A NEW METHOD OF PULSE-WISE SPECTRAL ANALYSIS OF GAMMA-RAY BURSTS

The basic necessity for a good spectral analysis of GRB pulses is wide band coverage to identify additional spectral components. The GBM on board the Fermi satellite, with its wide band width and excellent sensitivity, provides a good data base for such studies. It has two scintillation detectors: the sodium iodide (NaI) detector is sensitive in the ≳8 keV to ∼900 keV range while the bismuth germanate (BGO) energy range is ∼200 keV to ∼40 MeV (Meegan et al. 2009).

We examined the Nava catalog (Nava et al. 2011) of Fermi/GBM GRBs and found that there are 112 bright (fluence ⩾10⁻⁶ erg), long (δt ⩾ 15 s) GRBs and 11 of these GRBs have single/separable pulses. GRB 081221 is the brightest among them. In Figure 1, we have plotted the LC of this GRB with the Norris model (Norris et al. 2005) fitted for the two pulses. We have also made a systematic analysis of the other 10 GRBs and the results of the time-integrated spectral analysis for all of them are given later.

Zoom In Zoom Out Reset image size

We use the CSPEC data for time-integrated study and the time tag event (TTE) data for the time-resolved spectral analysis. We choose two or more NaI detectors having high count rate and one/both BGO detector(s). For source selection and background subtraction, we use the rmfit v3.3pr7 tool, developed by User Contributions of Fermi Science Support Center. The background exposure time is chosen before and after the burst. This background is modeled by a polynomial of different degrees, according to the need. The pulse-height analyzer (PHA) files are binned in energy channels so as to get a minimum count in each spectral bin. Typically, the NaI detectors are binned by minimum count ≳ 40, while the BGO detectors are binned by requiring a minimum count of ∼50–60. First, we perform time-integrated analysis for all the 11 bright, long GRBs having single/separable pulses. We implement both C-stat and χ² minimization methods in rmfit. We fit either Band or CPL model.

In Table 1, we report the best-fit parameter values along with the corresponding 3σ errors. The reduced C-stat and $\chi ^2_{{\rm red}}$ with degrees of freedom (dofs) are also reported. For comparison, we quote the results of Nava et al. (2011) for these GRBs. It is clearly seen that all these values are matching quite well. The sources of very minor deviations between the values of this work, done by C-stat minimization, and Nava et al. (2011), are (1) mismatch between actual start and stop time, (2) exact background selection and modeling, and (3) the exact number of detectors and the channels used. Comparing the deviation of the parameter values, it is clear that the deviation resulting from using different statistics other than C-stat, i.e., χ² minimization, is much less than that resulting from these other reasons. Hence, we conclude that the statistics plays a minimal role in actual parameter estimation. In fact, the GRBs taken in our analysis are all bright GRBs (fluence ⩾10⁻⁶ erg). Hence, by default, the χ² minimization is a correct technique for parameter estimation of GRBs with high count rates.

Table 1. Results of Time-integrated Spectral Analysis of the GRBs

GRB	t1, t2	This Work		Nava et al. (2011)
(Model)		C-stat	χ2
080904	−4.096, 21.504	$\alpha =-1.22^{+0.21}_{-0.20}$	$\alpha =-1.21^{+0.20}_{-0.19}$	α = −1.14 ± 0.05a
(CPL)		$E_p=40.1^{+3.92}_{-3.56}$	$E_p=39.8^{+3.68}_{-3.34}$	Ep = 39.24 ± 0.75
		Cb = 1.08  (597)	$\chi ^2_{{\rm red}}=1.23$ (597)	C = 1.14 (587)
080925	−3.840, 32.0	$\alpha =-1.06^{+0.11}_{-0.10}$	$\alpha =-1.06^{+0.11}_{-0.10}$	α = −1.03 ± 0.03
(Band)		$\beta ==-2.34^{+0.30}_{-1.13}$	$\beta ==-2.24^{+0.24}_{-0.74}$	β = = − 2.29 ± 0.08
		$E_p=158.9^{+31.6}_{-24.4}$	$E_p=157.3^{+33.5}_{-24.9}$	Ep = 156.8 ± 7.07
		C = 1.17 (712)	$\chi ^2_{{\rm red}}=1.14$ (712)	C = 1.13 (716)
081118	0.003, 19.968	$\alpha =-0.42^{+0.70}_{-0.48}$	$\alpha =-0.37^{+0.70}_{-0.49}$	α = −0.46 ± 0.10
(Band)		$\beta ==-2.18^{+0.16}_{-0.35}$	$\beta ==-2.14^{+0.15}_{-0.19}$	β = = − 2.29 ± 0.05
		$E_p=55.93^{+22.2}_{-12.5}$	$E_p=54.0^{+19.7}_{-12.0}$	Ep = 56.79 ± 2.77
		C = 1.17 (716)	$\chi ^2_{{\rm red}}=1.02$ (716)	C = 1.16 (601)
081207	0.003, 103.426	$\alpha =-0.58^{+0.10}_{-0.09}$	$\alpha =-0.58^{+0.12}_{-0.11}$	α = −0.58 ± 0.02
(Band)		$\beta ==-2.15^{+0.17}_{-0.33}$	$\beta ==-2.13^{+0.20}_{-0.41}$	β = = − 2.22 ± 0.7
		$E_p=363.4^{+70.7}_{-51.5}$	$E_p=364.5^{+82.8}_{-59.4}$	Ep = 375.1 ± 13.2
		C = 1.43 (713)	$\chi ^2_{{\rm red}}=1.02$ (713)	C = 1.74 (596)
081217	−28.672, 29.696	$\alpha =-1.09^{+ 0.15}_{ -0.14}$	$\alpha =-1.10^{+0.16}_{-0.14}$	α = −1.05 ± 0.04
(CPL)		$E_p=193.0^{+65.9}_{-37.3}$	$E_p=200.5^{+77.2}_{-41.7}$	Ep = 189.7 ± 11.2
		C = 1.19 (715)	$\chi ^2_{{\rm red}}=1.06$ (715)	C = 1.46 (599)
081221	0.003, 39.425	$\alpha =-0.84^{+0.06}_{-0.05}$	$\alpha =-0.84^{+0.06}_{-0.06}$	α = −0.82 ± 0.01
(Band)		$\beta ==-4.24^{+0.93}_{-10.2}$	$\beta ==-3.89^{+0.69}_{-7.1}$	β = = − 3.73 ± 0.20
		$E_p=85.25^{+2.89}_{-3.08}$	$E_p=85.09^{+3.23}_{-3.19}$	Ep = 85.86 ± 0.74
		C = 1.64 (595)	$\chi ^2_{{\rm red}}=1.49$ (595)	C = 1.67 (600)
081222	−0.768, 20.736	$\alpha =-0.89^{+0.14}_{-0.12}$	$\alpha =-0.89^{+0.14}_{-0.12}$	α = −0.90 ± 0.03
(Band)		$\beta ==-2.46^{+0.37}_{-1.37}$	$\beta ==-2.32^{+0.31}_{-0.98}$	β = = − 2.33 ± 0.10
		$E_p=169.2^{+37.3}_{-27.4}$	$E_p=168.9^{+39.1}_{-29.8}$	Ep = 167.2 ± 8.28
		C = 1.12 (595)	$\chi ^2_{{\rm red}}=1.07$ (595)	C = 1.23 (604)
090129	−0.256, 16.128	$\alpha =-1.43^{+ 0.19}_{ -0.16}$	$\alpha =-1.46^{+0.18}_{-0.16}$	α = −1.46 ± 0.04
(CPL)		$E_p=170.4^{+130.0}_{-48.5}$	$E_p=195.5^{+212}_{-63.5}$	Ep = 166.0 ± 15.1
		C = 1.09 (596)	$\chi ^2_{{\rm red}}=1.03$ (596)	C = 1.12 (602)
090709	0.003, 18.432	$\alpha =-1.04^{+ 0.38}_{ -0.32}$	$\alpha =-1.08^{+0.37}_{-0.31}$	α = −0.96 ± 0.08
(CPL)		$E_p=116.7^{+76.9}_{-30.6}$	$E_p=124.1^{+101}_{-34.7}$	Ep = 137.5 ± 12.5
		C = 1.05 (596)	$\chi ^2_{{\rm red}}=1.01$ (596)	C = 1.17 (602)
091020	−3.584, 25.088	$\alpha =-1.31^{+ 0.29}_{ -0.18}$	$\alpha =-1.32^{+0.22}_{-0.19}$	α = −1.20 ± 0.06
(CPL)c		$E_p=255.7^{+332.0}_{-92.0}$	$E_p=276.4^{+485.0}_{-107.0}$	β = = − 2.29 ± 0.18
		C = 1.03 (354)	$\chi ^2_{{\rm red}}=0.95$ (354)	Ep = 186.8 ± 24.8
				C = 1.18 (354)
091221	−2.048, 37.889	$\alpha =-0.62^{+0.27}_{-0.21}$	$\alpha =-0.62^{+0.34}_{-0.23}$	α = −0.57 ± 0.05
(Band)		$\beta ==-2.40^{+0.50}_{-3.15}$	$\beta ==-2.26^{+0.45}_{-2.80}$	β = = − 2.22 ± 0.10
		$E_p=191.3^{+67.4}_{-47.5}$	$E_p=189.5^{+76.8}_{-57.1}$	Ep = 194.9 ± 11.6
		C = 1.42 (474)	$\chi ^2_{{\rm red}}=1.12$ (474)	C = 1.44 (466)

Notes. ^aThe errors quoted from Nava et al. (2011) are symmetric errors. Errors for this work are 3σ errors. ^bC is the reduced C-stat value, the numbers in the parentheses are dofs. ^cThe Band spectrum showed unbound 3σ errors; we found better fit with CPL for this GRB.

Download table as:  ASCIITypeset image

We select four models for the spectral study: (1) Band model, (2) blackbody with a power law (BBPL), (3) a modified blackbody with a power law (mBBPL), and (4) two blackbodies with a power law (2BBPL). The Band function can be written in terms of the spectral indices (α and β) and the peak energy (E_peak) as

$$\begin{equation} I(E) = \left\lbrace \begin{array}{@{}ll}A_{b}\left[\frac{E}{100}\right]^{\alpha } \exp \left[\frac{-(2+\alpha)E}{E_{{\rm peak}}}\right] ;\; {\rm if}\;E\le \frac{(\alpha -\beta)}{(2+\alpha)}E_{{\rm peak}}\\ \\ A_{b}\left[\frac{E}{100}\right]^{\beta } \exp \left[\beta -\alpha \right]\left[\frac{\left(\alpha -\beta \right)E_{{\rm peak}}}{100\left(2+\alpha \right)}\right]^{(\alpha -\beta)} ;\;{\rm otherwise}. \end{array} \right. \!\! \end{equation} \tag{ 1 }$$

Here, A_b is the normalization constant. A model consisting of thermal (BB with temperature kT) and non-thermal (PL with index Γ) components has been used earlier (see, e.g., Ryde 2004; Ryde et al. 2006; Ryde & Pe'er 2009; Pe'er & Ryde 2011). We name this function as BBPL. This can be written as

$$\begin{equation} I(E)=\frac{K_{1}\times 8.0525E^{2}}{(kT)^{4}[\exp \left(E/kT\right)-1]}+K_{2}E^{-\Gamma }, \end{equation} \tag{ 2 }$$

where K₁ and K₂ are normalization constants. There are suggestions (e.g., Ryde et al. 2010) in the literature of a modified blackbody (mBB), which may exist due to angular dependence of the optical depth and the observed temperature (Pe'er 2008). Hence, we also investigate this model with a power law (mBBPL). The mBB model is a multi-color BB disk model; the local disk temperature kT(r) is proportional to r^−p. In several GRBs there are distinct additional thermal components (see, e.g., Shirasaki et al. 2008) and further, if the GRB spectrum is due to thermal IC of seed photons, then there may be, in principle, multiple photon baths. For brevity, we take one more BB component and call it 2BBPL model. This has essentially two BBs with two temperatures—kT_h and kT_l and two normalizations—K_1h and K_1l. We use all the four models for our subsequent analysis. We note here that it is also possible to have combinations of the above models (like Band+PL, Band + BB + PL), but, at this juncture, we consider only these four generic models. This is essentially because of the fact that the time-resolved spectra of GRBs generally consist of a broad peak with wings, which can be adequately captured by any of the above four models.

Time-resolved spectral studies require a large number of parameters. If n is the number of time bins, then a four-parameter model, such as Band or BBPL, requires 4n parameters for a full description. Our motivation, in this study, is to reduce the number of parameters with some reasonable assumptions. The basic assumption we make is that the temperature (kT) and the peak energy (E_peak) follow smooth time evolution. This evolution has a break at the peak of the pulse (see also Ryde & Pe'er 2009). In the following discussion we shall refer to them as two episodes—rising and falling part. We assume that the time evolution law of kT and E_peak is simple PL of time, i.e., ∼t^μ, with different μ in different episodes. If m and n are the number of time bins in these two episodes, respectively, then this parameterization reduces the number of free parameters by m + n − 2. Note that, ideally one should use one more parameter to account for the start time, but we have chosen it to be zero for all the cases, except for the rising part of GRB 090618, where the pulse starts from −1 s. Hence, only for this case, we have chosen this start time to be −10 s. As the parameterization is not corrected for the actual start time, the actual values of μ are not unique and hence, these values should not be used to compare different pulses. We also assume that for the Band model the photon indices (α and β) are constants in each episode. Their values can be determined by simultaneously fitting all the time-resolved spectra in an episode, with the photon indices tied. For BBPL model, we assume that the index (Γ) of the PL is constant in each episode. This index can be determined in the similar way as the photon indices of the Band. The parameterization of the norm of BBPL model is more complicated, as, unlike Band, we have two norms. In the Band model, norm is a free parameter. In order to have the equal number of parameters as Band, either we can use an overall free norm with suitable parameterization of BBPL norms, or, we can parameterize one of the norms and treat the other as a free parameter. Ryde & Pe'er (2009) have shown that the parameter ${\cal {R}} = (F_{{\rm BB}}/\sigma T^4)^{1/2}$ either increases with time or remains constant. If we assume that the observed flux of the BB varies as a simple function of time as F_BB ∼ t^ζ, then the variation of ${\cal {R}}$ can be parameterized as ∼t^{ζ/2 − 2μ}. For μ ⩽ ζ/4, ${\cal {R}}$ will show the expected behaviors. The BB flux variation can be translated to the norm variation of Equation (2) as $K_1 \sim t^{\nu _1}$. Now, if we parameterize the BB norm and treat the PL norm as a free parameter of the model, we have the same number of free parameters as Band model. But, this makes the BBPL more constrained, in the sense that the overall norm is not a free parameter, as in the case of Band model. To overcome this difficulty, let us assume (which will be justified later) that the PL component has norm variation as smooth as that of BB, i.e., $K_2 \sim t^{\nu _2}$. Then the overall norm (K) can be made a free parameter by parameterizing the ratio of the BB norm and PL norm as $K_1/K_2 \sim t^{{\nu _1}/{\nu _2}} \sim t^{\nu }$.

Thus, both the Band and BBPL models have the equal number of free parameters: m + n + 8. For the Band model, m+n are norms of the Band function in m+n time bins. The other eight parameters are α, β, μ, and the E_peak at the starting bin, E_peak(t₀), in the two episodes. Peak energy at any time t is determined by E_peak(t₀) × (t/t₀)^μ. For the BBPL model, m+n are overall norms (K) and the other four parameters are PL index (Γ), μ, ν, and kT at the initial bin, kT(t₀), in the two episodes. For the mBBPL model, we have an added parameter, namely, p, in each episode. Hence, the number of parameters is m + n + 10. For the 2BBPL model, the photons are boosted by the same material, hence, the BB parameters cannot be arbitrary. We assume that the ratio of temperatures and norms of these BBs are fixed, which should be determined by tying the ratios in all bins. Hence, the number of parameters in this model is m + n + 12. As an example, if there are five time bins preceding to the peak of a pulse and 10 bins afterward, then the parameterization for Band and BBPL reduces the number of free parameters from 60 to 23; for a total of 25 bins this factor is 100 to 33 and so on.

To compare the significance of a model with respect to another, we have performed the F-test. The F value is defined in the most general case as $F=(({\chi ^2_1/{\rm dof}_1})/({\chi ^2_2/{\rm dof}_2}))$, where index 1 is used for the primary model, while 2 is used for the alternative model. We compute the probably (p) of a given F value and thereby find the σ significance (and the % confidence level—CL) of the alternative model preferred over the primary model. If the primary model is a subset of the alternative model, then the F value is defined as, $F= ({(\chi ^2_1-\chi ^2_2)/({\rm dof}_1-{\rm dof}_2)}/({\chi ^2_2/{\rm dof}_2}))$ and the difference between the dofs (M) is deemed as the dof of the primary model.

We start with a time-resolved spectral analysis so that some of the assumptions sketched above can be examined and validated. The major challenge in time-resolved spectroscopy is to define the time bin size, which crucially depends on two factors: (1) timescale of spectral evolution and (2) the minimum bin size allowed by the data, which in turn depends on the subjective decision of signal-to-noise ratio (S/N). One cannot violate the latter condition for a given S/N. The peak count rate of this GRB is ∼4000, while we have demanded that each PHA bin should have at least 40 counts (S/N ∼ 6.3, i.e., total ∼5000 counts, for 128 channels). Hence, we cannot choose smaller than ∼1 s time bin. First, we choose uniform time bins of 3 s to extract time-resolved spectra. Later, we reduce the time bin to 1 s to check any improvement due to finer time bins.

3.1.1. Case I: Bin Size of 3.0 s

In Table 2, we report the results of the Band and BBPL fits to the time-resolved data of 3 s bin size. The time bin starts from −1 s, with 14 bins (numbered 0–13). Approximately, the first four bins belong to pulse 1, the last eight bins belong to pulse 2, and the two intermediate bins belong to the overlapping region. For the BBPL fit, we first fit the spectra with the PL index (Γ) free. We note that Γ is more or less constant for the major portion of the burst (bins 0–2 for pulse 1 and 6–11 for pulse 2). We take the average of Γ over these bins, separately for the two pulses, and found in both cases, Γ = 1.83 with standard deviation (σ) 0.14 and 0.10 for pulses 1 and 2, respectively. Γ has large error bars in the last bin of pulse 1 and last two bins of the second pulse. The values in the overlapping region (bins 4 and 5), apart from having large errors, may be ambiguous, and hence neglected. We freeze Γ to 1.83 for all the bins and redo the analysis. The corresponding values are also reported in Table 2. Note that, by doing this we are gaining 1 dof in each time bin. It is clear from Table 2 that for the second pulse, the time-resolved spectral fit with the Band model are much better than those of the BBPL fit in terms of the $\chi _{{\rm red}}^{2}$. On the contrary, these values are comparable in the first pulse. Hence, the first pulse may be dominated by thermal emission. Note, however, that $\chi _{{\rm red}}^{2}$ of BBPL is poor in bin 1 of the first pulse, which has, in fact, the highest flux. Hence, this points to a different radiation mechanism than a simple BB, which may appear in the high flux regions.

Table 2. Results of BBPL and Band Fitting of Time-resolved Data of GRB 081221

Bin	BBPL (Γ Free)					BBPL (Γ Frozen to 1.83)				Band
	kT	K1	Γ	K2	$\chi ^{2}_{{\rm red}}({\rm dof})$	kT	K1	K2	$\chi ^{2}_{{\rm red}}({\rm dof})$	α	β	Epeak	$\chi ^{2}_{{\rm red}}({\rm dof})$
0	$38.03_{-4.28}^{+4.91}$	$2.99_{-0.68}^{+0.67}$	$1.73_{-0.25}^{+0.42}$	$5.31_{-3.15}^{+12.72}$	1.03(67)	$38.28_{-3.97}^{+4.78}$	$3.14_{-0.46}^{+0.49}$	$7.18_{-2.16}^{+2.25}$	1.01(68)	$-0.28_{-0.30}^{+0.36}$	−10.0	$178.06_{-27.57}^{+42.39}$	1.03(67)
1	$16.26_{-2.21}^{+2.45}$	$1.76_{-0.36}^{+0.39}$	$1.77_{-0.13}^{+0.15}$	$14.27_{-5.98}^{+9.51}$	1.34(76)	$16.92_{-1.63}^{+1.87}$	$1.83_{-0.33}^{+0.34}$	$17.33_{-2.99}^{+3.08}$	1.33(77)	$-0.69_{-0.22}^{+0.39}$	$-3.76_{-\infty }^{+1.30}$	$77.92_{-15.58}^{+11.08}$	1.10(76)
2	$10.14_{-2.07}^{+3.12}$	$0.78_{-0.27}^{+0.28}$	$1.99_{-0.23}^{+0.27}$	$20.66_{-13.36}^{+28.87}$	1.02(67)	$9.07_{-1.33}^{+1.49}$	$0.84_{-0.25}^{+0.26}$	$10.94_{-3.10}^{+3.28}$	1.02(68)	$-0.24_{-1.10}^{+1.64}$	$-2.55_{-\infty }^{+0.29}$	$35.41_{-8.53}^{+18.41}$	0.98(67)
3	$10.98_{-2.14}^{+2.56}$	$0.74_{-0.27}^{+0.31}$	$2.16_{-0.25}^{+0.40}$	$32.99_{-20.23}^{+65.24}$	0.94(69)	$9.07_{-1.32}^{+1.39}$	$0.80_{-0.24}^{+0.26}$	$9.21_{-2.87}^{+3.02}$	0.99(70)	$-0.85_{-0.54}^{+0.80}$	$-3.07_{-\infty }^{+0.62}$	$39.74_{-7.77}^{+8.60}$	0.93(69)
4	$6.82_{-1.13}^{+1.69}$	$0.61_{-0.26}^{+0.28}$	$1.85_{-0.34}^{+0.27}$	$9.05_{-7.42}^{+17.99}$	0.92(134)	$6.76_{-1.08}^{+1.17}$	$0.63_{-0.20}^{+0.21}$	$8.13_{-2.71}^{+2.89}$	0.91(135)	$0.52_{-1.60}^{+3.16}$	$-2.47_{-0.54}^{+0.25}$	$24.48_{-5.39}^{+9.05}$	0.92(134)
5	$11.36_{-1.74}^{+2.01}$	$1.12_{-0.29}^{+0.30}$	$2.08_{-0.13}^{+0.16}$	$49.09_{-19.27}^{+31.10}$	0.91(147)	$9.18_{-1.02}^{+1.06}$	$1.19_{-0.26}^{+0.27}$	$18.14_{-3.07}^{+3.19}$	0.97(148)	$-1.06_{-0.32}^{+0.46}$	$-2.92_{-\infty }^{+0.45}$	$43.19_{-7.51}^{+7.34}$	0.88(147)
6	$22.61_{-0.99}^{+1.00}$	$8.52_{-0.65}^{+0.68}$	$1.77_{-0.06}^{+0.07}$	$35.88_{-7.30}^{+9.67}$	1.56(178)	$23.10_{-0.78}^{+0.82}$	$8.92_{-0.49}^{+0.50}$	$43.47_{-3.33}^{+3.37}$	1.56(179)	$-0.45_{-0.09}^{+0.09}$	−10.0	$102.66_{-4.34}^{+4.77}$	1.16(178)
7	$23.69_{-0.73}^{+0.74}$	$13.67_{-0.74}^{+0.77}$	$1.73_{-0.05}^{+0.06}$	$41.65_{-6.89}^{+8.52}$	1.86(182)	$24.34_{-0.59}^{+0.61}$	$14.53_{-0.58}^{+0.58}$	$55.80_{-3.53}^{+3.57}$	1.89(183)	$-0.31_{-0.08}^{+0.08}$	$-3.82_{-\infty }^{+0.52}$	$105.94_{-3.95}^{+4.20}$	1.28(182)
8	$19.77_{-0.82}^{+0.84}$	$8.93_{-0.58}^{+0.60}$	$1.76_{-0.04}^{+0.05}$	$53.13_{-7.89}^{+9.22}$	1.87(180)	$20.51_{-0.65}^{+0.67}$	$9.46_{-0.49}^{+0.50}$	$66.35_{-3.90}^{+3.95}$	1.90(181)	$-0.61_{-0.08}^{+0.09}$	$-3.30_{-1.09}^{+0.41}$	$91.76_{-4.95}^{+4.85}$	1.24(180)
9	$14.41_{-0.77}^{+0.81}$	$5.93_{-0.44}^{+0.44}$	$1.86_{-0.04}^{+0.05}$	$75.42_{-12.23}^{+14.09}$	1.69(175)	$14.06_{-0.54}^{+0.57}$	$5.87_{-0.42}^{+0.43}$	$67.70_{-4.14}^{+4.20}$	1.69(176)	$-0.88_{-0.08}^{+0.14}$	$-9.37_{-\infty }^{+19.37}$	$70.82_{-2.87}^{+3.00}$	1.09(175)
10	$12.64_{-1.05}^{+1.17}$	$3.36_{-0.37}^{+0.38}$	$1.84_{-0.06}^{+0.06}$	$54.40_{-11.52}^{+13.64}$	1.58(164)	$12.41_{-0.72}^{+0.78}$	$3.35_{-0.37}^{+0.37}$	$51.03_{-3.86}^{+3.92}$	1.57(165)	$-1.02_{-0.10}^{+0.11}$	$-9.37_{-\infty }^{+19.37}$	$66.62_{-3.22}^{+4.25}$	1.23(164)
11	$11.33_{-1.06}^{+1.19}$	$1.97_{-0.30}^{+0.31}$	$2.01_{-0.12}^{+0.14}$	$40.27_{-15.24}^{+22.81}$	1.32(149)	$10.27_{-0.70}^{+0.73}$	$2.03_{-0.29}^{+0.30}$	$19.73_{-3.23}^{+3.33}$	1.36(150)	$-0.61_{-0.31}^{+0.45}$	$-3.11_{-\infty }^{+0.44}$	$45.39_{-5.74}^{+5.17}$	1.26(149)
12	$10.03_{-1.07}^{+1.19}$	$1.42_{-0.27}^{+0.29}$	$2.21_{-0.24}^{+0.36}$	$37.04_{-21.85}^{+59.08}$	0.99(141)	$8.96_{-0.72}^{+0.74}$	$1.50_{-0.25}^{+0.26}$	$8.31_{-2.79}^{+2.91}$	1.03(142)	$-0.51_{-0.37}^{+0.43}$	−10.0	$39.19_{-3.41}^{+3.82}$	0.95(141)
13	$7.61_{-1.11}^{+1.35}$	$0.65_{-0.19}^{+0.17}$	$3.65_{-1.58}^{+3.10}$	164.92	1.13(127)	$7.29_{-0.98}^{+1.02}$	$0.65_{-0.20}^{+0.20}$	$0.44_{-0.44}^{+2.39}$	1.15(128)	$1.18_{-1.59}^{+1.72}$	$-4.48_{-\infty }^{+1.25}$	$27.93_{-3.58}^{+4.48}$	1.14(127)

Notes. For BBPL, values are quoted for both power-law index (Γ) free and frozen to the mean value at the high count rate region: 1.83. K₁ is BB normalization, while K₂ is that of the PL. The norms should be used as the relative normalization, as there is another constant multiplication due to detector effective area. Bins are 0–13 with 0 denoting −1 to 2 s and subsequently equal bin size of 3 s is applied. Errors in K₂ of bin 13 of BBPL (Γ free) could not be determined. In many cases, only the upper error in β could be determined.

Download table as:  ASCIITypeset image

In Figure 2, we have plotted the $\chi _{{\rm red}}^{2}$ of BBPL with Γ thawed, BBPL with Γ frozen, and Band fit with filled circles, open circles, and stars, respectively. The BBPL model is clearly inferior to the Band model for a major portion of the second pulse, especially in the regions where photon counts are high. In the first pulse (−1 to 11 s), the BBPL model except for the one bin is comparable to the Band model. We also fit mBBPL and 2BBPL, which are shown by filled boxes and pluses. The mBBPL and 2BBPL are as good as the Band model, in terms of $\chi _{{\rm red}}^{2}$. Hence, the correct model for the major portion of the burst is either of these three.

Zoom In Zoom Out Reset image size

3.1.2. Case II: Bin of Size 1.0 s

In the time-resolved analysis of Section 3.1.1, the Band and other models show superior fits compared with the BBPL model in terms of $\chi _{{\rm red}}^{2}$. But, as Zhang et al. (2011) pointed out for GRB 090902B, this can be an effect of evolution of the BBPL within a time bin. To investigate this, we make finer time bins of 1 s and redo the time-resolved spectroscopy. Note that this is the finest bin possible for an S/N ∼ 6.3. In Table 3, we report the average values of $\chi _{{\rm red}}^{2}$ for different model fittings, obtained for different bin sizes. Apparently, there are improvements in the $\chi _{{\rm red}}^{2}$ for finer bins, but we also see that these improvements are of the same orders for different models. Hence, it seems that the spectrum is not due to the evolution of BBPL, but one of these other models. But, all of these models are comparable. Hence, we cannot hope to find the correct model by taking finer bins. Presence of a simple BB is apparent in the first pulse, with the exception of the second bin, where one of the other models is correct. The second pulse is dominated either by an mBB or a fully non-thermal radiation (Band) or IC (2BBPL). In order to find the right answer, one should carefully parameterize the spectral evolution in each episode of the pulses separately and reduce the set of free parameters in the description.

Table 3. $\chi _{{\rm red}}^2$ of different Models for Time-resolved Spectral Analysis of GRB 081221

Method	3 s Time Bins		1 s Time Bins
	$\langle \chi _{{\rm red}}^2 \rangle$ of Full GRB	$\langle \chi _{{\rm red}}^2 \rangle$ (Second Pulse)	$\langle \chi _{{\rm red}}^2 \rangle$ of Full GRB	$\langle \chi _{{\rm red}}^2 \rangle$ (Second Pulse)
BBPL (Γ free)	1.31 ± 0.35	1.52 ± 0.32	1.11 ± 0.26	1.21 ± 0.28
BBPL (Γ frozen)	1.30 ± 0.35	1.50 ± 0.33	1.16 ± 0.27	1.22 ± 0.29
Band	1.09 ± 0.14	1.17 ± 0.11	1.00 ± 0.16	1.04 ± 0.18
mBBPL	1.15 ± 0.14	1.23 ± 0.13	1.07 ± 0.17	1.06 ± 0.19
2BBPL	1.09 ± 0.15	1.17 ± 0.13	1.02 ± 0.16	1.05 ± 0.17

Download table as:  ASCIITypeset image

The spectral evolution in the pulses is not arbitrary. For example, the temperature of the BB evolves with time in a smooth way. This can be seen from Figure 3, where we have plotted the kT of BBPL (both Γ free and frozen) by circles. The peak energy variation is shown by pluses. The parameters evolve smoothly as a function of time. For a single pulse, Ryde & Pe'er (2009) have shown that the temperature remains constant or slowly declines with an average PL index, 〈a_T〉 = −0.07 with σ(a_T) = 0.19, during the rise of the pulse and decays faster with an average index 〈b_T〉 = −0.68 with σ(b_T) = 0.24. The break time of this evolution has a strong positive correlation with the pulse peak time. We find a similar behavior for both kT and E_peak evolution. Hence, the spectral evolution can be described by a simple time evolution of temperature (kT ∼ t^μ) or the peak energy (E_peak ∼ t^μ). The index, μ, in principle may have two values in the two episodes, namely, the rising and the falling part.

Zoom In Zoom Out Reset image size

Liang & Kargatis (1996, hereafter LK96) showed for fast-rise exponential-decay pulses that the peak energy of the EF(E) (or νF_ν) spectrum follows a more complicated evolution. E_peak decreases exponentially with the running fluence as

$$\begin{equation} E_{{\rm peak}}(t)=E_{{\rm peak},0} \exp \left(-\frac{\phi _{{\rm Band}}(t)}{\phi _{{\rm Band},0}}\right), \end{equation} \tag{ 3 }$$

where E_{peak, 0} and ϕ_{Band, 0} are the constants of the E_peak evolution law, $\phi _{{\rm Band}}(t)=\int _{t_0}^t f(t^{\prime })dt^{\prime }$ is the fluence at time t, f(t') being the flux. In Figure 4 (upper panels), we have plotted the ln (E_peak) with the fluence. Note that the fluence of each pulse is calculated from their respective start time. Clearly, the E_peak evolution strictly follows the LK96 law in the first pulse. In the second pulse, however, the variation is not smooth throughout. In the falling part, the variation is clearly LK96 type, but in the rising part, the variation is rather "soft-to-hard." This effect may be the result of overlap between two pulses. In fact, the first two bins of the second pulse belong to the overlapping region. Hence, they might be contaminated with the preceding pulse. However, the third bin, where there should not be any effect of the first pulse, also deviates from the LK96 law. This might indicate that the second pulse is genuinely intensity tracking. Kocevski & Liang (2003) argued that the "intensity-tracking" pulses for which this evolution does not appear very prominent are rather made of more than one short hard-to-soft pulse. Ghirlanda et al. (2011), on the other hand, have analyzed time-resolved spectra of 11 long and 12 short Fermi GRBs and found that the long GRBs appear to follow a "soft-hard-soft" trend, tracking the flux of the GRB, rather than a strict "hard-to-soft" evolution. Lu et al. (2012) have categorized GRB 081221 as one having a strict "hard-to-soft" pulse followed by "intensity-tracking" pulse. They have simulated overlapping pulses to show that in the overlapping region the spectral evolution may appear to be "intensity tracking." However, they also found some single pulses to have "intensity-tracking" spectral evolution. Hence, the second pulse may be genuinely "intensity tracking."

Zoom In Zoom Out Reset image size

In Figure 4 (bottom panels), the temperature evolution is plotted against BB fluence. Here the same behavior is noticed. Hence, kT evolution of the first pulse and the falling part of the second pulse can as well be described by a similar exponential decay:

$$\begin{equation} kT(t)=kT_0 \exp \left(-\frac{\phi _{{\rm BB}}(t)}{\phi _{{\rm BB},0}}\right), \end{equation} \tag{ 4 }$$

where $\phi _{{\rm BB}}(t)=\int _{t_0}^t f(t^{\prime })dt^{\prime }$ is the running fluence of the BB component at time t, f(t') being the flux at t'. ϕ_{BB, 0} and kT₀ are the constants of the evolution law. LK96 law is empirical and a simpler version, in principle, can be used instead, such as a simple PL of time. As fluence is a monotonically increasing function of time, either of them can be used for evolution study.

The BBPL model has two components. Hence, in order to parameterize the norms of this model one has to see the flux evolution of the individual components. In Figure 5, we have plotted both photon and energy flux of the individual components calculated for 8–900 keV energy range. The flux evolutions look similar for both Γ free and frozen cases. Interestingly, the PL flux is as smooth as the BB flux, in each pulse. Hence, as argued in Section 2.3, we can safely assume that the ratio of their evolutions is a smooth function of time. In Figure 6, we have shown the evolution of α, β of Band, and Γ of BBPL. The parameter β, in many cases, has either large error bars or only an upper limit could be derived. In some cases, they peg to the value −10. It is clear from Figure 6 that the parameters remain reasonably constant at all episodes (rising and falling part) of a pulse. Hence, we tie them over all the time bins in a given episode to determine their values with greater accuracy. This reduces the number of free parameters of the description of spectral evolution to a great extent, as described in Section 2.3.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The fact that the parameters are well-behaved functions of time makes the time-resolved spectroscopy more tractable, as we can reduce the number of free parameters in the pulse-wise description (see Section 2.3). Following the parameterization and tying scheme of Section 2.3, we do the spectral analysis for the individual pulses of the GRB. In the following analysis, we use the TTE data of NaI (n0, n1, n2) and BGO (b0) for our analysis. The constant, which takes care of the relative normalization of the detectors, should not vary throughout the burst. Hence, we freeze them to the values obtained in the time-integrated analysis, i.e., 2.25, 2.32, 2.34, and 3.24, respectively. Additionally, we make the following changes compared with the time-resolved spectral analysis discussed earlier. We divide the data into spectra of equal total counts rather than equal time bins so that equal importance is given to all individual spectra. Further, the spectral data in each bin are regrouped into spectral channels to provide a uniform S/N. We also note that the 30–40 keV region of the spectrum of this GRB has the known calibration issues due to the K-edge of NaI (see, e.g., Guiriec et al. 2011). This does not matter much for parameter estimations, but, if one wants to compare different models in terms of χ² then it is wise to neglect these bins. In the following, we have done the spectroscopy by neglecting the 30–40 keV band.

3.3.1. Analysis of Pulse 2

This pulse constitutes the major portion of the burst. The count rate is ≳ 3 times higher than pulse 1. Hence, we can analyze this pulse with greater accuracy and later use our experience to analyze the other one. We perform the analysis for two cases as follows.

Case I: Analysis for count per time bin ≳3000. This analysis is done by dividing the second pulse from 17.0 s onward, requiring ≳3000 counts per time bin. We divide the pulse into two parts. 17–21.45 s is the rising part and the rest up to 40.45 s is the falling part. In the rising part, we get three time bins and the falling part has nine of them. The spectral bins in the energies >100 keV sometimes show less than 2σ count, while <15 keV show less than 3σ count. Hence, we merge the 8 keV to 15 keV bins to form one bin; 100–900 keV bins are merged into seven bins, with progressively higher binning at higher energies. Similarly, spectral bins of the BGO (200 keV to 30 MeV) are merged into five large bins. All the spectral fit parameters are listed in Table 4.

Table 4. Study of Spectral Evolution in Pulse 2 (17.0–40.55 s) of GRB 081221 (Neglecting 30–40 keV)

Model	χ2 (dof)	μ	ν	α	β	Epeaka	p	Γ	kTh/kTin/kTa	kTla
Case I: Count per time bin ≳3000—Rising part (17.0–21.45 s, 3 bins)
Band	364.41 (354)	1.0 ± 0.3	⋅⋅⋅	$-0.44_{-0.03}^{+0.06}$	$-7.61_{-\infty }^{+2.14}$	$98.59_{-3.02}^{+2.75}$	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅
BBPL	455.92 (354)	0.5 ± 0.3	3.2 ± 1.0	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	$1.89_{-0.04}^{+0.05}$	$24.16_{-0.63}^{+0.64}$	⋅⋅⋅
mBBPL	355.68 (353)	0.9 ± 0.2	0.8 ± 1.0	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	$0.81_{-0.04}^{+0.08}$	$1.81_{-0.11}^{+0.20}$	$40.02_{-2.04}^{+2.87}$	⋅⋅⋅
2BBPL	351.78 (352)	0.6 ± 0.1	4.7 ± 1.0	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	$1.94_{-0.11}^{+0.16}$	$28.73_{-1.44}^{+1.67}$	$9.75_{-1.03}^{+1.14}$
Case I: Count per time bin ≳3000—Falling part (21.55–40.55 s, 9 bins)
Band	1188.17 (993)	−2.1 ± 0.1	⋅⋅⋅	−0.68 ± 0.05	$-3.55_{-0.44}^{+0.26}$	$115.5_{-3.2}^{+3.0}$	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅
BBPL	1557.25 (993)	−1.9 ± 0.1	−3.2 ± 0.4	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	$2.02_{-0.02}^{+0.03}$	$26.81_{-0.57}^{+0.58}$	⋅⋅⋅
mBBPL	1223.39 (992)	−2.0 ± 0.2	3.5 ± 0.3	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	$0.74_{-0.03}^{+0.02}$	$2.03_{-0.06}^{+0.09}$	$49.93_{-1.44}^{+2.82}$	⋅⋅⋅
2BBPL	1147.53 (991)	−1.9 ± 0.1	−3.1 ± 0.4	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	$2.15_{-0.08}^{+0.09}$	$38.13_{-1.52}^{+1.63}$	$13.33_{-0.68}^{+0.71}$
Case II: Count per time bin ≳1000—Rising part (17.0–21.45 s, 10 bins)
Band	1247.91 (1187)	1.5 ± 0.3	⋅⋅⋅	$-0.44_{-0.06}^{+0.05}$	$-9.15_{-\infty }^{+4.02}$	$90.35_{-2.48}^{+2.96}$	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅
BBPL	1328.50 (1187)	1.0 ± 0.3	4.4 ± 0.8	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	$1.89_{-0.04}^{+0.05}$	$22.55_{-0.63}^{+0.64}$	⋅⋅⋅
mBBPL	1239.20 (1186)	0.9 ± 0.1	1.3 ± 0.5	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	0.87 ± 0.09	$1.96_{-0.14}^{+0.28}$	$39.01_{-1.92}^{+3.48}$	⋅⋅⋅
2BBPL	1222.76 (1185)	0.4 ± 0.6	9.0 ± 2.0	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	$2.07_{-0.16}^{+0.37}$	$30.30_{-1.54}^{+1.73}$	$9.96_{-0.96}^{+1.01}$
Case II: Count per time bin ≳1000—Falling part (21.55–40.55 s, 29 bins)
Band	3743.36 (3448)	−2.5 ± 0.1	⋅⋅⋅	$-0.75_{-0.05}^{+0.06}$	$-3.56_{-0.77}^{+0.31}$	125.7 ± 3.9	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅
BBPL	4133.63 (3448)	−2.1 ± 0.1	−3.6 ± 0.2	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	2.04 ± 0.03	$28.15_{-0.67}^{+0.68}$	⋅⋅⋅
mBBPL	3804.05 (3447)	−2.0 ± 0.1	3.0 ± 0.5	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	$0.72_{-0.02}^{+0.03}$	$2.22_{-0.08}^{+0.11}$	$53.00_{-2.39}^{+2.13}$	⋅⋅⋅
2BBPL	3690.21 (3446)	−2.0 ± 0.2	−3.6 ± 0.4	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	$2.31_{-0.11}^{+0.13}$	$41.40_{-1.61}^{+1.69}$	$13.72_{-0.62}^{+0.65}$

Note. ^aThe values quoted are for the first time bin.

Download table as:  ASCIITypeset image

(a) Rising part. In the rising part (first four rows of Table 4), the BBPL, compared with the other models is inferior with regard to χ² (dof): 455.92 (354). If we parameterize only the BB norm, and treat the PL norm as a free parameter, the corresponding χ² (dof) is 487.68 (354). A BBPL fitting with no constraint gives χ² (dof) = 451.09 (348). While we have gained dof, the $\chi ^2_{{\rm red}}$ remains of the same order, which confirms that the parameterization works. In comparison with BBPL, mBBPL is a better fit with χ² (dof) = 355.68 (353), and significance of 2.55σ (98.93% confidence). The Band model is better than the BBPL model, with χ² (dof) = 364.41 (354), and significance of 2.37σ (98.23% confidence). This suggests that the radiation mechanism in the rising part of the second pulse may be a photospheric emission, but the thermal part is an mBB rather than a simple BB. However, if we compare these values with a 2BBPL model, then we immediately see that this model is the best with χ² (dof) = 351.78 (352). As the set of parameters of BBPL model is a subset of 2BBPL, the significance of 2BBPL compared with BBPL is much higher: 9.29σ (100% confidence). Compared with the Band model, 2BBPL model has a significance of 0.86σ (60.94% confidence), which shows that 2BBPL is only marginally better than the Band model.

(b) Falling part. In the falling part (see Table 4), the Band model is better compared to the BBPL as well as the mBBPL model. Compared with the mBBPL model, the Band model has 35.22 less χ² with one more dof. A comparison with the 2BBPL model, on the other hand, shows that 2BBPL is better than the Band model at 1.03σ significance (69.71% confidence). Compared with the Band model, 2BBPL has 40.64 less χ² with two less dofs. The Band model does not show much difference in terms of residuals of individual spectral fit. But, it is only when we perform a parameterized joint fit that we realize that the 2BBPL model is marginally better than the Band model (Table 4). Hence, in this region either Band or 2BBPL is the best model, with 2BBPL marginally better. In Figure 7, we have shown the significance of 2BBPL fitting over the BBPL fitting as a case study. The residual of the BBPL model shows excess at various channels. No such structure is visible in the residual of the 2BBPL fit. Note that the NaI K-edge is present in both the residuals between 30 and 40 keV. We have done the fitting both by including and excluding this band. When all the channels are used, the χ² (dof) of BBPL and 2BBPL are 340.85 (217) and 239.82 (215), respectively. 2BBPL is preferred over BBPL at a significance of 8.42σ (100% confidence, p = 3.86 × 10⁻¹⁷). If we exclude the 30–40 keV bins, the corresponding χ² (dof) are 300.26 (197) and 193.17 (195), while 2BBPL is preferred at a significance of 9.01σ (100% confidence, p = 2.10 × 10⁻¹⁹).

Zoom In Zoom Out Reset image size

Case II: Analysis for count per time bin ≳1000. To check whether lowering the size of the time bins improves the BBPL fitting, we perform the same analysis for count per time bin ≳1000. As the count rate is lower, we merge the 100–900 keV of NaI detectors into five channels rather than seven. The rest of the binning remains the same. We obtain 10 bins in the rising and 29 bins in the falling part.

(a) Rising part. The values of χ² (dof) for BBPL, Band, mBBPL, and 2BBPL are 1247.91 (1187), 1328.50 (1187), 1239.20 (1186), and 1222.76 (1185), respectively. Compared with BBPL model, the Band model is preferred at 1.36σ (82.76% confidence), mBBPL is preferred at 1.56σ (88.17% confidence), while 2BBPL is preferred at 9.66σ (100% confidence). 2BBPL is preferred over Band model at 0.89σ (62.60% confidence). Hence, the conclusions of Case I remains unaltered.

(b) Falling part. Similarly, in the falling part, the finer bin makes equal impact on all the models and hence, the conclusion remains the same. Note that compared with mBBPL, the 2BBPL model has 113.84 less χ², with the cost of one more dof. Hence, the 2BBPL model is preferred over mBBPL in the falling part (1.31σ with 68.99% confidence). Compared with the BBPL model, Band, mBBPL, and 2BBPL are preferred at 3.12σ (99.82% confidence), 2.67σ (99.24% confidence), and 19.61σ (100% confidence), respectively. 2BBPL is marginally better than Band at 0.95σ (65.64%).

3.3.2. Analysis of Pulse 1

The time-resolved spectra of this pulse are extracted by requiring ≳ 1000 counts bin⁻¹, as the photon count is ∼1/3 of pulse 2. As before the spectral bins of NaI are binned in 8–15 keV and 100–900 keV, while BGO spectral channels are merged to form five broad channels. The 30–40 keV band is neglected. The results of different fits are reported in Table 5.

Table 5. Study of Spectral Evolution in Pulse 1 (−1.0 to 12.05 s) of GRB 081221

Model	χ2 (dof)	μ	ν	α	β	Epeaka	p	Γ	kTh/kTin/kTa	kTla
Rising part (−1 to 2.15 s, 1 bin)
Band	115.78 (116)	⋅⋅⋅	⋅⋅⋅	$-0.55_{-0.22}^{+0.26}$	−10.0	$170.3_{-22.7}^{+30.7}$	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅
BBPL	109.67 (116)	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	$1.93_{-0.21}^{+0.35}$	$38.27_{-3.76}^{+4.08}$	⋅⋅⋅
mBBPL	110.27 (115)	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	$0.98_{-0.28}^{+\infty }$	$2.15_{-4.46}^{+\infty }$	$62.78_{-7.22}^{+18.14}$	⋅⋅⋅
2BBPL	103.05 (114)	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	$1.74_{-3.04}^{+\infty }$	$38.47_{-3.66}^{+4.40}$	$6.57_{-1.64}^{+3.17}$
Falling part (2.25 to 12.05 s, 4 bins)
Band	544.65 (473)	−0.7 ± 0.1	⋅⋅⋅	$-0.86_{-0.19}^{+0.22}$	$-3.61_{-\infty }^{+0.69}$	$82.02_{-7.43}^{+7.53}$	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅
BBPL	571.47 (473)	−0.7 ± 0.1	−0.2 ± 0.4	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	$2.09_{-0.11}^{+0.13}$	$19.62_{-1.77}^{+1.88}$	⋅⋅⋅
mBBPL	548.22 (472)	−0.7 ± 0.2	2.5 ± 0.8	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	$0.63_{-0.03}^{+0.10}$	$1.51_{-\infty }^{+0.52}$	$41.63_{-6.76}^{+6.49}$	⋅⋅⋅
2BBPL	544.79 (471)	−0.7 ± 0.2	−0.1 ± 0.5	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	$2.04_{-0.14}^{+0.28}$	$28.59_{-4.45}^{+5.99}$	$9.95_{-1.89}^{+2.30}$

Notes. The bins are obtained by requiring ≳1000 counts bin⁻¹ (30–40 keV bins are neglected). ^aThe values quoted are for the first time bin.

Download table as:  ASCIITypeset image

(a) Rising part. The rising part has only one bin from −1.0 to 2.15 s. Interestingly, the BBPL model is marginally better than the Band model in the rising part (see Table 5) at 0.87σ (61.46% confidence). The mBBPL model has comparable χ² as BBPL, but with one more parameter. 2BBPl model has a significance of 1.04σ (70.17% confidence) compared with the Band model, while the same model has a significance of 2.19σ (97.12% confidence) compared with the BBPL model.

(b) Falling part. In the falling part of this pulse, the BBPL model is no longer the best model. Band is the best model with χ² (dof) = 544.65 (473). The mBBPL and 2BBPL models are comparable to Band with χ² (dof) 548.22 (472) and 544.79 (471). Hence, the spectrum may be still thermal, though the thermal part is no longer a simple BB, but either a multi-color BB (mBB) or has multiple spectral component (one more BB) or simply synchrotron dominated (Band). Note also that the low-energy photon index ($\alpha =-0.86_{-0.19}^{+0.22}$) in the falling part is within the regime of synchrotron model, which is clearly in contrast with the rising part ($\alpha =-0.55_{-0.22}^{+0.26}$). This phenomenon of softening of photon index at the falling part of a pulse can be seen for all the pulses (see Tables 4 and 5). In Table 6, we have listed all the significance levels (in terms of p-value, sigma level, and CL) of a model over another. Model₁ is the primary model, while Model₂ is the alternative model. It is clear from the table that the 2BBPL model is preferred over the Band model, though marginally, in some cases. The p values denote the probably that the alternative hypothesis is incorrect. Hence, the lower the value of p is, the better the alternative model will be over the primary model. Only in one case, namely the falling part of pulse 1, Band is preferred over 2BBPL. But the p-value of this case is 0.48, which signifies that they are only comparable. Interestingly, if we use finer bin size, the significance of Band and mBBPL over BBPL decreases in the second pulse. The significance of 2BBPL, however, increases.

Table 6. Comparison of Different Model Fits at Different Episodes of GRB 081221

Region	Model2/Model1	p	σ	CL
Pulse 1, Rising part	BBPL/Band	0.385	0.87	61.46%
	mBBPL/Band	0.415	0.81	58.50%
	2BBPL/Band	0.298	1.04	70.17%
	2BBPL/BBPL	0.029	2.19	97.12%
Pulse 1, Falling part	Band/BBPL	0.301	1.03	69.93%
	mBBPL/BBPL	0.334	0.965	66.57%
	2BBPL/BBPL	1.29× 10−5	4.36	99.99%
	Band/2BBPL	0.480	0.705	51.95%
Pulse 2, Rising part	Band/BBPL	0.018	2.37	98.23%
(≳3000 counts bin−1)	mBBPL/BBPL	0.011	2.55	98.93%
	2BBPL/BBPL	1.5× 10−20	9.29	100%
	2BBPL/Band	0.390	0.86	60.94%
Pulse 2, Falling part	Band/BBPL	1.04× 10−5	4.41	99.99%
(≳3000 counts bin−1)	mBBPL/BBPL	7.86× 10−5	3.95	99.99%
	2BBPL/BBPL	1.99× 10−66	17.22	100%
	2BBPL/Band	0.303	1.03	69.71%
Pulse 2, Rising part	Band/BBPL	0.172	1.36	82.76%
(≳1000 counts bin−1)	mBBPL/BBPL	0.118	1.56	88.17%
	2BBPL/BBPL	4.55× 10−22	9.66	100%
	2BBPL/Band	0.374	0.89	62.60%
Pulse 2, Falling part	Band/BBPL	0.0018	3.12	99.82%
(≳1000 counts bin−1)	mBBPL/BBPL	0.0075	2.67	99.24%
	2BBPL/BBPL	1.23× 10−85	19.61	100%
	2BBPL/Band	0.343	0.95	65.64%

Download table as:  ASCIITypeset image

3.3.3. Connection between the Rising and Falling Parts

The smooth variations of the parameters demand that the temperature, kT (of BBPL or mBBPL or 2BBPL), or peak energy, E_peak (of Band model), should be a continuous function of time, even during the pulse peak time. Hence, these parameters should match at the peak within errors to the one predicted by the empirical law. We follow the evolution law of the rising part to predict these parameters in the first bin of the falling part. We compare these values with the corresponding observed values. In Figure 8, we have plotted the observed values with respect to the predicted values. Note that the error bars of the observed values are much less compared with those of the predicted values. The sources of errors in the predicted values are errors in the evolution parameter, μ and the errors in the actual parameter at the starting bin of the rising part. Generally, the parameter μ has large errors, which affects the errors of the predicted values considerably. The data points are essentially the same for both the wider bin (open circles) and the finer bin (filled circles). The dot-dashed line, which shows the equality of the observation and prediction, goes through all the points.

Zoom In Zoom Out Reset image size

GRB 090618 is interesting in many aspects (for details see Ghirlanda et al. 2010; Rao et al. 2011; Basak & Rao 2012a), one being its very high fluence. The redshift (z) of this object is 0.54 and the total fluence is 3398.1  ±  62.0  ×  10⁻⁷ erg cm⁻², when integrated over its duration (182.27 s). In terms of fluence, this is the brightest among all GRBs detected by Fermi. This is a very long GRB with multiple peaks. It has four broad pulses as follows. Pulse 1 is rather a clean precursor from −1 s to 40 s. The second pulse is well separated from this precursor, and occurs from 50 s to 75 s, with two structures in 50–61 s. The third pulse, occurring from 75 s to 100 s, is contaminated with the falling part of the second pulse, and the rising part of the fourth pulse. The fourth pulse occurs from 100 s to 124 s. Though the secondary pulses (i.e., other than the precursor) are sometimes overlapping, we can still examine the spectral variation in the first pulse and in the major portions of the other pulses. GRB 081221 has only one secondary pulse, which makes it more convenient. However, in contrast with GRB 090618, the precursor of GRB 081221 has overlaps with the secondary pulse. In order to compare the results of GRB 081221, we shall take the precursor and the second pulse of GRB 090618.

3.4.1. Precursor Pulse

The time-resolved spectra of this pulse are obtained by requiring minimum of 1000 counts bin⁻¹. We obtain 10 spectra in the rising part (−1.0 to 14.15 s) and 11 spectra in the falling part (14.15 to 40.85 s). For the rising part, the parameters, μ and ν, are obtained by assuming the start time at −10.0 s (see Section 2.3). The results of spectral fitting by different models are reported in Table 7. The advantage of this pulse 1 over GRB 081221 is it is longer and brighter, enabling us to parameterize the rising part. Also, this pulse is fully separated from the secondary events. It is clear from Table 7 that the BBPL model is inferior to the mBBPL model for this pulse. In the case of GRB 081221, we found that the BBPL and mBBPL models are comparable to each other, and marginally better than the Band model. In this case, we definitely need an mBBPL rather than a BBPL for a fit comparable to that of Band. Note that 2BBPL is only comparable, but not better than mBBPL. Hence, the spectrum in the rising part may be mBB dominated. In the falling part, mBBPL is again comparable to Band. The 2BBPL model is superior to all the models in the falling part. The same conclusion was drawn for GRB 081221. Hence, there is hardly any difference of spectral evolution in the precursor pulse between these two GRBs.

Table 7. Study of Spectral Evolution in Pulse 1 (−1.0 to 40.85 s) of GRB 090618

Model	χ2 (dof)	μ	ν	α	β	Epeaka	p	Γ	kTh/kTin/kTa	kTla
Rising part (−1 to 14.15 s, 10 bins)
Band	623.94 (547)	−0.8 ± 0.2	⋅⋅⋅	$-0.46_{-0.09}^{+0.10}$	$-3.07_{-0.61}^{+0.32}$	$344.9_{-22.0}^{+22.1}$	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅
BBPL	661.77 (547)	−0.6 ± 0.2	−1.4 ± 0.4	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	$1.71_{-0.04}^{+0.05}$	$64.79_{-2.46}^{+2.50}$	⋅⋅⋅
mBBPL	621.93 (546)	−0.7 ± 0.2	−0.5 ± 2.0	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	$0.83_{-0.06}^{+0.09}$	$1.69_{-0.10}^{+0.19}$	$125.1_{-8.7}^{+12.5}$	⋅⋅⋅
2BBPL	624.55 (545)	−0.7 ± 0.2	−1.4 ± 0.8	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	$1.72_{-0.08}^{+0.11}$	$79.69_{-5.39}^{+9.19}$	$25.88_{-5.08}^{+9.45}$
Falling part (14.15 to 40.85 s, 11 bins)
Band	602.53 (571)	−1.0 ± 0.3	⋅⋅⋅	$-0.79_{-0.11}^{+0.13}$	$-3.02_{-1.10}^{+0.38}$	$186.2_{-19.2}^{+18.3}$	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅
BBPL	642.37 (571)	−0.9 ± 0.3	−1.4 ± 0.7	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	1.83 ± 0.05	$37.95_{-2.50}^{+2.67}$	⋅⋅⋅
mBBPL	602.30 (570)	−1.1 ± 0.2	1.5 ± 1.5	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	$0.69_{-0.03}^{+0.06}$	$1.70_{-0.19}^{+0.21}$	$85.81_{-7.67}^{+12.27}$	⋅⋅⋅
2BBPL	593.74 (569)	−0.3 ± 0.3	−2.9 ± 1.0	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	$2.11_{-0.12}^{+0.10}$	$50.88_{-4.66}^{+4.88}$	$15.33_{-1.65}^{+1.93}$

Notes. The bins are obtained by requiring ≳1000 counts bin⁻¹. ^aThe values quoted are for the first time bin.

Download table as:  ASCIITypeset image

3.4.2. The Second Pulse

This pulse is more difficult to analyze because it has two small structures in the rising part and two pulses at the peak. Hence, we ignore the 50–61 s of data and analyze only the 61–75 s of data. The falling part covers 64.35–74.95 s, where we obtain 24 time-resolved spectra requiring 2000 counts bin⁻¹. This, in principle, can be lowered as we have higher count rate, but, following the analysis of GRB 081221, we restrict ourselves to a moderate count per bin. The results of joint spectral fit with various models are reported in Table 8. It is clear from this table that BBPL and mBBPL are not the correct models in the falling part. In comparison, the Band model is much better. However, the 2BBPL model, which shows comparable or better fit than Band in all cases, is only comparable to mBBPL in this particular case. This may arise due to the fact that this pulse is actually a combination of two highly overlapping pulses (see Rao et al. 2011). Hence, we redo the analysis on 11 spectra from 69.25 s to 74.95 s, which covers only the falling part of the second pulse. We obtain the following χ² (dof): 1137.67 (991), 1179.13 (990), and 1156.25 (989) for Band, mBBPL, and 2BBPL, respectively. Hence, the Band and 2BBPL models are comparable, though we cannot rule out the possibility of contamination even in this falling part.

Table 8. Study of Spectral Evolution in Pulse 2 (61–75.0 s) of GRB 090618

Model	χ2 (dof)	μ	ν	α	β	Epeaka	p	Γ	kTh/kTin/kTa	kTla
Rising part (61–64.35 s, 1 bin)
Band	2012.93 (1707)	8.0 ± 3.0	⋅⋅⋅	$-0.68_{-0.05}^{+0.06}$	$-2.49_{-0.12}^{+0.09}$	$192.5_{-10.3}^{+11.3}$	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅
BBPL	2173.47 (1707)	5.0 ± 1.0	6.5 ± 5.0	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	1.64 ± 0.02	$42.82_{-1.16}^{+1.17}$	⋅⋅⋅
mBBPL	2099.21 (1706)	8.0 ± 1.5	10.0 ± 5.0	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	0.68 ± 0.02	1.29 ± 0.08	$97.1_{-4.9}^{+6.4}$	⋅⋅⋅
2BBPL	2030.83 (1705)	6.0 ± 1.5	3.0 ± 3.0	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	$1.78_{-0.04}^{+0.05}$	$115.56_{-12.09}^{+13.33}$	$33.19_{-1.73}^{+1.71}$
Falling part (64.35–74.95 s, 24 bins)
Band	1574.79 (1317)	−14.0 ± 1.0	⋅⋅⋅	−0.88 ± 0.03	$-2.74_{-0.10}^{+0.08}$	$262.8_{-7.9}^{+8.5}$	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅
BBPL	2402.54 (1317)	−5.3 ± 0.3	−6.9 ± 0.9	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	1.78 ± 0.01	$53.91_{-1.00}^{+1.01}$	⋅⋅⋅
mBBPL	1794.05 (1316)	−8.0 ± 3.0	15.0 ± 5.0	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	$0.65_{-0.005}^{+0.006}$	$1.58_{-0.10}^{+0.03}$	$157.36_{-5.68}^{+4.18}$	⋅⋅⋅
2BBPL	1794.06 (1315)	−5.5 ± 0.5	−4.0 ± 0.5	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	1.78 ± 0.02	$73.08_{-2.57}^{+3.04}$	$21.72_{-1.44}^{+1.68}$

Notes. The bins are obtained by requiring ≳2000 counts bin⁻¹. ^aThe values quoted are for the first time bin.

Download table as:  ASCIITypeset image

In the rising part (see Table 8), though mBBPL is better than BBPL, it is inferior to 2BBPL and Band. Hence, the rising part of this pulse is probably synchrotron dominated. 2BBPL is comparable to Band. In summary of this pulse, the whole episode can be described by Band model. This is in contrast with the second pulse of GRB 081221, where mBBPL clearly dominates the rising part, and then it is taken over by Band.