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A Heitler Model of Extensive Air Showers

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A Heitler Model of Extensive Air Showers Astroparticle Physics 22 (2005) 387–397 www.elsevier.com/locate/astropart A Heitler model of extensive air showers J. Matthews * Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803, USA Department of Physics, Southern University, Baton Rouge, LA 70813, USA Received 8 August 2004; received in revised form 3September 2004; accepted 13September 2004 Available online 26 October 2004 Abstract A simple, semi-empirical model is used to develop the hadronic portion of air showers in a manner analogous to the well-known Heitler splitting approximation of electromagnetic cascades. Various characteristics of EAS are plainly exhibited with numerical predictions in good accord with detailed Monte Carlo simulations and with data. Results for energy reconstruction, muon and electron sizes, the elongation rate, and for the effects of the atomic number of the primary are discussed. Ó 2004 Elsevier B.V. All rights reserved. PACS: 13.85.Tp; 95.85.Ry; 96.40.Pq Keywords: Cosmic rays; Extensive air showers; Simulations 1. Introduction hampered by limited knowledge of interaction cross-sections and particle production at high Extensive air showers develop in a complex way energies. as a combination of electromagnetic cascades and Before the era of high-speed computing, Heitler hadronic multiparticle production. It is necessary presented a very simple model of electromagnetic to perform detailed numerical simulations of air (EM) cascade development [1]. He and others in showers to infer the properties of the primary cos- that time (notably Rossi [2]) also developed more mic rays that initiate them. But simulations are a sophisticated analytical tools, which included challenge since the number of charged particles more physical effects. Such approaches, past and in a high energy shower can be enormous, perhaps present, are well described by Gaisser [3]. exceeding 1010. The design of algorithms is also We will consider here HeitlerÕs simplest concep- tion of EM cascades and extend it to the case * Address: Department of Physics and Astronomy, Louisi- of extensive air showers. The purpose of using ana State University, Baton Rouge, LA 70803, USA. a very simple model is to show plainly the physics E-mail address: [email protected] involved. It cannot replace fully detailed 0927-6505/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.astropartphys.2004.09.003 388 J. Matthews / Astroparticle Physics 22 (2005) 387–397 simulations. Nevertheless, HeitlerÕs EM model pre- sumed to decay to muons which are observed at dicted accurately the most important features of the ground. electromagnetic showers. This first approximation assumes that interac- HeitlerÕs model (Fig. 1a) has e+,eÀ, and pho- tions are perfectly inelastic, with all the energy tons undergoing repeated two-body splittings, going into production of new pions. We will study either one-photon bremsstrahlung or e+eÀ pair the more realistic case which includes a leading production. Every particle undergoes a splitting particle carrying away a significant portion of the after it travels a fixed distance related to the radi- energy later (Section 4). ation length. After n splittings there are 2n total The important difference between a hadronic particles in the shower. Multiplication abruptly cascade and a pure EM shower is that a third of ceases when the individual e± energies drop below the energy is ‘‘lost’’ from new particle production e the critical energy nc, where average collisional en- at each stage from p decay. Thus the total energy ergy losses begin to exceed radiative losses. of the initiating particle is divided into two chan- This simplified picture does not capture accu- nels, hadronic and electromagnetic. The primary rately all details of EM showers. But two very energy is linearly proportional to a combination important features are well accounted for: the final of the numbers of EM particles and muons. total number of electrons, positrons, and photons We examine the model in detail below. In par- Nmax is simply proportional to E and the depth of ticular, we will look at its predictions for measur- maximum shower development is logarithmically able properties of extensive air showers, proportional to E. attempting to assess which predictions are reliable We approximate hadronic interactions similarly and which may not be. First, we review the specif- [4]. For example, Fig. 1b shows a proton striking ics of HeitlerÕs electromagnetic shower model and an air molecule, and a number of pions emerging then develop the hadronic analogue. In all that fol- from the collision. Neutral pions decay to photons lows, the term ‘‘electron’’ does not distinguish be- almost immediately, producing electromagnetic tween e+ and eÀ. subshowers. The p± travel some fixed distance and interact, producing a new generation of pions. The multiplication continues until individual 2. Electromagnetic showers p pion energies drop below a critical energy nc , where it begins to become more likely that a p± As seen in Fig. 1a, an electron radiates a single will decay rather than interact. All p± are then as- photon after traveling one splitting length (a) γ (b) p n=1 n=1 + _ e e π +_ π o n=2 n=2 n=3 n=3 n=4 Fig. 1. Schematic views of (a) an electromagnetic cascade and (b) a hadronic shower. In the hadron shower, dashed lines indicate neutral pions which do not re-interact, but quickly decay, yielding electromagnetic subshowers (not shown). Not all pion lines are shown after the n = 2 level. Neither diagram is to scale. J. Matthews / Astroparticle Physics 22 (2005) 387–397 389 d = kr ln2, where kr is the radiation length in the 0.125 100 medium. (Note that d is in fact the distance over 30 GeV electron which an electron loses, on average, half its energy 0.100 incident on iron 80 by radiation.) After traveling the same distance, a plane + À g photon splits into an e e pair. In either instance, 0.075 60 the energy of a particle (electron or photon) is as- ) dE/dt Energy 0 sumed to be equally divided between two outgoing 0.050 40 particles. After n splitting lengths, a distance of (1/E Photons x 1/6.8 0.025 20 x = nkr ln2, the total shower size (electrons and Number crossin photons) is N ¼ 2n ¼ ex=kr . Electrons 0.000 0 Multiplication ceases when the energies of the 0 5 10 15 20 particles are too low for pair production or bre- t = depth in radiation lengths msstrahlung. Heitler takes this energy to be the critical energy ne, below which radiative energy Fig. 2. EGS4 simulation of electromagnetic showering in iron. c Points are particles or photons, the curve shows energy loss becomes less than collisional energy loss. In e deposition in each layer. air, nc ¼ 85 MeV. Consider a shower initiated by a single photon with energy E. The cascade reaches maximum size maximum. The model does not treat the loss of e N = Nmax when all particles have energy nc, so that particles as they range out. Also note that this sim- ulation does not display particles below 1.5 MeV. E ¼ neN : ð1Þ c max We conclude that the Heitler model predicts the shower size fairly well, if it is interpreted carefully. The penetration depth Xmax at which the shower reaches maximum size is obtained by determining The predicted depth of maximum from Eq. (2) is 7.4k , is in very good accord with the full the number nc of splitting lengths required for r e simulation. the energy per particle to be reduced to nc. Since nc Note that the final EM size N predicted by N max ¼ 2 , we obtain from Eq. (1) that max e the model may differ significantly from what is nc ¼ ln½E=nc= ln 2, giving actually measured by an experiment. The attenua- X c n k ln 2 k ln E =ne : 2 max ¼ c r ¼ r ½ c ð Þ tion of particle numbers is not accounted for in the The superscript ‘‘c’’ emphasizes that this expres- model, and in commonly used detectors such as sion is appropriate for purely electromagnetic scintillators, e± give much stronger signals than showers; the case for an air shower with hadronic photons do. It is useful therefore to differentiate components is considered later. The elongation e± and photons in the previous expressions. rate K is the rate of increase of Xmax with E, de- The model overestimates the actual ratio of fined as electrons to photons (as was noted by Heitler [1]). It predicts that after a few generations, the dX K max ; ð3Þ electron size approaches N 2 N . This is much dlog E e 3 max 10 too large for several reasons, mainly that multiple so that using Xmax from Eq. (2) yields photons are often radiated during bremsstrahlung. c À2 ± K = 2.3kr =85gcm per decade of primary en- Moreover, many e range out in the air. Such de- ergy for EM showers in air. tails of the development of the shower near and be- Fig. 2 shows a detailed EGS4 simulation [5] of yond its maximum are beyond the scope of this an electromagnetic cascade in iron (kr = model, requiring careful treatment of particle pro- À2 e 13.8 gcm , nc ¼ 22:4 MeV). HeitlerÕs model pre- duction and energy loss. dicts a maximum total size Nmax = 1340, about a Simulations confirm that photons greatly out- factor of two larger than the sum of electrons number electrons throughout the development of and photons in the figure. It can be seen that the the shower, as seen in Fig. 2. At their respective shower has begun to attenuate before reaching maxima, the number of photons is more than a 390 J.
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