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where m 0 , e and v are rest mass, electric charge and velocity of an electron,
while B is the magnetic flux density (Figure 28).
The gyro–frequency ν G does not depend on the velocity of the electron.
Therefore, for any electron velocity, cyclotron radiation is emitted as a line at
frequency ν G , which is modulated only by the spatial and temporal variation
of the magnetic field.

11.2 Relativistic Case

In the relativistic case, when the electron velocity v approaches light velocity
c, the radiation from an accelerated electron is emitted almost exclusively
in the direction of movement of the electron, due to the relativistic beaming
effect (Figure 29). A distant observer can detect this pulse–like radiation
only when the narrow beam is directed along or near the line of sight. Since
the beam direction rotates around the magnetic field at high speed, the
resulting high frequency pulses from a single electron produce a continuum–
like spectrum, with a peak frequency ν max :

ν G
ν max ∝ . (54)
1 − v 2
c 2
The energy distribution of high–energy electrons in active regions, such
as AGNs and SNRs, usually follows a power law:

N(E) ∝ E −γ , (55)

where E is the energy of the electron, N(E) is the number density of elec-
trons with energy E per unit energy range, and γ is the index of the energy
spectrum. Since the energy of an electron with velocity v:
m 0 c 2
2 , (56)
E = mc = q 2
1 − v
c 2
is proportional to the square root of its peak frequency ν max given in equation
1/2
(54) (i.e. E ∝ ν max ), a large number of electrons in a wide range of energy
yield a compound spectrum with energy density U ν , which is roughly given
by
1−γ
U ν ∝ ν 2 . (57)
Since the index γ of the energy spectrum of high–energy electrons, as ob-
served in the cosmic rays, is roughly γ ' 2.4, we can expect that the spectrum
of the synchrotron radiation is approximately
U ν ∝ ν −0.7 . (58)

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