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

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4. Results and discussion

We now present numerical solutions for the jet magnetosphere. The field distribution is calculated under the assumption of a small plasma loading, or in other words, in the force-free limit. Although the calculations are not entirely self-consistent, these are the first calculations of a collimated global jet magnetosphere in the context of Kerr metric.

Similar to the special relativistic, or even the Newtonian case, in a general relativistic treatment, the singularity of the stream equation at the Alfvén radius (or light surface) leads to a "kinky" structure of the magnetic field at this position, unless the regularity condition is properly satisfied.

In our previous work (Fendt 1994, Fendt et al. 1995) we investigated this problem for fast rotating, stellar magnetospheres. We found that a field matching could be achieved by adjusting both the shape of the jet boundary [FORMULA] and the poloidal current distribution (via the parameter b). This procedure may be interpreted as an adjustment of "magnetic pressure equilibrium" between the regions inside and outside the light cylinder.

In order to obtain smooth solutions at the outer light surface for the Kerr metric jet solutions presented here, we applied the same technique derived earlier (Fendt 1994, Fendt et al. 1995). Again, we emphasize that in the presented solutions the regularity condition at the outer LS determines the shape and the location of the jet boundary in the collimation region.

4.1. The global jet solution

Fig. 2 shows two examples of global jet solutions extending from the inner LS to an asymptotic jet collimated to a cylindrical shape. The flux surfaces pass the outer LS smoothly.

[FIGURE] Fig. 2a and b. Magnetic flux surfaces [FORMULA] of global Kerr jet solutions ([FORMULA], [FORMULA]). The asymptotic jet radius is 3 asymptotic LC radii [FORMULA]. Current distribution parameter a [FORMULA], [FORMULA], [FORMULA] ; b [FORMULA], [FORMULA], [FORMULA]. Units: [FORMULA] in [FORMULA], contour levels [FORMULA], [FORMULA] ; [FORMULA] in [FORMULA] (upper and right axis), or in M (lower and left axis)

The asymptotic jet radius is [FORMULA] corresponding to [FORMULA] for the parameters [FORMULA] and [FORMULA], or


The asymptotic jet radius is basically parameterized in terms of the outer LS, i.e. in terms of the rotational velocity of the field. We chose [FORMULA] under the assumption that the jet is launched within a distance of some [FORMULA] from the black hole.

If the field was rotating slower, when the flux surfaces originate further out in the disk, the asymptotic jet radius would be larger. With e.g. [FORMULA] and [FORMULA], we obtain an asymptotic jet radius 4 times larger. Note that the linear scaling in terms of [FORMULA] remains the same. It changes, however, in terms of [FORMULA].

We report that we were not able to obtain jet solutions with [FORMULA]. The numerical procedure converges in the asymptotic region for asymptotic 1D solutions with any jet radius. However, it does not for a lower boundary condition, which is even slightly different from the asymptotic solution, and thus implying a field curvature. This negative result is not caused by numerical problems. It does not depend on numerical parameters like element size.

Instead, we take this as an indication for an upper limit for the jet radius,


of a cylindrically collimated, rigidly rotating, force-free magnetic jet.

On the other hand, differential rotation or plasma inertia may open up the jet structure. Both effects lead to an increase of poloidal current. The question is whether the corresponding de-collimating toroidal field pressure will supersede the effect by collimating tension. We suppose that the differential rotation plays a minor role as long as all the flux originates within a small region in the disk.

The jet solutions presented here are not 'self-collimated'. The prescription of an asymptotic jet boundary may be interpreted as an external pressure from the surrounding material. However, the shape of the collimating jet radius [FORMULA] is determined by the regularity condition and/or the current distribution in the jet, and hence, is determined by the internal force-balance.

How the internal force-equilibrium affects the shape of the collimating jet can be seen in Fig. 2. In both solutions, the adjustment procedure (shape of jet boundary [FORMULA] distribution parameter b) was performed until the regularity condition is properly fulfilled. The two solutions have different poloidal current distributions, [FORMULA]. In comparison to the flux distribution with the broader current distribution ([FORMULA]), the solution with the more concentrated current distribution ([FORMULA]) involves an enhanced expansion (a slight de-collimation) beyond the outer light surface in order to obtain matching between the outer and inner solution.

4.2. The central domain and possible mass flow distribution

Fig. 3 shows subsets of the innermost part around the black hole from global solutions for different disk boundary conditions. These near-disk solutions might be interpreted as a disk corona. The overall picture could be summarized as follows.

  • There is magnetic flux outgoing towards the jet.
  • There is magnetic flux ingoing towards the black hole (which is hidden behind the inner LS).
  • The flux surfaces near the jet axis are not directly connected with the accretion disk.
  • Depending on the disk magnetic flux distribution, the curvature of the field lines close to the disk is rather different.
[FIGURE] Fig. 3a-c. Magnetic flux surfaces of Kerr jet solutions. Subsets of the inner most regions with different choices of disk distribution. Disk flux distribution parameters a [FORMULA], [FORMULA], [FORMULA] ; b [FORMULA], [FORMULA], [FORMULA]: c [FORMULA], [FORMULA], [FORMULA]. The disk flux distribution is [FORMULA] (see Appendix A.2). Units as in Fig. 2

If for the following we imagine a possible mass flow associated with the flux surfaces, we find three different flow regimes within the field distribution - an accretion region, an outflow region, and a region empty of a plasma flow.

The ingoing flux tubes would allow for magnetic accretion from outer parts directly into the black hole. It is however questionable, whether the field strength will be so strong that plasma is accreted along field lines or whether, on the contrary, accretion will be dominated by gravitation and will thereby determine the field topology. This question cannot be answered with the present, force-free approach and depends on parameters like field strength, mass flux, or magnetization.

Under the assumption of a finite flux distribution originating very close to the black hole, the inward-outward bending flux surfaces provide evidence for a hollow jet structure. Although we have to investigate the wind equation along the flux surfaces in order to gain detailed knowledge about the plasma flow behaviour, we believe that the following thoughts and considerations might be reasonable.

First, the slope of the flux surfaces is too small to allow for a 'centrifugal' acceleration of the plasma. Blandford & Payne (1982) obtained a minimum angle enclosed by the disk and flux surface of [FORMULA] for the onset of plasma acceleration. Although this result was specifically calculated for a self-similar differentially rotating field structure and a cold wind, we believe that we can use it as an estimate for our case. It seems to be obvious that along a flux surface perpendicular to the disk, a wind driven by centrifugal instability is not possible.

If we therefore consider a hot plasma, the thermal pressure in a hot disk corona has to accelerate the plasma from the disk to heights of about 4 horizon radii above the disk. Here, the slope of the flux surfaces becomes eventually less than the critical value, allowing for 'centrifugal' acceleration. Such strong thermal pressures would require a very hot corona. However, our own experience as well as results published in the literature, show that the slow magnetosonic point of a wind flow is always located very close to the injection point. Thus, thermal pressure is unlikely to be a driving force up to high altitudes above the disk.

Secondly, from our work on the cold relativistic wind equation (Fendt & Camenzind 1996), we know that the stationary character of the flow is very sensitive on the curvature of the poloidal field in the case of a high plasma magnetization. In regions where the slope of the field line changes, it is very likely that no stationary solutions of the wind equation are possible, indicating that here shocks and instabilities may arise. These shock waves could eventually be observed in the asymptotic AGN jet as the helically moving knots seen with the VLBI radio observations (e.g. Zensus et al 1995).

Thirdly, the flux surfaces near the axis are not connected with the disk boundary but with the inner LS. Blandford & Znajek (1977) argued that all particles moving along field lines passing the inner LS must travel inwards. Along these field lines no mass outflow from the disk is possible. We found no solutions with flux surfaces extending from the disk boundary towards the jet axis.

Being aware of the crudeness of the the preceding considerations, we conclude that in the solutions presented here the plasma will only flow within a thin layer of about [FORMULA] near the jet boundary, basically forming a hollow jet structure. The inner 90% of the jet cross-section will be empty of plasma flow from the disk. This picture is in good agreement with radio observations of AGN jets revealing moving knots along helical trajectories (Steffen et al. 1995, Zensus et al. 1995). It also fits within recent kinematic radiation models explaining the parsec scale motion of the jet knots by the lighthouse effect (Camenzind & Krockenberger 1992).

4.3. Angular momentum loss and Poynting flux from the black hole

The magnetosphere - poloidal current system is associated with an angular momentum flow and Poynting flux (or luminosity). The total Poynting flux in the jet [FORMULA] can be calculated, using the known current distribution,


(see Appl & Camenzind 1993), revealing a similar value for both field distributions in Fig. 2, [FORMULA].

The angular momentum loss [FORMULA] from the black hole into the jet follows from the integration of the current distribution for all flux surfaces leaving the horizon to the asymptotic jet,


and similarly for the Poynting flux (see Okamoto 1992). The outermost flux surface [FORMULA] leaving the black hole (or the inner light surface) to the asymptotic jet could be estimated from Fig. 2 and Eq. (18). The integration gives


For the parameters of the solutions in Fig. 2, this gives an angular momentum loss from the black hole [FORMULA] for the solution with a concentrated current distribution (Fig. 2a), and [FORMULA] for the other one (Fig. 2b). Since the coupling g is similar for both solutions, their [FORMULA] differ by a factor of 3.

What might be surprising with this result is that for two jet solutions with the same asymptotic jet radius, the same total magnetic flux and current flow, and also the same disk flux distribution, the angular momentum extraction from the black hole differs by a non-negligible value, which is determined by the internal structure of the jet magnetosphere. A similar calculation for the Poynting flux leads to [FORMULA] for the solution in Fig. 2a, and by the same factor 3 less for the other field distribution. The total angular momentum loss and Poynting flux in the jet differ only very few for both solutions.

We conclude this section with mentioning that the above estimates do not necessarely allow for an interpretation in terms of the dynamical evolution of the black hole. In our approach the flux surfaces emanating from the black hole/inner LS to the asymptotic jet also connect from the accretion disk to the black hole (with the same [FORMULA]). Thus, the same energy/angular momentum flow leaving the black hole also goes into the hole. What is important for the black hole evolution, are the total energy and angular momentum losses from the disk and the hole by the jet. However, the locally different structure of the current-magnetosphere system might affect the evolution of the accretion process and also radiative processes involved with the accretion.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998