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m ac and where the terms of order higher than u2/c2 are neglected. Substituting Eq. 4.4.31 in Eq. 4.4.29, we shall obtain, within the same accuracy, the expression

—(p) = 16 + 3 62 + —2 ^ + 2 4 = -2 + 2 4, (4.4.32)

where p = arccos(uv/uv) is the angle between v and u. Eq. 4.4.32 coincides with Eq. 4.4.18 obtained for non-relativistic case. Since in the relativistic case the relative velocity of particle and cloud is w = (v-u)/(1 -uv/c2); w(p) = (v2 -2uvcosp + u21 /(1 -uvcosp/c2), (4.4.33)

the relative change in particle energy averaged over all possible angles between v and u will be a E \ 1 2n n a e /

where AE (p) is determined by Eq. 4.4.32 and the distribution function of collision frequencies v(pp) is v(p) = w(p)/A(p); A(p) = V(l - uv cosp c2). (4.4.35)

Since the non-relativistic case was analyzed in detail in Section 4.4.1, only the case v >> u will be considered below. From Eq. 4.4.35 follows that

{v(p/ = — J dxJv(p)sinpdp = —-— J (v - 2uvx + u ) dx ~ v/A, (4.4.36) 4n 0 0 4n -1

and from Eq. 4.4.34 we obtain (including Eq. 4.4.32, 4.4.33, 4.4.35, and 4.4.36):

0 0