At l = 0, we shall obtain the maximum velocity

and the maximum kinetic energy of the particle

E = macvmax = mac^ + ZeHlo(u + vo) (4167) Ek max =-1-=-1-+-. (4.16.7)

4.16.2. Relativistic case of particle acceleration during magnetic collapse

If the particle velocity is sufficiently high, v >> 2u, the change of the distance between the planes during the collision time is relatively small. The energy increase will be

vmax nmac

Since

we obtain

where E is the total energy of the particle. The distance between the planes during the period between the collisions

will change by

From this dl/dp = -l/p -rnc/ZeH , (4.16.13) and the solution will be found in the form l = -ncpl2ZeH + C1/p ,

where the constant

Here pr and lr are respectively the particle momentum and the distance between the planes at a selected moment when the condition v >> 2u is already satisfied. Thus we find p= Pr \

ZeHl ncpr

2ZeHlr ncpr

ZeHl

7Wpr

The maximum value of the momentum will be obtained at l = 0:

V ncpr

4.16.3. The case of particle acceleration from very low energies up to relativistic energies

In this case the change of the velocity will be determined first by Eq. 4.16.5 and then, after satisfying the condition v >> 2u, by Eq. 4.16.8. Since in almost all cases

u < 10 cm/sec, it is expedient to take vr = 20u = 2 x 10 cm/sec as the boundary between the scopes of the effect of there expressions, for at v < vr particles are nonrelativistic and the Eq. 4.16.2 is valid, whilst at v > vr the condition v >> 2u is satisfied within a sufficient accuracy and the Eq. 4.16.8 is valid up to the highest energies.

According to Eq. 4.16.2, v = vr will be achieved if k = 10 is used (let us note that the condition ft < (1 + a1 )/9(11 + a1) or ft < (1 + a1 )/10(10 + a1) should be satisfied in this case, otherwise the magnetic clouds will collide earlier than the particle velocity reaches v = vr = 20u and the Eq. 4.16.7 has to be used to estimate Ek max . In this case pr = 20macu + po, where po = macvo . Thus we find that p = ((0macu + po)

ZeHl

1 + 2(ZeHl0 - 2ncmacu )(1 + a1 - 9ft (11 + a1)) nc(20macu + po )(19 + «1)

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