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Astron. Astrophys. 355, 818-828 (2000)

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2. The magnetic structure

2.1. General considerations

The collimation and acceleration of these outflows is most likely of magnetic origin (Blandford & Payne 1982; e.g. Begelman et al. 1984). This mechanism eventually leads to cylindrically collimated jets (Heyvaerts & Norman 1989; Lery et al.  1999) and can produce the high Lorentz factors required by the observations. Most present 2D stationary models of the collimation and acceleration are, however, making particular assumptions, such as self-similarity (e.g. Blandford & Payne 1982), force-free magnetic fields, (e.g. Fendt et al. 1995), about the shape of the magnetic surfaces (e.g. Lery et al. 1998), or specific launching conditions (e.g. Shu et al. 1994). Given our ignorance of the details of the collimation process, and taking into account the diversity in physical conditions at the base of the jet and the surrounding medium, we consider simple jet models with a wide variety of magnetic profiles. We assume that the underlying flow is approximately stationary. The propagation of the jet through the interstellar/intergalactic medium is modeled by a sequence of quasi-equilibria, provided the ambient medium varies smoothly enough that the jet can continuously adjust itself to the changing conditions.

A magnetohydrodynamic outflow can exhibit Kelvin-Helmholtz, current-driven, and pressure-driven instabilities. In general it is not possible to unambiguously separate these effects. Only for very simple and idealized configurations one can hope for physical understanding. In this paper we restrict ourselves to current-driven modes.

For the purpose of this paper we approximate the MHD jet by an infinitely long cylindrical outflow of a perfectly conducting plasma. We suppose that the jet possesses constant density and velocity, as well as negligible thermal pressure and rotation. This excludes pressure-driven instabilities, and Kelvin-Helmholtz instabilities arise only due to the vortex sheet at the jet boundary. The equilibrium is then described by a force-free field in the rest frame of the jet,

[EQUATION]

Cylindrical coordinates [FORMULA] are used, and [FORMULA]. We suppose that the jet is in equilibrium with the ionized surrounding medium across its boundary at [FORMULA]. For the study of stability the magnetic pitch, P,

[EQUATION]

is a crucial quantity (e.g. Bodin & Newton 1980). Consequently, we will characterize the equilibria in terms of the profile [FORMULA], which uniquely determines the magnetic field configuration.

A real jet possesses a small, but finite opening angle. Flux conservation implies that the poloidal field falls of as [FORMULA], while the azimuthal field scales as [FORMULA], such that the latter eventually becomes dominant at large radii (Begelman et al.1984). Parameterizing this naive scaling in terms of the total magnetic flux, [FORMULA], of [FORMULA] Gauss cm2 and the total electric current, I, of the order [FORMULA] Ampère (e.g. Blandford 1990; Camenzind 1990) which can be carried by the disc-jet system of an AGN, yields an estimate for the pitch for extragalactic jets,

[EQUATION]

Even if the jets are launched with a large pitch, the pitch necessarily decreases, as the jet propagates and simultaneously becomes wider. Hence most observable jets are expected to possess a small pitch, [FORMULA]. The values relevant for YSOs ([FORMULA] Gauss cm2, [FORMULA] Ampère, and [FORMULA] AU) lead to similar conclusions.

2.2. Equilibria of magnetized jets

A simple equilibrium which satisfies the scaling above is given by

[EQUATION]

This configuration possesses constant pitch, [FORMULA], which also defines the radial scale length for the magnetic field. The constant pitch model has been applied to jets e.g. by Appl & Camenzind (1993a, 1993b) and Eichler (1993).

Since the precise jet structure is unknown, we consider, in addition to the constant pitch field, equilibria with radially varying pitch. This will provide our results with the required robustness and generality. As a convenient profile we choose

[EQUATION]

where [FORMULA] is the value on the axis, n is a positive integer, and [FORMULA] and [FORMULA] represent equilibria with monotonically increasing and decreasing pitch, respectively. The corresponding equilibrium magnetic field components are calculated numerically. Without loss of generality we restrict ourselves to positive values of [FORMULA].

As the jet continues to propagate through the interstellar/intergalactic medium, it may undergo a magnetic relaxation process and attain a state of minimum energy (Taylor 1986; Königl & Choudhuri 1985; Appl & Camenzind 1992). Such a relaxation process is observed in laboratory plasmas and leads to magnetic configurations known as reversed field pinches (RFP). Equilibria of this type have previously been considered for CTR (e.g. Bodin & Newton 1980). Despite of [FORMULA], they possess favourable stability properties due to the high magnetic shear. A widely used simple model for such an equilibrium is the linear force-free field, [FORMULA], with [FORMULA] constant. The Bessel function model (BFM),

[EQUATION]

is the solution for cylindrical symmetry. The pitch profile is [FORMULA] with a value [FORMULA] on the axis.

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© European Southern Observatory (ESO) 2000

Online publication: March 9, 2000
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