An hypothesis on the origin of the cosmic radiation
SUPPLEMENTO
AL
VOLUME
VI,
~ERIE
IX
D~EL
NUOVO
CIMENTO
N.
3, 1949
An Hypothesis on the Origin of the Cosmic Radiation. E. FERMI
Chicago
Cosmic rays are more a n d more being recognised as a p h e n o m e n o n of cosmic i m p o r t a n c e . As an i n t r o d u c t i o n I would like to give a few figures t h a t stress this i m p o r t a n c e . W e k n o w the i n t e n s i t y of the cosmic r a d i a t i o n t h a t comes from the outside into the atmosphere. The n u m b e r of particles with an energy of t h e order of or g r e a t e r t h a n four billions of electron volt is a b o u t 0.] particles per square c e n t i m e t r e per steradian, l~rom this figure we can e s t i m a t e the e n e r g y present per cm 3 in t h e form of cosmic r a y s of over 4 GeV. One finds 6 . 1 0 -13 e r g / e m L Very p r o b a b l y , particles of lower energy are also present a n d m a y be cut off b y m a g n e t i c field action, p e r h a p s b y the m a g n e t i c field of t h e sun. By a r a t h e r u n c e r t a i n estimate one m a y be led to h~crease t h e previous figure b y a f a c t o r 3. The cosmic r a y s r e p r e s e n t therefore an energy d e n s i t y of 2 2 . 1 0 -1~ erg/cm 3. This energy should be c o m p a r e d with other a s t r o n o m i c a l or cosmic energies. ]f one assumes t h a t r a d i a t i o n with this average d e n s i t y occupies all t h e i n t e r s t e l l a r space of the g a l a x y , one obtains t h e result t h a t the overall e n e r g y of t h e cosmic r a d i a t i o n is of the same order of m a g n i t u d e as the k i n e t i c energy of the disordered motions of the stars. The a m o u n t of e n e r g y is so large t h a t one m i g h t l e g i t i m a t e l y d o u b t w h e t h e r or not it is possible to find a m e c h a n i s m c a p a b l e of p r o d u c i n g cosmic r a d i a t i o n in such a staggering a m o u n t . p F o r this reason TELLER has r e c e n t l y proposed a (, Non-Cosmic T h e o r y ~) of the cosmic r a d i a t i o n b y assuming t h a t the cosmic r a d i a t i o n instead of e x t e n d i n g to t h e i n t e r s t e l l a r space is confined to the i m m e d i a t e v i c i n i t y of the sun. This h y p o t h e s i s was l a t e r developed b y TELLER, ]~ICHT~IYER a n d A L P V ~ . I will not discuss it now because I believe t h a t Prof. A L F V ~ will do so next, a n d I will not even discuss the h y p o t h e s i s considered in ]~AGGE~S r e p o r t concerning a possible stellar origin of t h e cosmic r a d i a t i o n . I would like i n s t e a d to discuss a different possibility a c c o r d i n g to which cosmic r a y s acquire m o s t of their energy while t r a v e l l i n g t h r o u g h space. I w a n t to assume: first, t h a t t h e cosmic r a d i a t i o n is a galactic p h e n o m e n o n ,
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w h e r e b y I m e a n t h a t t h e c o s m i c r a d i a t i o n fills w i t h m o r e or less u n i f o r m e n e r g y d i s t r i b u t i o n all t h e s p a c e of o u r g a l a x y . This a s s u m p t i o n r e q u i r e s a m e c h a n i s m c a p a b l e of h o l d i n g t h e c o s m i c r a y p a r t i c l e s w i t h i n t h e g a l a x y . I t h a s b e e n o f t e n a s s u m e d t h a t this m a y be d u e to a g a l a c t i c m a g n e t i c field w i t h closed in lines of force. Before m a k i n g f u r t h e r assump.tions, I w o u l d like to invest i g a t e w h a t one c a n d e d u c e f r o m t h i s h y p o t h e s i s . O u r g a l a x y c o m p r i s e s s t a r s a n d m a t t e r . T h e diffuse m a t t e r h a s a n a v e r a g e d e n s i t y of a b o u t 10 -24 g / c m 3, ~ figure e a s y t o r e m e m b e r b e c a u s e i t c o r r e s p o n d s a p p r o x i m a t e l y to one h y d r o g e n a t o m p e r c m 3. A s i m p l e c a l c u l a t i o n shows t h a t t h e p r o b a b i l i t y of collision of a cosmic r a y a g a i n s t a s t a r is e x t r e m e l y small. H o w e v e r , t h e p r o b a b i l i t y t h a t t h e cosmic r a y p a r t i c l e m a y h a v e a n u c l e a r collision is n o t a t all negligible. I n d e e d , w e c a n m a k e a c r u d e estim a t e of t h i s p r o b a b i l i t y as follows: W e k n o w d i r e c t l y f r o m c o s m i c r a y e x p e r i m e n t s t h a t w h e n cosmic r a y p a r t i c l e s e n t e r f r o m t h e o u t s i d e i n t o t h e e a r t h ' s a t m o s p h e r e t h e y soon collide w i t h air nuclei. T h e m e a n free p a t h f o r t h i s collision is of t h e o r d e r of m a g n i t u d e of one h u n d r e d g r a m s p e r em:. Since t h e d e n s i t y is 10 -24, t h e c o r r e s p o n d i n g m e a n free p a t h for a c o s m i c r a y p a r t i c l e t r a v e l l i n g t h r o u g h t h e g a l a x y will b e a b o u t 102~ cm. Since t h e p a r t i c l e t r a v e l s w i t h a l m o s t t h e v e l o c i t y of light, t h e t i m e t a k e n f o r t r a v e r s i n g t h i s d i s t a n c e will b e 102~/3 • 101° =- 3 • 10 ~5 sec =- l 0 s y e a r s . I n t h e f o l l o w i n g c a l c u l a t i o n s I h a v e u s e d s l i g h t l y different figures y i e l d i n g : (1)
T = 7 • 107 y e a r s ,
for t h e a v e r a g e t i m e t h a t a cosmic r a y p a r t i c l e t r a v e l s b e f o r e a n u c l e a r collision h a p p e n s t h a t effectively d e s t r o y s it. This t i m e is r a t h e r s h o r t c o m p a r e d to t h e a g e of t h e u n i v e r s e e s t i m a t e d to b e t w o o r t h r e e b i l l i o n y e a r s : W e a r e t h e r e f o r e l e d to t h e c o n c l u s i o n t h a t o n l y v e r y few of t h e c o s m i c r a y p a r t i c l e s t h a t we n o w o b s e r v e c a n b e as o l d as t h e g a l a x y . I t s e e m s n e c e s s a r y , t h e r e f o r e , to a s s u m e t h e e x i s t e n c e of a mechanism that continuously produces new cosmic ray particles. W i t h o u t discussing y e t w h a t t h i s m e c h a n i s m m a y be, w e w a n t to i n t r o d u c e as a s e c o n d a s s u m p t i o n t h a t t h e p r o d u c t i o n is u n i f o r m in t i m e . Since a p a r t i c l e h a s a m e a n life of 70 m i l l i o n y e a r s , its p r o b a b i l i t y of s u r v i v a l a f t e r a t i m e , t, will b e (2)
exp (--
t/T).
This e x p r e s s i o n gives t h e age d i s t r i b u t i o n l a w of t h e p a r t i c l e s t h a t a r e n o w in e x i s t e n c e . O n e c a n n o w m a k e t w o a l t e r n a t e a s s u m p t i o n s : one is t h a t t h e c o s m i c r~d i c t i o n p a r t i c l e s a r e o r i g i n a l l y p r o d u c e d w i t h a n e n e r g y e q u a l to or h i g h e r t h a n t h e i r p r e s e n t e n e r g y . T h e o t h e r one is t h a t t h e c o s m i c r a y p a r t i c l e s a r e o r i g i n a l l y p r o d u c e d a t a r e l a t i v e l y low e n e r g y a n d a r e g r a d u a l l y a c c e l e r a t e d .
AN
tIYPOTHESIS
ON
THF.
ORIGIN
OF
THE
COSMIC
RADIATIO~ ~
31O
I n what follows we shall take this second point of view which has the advantage to require an injection mechanism less powerful than the one t h a t would be required for the first assumption. We assume, therefore, the existence of an accelerating process whereby vhe energy of a particle gradually increases as its age increases and is a function of the age. The dependence of the energy upon the age m a y then be determined from the knowledge of the energy distribution of the cosmic radiation. The experimentally known energy distribution of cosmic ray particles is rather complicated at low energies but takes the form of a simple power law for energies above a few GeV. We will assume this simplified law:
(3)
I ( E ) d E == k E -~.9 d E .
the exponent 2.9 is chosen to fit the observations. We have assumed t h a t the energy of a particle is a function of its age; t (4)
E =
](t).
F r o m t.he knowledge of the age distribution (2), and the energy distribution (3), one can determine ](t). Indeed, t h e n u m b e r of particles with age between t and t + dt is proportional t0 exp [--. t i T ! d r , and the n u m b e r of particles with energy between E and E + d E is proportional to
dE d/ E~.~ - - p . 9
:
we find, therefore, (5)
d/
/2.9 -- a exp [ - - t / T ! ,
where a is a proportional constant. (6)
I n t e g r a t i o n yields
1.9/1-9 -- T exp [-- t/:T] ,
where the integration constant has been set to equal 0 because for large t ] becomes infinite. This equation can be rewritten in the form: (7)
f(t) := Eo exp [ t / ( 1 . 9 T ) ] ,
~-here E , is a new constant t h a t represents the initial energy of the particle. F r o m our assumptions follows a very specific law (7) whereby the energy of the cosmic r a y particles must increase with time. According to (7), the energy must increase every year by a fraction of about 10 -8 of its value, so for a proton with energy equal to its rest energy, the energy will increase at the
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r a t e of a b o u t only 10 eV per y e a r a n d will increase c o r r e s p o n d i n g l y faster for p r o t o n s of higher energy. I t is clear, however, t h a t in a n y case t h e r a t e of increase of t h e energy will be quite slow since it takes a b o u t 100 million years to double the initial v a l u e of t h e energy. A v e r y simple process t h a t leads to the acceleration law (7), is due to t h e collision a g a i n s t large m o v i n g objects. W i t h o u t specifying y e t w h a t p a r t i c u l a r objects will be considered as likely obstacles against which the collisions t a k e place, we w a n t to assume t h a t a cosmic r a y f r e q u e n t l y collides against large m o v i n g obstacles. T h a t the energy of t h e cosmic r a y will on the a v e r a g e increase in such collisions is clear from t h e f a c t t h a t u l t i m a t e l y s t a t i s t i c a l e q u i l i b r i u m would be established with e q u i p a r t i t i o n of energy between the obstacles a n d the cosmic r a y p~rticles. This corresponds, of course, to an e x t r e m e l y high energy v e r y m a n y orders of m a g n i t u d e b e y o n d the m a x i m u m energy o b s e r v e d in cosmic rays. g~hat limits the efficiency of this process in increasing the energy of the cosmic r a y particles is, therefore, not the m a x i m u m energy a t t a i n a b l e which is effectively infinite, b u t r a t h e r the r a t e at which the energy is t r a n s m i t t e d . N o t all collisions will accelerate t h e particle. A c t u a l l y , head-on collisions will p r o d u c e an acceleration a n d o v e r - t a k i n g collisions will produce a deceleration. On the average, there is acceleration p r i m a r i l y because h e a d - o n collisions are somewhat more p r o b a b l e t h a n o v e r - t a k i n g collisions since t h e r e l a t i v e velocity is larger in the former case. One can c o m p u t e in an e l e m e n t a r y w a y the order of m a g n i t u d e of the ~verage increase, 3E, per collision of a particle of energy E (including rest energy) colliding a g a i n s t objects m o v i n g w i t h velocity, V. The result is: (8)
3E
~
EV2/c
~ .
If we assume t h a t the collision cross-section is i n d e p e n d e n t of the energy, t h e n u m b e r of collisions p e r u n i t time will also be a p p r o x i m a t e l y i n d e p e n d e n t of t h e energy since the v e l o c i t y of the cosmic r a y particles is p r a c t i c a l l y c o n s t a n t a n d equal to c. I t follows f r o m (8) t h a t the gain in energy p e r u n i t time is p r o p o r t i o n a l to t h e energy. The energy therefore increases according to an e x p o n e n t i a l law. W e shall t a k e V of the order of 30 km/sec. This g~ves (8) ~E ~ 1 0 - s E . T h a t is again an average gain of energy per collision of one p a r t in 10 s. A f t e r N collisions the energy will b e : (9)
E - - Eo exp [NB 2] ,
where B ~ --
V2
C2
__
lO-S .
AN
kI~-POTHESIS
ON
TItE
ORIGIN
OI:
Tile
COSMIC
RADIATION
321
I n order to estimate the number of collisions, we introduce a scattering mean free path ;~. The number of collisions after a time, t, since the initiation of the process will be ~r =: ct/;% since the particle travels practically at the velocity of light. Consequently we g e t : (10)
E = Eo exp [ctB~/'A].
Comparing (10) with (7) one obtains (11)
)~ ~
1.9B2eT
~-~1018 cm :
I light y e a r .
We will now further specialize our assumptions and introduce the hypothesis t h a t the collisions responsible for the increase in energy are ~g~inst moving irregularities in a cosmic m~gnetic field. The idea o~ the existence of such ~ moving magneti~ field is due to Dr. A L F V ~ who has made a thorough magneto h y d r o - d y n a m i c study of the influence that the extremely tenuous interstellar matter has on the propagation of a magnetic field t h a t penetrates it. Unfortunately, I do not have the time to explain in detail his v e r y stimulating ideas on this subject. I shall only mention that dne to the relatively high electrical conductivity of vhe interstellar medium, the lines of force ~re practically ~ttached to the ma~tter so t h a t they arc dragged b y the turbulent motion of the interstellar matter. A cosmic ray p~rticle will be deflected b y the action of the magnetic field and will gain energy in the process as previously discussed. I n order to obtain agreement with the experimentally observed spectrum, we must assmne that the size of the m i n i m u m vortices which drag the lines of force is of the order of one light year, a value which does not appear implausible. ~ o t h i n g has been said so far of the injection mechanism of the particles of relatively low energy which will be further accelerated b y the proposed me~hod. I n order t h a t a particle so injected m a y eventually become an energetic cosmic ray, it is necessary that its initial energy be above ~ certain limit which will be c~lled the injection threshold. Indeed, the accelerating mechanism will function only when the energy gained b y the accelerating mechanism is greater that the energy lost b y ionization. An estimate of the injection threshold yields for w.rious particles the following values: Protons Particles Oxygen nuclei Iron nuclei
- 100 MeV 1 GeV - . 1 GeV per nucleon 5 GeV per nucleon
I t is seen that the injection threshold is quite large for h e a v y nuclei and
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this r e p r e s e n t s t h e m o s t serious difficulty for t h e p r o p o s e d t h e o r y . Even w i t h o u t special a s s u m p t i o n as to t h e origin of t h e i n i t i a l p r o t o n s a n d u p a r ticles t h e i n j e c t i o n of t h e s e l i g h t c o m p o n e n t s of t h e cosmic r a d i a t i o n m a y be u n d e r s t o o d p e r h a p s as due to t h e collisions of t h e cosmic r a d i a t i o n itself w i t h t h e nuclei of t h e i n t e r s t e l l a r m a t t e r . F r o m t h i s p o i n t of view, cosmic r a d i a t i o n w o u l d be a self r e g e n e r a t i n g process. N o such s i m p l e e x p l a n a t i o n , h o w e v e r , seems a d e q u a t e for t h e i n j e c t i o n of h e a v i e r nuclei like~ for i n s t a n c e , o x y g e n or iron, since no k n o w n m e c h a n i s m could y i e l d a n i r o n nucleus w i t h 5 G e V p e r n u c l e o n w i t h o u t d e s t r o y i n g t h e nucleus itself. One m u s t a s s u m e , therefore, a special i n j e c t i o n m e c h a n i s m for t h e s e p a r t i c l e s , p e r h a p s like t h e one s u g g e s t e d r e c e n t l y b y SPITZER [1]. I n conolusion, t h e p r o p o s e d t h e o r y seems to be q u i t e a d e q u a t e f o r u n d e r s t a n d i n g t h e m~in f e a t u r e s of t h e p r o t o n c o m p o n e n t s of t h e cosmic r a d i a t i o n ~nd p e r h a p s also of t h e g p a r t i c l e c o m p o n e n t . I t does n o t s e e m a d e q u a t e to u n d e r s t a n d t h e presence in t h e cosmic r a d i a t i o n of ~ significant f r a c t i o n of h e a v i e r nuclei. I f t h e g e n e r a l f e a t u r e s of t h e p r e s e n t t h e o r y s h o u l d p r o v e correct, t h e r e s h o u l d be a n i n d e p e n d e n t u n d v e r y p o w e r f n l i n j e c t i o n m e c h a n i s m , of t h e h e a v y c o m p o n e n t of t h e cosmic r a d i a t i o n .
BIBLIOGRAPHY [1] LYMAN SPITZEI~, Jr.: Phys, Rev., 76, 583, (1949).
DISCUSSIONE E OSSERVAZIONI - - E. BACGE, Hamburg: Bei dem yon FERMI diskutiorten Mechanismus ftir die Beschleunigung und Aufsammlung der H6henstrahlungsteilchen innerhalb der MilchstraBo scheint es wichtig zu sein, die Fragc zu diskutieren, welcher Toil kosmischer Stralflung an der Oborfl~che des gMaktischen Systems in den Vqeltraum hinauswandert. Da das yon FERMI postulierte galaktische Magnetfeld gro2e riiumliche Schwankungen besitzen mu~, wenn der Beschleunigungsprozel~ iiberhaupt wirksam werden soll, wird es untor anderem auch nahezu feldfreie Beroiche geben und dies kann besondors an don Randbereichen der Milehstrai3e das Entweichon dor HShenstrahlungsteilcben ermSglichen. E. FERMI, Chicago: In order to avoid a large loss of particles out of the boundaries of the galaxy it is sufficient to assume that the lines of force are closed or at least that very few of them escape to the outer space. -
-
DISCUSSION]~
E
OSS]~RVAZIONI
L. JANOSSY, Dublin: T h e i n j e c t i o n process of p r o t o n s is h e l p e d b y t h e c i r c u m s t a n c e colliding w i t h a nucleus is likely to r e t a i n a n a p p r e c i a b l e f r a c t i o n of t h u s r e m a i n v e r y m u c h a b o v e t h e i n j e c t i o n energy. T h e q u e s t i o n is h e a v y f r a g m e n t s arising o u t of n u c l e a r collisions c a n serve to i n j e c t -
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that a nucleon its e n e r g y a n d raised whether h e a v i e r nuclei.
E. FERMI, Chicago: X a t u r a l l y I hope t h a t y o u are right. On t h e o t h e r h a n d it seems to me v e r y difficult to u n d e r s t a n d t h e fact f r o m t h e t h e o r e t i c a l p o i n t of view. But, of course, if t h e fact s h o u l d p r o v e t o be t r u e it will h a v e some t h e o r e t i c a l e x p l a n a t i o n . -
-
AL
VOLUME
VI,
~ERIE
IX
D~EL
NUOVO
CIMENTO
N.
3, 1949
An Hypothesis on the Origin of the Cosmic Radiation. E. FERMI
Chicago
Cosmic rays are more a n d more being recognised as a p h e n o m e n o n of cosmic i m p o r t a n c e . As an i n t r o d u c t i o n I would like to give a few figures t h a t stress this i m p o r t a n c e . W e k n o w the i n t e n s i t y of the cosmic r a d i a t i o n t h a t comes from the outside into the atmosphere. The n u m b e r of particles with an energy of t h e order of or g r e a t e r t h a n four billions of electron volt is a b o u t 0.] particles per square c e n t i m e t r e per steradian, l~rom this figure we can e s t i m a t e the e n e r g y present per cm 3 in t h e form of cosmic r a y s of over 4 GeV. One finds 6 . 1 0 -13 e r g / e m L Very p r o b a b l y , particles of lower energy are also present a n d m a y be cut off b y m a g n e t i c field action, p e r h a p s b y the m a g n e t i c field of t h e sun. By a r a t h e r u n c e r t a i n estimate one m a y be led to h~crease t h e previous figure b y a f a c t o r 3. The cosmic r a y s r e p r e s e n t therefore an energy d e n s i t y of 2 2 . 1 0 -1~ erg/cm 3. This energy should be c o m p a r e d with other a s t r o n o m i c a l or cosmic energies. ]f one assumes t h a t r a d i a t i o n with this average d e n s i t y occupies all t h e i n t e r s t e l l a r space of the g a l a x y , one obtains t h e result t h a t the overall e n e r g y of t h e cosmic r a d i a t i o n is of the same order of m a g n i t u d e as the k i n e t i c energy of the disordered motions of the stars. The a m o u n t of e n e r g y is so large t h a t one m i g h t l e g i t i m a t e l y d o u b t w h e t h e r or not it is possible to find a m e c h a n i s m c a p a b l e of p r o d u c i n g cosmic r a d i a t i o n in such a staggering a m o u n t . p F o r this reason TELLER has r e c e n t l y proposed a (, Non-Cosmic T h e o r y ~) of the cosmic r a d i a t i o n b y assuming t h a t the cosmic r a d i a t i o n instead of e x t e n d i n g to t h e i n t e r s t e l l a r space is confined to the i m m e d i a t e v i c i n i t y of the sun. This h y p o t h e s i s was l a t e r developed b y TELLER, ]~ICHT~IYER a n d A L P V ~ . I will not discuss it now because I believe t h a t Prof. A L F V ~ will do so next, a n d I will not even discuss the h y p o t h e s i s considered in ]~AGGE~S r e p o r t concerning a possible stellar origin of t h e cosmic r a d i a t i o n . I would like i n s t e a d to discuss a different possibility a c c o r d i n g to which cosmic r a y s acquire m o s t of their energy while t r a v e l l i n g t h r o u g h space. I w a n t to assume: first, t h a t t h e cosmic r a d i a t i o n is a galactic p h e n o m e n o n ,
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w h e r e b y I m e a n t h a t t h e c o s m i c r a d i a t i o n fills w i t h m o r e or less u n i f o r m e n e r g y d i s t r i b u t i o n all t h e s p a c e of o u r g a l a x y . This a s s u m p t i o n r e q u i r e s a m e c h a n i s m c a p a b l e of h o l d i n g t h e c o s m i c r a y p a r t i c l e s w i t h i n t h e g a l a x y . I t h a s b e e n o f t e n a s s u m e d t h a t this m a y be d u e to a g a l a c t i c m a g n e t i c field w i t h closed in lines of force. Before m a k i n g f u r t h e r assump.tions, I w o u l d like to invest i g a t e w h a t one c a n d e d u c e f r o m t h i s h y p o t h e s i s . O u r g a l a x y c o m p r i s e s s t a r s a n d m a t t e r . T h e diffuse m a t t e r h a s a n a v e r a g e d e n s i t y of a b o u t 10 -24 g / c m 3, ~ figure e a s y t o r e m e m b e r b e c a u s e i t c o r r e s p o n d s a p p r o x i m a t e l y to one h y d r o g e n a t o m p e r c m 3. A s i m p l e c a l c u l a t i o n shows t h a t t h e p r o b a b i l i t y of collision of a cosmic r a y a g a i n s t a s t a r is e x t r e m e l y small. H o w e v e r , t h e p r o b a b i l i t y t h a t t h e cosmic r a y p a r t i c l e m a y h a v e a n u c l e a r collision is n o t a t all negligible. I n d e e d , w e c a n m a k e a c r u d e estim a t e of t h i s p r o b a b i l i t y as follows: W e k n o w d i r e c t l y f r o m c o s m i c r a y e x p e r i m e n t s t h a t w h e n cosmic r a y p a r t i c l e s e n t e r f r o m t h e o u t s i d e i n t o t h e e a r t h ' s a t m o s p h e r e t h e y soon collide w i t h air nuclei. T h e m e a n free p a t h f o r t h i s collision is of t h e o r d e r of m a g n i t u d e of one h u n d r e d g r a m s p e r em:. Since t h e d e n s i t y is 10 -24, t h e c o r r e s p o n d i n g m e a n free p a t h for a c o s m i c r a y p a r t i c l e t r a v e l l i n g t h r o u g h t h e g a l a x y will b e a b o u t 102~ cm. Since t h e p a r t i c l e t r a v e l s w i t h a l m o s t t h e v e l o c i t y of light, t h e t i m e t a k e n f o r t r a v e r s i n g t h i s d i s t a n c e will b e 102~/3 • 101° =- 3 • 10 ~5 sec =- l 0 s y e a r s . I n t h e f o l l o w i n g c a l c u l a t i o n s I h a v e u s e d s l i g h t l y different figures y i e l d i n g : (1)
T = 7 • 107 y e a r s ,
for t h e a v e r a g e t i m e t h a t a cosmic r a y p a r t i c l e t r a v e l s b e f o r e a n u c l e a r collision h a p p e n s t h a t effectively d e s t r o y s it. This t i m e is r a t h e r s h o r t c o m p a r e d to t h e a g e of t h e u n i v e r s e e s t i m a t e d to b e t w o o r t h r e e b i l l i o n y e a r s : W e a r e t h e r e f o r e l e d to t h e c o n c l u s i o n t h a t o n l y v e r y few of t h e c o s m i c r a y p a r t i c l e s t h a t we n o w o b s e r v e c a n b e as o l d as t h e g a l a x y . I t s e e m s n e c e s s a r y , t h e r e f o r e , to a s s u m e t h e e x i s t e n c e of a mechanism that continuously produces new cosmic ray particles. W i t h o u t discussing y e t w h a t t h i s m e c h a n i s m m a y be, w e w a n t to i n t r o d u c e as a s e c o n d a s s u m p t i o n t h a t t h e p r o d u c t i o n is u n i f o r m in t i m e . Since a p a r t i c l e h a s a m e a n life of 70 m i l l i o n y e a r s , its p r o b a b i l i t y of s u r v i v a l a f t e r a t i m e , t, will b e (2)
exp (--
t/T).
This e x p r e s s i o n gives t h e age d i s t r i b u t i o n l a w of t h e p a r t i c l e s t h a t a r e n o w in e x i s t e n c e . O n e c a n n o w m a k e t w o a l t e r n a t e a s s u m p t i o n s : one is t h a t t h e c o s m i c r~d i c t i o n p a r t i c l e s a r e o r i g i n a l l y p r o d u c e d w i t h a n e n e r g y e q u a l to or h i g h e r t h a n t h e i r p r e s e n t e n e r g y . T h e o t h e r one is t h a t t h e c o s m i c r a y p a r t i c l e s a r e o r i g i n a l l y p r o d u c e d a t a r e l a t i v e l y low e n e r g y a n d a r e g r a d u a l l y a c c e l e r a t e d .
AN
tIYPOTHESIS
ON
THF.
ORIGIN
OF
THE
COSMIC
RADIATIO~ ~
31O
I n what follows we shall take this second point of view which has the advantage to require an injection mechanism less powerful than the one t h a t would be required for the first assumption. We assume, therefore, the existence of an accelerating process whereby vhe energy of a particle gradually increases as its age increases and is a function of the age. The dependence of the energy upon the age m a y then be determined from the knowledge of the energy distribution of the cosmic radiation. The experimentally known energy distribution of cosmic ray particles is rather complicated at low energies but takes the form of a simple power law for energies above a few GeV. We will assume this simplified law:
(3)
I ( E ) d E == k E -~.9 d E .
the exponent 2.9 is chosen to fit the observations. We have assumed t h a t the energy of a particle is a function of its age; t (4)
E =
](t).
F r o m t.he knowledge of the age distribution (2), and the energy distribution (3), one can determine ](t). Indeed, t h e n u m b e r of particles with age between t and t + dt is proportional t0 exp [--. t i T ! d r , and the n u m b e r of particles with energy between E and E + d E is proportional to
dE d/ E~.~ - - p . 9
:
we find, therefore, (5)
d/
/2.9 -- a exp [ - - t / T ! ,
where a is a proportional constant. (6)
I n t e g r a t i o n yields
1.9/1-9 -- T exp [-- t/:T] ,
where the integration constant has been set to equal 0 because for large t ] becomes infinite. This equation can be rewritten in the form: (7)
f(t) := Eo exp [ t / ( 1 . 9 T ) ] ,
~-here E , is a new constant t h a t represents the initial energy of the particle. F r o m our assumptions follows a very specific law (7) whereby the energy of the cosmic r a y particles must increase with time. According to (7), the energy must increase every year by a fraction of about 10 -8 of its value, so for a proton with energy equal to its rest energy, the energy will increase at the
320
E. fErMI
r a t e of a b o u t only 10 eV per y e a r a n d will increase c o r r e s p o n d i n g l y faster for p r o t o n s of higher energy. I t is clear, however, t h a t in a n y case t h e r a t e of increase of t h e energy will be quite slow since it takes a b o u t 100 million years to double the initial v a l u e of t h e energy. A v e r y simple process t h a t leads to the acceleration law (7), is due to t h e collision a g a i n s t large m o v i n g objects. W i t h o u t specifying y e t w h a t p a r t i c u l a r objects will be considered as likely obstacles against which the collisions t a k e place, we w a n t to assume t h a t a cosmic r a y f r e q u e n t l y collides against large m o v i n g obstacles. T h a t the energy of t h e cosmic r a y will on the a v e r a g e increase in such collisions is clear from t h e f a c t t h a t u l t i m a t e l y s t a t i s t i c a l e q u i l i b r i u m would be established with e q u i p a r t i t i o n of energy between the obstacles a n d the cosmic r a y p~rticles. This corresponds, of course, to an e x t r e m e l y high energy v e r y m a n y orders of m a g n i t u d e b e y o n d the m a x i m u m energy o b s e r v e d in cosmic rays. g~hat limits the efficiency of this process in increasing the energy of the cosmic r a y particles is, therefore, not the m a x i m u m energy a t t a i n a b l e which is effectively infinite, b u t r a t h e r the r a t e at which the energy is t r a n s m i t t e d . N o t all collisions will accelerate t h e particle. A c t u a l l y , head-on collisions will p r o d u c e an acceleration a n d o v e r - t a k i n g collisions will produce a deceleration. On the average, there is acceleration p r i m a r i l y because h e a d - o n collisions are somewhat more p r o b a b l e t h a n o v e r - t a k i n g collisions since t h e r e l a t i v e velocity is larger in the former case. One can c o m p u t e in an e l e m e n t a r y w a y the order of m a g n i t u d e of the ~verage increase, 3E, per collision of a particle of energy E (including rest energy) colliding a g a i n s t objects m o v i n g w i t h velocity, V. The result is: (8)
3E
~
EV2/c
~ .
If we assume t h a t the collision cross-section is i n d e p e n d e n t of the energy, t h e n u m b e r of collisions p e r u n i t time will also be a p p r o x i m a t e l y i n d e p e n d e n t of t h e energy since the v e l o c i t y of the cosmic r a y particles is p r a c t i c a l l y c o n s t a n t a n d equal to c. I t follows f r o m (8) t h a t the gain in energy p e r u n i t time is p r o p o r t i o n a l to t h e energy. The energy therefore increases according to an e x p o n e n t i a l law. W e shall t a k e V of the order of 30 km/sec. This g~ves (8) ~E ~ 1 0 - s E . T h a t is again an average gain of energy per collision of one p a r t in 10 s. A f t e r N collisions the energy will b e : (9)
E - - Eo exp [NB 2] ,
where B ~ --
V2
C2
__
lO-S .
AN
kI~-POTHESIS
ON
TItE
ORIGIN
OI:
Tile
COSMIC
RADIATION
321
I n order to estimate the number of collisions, we introduce a scattering mean free path ;~. The number of collisions after a time, t, since the initiation of the process will be ~r =: ct/;% since the particle travels practically at the velocity of light. Consequently we g e t : (10)
E = Eo exp [ctB~/'A].
Comparing (10) with (7) one obtains (11)
)~ ~
1.9B2eT
~-~1018 cm :
I light y e a r .
We will now further specialize our assumptions and introduce the hypothesis t h a t the collisions responsible for the increase in energy are ~g~inst moving irregularities in a cosmic m~gnetic field. The idea o~ the existence of such ~ moving magneti~ field is due to Dr. A L F V ~ who has made a thorough magneto h y d r o - d y n a m i c study of the influence that the extremely tenuous interstellar matter has on the propagation of a magnetic field t h a t penetrates it. Unfortunately, I do not have the time to explain in detail his v e r y stimulating ideas on this subject. I shall only mention that dne to the relatively high electrical conductivity of vhe interstellar medium, the lines of force ~re practically ~ttached to the ma~tter so t h a t they arc dragged b y the turbulent motion of the interstellar matter. A cosmic ray p~rticle will be deflected b y the action of the magnetic field and will gain energy in the process as previously discussed. I n order to obtain agreement with the experimentally observed spectrum, we must assmne that the size of the m i n i m u m vortices which drag the lines of force is of the order of one light year, a value which does not appear implausible. ~ o t h i n g has been said so far of the injection mechanism of the particles of relatively low energy which will be further accelerated b y the proposed me~hod. I n order t h a t a particle so injected m a y eventually become an energetic cosmic ray, it is necessary that its initial energy be above ~ certain limit which will be c~lled the injection threshold. Indeed, the accelerating mechanism will function only when the energy gained b y the accelerating mechanism is greater that the energy lost b y ionization. An estimate of the injection threshold yields for w.rious particles the following values: Protons Particles Oxygen nuclei Iron nuclei
- 100 MeV 1 GeV - . 1 GeV per nucleon 5 GeV per nucleon
I t is seen that the injection threshold is quite large for h e a v y nuclei and
322
E. FERMI
this r e p r e s e n t s t h e m o s t serious difficulty for t h e p r o p o s e d t h e o r y . Even w i t h o u t special a s s u m p t i o n as to t h e origin of t h e i n i t i a l p r o t o n s a n d u p a r ticles t h e i n j e c t i o n of t h e s e l i g h t c o m p o n e n t s of t h e cosmic r a d i a t i o n m a y be u n d e r s t o o d p e r h a p s as due to t h e collisions of t h e cosmic r a d i a t i o n itself w i t h t h e nuclei of t h e i n t e r s t e l l a r m a t t e r . F r o m t h i s p o i n t of view, cosmic r a d i a t i o n w o u l d be a self r e g e n e r a t i n g process. N o such s i m p l e e x p l a n a t i o n , h o w e v e r , seems a d e q u a t e for t h e i n j e c t i o n of h e a v i e r nuclei like~ for i n s t a n c e , o x y g e n or iron, since no k n o w n m e c h a n i s m could y i e l d a n i r o n nucleus w i t h 5 G e V p e r n u c l e o n w i t h o u t d e s t r o y i n g t h e nucleus itself. One m u s t a s s u m e , therefore, a special i n j e c t i o n m e c h a n i s m for t h e s e p a r t i c l e s , p e r h a p s like t h e one s u g g e s t e d r e c e n t l y b y SPITZER [1]. I n conolusion, t h e p r o p o s e d t h e o r y seems to be q u i t e a d e q u a t e f o r u n d e r s t a n d i n g t h e m~in f e a t u r e s of t h e p r o t o n c o m p o n e n t s of t h e cosmic r a d i a t i o n ~nd p e r h a p s also of t h e g p a r t i c l e c o m p o n e n t . I t does n o t s e e m a d e q u a t e to u n d e r s t a n d t h e presence in t h e cosmic r a d i a t i o n of ~ significant f r a c t i o n of h e a v i e r nuclei. I f t h e g e n e r a l f e a t u r e s of t h e p r e s e n t t h e o r y s h o u l d p r o v e correct, t h e r e s h o u l d be a n i n d e p e n d e n t u n d v e r y p o w e r f n l i n j e c t i o n m e c h a n i s m , of t h e h e a v y c o m p o n e n t of t h e cosmic r a d i a t i o n .
BIBLIOGRAPHY [1] LYMAN SPITZEI~, Jr.: Phys, Rev., 76, 583, (1949).
DISCUSSIONE E OSSERVAZIONI - - E. BACGE, Hamburg: Bei dem yon FERMI diskutiorten Mechanismus ftir die Beschleunigung und Aufsammlung der H6henstrahlungsteilchen innerhalb der MilchstraBo scheint es wichtig zu sein, die Fragc zu diskutieren, welcher Toil kosmischer Stralflung an der Oborfl~che des gMaktischen Systems in den Vqeltraum hinauswandert. Da das yon FERMI postulierte galaktische Magnetfeld gro2e riiumliche Schwankungen besitzen mu~, wenn der Beschleunigungsprozel~ iiberhaupt wirksam werden soll, wird es untor anderem auch nahezu feldfreie Beroiche geben und dies kann besondors an don Randbereichen der Milehstrai3e das Entweichon dor HShenstrahlungsteilcben ermSglichen. E. FERMI, Chicago: In order to avoid a large loss of particles out of the boundaries of the galaxy it is sufficient to assume that the lines of force are closed or at least that very few of them escape to the outer space. -
-
DISCUSSION]~
E
OSS]~RVAZIONI
L. JANOSSY, Dublin: T h e i n j e c t i o n process of p r o t o n s is h e l p e d b y t h e c i r c u m s t a n c e colliding w i t h a nucleus is likely to r e t a i n a n a p p r e c i a b l e f r a c t i o n of t h u s r e m a i n v e r y m u c h a b o v e t h e i n j e c t i o n energy. T h e q u e s t i o n is h e a v y f r a g m e n t s arising o u t of n u c l e a r collisions c a n serve to i n j e c t -
323
-
that a nucleon its e n e r g y a n d raised whether h e a v i e r nuclei.
E. FERMI, Chicago: X a t u r a l l y I hope t h a t y o u are right. On t h e o t h e r h a n d it seems to me v e r y difficult to u n d e r s t a n d t h e fact f r o m t h e t h e o r e t i c a l p o i n t of view. But, of course, if t h e fact s h o u l d p r o v e t o be t r u e it will h a v e some t h e o r e t i c a l e x p l a n a t i o n . -
-
