Kun Xu,Xiang-Dong Li,Zhe Cui,Qiao-Chu Li,Yong Shao,Xilong Liang,and Jifeng Liu,5

1School of Astronomy and Space Sciences,University of Chinese Academy of Sciences,Beijing 100049,China; xukun@smail.nju.edu.cn

2Key Laboratory of Optical Astronomy,National Astronomical Observatories,Chinese Academy of Sciences,Beijing 100101,China

3 Department of Astronomy,Nanjing University,Nanjing 210023,China

4 Key Laboratory of Modern Astronomy and Astrophysics,Nanjing University,Ministry of Education,Nanjing 210023,China

5 WHU-NAOC Joint Center for Astronomy,Wuhan University,Wuhan,China

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 has not 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 are in widely-separated 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.

Key words: stars: magnetars–stars: neutron–stars: rotation–(stars:) supergiants–accretion–accretion disks–X-rays: binaries

1.Introduction

Magnetars are thought to be a special population of neutron stars (NSs) with magnetic fields B ?1014G (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 (Kaspi & Beloborodov 2017; An & Archibald 2019;Beniamini et al.2019).However,low-field magnetars were found in observations (Rea et al.2010; Zhou et al.2014),and the dipole magnetic field of a few×1012G 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 Catalog6The url of the catalog is http://www.physics.mcgill.ca/～pulsar/magnetar/main.html.(Olausen & Kaspi 2014) with characteristic magnetic fields from～6×1012to～2×1015G.In addition,some central compact objects(CCOs)may be magnetar candidates,e.g.,the one located close to the center of RCW 103 (1E 161 348-5055) is thought to be a magnetar with B ?5×1015G (De Luca et al.2006;Tong et al.2016;Ho&Andersson 2017;Xu&Li 2019).All magnetars and candidates are probably isolated NSs (King & Lasota 2019).But in the theory of binary evolution,one cannot exclude the possibility that magnetars are in binaries.

So the question becomes,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 peak luminosity of M82 X-2 is～1040erg s-1,so its magnetic field has been estimated to be～1014G (Eksi et al.2015; Tsygankov et al.2016),while other estimation implies that its magnetic field is～1012–1013G (Dall’Osso et al.2015;Xu & Li 2017) or even lower (?109G,Kluzniak &Lasota 2015).Subsequently,more and more ULXPs were 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～1012G(Chen 2017).This is in accord with the current view that the magnetars’ ultrahigh fields decay on a timescale＜105–106yr by Hall draft and Ohmic diffusion (Turolla et al.2015).After decay,the magnetic fields of old magnetars (or low magnetic field magnetars,or post-magnetars) are on a 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(～2.6 hr)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～10 s by magnetic dipole radiation on a timescale of 104–105yr,then its magnetosphere started to interact with wind material from the companion and the NS spun down to～104s within 105yr.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 200 428 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.2021a; Tavani et al.2021).This indicates 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.,～16.35 days for FRB 180 916.J0158+65 (Chime/Frb Collaboration et al.2020) and a possible～160 days for FRB 121 102(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.2021b).Therefore,it also becomes a very interesting subject in the FRB field whether magnetars can exist in binary systems (Zhang & Gao 2020).Moreover,some γ-ray binaries are suggested to be magnetar binaries.Torres et al.(2012) reported a magnetar-like X-ray flare from another gamma-ray binary LS I+61° 303.Last year,Yoneda at al.(2020) showed that the gamma-ray binary LS 5039 maybe a magnetar binary,because the obtained spin period and its derivative are similar to those of magnetars.Additionally,the obtained spin-down luminosity is lower than its bolometric luminosity and its emissive energy should be supplied by the decay of the magnetic field of the magnetar.

Popov & Turolla (2012) modeled the formation channel for the NS in the Be/X-ray binary SXP 1062,which has a long spin period (Ps=1062 s) and short age,indicating that its initial magnetic field may be larger than 1014G and decayed to 1013G.Zhang et al.(2004) 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 freefall 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 windfed HMXBs.

In this paper,we perform the spin evolution of NSs in windfed HMXBs and compare the spin period 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 provided in Sections 4 and 5,respectively.

2.Model

2.1.Wind Accretion Regimes

We consider a wind-fed accretion binary system consisting of an NS of mass MNS=1.4 M⊙a(bǔ)nd a massive main sequence(MS) companion that does not fill its Roche lobe (RL).In general,the NS in a binary system is born with a short spin period (Ps～0.01–0.1 s),and then it can experience the following possible phases sketched in Figure 1.

Figure 1.Wind accretion regimes,see Section 2.1.

Figure 2.The magnetic field evolution with Bi=1012,Bi=1013,Bi=1014 and Bi=1015 G,where B＜1013 G remains unchanged in its life while Bi ≥1013 G decays in the form of Equation (15).

Figure 3.Results of the reference model in the Corbet diagram.The filled circles indicate the maximum spin periods Ps,max during the evolution process in each case.The green,blue,red and orange circles signify that the initial magnetic fields of the NSs are Bi=1012,Bi=1013,Bi=1014 and Bi=1015 G,respectively.The dashed lines string together the circles in the same color and the colored regions give the predicted Ps,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 respectively.

Figure 4.Results of the reference model with NSs’luminosity LX ＞1032 erg s-1 during the accretion phases.The gray filled and unfilled circles indicate the predicted maximum and minimum Ps of the NSs,respectively.Also,the gray regions give the predicted range of Ps of the NSs in each panel.From top-left to bottom-right panels,the magnetic fields are Bi=1012,Bi=1013,Bi=1014 and Bi=1015 G,respectively.There is no NS that enters accretion phases in the case of Bi=1012 G.

Figure 5.The spin evolution process of the reference model when Porb=10 days.The dotted,dashed,dotted–dashed and solid lines represent the evolution of NSs with magnetic field Bi=1012,Bi=1013,Bi=1014 and Bi=1015 G,respectively.In each line,the black,orange,green,red and blue parts indicate that the NSs are in phases a,b,c,d1 and d2 respectively.In phase c(the supersonic propeller phase)and d2(the subsonic settling accretion 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＜log t (yr) ＜7and the gray region covers the Pspin range of the NSs in HMXBs.

Figure 6.Same as Figure 3 but with the companion mass M2=10,20 (the reference model) and 30,from top to bottom respectively.

2.1.1.Phase a: The Pulsar-like Spindown Phase

where μ31and P0are the NS’s magnetic dipole moment in units of 1031G cm2and its spin period in units of 1 s,respectively.I=I45×1045g cm2is 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.,Rwind＜Reff; correspondingly the transitional spin periods Paband Pac(Davies & Pringle 1981) are

Here ρwis the stellar wind density at the orbit of the NS,which can be computed by assuming an isotropically expanding wind at a speed of vw

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.,Rg＜Rlcin this case and phase b occurs when Rwind＜Rlc,as well as P ＞Pabcommences before P ＞Pacif Pab＜Pac.The spin-down rate in phase b is (Zhang et al.2004)

When the outer boundary of the wind plasma reaches the Bondi radius (i.e.,Rwind=Rg),phase b stops at (Davies &Pringle 1981)

2.1.3.Phase c: The Supersonic Propeller Phase

When Rwind＜Rg,which is equivalent to P ＞Pbc,the supersonic propeller phase(phase c)arrives in the weak stellar wind case,while NSs will enter phase c directly from phase a if the stellar wind is strong enough(Illarionov&Sunyaev 1975),i.e.,phase c comes when Rwind＜Rgin the case of Rg＞Rlcas well as P ＞Pacoccurs before P ＞Pabif Pac＜Pab.The spindown rate in this phase is (Zhang et al.2004)

2.1.4.Phase d1 and Phase d2: The Accretion Phases

When the inner boundary of the wind material becomes smaller than the corotation radius(Rwind＜Rco)and the cooling is efficient,as indicated by Ps＞Peq,a bow shock can form at the Bondi radius around the NS and quasi-spherical accretion onto the NS occurs.The shocked matter falls toward 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＞while phase d2 occurs at(Shakura et al.2012),where(Arons & Lea 1976a,1976b;Elsner & Lamb 1977,1984).In phase d1,the accreted material transfers spin-up torque to the NS

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 binary systems.The simulation begins from the birth time of the NS when the optical star is thought to be in the zero age main sequence(ZAMS)stage,and ends when the optical/companion star starts to fill its RL or explodes as a supernova.

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

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)

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 M⊙NS and a 20 M⊙optical star with metallicity Z=Z⊙.The wind parameters of the companion η and β are set to be 1.0 and 0.8 respectively.Figure 3 features the results of the reference model in the Corbet diagram,where the filled circles indicate the maximum spin periodsPs,maxduring the evolution process in each case.From lower to upper,the green,blue,red and orange circles signify that the initial magnetic fields of the NSs are Bi=1012,Bi=1013,Bi=1014and Bi=1015G,respectively.From left to right in each color,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 days,respectively.The dashed lines string together the circles with the same color,predicting the possiblePs,maxwith the Porbranging from 3 to 165 days.The colored regions give the predictedPs,maxdistributions of the NSs with initial magnetic field between the upper and lower dashed lines.For example,the blue dashed line and region predict thePs,maxof the NSs with Bi=1013G and 1012＜Bi≤1013G.From this figure,one can ascertain 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 being captured and quickly lose their angular momentum (see Figure 5).There are significant drops in the blue and red dashed lines but not in the orange and green dashed lines in Figure 3,because cases with Bi=1012G and Bi=1015G end at phases a/b and d2,respectively,while cases with Bi=1013G and Bi=1014G 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 line.7This 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 do not discuss it in the following.

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

We compare our results with two subclasses of HMXBs with detected Psand Porb,which are the supergiant X-ray binaries(SgXBs) and the supergiant fast X-ray transients (SFXTs).The triangles and squares mark the SgXBs and SFXTs respectively.8Data from Martínez-Nú?ez et al.(2017) and Tauris et al.(2017).The SgXBs and SFXTs are two subclasses of HMXBs.SgXBs are persistent systems in X-rays (LX=1036–1038erg 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-Nú?ez et al.2017; Tauris et al.2017).However,SFXTs are not persistent,which stay in quiescence at LX=1032–1034erg s-1for most of the time (Romano et al.2015) and exhibit short outbursts lasting a few hours reaching 1037–1038erg s-1(Rampy et al.2009; Bozzo et al.2011).Figure 3 demonstrates that most of the observed HMXBs,which have Ps＜1000 s and Porb＜30 days,are distributed in the blue region,meaning that they are probably normal NSs.Five other sources with Ps＞1000 s or Porb＞30 days 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 can be seen as the separatrix between magnetars and other NSs.

Figure 4 displays the Psregions of NSs’ luminosity LX＞1032erg s-1during the accretion phases.The gray filled and unfilled circles indicate the predicted maximum and minimum Psof the NSs,respectively.The gray regions also give the predicted range of Psfor the NSs in each panel.From top-left to bottom-right panels,the magnetic fields are Bi=1012,Bi=1013,Bi=1014and Bi=1015G,respectively.In the case of Bi=1012G,no NS enters the accretion phase.

We plot the spin evolution process of the reference model with Porb=10 days in Figure 5.From lower to upper,the dotted,dashed,dotted–dashed and solid lines represent the evolution of NSs with magnetic field Bi=1012,Bi=1013,Bi=1014and Bi=1015G,respectively.In each line,the black,orange,green,red and blue parts indicate that the NSs are in phases a,b,c,d1 and d2,respectively.The right panel is the magnified part of the left panel with 5＜logt(y r) ＜7.

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

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

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 Bi=1016 G.

3.2.Parameter Study

The main configurable parameters in our model are the companion mass M2,and the wind parameters η and β.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 M2

We first change the mass of the companion M2.The results of maximum spin period with different companion mass are exhibited in Figure 6.From top to bottom,M2is taken to be 10 M⊙,20 M⊙(the reference model)and 30 M⊙respectively.It shows that more sources fall in the red and orange regions in the upper panel while less sources are 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 11.The distribution of magnetars and normal NSs in HMXBs predicted by our model.

3.2.2.The Wind Coefficient η

Figure 7 shows the dependence ofPmaxon the wind coefficient η,where η=0.5,1 (the reference model),2 and 3 from top-left to bottom-right.Compared with the reference model,we find that less sources are in the red and orange regions when η=0.5,while more sources are in these regions when η=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 β

Karino(2020)discussed the influence of parameter β within larger scale,where β=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 thePmaxresults with different values of wind power law index β.From top-left to bottomright,we take β=0.8 (the reference model),1,4 and 7.It shows that when β 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 β 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 Rlcto start the interaction with the wind material.

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 Porb=164.6 days and spin period Ps=186.78 s (Masetti et al.2006; Sidoli et al.2006,2007,2020).That means J11215 needs an initial magnetic field Bi＞1013G to explain its orbital and spin period,which implies that it is probably a magnetar.In a widely-separated 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 Rlcin 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 Porb=10.448 days and Ps=38.2 s and J18483-0311(hereafter J18483)with Porb=18.55 days and Ps=21 s.9The pulsation at 21 s was not confirmed in other observations and a reanalysis of the observations of the discovery showed the lack of any signal(Ducci et al.2013).We think these sources are or have been in the direct 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 parameter space in our model while J18483 needs a small η and massive companion.

4.1.3.4U 1954+31

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 be centered at the region where the spin periods and orbital periods are both short,while magnetars may be located at the region where either Psor Porbare long.As proposed in Li&van den Heuvel (1999),the reason is that a strong magnetic field makes an NS achieve a slower spin period in phase a by magnetic dipole radiation,as well as a larger effective radius,so that it is easy for the NS to capture the wind stellar material and it can enter the interaction phases (phase b,c,d1 and d2) earlier,then the interaction between the wind material and the magnetosphere could have a long time to spin the NS down.

5.Summary

In this work,we explore the possible parameter space for magnetars in HMXBs by performing spin evolution of NSs.We arrange and simplify the wind accretion regimes taking the direct and 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 widely-separated 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 be born as magnetars in HMXBs,then their magnetic fields decay to the order of normal NSs through a few Myr,but the spin periods record their evolution information and can provide some traces about their initial magnetic fields.

Acknowledgments

We thank the anonymous referee for helpful comments and suggestions.This work was supported by the Natural Natural Science Foundation of China (NSFC,Grant Nos.11933004,11988101,12041301,12063001,11773015),Project funded by China Postdoctoral Science Foundation No.2021M703168,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

Parameter Study of the Evolution of the Magnetic Field

Figure A1.The magnetic field(left)and spin(right)evolution tracks with different α.The gray region in the left panel covers the range of the estimated magnetic field and age of NSs in HMXBs from observations.

Figure A2.The magnetic field (left) and spin (right) evolution tracks with different τd.The gray region in the left panel covers the range of the estimated magnetic field and age of NSs in HMXBs from observations.

Figure A3.Same as Figure 3 but with α= 1,1.2,1.6 (the reference model) and 2 from top-left to bottom-right.

Figure A4.Same as Figure 3 but with τd=102,103(the reference model)and 104 yr,from top to bottom.

ORCID iDs