{'fraction_non_alphanumeric': 0.026230269266480966, 'fraction_numerical': 0.005571030640668524, 'mean_word_length': 4.464172479391249, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Magnetic fields in Accretion Discs around Neutron Stars - Consequences for the change of spin

Accretion disks are ubiquitous in the universe and it is generally accepted that magnetic fields play a pivotal role in accretion-disk physics. The spin history of millisecond pulsars, which are usually classified as magnetized neutron stars spun up by an accretion disk, depends sensitively on the magnetic field structure, and yet highly idealized models from the 80s are still being used for calculating the magnetic field components. We present a possible way of improving the currently used models with a semi-analytic approach. The resulting magnetic field profile of both the poloidal and the toroidal component can be very different from the one suggested previously. This might dramatically change our picture of which parts of the disk tend to spin the star up or down.


Introduction
Neutron stars (NSs) in binary systems can constantly attract matter from their companions. Depending on its angular momentum, matter can reach the NS through a spherically symmetric wind, or can instead create a disk-like structure around the star. If there were no viscosity in the disk, it could be in perfect Keplerian rotation and no matter would accrete onto the NS. However what we see from observations is that accretion does occur, therefore there must be some kind of viscosity in the disk which is responsible for transporting angular momentum outward and matter inward. Molecular viscosity proved to be too small to justify the required accretion rates. The best possible source for this viscosity is turbulence, which could be generated by the magnetic field through the magneto-rotational instability.
In addition to generating the turbulence, magnetic fields can have many other effects on the whole system, such as modifying the spectral properties and launching jets or winds. We are here interested in the spin history of the central object and we consider the recycled pulsar scenario, where an old NS is spun up by an accretion disk to become a millisecond pulsar. Within this scenario the magnetic field plays a very important role because it creates a strong link between part of the disk and the star. Therefore the whole disk contributes to angular momentum exchange with the NS and not only the accreted matter in the inner disk region. Moreover the magnetic torque can have both signs, i.e. it can both spin the star up or down.
Currently, simplified models from the 80s are used to calculate both the magnetic field structure and the related magnetic torque. Our goal is to improve on those models by following a step-by-step procedure, which can also help us in deconstructing what we see from large numerical simulations.

Current Models
Although large numerical calculations are presently available, simplified analytic models developed in the 80s are still being used. These take the poloidal component of the magnetic field B pol to be a perfect dipole, the disk to be thin and neglect all radial derivatives with respect to vertical ones. Moreover disk matter is assumed to have only toroidal motion with Keplerian angular velocity. By further making the kinematic approximation, i.e. no feedback from the magnetic field on the velocity field, the following analytic solution is found (Wang 11 and Campbell 1 ): where ̟ is the cylindrical radius; γ a is the amplification factor (which is is of the order of unity and depends on the steepness of the vertical transition from Keplerian rotation inside the disk to corotation with the star at the top of the disk); Ω disk and Ω s are the disk and stellar angular velocities, respectively; and τ d is the dissipation time scale. Equation (1) states that the disk rotation distorts the original magnetic field and creates a toroidal component, B φ , which is proportional both to B z and to the relative angular velocity ∆Ω between the disk and the central star.
Once the magnetic field structure is known, one can calculate the magnetic torque exerted by the disk on the star. This is usually done following Ghosh and Lamb 2,3 who, by making the same assumptions as for Eq. (1), obtained: This equation states that (i) the inner part of the disk (i.e. r < r cor a ) provides a spin-up torque and that (ii) the outer part of the disk instead gives spin-down.
Many investigators have simulated the magnetosphere-disk interaction, solving the full set of the MHD equations (see e.g. Romanova et al. 9 ). The work that we a The corotation radius rcor is defined as the radius where a Keplerian disk rotates with the same angular velocity as the central star.
are presenting here should not be seen as being in competition with these analyses, but rather as being complementary to them, by providing both a useful test case and a means for getting a better understanding of the underlying processes.

Our strategy
The conceptual papers from the 80s mentioned above, continue to be widely quoted and used as the basis for new research (see e.g. Kluzniak and Rappaport 5 ) and our work here stands in the tradition of refining these approaches. Our strategy can be summarized with the following four steps: (i) begin by defining an appropriate simple model; (ii) obtain a reduced set of equations for the model; (iii) solve for the magnetic field structure; (iv) solve for the torque exerted on the neutron star.

Initial model
Our initial model represents a rotating NS, with a dipole magnetic field aligned with the rotation axis. The disk is truncated at the Alfven radius and has a corona above and below it, treated as a boundary layer between the disk and surrounding vacuum. We impose dipole boundary conditions (BCs) on all boundaries, assume axisymmetry (∂ φ [. . . ] = 0), stationarity (∂ t [. . . ] = 0) and neglect the magnetic feedback.
The model is fully 2D (no vertical integration or Taylor expansion) and includes all components of v. We make no assumption about the magnetic field components and solve self-consistently for both the poloidal and toroidal components.
For the velocity field in our first model 6,7 we used the Shakura and Sunyaev profile 10 (SS), suitably modified to make it consistent with dipolar BCs and fully 2D. In our second model 8 we used a more general 2D profile, with a backflow in the equatorial region (Kluzniak and Kita 4 , KK).

Equations
We consider the induction equation for mean fields: where η T is the turbulent magnetic diffusivity and dynamo effects are neglected. We then write this equation in spherical coordinates and apply all of the hypotheses mentioned earlier. This gives the following set of equations: There is the partial decoupling in this set of equations, i.e. the equations describing the poloidal structure do not contain any azimuthal quantity. We can therefore solve Eqs. (3) and (4) first and then use the results to solve Eq. (5). We solve this set of partial differential equations numerically, by using the Gauss-Seidel relaxation method on a grid whose resolution is ∆r ∼ 0.7 r g and ∆θ ∼ 0.1 • , where r g ∼ 4 km is the Schwarzschild radius for a 1.4 M ⊙ NS.

Magnetic field structure
In the previous simplified models the poloidal component of the magnetic field was taken to be a pure dipole, however, we find that deformations of B pol can be quite substantial as a result of the poloidal motion of the matter. The left panel of Fig. 1 shows a plot of the magnetic field lines in a vertical section through the disk. The toroidal field can also be very different from the predictions of Eq. (1). As shown in the right panel of Fig. 1, it has a fully 2D structure with multiple zeros which are not only at the corotation radius. To facilitate comparison, in the left panel of Fig. 2 we plot the shell average of B φ for two of our models, and as given by Eq. (1).

Torque
The torque exerted on the NS by the disk is the moment of the Lorentz force: We calculate T B−disk without neglecting radial derivatives and using our previous results for the magnetic field. We studied ∼ 30 configurations and found that, even with the same values of the NS period and magnetic field, the rate of period changė P can be positive, negative or zero depending on value of the poloidal magnetic Reynolds number. We findṖ to be in the range:

Conclusions
Simplified models from the 80s are still being used for describing magnetic fields in accretion disks. We have studied more realistic models and found very different results. B pol shows large deviations away from the dipole that is assumed in the previous models: the field lines are pushed inward and accumulate in the inner part.