Invited, Rapporteur, and Highlight papers of ICRC 2001: 240

ICRC 2001c© Copernicus Gesellschaft 2002

The physics of the knee in the cosmic ray spectrum

K-H. Kampert 1,2, T. Antoni1, W. D. Apel2, F. Badea1,*, K. Bekk2, A. Bercuci2,*, H. Blumer2,1, E. Bollmann2,H. Bozdog3, I. M. Brancus3, C. Buttner2, A. Chilingarian 4, K. Daumiller 1, P. Doll2, J. Engler2, F. Feßler1, H. J. Gils2,R. Glasstetter1, R. Haeusler1, A. Haungs2, D. Heck2, J. R. Horandel1, T. Holst2, A. Iwan1,5, J. Kempa5,+, H. O. Klages2,J. Knapp1,¶, G. Maier2, H. J. Mathes2, H. J. Mayer2, J. Milke1, M. M uller2, R. Obenland2, J. Oehlschlager2,S. Ostapchenko1, M. Petcu3, H. Rebel2, M. Risse2, M. Roth2, H. Schieler2, J. Scholz2, T. Thouw2, H. Ulrich 1,B. Vulpescu3, J. H. Weber1, J. Wentz2, J. Wochele2, J. Zabierowski6, and S. Zagromski2

1Institut fur Experimentelle Kernphysik, University of Karlsruhe, 76021 Karlsruhe, Germany2Institut fur Kernphysik, Forschungszentrum Karlsruhe, 76021 Karlsruhe, Germany3National Institute of Physics and Nuclear Engineering, 7690 Bucharest, Romania4Cosmic Ray Division, Yerevan Physics Institute, Yerevan 36, Armenia5Department of Experimental Physics, University of Lodz, 90236 Lodz, Poland6Soltan Institute for Nuclear Studies, 90950 Lodz, Poland¶now at: University of Leeds, Leeds LS2 9JT, U.K.∗on leave of absence from IFIN-HH, Bucharest+now at: Warsaw University of Technology, 09-400 Plock, Poland

Abstract. Recent results from the KASCADE extensive airshower experiment are presented. After briefly reviewingthe status of the experiment we report on tests of hadronicinteraction models and emphasize the progress being madein understanding the properties and origin of the knee atEknee ∼= 4 · 1015 eV.

Analysing the muon- and hadron trigger rates in the KAS-CADE calorimeter as well as the global properties of highenergy hadrons in the shower core leads us to conclude thatQGSJET still provides the best overall description of EASdata, being superior to DPMJET II-5 andNEXUS 2, for ex-ample.

Performing high statistics CORSIKA simulations and ap-plying sophisticated unfolding techniques to the electron andmuon shower size distributions, we are able to success-fully deconvolute the all-particle energy spectrum into en-ergy spectra of 4 individual primary mass groups (p, He, C,Fe). Each of these preliminary energy distributions exhibitsa knee like structure with a change of their knee positionssuggesting a constant rigidity ofR ∼= 2-3 PV.

1 Introduction

The origin and acceleration mechanism of ultra-high energycosmic rays (E >∼ 1014 eV) have been subject to debatefor several decades. Mainly for reasons of the power re-quired to maintain the observed cosmic ray energy densityof εcr ≈ 1 eV/cm3, the dominant acceleration sites are gen-

Correspondence to:K.-H. Kampert (kampert@ik.fzk.de)

erally believed to be supernova remnants (SNR). Chargedparticles mainly originating from the surrounding interstel-lar medium of the pre-supernova star may get trapped at thehighly supersonic shock wave generated by the SN explo-sion. Repeatedly reflections on both sides of the shock frontlead to an acceleration by the so-called ‘first order Fermimechanism’. Naturally, this leads to a power law spec-trum dJ/dE ∝ E−γ as is observed experimentally. Sim-ple dimensional estimates show that this process is limited toEmax <∼ Z × (ρ × B), with Z being the atomic number ofthe cosmic ray (CR) isotope andρ, B the size and magneticfield strength of the acceleration region. A more detailed ex-amination of the astrophysical parameters suggests an upperlimit of acceleration ofEmax ≈ Z × 1015 eV (Drury, 1994;Berezhko and Ksenofontov, 1999). Curiously, the CR spec-trum steepens fromγ ' 2.75 to' 3.1 atE ' 4 × 1015 eVwhich is called the ‘knee’. The coincidence thus may indi-cate that the ‘knee’ is related to the upper limit of accelera-tion.

Alternative interpretations of the knee discuss a change inthe propagation of CRs from their sources to the solar sys-tem. Such kind of propagation effects are conveniently de-scribed by an ‘escape time’ from our galaxy,τesc. Extrap-olation of τesc from direct measurements in the GeV rangeto higher energies breaks down atE ≈ 3 · 1015 eV, becausecτesc ∼ 300 pc which is the thickness of the galactic disk(Gaisser, 2000). The value corresponds to just one cross-ing of the disk and would give rise to significant anisotropieswith respect to the galactic plane when approaching thisvalue. Similarly as to the process of acceleration at SNR

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shocks, the process of galactic containment is closely relatedto magnetic field confinement, i.e. in addition to anisotropiesone again expectsEgal

max ∝ Z.A picture related to both of these interpretations has been

proposed byErlykin and Wolfendale(1997). They considerthe knee as a superposition of a weakly energy dependentgalactic modulation with additional prominent structures inthe flux spectrum caused by a single near-by object. Thisso-called ‘’single source model” assumes that a shock waveof a recent nearby supernova which exploded some 10 000years ago at a distance of a few hundred parsecs, currentlypropagates (or has recently propagated) through the solarsystem causing distinct peaks of elemental groups in the en-ergy spectrum. The most recent update of this model can befound in the proceedings to this conference (p1804). How-ever, there is some controversy whether or not the statistics ofpresently existing data gives sufficient support to the model.

Besides those kind of astrophysical interpretations, alsoparticle physics motivated pictures of explaining the kneewere put forward. For example,Wigmans(2000) suggestedthe inverseβ-decay reactionp + νe → n + e+ with mas-sive relic neutrinos could destroy protons. Simple kine-matics shows that this channel is open forEp > 1.7 ·1015 eV/mν(eV). Thus, a knee energyEknee ' 4 PeVwould correspond to an electron neutrino mass ofmν ∼0.4 eV, a value presently not excluded by any other obser-vation or experiment. However, ‘eating’ sufficiently largeamounts of protons by such a process requires extraordinaryhigh local densities of relic neutrinos, which appears doubt-ful even if possible gravitational trapping is considered. Incase of a non-zero magnetic dipole moment of massive neu-trinos, also electromagnetic interactions with much largercross section would be possible, thus giving rise to inelasticGZK-like pν-interactions. Such a picture requiring neutrinosmasses of about 100 eV has recently been discussed byDova,Epele, and Swain(2001).

Nikolsky (1995) suggested that the knee is not a propertyof the primary energy spectrum itself, but may be caused bychanging high-energy interactions in the Earth’s atmosphere.Producing a new type of a heavy particle in the first inter-actions escaping unseen by air shower experiments, or anabrupt increase in the multiplicity of produced particles (seeproceedings to this conference, p1389) could, in principle,mimic a break in the spectrum. From the particle physicspoint of view this is not completely ruled out as the centre-of-mass (cms) energy available at the knee is above Tevatronenergies. However, a well worked out model simultaneouslydescribing accelerator and CR data without violating generalphysics, like the unitarity principle, is still to be presented.

Very recently, exotic scenarios with extra dimensions andTeV-scale quantum gravity have been discussed byKazanasand Nicolaidis(2001) and others. Again, the argument isthat the cms-energy of the knee (

√s ∼= 2 TeV) is just above

the highest energies reached by present accelerators and thatsome part of the CR energies is transferred into invisiblechannels, in this case gravitational energy. Some advantageof this picture is that the required fast growth of cross section

for graviton radiation does not necessarily violate S-waveunitarity.

A common feature of the particle physics interpretationsof the knee is the expectation of seeing the break of differentelements to be displaced by their mass numberA rather thanby their nuclear chargeZ. This is understood from the reac-tion mechanism being governed, at sufficiently high energy,by the energy per nucleonE/A of the incident particle.

To distinguish between these various ideas, better mea-surements of the energy spectrum and elemental compositionof cosmic rays in the knee region are essential. Most com-monly, the composition is expressed in terms of the meanlogarithmic mass,〈lnA〉. Obviously, such an average quan-tity appears to be rather insensitive to details of the individ-ual energy spectra. Clearly, the most sensitive testbed forcomparing different models would be given by CR energyspectra of different elemental groups. Direct measurementscould in principle provide such information. However, dueto their very limited collection area and exposure time, directmeasurements across the knee energy are practically impos-sible. On the other hand, the well known difficulties in theinterpretation of EAS data have not yet permitted such mea-surements either. Thus, beyond the knee energy very little isknown about CRs other than their all-particle energy spec-trum (Watson, 1998).

The primary goal of the KASCADE experiment is to pro-vide such information by measuring the electromagnetic,muonic, and hadronic EAS components with high precisionin each individual event and by applying advanced methodsof multi-parameter data analyses. Determining the energyspectrum and composition of cosmic rays in the knee re-gion with high precision also calls for detailed EAS simula-tions, as performed by the CORSIKA package (Heck, 1998).Testing and improving hadronic interaction models enteringthe EAS simulations is another objective of KASCADE. Asdiscussed below, this is realised mostly by investigating theproperties of high energy hadrons in the shower core. Re-constructing properties of the primary CRs from differentsets of EAS observables and comparing their results providesanother powerful means to judge the reliability of measure-ments and simulations.

2 Experimental

KASCADE (Karlsruhe Shower Core and Array Detector) islocated at the laboratory site of Forschungszentrum Karls-ruhe, Germany (at8◦ E, 49◦ N, 110 m a.s.l.). In brief, itconsists of three major components (see Fig.1);

1. A scintillator array comprising 252 detector stationsof electron and muon counters arranged on a grid of200 × 200 m2 and providing in total about 500 m2 ofe/γ- and 620 m2 of µ-detector coverage. The detectionthresholds for vertical incidence areEe > 5 MeV andEµ > 230 MeV.

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central detector

array cluster

detector station

electronic station

B B

B - B

muon tunnel

0 10m 20m

200 m

200

m13 m

sand shielding

streamertubes

concrete

steel

540 cmLength: 48 m

240

cm

MuonChambers

StreamerTubes

Concrete

Iron Plates

Trigger Layer

Top Cluster

Top Layer

TMS-Kammern

TMS / TMP-Chambers

20 m16 m

Fig. 1. Schematic layout of the KASCADE experiment (left), with its streamer tube tracking system (top right) and central detector (bottomright).

2. A central detector system (320 m2) consisting of ahighly-segmented hadronic calorimeter read out by44 000 channels of warm liquid ionization chambersdistributed over 9 read-out layers, another set of scin-tillation counters above the shielding (top cluster), atrigger plane of scintillation counters in the third irongap and, at the very bottom, 2 layers of positional sen-sitive MWPC’s, and a streamer tube layer with padread-out for investigation of the muon component atEµ > 2.4 GeV.

3. A 48×5.4 m2 tunnel housing three horizontal and a twovertical layers of positional sensitive limited streamertubes for muon tracking atEµ > 0.8 GeV.

More details about the experiment can be found inDollet al. (1990) and in Klageset al. (1997). First correlateddata have been taken with some parts of the experiment since1996 and with its full set-up since 2000. At present, morethan 500 Mio. events have been collected in a very stablemode and with a trigger threshold of the array correspondingtoE ∼ 4 · 1014 eV.

3 Tests of high-energy hadronic interaction models

The observation of EAS provides an opportunity to studyglobal properties of hadronic interactions in an energy rangenot accessible to man-made accelerators. For example, thecms-energy at the Tevatron collider corresponds to a fixedtarget energy in the nucleon-nucleon system ofEp ' 1.7 ·

1015 eV. Even more importantly, the diffractive particle pro-duction dominating the energy flux in the forward region in-fluences the EAS development most strongly, but has beenstudied experimentally only at comparatively low energies of√s ' 10 GeV (Kaidalov, 1979). Most of the beam energy in

present collider experiments remains unobserved. For exam-ple, the UA5 experiment could register up to 30 % of the to-tal collision energy at

√s = 0.9 TeV, while the CDF detector

registers only about 5 % at√s = 1.8 TeV. Hence, hadronic

interaction models applied to higher cms-energies and to par-ticle production in the very forward region rely on extrapola-tions and may cause systematic uncertainties in simulationsof EAS. Additional uncertainties arise from simulations ofp-nucleus and nucleus-nucleus collisions including a possi-ble formation of a quark-gluon plasma. Again, such data areimportant for EAS interpretations but have been studied onlyat low energies in the past (SPS and ISR at CERN). Onlyvery recently, RHIC data at

√s = 200 GeV have become

available and will be very helpful in this respect.

In KASCADE, hadronic interaction models have beentested by EAS data employing two basic approaches: Atlower primary CR energies, where data from direct measure-ments still exist, the energy spectra and rate of single hadronsnot accompanied by air showers were studied. At higher CRenergies, on the other hand, various features of the hadronicshower cores were investigated. In both approaches, the sen-sitivity to details of the hadronic interaction models arisesmostly from the high energy threshold applied to the hadronreconstruction in the calorimeter.

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estimation ofcommon systematics

KASCADE DPMJET

SIBYLL 1.6

QGSJET

VENUS

HDPM

neXus 2

20

25

30

35

40

45

0 2 4 6 8 10 12

hadron rate (1/min)

trig

ger

rate

(1/

min

)

KASCADE QGSJET

σCDF

σdiffr-3.5%

σinel+5%

σdiffr-6.5%

σdiffr-10%σinel+10%

16

18

20

22

24

26

28

30

32

34

0 1 2 3 4 5 6

Fig. 2. Trigger rate vs hadron rate in the KASCADE central detec-tor. The top panel compares different hadronic interaction modelsto the experimental data. The systematic uncertainty, mostly givenby the absolute flux uncertainty of the direct experiments, is indi-cated by the dotted line. The lower panel shows results obtainedfrom the QGSJET model with modified inelastic cross section anddiffraction dissociation (Risse, 2000; Antoni et al., 2001).

3.1 Tests at low primary energies: Study of trigger rates

An important source of uncertainty in hadronic interactionmodels originates from the energy dependent inelastic cross-sections (Block, Halzen, Stanev, 2000). Taking into accountthe deviations among different experiments at Tevatron en-ergies, the proton-antiproton cross section is not known tobetter than 5 % at best (Avila et al., 1999). As shown be-low, these uncertainties are amplified when predicting abso-lute hadron fluxes at sea-level by means of EAS simulations.Thus, one may take advantage of this strong dependence andperform stringent tests of models on the basis of EAS data.The idea is to use data ofabsoluteCR fluxes up to severalTeV of energy, as obtained from balloon- and satellite borneexperiments at the top of the atmosphere. Taking into ac-count the measured energy distribution and chemical compo-sition, these particles are then propagated through the atmo-sphere employing the CORSIKA simulation package. At thelevel of the KASCADE experiment, we then ask for triggersreleased by either high energy hadrons (Eh >∼ 90 GeV) or aminimum number of 9 muons (Eµ >∼ 0.49 GeV) detected inthe central detector of KASCADE. Such an energy cut ap-plied to the hadrons automatically selects particles originat-ing from the first high-energy interactions in the atmosphere.Employing different hadronic interaction models, these sim-ulated inclusive trigger rates are then compared to actual ex-perimental data (Antoni et al., 2001). Figure2 shows the

10-3

10-2

dN/d

xla

b

0 0.2 0.4 0.6 0.8 1

DPMJET 2.4HDPMQGSJET 98SIBYLL 1.6VENUS

Fig. 3. CORSIKA simulations of Feynman-x distributions of lead-ing baryons fromp + N interactions at1016 eV energy. Five dif-ferent interaction models (versions of 1997) are shown (Hecket al.,2001).

results. None of the predictions agrees well with the exper-imental data, particularly the hadron rates are overestimatedby up to a factor of 3. Furthermore, there are also large dif-ferences found among the models. This convincingly provesthe sensitivity of the experimental observable to details of theinteraction models.

Earlier investigations of high-energy hadrons observed inthe shower core (Antoni et al., 1999) lead us to the conclu-sion that QGSJET (Kalmykov and Ostapchenko, 1993) pro-vides the best overall prescription of EAS data. This was in-dependently concluded also from a consistency analysis in-cluding data from different EAS experiments (Erlykin andWolfendale, 1998). Therefore, several modifications wereapplied to the QGSJET model in order to study their influ-ence to the predicted trigger rates. Results are presented inthe lower panel of Fig.2. Increasing the inelastic proton-aircross section by 5 % (10 %) reduces the predicted hadron rateby approx. 27 % (54 %). Similar effects are obtained by low-ering the diffraction dissociation by up to 10 % of the inelas-tic cross section. Clearly, there is a strong sensitivity to theseparameters. Since the total inelastic cross section appearsto be the better known quantity, we conclude that the uncer-tainty arises mostly from the diffraction dissociation whichmay be overestimated in the simulated hadron-nucleus inter-actions by about 5 % of the inelastic cross sectionσp+air

inel . Infact, the trigger rates predicted by the different models shownin Fig. 2 (top) are strongly correlated to the simulated shapesof the Feynman-x distributions of leading baryons. This isdemonstrated in Fig.3 for CORSIKA simulations ofp + Ninteractions atE = 1016 eV; for the same total cross sec-tion, a large fraction of diffractive interactions leads to hightrigger rates and vice versa. In order to verify the aforemen-tioned conclusions about the observed deviations of the trig-ger rates it would be highly desirable to measure the protoninelastic cross section on nitrogen and oxygen at the highestenergies at accelerators and to study the kinematical regionof diffraction dissociation.

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-0.1 5

-0.1

-0.0 5

0

0.05

4 4 .5 5 5 .5 6 6.5lg(Ne)

rela

tive

devi

atio

n <

lg(Σ

Eh/

GeV

)>

p FeQGSJET

-0.3

-0.2

-0.1

0

0.1

3 3 .5 4 4 .5 5lg(Ntr

µ)

rela

tive

devi

atio

n <

lg(Σ

Eh/

GeV

)>

p FeQGSJETNEXUSDPMJET

NEXUSDPMJET

Fig. 4. Deviations of the simulated hadronic energy sum normal-ized to experimental KASCADE data for QGSJET,NEXUS 2, andDPMJET II-5. Data in the upper and lower panel are binned asfunction of electron- and truncated muon size, respectively. (Milke,2002)

3.2 Tests at high primary energies: Study of hadrons in theshower core

While the previous study selects mostly diffractive type in-teractions in the atmosphere, the following analysis samplesthe more ’typical’ nuclear interactions leading to a fully de-veloped EAS. Events have to fulfil standard cuts in electronand muon number for EAS (Ne ≥ 104,N tr

µ ≥ 103) and needto have at least one hadron withE >∼ 50 GeV reconstructedin the calorimeter. Details of this type of analysis were pre-sented byMilke et al. (2001) at this conference. An exam-ple employing only the three models which are closest to thedata in Fig.2 is depicted in Fig.4. Here, the total hadronicenergy observed in the calorimeter is compared with COR-SIKA simulations for proton and iron primaries employingthe QGSJET (Kalmykov and Ostapchenko, 1993), NEXUS

2 (Drescher et al., 2000), and DPMJET (version II-5)(Ranft, 1999) models. The upper panel shows the ra-tio (lg

∑Eh,sim− lg

∑Eh,data)/ lg

∑Eh,data for each of

these models as a function of electron size. The lower panelshows the same quantity as function of the truncated muon

size, defined asN trµ =

∫ 200m

40m2πρµ(r)dr. Considering the

steep primary energy distribution and the different EAS de-velopments for light and heavy primaries, binning the data asa function oflgNe tends to enrich showers originating fromlight primaries. Therefore, proton simulations (open symbolsin the upper panel of Fig.4) are expected to closely resemblethe experimental data. QGSJET simulations follow this ex-pectation best. With some exceptions at lower electron sizes,also DPMJET provides a reasonable description of the ex-perimental data. However,NEXUS 2 clearly fails in this test:there is too little hadronic energy at given electron size. Sincehadrons in an EAS continuously populate the electromag-netic component, this mismatch points to a general problemof the balance between electromagnetic and hadronic energyin the used version ofNEXUS. However, it should be men-tioned that this version ofNEXUS is still considered prelim-inary and will be subject of further improvements. The ob-served overall decrease of simulated/experimental hadronicenergies may be due to an increasingly heavier compositionin the experimental data at higher energies (see next section).

The truncated muon number provides an energy estimatorwhich is almost independent from primary mass. Thus, re-peating the analysis and plotting the same quantities now asa function of truncated muon size should yield results withthe proton and iron simulations being reasonably above andbelow the experimental data. Again, QGSJET exhibits theexpected behaviour (lower panel of Fig.4). Different fromabove, now theNEXUS model follows the QGSJET resultsvery closely and DPMJET fails the test; DPMJET II-5 pro-duces significantly too many hadrons so that a compositionbased on this model would appear too heavy. Such a compo-sition contradicts measurements from other experiments aswell as those from KASCADE (see below). AtlgN tr

µ>∼ 4.5,

i.e. energies corresponding toE >∼ 1016 eV, a compositioneven heavier than iron would be needed to describe the exper-imental data. The combined results of Fig.4 top and bottomthus suggest that EAS simulated with DPMJET II-5 penetratetoo deeply into the atmosphere. The conclusion is confirmedalso by studying the multiplicity and energy distribution ofhadrons in the calorimeter (Milke et al., 2001).

4 Determination of the primary energy spectra andmass composition

Results presented in the previous section have demonstratedthat QGSJET is still the model of choice when analysingEAS data in the knee region. Deviations between the MonteCarlo predictions and experimentally observed distributionsof high energy hadrons (Eh >∼ 90 GeV) have become rea-sonably small, however, efforts are still needed for furtheroptimization of interaction models. As pointed out before,selecting high energy hadrons amplifies the sensitivity to theactual modelling of high energy hadronic interactions. Lowenergy electrons and muons, on the contrary, should be lessaffected as they result from an average of many interactionsat mostly lower cms-energies. Thus, in the following we

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will concentrate on the reconstructed electron and truncatedmuon numbers.

4.1 Unfolding techniques

It is well known that for given zenith angle, i.e. fixed atmo-spheric thickness,Ne andN tr

µ depend simultaneously on themass,A, and energy,E, of the primary particle. Besides theirspecificaveragerelation(Ne, N tr

µ )↔ (E,A) also theirfluc-tuationsare known to change with energy and mass. Clearly,such effects are to be taken into account when reconstruct-ing the properties of the primary particles, particularly in thepresence of steeply falling energy spectra. Mathematically,the observed electron and muon shower size distributions ata given zenith angle bin can then be written as

dJ

d lgNe,µ=∑A

∫ ∞−∞

dJA(lgE)d lgE

pA(lgNe,µ| lgE) d lgE

where the sum runs over all primary massesA. The quan-tity pA(lgNe,µ| lgE) denotes the probability for a primaryparticle of massA and energylgE to be reconstructed as anair shower with electron- and muon sizelgNe and lgN tr

µ ,respectively. The equation thus represents a set of Fredholmintegral equations of 1st kind. In practice, not all primarymassesA will be taken into account, but only a representa-tive group of different primary particles, here protons, he-lium, carbon, and iron. Suitable methods to solve the in-verse equation, i.e. to infer the physical quantities of interest,dJA/d lgE, from the experimentally observeddJ/d lgNe,µdistributions, are unfolding algorithms. Here, we have cho-sen the iterative scheme of the Gold-algorithm (Gold, 1964)which tries to minimize theχ2 functional and allows onlynon-negative solutions (Ulrich et al., 2001).

The kernel functionpA needs to be determined from EASand detector simulations and has to account for all kindsof physical and experimental effects, such as fluctuations ofshower sizes, trigger and detection efficiencies or reconstruc-tion accuracies. It has been determined from high statisticsCORSIKA/QGSJET simulations generated with thinning op-tion for 3 different bins of zenith angle (0◦-18◦, 18◦-25.8◦,25.9◦-32.3◦), four primary masses and for a large numberof fixed primary energies. The individualNe andN tr

µ dis-tributions obtained for each primary energy and mass werethen parametrized and the variation of the parameters withshower size, zenith angle, primary energy, and primary masswas used in the analysis. Details of this method are describedin Ulrich et al. (2001) and byUlrich (2002).

Results based on the experimentallgNe and lgN trµ

shower distributions of all 3 zenith angle bins are presented inFig. 5 for 4 primary mass groups. Each of the reconstructedenergy distributions shows a knee like structure which isshifted towards higher energies with increasing mass. Theknee in the all-particle spectrum at about 4 PeV appears tome mostly caused by the break of the light elements protonand helium. This is well in agreement to earlier analysesof KASCADE data (Glasstetteret al., 1999; Antoni et al.,2002).

It should be stressed that no assumption about spectralshapes or mass abundances is made in the analysis presentedhere, only the measured electron and muon shower size dis-tributions at 3 zenith angles and the simulated response func-tion pA are used. The error bars in Fig.5 represent statisticalerrors only and are dominated by the limited statistics of thepresently available CORSIKA simulations. Statistical errorsof the experimental data correspond to about 20 months ofdata taking and are very small compared to that of the simu-lations.

In order to investigate the robustness of the reconstructedenergy spectra, the unfolding algorithm has been repeatedmany times employing different sets of input parameters tothe kernel functionpA. These parameters describing theshapes of the simulatedNe andN tr

µ distributions for fixedprimary energies (see above) have been randomly distributedaccording to their individual fit uncertainties assuming Gaus-sian error shapes. The result of this study is shown inFig. 6 for the reconstructed proton and helium energy spec-tra. The shoal of symbols, all obtained using the same ex-perimental data, clearly demonstrates the presence of a kneeat distinct energies in both of the reconstructed spectra. Acloser inspection of the spectra below the knee energy ex-hibits a steeper proton spectrum as compared to helium anda smoother turnover into a power law behaviour above theknee. However, it should be kept in mind that the recon-structed proton spectrum is significantly affected by the largeshower fluctuations limiting the experimental energy resolu-tion. In fact, simulations have shown that even a sharp breakwould result in a smoothness similar to the observed one (Ul-rich, 2002). The observed scattering of the reconstructedfluxes at a given energy has been used to estimate the sta-tistical uncertainties of the unfolding procedure. Adding thisin quadrature to the experimental statistical errors yields thevalues shown in Fig.5. Clearly, improving the statistics ofthe Monte Carlo simulations in the near future will reducethis error.

The still preliminary energy spectra support the picture ofa rigidity dependent scaling of the knee position. However,sinceZ/A ' 0.5 for all shown mass groups withA ≥ 4,the crucial test for distinguishing between aE/A andE/Zscaling of the knee is given by the comparison of the protonand helium spectra. As can be seen from Fig.6, the kneebetweenp and He is shifted by factor of≈ 2.5 favouring therigidity picture. However, because of the different spectralshapes, such a comparison is not straightforward and furtherinvestigations are needed to verify that conclusion. Also, thesensitivity to the used hadronic interaction model needs to bestudied. This may shift the overall energy scale and flux ofthe individual curves but is anticipated to have only minorinfluence to the relative positions of the observed knees.

4.2 Chemical composition and comparison to neural net-work approaches

Individual energy spectra of the type shown above are ide-ally suited for comparisons with cosmic ray source- and ac-

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Primary Energy (eV)10

1510

1610

17

1.5

GeV

-1 s

-1 s

r-2

m

2.5

E⋅)

0fl

ux I

(E 102

103

⋅

sum of allprotonheliumcarboniron

KASCADE preliminary

Fig. 5. Preliminary energy distributionsof four primary mass groups as obtainedfrom the unfolding procedure (Ulrich etal., 2001). The sum of the 4 individualdistributions represents the all-particleCR-spectrum and is shown by the blacksquares. The vertical error bars repre-sent the statistical uncertainties whichare dominated by the statistical uncer-tainties of Monte Carlo simulations (seealso Fig.6).

Primary Energy (eV)10

1510

16

]2 G

eV-1

s-1

sr

-2 [

m3 0

E⋅) 0

diff

. flu

x I(

E 105

106

proton

helium

Fig. 6. Results of repeated unfolding procedures using differentinput parameters from the Monte Carlo simulations. The scatteringof the symbols has been used to estimate the statistical uncertaintyof the Monte Carlo simulation. See also Fig.5 and text for details(Ulrich, 2002).

celeration models. However, up to now most EAS experi-ments have only been able to extract the all-particle energyspectrum and the mean logarithmic mass,〈lnA〉. As a con-sequence, most theoretical papers present their results justin terms of these rather insensitive and inclusive variables.Thus, for completeness we plot in Fig.7 the quantity〈lnA〉as a function of the primary energy. As expected from thestructure of the individual energy spectra, we find an increas-ingly heavier composition at energies above the knee. Thedata are compared to theoretical calculations byErlykin andWolfendale(1997), Berezhko and Ksenofontov(1999), andSwordy(1995). Also, other experimental results from KAS-CADE are included. Common to all models and data is anincreasingly heavier mass at energies above the knee. Only

the single source model ofErlykin and Wolfendale(1997)shows some structure at∼ 1016 eV due to the assumed oc-currence of an iron peak. As can be seen from that figure,the models differ significantly in their prediction of the meanmass and, except for the Swordy calculation, predict heaviermasses than extracted from the experimental data.

In addition to the results from the unfolding procedure, weinclude earlier results from an analysis of thelgNµ/ lgNeratio (Weberet al., 1999) and from a neural network algo-rithm applied to the reconstructed electron and muon num-bers at 3 different zenith angels (Roth et al., 2001). Withintheir errors, we find similar results from the unfolding pro-cedure and from thelgNµ/ lgNe ratios. However, the neu-ral network approach yields a similar trend only at energiesabove the knee. In contrast to the other results, the compo-sition shows a weak decrease in the mean logarithmic massby about 0.5 units in the energy range from about 1 PeV to4 PeV. Similar trends have been observed also by Cherenkovair shower measurements. (For a compilation of several datasets seeSwordyet al. (2002)). At present, it cannot be saidwhether the effect is real or an artefact of the data analysis,such as due to improperly accounted EAS fluctuations, neu-ral network training effects, etc. . For example, it is easy torealize that the occurrence of a break in the individual energyspectra combined with the well known stronger shower fluc-tuations for light primaries could qualitatively account forsuch an effect. Generally, this would produce a bias towardsa lighter composition the steeper the power law distributionof the true proton spectrum is.

The experimental data match well to direct measurementsrecently reported by the JACEE and RUNJOB experimentsup toE ' 1015 eV (Asakimoriet al., 1998; Apanasenkoetal., 2001). While RUNJOB reports an almost constant com-position of〈lnA〉 ' 1.6 in the energy range1013 - 1015 eV,that of JACEE tends to increase from 1.5 to 2.5 in the same

247

SwordyBerezhkoE&W

NN (0-18°, 18-25°, 25-32°)

Primary Energy (eV)10

1510

1610

17

< ln

A >

1.5

2

2.5

3

3.5

4

4.5

He

C

Si

FeKASCADE preliminary

Unfoldingµ/e-Ratioµ/e-Ratioµ/e-Ratio

Fig. 7. Mean logarithmic mass as afunction of the primary energy. The the-oretical calculations are adopted fromErlykin and Wolfendale(1997) (E&W),Berezhko and Ksenofontov(1999), andSwordy (1995). The KASCADE data(symbols) are based on the unfoldingprocedure, an earlier analysis of thelgNµ/ lgNe ratio (Weberet al., 1999),and are from a neural network algo-rithm applied to the reconstructed elec-tron and muon numbers at 3 differentzenith angels (Rothet al., 2001).

energy range. This may point to some methodical problemsin at least one of the two experiments. However, the statisti-cal and systematic uncertainties of both experiments are stillsufficiently large at1015 eV (〈lnA〉 ' 1.5+2.0

−1 for RUNJOBand2.5±0.6 for JACEE) to be compatible to one another andto the reported KASCADE data. Even though only 40 % ofthe available RUNJOB data have been included in that anal-ysis and some more data being available from JACEE, theunsatisfactory situation of lacking statistics in direct experi-ments around1015 eV will hardly change in the near future.

5 Discussion and outlook

We have presented results on tests of the latest versions ofcurrently available hadronic interaction models implementedinto CORSIKA. Only three of the models (QGSJET, DP-MJET II-5, andNEXUS 2) could describe the experimen-tally observed trigger rates at a reasonable level. The mod-els Sibyll 1.6, VENUS, and HDPM overestimate the hadrontrigger rates by up to a factor of 3. The differences in thepredicted trigger rates are strongly correlated to the amountof diffractive dissociation, as can be seen from the Feynman-x distributions of leading baryons in Fig.3. Clearly, thereis a need to improve the knowledge about diffractive inter-actions up to the highest energies by conducting either ded-icated experiments or by increasing the forward acceptanceof currently operating and planned accelerator experiments.

Investigating the properties of QGSJET,NEXUS, andDPMJET in EAS at energies around the knee reveals someproblems also for the latter two models.NEXUS 2 pre-dicts too little hadronic energy at given electron size, i.e. thebalance between electromagnetic and hadronic energy dis-agrees with experimental data. DPMJET II-5, on the otherhand, produces too many hadrons (and electrons) at givenmuon size, i.e. the EAS penetrate too deeply into the atmo-sphere. Thus, applying DPMJET II-5 for composition stud-ies would mimic a too heavy composition. Combining all

results, QGSJET still provides the best overall description ofEAS in the knee region. Nevertheless, a continuation of suchtests up to higher primary energies with optimized experi-mental observables and with tuned cuts to the hadron energy,appears mandatory in absence of adequate accelerator exper-iments.

Analysing the energy spectrum and composition of CRs inthe knee region calls for suitable analysis techniques whichproperly account for mass and energy dependent EAS fluc-tuations and which enable the simultaneous usage of many(correlated) EAS parameters. Here, we have presented re-sults from an ‘inclusive’ unfolding technique applied to theelectron and muon shower size distributions at differentzenith angles as well as results from an ‘exclusive’ neuralnetwork algorithm. The notations ‘inclusive’ and ‘exclusive’refer to the fact that the unfolding algorithm does not aim atproviding information about the primary energy and mass insingle events, but only for the full sample of events. In otherwords, the technique is applied to measuredshower size dis-tributions and it suppliesenergy distributions. This lack ofinformation in single events appears to be well compensatedfor by the sensitivity to details of the primary energy spectra.Employing this approach provides, for the first time, primaryenergy spectra for different mass species. Without makingany ‘a priory’ assumption about the individual energy spec-tra, we find power-law distributions with a knee-like struc-ture in each of the distributions. The observed shift of theknee energies supports the picture of a rigidity dependentcut-off, as expected from most astrophysical models of CRorigin and acceleration.

The neural network approach confirms these overall trendsbut quantitative differences to the unfolding technique areobserved and deserve further studies. Using only the electronand muon shower size as input does not allow to adequatelydistinguish more than two or at most three mass groups ona shower-by-shower basis. However, another advantage be-sides the shower-by-shower information is the simple inclu-

248

10-3

10-2

10-1

-2 -1 0 1 2 3 4

Radial Angle (deg)

arb.

uni

ts preliminaryFedatap

Fig. 8. Distribution of the muon production height, expressed interms of the radial angle for3.75 ≤ lgN tr

µ ≤ 4.0. The data arecompared to CORSIKA/QGSJET simulations of proton and ironprimaries (Buttneret al., 2001).

sion of more EAS observables and the automatic consid-eration of correlations among the parameters. Such multi-parameter analyses have revealed systematic effects in thereconstructed energy and mass depending of the EAS ob-servables used (Antoni et al., 2002), thereby demonstratingthe need for further improvements of the EAS simulations.

Clearly, much more work is still needed to verify and man-ifest the important finding of the observed break in the indi-vidual energy spectra. Despite the enormous progress madein recent years, parallel efforts are still required to (i) im-prove the modelling of EAS and to understand also the tailsin the distributions of EAS observables, (ii) to continue de-veloping advanced and complementary EAS analysis tech-niques, (iii) to provide additional EAS observables by theexperimental data, and (iv) to improve the statistics and re-construction accuracies particularly at the highest energies(∼ 1017 eV) in order to verify the occurrence of an ironknee. All of these aspects are presently addressed by theKASCADE collaboration. For example, besides improve-ments to CORSIKA and its interaction models, a new two-dimensional non-parametric unfolding algorithm has beendeveloped to account also for the shower-by-shower corre-lations of the electron and muon numbers (Roth, 2002; An-toni et al., 2002). The preliminary results obtained with thatmethod agree well with those presented in this article. Also,new observables just become available in KASCADE andwill be included in future analyses. As an example we showin Fig. 8 preliminary results of the muon production heightas extracted from the data of the streamer tube tracking de-tector in the tunnel north of the central detector (Buttneretal., 2001). This observable will become very powerful par-ticularly at energies>∼ 1016 eV, but further optimization ofthe tracking algorithm and cut parameters is still needed be-fore including the observable in the standard set of energyand composition parameters. Finally, in order to improvethe measurements of primary energy and mass up to energiesof about1018 eV, the KASCADE and EAS-TOP experimenthave been combined at the site of Forschungszentrum Karls-ruhe to form a new experiment, called KASCADE-Grande(Bertainaet al., 2001). After finishing the deployment of 37stations of 10 m2 detection area each over an area of∼ 0.5km2 by the end 2001 the new experiment will start to takedata in spring 2002.

Acknowledgements.This work has been supported by Forschungs-zentrum Karlsruhe, by the Ministry for Research of the FederalGovernment of Germany (BMBF), by a grant of the Romanian Na-tional Agency for Science, Research and Technology, by a researchgrant of the Armenian Government, and by the ISTC project A116.The collaborating group of the Cosmic Ray Division of the SoltanInstitute of Nuclear Studies in Lodz is supported by the Polish StateCommittee for Scientific Research. The KASCADE collaborationwork is embedded in the frame of scientific-technical cooperation(WTZ) projects between Germany and Armenia, Poland, and Ro-mania.

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