Giller, M

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29th International Cosmic Ray Conference Pune (2005)  7, 187-190  A new parameterisation of the longitudinal shower development  M.Gillera, A.Kacperczykb, J.Malinowskia and W.Tkaczyka (a) Department of Experimental Physics,University of Łódź, Poland (b) Department of Theoretical Physics,University of Łódź, Poland Presenter: M. Giller (mgiller@kfd2.fic.uni.lodz.pl), pol-giller-M-abs1-he14-oral  Using CORSIKA simulations of the highest energy extensive air showers we show that the total number of particles in a shower  can be described as  a function of age by two halves of a Gaussian function with two different widths, σ1 and σ2 fluctuating from one shower to another. Thus, a shower cascade curve can be fitted by 4 parameters: Xmax (to determine s(X)), Nmax , σ1 and σ2 . The both sigmas are quite well correlated with Xmax, and with each other. This makes the choice of initial parameters for a fitting procedure (in shower reconstruction from fluorescence data) quite easy.   1. Introduction  We have shown in [1] (see also Nerling et al [2]) that the shower age parameter s, as defined by Hillas [3] ( in analogy  to that for a pure electromagnetic cascade): max23)(XXXXs +=       (1) where X is the slant depth ( in g cm-2) of the shower level in the atmosphere, is the only parameter determining the shape of the energy spectrum of electrons. As a consequence of this the electron angular distribution is a function of the shower age as well [5,6]. Also the lateral distribution of electrons (when the lateral distance is expressed in Moliere units) depends on the shower age only [4]. As the fluorescence light emission, caused by an electron in a path element of the atmosphere, is believed to be proportional to its energy deposit in this path, the light flux depends (although weakly) on the electron energy spectrum. In all the above mentioned distributions it was their shape that is s-dependent. The absolute numbers depend, of course, on the total number of particles N. In this paper we study how the total number of charged particles N behaves as a function of s.  2. 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 [7] with four (or six) free parameters. As the age parameter s describes so well the energy, angular and lateral distributions of electrons, we thought it worth to check 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, 1020 eV)  N(s) can be described very well by two halves of Gaussian distributions: ( )( )( )> −−< −−=1sfor       21exp1sfor      21exp222max212maxσσsNsNsN   (2) 188         M. Giller et al.  Thus, the longitudinal development of any shower can be described again by 4 parameters: Nmax, Xmax (to determine s(X) , σ1 and σ2. A comparison of our new parameterization with that of the 4- and 6-parameter Gaisser-Hillas is shown in Figure1, 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 1019 eV and 1020 eV. 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 4-parameter Gaisser-Hillas, whereas for protons it is the opposite. Of course, the 6-parameter Gaisser-Hillas is the best.  01230 0.2 0.4 0.6 0.8 1 1.2 1.4sNsimN fit ours,  2 GaussiansGaisser - Hillas 4 par.Gaisser - Hillas 6 par.c)01230 0.2 0.4 0.6 0.8 1 1.2 1.4sNsimN fit ours,  2 GaussiansGaisser - Hillas 4 par.Gaisser - Hillas 6 par.b)01230 0.2 0.4 0.6 0.8 1 1.2 1.4sNsimN fit ours,  2 GaussiansGaisser - Hillas 4 par.Gaisser - Hillas 6 par.a)01230 0.2 0.4 0.6 0.8 1 1.2 1.4sNsimN fit ours, 2 GaussiansGaisser - Hillas 4 par.Gaisser - Hillas 6 par.d) Figure 1. Ratio of simulated number of particles to that fitted (see the inset) for four typical showers: (a) primary proton, 1019 eV; (b) proton, 1020 eV; (c) Fe, 1019 eV; (d) Fe, 1020 eV.  The observed light fluxes fluctuate much more than the simulated number of particles, so that fitting a 6-parameter N(X) is not always effective. One has to use as few parameters as possible, and the two-Gaussian form seems to be promising. Moreover, we studied the correlations between Xmax and the both sigma’s. The   0.130.150.170.190.210.23600 700 800 900 1000 1100Xmax [g/cm2]σ 20.130.150.170.190.210.23600 700 800 900 1000 1100Xmax [g/cm2]σ 1p 1e19p 1e20Fe 1e19Fe 1e20g 1e17g 1e18 Figure 2. Correlation of σ1 (left) and σ2 (right) with Xmax. g denotes the primary photon. The lines are best linear fits to hadronic showers only   both sigma’s are negatively correlated with Xmax. Figure2 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 sigma’s are positively correlated with each other, see Figure3. In Figure2 and 3 there are also values (denoted  as g) for pure electromagnetic cascades with  lower energies (1017 and 1018 eV). These are  the  components  of the higher energy showers constituted of many cascades. At first we thought that the                 A new parameterisation …                                 189  scatter in the correlation plots might be produced mainly by fluctuations in the hadronic processes. However, as one can see 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, σ1 and σ2 can be used in the reconstruction procedure of showers from the fluorescence detectors measurements. 0.150.170.190.210.230.250.15 0.17 0.19 0.21 0.23 0.25σ1σ 2p 1e19p 1e20Fe 1e19Fe 1e20g 1e17g 1e18                                           Figure 3. Correlation between σ1 and σ2 in individual showers.   3. Application to shower reconstruction  In a fluorescence detector the telescope camera consists of many pixels (PMT’s), each having a small field of view and observing a small element of a shower track (440 pixels, with ~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, can not 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 distribution of electrons at this point. With the Cherenkov (Ch) light, however, the situation is different. As it 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. 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 i-th pixel, seeing the shower track element ∆Xi at depth Xi, collecting light from the (small) solid angle ∆Ω at an angle θi to the shower axis, consists of  two components: the fluorescence and the Cherenkov light: ( ) πθ 4)(         ,,, iifliChiflii dXdEXNknwithnnn ∆Ω=∆∆+∆=∆   (3)  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 and <dE/dX>, the mean energy deposit rate per unit shower track length, is taken for the age si of the shower level Xi. The second term in equation (3) consists again of two components: the direct and the scattered Ch light discussed in some detail in [5,6]. 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 to 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:  190      M. Giller et al.  1. 2. 3. 4. 5. first guess the initial values of Nmax and Xmax for the minimizing procedure: Xmax - as the depth of the m-th pixel with the maximal signal, and Nmax calculated from (3), with i=m and <dE/dX> taken for s=1 (we neglect first Ch light), having Xmax  the dependence X(s) is determined from (1), from the straight line σ1(Xmax) in Figure2 (fitted to the points without any weights): σ1= 0.319 - 0.000141Xmax   we find the initial value of σ1, from the line in Figure3: σ2= 0.0340 + 0.7682 σ1   we find the initial value of σ2, 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 [5,6]. Using a minimising 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 procedures have been applied, working properly only if that fraction is small), can be an additional information, helping in finding shower cascade curves.   4. Conclusions  We have found that each shower curve N(s) can be very well described by a 4-parameter analytical curve-two halves of the Gaussian distributions with 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 by telescope detectors.  Thanks to the similarity of the showers [5], 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).    5. Acknowledgment  This work has been supported by the grant of the Łódź University no. 696.  References  [1] Giller M et al.,  J.Phys. G: Nucl. Phys.,  30, 97 (2004) [2] Nerling F et al., Proc. 28th ICRC, Tsukuba (2003) [3] Hillas A  M, J. Phys G: Nucl. Phys. 8, 1475 (1983) [4] Giller M, Stojek H, Wieczorek G, Int. Mod. J. Phys. in print (2005) [5] Giller M,  Kacperczyk A, Malinowski J, Tkaczyk W, Wieczorek G, J. Phys. G: Nucl. Part. Phys. 31, 947 (2005) [6] Giller M and Wieczorek G, 29th ICRC, Pune (2005) [7] Gaisser T K and Hillas A M, Proc. 15th ICRC, Plovdiv 8, 353 (1977) 

A new parameterisation of the longitudinal shower development

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