CC BY
36
UDK 537.591.15
R.I. Raikin, A. A. Lagutin
Mass composition of cosmic rays from lateral distribution of electrons in extensive air showers
Radial scale factors of lateral distribution of electrons in extensive air showers are applied for primary composition estimation in the framework of scaling formalism. Consistent evidence of increasing average primary mass with energy above the knee was obtained from the analysis of KASCADE and MSU shower arrays data with weak sensitivity to the exact form of the fitting lateral distribution function and to the hadronic interaction model.
1. Introduction
The key problem for primary cosmic rays chemical composition deduction from extensive air showers (EAS) measurements is sensitivity of observables to properties of hadronic interactions. Various techniques for mass composition analysis are based on the following primary mass indicators available for ground-based air shower arrays (see, e.g., reviews [1, 2]):
* depth of shower maximum and its distribution;
- electron size to muon size ratio;
- muon densities far from the shower core;
- lateral distribution shape parameters of different shower components.
It is also worth to mention multi-component approaches combining several shower observables taking into account their correlations and fluctuations.
Unfortunately, in the framework of the above mentioned techniques it is rather impossible to suppress the influence of mode! uncertainties sufficiently for reliable conclusions on both the primary particle type on event-by-event basis and the average mass at a certain energy. In case of individual showers the problem is even more complex because of large fluctuations in shower development.
In this paper we investigate the possibilities for the improvement of primary mass estimation with the use of radial scale factors of lateral distribution function (LDF) of electrons assuming the scale in-variance of the electron LDF found in our ealier works (3—5J.
This work is supported in part by Russian Foundation (or Basic Research, project N07-02-01154-a and MK-2873.2007.2 grant.
This paper is an extended version of contribution to XV International Symposiums for Very High Energy Cosmic Ray Interactions, Paris, 2008
2. Scaling formalism
One of the basic EAS quantities necessary for main shower parameters reconstruction is the lateral distribution of charged particles in some experimentally accessible radial distance range. The exact form of lateral distribution function is still uncertain. The majority of analytical parameter-izations of LDF of different EAS components is traditionally based on the well known Nishimura-Kamata-Greizen (NKG) function [6]:
p{r;E0,s) =
N(E0, s)_r(4.5 -s)
ñg 27rr(s)r(4.5 - 2s) X s—4,5
r
fio
r
l + Ro
0)
Here p(r;Eo,s) is the particle density at radial distance r from the core position in shower with primary energy Eq and the age parameter s, N(Eq,s) - total number of particles at the observation depth, fi0 - shower scale radius, which does not depend on primary particle type and energy (originally - the Moliere unit).
Various modifications of NKG form, such as introducing an additional fixed or age-dependent scale coefficient or a local age parameter s(r) and also generalizations of the function by using third power-law term were suggested.
A different theoretically motivated approach (scaling formalism) was proposed in our papers [3-5]:
^•"-^'Gwko)- (2)
where t is the observation depth. The scaling function F(x) can be represented in the following form:
F(x) = Cx~a{ 1 + x)-{0~a){l + (x/lO)^)-5. (3)
For electron LDF parameters of function (3) are determined according simulations as C = 0.28, a =
1.2, /3 = 4.53, 7 = 2.0, S = 0.6, R0 mean square radius of electrons:
Rn
root
RL(Eo,t)
2 7T
N(E0,t)
oo /
r3pe(r-E0,t)dr. (4)
According to our calculations scaling function allows to reproduce electron densities within 10% uncertainty for E0 = (1014 - 1020) eV, t = (600 -1030) g/cm2, x = (0.05 - 25). The last condition corresponds to the radial distance range from r ~ (5-10) m to r ~ (2.5-4) km depending on the shower age (see Table 1). Thought this limitation makes scaling approach inadequate for shower size and core position estimation, it describes well the shape of LDF measured by ground-based shower arrays in wide radial distance range and is useful to both compact and large ground-based air shower arrays data.
It is very important that the scaling property (2), as such, is almost insensitive to variations of basic parameters of hadronic interaction model implemented in simulations [5].
Scaling formalism was successfully applied to the description of experimental data of KAS-CADE [7] and AGASA [4, 5] while its applications to Yakutsk and MSU air shower arrays data face problems [8-10]. It is also worth to mention that function (3) is used by KASCADE-Grande Collaboration as muon LDF [11]. The corresponding set of parameters is C = 0.28, a - 0.69, (3 = 3.08, 7 - 2.0, 6 = 1.0, Rq = 320 m.
3. Radial scale factor as primary mass estimator
The method for mean primary mass deduction based on scaling formalism was introduced in [5, 12]. Root mean square radii at sea level for vertical air showers generated by protons and iron nuclei in the energy range 10165 — 1020 eV corresponding to different modern interaction models are shown in Fig.I. The calculations were made using the relationship [5] between the root mean square radius and the average depth of shower maximum imax based on the shower universality concept. Different models predictions [13] for £max were implemented. It is seen that the root mean square radius can be used as a primary mass indicator with relatively weak sensitivity to hadronic interaction model.
Unfortunately, the direct estimation of i?ms from the experimental data is difficult so far as it demands high-precision reconstruction of the lateral distribution of an electron component in a wide radial distance range.
160 150 140 130 120 110 100
QGSJET 01 -
QGSJET 11-3 -----------'
'"'"-vj^^jron SIBYLL 2.1 ..............
EPOS 1.6 -----------.
prolon^^^
16.5 17 17.5 18 16.5 19 19.5 20 lg E (eV)
Figure 1. Energy dependence of the root mean square radii for different hadronic models.
<
Figure 2. Functions A(x) (1) and dA{x)/dx (2). See text for details.
This is illustrated in figure 2, where the importance of particles spread at different x = r/iîms areas for the formation of the root mean square radius is shown by plotting function
A(x) =
•ñms
0
1/2
(5)
where Ris the truncated, root mean square ra-
Table 1
Values of r = x Rms (m) corresponding to x — 0.05, 0.24 and 25.0 for proton-induced EAS with E0 = 1014, 1018 eV at t = 614 g/cm2 (1) and t = 1030 g/cm2 (2).
X 1014 eV 1018 eV
1 2 1 2
0.05 7.7 8.3 5.0 6.0
0.24 37 40 24 29
25.0 3854 4135 2492 2982
ig£o, GeV
lgNe = (3.9 - 4.3)--\gNe = (4.7-5.1)------ lgNe = (5.9 - 6.3).............. lgNe = (6.7 - 7.1)-------
Figure 3. Primary energy distributions in four selected bins Ne, simulated according to KASCADE shower classification procedure [7] (distribution are renormalized to equal height of maximum for convenience of comparison).
dius, integrated only over the limited radial distance range, here from r\ = 0 to t'2 = xRms.
Our approximation of scaling function F(x) (3) is used to get the data. In table 1 we show core distances corresponding to x — 0.05, 0.24 and 25.0 for proton-induced EAS with E0 = 1014, 1018 eV at mountain level and sea level.
The value of x ~ 0.24 corresponds to the maximum of function dA(x)/dx, which is also shown in figure 2, and represents the distance formally most important for the root mean square radius without taking into consideration any concrete experimental conditions.
4. Influence of shower fluctuations in case of compact arrays
Another one problem for adequate Rms estimation, apart from the deficiency in radial distance range well covered by the detectors, is the influence of the experimental procedures of the shower classification and data processing on the shape of lateral distribution. This effect can also be described in the framework of scaling formalism (5), but the variation of the root mean square radius compared with the data obtained theoretically for the electron LDF should be taken into account.
In case of relatively low primary energies, when shower selection is made by the total number of electrons, e.g. for example for KASCADE and Moscow State University (MSU) air shower arrays the essential factor is influence of large fluctuations in individual shower development.
To examine this factor thoroughly we made simulations of extensive air showers initiated by protons and iron nuclei of vertical incidence assuming power-law differential energy spectrum of pri-
maries with exponent ai — 2.62. c*2 = 3.02 and also with sharp knee from ai to a<2 at Eq = 1065 GeV. We used the semi-analytical code [5] with full Monte-Carlo treatment of hadronic part of cascade based on quark-gluon string model and analytical expressions of pure electromagnetic sub-showers keeping al! the basic sources of fluctuations. The fluctuations of different EAS components calculated by our code are in good agreement with CORSlKA/QGSjet results.
According to [7] we simulated shower classification procedure used at KASCADE array and evaluated lateral distributions and root mean square radiuses of electrons in eight bins of shower size. The number of showers in each bin amounts from ~ 5000 for lower energies (lgJVe = 3.9 - 4.3) to ~ 1000 for higher energies (lgiVe = 6.7 - 7.1).
The primary energy distributions in four from eight bins of shower size is shown in fig. 3. One can see that the energies largely overlap in different bins though the selected bins are not neighboring.
We evaluated the correction factors defined as the ratio of root mean square radius calculated for certain shower size bin to that for corresponding average primary energy: K = These
correction factors for vertical proton initiated showers at sea level calculated with above mentioned assumptions about primary energy spectrum are shown in fig. 4. It is clear, that values of K approach to 1 with energy as shower fluctuations decrease. The same effect takes place for a heavier primary nuclear or for smaller shower size bins. At the same time the correction, which should be made for an adequate comparison of theoretical and experimentally estimated root mean square radiuses is essential for all considered shower size bins.
4 4.5 5 5.5 6 6.5 7
Ik N..
Figure 4. Correction factors K = (Rml/^ms) f°r proton initiated vertical EAS at sea level assuming different primary energy spectrum exponents: c*i = 2.62 (triangles with dotted approximation curve), c*2 = 3.02 (squares with dashed curve), spectrum with the knee at Eq — 106-5 GeV (solid circles with solid curve). See text for details.
0 50 100 150 200 r, m
lg Ne: 3.9-4.3 •—s—- 5.9-6.3 •
4.3-4.7 —»..........6.3-6.7' v
4.7-5,1 > ♦•••• 6.7-7.1 ■--»-
5.1-5.5 ■ Scaling function fit —— 5.5-5.9
Figure 5. KASCADE array data for electron LDFs |7] fitted by scaling function (2), (3) with Ne and Rq as free parameters.
5. Comparisons with experimental data
Assuming the validity of scaling approach we have performed fitting of experimental lateral distributions of electrons obtained by KASCADE [7] and MSU air shower array [9] using using expression (2) with three different scaling functions:
1) theoretically proved function (3) with mean square radius of electrons as radial scale parameter;
2) modified NKG-function [7] with fixed shower age parameter (s=1.65) and variable R0-,
3) polynomial function:
F(x) =C,exp|^at(lnx)l| , (6)
with n = 4 and parameters a* being fixed for all bins independently from Rq.
It is important that in all cases iteration procedure was implemented for discrimination of experimental data in order to use data at distances where scaling formalism is valid (see Section 2.) and also to use identical radial distance ranges with respect to scaling variable x = t/Rq for all energies.
Results of fitting are summarized in Tab. 2, 3. In Fig. 5, 6 we show the experimental data together with scaling functions (3).
All the fitting functions give satisfactory overall fit of experimental data of both KASCADE and MSU array. Though polynomial and NKG functions give better accuracy in considered radial distance ranges, they both lead to incorrect predictions for extremely large core distances, while the-
oretically motivated function (3) remains realistic up to r ~ 25R ms'
It is not surprising, that using of different fitting functions leads to significantly different values of Rq. An additional bias in R0 can be related with the insufficiency of radial distance range well covered by the array or some other systematic errors in data processing. So it is worth to compare the rate of change of radial scale factors with energy which obviously reflects the rate of change of mean primary mass.
1000 100 10
C4
E
ex
1
0.1
0.01
Figure 6. MSU array data for charged particles LDFs [8, 9] fitted by scaling function (2), (3) with Ne and Ro as free parameters.
Table 2
Radial scale factors (flo ± ¿Ro) obtained by fitting of KASCADE experimental LDF [7] by
different scaling functions.
IgWe Function (2), (3) Modified NKG (s=1.65) Polynomial (6)
3.9-4.3 146.8 ±2.5 29.79 ±3.0 io-3 174.4 ± 1.6
4.3 - 4.7 134.3 ± 2.2 26.93 ±2.1 10"1 156.6 ± 1.2
4.7 - 5.1 125.0 ± 1.5 25.17 ± 1.9 io-1 146.0 ± 1.2
5.1 - 5.5 122.6 ± 1.3 23.92 ± 1.3 10-i 138.6 ±0.9
5.5 - 5.9 122.8 ± 1.6 23.65 ± 1.1 10-1 137.8 ±0.5
5.9-6.3 122.6 ±2.1 24.03 ± 1.6 10-1 139.8 ±0.9
6.3 - 6.7 125.4 ± 2.3 24.70 ±2.1 10"1 141.0 ± 1.2
6.7 - 7.1 130.6 ± 1.8 25.55 ±3.5 10-1 145.1 ±2.1
Table Radial scale factors (Rq ± SRo) obtained by fitting of MSU array experimental LDF [9] by different scaling functions.
Ig^e Function (2), (3) Modified NKG (s=1.65) Polynomial (6)
5.2-5.4 189.3 ± 8.9 33.4 ± 1.2 195.0 ±6.2
5.6 - 5.8 182.9 ± 12.5 32.2 ± 1.7 188.2 ±9.4
6.0-6.2 192.9 ± 14.7 33.9 ± 1.7 198.0 ±9.3
6.4-6.6 201.1 ± 11.8 35.6 ± 1.9 207.0 ± 10.9
190
180
170
E 160 * 150
140 130 120
.5
Figure 7. Superposed radial scale factors obtained under different assumptions about electron LDF in comparison with theoretically calculated root mean square radii (QGSjet) for different primaries.
#
KASCADE: Scaling function fit N KG-fit (s=1.65) Polynomial fit MSU: Scaling function fit NKG-fit (s=1.65) Polynomial fit ms: proton iron
-e--e-
4.5
5.5 lg Ne
6.5
In fig. 7 values of Rq superposed with each other by the appropriate factors are shown in comparison with root mean square radius calculated for vertical proton and iron initiated showers at sea level taking into account the correction factor K for primary spectrum with the knee. As it is seen from the figure the variation of Rq with energy obtained using different functions gives a consistent evidence of increasing average primary mass above the knee.
6. Conclusions
Scaling approach for primary composition deduction from lateral distribution of charged particles was applied to experimental data of KASCADE and MSU arrays.
It is shown that irrespective of the exact form of lateral distribution function used for experimental data fitting scaling formalism remains valid and thus radial scale factors can be considered as primary mass indicators.
In case of relatively low primary energies correction by a factor K — (R^/R^) should be
made when comparison of lateral distributions of electrons measured by compact ground-based experimental arrays to LDFs calculated theoretically for fixed primary energy is carried out.
Absolute values of radial scale factor Rq contain systematic errors and could be biased depending on the form of lateral distribution function chosen for experimental data processing and final fitting. However, if one concerns the rate of change of Rq with primary energy, which is a good measure for primary mass composition variation, then different scaling functions used in the framework of scaling formalism do not contradict each other.
Basic analysis using different assumptions about the form of scaling LDF leads to consistent model insensitive conclusion that average primary particle mass above the knee increases with energy.
As charged particles LDF far from the shower core is a superposition of electron and muon distributions of different form, two-component method can be suggested for higher energies considering radial scale factor of electrons and parameters of muon LDF separately.
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