- #1
lakmus
- 23
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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
[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!
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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