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In the case a = const it follows from Eq. 4.2.6 that t = a-1 ln(E/Ej); dt = dEjaE . (4.2.8)

Substituting Eq. 4.2.8 in Eq. 4.2.7 we arrive at a power distribution over the energy of accelerated particles in the accelerated volume, obtained by Fermi (1949):

4.3. Development of the Fermi model: head-on and overtaking collisions

4.3.1. Non-relativistic case

In this case the particle's velocity is changed by ±2u in each particle-cloud collision for the head-on and overtaking collisions, respectively. Therefore, the energy variation is

AEk±=AE± = mac(v ± 2u)2/2 - macv1 ¡2 = ±2macvu + 2macu2. (4.3.1)

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