A detailed calculation of the production spectra of charged hadrons produced by interactions of cosmic rays in the interstellar medium is presented along with a thorough treatment of pion and muon decays. Newly parameterized inclusive cross sections of hadrons were used and exact kinematic limitations were taken into account. Single parametrized expressions for the production spectra of both secondary positrons and electrons in the energy range .1 to 100 GeV are presented. The results are compared with other authors' predictions. Equilibrium spectra using various models are also presented

Ng, L. K.

Wong, C. M.

English

NASA Technical Reports Server

Re-estimation of the Production Spectra of Cosmic Ray 
Secondary Positrons and Electrons in the ISM 
C.M. Wong and L.K. Ng 
Physics Department, University of Hong Kong 
Hong Kong 
Abstract 
The present work includes a detaile? cflculation of the 
production spectra of charged hadrons (T-,K-) produced by 
interactions of cosmic rays in the interstellar medium, and 
a thorough treatment of pion. and muon decays. Newly paramet- 
rised inclusive cross sections of hadrons were used and 
exact kinematic limitations were taken into account. Single 
parametrised expressions for the production spectra of bofh 
secondary positrons and electrons in the energy range 10- - 
18+3 GeV are presented. The results are compared with other 
authors' predictions. Equilibrium spectra using various 
models are also presented. 
1. Introduction. It is well known that cosmic ray secondary 
positrons and electrons in the ISM arise from the rue decay 
chain of secondary IT produced in the interactions of+cosmic 
rays with the ISM. Given the production spectra of e-, their 
equilibrium spectra can be calculated by model predictions. 
A+nd by combining with the available high-energy cosmic ray 
e-data, we can obtain a better understanding of cosmic ray 
propagation in the galaxy. However, there is no general 
agreement among the previous calculations of the production 
spectra of 4 .  In this work, we have re-estimated these 
spectra based on newly parametrised inclusive cross sections 
of hadrons which are applicable at energies from the ISR 
down to those near the production threshold. Moreover, exact 
kinematic limitations are taken into account. 
e 
2. Production spectra of hadrons. Re-estimation of the 
production spectra of hadrons is motivated by the existence 
of large divergence among various inclusive cross section 
formulae1, and the availability oE newly parametrised 
formulae based on more recent accelerator data2. 
The production spectra of hadrons in p-p - collisions is 
given by 27r 00 omax d30 
P (E,) =m j(Ep)dEp (~-3) pt dO 
P Et 0 dp 
where Et is the threshold energy of p-p collisions for a 
given hadron energy Eh, ha, is the maximum angle of hadron 
emission in the laboratory system and j(Ep) is the differen- 
tial flux of "interstellar protons taken from ~ef.3. Calcula- 
tion was done without applying any approximation to the 
integrations,and exact kinematic limitations were taken into A 
386 
OG 6.2 - 8 
e- t o h  t o t  i   
y si ron  l ro  h  SM 
. . o  . .  
si  e ar t  niversi y   
  
bstr  
 r ncludes  t i d a ion  h  
ion a g   TI± ,K±) 
n eractions i  a  n h  lar e  
 horough reatm  o   s. e l et
ised n usive  ions  er  
 n at  im ations er  ake  nto t   
etr ions  h  on a 91  
y ron  e ron  n h  g  ang   
0 3 V e t .  s  a  i  
 i t . ui brium a n o  
odel   so t .
. ro o   s el  ow  h  i  a  c y 
ron  e trons n h  SM se rom h  TI~   
n co y TI n h  eractions  s c 
a  i he S . i  h  on   ±   
ibrium a n  ted ode  i
nd  bi ng i  he a e c ay 
± ,   n  t anding  ay 
ion n he . ever  h e s   
eem   he o  ions  he t o  
ra  e±  n s r    estim ed h  
ra  l  etr e n usive s ions 
  i  e l ca e   rom he SR 
ow  o ho e  he t o h es  ore er   
n at c im ations e aken nto t  
. on a   e- im t o he 
on a   s oti he tence 
 arge v rgen   ious n usive o s ion 
orm ae l   he i i ty f l  etr sed 
orm ae  or   o  2 • 
 
ven 
 t s he hreshold gy  isions o   
ven on gy ~, Gmax s he imum e  on 
i on in ~e ab ory ystem  (Ep) s he feren
ia  lux  interstellar on  aken rom R .3. l
ion   i  ng  ion o he 
nteg ations,and  ne at c im ations  aken nto 
https://ntrs.nasa.gov/search.jsp?R=19850025794 2020-03-20T17:59:34+00:00Z
account. Results for rf and K' are shown in figure 1. The 
uncertainties in the input spectrum gives rise to the shaded 
areas at low energies. 
3. Production spectra of ef. Production spectra of ef in 
interstellar space is obtained by rigorous treatment of 
pion and muon decays taking into account of precise decay 
kinematics and muon decay asymmetry. Correction for the 
cosmic ray and the ISM compositions4 and the contribution 
from kaons5 are then added. The results are given in figures 
2 and 3. 
LOC 
LOG 
______-------- 
- Present work 
LOG - 1  
2 
., 
C: 
-% 
u 
Fig.2 Production spectra of Fig.3 Production spectra of 
.,.,+? p+ and e+. T-? P- and e'.
- 
1- 1 - 
'I 
t; 
m 
Z 
- 
,Y 0 -  
n 
., -
I 
W 
I 
Fig. 1 
%+ 
Production spectra 
of charged hadrons 
in p-p collisions. 
38·7 OG 6.2 - B 
account. Results for n± and K± are shown in figure 1. The 
uncertainties in the input spectrum gives rise to the shaded 
areas at low energies. 
L~;---______ -r ________ ~ __________ r-________ ~ 
-
-> 
• t:I 
.. 
';'e 1 
u 
III 
.. 
.. 
.. 
Ng 
u:; 0 
Ii: 
-
• N 
... 
+ 
'TC 
7( 
-~--------------------------------,p"'---
,.rIP' 
,l' 
~1l; 
j 
l/ /;' 
l , " " I 
,,/,, 
,,-' " 
" 
" 
" 
" 
+ --~---------------------------­
... .. -.. 
, ... -... 
}( ---------------------
-----
-----
~ -lr_71--------~· ~O--------~1~--------~2--------~' 
Enerty (GeV) 
Fig. 1 
Production spectra 
of charged hadrons 
in p-p collisions. 
3. Production spectra of e±~ Production spectra of e± in 
interstellar space is obtained by rigorous treatment of 
pion and muon decays taking into account of precise decay 
kinematics and muon decay asymmetry. Correction for the 
cosmic ray and the ISM compositions4 and the contribution 
from kaons5 are then added. The results are given in figures 
2 and 3. 
LOG 
2.r-----~-------r------~----~ 
> 
• t:I 
l" 
'e . 
u 
III 
.. 
.. 
.. 
... 
. ,/ / ' 
" 
" 
" 
7(+ 
",,:::.<····-···-····-··· .. ········e·+-······ .. 
Protheroe (1982~ 
Present work 
- 1 '--------''----__ ---'-______ -'-____ --' 
LOG -1 o . 1 
Energy(GeVI 
2 
Fig.2 Production spectra of 
n+, p+ and e+. 
3 
LOG 
2 
. 
> 
• ..,
, 
-... 1t N 
'Ii 
u 
at }-'-
.. 
.. 
.. 
~ e-
~ 
... 
ili: 
. 
.. 
... 
LOG -1 2 3 
Energy (GeV ) 
Fig.3 Production spectra of 
- p- and e-. n , 
The spectra can be parametrised as follows with the 
coefficients listed in Table I. 
~ ( 3 ~ )  = [C1EgC2]/[1 + CAEe-'~] fo r  0.1 GeV 4 Ee'&lOOO.O GeV 
2 
where CA = C3 + C41n Ee + C51n Ee 9 
. CB = C6 + C71n Ee + c81n2~,. 
7, 
- 
Above fo r  electrons. 
TABLE I 
Limit 
Upper 
Lower 
Above for  positrons. 
4. Results and discussion. ~igure' 4 compares the present 
calculated positron spectrum with the previous calculations. 
Our results are significantly higher than the others 
especially near the energy range 1 - 10 GeV. The differences 
are mainly due to different adoption of the inclusive cross 
section parametrisations and different assumed interstellar 
proton spectra. However, since in our calculation, more 
recent accelerator data have been used and rigorous kine- 
matic treatment has been done, we believe the production 
spectrum has been improved. 
C1 C2 C3 C4 C5 '6 C7 '8 ~. 
2.84 2.72 0.175 -0.146 -0.00992 0.892 0.00499 0.0389 
2.84 2.73 0.252 -0.148 -0.0454 1.09 -0.0127 0.0474 
* 
~ i m i t  
'upper 
Lower 
Fig.4 Production spectra of e+ predicted by various authors. 
c C2 c3 c4 c5 C6 c, C8 
1 . 8  2.72 0.371 -0.305 0.0698 0.768 0.0444 0.0536 
1.58 2.72 0.433 -0.312 . 0.0570 0.901 0.0688 0.0575 
388 
OG 6.2 - 8 
t r etri low  it
effi ste abl . 
Q(Ee) IEe- C ]/[1 
r    ;   e  
]       
Li it S C2 , 
Upper .  . 2 7; 
Lo er 2.84 .  2 
b f r positrons ove 0 . 
Limit Cl C2 C, 
Upper .5  .  
Lo er .  .  4;; 
bove for electrons. 
 " '-lOOO.   
C  , C6 C  C8 
.146 .00992 2 . 99 ;  
.148 454  .0127 .  
C.1 C" 6 C7 
.; 5 .Ob   .  . ;  
.; 2  .  
 I 
s l s nd sc  Figure"  o  the  
c ated tron spectrum i h the e ous io  
 re ts re ig ficantly g  than the h s 
i y  the rgy range 1 - 10 . e fferences 
are ai  due to fferent ion  the inclusive ross 
se tion etr tions and fferen  assumed in e tellar 
oton s  e , since in o r c c ation, e 
recent a erator d a have been used and rigorous n
at c treatment has been d , e ieve the pro ion 
spectrum has been im . 
lO'r-------------~------------~------------_,------------~ 
i 
'E 
u 
... 
~ 
.. 
'" 
§ 
a.. 
10' 
~ -I 
::; 10 
... 
-- --- R ••• " (1114)7 
-------- ----- 011 ...... " ht'" (1111)5 
-------_ 1111., ., II (Itn)8 
-------- ""hll. (1I1Z)6 
------ " ••••••• rt 
E nerpy IGeV) 
Fig.4 Production spectra of e+ predicted by various authors. 
, 
For the purpose of illustration, equilibrium spectra 
have been calculated using this positron production spectrum 
and various propagation models as shown in figure 5. It is 
seen that the predicted curves are all close to the recent 
datum due to Golden et all3. 
+ Fig.5 Equilibrium spectra of e based on the present 
calculated production spectrum. 
5. Acknowledgement. Authors are grateful to Dr. L.C. Tan 
and Dr. P.K. MacKeown for their helpful comments and to 
Professor D.J. Newrnan for his support and encouragement. 
References. 
1. Ghoshdastidar M R et al, Proc.17th ICRC -5- (1981) 1. 
2. Tan L C et al, J. Phys. G: Nucl.Phys. 9 (1983) 1289. 
3. Ormes J F et al, Ap. J. 272 (1983) 756: 
4. Tan L C et al, J. Phys. G: Nuc~. Phys. 1 (1981) 1135. 
5. Orth C D et al, Ap. J. 286 (1976) 312. 
6. Protheroe R J, Ap. J. - 254 (1982) 391. 
7. Ramaty R., High Energy Particles and Quanta in 
Astrophysics, MIT Press, Cambridge, (1974)122. 
8. Giler M et al, J. Phys. A Math.Gen., 10 (1977) 843. 
9. Fanselow J L et al, Ap. J. - 158 (19691771. 
10.Daugherty J K et al, Ap. J. 198 (1975) 493. 
1 9 9  (1975) 669. 11.Buffington A et al, Ap. J. - 
12.Hartman R C et a1,Ap. J. - 204 (1976)927. 
l3.Golden R L et al, Proc. 16th ICRC (Kyoto) - 1 (1979) 470. 
389 
OG 6.2 - 8 
For the purpose of illustration, equilibrium spectra 
have been calculated using this positron production spectrum 
and various propagation models as shown in figure 5. It is 
seen that the predicted curves are all close to the recent 
datum due to Golden et all~ 
10'r-------------,--------------,--------------, 
-2 10 
• fonltl ... (19&9) 9 
o Doughe", (1975)10 -.-.-
'" Buffinglon (1975)1r----
• Hmmo. (1976) 12 
• Gold.n (1979) 13 
CI ••• d Gol .. , _.1 
L .... ' 801 M.d.1 
Di ... Moi.1 
-J 
10 ~~----------~------------~~------------7 10-' 10' 10' 10' 
Fig.5 Equilibrium spectra of e+ based on the present 
calculated production spectrum. 
5. Acknowledgement. Authors are grateful to Dr. L.C. Tan 
and Dr. P.K. MacKeown for their helpful comments and to 
Professor D.J. Newman for his support and encouragement. 
References. 
1. Ghoshdastidar M R et aI, Proc.17th ICRC 2- (1981) 1. 
2. Tan L C et aI, J. Phys. G: Nucl.Phys • ..2.. (1983) 1289. 
3. Ormes J F et aI, Ap. J. 272 (1983) 756. 
4. Tan L C et aI, J. Phys. G: Nucl. Phys. 7 (1981) 1135. 
5. Orth C 0 et aI, Ap. J. 206 (1976) 312. 
6. Protheroe R J, Ap. J. 2 .54 (1982) 391. 
7. Ramaty R., High EnergyJParticles and Quanta in 
Astrophysics, MIT Press, Cambridge, (1974)122. 
8. Giler M et aI, J. Phys. A Math.Gen., 1£ (1977) 843. 
9. Fanselow J L et aI, Ap. J. 158 (1969) 771. 
10.Daugherty J K et al, Ap. J. 198 (1975) 493. 
11.Buffington A et al, Ap. J. 199 (1975) 669. 
12.Hartman R C et al,Ap. J. 204 (1976)927. 
13.Golden R L et aI, Proc. 16th ICRC (Kyoto) 1.. (1979) 470. 


Reestimation of the production spectra of cosmic ray secondary positrons and electrons in the ISM

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