It should be noted that the Eq. 4.4.22 may be used in approximate calculations over the entire energy range since the term with arctg in Eq. 4.4.21 varies comparatively little, namely from 0 at Ek = Eko to 0.187 at Ek >> Eko.
If a particle starts being accelerated at the instant t = 0 from the initial energy Eki > Eko, the law of change in Ek with time t will be determined by the relation
8tV2Epm: =4EJE:-4E-K-^arctan 5
3A VEk/5Ek, +V 5Ek/Eko or, for the energy range Ek >> Eko:
The Eq. 4.4.24 may be used over practically the entire energy range since the term with arctan in Eq. 4.4.21 varies comparatively little, namely from 0 at Ek = Eko to 0.187 at Ek >> Eko.
The accelerated particle spectrum will be determined using the particle acceleration time distribution (see Eq. 4.4.6); then it follows from Eq. 4.4.24 that n(Ek )dEk - exp
The spectrum described by Eq. 4.4.25 may be presented in the form n(Ek )dEk - Ek dEk, (4.4.25a)
4.4.2. Relativistic case
Let us lift the limitation v << c (but meet the condition u << c). If in the laboratory coordinate system the particle velocity prior to collision is v and the cloud velocity is u, then in the coordinate system related to the cloud for the particle velocity prior to collision we obtain v' _ v _
In the same coordinate system, after reflection from the cloud, the longitudinal (along the cloud front) component of particle velocity will not change, whereas the sign of the transverse component will reverse. If, after that, the laboratory coordinate system is used, we obtain v fin _
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