24.

Cosmic rays 1
24. COSMIC RAYS
Revised March 2002 by T.K. Gaisser and T. Stanev (Bartol Research Inst., Univ.
of Delaware); revised September 2005 by P.V. Sokolsky (Univ. of Utah) and R.E.
Streitmatter (NASA)

24.1. Primary spectra

The cosmic radiation incident at the top of the terrestrial atmosphere includes all
stable charged particles and nuclei with lifetimes of order 106 years or longer. Technically,
“primary” cosmic rays are those particles accelerated at astrophysical sources and
“secondaries” are those particles produced in interaction of the primaries with interstellar
gas. Thus electrons, protons and helium, as well as carbon, oxygen, iron, and other nuclei
synthesized in stars, are primaries. Nuclei such as lithium, beryllium, and boron (which
are not abundant end-products of stellar nucleosynthesis) are secondaries. Antiprotons
and positrons are also in large part secondary. Whether a small fraction of these particles
may be primary is a question of current interest.
Apart from particles associated with solar flares, the cosmic radiation comes from
outside the solar system. The incoming charged particles are “modulated” by the solar
wind, the expanding magnetized plasma generated by the Sun, which decelerates and
partially excludes the lower energy galactic cosmic rays from the inner solar system.
There is a significant anticorrelation between solar activity (which has an alternating
eleven-year cycle) and the intensity of the cosmic rays with energies below about 10 GeV.
In addition, the lower-energy cosmic rays are affected by the geomagnetic field, which they
must penetrate to reach the top of the atmosphere. Thus the intensity of any component
of the cosmic radiation in the GeV range depends both on the location and time.
There are four different ways to describe the spectra of the components of the cosmic
radiation: (1) By particles per unit rigidity. Propagation (and probably also acceleration)
through cosmic magnetic fields depends on gyroradius or magnetic rigidity, R, which is
gyroradius multiplied by the magnetic field strength:

pc
R= = rL B . (24.1)
Ze

(2) By particles per energy-per-nucleon. Fragmentation of nuclei propagating through

the interstellar gas depends on energy per nucleon, since that quantity is approximately
conserved when a nucleus breaks up on interaction with the gas. (3) By nucleons per
energy-per-nucleon. Production of secondary cosmic rays in the atmosphere depends
on the intensity of nucleons per energy-per-nucleon, approximately independently of
whether the incident nucleons are free protons or bound in nuclei. (4) By particles per
energy-per-nucleus. Air shower experiments that use the atmosphere as a calorimeter
generally measure a quantity that is related to total energy per particle.
The units of differential intensity I are [cm−2 s−1 sr−1 E −1 ], where E represents the
units of one of the four variables listed above.

CITATION: S. Eidelman et al., Physics Letters B592, 1 (2004)

available on the PDG WWW pages (URL: http://pdg.lbl.gov/) December 20, 2005 11:24
2 24. Cosmic rays
The intensity of primary nucleons in the energy range from several GeV to somewhat
beyond 100 TeV is given approximately by

nucleons
IN (E) ≈ 1.8 E −α , (24.2)
cm2 s sr GeV

where E is the energy-per-nucleon (including rest mass energy) and α (≡ γ + 1) = 2.7

is the differential spectral index of the cosmic ray flux and γ is the integral spectral
index. About 79% of the primary nucleons are free protons and about 70% of the rest are
nucleons bound in helium nuclei. The fractions of the primary nuclei are nearly constant
over this energy range (possibly with small but interesting variations). Fractions of both
primary and secondary incident nuclei are listed in Table 24.1. Figure 24.1 [1] shows the
major components as a function of energy at a particular epoch of the solar cycle. There
has been a series of more precise measurements of the primary spectrum of protons and
helium in the past decade [2–7].

Table 24.1: Relative abundances F of cosmic-ray nuclei at 10.6 GeV/nucleon nor-

malized to oxygen (≡ 1) [8]. The oxygen flux at kinetic energy of 10.6 GeV/nucleon
is 3.26 × 10−6 cm−2 s−1 sr−1 (GeV/nucleon)−1 . Abundances of hydrogen and
helium are from Ref. [5,6].

Z Element F Z Element F

1 H 540 13–14 Al-Si 0.19

2 He 26 15–16 P-S 0.03
3–5 Li-B 0.40 17–18 Cl-Ar 0.01
6–8 C-O 2.20 19–20 K-Ca 0.02
9–10 F-Ne 0.30 21–25 Sc-Mn 0.05
11–12 Na-Mg 0.22 26–28 Fe-Ni 0.12

Up to energies of at least 1015 eV, the composition and energy spectra of nuclei are
typically interpreted in the context of “diffusion” or “leaky box” models, in which the
sources of the primary cosmic radiation are located within the galaxy [9]. The ratio
of secondary to primary nuclei is observed to decrease approximately as E −0.5 with
increasing energy, a fact interpreted to mean that the lifetime of cosmic rays in the galaxy
decreases with energy. Measurements of radioactive “clock” isotopes [10,11] in the low
energy cosmic radiation are consistent with a lifetime in the galaxy of about 15 Myr.
The spectrum of electrons and positrons incident at the top of the atmosphere is
steeper than the spectra of protons and nuclei, as shown in Fig. 24.2. The positron
fraction decreases from ∼ 0.2 below 1 GeV [12–14] to ∼ 0.1 around 2 GeV and to ∼ 0.05
in at the highest energies for which it is measured (5 − 20 GeV) [15]. This behavior
refers to measurements made during solar cycles of positive magnetic polarity and at high

December 20, 2005 11:24

 24. Cosmic rays 3
10

0.1
He
Differential flux (m2 sr s MeV/ nucleon)−1

10 −2 H
C
10 −3

Fe
10 −4

10 −5

10 −6

10 −7

10 −8

10 −9
10 102 103 104 105 106 107
Ki neti c ener gy ( MeV/nucleon)

Figure 24.1: Major components of the primary cosmic radiation (from Ref. 1).

geomagnetic latitude. Ref. 14 discusses the dependence of the positron fraction on solar
cycle and Ref. 5 studies the geomagnetic effects.
The ratio of antiprotons to protons is ∼ 2 × 10−4 [16,17] at around 10–20 GeV, and
there is clear evidence [18–20] for the kinematic suppression at lower energy that is the
signature of secondary antiprotons. The p/p ratio also shows a strong dependence on the

December 20, 2005 11:24

4 24. Cosmic rays

E 3 dN/dE [m–2 s–1 sr–1 GeV 2]

500
electrons + positrons
300
200

100
70
50

10 100 1000
E [GeV]
Figure 24.2: Differential spectrum of electrons plus positrons multiplied by E 3
(data summary from Ref. 9). The dashed line shows the proton spectrum multiplied
by 0.01.

phase and polarity of the solar cycle [21] in the opposite sense to that of the positron
fraction. There is at this time no evidence for a significant primary component either
of positrons or of antiprotons. No antihelium or antideuteron has been found in the
cosmic radiation. The best current measured upper limit on the ratio antihelium/helium
is approximately 7 × 10−4 [22]. The upper limit on the flux of antideuterons around 1
GeV/nuleon is approximately 2 × 10−4 m2 s sr GeV/nucleon [23].

24.2. Cosmic rays in the atmosphere

Figure 24.3 shows the vertical fluxes of the major cosmic ray components in the
atmosphere in the energy region where the particles are most numerous (except for
electrons, which are most numerous near their critical energy, which is about 81 MeV in
air). Except for protons and electrons near the top of the atmosphere, all particles are
produced in interactions of the primary cosmic rays in the air. Muons and neutrinos are
products of the decay of charged mesons, while electrons and photons originate in decays
of neutral mesons.
Most measurements are made at ground level or near the top of the atmosphere,
but there are also measurements of muons and electrons from airplanes and balloons.
Fig. 24.3 includes recent measurements of negative muons [4,24–26]. Since µ+ (µ− ) are
produced in association with νµ (ν µ ), the measurement of muons near the maximum of
the intensity curve for the parent pions serves to calibrate the atmospheric νµ beam [27].
Because muons typically lose almost two GeV in passing through the atmosphere, the
comparison near the production altitude is important for the sub-GeV range of νµ (ν µ )
energies.
The flux of cosmic rays through the atmosphere is described by a set of coupled cascade
equations with boundary conditions at the top of the atmosphere to match the primary
spectrum. Numerical or Monte Carlo calculations are needed to account accurately for
decay and energy-loss processes, and for the energy-dependences of the cross sections and
of the primary spectral index γ. Approximate analytic solutions are, however, useful in

December 20, 2005 11:24

 24. Cosmic rays 5
Altitude (km)
15 10 5 3 2 1 0
10000

1000

[m–2 s–1 sr–1]

_
νµ + ν µ
100 µ+ + µ −

10 p+n
Vertical flux

1
e+ + e−

π+ + π−
0.1

0.01
0 200 400 600 800 1000
Atmospheric depth [g cm–2]
Figure 24.3: Vertical fluxes of cosmic rays in the atmosphere with E > 1 GeV
estimated from the nucleon flux of Eq. (24.2). The points show measurements of
negative muons with Eµ > 1 GeV [4,24–26].

limited regions of energy [28]. For example, the vertical intensity of nucleons at depth
X (g cm−2 ) in the atmosphere is given by
IN (E, X) ≈ IN (E, 0) e−X/Λ , (24.3)
where Λ is the attenuation length of nucleons in air.
The corresponding expression for the vertical intensity of charged pions with energy
Eπ  π = 115 GeV is

ZNπ X Eπ
Iπ (Eπ , X) ≈ IN (Eπ , 0) e−X/Λ . (24.4)
λN π
This expression has a maximum at t = Λ ≈ 120 g cm−2 , which corresponds to an altitude
of 15 kilometers. The quantity ZNπ is the spectrum-weighted moment of the inclusive
distribution of charged pions in interactions of nucleons with nuclei of the atmosphere.
The intensity of low-energy pions is much less than that of nucleons because ZNπ ≈ 0.079
is small and because most pions with energy much less than the critical energy π decay
rather than interact.

December 20, 2005 11:24

6 24. Cosmic rays
24.3. Cosmic rays at the surface

24.3.1. Muons: Muons are the most numerous charged particles at sea level (see
Fig. 24.3). Most muons are produced high in the atmosphere (typically 15 km) and
lose about 2 GeV to ionization before reaching the ground. Their energy and angular
distribution reflect a convolution of production spectrum, energy loss in the atmosphere,
and decay. For example, 2.4 GeV muons have a decay length of 15 km, which is reduced
to 8.7 km by energy loss. The mean energy of muons at the ground is ≈ 4 GeV. The
energy spectrum is almost flat below 1 GeV, steepens gradually to reflect the primary
spectrum in the 10–100 GeV range, and steepens further at higher energies because
pions with Eπ > π ≈ 115 GeV tend to interact in the atmosphere before they decay.
Asymptotically (Eµ  1 TeV), the energy spectrum of atmospheric muons is one power
steeper than the primary spectrum. The integral intensity of vertical muons above
1 GeV/c at sea level is ≈ 70 m−2 s−1 sr−1 [29,30], with recent measurements [31–33]
tending to give lower normalization by 10-15%. Experimentalists are familiar with this
number in the form I ≈ 1 cm−2 min−1 for horizontal detectors.
The overall angular distribution of muons at the ground is ∝ cos2 θ, which is
characteristic of muons with Eµ ∼ 3 GeV. At lower energy the angular distribution
becomes increasingly steep, while at higher energy it flattens, approaching a sec θ
distribution for Eµ  π and θ < 70◦ .

0.2
pµ2.7dN/dpµ [cm–2 s–1 sr–1 (GeV/c)1.7]

0.1

0.05

0.02

0.01

0.005

0.002
1 10 100 1000
pµ (GeV/c)
Figure 24.4: Spectrum of muons at θ = 0◦ ( [29],  [34], H [35], N [36], × and
+ [31], and θ = 75◦  [37]).

December 20, 2005 11:24

 24. Cosmic rays 7
Figure 24.4 shows the muon energy spectrum at sea level for two angles. At large angles
low energy muons decay before reaching the surface and high energy pions decay before
they interact, thus the average muon energy increases. An approximate extrapolation
formula valid when muon decay is negligible (Eµ > 100/ cos θ GeV) and the curvature of
the Earth can be neglected (θ < 70◦ ) is

dNµ 0.14 Eµ−2.7

≈
dEµ cm2 s sr GeV
 

 

1 0.054
× + , (24.5)

 1 + 1.1Eµ cos θ 1.1Eµ cos θ 

1+
115 GeV 850 GeV

where the two terms give the contribution of pions and charged kaons. Eq. (24.5) neglects
a small contribution from charm and heavier flavors which is negligible except at very
high energy [38].
The muon charge ratio reflects the excess of π + over π − in the forward fragmentation
region of proton initiated interactions together with the fact that there are more protons
than neutrons in the primary spectrum. The charge ratio is between 1.1 and 1.4 from
1 GeV to 100 GeV [29,31,32]. Below 1 GeV there is a systematic dependence on location
due to geomagnetic effects [31,32].

24.3.2. Electromagnetic component: At the ground, this component consists of

electrons, positrons, and photons primarily from electromagnetic cascades initiated
by decay of neutral and charged mesons. Muon decay is the dominant source of
low-energy electrons at sea level. Decay of neutral pions is more important at high
altitude or when the energy threshold is high. Knock-on electrons also make a small
contribution at low energy [39]. The integral vertical intensity of electrons plus positrons
is very approximately 30, 6, and 0.2 m−2 s−1 sr−1 above 10, 100, and 1000 MeV
respectively [30,40], but the exact numbers depend sensitively on altitude, and the
angular dependence is complex because of the different altitude dependence of the
different sources of electrons [39–41]. The ratio of photons to electrons plus positrons is
approximately 1.3 above a GeV and 1.7 below the critical energy [41].

24.3.3. Protons: Nucleons above 1 GeV/c at ground level are degraded remnants of
the primary cosmic radiation. The intensity is approximately represented by Eq. (24.3)
with the replacement t → t/ cos θ for θ < 70◦ and an attenuation length Λ = 123 g cm−2 .
At sea level, about 1/3 of the nucleons in the vertical direction are neutrons (up from
≈ 10% at the top of the atmosphere as the n/p ratio approaches equilibrium). The
integral intensity of vertical protons above 1 GeV/c at sea level is ≈ 0.9 m−2 s−1 sr−1
[30,42].

December 20, 2005 11:24

8 24. Cosmic rays
24.4. Cosmic rays underground
Only muons and neutrinos penetrate to significant depths underground. The muons
produce tertiary fluxes of photons, electrons, and hadrons.
24.4.1. Muons: As discussed in Section 27.6 of this Review, muons lose energy by
ionization and by radiative processes: bremsstrahlung, direct production of e+ e− pairs,
and photonuclear interactions. The total muon energy loss may be expressed as a function
of the amount of matter traversed as
dEµ
− = a + b Eµ , (24.6)
dX
where a is the ionization loss and b is the fractional energy loss by the three radiation
processes. Both are slowly varying functions of energy. The quantity  ≡ a/b (≈ 500 GeV
in standard rock) defines a critical energy below which continuous ionization loss is more
important than radiative losses. Table 24.2 shows a and b values for standard rock as
a function of muon energy. The second column of Table 24.2 shows the muon range in
standard rock (A = 22, Z = 11, ρ = 2.65 g cm−3 ). These parameters are quite sensitive
to the chemical composition of the rock, which must be evaluated for each experimental
location.
Table 24.2: Average muon range R and energy loss parameters calculated for
standard rock [43]. Range is given in km-water-equivalent, or 105 g cm−2 .
P
Eµ R a bbrems bpair bnucl bi
GeV km.w.e. MeV g−1 cm2 10−6 g−1 cm2

10 0.05 2.17 0.70 0.70 0.50 1.90

100 0.41 2.44 1.10 1.53 0.41 3.04
1000 2.45 2.68 1.44 2.07 0.41 3.92
10000 6.09 2.93 1.62 2.27 0.46 4.35

The intensity of muons underground can be estimated from the muon intensity in the
atmosphere and their rate of energy loss. To the extent that the mild energy dependence
of a and b can be neglected, Eq. (24.6) can be integrated to provide the following relation
between the energy Eµ,0 of a muon at production in the atmosphere and its average
energy Eµ after traversing a thickness X of rock (or ice or water):

Eµ = (Eµ,0 + ) e−bX −  . (24.7)

Especially at high energy, however, fluctuations are important and an accurate calculation
requires a simulation that accounts for stochastic energy-loss processes [44].
There are two depth regimes for Eq. (24.7). For X  b−1 ≈ 2.5 km water equivalent,
Eµ,0 ≈ Eµ (X) + aX, while for X  b−1 Eµ,0 ≈ ( + Eµ (X)) exp(bX). Thus at shallow

December 20, 2005 11:24

 24. Cosmic rays 9
depths the differential muon energy spectrum is approximately constant for Eµ < aX and
steepens to reflect the surface muon spectrum for Eµ > aX, whereas for X > 2.5 km.w.e.
the differential spectrum underground is again constant for small muon energies but
steepens to reflect the surface muon spectrum for Eµ >  ≈ 0.5 TeV. In the deep regime
the shape is independent of depth although the intensity decreases exponentially with
depth. In general the muon spectrum at slant depth X is

dNµ (X) dNµ dEµ,0 dNµ bX

= = e , (24.8)
dEµ dEµ,0 dEµ dEµ,0

where Eµ,0 is the solution of Eq. (24.7) in the approximation neglecting fluctuations.
Fig. 24.5 shows the vertical muon intensity versus depth. In constructing this
“depth-intensity curve,” each group has taken account of the angular distribution of the
muons in the atmosphere, the map of the overburden at each detector, and the properties
of the local medium in connecting measurements at various slant depths and zenith
angles to the vertical intensity. Use of data from a range of angles allows a fixed detector
to cover a wide range of depths. The flat portion of the curve is due to muons produced
locally by charged-current interactions of νµ .

100
electron-like muon-like

10
Rate [kt–1yr–1GeV–1]

0.1

0.01

0.001
0.1 1 10 1 10 100
E [GeV] E [GeV]
Figure 24.6: Sub-GeV and multi-GeV neutrino interactions from SuperKamiokande [50].
The plot shows the spectra of visible energy in the detector.

December 20, 2005 11:24

10 24. Cosmic rays
10 −6

Vertical intensity (cm−2 sr−1 s−1 )

10 −8

10 −10

10 −12

10 −14
1 10 100
Depth (km water equivalent)
Figure 24.5: Vertical muon intensity vs depth (1 km.w.e. = 105 g cm−2 of standard
rock). The experimental data are from: : the compilations of Crouch [45], :
Baksan [46], ◦: LVD [47], •: MACRO [48], : Frejus [49]. The shaded area at large
depths represents neutrino-induced muons of energy above 2 GeV. The upper line is
for horizontal neutrino-induced muons, the lower one for vertically upward muons.

24.4.2. Neutrinos: Because neutrinos have small interaction cross sections, measure-
ments of atmospheric neutrinos require a deep detector to avoid backgrounds. There are
two types of measurements: contained (or semi-contained) events, in which the vertex
is determined to originate inside the detector, and neutrino-induced muons. The latter
are muons that enter the detector from zenith angles so large (e.g., nearly horizontal or
upward) that they cannot be muons produced in the atmosphere. In neither case is the
neutrino flux measured directly. What is measured is a convolution of the neutrino flux
and cross section with the properties of the detector (which includes the surrounding
medium in the case of entering muons).
Contained and semi-contained events reflect neutrinos in the sub-GeV to multi-GeV
region where the product of increasing cross section and decreasing flux is maximum. In
the GeV region the neutrino flux and its angular distribution depend on the geomagnetic

December 20, 2005 11:24

 24. Cosmic rays 11
location of the detector and, to a lesser extent, on the phase of the solar cycle. Naively,
we expect νµ /νe = 2 from counting neutrinos of the two flavors coming from the chain
of pion and muon decay. This ratio is only slightly modified by the details of the decay
kinematics, but the fraction of electron neutrinos gradually decreases above a GeV as
parent muons begin to reach the ground before decaying. Experimental measurements
have to account for the ratio of ν/ν, which have cross sections different by a factor of 3 in
this energy range. In addition, detectors generally have different efficiencies for detecting
muon neutrinos and electron neutrinos which need to be accounted for in comparing
measurements with expectation. Fig. 24.6 shows the distributions of the visible energy
in the Super-Kamiokande detector [50] for electron-like and muon-like charged current
neutrino interactions. Contrary to expectation, the numbers of the two classes of events
are similar rather than different by a factor of two. The exposure for the data sample
shown here is 50 kiloton-years. The falloff of the muon-like events at high energy is a
consequence of the poor containment for high energy muons. Corrections for detection
efficiencies and backgrounds are, however, insufficient to account for the large difference
from the expectation.
Two well-understood properties of atmospheric cosmic rays provide a standard for
comparison of the measurements of atmospheric neutrinos. These are the “sec θ effect”
and the “east-west effect”. The former refers originally to the enhancement of the flux
of > 10 GeV muons (and neutrinos) at large zenith angles because the parent pions
propagate more in the low density upper atmosphere where decay is enhanced relative to
interaction. For neutrinos from muon decay, the enhancement near the horizontal becomes
important for Eν > 1 GeV and arises mainly from the increased pathlength through the
atmosphere for muon decay in flight. Fig. 24.7 from Ref. 51 shows a comparison between
measurement and expectation for the zenith angle dependence of multi-GeV electron-like
(mostly νe ) and muon-like (mostly νµ ) events separately. The νe show an enhancement
near the horizontal and approximate equality for nearly upward (cos θ ≈ −1) and nearly
downward (cos θ ≈ 1) events. There is, however, a very significant deficit of upward
(cos θ < 0) νµ events, which have long pathlengths comparable to the radius of the
Earth. This pattern has been interpreted as evidence for oscillations involving muon
neutrinos [50]. (See the article on neutrino properties in this Review.) Including three
dimensional effects in the calculation of atmospheric neutrinos may change somewhat the
expected angular distributions of neutrinos at low energy [52], but it does not change the
fundamental expectation of up-down symmetry, which is the basis of the evidence for
oscillations.
The east-west effect [53,54] is the enhancement, especially at low geomagnetic latitudes,
of cosmic rays incident on the atmosphere from the west as compared to those from
the east. This is a consequence of the fact that the cosmic rays are positively charged
nuclei which are bent systematically in one sense in the geomagnetic field. Not all
trajectories can reach the atmosphere from outside the geomagnetic field. The standard
procedure to see which trajectories are allowed is to inject antiprotons outward from
near the top of the atmosphere in various directions and see if they escape from the
geomagnetic field without becoming trapped indefinitely or intersecting the surface of
the Earth. Any direction in which an antiproton of a given momentum can escape is an

December 20, 2005 11:24

12 24. Cosmic rays
300 multi-GeV e-like multi-GeV µ-like + PC

Number of events
200

100

0
–0.8 –0.4 0 0.4 0.8 –0.8 –0.4 0 0.4 0.8
cos Θ cos Θ
Figure 24.7: Zenith-angle dependence of multi-GeV neutrino interactions from
SuperKamiokande [51]. The shaded boxes show the expectation in the absence of
any oscillations. The lines show fits with some assumed oscillation parameters, as
described in Ref. 51.

allowed direction from which a proton of the opposite momentum can arrive. Since the
geomagnetic field is oriented from south to north in the equatorial region, antiprotons
injected toward the east are bent back towards the Earth. Thus there is a range of
momenta and zenith angles for which positive particles cannot arrive from the east but
can arrive from the west. This east-west asymmetry of the incident cosmic rays induces
a similar asymmetry on the secondaries, including neutrinos. Since this is an azimuthal
effect, the resulting asymmetry is independent of possible oscillations, which depend on
pathlength (equivalently zenith angle), but not on azimuth. Fig. 24.8 (from Ref. 55) is a
comparison of data and expectation for this effect, which serves as a consistency check of
the measurement and analysis.
Muons that enter the detector from outside after production in charged-current
interactions of neutrinos naturally reflect a higher energy portion of the neutrino
spectrum than contained events because the muon range increases with energy as well
as the cross section. The relevant energy range is ∼ 10 < Eν < 1000 GeV, depending
somewhat on angle. Neutrinos in this energy range show a sec θ effect similar to
muons (see Eq. (24.5)). This causes the flux of horizontal neutrino-induced muons to
be approximately a factor two higher than the vertically upward flux. The upper and
lower edges of the horizontal shaded region in Fig. 24.5 correspond to horizontal and
vertical intensities of neutrino-induced muons. Table 24.3 gives the measured fluxes of
upward-moving neutrino-induced muons averaged over the lower hemisphere. Generally
the definition of minimum muon energy depends on where it passes through the detector.
The tabulated effective minimum energy estimates the average over various accepted
trajectories.

December 20, 2005 11:24

 24. Cosmic rays 13
Table 24.3: Measured fluxes (10−13 cm−2 s−1 sr−1 ) of neutrino-induced muons as a
function of the effective minimum muon energy Eµ .

Eµ > 1 GeV 1 GeV 1 GeV 2 GeV 3 GeV 3 GeV

Ref. CWI [56] Baksan [57] MACRO [58] IMB [59] Kam [60] SuperK [61]
Fµ 2.17±0.21 2.77±0.17 2.29 ± 0.15 2.26±0.11 1.94±0.12 1.74±0.07

100 e-like µ-like

80
Number of events

60

40

20

0
0 π/2 π 3π/2 2π 0 π/2 π 3π/2 2π
φ φ
Figure 24.8: Azimuthal dependence of ∼GeV neutrino interactions from Su-
perKamiokande [55]. The cardinal points of the compass are S, E, N, W starting
at 0. These are the direction from which the particles arrive. The lines show the
expectation based on two different calculations, as described in Ref. 55.

24.5. Air showers

So far we have discussed inclusive or uncorrelated fluxes of various components of the
cosmic radiation. An air shower is caused by a single cosmic ray with energy high enough
for its cascade to be detectable at the ground. The shower has a hadronic core, which acts
as a collimated source of electromagnetic subshowers, generated mostly from π 0 → γ γ.
The resulting electrons and positrons are the most numerous particles in the shower. The
number of muons, produced by decays of charged mesons, is an order of magnitude lower.
Air showers spread over a large area on the ground, and arrays of detectors operated
for long times are useful for studying cosmic rays with primary energy E0 > 100 TeV,

December 20, 2005 11:24

14 24. Cosmic rays
where the low flux makes measurements with small detectors in balloons and satellites
difficult.
Greisen [62] gives the following approximate expressions for the numbers and lateral
distributions of particles in showers at ground level. The total number of muons Nµ with
energies above 1 GeV is
 3/4
Nµ (> 1 GeV) ≈ 0.95 × 105 Ne /106 , (24.9)

where Ne is the total number of charged particles in the shower (not just e± ). The
number of muons per square meter, ρµ , as a function of the lateral distance r (in meters)
from the center of the shower is
 1.25 
1.25 Nµ 1 r −2.5
ρµ = r −0.75 1 + , (24.10)
2π Γ(1.25) 320 320

where Γ is the gamma function. The number density of charged particles is

ρe = C1 (s, d, C2 ) x(s−2) (1 + x)(s−4.5) (1 + C2 xd ) . (24.11)

Here s, d, and C2 are parameters in terms of which the overall normalization constant
C1 (s, d, C2) is given by

Ne
C1 (s, d, C2) = [ B(s, 4.5 − 2s)
2πr12

+ C2 B(s + d, 4.5 − d − 2s)]−1 , (24.12)

where B(m, n) is the beta function. The values of the parameters depend on shower size
(Ne ), depth in the atmosphere, identity of the primary nucleus, etc. For showers with
Ne ≈ 106 at sea level, Greisen uses s = 1.25, d = 1, and C2 = 0.088. Finally, x is r/r1 ,
where r1 is the Molière radius, which depends on the density of the atmosphere and hence
on the altitude at which showers are detected. At sea level r1 ≈ 78 m. It increases with
altitude.
The lateral spread of a shower is determined largely by Coulomb scattering of the
many low-energy electrons and is characterized by the Molìere radius. The lateral spread
of the muons (ρµ ) is larger and depends on the transverse momenta of the muons at
production as well as multiple scattering.
There are large fluctuations in development from shower to shower, even for showers of
the same energy and primary mass—especially for small showers, which are usually well
past maximum development when observed at the ground. Thus the shower size Ne and
primary energy E0 are only related in an average sense, and even this relation depends
on depth in the atmosphere. One estimate of the relation is [63]

E0 ∼ 3.9 × 106 GeV (Ne /106 )0.9 (24.13)

December 20, 2005 11:24

 24. Cosmic rays 15
for vertical showers with 1014 < E < 1017 eV at 920 g cm−2 (965 m above sea level).
Because of fluctuations, Ne as a function of E0 is not the inverse of Eq. (24.13). As E0
increases the shower maximum (on average) moves down into the atmosphere and the
relation between Ne and E0 changes. At the maximum of shower development, there are
approximately 2/3 particles per GeV of primary energy.
Detailed simulations and cross-calibrations between different types of detectors are
necessary to establish the primary energy spectrum from air-shower experiments [63,64].
Figure 24.9 shows the “all-particle” spectrum. The differential energy spectrum has
been multiplied by E 2.5 in order to display the features of the steep spectrum that are
otherwise difficult to discern. The steepening that occurs between 1015 and 1016 eV is
known as the knee of the spectrum. The feature around 1019 eV is called the ankle of the
spectrum.

104
Flux dΦ/dE × E 2.5 [m−2 s−1 sr−1 GeV1.5 ]

⊗⊗⊗
KNEE :
direct JACEE
✢✧✧✧ ⊗❄❄+∇
⊗ ⊗✕✕✕❄✕❄✕
⊗❄❄∇∇
⊗❄❄⊗ ⊗❄
∇✕✕⊗
⁄✕
❄∇
✢ ✧✧
✢ +✧
✢+✧✧ +✧+⊗∅ +⊗ ❄✕
⊗⊗ ∅
∇⁄❄✕∅
✕
⊗ ∇
❄✕ ✕∅
✢ ✢ ∅+∅ +❄++ ⊗⁄⁄⁄❄
✕∇❄∅
⁄✕
❄⊗
++ ⁄∅
✕∇
✕
⊗✕⊗∅
RUNJOB
✧✧ ✢❄⊗❄❄✕
+ AGASA ✢
✧ +∇
⊗
⁄❄
✕+✕ ∅
∇⁄⁄✕
❄⊗
✧✧ ❄✕ ∅
+❄⊗
∇⁄❄+✕∅
✕ ⁄+ SOKOL
+ Akeno 20 km2 ✧✧✢✧ ⊗
∇
❄✕❄⊗
∅
103 ✧✢✧✕✕∇++ ⁄ ❄⁄❄∅❄⊗
⁄❄∅❄⊗
⁄ ⁄❄⁄
∇∇ +⁄∅❄⊗+❄❄⊗ Grigorov
+ Akeno 1 km2 ✢ ∇∅∇⁄⁄∅ +
❄❄+⁄❄+❄❄⁄
✜ AUGER ∅ ⁄ ++⁄
⁄ ❄∅
⁄∅
∇∇∇ ⁄∅∇∅++++
BLANCA ⊕ HiRes/MIA ∅∅+++++ ++
∅
✡✡ + ++
✧ CASA-MIA ⁄ KASCADE (e/m QGSJET) ∅∅✡+✡+ +
⊕⊕⊕⊕⊕∅ ✡ + ++
102 ◊ DICE ⁄ KASCADE (e/m SIBYLL) ⊕⊕
⊕⊕+✡⊕
✢ BASJE-MAS KASCADE (h/m) ⊕✡✡+ ++++
✡
✕ EAS-Top ❄ KASCADE (nn) ⊕✡⊕ ++
✡ ++ +
⊕✡ ✡ +++
✡ Fly’s Eye ∅ MSU ✡⊕ ⊕✡✡+++✡+✡+ ++++
⊕ +
✜⊕✡ ⊕ ✡✡+ ++
✜⊕ ⊕✜ ⊕
✜✜⊕
Haverah Park Mt. Norikura +
✜⊕⊕⊕✜ ⊕
✜ ⊕✡ +++ +++
10 Haverah Park Fe ✲ SUGAR ✜ ✡⊕ ⊕⊕ ✜
Haverah Park p ⊗ Tibet AS γ ⊕ ⊕✡✜⊕ ✲✲
✜ ⊕✜ +
HEGRA ⊗ Tibet AS γ-III ✜✜
✜
⊕ HiRes-I ∇ Tunka-25 ⊕
⊕
⊕ HiRes-II Yakutsk ANKLE
1
104 105 106 107 108 109 1010 1011
Energy E [GeV]
Figure 24.9: The all-particle spectrum: for references see [65]. Figure used by
permission of author.

Measurements with small air shower experiments in the knee region differ by as much
as a factor of two, indicative of systematic uncertainties in interpretation of the data.
(For a recent review see Ref. 66.) In establishing the spectrum shown in Fig. 24.9, efforts
have been made to minimize the dependence of the analysis on the primary composition.
Ref. 67 uses an unfolding procedure to obtain the spectra of the individual components,
giving a result for the all-particle spectrum between 10 15 and 1017 eV that lies toward
the upper range of the data shown in Fig. 24.9. In the energy range above 1017 eV,
the Fly’s Eye technique [68] is particularly useful because it can establish the primary
energy in a model-independent way by observing most of the longitudinal development
of each shower, from which E0 is obtained by integrating the energy deposition in the
atmosphere.
If the cosmic ray spectrum below 1018 eV is of galactic origin, the knee could reflect
the fact that some (but not all) cosmic accelerators have reached their maximum energy.
Some types of expanding supernova remnants, for example, are estimated not to be able

December 20, 2005 11:24

16 24. Cosmic rays
to accelerate particles above energies in the range of 1015 eV total energy per particle.
Effects of propagation and confinement in the galaxy [69] also need to be considered.
It was previously thought that the most likely explanation for the dip in the spectrum
near 3 × 1018 eV called the ankle is a higher energy population of particles overtaking a
lower energy population, for example an extragalactic flux beginning to dominate over
the galactic flux. The situation now seems more complicated because of better evidence
for the existence of a break in the spectrum near 5 × 1017 eV called the second knee [70],
just below the ankle structure. There are clear predictions of a dip structure in the
region of the observed ankle produced by e+ /e− energy losses of extragalactic protons on
the 2.7 K cosmic microwave radiation (CMB) [71]. This dip structure has been claimed
to be a more robust signature of both the protonic and extragalactic nature of the
highest energy cosmic rays than the GZK cutoff (see below) itself. If this interpretation
is correct, then the most likely explanation of the second knee would be the termination
of the galactic cosmic ray flux. Composition changes across this energy region would be
important correlative evidence for this view.
If the cosmic ray flux above the second knee is cosmological in origin, then there should
be a rapid change in the spectral index (called the GZK cutoff) around 5 × 1019 eV,
resulting from the onset of inelastic interactions with the (CMB) [72,73]. While several
experiments have reported events that have been assigned energies above 1020 eV [74–77],
more recent experiments such as HiRes [78] have failed to confirm this; results are
consistent with the expected cutoff. An implication of a continued spectrum would be
that some sources of the highest energy particles must be relatively nearby. For example,
the attenuation length for protons at 2 · 1020 eV is 30 Mpc [73]. Both cosmic accelerators
(bottom up) and massive cosmological relics (top down ) have been suggested [74–77].
Figure 24.10 gives an expanded view of the high energy end of the spectrum, showing
only the more recent experiments. This figure and the previous one have shown the
differential flux multiplied by a power of the energy, a procedure that enables one to see
structure in the spectrum more clearly but amplifies small systematic differences in energy
assignments into sizable normalization differences. All existing experiments are actually
consistent in normalization if one takes quoted systematic errors in the energy scales into
account. However, the continued power law type of flux beyond the GZK cutoff claimed
by the AGASA experiment is contradicted by the HiRes data. New data from the Pierre
Auger experiment [79] are still too preliminary in energy scale and statistics to impact
this debate.
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December 20, 2005 11:24

 24. Cosmic rays 17
HiRes-2 Monocular

F lux × E 3 [m −2 s−1 sr −1 eV 2 ]
10 HiRes-1 Monocular
AGASA
Auger SD

0.1
17 17.5 18 18.5 19 19.5 20 20.5 21
log10(E ) [eV]

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