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

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4. Conclusions

In this paper the asymptotic force-balance across collimated magnetic flux surfaces was investigated. Relativistic effects due to rapid rotation of the field as well as differential rotation was included in the treatment.

The related astrophysical scenario is that of a highly collimated magnetic jet originating in an accretion disk, as observed in active galactic nuclei, galactic high energy sources with superluminal jets, and also protostellar jets with non-relativistic jet motion.

We presented numerical solutions of the asymptotic jet equilibrium for different assumptions of the field rotation. For a general assumption for the asymptotic field distribution we also derived an analytical solution.

The main results are the following

  • Differential rotation always leads to a decrease of the jet radius in terms of the asymptotic light cylinder radius.
  • This effect can be balanced by an increase of the poloidal current.
  • The inner structure of the jets remains more or less unchanged, the outer part becomes 'compressed' by differential rotation.
  • Jet expansion rates could be estimated under the assumption of a certain rotation law for the foot points of the field (e.g. Keplerian).
  • A general analytical solution was derived for the asymptotic flux distribution together with the rotation law of the field lines and the current distribution.

Depending on the steepness of the rotation law, the ratio in the jet radius between jets with and without differential rotation can be of the order of two. We also showed that differential rotation plays a role only for jets with low poloidal current and a broad field distribution.

In order to maintain jets with the same jet radius, but with a different gradient of field rotation, the strength of the poloidal current must be increased. In this sense, differential rotation may be considered as collimating effect and poloidal currents as de-collimating effect. However, compared to the rigid rotating field distribution, the minimum poloidal current required is decreased by a factor, which depends on the rotation rate of the outermost flux surface.

While within the asymptotic one-dimensional limit jets with arbitrary radius could be obtained, there are indications that 2D solutions of the relativistic GSS equation (but without differential rotation) only exist for asymptotic jet radii of the order of several light cylinder radii (Fendt et al 1995, Fendt 1996). It was impossible to obtain numerical solutions with jet radii larger than [FORMULA] light cylinder radii. This result was not caused by numerical effects. The results of the present paper indicate that the jet radii are even smaller.

A central question is therefore the scaling of light cylinder radius in terms of stellar (or black hole) radii. This, however, could only be inferred from a two-dimensional solution of the trans-field equation. We believe that inclusion of inertial effects would possibly widen the jet. However, one should keep in mind that in the case of self-similar jets Contopoulos & Lovelace (1994) and Ferreira (1997) have shown that centrifugal forces could be balanced by magnetic tension leading to a recollimation of the jet.

A critical point of the present investigation is that the interaction between the jet boundary and the ambient medium is not included in the force-balance. Hence, the question whether the jet is self-collimated or pressure collimated by the ambient medium cannot be answered. However, if we take a certain jet radius as given (by e.g. observational arguments), the results of this paper give examples of the local force-free force-balance of a jet with such a radius. In this picture the field pressure at the jet boundary must be balanced by the external pressure. Smaller or larger jet radii would change the jet parameters accordingly.

By comparing the field rotation near the foot points of the field lines (near a 'disk') and in the asymptotic regime, we were able to give some estimates on the expansion rate of the jets. Protostellar jets seem to have high expansion rates of the order of 1000, but these values are biased by the force-free assumption for the force-balance. Expansion rates of AGN jets are lower, a typical value might be 10. It can be said that high-mass fast-rotating AGN jet expansion rates are expected to be higher than those from low-mass slow rotating ones.

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© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998