For the 'temperature' radiation where N(co) = TeffJ^rn, the acceleration proves to be similar to the Fermi acceleration:

dt 3 me me

4.10.5. Electron acceleration by the radiation during their induced Compton scattering

Levich and Sunyaev (1971) showed that the induced Compton scattering of the radiation by the cold electrons brings about their effective heating to the relativistic temperatures. Of obvious interest is the further heating of already relativistic electrons with energies E > mec . The analysis of acceleration of such electrons owed to the induced Compton scattering was made in the papers Blandford (1973), Charugin and Ochelkov (1974, 1977), Kurlsrud and Arons (1975).

Statistical acceleration of ultra-relativistic electrons by random electromagnetic waves is considered in the work (Blandford, 1973) in which the classical treatment of the induced Compton effect is generalized for relativistic case. The behavior of the function of electron distribution in a stationary field of radiation has been studied. Kurlsrud and Arons (1975) have studied the statistical acceleration of particles up to relativistic energies when affected by a set of spherical electromagnetic waves. They have shown that if a particle moves among a great number of antennas (for example, pulsars) emitting electromagnetic waves, the particle's energy and value of momentum increase in a stochastic manner. The essence of the mechanism is as follows: the trajectory of a particle which moves between the antennas (pulsars), each of which emits a strong electromagnetic wave, comprises the points at which the phase difference of the particle's oscillatory motions when affected by the two waves is constant in the coordinate system where the particle is at rest on the average. After arriving at such a point the particle will be accelerated until the drift motion or acceleration carries it away from the resonance region. Such a kind of the nonlinear resonant acceleration in which a particle interacts with two waves is known from Landau's theory of nonlinear attenuation, and it may be shown that such acceleration is a particular case of the Compton scattering. If the antennas (pulsars) are randomly located, the summation over all resonances and all pulsars will result in stochastic acceleration of the particles.

In the paper of Charugin and Ochelkov (1977) it was dwell upon the discussion of the objects of relativistic electrons heating by the induced Compton scattering in the sources with relativistic brightness temperatures and isotropic distribution of the radiation.

4.10.6. Acceleration of charged particles by electromagnetic radiation pressure

Noerdlinger (1971) analyzes the motion of a particle ejected by a central body (star) when affected by radiation pressure. The expression for the accelerating force F affecting the particle which generalizes the corresponding results of (Chandrasekhar, 1934a,b) for the relativistic velocities of motion has been obtained. It was assumed when deriving the formula for F that the radiation field was purely radial and that the effective particle cross section o was a function of the radiation field frequency. It has been shown that the final velocity of the ejected particle vM depends on the value Foro/macc2 , where Fo is the force affecting the stationary particle at the initial moment t = to ; ro is the initial radial coordinate of the particle; mac is the accelerated particle mass. The plots of the dependences of the dimensionless velocity of particle motion 3 = c on Forojmacc2 for relativistic and non-relativistic cases were obtained. Noerdlinger (1974) has studied the effect of the finite size of the electromagnetic radiation source on charged particle acceleration by radiative pressure. The extreme value of the Lorentz-factor Yi has been found (as a function of the distance from the source) above which any radiation field, whatever strong, cannot accelerate particles. The final energy of particles decreases significantly for very strong sources if a source ejects the particles within a large angle (at the same initial acceleration). In the asymptotic extreme, the final energy of the particle at rest at the source surface is proportional to the fourth-power root of the source's luminescence for strong sources, whereas it is proportional to cube root of the luminescence for a point source.

Nakada (1973) examines the heavy ion acceleration in the case of resonant scattering of radiation near a bright source. The ion energy proves to be limited by the Doppler effect, aberration, photo-ionization, and ionization in the collision with the electrons of medium. It has been shown that the O and B stars may accelerate heavy ions up to 200 and 80 MeV/nucleon respectively, whilst the supernovae may accelerate them up to several hundreds of MeV/nucleon. Gordon (1975) has found the velocity distribution of fully ionized isolated atoms which are produced from partially ionized atoms and reside in the field of a point source of radiation with the power spectrum Iv(v) = AvY/r2 (where v is the frequency; r is the distance from the source; Iv(v) is the radiation intensity). The radiation pressure onto the atoms in case of the radiation absorption in the resonance lines and the atomic photo-ionization have been taken into account. It has been found that the velocity distribution of the fully ionized atoms is independent of the radiation intensity and is a function of only the value of the exponent y. It has been shown that for the exponent y typical of the astrophysical objects the atoms cannot be accelerated by the radiation pressure up to relativistic velocities.

4.11. Statistical acceleration of particles by the Alfven mechanism of magnetic pumping

4.11.1. Alfven's idea of particles acceleration by magnetic pumping

Alfven (1949, 1959) studied the acceleration of charged particles moving in an alternating magnetic field H and showed that a field enhancement would result in adiabatic acceleration of particles defined by the relation pHh = const, p// = const. (4.11.1)

If in this case the particles are scattered by magnetic inhomogeneities the field will return to the initial value without decelerating the particles down to an energy corresponding to the initial state. The final result of a single cycle will be some gain in the particle energy. For the stationary state, if the rate of energy variation is independent of particle energy, the resultant spectra will be of the form ^ p-, where y = 1. It is assumed in order to obtain a higher y in conformity to the experimental data that the spectrum is variable in space. Analysis of the assumption shows that the previously accelerated particles are injected in the vicinities of stars where they may be subsequently accelerated by the alternating magnetic field. The particles of relatively low energies are confined within the regions close to the active stars, whereas the high energy particles are distributed over a much extended region. The inclusion of the absorption of the accelerated particles gives a power spectrum similar to the observed spectral form.

It is of importance to note that Alfven's concept (1959) of a combination of the particle scattering and the betatron acceleration makes it possible to obtain a systematic increase of particle energy in a fluctuating magnetic field (even if the magnetic field intensity does not increase on the average). This phenomenon was subsequently called magnetic pumping. This mechanism, which is most probably of great importance to the acceleration processes in various objects, will be considered below in more details.

4.11.2. Relative change of the momentum, energy, and rigidity of particles in a single cycle of magnetic field variation in the presence of scattering

In accordance with the work (Alfven, 1959), we shall examine the following simple model. Let us assume that the charged particles are confined within some volume comprising the magnetic field inhomogeneities against the background of homogeneous magnetic field. The frequency of collisions with inhomogeneities is v _ v/A , where A is the transport scattering path of particles and v is the particle velocity. Let the field vary in time in the following manner:

(1) the field increases from Ho to H1 within a short time from t1 to t2 (here

(2) then, within time from t2 to t3 (here t3 -12 >> v_1), the field remains at the level H1 ;

(3) after that the field intensity falls rapidly down to the initial value Ho within time from t3 to t4 (here £4 -13 << v_1);

(4) during a period from t4 to t5 (here t5 -14 >> v_1) the field remains if at the level Ho .

Let us consider how the particle momentum and energy will vary during the above-mentioned cycle. Since, at the beginning of cycle (the same as period from

¿4 to ¿5), the field was equal to Ho during a long period (>>v-1) and the equilibrium distribution of the energy degrees of freedom has set in owing to the collisions with inhomogeneities; so that the momentum components across and along the field were determined respectively as ph _ 3 p2; p2/1 _ 3 p2, (4.11.2)

where p1 is the particle momentum at the instant . During the first interval the collisions with inhomogeneities may be neglected and, according to Eq. 4.11.1:

2 Hi 2 2 Hi 2 2 2 1 2 Pl2 =— Pl1 =-—Pi ; P// 2 = P//1 = - P1 ■

From Eq. 4.11.3 follows that the total change will be

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