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The implication of charged particle lateral distribution function for extensive air shower studies
Nuclear Physics B (Proc. Suppl.) 175–176 (2008) 334–337 www.elsevierphysics.com
The implication of charged particle lateral distribution function for extensive air shower studies Yu.A. Fomina , N.N. Kalmykova, J. Kempab , G.V. Kulikova , V.P. Sulakova a D.V. Skobeltsyn Institute of Nuclear Physics of M.V.Lomonosov Moscow State University, Moscow 119992, Russia b
Warsaw University of Technology Off Campus Plock, 09 400 Plock, Poland
The knowledge of charged particle lateral distribution function (LDF) is of prime importance in extensive air shower (EAS) investigations. This function is necessary for the determination of the total number of particles as well as some other classification parameters. The Nishimura-Kamata-Greisen (NKG) function is being actively employed by many researchers in spite of the fact that it was derived under rather crude assumptions (in so-called B Approximation of the electromagnetic cascade theory). Our paper discusses the dependence of the EAS size spectrum on the LDF form adopted and compares two LDFs: the traditional NKG-function and the scaling function suggested recently. Prominence is given to the EAS MSU data but the results of other EAS arrays (AGASA, Yakutsk and KASCADE) are also considered.
1. Introduction One of the important problems of extensive air shower (EAS) studies is the investigation of charged particle lateral distribution function (LDF). This function is necessary for determination of the experimental size spectrum of showers as well as ρ600 spectrum (ρ600 — charged particle density at 600 m from axis). The Nishimura-Kamata-Greisen (NKG) function is still in use (see [1]) in spite of the fact that it was derived in the framework of B Approximation of the electromagnetic cascade theory for infinitely high primary energy. One would think that at present there are no arguments in favour of the NKG-function. On the contrary, it was shown in [2–4] and other works that this function does not describe properly the electron LDF even in photon showers. Moreover, the nuclear-cascade process plays the main role in shower development and it should influence on the form of LDF. However the NKGapproximation describes experimental LDF with reasonable accuracy, although concrete parameters on which this function depends (shower age s and Molier radius r0 ) should be taken as ad0920-5632/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2007.11.025
justable values. On the other hand the analysis of theoretical LDF calculations performed with taking into account nuclear-cascade process [5] shows that the electron LDF in EAS may be described in the framework of scaling formalism obtained earlier for pure electromagnetic showers [6]. As shown in [5] the scaling formalism allows to reproduce the experimental LDF obtained at AGASA array rather well. The aim of this paper — is it possible to use the scaling formalism in form [5] for description of experimental data of the EAS MSU array? 2. Experiment The EAS MSU data were used to derive the experimental electron LDF, compare it with theoretical predictions, and investigate also an influence of the LDF form on the EAS size spectrum. The EAS MSU array is described in [7]. The array covers an area of approximately 0.5 km2 and includes 77 detectors of charged particle density ρ used for determination of the EAS size Ne . Each detector consists of Geiger counters of various areas to measure an interval of densities from 0.5
Yu.A. Fomin et al. / Nuclear Physics B (Proc. Suppl.) 175–176 (2008) 334–337
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up to ∼1500 particles m−2 . The arrival direction of a shower was determined with scintillator detectors. Our previous analysis showed that experimental LDF’s are described rather well by the function proposed by Greisen [8] for electron LDF’s in EAS. However, the best agreement can be achieved for the empirical LDF having the more complicated form (see [9]). In the paper presented we analyze experimental data using the NKG-function ρ(r) = Ne C(s)r0−2 (
r s−2 r ) (1 + )s−4.5 r0 r0
and also the scaling LDF function suggested in [5]. This function has the following form −1.2 −3.33 Ne · 0.28 r r ρ(r) = 1+ 2 Rms Rms Rms 2 −0.6 r 1+ 10 · Rms and is a function of the mean square radius Rms only. From calculations [7] it follows that the mean square radius depends on the average cascade curve maximum position and consequently on the particle primary energy. Normalized electron LDF’s in proton shower are given in Fig. 1 for primary energy range 1015 – 1017 eV which is investigated with the EAS MSU array. Fig. 1 shows that the LDF form depends rather weakly on primary energy. In particular, the difference of LDF’s does not exceed 20% if one considers the interval 3–300 m (the most essential one for the EAS MSU array). We also plot in Fig. 1 the NKG-function for s=1.2 which is more flat at small distances from shower axis. Then we compared theoretical LDF’s with experimental charged particle LDF’s for the wide range of sizes, Ne = 105 ÷ 3 × 107 . Showers in this range were divided into narrow size intervals ΔlgNe = 0.2. The size Ne of every shower was determined by maximum likelihood method with the NKG-function taken as an a priori LDF function. In Fig. 2 the experimental and calculated LDF’s are compared for several ΔlgNe intervals.
Figure 1. Normalized LDF’s; LDF NKG with s = 1.2 (1), LDF [5] for Rms = 153, 141, 129 m; (2, 3, 4) corresponding to E0 = 1015 , 1016 , 1017 eV.
The required value of Rms in each interval was determined in the following manner. Knowing the average Ne for NKG- function in each interval we obtained the primary energy, E0 , in the framework of the QGSJET-model [10] and then derived Rms according to [5]. Theoretical Ne values for each interval were found by the least-squares method. The results of our comparison show that the calculated functions agree with experimental data rather poorly, especially for small distances from the shower axis. It is clear from Fig. 2 that the NKG-functions agree with experimental LDF much better. To obtain the size spectrum using the LDF proposed in [5] it is desirable to determine Ne in each shower. The spread of Rms obtained is considerably greater than theoretical Rms predictions. In many cases the value Rms was not determined practically. On the other hand the parameter, s, in individual showers is defined with a good accuracy when the NKG-function is used. That is why we obtained the size spectrum for the LDF
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Yu.A. Fomin et al. / Nuclear Physics B (Proc. Suppl.) 175–176 (2008) 334–337
3. Discussion
Figure 2. LDF’s for different Ne LDF [5] — solid line; LDF NKG — dashed lines; 1 — lgNe = 5.2– 5.4; 2 — lgNe = 5.6–5.8; 3 — lgNe = 6.0–6.2; 4 — lgNe = 6.4–6.6.
with fixed value of Rms (the values of Rms corresponding to energies 1015 ÷ 1017 eV were taken). We compare the size spectra for nearly vertical showers with zenith angles less 18◦ . This comparison showed that the spectrum obtained for the theoretical LDF [5] has a form essentially differing from other data, that may be found in the literature, especially for the energy range before the knee. The spectrum obtained for the NKGfunction has the knee at Ne ∼ 4 × 105 , and the value of the spectral index change is 0.50–0.02 in agreement with our previous results [11]. The indices of power differential spectra before and after the knee are shown in the table below.
LDF [5] NKG
Before the knee -2.12 -2.37
After the knee -2.81 -2.87
Our analysis showed that the experimental data of the EAS MSU array are not described adequately by the LDF of charged particle in EAS obtained in [5] in the framework of scaling formalism. However this negative result should not be used as an argument against the idea of scaling behaviour of electron LDF in electron-photon showers. On the contrary, results [12] show the good agreement between the experimental LDF of KASCADE array and the calculated one obtained in the framework of the scaling approximation [6] for partial electron-photon cascades. It is worth remembering that in this case the scaling LDF is parametrized by Rms also, but Rms depends on age parameter, s, of electron-photon showers. As shown in [5] the scaling LDF describes AGASA data well but does not describe Yakutsk data. Similar conclusions were made in [13], where it was shown that the theoretical LDF calculated for the Yakutsk array disagrees with the experimental data, while for KASCADE, AGASA and EAS MSU arrays analogous calculations give satisfactory agreement with experimental LDF’s. It should be pointed out that the EAS MSU array data, obtained with Geiger-Muller counters, correspond to the LDF of charged particles more closely than for arrays with scintillator detectors, as one must take into account the difference between the LDF of energy deposit in scintillator and the LDF of charged particles. 4. Acknowledgement This work of Yu.A.F., N.N.K., G.V.K., and V.P.S has been supported by the Federal scientific and technical program, contract 02.452.11.7053 and Russian Foundation for Basic Research grant 05-02-16401.
Yu.A. Fomin et al. / Nuclear Physics B (Proc. Suppl.) 175–176 (2008) 334–337
REFERENCES 1. K.Greisen, Extensive Air Showers. In: Progress in Cosmic Ray Physics, v.III (Ed. J. Wilson), 1956, Amsterdam. 2. Allan H.R. et al., Proc. 14th ICRC, Munchen, 1975, V.6, P.3071. 3. Hillas A.M., Lapikens J., Proc. 15th ICRC, Plovdiv, 1977, V.8, P.460. 4. Lagutin A.A. et al., Proc. 16th ICRC, Kyoto, 1979, V.7, P.18. 5. Raikin R.I., Lagutin A.A., Inoue N. and Misaki A., Proc 27th ICRC, Hamburg, 2001, V.1, P.290. 6. Raikin R.I., Lagutin A.A., Inoue N. and Misaki A., Proc 27th ICRC, Hamburg, 2001, V.1, P.294.
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7. Vernov S.N.et al., Proc. 16th ICRC, Kyoto, 1979, V.8, P.129. 8. Greisen K., Ann. Rev. Nucl. Sci., 1966, V.10, P.63. 9. Fomin Yu.A. et al., Proc. 28th ICRC, Tsukuba, 2003, V.1, P.119. 10. Kalmykov N.N. et al., Nucl. Phys. B (Proc, Suppl.), 1997, V.52B, P.17 11. Fomin Yu.A. et al., Proc. 22nd ICRC, Dublin, 1991, V.2, P.85. 12. Kalmykov N.N. et al., Nucl. Phys. B (Proc, Suppl.), 2003, V.122B, P.380. 13. Sleptsova V.R. et al., Nucl. Phys. B (Proc, Suppl.), 2003, V.122B, P.255
The implication of charged particle lateral distribution function for extensive air shower studies Yu.A. Fomina , N.N. Kalmykova, J. Kempab , G.V. Kulikova , V.P. Sulakova a D.V. Skobeltsyn Institute of Nuclear Physics of M.V.Lomonosov Moscow State University, Moscow 119992, Russia b
Warsaw University of Technology Off Campus Plock, 09 400 Plock, Poland
The knowledge of charged particle lateral distribution function (LDF) is of prime importance in extensive air shower (EAS) investigations. This function is necessary for the determination of the total number of particles as well as some other classification parameters. The Nishimura-Kamata-Greisen (NKG) function is being actively employed by many researchers in spite of the fact that it was derived under rather crude assumptions (in so-called B Approximation of the electromagnetic cascade theory). Our paper discusses the dependence of the EAS size spectrum on the LDF form adopted and compares two LDFs: the traditional NKG-function and the scaling function suggested recently. Prominence is given to the EAS MSU data but the results of other EAS arrays (AGASA, Yakutsk and KASCADE) are also considered.
1. Introduction One of the important problems of extensive air shower (EAS) studies is the investigation of charged particle lateral distribution function (LDF). This function is necessary for determination of the experimental size spectrum of showers as well as ρ600 spectrum (ρ600 — charged particle density at 600 m from axis). The Nishimura-Kamata-Greisen (NKG) function is still in use (see [1]) in spite of the fact that it was derived in the framework of B Approximation of the electromagnetic cascade theory for infinitely high primary energy. One would think that at present there are no arguments in favour of the NKG-function. On the contrary, it was shown in [2–4] and other works that this function does not describe properly the electron LDF even in photon showers. Moreover, the nuclear-cascade process plays the main role in shower development and it should influence on the form of LDF. However the NKGapproximation describes experimental LDF with reasonable accuracy, although concrete parameters on which this function depends (shower age s and Molier radius r0 ) should be taken as ad0920-5632/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2007.11.025
justable values. On the other hand the analysis of theoretical LDF calculations performed with taking into account nuclear-cascade process [5] shows that the electron LDF in EAS may be described in the framework of scaling formalism obtained earlier for pure electromagnetic showers [6]. As shown in [5] the scaling formalism allows to reproduce the experimental LDF obtained at AGASA array rather well. The aim of this paper — is it possible to use the scaling formalism in form [5] for description of experimental data of the EAS MSU array? 2. Experiment The EAS MSU data were used to derive the experimental electron LDF, compare it with theoretical predictions, and investigate also an influence of the LDF form on the EAS size spectrum. The EAS MSU array is described in [7]. The array covers an area of approximately 0.5 km2 and includes 77 detectors of charged particle density ρ used for determination of the EAS size Ne . Each detector consists of Geiger counters of various areas to measure an interval of densities from 0.5
Yu.A. Fomin et al. / Nuclear Physics B (Proc. Suppl.) 175–176 (2008) 334–337
335
up to ∼1500 particles m−2 . The arrival direction of a shower was determined with scintillator detectors. Our previous analysis showed that experimental LDF’s are described rather well by the function proposed by Greisen [8] for electron LDF’s in EAS. However, the best agreement can be achieved for the empirical LDF having the more complicated form (see [9]). In the paper presented we analyze experimental data using the NKG-function ρ(r) = Ne C(s)r0−2 (
r s−2 r ) (1 + )s−4.5 r0 r0
and also the scaling LDF function suggested in [5]. This function has the following form −1.2 −3.33 Ne · 0.28 r r ρ(r) = 1+ 2 Rms Rms Rms 2 −0.6 r 1+ 10 · Rms and is a function of the mean square radius Rms only. From calculations [7] it follows that the mean square radius depends on the average cascade curve maximum position and consequently on the particle primary energy. Normalized electron LDF’s in proton shower are given in Fig. 1 for primary energy range 1015 – 1017 eV which is investigated with the EAS MSU array. Fig. 1 shows that the LDF form depends rather weakly on primary energy. In particular, the difference of LDF’s does not exceed 20% if one considers the interval 3–300 m (the most essential one for the EAS MSU array). We also plot in Fig. 1 the NKG-function for s=1.2 which is more flat at small distances from shower axis. Then we compared theoretical LDF’s with experimental charged particle LDF’s for the wide range of sizes, Ne = 105 ÷ 3 × 107 . Showers in this range were divided into narrow size intervals ΔlgNe = 0.2. The size Ne of every shower was determined by maximum likelihood method with the NKG-function taken as an a priori LDF function. In Fig. 2 the experimental and calculated LDF’s are compared for several ΔlgNe intervals.
Figure 1. Normalized LDF’s; LDF NKG with s = 1.2 (1), LDF [5] for Rms = 153, 141, 129 m; (2, 3, 4) corresponding to E0 = 1015 , 1016 , 1017 eV.
The required value of Rms in each interval was determined in the following manner. Knowing the average Ne for NKG- function in each interval we obtained the primary energy, E0 , in the framework of the QGSJET-model [10] and then derived Rms according to [5]. Theoretical Ne values for each interval were found by the least-squares method. The results of our comparison show that the calculated functions agree with experimental data rather poorly, especially for small distances from the shower axis. It is clear from Fig. 2 that the NKG-functions agree with experimental LDF much better. To obtain the size spectrum using the LDF proposed in [5] it is desirable to determine Ne in each shower. The spread of Rms obtained is considerably greater than theoretical Rms predictions. In many cases the value Rms was not determined practically. On the other hand the parameter, s, in individual showers is defined with a good accuracy when the NKG-function is used. That is why we obtained the size spectrum for the LDF
336
Yu.A. Fomin et al. / Nuclear Physics B (Proc. Suppl.) 175–176 (2008) 334–337
3. Discussion
Figure 2. LDF’s for different Ne LDF [5] — solid line; LDF NKG — dashed lines; 1 — lgNe = 5.2– 5.4; 2 — lgNe = 5.6–5.8; 3 — lgNe = 6.0–6.2; 4 — lgNe = 6.4–6.6.
with fixed value of Rms (the values of Rms corresponding to energies 1015 ÷ 1017 eV were taken). We compare the size spectra for nearly vertical showers with zenith angles less 18◦ . This comparison showed that the spectrum obtained for the theoretical LDF [5] has a form essentially differing from other data, that may be found in the literature, especially for the energy range before the knee. The spectrum obtained for the NKGfunction has the knee at Ne ∼ 4 × 105 , and the value of the spectral index change is 0.50–0.02 in agreement with our previous results [11]. The indices of power differential spectra before and after the knee are shown in the table below.
LDF [5] NKG
Before the knee -2.12 -2.37
After the knee -2.81 -2.87
Our analysis showed that the experimental data of the EAS MSU array are not described adequately by the LDF of charged particle in EAS obtained in [5] in the framework of scaling formalism. However this negative result should not be used as an argument against the idea of scaling behaviour of electron LDF in electron-photon showers. On the contrary, results [12] show the good agreement between the experimental LDF of KASCADE array and the calculated one obtained in the framework of the scaling approximation [6] for partial electron-photon cascades. It is worth remembering that in this case the scaling LDF is parametrized by Rms also, but Rms depends on age parameter, s, of electron-photon showers. As shown in [5] the scaling LDF describes AGASA data well but does not describe Yakutsk data. Similar conclusions were made in [13], where it was shown that the theoretical LDF calculated for the Yakutsk array disagrees with the experimental data, while for KASCADE, AGASA and EAS MSU arrays analogous calculations give satisfactory agreement with experimental LDF’s. It should be pointed out that the EAS MSU array data, obtained with Geiger-Muller counters, correspond to the LDF of charged particles more closely than for arrays with scintillator detectors, as one must take into account the difference between the LDF of energy deposit in scintillator and the LDF of charged particles. 4. Acknowledgement This work of Yu.A.F., N.N.K., G.V.K., and V.P.S has been supported by the Federal scientific and technical program, contract 02.452.11.7053 and Russian Foundation for Basic Research grant 05-02-16401.
Yu.A. Fomin et al. / Nuclear Physics B (Proc. Suppl.) 175–176 (2008) 334–337
REFERENCES 1. K.Greisen, Extensive Air Showers. In: Progress in Cosmic Ray Physics, v.III (Ed. J. Wilson), 1956, Amsterdam. 2. Allan H.R. et al., Proc. 14th ICRC, Munchen, 1975, V.6, P.3071. 3. Hillas A.M., Lapikens J., Proc. 15th ICRC, Plovdiv, 1977, V.8, P.460. 4. Lagutin A.A. et al., Proc. 16th ICRC, Kyoto, 1979, V.7, P.18. 5. Raikin R.I., Lagutin A.A., Inoue N. and Misaki A., Proc 27th ICRC, Hamburg, 2001, V.1, P.290. 6. Raikin R.I., Lagutin A.A., Inoue N. and Misaki A., Proc 27th ICRC, Hamburg, 2001, V.1, P.294.
337
7. Vernov S.N.et al., Proc. 16th ICRC, Kyoto, 1979, V.8, P.129. 8. Greisen K., Ann. Rev. Nucl. Sci., 1966, V.10, P.63. 9. Fomin Yu.A. et al., Proc. 28th ICRC, Tsukuba, 2003, V.1, P.119. 10. Kalmykov N.N. et al., Nucl. Phys. B (Proc, Suppl.), 1997, V.52B, P.17 11. Fomin Yu.A. et al., Proc. 22nd ICRC, Dublin, 1991, V.2, P.85. 12. Kalmykov N.N. et al., Nucl. Phys. B (Proc, Suppl.), 2003, V.122B, P.380. 13. Sleptsova V.R. et al., Nucl. Phys. B (Proc, Suppl.), 2003, V.122B, P.255