2

than the relative loss energy on 4u / c . What is the physical sense of this difference? In Fig. 4.3.1 trajectories of particles inside moved with velocity u magnetic cloud for the head-on and overtaking collisions are shown in the laboratory system of coordinates.

Fig. 4.3.1. Illustration of the derivation of the Eq. 4.3.2 in case of particle collision with a moving magnetic cloud in the laboratory coordinate system: a - overtaking collision, b -head-on collision. The magnetic field in the cloud H is perpendicular to the plane of the figure. The induced electric field E that changes the energy of particle during moving inside the cloud is also shown. The dashed curves show the trajectories suggested by Fermi (1949) and adopted in the scientific literature; the solid curves present the real trajectories.

Fig. 4.3.1. Illustration of the derivation of the Eq. 4.3.2 in case of particle collision with a moving magnetic cloud in the laboratory coordinate system: a - overtaking collision, b -head-on collision. The magnetic field in the cloud H is perpendicular to the plane of the figure. The induced electric field E that changes the energy of particle during moving inside the cloud is also shown. The dashed curves show the trajectories suggested by Fermi (1949) and adopted in the scientific literature; the solid curves present the real trajectories.

Let us note, that the difference 4u / c which follows from Eq. 4.3.2 and Fig.

4.3.1 is very small in comparison with the total relative energy gain or loss 2uv/c2 (at v >> u) and usually is neglected (starting from Fermi, 1949). However, after averaging, taking into account the frequencies of head-on and overtaking collisions, the second term in the right hand side of Eq. 4.3.2 gives two or more time bigger contribution to the total particle acceleration in the statistical mechanism than the first term usually used. If X is the transport scattering path of particles before their collision with magnetic clouds, the frequency of the head-on and overtaking collisions will be in the non-relativistic case (v << c, u << c):

which coincide with Eq. 4.2.2. The total variation of particle energy in unit time is dEk/dt = [AEk)+ v+ + [AEk)-V- , (4.3.4)

dEk/dt = dE/dt = 4macvu2/a = (4u2/A\j2macEk . (4.3.5)

It will be noted that the resultant dE/dt without including the second addend in the right hand side of Eq. 4.3.1 (as was done in Fermi, 1949; see Eq. 4.2.1) is twice as small (compare Eq. 4.2.3 and Eq. 4.3.5). By integrating of Eq. 4.3.5 for the initial condition Ek = Ea at t = 0, we obtain

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