# Superposition model test in small CORSIKA shower simulations

Tadeusz Wibig  Department of Theoretical Physics, Faculty of Physics and
Applied Informatics, University of Lodz,
Pomorska 149/153, PL-90-236 Łódź, Poland<EMAIL_ADDRESS>
###### Abstract

The calculation of the flux of particles reaching the earth’s surface, as well
as cases of simultaneous arrival at the observation level of groups of
particles – very small Extensive Air Showers, require simulation calculations
and taking into account a much greater extent of fluctuations of the
development of the showers generated by very low energy cosmic ray particles,
much greater than usual for showers with energies near the “knee” and above
due to the very steep cosmic ray energy spectrum. These fluctuation depend
significantly on the mass number of the primary cosmic ray particle. Nuclei
are sets of nucleons and it seems an obvious approximation to assume that the
extensive air showers initiated in the upper atmosphere by atomic nuclei are a
convolution of the showers generated by single nucleons. This is called the
superposition hypothesis. In this paper, we test this superposition model. We
will show, using results of a detailed simulation program CORSIKA, that the
superposition assumption is not fulfilled for higher moments of the
distributions of the main characteristics of extensive air showers. We will
discussed some extensions of the simply superposition model that ensure
consistency with the results of detailed simulations.

Keywords: Cosmic Rays, Extensive Air Showers, Simulations, Superposition,
Nuclei Interactions.

##  I Introduction

The primary cosmic radiation consists mainly of hydrogen nuclei (protons), but
alongside them, as we have known since the middle of the last century, there
are nuclei of heavier elements. Nuclei heavier than iron no longer make a
significant contribution to the total flux, although they are important
targets of studies on the origin and propagation of cosmic radiation.

The steep cosmic ray energy spectrum causes that the flux of secondary
particles reaching the earth’s surface is determined by the flux of primary
particles with relatively low energies. They interact with the nuclei of atoms
of the Earth’s atmosphere and when they exceed a certain, not very high
threshold energy, they may initiate cascades of secondary particles. And if in
the majority of cases these cascades fade away during the propagation through
the atmosphere, if there is only a slight probability that something from them
will reach the surface of the Earth, after having taken into account a
significant gradient of the energy spectrum, these highly fluctuated cases
constitute the majority of incoherent particles, or particles associated in
small groups - very small cosmic ray showers observed at the ground level.
Fluctuations in the development of Extensive Air Showers (EAS) depend on many
factors. One of these is the mass of the primary particle. Understanding how
it influences the development and fluctuations of EAS can greatly facilitate,
and increase the accuracy of calculations as a tool for the interpretation of
very small cosmic ray showers or single secondary cosmic ray particle flux
measurements.

The fundamental role in the shower development is played by inelastic
interactions of primary cosmic ray particles. As a result, secondary hadrons
are created, mainly pions, and among them about 1/3 are neutral pions, which
immediately decay into two gamma quanta initiating electromagnetic cascades.
Electrons (positrons) and photons make up the vast majority of particles in
showers. Their interactions, which are described by quantum electrodynamics,
are not a serious physical problem, although it would be a mistake to think
that we know everything about them. The main problem of the simulations of the
high energy primary cosmic cascades is the number of its constituents going
into billions, which makes precise simulation calculations difficult and
partly even impossible. Hadrons, nucleons in atomic nuclei interacting
strongly do not pose such a problem. In this case, an important limitation is
a lack of knowledge of physics of multiparticle production processes. Most
often we are dealing with a small four-momentum transfer soft secondary
particle creation. It is described by more or less phenomenological models of
particle production extrapolating available data from accelerator experiments
by many orders of magnitude in energy and to areas of momentum immeasurable in
collider experiments.

In the lowest energy region of the cosmic ray particle energy spectrum, which
is the field of our main interest in this work, both the description of the
electromagnetic component and the phenomenology of strong interactions, due to
the multitude of data collected over the years, the situation apparently looks
much better. Simulation description of physics of small showers (we will
define hereafter “small showers” those induced by particles of energies below
the “knee” $<10^{15}$ eV) does not seem to cause any particular difficulties.
However, there appear other specific problems connected with the very steep
slope of the cosmic ray energy spectrum. As it is commonly known, the spectrum
is power-law with index (differential) of about 3. This spectrum is imposed by
Nature as an a priori factor in Bayesian approach whenever we want to compare
the results of simulation calculations, which, out of obvious necessity, are
made for a fixed energy of the primary particle, with measurements, which are
just an average taking into consideration all a priori factors, and,
therefore, also the steepness of the energy spectrum.

This problem of course exists for all energies, but the influence of any
fluctuations for higher energies is taken into account relatively easily. By
virtue of the law of large numbers, we expect that the fluctuations will have
a normal (possibly log-normal) distribution with a small relative width and an
appropriate integration with a power spectrum gives a certain shift on the
energy scale without significantly changing the nature of the observed
relations. In the case of small and very small showers in extreme cases we
deal with distributions of observables which are small natural numbers or
distributions which are not normal and have additionally significant relative
width (dispersion). Due to the steepness of the energy spectrum in cases of
small showers, when integrating (averaging with the spectrum as a prior) we
practically do not operate on average values, and sometimes we are quite far
from them. To take such great fluctuations into account in simulation
calculations, it is necessary to simulate a sufficiently great number of
events and, although simulating a shower initiated by particles with energies
of the order of TeV does not take much computing time, however, when such
simulations must be repeated millions of times we stand in a situation similar
to the calculations for ultra high-energy cosmic rays.

##  II CORSIKA program

Modelling of strong interactions has a long history. Majority of models
currently available on the market were developed a long time ago and are
continuously refined and improved as new data become available. These models
have to be implemented into the geometrical structure of EAS particles
transport through the Earth’s atmosphere. Complementing them with a much
better understood formalism of electromagnetic cascades and descriptions of
other more and less relevant processes leads to processes that simulate the
development of EAS. One of the most commonly used programs for this purpose
today is the CORSIKA [1, 2] developed over 30 years ago in Karlsruhe for the
KASCADE experiment [3, 4]. Since then this program has been significantly
extended and developed. Over the years, new features have been added and the
computational capabilities have been extended to meet the computational needs
of new cosmic ray experiments and it is now used also for simulations at the
highest observed energies (even up to 1021 eV).

The CORSIKA program provides users with a number of options. They concern both
the structure of the simulation algorithm itself, including models of high-
and low-energy interactions, which should be attached to the basic simulation
scheme, as well as those connected with the parameters of the simulated
showers, which are of interest to the user (Cherenkov radiation, radio
emission, and atmospheric neutrinos). There are so many options that just
listing them with the minimum necessary description takes almost 200 pages in
the latest version of the CORSIKA manual [2]. The already linked program
requires run parameters to be set using the control cards. In principle, for a
typical simulations the default set embedded in the program itself is
selected. However, if you want to use the CORSIKA program for a purpose
slightly or more different than typical, you have to set the parameter values
particularly carefully. For the purposes of this work, we mainly want to use
the program to simulate small and very small showers. For example, in our
case, it is essential to determine precisely the size of the fluctuations we
are dealing with at the lowest energies, so we must not use any thinning
option.

##  III Superposition model

Most of the observed characteristics of EAS depend not only on the energy of
the primary particle, but also on its mass (mass number). The primary cosmic
ray flux is dominated by the nuclei of hydrogen and helium, the CNO group, a
widely blurred group of ’intermediate’ nuclei: Ne, Mg, and Si, and the iron
group, never the less we know that the whole Mendeleev table can come from
above. The mass composition of the primary cosmic radiation is known (in the
small shower energy region) with a good accuracy, but anyway the mass of the
primary particle is another factor to be taken into account when averaging
simulation calculations, which increases their calculation time several times.

In order to get rid of at least this problem, for almost half a century in the
EAS physics a quite obvious model called the superposition model has been
functioning [5, 6]. Since the atomic nucleus from the point of view of high-
energy processes of particle production is a collection of nucleons (protons
and neutrons, which do not differ at all in the aspect under discussion) it
can be treated this way and one can assume that the EAS initiated by the
nucleus of, for example, iron is a sum, a superposition, of 56 proton showers,
where each proton would have 56 times less energy than the nucleus of iron as
a whole.

This undoubtedly attractive model made it possible to formulate at once, as if
by definition, many theorems concerning the comparison of, e.g., iron and
proton initiated showers, which have been used and are still in use for the
analysis of the mass composition of the primary cosmic rays in experimental
studies.

##  IV Average EAS characteristics

###  IV.1 Shower size

First of all, from the very definition there is a relation between the number
of particles in proton and iron showers:

$N_{e,\mu}({\rm iron},E)=56\times N_{e,\mu}({\rm proton},E/56)$ (1)

(surely, an analogous relation holds for showers initiated by any nuclei with
other masses). $N_{e}$ and $N_{\mu}$ denote here the number of electrons (and
positrons) and muons at the observation level, which will be hereafter called
electron and muon shower size. This relationship is illustrated in Fig.1.

Figure 1: Average values of the number of electrons (circles) and muons
(squares) at the sea level in vertical shower obtained using CORSIKA as a
function of energy per nucleon of the primary particles the lower scale (the
total proton energy of the proton showers). The upper energy scale shows the
total energy of the iron nuclei. The graph shows at the same time results for
proton showers (filled symbols and the left axis for the number of particles)
and scaled-up iron showers (open symbols and the right axis). The solid lines
show, respectively, the results of CORSIKA fits to the proton shower sizes
used to obtain the results predicted by the superposition model for iron
showers.

As we can see, an obvious agreement occurs not only in the area of relatively
high energies, where we are dealing with purely power-law relations, but also
in areas where the average numbers of shower particles are smaller than 1,
which means that there is a significant number of showers that do not reach
the observation level (in our case the sea level) at all.

###  IV.2 Shower particle radial distributions

It has been known for many years that the transverse distributions of
particles in extensive air showers are well described by a simple formula
proposed by Greisen [7]. Its validity was confirmed by theoretical
considerations and numerical calculations with respect to electromagnetic
cascades by Kamata and Nishimura [8], hence its commonly accepted name:
Nishimura-Kamata-Greisen (NKG) function (which we adopted also for the muon
distribution):

$\rho_{e/\mu}(r)={N_{e/\mu}\over 2\pi
r_{0}^{2}}\>{\Gamma(4.5-s)\over\Gamma(s)\>\Gamma(4.5-2s)}\left({r\over
r_{0}}\right)^{s-2}\left({1+{r\over r_{0}}}\right)^{s-4.5}~{}~{}~{},$ (2)

where $N_{e/\mu}$ is the electron/muon shower size, $r_{0}$ is a radial scale
parameter called in the case of the electromagnetic cascade theory the Molière
unit equal to about 2 radiation lengths units above the observation level (for
muon distribution its value was adjusted and parametrized using the CORSIKA
results) and $s$ is the age parameter.

Figure 2: Average radial distribution of electrons (circles) and mouns
(squares) in CORSIKA showers of energy $10^{13}$ eV (left plot), $10^{14}$ eV
(middle plot), and $10^{15}$ eV (right plot) initiated by proton (black filled
symbols) and iron nuclus (open symbols). Solid lines for proton induced
showers, dashed for iron showers.

Examples of average radial distributions for the electron and muon components
of CORSIKA showers are shown in Fig.2 for primary protons and iron nuclei of
energies 10 TeV, 100 TeV and 1 PeV. The solid lines show the results obtained
from a smooth parametrization of the age parameter $s$ in Eq.(2) using the
average electron and muon distributions for proton showers over the whole
analysed energy range, also for very small showers.

The results for iron showers shown in Fig.2 with dashed lines have been
obtained according to the superposition principle from the fits to
correspondingly lower energy proton showers. The normalization is taken from
the results shown in Fig.1.

For showers containing on average about 1 particle, which corresponds to a
(mean) central density of 10-6 per m2 (100 GeV proton, – Fig.1) the NKG
function does not yet do a poor job of describing the mean distributions of
electrons and muons, but at central densities of 10-8 per m2, which is the
case, as seen in Fig.1 for 10 GeV protons and iron nuclei with a total energy
of about 1 TeV the distributions can be considered almost uniform and shower
particles can appear almost everywhere up to a distance of several hundred
meters from the shower axis.

For the purposes of this work, the lack of a statistically justified mean
distribution of particles in very small showers is not of particular
importance, since the integrals of these distributions we eventually aim for
will be normalized by the total number of particles in such showers, which
will be of the order of 0.01, as shown in Fig.1.

###  IV.3 Shower longitudinal profile

The average number of particles in the showers and their average lateral
distributions show in the low and very low energy region agreement with the
superposition principle.

Analyzing the application of the superposition model in the case of small
showers, it is worth to pay some attention to the longitudinal development of
the showers. The features of interest here in general are very difficult for
any precise observations and measurements. Nevertheless, the comparison of the
longitudinal development of showers resulting from the CORSIKA program in the
case of proton and iron induced showers may provide important information
regarding the considered superposition hypothesis. The longitudinal profile of
the shower provides an opportunity to follow the process of successive nucleon
interactions of the initial heavy nucleus and the superposition mechanism.

In the case of a shower initiated by a proton, its size at the maximum (the
largest number of particles that the developing shower reaches at a certain
point) is determined to a good approximation by the energy of the primary
particle.

$N_{\rm max}=\frac{1}{2}\frac{\langle n\rangle}{3}\>\langle K_{\rm
inel}\rangle\>\frac{E}{\epsilon_{c}}$ (3)

where $\epsilon_{c}$ is the critical energy of electron in air, $\langle
K_{\rm inel}\rangle$ effective interaction inelasticity coefficient and
$\langle n\rangle$ is the effective pion multiplicity [9]. It is a normalising
factor for the longitudinal shower development function describing the number
of particles at the depth $x$ in the atmosphere and can be described by
different phenomenological formulas [10]. We have decided to use the relation
called the Gaisser-Hillas function [11] in our superposition hypothesis
examination.

$N(x)=N_{\rm max}{\left(\frac{x-x_{0}}{x_{\rm
max}-x_{0}}\right)}^{\left({\frac{x_{\rm
max}-x_{0}}{\lambda}}\right)}\>\exp{\left(-\>\frac{x-x_{\rm
max}}{\lambda}\right)}$ (4)

In the case of heavy nucleus we are dealing with $A$ nucleons each of which
can act at a specific depth in the atmosphere $x_{i}$ ($i=1,2,3,\ldots,A$). If
$F_{A}(\\{x_{i}\\})$ describes the $A$-dimensional probability density of the
set of positions $x_{i}$, for the superposition model, when all $x_{i}$s are
independent of each other, we have

$F_{A}\left(\\{x_{i}\\}\right)=\prod\limits_{j=1}^{A}\frac{1}{\lambda_{p}}\>\exp\>(-x_{j}/\lambda_{p})$
(5)

where $\lambda_{p}$ is the interaction length of proton.

It may be noted, obviously, that the averaged longitudinal profile of a shower
initiated by a cosmic ray nucleus of mass $A$ is in this case equal to the sum
of $A$ average profiles of proton showers of correspondingly smaller ($E/A$)
energy only if interaction properties ($\langle K_{\rm inel}\rangle$, $\langle
n\rangle$) and cross section ($\lambda_{p}$) does not change with the particle
energy.

This leads to the fact that the average depth of the maximum of the iron
shower is exactly in the exact place of the maximum of the shower for the
proton shower with the initial energy 56 times smaller.

$x_{\rm max}({\rm iron},E)=x_{\rm max}({\rm proton},E/56)$ (6)

The situation for the longitudinal development of CORSIKA showers is
illustrated Fig.3. As can be seen, a general agreement with Eq.(6) is
maintained. A small deviation visible for very small showers is a result of
the strong energy dependence of the interaction characteristics of nucleons
with energies below a few TeV. This situation can be handled by the so-called
elongation-rate theorem in its full form [12, 13].

Figure 3: Average depth of the shower maximum obtained with the CORSIKA for
primary proton (black circles) and iron (open squares) as a function of the
primary particle energy per nucleon. The line is the fit for vertical proton
showers.

##  V EAS fluctuations – higher moments

###  V.1 Shower size fluctuations

For a fixed energy of the primary particle, the electron and muon size of EAS
is surely not the same every time and we expect a fluctuation. These
fluctuations are an intrinsic property of reality and of the simulations and
result from the probabilistic nature of multi-particle production processes.
Fig.4 shows examples of electron and muon CORSIKA shower size spreads for
showers initiated by protons with energies of 1013 eV and 1015 eV. The lines
correspond to the Gaussian distribution (actually Log-normal) fitted to the
histograms shown. As can be seen, in all cases the Log-normal distribution
describes the scatter of sizes quite well.

Figure 4: Examples showing spread of electron (a, c) and muon (b, d) numbers
in CORSIKA vertical showers initiated by protons of energy of 1013 eV (a, b)
and 1015 eV (c, d). Lines shows respective fits of the Gaussian (Log-normal)
distribution.

For showers smaller than those illustrated in Fig.4, when the sizes are of the
order of a few particles, the Log-normal distribution is obviously no longer
the best. For such a small number of particles, we should expect an important
correlation between them caused by their common origin from subcascades, which
developed accidentally just above the level of observation. In addition, of
course, in such cases, Poisson-type statistical fluctuations are superimposed
on fluctuations in their number caused by the probabilistic nature of the
phenomenon itself.

For the superposition model, we have, taking into account that 56 proton
showers consisting of one iron shower are independent from each other and each
of them fluctuates as in the examples shown in Fig.4, dispersions of the
distribution of the total number of electrons (and muons) for the proton and
iron EAS should satisfy the relation

$D_{e,\mu}({\rm iron},E)~{}=~{}\frac{1}{\sqrt{56}}~{}~{}D_{e,\mu}({\rm
proton},E/56)$ (7)

Figure 5: Dispersion of the number of electrons a) and muons b) in vertical
CORSIKA shower as a function of the energy per nucleon of the primary
particles. There are simultaneously shown results for dispersion of proton
showers (black dots, and left axis) and scaled-up iron showers (open symbols,
and right axis). The thick solid lines represent smooth fit for CORSIKA proton
showers. Thin lines represent iron shower results of the superposition model
with additional small statistics conditions taken into account.

Fig. 5 shows the CORSIKA proton shower results represented by black symbols
compared with iron shower results (empty symbols). The first thing that is
noticeable is that the scaling according to Eq.(7) does not work for CORSIKA
showers. Especially in the case of electron size, Fig.5(a), the points
corresponding to iron showers after rescaling corresponding to the relation
Eq.(7) lie well above the points for proton showers. The superposition model
regarding the CORSIKA simulations shows clearly that fluctuations of the total
number of electrons in showers initiated by iron nuclei are wider than one
could expect (almost about a factor of 2 [14, 15]). In the case of the muon
component this difference is also present, but it is less significant.

In Fig.5 the thick lines show smooth fits to the CORSIKA results of dispersion
in proton showers, while the thin lines show the results obtained from a
superposition model that relates dispersion in iron showers to dispersion in
proton showers (with additional small statistics requirements taken into
account). The superposition model (taking into account the necessary
modifications for showers with very small number of particles at the
observational level) tends, in general to the relation given in Eq.(7).
Rescaled lines for iron showers are approaching lines for protons for muons
already above the energies of iron nuclei of about a few times $10^{12}$ eV,
and for electrons the agreement appears higher. However, the superposition
model results clearly do not agree with the points CORSIKA simulations.

###  V.2 Shower longitudinal profile fluctuations

A similar, but even more significant inconsistency is observed in the
dispersion of the position of the shower maximum. The situation is shown in
Fig.6. The CORSIKA results for proton showers are shown by black circles and
at the same time the $\sqrt{56}$-scaled dispersions for showers initiated by
iron nuclei are shown by empty squares. The superposition model Eq.(7)
suggests that all these points for proton and iron showers should lie on the
same line, but in CORSIKA reality they clearly do not.

Figure 6: Dispersion of the depth of the maximum shower obtained with CORSIKA
for primary proton (filled circles and left vertical scale for dispersion) and
primary iron (empty squares and right-hand scale for the respective
dispersion) as a function of primary particle energy per nucleon. The thick
solid line is our fit to the proton shower results. The thin line represents
result of superposition model of the iron shower with additional small
statistics conditions taken into account.

The dispersion of proton showers $x_{\rm max}$ points are joined by a thick
solid line. It is our fit to the CORSIKA points. It is used to obtain results
for iron showers in the superposition model described mainly by the equation
Eq.(5) completed by the Gaisser-Hillas function Eq.(4). According to the
superposition model, the $\sqrt{56}$-scaled dispersion for showers initiated
by iron nuclei should agree with the results for proton showers, and this is
indeed the case. Superposition model results for iron showers shown in Fig.6
by the thin line clearly asymptotically tend towards the agreement with the
results for proton showers.

However, the problem is that CORSIKA shows a significant inadequacy of the
simple model in this case: the thin line is far from connecting the empty
squares in Fig.dysf.

The dispersion of $x_{\rm max}$ values obtained from the CORSIKA program for
iron showers are about three times larger than expected for the superposition
model (at the same energy per nucleon). This is a clear signal that the simple
superposition model needs to be modified.

##  VI Modifications of the simple superposition model

The simple superposition model in the case of average $N_{e}$ and $N_{\mu}$
must lead to the relation given by Eq(1) and for $x_{\rm max}$ by Eq.(6)
(neglecting the mentioned energy dependence of interaction parameters of
particles) with the regardless of whether the nucleon sub-showers comprising
the iron shower are in any way dependent on each other or not. From the
physical picture of the hadron cascade development process, it is clear that
the subshowers are correlated to some extent. The interaction cross section of
a heavy nucleus with the nucleus of the atmosphere is clearly larger than the
cross section of a single nucleon interaction. The widely accepted Glauber
model [16, 17] of nuclear collision and its further evolution to the form of
the “wounded nucleon” model [18] explain the difference in cross sections by
the fact that a single interaction of nuclei involves several individual
nucleon-nucleon interaction at a time. The average number of interacting,
wounded nucleons is given by the ratio of active cross sections:

$\langle w_{A}\rangle=A\>{\sigma_{p-{\rm air}}\over\sigma_{A-{\rm
air}}}=A\>{\lambda_{A-{\rm air}}\over\lambda_{p-{\rm air}}}$ (8)

Figure 7: Distribution of number of wounded nucleon for Fe-N interactions at
the energy of 1 TeV/nucleon calculated for microscopic Glauber picture [19,
20] (solid line) and the respective binomial distribution with the same
average number of wounded nucleons shown by the dashed line.

The number of wounded nucleons fluctuatesand its distribution is in principle
unknown. As the first approximation we used the assumption of independence of
interactions of nucleons in the nucleus. Thus, the actual number of wound
nucleons must follow a binomial distribution. We called this solution the
wounded nucleon superposition model.

In the geometrical concept of the interaction of compound nucleons, which in
some sense underlies the Glauber theory, the individual collisions of nucleons
from a beam nucleus with some nucleon of a target nucleus are not, and cannot
be, independent. The distribution of the number of wounded nucleons can be
attempted to be determined using a simple geometric picture, but the
assumption that the number of wounded nucleons depends on the integrated
transparency of the two colliding nuclei may not necessarily be correct and
accurate. Detailed calculations [19, 20] show that the actual width of the
distribution of the number of wounded nucleons is much wider than would be
implied by the model of independent interactions with the same mean defined by
the transfer Eq.(8). This is shown in Fig.7.

Taking into account these more realistic distributions we obtain an increase
in the spread of the some shower characteristics. This applies both to the
total shower sizes (muon and electron) and to the width of the distribution of
the depth of the shower development maximum $x_{\rm max}$. We called this
version of the superposition model genuine wounded nucleon superposition
model.

Following the wounded nucleon idea, we need to modify Eq.(5) to take into
account the microscopic mechanism of nuclei interaction. If in the particular
interaction the number of wounded nucleons is equal to $w$, then just $w$
nucleons will interact in the same place, at the same depth in the atmosphere,
leading to the multiplication of identical (or almost identical) subshowers.

We must also assume something about the interaction of the remaining not-
wounded nucleons. Let us assume, as a first considered model, that the nucleus
in the first interaction fragments itself completely and all nucleons of which
it was composed travel further in the atmosphere as free protons. Then

$F_{A}\left(\\{\overbrace{x_{0},x_{0},...,x_{0}}^{w\>times},x_{w+1},x_{w+2}...,x_{A}\\}\right)=\frac{1}{\lambda_{Fe}^{w}}\>\exp\>\left(-\>\frac{w\>x_{0}}{\lambda_{Fe}}\right)\prod\limits_{j=1}^{A-w}\frac{1}{\lambda_{p}}\>\left.{\exp\left(-\>\frac{(x_{j}-x_{0})}{\lambda_{p}}\right)}\right|_{x_{j}>x_{0}}$
(9)

and if we also consider Eq.(8) we immediately obtain, that the distribution of
the value of the depth of the first interaction of the single nucleon $x_{i}$
(we neglect the unimportant ordering occurring in Eq.(9)) is the same as in
the model of the simple superposition described by Eq.(5) [19]. It follows
that the position of the EAS maximum, as determined, first of all, by the
depths of the first interactions, will be the same in the simple superposition
model and in the wounded nucleon model assuming the first interaction of the
nucleus as common for $w$ its nucleons and independent cascades of remaining
$A-w$ non-wounded nucleons starting at the first nucleus interaction point.
The correlation introduced in Eq.(9) also, of course, does not affect the
average shower size values and the averaged electron and muon radial
distributions.

The correlation introduced by the wounded nucleon superposition model, which
binds independent, in the simple superposition model, sub-cascades of single
nucleons, however, must lead to an increase in the fluctuations of both
electron and muon average shower sizes and to an increase in the spread of the
position of the shower maximum in showers initiated by heavier nuclei. This
effect is shown in Fig.8 where we show the dispersion of electron and muon
sizes for iron induced showers. The results for the simple superposition are
shown by the solid line (same as in Fig.4) and for the wounded nucleon
superposition model by the dashed line.

The increase in dispersion for the position of the shower maximum resulting
from the introduction of the correlation described above is about 5 g/cm2 and
is shown by the dashed line in Fig.9a. It does not explain the effect observed
for CORSIKA showers.

Figure 8:  Dispersion of the electron and muon size of CORSIKA iron showers
(shown already in Fig.5) as a function of energy per nucleon. Thin solid line
represents results of the superposition model (as in Fig.5), dashed line
wounded nucleon superposition model, and dotted line genuine wounded nucleon
superposition model, respectively.

The assumption of a complete fragmentation of the nucleus in its first
interaction obviously does not have to be correct. Another extreme case, when
non-wounded nucleons remain as one smaller nucleus, does not change the
situation given in Eq.(9) substantially, if we only consider the average
values of the discussed shower parameters. Replacing the product in Eq.(9) by
a complex of appropriate exponential factors for successively smaller nuclei
remaining after successive interactions leads also to the same distribution of
the first interaction positions. The contribution to the development of EAS
from successive interactions of fragmented or unfragmented remnants of primary
nuclei has no significant effect either on the size of the shower or on the
depth of its maximum. However, some effects are expected for higher moments of
the distributions of some characteristics of the shower parameters.

Figure 9:  Dispersion of the depth of the shower maximum obtained with the
CORSIKA for primary iron as a function of the energy per nucleon for total
fragmentation of the primary nucleus in the first interaction (a), and
assuming no fragmentation of the nucleus consisting of non-wounded nucleons
(b). Solid lines are obtained using simple superposition model, dashed
represents the wounded nucleon superposition model and dotted lines are the
result of the genuine wounded nucleon superposition model.

The dispersion of the flux distributions for the model with no fragmentation
is shown in Fig.8 by the dashed lines. The changes in the widths of the shower
size distributions are not large. They reach 10-20% as it can be seen in
Fig.8. However, the change is great when considering the characteristics of
the longitudinal distribution of the iron showers. A comparison of Fig.9a and
Fig.9b shows that the differences are as high as factor 2.

Of course, both assumptions about fragmentation of heavy nuclei are extremes
and the truth most likely lies in the middle. Our calculations show that in
the CORSIKA program it is closer to the situation of reduced fragmentation.
The binomial distribution of the number of wounded nucleons even in the
absolute lack of fragmentation does not lead to the seen in CORSIKA showers
width of the distributions of $x_{\rm max}$. A more realistic distribution
following the Glauber model predictions provides a sufficient margin for
reduced fragmentation models.

##  VII Conclusions

We used the CORSIKA program to obtain shower characteristics from the lowest
primary energies allowed by the program. The average values of the most
important parameters for shower: the total number of particles, the shape of
their lateral and longitudinal distributions and their second moments, were
found and analysed. We did this separately for the soft component - electrons
and for the hard component - muons.

Analyzing the electron and muon shower sizes, radial and longitudinal shower
particle distributions for different atomic masses of the primary nuclei, we
have examined the superposition hypothesis, saying that the extensive air
shower produced in the interaction of a nucleus of mass $A$ is a simple
compound of $A$ showers initiated by nucleons (protons) of appropriately lower
energy. The behaviour of the average shower size values agrees with this
assumption. However, investigating fluctuations of the shower size (electron
and muon), we have noticed that the assumption that the shower from nucleons
composing according to the superposition principle, independently of each
other, does not correspond to reality, or at least does not correspond to the
reality of the CORSIKA program.

We have investigated possible modifications to remedy this inconsistency by
including a correlation between some of the constituent subshowers.

We have tested the possibility that the fluctuations in the number of wounded
nucleons are independent, and therefore described by a binomial distribution
and a more realistic case distribution that more closely approximates the
collective geometric picture of the interaction. In both cases we obtained
better agreement with the results for the CORSIKA showers, but the most
sensitive shower parameter for resolving between these possibilities turned
out to be the position of the shower maximum, or more precisely the dispersion
of this quantity.

The magnitude $x_{\rm max}$ for very small shower energies is unfortunately
difficult to achieve, or not achievable experimentally at all. Apart from the
characteristics of the interaction act itself and interaction cross sections,
it depends also on the fragmentation model of heavy nuclei initiating the
showers. We have tested two extreme possibilities: either the nucleus
completely fragments already in the first interaction, or the non-wounded
nucleons propagate further in the atmosphere as one smaller nucleus.The
CORSIKA results suggest that the latter case is closer to reality while
allowing for the possibility of limited fragmentation, which seems to be in
accordance with expectations.

## References

## References

  * Heck  _et al._ [1998] D. Heck, J. Knapp, J. N. Capdevielle, G. Schatz, and T. Thouw,  _CORSIKA: a Monte Carlo code to simulate extensive air showers._ (KZKA-6019, Karlsruhe, 1998).
  * Heck and Pierog [2020] D. Heck and T. Pierog, Extensive air shower simulationwith corsika: A user’s guide (version 7.7402), Karlsruher Institute fur Technologie Report from December 18, 2020 (2020).
  * et al. [2003] T. et al., The cosmic-ray experiment KASCADE, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 513, 490 (2003).
  * H. O. Klages et al. [1998] H. O. Klages et al., The extensive air shower experiment kascade - first results, in  _Proceedings, 25th International Cosmic Ray Conference (ICRC1997) Durban, South Africa, 1997_, Vol. 8 (World Scientific, 1998) pp. 297–306.
  * de Beer  _et al._ [1966] J. F. de Beer, B. Holyoak, J. Wdowczyk, and A. W. Wolfendale, A theoretical analysis of extensive air showers i. near-vertical showers at sea level, Proceedings of the Physical Society 89, 567 (1966).
  * de Beer  _et al._ [1968] J. F. de Beer, B. Holyoak, H. Oda, J. Wdowczyk, and A. W. Wolfendale, A theoretical analysis of extensive air showers II. fluctuations of muon and electron numbers, Journal of Physics A: General Physics 1, 72 (1968).
  * Wilson and Greisen [1956] J. G. Wilson and K. Greisen,  _Progress in cosmic ray physics_ , Vol. 3 (Amsterdam: North-Holland, 1956).
  * Kamata and Nishimura [1958] K. Kamata and J. Nishimura, The Lateral and the Angular Structure Functions of Electron Showers, Progress of Theoretical Physics Supplement 6, 93 (1958), https://academic.oup.com/ptps/article-pdf/doi/10.1143/PTPS.6.93/5270594/6-93.pdf .
  * Stanev [2011] T. Stanev, 13th International Conference on Elastic and Diffractive Scattering (Blois Workshop) - Moving Forward into the LHC Era (2011), arXiv:1002.3527 [hep-ph] .
  * Matthews  _et al._ [2010] J. A. J. Matthews, R. Mesler, B. R. Becker, M. S. Gold, and J. D. Hague, A parameterization of cosmic ray shower profiles based on shower width, Journal of Physics G: Nuclear and Particle Physics 37, 025202 (2010).
  * Gaisser and Hillas [1977] T. K. Gaisser and A. M. Hillas, Reliability of the Method of Constant Intensity Cuts for Reconstructing the Average Development of Vertical Showers, in  _International Cosmic Ray Conference_ , International Cosmic Ray Conference, Vol. 8 (1977) p. 353.
  * Linsley [1977] J. Linsley, Structure of Large Air Showers at Depth 834 G cm-2 Applications, in  _International Cosmic Ray Conference_ , International Cosmic Ray Conference, Vol. 12 (1977) p. 89.
  * McComb and Turver [1981] T. J. L. McComb and K. E. Turver, The Elongation Rate Theorem and its Application, in  _International Cosmic Ray Conference_ , International Cosmic Ray Conference, Vol. 6 (1981) p. 234.
  * Gaisser  _et al._ [1982] T. K. Gaisser, T. Stanev, P. Freier, and C. J. Waddington, Nucleus-nucleus collisions and interpretation of cosmic-ray cascades above 100 TeV, Phys. Rev. D 25, 2341 (1982).
  * Schatz  _et al._ [1994] G. Schatz, T. Thouw, K. Werner, J. Oehlschlager, and K. Bekk, Validity of the superposition model for extensive air showers, Journal of Physics G: Nuclear and Particle Physics 20, 1267 (1994).
  * Glauber [1970] R. J. Glauber, Theory of high energy hadron-nucleus collisions, in  _High-Energy Physics and Nuclear Structure_ , edited by S. Devons (Springer US, Boston, MA, 1970) pp. 207–264.
  * Glauber and Matthiae [1970] R. Glauber and G. Matthiae, High-energy scattering of protons by nuclei, Nuclear Physics B 21, 135 (1970).
  * Bialas  _et al._ [1976] A. Bialas, M. Bleszynski, and W. Czyz, Multiplicity Distributions in Nucleus-Nucleus Collisions at High-Energies, Nucl. Phys. B 111, 461 (1976).
  * Engel  _et al._ [1992] J. Engel, T. K. Gaisser, P. Lipari, and T. Stanev, Nucleus-nucleus collisions and interpretation of cosmic-ray cascades, Phys. Rev. D 46, 5013 (1992).
  * Wibig [1998] T. Wibig, Inelasticity and intranuclear cascading in geometrical multichain model, Journal of Physics G: Nuclear and Particle Physics 24, 567 (1998).