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13.1. Some Theories

The most important unanswered questions in the study of powerful radio galaxies are: what is the ultimate source of energy? And why does a significant fraction of the energy released take the form of two highly collimated plasma beams? There has been a tremendous amount of theoretical work on these two related issues. For an exhaustive early review of this subject see Begelman, Blandford, and Rees (1984), while a more recent treatment of these issues can be found in Wiita (1991), and Begelman (1993). We draw heavily upon these articles in what follows. Unfortunately, the observational constraints on these models remain mostly circumstantial and indirect. For the sake of completeness, we summarize a few of the basic theoretical points on central engine models, and present the observations of Cygnus A that may be relevant to these issues.

Most models for energy generation in AGN involve accretion onto a massive black hole at the center of the parent galaxy. The strongest argument for this idea remains that accreting black holes are extremely efficient at turning mass into energy. The gravitational energy released by in-falling material of mass, m, is roughly equal to the binding energy at the last stable orbit. Since this is roughly of order the Schwarzschild radius, rs = 2GM / c2, where M is the black hole mass, the energy released is a substantial fraction of the rest mass energy m c2. Calculations for mass-to-energy conversion efficiency for accretion onto black holes yield 6% for non-rotating (Schwarzschild) holes, and 32% for maximally rotating (Kerr) holes. For comparison, nuclear fusion has a mass-to-energy efficiency < 1% (Wiita 1991). The gravitational energy released during infall is thought to be thermalized through the macroscopic viscosity supplied either by magnetic fields and/or turbulence. Other evidence for massive black holes in AGN include: variability of X-ray, optical, and radio nuclei on time-scales less that one day indicating a very small scale for the energy releasing region, and relativistic velocities in pc-scale jets indicating relativistic potential wells. Spectacular evidence for very centrally condensed masses in AGN has come from recent VLBI observations of rotating gas in the nucleus of NGC4258 showing a mass of 3.6 x 107 Msun within 0.13 pc of the galaxy center (Miyoshi et al. 1995).

Models for the initial collimation of jets from accreting black holes typically involve either `nozzles', `funnels', or magnetically-focused beams. These different models depend on the form assumed by the accreting material, which in turn depends on the accretion rate. In all models the disks settle into orbits co-planar with the equator of the hole. The critical accretion rate is set by the Eddington limit = LE / c2, where LE is the standard Eddington luminosity, LE = 1.3 x 1038 MBH ergs sec-1, where MBH is the mass of the hole in Msun. For subcritical accretion rates the radiation can be released efficiently at the disk surface, and a thin, rotating disk forms (ie. the disk is `cold' relative to the virial temperature). For critical accretion rates the photons heat the disk substantially, forming a `thick disk', with temperature approaching the virial temperature. Such a disk will have narrow `funnels' in the polar regions of non-stationarity (where a particle must either flow out, or fall in). For super-critical accretion rates the optical depth becomes such that the diffusion velocity for the photons is less than the in-flow velocity. In this case the hole is thought to be `smothered', and the fluid and radiation field are in equilibrium. What may happen in this case is development of a quasi-spherical `wind', where a wind is defined simply as material that has been heated (via photons or interactions with MHD waves) to the point that it can escape the gravitational potential.

The original `beam' model of Blandford and Rees (1974) involved a hot, buoyant plasma injected into a flattened cloud of cooler gas. The hot gas `percolates' through the denser gas via Rayleigh-Taylor instabilities. Steady flow patterns can be formed where the hot gas escapes from the cooler gas in the direction of the minor axis of the pre-existing gas cloud, forming a convergent-divergent `de Lavall nozzle' with a minimum radius at the trans-sonic point (Norman et al. 1981). In fact, two oppositely directed outflows will form naturally, since the pressure in the hot `bubble' is maximized at the position opposite the first jet. Blandford and Rees applied this model to Cygnus A, and predicted a nozzle forming at 200 pc from the nucleus. This is obviously not correct, given the collimated jet at sub-pc scales. However, the nozzle idea remains attractive in the case of quasi-spherical outflow in the super-critical accretion regime, in order to establish some form of initial collimation of the outflow. Models where nozzles form at distances leq few pc from the hole have been discussed in Norman et al. (1981) and Smith et al. (1983). These short, fat nozzles lead to fairly broad opening angles, and hence require some further focusing for the case of highly collimated radio jets as in Cygnus A.

Outflows from a thin disk driven by radiation pressure will most likely be very broad, and hence not applicable to radio jets. A similar argument can be used against winds from thin disks (Wiita 1991).

The narrow, deep funnels formed in thick disk models are obvious candidates for the formation of highly collimated outflows (Abramowicz and Piran 1980). Radiation pressure in these funnels will be both focusing, and outwardly accelerating. Due to Compton drag such radiation pressure driven jets are limited to form at radii of order 100 Rs, and are limited to bulk Lorentz factors leq 1.6 for a proton-electron plasma, and 4 for a pair plasma. Whether wind-type solutions work in the context of funnels remains to be demonstrated. Also, Wiita (1991) points out that even in the cases of deep, narrow funnels, the beams will be fairly broad (opening angles geq 10°).

Magnetic fields may play a fundamental role in the formation of highly collimated outflow from accreting massive black holes. One simple model involves a centrifugally driven outflow along magnetic field lines which are anchored to a rotating disk. If the emergent field at the disk surface makes an angle leq 60° with respect to the outward radial direction, the charged particles will feel a centrifugal force larger than gravity, and fly outwards like beads on a string (Blandford and Payne 1982, Begelman 1993).

Another magnetic model for jet formation involves the creation of a thick `ion-torus' in the case of sub-critical accretion. Such a torus will form if the infall time-scale is less than the cooling time. In this case the temperatures of the ions and electrons will decouple, with the ions having a temperature close to virial, giving rise to a thick `ion-supported torus'. Since the accretion rate is low, the energy required to power the radio jets comes directly from the rotational energy of the hole, via the magnetic fields (Rees et al. 1982, Blandford and Znajek 1977). The polar fields are anchored to the hole, which behaves as a rotating conducting sphere. In the funnel region the fields `wind-up', and a net-outward Poynting flux is generated, maintained by surface currents in the funnel walls. This Poynting flux becomes `mass-loaded' (through e.g. pair production in the funnel zone), and the ensuing outflow may be collimated by the toroidal component of the magnetic field (Chan and Henriksen 1980, Blandford and Payne 1982). The attraction of this model for radio galaxies such as Cygnus A is that the rotational energy of the hole goes almost directly into the beam, and not into generating a strong optical or X-ray AGN. This model provides an alternative to the standard unification-by-orientation model: the quasar phase occurs when the accretion rate is high, while the radio galaxy phase occurs when the accretion rate is low.

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