Disk accretion onto a magnetized star

The problem of interaction of the rotating magnetic field, frozen to a star, with a thin well conducting accretion disk is solved exactly. It is shown that a disk pushes the magnetic field lines towards a star, compressing the stellar dipole magnetic field. At the point of corotation, where the Keplerian rotation frequency coincides with the frequency of the stellar rotation, the loop of the electric current appears. The electric currents flow in the magnetosphere only along two particular magnetic surfaces, which connect the corotation region and the inner edge of a disk with the stellar surface. It is shown that the closed current surface encloses the magnetosphere. Rotation of a disk is stopped at some distance from the stellar surface, which is 0.55 of the corotation radius. Accretion from a disk spins up the stellar rotation. The angular momentum transferred to the star is determined.


INTRODUCTION
The problem of interaction of an accretion disk with a magnetized star is very important for understanding of compact X-ray sources. The energy release for a variable X-ray sources and Xray pulsars is due to the accretion of the matter from a companion star onto the surface of a compact star, namely, on a neutron star. Falling onto a neutron star, a particle gains the energy per unit mass equal to the value of the gravitational potential on the stellar surface ϕ = 0.15c 2 (Ms/M⊙)(Rs/10km) −1 . where Ms and Rs are stellar mass and radius, respectively, and M⊙ is the solar mass. The energy release depends on how the accretion is realized. The magnetic field, which is frozen to a neutron star, is high enough to prevent a free falling of the matter on a star. The naive point of view is that the accretion is free outside the so called Alfvén surface, and then continues along the magnetic field lines down to the polar caps of a star (Lamb 1989). In a standard approach Alfvén surface is the surface where the magnetic field pressure B 2 /8π is equal to the pressure of the accreting matter. But here it is not clear what is the pressure of the accreting gas. Is it the total dynamic pressure ̺v 2 /2 or its radial part only, ̺v 2 r /2 ? The Alfvén radius is estimated for the spherical accretion, v = vr, but then is applied to the disk accretion. But in this case the gas velocity v comes mainly from the velocity of disk rotation and is of the order of Keplerian velocity vK , which is much larger than the radial velocity. The gradient of the magnetic pressure in the azimuthal direction is small and therefore does not oppose significantly the disk rotation. Besides it is clear that the disk must compress the magnetic field within the disk inner edge. These features do not help to answer the question: where is the real position of the Alfvén surface.
An accreting gas is a plasma consisting of ions and electrons, which have a quite different Larmor radii in a magnetic field. Moving across the magnetic field, the plasma gets polarized and creates the large scale electric field E. If this field is E = −v × B/c then the plasma continues to drift in the crossed B, E fields at the same velocity v not slowing down. The real value of E depends on the boundary conditions on a disk and a star surface, on electric currents in a plasma and on conditions of their closing. The important characteristics are the electric conductivity of a disk plasma and the neutron star conductivity in the surface layers. The ionized plasma of an accretion disk has the conductivity σ = 10 13 (Te/1eV ) 3/2 (Λ/10) −1 s −1 , which is high enough to consider a disk as an ideal conductor. Here Te is the temperature of electrons in a disk, which is higher than 10eV ; Λ is the Coulomb logarithm (Λ ≃ 20). At such a high conductivity σ the width of the skin layer λ sk = (τ c 2 /σ) 1/2 is less than the disk width H. The value of τ is the characteristic time of turbulent motion in the α-disk, τ = H/vK . The condition σ ≫ c 2 /HvK is well fulfilled in the inner parts of a disk. As for the conductivity of surface layers of a neutron star σNS, then it is as high as 10 21 s −1 (Blandford et al. 1983;Potekhin 1999). Therefore, σNS ≫ σ, and a neutron star can be considered as ideal conductor too.
As ideal conductor, a disk tends to exclude the stellar magnetic field, pushing it toward a star. Heavy disk ions tend to rotate with the Keplerian velocity, but magnetized electrons are frozen to the magnetic field lines, which rotate with the angular velocity of a star ωs. Thus, the point ρ = ρc where a disk corotates together with a star, ωsρc = (GMs/ρc) 1/2 , ρc = GMs ω 2 s 1/3 = 1.5 · 10 8 Ms M⊙

1/3
Ps 1s is the point beginning from which (ρ < ρc) motions of ions and electrons differ essentially. Here G is the gravitational constant and Ps = 2π/ωs is the period of the stellar rotation. The region of corotation is namely the region where the interaction of an accreting disk with the stellar magnetic field begins. The point of corotation is inside the stellar light cylinder radius RL = c/ωs for all rotating neutron stars: Ps > 3 · 10 −5 (Ms/M⊙)s. On the other hand, the corotation radius is larger than the neutron star radius Rs. Because of that we will consider the undisturbed magnetic field of a neutron star as dipolar. Also we assume that the axis of a dipole is parallel to the neutron star rotation axis. The paper is organized as follows. In section 2 we describe an arbitrary axisymmetric magnetic field. In section 3 we find the motion of ions in a disk. Section 4 describes electric currents in the stellar magnetosphere. In section 5 we find the structure of the magnetic field. Finally, in Section 6 we find the torque acting on a star and discuss the obtained results.

AXISYMMETRIC MAGNETIC FIELD
Since our aim is to find the magnetic field of the magnetosphere of a star, we initially introduce convenient variables, which describe it more simply. For an axisymmetric magnetosphere, when the angle between the axis of the magnetic dipole and axis of the rotation of a star equals zero, it is useful to introduce the flux of the poloidal magnetic field f (ρ, z). Here ρ and z are the cylindrical coordinates. Then the components of the magnetic field are The function g(ρ, z) describes the toroidal magnetic field produced by the poloidal electric currents flowing in a stellar magnetosphere. The relation f = constant is the equation for the magnetic surface on which the magnetic field lines are lying. It is also convenient to describe magnetic field lines of the poloidal magnetic field not only as f = f (ρ, z), but also ρ = ρ(z, f ). The poloidal magnetic field is For the dipole magnetic field the magnetic flux f (ρ, z) is where Bs is the magnitude of the magnetic field on the surface of a star on its equator, Bs = |Bz(r = Rs)|eq. For our problem it is important to know the value of magnetic flux in the equatorial plane, z = 0, f0(ρ) = f (ρ, z = 0). Then the vertical magnetic field on the equator is Bz(z = 0) = ρ −1 ∂f0/∂ρ.
It is useful to measure the magnetic field in the units of Bs and the distances ρ, z in terms of star radius Rs; then the units for the magnetic flux will be BsR 2 s . The relations (1) are the same in these dimensionless variables. For a dipole field, the dimensionless flux is f ′ d = ρ ′2 /(ρ ′2 + z ′2 ) 3/2 , f ′ 0d = 1/ρ ′ . In the star magnetosphere, electric currents flow along the magnetic field lines, j = aB, where a(r) is an arbitrary scalar. In this case, the poloidal components of the Maxwell equation, These relations mean that the toroidal magnetic field and the electric currents in an axisymmetric magnetosphere are functions of the poloidal magnetic flux, f . The important case is the absence of toroidal volume electric currents, j φ = 0. Then (∇ × B) φ = 0, and the equation for the magnetic flux f is the Laplace type The function f is neither even nor odd with respect to the coordinate z due to the electric currents flowing in accretion disk in the equatorial plane. Thus, we expand the function f over the exponential functions exp(−λz), for z > 0. The solution of the equation (2) is where J1 is Bessel function of the first kind. An arbitrary function ϕ(λ) in this integral is defined by boundary conditions for the equation (2). For the dipole magnetic field function ϕ is ϕ(λ) = 1 (in dimensionless units). The deviation of ϕ from the unity describes the distortion of the stellar magnetospheric magnetic field from dipolar. It is useful to express the function ϕ through the value of the magnetic flux on the boundary z = 0, f0(ρ). Because of f0(ρ) = ∞ 0 λρJ1(λρ)φ(λ)dλ, using the inverse transformation, we get The equations (3) and (4) determine the magnetic field through the boundary condition f0(ρ). Of course, we also take into account another boundaries: infinity and the axis ρ = 0. The quantity f (ρ, z) is finite there except the point

THE ACCRETION DISK
The disk is produced from the matter falling from a companion star to a neutron star. The rate of accretion is ofṀ . The centrifugal force prevents a disk matter from spreading over a magnetosphere, and this gives rise to an equatorial plasma disk. The width of this disk is small compared with its radial size. The strong magnetic field of a neutron star forces the disk plasma to rotate with the angular velocity of a star. We will describe motions of ions and electrons separately because their motions are quite different in the strong magnetic field of a star. Disk ions have radial, vρ, and azimuthal velocities, v φ . We consider a stationary and axisymmetric star magnetosphere. Due to axial symmetry, all quantities are independent of the azimuthal angle φ. Then, the continuity equation for ions is where ni is ion density, ρout is the outer radius of the disk, ρout ≫ ρc. Integrating Equation (5) over ρ and z, we obtain Here Σ is the surface disk density, Σ = nidz. Let us note the very important fact that the divergence of the ion flux at ρ < ρout equals zero. Now we write the equation of motion of ions The star acts on ions by gravitational force, Fρ = −miGMs/ρ 2 , which, neglecting other forces, must result in the rotation of ions with the Keplerian velocity, vK = (GMs/ρ) 1/2 . The ρ-component of Equation (7) gives We can see from Equation (8) that in the absence of the electromagnetic fields Bz and Eρ, the ions rotate with the Keplerian velocity, v φ = vK . In contrast, in a strong magnetic field Bz, ions have the electric drift velocity, v φ = −cEρ/Bz. For corotation, i.e. the rigid rotation with the angular velocity of the star, v φ = ωsρ, the plasma in the disk must be polarized to create the radial electric field, Eρ = −ωsρBz/c. The corotation velocity is less than the Keplerian velocity at the distance ρ < ρc. Because the Keplerian velocity goes down with increasing distance ρ, the corotation of a plasma disk means that the centrifugal force in the disk becomes less than the gravitation force at ρ < ρc, and ions must move inwards in the radial direction. But this is not easy, due to the conservation of angular momentum. That is described by the φ-component of the equation (7), Let us note that there is no the toroidal electric field, E φ = 0, because under stationary conditions the electric field must be potential. Because the ion radial velocity, vρ, is not equal to zero (see Equation (6)), the solution of Equation (9) is Equation (10) is the conservation law of the total angular momentum of an ion in magnetic field. But the dimesionless coefficient before the second term in Equation (10), written as (qiB/cmiωs)(ωsf0/B), is much greater than unity. It is proportional to the ratio of the ion cyclotron frequency of rotation in the magnetic field of a star, ωci = qiBs/cmi, to the frequency of rotation of star, ωs. We denote this ratio Ωc = ωci/ωs. This is the cyclotron frequency of ions in natural units of the stellar rotation frequency Ωc = 1.5 · 10 12 Ps 1sec Bs 10 9 G .
For the dipole magnetic field, the terms in the left hand side of Equation (10) become of the same order only at large distances ρ/Rs ≃ Ω 1/3 c . Angular momentum conservation prohibits ion radial motion in such a strong magnetic field as that of a star if we assume it is (close to) dipolar one. The only possibility to allow ions to move radially is an expulsion of magnetic field from the disk. Only if the magnetic flux changes slowly in the disk, ∆f0 ≃ Bsρ 2 Ω −1 c , does the motion of ions in radial direction become possible. But a small variation of the magnetic flux f0 with the radial distance ρ means a small value of the vertical magnetic field Bz in the equatorial plane (see formula (1)).
As for electrons, the parameter Ωc for them is at least 10 3 times greater than for ions. The plasma electrons are strongly magnetized and can move only along a magnetic surface f = const. They generate a current stream at f = fc = f0(ρ = ρc) towards a star. Thus, the disk plasma is separated at ρ < ρc: ions create the plasma disk in the equatorial plane, but electrons create the electric current Jc on the magnetic surface f = fc. This does not mean that the plasma disk is positively charged, it is neutral. The electrons from the 'sea' of electrons of a stellar magnetosphere neutralize any electric charge.

MAGNETOSPHERIC AND STAR SURFACE CURRENTS
We consider the magnetospheric plasma which has an infinite conductivity. Under stationary conditions it can rotate around the z axis with the angular velocity ω(ρ, z). In the frame rotating with this velocity the electric field E ′ is equal to zero. So, the electric field in the laboratory frame is The electric field must be potential E = −∇Ψ, where Ψ is the electric potential. This means that the frequency of rotation ω(ρ, z) is only function of the magnetic flux f, ω = ω(f ), and the electric The magnetospheric current Jc flows from a star surface to a disk along the magnetic surface f = fc. This current is closed by the star surface current Js. The surface current, Js, flowing on the stellar surface is connected with the volume magnetospheric current, j, by the continuity equation The divergence in this equation is taken along the surface where surface currents flow, and jn is the normal (to this surface) component of the volume current j. The analogous equation connects the ion surface current of the disk J d with the magnetospheric volume current j. But the divergence of the ion surface current, as we have already emphasized (see equation (6)), equals zero. The electrons of the disk can only rotate with the angular velocity of the magnetosphere ω(f ), and the divergence of their flux is also equal zero. Thus, the volume magnetospheric current j is absent except on the particular magnetic surfaces f = fc and f = fs. The value of f = fs is the magnetic flux where the disk stops rotation, fs > fc. The total current I produced by the accretion flow is I = qiṀ mi = 6 · 10 20 Ṁ 10 −10 M⊙/y A.
One half of it flows in the north hemisphere and another half flows in the south hemisphere. The magnetospheric currents flow along the magnetic field on two magnetic surfaces These currents are closed by the surface current flowing in the stellar polar region. The magnetospheric electric currents js, jc produce in the region fs > f > fc the toroidal magnetic field B φ , Then the φ-component of the Maxwell equation ∇ × B = 4πj/c gives the equation for the poloidal magnetic flux The right hand side of the equation (14) is not equal to zero only on the magnetic surfaces f = fs, fc, where the magnetospheric electric currents flow along the magnetic field. Due to the toroidal component of the magnetic field the longitudinal currents have a toroidal component, which distorts the poloidal magnetic field. Outside the surfaces fs, fc, Equation (14) coincides with Equation (2) and its solution is given by relations (3,4). However, the fields on both sides of the discontinuities are different. The connection between the magnetic fields at f > fs and f < fs, and also between f > fc and f < fc, can be obtained by integrating the equation (14) over f in the small region near surfaces f = fs and f = fc respectively. For that we need to go over to another independent variables, (z, f ), and consider the magnetic surfaces to be determined by the relation ρ = ρ(z, f ). Equation (14) becomes We take into account that on both sides of discontinuity the shapes of the magnetic surfaces are the same Then integrating the equation (15) we obtain This is the natural condition of equality of the magnetic pressures on both sides of a discontinuity, which follows exactly from our equations.

STRUCTURE OF MAGNETIC FIELD
Here and below we will use the dimensionless variables ω ′ = ω/ωs, r ′ = r/Rs, v ′ = v/Rsωs, B ′ = B/Bs, Further on we will omit the index ′. There appear several dimensionless parameters. The ratio of the ion cyclotron frequency to the frequency of star rotation, mentioned alrready in the section 3, is Ωc = qiBs cmiωs .
The second parameter is the magnetic field, created by the current I/2 (11) flowing on the star surface, in the units of Bs Moving toward a star a disk pulls out the star magnetic field lines. Vertical component of the magnetic field Bz decreases and the ions can move in the radial direction. But the electrons remain magnetized. They must move along magnetic field surfaces only. The single possibility for them to change their motion from the Keplerian rotation to the corotation with a star is to do that at the point of corotation ρ = ρc. Thus, a disk banks all dipole magnetic field lines crossing the equatorial plane at ρ > ρc into the region ρ < ρc. This must had happened because magnetic field lines were froze initially to the ideal conducting matter of a disk and had to move together with it. We conclue therefore that the stellar magnetosphere under the action of an ideal accretion disk is confined to the region ρ < ρc! Thus, the magnetic field flux at the point ρ = ρc equals zero, fc = 0. The magnetic field line f = 0 just comes from The velocity of rotation of plasma disk ions is given by Eq.
Here we consider that at the point ρ = ρc the velocity of ion rotation is equal to the Keplerian velocity vK . Eq. (20) fixes the value of magnetic flux at the stopping of the disk rotation v φ = 0, fs = ρ 2 c /Ωc. The electrons of the disk rotate with the angular velocity of rotation of the magnetic field lines ω(f ) = ωs, however. Thus, the plasma disk produces the surface toroidal electric current which gives the discontinuity of the radial magnetic field on the surface z = 0.
The value of (v φ − ρ) is a function of ρ, f0(ρ) and is determined by the relations (20) It is seen from (21,23) that the value of the radial magnetic field up to the stopping point ρ = ρs is negative, Bρ(ρ, z = +0) < 0. At the point ρ = ρc the radial magnetic field in the equatorial plane is equal zero, and the magnetic field line here is perpendicular to a disk. Then magnetic field lines bend toward a star. The value of the radial magnetic field Bρ(ρ, z = z + 0) determines the derivative of the magnetic flux over z on the equator Let us find this derivative as a function of f0(ρ). From the equations (3,4) we obtain The second integral is known (Oberhettinger 1972), and we get Here K(x) is the complete elliptic integral of the first kind. It equals π/2 at x = 0, but has the logarithmic singularity at x → 1, K(x) → ln[16/(1 − x 2 )]/2. Equation (25) must be supplemented with the condition of absence of the radial magnetic field in the region ρ < ρs, ∂f0/∂z = 0. Let us note that substitution of the dipole magnetic field, f0 = 1/ρ, into Eq. (25) satisfies this condition for all ρ. Also the square of the vertical magnetic field B 2 z behind the stopping point, ρ = ρs − 0, must be equal to the sum of the square of the the radial one B 2 ρ (Bz << Bρ) and the square of the toroidal one B 2 φ = B 2 /ρ 2 s before the stopping point (see the condition (16)). As a result we get Bz = −B(1 + ρ 2 s /v 2 ρ (ρ = ρs)) 1/2 /ρs. We see that the magnetic field is strongly compressed here by a disk, |Bz| >> |B dip | = ρ −3 s . The equations (24-26) close the system of equations which determine the function f0(ρ). We add also the energy conservation law following from the equations (8,9) The particle energy decreases due to the work of the radial electric field in the disk on ions. This electric field is generated in the disk by the rotating magnetic field. Considering that the radial disk velocity vρ is smaller than the rotation velocity v φ = vK at the point ρ = ρc, we find the radial velocity of a disk at any point The radial velocity vρ defines the surface density Σ(ρ) = 1/(ρ|vρ|). The radial velocity vρ at the stopping point ρ = ρs does not equal zero and is negative At the point ρ = ρs disk ions begin to move along the magnetic surface f = fs toward the star surface to its polar cap. Particles without the mechanical angular momentum (v φ = 0) fall down to a star freely but along curved fixed magnetic lines. They get the kinetic energy v 2 /2 = ρ 3 c − ρ 2 c on the stellar surface which corresponds to fall down from the height of the corotation point ρc.
The function f0(ρ) on all axis 0 < ρ < ∞ is different in three regions: f0 = f (1) 0 (ρ) at ρ < ρs; f0 = f (2) 0 (ρ) at ρs < ρ < ρc; f0 = 0 at ρ > ρc. Thus, the equation (25) Figure 2. Dimentionless poloidal magnetic flux in the equatorial plane, f 0 , versus distance ρ from the z axis. Four regimes of the ρ-dependance are distinguished: A Rs < ρ < ρs − ∆ρ (∆ρ << ρs) where the dipolar formula is valid, f 0 (ρ) = 1/ρ; B Narrow region ρs − ∆ρ < ρ < ρs where the poloidal field lines are strongly compressed by the disk, and f 0 (ρ) decreases rapidly from ≈ 1/ρs down to fs ≪ 1/ρs; C ρs < ρ < ρc. Here, f 0 (ρ) is small and decreases from fs to f 0 (ρc) = 0. D For ρ > ρc, f 0 (ρ) = 0. For details see section 5. (1) The value of the magnetic flux f (2) 0 is small. At the point ρ = ρs it equals f (2) 0 = fs << 1/ρs, then it decreases to zero at ρ = ρc. On the contrary, the function f (1) 0 (ρ) increases strongly from the value f (1) 0 = fs at ρ = ρs to the dipole one 1/ρs at ρ < ρs due to the large derivative ∂f The width of this transition region ∆ρ is small, ∆ρ ≃ 1/(ρsB) << ρs. Here the magnetic field is large, it is strongly compressed by the disk. Then, approaching to a star, the magnetic field becomes dipolar f Here E(x) is the complete elliptic function of the second kind. The root of this equation, except the trivial one, ρs = 0, is ρs = 0.55ρc. The point ρ = ρs can be called the Alfvén point. But it corresponds to the stopping of the disk rotation, not of the radial motion. And its position is fixed by the corotation radius and does not depend on the star magnetic field Bs. That is due to the strong compression of the dipole magnetic field by a disk in the region ρ < ρs.

DISCUSSION
We have shown that the structure of the rotating neutron star magnetic field, interacting with a thin accretion disk, can be exactly solved, and we obtained an analytical solution for this field. To achieve this we have used ideal approximations: an axisymmetric, well conducting magnetosphere with a thin plasma disk. These conditions are adequate to the reality. We have demonstrated that a disk compresses the stellar magnetic field, pushing it towards a star. The magnetic field is compressed in the region ρ < ρs, where ρs is the inner edge of the disk where it stops rotation in the laboratory frame, ρs = 0.55ρc. After that the matter of a disk, with zero mechanical angular momentum, falls down onto a star along the magnetic field surface f = fs. The loop of the electric current starts near the point of corotation ρ = ρc. The electric current I (11) flows along the disk at ρs < ρ < ρc, then half of it, I/2, flows along the magnetic surface f = fs to the stellar surface , then along the stellar surface in the polar region, and finally returns to a disk along the magnetic surface f = fc = 0. There is a symmetric current structure in the second hemisphere. Let us emphasize that the magnetosphere exists only inside the magnetic surface f = fc = 0. It is maintained by the pinching Ampére force fa = je × B φ /c acting on the electron current je flowing outside a star along the magnetic surface fc. This force just balances the gradient of the magnetic pressure ∇B 2 /8π inside the magnetosphere boundary.
The potential difference Ψ created by the rotating magnetic field is (Landau et al. 1984) Ψ = ωs c fs.
The electric current I, flowing along the star surface perpendicular the magnetic field, creates the Ampére force and spins up the star rotation. The torque K is It produces the acceleration of the star rotationṖṡ Ps = −1.5 · 10 −13 Ps Here Is is the moment inertia of a star. This value ofṖs corresponds to the observed one for the close X-ray binaries (Rappaport & Joss 1983). We see that the total angular momentum, which has a disk at the point of corotation, is transferred to a star. The answer does not depend on the strength of the stellar magnetic field Bs. It is due to the strong magnetization of a disk plasma. The magnetization parameter Ωc (18) is much greater than unity. And no matter whether it is very large (10 12 ), or not so much (10 5 ). The role of the stellar magnetic field consists in stopping of the disk rotation, i.e. transformation of the mechanical angular momentum of particles to the electromagnetic momentum. The angular momentum of a charged particle in a magnetic field consists of two parts: the mechanical one and the electromagnetic one (see the formula (10)). And approaching a star, a particle loses its mechanical part and increases the electromagnetic one. Therefore, if a matter element falls down onto a star,Ṁ > 0, then it transfers its angular momentum to a star, K > 0. No matter whether it is the mechanical angular momentum or the electromagnetic one. Thus, under the accretion, there must not be the negative part in the torque, which can be attributed to the electromagnetic torque (Ghosh et al. 1977). Appearance of the stellar spin down is connected with assumptions that, first, the toroidal magnetic field B φ changes its direction at some value of the radius, and, second, the value of the poloidal field Bz in the disk does not differ strongly from the dipole one (for example, see Kluzniak & Rappaport 2007). The toroidal magnetic field is determined by poloidal currents, and the zero value of it at some distance means the closure of currents. At larger distance the field B φ is equal to zero. Indeed, we see that the toroidal field exists only in the region ρs < ρ < ρc and B φ > 0 (13).
Let us note that at the small value of accretion rateṀ it is impossible to construct the self consistent solution for the stellar magnetosphere. When the magnetic field at the point ρs becomes less than the dipole one at this point it means that there appears the decompression of the magnetic field. This is impossible for the plasma moving toward a star because the magnetic field is frozen to the well conducting matter. Formally, in this case it is impossible to match the solution for the magnetic flux f0(ρ) with the dipole field. This takes the place when |Bz(ρ = ρs)| < ρ −3 s . Because |Bz| = B(1 + ρ 2 s /v 2 ρ ) 1/2 /ρs, ρs = 0.55ρc and vρ = −0.8ρc, this condition is B < 2ρ −2 c . It means that the expression (29) Here µs is the stellar magnetic moment, µs = BsR 3 s . ForṀ < Mmin a disk does not penetrate into the star magnetosphere.
In some sense, the stellar magnetic field does not prevent the accretion of a ionized matter onto a star, but, on the contrary, helps it. Without magnetic field an ideal matter element can not accrete due to the angular momentum conservation. We need to introduce an anomalous viscosity represented by α parameter in the α-disks. In a magnetic field a charged particle possesses two parts of the angular momentum, the mechanical one and the electromagnetic one. In the language of magneto hydrodynamics, we have the viscous stress tensor and the magnetic stress tensor, which is proportional to the product B φ Bz. Appearance of the toroidal magnetic field is impossible without poloidal electric currents, they must be closed on the stellar surface, transforming the mechanical angular momentum of a disk into the stellar angular momentum.
It is clear that if the inner edge of a disk ρ d is placed further than the the corotation point, then it will extract the angular momentum from a star. In this case the centrifugal force is larger than the gravitation force, and a disk must be pushed outward from a star. This is the propeller regime. It is stationary, or quasi stationary process, if we have a source of the matter at the inner edge of a disk. Such situation takes place in the Jupiter magnetosphere where the Jupiter's satellite Io supplies a gas to a plasma disk at the point ρ d > ρc. The electric current in the current loop has the direction opposite to that in the case of accretion (Istomin 2005). If there is no gas source in the stellar magnetosphere then a disk is ejected out to the light cylinder taking the angular momentum ∆J from a star of the order of Here M d is the mass of a disk and RL is the radius of the light cylinder, RL = c/ωs. Up to now we considered only the situation when a disk and a star rotate in the same direction (corotating disk). There exists the interesting case when an accretion disk rotates in the opposite direction than a star. A corotating disk had the same sign of the angular momentum as a star. The disk matter falling onto star transmits its angular momentum to a star. Then, the torque is positive. If the disk rotates in the opposite direction than a star (counterrotating disk), then, matter falling onto the stellar surface, will spin down the stellar rotation. The torque becomes negative. For the countercotating disk it is impossible to build up the stationary self consistent solution like we did for the coratation case. A star must destroy the inner parts of a disk approaching too close to it. It tends to re-rotate the inner part of a disk to the direction of its own rotation. In the other words, the magnetosphere-disk coupling tends to enforce corotation to the inner layers of the disk. The accretion becomes more quasispherical than disk-like. We can estimate with a good accuracy the torque acting on a star in this case. And the exact solution obtained above will help us. We saw that to reduce to zero its mechanical angular momentum, a falling particle must deflect from its position on the magnetic field surface f by the value of ∆f = ρmvm/Ωc (see formula (20)). Here values ρm and vm are the radial position and the azimuthal velocity of a particle on the magnetospheric boundary, respectively. For the rotating disk, the boundary of the magnetosphere is the corotation radius, ρ = ρc. For the counterrotation there is no a corotation point. The magnetospheric boundary for the quasi spherical accretion must be close to the Alfvén radius, ρm ≃ ρA. Falling onto a star, ions move almost along the magnetic field lines deflecting by a small value of ∆f . Electrons will deflect also, but their magnetization parameter Ωc is much larger than that for ions, and we can neglect electron deflection. Thus, ions and electrons will be on the stellar surface at different magnetic surfaces separated by the value of ∆f . The current I (11) will flow along the surface, it will produce the torque K (29). For the counterrotation of a star and a disk the value of ∆f will be negative, and the electric current on the stellar surface will have the opposite direction than for the case of corotation. As a result the torque acting on a star will be negative and equal to K = −Ṁ vmρm. That is the exact relation. But the position of the magnetosphere boundary is not known exactly. If we do not take into account the magnetic field compression on the boundary it is given by the traditional formula for the Alfvén radius ρm = ρA = (µ 2 s /2 1/2Ṁ ρ 3/2 c ωs) 2/7 (see, for example, Arons 1993). Because for the accretion onto a star the radius of the magnetosphere ρm must be smaller than the corotation radius ρc, the deceleration torque will be always less than the acceleration torque at the same value of the accretion rateṀ . Thus, the observed stellar spin down with noticeableṀ can be a result of the accretion of the matter from a counter rotating disk. This is quite possible because the disk formation depends on a companion star, on the stellar orbital motion and many other reasons.
In conclusion we compare the results obtained here with the numerical calculations. The numerical simulations were done in the frame of the MHD approximation by Romanova et al. 2011Romanova et al. , 2012. Though our treatment have been done in the frame of two-fluid hydrodynamics, the similar features were seen in numerical singlefluid simulations: the compression of the stellar dipole magnetic field by a disk, the falling of the disk matter along fixed magnetic field lines, the pushing out of magnetic field lines from a disk, and the proportionality of the torque K, acting on a star, to the accretion mass rateṀ .