Astroparticle Physics

68 5 Acceleration MechanismsOn average, however, the particle gains an energyE = 1 2 m(v2 1 + v2 2 + 2v(v 2 − v 1 )) . (5.11)Fermi mechanismof 1st orderacceleration to 100 TeVSince the quadratic terms can be neglected and because ofv 2 >v 1 , one getsE ≈ mvv ,EE≈ 2 vv . (5.12)This calculation followed similar arguments as in (5.6) and(5.7).Both presented shock acceleration mechanisms are linearin the relative velocity. Sometimes this type of shock accelerationis called Fermi mechanism of first order. Underplausible conditions using the relativistic treatment, maximumenergies of about 100 TeV can be explained in thisway.Fermi mechanism of second order (or more general Fermimechanism) describes the interaction of cosmic-ray particleswith magnetic clouds. At first sight it appears improba-ble that particles can gain energy in this way. Let us assumethat a particle (with velocity v) is reflected from a gas cloudwhich moves with a velocity u (Fig. 5.5).If v and u are antiparallel, the particle gains the energyFermi mechanismof 2nd ordercolliding magnetic clouds5.4 Fermi Mechanism“Results! Why man, I have gotten alot of results, I know several thousandthings that don’t work.”Thomas EdisonE 1 = 1 2 m(v + u)2 − 1 2 mv2 = 1 2 m(2uv + u2 ). (5.13)In case that v and u are parallel, the particle loses an energyE 2 = 1 2 m(v−u)2 − 1 2 mv2 = 1 2 m(−2uv+u2 ). (5.14)On average a net energy gain ofE = E 1 + E 2 = mu 2 (5.15)Fig. 5.5Energy gain of a particle by areflection from a magnetic cloudresults, leading to the relative energy gain ofEE= 2 u2v 2 . (5.16)