Chou, A S

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29th International Cosmic Ray Conference Pune (2005) 7, 319322An Universal Description of the Particle Flux Distributions in ExtendedAir ShowersAaron S. Chou , Katsushi Arisaka, Maximo David Ave Pernas, David Barnhill,Pierre Billoir, Arun Tripathiand Tokonatsu Yamamoto(a) Fermilab(b) Department of Physics, UCLA(c) Kavli Institute for Cosmological Physics, University of Chicago(d) L.P.N.H.E. Paris VI-VIIPresenter: A. Chou (achou@fnal.gov), usa-chou-A-abs1-he14-posterIt is shown that the electromagnetic and muonic fluxes in extended air showers (EAS) can be described using asimple model incorporating attenuation and geometrical dispersion. The model uses a reduced set of parame-ters including the primary energy  , the position of shower maximum 	 relative to the ground, and a muonflux normalization  . To a good approximation, this set of three physical parameters is sufficient to predictthe variability of the particle fluxes due to systematic differences between different models of composition andhadronic interactions, and due to statistical event-by-event differences in shower development. Measurementsof these three physical observables are therefore unbiased and very nearly model-independent, in contrast withstandard measurement techniques. The theoretical problem of determining primary composition is thus de-convolved from the measurement procedure, and may be approached in a subsequent analysis of the measureddistributions of (  ,  	 ,   ).1. IntroductionWhen analyzing EAS data collected by surface detector (SD) arrays, Monte Carlo simulations are often neededto extract information about the primary cosmic ray from the ground level particle fluxes. Because of uncer-tainties due to the modelling of primary composition and hadronic interactions, the interpretation of the datais somewhat ambiguous and the systematic uncertainties in the analyses are difficult to estimate. By studyingsimulations performed with AIRES [1] using various combinations of primary composition and hadronic inter-action model, we find that all of the model-dependent effects can be parameterized to a good approximation asfunctions of the position of shower maximum  	 , and a muon flux normalization   . Different simulationsgive systematically different predictions for the mean values and shower-to-shower statistical fluctuations inthese two quantities. One can therefore gain information concerning primary composition and interactions bymeasuring  and  via the spatial and temporal structure of the fluxes observed in SD arrays.For example, at ground level the shower can be approximated as a superposition of an electromagnetic (EM)shower composed of ﬁﬀ ; and of a muonic shower resulting from the decay of charged pions from ahadronic component of the original shower. As a shower evolves, the flux at any fixed core distance growsdue to transverse diffusion of the particles near the core but decreases due to ionization loss. At large traversaldistances, the ionization loss is dominant, and attenuates the EM shower much more than the muonic showerdue to the large number of particles produced in the EM cascade. The ratio of the fluxes is therefore dependenton both the core distance ﬂ of the detector and on the position of the shower relative to the ground, a quantityconveniently parameterized using ﬃ . It is also obviously directly dependent on  .Furthermore, because of radiative scattering, the EM shower front is delayed in time relative to the muonicshower front. As a result, the muonic flux peaks at earlier times than the EM flux. For the same reason,the spread of the arrival times of the particles is also distinct for EM particles and muons, as the latter ones320 A. Chou et al.follow more direct paths from their production sites. This phenomenon can be observed in the time structureof the signals which then provides more information about 	 and  . Asymmetries in both the signallevel and in the time structure also arise in inclined showers. Finally, the time of arrival of the signals can beanalyzed to infer a radius of curvature of the shower front which is also related to  	 . By parameterizingthe particle flux distributions at any observation position and time in terms of (  ,  	 ,  ), one can predictthe dependence of any ground observable as a function of these parameters. Measurements of these groundobservables can therefore be used to constrain these parameters, and to gain information about the cosmic rayprimary composition and hadronic interactions [2].2. The Universal Flux FunctionsFor this study we use AIRES 2.6.0 simulations, including a shower library consisting of 25,000 showers (gen-erated at Fermilab by Sergio Sciutto) with thinning 10 "! and thinning weight 0.15. The ground level is definedas the altitude of the Pierre Auger Observatory in Malargue, with a vertical atmospheric depth of 875g/cm # .The showers are generated using discrete values of log(  /EeV) = $ 18.0,18.5,19.0,19.5,20.0,20.5 % , zenith angle&= $ 0,25,36,45,53,60,66,72,78,84 % and azimuthal angle. The showers are simulated with either proton or ironprimaries, and with either the SIBYLL 2.1 or QGSJET01 interaction model.To investigate the evolution of the particle flux as the shower front traverses the atmosphere, we define ' (as the slant depth measured along the shower axis between the position of  	 and the detection positionon the ground. ')( serves as a convenient age parameter for the shower development. In order to facilitatecomparisons between showers at different zenith angles&, we define the particle fluxes as fluxes through planes(defined locally near each detection position) transverse to the shower axis. To construct the transverse planefluxes from the ground plane fluxes reported by AIRES, each particle record must be weighted by an acceptancecorrection factor *,+-&/.012.31412.51637 to account for the relative size and orientation of the collection areas in thetwo planes. The weighted particle counts are then divided by the accumulation area in the transverse plane toobtain the fluxes.12 ,14 , and16 are unit vectors along the particle direction, the downward shower axis, and thedownward vertical direction, respectively.In figure 1, the EM and muon particle number fluxes at a core distance of R=1.1km are plotted versus ' (for four combinations of primary and model. Each plot is comprised of a set of discrete bands, each onecorresponding to a fixed zenith angle&(larger&correspond to larger ' ( ). The horizontal dispersion of eachband is caused by shower-to-shower fluctuations in  	 . The vertical dispersion of the EM flux is causedby poisson fluctuations of the particle fluxes and, in the case of the muon flux, by the combination of poissonfluctuations and fluctuations in  . The simulations all fall along a common curve in the case of the EM flux.Only small differences can be seen, mainly caused by the contribution of muon decay to the EM particle fluxes.This coincidence is perhaps expected since approximately 90 8 of the shower energy is deposited into the EMcomponent, regardless of the model chosen, and the EM component attenuates predictably with depth.This coincidence is perhaps expected since approximately 90 8 of the shower energy is deposited into the EMcomponent, regardless of the model chosen, and the EM component attenuates predictably with depth. It isremarkable that the ‘universal’ curve is well described by a Gaisser-Hillas (GH) function, that is: the densityat a remote distance follows precisely the evolution of the core. The GH shape peaks at about 150-200g/cm #when the growth of the flux due to transverse diffusion from the shower core is balanced by the loss due toionization. The slight mismatches of the slopes of each constant&band are found to be due to the geometriceffect of screening of the particle flux by the earth. i.e. For more inclined showers, a larger fraction of theparticle flux at ground with12.914ﬃ:<; becomes upward going with12.316ﬃ=<; , and is hence blocked by the earth.The muon particle number flux also seems to have a universal shape (which looks similar to a GH function butAn Universal Description of the Particle Flux ... 321obviously arises from different underlying physics.) The main difference between simulations with differentprimary and interaction model assumptions is the overall normalization of the flux. The relative normalizationis found to be common to the fluxes at all R, and hence only a single normalization factor  is necessary,and can be defined in an arbitrary way. Therefore, a parameterization of the differences in the simulationpredictions in terms of 	 ( '>( , ?@"ABDC ) and  seems to be appropriate. For ease of comparison, > may bemeasured in units of the muon flux predicted by some arbitrarily chosen simulation, and can then be interpretedas a ratio of fluxes. Alternatively, it may be measured relative to the normalization of the universal EM fluxfunction and interpreted as the ‘muon richness’ of the shower.)2 (g/cmGD0 100 200 300 400 500 600 700 800 900)2m⁄EM number flux (#01020304050 Iron, QGSJETProton, QGSJETIron, SIBYLLProton, SIBYLL)2 (g/cmGD0 100 200 300 400 500 600 700 800 900)2m⁄Muon number flux (#0.20.40.60.811.21.41.61.82Figure 1. The EM and muon particle number ux plotted vs EGF for 4 combinations of primary particle and interactionmodel. The data are taken at a core distance H =1.1km from simulations with I =10EeV.)2 (g/cmmax-XcoreX0 200 400 600 800 1000)2m⁄EM number flux (#01020304050 = 0ζ = 90ζ = 180ζ)2 (g/cmGD0 200 400 600 800 1000 1200)2m⁄EM number flux (#01020304050 = 0ζ = 90ζ = 180ζFigure 2. (Left) The EM particle number ux at 3 values of azimuthal angle J , plotted versus the slant depth betweenKMLNOand KQPSRUTWV . The data are from simulations of 10EeV protons with QGSJET. (Right) The same, plotted versus E F .Signals measured at fixed core distance from inclined showers are also predicted to exhibit an azimuthal asym-metry about the core position. This asymmetry is partially due to the geometric acceptance of the detectors,and partially due to shower development because for inclined showers, 'ﬃ( varies with azimuthal angle. Onemay measure in the transverse plane the azimuthal angle X from the ‘younger’ side of the shower. Particlesemitted from a fixed point along the shower axis must travel a longer distance to reach a detector at X =180322 A. Chou et al.than to reach a detector at X =0. In figure 2(left), the EM number fluxes at ﬂ =1.1km for X =0,90,180 are plottedversus the slant depth between 	 and the core position of the shower on the ground. In figure 2(right), thesame quantities are plotted versus ')( of the detection position. The asymmetry in the first plot mostly dis-appears after correcting for the different tranversal distances. The slight residual discrepancies are again dueto the geometric earth-screening effect. Because the fluxes are symmetrically defined here in the transverseplane, an additional contribution to the asymmetry due to the geometric acceptance of non-spherical detectorsis not exhibited. For real-world detectors, large acceptance corrections may be needed because the distributionof particle incidence angles on the detectors are very different for different values of X .The particle number flux distributions at any detector position can be expressed as functions of '( whichis a function of  	 . The normalization of each function is as usual approximately linear in  , and/or  . The fluxes can also be binned in time Y , measured with respect to a planar shower front traveling at thespeed of light, and expressed again as functions of ' ( in each time bin. With such a parameterization, themean number fluxes of EM particles and muons can be predicted at any ground position from simple universalfunctions ZG[]\>^ "_ )`a 	 `a /bdc0?@ ABDC`UY7 where ?@ ABeC is measured relative to the shower geometry. With theapproximation that there is no correlated substructure in the shower, the fluctuations in the particle countsmeasured at each detector position are considered to be independent and Poissonian. These functions F maybe used to model the signals in thin scintillation detectors which mainly measure the particle number flux.However, to correct for the geometric acceptance of the detectors as well as for the earth-screening effect,one must also model the angular distribution of the shower particles about the shower axis, as a function of'>( . This can also be accomplished by parameterizing the attenuation losses as a function of this angle. Togeneralize this method for an arbitrary detector type, one may model the detector response as a function of aparticle’s type, energy and trajectory to form the corresponding signal flux functions f/[\^  .3. ConclusionsWe have shown that the composition/interaction model dependence of the predicted shower signal fluxes atany detection position relative to the shower geometry can be parameterized by universal functions dependingon (  , ) ,  ). These universal functions may be used to predict the dependence on these parametersof various surface detector observables such as signal asymmetries, signal risetime, and shower front curva-ture. By inferring the values of (  , 	 ,  ) from measurement of the various observables, one can makestringent tests of models of primary composition and hadronic interactions. An example of a procedure toperform simultaneous event-by-event measurements of these parameters, utilizing the universal flux functionsis described elsewhere[3].References[1] S. J. Sciutto, User’s Manual and Reference Guide (2002). www.fisica.unlp.edu.ar/auger/aires.[2] P. Billoir, Pierre Auger Observatory Internal Note GAP 2000-025M. Ave et al., 28th ICRC, Tsukuba (2003) 563.P. Billoir, Pierre Auger Observatory Internal Note GAP 2004-010.M. Ave and T. Yamamoto, Pierre Auger Observatory Internal Note GAP 2004-005.M. Ave et al., Pierre Auger Observatory Internal Note GAP 2004-028.K. Arisaka et al., Pierre Auger Observatory Internal Note GAP 2004-037.P. Billoir, Pierre Auger Observatory Internal Note GAP 2005-004.M.T. Dova et al., Pierre Auger Observatory Internal Note GAP 2005-032[3] A. Chou et al., 29th ICRC, Pune (2005), usa-chou-A-abs2-he14-oral.

An Universal Description of the Particle Flux Distributions in Extended Air Showers

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