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Astron. Astrophys. 323, 999-1010 (1997)

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1. Jet formation from disk magnetic fields

Observations of different kinds of jet sources give convincing evidence that jet formation is always connected to the presence of an accretion disk. This holds for various scales of energy output, jet velocity and nature of the jet emitting objects as there are active galactic nuclei (AGN), galactic superluminal jet sources, mildly relativistic jets from neutron stars (e.g. SS 433), and the numerous class of protostellar jets (e.g. Zensus et al. 1995; Mirabel & Rodriguez 1995; Mundt et al. 1990, Ray et al. 1996).

It is now generally accepted that magnetic fields play an important role in jet formation and propagation for all different kinds of jet sources. These jets are believed to originate very close to the central objects in the interaction region with the accretion disk or in the disk itself.

If the central object is a black hole as it is likely for AGN and galactic superluminal jet sources, the disk is the only possible location for a field generation (by dynamo action or/and advection of magnetic flux).

In the case of protostars and neutron stars the central object also carries a relatively strong magnetic field, and it is not yet clear, whether the jet magnetic field originates in the disk or in the star. However, there must clearly be a strong interaction between the stellar field and the accretion flow in a region, where the stellar field couples to the disk.

Plasma is ejected from the disk into the magnetosphere and becomes magnetically accelerated (see Ferreira & Pelletier 1995). Electric currents and inertia associated with the plasma flow collimate the jet. The observed degree of collimation is very high. The extragalactic jets, the galactic superluminal jets as well as protostellar jets are collimated almost to a cylindrical shape (Camenzind & Krockenberger 1992, Zensus et al. 1995; Ray et al. 1996).

While for extragalactic and galactic superluminal jets a fully relativistic description is obviously necessary, the case of protostellar jets is more complicated. The protostellar jet velocities of about [FORMULA] (Mundt et al. 1990) are clearly non-relativistic. However, if the field is anchored in the accretion disk, the rapid rotation of the inner disk may lead to field rotational velocities of the order of the speed of light (Camenzind 1990; see also Fendt et al 1995). In this case a relativistic treatment of the MHD would be required. We emphasise that there are no relativistic effects in the dynamics of the jet motion itself (since the Alfvén surface would be well inside the light surface, where the field rotational velocity equals the speed of light).

Appl & Camenzind (1993a, b; hereafter ACa, ACb) investigated the asymptotic trans-field equation in the case of constant field rotation. They were first to find a non-linear analytical solution for a cylindrically collimated asymptotic field distribution (ACb). They also derived relations between the interesting jet parameters jet radius, current strength, and the field and current distribution.

In previous papers these results where used as a boundary condition for the calculation of global two -dimensional jet magnetospheres (Fendt et al. 1995; Fendt 1996). As it was shown, the critical solution of the wind equation along the calculated field structure asymptotically approaches the analytical force-free result (Fendt & Camenzind 1996).

However, since jet motion is connected to an accretion disk, and since the accretion disk rotates differentially, the jet magnetosphere, if it is anchored in the disk, essentially obeys differential rotation. This feature should therefore be a natural ingredient for any magnetic jet structure. How differential rotation effects the asymptotic jet equilibrium, is not obvious, since it involves collimating and de-collimating terms in the force-balance equation. Ferreira (1997) showed that differential rotation plays a major role in recollimation of jets and their asymptotic behaviour.

As a principal problem for differentially rotating relativistic jet magnetospheres, the position and shape of the singular light surface is not known a priori, but have to be calculated iteratively in a non-trivial way together with the flux distribution.

A differentially rotating field distribution is further interesting near the jet boundary. Here, models with a rigid field rotation imply a sharp cut off of the field rotation in the jet and in the surrounding interstellar medium, while with a differentially rotating field a smoother transition is possible.

The structure of the paper is as follows. In Sect. 2 we recall some basic equations of the theory of relativistic magnetospheres and discuss several difficulties with the solution of the Grad-Schlüter-Shafranov (hereafter GSS) equation. We evaluate the GSS equation for asymptotic cylindrical jets, including differential rotation. In Sect. 3 we discuss our results. We investigate, whether current free cylindrical jets are possible. We solve the asymptotic GSS equation for different assumptions for the field rotation and finally present a general analytic relation between the current distribution and the rotation law.

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© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998