3. The limit on the size of the emission region
Even if the size at radio wavelengths - about 1.3´ rms - is used as a most extreme possibility for the size at TeV energies, the source size is still smaller than the angular resolution for the best subsets - about 2´ to 3´. Therefore, one cannot expect to generate a detailed map of the source. Instead, an extended emission region of (rms) size would primarily show up as a slight broadening of the angular distribution of gamma-rays, beyond the value determined by the experimental resolution:
For an intrinsic resolution of 3´ (in a projection) and a 1.5´ rms source size, one would find a 3.4´ wide angular distribution; for the 6´ resolution, the resulting width is 6.2´. In order to positively detect a finite source size, or to derive stringent upper limits, one has (a) to measure the width of the angular distribution of gamma-rays with sufficient statistical precision, and (b) to quantitatively understand the response function of the instrument at the same level, and to control systematic effects which influence the resolution.
For a given number n of events from the source, and ignoring for the moment the effect of background under the signal, the statistical error on the width of the (projected) angular distribution is . In case the angular distribution is consistent with a point source, Eqn. 2 then implies a (1 standard deviation) upper limit . Therefore, one will want to minimize by selecting a subset of events with particularly well-determined directions, even at the expense of event statistics. More critical are, in general, systematic errors. Pointing imperfections, changes in mirror alignment, etc. can cause differences in the angular resolution between data sets taken at different times, under different conditions, and between the data and the simulation. If, e.g., the intrinsic resolution of the instrument is known and reproducible to 10%, the minimum source size which can be reliably detected is , according to Eqs. 1 or 2. Once more, a selection towards small is preferred. Among gamma-ray events, we find that about 1% of the events have a predicted resolution below 1.8´ (), 6% below 2.4´ (), 15% below 3´ (), and 60% below 6´ (), respectively. In particular the samples with resolutions better than 2.4´ or 3´ combine good angular resolution with acceptable statistics. The cuts on angular resolution have the additional benefit on enhancing the gamma-ray sample relative to the cosmic-ray background; cosmic-ray showers generate more diffuse images and have a worse angular resolution. Fig. 2 shows the angular distribution of events retained after a cut at 3´ resolution in both projections, applying only very loose additional cuts on event shapes (a cut on the mean scaled width at 1.2, which retains over 80% of the gamma-ray events). The selection also biases the sample towards higher energies, since high-energy events produce more intense images, with smaller errors on the image parameters. In the overall data sample, the median energy of reconstructed events is 0.9 TeV; after a cut on the resolution at 3´, this value rises to 2.0 TeV.
While the evaluation of statistical errors is straight forward, the control of systematic errors is more difficult. The pointing of the telescopes is referenced to and corrected offline on the basis of star images (Pühlhofer et al. 1997), and the achievable pointing precision has been investigated in considerable detail. We are confident to achieve a pointing deviation of less than in each coordinate. Indeed, when the Crab data set was subdivided into seasonal subsets, the reconstructed source position was in all cases consistent with the nominal source location - the position of the Crab pulsar.
To evaluate the level at which the angular resolution is understood, we use the sample of gamma-rays from the AGN Mrk 501 as a reference set (see Aharonian et al. (1999b, 1999d) for details on this data set), assuming that Mrk 501 represents a point source. Table 1, Columns 2,3 compare the measured angular distribution for different subsets of events with the Monte-Carlo predictions. `Angular resolution' again refers to the Gaussian width of the projected angular distribution of events. Excellent agreement between data and Monte-Carlo is seen for all data sets. We note that the resolution estimates given by the reconstruction algorithm are low by 10% to 15%, for the samples selected for good resolution. The 2.4´ sample, e.g., should show a 2.2´ resolution, compared to the measured value of 2.46´. Given the relatively crude parametrization of image parameter errors used in the reconstruction (Hofmann et al., 1999), a deviation at this level is not unexpected, and in any case the effect is fully reproduced by the simulations.
Table 1. Width of the angular distribution of events relative to the source, comparing the Mrk 501 and Crab data sets with Monte Carlo simulations using the measured gamma-ray energy spectrum as an input. Data sets are selected according to the estimate of the angular resolution, as provided by the shower reconstruction algorithm. The quoted width values are derived using a Gaussian fit to the projected angular distribution. For the last two rows of the table, only the central part of the distribution is fit; there are significant non-Gaussian tails (both in the Monte Carlo and in the data).
After these preliminaries, we can now address the Crab data set. Fig. 3 a shows the background-subtracted angular distribution of reconstructed showers relative to the source direction, projected onto the axes of a local coordinate system, after a cut on the estimated error of less than 3´. Superimposed is the corresponding distribution of Monte-Carlo events, generated with the measured Crab spectrum. Fig. 3 b shows the corresponding comparison with gamma-rays from Mrk 501. In particular after the selection on good angular resolution, the two event samples are very similar in their characteristics (mean predicted resolution, mean number of telescopes contributing to the reconstruction, etc.) and can be compared directly, despite the differences in the energy spectrum of the two sources. Table 1, Columns 4, 5 list the widths of the distributions for the Crab gamma-rays, and the corresponding simulations. In general, we find, within the statistical errors, good agreement between the Crab and Mrk 501 data sets, and between Crab data and Monte-Carlo.
Another option to allow a direct comparison of the Mrk 501 and Crab data sets is not to look at the angular deviations between a gamma-ray and the source, but rather to normalize these quantities with the angular resolutions predicted event-by-event by the reconstruction algorithm. Ideally, one expects to see a Gaussian distribution of unit width, independently of the energy spectrum, the number of telescopes active for a given data set, etc. The observed distributions are indeed Gaussian; as mentioned above, their widths differ by up to 10% to 15% from unity, depending on the selection of events. Most importantly, however, these width values are consistent between all four sets of events (Mrk 501 MC, Mrk 501 data, Crab MC, Crab data), within errors of 1.5% or less for the large-statistics sets where all events are included. This agreement demonstrates that there are no uncontrolled systematic effects between the two experimental data sets, or between the experimental data and the simulations.
Since the width of the angular distribution of gamma-rays from the Crab Nebula is consistent with the expected width, and with the width observed for Mrk 501, we can only give an upper limit on the source size. Taking into account the statistical errors on the Crab sample and on the reference samples, we find - following Caso et al. (1998) - 99% confidence level upper limits of 1.0´ for the sample with a cut at 3´ resolution, and 1.3´ for the 2.4´ sample. To be conservative, and since it was always planned to use the 2.4´ sample as a safest compromise between statistical and systematic uncertainties, we adopt the 1.3´ limit. Adding in additional systematic uncertainties due to pointing precision, we quote a final limit of 1.5´ for the rms source size at a median energy of 2 TeV.
A possible contribution of gamma-rays from hadronic processes is generally expected to be most relevant at higher energies, in the 10 TeV to 100 TeV range. Therefore, the source size was also studied for an event sample with reconstructed energies above 5 TeV; data are consistent with a point source and the corresponding limit on the rms source size is 1.7´. Limited statistics prevent studies at even higher energies.
The elliptical shape of the X-ray emission region suggests to perform the same analysis in a rotated coordinate system, with its axes aligned along the major and minor axes of the X-ray profile. Results obtained for the width in the major and minor direction do not show any significant difference. Given the fact that the limits are large compared to the (rms-)size of the X-ray emission region, this observation is not surprising. The TeV source is reconstructed 0.2´ from the location of the pulsar, and is consistent both with the location of the pulsar, and with the center of gravity of the X-ray emission region, within the systematic errors in the telescope pointing (less than in each coordinate). While the statistical error alone is small enough to resolve a shift of 0.3´ as observed in the X-ray image, current systematic errors prevent such a measurement at TeV energies.
The limits obtained in this work are included in Fig. 1.
We note in passing that also the widths of the distributions obtained for the AGN Mrk 501 are consistent with MC expectations, indicating the absence of a halo on the arcminute scale. Potential halo types for AGNs include wide pair halos (Aharonian et al. 1994) or narrow halos caused by intergalactic magnetic fields (Plaga 1995). For a relatively near source such as Mrk 501, a pair halo would be much wider than the field of view of the camera and could not be detected as an increased apparent source size; the other type of halo speculated in (Plaga 1995) would be well below our resolution. We do not see any time-dependence of the source size, or any correlation with the TeV gamma-ray flux. Details will be given elsewhere.
© European Southern Observatory (ESO) 2000
Online publication: October 10, 2000