First and second Synchrotron functions.

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

  • First synchrotron function F ( x ) = x ∫ x ∞ K 5 3 ( t ) d t {\displaystyle F(x)=x\int _{x}^{\infty }K_{\frac {5}{3}}(t)\,dt}
  • Second synchrotron function G ( x ) = x K 2 3 ( x ) {\displaystyle G(x)=xK_{\frac {2}{3}}(x)}

where Kj 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 .
  • Rybicki, George B. (2004). (PDF). Alan P. Lightman. Weinheim. p. 191. ISBN 978-3-527-61817-0. OCLC .{{cite book}}: CS1 maint: location missing publisher (link)