Synchrotron function

Image:Synchrotron Functions.svg

In mathematics the synchrotron functions are defined as follows (for x ≥ 0):{{Cite journal |last1=Fouka |first1=M. |last2=Ouichaoui |first2=S. |date=2013-01-29 |title=Analytical Fits to the Synchrotron Functions |journal=Research in Astronomy and Astrophysics |volume=13 |issue=6 |pages=680–686 |language=en |doi=10.1088/1674-4527/13/6/007|arxiv=1301.6908 |bibcode=2013RAA....13..680F |s2cid=118480582 }}

First synchrotron function $F(x) = x \int_x^\infty K_{\frac{5}{3}}(t)\,dt$
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.

Synchrotron function

Use in astrophysics

References

Further reading