Synchrotron function

File:Synchrotron Functions.svg

First and second Synchrotron functions.

In mathematics the synchrotron functions are defined as follows (for x ≥ 0):<ref>Template:Cite journal</ref>

• First synchrotron function <math display="block">F(x) = x \int_x^\infty K_{\frac{5}{3}}(t)\,dt</math>
• Second synchrotron function <math display="block">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.

References

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Further reading

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