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where the last expression in Eq. 4.3.6 is valid at u21A = const (let us note that in general with increasing energy of accelerated particle, bigger magnetic clouds with bigger velocities became more effective for scattering, so this parameter can change during particle acceleration, even the conditions in the source are stationary; see in detail below Section 4.5).

4.3.2. Relativistic case

In this case, the particle velocity in the coordinate system related to the cloud will be v' =(v ± u)/(l ± uv/c2), (4.3.7)

where v is the velocity of particles before their collisions with the cloud; the + and -signs correspond to the head-on and overtaking collisions, respectively. In this coordinate system the particle velocity after reflection varies only in the direction having conservation of the modulus of the velocity. Returning to the laboratory coordinate system we find that the finite particle velocity after a head-on or overtaking collisions will be respectively vfm = (v'±u )/(l ± uV/ c2), (4.3.8)

or, considering Eq. 4.3.7 and Eq. 4.3.8, we obtain (taking into account that may be v

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