THERMAL EMISSIONS SPANNING THE PROMPT AND THE AFTERGLOW PHASES OF THE ULTRA-LONG GRB 130925A

To arrive at a correct spectral model, it is important to fix the continuum, and hence data above ∼10 keV are very crucial. We re-analyze the data whenever a higher energy observation is available (either BAT or NuSTAR observation). These epochs are: (I) 150–300 s during the prompt emission (joint BAT–XRT observations), (II) 1.8 days during the afterglow (joint NuSTAR-XRT observations), and (III) ∼9–11 days (joint Chandra–NuSTAR observations). The observation in epoch II provides the best possible data among all the observations due to the superior sensitivity and resolution of the NuSTAR detectors (and the higher flux compared to epoch III). In Figure 1, we show the spectrum of epoch II, fitted with various models. We follow the standard procedure described by B14 to extract the spectral data. The spectrum is fitted in XSPEC v12.8.2. We fit the following models: (A) PL, (B) PL with a Gaussian absorption (gabs×PL), (C) a blackbody with a power law (BBPL), and (D) two blackbodies with a power law (2BBPL). In all cases we have used a wabs and a zwabs model to account for the absorption in our Galaxy and in the source, respectively. P14 have fitted the time-resolved data during 20 ks–8 Ms with a wabs×zwabs×PL model. As the equivalent hydrogen column density (${N}_{{\rm{H}}}$) at the source should not change over time, they have linked this parameter in all time bins to obtain its precise value—$(2.1\pm 0.1)\times {10}^{22}$ cm⁻². We have used this value, along with the Galactic absorption of $1.7\times {10}^{20}$ cm⁻², to perform uniform fitting with all the models. Keeping the source absorption free gives a value close to P14, within the statistical errors. Note in Figure 1 (panel A) that a PL fit shows a "dip" near 5 keV. We obtain ${\chi }^{2}$/degrees of freedom (dof) = 158.7/124 for a PL fit. Incidentally, a Band function (Band et al. 1993) gives a similar unacceptable fit with ${\chi }^{2}$/dof = 153.0/121. A Gaussian absorber gives an immediate improvement with ${\chi }^{2}$/dof = 100.8/121. However, as noted by B14, we also see a shift in the centroid from 6.4 ± 0.8 keV to $\lt 4.0$ keV from epoch II to epoch III. Hence, the Gaussian absorber is unphysical. A BBPL model gives a reasonable fit with ${\chi }^{2}$/dof = 106.1/122 (see panel C). Finally, we fit the spectrum with a 2BBPL model and obtain ${\chi }^{2}$/dof = 96.7/120. The parameters of the blackbodies are shown in the first two rows of Table 1. We perform an F-test to find the significance of adding the second blackbody. We find a $2.9\sigma$ significance corresponding to a p-value $=3.8\times {10}^{-3}$. The temperature of the two blackbodies in this epoch are ${5.12}_{-1.52}^{+2.79}$ keV and ${0.42}_{-0.10}^{+0.11}$ keV. Note that these values are close to those reported by B14 and P14 (both within $1\sigma$), respectively. Based on the analysis of epoch II, we fit the spectrum of epoch III with a 2BBPL model. We freeze the PL index to the value obtained in epoch II (−3.74). B14 has combined the NuSTAR data of ${T}_{0}+8.8$ and ${T}_{0}+11.3$ days. As both blackbodies may evolve significantly during this time, we use only the data of ${T}_{0}+11.3$ days. We obtain a ${\chi }^{2}$/dof = 146.55/133. The parameters of the blackbodies are shown in the last two rows of Table 1. The temperature of the two blackbodies are ${3.0}_{-0.5}^{+0.6}$ keV and 0.15 ± 0.03 keV. We note that the temperature of the higher-temperature blackbody is somewhat lower than the temperature of the single blackbody as obtained by B14 (${4.0}_{-0.6}^{+0.7}$).

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Table 1.  Parameters of the Two Blackbodies in the Two NuSTAR Observations

Interval	kT	${F}_{\mathrm{BB}}$	${R}_{\mathrm{BB}}$
	(keV)	(erg cm−2 s−1)	108 cm
Observation 1 ($\sim {T}_{0}+1.8$ days)
BB1a	${5.12}_{-1.52}^{+2.79}$	$(5.46\pm 2.58)\times {10}^{-13}$	0.9 ± 0.3
BB2	${0.42}_{-0.10}^{+0.11}$	$(1.25\pm 0.92)\times {10}^{-12}$	$(2.0\pm 1.0)\times {10}^{2}$
Observation 2 ($\sim {T}_{0}+11$ days)
BB1	${3.0}_{-0.5}^{+0.6}$	$(1.3\pm 0.3)\times {10}^{-13}$	1.3 ± 0.2
BB2	0.15 ± 0.03	$(2.0\pm 1.4)\times {10}^{-12}$	$(2.0\pm 1.0)\times {10}^{3}$

Note.

^aBB₁=Higher-temperature blackbody, BB₂=Lower-temperature blackbody.

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In order to see how the temperature of the two blackbodies evolve during the prompt emission, we fit the joint BAT–XRT data in  the 150–300 s interval. We obtain three time-resolved spectra in 150–175, 175–225, and 225–295 s intervals. For each case, we obtain a reasonable fit with reduced ${\chi }^{2}$ close to 1 for 76 dof (see Table 2). In the first interval, the 2BBPL model shows significant improvement compared to a BBPL model with $\Delta {\chi }^{2}=18.6$ at the expense of two dof. The F-test gives a $3.3\sigma$ significance (p-value $=9.3\times {10}^{-4}$) for the addition of the second blackbody. It is worth mentioning that a Band function also gives an acceptable fit to the prompt emission data. However, as already mentioned the Band function is inadequate during epoch II, where we have good quality data. In addition to the joint BAT–XRT data, we also fit a BBPL model to the falling part of the final pulse (105–150 s) and obtain ${kT}={12.9}_{-3.7}^{+4.0}$ keV.

Table 2.  Parameters of the BBPL Model Fitted to the BAT Data in the 105–150 s Interval, and that of the 2BBPL Model Fitted to the Joint BAT–XRT Data in the 150–295 s Interval

Parameter	105–150 s	150–175 s	175–225 s	225–295 s
${n}_{{\rm{H}}}$ (1022 atoms cm−2)	⋯	2.1 ± 0.2	1.7 ± 0.2	1.7 ± 0.2
kT(keV)	${13.0}_{-3.7}^{+4.0}$	⋯	⋯	⋯
N	${0.11}_{-0.06}^{+0.09}$	⋯	⋯	⋯
kTha (keV)	⋯	${8.9}_{-1.7}^{+1.8}$	${7.0}_{-3.5}^{+5.8}$	${8.5}_{-7.1}^{+4.6}$
Nh	⋯	${0.11}_{-0.04}^{+0.05}$	$(3.5\pm 3.2)\times {10}^{-2}$	$({3.2}_{-2.5}^{+3.2})\times {10}^{-2}$
kTl (keV)	⋯	${2.1}_{-0.6}^{+0.5}$	2.1 ± 0.5	${2.3}_{-0.7}^{+0.5}$
Nl	⋯	${0.14}_{-0.06}^{+0.07}$	$({7.6}_{-3.5}^{+4.6})\times {10}^{-2}$	$({7.7}_{-7.4}^{+12.0})\times {10}^{-2}$
Γ	−2.3 ± 0.2	−2.2 ± 0.1	−2.1 ± 0.1	−2.1 ± 0.1
${\chi }_{\mathrm{red}}^{2}$ (dof)	0.93 (54)	1.2 (76)	1.3 (76)	1.1 (76)

Note.

^ah = Higher-temperature blackbody, l = Lower-temperature blackbody.

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In Figure 2, we show the temperature evolution of the two blackbodies (filled symbols—circles for the higher-temperature blackbody and squares for the lower-temperature blackbody). The light curve (15–50 keV flux in units of erg cm⁻² s⁻¹) of the GRB in the BAT and XRT detectors is shown in the background. The star represents the single blackbody temperature during the falling part of the last pulse in the BAT data. For comparison, we have plotted the temperature values obtained by P14 (open squares) and B14 (open triangles). We note that the values of the higher and the lower temperatures are in general agreement with those of B14 and P14, respectively. We fit the evolution of both blackbodies as a PL function of time. In doing this, we assume that all the blackbody temperatures given by P14 are that of the lower-temperature blackbody of the unified 2BBPL model. The slope of the evolution is −0.15 ± 0.05 and −0.20 ± 0.02 for the higher- and lower-temperature blackbody, respectively. The two evolutions can be considered to be either approximately similar or the higher-temperature blackbody to have a slower temperature evolution. Note that due to the large gap in the prompt and afterglow coverage of the high energy observation the error in the slope of the higher-temperature blackbody evolution is large. Hence, the comparison between the two temperature evolutions remains inconclusive. In the inset of Figure 2, we show the time evolution of the flux of the individual components of the 2BBPL model. We have shown only those points where we have high energy observation with either BAT or NuSTAR. In order to compare the flux evolution of the components, the blackbody flux is normalized to the PL flux of the initial time bin. We note that the thermal flux decreases more rapidly compared to the PL flux. The flux of the higher-temperature blackbody evolves even faster.

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It is interesting to calculate the apparent emission radius for a spherical emission region (${R}_{\mathrm{BB}}$) during the two epochs of NuSTAR observation. Both B14 and P14 find that the value of ${R}_{\mathrm{BB}}$ shows a contraction during this time. As we have two blackbodies, we calculate the value of ${R}_{\mathrm{BB}}$ for each of them as follows:

$$\begin{eqnarray}&&{R}_{\mathrm{BB}}=\sqrt{\displaystyle \frac{{d}_{{\rm{L}}}^{2}{F}_{\mathrm{BB}}}{\sigma {[(1+z)T]}^{4}}}\end{eqnarray} \tag{ 1 }$$

where ${d}_{{\rm{L}}}$ is the luminosity distance, σ is Stefan–Boltzmann constant $=\;5.6704\times {10}^{-5}$ erg cm⁻² s⁻¹ K⁻⁴, ${F}_{\mathrm{BB}}$ is the observed blackbody flux, and $(1+z)T$ is the temperature corrected for the redshift, z. The radius of the higher-temperature blackbody in the two epochs is found to be ${R}_{\mathrm{BB}}=(0.9\pm 0.3)\times {10}^{8}$ cm and ${R}_{\mathrm{BB}}=(1.3\pm 0.2)\times {10}^{8}$ cm, which shows a slow increment. Interestingly, the value of the lower-temperature blackbody ${R}_{\mathrm{BB}}$  is found to be ${R}_{\mathrm{BB}}=(2.0\pm 1.0)\times {10}^{10}$ cm and ${R}_{\mathrm{BB}}=(2.0\pm 1.0)\times {10}^{11}$ cm (see the last column of Table 1). Note that this result is markedly different from that obtained by B14 and P14. Our analysis shows a definite increment of the apparent radius at the later epoch.

In Paper I, we proposed a spine–sheath jet to explain the evolution of the two blackbodies. A fast moving spine with a slower sheath component is expected in a number of physical scenarios, e.g., a hot cocoon formed over the GRB jet as it pierces through the envelope of the progenitor (Mészáros & Rees 2001; Ramirez-Ruiz et al. 2002; Zhang et al. 2003, 2004), or a collimated proton jet along with a wider neutron sheath for a magnetic jet (Vlahakis et al. 2003; Peng et al. 2005). Such a structured jet is frequently invoked to explain a double jet break and optical re-brightening (e.g., Berger et al. 2003; Liang & Dai 2004; Holland et al. 2012). If the sheath component of the jet is in fact the cocoon, we expect it to be less collimated and the terminal bulk Lorentz factor (η) to be lower by a factor of ∼10 than the spine.

The observed physical sizes of the two blackbodies show very slow expansion rates and the corresponding sizes of the photospheres would also have slow speeds and hence they cannot be tied to the actual sizes of the outflowing plasma. Hence the photospheres have to be steady/gradually changing regions in the outflow. We note that the spine photosphere produces the higher-temperature blackbody while the lower-temperature blackbody is produced by the sheath photosphere. Note that the size of the emitting regions calculated above are considered to be spherical emission regions. The physical photosphere can be related to the observed size as ${r}_{\mathrm{ph}}=\eta {R}_{\mathrm{BB}}$. With reasonable values of η for the spine and the sheath (say, 10³ and 10², respectively), one should obtain larger values of the corresponding emission radius. The observed properties of the two photospheres can be used to constrain the properties of the spine and the sheath. For example, we have found that the higher-temperature blackbody has a much slower evolution and hence we can assume that the spine remains "steady" for a long time. The sheath photosphere, on the other hand, shows considerable evolution. For a cocoon sheath, a rapid photospheric expansion is indeed expected (Starling et al. 2012).

Evans et al. (2014) systematically analyzed 672 XRT GRBs, and found that after 3 ks of the individual trigger, the hardness ratio (HR) does not show any evolution for the majority of GRBs. For 12 GRBs (including GRB 130925A), which show $\gt 5\sigma$ significance of HR variation at late times, the PL index shows a continuous increment with time. They argue that the HR evolutions in these GRBs including GRB 130925A can be explained by a dust scattering model.

However, the hard X-ray data at late times require emission above 10 keV, which is not compatible with a steep PL model (as required by the XRT data alone). Hence, we explore here whether the spectral evolution of GRB 130925A could be explained by the 2BBPL model and whether this model is a generic feature of all GRBs. We have taken the temperature evolution as given in Figure 2, and verified that the count rate variation in two energy bands in the XRT data (0.3–1.5 and 1.5–10 keV) after 100 ks is compatible with the flux variation of the three components as given in the inset of Figure 2.

The rare occurrence of spectral variation after 3 ks could be explained by postulating that the blackbody components decay at different rates for different GRBs. We investigate this hypothesis by comparing the results of GRB 130925A with those of GRB 090618. In Figure 3, we plot the relevant parameters for these two GRBs, but the time axis of GRB 090618 is arbitrarily stretched (100–300 s data are stretched to 100–2 $\times$ 10⁷ s). The close similarity of the spectral variation of these two GRBs encourages us to hypothesize that the 2BBPL model is a generic feature of all GRBs and the blackbody components decay at different rates for different GRBs.

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