Mean Error Radius

In fig.  the distribution of the angular distance between the true direction of the 45 well localized GRBs and the direction derived with the GRBM, is shown. In spite of the small number of bursts, this subset is the most reliable, since the uncertainties ( $< 1\rm ^{\circ}$) on the true positions can be neglected, so that only the systematic and statistical errors due to the GRBM estimate have to be taken into account.

Figure: GRBM-true Position Angular Distance Distribution. The sample only includes 45 well localized GRBs (WFC, IPN, etc...).

About 90% true positions indeed lie within the GRBM error 90% CL regions, inclusive of 10$\rm ^{\circ}$ systematic. The same angular distance distributions in two cases have been taken into studied as well: in figg.  and the GRBM-BATSE 4B and GRBM-BATSE 4B+Kommers'+Stern's common GRBs distributions are shown, respectively; also in this case, only the GRBs, that could have been positioned by the GRBM, have been taken into consideration. In these figures, only the GRBM centroid coordinates and the BATSE positions have been used, with no care about their uncertainties; nevertheless, from these figures it can be easily realized that, from $0\rm ^{\circ}$ to $\sim 40\rm ^{\circ}$ angular deviation the number of bursts looks not to change significantly; on the other hand, a very few cases show angular distances $> 40\rm ^{\circ}$. This somehow gives an idea of a ``mean'' angular dimension of the GRBM error regions, though these often show oblong CL profiles, especially along the local elevation ($\theta$ angle) dimension (see some figures from section ).

Figure: GRBM-BATSE Position Angular Distance Distribution. The sample only includes 152 BATSE 4B GRBs, that have been localized also with the GRBM localization technique.

Figure: GRBM-BATSE Position Angular Distance Distribution. The sample only includes 203 BATSE (4B+Kommers'+Stern's) GRBs, that have been localized also with the GRBM localization technique.

In order to test the consistency of the GRBM and BATSE localizations, in fig.  the distribution of the discrepancies between the GRBM and the BATSE (4B only) positions of 152 common bursts is shown. The discrepancy used is the angular distances, expressed in terms of $\sigma$, where $\sigma$ is calculated according to the eq. :

$$\sigma \ = \ \frac{\mbox{angdist}\Big (\alpha_G,\delta_G,\al... ...a_{Gst}^2 + \sigma_{Gsy}^2 + \sigma_{Bst}^2 + \sigma_{Bsy}^2}}$$ (44)

where $\alpha_G,\delta_G$ are the GRBM centroid coordinates, $\alpha_B,\delta_B$ the BATSE center coordinates, and $\sigma_{Gst}, \sigma_{Gsy}, \sigma_{Bst}, \sigma_{Bsy}$ are the statistical ($st$) and systematic ($sy$) uncertainties of the GRBM ($G$) and BATSE ($B$) positions; the following values have been assumed: $\sigma_{Gsy} = 10\rm ^{\circ}$ and $\sigma_{Bsy} = 1.6\rm ^{\circ}$ ([Meegan et al., 1996,Paciesas et al., 1999]). The GRBM statistical error $\sigma_{Gst}$ must be interpreted as the 90% CL mean error radius $\rho$ discussed in section ).

Figure: GRBM-BATSE Position Discrepancy Distribution. The sample only includes 152 BATSE 4B GRBs, that have been localized also with the GRBM localization technique. The discrepancy takes into account both the total error (90% CL) owing to the GRBM localization and the total BATSE error, with $1.6\rm ^{\circ}$ systematic for the latter.

From eq.  it should be expected that $\sim$ 90% of the GRBM-BATSE bursts considered should have discrepancies lower than 1$\sigma$; this property can be more easily verified, when considering the integral distribution of the discrepancies, as shown in figg.  for both cases: 152 GRBM-BATSE (4B only) shared bursts, and 203 GRBM-BATSE (4B+Kommers'+Stern's) shared bursts, respectively. From these figures, it seems that the 90% of bursts have discrepancies lower than $\sim 1.3\sigma$ instead of $1.0\sigma$; nevertheless this looks like being quite consistent with the above expectation, although the uncertainties are not shown in these figures.

Figure: GRBM-BATSE Position Discrepancy Integral Distributions. The sample only includes 152 BATSE 4B (top panel) and 203 BATSE 4B+Kommers'+Stern's (bottom panel) GRBs, that have been localized also with the GRBM localization technique. The discrepancy takes into account both the total error (90% CL) owing to the GRBM localization and the total BATSE error, with $1.6\rm ^{\circ}$ systematic for the latter.

This argument indirectly confirms that the GRBM systematic error in positioning is around $10\rm ^{\circ}$.