Annu. Rev. Astron. Astrophys. 1981. 19:
373-410 Copyright © 1981 by . All
rights reserved |

We assume throughout that the radio emission is due to partially
opaque electron synchrotron radiation. For a uniform source of angular
size, , a red shift, *z*,
magnetic field, *B*, and for a power-law
distribution of particle energy given by *N(E) dE*
*E*^{} *dE*, the
frequency, _{m}, where the
flux density reaches a maximum value, *S*_{m},
( ~ 1) is given approximately
by ^{(2)}

The function
*f*() only
weakly depends on , and for = 2, *f*() ~ 8.
At frequencies >>
_{m}, the spectrum is given
by the usual power law *S*
^{}, where the spectral index,
= (1 -
) / 2. At
frequencies <<
_{m},
*S*
^{} with
= 2.5, a value characteristic
of a homogeneous opaque synchrotron source.

Variations in opacity throughout the source lead to an overall spectrum that can be considered as the superposition of many simple regions described by Equation (1) and can give rise to the so-called flat or undulating spectra typically observed over a wide range of frequency.

Synchrotron radiation losses lead to a characteristic electron
half-life at _{m} of

In very compact sources, in which the radiation energy density is
comparable to the magnetic energy density, inverse Compton scattering
causes additional electron energy losses, which lead to an upper limit
to the observable brightness temperatures of ~ 10^{12} K
(Kellermann &
Pauliny-Toth 1969).
In principle, the magnetic field, *B*, and energy
content in the form of relativistic particles, *E*_{p}, and
magnetic
fields, *E*_{m}, may be estimated from detailed observation of the
brightness distribution at many wavelengths which give the
distribution of opacity and brightness temperature. Because the
calculated values of *B*, *E*_{p}, and
*E*_{m} depend on the observables
and
_{m},
raised to a power ~ 10, accurate determinations are not feasible with
the limited data currently available. Rough estimates indicate *B* ~
10^{-3±1} gauss, *E*_{p} ~
10^{5.2-5.6} ergs, and *E*_{m} some 10 orders of
magnitude
less. Detailed formulae for calculating *B*, *E*_{p},
and *E*_{m} from the
observables ,
_{m}, and
*S*_{m} including the effect of complex geometries are given by
Pacholczyk (1970),
Jones et al. (1974),
and Moffet (1975).

^{2}
Throughout this paper values of flux densities, *S*, are given in Jy;
angular sizes, , in milliarcsec
= mas; magnetic fields, *B*, in gauss,
and frequencies, , in GHz. For
calculations involving distance, we
take *H* = 50 km s^{-1} Mpc^{-1} and
*q*_{0} = 0. Back.