THE ORIGIN OF COSMIC RAYS AND RADIO ASTRONOMY

V. L. Ginzburg

Submitted 1953 | SovietRxiv: ru-195301.77757 | Translated from Russian

Full Text

THE ORIGIN OF COSMIC RAYS AND RADIO ASTRONOMY

V. L. Ginzburg

Introduction ........................................................................ 343

§ 1. The electronic component of cosmic rays and cosmic radio emission ............................................................ 348

§ 2. Motion of charged particles in the interstellar medium ............ 359

§ 3. The statistical mechanism of particle acceleration in the interstellar medium and in stellar envelopes ................................ 368

§ 4. Remarks on the theory of the origin of cosmic rays ................. 377
a. Some consequences following from the experimental data. Critique of the hypothesis of the solar origin of cosmic rays .......... 377
b. Supernovae and novae as probable sources of cosmic rays .......... 385

§ 5. Conclusion ..................................................................... 388

References .......................................................................... 391

INTRODUCTION

The question of the origin of cosmic rays arose, in essence, more than 40 years ago, simultaneously with the discovery of cosmic rays. However, for a long period of time data on the primary cosmic rays reaching the boundary of the Earth’s atmosphere were entirely lacking, and for this reason it was impossible to discuss the problem of the origin of cosmic rays seriously. Suffice it to say that even 20 years after the discovery of cosmic rays it was customary to assume that the primary rays consisted of hard photons; subsequently the primary particles were considered to be electrons, and only about 10 years ago was it finally established that the greater part of the primary component consists of protons. Helium nuclei and the nuclei of other elements in the composition of the primary cosmic rays were discovered only in 1948, and thus only in the very last years has the composition of the primary component become clear; knowledge of it is absolutely necessary for constructing a theory of the origin of cosmic rays based on experimental data rather than on speculation.

We shall give here briefly the basic information on the primary component of cosmic rays near the Earth (for more detail see the collections 1, 2).

The primary component consists of protons (80–85%), $\alpha$-particles (15–20%), and nuclei of other elements (Li, Be, B, C, N, O, Ne, Fe, etc.); moreover nuclei with $Z>2$, taken together, are present in the primary flux in an amount of $\sim 1\%$ of all incident particles. It is significant that the composition of the nuclear component of cosmic rays does not coincide with the average composition of matter in the universe. This applies especially to the nuclei Li, Be, and B, whose percentage abundance in cosmic rays is about a million times greater than on the average in space.
The energy spectrum of primary protons and nuclei has the form

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

where $I$ is the intensity of the corresponding particles, $K$ is a certain constant, and $E$ is the total energy of a proton or of one nucleon in a nucleus (thus, the energy of a nucleus is equal to $AE$, where $A$ is the atomic weight of the nucleus). In the energy range $5\cdot 10^8\ \mathrm{ev}<E<3\cdot 10^{10}\ \mathrm{ev}$, both for protons and for nuclei, $\gamma=2.0\div 2.2$. At higher energies the spectrum is known only for the entire primary component, and not separately for protons and nuclei. For $5\cdot 10^{10}<E<10^{12}\ \mathrm{ev}$, $\gamma\simeq 2.5$; in the region $10^{13}<E<10^{18}\ \mathrm{ev}$, $\gamma\simeq 2.7\div 3$. It is not excluded that at high energies the composition of the primary component differs from its composition at $E<3\cdot 10^{10}\ \mathrm{ev}$.

At geomagnetic latitude $58^\circ$, where in the vertical direction only protons with energy $E>1.5\cdot 10^9\ \mathrm{ev}$ can reach the Earth ($E_{\mathrm{kin}}=E-Mc^2>5.6\cdot 10^8\ \mathrm{ev}$), the total intensity of the primary particles is

\[ I\simeq 0.3\, \frac{\text{particles}}{\mathrm{cm}^2\cdot \mathrm{sec}\cdot \mathrm{steradian}}. \]

Hence, by virtue of the isotropy of cosmic rays, it follows that the concentration of primary cosmic particles near the Earth is

\[ N(E>1.5\cdot 10^9\ \mathrm{ev}) \simeq \frac{4\pi}{c} I(E>1.5\cdot 10^9\ \mathrm{ev}) \simeq 10^{-10}\ \mathrm{cm}^{-3} \]

and in the energy region $5\cdot 10^8<E<3\cdot 10^{10}\ \mathrm{ev}$, approximately,

\[ N(E)\simeq \frac{0.1}{E^2}\ \mathrm{cm}^{-3}\,\mathrm{ev}^{-1}, \]

\[ N(E>E_0)=\int_{E_0}^{\infty} N(E)\,dE \simeq \frac{0.1}{E_0}\ \mathrm{cm}^{-3}, \tag{2} \]

where $E$ and $E_0$ are measured in electron-volts*).

*) More precisely, under isotropy $N(E)=\dfrac{4\pi}{v}I(E)$, where $v$ is the velocity of the particles. Since for $E_{\mathrm{kin}}\sim 5.6\cdot 10^8\ \mathrm{ev}$ still $v\sim c$ and we do not aim at great—

The energy density $W=\int E_{\rm kin}N(E)\,dE$, associated with cosmic rays, where $N(E)$ is taken according to (2), reaches the value $W\sim 1\ {\rm eV/cm^3}$. For comparison we note that the energy density of radiation near the plane of the Galaxy is approximately $0.3\ {\rm eV/cm^3}$, while the energy density of thermal motion in ionized regions of the interstellar gas is of the order of $1\ {\rm eV/cm^3}$ (this value is obtained for a gas temperature $\sim 10^4{}^\circ$ and concentration $n\sim 1$). Thus, under the assumption that the concentration and spectrum of cosmic rays in the Galaxy as a whole are close to those observed near the Earth, one must consider that the energy associated with cosmic rays is comparable to and even exceeds the energy of thermal radiation and the internal energy of the interstellar gas.

Electrons, positrons, and photons (i.e., particles of the soft component) have not been detected in the composition of the primary cosmic rays. It follows from experiment only that the flux of soft particles with $E>1.2\cdot 10^9\ {\rm eV}$ does not exceed 0.6% of the total flux of primary particles. Hence it follows that the concentration of relativistic electrons near the Earth is
$N_e(E>1.2\cdot 10^9\ {\rm eV})<5\cdot 10^{-13}\ {\rm cm^{-3}}$.
It is evident from experiment that the primary cosmic rays are isotropic in direction. Furthermore, a connection has been established between cosmic rays and solar activity. In this respect especially noteworthy are the strong variations in the intensity of cosmic rays associated with powerful solar eruptions (see 1, 2). Finally, one must also take into account, and find an explanation for, the cutoff of the spectrum at high latitude, i.e., the fact that in the spectrum of cosmic particles reaching the Earth there are no particles with momenta less than approximately $1.2\cdot 10^9\ {\rm eV}/c$ (for singly charged particles)$^{2,55}$.

It is obvious that any theory of the origin of cosmic rays deserves close attention only if it is in agreement with at least the main facts listed above. From this point of view the untenability of a whole series of old hypotheses on the origin of cosmic rays is immediately clear. For example, the annihilation hypothesis, which has been discussed many times, does not explain the presence of complex nuclei in the primary component.

The presence of electric charge in the primary particles, and the whole body of available data taken together, directly indicate that the acceleration of cosmic particles is somehow connected with electro-

with this accuracy, one may, as has been done in the text, put

\[ N(E>E_0)\simeq \frac{4\pi}{c}\, I(E>E_0). \]

by magnetic fields. In particular, from this point of view the presence in the primary component of various nuclei is natural, as Alfvén^4 pointed out as early as 10 years before such nuclei were detected experimentally. At the same time, the assertion of the electromagnetic character of the acceleration is so general that by itself it is still quite insufficient for constructing a theory of the origin of cosmic rays.

The task of a theory of the origin of cosmic rays is to indicate a concrete mechanism of acceleration under real astrophysical conditions and for quite definite objects (on the Sun, on stars of a definite class, etc.). Attempts in this direction have been made for a long time, beginning with the work of Swann,^5 who indicated that a change in time of the magnetic field near sunspots (and, possibly, starspots) may lead to the acceleration of particles to energies of $10^9 \div 10^{10}$ ev. Subsequently Alfvén^4 (see also^6) proposed a model of a “celestial cyclotron,” connected with the assumption of the presence of a magnetic moment in both components of binary stars. Later the question of the acceleration of particles on the Sun and in stellar atmospheres was considered by a number of authors, especially in detail by Ya. P. Terletskii^7 (who proposed, in particular, an acceleration mechanism connected with the noncoincidence of the magnetic moment and the axis of rotation of the star) and, recently, by Riddiford and Butler.^8

It is not our task to enter into any detailed discussion of the question of the induction mechanism of particle acceleration on the Sun and on stars (see works^3,4,5,7,8 and reviews^6,9), and in this connection we shall confine ourselves to only two remarks.

First, the theory of stellar and solar induction accelerators is still far from complete. The point here is above all that in the corresponding works the accelerated particles are regarded as moving in a vacuum (certain reservations made in this respect in^8 do not alter the essence of the matter). Meanwhile stellar atmospheres, which in a first approximation rotate together with the star, possess enormous conductivity, as a result of which the variable magnetic field—for example, the variable field of spots—must to some degree be screened by currents flowing in the envelope. In other words, the high conductivity, and also the mobility of the medium, make it incomparably more difficult to create in this medium a strong electric field than in a vacuum (see^6 § 7.6 and^1,12). Therefore, and also because of the presence of ionization losses, the theory of stellar accelerators cannot be based on a model in which it is assumed that the particle moves in a vacuum. Such an approximation may possibly prove admissible in regions sufficiently far removed from the photosphere of the star, where the gas density is small and the mean free path is very large, but in these regions the field may already be considerably weaker than is assumed in estimates of the energies of the accelerated particles. Thus, the question of stellar and solar induction accelera-

observers requires further investigation, taking into account the circumstance noted.

The second remark that must be made is perhaps even more essential. Let us even suppose that, under certain conditions, stellar induction accelerators can operate and can generate cosmic rays. But are these conditions realized in reality; which stars generate cosmic rays, in what quantity, and with what spectrum? To all these questions no answer is given in essence and, most importantly, if the Sun is excluded from consideration, no ways are visible by which such answers could be obtained.

Matters stand somewhat better also with the theory—or, more precisely, the hypothesis—of the solar origin of cosmic rays^(10, 11, 12, 13). With regard to the generation of cosmic particles on the Sun or near the Sun there are definite data; some assumptions that have to be made admit of verification. However, the most important element of the theory of the solar origin of cosmic rays is the wholly hypothetical assumption about the character of the magnetic field in the neighborhood of $\sim 0.1$ light-year around the Sun. In addition, this hypothesis encounters a number of other objections, which will be discussed in § 4a.

Thus, at the present time, in our opinion, there are no convincing arguments in favor of the assumption of induction generation of the main part of cosmic radiation on stars or on the Sun.

Another possibility, indicated by Fermi^(14), is that the acceleration of particles occurs in the interstellar medium. However, it is now clear that if the Fermi mechanism in interstellar space is at all effective, then it acts only at energies greater than approximately $10^{11}$ eV, but cannot explain the primary acceleration of cosmic rays.

From the brief survey made above, reflecting the state of the question of the origin of cosmic rays without taking into account radio-astronomical data, which will be discussed below, it is clear that in this field one cannot speak not only of the existence of some finished theory, but until recently it was not even possible to draw a reliably substantiated, and not purely hypothetical, picture of the origin of cosmic rays.

This circumstance cannot cause any particular surprise if one recalls that all the data mentioned about primary cosmic rays pertain to the vicinity of the Earth, whereas from the point of view of the origin of cosmic rays what is of primary interest is information about these rays in the places of their generation, in the expanses of the Galaxy or, in the best case, near the Sun.

Is there any hope of obtaining experimental data on cosmic rays far from the Earth? At first glance, only a negative answer can be given to this question, and therefore among

among physicists, a skeptical attitude toward the prospects for the development of theories of the origin of cosmic rays proved to be very widespread.

The main purpose of the present article is to illuminate the substantial change in the situation connected with the development of radio astronomy. As we shall see in § 1, radio-astronomical methods make it possible to determine the concentration and spectrum of the electronic component of cosmic rays in the spaces of the Galaxy, and make it possible to indicate and investigate the regions where cosmic rays are generated. Thus there opens up the possibility of clarifying precisely those points which until now remained completely obscure and which hindered the solution of the question of the origin of cosmic rays.

True, radio-astronomical data make it possible to judge directly only the electronic component of cosmic rays (below, for brevity, we shall speak of these electrons as cosmic rays). But information about electrons can be connected with information about cosmic protons and nuclei, and thus the known limitation of radio methods by no means deprives them of their value, quite apart from the fact that cosmic electrons themselves are of the same interest as protons and nuclei.

For the interpretation of experimental data on cosmic rays it is necessary to consider the question of various processes occurring with fast particles during their motion in the interstellar medium. This will be discussed in § 2 and, in part, in § 3, devoted to the question of the statistical mechanism of acceleration of particles both in the interstellar medium and in the envelopes of stars. Further, in § 4 we shall dwell on a number of questions connected with the origin of cosmic rays, and shall try to give a general scheme of the generation and evolution of cosmic rays in the Galaxy.

In the concluding § 5 brief conclusions will be drawn concerning the state of the problem of the origin of cosmic rays at the present time, and the tasks of further experimental and theoretical work in this interesting field of astrophysics and cosmic-ray physics will be indicated.

§ 1. THE ELECTRONIC COMPONENT OF COSMIC RAYS AND COSMIC RADIO EMISSION

As is known, on the Earth there is observed both solar radio emission and cosmic radio emission, which consists of several components: extragalactic radiation (radiation of extragalactic nebulae), radiation of individual galactic nebulae, and general galactic radiation.

The radiation of nebulae (galactic and extragalactic) reaches the Earth within a small solid angle; that is, here we are dealing with discrete sources of cosmic radio emission.

General galactic radio emission, on the contrary, changes relatively slowly with a change in the direction of observation and has, as it were, the character of a continuous background, against which individual discrete sources of radio emission are noticeable (they are often called “radio stars” or “radio nebulae”).

Part of the general galactic radio emission is thermal in character—it is due to the thermal radiation of ionized regions of interstellar gas (for more detail on this, as well as on all other radio-astronomical questions touched upon by us, see $^{15,16}$). However, in the meter range, at wavelengths longer than approximately $1.5\ \mathrm{m}$, thermal radiation is not the principal component for most directions. This is clear from the fact that the temperature of ionized regions of interstellar gas (the so-called HII regions) does not exceed $\sim 10^4$ degrees. Therefore the effective temperature $T_{\mathrm{eff}}$ of thermal general galactic radio emission also cannot exceed $10\,000^\circ$. Meanwhile, for example, at a wavelength of $16.3\ \mathrm{m}$ in the direction of the center of the Galaxy $T_{\mathrm{eff}} \approx 175\,000^\circ$, and in the direction of the galactic pole $T_{\mathrm{eff}} \approx 50\,000^\circ$.

Until quite recently, attempts were made to connect nonthermal galactic radiation with the aggregate radiation of discrete sources unresolved by the apparatus, which were considered to be stars of a new type. However, this explanation, which for a number of reasons always seemed to us very doubtful*), has now completely fallen away after the nonstellar nature of the discrete sources was established (see above).

Another assumption, at present the only known one, amounts to the statement that nonthermal galactic radio emission is the bremsstrahlung radiation of relativistic (cosmic) electrons in interstellar magnetic fields $^{17—20}$.

In a magnetic field an electron rotates (moves along a helical line) with cyclic frequency

\[ \omega_H=\frac{eH}{mc}\cdot\frac{mc^2}{E}, \tag{3} \]

where $E$ is the total energy and $H$, as everywhere below, is the component of the magnetic field perpendicular to the velocity. In the nonrelativistic case $\omega_H=eH/mc$, and the radiation of the electron has a dipole character, the radiation frequency being equal to $\omega_H$; but for $E/mc^2 \gg 1$ the radiation must have an entirely different character, as is already clear at once from the well-known fact that a relativistic particle emits electromagnetic waves practically only within an angle $\theta \sim mc^2/E$ with

*) It is sufficient to say that the hypothetical radio stars would have had to give radio emission with an effective temperature reaching $10^{18}$ degrees, and at the same time belong to some type of extremely numerous red or even infrared dwarfs.

direction of the velocity. Therefore a relativistic electron moving, for example, in a circle emits only in directions close to the plane of the orbit, and at all times along the tangent to its trajectory. As a result, an observer in the plane of the orbit will register separate “bursts” of radiation, repeated with frequency $\nu_H=\dfrac{\omega_H}{2\pi}$ and lasting a time $\Delta t\sim \dfrac{R\theta}{c}\sim \dfrac{mc^2}{E\omega_H}\sim \dfrac{mc}{eH}$, where $R\simeq \dfrac{c}{\omega_H}$ is the radius of the orbit and $\theta\sim \dfrac{mc^2}{E}$ is the angle within which the radiation is concentrated.

A rigorous analysis of the radiation of relativistic particles moving in a constant homogeneous magnetic field was carried out by Schott$^{21}$ more than forty years ago. Subsequently these results were partly rederived, and partly subjected to further discussion and interpretation, in papers by a number of authors$^{22-28}$. Below, as in$^{18-20}$, we shall base ourselves on the work of V. V. Vladimirsky$^{25*}$.

The energy radiated by an electron in 1 second in the frequency interval $d\nu$ is equal to $P(\nu)d\nu$, where

\[ \left. \begin{aligned} P(\nu,E)\equiv P(\nu)&=2\pi P(\omega) =16\frac{e^3H}{mc^2}\,p\!\left(\frac{\omega}{\omega_m}\right) \\ &=16\left(\frac{e^2}{c}\right)\omega_H \left(\frac{\omega}{2\omega_H}\right)^{1/3}Y(u), \\ \omega_H&=\frac{eH}{mc}\frac{mc^2}{E},\qquad \omega_m=\frac{eH}{mc}\left(\frac{E}{mc^2}\right)^2,\qquad u=\left(\frac{\omega}{2\omega_m}\right)^{2/3}. \end{aligned} \right\} \tag{4} \]

Here $H$ is the component of the magnetic field perpendicular to the velocity; it is assumed (as everywhere below) that $E/mc^2\gg 1$, and, finally, $p\!\left(\dfrac{\omega}{\omega_m}\right)$ and $Y(u)=\dfrac{p(2u^{3/2})}{u^{1/2}}$ are functions which, in limiting cases, have the form

\[ \left. \begin{aligned} \frac{\omega}{\omega_m}\ll 1:\qquad &p\!\left(\frac{\omega}{\omega_m}\right) =0.256\left(\frac{\omega}{\omega_m}\right)^{1/3}, \qquad Y(u)=0.256, \\[6pt] \frac{\omega}{\omega_m}\gg 1:\qquad &p\!\left(\frac{\omega}{\omega_m}\right) =\frac{1}{16}\left(\pi\frac{\omega}{\omega_m}\right)^{1/2} e^{-\frac{2\omega}{3\omega_m}}, \\[6pt] &Y(u)=\frac{(2\pi)^{1/2}}{16}\,u^{1/4}e^{-\frac{4}{3}u^{3/2}} . \end{aligned} \right\} \tag{5} \]

* Let us note that in the cases of interest to us the condition $\hbar\omega\ll E$ is certainly fulfilled, where $\omega$ is the angular frequency of the radiation, $\hbar=1.05\cdot 10^{-27}$ is Planck’s constant, and $E$ is the energy of the particle. Under this condition the radiation may be treated classically$^{25,28}$, as we do. Statements encountered in the literature$^{24}$ that quantum effects should be noticeable when conditions considerably weaker than the one indicated are not satisfied are erroneous.

The values of the functions $Y(u)$ and $uY(u)$, which we shall need below, are given in Table I.

Table I

$u$	$Y(u)$	$uY(u)$
0	0.256	0
0.2	0.204	0.041
0.4	0.156	0.062
0.6	0.115	0.069
0.8	0.081	0.065
1.0	0.055	0.055
1.2	0.036	0.0425
1.4	0.023	0.0325
1.6	0.014	0.023
1.8	0.0085	0.0154
2.0	0.0050	0.01
2.2	0.0028	—
2.4	0.0015	—
2.6	0.00083	—
2.8	0.00044	—
3.0	0.00023	0.0007
3.5	0.00004	—
4.0	0.000006	—

For $\omega \simeq 0.5\omega_m$ the function $p\left(\dfrac{\omega}{\omega_m}\right)$ is maximal and equal to 0.10. Thus, at the maximum,

\[ P(\nu_{\max}) = 1.6\,\frac{e^3H}{mc^2} = 2.15 \cdot 10^{-22} H\, \frac{\mathrm{erg}}{\mathrm{sec}\cdot\mathrm{cycle}}, \tag{6_1} \]

\[ \nu_{\max} = 0.5\,\frac{\omega_m}{2\pi} = 1.4 \cdot 10^6 H \left(\frac{E}{mc^2}\right)^2 \ \mathrm{cycle}. \tag{6_2} \]

Let us apply the formulas given above to the problem of interest to us (see $^{18\text{--}20}$).

The intensity of the radio emission observed on the Earth is equal to*)

\[ I_\nu = \frac{1}{4\pi}\int P(\nu,E)N_e(E,\mathbf r)\,dE\,dr, \tag{7} \]

where $N_e(E,\mathbf r)$ is the differential energy 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 field $\mathbf H$, the radiation along the line of sight may on average be regarded as isotropic (whence

*) We recall that, by definition, the intensity $I_\nu$ is the flux of radiation referred to unit solid angle and to the spectral interval through a unit area perpendicular to the direction of observation (the direction of propagation of the radiation). The flux of radiation from some source (for example, a discrete source of radio emission) is equal to

\[ F_\nu = \int_{\Omega} I_\nu\,d\Omega, \]

where $\Omega$ is the solid angle corresponding to the source under consideration.

the factor $\frac{1}{4\pi}$ in (7); the assumption made leads to the fact that the resulting formulas are correct, generally speaking, only up to a factor of order unity).

Moreover, in (7) the absorption of radiation in the interstellar medium is not taken into account, but this circumstance will tacitly be taken into consideration in the treatment of the experimental data. It should also be noted that we regard the radiation as occurring in vacuum, i.e. we take the refractive index of the medium in which the relativistic electron moves to be equal to unity. This is legitimate only if

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

where $n(\nu)$ is the refractive index for the radiation frequency $\nu$ under consideration [see $^{24,49}$ and $*$]. In the cases of interest to us (interstellar gas, expanding stellar envelopes) one may assume that the refractive index of the medium is equal to the refractive index of an electron-ion plasma in the absence of an external field, i.e. that

\[ n^{2}=1-\frac{e^{2}N_{0}}{\pi m\nu^{2}}, \]

where $N_{0}$ is the concentration of free electrons in the medium. Usually $|1-n|\ll 1$ and, consequently,

\[ 1-n=\frac{e^{2}N_{0}}{2\pi m\nu^{2}}. \]

Putting now $\nu\sim \nu_{\max}$, we see that the above formulas for the radiation intensity, in which $n=1$ has been put, are applicable if

\[ \left(\frac{E}{mc^{2}}\right)^{2}\gg \frac{10^{-5}N_{0}}{H^{2}}. \]

In the interstellar medium, for $H\sim 10^{-5}$ and $N_{0}\sim 1$

$*$ It is especially easy to obtain the inequality given in the text and to understand its meaning if one recalls that in formulas characterizing the radiation of a relativistic particle in vacuum, the denominators contain factors of the type

\[ (1-\beta\cos\theta), \]

where $\beta=\frac{v}{c}$, and $\theta$ is the angle between the direction of the velocity and the direction of observation. For example, the frequency of the light emitted by a moving oscillator is

\[ \nu(\theta)=\frac{\nu_{0}\sqrt{1-\beta^{2}}}{1-\beta\cos\theta}, \]

where $\nu_{0}$ is the proper frequency of the oscillator. If the radiation occurs not in vacuum but in a medium, then in the expression $1-\beta\cos\theta$ one must replace $\beta$ by $\beta n=\frac{vc}{n}$ (this replacement is clear, since the phase velocity of light in the medium is $\frac{c}{n}$), and the formula for the Doppler effect takes the form

\[ \nu(\theta)=\frac{\nu_{0}\sqrt{1-\beta^{2}}}{|1-\beta n\cos\theta|} \]

(for details see $^{285}$). Hence it is clear that for $n>1$ the radiation can be similar to that which occurs in vacuum in the extremely relativistic case ($\beta\gg 1$) already for $\beta<1$, and sometimes even for $\beta\ll 1$ (see $^{24}$). If, however, $n<1$, as is always the case in a plasma in the absence of, or in a sufficiently weak, constant magnetic field, then $1-\beta n\cos\theta$ does not tend to zero even for $\beta\to 1$, and therefore the radiation (provided only that $1-n$ is not too small) may lack the features typical of the relativistic case. The influence of $n$ may be neglected, evidently, if

\[ |(1-\beta n)-(1-\beta)|\ll 1-\beta, \]

i.e. under the condition

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

(in passing to the last expression it has been assumed that $1-\beta\ll 1$).

this inequality takes the form $\left(\dfrac{E}{mc^2}\right)^2 \gg 10^5$ and is satisfied even for waves with length $\lambda \sim 30$ m (the radiation of electrons is especially important, for which $\nu_{\max}\sim\nu=\dfrac{c}{\lambda}\sim 10^7$ and, consequently, $\left(\dfrac{E}{mc^2}\right)^2=\dfrac{\nu_{\max}}{1.4\cdot 10^6 H}\sim 10^6$). Similar estimates show that one may put $n=1$ in practically all cases of interest to us, but for sufficiently long waves or for large values of the parameter $\dfrac{N_0}{H^2}$, allowance for the fact that $n\ne 1$ leads to a sharp decrease in the radiation intensity.

If we are interested in radiation at some frequency $\nu$, then the minimum required number of electrons can be determined by assuming that all electrons possess the energy $E$, associated with the frequency $\nu=\nu_{\max}$ by the second of relations (6). In this case the radiation is maximal and

\[ I_{\nu_{\max}}^{\max}=\frac{P(\nu_{\max})}{4\pi}N_eR= \]

\[ =1.7\cdot 10^{-23}HN_eR\ \frac{\mathrm{erg}}{\mathrm{cm}^2\cdot\mathrm{sec}\cdot\mathrm{cycles}\cdot\mathrm{steradian}}, \tag{8} \]

where $N_e$ is the mean concentration of electrons along the line of sight (in the case under consideration, obviously, $N_e(E')=N_e\delta(E-E')$, where $E$ is related to $\nu$ by relation $(6_2)$), and $R$ is the size of the radiating region in the given direction.

In the case of thermal radiation with effective temperature $T_{\mathrm{eff}}$

\[ I_\nu=\frac{2kT_{\mathrm{eff}}}{\lambda^2} =\frac{2.76\cdot 10^{-16}T_{\mathrm{eff}}}{\lambda^2}\, \frac{\mathrm{erg}}{\mathrm{cm}^2\cdot\mathrm{sec}\cdot\mathrm{cycles}\cdot\mathrm{steradian}}. \tag{9} \]

Experimentally, for $\lambda=\dfrac{c}{\nu}\sim 10^3=10$ m, $T_{\mathrm{eff}}\sim 10^5$, whence $HN_eR\sim 10^6$, and for $R\sim 10^{23}$ (the dimensions of the Galaxy)

\[ HN_e\sim 10^{-17}. \tag{10} \]

According to data that will be discussed below, $H\lesssim 10^{-5}$, and thus $N\gtrsim 10^{-12}\ \mathrm{cm}^{-3}$. The value obtained is two orders of magnitude smaller than the number of cosmic protons observed near the Earth and does not contradict data on the concentration of cosmic electrons near the Earth. Indeed, for $\nu\sim 3\cdot 10^7$ ($\lambda\sim 10$ m) and $H\sim 10^{-5}$, according to $(6_2)$ $E\sim 10^8$ eV, whereas near the Earth $N_e(E>10^9)<5\cdot 10^{-13}\ \mathrm{cm}^{-3}$, and nothing at all is known about cosmic electrons with $E<10^9$ eV (electrons with $E<10^9$ eV are “cut off” by the magnetic field of the Sun or of the solar system; see $^{20}$ and, in more detail, the article by V. L. Ginzburg, G. G. Getmantsev, and M. I. Fradkin in $^{1}$).

Thus, if in the Galaxy there is a magnetic field $H\sim 10^{-6}\div 10^{-5}$ oersted, then the hypothesis of a connection between cosmic radio emission and the braking radiation of cosmic electrons in magnetic fields turns out ...

is quite reasonable and deserves careful verification. It is therefore necessary to dwell, at least briefly, on the question of the magnitude of the interstellar magnetic field.

In this respect there are the following indications (for more detail see $^{1, 6, 29, 30}$):

1) The isotropy of cosmic rays and the necessity of retaining them in the Galaxy (see § 3) force one to suppose that in the interstellar medium there is a magnetic field with $H > 10^{-8}$ oersted. Indeed, the radius of curvature of a relativistic singly charged particle with energy $E$ in a field $H$ is equal to

\[ r=\frac{E\ (\text{in ev})}{300\,H}. \tag{11} \]

Hence, for $E \sim 10^{17}$ ev (an energy close to the maximum observed), $r \sim R \sim 3 \cdot 10^{22}$ cm $\simeq 10\,000$ parsecs*) for $H \sim 10^{-8}$ oersted. In fact, one must require the fulfillment of the inequality $r \ll R$, but, on the other hand, isotropy has been proved $^{31}$ with a high degree of accuracy only for particles with $E \lesssim 10^{14}$ ev. Thus the resulting estimate of the field $H$ is rather crude; in any case, a field $\sim 10^{-6}$ oersted ensures isotropy and the slow escape of particles from the Galaxy.

2) Attempts are made to explain the experimentally observed polarization of starlight by the influence of cosmic dust oriented in magnetic fields of intensity $\sim 10^{-5}$ (it is true that both the experimental and the theoretical sides of the question of the polarization of starlight do not permit the indicated conclusion to be regarded as final; see $^{29}$).

3) General magnetohydrodynamic considerations speak in favor of the existence of interstellar magnetic fields. Interstellar gas is an excellent conductor (in most cases its conductivity may be regarded as infinite); on the other hand, this gas is in a state of turbulent motion, which in the case of a conducting medium must, by a number of considerations, be accompanied by the appearance of currents and of a magnetic field. In the stationary state one may expect equality of the magnetic energy and the kinetic energy of the gas,

\[ \frac{H^2}{8\pi}=\frac{\rho v^2}{2} \tag{12} \]

($\rho$ is the density of the gas and $v$ its velocity).

*) According to the very convincing considerations of S. B. Pikelner $^{30}$, the galactic rarefied interstellar gas (concentration $n \lesssim 0.1$) forms a “spherical subsystem” with dimensions $\sim 10\,000$ pc (probably, a system closer to an ellipsoidal one with a minor axis somewhat less than or close to $10^4$ pc, and a major axis $\sim 3 \cdot 10^4$ pc). The magnetic field and cosmic rays form a subsystem close to the subsystem of rarefied gas. Let us recall that the greater part of the “stellar population” and the clouds of interstellar gas ($n \sim 10$) form a flat subsystem with thickness $\sim 10^2$ pc and diameter $\sim 3 \cdot 10^4$ pc $\simeq 10^{23}$ cm (this is the diameter of the Galaxy).

The question of whether this equality is attained in interstellar space has not yet been definitively resolved; this problem is very important and interesting. In any case, from (12) one can estimate the field $H$, obtaining thereby, apparently, an upper limit. Using the known values of $\rho$ and $v$ for clouds of interstellar gas $\left(v \simeq 6\cdot 10^5 \frac{\mathrm{cm}}{\mathrm{sec}},\ \rho \simeq 2\cdot 10^{-23}\frac{\mathrm{g}}{\mathrm{cm}^3}\right)$, we obtain $H \sim 10^{-5}$. In the rarefied interstellar gas between clouds and above the plane of the Galaxy, $v \sim 5\cdot 10^6$, $\rho \sim 10^{-25}$, and also $H \sim 10^{-5}$ (see $^{30}$).

Let us note that the energy density of the field for $H = 3\cdot 10^{-6} \div 10^{-5}$ is equal to

\[ \frac{H^2}{8\pi} \simeq 0.3 \div 3 \frac{\mathrm{eV}}{\mathrm{cm}^3}, \]

i.e., is of the same order of magnitude as the energy density associated with cosmic rays (see the introduction).

We see that the value of the field $H$ which was adopted above for the interpretation of the radio-astronomical data does not contradict the estimates given. Therefore, in our opinion, the very existence of nonthermal galactic radio emission should also be regarded as one of the most convincing indications of the existence of an interstellar magnetic field.

The persuasiveness of this argument will become especially clear in the light of the whole further exposition; moreover, from this point of view it is essential that at present no possibilities are visible for explaining nonthermal cosmic radio emission in any other way except the one under discussion*).

Let us pass to a more detailed interpretation of the data on nonthermal galactic radio emission. These data can be summarized by indicating that in the range $1.5 < \lambda < 16.3\ \mathrm{m}$

\[ I_\nu = a\frac{c}{\nu} = a\lambda,\qquad a \simeq 5\cdot 10^{-21}\frac{\mathrm{erg}}{\mathrm{cm}^3\,\mathrm{sec}\cdot\mathrm{cycles}\cdot\mathrm{steradian}}, \tag{13} \]

*) It is easy to show that the bremsstrahlung radiation of electrons and protons in their collisions with particles of interstellar gas, and a number of other conceivable processes, are not able, under reasonable assumptions, to lead to the appearance of the observed radio emission. In $^{32}$ this emission is also associated with bremsstrahlung in magnetic fields, but with the radiation not of electrons, but of protons. Under such an assumption it is still necessary to admit the existence of magnetic fields. At the same time, as was noted already in $^{18}$, such an explanation appears to us completely unrealistic. The point is that a proton radiates in the same way as an electron if its energy

\[ E_p = \left(\frac{M}{m}\right)^2 E = 3.4\cdot 10^6 E, \]

where $E$ is the energy of the electron and $M$ is the mass of the proton. Therefore, to explain the observed radiation, the concentration of protons with $E_p \gtrsim 10^{14}$ would have to be several orders of magnitude greater than that observed in cosmic rays near the Earth. We shall not dwell in detail on the difficulties connected with such an assumption, since the preferential abundance of electrons in comparison with protons in the present case seems to us completely obvious.

where the accuracy of the determination of $a$ probably does not exceed 50% (the quoted value was obtained for the direction toward the galactic pole; for more detail see¹).

Expression (13) must be compared with (7), for which it is also necessary to specify the form of the spectrum. We shall assume that the spectrum of cosmic electrons, like the spectrum of other cosmic particles, has a power-law character:

\[ N_e(E)=\frac{K}{E^\gamma}. \tag{14} \]

Taking into account (4) and (14), and assuming that along the line of sight over the path $R$ the density $N_e(E)$ does not depend on the coordinates, we obtain from (7):

\[ \begin{aligned} I_\nu &=\frac{R}{4\pi}\int_0^\infty P(\nu,E)N_e(E)\,dE= \\ &=\frac{3}{\pi}(2\pi)^{\frac{1-\gamma}{2}} \frac{e^3H}{mc^2} \left[\frac{2eH}{m^3c^5}\right]^{\frac{\gamma-1}{2}} U(\gamma)KR\nu^{\frac{1-\gamma}{2}}\simeq \\ &\simeq 1.3\cdot10^{-22}(2.8\cdot10^8)^{\frac{\gamma-1}{2}}\times \\ &\quad \times U(\gamma)KH^{\frac{\gamma+1}{2}}R\lambda^{\frac{\gamma-1}{2}} \ \frac{\text{erg}}{\text{cm}^2\cdot\text{sec}\cdot\text{ster}\cdot\text{radian}}, \end{aligned} \tag{15} \]

where

\[ U(\gamma)=\int_0^\infty Y(u)\,u^{\frac{3\gamma-5}{4}}\,du; \]

for $\gamma=1;\ \frac{5}{3};\ 2;\ 3$ and 7 the function $U(\gamma)$ is respectively equal to $0.37;\ 0.163;\ 0.125;\ \frac{\pi}{36}=0.087$ and $\frac{7\pi}{144}=0.153$.

Experimentally, the intensity of the total nonthermal galactic radio emission $I_\nu$ is proportional to $\lambda$ (see (13)), and consequently, according to (13)—(15), $\gamma=3$ and

\[ I_\nu\simeq 3.1\cdot10^{-15}KH^2R\lambda\simeq 5\cdot10^{-21}\lambda, \tag{16} \]

whence $KH^2R\simeq 1.6\cdot10^6$*).

Taking the largest reasonable values $R\sim 5\cdot10^{22}\ \text{cm}$ and $H\sim10^{-5}$, we obtain the smallest value

\[ K_{\min}\sim 3\cdot10^{-19}\simeq 10^5\ \frac{(\text{eV})^2}{\text{cm}^3}. \]

In order to have a certain margin, let us put

\[ K=10^6\ \frac{(\text{eV})^2}{\text{cm}^3}, \]

as a result of which

*) Let us note that in the case under consideration (for $\gamma=3$) a more careful averaging over the angle between the direction of observation and the direction of the magnetic field $H_0$, isotropic on the average, leads only to the fact that in formula (16)

\[ H^2=\frac{2}{3}H_0^2. \]

we arrive at the spectrum

\[ \left. \begin{aligned} N_e(E) &\simeq \frac{10^6}{E^3}\ \text{ev}^{-1}\ \text{cm}^{-3},\\ N_e(E>E_0) &= \int_{E_0}^{\infty} N_e(E)\,dE \simeq \frac{10^6}{2E_0^2}\ \text{cm}^{-3}. \end{aligned} \right\} \tag{17} \]

Hence $N_e(E>10^9)=5\cdot 10^{-13}$, which is not in contradiction with the data on cosmic electrons at the Earth (see also below). At the same time the concentration $N_e(E>10^8\ \text{ev})=5\cdot 10^{-11}$, i.e. in order of magnitude it is equal to the concentration of primary cosmic particles (mainly protons with $E_{\text{kin}}>5.6\cdot 10^8\ \text{ev}$ at the Earth).

For $\gamma=3$, the maximum contribution to the radiation with frequency $\nu$ is made by electrons with energy $E$ corresponding to the value

\[ u=\left(\frac{\omega}{2\omega_m}\right)^{2/3} =\left(\frac{\nu}{\dfrac{eH}{\pi mc}\left(\dfrac{E}{mc^2}\right)^2}\right)^{2/3} \simeq 0.6 \quad \text{(see (4) and Table I).} \]

The radiation of particles with $u>2$ and $u<0.1$ is already quite negligible. It follows, for example, that for $H\sim 10^{-5}$ the radiation at wavelength $\lambda=16.3\ \text{m}$ is determined mainly by electrons with energy $E\simeq 5\cdot 10^8\ \text{ev}$, while the contribution of particles with $E<2\cdot 10^8\ \text{ev}$ and $E>2\cdot 10^9\ \text{ev}$ is already negligible. At a wavelength of $1.5\ \text{m}$, the most important electrons are those with $E\simeq 2\cdot 10^9\ \text{ev}$. Thus, the spectrum (17) pertains to the region $2\cdot 10^8<E<5\cdot 10^9\ \text{ev}$, the upper limit being inexact. The point is that at a wavelength of $1.5\ \text{m}$ the thermal galactic radiation is already very substantial; there are also indications that for $\lambda<1.5\ \text{m}$ the intensity of the nonthermal radiation decreases with decreasing wavelength faster than $\lambda$. Therefore, for $E>10^9\ \text{ev}$, for electrons located in the direction of the galactic pole, evidently $\gamma>3$ (see $^{19}$ and below). In this connection the value $N_e(E>10^9\ \text{ev})=5\cdot 10^{-13}$ appears overestimated, which is favorable from the point of view of the data on electrons at the Earth.

Further experimental investigation of the intensity and spectrum of the general galactic radio emission in different directions will make it possible substantially to refine and detail the data on cosmic electrons in the Galaxy.

The radio emission of most discrete sources (radio nebulae) also cannot be reduced to thermal radiation and is explained by bremsstrahlung radiation in magnetic fields. As regards those discrete sources which are other galaxies—for example, in the case of the great nebula M31 in the constellation Andromeda—this conclusion is especially closely connected with that made above. For example, when observing from the nebula M31, our Galaxy would also be noticeable in the radio range by virtue of the fact that the radiat—

the cosmic electrons escape, of course, beyond the limits of the Galaxy. As for the radio nebulae located in the Galaxy, the most important of them are remnants of supernova outbursts. This applies to the discrete source of radio emission in Taurus (the source Taurus A, or the Crab Nebula—Supernova of 1054), to the most powerful of the known discrete sources—the source in Cassiopeia (Cassiopeia A—Supernova of 369) and to the Supernova of 1572. The flux of radio emission from the source in Cassiopeia in the meter range is

$F_\nu \approx 2 \cdot 10^{-19}\ \dfrac{\text{erg}}{\text{cm}^2 \cdot \text{sec} \cdot \text{cycles}}$*),

for Taurus A in the range $7.5\,m \div 20\,cm$ the flux, to an accuracy of up to 25%, is constant and equal to $F_\nu = 2 \cdot 10^{-20}$. For the Supernova of 1572 at $\lambda = 1.9\,m$, $F_\nu = 1.7 \cdot 10^{-21}$. The last of the supernovae that flared up in our Galaxy—the Supernova of 1604—is certainly not a particularly powerful source, but it may radiate approximately as the remnant of the Supernova of 1572 does (the radio telescope of high sensitivity with which the radiation of the remnants of the Supernova of 1572 was observed could not be pointed at the Supernova of 1604, so that the question of its radiation remains open).

If the electron spectrum and the magnetic-field intensity are assumed constant throughout the whole source, then the flux from the source is equal to (see (7)):

\[ F_\nu = \int I_\nu\, d\Omega = \frac{V}{4\pi R^2}\int P(\nu,E)N_e(E)\,dE, \tag{18} \]

where $V$ is the volume of the source and $R$ is the distance to it.

Under the most favorable conditions (6)

\[ F_\nu = 1.6\,\frac{e^3H}{mc^2}\,\frac{N_eV}{4\pi R^2} = 1.7 \cdot 10^{-23} H\,\frac{N_eV}{R^2}\, \frac{\text{erg}}{\text{cm}^2 \cdot \text{sec} \cdot \text{cycles}}, \tag{19} \]

where $N_e$ is the concentration of electrons, all of which are assumed to possess an energy connected with the frequency by relation $(6_2)$. For Taurus A, $R \approx 1500\,\mathrm{pc} \approx 5 \cdot 10^{21}\,\mathrm{cm}$, the radius of the nebula $\rho \sim 1\,\mathrm{pc} \approx 3 \cdot 10^{18}\,\mathrm{cm}$ (angular size $\sim 5'$) and $V \approx \dfrac{4\pi}{3}\rho^3 \sim 10^{56}\,\mathrm{cm}^3$; for Cassiopeia A, $R \sim 10^{22}\,\mathrm{cm}$, $\rho \sim 10^{19}\,\mathrm{cm}$, $V \sim 10^{57}\,\mathrm{cm}^3$. The value of the field $H$ in the shells of supernovae, as follows from estimates based on relation (12), does not exceed $10^{-3}$ oersted, but, in all probability, the field is weaker. Therefore in works $^{18,33,34}$, whose results we use, the value $H \sim 10^{-4}$ is adopted. Substituting in (19) all the values given above, we obtain for Taurus A $N_eV \sim 10^{50}$ and for Cassiopeia

*) Let us note for comparison that this flux corresponds, for example, at $\lambda = 3\,m$, to radio emission from the Sun with an effective temperature $T_{\mathrm{eff}} \approx 10^6$ degrees, which is often observed in practice. Thus, in the meter range in the sky “shine” at once, as it were, “two Suns,” or, more precisely, even “three Suns,” since the extragalactic discrete source in Cygnus is only about twice weaker than the source in Cassiopeia.

$N_e V \sim 10^{52}$ or, respectively, $N_e \sim 10^{-6}\ \text{cm}^{-3}$ and $N_e \sim 10^{-5}\ \text{cm}^{-3}$. The values quoted are, obviously, minimal ones.

Using formulas (15) and (18), these results can be refined by determining the electron spectrum in the envelopes of supernovae[^34]. In this case, for Taurus A, $\gamma \simeq 1$, and for Cassiopeia, according to less accurate data, $\gamma \simeq 8/3$. At the same time, for the strongest extragalactic sources (Cygnus A, Centaurus, Leo), as well as for our Galaxy as a whole, $\gamma \simeq 3$. An estimate of the total energy and the total number of relativistic electrons in Taurus A, where the spectrum is best known, leads to values reaching $\sim 10^{50}\ \text{erg}$ and $N_e(E > 2.5 \times 10^8\ \text{eV})V \sim 10^{51}$, which corresponds to an average energy of $\sim 10^{11}\ \text{eV}$ per electron.

We point out that I. M. Gordon[^35], in application to the continuous optical spectrum of solar eruptions, and then I. S. Shklovsky[^36], in relation to the continuous optical radiation of the Crab Nebula (Taurus A), assumed that this radiation is also the braking radiation of fast electrons in magnetic fields. For Taurus A the flux of continuous radiation in the optical region is[^16] $F_\nu = 1.5 \cdot 10^{-23}$. Such radiation will be ensured if, in this source, the electron spectrum with $\gamma \simeq 1$ extends to energies $E \sim 5 \cdot 10^{11}\ \text{eV}$.

Thus, radio methods make it possible to detect the presence of relativistic electrons in various places in the universe. In particular, the data cited on the radio emission of supernova remnants indicate that, as a result of supernova explosions, a large number of relativistic particles is formed. The significance of this fact for the theory of the origin of cosmic rays, as was first emphasized by I. S. Shklovsky[^33], can hardly be overestimated. This will be discussed in more detail in § 46.

§ 2. MOTION OF CHARGED PARTICLES IN THE INTERSTELLAR MEDIUM

In investigating the question of the origin of cosmic rays it is necessary to clarify the character of their motion in interstellar space, where the particles both lose and, under certain conditions, acquire energy. Moreover, the corresponding analysis given below applies to a considerable extent also to the case when the particles move in denser regions of the universe (in nebulae, expanding envelopes of supernovae and novae, atmospheres of stars and of the Sun).

We begin by considering the loss of energy by protons and nuclei. The effective cross section for collision of two nuclei $i$ and $k$ with atomic weights $A_i$ and $A_k$ may be written in the form

\[ \begin{gathered} \sigma = \pi (r_i + r_k - 2\Delta r)^2, \qquad r_i = 1.45 \cdot 10^{-13} A_i^{1/3},\\ \Delta r \simeq 0.85 \cdot 10^{-13}, \qquad r_k = 1.45 \cdot 10^{-13} A_k^{1/3}. \end{gathered} \tag{20} \]

In collisions of protons with nuclei with $A \gtrsim 8$, expression (20) can with sufficient accuracy be replaced by

\[ \sigma=\pi r^2=6.6\cdot 10^{-23} A^{2/3}\ \text{cm}^2 . \tag{21} \]

For collisions of protons with protons, according to (20), $\sigma=4\cdot 10^{-26}\ \text{cm}^2$; sometimes in this case one also uses the cross section $\sigma=\pi\left(\dfrac{\hbar}{\mu c}\right)^2=6\cdot 10^{-26}$ $(\mu=276m$—the mass of the $\pi^+$-meson).

It is necessary, however, to consider what cross section is of interest to us. For what follows it is primarily important to know the cross section corresponding to the disappearance of a particle of the given kind. In the case of nuclei, cross section (20) corresponds precisely to this condition,² since in the collisions in question the nuclei are destroyed (in other words, nuclear stars are produced).

In collisions of protons with protons, in some cases even after the collision one of the protons possesses high energy, and the cross section corresponding to the disappearance of the fast proton is probably substantially smaller than the cross section obtained from (20). Unfortunately, there are no experimental data in this direction, and we shall adopt, as in $^{14}$, the cross section $\sigma=2.5\cdot 10^{-26}\ \text{cm}^2$.

The mean free paths of protons and nuclei in hydrogen, and in a mixture $90\%\ \mathrm{H}+10\%\ \mathrm{He}$, are given in Table II.

Table II

Nucleus	Cross section in hydrogen	Range in hydrogen $l=\dfrac{1.67\cdot 10^{-24}}{\sigma}\ \text{g}/\text{cm}^2$	Range in hydrogen $l=\dfrac{1}{\sigma n}\ \text{cm}$ at $n=0.1$	Time of free path in years at speed $v=c$; $T=l/c$ (hydrogen, $n=0.1$)	Mixture of hydrogen with $n=0.09$ and helium with $n=0.01$; $T$ in years	$T_{\text{proton}}/T_{\text{nucleus}}$
Proton . . . .	$2.5\cdot 10^{-26}$	67	$4\cdot 10^{26}$	$4\cdot 10^8$	$2.9\cdot 10^8$	—
$\alpha$-particle . . .	$1.3\cdot 10^{-25}$	13	$7.7\cdot 10^{25}$	$8\cdot 10^7$	$7.4\cdot 10^7$	4
$\mathrm{O}_{16}^{8}$ . . . .	$3.6\cdot 10^{-25}$	4.6	$2.8\cdot 10^{25}$	$3\cdot 10^7$	$2.8\cdot 10^7$	10
$\mathrm{Fe}_{56}^{26}$ . . . .	$9\cdot 10^{-25}$	1.9	$1.1\cdot 10^{25}$	$1.2\cdot 10^7$	$1.2\cdot 10^7$	25

As already mentioned in § 1, in the Galaxy on the average $n \lesssim 0.1$, and the accuracy of the available data is not such that one could choose between the values in the 4th and 5th columns of Table II. Below, for the lifetime of cosmic protons, we shall adopt the value $T=4\cdot 10^8$ years, and we shall use the values from the 5th column to determine the ratios

\[ \frac{T_{\text{proton}}}{T_{\text{nucleus}}}, \]

which are given in the last column of the table.

Ionization losses of protons and nuclei with charge $Z$ in atomic hydrogen are as follows (it is assumed that $E \ll \dfrac{M}{m}Mc^2$):

\[ -\left(\frac{dE}{dt}\right)_{\!и} = \frac{4\pi n e^4 Z^2}{m\upsilon} \left\{ \ln \frac{2m\upsilon^2}{I} \left(\frac{E}{Mc^2}\right)^2 - \frac{\upsilon^2}{c^2} \right\} = \]

\[ = 7.62\cdot 10^{-9} nZ^2 \frac{c}{\upsilon} \left\{ 22.2+4\ln\frac{E}{Mc^2} +2\ln\frac{\upsilon^2}{c^2} -2\frac{\upsilon^2}{c^2} \right\} \ \frac{\mathrm{eV}}{\mathrm{sec}}, \tag{22} \]

where $M$ is the mass of the nucleus (in the case of a proton $Z=1,\ M=1837m$), $E$ is its total energy, $I=15$ eV is the mean excitation energy, and $n$ is the concentration of atomic electrons (in the case of hydrogen, $n$, of course, is simultaneously the concentration of hydrogen atoms). In the nonrelativistic case

\[ \left(E_{\mathrm{kin}}\simeq \frac{M\upsilon^2}{2}\ll Mc^2\right): \]

\[ -\left(\frac{dE}{dt}\right)_{\!и} = 7.62\cdot 10^{-9} nZ^2 \sqrt{\frac{2Mc^2}{E_{\mathrm{kin}}}} \left\{ 11.8+\ln\frac{E_{\mathrm{kin}}}{Mc^2} \right\} \frac{\mathrm{eV}}{\mathrm{sec}}. \tag{23} \]

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

\[ -\left(\frac{dE}{dt}\right)_{\!и} = 7.62\cdot 10^{-9} nZ^2 \left\{ 20.2+4\ln\frac{E}{Mc^2} \right\} \frac{\mathrm{eV}}{\mathrm{sec}}. \tag{24} \]

If the hydrogen is completely ionized, then in the nonrelativistic case in (22) one must put

\[ I=\hbar\omega_0=\hbar\sqrt{\frac{4\pi e^2 n}{m}} = 1.2\cdot 10^{-12}\sqrt{n}\ \mathrm{eV}, \]

as a result of which in (23) the value $11.8$ must be replaced by $42-\dfrac{1}{2}\ln n$ ($n$ is the electron concentration). In the relativistic case the expression for the losses in an ionized gas has the form

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

\[ = 7.62\cdot 10^{-9} nZ^2 \left\{ \ln\frac{2W}{mc^2}-\ln n+74.6 \right\} \frac{\mathrm{eV}}{\mathrm{sec}}, \tag{25} \]

where $n$ is the electron concentration and $W$ is the maximum energy transferred to an electron

\[ \left( W=E \text{ for } E\gg \frac{M}{m}Mc^2;\quad W=2mc^2\cdot\left(\frac{E}{mc^2}\right)^2 \right. \]

\[ \left. \text{for } Mc^2\ll E\ll \frac{M}{m}Mc^2 \right). \]

Below, hydrogen will usually be assumed to be nonionized, as is the case in most regions of the Galaxy. In the case

for an ionized gas the losses may then be $n$ times greater by a factor of $2 \div 4$ than in the absence of ionization. Ionization losses associated with the presence in the interstellar medium of helium and other elements will not be taken into account below, since they do not exceed $20 \div 25\%$ of the losses in hydrogen.

Protons with energy $E = 10^{10}$ eV, according to (24), for $n = 0.1$, lose $2.3 \cdot 10^{-8}$ eV/sec, or over the lifetime $T = 4 \cdot 10^8$ years $= 1.25 \cdot 10^{16}$ sec lose $\sim 10^8$ eV, i.e., considerably less than their initial energy. For iron nuclei with energy $10^{10}$ eV/nucleon (i.e., with total energy $\sim 5 \cdot 10^{11}$ eV), the ionization losses over the time between nuclear collisions are also small. From the examples given it is clear that ionization losses of relativistic protons and nuclei can be neglected when estimating their lifetime (i.e., path length). This fact is well known from the physics of cosmic rays, and we have dwelt on ionization losses because their inclusion is essential in considering the conditions necessary for the acceleration of particles (see § 3).

No losses, apart from those considered in the case of protons and nuclei, need be taken into account (for example, losses associated with particles falling onto stars, as well as certain other losses, are negligibly small).

However, with regard to nuclei it is necessary also to emphasize the importance of the question of their mutual transformations (see$^2$, pp. 233 ff.). The existence of such transformations is clear in connection with the presence in the primary flux of Li, Be, and B nuclei, whose relative abundance in the universe is on average $\sim 10^6$ times smaller than in cosmic rays (the supposition that in the cosmic-ray source the abundance of Li, Be, and B nuclei differs by many orders of magnitude from the average seems extremely improbable for a number of reasons). To form the observed amount of Li, Be, and B in the primary flux, it is necessary that the path length of C, N, and O nuclei in interstellar hydrogen be, on average, not less than the mean free path for nuclear collisions, i.e., not less than $\sim 5$ g/cm$^2$ (see Table II). Thus the cosmic rays observed on the Earth have lived, on average, no less than $3 \cdot 10^7$ years; most likely, cosmic rays are generated on average uniformly in the Galaxy and have time to mix within it; in this case the mean “age” of the protons and nuclei reaching the Earth is determined simply by their “lifetime” (see Table II). Since nuclei, for example Fe, “live” 25 times less than protons, while in the cosmic-ray flux they are $\sim 5 \cdot 10^3$ times fewer$^1$, they must be generated in the source in approximately 200 times smaller numbers than protons. Meanwhile, on average in the universe Fe nuclei are $\sim 4 \cdot 10^4$ times fewer than protons. In addition, owing to the fragmentation of nuclei, a significant fraction of cosmic protons (and $\alpha$-particles) may have a secondary origin. It follows from this that, in the source, for one reason or another, fast nuclei are generated incomparably more easily than proto-

...*) and it is not excluded that a considerable fraction of cosmic protons and $\alpha$-particles is of secondary origin.

Let us proceed to consider the losses suffered by electrons.

In the nonrelativistic case, the ionization losses of electrons are determined with sufficient accuracy by formula (23) with $Z=1$ and $M=m$. In the relativistic case the losses of electrons in atomic hydrogen are as follows:

\[ -\left(\frac{dE}{dt}\right)_{\!и} = \frac{2\pi e^4 n}{mc}\ln\frac{E^3}{2mc^2 I^2} \]

\[ = 7.62\cdot 10^{-9} n\left\{20.1+3\ln\frac{E}{mc^2}\right\} \frac{\text{eV}}{\text{sec}} . \tag{26} \]

If the gas is ionized, then for electrons formula (25) is valid, with $W=\dfrac{E}{2}$ and, of course, with $Z=1$.

The radiative losses of electrons in atomic hydrogen (i.e., losses connected with the bremsstrahlung of electrons in collisions with protons) are determined by the expression

\[ -\frac{1}{E}\left(\frac{dE}{dt}\right)_{\!p} = 8.0\cdot 10^{-16} n\ \text{sec}^{-1}. \tag{27} \]

The adopted value corresponds to a $t$-unit equal in hydrogen to $62\ \mathrm{g/cm^2}$ (see $^{50}$). Along such a path the energy of an electron decreases on average by a factor $e$, and the emitted photons have an energy comparable with the energy of the electron. A path of $64\ \mathrm{g/cm^2}$ corresponds, for $n=0.1$, to a time $T\simeq 4\cdot 10^8$ years; within the accepted accuracy this time coincides with the lifetime of a proton (see Table II). The hard photons formed in the braking of electrons almost all leave the Galaxy, since the thickness of the gas in the Galaxy is only $\sim 10^{-25}\cdot 10^{23}=10^{-2}\ \mathrm{g/cm^2}$ of hydrogen**). Therefore, if one does not consider the question of the equilibrium of photons in large regions of the universe (this interesting question has not yet been discussed) and confines oneself to our Galaxy, then the emitted photons may, from the point of view of the particle balance, be completely neglected.

Of some interest are also the losses connected with the inverse Compton effect—the scattering of fast electrons by thermal photons$^{38,39}$ present in the Galaxy in a rather large

*) This does not mean, of course, that protons are necessarily generated in an absolute amount smaller than nuclei. On the contrary, they may be generated several times (approximately up to $10\div 20$) more than nuclei.

**) Let us recall that charged particles, or at any rate most of them, do not leave the Galaxy owing to the “entangling” action of the chaotic magnetic fields present in the interstellar medium.

number as a result of their emission by stars. If the spectrum of the thermal radiation corresponds to a temperature of $6000^\circ$, the mean photon energy is

$\bar{\varepsilon}=2.73\,kT=1.42\ \text{eV}$. We shall take the mean radiation density in the Galaxy, over the entire region where cosmic rays are present, to be

\[ \bar{\rho}=0.03\ \frac{\text{eV}}{\text{cm}^3}, \]

whence the mean photon concentration is

\[ \bar{n}=\frac{\bar{\rho}}{\bar{\varepsilon}}= \]

\[ =2\cdot 10^{-2}\ \text{cm}^{-3} \]

(in the plane of the Galaxy the values of $\bar{\rho}$ and $\bar{n}$ are probably an order of magnitude larger). In the coordinate system associated with the electron, the photon energy is equal to

\[ \varepsilon'=\frac{E}{mc^2}\,\varepsilon(1+\beta\cos\alpha), \]

where $\beta=\dfrac{v}{c}$, $\varepsilon$ is the photon energy in the terrestrial reference frame (in this system the electron energy is $E$, and the angle between the directions of motion of the electron and photon is $\alpha+\pi$). On the average, for isotropic radiation,

\[ \varepsilon'=\frac{E}{mc^2}\,\bar{\varepsilon}, \]

and $\varepsilon'\ll mc^2$ under the condition

\[ \frac{E}{mc^2}\frac{\bar{\varepsilon}}{mc^2}\ll 1. \tag{28} \]

Under this condition, which may practically be regarded as fulfilled as long as

\[ \frac{E}{mc^2}\frac{\bar{\varepsilon}}{mc^2}<\frac14 \]

(i.e., for $\bar{\varepsilon}=1.42\ \text{eV}$ as long as $E<5\cdot 10^{10}$), the cross section for scattering of photons by electrons may be regarded as Thomson, i.e. equal to

\[ \sigma_0=\frac{8\pi}{3}\left(\frac{e^2}{mc^2}\right)^2=6.6\cdot 10^{-25}\ \text{cm}^2. \tag{29} \]

For $\bar{n}=2\cdot 10^{-2}$ we obtain from this the mean free path

\[ l=\frac{1}{\sigma\bar{n}}\sim 10^{26}\ \text{cm} \]

and the time between scattering events

\[ T=\frac{l}{c}\sim 10^8\ \text{years}. \]

The mean energy imparted to a photon in scattering, in the case under consideration, is, to within a factor of order unity,

\[ \Delta E\simeq \frac{E}{mc^2}\varepsilon' =\left(\frac{E}{mc^2}\right)^2\bar{\varepsilon} =1.42\left(\frac{E}{mc^2}\right)^2\text{eV} \]

and, consequently,

\[ -\left(\frac{dE}{dt}\right)_k \simeq \sigma_0 n c\,\bar{\varepsilon} \left(\frac{E}{mc^2}\right)^2 \simeq 5.6\cdot 10^{-16} \left(\frac{E}{mc^2}\right)^2 \frac{\text{eV}}{\text{sec}}. \tag{30} \]

From comparing formula (27) with $n=0.1$ and formula (30), it follows that radiative losses exceed losses due to the inverse Compton effect up to energies $E\sim 5\cdot 10^{10}\ \text{eV}$, and, for example, at $E\sim 10^9\ \text{eV}$ the losses (30) are 40 times smaller than the losses (27). In their nature, losses of both types are also close—they occur by large-

“portions.” Therefore, for electrons with energies less than $10^{10}$ eV, which are of interest to us, losses due to the inverse Compton effect in the Galaxy may be neglected$^*$, as we shall do (such losses, on the contrary, may be substantial in some variants of the theory of the solar origin of cosmic rays${}^{39}$).

The last energy-loss mechanism for electrons that deserves attention is losses to bremsstrahlung in magnetic fields. In § 1 the radiation itself was considered; here we are interested only in the energy losses associated with it, equal to (see, for example, ${}^{27}$):

\[ -\left(\frac{dE}{dt}\right)_{\!M} = \frac{2c}{3} \left(\frac{e^2}{mc^2}\right)^2 H^2 \left(\frac{E}{mc^2}\right)^2 = \]

\[ = 0.98\cdot 10^{-3} H^2 \left(\frac{E}{mc^2}\right)^2 \frac{\text{eV}}{\text{sec}}, \tag{31} \]

where $H$ is the component of the field perpendicular to the electron velocity, and it is assumed that $E \gg mc^2$.

The energy losses of electrons are compared in Table III. With regard to radiative losses, it should be kept in mind that the values given are average ones, whereas the actual losses occur in large portions. Ionization and magnetic-bremsstrahlung losses, on the other hand, are continuous. The table also indicates the energy gain under some conditions, which will be discussed in § 3.

It follows from Table III (p. 366) that magnetic-bremsstrahlung losses at $H=3\cdot 10^{-6}$ are greater than ionization losses beginning with energies $E \approx 10^9$ eV, and at $H=10^{-5}$, beginning with energies $E \approx 3\cdot 10^8$ eV. Therefore, as a first approximation one can usually take into account only magnetic-bremsstrahlung losses. In this case, integrating equation (31), we obtain:

\[ \left. \begin{aligned} \frac{mc^2}{E} - \frac{mc^2}{E_0} &= \frac{2e^4 H^2}{3m^3 c^5}\,t,\\[6pt] \frac{E}{mc^2} &= \frac{1}{ 1.9\cdot 10^{-9} H^2 t_{\text{sec}}+\dfrac{mc^2}{E_0} }, \end{aligned} \right\} \tag{32} \]

where $E$ is the energy at the moment $t$, and $E_0$ is the energy at the moment $t=0$.

Suppose that electrons with energy $E_0$ enter the interstellar magnetic field at all times with equal probability. Then the number of electrons with energies in the interval from $E$ to $E+dE$, where $E<E_0$, is proportional to the time $dt$ spent by the electrons in this interval,

\[ N_e(E)\,dE = K_1\,dt = \frac{K\,dE}{E^2}, \tag{33} \]

$^*$ Note that at high energies the average losses due to the inverse Compton effect are comparable to or greater than the radiative losses (27), but are still smaller than the losses to radiation in a magnetic field $H \sim 10^{-6}\div 10^{-5}$ oersted.

where $dt$ is expressed in terms of $dE$ by means of (31), and $K_1$ and $K$ are certain constants$^*$. Thus, under the stated conditions, electrons decelerating in a magnetic field have a power-law spectrum (14) with $\gamma=2$.

Table III

Loss and gain of energy by electrons in eV/sec
(all values rounded)

Electron energy in eV	Ionization losses (formula (26)) $n=0.1$	Radiative losses (formula (27)) $n=0.1$	Losses due to radiation in a magnetic field (formula (31)) $H=3\cdot10^{-6}$	Losses due to radiation in a magnetic field (formula (31)) $H=10^{-5}$	Energy gain (formula (36)) $\alpha=5\cdot10^{-17}$	Energy gain (formula (36)) $\alpha=2\cdot10^{-13}$
$5\cdot10^7$	$2.7\cdot10^{-8}$	$4\cdot10^{-9}$	$10^{-10}$	$10^{-9}$	$2.5\cdot10^{-9}$	$10^{-5}$
$10^8$	$2.8\cdot10^{-8}$	$8\cdot10^{-9}$	$4\cdot10^{-10}$	$4\cdot10^{-9}$	$5\cdot10^{-9}$	$2\cdot10^{-5}$
$5\cdot10^8$	$3.2\cdot10^{-8}$	$4\cdot10^{-8}$	$10^{-8}$	$10^{-7}$	$2.5\cdot10^{-8}$	$10^{-4}$
$10^9$	$3.3\cdot10^{-8}$	$8\cdot10^{-8}$	$4\cdot10^{-8}$	$4\cdot10^{-7}$	$5\cdot10^{-8}$	$2\cdot10^{-4}$
$5\cdot10^9$	$3.7\cdot10^{-8}$	$4\cdot10^{-7}$	$10^{-6}$	$10^{-5}$	$2.5\cdot10^{-7}$	$10^{-3}$
$10^{10}$	$3.9\cdot10^{-8}$	$8\cdot10^{-7}$	$4\cdot10^{-6}$	$4\cdot10^{-5}$	$5\cdot10^{-7}$	$2\cdot10^{-3}$
$5\cdot10^{10}$	$4.3\cdot10^{-8}$	$4\cdot10^{-6}$	$10^{-4}$	$10^{-3}$	$2.5\cdot10^{-6}$	$10^{-2}$

If the spectrum of the electrons being admitted (injected) has the form $N_{e,\text{in}}=K_2/E^\alpha$ (in the preceding case $N_{e,\text{in}}=\mathrm{const}\cdot\delta(E-E_0)$), then the spectrum $N_e(E)$ in a magnetic field has the form$^\circ$

\[ N_e(E)=\frac{K}{E^2}\int_E^{E_0}\frac{K_2\,dx}{x^\alpha} =\frac{\mathrm{const}}{E^2}\left\{\frac{1}{E^{\alpha-1}}-\frac{1}{E_0^{\alpha-1}}\right\}, \]

where

\[ \int_E^{E_0}\frac{K_2\,dx}{x^\alpha} \]

is the number of injected electrons with energy greater—

$^*$ The derivation of formula (33) given here is equivalent to obtaining it from the equation of conservation of the number of particles in energy space (i.e., ultimately, from the kinetic equation). Denote the total number of particles

\[ \int_0^{E_0} N_e(E)\,dE \]

by $N_0$. Then

\[ \frac{dN_0}{dt}=\frac{dN_0}{dE}\frac{dE}{dt}=K_1, \]

where, under the stated conditions, $K_1$ is a constant equal to the number of particles admitted into the field per unit time. Since $dN_0=N_e(E)dE$, and, on the other hand, $dN_0=K_1dt$, expression (33) follows directly.

rays $E$, and $E_0$ is the maximum energy of the injected electrons. For $E \ll E_0$ and $\alpha > 1$

\[ N_e(E)=\frac{\mathrm{const}}{E^{\alpha+1}} . \tag{34} \]

If $\alpha<1$,

\[ N_e=\frac{\mathrm{const}}{E^2} \]

and for $\alpha=1$

\[ N_e=\frac{\mathrm{const}}{E^2}\ln\frac{E_0}{E}. \]

Expression (34) allows one to draw a conclusion${}^{40}$ which seems to us very important. In the introduction it was pointed out that at energies $E \lesssim 10^{10}\ \text{eV}$, both for protons and for nuclei, $\gamma=2$, which indicates the common nature of their origin. Suppose that the sources of primary cosmic rays generate electrons with the same spectrum. Then, under the influence of braking in the magnetic field, this spectrum is transformed so that $\gamma=3$ (in (34) one must put $\alpha=2$). But precisely such a spectrum was obtained in § 1 from the data on cosmic radio emission. Allowance for radiative and ionization losses in the first approximation does not change this conclusion. During the lifetime (the time between radiative collisions) $T\sim 10^{16}\ \text{s}$ in a field $H\sim 10^{-5}$, according to formula (32)

\[ \frac{E}{mc^2}\sim \frac{1}{2\cdot 10^{-3}+\dfrac{mc^2}{E_0}}, \]

i.e. the energy of electrons with any initial energy decreases at least to $3\cdot 10^8\ \text{eV}$. Hence it follows that a spectrum with $\gamma=3$ in general has time to become established also when radiative losses are taken into account. Ionization losses at small energies $E\lesssim 5\cdot 10^8$ somewhat change the spectrum (it has the form

\[ \frac{\mathrm{const}}{E(E^2+a)} \]

instead of

\[ \frac{\mathrm{const}}{E^3} \]

), but this change at energies greater than $(2\div 3)\cdot 10^8\ \text{eV}$ is evidently insignificant, as follows from the experimental data on the spectrum of radio emission and is in agreement with the corresponding estimates for $H\sim 10^{-5}$.

In concluding the present section, let us make one more remark concerning cosmic electrons in the interstellar medium. Even if the primary sources of cosmic rays did not emit electrons, fast electrons would have to appear in interstellar space primarily as a result of nuclear collisions. According to data obtained largely by Soviet physicists under the direction of D. V. Skobeltsyn, in collisions of cosmic nucleons electron–nuclear showers are formed: $\pi^\pm$-mesons are generated, which on decay ultimately give electrons (and positrons, which we do not distinguish from electrons). For $E>2\cdot 10^9 \div 10^{10}\ \text{eV}$

as a result of each collision, of the order of one $\pi^\pm$-meson is produced (see $^{2,41}$) and, consequently, of the order of one electron. Since the lifetimes of protons and electrons are approximately the same and equal to $\sim 4\cdot 10^8$ years, in equilibrium the concentration of electrons is $N_e \approx sN$, where $N$ is the concentration of protons in cosmic rays and $s$ is the average number of electrons born as a result of the disappearance of a cosmic proton (we neglect the role of nuclei). As was said, $s\sim 1$, and thus $N_e\sim N$, but the accuracy of the available data is still insufficient to exclude a value, say, $N_e\sim \frac{1}{10}N$. If the spectrum of the electrons produced is close to the proton spectrum (i.e. $\gamma=2$), then under the influence of braking in magnetic fields a spectrum with $\gamma=3$ will again be obtained, as is in agreement with experiment. For a number of reasons (see § 4) it seems to us, however, probable that the main mass of cosmic electrons with energies greater than $(2\div 3)\cdot 10^8\ \mathrm{eV}$ is generated by the primary sources of cosmic rays. In this case the electrons born in nuclear collisions must either constitute only a small, say $1/10$, part of all electrons, or, more likely, have on the average an energy $<2\cdot 10^8\ \mathrm{eV}$, which is quite plausible. As for photons formed in the decay of $\pi^0$-mesons, they leave the Galaxy practically freely, and we shall not take their role into account (see above).

As E. L. Feinberg$^\dagger$ has pointed out, in connection with the problem of the transformation of particles in the interstellar medium there arises the question of the number of antiprotons and positrons, as well as positronium, formed in the Galaxy. Determination of the concentration of all these particles (of positrons and antiprotons by investigating the composition of the primary cosmic rays at the Earth, and of positronium from its radiation, if such can be sufficiently intense) would make it possible to refine substantially our information on cosmic particles in interstellar space.

§ 3. STATISTICAL MECHANISM OF PARTICLE ACCELERATION IN THE INTERSTELLAR MEDIUM AND IN THE ENVELOPES OF STARS

The motion of charged particles in the interstellar medium resembles Brownian motion or the motion of molecules in a gas. Indeed, the presence of the interstellar magnetic field leads to the fact that, in a region where this field is quasi-uniform, the trajectory of a particle winds around a line of force of the field and, averaged over the period of revolution, is close to a straight line. But on passing into a region with another direction of the field, the trajectory changes and as a whole represents a broken line. If the size of the regions in which the field noticeably changes its direction is small in comparison with the sizes of regions with a quasi-uniform field, then the motion of the particle may be regarded as analogous to the motion of a molecule in a gas: in a region of uniform field the motion is free, while the change in the direction of the velocity at the boundary

ORIGIN OF COSMIC RAYS AND RADIO ASTRONOMY

of the region, like a collision with another molecule, and usually may be regarded as occurring instantaneously. The dimensions of the region with a quasi-uniform field thus play the role of the mean free path $l$. The mean free time is $\tau = \frac{l}{v_0}$, where $v_0$ is the velocity of the translational motion along the trajectory, of the same order of magnitude as, usually, the velocity of the particle itself (therefore below we shall take

\[ \tau \sim \frac{l}{v}, \]

where $v$ is the particle velocity). If the magnetic fields do not change with time, then the collision process described leads only to diffusion of the particles and to a “mixing” of their velocities with respect to directions, but not to a change in the particle energies. The mean-square distance $L$ traversed by a particle in time $t$, as is known from the theory of diffusion, is equal to

\[ L=\sqrt{6Dt}\sim\sqrt{lvt}, \tag{35} \]

where $D \sim \frac{lv}{3}$ is the diffusion coefficient (for a more detailed discussion of diffusion in interstellar space see ${}^{31,42—44}$). According to astronomical data, in the interstellar medium $l \gtrsim 10^{19}\ \text{cm}$ (see ${}^{30}$) and, for $v \sim c$ and $t \sim T \sim 10^{16}\ \text{sec}$ (the lifetime of protons), $L \sim 3 \cdot 10^{22}\ \text{cm}$, i.e., of the order of the dimensions of the Galaxy. Thus for $l \lesssim 10^{19}\ \text{cm}$ protons, and still more so nuclei, indeed do not have time in any appreciable number to leave the Galaxy.

Real interstellar magnetic fields are connected with interstellar gas, which is in a state of turbulent motion. Therefore, in a coordinate system in which the gas is moving, the fields are not constant in time. Taking this circumstance into account in the first approximation does not change the character of diffusion in space, but leads to the fact that the energy of cosmic particles may increase. This important fact, noted by Fermi ${}^{14}$, is easy to understand. In a region where the magnetic field is quasi-uniform, the velocity field of the gas is in all probability also homogeneous. Therefore such regions as a whole may be regarded, from the point of view of interest to us, as one gigantic particle moving with some velocity $u$. In a state of statistical equilibrium the energy falling on each degree of freedom must be the same; the kinetic energy of the moving region (for example, a cloud of interstellar gas) is enormous, and therefore a cosmic particle colliding with this cloud must on the average acquire energy (the energy of cosmic particles, as is easy to estimate, is incomparably smaller than the energy they would have in statistical equilibrium). In this case, just as, for example, in collisions of elastic spheres, it is sufficient to take into account the laws of conservation of energy and momentum, and there is no need to analyze the process of the collision itself, which in the case under consideration is connected with the action of the Lorentz force at the boundary

between regions*). In accordance with this the result does not depend on the intensity of the magnetic field, and the latter acts, as it were, as a “driving belt,” ensuring the transfer of energy from the moving gas to the particle. The acceleration mechanism described may with full justification be called statistical. In statistical acceleration the magnetic field may, on the average, neither increase nor decrease; thus statistical acceleration is fundamentally different from induction acceleration. It is also clear from this that statistical acceleration, for example in a mixture of gases, will occur even in the absence of a magnetic field.

Applying the laws of conservation of energy and momentum to the collision of a cosmic particle with energy

\[ E=\frac{Mc^2}{\sqrt{1-\frac{v^2}{c^2}}} \]

with a cloud (with practically infinite mass), moving with velocity $u$, one can show $^{14}$ that in one collision, on the average, the energy acquired is $\Delta E\sim \frac{u^2}{c^2}E$. In the nonrelativistic case, when $E\simeq Mc^2$, this result is very easy to obtain using the well-known formulas for an elastic collision of slow particles (see, for example, $^{45}$). In each collision the change of energy in absolute value is $\Delta E_1\sim Muv$, where $v$ is the velocity of the particle and $u$ is the velocity of the cloud. But in the isotropic case, upon averaging over directions, $\Delta E=u^2M=\frac{u^2}{c^2}Mc^2\*\*). Since under cosmic conditions \(\frac{u^2}{c^2}\ll 1$ and usually $v\sim c$, the fluctuations in the acceleration are very large, since the change of energy in each collision $\Delta E_1\sim Muv$ is much greater than the mean change of energy $\Delta E\sim Mu^2$. However, we shall not discuss here the question of fluctuations and shall dwell somewhat more fully only on systematic acceleration***).

*) Of course, in order to be able to speak of a collision in the ordinary sense of the word, it is necessary that the collision time be less than the mean free time; moreover, the radius of curvature of the particle in the magnetic field must be considerably smaller than the mean free path $l$, since otherwise the particle’s direction changes significantly during the collision (if the radius of curvature is comparable with or greater than $l$, then the effectiveness of collisions obviously decreases).

**) The relation $\Delta E\sim \frac{u^2}{c^2}Mc^2$ is obtained without any calculations by considering the case when $v=0$ (the particle is at rest before the collision). It is clear that in this case the velocity of the particle after collision with the cloud will be of the order of the cloud velocity $u$, and its energy $E=\Delta E\sim Mu^2=\frac{u^2}{c^2}Mc^2$.

***) The role of fluctuations was emphasized by V. I. Veksler and A. A. Logunov $^{1}$ and can be clarified on the basis of the work of Ya. P. Terletskii and A. A. Logunov $^{42}$. According to $^{42}$, allowance for fluctuations leads to replacement of formula (40) (see below) by the expression

\[ \gamma=-\frac{1}{2}+\sqrt{\frac{9}{4}+\frac{4}{\alpha T}}; \tag{40a} \]

The average change of a particle’s energy per unit time as a result of collisions is evidently as follows:

\[ \frac{dE}{dt}=\alpha E=\frac{\Delta E}{\tau}\sim \frac{u^2}{c^2\tau}E\sim \frac{u^2v}{c^2l}E, \tag{36} \]

where $\tau\sim \dfrac{l}{v}$ is the mean free time between collisions, and $l$ is the corresponding mean free path $\left(\dfrac{1}{\tau}\sim \dfrac{v}{l}\right.$ is the number of collisions per unit time$\left.\right)$. If $\alpha$ does not depend on time and, at $t=0$, $E=Mc^2$, then

\[ E=Mc^2 e^{\alpha t}\sim Mc^2 e^{\frac{u^2v}{c^2l}t}. \tag{37} \]

Let the particles begin to be accelerated with equal probability at any instant of time, and, in addition to acceleration, let only the possibility of their disappearance be taken into account, characterized by the lifetime $T$. Then the probability of finding a particle with “age” in the interval $t, t+dt$ is equal to

\[ dW=\frac{e^{-\frac{t}{T}}\,dt}{T}, \qquad \int dW=1, \tag{38} \]

But by virtue of relations (36) and (37), $dt=\dfrac{dE}{\alpha E}$ and $t=\dfrac{\ln \dfrac{E}{Mc^2}}{\alpha}$; consequently, expression (38) takes the form

\[ dW=\mathrm{const}\cdot N(E)\,dE=\frac{1}{\alpha T}(Mc^2)^{\frac{1}{\alpha}}\, \frac{dE}{E^{1+\frac{1}{\alpha T}}}, \tag{39} \]

where $N(E)$ is the density of particles in the interval $dE$.

Thus the differential spectrum has the form

\[ N(E)=\frac{K}{E^\gamma}, \qquad \gamma=1+\frac{1}{\alpha T}. \tag{40} \]

If ionization losses are taken into account, it becomes clear that the statistical mechanism can accelerate only particles with energy greater than some minimum energy (injection energy) $E_i$. This

when $\alpha T\gg \dfrac{16}{9}$, formula (40a) practically coincides with (40). In the case of interest to us, when $\gamma=2.7$, according to (40) $\alpha T\simeq 0.6$, and according to (40a) $\alpha T\simeq 0.5$, i.e. the difference is also quite insignificant. However, in some cases allowance for fluctuations may prove substantial.

the energy is determined, obviously, from the relation (see (36)),

\[ \alpha E_{\mathrm{i}}=\left(\frac{dE}{dt}\right)_{\mathrm{i}}, \tag{41} \]

where $\left(\dfrac{dE}{dt}\right)_{\mathrm{i}}$ are the ionization losses at energy $E_{\mathrm{i}}$.

If the statistical mechanism of acceleration in the interstellar medium is significant and determines the particle spectrum, then from the experimental data and formula (40) one can find the coefficient $\alpha$. Experimentally, for $E>10^{13}$ ev, $\gamma \simeq 2.7$, whence, according to (40), $\alpha T \simeq 0.6$. Taking for protons $T \simeq 4\cdot 10^8$ years $\simeq 1.25\cdot 10^{16}$ sec $(n\simeq 0.1)$, we obtain*):

\[ \alpha \simeq 5\cdot 10^{-17}\ \mathrm{sec}^{-1}. \tag{42} \]

For nuclei the lifetime $T$ is considerably smaller (see Table II), and therefore, if for protons $\gamma=2.7$, then with the same $\alpha$, for He $\gamma\simeq 8$, for O nuclei $\gamma\simeq 18$, and for Fe $\gamma\simeq 43$. Experimentally, at an energy per nucleon $E\leqslant 3\cdot 10^{10}$, the spectrum is the same both for protons and for nuclei, with $\gamma=2.0\div 2.2$ (see the Introduction). Hence it is clear that, if the statistical mechanism of acceleration in interstellar space is at all significant, it acts appreciably only at energies $E\gtrsim 3\cdot 10^{10}$. This important circumstance is, in general, in agreement with the conclusions that follow from estimates of the injection energy $E_{\mathrm{i}}$. Using formulas (22) and (41), for $n\sim 0.1$ and $\alpha\sim 5\cdot 10^{-17}$, we obtain the values $E_{\mathrm{i,kin}}=E_{\mathrm{i}}-Mc^2$ in atomic hydrogen indicated in Table IV (the figures are rounded).

Table IV

Nucleus	$E_{\mathrm{i,kin}}$ in ev	$E_{\mathrm{i,kin}}$ per nucleon in ev
Proton	$10^8$	$10^8$
$\mathrm{He}^{2}_{4}$	$4\cdot 10^8$	$10^8$
$\mathrm{O}^{8}_{16}$	$2\cdot 10^{10}$	$10^9$
$\mathrm{Fe}^{25}_{56}$	$3\cdot 10^{11}$	$5\cdot 10^9$

If the gas is ionized or has density $n\sim 1$ (both of these cases occur in the interstellar medium), then the energy $E_{\mathrm{i,kin}}$ is several times larger. In any case, for Fe $E_{\mathrm{i,kin}}\gtrsim 3\cdot 10^{11}$ ev, and thus it is clear that at least the heavy nuclei in cosmic rays are accelerated to energy $3\cdot 10^{11}$ ev not in the interstellar medium. It is natural to think that this also applies to protons.

As has already been pointed out, if at $E\gtrsim 10^{10}\div 3\cdot 10^{10}$ statistical acceleration is significant, then the spectra of protons and nuclei at high—

*) In 14 it was assumed that $T\simeq 3\cdot 10^7$ years $(n\sim 1)$, and therefore the value $\alpha\simeq 3\cdot 10^{-16}\ \mathrm{sec}^{-1}$ was adopted. For the Galaxy as a whole it is more correct to use the value (42).

ORIGIN OF COSMIC RAYS AND RADIO ASTRONOMY

at high energies must be quite different (if for protons $\gamma=2.7$, then for Fe nuclei $\gamma=43$). As a result, at energies $E \gtrsim 10^{13}\div 10^{14}$ eV the percentage of nuclei should be negligible. Experimentally, unfortunately, this question has not yet been clarified${}^{1}$, although some fragmentary data rather indicate the presence of fast nuclei in the composition of the primary component. The character of the spectrum of high-energy nuclei can be clarified both by the photographic-plate method and by observing correlated broad atmospheric showers${}^{46}$. It is hardly possible to resolve the question of the efficiency of acceleration in interstellar space by using astrophysical data to determine the coefficient

\[ \alpha \simeq \frac{u^{2}}{c^{2}\tau}. \]

In${}^{14}$ the values $u\sim 3\cdot 10^{6}$ and $l\sim c\tau \simeq 1.3\cdot 10^{18}$ cm $\simeq 0.4$ pc were adopted; the choice of these values in the case of clouds of interstellar gas is not justified${}^{47}$, but as applied to the rarefied interstellar gas the choice of the parameters $u\sim 5\cdot 10^{6}$ and $l\sim 2\cdot 10^{19}$ cm $\sim 7$ pc (see${}^{30}$), leading to $\alpha\simeq 5\cdot 10^{-17}$, meets with no objections. It may well turn out, however (there are no direct data on this point), that the effective parameters $u$ and $l$ have a somewhat different value and, in particular, one for which $\alpha T \ll 1$ and $\gamma \gg 3$. In this case, acceleration in interstellar space will be insignificant and the observed spectrum will be determined by the spectrum of particles generated in the primary sources. It is precisely this possibility that seems to us the most probable, in view of what was said above, the estimates of the particle energy that can be obtained in supernova outbursts (see below), and taking into account that the exponent $\gamma\simeq 3$ is obtained in interstellar acceleration only as the result of an accidental “fortunate” choice of parameters. Moreover, the data on the spectrum testify to a smooth, slow increase of $\gamma$ with energy from $\gamma\simeq 2$ to $\gamma\simeq 3$. It would be at least strange if two different mechanisms (primary sources at $E\lesssim 10^{11}$ eV and interstellar acceleration at $E\gtrsim 10^{11}$ eV) led to such a smooth “stitching together” of the spectra.

The mechanism considered and expressions (36)—(42), of course, also apply to electrons ($M=m$, $T$, as for protons, $\sim 4\cdot 10^{8}$ years). In this case, however, in addition to ionization losses, magneto-bremsstrahlung losses are also important, increasing rapidly with energy, and, as is clear from Table III, statistical acceleration of electrons is practically ineffective (at $\alpha\sim 5\cdot 10^{-17}$ and $H\sim 10^{-5}$ oersted an electron is not accelerated at any energy; at $H\sim 3\cdot 10^{-6}$ acceleration takes place in a very narrow energy interval near $E\sim 10^{9}$ eV, since at lower energy ionization losses prevail over the acquisition of energy, while at higher energies magneto-bremsstrahlung losses predominate). Thus, even if the statistical mechanism in interstellar space is effective as applied to protons, fast electrons will not appear as a result of its action. The absence in experiment of a significant number of fast

electrons thus makes it possible to give an independent estimate of the upper limit of the quantity $\alpha$, which turns out to be close to the value (42). If, however, one adopts, as was done in $^{43}$, the value $\alpha=2\cdot 10^{-13}$, then fast electrons would be observed in the primary flux in the same quantity as protons (see, in this connection, $^{1,20}$ and Table III)*). Thus, the statistical mechanism of particle acceleration in the interstellar medium has, at best, limited significance in determining the spectrum of protons and nuclei at high energies.

However, in denser regions—in the atmospheres of stars, in the expanding envelopes of novae and supernovae, in diffuse nebulae—this mechanism may be considerably more effective and is probably the basic one$^{40}$**). In dense regions the acceleration usually continues for a time much shorter than the lifetime $T$. For example, in a supernova envelope the maximum acceleration time is $t_m\sim 3\cdot 10^3$ years (the time before the breakup of the envelope) and $T\sim 4\cdot 10^5$ years (for $n\sim 10^3$; more likely even $n\sim 10^2$ and $T\sim 4\cdot 10^6$). In such a case one may take $T\to\infty$, and we arrive at the spectrum (see (37)—(40))

\[ \left. \begin{aligned} N(E)&=\frac{\mathrm{const}}{E}\quad &&\text{for } E<E_{\max}=Mc^2e^{\alpha t_m},\\ N(E)&=0 &&\text{for } E>E_{\max}. \end{aligned} \right\} \tag{43} \]

However, this spectrum, like the more general spectrum (40), is obtained only under the assumption that the parameter $\alpha\simeq \dfrac{u^2v}{c^2l}$ (see (36)) is constant in time. In the case of the interstellar medium such an assumption is quite natural. But if the matter concerns, for example, the expanding envelope of a supernova, there are no grounds without

*) In $^{43}$, and also in $^{48}$, it is assumed that the acceleration mechanism in the interstellar medium is not statistical but inductional, i.e. associated with the growth of the field in the region under consideration (if the field is homogeneous, then in the relativistic case $E\sim\sqrt{H}$). However, it seems to us extremely unlikely that over a time $\sim 4\cdot 10^8$ years the mean field in the Galaxy could increase by many orders of magnitude. If the field increases by 100–1000 times, as may occur in the envelopes of supernovae, in collisions of nebulae, etc., then the energy increases only by 10–30 times and the induction mechanism can have only auxiliary significance. In the case of the Galaxy as a whole the field probably grows still much more slowly and, therefore, the induction mechanism, at least without the involvement of statistical elements, will not be significant.

**) In note $^{37}$ it was pointed out that particles might be accelerated in the interstellar medium, and essentially also in denser ionized regions, as a result of the propagation of plasma waves (the corresponding mechanism is close to that existing in the case of linear accelerators). However, the question of the effectiveness of such a mechanism under cosmic conditions, as far as we know, has not been analyzed by anyone and thus remains completely open.

For further analysis we assume that $\alpha=\mathrm{const}$. If $\alpha=\alpha(t)$, then

\[ E=Mc^2\exp\left\{\int_0^t \alpha(t)\,dt\right\} \]

and the spectrum changes. For example, for $v\simeq c$ and $\dfrac{u^2}{l}=\dfrac{\mathrm{const}}{t}$, $\alpha=\dfrac{a}{t}$, the spectrum is a power-law one with exponent $\gamma=1-\dfrac{1}{a}$ and $E=Mc^2\left(\dfrac{t}{t_0}\right)^a$, where $t_0$ is some minimum time starting from which $\alpha\sim\dfrac{1}{t}$. In addition to the dependence of $\alpha$ on $t$, it is also necessary to take into account the possibility that an appreciable role is played not by a single parameter $\alpha$, but by several such parameters. In such a case, however, it is probably still possible to introduce one time-dependent effective quantity $\alpha$. Further, in deriving the spectra (40) and (43) it is assumed that the particles begin to accelerate with equal probability at any instant of time. In application to acceleration under nonstationary conditions this assumption may prove inadmissible.

For all these reasons it is not at present possible to specify theoretically the spectrum of particles generated during supernova outbursts (the rather modest experimental data available indicate, moreover, that in different cases this spectrum is different; see § 1). To determine the spectrum it is necessary first of all to develop the magnetohydrodynamic theory of turbulence and apply it to the case of an expanding envelope. In doing so it may prove necessary to take fluctuations into account (see above) and, under some conditions, to refine the mechanism by which energy is transferred from the moving gas to the cosmic particle, since, as was already indicated, the introduction of the concept of collisions of particles with regions of a quasi-uniform field is not always legitimate. However, the need to develop the theory and the existing uncertainty concerning the spectrum by no means deprive us of the possibility of clarifying a number of points concerning the character of statistical acceleration in the envelopes of novae and supernovae*). In the envelopes of supernovae $u\lesssim5\cdot10^8\ \mathrm{cm/sec}$, and the lifetime of the envelope before its mixing with the interstellar medium is $\lesssim3000$ years $\simeq10^{11}$ sec.; the radius of the envelope is $\rho\lesssim10^{19}\ \mathrm{cm}$. The field in the envelope $H$ is probably no stronger than $3\cdot10^{-4}$ oersted, and, consequently, the radius of curvature

\[ r\simeq\frac{E}{300H}\sim10^{12}\ \mathrm{cm} \]

for

*) We confine ourselves here to consideration of this case only. However, the statistical acceleration mechanism is also of interest in other relatively dense regions of the Galaxy (in collisions of diffuse nebulae$^{48}$, in collisions of a nebula with a star, etc.), as well as on the Sun and near it. Consideration of statistical acceleration on the Sun$^{40}$ deserves special attention, since it is important for understanding a whole series of manifestations of solar activity (see $^{1,2,6,35,49}$).

$E \sim 10^{11}$ ev and $r \sim 10^{14}$ cm at $E \sim 10^{13}$ ev. To estimate the maximum attainable energy, let us assume that $\alpha \simeq \dfrac{u^2 v}{c^2 l} = \mathrm{const}$, and take the values $u \sim 10^8$, $l \sim 10^{14}$, $v \sim 10^{10}$, and $t_m \sim 10^{10} \simeq 300$ years. Then $\alpha \sim 10^{-9}$, $\alpha t_m \sim 10$, and $E_{\max} \sim 10^4 Mc^2$. The chosen value of $l$ satisfies, for $E \lesssim 10^{12}$ ev, the conditions $l \gg r$ and $l \ll \rho$. Moreover, with the adopted values, during the time $t_m$ the diffusion path of the particle $L \sim \sqrt{lvt_m} \sim 10^{17}$ cm $\ll \rho$, and the particles do not leave the envelope. In view of the sensitivity of the maximum energy $E_{\max}$ to the choice of parameters, in principle it is possible to obtain an energy still substantially higher than in the example given; thus, for the same parameters, but $t_m \sim 3 \cdot 10^{10}$ sec $\simeq 1000$ years, $E_{\max} \sim 10^{12} Mc^2 \sim 10^{21}$ ev (for protons)*. Of course, such a high energy will not in fact be attained, since at $E \sim 10^{21}$ ev and $H \sim 3 \cdot 10^{-4}$, $r \sim 10^{22}$ cm $\gg \rho$. But the estimates given provide grounds for supposing that, as a result of supernova outbursts, under a favorable combination of circumstances particles with the highest observed energies, $\sim 10^{17} \div 10^{18}$ ev, may be generated. Further, under statistical acceleration the energy $E_{\max}$ is proportional to the particle mass $M$ (see, for example, (43)). Therefore the mean energy per nucleon for the nuclear component will be the same as for protons; the spectra of protons and nuclei should also be identical, which corresponds to experimental data.

In the Crab Nebula, as indicated in § 1, there are electrons with energy $E \sim 10^{11} \div 10^{12}$ ev and, consequently, protons should have been accelerated to energies $\sim 10^{14} \div 10^{15}$ ev, and Fe nuclei—to energies $\sim 10^{16} \div 10^{17}$ ev. Thus the supposition made above concerning the possibility of obtaining very large energies in supernova outbursts finds a certain experimental confirmation.

Statistical acceleration in the envelopes of stars and other dense regions deserves special attention also from the standpoint of the question of injection. For nonrelativistic particles, using formula (41) and formula (23), we obtain:

\[ E_{\mathrm{и,\,кин}} = 2.3 \cdot 10^{-22} \left\{ 42 + \ln \frac{E_{\mathrm{и,\,кин}}}{Mc^2} - \frac{1}{2}\ln n \right\}^2 \frac{m}{M} \frac{Z^4 n^2}{a^2} < \]

\[ < 4 \cdot 10^{-19} \frac{m Z^4 n^2}{M a^2} \ \text{ev}, \tag{44} \]

* When the particle velocity $v$ is decreased, the value of $\alpha$ falls. However, at small velocities the gyroradius also decreases, and smaller values $l \gg r$ may play a role (for $v \sim 3 \cdot 10^8$ for protons at $H \sim 10^{-4}$, $r \sim 3 \cdot 10^8$). Moreover, in the nonrelativistic region, where the induction mechanism is more effective, it can play a certain auxiliary role, since the field still increases by a factor of 100–1000.

where $E_{\mathrm{i,kin}}$ is the kinetic energy of the injected particle with mass $M$ and charge $Z$ ($m$ is the electron mass, $n$ is the electron concentration; the gas is assumed to be completely ionized, which is taken into account by replacing in (23) the coefficient 11.8 by the coefficient $42-\frac{1}{2}\ln n$). In the envelopes of supernovae, for $\alpha\sim 10^{-9}$ (see above) and $n\sim 10^2$, for electrons $E_{\mathrm{i,kin}}<4000$ eV, for protons $E_{\mathrm{i,kin}}<2$ eV, and for Fe nuclei $E_{\mathrm{i,kin}}<20000$ eV. The last two figures are not realistic, since formula (23) for ionization losses is applicable only so long as the velocity of the particle is considerably greater than the thermal velocity of the electrons in the gas, i.e. so long as $v\gg \sqrt{kT/m}$. It is clear, however, that the injection energy in the envelope is very small—this is a consequence of the relatively large value of $\alpha$. Thus, in the case of statistical acceleration in the envelopes of stars, the question of injection in the form in which it occurs in the interstellar medium does not arise, and one should rather be concerned that the injection energy not be too low, since in that case many particles would be accelerated and the entire process would quickly die out. A detailed analysis of this question has not yet been carried out, and we shall confine ourselves to just one further remark.

At small ion energies, when their velocity $v\lesssim 10^8$ and, consequently,

\[ E_{\mathrm{kin}}=\frac{Mv^2}{2}\ll \frac{M}{m}\ \text{eV}, \]

the charge of the ion is not $Z$ (the “bare” nucleus), but unity (or two). Therefore, at the initial stage of acceleration, where the question of injection is the essential one, acceleration of ions proceeds more readily than in the case of protons (see (44)). It is possible that this point is important for explaining the fact that in sources of primary cosmic rays nuclei are generated more readily than protons (see § 2). Of known significance from this point of view may also be the circumstance that, according to some data, supernovae are relatively poor in hydrogen (see $^{1}$).

§ 4. REMARKS ON THE THEORY OF THE ORIGIN OF COSMIC RAYS

a) Certain consequences following from the experimental data. Critique of the hypothesis of the solar origin of cosmic rays

The conclusions that can be drawn on the basis of the available experimental data and the considerations set out in §§ 2, 3 are as follows:

Cosmic electrons with energies $10^8 \div 10^9$ eV are distributed in the Galaxy approximately in the same way as the rarefied interstellar gas, i.e. they form a quasi-spherical subsystem with radius $R\sim 3\cdot 10^{22}\ \text{cm}\sim 10000\ \text{pc}$. The primary sources of cosmic rays are, in all probability, located in the region of the galactic plane (a layer with thickness $\sim 100\ \text{pc}$). In regions remote from this plane, …

particles arrive as a result of diffusion in the interstellar gas. Indeed, according to (35), during the lifetime of the electrons $T \simeq 4\cdot 10^8 \simeq 10^{16}\ \mathrm{sec}$, the diffusion path is $L \sim \sqrt{lct}\sim 2\cdot 10^{13}\sqrt{l}\sim R\sim 3\cdot 10^{22}\ \mathrm{cm}$ for $l\sim 10^{18}\ \mathrm{cm}$. Taking into account the uncertainty of the estimate, the value obtained for $l$ agrees with other astrophysical data. The spatial distribution of protons should be approximately the same as that of electrons, since the corresponding lifetimes are close. Nuclei, however, should form a somewhat denser subsystem (let us recall that the lifetime, say, for Fe nuclei is 25 times less than for protons). The same applies to fast electrons ($E>10^9\ \mathrm{eV}$), as a result of which a certain concentration of fast electrons in the galactic plane should be observed. This circumstance permits verification by radioastronomical methods $^{19}$.

The presence in cosmic radiation near the Earth of Li, Be, B nuclei indicates that the path length of cosmic protons reaching the Earth is, on the average, not less than $5 \div 10\ \mathrm{g/cm^2}$ of interstellar hydrogen (the traversal time $T \gtrsim 3 \div 6\cdot 10^7$ years). Most probably, as follows in particular from the preceding point 1, cosmic rays are distributed more or less uniformly in the Galaxy. In this case the mean “age” of the particles reaching the Earth is of the order of their lifetime *), which for protons is 25 times greater than for Fe nuclei ($T_{\mathrm{pr}}\sim 4\cdot 10^8$ years, $T_{\mathrm{Fe}}\sim 1.6\cdot 10^7$ years). In the primary flux at the Earth there are $\sim 3.5\cdot 10^3$ times fewer Fe nuclei than protons, but they should be generated in the sources only by

\[ \xi \sim 3.5\cdot 10^3\,\frac{T_{\mathrm{Fe}}}{T_{\mathrm{pr}}}\sim 150 \]

times less than protons. In reality $\xi$ is still smaller, since part of the protons and $\alpha$-particles in cosmic rays must be produced as a result of the decay of more complex nuclei. At the same time, in the universe, on average, there are $\sim 4\cdot 10^4$ times fewer Fe nuclei than hydrogen. We thus arrive at the conclusion that in the sources nuclei are generated with significantly greater efficiency than protons (if the chemical composition of the gas in the source coincides with the average one, then the probability of acceleration, for example, of an Fe nucleus must be at least $\sim 10^3$ times greater than the probability of acceleration of a proton). Taking into account the abundance of electrons in the sources, whose concentration in cosmic rays is comparable with the concentration of protons (see § 1), and also bearing in mind the possibility of the formation of electrons in interstellar space in nuclear collisions, we come to the conclusion that electrons are generated with an efficiency in any case no greater than protons.

*) If the particles were formed predominantly near the solar system and could leave this region, then the mean age of cosmic protons and light nuclei could turn out to be the same and, by virtue of what was said, equal to $\sim 5 \div 10^2\ \mathrm{g/cm^2}$.

At energies $E \lesssim 3 \cdot 10^{10}\ \dfrac{\text{eV}}{\text{nucleon}}$, the spectra of protons and nuclei are the same ($\gamma \simeq 2$), which indicates the commonality of their origin. For electrons $\gamma \simeq 3$, but, as was shown in § 2, this means that in the primary sources for electrons as well $\gamma \simeq 2$. Thus, all components of cosmic radiation are generated with a single spectrum with $\gamma \simeq 2$. Where does this generation occur? In interstellar space, because of the relatively high injection energy, only further acceleration of particles can occur, but not their primary acceleration. If the sources are distributed uniformly in space, then the spectrum of the accelerated particles has the form (40), and, as shown in § 3, interstellar acceleration can be significant only at energies $E \gtrsim 10^{11} \div 10^{12}\ \text{eV}$. In this case, if interstellar acceleration is significant, the spectrum of nuclei will differ strongly from the spectrum of protons, for which in this case at the highest energies $\gamma \simeq 2.7 \div 3.0$. For a number of reasons (see § 3), it seems more probable that interstellar acceleration is generally ineffective and that the spectrum is completely determined by the primary sources. In this case it may be that for protons and for high-energy nuclei $\gamma \simeq 2.7$, or even that for nuclei $\gamma \simeq 2.7$, while for protons $\gamma$ is larger than this value (with interstellar acceleration, conversely, $\gamma_{\text{nuclei}} > \gamma_{\text{protons}}$). A certain increase of the exponent $\gamma$ on going to higher energies may be connected not only with a change in the generation conditions in the source, but also with the role of diffusion of particles out of the Galaxy. The point is that at $E \sim 10^{15} \div 10^{17}\ \text{eV}$, the radius of curvature at $H \sim 10^{-5}$ is of order $r \sim 3 \cdot 10^{17} \div 3 \cdot 10^{19}$, the effective value of the mean free path $l$ increases, and the escape of particles from the Galaxy is accelerated, which will lead to an increase of $\gamma$ with increasing energy. It is more probable, however, that the determining factors are nevertheless the properties of the sources.
The question of the nature of the sources of cosmic rays will be discussed somewhat below. Here we shall indicate the requirements on these sources that follow from the experimental data. The energy density of cosmic particles in the Galaxy is $\sim 1\ \dfrac{\text{eV}}{\text{cm}^{3}}$ (see the Introduction), the volume of the region occupied by them is $V \sim \dfrac{4\pi}{3} R^{3} \sim 10^{68}\ \text{cm}^{3}$, and the total energy is $W \sim 10^{58}\ \text{eV}$. Since the composition of cosmic rays is renewed in a time $T \sim 4 \cdot 10^{8}$ years, for the power of the sources of cosmic rays we obtain $\dfrac{W}{T} \sim 10^{52}\ \dfrac{\text{eV}}{\text{sec}} \simeq 10^{40}\ \dfrac{\text{erg}}{\text{sec}}$. This value is probably overestimated by one or two orders of magnitude (since rather the maximal energy density and volume were taken), and thus,

\[ \frac{W}{T} \sim 10^{38} \div 10^{40}\ \frac{\text{erg}}{\text{sec}} . \tag{45} \]

An even cruder estimate for the number of generated particles is obtained—

is obtained from this by division by the mean energy of a cosmic particle

\[ \widetilde E \sim 10^9 \text{ eV} \]

and indicates the necessity of formation in the Galaxy of

\[ \frac{W}{\tau \widetilde E} \sim 10^{41} \div 10^{43}\ \frac{\text{particles}}{\text{sec}}. \]

Above we proceeded from the assumption of the “distant” origin of cosmic rays, that the distribution of cosmic rays in the Galaxy, or at any rate in the greater part of it, is approximately homogeneous. A number of data, already partly mentioned, speak in favor of this assumption. However, there also exists another point of view, which connects the formation of cosmic rays with the Sun or the stars nearest to us $^{10,11,13,43}$. The difference between the two hypotheses is, of course, radical. It is therefore necessary to dwell on the theory of the solar (or near-solar) origin of cosmic rays and to clarify to what extent it deserves attention.

According to the hypothesis of the solar origin of cosmic rays $^{10,11}$, cosmic rays are generated on the Sun or near the Sun (in $^{11}$ it is assumed that the acceleration takes place within the limits of the Earth’s orbit) and are retained by the magnetic field in some volume with radius $R$. In this case, of course, the particles arrive at the Earth, for the most part, not directly from the Sun, but from the entire region in which they accumulate after their acceleration near the Sun. There is essentially only one argument in favor of such a hypothesis: it has been established experimentally that during powerful eruptions on the Sun cosmic rays are sometimes formed; in smaller quantities cosmic rays are generated on the Sun (or near it), probably not only during eruptions, but also in connection with other processes (spots, prominences, etc.; see $^{1,2,51}$). Another argument in favor of the solar origin of cosmic rays consists in the fact that, in the case of their galactic origin, the energy going into the formation of cosmic rays is only $\sim 10^4$ times less than the energy radiated by all stars in the form of light$^*$, which is considered improbable $^{10}$. However, this assertion seems to us to be without any foundation. In supernova outbursts the energy going into the acceleration of cosmic particles is comparable with the energy going into radiation (see § 46). Hence, and from other considerations, it is completely clear that the transition into cosmic rays

\[ \text{ } \]

$^*$ The energy densities of radiation and of cosmic rays in the Galaxy are of one order. Cosmic particles live $\sim 4 \cdot 10^8$ years, while light is in the Galaxy for

\[ \sim \frac{R}{c} \sim 10^{12}\ \text{sec.} \simeq 3 \cdot 10^4\ \text{years} \]

($R$ is the size of the Galaxy). It follows from this that the energy going to maintain the intensity of cosmic rays is $4 \cdot 10^8 / 3 \cdot 10^4 \sim 10^4$ times less than the energy radiated in the form of light. In the theory of the solar origin, cosmic rays are present only in a relatively small region around the Sun, and therefore the fraction of energy going into their formation is greatly reduced.

energy equal to $\sim 10^{-4}$ of the energy radiated by the stars in the form of light is quite possible, and here one need not encounter any difficulties. As for the generation of particles by the Sun, in experiment only particles with energy $\sim 10^9$ eV have been observed. Therefore there are no direct grounds for assuming that the Sun can accelerate particles to higher energies. Further, in a field $H \leq 10^{-5}$ the radius of curvature for particles with $E \sim 10^{14}$ eV is $\gtrsim 3\cdot 10^{16}$ cm; the size of the region around the Sun where cosmic rays are retained is also $\sim 3\cdot 10^{16}$ (see $^{11}$ and below). It follows from this that particles with $E \gtrsim 10^{14}$ eV certainly do not have a solar origin, while their spectrum is a smooth continuation of the spectrum at $E < 10^{14}$ eV, which within the framework of the solar hypothesis is, at the very least, quite strange. Radio-astronomical data also speak against the solar hypothesis, indicating that cosmic rays exist far beyond the limits of the solar system, in the interstellar medium and in individual nebulae (see § 1). Therefore it seems to us that there are no weighty arguments in favor of the theory of the solar origin of cosmic rays, while a whole series of facts speaks against it.

This conclusion becomes still considerably more convincing if one examines more carefully the question of the conditions for the existence of the “cloud” of cosmic particles in the vicinity of the Sun. Two possibilities are conceivable: the radius of the corresponding region $R$ may either be of the order of the radius of the Earth’s orbit, $R_\oplus$, or be considerably larger. In the first case cosmic particles disappear primarily as a result of collisions with the planets and live for $\sim 5000$ years $\sim 10^{11}$ sec. $^{52}$. The gas density in the solar system is $<10^{-22}$ g/cm$^3$, and thus over its lifetime a particle traverses a path $<10^{11}\cdot 3\cdot 10^{10}\cdot 10^{-22}\sim 0.3$ g/cm$^2$, whereas for the formation of Li, Be, and B it is necessary that the path length exceed $5\div 10$ g/cm$^2$. This argument, as well as still more convincing*) estimates connected with the requirement that there be no anisotropy of the cosmic radiation on the Earth $^{11}$ and with taking account of the balance of the number of particles $^{47}$, lead to the conclusion that $R \gg R_\oplus$, and in particular $R\sim 10^{16}\div 10^{17}$ cm. We shall give the corresponding estimate of $R$, proceeding from considerations of the balance of the number of particles.

The intensity of cosmic rays on the Earth is $I\sim 0.3$ particles per cm$^2\cdot$sec$\cdot$steradian; the mean flux of cosmic radiation from the Sun directly reaching the Earth is $I_\odot \Delta\omega \simeq 7\cdot 10^{-4}$ particles (see $^{47}$). ($I_\odot$ is the intensity of the solar cosmic radiation and $\Delta\omega$ is the solid angle under which the Sun is seen from the Earth.) The density of cosmic rays in the region where they accumulate

*) The remark made about the amount of gas traversed by a particle cannot be decisive, since the density of dust in the solar system is probably somewhat greater than the gas density and therefore it might be possible to explain the appearance of the required number of Li, Be, and B nuclei.

(the radius of this region is $R$), of the order $N \simeq \dfrac{4\pi}{c} I$, and their total number is equal to $\dfrac{4\pi}{c} I \cdot \dfrac{4\pi}{3} R^3$. This number must be supplied by the Sun during the lifetime of a particle $T$, whence

\[ \frac{4\pi}{3} R^3 \frac{4\pi}{c} I = I_{\odot}^{\triangle\omega} T 4\pi R_{\odot}^{2}. \tag{46} \]

Substituting the values given above, we obtain:

\[ R \sim 0.1 \sqrt[3]{cTR_{\odot}^{2}}. \tag{47} \]

Since $R_{\odot}=1.5\cdot 10^{13}\ \text{cm}$, from (47), for $T \simeq 5000$ years we obtain $R \sim 10^{15}\ \text{cm}$, whereas in estimating the time $T$ it was assumed that $R \sim R_{\odot}$. Thus, $R \gg R_{\odot}$. But in this case the time $T$ is the lifetime of the particles in motion in the interstellar medium, that is, $T \sim 4\cdot 10^{8}$ years or, rather, $T \sim 4\cdot 10^{7}$ years, since in the region of interest to us probably $n \sim 1$, and not $n \sim 0.1$. Hence $R \sim 3\cdot 10^{16}\ \text{cm} \sim 0.01\ \text{pc}$.

In this estimate it is evidently assumed that cosmic particles do not leave the region under consideration and that their number can decrease only owing to nuclear collisions (or radiative losses in the case of electrons). But in order for the particles to be retained in a region with $R \sim 3\cdot 10^{16}\ \text{cm}$, very stringent conditions are necessary. In a homogeneous magnetic field the particles will practically immediately leave the region under consideration in the direction of the field lines. If the field is inhomogeneous and diffusion takes place, then under any reasonable assumption about the diffusion coefficient it is likewise impossible to prevent a strong escape of particles. Indeed, the mean free path (with respect to scattering of particles by the magnetic field) cannot be less than $l \sim 10^{13}\ \text{cm}$. It suffices to say that the radius of curvature for $H \sim 10^{-5}$ and $E \sim 10^{10}\ \text{eV}$ is equal to $3\cdot 10^{12}\ \text{cm}$, while the mean free path of interstellar hydrogen atoms at $n \sim 1$ is even $10^{14} \div 10^{15}\ \text{cm}$. Therefore the magnetic field, not to mention the field of the Sun itself, which for $R > R_{\odot}$ plays no role (for $R > R_{\odot}$ it is less than $10^{-6}$ oersted), cannot vary strongly over distances smaller than $10^{13}\ \text{cm}$. More likely, as is clear from astrophysical data, the field varies strongly only over distances $\sim 10^{18} \div 10^{19}\ \text{cm}$. Taking the very smallest value $l \sim 10^{13}\ \text{cm}$, for the diffusion coefficient we obtain:

\[ D \sim \frac{cl}{3} \sim 10^{23}\ \text{cm}^{2}\ \text{sec}^{-1} \]

and the path $R$ will be traversed in the time

\[ t \sim \frac{R^{2}}{6D} \sim \frac{(3\cdot 10^{16})^{2}}{6\cdot 10^{23}} \sim 10^{9}\ \text{sec} \sim 30\ \text{years}. \]

Thus it is quite clear that the theory of the solar origin of cosmic rays can be seriously discussed only if the magnetic field in the region with dimensions $\sim 10^{16}\div 10^{17}$ around the Sun has a toroidal character, that is, the magnetic field lines are closed, as Alfvén had to assume $^{11}$. But such an assumption is very far-reaching, has no real basis, and is made only in order to preserve the theory of the solar origin of cosmic rays. Taking this circumstance into account, as well as all the other weighty considerations given above, it seems to us that the hypothesis of the solar origin of cosmic rays is completely unfounded and, at least at the present time, cannot in any way compete with the hypothesis of the galactic origin of cosmic rays. The remarks made apply, in essence, also to the variant of the theory of the origin of cosmic rays proposed by Ya. P. Terletskii and A. A. Logunov $^{43}$. In $^{43}$ it is assumed that acceleration occurs in the interstellar medium, with $\alpha T \gg 1$ (see (40)); in order that the exponent $\gamma$ be not 1, as follows from (40) for $\alpha T \gg 1$, but, in agreement with experiment, close to 2.5, in $^{43}$ the distribution of cosmic rays is considered nonuniform in space. Then, if the diffusion coefficient is proportional to the energy, the faster particles more quickly leave the region near the primary sources of cosmic rays and in this region $\gamma \simeq 2.5$. In this case the sources must be located from the point of observation at a distance

\[ R \ll 2 \sqrt{\frac{\beta}{\alpha}(E-E_0)}, \]

where $E$ is the particle energy, $E_0$ its initial energy, $\alpha$ is the coefficient in equation (36), and the diffusion coefficient is $D=\beta E$. Further, in $^{43}$ it is put that $\alpha=2\cdot 10^{-13}$, $T=2\cdot 10^{15}$ (that is, $\alpha T=400$), $\beta \simeq \dfrac{c}{3eH}$, or

\[ D\sim \frac{cE}{3eH}=\frac{cr}{3}, \]

where

\[ r=\frac{E}{eH}=\frac{E\,(\text{ev})}{300H} \]

is the radius of curvature in the magnetic field. Hence, for

\[ E\sim 10^{10}\ \text{ev}\gg E_0 \]

\[ R\ll 2\sqrt{\frac{D}{\alpha}}\sim 6\cdot 10^{17}\ \text{cm} \]

in a field $H\sim 10^{-5}$, and

\[ R\ll 2\cdot 10^{18}\ \text{cm} \]

in a field $H\sim 10^{-6}$. Even if one takes for $\alpha$ the value $10^{-15}$ (see below), the length $R$ is of the order of, or less than, the distance to the nearest stars, equal to $\sim 1.3\text{ pc}\simeq 4\cdot 10^{18}$ cm. Thus it is clear that in $^{43}$ the question is, in essence, one of a theory of the solar origin of cosmic rays, perhaps supplemented by taking into account the radiation of several nearest stars, among which, as is known, there are no stars with especially anomalous properties (novae, magnetic stars, etc.). It seems to us that all the indicated assumptions made in paper $^{43}$ are either unfounded or inadmissible. In § 2 it was pointed out that the value $\alpha=2\cdot 10^{-13}$ is inadmissible, since in that case fast electrons would be present (as is clear from paper $^{39}$, and it is easy to show, using formula (30), the account of the inverse

of the Compton effect on solar radiation for $R \gg R_{\odot}$ does not change this conclusion). Even in a field $H \sim 10^{-5}$ (with $R \ll 6\cdot 10^{18}\ \mathrm{cm}$ for $a \approx 10^{-15}$), the value of $a$ must not exceed $5\cdot 10^{-16} \div 10^{-15}\ \mathrm{sec}^{-1}$ (see Table III). But even for $a=10^{-15}$, the parameter $aT=2$, whereas the condition $aT \gg 1$ may be regarded as practically fulfilled only for $aT>5 \div 10$. The situation could be saved only by increasing the value of $T$, which is hardly justified near the solar system (the adopted value $T=2\cdot 10^{15}$ corresponds to a density $n \sim 1$). Further, we see no grounds for taking the mean free path for scattering to be $l\sim r$, as is done in $^{43}$ (as indicated above, in $^{43}$

\[ D\sim \frac{cl}{3}\sim \frac{cr}{3}. \]

The case $l\sim r$ will occur in scattering by a magnetic dipole (see $^{1}$), when the region of uniform field is much smaller than the radius of curvature. But for $E\sim 10^{10}\ \mathrm{ev}$, $r\sim 3\cdot 10^{12}\ \mathrm{cm}$, and, as indicated above, the field cannot be nonuniform at smaller distances, while according to all the data it is more likely uniform even at distances $l\sim 10^{18}\div 10^{19}\ \mathrm{cm}$. Thus, the assumption that the diffusion coefficient $D$ is proportional to the particle energy $E$ appears completely unfounded, especially over a wide energy interval.

In addition to the difficulties noted, and also the radio-astronomical arguments already mentioned against any theory of the solar origin of cosmic rays, theory $^{43}$ also fails to satisfy the obligatory requirement that the number of particles be balanced. Indeed, the number of particles leaving the region under consideration surrounding the Sun is

\[ Y\sim 4\pi R^2 \frac{DN}{R}, \]

since the diffusion flux is

\[ j=-D\nabla N\sim \frac{DN}{R}, \]

where $R$ is a characteristic size and $N$ is the particle concentration. Taking

\[ D\sim \frac{cr}{3}\sim 10^{23} \]

(for $E\sim 10^{10}\ \mathrm{ev}$), $R\sim 10^{18}\ \mathrm{cm}$, and $N\sim 10^{-10}$, we obtain a flux

\[ Y\sim 10^{32}\ \text{particles/sec}. \]

From the Sun, however, there comes a particle flux

\[ Y_{\odot}\sim 4R_{\odot}^{2} I_{\odot}\Delta\omega \sim 10^{24}\ \text{particles/sec} \]

(see above). Thus the Sun gives $8$ orders of magnitude fewer cosmic particles than is required to maintain the balance. Even if one were, with difficulty, to “stretch” the size of the region filled with cosmic rays to $R\sim 10^{19}\ \mathrm{cm}$ and take into account the contribution of several nearest stars, the situation would not change, since these stars are in no way remarkable and there are absolutely no grounds for assuming that they emit cosmic rays many orders of magnitude more strongly than the Sun. If, however, scheme $^{43}$ is supplemented by the assumption of the presence of a toroidal field, then the particles will cease to leave the near-solar region, but at the same time their spectrum will be characterized by the exponent $\gamma=1$. In view of what has been said, we see no possibility whatever of “saving” the variant of the theory of the solar origin

of cosmic rays, proposed in $^{43}$. Thus, the negative conclusion concerning the hypothesis of a solar origin of cosmic rays remains fully in force.

This does not mean, of course, that cosmic rays generated by the Sun are of no interest.*) On the contrary, the study of the connection between solar activity and cosmic rays deserves every attention. But within the framework of the theory of the galactic origin of cosmic rays, the generation of fast particles by the Sun is of secondary importance and is significant only near the Sun and, in particular, on the Earth. In the overall balance of cosmic particles being produced and lost in the Galaxy, the contribution not only of the Sun, but even of all $10^{11}$ galactic stars, if they radiate in the same way as the Sun, is negligible. Indeed, $10^{41} \div 10^{43}$ particles/sec. must be produced in the Galaxy,**) whereas the Sun gives $10^{24}$ particles/sec., and $10^{11}$ stars of the solar type will give $10^{35}$ particles/sec., which is still at least 6 orders of magnitude less than required.

b) Supernovae and novae as probable sources of cosmic rays

Stars of the solar type cannot, as has just been shown, serve as the primary sources of cosmic rays in the Galaxy. Such sources must be some special objects, probably few in number in view of their very anomalous properties; but if this is so, then the sources must be very powerful and efficient. These are roughly the considerations that long ago led to the suggestion of the generation of cosmic rays during supernova outbursts (see $^{53}$). Until recently, however, this hypothesis had not been supported by anything, apart from the purely energetic considerations indicated below, which, without further investigation, cannot be regarded as convincing. In § 1 the change in the situation associated with the development of radio astronomy was already noted. From radio-astronomical data it follows that, as a result of a supernova outburst, relativistic electrons are formed in its expanding envelope, and there is every reason to believe (see § 3) that cosmic protons and nuclei are present in no smaller quantity. Thus, the experimental

*) In the case of the Sun, and also of stars, the generation not only of charged particles but also of neutrons may be significant (see $^{1,54}$).

**) This estimate of the number of particles that must be generated in the Galaxy was made above (see (45) and what follows) without taking into account the diffusive escape of particles from the Galaxy. Therefore it is appropriate to emphasize that allowing for diffusion does not change the estimate in order of magnitude: the diffusion flux of particles from the Galaxy

\[ Y \sim 4\pi R^{2}\frac{DN}{R} \sim 4\pi R\frac{cl}{3}N \sim 3\cdot 10^{41} \div 3\cdot 10^{42} \]

for $l \sim 10^{18} \div 10^{19}\,\mathrm{cm}$, $R \sim 3\cdot 10^{22}\,\mathrm{cm}$, and $N \sim 10^{-10}$. This estimate at the same time shows that, in a more exact analysis, allowance for the diffusion of particles from the Galaxy is necessary.

the data directly indicate the possibility of the generation of cosmic particles as a result of supernova outbursts, whereas there are no analogous indications whatever with respect to stars of other types. Therefore the hypothesis of supernovae as sources of primary cosmic rays is very attractive and deserves the most serious attention$^{1,33,34}$.

Let us discuss this hypothesis in more detail.

Can supernovae provide the release of energy necessary to maintain the observed intensity of cosmic rays, i.e. can supernovae give, on average, $10^{38} \div 10^{40}$ erg/sec to cosmic rays? In supernova outbursts, $10^{49} \div 10^{50}$ erg goes into visible radiation; the energy released in the form of kinetic energy is an order of magnitude smaller. Supernovae flare up in our Galaxy on average once every 300 years, $\simeq 10^{10}$ sec, and hence their mean energy release amounts to not less than $10^{39} \div 10^{40}$ erg/sec. It follows from this that supernovae can provide the required replenishment of the energy of cosmic radiation, if approximately as much energy goes into this as is released in the form of light or even in the form of kinetic energy of the expanding envelope. In any case the energy reserve is sufficient; no obvious contradiction arises. Moreover, knowing the spectrum of the radio emission of a supernova, one can find the spectrum of the electrons in the envelope and the total energy associated with them. Unfortunately, the data on the spectra of Taurus A and Cassiopeia A are still not sufficiently accurate and complete, but for Taurus A an estimate$^{34}$ gives an energy reaching $\sim 10^{50}$ erg. Even if this figure is overestimated by one or two orders of magnitude, it is clear that as a result of supernova outbursts energy in the required, order-of-magnitude amount does indeed pass into cosmic radiation. Estimates in this direction$^{33}$, proceeding from the balance of the number of particles, also lead to a favorable conclusion (up to $10^{52}$ particles are formed in an outburst, i.e. on average up to $10^{42}$ particles per second, whereas $10^{41} \div 10^{43}$ particles/sec are lost in the Galaxy).

The success of the energy estimate also makes it possible to put forward the supposition of a possible generation of cosmic rays in the outbursts not only of supernovae, but also of novae. Novae, apparently, do not differ in principle from supernovae, but for them the energy release is $\sim 10^{45} \div 10^{46}$ erg. Taking into account that in the Galaxy $\sim 100$ novae flare up per year, we arrive at a mean release of $3 \cdot 10^{39} \div 3 \cdot 10^{40}$ erg/sec. Consequently, if cosmic rays are also generated in nova outbursts, which is quite probable, then the contribution of novae may even exceed the contribution of supernovae*).

*) In this connection it is natural to suppose that novae give radio emission that is $10^{3} \div 10^{4}$ times weaker than in the case of supernovae. This corresponds to a flux reaching $F \sim 10^{-22}$ erg/cm$^2$·sec·Hz for a star at a distance $\sim 10^{3}$ pc. Such a value is two orders of magnitude smaller than

The second question: can particles of the required energy be formed in a flare? The answer to this question has already been given in § 3, where it is shown that in flares there is a fundamental possibility of accelerating particles (especially nuclei) up to the highest observed energies, $\sim 10^{17} \div 10^{18}$ eV. This conclusion is supported by the fact that data on the spectrum of supernovae testify to the presence (see $^{36,34}$ and § 1), for example, in Taurus A of electrons with energy $\sim 10^{12}$ eV, which for Fe nuclei corresponds to an energy $\sim 10^{17}$ eV (since the maximum energy is proportional to the rest mass; see (43)). In favor of the hypothesis of the generation of cosmic rays by supernovae and novae there is one more interesting argument, due to I. S. Shklovsky$^1$. The galaxy nearest to us—the Magellanic Clouds—does not give noticeable radio emission, and it may be asserted that its radio-emitting capacity is considerably lower than that of our Galaxy or of the nebula M31 (the large galaxy nearest to us, located in the constellation Andromeda). At the same time, in the Magellanic Clouds over several decades of observations only three novae have flared up, whereas in our Galaxy several thousand of them have flared up during this time. Hence it is natural to conclude that the Magellanic Clouds are poor in novae (supernovae flare up so rarely that no direct conclusions can be drawn with respect to them; of course, from the standpoint under discussion there should also be anomalously few of them in the Magellanic Clouds). Owing to the poverty in novae, the intensity of cosmic electrons in the Magellanic Clouds is small, and therefore the general radio emission in question is also weak.

To avoid misunderstandings we shall also point out that a supernova flare need not be accompanied by any increase in the intensity of cosmic rays on the Earth. First, cosmic particles are accelerated for a long time, considerably longer than the period of bright flare, and leave the expanding envelope of the star, apparently, chiefly after 1000–3000 years following the flare, when the envelope dissipates. Second, the observed intensity of cosmic radiation is connected with about a million supernova flares (the lifetime of particles in interstellar space is $\sim 4 \cdot 10^8$ years, the time between flares is $\sim 300$ years) or a billion nova flares. Therefore the contribution of a single supernova flare, even if all the particles immediately left its envelope, would amount to only $\sim 10^{-6}$ of the intensity of the observed cosmic radiation (the situation may change only in the exceptional case when the supernova flares up very close to the Sun). All that has been said leads

$^1$ For Taurus A too lies at the limits of experimental possibilities. Closer novae may give still greater radio emission, and in this way attempts to detect the radio emission of novae are quite real and deserve every attention$^{34}$.

leads us to the conclusion that the sources of cosmic rays are supernovae and, possibly, novae. This supposition is based on radio-astronomical observational data and is in agreement with, or at any rate does not contradict, all the other facts and general requirements. Can one therefore assert that only the contribution of supernovae and novae determines the intensity of cosmic radiation? This cannot be asserted definitively, although it is very probable. Stars of the solar type (and still less active ones) cannot, all taken together, be responsible for more than a $\sim 10^{-6}$ part of the intensity of cosmic radiation. There are likewise no indications that stable stars of any classes can generate a number of particles comparable with that formed in supernovae or novae. But, of course, it is not excluded that on some stars, or in collisions of diffuse nebulae, etc., there occurs generation that is noticeable in the overall balance of the number of cosmic rays. Such a possibility exists, but it is unlikely and, most importantly, does not change the situation—generation in supernovae takes place, generation in novae is very probable, and the number of cosmic particles formed in this process is, according to estimates, sufficient to maintain the observed intensity of cosmic rays. Therefore there is every reason to regard supernovae and novae as the sources of cosmic rays and to develop further the theory of the origin of cosmic rays on this basis.

§ 5. CONCLUSION

The general picture of the origin and evolution of cosmic radiation, according to what has been said above, is as follows.

Cosmic particles—electrons, protons, and nuclei with high energy—are generated as a result of outbursts of supernovae and, probably, novae. Acceleration occurs in the expanding, turbulent envelope ejected during the stellar outburst. The acceleration mechanism is statistical. With such a mechanism, particles in the envelope can apparently be accelerated up to the highest observed energies, $\sim 10^{17} \div 10^{18}$ eV, and the energy per nucleon in nuclei will be the same as the energy of protons. The spectra of protons and nuclei, if one does not speak of the maximum energy, must be identical. Owing to more favorable injection conditions, one may expect a relatively greater efficiency of acceleration of nuclei as compared with protons. All these conclusions are in agreement with experiment. As for the exponent in the spectrum $\gamma$, further work is needed for its theoretical determination, but it can already be said that the spectrum is indeed a power law and $\gamma$ may be equal to $2 \div 3$, as is found experimentally. It is quite possible (there are some indications of this; see §§ 1, 3) that in different outbursts somewhat different values of $\gamma$ are obtained, and that the observed spectrum of cosmic particles is a certain aver—

... over many bursts. The spectrum of the electrons accelerated in the envelopes must in general coincide with the spectrum of the protons and nuclei.

As the envelope ejected by the supernova expands and dissipates, the relativistic particles moving in the envelope begin ever more rapidly to escape into interstellar space; this process is practically completed in a time of $\sim 1000$–$3000$ years. In the interstellar medium cosmic particles wander, “colliding” with magnetic fields moving together with the interstellar gas. The corresponding mean free path of the particles is $\sim 10^{19}$ cm. Cosmic protons “live” in the Galaxy for $\sim 4 \cdot 10^8$ years and are sharply slowed down as a result of collisions with interstellar protons. Nuclei live less long—for example, Fe nuclei “live” 25 times less than protons. Cosmic electrons live, like protons, $\sim 4 \cdot 10^8$ years (they lose energy sharply as a result of radiative losses). During their lifetime cosmic particles manage to diffuse several thousand parsecs from the plane of the Galaxy, as a result of which cosmic particles form a quasi-spherical subsystem close to the subsystem of rarefied interstellar gas. During motion in the interstellar medium a transformation of the composition of the nuclear component of cosmic rays occurs; in particular, Li, Be, and B nuclei appear. The number of these nuclei is consistent with the assumption that the mean residence time of nuclei in the Galaxy is determined by their lifetime with respect to nuclear collisions. The spectrum of protons and nuclei near the Earth, apart from the very soft particles that are not observed because of high-latitude formation, at not too high energies coincides with their spectrum in the sources. With respect to the energy range up to $\sim 10^{11}$ eV/nucleon, this conclusion is confirmed experimentally (the spectra of protons and nuclei coincide, with $\gamma \simeq 2$). At higher energies the spectrum separately of protons and separately of nuclei has not been reliably established. Therefore the possibility is not yet excluded that high-energy particles are accelerated in the interstellar medium. In this case the spectrum of nuclei will fall with increasing energy much more steeply than the spectrum of protons. It is more probable that here too the spectra are determined by the spectrum of the generated particles and, possibly, at the very highest energies by an increase in the diffusive escape of particles from the Galaxy (§ 4a). The spectrum of electrons, during their diffusion in interstellar space, changes because of bremsstrahlung in magnetic fields, and the spectral index becomes equal to 3, which agrees with radio-astronomical data.

The picture described seems to us, in general, quite well founded, but, of course, a whole series of points require clarification and experimental verification. Let us list first of all the tasks of experiment.

Radio-astronomical observations of the general Galactic radio emission make it possible to refine the data on the quantity, spectrum, and pro-

spatial distribution of cosmic electrons in the Galaxy. Observation of the radio emission of supernovae will make it possible to obtain information on the spectrum of electrons in supernovae, on their quantity, and on their spatial distribution in the shell (for the latter purpose it is necessary to work with installations having high angular resolving power; in this connection it is also advisable to measure the polarization of the radiation $^{34}$). Radio observations must, as regards the continuous spectrum of supernovae, be supplemented by optical observations (it is necessary to determine the spectrum, intensity, and polarization of the radiation) $^{1,36}$.

It is necessary to find, by the photographic-plate method, or by using data on extensive atmospheric showers, the spectrum of the primary nuclear component of cosmic rays at the Earth at high energies. This will make it possible to settle the question of the effectiveness of particle acceleration in interstellar space. Further refinement of the available data on the composition of the nuclear component at the energies already investigated up to $3 \cdot 10^{10}$ eV/nucleon is also, of course, of interest.
Further refinement of information on the electron and photon components of the primary cosmic radiation at the Earth is advisable. With an increase in the accuracy of the experiment by even one order of magnitude, there are chances of detecting primary electrons, which would be of great importance.
Both from the point of view of the problem of the origin of cosmic rays and from the point of view of the physics of the Sun and the solar system, further study of the influence of the Sun on cosmic radiation is important (variations of intensity, correlation with eruptions and magnetic storms, etc.), as is clarification of the question of the nature of the high-latitude cutoff $^{1,2,20,55}$.

A number of problems also confront theory, among which the most important are:

Development of the magnetohydrodynamic theory of turbulent motion and its application to the interstellar medium and to the envelopes of stars (first of all to the envelopes of supernovae and novae).
Further analysis of the question of the acceleration of particles moving in a turbulent magnetized medium (mainly statistical acceleration with allowance for fluctuations is meant). Especially important is the application of the corresponding results to the case of supernovae and novae, with the aim of determining the spectrum of the cosmic rays generated. This also includes the problem of injection, which is essential, in particular, for clarifying the reason for the high efficiency of acceleration of nuclei as compared with protons.
A number of other questions are also of interest, some of which are closely connected with the interpretation of experimental data. Let us point to the question of the evolution of nuclei in cosmic rays and the clarification of the composition of nuclei generated in the sources. Further, it is necessary to determine what the density of $\gamma$-rays in our Galaxy is, taking into account the arrival of $\gamma$-rays from other galaxies. Questions concerning anti-

protons and positrons in the composition of the primary component near the Earth, on the isotropy of cosmic rays near the Earth in connection with the role of the magnetic field in the solar system and near it, on the magnetic moment of the solar system and the nature of the high-latitude cutoff (on all these questions see ¹).

Thus, it is still necessary to clarify a number of essential points before the problem of the origin of cosmic rays can be considered clear in all its aspects. But, as it seems to us, the main thing here has already been done, and the picture outlined above does not undergo fundamental changes, as was the case until recently, up to the use of radio-astronomical methods for elucidating this range of questions.

With the development of radio astronomy, as well as cosmic electrodynamics, the question of the origin of cosmic rays has become a truly astrophysical problem and has emerged from the stage of predominantly hypothetical constructions that could not be checked by observations. Therefore, and also taking into account the progress of cosmic-ray physics, one may be confident that the further development of the theory of the origin of cosmic rays will proceed forward in rapid strides.

CITED LITERATURE

Proceedings of the 3rd conference on questions of cosmogony: The Origin of Cosmic Rays. Publ. Acad. Sci. USSR, 1953.
Progress in cosmic ray physics, Amsterdam (1952).
H. Alfvén, Zeits. f. Physik 107, 579 (1937).
H. Alfvén, Nature 143, 435 (1939).
W. F. G. Swann, Phys. Rev. 43, 217 (1933).
H. Alfvén, Cosmical Electrodynamics, Foreign Literature Publishing House, 1952.
Ya. P. Terletskii, JETP 16, 403 (1946); 19, 1059 (1949); Vestnik MGU No. 1, 75 (1948).
L. Riddiford and S. T. Butler, Phil. Mag. 43, 447 (1952).
Ya. P. Terletskii, UFN 44, 46 (1951).
R. D. Richtmyer and E. Teller, Phys. Rev. 75, 1729 (1949).
H. Alfvén, Phys. Rev. 75, 1732 (1949); 77, 375 (1950).
K. O. Kiepenheuer, Phys. Rev. 78, 809 (1953).
E. McMillan, Phys. Rev. 79, 498 (1950).
E. Fermi, Phys. Rev. 75, 1169 (1949).
V. L. Ginzburg, UFN 32, 26 (1947); 34, 13 (1948);
G. G. Getmantsev, UFN 44, 527 (1951); UFN 50, 472 (1953);
M. Ryle, UFN 46, 508 (1952).
I. S. Shklovskii, Astr. Journal 29, 418 (1952); 30, 15 (1953).
H. Alfvén and N. Herlofson, Phys. Rev. 78, 616 (1950);
K. O. Kiepenheuer, Phys. Rev. 79, 738 (1950).
V. L. Ginzburg, DAN 76, 377 (1951).
G. G. Getmantsev, DAN 83, 557 (1952) and an article in the memorial volume for A. A. Andronov, Publ. Acad. Sci. USSR, 1953.
V. L. Ginzburg and M. I. Fradkin, DAN 92, No. 3 (1953).
G. Schott, Electromagnetic radiation, Ch. VII, Cambridge (1912).
L. A. Artsimovich and I. I. Pomeranchuk, JETP 16, 379 (1946).
L. Shiff, Rev. Sci. Instr. 17, 6 (1946).

V. L. Ginzburg, Izv. AN SSSR, Ser. Fizich., 11, 165 (1947).
V. V. Vladimirskii, ZhETF 18, 393 (1948).
D. Ivanenko and A. Sokolov, DAN 59, 1551 (1948).
L. Landau and E. Lifshitz, Field Theory, § 73, Gostekhizdat, 1948.
J. Schwinger, Phys. Rev. 75, 1912 (1949).
28a. A. A. Sokolov, DAN 67, 1013 (1949); G. Parzen, Phys. Rev. 84, 235 (1951).
28b. I. M. Frank, Izv. AN SSSR, Ser. Fizich., 6, 3 (1942); V. L. Ginzburg and I. M. Frank, DAN 56, 583 (1947).
Problems of Cosmic Aerodynamics, IIL, 1953.
S. B. Pikelner, DAN 88, 229 (1953). Izvestiya Krymskoi Astrofiz. Observatorii 10, 74 (1953).
G. Cocconi, Phys. Rev. 83, 1193 (1951).
A. A. Korchak and Ya. P. Terletskii, ZhETF 22, 507 (1952).
I. S. Shklovskii, DAN 91, 475 (1953).
V. L. Ginzburg, DAN 92, No. 6 (1953).
I. M. Gordon, DAN 92, [[unclear: crossed-out number]] (1953).
I. S. Shklovskii, DAN 90, 983 (1953).
D. Bohm and E. P. Gross, Phys. Rev. 74, 624 (1948).
E. Feenberg and H. Primakoff, Phys. Rev. 73, 449 (1948).
T. M. Donahue, Phys. Rev. 84, 972 (1951).
V. L. Ginzburg, DAN 92, No. 4 (1953).
V. I. Gol’danskii, A. L. Lyubimov and B. V. Medvedev, UPhN 49, 3 (1953).
Ya. P. Terletskii and A. A. Logunov, ZhETF 21, 567 (1951).
Ya. P. Terletskii and A. A. Logunov, ZhETF 23, 682 (1952).
C. Y. Fan, Phys. Rev. 82, 209 (1951).
L. Landau and L. Pyatigorskii, Mechanics, § 11, GTTI, 1940.
G. T. Zatsepin, DAN 80, 577 (1951).
A. Unsöld, Phys. Rev. 82, 857 (1951).
L. E. Gurevich, Problems of Cosmogony, vol. 2, 1953.
G. G. Getmantsev and V. L. Ginzburg, DAN 87, 187 (1952).
S. Z. Belen’kii, Cascade Processes in Cosmic Rays, Gostekhizdat, 1948.
V. Sarabhai and R. P. Kane, Phys. Rev. 90, 204 (1953).
E. O. Kane, T. J. Shanley and J. A. Wheeler, Rev. Mod. Phys. 21, 51 (1949).
D. ter Haar, Rev. Mod. Phys. 22, 119 (1950).
G. W. Hutchinson, Phil. Mag. 43, 847 (1952).
J. A. Van Allen, Nuovo Cimento 10, 630 (1953).

Submission history

[v1] 1953

Nucleus	Cross section in hydrogen	Range in hydrogen \(l=\dfrac{1.67\cdot 10^{-24}}{\sigma}\ \text{g}/\text{cm}^2\)	Range in hydrogen \(l=\dfrac{1}{\sigma n}\ \text{cm}\) at \(n=0.1\)	Time of free path in years at speed \(v=c\); \(T=l/c\) (hydrogen, \(n=0.1\))	Mixture of hydrogen with \(n=0.09\) and helium with \(n=0.01\); \(T\) in years	\(T_{\text{proton}}/T_{\text{nucleus}}\)
Proton . . . .	\(2.5\cdot 10^{-26}\)	67	\(4\cdot 10^{26}\)	\(4\cdot 10^8\)	\(2.9\cdot 10^8\)	—
\(\alpha\)-particle . . .	\(1.3\cdot 10^{-25}\)	13	\(7.7\cdot 10^{25}\)	\(8\cdot 10^7\)	\(7.4\cdot 10^7\)	4
\(\mathrm{O}_{16}^{8}\) . . . .	\(3.6\cdot 10^{-25}\)	4.6	\(2.8\cdot 10^{25}\)	\(3\cdot 10^7\)	\(2.8\cdot 10^7\)	10
\(\mathrm{Fe}_{56}^{26}\) . . . .	\(9\cdot 10^{-25}\)	1.9	\(1.1\cdot 10^{25}\)	\(1.2\cdot 10^7\)	\(1.2\cdot 10^7\)	25

Electron energy in eV	Ionization losses (formula (26)) \(n=0.1\)	Radiative losses (formula (27)) \(n=0.1\)	Losses due to radiation in a magnetic field (formula (31)) \(H=3\cdot10^{-6}\)	Losses due to radiation in a magnetic field (formula (31)) \(H=10^{-5}\)	Energy gain (formula (36)) \(\alpha=5\cdot10^{-17}\)	Energy gain (formula (36)) \(\alpha=2\cdot10^{-13}\)
\(5\cdot10^7\)	\(2.7\cdot10^{-8}\)	\(4\cdot10^{-9}\)	\(10^{-10}\)	\(10^{-9}\)	\(2.5\cdot10^{-9}\)	\(10^{-5}\)
\(10^8\)	\(2.8\cdot10^{-8}\)	\(8\cdot10^{-9}\)	\(4\cdot10^{-10}\)	\(4\cdot10^{-9}\)	\(5\cdot10^{-9}\)	\(2\cdot10^{-5}\)
\(5\cdot10^8\)	\(3.2\cdot10^{-8}\)	\(4\cdot10^{-8}\)	\(10^{-8}\)	\(10^{-7}\)	\(2.5\cdot10^{-8}\)	\(10^{-4}\)
\(10^9\)	\(3.3\cdot10^{-8}\)	\(8\cdot10^{-8}\)	\(4\cdot10^{-8}\)	\(4\cdot10^{-7}\)	\(5\cdot10^{-8}\)	\(2\cdot10^{-4}\)
\(5\cdot10^9\)	\(3.7\cdot10^{-8}\)	\(4\cdot10^{-7}\)	\(10^{-6}\)	\(10^{-5}\)	\(2.5\cdot10^{-7}\)	\(10^{-3}\)
\(10^{10}\)	\(3.9\cdot10^{-8}\)	\(8\cdot10^{-7}\)	\(4\cdot10^{-6}\)	\(4\cdot10^{-5}\)	\(5\cdot10^{-7}\)	\(2\cdot10^{-3}\)
\(5\cdot10^{10}\)	\(4.3\cdot10^{-8}\)	\(4\cdot10^{-6}\)	\(10^{-4}\)	\(10^{-3}\)	\(2.5\cdot10^{-6}\)	\(10^{-2}\)

Nucleus	\(E_{\mathrm{i,kin}}\) in ev	\(E_{\mathrm{i,kin}}\) per nucleon in ev
Proton	\(10^8\)	\(10^8\)
\(\mathrm{He}^{2}_{4}\)	\(4\cdot 10^8\)	\(10^8\)
\(\mathrm{O}^{8}_{16}\)	\(2\cdot 10^{10}\)	\(10^9\)
\(\mathrm{Fe}^{25}_{56}\)	\(3\cdot 10^{11}\)	\(5\cdot 10^9\)

Full Text

THE ORIGIN OF COSMIC RAYS AND RADIO ASTRONOMY

CONTENTS

INTRODUCTION

§ 1. THE ELECTRONIC COMPONENT OF COSMIC RAYS AND COSMIC RADIO EMISSION

§ 2. MOTION OF CHARGED PARTICLES IN THE INTERSTELLAR MEDIUM

§ 3. STATISTICAL MECHANISM OF PARTICLE ACCELERATION IN THE INTERSTELLAR MEDIUM AND IN THE ENVELOPES OF STARS

§ 4. REMARKS ON THE THEORY OF THE ORIGIN OF COSMIC RAYS

a) Certain consequences following from the experimental data. Critique of the hypothesis of the solar origin of cosmic rays

b) Supernovae and novae as probable sources of cosmic rays

CITED LITERATURE

Submission history

Access Paper

Citation

Share

Related Papers

Feedback

THE ORIGIN OF COSMIC RAYS AND RADIO ASTRONOMY