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3.1. The launching of extragalactic jets

While several implications from the observations of extragalactic jets based on fundamental theories have already been described, we now turn to matters less accessible to direct observation because they occur on such minute spatial scales. Here we have space to merely list some of the ever growing lines of evidence that have convinced essentially all astrophysicists that accreting SMBH's provide the prime mover for AGNs: accretion into a very deep potential well is the only mechanism that appears to be sufficiently efficient (easily > 5%) in converting matter into the energy to power the most luminous AGN; fast luminosity variations imply most of the emitted power must emanate from a very compact region, probably just a few rg = GM / c2 = 1.5 (MBH / Msun) km in extent; a SMBH, particularly a rotating one, is the only known way to produce the stable axis needed to produce jets extending for Mpc that must have remained active for > 107 years; VLBI measurements have shown collimated emission to have been produced within < 100rg in a few nearby radio loud AGN; stellar velocity dispersions in the inner cores of nearby AGN rise very steeply towards the center, implying immense densities of matter (> 107 Msun pc-3); maser emission lines have demonstrated clear Keplerian rotation about massive dark cores; the shapes of x-ray emission lines are best explained as emerging from the inner portions of accretion disks around SMBHs, where general relativistic effects play important roles.

We now proceed to a brief discussion of the various classes of models that have been proposed to produce jets from the environs of a SMBH. The fundamental types of models proposed through the 1980's for the origin of jets were reviewed extensively by Wiita (1991), and here we will very briefly summarize those scenarios and note some more recent results. All models involve the expulsion of a certain fraction of the matter being accreted by the SMBH. The key distinction between the major classes of jet launching scenarios is whether or not magnetic fields are assumed to be primarily responsible for the expulsion of jet plasma.

Purely hydrodynamical models based upon winds from standard thin accretion disks (e.g., Shakura & Sunyaev 1973) have difficulties in accelerating substantial amounts of matter to relativistic velocities. Crudely, the maximum velocity of an outflow depends upon the depth of the potential well in the region from which the matter is expelled, so if the matter is launched from a significant portion of the disk, then much of it will have relatively low velocities. Such a wind may be able to provide additional collimation to a jet propelled from the innermost region of the AGN (e.g., Sol et al. 1989) but is unlikely to provide either sufficient collimation or sufficient velocity to explain the observations of VLBI scale extragalactic jets.

At very high accretion rates (comparable to or above that required to produce the Eddington limit) radiation supported thick accretion disks can form (e.g., Paczynski & Wiita 1980). This type of accretion flow provides narrow funnels and a more centrally concentrated region from which to launch the plasma, so it had promise to be a reasonable way to produce powerful collimated beams. This type of flow could produce super-Eddington luminosities but involved quite low efficiencies, as much of the emitted radiation was swallowed by the central black hole in an extremely optically thick flow. However, once the matter gets to mildly relativistic speeds, the aberrated radiation actually produces a drag on the outflow and prevents Gamma values much in excess of 2 (e.g., Narayan et al. 1983). It might be possible to avoid this limit and attain Gamma ~ 10 if clouds of electrons are present in the flow region; if they can produce enough synchrotron self-absorption of the radiation from the funnel, the red-shifted photons will be unable to decelerate the flow (Ghisellini et al. 1990).

Over the past decade other versions of low efficiency accretion flows, which can occur for very low accretion rates, have been frequently discussed. Early proposals along these lines (e.g., Ichimaru 1977, Rees et al. 1982) have been developed along different directions by Chakrabarti (e.g., 1990, 1996) and by Abramowicz, Lasota, Narayan and collaborators (e.g., Narayan et al. 1998). Aside from its fundamental importance as a way of treating more general non-Keplerian accretion flows, the main rationale for the of study of these advection dominated flows was to explain the spectra of X-ray binaries in different states and to fit the spectrum of radiation emerging from non-active galactic nuclei, such as that in our Milky Way. Recently, variants on these models have been shown to accommodate substantial outflows (e.g., Begelman & Blandford 1999). When the velocity structure is sub-Keplerian, as should happen under a wide range of accretion flows in the near vicinity of the BH, then the flow can quite naturally lead to a centrifugal pressure supported boundary layer, which has been shown to be capable of launching significant outflows (Das 1998, Das & Chakrabarti 1999). This mechanism appears to be able to produce adequately relativistic outflows of reasonably good collimation. However all of these fundamentally hydrodynamical (HD) models make one or more critical simplifying assumptions, and their robustness and range of applicability remain to be tested.

While the last mentioned HD launch mechanisms do appear to be promising they have not yet been studied intensively, and at this point a large majority of workers in this area consider that magnetic fields have an important role to play in the ejection and initial collimation of flows from the vicinities of SMBH's. The fact that jets emit via the synchrotron mechanism makes it clear that magnetic fields are present, and several plausible ways to use magnetic fields to accelerate and collimate flows have long been known.

The advantage of magnetic acceleration mechanisms is that they can simultaneously and naturally produce relativistic velocities, narrow jets and large momentum fluxes. The idea that jets were predominantly a Poynting flux with little mass loading was proposed by Rees (1971), and the possibility that powerful currents could be generated in accretion flows which would then accelerate collimated outflows was first noted by Lovelace (1976). Other pioneering works in this area were by Bisnovatyi-Kogan & Ruzmaikin (1976) and Blandford (1976).

An enormous number of variants on magnetically accelerated jet models have been put forward over the past quarter-century, and we cannot even begin to summarize this literature here. But the majority of them fundamentally rely on either extracting energy and angular momentum through magnetic fields anchored in the disk (e.g., Blandford & Payne 1982; BP), or by extracting the spin energy of the black hole itself, through magnetic fields threading its horizon (e.g., Blandford & Znajek 1977; BZ).

An excellent introduction to the physics of magnetohydrodynamical (MHD) jet production mechanisms is the review by Spruit (1996). The vast majority of this research effort has naturally concentrated on ideal MHD models (where the plasma is tied to the field lines) and makes the simplifying assumptions of stationary and axisymmetry flows with infinite conductivity. While the MHD assumption does not always hold around pulsars, and the low temperatures around protostars imply that finite conductivity can be important there, for the conditions around BHs both of these assumptions should be excellent. Spruit (1996) shows that the centrifugal (beads-on-a-wire) approach of, e.g., BP, and the purely magnetic approach of, e.g., Lovelace et al. (1987) are completely equivalent.

The self-consistent computation of MHD flows is extremely difficult because of the possible presence of multiple critical points, each of which can be associated with a shock, which can be of either the slow- or fast-type (e.g., Heyvaerts & Norman 1989). The types and locations of these critical points depend sensitively on the assumptions made about boundary conditions and initial topology of the magnetic fields. Nonetheless, a variety of initial conditions and analytical approaches have been explored and do provide some general conclusions. MHD jets can collimate asymptotically to a cylindrical structure if they carry a sufficient net current, whereas they are very likely to attain a paraboloidal cross-section if they do not (e.g., Chiueh et al. 1991). Self-confined equilibria can be achieved and analytically described in sensible approximations (e.g., Appl & Camenzind 1993).

A recent generalization of the Blandford-Payne model allows for a hotter initial plasma and finds solutions which start with a sub-slow magnetosonic speed and subsequently cross all critical points, at the slow magnetosonic, Alfvén and fast magnetosonic separatrix surfaces (Vlahakis et al. 2000). These models tend to over-collimate toward the jet axis, as do many other MHD calculations, so it is clear that some of the assumptions going into these models must be relaxed. Such relaxation can best be accomplished through the numerical modeling to be discussed in Section 4.1.

It is worth recalling that extraction of the spin energy of the SMBH through very low accretion rates coupled to magnetic fields is an alternative to unified models as a way of understanding the differences between radio galaxies and quasars (Rees et al. 1982). Recently the underpinnings of the basic BZ mechanism have come under renewed investigation. Ghosh & Abramowicz (1997) have argued that magnetic field strengths in the inner parts of accretion disks are weaker than estimated earlier so that the strength of the BZ process for the extraction of the rotational energy of the black hole is lower than imagined previously. Livio et al. (1999) argue that the magnetic fields threading the BH should not be stronger than those in the inner parts of the disk, so that the BZ mechanism should always contribute less power than the disk feeding it. However, if the field strength continues to grow in the innermost disk region and if plasma plunging into the BH exerts a strong torque on the innermost portion of the disk, as has been suggested recently (Krolik 1999, Agol & Krolik 2000), then these caveats may be weakened. Furthermore, it is worth noting that the standard BZ flow is likely to be subject to a screw-instability which can limit the extent out to which it can produce plasma acceleration (Li 2000a). However, a related scenario, where magnetic field lines connect plasma particles inside the ergosphere of a Kerr BH with remote loads, is of real interest. Frame dragging twists the field lines so that energy and angular momentum are extracted from the plasma particles, and if the magnetic field is strong enough, then the particles can have negative energy as they fall in, thereby allowing extraction of the BH's rotational energy (Li 2000b). In all of these efforts certain crucial, and not yet adequately justified, assumptions must be made; therefore, none of the specific MHD launch mechanisms can be considered to be convincing, though many remain plausible.

3.2. The propagation and stability of extragalactic jets

Once launched, the key question becomes: can these theoretical jets survive to the distances demanded by observations, where jet hot-spots are often much narrower than 1% of the jet length? Stability analyses of hydrodynamical jets began with discussions of the Kelvin-Helmholtz (or two-stream) instability which showed that faster jets, particularly relativistic ones, could survive longer (e.g., Turland & Scheuer 1976). Important early contributions were made by Hardee (e.g., 1982, 1987), Birkinshaw (1984), Ferrari et al. (1980), Bodo et al. (1989), among others. Then the analyses were expanded from two-dimensional cylinders to two-dimensional slabs, which can mimic some three-dimensional effects (e.g., Hardee & Norman 1988), to three-dimensional hydrodynamics (HD); then, various assumptions about magnetic field geometry have been studied within an ideal MHD framework.

The ordinary mode (N = 0) is always excited whenever there is a boundary between a flow and a static region. However, various reflection modes (N > 0), which exhibit N pressure nodes within the jet, can also be excited if the walls of the cavity can vibrate coherently; effectively, the sound waves hitting the boundary at an angle can constructively interfere within the jet. Under plausible circumstances the N = 1 mode can grow faster than the N = 0 mode. Very typically, the dominant modes in 2-D are those with wavelengths of ~ 5Rj, and these lead to growth lengths of ~ 3M Rj, where Rj is the jet radius and M is the Mach number of the jet (with respect to its internal sound speed). The growth of both pinch (m = 0) and kink (m = 1) modes can be quite fast, but jets with smooth transverse velocity gradients are more stable.

The stability properties of MHD jets have been recently addressed by many authors. Jet magnetic field geometries that evolve into primarily concentric toroidal structures are usually most unstable to kink (m = 1) instabilities (Begelman 1998). A study of rotating jets confined by toroidal fields has shown that rigid rotation tends to stabilize, while differential rotational destabilizes, the jet in a way similar to the magneto-rotational instability which is now believed to dominate viscosity production in accretion disks (Hanasz et al. 2000). In this local analysis, if the azimuthal velocity exceeds the Alfvén azimuthal speed, the rigidly rotating part of the jet interior can be completely stabilized, while the strong shearing instability acts on the layer between the rotating jet interior and the external medium, perhaps thereby explaining the limb-brightening seen in some jets (Hanasz et al. 2000). Other rotating MHD jet models have been recently analyzed for stability by Lery & Frank (2000). These connect to a Keplerian disk and have a complex structure: a dense, current carrying central core; an intermediate magnetically dominated region; and a low density outer region carrying a return current. Another approach to the stability of rotating MHD jets has been taken by Kersalé et al. (2000), who use the ballooning ordering expansion to find that cylindrical configurations can be destabilized by a negative magnetic shear as well as by a favorable equilibrium pressure gradient. They note that rotating jets with vanishing current density along the axis, as well as most non-rotating MHD jet models, would be unstable.

The major shortcoming of all of these analytical models is that they can only compute the linear growth rates of various instabilities under initially regular conditions; if taken at face value, probably no analytically computed jet could propagate stably for 100 kpc or more, yet of course many such extended extragalactic sources are observed. Therefore high-resolution numerical simulations are required to explore the non-linear effects which can provide saturation of the linear instabilities.

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