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

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3. Results and discussion

We 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 [FORMULA]. Finally, using a general ansatz for the asymptotic field distribution we derive a relation between [FORMULA] and [FORMULA].

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,

[EQUATION]

with the formal solution

[EQUATION]

For physical reasons, [FORMULA] should be monotonous (since coupled to the disk rotation), and positive for all x. In order to be consistent with the chosen normalisation, we further require [FORMULA], and [FORMULA] for [FORMULA]. From the latter condition, it follows that the integration constant must vanish, [FORMULA]. Otherwise the rotational velocity [FORMULA] of the field would diverge for [FORMULA]. Note that, although the angular velocity may diverge with [FORMULA], [FORMULA], the rotational velocity remains finite for [FORMULA].

We can further see that for particular choice, a bounded current distribution with the core radius a,

[EQUATION]

and for [FORMULA] the current term in Eq. (21) does not diverge in the limit [FORMULA], leading to finite angular field rotation (since [FORMULA] must be finite), while for [FORMULA] the angular velocity diverges but not the rotational velocity, [FORMULA] finite value.

3.1. The case of constant or vanishing current

Now we take a look at the case of a vanishing poloidal current. A constant current, [FORMULA] const, would imply a divergence in the field rotation.

If [FORMULA], from the regularity condition (12) it follows that [FORMULA]. From Eq. (21) we conclude that a physical rotation law (which does not diverge at [FORMULA]) requires that the numerator [FORMULA] vanishes together with the denominator [FORMULA]. This, however, is in contradiction with the requirement of a decreasing, monotonous rotation law, as it can be derived from the following. A vanishing integral [FORMULA] requires that the integrand changes sign at a certain position. Thus, [FORMULA] has to have a maximum (a minimum is ruled out, since [FORMULA]), and also the term [FORMULA]. On the other hand, the integral has a maximum too, but not necessarily at the same position. This implies that the ratio of numerator and denominator passes a point of inflection, where both terms equal, and therefore [FORMULA]. Since also [FORMULA] by definition, this is in contradiction with a monotonous rotation law.

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 cylinder

The next case we will investigate is for a rotation law

[EQUATION]

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 [FORMULA], which is the light cylinder of a rigidly rotating magnetosphere. Here, [FORMULA].

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 [FORMULA]) the rotational velocity will diverge if [FORMULA]. Also, the surface [FORMULA] 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 y disappears in Eq. (10), we can immediately write down the solution

[EQUATION]

With a current distribution [FORMULA] the field distribution is

[EQUATION]

This gives a rotation law for the flux surfaces

[EQUATION]

We show the solution with bounded current distribution (22) and [FORMULA] in the Appendix. Fig. 1 displays both results in comparison with a field distribution resulting from a rigid rotation law, [FORMULA].

[FIGURE] Fig. 1a-d. a,c Magnetic pressure distribution [FORMULA] and b,d flux distribution [FORMULA] for a field rotation law [FORMULA] (solid, long-dashed) and [FORMULA] (dotted, short-dashed). a,b Current distribution [FORMULA], [FORMULA], c,d current distribution (22), [FORMULA], [FORMULA].

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 [FORMULA], Eq. (24) reveals [FORMULA], which is identical to the results of Contopoulos (1994). In the force free limit, his function [FORMULA] is identical to our poloidal current [FORMULA].

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, [FORMULA] (we assume here that the flux surfaces originating in the disk rotate with this velocity).

Since the field rotation near the disk [FORMULA] must be the same as in the asymptotic regime, [FORMULA], the flux distribution near the disk can be calculated,

[EQUATION]

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, [FORMULA] and the radius near the disk [FORMULA] is

[EQUATION]

We can further calculate the foot point of the outermost flux surface, [FORMULA], from Eq. (27), and with that and Eq. (28) the 'total expansion rate' of the jet

[EQUATION]

The first term in this equation varies rather weakly with g, and is of the order of unity (unless g is not much larger or much less than unity). For the second term we calculate for AGN ([FORMULA], [FORMULA]) a number value of about 2, which is surprisingly small, and for protostars ([FORMULA], [FORMULA]) a value of [FORMULA], respectively. This result may indicate on an intrinsic difference between the two jet sources. However, we should keep in mind that inertial forces may change the protostellar jet expansion rate and that the assumed current distribution might not be appropriate.

Comparing the field distribution near the disk (27) and in the asymptotic region (25) at small radii [FORMULA],

[EQUATION]

we may principally expect a recollimation of certain flux surfaces, depending on the source parameters [FORMULA] and the radius x. However, we believe that such kind of conclusions (e.g. 'recollimation predominantly for low mass AGN') might be exaggerated, since not very much is known about the disk field distribution and rotation, especially for small radii near the star, black hole, or disk boundary layer.

3.3. Numerical solutions of the asymptotic GSS including differential rotation

In this section numerical solutions to the asymptotic GSS equation with differential rotation are presented. Here, the current distribution is prescribed, and Eq. (10) is solved for different assumptions for the rotation law, [FORMULA].

In order to allow a comparison with rigid rotation solutions we chose a bounded current distribution (22) (with [FORMULA]) in parallel to the work of ACa, b. For the rotation law we require that (i) it is finite at [FORMULA], (ii) [FORMULA], in accordance with the normalisation, and (iii) [FORMULA]. These requirements are satisfied by e.g. the following functions,

[EQUATION]

where h plays the role of a steepness parameters (see Fig. 2).

[FIGURE] Fig. 2a-d. The rotation laws applied for the asymptotic jet magnetosphere (29) (a), (c), and (30) (b), (d). Square of the a, b angular velocity [FORMULA], and c, d the rotational velocity [FORMULA]. Parameters: [FORMULA] (solid), [FORMULA] (dotted), [FORMULA] (short-dashed), [FORMULA] (long-dashed), [FORMULA] (dotted-dashed).

There is a further condition (iv) for a rotation law. Rotation laws remaining valid for [FORMULA], have to be flatter than [FORMULA]. Otherwise the rotational velocity of the field lines will pass a maximum and finally decreases to values [FORMULA] (Fig. 2c). Note that ansatz (29) cannot be applied for arbitrarily large radii in the case of a high steepness parameter h. Since rotation law (29) is applied for a finite flux distribution, there is no serious problem as long as the turn-over of the rotational velocity is located beyond the jet radius. Ansatz (30) is more general, however, the analytical expressions look more complicated.

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.

[FIGURE] Fig. 3a-e. a Magnetic pressure distribution [FORMULA] and b flux distribution [FORMULA]. c Coupling constant as a function of the jet radius. The field rotation law is (29) with different steepness parameters, [FORMULA] (solid), [FORMULA] (dotted), [FORMULA] (short-dashed), [FORMULA] (long-dashed), [FORMULA] (dotted-dashed). The solid curves coincide with the analytical result from ACb. Note that the solid curves correspond to the dotted curves in Fig. 1c and 1d.

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 [FORMULA] and [FORMULA] or [FORMULA], we can derive the distribution of the conserved quantities [FORMULA] and [FORMULA] (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 [FORMULA]. The same behaviour is mirrored in Fig. 3e, if we compare the poloidal current at the jet boundary, [FORMULA], for different h.

The force-equilibrium is affected by differential rotation predominantly in the outer part of the jet. The field distribution within the core radius a of the asymptotic jet is not concerned very much by differential rotation, although a slight de-collimating effect can be observed. The behaviour changes beyond of [FORMULA], where the collimating effect is stronger than the de-collimation effect in the inner part.

Our results clearly show that differential rotation has a collimating influence. Depending on the steepness parameter, the asymptotic jet radius (defined by [FORMULA]) varies by a factor up to 2, which could be even larger for a lower coupling g. Note that the spatial scaling is in terms of the asymptotic jet radius [FORMULA]. This parameter, however, and thus the absolute scaling can only be inferred from a 2D solution. In Sect. 2.2 we gave arguments that, due to the rapid rotation of the accretion disk, [FORMULA] could be closer to the jet axis compared to solutions with constant rotation [FORMULA].

3.4. A non-linear analytical solution

In this section we derive a general analytical solution for the rotation law [FORMULA]. We assume a form of flux distribution parameterised as in Eq. (18). However, in contrary to the case of rigid rotation, the parameter [FORMULA] is not a priori coupled to the current distribution (e.g. Eq. 19). Then, the asymptotic GSS can be transformed into an ordinary differential equation for [FORMULA],

[EQUATION]

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

[EQUATION]

with the integration constant C. This solution diverges for [FORMULA] unless [FORMULA]. Thus, we obtain

[EQUATION]

and vice versa a relation for the current distribution in terms of [FORMULA]. In the limit [FORMULA] the solution approaches

[EQUATION]

For a current distribution (19) we end up with the result of ACb with constant angular velocity of the field, [FORMULA].

Since by definition [FORMULA], [FORMULA], we can derive an expression for the coupling constant

[EQUATION]

Eq. (35) is visualised in Fig. 4. We see that differential rotation plays a dominant role only for low-g / low-b jets, i.e. jets with low poloidal current and a broad field distribution (i.e. large core radius). Note that although a is shifted to lower values for steeper differential rotation, the magnetic flux [FORMULA] remains unchanged. In the limiting case of rigid rotation the parameter b describes steepness of the poloidal current distribution. We can rewrite Eq.(35) in terms of the core radius a of the field distribution

[EQUATION]

This shows that in order to obtain the same asymptotic magnetic jet structure (with the same parameters a, b, or [FORMULA] in Eq. (18)), the current has to be larger (parameterised by the coupling constant g) in the case of larger gradients of the rotation law. Similarly, for a fixed ratio [FORMULA] and g, but decreasing [FORMULA], also the core radius a (and thus [FORMULA]) is decreasing.

[FIGURE] Fig. 4. Interrelation between the jet parameters g, [FORMULA], [FORMULA] and the angular field velocity at the jet boundary, [FORMULA]. [FORMULA] (solid), [FORMULA] (dotted), [FORMULA] (short-dashed), [FORMULA] (long-dashed). The thick solid curve is the boundary of the forbidden regime, where no rigid rotating jet solutions are possible. The solid curves coincide with the result from ACb.

The thick line in Fig. 4 is the limiting value for the coupling constant g for rigid rotation, where the core radius a diverges (ACb). It corresponds to a minimum current required for rigid rotating magnetic jets, [FORMULA]. In the case of differential rotation, this value is decreased by a factor [FORMULA].

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 function [FORMULA]. Any solution [FORMULA], [FORMULA] has to lie within the limiting curves of [FORMULA] and [FORMULA] in Fig. 4. The ratio of the coupling constants for constant rotation ([FORMULA]) and for maximum differential rotation ([FORMULA]) is

[EQUATION]

Again we derive the 'total expansion rate' similar to Sect. 3.2 by comparison of the asymptotic solution with Keplerian disk rotation,

[EQUATION]

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,

[EQUATION]

we see that a stronger gradient in the field rotation (a lower value of [FORMULA]) 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

[EQUATION]

in the case of AGN, and

[EQUATION]

for protostellar objects.

We may assume that AGN jets are highly relativistic with [FORMULA], and therefore are strong differential rotators, [FORMULA]. Their expansion rate would then be of the order of 50. In the case of protostars [FORMULA], and thus [FORMULA]. The expansion rate would then be of the order of 600. The applied number values for [FORMULA] and [FORMULA] are only raw estimates, indicating 'steep' or 'flat' rotation laws and 'highly' or 'weakly' relativistic field rotation, respectively.

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 distribution

We 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. [FORMULA]. 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 [FORMULA] and [FORMULA], respectively. However, in this case the description of the poloidal current as a function [FORMULA], seems to fail. Instead it is supposed, that always [FORMULA], and one has to assume such kind of current distribution.

The problem is more serious for the other 'free' function, the field rotation [FORMULA]. Here, if we suppose an accretion disk as source for the magnetic flux, all foot points of the flux surfaces must rotate with monotonously decreasing angular velocity. Again, the description does not support a different field rotation for [FORMULA] and [FORMULA]. This statement is also valid for a non force-free description.

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 law

As an example for a current distribution appropriate for differential rotation we may chose

[EQUATION]

The steepness parameter d describes the variation from constant rotation. This leads to a field rotation

[EQUATION]

The jet radius is by definition at [FORMULA], and from Eq. (18) it follows [FORMULA]. Since [FORMULA], we calculate for the flux distribution parameter

[EQUATION]

Again, [FORMULA] gives the result derived by ACb. Otherwise, a and also [FORMULA] is decreased for fixed g and b.

The expression for the coupling constant is

[EQUATION]

The angular velocity of the outermost flux surface is

[EQUATION]

The interrelation of the parameters g, b, d and [FORMULA] is similar to Fig. 4. However, the parameter d has to be chosen such that [FORMULA], and [FORMULA] lies within the limiting curves of [FORMULA] and [FORMULA] in Fig. 4.

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© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998

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