arXiv:astro-ph/9803068v1 6 Mar 1998 lhuhohrojcsi h ru r crtn directly accreting are group the discs, in accretion possess objects to other pulsars. known X-ray are although brightest systems the the these of of of number measurements Some a daily of essentially made frequencies obtain has spin BATSE/CGRO to stars. possible neutron it accreting the are 1993, Campbell 1995). groups & Quaintrell several Cameron & by 1996, Campbell constructed Clarke Cameron, & been have Armitage the star T(e.g its between the the of interaction of and magnetic rid evolution disc the got rotational on has the based a of Tauri-stars star represent Models the to disc. when accretion believed stage, disc are evolutionary accretion Tauri-stars an later of that T signature Tauri-stars weak-lined the lack T The and (naked) are rotators and fast weak-lined discs are and accretion having rotators, of signs slow clear show that stars model Lamb & Ghosh the of necessary. re- modification is a thinking discs how current that accretion in show discuss with transport and in will momentum fits I angular model the 1997). garding Lamb al. & et Ghosh of Nelson the observations (e.g. recent by & pulsars challenged (Ghosh X-ray been Tauri-stars model has T adopted 1979a) as generally Lamb diverse The as pulsars. objects ac- X-ray understand- of and of for evolution importance aspects spin of the is understood it ing least time same the the of At surrounding cretion. one a is and disc star magnetic accretion a between interaction The INTRODUCTION 1 ue2021 June 6 0HA, CB3 Cambridge Road, Madingley Astronomy, of Institute Torkelsson Ulf stars and discs accretion between torques Magnetic o.Nt .Ato.Soc. Astron. R. Not. Mon. c 00RAS 0000 tteohredo h ag fmgei accretors magnetic of range the of end other the At Tauri- T Classical forms. two in come Tauri-stars T 000 0–0 00)Pitd6Jn 01(NL (MN 2021 June 6 Printed (0000) 000–000 , -as tr tr:pemi sequence pre-main stars: – stars X-rays: e words: th Key that in pulsars. reversals field X-ray torque magnetic disc-fed the the Lamb explain some diff if & e can in sign the mechanism therefore Ghosh changes by this may the torque that torque determined the in suggest magnetic and not as The star, is disc the parameter. field the down free and disc a star rather the is the the of mag between will direction of and velocity form The radius, angular latter increasing radii. The with large be field. slowly interaction at magnetic more the own strength from in disc’s accret coming we decreases the torque the the a and between is is field velocity there magnetic There angular addition accretor. in in appear but magnetic can difference star, a torques the magnetic and from of disc resulting types two accretion that an paper between this in show I ABSTRACT crto:aceindss–MD–antcfils–sas neutron stars: – fields –magnetic MHD – discs accretion accretion: . ntdKingdom United oiino h ne deo h crto icvre with varies disc accretion the of The edge radius. inner co-rotation the the of outside torque position disc spin-down accretion magnetic star, inside the the disc the from and accretion to radius, the disc co-rotation from the the torque from spin-up matter magnetic momentum accreting angular the the the between there by balance is a carried star of the of result whilst evolution the spin star, co-rotati fore The the the star. outside the up brake key disc radius spin accretion a the the plays radius penetrating penetrating same, co-rotation lines those the field the are magnetic inside The star disc model. the this ve- and in angular disc role the the where disc radius, of accretion of co-rotation locities the The because star. between direction, the difference toroidal and velocity the winds in angular which field the disc, The magnetic the disc. the diffusive penetrating a up is interaction with field the field magnetic magnetic describe dipolar stellar They dipolar (1979b). a Lamb of in & more Ghosh or by years observed 10 been for only stay has may 1626-67 thus state. 4U one and but once, days, X-3 switch 100 Cen to - of case 10 differ the is spends state in it other system systems, the different a to the switching time between before in of widely states time length the of of The one amount states. in two same and the the states, of about two each the spends in system sign) typical comparable the a in are difference derivatives the from that spin-u frequency (apart found of The have periods spin-down. between (1997) oscillating and al. are latter et pulsars the Nelson X-ray to attention paper. the any this pay not in will objects I wind. stellar a from h tnadmdlfrtesi hne a presented was changes spin the for model standard The A T E tl l v1.4) file style X te pnu rspin or up spin ither aebe observed been have nteinteraction the in o icadthe and disc ion we h stellar the tween icrvre.I reverses. disc e lkontorque ll-known eoedominate refore ei torque netic oe,but model, rnein erence on p – s - 2 U. Torkelsson the accretion rate such that it moves closer to the star when field. Some groups have started with a stellar dipole field the accretion rate increases. Thus we expect the star to spin that threads the disc (Hayashi, Shibata & Matsumoto 1996, up, or at least spin down more slowly, when the accretion Goodson, Winglee & B¨ohm 1997), whilst other groups have rate, or equivalently the luminosity, is high. taken into account that the disc may possess a magnetic A complicated, and not yet fully solved problem is the field on its own (Rast¨atter & Neukirch 1997, Miller & Stone position and physical properties of the boundary layer be- 1997). The simulations by Hayashi et al. (1996) and Good- tween the accretion disc and the stellar magnetosphere. son et al. (1997) both show that the stellar magnetic field Ghosh & Lamb (1979a) presented solutions for the boundary is wound up by the Keplerian accretion disc and becomes layer, but instead of solving for the magnetic field in the disc unstable. A reconnection event follows and mass is thrown given a certain magnetic diffusivity they assumed a magnetic out more or less along the rotational axis. Goodson et al. field and solved for the diffusivity. Heptinstall & Campbell suggest that this is the matter forming the stellar jet, but (1998) and Brandenburg & Campbell (1998) have recently they notice that there is also a slower outflow between the presented solutions for the magnetic field and velocity in the jet and the disc, which may form a disc wind. In general the disc given a certain model for the magnetic diffusivity. The simulations show more evidence for outflow than for accre- disc is terminated as the magnetic force becomes stronger tion onto the star, though Miller & Stone (1997) find a polar than the viscous force in the disc (cf. Campbell 1992, 1997). accretion flow when their disc contains a vertical magnetic The temperature and disc thickness increases dramatically field, and likewise do Rast¨atter & Neukirch (1997) find that close to the inner disc edge, and the disc is subject of a vis- reconnection causes matter to be accelerated along the field cous instability similar to the Lightman-Eardley instability lines leading to the polar caps. (Lightman & Eardley 1974, Lightman 1974). To summarise, the simulations are not yet mature Ghosh & Lamb (1979b) calculated the torque acting on enough to provide any information on the structure of steady the star by integrating over the surface of the disc. This is accretion discs around magnetic stars. The main reason for without a doubt the most convenient approach, but it does this is that they have concentrated on following transient not provide much information on how the torque is trans- events for a brief time interval. However the simulations ferred to the star. To understand this problem it is necessary suggest that the existence of a dynamo-generated magnetic to obtain a solution for the structure of the magnetosphere. field in the accretion disc is important for the interaction be- Simple solutions assuming no flows through the magneto- tween the disc and the stellar magnetosphere. The purpose sphere have been presented by Bardou & Heyvaerts (1996). of this paper is to explore how the dynamo may affect the The neglect of a flow through the magnetosphere may be un- exchange of angular momentum between the accretion disc realistic though, and Li, Wickramasinghe & R¨udiger (1996) and the star. I compare the torque generated by the winding have presented a class of funnel flow solutions in which the up of the stellar magnetic field with the torque generated by angular momentum is carried by the matter. Their solutions the coupling between a stellar magnetic field and a dynamo- include a slow magnetosonic shock, in case the flow is sub- generated magnetic field in the disc in Sect. 2. The existence Alfv´enic, close to the stellar surface. The shock is forced of two different forms of magnetic torques may influence the to be close to co-rotation with the star, which minimises spin evolution of the X-ray pulsars as suggested in Sect. 3. the torque between the shock and the star. In addition the Finally my conclusions are summarised in Sect. 4. toroidal magnetic field at the shock is small, which keeps the torque between the shocked and unshocked matter small. The authors thus conclude that most of the angular mo- mentum of the matter in the funnel has to be propagated 2 MAGNETIC TORQUES back to the accretion disc. Wang (1997) presents two major We can distinguish between two different ways in which the objections to this result. He points out that a small toroidal accretion disc and the star exchanges angular momentum; field is enough to produce a significant torque on the star. matter flowing over from the disc to the star carries with Furthermore if the angular momentum is propagated back it angular momentum, and the magnetic stress at the disc to the disc, it must be transported outwards through the disc surface exerts a torque on the disc. The torque due to the by the usual viscous torque, but he argues that the viscous angular momentum carried by the accreting matter from the torque is too weak to do this. inner edge of the disc is Safier (1998) goes a bit further and puts into question ˙ 1/2 whether the magnetic field of a T Tauri-star is a closed dipole τaccr = M (GMr0) , (1) field. He notes that the magnetic activity of a T Tauri star where M˙ is the accretion rate, G the gravitational constant, will heat up the atmosphere of the star and thus produce a M the mass of the accreting star, r0 the inner radius of the corona and a stellar wind. This stellar wind should not be disc. For the rest of this paper it is useful to write confused with the jets, that are typically much more massive, but it dominates the magnetic field of the star, which thus r0 = ξrA, (2) assumes an open topology as it is drawn out by the wind. The open field lines are mainly radial so fewer field lines where rA is the Alfv´en radius 1/7 penetrate the accretion disc than for a dipole magnetic field. 2π2µ4 The large uncertainties in the models make it interest- rA = , (3) GMM˙ 2µ2 ing to try to simulate the interaction between the accretion  0  disc and the magnetosphere numerically. Such simulations µ0 is the permeability of free space, and µ is the magnetic are in themselves sensitive to the assumptions used to con- dipole moment of the accreting star. To calculate ξ a de- struct the initial state. A critical issue is the initial magnetic tailed model of the accretion disc is needed (Ghosh & Lamb

c 0000 RAS, MNRAS 000, 000–000 Magnetic torques 3

2 3 3 1979a), but its value is typically around 0.5 (e.g. Frank, King where the fastness parameter ωs = (r0/GM)/(rco/GM) & Raine 1992). The torque can now be written as (Elsner & Lamb 1977) is introduced.

2 1/14 1/2 τaccr = 2π ξ τ0, (4) 2.2 The torque due to the coupling between the where
stellar dipolar magnetic field and a 3 1/7 GMM˙ 2 µ2 dynamo-generated toroidal field in the disc τ0 = . (5) µ0 "
# It is now widely believed that the accretion is driven by a magnetic stress that is generated by a dynamo in the ac- The magnetic torque is the result of the coupling be- cretion disc (e.g. Brandenburg et al. 1995, Stone et al. 1996, tween the vertical magnetic field of the star and the toroidal Matsumoto & Tajima 1995). The torque needed to drive the magnetic field in the disc. The torque acting on the upper accretion is surface of the disc can be written as ∞ M˙ √GMr. (9) BzBφ τmag = 2π r rdr, (6) µ0 The magnetic torque, on the other hand, can be written as Zr0 where B is the vertical magnetic field, B the toroidal field BφBr z φ 2πr2Hr , (10) and µ0 the magnetic permeability of free space. There is µ0 a similar contribution, but with the opposite sign from the where 2H is the thickness of the disc. I write Bφ = γdynBr, lower surface, thus angular momentum is exchanged between where γdyn 10 (Brandenburg 1998). Equating Eqs. (9) and ∼ the disc and the star only if Bφ changes sign from the up- (10) the dynamo-generated toroidal field is per to the lower surface. This is true if Bφ is generated by the winding up of the stellar magnetic field, but it is Mµ˙ 0γ B = dyn (GMr)1/4 , (11) not true for a quadrupolar magnetic field generated by a φ 4πr2H disc dynamo. Linear mean-field αΩ-dynamos with a positive r α-effect (e.g. Stepinski & Levy 1990, Torkelsson & Bran- which falls off more slowly with r than the field that is gen- denburg 1994a) generate preferentially quadrupolar mag- erated by the winding up of the stellar dipole field in Sect. netic fields, but nonlinear αΩ-dynamos can generate a large 2.1. This may be a lower limit to Bφ as the work by Bran- range of different magnetic field configurations (Torkelsson denburg & Campbell (1998) suggests that the torque in Eq. (10) must be much larger than the torque in Eq. (9) to re- & Brandenburg 1994b). It is not necessary that Bφ is coher- ent across the entire accretion disc, because, as we will see distribute the angular momentum exchanged with the star. later, the magnetic torque is concentrated to region inside The Shakura-Sunyaev (1973) model predicts that H/r r1/8 for a geometrically thin, optically thick disc 2r0. ∝ in which the gas pressure dominates over the radiation pres- sure. Because this is sensitive to the opacity and equation 2.1 The torque due to the winding up of the of state, and to keep things simple, I will take H/r to be a stellar field in the disc constant. The torque due to the coupling between the stellar field and the dynamo-generated toroidal field in the disc is A toroidal field is generated in the disc because the angular then velocities of the disc and the star match only at the co- ∞ rotation radius, rco. Inside rco the disc rotates faster than 1 µ τmag,dyn = 2 2π r 3 Bφrdr = the star so that BzB < 0 at the upper surface of the disc, × µ0 r φ Zr0 and the disc is losing angular momentum to the star, whilst −1/2 2 4√3 H 4πγdynµ 2 1/4 co ˙ BzBφ > 0 outside r and the disc is gaining angular mo- 3 GMM r0 (12) mentum from the star. To calculate B one must know the 5 r 3µ0r0 φ   r magnetic diffusivity in the disc (e.g. Campbell & Heptinstall of either sign. This can be re-written using
Eq. (2) as 1998, Brandenburg & Campbell 1998). A simpler approach −1/2 1/28 − is to write Bφ = γBz, where the azimuthal pitch γ 3 is 4 23 4 1/2 5/4 H ∼ τmag,dyn = 2 π γdynξ τ0 (13) assumed to be constant (e.g. Ghosh & Lamb 1979a). Obvi- 5 r ously this approach must fail close to rco, where the shear This is in a sense an
upper limit to the torque caused by vanishes. Neglecting this technical complication I calculate the disc dynamo, as it requires the toroidal magnetic field the torque acting on the star due to the interaction between to be aligned across the entire disc, but it is potentially the the stellar dipole field and the disc largest component of the total torque, and must therefore rco 2 ∞ 2 be taken into account. γµ 2 γµ 2 τmag = 2 2π 6 r dr 6 r dr = × µ0r − µ0r Zr0 Zrco  2 3 4πγµ r0 3 DISCUSSION: THE SPIN EVOLUTION OF 3 1 2 , (7) 3µ0r0 − rco DISC-ACCRETING X-RAY PULSARS     where µ is the magnetic dipole moment. Substituting r0 from X-ray pulsars are the best laboratories for studying the ex- Eq. (2) we get change of angular momentum between the accretion disc 1/7 and the accreting star, because the moment of inertia of a 2 (16π) −3 2 τmag = γξ 1 2ω τ0, (8) 3 − s neutron star is comparatively small, and an X-ray pulsar
c 0000 RAS, MNRAS 000, 000–000 4 U. Torkelsson can be timed accurately. The torque is measured from the X-ray flux in the 20 - 60 keV band with the spin-down rate change in spin frequency of GX 1+4. There is no clear correlation, and the largest τ spin-down rates seem to occur at the highest X-ray fluxes, ν˙ = , (14) 2πI which contradicts the Ghosh & Lamb model. Cen X-3 shows an altogether different behaviour. On where I is the moment of inertia of the neutron star. Adding − − average it is spinning up at a rateν ˙ = 8 10 13 Hz s 1, but Eqs. (5), (8) and (13) the spin change is × BATSE revealed a lot of fine structure in its spin evolution, −14 −1 −1 3/7 ˙ 6/7 2/7 and it is alternating between spin-up and spin-down phases ν˙ = 2.4 10 Hz s I40 m M14 µ20 τ,˜ (15) − − − − × withν ˙ = 7 10 12 Hz s 1 andν ˙ = 3 10 12 Hz s 1, respec- × − × where tively (Bildsten et al. 1997). Typically it spends 10 to 100 1/2 −1/2 days in one state before switching to the other state faster 1/2 γ 2 γdyn H τ˜ = 1.2ξ +1.2 1 2ωs 1.7 ,(16) than can be resolved by BATSE. OAO 1657-415 appears to ξ3 − ± ξ5/2 r     be a sister system of Cen X-3’s although the companion has 40 2
14 −1 I40 = I/10 kgm , m = M/M , M˙ 14 = M/˙ 10 kg s , not been identified. 20 3 and µ20 = µ/10 Tm . The Ghosh⊙ & Lamb (1979b) model, The existence of two states of comparable but oppo- which corresponds to the first two terms above, predicts that sitely directed torques is not expected from the Ghosh & at a sufficiently low accretion rate the torque is spinning Lamb model. It is, though, a natural consequence, ifτ ˜ is down the neutron star, but as the accretion rate increases dominated by its last term, the coupling between the stellar and the inner disc edge is pushed closer to the neutron star, magnetic field and the dynamo-generated field in the disc. the torque changes sign, and the spin up torque increases In that case we expect the spin-up torque to be somewhat with increasing accretion rate. At high accretion rates larger than the spin-down torque in agreement with the ob- servations. M˙ 6/7 ν˙ . (17) There are two time scales that must be explained, the ∝ ν2 time scale over which the torque is constant, and the time It is difficult to derive any form of similar relation when the scale for reversing the torque. These time scales are 10 - angular momentum exchange is dominated by the coupling 100 days, and less than 10 days, respectively, for Cen X-3. to a dynamo-generated magnetic field in the disc. In the cases of GX 1+4 and 4U 1626-67, on the other hand, The amount of timing data of X-ray pulsars has in- the torque remains constant for several years, and the sparse creased dramatically since the launch of BATSE/CGRO. sampling at the time of torque reversal provides a generous Bildsten et al. (1997) have recently presented a compila- upper limit for the time scale of the torque reversal. The tion of 5 years of monitoring of X-ray pulsars. Most X-ray time scale for reversing the torque should be the same as the pulsars are fed by stellar winds from their supergiant com- time scale for reversing the magnetic field via diffusion in my panions, and are therefore irrelevant for this paper. There model. I assume the diffusive time scale to be comparable are two groups of X-ray pulsars that have accretion discs. to the viscous time scale These are the Be/X-ray transients and an inhomogeneous 2 −2 r −1 H −1 group of binaries with steady accretion discs. The neutron tvis = αSS Ω , (18) ∼ νturb r stars in the Be/X-ray transients sometimes pick up an accre-   tion disc at the time of their periastron passage, which leads where νturb = αSSHcs is the turbulent viscosity, αSS the to an outburst of X-rays. The observations of transient X- Shakura-Sunyaev (1973) parameter, and cs the sound speed. ray pulsars, such as EXO 2030+375 (e.g. Reig & Coe 1998) The dynamo torque is so strongly concentrated to the inner and A0535+262 give some support to the Ghosh & Lamb part of the disc that half of the torque is due to the disc in- model as the X-ray pulsars spin up during the outbursts, side 1.74r0. The viscous time scale at this radius is typically and that the spin-up rate decreases as the X-ray flux goes less than 0.5 days. down, but a major uncertainty is the relation between the It is much less clear what sets the time scale over which observed X-ray flux and the accretion rate (Bildsten et al. the magnetic torque is constant. The ratios of the intervals 1997). between torque reversals and the orbital periods may be The persistent X-ray pulsars with accretion discs are roughly comparable for Cen X-3 and GX 1+4, though only comparatively few and make up an inhomogeneous set. The lower limits to the orbital period of the latter are known systems I will discuss in the following are 4U 1626-67, GX (Chakrabarty & Roche 1997). The time interval between 1+4, Cen X-3 and OAO 1657-415, in which the mass losing the torque reversals is comparable to the viscous time scale stars are a degenerate dwarf, a red giant, an O6-8 supergiant, for the entire disc, which suggests that the time scale is and an OB supergiant respectively. 4U 1626-67 had been set by the time over which the disc can support a given − − spinning up steadily at a rateν ˙ = 8.5 10 13 Hz s 1 from its field configuration against diffusion. This speculation fails × discovery by Uhuru (Giacconi et al. 1972) until the beginning completely in the case of 4U 1626-67, which is the smallest of the BATSE observations, but it is now spinning down just of the systems I have discussed, and yet the torque remains − − as steadily at a rateν ˙ = 7.2 10 13 Hz s 1 (Chakrabarty constant over a time scale of several years. − × et al. 1997a). GX 1+4 is similar in the sense that it was also Some other mechanisms have been proposed to explain − − discovered in a state of spinning up atν ˙ = 6.0 10 12 Hz s 1 the transitions between spin-up and spin-down states in X- × (Davidsen, Malina & Bowyer 1977, Nagase et al. 1989), but ray pulsars. van Kerkwijk et al. (1998) note that the ac- since the days of Ginga (Makishima et al. 1988) it has been cretion disc can be subject of a warping instability due to − − spinning down atν ˙ = 3.7 10 12 Hz s 1. Chakrabarty et the irradiation from the neutron star. This warp may be so − × al. (1997b) have studied the correlation between the pulsed extreme that the inner part of the accretion disc flips over

c 0000 RAS, MNRAS 000, 000–000 Magnetic torques 5 and rotates in the opposite direction, which would lead to Campbell C. G., Heptinstall P. M., 1998, MNRAS, in press a torque reversal. Yi, Wheeler & Vishniac (1997) suggest a Chakrabarty D., et al., 1997a, ApJ, 474, 414 modification of the Ghosh & Lamb model. The torque rever- Chakrabarty D., et al., 1997b, ApJ, 481, L101 sals are due to small changes in the accretion rate, but the Chakrabarty D., Roche P., 1997, ApJ, 489, 254 inner part of the accretion disc changes between a standard Davidsen A., Malina R., Bowyer S., 1977, ApJ, 211, 866 Elsner R. F., Lamb F. K., 1977, ApJ, 215, 897 thin disc and an advection-dominated flow at the same time. Frank J., King A. R., Raine D., 1992, Accretion power in astro- The advection-dominated flow is less efficient in radiating physics, Cambridge University Press energy, which can solve the problem of the lack of a corre- Fryxell B. A., Taam R. E., 1988, ApJ, 335, 862 lation between the torque and the observed X-ray flux. A Ghosh P., Lamb F. K., 1979a, ApJ, 232, 259 third possibility was suggested by Nelson et al. (1997). The Ghosh P., Lamb F. K., 1979b, ApJ, 234, 296 spin-down can be explained if the disc is retrograde, which Giacconi R., Murray S., Gursky H., Kellogg E., Schreier E., was first suggested by Makishima et al. (1988). It is difficult Tananbaum H., 1972, ApJ, 178, 281 to see how a retrograde disc can appear in a Roche-lobe Goodson A. P., Winglee R. M., B¨ohm K.-H., 1997, ApJ, 489, 199 overflow, but numerical simulations of accretion from winds Hayashi M. R., Shibata K., Matsumoto R., 1996, ApJ, 468, L37 have shown that a temporary disc rotating in the ’wrong’ Lightman A. P., 1974, ApJ, 194, 429 Lightman A. P., Eardley D. M., 1974, ApJ, 187, L1 direction may appear (e.g. Fryxell & Taam 1988). Li J., Wickramasinghe D. T., R¨udiger G., 1996, ApJ, 469, 696 Makishima K., et al., 1988, Nat, 333, 746 Matsumoto R., Tajima T., 1995, ApJ, 445, 767 4 SUMMARY Miller K. A., Stone J. M., 1997, ApJ, 489, 890 Nagase F., 1989, PASJ, 41, 1 I have shown that the torque acting between an accretion Nelson R. W., et al., 1997, ApJ, 488, L117 disc and an accreting star can be enhanced by the pres- Rast¨atter L., Neukirch T., 1997, A&A, 323, 923 ence of an intrinsic magnetic field in the accretion disc. The Reig P., Coe M. J., 1998, MNRAS, 294, 118 orientation of the magnetic field in the disc, and thus the Safier P. N., 1998, ApJ, 494, 336 direction of the magnetic torque between the disc and the Shakura N. I., Sunyaev R. A., 1973, A&A, 24, 337 star, is arbitrary, because the dynamo does not have any Stepinski T. F., Levy E. H., 1990, ApJ, 362, 318 information on the rotation of the neutron star. In particu- Stone J. M., Hawley J. F., Gammie C. F., Balbus S. A., 1996, ApJ, 463, 656 lar it is therefore possible to reverse the torque by reversing Torkelsson U., Brandenburg A., 1994a, A&A, 283, 677 the magnetic field in the disc. This mechanism may explain Torkelsson U., Brandenburg A., 1994b, A&A, 292, 341 the observed torque reversals in some X-ray pulsars that are van Kerkwijk M. H., Chakrabarty D., Pringle J. E., Wijers R. A. fed by accretion discs. The short time scale for the reversals M. J., 1998, ApJ, submitted compared to the long time scale over which the torque re- Wang Y.-M., 1997, ApJ, 487, L85 mains the same may be explained as the difference between Yi I., Wheeler J. C., Vishniac E. T., 1997, ApJ, 481, L51 the diffusive (viscous) time scales for the inner region of the disc, to which the torque is concentrated, and the entire ac- cretion disc. A problem with this connection is that it cannot explain the stability of the torque in the short-period X-ray binary 4U 1626-67.

ACKNOWLEDGEMENTS I acknowledge the support of an EU post-doctoral fellow- ship, and I thank Axel Brandenburg for reading a draft ver- sion of this paper.

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