Astron. Astrophys. 359, 337-346 (2000) 3. Extended source analysisThe analysis of extended sources has required the development of new techniques in TeV gamma-ray astronomy (Akerlof et al. 1991, Hess et al. 1997, Buckley et al. 1998, Connaugton et al. 1998). These methods use shape parameters such as width and length in combination with source position-dependent orientation and location cuts (e.g. alpha, asymmetry and dis ). The orientation and location parameters are recalculated at every trial source position and a skymap is created. In the analysis of CANGAROO Vela Pulsar data, Yoshikoshi et al. (1997) used a set of cuts based on alpha, length, width, concentration and dis . A continuous probability distribution was initially used to locate the position of the most significant point in the skymap. A gamma-ray flux at this point was then estimated from the ON-OFF excess obtained after using a combination of shape and location cuts with alpha . These cuts were optimised using Monte Carlo simulations and therefore are a priori decisions. We adopt the same a priori philosophy here in order to determine the significance of any result without the need to consider statistical penalties. The set of cuts described here were also used in the analysis of Centaurus A data described by Rowell et al. (1999). The Monte Carlo simulation package, MOCCA92 (Hillas 1995), was used to generate erenkov images from extensive air showers (EAS). Gamma-ray primaries were sampled from a power law above 0.8 TeV with integral spectral index -1.6. Cosmic ray primaries, represented by a combination of proton, helium and nitrogen primaries, were sampled from a power law of spectral index -1.65 above 1.5 TeV. In designing an analysis suited to off-axis and extended sources, it is important to consider the off-axis sensitivity of the CANGAROO camera, particularly considering that it has a relatively small field of view and operates at a gamma-ray threshold of about 1.5 TeV. The behaviour of various image parameters as a function of gamma-ray source position has been investigated. A gamma-ray point source was placed at six positions across the camera and each position considered independently. The resulting distributions of width and miss showed little variation with source distance from the camera centre, in contrast to those of length and dis (Fig. 2). For increasing source distances from the camera centre, larger values of length and dis are possible, behaviour which is readily understood in terms of camera edge effect reduction on one side as the source approaches the camera edge. In particular, the length parameter for gamma-rays is very similar to that for cosmic-rays for sources beyond about 1.0^{o} from the camera centre. Indeed, even for on-axis sources, the field of view of the 3.8 m camera imposes some edge effects, a consequence of operating at a relatively high gamma-ray energy threshold. Thus, for off-axis sources, a reduced gamma ray efficiency will result when using cuts derived for an on-axis source.
Firstly, there is a clear case for increasing the gamma-ray efficiency while maintaining the quality factor on the grounds of improved event statistics. An increase in gamma-ray efficiency entails an increase in background acceptance, thus improving an estimate of the background and diluting any unaccounted systematic effects. It is also possible to examine the effect of the gamma-ray efficiency on the ON-OFF significance after cut application. We can express Eq. (1) in another way, incorporating the gamma-ray efficiency, after application of image cuts: where the gamma-ray signal, (=ON-OFF) and the background, (OFF), represent those prior to image cuts. The efficiency for gamma-ray selection is given by and that for the background (CR), . The quality or Q-factor of the cuts is given by . Here, we set the normalisation factor from Eq. (1), . We can see that the significance obtained after image cuts is dependent on the quality factor and , somewhat slightly, on the gamma-ray cut efficiency. For sources with a high gamma-ray to CR flux ratio (e.g. 0.l) with 4 (as for the CANGAROO telescope), both denominator terms are then similar, the sensitivity of S to becomes more apparent. Such a situation may arise in the case of searches for bursts from AGN and/or gamma-ray bursts over short time scales. For signal to noise ratios expected of SNRs such as W28 however, we would expect only a minor improvement in S from the above arguement. Thus, the main motivation for increasing here is simply to work with increased statistics. The variable cuts on length and for dis were incorporated into the cut ensemble. Along with the 3rd moment of the image, asymmetry , we use an approximation of the distance between the assumed source position and calculated source position for each image, D. This distance, expressed in units of standard deviation, is given by:- where are the variances for miss and dis respectively. The variances, and represent the transverse and longitudinal errors in the most likely source position for an image. D can be characterised as the source density function (SDF). When used in combination with shape and asymmetry cuts, a cut on D provides a gamma-ray acceptance of 40% for an on-axis point source. D is similar to the normalised cluster, or Mahalonobis distance for miss and dis (Hillas & West 1991), although here we are neglecting cross-term variances. D can also be considered a discrete analogue of the probability distribution function used by Yoshikoshi et al. (1997). The longitudinal error in source location is greater than the transverse error by about a factor of two. Following Akerlof et al. (1991), some reduction in the longitudinal error can be achieved by making use of the elongation (defined as length/width ) of the image such that the expected longitudinal source distance is given by where g is an empirically derived constant. For the 3.8 m camera we derive a value of using simulations. Fig. 3 gives distributions of the parameter D for simulated gamma-rays and cosmic rays, and a comparison to real OFF source data. Linear fits were found for the cut on length and value of , respectively, as a function of source displacement from the camera centre. The total cut combination is listed in Table 1. These cuts (Table 1) provide a constant gamma-ray efficiency of 40% (and cosmic-ray efficiency) and quality factor 4 at the same value of D. Without the variable length and dis criteria, the gamma-ray acceptance quickly reduces to less than 30% for sources outside 0.5^{o} from the camera centre for no significant improvement in Q-factor. Table 2 gives the simulated performance of the cut D for various source positions. A comparison to the performance obtained by the set of cuts used in the Vela Pulsar analysis (Yoshikoshi et al. 1997) is included. These cuts were used to obtain a significant excess of gamma-ray-like events from a region displaced 0.14^{o} from the pulsar position. The Vela Pulsar cuts are based on the same noise cuts listed in Table 1, except that fixed values of length and dis are used and a cut on alpha is substituted for . The Vela Pulsar cuts do not provide a constant gamma-ray acceptance over the search region, and the background (cosmic ray) acceptance also decreases sharply.
Table 1. Image cuts used in the extended source analysis. See Eq. ( 3) for a definition of D. The parameter d is the assumed source displacement from the camera centre (in degrees). The length cut and value of are dependent upon the assumed source position, and are derived from linear fits. Table 2. Simulated performance of the image cuts of this work (assessed using the quality factor Q) compared with cuts used in the Vela Pulsar analysis. Since the statistical errors of Q are %, the Vela Pulsar cuts are not statistically different from those of this work. The Vela Pulsar cuts use the same noise cuts as defined in Table 1, but fixed values of length and dis (Yoshikoshi et al. 1997). The resulting flux or upper limit from the search is found by dividing the ON-OFF excess, N (or 3 upper limit thereof), by the position-dependent exposure (for an extended source, averaged over the integration region): where is the raw trigger efficiency for gamma-rays averaged over the integration region, is a constant gamma-ray selection efficiency for the SDF cut, A is the area over which gamma-rays are simulated ( cm^{2}) and T is the total observation time. We set to be the average of the simulated gamma-ray cut efficiencies out to , i.e. = 0.41. An unavoidable decline in for gamma-rays will occur as the source position is displaced further off-axis. Simulations show that decreases to less than half the on-axis value at the corners of the search. A linear fit was used to characterise at all points within such that as a function of the source displacement from the camera centre, d. We find (1995 data) at the camera centre above the minimum simulated energy of 0.8 TeV for the 1995 dataset. The raw gamma-ray trigger efficiency for 1994 data was estimated by scaling the 1995 value by the ratio of observed event rates after noise cuts, giving (1994 data). In characterising and , we are assuming that the gamma-ray flux of an extended source is isotropically distributed. To calculate the flux applicable to the energy threshold of 1.5 TeV (Roberts et al. 1998) we used the gamma-ray spectral index of -1.6 adopted in the simulations. For a point-like search, the flux was taken as that from the point of interest, using a single value of . The nature of our gamma-ray selection cuts naturally incorporates the gamma-ray point spread function (PSF). The cut on D accepts events with derived source positions within an optimal radius. A search for an extended source therefore does not require any extra area to account for the PSF in addition to the source area itself. In an extended source search, at a suitably high number of assumed source positions in the region of interest, we sum the events passing all cuts, taking care not to count an event more than once. Skymaps of the statistical significance, S, of a gamma-ray signal were generated over a area at 0.05^{o} steps. At each grid point, representing an assumed source position, image parameters were calculated and the number of events passing the cuts of Table 1 cumulatively summed for all data. Since the resolution of the grid (0.05^{o}), is smaller than the effective acceptance area of the cuts (the cut on D alone is more powerful than a cut on alpha 10^{o}), each skymap point value will not be fully independent of its neighbours. As a final check on data integrity, the distribution of S obtained on a run-by-run basis (i.e. run-by-run skymap) was quantitatively assessed for systematic effects. The most important systematic effect to consider in this type of analysis is the consistency of the trigger threshold over the entire camera between ON and OFF source runs. Such an effect is difficult to compensate for after the data are taken. The Kolmolgorov-Smirnoff (KS) test is used to examine the likelihood that the distribution of S of the skymap obtained on a run-by-run basis is derived from a normal distribution. Over the time scales of a single run (5 hours), we do not expect significant contributions from a steady source of TeV gamma-rays of the strength expected of a SNR or plerion. A relatively strong KS probability of 4 was used to reject pairs of data that appeared severely affected by such systematics. For unmatched pairs, a well-behaved OFF or ON comparison run was used. A total of 10 hours data were rejected using the KS test, representing three ON/OFF pairs from the 1994 dataset. © European Southern Observatory (ESO) 2000 Online publication: June 30, 2000 |