4 Discussion

4.1 Pulse shapes in GRBs

A complete study of the BATSE time profiles of the brightest 319 GRBs has been presented. The statistical analysis of the data reveal the ubiquitous nature of the lognormal distribution in GRB time profiles (Figs. 6-10). The means and variances of the best fit lognormal functions are given in Table 2. The results presented in Figs. 6-10 are for the isolated pulses from the four GRB categories. The data in Table 7 and the first half/second analysis show that the median values of the pulse properties vary with N. Lognormal distributions also apply to the spectral properties of GRBs. The BATSE spectroscopic detectors revealed that the break energies in GRB spectra are compatible with a lognormal distribution (Preece et al. 2000). The FWHM of the distributions that describe the pulses are in the range 14-45 (Table 2) whereas the value for $E_{\rm peak}$ is only $\approx$4 and extends from 137 keV to 535 keV. The unexpected narrowness of the $E_{\rm peak}$ distribution is a major problem in GRBs (Brainerd 2000). Furthermore, spectral fitting of 41 pulses in 26 GRBs showed that the spectral hardness parameter $E_{\rm peak}$decays linearly with energy fluence and that the distribution of the decay constants is roughly lognormal (Crider et al. 1999).

Figure 15: The median values of a) pulse area and b) pulse amplitudes versus the number of pulses.

The Spearman rank order correlation coefficients were obtained for a range of burst parameters (Table 3). There is good agreement with the results of Lee et al. (2000aa) who used a different method. N is strongly correlated with total fluence, T₉₀, HR and $E_{\rm peak}$ (Fig. 13). N is an important quantity in determining GRB properties and provides the link between total fluence and duration since both increase with N. The correlation coefficients between the parameters that describe the pulses are given in Table 4. The rise and fall times, FWHM and area of the pulses are highly correlated. There is a high probability that a pulse with a fast rise time will also have a fast fall time and a short FWHM. The pulse amplitude is negatively correlated with the rise and fall times and FWHM. The anticorrelation between the pulse amplitude and pulse width has been observed in other studies (Lee et al. 2000a; Fenimore & Ramirez-Ruiz 2001). The pulse width is also a function of energy and varies as E^-0.45 (Fenimore et al. 1995) and this effect has been attributed to synchrotron radiation (Piran 1999). The FWHM is strongly correlated with the preceding and subsequent time interval between pulses (Table 4). This result is in agreement with Nakar & Piran (2001) and the prediction of the internal shock model. A further comparison of the pulse timing parameters, the energy dependence and the spectral lag may reveal further interesting constraints on the emission process (Norris et al. 2001; Daigne & Mochkovitch 2001). The unique properties of the pulses in GRBs have been summarised by McBreen et al. (2002).

The lognormal distribution arises from a statistical process whose result depends on a product of probabilities arising from a combination of independent events. It therefore identifies the statistical process but not the combination of events that lead to the formation of the pulse shape and the peak energy. In the internal shock model the main factors contributing to the pulse shape include (Rees & Mészáros 1994; Piran 1999).

1.

The Lorentz factors, masses and thickness of the interacting shells.

2.

The distance from the central engine and curvature where the collisions occur.

3.

The energy conversion in the shock, the nature of the magnetic field and particle acceleration process.

4.

The synchrotron radiation mechanism possibly modified by self-absorption and pair production.

5.

The time scale for energy loss by the particles.

The resulting properties of the pulses in GRBs depend on a combination of many events and hence it is not surprising that the lognormal distribution gives an elegant description of all their properties. Random multiplicative processes abound in a variety of natural phenomena and a good example is the statistical description of the strokes in flashes of lightning. Almost all the properties of the strokes in the flashes and the flashes themselves are well described by lognormal distributions (Uman 1987; McBreen et al. 1994).

4.2 Time intervals between pulses in GRBs

It is often found that distributions that seem to be lognormal over a wide range change to an inverse power law distribution for the last few percent. An amplification model has been used to characterise the transition from a lognormal distribution to a power law that is often called a Pareto-Lévy tail (Montroll & Shlesinger 1982). The distribution of time intervals conform to the lognormal distribution over most of the range with the exception of about 5% of the time intervals longer than about 15 s (Fig. 11). The Pareto-Lévy tail of time intervals have an amplification process that is not available to most time intervals.

The origin of the nonrandom distribution of time intervals between pulses is an important clue to the GRB process. In the internal shock model there is almost a one to one correspondence between the emission of shells and pulses resulting from the collisions of shells (Kobayashi et al. 1997). Hence the time intervals between pulses is an almost direct measure of the activity of the central engine. The temporal behaviour of soft gamma-ray repeaters and young pulsars provide additional context in which to view the results of GRB time profiles. The time intervals between about 30 microglitches in the Vela Pulsar are consistent with a lognormal distribution with a mean of 50 days (Hurley et al. 1994; Hurley et al. 1995; Cordes et al. 1988). The amount of energy involved in the microglitches is about 10³⁸ ergs. The macroglitches in the Vela Pulsar are about a thousand times more powerful but occur too infrequently to determine the distribution of time intervals but they have a wide range and do not seem inconsistent with the lognormal distribution. More energetic outbursts have been recorded from SGR sources. The two most energetic events released about $5 \times 10^{44}$ ergs in $\gamma$-rays from SGR 0526-66 on 5 March 1979 (Mazets et al. 1979) and about 10⁴³ ergs from SGR 1900+14 on 27 August 1998 (Hurley et al. 1999). The SGR sources also generate a large number of smaller outbursts, and it has been shown that the time intervals between outbursts are distributed lognormally (Hurley et al. 1994; Gögüs et al. 2000). Hence lognormally distributed time intervals between outbursts and glitches are characteristic features of SGR sources and neutron star microglitches. It is widely accepted that these sources are rotating neutron stars with high magnetic fields. It is not unreasonable to argue that the coupled effects of rapid rotation and intense magnetic fields (Kluzniak & Ruderman 1998) are also involved in powering GRBs since the time intervals between pulses are also consistent with a lognormal distribution.

The possibility that a rapidly rotating neutron star with a surface magnetic field of $\sim$10¹⁵ Gauss could power a GRB has been suggested (Usov 1992). Once formed such a neutron star could lose its rotational energy catastrophically on a time scale of seconds. The rotation of the star decelerates because of the applied torques. Powerful transient fields may also occur in the merger of two neutron stars or a neutron star and a black hole. The energy stored in differential rotation of the collapsed object would be released in sub-bursts as toroidal magnetic fields are repeatedly wound up to $\sim$10¹⁷ Gauss (Kluzniak & Ruderman 1998). The emergence of a toroid is accompanied by huge spin down torques, the reconnection of new surface magnetic fields and rapid release of a sub-burst of energy of about 10⁵¹ ergs. The release of rotational energy in repeated sub-bursts could power the GRB. If the differentially rotating compact object forms a torus about a spinning black hole either in a merger or core collapse of a massive star, energy can be extracted by the magnetic field that threads the torus and the black hole (Mészáros 2001). As the torus builds up and ejects its magnetic toroids, the differential rotation of the torus could be maintained by the spin of the black hole.The models of GRBs with the coupled effects of rapid rotation and ultra intense magnetic fields are particularly attractive because the time intervals between pulses in GRBs are distributed lognormally and follow the pattern observed in non-catastrophic events in SGRs and pulsars.

The time intervals between the pulses are correlated with each other and the correlation decreases slowly with increase in the number of time intervals (Table 5). This effect had previously been observed in a small sample of GRBs (Nakar & Piran 2001) and attributed to the internal shock model. In addition the pulse amplitudes are also correlated with each other and this effect decreases more rapidly than the time intervals between pulses (Table 6). Similar correlations have been found between the pulse amplitudes and also time intervals between pulses in short GRBs (McBreen et al. 2001). In the internal shock model, these correlations originate in the central engine and provide strong constraints on any viable model of GRBs.

GRB models leave open many possibilities to account for the Pareto-Lévy tail of long time intervals. The excess of long time intervals have been noted in other studies (Ramirez-Ruiz & Merloni 2001; Nakar & Piran 2001). The properties of the GRBs with long time intervals will be covered in a separate publication.

4.3 Numbers of pulses and jets in GRBs

A detailed comparison has been made between the distributions of the properties of the pulses in the first half and second half for three categories of GRBs. There are no statistically significant differences between the median values of the time intervals between pulses, pulse amplitude, areas and FWHM in the first half and second half of GRBs (Table 8). There are two trends in the pulse rise times and fall times that should be noted: 1)  the median rise time is slower in the first half for the three categories of bursts (Table 8) and 2) the median fall time is faster in the first half for categories N and O. The combination of slower rise times and faster fall times gives a pulse asymmetry ratio with a significant difference between the first half and second half for category N and at a reduced significance level for O and P (Table 8). The effect could be caused by a clearing out process such as additional baryon loading or Compton drag in the first half of the bursts with small number of pulses. These results are also compatible with the constancy of the pulse widths observed by Ramirez-Ruiz & Fenimore (2000) using a peak aligned profile method on a small sample of GRBs with more than 20 pulses. In the internal shock model, the rapid variability in GRB time profiles is due to emission from multiple shocks in a relativistic wind (Piran 1999; Panaitescu et al. 1999; Downes et al. 2001). The temporal position of the pulse is unconnected to the collision parameters and in this way the little or no evolution of the pulses in GRBs can be explained (Fenimore & Ramirez-Ruiz 2001). The rise time and fall time may be determined by the hydrostatic time $\approx$d/c and the angular spreading time $\approx$D/c, where dand D are the width and separation of the shells (Kobayashi et al. 2001). The observed evolution of the pulses requires the shells to be narrower and farther apart later in the GRB. This prediction is in agreement with the data because the time intervals between pulses are longer in the second half of the burst (Table 8).

However, it is evident from Fig. 14 and Tables 7 and 8 that as the number of pulses in a burst increases, the median values of the rise and fall times, FWHM and time intervals all decrease and by about the same amount. The GRBs with more pulses also have on average significantly longer durations, higher fluences and hardness ratios (Table 7). The variability index of a GRB was taken to be the number of pulses $\geq$$5 \sigma$ divided by the time the GRB emission was also $\geq$$5 \sigma$. The median values are given in Table 9 for the four GRB categories. The GRBs with more pulses have a higher variability index. These results provide an interesting interpretation of the two correlations that have been reported for GRBs with known redshift: (1)  the more luminous GRBs to be more variable (Fenimore & Ramirez-Ruiz 2001) and (2)  there is an anticorrelation between the arrival times of high energy and low energy pulses in GRBs (Norris et al. 2000; Salmonson 2000). Recently Schaefer et al. (2001) showed that there is a relationship between the variability and spectral lag (Ioka & Nakamura 2001). There is a good correlation between the values of the variability obtained here and those of Fenimore & Ramirez-Ruiz (2001). GRBs with higher values of HR have lower values of $<V/V_{\rm max}>$ implying they are a more distant and luminous population (Schmidt 2001).

Our knowledge on the shape of the emitting region in GRBs is restricted because, due to relativistic beaming, only a small portion of angular size $\sim$ $\Gamma^{-1}$ is visible to the observer. Thus the observer is unable to distinguish a sphere from a jet as long as where $\theta$ is the radius of the opening angle of jet (Rhoads 1997). However as the source continues its rapid expansion, $\Gamma$will decrease, and when there will be a marked decrease in the observed flux. The steep time dependence of the afterglow emission, sometimes with changes in the slope of the spectrum, and radio emission have been widely interpreted as evidence for emission from jets (Castro-Tirado 2001; Mészáros 2001; Frail et al. 2001). The GRBs with more pulses appear to have higher values of the Lorentz factor $\Gamma$. The higher values of $\Gamma$ may come from a more efficient and more active central engine. In one variation of the internal shock model, it was assumed that the degree of collimation of the jet depended on the mass M at the explosion (Kobayashi et al. 2001). A wide jet involves a large mass that results in a flow with a lower $\Gamma$. The pulse properties depend strongly on $\Gamma_{\rm min}, \Gamma_{\rm max}$ and the radius of the photosphere $R\pm$. GRBs with faster pulses originate in collisions above $R\pm$ whereas GRBs with slower pulses have smaller values of $\Gamma$ and some collisions below the photosphere. While this homogeneous model may explain pulse properties in GRBs, the strong possibility of inhomogeneous jets with a variable $\Gamma$ should also be examined.

Baryon loading can be a major problem in GRB models and severely limit the attainable value of $\Gamma$(Rees 1999; Mészáros et al. 1998; Salmonson 2000). There maybe a broad range of $\Gamma$'s in the outflow with the highest value occurring close to the rotation axis where the baryon contamination should be at a minimum. At larger angles from the axis, there may be an increasing degree of contamination with a corresponding drop in $\Gamma$. The outcome of a collapse in a massive star whose iron core collapsed to a black hole have been computed (MacFadyen & Woosley 1999). The resulting jet that drives out through the star is probably powered by a MHD process which can in principle convert a large portion of the binding energy at the last stable orbit into jet energy. The large amount of energy dumped into the natural funnel-shaped channel creates a highly collimated jet, focused into a small region of the sky. The largest value of $\Gamma$ occurs on axis and decreases with increasing $\theta$because the material coming at the observer has less energy at larger angles. The emission is still beamed into an angle $\Gamma^{-1}$ but in this inhomogeneous model the angle varies across the opening angle of the jet (Rossi et al. 2001). In this situation the properties of the pulses in GRBs can be influenced by the jet. The BATSE sample of the brightest GRBs should contain a range of angles within the jet and hence different values of $\Gamma$. In this context it is reasonable to identify the complex GRBs with more pulses and higher values of $\Gamma$ with angles near the axis of jet. The GRBs that are viewed at larger angles from the jet axis have on average lower values of $\Gamma$, and develop at the greater distances from the central engine and should have slower pulses.

In this context it is interesting to note that the pulse evolution consisting of slower rise times and faster fall times in the first half, is more pronounced for GRBs in category N than either O or P. In a jet model with a variable $\Gamma$, the GRBs in category N would be on average farther from the axis and more sensitive to a clearing out effect such as additional baryon loading or Compton drag (Rees 1999; Ghisellini 2001) in the initial phase of the GRB.

The steep and variable slope of the decay of GRB afterglows have been widely interpreted as evidence for jets in GRBs (Mészáros 2001; Castro-Tirado et al. 2001). If the axis of the jet is pointed close to the observer, the GRB will be intense and the afterglow should contain evidence for good alignment. It is interesting that the two brightest GRBs detected by WFC on BeppoSAX also were the best aligned. The recent detection of a bright GRB with a fluence of 10^-4 ergs cm^-2 also had a very bright afterglow (Castro-Tirado et al. 2001). However many more GRBs and afterglows are required to verify the existence of a pattern between the strongest GRBs and their afterglows (Frail et al. 2001). The distribution of the number of pulses per GRB (Fig. 5) may broadly represent the beaming by the jet because bursts with large numbers of pulses and higher variability (Table 7) may be close to the axis and bursts with smaller numbers of pulses and less variability further off-axis.