8.3. Pressure Balance
An important question is pressure balance of the various gaseous components in the cluster center (Arnaud et al. 1984, Carilli et al. 1994b, Alexander and Pooley 1996). At the hotspot radius of 60" the pressure in the unperturbed ICM is 1 x 10-10 dyn cm-2. The minimum pressures in the hotspots are 3 x 10-9, while those in the higher surface brightness features in the heads of the radio lobes (beyond 45" from the nucleus) are few x 10-10. Hence the hotspots are highly over-pressured relative to the external medium, and must be ram pressure confined. The external Mach number, M, for the hotspot advance can be calculated from the formula: PHS / PICM = (5M2 - 1) / 4, derived from the standard Rankine-Hugoniot shock jump conditions in an ideal gas with ratio of specific heats = 5/3. For Cygnus A we obtain M = 5, implying an advance speed of 0.02c. The material in the heads of the lobes is mildly over-pressured (factor of a few), and hence still requires ram pressure confinement, explaining the very sharp edges for the entire leading-surface of the radio lobes.
Within 20" of the cluster center the gas pressure has risen to 5 x 10-10 dyn cm-2 (Reynolds and Fabian 1995). This is comparable to the pressure in the clumpy optical line emitting gas seen on kpc-scales for which pressures 8 x 10-10 dyn cm-2 have been derived (Osterbrock 1989, Carilli et al. 1989b). This is also similar to the minimum energy pressure in the kpc-scale radio jet 6 x 10-10 dyn cm-2. However, the minimum pressure in the radio lobes is only 8 x 10-11 dyn cm-2. Hence, the lobes within 15 h-1 kpc of the cluster center appear under-pressured relative to the thermal gas - a statement in conflict with the observed exclusion of the ICM from the radio lobes. The problem is worsened for an under-expanded lobe, in which case the sheath of shocked ICM enveloping the lobe would still be over-pressured relative to the unperturbed ICM (Begelman and Cioffi 1989). Carilli et al. discuss two possible solutions to this dilemma (see also Böhringer et al. 1993): either a significant departure from minimum pressure conditions in the radio lobes, or lobe pressures dominated by relativistic protons, with k 20. As pointed out above, such a departure from minimum energy conditions in the lobes could also solve the jet confinement problem.
While ROSAT has revealed evidence for the second dynamical component in Cygnus A as predicted by the jet model for powerful radio galaxies, significant questions remain concerning the interaction of the source with its environs, in particular concerning the bow shock at the leading edge of the lobe, and pressure balance between the radio bridge and the external medium. We expect that sensitive, spatially resolving spectroscopy with the next generation of X-ray satellites will delineate the density and temperature structure in the shocked ICM around Cygnus A in spectacular detail, thereby fullfilling the hopes of Begelman et al. (1984) of a fully constrained physical model for the dynamical evolution of a powerful radio source in a cluster atmosphere.