Definitions

# Synchrotron function

In mathematics the synchrotron functions are defined as follows (for x ≥ 0):

• First synchrotron function

$F\left(x\right) = x int_x^infty K_\left\{frac\left\{5\right\}\left\{3\right\}\right\}\left(t\right),dt$

• Second synchrotron function

$G\left(x\right) = x K_\left\{frac\left\{2\right\}\left\{3\right\}\right\}\left(x\right)$

where Kj is the modified Bessel function of the second kind. These functions are shown on the right, as the output from a plot in Mathematica.

## Use in Astrophysics

In astrophysics, x is usually a ratio of frequencies, that is, the frequency over a critical frequency (critical frequency is the frequency at which most synchrotron radiation is radiated). This is needed when calculating the spectra for different types of synchrotron emission. It takes a spectrum of electrons (or any charged particle) generated by a separate process (such as a power law distribution of electrons and positrons from a constant injection spectrum) and converts this to the spectrum of photons generated by the input electrons/positrons.

For further information, see High Energy Astrophysics, Malcolm S. Longair, CUP, 1990.

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