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Astron. Astrophys. 327, 662-670 (1997)

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3. Plasma flow in a monopole like magnetic field

We assume that a cold plasma with initial four-velocity [FORMULA] is ejected from the stellar surface. Here [FORMULA] and [FORMULA] are the initial Lorentz-factor and the initial velocity of the plasma. The distribution of the magnetic flux on the stellar surface corresponds to that from a magnetic monopole. Therefore in the absence of stellar rotation the plasma flow is spherically symmetric. The toroidal component of the magnetic field is generated by rotation. Spherical symmetry of the flow is violated. In what follows we assume that the rotation is uniform so that [FORMULA].

The problem of the plasma flow for slow rotation under the condition [FORMULA] was solved by Bogovalov (1992 ) analytically. [FORMULA] is the radius of the fast mode sound surface where the velocity of the plasma equals the local fast mode velocity. It follows from this solution that the perturbation of the poloidal magnetic field produced by rotation is proportional to [FORMULA] and the velocity of plasma in the subfast sonic region is constant. Below we consider the expansion of the exact solution on the parameter [FORMULA] and estimate the first corrections to the leading terms of the solution nonvanishing at [FORMULA]. We do not impose the condition [FORMULA] so that the star can rotate fast.

Let us assume at the beginning that for [FORMULA] the poloidal magnetic field goes to the field of the magnetic monopole at fixed other parameters. Below we check that this assumption is valid. In other words, we assume that the nonvanishing term in the expansion of the poloidal magnetic field in powers of [FORMULA] is the field of the magnetic monopole. For this field [FORMULA], where the index "0" denotes values at the stellar surface.

The estimation of the first nonvanishing term in the expansion of h in powers of [FORMULA] in the region where [FORMULA] follows from (7) and (8). In this region the four - velocity [FORMULA] is constant and equals [FORMULA]. Expression (7) gives the following equation for h.

[EQUATION]

The right and left parts of this equation must be equal to zero simultaneously at the Alfven surface. This condition gives the expression for the radius of this surface

[EQUATION]

Together with (14) it gives

[EQUATION]

It follows from definition W (3) that

[EQUATION]

and for h we have

[EQUATION]

In this approximation the condition [FORMULA] adopted above is valid everywhere. The first term in the expansion of [FORMULA] in powers of [FORMULA] starts with the first power. This term gives corrections to the poloidal magnetic field proportional to [FORMULA]. To obtain first corrections to h, we can neglect that correction to the poloidal field in (7). Keeping in Eq. (7) terms proportional to [FORMULA] we can obtain the following equation for h

[EQUATION]

It is easy to verify by direct calculation that the expression for h which is accurate up to terms proportional to [FORMULA] is as follows

[EQUATION]

[FORMULA] turns out to be equal to zero in this approximation. The expressions for W and M can be obtained from their definitions. They are

[EQUATION]

and

[EQUATION]

The first corrections to the poloidal field proportional to [FORMULA] appear in this approximation. Before we estimate these corrections let us make sure that the leading term in the expansion of the poloidal magnetic field is really the field of the magnetic monopole. Eluminating terms proportional to [FORMULA] from Eq. (13) for the leading term we obtain the following equation

[EQUATION]

For the given boundary conditions the field of the monopole is the solution of the equation above. The transfield equation accurate up to terms proportional to [FORMULA] is

[EQUATION]

The first corrections to the monopole-like solution are due to the term [FORMULA] which plays the role of a perturbation. The ratio of this term to the leading one is of order [FORMULA]. This ratio increases with distance and after some distance it exceeds 1. It is necessary to take into account however that we are discussing in this paper the flow in the subsonic region. The plasma flow in this region does not depend on the flow in the super sonic region since no MHD signal produced in the supersonic region is able to reach the subsonic region (Bogovalov 1994 ). Therefore it is sufficient for us to demand that the first corrections to the leading terms are small in the subsonic region. Mathematically it means

[EQUATION]

where [FORMULA] means the radius of the fast mode surface. For a cold plasma this radius is defined by the relationship (Bogovalov 1994 )

[EQUATION]

For the monopole-like magnetic field and under the condition [FORMULA] the fast mode surface radius is

[EQUATION]

Expression (27) together with condition (25) show that the corrections to the monopole like magnetic field in the subsonic region are proportional to [FORMULA], where [FORMULA], and a is the distance from the basis of the field line to the axis of rotation. The corresponding corrections to h and [FORMULA] are also to be proportional to powers of [FORMULA]. The exact solution is expanded in powers of the parameter [FORMULA] in the subsonic region. In the most interesting case of real pulsars this parameter is small.

So, under the condition [FORMULA] in the subsonic region the plasma moves radially with constant Lorentz-factor in the poloidal magnetic field. This field coincides with the field of the magnetic monopole with an accuracy [FORMULA]. The toroidal magnetic velocity is equal to zero. The toroidal magnetic field is defined by expression (20).

At distances [FORMULA] the energy of the plasma and the poloidal magnetic field begin to change. There is no doubt that a collimation of the plasma to the axis of rotation will take place. Analysis shows that finally a part of the plasma will be collimated along the axis of rotation (Heyvaerts & Norman 1989 ) in a jet with characteristic transversal dimension [FORMULA], where [FORMULA] is the Lorentz-factor of the plasma at infinity (Bogovalov 1995 ).

At first sight it sounds strange that the dynamics of the plasma and the poloidal magnetic field are only slightly pertubed, when the toroidal magnetic field [FORMULA] and the electric field [FORMULA] are comparable and even exceed the poloidal magnetic field. The explanation is simple. Actually these terms come into the equations in the combination [FORMULA]. This combination is small even when each of the two terms [FORMULA] and E are not small. It is easy to understand that the value [FORMULA] is the square of the toroidal magnetic field in a coordinate system comooving with the plasma. The poloidal magnetic field does not change in this system (Landau & Lifshitz 1975 ). This is why the condition that the perturbation of the flow by rotation is small coincides with the condition that the toroidal magnetic field is small in comparison with the poloidal magnetic field in the comooving coordinate system.

In conclusion of this section it is worth to compare obtained solution with that obtained by Michel (1973 ) in the massless approximation. It is seen that Michel's solution is the limit of our solution at the conditions [FORMULA], [FORMULA], [FORMULA], [FORMULA].

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© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998
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