Magnetic mirrors / Fermi acceleration

(F. F. Chen ~ 2.12, Fermi acceleration of cosmic rays).

A cosmic ray proton is trapped between two moving magnetic mirrors with mirror ratio \(R_m=5\). Initially its energy is \(W=1\) keV and \(v_\perp = v_\parallel\) at the midplane. Each mirror moves toward the midplane with a velocity \(v_m=10\) km/s and the initial distance between the mirrors is \(L=10^{10}\) km.

(a) Using the invariance of \(\mu\), find the energy to which the proton is accelerated before it escapes.

(b) How long does it take to reach that energy? Suggestions: i) suppose that the $B$ field is approxiamtely uniform in the space between the mirrors and changes abruptly near the mirrors, \textit{i.e.}, treat each mirror as a flat piston and show that the velocity gained at each bounce is \(2v_m\); ii) compute the number of bounces necessary; iii) assume that the distance between the mirrors does not change appreciably during the acceleration process.