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

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4. Time dependent problem

We have made sure that under the condition [FORMULA] the plasma is not accelerated in the subsonic region at the stationary outflow. The assumption about stationarity of the flow strongly limits the validity of this conclusion. The flow of a relativistic plasma can be nonstationary. To verify this hypothesis the numerical simulation of the time dependent axisymmetrical problem has been performed.

As above the plasma flow is considered in the magnetosphere produced by a magnetic monopole. Similar simulations were performed earlier by us for a nonrelativistic plasma (Bogovalov 1996 ). It is convenient to consider the system of equations defining the time dependent flow of plasma in dimensionless variables. The transition to the new dimensionless variables is performed according to the following rules: [FORMULA], [FORMULA], where [FORMULA] and [FORMULA] are the poloidal magnetic field of the magnetic monopole and the density of the plasma on the distance from the center of the star equal to the light cylinder, [FORMULA], [FORMULA], [FORMULA], [FORMULA]. [FORMULA], [FORMULA], [FORMULA]. Below, all notations of the dimensionless variables are accepted as usual dimension ones. The equations in these variables are as follows

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

where [FORMULA], [FORMULA], [FORMULA].

Eq. (28) and (29) express the frozen in condition. The first equation is written for the [FORMULA] function defined by (2). Here [FORMULA] depends not only on coordinates but also on time. This equation can be obtained easily from the well known equation [FORMULA].

Eq. (30, 31, 32) define the dynamics of the plasma. Eq. (33) expresses the conservation of matter fux. Gravity is neglected.

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© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998
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