3. (34 points) One of the possible mechanisms to produce high-energy protons in cosmic rays is the Fermi acceleration pr

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3 34 Points One Of The Possible Mechanisms To Produce High Energy Protons In Cosmic Rays Is The Fermi Acceleration Pr 1 (58.35 KiB) Viewed 24 times

3. (34 points) One of the possible mechanisms to produce high-energy protons in cosmic rays is the Fermi acceleration process. The maximum energy of these cosmic rays is of the order of 10% eV. Laboratory frame Wall frame Ü V. V. V vi' = V. (a) Determine the Lorentz factor of such cosmic rays. (b) Consider a proton moving with velocity Voin the +1-direction in the laboratory frame. This proton is bounced off by the wall and then moves with a velocity Ū in the ---- direction. Velocities here are in unit of the speed of light c. Find the velocity Vof the proton as observed in the reference frame in which the wall is at rest (wall frame). The Lorentz transformation of the velocity vectors is given by V V +Ü 1 + V U where the primed and unprimed denote the coordinates in laboratory and wall frame respectively. (c) Assuming that in the wall frame, the collision is elastic, and therefore the proton after reflection moves in the ----direction with the velocity V1 = V. Show that the ratio of the Lorentz factors after and before the reflection becomes = (1 +20V + U4) (Hint: Apply Lorentz transformation.) Express the velocity in the laboratory frame Vi as a function of U and Vo. (d) If the velocity of the wall is 50% of the speed of light, i.e., U = 0.5, estimate the number of reflections for the proton to acquire an energy above 1021 eV. Assume that the initial Lorentz factor of the proton is 4 = 10 so you may take V~1.