When the apparent source brightness temperature approaches the equivalent
kinetic temperature of the relativistic electrons, synchrotron
self-absorption becomes important, and part of the radiation is absorbed
by the relativistic electrons
along the propagation path. Below the self-absorption cutoff frequency,
the spectrum is just that of a blackbody with an equivalent temperature
*T*_{k} = *E* / *k*.

From Equation (1.8),
*E*
^{1/2}, so in an
opaque synchrotron source the flux density
*S*
^{2.5}, rather than
the ^{2} law found in
thermal sources. In other words,
an opaque synchrotron source may be thought of as a body whose equivalent
temperature depends on the square root of the frequency. Self-absorption
occurs below a frequency
_{c} where the kinetic
temperature

is equal to the brightness temperature

Assuming uniform source parameters, the magnetic field can then be
estimated from observation of
_{c} and surface
brightness from

(13.12) |

where *S*_{m} is the maximum flux density in janskys,
_{c}
the cutoff frequency in GHz,
and the angular
size in milliarcseconds. The quantity
is a correction for the
relativistic Doppler shift if the source is moving with high velocity
(see Section 13.3.7). If
<< *c*,
~ 1. The function
*f* (*p*) only weakly depends on geometry and the
value of *p*, and is about 8 for *p* ~ 2.5. Variations in opacity
throughout the source give an overall spectrum that can be considered as
the superposition of many simple regions described by Equation (13.2)
and can give rise to the so-called *flat* or
undulating spectra typically observed in compact radio sources (e.g.,
Condon and Dressel
1973).

The magnetic field in a compact radio source can be determined directly
from the observables
, *S*_{m},
and _{c} by Equation
(13.12). The magnetic energy, *E*_{B}, can be
estimated from Equation (13.8) to be

(13.13) |

Similarly, from Equation (13.6), the energy in relativistic electrons is given approximately by

(13.14) |

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

(13.15) |

In practice, the use of Equations (13:13) and (13.14) to derive the
magnetic and particle energy is difficult due to the strong dependence
on angular size and cutoff
frequency. However, in almost every case where measurements exist, the
particle energy appears to greatly exceed the magnetic energy. But note
the dependence on
and
_{c} to about the
tenth power; so small changes in the geometry may lead to
other conclusions. The ratio of
*E*_{c} / *E*_{B} may also be
reduced considerably with even modest values of
. See
Section 13.3.6.