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.