Synchrotron function

File:Synchrotron Functions.svg

First and second Synchrotron functions.

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

• First synchrotron function $$F(x)=x\underset{x}{\overset{\mathrm{\infty }}{\int }}{K}_{\frac{5}{3}}(t)dt$$
• Second synchrotron function $$G(x)=x{K}_{\frac{2}{3}}(x)$$

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.