**13.4.2. Jet Physics**

Since we have no direct way of estimating the density and velocity of a radio-emitting jet, a maze of indirect arguments and physical assumptions is made to deduce the nature of the phenomena. However, within the framework of a set of physical assumptions, many deductions can be made, and if we do not make a completely incorrect assumption, (for example, that the radio brightness is due to the incoherent synchrotron process), we can sometimes limit the physical conditions to a fairly narrow range of possibilities. Basically, we have the radio brightness and its linear polarization at one or more frequencies to work with. We also may have some knowledge of the external conditions from X-ray or optical observations.

*(a) Straight Jets*

Let us assume that a jet is a collimated flow consisting of thermal and
relativistic gas initially moving with some velocity v and some radius
*r*. The brightness is then
affected by (1) radiation losses, (2) adiabatic gains or losses, and (3)
other energy gains or losses by the relativistic electrons. As a jet
expands, the particles in the jet
will gain or lose energy, consistent with their equation of state. In
particular, the relativistic particle energy density,
, in a
volume *V* will change as
*V*^{-4/3}, or the
the total energy of a single particle will vary as
*E*
*V*^{-1/3}.

Thus, in the cylindrical geometry of a jet, as the radius of the jet,
*r*_{j}, increases, each relativistic electron should lose
energy to the expansion as
*E*
*r*_{j}^{-2/9}.

If magnetic flux is conserved then *B*_{||}
*r*_{j}^{-1}, and
*B*_{}
*r*_{j}^{-2}.

If the velocity of the jet, *v*_{j}, remains constant and
no energy is added to the particles or magnetic field from other
sources, the luminosity of all observed jets would decrease much faster
than is observed (e.g.,
Bridle and Perley
1984).
Thus one of these assumptions must be incorrect. If the velocity
decreases, then the density of particles and the perpendicular magnetic
field strength will increase, thus counteracting the
effects of any expansion. Combining both effects for a power law energy
spectrum, the intensity,
*I*_{} varies as

or

Thus, the jet can actually brighten with certain combinations of
parameters. However, if *v*_{j} decreases sufficiently,
then radiation losses
can become important. Also, particles lose energy through inverse
Compton scattering to the 3 K background, so the net rate at which
particles lose energy reaches a minimum at a
magnetic field strength of a few microgauss. Thus, it is not possible to
explain the brightness of jets by simply letting *v* decrease
indefinitely. If adiabatic effects alone
cannot explain the brightness distribution, then some nonadiabatic
effect must be contributing to the energy in the particles and/or
fields. The most obvious source
is probably the energy in the bulk flow of any thermal plasma in the
jet. This could
be transferred to the particles through interactions with shocks or
through plasma waves in a turbulent plasma. These processes, however,
seem to work best when adding energy to already relativistic
particles. Theoretical calculations and in situ
space observations show they are very inefficient in accelerating
thermal particles, especially electrons, to relativistic energies
(Lee 1983).
Since these and other processes are uncertain in their details, usually
it is simply assumed that a fraction,
, of the
kinetic energy in the jet is converted to relativistic electron
energy. Thus,

(13.31) |

where *L*_{rad} is the total emitted radiation and
_{j} is
the density of thermal particles in the jet.

Equation (13.31) is called the kinetic luminosity equation. Rough
estimates for jet or total-source requirements are often made by simply
using the total luminosity of (half) of the source as
*L*_{rad} and estimating
from the
observed Faraday
depolarization or the density of the background gas. Clearly, a better
approach would be to combine at least adiabatic effects with particle
acceleration but this has rarely been done.

*(b) Bent Jets*

If as in the wide-angle tails or narrow-angle tails, the jet is bent, an additional constraint exists, since the time-independent Euler's equation should apply or

(13.32) |

If *R* is the scale length over the jet bends, then
(*v* ^{.}
)*v*
*v*_{b}^{2} / *R*. Then

(13.33) |

A galaxy moving with velocity *v*_{g} through an
intracluster medium with density
_{icm}
experiences a ram pressure
_{icm}
*v*_{g}^{2}. This pressure is exerted over a scale
length *h*. If the jet is directly exposed to the intracluster
medium, then *h* = *r*_{j}. On the other
hand, the jet maybe inside the interstellar medium of a galaxy. Then
*h* is the pressure scale height in the galaxy. In any case, one
can write

(13.34) |

Combining the kinetic luminosity equation (13.31) with the Euler's
equation in the form (13.32), we can eliminate one of the common
variables. For example, eliminating *v*_{b} we can get

(13.35) |

For cases involving narrow-angle tails moving at 10^{3} km
s^{-1} with
respect to the external medium, one can find acceptable applications of
this equation. However, for the wide-angle tails, one has a higher
luminosity to explain and strong evidence in some cases that the parent
galaxy is moving very slowly or not at all with respect
to the intracluster medium. Thus, a simple picture of motion causing the
bending of wide-angle tails appears to fail. More complete models
including adiabatic effects or other energy sources for the particles
such as turbulence in the gas entrained from the intracluster medium
appear to be necessary.