2. NOZZLES, FUNNELS AND HOTPLATES

2.1. Nozzles

Early attempts to account for jets involved nozzles. An analogy was developed with a rocket exhaust in which an adiabatic flow must first converge and then diverge in order to pass through a sonic point. It was supposed that a nozzle is formed in the denser surrounding gas (e.g., Raga & Cantó 1989). The difficulty raised by this explanation, which is most acute in the case of AGN and YSO, is that the collimation is observed to occur on so small a scale that the pressure in the subsonic flow ( L_j/R²_coll V_j) is so large that catastrophic radiative cooling would ensue in the most powerful members of these two classes. Nevertheless, this model did have the merit that it drew attention to the need for a steady flow to pass through critical points. In addition, it is not ruled out for the weak (Type 1) radio sources, and a more complex version in which the flow remains roughly trans-sonic and turbulent while entraining the surrounding gas may well be appropriate (e.g., Bicknell 1984).

2.2. Funnels

An alternative explanation, unique to black holes associated with AGN or some Galactic X-ray sources, is made possible by the non-Euclidean geometry near the event horizon. If there is enough pressure to accelerate orbiting gas relative to a freely falling reference frame, then the surface of an accretion disk may form a torus which develops a narrow, axisymmetric funnel with axis along the spin of the black hole. The pressure has been envisaged to be either radiation pressure (e.g., Jaroszynski, Abramowicz & Paczynski 1980) or ion pressure, under the assumption that electrons can remain cool, (Rees et al. 1982). In principle, a relativistic outflow can be collimated along such a funnel.

Three problems have been identified with this explanation. The first is that Papaloizou & Pringle (1984) have found global, non-axisymmetric instabilities in these structures which may destroy them in a few dynamical timescales. (Though, see Hawley 1991 for a recent numerical discussion of the possible stabilizing effect of radial inflow.) Hydromagnetic stresses, which are now believed to be more generally relevant (see below), may only exacerbate the trouble. The second problem is that, in the case of radiation-supported tori, radiation drag limits the escape speeds to values well below those needed to account for superluminal motion. The third objection applies to AGN like Cen A, where there is a very energetic radio source requiring a central black hole with mass in excess of ~ 10⁹ M. There is no evidence for a bolometric luminosity comparable to the associated Eddington limit as would be needed to support a radiation-dominated torus.

A combined nozzle and funnel model has been developed by Begelman & Rees (1984) in the context of SS433. Here it is supposed that a super-Eddington outflow is powered by the spin of a central black hole or neutron star and that this is collimated by a pair of nozzles formed in an orbiting, radiation-dominated torus (cf. also Eggum, Coroniti & Katz 1988). A mildly relativistic flow speed may naturally be produced, though there is no quantitative rationalization of the speed 0.26c associated with SS433. The effects of radiative viscosity on a shear flow like this can be quite acute and a recent analysis by Arav & Begelman (1992, preprint) shows that the width of a laminar, radiation-dominated boundary layer increases M². This boundary layer may provide a protective sheath around the jet and allow it to escape without relinquishing much of its thrust.

2.3. Hotplates

Yet another explanation for jets, that has been more often sketched than computed, involves a thin accretion disk which is regarded as some type of hot surface that "naturally" expels gas along its symmetry axis. If radiation pressure is responsible, then, in order for this to occur without the disk itself being destroyed, the opacity of the wind must exceed that of the disk. One way for this to happen is if there is copious electron-positron pair production in an accretion disk corona. The opacity can then increase by as much as the proton-electron mass ratio. Radiation drag, though is still a problem at high speeds due to relativistic aberration limits the Lorentz factors to ~ 10, though good collimation seems more problematic (Melia & Königl 1989). A variation on this model, relevant to YSO accretion disks, relies on the formation of dust in the expanding wind to increase the opacity.

It may also be possible to drive gas away from the surface of a disk by heating it to greater than the escape velocity. In the context of an AGN, Compton heating by X-rays has been invoked (Begelman & McKee 1983). Only relatively low speed outflows are likely to result.