7.1. Minimum Energy
A fundamental parameter in the physical analysis of radio galaxies is the magnetic field strength. Most studies of cosmic radio sources use the `minimum energy' assumption to estimate field strengths. The minimum energy calculation, first presented by Burbidge (1956), relies on the fact that given a synchrotron emissivity, the summed energy density in relativistic particles and magnetic fields can be minimized by adjusting the magnetic field strength. The minimum energy condition implies rough equipartition of energy between fields and relativistic particles. Whether such a minimum configuration for the fields and particles is reached remains to be verified (Leahy 1991).
The minimum energy calculation involves a number of assumptions, as
outlined in
Miley (1980).
The values for minimum energy fields quoted
herein are calculated from Miley's equation 2, with unity filling factor,
and lower and upper cut-off frequencies of 10 MHz and 100 GHz,
respectively. The most important unknown parameter in this calculation
is the `k factor', where k is the ratio of energy density in
relativistic protons to that in relativistic electrons. We use a value
of k = 1, unless stated otherwise. We also assume fields which are
tangled on scales much smaller than our resolution element
(4)
Such a field configuration implies that the effective magnetic pressure
(including the tension term) is 1/3 times the magnetic energy density.
In this case the total pressure in relativistic particles and fields is
simply 1/3 times the total energy density, and minimizing total pressure
yields the same result for both fields and pressures as minimizing total
energy density
(Leahy 1991).
We also use h = 0.75 for deriving the
minimum energy fields, since the fields are insensitive to h (fields
h2/7).
For the hotspots in Cygnus A we calculate fields from the observed surface brightness distribution at 15 GHz at 0.1" resolution, assuming a disk-like geometry and a spectral index of -0.5. Minimum energy fields vary from 250 µG to 350 µG on the high surface brightness structures. We use a value of 300 µG as representative, with the corresponding minimum pressure of 3 x 10-9 dyn cm-2. For the lobes we calculate minimum energy fields from the observed surface brightnesses on the 327 MHz image at 4.5" resolution, assuming a cylindrical geometry with a width of 25" and a spectral index of -0.7. Fields vary from about 65 µG in the heads of the lobes to 45µG in the center of the radio bridge. Corresponding pressures are 1 x 10-10 dyn cm-2 and 6 x 10-11 dyn cm-2, respectively. For the kpc-scale radio jet we assume a spectral index of -0.8 and a cylindrical geometry with a width of 1". The fields in the jet knots are about 130µG, and the corresponding pressure is 6 x 10-10 dyn cm-2.
4 Maxwell's equations, in particular a
divergence-less field,
prohibit a truly isotropic microscopic field distribution. When we
discuss tangled fields, we mean tangled on scales less than the
resolution element. Back.