7.3.1 Energy Bias
Atmosphere & Detector Model
There are numerous systematic uncertainties associated with the atmospheric model and the detector simulation. These uncertainties are common with the gamma-ray analysis; it is assumed here that they have the same efect on the iron spectrum as on the gamma-ray spectrum.
The efect of the variability of the atmosphere was assessed by simulating gamma-ray showers with atmospheric models corresponding to a particularly warm and a particularly cold week of all winter seasons from 2009 to 2012. The resulting showers were analyzed with the nominal lookup tables and were found to have an energy scale shifted by ±10 % compared to nominal.
7.3 Sources of Systematic Uncertainty energy [TeV] 2 10 ] -1 sr 1.50 TeV -1 s -2 [cm Ω dN/dE dA d 2.50 E -2 10 -1 10
20 dead pixels, QGSjet 0 dead pixels, QGSjet 40 dead pixels, QGSjet 20 dead pixels, Sibyll
Figure 7.4: Efects of the number of simulated dead channels and on the hadronic interaction model on the cosmic-ray iron spectrum. The diferential Ćux has been multiplied by E2.5 to improve the visual clarity.
It is estimated that variations in the aerosol content can shift the energy by another ±10 % independently of the variations in the density proĄle.
The dominant uncertainties related to the detector model are due to the mirror reĆectivity. The reĆectivity changes over time: Mirrors age and get dusty as they are in use. VERITAS mirrors are washed several times per season and batches of mirrors are removed for re-coating and replaced by freshly coated ones, restoring the reĆectivity to the original levels. The average reĆectivity over the 3-year time period in question is used for the simulations. The variation in reĆectivity has been estimated to afect the energy scale by ±10 %. Other sources of uncertainty are due to the quantum eiciency of the PMTs, the collection eiciency of the Ąrst dynode in the PMT, the eiciency of the light cones, the modeling of the pulse shape, and the shadowing of the camera by the telescope structure. Altogether, the uncertainties listed here lead to a ±20 % systematic uncertainty on the energy scale.
For a power law spectrum, a constant bias of the energy scale would not afect the measured spectral index, just the normalization. Let E be the true energy and Erec = (1 + b) · E the reconstructed energy, where b is the constant bias. Then E = Erec
1+b and dEdErec =
1
1+b. Let the true spectrum be a power law with normalization constant N0 at energy E0 and index γ: dN
dE = N0· ⎞E
E0
⎡−γ
7 The Cosmic Ray Iron Spectrum dN dErec = dN dE · dE dErec = N0· ⎤ E rec E0· (1 + b) ⎣γ ·1 + b1 = (1 + b)γ−1· N0·⎤E E0 ⎣γ
For example, a 20 % uncertainty on the energy bias would leads to a 40 % uncertainty on the spectral normalization for a spectral index of 2.82.
It is also estimated that there is a systematic uncertainty on the spectral index of ±0.2 due to these efects.
Energy Bias of the Template Analysis
If the energy bias is not constant, the measured spectral shape can be afected. In this case, the energy bias is approximately linear in log(E): Erec = E ·
⎞ 1 + b · log⎞E E′ 0 ⎡⎡ for some normalization energy E′
0 and linear term b. For example, the energy bias for iron showers with 0◦ (20◦) zenith angle approximately follows this log-linear relationship with b = −0.2 and E′
0 = 200 TeV (50 TeV), see Fig. 6.2.
In this case, there is no closed expression for E (Erec). However, we can write
dN dErec = dN dE · dE dErec = dN dE · ⎤dE rec dE ⎣−1 = dN dE · ⎤ 1 + b + b · log ⎤E E0 ⎣⎣−1
This function can be plotted by using E as a parameter and expressing both Erec and dEdNrec in terms of E. See Fig. 7.5 for an example. The true spectrum was chosen as a power law with normalization constant N0 = 1, normalization energy E0= 50 TeV, and index γ = 2.5.
The resulting reconstructed spectra are Ąt well by a power law with index γ′ = 2.6 and normalization constant 1.45 (1.25) over the energy range considered here. The reconstructed spectrum is softer than the true spectrum by about ∆γ ≈ 0.1.
For a log-linear energy bias as described here, the shift in the measured spectral index depends on the linear term b, while the shift in the normalization depends on the energy at which the bias vanishes.
For the analysis presented here, the efect of the energy bias is partially mitigated by the efective area, which is determined according to the reconstructed energy, not the true energy. To be conservative, we still estimate that the energy bias due to the analysis method results in a systematic uncertainty of +0
−0.1 on the spectral index and +0 %−30 % on the normalization.
7.3 Sources of Systematic Uncertainty 0.01 0.1 1 10 100
Differential Flux [A.U.]
Energy [TeV]
true spectrum dN/dE = C*E-2.5 reco. spectrum dN/dErec, E0’=200 TeV
reco. spectrum dN/dErec, E0’= 50 TeV
Figure 7.5: Toy study to estimate the efect of a log-linear energy bias on the reconstructed spectrum. See text for details.
7.3.2 Dead Pixels in the Camera
Pixels in the VERITAS camera may be turned of either temporarily (high currents due to bright starlight) or for a longer time (broken PMTs/cables, issues with the FADC channel). These pixels are not used in the analysis and this can afect image parameters such as the length and width. Also, the identiĄcation of DC pixel candidates is afected: If the DC light lands on a ŚdeadŠ pixel, the correct DC pixel candidate cannot be identiĄed. In the data set studied here, the number of dead pixels per camera ranged from 6 to 49 per camera, with an average of 18 dead pixels per camera. In the simulations used for the nominal analysis (train- ing and evaluation of random forests, calculation of efective areas), 20 pixels were turned of per camera. The entire procedure was repeated with 0 and 40 dead pixels per camera in the simulation (look-up tables were not changed). The resulting energy spectra are plotted in Figs. 7.3 and 7.4 adn the results of the spectral Ąt can be found in Table 7.2. The efect on the energy spectrum is smaller than the statistical uncertainty of the Ąt. The normalization changes by ≤ 7 % and the index by ≤ 0.07. We use these values as a conservative estimate of the systematic uncertainties due to the number of dead channels.
7.3.3 Hadronic Interaction Model
Due to the large computing requirements, it was not possible to re-do all simulations with a diferent interaction model. Instead, a Sibyll-based spectrum was calculated as follows. First, in the BDT testing sample (used to determine the ON/OFF ratios), the proton, helium, and iron samples were replaced with simulations based on Sibyll, and new ON/OFF ratios were calculated (cf. Table 6.3). Second, the efective areas used in each energy bin were adjusted by the ratio of the number of Sibyll events passing the cuts over the number of QGSjet events passing cuts (c.f. Table 6.4). The resulting spectrum can be seen in Figs. 7.3 and 7.4.
7 The Cosmic Ray Iron Spectrum
Cause Efect on N0 Efect on γ
Atmosphere + detector model ±40 % ±0.2
Dead pixels ±7 % ±0.07
Intrinsic energy bias +0 %
−30 % +0.0 −0.1 Efective area ±10 % Ů Interaction Model ±12 % ±0.1 Remaining Background +0 % −15 % Ů Total +44 % −55 % +0.23 −0.25 Table 7.3
The index is harder than the nominal spectrum by about 0.1 and the normalization is lower than the nominal spectrum by about 12 %. We estimate the uncertainty due to the hadronic interaction model to be ∆γ = ±0.1 and ∆f0 = ±12 %.
7.3.4 Effective Area
As described in Section 6.2.1, the efective collection area is determined from Monte-Carlo simulations. A statistical uncertainty on the efective area exists due to limited shower statistics. It is below 10 % for all energy bins. To be conservative, we estimate a 10 % systematic uncertainty on the Ćux normalization due to this efect.
7.3.5 Remaining Background
Elements up to chromium (Z = 24) were accounted for in the background sample. According to Hörandel, J. R., (2003), manganese (Z = 25) has a Ćux of about 10 % of the iron Ćux in the energy range under study here, and nickel (Z = 28) has a Ćux of about 7 % of the iron Ćux. All other elements can be neglected (< 1 % of the iron Ćux). Assuming that showers induced by manganese and nickel have similar properties as iron does, the measured ŚironŠ Ćux may be contaminated by up to 15 % Manganese and Nickel. This is assigned as an additional systematic uncertainty.
7.3.6 Combined Systematic Uncertainty
All sources of systematic uncertainty investigated here are listed again in Table 7.3. They are independent of each other and can be added in quadrature. The resulting Ąnal measurement of the parameters of the cosmic ray iron spectrum are thus:
f0 =⎞4.82 ± 0.98(stat.)+2.12−2.65(syst.)⎡· 10−7cm−2s−1TeV−1sr−1 γ = 2.82 ± 0.30(stat.) ±+0.23−0.25(syst.)