**12.5.2. Effect of Energy Losses**

Even if relativistic electrons are initially produced with a power law distribution, differential energy losses can alter the energy spectrum. Relativistic electrons lose energy by synchrotron radiation and by the inverse Compton effect, which are both proportional to the square of the energy; by ordinary bremsstrahlung and adiabatic expansion, which are directly proportional to the energy; and by ionization; which is approximately proportional to the logarithm of the energy. Approximating the logarithmic term by a constant, the rate of energy loss may be written

(12.10) |

If electrons are being supplied to the
source at a rate *N*(*E*, *t*), then the equation of
continuity describing the time dependence of
the energy distribution *N*(*E*, *t*) is

(12.11) |

If at *t* = 0

(12.12) |

and if synchrotron and inverse Compton
losses dominate and there is no injection of
new particles [*N*(*E*, *t*) = 0], then

(12.13) |

where

Thus, even with an initial energy distribution extending to unlimited energy, there will be a cut-off at

(12.14) |

and a corresponding cut-off in the synchrotron
radiation spectrum. In the special case where
~ - 2,
*N*(*E*, *t*) can become very large for
energies slightly less than *E*_{c} because of the
piling up near *E*_{c} of electrons with large
initial energies as the result of their more rapid
rate of energy loss. In this case, if *E*_{2} is
sufficiently large so that
(*E*_{2}^{'} / *E*_{c}) ~ 1, then the
radiation spectrum will become flat just
below the upper cut-off frequency,
_{c} =
*cBE*_{c}^{2}. Above
_{c} the spectrum
sharply decreases rapidly for all values of
.

If the distribution of electron pitch angles is random, then the cut-off
frequency for each pitch angle differs. At low frequencies
where energy losses are not important, the
spectral index, ,
remains equal to its initial value
_{0} = (1 -
) / 2. But
at higher frequencies if the
pitch angle of each electron remains constant
(Kardashev, 1962),
= (4/3
_{0} - 1).
If on the other hand the pitch angle distribution is continuously made
random, e.g., by irregularities in the magnetic field, then all the
electrons see the same effective magnetic field
and the spectrum shows the same sharp cut-off
observed with a single pitch angle. The frequency at which synchrotron
radiation loss is important is given by

(12.15) |

If the relativistic electrons are continuously injected with
*Q*(*E*) =
*KE*^{-0}, then for
<
_{b} the spectral
index remains constant with
=
_{0}. But at
higher frequencies where the rate of energy loss is balanced by the
injection of new particles, the equilibrium
solution of Equation (12.11) with
(*N* /
*t*) = 0 gives
=
(_{0} - 1/2).