Magnetic and spin evolution of neutron stars in close binaries

© 1998 RAS

Mon. Not. R. Astron. Soc. 295, 907–920 (1997)

Magnetic and spin evolution of neutron stars in close binaries

V. Urpin,1,2 U. Geppert3 and D. Konenkov1

1A. F. Ioffe Institute of Physics and Technology, 194021 St Petersburg, Russia2 Department of Mathematics, University of Newcastle, Newcastle upon Tyne NE1 7RU3 Astrophysikalisches Institut Potsdam, An der Sternwarte 16, D-14482 Potsdam, Germany

Accepted 1997 November 28. Received 1997 July 14; in original form 1997 May 7

A B STR ACTThe evolution of neutron stars in close binary systems with a low-mass companion isconsidered, assuming the magnetic field to be confined within the solid crust. Weadopt the standard scenario for the evolution in a close binary system, in which theneutron star passes through four evolutionary phases (‘isolated pulsar’ – ‘propeller’– accretion from the wind of a companion – accretion resulting from Roche-lobeoverflow). Calculations have been performed for a great variety of parameterscharacterizing the properties of both the neutron star and the low-mass companion.We find that neutron stars with more or less standard magnetic field and spin periodthat are processed in low-mass binaries can evolve to low-field rapidly rotatingpulsars. Even if the main-sequence life of a companion is as long as 1010 yr, theneutron star can maintain a relatively strong magnetic field to the end of theaccretion phase. The model that is considered can account well for the origin ofmillisecond pulsars.

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

1 INTRODUCTION

In recent years, evidence has been obtained for a relativelyfast magnetic field decay in neutron stars undergoing accre-tion in binary systems (see, e.g., Taam & van den Heuvel1986 and Chattacharya & van den Heuvel 1991). The weaksurface magnetic fields of many binary radio pulsars(VR109 G), for which efficient accretion has taken place,strongly suggest the idea of the accretion-induced magneticfield decay in these neutron stars. Most of the low magneticfield pulsars are believed to be old neutron stars that haveexperienced mass transfer, in low-mass binary systemswhere accretion results from Roche-lobe overflow and maylast for a relatively long period.

Recently Geppert & Urpin (1994) and Urpin & Geppert(1995) have suggested a simple and efficient mechanism ofrapid field decay in neutron stars undergoing accretion.Accretion changes the thermal evolution of the neutron stardrastically if the accretion rate is sufficiently high and masstransfer lasts a sufficiently long time (see, e.g., Fujimoto etal. 1984). The energy release resulting from accretion heatsthe neutron star and reduces the crustal conductivity,because the latter depends on the temperature. A decreaseof the conductivity accelerates the ohmic decay of the mag-

netic field if this field is maintained by currents in the crust.If the mass transfer phase lasts as long as 106–107 yr thesurface field strength can be reduced by a factor 1102–104

depending on the magnetic configuration at the beginningof the accretion phase.

Accretion influences not only the field decay also but thespin evolution of pulsars, and the strength of the magneticfield is one of the dominating factors that determines thisevolution. If the total amount of accreted matter is largeenough (E0.01–0.1 M>), the neutron star in a low-massbinary can be ‘recycled’ to millisecond periods (Alpar et al.1982).

The accretion phase, however, does not exhaust all pos-sible stages in the evolution of neutron stars entering low-mass binary systems. The evolution of such systems isextremely long, because even the main-sequence lifetime ofa low-mass companion exceeds 109 yr. In accordance withthe standard scenario, the neutron star in a close binary canpass throughout several evolutionary phases (see, e.g., Prin-gle & Res 1972 and Illarionov & Sunyaev 1975).

(i) The initial obscured radio pulsar phase, in which thepressure of the pulsar radiation is sufficient to keep plasmafrom the stellar wind of the companion away from the

neutron star magnetosphere. The radio emission of thepulsar is partly (or completely) absorbed by the plasmacloud surrounding the binary system, so the pulsar is prac-tically unobservable; the pulsars spin down as a result ofmagnetodipole radiation.

(ii) The propeller phase, in which the radiation pressure,which is reduced because of field decay and spin-down,cannot prevent the infalling matter from interaction withthe magnetosphere. However, the spin rotation is still suffi-ciently rapid to eject the wind matter as a result of centri-fugal force and to transfer the angular momentum from theneutron star to the wind matter.

(iii) The wind accretion phase, in which the matter fromthe wind falls on to the surface of the neutron star. Nuclearburning of the accreted material heats the neutron star andcauses accretion-driven field decay; in the course of thisphase the neutron star can spin up slightly, because theaccreted matter carries a certain amount of angularmomentum.

(iv) The enhanced accretion phase, which starts whenthe secondary star leaves the main-sequence and fills itsRoche lobe. Vigorous mass transfer on to the neutron starheats the interior to a very high temperature, 1108–108.5 K(Fujimoto et al. 1984; Miralda-Escude, Haensel & Paczyn-ski 1990), and accelerates the accretion-driven field decay.A steady Keplerian disc is formed by the accretion flowoutside the neutron star magnetosphere, and the accretiontorque spins up the neutron star to a very short period.

The neutron star processed in the above evolutionarypicture can probably prolong its life as a radio pulsar withrelatively low magnetic field and short spin period afteraccretion resulting from Roche-lobe overflow is exhausted.The properties of pulsars produced by evolution in low-mass binaries have not been analysed in detail yet, and weaddress this problem in the present paper. Note that a simi-lar problem has been considered by Jahan Miri & Bhatta-charya (1994), who assumed that the magnetic evolution isdetermined by the expulsion of the field lines from theneutron star core because of interaction between thefluxoids in the proton superconductor and the vortices inthe neutron superfluid. The authors, however, restrictedthemselves to the case of wide low-mass binaries and con-sidered the evolution only during the main-sequence life-time.

The paper is organized as follows. We describe theadopted model in detail in Section 2. The results of calcu-lations are presented in Section 3. In Section 4, we discussour results and compare them with the available observa-tional data on the evolution of neutron stars in low-massbinaries.

2 DESCR IPTION OF THE MODEL

We consider the evolution of neutron star in a binary systemwith the secondary being a low-mass star. The binary isassumed to be relatively close; thus the companion can fillits Roche lobe after the end of its main-sequence life. Bothmagnetic and spin evolution of the neutron star may bestrongly influenced by accretion from the companion.During its main-sequence evolution [which lasts as long as(1–5)Å109 yr], the secondary loses mass because of the

stellar wind, and a certain fraction of the wind plasma cangenerally be captured and accreted by the neutron star.After the secondary leaves the main sequence and its Rochelobe, the rate of mass loss and hence the accretion rate canbe drastically enhanced.

We assume that the magnetic field of the newly bornneutron star is created by an unspecified mechanism eitherin the course of collapse or soon after the neutron star isborn. We also assume that the field is maintained by cur-rents in the crust and originally occupies only a fraction ofthe crust volume. The evolution of the crustal field is deter-mined by the conductive properties of the crust andmaterial motion throughout it (see Sang & Chanmugam1987 and Urpin & Geppert 1995 for details). Neglecting theanisotropy of conductivity, the induction equation govern-ing the evolution of the magnetic field reads

qB

qt\Ð

c2

4pHÅÅ 21s

HÅÅB3+HÅÅ (vÅÅB), (1)

where s is the conductivity and v is the velocity of materialmovement. The velocity v is caused by the flux of theaccreted matter throughout the crust and may be non-zeroonly during those evolutionary phases at which accretion onto the neutron star surface is allowed. The total amount ofthe accreted mass in most cases does not exceed 0.1–0.2 M>

(see Bhattacharya & van den Heuvel 1991), even forneutron stars undergoing a very extended accretion phase.Therefore, the mass and radius of the neutron star do notchange significantly and hence the corresponding changesin the crustal thickness, density profile, etc., are also negli-gible. The total mass flux is the same throughout the crustand should be taken to be equal to the accretion rate.Assuming spherical symmetry of the mass flow in deep crus-tal layers, one obtains the radial velocity of materialmovement.

v\M

4pr2r, (2)

where M is the accretion rate and r is the density. Thisinward-directed flow tries to push the magnetic field intothe deep crustal layers and can generally influence the evo-lution of the field at a high accretion rate.

Introducing the vector potential for a dipole magneticfield A\(0, 0, Ah), where Ah\s(r, t) sin y/r and (r, y, h) arespherical coordinates, the induction equation (1) can betransformed to

qs

qt\

c2

4ps 2q2s

qr2Ð

2s

r23Ðvqs

qr. (3)

At the surface r\R, the standard boundary condition for adipole field should be fulfilled, i.e. Rqs/qr+s\0. Weassume the core of the neutron star to be superconductive,so that the magnetic field cannot penetrate into the core.Therefore, the second boundary condition is s\0 at thecrust–core boundary, which lies at the density 2Å1014 gcmÐ3.

The conductive properties of the solid crust are deter-mined by the scattering of electrons on phonons and impur-ities (see, e.g., Yakovlev & Urpin 1980). Scattering on

908 V. Urpin, U. Geppert and D. Konenkov

© 1998 RAS, MNRAS 295, 907–920

phonons dominates the transport processes at relativelyhigh temperatures and low densities, whereas scattering onimpurities gives the main contribution to the conductivity atlow temperatures and large densities. The total conductivityof the crystallized region is given by

1

s\

1

sph

+1

simp

, (4)

where sph and simp are the conductivities resulting fromelectron–phonon and electron–impurity scattering mech-anisms, respectively. The phonon conductivity depends onthe temperature, decreasing when T increases, and henceheating caused by accretion accelerates the field decay. Theimpurity conductivity is practically independent of the tem-perature, but its magnitude is determined by the so calledimpurity parameter,

Q\1

ni

+np

np(ZÐZp)2, (5)

where the dominant background ion species has the numberdensity ni and charge Z, and np is the number density of animpurity species of charge Zp; the summation is over allspecies of impurities. In our calculations, we use thenumerical data for the phonon conductivity obtained byItoh, Hayashi & Kohyama (1993) and a simple analyticalexpression for the impurity conductivity derived by Yakov-lev & Urpin (1980). As the conductivity depends on thetemperature, the magnetic evolution of a neutron star isstrongly sensitive to its thermal history. According to thescenario suggested in Section 1, we divide the evolution intofour essentially different phases:

(i) the ‘isolated pulsar’ phase (phase I), in which theneutron star scarcely feels the influence of its companion,.

(ii) the ‘propeller’ phase (phase II), in which, because offast spin rotation, the neutron star ejects the infalling windmatter,

(iii) the wind accretion phase (phase III), in which accre-tion of the wind plasma is allowed, and

(iv) the accretion phase (phase IV), in which the second-ary fills its Roche lobe and the neutron star experiencesvigorous mass transfer.

Evidently, during phases I and II the thermal evolution ofthe neutron star in a binary system exactly follows that of anisolated neutron star. For the time dependence of the tem-perature during these two phases, we use the standard cool-ing model given by Van Riper (1991) for a 1.4-M> neutronstar with the Pandharipande–Smith equation of state.During phases III and IV, however, the neutron star can besubstantially heated by accretion. The thermal evolution ofaccreting neutron stars has been a subject of study of severalpapers (see, e.g., Fujimoto et al. 1984, Miralda-Escude et al.1990). It was argued that, after a relatively short initialstage, the temperature distribution within the neutron starreaches a steady state. The temperature is then almost uni-form in deep layers of the crust of neutron star withstandard neutrino emissivities. The temperature that thecrust will attain in the steady state depends on the accretionrate, M. For phase III, with a low accretion rate, we use thecrustal temperature computed by Zdunik et al. (1992) for an

accretion rate within the range 10Ð11EME10Ð16 M> yrÐ1.For phase IV, when accretion is caused by Roche-lobe over-flow, the internal temperature has ben calculated by Fuji-moto et al. (1984) for the accretion rate M\3Å10Ð10 M

> yrÐ1

and by Miralde-Escude et al. (1990) for M within the range10Ð10EME10Ð11 M> yrÐ1. We match their results andextrapolate the obtained dependence slightly in order toestimate T for M110Ð9 M> yrÐ1, the maximal accretion ratethat we use in calculations.

Note that the accretion rate during phases III and IV issensitive to the separation between the stars and theirmasses. In the present study we will not consider thedependence of the evolution on the orbital period of abinary, despite the importance of this dependence for theinterpretation of observational data (see van den Heuvel &Bitzaraki 1995). We are planning to address this problem ina forthcoming paper.

As the spin evolution of the neutron star is stronglycoupled with its magnetic evolution, it may be essentiallydifferent for different evolutionary phases. As already men-tioned, the neutron star is scarcely influenced by its com-panion during phase I, because the wind plasma of thesecondary is stopped by the pressure of magnetodipoleradiation behind the light cylinder. The stopping radius, Rs,is determined by the balance between the dynamical pres-sure of the wind and the radiative pressure of magnetodi-pole waves. The dynamical pressure of the wind is of theorder of rwV 2

w, where rw and Vw are the density and velocityof the wind plasma, respectively, near the stopping radius.The density, rw, at the distance s from the secondary may beestimated as rw2M0 /4ps2Vw, with M0 being the rate of massloss of the secondary. The pressure of the magnetodipoleradiation at the distance Rs from the neutron star is1(1/4pR2

s)(B2R6W4/6c4), where B is the surface fieldstrength at the magnetic pole and W is the spin angularvelocity of the neutron star. Equating this pressure to thedynamical pressure of the wind at the nearest point to theneutron star, and assuming that the stopping radius issmaller than separation between stars, a, i.e. s1a, oneobtains the expression for Rs,

Rs12B2R6W4

6c4Vw

·a2

M031/2

. (6)

In our calculations, it is more convenient to use the accre-tion rate, M, instead of the rate of mass loss M0. At aaRG,where RG22GM < /V 2

w is the radius of gravitational capture(Bondi 1952) and M is the neutron star mass, these quanti-ties are related by 4M/R2

G2M0 /a2; we assume that the windvelocity is larger than both the sound velocity of the windplasma and the velocity of orbital motion. Phase I lasts whilethe stopping radius is larger than the radius of gravitationalcapture, RsaRG.

With the assumption that the spin-down torque on theneutron star is caused by radiation of magnetodipole waves(Ostriker & Gunn 1969), the evolution of the spin period, P,is governed by

PP\8p2B2R6

3c3I, (7)

where I is the moment of inertia. Note that approximately

Magnetic and spin evolution of neutron stars 909

© 1998 RAS, MNRAS 295, 907–920

the same braking law applies if the torque is determined bycurrents, in the model suggested by Goldreich and Julian(1969). At RssRG, the pressure of the magnetodipoleradiation cannot prevent the wind plasma from penetratingbehind the light cylinder. The influence of the wind plasmaon the spin-down rate becomes appreciable, and furtherspin evolution of the neutron star departs from that of anisolated pulsar. In our simplified model, the conditionRs\RG determines the transtion between phases I and II.

During phase II, the wind matter can directly interactwith the neutron star magnetosphere. However, rotationstill is rather fast and the magnetosphere acts as a ‘pro-peller’, ejecting the infalling wind matter. During this phase,a certain fraction of the spin momentum of the neutron staris transferred to the matter being ejected, and the spinperiod may increase by a few orders of magnitude. In calcu-lations presented here, we adopt a very simplified model ofthe interaction between the accretion flow and magneto-sphere which is, nevertheless, qualitatively close to thatcommonly encountered in the literature (see, e.g., Pringle &Rees 1972 and Illarionov & Sunyaev 1975). We assume thatthe accreted matter interacts with the magnetosphere at theso-called Alfven radius, RA:

RA\2 R6B2

4MZGM32/7

21.2Å109 2 B212

MÐ1032/7

cm, (8)

where B12\B/1012 G and MÐ10\M/10Ð10 M> yrÐ1; thenumbers given are for a neutron star with M\1.4 M> andR\1.64Å106 cm. If the neutron star rotates rapidly and itsangular velocity is larger than the Keplerian angular velocitya the Alfven radius, WK(RA), the wind matter has to beexpelled by the magnetosphere, carrying the spin angularmomentum out too. In order to be expelled, the wind mattermust reach an angular momentum slightly larger than thatcorresponding to Keplerian rotation. Therefore, theamount of the spin momentum lost by the neutron star perunit time, Jp, may be estimated as

Jp34pR4A rwWKvr3MZGMRA , (9)

where vr is the radial velocity of the accreting flow near themagnetospheric boundary. The corresponding spin-downrate of the neutron star is

P3P2J

2pI3bP2B2/7M6/7, (10)

where b\(GMR2/8)3/7/pI. The neutron star rotation slowsdown until the angular velocity becomes comparable toWK(RA). We determine the critical period, Peq, at which thetransition between the ‘propeller’ and accretion phasestakes place from the condition W\WK(RA):

Peq\18.6B6/7

12

M3/7Ð10

s. (11)

This period can be large if the neutron star is accompaniedby a secondary star with a low rate of mass loss. Equation(11) determines the so-called spin-up line in the B–P plane.If the spin rotation slows down to the value given by equa-tion (11), it is generally accepted that the accreting mattercan flow through the magnetosphere and accrete on to the

surface, so that the neutron star enters phase III. Duringthis phase accretion from the stellar wind is allowed and, asmentioned earlier, nuclear burning of the accreted materialcauses some heating of the neutron star interior and inducesaccretion-driven field decay. In turn, the decay of the mag-netic field decreases the critical period, Peq. As accretionfrom the wind in a binary cannot be exactly spherical, theaccreting matter carries some of the angular momentum tothe neutron star (see Geppert, Urpin & Konenkov 1995)and, because of this, the neutron star can experience spin-up to a shorter period. A balance will probably be reachedbetween spin-up and the rate of field decay, so that theneutron star slides down the corresponding spin-up line(see, e.g., Bhattacharya & Srinivasan 1991). Phase II of theevolution lasts either until the end of the main-sequencelifetime of the companion (when it fills its Roche lobe andthe neutron star enters phase IV) or until the magnetic fieldbecomes weak to such an extent that the accreting angularmomentum is insufficient to maintain a balance betweenspin-up and the rate of field decay.

The amount of angular momentum carried by theaccreted matter can be estimated from the condition at themagnetospheric boundary, assuming that the angularmomentum there is characterized by its Keplerian value,rWkR

24, multiplied by some ‘efficiency’ factor j. This factor is

11 if the accreted wind matter forms a Keplerian discoutside the magnetosphere. If a disc is not formed, then theefficiency of spin-up is much lower and the parameter j isprobably of the order of 0.1–0.01 (see Jahan Miri & Bhatta-charya 1994). The amount of spin angular momentumadded to the neutron star per unit time, Jw, can be estimatedas

Jw3j ·4pR4A rWKvr3jMZGMRA. (12)

The corresponding spin-up rate is

P3ÐjbP2B2/7M6/7. (13)

If the magnetic field becomes very weak, P may be small andaccretion cannot maintain a balance between the spin upand field decay rates, so that the neutron star leaves thespin-up line.


© 1998 RAS, MNRAS 295, 907–920

After the companion fills its Roche lobe and phase IVstarts, accretion on to the neutron star is strongly enhancedand heating of the crust leads to rapid field decay. In low-mass binaries, accretion resulting from Roche-lobe over-flow can probably last as long as (1–5)Å107 yr. During thisphase, the accreted matter likely forms a Keplerian discoutside the neutron star magnetosphere. Therefore, therate of angular momentum transfer and the spin-up rate aregiven by equations (12) and (13), respectively, with the ‘effi-ciency’ parameter j11. The neutron star spins up inaccordance with these equations until it approaches thespin-up line corresponding to the Roche-lobe accretionrate. The accretion rate probably has to reduce when theneutron star reaches the spin-up line (Bhattacharya & Srini-vasan 1991) but, nevertheless, the field continues to decay asa result of accretion-driven mechanisms. During furtherevolution, a balance should be reached between spin-up andthe rate of field decay, as in the case of accretion from thewind. Because of this balance, the neutron star slides downthe corresponding spin-up line with a rate that is deter-mined by field decay. In reality, the evolution on the spin-up

line may be very complicated, but in our calculations we willadopt a simplified model. Namely, we will neglect possibledifferences in the accretion rate before and after theneutron star raches the spin-up line. As mentioned above,the amount of matter accreted on to the surface may bereduced when the star evolves on the spin-up line. However,this decrease of the accretion rate is unlikely to change theinternal temperature drastically because at high accretionrates the temperature is close to saturation and dependsrelatively weakly on the accretion rate (see Miralda-Escudeet al. 1990 and Fujimoto et al. 1984). Therefore we hopethat our assumption does not change the rate of the mag-netic field decay on a spin-up line substantially. Note that inour model the neutron star slides down the correspondingspin-up line with the maximal rate, but in reality the evolu-tion will be slower. The star can leave the spin-up line eitherif accretion is exhausted or if the field becomes too weak tomaintain a balance between spin-up and field decay.

3 NUMER ICA L R ESULTS A ND DISCUS SION

We consider the evolution of the neutron star for a widerange of accretion rates and initial magnetic fields. Calcula-tions are performed for a neutron star model with M\1.4M> based on the Pandharipande–Smith equation of state(Pandharipande & Smith 1975). This model is representa-tive of stiff equations of state, so the radius and thickness ofthe crust for the corresponding neutron star are relativelylarge, 216.4 and 4.2 km, respectively. We choose a stiffmodel for our calculation because the magnetic evolution ofisolated neutron stars based on such models is in goodagreement with the available observational data on pulsarmagnetic fields (Urpin & Konenkov 1997). The evolution ofneutron star models with equations of state softer than thatof Friedman–Pandharipande seems to be inconsistent withthe B–P distribution of radio pulsars.

The conductive properties of the crust depend on itschemical composition which, unfortunately, is rather uncer-tain for neutron stars. This dependence is not significantduring the initial evolution, when the magnetic field is con-fined to the layers with rather low density and the differencebetween the existing models of chemical compostion is notvery large. However, the effect of composition may beimportant during phase IV, when the neutron star isstrongly heated and the conductivity is determined by pho-non scattering, which is strongly sensitive to the chemicalcomposition. Therefore, in our calculations, we adopt thesimplest model, in accordance with which the crust is com-posed of nuclei processed in various nuclear transforma-tions in accreting neutron stars (see Haensel & Zdunik1990) during the whole evolution of the neutron star.

In calculations, the impurity parameter, Q, is assumed tobe constant throughout the crust during the whole evolutionand to range from 0.001 to 0.1. The initial spin rotation isassumed to be relatively fast, P0\0.01 s. Note, however,that the results are not sensitive to the value of P0. On thecontrary, the evolution is strongly dependent on the originalmagnetic configuration of the neutron star. The initial mag-netic field is assumed to be confined to the outer layers ofthe crust with densities rRr0. The calculations presentedhere are performed for r0 in the range 4Å1011 to 1013 g

cmÐ3. These values correspond to a depth from the surfaceof 2820 and 1100 m, respectively. The choice of r0 isimposed by comparing the calculated magnetic and spinevolution of isolated neutron stars with a crustal magneticfield with the available observational data on P and P forradio pulsars (see Urpin & Konenkov 1997). The initialfield strength at the magnetic pole, B0, is taken to be withinthe range 1013EB0E1011 G.

The present study is mainly addressed to neutron starsentering binary systems with a low-mass companion. Forsuch secondary stars, the main-sequence lifetime, tms, maylast as long as 109–1010 yr. During the main-sequence evolu-tion of the secondary, the accretion rate on to the neutronstar is determined by both the characteristics of the stellarwind and the separation between the stars, and may gener-ally vary within a wide range. In calculations, we supposethat M\10Ð13, 10Ð15 and 10Ð17 M> yrÐ1 during both thepropeller and wind accretion phases; Vw\500 km sÐ1,a\5Å1012 cm. We also assume j\0.1 for the efficiencyparameter. Note, however, that our calculations show aweak sensitivity of the evolutionary tracks to the particularchoice of j. After the secondary leaves the main-sequenceand fills its Roche lobe, the accretion rate has to be stronglyenhanced, M210Ð9–10Ð10 M> yrÐ1. We do not consider ahigher accretion rate because of the lack of calculations onthe thermal structure of accreting neutron stars withMa3Å10Ð10 M> yrÐ1.

Figs 1–4 show examples of the magnetic, spin andthermal evolution of a neutron star in a binary system fordifferent values of the parameters. Numbers in the toppanels indicate the evolutionary phase. Note a difference inthe horizontal scales for different phases. The evolutionarytracks are shown as solid lines if the accretion rate resultingfrom Roche-lobe overflow is M\10Ð10 M> yrÐ1 and asdashed lines if M\10Ð9 M> yrÐ1.

In Fig. 1, we plot the evolution for a relatively stronginitial magnetic field, B0\1013 G, and for r0\1013 g cmÐ3.The impurity parameter is Q\0.01 and the duration of themain-sequence lifetime of a low-mass companion is 3Å109

yr. Accretion resulting from Roche-lobe overflow isassumed to last as long as 108 yr; thus the total amount ofaccreted mass can reach 0.1 M> if M\10Ð9 M> yrÐ1.During phase III, the accretion rate is 10Ð15 M> yrÐ1. Thismodel gives an example of a neutron star that is processedin all evolutionary stages. Phase I, when the evolution of theneutron star is not influenced by a companion, lasts a littleless than 107 yr. During this sort stage the field does notdecay significantly, thus its surface strength is about2Å1012 G when the neutron star enters the ‘propeller’phase. An efficient spin-down caused by strong magneto-dipole radiation leads to a rapid increase of the spin periodfrom 10Ð2 to 26 s. During the next evolutionary stage,when the neutron star works as a propeller, the field con-tinues to decay slowly because, after 107 yr of the evolution,the temperature falls to a very low value and the conductiv-ity of crustal matter is high. Note that during this stage theconductivity is determined by impurity scattering anddepends on the impurity parameter Q. Phase II is relativelylong (2230 Myr), but the field is still very strong (11012 G)to the end of this phase. As a result of such a strong field, theneutron star can eject the wind matter, which penetratesinto the magnetosphere, as long as the spin period is shorter


© 1998 RAS, MNRAS 295, 907–920

than 24Å103 s. When spin rotation slows down to thisperiod, matter from the wind can fall on to the surface of theneutron star and it becomes a soft and not very bright X-raysource. Nuclear burning of the accreted material heats theneutron star interior to T22Å106 K. However, this heatingis too weak to accelerate the field decay appreciably; thus,the magnetic evolution during the wind accretion phasedoes not differ essentially from that of an isolated star. Notethat wind accretion can accelerate the decay appreciablyonly if the accretion rate is larger than 10Ð13 M> yrÐ1. Untilthe end of the wind accretion, the field strength remains ashigh as 2Å1011 G. The neutron star can slowly spin up,obtaining some angular momentum from the accreted windmatter, but this is not enough for the star to spin up to ashort period; the period therefore turns out to be as long as1103 s after 3Å109 yr, when the companion ends its main-sequence life. Phase III is the most extended for the modelconsidered, however, as will be shown, it may not be thecase for other models. The most dramatic changes areexperienced by the neutron star during phase IV, whenenhanced accretion resulting from Roche-lobe overflowstarts. Accretion with an accretion rate M\10Ð10–10Ð9 M>

yrÐ1 or higher heats the neutron star to a temperature

T2(2–3)Å108 K and decreases the crustal conductivitysubstantially. If the field is reduced only by a factor 150during the previous 3Å109 yrÐ1, the accretion-driven mech-anism leads to much more efficient decay. Thus, in the caseM\10Ð9 M> yrÐ1, the field weakens 230 times after 108 yrof accreting, when the neutron star has accreted 0.1 M>. Fora higher accretion rate, the decay is even stronger. As theaccreted matter carries a large amount of the angularmomentum with it, the neutron star can accelerate its spinrotation to relatively short periods during phase IV. Thespin evolution is non-monotonous (this is shown in the smallpanel in Fig. 1). At the start of accretion the neutron starspins up very rapidly, and after 1104 yr it reaches the corre-sponding spin-up line. As the magnetic field is still suffi-ciently strong [1(1–2)Å1011 G], the corresponding criticalperiod, Peq, is relatively long (11 s). Further spin evolutionproceeds slowly because the neutron star slides down thespin-up line, and the rate of this migration is determined bythe field decay. Finally, when accretion is exhausted after108 yr, the neutron star that has been processed in all theabove transformations will operate as a radio pulsar with alow magnetic field [1(1–3)Å1010 G] and a short period(10.04–0.3 s).


© 1998 RAS, MNRAS 295, 907–920

Figure 1. The magnetic, spin and thermal evolution of a neutron star with B0\1013 G, r0\1013 g cmÐ3 and Q\0.01. The duration of themain-sequence lifetime of the companion is tms\3Å109 yr; the rate of accretion from the stellar wind is 10Ð15 M> yrÐ1. The evolution isshown for two values of the accretion rate resulting from Roche-lobe overflow, M\10Ð10 (solid lines) and 10Ð9 M> yrÐ1 (dotted lines). Theduration of accretion is assumed to be 108 yr for both cases. The panel shows the spin evolution during phase IV, ta\tÐtms.

Fig. 2 shows another example of the evolution, which istypical for stars with a relatively weak initial magnetic field.The initial field is assumed to be 1011 G; other parametersare the same as in Fig. 1. In this case, the neutron star doesnot pass through all evolutionary stages: accretion of matterfrom the wind is impossible because rotation spins downvery slowly as a result of a weak magnetic field and, after3Å109 yr, the period is still shorter than the correspondingvalue of Peq. A weak initial field results in a very long dura-tion of phase I for this model: the period slows down to avalue of 10.4 s, which allows the infalling matter to interactwith the magnetosphere only after 1600 Myr. However, themagnetic field turns out to be reduced only by a factor 110and is 11010 G after this long phase. For the remainder ofthe main-sequence lifetime of the companion (which isabout 80 per cent of this lfietime), the neutron star works asa propeller, ejecting the wind plasma. Despite a long dura-tion for this phase, neither the magnetic field nor the spinrotation change significantly before the beginning of Roche-lobe overlofw: the surface field strength decreases approxi-mately by a factor of 3, whereas the period increases from0.4 to 0.46 s. Note the great difference in the periods justbefore the accretion phase for this model and the modelplotted in Fig. 1, where P2103 s a t\3Å109 yr. Enhancedaccretion, however, can accelerate a spin rotation veryrapidly; thus an accreting neutron star forgets quickly about

its period before the accretion phase. Recycling is particu-larly fast during the first 1(4–10) Myr of accreting, whenthe neutron star spins up to 10–30 ms depending on theaccretion rate and approaches the corresponding spin-upline. Further spin evolution on the spin-up line proceedsslower but, nevertheless, the star can reach millisecondperiods if a mass transfer lasts as long as 108 yr. For thismodel, the ‘recycled’ periods when accretion is exhaustedare 11 and 110 ms for M\10Ð9 and 10Ð10 M> yrÐ1,respectively. The magnetic field is also substantially reducedduring phase IV under the action of the accretion-drivenmechanism. To the end of this phase, the surface field maybe as low as 108 G if M\10Ð9 M> yrÐ1 (when the totalaccreted mass is 0.1 M>) and 3Å108 G if M\10Ð10 M> yrÐ1

(when the accreted mass is 0.01 M>). When accretion hasbeen exhausted, the neutron star can be observed as a milli-second radio pulsar with a period 11–10 ms and a magneticfield 1(1–3)Å108 G.

In order to illustrate the effect of the impurity content, weplot in Fig. 3 the evolution of a neutron star with a stronglypolluted crust, Q\0.1. We also assume that for this modelthe field is initially confined to the layer with the smallestdensity, r0\1012 g cmÐ3, corresponding to a depth from thesurface 2900 m. Like the model plotted in Fig. 1, this onealso passes through all evolutionary stages. However, thefield decays faster for this model because the crustal con-


© 1998 RAS, MNRAS 295, 907–920

Figure 2. Same as Fig. 1, but for B0\1011 G.

ductivity is lower as a result of a higher impurity content andthe field is initially anchored in the layers wth lower density(and hence a lower conductivity). A faster decay leads to aslower spin-down and longer duration of the first two phasesin comparison with Fig. 1. Thus the ‘isolated pulsar’ and‘propeller’ phases last 212 and 650 Myr, respectively,whereas the duration of these phases in Fig. 1 is 26 and 230Myr. The maximal spin period that is reached to the end ofthe propeller phase does not exceed, 2300 s however. Anincrease of the impurity parameter Q causes a particularlyrapid field decay during phases II and III, when the con-ductivity is determined by scattering on impurities. Duringthese two phases the field is reduced by two orders of mag-nitude, and it is a little lower than 1010 G when the com-panion fills its Roche lobe and enhanced accretion starts.Accretion of matter during 108 yr leads to the formation ofa millisecond pulsar with a period within the range 22–20ms and with a magnetic field 2(3–10)Å108 G, dependingon the accretion rate during phase IV.

Fig. 4 represents the evolution of a neutron star with anintermediate initial magnetic field, B0\1012 G, and with alonger main-sequence lifetime of the companion,tms\5Å109 yr. Other parameters are the same as in Fig. 1.The evolution does not differ in the main from that plottedin Fig. 1, except for a weaker magnetic field of the neutronstar at the beginning of the accretion phase. For this model,

B22Å1010 G at t\5Å109 yr. When accretion starts, such aweak magnetic field is probably unable to channel theinflowing matter towards the magnetic poles; thus the rota-tion of the neutron star will not cause the observed X-rayemission to have a pulsed appearance. Note that the same istrue of the models plotted in Figs 2 and 3. In contrast, theneutron star with the evolutionary scenario represented inFig. 1 can be observed (at least at the beginning of theaccretion phase) as a pulsating binary X-ray source. Theaccretion-driven field decay causes a further decrease of themagnetic field so that by the end of phase IV it may be asweak as 109 G if M\10Ð9 M> yrÐ1. If accretion lasts 109 yr,the neutron star can be easily recycled to a very shortperiod. For this model, the final spin periods are 210 and100 ms after accreting 0.1 and 0.01 M>, respectively. Notethat the field strengths and periods of the neutron starsprocessed in accordance with the scenarios shown in Figs2–4 are very similar to those of millisecond pulsars.

Figs 5–10 show the evolutionary tracks of neutron stars inthe B–P plane for different values of various parameters.We also plot in these figures the observational data onisolated pulsars (dots) and pulsars in binary systems (opendiamonds) taken from the catalogue by Taylor, Manchester& Lyne (1993). The dotted lines show spin-up lines corre-sponding to different accretion rates (numbers near theselines indicate the logarithm of the accretion rate). The solid


© 1998 RAS, MNRAS 295, 907–920

Figure 3. Same as Fig. 1, but for r0\1012 g cmÐ3 and Q\0.1.


© 1998 RAS, MNRAS 295, 907–920

lines represent the evolutionary tracks. All tracks are cal-culated for the case of accretion resulting from Roche-lobeoverflow with M\10Ð9 M> yrÐ1. Numbers near the tracksindicate the logarithm of the age; numbers with the sub-script (a) mark the time required for the neutron star toreach the corresponding spin-up line after the end of main-sequence evolution. Large crosses show the points on trackswhere a transition takes place between phases I and II;small crosses with the corresponding numbers mark the endof tracks after 108 yr of enhanced accretion.

Fig. 5 plots the evolution of neutron stars that accretematter from the stellar wind with different accretion rates.Calculations are performed for M\10Ð12 (curve 1), 10Ð13

(2), 10Ð14 (3), 10Ð15 (4) and 10Ð17 M> yrÐ1 (5); the durationof the main-sequence lifetime of the companion is 3Å109 yr.The initial field is B0\1013 G, the density penetrated by theinitial field is r01012 g cmÐ3 and the impurity parameter isQ\0.1. The neutron stars starting the evolution with suchparameters typically pass through all evolutionary stages.The only exception is model (5) with a very weak stellarwind, M\10Ð17 M> yrÐ1. This wind is insufficient to slowdown the spin period to the critical value (11) when accre-tion from the wind becomes possible and, therefore, theneutron star is still in the ‘propeller’ phase when the com-panion fills its Roche lobe and enhanced accretion starts.The duration of phases is obviously strongly dependent onthe rate of wind accretion. For instance, the initial phase, in

Figure 4. Same as Fig. 1, but for B0\1012 G and tms\5Å109 yr.

Figure 5. The evolutionary tracks of a neutron star for differentrates of accretion from the stellar wind. M\10Ð12 (curve 1), 10Ð13

(2), 10Ð14 (3), 10Ð15 (4) and 10Ð17 M> yrÐ1 (5). Other parametersare B0\10Ð13 G, r0\1012 g cmÐ3, Q\0.1, tms\3Å109 yr. The rateof accretion resulting from Roche-lobe overflow is 10Ð9 M> yrÐ1;the duration of this phase is 108 yr.

which the neutron star evolves like an isolated neutron star,lasts as long as 2160 Myr if M\10Ð17 M> yrÐ1 and only 105

yr if M\10Ð12 M> yrÐ1. To the end of this phase, the mag-netic field and period are ranged from 2Å1011 G and from 6to 1 s, respectively, for different models. After this phase,the field and spin are reduced sufficiently that the windmatter can penetrate into the magnetosphere, and theneutron star works as a propeller. The ‘propeller’ phaselasts until the neutron star, moving to the right and down inthe B–P plane, reaches the corresponding spin-up line. Evi-dently, a star that experiences stronger accretion from thewind requires a shorter time to approach the spin-up line.Thus, one has P\Peq after 54 and 630 Myr of evolution ifM\10Ð12 and 10Ð15 M> yrÐ1, respectively. As mentionedabove, in the case of very weak accretion with M\10Ð17 M>

yrÐ1 the spin-up line can never be reached. Accretion fromthe wind starts when the magnetic field is still rather high formodels (1)–(4), B11011–1012 G. This field is certainly ableto channel the accreting matter towards the magnetic poles;thus, probably, some neutron stars accreting matter fromwinds in low-mass binaries can be observed as weak pulsat-ing X-ray sources. Further evolution of the neutron star onthe spin-up lines is completely determined by the fielddecay, which can be accelerated as a result of accretion-induced heating for a relatively high accretion rate. Thewind accretion phase is the most extended phase for allmodels plotted in Fig. 5 with the exception of model (5).The parameters of stars can be drastically changed duringthis long phase if the accretion rate is sufficiently large. Forinstance, the field is reduced by a factor 1104 and is only1104 G for model (1) with accretion rate M\10Ð12 M>

yrÐ1. Note, however, that at the end of the main-sequencelifetime of the companion the magnetic fields of models(2)–(5), which experience accretion with MR10Ð13 M>

yrÐ1, lie within a relatively narrow range, 1(2–10)Å109 G,despite a large difference in the duration of different phasesfor these models. The main reason for this is the relativelyweak heating caused by accretion with such low accretionrates, which makes the accretion-driven mechanism ineffi-cient at MR10Ð13 M> yrÐ1.

When the companion fills its Roche lobe and enhancedaccretion starts, the neutron star initially spins up veryrapidly: it requires only 0.2 Myr to reach the spin-up line formodel (5) and 1.6 Myr for model (2). These time intervalsare too short for an appreciable decay of the magnetic field;therefore the corresponding pieces of tracks are repre-sented by approximately horizontal plateaus. Note that themagnetic field of model (1) is very weak after 3Å109 yr ofevolution (1108 G), and this model will never reach thespin-up line. Like the case of accretion from a stellar wind,in its further evolution the neutron star slides down thespin-up line and the rate of this sliding is determined by thefield decay. However, in the case of Roche-lobe overflow,the accretion-driven mechanism is much more efficient andhence the decay is much faster. During phase IV, the mag-netic field decreases by a factor 130 for models (2)–(5);thus, when accretion is exhausted, all these neutron starswill work as pulsars with weak magnetic fields,1(1–5)Å108 G, and short periods, 12–10 ms. Model (1) isalso spun up to a millisecond period, but its magnetic fieldis substantially smaller than 108 G.


© 1998 RAS, MNRAS 295, 907–920

Figure 6. Same as Fig. 5, but for r0\1013 g cmÐ3 and Q\0.01.

In Fig. 6, we plot the same as in Fig. 5 but for the mag-netic field confined initially to deeper layers, r0\1013 gcmÐ3, and for a crust with a smaller impurity content,Q\0.01. Evidently, the magnetic evolution is much slowerfor a less polluted crust. Like the case shown in Fig. 5,models (1)–(4) pass through all evolutionary phaseswhereas model (5), with very weak accretion from the wind(M\10Ð17 M> yrÐ1), avoids phase III. As a result of lowerfield decay and hence faster spin-down, the duration ofphases I and II is shorter for the models in Fig. 6 than forthe models with the same wind in Fig. 5. Therefore, windaccretion starts when the magnetic field of the neutron staris very strong, B12Å1012 G for all models. As in the pre-vious case, wind accretion is the most extended phase formodels (1)–(4). To the end of the main-sequence evolutionof the companion (tms\3Å109 yr), the magnetic fields ofneutron stars that experience wind accretion with a lowaccretion rate, MR10Ð13 M> yrÐ1, lie in a narrow rangefrom 4Å1010 to 3Å1011 G. Only accretion with Ma10Ð13

M> yrÐ1 can substantially change the magnetic evolutionduring phase III.

Owing to a higher magnetic field at t\3Å109 yr, themodels plotted in Fig. 6 reach the spin-up line on a muchshorter time-scale when enhanced accretion starts. Thus,only 4Å103 yr is required for model (5) and 21 Myr formodel (1). Note that in the previous case model (1) couldnever reach the spin-up line during the accretion phase. Ifaccretion resulting from Roche-lobe overflow lasts 108 yr,models (2)–(5), which experienced weak accretion from thewind, evolve to relatively weakly magnetized pulsars withB1(2–10)Å109 G, rotating with a period 10.03--0.1 s.Model (1), with a very high accretion rate during phase III,evolves to a millisecond pulsar with P23 ms and B2108 G.Obviously, the magnetic field and period are larger for allpulsars if the accretion phase is shorter.

Fig. 7 compares the evolution of neutron stars with dif-ferent initial magnetic fields and with the same r0 and Q.Calculations have been performed for r0\1013 g cmÐ3,Q\0.01 and tms\3Å109 yr. It is apparent from this figurethat the evolution is strongly sensitive to the initial fieldstrength. The weaker the magnetic field, the longer theduration of the ‘isolated pulsar’ and ‘propeller’ phases and,correspondingly, the shorter the wind accretion phase. Forinstance, the neutron star can work as an ‘isolated pulsar’for only 0.5 Myr if B0\1013 G, but this stage may be as longas 63 Myr if the initial field is weak, B0\1011 G. A stronglymagnetized star with B0\1013 G can accrete matter fromthe stellar wind after 212 Myr, whereas a weakly magnet-ized star with B0\1011 G can only accrete after 2600 Myr.To the beginning of wind accretion, magnetic fields for themodels considered lie within a wide range from 1010 to2Å1012 G. The corresponding periods are 110–1000 s. Thewind acccretion phase is the most extended phase for allmodels plotted in Fig. 7. During the two previous phases,the magnetic field is typically reduced by a factor 15–10,whereas the decrease during wind accretion is about 10–50times. When the companion fills its Roche lobe and vigor-ous mass transfer starts, the magnetic field of the neutronstar is 2109 and 24Å1010 G for the most weakly and moststrongly magnetized among the models presented in Fig. 7,respectively. These fields are probably too weak to channelthe accreting matter towards the magnetic poles; thus theneutron star is unable to work as a pulsating X-ray sourceduring phase IV. Enhanced accretion accelerates the fielddecay; thus, when the mass transfer is exhausted, the fieldand period of models (1)–(3) turn out to be quite suitablefor millisecond pulsars. Model (4), which initially has astrong magnetic field, B0\1013 G, may also be transformed

into a weakly magnetized and rapidly rotating pulsartowards the end of the accretion phase. However, the fieldof this puslsar (12Å109 G) is a little stronger and theperiod (130 ms) is a little longer than it is typical formillisecond pulsars. Evidently if accretion lasts longer than108 yr model (4) can also evolve to a millisecond pulsar withthe standard parameters.

In Fig. 8, the dependence of evolutionary tracks on theduration of main-sequence lifetime of the companion, tms, isshown. Since we address mainly the evolution of neutronstars entering binary systems with a low-mass companion,calculations have been performed for a sufficiently long tms,1010EtmsE109 yr. Despite the fact that we plot in this figuretracks for a neutron star with a relatively polluted crust,Q\0.03, and an initial magnetic field B0\3Å1012 G, whichis certainly weaker than the measured fields of many iso-lated pulsars, it is seen that crustal currents can maintain asufficiently strong magnetic field during the whole evolu-tion. This conclusion is in contrast to the widely acceptedopinion that neutron star models with a crustal magneticfield are not suitable to account for the relatively strongmagnetic fields of many long-lived pulsars. Even for a ‘pol-luted’ crust, the decay turns out to be very slow; thus theneutron star can possess a field stronger than 109 G after1010 yr of evolution and enhanced accretion.

During the ‘isolated pulsar’ phase, which lasts 220 Myr,the field weakens by a factor 25, whereas during the ‘pro-peller’ phase, which is as long as 2630 Myr, the field isreduced only by a factor of 2. Working as a propeller at2610 Myr, the neutron star spins down to a period of theorder of 103 s, which is slow enough to accrete wind matter.


© 1998 RAS, MNRAS 295, 907–920

Figure 7. The evolution of a neutron star with different initialmagnetic fields: B0\1011 (curve 1), 3Å1011 (2), 1012 (3) and 1013 (4).Other parameters are r0\1013 g cmÐ3, Q\0.01, tms\3Å109 yr.The rates of accretion from the stellar wind and resulting fromRoche-lobe overflow are 10Ð13 M> yrÐ1, respectively.

Figure 8. The evolution of a neutron star in a binary system withdifferent durations of the main-sequence lifetime of the com-panion: tms\1010 (curve 1), 5Å109 (2), 3Å109 (3), 2Å109 (4) and109 yr (5). Other parameters are r0\1013 g cmÐ3, B0\3Å1012 G,Q\0.03. The rates of accretion from the stellar wind and resultingfrom Roche-lobe overflow are 10Ð15 and 10Ð9 M> yrÐ1, respec-tively.

At this time the field is rather strong, B23Å1011 G, and sothe accreted matter has to be channeled towards the mag-netic poles. Perhaps during the wind accretion phase such aneutron star can be observed as a weak and pulsating X-raysource. When the companion fills its Roche lobe, the mag-netic field of the neutron star can range from 3Å1010 to2Å1011 G depending on the duration of the main-sequencelifetime. The corresponding spin periods lie within therange 100–500 s. Owing to a relatively strong magneticfield, a spin-up is initially very fast; thus the neutron starapproaches the spin-up line shortly after the beginning ofRoche-lobe overflow. The field does not change during thisshort time interval (2104–105 yr). In the course of furtherevolution alongside the spin-up line, the field is decreasedby a factor 230 under the action of the accretion-drivenmechanism; thus to the end of the accretion phase a pulsarcan be formed with B1(2--5)Å109 G and P120–70 ms. Ifaccreton lasts longer than 108 yr and the total amount ofaccreted mass is larger than 0.1 M>, all the models plottedin Fig. 8 can evolve to ‘standard’ millisecond pulsars withB1108–109 G and P11–10 ms. Note that if the mass of alow-mass companion is relatively large so that the main-sequence lifetime is closer to 109 yr rather than 1010 yr, thenthe field at the beginning of phase IV is sufficiently strong(B12Å1011 G) to channel the accreted matter towards themagnetic poles, and the neutron star can work as a pulsatingX-ray source. At the end of accretion, the field has to bereduced to a very low value, insufficient to produce X-raypulses but sufficient for a neutron star to work as a radiopulsar.

Fig. 9 shows the dependence of the evolution on theimpurity ‘pollution’ of the crust. Calculations have beenperformed for values of the impurity parameter, Q, withinthe range 0.001 to 0.3. The duration of the main-sequencelifetime of the companion is 3Å109 yr. During the initialevolution, the field decay does not depend on the impurityparameter. This is caused by the fact that the neutron star isrelatively hot initially and the crustal conductivity is deter-mined by phonon scattering, which is independent of Q.Later on, when the star cools down and impurity scatteringdominates the conductivity, the latter becomes sensitive to‘pollution’ of the crust and the decay may be essentiallydifferent for different Q.

The behaviour of all models plotted in Fig. 9 is more orless standard during the main-sequence evolution of thecompanion. The only exception is model (1), with extremelylarge Q. Owing to a large impurity content and, as a conse-quence, a low conductivity, the magnetic field of this modeldecays too fast. During phases I and II, which last together21.6Å109 yr, the field strength has to be reduced morethan 1000 times and, when the neutron star approaches thespin-up line corresponding to the wind accretion rateM\10Ð15 M> yrÐ1, its field is too low (B12Å109 G) tomaintain a balance between spin-up and the rate of fielddecay. Therefore, the neutron star does not slide down thespin-up line in its further evolution but moves down the B–Pplane much faster. To the beginning of Roche-lobe over-flow, the field of the model (1) weakens to a very low value1108 G. Enhanced accretion causes further field decay;thus when accretion is exhausted the field strength of thismodel is extremely low (1106 G) and the pulsar is almostunobservable.

Models (2)–(5), with Q\0.001–0.1, indicate more or lessreasonable behaviour. Eventually the neutron star passesthrough all evolutionary phases and, after all transforma-tions, it forms a pulsar with a weak magnetic field [B1(2–30)Å108 G] and a short period (P15–40 ms). Note that atthe end of accretion the field of model (5) turns out to belower than that of models (3) and (4), despite a smallerimpurity parameter for model (5). As a result of substan-tially higher conductivity, the field of model (5) diffusesinward very slowly; thus, after 3Å109 yr, the current main-taining the magnetic configuration is located at much lowerdensities than for models (3) and (4). When accretionresulting from Roche-lobe overflow heats the neutron starto a temperature 12Å108 K, the conductivity of layersoccupied by the current becomes essentially lower formodel (5) because of a low density. Therefore, the fielddecays much faster for this model during the accretionphase, and at the end of evolution it is even lower than formodels (3) and (4).

The dependence of the evolutionary tracks on the intialdepth penetrated by the field is shown in Fig. 10. We plotthe tracks for four values of r0: 1011 (curve 1), 1012 (2), 1013

(3) and 1014 g cmÐ3 (4). Evidently, the smaller the initialdepth penetrated by the field, the faster the decay. Formodel (1), with the magnetic field anchored initially in arelatively narrow surface layer with depth s600 m, thedecay is so rapid that the neutron star cannot even be pro-cessed in all evolutionary transformations: after 3Å109 yr ofevolution the rotation is still fast enough to prevent accre-tion from the stellar wind. Because of this, model 1 evolvesfrom the ‘propeller’ phase directly to the enhanced accre-tion phase, missing wind accretion. However, even for this


© 1998 RAS, MNRAS 295, 907–920

Figure 9. The evolution of a neutron star with different values ofthe impurity parameter: Q\0.3 (curve 1), 0.1 (2), 0.03 (3), 0.01 (4)and 0.001 (5). Other parameters are r0\1013 g cmÐ3,B0\3Å1012 G, tms\3Å109 yr. The rates of accretion from thestellar wind and resulting from Roche-lobe overflow are 10Ð15 and10Ð9 M> yrÐ1, respectively.

model the field at the end of the mass transfer is sufficientlystrong that when accretion is exhausted the neutron star canwork as a millisecond pulsar with B13Å108 G and P18 ms.Models (2)–(4) pass through all phases and, evidently, theirfinal magnetic fields are higher and their periods longer. Forexample, at the end of accretion B may be as high as11011 G for model (4), where the field occupies initiallyabout half a crustal thickness.

4 CONCLUSION

We examined the magnetic and spin evolution of neutronstars in close binary systems with a low-mass companion.Owing to the long lifetime of the companion, evolutionarytransformations of the neutron star may last as long as109–1010 yr or may be very complex. Likely, most neutronstars in such binary systems experience vigorous mass trans-fer as a result of Roche-lobe overflow. This mass transfercan be extremely extended, and the total amount of massaccreted by the neutron star can exceed 0.1–0.5 M> (see,e.g., van den Heuvel & Bitzaraki 1995). The evolution ofneutron stars in low-mass binaries must also be influencedby the stellar wind during the main-sequence life of thecompanion. As a result of this, the neutron star in the courseof its evolution passes through several essentially differentevolutionary phases.

Calculations presented here are performed in agreementwith the standard scenario of evolution of a neutron star ina close binary system (Pringle & Res 1972; Illarionov &Sunyaev 1975). According to this scenario, the neutron starcan be processed in four main evolutionary phases. Initially,when the star rotates rapidly and the field is strong, thepressure of the pulsar radiation does not allow wind matterto penetrate into the magnetosphere, and the neutron star

evolves like an isolated pulsar. The duration of this stagemay vary, depending on the initial field strnegth and the rateof field decay as well as on the intensity of the stellar wind ofthe companion. The initial phase is obviously shorter for aneutron star with a higher magnetic field and for a strongerwind. Thus, for the initial field as strong as 1013 G, phase Imay last only 10.1–1 Myr if the neutron star captures grav-itationally 110Ð12–10Ð13 M> yrÐ1 from the wind plasma.However, this phase may be extremely long (1100–300Myr) if the initial field is weak, B011011 G or if the rate ofgravitational capture is small, 110Ð15–10Ð17 M> yrÐ1. Formost models considered, the field is typically reduced by afactor 15–10 during this phase, except for the cases whenits duration is extremely long.

When the energy loss caused by magnetodipole radiationslows down rotation to such an extent that the wind mattercan interact with the magnetosphere, the neutron starbegins to work as a propeller, ejecting the wind plasma.Obviously, the propeller action is more efficient for astronger magnetic field. The ‘propeller’ phase is usuallymore extended than phase I. During the ‘propeller’ phase,the magnetic evolution of the neutron star in a binary doesnot differ from that of an isolated star. As the star is rela-tively cool at the beginning of phase II, the conductivity ofthe crustal matter is determined by impurity scattering andmay be high for a low impurity content. Typically, becauseof this, the field decays less during a mor extended ‘pro-peller’ phase than during a shorter ‘isolated pulsar’ stage.The spin evolution proceeds, however, more rapidlybecause the rate of angular momentum loss is larger for thepropeller effect than for magnetodipole braking. During the‘propeller’ phase, the neutron star can spin down to a verylong period, 1102–104 s. Note that this period may beshorter for stars with low magnetic fields and for a strongerstellar wind.

After further evolution, the spin period of the neutronstar can reach the critical value given by equation (11).When rotation becomes slow enough, the centrifugal forceis unable to eject the wind plasma that penetrates into themagnetosphere, and matter falls on to the neutron starsurface. Nuclear burning of the accreted material heats theneutron star, decreases the crustal conductivity and accel-erates field decay. Note that the efficiency of the accretion-driven mechanism of field decay is strongly sensitive to theaccretion rate. The effect of accretion is relatively small fora low accretion rate, Ms10Ð13 M> yrÐ1, but it may be ofgreat importance for higher accretion rates. During windaccretion, the neutron star slides down the correspondingspin-up line, and the rate of this sliding is determined by therate of field decay. Depending on the duration of this phaseand the rate of accretion, the field can be reduced by a feworders of magnitude. As the accreted wind matter carriessome amount of angular momentum, the neutron star canalso spin up slightly during this phase but, for most casesconsidered, the spin period is still longer than 1 s up to theend of the main-sequence evolution of the companion.Usually, the wind accretion phase is the most extended of allevolutionary phases. Note, however, that the star can missthis stage in some cases (low rate of mass loss by the second-ary star, rapid decay of the magnetic field), so that theevolution goes directly from the ‘propeller’ phase toenhanced accretion.


© 1998 RAS, MNRAS 295, 907–920

Figure 10. The evolution of a neutron star with the original mag-netic field confined to layers of different density: r0\1011 (curve 1),1012 (2), 1013 (3) and 1014 g cmÐ3 (4). Other parameters areB0\3Å1012 G, tms\3Å109 yr, Q\0.01. The accretion rates are thesame as in Figs 8 and 9.

In low-mass binaries, the neutron star experiences themost dramatic changes during vigorous mass transfer, whichstarts when the companion ends its main-sequence evolu-tion and fills its Roche lobe. Mass transfer in such systemscan probably last as long as 107–108 yr. Nuclear burning ofthe accreted material can heat the neutron star interior to atemperature T1(2–3Å108 K and can substantially reducethe crustal conductivity. As a result of this, the accretion-driven mechanism of field decay is especially efficientduring phase IV. Typically, the field may be reduced about20–30 times if accretion lasts 108 yr, but the decrease may belarger for a longer accretion. Depending on the initial mag-netic configuration and the duration of the main-sequenceevolution, the surface magnetic field can fall to the value1108–109 G by the end of mass transfer. As the matteraccreted from the disc carries a large angular momentum,the spin evolves very rapidly. The neutron star reaches aspin-up line corresponding to enhanced accretion on a shorttime-scale, 10.01–1 Myr. In its further evolution, the starmoves along the spin-up line, while the magnetic field is ableto maintain a balance between spin-up and the rate of fielddecay. During this phase, the accretion torque spins up theneutron star to a very short period, 11–100 ms.

Our calculations show that a neutron star with a crustalmagnetic configuration can maintain a relatively strong fieldduring its whole evolution in a low-mass binary. Even if themain-sequence life of the companion lats as long as 1010 yrand, after that, the neutron star experiences strong accre-tion with M110Ð9 M> yrÐ1 during 108 yr, the magnetic fieldcan remain sufficiently strong to the end of the accretionphase. When accretion is exhausted, such a neutron starworks as a radio pulsar with a weak magnetic field,1108–1010 G, and a short period, P11–100 ms. These para-meters are close to those of millisecond pulsars, and per-haps these objects are formed in accordance with theconsidered scenario. Note that if the main-sequence evolu-tion is shorter and a mass transfer is not so extended, theneutron star processed in the above evolutionary transfor-mations should have a stronger magnetic field and a longer

period at the end of accretion. This gives a natural explana-tion of the origin of those radio pulsars in binary systemsthat lie between millisecond pulsars and the main pulsarpopulation in the B–P plane.

ACKNOWLEDGMENT

This work was financially supported in part by theRussian Foundation of Basic Research under Grant97-02-18096(a).


© 1998 RAS, MNRAS 295, 907–920

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Magnetic and spin evolution of neutron stars in close binaries