The shower size parameter as estimator of extensive air shower energy in fluorescence telescopes
ABSTRACT The fluorescence technique has been successfully used to detect ultrahigh energy cosmic rays by indirect measurements. The underlying idea is that the number of charged particles in the atmospheric shower, i.e, its longitudinal profile, can be extracted from the amount of emitted nitrogen fluorescence light. However the influence of shower fluctuations and the very possible presence of different nuclear species in the primary cosmic ray spectrum makes the estimate of the shower energy from the fluorescence data analysis a difficult task. We investigate the potential of shower size at maximum depth as estimator of shower energy. The detection of the fluorescence light is simulated in detail and the reconstruction biases are carefully analyzed. We extend our calculations to both HiRes and EUSO experiments. This approach has shown some advantages to the reconstruction of the energy when compared to the standard analysis procedure.
arXiv:astro-ph/0511677v1 23 Nov 2005
The shower size parameter as estimator of extensive air shower
energy in fluorescence telescopes
Vitor de Souza,∗Gustavo Medina-Tanco,†and Jeferson A. Ortiz‡
Instituto de Astronomia, Geof´ ısica e Ciˆ encias Atmosf´ ericas, Universidade de S˜ ao Paulo
Dipartimento di Fisica, Universit´ a degli Studi di Milano e I.N.F.N.
(Dated: November 14th, 2005 - J.A.Ortiz)
The fluorescence technique has been successfully used to detect ultrahigh energy cosmic rays
by indirect measurements. The underlying idea is that the number of charged particles in the
atmospheric shower, i.e, its longitudinal profile, can be extracted from the amount of emitted
nitrogen fluorescence light. However the influence of shower fluctuations and the very possible
presence of different nuclear species in the primary cosmic ray spectrum makes the estimate of the
shower energy from the fluorescence data analysis a difficult task. We investigate the potential of
shower size at maximum depth as estimator of shower energy. The detection of the fluorescence
light is simulated in detail and the reconstruction biases are carefully analyzed. We extend our
calculations to both HiRes and EUSO experiments. This approach has shown some advantages to
the reconstruction of the energy when compared to the standard analysis procedure.
PACS numbers: 96.40.Pq
∗Electronic address: email@example.com
†Electronic address: firstname.lastname@example.org
‡Electronic address: email@example.com
§Electronic address: firstname.lastname@example.org
Cosmic rays (CR) are the highest energy particles present in nature with energies exceed-
ing 1020eV . Their origin, nature and possible acceleration mechanisms are still a mystery,
despite the efforts of many cosmic ray experiments in the last decades. Such challenge is in
part due to the very low flux of high energy and the consequent few observed events to be
Measuring extensive air showers (EAS) is currently the only way to study the cosmic
ray spectrum and chemical composition at energies around and above PeV. At energies
E≥1017eV the shower development can be directly observed by measuring the fluorescence
light from atmospheric nitrogen that is excited by the ionization of the secondary charged
shower particles (essentially electrons and positrons). Experiments applying this technique
can determine the depth of maximum air shower development (Xmax) and the corresponding
number of charged particles (Nmax). Presently, the HiRes , Pierre Auger Observatory ,
EUSO , OWL  and Telescope Array ) experiments are using or planning to use
fluorescence detectors to investigate the ultra high energy cosmic rays.
The total amount of emitted fluorescence light in a shower is a very good approximation
to the total number of charged particles N(X), where X is the atmospheric depth. In this
sense the number of particles at shower maximum can serve as an estimator of the shower
energy. The total energy that goes into electromagnetic charged particles is obtained by
integration of the shower longitudinal profile
where α is the average ionization loss rate , and the integral on the right-hand side
represents the total track length of all charged particles in the shower projected onto the
As an alternative proposal  the electromagnetic energy can also be calculated by using
the fluorescence light intensity and the fluorescence efficiency, without the need of recon-
structing the number of particles as a function of the atmospheric depth. Such approach is
taken as a very precise measurement of the primary shower energy because it is supposed to
be weakly dependent on the simulation models and on the primary particle type. However,
when the shower development details are taken into account the calorimetric measurement
can lead to high systematic uncertainties. Of no less concern is the important fact that the
fluorescence efficiency as a function of air pressure, density and humidity is only known to
a certain extent. On approach given by Eq. 1, the average ionization loss rate is used in the
air shower reconstruction and hence the energy spectrum of the electron shower particles
must be known via Monte Carlo simulation.
Although the electrons and positrons constitute the majority among the charged particles
in a shower and contribute most to the production of fluorescence light, an also important
fraction of the shower energy is carried by particles which cannot be measured by the fluo-
rescence technique, i.e., particles that are invisible to fluorescence telescopes. This so called
“missing energy” has been calculated by Monte Carlo air shower simulation and contributes
to the uncertainties involved in the method, being sensitive to primary composition.
Theoretical works have shown the existence of a clear relation between the primary energy
and the maximum number of particles in the shower. Recently, Alvarez-Mu˜ niz et al. 
have studied the Nmax shower quantity as an estimator for the primary shower energy,
confirming the efficiency of this technique. Such approach was analyzed for different primary
particles and energies using a fast one-dimensional simulation program. However, telescopes
particularities and reconstruction procedures must be considered due to the introduction of
biases and fluctuations in the calculation of Nmax.
The scope of this work is to explore the possibility of estimating the primary shower energy
based on Nmax, taking as case studies the HiRes  and EUSO  fluorescence experiments,
i.e., ground and space based experiments, respectively. The telescopes particularities and
the reconstruction procedures are included in our analysis configuring a very realistic context
for the application of the technique.
Fluctuations are intrinsic to any extensive air shower and are a cause of uncertainty in the
energy reconstruction. In addition to that, the energy of the shower is calculated without
the knowledge of the mass of the primary particle which initiated the shower. The nature of
the primary particle affects the longitudinal development of showers and can exert influence
on the reconstruction of the air shower energy, especially when primary photons are taken
On the other hand, reconstruction procedures must relay on simulation programs in order
to relate the measurable parameters to energy. In addition to the choice of the particular
simulation program to be used, there is a general agreement that most shower reconstruction
uncertainties originate on uncertainties on the hadronic interaction models.
The usual procedure used by the HiRes and Auger collaborations to reconstruct the
shower energy with fluorescence telescopes correlates the integral of the energy deposited in
the atmosphere to the total shower energy. As mentioned before, a certain amount of the
shower energy is carried away by particles which are invisible to the fluorescence technique,
i.e., muons, neutrinos and high energy hadrons which are not converted to fluorescence
photons. Such “missing energy” has been estimated by Monte Carlo simulations and shown
to be dependent on the hadronic interaction model, primary composition, shower energy
and arrival direction.
In reference , Song et al. have estimated that the fluctuations due to the type of primary
particle are about ∼ 5% for showers initiated by proton and iron nuclei and around ∼ 20%
if primary photons are taken into account. Meanwhile Barbosa et al.  and Alvarez-Muniz
et al.  have calculated similar values for hadronic primaries (∼5%) which decrease with
energy. According to , the hadronic interaction model is the main source of systematic
fluctuation in the estimation of the missing energy at the highest energies. Due to the
differences in the multiplicities of secondary particles simulated by QGSJET01  and
SIBYLL2.1  hadronic interaction models, the unseen energy calculated with QGSJET is
about 50% higher than the value predicted by SIBYLL at 1020eV, which can be translated
into an uncertainty of ∼ 5% in the shower energy reconstruction.
The dependence of the missing energy on the energy of the primary is a consensus among
the various studies made so far. In addition, Barbosa and co-workers  have studied the
dependence of the missing energy on the arrival direction of the shower and limited it to be
at most 0.7%.
Furthermore, it has been suggested in reference  that the discrepancies among such
studies are on the order of 1-3% and could be explained by different hadronic interaction
models for lower energy particles. It might be relevant to mention that both groups used
different simulation programs: Song et al. and Barbosa et al. used different versions of the
well tested CORSIKA , while Alvarez-Muniz et al. used a hybrid fast one-dimensional
simulation program .
Considering all the uncertainties in the determination of the missing shower energy listed
above, one can estimate that the total systematic fluctuation related to the primary energy
reconstruction is about 5% to 10%.
In order to investigate the fluctuations in the maximum number of particles due to the
natural fluctuations of the shower, the different primary particles and the artificial fluctua-
tions introduced by different hadronic interaction models, we have used the recently released
CONEX  shower simulation program. We have simulated 5,000 showers initiated by
proton, iron and gamma primaries at each considered energy and for the QGSJET01 and
SIBYLL2.1 interaction hadronic models. Shower have been simulated with the minimum
cuts available, 1 GeV for hadrons and 1 MeV for electromagnetic particles.
Fig. 1 shows the Nmax distribution for 5,000 showers initiated by protons, iron nuclei
and gamma at 1019.5eV simulated with the QGSJET hadronic model. The distribution
illustrates the previous discussion regarding the fluctuations due to the primary particle
type. If only proton and iron nuclei are considered, the Nmax distribution shows a very
narrow distribution with median value of 1.90×1010and a dispersion of 3.6% at the 68% of
confidence level. If gamma shower are considered, the median of the combined distribution
(gamma+proton+iron nuclei) increases slightly to 1.92 × 1010while the dispersion reaches
7% at 68% of confidence level.
The same calculation has been done for proton and iron nuclei at 1020.5eV and the
fluctuations presented the same level of 3.6%. Gamma air showers have not been studied
at the energy of 1020.5eV due to the fact that the CONEX program does not include
the pre-shower algorithm effect [15, 16] that takes into account photon interactions with
the geomagnetic field affecting the longitudinal development of showers with energy above
In order to investigate the uncertainty due to the hadronic interaction model we have
simulated 5,000 proton showers with the CONEX program using SIBYLL and QGSJET
hadronic interaction models. Fig. 2 illustrates the Nmax distribution for proton showers
simulated with QGSJET and SIBYLL at 1019.5eV and the corresponding fluctuation is 4.6%.
Fig. 2 also shows the same distribution for iron nuclei at 1020.5eV and the corresponding
fluctuation is 3.3%.
Comparing the numbers given above and Figs. 1 and 2, we are able to conclude that the
systematic uncertainties in Nmaxare dominated by the hadronic interaction models rather