Evolution of neutron stars in high-mass X-ray binaries

V. Urpin,1;2* D. Konenkov2 and U. Geppert3

1 Department of Mathematics, University of Newcastle, Newcastle upon Tyne NE1 7RU2 A. F. Ioffe Institute of Physics and Technology, 194021 St Petersburg, Russia3 Astrophysikalisches Institut Potsdam, An der Sternwarte 16, D-14482 Potsdam, Germany

Accepted 1998 April 15. Received 1998 April 3; in original form 1997 October 14

A B S T R A C TWe consider the evolution of neutron stars during the X-ray phase of high-mass binaries.Calculations are performed assuming a crustal origin of the magnetic field. A strong wind fromthe companion can significantly influence the magnetic and spin behaviour of a neutron stareven during the main-sequence life of the companion. In the course of evolution, the neutronstar passes through four evolutionary phases (‘isolated pulsar’, propeller, wind accretion, andRoche lobe overflow). The model considered can naturally account for the observed magneticfields and spin periods of neutron stars, as well as the existence of pulsating and non-pulsatingX-ray sources in high-mass binaries. Calculations also predict the existence of a particular sortof high-mass binary with a secondary that fills its Roche lobe and a neutron star that does notaccrete the overflowing matter because of fast spin.

Key words: accretion, accretion discs – binaries: close – stars: magnetic fields – stars:neutron – pulsars: general – X-rays: stars.

1 I N T RO D U C T I O N

By means of X-ray observations, two main classes of binarysystems with neutron stars have been detected: high- and low-mass X-ray binaries (hereafter HMXBs and LMXBs, respectively).Although one component of both classes is a neutron star, theydiffer considerably in a number of characteristics (see e.g. Bhatta-charya & van den Heuvel 1991). For LMXBs, the companion is alow-mass star of mass <1.2 M(, while the companions of HMXBsare luminous early-type stars with masses within the range 5–40 M(. The spatial distributions of these two classes are also quitedifferent: HMXBs are concentrated towards the Galactic plane,characteristic of a relatively young population (stars with masses>10 M( do not live longer than ,3 × 107 yr). On the other hand,LMXBs are concentrated towards the Galactic Centre and aredistributed much more widely around the Galactic plane, typicalof an old stellar population (,1010 yr). The finding that LMXBs arevery old objects is supported by the observation of a number of themin globular clusters. Conclusions about the surface magnetic field ofthe neutron star in the binary system can be drawn from X-ray timevariability. While the regular X-ray pulses of many HMXBsindicate a strong field (1011:5¹1012:5 G, confirmed by the observa-tion of cyclotron lines in the hard X- and g-ray spectra), theoccurrence of X-ray bursts in LMXBs hints at their low magneticfield.

According to the evolutionary state of the companion, the orbitalperiods and mass transfer regimes, HMXBs can be divided into twodifferent classes (see e.g. Bhattacharya & van den Heuvel 1991):

standard massive X-ray binaries and Be/X-ray binaries. The stan-dard systems are characterized by a very massive (>20 M() evolvedcompanion which fills (or almost fills) its Roche lobe. The orbitalperiods of such systems range from 1 to 10 d. Mass transfer iscaused either by atmospheric Roche lobe overflow (in very closesystems, Porb < 5 d) or by a strong stellar wind in wider systems. Forboth transfer mechanisms, the accretion rate can be as high as10¹8¹10¹10 M( yr¹1. The typical lifetime of HMXBs as strongX-ray sources does not exceed 104¹105 yr (Savonije 1978); in thecase of accretion from a stellar wind it may be as low as 104 yr.

In Be/X-ray binaries, the companion star shows the emissionfeatures of a luminous early-type B star that is unevolved and deepinside its Roche lobe. The neutron star rotates in an eccentric, ratherwide orbit (Porb from 15 d up to years) around the rapidly rotatingBe star which loses mass by a steady stellar wind and/or by rotation-driven mass ejection from its equatorial region. Many Be/X-raybinaries show transient behaviour which is reflected by long quiet‘off’-periods (months to years) and relatively short ‘on’-periods(weeks to months) when the system manifests itself as a strongX-ray source. Accretion from the wind can produce X-ray lumi-nosities of ,1032¹1034 erg s¹1. For transient sources, however, thisluminosity can drastically increase (up to a value of ,1036¹

1038 erg s¹1, comparable to the X-ray luminosity of the standardsystems) during the outburst phase of the B star and especiallywhen, passing through periastron passage, the neutron star hits thematerial expelled from the Be star. The masses of the Be stars are ofthe order of 8–20 M(, on average smaller than those of thecompanions in standard X-ray binaries. This results in a longerlifetime of such systems, and their X-ray phase can be as long as,107 yr.

Mon. Not. R. Astron. Soc. 299, 73–77 (1998)

q 1998 RAS

* E-mail: vadim.urpin@newcastle.ac.uk

In this paper we consider the evolution of a neutron star in a high-mass binary. Interaction with plasma lost by the secondary may beessential in high-mass binaries even while the companion is deepinside its Roche lobe, because of the strong stellar wind frommassive stars (,10¹5¹10¹8 M( yr¹1). Perhaps accretion from thewind is insufficient to manifest the neutron star as a bright X-raysource during the main-sequence life of the companion, but thisaccretion may nevertheless influence the magnetic and spin behav-iour of the neutron star. Of course, when the companion fills (oralmost fills) its Roche lobe and the system becomes a standardHMXB, the rate of mass transfer increases drastically and the effectof accretion becomes more appreciable.

Observations provide information on the magnetic field and spinperiod of neutron stars in HMXBs at different evolutionary stages.The measured periods lie within the range 0.07–835 s (Bhatta-charya & van den Heuvel 1991), much wider than that of isolatedradio pulsars. Cyclotron lines have been observed in the spectra ofeight stars (White, Nagase & Parmar 1995). The field strengthinferred from these lines ranges from 1012 to 3:4 × 1012 G. Thesefields seem to be weaker than the mean field of radio pulsars(,4 × 1012 G), particularly if we take into account the fact thatcyclotron lines provide an estimate of the polar field that is a factorof 2 higher for the dipole configuration than the equatorial fieldinferred from P and P for radio pulsars. A comparison of these datawith evolutionary calculations may be a good test of the consistencyof theoretical models.

In the present paper we treat the evolution of a neutron star withthe magnetic field maintained by currents in the crust. This modelfits well the behaviour of isolated radio pulsars (Urpin & Konenkov1997) and neutron stars in low-mass binary systems (Urpin,Geppert & Konenkov 1998). It will be shown that the characteristicsof neutron stars in HMXBs may well be understood in the frame-work of this model also.

The paper is organized as follows. In Section 2 we describe theadopted model. The results of evolutionary calculations are pre-sented in Section 3. In Section 4, we give a brief summary of ourresults.

2 D E S C R I P T I O N O F T H E M O D E L

The main-sequence lifetime of a massive star, tms, is approximatelygiven by

tms < 1010ðMc=M(Þ¹2:5 yr; ð1Þ

where Mc is the stellar mass (see e.g. Bhattacharya & van denHeuvel 1991). In high-mass binaries, the mass of a companion istypically $10 M(, and hence the main-sequence evolution lasts notlonger than 3 × 107 yr. Within this time the companion loses itsmass as a result of the stellar wind. The empirical expression for therate of mass loss of O and B stars, Mc, has been derived by Lamers(1981) who analysed more than 50 early-type stars within the massrange 15–123 M(. This expression has the form

log Mc ¼ ¹ 4:83 þ 1:42 logðLc=106 L(Þ

þ 0:61 logðRc=30 R(Þ ¹ 0:99 logðMc=30 M(Þ; ð2Þ

where Lc and Rc are the luminosity and radius, respectively. Fortypical O and B stars with 40 $ Mc $ 10 M(, the rate of mass lossranges from 10¹6 to 10¹9 M( yr¹1, and a substantial fraction of thisplasma can be gravitationally captured by the neutron star. Theradius of gravitational capture is RG < 2GM=V2

w (Bondi 1952),where M is the neutron star mass and Vw is the wind velocity

(we assume that Vw is much larger than the sound speed of thewind plasma and the velocity of orbital motion). Typically,Vw , 1000 km s¹1 for massive stars, therefore RG < 4 × 1010 cmfor M ¼ 1:4 M(. Then the rate of gravitational capture is

MG < McðRG=2aÞ2; ð3Þ

where a is the separation between the stars. For a massive binarywith Mc > M, we have

RG

2a

� �2

< 10¹3 Mc

10 M(

� �¹2=3 Porb

1 d

� �¹4=3

; ð4Þ

where Porb is the orbital period. Note that for Be stars the velocity ofthe wind may be lower (Vw , 107 cm s¹1) than for O and B giantsbecause of the specific mass loss mechanism (see e.g. Nagase 1989)and, owing to this, the fraction of captured plasma may be evenhigher. For binaries with an orbital period of a few days, the neutronstar may capture ,ð10¹3¹10¹6ÞMc and accrete (when accretion isallowed) ,10¹9¹10¹13 M( yr¹1 even during the main-sequenceevolution of the companion. This accretion rate is an order ofmagnitude lower, at least, than the Eddington accretion rate, andtherefore the majority of high-mass binaries cannot be bright X-raysources prior to the Roche lobe phase. Nevertheless, such accretionmay have an appreciable effect on the magnetic and spin evolutionof a neutron star.

The neutron star passes through different evolutionary stages in abinary system. Following Illarionov & Sunyaev (1975), thesephases are as follows.

(1) Phase I: the ‘isolated pulsar’ phase in which the radiativepressure is sufficient to keep the plasma of the stellar wind awayfrom the neutron star magnetosphere; the evolution of a neutron starfollows exactly that of isolated stars.

(2) Phase II: the propeller phase in which the wind plasmainteracts with the magnetosphere but the spin is still sufficientlyfast to accelerate the infalling matter outward; the thermal andmagnetic evolution of the neutron star is the same as for an isolatedstar, but spin-down is more efficient.

(3) Phase III: the wind accretion phase in which accretion of thewind plasma is allowed; nuclear burning of the accreted materialheats the neutron star (Zdunik et al. 1992) and causes accretion-driven field decay (Urpin & Geppert 1995); the neutron star canexperience spin-up during this phase, owing to the angular momen-tum carried by the accreted material.

(4) Phase IV: the enhanced accretion phase which stars when thecompanion fills (or almost fills) the Roche lobe; the neutron star isheated to a very high temperature, ,ð2¹5Þ × 108 K (see Fujimotoet al. 1984; Brown & Bildstein 1998); the accretion-driven fielddecay and spin-up are much more efficient than during phase III.

Calculations have been done with the numerical code describedin detail by Urpin et al. (1998). We consider the evolution for a1.4-M( neutron star model with the equation of state of Pandhari-pande and Smith (PS; see e.g. Pandharipande & Smith 1975) andwith the standard cooling scenario. Our choice is motivated by thefact that only models with sufficiently stiff equations of state (stifferthan that of Friedman and Pandharipande) and with standard cool-ing seem to be suitable to account for the available observationaldata on the magnetic evolution of isolated pulsars (Urpin &Konenkov 1997). For the PS model, the radius and crust thicknessare 16.4 km and 4200 m, respectively; we assume the crust bottomto be located at a density of 2 × 1014 g cm¹3.

Unfortunately, our numerical code does not allow us to model ahighly variable evolution of transient Be/X-ray binaries. Therefore

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q 1998 RAS, MNRAS 299, 73–77

the calculations presented here concern only standard HMXBs andthose of Be/X-ray systems that do not exhibit transient behaviour.We also do not consider the late evolution of a neutron star whenaccretion owing to Roche lobe overflow is exhausted. Probably, thefurther evolution leads to a common envelope phase (van denHeuvel & Bitzaraki 1995), which requires a very refined considera-tion of the thermal effects. We are planning to address this problemin a forthcoming paper.

3 N U M E R I C A L R E S U LT S

The evolution of the crustal magnetic field is sensitive to the initialdepth penetrated by the field. We assume the magnetic field to beinitially confined to the outer layers of the crust with densitiesr # r0. Calculations are performed for r0 within the range 1011¹

1013 g cm¹3. These values correspond to a depth from the surface of<620 to <1100 m. The choice of r0 is imposed by a comparison ofthe calculated evolutionary tracks of isolated neutron stars with theavailable observational data on P and P for radio pulsars (see Urpin& Donenkov 1997).

The magnetic evolution depends also on the conductive proper-ties of the crust, which are determined (see Yakovlev & Urpin 1980)by scattering of electrons on phonons (at a high temperature andrelatively low densities) and on impurities (at a low temperature andhigh densities). The impurity conductivity depends on the so-calledimpurity parameter, Q. The exact value of Q is uncertain, but it isusually assumed that Q , 0:1¹0:001. Note that, in HMXBs, avariation of Q shows no appreciable effect since the crustaltemperature is sufficiently high during the whole evolution andthe conductivity is mainly determined by scattering on phonons.

The initial field strength and spin period are assumed to be moreor less standard for young pulsars, B0 ¼ 1012¹1013 G and P0 ¼

0:01 s, respectively. Note, however, that the results are not sensitiveto a variation of P0. The accretion rate may vary within a widerange during the life of the neutron star. We use the same values(¼ 10¹9¹10¹13 M( yr¹1) for both the wind accretion rate and therate of gravitational capture during the propeller phase. Duringphase IV, the accretion rate is assumed to be higher, M ¼ 10¹8¹

10¹9 M( yr¹1.In our numerical code, the spin evolution during phases III and IV

is determined by the efficiency parameter, z . The amount of angularmomentum carried by accreted matter can be estimated from thecondition at the magnetospheric boundary assuming that the angu-lar momentum is characterized by its Keplerian value multiplied bythe factor z. This factor is ,1 if the accreted matter forms aKeplerian disc outside the neutron star magnetosphere. If the discis not formed then the efficiency of spin-up is much lower and theparameter z is probably of the order of 0.1–0.01.

In Fig. 1 we plot evolutionary tracks for the model that representsneutron stars in non-transient Be/X-binaries. We assume tms ¼

107 yr, Vw ¼ 107 cm s¹1, r0 ¼ 1013 g cm¹3 and Q ¼ 0:01. Theduration of the ‘isolated pulsar’ phase is very sensitive to the initialfield strength. For B0 ¼ 1013 G, this phase is extremely short(#3 × 105 yr) because of fast spin-down. In the case of a weakinitial field (B0 ¼ 1012 G), the spin-down rate is lower, thus radia-tion can prevent the wind plasma interacting with the neutron starmagnetosphere for longer. During phase I the magnetic fieldexperiences only minor changes, whereas the spin period canincrease by more than two orders of magnitude in some models.Phase II is more extended and can sometimes last until the end of themain-sequence life of the companion if the initial field is weak(#1012 G) and the accretion rate is low (#10¹13 M( yr¹1). Thus, in

some Be binaries, the neutron star may not pass through the X-rayphase at all (except, maybe, for relatively short outburst periods).The magnetic field at the end of the propeller phase is only a factorof ,3 weaker than the initial field for all models considered. Incontrast, the spin period experiences the most dramatic changesduring this phase. Owing to an efficient angular momentum loss, theneutron star spins down to very long periods ,100¹300 s. Themaximum period reached during phase II is longer for a neutron starwith a stronger initial magnetic field and a lower accretion rate.Note, however, that for very low accretion rates the neutron stardoes not have enough time to reach the maximum period during themain-sequence evolution of the companion. In the case B0 ¼

1013 G, the longest period is ,500 s, which the neutron star reachesat M , 10¹13 M( yr¹1. For B0 ¼ 1012 G, the longest period isshorter, ,30 s, and it can be reached at a higher accretion rate,M , 3 × 10¹12 M( yr¹1.

Owing to efficient propeller action, the neutron star enters thewind accretion phase with a long period but still with a strongmagnetic field, ,B0=3. This field seems to be sufficient to channelaccretion to the magnetic poles, and thus the neutron star canmanifest itself as a long-period pulsating X-ray source at thebeginning of phase III. However, nuclear burning of the accretedmaterial heats the neutron star and accelerates the accretion-drivenfield decay. For the maximal accretion rate, M ¼ 10¹10 M( yr¹1,the field strength at the end of the main-sequence evolution isB , 1011 and ,1010 G for B0 ¼ 1013 and 1012 G, respectively.These values are relatively low and, perhaps pulsations of X-rayemission may be smoothed (or even unobservable) at the end of theevolution, particularly for the case B0 # 1012 G. For a loweraccretion rate, the decay is less appreciable, and the magneticfield can channel accretion to the poles up to the end of the windaccretion phase. The angular momentum carried by the windplasma can spin up the neutron star. The final spin period after107 yr of evolution depends on the accretion rate and the initial fieldstrength. This period may be short (,0:3¹3 s) for weakly magne-tized stars with B0 # 1012 G, but it may also be rather long

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q 1998 RAS, MNRAS 299, 73–77

Figure 1. The B–P tracks of a neutron star during the main-sequenceevolution of the companion: tms ¼ 107 yr, Vw ¼ 107 cm s¹1, r0 ¼

1013 g cm¹3, Q ¼ 0:01, z ¼ 0:1. In the case of a strong initial magneticfield (B0 ¼ 1013 G), tracks are calculated for M ¼ 10¹10 (curve 1), 10¹11 (2)and 10¹12 M( yr¹1 (3); in the case of a weak field (B0 ¼ 1012 G) forM ¼ 10¹10 (4) and 10¹11 (5). Numbers near the tracks indicate the logarithmof the phase transition time. Filled circles mark the ends of tracks.

(,10¹100 s) for strongly magnetized stars that experience accre-tion at a low rate. Note that an increase of the efficiency parameter,z, from 0.1 to 1 yields the same evolution. The efficiency of angularmomentum transfer has to be reduced much more (z # 0:001) inorder to find a deviation from the evolution represented in Fig. 1.

An example of the evolution for a neutron star in a standard high-mass binary is shown in Figs 2 and 3. The main-sequence lifetime ofthe companion is 2 × 106 yr (corresponding to a star with a mass<30 M(). The calculations of phase IVare extended to 105 yr, moretypical of atmospheric Roche lobe overflow (see Bhattacharya &van den Heuvel 1991). The duration of phase IVis shorter (,104 yr)if accretion is caused by a strong wind; however, the results for ashorter accretion phase can be easily obtained from Fig. 2. Theaccretion rate during phase IV is assumed to be 10¹8 M( yr¹1. Suchstrong accretion heats the neutron star interior to <5 × 108 K (seeBrown & Bildstein 1998). The wind velocity may be rather high instandard systems, and therefore we assume Vw ¼ 108 cm s¹1.

Fig. 2 shows the temporal behaviour of the magnetic field andspin period for initially strongly (B0 ¼ 1013 G) and weakly (B0 ¼

1012 G) magnetized neutron stars. The difference in the wind

velocity compared with Fig. 1 results only in a slightly laterphase transition I–II for standard binaries. Owing to the shortermain-sequence evolution, however, the neutron star cannot passthrough the wind accretion phase for some models with a lowaccretion rate. For example, this phase is missed for B0 ¼ 1013 G ifM # 10¹12 M( yr¹1, and for B0 ¼ 1012 G if M # 10¹11 M( yr¹1.When Roche lobe overflow starts, the field ranges from 2 × 1012 to3 × 1011 G for a strongly magnetized star, and from 2 × 1011 to4 × 1010 G for a weakly magnetized star. It is possible that a field of#3 × 1011 G cannot channel accretion with M ¼ 10¹8 M( yr¹1 tothe magnetic poles (see e.g. Taam & van den Heuvel 1986), thus theneutron star with B0 , 1012 G will not exhibit pulsations of X-rayemission during the Roche lobe phase. In contrast, if the initial fieldis strong, B0 , 1013 G, the neutron star can maintain a sufficientlystrong field and channel Roche lobe accretion to the poles duringthe whole of phase IV. Note that the field can be reduced only by afactor of #2¹3 for all considered models because of the shortduration of this phase.

The spin period increases by factors of ,100 and ,10, respec-tively, during phase I for B0 ¼ 1013 and 1012 G. An efficientpropeller action leads to a further spin-down of the neutron star.For the model with a strong initial field, a maximum period of,300 s can be reached for relatively weak wind accretion withM , 10¹13 M( yr¹1. For a weakly magnetized star, the spin istypically faster, and the maximum period is ,6¹8 s. Note that themaximum spin period may be longer in binaries with less massivecompanions and more extended main-sequence evolution. Whenthe secondary fills its Roche lobe, the neutron star experiences avery fast spin-up in the case of near-Eddington mass transfer, andthus it can be a slowly rotating pulsating X-ray source only for avery short time. For a low accretion rate, spin-up to the correspond-ing spin-up line may take longer.

In Fig. 3, we plot the evolutionary tracks of the models presentedin Fig. 2. The main difference compared with Fig.1 is a weaker fielddecay caused by a shorter main-sequence lifetime. The mostremarkable behaviour is shown by model 5. We believe that thissort of behaviour may be typical of all weakly magnetized stars instandard binaries. Model 5 has an initial field of 1012 G which isweakened to <2 × 1011 G after phase I. This field is relatively weakfor an efficient propeller action, therefore spin-down proceeds veryslowly during phase II, and the neutron star does not pass throughphase III. At the end of main-sequence evolution, the spin period is<0:3 s. When Roche lobe overflow starts, the plasma of thecompanion forms a disc around the neutron star. However, thespin angular velocity is larger than the Keplerian angular velocity atthe Alfven radius even for near-Eddington mass transfer. Thetransferred matter cannot penetrate through the centrifugal barrier,and accretion on to the neutron star is prohibited. Therefore, at thebeginning of Roche lobe overflow, the neutron star continues towork as a propeller and to eject matter from the disc. This ‘discpropeller phase’ lasts approximately 8 × 103 yr, while the spinperiod is shorter than the corresponding corotation period(<0:67 s). When the spin slows down to corotation, accretion isallowed and the neutron star can manifest itself as a bright X-raysource. However, the field is rather low when the disc propelleraction ends (< 2 × 1011 G), thus this model should exhibit a non-pulsating behaviour. It appears that such evolution may be typical ofall weakly magnetized neutron stars if the wind accretion rate is#10¹11 M( yr¹1 during the main-sequence life of the companion.

Since the initial depth penetrated by the field is quite uncertain, inFig. 4 we plot for comparison the evolutionary tracks for r0 ¼

1012 g cm¹3. The evolution is qualitatively similar to that plotted in

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Figure 2. The magnetic (solid lines) and spin (dashed lines) evolution of aneutron star with tms ¼ 2 × 106 yr, r0 ¼ 1013 g cm¹3, Q ¼ 0:01 and z ¼ 0:1.Panels (a) and (b) show the evolution during the main sequence life of thecompanion and during the accretion phase, respectively. In the case of astrong initial field (B0 ¼ 1013 G), tracks are calculated for M ¼ 10¹10 (curve1), 10¹11 (2) and 10¹12 M( yr¹1 (3); in the case of a weak initial field(B0 ¼ 1012 G) for M ¼ 10¹10 (4) and 10¹11 M( yr¹1 (5).

Figure 3. Evolutionary tracks of the models represented in Fig. 2.

Fig. 3. The only difference is faster decay caused by lowerconductivity in less dense layers and, as a consequence, morerapid spin during the whole of the evolution. Model 5 with aweak initial field again exhibits the presence of the ‘disc propeller’phase. At the end of the main-sequence life of the companion, thespin period and the magnetic field of this model are <0:13 s and<1011 G, respectively, thus accretion is prohibited because of thecentrifugal barrier and the neutron star works as a propeller. The‘disc propeller’ phase is relatively long, <1:5 × 104 yr, and cangenerally be comparable to the duration of the Roche lobe overflow,thus accretion does not start at all in some binaries despite thesecondary filling its Roche lobe.

4 C O N C L U S I O N S

We have explored the magneto-rotational evolution of neutron starsthat are components of X-ray binaries with a high-mass companion.The evolution is modelled in accordance with the generallyaccepted scenario which implies that the neutron star passesthrough several evolutionary phases (‘isolated’ pulsar, propeller,wind accretion, and Roche lobe overflow). The evolution has beenconsidered for a neutron star model with a crustal magnetic field.This model seems to be able to account for the magneto-rotationalevolution of isolated neutron stars (Urpin & Konenkov 1997) andthose in LMXBs (Urpin et al. 1998). The results of the present studyshow that this model gives also a good quantitative agreement withthe observed properties of neutron stars in high-mass binaries.

Since the lifetime of high-mass binaries is typically shorter thansome 107 yr and the total amount of accreted mass is not very large(#10¹3 M(), the evolution of a neutron star in such a system differsqualitatively from that in a low-mass binary. For instance, spin-up tovery short periods, as for millisecond pulsars, is certainly impos-sible for neutron stars in high-mass binaries. Since accretion fromthe wind is much stronger in high-mass systems, the accretion-driven mechanism can accelerate the field decay even during phaseIII. Thus the surface field strength can be reduced by a factor of ,30during the main-sequence life of the companion if the wind issufficient to provide an accretion rate of ,10¹10 M( yr¹1. Ourcalculations show that a neutron star starting its life with a more orless standard field of ,1013 G can maintain B , 3 × 1011¹3 ×1012 G until the end of the X-ray phase. These values agree wellwith the measured field strength in HMXBs (White et al. 1995).Such fields are sufficient to channel the accreted matter to the

magnetic poles and to produce regular X-ray pulsations. In contrast,the field of an initially weakly magnetized neutron star(B0 # 1012 G) can be reduced to a value of ,4 × 1010¹2 × 1011 G,insufficient to channel plasma during the Roche lobe phase. Thusour model explains naturally the existence of pulsating and non-pulsating X-ray sources in high-mass binaries (see e.g. Bhatta-charya & van den Heuvel 1991).

The model considered can also account for a wide range ofneutron star spin periods observed in high-mass binaries. Depend-ing on the initial field strength and the rate of mass loss, the neutronstar can spin down to P , 102¹103 s during the propeller phase. Incontrast, accretion can accelerate rotation to P , 0:1 s duringphases III and IV, thus the range of periodicity of HMXBs shouldbe very wide, 0.1–1000 s.

Our model predicts the existence of a particular sort of objectamong high-mass binaries: systems with a evolved companionfilling its Roche lobe and a non-accreting neutron star. In binarieswith relatively weak wind accretion, the neutron star spins downvery slowly even during the propeller phase, B0 # 1012 G. When thecompanion ends its main-sequence evolution and fills the Rochelobe, the overflowing matter probably forms an accretion discaround the neutron star. However, the spin period of a weaklymagnetized neutron star can be shorter than is required by thecorotation condition at the magnetospheric boundary. Thus accre-tion turns out to be forbidden and the neutron star continues to workas a propeller. This ‘disc propeller’ phase seems to be unavoidablefor neutron stars with an initial magnetic field that is not very strong.The duration of this phase (,104 yr) may be comparable to theduration of the Roche lobe phase. Thus, for some systems, enhancedaccretion does not start at all. If Roche lobe overflow lastssufficiently long to spin down the neutron star, then accretion isallowed and such a system spends the rest of the Roche lobe phaseas a typical HMXB.

AC K N OW L E D G M E N T

This work has been supported by the Russian Foundation of BasicResearch (Grant 97-02-18086).

REFERENCES

Bhattacharya D., van den Heuvel E. P. J., 1991, Phys. Rep., 203, 1Bondi H., 1952, MNRAS, 112, 195Brown E., Bildstein L., 1998, ApJ, 496, 915Fujimoto M., Hanava T., Iben I., Jr, Richardson M., 1984, ApJ, 278, 813Illarionov A., Sunyaev R., 1975, A&A, 75, 18Lamers H., 1981, ApJ, 245, 593Nagase F., 1989, PASJ, 41, 1Pandharipande V. R., Smith R., 1975, Nucl. Phys. A, 237, 507Savonije G. J., 1978, A&A, 62, 317Taam R. E., van den Heuvel E. P. J., 1986, ApJ, 305, 235Urpin V., Geppert U., 1995, MNRAS, 275, 1117Urpin V., Konenkov D., 1997, MNRAS, 292, 167Urpin V., Geppert U., Konenkov D., 1998, MNRAS, 295, 907van den Heuvel E. P. J., Bitzaraki O., 1995, in van Paradijs J., Alpar A., eds,

The Lives of Neutron Stars. Kluwer, Dordrecht, p. 421White N. E., Nagase F., Parmar A. N., 1995, in Lewin W. H. G.,

van Paradijs J., van den Heuvel E. P. J., eds, X-ray binaries. CambridgeUniv. Press, Cambridge, p. 1

Yakovlev D., Urpin V., 1980, SvA, 24, 303Zdunik J., Haensel P., Paczynski B., Miralda-Escude J., 1992, ApJ, 384, 129

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Figure 4. As Fig. 2 but for r0 ¼ 1012 g cm¹3.