Properties of Short Gamma-ray Burst Pulses from a BATSE TTE GRB Pulse Catalog

The pulse-fitting model consists of two parts. The first is the general four-parameter empirical pulse model of Norris et al. (2005). The hypothesis is that a GRB emission episode can be modeled by the following asymmetric, monotonically increasing and decreasing intensity function:

$\begin{eqnarray}&&I(t)=A\lambda {e}^{[-{\tau }_{1}/(t-{t}_{s})-(t-{t}_{s})/{\tau }_{2}]},\end{eqnarray} \tag{ 1 }$

where t is time since trigger, A is the pulse amplitude, t_s is the pulse start time, τ₁ is the pulse rise parameter, τ₂ is the pulse decay parameter, and the normalization constant λ is given as $\lambda =\exp [2{({\tau }_{1}/{\tau }_{2})}^{1/2}]$. Poisson statistics and a two-parameter background counts model of the form B = B₀ + BS × t are assumed (where B is the background counts in each bin and B₀ and BS are constants denoting the mean background (counts) and the rate of change of this mean background [counts s^–1]). Observable pulse parameters obtained from this model include the pulse peak time τ_peak where

$\begin{eqnarray}&&{\tau }_{\mathrm{peak}}={t}_{s}+\sqrt{{\tau }_{1}{\tau }_{2}},\end{eqnarray} \tag{ 2 }$

along with the pulse duration w and the pulse asymmetry κ. As a result of the rapid smooth rise and fall of the pulse model, w and κ are measured relative to some fraction of the peak intensity. Using the fraction previously described (Hakkila & Preece 2011) as I_meas/I_peak = e⁻³ (corresponding to 4.98% I_peak),

$\begin{eqnarray}&&w={\tau }_{2}{[9+12\mu ]}^{1/2},\end{eqnarray} \tag{ 3 }$

where $\mu =\sqrt{{\tau }_{1}/{\tau }_{2}}$, and

$\begin{eqnarray}&&\kappa \equiv {[1+4\mu /3]}^{-1/2}.\end{eqnarray} \tag{ 4 }$

Asymmetries range from symmetric (characterized by κ = 0) to asymmetric having longer decay times than rise times (0 < κ ≤ 1). The Norris et al. (2005) pulse model cannot physically describe pulses in which asymmetries are characterized by longer rise than decay times, but it provides a good first-order fit to BATSE, Fermi GBM, and Swift pulses.

Residuals to the Norris et al. (2005) model can be produced by subtracting each best-fit model from an observed pulse light curve. Small yet distinct deviations in the residuals are found to be systematically in phase with the light curve (Hakkila & Preece 2014), and these deviations are needed to accurately describe GRB pulse shapes. Although the deviations are closely aligned with the pulse duration, they are not always contained within it. Thus, we have defined the larger fiducial time interval w_fid as

$\begin{eqnarray}&&{w}_{\mathrm{fid}}={\tau }_{\mathrm{end}}-{\tau }_{\mathrm{start}}=4.4{\tau }_{2}(\sqrt{1+\mu /2}+1)+\sqrt{{\tau }_{1}{\tau }_{2}},\end{eqnarray} \tag{ 5 }$

with the fiducial end time t_end given by

$\begin{eqnarray}&&{t}_{\mathrm{end}}=\displaystyle \frac{w}{2}(1+\kappa )+{t}_{s}+{\tau }_{\mathrm{peak}}\end{eqnarray} \tag{ 6 }$

and the fiducial start time t_start given by

$\begin{eqnarray}&&{t}_{\mathrm{start}}={t}_{s}-0.1\left[\displaystyle \frac{w}{2}(1+\kappa )-{\tau }_{\mathrm{peak}}\right].\end{eqnarray} \tag{ 7 }$

The strange, wavelike pattern of the residual variations can be fitted with an empirical function (Hakkila & Preece 2014):

$\begin{eqnarray}&&\mathrm{res}(t)\,=\\ \left\{\begin{array}{ll}{{aJ}}_{0}(\sqrt{{\rm{\Omega }}[{t}_{0}-t-0.005]}) & \mathrm{if}\ t\lt {t}_{0}-0.005\\ a & \mathrm{if}\ {t}_{0}-0.005\leqslant t\leqslant {t}_{0}+0.005\\ {{aJ}}_{0}(\sqrt{s{\rm{\Omega }}[t-{t}_{0}-0.005]}) & \mathrm{if}\ t\gt {t}_{0}+0.005.\end{array}\right.\end{eqnarray} \tag{ 8 }$

Here J₀(x) is an integer Bessel function of the first kind, t₀ is the time of the residual peak (measured from the trigger time), a is the amplitude of the residual peak, Ω is the Bessel function's angular frequency that defines the timescales of the residual wave (a large Ω corresponds to a rapid rise and fall), and s is a scaling factor that relates the fraction of time that the function before t₀ has been compressed relative to its time-inverted form after t₀. The time during which the pulse intensity is a maximum is required to be a plateau instead of a peak, with a duration of w_plateau ≈ 0.010w_fid. Since there is no evidence in the pulse shape that the Bessel function continues beyond the third zero (following the second half-wave), the function is truncated at the third zeros J₀ (x = ±8.654).

The fiducial values can be converted back to values in the measured time interval using

$\begin{eqnarray}&&{a}_{\mathrm{meas}}=a\end{eqnarray} \tag{ 9 }$

$\begin{eqnarray}&&{s}_{\mathrm{meas}}=s\end{eqnarray} \tag{ 10 }$

$\begin{eqnarray}&&{t}_{0;\mathrm{meas}}={t}_{0}({t}_{\mathrm{end}}-{t}_{\mathrm{start}})+{t}_{\mathrm{start}}\end{eqnarray} \tag{ 11 }$

and

$\begin{eqnarray}&&{{\rm{\Omega }}}_{\mathrm{meas}}={\rm{\Omega }}/({t}_{\mathrm{end}}-{t}_{\mathrm{start}}),\end{eqnarray} \tag{ 12 }$

where t_start and t_end are the real time values corresponding to the start and end of the fiducial duration.

A convenient way to describe the pulse residual amplitudes is to normalize them to the pulse fit amplitudes, producing an intensity quantity that is independent of the instrument's signal-to-noise ratio (S/N). The relative amplitude R is given by

$\begin{eqnarray}&&R=a/A.\end{eqnarray} \tag{ 13 }$

The 4 ms S/N is a measure of the peak brightness of each TTE GRB, relative to its background count measured with 4 ms temporal resolution. The S/N is

$\begin{eqnarray}&&{\rm{S}}/{\rm{N}}=({P}_{4}-B)/{\sqrt{P}}_{4},\end{eqnarray} \tag{ 14 }$

where P₄ is the 4 ms peak count and B is the mean background count. This S/N is primarily appropriate when analyzing GRB pulses fit on the 4 ms timescale.

The technique we use for extracting pulses from TTE light curves is a modification of that described previously (e.g., Hakkila & Preece 2014 and references therein). This is because we have changed our expectations about the monotonic nature of GRB pulses based on our previous analyses. The 4 ms TTE light curves we are using generally exhibit clearly defined, isolated emission episodes, and many of these episodes appear to have shapes that are consistent with those identified for long/intermediate GRB pulses. Our a priori expectation of multiple peaks, rather than of strict monotonicity, allows us to hypothesize that every emission episode contains a potential pulse, with light curves that might be improved with the addition of the Hakkila & Preece (2014) residual function. Overlapping pulses will be problematic, but these would have been problematic even if we had assumed that every bump in the light curve represented a monotonic, overlapping pulse. However, we can check our results for short GRB pulses against our prior results for long/intermediate GRB pulses as our analysis proceeds.

This GRB pulse analysis approach is more comprehensive and systematic than any of our previous studies involving BATSE, Swift, and Fermi GBM 64 ms data. Here we attempt to analyze all bursts that entirely or mostly fit in the TTE temporal window, in contrast to the prior selection criterion of only bursts that appear to be composed of single, isolated pulses (from the 64 ms studies). This TTE study thus depends primarily on the duration of the fitted pulse, and not on how easy it is to fit the light curve. This allows us to estimate completeness for our results.

The 2 s temporal limit of the BATSE TTE window suggests that most TTE bursts belong to the short GRB class. Rather than accepting this at face value using the "T₉₀ ≤ 2 s" assertion, we prefer to classify GRBs using a formal statistical clustering or data mining technique. As discussed previously (see Section 1), formal techniques commonly prefer three GRB classes over two. Since the assignment of any GRB to a specific class depends on the classification technique being used, the burst characteristics being assessed, and the data set being analyzed, we must choose a methodology for classifying the bursts in our sample. The three-class model obtained by Horváth et al. (2006), obtained through application of principal component analysis (PCA) to GRB duration, fluence, and hardness data (Bagoly et al. 1998; Balázs et al. 2003), provides a thoughtful systematic basis for classification. Unfortunately, Horváth et al. (2006) do not provide classifications for many of the BATSE GRBs in our sample.

We can extend the Horváth et al. (2006) results to our TTE data set using rules developed from supervised classification techniques. Our choice of a supervised classification algorithm is the decision tree C4.5 (Quinlan 1993), found in the WEKA freeware suite of data mining tools under the J48 implementation (Frank et al. 2016). We use the classification results of Horváth et al. (2006), combined with the durations, fluences, and spectral hardnesses of all bursts found in the BATSE Final Catalog (https://gammaray.msfc.nasa.gov/batse/grb/catalog/current/). J48 uses previously classified bursts (a training set) to identify simple IF THEN ELSE classification branch rules. Each IF THEN ELSE statement is a branch on this classification tree, and the terminal branches are referred to as leaves. A measure of entropy determines whether or not information is gained through the branching process. As it is being developed, a classification tree can be pruned if desired to eliminate sparsely filled leaves. Once a classification tree exists, the IF THEN ELSE rules can be used to classify unknown objects while assigning to each a probability that each has been placed in the appropriate class.

For this application of J48, we use the default parameter settings, which are known to generally provide a good performance (Frank et al. 2016). Pruning has been disabled because our goal is to extend rather than generalize the Horváth et al. (2006) classification scheme. However, in order to avoid having rules that apply to individual bursts, the minimum number of objects per leaf has been set to two.

The resulting J48 tree has 14 leaves developed from 1557 GRBs using the attributes of logarithmic duration, hardness (channel 3/channel 2), and fluence, with a 97.88% accuracy. The confusion matrix demonstrates the effectiveness of the resulting rule set: diagonal matrix elements indicate agreement between J48 and the original classification for 1524 long (L), short (S), and intermediate (I) GRBs, while off-diagonal elements show how the remaining 33 GRBs are misclassified.

The final J48 tree shows the rules for each leaf, with parentheses indicating the number of originally and reclassified GRBs placed in the leaf:

The long and short GRB classes are clearly identified at the ends of the duration distribution, as 97% (983/1009) of the long bursts have T₉₀ > 8.57 s and 68% (279/412) of the short bursts have T₉₀ ≤ 0.89 s. The rules for classifying short GRBs are more successful (97% of the time for 398/412) when spectral hardness is also included (119 additional short GRBs are characterized by 0.89 s < T₉₀ ≤ 2.41 s and hr > 2.03). However, both short and intermediate bursts are found in the time interval spanned by the TTE window (0.89 s < T₉₀ < 2 s).

We can assign probable class membership to each previously unclassified GRB using these J48 rules. Some bursts are more difficult to classify because they lack one or more of the classification attributes, so J48 assigns greater uncertainty to these classifications. However, bursts lacking one or more classification attribute are rare, so most of the probabilities that a burst belongs to a preferred class exceed 90%. As a result, we consider burst classification probabilities of less than 90% to represent questionable (denoted with a "?" in our catalog) class assignments.

The classification results, shown in Table 1, verify that most (≈90%) of the TTE GRBs belong to the short GRB class. Use of these classification values allows us to examine the characteristics of short GRB pulses with greater confidence than can be found by defining short GRBs only as those having T₉₀ < 2 s.

Ninety-one percent of the TTE complete bursts and 85% of the TTE partial bursts are in the short class. This difference results because TTE partial bursts (which do not necessarily fit within the TTE window) are typically longer than TTE complete bursts (which must fit within the TTE window). The large percentage of TTE partial bursts belonging to the short class suggests that our short GRB sample is incomplete: short bursts can be found with durations longer than the TTE window because at least some short GRBs are longer than can be accommodated by the 2 s cutoff.

We use an iterative, heuristic approach to GRB pulse classification in order to recognize our partial yet still incomplete understanding of GRB pulse behaviors. This approach allows us to find and account for pulse behaviors that we expect while also allowing us to search for behaviors that we may not anticipate. An approach of this type (part statistical, part data mining) is needed because GRB pulses do not act like pulses in the standard sense of the term.

The Oxford English Dictionary defines a pulse as "a single vibration or short burst of sound, electric current, light, or other wave." The standard assumption has been that that GRB pulses are monotonic structures overlaying stochastically varying backgrounds. Although this assumption has been applied almost universally in GRB pulse fitting, the residuals of isolated GRB pulses demonstrate that it is not always valid (Hakkila & Preece 2014; Hakkila et al. 2015). There are negative repercussions to the measurement of GRB pulse properties if pulse monotonicity is assumed but not present. The assumption of pulse monotonicity serves to fragment larger structures and replace them with separate monotonic pieces. Using a monotonic pulse model instead of one that accounts for possible structure thus results in the recovery of more pulses, having shorter durations, and separated by smaller separations. These pieces cannot themselves have structure because any nonstochastic structure will be fragmented into smaller pulses by the monotonicity assumption. Important temporal correlations, such as the hard-to-soft spectral evolution characteristic of triple-peaked pulses, will go unrecognized and will be replaced by the less pronounced spectral characteristics of individual monotonic pulse pieces.

The Oxford English Dictionary defines complexity as "the state or quality of being intricate or complicated." Complexity is used rather vaguely in GRB analysis to define structure in GRB light curves. We have demonstrated (Hakkila & Preece 2014; Hakkila et al. 2015) that triple-peaked residual functions are responsible for one component of GRB pulse complexity. Because of this, we are open to the idea that other, more structured variations might also be present and extractable from GRB pulses. A useful technique for classifying GRB pulses is thus one that recognizes known pulse complexities while presuming that other unknown complexities might also exist.

GRB pulse light curves of long and intermediate bursts exhibit various degrees of structure and complexity. The smooth Norris et al. (2005) pulse model works best at fitting faint long and intermediate GRB pulses; brighter pulses show evidence of more complex structures such as the triple-peaked residual function (Hakkila & Preece 2014; Hakkila et al. 2015). The existence of this nonmonotonically increasing and decreasing complexity is problematic, as pulse-fitting techniques sometimes have difficulty determining whether complexity in a GRB emission episode results from identifiable substructures or if it represents embedded fainter pulses. To complicate matters, bright emission episodes often exhibit additional chaotic variations not present in faint ones.

We note that some TTE bursts are characterized by what appear to be closely overlapping emission episodes. These are the most difficult events for us to fit because they could either represent two or more overlapping pulses or a single pulse that has an exceedingly complex temporal structure. We recognize that there will always be ambiguity in separating multipulsed emission episodes from multipeaked pulses, and we have made efforts to adequately document these ambiguous cases. It is fortunate for our analysis that the emission episodes of most TTE GRBs are clearly defined.

We approach the identification of GRB pulse complexity by assuming to first order that each isolated emission episode represents a single GRB pulse that can be fitted by the Norris et al. (2005) model. We further assume that the simplest form of complexity, representing a second-order variation in the monotonic pulse structure, is the smoothly varying triple-peaked Hakkila & Preece (2014) residual function. We use ${\chi }_{\nu }^{2}$ as our goodness-of-fit statistic to determine the effectiveness of these models, where χ² indicates the normalized deviation between the data and the model relative to the degrees of freedom ν. The value of ${\chi }_{\nu }^{2}$ is dependent on the amount of nonstochastic emission (signal) relative to the amount of stochastic emission (noise) and thus to the temporal interval selected for the analysis. The number of bins (and thus the value of ν) is sensitive to the bin size, as well as to the temporal interval. We choose the fiducial timescale for pulse analysis because it has been defined to contain most of the pulse emission.

Our previous analyses of long/intermediate BATSE, Fermi GBM, and Swift GRB pulses have found that this approach generally results in fits having ${\chi }_{\nu }^{2}\approx 1$ in low-S/N environments consistent with stochastic variations to the model. The model is less successful for pulses showing some complexity. Many of these less well fit pulses exhibit larger precursor and/or decay peaks or additional faint residual structures that occur in addition to the residual fit. Because the Hakkila & Preece (2014) function still contributes to these pulse residuals, these findings suggest that the model is not incorrect so much as it is incomplete in accounting for augmented pulse structures.

We choose to define pulse complexity in terms of the ${\chi }_{\nu }^{2}$ p-values. Here χ² is defined over the fiducial timescale, and ν from the Norris et al. (2005) model is the number of temporal bins minus the number of pulse- and background-fit parameters—two for the background and four for a single pulse. The p-value associated with ${\chi }_{\nu }^{2}$ has the conventional meaning: it is the probability that a χ² statistic having ν degrees of freedom is more extreme than the measured value. As will be seen, the binning most appropriate for fitting a GRB pulse depends on the pulse's flux distribution: an optimal fit has sufficient S/N over the duration of the pulse to identify and match available structures. The burst sample is limited to 64 ms resolution outside the TTE window, which makes it difficult to fit shorter TTE partial pulses and limits the number of bins being fitted. Inside the TTE window we have chosen to use 4 ms resolution. The number of bins available for a pulse fit depends on a pulse's duration and shape, as these quantities define the fiducial time interval. Thus, the optimal choice of a bin size is better made after a fit has been performed.

By limiting our comparisons to this fiducial interval, we minimize the chance that a good fit is obtained simply because it uses a large number of background bins, and this approach assures that the χ² fits are generally independent of pulse duration (exceptions can occur for fiducial timescales that extend beyond the TTE window; this can have the unfortunate effect of artificially increasing ${\chi }_{\nu }^{2}$ by decreasing the number of available background bins). We consider good fits (indicating relatively smooth light curves) to be those having best-fit p-values of p_best ≥ 5 × 10⁻³ (this is a standard choice for a "good" fit criterion). A Δχ² test is used to indicate that the residual function needs to be included in the fit: Δχ² is the difference in χ² obtained from the Norris et al. (2005) model minus that obtained from the Norris et al. (2005) model combined with the Hakkila & Preece (2014) residual model. The difference in the number of degrees of freedom between these fits is four per pulse. We require a Δχ² p-value of p_Δ ≤ 10⁻³ for the model to be improved. This is more stringent than the criterion we use for a "good" fit because we want to ensure that the residual function, rather than some other structure, is most likely to be responsible for the improvement in the fit.

We have classified the TTE pulses into four groups using complexity as our classification parameter. We identify simple pulses as those best characterized by the Norris et al. (2005) function alone. Blended pulses are explained by the Norris et al. (2005) function but also require the Hakkila & Preece (2014) residual function to obtain a best-fit value of p_best ≥ 5 × 10⁻³. Many pulse fits are significantly improved by the Hakkila & Preece (2014) residual function but exhibit additional unmodeled structures. This may in part result from the inability of our empirical models to explain the true evolution of the light curve. We define structured pulses as those having best-fit p-values of 10⁻⁵ ≤ p_best < 5 × 10⁻³; these pulses have many characteristics that can be explained by the pulse and residual models, but they also have statistically significant variations from these structures. The value p = 10⁻⁵ appears somewhat arbitrary but has been selected for its potential use as a data mining attribute. This p-value subdivides complex pulses into two groups, such that (1) structured pulses are expected to have characteristics common with blended pulses, but also (perhaps to a lesser extent) with more complex pulses, and (2) the number of structured pulses in our sample (50) is similar to the number of blended pulses (38). The remaining pulses have complicated light curves as defined by p_best < 10⁻⁵. Some of these complex pulses represent emission episodes having pronounced structures that might represent complex substructures, overlapping inseparable pulses, or a different physical phenomenon altogether. Although we have classified the TTE pulses according to these p_best values, the BATSE TTE GRB pulse catalog includes all p-values so that users may adjust these classification parameters as they wish.

In addition to finding some TTE emission episodes that upon reexamination appear to be overlapping pulses, we have found a small number that do not fit the existing pulse paradigm. The final pulses in these bursts are characterized by intensities that increase with time, producing asymmetric pulse shapes that are contrary to the intensity distribution function of Norris et al. (2005). We call bursts containing these pulse structures crescendo bursts. Some, but not all, crescendo bursts are preceded by a series of short, symmetric staccato pulses. Crescendo GRBs are discussed in greater depth in Section 4.8.

Two GRBs in this sample (BATSE triggers 1626 and 7427) have been identified by Norris & Bonnell (2006) as being short GRBs with extended emission. The residuals of both of these bursts show faint emission indicative of a brightening followed by a gradual decline, which is suggestive of a faint pulse structure rather than chaotic emission. However, the S/N of this extended emission is too faint to attempt to fit with the pulse model. An additional eight short GRBs (four TTE complete and four TTE partial) are listed in Bostancı et al. (2013) as having extended emission (TTE complete triggers 575, 1719, 5592, and 5634; TTE partial triggers 3611, 3940, 7063, and 7599). Of these bursts, only trigger 575 exhibits extended emission during the TTE readout. We note that the double-pulsed nature of BATSE trigger 575 also makes it unique among the sample of short GRBs with extended emission.

We have made the light curves obtained for the residuals and combined plots available using online figure sets. The residual light curves are shown in Figure 1, while the combined light curves (pulse plus residuals) are shown in Figure 2.

Examples of representative pulse fits are shown in Figures 2–5. Figure 2 shows an example of a simple pulse (trigger 373); the left panel shows the residual structure, while the right panel shows both the Norris et al. (2005) fit (dotted line) and the combined fit (solid line). Figure 3 similarly shows a blended pulse (trigger 2896), Figure 4 shows a structured pulse (trigger 5564), and Figure 5 shows a complex pulse (trigger 4955).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

View figure set (387 panels)

Tarball of figure set images

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The complexity classifications of all 434 pulses in our catalog are summarized in Table 2. Although the total numbers of TTE complete and TTE partial pulses are similar, the distributions of simple, blended, structured, and complex pulses are noticeably different. This is a direct result of the better temporal resolution of the 4 ms data relative to the 64 ms data. Whereas the 4 ms binning allows substructures (including separable pulses and the residual function) to be identified and fitted, the 64 ms binning merges these together and prevents them from being properly delineated for fitting. Thus, the lower-resolution partial TTE group contains more structured and complex pulses and fewer simple and blended pulses. Examination of the TTE data for the bursts in this group shows that these structures and separable pulses are present, but just not resolved in the 64 ms data (e.g., see the pulses in Figure 1).

The vast majority of the catalog bursts (345/387 = 89%) are single-pulsed. Most of the remainder are double-pulsed (33/387 = 9%), and a few (3/387 = 1%) are triple-pulsed. Six (6/387 = 2%) are crescendo bursts, having pulse structures similar to one another but different than that described by the standard pulse paradigm. We have likely underestimated the number of very short multipulsed GRBs having overlapping pulses because temporal resolution and pulse complexity make it hard to disentangle overlapping TTE pulses. On the other hand, our attempt to fit all sampled TTE pulses gives us confidence that we have identified most of the multipulsed TTE GRBs. Table 3 summarizes the multiplicity of TTE GRBs in this sample. A list of the multipulsed GRBs is provided in Table 4.

Although 90% of the catalog bursts are single-pulsed, we cannot unequivocally state that 90% of short GRBs are single-pulsed. Because our sample contains only short GRBs that entirely or mostly fit into the TTE window, we are missing the long-duration tail of the short GRB distribution (e.g., those with T₉₀ > 2 s), and these missing bursts might well contain multiple pulses. However, we also believe that this long-duration tail is small because short bursts are separated from the other GRB classes with the most clearly delineated boundary (see Horváth 1998; Mukherjee et al. 1998; Hakkila et al. 2003), and many bursts with T₉₀ > 2 s have the soft spectra of intermediate GRBs. Thus, we feel confident in stating that short GRBs are overwhelmingly single-pulsed.

The BATSE TTE GRB pulse catalog is contained in three separate online files. Part I (Table 5) contains information related to the Norris et al. (2005) model fit. Part II (Table 6) contains information pertaining to the Hakkila & Preece (2014) residual fit (if available), descriptions of the overall pulse fit, and ancillary information such as pulse fluence and energy hardness. Part III (Table 7) contains the names of the files containing both the residual fits and the total fits to the pulse light curves, as well as comments about the pulses. These three files may be merged by the user to create a larger table, if desired.

Table 5.  Information Contained in the BATSE TTE GRB Pulse Catalog (Part I)

Table Column	Header	Variable	Units	Description
1	pulse_id	BATSE trigger	...	number (+ letter)
2	resolution	resolution	...	4 ms for TTE complete or 64 ms for TTE partial
3	B	B0	counts	mean background per bin
4	B_err	σB0	counts	mean background uncertainty per bin
5	BS	BS	counts s–1	background rate change per bin
6	BS_err	σBS	counts s–1	background rate change uncertainty per bin
7	ts	ts	s	pulse start time, from Equation (1)
8	ts_err	σts	s	pulse start time uncertainty
9	A	A	counts	pulse amplitude, from Equation (1)
10	A_err	σA	counts	pulse amplitude uncertainty
11	tau1	τ1	s	pulse rise parameter, from Equation (1)
12	tau1_err	στ1	s	pulse rise parameter uncertainty
13	tau2	τ2	s	pulse decay parameter, from Equation (1)
14	tau2_err	στ2	s	pulse decay parameter uncertainty
15	w	w	s	pulse duration, from Equation (3)
16	w_err	σw	s	pulse duration uncertainty
17	kappa	κ	...	pulse asymmetry, from Equation (4)
18	kappa_err	σκ	...	pulse asymmetry uncertainty
19	tau_pk	τpeak	s	pulse peak time, from Equation (2)
20	tau_pk_err	στpk	s	pulse peak pulse time uncertainty
21	t_start	tstart	s	fiducial start time, from Equation (7)
22	t_end	tend	s	fiducial end time, from Equation (6)
23	chi∧2	χ2	...	goodness of fit for pulse + background model
24	nu	ν	...	degrees of freedom for pulse + background model
25	chi∧2_nu	${\chi }_{\nu }^{2}$	...	reduced goodness of fit for pulse + background model

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

Download table as:  DataTypeset image

Table 6.  Information Contained in the BATSE TTE GRB Pulse Catalog (Part II)

Table Column	Header	Variable	Units	Description
1	pulse_id	BATSE trigger	...	number (+ letter)
2	t0	t0	s	residual peak time, from Equation (5)
3	t0_err	σt0	s	residual peak time uncertainty
4	a	a	counts	residual amplitude, from Equation (5)
5	a_err	σa	counts	residual amplitude uncertainty
6	omega	ω	s−1	residual Bessel frequency, from Equation (5)
7	omega_err	σω	s−1	residual Bessel frequency uncertainty
8	s	s	...	Bessel function stretching parameter, from Equation (5)
9	s_err	σs	...	Bessel function stretching parameter uncertainty
10	chi∧2	χ2	...	goodness of fit for pulse + residual + background model
11	nu	ν	...	degrees of freedom for pulse + residual + background model
12	chi∧2_nu	${\chi }_{\nu }^{2}$	...	reduced goodness of fit for pulse + residual + background model
13	delta_chi∧2	Δχ2	...	goodness-of-fit improvement from residual model
14	delta_nu	Δν	...	difference in degrees of freedom
15	p_delta	pΔ	...	p-value of model improvement
16	include	include	...	"x" to include residuals, "o" to exclude, based on pΔ
17	R	R	...	ratio of A/a, from Equation (13)
18	4 ms_pk_cts	4 ms peak counts	counts	measured peak counts per bin
19	4 ms_S/N	4 ms S/N	...	S/N from Equation (14)
20	burst_class	burst class	...	GRB class from Section 3.1 and summarized in Table 1
21	p_best	pbest	...	best-fit p-value
22	pulse_class	pulse class	...	pulse classification, described in Section 3.2
23	S	S	erg cm−2	energy fluence from BATSE catalog and pulse fits
24	S_err	σS	erg cm−2	energy fluence uncertainty
25	HR	HR	...	energy hardness from Equation (15)
26	HR_err	σHR	...	energy hardness uncertainty

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

Download table as:  DataTypeset image

As indicated previously, this catalog strives to systematically create a characterization of short GRB prompt emission by attempting to fit all short GRB pulses that fit entirely or partially within BATSE's TTE window. We want to understand and characterize as many of our selection biases as we can. We believe that the selection of TTE bursts is random during the time of BATSE's operation. We have attempted to describe uncertainties in the pulse-fitting process, as well as characterizing uncertainties in the measured pulse parameters. We have created four descriptors based on our definitions of good fits: the temporal resolution (Table 5, Column (2)), the pulse complexity (Table 6, Column (22)), the GRB class (Table 6, Column (20)), and the decision whether or not to include the residual fit in the final best fit (Table 6, Column (16)).

The comments provided in Table 7 describe a variety of characteristics that are not covered in the main catalog. These include (a) instrumental reasons why pulses have been defined as TTE partial rather than TTE complete, (b) augmented descriptions of pulse characteristics, and (c) burst or pulse properties measured elsewhere. In Table 7 we have delineated some of these complex morphologies using visual descriptions rather than formal statistical ones:

• 1.  
  Crown pulses consist of a clearly defined single emission episode exhibiting many small peaks around the time of maximum emission;
• 2.  
  u-pulses have u-shaped or bowl-shaped double-peaked light curves characterized by short temporal spikes at the beginning and at the end of the main emission episode, and sometimes they also have a spike at the center of the pulse;
• 3.  
  noisy double-peaked pulses have asymmetric light curves with abnormally bright and long decay peaks;
• 4.  
  twin peaks indicate single emission episodes having two closely separated peaks overriding the main emission.

We note that some crown pulses and u-pulses have similar morphologies; it is possible that some crown pulses are merely unresolved u-pulses.

Pulse energy fluences and hardnesses have been provided in the BATSE Final Catalog (https://gammaray.msfc.nasa.gov/batse/grb/catalog/current/) and are available for almost all of the TTE GRBs (a few have been obtained from Goldstein et al. (2013) and are identified in Table 7). For this analysis we define energy hardness as

$\begin{eqnarray}&&\mathrm{HR}=({S}_{3}+{S}_{4})/({S}_{1}+{S}_{2}),\end{eqnarray} \tag{ 15 }$

where S_n refers to the fluence in BATSE energy channel n.

The BATSE catalog publishes fluences and hardnesses for bursts, not for pulses. However, as discussed in Section 3.3, most TTE bursts appear to contain only a single emission episode, and only a few have recognizable extended emission. Thus, pulse fluences and hardnesses are generally the same as the fluences and hardnesses of the bursts in which they are found. When bursts consist of multiple pulses, modeled pulse fits are used to extract both pulse durations and energy-dependent counts fluences of constituent pulses (see, e.g., Hakkila & Preece 2011). The counts fluences are then combined with BATSE catalog data to obtain energy fluences and hardnesses for individual pulses. This modeling approach seems to produce reasonably accurate measurements of pulse duration even if pulses contain considerable structure. In other words, large χ² uncertainty results more from poor matches to pulse structure than from difficulties in measuring pulse boundaries. The efficacy of the approach can be seen, for example, in the fits shown in Figures 2 through 6.

Zoom In Zoom Out Reset image size

Using the GRB class definitions from Section 3.1, we find that short GRB pulse durations, fluences, and hardnesses correlate with one another in manners consistent with those described in Hakkila & Preece (2011) for long and intermediate GRB pulses. As expected, pulses with longer durations have correspondingly larger fluences (Figure 7, left panel). Similarly, harder pulses have correspondingly larger fluences, as high-energy photons contain significantly more energy than low-energy photons (Figure 7, right panel). A Spearman rank-order correlation test (SC) indicates that pulse fluence and duration are highly correlated (SC = −0.31, p = 6 × 10⁻¹⁰) and that hardness and fluence are even more highly correlated (SC = 0.58, p = 7 × 10⁻³⁶). However, hardness and duration are uncorrelated (SC = 0.46, p = 0.37).

Zoom In Zoom Out Reset image size

The amplitudes of short GRB pulses are related to their durations and fluences (Figure 8). A Spearman rank-order correlation indicates that short GRB pulse amplitude and duration are highly anticorrelated (SC = −0.49, p = 6 × 10⁻²⁵), as are amplitude and fluence (SC = 0.31, p = 9 × 10⁻¹⁰). This relationship has been found previously for short GRBs (Hakkila & Preece 2011; Norris et al. 2011) and is itself an extension of the pulse amplitude versus duration anticorrelation found by Hakkila et al. (2008) for GRB prompt emission and known to extend to X-ray flares (Margutti et al. 2010). Using measurements of maximum count rates divided by minimum count rates on three different timescales, and comparing these measurements with pulse durations, Hakkila & Preece (2011) have demonstrated that this effect is real rather than due to a selection bias. In addition to duration and fluence, pulse hardness and amplitude (Figure 9) appear to be weakly correlated (SC = 0.11, p = 3 × 10⁻²) in short GRB pulses.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The short TTE complete and short TTE partial samples are not representative of the same underlying population owing to a sampling bias. Because TTE complete pulses fit completely within the TTE window whereas TTE partial pulses do not, TTE complete pulses tend to have shorter durations and smaller fluences than TTE partial bursts (Figure 10), as indicated by Student's T-tests (T) comparing the logarithmic distributions of duration (T = −12.0, p = 2 × 10⁻²⁸) and fluence (T = −3.6, p = 4 × 10⁻⁴) for short GRBs. As a result of the aforementioned anticorrelation between amplitude and duration, the difference between the TTE complete and TTE partial duration distributions reflects a difference between the amplitude distributions. This can be seen in Figure 11 and in the Student's T-test results (T = 5.8, p = 10⁻⁸). A similar result has been previously identified by Norris et al. (2011) for short Swift GRB pulses.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

These results demonstrate that the overall distribution of short GRB pulse properties cannot be described using TTE complete pulses alone. Instead, the different properties of TTE partial pulses must be included. Even when these have been included, the combined TTE sample is not entirely representative of the underlying short GRB pulse distribution. Only a small fraction of bursts in this sample have been formally classified as long/intermediate GRBs, suggesting that the long-duration end of the short GRB sample cannot be sampled by the durations of TTE windows.

The pulse classes defined in Section 3.2 in terms of complexity represent a continuum of characteristics; our definition of four discrete groups is somewhat arbitrary. Figure 12 demonstrates where the simple (crosses), blended (asterisks), structured (diamonds), and complex (triangles) groups are found in terms of their fit improvement by the residual function (p_Δ) and their final best fit (p_best) for both the TTE complete (blue) and TTE partial (red) samples. The vast majority of the pulses lie in the upper right corner (large best-fit p-values and large Δχ² p-values); these simple pulses are well characterized using only the Norris et al. (2005) pulse model. Blended TTE complete pulses along the top of the graph (large best-fit p-values but small Δχ² p-values) are best fit by the Norris et al. (2005) model combined with the Hakkila & Preece (2014) residual model. Structured TTE complete pulses (small best-fit p-values) are too complex to be completely characterized using combinations of the Norris et al. (2005) and Hakkila & Preece (2014) pulse models, but there is a gradual change in complexity from blended, to structured, to complex pulses. Many TTE partial pulse light curves cannot adequately be explained by the Norris et al. (2005) pulse function alone, but the temporal binning of these light curves provided too few data points for any residual structure to be identified.

Zoom In Zoom Out Reset image size

Eighty-five pulses have been fitted using both 4 ms and 64 ms data. Although 64 ms data provide inadequate temporal resolution for fitting most residuals (described previously), the bulk observable Norris et al. (2005) pulse properties (amplitude A, duration w, and asymmetry κ) have been measured using data from both timescales. These properties can be directly compared to provide an internal check on the reliability of the pulse-fitting process and also to provide insights into the measurement uncertainties of these properties as determined by the MPFIT.PRO nonlinear least-squares routine (Markwardt 2009).

The 64 ms timescale pulse amplitudes (A64) are compared to their 4 ms pulse counterparts (A4) in the left panel of Figure 13. The majority of these amplitudes are highly correlated, and a Spearman rank-order correlation test indicates that these amplitude measurements are highly correlated (SC = 0.77, p = 6 × 10⁻¹⁸). However, a few pulse amplitudes are found to differ systematically by large amounts, while a few others have abnormally large statistical uncertainties. The systematically different measurements all have uncertainties of σ_A = 0, and all but one of these values have been measured on the 64 ms timescale. In fact, both the large and small uncertainties are associated with large-amplitude pulses having short durations relative to the bin size. We conclude that the nonlinear least-squares routine has difficulty converging at the intensity inflection point when a limited number of data points are present to describe large intensity variations. Upon excluding 32 pulses having either very large (σ_A64 ≥ 5 × A₆₄ and σ_A4 ≥ 5 × A₄) or small (σ_A64 = 0 and σ_A4 = 0) measurement uncertainties, the expected relationship is recovered and shown in the right panel of Figure 13. Although most of these measurements are consistent with unity, many 64 ms amplitudes are slightly smaller than their 4 ms counterparts. This results because the 64 ms binning washes out some of the 4 ms pulse structure.

Zoom In Zoom Out Reset image size

The 64 ms timescale pulse durations (w₆₄) are compared to their 4 ms pulse counterparts (w₄) in the left panel of Figure 14. The duration measurements are highly correlated (SC = 0.92, p = 7 × 10⁻³⁶), even though the individual measurement uncertainties are large. In other words, there do not appear to be systematic differences between durations measurements made using fits on different timescales. As expected, relative uncertainties (σ_w/w) increase as pulse durations approach the temporal resolution. Limiting the sample to durations measured accurately on both timescales demonstrates the consistency of fitting on the two different timescales; this is shown in the right panel of Figure 14 for 42 pulses having σ_w ≤ w.

Zoom In Zoom Out Reset image size

Finally, the 64 ms pulse asymmetries (κ₆₄) are compared to their 4 ms counterparts (κ₄) in the left panel of Figure 15. Figure 15 demonstrates that asymmetry measurements are difficult to make on the short GRB timescales, as large uncertainties accompany the measurements for many of the pulses. However, a Spearman rank-order correlation test indicates that the asymmetry measurements are weakly correlated (SC = 0.24, p = 0.07). This correlation can be clarified by limiting the sample to accurately measured pulses. The right panel of Figure 15 shows that both 64 ms and 4 ms resolutions measure similar asymmetries when the sample is limited to those pulses having accurate measurements (σ_κ ≤ 0.15).

Zoom In Zoom Out Reset image size

It appears that the formal fitting process has led to overestimates of many uncertainties for σ_w and σ_κ. Through inspection it appears that many ${\sigma }_{{\tau }_{\mathrm{pk}}}$ measurements have also been overestimated. The uncertainties for these observables were propagated from the fitted values of σ_A, σ_ts, σ_tau1, and σ_tau2. Although the fitted variables are generally not observable (with the exception of A), the uncertainties in the measurement of many of these variables also seem to be inordinately large.

The mathematical expression describing the Norris et al. (2005) intensity model (Equation (1)) has several characteristics that make it difficult to fit. The largest signal exists at time τ_peak, when the pulse intensity equals the amplitude A and where exponentially increasing and decreasing intensity functions involving τ₁ and τ₂ are joined. The start and end of the pulse provide few additional helpful fitting constraints: t_s occurs when the pulse rise intensity equals the background at the beginning of the pulse (t_s prevents the intensity function from going to infinity prior to the pulse's beginning), the exponential rise τ₁ determines how fast the intensity increases from t_s, and the exponential decay of τ₂ describes the rate of intensity decrease while ensuring that this intensity will never quite reach the background. The interplay between t_s and τ₁ constrains the pulse rise, while the interplay between τ₁, τ₂, and A constrains the pulse peak. These pulse parameters are harder to fit when the temporal resolution is poor, as there are fewer intensity points available with which to describe the intensity function.

Very large and very small values of τ₁ and τ₂ are particularly hard to constrain. A small τ₁ value indicates a slow pulse rise, while a large value indicates a rapid rise producing a correspondingly early pulse start time t_s. A small τ₂ value indicates a rapid pulse decay, while a large τ₂ value indicates a slow pulse decay. Pulse fits resulting in these large and/or small τ₁ and τ₂ values are often accompanied by fitting uncertainties exceeding the measured value by an order of magnitude or more. This most often happens in pulses that are short relative to the temporal resolution, as these occur where the rates of intensity rise and fall are masked by the temporal bin size.

When the τ₁ rise and τ₂ decay components of this function are well behaved (10⁻³ ⪅ τ₁ ⪅ 10³ and 10⁻³ ⪅ τ₂ ⪅ 10³), smoothly varying functions result and the τ₁ and τ₂ distributions seem to be normally distributed. Very large or very small pulse rise and decay values produce uncertainty distributions that appear to be asymmetric.

Increased temporal resolution improves the quality of both the pulse fits and measured pulse parameters. For this reason, the formal pulse-fitting parameters obtained from the 4 ms TTE complete sample have smaller formal uncertainties than their 64 ms TTE partial counterparts. However, constraints are present for all pulses in the TTE pulse catalog as a result of poor counting statistics: the higher TTE resolution results in fewer counts per bin, which provides its own limits on pulse property measurement.

The formal duration errors obtained from MPFIT can be compared to the internal error distribution taken from the differences between w₄ and w₆₄. In other words,

$\begin{eqnarray}&&{\sigma }_{w-\mathrm{internal}}=| {w}_{4}-{w}_{64}| /\sqrt{2}.\end{eqnarray} \tag{ 16 }$

We find that the internal and external duration error distributions are consistent with one another such that

$\begin{eqnarray}&&{\sigma }_{w-\mathrm{internal}}^{2}\approx {\sigma }_{w4}^{2}+{\sigma }_{w64}^{2}\end{eqnarray} \tag{ 17 }$

upon excluding pulses with poorly measured durations (σ_w4 ≥ 10 s and σ_w64 ≥ 10 s). It should also be noted that duration uncertainties increase for faint pulses (as measured both by fluence and by peak flux). This is not surprising, as the duration definition (given in Equation (3)) is dependent on intensity.

Some pulse complexity appears to result from a selection bias stemming from inadequate temporal resolution; this can be found by examining the different numbers of events in each of the pulse complexity classes (see Table 2). Far more TTE complete pulses can be characterized by the Norris et al. (2005) pulse function plus the Hakkila & Preece (2014) residual function than TTE incomplete pulses. In other words, poor temporal resolution appears to have created false pulse structures by rebinning and smearing out the known triple-peaked pulse characteristics.

Once we exclude TTE partial pulses from our sample, we find that bright GRB pulses tend to have more complex structures than faint pulses, in agreement with previous results obtained for long/intermediate GRB pulses (Hakkila & Preece 2014; Hakkila et al. 2015). We characterize the TTE complete sample by a 4 ms definition of S/N (see Equation (14)). Figure 16 demonstrates that pulse complexity (characterized by the best-fit p-value p_best) increases as S/N increases. The correlation between these characteristics is significant (a Spearman rank-order correlation analysis finds SC = 0.48, p = 6 × 10⁻¹⁵). Figure 16 shows that simple and blended pulses are the faintest, structured pulses are brighter, and complex pulses are the brightest. We draw several conclusions from Figure 16:

• 1.  
  Short GRB pulses, like their long and intermediate burst counterparts, exhibit a triple-peaked structure.
• 2.  
  A smaller percentage of short GRB pulses seem to exhibit measurable residual structure (blended and structured) compared to long/intermediate GRBs, although it is difficult to imagine what a complete sample of long/intermediate burst pulses should look like based on existing analyses.
• 3.  
  The triple-peaked structure is less pronounced for low-S/N GRB pulses (see Figure 16), suggesting a sampling bias by which structure might be present in most or all pulses but cannot be resolved with low photon counts.
• 4.  
  More pronounced pulse structures are observed at high S/N (as denoted by the relative number of structured and complex pulses), suggesting that most or all GRB pulses contain complex structures, but these also are washed out at low S/N.

Zoom In Zoom Out Reset image size

GRB pulses are difficult to resolve and to fit at low S/N, resulting in less certain measurements of their properties relative to bright pulses. This can have the undesired effect of altering pulse properties near the S/N threshold. Figure 17 demonstrates that faint TTE pulse properties do indeed differ from those of bright pulses, as measured by R (the ratio of the residual fit amplitude to the pulse fit amplitude; see Equation (13)) relative to S/N. We find the following:

• 1.  
  Pulses characterized by large complexities (structured and complex) are observed at larger S/N than those having simpler structures (simple and blended). See Table 8.
• 2.  
  Pulses with large residual structures (R > 0.8) are primarily found near the minimum S/N threshold. See Figure 17.
• 3.  
  Pulses observed at the largest S/N have the smallest measured R values. See Figure 17.

Zoom In Zoom Out Reset image size

Table 8.  TTE S/N of TTE Pulses by Complexity

Pulse Complexity	$\langle {\rm{S}}/{\rm{N}}\rangle$	σS/N
Simple	5.8	2.7
Blended	7.3	3.3
Structured	8.5	4.0
Complex	10.9	5.1

Download table as:  ASCIITypeset image

The light curves of bright TTE pulses exhibit more pronounced structural complexity than the smooth light curves of fainter pulses. Some of this can be explained by the simple observation that noise is capable of washing out pre-existing pulse structures and making pulse light curves look smoother. However, the large S/N range spanned by pulses suggests that there might also be an intrinsic effect such that bright pulses exhibit larger temporal variabilities than faint ones. This can only be explained if bright pulses are also more luminous than faint ones. Such a conclusion is consistent if a pulse lag versus pulse luminosity relationship exists for short GRBs that is analogous to the relationship identified previously for long ones (e.g., Hakkila et al. 2008).

The addition of the residual function improves many of the short GRB pulse fits. The wavelike form of the residual function, which can be described by a modified Bessel function attached to a compressed mirror image of itself, produces a rippled or multipeaked shape to the otherwise monotonic underlying pulse. The multipeaked shape is common among the isolated pulses in long/intermediate bursts detected by BATSE, Swift, and Fermi GBM (Hakkila & Preece 2014; Hakkila et al. 2015), and the characteristics of the residual function correlate with a number of other pulse properties.

The residual function is generally confined to the temporal interval occupied by the underlying pulse: the duration of the residual function (characterized by the Bessel frequency Ω) correlates with the pulse duration (w), which is similar to results found for long/intermediate GRB pulses (Hakkila & Preece 2014; Hakkila et al. 2015). The left panel of Figure 18 demonstrates this correlation (SC = −0.80, p = 6 × 10⁻²⁰).

Zoom In Zoom Out Reset image size

For long/intermediate GRB pulses, the inherent asymmetry of the residual function (characterized by s) anticorrelates with the pulse asymmetry (κ), indicating that the residual structure is aligned with the underlying pulse shape. Unfortunately, a similar correlation cannot be verified for short GRB pulses (SC = −0.35, p = 2 × 10⁻² is found), as the low-S/N environment in which these pulses are found makes accurate κ measurements difficult.

An anticorrelation is found between Ω and s (demonstrated in the right panel of Figure 18, with a Spearman rank-order correlation of SC = 0.46, p = 8 × 10⁻⁶). This is surprising because this correlation suggests that duration (w) and asymmetry (κ) are related, whereas no correlation is found (p = 0.82). We suspect that this correlation is not entirely real; it might result from the low-S/N environment in which short GRB pulses are found, the potentially interdependent ways in which Ω and s contribute to the residual function in Equation (8), and the fact that our initial estimates of Ω and s are based on κ.

The peak time of the residual function t₀ is found to not always align with the peak time of the underlying pulse τ_peak. This is demonstrated in Figure 19, where the difference t₀−τ_peak has been normalized to a standard time by dividing it by the pulse duration w. Although this offset appears to be real, the reason for the offset (which is positive for some bursts and negative for others) is still not understood, because it implies that the pulse and the residual function are somewhat independent of one another.

Zoom In Zoom Out Reset image size

The amplitude of the residual function (a) varies from near zero to roughly the pulse amplitude (A); the ratio of these amplitudes is characterized by R. However, we find no obvious correlation between the normalized difference and the alignments of t₀ and τ_peak with other pulse parameters (e.g., R, HR, S).

Long/intermediate GRB pulse light curves evolve from hard to soft, with rehardening occurring at or just prior to each of the three pulse peaks (Hakkila et al. 2015). Asymmetric pulses are hard overall and have pronounced hard-to-soft evolution; these contrast with symmetric pulses that are softer and have weak hard-to-soft evolution. This weak evolution can result in softer precursor peaks than central peaks, giving pulses the appearance of having intensity tracking behaviors.

It is interesting to see whether short GRB pulses undergo similar spectral evolutions to long/intermediate GRB pulses. Finding that they do would independently validate our initial assumption that short GRB emission episodes are indeed individual pulses, because we made no spectrally dependent assumptions about spectral evolution in our pulse definition (see 2.2).

The TTE pulse light curves have been collected in the four energy channels, described previously. Although the count rates in each of these four channels are low, they provide some information that can be used to infer pulse spectral evolution. We define the counts hardness (hr) in each time bin i as

$\begin{eqnarray}&&{{hr}}_{i}=\displaystyle \frac{{C}_{3i}+{C}_{4i}}{{C}_{1i}+{C}_{2i}},\end{eqnarray} \tag{ 18 }$

where C_1i, C_2i, C_3i, and C_4i are the counts per bin in channels 1, 2, 3, and 4, respectively. We track 4 ms pulse spectral evolution by measuring hr_i in each bin between t_start and t_end, in a manner similar to that done in Hakkila et al. (2015) for BATSE and Swift data.

We sum the counts from many pulses to get summed light curves and spectral evolutionary averages; this approach overcomes limits imposed by small number counting statistics and allows us to examine spectral evolution as a function of pulse structure. (Note: we have excluded pulses with negative total hardness ratios, as well as pulses with trigger numbers between 3282 and 3940 having incorrectly transcribed channel 1 counts data). Figure 20 shows the normalized mean light curve (solid line) and hr evolution (dashed line) for all 159 BATSE TTE complete pulses (left panel). Also shown are normalized mean light curves (solid lines) and counts hardness evolutions (dashed lines) of 102 simple pulses (right panel). Figure 21 contrasts these with the normalized mean light curves (solid lines) and counts hardness evolution (dashed lines) of 25 blended pulses (left panel) and 32 structured/complex pulses (right panel). Figure 22 shows the normalized mean light curves (solid lines) and counts hardness evolutions (dashed lines) of long/intermediate BATSE pulses (left panel) and long/intermediate Swift pulses (right panel).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

It is not surprising that the summed light curves exhibit the triple-peaked structure, as the light curves have been co-added using this structure as a temporal template. However, this co-adding should not produce the observed hard-to-soft pulse evolution, with spectral rehardening occurring at or just before each of the three peaks. This behavior is similar to that seen in long/intermediate GRB pulses, independently demonstrating that these short GRB emission episodes are individual pulses.

The normalized mean light curves verify the hypothesis that each emission episode contains but a single pulse. This appears to be true even for structured and complex pulses, where highly variable light curves are co-added to produce smooth light curves exhibiting only the triple-peaked structure. The rapidly varying component does not alter the underlying hard-to-soft spectral evolution, which is similar to that found in simple and blended pulses. However, structured/complex pulses appear to be harder than smoother pulse types, suggesting that the highly variable component is responsible for this. This result leads us to two important conclusions: (1) despite their highly variable structures, complex emission episodes are also single pulses, and (2) the highly variable component found in structured/complex pulses contains higher-energy photons than what is found in the smoothly evolving component.

This verification leads us to draw an additional important conclusion: structure and complexity beyond the triple-peaked pulse shape represent an additional, randomly distributed emission component that is not present in all short GRB pulses. Summing together a large number of structured and complex pulses should itself produce a complex light curve rather than the triple-peaked structure seen in the right panel of Figure 21. As described in the previous paragraphs, this additional emission component is bright, hard, and variable.

Although multipulsed TTE bursts are uncommon (making up only 10% of the population), their light curves are interesting because they contain interpulse separations as well as pulse durations. For multipulsed GRBs we define the interpulse separation (w_sep) as

$\begin{eqnarray}&&{w}_{\mathrm{sep}}={\tau }_{\mathrm{peak}2}-{\tau }_{\mathrm{peak}1},\end{eqnarray} \tag{ 19 }$

where τ_peak1 and τ_peak2 are the times of maximum amplitude for pulses 1 and 2, respectively. The T₉₀ duration of a two-pulsed GRB is thus

$\begin{eqnarray}&&{T}_{90}\approx {\tau }_{1;\mathrm{rise}}+{w}_{\mathrm{sep}}+{\tau }_{2;\mathrm{decay}},\end{eqnarray} \tag{ 20 }$

where τ_1;rise is the rise time of the first pulse and τ_2;decay is the decay time of the second pulse. Since w_sep is generally larger than the durations of either pulse, the T₉₀ duration of a GRB is generally dominated by the interpulse separation (see, e.g., Hakkila et al. 2003). Measurements of w_sep allow us to explore relationships between emission times of pulses in multipulsed GRBs, as well as the pulsed emission itself.

Strong correlations exist between the times of the emission episodes and the intervals separating them. The left panel of Figure 23 demonstrates that interpulse separations strongly correlate with first pulse durations; a Spearman rank-order test finds SC = 0.78 and p = 6 × 10⁻⁸. This correlation indicates that longer energy release times in the first pulse introduce correspondingly longer wait times until energy is released in the next pulse. The right panel of Figure 24 shows that the second pulse's duration is also longer when the first pulse's duration is long; a Spearman rank-order correlation test finds SC = 0.60 and p = 2 × 10⁻⁴. This correlation indicates that the energy release time of the second pulse lasts longer when the energy release time of the first pulse is also long.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Even though few GRB redshifts were available during the BATSE era, the three durations we have measured independently in each burst (w₁, w₂, and w_sep) can provide us with sufficient information to develop two redshift-independent parameters. We define these parameters by dividing the second pulse's duration and the burst's interpulse separation by the corresponding first pulse's duration. Since all three parameters are time dilated by the same factor 1 + z (where z is the redshift), the ratios w₂/w₁ and w_sep/w₁ are redshift independent. Figure 24 demonstrates the intrinsic correlation between w₂/w₁ and w_sep/w₁ (SC = 0.67, p = 2 × 10⁻⁵). This correlation demonstrates a lengthening of the observed emission episodes coupled with a lengthening of the waiting time between these episodes.

Pulse durations and interpulse separations are not independent quantities: later pulses have memories of at least some properties of the initial pulses, as well as of the gaps separating the pulses. If pulses represent structures undergoing kinematic motion, then a long pulse duration indicates that the pulsed emission occurs over a large distance. Similarly large interpulse gaps indicate either a large distance between locations where a pulse occurs or a deceleration in the bulk flow velocity. One interpretation of the increase in duration between the second pulse and the first pulse would then be that the emitting material has slowed and/or lost energy. This could be the result of jet expansion and/or slowing of the bulk flow.

As described in Table 2, four of the TTE bursts have pulse structures that are inconsistent with the standard GRB pulse paradigm. These bursts are instead characterized as asymmetric structures that increase gradually in intensity and then end with an abrupt crescendo (see Table 9). The individual pulse structures leading to the crescendo are clearly visible for triggers 3735 and 5439 (Figure 25), whereas they are unresolved for triggers 1453 and 3173 (Figure 26) and trigger 7375 (Figure 27). Because the bursts all increase in intensity with time, we refer to these gamma-ray transients as crescendo bursts and the rapid-fire pulses as staccato pulses. Our limited temporal resolution, coupled with the fact that trigger 5439 is an intermediate GRB, prohibits us from determining whether there is more than one category of crescendo bursts.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

In reevaluating the pulses in the BATSE TTE pulse catalog with this new definition in mind, we notice that the pulses associated with triggers 218 and 7753 also exhibit possible crescendo behavior. We have identified these pulses as possible crescendo bursts in the Comments column of the catalog (Table 7).

Although GRB pulse structure provides minimal evidence that short and long GRBs have different progenitors, crescendo GRBs with staccato pulses exhibit signatures of emission predicted from neutron star–black hole mergers. Tidal disruption of the neutron star is expected in a coalescing system of this type, forming a torus around the black hole. The black hole spin should cause the torus to precess via Lense–Thirring torques (Stone et al. 2013), resulting in a signal consisting of a small number of quasi-periodic events with interpulse separations of around 30–100 ms. The predicted precession period T_p should increase as T_p ∝ t^4/3, leading to a corresponding increase in the interpulse separation. The separations between the staccato pulses in BATSE triggers 3735 and 5439 exceed the expected 30 to 100 ms window, and these separations do not increase as t^4/3, so it seems unlikely that these crescendo bursts are consistent with the neutron star–black hole merger model. However, the variable emission in crescendo bursts 1453, 3173, and 7375 is of a shorter timescale and may be consistent with the model, although this is undetermined owing to the unresolved temporal binning. Regardless, the rarity of bursts having nonpulsed emission of the type predicted by Stone et al. (2013) is in agreement with the results of Dichiara et al. (2013), who find that events having these predicted properties do not dominate the short GRB population.

Not all crescendo bursts necessarily belong to the short GRB class. At least one long GRB (BATSE trigger 1425; Figure 28) appears to exhibit crescendo behavior along with staccato pulses. However, it should be noted that pulses in this burst appear to have asymmetric shapes consistent with the Norris et al. (2005) pulse model, unlike the pulses in the crescendo GRBs 3735 and 7375.

Zoom In Zoom Out Reset image size