HOW LONG DOES A BURST BURST?

A number of gamma-ray bursts (GRBs; namely, GRBs 101225A, 111209A, 121027A, and the most recent GRB 130925A) were found to last much longer (∼hours instead of tens of seconds) than typical GRBs (Levan et al. 2014; Gendre et al. 2013; Virgili et al. 2013; Stratta et al. 2013). Such "ultra-long" GRBs were also seen historically in BATSE and Konus–Wind data (see, e.g., Connaughton et al. 1997; Connaughton 1998, 2002; Giblin et al. 2002; Nicastro et al. 2004; Levan et al. 2005; Pal'shin et al. 2008). Motivated by such long durations and other multi-wavelength properties (e.g., the faint host galaxy of GRB 101225A and its late time color consistence with Type II supernovae; SNe II), several groups (Levan et al. 2014; Gendre et al. 2013) have proposed that the unusually long durations of these GRBs may point toward a new type of progenitor stars with much larger radii, such as blue supergiants (Mészáros & Rees 2001; Nakauchi et al. 2013), in contrast to the well-accepted compact Wolf–Rayet star progenitor (Woosley & Bloom 2006). In this scenario, the stellar envelope of a large-radius massive star would fall back in an extended timescale to fuel the central engine and to power a relativistic jet. The expected cocoon emission can explain anomalies in the afterglow data (Nakauchi et al. 2013). If this is the case, then ultra-long GRBs may form a distinct new population from the traditional short (compact star merger type) and long (Wolf–Rayet collapsar) GRBs.

However, careful studies based on many more criteria (other than duration alone) are needed to claim a new population. While the short and long dichotomy has long been known (Kouveliotou et al. 1993), it was not until the discoveries of the afterglow, redshift, and host galaxies of both types of events that a firm claim was made about their distinct progenitor types. Indeed, based on a dozen multi-wavelength observational criteria (Zhang et al. 2009), one was able to establish robust evidence that long (collapsar/magnetar type) and short (compact star merger type) GRBs are very different from each other, not only in duration, but also, more importantly, in their host galaxy types, specific star formation rate, SN association, circumburst medium properties, spectral properties, empirical correlations, and derived jet opening angles. Any proposal to claim a new population of GRBs should be performed in a similar manner. Even though these multi-wavelength criteria are being paid attention to (e.g., Levan et al. 2014), a careful comparative study between the proposed "ultra-long" GRB population and the more classical long GRB population is needed.

Interestingly, not all claimed ultra-long GRBs have ultra-long durations in γ-rays. Only GRBs 111209A and 130925A have an exceedingly long γ-ray T₉₀, i.e., >10, 000 s (Golenetskii et al. 2011, 2013; Markwardt et al. 2013). GRB 101225A was first measured to have a T₉₀ of 1088 ± 20 s (Palmer et al. 2010; Grupe et al. 2013). Later studies measured a longer duration of up to 7000 s based on the analysis of γ-ray data from Burst Alert Telescope (BAT) in subsequent Swift orbits (Thöne et al. 2012). The γ-ray duration of GRB 121027A, on the other hand, is only 62.6 ± 4.8 s in Swift/BAT band (Barthelmy et al. 2012), which is very typical for long GRBs. The main supportive evidence that GRBs 121027A and 101225A were included in the ultra-long category was their long-lasting highly variable X-ray light curves (Levan et al. 2014). In other words, the "ultra-long" durations of GRBs 121027A ("T₉₀" ∼ 6000 s; Levan et al. 2014) and 101225A ("T₉₀" ∼ 7000 s; Levan et al. 2014) are both observed in the X-ray band as opposed to being seen in γ-ray band only. In fact, Swift observations over the years have revealed that the GRB central engine lasts much longer than indicated by T₉₀ (Zhang 2011), via the manifestation of both X-ray flares (Burrows et al. 2005; Zhang et al. 2006; Liang et al. 2006; Chincarini et al. 2007; Margutti et al. 2011) and the so-called "internal plateaus"—X-ray plateaus followed by an abrupt decay that cannot be interpreted with the external shock model (Troja et al. 2007; Liang et al. 2007). Some authors even suggested that the entire X-ray afterglow may be of an internal origin powered by central engine (Ghisellini et al. 2007; Kumar et al. 2008; Murase et al. 2011). The existence of an extended tail emission in most long GRBs was already hinted from the BATSE data through stacking long GRB light curves (Connaughton 2002). If we believe that GRB duration definition should invoke X-ray data, then the duration distribution of GRBs should be re-analyzed in a systematical manner.

In this paper, we perform a comprehensive study of Swift X-ray Telescope (XRT) data, focusing on the long-term central engine activities in the X-ray light curves, to address typically how long a burst lasts, and whether the claimed ultra-long GRBs are special. In Section 2, we propose a new definition, t_burst, from the physical point of view, to measure the true timescale of the central engine activity. We also introduce quantitative observational criteria to measure t_burst from data. In Section 3, we use the Swift data to systematically derive t_burst and its distribution. We discuss the results and theoretical implications in Section 4.

Mounting evidence supports the hypothesis that X-ray flares have the same intrinsic physical origin as γ-ray pulses, but just have a reduced flux and peak energy so that they can be below the sensitivity threshold of a γ-ray detector (see Figure 1 for illustration). For extremely bright X-ray flares, the tips of the flares can be registered by the γ-ray detector, and hence, included in T₉₀. Figure 2 gives an example of a GRB (090715B) whose early X-ray flare as detected by Swift XRT (red) was also recorded by Swift BAT (blue), but the later extended X-ray flares were not. Therefore, T₉₀ measurement is not a reliable quantity to describe how long a burst "bursts."

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In this paper, we give a physically motivated definition of the duration of a GRB: The burst duration t_burst is an observable quantity of a GRB, during which the observed (γ-ray and X-ray) emission is dominated by emission from a relativistic jet via an internal dissipation process (e.g., internal shocks or magnetic dissipation), not dominated by the afterglow emission from the external shock.

This definition is different from the traditional T₉₀ in that it considers multi-wavelength signatures in addition to γ-rays. The rationale of using such a definition is illustrated in an illustration in Figure 1. The GRB central engine continuously ejects energy but generally with a reduced power as a function of time. The peak energy of the spectrum E_p is positively correlated to its luminosity (e.g., Lu et al. 2012), so it decreases with time. At a certain epoch (∼T₉₀), the signal drops out from the γ-ray band, but it still continues in the X-ray band. On the other hand, the afterglow component sets in early on, peaking at t_{ag, p} and decays with time. It is initially over-shone by the internal-origin X-ray component (X-ray flares and plateaus). Since the decay of internal emission is typically very steep, the afterglow component will eventually show up. The X-ray light curve therefore displays a steep-to-shallow transition when the external shock component emerges. In principle, the central engine can activate again to power bright internal emission to outshine the afterglow component again later. So a secure lower limit of the central engine activity time should be defined by the last observed steep-to-shallow transition, and this is our definition of t_burst.

Such a definition is, however, not easy to quantify. This is because in order to claim an internal origin of X-ray emission, theoretical modeling is needed to exclude an external shock origin of the observed flux. The standard external shock afterglow model (see, e.g., Gao et al. 2013 for a review) generally predicts broken power-law light curves. The steepest decay can be achieved when the blast wave enters a void, during which emission is powered by the high-latitude emission (Zhang et al. 2007, 2009). The decay slope in this regime is α = 2 + β (convention F_ν∝t^−αν^−β; Kumar & Panaitescu 2000), which is typically smaller than three. Due to the equal-arrival-time surface effect, any variability in external shock emission should satisfy Δt/t ⩾ 1, where Δt and t are the variability timescale and the epoch of observation, respectively (e.g., Ioka et al. 2005). As a result, rapid variabilities with Δt/t  ≪  1 (as observed in X-ray flares) and any steep decay with slope steeper than −3 (as observed in "internal X-ray plateaus") are deemed as being due to an internal origin.

We therefore adopt the following procedure to define t_burst of a GRB: (1) calculate T₉₀ for the Swift/BAT light curve; (2) fit the Swift/XRT light curve as a multi-segment broken power law; (3) identify the steep-to-shallow transitions in the light curve, and record the decay slope before the transition; (4) identify the last transition with pre-break slope steeper than −3, and record the transition time.⁵ The burst duration t_burst is defined as the maximum of this transition time and T₉₀ of γ-ray emission.⁶

Note that this method identifies only the X-ray emission that must be of an internal origin, but may not necessarily catch the full duration of internal emission if some internal-origin emission does not show such a steep decline (e.g., Ghisellini et al. 2007; Kumar et al. 2008; Murase et al. 2011). Therefore, we may typically regard t_burst as the lower limit of GRB central engine activity.

As of 2014 January 22, 712 GRBs have X-ray afterglows detected by Swift/XRT. All the XRT light curves are directly taken from the Swift/XRT team Web site⁷ (Evans et al. 2009) at the UK Swift science Data Centre, which were processed using HEASOFT v6.12. Several example light curves are presented in Figure 3, including the four ultra-long GRBs and some typical GRBs with canonical X-ray light curve behavior. One can see that the central engine activity usually lasts much longer than T₉₀.

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In order to measure t_burst, we use only well-sampled XRT light curves with late-time observations. We select a "good" sample based on the following criteria: (1) the X-ray light curve must have at least six data points, excluding upper limits; (2) the X-ray light curve has at least one steep-to-shallow transitions (with the steeper slope <−3); or (3) if the X-ray light curve has no steep-to-shallow transition, the starting time of XRT observation, T_{X, 0}, is smaller than T₉₀. For this latter case, we take T₉₀ as t_burst. Our final good sample consists of 343 GRBs (Table 1). This "good" sample, despite having robust measurements of t_burst, is incomplete. A good fraction of GRBs (consisting of 304 GRBs), which we define as the "undetermined" sample, have at least 6 data points in the light curves, do not have a required steep-to-shallow transition (with steeper slope <−3), but have an observational gap between T₉₀ and T_{X, 0}. The t_burst of these GRBs likely fall into the gap between T₉₀ and T_{X, 0}, but are not included in the "good" sample. Therefore the "good" sample is biased against GRBs with a short t_burst.

The essential part of measuring t_burst is to identify a shallower break feature in the late segments of the X-ray light curve. This is tricky, since late-time X-ray data sometimes have too few photons, or the entire light curves lack time coverage.⁸ To maximize the use of the observational data, we apply a multivariate adaptive regression splines technique (e.g., Friedman 1991) to the observed light curves in the logarithmic scale, which can automatically detect and optimize breaks.⁹ By measuring the decay slope before the break, one can judge whether the pre-break emission is internal, and hence, to measure t_burst. Figure 4 shows several examples of such measurements. In several cases (e.g, GRBs 130925A, 121027A, 111209A, 090715B, and 051117A), such a break is clearly identified so that t_burst is measured. In a few cases (e.g., GRB 140102A), such a break is not identified, but there is overlap between γ-ray and X-ray observations, i.e., T_{X, 0} < T₉₀. For these cases, we take t_burst = T₉₀. In some other cases (e.g., GRBs 101225A and 050724), the emergence of the external shock afterglow component is lacking at the end of X-ray observation, so that only the lower limit of t_burst can be determined as the last XRT observation time. In some other cases (e.g., GRB 110503A), the X-ray light curve is dominated by the afterglow component from the beginning, and there is no overlap between T₉₀ and the XRT observation, we thus exclude them in them good sample but include them into the undetermined sample.

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The distribution of t_burst of the good sample is shown in Figure 5(a).¹⁰ The median value of t_burst of the good sample is 428 s, which is much longer than the peak of T₉₀ distribution in previous works (e.g., about 20 s for the BATSE sample; Preece et al. 2000). Within the entire sample, about 25.6% GRBs have t_burst > 10³ s and 11.5% GRBs have t_burst > 10⁴ s. Interestingly we found the traditional short GRBs (with T₉₀ ⩽ 2 s) in our good sample have similar values of t_burst (blue solid line in Figure 5(a)) to long GRBs.

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The distribution of the t_burst of the good sample can be fitted by a mixture of two normal distributions in log space,¹¹ with a narrow, significant peak at ∼355 s, and a wider, less significant peak at ∼2.8 × 10⁴ s, respectively.¹²

As discussed above, this apparent bimodal distribution is subject to strong selection effects due to observational biases. In the following, we address two strong selection effects in turn.

• 1.  
  First, there is a Swift slewing gap between γ-ray observations (i.e., T₉₀) and the first XRT observation time, T_{X, 0}. It is likely that t_burst falls into this gap for many GRBs in the undetermined sample (e.g., GRB 110503A in Figure 4). The inclusion of this sample (whose size is comparable to the good sample) would modify the t_burst distribution significantly. In order to check how this effect changes the t_burst distribution we perform the following tests.

  • (1)  
    We simply let t_burst = T₉₀ for the undetermined sample and plot the distribution of t_burst of the whole sample (good + undetermined) of 647 GRBs in Figure 5(b). By doing so, the values of t_burst in the undetermined GRB sample could be underestimated, so that Figure 5(b) may be still regarded as a biased illustration of the t_burst distribution. Under this treatment, these t_burst values are more consistent with a single component. However, a Gaussian model can only poorly fit the data: there appears a sudden drop of t_burst around 1000 s and a significant excess in the "ultra-long" regime with t_burst ⩾ 10⁴ s.
  • (2)  
    By assuming T₉₀ ⩽ t_burst ⩽ T_{X, 0}, we generate a uniformly distributed random value of t_burst between T₉₀ and T_{X, 0} in logarithmic scale and assign it to t_burst for each GRB in the undetermined sample. We then plot the t_burst distribution of the whole sample (good + undetermined) in Figure 5(c). A Gaussian fit is improved, but the excess of the ultra-long GRBs still exists.
• 2.  
  There is an orbital gap around thousands of seconds (Figure 4, e.g., GRB 110503A) due to various reasons such as geometry configuration between Swift orbital position relative to the GRB source position which is subject to Sun, Moon, and Earth observation constraints, instrumental temperature of Swift, and delay of observation in respect to the priority of other ongoing observations (target of opportunities; ToOs). All these factors act as a selection effect against finding t_burst values within this gap. This gap (starting from t_{gap, 1} and ending at t_{gap, 2}, which are measured in the observed light curves, see, e.g., GRB 110503A in Figure 5) has a typical value of ∼3200 s (Figure 6(a)). The existence of such a gap has two effects on the t_burst distribution. First, if t_burst falls into this gap, these values are not registered, so that one would expect a dip in the t_burst distribution. Second, for those bursts whose real t_burst falls into this gap, one would mistakenly take an earlier steep-to-shallow transition break as t_burst, giving rise to a pile-up effect before the beginning of the orbital gap (see Figure 6(b)), which may be responsible for the sharp drop of the t_burst distribution around 1000 s in Figure 5(b). In order to test these speculations, we perform a Monte Carlo simulation by assuming that the intrinsic t_{burst, int} distribution is a single-peak Gaussian distribution in logarithmic space. Guided by the fit in Figure 5(c), we assume that the Gaussian distribution has a mean value μ = log t_{burst, int} = 2.2 and a standard deviation σ = 0.6. We generate 10⁴ GRBs whose t_{burst, int} follows such a distribution as shown in Figure 7(a). Each simulated GRB has a parameter set of {t_{burst, int}, T₉₀, t_{gap, 1}, t_{gap, 2}}, where T₉₀, t_{gap, 1}, t_{gap, 2} are generated following their corresponding observed distributions, as shown in Figures 7(b) and (c). To take account of the orbital gap effect, we check whether each t_{burst, int} falls into the gap between t_{gap, 1} and t_{gap, 2} for each simulated GRB. If not, we take the "observed" value t_burst = t_{burst, int}. If yes, we then assign t_burst a random value between T₉₀ and t_{gap, 1} in the logarithmic scale. The distribution of the final simulated t_burst is shown as the solid line in Figure 7(d), where the intrinsic input distribution is also plotted as the red dotted histogram. The resulting simulated the t_burst distribution shows a significantly sharp drop around 1000–3000 s as well as dip afterward. All these signatures are similar to the t_burst distributions derived from the data (Figures 5(a)– (c)). Our simulation suggests that the hypothesis of one single t_burst distribution component cannot be ruled out by the data.

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In this paper, we investigate the true GRB central engine activity duration distribution by considering both γ-ray and X-ray data. By defining t_burst based on some physically motivated criteria, we robustly derived t_burst for 343 GRBs. The t_burst distribution of this "good" sample shows an apparent bimodal distribution. If this is true, ultra-long GRBs could be more common than suggested in the literature (e.g., Levan et al. 2014). However, by including a larger sample whose t_burst values are not measured but can be guessed (303 GRBs in the "undetermined" sample) and by addressing two important selection effects, we found that the intrinsic t_burst distribution can be consistent with one single component. The existence of a separate "ultra-long" category of GRBs (Levan et al. 2014; Gendre et al. 2013; Boer et al. 2013) is neither required nor excluded by the data. Our results suggest that the ultra-long GRBs could be just a tail of a single long-duration GRB sample (see also Virgili et al. 2013).

As shown in Figure 8, our result indicates that a large fraction of long GRBs are actually quite long, even though their T₉₀ values are not extremely long. Evidence that two such long GRBs (030329 and 130427A) have associated Type Ic supernovae (Stanek et al. 2003; Hjorth et al. 2003; Xu et al. 2013) suggest that their progenitor is likely a Wolf–Rayet star whose hydrogen and helium envelopes have been depleted. The fact that their T₉₀ values are much longer than 10 s, the typical timescale for the jet to penetrate through the stellar envelope, suggests that the burst duration is not necessarily related to the size of the progenitor. Hence, making a direct connection between ultra-long GRBs and blue supergiants progenitor lacks strong physical justification. Theoretical investigations show that it becomes much more difficult for a jet to successfully penetrate through the stellar envelope of a blue supergiant, so that a significant fraction of such collapsing stars may just lead to failed GRBs (Murase & Ioka 2013). Also, blue supergiants are very unstable and short-lived, and their final explosion properties, including the possibility of launching a jet remain unclear.

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How to prolong a GRB central engine duration with a compact progenitor star is an open question. For variable emission such as X-ray flares, fragmentation in the massive star envelope (King et al. 2005), fragmentation in the accretion disk (Perna et al. 2006), and the formation of a magnetic barrier around the accretor (Proga & Zhang 2006) have been proposed. If the engine is a millisecond magnetar instead of a black hole, the magnetic activity of the millisecond magnetar can power an extended emission (Metzger et al. 2011). The steady spin down of the magnetar (Zhang & Mészáros 2001) would also power an internal X-ray plateau (Troja et al. 2007). Alternatively, fall-back accretion of the stellar envelope onto a newly formed black hole (Kumar et al. 2008; Wu et al. 2013) can also make extended internal X-ray emission. All these mechanisms could also be applied to ultra-long GRBs without invoking a large progenitor star.

The wide peak of ultra-long GRB components may be also understood in a scenario where those GRB progenitor stars have a distribution of mass and size, ranging from Wolf–Rayet stars to blue supergiants. Further multi-wavelength data, especially the properties of associated SNe and host galaxies of GRBs with different t_burst, are needed to make further progress.

Bromberg et al. (2013) found a plateau in the dN/dT₉₀ distribution in the BATSE, Swift, and Fermi Gamma-ray Burst Monitor samples, and argued that it provides direct evidence of the collapsar model. Realizing that T₉₀ is no longer a good indication of central engine activity timescale, we apply our t_burst data in the good sample to carry out a dN/dt_burst analysis. The plateau found by Bromberg et al. using T₉₀ is not reproduced with t_burst (Figure 9). Admittedly, the jet power in most GRBs reduces with time, and the most energy is still released during T₉₀. In any case, the collapsar signature suggested by Bromberg et al. (2013) may need further investigation.

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We thank an anonymous referee for thoughtful comments, and David N. Burrows, Peter Mészáros, Xiao-Hong Zhao, Peter Veres, Kazumi Kashiyama, Xue-Wen Liu, Derek Fox, and Shaolin Xiong for helpful discussion and suggestions. We thank Dirk Grupe for the information about Swift operations. B.B.Z. thanks Jason Rudy for helpful comments on the codes of multivariate adaptive regression splines fitting. This work was partially supported by NASA/Fermi GI grant/NNX11AO19G (B.B.Z.). B.Z. acknowledges support from NASA NNX10AP53G. KM acknowledges the support by NASA through a Hubble Fellowship, grant No. 51310.01 awarded by the STScI, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract No. NAS 5-26555. We acknowledge the use of public data from the Swift data archive.

Facility: Swift - Swift Gamma-Ray Burst Mission