Synchrotron function

First and second Synchrotron functions.

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

• First synchrotron function

[math]\displaystyle{ F(x) = x \int_x^\infty K_{\frac{5}{3}}(t)\,dt }[/math]

• Second synchrotron function

[math]\displaystyle{ G(x) = x K_{\frac{2}{3}}(x) }[/math]

where K_j is the modified Bessel function of the second kind.

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.