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Astron. Astrophys. 319, 340-359 (1997)

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2. Keplerian accretion discs driving jets

2.1. Magnetohydrodynamic equations for MAES

In order to produce bipolar jets, it is natural to rely on a bipolar topology for the large scale magnetic field. The other alternative, a quadrupolar topology, could in principle be used too, but such a topology could only produce weak jets (if any, see Appendix A). In both cases, this large scale magnetic field has two distinct possible origins: advection of interstellar magnetic field with accreting matter or "in-situ" production through a disc dynamo (Pudritz 1981, Khanna & Camenzind 1994). Both scenarii raise unanswered questions. For example, on the degree of diffusion of the infalling matter, which strongly determines the field strength in the inner regions (Mouschovias 1991, FP95). On the other hand, dynamo theory is still kinematic and one cannot easily infer from these studies what would be the final stage of the magnetic topology, if the matter feedback on the field is taken into account (Yoshizawa & Yokoi 1993). Most probably, a realistic scenario would have to take into account advection of external magnetic field while its amplification by the local dynamo.

In what follows, we restrict ourselves to the bipolar topology and assume that this field, although it has a profound dynamical influence on the disc, is not strong enough to significantly perturb its radial balance. Thus, the disc is supposed to be geometrically thin (its half-width at a distance r verifies [FORMULA]), in a quasi Keplerian equilibrium. The magnetohydrodynamic equations describing stationary, axisymmetric MAES are then the following:


where [FORMULA] is the gravitational potential of the central object (disc self-gravity is neglected) and [FORMULA] is a viscous stress tensor (Shakura & Sunyaev 1973). Both disc velocity and magnetic field were decomposed into poloidal and toroidal components, namely [FORMULA] and [FORMULA] respectively. A magnetic bipolar topology imposes an odd [FORMULA] with respect to z, as well as a poloidal field


where [FORMULA] is an even function of z. A magnetic surface, which is a surface of constant magnetic flux, is directly labelled by [FORMULA]. The distribution of magnetic flux through the disc is a free (and unknown) function. We will therefore use a prescription consistent with the complete set of equations.

Since the magnetic field threads the disc, steady-state accretion requires that matter diffuses through the field. As usual in astrophysics, normal transport coefficients are far too small to account for the expected motions. Drift between ions and neutrals, known as ambipolar diffusion, could play such a role in accretion discs (Königl 1989, Wardle & Königl 1993). However, jets are observed in a wide variety of objects, which suggests that they are produced by a mechanism independent of the disc ionisation degree. Moreover, it is now well known that magnetized discs are prone to instabilities (e.g. Balbus & Hawley 1991, Tagger et al. 1992, Foglizzo & Tagger 1995, Curry & Pudritz 1995, Spruit et al. 1995). Thus, we assume that the disc is turbulent and that the non-linear evolution of this turbulence provides the required anomalous transport coefficients, such as the magnetic diffusivity [FORMULA], resistivity [FORMULA] and viscosity [FORMULA] (appearing in [FORMULA]). All these transport coefficients should achieve a comparable level (Pouquet et al. 1976). Nevertheless, we introduced in Eq. (5) a "toroidal" magnetic diffusivity [FORMULA] to account for anisotropy with respect to the "poloidal" diffusivity [FORMULA]. Indeed, the winding up of the field due to disc differential rotation can lead to strong instabilities and thus, to enhanced transport coefficients with respect to the toroidal field (see FP95).

Finally, in order to close the system, we use a polytropic approximation


with [FORMULA] constant along a magnetic surface, in the isothermal case ([FORMULA]). Magnetized jets driven by an accretion disc can be initially launched either by predominant magnetic effects (magnetically-driven) or by enthalpy alone (thermally-driven). For discs without a hot corona (cold MAES with negligible enthalpy), FP95 showed that jets could indeed be produced through magnetic effects alone. For these magnetically-driven jets, the vertical profil of the temperature has not a strong influence on plasma dynamics, therefore allowing a description with an isothermal structure. The reason lies in the fact that most of the accretion power goes into the jets as an MHD Poynting flux. Thus, magnetically-driven jets will always be associated with weakly dissipative discs, where thermal effects can be neglected (see below).

Such a cold MAES corresponds to a "clean" magnetic structure at the disc surface, with just one polarity. In fact, it is likely that the magnetic structure would consist of open field lines coexisting with small scale loops anchored at different radii (Galeev et al. 1979, Heyvaerts & Priest 1989). Such a situation would then most probably result in the formation of a hot corona. The treatment of such a hot MAES is postponed to future work.

The local state of a cold MAES is mainly characterized by the set of following parameters evaluated at the disc midplane,


namely, the disc aspect ratio [FORMULA], the magnetic Reynolds number [FORMULA], the turbulence level parameter [FORMULA], the disc magnetization µ, the ejection index [FORMULA] and the ratio [FORMULA] of the magnetic to the viscous torque (FP95). This ratio depends on the magnitude µ of the field, the magnetic shear q of order unity, defined as


and being a measure of the toroidal field at the disc surface (Ferreira & Pelletier 1993a, hereafter FP93a) and, of course, on the magnitude of the viscosity through the well known [FORMULA] parameter (Shakura & Sunyaev 1973). Under the sole assumption that the disc is in Keplerian balance, angular momentum conservation implies that


must be verified at the disc midplane. If large scale magnetic fields are irrelevant ([FORMULA]), one recovers that the Reynolds number [FORMULA] is of order unity, which is required in standard viscous discs. On the contrary, when they significantely brake the disc, one obtains that the degree of curvature of the field lines at the disc surface (measured by [FORMULA]) depends mostly on how strong is [FORMULA]. BP82 showed that in order to magnetically launch jets without a hot corona, the magnetic structure needs to be bent by more than [FORMULA] with respect to the vertical axis. This implies a magnetic Reynolds number of order [FORMULA], therefore [FORMULA]. Cold jets require then a dominant magnetic torque, imposing a corresponding value on both the field strength and shear.

Energy conservation equation (FP93a),


shows that the total available mechanical power [FORMULA] is shared by the disc luminosity [FORMULA], the outward MHD Poynting flux [FORMULA] and the flux of thermal energy [FORMULA] from each surface of the disc. The liberated power is such that [FORMULA] (see FP95, Ferreira 1996), where [FORMULA] is the disc inner radius and [FORMULA] is the accretion rate at the disc outer edge [FORMULA]. Around a compact object, this accretion power is


for [FORMULA], [FORMULA], where [FORMULA] is the Schwarzschild radius. Around a protostar, this power is


for [FORMULA], [FORMULA] (10 stellar radii for a typical T-Tauri star). Since the available energy is stored into rotation, energy conservation can be rewritten as


where [FORMULA] is the fraction of the internal energy that was transferred in the jet as enthalpy. The total jet power, which can be either radiated away or injected at the terminal shock (e.g. extragalactic radio lobes), arises from the sum of the MHD Poynting flux and the plasma thermal power. Thus, magnetized accretion discs without a hot corona ([FORMULA], [FORMULA]) radiate only a fraction of order [FORMULA] of the jet power.

2.2. From accretion to ejection

MAES are intricate structures where accretion and ejection are interdependent. Therefore, we will frequently make references to disc physics and quantities in our investigation of jet physics. Let us then, for the sake of completeness, summarize here results concerning magnetized disc physics.

Two simultaneous processes are responsible for accretion (FP93a): (1) a turbulent magnetic diffusivity [FORMULA] allowing matter to steadily diffuse through the field; (2) a dominant magnetic torque [FORMULA], which brakes the disc and stores into the field both angular momentum and mechanical energy of the plasma. A local quasi-magnetohydrostatic vertical equilibrium is achieved, the plasma pressure gradient hardly competing with both tidal compression and vertical magnetic pinching force. A radial magnetic tension slightly counteracts gravity, thus leading to a sub-Keplerian rotation rate.

As a result, plasma inside the disc (see Fig. 1) is being accreted, slightly converging towards the disc midplane (both [FORMULA] and [FORMULA] are negative). How then is ejection achieved ?

[FIGURE] Fig. 1. Side view of a magnetized accretion disc driving jets, with [FORMULA], [FORMULA] and an ejection efficiency [FORMULA]. Plasma (dotted lines) enters the structure at its outer edge and is accreted with a slight converging motion through the magnetic field (solid lines). Since the magnetic pinching force decreases vertically, plasma reaches a layer where the pressure gradient slowly expells it. Diffusion is still necessary at the disc surface in order to allow a transition between the accretion disc and the ideal MHD jet. This transition region is approximatively one disc scale height thick. Farther out, plasma is frozen in a particular magnetic field line and is accelerated through the Lorentz force. Note the scaling factor [FORMULA] applied to the vertical axis.

Ejection comes out naturally if the radial current density [FORMULA] decreases vertically on a disc scale height (Ferreira & Pelletier 1993b). Indeed, in a Keplerian disc, this is the only possibility to change the sign of the magnetic torque [FORMULA]. This torque must become positive at the disc surface in order to provide a magnetic acceleration. Such a situation requires that the counter current due to the disc differential rotation balances the current induced by the unipolar induction effect (FP95). If this is not fulfilled, the complete structure is unsteady: in fact, the requirement of stationarity provides the level of the "toroidal" diffusivity [FORMULA].

The vertical decrease of [FORMULA] leads to the following important effects:

(1) the vertical Lorentz force decreases, allowing the plasma pressure gradient to gently lift up matter from the disc surface;

(2) matter is azimuthally accelerated by the magnetic field, leading then to a radial centrifugal acceleration.

Note that, although it is plasma pressure that provides a positive vertical velocity, this process is purely magnetic. It arises naturally if the magnetic pinching force (that must be comparable to the plasma pressure gradient) decreases vertically.

In order to get an insight on the jet physical processes, it is worthwhile to project the Lorentz force parallel and perpendicular to any poloidal magnetic surface,


Here, [FORMULA] is the total current flowing within this magnetic surface, [FORMULA] and [FORMULA]. This shows directly that plasma is accelerated by the current leakage through this surface ([FORMULA]). This effect gives rise to both a poloidal ([FORMULA]) and a toroidal ([FORMULA]) magnetic force, the latter providing the centrifugal force. Thus, cold jets are better referred to as being magnetically-driven rather than centrifugally-driven (this was also pointed out by Contopoulos & Lovelace 1994). When the current I vanishes, or when it flows parallel to the magnetic surface, no magnetic acceleration arises anymore and the plasma reaches an asymptotic state. Note also that the way this current is distributed across the magnetic surfaces is of great importance for the jet transverse equilibrium ([FORMULA]). This shows that one has to be careful when dealing with a particular current distribution, since it is of so great importance in both jet collimation and acceleration (Appl & Camenzind 1993a, 1993b).

The global current topology is linked to the ejection efficiency, which is measured by the ejection index [FORMULA] ([FORMULA] in a viscous disc without jet). For small ejection efficiencies ([FORMULA]), ejection takes place against a pinching vertical Lorentz force, with a positive radial component. For those tenuous ejections, the current enters the disc at its inner edge, flows up inside the jet and closes back along the axis. High ejection efficiencies ([FORMULA]) are achieved when matter is lifted by both plasma and magnetic pressure gradients (positive vertical Lorentz force), the corresponding radial Lorentz force being negative. In this case, the current flows down the jet, enters the disc at its surface and closes in an outer cocoon (Ferreira 1994).

The set of MAES parameters (8) is diminished by the requirement that the overall structure is in steady-state. This implies that the flow has to smoothly cross the usual MHD critical points it encounters (see FP95). The constraint due to the first critical point, the slow-magnetosonic (SM) point, allows these two current topologies to be realized. In what follows, we will show that only tenuous ejections can produce trans-Alfvénic jets.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998