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

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6. Discussion

The results of our analysis presented in Sect. 3 close a long series of attempts to obtain acceleration of a relativistic plasma near the light cylinder in the model of the axisymmetrical rotator at the stationary outflow. We found that in a self - consistent solution of the problem of plasma outflow in the magnetosphere produced by a magnetic monopole plasma is not accelerated up to the distance [FORMULA] even under the conditions [FORMULA] if the condition [FORMULA] is fulfilled. It is important that these are the conditions that are fulfilled in real radio pulsars. For example, for the Crab pulsar we have [FORMULA] and [FORMULA] (Lyubarsky 1995 ; Daugherty & Harding 1982 ). Acceleration is possible at the stationary outflow at distance larger than [FORMULA]. It will be accompanied by collimation of plasma in jet with transversal dimension of the order [FORMULA] (Bogovalov 1995 ), where [FORMULA] is the Lorentz-factor of plasma at infinity.

Questions arise. How general is the conclusion about the absence of acceleration of plasma near the light cylinder at the stationary outflow? Does this conclusion remain valid for axisymmetrical dipole or for oblique rotator? And lastly, how is plasma accelerated in reality? Possible answers on these questions are discussed below.

Let us consider the value [FORMULA] given by formula (8) at the light cylinder. It is

[EQUATION]

where [FORMULA]. This value is defined totally by the bend of the field lines on the light cylinder in toroidal direction. We stress that this is valid for an arbitrary poloidal field, not only for a monopole like. It is seen from (35) that the less toroidal bend of the field lines the stronger the plasma acceleration on the light cylinder. Unfortunately it appears impossible to make the toroidal bend of field lines small even at infinitesimally small flux of plasma (we assume however that the density of plasma always strongly exceeds the Goldreich-Julian (1969 ) density [FORMULA]). The bend of the field lines in relativistic case is defined not only by the mass flux of matter, but it is also defined by the flux of energy which is equivalent to the mass flux. This is why even in the massless approximation the toroidal bend h is equal to 1 on the light cylinder for the monopole like field. In the dipole field we have to expect the same order of magnitude of h on the light cylinder. It follows from (7) that the dependence of h on x is defined by the dependence of [FORMULA] on x at the motion along a field line. For the monopole like magnetic field [FORMULA] is constant. For dipole magnetic field [FORMULA] depends on x, but rather weakly. According to (Bogovalov 1989 ) this value is twice less on the light cylinder than on the surface of the star if to move along the last open field line of the dipole magnetic field. The weak dependence of [FORMULA] on x is connected with the fact that this value is approximately equal to the flux of the poloidal magnetic field through a surface limited by the field line [FORMULA]. This flux is constant. Close dependence of [FORMULA] on x for the monopole like and for the dipole magnetic field says that the value of h on the light cylinder for the dipole field will be of the same order as for the monopole like field. No acceleration is expected in dipole field at the stationary outflow.

In the light of this analysis it becomes clear the reason why Beskin et al. (1983 ) have obtained strong acceleration of plasma for poloidal magnetic field which fall down with distance as [FORMULA]. For this field the value [FORMULA] is much more larger on the light cylinder than on the surface of the star. The toroidal bend h becomes very small and according to (35) plasma is strongly accelerated. Unfortunately the law [FORMULA] is not realized in reality. Therefore this mechanism of acceleration can not be realized in real pulsars.

We have not any accurate analysis of the plasma acceleration in the magnetosphere of an oblique rotator. Therefore the question about the role of the nonaxisymmetry of rotation of real pulsars for acceleration of plasma remains open. However, the numerical simulations of the time dependent plasma flow show that even in the magnetosphere of the axisymmetrical rotator the Poynting flux dominated stationary flow of relativistic plasma is likely unstable. The flow becomes nonstationary. A region of turbulent flow is formed beyond the fast mode surface. Plasma is accelerated in this region. There is no doubt that the phenomenon of instability plays a crucially important role in the magnetosphere of real pulsars.

It is necessary to keep in mind that only axisymmetrical perturbations were admissible in the numerical simulations. The question about the stability of the flow with respect to the nonaxisymmetrical perturbations is open. We can not exclude that the flow will be nonstationary in the region beyond the Alfven surface with respect to these perturbations. Further analysis is necessary to clarify this problem.

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© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998
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