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Astron. Astrophys. 349, 11-28 (1999)

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4. Determination of energy spectra and sources of systematic errors

Compared to single IACTs, stereoscopic IACT systems permit drastic reduction of systematic errors, in particular for tasks like the precision determination of energy spectra. Using the redundant information provided by the multiple views, essentially all relevant characteristics, such as the radial distribution of Cherenkov light or the trigger probabilities of the telescopes can be verified experimentally (see also Hofmann 1997). Simultaneous sampling of the intensity of the Cherenkov light front in different locations emphasizes the calorimetric nature of the energy determination, and reduces the effect of local fluctuations. Finally, given the unambiguous reconstruction of the shower geometry and the fact that at energies of one TeV virtually all events within 100 m from the central telescope trigger the system, and that above a few TeV almost all events within 200 m trigger, the effective detection area above 1 TeV can be basically defined by pure geometry, without relying on simulations. Only the threshold region requires a critical consideration.

Under ideal conditions, the differential energy spectrum of the incident radiation is determined as

[EQUATION]

where [FORMULA] is the measured rate of [FORMULA]-rays of energy E after background subtraction, [FORMULA] is the effective detection area and [FORMULA] is the efficiency of the cuts applied to isolate the signal and to suppress the background. The effective area and the cut efficiencies are usually derived from Monte Carlo simulations.

A complication arises from the finite energy resolution, described by the response function [FORMULA], the probability that a [FORMULA]-ray of energy E is reconstructed at an energy [FORMULA]. The measured rate is hence given by the convolution

[EQUATION]

Eq. 2 can no longer be trivially inverted to yield [FORMULA]. Options to find [FORMULA] include the explicit deconvolution using a suitable algorithm, which will usually make some assumption concerning the smoothness of the spectrum. Another approach is to assume a certain functional form for the shape of the spectrum, and to determine free parameters such as the flux and the spectral index from a fit of Eq. 2 to the data. Finally, one can absorb the effect of the energy smearing into a modified effective area A, defined such that Eq. 1 holds. The latter approach is the simplest, but has the disadvantage that now A depends on the assumed shape of the spectrum. However, with the [FORMULA] energy resolution provided by the HEGRA CT system, shape-dependent corrections are negligible for most practical purposes, and results are stable after one iteration. Therefore, while both other techniques were pursued, the final results are based on this third method.

The typical energy dependence of the effective area is shown in Fig. 1. Below 1 TeV, the effective detection area rises steeply with energy, and then saturates at around [FORMULA] m2. The saturation reflects the cut on a maximum distance from the central telescope of 200 m. The technical implementation of the energy reconstruction and flux determination is described in detail in Paper 1. Compared to the brief discussion given above, a main complication for real data arises from the dependence of A on the zenith angle [FORMULA], which varies during runs and from run to run. In order to be able to interpolate between Monte Carlo-generated effective areas at certain discrete zenith angles, a semi-empirical scaling law is exploited, which relates the effective areas at different zenith angles. The variation of zenith angles with time is accounted for through replacing Eq. 1 by the sum over all events recorded within the observation time T

[EQUATION]

where each event is weighted with the appropriate effective area, given its energy and zenith angle. Note that for each period of a certain hardware configuration a set of effective areas is used which has been determined from the Monte Carlo simulations which model in detail the specific hardware performance, i.e. which take into account the trigger configuration and the mirror point spread function (see Paper 1).

[FIGURE] Fig. 1. Effective area of the HEGRA IACT system as a function of energy, for vertical [FORMULA]-rays. The saturation at [FORMULA] m2 at high energies reflects the cut in impact distance at 200 m relative to the central telescope. The uncertainty of the effective area in the threshold region caused by a 5% variation of the detection threshold and by the interpolation in zenith angles is shown by the hatched area. The corresponding systematic error on the effective area is [FORMULA]% at 500 GeV and [FORMULA]% at 1 TeV.

The key aspect in a precise and reliable determination of [FORMULA]-ray spectra is the control of systematic errors. Sources of systematic errors include, e.g.,

  • Systematic errors in the determination of the absolute energy scale.

  • Deviations from the linearity of the energy reconstruction, caused e.g. by threshold effects at very low energies, and possible saturation effects in the PMTs or the electronics at very high energies.

  • Systematic errors in the determination of the effective area A; particularly critical is the threshold region, where A is a very steep function of E.

  • Systematic errors in the determination of the (energy-dependent) efficiency of the angular and image shape cuts.

As discussed in detail in Appendix A, a non-accurate modeling of the detector response or the atmospheric transmission possibly results in 1) a shift in the energy scale of the reconstructed [FORMULA]-ray spectrum, and 2) a distortion of the shape of the spectrum. Most systematic uncertainties, e.g. mirror reflectivities and PMT quantum efficiencies, exclusively contribute to an error of the energy scale. The calculations in Appendix A show that the curvature of the spectrum is reconstructed correctly provided that the Monte Carlo simulations accurately model the correlation between the detector threshold and the reconstructed energies including the fluctuations involved. If the Monte Carlo description of this correlation is incorrect, the energies reconstructed in data and the effective areas computed from Monte Carlo simulations do not match. A shift of the reconstructed energies by a factor [FORMULA] relative to the effective area A used for the evaluation of the spectrum results in a flux error of

[EQUATION]

The flux error is potentially large in the threshold region, where A varies quickly with E, [FORMULA] with [FORMULA] and [FORMULA], but is negligible at high energies, where A is constant, and is simply governed by the geometrical cuts (see Fig. 1).

The remainder of this section is dedicated to a discussion of the various sources of systematic errors.

4.1. Reliability of the determination of the shower energy

A crucial input for the determination of shower energies is of course the expected light yield as a function of core distance. Since the average core distance varies significantly with energy, inadequacies in the assumed relation will not only worsen the energy resolution, but will systematically distort the spectrum. With the redundant information provided by a system of Cherenkov telescopes, it is possible to actually measure the light yield as a function of the distance from the shower core and to verify the simulations (Aharonian et al. 1998). Briefly, the idea is to select showers with a fixed impact parameter relative to a telescope A, and with a fixed light yield in this telescope. This provides a sample of showers of constant energy, and now the light yield in other telescopes can be measured as a function of core distance. Fig. 2 shows the characteristic shape of the light pool for [FORMULA]-ray showers, which is well reproduced by the simulation; this holds also for the variation of the shape with shower energy and zenith angle (Aharonian et al. 1998).

[FIGURE] Fig. 2. Measured light yield as a function of distance to the shower core for [FORMULA]-ray showers in the energy range of about 0.9 to 1.8 TeV compared with Monte Carlo simulations (hatched band). Note that the light yield assigned to images depends on the field of view of the camera, and on the definition of "image" pixels; the data should not be compared to "raw" simulations not including such effects.

A second key ingredient in the energy determination is the reconstruction of the core location. Unlike in the case of the angular resolution, where it is easy to show that the simulation reproduces the data by comparing the distribution of shower directions relative to the source with the simulations, a direct check of the precision of the core reconstruction is not possible. However, noting that two telescopes suffice for a stereoscopic reconstruction, one can split up the four-telescope system into two systems of two telescopes and compare the results (Hofmann 1997). Fig. 3a illustrates the difference in core coordinates between the two subsystems for data and for the Monte Carlo simulations. As can be recognized the Monte Carlo simulations accurately predict the distribution of the distances between the two cores. Under the assumption that the two measurements are uncorrelated and that the reconstruction accuracy achieved with two telescopes is the same for two telescope and four telescope events the width of the difference distribution should be [FORMULA] times the resolution of a two-telescope system. By this means, the 2 telescope resolution is determined to be [FORMULA] m. The Monte Carlo simulations show, that this reconstruction accuracy does indeed agree nicely with the true reconstruction accuracy for 2 telescope events.

[FIGURE] Fig. 3. a  Difference in the x coordinate of the shower core as measured by independent subsystems of two telescopes for events where all four telescope triggered. Showers are selected to provide a minimum stereo angle of [FORMULA] in each subsystem, and a maximum core distance of 200 m from the central telescope. Points show the data, the dashed line the simulation, and the full line a Gaussian fit to the data with a width of 14 m. b  Comparison of the energies measured by the two subsystems. The width of the Gaussian fit is 25%. Same cuts as in a .

The same technique can be used to study the energy resolution (Fig. 3b; see also Hofmann 1997). Also here the data is in excellent agreement with the simulations. In this case, the Monte Carlo studies predict, that the energies measured with the two subsystems are considerably correlated: assuming uncorrelated estimates, an energy resolution of [FORMULA] is inferred; the Monte Carlo simulations predict a true energy resolution for two telescope events of 23%. On the basis of simulations, fluctuations in the shower height can be identified as the origin of this correlation. Work is ongoing to use the stereoscopic determination of the shower height to improve the energy resolution. The excellent agreement between data and Monte Carlo simulations confirms that the experimental effects entering the energy determination are well under control, and that consequently the estimate of the energy resolution based on the simulations is reliable.

4.2. The absolute energy scale

The absolute energy calibration of IACTs is a significant challenge, lacking a suitable monoenergetic test beam. Factors entering the absolute energy calibration are the production of Cherenkov light in the shower, the properties and the transparency of the atmosphere, and the response of the detection system involving mirror reflectivities, PMT quantum efficiencies, electronics calibration factors etc. So far, mainly three techniques were used to calibrate the HEGRA telescopes:

  • The comparison between predicted and measured cosmic-ray detection rates. Given the integral spectral index of 1.7 for cosmic rays, an error [FORMULA] in the energy scale results in an error of [FORMULA] in the rate above a given threshold. Apart from the slightly different longitudinal evolution of hadronic and of [FORMULA]-ray induced showers, this test checks all factors entering the calibration. The Monte Carlo simulations reproduce the measured cosmic-ray trigger rates within 10% (Konopelko et al. 1999). The IACT system was furthermore used to determine the flux of cosmic-ray protons in the 1.3 to 10 TeV energy range (Aharonian et al. 1999b). The measured flux of protons

    [EQUATION]

    [EQUATION]

    excellently agrees with a fit to the combined data of all other experiments, indicating that systematics are well under control and that the assigned systematic errors - partially related to the energy scale - are rather conservative.
  • Given the measured characteristics of the telescope components such as mirrors or PMTs, the sensitivity of the telescopes can be determined with an overall error of 22%.

  • Using a distant, calibrated, pulsed light source, an overall calibration of the response of the telescope and its readout electronics could be achieved, with a precision of 10% (Fraß et al. 1997).

The last three techniques have to rely on the Monte Carlo simulations of the shower and of the atmospheric transparency. While Monte Carlo simulations have converged and different codes produce consistent results, details of the atmospheric model and assumptions concerning aerosol densities can change the Cherenkov light yield on the ground by about 8% (Hemberger 1998).

Within their errors, all calibration techniques are consistent. Overall, we believe that a 15% systematic error on the energy scale is conservative, given the state of simulations and understanding of the instrument. Based on the comparison with cosmic-ray rates, one would conclude that the actual calibration uncertainty is below 10%.

4.3. The threshold region and associated uncertainties

As discussed earlier in detail, the threshold region is very susceptible to systematic errors. In the sub-threshold regime events trigger only because of upward fluctuations in the light yield, and energy estimates tend to be biased towards larger values. Fig. 4 shows the mean reconstructed energy as a function of the true energy for a sample of simulated events at typical zenith angles. Note that for the 1997 data set we have a noticeable number of detected [FORMULA]-rays events below 500 GeV. However in this energy region the bias is so strong that a reliable correction is no longer possible. Therefore spectra will only be quoted above this energy.

[FIGURE] Fig. 4. Mean reconstructed shower energy as a function of the true energy for showers incident under 20o zenith angle, also showing the rms errors of the energy reconstruction.

The first major source of systematic errors at low energies is the detailed modeling of the triggering process. In the simulation great emphasis was placed on the correct description of the pixel trigger probabilities as a function of signal amplitude. For each trigger configuration and for each telescope this dependence has been derived from air shower data using the recorded information about which pixels of a triggered telescope surpassed the discriminator threshold and which pixels did not. As discussed above and in Appendix A, the onset of the size -distribution provides a sensitive test of the quality of the simulation. Fig. 5 shows the measured and simulated size -distribution for two telescopes for a certain trigger configuration ("Data-period I" of Paper 1). For our current best simulations, fits to the rising edges of the size -distributions indicate for the individual telescopes and the different trigger configurations deviations of the size scale between data and Monte Carlo on the 5% level. For conservatively estimating the systematic error on the shape of the Mkn 501 spectrum, we allow for a [FORMULA]5% correlated shift of the thresholds of all four telescopes. The resulting uncertainties in the effective detection area A are indicated in Fig. 1; as expected, A significantly changes in the threshold regime, but remains constant well above threshold.

[FIGURE] Fig. 5. Size -distributions of [FORMULA]-ray induced air showers for data and simulations for two of the four telescopes. The data is from a period of a certain trigger configuration, namely "Data-period I" of Paper 1, and the Monte Carlo simulations used the telescope specific single pixel trigger probability as function of signal amplitude derived from this data (Mkn 501 data, background subtracted, Monte Carlo weighted according to the results in Sect. 5).

Random threshold variations (Gaussian-distributed with a width of 15%) between individual pixels were found to be of relatively small influence compared to the systematic threshold shifts.

As another source of systematic errors of special importance in the threshold region, the accuracy of the energy reconstruction for zenith angles between the discrete simulated zenith angles ([FORMULA], [FORMULA], [FORMULA], [FORMULA]) has been considered. Imperfections in the scaling law used to relate the Cherenkov light yield at different zenith angles [FORMULA] could result in a systematic shift of the reconstructed energy at intermediate values of [FORMULA]. To test the description, the energy of the Monte Carlo showers with [FORMULA] was determined with the light yield tables of the [FORMULA]- and [FORMULA]-showers and the result was compared to the result based on using the [FORMULA] light yield table. The systematic shift of the reconstructed energies was about 5% below 1 TeV and 2% above 1 TeV; based on this and other studies we believe that using the full set of simulations, systematic shifts in the reconstructed energy are below 5% and 2%, below and above 1 TeV respectively.

The modified effective area used for the determination of the spectrum slightly depends on the assumed source spectrum. At the lowest and at the highest energies the source spectrum can not be determined with high statistical accuracy. For energies below 1 TeV we conservatively estimate the corresponding uncertainty in the modified effective area, by varying the spectral index of an assumed source spectrum dN/dE[FORMULA][FORMULA] from [FORMULA] to [FORMULA]. At the highest energies above 15 TeV we vary the assumed source spectrum from a broken power law to a power law with an exponential cutoff, both specified by fits to the data. In addition we explore the statistical significance of the [FORMULA]-ray excess at the highest energies by a dedicated [FORMULA]-analysis (see Sect. 5).

In the threshold region of the detector the total systematic error is dominated by the systematic shift of the telescope thresholds and by the possible systematic shift in the reconstructed energy due to the zenith angle interpolation. The total systematic error is computed by summing up the individual relative contributions in quadrature.

4.4. Saturation effects at high energies

The PMTs and readout chains of the HEGRA system telescopes provide a linear response up to amplitudes of about 200 photoelectrons. At higher intensities, nonlinearities of the PMT become noticeable, and also the 8-bit Flash-ADC saturates. With typical signals in the highest pixels of 25 photoelectrons per TeV [FORMULA]-ray energy, the effects become important for energies around 10 TeV and above.

Saturation of the Flash-ADC can be partly recovered by using the recorded length of the pulse to estimate its amplitude, effectively providing a logarithmic characteristic (Heß et al. 1998). The saturation characteristics of the PMT/preamplifier assembly have been measured, and a correction is applied. These nonlinearities and the Flash-ADC saturation are included in the simulations. As a very sensitive quantity to test the handling of saturation characteristics, the dependence of the pulse height in the peak pixel on the image size emerged (Fig. 6). Data and simulations are in very good agreement up to pixel amplitudes exceeding 103 photoelectrons, equivalent to [FORMULA]50 TeV [FORMULA]-ray showers. Older versions of the simulations, which did not properly account for PMT/preamp nonlinearities, showed marked deviations for size -values above 103.

[FIGURE] Fig. 6. The mean signal in the peak pixel, hot1, as a function of the image size , for data (full points) and simulations (open points).

Independent tests of saturation and saturation corrections were provided by omitting, both in the data and in the simulation, the highest pixels in the image, and by comparing event samples in different ranges of core distance and zenith angle. The saturation effect is non-negligible only at very high energies, namely [FORMULA]. However, all tests show that given the quality of our current simulations, systematic errors induced by saturation effects even in this energy region are yet small compared to the statistical errors.

4.5. Efficiency of cuts

Among the sources of systematic errors, the influence of cuts is less critical. The large flux of [FORMULA]-rays from Mrk 501 combined with the excellent background rejection of the IACT system allows to detect the signal essentially without cuts. In the analysis, one can afford to apply only rather loose cuts, which keep over [FORMULA] of all [FORMULA]-rays; only at the lowest energies, a slightly larger fraction of events is rejected. Since only a small fraction of the signal is cut, the uncertainty in the cut efficiency is a priori small; in addition, the efficiency can be verified experimentally by comparing with the signal before cuts, see, e.g., Fig. 7. The cut efficiencies in Fig. 7 deviate from the results shown in Paper 1 on the 5%-level due to slightly improved Monte Carlo simulations. We conservatively estimate the systematic error the cut efficiencies, to be 10% at 500 GeV decreasing to a constant value of 5% at 2 TeV and rising above 10 TeV to 15% at 30 TeV. At low energies the acceptances are corrected according to the measured efficiencies.

[FIGURE] Fig. 7a and b. Efficiency of the cut on the shower direction relative to the Mrk 501 location after the software threshold of at least 40 photoelectrons in two or more telescopes a , and of the shape cuts used to enhance [FORMULA]-rays b , as a function of energy. Full points show the measured efficiencies, open points the results of the Monte Carlo simulations. Efficiencies determined from data can be larger than one due to background fluctuations.

4.6. Other systematic errors and tests

The precision in the determination of the effective areas is partly limited by Monte Carlo statistics. The universal scaling law used to relate Monte Carlo generated effective areas at different zenith angles involves a rescaling of shower energies. Statistical fluctuations in a Monte Carlo sample at a given energy and angle will hence influence the area over a range of energies. Because of the resulting slight correlation, Monte Carlo statistics is in the following included in the systematic errors.

To test for systematic errors, the data sample was split up into subsamples with complementary systematic effects, and spectra obtained for these subsamples were compared. Typical subsamples include

  • Events where 2, 3 or 4 telescopes are used in the reconstruction.

  • Events passing a higher `software trigger threshold', requiring e.g. a minimum size of 100 photoelectrons, or a signal in the two peak pixels of at least 30 photoelectrons.

  • Events with showers in a certain distance range from the center of the system (CT3), e.g. 0-120 m compared to 120-200 m. This comparison tests systematics in the light-distance relation as well as the correction of nonlinearities in the telescope response.

  • Events where all pixels are below the threshold for nonlinearities.

  • Events in different zenith angle ranges. The comparison of these spectra provides a sensitive test of the entire machinery, and also of nonlinearities, where the data at larger angles should be less susceptible because of the smaller size at a given energy.

For each of the subsamples, the effective area and cut efficiencies were determined, and a flux was calculated. In all cases, deviations between subsample spectra were insignificant, or well within the range of systematic errors. Among the variables studied, the most significant indication of remaining systematic effects is seen in the comparison of different ranges in shower impact parameter relative to the central telescope. The ratio of the spectra determined with the data of small ([FORMULA]120 m) and large (between 120 m and 200 m) impact distances is shown in Fig. 8. A fit to a constant gives a mean ratio of [FORMULA] with a [FORMULA]-value of 23.1 for 12 degrees of freedom, corresponding to a chance probability for larger deviations of 5%.

[FIGURE] Fig. 8. Ratio of the spectra computed with the events with small (0 to 120 m) and with large (120 m to 200 m) impact distances relative to the central telescope. A fit to a constant gives a ratio of [FORMULA] with a [FORMULA] of 23.1 for 12 degrees of freedom. The fit is restricted to the energy region of reasonable systematic errors, i.e. with an effective detection area for both event samples larger than [FORMULA] m2.

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Online publication: August 25, 1999
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