The difference of photon arrival time, which is known as spectral lag, is well known characteristics of gamma-ray bursts (GRBs). In particular, long duration GRBs show a soft lag which means that high energy photons arrive earlier than soft photons. The lag-luminosity relation is the empirical relationship between the isotropic peak luminosity and the spectral lag. We calculated the spectral lags for 40 known redshift GRBs observed by Swift addition to the previous 31 GRB samples. We confirmed that most of our samples follow the lag-luminosity relation. However, we noticed that there are some GRBs which show a significant scatter from the relation. We also confirm that the relationship between the break time and the luminosity of the X-ray afterglow (so-called Dainotti relation) extends up to the lag-luminosity relation.

1. Introduction

The time variability and the spectral evolution seen in the prompt emission of gamma-ray bursts (GRBs) are not well understood despite the large data samples collected by various GRB instruments. However, there are some clues to understand temporal and spectral variations seen in the prompt emission. The spectral lag which is known as the difference of an arrival time between high energy photons and low energy photons is one of such clues. It is known that high energy photons arrive earlier than low energy photons for long duration GRBs (hereafter long GRBs) [1, 2], while short duration GRBs (hereafter short GRBs), which have a duration shorter than 2 seconds, have a smaller or a negligible lag than that of long GRBs [3, 4].

There is an empirical relation between a spectral lag and an isotropic peak luminosity which is called lag-luminosity relation [5]. The relation is discovered based on the analysis of only 6 known- GRBs using the BATSE samples. Recently, Ukwatta et al. [6] calculated the spectral lag for 31 Swift known- GRBs. They found that the Swift GRBs follow the lag-luminosity relation with a significant scatter in their samples from the best fit relation.

Many theoretical models suggested explaining the lag-luminosity relation. Salmonson [7] suggests that this relation is caused by the variation in line-of-sight velocity of GRBs. Ioka and Nakamura [8] proposed that an observer’s viewing angle to the jet causes this relation, while Schaefer [9] suggested that a lag and a peak luminosity are related to a cooling time of a pulse. Although there are many theoretical models that have been proposed to explain the lag-luminosity relation, physical origin of this relation is still unclear.

On the other hand, Dainotti et al. [10] discovered the empirical relation between an X-ray break time and an X-ray luminosity from the Swift X-ray afterglow samples (hereafter Dainotti relation). The break time from a shallow to a normal decay in the X-ray afterglow light curve (e.g., [11]) is anticorrelated with the luminosity at the break time. The theoretical interpretations of this relation include long-term behavior of an external accretion disk [12] and a prior emission model for X-ray plateau phase [13]. Furthermore, Sultana et al. [14] suggested that the lag-luminosity relation at the prompt emission nicely connects to the Dainotti relation found from the X-ray afterglow data. Also, they proposed that both relations might be explained individually by the same kinematic effect, which is viewing the jet at “off-beam” line of sight [8, 15]. If viewing angle is large, an isotropic luminosity and a spectral lag are large in a prompt emission following Ioka and Nakamura [8]. Also, Eichler and Granot [15] proposed that the viewing angle which is slightly larger than an opening angle makes a long plateau and a low flux emission in an afterglow. Therefore, if this finding is true, we are probably seeing a clear physical connection between the prompt and the afterglow emission. However, since those findings are still based on a small number of GRBs, it is important to investigate using rich GRB samples.

The number of Swift known- GRBs is increasing and becoming large samples. We analyzed additional 40 Swift known- GRBs following the method described in Ukwatta et al. 2010 [6]. Our paper is organized as follows. The analysis method is described in Section 2. The results of analysis and application to the lag-luminosity relation are shown in Section 3. We discussed our results in Section 4. We concluded and summarized our results in Section 5.

2. Method

We calculated a spectral lag following the method described by Ukwatta et al., 2010 [6], using the data of Swift Burst Alert Telescope (BAT [16]). First, we made the mask-weighted (background subtracted) light curves using the BAT event data for four energy bands (15–25 keV, 25–50 keV, 50–100 keV, and 100–200 keV). We made 1000 simulated light curves for each energy band. Those simulated light curve bins are calculated by where is a random number from the standard normal distribution. We calculated cross correlation function (CCF) for all 6 combinations of energy bands and determined spectral lags. We used the Band’s CCF formula (Band 1997 [17]) which is expressed as for two series and with a delay , where . We fitted CCF with the Gaussian function to determine the CCF peak for each 1000 CCFs generated from simulated light curves. Then, we made the histogram of the CCF peaks and determined the center value as the best fit value of a lag and its standard deviation as an error of a lag.

To cross-check the validity of our code, we analyzed 5 GRBs whose lags have been reported in Ukwatta et al. 2010 [6]. The results of this cross-check are shown in Figure 1. The figure clearly shows that our lag values are consistent with those of Ukwatta et al. 2010 [6], including the size of the errors.

We selected 229 long GRB samples from Swift known- GRBs between March 2005 and May 2013 at the beginning. Three selections were applied for the initial samples. As the first selection, we selected 198 GRBs of which spectral lags have not been calculated in Ukwatta et al., 2010 [6]. In the second selection, the 92 GRBs are selected based on the peak flux threshold (1.5 peak photon flux in the 15–150 keV band). In the third selection, the GRBs which are possible to estimate the peak energy () using the relationship between and the photon index of the simple power-law fit to the BAT spectral data [18] are selected from the samples. Finally, our samples became 40 GRBs. There are 14 GRBs between 2005 and 2009 which are selected in our samples but not in Ukwatta et al. The difference between our samples and Ukwatta’s sample is that our samples include darker GRBs than Ukwatta’s samples. Those weak samples are needed in order to confirm the lag-luminosity relation for GRBs which have the low luminosities. The average peak flux of Ukwatta’s samples is ~13 counts/sec/cm2 in 15–150 keV, whereas the averaged peak flux of our 14 GRBs is ~5 counts/sec/cm2.

We used time interval as the time range of the analysis. We examined the time bin of the light curve from 1 ms to 1024 ms and reported the lag as the reliable lag values if the average is greater than ~0.5. This value corresponds to a reasonable signal to noise level in a light curve in this analysis. The validity of the correspondence between a and a signal to noise level is studied in Ukwatta et al. 2010 [6].

Most of the GRB spectra can be fitted by the Band function [19] which has four parameters (normalization, , and power-law photon indices and ):We calculated the following equation to determine a normalization :where is 1 sec peak photon flux in the 15–150 keV band and is when . The flux in the source flame energy band 1–10000 keV is calculated by The isotoropic peak luminosity is where is the luminosity distance: where is Hubble constant. We assumed that  km s−1 Mpc−1 =  s−1, , . The error of was estimated by performing the calculations for 1000 times taking into account the uncertainties in spectral parameters. We used reported Band function parameters , , , flux, and their errors from the BAT analysis (e.g., [20], GCN circulars) as the input parameters. However, if , , and are not constrained, we estimated from a photon index based on a single power-law fit [18]. In this case, and are used from the distributions by BATSE where and [21].

3. Results

We calculated spectral lags for 40 known- GRBs. Table 1 shows the results of spectral lag analysis and calculation of . The columns “LC st” and “LC et” are the start time and the end time of the light curve. The “CCF fit range min” and “CCF fit range max” are the fitting range of the CCF to the Gaussian function. The channel numbers from 1 to 4 correspond to the following energy bands: 15–25 keV, 25–50 keV, 50–100 keV, and 100–200 keV, respectively. We called two pieces of light curve data which are cross-correlated to calculate a lag in the channel numbers. For example, “” means the lag value obtained by cross-correlating the light curves between the 50–100 keV and the 15–20 keV band. We calculated spectral lags for all 6 combinations. However, if an average of 1000 is not larger than ~0.5 in the 1 ms to 1024 ms time binning, we did not report the lags. A few GRBs show the lags which have large errors or negative lags. Since the light curves of those GRBs generally have low statistics, we believe the lags were not measured with a good accuracy.

Figures 2 and 3 shows the relationship between lags and luminosities of 71 known- Swift GRBs based on the BAT data. Table 2 summarized linear correlation coefficients between the lag measured by different combinations of channels and the luminosity. We used a maximum likelihood method which was used to analyze a peak luminosity and variability correlation by Guidorzi et al. 2006 [22]. This method accounts for and errors and samples variance [23]. The parameters are the normalization and index of the power-law and the variance of (). Table 3 shows the results of fittings. In the calculation and fittings, the results of GRB060614A were not used. This is because GRB060614A has smaller lag than that expected from the lag-luminosity relation [24]. The results show that an isotropic peak luminosity negatively correlates with a spectral lag for all combinations of the channels. The best fit power-law index of the lag between the 50–100 keV band and the 15–25 keV band and the luminosity is with logarithm variance of 0.78. The linear correlation coefficient in this case is with sample size of 62 (null probability of 0.2%). Therefore, we concluded that there is a correlation between the lag and the luminosity with the Swift GRB samples.

Figure 4 shows the lag-luminosity relation overlaid with the break time and the luminosity of the X-ray afterglow [14]. We used the results of Lag31 for the plot. This is because the lags between consecutive energy bands show low significant values. Also, the sample size of the lags between channel 4 and other channels is small due to a low signal to noise in the light curve of channel 4 in general.

We increased the samples which have both the lag and the luminosity values of the prompt emission, and the X-ray afterglow properties by combining our results and the latest report by Dianotti et al. 2013 [25]. As a result of power-law fitting, best fit parameters are with . The correlation coefficient is −0.96 with a sample size of 32. We confirmed that the two relations are connected with a power-law index of −1 as suggested by Sultana et al. 2012 [14], with larger samples.

4. Discussion

Our large known- samples confirmed that the isotropic luminosity negatively correlates with the spectral lag (Figures 2 and 3). The linear correlation coefficients are between −0.4 and −0.5. These results suggest that the most of the long GRBs are consistent with the lag-luminosity relation. However, the results of fitting show the samples variance between 0.7 and 0.9. This suggests that samples show a significant scatter from the best fit line.

Even if we considered scatter, GRB060614A is far away from the best fit line. Our result confirmed that GRB 060614 is a real outlier to the relation [24].

Combining the fact that the duration itself might not be sufficient enough to distinguish the different classes of GRBs [26], we think multiple aspects of the prompt and the afterglow properties are needed to investigate and classify GRBs appropriately.

We estimated the bulk Lorentz factor from the averaged Lag31 value of ~0.6 seconds following the argument of Lu et al. [27]: where and is shown by The bulk Lorentz factor is estimated to be 115 assuming a source frame peak energy of 240 keV (typical observed peak energy of 80 keV for the Swift GRBs), a source frame energy of 75 keV, and redshift of 2.

Figure 4 suggests the clear extension of the lag-luminosity relation to the Dianotti relation. Although the connection of those two relations is still puzzling, this might imply an important clue to understand the physical relationship between the prompt and the X-ray afterglow emission.

5. Conclusion and Summary

We calculated spectral lags of 40 known- Swift GRBs in addition to the previously calculated 31 GRBs. We found that most of our samples follow the lag-luminosity relation with a scatter of samples. We also confirmed that the negative relation between the break time and the luminosity in the X-ray afterglow extends nicely to the lag-luminosity relation of the prompt emission.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.


The authors would like to thank the anonymous referee for comments and suggestions that materially improved the paper. This work was supported by the Fermi Guest Investigator Program and also partially supported by the Grant-in-Aid for Scientific Research KAKENHI Grant no. 24103002.