- #1
unscientific
- 1,734
- 13
Homework Statement
An unstable ion of mass ##M##, energy ##E## emits a massless particle of energy ##E_\nu## at angle ##\theta##. In the rest frame of the ion, find ##E_\nu^*## and ##cos \theta^*##.
Ions are now accelerated to ##\gamma=100## and a detector with radius ##r=20m## is placed ##D=200 km## away coaxially. Show that ##\cos \theta^* = \frac{1-\gamma^2 \theta^2}{1+\gamma^2 \theta^2 - \frac{\theta^2}{2}}## where ##\theta \simeq \frac{r}{D}##. Assuming in rest frame of ion, neutrinos are emitted equally in all directions, find the fraction of neutrinos that get through the detector.
Homework Equations
The Attempt at a Solution
Part (a)
By boosting to CM frame, we have
[tex]E_\nu^* = \gamma E_\nu \left( 1 - \beta \cos \theta\right) [/tex]
[tex]\cos \theta^* = \frac{\cos \theta - \beta}{1-\beta \cos \theta}[/tex]
Part (b)
Expanding small angles, we have
[tex]\cos \theta^* = \frac{1-\gamma^2 \theta^2}{1+\gamma^2 \theta^2 - \frac{\theta^2}{2}}[/tex]
How do I find the fraction of particles getting through? I did this in the rest frame:
[tex]f = \frac{4\int_0^{\theta^*} \sin \theta d\theta \int_0^{\phi^*} d\phi}{4\pi} [/tex]
[tex]f = \frac{1}{\pi} \left(1 - cos \theta^*\right) \theta^*[/tex]
Do I divide this by ##\gamma## to account for time dilation to find rate of particles getting through?