## 2. MethodIn 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 ## 2.1. Simultaneous burstsThe positions of 27 simultaneous bursts detected by WATCH and BATSE are in good agreement. If BATSE error boxes are considered, there are 20 overlaps with WATCH boxes. Instead, if 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" estimateRecurrence, 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 where with , and the
distance between the integration point and the centre of the
is analogous to 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 to 3. 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
by considering only overlaps between 3 error
boxes, instead of assuming unlimited error boxes, is less than 0.1%,
irrelevant for our final conclusions. In the approximation that
and (which is quite
accurate, since typical values are and
a few degrees),
approximately depends on like
, so it decreases rapidly when both probability
distributions are not close to each other.
provides a measurement of whether both GRBs originated from the same
source or not. Based on the former arguments, we define the "added
correlation"
## 2.3. Simulation of WATCH cataloguesMonte Carlo simulations of 1500 WATCH-like catalogues have been
performed. They provided 1500 values for ## 2.4. Random and real overlapsWe call random overlaps to overlaps between the BATSE bursts and the simulated WATCH events. The random overlaps provide a set of that follows a distribution so-called . In order to estimate such distribution, the value of the overlapping functions are calculated for all the overlaps between the BATSE sample and 50 simulated WATCH catalogues. It shows a mean value and a deviation . 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, , for each of the 27 BATSE-WATCH simultaneous pairs is calculated. The distribution of these 27 values of is called . The mean value of the real overlaps, , and the deviation . As expected is greater than . 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 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 larger than . The next step is to consider as the expected distribution of for repeaters. This consideration is based on the two following assumptions: -
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 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. -
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 repeatersThe set of 1500 's calculated using mock
WATCH catalogues follows a Gaussian probability distribution
(hereafter called , 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
gives us the expected value of the "added correlation" when WATCH and
BATSE catalogues do not share any source. Assuming that
represents the expected value of the
overlapping function for repeater sources, we can introduce trial
repeaters and construct the 's probability
distributions for different number of
repeaters,
The intersection between and the
distributions provides 9 values called
. Based on the 9 values of
we can obtain the distribution of
for . Taking into account
that the maximum number of allowed coincidences is
, the distribution of 's
can be normalized by imposing . Then,
provides the probability that the BATSE and
WATCH catalogues share
© European Southern Observatory (ESO) 1998 Online publication: July 7, 1998 |