Spectra of cosmic ray electrons and positrons in the Galaxy
A new study of the cosmic ray electron and positron spectra is presented, using an anomalous diffusion model to describe the particles propagation in the Galaxy. The parameters defining the anomalous diffusion have been recently determined from the study of nuclei propagation. The computed electron...
Збережено в:
| Дата: | 2001 |
|---|---|
| Автори: | , , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
|
| Назва видання: | Вопросы атомной науки и техники |
| Теми: | |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/79890 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Spectra of cosmic ray electrons and positrons in the Galaxy / Spectra of cosmic ray electrons and positrons in the Galaxy // Вопросы атомной науки и техники. — 2001. — № 6. — С. 209-213. — Бібліогр.: 33 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-79890 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-798902015-04-07T03:01:50Z Spectra of cosmic ray electrons and positrons in the Galaxy Lagutin, A.A. Osadchiy, K.I. Gerasimov, V.V. Anomalous diffusion, fractals, and chaos A new study of the cosmic ray electron and positron spectra is presented, using an anomalous diffusion model to describe the particles propagation in the Galaxy. The parameters defining the anomalous diffusion have been recently determined from the study of nuclei propagation. The computed electron and positron spectra under assumption that positrons, as well as electrons, are accelerated by a galactic source, are in a good agreement with the measurements. The source spectral index, found from experimental data, in this approach turns out to be equal to 2.95 for electrons and positrons. The predicted positron fraction e⁺/(e⁺+e⁻) in high energy region E≈10² ÷10³ GeV is ~0.06. 2001 Article Spectra of cosmic ray electrons and positrons in the Galaxy / Spectra of cosmic ray electrons and positrons in the Galaxy // Вопросы атомной науки и техники. — 2001. — № 6. — С. 209-213. — Бібліогр.: 33 назв. — англ. 1562-6016 PACS 95.30.Jx, 95.85.Ry http://dspace.nbuv.gov.ua/handle/123456789/79890 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Anomalous diffusion, fractals, and chaos Anomalous diffusion, fractals, and chaos |
| spellingShingle |
Anomalous diffusion, fractals, and chaos Anomalous diffusion, fractals, and chaos Lagutin, A.A. Osadchiy, K.I. Gerasimov, V.V. Spectra of cosmic ray electrons and positrons in the Galaxy Вопросы атомной науки и техники |
| description |
A new study of the cosmic ray electron and positron spectra is presented, using an anomalous diffusion model to describe the particles propagation in the Galaxy. The parameters defining the anomalous diffusion have been recently determined from the study of nuclei propagation. The computed electron and positron spectra under assumption that positrons, as well as electrons, are accelerated by a galactic source, are in a good agreement with the measurements. The source spectral index, found from experimental data, in this approach turns out to be equal to 2.95 for electrons and positrons. The predicted positron fraction e⁺/(e⁺+e⁻) in high energy region E≈10² ÷10³ GeV is ~0.06. |
| format |
Article |
| author |
Lagutin, A.A. Osadchiy, K.I. Gerasimov, V.V. |
| author_facet |
Lagutin, A.A. Osadchiy, K.I. Gerasimov, V.V. |
| author_sort |
Lagutin, A.A. |
| title |
Spectra of cosmic ray electrons and positrons in the Galaxy |
| title_short |
Spectra of cosmic ray electrons and positrons in the Galaxy |
| title_full |
Spectra of cosmic ray electrons and positrons in the Galaxy |
| title_fullStr |
Spectra of cosmic ray electrons and positrons in the Galaxy |
| title_full_unstemmed |
Spectra of cosmic ray electrons and positrons in the Galaxy |
| title_sort |
spectra of cosmic ray electrons and positrons in the galaxy |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2001 |
| topic_facet |
Anomalous diffusion, fractals, and chaos |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/79890 |
| citation_txt |
Spectra of cosmic ray electrons and positrons in the Galaxy / Spectra of cosmic ray electrons and positrons in the Galaxy // Вопросы атомной науки и техники. — 2001. — № 6. — С. 209-213. — Бібліогр.: 33 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT lagutinaa spectraofcosmicrayelectronsandpositronsinthegalaxy AT osadchiyki spectraofcosmicrayelectronsandpositronsinthegalaxy AT gerasimovvv spectraofcosmicrayelectronsandpositronsinthegalaxy |
| first_indexed |
2025-07-06T03:50:18Z |
| last_indexed |
2025-07-06T03:50:18Z |
| _version_ |
1836867983621750784 |
| fulltext |
P A R T 2
A N O M A L O U S D I F F U S I O N , F R A C T A L S , A N D C H A O S
SPECTRA OF COSMIC RAY ELECTRONS AND POSITRONS
IN THE GALAXY
A.A. Lagutin, K.I. Osadchiy, V.V. Gerasimov
Altai State University, Barnaul, Russia
e-mail: gerasimov@theory.dcn-asu.ru
A new study of the cosmic ray electron and positron spectra is presented, using an anomalous diffusion model
to describe the particles propagation in the Galaxy. The parameters defining the anomalous diffusion have been
recently determined from the study of nuclei propagation. The computed electron and positron spectra under
assumption that positrons, as well as electrons, are accelerated by a galactic source, are in a good agreement with the
measurements. The source spectral index, found from experimental data, in this approach turns out to be equal to
2.95 for electrons and positrons. The predicted positron fraction e+/(e++e–) in high energy region E≈102 ÷103 GeV is
~0.06.
PACS 95.30.Jx, 95.85.Ry
1. INTRODUCTION
Observations of non-thermal radiation of the Galaxy
stimulated investigations of propagation of cosmic ray
electrons through the interstellar medium. Since basic
paper [1], the problem of calculation of electron
spectrum was considered in series of papers (see, for
example, [2-10]). The normal diffusion equation for
concentration of the electrons with energy E, N(r,t,E),
generated by sources distribution with density function
S(r,t,E),
+
∂
∂+∆=
∂
∂ )),,()((),,( EtrNEb
E
EtrND
t
N
+ ),,,( EtrS
(1)
has been used to study the electron energy spectrum
modifications in the interstellar medium (ISM). In the
equation (1) D is the diffusivity, b(E) describes the
energy-loss rate of electrons.
Recently, in the papers [11-13], new view of the
cosmic ray propagation problem was presented. It has
been shown that the ``knee'' in the primary cosmic ray
spectrum is due to large free paths (the so called “Lévy
flights”) of cosmic rays particles between magnetic
domains - traps of the returned type. As the “Lévy
flights” distributed according to inverse power law
,2,,3 <∞→∝ −− αα rAr is an intrinsic property of
fractal structures, in the fractal-like medium the normal
diffusion equation (1) certainly does not hold.
Based on this argument in [14] an anomalous
diffusion (superdiffusion) model for describing of
electrons transport in the fractal-like ISM was proposed.
This superdiffusion equation for concentration of the
electrons has been presented in the form
),,,(),,()((
),,())(,( 2/
EtrSEtrNEb
E
EtrNED
t
N
+
∂
∂+
+∆−−=
∂
∂ αα
(2)
where ),( αED is the anomalous diffusivity and
2/)( α∆− is the fractional Laplacian (called “Riss
operator” [15]).
The solution of superdiffusion equation (2) in the
case of point impulse source with inverse power
spectrum and the behavior of energy spectrum of
electrons in high energy region were found.
The main goal of this paper is to calculate the
spectra of electrons and positrons from sub-GeV to TeV
energies in the framework of anomalous diffusion
model. We don't use the assumption made in [14] that
the mean time of particle staying in a trap <τ> is finite.
In this paper, similarly to [13], we suppose that a
particle can spend anomalously a long time in a trap. An
anomalously long time means that
∞== ∫
∞
0
)(ττ ττ qd ,
so the distribution of particles staying in traps, q(τ), has
a tail of power law type ∝ B t-β-1 , t→ ∞ with β<1 (the so
called “Lévy trapping time”).
2. FLUX OF ELECTRONS FROM POINT
SOURCE
The flux of electrons, J(r,t,E), is related to the source
S(r0,t0,E0) by the propagator G(r,t,E;r0,t0):
).(),,(
),;,,(),,(
0000
00000
τδ −−×
= ∫ ∫∫
∞−
∞
ttEtrS
trEtrGdtdErdEtrJ
t
E
(3)
Here
∫=
0
,
)( '
'E
E Eb
dEτ (4)
)( 0 τδ −− tt reflects the law of energy conservation in
the continuous losses approach.
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 209-213. 209
210
Fig. 1. L- and G-components and total energy spectrum of electrons in ISM
Fig. 2. Modulated energy spectrum of electrons near the solar system
The propagator in the anomalous diffusion model
under consideration has the form [13]
),)),,(((
)),,((
4
),;,,(
/1),(
3
/3
00
αββα
αβ
βα
βα
π
−
−
Ψ×
×=
tEDr
tEDctrEtrG
(5)
where
.)()()( /3)1,(
1
)(
3
0
),(
3 ττττ αβββαβα dqrqr ∫
∞
=Ψ (6)
Here )()(
3 rq α is the density of three-dimensional
spherically-symmetrical stable distribution with
characteristic exponent 2<α ([16]) and )()1,(
1 τβq is
one-sided stable distribution with characteristic
exponent β [17]. The parameters α, β are determined
by the fractional structure of ISM and by trapping
mechanism correspondingly, the anomalous diffusivity
),,( βαED - by the constants A and B in the asymptotic
behaviour for “Lévy flights” (A) and “Lévy waiting
time” (B) distributions:
).,(/),(),,( βαβα EBEAED =
The energy-loss rate of relativistic electrons is
described by the equation (14) from [9]
),)((
)(
212
2
210
EEEEb
EbEbbEb
dt
dE
++≈
≈++==−
(7)
where b0=3.06∗10-16 n (GeV s-1) is for the ionization
losses of the electrons in ISM with number density n
(cm−3), b1E with b1=10-15 n (s-1) corresponds to the
bremsstrahlung energy losses, and b2E2 with b2=1.38 10−
16 (GeV s)-1 represents synchrotron and inverse Compton
losses (for B≈5µG and ω≈1(eV/cm3)), E1≈b0/b1, E2≈
b1/b2. Using (7), the solution of the equation (4) relative
to E can be presented in the form [9]:
.
)/())(1(1
)( 1
122
1
0 1
E
EEEEe
EEE b −
−+−−
+= − ττ
Taking into account that b1τ ≤ 3.15 10-8 n 105 =3.15 n
10-3 << 1, we derive from the later equation
.
)/()(1
)( 1
1221
1
0 E
EEEEb
EEE −
−+−
+=
τ
τ (8)
With help of the equations (3), (5), (8) it's easy to
calculate the flux of electrons for the sources interesting
for astrophysics. For example, for point impulse source
<
>
=Θ
Θ−Θ= −
,0,0
,0,1
)(
),()()(),,( 0
x
x
x
ttTrESEtrS p δ
we have
),)),(((
))(1()),((
)(),,(
/11),(
3
2
22
/11
)](/1,min[
],0max[
00
22
αββα
αβ
ττλ
τττλ
ττ
−−
−−−
+
−
−
Ψ×
×+−×
×= ∫
Er
EEbE
EdSEtrJ
EEbt
Tt
p
T
(9)
where
∫=
)(
'
'
'0
.
)(
),,(),(
tE
E
dE
Eb
EDtE βαλ
It should be noted that in the case β = 1, the equation
(9) comes to the solution, obtained earlier in [14]. If α =
2, β=1, we have the standard solution [1].
3. ENERGY SPECTRUM OF ELECTRONS
The flux J of electrons due to all sources of Galaxy
can be separated into two components:
).1(
)1(
1
1
0
kpcrJ
kpcrJJ
G
L
R
kpc
kpc
>+
+≤=+= ∫∫
(10)
The similar separation is frequently used in the studies
of cosmic rays (see, for example, [9] and references
therein).
The first component (L) in (10) describes the
contribution of the nearby sources (at distance r ≤ 1 kpc)
to observed flux J. The second component (G) is the
contribution of the distant sources (r> 1 kpc) to J.
The nearby sources used in our calculations are
presented in Table 1. Based on result (9) we suppose
∑
≤
=
kpcr
iiTL EtrJJ
1
),,,( (11)
where injection time T ≈ 104 ÷ 105 y.
Table 1. List of the nearby supernova remnants
Name R, pc t, 105 Source
Lopus Loop 400 0.38 [7]
Monoceros 600 0.46 [7]
Vela 400 0.11 [7]
Cyg. Loop 600 0.35 [7]
CTB 13 600 0.32 [7]
S 149 700 0.43 [7]
STB 72 700 0.32 [7]
CTB 1 900 0.47 [7]
HB 21 800 0.23 [7]
HB 9 800 0.27 [7]
Monogem 300 0.86 [18]
Geminga 400 3.4 [18]
Loop I 100 2.0 [18]
Loop II 175 4.0 [18]
Loop III 200 4.0 [18]
Loop IV 210 4.0 [18]
The second component (G) is evaluated under
assumption that the distant sources (r > 1 kpc) are
distributed uniformly both in space and time in the
Galaxy.
The parameters defining the anomalous diffusivity
and used in our calculations have been recently derived
from the study of nuclei propagation [12]: α =1.7; β=
0.8; D(E,α,β)=D0(E/1 GeV)δ with D0 ≈ (1 ÷ 4) 10-3
pc1.7 y-0.8 and δ = 0.27. Only one parameter p defining
injection spectrum of electrons in the sources is found
by fit. Extensive calculations show that the best fit of
experimental data may be get at p ≈ 2.95.
The spectra of L- and G- components and the total
spectrum in ISM are demonstrated in Fig. 1. The
experimental energy spectrum of synchrotron electrons
in ISM obtained in [19,20] is presented in this figure.
To describe the influence of the solar modulation,
the force model of [21] is used:
)].([
)()]([
)(
)( 222
222
mod tEJ
cmtE
cmE
EJ
e
e Φ+
−Φ+
−
=
The modulated spectrum near solar system under
condition that Φ (t) = 600MeV is shown in Fig. 2. We
conclude from Fig. 1 and Fig. 2 that experimental data
have a good agreement with our theoretical calculations.
4. ENERGY SPECTRUM OF POSITRONS
AND POSITRON FRACTION
It is commonly believed that the cosmic ray
positrons are a secondary component resulting from the
decay of π+, κ+ produced in the nuclear interactions of
cosmic rays with the ISM. Previous calculations of the
Fig. 3. Modulated energy spectrum of positrons near the solar system
Fig. 4. Positron fraction e+/(e++e-) near the solar system
secondary positron flux made in [10,22] have shown that
the predicted positron fraction is in good agreement with
the measurements up to 10 GeV, beyond which the
observed flux is higher than that calculated. To reproduce
the positron observations above 10 GeV either a primary
positron component or a harder interstellar nucleon
spectrum is required [9,10,22]. As the harder interstellar
nucleon spectrum is not consistent with direct proton
measurements, in this paper we suppose that high-energy
positron excess is due to contribution of a primary
positron component.
We demonstrate in Fig. 3 and Fig. 4 that a primary
positron component as large as 6% of the primary
electron spectrum allows us to reproduce the observed
positron spectrum as well as the positron fraction.
5. CONCLUSION
We have carried out a new study of the cosmic ray
electron and positron spectra using an anomalous
diffusion model to describe the particles propagation in
the Galaxy. The parameters defining the anomalous
diffusion have been recently determined from the study of
nuclei propagation. The computed electron and positron
spectra under assumption that positrons, as well as
electrons, are accelerated by a galactic source, are in a
good agreement with the measurements.
We have shown that the sources of high-energy (E ≥
100 GeV) electrons and positrons, observed in the solar
system are relatively young local sources (r ≤ 200 pc, t ∼
105 y), injecting particles during the time T ∼ 104 ÷ 105
y. The behavior of spectra in the low-energy region is
defined by distant (r ≥ 1kpc) sources.
The source spectral index, found from experimental
data, in this approach turns out to be equal to 2.95 for
electrons and positrons. The proximity of this exponent to
one obtained earlier [12] for nuclei components (p ≈ 2.9)
can indicate the same mechanism of particles
acceleration. The predicted positron fraction e+/(e++e-) in
high energy region E≈102 ÷103 GeV is ~0.06.
REFERENCES
1. S.I. Syrovatskii. Distribution of the relativistic
electrons in the Galaxy and a spectrum of
synchrotron radio emission // Astr. J. 1959, v.36, p. 17-
28 (in Russian).
2. V.L. Ginzburg, S.I. Syrovatskii. Origin of
cosmic rays. Pergamon Press, 1964, 384p.
3. G.B. Berkey, C.S. Shen. Origin and propagation
of cosmic-ray electrons // Phys. Rev. 1969, v. 188,
p. 1994-2010.
4. C.S. Shen. Pulsars and very high-energy
cosmic-ray electrons // Ap. J. 1970, v. 162, p. 181-
186.
5. S.V. Bulanov, V.A. Dogel, S.I. Syrovatskii.
Electron component of the cosmic rays //
Kosmicheskie issledovanija. 1972, v. 10, p. 532-545.
6. R. Cowsik, M.A. Lee. On the sources of
cosmic ray electrons // Ap. J. 1979, v. 228, p. 297-
301.
7. J. Nishimura, M. Fujii, T. Taira. Electron
spectrum at the high energy side // Proc. of the 16th
ICRC, 1979, v. 1, p. 488-493.
8. V.S. Berezinsky, S.V. Bulanov, V.L. Ginzburg
et al. Astrophysics of cosmic rays. North Holland,
Amsterdam, 1990, 528 p.
9. A.M. Atoyan, F.A. Aharonjan, H.J. Völk.
Electrons and positrons in the galactic cosmic rays //
Phys. Rev. D. 1995, v. 52, p. 3265-3275.
10. I.V. Moskalenko, A.W. Strong. Production
and propagation of cosmic-ray positrons and
electrons // Ap. J. 1998, v. 493, p. 694-707.
11. A.A. Lagutin, Yu.A. Nikulin, V.V. Uchaikin.
The “knee” in the primary cosmic rays spectrum as
consequence of the anomalous diffusion of the
particles in the fractal interstellar medium // Nucl.
Phys. B. (Proc. Suppl.), 2001, v. 97, p. 267-270.
12. A.A. Lagutin, D.V. Strelnikov, and
A.G. Tyumentsev. Mass composition of cosmic rays
in anomalous diffusion model: comparison with
experiment. Proc. of the 27th ICRC. 2001, v. 5,
p. 1896-1899.
13. A.A. Lagutin, V.V. Uchaikin. Fractional diffusion
of the cosmic rays. Proc. of the 27th ICRC. 2001, v. 5,
p. 1900-1902.
14. A.A. Lagutin, Yu.A. Nikulin, V.V. Uchaikin.
Anomalous diffusion of the cosmic rays // Izv. RAN.
Ser. fiz. 2002 (to be published).
15. S.G. Samko, A.A. Kilbas, O.I. Marichev.
Fractional integrals and derivations and some
applications. Minsk: “Nauka”, 1987, 688 p. (in
Russian).
16. V.M. Zolotarev, V.V. Uchaikin,
V.V. Saenko. Superdiffusion and stable laws //
ZhETF. 1999, v. 115, p. 1411-1425 (in Russian).
17. V.V. Uchaikin, V.M. Zolotarev. Chance and
Stability. VSP. Netherlands, Utrecht, 1999, 570 p.
18. T.A. Lozinskaya Supernova and star wind:
interactions with galactic gas. M.: “Nauka”, 1986, 304 p.
(in Russian).
19. W.R. Webber, G.A. Simpson and H.V. Cane.
Radio emission, cosmic ray electrons, and the
production of γ-rays in the Galaxy // Ap. J. 1980,
v. 236, p. 448-459.
20. J.D. Peterson, P.R. Higbie, J.M. Rockstroh
and W.R. Webber. A new look at galactic polar
radio emission and the local interstellar electron
spectrum. Proc. of the 26th ICRC. 1999, OG 3.2,
p. 17-20.
21. L.J Gleeson, W.I. Axford. Solar modulation
of Galactic cosmic rays // Ap. J. 1968, v.154,
p. 1011-1026.
22. R.J. Protheroe. On the nature of the cosmic
ray positron spectrum // Ap. J. 1982, v. 254, p. 391-
397.
23. J. Nishimura, T. Kobayashi,Y. Komori and
N. Tateyama. Observation of primary electron
spectrum and its astrophysical significance. Proc of
the 25th ICRC. 1997, v. 4, p. 233-236.
24. R.L. Golden, C. Grimani, B.L. Kimbell et al.
Observations of cosmic-ray electrons and positrons
using an imaging calorimeter // Ap. J. 1994, v. 436,
p. 769-775.
25. R.L. Golden, S.A. Stephens, B.G. Mauger et
al. Observations of cosmic ray positrons in the
region from 5 to 50 GeV // Ap. J. 1987, v. 188,
p. 145-154.
26. A.A. Lagutin, Yu.A. Nikulin, V.V. Uchaikin.
The “knee” in the primary cosmic rays spectrum as
consequence of fractal structure of the galactic
magnetic field. Preprint ASU-2000/4, 2000, Barnaul,
16 p.
27. D. Műller, S.V. Barwick, J.J. Beatty et al.
Energy spectra of electrons and positrons from 5 to
100 GeV. Proc. of the 25th ICRC. 1997, v. 4, p. 237-
240.
28. G. Barbiellini, G. Basini, R. Belotti et al.
Measurements of the positron and electron spectra
with the CAPRICE experiment. Proc of the 25th
ICRC, 1997, v. 4, p. 221-224.
29. S.W. Barwick, J.J. Beatty, C.R. Bower et al.
The energy spectra and relative abundances of
electrons and positrons in the galactic cosmic
radiation // Ap. J. 1998, v. 498, p. 779-789.
30. J. Nishimura, M. Fujii, T. Taira et al.
Emulsion chamber observations of primary cosmic-
ray electrons in the energy range 30-1000 GeV // Ap. J.
1980, v. 238, p. 394-409.
31. K. Tang. The energy spectrum of electrons
and cosmic-ray confinement: a new measurements
and its interpretation // Ap. J. 1984, v. 278, p. 881-
892.
32. M. Boezio, P. Carlson, T. Francke et al. The
cosmic-ray electron and positron spectra measyred at
1 1 AU during solar minimum activity // Ap. J. 2000,
v. 532, p. 653-669.
33. S.W. Barwick, J.J. Beatty, A. Bhattacharyya
et al. Measurements of the cosmic-ray positron
fraction from 1 to 50 GeV // Ap. J. 1997, v. 482,
L. 191-194.
Altai State University, Barnaul, Russia
1. Introduction
2. Flux of electrons from point source
3. Energy spectrum of electrons
4. ENERGY SPECTRUM OF POSITRONS AND POSITRON FRACTION
5. CONCLUSION
References
|