7. Astrophysical consequences
We have studied the kink instability in magnetized jets. It has been found that the constant pitch model provides good estimates for growth rates and wavelengths in the small pitch regime. The previous results will be applied to VLBI jets from AGN, which are characterized by relativistic velocities. The linear evolution of the instability is described by a non-relativistic treatment in the rest-frame of the bulk flow, if the additional inertia due to the magnetic field and the displacement current (Begelman 1998) is neglected. The opposite limit, where the rest mass has been assumed to be negligible compared to the electromagnetic energy, has been considered by Istomin & Pariev (1994, 1996), and Lyubarskii (1999). The Lorentz factor , as well as the forces due to the electric field, are second order in the perturbed quantities and do not contribute in a linear analysis.
The equilibrium magnetic field is related to the jet velocity by the ideal MHD condition, , where now and are the laboratory fields, measured in a system at rest in the external medium, is the angular velocity of the footpoint where the poloidal field lines are anchored, is the mass flux per magnetic flux, and n the proper particle density. For large radii the rotation of the plasma can be neglected, and eliminating from the MHD condition we obtain for the jet velocity . The corresponding rest frame fields are and , respectively, and the relation between the jet velocity and the magnetic pitch in the rest frame becomes (Appl & Camenzind 1993b). Choosing for the Keplerian rotation at the marginally stable orbit, , of a Schwarzschild black hole (), we obtain . Among the observable jets only narrow, powerful objects such as quasars can therefore be expected to have a dominantly longitudinal magnetic field with large pitch, while less powerful jets (e.g. from BL Lac objects), correspond to very small values of the pitch. Even if at the base jets have , as they open up and become wider, their pitch necessarily decreases to small values. The current-driven instabilities do not propagate in the rest frame of the plasma, but are advected with the relativistic jet velocity. The jet propagates a distance during an e-folding time, where is the growth time in the observer's frame. We find for the distance propagated during one e-folding time, , or putting numbers appropriate for the VLBI jets,
According to this estimate only in powerful relativistic jets from massive objects the magnetic structure can survive over distances of many jet radii. The growth rate possesses a maximum at a well defined wavelength, and the instability consists of a non-axisymmetric perturbation advected with the bulk flow. We can speculate that dissipative processes resulting from the kink instability are responsible for the necessary in situ acceleration of synchrotron electrons, or are related to the formation of the VLBI knots, which apparently are non-axisymmetric features within the flow channel (Zensus et al. 1995). Observable features in large scale jets, such as the helical filaments in the jets of M87 (Biretta 1996) and 3C273 (Bahcall et al. 1995) are, however, unlikely to be due to kink instabilities of equilibria such as the constant pitch field. The simple scaling arguments preceding Eq. (3) indicate that, when the jet opens to radii of the order of several pc, the pitch becomes very small, and consequently, the fast kink mode will alter the magnetic structure long before the plasma has reached the distances where these filaments are observed. We conclude that, if not stabilized nonlinearly, the magnetic field configurations, as they result from the collimation zone, will not survive unaltered out to large distances. The instability would continue to operate until a large fraction of the electric current is dissipated, and the pitch has increased to a much smaller value. This does not mean, however, that the jet is disrupted (see below). Unstable modes are excited over a large wavelength regime, where the dominant mode possesses a wavelength much smaller than the jet radius. This may give rise to turbulent dissipation. For magnetically confined outflows this can result in a loss of collimation due to current dissipation. Such a jet would continuously become wider and eventually become confined by the pressure of the ambient medium. The internal structure of the jet is expected to undergo a magnetic relaxation process (Taylor 1986) and settle into a minimum energy configuration, which for sufficiently wide jets possesses a non-axisymmetric contribution (Königl & Choudhuri 1985). The helical filaments may then be kink instabilities triggering the transition to the non-axisymmetric minimum energy state (Appl 1996b), given that the growth rate for the linear force-free field are about an order of magnitude smaller that for the other examples.
Königl & Choudhuri (1985) argued that supersonic motion would suppress ripples at the interface of the jet, and that therefore the relaxation of the magnetic field would proceed as if the jet were bounded by a rigid wall. Eichler (1993) criticized that according to this line of reasoning the Kelvin-Helmholtz instability would not develop either. Our results lead us to conclude that the CDI in jets are internal modes, which do not deform the surface. If this remains true in the nonlinear regime, these jets will not be disrupted by external kink instabilities, as has been argued by Eichler (1993) and Spruit et al. (1997). However, the internal kink appears to behave essentially the same way, whether the jet is confined by any external pressure or by the hoop stresses of its own magnetic field. This is likely to lead to the dissipation of most of the electric current. In this case magnetic confinement would not be maintained over very large distances. The same mechanism may also put a limit on magnetic collimation at the base by magnetic fields alone.
For jets with the growth rate of the CDI is only a fraction of the Alfvén crossing time and therefore generally much shorter than for the KHI. Since in supermagnetosonic jets the CDI does not involve the jet boundary, and the kinetic energy dominates the magnetic energy we have argued that the CDI will not destroy such jets, but lead to dissipation of the electric current, thereby altering the magnetic structure. The gross dynamics of a pressure confined jet may then be governed by the KHI of a magnetized jet with a modified magnetic configuration. The KHI of MHD jets has been shown not to be too sensitive to the details of the magnetic profile, if instead of the hydrodynamic Mach number the fast magnetosonic Mach number is taken (Appl & Camenzind 1992). It can therefore be misleading to conclude on the fate of the jet by considering growth rates alone. Relativistic motion is favourable for stability in both cases. High Lorentz factors reduce the growth of CDI kinematically through time dilatation, and the KHI are in addition stabilised dynamically by an enhanced inertia.
The results of this paper, as well as most consequences, also apply to the case of jets from young stellar objects. Due to their non-relativistic velocities the CDI in jets from YSO are expected to grow even more rapidly. Dissipation of the electrical current may provide the necessary heating to explain the observed line emission. The regularly spaced knots (e.g. Zinnecker & McCaughrean 1997) however appear to be on the jet axis, and are therefore not likely to be due to non-axisymmetric CDI as they are discussed here. Unlike in extragalactic jets, thermal effects and radiative losses are dynamically important. Jet models taking into account typical conditions at the jet base are characterized by large negative pressure gradients, which can destabilize axisymmetric modes (Lery et al. 1998, 1999).
We have discussed and presented results for highly idealized jet models. Real jets certainly possess a velocity and density gradient, which makes a similar analysis impractical and less clear-cut. Since our results are pretty robust in the regime relevant to jets, they provide a useful guideline for the properties of current-driven instabilities in jets.
© European Southern Observatory (ESO) 2000
Online publication: March 9, 2000