Energy spectra of electrons in the extensive air showers of ultra-high energy
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2004 J. Phys. G: Nucl. Part. Phys. 30 97 (http://iopscience.iop.org/0954-3899/30/2/009)
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J. Phys. G: Nucl. Part. Phys. 30 (2004) 97–105 PII: S0954-3899(04)66710-7
Energy spectra of electrons in the extensive air showers of ultra-high energy
M Giller1, G Wieczorek1, A Kacperczyk2, H Stojek1and W Tkaczyk1
1Department of Experimental Physics, University of Lodz, Pomorska 149/153, 90-236 Lodz, Poland
2Department of Theoretical Physics, University of Lodz, Pomorska 149/153, 90-236 Lodz, Poland
E-mail: mgiller@kfd2.fic.uni.lodz.pl
Received 28 July 2003 Published 8 January 2004
Online atstacks.iop.org/JPhysG/30/97(DOI: 10.1088/0954-3899/30/2/009) Abstract
We show that the shape of the energy spectrum of electrons in an extensive air shower with an ultra-high energy (E 1019 eV) at a given level in the atmosphere depends only on the shower age at this level. It depends practically neither on the primary particle mass nor on its energy. This fact considerably simplifies interpretation of data from the experiments (e.g., Fly’s Eye, HiRes and Auger) determining cascade curves of single showers by the fluorescence technique. In particular, the contribution of the scattered Cherenkov light to the total flux produced by a shower can be easily taken into account. Our conclusion has been drawn by analysing results of Monte Carlo simulations of showers with the CORSIKA (Heck et al 1998 Report FZKA 6019) program, using the QGSJet interaction model.
1. Introduction
A determination of the cosmic ray energy spectrum at the ultra-high energy region (E0 > 1019 eV) has recently become one of the most urgent goals for the cosmic ray community. The question of the existence of the Greisen–Zatsepin–Kuzmin cut-off at E0 5 × 1019eV cannot be resolved on the basis of the present measurements. Moreover, there is a discrepancy between the results of two experiments, AGASA [2] and HiRes [3], such that, roughly speaking, the latter does see the cut-off, whereas AGASA data show a turn up rather than a cut-off. The two experiments use quite different methods to determine the primary energy of a shower. AGASA registers charged particles on the ground whereas HiRes measures fluorescence light emitted by the excited atmosphere and is able (in principle) to determine the whole cascade curve (number of charged particles as a function of depth in the atmosphere) of a shower. The discrepancy between the two experiments proves that there are
0954-3899/04/020097+09$30.00 © 2004 IOP Publishing Ltd Printed in the UK 97
98 M Giller et al
-3 -2 -1 0 1 2 3
log(E/GeV) 1e-03
1e-02 1e-01
1 Ne
dN dlnEe 0.7
1.3
1.0
Figure 1. Energy spectra of electrons at different shower ages s: 0.7, 0.8, . . . ,1.3. Each curve is an average of 10 proton showers with E0= 1019eV.
systematic errors in determining the cosmic ray flux as a function of energy and one of the reasons may be the determination of the primary energy itself.
Our investigation here refers to the problem of the shower primary energy as measured by the fluorescence light method. The transition from the measured light curve to the cascade curve is not quite trivial. It is mainly dependent on the accuracy of the assumed scattering characteristics of the atmosphere, this being, however, a separate problem, not discussed in this paper.
Another important factor here is contamination of the fluorescence flux by the Cherenkov light emitted by the shower particles and scattered in the atmosphere sideways. In this paper we show that its contribution can be treated in an unique way for all showers. It is a consequence of the fact that the energy spectrum of electrons at a given level in the atmosphere depends on the stage of the shower development only, i.e. on the age parameter s at this level.
2. Energy spectra of electrons
The shape of the electron energy spectrum at a given level should depend on the stage of the shower development there. A fixed depth (in g cm−2) in the atmosphere is not a good indication of this stage, even for showers produced by primary particles with the same parameters (mass, energy, direction) because of fluctuations in the shower development (mainly of the position of the first interaction and energy deposited for production of secondary particles). In the pure electromagnetic cascade theory it is the age parameter s, growing monotonically with the slant depth, that describes the cascade development. We will show that it can also be used to characterize the ‘age’ of hadronic showers, i.e. that the electron energy spectra can be described by the s parameter only, defined as
s= 3X/(X + 2Xmax) (1)
where X is the atmospheric path (in g cm−2) traversed by a shower, and Xmaxis the path to the shower maximum.
Figure 1 shows the energy spectra of electrons (normalized to 1) in proton induced showers, for various ages s, each averaged over 10 showers, simulated with the CORSIKA
-3 -2 -1 0 1 2 3
log(E/GeV)
1e-03 1e-02 1e-01
1 NdN dlnE
e e
1.3 0.7
1.0
Figure 2. Electron energy spectra in individual showers (three randomly chosen showers for each s= 0.7, 1, 1.3).
-3 -2 -1 0 1 2 3
log(E/GeV)
1e-03 1e-02 1e-01
0.7
1.3 1dN NdlnEe 1.0
e
p Fe
Figure 3. Comparison of the electron energy spectra for proton and iron initiated showers, at s= 0.7, 1, 1.3 for E0= 1019eV.
program [1] with the QGSJet model of nuclear interactions. For each shower Xmaxhas been determined separately. It is seen that the spectra become steeper while s increases, although the low energy part does not change much, particularly at later stages of shower development.
In the fluorescence experiments it is single showers that one has to analyse and one could worry whether fluctuations from shower to shower could change the shape of the spectrum.
Figure2illustrates this effect where spectra are presented for three randomly chosen showers for s = 0.7, 1 and 1.3. In the energy region where there are most particles the differences between the spectra are really small.
Next we check whether the spectra depend on the primary particle: on its mass and energy.
This is illustrated in figures3and4. From figure3it can be seen that for iron initiated showers
100 M Giller et al
-3 -2 -1 0 1 2 3 4
log(E/GeV) log(E/GeV)
5 10 15 20
0.7 1.01.3
protons s =
-3 -2 -1 0 1 2 3 4
5 10 15 20
s = 0.7 1.01.3
Fe
Figure 4. Ratio of electron energy spectra for E0= 1020eV to that for E0= 1019eV at different stages of shower development.
the spectrum is practically the same as that for a primary proton. It becomes slightly steeper only above∼100 GeV where the number of electrons constitutes only a small fraction of the total at this level.
Figure 4 shows that the shapes of the energy spectra at various stages of shower development are practically independent of the primary energy. In this figure there are plotted ratios of the energy spectrum for E0= 1020eV to that for E0= 1019eV, for three values of s, for proton (left) and Fe (right) showers. It is seen that a deviation from the flat line occurs only at high electron energies, but it exceeds 10% for energies larger than 103GeV where there are negligibly few particles. The scatter of points at low energies is probably caused artificially by the thinning method, used in simulations of showers with such high energies.
Thus, we have demonstrated that the shapes of the electron energy spectra do not depend either on mass or energy of the primary particle but do depend on the age parameter s only.
The idea that the shapes of particle spectra in the shower can be described by the age parameter comes from the analytical solutions for a pure electromagnetic cascade. The particle energy spectrum for energies Ecr E E0, where Ecris the critical energy of the medium in which the cascade develops and E0is the primary energy of electron or photon, is:
dNe
d lnE ∼ E−s (2)
with the same definition of the age parameter. As the extensive air shower is a sum of many electromagnetic cascades of different primary energies, initiated at different levels in the atmosphere, it is not obvious that the s parameter as defined by (2) should describe the shapes of the energy spectra in this case as well.
A fit to the electron energy spectra depending on s only, in showers of much lower primary energies, was proposed by Hillas [4]. We have checked that it does not describe the CORSIKA results for E0 = 1019–1020eV sufficiently well so that a new formula would be useful. We have fitted the following analytical formula describing quite well the shapes of the electron energy spectra for different ages:
1 Ne
dNe
d lnE = C(s)
1− a exp
−d(s) E Ecr
1 + E
Ecr
−[s+b·ln( E
cEcr)]
(3) where a = 1.005, b = 0.06, c = 189, d(s) = 7.06s + 12.48 and C(s) = 0.111s + 0.134 in 0.7 s 1.3 region.
-3 -2 -1 0
0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02
E /Ecrit
E /E
10-2 10-1 1 10 102
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02
s=0.7
-3 -2 -1
edN
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.022
crit
10-2 10-1 1 10 102
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.022
s=1.0
-3 -2 -1
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.022 0.024
crit
10-2 10-1 1 10 102
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.022 0.024
s=1.3
E /E
Figure 5. Comparison of the calculated energy spectraN1
e dNe
dlnE (histogram) with the analytical formula (dotted line).
We have adopted here Ecr = 80 MeV, as close to the critical energy of the air. Neis the total number of electrons at the level of age s.
The actual values of the parameters differ from our preliminary fit, presented at the ICRC in Japan [5]. (The fit presented here has been done to new simulations where the upper limit for the weight, ascribed to an electron in the thinning procedure, was much lower (105)than that assumed before (1030).)
The analytical form of this formula has been chosen such as to keep the power law (3) for E Ecr(thus the factor in brackets). For a hadronic shower the spectrum is, however, a bit concave (see figure1), so to allow for this we have introduced a small term in the power index, with constants b and c. For E ∼ Ecr and smaller the spectrum turns down and the factor responsible for this is that with the exponent.
A comparison of the calculated spectra with the fitted formula is represented in figure5.
In the region s= 0.7–1.3 the fit describes very well the electron spectra in the most important energy range EE
cr <100.
Another analytical form for the electron energy spectra has been independently proposed by Nerling et al [6]. It describes the results from CORSIKA simulations practically equally well as ours.
3. Light emitted by shower particles and a determination of the shower cascade curve In the highest energy showers practically all particles are electrons (of both signs), unless much below maximum shower where the contribution of muons becomes larger. Thus, we shall be concerned with electrons only.
102 M Giller et al
While traversing the atmosphere, the electrons excite the nitrogen molecules, which within∼10 ns emit fluorescence light, mainly in the violet region between 300 and 400 nm.
The number kflof the fluorescence photons emitted (isotropically) per unit length (kfl∼ 4–5 photons per metre) depends only weakly on the position in the atmosphere, as well as on particle energy (but see later), being thus a good signature of the number of particles at the atmospheric depth from where the light arrives at a detector. (To find the particle number the collecting area and solid angle of the detector have to be known, as well as the geometry of the shower.)
The detector registers, however, all the infalling light. Another important source of shower emission is the Cherenkov light produced by electrons in the atmosphere. Although the total Cherenkov flux produced by shower particles is much larger than that from fluorescence, it is collimated forwards (along the shower axis, unlike fluorescence, being emitted isotropically).
This forward Cherenkov flux is scattered sideways by atmospheric molecules (the Rayleigh scattering) as well as by aerosols (Mie scattering).
The main goal of the experiments based on the fluorescence technique is to detect showers of the highest energies, E0 1019 eV. As these are extremely rare they can be seen only at large distances (typically 10–20 km). Such a distant shower has a small lateral width and can be treated as a point light source moving along the straight line (shower axis) with the velocity of light.
In the experiments with the fluorescence detectors most showers are seen at rather large angles to their axes, so that it is the scattered Cherenkov light that arrives together with the fluorescence. In showers looked at angles20◦or so, the contribution of the directly produced Cherenkov light becomes large but we shall not discuss those here.
The number of photons n produced by a shower at a given slant depth x (measured from the top of the atmosphere in g cm−2), along a small path l (in metres) in a small solid angle
at angle θ to the axis, is a sum of fluorescence and Cherenkov light and equals
n(x)= nfl(x)+ nch(x)
= kflNe(x)l
4π + nch(x)ρ(x)l
λa(x) f (θ ) (4)
where nchis the number of Cherenkov photons at level x, ρ(x) is the atmosphere density at this level, λa(x)is the light attenuation length (in g cm−2) and f (θ ) is the fraction of all scattered photons that goes into the solid angle inclined by θ to the shower axis.
Were it not for the second term, the determination of the total number of electrons Ne(x) would be trivial (once the atmospheric attenuation from the shower to the detector is known).
The second term, though smaller than the first one, should not be neglected [7]. The number of Cherenkov photons nch(x)at a given level can be represented by the following integral
nch(x)=
x 0
T (x, x)dnch(x)
dx dx (5)
where
dnch(x) dx =
∞
Ethr(x)
dNe(E, x) dE
dn(1)ch(E)
dx dE. (6)
Formula (6) represents the number of Cherenkov photons produced at xby all electrons (above the threshold) per unit path. T (x, x)is the fraction of photons produced at x, arriving at level x. The number of photons produced by one electron with energy E per unit path is denoted by dn(1)ch(E)
dx and Ethr(x)is the electron threshold energy at this level, depending only on the refractive index of the atmosphere at x
Ethr √675 MeV
x/g cm−2
. Note that dn(1)ch(E) dx,
0.7 0.8 0.9 1 1.1 1.2 1.3 shower age
0.2 0.25 0.3 0.35 0.4 0.45
1 km 3 km 5 km 7.5 km 10 km height above
Auger level
Figure 6. Effective fraction of electrons F (s, h) emitting Cherenkov light as a function of shower age s and height h above the Auger level. The thick, almost horizontal lines correspond to typically developed showers for four cases (from top to bottom): proton 1020eV, proton 1019eV, Fe 1020eV and Fe 1019eV.
if dxis in g cm−2, does not depend on the height in the atmosphere. Its dependence on the electron energy is strong close to the threshold only: the Cherenkov flux saturates quickly with the electron energy, reaching∼90% of its maximum value at E 3Ethr, where kch = 172 photons/(g cm−2) in the (300–400) nm wavelength band.
It is convenient to introduce here the effective fraction of electrons, F, emitting Cherenkov light at a given level x, as
F = dnch(x)/dx
kchNe(x) . (7)
Then we have from (6) that F =
∞
Ethr(x)
dNe(E, x) NedE
dn(1)ch(E)
kchdx dE. (8)
As we have shown in the previous paragraph that the normalized to unity energy spectra,
1 Ne
dNe
dE, depend on the shower age s only, and dn(1)ch(E)
dx depends on E only, thus fraction F is a function of s and the height h in the atmosphere (because of Ethr(x)). Once an atmospheric model is adopted F (s, h) can be calculated from (8).
The result is presented in figure6. for the standard US atmosphere. Fraction F is shown as a function of shower age, for several heights above the level of the southern site of the Pierre Auger Observatory (Malargue in Argentina, 1452 m a.s.l). Analysing a shower light curve one has to know only the position of the shower maximum to determine the fraction F (s, h), without worrying about the shower primary energy or mass. A useful analytical form fitted to the calculated values is
F (s, h)= 0.66 − 0.26s − 0.0129h(km). (9)
For the real showers there is, however, a strong correlation between s and h, so that not all regions in figure6 are equally important. The thick, almost horizontal, lines correspond to four typically developed showers with E0 = 1019eV and 1020eV, both for primary proton
104 M Giller et al
and iron nucleus. It is seen that the actual fraction of ‘Cherenkov electrons’ does not vary much along the shower, decreasing by less than 15%. This is a result of two effects working in opposite directions. As the shower develops the energy spectrum becomes steeper, so that the number of electrons above a fixed energy decreases. At the same time, however, the threshold energy for Cherenkov radiation goes down, almost compensating the first effect.
Formula (4) for the number of photons emitted towards the detector by a shower path element l can now be rewritten in the following way
n(x)= Ne(x)
kfl
4π + ρ(x) λa(x)f (θ )
x 0
T (x, x)Ne(x)
Ne(x)F[x(s, h)] dx
l. (10)
It should be mentioned, however, that measurements show [8] that the number of fluorescence photons is proportional to the energy released for ionization rather than to the number of electrons. This effect is rather small as most electrons have relativistic energies within some two order of magnitudes. It is, however, easy to allow for it once the shower age s is assigned to the position in the atmosphere x. The first term kfl/4π in formula (10) should then be multiplied by a correction factor dependent on s only and equal to
∞
0
dNe
NedE
dWion(E) dx
dWion(E1)
dx dE (11)
where dWiondx(E) is the energy loss rate of an electron with energy E, and the energy E1
corresponds to the electron emitting exactly kflphotons per unit length.
Assuming that all the necessary parameters of the atmosphere (ρ(x), λa(x), f (θ ), T (xx)) are known, one can calculate Ne(x)from (10) once n(x) is determined by the experiment and the shape Ne(x)/Ne(x)of the cascade curve above level x is known.
The shape of the total number of electrons Ne/Nmaxwas studied by the HiRes group [10]
with the CORSIKA program for constant E0 = 1018eV with the conclusion that it does not fluctuate much. Our studies show, however, that it changes systematically with E0. We find that the shapes expressed as a function of x− xmaxvary little with primary mass, energy and fluctuations of xmax. Nevertheless, assuming their unique shape would be perhaps too a crude approximation, the more so as this shape depends on the nuclear interaction model.
The information about Ne(x)/Ne(x)can be obtained by the following methods.
One can adopt a Gaisser–Hillas function for Ne(x)(as has already been done in the Fly’s Eye [9] and HiRes experiment) with four free parameters and fit the best parameters to n(x).
Another way is to assume that the contribution of Cherenkov light from the highest bin
x is negligible. Regarding the integral over x in formula (10) as a sum of the integrand over all highest bins x, it is possible to find Ne(x1), Ne(x2)and so on from the consequent equations for n(x1), n(x2), and so on.
A method used so far by the Auger Collaboration starts by assuming no Cherenkov contribution anywhere in the light curve. Then Ne(1)(x) is derived and the Cherenkov contribution calculated from it for different x. This contribution is then subtracted from the observed flux and the whole procedure is repeated (Ne(2)(x)derived and so on). This procedure is convergent, however, only if the Cherenkov flux is small when compared to the fluorescence. Although most often it does work, it sometimes fails.
This paper shows that the scattered Cherenkov contribution can be taken into account right from the start of the analysis, and in a unified way for all showers. Our approach does apply to all showers independent of the value of the Cherenkov contribution.
4. Conclusions
We have shown that the shape of the electron energy spectrum in the ultra-high energy extensive air shower depends only on the age parameter s of the shower at the level considered. The spectrum shape does not depend on primary particle mass or on its energy. It can be fitted quite well by an analytically represented function of s.
The effective fraction of electrons F contributing by the Cherenkov radiation to the total light emitted can be expressed as a function of shower age and height in the atmosphere only (for the actual showers it is almost constant). It simplifies much the reconstruction of the cascade curves of the extensive air showers detected by the fluorescence technique. An accurate determination of the cascade curve allows one to determine well the total primary energy released into the electromagnetic cascades. Unless the ultra-high energy interactions are not far from model assumptions about multiparticle production, the fraction of the primary energy arriving at the earth in another form (muons, neutrinos, hadrons) can be allowed for (it is, however, small). Thus, the primary particle energy can be determined in a practically model independent way.
Acknowledgments
This work was supported by Polish Ministry of Scientific Research and Information Technology under the grant no 2 PO3D 011 24. The work of GW was supported by grant 5 PO3D 025 21.
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