**5.3.5. Heating by relativistic electrons**

Clusters of galaxies often contain radio sources having steep spectral
indices (Section 3.1); the radio emission
from these sources is believed to
arise from synchrotron emission by relativistic electrons. These electrons
(particularly the lower energy ones) can interact with the intracluster gas
and may heat this gas
(Sofia, 1973;
Lea and Holman, 1978;
Rephaeli, 1979;
Scott *et al.*,
1980).
A relativistic electron passing through a plasma
loses energy through Coulomb interactions with electrons within a Debye
length of the particle, and through interactions with plasma waves on
larger scales. The rate of loss by an electron with total energy
*m*_{e} *c*^{2} is

(5.30) |

where _{p} is
the plasma frequency
(_{p}^{2}
4 *n*_{e}
*e*^{2} / *m*_{e}), *r*_{e}
*e*^{2} / (*m*_{e} *c*^{2}) is the
classical electron radius, and *n*_{e} is the electron
density in the plasma. The term
in square brackets is
40 for most values of
and
*n*_{e} of interest, and does not
vary significantly with either parameter. Numerically, the heating rate is
10^{-18}
*n*_{e} erg/s, ignoring the variation of the term in square
brackets. The total
heating rate is determined by multiplying this rate per electron by the
total
number of relativistic electrons in the intracluster gas. If the
relativistic electrons
have a power-law spectrum (equation 5.2), then the heating rate can be
determined from the synchrotron radio emission rate given by equation (5.7).
The heating rate is

(5.31) |

Here *L*_{r} is the radio luminosity at a frequency
_{r},
_{r} is the
radio spectral
index, *b* is the intracluster magnetic field, *a*(*p*)
is the function given by equations (5.4) and (5.8), and
_{l}
is the lower limit to the electron spectrum
(equation 5.2). Cgs units are to be used for *L*_{r} and
*B*. The first quantity in parentheses is independent of the frequency
_{r} at which the
radio source is
observed. Equation (5.31) includes only electrons and should be increased
to include the heating by ions, which do not produce observable radio
emission. While the radio flux and spectrum can be measured directly, the
magnetic field strength must be estimated from the radio observations
(Section 3.6), and the lower limit to
the electron spectrum is generally unknown.

This heating rate is significant only for radio sources with steep spectra
which extend to very, very low frequencies.
Lea and Holman (1978)
used low frequency radio observations of clusters to determine the radio
flux and spectral index, and found that the electron spectrum must extend
down to
_{l}
10(*B* /
*µ*G)^{-1} if the heating rate is to be comparable to
the X-ray luminosity of the cluster. These low energy electrons would
produce radio emission at a frequency of about 400 Hz, which is about
10^{5}
times too low to be observed. Extrapolating the radio spectrum from the
lowest observed frequencies
( 26 MHz) down to
these low frequencies
increases the total number of relativistic electrons and the corresponding
heating rate by about 10^{5}. Thus the hypothesis that
relativistic electrons
provide significant heating to the intracluster gas according to equation
(5.31) requires an enormous and untestable extrapolation of cluster radio
properties.

Several authors have argued that the heating rate of equation (5.31) should
be increased by collective plasma interactions between the relativistic
electrons and the intracluster gas
(Lea and Holman, 1978;
Rephaeli, 1979;
Scott *et al.*,
1980).
In these models, it is assumed that the relativistic electrons are streaming
away from a powerful radio source at the center of the cluster.
Rephaeli (1979)
assumed that the streaming speed is limited by the Alfvén
velocity, as has generally been argued
(Jaffe, 1977;
Section 3.4). Then he finds that the
relativistic electrons excite Alfvén waves and lose energy at a
rate of
-*d*
/ *dt*
*v*_{A}
/
*L*_{e}, where *L*_{e} is the scale length of
the relativistic electron distribution. For *B*
1 *µ*G,
*L*_{e}
1 Mpc, and *n*_{e}
10^{-3}
cm^{-3}, this gives a heating rate about 10^{-2}
_{l}
of that in equation (5.31), which is never very significant.

Alternatively,
Scott *et al.*
(1980)
have assumed that the relativistic electrons
stream at nearly the speed of light
(Holman *et al.*,
1979).
As discussed in Section 3.4, this
hypothesis is controversial. They discuss a number of plasma instabilities
that greatly increase the heating rate under these circumstances. The
increase could be as much as a factor of 10^{5}, which would
allow the electrons that
produce the observed radio emission in clusters to heat the intracluster
gas.

Models in which relativistic electrons heat the intracluster gas suffer
from two general problems. First, the total energy requirements of
10^{63-64} erg are extreme
for a single radio source, although it is possible that many cluster sources
contribute to the heating over the lifetime of the cluster. Second, the
radio
sources generally occupy only a small fraction of the cluster;
cluster-wide radio haloes are rare
(Section 3.4).
It is difficult to see how several discrete
radio sources would heat the intracluster gas without producing very
strong, observable variations in the X-ray surface brightness, which are not
seen.

Vestrand (1982) has suggested that heating of the intracluster gas by relativistic electrons is important only in clusters having radio haloes. He noted that Coma, which has a prominent halo, also has an unusually high gas temperature for its velocity dispersion (Section 4.5.1).