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Here the frequency of particle collisions with magnetic clouds is v(p) = w(p)/A, (4.4.16)

where À is the particle transport path for collisions with clouds (e.g., if l is the characteristic size of clouds, Ncl = d-3 is the clouds concentration, d is the mean distance between clouds, then À = d3/l2 ). Taking into account Eq. 4.4.8 we obtain dEk dt lu2mac f

2u2ma

2u2m

15Àv or, after expressing the velocities v and u in terms of Ek and Eko, we find that dEk 16E;

dt ko

4 a/2 Ekomac 10Eko 42Ekomac pEk/mac + 2Eko if Ek > Eko, (4.4.18)

Let the particles be accelerated from some initial energy Eki at the instant t = 0 . Consider first the case in which Eki < Eko. In this case we obtain from Eq. 4.4.18 for the energy range Ek < Eko , i.e. the time interval 0 < t < t1

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