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

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1. Introduction

The discovery of radio pulsars has initiated considerable interest into the problem of the flow of relativistic plasmas in the magnetosphere of fast rotating neutron stars. This problem has become of more interest in recent years also in connection with the discovery of relativistic jets from some galactic astrophysical objects (Mirabel & Rodrigues 1996 ). The discovery of high energy and ultra high energy gamma - ray emission from AGN's (Wagner 1995 ) shows that the acceleration of plasma to ultrarelativistic energies occurs also in these objects (Ferrari et al. 1996 ; Pelletier et al. 1996 ; Melia & Königl 1989 ). It is clear now that an analysis of the relativistic plasma flow is necessary for understanding the processes taking place in compact galactic objects and also in AGN's. However, in the present paper we concentrate our attention totally on the problem of relativistic plasma flow in the magnetosphere of radio pulsars.

In spite of systematic research in the field of the physics of radio pulsars in the last years, the structure of the magnetosphere and the mechanisms for the generation of electromagnetic emission in these objects remain vague to large extent (Lyubarsky 1995 ). Up to now, all that is generally agreed upon is the mechanisms of plasma production in radio pulsar magnetosphere. The basis of the theory of plasma production in radio pulsar magnetospheres was initiated by Sturrock (1971 ). This theory was developed in more detail for different conditions on the stellar surface by Ruderman & Sutherland (1975 ) and by Arons (1981 ). According to them, primary leptons (electrons or positrons) produced via [FORMULA] pair production processes or extracted from the surface of the star are accelerated near the stellar surface by electric fields in the so called "inner gap". At Lorentz-factors [FORMULA], the accelerated leptons generate curvature photons which are able to further produce new leptons in the curved magnetic field. The electromagnetic cascade is developed and the density of plasma increases. When the density becomes high enough to screen the electric field parallel to the magnetic field the acceleration of the plasma is stopped. Plasma with Lorentz-factors [FORMULA] and densities [FORMULA] is formed. The distance from the stellar surface where this happens is of the order of several stellar radii. The flux of the kinetic energy of this plasma is much less than the total rotational losses of the fast rotating radio pulsars. Almost all energy is carried out by the electromagnetic field. The ratio of the Poynting flux to the flux of the kinetic energy of plasma is (Lyubarsky 1995 )

[EQUATION]

where [FORMULA] is the magnetization parameter introduced by Michel (1969 ), [FORMULA] is the radius of the star, [FORMULA] and P are the angular velocity and the period of rotation, [FORMULA] is the magnetic field on the surface of the star.

This ratio is very large near the stellar surface for fast rotating pulsars. At the same time we know from observations of the Crab Nebula, for example, that this ratio at large distances from the pulsar [FORMULA] (Kundt & Krotschek 1980 ). Moreover the theoretical analysis of the conditions in the Crab Nebula performed by Kennel & Coroniti (1984 ) shows that the ratio of the Poynting flux to the flux of the kinetic energy in the relativistic wind produced by the pulsar can be [FORMULA] at the distance from the pulsar [FORMULA]. It follows from the observations that there exists some mechanism for the transformation of the Poynting flux into the flux of the kinetic energy as the plasma travels from the star to infinity (Camenzind 1989 ). This mechanism is apparently the basic mechanism for plasma acceleration since it ensures the transformation of at least 50% of the rotational energy of the neutron star in to the kinetic energy of the plasma. Determination and investigation of this mechanism is one of the key problems in the physics of radio pulsars. The solution of this problem is inevitably necessary for understanding the mechanisms for the generation of the electromagnetic emission in radio pulsars.

The axis of rotation of real radio pulsars is not directed along the magnetic moment. In spite of this an axisymmetrically rotating star can be considered as an appropriate model for an investigation of the mechanism of the Poynting flux transformation in radio pulsar's magnetosphere. It appears that the axisymmetrically rotating star ejecting relativistic plasma has rotational energy losses comparable with the rotational losses of real pulsars. The energy of rotation is carried out near the surface of the star by the Poynting flux. Therefore the problem of the Poynting flux transformation can be considered in this model. At the same time this problem is remarkably simplifield in the model of the axisymmetrical rotator since the plasma flow can be considered as stationary and axisymmetric. This is why many attempts have been made to investigate the flow of a relativistic plasma in the model of the axisymmetrical rotator.

The density of the relativistic plasma produced in the electromagnetic cascade is high enough to screen the electric field parallel to the magnetic field. The plasma can be considered as cold to a first approximation (Lyubarsky 1995 ). The magnetohydrodynamical approximation can be used for the description of the flow of this plasma. Michel (1969 ) was the first to use this approach for the investigation of the relativistic plasma flow in the magnetosphere of an axisymmetrical rotator. He obtained the solution to the problem for the plasma flow in a prescribed monopole like magnetic field and It was found that the acceleration of the plasma in this system is not effective. Moreover the relativistic plasma appeared subsonic in the whole space up to infinity. It is obvious that the strongest limitation of Michel's solution is the assumption that the poloidal magnetic field is not affected by the moving plasma. The problem of the acceleration of a relativistic plasma in the magnetosphere of a rotating star taking into account the effect of the plasma on the poloidal magnetic field is not solved up to now.

The solution of the full selfconsistent problem of the stationary plasma flow in the magnetosphere of the axisymmetric rotator is connected with the solution of a set of nonlinear equations of mixed type. Several attempts to solve this problem have been made by several authors. Sakurai (1985 ) solved this problem numerically for a nonrelativistic plasma. For the relativistic case this problem was numerically investigated by Camenzind (1986a , 1986b , 1989). For slow rotation, a solution was obtained analytically by Bogovalov (1992 ). Unfortunately, the solution of the selfconsistent problem for fast rotation of the star encounters severe difficulties and a few approximations are needed. Several attempts have been made to solve the problem in the force - free approximation (Michel 1973 ; Mestel 1979 ; Scharlemann & Wagoner 1973 , Beskin et al. 1983 ; Lyubarsky 1990 ; Sulkanen & Lovelace 1990 ). Another direction is the investigation of the self - similar problem (Blandfrod & Payne 1982 ; Lovelace et al. 1991 ; Contopoulos 1994 ; Tsinganos et al. 1993 ). These investigations have clarified the problem to a large extent. However, the problem of the relativistic plasma acceleration remains largely unsolved.

The plan of this paper is as follows. In Sect. 2 the equations describing the stationary flow of a cold relativistic plasma are presented. Sect. 3 is devoted to the analytical study of the problem of the stationary plasma outflow in the monopole like magnetic field. Special attention is directed to the Poynting flux dominated plasma flow under the conditions [FORMULA] and [FORMULA] typical for rapidly rotating radio pulsars. In Sect. 4 the equations and the method of solution of the nonstationary problem of the relativistic plasma flow are discussed. The results of the numerical solution of the time dependent problem of relativistic plasma outflow are presented in Sect. 5. The basic results of the work are discussed in the last section.

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© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998
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