Astroparticle Physics 60 (2015) 92–104

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An extended universality of electron distributions in cosmic ray showersof high energies and its application

http://dx.doi.org/10.1016/j.astropartphys.2014.04.0030927-6505/� 2014 Elsevier B.V. All rights reserved.

⇑ Corresponding author. Tel.: +48 42 365 5665.E-mail address: maria.giller@kfd2.phys.uni.lodz.pl (M. Giller).

Maria Giller ⇑, Andrzej Smiałkowski, Grzegorz WieczorekThe University of Lodz, Faculty of Physics and Applied Informatics, Pomorska 149/153, 90-236 Lodz, Poland

a r t i c l e i n f o

Article history:Received 3 January 2014Received in revised form 4 April 2014Accepted 11 April 2014Available online 30 April 2014

Keywords:High energy extensive air showersUniversal electron distributionsFluorescence and Cherenkov lightAir shower reconstruction

a b s t r a c t

It is shown that the shape of any electron distribution in a high energy air shower is the same in all suchshowers, if taken at the same age, independently of the primary energy, mass and thus, of the interactionmodel. A universal behaviour has been also found within a single shower, such that the lateral distribu-tions of electrons with fixed energies, at various shower ages, can be described by a single function ofonly one variable. The angular distributions of electrons with a fixed energy can be represented, at a givenlateral distance, by a function of the product h � Ea only, which is explained by a model of small angle elec-tron scattering with simplified energy losses. These results have been obtained by Monte Carlo simula-tions of the extensive air showers. The electron universality can be used as a method for determiningthe longitudinal profile of any single shower from its optical images measured by the fluorescence lighttechnique, which is particularly useful with showers observed with large fraction of the Cherenkov light.

� 2014 Elsevier B.V. All rights reserved.

1. Introduction

The subject of this paper is a study of various distributions ofextensive air shower electrons, of both signs, at different levels ofshower development. We concentrate on the highest energy endof the primary energy spectrum, at E0 J 1016�17 eV. This energyregion is of particular interest for several reasons. It is mainly thefact that the measured energy spectrum displays several struc-tures, as compared to the straight power-law line in the log–logscale, such as the second knee, the ankle and the apparent GZKcut-off [1]. Each of these structures may be, and probably is, animportant imprint of the acceleration and/or propagation mecha-nisms of cosmic rays. In addition, the arrival directions of theprimary particles, i.e. the shower axes, should indicate towardsthe particle sources, however, probably only for E0=Z > 1019 eV,where Z is the primary particle charge.

The only way to study high energy cosmic rays is to observe theextensive air showers produced by them in the atmosphere. Themost precise method for a determination of the primary energyE0 on the event by event basis is to measure the optical lightemitted by the secondary particles of the shower while theypropagate through the atmosphere. They excite the atmosphericnitrogen which in turn emits isotropically the fluorescence light,so that the air showers can be observed from the side at distances

of several to some tens of kilometres. The fluorescence techniqueenables measurements of light images of a shower from whichthe longitudinal shower profile – the number of particles, or theirenergy deposit, as a function of depth in the atmosphere – can bededuced. This is possible because experiments show that the fluo-rescence light emitted by an air shower path element is propor-tional to the energy loss of shower particles along this element[2]. Having determined the longitudinal shower profile one cancalculate the energy of the primary particle by integrating theenergy deposit along the shower track in the atmosphere in away almost independent of its mass and the actual, althoughunknown, high energy interaction characteristics.

The fluorescence technique was used successfully for the firsttime by the Fly’s Eye experiment [3] and by its successors, thevarious set-ups of HiRes (see e.g. [4]). More recently it has beenapplied in the Pierre Auger Observatory [5,6] and the TelescopeArray [7], together with large arrays of particle detectors.

A reconstruction of the primary energy E0 of an air shower is,however, complicated by the fact that about one third of showerelectrons also emit Cherenkov (Ch) light [8]; the number of Chphotons produced by a high energy particle is 5 � 6 times largerthan its fluorescence photon yield. Since the direction of the Chemission is almost the same as that of the electron velocity, andthe electron velocity is almost that of the shower, most Ch photonsaccumulate and propagate together with shower particles. Thosescattered in the atmosphere to the side by the Rayleigh and Mieprocesses constitute a non-negligible fraction of the total light

M. Giller et al. / Astroparticle Physics 60 (2015) 92–104 93

registered by a telescope. Moreover, when an air shower isobserved at an angle to its axis d < 30�, the flux of the so calleddirect Ch photons, i.e. those produced at the observed level,becomes also important [9]. In addition, the photons from all theabove processes are subject to scattering in the atmosphere ontheir way from the shower to the detector, causing blurring ofthe image. An analytical treatment of this effect [10], as well asthat based on simulations [11], have been worked out.

Thus, the question arises how to reconstruct correctly the longi-tudinal shower profile i.e. the depth of the shower maximum Xmax

– correlated with the mass of the primary particle – and its energyE0. The reconstruction method used by the Pierre Auger Collabora-tion [12] is based on assuming as a first step an approximate rela-tion between the Ch component and the energy deposit. Theobtained shower profile is then subject to iterations to allow fora more accurate treatment of the Ch light. This method works verywell for the air showers registered by the hybrid detector i.e. by atleast one telescope of the Fluorescence Detector and one station ofthe Surface Detector, an array of 1660 water-Cherenkov tanks.However, for this approach to work correctly the fraction of theCh contribution in the telescope should not exceed (50–70%) ofthe fluorescence light.

The method for the reconstruction of the longitudinal showerprofiles proposed in this paper is based on the universality of elec-tron distributions in the air showers. By the universality of electrondistributions we mean two things:

1. the distributions of electrons within one shower can bedescribed by less than five parameters, being combinations ofthe five variables: shower age s, electron energy E, lateral dis-tance r and the two component angle ~h to the shower axis;

2. the shape of any electron distribution is the same in any high-energy shower if taken at the same shower age s, independentlyof the primary energy and mass.

The word universality was also used with different meanings,e.g. concerning the muon component [13,14] or a particular detec-tor of shower particles, as in e.g. [15]. Here, however, we are con-cerned with shower electrons only.

In this paper we show that the electron distributions in the highenergy showers are indeed universal in both above meanings. Toprove this we study, by full simulations of proton and iron initiatedshowers with E0 ¼ 1016 and 1017 eV, the electron lateral distribu-tions with fixed energies and their angular distributions as a func-tion of the lateral distance. We find that the lateral distributions ofelectrons with different energies at various shower ages can berepresented with a good accuracy by a single function, i.e. a univer-sal one in the first sense, of a variable proportional tor � E0:53 � e�0:893ðs�1Þ, where r is the electron lateral distance ing cm�2. In addition, the angular distributions of electrons withenergy E at a given distance r are functions of r and ~h � E0:73 only.These new universal characteristics, determined in this paper, con-stitute an extension of the meaning of the electron universality asdescribed in point 1. We show that these distributions are thesame for any simulated shower, so that they are universal in thesecond sense for E0 ¼ 1016 � 1017 eV.

Next, we have extended the energy range of this universality upto E0 ¼ 1020 eV by comparing the lateral distributions with earlierstudies [22–24] based on Monte-Carlo shower simulations withthe thinning procedure.

Universality can be applied to the reconstruction of showerswithout any restriction on the Cherenkov contribution to the totallight. It enables an exact prediction of a shower image on the tele-scope camera at any time interval, once the shower developmentstate, i.e. its age s and the total number of electrons Ne at this timeare known. We show that to correctly reconstruct the longitudinal

profile of an air shower with any geometry from its optical images,in principle all the electron distributions have to be known. Thiswork simplifies their description and shows that our predictionscan be applied to any high-energy shower.

We also give formulae connecting the lateral distance and theangle of an emitting electron in the air shower with the positionof the photons on the camera of an imaging telescope.

2. Universality of the electron distributions in the extensive airshowers

2.1. The universality so far

Hillas [17] suggested that by analogy to the pure electromag-netic cascade an age s of the hadronic shower at a slantatmospheric depth X in g cm�2, defined as

s ¼ 3X=ðX þ 2XmaxÞ ð1Þ

might be useful for a description of shower particles. The advent offast computers made it possible to use Monte-Carlo methods forsimulations of individual hadronic showers and to obtain all the rel-evant particle distributions at different ages. By using the showersimulation tool, CORSIKA [18], it was shown [8] that the shapes ofthe energy spectra of electrons on a given level of shower develop-ment depend on the age s of this level, as it is in the purely electro-magnetic cascades. They do not depend on the energy or mass ofthe primary particle. (Of course, the highest energy end of the spec-trum does depend on the primary energy E0, but the independenceconcerns the bulk of electrons, i.e. those with energies around thecritical energy of the air, �80 MeV, up to, say, �1 GeV).

It is important to note that if any shower characteristic is inde-pendent of the primary particle mass it must be also independentof the interaction model adopted. Indeed, it can be imagined thatan iron shower is actually initiated by a proton with drasticallychanged interaction model: the first interaction cross-section ismuch larger – such as that for an iron nucleus, the dissipation ofthe primary energy is much quicker – as it is in an iron shower,and so on. Thus, if a shower characteristic is unchanged despitesuch drastic changes in the proton interaction model it will be evenless affected by the differences between various models of the pro-ton interaction.

Moreover, what is important for our purposes here, it wasshown in [8] that for big showers, where the number of electronsin a given energy interval is large, the electron energy spectrumfluctuates very little from shower to shower.

Other universal characteristic, determined first in [19], is thatthe angular distribution of shower electrons with a given energy ata particular age s depends of this energy only. It does not dependeven on the age s. This universal feature simplified a calculation ofthe angular distribution of the Cherenkov light emitted by showerelectrons as a function of s and height in the atmosphere, done bythe integration of such a distribution for a given E subject to weight-ing by the energy spectrum and the Ch yield at this level [20].

Yet another universal feature of electrons in the air showers istheir lateral distribution at a given level s. When electrons withall energies are considered, the shape of their lateral distributions,with the lateral distance taken in Molière units, depends on s only[21]. The lateral distributions of electrons with fixed energies werestudied first in [22] showing that also they depend on s only(Fig. 1).

The idea of the shower universality was explored by otherauthors as well. Nerling et al. [23] fitted the electron energy distri-butions as a function of s by a relatively simple function. They stud-ied and parametrised the angular distribution of electrons and that

Fig. 1. Independence of the lateral distributions of electrons with two fixedenergies E of the primary particle [22]. Shower maximum. Each curve is an averageof 20 showers.

94 M. Giller et al. / Astroparticle Physics 60 (2015) 92–104

of the produced Ch light as functions of height in the atmosphereand s.

In a later study Lafebre et al. [24] analysed further the angularand lateral distributions for fixed electron energies, providinganother parametrisation (but see Section 2.2.1). They also analysedfluctuations from shower to shower of the distributions of electronenergy, angle and lateral distance. They found that for the bulk ofelectrons they were very small, confirming their universal nature.Thus, the electron universality has been used so far in the secondsense i.e. as an independence of the primary particle parameters.

As it will be shown below all the above described distributionsare not sufficient for our final purpose which is to predict, as accu-rately as possible, optical shower images.

2.2. Extending the air shower universality

We start with showing that the universality in the first senseholds for one 1017 eV iron shower. Then, we will demonstrate thatit holds also in the second sense.

Thus, our first aim is to describe, in a possibly simple way, thefunction f ð~h; r; E; sÞ, a 5-dimensional distribution of electrons. Wecan represent this function as follows

f ð~h; r; E; sÞ ¼ fEðE; sÞ � frðr; E; sÞ � fhð~h; r; E; sÞ: ð2Þ

The variables on the r.h.s. of the semicolons are just the parametersof the functional dependence on the variable before the semicolon.The normalisations of the above functions are such thatZ E0

0fEðE; sÞdE ¼ 1;

Z 1

0frðr; E; sÞdr ¼ 1;

Z4p

fhð~h; r; E; sÞdX ¼ 1: ð3Þ

Thus, the normalisation of the function f at any level s isZZZf ð~h; r; E; sÞdXdrdE ¼ 1 ð4Þ

and the number of electrons dNe on level s with energies ðE; Eþ dEÞ,at the lateral distance ðr; r þ drÞ and with angles~h within dX equals

dNe ¼ Ne f EðE; sÞ � frðr; E; sÞ � fhð~h; r; E; sÞ dXdrdE; ð5Þ

where Ne is the total number of electron at age s. It is only fh whichis still unknown. However, we start with studying frðr; E; sÞ againsince we have found an interesting description of it.

2.2.1. Lateral distributions of electrons with fixed energiesAs the lateral distribution (LD) of electrons with all energies, as

well as the energy distribution fEðE; sÞ depend on s only it isexpected that the only parameters on which the LD of electronswith a fixed energy E should depend are E and s. Indeed, it is thecase, as it was shown in [22] by using shower simulations withCORSIKA. The distance scale, rE, was chosen to be inversely propor-tional to E.

Now we propose a better scale

r0E ¼ X021 MeV

E

� �a

with a < 1; ð6Þ

where X0 is the radiation length of the air.First we study LD of electrons with fixed E around the shower

maximum. Analysing full simulation results of one iron showerwith E0 ¼ 1017 eV we have found that if the LD of electrons witha fixed E is presented as a function of r=r0E, with a ¼ 0:53, it isalmost independent of E. Fig. 2 illustrates this situation for theshower level s ’ 1 (the actual level studied is s ¼ 0:95, but to stressthat the level is very close to shower maximum we will often quoteit as s ’ 1). Each of the four graphs refers to a different fixed elec-tron energy E from the most populated regions. The histogram rep-resents the true distribution obtained from the shower simulation,and the solid line is the distribution of r=r0E i.e. that of r � E0:53 for allelectrons on this level, so it is the same for all four graphs. Theindex a ¼ 0:53 was chosen to best fit the distributions of r � Ea forvarious E. It can be seen that the curve for all electrons (solid)describes well the histograms for particular values of E. Thus,instead of frðr; E; s ’ 1Þ describing many distributions for manyenergies, we have a single function Frðr=r0E; s ’ 1Þ determined forall electrons on this level and normalised to unity afterwards.

We fit it by an analytical function of the Nishimura–Kamata–Greisen type [16], as it was done in [22]:

frðr; E; s ¼ 1Þdr ¼ Frðr=r0E; s ¼ 1Þd log r=r0E

¼ Crr0E

� �a

1þ krr0E

� ��b

� d logrr0E; ð7Þ

where C ¼ lnð10Þ � ka CðbÞCðaÞ�Cðb�aÞ, and the parameters for s ¼ 0:95 are

k ¼ 10:0; a ¼ 1:694; b ¼ 3:675.Since the fit differs a bit from the actual distributions for differ-

ent E, we have also calculated the best fitting parameters a and b asfunctions of E and approximated them as polynomials oflog EðGeVÞ. The corresponding coefficients are given in Table 1and the dotted curves in Fig. 2 refer to these values.

Next, we move to levels away from the shower maximum. In[22] it was shown that the LD at a level s – 1 is almost the sameas that for s ¼ 1 but shifted in the log r=rE scale by DðsÞ, where

DðsÞ ¼ 0:388ðs� 1Þ ð8Þ

was independent of E. Such a shift is equivalent to the rescaling ofthe r0E by introducing a new scale rE;s that includes different levels s

rE;s ¼ r0E � 100:388ðs�1Þ ¼ X021 MeV

E

� �0:53

� e0:893ðs�1Þ: ð9Þ

Thus, we find that LD of electrons with any energy E at the level scan be described, in good approximation, by a single function FðuÞwhere

u ¼ r=rE;s ð10Þ

and

frðr; E; sÞdr ¼ FðuÞd log u

’ 5:1 � 102 u1:694ð1þ 10uÞ�3:675d log u ð11Þ

with the same values of the parameters as those for Fr , Eq. (7).

Fig. 2. Independence of lateral distributions Frðr=r0E ; s ’ 1Þ of electron energy E, for four values of E. One 1017 eV iron shower.

Table 1Dependence of parameters a and b on x ¼ log E (GeV) asp ¼ a x3 þ bx2 þ c xþ d; k ¼ 10.

Parameter Coeff

a b c d

a 0.117 0.016 �0.5 1.35b 0.090 0.110 �0.5 3.20

Fig. 3. Comparison of the universal distribution FðuÞ with parameters fitted to allelectrons at s ¼ 0:95 with distributions of u ¼ r=rE;s at s – 1. Values of E and schosen from edges of their distributions. FðuÞ determined from one 1017 eV ironshower. Other curves – averages from 20 showers each, initiated by 1019 and1020 eV proton and iron nuclei [22].

M. Giller et al. / Astroparticle Physics 60 (2015) 92–104 95

Fig. 3 shows a comparison of FðuÞ (thick, solid line) with the sim-ulation results for two electron energies, E ’ 22:4 MeV and282 MeV from D log E ¼ 0:1, at two ages s ¼ 0:7 and 1.3, thereforefar from shower maximum, and at both ends of the energy distribu-tion, taken from [22]. It is seen that even at these rather extreme val-ues of E and s the universal curve describes the actual distributionapproximately well. A better description is achieved by allowingfor the energy dependence of the parameters a and b (Table 1). Alsoa better agreement of the curves is achieved for js� 1j 6 0:2 andenergies (40� 60) MeV, where there are most of shower electrons.

It should be noted that in Fig. 3 we compare average results1 forE0 ¼ 1019 and 1020 eV proton and iron showers from [22], with oursingle 1017 eV iron shower. The agreement confirms the universalcharacter of FðuÞ in the second sense.

1 The aim of paper [22] was not to study fluctuations from shower to showerAnyway, since the primary energy used was ultra-high it was necessary to applythinning in shower simulations which causes an increase of the electron distributionsfluctuations. Thus, several showers were simulated for each primaries.

.

Lafebre et al. [24] also studied electron distributions in the airshowers. They parametrised the distribution of r=rM for fixed E asfunctions of the absolute distance t ¼ ðX � XmaxÞ=X0 from theshower maximum. Thus, to compare their results with ours wehave to adopt some value of Xmax. We choose two values:Xmax ¼ 770 g cm�2 – referring to proton showers with E0 ’ 1019 eV,

96 M. Giller et al. / Astroparticle Physics 60 (2015) 92–104

and Xmax ¼ 580 g cm�2 – to iron with 1017 eV. Their parametriseddistributions for the two values of Xmax are shown in Fig. 4 togetherwith the simulations (solid lines, from [22]) and our best fittedcurves (Eq. (7)) rescaled to s – 1. It can be seen that their descrip-tion is slightly worse for s ¼ 1:3 and E ¼ 282 MeV. This may becaused by their using the variable t to describe the universality,i.e. the independence of E0. We claim that the universal distribu-tions are functions of s, thus, of ðX � XmaxÞ=Xmax, not those ofX � Xmax. Certainly, at a level several hundreds g cm�2 away fromthe maximum of a 1020 eV shower the electron distributions arecloser to those at the maximum than the distributions at the samedistance in g cm�2 in a 1016 eV shower.

Finding FðuÞ (Eq. (11)) which describes the lateral distributionsof the bulk of electrons with different energies at various ageswithin one shower we have confirmed the universality in the firstsense, defined in the Introduction. But at the same time we haveagain confirmed the universality of Frðr=r0E; s ’ 1Þ in the secondsense, i.e. its independence of E0 and mass. Indeed, in Fig. 4 Lafebreproton showers have E0 ¼ 1018 eV and ours is an iron one withE0 ¼ 1017 eV rescaled to s – 1 based on simulations for E0 ¼ 1019

and 1020 eV proton and iron showers [22].

2.2.2. Angular distributions of electrons with fixed energies at variouslateral distances

We present results concerning levels close to the showermaximum, i.e. s ’ 1.

Fig. 4. Comparison of the distributions of r=rM obtained by three ways for two

To study fhðh; r; E; sÞ we have used the fully simulated ironshower with E0 ¼ 1017 eV. The number of electrons in the 4-dimen-tional intervals DX

!� Dr � E is much smaller than in the previously

discussed distributions. Thus a full shower simulation was neces-sary to avoid the artificial fluctuations introduced by thinning.The highest primary energy of an air shower to be fully simulatedby us in a reasonable time is 1017 eV.

The angular distribution at some distance r from the showeraxis is not symmetric around it. For its description we have chosenthe two angles: hr , called radial angle, being the projection of theparticle angle~h on the plane containing the shower axis and vector~r, and ht – the tangential angle being the projection of ~h on theplane perpendicular to~r.

A clear demonstration of a dependence of the electron angulardistribution on the lateral distance is shown in Fig. 5, where themean angle hhri is presented as a function of the lateral distancer=rM in the Molière units for electrons with energies E > 22 MeV(hhti ¼ 0). Also shown are the dispersions of both angles, increasingwith the distance as well.

Next, we notice that the distributions of hr for different E andany r scale in such a way that the distribution of q ¼ hr � Ea isalmost independent of E. For s = 0.95 we obtain a = 0.73. This isillustrated in Fig. 6, where the distributions of q ¼ hr � E0:73 arepresented for three electron energies, from the region, where thereare most of them, for some radial distances r=rM and for all r. (How-ever, a seems to decrease slightly with r, but at this stage we adoptit as independent of r.)

electron energies at two ages. Lafebre fits done to averages of 20 showers.

Mlog r/r

-2 -1.5 -1 -0.5 0 0.5 1

Ang

le (d

eg)

0

5

10

15

20

25

1≈E>22 MeV; s⟩rrθσ

tθσ

rθ⟨

rθσ

tθσ

Fig. 5. Dependence of the mean electron angle hhri, its dispersion rhr and thedispersion rht of ht on the radial distance r=rM .

M. Giller et al. / Astroparticle Physics 60 (2015) 92–104 97

To parametrise the distributions of hr � E0:73 for different r=rM asuitable function seems again that of the NKG form

Fq x;r

rM; s ’ 1

� �dx ¼ bl=Bðl; m� lÞ � xl�1ð1þ bxÞ�m dx; ð12Þ

where Bðl; m� lÞ is the Euler beta function. Since hr and q may benegative we add to the variable q a value 3 deg � GeV0:73 and fit thedistributions of the above form with x ¼ qþ 3. The results are illus-trated in Fig. 7, where the actual distributions of x ¼ hrE

0:73 þ 3 arepresented for electrons at four distance intervals, together with thefitted curves. We see that the fits describe well the histograms. (Thefits with a gamma function were worse.) The integrals over x of alldistributions are equal to unity. The dependence of l and b ony ¼ logðr=rMÞ have been parametrised as 3-degree polynomials:ay3 þ by2 þ cyþ d, and the coefficient values are given in Table 2.Value of m has been set constant: m ¼ 171.

Next, it is necessary to determine the distributions of the tan-gential angle ht . At this stage of the analysis we assume that thereis no correlation between hr and ht at a particular distance interval.Indeed, our study shows that the angular distributions, at a fixed rand E, depend roughly on

g ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðhr � hhriÞ2 þ h2

t

q: ð13Þ

If the distribution of g per unit solid angle had been Gaussian,which is only approximately true, then the distributions of hr andht would have been independent.

We first check whether the distributions of s ¼ ht � E0:73 areindependent of E for each interval of logðr=rMÞ. This is found tobe the case and the independence is even better fulfilled than thatfor q ¼ hr � E0:73. In Fig. 8 we present the results for three values ofE, as in Fig. 6, for three distance intervals and for all r.

We fit the distributions of s for each distance interval with asum of two Gaussian functions:

Fs s;r

rM; s ’ 1

� �¼ 1ffiffiffiffiffiffiffi

2pp p

r1� exp

�s2

2r21

� �þ 1� p

r2� exp

�s2

2r22

� �� �:

ð14Þ

Each fit has three free parameters: r1;r2 – the widths of eachGaussian and p – the weight of the first one; the both means are

zero. The parameters have been represented as functions ofy ¼ logðr=rMÞ with 2-degree polynomials by2 þ cyþ d; the valuesof the coefficients are given in Table 2. In Fig. 9 we compare theactual distributions of s with the fitted functions for four distanceintervals. The fits are satisfactory.

Next, we check whether the angular distribution fhð~h; r; E; sÞobtained here for one 1017 eV iron shower is universal in the sec-ond sense, i.e. whether it does not fluctuate from one 1017 eV ironshower to another and whether it is independent of E0 and theprimary mass. Fig. 10 illustrates that it is really the case. It showsthe distributions Fq and Fs at two distances r=rM for four showers:initiated by proton or iron nucleus, each with E0 ¼ 1016 or 1017 eV,at s ¼ 0:95. It is seen that the curves are almost identical even forsingle showers with different primary parameters.

For higher E0 it is difficult to obtain f ð~h; r; E; sÞ from full showersimulations – although the thinning procedure cuts the computingtime, it introduces artificial fluctuations in the number of electronsin the 4-dimensional volumes DhrDhtDrDE, at the ages s, where thedistributions are studied. However, we do not think it is actuallynecessary to get f ð~h; r; E; sÞ for higher E0. Indeed, since we haveshown that frðr; E; sÞ is universal for E0 ¼ 1016 � 1020 eV in a broadrange of r; E and s then also f ð~h; r; E; sÞ must be universal. If it wasnot true then it would be difficult to imagine that frðr; E; sÞ wouldbe universal, since the lateral and angular distributions dependon each other.

Concerning levels s – 1 it is only the functions f ð~h; r; E; s – 1Þthat need to be determined yet. Since the angular distribution fora fixed E, integrated over the lateral distance r, does not depend onthe age s, and our model explaining the universality of ~h � E0:73

holds for any s, we have reasons to postulate that at any s

fhð~h; r; E; sÞ � f 0 ~h � E0:73;r

r0M

� �; ð15Þ

where

r0M ¼ rM e0:893ðs�1Þ: ð16Þ

This means that by rescaling the Molière radius according toEq. (16) the distributions of q and s should stay almost the same.However, this has to be checked by simulations.

Finally, to understand the universality of the distribution of~h � E0:73 let us consider a simplified model of electron propagation.In the small-angle scattering approximation we have that

dhh2r i ¼

12

21 MeVE

� �2

dt; ð17Þ

where dt ¼ dX=X0 is the element of the electron path length in radi-ation units. Assuming that the electron energy loss rate for brems-strahlung and ionisation equals

� dEdt¼ Eþ e; ð18Þ

where e is the critical energy of the air, and that the initial electronenergy Ei � E, we obtain thatffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihh2

r ðEÞiq

’ 21 MeVe

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffieE� ln 1þ e

E

� �r: ð19Þ

This means that the variance hh2r i of an electron with energy E

depends on its final energy only. In the energy region0:2 6 E=e 6 2, where there are most electrons, the r. h. s. can be verywell approximated by �E�0:73 (Fig. 11). As it turns out from simula-

tions it is not onlyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihh2

r ðEÞiq

� E0:73 that is almost independent of E,

but also the distributions of hr � E0:73 (Fig. 6).Unfortunately, it is not so easy to explain the universality of the

distribution of r � E0:53. The electron lateral distance r, unlike itsangle, does depend on the angles of the parent photons and

ρ-3 -2 -1 0 1 2 3 4 5

1)≈,s

M ;r

/rρ( ρF

0

0.2

0.4

0.6

0.8

1

<-1.36M

-1.4<log r/r

E=22 MeV

67 MeV

200 MeV

ρ-3 -2 -1 0 1 2 3 4 5

1)≈,s

M ;r

/rρ( ρF

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7<-0.86

M-0.9<log r/r

ρ-3 -2 -1 0 1 2 3 4 5

1)≈,s

M ;r

/rρ( ρF

0

0.1

0.2

0.3

0.4

0.5

0.6<-0.36

M-0.4<log r/r

ρ-3 -2 -1 0 1 2 3 4 5

1) ≈,s

M ;r

/rρ( ρF

0

0.1

0.2

0.3

0.4

0.5

0.6 all r

Fig. 6. Independence of the distributions Fq of q ¼ hr � E0:73 (hr in degrees, E in GeV) of electron energy E; s ’ 1.

98 M. Giller et al. / Astroparticle Physics 60 (2015) 92–104

electrons, so that the electron history is important. Thus, the lateraldistribution of electrons with fixed E does depend on the showerage.

2.2.3. Approximate formula for the full electron distribution at showermaximum

Summarising, at the considered level (s ’ 1) the number ofelectrons, dNe, with energy ðE; Eþ dEÞ at a lateral distanceðr; r þ drÞ with angles ðhr ; hr þ dhrÞ and ðht ; ht þ dhtÞ equals(Eq. (5)) with our approximations

dNe ’ Ne f E E; s ’ 1ð ÞdE � Frrr0E

; s ’ 1� �

d logrr0E

� Fq hrE0:73 þ 3;

rrM; s ’ 1

� �E0:73dhr

� Fs htE0:73;

rrM; s ’ 1

� �E0:73dht; ð20Þ

where we have assumed that the distributions of hr and ht areindependent.

3. Shower reconstruction

3.1. The general idea of the method

We will show that the above described universality of extensiveair showers enables an accurate prediction of their optical imagesregistered in the consecutive time intervals while the shower

develops in the atmosphere. The accuracy depends, of course, alsoon the knowledge of the atmospheric optical conditions and of thefate of the photons in the detector. The prediction becomes possi-ble because the shapes of all the relevant electron distributions arethe same for any high-energy shower, depending only on theshower age s. A high-energy shower means that the number ofelectrons at a given level s in each 4-dimensional binDE � Dr � Dhr � Dht is big enough as not to fluctuate much fromshower to shower; as we have checked even E0 ’ 1016 eV fulfilsthis condition.

This work has been stimulated by our participation in the PierreAuger Observatory (e.g. [5,6]), so we shall assume that the opticaldetector is an imaging one, i.e. it consists of a mirror and a camerain the focal plane of it enabling one to measure the angular distri-bution of the arriving photons in a given time interval i.e. theinstantaneous image of a shower. The telescope integration timeis relatively short (100 ns in Auger) as compared to the total timewhile individual showers are in the telescope field of view. Weassume that the shower geometry is known i.e. its distance fromthe telescope and the arriving direction. Thus, our data for oneshower would consist of its consecutive optical images i.e. theangular distributions of photons at the telescope within successive100 ns time intervals, as the air shower travels through the atmo-sphere. The angular distribution of the photons arriving at the tele-scope translates, thanks to the concave mirror, to given numbers ofphotons in the particular pixels (PMTs) of the telescope camera.

Finding from air shower simulations the shapes of all relevantelectron distributions on various levels one will be able, by

+3ρ0 1 2 3 4 5 6 7 8

1)≈,s

M+3

;r/r

ρ( ρF

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8 <-1.36M

-1.4<log r/r

NKG fits

histograms - MC

+3ρ0 1 2 3 4 5 6 7 8

1)≈,s

M+3

;r/r

ρ( ρF

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7<-0.96

M-1.0<log r/r

+3ρ0 1 2 3 4 5 6 7 8

1)≈,s

M+3

;r/r

ρ( ρF

0

0.1

0.2

0.3

0.4

0.5

0.6 <-0.76M

-0.8<log r/r

+3ρ0 1 2 3 4 5 6 7 8

1) ≈,s

M+3

;r/r

ρ( ρF

0

0.1

0.2

0.3

0.4

0.5 <-0.36M

-0.4<log r/r

Fig. 7. Fitting the simulated distributions Fqðqþ 3; r=rM ; s ’ 1Þ with Nishimura–Kamata–Greisen functions for four distance intervals.

Table 2Dependence of the parameters l and b (Eq. (12)) and r1;r2 and p (Eq. (14)) aspolynomials ay3 þ by2 þ cyþ d, where y ¼ log r=rM m ¼ 171.

Parameter Coeff

a b c d

l � 26.3 1.4 21.4b �0.085 �0.036 �0.017 0.041r1 – – 0.138 0.905r2 – �0.092 �0.07 0.443p – – 0.168 0.463

M. Giller et al. / Astroparticle Physics 60 (2015) 92–104 99

assuming a particular absolute number of electrons NeðXðsÞÞ, topredict the exact number of photons that should arrive at eachcamera pixel at any time interval. Allowing for the detector effects,connected with corrector ring, reflection efficiency of the mirror,PMT real field of view and its efficiency, and some others, one willbe able to compare the predictions with the measurements and, bychanging NeðXðsÞÞ, find the best fitting function. The method pre-dicts equally well the Cherenkov and the fluorescence images soit is applicable to showers with any contribution of the former.

Our method can be applied to any individual air shower withE0 J 1017 eV emitting enough light to be detected from the side.

3.2. The instantaneous optical image of an air shower

It was shown by Sommers [25] that an instantaneous image of ashower has circular symmetry resulting from the axially symmetric

lateral distribution of electrons. If the angle between the showeraxis and the line of sight is d then photons arriving simultaneouslyat the camera are those produced by particles at their crossing theplane inclined at an angle d=2 to the axis and perpendicular to theshower-detector plane (Fig. 12). As the lateral extension of the elec-trons is of the order of 1 Molière unit ’10 g cm�2 the electrons atthe upper end of the cross-section are almost at the same age s asthose at the lower end what results in the mentioned circularsymmetry.

From Fig. 12 it is clear that we can assume that all electronswhich give contribution to the shower image while the showercore is at the point O are those lying at this level i.e. on the planeperpendicular to the shower axis at point O. The angular extentof the shower image in the light emitted at the observed level willdepend, in general, on the lateral and angular distributions of theemitting electrons, and on the shower distance, of course. Eachsmall surface element on the camera will receive photons emitted,or scattered, within a particular angle dX

!by electrons at a partic-

ular distance element r dr du. To show the exact relations wedefine on the telescope camera two angles: a as the angular dis-tance between the centre of the shower image and the point P00,where we want to predict the number of photons, and u the azi-muthal position of point P00. It can be derived that photons arrivingat point P00 must be produced at a lateral distance r ¼ tan a � d,where d is the distance from the telescope to the shower, and azi-muth u at angles hr and ht determined from the formulae

tan hr ¼ tan h � cos ðuþ bÞ ð21Þtan ht ¼ tan h � sin ðuþ bÞ;

τ-3 -2 -1 0 1 2 3 4 5

1)≈,s

M ;r

/rτ ( τF

0

0.2

0.4

0.6

0.8

1<-1.36

M-1.4<log r/r

E=22 MeV

67 MeV

200 MeV

τ-3 -2 -1 0 1 2 3 4 5

1)≈,s

M ;r

/rτ ( τF

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8<-0.86

M-0.9<log r/r

τ-3 -2 -1 0 1 2 3 4 5

1) ≈,s

M ;r

/rτ ( τF

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7<-0.36

M-0.4<log r/r

τ-3 -2 -1 0 1 2 3 4 5

1)≈,s

M ;r

/rτ ( τF

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

all r

Fig. 8. Independence of the distributions Fs of s ¼ ht � E0:73 of electron energy E; s ’ 1.

100 M. Giller et al. / Astroparticle Physics 60 (2015) 92–104

where

cos h ¼ cos d � cos a� sin d � sin a � cos u; ð22Þcos b ¼ ðcos a� cos d � cos hÞ=ðsin d � sin hÞ:

Thus, we see that it is not only the lateral and the angular distribu-tions of electrons that have to be known to predict where exactly onthe camera the produced photons fall. We would also need to knowthe angular distributions of electrons at various distances (seeabout the Cherenkov image below).

3.2.1. The fluorescence imageThe fluorescence light is emitted according to the lateral distri-

bution of the energy deposit, isotropically from each point of theair shower. Thus, the angular distribution of this light is deter-mined by the lateral distribution of the energy deposit, projectedon the plane containing point P0 in Fig. 12, perpendicular to the lineof sight OO00, as seen from the telescope.

The lateral distribution of the energy deposited, as function ofr=rM , depends only on s. It was parametrized in [26] and can beused for a prediction of the fluorescence image of an air showeronce the shower age s and the total number of electrons NeðXÞ(see Section 3.2.4) at the considered level are known.

3.2.2. The image in the scattered Cherenkov lightThe Cherenkov (Ch) photons travelling together with the

shower particles are being scattered to the side in the atmosphere.Thus, the instantaneous shower image in this light depends on the

lateral distribution of the Ch photons at the observed shower leveland on their angular distribution after the scattering.

This problem was studied in [20], where it was shown thatbeyond the shower maximum the angular extent of the total opti-cal image is dominated by Ch photons scattered at the observedlevel rather than by the fluorescence light. It was also shown thatthe lateral distribution of the Ch light scattered at the shower levels depends mainly on the angular distributions of the emitting elec-trons above the scattering level. The electron lateral distributionsthere do not play an important role and were taken into accountin an approximate way. It was assumed that the angular and lateraldistributions of electron are independent of each other, which, aswe have seen, is not true.

Strictly speaking, to accurately predict the lateral distribution ofthe Ch light arriving at some observed level s it is necessary toknow all the electron distributions discussed above, at the levelss0 higher that the observed one. Indeed, the number of Ch photons,dNCh produced at the level s0 per unit path at a distance ðr; r þ drÞ atan angle ~h within dX equals

dNChð~h; r; s0Þ ¼ Ne dr dX �Z E0

EthðhÞYCh ðEÞfEðE; s0Þ f rðr; E; s0Þ f hð~h; r; E; s0ÞdE;

ð23Þ

where YChðEÞ is the number of Ch photons emitted by one electronwith energy E per unit path. Here we have assumed that the Ch lightis produced in the direction of the electron. Approximate values ofdNCh, assuming hr independent of ht , are obtained with the help ofEq. (20).

τ-3 -2 -1 0 1 2 3 4 5

1)≈,s

M ;r

/rτ( τF

0

0.2

0.4

0.6

0.8

1<-1.36

M-1.4<log r/r

two gauss fits

histograms - MC

τ-3 -2 -1 0 1 2 3 4 5

1)≈,s

M;r

/rτ( τF

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

<-0.96M

-1.0<log r/r

τ-3 -2 -1 0 1 2 3 4 5

1)≈,s

M;r

/rτ( τF

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

<-0.76M

-0.8<log r/r

τ-3 -2 -1 0 1 2 3 4 5

1) ≈,s

M;r

/rτ( τF

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7<-0.36

M-0.4<log r/r

Fig. 9. Fitting the simulated distributions Fsðs; r=rM ; s ’ 1Þ with a sum of two Gaussians for four distance intervals.

M. Giller et al. / Astroparticle Physics 60 (2015) 92–104 101

Knowing the angular and lateral distributions of the producedCh photons at any level s0 < s and allowing for the atmosphericattenuation is sufficient to obtain the angular and lateral distribu-tions of the Ch photons arriving at the observation level s. Finally,using formula (21) and (22) and knowing the angular distributiondue to the scattering, one obtains, where at the camera the scat-tered Ch photons should arrive.

3.2.3. The image in the direct Cherenkov lightThe direct Ch light is that produced at the observed shower ele-

ment DX towards the direction of the telescope. For most air show-ers detected by the Pierre Auger Observatory the contribution ofthis light to the optical image is very small. The elevation anglesof the telescopes are not large: 0 < a < 30�, which, together withthe limit on the zenith angle hz < 60� of the well reconstructedshowers, results in rather large typical viewing angles d. Sincethe electron directions are concentrated around the shower axisand the Ch light is emitted at very small angles the direct light issmall indeed.

However, the situation is different for the telescopes of theHEAT extension to the Auger Observatory [27], where the elevationangles reach 60�. We have derived the distribution of the viewingangles d for a given elevation angle a assuming the isotropic zenithangle distribution of the shower directions, FðhzÞ, cut above 60�.We have

FðhzÞ ¼2

1� cos2 h0sin hz cos hz; ð24Þ

where h0 ¼ 60�. The resulting distribution f ðxÞ, where x ¼ cos d isthe following

(a) for a 6 30�

f ðxÞ ¼ 83p

sin a � x p2þ arcsin

sin a � x� 12

cos affiffiffiffiffiffiffiffiffiffiffiffiffiffi1� x2p

� �� �

þ cos2 a� 14þ sin a � x� x2

r ; ð25Þ

(b) for a P 30�

f ðxÞ ¼ 83

sin a � x for 0 6 d 6 a� 30�; ð26Þ

f ðxÞ ¼ as in case ðaÞ for a� 30� 6 d 6 150� � a:

Fig. 13 shows the distribution f ðxÞ for a ¼ 15� – the elevation of thecentre of an Auger telescope, a ¼ 45� – that of the centre of a HEATtelescope and a ¼ 60� – the HEAT maximum elevation. A dramaticdifference between the d distributions for Auger and the HEAT tele-scopes is seen for small angles d. For d < 30� the corresponding frac-tions of showers are �3.8%, 21% and 29%. (This is only a roughillustration of the differences since many selection criteria in dataprocessing will surely influence these numbers). Nevertheless,Fig. 13 shows that the effect of the direct Ch light plays a consider-ably bigger role in the air showers registered by HEAT and needs tobe treated in an accurate way.

For distant air showers, when all Ch light falls into one pixel, theCh signal is / dNchðdÞ=dX, which is almost equal to the angular

ρ-3 -2 -1 0 1 2 3 4 5

1)≈,s

M ;r

/rρ( ρF

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

<-0.96M

-1.0<log r/r

ρ-3 -2 -1 0 1 2 3 4 5

1)≈,s

M ;r

/rρ( ρF

0

0.1

0.2

0.3

0.4

0.5<-0.36

M-0.4<log r/r

τ-3 -2 -1 0 1 2 3 4 5

1) ≈,s

M ;r

/rτ ( τF

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

<-0.96M

-1.0<log r/r

τ-3 -2 -1 0 1 2 3 4 5

1)≈,s

M ;r

/rτ ( τF

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

<-0.36M

-0.4<log r/r

Fig. 10. Independence of Fqðq; r=rM ; s ’ 1Þ – two upper graphs for two distance intervals and Fsðs; r=rm; s ’ 1Þ – two lower graphs – of primary energy and mass. On eachgraph there are 4 curves, each referring to proton or iron shower with E0 ¼ 1016 or 1017 eV.

Fig. 11. Mean square radial angleffiffiffiffiffiffiffiffiffihh2

r iq

as function of electron energy E in units ofthe critical energy e.

Fig. 12. Schematic presentation of emission of light arriving simultaneously atcamera.

102 M. Giller et al. / Astroparticle Physics 60 (2015) 92–104

distribution of the electrons emitting Ch light integrated over thelateral distance r. However, if the shower is close enough so that

its lateral extent can be measured by the telescope camera, thenumber of Ch photons registered by an individual pixel has to becalculated with the use of Eqs. (20)–(22).

To illustrate the importance of this detailed approach we havecalculated the value of the direct Ch signal produced at s � 1 andobserved from two distances such that the total image is containedwithin angle f. Fig. 14 presents the ratio of the more accurately cal-culated Ch signal, i.e. using Eqs. (20)–(22) to the approximate one

-1 -0.5 0 0.5 1x = cos

0

0.5

1

1.5

2

2.5

f(x)

=15 , Auger 45 , HEAT 60 , HEAT maximum

δ

α Ο

Ο

Ο α

Fig. 13. Distribution f ðcos dÞ of shower viewing angle d, for three values of theelevation angle a. Big difference is seen between an Auger telescope and that ofHEAT.

5 10 15 20 25 30 350.6

0.7

0.8

0.9

1

1.1

1.2

[deg]δ

R

°=3.6ζ

°=7.2ζ

Fig. 14. Ratio R of the accurately calculated direct Cherenkov signal to theapproximate one, dNChðdÞ=dX, for two angular radii f of the Ch image as functionof viewing angle d.

M. Giller et al. / Astroparticle Physics 60 (2015) 92–104 103

dNchðdÞ=dX. It is seen that even for f ¼ 3:6� the difference may be asbig as �20%.

3.2.4. The total number of electronsSo far we have discussed only the shapes of the electron distri-

butions. However, to predict the actual numbers of the photonsarriving at the detector from a depth X one needs to know the totalnumber of electrons, NeðXÞ, at this depth. It is well known thatNeðXÞ can be described by a 4-parameter, Nmax; Xmax; X1 and K,function of X, called the Gaisser–Hillas function [28]. Thus, our finalprocedure of a shower reconstruction process is to find such valuesof the four parameters describing NeðXÞ with which the predictednumbers of photons in all hit pixels, at all time intervals, fit bestthe measured ones. The age s of a level X is determined solely bythe ratio X=Xmax (Eq. (1)).

4. Conclusions

In this paper we have extended the meaning of the universalityof the electron distributions by showing that in the primary energyregion E0 ¼ 1016 � 1017 eV the lateral distributions of electronswith any energy E (20 MeV < E < 1 GeV) on any level s of theshower development ð0:7 < s < 1:3Þ can be described by just oneuniversal function Fðr=rE; sÞ (Eqs. (9)–(11)). We have called it theelectron universality in the first sense.

We have also shown, by comparing our results with those ofother authors, that the same function describes well the corre-sponding distributions in a broad range of the primary energy:E0 ¼ 1016 � 1020 eV, independently of the primary mass and thus,of the interaction model. This confirms the universality of the lat-eral distribution in the second sense.

Concerning the angular distributions f ð~h; r; E; sÞ we have foundanother universal (in the first sense) behaviour: at s ’ 1 theydepend on h � E0:73 only rather than on h and E separately. We haveexplained this by considering a model of the small angle scatteringwith simplified energy losses. We have checked that this is true forany shower with E0 ¼ 1016 � 1017 eV and we give arguments thatthis universality should extend up to 1020 eV.

Thus, we can state that the function f ð~h; r; E; sÞ describing fullythe bulk of electrons in a high-energy shower is the same in anysuch shower independently of the primary energy and mass. Theprimary energy only has to be high enough, i.e. E0 P 1016 eV, sothat the numbers of electrons in the 4-dimensional volumesDhrDhtDrDE, at the ages s, where f ð~h; r; E; sÞ is studied, be large.

It should be, however, noted that the description of the electrondistributions presented in this paper is not complete yet. Thereremains a better approximation of the angular distributions at par-ticular distances r, since hr and ht are not entirely independent aswe have assumed. Moreover, our hypothesis in Eq. (15) has to bechecked for various shower ages s. This, however, does not under-mine the universal character of the function f ð~h; r; E; sÞ in the sec-ond sense. It only means that the minimum number of theindependent variables describing all electron distributions in ashower has yet to be worked out.

The electron universality in the air showers can be used forreconstruction of longitudinal shower profiles from their opticalimages. The images can be predicted for any single shower, oncethe NeðXÞ is adopted because the shapes of the electron distribu-tions do not fluctuate. A detailed knowledge of the functionf ð~h; r; E; sÞ, simplified here, is necessary for a correct prediction ofthe shower image in the Cherenkov light, which is of particularimportance for air showers observed at viewing angles d < 30�,as those registered by HEAT at Auger. For the fluorescence imageit is only frðr; E; sÞ that matters due to the isotropic emission of thislight.

Our choice to determine electron distributions for fixed ener-gies E enables an easy allowing for the fact that air showersdevelop at various depths, thus at various heights h. Then, for thesame shower age s the energy threshold for the Cherenkov emis-sion, EthðhÞ, changes from shower to shower. To find the emittednumber of Cherenkov photons from a given distance r, at someangle ~h one needs only to integrate the contributions from elec-trons with E > EthðhÞ.

Another way of treating the Ch light would be to parametrise itsemission as a function Fðh; r;h; sÞ. Nerling et al. [23] chose this wayby parametrising the distributions as Fðh; h; sÞ, thus, integratedover the lateral distance r and electron energy E. However, wethink that the distributions f ð~h; r; E; sÞ may be also of a more gen-eral interest than that connected with the Cherenkov light. It isby analysing them that we have found universal features of theelectron distributions in a single air shower.

Acknowledgements

The authors thank the Pierre Auger Collaboration for inspiringus to this study, teaching us the optical method and for many help-ful discussions. We also thank Andrzej Kacperczyk for taking partin the determination of the lateral distributions for electrons withfixed energies. This work has been supported by the Grant No. NN202 200239 of the Polish National Science Centre.

104 M. Giller et al. / Astroparticle Physics 60 (2015) 92–104

References

[1] R.U. Abbasi et al., Phys. Rev. Lett. 92 (2004) 151101;Pierre Auger Collaboration, Phys. Lett. B 685 (2010) 239–246;W.D. Apel et al., Phys. Rev. Lett. 107 (2011) 171104;R. Abbasi et al., Astropart. Phys. 44 (2013) 40–58;J.J. Beatty, S. Westerhoff, Annu. Rev. Nucl. Part. Sci. 59 (2009) 319 (see e.g.).

[2] F. Kakimoto et al., Nucl. Instrum. Methods A 372 (1996) 527;P. Colin et al., Astropart. Phys. 27 (2007) 317.

[3] R.M. Baltrusaitis et al., Nucl. Instrum. Methods A 240 (1985) 410.[4] T. Abu-Zayyad et al., Ap. J. 557 (2001) 686;

R.U. Abbasi et al., Astropart. Phys. 23 (2005) 157.[5] The Pierre Auger Collaboration, Nucl. Instrum. Methods A 523 (2004) 50.[6] Pierre Auger Collaboration, Nucl. Instrum. Methods A 620 (2010) 227.[7] H. Kawai et al., Nucl. Phys. B (Proc. Suppl.) 175 (2008) 221.[8] M. Giller, G. Wieczorek, A. Kacperczyk, H. Stojek, W. Tkaczyk, J. Phys. G: Nucl.

Part. Phys. 30 (2004) 97.[9] M. Nagano, A.A. Watson, Rev. Mod. Phys. 72 (2000) 689.

[10] M. Giller, A. Smiałkowski, Astropart. Phys. 36 (2012) 166.[11] J. Pe�kala et al., Nucl. Instrum. Methods A 605 (2009) 388.[12] M. Unger, B.R. Dawson, R. Engel, F. Schuessler, R. Ulrich, Nucl. Instrum.

Methods A 588 (2008) 433.

[13] A.S. Chou, M.D. Ave Pernas, T. Yamamoto, in: 29th Int. Cosmic Ray Conf., 2005,p. 101.

[14] A. Yushkov et al., Phys. Rev. D 81 (2010) 123004.[15] F. Schmidt, M. Ave, L. Cazon, A. Chou, Astropart. Phys. 29 (2008) 355.[16] J. Nishimura, K. Kamata, Progr. Theor. Phys. 6 (1958) 93.[17] A.M. Hillas, J. Phys. G: Nucl. Part. Phys. 9 (1983) 1433.[18] D. Heck, J. Knapp et al., Technical Report 6019, Forchungszentrum, Karlsruhe,

1998.[19] M. Giller, A. Kacperczyk, J. Malinowski, W. Tkaczyk, .G. Wieczorek, J. Phys. G:

Nucl. Part. Phys. 31 (2005) 947.[20] M. Giller, G. Wieczorek, Astropart. Phys. 31 (2009) 212.[21] M. Giller, H. Stojek, G. Wieczorek, Int. J. Mod. Phys. A 20 (2005) 6821.[22] M. Giller, A. Kacperczyk, W. Tkaczyk, in: Proc. 30th Int. Cosmic Ray Conf.

Merida, 2007.[23] F. Nerling, J. Bluemer, R. Engel, M. Risse, Astropart. Phys. 24 (2006) 421.[24] S. Lafebre, R. Engel, et al., Astropart. Phys. 31 (2009) 243.[25] P. Sommers, Astropart. Phys. 3 (1995) 349.[26] D. Góra, R. Engel, D. Heck, et al., Astropart. Phys. 24 (2006) 484.[27] C. Meurer, N. Scharf, Pierre Auger Collaboration, Astrophys. Space Sci. Trans. 7

(2011) 183.[28] T.K. Gaisser, A.M. Hillas, in: Proc. 15th Int. Cosmic Ray Conf. Munich, 1975, p.

353.