Astroparticle Physics 59 (2014) 4–11

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Astroparticle Physics

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An extended Heitler–Matthews model for the full hadronic cascadein cosmic air showers

http://dx.doi.org/10.1016/j.astropartphys.2014.03.0100927-6505/� 2014 Elsevier B.V. All rights reserved.

⇑ Tel.: +31 (0)20 592 5126.E-mail addresses: hans.montanus@wxs.nl, hansm@nikhef.nl

J.M.C. Montanus ⇑Nikhef, Science Park 105, 1098 XG Amsterdam, The Netherlands

a r t i c l e i n f o a b s t r a c t

Article history:Received 30 December 2013Received in revised form 24 March 2014Accepted 27 March 2014Available online 5 April 2014

Keywords:Cosmic raysAir showersHeitler–Matthews modelHadronic cascade

The Heitler–Matthews model for hadronic air showers will be extended to all the generations of electro-magnetic subshowers in the hadronic cascade. The analysis is outlined in detail for showers initiated byprimary protons. For showers initiated by iron primaries the part of the analysis is given for as far as itdiffers from the analysis for a primary proton. Predictions for shower sizes and the depth of maximumshower size are compared with results of Monte Carlo simulations. The depth of maximum as it followsfrom the extrapolation of the Heitler–Matthews model restricted to the first generation of electromag-netic subshowers is too small with respect to Monte Carlo predictions. It is shown that the inclusionof all the generations of electromagnetic subshowers leads to smaller predictions for the depth of max-imum and to smaller predictions for the elongation rate. The discrepancy between discrete model predic-tions and Monte Carlo predictions for the depth of maximum can therefore not be explained from thenumber of generations that is taken into consideration. An alternative explanation will be proposed.

� 2014 Elsevier B.V. All rights reserved.

1. Introduction

A simplified description of the longitudinal evolution of electro-magnetic showers is given by the Heitler model [1]. Starting with aprimary particle of energy E0, the number of particles doublesevery splitting length d ¼ kr ln 2, where the radiation length kr isabout 37 g cm�2. The doubling stops when the energy per particleis equal to the critical energy ne

c � 85 MeV. The resulting Heitlerprofile is

NðXÞ ¼ 2X=d; X 6 necd;

0; X > necd;

(ð1Þ

where nec is maximum the number of steps: ne

c ln 2 ¼ lnðE0=necÞ.

A Heitler model for the hadronic cascade in air showers hasbeen constructed by Matthews [2]. The Heitler–Matthews modelis useful for the explanation of hadronic cascades as well as forthe analytical derivation of relations between quantities asprimary energy, muon number, electron number and depth ofmaximum shower size [3–5]. For the prediction of the number ofcharged particles it is assumed that each hadronic interactionresults in Mch ¼ 10 charged pions and 1

2 Mch ¼ 5 neutral pions. Thatis, the total multiplicity M is equal to 15. The neutral pions initiate

electromagnetic subshowers when they decay into photons. Forthe prediction for the depth of maximum shower size, restrictedto the first generation of electromagnetic subshowers, the multi-plicity and interaction length are parameterized by the energy ofthe interaction.

The atmosphere is divided into layers of atmospheric thicknessdI. After the traversing of each layer the number of charged pions isassumed to be Mch times larger if dI ¼ kI ln 2, wherekI ¼ 120 g cm�2 is the interaction length of strongly interactingpions. Consequently, after n layers the number of charged pionsis ðMchÞn. The energy per pion is

Ep;n ¼E0

Mn : ð2Þ

The stopping energy is estimated on the basis of the finite lifetimeof the pions in the atmosphere. For this it suffices to consider theapproximate relation between atmospheric depth and height:

XðhÞ ¼ 1030 � e�h=8 $ hðXÞ ¼ 8 lnð1030=XÞ; ð3Þ

where X is the depth in g cm�2 and h is the height in km. Neutralpions decay almost immediately into two photons, cs ¼ 25 nm[6]. Each resulting photon starts an electromagnetic shower. Thedecay length of the charged pions is ccs, where cs ¼ 7:8 m [6].The decay length is of the order of a kilometer because of the rela-tivistic time dilation. As a consequence charged pions may interactwith the atmosphere and propagate the hadronic shower, before

J.M.C. Montanus / Astroparticle Physics 59 (2014) 4–11 5

decay. If the probability for decay in the next layer is larger than theprobability of a hadronic interaction, the pions are assumed todecay and the cascade stops. This happens after nc layers. The cor-responding energy of the decaying charged pions, the stoppingenergy np

c , follows from

npc ¼

E0

Mnc: ð4Þ

The stopping energy turns out to be around 20 GeV.

2. Model parameters

In this paper the Heitler–Matthews model is extended to all thegenerations of pions in the shower. The complete analysis will beimproved by consequently taking the multiplicity M and interac-tion length kI to depend on the energy of the hadron in the shower.One of the consequences is that the thickness of the cascade layersincreases with depth, see Fig. 1.

We will take the energy dependence of the p–air multiplicityand interaction length to be given by Monte Carlo event generatorsbased on QCD and parton models. The calculated p–air chargedmultiplicity, see Fig. 5 of [7], Fig. 7 of [8] and Fig. 5 of [9], suggeststhe relation

Mch � 0:1 � E0:18; ð5Þ

where E is the energy in eV. Taking the ratio of the charged and neu-tral pions as 2 : 1, we have for the total multiplicity

M � 0:15 � E0:18: ð6Þ

It should be emphasized that the relation between multiplicity andenergy is rather uncertain since different models predict differentmultiplicities. In particular for large energies the differences canbe large, even more than 100%. From Fig. 5 of [7] and Fig. 7 of [8]we see that the pion multiplicity does not differ substantially fromthe proton multiplicity. We therefore will use the relations (5) and(6) for both the multiplicity in proton–air (p–air) and pion–air(p–air) interactions. A parameterization with other values for theconstants will, of course, affect the results quantitatively. It does,however, not alter the results qualitatively.

Both the p–air and p–air inelastic cross sections grow withenergy. The p–air inelastic cross section at large energies obtainedfrom observations of extensive air showers are in good agreementwith QGSJET predictions [10–13]. For the present analysis we will

Fig. 1. The hadronic cascade for energy dependent interaction lengths.

therefore use the QGSJET predictions for the p–air inelastic crosssection. We will also use the QGSJET predictions for the p–airinelastic cross section [8,14]. From these cross sections approxima-tions for the energy dependent interaction lengths can be derivedwhich are sufficiently accurate for our purpose. For p–air this is:

kp—air ½g cm�2� � 200� 3:3 lnðE½eV�Þ: ð7Þ

For p–air this is

kp—air ½g cm�2� � 145� 2:3 lnðE½eV�Þ: ð8Þ

For the hadronic cascade Matthews and Hörandel take dI ¼ kI ln 2 asthe relation between layer thickness and interaction length [2,3].This might have been motivated by the expression kr ln 2 for thesplitting length in the electromagnetic cascade. There the ratioln 2 results from the translation of the radiation length to the split-ting length. In an intermediate model for electromagnetic showersthe splitting length, kr ln 2, is effectively used as the electromag-netic interaction length [15]. For the hadronic cascade, however,there is no reason for the ratio ln 2 since kI is already an interactionlength. As a consequence the thickness of the interaction layer inhadronic cascades is equal to the interaction length. For hadronicshowers we will therefore use the relation dI ¼ kI.

3. The hadronic cascade for a primary proton

Now we consider a hadronic cascade where the hadronic parti-cles interact after having traversed a layer of atmosphere. Thethickness of each layer will be taken equal to the actual interactionlength as given by (7) or (8). After each interaction M pions are pro-duced as given by (6). In accordance with the Heitler model forelectromagnetic showers, the energy is assumed to be equallydivided over the particles produced. After each interaction thenew energy of the charged hadrons then follows from a successiveapplication of the equation

Ejþ1 ¼Ej

MðEjÞ: ð9Þ

Starting with a primary proton with energy E0 the energy of theparticles after the first interaction is

E1 ¼E0

0:15 � E0:180

� 6:7 � E0:820 : ð10Þ

After the second interaction this is

E2 ¼E1

0:15 � E0:181

� 6:7 � E0:821 � 6:71:82 � E0:822

0 : ð11Þ

Repeating the iteration we find for the energy per particle after ninteractions

En ¼ 6:7an � Ebn0 ; ð12Þ

where

an ¼1� 0:82n

1� 0:82; bn ¼ 0:82n: ð13Þ

For the interaction lengths we obtain for the primary proton, n ¼ 0,

k0 ¼ kp—airðE0Þ ¼ 145� 2:3 lnðE0Þ ð14Þ

and for the produced pions, n P 1,

kn ¼ kp—airðEnÞ ¼ 200� 3:3 lnðEnÞ: ð15Þ

With the substitution of (12) this is

kn ¼ 200� 6:3an � 3:3bn lnðE0Þ; n P 1: ð16Þ

Table 2Subsequent interaction lengths in hadronic cascades for a primary proton. The kn andthe cascade average �k are in g cm�2.

nc E0 (eV) k0 k1 k2 k3 k4 �k

1 1:5 � 1012 81 – – – – 81

2 2:9 � 1013 74 110 – – – 92

3 1:4 � 1015 65 99 111 – – 92

4 1:9 � 1017 53 86 100 112 – 88

5 8:2 � 1019 40 70 87 101 112 82

Table 3Subsequent multiplicities and their geometric mean in hadronic cascades for aprimary proton.

nc E0 (eV) M0 M1 M2 M3 M4 �M

1 1:5 � 1012 23 – – – – 23

2 2:9 � 1013 40 20 – – – 28

3 1:4 � 1015 80 36 19 – – 38

4 1:9 � 1017 193 75 34 18 – 55

5 8:2 � 1019 576 183 72 33 18 85

6 J.M.C. Montanus / Astroparticle Physics 59 (2014) 4–11

The atmospheric depth of the ncth interaction is given by

XðncÞ ¼Xnc�1

n¼0

kn; nc P 1: ð17Þ

Substitution of (14) and (16) leads to

XðncÞ ¼ 200nc � 55� 6:3Xnc�1

n¼1

an � 2:3þ 3:3Xnc�1

n¼1

bn

!lnðE0Þ: ð18Þ

With the approximate relation (3) between height and atmosphericdepth, we obtain for the difference in height between the ncth andðnc þ 1Þth interaction:

Dh ½km� ¼ 8 lnXðnc þ 1Þ

XðncÞ

� �: ð19Þ

For the decay length of the charged pions after the ncth interactionwe find

csc½km� � 7:8 � 10�3 Enc

mp�� 5:6 � 10�11 � 6:7anc � Ebnc

0 ; ð20Þ

where we have taken 140 MeV/c2 for the mass of the charged pions[6]. We will follow Matthews with the reasonable assumption thatthe pions will decay when the decay length is half the layer thick-ness: ccs � 1

2 Dh [2]. That is

5:6 � 10�11 � 6:7anc � Ebnc0 ¼ 4 ln

Xðnc þ 1ÞXðncÞ

� �: ð21Þ

In this equation we substitute integer values for nc and solvenumerically for the primary energy E0. Results of interest are shownin Tables 1–3. In Table 1 the height h is calculated with Eq. (3). InTables 2 and 3 the interaction lengths respectively multiplicitiesare given for all interactions up to the final one after which decayoccurs. Of course, the number of interactions in the cascade andthus also the penetration depth of the shower increases with theenergy of the primary particle. Since a larger atmospheric depthat lower altitudes corresponds to a larger probability for a chargedpion to interact before decay, we expect a smaller stopping energy.From the last entry in Table 1 we see the stopping energy indeeddecreases for increasing primary energy.

The number of muons is given by

Nl ¼23

� �nc

�Ync�1

n¼0

Mn: ð22Þ

For a primary proton with energy 1:4 � 1015 eV, as an example, thenumber of muons is about 1:6 � 104. The energy of the final pionsis about 2:6 � 1010 eV, see last entry of Table 1. For the quantity

b ¼ ln Nl

ln E0=npc

� � ð23Þ

we then obtain the value 0.89. For other primary energies the valueof b is found to increase slightly from 0.88 through 0.92 for aprimary energy increasing from 1012 eV through 1020 eV. Since wework with larger multiplicities, these values are slightly larger thanthe one obtained by Matthews [2].

Table 1Characteristics of hadronic cascades for a primary proton.

nc E0 (eV) XðncÞ Xðnc þ 1Þ hðncÞ (km) Dh (km) npc (eV)

1 1:5 � 1012 81 198 20:4 7:2 6:4 � 1010

2 2:9 � 1013 184 303 13:8 4:0 3:6 � 1010

3 1:4 � 1015 275 396 10:6 2:9 2:6 � 1010

4 1:9 � 1017 352 473 8:6 2:4 2:1 � 1010

5 8:2 � 1019 409 531 7:4 2:1 1:9 � 1010

From the last column in Table 3 we find the following approx-imate relation between the effective multiplicity (the geometricmean multiplicity) and the primary energy: ln �M � 1:05þ0:074 ln E0. The effective charged multiplicity then is given byln �Mch � 0:65þ 0:074 ln E0. Substituting the latter in theMatthews’ expression [2]

b ¼ 1� j3 ln Mch

; ð24Þ

where j is the inelasticity, we obtain a refinement for the energydependence:

b ¼ 1� j1:9þ 0:22 ln E0

: ð25Þ

4. Hybrid Heitler scheme

The Heitler line of reasoning can also be applied to electromag-netic shower profiles other than the Heitler profile (1). Let NðX; EÞbe an electromagnetic shower profile which for a primary energy Ehas its maximum NmaxðEÞ at depth XmaxðEÞ. For twice the primaryenergy the particle will split into two particles with energy E afterone splitting length d ¼ kr ln 2. The corresponding shower profilecan be regarded as twice the shower profile for E shifted with dtowards a larger depth. As a consequence Nmaxð2EÞ ¼ 2NmaxðEÞand Xmaxð2EÞ ¼ XmaxðEÞ þ d. The latter corresponds to the followingelongation rate:

dXmax

dlog10 E� DXmax

Dlog10 E¼ d

log10 2¼ kr ln 10: ð26Þ

This scheme will be applied to the present hadronic cascades. Eachtime when neutral pions decay into photons an electromagneticsubshower is initiated and the corresponding longitudinal profileis substituted. It is similar to what is done in the hybrid Monte Carlomodel CONEX [16]. For the electromagnetic shower profile we takethe Greisen function [17]:

NeðXÞ ¼0:31ffiffiffiffiffi

ycp � eX

krð1�1:5 ln sÞ; ð27Þ

where yc ¼ lnðE0=necÞ and where

s ¼ 3XX þ 2Xmax

ð28Þ

is the age parameter. For showers initiated by a photon a goodprediction for the depth of maximum is given by

12

J.M.C. Montanus / Astroparticle Physics 59 (2014) 4–11 7

Xmax;c ½g cm�2� ¼ nc � d ¼ yc � kr � 85log10 E0 � 675: ð29Þ

electrons

muons

12 14 16 18 200

2

4

6

8

10

Log 10 E 0 eV

Log

10N

Fig. 3. The maximum number of electrons (�) and muons as a function of theenergy of the primary proton (dots) and the linear fits (dashed).

For energies larger than 1 EeV, the depth of maximum for photonshowers is larger than predicted by (29) because of the Landau–Pomeranchuk–Migdal (LPM) effect [19–21]. See for instance thecorresponding curve in Fig. 13 of [18]. By means of a hybrid MonteCarlo model the consequences of the LPM effect for hadronic airshowers are found negligible for primary energies below3 � 1020 eV [8]. For such extremely high primary energies the largemultiplicity causes the energy of the decay photons after the firstinteraction to be below 1 EeV, outside the LPM regime. We willtherefore conveniently take (29) for the depth of maximum showersize of electromagnetic (sub) showers. The total electromagneticshower profile is obtained by adding the profiles of the electromag-netic subshowers. As an example we consider a shower caused by a1:4 � 1015 eV primary proton. The first interaction at depth 65 g/cm2

(Table 2) produces an expected number of 80 pions (Table 3). Equalenergy division over the secondaries leads to 1:75 � 1013 eV for eachpion. One-third of the pions are assumed to be neutral. Also for thegamma production we will assume equal energy division. Theneach neutral pion decays into two gammas with energy8:8 � 1012 eV. Gamma showers with this initial energy have a depthof maximum at about 425 g/cm2. The depth of maximum of thiselectromagnetic subshower is therefore at 490 g/cm2. The chargedpions survive the next interaction layer of 99 g/cm2 to cause a sec-ond generation of gammas. In a similar way as for the first genera-tion the depth of maximum of the second and third generation arefound to be both about 460 g/cm2. The total energy contents in thethree subsequent electromagnetic subshowers are 1

3 E0;29 E0 and 4

27 E0

respectively. After the third interaction the charged pions areassumed to decay into a muon. The contribution of all three gener-ations of electromagnetic showers to the total shower is illustratedin Fig. 2.

Since the depth of maximum for the second and third genera-tion is smaller than for the first generation, the depth of maximumfor the total electromagnetic shower will also be smaller than pre-dicted by the first generation. For the example above: the depth ofmaximum of the total shower is 470 g/cm2, which is 20 g/cm2

smaller than the value 490 g/cm2 for the first generation. Thesedifferences are even larger for larger primary energies.

The depth of maximum and the maximum number of electronsand positrons of the total electromagnetic shower is found bynumerical inspection. This semi-analytical approach is utilized tobe able to consider all the generations of subshowers. The numberof muons is found by means of Eq. (22). In Fig. 3 we have plotted

1

2

3

0 200 400 600 800 1000

104

105

106

X g cm 2

Ne

Fig. 2. Total electromagnetic shower profile (solid) as it results from the addition ofthe profiles of first, second and third (labeled as 1,2 and 3) generation ofelectromagnetic subshowers (dashed) for a 1.4 PeV proton primary.

the maximum size of the total electromagnetic shower and thenumber of muons as a function of initial energy.

A linear fit, see the dashed lines in Fig. 3, for the maximumnumber of electrons (�) and number of muons yields

Ne � 0:57 � ðE0 ½GeV�Þ1:019 ð30Þ

and

Nl � 0:015 � ðE0 ½GeV�Þ0:975 ð31Þ

respectively. The relation between log10ðNe þ 25NlÞ and log10 E0 ispractically linear. It is of interest for energy parameterization ofcosmic air showers [22]. The present model values of Ne þ 25Nl

as a function of primary proton energy E0 are close to the onesobtained with Monte Carlo models, e.g. Fig. 3 of [22].

The depth of the maximum shower size of the total electromag-netic shower is also determined by numerical inspection. In Fig. 4the depth of maximum shower size is plotted as a function of theenergy of the primary proton.

A cubic fit, see the dashed line in Fig. 4, yields

Xmax;p � �1796þ 314:3 lg E0 � 14:9ðlg E0Þ2 þ 0:265ðlg E0Þ3; ð32Þ

where the common notation lg for log10 is used for abbreviation.The latter expression is the result for a primary proton as indicatedby the subscript ‘p’. It parameterizes the energy dependence of the

proton

12 14 16 18 20

300

400

500

600

700

Log 10 E 0 eV

Xmax

gcm

2

Fig. 4. The depth of maximum shower size as a function of energy of a protonprimary for the situation with complete inelasticity (dots) and the correspondingcubic fit (dashed).

Table 4Characteristics of hadronic cascades for a primary iron.

nc E0 (eV) XðncÞ Xðnc þ 1Þ hðncÞ (km) Dh (km) npc (eV)

1 1:9 � 1013 12 127 35.6 18.9 1:7 � 1011

2 1:8 � 1014 121 239 17.2 5.5 4:9 � 1010

3 7:7 � 1015 221 341 12.3 3.5 3:1 � 1010

4 9:4 � 1017 309 430 9.6 2.6 2:4 � 1010

5 3:8 � 1020 380 502 8.0 2.2 2:0 � 1010

8 J.M.C. Montanus / Astroparticle Physics 59 (2014) 4–11

elongation rate. For energies above 1014 eV the curve is almostlinear:

Xmax;p � �140þ 40 lg E0: ð33Þ

In this energy region the elongation rate is constant. Clearly, thedepth of maximum on the basis of all the generations of gammashowers is small in comparison with Monte Carlo simulations: 40instead of 58 g cm�2 per decade of energy. An analytical estimatecan be obtained by considering solely the first generation of c’s[2,3]. Then, with the present model parameters,

Xmax;p � kp—air þ Xmax;cðE0=ð2MÞÞ � �485þ 64:4 lg E0: ð34Þ

That the corresponding elongation rate is close to value predictedby Monte Carlo simulations should be considered as coincidental.At this point it is instructive to consider also the second generationin the analytical estimation. To this end we leave the constants inthe expression for the multiplicity unspecified: M ¼ q � Ep. Thedepth of maximum of the first and second generation of gammashowers then respectively read

Xmax;p ¼ kp—airðE0Þ þ Xmax;cðE0=ð2qEp0ÞÞ ð35Þ

and

Xmax;p ¼ kp—airðE0Þ þ kp—airðE1Þ þ Xmax;cðE1=ð2qEp1ÞÞ: ð36Þ

With the substitution of E1 ¼ E0=ðqEp0Þ these expression reduce to

Xmax;p � ð79:7� 85pÞ lg E0 � 556� 85 lg q ð37Þ

respectively

Xmax;p � ð72� 162pþ 85p2Þ lg E0 � 356þ ð85p� 162Þ lg q: ð38Þ

For reasonable values for p and q, in the neighborhood of p � 0:18and q � 0:15, the depth of maximum of the second generation issmaller than the one from the first generation. The inclusion ofthe second generation in the analysis will therefore decrease thedepth of maximum.

5. Iron primary

The hadronic cascade for an iron primary differs from the onefor a proton primary by a smaller depth of first interaction andby a larger multiplicity in the first interaction. The iron–air crosssection is about 2000 mb, see Fig. 54 of [23]. The correspondinginteraction length is kFe—air � 12 g cm�2. The relatively smallenergy dependence of the iron–air cross section will only lead toa negligible difference of a few g cm�2. According to the superposi-tion model [17] the multiplicity of a composite nucleus withatomic mass A is equal to A times the multiplicity of a proton witha A times smaller energy:

Mch ¼ 0:1A � E0

A

� �0:18

: ð39Þ

For iron–air, A ¼ 56, this is Mch ¼ 2:7 � E0:180 . Since not all nucleons

will participate in the same rate as a single proton, the latter shouldbe multiplied by a factor smaller than unity. If this factor is (almost)independent on energy, the superposition model predicts a constantrate between the iron–air and proton–air multiplicity. This isindeed what is seen from QCD based models and from a color glasscondensate approach, see Fig. 5 of [7] and Fig. 7 of [24]. On the basisof Fig. 5 of [7] we take in our model the iron–air multiplicity as

Mch ¼ 0:3E0:180 ; M ¼ 0:45E0:18

0 : ð40Þ

In the absence of elasticity the iron–air multiplicity is only presentin the first interaction. Except for these two adaptions the analysisis identical to the one for a primary proton. Starting with a primary

iron with energy E0 the energy of the particles after the first inter-action is

E1 ¼E0

0:45 � E0:180

� 2:2 � E0:820 : ð41Þ

Without elasticity the subsequent interactions are governed by thepion multiplicity:

E2 ¼E1

0:15 � E0:181

� 6:7 � E0:821 : ð42Þ

Repeating the iteration we find for the energy per particle after ninteractions

En ¼ 6:7an�1 � Ebn�11 ; ð43Þ

where an and bn are as defined in Section 3. Substitution of (41)gives

En ¼ 6:7an�1 � 2:2bn�1 � Ebn0 ; ð44Þ

For the interaction lengths we have for the primary iron, n ¼ 0,

k0 ¼ kFe—airðE0Þ ¼ 12: ð45Þ

The interaction lengths for the produced pions, n P 1, are as givenbefore.

kn ¼ kp—airðEnÞ ¼ 200� 3:3 lnðEnÞ: ð46Þ

With the substitution of (44) this is

kn ¼ 200� 6:3an�1 � 2:6bn�1 � 3:3bn lnðE0Þ; n P 1: ð47Þ

The atmospheric depth of the ncth interaction, nc P 1, becomes

XðncÞ ¼ 200nc � 188� 6:3Xnc�1

n¼1

an�1 � 2:6Xnc�1

n¼1

bn�1

� 3:3 lnðE0ÞXnc�1

n¼1

bn: ð48Þ

For the decay length of the charged pions after the ncth interactionwe now find

csc ½km� � 5:6 � 10�11 � 6:7anc�1 � 2:2bnc�1 � Ebnc0 : ð49Þ

So, for the depth of the decay of the pions we obtain the followingequation:

5:6 � 10�11 � 6:7anc�1 � 2:2bnc�1 � Ebnc0 ¼ 4 ln

Xðnc þ 1ÞXðncÞ

� �: ð50Þ

As before, we substitute integer values for nc and solve numericallyfor the primary energy E0. Results of interest are shown in Tables 4–6, the equivalents of Tables 1–3.

For the energy entries in Table 4 the maximum number ofelectrons and muons for an iron initiated air shower is calcu-lated in a similar way as for the proton initiated air showers.The resulting curves, see Fig. 5, are practically identical to theones for a proton primary in Fig. 3. A linear fit, see the dashedlines in Fig. 5, for the maximum number of electrons (�) andnumber of muons yields

Table 5Subsequent interaction lengths in hadronic cascades for a primary iron. The kn and thecascade average �k are in g cm�2.

nc E0 (eV) k0 k1 k2 k3 k4 �k

1 1:9 � 1013 12 – – – – 12

2 1:8 � 1014 12 109 – – – 60

3 7:7 � 1015 12 98 110 – – 74

4 9:4 � 1017 12 85 100 111 – 77

5 3:8 � 1020 12 69 86 101 112 76

Table 6Subsequent multiplicities and their geometric mean in hadronic cascades for aprimary iron.

nc E0 (eV) M0 M1 M2 M3 M4 �M

1 1:9 � 1013 110 – – – – 110

2 1:8 � 1014 165 22 – – – 60

3 7:7 � 1015 325 38 20 – – 63

4 9:4 � 1017 773 78 36 19 – 80

5 3:8 � 1020 2274 189 73 34 18 114

iron

12 14 16 18 20

200

300

400

500

600

Log 10 E 0 eV

Xmax

gcm

2

Fig. 6. The depth of maximum shower size as a function of energy of an ironprimary for the situation with complete inelasticity (dots) and the correspondingcubic fit (dashed).

proton

iron

12 14 16 18 20

200

300

400

500

600

700

Log 10 E 0 eV

Xmax

gcm

2

Fig. 7. The depth of maximum shower size as a function of energy for proton andiron primaries for the situation with complete inelasticity (dots) and thecorresponding cubic fits (dashed).

J.M.C. Montanus / Astroparticle Physics 59 (2014) 4–11 9

Ne � 0:59 � ðE0½GeV�Þ1:010 ð51Þ

and

Nl � 0:0055 � ðE0½GeV�Þ1:016 ð52Þ

respectively. As for proton primaries, the present model values ofNe þ 25Nl as a function of primary proton energy E0 are close tothe ones obtained with Monte Carlo models, see Fig. 3 of [22]. Thecompositional insensitivity of the relation between Ne þ 25Nl andE0 is of interest for energy parameterization [22].

Also the depth of maximum shower size for an iron primary iscalculated in a similar way as for a proton primary. The result isshown in Fig. 6.

Both the smaller interaction length and the larger multiplicityhave reduced the depth of maximum shower size with respect toa proton initiated shower. A cubic fit, see the dashed line inFig. 6, yields

Xmax;Fe � �2116þ 329:4 lg E0 � 14:43ðlg E0Þ2

þ 0:237ðlg E0Þ3; ð53Þ

where the subscript Fe identifies the primary particle. As for protonshowers the curve is almost linear for energies above 1014 eV:

electrons

muons

12 14 16 18 200

2

4

6

8

10

12

Log 10 E 0 eV

Log

10N

Fig. 5. The maximum number of electrons (�) and muons as a function of theenergy of the primary iron (dots) and the linear fits (dashed).

Xmax;Fe � �305þ 45 lg E0: ð54Þ

As for the proton an analytical estimate can be obtained by consid-ering solely the first generation of c’s:

Xmax;Fe � kFe—air þ Xmax;cðE0=ð2MÞÞ � �659þ 69:8 lg E0: ð55Þ

Also here the actual elongation rate is substantially smaller whenfurther generations are taken into account.

6. Energy distribution

In the Heitler–Matthews model as well as in the present exten-sion the energy of the interaction is assumed to be equally dividedover the secondary particles. In reality the distribution of energy ishighly inhomogeneous. This is observed in proton-proton colli-sions, see Fig. 48 of [25]. For proton–air collisions it is predictedby Monte Carlo models, see Fig. 6 of [18]. Many secondaries obtaina small part of the energy while a few particles obtain a larger part.The elasticity effect, where a substantial part of the energy is takenby the leading particle, can be regarded as the most profound man-ifestation of the inhomogeneous energy distribution. As will beargued both the elasticity and the inhomogeneous energy distribu-tion over the non-leading secondaries increase the depth of maxi-mum shower size. In our opinion the equal division of energy istherefore responsible for a substantial part of the discrepancy withrespect to shower simulators which have the inhomogeneous

10 J.M.C. Montanus / Astroparticle Physics 59 (2014) 4–11

energy distribution incorporated. To give some foundation to theidea we consider the following relation [26]:

DXmax

Xmax� �1

2DMM� 1

10Djj: ð56Þ

In this relation DXmax is the shift of the depth of maximum showersize, M is the multiplicity and j is the inelasticity. For protons andpions the inelasticity is roughly about 2

3, see Fig. 6 of [8]. Startingfrom the inelastic situation, j ¼ 1, the change in inelasticity isDj ¼ � 1

3. According to the second part in the right hand side ofEq. (56) this corresponds to a shift

DXmax �1

30� Xmax: ð57Þ

That is, about one seventh of the discrepancy can be explained byelasticity.

Also the inhomogeneous energy distribution over the non-lead-ing secondaries increase the depth of maximum shower size.Among the secondary charged pions there will be a few with rela-tively large energy who penetrate deeper into the atmospherethereby contributing to the depth of maximum shower size in asimilar way as elasticity does. At the same time there will be manysecondary charged pions with energies so low that they will decaybefore they reach final generation of the cascade. In effect thisreduces the cascade mean multiplicity. From the first part of theright hand side of Eq. (56) it follows that a substantial contributionto the shift DX can be expected. Since the effective reduction ofmultiplicity will be larger for large multiplicities and thus for largeenergies, the relative shift will be larger for large energies. As aconsequence the inhomogeneous energy distribution over thenon-leading secondaries does increase the depth of maximumshower size as well as the elongation rate. To quantify these state-ments we consider the first generation prediction (37) and investi-gate its sensitivity for the multiplicity. A change in the multiplicityM ¼ q � Ep can be obtained by a change of q and a change of p. Fromthe derivative of (37) with respect to q we obtain

@Xmax;p

@q� �37

q: ð58Þ

A decrease of the multiplicity by a decrease of q leads to an increaseof Xmax;p independent of the energy E0. Alternatively, a decrease of qincreases solely the absolute level of Xmax;p. Variations of q do notaffect the elongation rate. From the derivative of (37) with respectto p we obtain

@Xmax;p

@p� �85 lg E0: ð59Þ

Here a decrease of the multiplicity by a decrease of p leads to anincrease of Xmax;p proportional to lg E0. This is slightly suppressedby second and further generations. The important conclusion how-ever is that the variation of p affects both the absolute value ofXmax;p and the elongation rate.

Since it makes a difference whether the multiplicity is varied byq or by the power p it is better to distinguish the sensitivity ofXmax;p for it. That is, instead of expressing DXmax=Xmax in terms ofDM=M we express it in terms of Dp=p and Dq=q. The substitutionof (58) and (59) in

DXmax;p

Xmax;p¼ 1

Xmax;p

@Xmax;p

@pDpþ 1

Xmax;p

@Xmax;p

@qDq; ð60Þ

leads to

DXmax;p

Xmax;p¼ �85 lg E0

Xmax;pDp� 37

Xmax;p

Dqq: ð61Þ

From M ¼ qEp it follows that DM=M ¼ Dq=q andDM=M ¼ 2:3 lg E0Dp. So, if the multiplicity is varied solely by eitherq or p, we can write (61) as

DXmax;p

Xmax;p¼ � 37

Xmax;p

DMM

: ð62Þ

For a depth of maximum of, say, Xmax;p � 550 g cm�2 the latter is ofthe order

DXmax;p

Xmax;p� � 1

15DMM

: ð63Þ

The latter suggests that the factor 12 in (56) is an overestimation of

the sensitivity of the depth of maximum for variations of themultiplicity.

7. Conclusions and discussion

Hadronic cascades in cosmic air showers are analyzed by meansof a Heitler–Matthews model extended to all generations of pions.For all the predictions a multiplicity and interaction length isapplied parameterized for energy. It is argued that the thicknessof the interaction layers should be taken equal to the interactionlength and not a fraction ln 2 of it. Although this increases the pre-diction for the depth of maximum shower size with a few tens ofg cm�2, the value for Xmax still is too small in comparison withMonte Carlo simulations. It is shown that an analysis based onall the generations in the hadronic cascade does lead to smallerelongation rates than an analysis solely based on the first genera-tion of c showers. The agreement of the latter with the Monte Carloprediction for the elongation rate can therefore be considered ascoincidental. The depth of maximum curves for proton and ironprimaries as predicted by the present model are both shown inFig. 7.

The curves do show some similarities to the ones resulting fromMonte Carlo simulations [9,18,27,28]: they are almost linear forenergies larger than 1014 eV, at low energies the elongation ratetends downwards for increasing energies and the separationbetween the proton and iron curve is in agreement with MonteCarlo simulations. Also the elongation rate for iron being largerthan for proton, and the corresponding decrease of the separationfor increasing energy is in agreement with Monte Carlo simula-tions. From the present model this can be understood as follows.The smaller interaction length causes a lower level for the ironcurve. If the iron and proton multiplicities would be identical thecurves would run parallel. The larger iron multiplicity has twoeffects. It reduces the energy of the first generation of subshowersand a larger initial energy is required for the shower to survive thesame number of interactions. The first effect decreases the level ofthe iron curve, while the second effect shifts the curve to largerenergies. Since the second and further generations are governedby pion multiplicities, the cascade average multiplicity for irontends towards the one for a proton primary. As a consequencethe separation between corresponding dots in Fig. 7 becomes smal-ler for larger energies, resulting in an iron curve that tends to theproton curve for increasing energy.

Despite the agreements with Monte Carlo simulations there aretwo important differences: the absolute levels of the curves are toosmall and the elongation rates are too small. For a proton and ironprimary the depth of maximum shower size as predicted by MonteCarlo models, see for instance the right panel of Fig. 10 of [9],Fig. 13 of [18] or Fig. 9 of [27], is about

Xmax;p � �310þ 58 lg E0 ð64Þ

and

Xmax;Fe � �580þ 67 lg E0: ð65Þ

J.M.C. Montanus / Astroparticle Physics 59 (2014) 4–11 11

respectively. The comparison with the present model predictions(33) and (54) learns that the Monte Carlo predictions are largerby about 100 g/cm2 at 1014 eV up to more than 150 g/cm2 at1020 eV. In our opinion a substantial part of the discrepancy maybe caused by the homogeneous energy distribution of the secondar-ies in the discrete model. An inhomogeneous energy distributionwill effectively reduce the multiplicity. As argued in the previoussection a decrease of the power p in the relation M ¼ qEp doesincrease both the depth of maximum and the elongation rate. Bothare needed to explain the discrepancy with Monte Carlo simula-tions. The fact that the present model prediction for the elongationrate also needs to be increased in order to match with Monte Carlosimulations can even be regarded as a support of the present inclu-sion of all the generations of electromagnetic showers.

Of course one can think of several other possible contributionsto the discrepancy. For instance, the present model is discrete. Theinteractions occur at discrete intervals and the stopping occurs rig-idly when the decay length is half the interaction length. It cannotbe excluded that the inclusion of statistics in these processes willaffect the prediction for depth of maximum. This is beyond thescope of the present discrete model.

Acknowledgments

I am grateful to the reviewers for their useful comments. I wishto thank Dr. J.J.M. Steijger for his detailed and useful comments onan earlier draft of this paper. I wish to thank prof. J.W. van Holtenand Prof. B. van Eijk for their useful comments and encouragement.I wish to thank Nikhef for its hospitality. The work is supported by

a grant from NWO (Netherlands Organization for ScientificResearch).

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