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Astron. Astrophys. 319, 1025-1035 (1997)

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1. Jet formation around black holes

Jet motion originating in the close environment of a rotating black hole is observationally indicated for two classes of sources concerning mass and energy output.

The first class are the active galactic nuclei (hereafter AGN). Following the standard model, AGN jet formation develops in the magnetized environment around rotating, super massive black holes with a mass of the order of [FORMULA] (Sanders et al. 1989, Blandford & Payne 1982, cf. Blandford et al. 1990, Kollgaard 1994). From evolutionary arguments (accretion of angular momentum) these black holes are believed to be very rapid rotators.

In some quasars and BL Lacertae objects, the jet knots are observed to follow helical trajectories on parsec-scale with a de-projected highly relativistic speed. The high jet velocity together with a small angle between the line of sight and the propagation vector involves a time shift from knot time to observer time, and thus the projected jet motion appears as superluminal motion. Examples are 3C 273 (Schilizzi 1992, Abrahan et al. 1994) and 3C 345 (Zensus et al. 1995).

Radio observations have also detected superluminal motion in the Galaxy. Examples are the high energy source 1915+105 (Mirabel & Rodriguez 1994) and the X-ray source GRO J1655-40 (Tingay et al. 1995). The de-projected jet speed of both sources is surprisingly similar ([FORMULA]). This velocity may correspond to the escape velocity from a point near the black hole (Mirabel & Rodriguez 1995). However, there are not many details known about the intrinsic sources.

In both cases, the jets are detected in non-thermal radio emission, clearly indicating a magnetic character of the jet formation and propagation.

From the introductionary remarks above, it is clear that a quantitative analysis of the jet structure in these sources must take into account both magnetohydrodynamic (hereafter MHD) effects and general relativity.

In this paper we will numerically investigate the structure of a jet magnetosphere in Kerr geometry. The calculated field distributions represent global solutions to the local cross-field force-balance equation.

The first theoretical formulation of the electromagnetic force-equilibrium in Kerr space-time around fast rotating black holes was given by Blandford & Znajek (1977) and Znajek (1977). They presented the first solutions of the problem and discovered the possibility of extracting rotational energy and angular momentum from the black hole electromagnetically (the so called Blandford-Znajek process). Okamoto (1992) investigated the black hole magnetic field structure and the black hole evolution under influence of the Blandford-Znajek process. Examining the fast magnetosonic points of the wind and the accretion, he found analytical expressions for the poloidal current and the field rotation law.

With the development of the 3+1 split of Kerr space-time (Thorne & Macdonald 1982, Macdonald & Thorne 1982, Thorne et al. 1986) the understanding of the electrodynamics of rotating black holes became more transparent. For a chosen global time, the tensor description splits up in the usual fields [FORMULA], [FORMULA], current density [FORMULA], and charge density [FORMULA] . The formulation of the MHD equations becomes very similar to that of flat Minkowski space, which are used in pulsar electrodynamics.

Using this powerful tool, Macdonald (1984) calculated first the numerical solutions for the magnetic field force-balance around rotating black holes. Three models (magnetic field distribution roughly radial, uniform, or paraboidal) of differentially rotating magnetospheres were investigated, however, the integration region was limited to [FORMULA] horizon radii.

Camenzind (1986, 1987) formulated a fully relativistic description of hydromagnetic flows, basically applicable to any field topology. The so-called wind equation considers the stationary force-balance of the plasma motion along the magnetic field. The (flat space) transfield equation was solved by using the method of finite elements.

Haehnelt (1990) extended this procedure for Kerr space-time in the 3+1 description. The solutions explicitly show the interrelation between the poloidal current strength and the collimation of the flux surfaces. They were calculated on separate integration domains inside and outside the light cylinder (see Camenzind 1990). However, there was a mismatch between the inner and outer solution at the light cylinder, which is a singular surface of the relativistic transfield equation. This matching problem is well known from in the literature of pulsar magnetospheres (Michel 1991).

So far, no global magnetic field solutions could yet be found originating in the accretion disk close to the rotating black hole and, passing through the outer light surface, eventually reaching the asymptotic regime of a collimated jet.

The matching problem of relativistic force-free magnetospheres was investigated in the context of stellar jets (Fendt 1994, Fendt et al. 1995). It then became clear that a mismatch at the light cylinder could be removed by a proper adjustment of the current distribution and the outer boundary condition, which could be interpreted as an adjustment of the "magnetic pressure equilibrium" between the regions inside and outside the light cylinder.

In this paper, we like to extend the results from Fendt et al. (1995) to the general relativistic context. The solutions presented here are global solutions for the stationary black hole force-free electrodynamics in the sense that they smoothly pass the singular surface of the outer light surface. The field lines originate near the inner light surface close to a rotating black hole and collimate to an asymptotic jet of finite radius of several (asymptotic) light cylinder radii.

The structure of this paper is as follows. In Sect. 2, basic equations of the theory of relativistic magnetospheres in the context of Kerr metric are reviewed. In Sect. 3, the model underlying the numerical calculations is discussed. We present our numerical results in Sect. 4 and discuss solutions with different topologies and jet parameters.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998