P H YS I CAL R EVI EW VOLUM E 75, NUMBER 8 APRIL 1S, 1949

On the Origin of the Cosmic Radiation

ENRICO FERMIInstitute for Nuclear Studies, University of Chicago, Ckicago, ILlinois

{Received January 3, 1949)

A theory of the origin of cosmic radiation is proposed according to which cosmic rays are originatedand accelerated primarily in the interstellar space of the galaxy by collisions against moving mag-metic 6elds. One of the features of the theory is that it yields naturally an inverse power law for thespectral distribution of the cosmic rays. The chief difhculty is that it fails to explain in a straight-forward way the heavy nuclei observed in the primary radiation.

I. INTRODUCTION

N recent discussions on the origin of the cosmicradiation E. Teller' has advocated the view

that cosmic rays are of solar origin and are keptrelatively near the sun by the action of magneticfields. These views are amplified by Alfvhn, Richt-myer, and Teller. ' The argument against the con-ventional view that cosmic radiation may extendat least to all the galactic space is the very largeamount of energy that should be present in form ofcosmic radiation if it were to extend to such a hugespace. Indeed, if this were the case, the mechanismof acceleration of the cosmic radiation should beextremely efficient.

I propose in the present note to discuss a hy-pothesis on the origin of cosmic rays which attemptsto meet in part this objection, and according towhich cosmic rays originate and are acceleratedprimarily in the interstellar space, although theyare assumed to be prevented by magnetic fieldsfrom leaving the boundaries of the galaxy. Themain process of acceleration is due to the interactionof cosmic particles with wandering magnetic fieldswhich, according to Alfvbn, occupy the interstellarspaces.

Such fields have a remarkably great stabilitybecause of their large dimensions (of the order ofmagnitude of light years), and of the relatively highelectrical conductivity of the interstellar space.Indeed, the conductivity is so high that one mightdescribe the magnetic lines of force as attached tothe matter and partaking in its streaming motions.On the other hand, the magnetic field itself reactson the hydrodynamics' of the interstellar mattergiving it properties which, according to Alfvkn, canpictorially be described by saying that to each lineof force one should attach a material density due tothe mass of the matter to which the line of force islinked. Developing this point of view, Alfthn isable to calculate a simple formula for the velocityV of propagation of magneto-elastic waves:

V=H/(4s p) &, (1)' Nuclear Physics Conference, Birmingham, 1948.~Alfvdn, Richtmyer, and Teller, Phys. Rev. , to be pub-

lished.I H. Alfv4n, Arkiv Mat, f. Astr. , o. Fys. 298, 2 (1943).

where H is the intensity of the magnetic field andp is the density of the interstellar matter.

One finds according to the present theory that aparticle that is projected into the interstellarmedium with energy above a certain injectionthreshold gains energy by coll'isions against themoving irregularities of the interstellar magneticfield. The rate of gain is very slow but appearscapable of building up the energy to the maximumvalues observed. Indeed one finds quite naturallyan inverse power law for the energy spectrum of theprotons. The experimentally observed exponent ofthis law appears to be well within the range of thepossibilities.

The present theory is incomplete because nosatisfactory injection mechanism is proposed exceptfor protons which apparently can be regenerated atleast in part in the collision processes of the cosmicradiation itself with the diffuse interstellar matter.The most serious difficulty is in the injectionprocess for the heavy nuclear component of theradiation. For these particles the injection energyis very high and the injection mechanism must becorrespondingly efficient.

II. THE MOTIONS OF THE INTERSTELLAR MEDIUM

It is currently assumed that the interstellar spaceof the galaxy is occupied by matter at extremelylow density, corresponding to about one atom ofhydrogen per cc, or to a density of about 10 "g/cc.The evidence indicates, however, that this matteris not uniformly spread, but that there are conden-sations where the density may be as much as tenor a hundred times as large and which extend toaverage dimensions of the order of j.o parsec.(1 parsec. =3.1)&10'8 em=3. 3 light years. ) Fromthe measurements of Adams4 on the Doppler effectof the interstellar absorption lines one knows theradial velocity with respect to the sun of a sampleof such clouds located at not too great distance fromus. The root mean square of the radial velocity,corrected for the proper motion of the sun withrespect to the neighboring stars, is about 15 km/sec.We may assume that the root-mean-square velocity

4 W. S. Adams, A.p.J. 9'7, 105 (1943).ii69

E N R I CO FERM I

is obtained by multiplying this figure by the squareroot of 3, and is therefore about 26 km/sec. Suchrelatively dense clouds occupy approximately 5percent of the interstellar space. '

Much less is known of the much more dilutematter between such clouds. For the sake ofdefiniteness in what follows, the assumption will bemade that this matter has a density of the order of10 ", or about 0.1 hydrogen atoms per cc. Evenfairly extensive variations on this figure would notvery drastically alter the qualitative conclusions.If the assumption is made that most of this materialconsists of hydrogen atoms, it is to be expected thatmost of the hydrogen will be ionized by the photo-electric effect of the stellar light. Indeed, one canestimate that some kind of dissociation equilibriumis established under average interstellar conditions,outside the relatively dense clouds, for which

n+'/no = (Ti) &,

where n+ and no are the concentrations of ions andneutral atoms per cc, and T~ is the absolute kinetictemperature in degrees K. Putting in this formulan+=0. 1, one 6nds that the fraction of undissociatedatoms is of the order of 1 percent, even assuminga rather low kinetic temperature of the order of100'K.

It is reasonable to assume that this very lowdensity medium mi11 have considerable streamingmotions, since it will be kept stirred by the movingheavier clouds passing through it. In what follows,a root-mean-square velocity of the order of 30km/ ec. will be assumed. According to Alfvhn'spicture, we must assume that the kinetic energy ofthese streams will be partially converted into mag-netic energy, that indeed, the magnetic field willbuild up to such a strength that the velocity ofpropagation of the magneto-elastic waves becomesof the same order of magnitude as the velocity ofthe streaming motions. From (1) it follows thenthat the magnetic field in the dilute matter is ofthe order of magnitude of 5 &10 6 gauss, while itsintensity is probably greater in the heavier clouds.The lines of force of this field will form a verycrooked pattern, since they will be dragged in alldirections by the streaming motions of the matterto which they are attached. They will, on the otherhand, tend to oppose motions where two portionsof the intersteIIar matter try to Row into eachother, because this mould lead to a strengtheningof the magnetic held and a considerable increase ofmagnetic energy. Indeed, this magnetic eR'ect willhave the result to minimize what otherwise wouldbe extremely large friction losses which would dampthe streaming motions and reduce them to dis-ordered thermal motions in a relatively short time.

' B. Stromgrcn, A.p.J. 108, 242 (1948).

III. ACCELERATION OF THE COSMIC RAYS

We now consider a fast particle moving amongsuch wandering magnetic 6elds. If the particle isa proton having a few Bev energy, it will spiralaround the lines of force with a radius of the orderof 10"cm until it "collides" against an irregularityin the cosmic 6eld and so is reflected, undergoingsome kind of irregular motion. On a collision botha gain or a loss of energy may take place. Gain ofenergy, however, wi11 be more probable than loss.This can be understood most easily by observingthat ultimately statistical equilibrium should beestablished between the degrees of freedom of thewandering fields and the degrees of freedom of theparticle. Equipartition evidently corresponds to anunbelievably high energy. The essential limitation,therefore, is not the ceiling of energy that can beattained, but rather the rate at which energy isacquired. A detailed discussion of this process ofacceleration will be given in Section VI. An ele-mentary estimate can be obtained by picturing the"collisions" of the particles against the magneticirregularities as if they were collisions againstreflecting obstacles of very large mass, movingwith disordered velocities averaging to V= 30km/sec. Assuming this picture, one finds easilythat the average gain in energy per collision is givenas order of magnitude by

Re =8'm,

where m represents the energy of the particleinclusive of rest energy, and 8= U/c=10 '. Thiscorresponds, therefore, for a proton to an averagegain of 10 volts per collision in the non-relativisticregion, and higher as the energy increases. Itfollows that except for losses the energy will increaseby a factor e every 10' collisions. In particular, aparticle starting with non-relativistic energy willattain, after N collisions, an energy

w = Mc' exp (B'A').

Naturally, the energy can increase only if the lossesare less than the gain in energy. An estimate to begiven later (see Section VII) indicates that theionization loss becomes smaller than the energygain for protons having energy of about 200 Mev.For higher energy the ionization loss practicallybecomes negligible. %e shall discuss later theinjection mechanism.

IV. SPECTRUM OF THE COSMIC RADIATION

During the process of acceleration a proton maylose most of its energy by a nuclear collision. Thisprocess is observed as absorption of primary cosmicradiation in the high atmosphere and occurs witha mean free path of the order of magnitude of 70

OR I 6 I N OF COSM I C RA D I ATION

w(t) =Mc' exp(B't/r) (9)

Combining this relationship between age and energywith the probability distribution of age givenpreviously, one finds the probability distribution ofthe energy. An elementary calculation shows thatthe probability for a particle to have energybetween m and m+dm is given by

s(w)dw=(7/B'T)(Mc')'s'rdw/w'+' 'r. (10)

It is gratifying to find that the theory leads natu-rally to the conclusion that the spectrum of thecosmic radiation obeys an inverse power law. Bycomparison of the exponent of this law with theone known from cosmic-ray observations, that is,about 2.9, one finds a relationship which permitsone to determine the interval of time v betweencollisions. Precisely, one finds: 2.9 = 1+r/B'T,from which follows

v = j..98'T.

Using the previous values of 8 and T, one findsv=4&10~=j..3 years. Since the particles travelwith approximately the velocity of light, thiscorresponds to a mean distance between collisions

g/cm', corresponding to a cross section of about

o =25X10 "cm' (5)per nucleon.

In a collision of this type most of the kineticenergy of the colliding nucleons is probably con-verted into energy of a spray of several mesons.

It is reasonable to assume that the cosmic rayswill occupy with approximately equal density allthe interstellar space of the galaxy. They will beexposed, therefore, to the collisions with matter ofan average density of 10 '4, leading to an absorptionmean free path

4=7&10"cm. (6)

A particle traveling with the velocity of light willtraverse this distance in a time

T=ti/c=2X10" sec.

or about 60 million years.The cosmic-ray particles now present will there-

fore, in the average, have this age. Some of themwill have accidentally escaped destruction and beconsiderably older. Indeed, the absorption processcan be considered to proceed according to anexponential law. If we assume that original particlesat all times have been supplied at the same rate,we expect the age distribution now to be

exp( —t/T) dt/T. (8)

During its age t, the particle has been gainingenergy. If we call v. the time between scatteringcollisions, the energy acquired by a particle of aget will be

of the order of a light year, or about 10"cm. Sucha collision mean free path seems to be quitereasonable.

The theory explains quite naturally why noelectrons are found in the primary cosmic radiation.This is due to the fact that at all energies the rateof loss of energy by an electron exceeds the gain.At low energies, up to about 300 Mev, the loss ismainly due to ionization. Above this energy radi-ative losses due to the acceleration of the electronsin the interstellar magnetic field play the dominantrole. This last energy loss is instead quite negligiblefor protons. Also, the inverse Compton e8ectdiscussed by Feenberg and Primako6' will con-tribute to eliminate high energy electrons.

V. THE IN JECTION MECHANISM. DIFFICULTIESVfITH THE INJECTION OF HEAVY NUCLEI

In order to complete the present theory, theinjection mechanism should be discussed.

In order to keep the cosmic radiation at thepresent level it is necessary to inject a number ofprotons of at least 200 Mev, to compensate forthose that are lost by the absorption process.According to recent evidence, ' the primary cosmicradiation contains not only protons but also somerelatively heavy nuclei. Their injection energy ismuch higher than that of protons, primarily onaccount of their large ionization loss. (See furtherSection VII.) Such high energy protons and heaviernuclei conceivably could be produced in the vicinityof some magnetically very active star. ' To statethis, however, merely means to shift the difficultyfrom the problem of accelerating the particles tothat of injecting them unless a more precise esti-mate can be given for the efficiency of this or ofsome equivalent mechanism. With respect to theinjection of heavy nuclei I do not know a plausibleanswer to this point.

For the production of protons, however, onemight consider also a simple mechanism which, ifthe present theory is at all correct in its generalfeatures, should be responsible for at least a largefraction of the total number of protons injected.According to this mechanism the cosmic radiationregenerates itself as follows. When a fast cosmic-rayproton collides in the interstellar space against aproton nearly at rest, a good share of the energywi11 be lost in the form of a spray of mesons, andtwo nucleons wi11 be left over with energy muchless than that of the original cosmic ray. Estimatesindicate that in some cases both particles may havean energy left over above the injection threshold of

6 E. Feenberg and H. PrimakoE, Phys. Rev. 73, 449 (1948).'Freier, Lofgren, Ney, and Oppenheimer, Phys. Rev. 74,

1818 (1948); H. L. Bradt and B. Peters, Phys. Rev. 74, 1828(1948).

8See for examp1e W. F. G. Swann, Phys. Rev. 43, 217(1933)and Horace %. Babcock, Phys. Rev. 74, 489 (1948).

FERM I

200 Mev, in some cases one and in some cases none.We can introduce a reproduction factor k, definedas the average number of new protons above theinjection energy arising in a collision of an originalcosmic-ray particle. As in a chain reaction, if k isgreater than one the over-all number of cosmic rayswill increase; if k is less than one it will decrease;if k is equal to one it will stay level.

Apparently the reproduction factor under inter-stellar conditions is rather close to one. This isperhaps not a chance, but may be due in part tothe following self-stabilizing mechanism. The mo-tions of the interstellar matter are not quite con-servative, in spite of the reduced friction, caused bythe magnetic 6elds. One should assume, therefore,that some source is present which steadily deliverskinetic energy into the streaming motions of theinterstellar matter. Probably such a source ofenergy ultimately involves conversion of energyfrom the large supplies in the interior of the stars.The motions of the interstellar medium are in adynamic equilibrium between the energy deliveredby this source and the energy losses caused byfriction and other causes. In this balance theamount of energy transferred by the interstellarmedium to cosmic radiation is by no means ir-relevant, since the total cosmic ray energy iscomparable to the kinetic energy of the streaming,irregular motions of the galaxy. One should expect,therefore, that if the general level of the cosmicradiation should increase, the kinetic energy of theinterstellar motion would decrease, and vice versa.The reproduction factor depends upon the density.As the density increases, the ionization losses willincrease proportionally to it. This tends to increasethe injection energy and consequently to decreasethe reproduction factor. On the other hand, also,the rate of energy gained will change by an amountwhich is hard to de6ne unambiguously. One mightperhaps assume, however, that the velocity of thewandering magnetic 6elds increases with thepower of the density, as would correspond to thevirial theorem, and that the collision mean freepath is inversely proportional to the -', power of thedensity, as one might get from geometrical simili-tude. One would 6nd that the rate of energy increaseis proportional only to the -', power of the density.The net effect is an increase of the injection energyand a decrease of the reproduction factor withincreasing density. If the reproduction factor hadbeen initially somewhat larger than one, the generallevel of the cosmic radiation would increase, drain-ing energy out of the kinetic energy of the galaxy.This would determine a gravitational. contractionwhich would increase the density and decrease kuntil the stable value of one is reached. Theopposite would take place if k initially had beenconsiderably less than one.

But even if this stabilizing mechanism is notadequate to keep the reproduction factor at thevalue one, and therefore an appreciable change inthe general level of the cosmic radiation occursover periods of hundreds of millions of years, thegeneral conclusions reached in Section IV wouldnot be qualitatively changed. Indeed, if k weresomewhat different from one, the general level ofthe cosmic radiation would increase or decreaseexponentially, depending on whether k is largerthan or less than one. Consequently the number ofcosmic particles injected according to the mecha-nism that has been discussed will not be constantin time but will vary exponentially. Combiningthis exponential variation with the exponentialabsorption (8), one still finds an exponential lawfor the age distribution of the cosmic particles atthe present time, the only difference being that theperiod of this exponential will be changed by asmall numerical factor.

The injection mechanism here proposed appearsto be quite straightforward for protons, but utterl&inadequate to explain the abundance of the heavy.nuclei in the primary cosmic radiation. The injec-tion energy of these particles is of several Bev, andit is dif6cult to imagine a secondary effect of thecosmic radiation on the diffuse interstellar matterwhich might produce this type of secondary withany appreciable probability. One might perhapsassume that the heavy particles originate at thefringes of the galaxy where the density is probablylower and the injection energy is therefore probablysmaller. This, however, would require extremeconditions of density which are not easily justihable.It seems more probable that heavy particles areinjected by a totally different mechanism, perhapsas a consequence of the stellar magnetism.

If such a mechanism exists one mould naturallyexpect that it would inject protons together withheavier nuclei. The protons and perhaps to asomewhat lesser extent the n-particles would befurther increased in numbers by the "chain reac-tion" which in this case should have k &1. Indeedtheir number would be equal to the number injectedduring the lifetime T increased by the factor1/(1 —k). Heavy particles instead would slowlygain or slowly loose energy according to whethertheir initial encl;gy is above or below the injectionthreshold. They would, however, have a shorterlifetime than protons because of the presumablylarger destruction cross section. Their numbershould be approximately equal to the numberinjected during their lifetime.

One should remark in this connection that aconsequence of the present theory is that the energyspectrum of the heavy nuclei of the cosmic radiationshould be quite different from the spectrum of theprotons, since the absorption cross section for a

ORIGIN OF COSMIC RADIATION j.i 73

sin%/II =constant, (12)

where 8 is the angle between the direction of theline of force and the direction of the velocity of theparticle, and H is the local field intensity. As theparticle approaches a region where the 6eld intensityis larger, one will expect, therefore, that the angle8 increases until sin8 attains the maximum possiblevalue of one. At this point the particle is rejectedback along the same line of force and spirals back-wards until the next region of high field intensity isencountered. This process will be called a "Type A"reAection. If the magnetic field were static, such areAection would not produce any change in thekinetic energy of the particle. This is not so,however, if the magnetic field is slowly variable.It may happen that a region of high field intensitymoves toward the cosmic-ray particle which collidesagainst it. In this case, the particle will gain energyin the collision. Conversely, it may happen thatthe region of high field intensity moves away fromthe particle. Since the particle is much faster, itwill overtake the irregularity of the 6eld and bereAected backwards, in this case with loss of energy.The net result will be an average gain, primarily forthe reason that head-on collisions are more frequentthan overtaking collisions because the relativevelocity is larger in the former case.

Somewhat similar processes take place when thecosmic-ray particle spirals around a curve of theline of force as outlined in Fig. 1 ("Type 8"

heavy particle is presumably several times largerthan that of a proton. One would expect, therefore,that the average age of a heavy particle is shorterthan the age of a proton, which leads to an energyspectrum decreasing much more rapidly with energyfor a heavy particle than it does for protons. Anexperimental check on this point should be possible.

VI. FURTHER DISCUSSION OF THEMAGNETIC ACCELERATION

In this section the process of acceleration of thecosmic-ray protons by collision against irregularitiesof the magnetic field will be discussed in somewhatmore detail than has been done in Section III.

The path of a fast proton in an irregular magneticfield of the type that we have assumed wi11 berepresented very closely by a spiraling motionaround a line of force. Since the radius of thisspiral may be of the order of 10" cm, and theirregularities in the field have dimensions of theorder of 10"cm, the cosmic ray will perform manyturns on its spiraling path before encountering anappreciably different 6eld intensity. One finds byan elementary discussion that as the particleapproaches a region where the 6eM intensityincreases, the pitch of the spiral will decrease. Onefinds precisely that

FCear. um

=b

FiG. 1. Type 8 reHection of a cosmic-ray particle.

reflection). Here again, the energy of the particlewould not change if the magnetic held were static.On the other hand, the lines of force partake ofthe streaming motions of the matter, and it mayhappen that the line of force at the bottom of thecurve moves in the direction indicated by thearrow a, or that it moves in the direction indicatedby the arrow b. In the former case there will bean energy gain (head-on collision) while in case b

(overtaking collision) there will be an energy loss.Gain and loss, however, do not average out com-pletely, because also in this case a head-on collisionis slightly more probable than an overtaking colli-sion due to the greater relative velocity.

The amount of energy gained or lost in a collisionof the two types described can be estimated with asimple argument of special relativity, without anyreference to the detailed mechanism of the collision.In the frame of reference in which the perturbationof the 6eld against which the collision takes placeis at rest, there is no change of energy of theparticle. The change of energy in the rest frame ofreference is obtained, therefore, by 6rst transform-ing initial energy and momentum from the restframe to the frame of the moving perturbation. Inthis frame an elastic collision takes place wherebythe momentum changes direction and the energyremains unchanged. Transforming back to theframe of reference at rest, one obtains the finalvalues of energy and momentum. This procedureapplied to a head-on collision, gives the followingresult,

1+2BP cos6+8'K' 1 —B'

where pc is the velocity of the particle, 0 is theangle of inclination of the spiral, and Bc is thevelocity of the perturbation. It is assumed that thecollision is such as to produce a complete reversalof the spiraling direction by either of the twomechanisms outlined previously. For an overtakingcollision, one finds a similar formula except thatthe sign of 8 must be changed. We now averagethe results of head-on and overtaking collisions,taking into account that the probabilities of thesetwo types of events are proportional to the relativevelocities and are given therefore by (P cos8+8/2P cosO) for a head-on collision and (P cos8 8/—2P cos8) for an overtaking collision, The result for

E N R I CO FERM I

TABLE I. Energy loss per g/em~ of material traversed.

Energy

10' ev10'109101,0

Loss/g/cm~

94 X10' ev15 Xio'4.6X10'4 6X 106

Gain/g/cm~

7.8 X10' ev8.6X106

16.1X10691 X10'

VII. ESTIMATE OF THE INJECTION ENERGY

Acceleration of a cosmic-ray particle will not bepossible unless the energy gain is greater than theionization loss. Since this last is very large forprotons of low velocity, only protons above acertain energy threshold will be accelerated. In

the average of ln(w'/w) up to terms of the order of8 is'

(In(w'/w))A, ——4B' —28'P' cos'8 (14)

which confirms the order of magnitude for theaverage gain of energy adopted in Section III.

As a result of the extreme complication of themagnetic field and of its motion, it does not appearpractical to attempt an estimate by more than theorder of magnitude.

One might expect that after a relatively shorttime the angle 0 will be reduced to a fairly lowvalue so that Type A reflections will becomeinfrequent. This is due to the fact that when 8 islarge, fairly large increases in energy and decreasesof 8 may occur, if the particle should be caughtbetween two regions of high field moving againsteach other along a force line. One can prove thatType 8 reRections change gradually and ratherslowly the average pitch of the spiral. It appears,therefore, that except for the beginning of theacceleration processes the Type A will not give aslarge a contribution as one otherwise might expect.

Section III a value of 200 Mev for this "injectionenergy" has been given; a justihcation for thisassumed value wil. l be given now. In estimatingthe injection energy we will assume that theparticle, during its acceleration, 6nds itself bothinside relatively dense clouds and in the moredilute material outside of the douds for lengths oftime proportional to the volumes of these tworegions. The ionization loss will be due, therefore,to a material of an average density equal to theaverage density of the interstellar matter, whichhas been assumed to be 10 " g/cm', consistingmostly of hydrogen. In Table I the energy loss perg/cm' of material traversed is given a,s a functionof the energy of the proton. In the third column ofthe table the corresponding energy gain is given.It is seen that the loss exceeds the gain for particlesof energy less than about 200 Mev, as has beenstated.

A similar estimate yields for the acceleration ofn-particles an injection energy of about 1 Bev, forthe acceleration of oxygen nuclei the initial energyrequired is about 20 Bev, and for an iron nucleusit would amount to about 300 Bev. As alreadystated, it does not appear probable that the heavynuclei found in the cosmic radiation are acceleratedby the process here described, unless they shouldoriginate at some place in the galaxy where theinterstellar material is extremely dilute.

I would like to acknowledge the help that I hadfrom several discussions with E. Teller on therelative merits of the two opposing views that weare presenting. I learned many facts on cosmicmagnetism from a discussion with H. Alfvbn, onthe occasion of his recent visit to Chicago. Theviews that he expressed then were quite materialin inAuencing my own ideas on the subject.