Next: The effect of dielectric
Up: Relativity and electromagnetism
Previous: The angular distribution of
Synchrotron radiation (i.e., radiation emitted by a charged particle
constrained to follow a circular orbit by a magnetic field) is of
particular importance in astrophysics, since much of the
observed radio frequency
emission from supernova remnants and active galactic nuclei is thought to
be of this type.
Consider a charged particle moving in a circle of radius with constant
angular velocity . Suppose that the orbit lies in the 
plane. The radius vector pointing from the centre of the orbit
to the retarded position of the charge is defined

(395) 
where
is the angle subtended between this vector
and the axis. The retarded velocity and acceleration of the
charge take the form
where and
.
The observation point is chosen such that the radius vector ,
pointing from the retarded position of the charge to the observation point,
is
parallel to the  plane. Thus, we can write

(398) 
where is the angle subtended between this vector
and the axis.
As usual, we define as the angle subtended between the retarded
radius vector and the retarded direction of motion of
the charge . It follows that

(399) 
It is easily seen that

(400) 
A little vector algebra shows that

(401) 
giving

(402) 
Making use of Eq. (2.320), we obtain

(403) 
It is convenient to write this result in terms of the angles
and , instead of and . After a
little algebra we obtain

(404) 
Let us consider the radiation pattern emitted in the plane of the
orbit; i.e., , with
. It is easily seen that

(405) 
In the nonrelativistic limit the radiation pattern has a
dependence. Thus, the pattern is like
that of dipole radiation where the axis is
aligned along the instantaneous direction of acceleration. As the
charge becomes more relativistic the radiation lobe in the forward
direction
(i.e.,
)
becomes more more focused and more intense. Likewise, the
radiation lobe in the backward direction
(i.e.,
) becomes more diffuse.
The radiation pattern has zero intensity at the angles

(406) 
These angles demark the boundaries between the two radiation
lobes.
In the nonrelativistic limit
, so the two lobes are
of equal angular extents. In the highly relativistic limit
, so the forward
lobe becomes highly concentrated about the forward direction ().
In the latter limit Eq. (2.336) reduces to

(407) 
Thus, the radiation emitted by a highly relativistic charge is focused into
an intense beam of angular extent pointing in the instantaneous
direction of motion. The maximum intensity of the beam scales like .
Integration of Eq. (2.335) over all solid angle (using
) yields (not very easily!)

(408) 
which agrees with Eq. (2.309) provided that
. This expression can also be written

(409) 
where
meters
is the classical electron radius, is the rest mass of the charge,
and . If the circular motion takes place in an orbit of radius
perpendicular to a magnetic field , then satisfies
. Thus, the radiated power is

(410) 
and the radiated energy
per revolution is

(411) 
Let us consider the frequency distribution of the emitted radiation
in the highly relativistic limit. Suppose, for the sake of
simplicity, that the observation point lies in the plane of
the orbit (i.e., ). Since the radiation emitted
by the charge is beamed very strongly in the
charge's instantaneous direction
of motion, a fixed observer is only going to see radiation (at some
later time)
when this direction points almost directly towards the point of observation.
This occurs once every rotation period when , assuming that
. Note that the point of observation is located many
orbit radii away from the centre of the orbit along the positive
axis. Thus, our observer sees short periodic
pulses of radiation
from the charge. The repetition
frequency of the pulses (in radians per second)
is . Let us calculate the duration of each pulse. Since the
radiation emitted by the charge is focused into a narrow beam of
angular extent
, our observer only sees
radiation from the charge when
. Thus,
the observed pulse is emitted during a time interval
. However,
the pulse is received in a somewhat shorter time interval

(412) 
because the charge is slightly closer to the point of observation at the
end of the pulse than at the beginning. The above equation
reduces to

(413) 
since and
.
The width
of the pulse in frequency space obeys
. Hence,

(414) 
In other words, the emitted frequency spectrum contains harmonics
of frequency up to times that of the fundamental, .
More involved calculations^{8} show that in the ultrarelativistic limit
the power radiated in the th harmonic (whose
frequency is
) is given by

(415) 
for
, and

(416) 
for . Note that the spectrum cuts off approximately
at the harmonic order , as predicted earlier. It can also be
demonstrated^{9} that seven times as much energy is
radiated with a polarization parallel to the orbital plane
than with a perpendicular polarization. A
power spectrum at low frequencies coupled with
a high degree of polarization are the hallmarks of synchrotron radiation.
In fact, these two features are used in
astrophysics to identify synchrotron radiation from supernova remnants,
active galactic nuclei, etc.
Next: The effect of dielectric
Up: Relativity and electromagnetism
Previous: The angular distribution of
Richard Fitzpatrick
20020518