**3.2. Radiative transfer**

So far we have assumed implicitly that the observed radiation is unchanged in spectrum and polarisation from what was emitted. In fact there are a variety of propagation effects which can substantially modify the emergent radiation and indeed in some sources we can obtain important physical constraints on the conditions within and around the radio components simply from our failure to detect these changes.

Free-free absorption will effectively suppress the emission below a frequency

(8) |

where *EM* = (*n*_{e} / 1
cm^{-3})^{2}(*dl* / 1 pc) is the
emission measure of the
thermal plasma, of temperature 10^{4}*T*_{4} K,
along the line of sight. In
those sources, e.g., associated with quasars displaying prominent
emission line spectra, typical values are
*EM* ~ 10^{12}, *T*_{4}
1 and so the
emitting filaments cannot completely occult the radio source.

A second (fairly *un*important) limit on the thermal electron density
can be set from the manifest absence of Razin suppression of the
synchrotron emission. This gives

(9) |

Of more interest are the limits derived from polarisation observations. Most sources display 1-10 per cent linear polarisation and 0.1 per cent circular polarisation. The fact that this much linear polarisation emerges and that its plane of polarisation does not rotate during the evolution of a source places an upper bound on the internal Faraday rotation of the source. Both thermal and relativistic electrons can contribute to the rotation and the expected depolarisation. The equation of radiative transfer has been solved in a variety of source models.

For a homogeneous source the Faraday rotation due to thermal plasma
is
*n*_{e}*BR* / *v*^{2}, where
*n*_{e} is the thermal electron density; and the
brightness temperature where the source is just optically thin is
*T*
*n*_{rel}*Bl* / *v*^{2} where
*n*_{rel} is the density of relativistic electrons emitting at
this frequency. We can therefore obtain a limit on
*n*_{e} / *n*_{rel}
(3 × 10^{9} /
*T*) *independent* of *B*, as most sources have
*T*
10^{12} K and
1
(otherwise they would be strongly depolarized). This limit is typically
~ 10^{-3} - 10^{-4} and is so small that the Faraday
rotation due
to relativistic electrons must be investigated. The rotation due to
the relativistic plasma is
*n*_{rel}*BR* / *v*^{2}
<ln /
^{2}>; and for the brightest
sources, not showing changes in
, the electron distribution
function must cut off below energies ~ 10-100 MeV
[31,
32].

As radio components have a typical degree of polarisation
10 per
cent of the theoretical maximum for a synchrotron source (~ 70 per
cent) they must exhibit some structural de-polarisation, which will
increase the limits on *n*_{e} and
*n*_{rel}, as the field will reverse several
times along the line of sight, reducing the rotation measure. It is in
principle possible to reduce
to a negligible value without
destroying the polarisation simply by shearing a source with small
scale magnetic turbulence, but in practice this is probably difficult
to maintain for any length of time since the timescale for
isotropising the field is only that taken by an Alfven wave to
traverse the scale of an irregularity. A second type of inhomogeneity
occurs if a lot of the mass is contained in filaments that do not
completely occult the source. In this case the radio-emitting plasma
is probably dynamically decoupled from the filaments. The natural
inference is then that the radio emitting regions contain
predominantly relativistic plasma.

A circular polarisation can also be produced as a consequence of
propagation through a relativistic plasma
[33,
34].
This arises because
the normal modes of propagation through a plasma are not purely
circularly polarised; and so, by a straightforward modification of the
mechanism responsible for Faraday rotation, a linearly polarised wave
emitted by the synchrotron process will be partially converted into a
circularly polarised wave. If the relativistic electron distribution
function is a power-law with
*s* 2 and
extends down to some energy
_{min}
*mc*^{2}, then the expected degree of circular polarisation
_{c} produced
by radiative transfer near turnover is approximately related to the
Faraday rotation by

(10) |

where
_{n}
*mc*^{2} is the energy of electrons radiating at the
turnover frequency. The sign of the circular polarisation generally reverses
near turnover.

Circular polarisation is also produced intrinsically in the synchrotron process [35 - 37] and it too reverses near turnover. The expected magnitude is

(11) |

In an inhomogeneous source these estimates must be reduced like
.
_{c}
can in principle be reduced to a negligible value without seriously
diminishing the linear polarisation.

Reported values of
_{c} are in the range
0.1-1 per cent. If reliable
upper limits < 0.1 per cent turn out to be very common, then they are
probably too small to be understood in terms of a relativistic
electron synchrotron source, and one intriguing possibility is that
the source comprise equal numbers of electrons and positrons. If
observations could be pushed to the point where this seemed the most
attractive solution, then it would have important consequences for
isolating the particle acceleration mechanism. The sources where one
might expect detectable circular polarisation are those such as the
low-power compact component of 3C 84 where the field may be
0.1 G.

Another propagation effect that can profoundly influence the
character of the emergent radiation is *induced* Compton scattering. If
we consider scattering off free electrons, then induced scatterings
[38,
39]
are more important than spontaneous scatterings if the
brightness temperature *T* exceeds
*m*_{e}*c*^{2} / *k*
10^{10}
K. (The rate of induced
scattering out of a photon state exceeds the spontaneous rate by a
factor of order the photon occupation number
~ *kT* / *h*.
However there
are also induced scatterings into this state and so the net induced
scattering rate is reduced by the relative Compton shift,
/
(*h* /
*m*_{e}*c*^{2})).
Typical brightness temperatures are
*T*
10^{12} K and so if
the optical depth to spontaneous Thomson scattering,
_{T}
traversed by the emergent radiation lies in the range
10^{-2}
_{T}
1, then induced
effects can influence the spectrum without increasing the source size
by spontaneous scattering. (It is necessary that the thermal plasma
be close to the source as the induced scattering rate
^{4}
where is
the angle subtended by the source
[40,
41].
This is because induced
scatterings only occur between states that contain a high number of
photons already. As the photons are "hotter" than the thermal plasma
they will fall in frequency whilst conserving their total number, and
if (*kT* / *m*_{e}*c*^{2})
_{T}^{2}
1 the spectrum
will be severely distorted. Linear
polarisation can also be created and destroyed by this mechanism
[41].
However, comparison with eq. (8) shows that free-free absorption
at ~ 1 GHz in a source of size ~ 1 pc is likely to occur before
induced scattering unless the electron temperature is
10^{5} K. (If
free-free absorption became important it would in any case raise the
temperature
[42].)
A second constraint on the physical conditions in
such a scattering region is that the magnetic field strength be much
less than in the source in order that the Faraday rotation not be
greater than observed.