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A New Theory of the Origin of Primary Cosmic Rays
1. Introduction
The question of the origin of primary cosmic radiation, which has confronted nuclear physics and astrophysics for decades, remains open to this day. A number of the simplest assumptions and theories have not withstood experimental verification. However, the factual material, which has become extraordinarily extensive in our time—largely thanks to the successes of Soviet physicists—has made it possible to advance new theories that seem more substantiated and attainable for experimental testing.
Questions concerning the spectrum of primary radiation by mass and energy, variatrons, mesons, and shower processes have been considered in a number of monographs and reviews [for example, the collection Mesons¹, the series of works by the group of Alikhanov and Alikhanyan², the works of Wexler’s group⁴, Rossi’s review⁵, and also numerous articles in DAN SSSR and JETP].
Below we shall discuss Fermi’s theory of the origin of primary cosmic rays, published in the summer of 1949⁶. It must be noted that as early as 1945 Ya. Terletsky put forward a theory of the origin of fast charged particles in the magnetic field of stars with a rotation axis and direction of magnetic moment that do not coincide⁷. Fermi bypasses the work of the Soviet scientist in silence, although Terletsky’s conclusion contains a necessary and missing link for Fermi’s theory. Namely, the principal gap in the latter is the mechanism of the origin of protons and other heavier nuclei with energy exceeding a certain threshold necessary for subsequent acceleration in “hydrodynamic” wandering interstellar magnetic fields.
Fermi refers here to the experimentally unverified theory of Swann⁸ and to the works of Teller and Richtmyer⁹ and Alfvén¹⁰, published in 1949, which, on the question of the origin of fast charged particles, essentially repeat Terletsky’s idea four years after him.
A strong argument against the widely held view that cosmic radiation fills at least the entire volume of the galaxy is the enormous quantity of energy associated with the particles of radiation in so vast a volume. If the particles of primary radiation do indeed fill the volume of the galaxy, then the process of their origin (i.e., acceleration to the observed energies) must be extremely intensive.
Fermi’s hypothesis may serve as a partial answer to this objection. According to his theory, cosmic rays arise and are accelerated in interstellar space; magnetic fields hinder the escape of particles beyond the limits of the galaxy. The acceleration process proceeds by means of the interaction of particles with magnetic fields “wandering” in space.
Such magnetic fields, having an extent on the order of light-years, owing to their size and relatively high electrical conductivity of interplanetary space, must be very stable. According to Alfvén¹¹, one may assume that magnetic lines of force are “attached” to, and follow, the streams of matter filling interstellar space, or, conversely, that with each line of force of the magnetic field there is associated some density of matter.
If a particle (a proton or a heavier nucleus) with energy above a certain threshold enters the interstellar medium, then, encountering on its way the moving inhomogeneities of the interstellar magnetic fields, it may gradually accumulate energy. The rate of accumulation of energy is insignificant, but energies on the order of the maximum observed values can be attained.
2. ON THE MOTION OF INTERSTELLAR MATTER
The interstellar space of the galaxy is filled with matter of unusually low density, corresponding to approximately one hydrogen atom per cubic centimeter (i.e., about \(10^{-24}\ \text{g}/\text{cm}^3\)). There are data indicating that denser “clouds” exist in space, with densities increased by tens and hundreds of times, whose average extent is on the order of 30 light-years. Measurements of the Doppler effect¹² in absorption lines by interstellar matter give the radial velocity of a cloud relative to the Sun. The mean square radial velocity with a correction for the Sun’s own motion relative to the nearest stars reaches approximately \(15\ \text{km}/\text{sec}\).
Clouds of the indicated type occupy about 5% of the interstellar space of the galaxy.
It may be assumed that the density of matter outside the clouds is on the order of \(10^{-25}\ \text{g}/\text{cm}^3\), i.e., 0.1 hydrogen atom per \(1\ \text{cm}^3\). The greater part of the matter is evidently composed of hydrogen atoms, and, moreover, they are all ionized as a result of the photoelectric effect caused by quanta of starlight. It can be shown that the fraction of undissociated atoms does not exceed one or two percent.
It is reasonable to suppose that the rarefied medium described executes various motions as a result of disturbances caused by the denser clouds of matter moving in it. The mean velocity of such streams, evidently, also reaches tens of \(\text{km}/\text{sec}\). Alfvén showed that the intensity of the magnetic field in places of greater rarefaction of matter may reach \(5 \cdot 10^{-6}\) gauss and attain larger values in denser clouds of matter.
The lines of force of such fields must form irregular loops, bending with the flow of matter with which the magnetic field is associated. However, these fields must hinder the flow of one accumulation of interstellar matter into another, for this would lead on the average to strengthening of the magnetic field and increasing its energy. One of the results of the existence of interstellar wandering magnetic fields, therefore, will be a reduction to a minimum of losses of the friction type, which could gradually retard the flows of matter and reduce them to disordered thermal motions.
3. ACCUMULATION OF ENERGY BY COSMIC-RAY PARTICLES
Let a particle possessing considerable energy move in the wandering magnetic fields described above. If this particle is a proton with an energy of the order of several billion electron-volts, it will fly in a “spiral” around the lines of force, curving with a radius of the order of \(10^{13}\) cm, until it “runs into” some irregularity of the cosmic magnetic field and is “reflected” by it.
In such a “collision” there may occur either a loss or an accumulation of energy. However, accumulation of energy is somewhat more probable than its loss. The reason for this can be understood if one observes that, ultimately, statistical equilibrium must be established between the degrees of freedom of the particle and the degrees of freedom of the wandering fields. Equipartition of energy among the degrees of freedom also takes place for very large values of the energy.
An elementary estimate of the change in the energy of a particle may be made by supposing that the particles “collide” with irregularities of the magnetic field as they would collide with obstacles of very large mass, moving chaotically with mean velocities of the order \(\beta c = 30\) km/sec. The mean increment of energy in one collision, in order of magnitude, may be estimated as
\[ \delta w = \beta^{2} w, \tag{1} \]
where \(w\) is the energy of the particle, including its rest energy, and \(\beta = \frac{v}{c} \simeq 10^{-4}\).
For a nonrelativistic proton this gives about \(10\) ev per collision; for a relativistic one, correspondingly more.
If all collisions led to an increase of energy, it would increase by a factor \(e\) every \(10^{8}\) collisions. In particular, a nonrelativistic particle after \(N\) collisions will acquire an energy
\[ w = Mc^{2} e^{\beta^{2}N}. \tag{2} \]
Of course, the increase of energy will occur only if its losses proceed at a slower rate than the accumulation. Fermi succeeded in showing that losses to ionization become smaller than the increase of energy for protons with energies greater than \(200\) Mev. For very large energies, losses to ionization become negligible.
4. ON THE SPECTRUM OF PRIMARY COSMIC RADIATION
In the process of energy increase, an accelerated particle (for example, a proton) may at once lose a considerable part of its energy as a result of a nuclear collision. This process is observed as the absorption of primary cosmic-ray particles by the atmosphere, and to it there corresponds a mean free path of the order of \(70\) g/cm\(^2\), with an effective cross section
\[ \sigma = 2.5 \cdot 10^{-26}\ \text{cm}^{2} \tag{3} \]
per 1 nucleon.
In a collision of this type, the greater part of the kinetic energy probably passes into the energy of several mesons.
Let us suppose that cosmic radiation fills the approximately identical densities of the volume of the Galaxy. In this case collisions with matter of mean density \(10^{-24}\) g/cm\(^3\) will occur, giving a mean free path
\[ \Lambda = 7 \cdot 10^{25}\ \text{cm}. \tag{4} \]
at velocities of the order of \(c\). To this path there corresponds the time
\[ T=\frac{\Lambda}{c}=2\cdot 10^{15}\ \text{sec} \tag{5} \]
(i.e. about 60 million years). This is the mean age, at the present moment, of particles of primary radiation. Some of them are considerably “older.”
If it is assumed that the accelerated particles entered the interstellar fields throughout the whole time at one rate, the distribution by ages is represented by the function
\[ \frac{e^{-\frac{t}{T}}\,dt}{T}; \tag{6} \]
during its “life” \(t\) a particle accumulates energy.
If \(\tau\) denotes the time between collisions leading to scattering, then the energy of a particle of age \(t\) is equal to
\[ w(t)=Mc^2 e^{\frac{\beta^2 t}{\tau}}. \tag{7} \]
Combining this expression with the preceding one (the age-distribution function), one can obtain the probability distribution of particles with various energies: the probability that a particle will have energy between \(w\) and \(w+dw\)
\[ \pi(w)\,dw=\frac{\tau}{\beta^2 T}(Mc^2)^{\frac{\tau}{\beta^2 T}}\,dw\, \frac{1}{w^{1+\frac{\tau}{\beta^2 T}}}. \tag{8} \]
It is very interesting that the spectrum of the primary cosmic radiation, derived from Fermi’s assumptions, obeys an inverse-power law, which, as is known, is confirmed by experiment.
The theory gives a natural explanation for the absence of electrons in the primary radiation: at no energy can the accumulation of energy by an electron exceed the losses. Up to approximately \(300\) Mev, the principal share of the energy losses by electrons is ionization. At higher energies, radiation losses predominate. For protons the latter are extremely small.
5. MECHANISM OF THE ORIGIN OF PARTICLES WITH ENERGIES SUFFICIENT FOR SUBSEQUENT ACCELERATION
To maintain a constant intensity of cosmic radiation, i.e. to compensate for the absorption process, it is necessary that protons with energies not lower than \(200\) Mev continuously enter interstellar space.
However, there are direct experimental data\(^{4,13,14,15,16}\) that the primary particles include not only protons, but also comparatively heavy nuclei. The energy threshold (the minimum value for the possibility of subsequent acceleration), owing to the large ionization losses, is considerably higher for heavy nuclei than for protons. As the author of the theory points out, its principal shortcoming is the absence of an explanation of the mechanism of acceleration of nuclei of considerable mass. One of the possible suppositions is that heavy nuclei of sufficient energy may
appear near stars surrounded by very strong magnetic fields.^7
For protons Fermi proposes the following simple mechanism for the conservation of the total number of particles.
In the collision of a fast cosmic proton with a slow one, a large part of the energy will go into the formation of several mesons; however, in some cases the remaining energy of the protons will still exceed the acceleration threshold (about 200 MeV). One may introduce a multiplication coefficient \(k\) (the mean number of protons with energy above the threshold arising after the collision of a fast proton). Just as in the case of a chain nuclear reaction, if \(k>1\), the total number of cosmic rays increases; if \(k<1\), it decreases, and for \(k=1\) it remains constant. Evidently, in reality the multiplication coefficient in the interstellar medium is very close to unity. This may be a consequence of the following self-regulating process: the motion of interstellar matter, despite the small friction, is not conservative. Therefore one must assume that some source continuously replenishes the kinetic energy of the flows of matter. It is quite probable that the energy reserves are drawn in this case from the internal regions of stars; the motion of interstellar matter is in dynamic equilibrium between losses due to friction and other losses and the incoming energy.
A significant role in the energy balance must also be played by the fraction transmitted to accelerated particles of cosmic rays. Evidently, if the total energy of cosmic rays increases, then the kinetic energy of the slow flows of interstellar matter will decrease, and conversely.
The multiplication coefficient \(k\) depends on the density of matter: as the density increases, ionization losses also increase, which leads to an increase of the threshold of acceleration and a decrease of \(k\).
Fermi regards the velocity of the wandering magnetic fields as increasing proportionally to the cube root of the density; the mean free path of particles from collision to collision is inversely proportional to the cube root of the density; the rate of accumulation of particle energy is proportional to the density to the power \(2/3\).
If the multiplication coefficient is greater than unity, the total energy of cosmic rays will increase, while the kinetic energy of the currents in the galaxy will decrease, which will cause gravitational contraction, increasing the density and lowering \(k\) until equilibrium is reached, i.e. \(k=1\). If initially \(k<1\), the process will proceed to \(k=1\) in the opposite direction.
As was said above, the theory cannot give a direct explanation for the presence in primary cosmic radiation of a large number of heavy nuclei. The energy threshold for their acceleration by interstellar magnetic fields reaches \(10^9\) eV and more. One might suppose that heavy particles begin to be accelerated at the boundary of the galaxy, where the density of matter is still lower and, consequently, the energy threshold is lower.
If, however, energetic heavy particles arise as the result of another process (for example, in the strong magnetic fields of stars), then protons must arise together with them as well; the number of the latter must increase, since at sufficiently high density of matter \(k<1\). Heavy particles must have a lifetime shorter than protons, owing to the large relative losses to ionization.
The energy spectrum of heavy particles, if the theory is correct, must be quite different from that for protons, because of the greater effective absorption cross section. If the mean “age” of heavy particles is less than that of protons, then their number must decrease with increasing energy more rapidly, which can be tested experimentally.
6. ON THE “COLLISIONS” OF CHARGED FAST PARTICLES WITH MAGNETOHYDRODYNAMIC FIELDS
The trajectory of a fast proton in an interstellar magnetic field of the type described above must be very close to a spiral encircling the direction of the lines of force. The radius of such a spiral is of the order of \(10^{12}\) cm, and since the extent of the regions of field variation reaches \(10^{18}\) cm, the particle will make many revolutions before it encounters a change of field on its path. If the field increases, the pitch of the spiral gradually decreases, while the ratio
\[ \frac{\sin^2 \theta}{H} \tag{9} \]
remains constant (\(\theta\) is the angle between the vector of the magnetic-field intensity \(\mathbf H\) and the velocity of the particle). \(\theta\) will increase until \(\sin \theta\) becomes equal to unity; then the particle is “reflected” back along the same line of force.
In the case of a static magnetic field, no change in the kinetic energy of the particle occurs. However, if the region with higher \(H\) moves toward the particle, the latter will acquire, after the “collision,” a certain increment of energy. In the case where the fast particle is “reflected” from a region with higher \(H\) “catching up” from behind, the energy of the particle will decrease. Head-on collisions, however, are somewhat more probable, owing to the fact that the relative velocity in this case is greater.
An analogous process will occur also in the case when the particle flies in a spiral along the bend of a line of force. Here too, in the case of a static field, the energy of the particle will not change; however, if the place of bending of the line of force moves toward the particle, the latter’s energy will increase somewhat (there will be a “head-on collision with the bend”); in the opposite case the energy of the particle will decrease.
The amount of energy gained or lost by a particle in “collisions” of the two types described may be estimated in the following way.
If the coordinate system in which the perturbation of the field with which the particle collides occurs is at rest, the energy of the particle does not change.
The change of energy in the resting frame of reference is obtained by transforming the initial energy and momentum in the resting frame of reference to the frame of reference of the moving perturbation. In the latter an elastic collision takes place, in which, as is known, the momentum changes sign, while the energy is conserved. Performing the transformation back to the resting frame, one may obtain the final values of the energy and momentum. For a head-on collision the ratio of the energy after the collision to the initial energy is expressed as
\[ \frac{w'}{w}=\frac{1+2B\beta\cos\theta+\beta^2}{1-\beta^2}, \tag{10} \]
where \(\beta c\) is the velocity of the particle,
\(\theta\) is the angle of inclination of the spiral,
\(Bc\) is the velocity of the perturbation.
For the collision of a particle with a receding perturbation in expression (10) it is necessary to change the sign of \(B\).
7. ESTIMATE OF THE ENERGY THRESHOLD NECESSARY FOR THE FURTHER ACCELERATION OF PARTICLES WANDERING THROUGH INTERSTELLAR MAGNETIC FIELDS
Acceleration of particles cannot take place if the accumulation of energy does not exceed the loss to ionization. Since the latter is very large for protons with small velocities, only protons with energies above a certain threshold will be accelerated.
It may be assumed that during the time of acceleration the particle will be located in comparatively dense clouds of matter and in rarefied matter for intervals of time proportional to the volumes of these regions.
Therefore the ionization losses will be determined by matter of the mean density of interstellar matter, i.e. about \(10^{-24}\,\mathrm{g/sec^3}\); Fermi gives a table of losses per \(1\,\mathrm{g/cm^3}\) of substance (hydrogen) as a function of the energy of a fast proton. The same table also gives the corresponding increments of energy due to the three types of collisions discussed above. From the table it is evident that, beginning with energies of the order of \(200\,\mathrm{MeV}\), protons are accelerated.
| Proton energy in eV | Losses per \(\mathrm{g/cm^2}\), in eV | Increase of energy per \(\mathrm{g/cm^2}\), in eV |
|---|---|---|
| \(10^7\) | \(94\cdot 10^6\) | \(7.8\cdot 10^6\) |
| \(10^8\) | \(15\cdot 10^6\) | \(8.6\cdot 10^6\) |
| \(10^9\) | \(4.6\cdot 10^6\) | \(16.1\cdot 10^6\) |
| \(10^{10}\) | \(4.6\cdot 10^6\) | \(91\cdot 10^6\) |
A similar estimate for \(\alpha\)-particles gives an acceleration threshold of about \(10^9\,\mathrm{eV}\), and for an oxygen nucleus about \(40\cdot 10^9\,\mathrm{eV}\); for an iron nucleus the threshold is \(300\cdot 10^9\,\mathrm{eV}\). It is therefore unlikely that heavy nuclei of primary cosmic rays are accelerated as a result of Fermi’s process; acceleration would be possible only in regions of the galaxy where the density of matter is considerably lower.
V. S. Vavilov.
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