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Astron. Astrophys. 342, 69-86 (1999)

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3. Analysis of the IACT system data

3.1. Monte Carlo simulations

The Monte Carlo simulation (Konopelko et al. 1998) is divided into two steps. First, the air showers are simulated and the Cherenkov photons hitting one telescope are stored on mass storage devices. Thereafter the detector simulation is carried out. This method has the advantage that it is possible to use the same simulated showers with different detector setups. The air shower simulation is based on the ALTAI code. The results of the code have been tested against results of the CORSIKA code which gave for hadron induced air showers excellent agreement in all relevant observables, e.g., in the predicted detection rates and in the distribution of the image parameters (Hemberger 1998). The detector simulation (Hemberger 1998) accounts for the absorption of Cherenkov photons in the atmosphere due to ozone absorption, Rayleigh scattering and Mie scattering. Furthermore, the mirror reflectivity, the mirror point spread function, and the acceptances of the plexiglass panels and the light collecting funnels in front of the cameras are taken into account. See Table 2 for a summary of the efficiencies which are relevant for the simulations. In the table only the mean values averaged over the wavelength region from 300 to 600 nm are given, in the simulations the efficiencies depend on the wavelength. The point spread functions of the telescope mirrors are extracted from the point runs. The time-resolved photon to photoelectron conversion by the PMTs is modeled using a measured PMT pulse shape and a measured single photoelectron spectrum. Finally, the trigger processes and the digitization of the PMT pulses are simulated in detail.


[TABLE]

Table 2. Efficiencies averaged over the wavelength region from 300 to 600 nm.


The simulated events are stored in the same format as the raw experimental data and are processed with the same event reconstruction and analysis chain as the experimental data. Showers induced by photons as well as by hydrogen, helium, oxygen, and iron nuclei were simulated for the zenith angles [FORMULA], [FORMULA], [FORMULA], and [FORMULA]. For the purpose of comparing the experimental data with the Monte Carlo predictions, the Monte Carlo events are weighted to generate the appropriate spectrum. The events induced by the proton, helium, oxygen, and iron nuclei are weighted according to the cosmic ray abundances of the corresponding groups from (e.g. Wiebel et al. 1998). For each type of primary particle, and for each zenith angle approximately 2 [FORMULA] 105 showers have been generated.

The excellent agreement of the observable quantities in the experimental data and the Monte Carlo data for cosmic ray-induced showers as well as for photon-induced showers is described in detail in Konopelko et al. (1998) and in Aharonian et al. (1998). The comparisons between data and Monte Carlo which are relevant for the analysis of the Mkn 501 data of this paper are discussed in the following 4 subsections.

Three different Monte Carlo event-samples have been generated, with different trigger settings and mirror point spread functions, corresponding to the three data-taking periods.

3.2. Data sample and data cleaning

The analysis described in this paper is based on 110 hours of Mkn 501 data acquired between March 16th, 1997 and October 1st, 1997 under optimal weather conditions (i.e. a clear sky and a humidity less than 90%), with the optimal detector performance, and with Mkn 501 being more than [FORMULA] above the horizon. Only data runs where all 4 IACTs were operational and in which not more than 20 pixel were defective in any IACT have been admitted to the analysis. Furthermore the data runs had to satisfy the requirements of the mean cosmic ray rate deviating by less than 15% from the zenith angle dependent expectation value, and the width parameter averaged over all events and all telescopes deviating by less than 6% from the zenith angle dependent expectation value.

The Mkn 501 data were acquired in the so called "wobble mode" (Daum et al. 1997). In this mode the telescopes are pointed into a direction which is shifted by [FORMULA] in declination with respect to the source direction. The direction of the shift is reversed for each data run of 20 minutes duration. For each run, the solid-angle region located [FORMULA] from the Mkn 501 location on the opposite side of the camera center is used as OFF region for estimates concerning the background contamination of the ON region by cosmic ray-induced showers. The large angular distance between the ON solid-angle region around the Mkn 501 direction and the OFF solid-angle region assures a negligible contamination of the OFF data with Mkn 501 [FORMULA]-rays. The symmetric location of the ON and the OFF region in the camera with respect to the optical axis and the camera geometry assures almost equal background characteristics for both regions. The zenith angle dependence of the background rate is to first order compensated by using as many runs with lower-declination OFF regions as with higher-declination OFF regions.

The overall stability of the detector and the understanding of the detector performance during the three data periods has been tested by comparing several key Monte Carlo predictions for hadron-induced showers with the experimental results. In the following 3 subsections tests concerning photon-induced showers, based on [FORMULA]-rays from Mkn 501, will be described.

The observed cosmic ray detection rates have been compared with the detection rates as inferred from the Monte Carlo simulations together with the cosmic ray fluxes from the literature. In Fig. 1 the dependence of the cosmic ray detection rate on the zenith angle is shown for the first data-period and the corresponding Monte Carlo data-sample. The Monte Carlo describes the dependence with an accuracy of 10%. In Fig. 2 the measured and predicted rates are shown for the whole 1997 data-base. The measured rates have been normalized to a zenith angle of [FORMULA] according to the empirical parameterization shown in Fig. 1. For all three data periods the Monte Carlo simulations predict the measured rates with an accuracy of 10%, and they accurately describe the relative rate differences between the data-taking periods. The measured rates within each data-period show a spread of [FORMULA] FWHM, after correcting for the zenith angle dependence of the rate. The origin of this small spread is still unclear. The rate deviations do not correlate with the temperature, the pressure, or the humidity, as measured at the Nordic Optical Telescope which is localized within several hundred meters from the HEGRA site. Neither is a correlation found with the V-band extinction measured with the Carlsberg Meridian Circle which is situated at a distance of [FORMULA]500 m from the HEGRA site.

[FIGURE] Fig. 1. The cosmic ray detection rate of the IACT system (dots) as a function of zenith angle (data from data-period I). Monte Carlo rate predictions are superimposed (solid line). The measured and the Monte Carlo data agree with an accuracy of 10% (hardware threshold, no cuts).

[FIGURE] Fig. 2. The measured cosmic ray detection rate (dots) and the Monte Carlo based predictions (solid line) are shown for all the cosmic ray data of the Mkn 501 runs of 1997. The measured rates have been normalized to a zenith angle of [FORMULA] using an empirical parameterization. The Monte Carlo simulations for [FORMULA] zenith angle have been used (hardware threshold, no cuts).

To summarize, the measurements of the cosmic-ray event rate prove the stability of the IACT system at a level of 5% and the event rate is correctly predicted by the Monte Carlo simulations using the cosmic ray abundances from the literature with an accuracy of 10%.

3.3. The stereoscopic reconstruction of the direction of primary particles

Based on the stereoscopic images of the shower, the shower axis is reconstructed accurately and unambiguously using a simple geometric method (Aharonian et al. 1997b). The reconstruction permits to determine the distances of the telescopes from the shower axis and consequently, to accurately reconstruct the shower energy and to efficiently suppress the background of cosmic ray-induced showers.

The reconstruction method uses the standard second moment parameterization (Hillas 1985; Fegan 1996) of the individual images. Each image is described by an ellipsoid of inertia computed from the measured Cherenkov light intensities in the camera. The intersection of the major axes of two images superimposed in the "common focal plane", i.e. in directional space, yields one estimate of the shower direction. If more than two telescopes observed a shower, the arrival directions computed for all pairs of images are combined with a proper weighting factor to yield the common estimate of the arrival direction. The weighting factor is chosen proportional to [FORMULA], where [FORMULA] is the angle between the two major axes. Taking into account the shower direction, the shower core is reconstructed using a very similar geometric procedure. Note, that this method is based exclusively on the geometry of the imaging systems and of the shower axis and does not rely on any Monte Carlo predictions.

The angular resolution achieved with this method has been determined using both Mkn 501 [FORMULA]-ray data and the Monte Carlo simulations. In the case of the Mkn 501 data this is done as follows. The squares of the angular distances [FORMULA] of the reconstructed shower directions from the Mkn 501 position are histogrammed. The subtraction of the corresponding distribution of the fictitious OFF source yields the background-subtracted distribution of the [FORMULA]-ray events. In order to reduce the background-induced fluctuations, the analysis is performed with the [FORMULA]/h-separation cut [FORMULA] (see next subsection for the definition of [FORMULA]). In the following, the Monte Carlo photon-induced showers are weighted according to a power law spectrum with differential spectral index of -2.2.

On the left side of Fig. 3 the [FORMULA]-distributions for the ON and the OFF regions are shown for the zenith angle interval [FORMULA]-[FORMULA]. On the right side of Fig. 3 the distribution obtained after background subtraction is compared to the distribution for the [FORMULA] Monte Carlo showers. There is good agreement between the data and the Monte Carlo. The projected angular resolution is [FORMULA] for showers near the zenith and is slightly worse for the [FORMULA]- and the [FORMULA]-showers i.e. [FORMULA] and [FORMULA] respectively. In the analysis presented in this paper, only a loose cut of [FORMULA] is used which accepts, after softwarethreshold, 85% of the photon induced showers and rejects 99% of the hadron-induced showers. By this loose cut the systematic uncertainties caused by the energy-dependent [FORMULA]-ray acceptance of the cut are minimized.

[FIGURE] Fig. 3. On the left side, the squared angular distances [FORMULA] of the reconstructed event directions from the Mkn 501 direction (full line) and from the OFF source direction (dotted line) are shown for zenith angles below [FORMULA]. On the right side, the measured background subtracted [FORMULA]-distribution (full circles) is compared to the [FORMULA] zenith angle Monte Carlo simulations (open circles) All distributions have been computed using the software threshold of at least 2 IACTs with [FORMULA] and the loose [FORMULA]/h-separation cut [FORMULA].

In Fig. 4 the [FORMULA]-ray acceptance ([FORMULA]) of the angular cut is shown as a function of the reconstructed shower energy, as determined both from the Mkn 501 and from Monte Carlo data. In the case of the Mkn 501 data the same background subtraction technique is used as described above. For the lowest energies ([FORMULA] 1 TeV) the [FORMULA]-ray acceptance is slightly lower than for higher energies, i.e. the angular resolution is slightly worse. This is a consequence of the photon statistics per image and the corresponding uncertainty of the images' major axes. At higher energies the angular resolution improves less than expected from the increase in photon statistics. This is a consequence of an increasing fraction of showers which at higher energies are still able to fulfill the trigger criteria, albeit having impact points far away from the telescope system. With increasing distance of the impact point from the telescope system, the accuracy of the direction reconstruction decreases as a consequence of smaller angles between the image axes of the different telescopes.

[FIGURE] Fig. 4. The [FORMULA]-ray acceptance of the cut [FORMULA] as a function of the reconstructed primary energy, computed with the Mkn 501 gamma-rays (full circles) and with the Monte Carlo simulations (open circles). The computation of the cut acceptances is based on the number of excess events found in the ON region of angular radius of [FORMULA] (cuts as in Fig. 3).

3.4. Image analysis gamma/hadron separation

The IACT technique permits the suppression of the background of cosmic ray-induced showers by the directional information and, additionally, by analysis of the shapes of the shower images. In the case of stereoscopic IACT systems, the gamma/hadron separation power of the parameter [FORMULA] of the standard second moment analysis, which describes the transverse extension of the shower image in one camera, can be increased substantially, due to two facts: First, the location and the orientation of the shower axis is known from the three-dimensional event reconstruction and cuts can be optimized accordingly. Second, a telescope system provides complementary information about the transversal extension of the shower obtained from different viewing angles.

The parameter "mean scaled width" [FORMULA] (Konopelko 1995; Daum et al. 1997) has been used for gamma/hadron-separation. It is defined according to:

[EQUATION]

where the sum runs over all [FORMULA] telescopes which triggered. [FORMULA] is the [FORMULA]-parameter measured with telescope i and [FORMULA] is the [FORMULA]-value expected for photon-induced showers, given the telescope distance [FORMULA] from the shower axis, the total number of photoelectrons, [FORMULA], observed in the telescope, and the zenith angle [FORMULA] of observations. The [FORMULA]-values are computed from a Monte Carlo table, using an empirical function for interpolation between the simulated zenith angles. By using a scaled [FORMULA]-parameter, it is possible to take into account that on average the widths of the shower images widen with increasing telescope distance from the shower axis and with the total number of photoelectrons recorded in a telescope. By averaging over the values computed for each telescope, the statistical accuracy of the parameter determination improves and the information about the shower gained from different viewing angles is combined.

Fig. 5 shows the distribution of the [FORMULA] parameter for the Mkn 501 [FORMULA]-rays and for the Monte Carlo photon data-sample. The distribution for the Mkn 501 [FORMULA]-rays has been obtained as follows. The [FORMULA]-values of the events satisfying the loose cut on the angular distance [FORMULA] from the Mkn 501 location [FORMULA] are histogrammed. The subtraction of the corresponding OFF distribution yields the background-free distribution of the [FORMULA]-ray events. As can be seen in Fig. 5 the experimental distribution and the Monte Carlo distribution are in excellent agreement.

[FIGURE] Fig. 5. On the left side, the mean scaled width distribution is shown for the ON region (full line) and for the OFF region (dotted line). On the right side, the background subtracted distribution (full circles) is compared to the Monte Carlo distribution (open circles). (all distributions: software threshold: at least 2 IACTs with [FORMULA], and, cut: [FORMULA], data: all zenith angles, Monte Carlo: zenith angle = [FORMULA]).

In the analysis of this paper only a loose cut of [FORMULA] which accepts, after softwarethreshold, 96% of the photons and rejects 80% of the cosmic ray-induced air showers is used. With this loose cut the systematic uncertainties caused by the energy dependent [FORMULA]-ray acceptance of the cut are minimized. In Fig. 6 the [FORMULA]-ray acceptances [FORMULA] of the shape cut as determined from data and as determined from Monte Carlo as a function of the reconstructed energy are compared to each other. The results are in excellent agreement with each other. Due to background fluctuations, the determination of the cut acceptance from experimental data can yield values larger than one.

[FIGURE] Fig. 6. The [FORMULA]-ray acceptance of the cut [FORMULA] as a function of the reconstructed primary energy, computed with the Mkn 501 gamma-rays (full circles) and with the Monte Carlo simulations (open circles) (cuts as in Fig. 5).

3.5. Reconstruction of the primary energy and determination of differential spectra

The determination of a differential [FORMULA]-ray spectrum is performed in several steps. In the first step, for all events of the ON and the OFF region the primary energy E is reconstructed under the assumption that the primary particles are all photons. The reconstruction is based on the fact that, for a certain type of primary particle and a certain zenith angle [FORMULA], the density of atmospheric Cherenkov light created by the extensive air shower at a certain distance from the shower axis is to good approximation proportional to the energy of the primary particle. Presently two algorithms are used:

  1. The first method is based on the functions [FORMULA] and
    [FORMULA]
    which describe the expected sum of photoelectrons, [FORMULA], and its variance as a function of the distance r of a telescope from the shower axis, the primary energy E of the photon, and the zenith angle [FORMULA]. Both functions are computed from the Monte Carlo event-sample and are tabulated in r-, E-, and [FORMULA]-bins. Given, for the ith telescope, the shower axis distance [FORMULA] from the stereoscopic event analysis and the sum of recorded photoelectrons [FORMULA], an estimate [FORMULA] of the primary energy is made by numerical inversion of the function [FORMULA]. Subsequently the energy estimates from all triggered telescopes are combined with a proper weighting factor proportional to [FORMULA] to yield a common energy value.

  2. In a very similar approach E is estimated from the [FORMULA] and the [FORMULA] using a Maximum Likelihood Method which takes the full probability density functions (PDFs) [FORMULA] of the [FORMULA]-observable into account. The PDFs are determined from the Monte Carlo-simulations for certain bins in r, E, and [FORMULA]. The common estimate for the energy E maximizes the joint a posteriori probability function [FORMULA] that the [FORMULA]-values have been observed at the zenith angle [FORMULA] and at the distances [FORMULA].

Monte Carlo showers have been simulated for the 4 discrete zenith angles [FORMULA], and [FORMULA]. The energies for arbitrary [FORMULA]-values between [FORMULA] and [FORMULA] are determined by interpolation of the two energy estimates computed with the Monte Carlo tables of the adjacent zenith angle values below and above [FORMULA]. Hereby an interpolation linear in [FORMULA] is used, where [FORMULA] is derived as described below. Very small images with [FORMULA] are excluded from the analysis. Both methods yield the same energy reconstruction accuracy of [FORMULA] for photon-induced showers, almost independent of the primary energy. In the analysis presented below the Maximum Likelihood Method is used. In Fig. 7 the relative reconstruction error [FORMULA] is shown for the second method for all triggered [FORMULA]-ray showers which pass the loose [FORMULA]/h separation cuts and which produce at least two or three images with [FORMULA]. Increasing the requirement on the minimum number of images improves the energy resolution slightly but reduces the [FORMULA]-ray statistics. In the following we are interested in one-day spectra with sparse photon statistics; consequently we will use the weaker condition of only two telescopes with [FORMULA].

[FIGURE] Fig. 7. The relative error of the energy reconstruction [FORMULA] (with [FORMULA] if the reconstructed energy is higher than the true energy), shown for [FORMULA]-ray induced showers. The full line shows the distribution for all showers with at least 2 telescopes with [FORMULA] and the dotted line shows the distribution for all showers with at least 3 telescopes with [FORMULA] (Monte Carlo, zenith angle [FORMULA], after loose [FORMULA]/h-separation cuts, weighting according to [FORMULA]).

In the second step, the differential photon flux per energy channel is computed using the formula

[EQUATION]

where [FORMULA] is the observation time and [FORMULA] is the width of the ith energy bin. The first sum runs over all ON events reconstructed in the ith energy bin. The second sum runs over all OFF events reconstructed in the ith energy bin. [FORMULA] is the reconstructed energy of the jth event and the parameter [FORMULA] is the zenith angle under which the source was observed when the jth event was recorded. The second term subtracts on a statistical basis the background contamination of the ON region. The effective area [FORMULA] accounts for the acceptance of the detector and its energy resolution. The factors [FORMULA] and [FORMULA] account for the [FORMULA]-ray acceptances of the angular cut and the image cut, respectively. Given the differentail flux, the integral flux can be computed easily by integrating Eq. 2 over the relevant energy range.

Generally, the effective area is computed from

[EQUATION]

where [FORMULA] is the number of Monte Carlo [FORMULA]-ray-induced showers generated for a certain energy and zenith angle bin, [FORMULA] is the number of these showers which trigger the detector and pass the selection cuts, and [FORMULA] is the area over which the Monte Carlo showers were thrown. The area [FORMULA] is chosen sufficiently large (depending on the primary energy and the simulated zenith angle between [FORMULA] and [FORMULA] m2), so that virtually no exterior events trigger the experiment.

The energy resolution of the detector is taken into account by using a slightly modified effective area [FORMULA] which takes, for a given power law spectrum [FORMULA], the response function of the energy reconstruction [FORMULA] (properly normalized) into account:

[EQUATION]

In practice, [FORMULA] is computed with Eq. 3, weighting the events according to an incident power law spectrum with differential spectral index [FORMULA] and using for E the reconstructed energy and not the true energy. Hereby the cut on the distance r of the shower axis from the center of the telescope system [FORMULA]195 m which is also used in the spectral analysis is taken into account.

Eq. 2 permits, by definition, to reconstruct accurately a differential power law spectrum with index [FORMULA]. Due to the good energy resolution of 20[FORMULA] of the IACT system, [FORMULA] depends only slightly on [FORMULA]. Monte Carlo studies prove that power law spectra with spectral indices between -1.5 and -3 are reconstructed with a systematic error smaller than 0.1 using [FORMULA] with a fixed value of [FORMULA]. Furthermore we have tested this method with several other types of primary spectra, i.e. with broken power law spectra and with power law spectra with exponential cut-offs. The method, used with [FORMULA], reproduces the input-spectra with good accuracy.

Differential Mkn 501 spectra obtained with this method, as well as the method for fitting model spectra to the data, will be discussed below. Alternatively we have tested the standard forward folding technique and more sophisticated deconvolution methods. The deconvolution methods yield a slightly improved effective energy resolution at the expense of a heavier use of detailed Monte Carlo predictions and/or a larger statistical error of the individual differential flux estimates.

On the left side of Fig. 8, [FORMULA] and on the left side of Fig. 9, [FORMULA] are shown, computed for [FORMULA], and [FORMULA]. The zenith angle dependence of the [FORMULA]-curves can be described with the following empirical formula:

[FIGURE] Fig. 8. On the left side the effective areas of the HEGRA system of 4 IACTs for [FORMULA]-ray detection as functions of the primary energy are shown for the 4 different zenith angles [FORMULA] and [FORMULA] (Monte Carlo). The right side shows the effective area for vertically incident showers [FORMULA] calculated with the effective areas at the zenith angles [FORMULA] and [FORMULA] according to Eq. 5 (hardware threshold of at least 2 triggered telescopes, no cuts).

[FIGURE] Fig. 9. On the left side the effective areas as function of the reconstructed primary energy are shown for the 4 different zenith angles (Monte Carlo). The same cuts as in the spectral studies have been used, i.e. the software threshold of two telescopes with a [FORMULA]-value above 40 and a distance of the shower axis from the center of the telescope system smaller than 195 m. On the right side it is shown, how the effective area for vertically incident showers can be computed from the effective areas computed for the other three zenith angles using Eq. 5.

[EQUATION]

with [FORMULA] and [FORMULA] (see Fig. 8, right side). The same formula with the exponents [FORMULA]0.4 and [FORMULA]2.2 describes the zenith angle dependence of the [FORMULA]-curves (see Fig. 9, right side). This formula is used in the data analysis to interpolate [FORMULA] between the simulated zenith angles.

The Monte Carlo reproduces nicely the following properties of [FORMULA]-ray-induced showers:

  • the shape of the lateral Cherenkov light distribution as a function of the primary energy (Aharonian et al. 1998),

  • the single telescope trigger probability as function of shower axis distance and primary energy, and

  • the distribution of the shower cores,

all determined with the Mkn 501 [FORMULA]-ray data-sample. Hence, we are confident that the Monte Carlo correctly predicts the [FORMULA]-ray effective areas, except for a possible scaling factor [FORMULA] in the energy which derives from the accuracies with which the atmospheric absorption and the Cherenkov photon to ADC counts conversion factor are known. Note, that the possible scaling factor a introduces an uncertainty in the absolute flux estimates, but not in the measured differential spectral indices.

The Crab Nebula is known to be a TeV source with an approximately constant TeV emission (Buckley et al. 1996). We have tested the full analysis chain described above and the stability of the IACT system directly with [FORMULA]-rays from this source. Within statistical errors the Crab observations prove that the IACT system runs stably and that the analysis based on the Monte Carlo simulations accounts correctly for the hardware changes performed during 1997 as well as for the IACT system's zenith angle dependence of the [FORMULA]-ray acceptance (Aharonian et al. 1999 in preparation).

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Online publication: December 22, 1998
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