Torino, February 10th 2009

Energy spectrum from Ne - Nµ

Mario Bertaina, University of Torino

1 Introduction

In this report I will describe the technique employed to measure the en-ergy spectrum by means of the Ne and Nµ information got at level 3 ofthe KASCADE-Grande analysis. The technique here described refers to theanalysis of the data processed with KRETA version V11702. However, inthe second part of the document I will also describe the results, presentlyobtained, using version V11803 in order to have some reference numbers forcomparisons with the methods developed by our colleagues. The same tech-nique will be applied also to the new stable version of KRETA as soon asit will be delivered. Most of the discussion presented in this document forversion V11702 is summarized in my talk at the KASCADE-Grande meetingin Freudenstadt (July, 2008).In this updated version I report the results of the same analysis using data

processed with KRETA version V11805. The updated version of the report

starts from section 12.

2 Concept

The main idea of this technique is to use the electron size Ne as the energyestimator and the ratio Ne/Nµ as the mass estimator, event by event. This isthe option exploited so far. A quick trial adopting Nµ as the energy estimatorhas been tried and it gave similar results (see section 5). The analysis isperformed using only reconstructed variables, instead of the true ones, inorder to include already possible systematics effects in the reconstructionprocedure. The technique is defined by means of the simulated data, andthen applied to the real data.

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3 Technique

The data are right now divided in bins of equal ∆secθ = 0.05. The analysisis conducted independently in each angular bin. At the end, the results ofeach angular bin is merged together with the others. The analysis has beenchecked in the first 2 angular bins (0◦-17.75◦ and 17.75◦-24.62◦) for versionV11702.First of all, the electron size Ne reconstructed at angle θ is shifted to thevertical direction by applying the following expression:

log10Ne(θ = 0) = log10Ne(θ) + secθ − 1 (1)

In the following, Ne will refer always to the vertical direction. As Nµ hasa very week dependence on the angular direction, at least for the angularregion used in this analysis, no correction Nµ(θ) is being applied.In the second step, the relations Ne/Nµ(Ne) and E(Ne) for the two extremecases of protons and Iron are extracted, assuming the following dependencies:

log10(Ne/Nµ)p,Fe = cp,Fe · log10Ne + dp,Fe (2)

log10(E)p,Fe = ap,Fe · log10Ne + bp,Fe (3)

Fig. 1 shows the scatter plots of Ne/Nµ(Ne) (left) and E(Ne) (right) in case ofprotons, while Fig. 2 shows the same distributions in case of Iron. Figs. 1 and2 show two different fits for each plot. The linear fit which really matters inthe present analysis is the one using sizes Ne > 5.4 which corresponds to Ironenergies EFe > 4 · 1015 eV (well below the energy threshold of KASCADE-Grande). The main reason of using a linear dependence in the log-log scale,at present, is the limited statistics available in the highest energy bins andthe associated re-sampling of the simulated data. Different expressions mightbe considered in future, which better follow the evolution of the full dots ineach plot, if the above concerns will be removed.

In order to estimate the energy of each single event on an event-by-eventbasis, an expression E(Ne, Nµ) which takes into account the different energydependence of the electron and muon sizes of each element has to be takeninto account. For this reason, first of all, a new parameter k is definedthrough the following expression:

k =log10(Ne/Nµ) − log10(Ne/Nµ)p

log10(Ne/Nµ)Fe − log10(Ne/Nµ)p

(4)

2

Figure 1: Scatter plots of Ne/Nµ vs Ne (left) and E vs Ne (right) in case ofprotons. The small dots indicate single events, while full ones refer to theaverage values in each interval on the x axis (∆Ne = 0.1). The error bar ofthe full dots indicates the σ of the distribution of the small dots in each Ne

interval. The linear fits are performed on the dots and their errors. Such fitsare used to obtain the parameters a, b, c, d of expressions 2 and 3.

Figure 2: Same as fig. 1 in case of Iron data.

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where log10(Ne/Nµ)p,Fe represent the linear expressions of 2 for protonsand Iron, while log10(Ne/Nµ) is the value of the specific event in analysis.Therefore, k is defined in a way that all the elements would provide on average0 < k < 1 ( k = 0 for protons, while k = 1 for Iron) in bins of Ne, appearingas a mass parameter. The expression of the energy dependence (E) on theelectron size is finally parametrized by the following expression:

log10(E) = (ap + ∆a · k) · log10(Ne) + (bp + ∆b · k) (5)

with ∆a = aFe − ap and ∆b = bFe − bp.

4 Results assuming spectrum with γ = −2

The technique explained in 3 has been applied to the proton and Iron sim-ulated data in the first angular bin (0·-17.75·) assuming a differential energyspectrum with slope γ = −2 obtaining the following epxressions for k(Ne)and E(Ne):

k =log10(Ne/Nµ) − (−0.023 + 0.196 · log10(Ne))

(−0.531 + 0.219 · log10(Ne) − (−0.023 + 0.196 · log10(Ne)(6)

log10(E) = (0.894 − 0.023 · k) · log10(Ne) + 1.511 + 0.376 · k (7)

In order to check the capability of reproducing the original energy spectrum,the above formula has been applied to the same data it has been derivedfrom, and the original and reconstructed spectra have been compared eachother. The results are shown in fig. 3. A discussion on the quality of thereconstructed spectrum in terms of systematics and uncertainties comparedto the original one is postponed to section 7.

5 Energy spectrum assuming Nµ as the en-

ergy estimator

As mentioned in the introduction another option as anergy estimator is Nµ.For this reason the analysis was repeated using the same set of data but thistime the correlations were searched according to the following expressions:

log10(Ne/Nµ)p,Fe = cp,Fe · log10Nµ + dp,Fe (8)

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Figure 3: Original (red line) and reconstructed (blue line) energy spectrumof protons (left) and Iron (right) according to expression 3.

log10(E)p,Fe = ap,Fe · log10Nµ + bp,Fe (9)

Figs. 4 and 5 show the scatter plots Ne/Nµ(Nµ) (left) and E(Nµ) (right),respectively, for protons and Iron.

This technique applied to the first angular bin of the energy spectrumwith slope γ = −2 gave the following expressions for k(Nµ) and E(Nµ):

k =log10(Ne/Nµ) − (−0.663 + 0.348 · log10(Nµ))

(−0.906 + 0.318 · log10(Nµ) − (−0.663 + 0.348 · log10(Nµ)(10)

log10(E) = (1.131 + 0.019 · k) · log10(Nµ) + 1.355 − 0.269 · k (11)

According to the fits of figs. 4 and 5 these expressions are valid for log10Nµ >4.7 which corresponds to Ep > 5 · 1015, well below the energy threshold ofKASCADE-Grande. Fig. 6 shows a comparison between using Ne (see eq. 7)or Nµ (see eq. 11) as the energy estimator. The good agreement betweenthe two energy spectra indicates that both parameters could be used, in anequivalent way, as energy estimators.

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Figure 4: Scatter plots of Ne/Nµ vs Nµ (left) and E vs Nµ (right) in caseof protons. The small dots indicate single events, while full ones refer to theaverage values in each interval (∆Nµ = 0.1). The error bar of the full dotsindicates the σ of the distribution of the small dots in each Nµ interval. Thelinear fits are performed on the dots and their errors. Such fits are used toobtain the parameters a, b, c, d of expressions 8 and 9.

Figure 5: Same as fig. 4 in case of Iron data.

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Figure 6: Energy spectrum using the data of the first angular bin, accordingto eq. 7 (black squares) or eq. 11 (red squares).

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Figure 7: Original (red line) and reconstructed (blue line) energy spectrumwith γ = −3 of protons (left) and Iron (right) according to expression 7.

6 The role of fluctuations with a steep energy

spectrum

The studies reported in the previous paragraphs were done assuming a γ =−2 spectrum. The role of fluctuations with a more realistic spectrum (γ =−3), were investigated in a second step. Fig. 7 shows the reconstructedenergy spectra starting from eq. 7, compared to the true ones for protonsand Iron. When a steeper energy spectrum is assumed, the reconstructedspectra do not match so well to the original ones as in the case of fig. 3. Forthis reason a new expression for the energy dependence E(Ne) and for theparameter k was determined starting from a spectrum already sampled withγ = −3.

Figs. 8 and 9 show the results of the fit of the scatter plots Ne/Nµ vsNe and E vs Ne, compared to the same fits obtained assuming a γ = −2spectrum. There is a clear difference in the fits that should be taken intoaccount. The new parametrization for the first angular bin becomes:

k =log10(Ne/Nµ) − (−0.273 + 0.166 · log10(Ne))

(−0.460 + 0.217 · log10(Ne) − (−0.273 + 0.166 · log10(Ne)(12)

log10(E) = (0.938 − 0.061 · k) · log10(Ne) + 1.100 + 0.673 · k (13)

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Figure 8: Distributions of Ne/Nµ vs Ne (left) and E vs Ne (right) in case ofprotons. The small dots indicate single events, while full ones refer to theaverage values in each interval. The error bar of the full dots indicates theσ of the distribution of the small dots in each Ne interval. The red linearfits are performed on the dots and their errors. Such fits are used to obtainthe parameters a, b, c, d of expressions 2 and 3. The pink fits represent thelinear fits already shown in fig. 1, where a γ = −2 is assumed for the energyspectrum.

The comparison between true and reconstructed energy spectra usingexpression 13 is shown in figs. 10, 11, and 12.

7 Systematics / uncertainties of the technique

The uncertainties and systematics of this technique have been checked usingdifferent methods which are listed and explained here in the following:

a) fit of the true and reconstructed energy spectra to verify if the techniqueintroduces a systematic error on a single component;

b) ratio of the reconstructed over true spectrum for proton and Iron tocheck systematics between different components;

c) χ2 of the contents of each bin to check bin by bin fluctuations;

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Figure 9: Same as fig. 8 in case of Iron data.

Figure 10: Original (red line) and reconstructed (blue line) energy spectrumof protons (left) and Iron (right) according to expression 13.

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Figure 11: Original (red line) and reconstructed (blue line) energy spectrumof Helium (left) and CNO (right) according to expression 13.

Figure 12: Original (red line) and reconstructed (blue line) energy spectrumof Silicon (left) and all five components mixed 20% each (right) according toexpression 13.

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d) systematic uncertainty in the flux.

7.1 Fit of the true and reconstructed energy spectra

Figs. 10, 11, and 12 present also a linear fit on the slope of the energyspectra. Two linear fits are plotted, red for the original spectrum and bluefor the reconstructed one. It is difficult to see both of them because theyoverlap on the entire energy range. In fact the differences of the absoluteflux between true and reconstructed energy spectrum foreseen by the linearfit differs in every energy bin by less than 5% for all components. It should behowever underlined that the fit is dominated by the first energy bins whichhave higher statistics and therefore this check concerns essentially the firstpart of the spectrum.

7.2 Ratio of the reconstructed and true spectra

In order to verify systematics in the procedure between different components,the two extreme cases of protons and Iron have been considered. Startingfrom the fit on the true and reconstructed spectra, the following ratio R hasbeen examined:

R =(Φ(E)rec/Φ(E)true)p

(Φ(E)rec/Φ(E)true)Fe

(14)

The evolution of the ratio R as a function of energy is displayed in fig. 13 bythe red points. The blue points represent the same data when the systematicsin flux is converted into systematics in energy assignment. The ratio Roscillates inside ±4% around the average value, indicating that the techniqueintroduces a very small systematic effect between the 2 extreme primaries.

7.3 χ2 test

In order to check the bin by bin fluctuations, a χ2 test has been used. Thevariable χ2 is defined through the expression:

χ2 = Σ(O/w − E/w)2

E/w2(15)

where:O = reconstructed number of events in each bin;

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Figure 13: The value of the ratio R as a function of energy is expressed bythe red points (flux). The blue points represent the same ratio when the fluxis converted into the energy spectrum.

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p He CNO Si Fe Mixχ2 1.174 1.440 0.800 0.802 1.452 1.214

Table 1: χ2 value for each element, according to expression 15.

p He CNO Si Fe Mix Aver.R’ -0.033 0.035 0.043 0.013 -0.007 0.007 0.010

σ(R′) 0.144 0.099 0.175 0.136 0.135 0.056 0.138

Table 2: Estimation of the R′ value as defined in eq. 16 for the differentelements.

E = true number of events in each bin;w = weight applied to each bin to get a spectrum with γ = −3.The results of the χ2 test for each element are summarized in table 1.Fluctuations are dominated by Poisson statistics, in fact the values of χ2

∼ 1.

7.4 Estimation of the uncertainties in the energy as-

signment

From the χ2 test it has been learnt that the fluctuations are dominated byPoisson. It has been tried then to check the region between 7 < log10(E) < 8(where the high statistics allows to check other effects a part from the Pois-sonian ones) in order to see if there where systematic errors in the energyassignment on the different bins, and which was the entity of the non Pois-sonian fluctuations. For this reason the following expression has been used:

R′ =O − E

E(16)

It is expected that the average values of R′ are centered around 0, while thedispersion of such values will give an estimation of the uncertainties in theenergy assignment. Results are reported in table 2. As expected the R′

values are centered around 0. The < R′ > fluctuates by few % among allthe elements proving that the formula is valid not only for the the two onesused to define the expression (p, Fe) but also for all the others. The averageflcutuation in the energy assignment is on the order of < σ(R′) > 13.8%.

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Figure 14: Differential flux multiplied by E2.5 as a function of the log10E(GeV) for the two first angular bins. Red points refer to 0◦ − 17.75◦ whileblue ones to 17.75◦ − 24.62◦.

8 Analysis of the second angular bin

The above analysis has been repeated on the second angular bin (17.75◦

- 24.62◦) giving as a result of the fits, the following expressions valid forlog10Ne > 5.5 (EFe > 4.5 · 1015):

k =log10(Ne/Nµ) − (−0.266 + 0.170 · log10(Ne))

(−0.501 + 0.219 · log10(Ne) − (−0.266 + 0.170 · log10(Ne)(17)

log10(E) = (0.911 − 0.033 · k) · log10(Ne) + 1.304 + 0.518 · k (18)

As the analysis has been conducted independently on the two first angularbins, the results on the absolute flux have been compared (see fig. 14).

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9 Results

The results obtained using the method here presented have been comparedwith the results obtained with the other techniques developped inside thecollaboration. A comparative result of the different methods is presented infig. 15 and shows a reasonable agreement.

The parameter k defined in this technique can give some information onthe evolution of the average mass of the cosmic radiation in the energy rangein analysis. Fig. 16 shows the evolution of the parameter k as a function ofenergy. It has to be underlined that the k parameter was defined to be onaverage 0 < k < 1 as a function of Ne, while here an off-set exists. Thisis due to the fact that this time the variable is expressed as a function ofenergy (k(E)) instead of Ne. This could be also the reason why k doesn’tshow a constant behavior as a function of energy. It might reflect differentthings such as the effect of shower fluctuations which decrease as a functionof energy, or of the energy parametrization introduced in eq. 3, or to otherreasons that should be investigated. Anyway, a good agreeement betweenthe results of the two angular bins can be noticed. The average value of kof the experimental data lies inside the results foreseen by the simulation ofpure CNO and Silicon components.

10 Analysis of the data analyzed using KRETA

version V11803

The analysis has been repeated on the entire angular region 0-40◦ for thedata processed using KRETA version V11803. Data have been subdivided inthe following 4 angular bins: 0◦-17.75◦, 17.75◦-24.62◦, 24.62◦-29.52◦, 29.52◦-39.72◦ (the 4th bin includes in reality all the data between 1.15< secθ < 1.30).The analysis is on the way, further checks are necessary. Here I report themain results obtained so far in order to have results more easily comparablewith those obtained by the other colleagues. A more detailed re-analysis ofthe data will be done when a more stable version of KRETA will be provided.

The expressions obtained for the 4 angular bins have been obtained usingsizes log10Ne > 5.8 (EFe > 1016). Results of the 4 angular bins in terms ofcapability of reproducing the original energy spectrum are reported in figs. 17and 18. Systematics of these results have been checked using the ratio R′

obtaining the results of table 3 for the 4 angular bins. The technique shows

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Figure 15: Differential flux multiplied by E2.5 as a function of energy obtainedwith the KASCADE-Grande data, employing different techniques. Blue starsrefer to Juan Carlos method (muons); red stars to the CIC method employedby Dirk; Green stars to the present method and black dots Dongwa’s analysis.It has to be underlined the all the other methods refer to a wider angularrange (usually 0 - 40 ◦) compared to the present one.

bin 1 bin 2 bin 3 bin 4 Aver.R’ -0.015 -0.012 -0.036 0.011 -0.013

σ(R′) 0.056 0.046 0.067 0.051 0.055

Table 3: Estimation of the R′ value as defined in eq. 16 for the mixed spec-trum (all five components) in the 4 different angular bins.

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Figure 16: Evolution of the parameter k as a function of log10E (GeV).Full (open) dots refer to the first (second) angular bin. Black dots refer tothe measured data, while the other colors to the expectation from a single-element composition.

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Figure 17: Original (red line) and reconstructed (blue line) energy spectrumof all 5 mixed components for bin 1 (left) and bin 2 (right) using simulateddata and expressions of V11803.

Figure 18: Original (red line) and reconstructed (blue line) energy spectrumof all 5 mixed components for bin 3 (left) and bin 4-6 (right) using simulateddata and expressions of V11803.

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Figure 19: The energy spectrum (differential flux multiplied by E2.5) as afunction of log10E obtained with version V11803 in the 4 different angularbins. Only statistical errors are plotted.

no significant systematics (% level). The bin-to-bin systematics is on theorder of 5-6%. Applying the expressions for k and E(Ne, k) obtained withversion V11803 to the experimental data, the fluxes shown in fig. 19 have beenobtained. From the dispersion of the points of the 4 different spectra it ispossible to extract a systematic error in each energy bin. This error includesall statistical and systematic uncertainties except the common systematicsof all the expressions used in the four bins (i.e. the slopes obtained in the 4bins are all steeper of softer than the true ones, the linear dependence in thelog-log scale is not correct, etc...) or the systematics due to specific hadronicgenerator (QGSjet2 model) in use. The flux (multiplied by E2.5) and itssystematic error in each bin is summarized in table 4. The data of table4 are plotted in fig. 20 together with those obtained with version V11702.Fig. 21 shows the behavior of the parameter k in the 4 different angular bins

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Energy J(E) × E2.5

(1016eV ) (1016· m−2s−1sr−1eV 1.5)

1.00 3.50 ± 0.021.26 3.12 ± 0.071.58 2.72 ± 0.062.00 2.35 ± 0.022.51 2.06 ± 0.063.16 1.80 ± 0.033.98 1.64 ± 0.035.01 1.49 ± 0.076.31 1.29 ± 0.057.94 1.12 ± 0.0910.0 0.96 ± 0.0612.6 0.74 ± 0.1115.8 0.68 ± 0.0620.0 0.62 ± 0.0325.1 0.42 ± 0.0531.6 0.41 ± 0.1639.8 0.44 ± 0.1050.1 0.45 ± 0.1063.1 0.29 ± 0.1479.4 0.37 ± 0.14100. 0.34 ± 0.32

Table 4: The flux (multiplied by E2.5) and its systematic error in each bin.These numbers have to be considered very preliminary results, here writtento be easily compared with the results of the other methods.

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Figure 20: Differential flux multiplied by E2.5 as a function of energy obtainedusing data of version V11803 in the angular region 0-40◦ (black points) to-gether with those obtained using version V11702 in the angular region 0-25◦

(red points), adopting the Ne − Nµ technique.

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Figure 21: Evolution of the parameter k as a function of log10E (GeV) forall 4 angular bins for the different primary elements and experimental data(left plot). The average of the 4 angular bins is plotted in the right plot.

and the average behavior. No significant differences with version V11702 areobserved.

11 To be done

There are still different checks that have to be done.

a) Understand the systematic related to the average slope of the E(Ne)dependence assumed.

b) Understand which should be the relation between the linear fits in thedifferent angular bins.

c) Try other dependencies (quadratic instead of linear?) to see which arethe systematics due to the assumption of a linear dependence.

d ) Try to find a unique relationship for the whole angular range.

e) Understand the error related to the event-by-event energy assignment.

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f) Understand the role of fluctuations (we know that fluctuations in cur-rently simulated data are much higher than in the real data).

g) Study the effect of the threshold in Ne to start the energy conversionof the data. Changing the threshold, how does it affect the first pointsof the spectrum?

12 Update using KRETA version V11805

The analysis has been repeated on the entire angular region 0-40◦ for the dataprocessed using KRETA version V11805. Data have been subdivided in thesame 4 angular bins as in version V11803: 0◦-17.75◦, 17.75◦-24.62◦, 24.62◦-29.52◦, 29.52◦-39.72◦ (the 4th bin includes in reality all the data between1.15< secθ < 1.30). The analysis is on the way, further checks are necessary.The main difference compared to the past versions is a modification of theexpression used to report all measured sizes to the vertical direction (eq. 1).Such equation has been replaced by:

log10Ne(θ = 0) = log10Ne(θ) + 1.57 · (secθ − 1) (19)

The number 1.57 is derived from x0/Λe · log10e where x0 = 800g/cm2 andΛe = 219g/cm2 have been used for depth and attenuation length (resultsobtained at EAS-TOP), numbers coming from the relation:

Ne(0) = Ne(θ) · 10(secθ−1)·x0/Λe·log10e (20)

In future, the idea is to report the sizes in each angular bin to the centerof the corresponding angular bin. However, I should stress the point thatthe systematic effect due to reporting all sizes to the vertical is not too big(because independent expressions for the energy relation in each angular binare used) and it’s kept under control by introducing a systematic error inthe flux assignment by looking at the systematic differences in flux in eachangular bin for every energy bin (see next section).The second difference compared to version V11803 is to have defined expres-sions for the k parameter and the energy relation (E(Ne)) valid starting fromlog10Ne > 5.5 (EFe > 5 ·1015) instead of log10Ne > 5.8 (EFe > 1016), in orderto take better into account the flux in the first energy bins.The energy and k expressions defined in the 4 angular bins are reported in

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the following (Ne indicates the electron size reported to the vertical directionby using expression 19).Bin 1 (0◦-17.75◦):

k =log10(Ne/Nµ) − (−0.0865 + 0.2043 · log10(Ne))

(−0.3444 + 0.2065 · log10(Ne) − (−0.0865 + 0.2043 · log10(Ne)(21)

log10(E) = (0.9426 − 0.0355 · k) · log10(Ne) + 0.9810 + 0.5609 · k (22)

Bin 2 (17.75◦-24.62◦):

k =log10(Ne/Nµ) − (0.2404 + 0.1823 · log10(Ne))

(−0.0651 + 0.1664 · log10(Ne) − (0.2404 + 0.1823 · log10(Ne)(23)

log10(E) = (0.9764 − 0.0228 · k) · log10(Ne) + 0.7950 + 0.4684 · k (24)

Bin 3 (24.62◦-29.52◦):

k =log10(Ne/Nµ) − (0.3716 + 0.1645 · log10(Ne))

(−0.1640 + 0.1781 · log10(Ne) − (0.3716 + 0.1645 · log10(Ne)(25)

log10(E) = (0.9514 − 0.0045 · k) · log10(Ne) + 0.9512 + 0.4036 · k (26)

Bin 4 (29.52◦-39.72◦):

k =log10(Ne/Nµ) − (0.7656 + 0.1062 · log10(Ne))

(−0.3466 + 0.2075 · log10(Ne) − (0.7656 + 0.1062 · log10(Ne)(27)

log10(E) = (1.0506 − 0.1375 · k) · log10(Ne) + 0.3931 + 1.2514 · k (28)

Results of the 4 angular bins in terms of capability of reproducing the originalenergy spectrum are reported in figs. 22 and 23.

Applying the expressions for k and E(Ne, k) obtained with version V11805to the experimental data, the fluxes shown in fig. 24 have been obtained.

13 Errors in the flux

The following errors have been considered:

a) statistical errors;

b) systematic error on the flux;

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Figure 22: Original (red line) and reconstructed (blue line) energy spectrumof all 5 mixed components for bin 1 (left) and bin 2 (right) using simulateddata and expressions of V11805.

Figure 23: Original (red line) and reconstructed (blue line) energy spectrumof all 5 mixed components for bin 3 (left) and bin 4-6 (right) using simulateddata and expressions of V11805.

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Figure 24: The energy spectrum (differential flux multiplied by E2.5) as afunction of log10E obtained with version V11805 in the 4 different angularbins. Only statistical errors are plotted.

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Figure 25: The energy spectrum (differential flux multiplied by E2.5) as afunction of log10E obtained with version V11805 for the first 2 angular bins(red points) and the last two (blue points).

c) systematic error on the core location from the center of KASCADE;

d) systematic error on the E(Ne) relation.

Concerning the systematic error on the flux, the fluxes of bin 1 and 2 plottedin fig. 24 have been summed together, and in the same way those of bin 3 and4-6. In this way the statistical error has been reduced. The result is plottedin fig. 25. The semi-difference of the flux in each energy bin gives the system-atic error (after the subtraction of the statistical error) due to the followingreasons: relative uncertainty between the energy and k relations defined inthe 4 angular bins, and uncertainty due to reporting all the electron sizesto the vertical. The small differences among the blue and red points clearlyindicate that the technique is self-consistent in the different angular bins.However, other two types of uncertainties have been considered. The first

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one is connected with the previous one. The error on the flux in the fourangular bins gives an indication on the relative uncertainty between energyan k relations in the different angular bins, but not an absolute error (i.e.the slopes of the E(Ne) are all steeper of softer than the true ones). In or-der to take into account such error, the energy relations in the 4 angularbins have been artificially modified (essentially changing the slopes) and thereconstructed energy spectra from simulation have been compared with thetrue ones. When the reconstructed energy spectra showed clearly a system-atic difference from the original ones (the check is performed ’by eye’) theiterative procedure is stopped and those parameters are used for determin-ing a new ’biased’ energy spectrum. In order to better clarify the technique,fig. 26 compares the reconstructed simulated energy spectra using the cor-rect (left) or biased (right) parameters for the first angular bin mixing all5 primaries together. Using the biased parametrization, the blue curve issistematically higher than the red one in the first, fully efficient, energy binswhile it’s sistematically lower in the highest energy bins, which means thatthis energy relation doesn’t reproduce properly the original spectrum. Thesame kind of plots have been created assuming proton or iron primaries only,and for all angular bins. Fig. 27 shows the comparison between the energyspectrum obtained using the correct and biased energy relation (average ofthe four angular bins). The difference of the flux in each energy bin gives thesystematic error (after the subtraction of the statistical error), in this caseon the absolute E(Ne) relation.

A further check on systematic effects (still under study) has been doneseparating the showers as a function of the distance of their cores from thecenter of KASCADE. This is motivated by the left side of fig. 28 where thelateral distributions of muons are plotted. In general, the central part of thelateral distribution of µ (250 < R < 470 m) is better fitted by KASCADE-Grande muon l.d.f.. Therefore, the data have been divided in two groups:250 < R < 470 m, and R < 250 m - R > 470 m (almost half number ofevents in each subset of data) and the systematic effect on the core distancehas been defined again as the semi-difference between the flux measured bythe two subsets of data. Further checks are going on separating even furtherthe data as a function of the distance (R < 250 m, R > 470 m).The flux (multiplied by E2.5) and its total error (as a squared sum of all

the systematic and statistical errors defined in this section) in each bin issummarized in table 5. The data of table 5 are plotted in fig. 29 togetherwith other experimental results. A check of the data in the highest energy

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Figure 26: Original (red line) and reconstructed (blue line) energy spectrumof all 5 mixed components for bin 1 using the correct (left) or biased (right)parametrization of the E(Ne) relation. Using the biased parametrization, theblue curve is sistematically higher than the red one in the first, fully efficient,energy bins while it’s sistematically lower in the highest energy bins.

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Figure 27: The energy spectrum (differential flux multiplied by E2.5) as afunction of log10E obtained using the correct energy relation (red points) andthe biased one (blue points).

31

Energy J(E) × E2.5

(1016eV ) (1016· m−2s−1sr−1eV 1.5)

1.00 4.59 ± 1.031.26 3.63 ± 0.561.58 2.96 ± 0.322.00 2.46 ± 0.112.51 2.08 ± 0.113.16 1.85 ± 0.133.98 1.67 ± 0.175.01 1.45 ± 0.096.31 1.31 ± 0.207.94 1.20 ± 0.1710.0 1.00 ± 0.2712.6 0.83 ± 0.2215.8 0.76 ± 0.1220.0 0.64 ± 0.0425.1 0.55 ± 0.1231.6 0.49 ± 0.0639.8 0.48 ± 0.0650.1 0.47 ± 0.1663.1 0.42 ± 0.1879.4 0.30 ± 0.19100. 0.34 ± 0.10

Table 5: The flux (multiplied by E2.5) and its total error in each bin.

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Figure 28: Lateral distributions of muons measured with KASCADE-Grande, as taken from A. Haungs’s talk at ISVHECRI 2008 (left), and effectof the selection based on the distance of the shower core from KASCADE onthe flux measured using the Ne − Nµ technique (right).

bins is under the way. Fig. 30 shows the behavior of the parameter k inthe 4 different angular bins and the average behavior. A systematic trend ofdecreasing of k as a function of energy is observed for all the primaries anddata. It is probably related to the new parametrization of k, and it needsfurther checks. However, it is still true that the experimental data lie aroundthe Silicon data. Finally, a comparison between the results obtained withthe different versions of KRETA is presented in fig. 31.

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Energy (eV/particle)

1310 1410 1510 1610 1710 1810 1910 2010 2110

)1.5

eV

-1 s

r-1

sec

-2 J

(E)

(m2.5

Flu

x E

1310

1410

1510

1610

1710

1810

1910

(GeV)ppsEquivalent c.m. energy 310 410 510 610

KASCADE - Grande Pierre Auger Exp.

direct data

PROTON

RUNJOB

KASCADE (QGSJET 01)

KASCADE H

KASCADE He

KASCADE heavy

KASCADE (SIBYLL 2.1)

KASCADE H

KASCADE He

KASCADE heavy

KASCADE-Grande J.C. Arteaga

KASCADE-Grande Ne-Nm Mario (0-40 deg) V11805

EAS-TOP

Akeno

HiRes I

HiRes II

AGASA

AUGER 2007 (ICRC 07)

RUNJOB

JACEE

ATIC

MUBEE

SOKOL

KASCADE-single h

Figure 29: Differential flux multiplied by E2.5 as a function of energy obtainedusing data of version V11805 in the angular region 0-40◦ (blue points).

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Figure 30: Evolution of the parameter k as a function of log10E (GeV) forall 4 angular bins for the different primary elements and experimental data(left plot). The average of the 4 angular bins is plotted in the right plot.

35

Energy (eV/particle)

1310 1410 1510 1610 1710 1810 1910 2010 2110

)1.5

eV

-1 s

r-1

sec

-2 J

(E)

(m2.5

Flu

x E

1310

1410

1510

1610

1710

1810

1910

(GeV)ppsEquivalent c.m. energy 310 410 510 610

KASCADE - Grande Pierre Auger Exp.

direct data

PROTON

RUNJOB

KASCADE (QGSJET 01) KASCADE H KASCADE He KASCADE heavyKASCADE (SIBYLL 2.1) KASCADE H KASCADE He KASCADE heavy KASCADE-Grande J.C. Arteaga KASCADE-Grande Ne-Nm Mario (0-25 deg) V11702 KASCADE-Grande Ne-Nm Mario (0-40 deg) V11803

KASCADE-Grande Ne-Nm Mario (0-40 deg) V11805 EAS-TOP

Akeno

HiRes I

HiRes II

AGASA

AUGER 2007 (ICRC 07)

RUNJOB

JACEE

ATIC

MUBEE

SOKOL

KASCADE-single h

Figure 31: Comparison among the differential fluxes multiplied by E2.5 asa function of energy obtained using data of versions: V11805 (blue points),V11803 (black points) and V11702 (red points).

36