Integrating the latter along the track would give a good estimation of the pri- mary energy

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December 1, 2005 14:32 WSPC/Guidelines-IJMPA 03019

International Journal of Modern Physics A Vol. 20, No. 29 (2005) 6821–6824

c

World Scientific Publishing Company

EXTENSIVE AIR SHOWER CHARACTERISTICS AS FUNCTIONS OF SHOWER AGE

MARIA GILLER, HUBERT STOJEK and GRZEGORZ WIECZOREK^∗

Department of Experimental Physics, University of Lodz, Pomorska 149/153, Lodz, 90-236 Poland gjw@kfd2.fic.uni.lodz.pl

Received 29 October 2004 Revised 5 November 2004

We show that extensive air showers (EAS) are all very similar when described by shower age and Moli`ere length unit. This allows to analyze fluorescence and Cherenkov light emitted by showers in a unified and simple way.

Keywords: ultra-high energy cosmic rays; extensive air showers

1. Introduction

(EAS) induced by the ultra-high energy cosmic rays can be observed not only at the Earth level, where particle densities at ground detector positions can be measured (as in the AGASA experiment). In principle, the whole cascade curve N(X) (number of particles versus slant depth in the atmosphere) can be measured by detecting the fluorescence light emitted isotropically by any element of the excited atmosphere (as in the HiRes and Auger experiments). The image of a shower looked at from a side is a rather thin line, with fluorescence light intensity profile almost proportional to the ionisation energy loss along the shower track. Integrating the latter along the track would give a good estimation of the primary energy. However, the detected light is normally contaminated by a non-negligible flux of the Cherenkov light, scattered aside or emitted towards the light detector. The Cherenkov light has to be subtracted from the detected light fluxes. It is then necessary to know the shower characteristics to do so.

2. The angular distribution of shower electrons

We have already shown,¹basing on shower simulations with CORSIKA, that the shape of electron energy spectrum at a given level in the atmosphere depends on the shower age s at this level only. It does not depend neither on the primary particle mass nor on its energy.

Here we show further that the angular distribution of shower electrons also depends on s only. Fig. 1 shows its independence of primary energy and mass. It allows to predict the relative contributions of the fluorescence and Cherenkov light at any level of any shower – independently of the unknown primary mass and energy.

6821 Int. J. Mod. Phys. A 2005.20:6821-6824. Downloaded from www.worldscientific.com by UNIVERSITY OF LODZ BIBLIOTEKA FIZYCZNA on 11/21/12. For personal use only.

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6822 M. Giller, H. Stojek & G. Wieczorek

0 10 20 30 40 50

angle [deg]

1e-05 1e-04 1e-03 1e-02 1e-01

fraction

p, 10^19eV p, 10^20eV Fe, 10^19eV Fe, 10^20eV

0.7 1

1.3

60

Fig. 1. The angular distribution of electron for two different E₀(10¹⁹and 10²⁰eV) and mass (proton and Fe) for three ages s = 0.7, 1 and 1.3.

Adopting some cascade curve N(X) one can find the fluorescence light emitted in the solid angle∆Ω from the shower track element ∆X

∆n_fl=kdE_ion dX

N(X)∆X∆Ω

4π ⁽¹⁾

whereD_dE

dXion

E is the average energy loss per one electron per unit shower track length and k is the proportionality constant. We will show thatD_dE

dXion

Ecan be found as a simple function of s, to be applied to any shower.

As both, energy and angular, distributions depend on s only it is then obvious that also the two-dimensional electron distribution _dE·dθ^d²^f (normalized to unity at any level) is a function of s only. Thus, the average ionisation energy rateD_dE

dXion

Eshould depend on s only (as the ionisation energy lost by each electron depends on its energy and its inclined by angleθ path length to the shower axis). Our calculations show that it weakly decreases with s. In Ref. 1 we have calculated the fraction of electrons, F(s,h), emitting Cherenkov light (Cherenkov electrons), as a function of the shower age s and the atmospheric height h only. By integrating the number of Cherenkov electrons over the shower track above a given level (and allowing for atmospheric absorption) one can calculate the total flux of Cherenkov photons at this level and then – the flux scattered towards the detector (by Rayleigh and Mie processes).

Some showers can be inclined to the detector line of sight by an angleθsmaller than

∼ 30^◦. Then the Cherenkov light produced along the track just seen (direct Cherenkov light) has to be taken into account. To calculate this one has to know the angular distribution Int. J. Mod. Phys. A 2005.20:6821-6824. Downloaded from www.worldscientific.com by UNIVERSITY OF LODZ BIBLIOTEKA FIZYCZNA on 11/21/12. For personal use only.

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Extensive Air Shower Characteristics as Function of Shower Age 6823

s=0.9 s=1.0 s=1.0 s=0.8 s=0.7

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

1

0.1

0.01

0.0001 0.001

Fig. 2. Lateral distribution of electrons vs. r/r_Mfor various s.

of the Cherenkov electrons, defined as

f_ch(θ;s,h) =^Z ^E⁰

E_th(h)

d²f

dEdθ^Y^ch^(E)dE ⁽²⁾

where Y_ch(E) is the Cherenkov light yield of one electron with energy E relative to the maximum yield k_ch=172 photons/g·cm⁻²). It is obvious that f_chdepends only on s and h of the level seen.

For a distant shower the lateral distribution of electrons can be neglected and the direct Cherenkov light contribution equals

∆n_dir=k_chN(X) f_ch(θ;s,h)∆(Ω)∆X (3) We have neglected the small emission angle of the Cherenkov light.

3. Lateral distribution of electrons

For close showers, when the lateral thickness of the image can not be neglected (it is contained in more than 1 angular pixel of the camera) some general treatment of all such showers is possible as well. We have found that the lateral distribution of electrons at a given level, with the lateral distance expressed in Moli`ere units at this level, is again a function of s only (Fig. 2). Then fitting some N(X) to the data, we can predict the lateral width of the shower image, which should depend on s and h only.

Int. J. Mod. Phys. A 2005.20:6821-6824. Downloaded from www.worldscientific.com by UNIVERSITY OF LODZ BIBLIOTEKA FIZYCZNA on 11/21/12. For personal use only.

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6824 M. Giller, H. Stojek & G. Wieczorek

Acknowledgements

This work was supported by Polish Ministry of Scientific Research and Information Tech- nology under the grant no. 2 PO3D 011 24.

References

1. M. Giller et al., J. Phys. G: Nucl. Part. Phys.30 97, (2004); see also F. Nerling et al., in Proc.

28th Int. Cosmic Ray Conf., Tsukuba2 611, (2003).

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