Examine separately the cases where 28 _ ¡3 = 0 and 28 _ ¡3 ^ 0 .

(1) The case 28 - P = 0. In this case, at m2acc4/E2 << 1 it follows from Eq. 4.5.22 that the acceleration parameter a = a, = 8u213c A, = const, (4.5.24)

i.e. the relative rate AE/E of particle energy gain will be constant. Here, using Eq. 4.2.7 and taking account of Eq. 4.5.3, we shall obtain an expression of the form Eq. 4.2.9, i.e. n(E)- E~Y, but with the power exponent being a function of E:


Thus, in this case the spectrum may be described by a power law with variable exponent y increasing with particle energy (if ¡3> 0). It can be easily seen that at ¡3^ 0 the Eq. 4.5.25 for y turns out to be Eq. 4.2.9 but with higher parameter of acceleration a.

(2) The case 28 - 3 # 0. If 28 - 3 # 0 then, according to Eq. 4.5.23 and using the notations of Eq. 4.5.24 we obtain

Since in this case t = [1 -(E/Ei(28-ß) dt = dE(E/Ei Y^-^laEi, (4.5.27)

we shall obtain for the accelerated particle density differential energy spectrum inside the source using Eq. 4.5.3:

n(E)« E"(1+28-2ß) exp[- (EE, )ß (l - (EE, )-(28-%T ( - ß))-1 J. (4.5.28)

It can be easily seen that when 28-ß ^ 0, the Eq. 4.5.28 turns out to be a power spectrum of the form n(EE, where y is determined by the Eq. 4.5.25.

If 28 - ß > 0, then in accordance with Eq. 4.5.26 and Eq. 4.5.28 the rate of the particle energy gain will be more rapid and the generated spectrum at E >> Ei will be the product of the power function E~Y with y = 1 - 2ß + 28 by an exponential function of the form exp[-(e/E;- )ß(aizi (28 -ß))-1 J. The exponential factor will result in the spectrum cutoff on the high energy side at E > E( ((28-ß)ß.

If 28 - ß < 0, then in accordance with Eq. 4.5.26 and Eq. 4.5.28 the rate of the particle energy gain will be rather slower than exponential, while the spectrum at E >> Ei will be expressed as the product of a power function of the form E~Y with exponent y = 1 - 2ß + 28 by the exponential factor exp[-(E/Ej )2ß-28 joiTi (ß-28)]. The spectrum cutoff on the high energy side is expected at E > Ei ((ß - 28))1 (2ß-28).

4.5.4. The nature of the constraint of the accelerated particle's energy

It follows from the expressions presented above for the rate of particle energy gain that an infinite increase in particle energy with time should be expected in all cases. For example, in the cases described by the Eq. 4.2.9 and 4.5.25 at ) = 0, E exponentially as exp(at) with time t. In the case 28 - )> 0, E within a finite time t = ( (28-)))-1. In the case 28 - 0 < 0, E ^ ra as x t .

Actually, however, these conclusions are erroneous.

The fact is that any source always comprises some maximum scale of inhomogeneities, the effective scattering by which corresponds to some particle energy Ecr. If /max and H((max) are the effective size and the mean intensity of the magnetic field in inhomogeneities of the largest scale, then

The value of Ecr can be reached in the case described in Section 4.2 at tcr = ln{Ecr/Ei )a; (4.5.30)

in the case described in Section 4.5.3 for 28 - 0> 0

and in the case described also in Section 4.5.3 but at 28- 0< 0

Consider, for example, the acceleration up to high energies under the condition of constancy of the acceleration parameter, i.e. that 28-0 = 0. It can be easily seen that the condition 28-0 = 0 cannot be satisfied once E ~ Ecr is reached. In fact, as was shown in Section 1.9, for any type of magnetic inhomogeneities, when the condition E > Ecr is satisfied the transport scattering path should increase with energy as A ^ E or even more rapidly, i.e. it is explicit that 0cr > 2 . On the other hand, since inhomogeneities exceeding the maximum scale are absent, it must be that ucr ~const, i.e. 8~ 0 for the energy range E > Ecr . Thus at E > Ecr it is explicit that 28 - 0 < 0; here, however, we obtain the case 28-0 + 0 described above and the energy gain rate at E > Ecr will be determined, according to Eq. 4.5.26, by

Here Ecr is determined by Eq. 4.5.29, tcr is determined by Eq. 4.5.30 and acr = u2r IcAcr , (4.5.34)

where Acr is transport path of particles with energy Ecr. In accordance with Eq. 4.5.28 the particle spectrum at E > Ecr will be determined by the expression n(E>Er ~ (E/Ecr ))0cr-1 exp{- (E/E^) ((E/Ecr) - l)(acrTcr0cr )-1}, (4.5.35)

where Tcr ~ L¡2cAcr is the mean lifetime of particles in their source with energy Ecr. It follows from Eq. 4.5.35 that an abrupt cutoff of the spectrum should take place at E > Ecr (acrTcrPcr ))20cr . The same will also take place at 28-0 + 0. Once the particle energy reaches the value E = Ecr determined by Eq. 4.5.29, the energy gain rate and the accelerated particle spectrum will be expressed at E > Ecr by the Eq. 4.5.33 and 4.5.35; in these relations, however, tcr will be determined from Eq. 4.5.31 in case 28- f3 > 0 and by Eq. 4.5.32 in the case 28 - 3 < 0 .

4.6. Formation of the particle rigidity spectrum during statistical acceleration

4.6.1. General remarks and basic relations

The mode of particle motion in magnetic fields is determined by particle rigidity

where p is the momentum and Ze is the charge of particle. It is of interest, therefore, to determine how the particle rigidity varies in time during the acceleration processes and what is the generated spectrum of the accelerated particle rigidity. This can be done on the basis of the relations obtained in Sections 4.2 - 4.5 using the relativistic expressions determining the relationship between E, dE and v with R and dR:

E = Ze( R2 + (Ampc2 /Ze^j ; dE = ZeR\^ R2 + (Ampc2/Ze^j dR;

where mp is the proton mass; for nuclei A is the atomic weight (mac = Amp ); for electrons A = 5.45 x 10_4).

Let the spectrum and motion of inhomogeneities be such that the effective values of A and u depend on R over a sufficiently wide range of rigidities as

where Ai and u are respectively the transport scattering path and the chaotic velocity of inhomogeneity motion which are effective for particles with rigidity Ri . Then, taking into account Eq. 4.6.2 and 4.6.3 we shall obtain for the mean time of particle acceleration in the source:

t = t(R/R,)_(3+1^R2 + (Ampc2/Zef^j (R2 + (Ampc2/Zefj 1 , (4.6.4)

where t = l}/2viAi is the mean time of acceleration in the source of particles with rigidity R ; here the particle's velocity vf = cRfI R

4.6.2. Non-relativistic range; X and u are independent of R

If R << Ampc2!Ze it follows from Eq. 4.6.2 that

Ek = (ZeR)12Ampc2; dEk = (Ze)2RdRAmpc2; v/c = ZeRAmpc2 . (4.6.5)

Then for the non-relativistic range Eq. 4.6.3 will remain unchanged, and Eq. 4.6.4 will turn out to be

T = Ti (R/Rf )-1. Substituting Eq. 4.6.5 in Eq. 4.4.18 we shall obtain d (R/R ) = 8u dt ~ 3à(R/R0 )

R0 = AmpcujZe

is the rigidity of particles with velocity u.

If Rf < Ro (i.e. vf < u ), then, after integrating Eq. 4.6.7 at the initial condition

R|f=0 =Rf we shall obtain

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