THE ORIGIN OF COSMIC RAYS

V. L. Ginzburg

CONTENTS

Introduction . . . 37
1. Primary cosmic rays near the Earth . . . 38
    a) Composition of cosmic rays . . . 38
    b) Energy spectrum . . . 41
    c) Isotropy. Number of electrons . . . 43
2. Magnetobremsstrahlung nature of cosmic radio emission and the distribution of cosmic rays in the Galaxy . . . 44
    a) Nature of nonthermal cosmic radio emission . . . 44
    b) Electron component of cosmic rays in the Galaxy . . . 47
    c) Cosmic (relativistic) electrons in the shells of supernovae . . . 52
3. Motion of cosmic particles in the interstellar medium . . . 56
    a) Energy losses in the case of protons and nuclei. Formation of secondary electrons and positrons . . . 56
    b) Energy losses in the case of electrons. Change in the energy spectrum during the motion of particles in the interstellar medium . . . 61
    c) Diffusion and statistical acceleration of particles in the Galaxy . . . 67
4. Supernovae and novae as sources of cosmic rays . . . 73
    a) Energy balance. Acceleration of particles in the shells of supernovae . . . 74
    b) Composition of cosmic rays near the Earth . . . 77
    c) Spatial distribution and isotropy of cosmic rays . . . 84
    d) Critique of alternative views . . . 87
Concluding remarks . . . 94
References . . . 96

INTRODUCTION

The question of the origin of cosmic rays naturally arose immediately after these rays were discovered. For a long time, however, only purely hypothetical constructions were possible in this respect, since data on cosmic rays were completely lacking even at the boundary of the Earth’s atmosphere, not to mention the Solar System or the interstellar medium. In recent years the situation has changed radically: in 1948–1950 the composition of primary cosmic rays near the Earth was clarified to a first approximation; then in 1950–1953 it became possible, on the basis of radio-astronomical data, to draw certain conclusions about the distribution of cosmic rays in the Galaxy and beyond it.

The possibility of using radio-astronomical methods is connected with the fact that the greater part of cosmic radio emission has a magnetobremsstrahlung nature, i.e., it is radiation from relativistic electrons moving in interstellar magnetic fields. Thus data on the spectrum and intensity of radio emission make it possible to determine the energy spectrum and the number of relativistic electrons that form the electron component of primary cosmic rays. At the same time

it turns out that cosmic electrons*) are present throughout the entire Galaxy, filling a quasi-spherical volume with radius \(R \sim 5 \cdot 10^{22}\) cm. Outside this region, i.e., between galaxies, there are considerably fewer cosmic electrons, and perhaps almost none at all. On the other hand, within the Galaxy itself the region of our solar system is not in any way exceptional. Hence it follows that cosmic rays are mainly of galactic origin, while the Sun and the solar system can be responsible for the appearance of only some small part of the cosmic rays of very low energy, and also for various variations in the intensity of cosmic rays. As for cosmic rays of metagalactic origin, if they can play any role at all, it is only in the region of very high energies. Further, a number of powerful radio nebulae (discrete sources of cosmic radio emission) are located in the Galaxy, and the radiation of these objects is undoubtedly of a magnetic-bremsstrahlung character. It follows that in the radio nebulae in question, which are the envelopes of supernova stars, there is a large number of cosmic electrons. It is therefore natural to think that the sources of cosmic rays in the Galaxy are supernovae, and possibly also novae. It is true that radio-astronomical data pertain only to cosmic electrons, and not to cosmic protons and nuclei. Therefore, in passing to the proton and nuclear components of cosmic rays, one has to make certain additional assumptions. It seems to us, however, that the element of uncertainty that arises here cannot bear any comparison with the wholly arbitrary identification of the sources of cosmic rays with magnetic stars or with some new class of stars—red dwarfs.

The purpose of the present article is to set forth a theory of the origin of cosmic rays based on radio-astronomical data and, of course, on all other information of an experimental character. Therefore, and in accordance with what has been said above, we shall not dwell in detail on theories of the solar and metagalactic origin of cosmic rays, nor on possible mechanisms of particle acceleration near stars.

In general, the present article in no way claims completeness, either with respect to covering the history of the question or from the standpoint of presenting experimental data and references to the literature. At the same time, an attempt has been made to present with sufficient fullness and detail that theory of the origin of cosmic rays in whose development the author is participating and which was previously most fully reflected in articles \(^{1-3}\). In doing so, for the sake of coherence of exposition, and also for convenience, material already set forth in \(^{1**}\) will be partly reproduced.

1. PRIMARY COSMIC RAYS NEAR THE EARTH

Let us briefly consider the data on primary cosmic rays near the Earth, i.e., cosmic rays beyond the boundary of the Earth’s atmosphere.

a) Composition of cosmic rays

Basically, primary cosmic rays consist of protons and \(\alpha\)-particles; the nuclei of all the remaining elements, taken together, are present

*) The terms cosmic electrons, cosmic protons, and cosmic nuclei are used below only with respect to particles entering into the composition of cosmic rays.

) The present article is also being published in English in volume 4 of the series of collections Progress in Cosmic Ray Physics, edited by J. Wilson.

in an amount of the order of 1% of all incident particles. More complete information on the composition of primary cosmic rays is given in Table I.

Table I

Relative abundance of nuclei in primary cosmic rays and on average in nature

Nuclei	In primary cosmic rays: number of nuclei per \(10^5\) protons	In primary cosmic rays: percent of the number of primary particles	In primary cosmic rays: percent of the number of nucleons in the composition of primary particles	On average in nature, per \(10^5\) hydrogen atoms
Protons \((p)\)	100 000	91.5	69	100 000
\(\alpha\)-particles \((Z=2)\)	10 000 (8000)	7.8	23	7 700
Group \(L\) \((\mathrm{Li}, \mathrm{Be}, \mathrm{B};\ Z=3\text{--}5)\)	\(\lesssim 50 \div 200\)	—	—	\(3.6 \cdot 10^{-4}\)
Group \(M\) \((\mathrm{C}, \mathrm{N}, \mathrm{O}, \mathrm{F};\ Z=6\text{--}9)\)	520 (400)	0.4	4.5	80
Group \(H\) \((Z \geqslant 10)\)	160	0.15	3.5	30
Iron \((\mathrm{Fe},\ Z=26)\)	30	—	—	1.5
All nuclei with \(Z>30\)	\(<1\)	\(<0.001\)	—	\(10^{-3}\)

In the second column of the table, data available in \(^{4}\) have been used; in parentheses are also indicated other values for the abundance of \(\alpha\)-particles and nuclei of group \(M\), which are adopted in \(^{5}\). The third and fourth columns of the table are compiled on the basis of these data \(^{5}\). The last column gives the average abundance of various elements in nature according to \(^{6}\).

The energy spectrum of protons and nuclei is such (see below) that the relative number of particles of different types for which the momentum per nucleon \(p/A\) is greater than some value \((p/A)_0\) does not depend on \((p/A)_0\), where \(p\) is the momentum of the particle and \(A\) is its atomic weight. Table I gives the relative numbers of particles with momentum per nucleon (i.e., velocity) greater than some value, which corresponds to the relative abundance of nuclei outside the action of the Earth’s magnetic field*). It should be noted that the relative abundance of protons and various nuclei in the primary component has been studied more or less in detail only for energies less than approximately \(10^{10}\ \mathrm{eV}/\text{nucleon}\). At higher energies the data are incomplete, but there is reason to think that up to energies \(\sim 10^{13}\ \mathrm{eV}/\text{nucleon}\) the spectrum of primary particles by charge remains unchanged (more precisely, up to an energy

* At the boundary of the Earth’s atmosphere (at an altitude greater than 30–40 km) the relative abundance of protons and nuclei differs from that indicated in Table I, since the ratio of ordinal number to atomic weight is equal to unity for protons, whereas for nuclei \(A/Z \simeq 2\); therefore the geomagnetic “cutoff” of the spectrum is different for protons and nuclei (for more detail see, for example, \(^{5}\)).

$\sim 10^{13}$ eV/nucleon, protons certainly remain the most numerous group of particles, so that the number of all other nuclei does not exceed $\sim 10\%$ of the total number of particles). The question of the composition of cosmic rays at energies $E>10^{13}$ eV and up to the highest observed values $E\sim 10^{18}—10^{19}$ eV remains open, although it is very important from various points of view. Of course, what is most essential here is to clarify the qualitative side of the matter, i.e., to establish whether the particles with the greatest energies are mainly protons or nuclei.

Another topical and insufficiently clarified point concerning the composition of the primary component is the question of the number of nuclei of group $L$ (Li, Be, and B). The ratio of the flux of nuclei of group $L$ to the flux of nuclei of group $M$, according to $^{4,7}$, is $N_L/N_M \lesssim 0.1$. At the same time, according to $^{8}$, $N_L/N_M \simeq 0.4 \div 0.5$, while according to $^{9}$ $N_L/N_M < 0.25$, and according to the data of the latest works $^{71,72}$, respectively, $N_L/N_M \simeq 0.35$ and $N_L/N_M \sim 0.7$. Below, the value $N_L/N_M \simeq 0.1$ will mainly be used, since within the framework of the ideas being developed such a value of the ratio $N_L/N_M$ is harder to explain than the ratio $N_L/N_M>0.1$. In Table I the value $N_L/N_M \lesssim 0.1 \div 0.4$ is adopted.

Table II

Relative concentrations of protons and nuclei

$p$ — protons; $\alpha$ — helium nuclei;	$L,\,M,\,H$ — nuclei of groups	$L,\,M$ and $H$
$N_p/N_\alpha = 10$	$N_p/N_M = 1.9\cdot 10^3$	$N_p/N_H = 6.2\cdot 10^3$
$N_\alpha/N_M = 19$	$N_\alpha/N_H = 62$
$N_L/N_M \lesssim 0.1 \div 0.4$	$N_L/N_H \lesssim 0.3 \div 0.4$
$N_H/N_M = 0.31$	$N_M/N_H = 3.2$

Since the relative content of nuclei given above is for particles with a velocity greater than some, within known limits arbitrary, velocity that is the same for all particles, the data of Table II can be used directly to find the concentration of particles. Indeed, if $N_j$ is the concentration of particles of type $j$, and $I_j$ is the corresponding intensity*), then $N_j=\int I_j/v_j d\Omega$, where $v_j$ is the velocity of the particles. Hence it is clear that, for the same distribution of particles in velocities and angles, $N_j/N_k=I_j/I_k$. It is precisely for this reason that the ratio of intensities $I_L/I_M$ was replaced above by the ratio of concentrations $N_L/N_M$. The various ratios $N_j/N_k$ from the data of Table I are given in Table II.

Let us note that for an isotropic distribution of particles with velocity $v_j$,

$$ \begin{aligned} N_j &= \frac{4\pi}{v_j} I_j,\\ F_j &= \pi I_j = \frac{1}{4} v_j N_j, \end{aligned} \tag{1,1} $$

where $F_j=\int I_j\cos\theta\,d\Omega$ is the flux of particles of sort $j$ in a given direction through a unit area ($\theta$ is the angle between the normal to the area and the direction corresponding to the solid angle $d\Omega$; the integration is carried out over the hemisphere, i.e., within $0\leq \theta \leq \pi/2$).

*) The quantity $I\,d\Omega\,dS$ is the number of particles passing per unit time in the perpendicular direction through an area $dS$ within the solid angle $d\Omega$.

b) Energy spectrum

The energy spectrum of primary protons and nuclei in a small energy interval may be represented in the form*)

\[ I_j(E)=\frac{K_j}{E^\gamma},\qquad I_j(E>E_0)=\int_{E_0}^{\infty} I_j(E)\,dE =\frac{K_j}{(\gamma-1)E_0^{\gamma-1}}, \tag{1,2} \]

where \(K_j\) and \(\gamma\) are certain constants, and \(E\) and \(E_0\) are the total energy of the proton or of one nucleon in the nucleus (thus, the total energy of a nucleus with atomic weight \(A\) is equal to \(AE\)).

For protons with kinetic energy \(E_k=E-Mc^2\) in the interval \(0.5\leq E_k\leq 10\) Bev, according to\(^4\)

\[ I_p(E_k>E_{k0})=\frac{3800}{(1+E_{k0})}\, \text{m}^{-2}\,\text{sec}^{-1}\,\text{steradian}^{-1}, \tag{1,3} \]

where \(E_k\) is measured in Bev. Other expressions for the spectrum are also known in the literature and, in particular, the value \(\gamma=1.9\) is given.

For nuclei in the interval \(0.35\leq E_k\leq 10\div 20\) Bev/nucleon we have:

\[ \left. \begin{aligned} I_j(E_k>E_{k0})&=\frac{K_j}{(1+E_{k0})^{1.2}}\, \text{m}^{-2}\,\text{sec}^{-1}\,\text{steradian}^{-1},\\ K_\alpha&=380,\qquad K_M=20,\qquad K_H=6. \end{aligned} \right\} \tag{1,4} \]

Thus, for protons and nuclei \(\gamma=1.9\div 2.2\) (in the interval \(0.35\div 0.5\leq E_k\leq 10\div 20\) Bev/nucleon; the difference in the values of \(\gamma\) for protons and nuclei can hardly be ascribed real significance**). With increasing energy the exponent \(\gamma\) slowly increases and is equal to \(\gamma\simeq 2.5\) for \(5\cdot 10^{10}<E<10^{12}\) ev/nucleon and \(\gamma=2.7\div 3\) for \(10^{13}<E<10^{18}\) ev (as already noted, for \(E>10^{13}\) ev the composition of the primary component is unknown). Let us note that in\(^5\) over the entire energy interval \(E_k>2\) Bev a spectrum of the type (1,2) with \(\gamma=2.5\) is used. For the further exposition these data are sufficient, and we shall confine ourselves to only a few more values.

*) Obviously, \(I(E)\,dE\) is the intensity for particles with energy between \(E\) and \(E+dE\), and \(I(E>E_0)\) is the intensity for particles with total energy \(E>E_0\). For the proton \(E=E_k+Mc^2=E_k+0.938\) Bev, where \(E_k\) is the kinetic energy.

) Note added in proof.** According to data\({}^{73}\)

\[ I_\alpha(E_k>E_{k0})=\frac{431}{(1+E_{k0})^{1.4\pm0.2}}, \]

whereas in the abstract\({}^{74}\) the spectra are given as:

\[ I_p(E>E_0)=\frac{4000}{E_0^{1.15}},\qquad I_\alpha=\frac{460}{E_0^{1.6}},\qquad I_M=\frac{40}{E_0^{2.0}},\qquad \text{and}\qquad I_H=\frac{30}{E_0^{2.3}}, \]

where \(E_0\) is the total energy per nucleon (the discussion concerns the region of the spectrum sensitive to geomagnetic effects). If such a difference in the spectra of protons and nuclei, contradicting other data, were in fact to exist, and if it also applied to the region of high energies, it would be significant for detailing the theory of the origin of cosmic rays and, in particular, the acceleration mechanism and the character of nuclear collisions in the interstellar medium. Lacking more detailed information on this point, we confine ourselves to the observation that in the scheme developed below the proximity of the spectra of protons and nuclei is, to some extent, not obligatory. Therefore, taking account of the difference in the spectra will require introducing only secondary changes into the theory.

At low energies \((E_k < 0.35 \div 0.5\ \text{BeV/nucleon})\) the spectrum of primary particles has been studied quite inadequately because of the so-called high-latitude cutoff of the spectrum. Quite recently it was believed that such a cutoff occurs approximately at geomagnetic latitude \(58^\circ\), which for protons corresponds to an energy \(E_k \simeq 5.6 \cdot 10^8\) (momentum \(cp = 1.2 \cdot 10^9\ \text{eV}\)*) . However, it has recently been found\({}^{11}\) that the high-latitude cutoff differs in different years and, apparently, is absent during the minimum of solar activity. A number of considerations (see, for example,\({}^{12}\)), as well as experimental data\({}^{13}\), indicate that the high-latitude cutoff is due to magnetic fields acting within the solar system, and not to effects in the sources of cosmic rays themselves or along the path of the rays in interstellar space. Various conjectures\({}^{12,14,15,16}\) are made concerning the character and nature of the corresponding magnetic fields. This question has not yet been clarified. It is undoubtedly closely connected with the problem of variations in the intensity of cosmic rays\({}^{2,14-16}\), the study of cosmic rays of solar origin (see\({}^{5,16}\)), and the determination of an effective geomagnetic equator for cosmic radiation\({}^{17}\). Below, however, we shall not touch on this whole range of questions, which are of great interest for the physics of the Sun and of the whole solar system, but have only an indirect relation to the problem of the origin of cosmic rays (see the introduction and the remarks in Section 4 g).

Fig. 1.

At the same time, taking into account the existence of the high-latitude cutoff and, in general, the influence of “local” magnetic fields, for the present it is necessary to discuss the properties of the primary component only for momentum \(cp > 1 \div 1.5 \cdot 10^9\ \text{eV}\).

If one starts from (1.3), then \(I(E_k > 10^9\ \text{eV}) = 0.19\ \text{cm}^{-2}\ \text{sec}^{-1}\ \text{steradian}^{-1}\). According to data\({}^{10,11}\), for the intensity of all particles with allowance for albedo \(I(E_k > 10^9\ \text{eV}) \simeq 0.3\ \text{particles}/\text{cm}^2\,\text{sec}\,\text{steradian}\), and without allowance for albedo \(I \simeq 0.5\). Other close values are also found in the literature. The integral spectrum of primary cosmic rays over a fairly wide energy interval is presented in Fig. 1 (on the abscissa is plotted the kinetic energy of protons in BeV).

Taking for \(I\) the value \(0.3 \div 0.4\) and, since the discussion concerns practically protons, putting the velocity \(v = c = 3 \cdot 10^{10}\), for the concentration of particles with \(E_k > 10^9\ \text{eV}\) we obtain

\[ N(E_k > 10^9\ \text{eV}) \simeq \frac{4\pi}{c} I \simeq 1.5 \cdot 10^{-10}\ \text{cm}^{-3}. \tag{1.5} \]

Using this value, in the energy region \(10^9 < E_k < 2 \div 4 \cdot 10^{10}\ \text{eV/nucleon}\) one may approximately put

\[ N(E) \simeq \frac{0.3}{E^2}\ \text{eV}^{-1}\text{cm}^{-3}, \qquad N(E > E_0) \simeq \frac{0.3}{E_0}\ \text{cm}^{-3}, \tag{1.6} \]

where the total energy \(E = M_p c^2 + E_k\) is measured in eV.

* According to\({}^{10}\), the cutoff occurred at geomagnetic latitude \(\sim 54^\circ\), which corresponds to the value \(cp = 1.7 \cdot 10^9\ \text{eV}\).

For the energy density contained in cosmic radiation, we obtain*)

\[ w_{cr}=\int_{E_0}^{\infty} E_k N(E)\,dE \sim 1\ \text{eV}/\text{cm}^3 . \tag{1.7} \]

For comparison, let us note that the energy density of light radiation near the plane of the Galaxy is \(0.3\ \text{eV}/\text{cm}^3\), the density of the kinetic energy of turbulent motion in the interstellar medium is \(\rho u^2/2 \sim 1\ \text{eV}/\text{cm}^3\) \((\rho \sim 10^{-24}\ \text{g}/\text{cm}^3,\ u \sim 10^6\ \text{cm}/\text{s})\), and the density of magnetic energy is \(H^2/8\pi \sim 1\ \text{eV}/\text{cm}^3\) \((H \sim 3\cdot 10^6 \div 10^{-5}\ \text{oersted})\).

Thus, the energy density of cosmic rays is of the same order as the density of other forms of energy in the Galaxy.**)

c) Isotropy. Number of electrons

Among the essential features of the primary cosmic radiation at the Earth, one should note its isotropy. In the energy range up to \(10^{15}\ \text{eV}\), according to \(^{18}\), the degree of anisotropy is \(\delta < 10^{-3}\), where

\[ \delta=\frac{\Delta F}{F}, \tag{1.8} \]

where \(F\) is the flux of cosmic rays in some direction, for example in the direction toward the center of the Galaxy, and \(\Delta F=F_{\max}-F_{\min}\) is the difference between the maximum and minimum fluxes in any directions, for example toward the galactic center and anticenter. For energies \(E \sim 10^{17}\ \text{eV}\), according to \(^{18}\), \(\delta < 0.1\), and, apparently, even for \(E \lesssim 10^{18}\) the asymmetry is \(\delta < 0.01 \div 0.02\) (see \(^{19}\); we shall not dwell on anisotropies of solar origin).

In conclusion we point out that electrons (positrons) and photons have not been detected among the primary cosmic rays. At the same time, according to \(^{20}\), for \(E>10^9\ \text{eV}\) in the primary flux of cosmic rays, electrons make up no more than \(0.6\%\). This means that

\[ I_e(E>10^9\ \text{eV}) < (0.3 \div 0.4)\,6\cdot 10^{-3} \simeq 2\cdot 10^{-3}\ \text{cm}^{-2}\ \text{s}^{-1}\ \text{steradian}^{-1} \]

and

\[ N_e(E>10^9\ \text{eV})=\frac{4\pi}{c} I_e < 1\cdot 10^{-12}\ \text{cm}^{-3}. \tag{1.9} \]

As will become clear from what follows (see Section 26), for electrons with \(E>10^9\ \text{eV}\) the value \(N_e \sim 10^{-12}\ \text{cm}^{-3}\) is, in a certain sense, quite large. It is therefore very important to substantially lower the attained accuracy limit in determining the flux of the primary electron component. In addition, it is no less important to try to detect softer electrons, which should be significantly more numerous (let us note, for example, that \(N_e(E>3\cdot 10^8\ \text{eV}) \simeq 10N_e(E>10^9\ \text{eV})\); see Section 26). During periods of minimum solar activity, when the geomagnetic cutoff threshold decreases, such measurements are apparently entirely possible. Unfortunately, the deep minimum of solar activity that occurred in 1954, as far as we know, was not used for this purpose.

*) Taking into account that the spectrum (1.6) is suitable only up to values \(E=1 \div 2\cdot 10^{10}\ \text{eV}\), the energy density \(w_{cr}\) can be estimated as

\[ \int \frac{0.3\,E_k}{E^2}\,dE \]

over the range \(2\cdot 10^9 \le E \le 2\cdot 10^{10}\), which also leads to (1.7). The calculations can be carried out more accurately, but there is no need for this here.

**) In ionized regions of the interstellar medium, where the temperature is \(T \sim 10^4\) degrees and the concentration of gas particles is \(n \sim 1\ \text{cm}^{-3}\), the energy density \(nkT\) is also of the order of \(1\ \text{eV}/\text{cm}^3\).

2. THE MAGNETIC-BREMSSTRAHLUNG NATURE OF COSMIC RADIO EMISSION AND THE DISTRIBUTION OF COSMIC RAYS IN THE GALAXY

It is quite obvious that even very complete information about primary cosmic rays near the Earth is still insufficient for solving the question of their sources and distribution in space. It is therefore difficult to overestimate the importance of data on the electronic component of cosmic rays far from the Earth, which can be obtained on the basis of investigation of cosmic radio emission.

a) The nature of nonthermal cosmic radio emission

Cosmic radio emission, which possesses a continuous spectrum, consists of radiation from discrete sources (radio nebulae) and of “general” cosmic radio emission, whose intensity depends only weakly on galactic coordinates. The general radiation is mainly of galactic origin and, in turn, is composed of thermal and nonthermal components. The first of these components is the thermal radio emission of interstellar gas, which, like this gas itself, is concentrated mainly in the galactic plane. In this case, of course, the effective temperature*) of the thermal radiation cannot exceed the kinetic temperature of the gas itself, which even in completely ionized regions is not higher than \(10\,000^\circ\). If, however, one is speaking of directions toward the galactic anticenter or pole, then in the meter range the effective temperature \(T_{\mathrm{eff}}\) of the thermal radiation is still considerably smaller, since the optical thickness of the gas in these directions is small. For metagalactic radiation \(T_{\mathrm{eff}} < 1000^\circ\). At the same time, for the general cosmic radiation in the region of the galactic center, \(T_{\mathrm{eff}}\) reaches values of \(2.7 \cdot 10^5\) degrees for the wave \(\lambda = 16.3\ \mathrm{m}\), about \(10^6\) degrees for \(\lambda = 32.8\ \mathrm{m}\), and even \(10^8\) degrees for \(\lambda = 140\ \mathrm{m}\); in the region of the galactic pole, at the wavelength \(16.3\ \mathrm{m}\), \(T_{\mathrm{eff}} \simeq 7.5 \cdot 10^4\) (the latest summary of radio-astronomical data \(^{21}\)).

It is clear from this that there exists a certain nonthermal galactic radio emission, which plays the dominant role in the range of meter and longer waves. From an analysis of the corresponding radio isophotes one may conclude \(^{22,21}\) that the sources of nonthermal galactic radio emission form a quasi-spherical subsystem with radius \(R \sim 3 \div 5 \cdot 10^{22}\ \mathrm{cm}\). Approximately the same spatial distribution is possessed by diffuse interstellar gas with a very small concentration \(n \lesssim 0.1\ \mathrm{cm}^{-3}\) (see \(^{23}\)). The existence in the Galaxy of the indicated distribution of radio-emission sources is now generally recognized, especially after an analogous “radio corona” was discovered \(^{24}\) in the nebula M31 in Andromeda, which is related in structure to our Galaxy.

What is the nature of the nonthermal galactic radio emission? For a long time attempts were made to suppose that this radiation is generated in the envelopes of an enormous number of hypothetical radio stars, which possess unusual properties and are not visible in the optical part of the spectrum \(^{22,25}\). Such an assumption, which has always seemed to us improbable, now appears completely untenable after it was established that all the discrete sources of cosmic radio emission studied from this point of view are nebulae, not stars. At the same time

*) The effective temperature \(T_{\mathrm{eff}}\) of radiation of any nature is, by definition, equal to the temperature of a black body whose radiation intensity is equal to the intensity of the radiation under consideration in the given, sufficiently narrow, frequency interval.

all the features of nonthermal cosmic radio emission are explained on the basis of the “magnetic bremsstrahlung hypothesis,” according to which this radiation is radiation emitted by relativistic electrons moving in interstellar magnetic fields \(^{26-28, 1-3, 12}\).

As is known, in a magnetic field \(H\) an electron moves along a helical line with angular frequency

\[ \omega_H=\frac{eH}{mc}\frac{mc^2}{E}, \tag{2,1} \]

where \(E\) is the total energy of the electron, and \(e\) and \(m\) are its charge and mass. In the ultrarelativistic case, which alone is of interest here, \(E\gg mc^2\), and the electron emits electromagnetic waves almost exclusively in the direction of its instantaneous velocity, within a narrow cone with angle

\[ \alpha\sim \frac{mc^2}{E}\ll 1. \]

Therefore, if the electron moves in a circle, an observer situated in the plane of the orbit will record, one after another at intervals \(\tau=\dfrac{2\pi}{\omega_H}\), radiation pulses with duration

\[ \Delta t\sim \frac{r\alpha}{c}\left(\frac{mc^2}{E}\right)^2 \sim \left(\frac{mc}{eH}\right)\left(\frac{mc^2}{E}\right)^2, \tag{2,2} \]

where \(r=\dfrac{c}{\omega_H}\) is the radius of the orbit, and the factor \(\left(\dfrac{mc^2}{E}\right)^2\) appears because of the Doppler effect. In the case, however, when the electron moves along a helical line, and the angle \(\theta\) between the field and the velocity satisfies the condition

\[ \theta\gg \alpha\sim \frac{mc^2}{E}, \]

we have

\[ \Delta t\sim \left(\frac{mc}{eH_\perp}\right)\left(\frac{mc^2}{E}\right)^2, \]

where \(H_\perp\) is the component of the magnetic field perpendicular to the direction of motion, i.e. to the velocity of the electron. Accordingly, the radiation spectrum consists of overtones of the frequency \(\omega_H\), but is practically continuous, with the maximum in the spectrum corresponding to the frequency

\[ \omega_{\max}\sim \frac{1}{\Delta t}\sim \left(\frac{eH_\perp}{mc}\right)\left(\frac{E}{mc^2}\right)^2 . \]

Detailed calculations show (see, for example, \(^{29}\)) that the energy emitted by an electron per second in the frequency interval \(d\nu=\dfrac{d\omega}{2\pi}\) is equal to \(P(\nu,E)\,d\nu\), where

\[ \left. \begin{aligned} P(\nu,E)=P(\nu) &=16\,\frac{e^3H_\perp}{mc^2}\, p\!\left(\frac{\omega}{\omega_n}\right) =16\,\frac{e^2}{c}\,\omega_{H_\perp} \left(\frac{\omega}{2\omega_m}\right)^{1/3}Y(u),\\ \omega_{H_\perp} &=\frac{eH_\perp}{mc}\frac{mc^2}{E},\qquad \omega_m=\frac{eH_\perp}{mc}\left(\frac{E}{mc^2}\right)^2,\qquad u=\left(\frac{\omega}{2\omega_m}\right)^{2/3},\\ Y(u)&=u^{-1/2}p(2u^{3/2}). \end{aligned} \right\} \tag{2,3} \]

The values of the functions \(Y(u)\) and \(p\!\left(\dfrac{\omega}{\omega_m}\right)\) are given in Table III; in limiting cases:

\[ \left. \begin{aligned} &\frac{\omega}{\omega_m}\ll 1;\qquad Y=0.256,\\ &p\!\left(\frac{\omega}{\omega_m}\right) =0.256\left(\frac{\omega}{2\omega_m}\right)^{1/3},\\ &\frac{\omega}{\omega_m}\gg 1;\qquad Y(u)=\frac{(2\pi)^{1/2}}{16}\,u^{1/4} \exp\!\left[-\frac{4}{3}u^{3/2}\right],\\ &p\!\left(\frac{\omega}{\omega_m}\right) =\frac{1}{16}\left(\pi\frac{\omega}{\omega_m}\right)^{1/2} \exp\!\left[-\frac{2}{3}\frac{\omega}{\omega_m}\right]. \end{aligned} \right\} \tag{2.4} \]

Table III

Values of the functions \(Y(u)\) and \(p\!\left(\dfrac{\omega}{\omega_m}\right)\)

\(u\)	\(\dfrac{\omega}{\omega_m}=2u^{3/2}\)	\(Y(u)\)	\(p\!\left(\dfrac{\omega}{\omega_m}\right)\)	\(u\)	\(\dfrac{\omega}{\omega_m}=2u^{3/2}\)	\(Y(u)\)	\(p\!\left(\dfrac{\omega}{\omega_m}\right)\)
0.0	0.0000	0.256	0.0000	2.2	6.5263	0.00281	0.00416
0.2	0.1789	0.204	0.0912	2.4	7.4361	0.00154	0.00238
0.4	0.5060	0.1562	0.0986	2.6	8.3847	0.00083	0.00134
0.6	0.9295	0.1150	0.0891	2.8	9.3706	0.00044	0.00074
0.8	1.4311	0.0813	0.0727	3.0	10.3923	0.00023	0.00040
1.0	2.0000	0.0554	0.0554	3.5	13.0958	0.000037	0.000069
1.2	2.6291	0.0364	0.0399	4.0	16.0000	0.0000055	0.000011
1.4	3.3130	0.0232	0.0274
1.6	4.0477	0.0143	0.0181
1.8	4.8299	0.00855	0.0115
2.0	5.6568	0.00497	0.00703

The graph of the function \(p\!\left(\dfrac{\omega}{\omega_m}\right)\) is presented in Fig. 2. At \(\omega \simeq 0.5\omega_m\) the function \(p\!\left(\dfrac{\omega}{\omega_m}\right)\) is maximal and is equal to 0.10. Thus, at the maximum

Fig. 2.

\[ \left. \begin{aligned} P(\nu_{\max})&=1.6\,\frac{e^3H_{\perp}}{mc^2} =2.16\cdot 10^{-22}H_{\perp}\ \text{erg/sec hertz},\\ \nu_{\max}&=0.5\,\frac{\omega_m}{2\pi} =1.4\cdot 10^{6}H_{\perp}\left(\frac{E_m}{mc^2}\right)^2\ \text{hertz}. \end{aligned} \right\} \tag{2.5} \]

The intensity*) of the bremsstrahlung magnetic radiation observed on Earth is equal to

\[ I_\nu=\frac{1}{4\pi}\int P(\nu,E)\,N_e(E,\mathbf r)\,dE\,d\mathbf r, \tag{2.6} \]

where \(N_e(E,\mathbf r)\) is the differential spectrum of electrons at the point \(\mathbf r\); the integration is carried out along the line of sight, and it is assumed that, owing to the chaotic character of the orientation of the field \(\mathbf H\) relative to the line of sight, the radiation is on the average isotropic (whence the factor \(\frac{1}{4\pi}\) in (2.6)). The assumptions made are such that formula (2.6) is valid only to within a factor of order unity.

In (2.6) the absorption of radiation in the interstellar medium has not been taken into account. Moreover, the radiation is regarded as occurring in vacuum, which is lawful only if

\[ |1-n(\nu)|\ll \left(\frac{mc^2}{E}\right)^2, \]

where \(n(\nu)\) is the refractive index of the medium (in the present case, of the interstellar gas) in which the electron moves\(^1\). From the point of view of the further exposition these restrictions are not essential.

b) The electronic component of cosmic rays in the Galaxy

If we are interested in the radiation at some frequency \(\nu\), then the minimum necessary number of electrons can be determined by assuming that all electrons have the energy \(E_m\), connected with the frequency by the second of relations (2.5). In this case the radiation is maximal and

\[ I_{\nu_{\max}}^{\max}= \frac{P(\nu_{\max})}{4\pi}\,N_e R = 1.7\cdot 10^{-23} H_\perp N_e R\, \frac{\text{erg}}{\text{cm}^2\,\text{sec}\,\text{cps}\,\text{steradian}}, \tag{2.7} \]

where \(N_e\) and \(H_\perp\) are certain mean values of the electron concentration and of the field \(H_\perp\) along the line of sight**), and \(R\) is the size of the radiating region in the given direction.

In the case of thermal radiation with effective temperature \(T_{\mathrm{eff}}\):

\[ I_\nu= \frac{2k\nu^2}{c^2}T_{\mathrm{eff}} = \frac{2kT_{\mathrm{eff}}}{\lambda^2} = \frac{2.76\cdot 10^{-16}T_{\mathrm{eff}}}{\lambda^2}\, \frac{\text{erg}}{\text{cm}^2\,\text{sec}\,\text{cps}\,\text{steradian}}. \tag{2.8} \]

Experimentally, for \(\lambda=16.3\ \text{m}\) in the direction of the galactic pole \(T_{\mathrm{eff}}=7.5\cdot 10^4\), and therefore from (2.7) and (2.8) it follows that

\[ N_eH_\perp R= \frac{2kT_{\mathrm{eff}}}{\lambda^2\cdot 1.7\cdot 10^{-23}} = 4.6\cdot 10^5. \tag{2.9} \]

On the other hand, various data and considerations leave no doubt that in the interstellar medium there are magnetic fields with intensity \(H\sim 3\cdot 10^{-6}\div 10^{-5}\) oersted (see, for example, \(^{30}\)). Taking for the galactic “corona” (see above) the values \(H_\perp\sim 10^{-5}\) and \(R\sim 5\cdot 10^{22}\), we see that

\[ N_e\sim 10^{-12}\ \text{cm}^{-3}, \tag{2.10} \]

*) The radiation intensity \(I_\nu\) is the flux of radiation, referred to unit solid angle and spectral interval, through unit area perpendicular to the direction of observation (the direction of propagation of the radiation).

**) In the present case \(N_e(E)=N_e\delta(E-E_m)\), where \(E_m\) is connected with \(\nu\) by the second of relations (2.5).

with the electron energy

\[ E_m=\sqrt{\frac{\nu_{\max}}{1.4\cdot 10^6 H_\perp}}\,mc^2\simeq 6\cdot 10^8\ \text{eV}. \]

This result obviously does not contradict the available experimental data on the number of electrons near the Earth (see (1.9)). Thus, the assumption of the magnetobremsstrahlung nature of the nonthermal galactic radio emission is in agreement with the requirements that, on quite different grounds, may be imposed on the values of the field \(H\) and the electron concentration \(N_e\).

The monochromatic spectrum used and formula (2.7), however, may serve only for orientation. Let us therefore proceed to consider radio-emitting electrons possessing the spectrum

\[ N_e(E)=\frac{K}{E^\gamma}. \tag{2.11} \]

In this case, from (2.3) and (2.6) we obtain (assuming that along the ray of vision the density \(N_e(E)\) is constant):

\[ I_\nu=\frac{2k\nu^2}{c^2}\,T_{\text{eff}} =\frac{R}{4\pi}\int_0^\infty P(\nu,E)N_e(E)\,dE = \]

\[ =\frac{3}{\pi}(2\pi)^{\frac{1-\gamma}{2}} \left(\frac{e^3H_\perp}{mc^2}\right) \left(\frac{2eH_\perp}{m^3c^5}\right)^{\frac{\gamma-1}{2}} U(\gamma)K R \nu^{\frac{1-\gamma}{2}}\simeq \]

\[ \simeq 1.3\cdot 10^{-22}(2.8\cdot 10^8)^{\frac{\gamma-1}{2}} U(\gamma)K H_\perp^{\frac{\gamma+1}{2}}R\lambda^{\frac{\gamma-1}{2}} \frac{\text{erg}}{\text{cm}^2\,\text{sec}\,\text{hertz}\,\text{steradian}}, \tag{2.12} \]

where

\[ \left. \begin{gathered} U(\gamma)=\int_0^\infty Y(u)u^{\frac{3\gamma-5}{4}}\,du,\qquad U(1)=0.37;\qquad U\!\left(\frac{5}{3}\right)=0.163,\\ U(2)=0.125;\qquad U(3)=\frac{\pi}{36}=0.087;\qquad U(7)=\frac{7\pi}{144}=0.153. \end{gathered} \right\} \tag{2.13} \]

The most important feature of formula (2.12), which must be emphasized here, is the independence of the spectral distribution of the radiation from the magnetic-field strength \(H_\perp\). Indeed, the intensity is proportional to \(H_\perp^{\frac{\gamma+1}{2}}\), but its dependence on frequency \(\nu\) is determined only by the factor \(\nu^{\frac{1-\gamma}{2}}\). Thus, knowing from experiment the dependence \(I_\nu(\nu)\), one can immediately find the exponent \(\gamma\) in the spectrum (2.11).

Until recently it was believed that, within the accuracy achieved, for nonthermal galactic radiation one could put \(I_\nu=\mathrm{const}\,\nu^{-1}\), i.e. \(T_{\text{eff}}\sim \nu^{-3}\) (see (2.8)). Hence it follows that

\[ \gamma=3. \tag{2.14} \]

According to \(^{21}\) \(I_\nu=\mathrm{const}\cdot \nu^{-\alpha}\), where \(\alpha=0.82\) in the range \(18.3\lesssim \nu\lesssim 300\ \text{Mc}\) \((1\lesssim \lambda\lesssim 16.3\ \text{m})\) and \(\alpha=0.86\) in the range \(2.1<\nu<18.3\ \text{Mc}\) \((16.3<\lambda<140\ \text{m})\). For \(\alpha=0.82\)

\[ \gamma=1+2\alpha=2.64. \tag{2.15} \]

To determine the magnitude \(I_\nu\) itself, we use the value already given above, \(T_{\mathrm{eff}}=7.5\cdot 10^{4}\) at \(\nu=18.3\ \mathrm{MHz}\) (\(\lambda=16.3\ \mathrm{m}\)). Then, taking (2.12)—(2.13) into account, for \(\gamma=3\) and \(\gamma=2.64\) we have:

\[ \left. \begin{aligned} I_\nu &= 7.8\cdot 10^{-18}=c^{(a)}\nu^{-1} =1.4\cdot 10^{-10}\nu^{-1} \\ &=0.95\cdot 10^{-4}\,K^{(a)}H_\perp^{2}R\nu^{-1},\\ I_\nu &= 7.8\cdot 10^{-18}=c^{(b)}\nu^{-0.82} =7\cdot 10^{-12}\nu^{-0.82} \\ &=4\cdot 10^{-8}\,K^{(b)}H_\perp^{1.82}R\nu^{-0.82}. \end{aligned} \right\} \tag{2.16} \]

Hence

\[ K^{(a)}H_\perp^{2}R \simeq 1.5\cdot 10^{-6}, \qquad K^{(b)}H_\perp^{1.82}R \simeq 2\cdot 10^{-4} \tag{2.17} \]

and, for \(H_\perp\sim 10^{-5}\) and \(R\sim 5\cdot 10^{22}\), we obtain \(K^{(a)}\sim 3\cdot 10^{-19}\) and \(K^{(b)}\sim 5\cdot 10^{-18}\). The adopted values of \(H_\perp\) and \(R\) are probably the largest possible ones. Therefore, in order to have a known margin, we shall use values of \(K^{(a)}\) and \(K^{(b)}\) five times larger. Thus the electron spectrum has the form:

\[ N_e^{(a)}(E)\simeq \frac{5\cdot 10^{5}}{E^{3}}\ \mathrm{eV}^{-1}\ \mathrm{cm}^{-3}, \qquad N_e^{(a)}(E>E_0)=\int_{E_0}^{\infty}N_e^{(a)}\,dE\simeq \]

\[ \simeq \frac{2.5\cdot 10^{5}}{E_0^{2}}\ \mathrm{cm}^{-3}, \tag{2.18a} \]

\[ N_e^{(b)}(E)\simeq \frac{5\cdot 10^{3}}{E^{2.64}}\ \mathrm{eV}^{-1}\ \mathrm{cm}^{-3}, \qquad N_e^{(b)}(E>E_0)\simeq \frac{3\cdot 10^{3}}{E_0^{1.64}}\ \mathrm{cm}^{-3}, \tag{2.18b} \]

where the energies \(E\) and \(E_0\) are measured in eV (in (2.16)—(2.17) CGS units were used).

According to (2.18),

\[ \left. \begin{aligned} N_e^{(a)}(E>10^{9}\ \mathrm{eV}) &\simeq 2.5\cdot 10^{-13}\ \mathrm{cm}^{-3},\\ N_e^{(b)}(E>10^{9}\ \mathrm{eV}) &\simeq 5.5\cdot 10^{-13}\ \mathrm{cm}^{-3}, \end{aligned} \right\} \tag{2.19} \]

which is not inconsistent with the data on cosmic rays near the Earth (see (1.9)), especially if one takes into account the possibility of reducing the values (2.19) by several times. At the same time, the concentration

\[ N_e^{(a)}(E>10^{8}\ \mathrm{eV})\simeq N_e^{(b)}(E>10^{8}\ \mathrm{eV})\simeq 2.5\cdot 10^{-11}\ \mathrm{cm}^{-3}, \]

i.e. only a few times smaller than the concentration of all primary cosmic particles reaching the Earth. For \(\gamma=3\), the greatest contribution to the radiation of frequency \(\nu\) is made by electrons with energy \(E\) corresponding to the value

\[ u=\left(\frac{\omega}{2\omega_m}\right)^{2/3} =\left(\frac{\nu}{\dfrac{eH_\perp}{\pi mc}\left(\dfrac{E}{mc^{2}}\right)^{2}}\right)^{2/3} \simeq 0.6, \]

i.e. with energy

\[ E=\sqrt{\frac{\nu}{2.6\cdot 10^{6}H_\perp}}\ mc^{2}. \]

Radiation from particles with \(u>2\) and \(u<0.1\) is already insignificant. Hence it follows that, for \(H_\perp\sim 10^{-5}\), the radiation at the wavelength \(16.3\ \mathrm{m}\) is determined mainly by electrons with energy \(E\simeq 4\cdot 10^{8}\), while the contribution of particles with \(E<2\cdot 10^{8}\ \mathrm{eV}\) and \(E>2\cdot 10^{9}\ \mathrm{eV}\) is already of little significance. At the wavelength \(1.5\ \mathrm{m}\) the greatest role is played

electrons with energy \(E \simeq 1.2\cdot 10^9\). Thus, the spectra (2.18) refer to the energy interval \(10^8 < E < 5\cdot 10^9\) eV, and variant b), for which \(\gamma = 2.64\), apparently corresponds more closely to the radio-astronomical data. Further, the radio isophotes indicate that cosmic electrons, in contrast to the stellar population of the Galaxy, are not concentrated in the galactic plane. On the contrary, as was already mentioned above, cosmic electrons form around the galactic center a gigantic “corona” with a radius up to \(5\cdot 10^{22}\) cm. In this case the surfaces of equal intensity of nonthermal radio emission at small distances from the center are close to ellipses with an axial ratio \(1:2.5\) (naturally, the major axis lies in the galactic plane); with distance from the center, the surfaces of equal intensity approach spherical ones. The density of radiation sources at a distance of \(6\cdot 10^{22}\) from the center is 25 times, and at a distance of \(\sim 9\cdot 10^{22}\) already 100 times, smaller than at the center\(^{21}\). This does not yet mean that the concentration of cosmic electrons falls by the same factor. The point is that the radiation intensity, for example, at \(\gamma = 3\) is proportional to \(H_\perp^2\), and therefore the decrease in intensity is partly also connected with a decrease in the strength of the magnetic field. A certain role may also be played by a lowering of the energy of the electrons as they move away from the galactic plane. At the same time, everything cannot be reduced to a weakening of the field, since a decrease in the field strength automatically leads to a fall in the concentration of cosmic particles, which move around the lines of force along a helical trajectory with radius

\[ r=\frac{E}{300 ZH}\sin\theta, \tag{2.20} \]

where the total energy \(E\) of a relativistic particle with charge \(Z\) is measured in eV, the radius \(r\) in centimeters, and \(\theta\) is the angle between the velocity (the particle trajectory) and the field (line of force). Even for \(H=3\cdot 10^{-6}\), \(\sin\theta=1\), and \(E=10^{10}\), the radius \(r\sim 10^{13}\), i.e. is negligibly small in comparison with the scales over which the field changes. For the latter reason cosmic electrons (the same applies, of course, to other particles of not too high energy) practically cannot leave the region occupied by the field.

The question of the character of the “boundary” between the galactic “corona” and intergalactic space is not sufficiently clear and requires special consideration. It is difficult, however, to doubt that such a boundary exists in some sense. Indeed, in intergalactic space the density of matter is \(\rho \simeq 10^{-29}\div 10^{-30}\ \mathrm{g/cm^3}\), while in the galactic “corona” \(\rho \simeq 10^{-25}\div 10^{-26}\ \mathrm{g/cm^3}\), which corresponds to a hydrogen concentration \(n=0.1\div 0.01\ \mathrm{cm^{-3}}\)*).

The strength of the magnetic field in intergalactic space can be estimated from the equality

\[ \frac{\rho u^2}{2}\sim \frac{H^2}{8\pi}, \]

where \(\rho\) is the density of the medium and \(u\) its velocity (as is known, such an estimate gives rather an upper limit for \(H\)). Taking \(\rho\sim 10^{-29}\div 10^{-30}\) and \(u\sim 3\cdot 10^7\) (the relative velocity of galaxies), we obtain \(H\sim (1\div 3)\cdot 10^{-7}\) oersted, whereas in the Galaxy \(H\sim 3\cdot 10^{-6}\div 3\cdot 10^{-5}\). Thus, undoubtedly, the properties of the galactic “corona” and of the intergalactic medium differ strongly from one another, and the question can only be how much

*) According to \(^{23,31}\) one may consider that in the “corona” near the plane of the Galaxy \(n\sim 0.1\), and at a distance \(R\sim 5\cdot 10^{22}\) already \(n\approx 0.01\). As for the work\(^{32}\), according to which in the “corona” \(n\sim 5\cdot 10^{-4}\ \mathrm{cm^{-3}}\), it meets serious objections\(^{31}\).

there is a smooth transition between them. Data on the radio brightness of our Galaxy and of the galaxy M31 give the impression that this transition is rather sharp, so that in the region with radius \(R>5\cdot10^{22}\ \mathrm{cm}\) from the center of the Galaxy there are already few cosmic particles. The existence of a boundary of the galactic “corona” is also supported by the well-known considerations concerning the need to retain cosmic rays in the Galaxy. Finally, the idea of galaxies as, in essence, systems of the type of a “tangle of lines of force,” closed with respect to the magnetic field, seems natural for other reasons as well.

In view of what has been said, we shall assume that, for cosmic rays, the Galaxy represents, as it were, a reservoir with radius \(R=3\div5\cdot10^{22}\ \mathrm{cm}\) and with practically completely reflecting walls*). The volume of the system is

\[ V\sim \frac{4\pi}{3}R^3\sim 1\div5\cdot10^{68}\ \mathrm{cm}^3. \tag{2,21} \]

The total energy of the cosmic rays in the Galaxy is equal to the volume multiplied by the energy density \(w_{cr}\), which near the Earth, according to (1,7), is of the order of \(1\ \mathrm{eV/cm^3}\). Taking into account that at the periphery of the Galaxy the density \(w_{cr}\) probably decreases somewhat, for the total energy we shall adopt the value

\[ W_{cr}=w_{cr}V\sim10^{67}\div10^{68}\ \mathrm{eV}\sim10^{55}\div10^{56}\ \mathrm{erg}. \tag{2,22} \]

Let us note that the energy associated with cosmic electrons is approximately 100 times smaller than the total energy of cosmic rays. Indeed, according to (2,18), the energy density of the electrons is

\[ w_e=\int_{E_0}^{\infty} EN_e(E)\,dE =\frac{5\cdot10^5}{E_0}+\frac{8\cdot10^2}{E_0^{0.64}}\ \mathrm{eV/cm^3}. \tag{2,23} \]

Hence, for \(E_0=10^8\ \mathrm{eV}\), the density \(w_e=5\cdot10^{-3}\div1\cdot10^{-2}\ \mathrm{eV/cm^3}\), in comparison with the total density \(w_{cr}\sim1\ \mathrm{eV/cm^3}\) (see (1,7)); to extend the spectrum (2,18) to energies smaller than \(E_0\sim10^8\ \mathrm{eV}\) is obviously unjustified, because of the need to take ionization losses into account (see Section 3b).

The picture of the distribution of cosmic particles in space to which we arrive on the basis of radio-astronomical data bears decisive witness in favor of a galactic origin of cosmic rays. It is true that these data do not refer to particles of very high energy, which might not have a strong galactic concentration. Against such an assumption, however, there is already the fact that a magnetic field \(\sim10^{-5}\), generally speaking, is sufficient to retain in the Galaxy not only the main mass of cosmic particles, but also particles with energy \(\sim10^{18}\ \mathrm{eV}\). Indeed, in the latter case the radius of curvature \(r\lesssim3\cdot10^{20}\ll(3\div5)\times10^{22}\ \mathrm{cm}\). If the particles of the greatest occurring energy are nuclei, which is quite possible, then the radius \(r\) is smaller by another one or two orders of magnitude. We shall discuss other arguments against a metagalactic origin of even part of the cosmic rays in Section 4g. The presence of cosmic electrons throughout the entire Galaxy also completely contradicts theories of the solar origin of cosmic rays; these theories, moreover, encounter other serious difficulties as well (see Section 4g).

*) Assuming that in the Galaxy \(H\sim10^{-5}\), while outside it, beyond a relatively sharp boundary, \(H=10^{-7}\), one may suppose that \(\sim1/100\) of all field lines leave the Galaxy. Hence, for the coefficient of transmission through the boundary we likewise obtain a value \(\sim1/100\), if one does not take into account the drift of particles caused by the inhomogeneity of the magnetic field (see Section 3b).

c) Cosmic (relativistic) electrons in the envelopes of supernovae

The significance of radio-astronomical data and of the magnetobremsstrahlung theory of the origin of cosmic radio emission is by no means limited to what has been said. Sources of cosmic radio emission, as already mentioned, include in particular various nebulae (radio nebulae). The radio emission of some of these nebulae is thermal in character; however, the radiation of the most powerful sources undoubtedly has a magnetobremsstrahlung nature. Indeed, the radiation flux \(F_\nu=\int I_\nu\,d\Omega\) from the sources Cassiopeia A and Taurus A (the Crab Nebula), for example, at wavelength \(\lambda=3\) m is respectively \(F_\nu=2.2\cdot10^{-19}\ \mathrm{erg}/\mathrm{cm}^{2}\ \mathrm{sec}\ \mathrm{cps}\) and \(F_\nu=1.8\cdot10^{-20}\ \mathrm{erg}/\mathrm{cm}^{2}\ \mathrm{sec}\ \mathrm{cps}\). This corresponds to values \(T_{\mathrm{eff}}\simeq3\cdot10^{7}\) degrees and \(T_{\mathrm{eff}}\simeq3\cdot10^{6}\) degrees, since (see (2,8))

\[ F_\nu=\int I_\nu\,d\Omega\simeq \frac{2kT_{\mathrm{eff}}}{\lambda^{3}}\,\Omega, \tag{2,24} \]

where \(\Omega\) is the solid angle under which the source is seen (for Cassiopeia A and Taurus A, \(\Omega\sim2\cdot10^{-6}\), which corresponds to angular dimensions of \(\sim5'\)). At the same time, the gas temperature in the nebulae does not exceed several thousand degrees, not to mention the fact that the radio-emission spectrum of both sources has nothing in common with the spectrum of thermal radiation. To explain the radio emission of Cassiopeia A and Taurus A, which are the envelopes in the Galaxy of the supernovae of 369 and 1054, by the combined radiation of some hypothetical “radio stars” would be simply absurd, and the only possibility consists in assuming the magnetobremsstrahlung nature of the radio emission of these sources \(^{26,27,21}\). In the case of the Crab Nebula (Taurus A), relativistic electrons are also responsible for the optical radiation with a continuous spectrum \(^{33}\).

One of the additional arguments in favor of this conclusion is provided by data on the polarization of the radiation. The point is that the radiation of relativistic electrons is strongly polarized (the electric vector lies in the plane of the orbit; for a spectrum of type (2,18a) the degree of polarization is \(76\%\)). Therefore, if the magnetic field in a nebula is quasihomogeneous over sufficiently large regions, one should expect polarization of the optical radiation, as was first indicated in \(^{34,35}\) (see also \(^{1}\)). Subsequently such polarization was indeed found both for the Crab Nebula and for the extragalactic source Virgo A (the nebula NGC 4486, or in another terminology M87), which possesses a continuous spectrum of visible radiation. No radiation mechanism other than magnetobremsstrahlung leads to the appearance of strong polarization, and thus the observation of polarization is very significant for confirming the concept being developed*). At the same time, the magnetobremsstrahlung nature of the radio emission of powerful discrete sources is established quite reliably even without invoking polarization measurements.

If the electron spectrum and the absolute value of the magnetic-field strength are taken to be the same at all points of the source, then the radiation flux received on the Earth from the source is equal to

\[ F_\nu=\int I_\nu\,d\Omega=\frac{V}{4\pi R^{2}}\int P(\nu,E)N_e(E)\,dE, \tag{2,25} \]

* In the case of magnetobremsstrahlung radio emission, polarization can also in principle be expected \(^{1,34-35}\). However, in this case the effect is masked owing to the rotation of the plane of polarization of the radiation in the interstellar medium \(^{3,37}\) and may not always be observed, and only with special equipment (preliminary data indicate the presence of weak polarization of the general Galactic radio emission at high Galactic latitudes).

where \(V\) is the volume of the source, \(R\) is the distance to it, and, for a power-law spectrum, \(\int PN_e\,dE\) is determined by expression (2.12)—(2.13).

Under the most favorable conditions (2, 5):

\[ F_{\nu_{\max}}^{\max} = P(\nu_{\max})\,\frac{N_eV}{4\pi R^2} = 1.7\cdot 10^{-23} H_\perp \frac{N_eV}{R^2} \frac{\mathrm{erg}}{\mathrm{cm}^2\,\mathrm{sec}\,\mathrm{cycle}}, \tag{2.26} \]

where \(N_e\) is the concentration of electrons which, by assumption, have energy

\[ E_m = \sqrt{\frac{\nu_{\max}}{1.4\cdot 10^6 H_\perp}}\;mc^2 . \tag{2.27} \]

The minimum energy of relativistic electrons in the source is

\[ W_{e,\min} = N_eE_mV = \frac{4\pi^{3/2}m^{5/2}F_\nu R^2\nu^{1/2}} {1.6e^{7/2}H_\perp^{1/2}} = 4.0\cdot 10^{13}\, \frac{F_\nu R^2\nu^{1/2}}{H_\perp^{1/2}}\ \mathrm{erg}, \tag{2.28} \]

where \(F_\nu\) is the flux at frequency \(\nu\), and formulas (2.3), (2.5), and (2.26) have been used. Knowing the spectrum of the radio emission of the source, one can, evidently, find the spectrum of electrons in the nebula, and then also the total energy of the electrons in it (see (2.25), (2.12)—(2.13)):

\[ \left. \begin{aligned} W_e &= V\int_{E_{e0}}^{E_{em}} EN_e(E)\,dE = \frac{KV}{2-\gamma} \left( \frac{1}{E_{em}^{\gamma-2}} - \frac{1}{E_{e0}^{\gamma-2}} \right), \quad \gamma\ne 2,\\[6pt] W_e &= KV\ln\frac{E_{em}}{E_{e0}}, \quad \gamma=2, \end{aligned} \right\} \tag{2.29} \]

where it is assumed that a spectrum of the type (2.11) extends from the energy \(E_{e0}\) to the energy \(E_{em}\), and outside this interval \(N_e(E)=0\) (for the insignificant further changes that must be introduced into formulas (2.12)—(2.13) in the case of a power-law spectrum bounded by the energies \(E_{e0}\) and \(E_{em}\), see \(^{36}\)).

The total number of electrons in the nebula is

\[ \left. \begin{aligned} N_t &= V\int_{E_{e0}}^{E_{em}} N_e(E)\,dV = \frac{KV}{1-\gamma} \left( \frac{1}{E_{em}^{\gamma-1}} - \frac{1}{E_{e0}^{\gamma-1}} \right), \quad \gamma\ne 1,\\[6pt] N_t &= KV\ln\left(\frac{E_{em}}{E_{e0}}\right), \quad \gamma=1. \end{aligned} \right\} \tag{2.30} \]

Some data \(^{39}\) for Cassiopeia A and Taurus A are given in Table IV, where \(R\) is the distance to the nebula and \(r_0\) is its radius. The minimum electron energy \(W_{e,\min}\) was calculated by formula (2.28) using the values of \(F_\nu\) and \(H_{\max}\) given in Table IV*). For Cassiopeia A the field

*) We note that, according to radio-astronomical data, the distance to Cassiopeia A is 5–6 times greater than the more reliable value of \(R\) indicated in Table IV, obtained by the optical method. Since the energy of the electrons in the nebula, determined by formula (2.28), is proportional to \(R^2\), the energy values given in Table IV may in this connection turn out to be underestimated by a factor of 20–40.

Table IV

Parameters of radio nebulae (shells of supernovae)

Source (radio nebula)	\(R,\ \mathrm{cm}\)	\(r_e,\ \mathrm{cm}\)	\(F_\nu,\ \dfrac{\mathrm{erg}}{\mathrm{cm}^2\,\mathrm{sec}\,\mathrm{Hz}}\), \(\lambda=3\ \mathrm{m}\)	\(W_{e,\min},\ \mathrm{erg}\)	\(W_e,\ \mathrm{erg}\)	\(H_{\max}\)	\(W_k,\ \mathrm{erg}\)	\(W_t,\ \mathrm{erg}\)
Cassiopeia A	\(1.2\cdot 10^{21}\)	\(1.0\cdot 10^{18}\)	\(2.2\cdot 10^{-19}\)	\(3.8\cdot 10^{44}\)	\(1.3\cdot 10^{45}\)	\(5\cdot 10^{-3}\)	—	\(10^{47}\)
Taurus A (Crab Nebula)	\(4.4\cdot 10^{21}\)	\(3.2\cdot 10^{18}\)	\(1.8\cdot 10^{-20}\)	\(1.4\cdot 10^{47}\)	\(4\cdot 10^{47}\)	\(10^{-4}\)	\(1.7\cdot 10^{48}\)	\(10^{47}\)

\(H_{\max}\) was determined from the relation

\[ \frac{\rho u^2}{2}=\frac{H^2}{8\pi}, \]

where it was assumed that \(\rho\sim 10^{-22}\ \mathrm{g\cdot cm^{-3}}\) and \(u\sim 10^8\ \mathrm{cm/sec}\). In reality the field strength is apparently smaller, and, consequently, \(W_{e,\min}>3.8\cdot 10^{44}\ \mathrm{erg}\). For the Crab Nebula the field \(H_{\max}\) was estimated from the requirement that electrons with energy \(E\sim 10^{12}\), responsible for the optical radiation, should not have had time over 900 years to lose a substantial part of their energy. The total energy of the electrons in the nebulae \(W_e\) was calculated on the basis of data on the spectrum of their radio emission. In this case the exponent \(\gamma\) in the electron spectrum (2.11) is immediately determined, as was already emphasized, from the data on the dependence of the flux \(F_\nu\) on wavelength, since for

\[ F_\nu\sim \lambda^\alpha \sim \nu^{-\alpha} \]

\[ \gamma=1+2\alpha. \tag{2.31} \]

In Table IV the values \(\gamma=3\) for Cassiopeia A and \(\gamma=1.5\) for Taurus A are used. The quantities \(W_k\) and \(W_t\), appearing in the last columns of Table IV, are respectively the total kinetic energy and the kinetic energy of turbulent motion in the nebulae (for more detail see \(^{39}\)).

It must be emphasized that all the values given in Table IV have only a purely approximate significance and may be substantially changed. Thus, in Cassiopeia A, for the electrons responsible for radio emission in the meter and decimeter ranges, it is apparently more correct to use the values \(\gamma=3.4\) and \(\gamma=2.2\), respectively. In the case of Taurus A, for the electrons producing the optical radiation, according to \(^{21}\) \(\gamma=3\), and according to \(^{40}\) \(\gamma=3.3\). There is still greater uncertainty in the values of the field strength. In particular, for Taurus A there are grounds to believe that acceleration of particles is also occurring at the present time \(^{40}\), as a result of certain nonstationary or fluctuation processes in the shell or in the ejections of the central star. Therefore the criterion used above for estimating the field \(H_{\max}\) is by no means obligatory, and in \(^{40}\) the value \(H\sim 10^{-3}\) is used, while in \(^{40a}\) the value \(H\sim 3\cdot 10^{-4}\) is used. For the total energy that was imparted to the electrons in the nebula since the outburst of the supernova in 1054, \(^{40}\) gives the value \(1.5\cdot 10^{48}\ \mathrm{erg}\).

As already mentioned, Cassiopeia A and Taurus A (the Crab Nebula) are expanding shells formed in the outbursts of supernovae in 369 and 1054. The shells of the supernovae of 185,

1572 and 1604 are also sources of powerful nonthermal radio emission, although weaker than in the case of Cassiopeia A and Taurus A. Moreover, of all reliably known supernovae, only the supernova of 1006 still cannot be unambiguously identified with known sources of radio emission ^21. Thus, radio astronomy has led to an important discovery, namely that during outbursts of supernova stars *) a large number of relativistic (cosmic-ray) electrons are formed in their expanding envelopes. The spectrum of these electrons varies within rather wide limits (from \(\gamma = 3.4 \div 3\) to \(\gamma = 1.5\)). The total energy contained in relativistic electrons lies in the range \(10^{45} \div 10^{48}\) erg. In the Crab Nebula there is, moreover, a significant fraction of electrons with energies up to \(10^{12}\) eV, responsible for the optical radiation (from (2.27), with \(\nu_{\max}\sim 10^{15}\) and \(H\sim 5\cdot 10^{-4}\), we obtain \(E_m\sim 5\cdot 10^{11}\)). It is natural to suppose that relativistic electrons are also produced as a result of outbursts of novae ^36. Since the energy release in novae is \(10^3 \div 10^4\) times smaller than in supernovae, one may expect that the energy of fast electrons in novae is \(10^{42} \div 10^{44}\) erg. Owing to the lower luminosity, the radio emission of novae has not yet been reliably established, although in the case of Nova Aquilae 1918 there are certain indications in this regard.

In our Galaxy supernovae flare up on average no less than once every 30–60 years, although they are recorded approximately 10 times less often because of interstellar absorption of light ^21. Novae flare up in the Galaxy in a number \(\sim 100\) per year. Relativistic electrons formed in the expanding envelopes of all these stars must, in considerable numbers, enter interstellar space (the electrons diffuse out of the envelopes and, most importantly, the envelopes themselves disperse and even in the case of supernovae cannot retain the electrons for more than several thousand years after the explosion). The same, of course, applies to fast protons and nuclei, which, in all probability, are formed in the envelopes as a result of the explosion. It follows from this that in the plane of the Galaxy and near it, where novae and supernovae are chiefly located, there must be powerful sources of cosmic rays. If for the moment we speak only of electrons, about which we have direct radio-astronomical data, then supernova outbursts lead to the generation of relativistic electrons with a mean power

\[ U_{e,\ \mathrm{sn}}=\frac{W_{\mathrm{sn}}}{\tau_{\mathrm{sn}}}\sim \frac{10^{45}\div 10^{48}}{10^9}\sim 10^{36}\div 10^{39}\ \text{erg/sec}, \tag{2.32} \]

where \(W_{\mathrm{sn}}\) is the mean energy release per supernova and \(\tau_{\mathrm{sn}}\sim 30\) years \(\simeq 10^9\) sec is the mean time between outbursts.

For novae, analogously (with \(\tau_{\mathrm{n}}\sim 1/100\) year),

\[ U_{e,\ \mathrm{n}}=\frac{W_{\mathrm{n}}}{\tau_{\mathrm{n}}}\sim \frac{10^{42}\div 10^{44}}{3\cdot 10^5}\sim 3\cdot 10^{36}\div 3\cdot 10^{38}\ \text{erg/sec}. \tag{2.33} \]

However crude these estimates may be, they are of great importance for the theory of the origin of cosmic rays, since they are based on observations and point directly to probable sources of cosmic rays.

*) The number of supernova stars should apparently also include the star \(\eta\)-Carinae, which flared up in 1843. According to recent data, kindly communicated to the author by I. S. Shklovsky, the optical radiation with a continuous spectrum emitted by this supernova (or, more precisely, by its envelope) is partially polarized. Therefore there is every reason to suppose that relativistic particles were formed as a result of the outburst of the star \(\eta\)-Carinae. Among supernovae, perhaps, should also be included the star that flared up in 1942 in the constellation Puppis. The question of the radio emission of both these objects is not yet clear.

3. MOTION OF COSMIC PARTICLES IN THE INTERSTELLAR MEDIUM

In discussing the problem of the origin of cosmic rays it is necessary to consider the processes occurring during the motion of cosmic rays (i.e., particles of high energy) in the interstellar medium. The corresponding analysis will also apply here to the case of the motion of particles in denser regions, such as, for example, the envelopes of supernovae.

a) Energy losses in the case of protons and nuclei. Formation of secondary electrons and positrons

Fast protons and nuclei lose energy as a result of nuclear collisions and ionization losses.

The effective cross section for the collision of nuclei \(i\) and \(k\) with atomic weights \(A_i\) and \(A_k\) may be represented in the form\(^4\)

\[ \left. \begin{aligned} \sigma &= \pi (r_i+r_k-2\Delta r)^2, \qquad r_{i,k}=1.45\cdot 10^{-13} A_{i,k}^{1/3}\ \mathrm{cm},\\ \Delta r &= 0.85\cdot 10^{-13}\ \mathrm{cm}. \end{aligned} \right\} \tag{3,1} \]

In collisions of protons with nuclei with \(A>8\), one may with sufficient accuracy put

\[ \sigma=\pi r^2=6.6\cdot 10^{-26} A^{2/3}\ \mathrm{cm}^2 . \tag{3,2} \]

For collisions of protons with protons, according to (3,1),

\[ \sigma_{pp}=4\cdot 10^{-26}\ \mathrm{cm}^2 . \tag{3,3} \]

However, for the case when one or both of the colliding nuclei are protons, formula (3,1) has not been verified experimentally and is therefore unreliable. This applies especially to proton–proton collisions, which play the principal role in the motion of cosmic particles in the interstellar medium. Since in proton–proton collisions, in the majority of cases, one of the protons has a large energy both before and after the collision, the value often used as the cross section for disappearance of the fast proton is

\[ \sigma=2.5\cdot 10^{-26}\ \mathrm{cm}^2 . \tag{3,4} \]

In Table V, in the 2nd column, are given the path lengths of protons and nuclei in hydrogen when using the value (3,4) for protons and formula (3,1) for

Table V

Nuclear path length in the interstellar medium

— Nucleus (\(A\) = mean atomic weight)	Path length in \(\mathrm{g/cm^2}\) (variant a)	Path length in \(\mathrm{g/cm^2}\) (variant b)	Path length in cm at \(n=0.1\) (variant a)	Free-path time \(T\) in years (variant a)	Mixing time \(T\) for \(n=0.09[\mathrm{H}]+0.01[\mathrm{He}]\) (variant b)
Proton	67	42	\(4\cdot 10^{26}\)	\(4.2\cdot 10^8\)	\(3.0\cdot 10^8\)
\(\alpha\)-particle	12.5	16.5	\(7.5\cdot 10^{25}\)	\(8\cdot 10^7\)	\(7.3\cdot 10^7\)
Group \(L\) (\(\bar A=8\))	7.5	—	\(4.5\cdot 10^{25}\)	\(4.8\cdot 10^7\)	\(4.5\cdot 10^7\)
Group \(M\) (\(\bar A=14\))	5.0	5.8	\(3\cdot 10^{25}\)	\(3.2\cdot 10^7\)	\(3.0\cdot 10^7\)
Group \(H\) (\(\bar A=30\))	3.0	3.3	\(1.8\cdot 10^{25}\)	\(1.9\cdot 10^7\)	\(1.8\cdot 10^7\)
Iron (\(A=56\))	1.9	—	\(1.15\cdot 10^{25}\)	\(1.2\cdot 10^7\)	\(1.2\cdot 10^7\)

nuclei (variant a). In the 3rd column the path length in hydrogen is also given, but using the cross section (3.3) for protons and certain cross sections selected in [5] for nuclei (variant b). Columns 4 and 5 give the length and time of the mean free path for variant a under the assumption that the concentration of interstellar hydrogen is \(n=0.1\ \mathrm{cm}^{-3}\). Finally, the last column gives the path time for motion in a mixture with a hydrogen concentration of \(0.09\ \mathrm{cm}^{-3}\) and helium \(0.01\ \mathrm{cm}^{-3}\) (variant v). Table VI gives various ratios of the nuclear path times; variant g is added, giving the ratio \(T_L:T_M:T_H=3:2:1\). This variant corresponds to choosing a larger value, in comparison with variants a and b, for the mean atomic weight of the nuclei of group \(H\). Taking into account the role of iron nuclei in group \(H\), such a procedure seems reasonable.

Table VI

Ratios of mean free times

Nucleus	\(T_j/T_p^{\text{nuc}}\) (variant a)	\(T_j/T_p^{\text{nuc}}\) (variant b)	\(T_j/T_p^{\text{nuc}}\) (variant v)	Variant	\(T_L/T_M\)	\(T_L/T_H\)	\(T_M/T_H\)
\(\alpha\)-particle	5.4	2.5	4.1	a	1.45	2.4	1.7
Group \(L\)	9	—	6.7	b	—	—	1.8
Group \(M\)	13	7.2	10	v	1.5	2.5	1.65
Group \(H\)	22	1.3	16.5	g	1.5	3.0	2.0
Iron	35	—	25	g	\(T_p/T_\alpha=4;\ T_p/T_H=20;\ T_\alpha/T_H=5;\ T_p/T_M=10;\ T_\alpha/T_M=2.5;\ T_p/T_L=20/3\)	\(T_p/T_\alpha=4;\ T_p/T_H=20;\ T_\alpha/T_H=5;\ T_p/T_M=10;\ T_\alpha/T_M=2.5;\ T_p/T_L=20/3\)	\(T_p/T_\alpha=4;\ T_p/T_H=20;\ T_\alpha/T_H=5;\ T_p/T_M=10;\ T_\alpha/T_M=2.5;\ T_p/T_L=20/3\)

For nuclei, the mean free time, or lifetime, \(T_j\) has an unambiguous meaning: after this time the number of fast nuclei of species \(j\) decreases by a factor of \(e\). In the case of protons one can distinguish the lifetime for interaction \(T_{pi}\) and the lifetime for energy loss \(T_{pE}\). The cross section for interaction is probably close to (3.3), and in each act of interaction the proton loses \(15\div 30\%\) of its energy \(^{41,42}\). In accordance with this cross section, the cross section corresponding to the proton’s loss of a significant fraction of its energy is \(2\div 4\) times smaller than the cross section for interaction, i.e. close to or even half as small as the cross section (3.4). Thus, in hydrogen, approximately \(T_{pi}=42\ \mathrm{g/cm^2}\) and \(T_{pE}=70\div 120\ \mathrm{g/cm^2}\). Below, for an interstellar medium (90% hydrogen + 10% helium) with particle concentration 0.1, the following values will be adopted for definiteness:

\[ T_{pi}=2\cdot 10^8\ \text{years}=6.3\cdot 10^{15}\ \text{sec.}, \tag{3.5} \]

\[ T=6\cdot 10^8\ \text{years}=1.9\cdot 10^{16}\ \text{sec.} \]

Some mean time will also often be used,

\[ T_p=4\cdot 10^8\ \text{years}=1.25\cdot 10^{16}\ \text{sec.} \tag{3.6} \]

As the times \(T_j\) for nuclei we shall mainly use the values corresponding to variant g (see Table VI) with the time \(T_p=4\cdot 10^8\) years. The concentration chosen above, \(n=0.1\ \mathrm{cm}^{-3}\), is, by its very ...

in this sense, the mean concentration of interstellar-gas particles in the region occupied by cosmic rays. Near the galactic plane, in gas clouds occupying a volume \(\sim 10^{65}\ \mathrm{cm}^3\), the concentration is \(n\sim 10\). Between clouds near the plane of the Galaxy \(n\sim 0.1\), and on the periphery of the galactic corona, with a total volume \(V\sim 1 \div 5\cdot 10^{68}\ \mathrm{cm}^3\), \(n\sim 0.01 \div 0.03\) (see Sec. 3b) and \(^{23,31}\). Therefore the mean value \(n\sim 0.1\) is in fact more or less justified, although the use of a mean concentration \(n\sim 0.03\) is also quite permissible. In the latter case the times \(T_{pi}\), \(T_{pE}\), and \(T_p\) must be increased by a factor of 3.

According to (2.22), the total energy contained in cosmic rays in the Galaxy is \(W_{cr}\sim 10^{55}\div 10^{56}\ \mathrm{erg}\), with accuracy up to a factor approximately equal to two; in what follows one may assume that this energy is concentrated in cosmic protons. Therefore the energy lost by cosmic rays in the Galaxy per unit time (see also (1.7)) is:

\[ U_{cr}=\frac{W_{cr}}{T_{pE}}\sim 10^{39}\div 10^{40}\ \mathrm{erg/sec}, \tag{3.7} \]

\[ w_{cr}=\frac{W_{cr}}{V}\sim 1\ \mathrm{eV/cm^3},\qquad u_{cr}=\frac{w_{cr}}{T_{pE}}\sim 10^{-28}\ \mathrm{erg/cm^3\,sec}. \]

At each collision (act of interaction), protons colliding with nitrogen and oxygen nuclei lose on average about \(30\%\) of their energy, which goes predominantly into the formation of \(\pi\)-mesons \(^{41}\). The energy of the secondary particles \(E_2\) is thus proportional to the energy of the primary particle (proton) \(E_1\),

\[ E_2=\xi E_1. \tag{3.8} \]

The fast proton after the act of interaction therefore carries away the energy

\[ E'_1=(1-\xi)E_1. \tag{3.9} \]

As stated, for air \(\xi\approx 0.3\), while in hydrogen \(\xi\) may be two or three times smaller, although it should be noted that for hydrogen the very equality (3.8) has not yet been confirmed experimentally.

If the quantity \(\xi\) is independent of, or depends only weakly on, the energy, which apparently is the case, then the spectrum of secondary particles and the spectrum of protons after collision do not change (they coincide with the spectrum of the primary particles). Indeed, the transformations (3.8)—(3.9) are scale transformations; in a collision, an element of the spectrum of primary particles \(N_1(E_1)\,dE_1\) passes, as it were, into an element of the spectrum of secondary particles, and \(N_1(E_1)\,dE_1=\frac{1}{s}N_2(E_2)\,dE_2\), where \(s\) is the number of secondary particles formed in the collision. Hence, and from equalities (3.8)—(3.9), it follows that

\[ N_2(E_2)=\frac{s}{\xi}N_1\left(\frac{E_2}{\xi}\right),\qquad N'_1(E'_1)=\frac{1}{1-\xi}N_1\left(\frac{E'_1}{1-\xi}\right). \tag{3.10} \]

In particular, if the spectrum is a power law, then the exponent \(\gamma\) for the secondary particles and for the fast protons present after the collision remains unchanged. In collisions of cosmic nuclei with particles of the interstellar medium, relation (3.8) probably also holds, i.e., the energy of the products (fragments) is proportional to the energy of the primary nuclei. Thus, nuclear collisions of cosmic protons and nuclei, under assumptions that are rather natural and justified within known limits, should not change the form of the energy spectrum of cos-

-mic rays. Of course, if relation (3.8) does not hold for nuclear fragments, which is quite possible, then the secondary products (for example, nuclei of group \(L\)) will have a spectrum different from the spectrum of the primary particles.

The \(\pi\)-mesons*) produced in collisions of cosmic protons carry away, in each collision, \(10 \div 30\%\) of the energy of these protons (as indicated above, we assume that for hydrogen \(\xi=0.1 \div 0.3\)). It may be thought that one third of these mesons are \(\pi^0\)-mesons, which produce \(\gamma\)-rays that subsequently leave the Galaxy (see Section 36). As for \(\pi^\pm\)-mesons, approximately \(1/4\) of the energy associated with them will pass, after the well-known chain of decays, into electrons and positrons (below we shall speak simply of electrons; neutrinos, of course, leave the Galaxy and are of no interest). Thus, in one collision \(2.5 \div 7.5\%\) of the energy of the primary proton will pass to electrons. In all the \(2 \div 4\) collisions needed for a proton to lose a substantial fraction of its energy, roughly speaking, \(5 \div 15\%\) of the proton energy will pass to electrons. This means that in the Galaxy the energy transferred to cosmic electrons per unit time is (see (3.7)):

\[ U_e=(5 \div 15)\cdot 10^{-2}U_{cr} =(5 \div 15)\cdot 10^{-2}\frac{W_{cr}}{T_p\bar E} \sim 5\cdot 10^{37}\div 1.5\cdot 10^{39}\ \text{erg/sec}. \tag{3.11} \]

The energy of secondary particles, especially electrons, is lower by \(1 \div 2\) orders of magnitude than the energy of the protons generating them, i.e. mainly protons with energy \(10^9 \div 10^{10}\) ev. Therefore a considerable fraction of the electrons with total energy (3.11) probably has a comparatively small energy \(10^7 \div 10^8\) ev. Bearing in mind the electrons responsible for the observed cosmic radio emission (see Section 26), one may assume that the energy transferred to them per unit time is

\[ U_e\sim 3\cdot 10^{36}\div 3\cdot 10^{38}\ \text{erg/sec},\quad u_e=\frac{U_e}{V}\sim 3\cdot 10^{-30}\div 3\cdot 10^{-32}\ \text{erg}/\text{cm}^3\text{sec}, \tag{3.12} \]

where the volume \(V\sim 10^{68}\ \text{cm}^3\) (see (2.21)).

Let us proceed to the calculation of ionization losses for protons and nuclei. When a particle with charge \(Z\) and mass \(M\) moves in atomic hydrogen,

\[ -\left(\frac{dE}{dt}\right)_i =\frac{4\pi e^4 Z^2 n}{mv} \left\{ \ln\left[ \frac{2mv^2}{I} \left(\frac{E}{Mc^2}\right)^2 \right] -\frac{v^2}{c^2} \right\} = \]

\[ =7.62\cdot 10^{-9}nZ^2\frac{c}{v} \left\{ 22.2+4\ln\frac{E}{Mc^2} +2\ln\frac{v^2}{c^2} -2\frac{v^2}{c^2} \right\}\ \text{ev/sec}, \tag{3.13} \]

where it is assumed that the total energy of the particle \(E \ll \dfrac{M}{m}Mc^2\), \(v\) is the velocity of the particle, \(e\) and \(m\) are the charge and mass of the electron, \(I\simeq 15\) ev is the mean excitation energy, and \(n\) is the concentration of atomic electrons, which for hydrogen coincides with the previously introduced concentration of hydrogen.

*) The number of antiprotons formed is very small (it cannot amount to more than \(0.1\%\) of the total proton flux)\(^{43}\).

In the nonrelativistic case \(\left(E_k=E-Mc^2\simeq \dfrac{Mv^2}{2}\ll Mc^2\right)\)

\[ -\left(\frac{dE}{dt}\right)_i =7.62\cdot 10^{-9}nZ^2\sqrt{\frac{2Mc^2}{E_k}} \left\{11.8+\ln\frac{E_k}{Mc^2}\right\}\ \text{eV/sec}. \tag{3.14} \]

In the relativistic case (for protons, practically, at \(E>2\div 3\cdot 10^9\) eV; at the same time, by assumption, \(\left.E\ll \dfrac{M}{m}Mc^2\right)\)

\[ -\left(\frac{dE}{dt}\right)_i =7.62\cdot 10^{-9}nZ^2 \left\{20.2+4\ln\frac{E}{Mc^2}\right\}\ \text{eV/sec}. \tag{3.15} \]

If hydrogen is completely ionized, then in the nonrelativistic case one must put
\(I=\hbar\omega_0=\hbar\sqrt{\dfrac{4\pi e^2 n}{m}}=1.2\cdot 10^{-12}\sqrt{n}\) eV, where \(n\) is the electron concentration, whence

\[ -\left(\frac{dE}{dt}\right)_i =7.62\cdot 10^{-9}nZ^2\sqrt{\frac{2Mc^2}{E_k}} \left\{42-\frac{1}{2}\ln n+\ln\frac{E_k}{Mc^2}\right\}\ \text{eV/sec}. \tag{3.16} \]

In the relativistic case, for motion in ionized hydrogen

\[ -\left(\frac{dE}{dt}\right)_i =\frac{2\pi e^4 n}{mc}Z^2 \left[\ln\frac{m^2c^3W}{4\pi e^3 n\hbar^2}+1\right] = \]

\[ =7.62\cdot 10^{-9}nZ^2 \left\{\ln\frac{2W}{mc^2}-\ln n+74.6\right\}\ \text{eV/sec}, \tag{3.17} \]

where \(n\) is the electron concentration and \(W\) is the maximum energy transferred to an electron
\(\left(W=E\right.\) for \(E\gg \dfrac{M}{m}Mc^2\) and
\(W=2mc^2\left(\dfrac{E}{Mc^2}\right)^2\) for
\(\left.Mc^2\ll E\ll \dfrac{M}{m}Mc^2\right)\).

We shall not take into account below the ionization losses associated with the presence of helium and other elements in the interstellar medium, since they do not exceed \(20\div 25\%\) of the losses in hydrogen. For protons with energy \(E=10^{10}\) eV, the losses at \(n=0.1\), according to (3.15) and (3.17), are respectively \(2.3\cdot 10^{-8}\) eV/sec and \(6.2\cdot 10^{-8}\) eV/sec. At the same time the average loss of energy due to nuclear collisions is considerably greater and in the present case is equal (see (3.5))

\[ -\left(\frac{dE}{dt}\right)_n =\frac{E}{T_{pE}} \sim \frac{10^{10}}{1.9\cdot 10^{16}} \simeq 5\cdot 10^{-7}\ \text{eV/sec}. \]

What has been said reflects, of course, the well-known possibility of neglecting, for relativistic protons, ionization losses in comparison with nuclear ones. In the case of nuclei, at the same energy per nucleon \(E\), the ionization losses are \(Z^2\) times greater than for protons. At the same time the nuclear losses also increase approximately by a factor of \(Z^2\), since they are now equal to \(\dfrac{AE}{T_j}\sim \dfrac{Z^2E}{T_{pE}}\), because \(A\simeq 2Z\), and the lifetime of nuclei \(T_j\sim T_p/Z\) (see Table VI). Therefore the relative importance of ionization losses for relativistic protons and nuclei may be regarded as the same.

Ionization losses become significant in passing to the nonrelativistic case, or even at kinetic energy \(E_k\sim Mc^2\). Allowance for these losses plays a major role in discussing the question of the injection of cosmic rays into interstellar space.

b) Energy losses in the case of electrons.
Change of the energy spectrum
in the motion of particles in the interstellar medium

Electrons moving in the interstellar medium experience ionization, radiation, Compton, and magnetic-bremsstrahlung losses.

The ionization losses of nonrelativistic electrons are not of interest here, and we shall limit ourselves to pointing out that in this case, to a first approximation, one may use formulas (3.13)—(3.14), (3.16) with \(Z=1\) and \(M=m\). In the relativistic case the ionization losses of an electron with energy \(E\) in atomic hydrogen are as follows:

\[ -\left(\frac{dE}{dt}\right)_i = \frac{2\pi e^4 n}{mc}\ln\frac{E^3}{2mc^2 I} = 7.62\cdot 10^{-9}n\left\{20.1+3\ln\frac{E}{mc^2}\right\}\ \text{eV/sec}. \tag{3.18} \]

In ionized hydrogen the losses are equal to

\[ -\left(\frac{dE}{dt}\right)_i = \frac{2\pi e^4 n}{mc} \left\{ \ln\frac{m^2c^2E}{8\pi e^2 n\hbar^2}+1 \right\} = \]

\[ = 7.62\cdot 10^{-9}n \left\{ \ln\frac{E}{mc^2}-\ln n+74.6 \right\}\ \text{eV/sec}. \tag{3.19} \]

Radiative energy losses are due to the emission by electrons of photons occurring when electrons collide with particles of the interstellar medium. The corresponding expressions are discussed in detail in \(^{12}\), and here we shall confine ourselves to giving the final formula, which for hydrogen is sufficiently accurate:

\[ -\frac{1}{E}\left(\frac{dE}{dt}\right)_r = 8.0\cdot 10^{-16}n\ \text{sec}^{-1}. \tag{3.20} \]

The adopted value corresponds to a \(t\)-unit equal in hydrogen to \(62\ \text{g}/\text{cm}^2\). Taking account of the role of helium and other nuclei leads to the fact that the effective value is \(t\simeq 52\ \text{g}/\text{cm}^2\). Along such a path the energy of the electrons decreases on the average by a factor of \(e\), the emitted photons having an energy comparable with the energy of the electron. For the latter reason one may evidently speak of a mean free path length and time, analogously to how this is done in the case of nuclear losses. For \(n=0.1\) the corresponding lifetime of an electron is \(T_e=3.3\cdot 10^8\) years. However, for convenience, below the value

\[ T_e\simeq T_p\simeq 4\cdot 10^8\ \text{years} = 1.25\cdot 10^{16}\ \text{sec}, \tag{3.21} \]

will be used, which, at the accuracy attainable in the region under consideration, is entirely equivalent to the preceding one.

Let us note that the \(\gamma\)-rays formed in the braking of electrons practically freely leave the Galaxy, since the thickness of the gas in their path amounts to only \(\sim 10^{-25}\cdot 10^{23}=10^{-2}\ \text{g}/\text{cm}^2\) of hydrogen.

The losses mentioned above as “Compton” are associated with the inverse Compton effect—the scattering of electrons on thermal photons \(^{44,45}\), which are present in the Galaxy in rather large quantity. These losses are analogous to radiative ones in the sense that they have a “catastrophic character” (they occur in large portions). The role of Compton losses is therefore reduced to a decrease in the lifetime of electrons \(T_e\). Without dwelling here on detailed calculations \(^{1,12}\), we indicate that the Compton losses may be taken equal to

\[ -\left(\frac{dE}{dt}\right)_k = \sigma_0\rho c\left(\frac{E}{mc^2}\right)^2 \simeq 5.6\cdot 10^{-16} \left(\frac{E}{mc^2}\right)^2\ \text{eV/sec}, \tag{3.22} \]

where

\[ \sigma_0=\frac{8\pi}{3}\left(\frac{e^2}{mc^2}\right)^2=6.6\cdot 10^{-25}\ \text{cm}^2 \]

and \(\rho\) is the mean energy density of the light radiation; in passing in (3.22) to the numerical value, \(\rho=0.03\ \text{eV}/\text{cm}^3\) was used, taking into account that the radiation in the entire galactic corona is meant (in the plane of the Galaxy it is usually assumed that \(\rho=0.3\ \text{eV}/\text{sec}\)). From formula (3.20), for \(n=0.1\), and from expression (3.22), it follows that the Compton losses become comparable with the radiation losses at electron energy \(E=3.5\cdot 10^{10}\ \text{eV}\) and, for example, at \(E=10^9\ \text{eV}\) are 35 times smaller than the latter. Thus, within the limits of the accuracy now attainable, Compton losses may be neglected. Of course, this applies to the concepts developed here, whereas when considering the motion of electrons near the Sun or stars, Compton losses may turn out to be appreciable.

The last, and in this case very important, mechanism of energy loss by electrons is magnetobremsstrahlung losses. The presence of these losses is connected with the radiation arising when electrons move in interstellar magnetic fields, which was discussed in detail in Section 2. The magnetobremsstrahlung losses are as follows:

\[ -\left(\frac{dE}{dt}\right)_m = \frac{2c}{3}\left(\frac{e^2}{mc^2}\right)^2 H_\perp^2 \left(\frac{E}{mc^2}\right)^2 = 0.98\cdot 10^{-3}H_\perp^2 \left(\frac{E}{mc^2}\right)^2 \text{eV/sec}, \tag{3.23} \]

where it is assumed that \(E\gg mc^2\).

The various energy losses by electrons are compared in Table VII.

Table VII

Energy losses by electrons (in eV/sec)

Electron energy \(E\) in eV	Ionization losses at \(n=0.1\), formula (3.18)	Ionization losses at \(n=0.1\), formula (3.19)	Radiation losses at \(n=0.1\) (formula (3.20))	Magnetobremsstrahlung losses (formula (3.23)), \(H_\perp=3\cdot 10^{-6}\)	Magnetobremsstrahlung losses (formula (3.23)), \(H_\perp=10^{-5}\)
\(5\cdot 10^7\)	\(2.6\cdot 10^{-8}\)	\(6.2\cdot 10^{-8}\)	\(4\cdot 10^{-9}\)	\(10^{-10}\)	\(10^{-9}\)
\(10^8\)	\(2.7\cdot 10^{-8}\)	\(6.3\cdot 10^{-8}\)	\(8\cdot 10^{-9}\)	\(4\cdot 10^{-9}\)	\(4\cdot 10^{-8}\)
\(5\cdot 10^8\)	\(3.1\cdot 10^{-8}\)	\(6.4\cdot 10^{-8}\)	\(4\cdot 10^{-8}\)	\(10^{-8}\)	\(10^{-7}\)
\(10^9\)	\(3.3\cdot 10^{-8}\)	\(6.4\cdot 10^{-8}\)	\(8\cdot 10^{-8}\)	\(4\cdot 10^{-8}\)	\(4\cdot 10^{-7}\)
\(5\cdot 10^9\)	\(3.6\cdot 10^{-8}\)	\(6.6\cdot 10^{-8}\)	\(4\cdot 10^{-7}\)	\(10^{-6}\)	\(10^{-5}\)
\(10^{10}\)	\(3.8\cdot 10^{-8}\)	\(6.6\cdot 10^{-8}\)	\(8\cdot 10^{-7}\)	\(4\cdot 10^{-6}\)	\(4\cdot 10^{-5}\)
\(5\cdot 10^{10}\)	\(4.4\cdot 10^{-8}\)	\(6.7\cdot 10^{-8}\)	\(4\cdot 10^{-6}\)	\(10^{-4}\)	\(10^{-3}\)

Taking into account ionization and magnetobremsstrahlung losses,

\[ \frac{dE}{dt}=\varphi(E)=-a-bE^2 = -4\cdot 10^{-7}n-4\cdot 10^{-15}H_\perp^2E^2\ \text{eV/sec}, \tag{3.24} \]

where \(E\) is measured in eV, and small simplifications have been made, clear from comparison with formula (3.23) and Table VII.

If a particle begins to move in the medium at the moment \(t=0\) with energy \(E_0\), then at the moment \(t\) its energy \(E\), according to (3.24), is determined from

relations

\[ t=\frac{\operatorname{arctg}\sqrt{\frac{b}{a}}\,E_0-\operatorname{arctg}\sqrt{\frac{b}{a}}\,E}{\sqrt{ab}} . \tag{3,25} \]

As long as \(\sqrt{b}\,E \gg \sqrt{a}\), one may take \(\operatorname{arctg}x=\frac{\pi}{2}-\frac{1}{x}\), and

\[ \left. \begin{aligned} E&=\frac{E_0}{1+bE_0t},\\ E&\gg \sqrt{\frac{a}{b}}=10^4\sqrt{\frac{n}{H_\perp^2}}\ \text{eV}, \end{aligned} \right\} \tag{3,26} \]

as is also immediately clear from (3,24) if ionization losses are neglected. In the opposite limiting case

\[ E=E_0-at,\qquad E_0\ll \sqrt{\frac{a}{b}}=10^4\sqrt{\frac{n}{H_\perp^2}}\ \text{eV}. \tag{3,27} \]

In case (3,26), the energy \(E\) ceases to depend on \(E_0\) if \(bE_0t\gg 1\). On the other hand, in (3,24)—(3,27) radiation losses have not been taken into account, since they are not continuous in character. This can be done only for \(t<T_e\), i.e., when considering time intervals shorter than the lifetime of the electron. Over the time \(T_e\), the energy \(E\) ceases to depend on \(E_0\) if

\[ bE_0T_e\simeq 50\cdot H_\perp^2 E_0\gg 1. \tag{3,28} \]

For \(H_\perp\sim 10^{-5}\) this inequality is practically satisfied for \(E_0>5\cdot 10^8\div 10^9\ \text{eV}\).

Under conditions of interest from the point of view of the theory of the origin of cosmic rays, one has to deal not with monoenergetic electrons but with an electron spectrum of the type \(N_e(E)=K_eE^{-\gamma}\). Owing to the presence of losses of various types, the particle spectrum changes, and it is necessary to find this spectrum under conditions in which certain sources supply \(q(E,t)\,dE\) particles per unit time in the energy interval \((E,E+dE)\). The particle spectrum is most conveniently obtained by using the continuity equation in “energy space”:

\[ \frac{\partial N(E,t)}{\partial t}+\operatorname{div}_E j(E,t) =\frac{\partial N}{\partial t}+\frac{\partial j}{\partial E}=q(E,t), \tag{3,29} \]

where \(j(E)\) is the flux of particles whose energy “passes” per unit time through the value \(E\).

If only the systematic change of energy\(^*\) is taken into account, then \(\frac{dE}{dt}=\varphi(E)\) and \(j(E)=N(E)\frac{dE}{dt}=N(E)\varphi(E)\), since \(\frac{dE}{dt}\) is the “velocity in energy space.” Hence

\[ \frac{\partial N(E,t)}{\partial t}+\frac{N(E,t)}{T} +\frac{\partial}{\partial E}\,[\varphi(E)N(E,t)]=q(E,t), \tag{3,30} \]

where, in contrast to expression (3,29), collisions leading to an abrupt change in energy have also been taken into account. The mean free time for these collisions is \(T\), and, of course, in the case of electrons this time corresponds to radiation losses and is equal to \(T_e\) (see (3,21)).

\[ \text{――――――} \]

\(^*\) We follow here \({}^{43}\). A more general equation, taking fluctuations into account, was considered in \({}^{46,47}\) (see also Section 3 B). An analogous equation was investigated in \({}^{47a}\).

Over times of order \(T_e \simeq T_p\) in the Galaxy, apparently no substantial changes occur and, in any case, these changes would be difficult to take into account*). Therefore below only the stationary case will be considered, when \(\dfrac{\partial N}{\partial t}=0\). Thus, to determine the spectrum we arrive at the equation

\[ \left. \begin{aligned} &\frac{\partial}{\partial E}\,[\varphi(E)N(E)]+\frac{N(E)}{T}=q(E),\\ &\varphi(E)=-a+aE-bE^2, \end{aligned} \right\} \tag{3.31} \]

where the function \(\varphi\) is written in the form to which one may restrict oneself in the subsequent discussion.

Without discussing the general solution of equation (3.31), we shall confine ourselves to separate particular cases (\(a \gtrless 0,\ b \gtrless 0\)):

\[ a=0,\ \alpha=0:\quad N(E)=\frac{\exp\left(-\dfrac{1}{bTE}\right)}{bE^2} \int_E^\infty q(\eta)\exp\left(\frac{1}{bT\eta}\right)\,d\eta, \tag{3.32} \]

\[ a=0,\ \alpha=0,\ bTE\gg 1:\quad N(E)=\frac{\displaystyle\int_E^\infty q(\eta)\,d\eta}{bE^2} =\frac{A}{(\gamma_0-1)bE^{\gamma_0+1}}, \tag{3.33} \]

\[ \alpha=0,\ b=0:\quad N(E)=\frac{1}{a}\exp\frac{E}{aT}\int_E^\infty q(\eta)\exp\left(-\frac{\eta}{aT}\right)\,d\eta, \tag{3.34} \]

\[ \alpha=0,\ b=0,\ E\ll aT:\quad N(E)=\frac{1}{a}\int_E^\infty q(\eta)\,d\eta =\frac{A}{(\gamma_0-1)aE^{\gamma_0-1}}, \tag{3.35} \]

\[ \alpha=0,\ b=0,\ E\gg aT:\quad N(E)=Tq(E)=\frac{AT}{E^{\gamma_0}}, \tag{3.36} \]

\[ a=0,\ b=0,\ \alpha<0:\quad N(E)=-\frac{1}{aE^{\left(1+\frac{1}{\alpha T}\right)}}\int_T^\infty q(\eta)\eta^{\frac{1}{\alpha T}}\,d\eta = \frac{A}{\left(1+\frac{1}{\alpha T}-\gamma_0\right)aE^{\gamma_0}}, \tag{3.37} \]

\[ a=0,\ b=0,\ \alpha>0:\quad N(E)=\frac{1}{aE^{\left(1+\frac{1}{\alpha T}\right)}}\int_0^E q(\eta)\eta^{\frac{1}{\alpha T}}\,d\eta, \tag{3.38} \]

where at the last stage in (3.33), (3.35), and (3.37) it has been put that \(q(E)=AE^{-\gamma_0}\) with \(\gamma_0>1\). The choice of the integration constant is determined from the obvious

*) The lifetime of the Galaxy is \(T_{\mathrm{gal}}\sim 6\cdot 10^9\) years. Therefore, for \(n\approx 0.03\) the time \(T_pE\) is only 3 times smaller than \(T_{\mathrm{gal}}\). If, however, \(n\approx 0.01\), which still cannot be categorically excluded, then \(T_pE\sim T_{\mathrm{gal}}\). Under such conditions the assumption of quasistationarity of the distribution of cosmic rays in the Galaxy is already, generally speaking, inadmissible. It would be, however, somewhat premature to consider a nonstationary picture. The main point is that at \(n\sim 0.01\) cosmic rays cannot be regarded as having been formed mainly during the period of formation of the Galaxy (see Sec. 4g).

requirements that, when particles are decelerated, the function \(N(E)\) be determined by the values \(q(\eta)\) with \(\eta \gg E\). Therefore, for \(a \geqslant 0,\ b \geqslant 0\) and \(\alpha \leqslant 0\), which corresponds to deceleration (see (3.34) and (3.31)), the integrals also extend over the limits from \(E\) to \(\infty\). If, however, \(a>0\), as is the case for the statistical acceleration considered below, then integration must already be carried out over the region of energies smaller than \(E\) (see (3.38)). In the case of cosmic protons and nuclei the magnetic-bremsstrahlung losses are negligible, while in the relativistic region the ionization losses are also small. This corresponds to case (3.36), and the particle spectrum remains unchanged—the same as their spectrum \(q(E)\) in the sources. The presence of noncatastrophic nuclear losses, in which after a collision the particle energy is proportional to its initial energy, as indicated in Section 3a, does not alter this conclusion. The same also follows from (3.37), since the case \(a=0,\ b=0,\ \alpha<0\) corresponds to losses
\[ \frac{dE}{dt}=-|\alpha|E, \]
which are proportional to the energy.

For high-energy electrons the principal losses are magnetic-bremsstrahlung losses (below in this section we assume that \(\alpha=0\)), and solution (3.33) is applicable. In this case the exponent in the spectrum \(N_e(E)=K_eE^{-\gamma}\) is equal to
\[ \gamma=\gamma_0+1, \tag{3.39} \]
where \(\gamma_0\) is the exponent in the expression for the source power \(q_e(E)=A_eE^{-\gamma_0}\). This conclusion, as is clear from (3.33), is valid under condition (3.28), i.e. in a field \(H_\perp\sim 10^{-5}\) for \(E>5\cdot 10^8 \div 10^9\) eV. With decreasing energy the spectrum becomes less steep, and in the low-energy region, when
\[ E\ll aT_e\sim 5\cdot 10^8,\qquad E\ll \sqrt{\frac{a}{b}}=10^4\sqrt{\frac{n}{H_\perp^2}}\sim 3\cdot 10^8\ \text{eV}, \tag{3.40} \]
already
\[ \gamma=\gamma_0-1. \tag{3.41} \]

According to radio-astronomical data (see Section 26), for electrons with energies \(10^8<E<5\cdot 10^9\) eV the value of \(\gamma\) is approximately constant and equal to \(\gamma=2.64\), although the exponent \(\gamma=3\) also, apparently, is not yet excluded. The constancy of \(\gamma\) upon transition to soft electrons with \(E\sim 1\div 2\cdot 10^8\) eV, responsible for radiation at wavelength \(\lambda\sim 100\) m, is very important. If this result is confirmed, it will indicate a smaller role of radiative and ionization losses than was assumed above. Such a situation is quite possible, since the adopted value \(n\sim 0.1\), as has already been noted, may quite possibly be overestimated. If \(n\sim 0.03\), then for \(H_\perp\sim 10^{-5}\) agreement with the data on the radio-emission spectrum is restored (condition (3.28) takes the form \(E\gg 7\cdot 10^7\)). An increase of the field \(H_\perp\) by approximately a factor of 1.5 leads to the same result. Such a possibility is not excluded, and at the same time it is clear that using values of \(H_\perp\) substantially smaller than \(10^{-5}\) is impermissible.

Thus, according to the available data, which require further refinement, in the electron energy region \(10^8<E<4\cdot 10^9\) magnetic-bremsstrahlung losses predominate and, consequently, relation (3.39) holds. In other words, the spectrum of the sources of cosmic electrons has the form
\[ q_e(E)=\frac{A}{E^{\gamma_0}},\qquad \gamma_0=1.64\div 2. \tag{3.42} \]

For protons in the energy region \(0.5\cdot 10^9<E_k<10^{10}\) eV approximately \(\gamma_p=1.9\div 2.2\). It is not clear to us to what extent a certain difference between \(\gamma_0\)

and \(\gamma_p\) can at present be assigned a real value. In any case, there is no doubt that protons and electrons are generated, if not with identical, then with very similar spectra. Such a result appears quite natural if electrons are accelerated in some sources. Electrons of secondary origin, arising in nuclear collisions, also apparently have a spectrum close to the proton spectrum (see above). It would be possible, in principle, to distinguish between these two possibilities (acceleration in sources or formation in nuclear collisions) by studying the primary electron–positron component of cosmic rays near the Earth\(^{42}\). The point is that secondary particles should with equal probability be electrons and positrons. If, however, acceleration takes place in the envelopes of supernovae, then positrons can appear in significant numbers only under a nuclear mechanism of injection\(^{39}\). In any case, if the primary light particles were mainly electrons, this would testify to the predominant role of acceleration of light particles in some sources where, moreover, the injection is not of nuclear origin. The values \(N_{e^+}\) given earlier (see (1.9) and (2.19)) indicate that this problem can be solved by increasing by only one order of magnitude the already achieved accuracy of determining the content of electrons and positrons in primary cosmic rays. Such experiments are, of course, very important also in the event that it proves impossible to distinguish positrons from electrons*).

The relative weight of the processes of electron acceleration in sources and their formation in nuclear collisions can, in principle, also be clarified from energy considerations. In radiation in a magnetic field, electrons lose per unit time an energy (see (3.23))

\[ u_{em}=\int_0^\infty \left(\frac{dE}{dt}\right)_m N_e(E)\,dE \simeq \frac{10^{-3}H_\perp^2}{(mc^2)^2} \int \frac{5\cdot 10^5}{E}\,dE \simeq 2\cdot 10^{-19}\ln\frac{E_{\max}}{E_{\min}} \sim \]

\[ \sim 8\cdot 10^{-19}\ \frac{\text{ev}}{\text{cm}^3\text{sec}} \sim 10^{-30}\ \frac{\text{erg}}{\text{cm}^3\text{sec}}; \qquad U_{em}=u_{em}\cdot V\sim 10^{38}\ \text{erg/sec}, \tag{3.43} \]

where the spectrum (2.18a) has been used.

For the spectrum (2.18b), which is closer to the radio-astronomical data, the result obtained is of the same order of magnitude. In (3.43) it is taken that \(E_{\max}=5\cdot 10^9\) and \(E_{\min}=10^8\) ev, which corresponds to the energy range of electrons responsible for the observed radio emission. Since in this range of energies magnetic-bremsstrahlung losses are dominant, the value (3.43) approximately corresponds to the total energy lost by electrons. This conclusion is also confirmed by the following calculation. According to (2.23), the energy density contained in electrons is \(w_e\approx 10^{-14}\ \text{erg}/\text{cm}^3\); therefore the effective time for energy loss by electrons is

\[ T_{e,\mathrm{eff}}=\frac{w_e}{u_{em}}\sim \]

\[ \sim \frac{10^{-14}}{10^{-30}}=10^{16}\ \text{sec.}, \]

whereas for radiative losses, according to

*) The totality of all the available data leaves, in our opinion, no doubt as to the existence of an electronic component of cosmic rays in the Galaxy. However, determining the number of primary electrons near the Earth would make it possible to refine the coefficient in the adopted electron spectra (2.18). Such a refinement is important not only in itself; it is also essential in order to be convinced that there are no additional factors in the solar system (for example, a relatively strong magnetic field) capable, in principle, of leading to a significant reduction of the intensity of primary cosmic electrons near the Earth. Incidentally, let us note that, in order to determine the relative weight of the process of meson formation and, subsequently, of positrons, it would also be possible to attempt to observe the best and central positron lines. At present, however, carrying out the corresponding measurements is most likely practically impossible.

(3.21) \(T_e \sim 1.25 \cdot 10^{16}\), and, probably, this time must still be increased by a factor of \(2 \div 3\) (see above). Thus, the radiative losses are indeed somewhat smaller than the bremsstrahlung losses\(^*\), for which \(u_{em} \sim 10^{-30}\ \mathrm{erg}/\mathrm{cm}^3\,\mathrm{sec}\). On the other hand, in nuclear collisions, according to (3.12), the electron component receives a power \(u_e \sim 3 \cdot 10^{-30} \div 3 \cdot 10^{-32}\ \mathrm{erg}/\mathrm{cm}^3\,\mathrm{sec}\), \(U_e = u_e V \sim 3 \cdot 10^{36} \div 3 \cdot 10^{38}\ \mathrm{erg}/\mathrm{sec}\). Comparing the bremsstrahlung losses (3.43) with this value, we see that the balance can be maintained, i.e. the hypothesis of a secondary (nuclear) origin of the electron component of cosmic rays in the Galaxy appears possible. However, the figures already given indicate rather that, as a result of nuclear reactions (the birth of mesons in nuclear collisions), the electrons may nevertheless receive \(10 \div 30\) times less energy than is required to maintain the balance. Since all the calculations in question obviously cannot claim an accuracy greater than one or two orders of magnitude, no more definite conclusions can yet be drawn in this respect. In any case, the hypothesis that the greater part of cosmic electrons is accelerated in some sources meets with no objections, although, in view of what has been said, it is not obligatory. (See the addition in proof on p. 96.)

c) Diffusion and statistical acceleration of particles in the Galaxy

In regions with a homogeneous magnetic field constant in time, cosmic particles move along helical lines with radius (2.20); the velocity of motion of particles along the field is

\[ v = c \cos \theta, \tag{3.44} \]

where the particle velocity is taken equal to \(c = 3 \cdot 10^{10}\), and \(\theta\) is the angle between the velocity and the field.

If the field is constant but inhomogeneous, the energy of the particle still remains constant, while the trajectory is close to a helical line only for a sufficiently slow dependence of the field strength \(H\) on the coordinates. In the latter case the particle moves along a field line in such a way that

\[ \frac{\sin^2 \theta}{H} = \frac{\sin^2 \theta_0}{H_0} = \mathrm{const}. \tag{3.45} \]

Obviously, the particle moves along the field line in one direction only so long as

\[ H < \frac{H_0}{\sin^2 \theta_0}. \]

In the case, however, when the field increases so that the value

\[ H = \frac{H_0}{\sin^2 \theta_0} \]

is reached, the angle \(\theta\) becomes equal to \(\pi/2\), the velocity \(v = 0\) (see (3.44)), and the particle is reflected, after which it moves along the field line in the opposite direction.

Thus, knowing the configuration of the magnetic field in the Galaxy, one can form a fairly complete picture of the spatial distribution and character of the motion of cosmic particles. Unfortunately, the available information about the Galactic magnetic field is still far from complete, and we shall have to make certain assumptions. We shall

\(^*\) At the same time, the closeness of the values \(T_{e,\mathrm{eff}}\) and \(T_e\) makes it possible, when composing the energy balance in the first approximation, not to consider separately those electrons that move at a small angle to the field lines (for these electrons the component of the field \(H_\perp\) is small, and consequently the bremsstrahlung losses are weak).

one assumes that the magnetic field in the Galaxy is chaotic and is characterized by some length \(l\), over which the field is approximately constant, after which the direction of the field changes substantially. In fact this is certainly not everywhere so and, for example, in the arms of the galactic spiral there is a certain regular field; but, on the other hand, the volume of these arms is[^30] only about \(1\%\) of the volume of the quasi-spherical Galaxy filled with cosmic rays and having radius \(R \sim 3 \div 5 \cdot 10^{22}\) cm. Therefore, if outside the spiral arms there is no regular field at all, or if the tendency toward ordering of the field is small, the assumption of chaos will be justified. Such an assumption has a number of grounds, in particular of a radio-astronomical character. As for the length \(l\) over which the field is quasi-uniform, it is natural to suppose that this length is, in order of magnitude, equal to the distance between gas clouds,[^30] i.e. reaches \(100\) parsecs \(\simeq 3 \cdot 10^{20}\) cm. In reality, of course, there is a whole set of values of \(l\), and here we are evidently speaking of some effective length. For what follows, it is primarily such an effective length \(l\) that is important, determining the diffusion coefficient of cosmic particles:

\[ D=\frac{lv}{3}\sim 3\cdot 10^9\,l \sim 10^{30}\ \text{cm}^2/\text{sec} \simeq 3\cdot 10^{37}\ \text{cm}^2/\text{year}, \tag{3,46} \]

where the velocity of motion along the lines of force \(v\) (see (3,44)) has been put equal to \(\sim 10^{10}\) cm/sec.

In the disk of the Galaxy the length \(l\) and the coefficient \(D\) are evidently smaller than the adopted values. However, for the galactic corona and for the Galaxy as a whole, the value (3,46) seems quite admissible. The effective “mean free path” \(l\) and the diffusion coefficient \(D=\dfrac{lv}{3}\) may, of course, be regarded as independent of the energy only if the radius of curvature \(r\) satisfies the inequality

\[ r=\frac{E\sin\theta}{300\,ZH}\ll l. \]

For \(Z=1\), \(H\sim 10^{-5}\), \(\sin\theta\sim 1\), and \(l\sim 3\cdot 10^{20}\), this condition is fulfilled up to energies \(E\sim 10^{17}\) eV; for nuclei the situation is still more favorable. If, however, among cosmic rays there exist protons with energy \(E>10^{17}\) eV, then for them the diffusion coefficient will depend on the energy and in magnitude will exceed the value (3,46). Since even at \(E\sim 10^{18}\) the radius \(r\sim l\sim 3\cdot 10^{20}\), this effect will not be especially large, quite apart from the fact that the dependence of \(D\) on \(E\) can be taken into account and does not alter the diffusion picture.

Below, without further qualification, we shall use diffusion concepts, as applied to cosmic rays in the Galaxy, analogous to those used when considering the diffusion of molecules in a gas or liquid. It is therefore necessary to emphasize here that the validity of such an approximation is not obvious, and the question of its accuracy will still have to attract close attention.

The diffusion of cosmic particles in the Galaxy will be considered below in Section 4b. Here we shall make only two remarks on this matter. First, it must be noted that diffusion proceeds rapidly enough for the entire gigantic galactic corona to be filled with cosmic particles accelerated near the galactic plane. Indeed, the mean distance \(L\) traveled by a particle in time \(t\) as a result of diffusion is

\[ L=\sqrt{2Dt}\sim \sqrt{lvt}\sim 10^{15}\sqrt{t}, \]

where the value (3,46) has been used.

During the lifetime of cosmic protons \(T_p \sim 10^{16}\) sec, the path \(L \sim 10^{23}\), i.e., is sufficiently large to fill a region with radius \(R \sim 3 \div 5 \cdot 10^{22}\) cm.

The second remark concerns the question of the escape of particles from the Galaxy as a result of diffusion. In all, in the Galaxy, with volume \(\sim 10^{68}\ \mathrm{cm}^3\), there are \(N_t \sim N(E > 10^9\ \mathrm{eV}) \cdot V \sim 10^{58}\) cosmic particles (see (1,5) and (2,21)). If the boundaries of the Galaxy were not reflecting, then the diffusion flux through these boundaries would be equal to

\[ S \sim 4\pi R^2 D \frac{\partial N}{\partial r} \sim (0.1 \div 1)4\pi R^2 D \frac{N}{R} \sim \]

\[ \sim 4RlvN \sim 10^{42} \div 10^{43} \]

particles per second, where it is assumed that \(R \sim 5 \cdot 10^{22}\), \(v \sim 10^{10}\), \(N \sim 10^{-10}\), and \(l \sim 10^{20}\) cm. Taking into account reflection from the boundaries, this value may decrease very strongly, so that for a transmission coefficient \(\sim 1\%\) the value \(S \sim 10^{40} \div 10^{41}\) does not appear to be underestimated*). At the same time, owing to nuclear collisions in the Galaxy,

\[ \frac{N_t}{T_p} \sim \frac{10^{58}}{10^{16}} \sim 10^{42} \]

particles per second disappear. Thus it appears probable or, in any case, quite possible, that the lifetime of particles in the Galaxy is indeed determined by nuclear losses, as was assumed above. At the same time the estimates given are very rough, and the transmission coefficient through the galactic boundaries is unknown. Therefore, undoubtedly, in neglecting the escape of cosmic particles from the Galaxy, we are making a certain assumption which, although very plausible, still cannot be regarded as completely justified.

In diffusion in the interstellar medium carrying magnetic fields, the energy of cosmic particles changes not only because of the presence of various losses, but also as a result of the motion of the medium itself. For the latter reason there is a time-variable component of the magnetic field, which leads to the appearance of an electric field and to a change in the particle energy. This process can be divided into two: energy changes connected with systematic strengthening or weakening of the magnetic field, and energy changes occurring even when the mean field strength is unchanged. The first acceleration mechanism is inductive in the usual sense of the word and is entirely analogous to that occurring in a betatron. Since in the Galaxy at present the field as a whole is probably stationary on the average or changes very slowly, such systematic inductive acceleration should not occur. As for acceleration of any nature, but not connected with a change in the mean magnetic-field strength, we shall call it statistical acceleration.

In its clearest form, the statistical mechanism \(^{48}\) of acceleration operates in the accepted diffusion model, in which charged particles move chaotically between magnetic clouds, which act as scattering centers. In a “collision” with a cloud,

*) Without taking into account the drift of particles caused by the inhomogeneity of the magnetic field, the transmission coefficient is equal to the ratio of the field strengths outside and inside the system (i.e., in the present case the Galaxy). The drift velocity, occurring perpendicular to the gradient of the magnetic field, is, in order of magnitude, equal to \(v_d \sim \dfrac{r}{L}v\), where

\[ r = \frac{E}{300H} \]

is the radius of curvature of the particle, \(v\) is its velocity, and \(L\) is the characteristic distance over which the magnetic field changes appreciably. From this it can be seen (the author is grateful to S. I. Syrovatskii for discussing this question) that for the main mass of cosmic particles with energy \(E \sim 10^9 \div 10^{13}\ \mathrm{eV}\) their escape from the Galaxy connected with drift is in all probability very small and plays no role. However, for particles with energy \(E > 10^{16}\ \mathrm{eV}\), the escape of particles caused by drift may already be very substantial. This circumstance may perhaps be connected with the steeper fall of the energy spectrum of cosmic particles in the region of very high energies.

having some velocity \(u \ll c\), the energy of the particle \(E\) changes by an amount \(\Delta E\) of order \(\dfrac{u v}{c^2}\), where \(v\) is the velocity of the particle’s translational motion. The quantity \(\Delta E\) depends on the angle between the velocities \(u\) and \(v\), and, for their isotropic distribution, as a result of averaging,

\[ \overline{\Delta E}\sim \frac{u^2}{c^2}E. \tag{3.47} \]

Here the existence of a nonzero quadratic effect (3.47) is connected with the fact that the relative velocity, and hence also the number of collisions, in the case where the particle and the cloud move toward one another is greater than the relative velocity in the case where the particle overtakes the cloud (for more detail see \(^{48}\), or, for example, \(^{5}\); for complete isotropy \(\overline{\Delta E}=\dfrac{4}{3}\dfrac{u^2}{c^2}E\)).

Exactly the same process of energy transfer would occur in the case of a mixture of two gases or of spheres of two kinds under conditions where the average energy of the heavy molecules or spheres is considerably greater than the energy of the light molecules or spheres. Cosmic particles evidently play the role of the light spheres, and magnetic clouds the role of the heavy ones. Hence it is quite clear that statistical acceleration should not be identified with induction acceleration, despite the fact that in the very process of a particle’s “collision” with a cloud an induction electric field does, of course, act. The features of statistical acceleration and, specifically, expression (3.47), over a wide range do not depend on the details of the adopted model with magnetic clouds and apparently occur under very general assumptions about the character of the chaotic motion of a conducting medium carrying magnetic fields (see in this connection \(^{49}\)).

The statistical mechanism of acceleration leads to the following increase in the mean energy of a particle per unit time:

\[ \frac{dE}{dt}=\alpha E=\frac{u^2}{c^2\tau}E=\frac{u^2 v}{c^2 l}E, \tag{3.48} \]

where \(\tau=\dfrac{l}{v}\) is a certain effective free time between collisions and \(l\) is the corresponding mean free path.

In passing from (3.47) to (3.48), in the last of these expressions an equality sign has been put (and not equality in order of magnitude), since all numerical factors can be referred to the effective time or mean free path \(\tau\) and \(l\). The effective length \(l\) in (3.48), generally speaking, is different from the length \(l\) determining the diffusion coefficient (3.46). However, within the limits of the accuracy now attainable, it is hardly possible to make any distinction here.

The velocity of the gas (clouds) \(u\) can be estimated from the relation \(\dfrac{H^2}{8\pi}\sim \dfrac{\rho u^2}{2}\); taking for the galactic corona \(H\sim 10^{-5}\) and \(\rho\sim 10^{-25}\ \mathrm{g/cm^3}\) (i.e., \(n\sim 0.1\ \mathrm{cm^{-3}}\)), we obtain \(u\sim 10^7=100\ \mathrm{km/sec}\). Hence, for \(l\sim 3\cdot 10^{20}\) and \(v\sim 10^{10}\), we obtain \(\alpha\sim 3\cdot 10^{-18}\ \mathrm{sec^{-1}}\). Using astronomical data \(^{50}\) on the velocities of clouds of interstellar gas in the plane of the Galaxy, we arrive at the value \(\alpha\sim 10^{-18}\). As for the value \(\alpha\sim 10^{-14}\), which is used in \(^{51}\), it not only is not based on any observational data, but also contradicts them, if one speaks of acceleration in the interstellar medium as a whole and not in some distinguished regions of it.

Let us now see what requirements may be imposed on the value of \(\alpha\) on the basis of data on the energy spectrum of cosmic rays. For protons and nuclei of sufficiently high energy, when ionization losses are insignificant, the differential spectrum (distribution function with respect to energies) \(N(E)\), in the presence of a statistical mechanism, is determined

from the equation

\[ \frac{\partial}{\partial E}[\alpha E N(E)] + \frac{N(E)}{T} = q(E). \tag{3.49} \]

This equation is, obviously, a special case of equation (3.31) for \(a=b=0\) (therefore we shall not give any further explanations here). If, for simplicity, it is assumed that certain sources (injectors) accelerate particles only up to some energy \(E_0\), after which these particles enter the interstellar medium, then \(q(E>E_0)=0\). Then for \(E>E_0\) the solution of equation (3.49) has the form (see (3.38)):

\[ N(E)=\frac{K}{E^{\left(1+\frac{1}{\alpha T}\right)}} , \qquad K=\frac{1}{\alpha}\int_0^{E_0} q(\eta)\eta^{\frac{1}{\alpha T}}\,d\eta, \tag{3.50} \]

i.e. the spectrum is a power-law one \(\left(N=\dfrac{K}{E^\gamma}\right)\) with exponent

\[ \gamma = 1+\frac{1}{\alpha T}. \tag{3.51} \]

For primary cosmic rays near the Earth, in a wide energy interval \(\gamma=2\div 3\) (see Section 16). Hence, if one assumes that cosmic rays are accelerated mainly in the interstellar medium\(^{48}\),

\[ \alpha T \sim 1. \tag{3.52} \]

For protons \(T\simeq T_p\sim 10^{16}\) sec. (see (3.5)), whence

\[ \alpha \sim 10^{-16}\ \text{sec}^{-1}. \tag{3.53} \]

Such a value is large from an astronomical point of view (see above), but nevertheless is probably admissible, taking into account the inaccuracy of the available data on the parameters \(l\) and \(u\) in the galactic corona (one must also keep in mind the possibility of the known decrease of \(\alpha\) due to an increase in the time \(T_p\)). A serious difficulty, however, is that condition (3.52) cannot be fulfilled simultaneously for both protons and nuclei (thus, for iron nuclei \(T_{\mathrm{Fe}}\simeq \frac{1}{25}T_p\)). At the same time, because of the closeness of the spectra of protons and nuclei, fulfillment of condition (3.52) for nuclei must be required at least for the main mass of cosmic rays with energy \(E\lesssim 10^{10}\) eV/nucleon; this also applies, according to less reliable data, to the region \(E\lesssim 10^{13}\) eV/nucleon (see Section 16).

The conclusion drawn, as has already been emphasized earlier in \(^{1}\), remains valid also when one takes into account not only systematic but also fluctuational growth of particle energy under the action of the statistical mechanism. The systematic statistical acceleration determined by formulas (3.47)—(3.48) is a second-order, quadratic effect. Particles, however, can acquire a large energy even in the absence of this quadratic effect, owing to fluctuations or, if one likes, Brownian motion or diffusion in energy space. In fact, at each collision a particle, as noted, receives an energy of order \(\Delta E\sim \dfrac{uv}{c^2}E\). The quantity \(\Delta E\) plays the role of a mean free path \(l_E\) in “energy space” and can have either sign, in consequence of which, upon averaging, only the quadratic effect (3.47) remains. There is, however, a certain probability that the number of collisions with \(\Delta E>0\)

will be greater than the number of collisions with \(\Delta E<0\), as a result of which the energy increases, just as the displacement of a Brownian particle increases with time. In order to take into account the fluctuation change of energy in an equation of the type (3.30), let us note that this change may be regarded as diffusion in energy space, with the corresponding diffusive particle flux

\[ j_D(E)=-D_E\frac{\partial N}{\partial E}, \]

where

\[ D_E\sim \frac{l_E^2}{\tau}\sim \frac{\overline{(\Delta E)^2}}{\tau}\sim \frac{u^2v^2E^2}{c^4\tau}\sim \alpha \frac{v^2}{c^2}E^2 \]

(here \(\tau=\dfrac{l}{v}\) is the free-path time, and \(\alpha\) is the quantity defined according to (3.48)).

As a result, the equation for the function \(N(E)\), with allowance for systematic and fluctuation acceleration, takes the form (see (3.29), (3.30), and (3.49)):

\[ \frac{\partial}{\partial E}(\alpha EN)-\frac{\partial}{\partial E}\left(\beta E^2\frac{\partial N}{\partial E}\right)+\frac{N}{T}=q, \tag{3.54} \]

where \(\beta E^2=D_E\).

In the general case the coefficients \(\alpha\) and \(\beta\) in this equation are independent, and in principle it is possible that \(\beta\gg\alpha\), or even that \(\beta\ne0\) and \(\alpha=0\). This will occur, for example, if a particle successively passes through regions with increasing and decreasing magnetic field, while on average over the entire region the field does not increase. However, within the framework of the diffusion model adopted, as shown above, \(\beta\sim \alpha \dfrac{v^2}{c^2}\lesssim \alpha\).

Starting from (3.54), we obtain a solution of the type (3.50), and for \(\beta\lesssim\alpha\) it is still necessary to require that condition (3.52) be satisfied; for example, when \(\alpha=\beta\), instead of (3.51) we obtain \(\gamma=\sqrt{1+\dfrac{1}{\alpha T}}\), which for \(\gamma=2\div3\) is almost equivalent to (3.51). If the value \(\alpha T_p\) is chosen so that \(\gamma=2\), then for iron nuclei \(\gamma=\sqrt{1+\dfrac{25}{\alpha T_p}}=8.7\). This result is in decisive contradiction with reality.

Thus, if the lifetime of cosmic particles is determined by nuclear losses, the statistical mechanism of acceleration in the interstellar medium proves unacceptable. Therefore, in \(^{51,52}\) an attempt was made to use a model in which the time \(T\) is determined by the escape of particles from the system. In this case one must assume that \(T<T_{\mathrm{Fe}}\), where \(T_{\mathrm{Fe}}\) is the nuclear lifetime for iron. Such a model, as applied to the entire Galaxy, contradicts the ideas developed above and also encounters a number of other difficulties, which will be discussed further in section 4г (see also \(^{42,53}\)).

Statistical acceleration in the interstellar medium is ineffective if

\[ \alpha T\ll1,\qquad \alpha\ll\frac{1}{T}\sim10^{-16}. \tag{3.55} \]

The estimates of \(\alpha\) given above show that fulfillment of condition (3.55) is very probable. Under such conditions the spectrum of cosmic rays is determined by the spectrum of particles generated in the sources. In order to see clearly what role statistical acceleration in the interstellar medium plays in this case, let us put: \(q(E)=\dfrac{A}{E^{\gamma_0}}\) for \(E>E_0\) and \(q(E)=0\) for

\(E < E_0\). Then the solution of equation (3.49) has the form (see (3.38))

\[ N(E)= \frac{A}{\left(1+\frac{1}{aT}-\gamma_0\right)aE^{\gamma_0}} + \frac{AE_0^{\left(1+\frac{1}{aT}-\gamma_0\right)}}{\left(1+\frac{1}{aT}-\gamma_0\right)aE^{\left(1+\frac{1}{aT}\right)}}. \tag{3.56} \]

Under condition (3.55) and \(\gamma_0 \sim 1\), in the region \(E \gg E_0\), evidently \(N(E)=\dfrac{AT}{E^{\gamma_0}}=q(E)T\), as is also immediately clear from (3.49).

The ineffectiveness of acceleration in the interstellar medium is also indicated by radio-astronomical data, according to which the spectrum of cosmic electrons remains unchanged in the energy range \(10^8 < E < 5\cdot 10^9\) eV. It follows from this, as was shown in Section 3b, that up to energies \(E\sim 10^8\) eV magnetobremsstrahlung losses dominate. Meanwhile, if there were strong statistical acceleration, the same for electrons as for protons and nuclei (what is meant is the value of the coefficient \(a\) in (3.48)), this could no longer be the case. Indeed, with simultaneous statistical acceleration and magnetobremsstrahlung losses (the energy \(E\) is measured in eV),

\[ \frac{dE}{dt}=aE-bE^2=aE-4\cdot 10^{-15}H_{\perp}^{2}E^2 . \tag{3.57} \]

In the region where \(aE>bE^2\), electrons are accelerated; in the region \(aE<bE^2\), they are slowed down. Therefore the electron spectrum will have a condensation point at the energy \(E_c=\dfrac{a}{b}\), which would also be reflected in the radio-emission spectrum. The known inaccuracy of radio-astronomical data and, most importantly, the variability of the quantity \(b\sim H_{\perp}^{2}\) along the line of sight do not yet make it possible to reach strict conclusions in this respect. However, it is most probable that \(E_c<10^8\) eV, whence

\[ a<bE=4\cdot 10^{-15}H_{\perp}^{2}E\sim 4\cdot 10^{-17}\ \mathrm{sec}^{-1}, \tag{3.58} \]

which is in agreement with (3.55).

Further refinement of radio-astronomical measurements can thus lead to a quite reliable estimate of the upper limit of the parameter \(a\), although even now this estimate seems to us, as applied to the galactic corona, quite convincing.

4. SUPERNOVAE AND NOVAE AS SOURCES OF COSMIC RAYS

A theory of the origin of cosmic rays must obviously indicate those sources in which particles are accelerated to relativistic energies. It is further necessary to explain the experimental data concerning the energy spectrum, composition, and isotropy of cosmic rays.

The totality of available information makes natural the hypothesis that cosmic rays are formed in the expanding envelopes of supernovae and, probably, novae. Leaking out from the envelopes of these stars, located near the galactic plane, cosmic particles fill the entire quasi-spherical volume of the Galaxy, where they lose their energy mainly as a result of nuclear collisions. We shall now proceed to a detailed justification of this point of view.

a) Energy balance.
Acceleration of particles in the envelopes of supernova stars

The most essential requirement that must be imposed on the sources of cosmic rays follows from energetic considerations. As we have seen, the energy lost by cosmic rays in the Galaxy per unit time is equal to (see (3.7))

\[ U_{cr}\sim 10^{39}\div 10^{40}\ \text{erg/sec}. \]

This value refers to protons and nuclei. In the case of electrons, the energy loss is equal to (see (3.43))

\[ U_e\sim 10^{38}\ \text{erg/sec}. \]

These powers are very large. For example, on the Sun, on average, \(\sim 10^{23}\div 10^{24}\) particles per second are produced with energies of the order of several Bev. This means that on the Sun no more than \(10^{21}\div 10^{22}\ \text{erg/sec}\) on average is converted into cosmic rays, and all \(10^{11}\) stars of our Galactic system, emitting cosmic rays with the same power, would yield \(10^{32}\div 10^{33}\ \text{erg/sec}\). The discrepancy between this value and the required power (3.7) amounts to 6–8 orders of magnitude, which speaks quite clearly of the severity of the energetic requirements on the sources of cosmic rays.

Supernovae and novae satisfy these requirements. In supernova outbursts, an energy \(\sim 10^{48}\div 10^{50}\) erg is converted into visible radiation. With a frequency of outbursts \(\sim 1/30\) per year, this gives a power of \(10^{39}\div 10^{41}\ \text{erg/sec}\). The very fact of such a large release of energy already attracts attention and served as the reason for proposing the hypothesis of the formation of cosmic rays during supernova outbursts (see \(^{54}\)). But, of course, the release of energy in the form of light does not yet in any way guarantee its conversion into cosmic rays. Therefore the hypothesis of cosmic-ray generation in supernovae attracted serious attention not only in light of radio-astronomical data \(^{55,36,1,12}\). Indeed, as was emphasized in section 2b, the envelopes of almost all known supernovae are powerful sources of radio emission, which testifies to the presence of relativistic electrons in these envelopes. The power converted into relativistic electrons, according to observations, is (see (2.32))

\[ U_{e,sn}\sim 10^{36}\div 10^{39}\ \text{erg/sec}. \]

In addition, novae probably supply \(3\cdot 10^{36}—3\cdot 10^{38}\ \text{erg/sec}\). After a time of the order of \(1000\div 3000\) years the envelopes of supernovae break up (are dispersed) in interstellar space. The same happens with the cosmic particles contained in the envelopes, which, however, may leave the envelope also at an earlier stage as a result of diffusion. Thus, supernovae and novae must supply the Galaxy with cosmic electrons with a power of \(10^{37}\div 10^{39}\ \text{erg/sec}\); this power is sufficient to maintain the energy balance, since the energy losses by electrons amount to \(\sim 10^{38}\ \text{erg/sec}\).

There are no direct data on the energy that is converted during stellar outbursts into protons and nuclei. However, the possibility of conversion into these particles of at least the same energy as is converted into electrons appears obvious even without special analysis. If, however, one considers the mechanism of particle acceleration in supernova envelopes in more detail, it becomes clear that it is possible and probable for energy to be converted into protons and nuclei in an amount 10–1000 times greater than the energy imparted to electrons (see below). Thus, one may expect that

on average, as a result of supernova and nova outbursts, the power transferred to cosmic rays will be

\[ U_{sn}\sim 10^{38}\div 10^{42}\ \text{erg/sec}. \tag{4.1} \]

Such an energy release is already sufficient for the energy balance in the Galaxy to be satisfied (see above).

Thus, supernovae and novae do indeed satisfy the very stringent energy requirements imposed on the sources of cosmic rays.

The identification of cosmic-ray sources with the envelopes of supernovae is based on evidence for the presence of relativistic particles in these envelopes. Therefore the question of the mechanism of particle acceleration as a result of supernova outbursts constitutes, to a certain extent, an independent problem, whose consideration is important and interesting but cannot substantially affect our conclusions*). In this connection, and also in view of the absence of sufficiently reliable information on the ejection and early phases of expansion of supernova envelopes, we shall restrict ourselves, with regard to the mechanism of particle acceleration, to only a few remarks.

In interstellar space statistical acceleration is rather inefficient (see Section 3b) for two reasons. First, the velocity of motions in the interstellar medium is relatively small \((u \lesssim 10^7\ \text{cm/sec})\). Second, the mean free path \(l\) is very large and reaches \(3\cdot 10^{20}\ \text{cm}\). At the same time the coefficient \(\alpha\), which determines the rate of increase of the energy (see (3.48)), is proportional to the ratio \(\dfrac{u^2}{l}\). The situation changes radically\(^{35,56}\) in regions with small-scale turbulence and, in particular, in the envelopes of supernovae and novae. In the envelopes of supernovae velocities \(u\sim 1\div 3\cdot 10^8\ \text{cm/sec}\) are encountered, while the scale of the turbulence in the early stages may quite well be equal to \(l\sim 10^{13}\ \text{cm}\). Hence

\[ \alpha=\frac{u^2 v}{c^2 l}\sim 10^{-7}\div 10^{-8}, \]

instead of values \(\sim 10^{-17}\div 10^{-18}\) in the interstellar medium. A more detailed consideration\(^{39}\), under the assumption of the existence of locally isotropic turbulence in the envelope, also leads, for the early stages of the envelope evolution, to values \(\alpha\sim 10^{-8}\). At later stages it is also necessary to take into account the presence of the general systematic expansion of the envelope. For the latter reason, instead of formula (3.48) one must use the expression\(^{39}\):

\[ \frac{dE}{dt}=\alpha_{\mathrm{eff}}E=\left[u^2-\frac{h v V l}{r}\right]\frac{v}{c^2 l}\,E, \tag{4.2} \]

where \(V\) is the expansion velocity of the envelope with radius \(r\), \(u\) and \(l\) are the velocity and characteristic length for turbulent (chaotic) motions, and, finally, \(h\) is a coefficient of order unity.

The presence in (4.2) of a negative second term leads to the result that, for a large velocity \(V\), at sufficiently late stages of expansion the particles are no longer accelerated, but are decelerated. On the other hand, when several envelopes are ejected, or when there is an additional outflow of gas masses from the central star, there are regions where not expansion but compression of the gas occurs. Under such conditions, in formula (4.2) the sign of the second term must be changed and \(r\) must be regarded as the characteristic size of the compressing

*) The situation is entirely different if one is speaking of some hypothetical sources of relativistic particles. In that case, with no observational data on the number and energy of the accelerated particles, it is necessary at least to show theoretically that the corresponding acceleration is not only possible, but also sufficiently effective.

regions. For \(V \sim 10^8\) and \(r \sim 10^{14}\), under such conditions we obtain \(\alpha_{\mathrm{eff}} \sim \dfrac{v^2 V}{c^2 r} \sim 10^{-7}\). In the Crab Nebula an additional ejection of matter may also be taking place at the present time\(^{40}\), i.e., 900 years after the stellar outburst. It is likely that in the early stages it was incomparably more intense. The ejection of additional shells has also been observed in a number of novae. Finally, there are some grounds for supposing\(^{57,40a}\) that the acceleration of particles to relativistic energies must occur in the course of the explosion of the central star itself, before the separation of its expanding shell.

The physical conditions and gas dynamics in the explosion of supernovae and novae still cannot be regarded as sufficiently studied. Nevertheless, on the basis of what has been said it is clear that, in stellar explosions, the statistical acceleration mechanism must indeed be very effective; induction acceleration associated with the very probable growth of the magnetic field may also play a certain role.

For statistical acceleration that began at the moment \(t=0\), the energy of a particle at the moment \(t\) is equal to

\[ E(t)=Mc^2\exp\left\{\int_0^t \alpha_{\mathrm{eff}}(t)\,dt\right\}, \tag{4.3} \]

where it is assumed that initially the particle with rest mass \(M\) was nonrelativistic (therefore \(E(0)=Mc^2\)).

With \(\alpha_{\mathrm{eff}}\sim 10^{-7}\) per year \((t\simeq 3\cdot 10^7)\), the energy of the particle reaches the value \(E\sim 20\,Mc^2\), i.e., it will already be relativistic. The exponential character of the dependence (4.3) leads to the fact that the energy is very sensitive to the choice of parameters and, for example, for \(\alpha\sim 10^{-9}\) and \(t\sim 1000\) years \(\simeq 3\cdot 10^{10}\), the energy is \(E\sim 10^{13}Mc^2\), which for protons corresponds to the energy \(E\sim 10^{21}\) eV (!). For Cassiopeia A, the remnant of the supernova of 369, the value \(t\sim 1000\) years is evidently admissible. Further, the shell of this star is expanding comparatively slowly, as a result of which the second term on the right-hand side of equation (4.2) is probably insignificant. Finally, the turbulent motions in the shell are very intense, and for them \(u\sim 10^8\) cm/sec. Hence the value \(\alpha_{\mathrm{eff}}\sim \alpha\sim 10^{-9}\) is obtained for \(l\sim 10^{14}\). Such a value does not seem unreasonable, at least during a certain period of time when the radius of the shell was smaller than the present radius \(r_0\sim 10^{18}\) cm.

In the statistical mechanism, the energy reached by a particle is proportional to its mass (see (4.3)). Therefore, if electrons are accelerated in the shell to the energy \(E\), then protons are accelerated to an energy at least

\[ \frac{M}{m}=1836 \]

times greater. The gap in energies may be still larger owing to magnetic-bremsstrahlung losses, which hinder the increase of electron energies.

In the Crab Nebula (Taurus A) there are certainly electrons with energies \(10^{11}\div 10^{12}\) eV (see \(^{21,40}\)) and, consequently, protons will have an energy greater than \(\sim 10^{14}\div 10^{15}\) eV, and, for example, iron nuclei an energy of \(10^{16}\div 10^{17}\) eV. The spectrum of electrons with energy in the region \(10^{11}\) eV has, according to \(^{21,40}\), the index \(\gamma=3\div 3.3\). In this region magnetic-bremsstrahlung losses are very substantial, whence it follows that for protons and nuclei one should expect a spectrum with degree \(\gamma=\gamma_0=2\div 2.3\) (see (3.39)). The latter value, within the achieved accuracy, agrees precisely with the spectrum of cosmic rays (see Section 16). In Cassiopeia A the spectrum of electrons with energies \(10^8\div 10^{10}\) eV is different from the spectrum of electrons in Taurus A. Already

It follows from this that one should not expect some single spectrum for all supernova shells. It is only important that in all known cases \(\gamma \simeq 2 \div 3\). Therefore it is quite possible to assume that the resulting mean spectrum of all cosmic radiation is formed as a result of the superposition of the spectra of cosmic particles accelerated in supernova outbursts (the energy of cosmic rays in the Galaxy, equal to \(\overline W_{cr}\sim 10^{55}\div 10^{56}\) ergs, corresponds to the energy converted into cosmic rays as a result of \(\sim 10^6 \div 10^7\) supernova outbursts). Since the energy of particles in the shell is proportional to their mass, the total energy contained in protons, for an equal number of accelerated protons and electrons, will be

\[ \frac{M}{m}=1836 \]

times greater than in electrons (energy losses are neglected; the conclusion reached is easily checked for any power-law spectrum \(^{39}\)). Such an assumption, however, encounters certain difficulties \(^{58,39}\), since the energy contained in relativistic particles turns out to be, in order of magnitude, equal to or even greater than all the energy of radiation released in the outburst. The energy of the particles is then greater than the kinetic energy in the shells, which is incompatible with the statistical mechanism of acceleration in these shells, in which the particles’ energy is drawn from the kinetic energy of the gas masses. This difficulty can be avoided if one takes into account the possibility of the generation of cosmic particles in the explosion of the central star itself \(^{57}\) or as a result of its subsequent activity. Furthermore, the injection conditions may be more favorable for electrons, which will lead to a larger number of them being accelerated in comparison with protons. Finally, the total energy of the electrons may be “brought up” to the total energy of the protons if one takes into account the possibility of accelerating electrons starting from relativistic energies \(^{39}\).

Thus we see that it is necessary to explain specially why the total energy of electrons is only comparatively slightly less than the total energy of protons and nuclei. The opposite situation, in which fast protons and nuclei contain substantially (thousands of times) more energy than electrons, is quite natural and probable. But precisely such an assumption—that the total energy of protons and nuclei is greater than the energy of electrons—was made above and is therefore justified.

The totality of the considerations presented shows that, as a result of supernova outbursts, cosmic rays satisfying the appropriate requirements with respect to total energy and energy spectrum can quite well be produced.

b) Composition of cosmic rays near the Earth

Novae are rather strongly concentrated near the galactic plane and are found, mainly, in a region of thickness \(\sim 2\cdot 10^{21}\) cm near it; the concentration of novae toward the galactic center is apparently not very strong, but it may nevertheless be assumed that the overwhelming majority of them are located closer to the center of the Galaxy than the solar system (the distance of the Sun from the galactic center is \(R_0 \approx 2.5\cdot 10^{22}\) cm). The spatial distribution of supernovae is unknown. It probably differs little from the distribution of novae. The chemical composition of supernova shells is also insufficiently clarified. There are, however, indications \(^{59,60}\) that in this case the content of heavy elements is substantially greater than the average in nature. As for the hydrogen content, this element in supernovae nevertheless, in all probability, remains rather abundant.

V. L. GINZBURG

Despite the paucity of all these data, it is necessary to use them to clarify questions concerning the composition and isotropy of cosmic rays near the Earth. Let us begin by considering the first of these problems[^61].

To determine the composition of cosmic rays at some point (for example, near the Earth), one must use the system of equations:

\[ \frac{\partial N_i}{\partial t} = \nabla \left(D_i \nabla N_i\right) - \frac{N_i}{T_i} + \sum_{j>i} p_{ij}\frac{N_j}{T_j} + q_i, \tag{4,4} \]

where \(N_i(\mathbf r,t)\) is the concentration of nuclei of type \(i\), \(q_i(\mathbf r,t)\) is the number of nuclei of the same type produced in sources, per unit volume and unit time, \(D_i(\mathbf r)\) is the diffusion coefficient, and \(T_i(\mathbf r)\) is the lifetime of a nucleus of type \(i\) before its disintegration as a result of collision with nuclei of atoms of the interstellar medium; finally, \(p_{ij}\) is the probability of formation of a nucleus of type \(i\) upon the disintegration of a nucleus of type \(j\).

In the theory of the origin of cosmic rays under discussion, the sources \(q_i\) are located near the galactic plane, with a known concentration toward the galactic center (see above). Diffusion, however, occurs throughout the entire galactic corona with radius \(R \sim 3\div 5\cdot 10^{22}\) cm. In this case the gas density near the galactic plane is greater than at the periphery, as a result of which the quantities \(D_i\) and \(T_i\) depend on the coordinates. However, in a first approximation) we shall not take this dependence into account, assuming that \(D_i = D \sim 3\cdot 10^{37}\ \mathrm{cm^2/year}\) (see (3,46)). The times \(T_i\) will also be regarded as independent of the coordinates and related to one another in accordance with variant в* in Table VI, when \(T_L:T_M:T_H = 3:2:1\).

Assuming that during the lifetime of cosmic rays the structure of the Galaxy does not change, we shall put \(\partial N_i/\partial t = 0\). As for the coefficients \(p_{ij}\), they are given in Table VIII (see [^8,^61]); here the coefficient \(p_{LM}\) is the probability of formation of a nucleus of group \(L\) when a nucleus of group \(M\) collides with nuclei of the interstellar medium, etc.

Table VIII

Values of the coefficients \(p_{ij}\)

Variant A	\(p_{LM}=p_{LH}=0.23,\quad p_{MH}=0.27\)
Variant B	\(p_{LM}=0.42,\quad p_{LH}=0.48,\quad p_{MH}=0.27\) \(p_{LL}=0.13,\quad p_{MM}=0.13,\quad p_{HH}=0.25\)

The values of \(p_{ij}\) according to variant A are, in all likelihood, minimal**). The data corresponding to variant B, as applied to the interstellar medium, are also of an indicative character (for several other values of \(p_{ij}\), see [^8,^59]). With regard to the coefficients \(p_{pj}\) and \(p_{\alpha j}\), we shall confine ourselves to the obvious remark that \(p_{pj} \leq A_j\) and \(p_{\alpha j} \leq \frac{1}{4}A_j\), where \(A_j\) is the atomic weight of nucleus \(j\). Since \(\alpha\)-particles in cosmic rays are considerably

*) In the next approximation it would be reasonable to consider diffusion in a medium having one (greater) density in the region near the galactic plane and another (smaller) density in the galactic corona.

**) In the most recent work on this subject, which has appeared [^75], values of \(p_{LN}\) and \(p_{LM}\) are given that are smaller by another factor of one and a half than those corresponding to variant A.

more than for all other nuclei, one may think that \(\overline{p}_{pj}\lesssim 4\); analogously, \(\overline{p}_{\alpha j}\lesssim 2\div 4\), where the bar denotes the average over all nuclei, respectively heavier than protons or \(\alpha\)-particles.

For the nuclei of a given isotope, collisions with particles of the interstellar medium are basically “catastrophic,” i.e., they lead to the formation of nuclei with a different atomic weight. However, when nuclei are divided into the groups \(L\), \(M\), and \(H\), this is no longer so, since, for example, all nuclei with \(Z>10\) belong to group \(H\). For this reason, in Table VIII values of the coefficients \(p_{LL}\), \(p_{MM}\), and \(p_{HH}\) different from zero are given. On the other hand, taking into account the transformation of nuclei of some group into nuclei of the same group is equivalent to changing the lifetime of the nuclei of this group. The latter is especially true under conditions when the energy per nucleon of primary and secondary nuclei is the same (this, apparently, corresponds to reality, provided only that the spectra of protons and nuclei do not differ greatly from one another (see Section 16). As a result, with the accuracy presently achieved in determining the coefficients \(p_{ij}\) and the times \(T_i\), it would be impossible to take into account transitions within individual groups of nuclei. Thus, below we shall assume that \(p_{ii}=0\).

Under the assumption of constancy of the energy per nucleon for primary and secondary nuclei, the concentrations \(N_i\) in equations (4.4) may be understood as the concentration of nuclei in any energy interval and, in particular, their total concentration with energies greater than some energy \(E_0\), corresponding to the threshold of geomagnetic or high-latitude cutoff of the spectrum.

In order to clarify the necessity of taking diffusion into account and the character of the distribution of the sources of cosmic rays, let us now assume that these sources are distributed uniformly over the entire volume occupied by cosmic rays. In this case the diffusion term vanishes, and the system of equations (4.4) takes the form

\[ N_i=\sum_{j>i} p_{ij}\frac{N_jT_i}{T_j}+T_iq_i . \tag{4.5} \]

Hence, for variant \(A\) (see Table VIII) we obtain

\[ \frac{q_M}{q_H}=\frac{N_M}{N_H}\cdot \frac{T_M}{T_H}-p_{MH}=1.33, \]

\[ \frac{N_L}{N_H}=\frac{T_L}{T_H}\left[p_{LH}+p_{LM}\left(\frac{q_M}{q_H}+p_{MH}\right)\right]=1.8;\qquad \frac{N_L}{N_M}=0.56. \tag{4.6a} \]

When the coefficients \(p_{ij}\) are chosen in accordance with variant \(B\),

\[ \frac{q_M}{q_H}=1.33;\qquad \frac{N_L}{N_H}=3.5;\qquad \frac{N_L}{N_M}=1.1. \tag{4.6b} \]

In (4.6a)—(4.6b) data from Tables II and VI (variant \(a\)) have also been used, according to which \(\dfrac{N_M}{N_H}=3.2\) and \(T_L:T_M:T_H=3:2:1\). In addition, it is assumed that \(q_L=0\), since the concentration of Li, Be, and B in nature is very small (see Table I). If in the sources \(q_L\ne 0\), then the values of \(\dfrac{N_L}{N_M}\) and \(\dfrac{N_L}{N_H}\) can only increase. Further, even if all protons are primary, i.e., \(p_{pj}=0\), then (see Tables II and IV, variant \(a\))

\[ \frac{q_p}{q_H}=\frac{N_p}{N_H}\frac{T_H}{T_p}\simeq \frac{1}{20}\frac{N_p}{N_H}\simeq 30. \tag{4.7} \]

If, however,

\[ \bar p_{pj}=4,\qquad \bar N_j=\frac{1}{10}N_p \quad \text{and} \quad \frac{\bar T_j}{T_p}=\frac{1}{4} \]

(the bar denotes averaging over all nuclei), then

\[ \frac{q_p}{q_j}=\frac{N_p}{\bar N_j}\frac{\bar T_j}{T_p}-\bar p_{pj}<0. \]

The last result means that even with complete inefficiency of the sources from the point of view of the generation of fast protons, in the case under consideration there would have to be more of them than is observed experimentally. But even the value

\[ \frac{q_p}{q_H}\simeq 30, \]

which is an upper limit, means that nuclei of group \(H\) are generated in the sources approximately 100 times more efficiently than protons. Here it is assumed that in the sources, as on average in nature, there are \(3\div 4\cdot 10^3\) times more protons than nuclei of group \(H\) (see Table I). Meanwhile there are no grounds whatever for believing that nuclei can be accelerated more efficiently (with greater probability) than protons\({}^{39}\). To assume, on the other hand, that in the sources the proton content is several orders of magnitude smaller than on average in nature also seems unlikely. This is one difficulty. If the number of nuclei of group \(L\) (Li, Be, B) in the cosmic rays near the Earth is indeed at least 10 times smaller than the number of nuclei of group \(M\), then the results (4.6) directly contradict experiment. Thus the adopted uniform distribution of sources appears, most likely, unacceptable, as also follows from the identification of cosmic-ray sources with supernovae and novae.

As another limiting case of the distribution of sources, in comparison with their uniform distribution, let us assume that these sources are point-like, i.e. \(q_i=Q_i\delta(\mathbf r)\), where \(\delta\) is the delta function. Then, under the remaining assumptions adopted, the system (4.4) takes the form

\[ \frac{d^2N_i}{dr^2}+\frac{2}{r}\frac{dN_i}{dr}-\frac{N_i}{DT_i} =-\frac{Q_i}{D}\delta(\mathbf r)-\sum p_{ij}\frac{N_j}{DT_j}. \tag{4.8} \]

The solution of this system for \(N_H\), \(N_M\), and \(N_L\) is as follows \((q_L=0,\ p_{Hj}=0)\):

\[ N_H=\frac{Q_H}{4\pi Dr}\exp\left\{-\frac{r}{\sqrt{DT_H}}\right\}, \tag{4.9} \]

\[ N_M=\frac{Q_H}{4\pi Dr}\left\{ \frac{Q_M}{Q_H}\exp\left[-\frac{r}{\sqrt{DT_M}}\right] +\frac{p_{MH}T_M}{T_M-T_H} \left[ \exp\left(-\frac{r}{\sqrt{DT_M}}\right) -\exp\left(-\frac{r}{\sqrt{DT_H}}\right) \right]\right\}, \tag{4.10} \]

\[ \begin{aligned} N_L=\frac{Q_H}{4\pi Dr}\Bigg\{& \frac{T_L}{T_L-T_H} \left[ p_{LH}-\frac{p_{LM}p_{MH}}{T_M-T_H}T_H \right] \left[ \exp\left(-\frac{r}{\sqrt{DT_L}}\right) -\exp\left(-\frac{r}{\sqrt{DT_H}}\right) \right] \\ &+\frac{p_{LM}T_L}{T_L-T_M} \left[ \frac{Q_M}{Q_H}+\frac{p_{MH}T_M}{T_M-T_H} \right] \left[ \exp\left(-\frac{r}{\sqrt{DT_L}}\right) -\exp\left(-\frac{r}{\sqrt{DT_M}}\right) \right]\Bigg\}, \end{aligned} \tag{4.11} \]

where reflection from the galactic boundaries is not taken into account.

From (4.9)—(4.11) we obtain

\[ \frac{Q_M}{Q_H}= \left[ \frac{N_M}{N_H}+\frac{p_{MH}T_M}{T_M-T_H} \right] \exp\left[ -\frac{r}{\sqrt{DT_H}} \left(1-\sqrt{\frac{T_H}{T_M}}\right) \right] -\frac{p_{MH}T_M}{T_M-T_H}, \tag{4.12} \]

\[ \begin{aligned} \frac{N_L}{N_H}={}& \frac{T_L}{T_L-T_H} \left[ p_{LH}-\frac{p_{LM}p_{MH}T_H}{T_M-T_H} \right] \left\{ \exp\left[ -\frac{r}{\sqrt{DT_H}} \left(1-\sqrt{\frac{T_H}{T_L}}\right) \right]-1 \right\} \\ &+\frac{p_{LM}T_L}{T_L-T_M} \left[ \frac{N_M}{N_H}+\frac{p_{MH}T_M}{T_M-T_H} \right] \left\{ \exp\left[ -\frac{r}{\sqrt{DT_H}} \left(\sqrt{\frac{T_H}{T_M}}-\sqrt{\frac{T_H}{T_L}}\right) \right]-1 \right\}. \tag{4.13} \end{aligned} \]

Using the experimental values at the Earth:

\[ \frac{N_M}{N_H}=3.2 \]

and

\[ \frac{N_L}{N_H}\ll 0.32\left(\frac{N_L}{N_M}\ll 0.1\right) \]

for variant \(A\) (see Table VIII), we obtain

\[ \frac{r}{\sqrt{DT_H}}\ll 0.7;\qquad 3.2>\frac{Q_M}{Q_H}>2.5. \tag{4.14} \]

For \(D=3\cdot 10^{37}\ \mathrm{cm}^2/\mathrm{year}\) and \(T_H=2\cdot 10^7\) years, the parameter \(\sqrt{DT_H}\simeq 2.5\cdot 10^{22}\ \mathrm{cm}\), and the distance from the source to the Earth is \(r\ll 0.7\sqrt{DT_H}=1.8\cdot 10^{22}\ \mathrm{cm}\). This distance is only 1.4 times smaller than the distance from the Sun to the center of the Galaxy, equal to \(R_0\simeq 2.5\cdot 10^{22}\ \mathrm{cm}\). With the accuracy with which all the parameters are known, it is in fact hardly possible even to speak of a difference here. Thus, if the concentration of hydrogen in the galactic “corona” responsible for diffusion is less than 0.1 (this, as already noted, appears probable), then \(T_H>2\cdot 10^7\) years. For \(n=0.03\) we have \(T_H=6\cdot 10^7\) years, \(\sqrt{DT_H}=4.5\cdot 10^{22}\), and \(r\ll 3.2\cdot 10^{22}\ \mathrm{cm}\). On the other hand, for variant \(B\), when the coefficients \(p_{ij}\) are larger (see Table VIII), it is necessary that \(r\ll 0.38\sqrt{DT_H}\), and even for \(T_H=6\cdot 10^7\) years \(r\ll 1.7\cdot 10^{22}\ \mathrm{cm}\).* Thus, in the case of a point source located at a distance \(r=1.7\div 3.2\cdot 10^{22}\ \mathrm{cm}\) and emitting no nuclei of group \(L\) at all, at the Earth the number of nuclei of this group would be no more than \(1/10\) of the number of nuclei of group \(M\).

In reality, of course, the sources are not concentrated in a small region, and even if they are appreciably concentrated toward the galactic center, the distance to the effective point source is \(r<R\simeq 2.5\cdot 10^{22}\ \mathrm{cm}\).

Starting from the solution (4.9)—(4.11), it is easy to obtain the result for various distributions of sources. For example, the most interesting

*) In a recently published paper \(^{71}\) the value

\[ \frac{N_L}{N_M}=0.35\pm 0.13, \]

is given, whence \(\frac{N_L}{N_H}\approx 1\). Hence, from formula (4.13), using the values of \(p_{ij}\) in accordance with variant \(B\), we arrive at the conclusion that \(r\approx 1.2\sqrt{DT_H}\). Since, further, \(r\lesssim 2.5\cdot 10^{22}\), it follows that \(\sqrt{DT_H}\lesssim 2\cdot 10^{22}\) and \(D\lesssim 2\cdot 10^{37}\ \mathrm{cm}^2/\mathrm{year}\) (for \(T_H=2\cdot 10^7\) years); for \(T_H=6\cdot 10^7\) years in this case \(D<10^{37}\ \mathrm{cm}^2/\mathrm{year}\). Thus, if \(\frac{N_L}{N_H}\sim 1\), then this value can be explained by using a diffusion coefficient \(D\) three times smaller than its maximum value \(\sim 3\cdot 10^{37}\) adopted above. This circumstance is, of course, favorable.

The presence of a large number of nuclei of group \(L\) is also confirmed by paper \(^{72}\), in which the value \(\frac{N_L}{N_M}\sim 0.7\) was obtained. A large value of \(N_L/N_M\) is also indirectly indicated by paper \(^{75}\).

the ratio \(\dfrac{N_L}{N_H}\) takes the form:

\[ \left. \begin{gathered} \frac{N_L}{N_H} = \frac{T_L}{T_L-T_H} \left[ p_{LH} - \frac{p_{LM}p_{MH}T_H}{T_M-T_H} \right] \left[ \frac{F_L}{F_H}-1 \right] + \frac{p_{LM}T_L}{T_L-T_M} \left[ \frac{N_M}{N_H} + \right. \\ \left. + \frac{p_{MH}T_M}{T_M-T_H} \right] \left[ \frac{F_L}{F_M}-1 \right], \\[6pt] F_i=\int \frac{1}{r}\exp\left(-\frac{r}{\sqrt{DT_i}}\right)\,dr, \end{gathered} \right\} \tag{4.15} \]

where the density of sources at a point \(r\), located at a distance \(r\) from the point of observation, is assumed to be constant throughout the entire region of integration and equal to zero outside it.

For simplicity alone, let us now consider the following case: the sources lie in the plane in which the point of observation also lies; the source density is constant in the coordinate \(r\) in the region \(r_{\min}\leq r\leq r_{\max}\) and is equal to zero outside this region; in the polar angle \(\varphi\) the distribution of sources may be arbitrary, but the same for nuclei of all kinds. Under such conditions

\[ F_i=\operatorname{const}\sqrt{DT_i} \left[ \exp\left(-\frac{r_{\min}}{\sqrt{DT_i}}\right) - \exp\left(-\frac{r_{\max}}{\sqrt{DT_i}}\right) \right] \]

and, if, for example, \(r_{\min}\ll \sqrt{DT_H}\simeq 2.5\cdot 10^{22}\) and \(p_{LM}=p_{LH}=0.23\) (variant \(A\)), then \(\dfrac{N_L}{N_M}\simeq 0.1\) for \(r_{\max}\simeq 4\cdot 10^{22}\). For smaller values of \(r_{\max}\) the ratio \(\dfrac{N_L}{N_M}\) is still smaller.

Let us turn to the question of proton diffusion. In this case, taking into account the formation of secondary protons, we write the system of diffusion equations in the form

\[ \left. \begin{aligned} \frac{d^2N_p}{dr^2} +\frac{2}{r}\frac{dN_p}{dr} -\frac{N_p}{DT_p} &= -\frac{Q_p}{D}\delta(r) - \bar p_{pj}\frac{\bar N_j}{D\bar T_j}, \\ \frac{d^2\bar N_j}{dr^2} +\frac{2}{r}\frac{d\bar N_j}{dr} -\frac{\bar N_j}{D\bar T_j} &= -\frac{\bar Q_j}{D}\delta(r), \end{aligned} \right\} \tag{4.16} \]

where \(\bar N_j,\ \bar Q_j,\ \bar p_{pj}\), and \(\bar T_j\) are the average values, already used above, for all nuclei taken together (including, of course, \(\alpha\)-particles and excluding protons).

For protons the collisions are not catastrophic, but, choosing for \(T_p\) some effective value, for estimates it seems possible to use equations (4.16), understanding by \(N_p\), for example, the total number of protons with energy greater than some threshold energy \(E_0\). The solution of the system (4.16) is analogous to the solution (4.9)—(4.10) of the system of equations (4.8) for \(N_H\) and \(N_M\). As a result

\[ \frac{Q_p}{\bar Q_j} = \left[ \frac{N_p}{\bar N_j} + \frac{\bar p_{pj}T_p}{T_p-\bar T_j} \right] \exp\left[ -\frac{r}{\sqrt{D\bar T_j}} \left( 1-\sqrt{\frac{\bar T_j}{T_p}} \right) \right] - \frac{\bar p_{pj}T_p}{T_p-\bar T_j} \simeq 7.7 \simeq 0.77\frac{N_p}{\bar N_j}, \tag{4.17} \]

where it is assumed that

\[ \overline{N}_j=\frac{1}{10}N_p,\quad \overline{T}_j=\frac{1}{4}T_p=10^8\ \text{yr},\quad D=3\cdot 10^{37}\ \text{cm}^2/\text{yr}, \]

\[ \overline{p}_{pj}=4\quad \text{and}\quad r=1.8\cdot 10^{22}\ \text{cm}. \]

If, however, secondary protons are completely neglected, then for

\[ T_\alpha=\frac{1}{4}T_p\quad \text{and}\quad \frac{T_H}{T_p}=\frac{1}{20} \]

(the remaining parameters as before):

\[ \begin{aligned} \frac{Q_p}{Q_\alpha} &=\frac{N_p}{N_\alpha} \exp\left[-\frac{r}{\sqrt{DT_\alpha}} \left(1-\sqrt{\frac{\overline{T}_\alpha}{T_p}}\right)\right] \simeq 0.85\,\frac{N_p}{N_\alpha},\\[6pt] \frac{Q_p}{Q_H} &=\frac{N_p}{N_H} \exp\left[-\frac{r}{\sqrt{DT_H}} \left(1-\sqrt{\frac{T_H}{T_p}}\right)\right] \simeq 0.56\,\frac{N_p}{N_H}, \end{aligned} \tag{4.18} \]

whereas for a uniform distribution of sources

\[ \frac{q_p}{q_\alpha}=\frac{N_p}{N_\alpha}\frac{T_\alpha}{T_p} =0.25\,\frac{N_p}{N_\alpha} \quad \text{and}\quad \frac{q_p}{q_H}=5\cdot 10^{-2}\frac{N_p}{N_H}. \quad (\text{see }(4.7)). \]

Thus, taking into account the nonuniform distribution of sources and the diffusion of cosmic particles, one can explain the presence near the Earth of only a relatively small number of Li, Be, and B nuclei, and it is unnecessary to suppose that the sources are in an entirely anomalous way poor in hydrogen. To establish agreement with the data on the composition of cosmic rays near the Earth (see Table 1), it is sufficient merely to assume that in the sources the relative number of nuclei of the groups \(M\) and \(H\) is 5–10 times greater than on the average in nature. Such an assumption, as has already been mentioned, is not only acceptable, but is also in agreement with certain ideas about the nature of supernova stars.

The reason why taking diffusion into account in our case sharply changes the situation is as follows. From the solution of the diffusion equation it is clear (see, for example, (4.9)) that in the region where \(\dfrac{r}{\sqrt{DT}}\ll 1\), the concentration of particles does not depend on the lifetime of the particles \(T\) (this conclusion is, of course, physically obvious, since the condition \(\dfrac{r}{\sqrt{DT}}\ll 1\) is the condition of proximity to the sources). Therefore, in the indicated region \(\dfrac{N_i}{N_j}\simeq \dfrac{Q_i}{Q_j}\), whereas for a uniform distribution of sources \(\dfrac{N_i}{N_j}=\dfrac{q_iT_i}{q_jT_j}\). The conclusion drawn remains qualitatively valid also for \(\dfrac{r}{\sqrt{DT}}\lesssim 1\), as is the case considered by us (\(r\sim 2\cdot 10^{22}\), \(\sqrt{DT_p}\simeq 10^{23}\), \(\sqrt{DT_H}\simeq 2.5\cdot 10^{22}\)). In this connection it is important to choose large values of \(D\) and \(T\), which was justified earlier*). In view of the still considerable uncertainty in the choice of these parameters, increasing by several times

*) In the literature one encounters the assertion (see, for example, \(^{5,51}\)) that the lifetime of cosmic particles, because of the data on their composition, cannot exceed the lifetime of nuclei of the group \(H\). Such a conclusion is obviously incorrect in the case of the model used here, which is significantly closer to reality than that adopted in \(^{5,51}\).

the ratio $\dfrac{N_L}{N_M}$ would not be difficult. On the contrary, to make this ratio significantly less than 0.1 does not seem possible, at least without introducing into the theory some new elements. Therefore, if in experiment it turned out that $\dfrac{N_L}{N_M} \ll 0.1$, this would require some revision of the concepts being developed. At present, however, there are no indications of such a very low content of Li, Be, and B in cosmic rays. On the contrary, the latest data $^{71,72}$, as has already been noted, speak in favor of $\dfrac{N_L}{N_M}$ being even greater than the value 0.1.

c) Spatial distribution and isotropy of cosmic rays

The solution of the diffusion equations considered in the preceding section determines not only the composition, but also the spatial distribution of cosmic rays in the Galaxy. Bearing in mind measurements on Earth, in this respect the greatest interest is the determination of the degree of anisotropy
$\delta = \dfrac{\Delta F}{F}$ (see (1.8)).

In the diffusion approximation the difference of the fluxes $\Delta F$ is simply the diffusion flux, $-D \nabla N$. As for the total flux of cosmic particles going in one direction, in the case of isotropy $F = \dfrac{1}{4} vN$ (see (1.1)). Thus,

$$ \delta = \frac{4D}{c}\frac{\left|\dfrac{dN}{dr}\right|}{N}, \tag{4,19} $$

where, obviously, for simplicity no attention is paid to the sign of the diffusion flux, the distribution is regarded as spherically symmetric, and the particle velocity $v$ is put equal to $c$. For the simplest solution of type (4.9), when
$N = \dfrac{Q}{4\pi Dr}\exp\left(-\dfrac{r}{\sqrt{DT}}\right)$, we have

$$ \delta = \frac{4D}{c}\left(\frac{1}{r}+\frac{1}{\sqrt{DT}}\right). \tag{4,20} $$

For $D = 10^{30}\ \mathrm{cm}^2/\mathrm{sec} = 3 \cdot 10^{37}\ \mathrm{cm}^2/\mathrm{yr}$ and $r = R_0 = 2.5 \cdot 10^{22}$, for nuclei of group $H$ ($T_H = 2 \cdot 10^7$ years) $\delta_H = 5 \cdot 10^{-2}$ and for protons ($T_p = 4 \cdot 10^8$ years) $\delta_p = 6 \cdot 10^{-3}$.

The values obtained are, in all probability, too high. First, reflection from the boundaries of the Galaxy was not taken into account above, which is legitimate only if

$$ \frac{R}{\sqrt{DT_i}} \gg 1, \tag{4,21} $$

where $R = 3 \div 5 \cdot 10^{22}$ is the radius of the Galaxy. For nuclei of group $H$ the ratio $\dfrac{R}{\sqrt{DT_H}} \lesssim 2$, for $\alpha$-particles $\dfrac{R}{\sqrt{DT_\alpha}} \lesssim 0.9$ and for protons $\dfrac{R}{\sqrt{DT_p}} \lesssim 0.45$ ($\dfrac{R}{\sqrt{DT_p}} = 0.45$ for $R = 5 \cdot 10^{22}$ and $T_p = 4 \cdot 10^8$ years;

if, however, \(R=3\cdot 10^{22}\) and \(T_p=10^9\) years, then \(\dfrac{R}{\sqrt{DT_p}}\simeq 0.17\). One may convince oneself that for nuclei of the groups \(L\), \(M\), and \(H\) the corrections connected with taking reflection into account are small; for the purposes pursued in Section 3b, these corrections are insignificant also for protons. But in calculating the asymmetry, since we wish to obtain a result with greater accuracy, it is expedient to take reflection into account.

With complete reflection of particles from a spherical boundary on which, consequently, \(\dfrac{dN}{dr}=0\), the solution of the first of equations (4.16) is as follows:

\[ N_p=\frac{Q_p e^{-\frac{r}{\sqrt{DT_p}}}}{4\pi Dr} \left\{ \frac{ \left(1+\frac{R}{\sqrt{DT_p}}\right) e^{-\frac{2(R-r)}{\sqrt{DT_p}}} - \left(1-\frac{R}{\sqrt{DT_p}}\right) }{ \left(1+\frac{R}{\sqrt{DT_p}}\right) e^{-\frac{2R}{\sqrt{DT_p}}} - \left(1-\frac{R}{\sqrt{DT_p}}\right) } \right\}; \tag{4.22} \]

where secondary protons are not taken into account (i.e. \(\overline p_{pj}=0\)) and the point source is assumed to be located at the center of the sphere. In particular, for \(R=\sqrt{DT_p}\) the solution is especially simple:

\[ N_p=\frac{Q_p}{4\pi Dr}e^{-\frac{r}{\sqrt{DT_p}}}, \tag{4.23} \]

and for

\[ \frac{R}{\sqrt{DT_p}}\ll 1 \]

we have

\[ N_p=\frac{Q_p}{4\pi D} \left\{ \frac{1}{r}+\frac{3DT_p}{R^3}+\frac{1}{2}\frac{r^2}{R^3} \right\}. \tag{4.24} \]

In this expression, for \(r\sim R\) the second term is the principal one, and thus

\[ N_p\simeq \frac{Q_pT_p}{\frac{4\pi}{3}R^3}, \]

which corresponds to a uniform distribution of sources. Already from this, and also directly on physical grounds, it is clear that reflection leads to a smoothing of the concentration and to a decrease of the asymmetry. For example, in the case (4.23) the degree of asymmetry \(S\) is determined by formula (4.20), but with a minus sign before the second term.

According to (4.19) and (4.22), when reflection is taken into account,

\[ \delta_p=\frac{4D}{c} \left\{ \frac{1}{r} + \frac{1}{\sqrt{DT_p}} \left[ 1- \frac{2}{ 1-\frac{R}{\sqrt{DT_p}} - \frac{1-\frac{R}{\sqrt{DT_p}}}{1+\frac{R}{\sqrt{DT_p}}} e^{\frac{2(R-r)}{\sqrt{DT_p}}} } \right] \right\}. \tag{4.25} \]

Hence, for \(R=5\cdot 10^{22}\ \mathrm{cm}\), \(r=2.5\cdot 10^{22}\), \(D=10^{30}\ \mathrm{cm^2/sec}\), and \(T_p=4\cdot 10^8\) years, we obtain \(\delta_p=6\cdot 10^{-4}\), which is 10 times smaller than the value obtained without taking reflection into account (i.e. by formula (4.20)).

The asymmetry may also be reduced for another reason. The point is that near the solar system, which is located in an arm of the galactic spiral, the diffusion coefficient \(D\) is most likely smaller than the adopted

values \(D \sim 10^{30}\ \mathrm{cm}^2/\mathrm{sec}\), associated with the choice of path length \(l \sim 3 \cdot 10^{20}\ \mathrm{cm}\). The same also applies to the entire region near the galactic plane. In analyzing the question of the composition of cosmic rays and their distribution in the Galaxy as a whole, this circumstance, generally speaking, is immaterial, and one must use the value of \(D\) referring to the main part of the sphere. But in finding the diffusion flux at a given point it is important to know the diffusion coefficient and the concentration distribution near this point. As a result, a very strong local variation of the diffusion flux may occur.

For \(E \lesssim 10^{15}\ \mathrm{eV}\) the radius of curvature of proton trajectories in a field \(H \sim 10^{-5}\) is less than \(3 \cdot 10^{17}\ \mathrm{cm}\). Therefore in the galactic plane it seems permissible for these particles to use values greater than \(l \sim 3 \cdot 10^{18}\) and \(D \sim 10^{28}\ \mathrm{cm}^2/\mathrm{sec}\). Owing to the decrease of the diffusion coefficient near the galactic plane, diffusion will occur mainly in the region of the galactic corona, and particles will reach the solar system to a considerable extent not from the direction of the galactic center, but from above and below (i.e. from the direction of the poles of the Galaxy). This, obviously, will lead to a reduction of the asymmetry*).

Taking all that has been said into account, we believe that obtaining a value consistent with experiment, \(\delta \lesssim 10^{-3}\), for particles with \(E \lesssim 10^{15}\ \mathrm{eV}\) presents no problem. For protons with energy \(E \sim 10^{18}\ \mathrm{eV}\), the radius of curvature is \(r \lesssim 3 \cdot 10^{20}\), and therefore one must use, say, the coefficient \(D \sim 3 \cdot 10^{30}\ \mathrm{cm}^2/\mathrm{sec}\), which even in the case (4.19) leads to the value \(\delta_p \approx 2 \cdot 10^{-2}\); for nuclei with \(E \simeq 10^{18}\) one may still assume that \(D \sim 10^{30}\) and \(\delta \lesssim 10^{-2}\). At the same time, in experiment at \(E \sim 10^{17}\ \mathrm{eV}\) the asymmetry is \(\delta < 0.1\), or, according to other data, at \(E \lesssim 10^{18}\ \mathrm{eV}\) \(\delta < 1 \div 2 \cdot 10^{-2}\) (see Section 1b).

Thus, from the point of view of the available information on the isotropy of cosmic rays, the model we have adopted meets with no objections.

In conclusion let us note that, in analyzing the question of the spatial distribution of electrons, one can no longer, generally speaking, confine oneself to the study of a diffusion equation of the type of equation (4.4). This is explained by the fact that electrons are constantly losing energy as a result of the presence of synchrotron losses. Therefore one must use an equation for the function \(N_e(\mathbf r, E)\), i.e. the concentration of electrons in a unit energy interval near the point \(\mathbf r\) and energy \(E\). Such an equation is a combination of expressions of the type (3.30) or (3.51) and (4.4); its solution for the purpose of finding the electron spectrum as a function of galactic coordinates, taking account of the adopted distribution of sources, has not yet been carried out. Nevertheless it is clear that near sources the electrons must be harder, and far from the galactic plane softer. This will lead to a change in the spectrum of radio emission as a function of galactic coordinates (the same applies, of course, to the nebula M31 and to other galaxies). Unfortunately, this point\(^{3,62}\) still remains unclarified**).

*) As mentioned in Section 4g, if, in accordance with the latest data, the ratio \(N_L/N_M\) is considered large (i.e. if one puts \(N_L/N_M \sim 0.5 \div 1\)), then this also indicates the possibility of choosing, in the corona, a coefficient \(D < 10^{30}\ \mathrm{cm}^2/\mathrm{sec}\); for this reason the asymmetry also decreases.

**) It should, however, be stipulated that the conclusion about a change in the radio-emission spectrum is obtained under the assumption of constancy of the quantity \(H_\perp\) (the component of the magnetic field perpendicular to the electron velocity). In reality the quantity \(H_\perp\) may change both because of a change in the field strength and as a result of an increase or decrease of the angle \(\theta\) between the electron velocity and the direction of the field. Therefore the analysis of the problem touched upon becomes more complicated.

d) Critique of Alternative Views

The theory of the origin of cosmic rays set forth above is, in our opinion, well substantiated. Nevertheless, it still contains certain hypothetical elements. In this connection it is especially important to dwell on the analysis of other possibilities, and to see whether there exist alternative views deserving attention.

Comparatively recently, attention was drawn to the theory of a solar origin of cosmic rays63, 64. There are two arguments in favor of such a hypothesis. First, if cosmic rays occupy only a relatively small volume with radius \(10^{16} \div 10^{17}\) cm near the Sun, then the energy contained in them is comparatively small. Secondly, the possibility of generating soft cosmic particles on the Sun or near the Sun has been proved as a result of observations of an increase in the intensity of cosmic rays accompanying certain chromospheric eruptions.

As for the first of these arguments, at the present time it appears completely unconvincing, since even the assumption of a uniform filling by cosmic rays of the entire quasi-spherical Galaxy does not encounter any contradictions of an energetic character (see Section 4a). The second argument, although more substantial, is in itself still quite insufficient. The flux of cosmic rays reaching us directly from the Sun amounts, on the average, to only about \(0.1\%\) of the total flux of cosmic rays at the Earth. Therefore the theory of a solar origin of cosmic rays is connected with the supposition of an accumulation of cosmic particles in the aforementioned comparatively small volume around the Sun. It can be shown, however, that in order to achieve agreement with the experimental data it is necessary to assume that the region in which the cosmic rays are enclosed possesses practically ideally reflecting walls1, 5, 50, 64. But there are no grounds whatsoever for such a far-reaching assumption, i.e., the assumption of a very peculiar and special configuration of the magnetic field near the solar system*). Furthermore, in a field \(H \sim 10^{-4} \div 10^{-5}\) oersted, the radius of curvature of the trajectory of particles with energies \(E > 3 \cdot 10^{14} \div 3 \cdot 10^{15}\) eV is already greater than the supposed radius of the region filled with cosmic rays. Thus, high-energy particles cannot be retained in this region and must have an origin different from that of the main mass of cosmic rays. At the same time there is no “break” in the energy spectrum of cosmic rays at high energies, which would be strange in the case of a completely different origin of particles of different energy (not to mention that there are no direct indications testifying to the acceleration on the Sun of particles to energies even greater than \(\sim 10^{10}\) eV). Finally, radio-astronomical data permit the conclusion that cosmic electrons are present far beyond the limits of the solar system, throughout the entire Galaxy, in the envelopes of supernovae, and in other galaxies.

All these considerations, taken together, leave no doubt that the theory of a solar origin of cosmic rays does not correspond to reality (see also1, 5, 65**).

*) Owing to the drift of particles in an inhomogeneous magnetic field, these particles will escape even from a system with closed lines of force. Therefore the possibility of creating ideally reflecting “walls” is unclear even in principle.

**) This conclusion, of course, by no means signifies that the study of solar variations in the intensity of cosmic rays is devoid of interest. On the contrary, this latter problem deserves every attention for a number of reasons, but not for the solution of the fundamental questions of the theory of the origin of cosmic rays.

At present one may also regard as having been abandoned the assumption of a metagalactic origin for the bulk of cosmic rays. A hypothesis has, however, been put forward \(^{18}\) that in metagalactic space cosmic particles of high energy are accelerated, for example beginning with an energy of \(10^{10}\) ev or greater. In favor of this point of view \(^{18}\) considerations are adduced which indicate that, within the framework of the theory of the galactic origin of cosmic rays, it is difficult to explain the isotropy of cosmic rays in combination with the fact that heavy nuclei are present near the Earth. But in the light of what has been said above (see Sections 4б and 4в), this argumentation, in our opinion, falls away completely: it is based on the use of a model that does not take into account diffusion and the spatial distribution of the sources (in \(^{18}\) it is assumed that the acceleration of particles takes place in interstellar space).

At the same time, a whole series of arguments can be advanced against the assumption of a metagalactic origin for a substantial part of the cosmic radiation. A magnetic field \(H \sim 10^{-5}\) can confine in the Galaxy with radius \(R \sim 5 \cdot 10^{22}\) cm even cosmic particles of high energy (see Section 2б). The spectrum of cosmic rays, as is known, is smooth and has no “break” of any kind. Meanwhile, as has already been emphasized, if cosmic rays in different energy intervals have different origins, such an absence of a “break” could occur only by virtue of an accidental coincidence. Further, both general energy considerations and radio-astronomical data indicate that in intergalactic space the concentration of soft cosmic rays is considerably smaller than in the Galaxy. But this fact, in combination with the assumptions of a metagalactic origin of cosmic rays of high energy, means that metagalactic cosmic rays have a different spectrum from galactic ones and, specifically, are poor in soft particles. For such an assumption, however, no other grounds are visible. Finally, in \(^{18}\) it is quite correctly pointed out that the statistical mechanism in metagalactic space is not sufficiently effective for accelerating particles to the required energies. If, however, one assumes that cosmic rays enter metagalactic space from individual galaxies, then the whole problem of the origin of high-energy cosmic rays in our Galaxy turns out to have been transferred to other galaxies. Logically this is permissible, but in view of the complete absence of convincing arguments in its favor such a hypothesis appears extremely unattractive. To all that has been said it must be added that, if there is a reflecting galactic boundary (see Section 2б), the entry of particles from metagalactic space into the Galaxy will be greatly hindered. The same is true also in the absence of boundaries, but with the decrease of the field that, in all probability, occurs on passing from the Galaxy into the region between galaxies. Indeed, in such a case, in motion from metagalactic space toward the galactic plane, the field along the trajectory of a particle increases and, consequently, some of the particles will be reflected back, as follows from relation (3.45).

As is clear from what has been said, when above we spoke of the metagalactic origin of cosmic rays, it was assumed that particles from the Metagalaxy enter our Galaxy. Among the metagalactic variants of the theory of the origin of cosmic rays, however, one may also include another variant, in which cosmic rays are considered to have formed in the process of formation of the Galaxy itself (during this period, obviously, a clear separation of the Galaxy from the Metagalaxy was not possible). Such an assumption is permissible only if the lifetime of cosmic rays is not much less than the time of the exist-

of the Galaxy, \(T_{\mathrm{gal}}\sim 6\cdot 10^9\) years. For protons this condition may prove to be fulfilled, since, as we have seen, at \(n\sim 0.1\ \mathrm{cm}^{-3}\), \(T_{pE}\sim 6\cdot 10^8\) years, but at \(n\sim 0.01\) already \(T_{pE}\sim 6\cdot 10^9\) years (the value \(n\sim 0.01\) is, according to \(^{31}\), the lower limit for \(n\); it may be thought that actually \(n\sim 0.03\)). But, on the other hand, for iron nuclei even at \(n\sim 0.01\) the time \(T_{\mathrm{Fe}}\sim 2\cdot 10^8\) years, i.e., considerably less than \(T_{\mathrm{gal}}\). At the same time there are absolutely no grounds for regarding cosmic protons and nuclei as having different origins. Therefore the variant under discussion of the theory of a metagalactic origin of cosmic rays also does not appear to deserve serious attention.

Thus, at the present time we see no grounds for the assumption of a metagalactic origin of any substantial part of the cosmic rays in the Galaxy, since such an assumption, first, is unnecessary and, second, encounters weighty objections*).

Rejecting the hypotheses of a solar and a metagalactic origin of cosmic rays, we automatically arrive at the assertion of their galactic origin, which is in accord with the ideas set forth above. Galactic theories of the origin of cosmic rays, however, may still differ very strongly from one another. The first difference is connected with the spatial distribution of cosmic rays: above it was assumed that cosmic rays fill a quasi-spherical region with radius \(R\sim 5\cdot 10^{22}\ \mathrm{cm}\), whereas in a whole series of papers (see, for example, \(^{5,51}\)) this region is regarded as immediately adjoining the galactic plane. We shall not dwell on this point again here, since it was discussed in detail earlier and, as one may hope, the use of a quasi-spherical distribution is sufficiently justified.

The second difference between galactic theories consists in the different assumptions about the sources and the mechanism of acceleration of cosmic rays. Thus, in a whole series of works (see, for example, \(^{66-68,50}\)) it was assumed that cosmic rays are formed in stellar atmospheres, in particular, of magnetic stars or stars with enhanced activity. In this case, however, no attention was paid to the question of the energy balance, and the acceleration mechanism itself was not considered sufficiently correctly and in detail. Meanwhile, as we saw in section 4a, the requirements of an energetic character imposed on sources of cosmic rays are very serious: if all \(\sim 10^{11}\) stars composing the Galaxy emitted cosmic rays in the same way as the Sun, this would lead to the release of energy \(10^6 \div 10^8\) times smaller than required. Of course, it is possible that on some stars cosmic rays are formed with a much greater intensity than on the Sun. But, on the other hand, there must be considerably fewer such stars than the total number of stars, the majority of which, as far as is known, do not possess any properties favoring the generation of cosmic rays. As a result, the noted difficulty undoubtedly remains in force. To this it must be added that there are no direct indications, for example of a radio-astronomical nature, of an increased generation of cosmic

*) From this it should not be concluded that there are no cosmic rays in intergalactic space. On the contrary, they are, in all probability, present as a result of the leakage of particles from individual galaxies, but their intensity is comparatively small, and the energy spectrum is most likely more or less close to the spectrum of cosmic rays in the Galaxy. Metagalactic cosmic-ray electrons are of greater interest from the point of view of studying the radio emission of the metagalaxy \(^{21}\).

electrons on stars of any type (shells of novae and supernovae are, of course, not meant here).

Theoretical considerations in this respect likewise give nothing. The point is that the theory of particle acceleration in stellar shells is only at the very initial stage of its development, since the calculations have been carried out without taking into account the high conductivity and mobility of stellar atmospheres (for example, in \(^{68}\) particles moving in a vacuum near a rotating star with a magnetic moment not parallel to the axis of rotation were considered). Therefore, for some models of stellar accelerators even the very existence of acceleration remains unclear and, as far as we know, in no case have either the total energy, or the energy spectrum, or the composition of the generated cosmic particles been found. It should be especially noted that one can hardly even hope that in the shells of stars cosmic rays can be accelerated to the highest observed energies \(E \sim 10^{18}\) ev.

The absence of a theory of particle acceleration on stars is not, of course, in itself an additional argument against the possibility of regarding stars as the main sources of cosmic rays. However, under conditions in which this assumption encounters the objection indicated earlier and is not confirmed by any experimental data, the impossibility of relying on reliable theoretical considerations is undoubtedly a negative factor.

Difficulties of an energetic character are mitigated in the combined model, in which stars play only the role of injectors, while the further acceleration of cosmic particles occurs in the interstellar medium \(^{48,52,51,5,47,49,69}\). Such a model also has a number of other advantages (the possibility of obtaining the required energy spectrum extending to the very highest energies, etc.). However, the mechanism of interstellar acceleration also encounters very serious difficulties. The chief of these was already noted in Section 3c and consists in the fact that different energy spectra are obtained for protons and nuclei if only the lifetime of the particles is determined by nuclear collisions. Therefore one has to suppose that the lifetime \(T\) is determined by the escape of particles from the accelerating region and, thus, is the same for all particles \(^{51,52}\). In this case, as was already mentioned in Section 3c, the time \(T\) must be less than the lifetime for the heaviest nuclei, i.e. \(T < 10^7\) years (for Fe nuclei the nuclear lifetime is \(T \simeq 10^7\) years). Below, for definiteness, we use the value \(T = 4 \cdot 10^6\) years \(= 1.25 \cdot 10^{14}\) sec. Hence, by virtue of condition (3.55),

\[ \alpha \sim 10^{-14}\ \mathrm{sec}^{-1}, \tag{4.26} \]

as is also adopted in \(^{51}\).

If the Galaxy were flat (disk-shaped), particles could leave it in a time \(T \sim 4 \cdot 10^6\) years, but in the case of a spherical Galaxy this is considerably more difficult. Quite apart from the very probable reflection of particles from the galactic boundaries, the time necessary for the diffusion of cosmic particles over a distance \(R \sim 5 \cdot 10^{22}\) cm is approximately \(10^8\) years for a mean free path \(l \sim 3 \cdot 10^{20}\) cm (see Section 3c). Therefore, for \(T \sim 4 \cdot 10^6\) years the mean free path must be increased still more. But already for \(l \sim 3 \cdot 10^{20}\) and the enormous velocity of turbulent motions in the galactic corona \(u \sim 100\) km/sec, the coefficient \(\alpha \sim 3 \cdot 10^{-18}\), i.e. at least three orders of magnitude smaller than the value (4.26). In general, it is obvious that increasing the escape of particles from the system is coupled with the assumption of an increase in the mean free path. The latter, however, leads to a lowering of the efficiency of interstellar particle acceleration and, as shown above (see also \(^{53}\)), leads to a contradiction with the assumptions made in the scheme \(^{51}\).

requiring that the value of \(\alpha\) increase to \(10^{-14}\ \mathrm{sec}^{-1}\). As a result, in the model of a spherical Galaxy, the necessity of using which can now hardly be doubted\(^*\), interstellar acceleration can be reconciled with the facts only by assuming that it takes place not throughout the entire Galaxy, but in some part of it or in separate regions. Then the parameters \(\alpha\) and \(T\) determining the spectrum apply only to these regions, from which, in this way, the particles must escape in a time \(4\cdot 10^6\) years.

Let us first suppose that the acceleration takes place in the galactic corona. Here, as already mentioned, \(u\sim 10^7\ \mathrm{cm/sec}\), and for \(l\sim 3\cdot 10^{20}\ \mathrm{cm}\)

\[ \alpha=\frac{u^2 v}{c^2 l}\sim 3\cdot 10^{-18}. \]

The possibility of increasing this value by a factor of 3000 by a further increase of \(u\) or decrease of \(l\) seems unlikely, since both these parameters are constrained by a number of requirements (for example, with a decrease of \(l\) the diffusion coefficient falls, while an increase of \(u\) is associated with the assumption of a further strengthening of the magnetic field or a decrease of the gas density). Further, on the basis of data on the radio emission of the galactic “corona” we arrived at the estimate \(\alpha<4\cdot 10^{-17}\) (see (3.58)). If, however, \(\alpha\sim 10^{-14}\), then in a field \(H\sim H_\perp\sim 10^{-5}\) electrons would be accelerated to an energy \(E\sim 2\cdot 10^{10}\ \mathrm{eV}\), which is in decisive contradiction with the information on the spectrum of cosmic radio emission.

The escape of particles from the corona in a time \(T\sim 4\cdot 10^6\) years likewise should not occur, as one may verify by repeating the estimates already given. Finally, the interstellar acceleration mechanism requires the injection of slower particles. Such injectors in the corona either do not exist at all, or are very few (obviously, the question is of certain stars). Therefore one must suppose that particles are injected near the galactic plane, reach (despite their low energy) the region of the corona, are then accelerated, and, entering the region of the galactic spiral again, reach the Earth.

All this makes the hypothesis of particle acceleration in the galactic corona more than doubtful and unattractive.

The situation would change somewhat if the gas concentration in the corona were much smaller than assumed above and were equal to \(n\sim 10^{-3}\ \mathrm{cm}^{-3}\) (see \(^{32}\)). In that case the nuclear lifetime would be so large that the escape of cosmic particles from the corona into the region of the galactic spiral could occur in a time \(T\sim 10^8\) years\(^ {**}\), less than the nuclear lifetime, but not leading to very large values of \(\alpha\)\(^{***}\).

\(^*\) For this reason we shall not analyze here the mechanism of interstellar acceleration as applied to the model of a disk-shaped galaxy. Let us note only that the difficulties of the corresponding theoretical scheme \(^{51}\) in this case are by no means reduced (see \(^{42,53}\)).

\(^ {**}\) Let us note that in paper \(^{70}\) it is assumed that in the corona \(n\sim 10^{-2}\div 10^{-3}\), but apparently it is not taken into account that in this case the lifetime of cosmic particles will be determined by their escape from the accelerating region. In any event, in \(^{70}\) no attempt is made to obtain the required spectrum of cosmic particles by selecting the parameters \(\alpha\) and \(T\), and the very dense and interesting aspect of the idea of interstellar acceleration is thereby discarded. If, as is apparently done in \(^{70}\), one assumes that the particles are accelerated in the corona, while their spectrum in charges and energies is established in denser regions near the galactic plane, then it is still necessary to demonstrate the possibility of such a transformation for concrete values of the various parameters. Since this is not done in \(^{70}\), we can in no way speak here of any realistic scheme for the origin of cosmic rays.

\(^ {***}\) If \(\alpha \lesssim 5\cdot 10^{-17}\), then during the lifetime of the Galaxy \(T_{\rm gal}\sim 6\cdot 10^9\ \text{years}\sim 2\cdot 10^{17}\ \mathrm{sec}\), a particle can, as a result of statistical acceleration, increase its energy by no more than \(\exp \alpha T_{\rm gal}\lesssim 10^4\) times. Under such conditions, acceleration in the interstellar medium would be of no interest even independently of all the other considerations discussed earlier (for this remark the author is indebted to V. A. Razin).

However, the use of a concentration \(n \sim 10^{-3}\ \mathrm{cm}^{-3}\) meets serious objections\(^{31}\), and the problem of injection still remains unclear even for \(n \sim 10^{-3}\). In addition, acceleration of cosmic rays at the expense of the kinetic energy of gas masses in the “corona” requires a substantial replenishment of this energy over a time of the order of the lifetime of cosmic rays \(T\) (the density of kinetic energy and the energy density of cosmic rays are approximately the same). The mechanism of such replenishment, especially as applied to the corona, is still unclear, and it remains to be proved that it is really possible to supply the energy needed to turbulize the gas in the corona in the presence of the strong friction caused by the generation of cosmic rays.

All that has been said compels us, at least at present, to reject the assumption that particles are accelerated predominantly in the galactic corona. We shall therefore make the alternative and considerably more plausible assumption that acceleration occurs only near the galactic plane, for example in the galactic spiral or in its individual parts. In this case the escape of particles from the accelerating region in a time \(T \sim 4 \cdot 10^{6}\) years can be ensured without any difficulty. The problem is worse with ensuring a high efficiency of acceleration. Use of data on the velocities of clouds of interstellar gas leads to the value \(a \sim 10^{-18}\) (see Sec. 3b), which is 4 orders of magnitude smaller than is needed (see (4.46)). Therefore, in order to increase the efficiency of acceleration, one has to make special assumptions, without any other basis, about a large role within the galactic spiral of systematic acceleration in “traps” (see\(^{52}\)) or of fluctuational acceleration in standing magnetohydrodynamic waves\(^{47}\). But even if the corresponding assumptions about the efficiency of acceleration are made, we encounter difficulties of another kind. The volume of the spiral is about \(1\%\) of the volume of the entire spherical Galaxy and is equal to \(\sim 10^{66}\ \mathrm{cm}^{3}\). The energy density of the field and the density of kinetic energy of the gas motion in the spiral are \(\sim 1\ \mathrm{eV}/\mathrm{cm}^{3} \sim 10^{-12}\ \mathrm{erg}/\mathrm{cm}^{3}\). Hence the whole energy reserve is \(\sim 10^{54}\ \mathrm{erg}\). Into cosmic rays, in order to maintain the balance, there must pass \(\sim 10^{39}\div 10^{40}\ \mathrm{erg}/\mathrm{sec}\) (see (3.7)), whence it is clear that the energy reserve of the gas in the spiral must be substantially replenished in \(10^{14}\div 10^{15}\ \mathrm{sec} \sim 3 \cdot 10^{6}\div 3 \cdot 10^{7}\) years. In fact this time is probably still smaller, since acceleration most likely occurs only in part of the volume occupied by the spiral. Providing such a very strong influx of energy constitutes a large, difficult, and completely unsolved problem (in work\(^{49}\) an even more pessimistic conclusion is drawn in this respect).

It thus turns out that, in order to explain the supply of the required energy into cosmic rays, one has to make an assumption which raises a new and no less serious energy problem.

A large value of \(a\) also leads, as was already indicated, to a difficulty connected with the acceleration of electrons, although in the case of the spiral the radio-astronomical data here still apparently do not permit decisive statements.

Finally, there remains a definite difficulty in the question of injection. For definiteness let us suppose that the injection energy*) \(E_{\mathrm{in}}\) for \(a \sim 10^{-14}\) is \(\sim 10^{6}\ \mathrm{eV}\) for protons and \(3 \cdot 10^{8}\ \mathrm{eV}\) for nuclei of group \(H\) (it is precisely such values that are adopted in\(^{51}\)). The presence of nuclei of group \(H\) (mean charge \(Z = 15\)) in cosmic rays indicates that injectors can accelerate them at least up to the indicated energy \(3 \cdot 10^{8}\ \mathrm{eV}\), and hence accelerate protons up to an energy \(\sim 10^{7}\ \mathrm{eV}\). It follows from this—

*) Let us recall that only starting with this energy does acceleration in the interstellar medium exceed ionization losses.

means that the injectors must supply of the order of \(0.1 \div 1\%\) of all the energy passing into cosmic particles, whose effective energy is \(10^9 \div 10^{10}\) eV.

If, thus, one assumes that the injectors must supply even only a \(10^{-3}\) part of all the necessary energy, as is also assumed in \(^{51}\), then they must still account for the entire power of cosmic-ray radiation, amounting to \(10^{36} \div 10^{37}\) erg/sec. At the same time we saw in section 4a that \(10^{11}\) stars of the solar type give only \(10^{32} \div 10^{33}\) erg/sec, i.e. 3–5 orders of magnitude less than is required. Thus the question of injection remains very acute, especially if acceleration occurs not throughout the whole volume of the stellar Galaxy, but only in a part of it, where there should also be fewer stars.

The difficulties and obscurities that arise in the questions of the efficiency of acceleration, the energy balance, and injection still do not, properly speaking, permit one to speak of the existence of any definite model and, still less, of a theory of interstellar acceleration of cosmic rays. Here, rather, there are only certain possibilities. Among them belongs, in particular, the assumption that has appeared already in recent works that interstellar acceleration occurs only in a few regions with conditions especially favorable for it \(^{49,69}\).

Such an assumption, taken by itself, leads, however, only to an aggravation of the energy difficulties. At the same time, the well-known closeness of the model of interstellar acceleration of particles in the region of the galactic spiral, and in particular in its separate regions, to the theory of the origin of cosmic rays set forth in sections 4a, b, and c, is striking. In fact, the distribution of sources is in this case approximately the same, while the spectrum of the particles in energies and charges is determined by the sources or, if one wishes, by a comparatively small region of the medium in which the acceleration takes place. The mechanism of acceleration is, in all probability, also the same in both cases.

The radical difference consists, nevertheless, in the fact that we regard as sources of cosmic rays quite definite objects—the shells of supernovae and, possibly, novae. The acceleration of particles in these objects is not only possible in principle, but also actually takes place. The necessary energy is supplied ultimately at the expense of the nuclear energy released in the explosion of the star. And if in this scheme there are insufficiently clarified aspects, then, so far as we know, there are in it no contradictions or difficulties connected with the interpretation of already known data (the points that remain obscure are connected chiefly with the absence of sufficient information about the early stages of supernova explosions).

At the same time, in the scheme of interstellar acceleration, as we see, it is necessary in the end not only to use the same distribution and certain general properties of the sources as in the theory of the generation of cosmic rays by supernovae, but, in addition, to make further substantial and as yet unsubstantiated assumptions.

In this connection, the continuous and very strong transformation of ideas about interstellar acceleration that has taken place in recent years \(^{51,52,47,49,69}\) does not seem accidental to us. At the same time it is easy to convince oneself that the theory of the origin of cosmic rays which was developed in 1953 on the basis mainly of radio-astronomical data \(^{12,22,23,28,35,55,56}\) and was then set forth in detail in article \(^{1}\), has not undergone any substantial changes since that time.

In summary, one may assert the following. The assumption of the acceleration of cosmic rays as a result of supernova-star flares.

within the limits of the available information, is admissible and sufficient for explaining all the known facts.

But, of course, it still cannot be guaranteed that supernovae and novae supply all the necessary energy, that there are no other sources, or that no additional acceleration of particles in the interstellar medium takes place. In this respect, if we are not speaking of altogether hypothetical constructions, it seems possible, in particular, that supernovae and novae play the role of injectors for subsequent interstellar acceleration. In this case, as is easy to see, all the difficulties with injection disappear. At the same time, all the other difficulties associated with an interstellar acceleration mechanism remain in force, and there are no convincing arguments in favor of the effectiveness of this acceleration. Thus, although it is still impossible to exclude the possibility of interstellar acceleration satisfying a number of the requirements mentioned above, it is natural to think that this possibility is not realized in actuality. Let us also note that if it nevertheless proves necessary to take into account further interstellar acceleration of the cosmic particles supplied by supernovae, this will change little in the theory we are developing. At present, however, as has been said, we do not yet see any special grounds for assuming such two-stage acceleration*).

CONCLUDING REMARKS

In conclusion we should like to note and bring together the problems of further investigations which may lead to a clarification of the picture of the origin of cosmic rays.

By the methods used in cosmic-ray physics it is necessary to clarify the following questions, most of which are well known and have long attracted attention.

1. Eliminate the discrepancies concerning the content of Li, Be, and B nuclei in the flux of primary cosmic rays near the Earth. Of course, other refinements of the composition of the primary component are also of interest and, in particular, the detection of nuclei substantially heavier than the iron nucleus.

2. Determine the composition of cosmic rays at very high energies. Here it is especially interesting to learn whether protons retain their dominant position also at energies \(E \gtrsim 10^{18}\) ev.

3. Refine the energy spectrum of cosmic rays. Is this spectrum in fact a power law over the entire interval from energies \(E \sim 10^9\) ev to \(E \sim 10^{18}\) ev? The presence of some break in the spectrum, as is clear from the remarks made in Sec. 3g, would be very significant. It would be very interesting to determine the spectrum at kinetic energy \(E_k < 5 \cdot 10^8\) ev/nucleon, which is possible during the period of minimum solar activity. Also important is the question of the degree of closeness of the spectra of protons and of nuclei of different groups.

4. Determine, or substantially lower, the upper limit of the degree of anisotropy of cosmic rays \(\delta\), especially at the very highest energies \(E \gtrsim 10^{18}\) ev (the discussion is, obviously, of variations of intensity in sidereal time).

*) The assumption of two-stage acceleration is tempting as applied to particles of the very highest energy. However, if supernovae accelerated particles only, say, up to an energy \(E \sim 10^{10}\) ev/nucleon, then only by pure chance could one explain the fact that the spectrum at this energy apparently has no break and changes smoothly in passing to higher energies. If, on the other hand, supernovae supply only still slower particles, then major difficulties of an energetic character remain, to say nothing of the fact that this assumption contradicts the data on the Crab Nebula, in which there are certainly electrons with energy \(E \sim 10^{11} \div 10^{12}\) ev.

1. To detect and investigate the spectrum of electrons and positrons in the composition of primary cosmic rays near the Earth. Even an improvement of the attained accuracy by an order of magnitude is promising here. It is important to determine the relative number of electrons in comparison with positrons.

During the period of minimum solar activity, it may be possible to detect soft electrons (energy \(E<10^9\) eV), of which there should be considerably more than hard ones.

1. To refine, both in cosmic rays and, chiefly, on accelerators, the cross sections for the interaction and absorption of fast protons and nuclei in hydrogen and helium. To determine the energy of the secondary electrons arising in collisions and the transformation coefficients of nuclei \(p_{ji}\). In other words, to refine all those, often quite approximate, values of cross sections and other parameters characterizing the interaction that were used in Sections 3a, 5, and 4b.

We shall not dwell here on problems connected with the study of cosmic rays of solar origin, nor on various variations in the intensity of cosmic rays (apart from the stellar variations mentioned).

Radio-astronomical methods can accomplish the following:

1) To refine, and partly to determine anew, the spectrum of cosmic radio emission over as wide a wavelength interval as possible and as a function of galactic coordinates. The same applies to other galaxies, and first of all to the nebula M31.

2) To determine and refine the spectra of as large a number as possible of galactic discrete sources, and especially of the remnants of supernovae and novae. In individual cases these investigations may be supplemented by determining the spectrum of continuous (magnetobremsstrahlung) radiation in the infrared and optical regions.

3) Measurement of the weak polarization of cosmic radio emission will apparently make it possible to draw important conclusions about the configuration of magnetic fields in the galactic “corona.” The importance of determining the polarization of the optical radiation of the Crab Nebula and, perhaps, of other objects is also evident.

It may be hoped, moreover, that a whole series of data valuable for the theory of the origin of cosmic rays can be obtained by astronomical and astrophysical means. These include the value, very important for a number of estimates, of the density of interstellar gas in the Galaxy and, in particular, in the galactic “corona”; further, as an object of study one must indicate the configuration and intensity of the magnetic field and, in this connection, the effective mean free path of cosmic particles, which determines the diffusion rate. Finally, refinement of the available data on velocities of motion in the interstellar medium, together with knowledge of the effective length over which the field changes its direction, is decisive for judging the effectiveness of interstellar acceleration of particles. It is also especially necessary to note observation of the evolution, various features, and chemical composition of the envelopes of supernovae and novae.

In the field of theory we shall point to the necessity of investigating the question of reflection from galactic boundaries and of the limits of applicability of the diffusion approximation when considering the motion of cosmic particles in the interstellar medium carrying magnetic fields. This also includes elucidation of the character of electron diffusion in the Galaxy with allowance for magnetobremsstrahlung and other losses, and a comprehensive analysis of the problem of injection and acceleration of particles in the envelopes of supernovae and novae.

The list given above, which could still be expanded somewhat, shows that there are still many unresolved or insufficiently clarified questions. Therefore, naturally, the theory of the origin of cosmic rays will continue to develop and be refined. The author, however, will allow himself in conclusion to express confidence that the theory of the origin of cosmic rays set forth in the present article will not in the future undergo substantial changes in its foundations and, thus, will not share the fate of all the preceding constructions in this field.

Addition in proof

As was indicated at the end of Section 3b, the question still remains unclear whether cosmic electrons have predominantly a secondary origin or come mainly from primary sources of cosmic rays. Recently I. S. Shklovskii has advanced considerations in favor of the first of these possibilities, pointing to the similarity of the radio-emission spectra in most powerful galactic discrete sources and in the galactic corona. This fact is not in agreement with relation (3.39). However, such an argument, though interesting, does not yet seem to us sufficiently convincing without further investigation. The point is that formula (3.39) applies directly only to the spatially homogeneous case. In the accepted model, however, the generation of cosmic rays apparently takes place mainly in the early stages of the expansion of the shell of a supernova star; then these electrons wander in the expanding shell and only after this emerge into interstellar space. Under such conditions, if the exponent in the spectrum of accelerated particles is \(\gamma_0\), this exponent already at the second stage will be close to \(\gamma=\gamma_0+1\), provided only that this stage is sufficiently long (the second stage is wandering in the shell in the presence of magnetic fields). In the transition to the third stage (wandering in interstellar space) the spectrum may change little further. A reliable answer to the question of the spectrum under the conditions under discussion will be possible only by solving the problem mentioned at the end of Section 4v.

In addition, it must be borne in mind that electrons of quite different energies are responsible for the radio emission in the corona and in discrete sources in the given frequency interval (this is connected with the large difference between the mean values of the fields \(H\) and \(H_{\perp}\) in the corona and in the sources). Therefore, as V. A. Razin emphasized, the similarity of the spectra of the sources and the corona in the range of meter and longer waves does not yet in general indicate similarity, in the indicated regions, of the energy spectra of electrons with energy \(E > 10^9\) ev.

In addition to what was said in Section 3v, let us note that the possibility of neglecting the escape of cosmic particles from the Galaxy becomes doubtful if the nuclear lifetime is \(T_p \sim 6\cdot 10^9\) years (see Section 4g). If the escape of cosmic particles is substantial, this also argues against the assumption of their acceleration in the process of formation of the Galaxy (we are speaking of the variant in which \(T_p \sim 6\cdot 10^9\) years; see Section 4g).

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