2.2
Gamma-ray emission from Galactic cosmic rays
The radio emission provides a view of the Galactic population of energetic
electrons up to few GeV. However, the nucleonic component of cosmic
rays remains largely invisible at radio frequencies. Gamma-ray observations at energies above ≈ 100 MeV are the most powerful tracers of the
nucleonic component of cosmic rays.
Energetic nuclei propagating through the Galactic medium will occasionally interact with the background gas. The inelastic cross section for ppinteractions is above the ∆-resonances roughly constant, increasing logarithmically with the center of momentum energy. Assuming a constant
cross section of σpp ≈ 30 mbarn, the interaction length λ = (nσ )−1 is
λ = 10 Mpc(n/1 cm−3 )−1 , which is obviously much larger than the spatial
size of the Galaxy. On average, the proton loses half of the initial energy
per interaction (inelasticity K ≈ 0.5). The energy loss time is therefore
tpp =
1
' 7 × 107 yrs(n/1 cm−3 )−1 .
cnσK
(97)
For heavier nuclei, the interaction cross section scales roughly with the geometric cross section of the nucleus A2/3 .
The production
p of π-mesons proceeds for a proton’s kinetic energy above
Tth = 2mπ c2 1 + 4mπ /m p ≈ 280 MeV with mπ c2 = 134.97 MeV.
The inelastic scattering of protons of sufficient energy leads to the formation of charged and neutral mesons which ultimately decay into electron/positrons, neutrinos, and photons. The branching ratios for π + , π − ,
and π 0 production in the final state are roughly 1/3 each, such that the inelasticity for the pion produced is κπ ≈ 0.5/3 ≈ 0.17. Given that the center of mass moves relativistically in the lab frame, the interaction products
will be emitted in the forward direction. Most of the energy will be carried
away by a leading particle which subsequently decays. For gamma-ray
production, the neutral meson π 0 is most important. Since so far, no neutrinos from Galactic cosmic rays have been detected, we will focus in the
following on gamma-rays produced by Galactic cosmic rays.
With the simplifying assumptions that the projectile and target are protons
and that the projectile is highly relativistic, a simple treatment of the neutral pion-production and subsequent decay π 0 → γγ will suffice9 . The
three body decay π 0 → e+ e− γ with a branching ratio of 1.2 % is included by
setting the branching ratio for π 0 → γγ to unity.
9 The
56
decay of the π 0 will produce two photons with energy E∗ = mπ0 /2. The
photons are produced isotropically (back-to-back) in the center-of-mass
system, the distribution of cos θ ∗ (with θ ∗ the rest-frame angle) is flat.
The observed (laboratory frame) photon energy Eγ is kinematically constrained to be
s
s
mπ 2 1 + β
mπ 2 1 − β
c
≤ Eγ ≤
c
,
(98)
2
1+β
2
1−β
which is derived from the energy of a photon emitted at an angle θ ∗ :
q
mπ 2
∗
Eγ =
c [1 + β cos θ ] 1 − β2 .
(99)
2
When looking at the distribution of photon energies on a logarithmic scale,
the peak is located at mπ /2 with a symmetric shape to lower and higher
energies (corresponding to a Doppler-broadened line of a two-body decay).
The volume emissivity jν from pp-interactions is then given by:
jν (r) =
Z
dE p
dσpp→γ ( E p , hν)
n (r) I ( E p , r)
d(hν)
(100)
with dσ/d(hν) the differential cross section for the production of a photon
with energy hν, n(r) the number density of gas, and I ( E p , r) the (isotropic)
cosmic-ray intensity. The observable gamma-ray intensity is then calculated by the line-of-sight integration
Iν =
Z
drjν .
(101)
neglecting absorption effects. Absorption in the Galaxy is only relevant
for photon energies of PeV (1015 eV) energies, where pair production with
the cosmic microwave background leads to an optical depth τ ≥ 1 for
r ≈ 10 kpc.
For the simple case of a homogeneous gas density as well as homogeneous
−γ
cosmic ray intensity following a power-law with I ( E p ) ∝ E p within a
fixed volume (outside the volume, the density drops to zero), the total
gamma-ray flux in the direction (ϑ, ϕ) is simply given by:
Iν (ϑ, ϕ) =
2Z pπ
n
R(ϑ, ϕ) I ( E p = hν) −3
γ+1
cm
57
(102)
with Z pπ a so-called spectral weight (Gaisser 1990) that depends on the
power-law index of the cosmic-ray intensity (γ ' 2.7), Z pπ ≈ 0.03. Note,
that the spectral shape of the cosmic-ray primary spectrum is repeated in
the gamma-ray intensity. The ratio of the intensities is then simply given
by
R(ϑ, ϕ)
Iν (ϑ, ϕ)
−6 n
= 3 × 10
.
(103)
I ( E p = hν)
1 kpc
cm−3
The actually observed gamma-ray spectrum and its distribution in the sky
is therefore a line-of-sight measure of the cosmic-ray spectrum φ(r, E) and
the distribution of gas n(r).
The first sensitive measurements of the gamma-ray sky at energies above
100 MeV were carried out with the ESA10 satellite COS-B. The spark chamber pair telescope onboard COS-B mapped the emission from the Galactic plane and detected 25 individual sources. The very succesfull EGRET
(again a spark chamber pair telescope) instrument onboard the Compton Gamma-Ray Observatory reached a sensitivity one order of magnitude better than COS-B. The EGRET measurements of the Galactic plane
emission have stipulated intense studies of the gamma-ray emission from
Galactic cosmic-rays. The most puzzling result achieved was that the measured energy spectrum deviated significantly from the energy spectrum
expected from the locally measured cosmic-ray spectrum. Since July 2008,
the GLAST mission (later re-named to Fermi) has been taken data with a
silicon tracker with tungsten converter material. The silicon based tracker
unit provides improved angular resolution and stable operation without
use of expendibles like the gas filling of the spark chamber. The Fermi results confirm the presence of a prominent band of emission in the Galactic
plane (see Fig. 10).
2.3
Cosmic-rays seen from Earth
The original discovery of cosmic rays was the result of the measurement of
the electrical conductivity of air at different altitudes. This series of experiments was carried out by Victor Hess during ballon flights in 1912. While
Hess did not observe directly cosmic rays, he concluded that the origin
10 European
Space Agency
58
Figure 10: All-sky view above 100 MeV in galactic coordinates in a
Hammer-Aitoff projection. The Galactic center is in the center, north up.
The data were taken with the Fermi satellite (Porter, 2009)
of the change in conductivity must be related to a phenomenon originating from outside the atmosphere. Following up on the possibility that the
sun was responsible, he carried out flights in darkness without noticable
changes of the measurements. This left as a possible explanation a form
of energetic radiation entering the atmosphere from above. Today, the detection of cosmic rays is not done with manned ballon flights but rather
in two distinctly different ways: either by measuring cosmic rays at the
top or beyond the atmosphere with ballons, rockets, or orbiting satellites
(direct methods) or via the detection of air showers (indirect methods).
2.3.1
Direct measurement methods
The atmosphere is an inpenetrable barrier for cosmic ray particles (with
the exception of neutrinos). With a total column depth of 1054 g/cm2 , nuclei (λ pp ≈ 70 g/cm2 ) as well as electrons (X0 ≈ 37 g/cm2 ) interact in the
upper layer of the atmosphere are not directly observable from ground.
Three different types of platforms are commonly used to carry instrumen-
59