Ing

where

G* (z )= 1 - z - 2z2 exp(z)Ei(- z); z = %2 (19) , (i.n.22)

The plot of the function G* (z) is also shown in Fig. 1.11.1. The maximum value of da is at 6 = nj2 and decreases down to 0 at 9 = 0; this is quite understandable since in this case the observer receives only the photon from the Planck distribution, but not the scattered photons. If the electron spectrum is of the power form described by Eq. 1.11.12, we shall obtain taking account of Eq. 1.11.21:

V mec j

Eph2 2Tph

where

It follows from Eq. 1.11.23, including Eq. 1.11.22, that at zo << 1 dW « Eph2jTph and at zo >> 1 dW (1 - cose)"^2 (ph2/Tph Z^2 . In this case the energy

Ephax , at which dW is a maximum, will be proportional to (1 - cos/)) [fjmec2).

It also follows from Eq. 1.11.23 that in the high enrgy photon range (when

((ph2 /4Tph ))ec2/Eo )>> 1) the emitted power depends on angle e as (sin 0/2)+1, where e is the angular distance between the emitting region and the central meridian (for solar flares), i.e. the most considerable flux of hard X-rays generated by accelerated relativistic electrons will be from the flares at the edge of the solar limb. The probability of Compton scattering W (E,vph1,vph2) of isotropic radiation by relativistic electron with energy E is generalized in (Charugin and Ochelkov, 1974) for the case where the frequency vph2 of the scattered quantum is smaller than the frequency vph1. It has been shown that the intensity of the scattered radiation «vph2 at vph2 > vph1 and is proportional to vph2 at vph2 < vph1 for a mono-energetic electron beam. In case of induced Compton scattering the relativistic electrons always gains energy. If the brightness temperature of radiation

Tph is such that kTph >> mec2 in some range of frequencies, the rate of heating

-(kTph/mec2)mec2/EJ caTp. Since the cooling owed to spontaneous Compton loss « E2, the electrons even in intense sources may be heated only up to energy E - mec2 (kTph/mec2 ^ . Despite the fact that the effectiveness of the electron heating owing to induced Compton scattering in real sources is small, considerable distortions of the spectra are expected in quasars if the density of electrons with energies E < 10mec2 exceeds 106 cm 3. Sweeney and Stewart (1974) have studied the nonlinear Compton radiative group deceleration. The equations of electron motion in a strong, plane, and circularly polarized electromagnetic wave have been numerically integrated by taking account of radiative deceleration. The principal attention was paid to the electron interactions with the electromagnetic wave whose phase velocity c¡a (where 0 < a < 1) exceeds the velocity of light in vacuum. It has been shown that the radiative deceleration accounts for the rapid evolution of charged particles. Irrespectively of the initial energy, the evolution of charged particles is rapid and their energies approach the asymptotic value

E ~ mec2//Vl - d2 at which the energy loss for radiative deceleration and the energy gain under the effect of the electric field of the strong wave are equalized. For the pulsar in the Crab nebula, the parameter of wave force / ~ 106 and the electrons acquire their asymptotic energies already at distances of ~ 2x1013 cm from neutron star.

Milton et al. (1974), Hari Dass et al. (1975) have studied the Compton scattering of photons by charged particles in the presence of homogeneous magnetic field whose value is comparable with critical (for electrons, 4x1013 Gs). In this case the effect of such a field of all orders should be taken into account or else the charge should be considered as bound. The calculation scheme and the expression for the cross-section are presented. The extreme cases are examined. Ochelkov and Prilutsky (1974) study the effect of the energy loss for the Compton radiation on the electron spectrum in the 'plasma kettles', i.e. the regions with high density of electromagnetic radiations, which may probably exist in the galactic nuclei and quasars. It was shown earlier that a power spectrum of electrons with exponent y = 3 which is universal, independent of specific size of a kettle, was generated in turbulent plasma in homogeneous magnetic field. It is emphasized, however, that the results obtained are inapplicable at sufficiently high densities of electromagnetic radiation in a kettle when the Compton scattering of the radiation by relativistic electrons becomes to be of importance. It has been shown that the power spectra of electrons with exponent y + 3 are generated in the kettles with high density of radiation (of the order of the energy density of plasmons or magnetic field); the value of y is already dependent on specific parameter of a kettle, for example the size, the relativistic electron density, etc. It is indicated that the value may be unambiguously determined from observations on the basis of the slope of the spectrum of X-rays from a kettle of Compton nature. For practical determination of y, however, the spectrum of X-rays from galactic nuclei (supposedly the Compton X-rays) has been insufficiently studied as yet. Similar results in the study of the comptonization and the generation of the relativistic electron spectrum in the plasma kettles have been obtained in (Nikolaev and Tsytovich, 1976; see Section 4.10.1) taking account of the Compton scattering of reabsorbed radiation. The universality of the plasma turbulent kettle as a source of relativistic electrons with a power spectrum under the conditions close to the real situation in the space in the presence of magnetic fields and magnetic turbulent modes of pulsations has been demonstrated. The dependence of the spectrum exponent y on the parameters characterizing the plasma of the turbulent reactor has been studied for various types of turbulence. The found y < 3 correspond to the range of the most probable values obtained in the studies of cosmic radio sources.

1.11.2. The influence of nuclear photo effects on accelerated particles

Gerasimova and Rozental (1961) have estimated the variations of the CR spectrum in case of a nuclear photo effect on stellar photons. The values have been obtained for iron nuclei which undergo a photo-effect on the photons whose spectrum is determined by the Planck function of black body radiation at T = 5800 °K. The nuclei of galactic origin fail to disintegrate completely; only their isotopic composition changes, since the (y,n) and (y,2n) photo-neutron reactions are predominant. The change of the exponent of the integral spectrum of galactic CR has been estimated. The nuclei of intergalactic origin produced more than 1010 years ago underwent the photo-effect completely. The nuclei that are being produced may enter the Galaxy from a region of 5 x1025 cm size which fails to give any significant contribution from CR to the spectrum observed on the Earth.

Pollack and Shen (1969a,b) have noted that, because of the Doppler effect, the photons of moderate energy in the coordinate system of their sources are shifted to the y-ray region in the coordinate system of highly energetic CR. Therefore such photons may knock out individual nucleons from compound nuclei and decrease the proton energy. The calculations show that the photon density during supernova explosions, in quasars, and in some pulsar models is sufficiently high to ensure a splitting of a-particles with total energies above 4 x 1015 eV when they are ejected from these potential sources of CR. The corresponding value of energy for nuclei of group VH is some 2 x1017 eV (see Table 1.11.1).

Table 1.11.1. Critical energies (in eV) of various nuclei for an essential energy loss in photon field. According to Pollack and Shen (1969a,b).

Source

Total photodisintegration

Significant energy loss

a-particles

very heavy nuclei, A > 50

protons

Supernova, type II

4x1015

9x1016

7x1016

Quasars

2x1015

8x1016

8x1016

Pulsars, r = 10 km

2x1015

1.2x1017

2x1017

Pulsars, r = 1000 km

7x1018

5x1020

2x1022

It follows from the results presented in Table 1.11.1 that the 1017^1019 eV CR must be almost completely protons. Besides that, the order of magnitude of the photon field is sufficient to result in a significant energy loss at E > 1016 eV. This is a probable reason for the decrease in the number of such high energy particles as observed in the spectrum.

Attention is paid in (Rengarajan, 1973) to the fact that during the early stage of a pulsar when an appreciable portion of the energy of neutron star rotation may be converted into high-energy CR and y-quanta, the surface of the star is very hot (with T ~ 107 °K), and therefore it is necessary to take account of the CR and y -quantum interactions with the photons of this radiation. Considering the photon emission to be black dark with T = 107 °K, the author has calculated the y -quantum absorption as a result of y-y collisions and the heavy nucleus photo-disintegration. It has been found that during the initial stage of a pulsar, when T ~ 107 °K on its surface, all the y -quanta with energy 108 - 1012 eV are completely absorbed and all the nuclei of the iron group with energy 1013 - 1015 eV/nucleon disintegrate almost completely. The calculations has been also carried out for T = 2x106 °K and T = 5x106 °K. At T = 5x106 °K, the y - quantum absorption is still significant, the optical thickness zYf~ 10 (for the 109^1010 eV energy range), whereas at T = 2x106 °K Tyy < 1. The iron nucleus photo-disintegration decreases pronouncedly with decreasing T, so that at already T = 5x106 °K the optical thickness t^ ~ 1 only at E ~ 5x1013 eV/nucleon and decreases abruptly at either side of this value of energy.

1.11.3. Effect of the universal microwave radiation on accelerated particles

Daniel and Stephens (1966) studied the effect of the isotropic thermal radiations of the Universe at T = 2.7 °K on the energy spectrum of high energy electrons by measuring the differential energy spectrum of electrons with energies E > 12 GeV in primary CR. Analysis of 28 detected electrons has shown that the total flux of primary electrons with effective energy E > 12 GeV is 0.51 ± 0.10 (m2sec.ster)-1. The differential energy spectrum of the 12^350 GeV electrons is of the form n (E )dE = 12.7E Y dE, where E is the electron energy in GeV; y = 2.1 ± 0.2.

The positron share in the total number of electrons and positrons is 0.70 ± 0.20, which is indicative of a positron excess at E > 12 GeV in contrast to a negative excess of electrons at lower energies. The measured spectrum of the 12^350 GeV electrons and the 1^10 GeV electron spectrum obtained from other experiments were compared with the calculated spectrum obtained on the assumption of electron equilibrium in the galactic halo and including the Compton backscattering of electrons by the luminescence photons and by the photons of black-body radiation at T = 2.7 °K. The comparison has shown that at E < 2 GeV the experimental and theoretical spectra fail to be in agreement, which may be explained by the solar modulation effect on the low-energy electron fluxes. On the other hand, at E < 12 GeV the observed appreciable excess of the experimental spectrum over the theoretical one is far beyond the possible errors. In this energy range the exponent of the theoretical spectrum y = 3.4 in comparison with the experimental y = 2.1 ± 0.2. The observed discord indicates that either the black-body radiation at T = 2.7 °K does not exist in the Universe or the adopted model of galactic halo is invalid. However, the theoretical spectrum calculated in terms of the extragalactic model for electrons gives even larger disagreement with the experimental data. For the subsequent experiments have confirmed existence of the Planck radiation in the Universe, the extragalactic model have been completely rejected. In this case a dual explanation may be given to the available experimental data on the electron spectrum at E > 12 GeV, namely (1) the electrons are probably not in the equilibrium state in the halo and (2) the entire observed spectrum of the 1-350 GeV electrons consists of two different components in the galactic halo model. One of the components, which accounts for the existence of the electrons with energies of up to ~ 10 GeV and has the spectrum n(E)dE = 50E~2AdE , comprises the directly accelerated electrons and the secondary electrons produced in nuclear interactions of CR when traversing a 2.5 g/cm2 path in the interstellar hydrogen. The second component accounts for the existence of the 10^350 GeV electrons and has the spectrum n(E)dE = 0.54E_11dE . Because of the Planck radiation this spectrum begins to fall more steeply at E > 20 GeV and reaches the exponent y = 2.1 at high energies.

Cowsik et al. (1966) emphasize that the universal radiation at T = 2.7 °K makes it possible to estimate the upper limit of the leakage lifetime of the primary CR electrons. For this purpose the equilibrium differential energy spectrum of electrons was calculated for various electron leakage lifetimes t and the results were compared with the corresponding experimental spectrum in the 1^350 GeV energy range. The exponent of the power energy spectrum of electron injection calculated disregarding the energy loss was assumed to be y = 2.4. The comparison showed that the spectrum calculated for t = 107 years was in a good agreement with experimental data if the possible errors are included, whereas the spectrum with t > 108 years gives too low an electron flux at energies of 100 GeV and higher. Thus, the leakage lifetime of electrons cannot exceed 107 years if the universal T = 3°K radiation exists and if the electrons with energies of up to several hundreds of GeV are generated in but a single source.

1.11.4. Effect of infrared radiation on accelerated particles

Shen (1970) notes that if the recently discovered (Shivanandan et al. 1968), IR radiation exists actually in the Galaxy, Vela X is probably the sole source of the very high energy CR electrons measured on the Earth (Shivanandan et al., 1968).

An abrupt cutoff of the energy spectrum of CR electrons is predicted at E ~ 2x103 GeV.

1.12. CR interaction with matter of space plasma as the main source of cosmic gamma radiation

1.12.1. The matter of the problem

The interaction of CR particles (protons, nuclei, and electrons) with matter determine the main processes of high energy gamma ray generation through neutral pions decay and bremsstrahlung emission. In Dorman (1996) there was estimated the expected gamma ray intensity generated by local and outer CR in different astrophysical objects for outer and inner observers. Any astrophysical object containing CR (of local and outer origin), magnetic fields and matter must generate gamma rays by neutral pion's decay (generated in interactions of CR protons and nuclei with matter), and by the generation of bremsstrahlung, synchrotron and curvature radiation of relativistic electrons, and by inverse Compton scattering of relativistic electrons on optical, infrared and relict 2.7 °K photons. The intensity and spectrum of gamma radiation depend on the CR spectrum, on the CR space-time distribution function, as well as on the spacial distribution of matter, magnetic fields and small energy background photons. Below we shall consider general formulas for gamma ray generation through neutral pion's decay (generated in nuclear interactions of proton-nuclear CR component with the matter of space plasma; see below Section 1.12.2), and gamma ray generation through interactions of CR electrons with matter and low energy photons in space plasmas (bremsstrahlung and inverse Compton radiation, respectively; see Section 1.12.3). on the basis of these formulas we shall make several estimations of expected gamma ray generation by flare CR from the Sun by their interactions with the matter of solar corona and solar wind as well as gamma ray generation by flare CR from other stars by interactions with the matter of stellar winds (Section 1.13). The same will be made for gamma ray generation by galactic CR interactions with matter of solar and stellar winds (Section 1.14).

1.12.2. Gamma rays from neutral pions generated in nuclear interactions of CR with space plasma matter

Let the distribution of space plasma matter in the spherical system of coordinates r,6,$ be determined by n(r,6,$)in units of atom.cm-3. Let us suppose that Npn (e, r,6,$) is the space distribution of the differential intensity of the proton-

nuclear component of CR, where E is the total CR particle energy in GeV/nucleon. The gamma ray intensity from some space plasma volume boundared by the surface ro (6,$) from neutral pions decay in this volume at the distance robs >> ro (6,$) will then be i \ -2 n2 ro(e,^)2n ~ t \ t W \

Fr,pn ^obs, Er)= robs 1 cos9d9 1dr 1 d0\dEapn ( Ey >>pn ,(1-12-1)

where, according to Stecker (M1971), Dermer (1986a,b), a pn

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