# Synchrotron spectrum

by lakmus
Tags: spectrum, synchrotron
 P: 23 Hi! 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 $\frac{I'}{(\nu')^3}=\frac{I}{\nu^3}$, where $I$ is the specific intensity and $\nu$ is the frequency. So the radiated intensity from inertial observe frame is $I'=\frac{I (\nu')^3}{\nu^3}$, using Doppler effect fromula $I'=\frac{I}{\gamma^3\left(1-\frac{v}{c}\cos{\theta}\right)^3}$, where $\theta$ is angle possition on the circular trajectory. I used $\theta = \frac{\omega_{cyclotron} t}{\gamma}$ . 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/teachin...s/node133.html) I found, that the maximum radiadion should be emmited at frequency $\propto \gamma^2 \omega_{cyclotron}$ , blue line at the picture. Thanks a lot for each advice! Attached Thumbnails