HR Distribution

In section  the effectiveness of the automatic threshold (set to $0.3$) on the HR for rejecting false GRBs has been emphasized. In fig.  the HR distribution of the 891 GRBs, used for the duration distribution discussed in the previous section, is shown.

Figure: HR Distribution of 891 GRBs from the GRBM catalog.

The distribution is satisfactorily fitted with a normal distribution plus a constant function (eq. ); the gaussian parameters so found are the following: $HR_0 = 0.72$ (center), $\sigma_{HR} = 0.12$ (sigma), with a $\chi^2 = 0.66$. From fig.  it is apparent that the HR condition (eq. ) does not reject any true cosmic GRBs.

The HR distribution for the only GRBs, whose HR is at least 3$\sigma$ significant, has been extracted and shown in fig. . This subset includes 790 GRBs and the derived distribution is well fitted with a normal distribution as well; the parameters gaussian parameters are $HR_0 = 0.71$ (center), $\sigma_{HR} = 0.12$ (sigma), with a $\chi^2 = 0.29$ (eq. ).

Figure: HR Distribution of 790 GRBs from the GRBM catalog, with at least 3$\sigma$ significant HRs.

It is worth mentioning: the width of the HR distribution somehow depends on the GRBM response as a function of the GRB arrival direction. In order to reduce this effect, the HRs used are the weighted averages of the HRs of the single GRBM units. It might be possible that the HR distribution deconvolved by the GRBM response function could be even narrower. In table  the parameters of the best fits of the distributions are reported.

$$\frac{dN_{fit}}{d(HR)} \ = \ Const. + N_0 \cdot \exp{\Big [-\frac{(HR - HR_0)^2}{2\sigma_{HR}^2}\Big ]}$$ (43)

Table: Parameters of the Gaussian + Constant functions, best fitting the HR distribution.

GRB set	Center	Sigma	Const.	Normaliz.	Red.
	($HR_0$)	$\sigma_{HR}$		$N_0$	$\chi^2$
All (891)	0.72	0.12	1.48	182.7	0.66
$>3\sigma$ (790)	0.71	0.12	0.63	166.3	0.29