3. The model assumptions
We now describe the model assumptions underlying the numerical calculations. The model topology basically follows the standard model for AGN (cf. Blandford 1990).
There is not very much known about the central sources of galactic superluminal jets. Since observationally the jet phenomenon of AGN and of young stellar objects as well is always connected to the signatures of an accretion disk, we assume a similar disk-jet scenario for the jet formation in galactic superluminal jet sources.
In the following, we discuss the three main components of the applied model - a central black hole, a surrounding accretion disk, and the asymptotic jet. A schematic overview of the model is shown in Fig. 1.
3.1. The central black hole
In the standard model for AGN the driving engine responsible for the activity is a rotating super massive black hole with a mass of about (Sanders et al 1989).
In the case of galactic superluminal sources, there is evidence that the central object is a black hole as well (Mirabel & Rodriguez 1995).
For the calculation of the field structure from the force-free stream equation, gravitational effects of the collapsed object play a non-obvious role. They appear in the stream equation in the description of the gravitogeometric background and in the two light surfaces.
The differential rotation of the space around Kerr black holes leads to the formation of two light surfaces (hereafter LS). Here, the rotational velocity of the field lines relative to the ZAMO equals the speed of light (see Blandford & Znajek 1977). This could be understood from the following. The concept of field lines implies the rigid rotation of each field line. Far from the hole, where is small, the outer LS describes the point where the field line velocity equals the speed of light seen from a static observer in the non-rotating space. Close to the hole, the space and the co-moving ZAMO is forced to (differentially) rotate. Since the field line is rigidly rotating, at a certain position (the inner LS) it will reach the speed of light in opposite direction seen from a ZAMO. Here, , and the 'field velocity' equals .
In terms of the global jet solution, the inner LS is very close to the horizon. With the numerical method applied, there is no fundamental hindrance for a solution between the inner LS and the horizon. However, since our main interest is the jet solution, we take the inner LS as inner boundary for the integration region. The black hole horizon itself remains hidden behind the inner LS.
The Schwarzschild radius defines the typical length scale for general relativistic effects,
For rotating black holes the event horizon is changed to . The angular velocity of the hole in terms of the Kerr parameter a and mass M is
Here, we choose .
Mirabel & Rodriguez (1995) mentioned that the de-projected jet speed of for the galactic superluminal jets could be related to the escape velocity from a region close to a black hole. Further, in contrast to other mildly relativistic jets () this is indicating both a black hole as central source and, also, that the jet origin is very close to the hole at a distance of several horizon radii (Mirabel & Rodriguez 1995).
3.2. The accretion disk
An accretion disk surrounding the central black hole seems to be the essential component concerning magnetic jet formation. It is considered to be responsible for the following necessary ingredients for jet formation, propagation, and collimation.
The accretion disk physics would further determine the rotation law of the jet magnetic field, (see Sect. 3.5).
The disk evolution is definitely influenced by the evolution of the jet and vice versa (cf. Ferreira & Pelletier 1995). However, since this global problem is literally unresolved, in this paper we take into account the accretion disk only as source for the magnetic flux, i.e. as a boundary condition for the flux function .
3.3. The asymptotic jet
We assume that the asymptotic jet is collimated to a cylindrical shape. This is in agreement with VLBI observations of the knot motion in e.g. 3C 345 (Zensus et al. 1995), and also with kinematic models explaining the short period optical variations by a geometrical lighthouse effect (Camenzind & Krockenberger 1992).
In the case of the parsec-scale jet in the quasar 3C 345, the best model fits give a very small intrinsic opening angle of (Zensus et al. 1995). The innermost region of the jet is not resolved observationally. In the radio VLBI measurements mentioned above, the angular beam resolution is corresponding to . This is comparable to .
The lighthouse model of Camenzind & Krockenberger (1992) reveals a jet radius of for both 3C 273 and BL Lacertae objects under the assumption that the field rotates with the angular speed of the marginally stable orbit. Further, the black hole parameters were assumed to be () and () for 3C 273 (BL Lacertae objects). The degree of jet collimation is very high and initial opening angles of () are found.
In the asymptotic regime, the metric simplifies to that of flat Minkowski space. Here, the special relativistic, one-dimensional jet equilibrium of Appl & Camenzind (1993) can be applied. They were first to find a non-linear analytical solution for a cylindrically collimated, asymptotic flux distribution,
where is defined as the cylindrical radius normalized by the asymptotic light cylinder radius. The flux distribution (17) corresponds to a core-envelope structure with core radius d. The poloidal current is concentrated within the jet core (see below). With Eq. (17) the asymptotic jet radius is defined by .
Using the method of finite elements for a numerical solution implies that we solve the boundary value problem. It is hence suitable to prescribe the asymptotic jet boundary and adjust later for the current profile parameter b (see Fendt et al. 1995 for details). This adjustment is the essential procedure in order to satisfy the regularity condition at the light surface.
In conclusion, the poloidal current distribution and the strength of the current are determined by the asymptotic jet. We choose the asymptotic jet radius in terms of asymptotic LC radii and .
3.4. The current distribution
The poloidal current distribution may be considered as a free function for the force-free stream function (although it is constrained by the regularity condition). In particular, since in the 3+1 description too, it is possible to apply the same current distribution for the region near the black hole as for the asymptotic, special relativistic region.
Here, we choose the analytical, non-linear solution for special relativistic (asymptotically cylindrical) pinches given by Appl & Camenzind (1993),
The parameter b describes the shape of the current profile. Together with the flux distribution (17) this current distribution simultaneously satisfies the asymptotic transfield equation (Appl & Camenzind 1993). The current flow is concentrated within the core radius d (see Eq. (17)). The strength of the current, , and the shape of the profile ( diffuse pinch, sharp pinch) control the magnetic structure and the kinematics of the jet. In particular, they determine the asymptotic jet radius and velocities.
As discussed above, in our approach we choose the jet radius as parameter and adjust the parameters and b in such a way that we obtain smooth solutions across the LS with an asymptotic jet radius of a few LC radii.
Eq. (18) represents a monotonous current distribution with no return current within the jet. A closure of the poloidal current flow would be achieved via the jet hot spots terminating the jet and the interstellar/intergalactic medium. In this picture the current is generated in the disk, flows along the jet to the hot spots, and returns back to the accretion disk in the surrounding medium outside of the jet. A return current within the jet might be a more realistic concept. However, serious difficulties for a two dimensional solution are involved with such a current distribution. These are e.g. a proper satisfaction of the regularity condition, as well as the need for a proper boundary condition for the asymptotic jet. We will address this topic to our future work. The solutions presented in this paper may be interpreted as the inner part of such a return current jet.
3.5. The rotation law
As was the case with the current distribution, the rotation law for the flux surfaces is a free function of the force-free stream equation.
In general, this rotation law follows from a detailed examination of the accretion process and the dynamo action in the disk. This is far beyond the scope of this paper and the complex physics of magnetized accretion disks is not yet fully understood. Although there are several models available for the different physical processes involved, such as (magnetic) viscosity, convection, advection, diffusion, kinematics, dynamo action, or relativistic effects, for a combined treatment of all the effects the problem is far from being resolved (not to mention that the jet itself provides an important boundary condition for the disk dynamo).
Having such a solution available, the calculated rotation law, , and flux generation, , would determine the rotation of the magnetic field .
As an example we discuss an approximate steady state solution for the flux distribution of a thin accretion disk around a black hole, , where is the diffusion parameter of the diffusion equation (Khanna & Camenzind 1992). For radii larger than the marginally stable orbit, , and the integration gives . Here, is the outer disk radius and the constant A is of the order of unity. Assuming a Kepler law for the disk rotation and additionally that the foot points of the flux surfaces rotate with Kepler speed, a possible rotation law for the field lines can be derived. Inverting the above disk flux distribution then leads to the rotation law .
As a consequence of differential rotation, the shape and position of the light surface would become a priori unknown quantities. Since these are singular surfaces and have to be considered numerically like boundaries, this would involve serious numerical complications. For this reason, for the time being, we consider . For the rotational velocity of the field, a fraction of the black hole rotation is assumed,
For , , the position of the asymptotic light cylinder is at .
There is strong indication that the superluminal jets originate very close to a central black hole at distances of a few horizon radii. For galactic sources, the argument is that the jet speed is close to the escape velocity near the horizon (see Mirabel & Rodriguez 1995). Standard models for BL Lacertae objects also put the jet formation close to the central source (see Kollgaard (1994) and references therein).
Thus, for these highly relativistic jet sources, the assumption of a constant rotation of the flux surfaces may well be applied. A problem might exist for only mildly relativistic jet motion.
© European Southern Observatory (ESO) 1997
Online publication: July 3, 1998