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

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3. Non-relativistic, magnetically-driven jets from Keplerian discs

3.1. Governing equations

Above the disc, both the decay of the turbulent diffusivity and the action of the poloidal Lorentz force leads matter into ideal MHD state (Ferreira 1996). In this state, matter is frozen in the field, the poloidal motion occurring along a magnetic surface. Together with axisymmetry, this allows to describe jets as a bunch of magnetic surfaces, nested on the accretion disc at an anchoring radius [FORMULA]. In this section, we write the well-known governing equations for self-confined jets in a form similar to the one Pelletier & Pudritz (1992, hereafter PP92) used. Written in such a form, these equations will allow us to derive general properties and conditions for non-relativistic, magnetically-driven jets when one takes into account the underlying disc. In Sect. 4, we will come back to the set of equations as described in Sect. 2 and solve them from the disc equatorial plane to the "jet end".

In the ideal MHD jet, Eq. (4) becomes

[EQUATION]

where [FORMULA] is a constant along a particular magnetic surface. Its value, [FORMULA], is directly evaluated at the Alfvén point (labelled with the cylindrical coordinates [FORMULA] and [FORMULA]), where the jet poloidal velocity reaches the local Alfvén speed. Eq. (5) becomes

[EQUATION]

and defines the rotation rate [FORMULA] of a magnetic surface. Since the field lines are anchored on the accretion disc, they rotate with roughly the same rotation rate as matter, namely [FORMULA]. Thus, one can look at this equation as providing the amount of toroidal field required to maintain matter, which is frozen in the field, rotating in the jet at a different rate than the field.

Using these two equations, the steady-state angular momentum conservation equation can then be written as

[EQUATION]

where [FORMULA] is the total specific angular momentum carried away by both matter and field in a particular magnetic surface. The radial and vertical momentum conservation equations are usually replaced by their projection along (Bernoulli equation) and perpendicular (Grad-Shafranov equation, also known as transfield equation, Tsinganos 1981) to such a magnetic surface. For polytropic flows, Bernoulli equation reads

[EQUATION]

where [FORMULA] is the constant specific energy carried by the jet and the jet enthalpy H is defined as [FORMULA]. For magnetically-driven jets (i.e. cold), this enthalpy plays no role neither in jet formation nor in its collimation. Indeed, it is only a fraction of order [FORMULA] of the Keplerian speed and so it will be simply dropped in the following section (but note that the solutions in Sect. 4 are obtained with the full set of self-similar MHD equations). As shown in FP95, ejection of matter from the disc arises naturally due to the vertical decrease, on a disc scale height, of the radial current density. This is a pure magnetic process, without any help from thermal effects. On the contrary, thermally-driven jets require a high enthalpy with a polytropic index [FORMULA] variable along a magnetic surface, namely close to unity near the disc to account for coronal heating and closer to the adiabatic value further away (Weber & Davis 1967, Tsinganos & Trussoni 1991, Sauty & Tsinganos 1994). This last possibility is postponed to future work. Bernoulli equation (19) describes how the total energy carried by the flow is transformed into kinetic energy, for a given magnetic configuration. Hence, one can interprete this equation as providing the velocity matter reaches in a given "magnetic funnel". The shape of this funnel, or more precisely the jet transverse equilibrium, is provided by Grad-Shafranov equation

[EQUATION]

where we introduced the Alfvénic Mach number [FORMULA].

The set of Eqs. (16) to (20), together with Eq. (7), completely describe magnetic jets where plasma pressure plays no dynamical role. Hereafter, we describe the leading parameters of these equations.

3.2. An unique parameter for cold jets

We choose to characterize our jet solutions by using parameters as defined in the pioneering work of BP82. Since each parameter is strictly defined at the footpoint [FORMULA] of a magnetic surface, it characterizes the jet state only locally. Thus, in a realistic 2-D case, one would have to prescribe them for a range of anchoring radii in the disc. However, if we assume that jets are produced from a large radial extension in the disc, then we can look for a jet solution defined with parameters that are constants (as in self-similar solutions), or slowly varying with the radius. Through all this paper, we will label with a subscript "o" any quantity evaluated at the disc midplane and a subscript "A" at the Alfvén surface. In particular, [FORMULA] will be the midplane density and [FORMULA] the accretion velocity at a radius [FORMULA].

The first parameter describes the magnetic configuration, namely

[EQUATION]

with [FORMULA] (FP93a). Jets with constant [FORMULA] have a magnetic field varying with a power law of the radius. BP82's self-similar solutions were obtained with the prescription [FORMULA]. The second parameter, defined as

[EQUATION]

is a measure of the magnetic lever arm that brakes the underlying accretion disc. Since the available energy is stored in the disc as rotational energy, a constraint on this lever arm arises through the requirement that jets are powered with a positive energy. Indeed, the total specific energy carried away by the (cold) magnetic structure writes

[EQUATION]

where [FORMULA]. Jets become free from the potential well if the "rotator" energy, namely [FORMULA], overcomes the generalized pressure [FORMULA] (PP92). This is fulfilled provided [FORMULA]. The last parameter,

[EQUATION]

measures the mass load on a particular magnetic surface. Since cold jets carry away the whole disc angular momentum, one expects to find a systematic relation between mass load and lever arm.

These parameters must be linked to the ejection index, since it is a local measure of the ejection efficiency. A way to find such a link is to look at the ratio of the MHD Poynting flux to the kinetic energy flux along a magnetic surface,

[EQUATION]

Such a ratio measures the amount of energy stored as magnetic energy with respect to the kinetic energy, being therefore a measure along the jet of the acceleration efficiency. At the jet basis, identified here as the SM-surface, the disc provides

[EQUATION]

where we used mass ([FORMULA]) and angular momentum conservation. Thus, the ejection index is also quite accurately the injection of energy into the jets. While the magnetic lever arm is directly [FORMULA], the mass load writes

[EQUATION]

which provides [FORMULA] (see Appendix B). Thus, the leading parameters for magnetically-driven jets can be simply expressed as

[EQUATION]

The expression for the magnetic configuration parameter [FORMULA] is general for any Keplerian disc, allowing to take into account simultaneously magnetic and gravitational terms (FP93a, FP95). All three parameters for cold jets can be quite accurately replaced by [FORMULA]: magnetic configuration and both mass load and lever arm are therefore tightly related. As a consequence, the parameter space for cold jets is mostly controlled by the range of allowed ejection indices.

By using (28), we are now able to see that any jet model that would allow ejection with a high efficiency (namely, [FORMULA]) would not obtain free jets ([FORMULA]). Nevertheless, jet solutions that successfully cross the SM-point with [FORMULA] were found to be possible (Ferreira 1994). Such a situation is therefore a transient feature, matter failing to reach the Alfvén surface and falling down to the disc after having been ejected out.

3.3. Constraints on the ejection index

3.3.1. Minimum ejection index: disc vertical equilibrium

A look at the parameter space in FP95, derived with the sole constraint due to the slow-magnetosonic point, indicates that there is a minimum ejection index [FORMULA]. Here, we show that this minimum ejection efficiency is related to the existence of a quasi-magnetohydrostatic (MHS) equilibrium inside the disc. Such an equilibrium, described by

[EQUATION]

is obtained when plasma pressure gradient balances both tidal compression and magnetic squeezing. This magnetic squeezing depends on the value of the magnetic field, measured by the parameter µ (see Eq. (8)), and on both curvature (for [FORMULA]) and shear (for [FORMULA]) effects. When the ejection efficiency decreases, the lever arm increases and so does the magnetic compression due to field curvature. The only way to find out another equilibrium state is then to decrease the value of the field itself (that is, µ). However, this results in an enhancement of the magnetic squeezing due to shear.

This can be qualitatively shown like follows (for more details, see Appendices B and C). The induction equation (5) can be written in the disc as

[EQUATION]

where the last term describes the effect of advection. In the resistive disc, this term is negligible in comparison with the others. Thus, the toroidal field at the disc surface,

[EQUATION]

is mainly the sum of two contributions, [FORMULA] due to the unipolar induction effect and [FORMULA], the counter current (of positive sign) provided by the disc differential rotation. If the disc were rigidly rotating (i.e., a Barlow Wheel), it would generate the first contribution but not the second one, therefore producing no jet. When µ decreases, the magnitude of this counter current decreases, thereby increasing the toroidal field at the disc surface and with it, the pinching effect of the corresponding magnetic pressure gradient.

There is thus a [FORMULA] below which no quasi-MHS equilibrium can be found anymore. The exact value of this minimum ejection index is impossible to find analytically, since it depends on a subtle equilibrium between terms that are all of the same order of magnitude. When such equilibrium is crudely treated, one can in principle still obtain a matching with jet solutions but the disc would certainly not survive an overwhelming magnetic squeezing.

In Sect. 4, we show the parameter space obtained for self-similar solutions, thus providing a numerical value for [FORMULA]. Because self-similarity does not influence solutions close to the Keplerian disc but allows instead to take into account all dynamical terms, we believe that this value is general. Since what limits the minimum ejection index is the increasing toroidal field, one can only significantly lower [FORMULA] by decreasing the value of [FORMULA], that is, by lowering [FORMULA]. Therefore, only discs where both viscosity and magnetic torques are relevant ([FORMULA], requiring a hot corona to produce jets), should allow ejection indices much smaller than those displayed here.

3.3.2. Maximum ejection index: jet acceleration

Conservation of angular momentum (Eq. (18)) implies that once a large scale magnetic field steadily brakes the disc with a significant torque, matter must be accelerated up to the Alfvén point. This severily constrains the mass load in the jet, thus providing an upper limit on the ejection index. When gravity becomes negligible, Bernoulli equation can be rewritten (FP93a) as

[EQUATION]

where the function g, defined as [FORMULA], can be expressed as

[EQUATION]

It measures the discrepancy between the angular velocities of matter and magnetic surface on which matter flows (PP92). At the jet basis, its value is approximately zero, then it increases and reaches unity for highly super-Alfvénic jets. The Alfvénic Mach number becomes then simply

[EQUATION]

A necessary condition for trans-Alfvénic jets is obtained by requiring that the above expression reaches or becomes much bigger than unity. At this stage, it is convenient to introduce the ratio, measured at the Alfvén point, of the rotation velocity of the magnetic surface to the poloidal Alfvén speed (Michel 1969, PP92)

[EQUATION]

This fastness parameter is a useful quantity since it encloses an information that we cannot have without solving first the whole structure, namely the angle of the poloidal magnetic field with respect to the vertical axis. This ratio, which must verify

[EQUATION]

will allow us to derive a general condition for trans-Alfvénic jets, in two extreme cases.

For "powerful" jets (high [FORMULA]), matter is expelled off the disc carrying almost no angular momentum. Acceleration takes place only if [FORMULA], with [FORMULA]. Thus, these jets require

[EQUATION]

This corresponds to the extreme case where it takes almost no power from the magnetic structure to accelerate plasma up to the Alfvén surface. Such a configuration could be obtained in the low mass load limit.

For "weak" jets (low [FORMULA]), the Alfvénic Mach number has to reach at least unity. At the Alfvén point, Bernoulli equation writes [FORMULA]. This shows that matter reaches its maximum velocity when [FORMULA], namely when the toroidal field is almost zero. Using Eq. (34), it can be seen that these jets require

[EQUATION]

This general condition shows that magnetically-driven jets are fast rotators ([FORMULA]) in order to successfully reach the Alfvén surface (BP82, PP92, Rosso & Pelletier 1994). This is required even in this case, where matter barely reaches it despite the almost complete transfer of energy from the magnetic structure. High mass loads could give rise to such jets.

Compiling criteria (37) and (38), one gets the necessary condition for trans-Alfvénic jets

[EQUATION]

which must be satisfied at each magnetic surface. Here, we get a much more precise constraint since it requires

[EQUATION]

This implies that higher ejection efficiencies are completely inconsistent with a steady trans-Alfvénic regime. Note that this value should be viewed as an upper limit for [FORMULA], the maximum ejection index [FORMULA] being in fact smaller. This is due to the weak constraint we used in the high- [FORMULA] case. Contopoulos (1995) proposed a simple jet model where the magnetic field has only a toroidal component. In such a case (obtained in the limit [FORMULA] very large), the Alfvén singularity disappears and the driving mechanism for jet acceleration is the magnetic pressure gradient. We would like to stress that such a configuration is, in principle, possible as a transient outburst from thin discs. Indeed, FP95 showed that magnetically-driven jets can be viewed as being either "centrifugally-driven" for small ejection indices ([FORMULA], the vertical component of the Lorentz force pinching the disc) or "magnetic pressure-driven" for high ejection indices ([FORMULA], the toroidal magnetic pressure lifting matter up, consistent with the large [FORMULA] limit used). However, only the first situation allows a steady state with respect to the Alfvén surface. The possibility remains that such a configuration would be steadily produced by a thick disc. In the next section, we compute global self-similar solutions, derive both [FORMULA] and [FORMULA] numerically and obtain the full parameter space for cold MAES.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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