Synchrotron function: Difference between revisions

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[[Image:Synchrotron Functions.svg|thumb|First and second Synchrotron functions.]]

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

In [[mathematics]] the '''synchrotron functions''' are defined as follows (for ''x'' &ge; 0):<ref>{{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 }}</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. These functions are shown on the right, as the output from a plot in Mathematica.

:<math>F(x) = x \int_x^\infty K_{\frac{5}{3}}(t)\,dt</math>

== Use in Astrophysicsastrophysics ==

* Second synchrotron function

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.

:<math>G(x) = x K_{\frac{2}{3}}(x)</math>

== References ==

where ''K''_''j'' is the modified [[Bessel function]] of the second kind. These functions are shown on the right, as the output from a plot in Mathematica.

<references />

== Further reading ==

[[Image:Firstsynchfunction.jpg|thumb|First synchrotron function, F(x)]]

*{{Cite book |last=Longair |first=Malcolm S. |title=High energy astrophysics |date=2011 |publisher=Cambridge University Press |isbn=978-0-511-93059-1 |edition=3rd |___location=Cambridge |oclc=702125055}}

<!-- Image with unknown copyright status removed: [[Image:secondsynchfunction.jpg|thumb|Second synchrotron function, G(x)]] -->

*{{Cite book |last=Rybicki |first=George B. |url=https://www.bartol.udel.edu/~owocki/phys633/RadProc-RybLightman.pdf |title=Radiative processes in astrophysics |date=2004 |others=Alan P. Lightman |isbn=978-3-527-61817-0 |___location=Weinheim |pages=191 |oclc=212140606}}

== 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.

For further information, see ''High Energy Astrophysics'', Malcolm S. Longair, CUP, 1990.

{{math-stub}}

[[Category:Special functions]]