The structure of thin accretion discs around magnetised stars
S. B. Tessema^{1,2}  U. Torkelsson^{2}
1  Department of Physics, Addis Ababa University,
PO Box 1176, Addis Ababa, Ethiopia
2  Department of Physics, University of Gothenburg, SE 412 96 Gothenburg, Sweden
Received 13 July 2009 / Accepted 30 September 2009
Abstract
Aims. We determine the steadystate of an axisymmetric thin accretion disc with an internal dynamo around a magnetised star.
Methods. Starting from the vertically integrated equations of
magnetohydrodynamics we derive a single ordinary differential equation
for a thin accretion disc around a massive magnetic dipole and
integrate this equation numerically from the outside inwards.
Results. Our numerical solution shows that the torque between
the star and the accretion disc is dominated by the contribution from
the dynamo in the disc. The location of the inner edge of the accretion
disc varies between
and
depending mainly on the strength and direction of the magnetic field generated by the dynamo in the disc
Key words: accretion, accretion discs  magnetohydrodynamics (MHD)  magnetic fields  stars: neutron  Xrays: stars  stars: premain sequence
1 Introduction
In this paper we present a new solution for an accretion disc around a magnetic star. This star could be a neutron star, a white dwarf, or a T Tauristar, but we assume that it is a neutron star since it is easy to measure the torque between the neutron star and the accretion disc by timing the Xray pulses from the neutron star. The new feature of our solution is to include the effect of an internal dynamo in the accretion disc. By doing this we hope to be able to explain the torque reversals that have been observed in some Xray pulsars.
Shakura & Sunyaev (1973) formulated the standard model of a geometrically thin, optically thick accretion disc. They were able to obtain an analytical solution of the heightintegrated hydrodynamic equations, after having introduced the prescription for the turbulent stress, which transports the angular momentum outwards through the disc; however, they did not explain why the disc is turbulent in the first place, since a disc in Keplerian rotation is stable according to Rayleigh's criterion. Balbus & Hawley (1991) instead showed that it is unstable if there is a weak magnetic field in the disc. Subsequent numerical simulations (e.g. Hawley et al. 1995; Balbus & Hawley 1998) confirmed that this instability generates turbulence and that the resulting turbulent stresses transport angular momentum outwards.
The interaction between a magnetised star and a surrounding accretion disc is one of the most poorly understood aspects of accretion. At the same time, it is central for our understanding of the spin evolution of objects as diverse as T Tauri stars and Xray pulsars. The magnetic field of the star penetrates the surrounding accretion disc and couples the two. According to the Ghosh & Lamb (1979b,a) model, the part of the accretion disc that is located inside the corotation radius provides a spin up torque on the star, since it is rotating faster than the star, while the more slowly rotating outer part of the accretion disc brakes the star. The net torque is determined by the location of the inner edge of the disc, which moves inwards as the accretion rate increases, thereby increasing the spin uptorque on the star.
Campbell (1997) proposes physical descriptions of the magnetic diffusivity in terms of turbulence or buoyancy, and Campbell & Heptinstall (1998a,b) solve the resulting equations numerically. For both forms of diffusivity, the magnetic coupling between the disc and the star leads to an enhanced dissipation in the inner part of the accretion disc compared to the standard Shakura & Sunyaev (1973) model. This raises the temperature such that electron scattering dominates Kramer's opacity at larger radii than is otherwise the case, thus increasing the fraction of the disc that is subject to the (Lightman & Eardley 1974) instability. Brandenburg & Campbell (1998) considered a form of magnetic diffusivity that allows further analytical progress to be made, but the qualitative results remain the same.
All the models predict a positive correlation between the accretion rate and the torque on the neutron star and even predict a negative torque on the neutron star at very low accretion rates. Timing of Xray pulsars during outbursts of Be/Xray transients have provided at least some qualitative support for such a correlation (e.g. Parmar et al. 1989). The BATSE instrument on the Compton Gamma Ray Observatory made it possible to extend this database significantly (Bildsten et al. 1997). In particular there are a few Xray pulsars with permanent discs that are oscillating between phases of constant spinup and constant spindown without a significant difference in the Xray luminosity between these states, which appears to contradict the standard model for a discaccreting Xray pulsar.
Nelson et al. (1997) propose that these torquereversals can be the result of transitions between corotating and counterrotating accretion discs, though several other models have also been proposed. Torkelsson (1998) argue that the torque between an accreting star and its disc can be enhanced by the presence of a magnetic field generated by the turbulence in the accretion disc. The torque reversals are then the result of a reversal of the magnetic field generated by this dynamo. However, he did not construct a selfconsistent model of the accretion disc. The aim of this paper is to construct such a model of an accretion disc with an internal dynamo around a magnetic star. We work in the spirit of Shakura & Sunyaev (1973) and assume that the disc is geometrically thin. In Sect. 2 we start from the equations of magnetohydrodynamics (MHD) and derive a single ordinary differential equation for the radial structure of the accretion disc. We then present numerical solutions of this equation in Sect. 3 and discuss the properties of these solutions in Sect. 4. Finally we summarise our conclusions in Sect. 5.
2 Mathematical formulation
We study a steady, thin axisymmetric Keplerian disc around a star with a magnetic dipole field. The basic equations describing the structure of the thin accretion disc can be derived from the equations of magnetohydrodynamics.
2.1 Conservation of mass
In steady state the continuity equation takes the form
where is the density and is the fluid velocity with radial, azimuthal, and vertical components, respectively. For a thin axisymmetric disc and after neglecting a vertical outflow from the disc, we get
where is the surface density
and H is the halfthickness of the disc. For a steady disc the integral of Eq. (2) gives
which is the accretion rate.
2.2 Conservation of momentum
If assuming a steady state the NavierStoke's equation can be written as
where P is pressure, kinematic viscosity, the gravitational potential the current density, and the magnetic field. The viscosity is in general low, and we only retain it where it plays a crucial role.
The radial component of Navier Stoke's equation is
For a thin accretion disc as shown below and the dominant terms of the equation give us
which shows that the disc rotates in a Keplerian fashion.
In similar manner, the vertical component of the momentum equation for a
steady flow is
Neglecting vertical outflows and assuming the magnetic field to be weak the equation reduces to the equation of hydrostatic equilibrium
Using H as the halfthickness of the disc, the pressure at the midplane of the disc is
but the hydrostatic equilibrium can also be expressed as
which shows that the Keplerian velocity is highly supersonic in a thin accretion disc, as assumed above.
The azimuthal component of NavierStoke's equation
reduces to
We neglect because the radial length scale is much longer than the vertical length scale in a thin accretion disc. Integrating Eq. (12) vertically across the disc and multiplying both sides by R, we get
where the specific angular momentum The magnetic term describes the exchange of angular momentum between the disc and the star via the magnetosphere. This term vanishes if is an even function of z, but the shear between the disc and the stellar magnetosphere generates an odd whose value in the upper half of the disc is
where , is the angular velocity of the star, and is a dimensionless parameter of a few (Ghosh & Lamb 1979b,a).
In this paper we consider the effect of adding
a largescale toroidal field
that is generated by an internal dynamo in the accretion disc. Such
a dynamo is a natural consequence of the magnetohydrodynamic turbulence in the
accretion disc (e.g. Balbus & Hawley 1998). To estimate the size of
,
we assume for the moment that the viscous stress in the accretion disc is due to the internal magnetic stress
where we use the Shakura & Sunyaev (1973) prescription for the viscosity in the last equality. Based on the results of numerical simulations of magnetohydrodynamic turbulence in accretion discs (e.g. Brandenburg et al. 1995) Torkelsson (1998) argues that
where . However, this is the sum of the largescale field and a smallscale turbulent field, that is also contributing to the stress through its correlation with a turbulent B_{R}field. Since the largescale field might be a small fraction of the total field we multiply with a factor to get an estimate for
where , and a negative value describes a magnetic field which is pointing in the negative direction at the upper disc surface.
We can now estimate the magnetic pressure in the accretion disc, to which
the toroidal magnetic field is the main contributor. According
to Eq. (17), the
pressure is approximately
(18) 
Since we use , and in our models, we see that the magnetic pressure will not significantly affect the vertical structure of the accretion disc.
The vertical magnetic field can likewise be split up into two
components; (i) the stellar dipolar magnetic field, whose value in the stellar
equatorial plane is
(19) 
where is the magnetic dipole moment; and (ii) a dynamo component . There is no obvious way to model within our onedimensional model, but numerical simulations like those by Brandenburg et al. (1995) suggest that B_{z} and B_{R} are comparable, so we expect that . In fact, will modify the structure of the poloidal magnetic field, but this can happen even if there is no internal dynamo in the disc because of the currents that are induced in the disc (e.g. Bardou & Heyvaerts 1996).
We can now expand the product
as
=  
=  
(20) 
As we see below the term is significantly greater than almost everywhere in the disc. The term is smaller in size than by a factor , and since it is difficult to model it, we ignore it. Finally the term does not contribute directly to the exchange of angular momentum between the accretion disc and the accretor, though it does affect it indirectly by contributing to the radial transport of angular momentum through the disc. Although this term can be important, we have decided to ignore it since there is no obvious way to model it in our onedimensional approach. One should notice here that this does not change the main qualitative conclusion of this paper that an internal dynamo in the accretion disc makes a significant contribution to the exchange of angular momentum between the disc and the star; on the contrary, we would see a stronger effect if we were to keep the term. We now expand the term in Eq. (13)
2.3 Conservation of energy
For a slow inflow of matter through an optically thick disc,
the local viscous dissipation
is balanced by the
radiative losses
This gives us
where is the temperature at the midplane of the disc, and the StefanBoltzmann constant. The optical depth of the disc is given by
and we assume that the opacity is given by Kramer's law
where m^{5} K^{7/2}.
Equation (22) should also contain a term describing the
magnetic dissipation
(25) 
where is the magnetic diffusivity, and we have used Eq. (17) in the last equality. This should be compared to
(26) 
so we see that the magnetic dissipation is negligible as long as .
2.4 Structure equations
We reduce these equations to a single ordinary differential equation for the
radial structure of the accretion disc. First we assume the equation of state
of an ideal gas,
where is the Boltzmann constant, the mean molecular weight, and the mass of a proton, but the pressure can also be expressed using the equation of hydrostatic equilibrium
which gives us a relation between H and
The viscous stress tensor gives us the equation
which we solve for the density of the gas
The optical depth of the disc is
Using Eqs. (22) and (32) we get
where and
The pressure is then given by
where
The component of the magnetic field generated by the internal dynamo can be expressed using Eqs. (17) and (35) as
where
The magnetic field due to the shear can be written as:
where
is the corotation radius, at which the Keplerian angular velocity is the same as the stellar angular velocity. Here is the spin period of the star, and . Equations (37) and (39) show that the magnetic field generated by the internal dynamo varies more slowly with radius than the magnetic field due to shear, . Thus the dynamo component dominates at large radii.
Equation (21) gives us an ordinary differential equation for y
where
and
The solution of Eq. (41) approaches the ShakuraSunyaev solution at large radii, thus giving us the boundary condition as
We introduce the dimensionless variable
through
and a dimensionless radial coordinate through
where is the Alfvén radius,
which is given by putting the magnetic pressure equal to the ram pressure of the accreting fluid (e.g. Frank et al. 2002). Here represents the mass transfer rate in units of 10^{13} kg s^{1}, and is the stellar magnetic dipole moment in units of 10^{20} T m^{3}. The boundary condition is then as , and Eq. (41) can be written as
where
and
is the fastness parameter.
3 Numerical solution
3.1 Global solutions
We integrate Eq. (47) inwards from a large radius, usually , to which we impose the boundary condition that , which is similar to the approach by Brandenburg & Campbell (1998), but most studies of accretion discs have rather applied a boundary condition at the inner edge of the disc, and then integrated the equations outwards.
There are two possible boundary conditions that can be applied at the inner
edge of the accretion disc, either that
,
which corresponds
to that
and
at the inner edge (case D), or that
(50) 
which means that the viscosity does not at all contribute to driving the accretion at this radius. Case D is the boundary condition that has been most widely used and that was adopted for instance by Shakura & Sunyaev (1973). In this case the density drops to zero because the inflow velocity becomes infinite, which is of course not realistic, and a more accurate treatment shows that the inflow becomes transonic close to this position Paczynski & BisnovatyiKogan (1981). In case V the inflow at the inner edge of the accretion disc is driven completely by the transfer of excess angular momentum from the accreting matter to the stellar magnetic field (e.g. Wang 1995).
For our fiducial model, we take a neutron star of and a magnetic moment of 10^{20} T m^{3}, which is accreting at . The dimensionless parameters and are, respectively, 1 and 10 in our fiducial model, while . The exact values of and are unimportant, since we vary the parameter below, but influences the solutions in its own way as is shown at the end of this section. We consider three different spin periods with corresponding corotation radii and fastness parameters (see Table 1). The system goes into the propeller regime for (e.g. Illarionov & Sunyaev 1975; Ghosh & Lamb 1979a,b).
Table 1: Spin parameters of the models.
Firstly, for a spin period of 7 s and and  1, we get the solutions shown from the top to the bottom of Fig. 1. The and 0.1 solutions have case D inner boundaries at 4.7 and 1.6 , respectively, while the other three solutions have case V inner boundaries (Fig. 2). The inner boundary is close to if which corresponds to the absence of an internal disc dynamo, but moves outwards as increases. In case V, the solution continues inside the inner edge of the accretion disc, but the viscosity counteracts the accretion, which is instead driven by the magnetic stresses. This regime has been described by Campbell (1998), who discusses how the disc is disrupted in this region, we assume that this region belongs to a boundary layer that we do not attempt to model in this paper. On the other hand, all solutions approach the ShakuraSunyaev solution at large radii, as required by our boundary condition.
Figure 1: for our fiducial neutron star with a spin period of 7 s and from the top to the bottom. 

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Figure 2: , as a function of r for the fiducial neutron star with a spin peroid of 7 s and with , and 0 from the bottom to the top. 

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By increasing the spin peroid to 100 s, we see that the solution also develops a case V inner boundary (Fig. 3), but there are only small quantitative changes for the and +1 solutions. (The inner boundary of the solution is now located at .) The dependence of the solution on the spin period can be better seen in Fig. 4, where we vary the spin period, when is fixed to 0. Then has a local minimum for s, but this minimum weakens and disappears as the spin period is increased. This is similar to how depends on . For has an unphysical negative local minima at a small radius, such that the physical solution has a case D boundary at several Alfvén radii. As is increased, the local minimum grows and the solution develops a case V inner boundary when the minimum becomes positive. Increasing removes the local minimum further and the solution becomes a strictly decreasing function of r, which asymptotically approaches as required by our boundary condition. Increasing has a similar effect to decreasing (Fig. 5).
3.2 The structure of an accretion disc
The physical structure of the accretion disc can now be expressed
using
and
We have that
In Fig. 6 we show the surface density as a function of radius for the fiducial neutron star with a spin peroid of 7 s and , 0, and 1, respectively. For the surface density attains a local maximum, while the other models have surface densities that are strictly decreasing functions of r. Figure 7 shows the radial velocity as a function of r. Since , it becomes infinite as the inner edge of the disc for , but it stays finite for and 1 and eventually goes to 0 in the boundary layer. Increasing the spin period of the neutron star has a very marginal effect on the accretion disc, though , which is proportional to , increases somewhat (Fig. 8).
Figure 3: for our fiducial neutron star with a spin period of 100 s and from the top to the bottom. 

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Figure 4: for our fiducial neutron star with and spin periods of 100, 18.7, and 7 s from the top to the bottom. 

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Figure 5: for our fiducial neutron star with spin period of 7 s and and and 5 (upper dashed, dotted, and solid lines, respectively), and , and 5 (lower dashed, dotted, and solid lines, respectively). 

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Figure 6: as a function of r for the fiducial neutron star with a spin period of 7 s and (solid line), (dashed line), and (dotted line). 

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Figure 7: V_{R} as a function of r for the fiducial neutron star with a spin period of 7 s and (solid line), (dashed line), and (dotted line). 

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We plot the magnetic fields for our fiducial model with varying and spin periods of 7 and 100 s, respectively, in Figs. 9 and 10, respectively. The corotation radius, at which changes sign, occurs inside of the inner edge of the accretion disc for and 1, and for these values of , is the dominant magnetic field everywhere inside the disc.
In general the ratio of the magnetic field due to shear to that generated by the internal dynamo is
It is only during rather extreme conditions that the shearinduced field can dominate at small radii, and the dynamo is always dominant at large radii.
4 Discussion
4.1 The inner edge of the accretion disc
We have shown that there are two different forms of inner disc boundaries that can be found among our solutions. We denote these as case D and V, respectively. Case D occurs only for sufficiently negative , while case V covers a larger part of the parameter domain that we have studied. We summarise our values for the inner radius of the accretion disc in Table 2. We see that the inner radius can be significantly larger than the Alfvén radius when is close to unity. This is the result of the dynamo enhancing the coupling of the stellar magnetic field to the accretion flow, such that it dominates the viscous torque at larger radii than would otherwise be the case.
4.2 The angular momentum balance
Figure 8: as a function of r. The two lower curves show discs with around neutron stars with spin periods of 7 s (dashed line) and 100 s (solid line), respectively. The two upper curves show discs with around neutron stars with spin periods of 7 s (dashed line) and 100 s (dotted line), respectively. 

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Table 2: The inner edge of the accretion disc and its torque on the fiducial neutron star.
To understand the exchange of angular momentum between the accretion disc and its environment we multiply Eq. (21) by and integrate it from R_{0}, the inner radius of the disc, to R_{1} the outer edge of the disc(61) 
The lefthand side is the difference between the angular momentum that is advected out of the inner edge of the accretion disc and that which is fed into the disc at its outer edge and the righthand side describes the contribution of magnetic and viscous torques to the angular momentum balance. The term describes the viscous tension at the outer edge of the disc, which is not considered further in this paper. Rather, we consider the exchange of angular momentum between the accretion disc and the neutron star. Firstly, we have the angular momentum which is advected from the disc to the star
Secondly, we have the magnetic torques on the neutron star, which we divide into one part due to the shear
and a second part due to the dynamo
Finally the viscous stress at the inner edge of the accretion disc transports angular momentum outwards away from the neutron star resulting in a torque
This torque vanishes for a case D inner boundary. We can now calculate the torques on our fiducial neutron star for our choices of spin periods and . These results are summarised in Table 2. Unless we see that the magnetic torque due to the dynamo is always significantly stronger than the magnetic torque due to the shear, and both are stronger for than for . The reason for this effect is that the central hole in the disc grows too large when . The dominant torque at is therefore the viscous torque at the inner boundary, which has usually been ignored.
Figure 9: The toroidal magnetic field for the fiducial neutron star with a spin period of 7 s. The solid and dashed lines show the field generated by the dynamo for and 1, respectively, while the dotted line shows the magnetic field generated by the shear. 

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Figure 10: The toroidal magnetic field for the fiducial neutron star with a spin period of 100 s. The solid and dashed lines show the field generated by the dynamo for and 1, respectively, while the dotted line shows the magnetic field generated by the shear. 

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Let us now compare our results with the BATSE data (Bildsten et al. 1997).
The 7.6 s Xray pulsar 4U162667 was observed to spin down at a rate
and spin up at
.
These spin
variations correspond to torques
(66) 
where I is the moment of inertia of the neutron star, which we measure in 10^{38} kg m^{2}, and we measure the spin acceleration in units of 10^{13} Hz s^{1}. The required torque is significantly larger than what is produced by in any of our models, but is even an order of magnitude greater than this value if . This overestimate can be explained by that we have assumed a unidirectional across the disc surface, while it might be more realistic to expect that the toroidal field is organised in magnetic annuli with opposite polarities, or that is significantly smaller fraction of the total turbulent magnetic field.
We explain the torque reversals in the same way as in Torkelsson (1998), where the dynamo undergoes a field reversal, which in our model corresponds to changing sign. One might speculate that, during this field reversal, the disc passes through a state corresponding to , in which the disc will have a smaller inner radius than during the states with an active dynamo. The closer to the surface of the neutron star that the accretion disc extends, the more of the Xray emission from the accretion columns it can absorb and reprocess. Torkelsson (1998) showed that the observed time scales on which the torques remain constant in the sources 4U162667, 4U162667, and 4U162667 are comparable to the global viscous time scales of their accretion discs, which constrains the mechanism that is responsible for the reversals of the magnetic field.
5 Conclusions
We have investigated the interaction between a magnetic neutron star and its surrounding accretion disc in the case where the accretion disc is supporting an internal dynamo. The magnetic field that is produced by the dynamo can lead to a significant enhancement of the magnetic torque between the neutron star and the accretion disc, compared to what is seen in the model by Ghosh & Lamb (1979b,a). This extra magnetic torque can explain the large variations in spin frequencies of 4U162667 and 4U162667 (Bildsten et al. 1997). Furthermore, a reversal of the magnetic field that is generated by the dynamo, similar to the reversals that we see of the magnetic fields on the Sun, could explain the torque reversals in these objects.
From the way that we calculate the structure of the accretion disc, we find two kinds of solutions with different behaviours at the inner edge. A few of our solutions have case D boundaries at which the density and temperature go to 0 at finite radius, while most of our solutions have case V boundaries at whichthe accretion is driven entirely by the magnetic tension between the accreting matter and the neutron star. In this case there is a viscous stress between the accretion disc and the boundary layer, which can transfer angular momentum outwards at a rate that is comparable to the one at which it is advected inwards by the accreting matter itself.
We have also found that the dynamo leads to that the inner edge of the accretion disc occurs at a radius that is larger than the traditional Alfvén radius. This effect is weak, though, for a realistic value of the dynamogenerated magnetic field.
AcknowledgementsS.B.T. thanks the Department of Physics at the University of Gothenburg for hospitality and support during this project. SBT is supported in part by the Swedish Institute (SI) under their Guest Scholarship Programme. UT thanks the Department of Physics at Addis Ababa University for their hospitality. This research has made use of NASA's Astrophysics Data System. We thank an anonymous referee for comments that have improved the quality of the paper.
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All Tables
Table 1: Spin parameters of the models.
Table 2: The inner edge of the accretion disc and its torque on the fiducial neutron star.
All Figures
Figure 1: for our fiducial neutron star with a spin period of 7 s and from the top to the bottom. 

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In the text 
Figure 2: , as a function of r for the fiducial neutron star with a spin peroid of 7 s and with , and 0 from the bottom to the top. 

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In the text 
Figure 3: for our fiducial neutron star with a spin period of 100 s and from the top to the bottom. 

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In the text 
Figure 4: for our fiducial neutron star with and spin periods of 100, 18.7, and 7 s from the top to the bottom. 

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In the text 
Figure 5: for our fiducial neutron star with spin period of 7 s and and and 5 (upper dashed, dotted, and solid lines, respectively), and , and 5 (lower dashed, dotted, and solid lines, respectively). 

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In the text 
Figure 6: as a function of r for the fiducial neutron star with a spin period of 7 s and (solid line), (dashed line), and (dotted line). 

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In the text 
Figure 7: V_{R} as a function of r for the fiducial neutron star with a spin period of 7 s and (solid line), (dashed line), and (dotted line). 

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In the text 
Figure 8: as a function of r. The two lower curves show discs with around neutron stars with spin periods of 7 s (dashed line) and 100 s (solid line), respectively. The two upper curves show discs with around neutron stars with spin periods of 7 s (dashed line) and 100 s (dotted line), respectively. 

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In the text 
Figure 9: The toroidal magnetic field for the fiducial neutron star with a spin period of 7 s. The solid and dashed lines show the field generated by the dynamo for and 1, respectively, while the dotted line shows the magnetic field generated by the shear. 

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In the text 
Figure 10: The toroidal magnetic field for the fiducial neutron star with a spin period of 100 s. The solid and dashed lines show the field generated by the dynamo for and 1, respectively, while the dotted line shows the magnetic field generated by the shear. 

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In the text 
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