Chapter Questions
What is the velocity of the particles in a gamma-ray burst jet exhibiting a Lorentz $\gamma$-factor of 200? In the first upgrade of the Large Hadron Collider each of the colliding proton beams was designed to reach an energy of $7 \mathrm{TeV}\left(7 \times 10^{12} \mathrm{eV}\right)$, which corresponds to a Lorentz $\gamma$-factor of 7460 . What is $v / c$ for a proton beam with this energy?
Verify that $\gamma m c^2$, where $m$ is the rest mass and $\gamma$ is the Lorentz $\gamma$ factor, behaves like the total energy in the limit of low velocity. Hint: The total energy is the potential (rest mass) energy plus the kinetic energy.
In a particular gamma-ray burst the intensity is observed to vary with a period of approximately $10 \mathrm{~ms}$. Estimate an upper limit for the size of the central engine powering the burst.
Assume for a typical gamma-ray burst that the distance is $\sim 3000$