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{\bf Introduction to Astrophysics II, 29:120\\
     Spring, 2009\\
     Steve Spangler's Whirlwind Guide to Synchrotron Radiation}
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Synchrotron radiation is electromagnetic radiation produced by  relativistic electrons moving in a magnetic field.  The following formulas are taken from the classic book {\em Radio Astrophysics} by A. Pacholczyk.  {\em These formulas use the cgs system of units.}

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{\bf Frequency at which emission occurs}
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Although synchrotron radiation can be produced by any charged particle, in astrophysics it is almost always electrons that are radiating.  A single electron (or group of electrons with the same energy and pitch angle $\theta$) emit radiation in a range of frequencies around a characteristic frequency $f_c$ given by
\begin{equation}
f_c = \left( \frac{3 e}{4 \pi m^3 c^5} \right) H \sin \theta E^2 = c_1 H \sin \theta E^2
\end{equation}
In equation (1) $H$ is the magnetic intensity in Oersteds.  However, since in the cgs system of units the permeability of a vacuum is 1 (``saints be praised!''), one can also use the value of the magnetic induction in Gauss.  $\theta$ is the pitch angle of the particle (angle between the velocity and the direction of the magnetic field), and $E$ is the particle energy (ergs). The value of the constant $c_1 = 6.27 \times 10^{18}$.  

Note that the higher the energy of the particle, the higher the frequency at which it emits.  Turning this around, observations of a source at a higher frequency are showing radiation from electrons of higher energy.  {\em X-ray synchrotron radiation} is observed in some cosmic sources, and is an indication of extremely high energy electrons. 

Radiation is emitted in a range of frequencies about $f_c$.  The shape of the spectrum is given by the function
\begin{eqnarray} 
F(x) \equiv x \int_x^{\infty}K_{5/3}(z)dz \\
F(x) \simeq \frac{4 \pi x^{1/3}}{\sqrt3 \Gamma (1/3) 2^{1/3}} \mbox{  ;  } x \ll 1 \\
F(x) \simeq \sqrt \frac{\pi x}{2} e^{-x}
 \end{eqnarray}
where $K_{5/3}(z)$ is a K-Bessel function of fractional order.  The argument of the function $F(x)$ is the ratio of the frequency to the characteristic frequency, 
$x=\frac{f}{f_c}$.  

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{\bf Spectrum of radiation from an ensemble of electrons}
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The incremental volume emissivity of synchrotron radiation from an incremental number of electrons $dN$ with energies in the range $E \rightarrow E + dE$ is
\begin{equation}
d \epsilon (f) = dN \left[  \frac{\sqrt 3 e^3}{4 \pi m c^2}  H \sin \theta F(x)  \right]  
\end{equation}
or
\begin{equation}
d \epsilon (f) = dN \left[ c_3 H \sin \theta F(x)  \right] \mbox{  with  } c_3 = 1.87 \times 10^{-23}
\end{equation}
The energy dependence of this expression is contained in $x$.
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{\bf Energy loss due to synchrotron radiation}
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A particle radiating synchrotron radiation is losing energy at a rate
\begin{equation}
\frac{dE}{dt} = - \left[ \frac{2e^4}{3m^4 c^7} \right] H^2 \sin^2\theta E^2 = c_2 H^2 \sin^2\theta E^2
\end{equation}
with $c_2 = 2.37 \times 10^{-3} $.  Note that the higher the energy of the particle, the faster it loses energy.

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{\bf Radiation from a power law spectrum of electrons}
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Observations of Galactic cosmic ray electrons show that they have a power law spectrum of the form\begin{equation}
\frac{dN}{dE} = N_0 E^{-\gamma}
\end{equation} 
so 
\begin{equation}
dN = N_0 E^{-\gamma} dE
\end{equation}
If we substitute this into Equation (5) or (6) and integrate over energy, we find that (see details in Pacholczyk)
\begin{equation}
\epsilon(f) \propto f^{-\alpha}
\end{equation}
with $\alpha = \frac{\gamma -1}{2}$.  A good graphical illustration of this integration and its result is given in Figure 28.5 of Carroll and Ostlie.  This result is in good agreement with the overwhelmingly common observation of power law radio spectra, with a narrow dispersion in $\alpha$.  The observed values of  $\alpha$ are close to that which characterizes the Galactic cosmic ray electrons. 
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