An extensive program for the calculation of galactic cosmic-ray propagation
has been developed. This is a continuation of the work described in Strong &
Youssefi (1995). The main motivation for developing this code is the prediction
of diffuse Galactic gamma rays for comparison with data from the CGRO
instruments EGRET, COMPTEL, and OSSE. The basic spatial propagation mechanisms
are (momentum-dependent) diffusion, convection, while in momentum space energy
loss and diffusive reacceleration are treated. Primary and secondary nucleons,
primary and secondary electrons, and secondary positrons are included.
Fragmentation and energy losses are computed using realistic distributions for
the interstellar gas and radiation fields.
  This study indicates that it is possible to construct a model satisfying a
wide range of observational constraints and provides a basis for future
developments.Comment: 5 pages including 7 figures, latex, aipproc.sty, aipproc.cls,
  epsfig.sty. To be published in Proc. 4th Compton Symp., 1997 (27-30 April,
  Williamsburg, Virginia). Details can be found at
  http://www.gamma.mpe-garching.mpg.de/~aws/aws.htm

Moskalenko, I. V.

Strong, A. W.

English

arXiv

Andrew W. Strong

Igor V. Moskalenko

Crossref

Modelling cosmic rays and gamma rays in the Galaxy

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Modelling cosmic rays
and gamma rays in the Galaxy
Andrew W. Strong⋆ and Igor V. Moskalenko⋆†
⋆Max-Planck-Institut für extraterrestrische Physik, D-85740 Garching, Germany
†Institute for Nuclear Physics, Moscow State University, 119 899 Moscow, Russia
Abstract. An extensive program for the calculation of galactic cosmic-ray prop-
agation has been developed. This is a continuation of the work described in [1].
The main motivation for developing this code [2] is the prediction of diffuse Galac-
tic gamma rays for comparison with data from the cgro instruments egret,
comptel, and osse. The basic spatial propagation mechanisms are (momentum-
dependent) diffusion, convection, while in momentum space energy loss and dif-
fusive reacceleration are treated. Primary and secondary nucleons, primary and
secondary electrons, and secondary positrons are included. Fragmentation and
energy losses are computed using realistic distributions for the interstellar gas
and radiation fields.
INTRODUCTION
We are developing a model which aims to reproduce self-consistently ob-
servational data of many kinds related to cosmic-ray origin and propagation:
direct measurements of nuclei, electrons and positrons, gamma rays, and syn-
chrotron radiation. These data provide many independent constraints on any
model and our approach is able to take advantage of this since it must be
consistent with all types of observation. We emphasize also the use of real-
istic astrophysical input (e.g., for the gas distribution) as well as theoretical
developments (e.g., reacceleration). The code is sufficiently general that new
physical effects can be introduced as required. The basic procedure is first to
obtain a set of propagation parameters which reproduce the cosmic ray B/C
ratio, and the spectrum of secondary positrons; the same propagation con-
ditions are then applied to primary electrons. Gamma-ray and synchrotron
emission are then evaluated. Models both with and without reacceleration are
considered. The models are three dimensional with cylindrical symmetry in
the Galaxy, the basic coordinates being (R, z, p) where R is Galactocentric
radius, z is the distance from the Galactic plane and p is the total particle
momentum. The numerical solution of the transport equation is based on a
c© 1997 American Institute of Physics 1
2
Crank-Nicholson implicit second-order scheme. In the models the propagation
region is bounded by z = zh beyond which free escape is assumed. A value
zh = 3 kpc has been adopted since this is within the range which is consis-
tent with studies of 10Be/Be and synchrotron radiation. For a given zh the
diffusion coefficient as a function of momentum is determined by B/C for the
case of no reacceleration; with reacceleration on the other hand it is the reac-
celeration strength (related to the Alfvén speed vA) which is determined by
B/C. Reacceleration provides a natural mechanism to reproduce the B/C ratio
without an ad-hoc form for the diffusion coefficient [3,4]. The spatial diffusion
coefficient for the case without reacceleration is D = βD0 below rigidity ρ0,
βD0(ρ/ρ0)
δ above rigidity ρ0, where β is the particle speed. The spatial diffu-
sion coefficient with reacceleration assumes a Kolmogorov spectrum of weak
MHD turbulence so D = βD0(ρ/ρ0)
δ with δ = 1/3 for all rigidities. For this
case the momentum-space diffusion coefficient is related to the spatial one
[4]. The injection spectrum of nucleons is assumed to be a power law in mo-
mentum. The interstellar hydrogen distribution uses HI and CO surveys and
information on the ionized component; the Helium fraction of the gas is taken
as 0.11 by number. The interstellar radiation field for inverse Compton losses is
based on stellar population models and iras and cobe data, plus the cosmic
microwave background. Energy losses for electrons by ionization, Coulomb,
bremsstrahlung, inverse Compton and synchrotron are included, and for nu-
cleons by ionization and Coulomb interactions. The distribution of cosmic-ray
sources is chosen to reproduce the cosmic-ray distribution determined by anal-
ysis of egret gamma-ray data [5]. The bremsstrahlung and inverse Compton
gamma rays are computed self-consistently from the gas and radiation fields
used for the propagation. The π0-decay gamma rays are calculated explicitly
from the proton and Helium spectra using [6]. The secondary nucleon and
secondary e± source functions are computed from the propagated primary
distribution and the gas distribution, and the anisotropic distributions of e±
in the µ± system was taken into account [7].
ILLUSTRATIVE RESULTS
Some results obtained are shown in the Figures. The energy dependence
of the B/C ratio, and local proton and Helium spectra are shown in Figs. 1
and 2. The spectrum of primary electrons is shown in Fig. 6. The adopted
electron injection spectrum has a power law index –2.1 up to 10 GeV; this
is chosen using the constraints from synchrotron and from gamma rays. The
electron spectrum is consistent with the direct measurements around 10 GeV
where solar modulation is small and it also satisfies the constraints from e
+
e−+ e+
.
Above 10 GeV a break is required in the injection spectrum to at least –2.4 for
agreement with direct measurements (which may however not be necessary if
local sources dominate the directly measured high-energy electron spectrum).
3
The synchrotron spectrum at high Galactic latitudes (Fig. 3) is important
since its shape constrains the shape of the 1–10 GeV electron spectrum. An
injection index –2.1 (without reacceleration) is the steepest which is allowed
by the radio data over the range 38 to 1420 MHz. As illustrated, an index –2.4
as often used (e.g., [8]) gives a synchrotron spectrum which is too steep.
The modelled gamma-ray spectrum for the inner Galaxy, shown here for the
case of no reacceleration (Fig. 4), fits well the comptel [9] and egret [5]
data between 1 MeV and 1 GeV beyond which there is the well-known excess
not accounted for by π0-decay with the standard nucleon spectrum [10,11].
The electron spectrum used here is increased by a factor 2 over that shown
in Fig. 6; this is required to reproduce well the observed gamma intensities.
It is still within the range allowed by the positron fraction (see below), and
the shape accords well with the flatter electron injection spectrum required
by synchrotron data (Fig. 3). Inverse Compton is dominant below 10 MeV,
bremsstrahlung becomes important for 3–200 MeV. The lower bremsstrahlung
combined with π0-decay leads to a good fit to the flat spectrum observed in
this range, in contrast to previous attempts to model the spectrum with a
steeper bremsstrahlung spectrum [8]. The 10–30 MeV γ-ray longitude profile
(Fig. 5) at low latitudes from this model can be compared with that from
comptel [9]. The e+ spectrum (Fig. 6) and the positron fraction (Fig. 7)
agree well with the most recent data compilation for 0.1–10 GeV [7].
This study indicates that it is possible to construct a model satisfying a
wide range of observational constraints and provides a basis for future devel-
opments.
More details can be found on http://www.gamma.mpe–garching.mpg.de/∼aws/aws.html
REFERENCES
1. Strong A.W., Youssefi G., Proc. 24th ICRC 3, 48 (1995)
2. Strong A.W., Moskalenko I.V., SchönfelderV.,Proc. 25th ICRC,4, 257 (1997)
3. Heinbach U., Simon M., ApJ 441, 209 (1995)
4. Seo E.S., Ptuskin V.S., ApJ 431, 705 (1994)
5. Strong A.W., Mattox J.R., A&A 308, L21 (1996)
6. Dermer C.D., A&A 157, 223 (1986)
7. Moskalenko I.V., Strong A.W., ApJ, 493, in press (1998)
8. Strong A.W. et al., A&AS 120, C381 (1996)
9. Strong A.W., et al., in AIP Proc., 1997, this Symposium
10. Hunter S.D., et al., ApJ 481, 205 (1997)
11. Mori M., ApJ 478, 225 (1997)
12. Webber W.R., et al., ApJ 457, 435 (1996)
13. Seo E.S., et al., ApJ 378, 763 (1991)
14. Engelmann J.J., et al., A&A 233, 96 (1990)
15. Protheroe R.J., ApJ 254, 391 (1982)
16. Barwick S.W., et al., ApJL 482, L191 (1997)
4
FIGURE 1. The energy dependence
of the B/C ratio can be reproduced
with D0 = 2.0×10
28 cm2s−1, δ = 0.6,
ρ0 = 3 GV/c without reaccelera-
tion (thick line) and D0 = 4.2×10
28
cm2s−1, vA = 20 km s
−1 with reac-
celeration (thin line). Dashed lines are
modulated to 500 MV. Data from [12].
FIGURE 2. Left panel: the local proton spectrum for injection index 2.15 (thin solid
line), 2.25 (thick solid line) without and with reacceleration respectively, compared with the
measured ‘interstellar’ spectrum (dashed [13] and dashed-dot [11] lines). Right panel: the
Helium spectrum with injection index 2.25 (thin solid line), 2.45 (thick solid line) without
and with reacceleration respectively, compared with the measured ‘interstellar’ spectrum
(dashed [13] and dashed-dot [14] lines).
FIGURE 3. The synchrotron spectrum at
the NGP, and predictions for electron injection
indices –2.1 (dashed line) and –2.4 (solid line).
  0.5<l< 30.0 , 330.0<l<359.0
 -5.0<b<  5.0
FIGURE 4. The γ-ray spectrum for the
inner Galaxy, 330◦ < l < 30◦, |b| < 5◦.
egret data [5], comptel data [9].
5
10-30 MeV
-5.0<b<5.0
FIGURE 5. Longitude distribution of gamma rays in energy range 10-30 MeV. Histogram:
comptel [9], dashed: inverse Compton, thin line: bremsstrahlung, thick line: total.
E
2
I , MeV cm 2 sr 1 s 1
0.001
0.01
0.1
1
10
100
10 10 10 10 10 10 10 0 1 2 3 4 5 6
electrons
Buffington 75
Golden 84
Golden 94
Taira 93
Ulysses
positrons
Fanselow 69
Buffington 75
Golden 87
Golden 94
>
>
v
v
v
v
v
Total
sec. e+
sec. e 
Energy, MeV
E
2
I , MeV cm 2 sr 1 s 1
0.001
0.01
0.1
1
10
100
10 10 10 10 10 10 10 0 1 2 3 4 5 6
electrons
Buffington 75
Golden 84
Golden 94
Taira 93
Ulysses
positrons
Fanselow 69
Buffington 75
Golden 87
Golden 94
>
>
v
v
v
v
v
Total
sec. e+
sec. e 
Energy, MeV
FIGURE 6. Spectra of secondary e±, and of primary electrons. Full lines: our model with
no reacceleration (left) and with reacceleration (right). Dash-dotted lines by Protheroe [15]:
lower is his leaky-box prediction for e+, upper is his adopted electron spectrum.
e
+
=(e
+
+ e
 
)
0.1 1 10 100
0.01
0.1
Fanselow 69
Agrinier 69
Daugherty 75
Buffington 75
Hartman 76
Muller 87
Golden 87
Golden 94
Golden 96
Clem 96
MASS2
Barbiellini 96
Barwick 97
Energy, GeV
e
+
=(e
+
+ e
 
)
0.1 1 10 100
0.01
0.1
Fanselow 69
Agrinier 69
Daugherty 75
Buffington 75
Hartman 76
Muller 87
Golden 87
Golden 94
Golden 96
Clem 96
MASS2
Barbiellini 96
Barwick 97
Energy, GeV
FIGURE 7. Positron fraction for model with no reacceleration (left panel) and with reac-
celeration (right panel). Dashed and dash-dotted lines by Protheroe [15]: his predictions of
the leaky-box and diffusive halo models respectively. The data collection is taken from [16].


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