In this Section we compare the reliability of the composition determination using S3 and
other mass sensitive parameters commonly used in composition analysis. We select the two most useful parameters, one from the surface technique, i.e. the rise time at 1000 m from the shower core, and other from the fluorescence technique, Xmax, the atmospheric
depth at which the maximum development of the cascade is reached. A brief discussion follows on specific details about the determination of these two parameters:
• Rise time at r0 = 1000 m from core, t1/2(r0) [ns]: The procedure followed by Auger
to calculate t1/2(r0) was explained earlier in Section 3.4.2. First, the rise time of each
at 1000 m is obtained by fitting the corrected rise time of each triggered station using the function t1/2(r) = (40 + ar + br2) ns. Parameters a and b are free in
the fit. Only the stations in the range from 600 to 1500 m from the shower axis and signal greater than 10 VEM, are included in the fit. At least three stations are required. Therefore, in case of showers at large zenith angles is not unusual that there are not enough stations passing the cuts, which reduces significantly the statistics available. Consequently, although the zenith angle distribution of our simulation set is isotropic (peaked at 45◦), there are more events whose t
1/2(r0) is
available at lower zenith angles.
• Xmax[g/cm2]: In order to assign a realistic Xmax value to our simulations, including
the response of the detector and the effects of the reconstruction method, we use the value simulated internally in AIRES and fluctuate it with a Gaussian distribution with standard deviation σ[Xmax] = 20 g/cm2, which is the Xmax resolution achieved
by Auger [154].
We use a maximum likelihood method to compare the reliability of the composition determination using the three parameters. We need samples with large statistics for this method. S3 and Xmax are almost independent on the zenith angle, so that it is possible
to combine events with different θ in the same sample. Obviously, that is not the case for t1/2(r0). Thus, a quadratic fit is performed, t1/2(r0) vs. sec(θ), for each primary and
hadronic model (see Fig. 5.13-left), and using the average values of the fitted parameters, we correct, in a simple way, the zenith angle dependence of t1/2(r0):
tcorr1/2(r0, sec θ) = tmeas1/2 (r0, sec θ) +
h
tf it1/2(r0, 1.05) − tf it1/2(r0, sec θ)
i
(5.21) The correction does not increase the fluctuations and tcorr
1/2(r0) shows a strong reduction
on the zenith angle dependence as shown in Fig. 5.13-right. For the subsequent analysis, we consider the lowest energy bin (from 1019 to 1019.1 eV) where we have larger statistics.
The sample for each set [primary, HIM, parameter] considered is binned. Let us call hp(i)
) θ sec( 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 RT1000 [ns] 0 100 200 300 400 500
Sibyll 2.1. Iron. With correction
Figure 5.13: t1/2(1000) as function of sec(θ) for Sibyll 2.1 and iron primaries. Error
bars are the RMS. Left panel: tmeas
1/2 (1000) without correction, the data is fitted with a
quadratic function. Right panel: tcorr
1/2 (1000) after correction, the data is fitted with a
linear function.
normalized to the total number of the events in the sample. The histograms hp and hf e
are assumed to be the distribution of the universe. The proton abundance of a sample is defined as Cp = Np/(Np + Nf e), where Np and Nf e are the number of protons and iron
nuclei in the given sample. We create samples of Ctrue
p from 0 to 1 in steps of 0.1. For
each value of Ctrue
p , we generate 300 sub-samples of Ns = 300 events each by taking them
randomly from hp and hf e. For each sub-sample, we generate a histogram Hs with the
same binning used in hp and hf e. Hs is not normalized so that
P
iHs(i) = Ns. Thus,
assuming Poisson statistics, the probability of the sub-sample Hs is given by
P ({Hs(i)}i) =
Y
i
·
exp (−Ht(i)) × Ht(i) Hs(i)
Hs(i)!
¸
(5.22) where
Ht(i) = Ns[Cpinfhp(i) + (1 − Cpinf)hf e(i)] (5.23)
Cinf
p represents the inferred proton abundance and it is obtained by maximizing ln P ({Hs(i)}).
Fig. 5.14 (Fig. 5.15) shows the inferred composition as a function of the true one corresponding to QGSJET-II (Sibyll 2.1). Similar results are obtained for both models but smaller error bars (which represent the 68% and 95% C.L.) for the latter, in agreement with the fact that the merit factor for all the parameters is greater for this HIM. The
number of events available for each mass sensitive parameter corresponding to a given exposure time, is a key value to compare their discrimination capabilities. Due to the limited duty cycle of the fluorescence telescopes, only 10% of the events are detected, so the statistics for Xmax are significantly lower than that of surface parameters. From Figs.
5.14 and 5.15, we see that considering the same exposure time, S3 gives the most accurate
results, i.e. its discrimination capability is greater than that of t1/2(r0) and Xmax. In
order to illustrate the significance of taking into account the limited statistics for Xmax
when doing composition studies, it is also shown the result for Xmax if the same statistics
as the SD parameters were available. The error bars are reduced becoming the smallest ones, but ten times more exposure would be required.
A second study has been performed in order to extend previous results to a larger energy range. Now, a fix true proton fraction CT rue
p = 0.5 is assumed and the inferred
proton fraction is calculated in the energy range from 1019.0 to 1019.6 eV. In order to
improve the small statistics in the higher energy bins, the distributions for each [primary, energy, HIM, parameter] are fitted using the Asymmetric Generalized Gaussian (AGG) function, defined as PAGG(y) = c γa Γ(1/c) exp[−γlc (−y + µ)c] if y < µ c γa Γ(1/c) exp[−γrc (y − µ)c] if y ≥ µ where γa= 1 σl+ σr µ Γ(3/c) Γ(1/c) ¶1/2 γl = 1 σl µ Γ(3/c) Γ(1/c) ¶1/2 γr = 1 σr µ Γ(3/c) Γ(1/c) ¶1/2 .
which has been already used in Chapter 4.
Fig. 5.16 shows examples of the fits performed for the three parameters considered. It can be seen that it is possible to fit asymmetric distributions with longer tails compared to Gaussian distributions. The fits are very accurate, so that we can extract samples from them and it is feasible to extent the previous study to a larger energy range.
For each set of [primary, energy, HIM, parameter], the samples are generated by randomly sampling the corresponding fitting function. Thus, we generate the histograms
Figure 5.14: Inferred vs. true proton fraction using QGSJET-II for t1/2(r0) (top-left), S3 (top-right) and Xmax (bottom-left) for the same exposure time (the 10% duty cycle
of the fluorescence telescopes is taking into account). The bottom-right panel shows the inferred proton fraction obtained using Xmax and samples with the same statistics as SD
Figure 5.15: As Fig. 5.14 but using Sibyll 2.1.
hp and hf e, that represent the universe, with 1000 events each. We also generate 200 sub-
samples for each case, but the number of events in the sub-samples varies as a function of primary energy because of the steepness of the spectrum. The number of events expected by Auger in 1 and 5 years of full operation are considered (for example, in one year and considering the spectrum reported in [29], around 500 events are expected at 1019.0 eV
and 70 at 1019.6 eV). To reproduce real conditions, the number of events with available
Xmax is 10% of the total in the sample. The procedure to infer the composition is the
same as explained before.
As in the previous case, there is no significant bias in the inferred proton abundance. However, Fig. 5.17 shows the uncertainty (the C.L. at 68%) on the determination of the proton abundance as a function of the primary energy. It can be seen that the best results are obtained by using S3. As mentioned before, the uncertainties corresponding to Sibyll
2.1 are smaller than that for QGSJET-II because the shower-to-shower fluctuations are in general smaller for Sibyll 2.1.
Figure 5.16: Examples of the fits with AGG function for the three parameters considered, different energy bins and hadronic interaction models.