If antiprotons in cosmic rays are produced as secondary particles in sources, it is expected that positrons are also created by the same process. The interstellar spectra of positrons and electrons are calculated by taking into account such sources. Spectra are then compared with observations

Stephens, S. A.

English

NASA Technical Reports Server

ELECTRONS AND POSITRONS FROM EXPANDING SUPERNOVA 
ENVELOPES I N  DENSE CLOUDS 
S.A. Stephens 
Tata I n s t i t u t e  of Fundamental Research 
Homi Bhabha Road, Bombay 400005, India 
ABSTRACT 
I f  antiprotons (5) i n  cosmic rays a re  produced a s  secondary 
p a r t i c l e s  i n  sources, it is expected t h a t  positrons a r e  a l s o  
created by the same process. In t h i s  paper we calculate the  
i n t e r s t e l l a r  spectra of positrons and electrons by taking 
in to  account such sources, and compare them with observations. 
1. Introduction. Large f lux of 5, i n  excess of tha t  expected on the 
bas is  of a few g.cm-2 of matter traversed by cosmic rays, has been obser- 
ved [I-31.  his excess has been explained a s  being produced i n  the  
expanding envelopes of supernova (SN), which explodes i n  dense cloudlets  
i n  the  Galaxy [4]. It i S  shown that  i f  30% of the  observed cosmic ray 
nucleons come from such sources, f, observations can be explained sa t i s -  
f ac to r i ly  [5]. A s  f, being produced i n  in terac t ions  of cosmic ray nucle- 
ons with gas, it i s  expected tha t  pions and kaons a re  a l so  produced i n  
these in terac t ions ,  which decay t o  electrons and positrons. In these 
sources i n i t i a l l y  accelerated electrons a r e  depleted a t  high energies 
due t o  synchrotron process. A s  a r e su l t ,  high energy spectrum of the  
electron component from these sources a re  expected t o  be r i ch  i n  posi- 
trons. In  t h i s  paper, we evaluate the equilibrium spectra of positrons 
and electrons i n  the Galaxy by including the  electron component from 
these sources. 
2. Electron Spectrum i n  the Same Source. The evolution of electron com- 
ponent inside SN envelope i s  coupled t o  tha t  of nucleonic component and 
therefore,  one needs t o  solve the  following coupled equations. 
In the above equations, 1st term on the  RHS describes continuous 
energy loss  of pa r t i c l e s .  In the  case of protons, t h i s  energy loss  cor- 
responds t o  ionization and adiabatic  cooling. For electrons, i n  addition 
t o  the  above, one needs t o  include bremsstrahlung and synchrotron pro- 
cesses; the  l o s s  due t o  inverse Compton process is small compared t o  
synchrotron process except a t  a very ear ly  phase of SN. 2nd term i n  
Eqn. 1 is the  loss  of nucleons due t o  in terac t ion  and 3rd term takes care 
390 
  I  
 IN 
. t ens 
it te t l  
,  
OG 6.2-9 
 ti r tons (p) i  i   re s r  
ti les i  , is t  t itr  re lso 
r t  e e I  t is r l l t  e 
t llar tr  itr s  l trons t i  
i to t  r s,  ar   
. r  flux p, i  ess t t   t e 
is  f  - 2 t r rse  s ic , s 
 [1 ] Th  ss s l i ed s i  in t e 
i  l s s r  ( ), i  l s in se l l ts 
in t e al  [ ]. ~ s n t t  0% t e s r  i   
le s c e. fr  s c  s r s, p r ations   l i  
t rily [ ]. s p i  r  in i teractions c s ic r  
ns it  s, t is ecte  t t i s a  s re lso r in 
t se i t r tions, i   o l trons  t . I  t se 
rces i itially l t  l ctrons re leted t i  r ies 
e to r tr   s a r , i  er  s tr  the 
lectron ponent fr  t ese rces re te  to  rich in 
t  In t is l ate t e ilibriu  s tra itr  
a  lectrons in the ala  i l i  t e l tr  c ponent fr  
t ese s r . 
l ctron ectr  in the a e rc . he e olution lectron c
t i side S e el e is coupled to t at l ic c ponent and • 
t f r , e s to s lve t e f ll i  le  ti . 
dJ 
d dE dE' 
-..E (J P - ~ J + f ~J = dt ) dt dE p A p I A p I E E 
(1) 
dJ + d <lE+ e- (J + e-= at) + Q + dt dE e- e- (2) 
In the above e ti s, 1st ter  on the  escribes continuous 
e er  loss rticles. In the case r t , this energy loss c
responds to i ization a  a iabatic c li . r el tr , in a ition 
to the above, one nee.ds to include re sstra lung and s c r tr  
cesses; the loss due to inverse o pton r cess is s all co pared to 
synchrotron r cess except at a ery early ase o S . 2n  ter  in 
q . 1 is the loss o nucleons due to interaction and 3rd ter  takes care 
https://ntrs.nasa.gov/search.jsp?R=19850025795 2020-03-20T17:59:37+00:00Z
of the finite inelasticity during collision. In these terms, v is the 
velocity of interacting particle, p the density and X the interaction 
mean free path. The production term Qef is given as 
In this equation, integral over 8 describes the production of pionq, in 
which pL and8 are the transverse momentum and angle of emission of 
pions; the invariant cross-section E(d%/d 3) depends upon E and hence 
the integral over Ep. The integrals over 4 and Fi, take carg of the 
energy distribution of decay products as pion decays to muon and muon to 
electron. The parameters relating to the production of pions and the 
evolution of SN are described elsewhere [ 5,6] . 
While evaluating Eqns. 1 and 2, we have set the initial condition 
that the energy spectra of nucleons and negatrons are simple power law in 
rigidity of the type A R - ~ . ~ ~ ,  where A = 2.5~10~ and 150 particles/cm2.sr. 
s.GV/c) for nucleons arid electrons respectively. It is assumed that the 
initial acceleration is complete at the onset of the adiabatic phase and 
the remnant leaves the cloudlet when cosmic rays traverse -50 g.cm-2 of 
matter. The magnetic field inside the cloud is expected to scale as 
~ 2 ~ ~ n ~ ,  with B~ = 4 JG for n~ = 1 atom.cm-3. The photon density corres- 
ponds to an outburst of optical photons of 1043 erg/s soon after the 
explosion, which decay with an e-folding time of 0.2 yrs. 
3. Equilibrium Spectrum in Interstellar Space. The equilibrium spectrum 
of ef in interstellar space is evaluated by solving the continuity equa- 
t ion 
In this equation, the source term Q consists of 3 terms. (1) The elec- 
e f trons from SN, which explode in dense clouds. This is obtained from 
Eqn. 2 and normalized to the observed g5 flux value at 9 GeV [I]. The 
production term is then given by J /T. (2) The second term comes from 
e f 
sources in which cosmic rays traverse matter in an energy dependent 
manner [71. It is assumed that at energies <2 Gv/c, the matter traversed 
is a4 g.cm-2 and above this energy it is proiortional to R-O.~. The 
secondary production of electrons is calculated using the equilibrium 
nucleon spectrum in the source and Q + is obtained after allowing for the 
energy dependent leakage. (3 )  The tgird term comes from the production 
of secondary electrons in interstellar space. In the case of negatrons, 
we have also included a t e n  which is 70% of 150 E-~.~~/T. The resident 
time T in Eqn. 4 is considered to be free parameter and is adjusted to 
fit the observed electron spectrum. 
4. Results. We have plotted in the lower part of Fig. 1, the observed 
flux values of electrons from recent experiments above 1 GeV [8-101. For 
convenience, the flux scale is multiplied by E3. The curve shown here 
is the calculated spectrum using a value of 5x106 yrs for T. One finds 
391 
h  in te n c ty ng l sio  n hese 
 nteracting  cle,  h i A 
e ree  t o erm  + s ven  
e-
dE 
= Iff I -1! 
l1'e 
dE 
--.1I 
11')..1 
de 
OG 6.2-9 
erm ,  s h  
h  n eraction 
(3) 
n h s ion, n e   e ibes h  ion  s  n
i P.J.. nd e  h  ransv rse e tum   i on  
 h  n a  ion ~/d     
h  n e   .  n e  ~ ~ ak   h  
ribution     o  o   o 
e ro   et ating o h t o  h  
ion N  be sew  6 . 
hil  ing s.    h ia  i ion 
h  a eo  o   im e er law n
g h yp  -2.75, er   .5x104 0 rt c 2 .sr. 
.G /  eo   e trons v y.  s ssum h  h  
a  ion s pl  h  h  t  
h e  eav  h  l   i  a  rav se 50 c -2 
a t r.  agnet  ield n ide h ou  s o e  
B2apnH , i o )..1G  H  o . - 3 •   si
 o  t r  t    oo  h  
on, i i  olding im    
i ibrium rum n    i ibrium trum 
 ± n lar  s e  solving h  t ty 
ion
dJ + e-
dt 
J + 
e-
T 
(4) 
n h s o , h   erm + si   erm  1)  
ron  rom  i o  n ~  o  i  s n rom 
.  al o h  ~  p lux    1].  
t o  erm s he   + . 2)  eco  erm  rom 
 hi i  a  rav fse att n  
anner ]   sum  2 V/  h  att rav rsed 
s ~4 c -    p rt o 0 6.  
t o ron   ed n h i ibrium
eo  trum h    t n low  h  
 ea . )  fii erm  rom h t
ro n a   t  
  s n lude   erm hi  0%  - 2 . 75 /T.  d  
im n .  o ree et  o 
 h  v e tron rum  
 esult e  n h  o  r i  h  
lux  ron  rom  t     ]   
i  h lux  ulti l .   how  
 h e trum n  l    .  ind  
NULLKR L r A N G  
1 , . 1 . I  I . . . .  , I d  
lo0 10' 10 
ENERGY IN GcV 
Fig. 1 The spectrum of elec- Fig. 2 Positron fraction is 
trons is shown on the plotted as a function 
lower part of this of energy 
figure and that of e'
is shown on the upper 
part 
a good fit to the observed data points. We make use of the same inter- 
stellar parameters to calculate the positron spectrum from Eqn. 4. 
The calculated positron spectrum in interstellar space is plotted 
in the upper part of Fig. 1. We have pointed out earlier that high 
energy electrons in SN envelopes are depleted by synchrotron energy loss 
process, and the positron spectrum is quite sensitive to the Bo value 
used in Eqn. 2. Three curves shown in Fig. 1 is based on different / 
values of Bo. The spectral steepening noticed in this figure is due to 
the combined effect of energy loss processes and the energy dependent 
matter traversal in sources. The data points plotted here are from 
Golden et al. [ I l l .  One notices from Fig. 1 that the positron spectrum 
is quite consistent with the model that major contribution comes from 
sources which produce antiprotons in cosmic rays. The spread in the 
data points is such that one cannot obtain a precise value of Bo, though 
a value between 4 and 8 VG could fit the data points well. 
We have shown in Fig. 2, the fraction of positrons as a function 
of energy for 3 different values of Bo as in Fig. 1. The increase in 
this ratio with energy arises from the fact that initially accelerated 
electrons in SN, which explode in dense clouds are depleted due to 
synchrotron process and that the electron component at higher energies 
is dominated by secondary particles. The initial decrease in the ratio 
is primarily due to the depletion of secondary electrons through synchro- 
I 
,.. 10-l 
, .. 
I:; 
,~ 
a 
...... 10 
o' 
10 
i   
.I'II1L.LRIli 'TANG 
• N IINI 1'111"11 et <1/ 
• $O£.W" • .,. 
 trum
ro   how  h
low  rt f h  
igure h f + 
s how  h
rt
392 
'14 
·12-
f-
·10 
" ~ ·08 
~ 
.. (U 
v 
" '06 : 
+ 
<lJ 
i 
'~i 
i   
A. 
I. 
c. 
OG 6.2-9 
1 
110 
zps 
4)'6 
r )"6 
si ro ra ion  
  ion 
  o h i t .   h  am  n -
et o e h si ro trum rom .
 e i ro trum n a   
h r i  e  
e ron     l ron g o  
 h si o trum i  ive h  
n . .   how  i .  fe e  
.  t  teepe ng t c igure  
h  bi o   h  
att rav  n   i  rom 
l  11]   t  rom   h i ro trum
s i  si e  i h  odel h aj iQ ion o  rom 
 hi  proton  i    h  
i    t   hou  
  w   G h  i  ell  
 how  n   h raction ro   u ion 
g   e e     n i .  n ea e n
 io i se  rom h  ia ly e
ectrons n  i o  n  loud  l  o
ron h e ron   
s i  eco y rt  ia   in h  io 
s ari  o h  l ion eco  ectron  hroug  y
.. 
tron process and later by the energy dependent source term in Eqn. 4. 
One also noticesthat the ratio attains an asymptotic value close to 
about 2% at very high energies. The absolute value of this ratio can be 
easily adjusted by about 20% by increasing the primary electron flux 
values by the same factor. In that case a small adjustment in the value 
of T could reproduce the good fit obtained for electrons in Fig. 1. We 
have also plotted the observed values of the fraction of e+ [11,121. 
One noticesthat the general trend of the data points are similar to the 
calculated values except for a normalization of about 20%, which is con- 
sistent with the errors in the results. 
Thus we conclude that SN exploding in dense clouds seems to 
explain not only the 6 results but also the e+ data. We have also shown 
elsewhere that the observed Cos-B Y-ray sources could also be explained 
[131. 
References 
R.L. Golden et al., Phys. Rev. Letters, 43, 1196 (1979) 
E.A. ~ogomolov et al., Proc. 17th ICRC, 9, 146 (1981) 
A. Buffington et al., A~.J., 248, 1179 (1981) 
B.G. Mauger and S.A. Stephens, Proc. 18th ICRC (Bangalore), 
9, 171 (1983) 
- 
S.A. Stephens and B.G. Mauger, Proc. 19th Rencentre de Moriond 
Astrophysics Meeting, p.217 (1984); Ap.Sp.Sci. (1985) in press. 
G.D. Badhwar and S.A. Stephens, Proc. 15th ICRC, 11, 149 (1977) 
S.A. Stephens, Proc. 17th ICRC (Paris) , 2, 214 (lzl) 
D. Muller and J. Tang, Proc. 18th ICRC (Bangalore) , - 2, 60 (1983) 
J. Nishimura et al., Ap. J., 238, 94 (1980) 
R.L. Golden et al., Ap. J., 287, 622 (1984) 
R.L. Golden et al., this Conference 
A. Buffington et al., Ap. J., 199, 669 (1975) 
S.A. Stephens, This Conference, OG 2.5-3 
393 OG 6.2-9 
ro  t  erm qn. .
t  o a n ptoti
t  r i  l
i j  t 0%  n ro u
l  . t  all t
od  t  ro i . . e
l t f ion , ]  
 t ces er l re t i t il
l t  r ali t t  hi
e it lt
 l o  e  
l t l p l t t . e ow  
sew s -r  l l
[1 ] • 
ef
[1] . . ol t l., s. ev. ett    
[2] . . B m lov t 1., r . t C  ~, 1 ) 
[3] . uffi t l., p.J.   1981) 
[4] . . auger . . t :- r . t CR  gal r )  
   
[5] . .  . . auger  r . t encent ori
st ysi eti  .  ) i p.Sp . Sci.  .
[6] . . ar . .  r . t C   1 ) 
[7] . . s ep , r . CRC ri  ~,  1981  
[8] . u ler  r . CR  gal r  ~,  ) 
[9] . i r t l. p. . , ,   
[10] . . ol l . , . .,  1984) 
[11] . . ol t l., onf
[12] . uff l., p. . , ,66  1975) 
[13] . .  i onfer  .  


Electrons and positrons from expanding supernova envelopes in dense clouds

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server