6. Summary and conclusion
In this paper, we have covered all dynamical processes at work in a stationary, weakly dissipative MAES of bipolar topology (see Appendix A for quadrupolar), by constructing continuous solutions from the accretion disc to super-Alfvénic jets. We summarize below our findings and discuss their implications.
(1) We have demonstrated that the usual parameters for magnetically-driven (cold) jets, namely the magnetic configuration, lever arm and mass load, are intrinsically related to the disc ejection efficiency . Thus, realistic jets would be described by a distribution of with the magnetic surfaces (or their anchoring radius).
(2) We showed that this parameter lies in a very narrow range for cold jets, namely . A minimun ejection efficiency is required for the disc to find a quasi-MHS equilibrium. A crude modelling of this sensitive equilibrium leads most probably to unstable regimes, the disc being too much pinched by the Lorentz force. On the other hand, a maximum mass load arises from the constraint of accelerating matter up to super-Alfvénic speeds. The threshold for the minimum ejection index is raised for decreasing disc aspect ratio and turbulence parameter . Both constraints impose that these parameters must verify and .
(3) As a result, the current must enter the disc at its inner edge, flow back in the jet and close by flowing down along the axis. This has two important consequences. First, although the disc can afford larger ejection efficiencies (Ferreira 1994), they would be non-steady, the flow not being able to reach the Alfvén surface. Second, this implies a strong influence on what's going on at the axis, thus suggesting a coupling with the inner central object magnetosphere. This work is under progress.
(4) The existence of a minimum ejection index has strong implications on jet energetics. Indeed, it forbids to construct jet models where the mass load is arbitrarily small (and thus, an asymptotic velocity arbitrarily high). Nevertheless, in the range allowed for , the ratio of the total ejection rate (in one jet) to the accretion rate, , would vary between and , depending on both and the radial extension of the magnetized disc (Ferreira 1996). This is in complete agreement with recent estimates made by Hartigan et al. (1995), who find a typical value of for jets from young stars.
shows that relativistic jets could be possible around compact objects. If Compton drag can be avoided (Phinney 1987), the mean bulk Lorentz factor that such jets could achieve,
strongly depends on how much power the magnetic structure keeps stored in its asymptotic regime (see FP95 for the definition of ). As an example, for (final kinetic energy flux comparable to the Poynting flux, Li et al. 1992), one gets between 1 (, ) and 8 (, ), with typical values lying between 1 and 4. Thus moderate mean bulk Lorentz factors are likely to be achieved by cold jets from compact objects.
In the context of YSOs, the mean jet velocity writes
where is the angular velocity at the inner disc radius. For , jets can reach a velocity between 1 (, ) and 10 (, ) times this velocity, with typical factors between 2 and 6. For structures settled at (10 stellar radii for a typical T-Tauri star), this provides jets with a mean velocity between 100 and 500 kms-1, in agreement with observations.
(5) All the above results are general to magnetically-driven jets from Keplerian accretion discs. We constructed global solutions, from the accretion disc to a super-Alfvénic jet, using a self-similar ansatz. It has been shown that the jet asymptotic behaviour strongly depends on the fastness parameter , which describes how fast is the magnetic "rotator" and is a measure of the amount of current still available at the Alfvén surface. This parameter must be bigger than (but of the order of) unity and is very sensitive to the physical conditions inside the disc. All our solutions display the same behaviour: the jet widens until a very strong acceleration efficiency is achieved, all the available power being eventually transferred into kinetic power; the centrifugal force decreases so much that the Lorentz force pinches the jet, making it to recollimate. The bigger and the larger the maximum radius reached by the jet. Inside our parameter range (which is weakly affected by our self-similar assumption), this behaviour stems from having a constant ejection index . Therefore, we expect that non self-similar jets from Keplerian discs described with constant parameters would also display recollimation, unless realistic 2-D boundary conditions are taken into account. Such boundaries should concern the transition from the outer viscous disc to the inner magnetized one, as well as a possible interaction with the central object.
(6) If magnetized discs are driving jets over a wide range of radii, then they can probably be described by an almost constant ejection index (and self-similar solutions are not too bad an approximation). In those circumstances, one would expect that these jets undergo a recollimation and then a shock, making the whole structure unsteady. Whether or not such a shock is terminal or a "magnetic focal point" (Gomez de Castro & Pudritz 1993, Ouyed & Pudritz 1993) remains to be carefully worked out. Besides the shock signature, such a structure could be detected through its apparent lack of radiation coming from the magnetized disc itself (FP95). However, one has to bear in mind that the fate of jets could be strongly modified by the external medium (Appl & Camenzind 1993a). Indeed, an external confinement could forbid a natural "over-widening", thus enforcing a redistribution of the current density inside the jet. By this way, asymptotic solutions with but with non-vanishing current I (in contrast with what is obtained here) could perhaps be achieved. Such an hypothesis should deserve further investigation.
(7) Finally, a major challenge remains the question of the source for the required magnetic diffusivity. Indeed, Heyvaerts et al. (1996) showed that in a disc braked by viscous stresses, any instability (possibly magnetic), triggering a turbulence with an injection scale of order the disc thickness, would provide . Such a situation is therefore incompatible with magnetically-driven jets. As a consequence, these powerful jets require physical conditions that were not yet investigated in discs like those, for example, met at the interaction with the central object magnetosphere. On the other hand, thermally-driven jets (but magnetically confined) should be viewed as a possible alternative. The jet power could be reduced down to a level comparable to the disc luminosity (see Eq. (14), with ). Such a picture is appealing because it allows jets along with radiating discs and offers a smooth transition between "viscous-like" discs and MAES (Ferreira, in preparation).
© European Southern Observatory (ESO) 1997
Online publication: July 3, 1998