## 3. Plasma flow in a monopole like magnetic fieldWe assume that a cold plasma with initial four-velocity is ejected from the stellar surface. Here and 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 . The problem of the plasma flow for slow rotation under the condition was solved by Bogovalov (1992 ) analytically. 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 and the velocity of plasma in the subfast sonic region is constant. Below we consider the expansion of the exact solution on the parameter and estimate the first corrections to the leading terms of the solution nonvanishing at . We do not impose the condition so that the star can rotate fast. Let us assume at the beginning that for 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 is the field of the magnetic monopole. For this field , where the index "0" denotes values at the stellar surface. The estimation of the first nonvanishing term in the expansion of
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 Together with (14) it gives It follows from definition W (3) that In this approximation the condition adopted
above is valid everywhere. The first term in the expansion of
in powers of starts with
It is easy to verify by direct calculation that the expression for
turns out to be equal to zero in this
approximation. The expressions for and The first corrections to the poloidal field proportional to 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 from Eq. (13) for the leading term we obtain the following 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 is The first corrections to the monopole-like solution are due to the term which plays the role of a perturbation. The ratio of this term to the leading one is of order . 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 where means the radius of the fast mode surface. For a cold plasma this radius is defined by the relationship (Bogovalov 1994 ) For the monopole-like magnetic field and under the condition the fast mode surface radius is Expression (27) together with condition (25) show that the
corrections to the monopole like magnetic field in the subsonic region
are proportional to , where
, and So, under the condition 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 At distances 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 , where 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 and the electric field
are comparable and even exceed the poloidal
magnetic field. The explanation is simple. Actually these terms come
into the equations in the combination . This
combination is small even when each of the two terms
and 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 , , , . © European Southern Observatory (ESO) 1997 Online publication: April 6, 1998 |