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

2. Basic equations of the stationary flow

The MHD equations governing the dynamics of a steady axisymmetric magnetized wind have been widely studied in several papers (see previous Section). We outline now the general properties of the relativistic MHD system following the formalism of Bogovalov (1994).

In an axisymmetric flow the magnetic field can be expressed as a sum of the poloidal component and the toroidal component . The poloidal magnetic field can be expressed as

where is the distance to the axis of rotation and is the unit vector corresponding to the azimuthal direction around the axis z. The function is proportional to the total flux of the poloidal magnetic field through a surface at the radius . In the frozen in approximation the relationship between the electric field and the poloidal magnetic field is (Weber & Davis 1967 ). The function is constant along the poloidal field lines and describes their differential rotation.

The first equation defining the dynamics of the plasma along the poloidal field lines is the equation for the conservation of the specific energy flux

We neglect gravitation of the neutron star since the gravitational energy of leptons is much less than their kinetic energy. The second equation is the equation for the conservation of the specific angular momentum flux

Projection of the frozen-in condition on the electric field gives

Here is the four-velocity of the plasma along a field line, is the toroidal component of the four-velocity of the plasma, , n is the density of the plasma, m is the mass of the particles, . The functions and , proportional to the energy and to the angular momentum flux per one particle, are constant along the field lines. Therefore they depend only on .

The relativistic relation between the components of the four-velocity is,

It is easy to obtain from these algebraic equations the following relationships which will be useful below

and

where , .

The transfield equation describing the balance of forces across the field lines has been investigated in canonical form by a number of authors (Ardavan 1979 ; Sakurai 1985 ; Bogovalov 1994 ). This is a complicated mixed-type second order equation in partial derivatives. For our purposes it is convenient to use this same equation, but in another form. We introduce a curvilinear orthogonal coordinate system formed by the poloidal magnetic and electric fields. Since the magnetic and electric field line of force pass through every point in space, we can always define such a coordinate system in the regions occupied by plasma flows. The introduction of such a coordinate system is not unique. It is convenient to choose the function as one of the coordinates. We define the other coordinate to be . When moving along a field line, remains constant, but varies. For the sake of simplicity, we consider to increase monotonically to infinity when moving along a field line of force. A geometrical interval in these variables is equal to

where and are components of the metric tensor. In accordance with Landau & Lifshitz (1975 ), the equation , where is the energy-momentum tensor for the plasma, has the following form in these units

The choice of the coordinate is not unique. It is possible to remedy this nonuniqueness by noting that

where is the radius of curvature of a poloidal field line of force. It is possible to obtain expression (11) by direct calculation. is positive if the radius of curvature is directed from the line of force to the rotational axis and negative if it is directed in the opposite sense. In addition

With these relations and the notation introduced earlier, we may obtain after some relatively strightforward manipulations the following form for the equation for the poloidal field:

© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998