Synchrotron function

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.

Further reading

• Longair, Malcolm S. (2011). High energy astrophysics (3rd ed.). Cambridge: Cambridge University Press. ISBN 978-0-511-93059-1. OCLC 702125055. 
• Rybicki, George B. (2004). Radiative processes in astrophysics. Alan P. Lightman. Weinheim. pp. 191. ISBN 978-3-527-61817-0. OCLC 212140606. https://www.bartol.udel.edu/~owocki/phys633/RadProc-RybLightman.pdf.