3.2. Opening Angle, Stability, and Confinement
An important constraint on jet physics is the opening angle. For a free jet the opening angle, will equal 2, where is the Mach angle defined by: tan() = ( Mj)-1, where Mj = jet Mach number, and = jet Lorentz factor. Any oblique shock in the jet due to a disturbance at the surface will also have the Mach angle relative to the jet axis. Likewise, turbulent boundary layers grow at the Mach angle unless prohibited for example by internal magnetic fields. Hence, a high Mach number jet has an effective `rigidity' since edge-effects are limited to propagate into the jet at the Mach angle (DeYoung 1991).
On pc-scales the jet is resolved transversely with a constant (deconvolved) FWHM = 2.2 mas from 4 mas to 20 mas, suggesting a confined jet (Carilli, Bartel, and Linfield 1991). The minimum pressure in jet knots at radii of a few pc are of order 10-6 dyn cm-2, which is below the expected pressures in the ambient medium in these regions, as discussed below. Hence the pc-scale jet could be pressure confined although the possibility of magnetic focusing remains attractive in order to explain such a well collimated outflow.
On kpc-scales the opening angle of the jet in Cygnus A can be determined either from the evolution of the FWHM of the knots in the jet out to 20", or from the edges defined by the low surface brightness inter-knot emission in the high pass filtered image. In both cases the opening angle derived is 1.6° (Carilli 1996). For a free non-relativistic jet, the implied Mach number is 72. For a free relativistic jet filled with relativistic fluid the internal sound speed is c / 3, hence Mj = 3 and for Cygnus A the implied Lorentz factor is: = 40. This is well above the range measured for superluminal quasars (e.g., Porcas 1987).
Perhaps the most reasonable conclusion is that the Cygnus A jet is confined. Perley et al. (1984) point out that the Cygnus A jet morphology is consistent with regions of larger opening angle ( 5°), and a constant width jet in between (full width to 10% of peak 1.1"), implying a confined jet. Even for a confined jet, oblique ridges may indicate the Mach number of the flow if they arise due to disturbances at the jet wall. The Cygnus A jet knots are oriented at roughly 7° to the jet axis, which if taken as the Mach angle, implies: Mj = 8. Lastly, there is the simple fact that the jet must widen from pc- to kpc-scales. Using the first resolved jet knot as an indicator of opening angle from the core leads to: Mj 13.
The simplest means of jet confinement is external pressure. Perley et al. (1984) point out that minimum energy pressures in the jet are about an order of magnitude greater than in the lobes, thereby requiring an alternative means for jet confinement, if minimum conditions apply. Below we review X-ray and optical observations which suggest that the lobes may be over-pressured relative to equipartition values by just this amount and hence that pressure confinement may still be viable.
A number of authors have suggested confinement of current carrying jets via the `plasma-pinch' mechanism due to toroidal fields surrounding the radio jet (Chan and Henriksen 1980, Bridle, Chan and Henriksen 1981, Bicknell and Henriksen 1980, Benford 1985, Siah 1985). This mechanism is notoriously unstable although it might be stabilized by longitudinal fields in the jet, or by a hyper-Alfvenic jet velocity.
One of the more interesting observational results concerning the jet in Cygnus A comes from the optical imaging of the inner 5" of the galaxy with HST by Jackson et al. (1994). They find that the jet passes through a `channel' in the line emitting gas about 1" from the nucleus. This channel is also seen in deconvolved ground-based images (Vestergaard 1992, Stockton et al. 1994), and it has been confirmed with more recent HST images of Cygnus A (see Fig. 6) presented in Cabrera-Guerra et al. (1996). These authors do not find evidence for increased reddening at the position of the gap, and hence argue that the gap is not simply a fortuitous projection of a dust filament at the position of the radio jet. Jackson et al. (1994) and Cabrera-Guerra et al. (1996) suggest that the jet has `blasted' a path through the emission line clouds. The shocked clouds cool quickly, and are subsequently photo-ionized by radiation from the (hidden) active nucleus. This morphology implies that the radio source has not completely evacuated its environments, at least within a few arcseconds of the nucleus. This is consistent with the simulations of Williams (1991) of a jet propagating through an atmosphere of decreasing density, in which case the inner part of the lobe gets `squeezed' by the high pressures in the central regions, leaving a `naked jet' in the inner regions, in direct contact with the external medium. The pressure in the line emitting gas is comparable to the minimum energy value in the jet allowing for pressure confinement of this `naked jet'.
Figure 6. The contours show the inner kpc-scale jet in Cygnus A at 5 GHz, 0.4" resolution. The grey-scale shows an HST image (resolution 0.1") of Cygnus A using the WFPC2 F622W filter, reproduced from Cabrera-Guerra et al. (1996). Note that the radio jet passes through a gap in the optical line emission.
However, there are a number of problems with the above model. First is the general problem of stability of a light, pressure confined jet (Birkinshaw 1991). Second, if the line clouds are moving at few hundred km sec-1 they will cross the jet on a time-scale of order 106 yrs, thereby further complicating the issue of jet stability. And third, in order to observe a gap due to the jet in the projected line emission the gas distribution must have a rather contrived geometry. A flattened, or disk-like, geometry for the line emitting gas is precluded since the expected deficit due to a small hole in a large disk, as seen in projection, would be small. The observed gap requires the gas be in a linear filament which happens to get `cut' by the jet.
Overall, it may be simplest to assume that the jets in Cygnus A are external pressure confined on pc- and kpc-scales with a free-expansion zone in between. The pc-scale jet is under-dense relative to its environments, and hence stability becomes a concern. The same concern holds for the jet-cloud interaction region seen at a distance of 1 h-1 kpc from the nucleus. However, beyond 10 h-1 kpc from the nucleus the jet would be over-dense relative to the confining medium (the radio lobes), leading to the relatively stable situation of a supersonic, pressure matched, ballistic jet from 10 h-1 kpc to the hotspots. The single requirement of this model is that the lobes be over-pressured relative to minimum energy - an assumption that agrees with other observational aspects of the Cygnus A galaxy, as discussed below.