Dead Time Correction

The expression  gives the dead time correction of the GRBM light curves, taking into account the GBRM ratemeter and the fixed dead time $\tau = 4\, \mu$s (table ):

$$n_i \ = \ \frac{m_i}{1 - m_i \cdot \tau}$$ (40)

In the above expression, $m_i$ and $n_i$ are the measured and the dead time corrected count rates for the $i$-th bin, respectively. This correction becomes relevant when $m_i$ is comparable with $1/\tau$.

Table: Some Ratemeter Dead Time Corrections

$m_i$ (c/s)	$n_i$ (c/s)	Correction (%)
6000.0	6147.5	2.5
12000.0	12605.0	5.0
18000.0	19396.6	7.8
24000.0	26548.7	10.6
30000.0	34090.9	13.6
36000.0	42056.1	16.8
42000.0	50480.8	20.2
48000.0	59405.9	23.8
54000.0	68877.6	27.6
60000.0	78947.4	31.6

In table  the correction factor vs. the measured count rate is reported: clearly, a correction of the order of $\sim$ 10% occurs for count rates of at least few $\sim 10^4$ c/s, i.e. for the brightest bursts.

After dead time correction, the background has been modelled by means of a least-square polynomial fit around the burst, excluding the central interval including the burst itself; the time boundaries of the interval used for fitting vary from case to case, mainly depending on the burst duration and time profile. A 2nd degree polynomial sufficed for nearly all the GRBs, except a few cases, in which higher degree polynomials are needed to account for delicate background variations.

The summed light curves of the two brightest GRBM units, one for each energy band, are taken into account as well.

In fig.  the example of background fit applied to the light curves of the GRB000830, UT 11:39:08, is shown.

Figure: Background Fit for GRB000830, UT 11:39:08. From top to bottom: GRBM1, ..., GRBM4, AC1, ..., AC4.

Once the background level has been determined from the fit, a set of 10 reduced $\chi^2$ is reported (two light curves -one for each energy band- for each GRBM units plus other two, concerning the summed light curves of the two brightest GRBM units), to evaluate the goodness of the fit. In the case of GRB000830, the GRBM units 3 and 2 were the brightest units; the 10 $\chi^2$ are the following: $1.080, 0.986, 1.241, 1.014, 1.151$ for the GRBM1, ..., GRBM4, GRBM3+2, and $1.248, 1.073, 1.020, 1.012, 1.285$ for the AC1,..., AC4, AC3+2, respectively.

In this example, the background fit interval used has the following boundaries: [-130 s, -40 s] $\cup$ [150 s, 250 s], where t=0 s is the GRB off-line trigger time.