Rise and fall of the X-ray flash 080330: an off-axis jet?

The BAT data were processed with the heasoft package (v.6.4) adopting the ground-refined coordinates provided by the BAT team (Markwardt et al. 2008). The BAT detector quality map was obtained by processing the nearest-in-time enable/disable map of the detectors.

The top panel of Figure 1 shows the 15–150 keV mask-weighted light curve of XRF 080330 as recorded by BAT, expressed as counts per second per fully illuminated detector for an equivalent on-axis source. The solid line displayed in Fig. 1 corresponds to the result of fitting the profile from $-1.2$ to $100$ s with a combination of four pulses (Markwardt et al. 2008) as modelled by Norris et al. (2005; hereafter N05 model). Table 1 reports the corresponding derived parameters: $t_{\rm p}$ (peak time), $A$ (15–150 keV peak flux), $\tau_{\rm r}$ (rise time), $\tau_{\rm d}$ (decay time), $w$ (pulse width), $k$ (pulse asymmetry) and the model fluence in the 15–150 keV band. The goodness of the fit is $\chi^{2}/{\rm dof}=375/379$. Parameter uncertainties were derived by propagation starting from the best-fit parameters and taking into account their covariance. We tried to apply the same analysis to the light curves of the resolved energy channels to investigate temporal lags and, more generally, the dependence of the parameters on energy; however, because of the faintness and softness of the signal, we could not constrain the parameters in a useful way.

In addition, the energy spectra in the 15–150 keV band were extracted using the tool batbinevt. We applied all the required corrections: we updated them through batupdatephakw and generated the detector response matrices using batdrmgen. Then we used batphasyserr in order to account for the BAT systematics as a function of energy. Finally we grouped the energy channels of the spectra by imposing a 3$\sigma$ (or 2-$\sigma$ when the S/N was too low) threshold on each grouped channel. We fitted the resulting photon spectra, $\Phi(E)$ (ph cm^-2s^-1keV^-1), with a power law with pegged normalisation (pegpwrlw model under xspec v.11.3.2). We extracted several spectra in different time intervals: over $T_{90}$, total, spanning the bunch of the first three pulses, the fourth pulse alone and that around the peak, determined on a minimum significance criterion. The results are reported in Table 2. The time-averaged spectral index is $\beta_{\gamma}=1.65\pm 0.51$ with a total fluence of $S(15-150\leavevmode\nobreak\ {\rm keV})=(3.6\pm 0.8)\times 10^{-7}$ erg cm^-2 and a $0.448$-s peak photon flux of $(1.0\pm 0.2)$ ph cm^-2 s^-1, in agreement with previous results (Markwardt et al. 2008). The bottom panel of Fig. 1 shows marginal evidence for a soft–to–hard evolution: $\beta_{\gamma}$ passes from $2.0\pm 0.5$ (first three pulses) to $1.4\pm 0.5$ (fourth pulse).

Following Sakamoto et al. (2008), a GRB is classified as an XRR (XRF) depending on whether the fluence ratio $S(25-50\leavevmode\nobreak\ {\rm keV})/S(50-100\leavevmode\nobreak\ {\rm keV})$ is lower (greater) than $1.32$. The fluence ratio of XRF 080330, $1.5_{-0.3}^{+0.7}$, places it among the XRFs, although still compatible with being an XRR burst. Although from BAT data alone we could not measure the peak energy of the time-integrated $\nu\,F_{\nu}$ spectrum, we tried to fit with a smoothed broken power law model (Band et al. 1993) by fixing the low-energy index $\alpha_{\rm B}$ to -1 (Kaneko et al. 2006), given that $\Gamma=\beta_{\gamma}+1>2$ and is very likely dominated by the high-energy index $\beta_{\rm B}$. This way we derived the following constraint: $E_{\rm p}<35$ keV, in agreement with the upper limit to its rest-frame (intrinsic) value, $E_{\rm p,i}=E_{\rm p}\,(1+z)<88$ keV, obtained by Rossi et al. (2008).

Recently, Sakamoto et al. (2009) calibrated a method aimed at estimating $E_{\rm p}$ from the $\Gamma_{\gamma}$ as measured with BAT, provided that $1.3$$<$$\Gamma$$<$$2.3$. In the case of XRF 080330, the confidence interval on $\Gamma$, $2.65\pm 0.51$, marginally overlaps with the allowed range; however, since $E_{\rm p}$ is anti-correlated with $\Gamma$ and the lower limit on $\Gamma$ lies within the usable range, we can derive an upper limit to $E_{\rm p}$ from the relation of Sakamoto et al. (2009), which turns out to be $30$ keV, in agreement with our previous value. These results are fully consistent with a previous preliminary analysis (Markwardt et al. 2008).

We constrained $E_{\rm iso}$ in the rest-frame 1–$10^{4}$ keV band using the upper limit on $E_{\rm p}$ of $35$ keV. Following the prescriptions by Amati et al. (2002) and Ghirlanda et al. (2004), we found $E_{\rm iso}<2.2\times 10^{52}$ ergs. Combined with $E_{\rm p,i}<88$ keV, this places XRF 080330 in the $E_{\rm p,i}$–$E_{\rm iso}$ space consistently with the Amati relation (Amati et al. 2002; Amati 2006).

The XRT data were processed using the heasoft package (v.6.4). We ran the task xrtpipeline (v.0.11.6) applying calibration and standard filtering and screening criteria. Data from 77 to 134 s were acquired in Windowed Timing (WT) mode and the following in Photon Counting (PC) mode due to the faintness of the source. Events with grades 0–2 and 0–12 were selected for the two modes, respectively. XRT observations went on up to $5.9\times 10^{5}$ s, with a total net exposure time of $30.6$ ks. The XRT analysis was performed in the 0.3–10 keV energy band.

Source photons were extracted from WT mode data in a rectangular region 40 pixels along the image strip (20 pixel wide) centred on the source, whereas the background photons were extracted from an equally-sized region with no sources.

Firstly, we extracted the first orbit PC data from 136 to 331 s, where the point spread function (PSF) of the source looked unaffected by the spacecraft drifting, and extracted the following refined position: RA$=11^{\rm h}$ $17^{\rm m}$ $04\aas@@fstack{s}68$, Dec. $=+30^{\circ}$ $37^{\prime}$ $24\aas@@fstack{\prime\prime}8$ (J2000), with an error radius of $4.0$ arcsec (Mao & Guidorzi 2008). We corrected these data for pile-up by extracting source photons from an annular region centred on the above position and with inner and outer radii of 4 and 30 pixels (1 pixel$\mbox{}=2\aas@@fstack{\prime\prime}36$), respectively. The background was estimated from a three-circle region with a total area of $30.3\times 10^{3}$ pixel² away from any source present in the field. Finally, we re-extracted the source photons over the entire first orbit within a larger circular region centred on the same position and with a radius of 40 pixels, to compensate for the drifting. The light curve of the full first orbit data (PC mode) was then corrected so as to match the previous one correctly produced in the 136–331 s sub-interval.

The resulting 0.3–10 keV light curve is shown in Fig. 3 (black empty triangles). It was binned so as to achieve a minimum signal to noise ratio (SNR) of 3. The data taken in following orbits were not enough to provide a significant detection and only a 3$\sigma$ upper limit was obtained.

The X-ray curve can be fitted with a broken power law, with the following parameters: $\alpha_{\rm x,1}=4.8\pm 0.4$, $t_{\rm b}=163_{-10}^{+9}$ s, $\alpha_{\rm x,2}=0.26\pm 0.10$ ($\chi^{2}/{\rm dof}=66/70$). The last upper limit clearly requires a further break. We set a lower limit on $\alpha_{\rm x,3}$ by connecting the end of first orbit data with the late upper limit under the assumption that the second break occurred at the beginning of the data gap. This turned into $\alpha_{\rm x,3}>1.3$ ($\geq$3$\sigma$ confidence). The later the second break time, the steeper the final decay.

We extracted the 0.3–10 keV spectrum in two different time intervals: i) “XRT-WT” interval, from $77$ to $134$ s (WT mode), corresponding to the initial steep decay; ii) “plateau” interval, from $423$ to $1507$ s (PC mode) corresponding to the following flat decay (or “plateau”) phase. Source and background spectra were extracted from the same regions as the ones used for the light curve for the corresponding time intervals and modes. The ancillary response files were generated using the task xrtmkarf. Spectral channels were grouped so as to have at least 20 counts per bin. Spectral fitting was performed with xspec (v. 11.3.2).

We modelled both spectra with a photoelectrically absorbed power law (model wabs$\cdot$zwabs$\cdot$pow), adopting the photoelectric cross section by Morrison & McCammon (1983). The first column density was frozen to the weighted average Galactic value along the line of sight to the GRB, $N_{\rm H}^{\rm(Gal)}=1.23\times 10^{20}$ cm^-2 (Kalberla et al. 2005), while the second rest-frame column density, $N_{{\rm H},z}$, was left free to vary. While during the steep decay we found no evidence for significant rest-frame absorption, with a 90% confidence limit of $N_{{\rm H},z}<1.4\times 10^{21}$ cm^-2, in the plateau spectrum we found only marginal evidence for it, $N_{{\rm H},z}=1.6_{-1.5}^{+1.8}\times 10^{21}$ cm^-2. The spectral index, $\beta_{\rm x}$, varies from $1.06_{-0.09}^{+0.10}$ to $0.80_{-0.15}^{+0.16}$: the significance of this change is $\sim 2.3$ $\sigma$. The best-fit parameters are reported in Table 2.

The Swift UVOT instrument started observing on 2008 March 30 at 03:42:19 UT, 63 s after the BAT trigger, with a $9.37$-s settling exposure. Since the detectors are powered up during this exposure, the effective exposure time may be less than reported. We checked the brightness in this exposure with later exposures, to confirm that no correction was needed. The first $99.7$-s finding chart exposure started at 03:42:39 UT in the white filter in event mode followed by a $399.8$-s exposure in the $V$ filter, also in event mode. Due to the loss of lock by the spacecraft star trackers, the attitude information was incorrect. In order to process the data, xselect was used to extract images for short time intervals. The length of the interval was chosen short enough that the drift of the spacecraft was mostly within $7\arcsec$, and at most $14\arcsec$, but long enough to get a reasonably accurate measurement. A source region was placed over the position of the source, making checks for consistency with the position of nearby stars, and the magnitudes were determined using the ftool uvotevtlc. In most cases, an aperture with a radius of $5\arcsec$ was used, and three measurements used a slightly larger aperture. No aperture correction was made, since the source shape in those cases was very elongated. The measured magnitudes were converted back to the original count rates, using the UVOT calibration (Poole et al. 2008). These were subsequently converted to fluxes using the method of Poole et al. (2008), but for an incident power-law spectrum with $\beta$$=$$0.8$ and for a redshift $z$$=$$1.51$.

Robotically triggered observations with the LT began at 181 s leading to the automatic identification by the GRB pipeline LT-TRAP (Guidorzi et al. 2006) of the optical afterglow at the position RA $=11^{\rm h}$ $17^{\rm m}$ $04\aas@@fstack{s}48$, Dec. $=+30^{\circ}$ $37^{\prime}$ $23\aas@@fstack{\prime\prime}8$ (J2000; 1$\sigma$ error radius of $0.2$ arcsec). This is consistent within $1.6$ $\sigma$ with the refined UVOT position. The afterglow was seen during the end of the rise with $r^{\prime}=17.3\pm 0.1$ and subsequently decay (Gomboc et al. 2008a). Following two initial sequences of $3\times 10$ s each in the $r^{\prime}$ filter during the detection mode (DM), the multi-colour imaging observing mode (MCIM) in the $g^{\prime}r^{\prime}i^{\prime}$ filters was automatically selected. Observations carried on up to $4.9$ ks.

The FTN observations of XRF 080330 were carried out from $8.4$ to $9.1$ hr and again from $31.8$ to $33.9$ hr with deep $r^{\prime}$ and $i^{\prime}$ filter exposures as part of the RoboNet 1.0 project³³3http://www.astro.livjm.ac.uk/RoboNet/ (Gomboc et al. 2006).

Calibration was performed against five non-saturated field stars with preburst SDSS photometry (Cool et al. 2008), by adopting their PSF to adjust the zero point of the single images. Photometry was carried out using the Starlink GAIA software. Magnitudes were converted into flux densities (mJy) following Fukugita et al. (1996). Results are reported in Table LABEL:tab:photom.

Optical $R$-band observations of the afterglow of GRB 080330 were carried out with the REM telescope equipped with the ROSS optical spectrograph/imager on 2008 March 30, starting about 55 seconds after the burst (D’Avanzo et al. 2008). We collected 38 images with typical exposures times of 30, 60 and 120 s, covering a time interval of about $0.5$ hours. Image reduction was carried out by following the standard procedures: subtraction of an averaged bias frame, division by a normalised flat frame. The astrometry was fitted using the USNOB1.0⁴⁴4http://www.nofs.navy.mil/data/fchpix/ catalogue. We grouped our images into 18 bins in order to increase the signal-to-noise ratio (SNR) and performed aperture photometry with the SExtractor package (Bertin & Arnouts 1996) for all the objects in the field. In order to minimise the systematics, we performed differential photometry with respect to a selection of local isolated and non-saturated standard stars.

The calibration of NOT images taken with the $R$ filter was performed with respect to the converted magnitudes in the $R$-band of the selected set of stars used for the calibration of LT and GROND images in the SDSS passbands. We transformed the $r^{\prime}$ and $i^{\prime}$ magnitudes of the calibration stars (Cool et al. 2008) into $R$ and $I$ magnitude following the filter transformations of Krisciunas et al. (1998).

Hereafter, the magnitudes shown are not corrected for Galactic extinction, whilst fluxes as well as all the best-fit models are. When the models are plotted together with magnitudes, the correction for Galactic extinction is removed from the models.

Starting at $\approx$$46$ min we obtained a 1800 s spectrum with a low resolution grism and a 1.3 arcsec wide slit covering the spectral range from about 3500 to 9000 Å at a resolution of 14 Å with the NOT (Fig. 2). The airmass was about 1.8 at the start of the observations. The spectrum was reduced using standard methods for bias subtraction, flat-fielding and wavelength calibration using an Helium-Neon arc spectrum. The rms of the residuals in the wavelength calibration were about 0.3 Å. The spectrum was flux-calibrated using an observation with the same setup of the spectrophotometric standard star HD93521. Table 6 reports the identified lines.

Figure 3 displays the light curves of the prompt emission (15–150 keV) and of the 0.3–10 keV and NIR/visible/UV afterglow derived from our data sets plus some points taken from RAPTOR (Wren et al. 2008). High-energy fluxes (magnitudes) are referred to the right-hand (left-hand) y-axis. First of all, we note that the peak time of the last $\gamma$-ray pulse (Table 1) is contemporaneous with the optical flash detected by RAPTOR, reported at $58.9\pm 2.5$ s (Wren et al. 2008).

The initial steep decay observed by XRT is a smooth continuation of the last $\gamma$-ray pulse and is thus the tail of the prompt GRB emission, and likely to correspond to its high-latitude emission. Most notably, during the X-ray steep decay the optical flux is seen to rise up to $\sim$$300$ s and finally a simultaneous plateau is reached at both energy bands, lasting up to $\sim$$1500$ s, when the X-ray observations stopped. This strongly suggests that the plateau is emission from a region which is physically distinct from that responsible for the prompt emission and its tail (the rapid decay phase).

As discussed in Sect. 4.2, the afterglow does not show evidence for spectral evolution throughout the observations, except for late epochs ($t\sim 10^{5}$ s), when there is evidence for reddening. The achromatic nature of the afterglow light curve allows for a multi-wavelength simultaneous fit of twelve light curves, where only the normalisations are left free to vary independently from each other. We consider all of the available passbands: $K$, $H$, $J$, $z^{\prime}$, $i^{\prime}$, $r^{\prime}$, $V$, $g^{\prime}$, $B$, $U$, $UWV1$ and X-ray, respectively. The latter curve is fitted from 300 s onward, so as to exclude the initial steep decay. Hereafter we present two alternative combinations of models, both providing a reasonable description of the flux temporal evolution. In both cases we had to add a 2% systematics to all of the measured uncertainties to account for some residual variability with respect to the models, in order to have acceptable $\chi^{2}$ values and correspondingly acceptable parameters’ uncertainties.

Figure 6 displays four SEDs we derived in as many different time intervals (see shaded bands in Fig. 3):

1. 1.

   SED 1 includes the last $\gamma$-ray pulse and the optical flash detected by RAPTOR (Wren et al. 2008), around 60 s;

2. 2.

   SED 2 corresponds to the final part of the optical rise, coinciding with the final part of the X-ray steep decay, spanning from $186.8$ to $269.4$ s;

3. 3.

   SED 3 has the broadest wavelength coverage and corresponds to the plateau phase, from $\sim$$400$ to $\sim$$1500$ s.

4. 4.

   SED 4 includes NIR/visible measurements around the possible late time break in the light curve (Fig. 4), at $\sim$$10^{5}$ s.

To construct SED 1 we made use of the RAPTOR measurement (Wren et al. 2008), a UVOT upper limit in the $V$ band and of the BAT spectrum of the fourth pulse. Figure 7 displays this SED: the solid line shows the best fit with a smoothed broken power law used to fit the high-energy photon spectra of the prompt emission of GRBs (Band et al. 1993).

The best-fitting parameters are the following: $\alpha_{\rm B}=-1.12$, $\beta_{\rm B}=-2.35$ and $E_{\rm p,i}=71$ keV ($\chi^{2}/{\rm dof}=5.8/5$) consistent with the limit on $E_{\rm p,i}$ derived in Sect. 3.1. The Band indices are photon indices, so the corresponding energy indices are $0.12$ and $1.35$, respectively. While $\beta_{\rm B}$ was constrained by the BAT data themselves, we solved the coupled indetermination $\alpha_{\rm B}$–$E_{\rm p,i}$ by initially freezing the low-energy index $\alpha_{\rm B}$ to the typical value of $-1$ (Kaneko et al. 2006) and then leaving it to vary. The above minimum $\chi^{2}$ was so found. Although this does not break the degeneracy of both parameters (for every $E_{\rm p,i}<88$ keV there is a value of $\alpha_{\rm B}$ for which an acceptable fit is given), here it is shown that the extrapolation of a typical Band model fitting the spectrum of the last pulse matches the optical flux observed during the flash. However, because of the lack of measurement of $\alpha_{\rm B}$ from $\gamma$-ray data, the optical flux matched by the extrapolation of the Band model may still be accidental.

The time interval of SED 2 corresponds to the first $JHK$ GROND frames and spans from $186.8$ to $269.4$ s (see Fig. 3). It consists of contemporaneous $Vr^{\prime}JHK$ frames as well as X-rays.

The NIR-to-X SED 2 can be fitted with a single power-law with $\beta_{\rm ox}=0.74\pm 0.03$ and negligible dust extinction. The optical data alone can be fitted with an unextinguished power law with $\beta_{\rm o}=0.61\pm 0.13$ (Table 2).

SED 3, taken during the plateau, is the richest one including all of the passbands considered in this work, but the $\gamma$-rays (see Fig. 3). Our NIR values are consistent with the $JHK$ points of Bloom & Starr (2008). Given the steadiness of the light curve and the evidence for no significant colour change along the plateau, the SED so obtained is fairly robust.

The multi-wavelength fitting of the light curves of Figs. 4 and 5 (Sect. 4.1) is dominated by the data points along the plateau phase. Thus, we built a SED using the best-fit normalisations (Sects. 4.1.1 and 4.1.2) and calculated at the time of $10^{3}$ s. We found the same results with improved uncertainties, due to the stronger constraints imposed by a multi-band fitting. The result of this SED is displayed in Figure 9. The solid line shows the best-fit model obtained adopting a rest-frame SMC-extinguished (Pei 1992), X-ray photoelectrically absorbed power law with $\beta_{\rm ox}=0.79\pm 0.01$. We note that the point corresponding to the UVW1 filter nicely agrees with the Lyman absorption at the GRB redshift. The rest-frame optical extinction was found to be negligible and a very tight limit could be derived, $A_{V,z}<0.02$. Thanks to the more precise estimate obtained on $\beta_{\rm ox}$, the estimate of $N_{{\rm H},z}$ improved correspondingly: $(2.7\pm 0.8)\times 10^{21}$ cm^-2. These results are consistent with what was obtained from the 0.3–10 keV spectrum alone and the accuracy of the estimates benefited significantly from the inclusion of NIR/visible data. Fitting the optical data alone, the result is similar: no need for a significant amount of extinction and, more importantly, the same index: $\beta_{\rm o}=\beta_{\rm ox}=\beta_{\rm x}$ (Table 2), thus ruling out the possibility of a significant reddening due to dust along the line of sight within the host galaxy.

As a consequence, two properties are inferred:

• •

  a negligible dust column density in the circumburst environment and along the line of sight to the GRB through the host galaxy;

• •

  a single power-law component accounting for the (observed) NIR to X-ray radiation, pointing to a single emission mechanism with no breaks in between. Furthermore, a single power-law spectrum implies an achromatic evolution, consistently with the observations, while the other way around is not true.

The epoch of SED 4 is $\sim$$80$ ks, i.e. around the final break (aftermath of the late brightening) following the light curve description given in Sect. 4.1.1 (Sect. 4.1.2) and shown in Fig. 4 (Fig. 5). This includes optical data and is shown in Fig. 6 (diamonds). Data can be fitted either with a single unextinguished power-law with $\beta_{\rm o}=1.05\pm 0.06$ ($A_{V,z}$ fixed to 0) or, alternatively, with $\beta_{\rm o}=0.85\pm 0.30$ and some extinction, $A_{V,z}=0.10_{-0.06}^{+0.14}$. An increase of the extinction along with time seems hard to explain physically, so we are led to favour a true reddening at this time. Compared with the previous SEDs (Table 2), SED 4 is redder by $\Delta\beta_{\rm o}=0.26\pm 0.06$ (significance of $\sim$$6\times 10^{-5}$). We point out that the reddening is independent of the fit choice, as demonstrated from the comparison of the bare power-law indices with no dust correction between the earlier and later spectra (Table 2).

In the context of the fireball model (e.g. Mészáros 2006 and references therein), the possibility that the peak of the optical afterglow emission corresponds to the passage of the peak synchrotron frequency is ruled out by the lack of spectral evolution: $\beta_{\rm o}$ should evolve from negative ($-1/3$) to positive values, while we find no evidence for $\beta_{\rm o}$ changing before $\sim$$8\times 10^{4}$ s.

Another possible interpretation of the optical peak is the onset of the afterglow, as for GRB 060418 and GRB 060607A (Molinari et al. 2007; Jin & Fan 2007) and, possibly, for XRF 071010A as well (Covino et al. 2008). In the case of XRF 080330, the rise during the pre-deceleration of the fireball within an ISM is much shallower than $\alpha\sim-3$, expected at frequencies between $\nu_{\rm m}$ and $\nu_{\rm c}$ (Sari & Piran 1999; Granot 2005; Jin & Fan 2007). A wind environment would fit in a better way the slow rise of XRF 080330. Under these assumptions and in the thin shell case as the duration of the GRB is much shorter than the deceleration time, we can estimate the initial Lorentz factor, $\Gamma_{0}$ (approximately, twice as large as the Lorentz factor at the peak), in a wind-shaped density profile, $n(r)=A\,r^{-s}$ ($A$ is constant), with $s=2$, from the peak time and the $\gamma$-ray radiated energy, $E_{\rm iso}$ (Chevalier & Li 2000; Molinari et al. 2007). For consistency, only the two-component model (Sect. 4.1.2; Fig. 5) must be considered. In this case, we take the peak time of the first component, $t_{p1}=600$ s: assuming $\eta=0.2$ (radiative efficiency), $A=3\times 10^{35}$ cm^-1, it turns out $\Gamma_{0}<80$, and a corresponding deceleration radius smaller than $7\times 10^{16}$ cm. As in the case of XRF 071010A, the initial Lorentz factor is smaller than those found for classical GRBs. There is no evidence for the presence of a reverse shock; should the injection frequency of the reverse shock lie within the optical passbands, it would dominate the optical flux and exhibit a fast ($\sim$$t^{-2.1}$) decay (Kobayashi & Zhang 2007), not observed here. Nonetheless, this can still be the case if the injection frequency lies below the optical bands (Jin & Fan 2007; Mundell et al. 2007). A weak point of this interpretation is that the case $s=2$ is ruled out: $\alpha=s(p+5)/4-3=0.79\pm 0.01$ (Granot 2005). Inverting this relation, a value of $s=1.4\pm 0.1$ is required to explain the observed $\alpha=-0.4\pm 0.2$. This argument, together with the absence of reverse shock, whose $F_{\nu,{\rm max}}$ should be much larger compared with the forward shock by a factor of $\sim\,\Gamma$ (although see above), makes the interpretation of a deceleration through a wind environment somewhat contrived.

In the context of a single jet viewed off-axis (Granot et al. 2005), the rising part of the XRF 080330 curve is explained by the emission coming from the edge of the jet: as the bulk Lorentz factor $\Gamma$ decreases, the beaming cone gets progressively wider, thus resulting in a rising flux. The peak in the light curve is reached when it is $\Gamma\,\sim\,1/(\theta_{\rm obs}-\theta_{0})$, where $\theta_{\rm obs}$ and $\theta_{0}$ are the viewing and jet opening angles, respectively. According to the optical afterglow classification given by Panaitescu & Vestrand (2008), XRF 080330 belongs to the class of slow-rising and peaking after 100 s events. Those authors found a possible anti-correlation between the peak flux and the peak time for a number of fast-rising afterglows, followed also by the slow-rising class and, in this respect, XRF 080330 is no exception. The suggested interpretation of the rise is either the pre-deceleration synchrotron emission or the emergence of a highly collimated outflow seen off-axis. In the latter case, assuming a power-law angular distribution of the kinetic energy, $\mathcal{E}(\theta)\propto(\theta/\theta_{\rm c})^{-q}$ ($q>0$), high values for $q$ correspond to slower rises and dimmer peak fluxes, for a fixed off-axis viewing angle ($\theta_{\rm obs}=2\,\theta_{\rm c}$). In the former case, the anti-correlation is ascribed to different circumburst environment densities for different events: XRF 080330, because of the negligible dust extinction, would lie in the high-peak flux region, which is not the case. This favours the interpretation of an off-axis jet whose angular distribution of energy quickly drops away from the jet axis.

An example of another XRF whose optical counterpart showed a very similar behaviour is XRR 030418. The rise of this XRR, for which only an upper limit to its redshift ($z<5$) was obtained (Dullighan et al. 2003), has been interpreted as due to the decreasing extinction along with time, caused by the dust column density crossed by the fireball during its expansion (Rykoff et al. 2004). Figure 11 shows the light curve compared with that of XRF 080330.

The solid line shows the best fit to the $r^{\prime}$ curve of XRF 080330 of Sect. 4.1.1, while the dashed line shows the same model applied to the XRR 030418 data. XRR 030418 displays a steeper rise ($\alpha_{1}=-1.5$), which strongly depends on the zero time and could be the same as that of XRF 080330 if the time origin was moved by $(190\pm 50)$ s forward in time (lab frame). However, there is nothing around this time in the $\gamma$-ray light curve of XRR 030418. Apart from the different slopes of the rise and the lack of a late-time steepening in the case of XRR 030418, the plateau and post-plateau decay look very similar. If both XRFs are caused by the same process, from the spectral (lack of) evolution XRF 080330 during the rise-plateau-initial decay phases we can rule out the decreasing dust column density hypothesis.

From the SED extracted around the plateau no break is found between optical and X-ray frequencies, with $\beta_{\rm ox}=0.79\pm 0.01$. In the regime of slow cooling it is reasonable to assume that both optical ($\nu_{\rm o}$) and X-ray (${\nu_{\rm x}}$) frequencies lie between the injection ($\nu_{\rm m}$) and the cooling ($\nu_{\rm c}$) frequencies: $\nu_{\rm m}<\nu_{\rm o}<\nu_{\rm x}<\nu_{\rm c}$ (Sari et al. 1998). The power-law index of the electron energy distribution, $p$, is given by $\beta_{\rm ox}=(p-1)/2$, yielding $p=2.58\pm 0.02$, fully within the range of values found for other bursts (e.g. Starling et al. 2008). The temporal decay index depends on the density profile: the ISM (wind) case predicts a value of $\alpha=3(p-1)/4=1.18\pm 0.02$ ($\alpha=3p/4-1/4=1.68\pm 0.02$). After the plateau, depending on the light curve modelling, the measured decay index is either $\alpha=\alpha_{3}=1.08\pm 0.02$ (Sect. 4.1.1; Fig. 4) or $\alpha=\alpha_{2}=2.0\pm 0.8$ (Sect. 4.1.2; Fig. 5). While the multiple smoothly broken power-law description (Sect. 4.1.1) definitely rules out the wind environment, both environments are still possible in the two-component model (Sect. 4.1.2), mainly because of the poorly measured decay index, $\alpha_{2}$. If one interprets the flat decay as due to energy injection (Nousek et al. 2006; Zhang et al. 2006), the corresponding index would be $q\sim 0.3$.

In the off-axis jet interpretation, even if we consider an initially uniform sharp-edged jet, the shocked external medium at the sides of the jet has a significantly smaller Lorentz factor than near the head of the jet, and therefore its emission is not strongly beamed away from off-beam lines of sight. As a result, either an early very shallow rise or decay is expected for a realistic jet structure and dynamics (Eichler & Granot 2006). In the case of XRF 030723, Granot et al. (2005) showed that, for $\theta_{\rm obs}\sim\,2\,\theta_{0}$, an initial plateau is expected in the light curve.

Our observations of the afterglow rise of XRF 080330 rule out the interpretation proposed by Yamazaki (2009) of the plateau as due to an artifact of the choice of the reference time, as all the other rising curves do.

According to the light curve description of Sect. 4.1.1 shown in Fig. 4, for which only an ISM environment is possible (Sect. 5.2), after the plateau phase the light curve is expected to approach the on-axis light curve with $\alpha=3(p-1)/4$. The late-time steepening observed around $8\times 10^{4}$ s, estimated to be $\Delta\alpha=\alpha_{4}-\alpha_{3}=2.4\pm 0.4$, cannot be produced by the passage of the cooling frequency $\nu_{\rm c}$ through the optical, as that is expected to be as small as $\Delta\alpha=1/4$ (ISM/wind).

In the off-axis jet interpretation, assuming a value for $\theta_{0}$ of a few degrees, another advantage of this interpretation is the steep late time decay (at $\gtrsim$ 1 day) as a consequence of joining the post jet break on-axis light curve. According to the light curve modelling given in Sect. 4.1.2 shown in Fig. 5, the gradual steepening following the plateau corresponds to the post-jet break emission: the observed power-law decay index, $\alpha_{2}=2.0\pm 0.8$ is consistent with the expected $\alpha$ $=$ $p$ $=$ $2.6$. We note that the relatively sharp jet break favours the ISM environment. Overall, in the context of an off-axis viewing angle interpretation, the light curve suggests that $\theta_{\rm obs}\sim(1.5-2)\theta_{0}$ as well as an early jet break (at $\lesssim\,1\;$day), which in turn implies a narrow jet with a half-opening angle of the order of a few degrees, $\theta_{0}\,\sim\,0.05$. As a simple feasibility check, we note that for $\theta_{\rm obs}<2\theta_{0}$ the ratio of the on-axis to off-axis $E_{\gamma,{\rm iso}}$ is equal to $\delta^{2}$ (assuming that the observed energy range includes $E_{\rm p}$ where most of the energy is radiated), where $\delta$ is the ratio of their corresponding Doppler factors and therefore of their $E_{\rm p}$ (Eichler & Levinson 2004). In our case, for an observed off-axis $(1+z)E_{\rm p}\sim 60\;$keV, an on-axis value of $\sim 1\;$MeV would require $\delta=1+[\Gamma_{0}(\theta_{\rm obs}-\theta_{0})]^{2}\sim 17$ and $\Gamma_{0}(\theta_{\rm obs}-\theta_{0})\sim 4$, which for $\theta_{\rm obs}-\theta_{0}\sim(0.5-1)\theta_{0}$ and $\theta_{0}\sim 0.05$ gives $\Gamma_{0}\sim 80-160$. Here $\Gamma_{0}$ is the initial Lorentz factor at the edge of the jet. More realistically, the jet would not be perfectly uniform with extremely sharp edges, and instead $\Gamma_{0}$ is expected to be lower at the outer edge of the jet and larger near its center (where it could easily reach several hundreds in our illustrative example here). In this case $\delta^{2}\sim 300$ so that the observed $E_{\gamma,{\rm iso}}$ in the $15-150\;$keV range, which is $2\times 10^{51}\;$erg, would imply an on-axis value for $E_{\rm\gamma,iso}$ of $\sim 10^{54}\;$erg, which for a narrow jet with $\theta_{\rm obs}\sim 0.05$ would correspond to a true energy of the order of $10^{51}\;$erg. This demonstrates that this scenario can work for reasonable values of the physical parameters. We point out that the estimate of the on-axis $E_{\gamma,{\rm iso}}$ of $\sim 10^{54}$ erg is for a wide energy range containing $E_{\rm p}$, since in our illustrative example most of the energy is released within the observed range. A more accurate estimate of the break time and of the corresponding opening angle is difficult, due to the degeneracy involved in the light curve modelling (Figs. 4 and 5). Table 4 summarises the main properties of XRF 080330.

Overall, the two-component description of the light curves of Sect. 4.1.2 shown in Fig. 5 appears to be slightly favoured over the multiple smoothly broken power-law of Fig. 4. So, irrespective of the nature of the rise and plateau, we speculate on the possible nature of the second component, modelled in Fig. 5 as a late time bump. Clearly, a SN bump, such as that possibly observed in the light curve of XRF 030723 (Fynbo et al. 2004), is ruled out mainly because that is expected to peak days later, which is incompatible with one single day after XRF 080330; not to mention the too high redshift of XRF 080330 for a 1998bw-like SN to be detected.

Alternatively, a density bump seems a viable solution, given that $\nu_{\rm o}<\nu_{\rm c}$ (e.g. Lazzati et al. 2002; Guidorzi et al. 2005), although the explanation of the observed contemporaneous reddening requires ad hoc assumptions, such as the case of GRB 050721, which showed similar properties to XRF 080330 (same $\beta_{\rm ox}$ with no breaks between optical and X-ray, late time redder optical bump). In that case, the observed reddening was explained as due to the presence of very dense clumps surviving the GRB radiation and with a small covering factor (Antonelli et al. 2006).

If the late reddening is due to the passage of $\nu_{\rm c}$ through the optical bands, in addition to what argued in Sect. 5.3, another weak point of the multiple smoothly broken power-law description of Fig. 4 (Sect. 4.1.1) is the chromatic change of $\Delta\beta_{\rm o}=0.26\pm 0.06$ we observe in the optical bands around $8\times 10^{4}$ s. The passage of the cooling frequency does not explain it: the observed reddening would be 0.5, i.e. twice as much. This could still be the case, if our measurement might have taken place in the course of the spectral change, as the broken power-law spectrum is a simple approximation. However, since it is $\nu_{\rm c}>\nu_{\rm x}$ during the plateau because of the unbroken power-law spectrum between optical and X-ray, $\nu_{\rm c}$ would have decreased very rapidly, thus making this option not reasonable: if at $10^{3}$ s it is $\nu_{\rm c}>\nu_{\rm x}=10^{18}$ Hz, at $10^{5}$ s it should be $\nu_{\rm c}>10^{17}$ Hz $\gg\,\nu_{\rm o}$, because $\nu_{\rm c}\propto\,t^{-1/2}$ (ISM case), thus this possibility is to be ruled out. Likewise, in the wind case it is $\nu_{\rm c}\propto\,t^{1/2}$. The times $t_{K}$ and $t_{g}$, at which $\nu_{\rm c}$ would cross the most redward and blueward filters, $K$ and $g^{\prime}$, would differ by a factor of $(\nu_{g}/\nu_{K})^{2}\sim 20$, which looks incompatible with the light curves. Furthermore, the observed reddening rules out the wind case, as $\beta_{\rm o}$ should decrease from $p/2$ to $(p-1)/2$.

Although an energy injection to the blast wave (forward shock) can explain the bump feature in the light curve, it is difficult to explain the reddening if we consider only the forward shock emission, as we have discussed. A possible explanation for the reddening is that the rebrightening is due to the short-lived ($\Delta\,t\sim\,t$) reverse shock of a slow shell which caught up with the shock front and increased its energy through a refreshed shock (e.g. Kumar & Piran 2000; Granot et al. 2003; Jóhannesson et al. 2006): since that shock is going into a shell of ejecta, rather than the external medium, it can have a much larger $\epsilon_{B}$ (magnetised fireball: Zhang et al. 2003; Kumar & Panaitescu 2003; Gomboc et al. 2008b) and, therefore, a lower $\nu_{\rm c}$, quite naturally; for an ISM where $\nu_{\rm c}$ decreases with time, then $\nu_{\rm c}$ of the reverse shock could be around the optical for reasonable model parameter values. The need for a separate component to explain a chromatic break in the light curve was also suggested in the case of GRB 061126 (Gomboc et al. 2008b). Notably, the final steepening after $10^{5}$ s, which in the modelling of Sect. 4.1.1 (Fig. 4) is described with $\alpha_{4}=3.5_{-0.3}^{+0.4}$, is compatible with the high-latitude emission of the reverse shock: $\alpha=\beta_{\rm o,late}+2=3.05\pm 0.06$. The jet break might happen slightly earlier than the break time in the optical light curve.

A somewhat more contrived way to explain the bump is the appearance of a second narrower jet in the two-component jet model, as proposed for XRF 030723 (Huang et al. 2004). In this model, the viewing angle, is within or slightly off the cone of the wide jet and outside the narrow jet. The plateau phase would reflect the deceleration of the wide jet (Granot et al. 2006). Depending on the isotropic-equivalent kinetic energy of the wide and narrow jet, $E_{\rm w,iso}$ and $E_{\rm n,iso}$, on the jet opening angles, $\theta_{0,{\rm w}}$ and $\theta_{0,{\rm n}}$, on the initial bulk Lorentz factors, $\gamma_{0,{\rm w}}$ and $\gamma_{0,{\rm n}}$, respectively, as well as on the viewing angle, the afterglow emission of either component is dominant at different times. According to the results of Huang et al. (2004), the light curve of XRF 080330 could be qualitatively explained as follows: the first component obtained in Sect. 4.1.2 (Fig. 5) represents the contribution of the wide jet dominating at early times: the rise could be due either to the afterglow onset (in a wind environment) or to a viewing angle slightly beyond the wide jet opening angle: $\theta_{\rm obs}\gtrsim\theta_{0,{\rm w}}$. The appearance of the second component would mark the deceleration and lateral expansion of the narrow jet, the peak time corresponding to the case $\gamma_{\rm n}\sim 1/\theta_{\rm obs}$. Unlike XRF 030723, which showed a relatively sharp late-time peak, the bump exhibited by XRF 080330 looks less sharp and pronounced. Although this might suggest a relatively lower energy of the narrow jet compared to XRF 030723, yet we cannot exclude the case $E_{\rm w,iso}\ll\,E_{\rm n,iso}$. The ratio of the observed energies of XRF 080330, of about $0.6$ according to the modelling of Sect. 4.1.2, corresponds to the ratio between the early and the late time emissions of the wide and narrow jets, respectively. Depending on the values of $\theta_{\rm obs}$, $\theta_{0,{\rm w}}$, $\theta_{0,{\rm n}}$, a comparable ratio of observed energies, such as that observed for XRF 080330, can still be obtained in the case of a much more energetic narrow jet, $E_{\rm w,iso}\ll\,E_{\rm n,iso}$.

Such a model turned out to be successful in accounting for the naked-eye GRB 080319B (Racusin et al. 2008): in that case, the two collimation-corrected energies were comparable, while the isotropic-equivalent energy of the narrow jet ($\theta_{0,{\rm n}}=0.2^{\circ}$) was about 400 times larger than that of the wide jet ($\theta_{0,{\rm w}}=4.0^{\circ}$).

However, in the context of the two-component model the explanation of the late time reddening simultaneously with the appearance of the narrow component emission requires two different values of $p$ for the two jets, which does not look reasonable on a physical ground. Another option is that the cooling frequency of the second jet might be around the optical at the time of the bump: however, if the shock micro-physics parameters ($p$, $\epsilon_{B}$, $\epsilon_{e}$), are the same for the two shocks, as expected on physical grounds, and obviously the external medium is the same, then the only thing that is different as far as $\nu_{\rm c}$ is concerned, is $E_{\rm iso}$. Since the presence of the bump requires $E_{\rm n,iso}>E_{\rm w,iso}$, then this would not work for the wind case, where $\nu_{\rm c}\propto\,E_{\rm iso}^{1/2}$. Even in the ISM case, where $\nu_{\rm c}\propto\,E_{\rm iso}^{-1/2}$, the fact that $\nu_{\rm c}>\nu_{\rm x}$ at $10^{3}$ s, and therefore $\nu_{\rm c}>10^{17}$ Hz at $10^{5}$ s for the wide jet, requires $E_{\rm n,iso}/E_{\rm w,iso}\gtrsim 10^{4}$ (which is very extreme) in order for $\nu_{\rm c,n}(10^{5}\leavevmode\nobreak\ {\rm s})$ to be around $10^{15}$ Hz (required by the observed reddening). Therefore, while this could work in principle, in practise it requires extreme parameters. In particular, the amplitude of the bump suggests that $E_{\rm n}\sim\,E_{\rm w}$, and therefore from the required $E_{\rm n,iso}/E_{\rm w,iso}\gtrsim 10^{4}$ it would follow $\theta_{\rm n}/\theta_{\rm w}\lesssim\,10^{-2}$, which seems pushed to the extreme.