GRB 050410 and GRB 050412: are they really dark GRBs?

GRB 050410 triggered Swift twice as a result of an anomalous time-out of the triggering code (Fenimore et al. 2005). In our analysis, we consider as reference time the time of the first trigger (T₀=12:14:25.36 UT).

The $15-150$ keV light curve of the prompt emission, shown in Fig. 1, is characterised by a broad peak with T₉₀=44$\pm$1 s. The burst shape can be modelled by two Gaussians of the same width (11$\pm$2 s), the first centered at 5$\pm$1 s and the second at 32$\pm$2 s.

The $15-150$ keV energy distribution of the burst, modelled with a simple power law of photon index $\Gamma=1.66\pm 0.07$, gave a marginally acceptable fit with a $\chi^{2}_{\rm red}$ of 1.67 for 36 dof (see Table 2). Better fits were obtained using a cut-off power law or a Band model (Band et al. 1993) with the high energy photon index, $\beta\,_{\rm Band}$, fixed to $-$2.3 according to the expected value (Preece et al. 2000). The best fit values of the spectral parameters are shown in Table 2. The improvement with respect to the power law fit is significant for both models, with a chance random probability of $\sim$7$\times$10^-5, assuming that the errors are normally distributed. The fluence in the $15-150$ keV energy range was (4.6$\pm$0.1)$\times$10^-6 erg cm^-2.

The XRT observed GRB 050410 for 10 orbits for a total exposure of $\sim$9 ks in WT mode and $\sim$71 ks in PC mode. The position of the burst, determined with astrometry solutions, was RA${}_{\rm J2000}=$05^h 59^m 13^s.94   Dec${}_{\rm J2000}=+$79°  36′  11$\aas@@fstack{\prime\prime}$7, with an uncertainty of 2$\aas@@fstack{\prime\prime}$3 (Butler 2007). These coordinates lie outside the error circle (3$\aas@@fstack{\prime\prime}$7 radius) of the position derived with the tool XRTCENTROID and 42$\arcsec$ from the BAT position reported by Fenimore et al. (2005).

The GRB 050410 XRT light curve shows a clear decay with time that can be well modelled ($\chi^{2}_{\rm red}$=0.87 with 13 d.o.f.) by a single power law with a slope $\alpha=$1.06$\pm$0.04. The light curve, shown in Fig. 2, was converted to flux in the $0.2-10$ keV band using a conversion factor derived from the best fit spectral model (see below). In the same figure, the BAT light curve extrapolated to the $0.2-10$ keV energy range is also plotted. The extrapolation was obtained by multiplying the $15-150$ keV rate by the average rate-to-flux conversion factor for GRB 050410 derived from the Band model parameters.

WT and PC spectra were fitted simultaneously because no spectral evolution with time was detected. The two spectra normalisations were left free to vary, thus accounting for the different average flux levels in the two observing periods. An absorbed power law reproduces well the emission ($\chi^{2}_{\rm red}$ of 0.64 for 13 d.o.f.) with the best fit parameters (see also Table 2) $\Gamma_{\rm x}=2.4\pm 0.4$ and $N_{\rm H}$=4${}^{+3}_{-2}\times$10²¹ cm^-2. Note that the absorbing column is non-zero at the 4-sigma level, and the best-fit value of the absorbing column is $\sim$ 5 times larger than the Galactic one (7.53$\times$10²⁰ cm^-2; Dickey & Lockman 1990).

The GRB 050412 prompt emission showed a double-peaked structure with the first peak centered at $-$1.1$\pm$0.4 s from the trigger (width of 4.7$\pm$0.6 s), and the second fainter and wider (11$\pm$3 s) peak centered at 16$\pm$5 s (see Fig.3). The estimated T₉₀ in the $15-150$ keV energy band was 27$\pm$1 s. The hardness ratio ($50-150$ keV and $15-50$ keV) does not show any significant evidence of spectral variation during the burst evolution.

The $15-150$ keV energy distribution was well described ($\chi^{2}_{\rm red}=0.55$ for 15 d.o.f.) by a single power law with an hard photon index of $\Gamma=0.7\pm 0.2$. The T₉₀ burst fluence in the $15-150$ keV energy range was (6.3$\pm$0.3)$\times$ 10^-7 erg cm^-2. A Band model and a cut-off power law were not able to reproduce this spectrum because some of the fitting parameters are not constrained.

The XRT detected a rapidly fading source at RA${}_{\rm J2000}=$12^h 04^m 25^s.2   Dec${}_{\rm J2000}=\,$-01°  12′  00$\aas@@fstack{\prime\prime}$4, with an uncertainty of 4$\aas@@fstack{\prime\prime}$2. This position, derived with the tool XRTCENTROID, is 60$\aas@@fstack{\prime\prime}$8 from the BAT position (Tueller et al. 2005) and it is in agreement with the refined value derived with astrometry technique (Butler 2007).

The XRT light curve is not consistent with a single power law ($\chi^{2}_{\rm red}=2.3$ with 18 d.o.f.). A better fit ($\chi^{2}_{\rm red}=0.7$ with 14 d.o.f.) can be obtained with a broken power law whose best fit parameters are: $\alpha_{1}$=0.7$\pm$0.4, $\alpha_{2}$=2.8${}_{-0.8}^{+0.5}$ and T_break=254${}_{-41}^{+79}$ s. The GRB 050412 light curve was converted to the $0.2-10$ keV flux with a procedure similar to the one used for GRB 050410, and plotted in Fig. 4 together with the BAT light curve extrapolated to the XRT energy range. In the same figure, the 3$\sigma$ upper limit obtained by Chandra is also plotted (Berger & Fox 2005). The upper limits derived from Swift observations 2 (2005-04-14) and 3 (2005-04-17) are not plotted in the figure, since they are less constraining than the Chandra one.

The first orbit of the GRB 050412 observation is affected by a continuous switching between PC and WT mode (caused by flickering pixels produced by the high CCD temperature and by the bright Earth contamination); thus the two operational mode spectra are almost contemporaneous. The WT spectrum only employed the first 1400 s of data to avoid a background flare present in the second part of the observation. WT and PC spectra were fitted simultaneously with an absorbed power-law model, leaving the normalisation free in order to take into account differences in the flux levels. The measured absorption column was consistent with the Galactic (2.21$\times$10²⁰ cm^-2; Dickey & Lockman 1990) and was fixed to this value. The best fit power-law spectral index was $\Gamma_{\rm x}=1.3\pm 0.2$ with a reduced $\chi^{2}$ of 1.18 (17 d.o.f.). Results of the spectral analysis are shown in Table 2.

The X-ray afterglow light curves of many bursts manifest a similar behaviour (O’Brien et al. 2006; Nousek et al. 2006; Willingale et al. 2006). The canonical X-ray light curve has an initial steep decay (usually interpreted as emission from the tail of the prompt GRB), followed by a flatter decay phase that can last up to 10⁵ s (interpreted as a refreshing of the forward shock), and a final steeper decay phase with power law indices consistent with the values measured before the launch of Swift (Frontera 2003). This canonical light curve is consistent with about 60% of the Swift afterglows. In many of the non-conforming cases, the first, or the first and the second branches are missing. The lack of detection of these portions of the light curve cannot always be attributed to missing observational data. Moreover, in about half of the afterglows, late X-ray flares, probably due to continued activity of the central engine, are observed (O’Brien et al. 2006; Nousek et al. 2006; Zhang et al. 2006).

The light curve of GRB 050410, starting from about 2000 s from the trigger, shows a constant decay slope that can be interpreted as the third branch of the canonical light curve. The synchrotron radiation theory applied to the fireball model predicts that the temporal decay index, $\alpha$, of a GRB afterglow and the spectral slope, $\beta$, are linked by relations that depend (i) on the density profile of the external medium (Meszaros & Rees 1997; Chevalier & Li 1999); (ii) on the observer’s perception of the geometry of the expansion: spherical whenever the expansion velocity corresponds to a Lorentz factor, $\Gamma_{\rm e}$, such that $\theta_{0}>\Gamma_{\rm e}^{-1}$, and beamed for $\theta_{0}<\Gamma_{\rm e}^{-1}$ with $\theta_{0}$ being the half opening angle of the jet (Sari et al. 1999; Rhoads 1999); (iii) on the observational frequency and its relation to the typical synchrotron frequency of newly shocked electrons, $\nu_{m}$, and to the cooling frequency $\nu_{c}$ corresponding to a synchrotron cooling time equal to the hydrodynamical expansion time (Sari et al. 1998). In particular, for fireball expansion in a uniform interstellar medium we are usually in the $\nu_{x}>\nu_{c}$ regime and expect $\alpha=(3p-2)/4$ and $\beta=p/2$, with the energy distribution of the electrons, $p$, greater than 2, before the jet edge effect is manifested. This relation is satisfied by GRB 050410 with a value $p=2.08\pm 0.05$. This value is lower than $p$ values predicted by numerical simulations (2.2-2.5), but still consistent for particle acceleration at ultraluminous shocks. From this result, we can derive a lower limit of 10 d to the time after the trigger of any jet break. The XRT light curve of GRB 050410 is therefore consistent with a pre-Swift X-ray afterglow light curve, or with the third branch of the canonical Swift afterglow light curve. The late start of XRT observations ($\sim$2000 s after the trigger) might have prevented the detection of the previous branches.

According to the typical Swift afterglow behaviour, the early break in the X-ray light curve of GRB 050412 may be interpreted as the flat to steep transition corresponding to the end of the refreshing of the forward shock. However the measured spectral index $\beta_{\rm x}$=0.3$\pm$0.2 would imply a flat electron energy distribution with $p=0.7\pm 0.3$. The expected temporal slope, either for spherical expansion in a uniform medium or for a standard isotropic wind model in the $p<2$ and $\nu_{x}>\nu_{c}$ regime, is inconsistent with $\alpha_{2}$.

The values of $\alpha_{1}$ ($0.7\pm 0.4$), $\alpha_{2}$ ($2.8_{-0.8}^{+0.5}$) and $\beta_{\rm x}$ ($0.3\pm 0.2$) of GRB 050412 fit the closure relations corresponding to a jet break occurring in the $\nu_{m}<\nu_{x}<\nu_{c}$ regime for $p=(2\beta_{\rm x}+1)=1.7\pm 0.3$ with $\alpha_{1}=3\,(2\beta_{\rm x}+3)/16$ and $\alpha_{2}=(2\beta_{\rm x}+7)/4$ according to Dai & Cheng (2001). An early achromatic break has also been detected in the X-rays and optical light curve for GRB 050801 (Rykoff et al. 2006; Covino et al. 2006), while achromatic breaks usually occur at T$>$10⁴ s (Blustin et al. 2006; Romano et al. 2006; Soderberg et al. 2006; Bloom et al. 2003; Frail et al. 2001; Covino et al. 2006). However, in the case of GRB 050412, this explanation is very unlikely because such an early time for the temporal break would require a very small value for the jet angle ($\theta_{0}<1$ deg) which is at variance with the beaming angle ($\theta_{0}>3$ deg) computed, assuming a peak energy greater than 150 keV, from the Amati (2006) and Ghirlanda et al. (2004) relations, that holds for most of the bursts.

Alternatively, the first part of the XRT light curve can be considered as the last peak of the prompt emission, and the decay rate beyond 250 s can be interpreted as due to curvature effect after an instantaneous turn off of the source (Kumar & Panaitescu 2000). Within this model the relation that should be satisfied is $\alpha_{2}=\beta_{\rm x}+2$ which is in agreement with our results. After the final detection at 1 ks, the X-ray light curve may start following a typical afterglow decay without violating the later upper limits, however a drop of more than three order of magnitudes in flux within the first 10⁴ s from the trigger and before the emergence of the afterglow is uncommon to Swift GRBs. It is then possible that GRB 050412 was a burst without afterglow (a naked burst) as GRB 050421 (Godet et al. 2006).

GRB 050410 and GRB 050412 do not have identified optical/infrared afterglows, therefore, it is worth considering how dark these bursts are. Both bursts are located in regions with moderate or small Galactic extinction ($E_{B-V}\sim 0.11$ and $0.02$ for GRB 050410 and GRB 050412, respectively). The available optical upper limits for GRB 050410 are too shallow for any meaningful claim, as shown in Fig. 5 where the broad band spectral energy distribution at the time of the Misra et al. (2005) optical measurement is plotted. The comparison with the X-ray flux ($\beta_{\rm ox}\sim 0.8$ computed at 11 h from the burst) indicates that it was a rather regular burst regards to its gamma-ray to X-ray ratio (see also Roming et al. 2006). However, the 2-10 keV flux computed at 1 h (2.4$\times$ 10^-12 erg cm^-2 s^-1) and at 11 h (1.9$\times$ 10^-13 erg cm^-2 s^-1) shows that this burst has a relatively faint X-ray afterglow (see Fig.4 in Roming et al. 2006).

A possible interpretation of a low afterglow flux level is a low density medium (Groot et al. 1998; Frail et al. 1999; Taylor et al. 2000). Assuming that the low X-ray flux is due to a low-density circum-burst medium, we derived an estimate of the interstellar medium near the GRB under some reasonable assumptions. From the detected peak energy, $E_{p}$, in the BAT spectrum, and using the Amati relation (Amati 2006) the values of the energy of the afterglow $E_{a}$ ($\sim$ 2$\times$10⁵¹ erg) and of the redshift $z$ ($\sim$ 0.4) were inferred. Assuming an electron index of $p=2.5$, taking into account the decay of the afterglow and assuming that the observing frequency is between the peak frequency $\nu_{\rm m}$ and the cooling frequency $\nu_{c}$, we derived $n<0.01$ cm^-3. The estimate of $n$, subject to the uncertainty on $z$, $E_{a}$ and $p$ (which increases with $z$ and decreases with $E_{a}$ and $p$) is lower than the typical value of GRBs with optical afterglow ($\sim$ 1 cm^-3), but similar to values which have already been derived for other GRBs (Panaitescu & Kumar 2002).

Things are considerably different in the case of GRB 050412: about 2.3 hr after the burst the optical afterglow was not detected with an upper limit of $R_{c}=24.9$ (Kosugi et al. 2005). In the Rol et al. (2005) Fig. 2 the GRB 050412 representative point would immediately classify it as a very dark burst. However, while any possible optical afterglow for this event is undoubtedly faint already at about 2 hr after the high energy event, to discuss the darkness of the burst a comparison with the detected X-ray flux is mandatory. The X-ray light curve of GRB 050412 shows a break at about 4 minutes after the burst and then a steep decay with a temporal index of about 2.8 (see Sect. 3.2). In computing the optical-to-X-ray spectral index at 11 hr after the burst,we may either assume that the optical upper limit went on fading with $T^{-1}$ (as assumed in Jakobsson et al. 2004), or that the optical decay follows the X-ray light curve. In a similar way, the X-ray density flux computed at 11 hr from the burst can be evaluated from the extrapolation of the decay curve, or assuming that, at that time, the burst flux reached the level of the Chandra upper limit. The resulting optical-to-X-ray spectral index $\beta_{\rm ox}$ would not clearly identify this event as dark because it ranges from $\sim$0.5 to $\sim$1 depending on how the optical and X-ray flux extrapolation are performed. Of course, the brighter the extrapolated X-ray flux, the more convincing is the classification of GRB 050412 as a dark event.

Figure 6 presents the optical upper limit by Kosugi et al. (2005) at 2.3 hr from the trigger together with the X-ray spectrum computed extrapolating the XRT flux with the measured decay rate ($\alpha_{2}$=2.8) and assuming no spectral variation with time (see label a). In the same figure, the X-ray spectrum is also computed assuming that the flux at 2.3 hr was as high as the upper limit measured by XRT at 7 ks from the burst (see label b) and that the light curve became flat at the level of the Chandra upper limit (see label c; Berger & Fox 2005). However, we have seen in section 4.1 that the most likely interpretation of the XRT light curve of GRB 050412 is not an afterglow, but the tail of the prompt emission from curvature effect. If this is the case, the possible underlying X-ray afterglow at 2.3 hr is expected to be softer than the measured spectrum (with a photon index $\Gamma\sim$2). This would make the X-ray spectra in Fig. 6 steeper, bringing the expected optical emission well above the observed upper limit, but it would not lead to a firm conclusion about the darkness of GRB 050412.

As last point, looking at Fig. 3 in Roming et al. (2006) both GRB 050410 and GRB 050412 appear fairly normal considering the X-ray flux at 1 hour compared to their prompt $\gamma$-ray flux.