Event reconstruction

γ-ray induced shower images obviously exhibit an elliptical shape and can be described by the first and second moments of an ellipse. This representation was first introduced by Hillas (1985), and therefore the parameters are also referred to as Hillas parameters. The first moment comprises the position of the ellipse or centre of gravity (COG), whereas from the matrix of second moments one obtains the length of the minor and major axis of the image, the width and length parameter, as well as its orientation ϑ. An additional quantity used in the reconstruction is the total intensity stored in the ellipse (size parameter). A schematic view of the Hillas parameters can be found in Fig. 1.7.

1 TeV gamma at a distance of 116 m

0 6 15 30 60 150 300 p.e.

(a)

2.3 TeV proton at a distance of 58 m

0 6 15 30 60 150 300 p.e.

(b)

Figure 1.6: ˇCerenkov-light distribution in the camera for a simulated γ-ray of 1 TeV energy (a) and for a simulated proton of 2.3 TeV energy (b). Pixels store intensities of up to 300 p.e. Pixels with an intensity of more than 5 p.e. or 10 p.e. are marked with a yellow and/or green cross and pixels which do not pass the pedestal RMS criterion are marked with a purple cross.

These five quantities inherently store information about the shower geometry, its spatial intensity distribution and information about the origin and energy of the primary particle. Width and length of the shower images contain information of the interaction processes at work during the shower development and can be used for γ/hadron separation. On the other hand is the size of the shower image connected to the primary particle energy. Combining the positional information COG and ϑ from multiple telescopes allows to geo- metrically reconstruct the incident direction and the shower impact point on ground. As soon as a shower is observed with multiple telescopes from different directions, each

pair of major image axes can be intersected in a common coordinate system5 _{to obtain the}

shower direction. In case N telescopes are reconstructing the direction, all N (N − 1)/2 possible estimated directions get weighted by the sine of the stereo angle between image axes, the ratio of width over length and by the size of the shower image. Thereby the fact is taken into account that bright, elongated images that are observed under larger angles allow a more precise determination of the shower direction. By averaging over all estimated positions, the final shower direction can be calculated. With this geometrical

method6 _{the achieved accuracy of the direction reconstruction is better than ≈0.1}◦ _per

5

This is the coordinate system in which all cameras are overlaid and which is perpendicular to the telescope pointing directions.

6

1.3 H.E.S.S. data analysis

Figure 1.7: Sketch illustrating the Hillas parametrisation of a γ-ray induced air shower detected in two telescopes. Width and length as well as the distance between camera centre and COG are later used in the analysis to select γ-ray like events. Intersecting the major axes in a common coordinate system allows to reconstruct the shower direction on the sky as well as the shower impact point on ground (see text for further details). The Figure was taken from (Aharonian et al. 2006a).

event. The shower impact position on ground is reconstructed in a similar fashion but in an array-wide coordinate system with the telescope positions as reference. Also here the accuracy of the reconstruction is remarkable with less than ≈10 m for showers with impact distances of < 200 m away from the array centre. Additionally, the length and

COG parameters provide information about the shower maximum Xmax, which is also

reconstructed stereoscopically. The geometrical reconstruction approach is illustrated in Fig. 1.7. The overall reconstruction accuracy can be further improved by taking into account not only the width over length ratio, the size and angle between shower images, but else the errors on all Hillas parameters. This approach corresponds to Algorithm 3 introduced by Hofmann et al. (1999) but is not utilised in this work.

The energy of the primary particle is – for constant zenith angle, distance between re- constructed shower direction and camera centre (this distance is henceforth referred to as offset) and for a given shower impact distance to the telescope – to first order lin- early dependent on the measured size of the corresponding shower image. In contrast to the direction reconstruction, where the shower images stored all the necessary informa- tion needed for reconstruction, the energy reconstruction needs further input from Monte Carlo γ-ray simulations. For every reconstructed distance, offset, size and zenith angle of an event, look-up-tables filled with Monte Carlo γ-ray simulations are used to predict the primary particle energy under the a γ-ray hypothesis. Since the optical efficiency of the telescopes change with time, the simulated optical efficiency has to be scaled to the actual optical efficiency of the telescopes. This is done on the basis of muon images as discussed in Section 1.3.2. The energy resolution obtained with this method is on average O(15%). However, the resolution changes with primary particle energy. Shower images of small intensity originate in general from showers of smaller energy which experience larger shower fluctuations. Hence, this results in a worse energy resolution for such events.