## 3. Results and discussionWe now discuss different solutions of the asymptotic GSS equation including differential rotation. We first consider the case of vanishing poloidal current. We give an analytical solution for a special rotation law leading to a 'degeneration' of the asymptotic light cylinder. Then, Eq. (10) is solved numerically for different assumptions for the asymptotic field rotation . Finally, using a general ansatz for the asymptotic field distribution we derive a relation between and . In general, the differential equation for the field pressure (10) can be rewritten as a differential equation for the angular velocity of the field lines, with the formal solution For We can further see that for particular choice, a bounded current
distribution with the core radius and for the current term in Eq. (21) does not diverge in the limit , leading to finite angular field rotation (since must be finite), while for the angular velocity diverges but not the rotational velocity, finite value. ## 3.1. The case of constant or vanishing currentNow we take a look at the case of a vanishing poloidal current. A constant current, const, would imply a divergence in the field rotation. If , from the regularity condition (12) it
follows that . From Eq. (21) we conclude that a
physical rotation law (which does not diverge at
) requires that the numerator
vanishes together with the denominator
. This, however, is in contradiction with the
requirement of a decreasing, We conclude only from asymptotic considerations that cylindrically collimated differentially rotating jets always carry a non-constant, net poloidal current. This is in agreement with previous results (Heyvaerts & Norman 1989, Chiueh et al. 1991). ## 3.2. A solution with degenerate light cylinderThe next case we will investigate is for a rotation law Now all asymptotic field lines rotate with the speed of light, and the light cylinder degenerates. Note that this does not contradict with our choice of normalisation. The length scale is measured in units of , which is the light cylinder of a rigidly rotating magnetosphere. Here, . The rotation law (23) and the corresponding field distribution may be considered as a somehow 'limiting case' for a physical field rotation. For a rotation law with a steeper slope (e.g. for ) the rotational velocity will diverge if . Also, the surface then plays the role of a somehow 'inverted' light cylinder since all field lines within (outside) the light cylinder rotate faster (slower) than the speed of light. Whether this behaviour could be considered as appropriate for astrophysical application also depends on the 2D field distribution. Since for assumption (23) the derivative term of With a current distribution the field distribution is This gives a rotation law for the flux surfaces We show the solution with bounded current distribution (22) and in the Appendix. Fig. 1 displays both results in comparison with a field distribution resulting from a rigid rotation law, .
We note that Contopoulos (1994) applied a similar rotation law for self-similar solutions of the 2D GSS equation, which take self-consistently into account also plasma inertia effects. With a current distribution , Eq. (24) reveals , which is identical to the results of Contopoulos (1994). In the force free limit, his function is identical to our poloidal current . As a simple application of this differentially rotating field distribution, the asymptotic solutions (25) and (26) are connected to an accretion disk with Keplerian rotation, (we assume here that the flux surfaces originating in the disk rotate with this velocity). Since the field rotation near the disk must be the same as in the asymptotic regime, , the flux distribution near the disk can be calculated, From the comparison of the disk flux distribution with the asymptotic flux distribution, it follows that for a certain flux surface the ratio between it's asymptotic radius, and the radius near the disk is We can further calculate the foot point of the outermost flux surface, , from Eq. (27), and with that and Eq. (28) the 'total expansion rate' of the jet The first term in this equation varies rather weakly with Comparing the field distribution near the disk (27) and in the asymptotic region (25) at small radii , we may principally expect a recollimation of certain flux surfaces,
depending on the source parameters and the
radius ## 3.3. Numerical solutions of the asymptotic GSS including differential rotationIn this section In order to allow a comparison with rigid rotation solutions we chose a bounded current distribution (22) (with ) in parallel to the work of ACa, b. For the rotation law we require that (i) it is finite at , (ii) , in accordance with the normalisation, and (iii) . These requirements are satisfied by e.g. the following functions, where
There is a further condition (iv) for a rotation law. Rotation laws
remaining valid for , have to be flatter than
. Otherwise the rotational In Fig. 3 we display the numerical solutions of the asymptotic force balance for ansatz (29). A solution with ansatz (30) looks very similar, we therefore omitted the plot. The solid curves show the field distribution with constant field rotation coinciding with the result of ACb, the other curves the result with increasing steepness of the rotation law, respectively.
The small peak in the field pressure (Fig. 3a) along the solution with the very steep rotation law results from numerical difficulties with the above mentioned decrease of rotational velocity for large radii and does not appear for the other ansatz. From the solutions and or , we can derive the distribution of the conserved quantities and (Fig. 3d, 3e), which could be applied for force-free 2D calculations. Fig. 3c shows the relation between the coupling constant
(measuring the strength of the poloidal current) and the jet radius.
In order to obtain jets with the same radius, the current strength has
to be increased with increasing steepness of .
The same behaviour is mirrored in Fig. 3e, if we compare the poloidal
current at the jet boundary, , for different
The force-equilibrium is affected by differential rotation
predominantly in the outer part of the jet. The field distribution
within the core radius Our results clearly show that differential rotation has a
## 3.4. A non-linear analytical solutionIn this section we derive a general analytical solution for the
rotation law . We assume a form of flux
distribution parameterised as in Eq. (18). However, in contrary to the
case of rigid rotation, the parameter is not
Now we investigate, whether a combination of current distribution and rotation law can be found, which is consistent with the chosen flux distribution. The general solution of Eq. (31) is with the integration constant and vice versa a relation for the current distribution in terms of . In the limit the solution approaches For a current distribution (19) we end up with the result of ACb with constant angular velocity of the field, . Since by definition , , we can derive an expression for the coupling constant Eq. (35) is visualised in Fig. 4. We see that differential
rotation plays a dominant role only for low- This shows that in order to obtain the same asymptotic magnetic jet
structure (with the same parameters
The thick line in Fig. 4 is the limiting value for the coupling
constant Eqs. (35) and (36) are a general result resting only on the
assumption of the field distribution (18). No assumption was yet made
about the Again we derive the 'total expansion rate' similar to Sect. 3.2 by comparison of the asymptotic solution with Keplerian disk rotation, where no assumption was made about a specific field rotation. Strong currents and large asymptotic jet radii imply a strong opening of the flux surfaces. If we rewrite Eq. (38) in terms of the field rotation, we see that a stronger gradient in the field rotation (a lower value of ) leads to a lower expansion rate. A vanishing field rotation of the outermost flux surface leads to a vanishing, unphysical, expansion rate. With reasonable numerical parameters the different central objects (see Sect. 2.2), the numerical values for the expansion rate are in the case of AGN, and for protostellar objects. We may assume that AGN jets are highly relativistic with
, and therefore are strong Keeping all the uncertainties in mind, we may generally expect lower expansion rates for the AGN. Especially the expansion rates for protostars have to be taken with care (see also discussion end of Sect.3.2). However, a rather general conclusion might be that high mass, fast rotating AGN have higher jet expansion rates than their low mass slower rotating counterparts. If we rewrite Eq. (35) we find an expression for the ratio of the jet radii in terms of the field rotation of the outermost flux surface. ## 3.4.1. The question of non-monotonous flux distributionWe note a general difficulty with non-monotonous flux distributions. In this case the jet magnetosphere would consist of flux surfaces with different foot points, but with the same absolute flux, e.g. . These flux surfaces are not directly connected within the integration domain. There is no physical reason, why they should not carry a different
poloidal current, as long it is conserved along
and , respectively.
However, in this case the description of the poloidal current as a
function , seems to fail. Instead it is
supposed, that always , and one The problem is more serious for the other 'free' function, the
field rotation . Here, if we suppose an
accretion disk as source for the magnetic flux, all foot points of the
flux surfaces We conclude that the monotonous disk rotation could only support monotonous flux distributions. Therefore, assumption (18) for the analytical solution seems to be rather general. ## 3.4.2. A special analytical rotation lawAs an example for a current distribution appropriate for differential rotation we may chose The steepness parameter The jet radius is by definition at , and from Eq. (18) it follows . Since , we calculate for the flux distribution parameter Again, gives the result derived by ACb.
Otherwise, The expression for the coupling constant is The angular velocity of the outermost flux surface is The interrelation of the parameters © European Southern Observatory (ESO) 1997 Online publication: May 26, 1998 |