3 Results

3.1 The Lognormal distribution

It has been shown previously that the lognormal distribution can adequately describe the properties of GRBs (Quilligan et al. 1999). The lognormal distribution is generated by statistical processes whose results depend on a product of probabilities arising from a combination of events (Aitchison & Brown 1957). A positive random variable X is said to be lognormally distributed if $Y=\log(X)$ is normally distributed with mean $\mu$ and variance $\sigma^2$. The probability density function is:

$$% f(x)= \left \{ \begin{array}{ll} {\frac{1}{\sqrt{2\pi} \sigma x... ...{(-(\ln{x}-\mu)^2/2\sigma^2)} & x > 0 \\ 0 & x \leq 0 \\ \end{array}\right. .$$ (1)

The median of the distribution occurs at $x={\rm e}^\mu$. Many examples of lognormal distributions occur in nature, such as the propagation of a laser beam in a turbulent medium, the size of cumulus clouds in the atmosphere and terrestrial lightning (Uman 1987; McBreen et al. 1994). In the case of terrestrial lightning, the durations, peak currents, intervals between the strokes in the flashes, and the flash charges are all lognormally distributed. The statistical properties of strokes in lightning discharges in the Earths atmosphere have a close resemblance to the statistical properties of pulses in GRBs. In both cases the result depends on a multiplicative process arising from a combination of independent events.

3.2 Pulse analysis

3.2.1 FWHM of pulses

The distribution of the FWHM of the pulses from 319 GRBs and the best lognormal fit are given in Fig. 6a. Figure 6b shows the same data plotted as a cumulative percent on logarithmic probability paper such that a lognormal distribution yields a straight line. All pulses have ${\rm\tau _\sigma } \geq 5$ and ${\rm\tau _I} \geq 50\%$. The distribution is very broad with a maximum at 0.7 s and half widths at 0.14 s and 3.5 s. The value of the reduced $\chi ^{2}$ is 0.3 showing the data is well fit by the lognormal distribution. In obtaining $\chi ^{2}$, the part of the fit below 0.1 s was not included because of the distortion caused by the limited time resolution that is apparent in Fig. 6b. The value of the reduced $\chi ^{2}$ for the best lognormal fit as a function of ${\rm\tau _\sigma }$ and ${\rm\tau _I}$ are given in Figs. 6c an 6d. The fits are acceptable over most of the range with the largest departures occurring at the lowest values of ${\rm\tau _\sigma }$ and ${\rm\tau _I}$ because of the serious effects of pulse pile up. The lognormal distribution is a better fit when the effects of the overlapping pulses are reduced. The parameters of the best lognormal fit for ${\rm\tau _\sigma } \geq 5$ and ${\rm\tau _I} \geq 50\%$ are given in Table 2.

Figure 6: a) The distribution of the FWHM of pulses with ${\rm\tau _\sigma } \geq 5$ and ${\rm\tau _I} \geq 50\%$ and the best lognormal fit to the data. b) The same data plotted as a cumulative percent such that a lognormal distribution yields a straight line. The large count in the first bin is due to the 64 ms time resolution. c)-d) The values of reduced $\chi ^{2}$ for the best lognormal fit as a function of ${\rm\tau _\sigma }$ and ${\rm\tau _I}$.

 

 

Table 2: The parameters of the best lognormal fit. The parameters are expressed as natural logarithms, with the width of the distributions shown in normal space.

Property	$\mu$	$\sigma$	$\chi ^{2}$	Width ($\pm$50%)
FWHM	-0.36	1.37	0.3	0.14-3.5
Rise time	-0.44	1.59	1.3	0.1-4.2
Fall time	-0.07	1.59	1.5	0.14-6.1
Pulse Amp.	9.0	1.12	0.3	2.2 $\times10^{3}{-} 30\times10^{3}$
Area	11.9	1.2	1.1	35 $\times 10^{3} {-} 600\times 10^{3}$
Time Int.a	0.21	1.03	-	0.37-4.14
Peak Energyb	5.6	0.58	-	137-535

^a See Sect. 3.2.4.
^b See Sect. 3.2.5.

3.2.2 Rise time and fall times of the pulses

The distribution of the rise times of the pulses and the best lognormal fit are given in Fig. 7a. Figure 7b shows the same data plotted as a cumulative percent and the large count in the first bin is due to the 64 ms resolution of BATSE. The parameters of the best lognormal fit to the broad distribution are listed in Table 2. The value of the reduced $\chi^{2} = 1.3$ show the data is compatible with the lognormal distribution.

Figure 7: a) The distribution of the pulse rise times with ${\rm\tau _\sigma } \geq 5$ and ${\rm\tau _I} \geq 50\%$ and the best lognormal fit to the data. b) The same data plotted as a cumulative percent. The large count in the first bin is due to the 64 ms time resolution.

Figure 8: The distribution of the fall times of the pulses with ${\rm\tau _\sigma } \geq 5$ and ${\rm\tau _I} \geq 50\%$ and the best lognormal fit to the data.

Figure 9: The distribution of the pulse amplitudes with ${\rm\tau _\sigma } \geq 5$ and ${\rm\tau _I} \geq 50\%$ and the best lognormal fit to the data.

The distribution of the fall times of the pulses and the best lognormal fit are given in Fig. 8. The parameters of the best lognormal fit to the data are given in Table 2. The fall times are also compatible with the lognormal distribution and have a wider range with a slightly larger mean than the rise times.

3.2.3 Areas and amplitudes of the pulses

The distributions of the amplitudes and areas of the pulses with ${\rm\tau _\sigma } \geq 5$ and ${\rm\tau _I} \geq 50\%$ from 250 GRBs, summed over two detectors, and the best lognormal fits are given in Figs. 9 and 10. The distributions are very broad and the values of the best lognormal fits to the data are listed in Table 2. The lognormal distribution is compatible with the pulse areas and the amplitudes.

   
3.2.4 Time intervals between pulses

The distribution of the time intervals between the pulses is given in Fig. 11. The peak in the distribution occurred at about 1.0 s and was truncated at short time intervals by the 64 ms resolution of the data (Fig. 11a). A minimum time interval of 128 ms is required because two maxima must be separated by at least one time bin. There could be a large additional excess of pulses with separations below the 128 ms resolution of the data that were not resolved. Other studies using different statistical methods from those employed here (Panaitescu et al. 1999; Lee et al. 2000a,b; Spada et al. 2000) have also noted a deficit of time intervals below one second. A parent lognormal distribution of time intervals with parameters similar to the observed distribution was simulated and the time intervals between the pulses recorded with an accuracy of 64 ms. The resulting distribution is given in Fig. 11a and the values of $\mu$ and $\sigma$ for the parent distribution are given in Table 2. The measured distribution of time intervals in GRBs is consistent with the parent lognormal distribution provided a small (5%) excess of time intervals longer than 15 s is not included. This excess is clearly visible in Fig. 11a. The time intervals greater than 15 s are plotted in Fig. 11b. The data is well fit by a power law of slope -1.2.

Figure 10: The distribution of the pulse areas with ${\rm\tau _\sigma } \geq 5$ and ${\rm\tau _I} \geq 50\%$ and the best lognormal fit to the data.

   
3.2.5 Peak energies

The values of the peak energy, $E_{\rm peak}$, of a large sample of GRBs are given by Preece et al. (2000). There is an overlap of 77 GRBs with our sample, and the distribution of the values of $E_{\rm peak}$ for each section of these bursts are given in Fig. 12. The distribution was well fit by a lognormal distribution with a small tail noticeable at low values of $E_{\rm peak}$. The values of $\mu$ and $\sigma$ are given in Table 2. The distribution of $E_{\rm peak}$ is noticeably narrower than that of the pulse parameters and spans a range of about 4 in width.

Figure 11: a) The measured distribution of time intervals between all the pulses and the best fit lognormal (dashed line) with allowance for the BATSE resolution of 64 ms. The excess of time intervals >15 s is called the Pareto Lévy tail. b) The Pareto tail of the time intervals is well fit by a power law (dashed line) of slope $\sim$-1.2.

Figure 12: The distribution of the peak energies and the best lognormal fit to the data (dashed line).

3.3 Summary

The major result of this part of the analysis is that the distributions of the rise times, fall times, FWHM, pulse amplitudes, pulse areas and the time intervals between the pulses are all very similar. The frequency distributions are very broad and cover about three orders of magnitude and all are compatible with the lognormal distribution. Li & Fenimore (1996) also showed that the pulse fluences and the time intervals between pulses are lognormally distributed for individual bursts in a small sample of bursts with more than 20 pulses. They also scaled the bursts to the same $\mu$ and $\sigma$ and showed that the summation of all the peak fluences and time intervals looked lognormal, although no significance level was given for the result. No normalisation of pulse properties was applied to the GRBs in this analysis, because only the brightest 319 bursts which had the best signal to noise ratio were analysed. From the data available for those bursts with known redshifts (Atteia et al. 1999), there does not appear to be any dependable standard by which to scale the bursts because of the broad range of intrinsic luminosities and their comparatively small range of distances. Therefore, to avoid introducing further biases, and to use all of the pulse information available, the data were analysed without scaling. However, as a test of this process, the data were also scaled and the same analysis performed on the scaled data, and no significant differences were found between the two data sets.

3.4 Correlations between burst and pulse parameters

It is important to determine how N relates to the other parameters of the GRB. In Fig. 13 N is plotted versus burst duration (T₉₀), total fluence and the median value of $E_{\rm peak}$. Spearman rank order correlation coefficients $\rho$ and associated probabilities were obtained for the quantities in Fig. 13. The values are listed in Table 3 which also includes an additional range of burst parameters. The parameter $C_{\rm max}$is the maximum value of the peak amplitude in that burst. The high values of $\rho$ show a strong correlation between N and the total fluence, T₉₀ and $E_{\rm peak}$.

Figure 13: The number of pulses in a GRB as a function of a) T90 b) total fluence and c) the median value of the peak energy of the GRB. Quantisation occurs in the figures for low values of N.

 

 

Table 3: Spearman rank order correlation coefficients between a range of burst parameters.

Properties	$\rho$	Probability
N vs. T90	0.45	$5 \times 10^{-17}$
N vs. Total Fluence	0.58	$3 \times10^{-30}$
N vs. C $_{\rm max}$	0.27	$6 \times 10^{-7}$
N vs. Hardness Ratio	0.29	$1.2 \times 10^{-7}$
N vs. E $_{\rm peak}$	0.49	$7 \times 10^{-6}$
T90 vs. Total Fluence	0.52	$6 \times 10^{-24}$
T90 vs. C $_{\rm max}$	-0.08	0.16
T90 vs. Hardness Ratio	0.11	0.05
Total Fluence vs. C $_{\rm max}$	0.48	$1.3 \times10^{-15}$
Total Fluence vs. Hardness Ratio (HR)	0.56	$3 \times 10^{-27}$
$C_{\rm max}$ vs. HR	0.25	$4.4\times10^{-6}$

 

 

Table 4: Spearman rank order correlation coefficients between a range of pulse parameters.

Properties	$\rho$	Probability
Rise Time vs. Fall Time	0.64	<10-48
Rise Time vs. FWHM	0.65	<10-48
Rise Time vs. Pulse Area	0.34	$2.9\times10^{-34}$
Rise Time vs. Pulse Amplitude	-0.27	$1.9\times10^{-21}$
Fall Time vs. FWHM	0.70	< 10-48
Fall Time vs Pulse Area	0.42	<10-48
Fall Time vs. Pulse Amplitude	-0.22	$1.4 \times 10^{-14}$
FWHM vs. Pulse Area	0.44	<10-48
FWHM vs. Pulse Amplitude	-0.27	$2.7 \times 10^{-21}$
Pulse Area vs. Pulse Amplitude	0.63	<10-48
FWHM vs. Time Interval	0.58	10-48

The values of $\rho$ are not always uniformly distributed within each burst category. T₉₀ versus fluence is much better correlated for category N than either O or P. N versus $C_{\rm max}$ and N versus HR are better correlated for category P than either N or O.

The Spearman rank order correlation coefficients and probabilities were evaluated for isolated pulses with the range of pulse parameters given in Table 4. The pulse parameters are strongly correlated with each other. The pulse amplitude is negatively correlated with the pulse rise and fall times and FWHM. In general the correlations are stronger for categories M and N than either O or P. The only significant exception to this trend is the pulse amplitude versus area which also has the highest values of $\rho$ for categories O and P.

 

 

Table 5: Spearman rank order correlation coefficients $\rho$for time intervals between pulses. The two values for $\rho$ and the probability are for unnormalised/normalised time intervals.

Number of	Total
Intervals	Number	$\rho$	Probability
1	2751	0.42/0.56	<10-48
2	2499	0.34/0.48	<10-48
5	1929	0.24/0.37	$5\times10^{-26}/{<}10^{-48}$
10	1395	0.20/0.29	$3\times10^{-13}/6\times10^{-27}$
15	890	0.16/0.25	$3\times10^{-6}/4\times10^{-14}$
20	634	0.10/0.23	$8\times10^{-3}/3\times10^{-9}$
25	459	0.08/0.22	$7\times10^{-2}/1\times10^{-6}$
30	322	0.03/014	$3\times10^{-2}/10^{-2}$

 

 

Table 6: Spearman rank order correlation coefficients $\rho$ and associated probabilities for the pulse amplitudes. The two values for $\rho$ and the probability are for unnormalised/normalised amplitudes.

Number of	Total
Amplitudes	Number	$\rho$	Probability
1	3039	0.72/0.57	<10-48
3	2499	0.55/0.32	<10-48
5	2098	0.52/0.24	< $10^{-48}/ 3\times10^{-29}$
7	1777	0.48/0.15	< $10^{-48}/ 6\times10^{-11}$
9	1510	0.43/0.08	< $10^{-48}/2 \times 10^{-3}$
10	1395	0.44/0.08	< $10^{-48}/3 \times 10^{-2}$

3.5 Correlations between the time intervals between pulses and pulse amplitudes

Spearman rank order correlation coefficients and probabilities were evaluated for the time intervals between pulses ($\Delta T$) with ${\rm\tau_\sigma} > 5$. The results are presented in Table 5 for two cases (1) the time intervals were not normalised and (2) the time intervals were normalised to T₉₀. There is a good correlation between the time intervals in both cases that declines slowly with increase in the number of time intervals. The largest values of $\rho$ occured in category N.

The Spearman correlation coefficients were also evaluated between pulse amplitudes and the results are given in Table 6 for two cases (1) the amplitudes were not normalised and (2) normalised to the largest amplitude pulse in the burst. The normalised pulse amplitudes are less strongly correlated over many pulses than the time intervals. These results were obtained for all pulses with ${\rm\tau _\sigma } \geq 5$ and without satisfying any selection based on pulse isolation. The role of pulse pile-up has yet to be investigated.

Figure 14: The median values of a) rise time, b) fall time, c)  FWHM and d) the time intervals between pulses versus the number of pulses.

3.6 The properties of the pulses as a function of, N, the number of pulses in the GRB

It was noticed early in this analysis that pulse properties depended strongly on N (Quilligan et al. 2000). The median value of the isolated pulse timing parameters were determined for all GRBs with the same value of N. The median values of the rise time, fall time, FWHM and time interval between pulses are plotted versus N in Fig. 14 (a-d). The largest value usually occurred for N = 1 or 2 and subsequently declined significantly as Nincreased. There are some values that are well removed from the general trend but they usually have a small number of pulses. The median values of the area and amplitude for isolated pulses are given in Figs. 15a,b. The trend is quite different from Fig. 14. The amplitude is reasonably constant up to $N \sim 35$with a clear increase for higher values of N. There is a similar but weaker trend for the pulse area which has the largest value at N = 1.

The properties of the four categories of GRBs are summarised in Table 7. The median values of the pulse timing parameters all decrease by at least a factor of four from category M to P. In contrast the median values of T₉₀, total fluence, hardness ratio and maximum pulse amplitude all increase significantly. The median variability, is defined as the number of pulses $\geq$5 $\sigma$ divided by the time the emission is $\geq$5 $\sigma$, also increases from category M to P.

 

 

Table 7: The properties of the four categories of pulses in GRBs. The last five entries are for the 250 GRBs that were summed over two detectors.

GRB Category	M	N	O	P
Number of Pulses per GRB	1-2	3-12	13-24	25+
Number of GRBs	67	162	56	34
Total number of pulses	103	981	933	1341
Number of isolated pulses at 50% level	83	522	476	494
Median T90 (s)	18.1	20.4	45.7	58.7
Median Total Fluence (ergs/cm2)	8.8 $\times 10^{-6}$	1.7 $\times 10^{-5}$	4.2 $\times 10^{-5}$	$1.2\times 10^{-4}$
Median hardness ratio (Chan $\frac{4+3}{2+1}$)	3.4	4.1	6.5	8.3
Median $C_{\rm max}$ (ph/cm2/s)	4.7	6.1	9.3	12.6
Median Variability	0.09	0.29	0.39	0.53
Median Rise Time (s)	1.7	0.8	0.64	0.45
Median Fall Time (s)	5.2	1.5	1.0	0.7
Median FWHM (s)	1.8	0.7	0.64	0.45
Median Time Interval (s)	4.8	1.9	1.5	1.0
Number of GRBs	55	130	38	27
Total number of pulses	87	778	648	1081
Number of isolated pulses at 50% level	70	416	319	404
Median Pulse Amplitude (Iso. pulses) (counts)	13 $\times 10^{3}$	7.3 $\times 10^{3}$	7.3 $\times 10^{3}$	9.5 $\times 10^{3}$
Median Area (counts)	560 $\times 10^{3}$	180 $\times 10^{3}$	140 $\times 10^{3}$	140 $\times 10^{3}$

3.7 First Half/Second Half analysis

 

 

Table 8: Summary of the first half/second half analysis of the three categories of the GRBs.

GRB Category	N	O	P
Number of Pulses per GRB	3-12	13-24	25+
Total Number of Pulses (1st/2nd half)	404/415	384/493	679/628
Total Number of Isolated Pulses (50%) (1st/2nd half)	193/220	182/261	203/271
Total Number of Isolated Pulses (75%) (1st/2nd half)	79/126	77/101	71/83
Median Rise Time (1st/2nd half)	1.02/0.70	0.77/0.58	0.51/0.45
K-S Statistic/Probability	0.13/5%	0.12/9%	0.11/11%
Median Fall Time (1st/2nd half)	1.09/1.89	0.90/1.22	0.70/0.70
K-S Statistic/Probability	0.20/.02%	0.15/2%	0.11/10%
Median Asymmetry Ratio (1st/2nd half)	1.0/0.4	0.75/0.49	0.75/0.54
K-S Statistic/Probability	0.27/ $3\times 10^{-5}\%$	.17/0.3%	0.14/1.5%
Median FWHM (1st/2nd half)	0.77/0.67	0.64/0.64	0.51/0.45
K-S Statistic/Probability	0.08/55%	0.08/52%	0.11/11%
Median Time Interval (1st/2nd half)	1.60/2.10	1.41/1.54	1.02/1.02
K-S Statistic/Probability	0.13/2%	0.06/35%	0.05/30%
Median Pulse Amplitude ( $\times 10^{3}$)(1st/2nd half)	6.2/5.4	7.3/6.8	18.1/11.2
K-S Statistic/Probability	0.09/41%	0.08/52%	0.09/25%
Median Area ( $\times 10^{3}$)(1st/2nd half)	151/127	133/118	152/130
K-S Statistic/Probability	0.08/50%	0.07/59%	0.14/2%

To study the evolution of the time profile as the GRB progresses, each GRB was divided into two, to include the pulses that occur before and after the strongest pulse in the burst. Only GRBs with more than two pulses are included, resulting in a reduced total sample of 252. The first half (pre-main pulse) of the GRB was compared with the second half (post-main pulse). The bursts are also sub-divided into three categories. A summary of the properties of the GRBs used is given in Table 8.

The first half/second half analysis was performed on the three timing parameters of the pulses, time intervals between the pulses, amplitude, area and the pulse asymmetry ratio which is defined as the ratio of the pulse rise time to the pulse fall time. The median values of the distributions in the three categories in the first half and second half analysis are given in Table 8 along with the results of the Kolmogorov-Smirnov (KS) tests. The KS probability is a measure of whether the two distributions (first/second half) are drawn from the same parent distribution.

The first result is that the median values of the timing parameters of the pulses and the time intervals between the pulses all decrease by an average of 1.8, from the category N to P including the first half and second half of the GRBs. In the case of the rise times, the median values of the distributions decrease from 1.02 to 0.51 in the first half and 0.70 to 0.45 in the second half. The trend in the median value of the pulse amplitude is in the opposite direction with larger pulses in category P than either O or N.

In the first/second half analysis there is a trend in the three categories for the median rise time to be slower in the first half of the burst (1.02 versus 0.70 for category N). The difference could be caused by an additional clearing out effect at the start of the GRB. There is also a clear indication at the 0.02% level that the pulse fall time is faster in the first half than the second for category N (1.09 versus 1.89 for category N) and this effect weakens for categories O and P. The median values of the pulse asymmetry ratio also show the most significant differences for category N where the median values are 1.0 and 0.4 for the first and second halves. The KS test gives good agreement between the first half and second half for the FWHM, time intervals between pulses, the pulse amplitudes and areas. The median values of the pulse amplitude and areas are however larger in the first half than the second half for the three categories of GRBs.