TIME RESOLVED SPECTROSCOPY OF SGR J1550−5418 BURSTS DETECTED WITH FERMI/GAMMA-RAY BURST MONITOR

We started by fitting the COMPT model to all selected 4 ms time bins of each burst. The fit parameters of some time-bins within the bursts were not well constrained due to a low number of counts. In such cases, we binned the data with a coarser time resolution to accomplish a minimum constraint of 3σ on E_peak, which is the peak energy of the COMPT spectrum. As a result, 14 bursts have less than 5 time bins for spectral fitting, and we removed these bursts from our sample. Our final sample for spectral analysis with the COMPT model is, therefore, 49 bursts with a total number of 1393 time bins, for which we studied the correlations of the COMPT model parameters (E_peak and PL index) with the 8−200 keV flux.

We first studied the spectral evolution with the flux of each burst separately, plotting E_peak as a function of flux. To identify the trend of the evolution, we fit this correlation with a PL and a broken PL (BPL) model. The choice for a BPL model is based on the time-integrated spectroscopy results of SGR J1550−5418 (V12), and the time-resolved spectroscopy of SGR J0501+4516 (Lin et al. 2011) which indicated a more complex relation between E_peak and flux than a PL (e.g., see Figure 1). An F-test showed an improvement in the fit, with a >99.99% non-chance occurrence, for 19 out of the 49 bursts. A more relaxed F-test criterion of >95% results in 37 out of the 49 bursts being better represented by a BPL than a PL (see, e.g., Figure 1). Interestingly, the flux values at which the break occurs for these 37 bursts seem to cluster around a common value and do not depend on any of the burst parameters, e.g., T₉₀, average flux, etc. A Gaussian fit to the distribution of these break flux values results in F ≈ (1.0  ±  0.2) × 10⁻⁵ erg s⁻¹ cm⁻² (Figure 2, upper left panel).

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For the remaining 12 bursts, a single PL model was a better fit to the E_peak versus flux evolution. However, examining these bursts closely, we noticed that none of them had more than five data points above (7 out of 12) or below (5 out of 12) the break flux level of the majority of events. Figure 2 displays these single PL trends of the seven and five events in the lower left and lower right panels, respectively. It is unclear whether these events would have followed the same break trend described above had they had more intense or faint data points, respectively. Finally, we studied the correlation between E_peak and flux for all 49 bursts simultaneously, and we show this result as dark-gray points in the upper right panel of Figure 2. Here we binned the data in flux requiring 40 individual data points in each bin, with the last bin (at the highest fluxes) having a slightly lower number of points. Each black dot in Figure 2 represents the weighted averages (in flux and E_peak) of these 40 individual measurements, while the error bars correspond to their 1σ deviations. Although not shown here, we also binned the data sorted by E_peak and tried a different number of data points per bin (from 20 to 60); we obtained consistent results. A BPL fit to the binned data is preferred at the >99.99% level over a single PL model. The BPL fit has a low-flux PL index of −0.18 ± 0.02 and a high-flux PL index of 0.12  ±  0.02, with the break between the two PL regimes at a flux of (9.1 ± 0.7) × 10⁻⁶ erg s⁻¹ cm⁻² (8−200 keV) and a break E_peak of 32 ± 2 keV, consistent with the 37 burst only fit. Finally, our PL indices and the break flux and E_peak values are similar to the values found in the time-integrated spectroscopy of SGR J1550−5418 bursts, and also in the time-resolved spectroscopy of SGR J0501+4516 bursts (Lin et al. 2011). These results are further discussed in Section 4.

V12 have shown that the index of the COMPT model in time-integrated spectroscopy is distributed narrowly around −1. Therefore, the OTTB model was an equally good fit to their time-integrated spectra, since it is similar to a COMPT model with an index of −1. We show in the top panel of Figure 3 the distribution of the index for our time-resolved analysis. The distribution is broader than for the V12 time-integrated study and the peak has now shifted to a higher value of −0.55 ± 0.58. To understand what prompted this shift, we plot in the bottom panel of Figure 3 the index as a function of flux. Below the flux value of ∼10⁻⁵ erg s⁻¹ cm⁻², identified as the break point in the E_peak−flux correlation, the index appears constant with flux, with a value around ∼ − 0.8. It is only above the break where the indices become gradually steeper, following the same trend observed between the E_peak and flux values. We also note that, fitting some of the time bins with the highest fluxes with the OTTB model, results in a bad fit, as one would expect if the index were deviating significantly from −1.

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We fit the initially selected 63 bursts (see Section 2) with a 2BB model, using our 4 ms time bins. We binned parts of the light curves for significance, i.e., to different time resolutions to accomplish a 3σ constraint on the low and high BB temperatures. Since the 2BB model has more free parameters than the COMPT one, coarser time bins were required. Again, we removed bursts from the initial sample with less than 5 time bins after re-binning, resulting in 48 bursts and 994 individual time bins.

We examined several correlations between the fit parameters of the 2BB model, i.e., (1) temperatures of the low temperature (low-kT) BB versus high temperature (high-kT) BB, (2) the individual fluxes of the 2 BBs, (3) the emitting areas corresponding to the low-kT and high-kT BBs, and (4) the radii of the emitting areas versus temperature for both BBs simultaneously. The effective radii of the emitting regions were calculated as in Lin et al. (2011), R² = (FD²/σT⁴) km², where F is the flux in erg s⁻¹ cm⁻², D is the distance to the source in km (here assumed as 5 kpc; Gelfand & Gaensler 2007; Tiengo et al. 2010), σ is the Stefan–Boltzmann constant, and T is the temperature in Kelvin. We used the same approach as for the COMPT model parameters to identify the trends of these different correlations, fitting them for each burst separately to a PL and a BPL model and performing F-tests to evaluate the improvement in the fits.

For 26 of the 48 bursts, we find that the temperatures of the low-kT BB versus high-kT BB are best fit with a BPL (>95% probability for an improvement from a PL fit to occur by chance) with indices and breaks consistent with each other. The rest of the bursts follow similar but not statistically significant trends, due to a low number of data points. Figure 4, upper left panel, shows the relation between low-kT and high-kT values for all 48 bursts simultaneously. Here (and in all panels of Figure 4), we binned the data at 50 points per bin (black dots). We find that a BPL is a better fit to the data (black solid line) compared to a single PL, at the >99.99% confidence level. Above the break, the relation is much steeper than below the break with slopes of 0.7 ± 0.2 and 0.2 ± 0.1, respectively. The corresponding break is at (4.8 ± 0.3, 13.1 ± 0.4) keV. These results are different than the time-integrated spectroscopy of these bursts, which showed only a positive correlation between the low- and high-kTs, with a very weak hint for a possible break at low temperatures (V12); in that work, the slope of the single PL fit was much steeper, i.e., 1.86 ± 0.09.

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To rule out any systematic effect as the origin of the break, and in particular the fact that the temperature of the low-kT BB falls below the Fermi/GBM energy coverage, we looked at Lin et al. (2012), who studied the combined spectra of the SGR J1550−5418 bursts simultaneously seen with Swift/XRT and GBM, thus extending the energy coverage down to 1 keV (see also Section 3.3). The authors found that the low- and high-kT temperatures derived solely with GBM are equal to the ones derived from the joint XRT/GBM fits. The only deviations (<1σ) between the two results are noticeable at very low temperatures (below 4.8 keV for the low-kT), where the GBM-only fits gave slightly (but consistently) higher low-kT temperatures than the joint ones. Therefore, if the break we see in our relation is due to systematic effects, one would also expect another break below a certain low-kT, where the relation becomes steeper.

The areas of the emitting regions as well as the fluxes of the low- and high-kT BBs are positively correlated, with consistent single PL slopes for all 48 bursts in our sample. These correlations are shown at the upper right and lower left panels of Figure 4 (the areas are represented as R², with R the radius). The PL slope of the correlation between the areas is 1.2 ± 0.3, while for the fluxes it is 0.7 ± 0.1. These slopes are consistent with the slopes for the time-integrated analysis of SGR J1550−5418 GBM bursts, which were 1.34 ± 0.04 and 0.83 ± 0.02, respectively. The slope of the correlation between the low- and high-kT BB fluxes is close to the 1 to 1 relation (dashed line), indicating that the BBs contribute almost equally to the total burst flux. We note that in the case of time-resolved spectral analysis of SGR J0501+4516 bursts this slope is 1.1 ± 0.1 (Lin et al. 2011), while for SGR 1900+14, it is 0.70 ± 0.03 (Israel et al. 2008).

Besides comparisons between the two BB components, we investigated the relation between the emission area and temperature, for both the low- and high-kT BB simultaneously. Most of the bursts showed a similar trend between these two parameters, namely, that the area of the low-kT BB decreases with temperature at a different pace than the area of the high-kT BB. This same trend is seen when combining all the individual time bins, as shown in the lower-right panel of Figure 4. A BPL fit to the binned data has slopes of −1.0 ± 0.3 and −6.4 ± 0.5 for the low-kT and high-kT correlations, respectively, and a break at a temperature of ∼7.9 keV.

To determine the significance of this last relation, we calculated the Spearman rank correlation coefficients of the low- and high-kT BB areas versus their corresponding temperatures. We found a correlation coefficient r ≈ −0.7 and a probability of ∼99.9% that the low-kT area is correlated to the temperature, while for the high-kT BB, we found a stronger correlation r ≈ −1.0 and a probability >99.99%. The time-integrated spectroscopy of the SGR J1550−5418 bursts (V12) hint at a similar relation for the high-kT BB; however, the low-kT BB did not show any correlation between area and temperature.

Finally, we investigated the evolution of temperature and area with flux, F. The upper left panel of Figure 6 (here we binned the data at 40 points per bin, i.e., black dots) shows the relation between the low- and high-kT areas (corresponding to the high- and low-kT BB temperatures) and fluxes. We again fit these correlations with a PL and a BPL to look for any possible breaks. It is clear that the area of both the low and high temperature BB increases with increasing flux. There is, however, a tendency for the low-kT area to flatten at the highest fluxes, while the high-kT area increases steadily. Indeed, the low-kT area as a function of flux is best fit with a BPL with slopes of 1.3 ± 0.2 and 0.5 ± 0.1, respectively, and a break at F = (3 ± 1) × 10⁻⁶ erg s⁻¹ cm⁻², whereas the high-kT area is best fit with a single PL with a slope of 1.3 ± 0.1. A Spearman rank test shows that the correlation coefficients and probabilities for the low- and high-kT areas as a function of flux are r = 0.8, p > 99.99% and r = 0.7, p > 99.99%, respectively. We estimate that about 92% of the low-kT-area data points with F > 10^−4.5 erg s⁻¹ cm⁻² land at ≲ 10³ km², which corresponds to the 1σ limit of the highest-flux binned data point.

The temperatures of the 2 BBs follow opposite trends with respect to flux, as can be seen in the upper right panel of Figure 6. The low-kT appears to positively correlate with flux, only above a break limit. This relation is best fit with a BPL, with slopes of 0.0 ± 0.1 and 0.13 ± 0.02, below and above F = 4 ± 2) × 10⁻⁶ erg s⁻¹ cm⁻². The high-kT, on the other hand, shows a mild negative correlation with flux, and it is best fit with a single PL with a slope of −0.04 ± 0.02. The Spearman rank correlation coefficients and probabilities are r = 0.5, p > 99.99%, and r = −0.3, p > 99.99%, for the low- and high-kT as a function of flux, respectively.

Given the correlations between the area and temperature with flux, next we examined the effect of the flux on the relation between area and kT. We created a color-coded plot by flux of these two parameters, which is shown in the lower left panel of Figure 6. We note that the relation between area and kT remains the same across all flux values, decreasing more steeply for the high temperatures compared to low temperatures. We show this also by separating our data into four flux ranges (F < 10^−5.5, 10^−5.5 < F < 10^−5.0, 10^−5.0 < F < 10^−4.5 and F > 10^−4.5) and fitting a PL and a BPL to the data in each flux range separately (Figure 6, lower right panel; flux groups have been shifted for clarity). For the three highest flux ranges, we find that a BPL is preferred over a PL fit at the >99.99% level, while for F < 10^−5.5 a BPL is only required by the data at the 90% level. Table 1 shows all fit results.

We performed simulations with RMFIT (rmfit4.3) to determine the parameter space for the COMPT and 2BB models, for a typical magnetar burst spectrum detected with GBM.

We first checked whether we can recover the different spectral parameters we get from the real data through simulations of a large set of synthetic spectra. We chose three bursts with at least 10 bins at low fluxes (<5 × 10⁻⁶ erg s⁻¹ cm⁻²). We then generated 1000 synthetic spectra for each bin and each relevant detector. Each synthetic spectrum was based on the sum of the predicted source counts and the measured background counts in each energy channel. The former counts were computed from the analytical function used to fit the real data (COMPT or 2BB) folded with the instrumental response function of the relevant detector. The background counts were estimated for each detector from the real data. Next, we applied Poisson fluctuations to the summed counts to obtain the final synthetic spectrum (see also V12; Guiriec et al. 2011). The resulting synthetic spectra were fitted with the model used to generate them. All input spectral parameters of both models were well recovered in these fits by comparing them and their statistical errors to the simulated parameter distributions and their uncertainties.

Next, we generated 1000 synthetic spectra using random input spectral parameters for the COMPT and 2BB models, decreasing the 8–200 keV flux from 4 × 10⁻⁶ erg s⁻¹ cm⁻² down to 10⁻⁷ erg s⁻¹ cm⁻², in steps of 0.2 dex. We then checked if the input spectral parameters were recovered within 1σ, at the 90% confidence level.

For the COMPT model, we varied the index between three different values (α = −0.5, −1.0, −1.5), in order to have a better handle on E_peak. The COMPT parameter space is shown in the upper right panel of Figure 2, for all three values of the index; for each value, the area below the line cannot be retrieved with the instrumental sensitivity of GBM. It is clear from the simulations that even with a soft spectrum (α = −1.5), we should have been able to detect bursts with E_peak as low as 10 keV for an 8–200 keV flux of ∼10⁻⁶ erg s⁻¹ cm⁻². The flux level goes down to 3 × ∼10⁻⁷ erg s⁻¹ cm⁻² for a burst with a hard spectrum, α = −0.5. Hence, we can conclude that the break we see in the E_peak −flux (and α −flux) relation at ∼10⁻⁵ erg s⁻¹ cm⁻² is not due to any systematic effects. We note that a very soft burst with α = −2.0, would have been detected down to an E_peak ≈ 16 keV had it had a flux of ∼10⁻⁶ erg s⁻¹ cm⁻².

For the 2BB model, and similar to COMPT, we varied high-kT in a pre-defined set of values of 10, 15, and 20 keV, in order to have a good handle over the low-kT spectral parameters, which are harder to determine with GBM. We made sure to vary the normalizations of the two components in a way that both BBs contribute equally to the total flux, according to the well-established relation between the low- and high-kT fluxes (Figure 4; Israel et al. 2008). We first found that the temperature of the high-kT component has little effect on the constraint of the low-kT spectral parameters. In the upper-right panel of Figure 6, we show the parameter space of the low-kT temperature with total flux. The parameter space in the case of the 2BB model is smaller than in the case of COMPT, which is expected considering the extra free parameter. Nonetheless, according to our simulations, we should have been able to detect bursts with temperatures falling in the area of the extrapolation of the low-kT relation with flux, seen above the break, hence concluding that the break we see in the kT −flux relation is real, and not due to systematic effects.

A large number of bursts were emitted by SGR J1550−5418 during the 2008–2009 active episode, especially on 2009 January 22 when the source emitted hundreds of bursts in nearly 24 hr. These bursts were detected by a multitude of high-energy space instruments. Savchenko et al. (2010; see also Mereghetti et al. 2009) analyzed about 100 bursts that were detected on 2009 January 22 with the INTEGRAL SPI−ACS and ISGRI detectors. Scholz & Kaspi (2011) presented a detailed time-averaged spectral study of hundreds of bursts from SGR J1550−5418 detected with Swift/XRT. Finally, Enoto et al. (2012) studied 13 very weak bursts detected with Suzaku from SGR J1550−5418 (F ≲ 10⁻⁸ erg cm⁻²). The better energy coverage and sensitivity of Fermi/GBM compared to these instruments allowed for a more detailed time-averaged spectroscopy of hundreds of bursts from SGR J1550−5418 (V12; see also von Kienlin et al. 2012). A COMPT model and a 2BB model were used to fit the time-integrated spectra, and V12 looked for correlations between the different fit parameters. For the COMPT model, V12 found a hint for a weak double correlation between E_peak and the flux as opposed to the strong one we find here (Figure 2). This could be attributed to the fact that only very few bursts showed an average flux above the flux break limit that we establish here (i.e., F ∼ 10⁻⁵ erg s⁻¹ cm⁻²). On the other hand, for the 2BB model, V12 found a strong positive correlation between the low-kT fluence and high-kT fluence with a slope of 0.83 and between the low-kT area and the high-kT area with a slope of 1.34. These results are in perfect agreement with the results we derive for our time-resolved spectroscopy (Figure 4, but with fluxes instead of fluences). However, the area−kT relation is clearer in our analysis than in V12, where we show that the low-kT area decreases with temperature, only at a slower pace than the high-kT area. This could be due to a combination of better statistics (more data) when performing time-resolved spectroscopy and/or the fact that the relation might be smeared out when performing time-integrated spectroscopy. Lin et al. (2012) studied the bursts of SGR J1550−5418 seen simultaneously with Swift/XRT and GBM, hence, broadening the energy coverage down to ∼1 keV. Interestingly, fitting the time-integrated spectra of these bursts with the 2BB model, Lin et al. (2012) recovered, although at a much lower significance (most likely due to much lower statistics), the trend we see here between the areas and temperatures of the 2BBs (Figure 4, lower right panel). We also note that the values of our fit parameters are comparable to results derived from time-integrated spectroscopy of the bursts of other sources, e.g., SGR 1900+14, SGR 1806–20, and SGR J0501+4516 (Feroci et al. 2004; Nakagawa et al. 2007; Lin et al. 2011).

Detailed time-resolved spectroscopy on a large number of bursts with good statistics have been performed for SGR 1900+14 (Israel et al. 2008), and SGR J0501+4516 (Lin et al. 2011). Israel et al. (2008) have studied the 2006 March 29 burst forest emitted by SGR 1900+14, when more than 40 bursts were detected by Swift/BAT, 7 of which were intermediate flares (see also Olive et al. 2004; Lenters et al. 2003; Ibrahim et al. 2001). The authors found that a 2BB or a COMPT model explains the spectra best. For the 2BB model, they found average temperatures of 4.8 ± 0.3 and 9.0 ± 0.3 keV, and radii of 30 ± 2 and 5.7 ± 0.5 km, for the low- and high-kT BB, respectively. The average temperatures and radii for SGR J1550−5418 that we find are similar within ∼2σ uncertainties, 〈low-kT〉 = 4.8 ± 0.7 keV, 〈high-kT〉 = 14 ± 2 keV, and $\langle R_{\rm low{\rm -}kT}\rangle =20\pm 4$ km, $\langle R_{\rm high{\rm -}kT}\rangle =3.0\pm 1.0$ km.

However, the correlations between the 2BB-model parameters that Israel et al. (2008) found for SGR 1900 + 14 differ slightly from the one we find here for SGR J1550−5418. For instance, Israel et al. (2008) found that the low- and high-kT luminosities contributed almost similarly to the total energy up to a luminosity of ∼3 × 10⁴⁰ erg s⁻¹, above which the low-kT luminosity appears to saturate, and the high-kT luminosity increases steadily to ∼3 × 10⁴¹ erg s⁻¹. We do not see a similar effect for SGR J1550−5418 in the lower-right panel of Figure 4, where the luminosities of the low- and high-kT components are positively correlated up to ∼3 × 10⁴¹ erg s⁻¹, an order of magnitude higher than the ∼3 × 10⁴⁰ erg s⁻¹ limit (red dotted-dashed line) derived with Israel et al. (2008).

Finally, the R² − kT correlation that Israel et al. (2008) found is similar to the one we find here but only for their brightest events with luminosities >3 × 10⁴⁰ erg s⁻¹, where the area of the low-kT decreases with temperature, at a slower pace than the area of the high-kT component. Below this luminosity, the relation for both the low- and high-kT areas changes sharply. Although we see a change in the area−kT correlation with flux for SGR J1550−5418 (Figure 6, lower panels), it is more gradual than the sharp turnover that Israel et al. (2008) reports for SGR 1900 + 14.

Lin et al. (2011) studied the time-resolved spectra of the five brightest Fermi/GBM bursts emitted by SGR J0501+4516, fitting them with a COMPT model and a 2 BB model. The authors found a clear double correlation between E_peak and the flux with a break at a flux value of ∼10⁻⁵ erg s⁻¹ cm⁻². Although this value is consistent with the one we derive for SGR J1550−5418 (Figure 2), this could be a pure coincidence, considering the difference in distance between the two sources (2 and 5 kpc for SGR J0501+4516 and SGR J1550−5418, respectively¹²). Moreover, the slopes of the BPL of the E_peak–flux relation for SGR J0501+4516 are in agreement, within uncertainties, to the values we report here for SGR J1550−5418 (Section 3.1). For the 2BB model, Lin et al. (2011) found (similar to Israel et al. 2008 and our results) that the high-kT area decreases faster than the low-kT area as a function of temperature. The authors did not find a correlation between kT or area and flux, although, the quality of the data was not constraining.

Finally, we note that the large amount and superb quality of data that we have in hand helped reveal new trends between the fit parameters of the COMPT and, especially, the 2BB model, showing the spectral evolution during short bursts with unprecedented detail. For the COMPT model, we established for the first time a strong positive correlation between the index and flux, above the break flux value of ∼10⁻⁵ erg cm⁻² s⁻¹; the exact value of the E_peak–flux relation turnover. Noteworthy new trends for the 2BB model are (1) the increase in areas with flux, accompanied with a decrease in temperature, for the high-kT component; (2) the increase in low-kT areas with flux while kT remains constant, until they reach a common break point in flux where the area starts to saturate and kT starts increasing; and (3) the smooth change in the slopes of the BPL fit to the area−kT relation, and disappearing at the lowest fluxes where a PL is sufficient to explain the area−kT relation, with a slope of ∼ − 4. We discuss the physical interpretation of these new results in the context of the magnetar model in the next section.

In the following subsections, we interpret our results in the context of the magnetar model. We note first that the similarity in the evolution of the 2BB and COMPT model parameters with flux for all SGR J1550−5418 bursts indicates that they were all triggered, more or less, by the same mechanism. Moreover, all the correlations that we find here do not seem to depend on the intrinsic parameters of a given burst, i.e., independent of the average flux, T₉₀, fluence, etc., implying that the radiative region is developing, and subsequently cooling, in a similar fashion during all bursts.

4.2.1. Two Black Body Model

There are several models that describe the triggering mechanism of magnetar short bursts, most famously, the model of Thompson & Duncan (1995, TD95 hereafter; see also Heyl & Hernquist 2005). TD95 considered the possibility that departure from magnetostatic equilibrium in the internal magnetic field of a magnetar, for instance, due to Hall drift and ambipolar diffusion, will cause a build-up of crustal stresses. These stresses, with the presence of the ultra-strong magnetic field, will result in the cracking of the crust, and the injection of an Alfvén wave into the magnetosphere. Energetic particles are accelerated throughout the region of the Alfvén wave, creating a trapped fireball of photon-pair plasma in the magnetosphere. Another model for triggering SGR bursts involves a magnetospheric reconnection caused by magnetic instability in the magnetosphere (Lyutikov 2003; Gill & Heyl 2010). This model can lead to the formation of a plasma fireball trapped inside the magnetospheric flux lines (Gill & Heyl 2010) and also to thermal emission from heating of the magnetar surface. We note that the crust cracking model can lead to significant magnetospheric reconnection, since the external magnetic field will feel any instability in the internal one.

Regardless of the triggering mechanism, the trapped fireball formed during the SGR bursts is hot, T ≳ 50 keV, with an extremely high density (TD95). Hence, the radiative mechanism of short SGR bursts is the cooling of a hot, optically thick plasma, confined by an ultra-strong magnetic field (B > B_QED). In this regime, the dominant energy exchange process is Compton scattering, and due to the high optical depth in the fireball, photons will most likely be thermalized. The diffusion time scale across the fireball at a distance R from the NS surface is much larger (∼10⁴ s; TD95) than the typical SGR burst time scale (∼0.2 s) due to the enormous optical depth through the pair plasma. Hence, the fireball likely loses energy as its cool surface layer propagates inward (TD95). A gradient in the magnetic field is expected throughout the fireball due to its large coronal volume (B∝R^{−(2 + p)}, 0 < p < 1 depending on the twist of the external dipole field, and R is the distance from the NS surface, Thompson et al. 2002), causing a gradient in the effective temperature of the fireball at different heights above the NS surface (TD95). The true spectrum would then be a distorted multi-color BB.

Alternatively, Israel et al. (2008; see also Kumar et al. 2010) attributed the 2BB model, which they used to fit the SGR 1900 + 14 bursts, to the effect of the strong magnetic field (B > B_QED) on the scattering cross-sections of the two polarization states of the photons (Herold 1979; Meszaros et al. 1980). In super strong magnetic fields, E-mode photons (photons with electric vector perpendicular to B) have a much reduced cross-section and can potentially diffuse out from deep inside the magnetosphere very close to the NS surface, equivalent to the high-kT component. O-mode photons, on the other hand, have a much higher cross-section and diffuse out from further out in the magnetosphere at large radii, corresponding to the low-kT component (TD95; Lyubarsky 2002). However, van Putten et al. (2013), modeling hydrostatic atmospheres in super-strong magnetic fields, showed that the E- and O-mode photospheres are very close in both temperature ($\Delta kT_{\rm {\rm {E}}-{\rm {O}}}\sim 1$ keV) and location, in sharp contrast to our findings (and those of Israel et al. 2008) between the low- and high-kT components. We note that the true picture is even more complicated due to the fact that the evolving fireball during a given burst will sample a large range of field strengths and orientations, thereby mixing the radiative transfer elements of Compton opacity.

One way of overcoming the complications mentioned above, is if the 2BB components represent two physically, and geographically distinct emitting regions, each with their own E-mode and O-mode photospheres. According to the distribution of areas and temperatures of our 2BB modeling, one would expect a hot one located close to the NS surface (or a hot spot on the surface), and a cooler one located further out in the magnetosphere. The large spatial separation between the two emitting areas (∼2 orders of magnitude in R²) will most likely induce a profound difference in magnetic field morphology between these two zones. Nonetheless, the strong positive correlations that we find between the areas and fluxes of these two regions indicate that they are very strongly connected and are emitting at similar rates.

How do these two emitting regions evolve during a burst? For the high-kT emitting region, the decrease in temperature (although small) and increase in areas with flux (Figure 6, upper panels, which is also true within each burst) is a clear indication of an adiabatically expanding emitting region, in line with the fireball model. The area of this emitting region ranges from a small hot spot on the NS surface (∼0.2 km) up to a radius of ∼5 km. This radius is consistent with the size of the effective radiative zone of the footpoints of the plasma fireball according to TD95 (considered to be the E-mode photosphere), which is about R_NS/2. Also, the temperature of this emitting region varies slightly, ranging on average from 10 keV to 15 keV (Figures 5 and 6), although fluxes vary by more than three orders of magnitude. This is also in line with the cooling fireball model, which predicts a slight spectral variation of the emergent spectrum from the emitting region (TD95).

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Interpreting the evolution of the low-kT component with flux is not straightforward. The temperature of this emitting region appears constant at very low fluxes while the area increases at a similar rate as the high-kT area (i.e., with the same slope of ∼1, Figure 6, upper panels). At F ≈ 10^−5.5 erg s⁻¹ cm⁻², the area starts to saturate while the temperature starts increasing. Since both BB components emit radiation at the same rate (Figure 4, lower left panel), an increase in temperature for the low-kT component with increasing flux, while its area starts saturating, is inevitable in order to maintain the same energy density as the high-kT component.

At the highest fluxes, the radius of the low-kT emitting area seems to reach a maximum of R_max ∼ 30 km (see Section 3.2; and upper left panel of Figure 6). This radius is much bigger than the NS radius (assuming surface emission), even if we include the effect of gravitational lensing, R_app = R(1–2GM/Rc²)^−1/2, where R_app is the apparent radius of the NS at infinity, R and M are its radius and mass (Pechenick et al. 1983; Psaltis et al. 2000; Özel 2013). In fact, considering even the largest and most massive NSs allowed by theoretical equation of states (Lattimer & Prakash 2001), we have R_app ≈ 20 km (Figure 3 in Lattimer 2012), much lower than the saturation radius of the low-kT emitting region. On the other hand, in the context of the fireball model, where the surface area $A_{\rm max}=\pi R_{\rm max}^2$ is a closed magnetospheric magnetic field bundle that anchors the emitting plasma, the saturation at such a level is expected since, for obvious geometrical reasons, it is difficult for the emitting region to be significantly larger than the NS area.

In the crust-cracking model for triggering SGR bursts (TD95), this saturation radius, R_max ∼ 30 km, of the high-kT component has very important consequences. The frequency of the Alfvén wave responsible for the creation of the photon-pair plasma fireball is, ν_max = c/R_max ≈ 10⁴ Hz¹³ (which is consistent with a characteristic seismic mode frequency for NS crusts; Blaes et al. 1989). This harmonic excitation of the magnetosphere, like any other, has a minimum characteristic excitation radius, R_ν (TD95). This excitation radius cannot exceed the radius of the magnetic loop, R_Loop, trapping the plasma fireball, R_ν < R_Loop, as this will result in an inefficient excitation (TD95).

The maximum effective area of the low-kT component, $\pi R_{\rm max}^2$, could be assumed to be a fraction, l₅/R_*, of the expected projected area of the relevant magnetic field loop, $\pi R_{\rm Loop}^2$, where l = l₅ × 10⁵ cm is the length scale of the surface crack (TD95), and R_* is the NS radius (assumed to be 10 km). This implies that

$$\begin{equation} R_{\rm Loop} \sim (R_*/l_5)^{1/2} R_{\rm max} \sim 100\, {\rm km,} \end{equation} \tag{ 1 }$$

assuming R_max = 30 km. This l₅/R_* scaling comes from the fact that the derived radius, R_max, of the low-kT effective area spans magnetic field lines anchored over a large angle of order unity on the stellar surface, whereas in the case of a magnetic loop with small area footpoints (corresponding to the hot BB effective area), a smaller angle of order ∼l₅/R_* is expected. Finally, according to the star-quake model, the minimum excitation radius of the Alfvén wave is (TD95),

$$\begin{equation} R_\nu \sim 10 B_{15}^{-2} \left(\frac{\theta _{\rm max}}{10^{-3}}\right) \left(\frac{V_\mu }{1.4\times 10^8\, {\rm cm\; s^{-1}}}\right)^{-1} l_5\, {\rm km}, \end{equation} \tag{ 2 }$$

where B = B₁₅ × 10¹⁵ G is the core magnetic field strength, θ_max ∼ 10⁻³ − 2 × 10⁻¹ is the yield strain of the crust (Horowitz & Kadau 2009), and V_μ ∼ (0.4–1.4) × 10⁸ cm s⁻¹ is the shear wave velocity (Steiner & Watts 2009; Douchin & Haensel 2001). As mentioned above, since R_ν < R_Loop ∼ 100 km, this yields a crustal internal magnetic field,

$$\begin{eqnarray} B & \gtrsim & 3.2\times 10^{14} \left(\frac{R_{\rm Loop}}{100\, {\rm km}}\right)^{-1/2} \left(\frac{\theta _{\rm max}}{10^{-3}}\right)^{1/2} \nonumber\\ && \times \left(\frac{V_\mu }{1.4\times 10^8\, {\rm cm \,s^{-1}}}\right)^{-1/2} l_5 ^{1/2}\, {\rm G}. \end{eqnarray} \tag{ 3 }$$

We note that we used the limits on V_μ and θ_max that yield the stringent lower limit on the crustal internal magnetic field B. A more reasonable value for V_μ (=0.7 × 10⁸ cm s⁻¹, Steiner & Watts 2009), and θ_max (=0.1, Horowitz & Kadau 2009), and substituting R_Loop with R_max results in

$$\begin{eqnarray} B & \gtrsim & 4.5\times 10^{15} \left(\frac{R_{\rm max}}{30\, {\rm km}}\right)^{-1/2} \left(\frac{\theta _{\rm max}}{0.1}\right)^{1/2} \nonumber\\ && \times \left(\frac{V_\mu }{0.8\times 10^8\, {\rm cm\, s^{-1}}}\right)^{-1/2} l_5 ^{1/2}\, {\rm G}. \end{eqnarray} \tag{ 4 }$$

We could also derive an upper limit on the crustal internal magnetic field considering that the maximum observed energy, E_max = L_max × 4_ms ∼ 1.2 × 10³⁹ erg, where 4_ms is the bin time interval, does not exceed the maximum magnetic energy possibly released in an area $l_5^2$ on the crust, which is $E_{\rm max}\sim 4\times 10^{40} l_5^2 B_{15}^{-2} (\theta _{\rm max}/10^{-3})^2$ erg (TD95). Taking θ_max = 0.1, the above implies

$$\begin{equation} B \lesssim 5.8\times 10^{17} \left(\frac{\theta _{\rm max}}{0.1}\right) l_5\, {\rm G}. \end{equation} \tag{ 5 }$$

Another interesting finding in our analysis is the R² − kT relation, and its dependence on flux, which is once again a complicated result to interpret qualitatively (Figure 6, lower panels). For instance, at the lowest fluxes (F < 10^−5.5 erg s⁻¹ cm⁻²), we find that R²∝kT^{−3.8 ± 0.4}, i.e., following lines of constant luminosities (F∝R²kT⁴). The relation breaks from the above with increasing flux. This indicates that at the lowest fluxes, photons emitted by both components follow a perfect Planckian distribution and start deviating from it with an increasing flux. One possible way to achieve a purely thermal spectrum is if photon splitting (Adler et al. 1970; Baring 1995; Baring & Harding 2001; Usov 2002; Chistyakov et al. 2012) is the dominant source of photon number changing (TD95). At the lowest fluxes, the size of the high-kT component is extremely small (some tens of meters), and very likely most of the emitted radiation is in the form of E-mode photons. At such small distance from the NS, the magnetic field is super critical, and, due to the high temperature of the high-kT component at these lowest fluxes, photon splitting is very efficient (low-kT ∼ 14 keV >kT_sp ∼ 11 keV, where kT_sp is the BB temperature at which the photon splitting rate is sufficient to maintain LTE; TD95), leading to complete thermalization of the high-kT component. With increasing flux, the high-kT temperature starts dropping and the spectrum starts deviating from a pure Planckian, perhaps due to other radiative processes coming into play, e.g., double Compton scattering (e + γ → e + γ + γ) and Bremsstrahlung. For the low-kT component, the picture is not clear, mostly due to the variation in field morphology and opacity for both scattering and photon splitting at high altitudes. However, since these two components appear to be strongly coupled (Figure 4), we speculate that a deviation in the high-kT photon distribution from a pure Planckian model would lead to the deviation in the low-kT component as well.

4.2.2. Comptonized Model