Fluence Distribution

To date, the fluences of the GRBs of this GRBM catalog, are expressed in terms of counts, since the GRBM response matrix has become available only recently (Dec 2001). Nevertheless, four distributions have been extracted: two fluence distributions, one for each energy bands, and two peak count rate distributions for both energy bands.

The fluences (counts) used for these distributions are the total counts summed over the four GRBM units; the same operation has been repeated for the peak count rates, expressed in counts/s. This way, the derived distributions have not been corrected for the different effective areas, depending on the GRB arrival direction with respect to the BeppoSAX local frame of reference; on the other side, the sum over the four GRBM units somehow accounts for this directional effect; furthermore, the high number of GRBs contributes to average this effect.

Figure: Top panel: cumulative fluence distribution (40-700 keV). The solid line shows the best powerlaw fit $N(>S) \propto S^{-\alpha}$, with $\alpha = 1.01 \pm 0.01 (1\sigma)$; the dashed line shows the case $\alpha = -3/2$. Bottom panel: cumulative peak countrate distribution (40-700 keV); the best fit gives $\alpha = 1.05 \pm 0.01 (1\sigma)$.

Figure: Top panel: cumulative fluence distribution ($>$ 100 keV). The solid line shows the best powerlaw fit $N(>S) \propto S^{-\alpha}$, with $\alpha = 0.98 \pm 0.01 (1\sigma)$; the dashed line shows the case $\alpha = -3/2$. Bottom panel: cumulative peak countrate distribution ($>$ 100 keV); the best fit gives $\alpha = 1.08 \pm 0.01 (1\sigma)$.

In figg.  the two cumulative distributions of the fluences and of the peak count rates in the 40-700 keV are shown. In the case of a homogeneous distribution within an euclidean space, it should be expected that $N(>S) \propto S^{-\alpha}$ and $N(>P) \propto P^{-\alpha}$, $\alpha = 3/2$, where $S$ is the fluence (counts) and $N(>S)$ is the number of GRBs with fluence greater than $S$, $P$ is the peak count rate (counts/s) and $N(>P)$ is the number of GRBs with peak count rate greater than $P$. In figg  the same distributions concerning the $>$ 100 keV band are shown. A power law fit has been performed in the following ranges: $10^4$ cts $< S < 4\cdot 10^4$ cts for the fluence distributions, and $5 \cdot 10^2$ cts $< P < 3\cdot 10^3$ for the peak count rate distributions; these ranges have been chosen by visual inspection.

The parameter $\alpha$ estimated by means of power law fits for the various distributions are the following: $1.01 \pm 0.01$ (fluence, 40-700 keV), $0.98 \pm 0.01$ (fluence, $>$ 100 keV), $1.05 \pm 0.01$ (peak count rate, 40-700 keV), $1.08 \pm 0.01$ (peak count rate, $>$ 100 keV); the uncertainties are $1\sigma$. These distributions suggest a different behavior of the GRB population from the case $\alpha = 3/2$, and this property agrees with the well known properties of the BATSE catalogs ([Meegan et al., 1996,Paciesas et al., 1999]).