**13.1.4. Energy Considerations**

The problem of the origin and evolution of extragalactic radio sources
is a formidable one; in particular, the source of energy needed to
account for the large power
output and the manner in which this energy is converted to relativistic
particles and magnetic flux is a subject of considerable
debate. Assuming only that synchrotron
radiation from ultra-relativistic electrons is responsible for the
observed radiation, the necessary energy requirements were shown by
Burbidge (1958)
to be as much as 10^{60} ergs or more.

Following Burbidge, if the relativistic particles have a power law
distribution with an index *p* between *E*_{l} and
*E*_{2}, then
for *p* 2, the energy
contained in relativistic electrons is

(13.4) |

The constant *K* can be evaluated if the distance to the source is
known. The total luminosity, *L*, is given by integrating Equation
(1.10), or

(13.5) |

For *p* = 2.5, and
*q*_{0} = + 1, *L* ~ 10^{44}
*z*^{2}*S*, where *S* is the
flux density at 1 GHz. ^{2}

Eliminating *K* between Equations (13.4) and (13.5) we have

(13.6) |

Using Equation (1.8) to relate *E*_{2} and
*E*_{1} to the cutoff frequency
and grouping all the constant terms together,

(13.7) |

The total luminosity *L* of the source may be estimated by integrating
the observed spectrum between 10 MHz and 100 GHz. The magnetic energy is
obtained from

(13.8) |

The total energy in fields and particles
(*E*_{c} = *E*_{e} + *E*_{B}) is
minimized when *dE* / *dB* = 0 or when

(13.9) |

The value of *B* estimated in this way must be treated with caution. It
depends almost entirely on the angular size,
, and is relatively
insensitive to the flux density, distance, or spectral index.

From Equations (13.7), (13.8), and (13.9), if is expressed in arcseconds, then

(13.10) |

and depends only weakly on *p*. Thus, the minimum energy is given when
the energy is nearly equally distributed between relativistic particles
and the magnetic field, and this is usually referred to as the
*minimum energy* or *equipartition* case.

Typically, the total energy contained in the extended sources is
estimated to be
in the range of 10^{57} to 10^{61} ergs and the magnetic
field between 10^{-6} and 10^{-4} gauss.
Under these near equilibrium conditions
(*E*_{e} ~ *E*_{B}),
the total energy depends to a large extent on the size of the source
(*E*
*r*^{9/7}). Thus the
larger sources with low surface brightness and low luminosity, such as
Centaurus A, are calculated
to contain almost as much energy as the smaller but much more powerful
high-surface-brightness objects such as Cygnus A or 3C295. However, high-resolution
observations indicate that in many sources the observed radio emission
comes from only a small fraction,
, of the projected
volume. The minimum total energy calculated from Equations (13.9) and
(13.10) is then multiplied by a factor of
^{3/7},
and the corresponding magnetic field is increased by the factor
^{-2/7}. It is
by no means clear that minimum energy, or equivalently equipartition,
conditions hold in extragalactic radio sources. Moreover, the value of
the fill-in-factor , is
very uncertain, and calculation of energy content or magnetic field
strength based on minimum energy or equipartition arguments must be
treated with caution.

For some years it was widely thought that the relativistic electrons
were secondary particles produced as the result of collisions between
high-energy protons. If
the ratio of the number of protons to electrons is *k*, then the
minimum total energy is increased by a factor of
(1 + *k*)^{4/7} and the magnetic field by
(1 + *k*)^{2/7}. Some
estimates of the value of *k* were as high as 100, with a corresponding
increase in energy requirements by about an order of
magnitude. Elimination of the factor *k*
and inclusion of the fill-in factor
, can easily reduce the
minimum-energy estimates by two or more orders of magnitude.

A more direct, although not necessarily more accurate, method of determining the magnetic field in extended extragalactic radio sources is based on the scattering of the microwave background radiation by the relativistic plasma. The ratio between the synchrotron radio flux and Compton scattered X-ray flux depends on the magnetic field strength and is given approximately by (Miley 1980)

(13.11) |

where *S*() is the radio
flux density at frequency
and *S*(*x*)
is the X-ray flux density
at energy *E*(*x*). Values of *B* estimated from
Equations (13.9) and (13.11) are in the
range of 10^{-6} to 10^{-4} gauss, so that the energy
content appears to be close to the
minimum requirement. But estimates of *B* from Equation (13.11) are
subject to error
if the thermal X-ray flux is not negligible compared with the Compton
scattered X-ray flux.

In a 10^{-4}-gauss field, electrons radiating at
> 1 GHz are
expected to decay in about 10^{6} years (Equation 1.8). Thus,
the absence of any observed spectral cutoff even at
>> 10 GHz suggests
continued acceleration of relativistic
particles. Even more restrictive limits are imposed by the observation
of both radio and optical
( ~ 10^{14} Hz)
synchrotron emission from the M87 jet at least 10 kpc away from the nucleus
(Biretta et al. 1983a).

Because an adiabatically expanding cloud of relativistic particles loses
energy as *R*^{-2p} (e.g.
Kellermann and
Pauliny-Toth 1968),
it has been argued that if the
observed relativistic electron clouds had expanded adiabatically from a
much smaller region, the initial energy requirements would be
prohibitively large. Although a variety of models have been discussed
which allow some of the kinetic energy of expansion to be converted into
relativistic particle energy, the adiabatic
loss problem has been a prime motivation for models which consider the
primary energy source to be in a compact region at the quasar or AGN
with the energy being transported from this "central engine" to the
outer lobes by a relativistic *beam*.
(Section 13.4.1). However, it has also
been pointed out by
Bicknell (1986)
and others that if the source has expanded to some final size and
another burst of particles
occurs, without significantly expanding the source, then no further
adiabatic losses occur.

^{2} Numerical expressions given in this
chapter are evaluated for *H* = 100 km s^{-1}
Mpc^{-1}, *q*_{0} = 1, and
*p* = 2.5( =
0.75). More general formulations for other values of
and more
complex geometries are given in
Moffet (1975),
Jones et al. (1974a,
b),
and Marscher (1983).
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