A comprehensive study of shower to shower fluctuations

doi:10.1016/j.astropartphys.2010.11.001

Astroparticle Physics

Volume 34, Issue 6, January 2011, Pages 503-512

https://doi.org/10.1016/j.astropartphys.2010.11.001 Get rights and content

Abstract

By means of Monte Carlo simulations of extensive air showers (EAS), we have performed a comprehensive study of the shower to shower fluctuations affecting the longitudinal and lateral development of EAS. We split the fluctuations into physical fluctuations and those induced by the thinning procedure customarily applied to simulate showers at EeV energies and above. We study the influence of thinning on the calculation of the shower to shower fluctuations in the simulations. For thinning levels larger than R_thin = 10⁻⁵ − 10⁻⁶, the determination of the shower to shower fluctuations is hampered by the artificial fluctuations induced by the thinning procedure. However, we show that shower to shower fluctuations can still be approximately estimated, and we provide expressions to calculate them. The influence of fluctuations of the depth of first interaction on the determination of shower to shower fluctuations is also addressed.

Research highlights

► Influence of thinning procedure on shower fluctuations in simulations. ► Expression to estimate physical and artificial (thinning) shower fluctuations. ► Study of physical shower fluctuations at ground.

Introduction

Extensive air showers (EAS) have been studied over the last 70 years [1]. They result from the interaction in the atmosphere of high-energy protons and nuclei arriving from space. The product of these collisions are a set of secondary particles carrying a fraction of the primary energy. These secondaries move through the atmosphere and interact again generating new secondaries. The process continues, increasing the number of secondary particles, until their energies are too low to contribute to the generation of new particles. Particles reaching ground are sampled with arrays of detectors, and their properties are used to infer the properties of the primary initiating the shower. Measurements of the electron and muon density, of the arrival time of the particles at ground, and of the depth at which the shower has the maximum number of particles (X_max), give information on the arrival direction, primary energy, and on the mass of the primaries [1].

The complexity of the cascade phenomena, and the poor knowledge of the hadronic interactions at very high energy [2], make the experimental determination of the properties of the primaries very difficult. Moreover, primary particles with the same energy, mass and direction produce secondary particles with parameters that vary from shower to shower. This feature is called “shower to shower fluctuations”. An understanding of the shower to shower fluctuations will help to improve the interpretation of cosmic-ray data.

The calculation of shower to shower fluctuations can in principle be addressed with Monte Carlo simulations of extensive air showers. However, the number of particles that are produced in an air shower at ultra high energy (above ∼10¹⁸ eV) is so large (∼10¹⁰), that it is almost impossible to follow the propagation to ground level of all the secondaries in the Monte Carlo in a reasonable amount of time, or even to store the large amount of information produced. For this reason, a statistical sampling procedure called “thinning” [3] is used in the simulations. Thinning algorithms typically consist on propagating only a small, representative fraction of the total number of particles in the shower, assigning statistical weights to the sampled particles to compensate for the rejected ones. However, thinning algorithms introduce artificial fluctuations in the simulated showers, hampering the determination of the intrinsic, physical shower to shower fluctuations with Monte Carlo simulations. For this reason the study of fluctuations using Monte Carlo simulations is quite difficult and uncertain. This is of utmost importance in cosmic-ray physics, since an incorrect assumption on the shower to shower fluctuations can lead to systematic errors on the determination of the parameters of the primary particles.

In this work we address the problem of determining the true, physical shower to shower fluctuations in Monte Carlo simulations, and quantify the effect of thinning on their determination. We give expressions that allow the estimation of physical fluctuations from Monte Carlo simulations, even in the case of relatively strongly thinned showers. Other recent approaches which study the effect of artificial fluctuations due to the thinning procedure are given in [4], [5]

The paper is organized as follows: in Section 2 we describe the simulations performed in this work, and the thinning algorithm adopted. In Section 3 we identify the different sources of fluctuations in shower simulations. In Section 4 we perform a comprehensive study of the fluctuations in the longitudinal and lateral shower development, and give expressions that allow to separate physical shower to shower fluctuations from the artificial fluctuations induced by the thinning procedure, and we quantify for which thinning levels these expressions can be applied. In Section 5 we apply our results to study the behaviour of the physical fluctuations at ground. In Section 6, we quantify the influence on the shower to shower fluctuations of the depth of first interaction of the primary initiating the shower. Finally, we summarize our conclusions in Section 7. In the Appendix we give an explicit mathematical derivation of the expressions presented in Section 4.

Section snippets

The simulations

In this work we have used the air shower simulation program, AIRES [6], [7], along with the hadronic model QGSJET01 [8] to simulate proton and iron-induced showers with primary energy 10¹⁹ eV. As explained above, due to the large number of particles that are created in the simulation, AIRES includes a statistical sampling algorithm, that consists on propagating a small, representative fraction of the total number of particles, assigning a statistical weight w to the sampled particles to

Fluctuations in EAS

In a real shower or in a simulation of an EAS, there are a number of different fluctuations that can occur. Rather generally, we can make a simple classification as shown in Table 1.

“Physical fluctuations” are those due to physical processes in the shower. Here we split them into those due to the first interaction, and those occurring in the secondary interactions, as is customary, and because it has recently been suggested that “universal” shower properties may emerge when considering only the

Fluctuations of the longitudinal profile

In Fig. 1 we show the average longitudinal profile of the number of electrons (left panels) and muons (right panels) $\bar{N}$ , obtained in simulations of 100 proton-induced showers with E_p = 10¹⁹ eV, for thinning levels R_thin = 10⁻⁶ and 10⁻⁷ and θ = 0°, 60°. Also shown are the relative shower to shower fluctuations $σ / \bar{N}$ for electrons and muons. As it is well known [13], the relative fluctuation has a minimum close to the depth of shower maximum. Also and as it is apparent from the figures, the dependence of

Dependence of fluctuations at ground on the number of particles

Let us now consider the dependence of fluctuations on the size of the ring around a distance to the shower core r, where particles are collected in the simulation. Let us first consider how the density of particles is evaluated. If $\bar{N} (r, Δ r)$ is the average number of particles at distance r in a small bin Δr (here we assume cylindrical symmetry around the shower axis, but the argument does not depend on this simplification), then the density of particles ρ(r) can be defined as: $ρ (r) = {lim}_{Δ r \to 0} \bar{N} (r, Δ r)$

Dependence of shower to shower fluctuations on composition and depth of first interaction

In this Section we study the influence of the fluctuations in the depth of first interaction on the overall shower to shower fluctuations of the number of particles. In Fig. 11 we plot the relative fluctuations $σ / \bar{N}$ in 10¹⁹ eV proton and iron-induced showers. In all panels we show the results of our regular simulations, together with the results of a special set of simulations performed by fixing the depth of first interaction of the primary particle (proton or iron) at the value of its mean

Conclusions

In this work we have performed a comprehensive study of shower to shower fluctuations by means of Monte Carlo simulations of extensive air showers. An understanding of the shower to shower fluctuations will help to improve the interpretation of cosmic-ray data.

The determination of the true, physical shower to shower fluctuations is hampered by the thinning procedure necessary to simulate in a practical manner air showers at EeV energies and above. However, we have shown that the artificial

Acknowledgements

We thank V. Canoa, G. Rodriguez–Fernandez, T. Tarutina, I. Valiño and E. Zas for discussions and comments. P.M.H. was supported by Juan de la Cierva grant. We thank Centro de Supercomputación de Galicia (CESGA) for computer resources. This work was made possible with support from the Ministerio de Ciencia e Innovación, Spain under grant FPA 2007-65114 and Consolider CPAN; and of ALFA-EC funds in the framework of the HELEN (High Energy Physics Latin-American-European Network) project. J. A-M

References (18)

V.A. Kuzmin et al.
JETP Lett.
(2007)
M. Ave
Nucl. Instrs. Methods Phys. Res. A
(2007)
T.K. Gaisser
Cosmic Rays and Particle Physics
(1992)
M. Risse, et al., Proc. of 27th ICRC, Hamburg, Germany, 2001, pp....
J. Kempa et al.
J. Phys. G: Nucl. Part. Phys.
(1998)
M. Nagano et al.
Rev. Mod. Phys.
(2000)
C.A. Garcı´a Canal
Phys. Rev. D
(2009)
A.M. Hillas, Proc of the Paris Workshop on Cascade simulations, in: J. Linsley, A.M. Hillas (Eds.), 1981, pp....A.M. Hillas
Nucl. Phys. B (Proc. Suppl.)
(1997)
D.S. Gorbunov et al.
Phys. Rev. D
(2007)

There are more references available in the full text version of this article.

Cited by (7)

A likelihood method to cross-calibrate air-shower detectors
2015, Astroparticle Physics
Citation Excerpt :
Air showers with the same geometry a and energy E show a fluctuating size S at the ground. These fluctuations [9–11] are caused by random outcomes of the first few interactions of the air-shower development and possibly from sampling a random mass A from the mass-distribution of cosmic rays. The mass A is never exactly known event-by-event and therefore the dependency S(A) adds to the observed fluctuations of S.
We present a detailed statistical treatment of the energy calibration of hybrid air-shower detectors, which combine a surface detector array and a fluorescence detector, to obtain an unbiased estimate of the calibration curve. The special features of calibration data from air showers prevent unbiased results, if a standard least-squares fit is applied to the problem. We develop a general maximum-likelihood approach, based on the detailed statistical model, to solve the problem. Our approach was developed for the Pierre Auger Observatory, but the applied principles are general and can be transferred to other air-shower experiments, even to the cross-calibration of other observables. Since our general likelihood function is expensive to compute, we derive two approximations with significantly smaller computational cost. In the recent years both have been used to calibrate data of the Pierre Auger Observatory. We demonstrate that these approximations introduce negligible bias when they are applied to simulated toy experiments, which mimic realistic experimental conditions.
Effects of massive photons from the dark sector on the muon content in extensive air showers
2013, Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics
Inspired by recent astrophysical observations of leptonic excesses measured by satellite experiments, we consider the impact of some general models of the dark sector on the muon production in extensive air showers. We present a compact approximative expression for the bremsstrahlung of a massive photon from an electron and use it within Monte Carlo simulations to estimate the amount of weakly interacting photon-like massive particles that could be produced in an extensive air shower. We find that the resulting muon production is by many orders of magnitude below the average muon count in a shower and thus unobservable.
Sensitivity of the correlation between the depth of shower maximum and the muon shower size to the cosmic ray composition
2012, Astroparticle Physics
Citation Excerpt :
Furthermore, an accurate simulation of the lateral development of the shower is not required. This removes any potential inaccuracies introduced by thinning algorithms [13,14] (common to many 3-dimensional air shower simulation codes). The muon shower size must also be specified at a certain atmospheric depth.
The composition of ultra-high energy cosmic rays is an important issue in astroparticle physics research, and additional experimental results are required for further progress. Here we investigate what can be learned from the statistical correlation factor r between the depth of shower maximum and the muon shower size, when these observables are measured simultaneously for a set of air showers. The correlation factor r contains the lowest-order moment of a two-dimensional distribution taking both observables into account, and it is independent of systematic uncertainties of the absolute scales of the two observables. We find that, assuming realistic measurement uncertainties, the value of r can provide a measure of the spread of masses in the primary beam. Particularly, one can differentiate between a well-mixed composition (i.e., a beam that contains large fractions of both light and heavy primaries) and a relatively pure composition (i.e., a beam that contains species all of a similar mass). The number of events required for a statistically significant differentiation is ∼200. This differentiation, though diluted, is maintained to a significant extent in the presence of uncertainties in the phenomenology of high energy hadronic interactions. Testing whether the beam is pure or well-mixed is well motivated by recent measurements of the depth of shower maximum.
On the chemical composition of cosmic rays of highest energy
2011, Journal of Physics G: Nuclear and Particle Physics
The particle-shower simulation code CORSIKA 8
2023, arXiv
Computer time optimization in Extensive Air Shower simulations
2017, Proceedings of Science

View all citing articles on Scopus

View full text

A comprehensive study of shower to shower fluctuations

Abstract

Research highlights

Introduction

Section snippets

The simulations

Fluctuations in EAS

Fluctuations of the longitudinal profile

Dependence of fluctuations at ground on the number of particles

Dependence of shower to shower fluctuations on composition and depth of first interaction

Conclusions

Acknowledgements

JETP Lett.

Nucl. Instrs. Methods Phys. Res. A

J. Phys. G: Nucl. Part. Phys.

Rev. Mod. Phys.

Phys. Rev. D

Nucl. Phys. B (Proc. Suppl.)

Phys. Rev. D