X-RAY SPECTRAL COMPONENTS OBSERVED IN THE AFTERGLOW OF GRB 130925A

GRB X-ray afterglow spectra are usually well-fit by absorbed PL models. We froze a Galactic N_H component of 1.7 × 10²⁰ cm⁻² (Kalberla et al. 2005; Evans et al. 2013) and allowed a varying N_H component at the reported redshift of z = 0.347.

A PL fit to the first-epoch NuSTAR data shows a clear deficit in the residuals in the 5–6 keV region (Figure 2). A joint PL fit including the Swift-XRT data improves the parameter constraints, particularly for N_H, but the residual structure remains. The goodness of fit is poor, with $\chi ^2_\nu = 1.6$ (Table 1). A PL fit to the Chandra data and a joint NuSTAR–Chandra PL fit also show residual structure (Figure 3) and poor goodness of fit, with $\chi ^2_\nu =2.2$. Additional components (Sections 4.2–4.5) improve these fits.

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Table 1. Best-fit Parameters of Spectral Models

Parameter	NuSTAR and Swift Epoch 1				NuSTAR and Chandra Epoch 2
	PL	gabs×PL	BB+PL	PL+PL	PL	gabs×PL	BB+PL	PL+PL
NH (1022 cm−2)	2.55$^{+0.24}_{-0.23}$	2.71$^{+0.41}_{-0.30}$	3.32$^{+0.35}_{-0.34}$	3.35$^{+0.40}_{-0.37}$	1.98±0.14	2.82$^{+0.31}_{-0.439}$	2.74$^{+0.21}_{-0.20}$	3.02$^{+0.29}_{-0.27}$
Γ	3.33±0.13	3.29$^{+0.17}_{-0.14}$	3.96$^{+0.24}_{-0.23}$	4.02$^{+0.33}_{-3.47}$	3.06±0.11	3.95$^{+0.55}_{-0.87}$	3.86$^{+0.20}_{-0.19}$	4.37$^{+0.40}_{-0.12}$
Γ2				1.28$^{+0.65}_{-0.73}$				1.65$^{+0.26}_{-0.29}$
E0 (keV)		5.89$^{+0.53}_{-0.33}$				<4.09
σ (keV)		1.75$^{+1.97}_{-0.70}$				5.15$^{+1.97}_{-2.97}$
τ(E = E0)		0.73$^{+0.12}_{-0.18}$				2.8$^{+1.3}_{-1.8}$
kT (comoving frame, keV)			5.58$^{+2.12}_{-1.15}$				4.02$^{+0.73}_{-0.56}$
χ2/ν	146.8/92	98.1/89	103.1/90	105.9/90	288.8/132	158.9/129	157.7/130	161.0/130
Pχ(X > χ|ν)	2.5E-4	0.23	0.16	0.12	1.0E-13	0.04	0.05	0.03

Note. Errors are 90% C.L.

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Multiplying by a Gaussian absorber (gabs×PL) in the first epoch markedly improves the fit residuals relative to a PL fit (Figure 2). The centroid of the Gaussian absorber is at 5.9$^{+0.4}_{-0.3}$ keV and $\sigma =0.9^{+0.6}_{-0.3}$ keV, both in the observer frame. The Swift data show similar residual structure, and in a joint fit the Gaussian absorber gives a similar centroid (6.0$^{+0.5}_{-0.3}$ keV) but greater width (1.8$^{+1.9}_{-0.7}$ keV; Table 1). In the joint fit, $\chi ^2_\nu$ improves to 1.1 from 1.6 for three additional parameters.

In the second epoch, a Gaussian absorber again improves the fit relative to a PL ($\chi ^2_\nu = 1.2$ from 2.2), but the parameters are poorly constrained. The joint NuSTAR and Chandra fit provides only an upper limit (4.1 keV) on the line centroid. This value is inconsistent with that of the first epoch, and the required line width is substantially larger ($\sigma =5.2^{+2.0}_{-3.0}$ keV, Figure 4). The large shift in the line centroid is difficult to explain with absorption by a single species. If the large linewidth is interpreted as turbulent velocity broadening, this implies relativistic velocities ≳ 0.1c that increase from the first epoch to the second, an unlikely scenario.

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We obtained good fits ($\chi ^2_\nu \sim 1.1$) with an absorbed bremsstrahlung plus PL model (Bremss+PL). The component is well-constrained in both epochs, with best-fit temperatures of 1.3±0.2 and 0.83$^{+0.12}_{-0.11}$ keV in the comoving frame. The fit emission measures are 1.1$^{+0.5}_{-0.3}$ × 10⁶⁹ cm⁻³ and 2.3$^{+0.9}_{-0.6}$ × 10⁶⁸ cm⁻³. These extreme emission measures, if produced by a constant-density medium, would require densities of order 10¹⁰(R/10¹⁶ cm)^−3/2 cm⁻³. However, a circumstellar medium this dense would be optically thick to electron scattering, violating the assumptions of the optically thin bremsstrahlung model. The emitting region would be optically thin only if the radius of the region were >10²⁰ cm, much larger than typical afterglow radii. More complex density profiles would require even higher densities at some locations. Thus while the addition of an optically thin bremsstrahlung spectral component improves the fit to the data, we are unable to construct a self-consistent physical interpretation for it. This problem persists even if instead we require a higher temperature for the bremsstrahlung component in order to fit the high-energy excess. The fit is worse (χ² increases by 5.9 in both epochs) and provides only a lower limit on the temperature (kT ≳ 25 keV in the comoving frame). The emission region must still be larger than 10¹⁸ cm to be optically thin.

Motivated by the presence of possible additional residual structure in the Chandra data in the 1–3 keV range, we attempted to fit mekal and apec plasma emission models to the second-epoch data. With standard abundances, these models fit metallicity values of zero, reproducing the unphysical Bremss+PL model. Even with highly variable abundances, single-temperature plasmas did not provide clear improvements in the fit.

We also fit a blackbody plus PL model (BB+PL). The χ² surface shows two minima for the blackbody temperature in both epochs, one near 5 keV and the second near 0.5 keV. In the first epoch the higher temperature is preferred ($\chi ^2_{\rm low} = 115.3$ versus $\chi ^2_{\rm high} = 103.1$ for 90 degrees of freedom (dof)), while in the second epoch the goodness of fit is closer to equivalent ($\chi ^2_{\rm low} = 156.2$ versus $\chi ^2_{\rm high} = 157.7$ for 130 dof). We argue that the higher-temperature blackbody fit is more plausible due to its relative consistency with the component observed in the first epoch and with theoretical expectations (Section 5).

The blackbody components provide 11% (29%) of the total 0.3–30 keV flux in the first (second) epoch. The implied radii for a spherical emission region are small and consistent with constant size: $1.1^{+0.5}_{-0.8}\times 10^8$ cm and $1.5^{+0.5}_{-0.6}\times 10^{8}$ cm. (The radii for the disfavored low-temperature blackbodies are larger but also relatively compact. However, they imply a physically unlikely contraction of the emitting region from (3.2 ± 0.8) × 10¹⁰ cm to (1.7 ± 0.4) × 10¹⁰ cm.)

While blackbody components have been reported in other GRB afterglow spectra, none have been observed at such late times, with such high temperatures, or with such small radii. At 1–10 days after the burst, the blackbody radius inferred from GRB 101225A was over 10¹⁴ cm and could be explained by the jet interaction with the circumstellar medium (Thöne et al. 2011). The inferred radius of 10⁸ cm for GRB 130925A is much harder to explain with a jet interaction model. This size scale is instead on par with the radius of the fallback accretion disks expected in stellar collapse (Fryer 2009).

If we assume we are observing this disk, the fit temperature can place constraints on the progenitor by constraining the conditions in the disk. The luminosity of an accretion disk is roughly equal to the potential energy released in the accretion. If we consider material at radius r, the luminosity (L) is given by $L=G M_{\rm BH} \dot{m} dr/r^2$ where $\dot{m}$ is the accretion rate and dr denotes a small annulus of material at radius r (integrating over dr would produce the total luminosity). The blackbody emission for such an annulus is L = σAT⁴ = σ2πrdr, where σ is the Stefan–Boltzmann constant and T is the blackbody temperature. If we know the temperature, we can then derive the accretion rate $\dot{m} = (2 \pi r^3 \sigma T^4)/(G M_{\rm BH})$. For our observed temperatures of 4–5.6 keV, the corresponding accretion rate is 10⁻⁹–10⁻¹⁰ M_☉ s⁻¹. Fallback 10⁵–10⁶ s after an SN or GRB explosion has been calculated for a range of progenitors and explosion energies (MacFadyen et al. 2001; Wong et al. 2014). Fallback at late times follows a simple PL (Chevalier 1989) and depends on the progenitor and the explosion energy of the SN associated with the GRB. Most fallback calculations (MacFadyen et al. 2001; Wong et al. 2014) predict fallback rates of 10⁻⁷–10⁻¹⁰ M_☉ s⁻¹ at 10⁵–10⁶ s for SN explosions of 1–3 × 10⁵¹ erg.

Our accretion rates imply a luminosity near 10⁵ times the Eddington limit for a stellar mass black hole. Although such extreme super-Eddington emission rates have been invoked from fallback (Dexter & Kasen 2013), the exact nature of such transient accretion is not well known. Steady-state solutions of disk accretion find that maintaining emission rates even an order of magnitude above Eddington is difficult (Jaroszynski et al. 1980). Whether such steady-state limits apply in transient situations like our fallback disk remains to be seen (Abramowicz 2005). Thus without a full model of these transient events, we are not able to establish a self-consistent explanation for the blackbody emission.

Finally, we considered a two PL model (PL+PL) like that reported for GRB 111209A (Stratta et al. 2013). This model is a slightly worse fit in both epochs than the BB+PL model for the same number of free parameters (Table 1).

Stratta et al. (2013) interpret the very hard (Γ ∼ 0) second PL component they report for GRB 111209A at 70 ks after the burst as the tail of the hard PL emission sometimes observed by Fermi-LAT (e.g., Zhang et al. 2011 and references therein). This component is detected in the late prompt and early afterglow phases and decays according to a PL; its physical origin remains uncertain. The non-detection by LAT of both GRBs complicates this interpretation. An extrapolation of our epoch 1 PL flux to the 0.1–10 GeV band gives a photon flux of 3 × 10⁻⁶ photons cm⁻² s⁻¹, a value higher than the upper limit of 7 × 10⁻⁷ photons cm⁻² s⁻¹ reported by Kocevski et al. (2013) in the first 2 ks after the burst, when the afterglow—and thus presumably the hard component—was much brighter. The problem is even more severe for the component reported by Stratta et al. (2013): its higher flux and much harder spectral index extrapolate to a 0.1–10 GeV photon flux of 1.5 photons cm⁻² s⁻¹, an extremely high value sufficient to trigger the LAT. We examined the late-time LAT data for both bursts and confirm no excess emission. Consistency with the nondetection by LAT in both cases thus requires a cutoff above the NuSTAR and XMM bandpasses but below the LAT bandpass at 30 MeV. This phenomenological model is plausible, but the connection of these components to the early-time hard PL components detected by LAT in other GRBs therefore remains speculative.

We verified the significance of the additional spectral components using Monte Carlo simulations according to the method of posterior predictive p-values (Protassov et al. 2002). We initialized each fit by stepping the additional feature through a grid in energy and finding the largest relative improvement in χ² (cf. Hurkett et al. 2008). This procedure accounts for the "look-elsewhere" effect of multiple trials, as we have no a priori expectation of the observed line energy or component temperature. In none of our 10⁴ simulated realizations of a null PL model did fits with alternative models (gabs×PL, BB+PL, or PL+PL) produce improvements in χ² as large as observed in the real data. This implies that the spectral features are significant at >3.9σ in both epochs: the χ² improvement for each model fit is extremely unlikely to be due to chance if the true underlying model were simply an absorbed PL.