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Astron. Astrophys. 336, 57-62 (1998)

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2. Method

In this section we outline the methodology proposed to carry out the study. First, we exclude the simultaneous bursts (Sect. 2.1) and calculate the so-called "added correlation" function between the WATCH and BATSE samples (Sect. 2.2). Afterwards, 1500 WATCH catalogues are simulated (Sect.  2.3) in order to calculate the expected value of the "added correlation" function. Then the distribution of the overlapping function for real and random overlaps is obtained (Sect. 2.4) and finally the probability of having different number of repeaters (Sect. 2.5) is found.

We consider that there is a common source in both samples when the emission of a repeater is detected at least twice, once by each experiment and the detections are separated in time. Thus, the same GRB detected simultaneously by both experiments is not considered as a common source. Our study is aimed at searching common sources detected by both WATCH and BATSE experiments.

2.1. Simultaneous bursts

The positions of 27 simultaneous bursts detected by WATCH and BATSE are in good agreement. If BATSE [FORMULA] error boxes are considered, there are 20 overlaps with WATCH [FORMULA] boxes. Instead, if [FORMULA] error boxes are taken into account there is only one burst (GRB 920714) that does not overlap. These 27 bursts were excluded from the BATSE sample of 1905 sources, because they are obviously the same sources detected by WATCH. Therefore the sample was reduced to 1878 bursts. Nevertheless, the simultaneous bursts were considered in further calculations (Sect. 2.4), because they provide information on the overlapping expected for a repeater detected by both BATSE and WATCH.

2.2. The "added correlation" estimate

Recurrence, even in a single case, would be immediately obvious if we had locations with no errors. However, the locations provided by BATSE and WATCH, while numerous, have inaccuracies and consequently a statistical analysis is required to demonstrate, or limit, the presence of common sources. If any repeater is present in both catalogues, an excess in the overlap between the error boxes of both catalogues would be expected. We define the overlapping function between the i-th WATCH and the j-th BATSE error boxes as the following integral over the galactic coordinates l and b:

[EQUATION]

where A is a normalization factor computed in such a way that [FORMULA] remains between 0 and 1. [FORMULA] is the BATSE exposure correction for the BATSE j-th burst. [FORMULA] and [FORMULA] are the galactic coordinates of the centre of the i-th WATCH and the j-th BATSE error boxes, [FORMULA] is the distance between the i-th WATCH and the j-th BATSE burst, and [FORMULA] are the [FORMULA] radii of the i-th WATCH and the j-th BATSE error boxes. [FORMULA] is a Gaussian-like normalized probability distribution given by the following expression:

[EQUATION]

with [FORMULA], and [FORMULA] the distance between the integration point and the centre of the j-th BATSE burst;

[EQUATION]

[FORMULA] is analogous to [FORMULA] based on WATCH coordinates. Although we are aware that the errors of BATSE locations do not follow a single Gaussian distribution (see Briggs et al. 1998), we consider that, for our purposes, we can extend the Gaussian approximation from 1[FORMULA] to 3[FORMULA]. This is a very appropriate and useful approximation which has been frequently used in the past (Fisher et al. 1987, Bennett and Rhie 1996), providing stringent upper limits on the 3B catalogue (Tegmark et al. 1996).

On the other hand, the error introduced in [FORMULA] by considering only overlaps between 3[FORMULA] error boxes, instead of assuming unlimited error boxes, is less than 0.1%, irrelevant for our final conclusions. In the approximation that [FORMULA] and [FORMULA] (which is quite accurate, since typical values are [FORMULA] and [FORMULA] a few degrees), [FORMULA] approximately depends on [FORMULA] like [FORMULA], so it decreases rapidly when both probability distributions are not close to each other. [FORMULA] provides a measurement of whether both GRBs originated from the same source or not. Based on the former arguments, we define the "added correlation" C as follows:

[EQUATION]

C is a parameter which is very sensitive to the presence of common sources in both catalogues. The larger the number of common sources, the higher the value of C obtained. Our study is based on the comparison of the "added correlation" C calculated for the real WATCH catalogue (renamed as [FORMULA]) with those obtained for 1500 WATCH simulated catalogues (renamed as [FORMULA]). C is the generalization for two probability distributions (WATCH and BATSE) of the R statistics introduced by Tegmark et al. (1996). [FORMULA] is corrected by the BATSE and WATCH exposure maps, the first one is taken into account in the term [FORMULA] included in the definition of [FORMULA], whereas the second one is considered to simulate the WATCH catalogues for which [FORMULA] are calculated.

2.3. Simulation of WATCH catalogues

Monte Carlo simulations of 1500 WATCH-like catalogues have been performed. They provided 1500 values for C called [FORMULA]. In order to determine reliable values for them, the exposure maps of the WATCH/GRANAT and WATCH/EURECA instruments were taken into account. The failure of unit number 2 on board GRANAT , and the limited field of view and the Earth blockage of WATCH/EURECA, made it that none of the experiments covered uniformly the sky. The WATCH/GRANAT map shows larger exposures towards the Galactic centre whereas the WATCH/EURECA one is under exposured towards the equatorial poles (Brandt 1994, Castro-Tirado 1994). If we assume that GRBs occur randomly both in space and time, the probability of detecting a GRB in a given direction is proportional to the exposure time spent on that region. Therefore for each simulated set of 57 bursts, 45 of them follow the WATCH/GRANAT exposure map, 10 the WATCH/EURECA exposure map and the remaining two bursts (representing GRB 920814 and GRB 921022) follow both exposure maps simultaneously. The simulated WATCH-like sets have the same error radii than the real WATCH catalogue.

2.4. Random and real overlaps

We call random overlaps to overlaps between the BATSE bursts and the simulated WATCH events. The random overlaps provide a set of [FORMULA] that follows a distribution so-called [FORMULA]. In order to estimate such distribution, the value of the overlapping functions [FORMULA] are calculated for all the overlaps between the BATSE sample and 50 simulated WATCH catalogues. It shows a mean value [FORMULA] and a deviation [FORMULA]. [FORMULA] provides the expected value of the overlapping function when there is a casual overlap between two boxes (not due to arise from the same source). The majority of the random overlaps shows very low values of the overlapping function because they tend to occur at the border of the error boxes in the tail of the probability distribution.

On the other hand, the overlapping function, [FORMULA], for each of the 27 BATSE-WATCH simultaneous pairs is calculated. The distribution of these 27 values of [FORMULA] is called [FORMULA]. The mean value of the real overlaps, [FORMULA], and the deviation [FORMULA]. As expected [FORMULA] is greater than [FORMULA]. This fact can be explained taken into account that the probability distributions due to a single GRB detected by both experiments tend to be close to each other, compared with two GRBs randomly located in the same zone in the sky. Thus, the random overlaps tend to occur in the tail of the probability distribution, thus forcing [FORMULA] to be very low. Moreover, the lower sensitivity of WATCH in comparison to BATSE implies that the 27 simultaneous bursts are brighter than the average BATSE bursts, (as the radii of the error boxes depend on the intensity) and therefore they have smaller error boxes, thus making [FORMULA] larger than [FORMULA].

The next step is to consider [FORMULA] as the expected distribution of [FORMULA] for repeaters. This consideration is based on the two following assumptions:

  1. There is little variation with time on the sensitivity of both experiments. A change in the sensitivity imply into differences in the sizes of error boxes and thus in the [FORMULA] values. If more accuracy is desirable, then it is necessary to know how the sensitivity of both instruments evolves, in order to correct the sizes of the error boxes depending on the date of detection.

  2. The intensities of different bursts from a repeater source do not change significantly in time. Therefore the sizes of the repeater error boxes remain approximately the same. A more complicated study would deal with the time evolution of the repeater sources.

2.5. Quantification of the number of repeaters

The set of 1500 [FORMULA]'s calculated using mock WATCH catalogues follows a Gaussian probability distribution (hereafter called [FORMULA], see Fig. 2). The simulated WATCH catalogues were generated only using the exposure maps and they only contain accidental overlaps, because the simultaneous GRBs were excluded from the sample. Therefore [FORMULA] gives us the expected value of the "added correlation" when WATCH and BATSE catalogues do not share any source. Assuming that [FORMULA] represents the expected value of the overlapping function for repeater sources, we can introduce trial repeaters and construct the [FORMULA]'s probability distributions [FORMULA] for different number of repeaters, N, by the following symbolic expression :

[EQUATION]

N being the number of repeaters. If we take any "added correlation" [FORMULA] of [FORMULA] repeaters with a probability given by [FORMULA], and then we add the contribution to [FORMULA] of any repeater with overlapping function given by [FORMULA] and subtract the contribution of any random overlap given by [FORMULA], we get a new "added correlation" [FORMULA]. This process can be repeated by introducing other real and random overlaps, providing a new set of [FORMULA]'s. Once a trial repeater has been introduced, this set of [FORMULA]'s will follow a different probability distribution from [FORMULA], called [FORMULA]. Thus, [FORMULA] provides the expected values of [FORMULA] when BATSE and WATCH share one source. Similarly, this method can be applied for [FORMULA] repeaters, in order to obtain the distributions of the "added correlations" for different number of repeaters, [FORMULA]. Fig. 3 shows the probability distributions [FORMULA] obtained using this procedure.

[FIGURE] Fig. 2. The values of the "added correlation" [FORMULA], for the simulated catalogues [FORMULA]. The solid line represents [FORMULA], the mean value of the "added correlation" for the simulated catalogues, the long-dashed lines are the [FORMULA] limits. As it is clearly seen the real value of the "added correlation" [FORMULA] (represented by the square) is below the [FORMULA] limit. Our results are not compatible with the presence of common sources, as expected from the graph.

[FIGURE] Fig. 3. The Gaussian-like curves represent the probability distributions [FORMULA], and the vertical dashed line shows the value of the BATSE-WATCH "added correlation". The intersection of [FORMULA] with the probability distributions [FORMULA] provides the set [FORMULA].

The intersection between [FORMULA] and the distributions [FORMULA] provides 9 values called [FORMULA]. Based on the 9 values of [FORMULA] we can obtain the distribution of [FORMULA] for [FORMULA]. Taking into account that the maximum number of allowed coincidences is [FORMULA], the distribution of [FORMULA]'s can be normalized by imposing [FORMULA]. Then, [FORMULA] provides the probability that the BATSE and WATCH catalogues share N sources (see Fig. 4).

[FIGURE] Fig. 4. The values of [FORMULA] for different number of repeaters. The probability of having N repeaters reaches its maximum at [FORMULA] (no common sources) and decreases rapidly with the number of repeaters.

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© European Southern Observatory (ESO) 1998

Online publication: July 7, 1998
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