J. Astrophys. Astr. (2017) 38: 4 c Indian Academy of Sciences DOI 10.1007/s12036-017-9424-2

Application of CORSIKA Simulation Code to Study Lateral and Longitudinal Distribution of Fluorescence Light in Extensive Air Showers

ZAHRA BAGHERI1,∗, PANTEA DAVOUDIFAR1, GOHAR RASTEGARZADEH2 and MILAD SHAYAN1

1Research Institute for Astronomy and Astrophysics of Maraghe, Maraghe, Iran. 2Departments of Physics, Semnan University, Semnan, Iran. ∗Corresponding author. E-mail: [email protected]

MS received 21 May 2016; accepted 2 December 2016; ePublication: 15 March 2017

Abstract. In this paper, we used CORSIKA code to understand the characteristics of cosmic ray induced showers at extremely high energy as a function of energy, detector distance to shower axis, number, and density of secondary charged particles and the nature particle producing the shower. Based on the standard proper- ties of the atmosphere, lateral and longitudinal development of the shower for photons and has been investigated. Fluorescent light has been collected by the detector for protons, helium, oxygen, silicon, calcium and iron primary cosmic rays in different energies. So we have obtained a number of electrons per unit area, distance to the shower axis, shape function of particles density, percentage of fluorescent light, lateral distri- bution of energy dissipated in the atmosphere and visual field angle of detector as well as size of the shower image. We have also shown that location of highest percentage of fluorescence light is directly proportional to atomic number of elements. Also we have shown when the distance from shower axis increases and the shape function of particles density decreases severely. At the first stages of development, shower axis dis- tance from detector is high and visual field angle is small; then with shower moving toward the Earth, angle increases. Overall, in higher energies, the fluorescent light method has more efficiency. The paper provides standard calibration lines for high energy showers which can be used to determine the nature of the particles.

Keywords. Cosmic ray—fluorescence light—extensive —Monte Carlo method—CORSIKA.

1. Introduction be seen by detectors from a distance of several kilo- meters (G’ora et al. 2001). Since the fluorescent light One of the important aspects of cosmic ray experi- entirely depends on the number of particles in shower, ments is determining energy of primary particles. But this method provides a good standard for measuring at very high energies, due to very low flux of radiation, primary energy. direct detection method is not possible. Instead, in these Air shower detection by fluorescence method (air energy ranges indirect detection methods should be glow) with sufficient sensitivity, in good climatic con- used; such as the study of particles and light in Exten- dition and favorable background light conditions, can sive Air Showers (EAS) induced by the interaction of be useful to follow large showers that pass through its primary particles with the Earth’s atmosphere. vision field. One of these methods is the detection of fluorescent The air fluorescence method offers several important advantages when compared with other particle detector light induced by secondary cosmic ray particles when type shower arrays: passing through the Earth’s atmosphere and interacting with nitrogen in the air. Fluorescent method is based (1) Instead of sampling the showers only at a few points on collecting emitted light from excited molecules by in an array on the ground at a particular stage of charged particles. These methods are used by the Fly’s development and requiring that the shower axis be Eye array. within the detector array for reliable data interpre- For primary particles with Ultra High Energy tation, the atmospheric fluorescence method allows (UHE), enough light will be produced by a large num- observing the shower from the outside, at some ber of secondaries in shower (EAS) so that the light can distance and angle. 4 Page 2 of 8 J. Astrophys. Astr. (2017) 38: 4

(2) Fluorescence techniques are capable of revealing About 25% of the emission occurs at short wave- almost the entire history of each of the recorded length that is very poorly transmitted by the atmo- events, unlike particle detector arrays. It records the sphere. Hardly any of the light is found at wavelengths longitudinal profile of the showers as well. longer than 428 nm, hence, the signal-to-noise ratio can (3) This method does not subject to large observational be improved by using a filter that transmits only in the fluctuations because of the abundant number of deep blue and ultraviolet. Taking into account the trans- photons that are detected, even from distant events, mission of a normal atmosphere as well as the energy as compared to the small number of particles that loss of fast electrons in air, the effective yield at the are usually recorded by the detectors of an array per detector is approximately one to two photons per meter shower. of track length for a large shower that strikes at a dis- (4) Like particle detector arrays, it is capable of deter- tance of about 10 km. The background light from the mining the direction of incidence of the primary dark (moonless) night sky is at best about 105 ph m−2 and the point of impact of the axis on the ground. μs−1 sr−1. There are, however, several disadvantages and limita- tions to this method that restrict its application, but flu- 2. Longitudinal development of fluorescent light orescence method is now being used very successfully. To determine the type, direction and energy of pri- To study the longitudinal development of extensive air mary particles, energy of secondary particles are used. showers, one can measure the fluorescence radiation by At very high particle energies sometimes the number air molecules when the charged particles are passing of secondaries reach millions of particles at the obser- through. Shower particles excite nitrogen molecules vation level. So using the equations is difficult, and and ions in the atmosphere and as a result, 10Ð50 even in some cases, impossible. To solve this problem, nanoseconds after the excitation; they export fluores- we considered proper approximations to describe the cence photons in average of 4Ð5 photons per meter for air showers, which implies to simulate them using the each (Shellard & Diazy 1999). Monte Carlo method. Approximately 80% of the emitted fluorescence The four main components of light can be dis- light is in the range of 300Ð450 nm (three main lines tinguished as: (i) fluorescence light, with isotropic are 337.1, 357.7 and 391.4 nm) for which the atmo- emission; (ii) direct , emitted pri- sphere is quite transparent (Shellard & Diazy 1999). marily in the forward direction; (iii) Rayleigh-scattered This fluorescent light will be detected by optical detec- Cherenkov light; (iv) Mie-scattered Cherenkov light. tors installed on the telescope when the light source The relative contributions of these components depend moves along the axis of shower. on the geometry of the shower with respect to the detec- As the number of fluorescent photons generated at tor, but in most cases the fluorescent light dominates each depth of X is proportional to the number of elec- the recorded signal. Assuming only minor effects on trons in that depth, with longitudinal study of the devel- the shower width by absorption and scattering pro- opment of these photons, one can obtain the number of cesses during the fluorescence light propagation from electrons in that depth and one may use their number to the shower to the detector, the light fraction F(α) measure the primary energy (Song et al. 2000). which is the total light recorded by fluorescent detec- tor, is mainly determined by the corresponding light 2.1 Fluorescence yield fraction F(r) (light fraction) which is the portion of particles or the proportional part of the fluorescent light If we define fluorescence yield (Y ) as the number of that is located at a distance of 0 to r, emitted around generated photons per electron per unit of length, then the shower axis where f(r) is the lateral distribution we will have of fluorescent light emitted, in the other words, it is the shape function or lateral distribution of photons while Nγ Y = . (1) r is the distance from the shower axis. Nel Since this work is intended as a general study, the resulting photon distribution after light propagation So the intensity of observed light is directly propor- is assumed to be recorded by an ideal detector and tional to the number of charged particles in each spot simulation is done for standard atmosphere default by of longitudinal development of extensive air shower, CORSIKA. and the number of photons per unit solid angle per J. Astrophys. Astr. (2017) 38: 4 Page 3 of 8 4 unit length can be obtained from the following equation this wavelength range, values of 4.4 photons per meter (G’ora et al. 2004): will be obtained by dividing the length over the min-   imum particle ionization at sea level (4.4 ph m−1e−1) 2 d Nγ YN photon ≈ e . (2) (Grieder 2010). dl d 4π sr m The fluorescence yield is primarily a function of the electron energy, temperature and atmospheric pressure. The total number of particles is given by the GaisserÐ But in most calculations it will be considered constant Hillas equation (Gaisser & Hillas 1977):   and equal to 4.02 photons per meter. (Xmax−X0)/λ X − X0 Ne(X) = Nmax Xmax − X0 − exp((Xmax X)/λ), (3) 3. Lateral development of fluorescent light in which X is the inclined atmospheric depth, X0 is the For primary particles with very high energy, sufficient depth of first interaction, Xmax is the shower maximum fluorescent light will be produced in shower process, i.e. the depth at which number of particles in shower due to the large number of secondary particles. So reaches a maximum value, λ is the hadronic interaction the shower can be recorded from a distance of several 2 length in the air (generally equals 70 g/cm )andNmax kilometers by a suitable optical detector. is the maximum number of particles in the shower that Since shower particles are close to shower axis, large is given by the following equation (Baltrusaitis et al. percentage of the fluorescent light is at a distance of 1985):   less than several ten meters and the observed fluores- E [GeV] 1.010 cent light will be seen as a spot, so studying fluorescent N = 0.7597 0 ∗ 109, (4) max 109 light as a point will be sufficient. With an optical imaging system for recording the where E is the primary energy of the shower. light emitted by a shower, the width of the shower is defined as a minimum angular diameter α of the image 2.2 Constant yield to include a specific part of F(α) of the total light recorded by the fluorescence detector. Measurements have shown that changes in fluorescence If ρN (X, r) be the number of electrons per square yield (Y ) as a function of height for electrons with con- meter in depth X and distance r from the shower axis, stant energy, is quite small. For example, for an electron then the shape function f (r) will be defined as f(r) = ρN (X,r) . It is called shape function because if fluores- with energy of 80 MeV, the measured fluorescence Ne(X) yields over 20 km effective height in the atmosphere cence yield is considered constant, then this function which will change lower than 12% around the mean gives us a lateral distribution of photons. So in this value of 4.8 photons/m (G’ora et al. 2004). case, F(r) (light fraction) is the portion of particles or These slight changes motivate us to use a constant the proportional part of the fluorescence light that is that is the average fluorescence yield of the shower located at a distance of 0 to r: particles. On the other hand, since the fluorescence is r = caused by ionization and excitation of molecules of F(r) f(r)2πr dr. (5) 0 limited air, it is expected that the fluorescence yield The final image of the shower will be made by the pho- depends on the ionization density in the particle path. tons that reach the detector at the same time (G’ora Most particles in the shower which were involved et al. 2001). These photons that make an instantaneous in energy deposition in air have a kinetic energy lower image of the shower come from the process of devel- than MeV to several hundred MeV and the energy opment of the shower. If dE is energy deposit per is in the range where the ionization density depends dX unit length that electron leaves when it passes through significantly on particle density. the atmosphere, and if this value has been estimated The fluorescence yield per unit length is almost inde- by its mean value  dE , then in case of the NKG pendent (about =∼10%) of pressure (or height) and dX approximation, lateral distribution of energy in the temperature. Air fluorescence decay time is generally atmosphere, ρ (X, r) can be obtained from the follow- less than 10 ns. E ing equation (G’ora et al. 2004): In the spectral region between 300 nm to 430 nm, the   dE total measured yield divided by energy loss (in MeV) ρE(X, r) = NmaxρN (X, r). (6) in air at 1 atmospheric pressure is about 15 photons. In dX 4 Page 4 of 8 J. Astrophys. Astr. (2017) 38: 4

3.1 Angular size of shower image series (Abramowitz & Stegun 1965), we can determine the distribution of fluorescent light by the following For pure electromagnetic showers, density of electron equation: at the shower (ρN (X, R)) is given by the NKG formula   4.5−s (G’ora et al. 2004; Kamata et al. 1958): 1 1 F(r) =     1 + rM sB (s , 4.5 − 2s) s−2 (s−4.5) r   Ne(X) r r ∞ n+1 ρ (X, r) = 1+ B(s+1,n+1) 1 N 2 r r × 1+ . rM M M B(4.5−s,n+1) 1 + rM n=0 r (4.5−s) (10) × , − (7) 2π(s)(4.5 2s) For s = 1(maximum of shower), the result is reduced to the following formula: ≡ 3X/Xl = Xv i.e., s X/X +2ln(E / ) and X cos θ ,where   ∞ l 0 0 −2.5 X ≡ ρ(h)dh is the vertical depth in atmosphere, r v h F(r) = 1 − 1 + . (11) θ is the zenith angle of shower, E0 is the primary rM energy of the shower, is its critical energy and 0 Using the above reversed equation, if at a moment, X is the radiation length in the air. ρ(h) is the air l detector distance to shower is R,forα angle which is density at an altitude h and r is Moliere radius. M a specific portion of the total fluorescent light (F (r)), The age parameter (s), specifies the current stage of we have shower evolution (s = 1, and means shower is at its   α r − r maximum). tan = ,α= 2tan 1 The Moliere radius is a natural horizontal scale that 2 R R  −1 rM −0.4 is caused by multiple scattering, and determines the = 2tan (1 − F(r)) − 1 . (12) r lateral distribution of the shower. Since the electron’s radiation length in the air depends on temperature and This angle gives a visual field which corresponds to the pressure, Moliere radius varies along the shower. Parti- percentage of light located at a distance of 0 to r. cle distribution in a shower at a certain depth relatively depends on the variation rM along the shower instead of value of rM in a certain depth. To consider it in a 4. Details of simulated samples certain atmospheric depth, rM value at twice the length of electron’s radiation length, is calculated as follows 4.1 Simulations for protons with energy 10 eV (G’ora et al. 2004):   At this point, using CORSIKA (Heck et al. 1998), 1/5.25588 P [mb]−37.94 cos θ initially primary beam of protons was simulated and T [K] P [mb] lateral and longitudinal distribution of fluorescence r [m]=272.5 , M P [mb]−37.94 cos θ light induced by excitation of nitrogen molecules in the (8) atmosphere has been studied and obtained, and then the angular size of shower image for simulated data where T is the temperature and P is the atmospheric was calculated. The height is CORSIKA default for pressure at height h. KASKADE, 110 m above sea level. As HiRes Group, we use a constant value for the flu- First ρ (X, r) (i.e. the number of electrons per unit = N orescence yield Y 4.02 photon/m. Spatial distribution area in the depth of (X)) and the distance to the shower of the light emitted will be defined by NKG formula. axis, r, have been obtained from simulations and are As a result, fluorescence distribution F(r) is deter- shown in Fig. 1(a). Then to obtain the shape func- mined by the normalized incomplete beta function, tion of particles density, f(r) (i.e. lateral distribution analytically: of photons), ρN (X, r) curve data have been divided to x the number of electrons at the observation level. As 1 a−1 b−1 N = × 8 F(r) = Ix(a, b) = u (1 − u) du, (9) a result, the value of e 75 10 is obtained by B(a,b) simulation; results can be seen in Fig. 1(b). To obtain 0 the percentage of particles (fluorescence light) located i.e., x = 1/(1+rM/r), a = s, b = 4.5−2s and B(a, b) at a distance of 0 to r, F(r), first we fitted the lateral is the Euler beta function. Using expansion of I (a, b) b x distribution curve with a function of ex+c and reached J. Astrophys. Astr. (2017) 38: 4 Page 5 of 8 4

(a) (b) (c) 1e+8 1e-2 120

) 1e-3 100 2 1e+7

1e+6 1e-4 80 60 1e+5 1e-5 40 1e+4 1e-6 20 1e+3

Particle Density (m 1e-7 0 1e+2 1e-8

1 10 100 1000 1 10 100 1000 Fraction particle of density (F) (%) 0.1 1 10 100 1000 Shape function of particle density particle of (f) function Shape r(m) r (m) r (m) (d) (e) 1e+8 10

1e+7 (deg)

1e+6

1e+5 1

1e+4

1e+3 Energe DepositeEnerge Density

1e+2 shower image size of 0.1 1 10 100 1000 0.1 1 10 100 r(m) R (Km) Figure 1. Simulation results for protons with energy 10 eV. (a) Number of electrons per unit area in the depth (X) and distance to the shower axis (r)(ρN (X, r)). (b) Percentage of particles (fluorescence light) located at a distance of 0 to r(F(r)). (c) Shape function of particles density f(r) (lateral distribution of photons). (d) Lateral distribution of energy dissipated in the atmosphere ρE(X, r), and (e) Detector visual field angle for collecting 90% of fluorescent light for proton in 10 eV.

(a) (b) (c) 1e+5 1e-1 120 1e+4 1e-2 1e+3 1e-3 1e+2 1e-4 100 1e+1 1e-5 1e+0 1e-6 1e-1 1e-7 80 1e-8 1e-2 1e-9 1e-3 1e-10 60 1e-4 1e-11 1e-5 1e-12 1014 eV 1e-6 14 1014 eV 10 eV 1e-13 40 15 15 10 eV

Particle Density 1e-7 1015 eV 1e-14 10 eV 1e-8 16 16 16 1e-15 10 eV 10 eV 1e-9 10 eV 1e-16 20 1e-10 1e-17 1e-11 1e-18 0

110100 110100 density (%) (F) particle Fraction of 10 100 Shape function of Particle Density (f) Density (f) Particle of Shape function r(m) r (m) r (m) (d) (e) 1e+12 1e+11 10 1e+10 1e+9 1e+8 1e+7 1e+6 1e+5 1 1e+4 1e+3 1e+2 1e+1 1e+0 14 1e-1 0.1 10 eV 1e-2 14 1e-3 10 eV 1015 eV 1e-4 1015 eV 1016 eV 1e-5 16 1e-6 10 eV Energy DepositeDensity 1e-7 1e-8 (deg) shower image of Size 0.01 1101000.1 1 10 100 r(m) R (Km) Figure 2. Simulation results for proton primaries with other energies. (a) Number of electrons per unit area in the depth (X) and distance to the shower axis (r)(ρN (X, r)). (b) Shape function of particles density f(r) (lateral distribution of photons). (c) Percentage of particles (fluorescence light) located at a distance of 0 to r(F(r)).(d) Lateral distribution of energy dissipated 14 15 in the atmosphere ρE(X, r). (e) Detector visual field angle for collecting 90% of fluorescent light for proton in 10 eV, 10 eV and 1016 eV. 4 Page 6 of 8 J. Astrophys. Astr. (2017) 38: 4

(a) 3 (b) (c) e e-6 2 120 e -7 ) 1 e 2 e -8 0 e 100 e -9 e-1 e -10 80 e-2 e -3 e-11 60 e 14 Fe , 10 eV -12 14 e-4 e Fe , 10 eV Fe , 1014 eV He 40 e-5 e-13 He He -6 O -14 e e O 20 O e-7 P e-15 P P Particle Density (m Si e-8 e-16 Si 0 Si Ca Ca Ca e-9 e-17 1 10 100 1000 1101001000Fraction particle of density (F) (%) 1101001000 Shape function of particle density (f) (f) density particle of Shape function r (m) r(m) r(m) (d) e14 (e) e13 100 14 12 Fe , 10 eV e He e11 e10 O 10 P 9 e Ca e8 Fe , 1014 eV Si e7 He 6 1 e O e5 P 4 Si

Energy deposite density Energy deposite e Ca e3 (deg) image shower of Size 0.1 1 10 100 1000 0.1 1 10 100 r (m) R (Km)

Figure 3. Simulation results for other cosmic ray species. (a) Number of electrons per unit area in the depth (X) and distance to the shower axis (r)(ρN (X, r)). (b) Shape function of particles density f(r) (lateral distribution of photons). (c) Percentage of particles (fluorescent light) located at a distance of 0 to r(F(r)). (D) Lateral distribution of energy dissipated in the atmosphere ρE(X, r). (e) Detector visual field angle for collecting 90% of fluorescent light for oxygen, calcium, protons, silicon, helium and iron in 1014 eV. the fit parameters a = 110/7, b = 1070 and c = 92/3. 4.2 Simulations for proton primaries Then we got integral from this function to achieve with other energies F(r). The results are shown in Fig. 1(c). To obtain the lateral distribution of energy dissipated in the atmo- At this point, using CORSIKA and the same approach, 14 15 sphere ρE(X, r), which is a measure of the fluorescence primary beam of protons with energies 10 ,10 and  dE  1016 eV were simulated to investigate the effects of photon’s intensity, we have considered dX equal to 2 MeV/(gr/cm2) because of constant fluorescence yield primary energy. The height is equal to 180 m above (4.02 photons per meter) (G’ora et al. 2004). The sea level. First ρN (X, r) have been calculated and are results for the lateral distribution of energy left in the shown in Fig. 2(a). Then to obtain f(r), ρN (X, r)curve atmosphere ρE(X, r), are shown in Fig. 1(d). Finally data have been divided to the number of electrons in visual field angle of detector for different distances to the observation level that had been obtained Ne = shower axis were calculated for our data and are shown 5617.97, Ne = 105814.88, Ne = 1549639.16 for ener- in Fig. 1(e). gies 1014,1015 and 1016 eV by simulation. The results The results are derived from CORSIKA with ver- can be seen in Fig. 2(b). Then we fitted the lateral distri- b tical radiation angle at a height 110 m above sea bution curve with function ex+c and we reached the fit level. In the CORSIKA simulations performed for this parameters a14 = 15110, b14 = 2059, c14 =−125.5, analysis, High-energy interactions are calculated by a15 = 177500, b15 = 2299, c14 =−125.5, a16 = the QGSJET02 model and low-energy interactions are 31.86, b16 = 1537 and c16 =−124.2. Then we got an calculated by the GHEISHA2002 model. Since each integral from this function to achieve F(r) (Fig. 2(c)). simulation lasts about more than 10 days for each The results for the lateral distribution of energy left in shower, only 2 showers were simulated for any the atmosphere ρE(X, r) is shown in Fig. 2(d). Finally energy. visual field angle of detector for different distances to J. Astrophys. Astr. (2017) 38: 4 Page 7 of 8 4

(a) (b) (c) e6 1e-2 120 e5

) 4 100

2 e 1e-3 e3 80 e2 1e-4 e1 60 0 e Fe , 1015 eV 1e-5 Fe , 1015 eV Fe , 1015 eV -1 e He He 40 He e-2 P 1e-6 P P e-3 O Ca 20 Ca -4 Na Na Na

Particle density (m e 1e-7 O -5 Si O 0 e Ca Si Si e-6 1e-8

11010010001 10 100 1000 Fraction particleof density (F) (%) 1 10 100 1000 Shape functionparticle of density (f) r (m) r (m) r (m) (d) e20 (e) 2 e19 e e18 e17 e1 e16 15 e e0 14 e 15 13 Fe , 10 eV Fe , 1015 eV e He -1 12 e He e P P e11 Ca Ca -2 e10 O e Na 9 Si O Energy deposite density density deposite Energy e Na Si e8 (deg) shower image of Size e-3 1 10 100 1000 0.1 1 10 100 r (m) R (Km)

Figure 4. Simulation results for other cosmic ray species. (a) Number of electrons per unit area in the depth (X) and distance to the shower axis (r)(ρN (X, r)). (b) Shape function of particles density f(r) (lateral distribution of photons). (c) Percentage of particles (fluorescent light) located at a distance of 0 to r(F(r)).(d) Lateral distribution of energy dissipated in the atmosphere ρE(X, r). (e) Detector visual field angle for collecting 90% of fluorescent light for oxygen, calcium, protons, silicon, helium and iron in 1015 eV. 14 15 Table 1. Number of electrons in the observation level (Ne) and fit parameters for different species in 10 eV and 10 eV.

Species Oxygen Calcium Proton Silicon Helium Iron

Ne14 59820.45 44194.89 105814.9 49550.26 87389.11 39103.26 −7 −7 −7 −21 −7 −31 a14 1.088 × 10 1.551 × 10 5.199 × 10 1.06 × 10 6.548 × 10 1.075 × 10 b14 708 668.1 335.5 75.72 310.7 814 c14 91.76 95.71 48.34 −47.19 46.68 18.13

Ne15 2556.77 1945.6 5617.97 2294.47 4611.45 1660.68 −6 −7 −6 −19 −7 −48 a15 1.346 × 10 6.896 × 10 1.52 × 10 1.566 × 10 7.181 × 10 2.208 × 10 b15 220.8 316.2 198.5 1704 277.3 801.5 c15 36 49.34 30.43 65.49 39.63 10.97 shower axis were calculated for our data and are shown photons), ρN (X, r) curve data have been divided to the in Fig. 2(e). number of electrons in the observation level that had been obtained as Ne in Table 1. The results can be seen 4.3 Simulations for other cosmic ray species in Figures 3(b) and 4(b). Then to obtain F(r),firstwe b fitted the lateral distribution curve with function ex+c In the next step, using CORSIKA and the same and we reached fit parameters a14, b14 and c14, a15, b15 approach, simulation was done for primary beam of and c15 for different species that are shown in Table 1. oxygen, calcium, proton, silicon, helium and iron with Then we got an integral from this function to achieve 14 15 energies 10 and 10 eV. The height is equal to 180 m F(r) (Figures 3(c) and 4(c)). The results for ρE(X, r) above sea level. are shown in Figures 3(d) and 4(d). Visual field angle First ρN (X, r) have been calculated and are shown in of detector for different distances to shower axis were Figures 3(a) and 4(a). Then to obtain the shape func- calculated for our data and are shown in Figures 3(e) tion of particles density f(r) (lateral distribution of and 4(e). All elements with the same energy were 4 Page 8 of 8 J. Astrophys. Astr. (2017) 38: 4 plotted on one graph to compare the composition of the and have been shown in Figures 1(e) to 4(e). As can various elements with each other. be seen in figures at the first stages of development (when R is great) shower axis distance from detector is high, and hence this angle is low. Then with the shower 5. Conclusions moving toward the Earth (when R is little) this angle increases. We have shown that with increasing shower We obtained the number of electrons per unit area in axis distance from detector, the angle would reduce. the depth (X) and distance to the shower axis (r), In general, the number and density of charged parti- ρN (X, r). The shape function of particle density was cles reduce when they are away from the shower axis, f(r) with the percentage of fluorescence light F(r). and so the density of fluorescent light reduces too. Lateral distribution of energy dissipated in the atmo- In simulation results for other cosmic ray species, as sphere is ρE(X, r) and visual field angle of detector is seen in Figures 3(c) and 4(c) about 90% of fluorescent α. These parameters were calculated for primary beams light in 1014 eV were located at distances of proton: of oxygen, calcium, proton, silicon, helium and iron, in 21 m, helium: 24 m, oxygen: 28 m, silicon: 30 m, cal- different energies and in this way we were able to calcu- cium: 37 m and iron: 38 m from the shower axis. And late lateral and longitudinal profiles of fluorescent light. in 1015 eV, they were located at distances of proton: In this method, the image size of shower, α, is related 29 m, helium: 35 m, oxygen: 53 m, silicon: 66 m, cal- to the width of shape function of particle density f(r), cium: 68 m and iron: 76 m from the shower axis. And and it can be calculated in the maximum of shower as the atomic number of these elements are protons: using the above equations for fixed Moliere radius, 1, helium: 2, oxygen: 8, silicon: 14, calcium: 20 and fixed fluorescence distribution F(r) and fixed detector iron: 26; it can be concluded that 90% of the fluorescent distance to shower (R). In simulation results for pro- light for elements with lower atomic number, has been tons with energy 10 eV, as seen in Fig. 1(c), about 90% located at lesser distances from the axis of the shower. of fluorescent light in 10 eV was located at a distance of 50 m from the shower axis. In simulations results Acknowledgement for proton primaries with other energies, as seen in Fig. 2(c), about 90% of fluorescent light in 1014,1015 This work has been supported financially by the and 1016 eV were located respectively at distances of Research Institute for Astronomy and Astrophysics 22, 17 and 15 m from the shower axis. of Maragha (RIAAM) under Research Project No. In simulation results for other cosmic ray species, as 1/3720-69. seen in Figs. 3(c) and 4(c), about 90% of fluorescent light in 1014 eV were located at distances of 28, 37, 21, 30, 24 and 38 m respectively from the shower axis of References oxygen, calcium, protons, silicon, helium and iron and in 1015 eV, were located at distances of 53, 68, 29, 66, Abramowitz, M., Stegun, I. A. (eds) 1965, Handbook of Mathematical Functions (New York, Dover Publications, 35 and 76 m from the shower axis in the same way. Inc.). By comparing the figures, we understood that this Baltrusaitis, R. M. et al. 1985, Proc. 19th ICRC, La Jolla, method in higher energy gives a better answer. With vol. 7, p. 159. energy decrease, number of charged secondary parti- Heck, D. et al. 1998, Forschungszentrum Karlsruhe Report cles and the number of fluorescence photons is also FZK, 6019. reducing. As a result, using this method in the detec- Grieder, P. K. F. 2010, Extensive Air Showers, High Energy tion of cosmic rays at higher energies is more effective. Phenomena and Astrophysical Aspects (Springer, Heidel- Actually in the present work we have showed that as berg Dordrecht London New York) p. 883. th energy increases, fluorescent light yield also increases. Gaisser, T. K., Hillas, A. M. 1977, Proc. 15 ICRC, Plovdiv, As can be seen from Figures 1(b) to 4(b), when dis- vol. 8, p. 353. tance from the shower axis increases then the value G’ora, D. et al. 2001, Astropart. Phys., 22(16), 129Ð136. G’ora, D. et al. 2004, Astropart. Phys., 22(1), 29Ð45. of the shape function of particle density f(r) strongly Kamata, K. et al. 1958, Suppl. Proger. Theor. Phys., 6, 93. reduces for all stages. Shellard, R. C., Diazy, J. C. 1999, Simulation and perfor- The angle of detector’s field-of-view is 90% of flu- mance of a cosmic ray fluorescence detection, SicaApli- orescence light located in it, for different detector dis- cada e Instrumenta_c∼ao, vol. 14, no. 3, pp. 86Ð95. tances from the shower axis for our data was obtained Song, C. et al. 2000, Astropart. Phys., 14(1), 7Ð13.