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

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 detailed calculation of the production spectra of charged hadrons (TI± ,K±) produced by interactions of cosmic rays in the interstellar medium, and a thorough treatment of pion . and muon decays. Newly paramet­rised inclusive cross sections of hadrons were used and exact kinematic limitations were taken into account. Single parametrised expressions for the production spectra of b91h secondary positrons and electrons in the energy range 10 -10+ 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 TI~e decay chain of secondary TI 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. And 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 e±. 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.

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 formulae l , and the availability of newly parametrised formulae based on more recent accelerator data2 •

The given by

where Et is the threshold energy of p-p collisions for a given hadron energy ~, Gmax is the maximum angle of hadron emission in t~e laboratory system and j(Ep) is the differen­tial flux of interstellar protons taken from Ref.3. Calcula­tion was done without applying any approximation to the integrations,and exact kinematic limitations were taken into

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

The spectra can be parametrised as follows with the coefficients listed in Table I.

Q(Ee) = [CIEe- C2]/[1 +

where CA = C; + C41n Ee +

C] = C6 + C71n Ee +

Limit S C2 C, Upper 2.84 2.72 0.17; Lower 2.84 2.73 0.252 Ab f r positrons ove 0 . Limit Cl C2 C, Upper 1.58 2.72 0.371 Lower 1. 58 2.72 0.4;; Above for electrons.

GeV " Ee'-lOOO.O GeV

C4 C, C6 C7 C8 -0.146 -0.00992 0.892 0.00499 0.0;89 -0.148 -0.0454 1.09 -0.0127 0.0474

C.1 C" C6 C7 C8 -0.;05 0.Ob98 0.768 0.0444 0.05;6 -0.;12 0.0570 0.901 0.0688 0.0575

TABLE I

4. Results and discussion. Figure" 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 kine­matic treatment has been done, we believe the production spectrum has been improved.

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.