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I try to construct the emission spectrum from relativistic electron rotating in homogeneous magnetic field - synchrotron. In my lecture notes a found out one really easy derivation using the invariance of

[itex]\frac{I'}{(\nu')^3}=\frac{I}{\nu^3}[/itex], where [itex] I [/itex] is the specific intensity and [itex] \nu[/itex] is

the frequency. So the radiated intensity from inertial observe frame is

[itex]I'=\frac{I (\nu')^3}{\nu^3}[/itex], using Doppler effect fromula

[itex]I'=\frac{I}{\gamma^3\left(1-\frac{v}{c}\cos{\theta}\right)^3}[/itex], where [itex] \theta [/itex] is angle possition on the circular trajectory. I used [itex] \theta = \frac{\omega_{cyclotron} t}{\gamma} [/itex] . Then I plotted the resulting

intensity, which looked ok (at least similar to some I found on the internet). I also did the fourier transformation (picture uploded). But the critical frequency is too hight, also the peaks are to widt - here (http://farside.ph.utexas.edu/teaching/em/lectures/node133.html) I found, that the maximum radiadion should be emmited at frequency [itex] \propto \gamma^2 \omega_{cyclotron} [/itex] , blue line at the picture.

Thanks a lot for each advice!

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# Synchrotron spectrum

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