Automatic GRB directions with GRBM

Concerning the positioning of bursts discovered during the off-line GRB quest, some observations must be pointed out: given that the localization technique makes use of the total counts left by each burst in the GRBM detector units, while in the on-line case (sec. ) there is a need to check the reliability of the total counts as they are automatically computed, in this case the same requirements are not taken into account, provided that the estimates of the total counts have been already refined from their original values automatically derived from the quest results.

The selection and the refinement of the off-line trigger sample resulted from the off-line quest, are among the topics treated in the next chapter; therefore, here we suppose that the total counts, or fluences, that are used for localizing the selected bursts, have been already refined and are now available.

After getting the measured fluences of a given burst, the localization technique searches for all the relative minima of the $\chi^2$ function, according to the procedure already described in section . When a certain number of directions is found, each corresponding to a local minimum, the uncertainties on both the local coordinates $\phi(i)$ and $\theta(i)$ of the $i$-th direction are estimated as follows: first, the 90% confidence level (hereafter CL) region is searched for; when this is not possible, then lower CL regions are looked for: namely 68% and 50% CLs; when these cannot be found either, the candidate direction is considered as an indefinite solution. This way, for every direction ($\phi(i)$,$\theta(i)$) a region defined by the two intervals $[\phi(i)-\delta^-_\phi(i),\phi(i)+\delta^+_\phi(i)]$ and $[\theta(i)-\delta^-_\theta(i),\theta(i)+\delta^+_\theta(i)]$ is found, in combination with each CL value (when available) and $\chi^2(\phi(i),\theta(i))$, i.e. the value of the $\chi^2$ function at the $i$-th relative minimum. The confidence regions so found are inclusive of both statistical and systematic (taken as $10\rm ^{\circ}$) uncertainties. At this stage, for each direction a mean error radius $\rho(i)$ is defined as the mean angular distance between the direction and its corners according to eq. :

$$\rho(i) \ = \ \Big [ \mbox{angdist}\Big(\phi(i),\theta(i),\phi(i)-\delta^-_\phi(i),\theta(i)-\delta^-_\theta(i)\Big) +$$

$$\mbox{angdist}\Big (\phi(i),\theta(i),\phi(i)+\delta^+_\phi(i),\theta(i)-\delta^-_\theta(i)\Big ) +$$

$$\mbox{angdist}\Big (\phi(i),\theta(i),\phi(i)-\delta^-_\phi(i),\theta(i)+\delta^+_\theta(i)\Big ) +$$

$$\mbox{angdist}\Big (\phi(i),\theta(i),\phi(i)+\delta^+_\phi(i),\theta(i)+\delta^+_\theta(i)\Big ) \ \Big ] \, / \, 4$$ (33)

where the function angdist is defined as follows:

$$\mbox{angdist}\Big(\phi_1,\theta_1,\phi_2,\theta_2\Big) \ = \... ...e between} \ (\phi_1,\theta_1) \ \mbox{and} \ (\phi_2,\theta_2)$$

When two or more directions show uncertainty regions that overlap one another, it means the number of distinct solutions is lower than the number of formal ones; in order to eliminate this potential redundancy, a parameter $q(i,j)$ is defined $\forall i,j$ ($i\ne j$):

$$q(i,j) \ = \ \frac{\mbox{angdist} \Big (\phi(i),\theta(i),\phi(j),\theta(j) \Big )}{\sqrt{\rho^2(i) + \rho^2(j)}}$$ (34)

When $i$ is fixed, all the $j$-th solutions with $q(i,j) < 1.0$ are not taken into account any longer, since they are believed to lie inside the error region of th $i$-th direction; this has to be repeated for all the $i$-th directions that have not been discarded yet. Now, when either the number of accepted position is greater than two, or at least one of the CLs is not defined, then no further step is done and the localization technique turns to be not feasible; otherwise, the one or two error region are checked against Earth blockage: when the whole error circle, centered on the direction $(\phi(i),\theta(i))$, and whose radius is equal to $\rho(i)$ is entirely Earth-blocked, then the $i$-th direction is rejected. At this point, after this selection procedure, only if a single direction has remained, the localization technique returns a reliable position for the original burst; this last selection rule works only for the off-line quest, that is for the only archive bursts. In the on-line case, the last requirement is little weaker, since it accepts also the case of two distinct solutions; the motivation of this choice has been already explained; the basic concept is that, while in the on-line case also different positions could be somehow useful for robotic searches for optical counterparts, on the other side, in the off-line case a more confident localization is needed, in order to perform some statistical analysis on the sample of GRBs found throughout the GRBM archive, especially when a single estimate for the direction of a given burst is required.