-sequence (MS) companion without
filling its Roche lobe (RL). In general, the NS in a binary system is born
with a short spin period ($P_{\rm s}\sim 0.01-0.1$ s), and then it can
experience the following possible phases sketched in Figure 1.

Figure 1: Wind accretion regimes, to see the section 2.1.

####  2.1.1 Phase a: the Pulsar-like Spindown Phase

First, an NS appears as a radio pulsar after its birth in supernova explosion
if its magnetic or radiation pressure is high enough to boot the wind plasma
outside the Bondi radius $R_{\rm g}=2GM_{\rm NS}/v_{\rm rel}^{2}$ (Bondi &
Hoyle 1944) or the radius of the light cylinder $R_{\rm lc}=cP_{\rm s}/2\pi$.
Here $v_{\rm rel}=(v_{\rm orb}^{2}+v_{\rm w}^{2})^{1/2}$ is the relative speed
towards the NS, where $v_{\rm orb}$ is the NS velocity at the orbit and
$v_{\rm w}$ is the stellar wind velocity. $G$ and $c$ are the gravitational
constant and the speed of light, respectively. For clarity, we define $R_{\rm
wind}$ as the wind termination radius from the neutron star and $R_{\rm
eff}=max(R_{\rm g},R_{\rm lc})$ as the effective radius of the NS, so can get
$R_{\rm wind}>R_{\rm eff}$ in phase a. The spin-down rate induced by magnetic
dipole radiation in this phase is (Zhang et al. 2004)

$\dot{P}_{\rm a}=9.73\times 10^{-18}\mu_{31}^{2}P_{\rm 0}^{-1}I_{45}^{-1}{\rm\
s\ s^{-1}},$ (1)

where $\mu_{31}$ and $P_{\rm 0}$ are the NS’s magnetic dipole moment in units
of $10^{31}$ G cm2 and its spin period in units of 1 s, respectively.
$I=I_{45}\times 10^{45}{\rm~{}g~{}cm}^{2}$ is the moment of inertia of the NS.

Phase a ends when either the wind material penetrates the light cylinder
radius or it enters the Bondi radius, i.e., $R_{\rm wind}<R_{\rm eff}$,
correspondingly the transitional spin periods $P_{\rm ab}$ and $P_{\rm ac}$
(Davies & Pringle 1981)

$P_{\rm
ab}=0.37\mu_{31}^{1/3}\dot{M}_{15}^{-1/6}v_{8}^{-5/6}\left(\frac{M_{\rm
NS}}{M_{\odot}}\right)^{1/3}{\rm\ s},$ (2)

and

$P_{\rm ac}=0.38\mu_{31}^{1/2}\dot{M}_{15}^{-1/4}v_{8}^{-1/2}{\rm\ s},$ (3)

where $v_{8}$ is the relative speed towards the NS in units of $10^{8}$ cm s-1
and $\dot{M}=\dot{M}_{15}\times 10^{15}$ g s-1 is the accretion rate onto the
NS given by Bondi & Hoyle (1944)

$\dot{M}=\pi R_{\rm g}^{2}\rho_{\rm w}v_{\rm rel}.$ (4)

Here $\rho_{\rm w}$ is the stellar wind density at the orbit of the NS, which
can be computed by assuming an isotropically expanding wind at a speed of
$v_{\rm w}$

$\rho_{\rm w}=-\dot{M}_{2}/(4\pi a^{2}v_{\rm w}),$ (5)

where $a$ is the orbital separation of the system and $\dot{M}_{2}$ is the
mass loss rate of the donor star.

####  2.1.2 Phase b: the Rapid Rotator Phase

The rapid rotator phase (phase b) comes if the wind material from the donor
star enters the light cylinder of the NS in front of the Bondi radius in the
weak stellar wind case (Illarionov & Sunyaev 1975), i.e., $R_{\rm g}<R_{\rm
lc}$ in this case and phase b occurs when $R_{\rm wind}<R_{\rm lc}$, as well
as $P>P_{\rm ab}$ commences before $P>P_{\rm ac}$ if $P_{\rm ab}<P_{\rm ac}$.
The spin-down rate in phase b is (Zhang et al. 2004)

$\dot{P}_{\rm b}=8.59\times
10^{-17}\mu_{31}\dot{M}_{15}^{1/2}v_{8}^{5/2}P_{\rm 0}^{2}{\rm\ s\ s^{-1}}.$
(6)

When the outer boundary of the wind plasma reaches the Bondi radius ( i.e.,
$R_{\rm wind}=R_{\rm g}$), phase b stops at (Davies & Pringle 1981)

$P_{\rm bc}=0.22\mu_{31}\dot{M}_{15}^{-1/2}v_{8}^{1/2}\left(\frac{M_{\rm
NS}}{M_{\odot}}\right)^{-1}{\rm\ s}.$ (7)

####  2.1.3 Phase c: the Supersonic Propeller Phase

When $R_{\rm wind}<R_{\rm g}$, which is equivalent to $P>P_{\rm bc}$, the
supersonic propeller phase (phase c) arrives in the weak stellar wind case.
While neutron stars will enter phase c directly from phase a if the wind
stellar is strong enough (Illarionov & Sunyaev 1975), i.e., phase c comes when
$R_{\rm wind}<R_{\rm g}$ in the case of $R_{\rm g}>R_{\rm lc}$ as well as
$P>P_{\rm ac}$ occurs before $P>P_{\rm ab}$ if $P_{\rm ac}<P_{\rm ab}$. The
spin-down rate in this phase is (Zhang et al. 2004)

$\dot{P}_{\rm c}=1.6\times 10^{-14}\mu_{31}^{8/13}\dot{M}_{15}^{9/13}P_{\rm
0}^{21/13}{\rm\ s\ s^{-1}}.$

When the inner boundary of the wind plasma reaches the corotation radius
$R_{\rm co}=(GM/\Omega_{\rm s}^{2})^{1/3}$, phase c ends at the equilibrium
spin period (Davies & Pringle 1981)

$P_{\rm eq}=5.0\mu_{31}^{2/3}\dot{M}_{15}^{-1/3}v_{8}^{-2/3}{\rm\ s}.$ (8)

####  2.1.4 Phase d1 and Phase d2: the Accretion Phases

When the inner boundary of the wind material becomes smaller than the the
corotation radius ($R_{\rm wind}<R_{\rm co}$) and the cooling is efficient,
indicated by $P_{\rm s}>P_{\rm eq}$, a bow shock can form at the Bondi radius
around the NS and quasi-spherical accretion onto the NS occurs. The shocked
matter falls towards the magnetosphere of the NS at a supersonic speed if it
cools down rapidly, or it forms a quasi-static shell around the magnetosphere
and settles down subsonically if the matter remains hot, corresponding to the
Bondi-Hoyle-Littleton (BHL) accretion phase (phase d1) or the subsonic
settling accretion phase (phase d2). Phase d1 takes place at
$\dot{M}>\dot{M}_{\rm crit}$ while phase d2 occurs at $\dot{M}<\dot{M}_{\rm
crit}$(Shakura et al. 2012), where $\dot{M}_{\rm crit}=8.4\times
10^{15}\mu_{31}^{1/4}(M/M_{\odot})^{-1/2}R_{6}^{7/8}{\rm~{}g~{}s}^{-1}$ (Arons
& Lea 1976b, a; Elsner & Lamb 1977, 1984).

In phase d1, the accreted material transfers spin-up torque to the NS

$N_{\rm super}=I\dot{\Omega}=\dot{M}R_{\rm M}^{2}\Omega_{\rm S},$ (9)

where $R_{\rm M}=[\mu^{4}/(2GM_{\rm NS}\dot{M}^{2})]^{1/7}$ is the
magnetospheric radius. This phase ends at the spin equilibrium state with
$P_{\rm s}=P_{\rm eq}$. In phase d2, besides the material torque, the
interaction between the magnetosphere and the plasma exerts an additional one
on the NS. Therefore the total torque that governs the NS spin evolution is

$N_{\rm sub}=A_{\rm const}\dot{M}_{\rm X,16}^{7/11}-B_{\rm const}\dot{M}_{\rm
X,16}^{3/11},$ (10)

where

$A_{\rm const}\simeq 3.73\times
10^{31}K_{1}\mu_{31}^{1/11}v_{8}^{-4}\left(\frac{P_{\rm orb}}{10{\rm\
day}}\right)^{-1},$ (11) $B_{\rm const}\simeq 3.61\times
10^{31}K_{1}\mu_{31}^{13/11}\left(\frac{P_{\rm s}}{100{\rm\ s}}\right)^{-1}.$
(12)

Here $K_{1}\sim 40$ is a dimensionless numerical factor (Shakura et al. 2012)
and $\dot{M}_{\rm X,16}=\dot{M}_{\rm X}/10^{16}{\rm~{}g~{}s}^{-1}$, where
$\dot{M}_{\rm
X}=10^{7}{\rm~{}g~{}s}^{-1}\cdot\mu_{30}^{2/21}(\dot{M}/10^{10}{\rm~{}g~{}s}^{-1})^{9/7}$
(Popov et al. 2015). The equilibrium spin period in phase d2, where this phase
stops, reads (Li et al. 2016)

$P_{\rm eq,sub}\simeq
96.8\mu_{31}^{12/11}v_{8}^{4}\dot{M}_{16}^{-4/11}\left(\frac{P_{\rm
orb}}{10{\rm\ day}}\right){\rm\ s}.$ (13)

In observation, the system in accretion phases (phase d1 and phase d2) can be
seen in X-ray band if the luminosity is high enough.

###  2.2 The Evolution of the Binary

We use the binary star evolution (BSE) code (Hurley et al. 2000, 2002; Kiel &
Hurley 2006; Shao & Li 2014, 2015, 2021) to simulate the evolution of the
binary systems. The simulation begins from the birth time of the NS when the
optical star is thought to be in the zero-main-sequence stage, and ends when
the optical/companion star starts to fill its Roche lobe or explodes as a
supernova.

One can estimate the wind velocity adopting the standard formula from Castor
et al. (1975)

$v_{\rm w}=\eta v_{\rm esc}(1-R_{2}/a)^{\beta},$ (14)

where $v_{\rm esc}=\sqrt{2GM_{2}/R_{2}}$ is the escape velocity at the surface
of the optical star. Here we take the coefficient $\eta\sim 0.5-3$ and the
power law index $\beta\sim 0.8-7$ (Waters & van Kerkwijk 1989; Owocki 2014;
Karino 2020).

###  2.3 The Evolution of the Magnetic Field

The magnetic field of an NS decays in the following form (Aguilera et al.
2008; Dall’Osso et al. 2012; Fu & Li 2012)

$B(t)=B_{\rm i}(1+\alpha t/\tau_{\rm d,i})^{-1/\alpha},$ (15)

where $\alpha=1.6$ and the initial field decay time $\tau_{\rm
d,i}=10^{3}{\rm~{}yr}/(B_{\rm i}/10^{15}{\rm~{}G})^{\alpha}$ (Fu & Li 2012) in
our reference model. We plot the the magnetic fields evolution with four
initial values in Figure 2, which are $B_{\rm i}=10^{12}$, $B_{\rm
i}=10^{13}$, $B_{\rm i}=10^{14}$ and $B_{\rm i}=10^{15}$ G.

Figure 2: The magnetic field evolution with $B_{\rm i}=10^{12}$, $B_{\rm
i}=10^{13}$, $B_{\rm i}=10^{14}$ and $B_{\rm i}=10^{15}$ G, where $B<10^{13}$
G remains unchanged in its life while $B_{\rm i}\geq 10^{13}$ G decays in the
form of Eq(15).

##  3 Results

###  3.1 The Reference Model

We first consider a series of circular orbital binaries as the reference
model, in which each one consists of a $1.4{\rm~{}}M_{\odot}$ neutron star and
a $20{\rm~{}}M_{\odot}$ optical star with metallicity $Z=Z_{\odot}$. The wind
parameters of the companion $\eta$ and $\beta$ are set up to be $1.0$ and
$0.8$ respectively. Figure 3 shows the results of the reference model in the
Corbet diagram, where the filled circles indicate the maximum spin periods
$P_{\rm s,max}$ during the evolution process in each case. From lower to
upper, the green, blue, red and orange circles represent that the initial
magnetic fields of the NSs are $B_{\rm i}=10^{12}$, $B_{\rm i}=10^{13}$,
$B_{\rm i}=10^{14}$ and $B_{\rm i}=10^{15}$ G, respectively. From left to
right in each colour, the circles symbolize the orbital periods of the
binaries, which are 3, 5, 8, 10, 15, 20, 25, 30, 40, 50, 60, 70, 80, 90, 100,
120, 165 day, respectively. The dashed lines string the circles in the same
colour, predicting the possible $P_{\rm s,max}$ with the $P_{\rm orb}$ ranging
from $3$ to $165$ day. The coloured regions give the predicted $P_{\rm s,max}$
distributions of the NSs with initial magnetic field between the upper and
lower dashed lines. For example, the blue dashed line and region predict the
$P_{\rm s,max}$ of the NSs with $B_{\rm i}=10^{13}$ G and $10^{12}<B_{\rm
i}\leq 10^{13}$ G. From this figure one can know that the NSs with high
initial magnetic field and in short orbital period systems can spin down to
very slow rotation, because the strong magnetic fields and the small
separations make the NS easy to capture stellar wind from the companion. The
NSs soon enter phase c or d2 after captured and quickly lose their angular
momentum (to see Figure 5). There are significant drop in the blue and red
dashed lines but not in the orange and green dashed lines in Figure 3, because
cases with $B_{\rm i}=10^{12}$ G and $B_{\rm i}=10^{15}$ G end at phase a/b
and d2, respectively, while the cases with $B_{\rm i}=10^{13}$ G and $B_{\rm
i}=10^{14}$ G in short orbital period systems end at phase d2 and those in
long orbital period systems end at phase a/b. The second green circle ends at
phase b while others end at phase a, which leads to a small bump in the green
dashed line222This may be a numerical mistake because it jumps from phase a to
phase b in the last step. Considering there is no significant influence on the
results, we don’t discuss it in the following..

Figure 3: Results of the reference model in the Corbet diagram. The filled
circles indicate the maximum spin periods $P_{\rm s,max}$ during the evolution
process in each case. The green, blue, red and orange circles represent that
the initial magnetic fields of the NSs are $B_{\rm i}=10^{12}$, $B_{\rm
i}=10^{13}$, $B_{\rm i}=10^{14}$ and $B_{\rm i}=10^{15}$ G, respectively. The
dashed lines string the circles in the same colour and the coloured regions
give the predicted $P_{\rm s,max}$ distributions for the NSs with initial
magnetic field between the upper and lower dashed lines. The squares and
diamonds mark two subclasses of HMXBs, which are SgXBs and SFXTs.

We compare our results with two subclasses of HMXBs detected $P_{\rm s}$ and
$P_{\rm orb}$, which are the supergiant X-ray binaries (SgXBs) and the
supergiant fast X-ray transients (SFXTs). The triangles and squares mark the
SgXBs and the SFXTs respectively 333data from Martínez-Núñez et al. (2017) and
Tauris et al. (2017). The SgXBs and SFXTs are two subclasses of HMXBs. SgXBs
are persistent systems in X-rays ($L_{\rm X}=10^{36}-10^{38}$ erg s-1), one of
which consists of a compact accretor and a supergiant mass donor of spectral
type O8-B1 I/II with strong stellar wind (Martínez-Núñez et al. 2017; Tauris
et al. 2017). While SFXTs are not persistent, which stay in quiescence at
$L_{\rm X}=10^{32}-10^{34}$ erg s-1 for most of the time (Romano et al. 2015)
and exhibit short outbursts lasting a few hours reaching $10^{37}-10^{38}$ erg
s-1 (Rampy et al. 2009; Bozzo et al. 2011). Figure 3 shows that most of the
observed HMXBs, which have $P_{\rm s}<1000$ s and $P_{\rm orb}<30$ day,
distribute in the blue region, meaning that they are probably normal neutron
stars. Other five sources with $P_{\rm s}>1000$ s or $P_{\rm orb}>30$ day are
in the red or orange regions, indicating that they may have strong magnetic
fields. One of them is in the orange region, which is probably a magnetar. The
red dashed line could be seen as the separatrix between magnetars and other
NSs.

Figure 4 shows the $P_{\rm s}$ regions of NSs’ luminosity $L_{\rm
X}>10^{32}{\rm~{}erg~{}s}^{-1}$ during the accretion phases. The gray filled
and unfilled circles indicate the predicted maximum and minimum $P_{\rm s}$ of
the NSs, respectively. And the gray regions give the predicted range of
$P_{\rm s}$ of the NSs in each panel. From top-left to bottom-right panels,
the magnetic fields are $B_{\rm i}=10^{12}$, $B_{\rm i}=10^{13}$, $B_{\rm
i}=10^{14}$ and $B_{\rm i}=10^{15}$ G, respectively. In the cases of $B_{\rm
i}=10^{12}$ G, no NS enters accretion phases.

Figure 4: Results of the reference model with NSs’ luminosity $L_{\rm
X}>10^{32}{\rm~{}erg~{}s}^{-1}$ during the accretion phases. The gray filled
and unfilled circles indicate the predicted maximum and minimum $P_{\rm s}$ of
the NSs, respectively. And the gray regions give the predicted range of
$P_{\rm s}$ of the NSs in each panel. From top-left to bottom-right panels,
the magnetic fields are $B_{\rm i}=10^{12}$, $B_{\rm i}=10^{13}$, $B_{\rm
i}=10^{14}$ and $B_{\rm i}=10^{15}$ G, respectively. There is no NS enters
accretion phases in the cases of $B_{\rm i}=10^{12}$ G.

We plot the spin evolution process of the reference model with $P_{\rm
orb}=10$ day in Figure 5. From lower to upper, the dotted, dashed, dot-dashed
and solid lines represent the evolution of NSs with magnetic field $B_{\rm
i}=10^{12}$, $B_{\rm i}=10^{13}$, $B_{\rm i}=10^{14}$ and $B_{\rm i}=10^{15}$
G, respectively. In each line, the black, orange, green, red and blue parts
indicate that the NSs are in phase a, b, c, d1 and d2. The right panel is the
magnified part of the left panel with $5<{\rm log}t~{}({\rm yr})<7$.

Figure 5: The spin evolution process of the reference model when $P_{\rm
orb}=10$ day. The dotted, dashed , dot-dashed and solid lines represent the
evolution of NSs with magnetic field $B_{\rm i}=10^{12}$, $B_{\rm i}=10^{13}$,
$B_{\rm i}=10^{14}$ and $B_{\rm i}=10^{15}$ G, respectively. In each line, the
black, orange, green, red and blue parts indicate that the NSs are in phase a,
b, c, d1 and d2. In phase c (the supersonic propeller phase) and d2 (the
subsonic settling accreion phase) before it reaches equilibrium, the NS loses
angular momentum rapidly so that its spin period rises steeply. The right
panel is the magnified part of the left panel with $5<{\rm log}t~{}({\rm
yr})<7$ and the gray region covers the $P_{\rm spin}$ range of the NSs in
HMXBs.

###  3.2 Parameter Study

The main configurable parameters in our model are the companion mass $M_{2}$,
the wind parameters $\eta$ and $\beta$. In this subsection, we vary these
parameters based on the reference model to see how they can influence the
results.

####  3.2.1 The Companion Mass $M_{2}$

We first change the mass of the companion $M_{2}$. The results of maximum spin
period with different companion mass are exhibited in Figure 6. From top to
bottom, $M_{2}$ is taken to be $10~{}M_{\odot}$, $20~{}M_{\odot}$ (the
reference model) and $30~{}M_{\odot}$. It shows that more sources fall in the
red and orange regions in the upper panel while less sources in these regions
in the lower panel compared with the reference model. This may imply that
systems with less massive companions are more likely to dedicate HMXBs with
magnetars.

Figure 6: Same as Figure 3 but with the companion mass $M_{2}=$ 10, 20 (the
reference model) and 30, from top to bottom.

####  3.2.2 The Wind Coefficient $\eta$

Figure 7 shows the dependence of $P_{\rm max}$ on the wind coefficient $\eta$,
where $\eta=0.5$, 1 (the reference model), 2 and 3 from top-left to bottom-
right. Comparing with the reference model, we find that less sources in the
red and orange regions when $\eta=0.5$, while more sources in these regions
when $\eta=$ 2 or 3. This may imply that the larger the wind coefficient, the
more magnetars are in HMXBs.

Figure 7: Same as Figure 3 but with the wind coefficient $\eta=$ 0.5, 1 (the
reference model), 2 and 3 from top-left to bottom-right. Comparing with the
reference model, there are less sources in the red and orange regions when
$\eta=0.5$, while more sources in these regions when $\eta=2$ or 3. This may
imply that the larger the wind coeﬀicient, the more magnetars are in HMXBs.

####  3.2.3 The Wind Power Law Index $\beta$

Karino (2020) discussed the influence of parameter $\beta$ within larger
scale, where $\beta=1$ and $7$ indicate the fast and slow wind cases,
respectively. They found that the NSs spin down rapidly in the slow wind cases
due to the propeller effect and settling accretion shell, while magnetic
inhibition causes spin-down in the fast cases. Figure 8 presents the $P_{\rm
max}$ results with different values of wind power law index $\beta$. From top-
left to bottom-right, we take $\beta=0.8$ (the reference model), 1, 4 and 7,
respectively. It shows that when $\beta$ is much larger, some sources with
short orbital periods need strong magnetic fields to explain their long spin
periods. But we are not sure whether $\beta$ can be as large as 7. This may
indicate that in the slow wind cases (Karino 2020), an NS needs a strong
magnetic field to get a large $R_{\rm lc}$ to start the interaction with the
wind material.

Figure 8: Same as Figure 3 but with the wind power index $\beta=$ 0.8 (the
reference model), 1, 4 and 7 from top-left to bottom-right. It shows that when
$\beta$ is much larger, some sources with short orbital periods need stronger
magnetic fields to explain their long spin periods.

##  4 Discussion

###  4.1 Special Sources

####  4.1.1 J11215$-$5952

From Figures 3, 7 and 8, we know that only one source is always in the red or
orange regions, which is J11215$-$5952 (hereafter J11215) with orbital period
$P_{\rm orb}=164.6$ day and spin period $P_{\rm s}=186.78$ s (Sidoli et al.
2006, 2007, 2020; Masetti et al. 2006). That means J11215 needs an initial
magnetic field $B_{\rm i}>10^{13}$ G to explain its orbital and spin period,
which implies that it’s probably a magnetar. In a wide-separation binary, the
NS may need a large effective radius to let the wind be captured. So J11215
needs a strong magnetic field to get a long spin period as well as a large
$R_{\rm lc}$ in phase a.

####  4.1.2 OAO 1657$-$415 and J18483$-$0311

There are two sources not covered in Figure 4 because their spin periods are
too small, which are OAO 1657$-$415 (hereafter OAO 1657) with $P_{\rm
orb}=10.448$ day and $P_{\rm s}=38.2$ s and J18483$-$0311 (hereafter J18483)
with $P_{\rm orb}=18.55$ day and $P_{\rm s}=21$ s. We think these sources are
or have been in directly accretion phase (phase d1) so that they can spin up
to a very fast rotation. We show two groups of parameters that can cover OAO
1657 and J18483 in Figure 9, in which we can explain OAO 1657 within the
parameters space in our model while J18483 needs a small $\eta$ and massive
companion.

Figure 9: Parameters that can cover OAO 1657 (the left panel) and J18483 (the
right panel).

####  4.1.3 4U 1954$+$31

Recently, 4U 1954$+$31 has been re-classified as a HMXB containing a late-type
supergiant with mass of $9^{+6}_{-2}~{}M_{\odot}$ (Hinkle et al. 2020). While
the companion mass arrived according to its position in the
Hertzsprung–Russell Diagram (HRD) is model depended, as there are still many
uncertainties plague the evolutionary tracks of stars, for instance the
stellar wind and mixing. As a comparison, we also evolve some models using the
stellar evolutionary code Modules for Experiments in Stellar Astrophysics
(MESA) (Paxton et al. 2011) and compare their evolutionary tracks with
4U1954+31 in HRD (the upper panel of Figure 10). In our models, the convection
is calculated following the standard mixing-length theory (Böhm 1958) with the
scale parameter $\alpha_{\rm MLT}=1.5$, and the convective regions are
determined by the Ledoux criterion. The semi-convection parameter are assumed
to be $\alpha_{\rm SC}=1.0$. Besides, the mass loss due to stellar wind are
calculated according to their effective temperature and surface Hydrogen (H)
mass fraction. That is, the prescriptions of Vink et al. (2001) and Nugis &
Lamers (2000) are used for hot H-rich ( $T_{\rm eff}>10^{4}\rm K$ and $X_{\rm
s}\geq 0.4$) and H-poor ($T_{\rm eff}>10^{4}\rm K$ and $X_{\rm s}<0.4$) stars,
respectively. While for $T_{\rm eff}<10^{4}\rm K$ we change to that of de
Jager et al. (1988). We simulate their evolution from zero age main sequence
(ZAMS) to core-Carbon (C) ignition, with initial mass $M_{\rm i}$ ranging from
12 to 19 $M_{\odot}$. Our predicted mass of 4U 1954+31 is $15^{+3}_{-3}\rm
M_{\odot}$, which is $\sim 6\rm M_{\odot}$ heavier than that of Hinkle et al.
(2020). The spin period of the NS is $\sim 5$ hr (Corbet et al. 2008; Marcu et
al. 2011; Enoto et al. 2014) while the orbital period is not confirmed. Only a
lower limit of 3 yrs is given (Hinkle et al. 2020), which is much longer than
the orbital periods of SgXBs and SFXTs in Figure 3. And we can’t explain it
within the parameter space used above, but the results in the lower panel of
Figure 10 can cover it with the following parameters: $\eta=0.5$, $\beta=0.8$,
$B_{\rm i}=10^{16}$ G and $M_{2}=15~{}M_{\odot}$. Since the separation of this
system is very wide, another possible model to explain its long spin period is
that the NS is firstly spun down by a fallback disk and then interacts wind
material.

Figure 10: Upper: Location of the companion of 4U 1954 on the HRD. Lower: A
group of parameters that can cover 4U 1954, in which the initial magnetic
field is $B_{\rm i}=10^{16}$ G.

###  4.2 The distribution of NSs in HMXBs

Figure 11 depicts the predicted distribution of magnetars and normal magnetic
field NSs in HMXBs. In the Corbet diagram, the normal NSs may center at the
region where the spin periods and orbital periods are both short, while
magnetars may locate at the region where either $P_{\rm s}$ or $P_{\rm orb}$
are long. As proposed in Li & van den Heuvel (1999), the reason is that a
strong magnetic field makes a NS achieving a slower spin period in phase a by
magnetic dipole radiation, as well as a larger effective radius, so that the
NS is easy to capture the wind stellar material and enters the interaction
phases (phase b, c ,d1 and d2) earlier, then the interaction between the wind
material and the magnetosphere could have much time to spin the NS down.

Figure 11: The distribution of magnetars and normal NSs in HMXBs predicted by
our model.

##  5 Summary

In this work, we explore the possible parameter space for magnetars in high-
mass X-ray binaries by performing spin evolution of neutron stars. We arrange
and simplify the wind accretion regimes taking the direct and the subsonic
settling accretion into account. Compared with previous studies, we use grid
methods combining the individual and population synthesis method and simulate
a more complete evolution process, including the evolution of winds from the
companions and the decay of the magnetic fields of the NSs. Our results show
that some NSs in the right and upper region in the Corbet diagram, which are
in wide-separation systems or have long spin periods, need magnetic fields in
magnetar magnitude to reproduce their relatively long spin periods. This
implies that some NSs may born as magnetars in HMXBs, then their magnetic
fields decay to the order of normal NSs through a few Myrs, but the spin
periods record their evolution informations and can provide some traces about
their initial magnetic fields.

In the calculations we employ the magnetic field evolution model modified by
Fu & Li (2012), where the parameters are taken as $\alpha=1.6$ and $\tau_{\rm
d,i}=10^{3}{\rm~{}yr}/(B_{\rm i}/10^{15}{\rm~{}G})^{\alpha}$. There are some
debates on these parameters (Dall’Osso et al. 2012; Fu & Li 2012; Gullón et
al. 2014) and we study them in the appendix. The simple $\beta$-velocity law
used in our work may not accurately describe the wind profile of the companion
since some self-consistent calculations indicate that the wind velocity field
is more complicated (Gräfener & Hamann 2005, 2008; Sander et al. 2015). We
model circular orbits only in the calculations while most of the HMXBs have
eccentric orbits because the NSs receive a natal kick at the moment of their
formation. The separation becomes variable in eccentric orbits, which leads to
parameters more complicated, including the wind velocity, the wind density,
the accretion rate, the NS orbital velocity and so on. Because the orbital
period is far shorter than the lifetime of the system, so for simplicity, we
can use a circular orbit to replace the influence of an eccentric orbit and
there is no significant impact on the results.

###### Acknowledgements.

We thank the anonymous referee for helpful comments and suggestions. This work
was supported by the Natural Science Foundation of China under grant No.
11933004, 11988101, 12041301, 12063001, 11773015, Project U1838201 supported
by NSFC and CAS, the program A for Outstanding PhD candidate of Nanjing
University, the National Key Research and Development Program of China
(2016YFA0400803) and the Fundamental Research Funds for the Central
Universities.

##  Appendix A Parameter study of the evolution of the magnetic field

We take $\alpha=1.6$ and $\tau_{\rm d,i}=\tau_{\rm d}/(B_{\rm
i}/10^{15}{\rm~{}G})^{\alpha}$ in the reference model, where $\tau_{\rm
d}=10^{3}{\rm~{}yr}$. There are some debates on the range of $\alpha$, e.g.,
Dall’Osso et al. (2012) takes $0\leq\alpha\leq 2$ while Fu & Li (2012) thinks
$1.5\lesssim\alpha\lesssim 1.8$ is most likely. We plot the magnetic field and
spin evolution tracks with different $\alpha$ and $\tau_{\rm d}$ in Figure 12
and 14, where the gray region covers the range of the estimated magnetic
fields (Caballero & Wilms 2012) and ages of NSs in HMXBs from observations.
They show in the cases with large $\alpha$ and $\tau_{\rm d}$, which means the
magnetic fields evolve slowly, the NSs can spin down to long periods. The
results of the longest spin periods with different $\alpha$ and $\tau_{\rm d}$
are exhibited in Figure 13 and 15, respectively. They show that more NSs need
strong magnetic fields to explain their long spin periods in the rapid
evolution cases (with small $\alpha$ and $\tau_{\rm d}$), while in the slowly
evolution cases, magnetar model is dispensable for the observed sources.

Figure 12: The magnetic field (left panel) and spin (right panel) evolution
tracks with different $\alpha$. The gray region in the left panel covers the
range of the estimated magnetic field and age of NSs in HMXBs from
observetions.

Figure 13: Same as Figure 3 but with $\alpha=$1, 1.2, 1.6 (the reference
model) and 2 from top-left to bottom-right.

Figure 14: The magnetic field (left panel) and spin (right panel) evolution
tracks with diffirent $\tau_{\rm d}$. The gray region in the left panel covers
the range of the estimated magnetic field and age of NSs in HMXBs from
observations.

Figure 15: Same as Figure 3 but with $\tau_{\rm d}=10^{2}$, $10^{3}$ (the
reference model) and $10^{4}$ yr, from top to bottom.

## References

  * Aguilera et al. (2008) Aguilera, D. N., Pons, J. A., & Miralles, J. A. 2008, A&A, 486, 255
  * An & Archibald (2019) An, H., & Archibald, R. 2019, ApJ, 877, L10
  * Arons & Lea (1976b) Arons, J., & Lea, S. M. 1976b, ApJ, 207, 914
  * Arons & Lea (1976a) Arons, J., & Lea, S. M. 1976a, ApJ, 210, 792
  * Bachetti et al. (2014) Bachetti, M., Harrison, F. A., Walton, D. J., et al. 2014, Nature, 514, 202
  * Beniamini et al. (2019) Beniamini, P., Hotokezaka, K., van der Horst, A., & Kouveliotou, C. 2019, MNRAS, 487, 1426
  * Beniamini et al. (2020) Beniamini, P., Wadiasingh, Z., & Metzger, B. D. 2020, MNRAS, 496, 3390
  * Bochenek et al. (2020) Bochenek, C. D., Ravi, V., Belov, K. V., et al. 2020, Nature, 587, 59
  * Böhm (1958) Böhm, K. H. 1958, ZAp, 46, 245
  * Bondi & Hoyle (1944) Bondi, H., & Hoyle, F. 1944, MNRAS, 104, 273
  * Bozzo et al. (2011) Bozzo, E., Giunta, A., Cusumano, G., et al. 2011, A&A, 531, A130
  * Caballero & Wilms (2012) Caballero, I., & Wilms, J. 2012, Mem. Soc. Astron. Italiana, 83, 230
  * Castor et al. (1975) Castor, J. I., Abbott, D. C., & Klein, R. I. 1975, ApJ, 195, 157
  * Chen (2017) Chen, W.-C. 2017, MNRAS, 465, L6
  * Chime/Frb Collaboration et al. (2020) Chime/Frb Collaboration, Amiri, M., Andersen, B. C., et al. 2020, Nature, 582, 351
  * CHIME/FRB Collaboration et al. (2020) CHIME/FRB Collaboration, Andersen, B. C., Bandura, K. M., et al. 2020, Nature, 587, 54
  * Corbet et al. (2008) Corbet, R. H. D., Sokoloski, J. L., Mukai, K., Markwardt, C. B., & Tueller, J. 2008, ApJ, 675, 1424
  * Cordes & Chatterjee (2019) Cordes, J. M., & Chatterjee, S. 2019, ARA&A, 57, 417
  * Cruces et al. (2021) Cruces, M., Spitler, L. G., Scholz, P., et al. 2021, MNRAS, 500, 448
  * Dai et al. (2016) Dai, Z. G., Wang, J. S., Wu, X. F., & Huang, Y. F. 2016, ApJ, 829, 27
  * Dai & Zhong (2020) Dai, Z. G., & Zhong, S. Q. 2020, ApJ, 895, L1
  * Dall’Osso et al. (2012) Dall’Osso, S., Granot, J., & Piran, T. 2012, MNRAS, 422, 2878
  * Dall’Osso et al. (2015) Dall’Osso, S., Perna, R., & Stella, L. 2015, MNRAS, 449, 2144
  * Davies & Pringle (1981) Davies, R. E., & Pringle, J. E. 1981, MNRAS, 196, 209
  * de Jager et al. (1988) de Jager, C., Nieuwenhuijzen, H., & van der Hucht, K. A. 1988, A&AS, 72, 259
  * De Luca et al. (2006) De Luca, A., Caraveo, P. A., Mereghetti, S., Tiengo, A., & Bignami, G. F. 2006, Science, 313, 814
  * Deng et al. (2021) Deng, C.-M., Zhong, S.-Q., & Dai, Z.-G. 2021, arXiv e-prints, arXiv:2102.06796
  * Duncan & Thompson (1992) Duncan, R. C., & Thompson, C. 1992, ApJ, 392, L9
  * Elsner & Lamb (1977) Elsner, R. F., & Lamb, F. K. 1977, ApJ, 215, 897
  * Elsner & Lamb (1984) Elsner, R. F., & Lamb, F. K. 1984, ApJ, 278, 326
  * Enoto et al. (2014) Enoto, T., Sasano, M., Yamada, S., et al. 2014, ApJ, 786, 127
  * Fu & Li (2012) Fu, L., & Li, X.-D. 2012, ApJ, 757, 171
  * Eksi et al. (2015) Eksi, K. Y., Andac, I. C., Cikintoglu, S., et al. 2015, MNRAS, 448, L40
  * Geng & Huang (2015) Geng, J. J., & Huang, Y. F. 2015, ApJ, 809, 24
  * Geng et al. (2021) Geng, J., Li, B., & Huang, Y. 2021, The Innovation, 2, 100152
  * Gräfener & Hamann (2005) Gräfener, G., & Hamann, W. R. 2005, A&A, 432, 633
  * Gräfener & Hamann (2008) Gräfener, G., & Hamann, W. R. 2008, A&A, 482, 945
  * Gu et al. (2020) Gu, W.-M., Yi, T., & Liu, T. 2020, MNRAS, 497, 1543
  * Gullón et al. (2014) Gullón, M., Miralles, J. A., Viganò, D., & Pons, J. A. 2014, MNRAS, 443, 1891
  * Ho & Andersson (2017) Ho, W. C. G., & Andersson, N. 2017, MNRAS, 464, L65
  * Hurley et al. (2000) Hurley, J. R., Pols, O. R., & Tout, C. A. 2000, MNRAS, 315, 543
  * Hurley et al. (2002) Hurley, J. R., Tout, C. A., & Pols, O. R. 2002, MNRAS, 329, 897
  * Hinkle et al. (2020) Hinkle, K. H., Lebzelter, T., Fekel, F. C., et al. 2020, ApJ, 904, 143
  * Igoshev & Popov (2018) Igoshev, A. P., & Popov, S. B. 2018, MNRAS, 473, 3204
  * Illarionov & Sunyaev (1975) Illarionov, A. F., & Sunyaev, R. A. 1975, A&A, 39, 185
  * Ioka & Zhang (2020) Ioka, K., & Zhang, B. 2020, ApJ, 893, L26
  * Karino (2020) Karino, S. 2020, PASJ, 72, 95
  * Kaspi & Beloborodov (2017) Kaspi, V. M., & Beloborodov, A. M. 2017, ARA&A, 55, 261
  * King & Lasota (2019) King, A., & Lasota, J.-P. 2019, MNRAS, 485, 3588
  * Kiel & Hurley (2006) Kiel, P. D., & Hurley, J. R. 2006, MNRAS, 369, 1152
  * Kluzniak & Lasota (2015) Kluzniak, W., & Lasota, J. P. 2015, MNRAS, 448, L43
  * Kuerban et al. (2021) Kuerban, A., Huang, Y.-F., Geng, J.-J., et al. 2021, arXiv e-prints, arXiv:2102.04264
  * Levin et al. (2020) Levin, Y., Beloborodov, A. M., & Bransgrove, A. 2020, ApJ, 895, L30
  * Li et al. (2021b) Li, C. K., Lin, L., Xiong, S. L., et al. 2021b, Nature Astronomy, 5, 378
  * Li & Zanazzi (2021) Li, D., & Zanazzi, J. J. 2021, ApJ, 909, L25
  * Li et al. (2021a) Li, Q.-C., Yang, Y.-P., Wang, F. Y., et al. 2021a, ApJ, 918, L5
  * Li et al. (2016) Li, T., Shao, Y., & Li, X.-D. 2016, ApJ, 824, 143
  * Li & van den Heuvel (1999) Li, X. D., & van den Heuvel, E. P. J. 1999, ApJ, 513, L45
  * Lyubarsky (2014) Lyubarsky, Y. 2014, MNRAS, 442, L9
  * Lyutikov et al. (2020) Lyutikov, M., Barkov, M. V., & Giannios, D. 2020, ApJ, 893, L39
  * Marcu et al. (2011) Marcu, D. M., Fürst, F., Pottschmidt, K., et al. 2011, ApJ, 742, L11
  * Martínez-Núñez et al. (2017) Martínez-Núñez, S., Kretschmar, P., Bozzo, E., et al. 2017, Space Sci. Rev., 212, 59
  * Masetti et al. (2006) Masetti, N., Pretorius, M. L., Palazzi, E., et al. 2006, A&A, 449, 1139
  * Mereghetti et al. (2020) Mereghetti, S., Savchenko, V., Ferrigno, C., et al. 2020, ApJ, 898, L29
  * Nugis & Lamers (2000) Nugis, T., & Lamers, H. J. G. L. M. 2000, A&A, 360, 227
  * Olausen & Kaspi (2014) Olausen, S. A., & Kaspi, V. M. 2014, ApJS, 212, 6
  * Owocki (2014) Owocki, S. 2014, arXiv e-prints, arXiv:1409.2084
  * Paxton et al. (2011) Paxton, B., Bildsten, L., Dotter, A., et al. 2011, ApJS, 192, 3
  * Petroff et al. (2019) Petroff, E., Hessels, J. W. T., & Lorimer, D. R. 2019, A&A Rev., 27, 4
  * Platts et al. (2019) Platts, E., Weltman, A., Walters, A., et al. 2019, Phys. Rep., 821, 1
  * Popov & Turolla (2012) Popov, S. B., & Turolla, R. 2012, MNRAS, 421, L127
  * Popov et al. (2015) Popov, S. B., Postnov, K. A., & Shakura, N. I. 2015, MNRAS, 447, 2817
  * Rajwade et al. (2020) Rajwade, K. M., Mickaliger, M. B., Stappers, B. W., et al. 2020, MNRAS, 495, 3551
  * Rampy et al. (2009) Rampy, R. A., Smith, D. M., & Negueruela, I. 2009, ApJ, 707, 243
  * Rea et al. (2010) Rea, N., Esposito, P., Turolla, R., et al. 2010, Science, 330, 944
  * Romano et al. (2015) Romano, P., Bozzo, E., Mangano, V., et al. 2015, A&A, 576, L4
  * Sander et al. (2015) Sander, A., Shenar, T., Hainich, R., et al. 2015, A&A, 577, A13
  * Sanjurjo-Ferrrín et al. (2017) Sanjurjo-Ferrrín, G., Torrejón, J. M., Postnov, K., et al. 2017, A&A, 606, A145
  * Shakura et al. (2012) Shakura, N., Postnov, K., Kochetkova, A., & Hjalmarsdotter, L. 2012, MNRAS, 420, 216
  * Shao & Li (2014) Shao, Y., & Li, X.-D. 2014, ApJ, 796, 37
  * Shao & Li (2015) Shao, Y., & Li, X.-D. 2015, ApJ, 802, 131
  * Shao & Li (2021) Shao, Y., & Li, X.-D. 2021, arXiv e-prints, arXiv:2107.03565
  * Sidoli et al. (2006) Sidoli, L., Paizis, A., & Mereghetti, S. 2006, A&A, 450, L9
  * Sidoli et al. (2007) Sidoli, L., Romano, P., Mereghetti, S., et al. 2007, A&A, 476, 1307
  * Sidoli et al. (2020) Sidoli, L., Postnov, K., Tiengo, A., et al. 2020, A&A, 638, A71
  * Smallwood et al. (2019) Smallwood, J. L., Martin, R. G., & Zhang, B. 2019, MNRAS, 485, 1367
  * Tauris et al. (2017) Tauris, T. M., Kramer, M., Freire, P. C. C., et al. 2017, ApJ, 846, 170
  * Tavani et al. (2021) Tavani, M., Casentini, C., Ursi, A., et al. 2021, Nature Astronomy, 5, 401
  * Tong & Xu (2012) Tong, H., & Xu, R. X. 2012, ApJ, 757, L10
  * Tong et al. (2016) Tong, H., Wang, W., Liu, X. W., & Xu, R. X. 2016, ApJ, 833, 265
  * Tong & Wang (2019) Tong, H., & Wang, W. 2019, MNRAS, 482, 4956
  * Tong (2015) Tong, H. 2015, Research in Astronomy and Astrophysics, 15, 517
  * Tsygankov et al. (2016) Tsygankov, S. S., Mushtukov, A. A., Suleimanov, V. F., & Poutanen, J. 2016, MNRAS, 457, 1101
  * Turolla et al. (2011) Turolla, R., Zane, S., Pons, J. A., Esposito, P., & Rea, N. 2011, ApJ, 740, 105
  * Turolla et al. (2015) Turolla, R., Zane, S., & Watts, A. L. 2015, Reports on Progress in Physics, 78, 116901
  * Vink et al. (2001) Vink, J. S., de Koter, A., & Lamers, H. J. G. L. M. 2001, A&A, 369, 574
  * Wada et al. (2021) Wada, T., Ioka, K., & Zhang, B. 2021, arXiv e-prints, arXiv:2105.14480
  * Waters & van Kerkwijk (1989) Waters, L. B. F. M., & van Kerkwijk, M. H. 1989, A&A, 223, 196
  * Xiao et al. (2021) Xiao, D., Wang, F., & Dai, Z. 2021, Science China Physics, Mechanics, and Astronomy, 64, 249501
  * Xu et al. (2021) Xu, K., Li, Q.-C., Yang, Y.-P., et al. 2021, ApJ, 917, 2
  * Xu & Li (2019) Xu, K., & Li, X.-D. 2019, ApJ, 877, 138
  * Xu & Li (2017) Xu, K., & Li, X.-D. 2017, ApJ, 838, 98
  * Yang & Zhang (2018) Yang, Y.-P., & Zhang, B. 2018, ApJ, 868, 31
  * Yang & Zhang (2021) Yang, Y.-P., & Zhang, B. 2021, ApJ, 919, 89
  * Zhang (2017) Zhang, B. 2017, ApJ, 836, L32
  * Zhang (2020) Zhang, B. 2020, Nature, 587, 45
  * Zhang et al. (2004) Zhang, F., Li, X.-D., & Wang, Z.-R. 2004, Chinese J. Astron. Astrophys., 4, 320
  * Zhang & Gao (2020) Zhang, X., & Gao, H. 2020, MNRAS, 498, L1
  * Zhou et al. (2014) Zhou, P., Chen, Y., Li, X.-D., et al. 2014, ApJ, 781, L16