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to vary in the course of acceleration. At v = 3 the solution of the equation Eq. 4.11.41 for the case of point source is of the form f (p, t) = P 3/2exP

2V nT

where po is the particle momentum during injection. The Eq. 4.11.46 also describes the particle distribution in the ultra-relativistic case. For this purpose the replacement v ^ v + 1 should be made in this expression. In accordance with this the Eq. 4.11.46 describes the distribution function in the ultra-relativistic case for v = 2. The characteristic time of acceleration t is determined in both cases by the relation:

In the case of stationary acceleration we obtain the power spectra of charged particles. According to Bakhareva et al. (1970a), the magnetic pumping mechanism including the loss for synchrotron radiation may ensure the observed power of the synchrotron X-radiation from the Crab nebula. The kinetic theory of particle acceleration by magnetic pumping was further developed in the work (Bakhareva et al., 1973) which gives a more general derivation of the equation of particle diffusion in the momentum space for quasi-linear approximation. The equation describes the evolution of the averaged distribution function f in the variable external magnetic field (pumping field) subject to strong scattering by turbulent pulsations during the pumping period. To close the set of equations describing self-consistently the evolution of the particle and wave spectra (hydromagnetic turbulence), the exact equation of the quasi-linear theory for the rate of the increase in the spectral function of waves dO/dt = 2yk O (4.11.50)

is used here (where Yk is the increment of the cyclotron instability owed to the distribution function deformation). The deformation is associated with particle acceleration by the inductive electric field of pumping. The increment Yk is calculated as a function of the averaged distribution f described by the diffusion equation. The stationary solution of the set of equations so obtained has been studied for the case of ultra-relativistic electrons taking account of the synchrotron radiation forming a sink of the energy pumped by the variable magnetic field. The region of the momentum space where the acceleration is most effective has been found, which makes it possible to construct simple formulas for estimating the electron energy density and the synchrotron radiation intensity.

4.12. Accelerated particle flux from sources

4.12.1. Particle flux from a source in stationary case

We found above the spectrum of accelerated particles in their source for different modes of statistical acceleration mechanism. Since the probability of particle ejection from the source may be energy-dependent, the spectrum of the outgoing flux may be appreciably different from the particle spectrum in the source. Consider a simple model. Let the source be a sphere of radius L, the transport scattering path inside the source be -, the particle velocity be v; then the diffusive particle flux from the source I(E) will be determined by the expression

where

and the parameter a ~ 1 is determined by the details of the diffusion model. Let the particle concentration and transport scattering path in the source be determined by the expressions n(E) = n0(E/E0 )-Y; - = -0(E/E0 Then, including Eq. 4.12.1, we obtain

In the non-relativistic energy range we get:

If n and - are determined by the power functions from R with exponents y and ft (see Section 4.4), then

rY+P

4.12.2. Particle flux from the source in non-stationary case

Assume that the particles are accelerated in the source within time from to to t1 . At the instant t1 the particle concentration in the source is n(E,h) = nx(EfE0) . (4.12.7)

If A = Ao (E/Eo) is the transport scattering path the change of the particle number inside the source at t > fj will be determined, taking account of Eq. 4.12.3, by the equation

the solution of which including the initial condition at t = f1 according to Eq. 4.12.7

Taking account of Eq. 4.12.1, it is easy now to determine the time variations of the particle flux from the source at t > ^ :

It can be seen from Eq. 4.12.10 that the energy spectrum of the particles ejected from the source varies significantly in time. If ¡3> 0 then at t -11 > L2avAo (E/E0 )

the exponential factor is already of significant importance and the spectrum becomes even softer in time.

4.12.3. Accelerated particles in the space beyond the stationary sources

Consider some volume of space in the form of a sphere of radius ro which contains stationary sources of accelerated particles. Let the total particle input from all sources to the considered volume per unit time be F(E). If the transport scattering path of particles within this volume is A (E) the total diffusive particle flux from the space volume is

where n (E, r) is the concentration of accelerated particles in the volume beyond the sources. If other losses of particles may be neglected we shall obtain from the balance equation

F(E ) = -4nr0 V- gradn(E, r )\r=r<} • (4.12.13)

Let us consider that approximately a^n (E, r)

where a1 ~ 1 is the parameter determined by the details of the problem. Then the averaged spectrum (over the considered volume) of accelerated particles in the space beyond the sources will be

It can be seen from Eq. 4.12.15 that the accelerated particle spectrum in the space beyond the sources may be significantly different both from the particle spectrum in the sources and from the spectrum of the particles ejected from the sources. If

A(e) — we obtain, considering that according to Eq. 4.12.3 F(E) — E~Y+P for relativistic particles with energy E and that according to Eq. 4.12.4

F(Ek) — for non-relativistic particles with energy Ek , that beyond the sources will be

4.12.4. The accelerated particle spectrum beyond non-stationary sources

Consider first the case where the flux of particles ejected from their sources may be presented in the form of product of 8-functions, i.e. F(E)8(r - ro )S(t - to). Then the energy spectrum of particles beyond the source in case of isotropic diffusion will be n (E, r, t ) = F (E)

For the particles with energy E, the peak at point r is reached at the moment

It follows from Eq. 4.12.17 that, at first, the accelerated particle spectrum increases rapidly, reaching the value