2 Results

The peak flux distributions for the 2 samples are displayed in Fig. 1. The slopes of the observed distributions appear compatible with the value expected for sources homogeneously distributed in Euclidean space (indicated by dotted lines). In order to obtain a more quantitative statement we have computed the maximum likelihood estimate of the slope (s) and the probability (P_-3/2) for the data to come from a "Euclidean'' distribution. P_-3/2 is computed with a Kolmogorov-Smirnov test.

* Triggers: $s = -1.59 \pm 0.11$   (P_-3/2 = 0.48).

* Real time GRBs: $s = -1.41 \pm 0.07$   (P_-3/2 = 0.12).

* Real time GRBs with a peak flux $\rm \geq 5\, ph\, cm^{-2}\, s^{-1}$

(231 bursts): $s = -1.51 \pm 0.10$   (P_-3/2 = 0.81)

(this threshold is indicated by the vertical line in Fig. 1.)

We conclude that the peak flux distribution of GRBs with $P \geq 5\, {\rm ph}\, {\rm cm}^{-2}\, {\rm s}^{-1}$ ($C_{\rm max} \geq 135\, {\rm c}\,{\rm s}^{-1}$)measured by ULYSSES shows no deviation from the "Euclidean'' case. This does not necessarily imply that the true peak flux distribution is a power law (a smoothly changing slope could also accomodate the data). This is rather an indication that the deviations we measure from the "Euclidean'' case are fully compatible with random fluctuations. Even if ULYSSES provides the best constraints to date on the slope of the bright end of the distribution, it cannot constrain it beyond the quoted value of $-1.51 \pm 0.10$.Combining ULYSSES and PVO data could probably reduce the uncertainty on this number. However, this is difficult due to the almost non-overlapping energy ranges of these 2 instruments.

A careful computation of the ULYSSES livetime for GRB detection allows us to measure the rate of "Euclidean GRBs'' with a good confidence. We find that they occur at a rate of $\rm \sim\! 65 \,yr^{-1}$(for GRBs longer than 2 seconds).