Similarity of extensive air showers with respect to the shower age
This article has been downloaded from IOPscience. Please scroll down to see the full text article.
2005 J. Phys. G: Nucl. Part. Phys. 31 947 (http://iopscience.iop.org/0954-3899/31/8/023)
Download details:
IP Address: 212.191.70.8
The article was downloaded on 21/11/2012 at 12:37
Please note that terms and conditions apply.
View the table of contents for this issue, or go to the journal homepage for more Home Search Collections Journals About Contact us My IOPscience
J. Phys. G: Nucl. Part. Phys. 31 (2005) 947–958 doi:10.1088/0954-3899/31/8/023
Similarity of extensive air showers with respect to the shower age
M Giller1, A Kacperczyk2, J Malinowski1, W Tkaczyk1and G Wieczorek1
1Department of Experimental Physics, University of Lodz, Poland
2Department of Theoretical Physics, University of Lodz, Poland E-mail:mgiller@kfd2.fic.uni.lodz.pl
Received 18 April 2005 Published 24 June 2005
Online atstacks.iop.org/JPhysG/31/947 Abstract
Using CORSIKA simulations of the highest energy extensive air showers we show that all showers are similar when described by the shower age parameter:
the angular and energy spectra of electrons at a given level in the atmosphere depend only on the shower age at this level. Moreover, electrons with a given energy have the same angular distributions at any level (age) of the shower. We have calculated these distributions and found analytical functions describing them quite well. The total number of particles can also be described in a simple way as a function of age by two halves of a Gaussian function with the widths, however, fluctuating from one shower to another. The description of large showers in terms of age (instead of depth in the atmosphere) is very useful in interpreting data from experiments observing fluorescence light, with admixture of Cherenkov, induced by the showers in the air.
1. Introduction
Cosmic rays (CR) of the highest energies can be studied only by observations of the extensive air showers (EAS) induced by them in the atmosphere. Of particular interest is the ultra- high energy region (E > 1019 eV) where more information can be deduced from the energy spectrum and/or CR arrival directions than it can be done at lower energies: an observation of a cutoff in the energy spectrum at about 5× 1019eV would practically settle the problem of CR origin at these energies—they should be extragalactic; moreover, if CR are protons or light nuclei they should eventually point out towards their sources. So far, however, neither of the two questions has been solved: the energy spectra measured by the two recent experiments, AGASA [1] and HiRes [2], are not conclusive; AGASA observation of some clustering of the shower arrival directions [3] has not been confirmed by HiRes. More data are needed and these will be supplied by the Pierre Auger Observatory [4] in construction, which by 2006 should have an area of 3000 km2covered by detectors, i.e. about 30 times bigger than that of AGASA.
0954-3899/05/080947+12$30.00 © 2005 IOP Publishing Ltd Printed in the UK 947
This work has been motivated by an attempt to understand the EAS and to be able, despite their fluctuating development in the atmosphere, to reconstruct properly the primary energy on the shower by shower basis. Observations of the fluorescence light excited in the atmosphere by the charged particles of a shower, allow us to reconstruct its cascade curve N (X), i.e. the particle number N as a function of the slant depth X in the atmosphere. However, the transition from the observed light to the number of particles is not quite straightforward: the fluorescence light excited by an electron depends (although weakly) on its energy, also the light detected by a telescope contains a non-negligible fraction of Cherenkov photons. To take these effects into account one has to know the energy- and angular distribution of the shower electrons at any (observed) level in the atmosphere. In our earlier paper [5] we have shown that the shape of the electron energy spectrum at a given level in the atmosphere depends only on the shower age parameter at this level. This allows us to predict exactly the number of fluorescence and Cherenkov photons per one electron produced in a unit path length, once the shower age of the observed level is known. The fluorescence light depends on the shower age only, whereas the Cherenkov light—also on the height in the atmosphere (as the threshold for this effect does depend on it). These yields do not fluctuate from shower to shower in such large showers, so that they can be calculated once forever and applied to any observed shower to deduce N (X).
In this paper we study further the shower properties as functions of the shower age s, such as the total number of charged particles N (s), and the angular distribution of shower electrons, having in mind applications of our findings to the shower reconstruction in experiments such as the Pierre Auger Observatory and HiRes. The aim of this paper is just to show that due to the similarity of showers, this task is, in principle, easy to do.
2. Energy spectra of electrons
The energy and angular distributions of charged particles (electrons and positrons, to be called simply electrons in what follows) in pure electromagnetic cascades were studied in some detail by Hillas [7], in the context of understanding the Cherenkov radiation emitted in air by primary gamma rays with energies (0.1–1) TeV. The advent of fast computers made it then possible to use Monte Carlo method to simulate development of individual cascades with exact cross sections for all relevant processes, what, of course, is not possible by analytical approaches.
From the latter it was known that the energy spectra of electrons in an electromagnetic cascade, as well as their angular distributions, should depend only on the age parameter sg, defined as: sg = t+2 ln(E3t0/β), where t is the slant depth in the atmosphere in cascade units, E0 the energy of the primary photon and β the critical energy of the air. The exact shapes had to be obtained, however, by simulations. It was not clear [8] how these distributions would behave in hadronic extensive air showers, being a superposition of many such cascades initiated on various depths by photons with various energies. It has turned out that the age parameter s, defined for depth X of a hadronic shower in an analogous way as sg[8], namely
s= 3X/(X + 2Xmax) (1)
where Xmaxis the depth of the shower maximum, describes very well the shower as a whole.
It has been shown by us [5] and independently by Nerling et al [6] that for large showers the shape of the energy spectrum of electrons at a given level X in the atmosphere depends only on the shower development stage at this level, i.e. on its age s. The shape of the spectrum at a given age s is the same independently of the nature of the primary particle (proton or iron) and its primary energy. For big showers, where the number of particles in a given energy bin is large, the spectrum practically does not fluctuate from shower to shower. What does fluctuate (for a fixed primary energy) is the total number of particles N (s) (see section 4) but not the
shape of the energy spectrum. Thus, in the sense of the energy spectra of electrons, all large showers are similar.
We have shown [5] that this fact turns out to be very useful in the air shower experiments based on observations of the fluorescence light excited in the atmosphere by shower charged particles: as it is believed that the fluorescence yield of a particle is proportional to its energy loss for ionization (what in turn depends on the particle energy), the total fluorescence flux (as observed by a pixel of the telescope camera) would depend on the energy spectrum of particles at the observed level s. These spectra have been fitted by an analytical function (different in [5,6]) with some s-dependent parameters, so that they can be easily used.
3. Angular distribution of electrons
As the electron angles with respect to the shower axis depend mainly on the Coulomb scattering process, where the scattering angles are inversely proportional to the particle energy, one would expect that also the angular distribution of electrons at a given level depends only on the shower age at this level. This is actually the case—we have already shown in [9] that the two-dimensional energy and angle distribution of electrons, F (E, θ; s), (normalized to 1) at a given level depends on the shower age s at this level only. Again, it does not depend on the primary particle mass or on its energy.
What we show here is something more: the whole dependence of the distribution F (E, θ; s) on the shower age s is contained in the energy distribution of electrons f (E; s), so that
F (E, θ; s) = f (E; s) g(θ; E) (2)
where g(θ; E) is the angular distribution of electrons of a given energy, defined as follows:
π 0
g(θ; E) 2π sin θ dθ = 1. (3)
This means that electrons of a given energy E have the angular distribution independent of the shower age. Actually, it is not so surprising. In a large shower any electron with a given energy E arises from a particle (electron or photon) with a much larger energy. The angles of the parents are much smaller than those of their daughters. As, on average, the angles add in quadrature, the angular distribution of the parent particles is irrelevant to that of the daughter with a much smaller energy E, the latter depending on the latest Coulomb scatterings, bremsstrahlung and/or pair production.
Figure1shows the angular distribution of electrons, g(θ; E), obtained with the CORSIKA [10] simulations, for three various energy bins ( log E = 0.1), corresponding to energies around 20 MeV, 200 MeV and 2 GeV. For each energy three curves are drawn, for s= 0.8, 1.0 and 1.2. In the regions where there are many particles the curves for different s are not to be distinguished from each other.
The knowledge about the angular distributions of particles in the shower is necessary in experiments detecting the showers by the light they produce when passing through the atmosphere (as it has been discussed in the introduction), particularly when it is necessary to predict the contribution of the direct Cherenkov light to the fluorescence flux (see section 5).
To this end we have fitted the angular distributions g(θ; E) by some analytical functions of x= θrmsθ(E), with four parameters, as follows,
g(θ; E) =
a1e−(c1x+c2x2) for x <2.7
a2x−α for x >2.7. (4)
10 20 30 40 50 60 70 80 [deg]
1e-03 1e-02 1e-01 1e+00 1e+01 1e+02
0.8 1.0 1.2
θ
s=
20 MeV 200 MeV
2 GeV g( ;E)θ
Figure 1. The angular distribution of shower electrons g(θ; E) for three values of electron energy (log E= 0.1). For each E there are three distributions for s = 0.8, 1.0 and 1.2.
-1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2
log(E/GeV)
0 2 4 6 8 10 12 14 16 18
θ(E) [deg]rms
Figure 2. Root mean square of the angle between the shower axis and the velocity of electron with energy E. Points—simulations; line—polynomial fit (see the appendix).
The root mean square angle, θrms, as a function of energy is shown in figure2. The flattening of the θrmsfor E > 1 GeV is an artefact, as our smallest angular bin is θ 1◦. The four independent parameters a1, c1, c2and α (a2 is determined by the matching condition of the two functions at x= 2.7) have been found for the consequent energy bins ( log E = 0.1, starting from log(E/GeV)= −1.65 (E = 22 MeV), up to −0.05 (891 MeV)) as fitting best the above analytical form to the simulations. Their values as function of energy are presented in figure3as the points. The scatter around a smooth behaviour is caused by small calculation inaccuracies due to the numbers of particles (in bins at large angles) being not large enough.
For practical reasons we have found the polynomials of log E fitting best the points (see the
-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0
log(E/GeV)
1 1.5 2 2.5
-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0
log(E/GeV)
2 2.5 3 3.5 4 4.5
-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0
log(E/GeV)
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2
-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0
log(E/GeV)
3 3.5 4 4.5 5 5.5 6
a
1c
12
α c
Figure 3. Parameters of the angular distribution of electrons as a function of their energy.
appendix). Figure4represents the angular distribution of electrons, g(θ; E), averaged over
log E = 0.1, for several energy bins. The histograms correspond to simulations, the lines are the analytical curves with the parameters described by the polynomials (so that for a given histogram, the lines do not correspond to the best parameters for this histogram). Our aim is to describe the electron angular distributions well not only for those angles where there are most particles (per unit solid angle), i.e. for small angles, but also for larger angles (say x >2), so in our best fitting procedure we assumed ‘the error’ in the fraction of particles in each angular bin (1◦)as 10%. Thus, the value of ‘χ2’ given at each graph in figure4gives an idea how good the fit is with the parameters taken from the polynomials. We can see that the fit is quite good, hoping that it is good enough for most applications (otherwise one can use the histograms themselves). The fit could be even better if we relaxed the fixed value x= 2.7 in (4), the same for any energy bin. However, for simplicity reasons, in order to avoid too many parameters, we have fixed it at this value chosen ‘by eye’.
The energy range we are dealing with is determined by the application concerning the Cherenkov contribution in the fluorescence air shower experiments, as described in the introduction. The energy threshold for the Cherenkov light to be emitted in the air by an electron is about 21 MeV at see level (increasing as X−1/2), thus our lower limit in energy.
The upper limit is determined by the electron angles of interest (larger than about 1◦) which is reached, as is seen from figure2, at about 1 GeV.
angular dist.
Entries 0
Mean 0.4168
RMS 0.393
/ ndf
χ2 35.21 / 38
a1 0.9593 ± 0.01543 c2 0.1662 ± 0 alpha 4.438 ± 0 Ankle 2.7 ± 0 c1 2.262 ± 0
θs θ /
0 1 2 3 4 5 6 7
10-5 10-4 10-3 10-2 10-1 1
angular dist.
Entries 0
Mean 0.4168
RMS 0.393
/ndf
χ2 35.21/38
a1 0.9593 ± 0.01543 c2 0.1662 ± 0 alpha 4.438 ± 0 Ankle 2.7 ± 0 c1 2.262 ± 0
angular dist.
Entries 0
Mean 0.3947
RMS 0.3611
/ ndf
χ2 31.89 / 27
a1 1.111 ± 0.02111 c2 0.0667 ± 0 alpha 3.884 ± 0 Ankle 2.7 ± 0 c1 2.684 ± 0
θs θ /
0 1 2 3 4 5 6 7 8 9
10-5 10-4 10-3 10-2 10-1 1
angular dist.
Entries 0
Mean 0.3947
RMS 0.3611
/ndf
χ2 31.89/27
a1 1.111 ± 0.02111 c2 0.0667 ± 0 alpha 3.884 ± 0 Ankle 2.7 ± 0 c1 2.684 ± 0
angular dist.
Entries 0
Mean 0.6345
RMS 0.2225
/ ndf
χ2 3.932 / 9
a1 2.315 ± 0.07337 c2 -0.3591 ± 0 alpha 3.404 ± 0 Ankle 2.7 ± 0 c1 4.219 ± 0
θs θ /
0 2 4 6 8 10 12 14
10-5 10-4 10-3 10-2 10-1
angular dist.
Entries 0
Mean 0.6345
RMS 0.2225
/ndf
χ2 3.932/9
a1 2.315 ± 0.07337 c2 -0.3591 ± 0 alpha 3.404 ± 0 Ankle 2.7 ± 0 c1 4.219 ± 0 angular dist.
Entries 0
Mean 0.44
RMS 0.411
/ ndf
χ2 45.39 / 49
a1 0.7939 ± 0.01128 c2 0.211 ± 0 alpha 5.34 ± 0 Ankle 2.7 ± 0 c1 1.992 ± 0
θs θ /
0 1 2 3 4 5 6 7
10-5 10-4 10-3 10-2 10-1 1
angular dist.
Entries 0
Mean 0.44
RMS 0.411
/ndf
χ2 45.39/49
a1 0.7939 ± 0.01128 c2 0.211 ± 0 alpha 5.34 ± 0 Ankle 2.7 ± 0 c1 1.992 ± 0
angular dist.
Entries 0
Mean 0.4662
RMS 0.2856
/ ndf
χ2 8.008 / 12
a1 1.84 ± 0.05119 c2 -0.2161 ± 0 alpha 3.47 ± 0 Ankle 2.7 ± 0 c1 3.726 ± 0
θs θ /
0 2 4 6 8 10
10-5 10-4 10-3 10-2 10-1
angular dist.
Entries 0
Mean 0.4662
RMS 0.2856
/ndf
χ2 8.008/12
a1 1.84 ± 0.05119 c2 -0.2161 ± 0 alpha 3.47 ± 0 Ankle 2.7 ± 0 c1 3.726 ± 0
angular dist.
Entries 0
Mean 0.3955
RMS 0.3234
/ ndf
χ2 8.991 / 18
a1 1.346 ± 0.03095 c2 -0.06748 ± 0 alpha 3.593 ± 0 Ankle 2.7 ± 0 c1 3.193 ± 0
θs θ /
0 1 2 3 4 5 6 7 8 9
10-5 10-4 10-3 10-2 10-1 1
angular dist.
Entries 0
Mean 0.3955
RMS 0.3234
/ndf
χ2 8.991/18
a1 1.346 ± 0.03095 c2 -0.06748 ± 0 alpha 3.593 ± 0 Ankle 2.7 ± 0 c1 3.193 ± 0
Figure 4. Angular distributions of shower electrons g(θ; E) × θrms2 (E)for several values of their energy: for the upper-left plot, log E= −(1.6–1.5); each next graph corresponds to an increase of log E by 0.3 (∼ factor of 2 in energy), so that the bottom-right is for log E = −0.1–0.
Summarizing this section, as it concerns the angular distribution of electrons with a given energy, all large showers are the same—this distribution does not even depend on the shower development stage (age). The angular distribution of all electrons does depend on the shower age (only), as the contribution of g(θ; E) with various energies changes, of course, with the shower age (the energy spectra become steeper). The age parameter s describes uniquely the energy- and angular distributions of electrons in large showers.
4. Total number of particles N (s)
The total number of particles N (X), which are mainly electrons (in the largest showers, here in consideration, the fraction of muons is about (2–3)% ) has been usually described by the analytical (gamma) function of the slant depth X in the atmosphere, proposed by Gaisser and Hillas [11]:
N (X)= Nmax
X− X1
Xmax− X1
(Xmax−X1λ )
exp
−X− Xmax
λ
(5) with four free parameters: Nmax, Xmax, X1and λ. In principle X1should be the depth where the shower starts to develop, i.e. the depth of the first interaction of the primary particle.
However, the best fit to the simulated showers gives mainly negative values of X1, so that a good description near shower maximum does not give a good fit at the shower beginning.
Moreover, sometimes the four parameters do not describe well enough the simulated showers and the parameter λ has been assumed to be a quadratic function of X, increasing the number of the fitted parameters to 6.
As the age parameter s describes so well the energy- and angular distributions of electrons, we thought it worth checking how the total number of particles depends on it. It is not so that the shape of N (s) dependence is unique; it does fluctuate from shower to shower. However, we have found that for any shower (primary proton or iron, E= 1019,1020eV) N (s) can be described very well by two halves of Gaussian distributions:
N (s)=
Nmaxexp
−(s− 1)2 2σ12
for s <1
Nmaxexp
−(s− 1)2 2σ22
for s >1.
(6)
Thus, the longitudinal development of any shower can be described again by four parameters: Nmax, Xmax (to determine s(X)), σ1 and σ2. A comparison of our new parametrization with that of the four- and six-parameter Gaisser–Hillas is shown in figure5, where we have drawn the ratios of simulated N (s) to the fitted values, for four more or less typical showers, each one for primary proton or iron, both with energies 1019eV and 1020eV.
The differences between the fitted and simulated N (s) are seen only there where the number of particles is very small. Our two-Gaussian fit seems to be a little better for primary iron nuclei than the four-parameter Gaisser–Hillas, whereas for protons it is the opposite. Of course, the six-parameter Gaisser–Hillas is the best. These differences, however, do not play an important role in the final aim to study the cascade curves—the determination of the primary energy, or rather of its part, Eem, going to the electromagnetic component (which, for the largest showers in consideration here, is 80–90% of the primary energy), to be calculated as follows:
Eem=
N (s(X)) dE
dX
dX (7)
where dE
dX
is the mean energy deposit rate (per one particle) in the atmosphere at level s. It is a (very weak) function of s only (as the energy- and angular distributions of electrons depend on s only), so that it can be calculated once forever and applied to any shower. The above integrals, with the curves N (s(X)) fitted to simulations with any of the three possibilities considered, differ from the integrals over the simulated values by much less than 1%, what is much too well considering other experimental inaccuracies. However, one has to fit some analytical curves not to the simulated showers but to the observed ones, in the form of light arriving at the telescope from the consequent parts of the shower track. These observed light fluxes
(a)
(c)
(b)
(d)
Figure 5. Ratio of simulated number of particles to that fitted (see the inset) for four typical showers: (a) primary proton, 1019eV; (b) proton, 1020eV; (c) Fe, 1019eV; (d) Fe, 1020eV.
Figure 6. Correlation of σ1(left) and σ2(right) with Xmax. g denotes the primary photon. The lines are best linear fits to hadronic showers only.
fluctuate much more than the simulated number of particles, so that fitting a six-parameter N (X)is no good. One has to use as few parameters as possible, and the two-Gaussian form is our proposition.
Moreover, we studied the correlations between the Xmaxand the both sigmas. The both sigmas are negatively correlated with Xmax. Figure6 shows values of the fitted σ1 and σ2
for several proton and iron initiated showers versus Xmax. The deeper a shower develops the narrower is the dependence N (s), as the two sigmas are positively correlated with each other (see figure7). In figures6and7there are also values for pure electromagnetic cascades of lower energies (1017 and 1018 eV). These are the components of the higher energy showers constituted of many cascades. First we thought that the scatter in the correlation plots (figure6) might be produced mainly by fluctuations in the hadronic processes. However, as one can see
Figure 7. Correlation between σ1and σ2in individual showers.
from the result for the gamma initiated cascades this is not quite so, as the fluctuations in these pure electromagnetic cascades are not much smaller than those in the hadronic showers.
The correlations between Xmax, σ1and σ2can be used in the reconstruction procedure of showers (see below).
5. Application to shower reconstruction
In a fluorescence detector the telescope camera consists of many pixels (PMTs), each having a small field of view and observing a small element of a shower track (440 pixels, with roughly 1.5◦diameter in the Auger experiment). A distant shower is then seen as a line of ‘fired’ pixels on the camera, each detecting the light flux from its field of view. Usually, most of the light is fluorescence, with some admixture of the Cherenkov which, however, cannot be neglected.
The fluorescence light emitted by any small path element of a single electron is isotropic, so that the light emitted by all electrons in a shower track element will also be isotropic, independently of the angular distributions of electrons at this point. With the Cherenkov (Ch) light, however, the situation is different. As is well known, it is emitted at very small angles with respect to the particle direction (<1◦in the air). The main Ch problem is that this light is scattered by the atmosphere aside, so that it adds to the fluorescence light observed at large angles to the shower axis. As the scattering is mainly by the Rayleigh process (at least high in the atmosphere), i.e. by large angles, and the particles propagate at rather small angles with respect to shower axis (see figure2), the scattered Ch light has practically the same angular distribution as if all particles travelled exactly along the shower axis (the mean square angle of particles and Rayleigh distribution add to get that of the scattered light). There are, however, a fraction of showers with axes inclined by angles, say, less than 30◦ to the telescope line of sight, and then the Ch light just produced at the observed shower path element (the so-called, direct Ch light), may even exceed the fluorescence flux. This directly produced Ch flux has the angular distribution (down to angles less than, say, 3◦) practically the same as that of the electrons. To reconstruct a shower, i.e. to find N (X), one has to be able to predict the amount of the Ch contribution for any track element of a shower. Thus, the number of photons
ni emitted towards the camera ith pixel, seeing the shower track element Xi at depth Xi,
collecting light from the (small) solid angle at an angle θito the shower axis, consists of the two components: the fluorescence and the Cherenkov light:
ni = ni,f l+ ni,Ch with ni,f l = kN(Xi)
dE dX
Xi (θi)/4π (8) where k is the proportionality constant between the number of fluorescence photons emitted per unit path (in g cm−2) and the energy loss rate for ionization (see [12] for details):
k= 4.5× 10−2ph cm−1
ρi(g cm−3)× 2.18 MeV/(g cm−2) (9)
where ρi is the air density and dE
dX
is taken for the age siof the shower level Xi. The second term in (8) consists again of two components: the direct and the scattered Ch light. The number of the direct Ch photons (produced in the field of view) equals
ni,Ch-d= N(Xi)
E0 Eth(h(Xi))
Ych(E)f[E; s(Xi)] g(θi; E) dE × Xi
cos θi
(θi) (10)
where Ych(E)is the number of Ch photons emitted per unit path by an electron with energy E, equal to
Ych(E)= Y0
1−
Eth E
2
(11) where Y0 = 172 photons per g cm−2for wavelengths λ= (300–400) nm.
The scattered Ch light has been discussed by us in detail in [5], and equals
ni,Ch-sc= Y0
ρ(Xi) λa(Xi)fs(θi)
Xi
0
T (X, Xi)N (X)F r[X(s, h)] dX× liav (12) where ρ(Xi)is the air density at Xi, λa(Xi)is the light attenuation length, fs(θ )is the angular distribution of the scattered light, T (X, X)is the fraction of photons produced at Xarriving at Xi, F r[X(s, h)] is the effective fraction of electrons emitting Ch light at given level and
liavis the average path length (in metres) of Ch photons in shower length element li(being slightly larger than li).
The procedure proposed here to find the shower cascade curve N (X(s)), having as data
ni (assuming that the atmosphere scattering properties are known one can deduce these values from the number of photons arriving at the individual pixels) and assuming that the shower geometry is known, i.e. the distance of its core to the detector and the zenith and azimuth angles (knowing atmosphere and shower geometry are separate experimental problems), is the following:
(i) first guess the initial values of Nmaxand Xmaxfor the minimizing procedure: Xmaxas the depth of the mth pixel with the maximal signal, and Nmaxcalculated from (8), with i= m and dE
dX
taken for s= 1 (we neglect first Ch light);
(ii) having Xmax, the dependence X(s) is determined from (1);
(iii) from the straight line σ1(Xmax)in figure6(fitted to the points without any weights)
σ1 = 0.319 − 0.000 141Xmax (13)
we find the initial value of σ1; (iv) from the line in figure7
σ2 = 0.0340 + 0.7682 σ1 (14)
we find the initial value of σ2;
(v) and finally, having all the necessary initial parameters, one can calculate the expected number of photons, to be detected by the individual pixels, and compare them with the measured ones. Using a minimizing procedure the four best fitting parameters can be found.
In principle the reconstruction possibilities of the above procedure should not depend on the fraction of Ch admixture, as it can be very well predicted (its both components—direct and scattered) once the adopted shower age is correct. So, it can be used to the showers with a large Ch fraction, i.e. to their parts deep in the atmosphere and/or to those inclined by (relatively) small angles to the telescope line of sight. So, the Ch light, treated so far as a nuisance in the fluorescence experiments (to get rid of it an iteration procedure has been applied, working properly only if that fraction is small), can be an additional information, helping in finding shower cascade curves.
6. Conclusions
Based on shower simulations (with the CORSIKA program) we show that all large showers are similar when described by the shower age parameter s. This means that electrons in all of them have the same shapes of the energy distributions at levels with the same age s. They also have the same shapes of the angular distribution, this being, however, a consequence of the fact that electrons of a given energy have an angular distribution independent of s (i.e. the same everywhere in the shower). These distributions do not fluctuate from shower to shower as long as the number of particles in the energy- and angle-bins is large, what is well fulfilled in showers in the consideration here (above, say 1018eV). They do not depend on the primary particle, neither on its mass nor on energy. The independence of the primary mass means that our conclusion about the similarity of showers is independent of the interaction model (unless, perhaps, the reality is completely different from our models).
The total number of particles (for a fixed primary energy) does, however, fluctuate to some extent from one shower to another, but when expressed as a function of s, these fluctuations are quite small (e.g. Nmaxfluctuates very little, mainly due to the small, changing admixture of muons). We have found that each shower curve N (s) can be very well described by a four-parameter analytical curve—two halves of the Gaussian distributions with two different widths. Finding the correlations between the parameters (σ1 with Xmax, and σ2 with σ1) we are able to propose a procedure for reconstructing large showers observed from the side by telescope detectors, designed to measure the fluorescence light excited in the atmosphere by the shower particles. Thanks to the similarity of the showers, both the fluorescence and the direct and scattered Cherenkov light fluxes can be accurately predicted for any adopted N (X(s))and compared with experimental results in order to find the best N (X).
We have also studied the lateral distributions of electrons [9] and found that, when the lateral distance is expressed in the Moliere unit at the considered level, these distributions depend also on the age parameter only. This problem will be studied in our future work.
Appendix
The dependence of parameters θrms, a1, c1, c2and α in formula (4) on y = log(E/GeV) is approximated by the following polynomials:
Parameter= b0+ b1y+ b2y2+· · · (A.1)
with:
Parameter b0 b1 b2 b3 b4
θrms 1.501 2 −1.964 7 1.473 1 −1.577 1
a1 2.180 1 1.391 71 0.327 08
c1 4.292 4 1.440 73 −0.642 80 −0.396 55 c2 −0.380 98 −0.429 41 0.162 31 0.124 49
α 3.390 3 −0.298 16 −0.480 17 −0.815 85 −0.068 69
Acknowledgment
This work has been supported by Polish Ministry of Scientific Research and Information Technology under the grant no 2 PO3D 011 24.
References
[1] Takeda M et al 2003 Astropart. Phys. 19 447 [2] Abbasi R et al 2005 Astropart. Phys. 23 157 [3] Uchihori Y et al 2000 Astropart. Phys. 13 151
[4] Abraham J et al (The Auger Colaboration) 2004 Nucl. Instrum. Methods A 523 50 [5] Giller M et al 2004 J. Phys. G: Nucl. Part. Phys. 30 97
[6] Nerling F et al 2003 Proc. 28th Int. Cosmic Ray Conf. (Tsukuba) [7] Hillas A M 1982 J. Phys. G: Nucl. Part. Phys. 8 1461
[8] Hillas A M 1982 J. Phys. G: Nucl. Part. Phys. 8 1475 [9] Giller M et al 2005 Int. J. Mod. Phys. A at press
[10] Heck D et al 1998 Report FZKA 6019 (Forschungszentrum Karlsruhe)
[11] Gaisser T K and Hillas A M 1977 Proc. 15th Int. Cosmic Ray Conf. (Plovdiv) vol 8, p 353 [12] Nagano M and Watson A A 2000 Rev. Mod. Phys. 3 689
