As with the extended sources, the primary radio emission is usually attributed to synchrotron radiation by relativistic electrons in a partially ordered field, although alternatives have been investigated. The evidence for this is perhaps not so good as in the extended sources, as the observed degrees of polarisation are much lower; however the majority of the sources can be satisfactorily interpreted using this hypothesis [26].

The source energetics are critically dependent on the angular size
, which can be estimated
in several different ways. As described by
Dr Kellerman, VLBI can provide angular information on milliarcsecond
structure, which can (with the aid of a cosmological model) be
converted into a linear size and shape; and an expansion speed that
apparently often exceeds *c*. Searches for interstellar scintillation,
as yet undetected, provide lower limits on
in the range ~ 1 - 10
*micro*arcsecond.

Secondly, and especially for sources that are too weak for VLBI, an
upper limit on the angular size can be set from the variability
timescale
_{var} (ignoring
kinematic effects discussed below).

(4) |

where *D* is the *luminosity distance* to the source
(*D* = *H*_{0}^{-1}
*q*_{0}^{-2}[*z q*_{0} + (1 -
*q*_{0})(1 - sqrt{1 + 2 *q*_{0} *z*})]
in a Friedmann cosmology).

Thirdly, in some sources a low-frequency turnover, interpretable as
synchrotron self-absorption, yields an estimate of the brightness
temperature and hence of the angular size. (If a turn-over is not
observed, the synchrotron hypothesis leads to a lower limit on the
angular size.) Essentially the limit arises because when the source
becomes self-absorbed the radiation brightness temperature
*S*
^{-2} is
comparable with the kinetic temperature of the electrons
(
(*v*_{n} / *B*)^{1/2})
emitting at the frequency *v*_{n} where the flux turns
over. Numerically,

(5) |

where *S*_{n} =
*S*(_{n}). More
detailed expressions correcting for
cosmological and spectral effects are given by Jones, O'Dell and Stein
[27].
More serious modifications are required with alternative
radiation processes. For example with proton synchrotron radiation
the brightness temperature in a source of fixed size and field
strength is increased by
~ (*m*_{p} / *m*_{e})^{1/2} but only at
the expense of
imposing severe constraints on the relativistic electron density
[28].
With inverse Compton scattering, the brightness temperature is
decreased. Coherent mechanisms of course permit still higher temperatures.

A fourth limit on the angular size comes from considering the
consequences of inverse Compton scattering of the radio photons by the
same electrons that produced them. If the radiation energy density
*U*_{rad}
exceeds the magnetic field energy density *B*^{2} /
8, then a relativistic
electron will lose more energy producing Compton photons, typically in
the infrared or optical range, than radio. These Compton photons can
themselves be scattered, and so on until their unscattered frequency
satisfies
*h*
*m*_{e} *c*^{2} with
the Lorentz
factor of the electron that
produced the original radio photon. At these frequencies, electron
recoil is important and the cross-section is diminished, thus
suppressing further scattering. Large Compton fluxes are undesirable
for two reasons: they greatly increase the energy requirements of the
source and they may come into conflict with existing observational
upper limits, particularly at X-ray energies. This therefore implies
that *U*_{rad}
*B*^{2} / 8.

If we assume that the radiation is produced in a homogeneous
spherical source of angular diameter
, and that
*S*() is
maximised at some frequency
_{u} with flux
*S*(_{u}),
typically in the mm range, then this condition becomes

(6) |

Strictly, we have no guarantee that the flux *S*_{u} arises
from the
same location and at the same time as *S*_{n} and so we can
obtain a
definite lower bound on the angular size by equating
_{n} with
_{u}

(7) |

or equivalently in terms of the observed brightness temperature,
*T*_{n}
10^{12}
K [29].
(It is a curious coincidence that this is about the
maximum brightness temperature that can be detected from a ~ 1 Jy
source using VLBI with an intercontinental baseline.)

As discussed in detail by Burbidge, Jones and O'Dell
[26], and by
O'Dell [30], the angles
_{VLB},
_{var} and
_{c} are in
general comparable, although the upper limit
_{var} tends to
be smaller than
_{VLBI} and the
lower limit
_{c} tends to be
larger. This encourages confidence in the
electron synchrotron hypothesis. A second conclusion is that generally
*B*_{eq} > *B*_{c}, and so the particle
pressure exceeds the magnetic
stresses. It is in general impossible to derive precise estimates of
magnetic field strengths, energy contents, etc because all such
quantities depend on very high powers of
.

The estimated energy content of compact components is sometimes as
large as
~ 10^{56} erg, assuming isotropic emission. These estimates
(which are larger than could be obtained from a single supernova-style
explosion) could be reduced by adopting some special geometry. But
they are in any case much smaller than the energies involved in
extended sources, which is consistent with the view that the latter
are built up by nuclear activity over periods
10^{6} yr.