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. 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 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+,e, 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+e 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 e. 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 + 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 simulation 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 2 ± 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- 2 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. Matthews / Astroparticle Physics 22 (2005) 387–397 factor of six larger than the number of e±. The We first develop the model for air showers initi- maximum electron size is an order of magnitude ated by cosmic ray protons. These results will then less than HeitlerÕs predicted Nmax. These ratios be extended to include heavier nuclei as the hold true in other media and at much higher ener- primaries. gies as well, with nearly identical values found in simulations of 1014 eV c-rays in air [6] and in 3.1. Model parameters PeV proton air showers (described in the next section). Following HeitlerÕs EM shower approach, we To extract the number of electrons Ne from attempt to keep parameters as constants, as far HeitlerÕs overall size N, we will simply adopt a cor- as possible. For the sake of concreteness, we will rection factor: consider numerical factors appropriate for a range of energies bracketing the ‘‘knee’’ region of the pri- N e ¼ N=g; ð4Þ 14 17 mary spectrum, E =10 eV to 10 eV. with a constant value of g = 10. This should be con- We also will use the same electron reduction sidered an order of magnitude estimate. Further factor g = 10 (as in Eq. (4)) to estimate the fraction adjustments can be made to g when comparing of the particle size which is actually observed. De- the model to actual experimental measurements. tailed simulations and experiments show that the Despite its limitations, the Heitler model ratio of photons to electrons in an air shower is reproduces two basic features of EM shower devel- very similar to that seen in a purely EM cascade. opment which are confirmed by detailed simula- Fig. 3 shows this ratio at ground level, from a full tions and by experiments: simulation [7] of proton induced extensive air showers (E = 1 PeV), as a function of the photon • The maximum size of the shower is propor- or electron energy at the ground. Very similar re- tional E, sults have been obtained elsewhere for showers ini- • The depth of maximum increases logarithmi- tiated by 1015 eV gamma rays [6]. 2 cally with energy, at a rate of 85 gcm per dec- The multiplicity Nch of charged particles pro- ade of primary energy. duced in hadron interactions increases very slowly with laboratory energy, growing as E1/5 in pp and

3. Hadronic showers

Air showers initiated by hadrons (Fig. 1b) are modeled using an approach similar to HeitlerÕs. The atmosphere is imagined in layers of ﬁxed thickness kI ln2, where kI is now the interaction length of strongly interacting particles. kI is assumed to be constant, a fairly good approximation for interactions in the range 10–1000 GeV. For 2 pions in air, kI 120 gcm [3]. Hadrons interact after traversing one layer, 1 producing Nch charged pions and 2 N ch neutral pions. A p immediately decays to photons, initiating electromagnetic showers. Charged pions con- tinue through another layer and interact. The Fig. 3. Ratio of photons to electrons, and photons to all process continues until the p± fall below the criti- p charged particles, at ground level from a simulated proton cal energy nc where they then are all assumed to induced, 1 PeV extensive air shower. E is the photon or electron decay, yielding muons. energy at the ground. J. Matthews / Astroparticle Physics 22 (2005) 387–397 391

p pp data [8]. Since the multiplicity in p-nucleon col- nc slowly decreases with increasing primary en- lisions is similar [10], we adopt a constant value ergy. Hereafter we adopt a constant value p Nch = 10, a value accurate to within a factor of nc ¼ 20 GeV. two for pion kinetic energies from about 1 GeV The number of interactions needed to reach p through 10 TeV. But we will remain cautious of Ep ¼ nc is, from Eq. (5), the validity of this approximation. The great ln E =np majority of interactions in a shower occur with ½ c p nc ¼ 3 ¼ 0:85log10½E=nc ; ð6Þ pion energies of order 100 GeV, much lower than ln 2 N ch the primary energy E . However, there are circum- giving n =3,4,5,6forE =1014,1015,1016,1017 eV stances—study of X for example—where the c max respectively. Eq. (6) does not depend strongly on properties of the first interaction have more moderate variations of the value chosen for N . importance. ch For example, using Nch = 20 changes nc only The other important parameter in our model is 16 p above E =10 eV, reducing it by one. nc , the energy at which we assume that further par- p As a last check of the self-consistency of our ticle production by p± ceases. We estimate n as c choices for N and np, we look at the properties the energy at which the decay length of a charged ch c of the interactions in the cascade. In the above pion becomes less than the distance to the next examples, the average kinetic energy of all the interaction point. interacting pions in a shower is about 250 GeV, Consider a single cosmic ray proton entering nearly independently of E . This energy corre- the atmosphere with energy E . After n interac- pffiffi sponds to s 22 GeV for pions colliding with tions (or atmospheric layers) there are N =(N )n ¼ p ch stationary nucleons. The mean pp charged multi- total charged pions. Assuming equal division of plicity at this energy is about 8 [5]. Allowing for energy during particle production, these pions car- n multiple interactions of a pion colliding with a tar- ry a total energy of (2/3) E . The remainder of the get air nuclei, our selection of N = 10 seems quite primary energy E has gone into electromagnetic ch reasonable. showers from p decays. The energy per charged In the following sections we implement the pion in atmospheric layer n is therefore hadronic shower model adopting the following E E : 5 (constant) values for the parameters: Nch = 10, p ¼ 3 n ð Þ e p 2 2N ch nc ¼ 85 MeV, nc ¼ 20 GeV, kI = 120g/cm , kr =37 g/cm2. After a certain number nc of generations, Ep becomes less than np. c 3.2. Primary energy For example, in a shower initiated by a 1015 eV primary proton, individual pion energies after four 4 The primary energy is finally divided between interactions are E ¼ 1015= 3 10 ¼ 20 GeV, using p 2 N pions and N electromagnetic particles in N = 10. The decay length of a pion with this en- p max ch subshowers. The number of muons is N = N . ergy is ccs = 1.1 km. Assuming an exponential l p By analogy to Eq. (1), the total energy in this case atmospheric density profile with scale height is 8 km, the linear distance between the altitudes of e p the beginning and the end of the fourth interaction E ¼ ncN max þ nc N l: layer is 1.8 km. This is the first layer that pions Scaling to the electron size N = N /g as before, have encountered where their probability of a de- e max p cay exceeds that of arriving at the next interaction e nc E ¼ gnc N e þ e N l point. The critical energy in this shower is then gnc np ¼ 20 GeV. c 0:85 GeV ðN þ 24N Þ: ð7Þ If we repeat the above exercise at other primary e l p 14 energies, we find that nc ¼ 30 GeV at E =10 eV Eq. (7) represents energy conservation. The rela- p 17 and nc ¼ 10 GeV at E =10 eV. It is evident that tive magnitude of the contributions from Nl and 392 J. Matthews / Astroparticle Physics 22 (2005) 387–397 from Ne is determined by their respective critical 3.3. Muon and electron sizes energies—the energy scales at which electromagnetic and hadronic multiplication cease. Different The number of muons in the shower is obtained nc primaries allocate energy differently between the using N l ¼ N p ¼ðN chÞ . The energy dependence electromagnetic and hadronic components, as do of the muon size is obtained by using Eq. (6) to statistical fluctuations. Eq. (7) implicitly accounts cast Nl in the form for these differences. p ln N l ¼ ln N p ¼ nc ln N ch ¼ b ln½E=nc ; The importance of the result in Eq. (7) is that E is simply calculable if both Ne and Nl are meas- where ured. The relation is linear and insensitive both to fluctuations and to primary particle type. ln½N ch b ¼ 3 ¼ 0:85: ð8Þ The practical use of Eq. (7) requires adjust- ln 2 N ch ments to account for experimental details. One effect is that measurements of particles are usually Note that although Nch in fact changes as the made after shower maximum. Another is that the shower develops, b depends only logarithmically relative sensitivity of an experiment to photons on its value. If Nch were an order of magnitude lar- and electrons affects the interpretation of meas- ger, b = 0.92. So our assumption that Nch is constant has little impact. The muon size of the ured value of Ne. Fig. 4 shows the energy reconstruction pub- shower is then lished by the CASA-MIA group [11]. Their result E b E 0:85 N ¼ 104 : ð9Þ has E = 0.8 GeV (Ne +25Nl), where Ne and Nl l np 1 PeV are measured shower sizes. Note that the relation c is linear and independent of the primary particle Recent studies using several detailed Monte Carlo type. In light of the caveats above, it must be con- simulations report values b = 0.85 ! 0.92 [9,10]. sidered somewhat accidental that the numerical Our present model will match these values very coefficients in their result agree so precisely with well when we include the effects of leading particles Eq. (7). in hadron interactions (see Section 4). In any case, the less-than-linear growth of Nl with primary energy has important consequences for modeling showers initiated by nuclei heavier than protons, 7 described later. 6.75 protons We estimate the electron size of the shower by iron assessing how much energy has gone into electro- 6.5 magnetic channels, assume it appears as EM sub- 6.25 showers, and then use Eq. (1) to obtain N. This ) µ approximation simply adds all the EM subshowers 6 from each interaction generation to determine the +25N

e* 5.75 total number of electrons produced. Conservation of energy implies that the primary 5.5

log10(N energy is split into electromagnetic and hadronic 5.25 parts E = Eem + Eh. The hadronic energy appears p in the muon component as E ¼ N ln , so that the 5 h c fraction into the EM component is 4.75 p b1 Eem E N lnc E 4.5 ¼ ¼ 1 p ; ð10Þ 2 2.2 2.4 2.6 2.83 3 3.2 3.4 3.6 3.8 4 E E nc log10(Energy TeV) where b is from Eq. (9). The EM fraction is 72% at 14 17 Fig. 4. Energy reconstruction from CASA-MIA (from [11]). E =10 eV, rising to 90% at E =10 eV. Eq. J. Matthews / Astroparticle Physics 22 (2005) 387–397 393

(10) can be well approximated by a power law particles in the first interaction. Each tends to raise a Eem / E in the energy region of interest here. the estimated altitude of shower maximum when After comparing series expansions near E ¼ compared to our previous method which used con- 105np, we obtain stant parameters. The interaction cross-section c a rises with energy, causing air showers to begin 1 Eem 6 E N e ¼ e 10 ; higher. The multiplicity of produced pions also in- g nc 1 PeV creases, giving individual p lower energy and where hence the EM subshowers from their decays will have shorter development lengths. 1 b a ¼ 1 þ 1:03: The first interaction occurs at an atmospheric 105ð1bÞ 1 depth X = kI ln2, where kI is in this case the inter- Inverting this expression gives action length of the primary proton. Using the ine- 0:97 lastic p-air cross-section from [10], we approximate E ¼ð1:5 GeVÞN e ; ð11Þ kI to obtain in good agreement with full simulations of Engel 0:99 X ¼ kI ln 2 et al. [9], which gave E ¼ð1:6 GeVÞN e .It 2 should be emphasized that Ne in Eq. (16) is to be ¼ð61 gcm Þð1:0 0:1ln½E=PeVÞ ln 2: interpreted as the electron size at shower 1 The first interaction yields 2 N chp ! N ch photons. maximum. Each photon initiates an electromagnetic shower of energy E/(3Nch), developing in parallel with 3.4. Depth of shower maximum the others. We parameterise the charged particle production in the first interaction as Nch = Xmax is the atmospheric depth at which the elec- 1/5 41.2(E/1 PeV) , based on pp data [8]. trons and photons of the air shower reach their The depth of maximum is now obtained as in maximum numbers. The EM component is gener- Eq. (2), for an EM shower of energy E /(3N ) ated by photons from p decays. The first interac- ch 1 starting at depth X, and using the energy-depend- tion diverts E into these channels. This is 3 encies of X and Nch: followed by additional showers from each subse- p e quent interaction point. X max ¼ X þ kr ln E=ð3N chncÞ We will use only the first generation c showers ¼ ðÞ470 þ 58log ½E =1 PeV gcm2: ð12Þ to estimate the depth where the shower reaches 10 p maximum size. The method previously employed As we expected, these values of X max are low when to calculate Ne (Eq. (16)) is inappropriate here. compared to detailed simulations [12,13], by about 2 In that case we summed all the electromagnetic en- 100 gcm or a bit less than 2kI. Presumably this is ergy and treated it as a single EM shower. For the consequence of neglecting the contributions of proper evaluation of Xmax, it would be necessary the next one or two generations of p production. to sum each generation of subshowers carefully Also, including the effects of leading particle pro- from their respective points of origin, accounting duction (Section 4 below), systematically raises p 2 for their attenuation near and after their maxima. X max by about 17 gcm . As we have noted, such treatment is beyond the Fig. 5 shows the model predictions alongside scope of the simple model we are using. the simulation results. The solid lines are adapted Using only the first interaction will certainly from simulations [12,13] using several modern underestimate Xmax, since it neglects the following interaction generators (QGSJET, SIBYLL, neXus, subshowers, but it will capture well the elongation DPMJET). Dashed lines are proton and iron p rate K (the rate of increase of Xmax with E). showers from the model. The dotted line repre- We shall take extra care here to account for the sents the photon induced EM showers. Note that energy dependence both of the primary interaction both the proton and the iron lines have been cross-section and of the multiplicity of produced shifted upward by 100 gcm2 to more closely 394 J. Matthews / Astroparticle Physics 22 (2005) 387–397

energy. (This illustrates LinsleyÕs elongation rate theorem [14], which pointed out that Kc for electromagnetic showers represents an upper limit to the elongation rate for hadron showers.)

3.5. Nuclear primaries

The superposition model is a simpliﬁed view of the interaction of a cosmic ray nucleus with the atmosphere. A nucleus with atomic number A and total energy E is taken to be A individual single nucleons, each with energy E/A, and each act- ing independently. We treat the resulting air shower as the sum of A separate proton air showers all starting at the same point. We can produce observable shower features Fig. 5. Depth of maximum vs. primary energy for air showers. by substituting the lower primary energy into the Dotted: photon induced showers; Dashed: proton and iron various expressions derived previously for pro- induced showers, each uniformly shifted higher by 100 gcm2. ton showers and summing A such showers Solid lines are from full simulations of p and Fe showers. where appropriate. The resulting nuclear-initiated shower properties are easily expressed in terms of

14 the corresponding quantities of a proton shower match the others at 10 eV, but the curve for EM with the same total energy E : showers is taken directly from Eq. (2). The con- A p 0:15 stant separation between the iron and proton lines N l ¼ N lA ; ð13Þ is what the model predicts, as will described in the A p next section. X max ¼ X max kr ln A; ð14Þ The predicted elongation rate Kp =58gcm2 per decade of energy is in excellent accord with E ¼ 0:85 GeV ðN e þ 25N lÞ: ð15Þ simulations [10]. One consequence is that nuclear showers have It is instructive to compare proton induced air more muons than proton showers, at the same to- showers to purely electromagnetic cascades by tal primary energy. This results from the less-than- rewriting Eq. (12) as linear growth of the muon number with energy. p c X max ¼ X max þ X kr ln½3N ch; The lower energy nucleons which initiate the shower generate fewer interaction generations, where X c is the depth of maximum of EM show- max and so lose less energy to electromagnetic compo- ers from Eq. (2). The elongation rate for proton nents. An iron shower will have (56)0.15 = 1.8 times showers is then as many muons as a proton shower of the same

p c d energy. K ¼ K þ fgX kr ln½3N ch dlog10E The showers from lower energy primaries also do not penetrate as deeply. X of iron showers ¼ 58 gcm2 per decade; max is higher than proton showers by kr ln(56) = as in Eq. (12). It is now easy to see that Kp is re- 150 gcm2 at all energies. This is in good accord duced from Kc for EM showers by two eﬀects: with simulations [13], as was seen in Fig. 5. increasing multiplicity Nch and increasing cross- As discussed earlier, the energy assignment (Eq. section (decreasing X). The multiplicity and (15) or Eq. (7)) is unaﬀected by A because the cross-section reductions are respectively 17 gcm2 expression intrinsically accounts for all of the pri- and 10 gcm2 from Kc =85gcm2 per decade of mary energy being distributed into a hadronic J. Matthews / Astroparticle Physics 22 (2005) 387–397 395 channel (seen as muons) and into electromagnetic showers.

4. Inelasticity

The simple model has neglected one aspect of hadronic interactions which has observable consequences. When two hadrons interact, a significant fraction of the total energy is carried away by a single ‘‘leading’’ particle. This energy is unavailable immediately for new particle production. Treatment of this effect is best done by detailed simulations. Nevertheless, we will try here to include it approximately in our model to understand the effects qualitatively. The inelasticity of a single interaction is de- Fig. 6. Variation of muon production index b, from Eq. (16),as scribed by a parameter j, defined as the fraction a function of inelasticity and charged particle multiplicity Nch. of the total energy directed into new pion production (both p± and p). The value of j has a value of except now the index is perhaps 0.5, but is not well known at high energy. ln½1 þ N ch Neglecting leading particle effects (as we have done b ¼ ln ð1 þ N Þ= 1 1j so far) corresponds to j = 1.0. ch 3 j Our estimation of the elongation rate is not af- 1 ¼ 1 0:14j: ð16Þ 3ln N fected by j < 1, since the energy dependence of the ½ ch multiplicity and cross-sections do not depend on j. The previous relation (Eq. (8)) is simply regained Our energy reconstruction formula (Eq. (7)) is also by substituting (Nch +1)! Nch and j = 1 in the robust against altering the division of energy be- above expression. tween hadronic and electromagnetic components. For inelasticity j = 0.5, the muon size increases We expect that including inelasticity will alter at a faster rate (b = 0.93) than in the original the depth of maximum. If a significant fraction model (b = 0.85). The effect of inelasticity on the of energy is taken away by a single leading particle growth of muon size with E is not large, but it then it is unavailable for the first EM subshowers. is comparable to the effect of assuming a larger For example, if j = 0.5, then Eq. (12) suggests that average multiplicity Nch. Fig. 6 exhibits Eq. (16), p X max will be higher by approximately 58log10½2¼ showing the nearly linear dependence of b with 2 17 gcm . j, for several assumed values of Nch. The resulting We also expect that the total muon production expectation that b ’ 0.90–0.95 is in better accord in a shower will increase more quickly with E with full simulation results [10]. than in our simple model, since less energy is ‘‘lost’’ to the electromagnetic channel in each interaction. 5. Summary In Appendix A, we develop our model in the same fashion as before, but adjusting the method A basic model of extensive air showers, similar to account for j < 1.0. We find that, once again, to HeitlerÕs model of electromagnetic cascades, the muon size increases with energy as gives several predictions which are in good agree- b ment with detailed simulations and with data. E The primary energy is proportional to a combi- N l ¼ p ; nc nation of the muon and electron sizes, 396 J. Matthews / Astroparticle Physics 22 (2005) 387–397

2 E (Ne +25Nl). The relative weighting depends (1 j)E is taken by a single leading particle, 3 jE 1 mainly on the characteristic energy scales at which is used to produce Nch charged pions, and 3 jE is 1 hadronic cascading and electromagnetic showering carried by 2 N ch neutral pions, subsequently cease. appearing in electromagnetic subshowers. Muon size grows with primary energy more We will assume that inelasticity effects will be b slowly than proportionally, N l E , with b = present in all hadron interactions and that j is con- 0.85. The exponent depends on the fraction of stant throughout. We will also assume that in each energy given to charged pions in each interaction. interaction the total multiplicity of produced If the model is extended to include significant charged particles is (1 + Nch). This is higher than inelasticity in pion interactions, then b = 0.90–0.95. in the model described before (by one particle), The elongation rate for hadronically induced air but this formulation is useful for making compar- showers is 58 gcm2, less than that from purely isons. Our simple model is recovered by taking the electromagnetic showers (85 gcm2). The differ- limit j ! 1, and only counting particles with non- ence arises from the increase of multiplicity with zero energy. energy in hadronic interactions and on the growth The fraction of the energy of an interaction of the p-air inelastic cross-section. which goes to particles which may interact again 1 Showers initiated by nuclei with atomic number (i.e, all except p )isð1 3 jÞ. As before, these par- A will have the same relationship between E and ticles travel a distance kI ln2 and then interact, their muon and electron sizes that protons do. producing more particles. In a cascade initiated Their depth of maximum will be higher than pro- by a primary with energy E, the total energy car- ton showers by kr lnA but the elongation rate is ried by interacting hadrons after n interactions (or 1 n quite similar. The muon size from from nuclear atmospheric layers) is ð1 3 jÞ E, distributed over n showers will be larger than that from proton show- (1 + Nch) charged pions. ers by a factor of A0.15. However, unlike the simple cascade model, this energy is not evenly distributed among all the hadrons, but instead is carried by groupings of parti- Acknowledgments cles whose energies are determined by the value of j. It is useful to consider the hadronic energy in This work was informed by very helpful discus- layer n by explicitly separating the part due to sions with M. Glasmacher, A.A. Watson, P. Som- the leading particle (1 j) from the inelastic, mers, and M.L. Cherry. The first presentation of pion-producing component ð2 jÞ: 3 an approach similar to this was given in M. Glas- 1 n 2 n macherÕs Ph.D. dissertation [15]. This work was 1 j ¼ð1 jÞþ j 3 3 supported in part by the US Department of Xn r Energy. n r 2 ¼ C ð1 jÞ j ; ðA:1Þ n;r 3 r¼0 where C is the binomial coefficient n AppendixA. Inelasticity n,r r ¼ n!=ðn rÞ!r!. The number of particles produced When two hadrons interact, a significant frac- which contribute to hadron production in an inter- tion of the total energy is carried away by a single action can be similarly expressed in terms of one leading particle. This energy is unavailable for leading particle and Nch charged pions. Then, in immediate new particle production. The inelastic- layer n, the number of such particles is ity can be described by a parameter j, which we Xn n nr define as the fraction of the total energy directed ð1 þ N chÞ ¼ Cn;rN ch : ðA:2Þ into pion production. r¼0 In an interaction initiated by a particle with en- The terms in Eq. (A.1) and Eq. (A.2) correspond ergy E, the energy is apportioned as follows: to one another, as can be seen by (tedious) count- J. Matthews / Astroparticle Physics 22 (2005) 387–397 397

Table A.1 Properties of the particle groups in the n = 4 layer of a cascade. The last two columns give speciﬁc numbers for the case where the 15 primary energy is E =10 eV, Nch = 10, and j = 0.5

Pion group r Number in group Total energy in group Number (Nch = 10) Energy per pion (j = 0.5) 4 01 (1 j) E 1 63TeV 3 14Nch 4(2/3j)(1 j) E 40 4.2 TeV 2 2 2 26N ch 6(2/3j) (1 j) E 600 280 GeV 3 3 34N ch 4(2/3j) (1 j)E 4000 19 GeV 4 4 4 N ch (2/3j) E 10,000 1.2 GeV ing as the cascade develops. In interaction genera- with tion n, it turns out that there are n + 1 separate groups of pions with common energies. For exam- ln½1 þ Nch b ¼ 1 ple, in layer n = 4, there are ﬁve distinct groupings ln ð1 þ N chÞ= 1 3j j of particles. The properties of the groups are given 1 ¼ 1 0:14j: ðA:3Þ in Table A.1. 3ln ½N ch There are more particles in this cascade than in the simple model, beyond what would be expected from simply incrementing Nch by one. For example, when E = 1 PeV, there are almost 50% more particles in the n = 4 layer of this cascade. But most References (>95%) of the particles are in the two groups with [1] W. Heitler, The Quantum Theory of Radiation, third ed., the lowest energy per pion. So we are not too far Oxford University Press, London, 1954, p. 386 (Section wrong if we proceed as before, and approximate 38). the terminal interaction layer nc as the layer where [2] B. Rossi, High Energy Particles, Prentice-Hall, Englewood the average energy per pion is the critical energy: Cliﬀs, NJ, 1952. [3] T.K. Gaisser, Cosmic Rays and Particle Physics, Cam- 1 nc E 1 j bridge University Press, Cambridge, 1990. np ¼ 3 ; c ð1 þ N Þnc [4] J. Matthews, in: Proc. 27th ICRC, Hamburg, 2001, p. 261. ch [5] Particle Data Group, Review of particle physics, Phys. which implies Rev. D 66 (2002) 203, Fig. 26-17. [6] N.N. Kalmykov, et al., in: Proc. 28th ICRC, Tsukuba, vol. ln½E =np n c : 2, 2003, p. 511. c ¼ 1 [7] S.J. Sciutto, in: Proc. 27th ICRC, Hamburg, 2001, p. 237. ln ð1 þ N chÞ= 1 j 3 [8] Particle Data Group, Review of particle physics, Phys. This is the same as Eq. (6) with the substitutions Rev. D 66 (2002) 258, Fig. 39-5. 3 1 [9] R. Engel, et al., in: Proc. 26th ICRC, vol. 1, Salt Lake City, 2 !ð1 j=3Þ and Nch ! (1 + Nch). As before, we obtain the relation between muon number 1999, p. 415. [10] J. Alvarez-Mun˜iz, et al., Phys. Rev. D 66 (2002) 033011. and primary energy by following the method lead- [11] M.A.K. Glasmacher, et al., Astropart. Phys. 12 (2000) ing to Eq. (9): 1–17. p [12] D. Heck, et al., in: Proc. 27th ICRC, Hamburg, 2001, ln N l ¼ ln N p ¼ nc ln½1 þ N ch¼b ln½E=nc ; p. 233. so that [13] J.W. Fowler, et al., Astropart. Phys. 15 (2001) 49–64. [14] J. Linsley, in: Proc. 15th ICRC, vol. 12, Plovdiv, 1977, b p. 89. E N l ¼ p [15] M.A.K. Glasmacher, Ph.D. thesis (unpublished), Univ. of nc Michigan, 1998.