IL NUOVO CIMENTO VOL. 8 B, N. 2 11 Aprile 1972

Antiproton Spectrum in Cosmic Rays (*).

M. C. C ~

Department o/ Physics and Astrophysical Sciences University o] 1Vevada - Las Vegas, ~ev.

(rieevuto il 6 Marzo 1971)

S u m m a r y . - The steady-state antiproton spectrum in cosmic rays as a result of the collision between primary cosmic protons and inter- stellar hydrogen is calculated by means of the statistical model of two fire- balls in multiple-particle production. Results based on Fermi's original one-fire-ball model are also presented for comparison. We found that the antiprotons obey approximately a power law with an exponent about 2.6 which is about (the same magnitude) equal to the exponent of the general cosmic rays spectrum. The numerical value of the exponent is found to be determined principally by the exponent of the cosmic protons, and very insensitive to the parameters of the two-fire-ball model intro- duced in the calculation. A closed expression of the ratio of the number densities of these secondary antiprotons to cosmic protons is expressed in terms of the proton exponent, interstellar hydrogen density, total p-p inelastic cross-section, antiproton lifetime in the Galaxy and the two- fire-bali parameters. The numerical values of the ratio range from about 3.8.10 -a at 10 TM eV to 6.8.10 -4 at 10 TM eV.

1. - Introduct ion.

Pr imary cosmic rays in our Galaxy consist of mainly high-energy protons ranging f rom 2.10 0 eV to 10 20 eV total energy which have been observed

experimentally. The interstellar gases consist of mainly low-energy hydrogen

atoms. When the high-energy protons collide with the interstellar gas nuclear

reactions take place, and various kinds of particles are created. I n particular, ant iprotons are created at different energies. These ant iprotons gradually

lose number by escaping from our Galaxy and by proton-ant iproton annihilation,

and lose energy through different processes. When a steady state is approached,

(*) This work is partially supported by the Research Committee of the University of Nevada, Las Vegas, Nev.

343

344 )L c. CI~EN

the ant iprotons a t ta in a definite shape in their energy-spectrum distribution.

I t is the purpose of this article to calculate this spectrum. .~V[ILFORD and RosE>- (1) calculated the upper limit of ~. They considered

only the reaction p + p - ~ p + p + p +~ . Total cosmic radiation, i.e. the total i ty

of components of cosmic rays, has been measured up to 10 ~~ eV (~). A cosmic

proton of 10 ~7 eV, which corresponds to a total center-of-mass energy of

10 ~a eV in a p-p system, can create hundreds of ~ when it collides with an

interstellar hydrogen atom. I t is obvious tha t to have a more accurate account

of ~ intensity, processes of multiple-~ creation should be considered.

VVe shall use ~ statistical model originated by FER)II (a) in the calculation

of multiple-particle production. FER~I proposed tha t a fire-ball, or highly-

excited nucleon-pion cloud, is formed after collision and immediately breaks

up into pions and (anti-) nucleons. The multiplicity of the reaction and the

Lorentz factor of the fire-ball as predicted ill Fermi 's model have been verified

by experiments (4) at low energies. The angular distributions predicted from

the theory also are also in general agreement with experimental data at low

energies. As energies increase, Fermi ' s model gives poorer bits of the angular

distr ibution (5). To remove this difficulty, TAJCAGI (~) proposed a two-fire-ball

model in which two fireballs are created from the collision and then break

off into multiple particles. This model readily interprets the two-center feature

of the angular distr ibution of secondary particles (5). t~owever, the elasticity

of the reactions predicted by the model is too low when compared with experi- ment . ~NIU (7), CIoK et al. (s) and CocooNI (9) independent ly proposed a model

with two fire-balls and two nucleons to fit experimental data. To obtain some

particles of much higher energies than the rest, HASEGAWA (~0) proposed a

multiple-fire-ball model. I n wide ranges of cosmic-ray energies, many secondary particles are created.

So we expect the two-fire-ball and the two-fire-ball- two-nucleon models to

(1) S. N. ~[ILFOI~D and S. ROSEN: Nature, 205, 582 (1965). (") D.A.',-Dt~EWS, A. C. EVA_XS, It. J. D. ]21';ID, ]2. M. TENNENT, A. A. WATSOn- and J. G. WILSON': Xature, 219, 343 (1968). (3) E. FJ~RMI: Progr. Theor. Phys., 5, 570 (1950). (4) V. V. Gt:SE~'A, N. A. DOBROTIN, N. G. ZELEVINSKEGA, K. A. KOSTELNIKOV, A. ~I. LE~EDEV and E. A. SnAWtTINSKY : Jourr,. Phys. Soc. Japan, 17, Suppl. I I I -A , 375 (1962); S. KANEKO and S. 0KAZAKI: _'VUOrO Cimento, B, 521 (1958). (5) J. J. LOI~D, J. FAINBERG and 3[. SCHEIN: Phys. Rer., B0, 970 (1950). (~) S. TAKAGI: Progr. Theor, Phys., 7, 123 (1952). (7) K. NIl:: Nuoco Cime~do, 10, 944 (1958). (8) 1 ). CIOK, T. COGHEN, J. CIILRULA, R. HOLYNSKI, A. JURAK, ~'[. 3[IESOIZRITZ, T. SA- NIEWSKA and O. STANISZ: ~Vuovo Cimento, B, 166 (1958). (9) G. CoceoNi: Phys. Rer., 111, 1699 (1968). (~0) S. HASEGAWA: Progr. Theor. Phys., 26, 150 (1961); 29, 128 (1963); see also S. FRAIYTSCHI: Y'ttOVO Cimento, 28, 409 (1963); A. AGNESE, M. LA CAMERA and A. WA- TAGt[IN: ~'ltOVO Cimento, 59A, 71 (1969).

A N T I P R O T O N SPECTRUM IN COSMIC RAYS ~

exhibit small differences at these energies. Therefore, we shall use the two-fire-ball model in our calculations. This contributes great simplifications in computation works. The many-fire-ball model may give more accurate results than ours. I t would be interesting to have calculation based on that model and compared with ours.

We shall calculate the spectrum in Fermi's original model first, not only because we wish to compare the results from these two models, but also because many expressions in the two-fire-ball model can be taken over directly from Fermi's model. In Fermi's calculation (..11), he neglected the conser- vation of charge, baryon number and isotopic spin. These effects shall be con- sidered in our calculations. As with FErmi {") in discussing nucleon and pion creation, we shall make the approximation that the secondary (anti) nucleons and pions are extremely relativistic. Our result of the ~ multiplicity (21b) does not differ appreciably from Fermi's simple calculation based on Planek's distribution. The calculation in Sect. 2"1 serves to show that the conservation laws are relatively unimportant and, therefore, may be omitted in rea~ling.

The antiproton spectrum as a result of a p-p collision is calculated in Sect. 2. The production rate of antiprotons in the Galaxy is presented in Sect. 3. The steady-state antiproton number-density spectrum in the Galaxy is obtained in Sect. 4 and the results are discussed in Sect. 5.

2. - Ant iproton spec trum f r o m proton-proton col l i s ion.

We consider the reaction

(1) p + p -+ xlp § x3n ~- x . ~ ~- x4~ § x s ~ + -~ xe~ ~ q- x ~ - ,

where the x's are the numbers of p, n, p, n, ~:+, ~:o and ~:-, respectively, in the final state. Processes involving the creation of strange particles and (anti-) hyperons which in tu rn decay into (anti-) nucleons are of smaller order and shall be neglected. FERMI assumed that a highly-excited pion-nucleon ball (fire-ball) is formed from two colliding protons immediately after collision and then breaks up into different kinds of particles. The probability that reaction (1) O c c u r s iS (3,11)

P(Eo, ~1, n3, . . . , n~) ---- K(2~)-a("-I~(yQ) "-l-

�9 g(T, T.)2'~+".+ ..... n~! ~(lr, o, O, n ) ,

(2) D 2M ~ Ec 3re a '

7 3 3 ~ Z" .

(11) E. FERMI: Phys. Rev., 92, 452 (1953); 93, 1434 (1954). See also M. KRETZSCHMAR: in Annual Review o] Nucleon Science (Palo Alto, 1961).

3 4 6 M . c . CHiN

7 I n the above expression, n = ~ n, is the to ta l numbe r of part icles in the

i = I

final s tate, E~ is the to ta l center-of-mass energy of the colliding protons, M

and ff the p ro ton and pion masses, respect ively, ~(E~,p, n) the phase-space integral corresponding to the to ta l l inear m o m e n t u m p . y~2 is the contracted

volume of the fire-ball, y being a dimensionless pa ramete r . F E I ~ I took the value 1 for y. He also suggested t ha t y be t aken as an adjus table p a r a m e t e r to be de-

t e rmined f rom exper imenta l da ta for best results . In this Section, the units for which ~ ~ c = 1 is used. The fac tor 2 "~+~+'*+'' is ~ weight factor due to the spin

of the nucleons. The factor n,! is to account for the shuffling among n~ iden- i=1

t ical protons, n2 ident ical ant iprotons , etc. g(T, T3) is a weight factor f rom isotopic spin considerat ion. I t is the num ber of isospin s ta tes which can be constructed f rom (n~-{-n2q-nsd-n4) par t ic les of isospin �89 and (nsq-n~q-nT) part icles of isospin 1 to fo rm a s ta te of to ta l and 3rd componen t of isospin T and T3, respect ively. Our ini t ial s ta te consists of two protons, so the to ta l and 3rd component of

isospin are bo th 1. To conserve isospin in the react ion, T and T3 in the final s ta te mus t also be 1. K is a normal iza t ion cons tant such t ha t

(3) ~_, ~_, P(E~, n~, n2, .. . , ~7) ----- i .

In tile summat ion over +h their values ~re res t r ic ted by charge and baryon- num ber conservat ion. The upper l imit of the summat ion over n is actual ly not s t r ic t ly defined in our ex t remely relat ivis t ic approximat ion . However , i t is not

i m p o r t a n t in our calculation. Therefore, we leave i t unspecified.

2"1. Antgproton spectrum ]rom a p-p collision in Fermi ' s model. - Let q~(E, E ) d E - - n u m b e r of ~ produced whose to ta l l abora to ry energies lie be- tween E and dE as the resu l t of a p-p collision with to ta l c.m. e n e r g y / ~ . The subscr ip t I indicates 1 fire-ball. We shall app rox ima te q~ by a (~-function

(~) q~(Eo, E) = n : : 5 ( E - - E : : ) ,

where n, , is the average num ber and E~v the average energy of an ~ produced in a collision.

Since t, he p r i m a r y cosmic protons are isot.ropic, any anisot ropy in the distri- bnt ion of the secondary 1~ will be smoothed out by the p r i m a r y protons. Therefore, the secondary ant ipro tons created by all cosmic protons arc Jsotropie in angular dis t r ibut ion, and the angular dis t r ibut ion of ~ product ion in a single p-p collision event will not be analysod. Even if the 6-function approx imat ion in (4) is not a good approx ima t ion for isolated values of E~, it is expected to give good resul ts when in tegra t ion is carr ied out la ter for E~ through the whole spec t rum

of cosmic protons. This is because of the over lapping of q~(E, E) when eontin-

A N T I P R O T O N S P E C T R U M IN COSMIC RAYS 3 4 7

uous values of E~ are considered. Although numerical values can be obtained by the use of a computer in various steps during the calculation, we feel i t is more advantageous at this t ime to get approximate closed expressions ra ther t han numerical values because the effect of different parazneters on the spectrum can be examined explici t ly through a closed expression. Fur thermore , we feel t ha t general features of the ~ spectrum as predicted by Fermi ' s model are to be verified before any detailed calculations can become meaningful.

The probabi l i ty t h a t a to ta l of n part icles of (anti) nucleons and pions emerge in the final s tate is f rom (2)

(5) P(~o, n) = ~: ~(E~, n~, . . . , n,) .

We shall consider the react ion (1) to be ex t remely relativistic, so t h a t the difference between the nucleon and pion masses is negligible. Therefore, a nucleon of e i ther spin s ta te emerges in the final state wi th the same probabi l i ty tha t a pion does. So among the final n particles, the average number of ~ is 2n/(2.4+3) ~ 2 n / 1 1 . The average number of ~) produced in a collision is t hen

(6) n , , = ~ (2n/ll)P(E~, n). n

The funct ion P(E~, n) is small for small n, t h en increases with n unt i l i t reaches a max imum value for cer ta in value n ~ of n. After tha t , P(E~, ~) decreases wi th increasing n. Again, we approximate .P(E~, n) by a (~-funetion in n, and replace n in (6) by n ~ , t hen n,~ ---- ( 2 n ~ J 1 1 ) ~ . P ( E ~ , n)= 2 n ~ / l l .

We shall evaluate E , , first. When there are n part icles in the final state, the average energy of ~ part icle is E*/n, where ~* is the to ta l p-p energy in the labora tory frame:

(7) ,E* : (2M)-~E,~ .

Therefore, when all possible n's are averaged, we have

(8) ~ , , = Z (~/n)P(~o, ~). n

Using the same &funct ion approximat ion for P(E~, n), we get

To evaluate n ~ . we have to use (2) to sum (5). The phase integral P(E~, 0, n) in (2) has been carr ied out by LEPORE and STUART (1~) and by R0ZE~T~L (13)

(13) j . V. LWPORE and R. N. STUART: Phys. Rev., 94, 788 (1954). (la) I. L. ROZENTAL: SOY. Phys. JETP, l , 166 (1955).

3 4 8 ~1. c . CHnX

in the ex t reme relat ivis t ic l imit

(10) (2n - - 1 ) !(4n - - 4) ! E~"-* p ( ~ , 0, n) = ((2~ - 1 ) ! ) ~ ( 3 n - ~ ) ! 2,--,

The isospin weight factor g(T, Ts) has been obta ined by YEIWh" and DE- St]~.~XT (14),

(11) g(T, T~) = (22' + 1) 7 ~- - > ~ ' 2 i : s = - l V ] \ i + 8/2 ~ T ~ ~

(i) t! = ) ! ( t - - i ) ! '

where s and t are the num ber of part icles in the final s ta te with isospin �89 and 1,

respect ively , and the s u m m a t i o n over i is ex tended only to those nonnegat ive integers which m a k e the t e rms on the r ight side of (11) meaningful . We are concerned mginly with large s and t, so the factorial can be evalua ted by means of Ster l ing 's formula :

(12) n! ~ V ~ e - " n "+�89 .

For T = 1, the last fac tor in (11) is a p p r o x i m a t e l y

Thus,

(13)

(2z)-�89 ='+' "

~ / ~ 2 7 ~ 7 ~ i v ~ = ~ 2i + , + 1 "

By replacing (2i + s + 1)-* by an average value (t § s + 1)-*, the sum- mat ion is seen to be just a b inomial expansion, so we obtain

g(1, 1) ~ (2n)-�89 + s + 1) - '3 * .

We pu t the above into (2):

3 1 (15) _P(Ec, nl, n 2 , . . . , n T ) ~ _ K v ~ , ~ n + 1

4n,+n,+n,+n, 3n,+n,+n,

y [ ni! i

I ff~\n--1 o(Eo, O, n) .

(14) Y. Y~IVIN ~I]d A. DE-SHALIT: ~OVO C4~e~do, 1, 1146 (1955).

ANTIPROTON SPECTRUM IN COSMIC RAYS ~ 9

Approximat ing 4~,+~,+~,+~,3-,+~,+~, in (15) by 3.6~,+",+'~+",3.6 ~.+~.+~, = 3.6 ~, we get

K 3 3.6 {3.6y~'~ "-x ~(E~, O, n) (16) P(E~, nx, ..., n,) ~ , V ~ n -~ l \ - ~ a / ~ ~ ! .

i

I f the conditions of conservat ion of charges and ba ryon numbers are relaxed, the summat ion over n~ can be carried out wi th the help of the formula of mult inomial expansion

(xl + x~ + . . . + x~)" =

By set t ing all x~'s = 1 , we obta in

~_, n! z , , , . , , 1 - i n j x'~" ~ ' " " x ~ ' "

i

Inser t ing the above expression and (16) into (5), we obtain

(17) P(Eo n) -" ~K 3 3.6 [ 3 . 6 y ~ ~-1 7" ' ~ ~ ~ - 1 ~ z - ~ / ~., e(~o, o, ~).

Here a factor ~ < I is in t roduced to account for the conservat ion of charges and ba ryon numbers , a is a funct ion of n only; even its approximate form is difficult to find. F r o m its physical meaning, i t is seen t ha t ~ is a finite number for any n. Then i t is reasonable to p u t a(n) in the following form for large n:

(18) a t ~ i

! where ~o is a nonvanishing fract ional number and the exponents W~> 0, ~o and ~ are constants.

I f ~ varies with n as in (18), i t is not necessary to know ~ explici t ly in our approximate calculation as described below.

Using Sterling's formula again in the phase integral (10), we get

lO.S 7 ~ 13.6yQ 3 q~ 1 (19) P(Ec'n+l)'~~ "2-~Eo(n-{-2)(n-]-l)!\ ~ "] ha,+1 ~

87c ~ E,(n Jr- 2)(n -~ 1) k 2167~3 ] n~"+~;

is de te rmined by (~/~n)P(E~.n)= 0, or nms, x

(20) 1 1

n + 2 n + l

r ~ 25e4yY2E~(216~z~) -~ .

- - q - l n r 4n-{-~ ~ l n n = O ,

3 5 0 M. C, CHGN

I t can easily be seen from (18) tha t (~c/~cn) In ~ at most varies with n as n-L

I t can be neglected along with other terms of the order n -~ compared with 1

in a first approximat ion solution for large n. The solution of (20) is easily found to be

( 2 1 . )

Therefore,

(21b)

n . . . . \ 216ye~ ] -- \ 8 1 ~ u a ] �9

2 [25yME2~] ~. ~ -~ 1-i-\ 8 1 ~ 2 ~ ~ ! "

Since E ~. = 2ME*, eq. (21a) or (21b) shows the multiplici ty varies as

E , which is the characterist ic of Ferret 's model. Subst i tut ing (21) and (11)

into (4), we obtain the final expression for the ~ spectrum from a p-p collision in Ferret 's model

(22a) q~(E~,E)~_ ~ \ ~ ] 5 E--)~3 ~ ~ ,

or in terms of laboratory energy E* of the cosmic proton,

(22b)

q~(E*, E) ~ - AE �9 ~ ( s A-1E *~') , 11

(50yM2~ -~ A ~ \ ~ 3 ] = 0.012y~.

Equat ions (22) have been derived by using units for which h = c = 1. The con- version to practical units can be carried out by observing that A E *~" is a dimen- sionless quanti ty, therefore, if certain a energy unit, say eV, is used for all the

parameters involved, M, #, E and E*, then q~(E*,/~) will be in (eV)-L

Electron volt shall be used as the unit of energy and mass throughout this article.

2"2. Antiproton spectrum /rom a p-p collision in the two-/ire-ball model.- In this model of TAKAOI (6), two fire-balls of equal rest masses arc created

as a result of collision. These fire-balls t ravel along the line of the incident proton

with equal and opposite velocities in the c.m. frame. Soon after created, these

balls break up into (anti) nucleons and pions. VVe shall assume tha t each ball breaks up into secondary particles in the same way ~nd following the same

laws as the single fire-bull does in the Fermi model in the rest frames of these balls. Let Mbl and M~2 be the rest masses of the two balls in their respective

rest frames, and Mbl and Mb2 move forward and backward, respectively, in the

c.m. frame. Wi th our assumption, it is seen tha t Mb~ and Mb~ separately play

ANTIPROTO:N SPECTRUM IN COSMIC RAYS ~ 5 1

the same role as the total c.m. energy Ec does in Fermi's model. Therefore, we shall take over the formula for n=~ and n~, in eqs. (21) for each of the fire-balls by simply replacing/~ by Mb, and Mb~ respectively.

The average laboratory energy of an emerging particle shall have to be evaluated separately. Let these energies be denoted by/~,~,and E2~ , for par- ticles from the first and second balls respectively. These energies have different expressions from Fermi's model. Let 7b~ (i----1, 2) be the Lorentz 7-value of the i-th ball in the laboratory frame and the corresponding value in the c.m. frame. Since Mb~ = Mb~(-~ Mb), then ~b, = ~b~(---- ~b) simply from symmetry considerations. The ~,-value of the motion of the center of mass relative to the laboratory frame is ~ = ~*/E~ = (~*/2M) �89 where, as before, ~* and Ec are the total laboratory energy and c.m. energy of the colliding protons. The total c.m. energy is given by

(23) .E~ ~-- 2M7~ , ~ -~ 2Mb'~b.

In the primary energy ranges of the cosmic rays we are considering, each ball is capable of breaking up into a large number of secondary particles, i .e.

Mb >>M, so y~ >>~b: From the relativistic addition of velocities, it can be derived that for the forward-moving ball

(24) ~'b~ ~ 2~7b ---- JE~(MMb) -~ ,

and for the backward-moving ball

(25) ~b2 ~ ~ ~- -- 2~7b 2 M '

where only leading terms in y~ or ~-~b ~ have been retained as 0~r >>1, ~b >>1. The laboratory energies of the balls are M~)~b~ (1 = 1, 2). Therefore, the

average laboratory energies of an emerging particle are

(26b)

Inserting (26) into (4) and replacing ~c in (21) by M~, we get the ~ spectrum from a p-p collision in the two-fire-ball model:

(27a) qn(Ec ~- 11 n~ .~ E ' 2Mn.~,x] '

(27b) nmax-- \ 8 1 ~ # 3 ] �9

352 M.C. CH~-\

The first and second t e rms in (27a) are the cont r ibu t ion f rom the forward- and backward -mov ing balls respect ive ly .

The res t mass of each bal l Mb is expected to increase wi th increasing p r i m a r y energy in order t h a t the mul t ip l ic i ty increase wi th increasing p r i m a r y energy. CoccoNI (9) selected 16 events of mul t ip le-par t ic le product ion in p-p collision

f rom cosmic shower events and de te rmined the res t masses of the ~b-values of these balls in t e rms of the cosmic-ray energies. His model is two fire-balls plus two nucleons. However , because of the large mul t ip l ic i ty , about 30, in those events , two nucleons ha~ing ve ry small effect on the values of Mb and ~b, therefore , we shall ignore the cont r ibut ion f rom the two nucleons and take his values of $~ to obta in an empir ical re la t ion be tween :~b and ~* in our t w o -

fire-ball model. We t r y to fit the re la t ion be tween ~ and J~* by the power law

(28) $b -- bE *~ ,

where eV is to be used for E*. Cocconi's resul ts were l is ted in Table I of his

article. We use all events in this Table except the one with an except ional ly large value of ~b ( = 48). An average fit is ob ta ined wi th

(29)

F r o m (23) we get

(30)

b _~ 0.135, f l _ 0 . 1 .

Mb I-- ~/2~(2b)-~/~ *�89 -- 1.6 .:105j~ *~ ,

where bo th M~ and E* ~.re in eV. I n s e r t i n g (30) into (27b) we get

/2y M2 E*I -~ \ ~ | . . . . | : B E *a-2~)/~

(31) B~_ ( 2YM~ ~ ' = 0.025y~.

\9~2#3b 2]

I n t e rms of the lnbora tory energy E*, (27a) t~kes the fo rm

2 [ 2 (32) q~(E*, E) = ~ B E *<~-~1~ (~(E - - B - 1 E *<a+2~)/4) -4- B E *(1-2/bl4 (~ ( E

To conver t to prac t ica l uni ts , eV is to be used for M, /~, ~ and E* in (27), (31) and (32), and q~(E*, .E) is then in (eV) -1.

3. - Product ion rate of ant iprotons f r o m the col l i s ion b e t w e e n c o s m i c protons

and interstel lar hydrogen.

The spectral n u m b e r dens i ty np(E*) of the cosmic protons is defined such t h a t np(E*)dE* is the n u m b e r of p ro tons per cm 3 whose to ta l energies lie

ANTIPROTON SPECTRUM IN COSMIC RAYS ~

between ~* and ( /7*+ dE*)eV, n~,(.E,*) can be expressed apprpximate ly in a power law (~):

(33) nr(E*) = ~YrE *-r , ,

where n~ and the exponent F~ are energy-dependent , b u t approximate ly con- s tant within wide ranges of energies. The interstel lar hydrogen can be considered at res t in our discussion. The collision f requency between the cosmic protons whose energies lie wi thin ~* and the inters te l lar hydrogen atoms is ca(E*)iVHn~(~*)d~* per second per cm a, where Zr~ is the H-number dens i ty and a(E*) the to ta l inelastic cross-section in p-p collision, and e the veloci ty of cosmic protons, approximate ly the veloci ty of light, a(E*) is energy dependen t bu t approaches a constant as E* increases indefinitely. Denote q~(~)dE as the number of ~ per cm 8 whose energy is wi thin 4 E created b y all cosmic protons per second. Then because in each collision of the hydrogen b y a cosmic p ro ton of energy J~*, q~(E*, J~) dE (i = X, I I ) an t ipro tons are created within the energy in terva l dE, we have

qTi(E) d E = dEfeq,(E*, where i = I , I I for the one - and two-fire-ball models. So

(34) qL =

Inser t ing (22) and (32) into (34) we get the spectral product ion ra te of in the Fermi model:

(35) q~,(E)-~vlv: fa(g*) 2 A E * ~ ( E - - A - i E * ~ ) 2 V , E*-r, dE * =

8 IV~[~ eaA(e-ar.I)sE-(ar~ -~>/s =3-5

* ( A E ) ~ . where ~, ~ and F~ are to be evaluated at E~

In the two-fire-ball model, we inser t (32) into (34) and find

(36) q~ii(E) = 8o

11(3 § 2fl)~r ~r , r~ r ~v~aD(8-4~l(3+,~) -~(_ -1+2~ms+2~> §

32b ~ ~- 33(1 - - 2/~ i c]~]~Bd(2-rp-2~l(a-s~)(Edb2)-2c~rD-1-2~t(a-e~ '

(is) See for e x a m p l e V. L. GINZBURG a n d S. I . SYROVATSKII: The Or ig in o] Cosmic R a y s (New York , 1964).

23 - I1 Nuovo Uimento n.

3 5 4 ~ . c . c n E N

* (BB)~/(3+~, where a, _~r and F~ in the first t e rm are to be evaluated at E~ *' (4b~BE) (~-~)~ For /'~ ~ 2.5, the rat io of and in the second t e rm at ~ ~

the 2nd to the 1st t e rm is ~ 9 0 0 0 y - ~ E -~ For y ~ 1 , the rat io is about 0.3 a t ~----1010eV and decreases rapidly with increasing energies. When the second t e r m is neglected, the spectrum takes the simple form

(37) 8 q~H(E) - - 11(3 Jr 2fl) eN~NPqB(S-arv)/(s+2fl)E-2(2rp-l+2~)/(a+efl)

However , if y is found to be small, bo th terms in (36) must be retained at low evergy. At sufficiently high energies only the first t e rm contributes.

4. - Steady-state antiproton spectrum in the Galaxy.

The spectral number dens i ty n~ of the cosmic ant iprotons is defined such t ha t n~(E, t ) d E ~ number of ~ per cm ~ at t ime t whose energies lie between E and E + dE. n~(E, t) is governed by the equat ion of cont inui ty for the conservat ion of part icle numbers

(38) n~(E, t) + ~(E, t) -~ ~--t ~ = q-~(E, t ) ,

where - - d E / d t is the energy-loss ra te of un ~ due to different processes, and q~(E, t) is the source of ~, or the product ion rate of ~ derived in Sect. 3. The only appreciable losses of cosmic ~ are the leakage out of our Galaxy and the p-~ annihilat ion with the interstel lar hydrogen atoms. Denote the lifetime of ~ for the two processes by T,,0 and T~= respectively, then the equivalent loss ra te is

dE E (39) dt T ~ ' T ~ = T .1 + T -1 eBc ~ n n �9

Subst i tu t ing (35) and (39) into (38), we get for Fermi ' s model a t s teady state ~n~/~t = O,

s_ v~' ] t Np(~A (s-4ri,)/a E-(4rv-2)/a "

The solution is

2 c N ~ N ~ (40) n ~ ( E ) =

3 - - Te" A (8-4r')I3 E-(4rP-~)Ia = 41"p - - 5 To,~q~(E) ,

where as before, hr,, /'D and a are to be evaluated at E* ---- (AE)r

ANTIPROTON SPECTRUM IN COSMIC RAYS

Similarly, by subst i tu t ing (37) and (39) into (38), we found for the two- fire-ball model

5 ~)-1 (41) ~ i ( E ) = (2/11) F , - - ~ + oNHN, aT.,, B'(~-r~"(~+~)E-~(~r~ -~+~'''~+~' =

=(3 + 2fl)(~F.--5 + 2fl)-~q~(E),

where s /'p and a axe to be evaluated at E* = (BE) '/(3+*~).

5. - D i s c u s s i o n and conc lus ion .

Some numerical results shall be presented in this Section by put t ing ap- propriate numbers in the formulae derived in the previous Sections.

i) The exponent o] the ~ spectrum. - Take /'p----2.5; the exponent in Fermi 's model is, from (35), (4Fp- -2 ) /3=2 .67 , while in the two-fire-ball model i t is, f rom (41), 2(2/ 'p--1 -~ 2fl)/(3 + 2fl) = 2.62.

Both models predict essentially the same result, which is also about the same as the exponent of the over all cosmic rays. The exponent in the two- fire-ball model is very insensitive to fl, since fl is very small, and is pr imari ly determined by / '~ . Errors or uncertaint ies in fl change very li t t le the ~ exponent. The exponents in both models are independent of the parameter y.

ii) Densit ies o/ ~. - From (33) and (40) the ratio of spectral-number densities of ~ to p is in Fermi 's model

(42) n;~(E) 1 8 rip(E) -- 4/~p -- 5 11 cY~aTafA(8-~r~laE(~-r~)l$ "

Strict ly speaking, N D and /~p in eqs. (33) and (40) are evaluated at different energies, as was discussed before. Equat ion (42) is obtained when N~ and/~D are considered constant a t these energies. However, if ~p a n d / ~ vary appreciably at these energies or more accurate results are needed, Np and T: should be evaluated at energies as discussed in Sect. 3 and 4. For ~ H : 1, T~f = ---- 0.9.10 ~" s (1), a = 31mb (~e), / , _-- 2.5, the rat io is

(43) ~x[n~ : 0.23(yE) -} .

For y = I , i t ranges from 2.3.10 -8 at E = 10 ~ eV to 2.3.10- ' a t 10 ~8 eV. For the two-fire-ball model, from (33) and (41),

(44)

(16) I. V. VOLKOVA: Izv. Akad. Nauk SSStr Ser. Fiz., 31, 1472 (1967).

3 5 6 ~ . c . CHE~

By using the same num ber as in F e rm i ' s model the ra t io is simplified to 0.12y-~176 For y ~-- 1, i t ranges f rom 3.8.10 -3 a t 10 ~2 eV to 6.8.10 -a

a t 10 ~8 eV. The ~ intensi t ies we obta ined here in general, agree wi th the resul ts in

ref. (~.~:). The two-fire-bull model predic ts larger ~ densities t han Fe rmi ' s model does, abou t 2 to 3 t imes higher f rom 10 ~2 to 10 ~s eV. Since the fire-b~ll model gives be t t e r fits of exper imenta l data. of secondary part icles in p-p colli- sion at higher energies, we expe.~t it to give a be t t e r predict ion of the an t ip ro ton

spec t rum. When sufficient da ta in mul t ip le pion-nucleon product ion with ex t reme

relat ivis t ic secondary nucleons are available, the p a r a m e t e r y can be de te rmined from, for example , t im expression for n~,. in eq. (21a) or (31) by experimenta~lly de te rmin ing the mos t p robable num ber b~ . a t ~ cer ta in energy 1~*. However ,

our results a.re not very sensi t ive to the var ia t ion of y as can be seen f rom (43) or (44). An increase of y by ~ factor 100 only decreases the ra t io by a factor

of about 2. Finally, the numerical resul ts given in this Section ~re subject to changes

of the in ters te l lar gas densi ty, p-p to ta l the exponent of cosmic proton, and the pa rame te r s in the exper imenta l ly de te rmined ~b-E relation. However , our resul ts have been expressed in closed forms containing these paramete rs , so t h a t numerical resul ts can be easily obta ined when different values of these

pa rame te r s are used.

The au thor wishes to th~nk Dr. S. D. D~nLL for an enl ightening conversat ion, and M. M0nELL for some compute r p rog ramming in the ear ly p~r t of the

calcula.tion. After the complet ion of this paper , Dr. S. DRELL brought to our a t t en t ion

the work of JOyES et. al. (is). They found t h a t the mul t ip l ic i ty v~ries approxi- m a t e l y as lnE* up to abou t 700 GeV. I f this logar i thmic dependence of E* is val id in the whole range of cosmic energy, then probably the mult i - f i re-b~l l model (~~ which predicts the correct energy dependence of mult ipl ici ty, would

give be t t e r resul ts .

(~) S. ROS~N: PJ~ys. Rev., 1511, 1227 (1967). (is) L. ~V. JoN~s, A. E. BussIAx, G. D. DE ME:ESTER, ]~. YV. Loo, D. E. LYON jr., P . V. RA~IA:NA ~[URTHY and R. F. ROTH: Phys. t~ev. Lett., 25, 1679 (1970).

�9 R I A _ S S U N T O (*)

Si calcola lo spcttro dei protoni dello stato stazionario m'i raggi cosmici ristfltanti della collisione fra protoni cosmici primari e l'idrogtmo interst~'llarc sulla base dcl modello statistic() di due fireball nella produzionc multiple di particelle. Si riportano per oonfronto

(*) Traduzione a ~.ura della Redazione.

ANTIPROTON SPECTRUM IN COSMIC RAYS ~

i r i s u l t a t i b a s a t i sul model lo or ig inale di F e r m i ad u n a sola fireball . Si ~ t r o v a t o t h e gli a n t i p r o t o n i obbed iscono a p p r o s s i m a t i v a m e n t e a d u n a legge esponenz ia le con u n espo- n e n t e di c irca 2.6 ehe ~ pressoch~ (stesso o rd ine di g randezza ) ugua le a l l ' e sponen te dei ragg i eosmiei in eomplesso. Si t r o v a ehe fl va lo re numer i co de l l ' e sponen te ~ d e t e r m i n a t o p r i n c i p a l m e n t e da l l ' e sponen t e dei p r o t o n i cosmici e r i s cn t e mol to poco dei p a r a m e t r i del model lo di due f i rebal l i n t r o d o t t i ne l calcolo. Un ' e sp res s ione eh iusa del r a p p o r t o t r a le dens i t~ n u m e r i c h e di ques t i a n t i p r o t o n i eosmici e dei p r o t o n i eosmiei si f o r m u l a in t e r m i n i de l l ' e sponcn te dei p ro ton i , del la dcns i t~ de l l ' id rogeno in te r s te l l a re , del la sezione d ' u r t o ane la s t i ea p -p to ta le , del la v i t a m e d i a dei pro~oni ne l la Galass ia e dei p a r a m e t r i delle due fireball . I va lo r i n u m e r i e i del r a p p o r t o v a r i a n o f r a circa 3 .8-10 -~ a 10 ~z e V a 6 .8 .10 -a a 10 ~8 eV.

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