24. Cosmic rays 1

24. COSMIC RAYS

Revised August 2011 by J.J. Beatty (Ohio State Univ.) and J. Matthews (LouisianaState Univ. and Southern Univ.); revised August 2009 by T.K. Gaisser and T. Stanev(Bartol Research Inst., Univ. of Delaware)

24.1. Primary spectra

The cosmic radiation incident at the top of the terrestrial atmosphere includes allstable 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 interstellargas. Thus electrons, protons and helium, as well as carbon, oxygen, iron, and other nucleisynthesized in stars, are primaries. Nuclei such as lithium, beryllium, and boron (whichare not abundant end-products of stellar nucleosynthesis) are secondaries. Antiprotonsand positrons are also in large part secondary. Whether a small fraction of these particlesmay be primary is a question of current interest.

Apart from particles associated with solar flares, the cosmic radiation comes fromoutside the solar system. The incoming charged particles are “modulated” by the solarwind, the expanding magnetized plasma generated by the Sun, which decelerates andpartially excludes the lower energy galactic cosmic rays from the inner solar system.There is a significant anticorrelation between solar activity (which has an alternatingeleven-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 theymust penetrate to reach the top of the atmosphere. Thus the intensity of any componentof 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 cosmicradiation: (1) By particles per unit rigidity. Propagation (and probably also acceleration)through cosmic magnetic fields depends on gyroradius or magnetic rigidity, R, which isgyroradius multiplied by the magnetic field strength:

R =p c

Z e= r

LB . (24.1)

(2) By particles per energy-per-nucleon. Fragmentation of nuclei propagating throughthe interstellar gas depends on energy per nucleon, since that quantity is approximatelyconserved when a nucleus breaks up on interaction with the gas. (3) By nucleons perenergy-per-nucleon. Production of secondary cosmic rays in the atmosphere dependson the intensity of nucleons per energy-per-nucleon, approximately independently ofwhether the incident nucleons are free protons or bound in nuclei. (4) By particles perenergy-per-nucleus. Air shower experiments that use the atmosphere as a calorimetergenerally measure a quantity that is related to total energy per particle.

The units of differential intensity I are [m−2 s−1sr−1E−1], where E represents the unitsof one of the four variables listed above.

K. Nakamura et al.(PDG), JP G 37, 075021 (2010) and 2011 partial update for the 2012 edition (pdg.lbl.gov)February 16, 2012 14:07

2 24. Cosmic rays

The intensity of primary nucleons in the energy range from several GeV to somewhatbeyond 100 TeV is given approximately by

IN (E) ≈ 1.8 × 104 (E/1 GeV)−α nucleons

m2 s sr GeV, (24.2)

where E is the energy-per-nucleon (including rest mass energy) and α (≡ γ + 1) = 2.7is the differential spectral index of the cosmic ray flux and γ is the integral spectralindex. About 79% of the primary nucleons are free protons and about 70% of the rest arenucleons bound in helium nuclei. The fractions of the primary nuclei are nearly constantover this energy range (possibly with small but interesting variations). Fractions of bothprimary and secondary incident nuclei are listed in Table 24.1. Figure 24.1 shows themajor components for energies greater than 2 GeV/nucleon.

Figure 24.1: Fluxes of nuclei of the primary cosmic radiation in particles perenergy-per-nucleus are plotted vs energy-per-nucleus using data from Refs. [1–12].The figure was created by P. Boyle and D. Muller. Color version at end of book.

The composition and energy spectra of nuclei are typically interpreted in the contextof propagation models, in which the sources of the primary cosmic radiation are located

February 16, 2012 14:07

24. Cosmic rays 3

within the galaxy [16]. The ratio of secondary to primary nuclei is observed to decreasewith increasing energy, a fact interpreted to mean that the lifetime of cosmic rays in thegalaxy decreases with energy. Measurements of radioactive “clock” isotopes in the lowenergy cosmic radiation are consistent with a lifetime in the galaxy of about 15 Myr.

Table 24.1: Relative abundances F of cosmic-ray nuclei at 10.6 GeV/nucleon nor-malized to oxygen (≡ 1) [6]. The oxygen flux at kinetic energy of 10.6 GeV/nucleonis 3.26 × 10−2 (m2 s sr GeV/nucleon)−1. Abundances of hydrogen and helium arefrom Refs. [2,3]. Note that one can not use these values to extend the cosmicray flux to high energy because the power law indicies for each element may differslightly.

Z Element F

1 H 540

2 He 26

3–5 Li-B 0.40

6–8 C-O 2.20

9–10 F-Ne 0.30

11–12 Na-Mg 0.22

Z Element F

13–14 Al-Si 0.19

15–16 P-S 0.03

17–18 Cl-Ar 0.01

19–20 K-Ca 0.02

21–25 Sc-Mn 0.05

26–28 Fe-Ni 0.12

Cosmic rays are nearly isotropic at most energies due to diffusive propagation inthe galactic magnetic field. Milagro [13], IceCube [14], and The Tibet-III air showerarray [15] have observed anisotropy at the level of about 10−3 for cosmic rays with energyof a few TeV, possibly due to nearby sources.

The spectrum of electrons and positrons incident at the top of the atmosphere isexpected to steepen by one power of E at an energy of ∼5 GeV because of the strongsynchrotron energy loss in the galactic magnetic fields. The ATIC experiment [21]measured an excess of electrons above 100 GeV followed by a steepening above 1,000 GeV.The Fermi/LAT γ-ray observatory confirmed the relatively flat electron spectrum [23]without confirming the peak of the ATIC excess at ∼800 GeV.

The PAMELA satellite experiment [24] measured the positron to electron ratio toincrease above 10 GeV instead of the expected decrease [25] at higher energy. Thestructure in the electron spectrum as well as the increase in the positron fraction couldbe related to contributions from individual nearby sources emerging above a backgroundsuppressed at high energy by synchrotron losses [26]. The low positron to electron ratiobelow 10 GeV is due to the new solar magnetic field polarity after the year 2001.

The ratio of antiprotons to protons is ∼ 2× 10−4 [27] at around 10–20 GeV, and thereis clear evidence [28] for the kinematic suppression at lower energy that is the signatureof secondary antiprotons. The p/p ratio also shows a strong dependence on the phaseand polarity of the solar cycle [29] in the opposite sense to that of the positron fraction.

February 16, 2012 14:07

4 24. Cosmic rays

0.10

0.05

0.02

positron/electron ratio

1

10

100

1 10 100 1000 10000

E3 d

N/d

E, m

-2s-1

sr-1

GeV

2

Energy, GeV

Figure 24.2: Differential spectrum of electrons plus positrons multiplied by E3

(data from [17–23]) . The line shows the proton spectrum multiplied by 0.01. Theinset shows the positron to electron ratio measured by PAMELA [24] compared tothe expected decrease [25].

There is at this time no evidence for a significant primary component of antiprotons. Noantihelium or antideuteron has been found in the cosmic radiation. The best measuredupper limit on the ratio antihelium/helium is currently approximately 1 × 10−7 [30]The upper limit on the flux of antideuterons around 1 GeV/nucleon is approximately2 × 10−4 (m2 s sr GeV/nucleon)−1 [31].

24.2. Cosmic rays in the atmosphere

Figure 24.3 shows the vertical fluxes of the major cosmic ray components in theatmosphere in the energy region where the particles are most numerous (except forelectrons, which are most numerous near their critical energy, which is about 81 MeV inair). Except for protons and electrons near the top of the atmosphere, all particles areproduced in interactions of the primary cosmic rays in the air. Muons and neutrinos areproducts of the decay chain of charged mesons, while electrons and photons originate indecays 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 [32–36]. Since µ+(µ−) areproduced in association with νµ(νµ), the measurement of muons near the maximum of

February 16, 2012 14:07

24. Cosmic rays 5

15 10 5 3 2 1 0

0 200 400 600 800 10000.01

0.1

1

10

100

1000

10000

Atmospheric depth [g cm–2]

Vert

ica

l fl

ux

[m

–2 s

–1 s

r–1]

Altitude (km)

µ+ + µ−

π+ + π−

e+ + e−

p + n

νµ + νµ_

Figure 24.3: Vertical fluxes of cosmic rays in the atmosphere with E > 1 GeVestimated from the nucleon flux of Eq. (24.2). The points show measurements ofnegative muons with Eµ > 1 GeV [32–36].

the intensity curve for the parent pions serves to calibrate the atmospheric νµ beam [37].Because muons typically lose almost 2 GeV in passing through the atmosphere, thecomparison near the production altitude is important for the sub-GeV range of νµ(νµ)energies.

The flux of cosmic rays through the atmosphere is described by a set of coupled cascadeequations with boundary conditions at the top of the atmosphere to match the primaryspectrum. Numerical or Monte Carlo calculations are needed to account accurately fordecay and energy-loss processes, and for the energy-dependences of the cross sections andof the primary spectral index γ. Approximate analytic solutions are, however, useful in

February 16, 2012 14:07

6 24. Cosmic rays

limited regions of energy [38,39]. For example, the vertical intensity of charged pionswith energy Eπ ≪ ǫπ = 115 GeV is

Iπ(Eπ, X) ≈ZNπ

λNIN (Eπ, 0) e−X/Λ X Eπ

ǫπ, (24.3)

where Λ is the characteristic length for exponential attenuation of the parent nucleonflux in the atmosphere. This expression has a maximum at X = Λ ≈121±4 g cm−2 [40],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 nucleonswith nuclei of the atmosphere. The intensity of low-energy pions is much less than thatof nucleons because ZNπ ≈ 0.079 is small and because most pions with energy much lessthan the critical energy ǫπ decay rather than interact.

24.3. Cosmic rays at the surface

24.3.1. Muons : Muons are the most numerous charged particles at sea level (seeFig. 24.3). Most muons are produced high in the atmosphere (typically 15 km) andlose about 2 GeV to ionization before reaching the ground. Their energy and angulardistribution reflect a convolution of the production spectrum, energy loss in theatmosphere, 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 groundis ≈ 4 GeV. The energy spectrum is almost flat below 1 GeV, steepens gradually toreflect the primary spectrum in the 10–100 GeV range, and steepens further at higherenergies because pions with Eπ > ǫπ tend to interact in the atmosphere before theydecay. Asymptotically (Eµ ≫ 1 TeV), the energy spectrum of atmospheric muons is onepower steeper than the primary spectrum. The integral intensity of vertical muons above1 GeV/c at sea level is ≈ 70 m−2s−1sr−1 [41,42], with recent measurements [43–45]favoring a lower normalization by 10-15%. Experimentalists are familiar with this numberin the form I ≈ 1 cm−2 min−1 for horizontal detectors. The overall angular distributionof muons at the ground is ∝ cos2 θ, which is characteristic of muons with Eµ ∼ 3 GeV. Atlower energy the angular distribution becomes increasingly steep, while at higher energyit flattens, approaching a sec θ distribution for Eµ ≫ ǫπ and θ < 70.

Figure 24.4 shows the muon energy spectrum at sea level for two angles. At large angleslow energy muons decay before reaching the surface and high energy pions decay beforethey interact, thus the average muon energy increases. An approximate extrapolationformula valid when muon decay is negligible (Eµ > 100/ cos θ GeV) and the curvature ofthe Earth can be neglected (θ < 70) is

dNµ

dEµdΩ≈

0.14 E−2.7µ

cm2 s sr GeV

×

1

1 +1.1Eµ cos θ

115 GeV

+0.054

1 +1.1Eµ cos θ

850 GeV

, (24.4)

February 16, 2012 14:07

24. Cosmic rays 7

where the two terms give the contribution of pions and charged kaons. Eq. (24.4) neglectsa small contribution from charm and heavier flavors which is negligible except at veryhigh energy [50].

100101 1000

100

1000

pµ [GeV/c]

p µ2.

7 dN/d

p µ

[m−2

s−1 s

r−1(G

eV/c

)1.7 ]

Figure 24.4: Spectrum of muons at θ = 0 (¨ [41], ¥ [46], H [47], N [48], ×,+ [43], [44], and • [45] and θ = 75 ♦ [49]) . The line plots the result fromEq. (24.4) for vertical showers.

The muon charge ratio reflects the excess of π+ over π− and K+ over K− in theforward fragmentation region of proton initiated interactions together with the fact thatthere are more protons than neutrons in the primary spectrum. The increase with energyof µ+/µ− shown in Fig. 24.5 reflects the increasing importance of kaons in the TeVrange [51] and indicates a significant contribution of associated production by cosmic-rayprotons (p → Λ + K+). The same process is even more important for atmosphericneutrinos at high energy.

24.3.2. Electromagnetic component : At the ground, this component consists ofelectrons, positrons, and photons primarily from cascades initiated by decay of neutraland charged mesons. Muon decay is the dominant source of low-energy electrons at sealevel. Decay of neutral pions is more important at high altitude or when the energythreshold is high. Knock-on electrons also make a small contribution at low energy [52].The integral vertical intensity of electrons plus positrons is very approximately 30, 6,and 0.2 m−2s−1sr−1 above 10, 100, and 1000 MeV respectively [42,53], but the exactnumbers depend sensitively on altitude, and the angular dependence is complex becauseof the different altitude dependence of the different sources of electrons [52–54]. Theratio of photons to electrons plus positrons is approximately 1.3 above 1 GeV and 1.7below the critical energy [54].

February 16, 2012 14:07

8 24. Cosmic rays

1.2

1.1

1.3

1.4

1.5

1.6

102101.0 103 104

BESSL3C

MINOS

pµ [GeV/c]

Fµ+

/Fµ−

Figure 24.5: Muon charge ratio as a function of the muon momentum fromRefs. [44,45,51].

24.3.3. Protons : Nucleons above 1 GeV/c at ground level are degraded remnants ofthe primary cosmic radiation. The intensity is approximately IN (E, 0)× exp(−X/ cos θΛ)for θ < 70. At sea level, about 1/3 of the nucleons in the vertical direction areneutrons (up from ≈ 10% at the top of the atmosphere as the n/p ratio approachesequilibrium). The integral intensity of vertical protons above 1 GeV/c at sea level is≈ 0.9 m−2s−1sr−1 [42,55].

24.4. Cosmic rays underground

Only muons and neutrinos penetrate to significant depths underground. The muonsproduce tertiary fluxes of photons, electrons, and hadrons.

24.4.1. Muons : As discussed in Section 27.6 of this Review, muons lose energy byionization and by radiative processes: bremsstrahlung, direct production of e+e− pairs,and photonuclear interactions. The total muon energy loss may be expressed as a functionof the amount of matter traversed as

−dEµ

dX= a + b Eµ , (24.5)

where a is the ionization loss and b is the fractional energy loss by the three radiationprocesses. Both are slowly varying functions of energy. The quantity ǫ ≡ a/b (≈ 500 GeVin standard rock) defines a critical energy below which continuous ionization loss is moreimportant than radiative losses. Table 24.2 shows a and b values for standard rock, and

February 16, 2012 14:07

24. Cosmic rays 9

b for ice, as a function of muon energy. The second column of Table 24.2 shows themuon range in standard rock (A = 22, Z = 11, ρ = 2.65 g cm−3). These parameters arequite sensitive to the chemical composition of the rock, which must be evaluated for eachlocation.

Table 24.2: Average muon range R and energy loss parameters a and b calculatedfor standard rock [56] and the total energy loss parameter b for ice. Range is given

in km-water-equivalent, or 105 g cm−2.

Eµ R a bbrems bpair bnucl∑

bi∑

b(ice)

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 1.66

100 0.41 2.44 1.10 1.53 0.41 3.04 2.51

1000 2.45 2.68 1.44 2.07 0.41 3.92 3.17

10000 6.09 2.93 1.62 2.27 0.46 4.35 3.78

The intensity of muons underground can be estimated from the muon intensity in theatmosphere and their rate of energy loss. To the extent that the mild energy dependenceof a and b can be neglected, Eq. (24.5) can be integrated to provide the following relationbetween the energy Eµ,0 of a muon at production in the atmosphere and its averageenergy Eµ after traversing a thickness X of rock (or ice or water):

Eµ,0 = (Eµ + ǫ) ebX− ǫ . (24.6)

Especially at high energy, however, fluctuations are important and an accurate calculationrequires a simulation that accounts for stochastic energy-loss processes [57].

There are two depth regimes for which Eq. (24.6) can be simplified. For X ≪

b−1 ≈ 2.5 km water equivalent, Eµ,0 ≈ Eµ(X) + aX , while for X ≫ b−1 Eµ,0 ≈

(ǫ + Eµ(X)) exp(bX). Thus at shallow depths the differential muon energy spectrum isapproximately constant for Eµ < aX and steepens to reflect the surface muon spectrumfor Eµ > aX , whereas for X > 2.5 km.w.e. the differential spectrum underground isagain constant for small muon energies but steepens to reflect the surface muon spectrumfor Eµ > ǫ ≈ 0.5 TeV. In the deep regime the shape is independent of depth althoughthe intensity decreases exponentially with depth. In general the muon spectrum at slantdepth X is

dNµ(X)

dEµ=

dNµ

dEµ,0

dEµ,0

dEµ=

dNµ

dEµ,0ebX , (24.7)

where Eµ,0 is the solution of Eq. (24.6) in the approximation neglecting fluctuations.

Fig. 24.6 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

February 16, 2012 14:07

10 24. Cosmic rays

in the atmosphere, the map of the overburden at each detector, and the properties of thelocal medium in connecting measurements at various slant depths and zenith angles tothe vertical intensity. Use of data from a range of angles allows a fixed detector to covera wide range of depths. The flat portion of the curve is due to muons produced locally bycharged-current interactions of νµ. The inset shows the vertical intensity curve for waterand ice published in Refs. [59–62]. It is not as steep as the one for rock because of thelower muon energy loss in water.

1 10 100

1 102 5

Figure 24.6: Vertical muon intensity vs depth (1 km.w.e. = 105 g cm−2of standardrock). The experimental data are from: ♦: the compilations of Crouch [58], ¤:Baksan [63], : LVD [64], •: MACRO [65], ¥: Frejus [66], and : SNO [67].The shaded area at large depths represents neutrino-induced muons of energy above2 GeV. The upper line is for horizontal neutrino-induced muons, the lower onefor vertically upward muons. Darker shading shows the muon flux measured bythe SuperKamiokande experiment. The inset shows the vertical intensity curve forwater and ice published in Refs. [59–62].

February 16, 2012 14:07

24. Cosmic rays 11

24.4.2. Neutrinos :

Because neutrinos have small interaction cross sections, measurements of atmosphericneutrinos require a deep detector to avoid backgrounds. There are two types ofmeasurements: contained (or semi-contained) events, in which the vertex is determinedto originate inside the detector, and neutrino-induced muons. The latter are muons thatenter the detector from zenith angles so large (e.g., nearly horizontal or upward) thatthey cannot be muons produced in the atmosphere. In neither case is the neutrino fluxmeasured directly. What is measured is a convolution of the neutrino flux and crosssection with the properties of the detector (which includes the surrounding medium inthe case of entering muons).

Contained and semi-contained events reflect neutrinos in the sub-GeV to multi-GeVregion where the product of increasing cross section and decreasing flux is maximum. Inthe GeV region the neutrino flux and its angular distribution depend on the geomagneticlocation 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 ofpion and muon decay. Contrary to expectation, however, the numbers of the two classesof events are similar rather than different by a factor of two. This is now understoodto be a consequence of neutrino flavor oscillations [70]. (See the article on neutrinoproperties in this Review.)

Two well-understood properties of atmospheric cosmic rays provide a standard forcomparison of the measurements of atmospheric neutrinos to expectation. These arethe “sec θ effect” and the “east-west effect” [69]. The former refers originally to theenhancement of the flux of > 10 GeV muons (and neutrinos) at large zenith angles becausethe parent pions propagate more in the low density upper atmosphere where decay isenhanced relative to interaction. For neutrinos from muon decay, the enhancement nearthe horizontal becomes important for Eν > 1 GeV and arises mainly from the increasedpathlength through the atmosphere for muon decay in flight. Fig. 24.7 from Ref. 68 showsa comparison between measurement and expectation for the zenith angle dependence ofmulti-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 significantdeficit of upward (cos θ < 0) νµ events, which have long pathlengths comparable to theradius of the Earth. This feature is the principal signature for oscillations [70].

Muons that enter the detector from outside after production in charged-currentinteractions of neutrinos naturally reflect a higher energy portion of the neutrinospectrum than contained events because the muon range increases with energy as wellas the cross section. The relevant energy range is ∼ 10 < Eν < 1000 GeV, dependingsomewhat on angle. Neutrinos in this energy range show a sec θ effect similar tomuons (see Eq. (24.4)). This causes the flux of horizontal neutrino-induced muons tobe approximately a factor two higher than the vertically upward flux. The upper andlower edges of the horizontal shaded region in Fig. 24.6 correspond to horizontal andvertical intensities of neutrino-induced muons. Table 24.3 gives the measured fluxes ofupward-moving neutrino-induced muons averaged over the lower hemisphere. Generallythe definition of minimum muon energy depends on where it passes through the detector.

February 16, 2012 14:07

12 24. Cosmic rays

multi-GeV µ C

0

5

100

150

200

250

300

Num

ber

of e

vent

s

cos θ

multi-GeV e-like

−1 −0.5 0 0.5 −1 −0.5 0 0.5 1cos θ

-like + P

Figure 24.7: Zenith-angle dependence of multi-GeV neutrino interactions fromSuperKamiokande [68]. The shaded boxes show the expectation in the absence ofany oscillations.

Table 24.3: Measured fluxes (10−9 m−2 s−1 sr−1) of neutrino-induced muons as afunction of the effective minimum muon energy Eµ.

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

Ref. CWI [71] Baksan [72] MACRO [73] IMB [74] Kam [75] SuperK [76]

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

The tabulated effective minimum energy estimates the average over various acceptedtrajectories.

24.5. Air showers

So far we have discussed inclusive or uncorrelated fluxes of various components of thecosmic radiation. An air shower is caused by a single cosmic ray with energy high enoughfor its cascade to be detectable at the ground. The shower has a hadronic core, which actsas a collimated source of electromagnetic subshowers, generated mostly from π0 → γ γdecays. The resulting electrons and positrons are the most numerous charged particles inthe shower. The number of muons, produced by decays of charged mesons, is an orderof magnitude lower. Air showers spread over a large area on the ground, and arrays ofdetectors operated for long times are useful for studying cosmic rays with primary energyE0 > 100 TeV, where the low flux makes measurements with small detectors in balloonsand satellites difficult.

February 16, 2012 14:07

24. Cosmic rays 13

Greisen [77] gives the following approximate expressions for the numbers and lateraldistributions of particles in showers at ground level. The total number of muons Nµ withenergies above 1 GeV is

Nµ(> 1 GeV) ≈ 0.95 × 105(

Ne/106)3/4

, (24.8)

where Ne is the total number of charged particles in the shower (not just e±). Thenumber of muons per square meter, ρµ, as a function of the lateral distance r (in meters)from the center of the shower is

ρµ =1.25 Nµ

2π Γ(1.25)

(

1

320

)1.25

r−0.75(

1 +r

320

)−2.5, (24.9)

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 + C2xd) . (24.10)

Here s, d, and C2 are parameters in terms of which the overall normalization constantC1(s, d, C2) is given by

C1(s, d, C2) =Ne

2πr21

[ B(s, 4.5− 2s)

+ C2 B(s + d, 4.5 − d − 2s)]−1 , (24.11)

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 withNe ≈ 106 at sea level, Greisen uses s = 1.25, d = 1, and C2 = 0.088. Finally, x is r/r1,where r1 is the Moliere radius, which depends on the density of the atmosphere and henceon the altitude at which showers are detected. At sea level r1 ≈ 78 m. It increases withaltitude as the air density decreases. (See the section on electromagnetic cascades in thearticle on the passage of particles through matter in this Review).

The lateral spread of a shower is determined largely by Coulomb scattering of themany low-energy electrons and is characterized by the Moliere radius. The lateral spreadof the muons (ρµ) is larger and depends on the transverse momenta of the muons atproduction as well as multiple scattering.

There are large fluctuations in development from shower to shower, even for showers ofthe same energy and primary mass—especially for small showers, which are usually wellpast maximum development when observed at the ground. Thus the shower size Ne andprimary energy E0 are only related in an average sense, and even this relation dependson depth in the atmosphere. One estimate of the relation is [84]

E0 ∼ 3.9 × 106 GeV (Ne/106)0.9 (24.12)

for vertical showers with 1014 < E < 1017 eV at 920 g cm−2 (965 m above sea level). AsE0 increases the shower maximum (on average) moves down into the atmosphere and the

February 16, 2012 14:07

14 24. Cosmic rays

relation between Ne and E0 changes. Moreover, because of fluctuations, Ne as a functionof E0 is not correctly obtained by inverting Eq. (24.12). At the maximum of showerdevelopment, there are approximately 2/3 particles per GeV of primary energy.

There are three common types of air shower detectors: shower arrays that studythe shower size Ne and the lateral distribution on the ground, Cherenkov detectorsthat detect the Cherenkov radiation emitted by the charged particles of the shower,and fluorescence detectors that study the nitrogen fluorescence excited by the chargedparticles in the shower. The fluorescence light is emitted isotropically so the showers canbe observed from the side. Detailed simulations and cross-calibrations between differenttypes of detectors are necessary to establish the primary energy spectrum from air-showerexperiments.

Figure 24.8 shows the “all-particle” spectrum. The differential energy spectrum hasbeen multiplied by E2.6 in order to display the features of the steep spectrum that areotherwise difficult to discern. The steepening that occurs between 1015 and 1016 eV isknown as the knee of the spectrum. The feature around 1018.5 eV is called the ankle ofthe spectrum.

[eV]E

1310 1410 1510 1610 1710 1810 1910 2010

]-1

sr

-1 s

-2 m

1.6

[GeV

F(E

)2.

6E

1

10

210

310

410

Grigorov

JACEE

MGU

Tien-Shan

Tibet07

Akeno

CASA-MIA

HEGRAFly’s Eye

Kascade

Kascade Grande 2011

AGASA

HiRes 1

HiRes 2Telescope Array 2011

Auger 2011

Knee

Ankle

Figure 24.8: The all-particle spectrum as a function of E (energy-per-nucleus)from air shower measurements [79–90,100–104]. Color version at end of book.

Measurements of flux with small air shower experiments in the knee region differ byas much as a factor of two, indicative of systematic uncertainties in interpretation of thedata. (For a review see Ref. 78.) In establishing the spectrum shown in Fig. 24.8, efforts

February 16, 2012 14:07

24. Cosmic rays 15

have been made to minimize the dependence of the analysis on the primary composition.Ref. 87 uses an unfolding procedure to obtain the spectra of the individual components,giving a result for the all-particle spectrum between 1015 and 1017 eV that lies towardthe upper range of the data shown in Fig. 24.8. In the energy range above 1017 eV,the fluorescence technique [89] is particularly useful because it can establish the primaryenergy in a model-independent way by observing most of the longitudinal developmentof each shower, from which E0 is obtained by integrating the energy deposition inthe atmosphere. The result, however, depends strongly on the light absorption in theatmosphere and the calculation of the detector’s aperture.

Assuming the cosmic ray spectrum below 1018 eV is of galactic origin, the knee couldreflect the fact that most cosmic accelerators in the galaxy have reached their maximumenergy. Some types of expanding supernova remnants, for example, are estimated not tobe able to accelerate protons above energies in the range of 1015 eV. Effects of propagationand confinement in the galaxy [91] also need to be considered. The KASCADE-Grandeexperiment [90] has reported observation of a second steepening of the spectrum near8 × 1016 eV, with evidence that this structure is accompanied a transition to heavyprimaries.

Concerning the ankle, one possibility is that it is the result of a higher energypopulation of particles overtaking a lower energy population, for example an extragalacticflux beginning to dominate over the galactic flux (e.g. Ref. 89). Another possibility isthat the dip structure in the region of the ankle is due to γp → e+ + e− energy lossesof extragalactic protons on the 2.7 K cosmic microwave radiation (CMB) [93]. This dipstructure has been cited as a robust signature of both the protonic and extragalacticnature of the highest energy cosmic rays [92]. If this interpretation is correct, then thegalactic cosmic rays do not contribute significantly to the flux above 1018 eV, consistentwith the maximum expected range of acceleration by supernova remnants.

The energy-dependence of the composition from the knee through the ankle is usefulin discriminating between these two viewpoints, since a heavy composition above 1018

eV is inconsistent with the formation of the ankle by pair production losses on theCMB. The HiRes and Auger experiments, however, present very different interpretationsof data on the depth of shower maximum Xmax, a quantity that correlates stronglywith the interaction cross section of the primary particle. If these results are interpretedusing standard extrapolations of measured proton and nuclear cross sections, then theHiRes data [94] is consistent with the ultrahigh-energy cosmic ray (UHECR) compositiongetting lighter and containing only protons and helium above 1019 eV, while Auger [95,96]sees a composition getting lighter up to 2 × 1018 eV and becoming heavier afterthat, intermediate between protons and iron at 3 × 1019 eV. This may mean that theextragalactic cosmic rays have a mixed composition at acceleration similar to the GeVgalactic cosmic rays. It is important to note that the measurements of Xmax may beinterpreted with equal validity in terms of a changing proton-air cross-section and nochange in composition.

If the cosmic ray flux at the highest energies is cosmological in origin, there should be arapid steepening of the spectrum (called the GZK feature) around 5 × 1019 eV, resultingfrom the onset of inelastic interactions of UHE cosmic rays with the cosmic microwave

February 16, 2012 14:07

16 24. Cosmic rays

[eV]E

1810 1910 2010

]-1

sr

-1 s

-2 m

1.6

[GeV

F(E

)2.

6E

1

10

210

310

HiRes 1

HiRes 2

Telescope Array 2011

Auger 2011

E/E=20%∆

Figure 24.9: Expanded view of the highest energy portion of the cosmic-rayspectrum from data of HiRes 1&2 [101], the Telescope Array [103], and the AugerObservatory [104]. The HiRes stereo spectrum [112] is consistent with the HiRes1&2 monocular results. The differential cosmic ray flux is multiplied by E2.6. Thered arrow indicates the change in the plotted data for a systematic shift in theenergy scale of 20%.

background [97,98]. Photo-dissociation of heavy nuclei in the mixed compositionmodel [99] would have a similar effect. UHECR experiments have detected events ofenergy above 1020 eV [89,100–102]. The AGASA experiment [100] did not observethe expected GZK feature. The HiRes fluorescence experiment [101,112] has detectedevidence of the GZK supression, and the Auger observatory [102–104] has presentedspectra showing this supression based on surface detector measurements calibratedagainst its fluorescence detector using events detected in hybrid mode, i.e. with both thesurface and the fluorescence detectors. Recent observations by the Telescope Array [103]also exhibit this supression.

Figure 24.9 gives an expanded view of the high energy end of the spectrum, showingonly the more recent data. This figure shows the differential flux multiplied by E2.6.The experiments are consistent in normalization if one takes quoted systematic errors inthe energy scales into account. The continued power law type of flux beyond the GZKcutoff previously claimed by the AGASA experiment [100] is not supported by the HiRes,Telescope Array, and Auger data.

One half of the energy that UHECR protons lose in photoproduction interactions that

February 16, 2012 14:07

24. Cosmic rays 17

[eV]νE

1610 1710 1810 1910 2010 2110

]-1

sr

-1 s

-2[m ν

dN

/dE

νE

-1610

-1510

-1410

-1310

-1210

-1110

-1010

-910

-810

-710

= 1:1:1)τν: µν: eνAll flavor limits(

IceCube40 (2011)

Auger (2011)

RICE (2011)

ANITA-2 (2010)

WB

ESS

Figure 24.10: Differential limits on the flux of cosmogenic neutrinos set byfour neutrino experiments. The curves show the Waxman-Bahcall benchmarkflux (WB, [111]) and a representative midrange model for the expected flux ofcosmogenic neutrinos (ESS, [110]) . The expected flux is uncertain by over an orderof magnitude in either direction.

cause the GZK effects ends up in neutrinos [105]. Measuring this cosmogenic neutrinoflux above 1018 eV would help resolve the UHECR uncertainties mentioned above. Themagnitude of this flux depends strongly on the cosmic ray spectrum at acceleration, thecosmic ray composition, and the cosmological evolution of the cosmic ray sources. Inthe case that UHECR have mixed composition only the proton fraction would producecosmogenic neutrinos. Heavy nuclei propagation produces mostly νe at lower energy fromneutron decay.

The expected rate of cosmogenic neutrinos is lower than current limits obtained byIceCube [106], the Auger observatory [108], RICE [107], and ANITA-2 [109], which areshown in Figure 24.10 together with a model for cosmogenic neutrino production [110] andthe Waxman-Bahcall benchmark flux of neutrinos produced in cosmic ray sources [111].At production, the dominant component of neutrinos comes from π± decays and hasflavor content νe : νµ : ντ = 1 : 2 : 0. After oscillations, the arriving cosmogenic neutrinos

February 16, 2012 14:07

18 24. Cosmic rays

are expected to be an equal mixture of all three flavors. The sensitivity of each experimentdepends on neutrino flavor. IceCube, RICE, and ANITA are sensitive to all three flavors,and the sensitivity to different flavors is energy dependent. The limit of Auger is only forντ and ντ which should be about 1/3 of the total neutrino flux after oscillations, so thislimit is plotted multiplied by a factor of three for comparison with the other limits andwith the theoretical estimates.

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