As we mentioned above, the Eq. 1.3.2 is valid for diffusion propagation of CR in space plasma (see Chapter 2 in more detail). In another case, when the propagation of CR is along magnetic field lines from some stationary source and there are no other CR sources in space, the contents of CR on the matter depth 5 from the stationary source (in units g/cm2) whilst taking into account fragmentation of nuclei will be determined by the following equation
ds Ai j>i Aj with the boundary condition
where nio is the CR contents at the exit from the source.
1.3.3. Expected fluxes of secondary electrons, positrons, y-quanta, and neutrinos
The nuclear interactions of CR with the space plasma matter in sources and during CR propagation also produce unstable secondary particles (K mesons, no mesons, mesons, neutrons) whose decay eventually gives y-quanta, electrons, and neutrinos. Since in this case the processes of generation and the kinematics of decay (no ^y + y, e±, n ^ p + e-, etc) are absolutely the same as those in the Earth's atmosphere, it is possible to use the result of (Greisen, 1960) and write, as a first approximation, in accordance with (Ginzburg and Syrovatsky, M1963) that the secondary particle generation by, for example, galactic CR will give the fluxes (in cm-2ster-1sec-1):
where K = 3.27 for y-quanta, 3.05 for neutrinos, 1.11 for electrons. In Eq. 1.3.5 the energy of secondary particles and gamma rays E is measured in GeV; X is the thickness of matter in g/cm2 at the beam along the line of sight in the region filled with CR. In the Galaxy the > 1 GeV secondary particle fluxes prove to be inconsiderable; even the largest expected flux (from the galactic center at X ~ 0.1 g/cm2) is about 2x10-5 cm-2ster-1sec-1 for y-quanta and neutrinos. According to Eq. 1.3.1, the estimate of generation rate for secondary electrons with E > 1 GeV in the halo (at a = 10-2 atom/cm3 concentration) gives an emissivity ~ 2x10-29 electron.cm-3sec-1, the value two orders smaller than that required to explain the observed flux of synchrotron radiation and the galactic electron flux observed on the Earth (Ginzburg and Syrovatsky, M1963). It can be seen below, however, that the inclusion of secondary particle generation is sometimes absolutely necessary in the problems of the study of CR generation and propagation. A lot of calculations of expected fluxes of secondary electrons, positrons, y-quanta and neutrinos in space and in the atmosphere (including comparison with experimental data) were made by Daniel and Stephens (1974), Orth and Buffington (1974), Ling (1975), Verma (1977a,b), Badhwar and Stephens (1977).
In particular, Daniel and Stephens (1974) made calculations of the expected intensity of electrons, positrons and y-quanta of various energies depending on the thickness of the atmosphere passed from 0 to 1000 g/cm2; the data on nucleus-meson cascade of cosmic radiation in the Earth's atmosphere and energy
transmission from the hadron component to the electron-photon component were included in the calculations. The differential energy spectra of secondary electrons, positrons, y-quanta and their variation with a depth of the atmosphere obtained are of primary importance in determining the respective corrections to the measurements of the fluxes of these particles from space. To estimate the effect of geomagnetic latitude the calculations were carried out for several values of geomagnetic cutoff rigidity: 0; 2; 4.5; 10 and 17 GV. To include approximately a chemical composition of primary CR for the free path to absorption, the following values were fixed: 120; 52; 33; 33, 32, 29 and 20 g/cm2, respectively, for protons, a-particles, and nuclear groups L, M, H1, H2, H3. A comparison of calculated results with available experimental data made it possible to refine the general parameters of elementary acts, which are included in a theory. The results of calculations from (Daniel and Stephens, 1974) for space-energy distribution of electrons, positrons and y-quanta in the Earth's atmosphere are presented in the form of three-dimensional diagrams, shown in Dorman, M2004 (Fig. 2.11.1-2.11.6 in Chapter 2).
Orth and Buffington (1974) computed by the Monte-Carlo method a meson-nucleus cascade of primary CR in the Earth's atmosphere and interstellar space at the depth < 10 g/cm2; the expected fluxes and energy spectra of secondary electrons and positrons were found in the interval of energies 1^100 GeV, arising as a result of the processes of decay of pions and muons: a development of subsequent electromagnetic cascades was also included. An approximate analytical expression was presented which makes it possible to determine easily the fluxes of secondary electrons and positrons with various energies, medium parameters and spectra of CR. The calculations were based on the following data: differential spectrum of primary protons of galactic CR depending on the proton energy Ep (in interstellar space: 2.0x104E-2'75 m-2ster-1sec-1GeV-1; the spectrum at the boundary of the
Earth's atmosphere, including solar modulation, was taken in the form
8.6x103E-2'55m-2ster-1sec-1GeV-1); the proton path with respect to inelastic interaction was taken to be 53.6; 57.3 and 100 g/cm2 in hydrogen, interstellar medium and the air, respectively; the ratios of a number of electron and positrons, generated by all of CR nuclei, to their number, generated by the protons only: 1.36, 1.34 and 1.20 in hydrogen, interstellar medium, and in the air; the conversion path of y-rays in the air, 48 g/cm2; the path lengths of ju± and n± mesons to decay, 6.24 E^ km and 0.0559 En km (where EM and En are the energies of muons and pions, in GeV). It was shown that to account for the recent measurements of positron flux with the energy > 4 GeV, one has to assume that primary CR pass, on average, (4.3-J;f) g/cm2 of the interstellar medium from the moment of their generation.
It is of especial interest to study high energy neutrinos. Silberberg and Shapiro (1977) studied a diffusion background of neutrinos which is caused by the three sources: a) generated by CR in the Earth's atmosphere; b) generated in our Galaxy; c) generated out of the Galaxy. The flux of atmospheric neutrinos formed as a result of a decay of n± and k ± -mesons generated in interaction of CR with the Earth's atmosphere, exceeds the flux of neutrinos of extra-terrestrial origin up to the energies ~1014 eV. At E > 1015 eV the direct generation of lepton pairs could be a prevailing source of atmospheric neutrinos. At higher energies (> 1016 eV) a considerable uncertainty and a strong sensitivity to a model of neutrino source were observed in the value of diffusion flux. At these energies the following processes of neutrino generation take place: a) direct production of neutrino in the atmosphere; d) neutrinos from radio galaxies; c) neutrinos from galaxies; d) neutrinos from pulsars. To determine what of the mechanisms gives a prevailing contribution to neutrino fluxes, the direct observations are necessary. According to Margolis and Schramm (1977), a study of super high energy neutrinos gives important information about CR in the Universe, about physics of high energies, and about the physics of weak interactions. Measurements of neutrino flux from point sources make it possible to study acceleration of CR by such astrophysical objects as pulsars, Supernovas, and radio galaxies. In Margolis and. Schramm (1977), an estimate was made of the flux value of neutrinos with the energies E >10n eV from several types of astronomical sources. Berezinsky (1977) considered some possible sources of neutrinos of superhigh energies: Supernovae and interactions of CR protons with atomic nuclei and microwave photons in interstellar and inter galactic space. The author presented energy spectra of neutrinos calculated for various mechanisms of their generation. Beresinsky (1977) emphasized the following important aspects of astrophysics of neutrinos of super-high energies (E > 1015 eV): a search for bursts of CR in remote cosmological epochs; a search for point sources of high-energy neutrinos; a study of interaction of neutrinos at the energies, unattainable for accelerators, a search for the W-bosons with a mass 30^100 GeV, and the measurement of a cross-section of neutrino interaction at the energy E > 1015 eV. Berezinsky and Zatsepin (1977) assumed that in the epoch of galactic and early-class stars formation, a burst took place with an energy output ~ 5x1060 erg in CR for our Galaxy. It is considered that the observed diffusion X- and y-radiation in the range 1 keV^30 MeV is caused by high energy CR from this burst. Neutrinos with the energy E > 3 x1015 eV should be produced as a result of interaction of high energy protons with microwave photons. The estimate was obtained for the flux value of these neutrinos. This estimate, as well as the assumption that neutrinos generated in early cosmological epoch did not undergo interactions in later stages of the Universe's expansion, makes it possible to register such neutrinos by means of a detector with the volume ~ 109 m3.
1.3.4. Expected fluxes of secondary protons and antiprotons
The problem of low energy secondary protons generated in nuclear reactions of CR with interstellar medium, was studied in the work of Wang (1973). The refined cross sections of (pH) and (pHe) interactions were used to obtain the intensities of secondary protons with an energy lower than 100 MeV, non-included processes were estimated. The rates of producing secondary protons were calculated and then a solution of the equation for the stationary density of secondary particles in the Galaxy was found. The intensity obtained of protons in interstellar medium for the energy range 10^100 MeV appeared to be by 3^5 times higher than the observed proton intensity near the Earth (this weakening is related to particle interaction with solar wind in interplanetary space. The value of intensity of protons obtained, born in nuclear interactions of high energy components of CR, is the lower limit of the actual intensity of protons near these energies in interstellar medium. Ganguli and Sreekantan (1976) calculated the expected fluxes of various secondary particles with the energy > 10 GeV produced in nuclear interactions of CR with interstellar medium in the Galaxy, based on accelerator data on the effective cross-sections of the generation of n± mesons, antiprotons, and deuterons in proton-proton interactions up to the energies ~ 1500 GeV in nuclear interactions of CR with interstellar medium in the Galaxy. It was found that of y-quanta with E > 10 GeV from the Galactic center on the boundary of the Earth's atmosphere should be = 10-4 and the flux of antiprotons (2 ■ 3)x10-4 of the proton flux in CR; the expected deuterium flux appeared to be negligibly small compared to the flux arising as a result of fragmentation of a-particles in their interaction with interstellar hydrogen.
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