ENERGY INJECTION IN GAMMA-RAY BURST AFTERGLOWS

In this scenario, the peak of the light curve emission corresponds to emission from a RS due to deceleration of the ejecta by the surrounding material. The ejecta are assumed to be composed of a conical shell segment at a single Lorentz factor. The RS light curves before and after the deceleration time depend on the properties of the environment (ISM or wind) and the ejecta. RS light curves for a constant density environment depend upon whether the ejecta shell is thick (${{\rm{\Delta }}}_{0}\gt l/2{{\rm{\Gamma }}}_{0}^{{}^{8}{/}_{3}}$) or thin (${{\rm{\Delta }}}_{0}\gt l/2{{\rm{\Gamma }}}_{0}^{{}^{8}{/}_{3}}$), where ${{\rm{\Delta }}}_{0}$ is the initial shell width, ${{\rm{\Gamma }}}_{0}$ is the initial Lorentz factor of the ejecta and $l={(3E/4\pi {n}_{0}{m}_{{\rm{p}}}{c}^{2})}^{1/3}$ is the Sedov length (Kobayashi 2000). Similarly, when the circumburst medium has a wind-like profile, the light curves again depend on whether ${{\rm{\Delta }}}_{0}\gt E/4\pi {{Ac}}^{2}{{\rm{\Gamma }}}_{0}^{4}$ (thick shell) or ${{\rm{\Delta }}}_{0}\lt E/4\pi {{Ac}}^{2}{{\rm{\Gamma }}}_{0}^{4}$ (thin shell; Kobayashi & Zhang 2003).

In the thick shell case, the RS crosses the ejecta in a time comparable to the burst duration, ${t}_{\gamma }$, and the light curves at all frequencies are expected to peak at that timescale. In the case of GRB 120326A, T₉₀ in the Swift/BAT band is $\approx 27$ s, which is much shorter than the observed time at which the light curves peak, ${t}_{\mathrm{peak}}\approx 0.4$ d. Thus, the re-brightening cannot be explained by RS emission in the thick shell case.

In the thin shell case, temporal separation is expected between the GRB itself and the peak of the RS, which occurs when the RS crosses the shell at ${t}_{\gamma }=90{E}_{{\rm{K}},\mathrm{iso},52}^{1/3}{n}_{0}^{-1/3}{{\rm{\Gamma }}}_{0,100}^{-8/3}$ s (ISM environment; Kobayashi 2000) or $t=2.9\times {10}^{3}(1+z){E}_{{\rm{K}},\mathrm{iso},52}{{\rm{\Gamma }}}_{0,100}^{-4}{A}_{*}^{-1}$ s (wind environment; Zou et al. 2005). A RS peak at 0.4 d would imply a rather low burst Lorentz factor of ${{\rm{\Gamma }}}_{0}\approx 17$. Additionally, the light curves at all frequencies would be expected to rise until ${t}_{\gamma }$ and decline thereafter (except above ${\nu }_{{\rm{c}}}$ where the flux is constant until ${t}_{\gamma }$ and disappears after ${t}_{\gamma }$). The observed R-band light curve, on the other hand, declines between 100 and 500 s, followed by a flat segment until 3500 s, followed by the rise into the bump at 0.4 d. This combination of a flat portion followed by a rise cannot be explained as being due to an RS.

Thus the onset of the afterglow does not provide a viable explanation for the re-brightening regardless of density profile, and whether the ejecta are in the thick- or thin-shell regime.

We now investigate the re-brightening in the context of viewing geometry effects. We consider a jet with an opening half-angle, ${\theta }_{\mathrm{jet}}$, viewed from an angle ${\theta }_{\mathrm{obs}}$. While small offsets in the observer's viewing angle relative to the jet axis do not cause significant changes in the light curves provided that ${\theta }_{\mathrm{obs}}\lesssim {\theta }_{\mathrm{jet}}$, it is possible to obtain a rising light curve when ${\theta }_{\mathrm{obs}}\gtrsim 2{\theta }_{\mathrm{jet}}$ (Granot et al. 2001; 2002). In this case, the time of the peak of the light curve, ${t}_{{\rm{p}}}$ is related to the jet break time for an on-axis viewer as ${t}_{{\rm{p}}}\approx (5+2\mathrm{ln}{\rm{\Theta }}){{\rm{\Theta }}}^{2}{t}_{\mathrm{jet}}\gtrsim 5{t}_{\mathrm{jet}}$, where ${\rm{\Theta }}\equiv {\theta }_{\mathrm{obs}}/{\theta }_{\mathrm{jet}}-1\gtrsim 1$ (Granot et al. 2002). This implies that for a light curve that peaks at ≈0.4 d, the on-axis jet break time must have occurred earlier, at ≈2 hr.

In this case, the radio light curves are expected to rise until the observer's line of sight enters the beaming cone of the jet and then approximately converge with the predicted on-axis light curves after 0.4 d. After the jet break, the flux density declines as ${t}^{{}^{-1}{/}_{3}}$ for $\nu \lt {\nu }_{{\rm{m}}}$ and as ${t}^{-p}$ for $\nu \gt {\nu }_{{\rm{m}}}$ (for an on-axis observer; Sari et al. 1999). Thus, the radio light curves should be declining at all frequencies after 0.4 d. However, the flux density at the 15.75 GHz AMI-LA band rises from (0.34 ± 0.14) mJy at 0.31 d to (0.77 ± 0.08) mJy at 7.15 d, which is inconsistent with this expectation. Further, relativistic beaming of the prompt radiation within the jet cone would typically result in an orphan afterglow (e.g., Huang et al. 2002) or require a large intrinsic energy for the GRB to be visible off-axis (e.g., Margutti et al. 2010a). This argues against the off-axis model as an explanation for the X-ray/UV/optical re-brightening.

If the blastwave encounters an enhancement in the local density as it propagates into the circumburst environment, an increase in the flux density is expected. Bumps in afterglow light curves have been ascribed to this phenomenon in the past (e.g., Berger et al. 2000; Lazzati et al. 2002; Schaefer et al. 2003). However, the flux density at frequencies above ${\nu }_{{\rm{c}}}$ (for slow-cooling spectra) and above ${\nu }_{{\rm{m}}}$ (for fast cooling spectra) are insensitive to variations in the circumburst density (Nakar & Granot 2007). In the case of GRB 120326A, a re-brightening is observed both in the UV/optical and in the X-rays. The X-rays are located above both ${\nu }_{{\rm{c}}}$ and ${\nu }_{{\rm{m}}}$ in the $p\approx 2.1$ (fast cooling) model. Thus in our favored afterglow model, the X-ray re-brightening cannot be due to a density enhancement. Additionally, Geng et al. (2014) find that a density jump in the circumburst medium usually does not lead to a significant rebrightening, even when emission from the resulting RS is included.

An alternative model for a re-brightening of the afterglow is the injection of energy into the blastwave shock due to prolonged central engine activity, deceleration of a Poynting flux dominated outflow, or the presence of substantial ejecta mass (and hence kinetic energy) at low Lorentz factors (Dai & Lu 1998; Rees & Meszaros 1998; Kumar & Piran 2000; Sari & Mészáros 2000; Zhang & Mészáros 2001, 2002; Granot & Kumar 2006; Zhang et al. 2006; Dall'Osso et al. 2011; Uhm et al. 2012). This mechanism has been invoked to explain plateaus in the observed light curves of a large fraction of GRB X-ray afterglows (Nousek et al. 2006; Liang et al. 2007; Yu & Dai 2007; Margutti et al. 2010a; Hascoët et al. 2012; Xin et al. 2012). In this section, we explore the energy-injection model as a possible explanation for the X-ray/UV/optical re-brightening at 0.4 d.

Models involving a transfer of energy from braking radiation of millisecond magnetars into the forward shock predict plateaus in X-ray light curves but do not generally lead to a re-brightening. In particular, they require an injection luminosity, $L\propto {t}^{q}$ (corresponding to an increase in the blastwave energy with time as $E\propto {t}^{1+q}$), with $q\leqslant 0$. In our $p\approx 2.1$ model, the X-rays are located above the cooling frequency. In this regime, the flux density above ${\nu }_{{\rm{c}}}$ is ${F}_{\nu \gt {\nu }_{{\rm{c}}}}\propto {E}_{{\rm{K}},\mathrm{iso}}^{(2+p)/4}{t}^{(2-3p)/4}$ (Granot & Sari 2002). For $p\approx 2.1$, this reduces to ${F}_{\nu \gt {\nu }_{{\rm{c}}}}\propto {E}_{{\rm{K}},\mathrm{iso}}^{1.03}{t}^{-1.1}$. During energy injection, $E\propto {t}^{m}$, such that ${F}_{\nu \gt {\nu }_{{\rm{c}}}}\propto {t}^{1.03m-1.1}$. The steep rise (${\alpha }_{1}=0.85\pm 0.19$; Section 3.1) requires $m=1.89\pm 0.18$ or $q\equiv m-1=0.89\pm 0.18$. Energy injection due to spin-down or gravitational wave radiation from a magnetar is expected to provide at best a constant luminosity ($q\leqslant 0$). Thus the observed re-brightening cannot be explained by energy injection from a magnetar. In the case of energy injection due to fall-back accretion onto a black hole, the expected accretion rate is $\dot{M}\propto {t}^{-5/3}$ (Phinney 1989), which is also insufficient to power the observed plateaus. Similarly, central engine activity is usually associated with flaring behavior in X-ray (Burrows et al. 2005a; Fan & Wei 2005; Falcone et al. 2006; Perna et al. 2006; Proga & Zhang 2006; Zhang et al. 2006; Chincarini et al. 2007, 2010; Margutti et al. 2010b, 2011a; Bernardini et al. 2011) and optical (Li et al. 2012) light curves, but it involves shorter characteristic time scales $({\rm{\Delta }}t/t\ll 1)$ than observed for GRB 120326A at 0.4 d.

In the standard afterglow model, the ejecta are assumed to have a single Lorentz factor. We now relax this assumption and consider models including a distribution of Lorentz factors in the ejecta as a possible explanation for the late re-brightening in GRB 120326A. Provided they are released over a time range small compared to the afterglow timescale, the ejecta arrange themselves in homologous expansion with the Lorentz factors monotonically increasing with distance from the source (Kumar & Piran 2000). We follow the formalism of Sari & Mészáros (2000), where the ejecta are assumed to possess a continuous distribution of Lorentz factors such that a mass, $M(\gt {\rm{\Gamma }})\propto {{\rm{\Gamma }}}^{-s}$ is moving with Lorentz factors greater than Γ with corresponding energy, $E(\gt {\rm{\Gamma }})\propto {{\rm{\Gamma }}}^{-s+1}$ down to some minimum Lorentz factor, ${{\rm{\Gamma }}}_{\mathrm{min}}$ (also known as the "massive-ejecta model"). Additionally, we posit a maximum Lorentz factor for this distribution, ${{\rm{\Gamma }}}_{\mathrm{max}}$, corresponding to the Lorentz factor of the blastwave ${{\rm{\Gamma }}}_{\mathrm{BW}}$ at the onset of energy injection. In this model, there is a gap between the blastwave, which is powered by an initial shell of fast-moving ejecta, and the leading edge of the remaining ejecta traveling at ${{\rm{\Gamma }}}_{\mathrm{max}}$. When these slower ejecta shells catch up with the blastwave at a time when ${{\rm{\Gamma }}}_{\mathrm{BW}}(t)\approx {{\rm{\Gamma }}}_{\mathrm{max}}$, they begin depositing energy into the blastwave, and energy injection commences. This proceeds until the lowest energy ejecta located at ${{\rm{\Gamma }}}_{\mathrm{min}}$ have transferred their energy to the blastwave, and the subsequent evolution proceeds like a standard afterglow powered by a blastwave with increased energy. The Lorentz factor of the blastwave at the end of energy injection is therefore ${{\rm{\Gamma }}}_{\mathrm{min}}$, which can be determined from modeling the subsequent afterglow evolution and invoking the standard hydrodynamical framework.

For an ambient medium with a density profile, $n\propto {r}^{-k}$, the energy of the blastwave increases as $E\propto {t}^{m}$ during the period of energy injection, where¹⁴ $m=\frac{(3-k)(s-1)}{7+s-2k}$. For a constant density medium (k = 0) this translates to $m=\frac{3(s-1)}{7+s}$, which is bounded $\left(0\lt m\lt 3\;\mathrm{for}\;s\in [1,\infty )\right)$ while for a wind-like environment (k = 2) we have $m=\frac{s-1}{s-3}$, which is also bounded $\left(0\lt m\lt 1{\rm{}}\;{\rm{for}}\;{\rm{}}s\in [1,\infty )\right)$.

For GRB 120326A, the optical and X-ray light curves are located above both ${\nu }_{{\rm{m}}}$ and ${\nu }_{{\rm{c}}}$ and the spectrum is fast cooling (${\nu }_{{\rm{c}}}\lt {\nu }_{{\rm{m}}}$; Section 5.1). In this regime, the flux density depends on the blastwave energy and time as, ${F}_{\nu }\propto {E}^{(2+p)/4}{t}^{(2-3p)/4}$. Writing $E={E}_{0}{(t/{t}_{0})}^{m}$ for the energy injection episode, this implies ${F}_{\nu }\propto {t}^{[(2+p)m+2-3p]/4}\propto {t}^{m-1}$, for p = 2. Since the optical and X-ray light curves rise with temporal index, $\alpha \approx 0.6$ (Section 3.1) and $p\approx 2$ (Section 5.1), this implies $m\approx 1.6$, which corresponds to $s=\frac{7m+3}{3-m}\approx 10$.

For a detailed analysis, we use the afterglow parameters for the maximum likelihood model listed in Section 5.1 as our starting model (which explains all available multi-wavelength data after the re-brightening). We then reduce the energy at earlier times as a power law with time,

$$\begin{eqnarray}{E}_{{\rm{K}},\mathrm{iso}}(t)=\left\{\begin{array}{ll}{E}_{{\rm{K}},\mathrm{iso},{\rm{f}}}, & t\geqslant {t}_{0}\\ {E}_{{\rm{K}},\mathrm{iso},{\rm{f}}}{\left(\displaystyle \frac{t}{{t}_{0}}\right)}^{m}, & {t}_{1}\lt t\lt {t}_{0}\\ {E}_{{\rm{K}},\mathrm{iso},{\rm{i}}}\equiv {E}_{{\rm{K}},\mathrm{iso},{\rm{f}}}{\left(\displaystyle \frac{{t}_{1}}{{t}_{0}}\right)}^{m}, & t\leqslant {t}_{1},\end{array}\right.\end{eqnarray} \tag{ 3 }$$

where ${E}_{{\rm{K}},\mathrm{iso},{\rm{i}}}$ and ${E}_{{\rm{K}},\mathrm{iso},{\rm{f}}}$ are the initial and final energy of the blastwave, respectively, with the latter fixed to the value determined by our MCMC analysis of the data at $\geqslant 0.4$ d. Energy injection into the blastwave at the rate $E(t)\propto {t}^{m}$ begins at t₁ and ends at t₀, with t₀, t₁, and m being free parameters in this model. We compute the spectrum at each time according to the instantaneous energy in the blastwave, adjusting the parameters t₀ and t₁ by hand until the resulting light curves match the observations.

We find that the X-ray and optical light curves can be modeled well by a single period of energy injection, with ${E}_{{\rm{K}},\mathrm{iso}}=1.4\times {10}^{53}\;\mathrm{erg}{(t/0.38\;{\rm{d}})}^{1.44}$ between ${t}_{1}=4.3\times {10}^{-2}$ d and ${t}_{0}=0.38$ d (Figure 4). In this model, the blastwave energy at the start of energy injection is ${E}_{{\rm{K}},\mathrm{iso},{\rm{i}}}\approx 5.9\times {10}^{51}$ erg. Thus $\approx 96\%$ of the final energy of the blastwave is injected during this episode. In comparison, ${E}_{\gamma ,\mathrm{iso}}=(3.15\pm 0.12)\times {10}^{52}$ erg (Section 3). The blastwave Lorentz factor can be computed from the expression ${\rm{\Gamma }}=3.65{{E}_{{\rm{K}},\mathrm{iso},52}}^{1/8}{n}_{0}^{-1/8}{t}^{-3/8}{(1+z)}^{3/8}$ (Granot & Sari 2002) and is ${\rm{\Gamma }}\approx 19$ at the start of energy injection, decreasing to ${\rm{\Gamma }}\approx 13$ at 0.38 d. Interpreted in the context of the massive-ejecta model, the energy injection rate of m = 1.44 would imply s = 8.4, corresponding to an ejecta distribution with $E(\gt {\rm{\Gamma }})\propto {{\rm{\Gamma }}}^{1-s}\propto {{\rm{\Gamma }}}^{-7.4}$ for $13\lesssim {\rm{\Gamma }}\lesssim 19$.

GRB 100418A was detected and localized by the Swift BAT on 2010 April 18 at 21:10:08 UT (Marshall et al. 2010). The burst duration is ${T}_{90}=8.0\pm 2.0$ s, with a fluence of ${F}_{\gamma }=(3.4\pm 0.5)\times {10}^{-7}$ erg cm⁻² (15–150 keV observer frame; Marshall et al. 2011). The afterglow was detected in the X-ray and optical bands by XRT, UVOT, and various ground-based observatories (Filgas et al. 2010; Siegel & Marshall 2010; Updike et al. 2010c), as well as in the radio by WSRT, VLA, the Australia Telescope Compact Array (ATCA), and the Very Long Baseline Array (VLBA; Chandra & Frail 2010b; Moin et al. 2010; van der Horst et al. 2010a). Spectroscopic observations with the ESO Very Large Telescope (VLT) yielded a redshift of z = 0.6235 (Antonelli et al. 2010). By fitting the γ-ray spectrum with a Band function, Marshall et al. (2011) determined the isotropic equivalent γ-ray energy of this burst to be ${E}_{\gamma ,\mathrm{iso}}=\left({9.9}_{-3.4}^{+6.3}\right)\times {10}^{50}$ erg (1–10⁴ keV; rest frame).

The X-ray light curve for this burst gently rises to 0.7 d, while the optical light curves exhibit a slow brightening at the same time. Both the X-ray and optical light curves break into a power-law decline following the peak. Using XRT and UVOT White-band observations, Marshall et al. (2011) find that the X-ray and optical bands are located on the same part of the synchrotron spectrum after the peak at 0.7 d. By fitting the X-ray and optical decline rates and X-ray-to-optical SED, they find that both ${\nu }_{{\rm{m}}}$ and ${\nu }_{{\rm{c}}}$ are located below the optical, requiring $p\approx 2.3$. In this regime, the light curves are insensitive to the density profile of the circumburst environment. In the absence of radio data, Marshall et al. (2011) assume an ISM model and determine that the non-detection of a jet break out to 23 d requires a high beaming-corrected energy, ${E}_{{\rm{K}}}\geqslant 3\times {10}^{52}{E}_{{\rm{K}},\mathrm{iso},54}^{3/4}{n}_{0}^{1/4}$. They investigate the off-axis model, and compute an intrinsic burst duration for an on-axis observer of ${T}_{90}^{\mathrm{on}}={\mathcal{D}}(\theta ={\theta }_{\mathrm{obs}}-{\theta }_{\mathrm{jet}})/{\mathcal{D}}(\theta =0){T}_{90}\approx 10$ ms, where ${\mathcal{D}}\equiv {[{\gamma }_{0}(1-{\beta }_{0}\mathrm{cos}\theta )]}^{-1}$ is the Doppler factor, ${\theta }_{\mathrm{obs}}$ is the off-axis viewing angle, ${\theta }_{\mathrm{jet}}$ is the jet opening half-angle, and T₉₀ is the observed burst duration. Based on this short on-axis ${T}_{90}^{\mathrm{on}}$ and large intrinsic energy output, they argue against the off-axis scenario.

Marshall et al. (2011) also consider the energy injection model. For the magnetar model involving injection of energy into the blastwave via Poynting flux, they find $q={-0.23}_{-0.14}^{+0.13}$, which is marginally consistent with the q = 0 case. For models with a Lorentz factor distribution, they use their measured value of q to calculate $s={6.4}_{-1.0}^{+1.3}$ (ISM model), or $s\lt 0$ (wind model, which is unphysical). They argue that since the ISM value is too different from the value $s\approx 2.5$ indicated by X-ray plateaus in Swift observations (Nousek et al. 2006), this mode of energy injection is unlikely.

Moin et al. (2013) reported multi-wavelength radio observations of GRB 100418A from 5.5 to 9.0 GHz from ATCA, VLA and VLBA. Using upper limits on the expansion rate of the ejecta derived from the VLBA observations, they report an upper limit on the average ejecta Lorentz factor of Γ ≲ 7 and an upper limit on the ejecta mass of ${M}_{\mathrm{ej}}\lesssim 0.5\times {10}^{-3}\;{M}_{\odot }$. They use this limit on Γ to suggest that the contribution of low-Lorentz factor ejecta to the blastwave energy must be negligible during the period of energy injection, and that a separate injection mechanism is required. We include their radio observations in our analysis (Section 6.1.2) and address the constraints derived from the VLBA data in Section 6.1.3.

Marshall et al. (2011) fit multi-band UVOT and XRT spectra at three different epochs (7 × 10⁻², 0.8, and 7.3 d) and found that a single power law with spectral index β ≈ −1.0 fit all three epochs well. We follow the procedures outlined by Evans et al. (2007); (2009), and Margutti et al. (2013) to obtain time-resolved XRT spectra for this burst. The spectra are well fit by an absorbed single power-law model with ${N}_{{\rm{H}},\mathrm{Gal}}=4.78\times {10}^{20}\;{\mathrm{cm}}^{-2}$ (Kalberla et al. 2005), and intrinsic hydrogen column ${N}_{{\rm{H}},\mathrm{int}}=({0.7}_{-0.7}^{+1.5})\times {10}^{21}\;{\mathrm{cm}}^{-2}$, with ${\rm{\Gamma }}=3.35\pm 0.09$ (85–130 s; ${\chi }^{2}=1.74$ for 123 degrees of freedom) and ${\rm{\Gamma }}=3.9\pm 0.3$ (130 s–180 s; ${\chi }^{2}=0.97$ for 55 degrees of freedom) during the early and late WT-mode steep decay, respectively, followed by ${{\rm{\Gamma }}}_{{\rm{X}}}=2.08\pm 0.18$ ($2.3\times {10}^{-3}$ d–1.8 d; ${\chi }^{2}=0.75$ for 79 degrees of freedom) during the plateau phase(, and ${{\rm{\Gamma }}}_{{\rm{X}}}=1.81\pm 0.20$ (1.8–34.5 d; ${\chi }^{2}=0.70$ for 89 degrees of freedom) for the remainder of the observations. We obtain a flux light curve in the 0.3–10 keV XRT band following the methods reported by Margutti et al. (2013), which we convert to a flux density light curve at 1 keV using the above time-resolved spectra. We also analyze all UVOT photometry for this burst and report our results in Table 6.

Table 6.  Swift/UVOT Observations of GRB 100418A

$t-{t}_{0}$	Filter	Frequency	Flux Density	Uncertainty	Detection?
(days)		(Hz)	(μJy)	(μJy)	(1 = Yes)
0.00188	White	8.64e+14	11.5	2.47	1
0.00499	White	8.64e+14	10.6	2.81	1
0.00591	u	8.56e+14	8.41	2.74	0
0.00753	b	6.92e+14	74.7	22.3	0
0.011	White	8.64e+14	10.8	3.25	1

Machine-readable versions of the table is available.

Download table as:  DataTypeset image

The X-ray light curve up to 8 × 10⁻³ d declines steeply as ${\alpha }_{{\rm{X}}}=-4.52\pm 0.14$, transitioning to a plateau, at about 0.01 d, followed by a re-brightening. The X-ray observations between 0.02 d and 5.5 d can be well fit with a broken power-law model, with ${t}_{{\rm{b}}}=0.54\pm 0.18$ d, ${F}_{\nu ,{\rm{X}}}({t}_{{\rm{b}}})=0.32\pm 0.08\;\mu \mathrm{Jy}$, ${\alpha }_{1}=0.46\pm 0.20$, and ${\alpha }_{2}=-1.15\pm 0.10$, with the smoothness parameter fixed at y = 5.0. A broken power-law fit to the UVOT White-band data between 6 × 10⁻³ d and 3.1 d yields ${t}_{{\rm{b}}}=0.64\pm 0.17$ d, ${F}_{\nu ,\mathrm{White}}({t}_{{\rm{b}}})=32\pm 4\;\mu \mathrm{Jy}$, ${\alpha }_{1}=0.34\;\pm$ 0.09, and ${\alpha }_{2}=-1.02\pm 0.18$, with y = 5.0 (Figure 7). The decline rate in the White band following the re-brightening is consistent with the decline rate in the R-band¹⁵ between 0.4 d and 10 d (${\alpha }_{{\rm{R}}}=-0.97\pm 0.08$) and in the H-band between the PAIRITEL observations at 0.46 d and 1.49 d (${\alpha }_{{\rm{H}}}=-1.0\pm 0.1$). Thus, the break time and rise and decay rates of the re-brightening in the X-ray and optical bands are consistent. Finally, we fit the X-ray and UVOT White-band data jointly, where we constrain the model light curves in the two bands to have the same rise and decay rate and time of peak with independent normalizations. Using a Markov Chain Monte Carlo simulation using emcee (Foreman-Mackey et al. 2013), we find ${t}_{{\rm{b}}}={0.58}_{-0.34}^{+0.37}$ d, ${\alpha }_{1}={0.42}_{-0.13}^{+0.31}$, and ${\alpha }_{2}=-{1.10}_{-0.21}^{+0.15}$, with a fixed value of y = 5.0.

Zoom In Zoom Out Reset image size

We extract XRT spectra at 0.06–0.08 d (mean photon arrival 0.07 d), and at 0.58–1.81 d (mean photon arrival 0.88 d). We extrapolate the second SED to 1.5 d and show the complete optical to X-ray SED at 0.07 d and 1.5 d in Figure 8. The spectral index between the UVOT B-band observation at 0.07 d and the X-rays is ${\beta }_{\mathrm{opt}-{\rm{X}}}=-0.93\pm 0.09$, which is consistent with the X-ray spectral index of ${\beta }_{{\rm{X}}}=-1.08\pm 0.18$ over this period, suggesting that the optical and X-ray are located on the same part of the afterglow SED at 0.07 d. The spectral index between the NIR and X-rays at 1.5 d is ${\beta }_{\mathrm{NIR}-{\rm{X}}}=-0.94\pm 0.04$, which is consistent with the spectral index in the NIR and optical alone, ${\beta }_{\mathrm{NIR}-\mathrm{opt}}=-0.8\pm 0.1$, once again suggesting that the NIR/optical and X-rays are located on the same part of the afterglow SED at 1.5 d and that intrinsic extinction is negligible. This implies that either ${\nu }_{{\rm{m}}}\lt {\nu }_{\mathrm{NIR}}\lt {\nu }_{{\rm{X}}}\lt {\nu }_{{\rm{c}}}$ with $p\approx 3$ or ${\nu }_{{\rm{m}}},{\nu }_{{\rm{c}}}\lt {\nu }_{\mathrm{NIR}}\lt {\nu }_{{\rm{X}}}$ with $p\approx 2$ at both 0.07 d and 1.5 d. If ${\nu }_{{\rm{m}}}\lt {\nu }_{\mathrm{NIR}}\lt {\nu }_{{\rm{X}}}\lt {\nu }_{{\rm{c}}}$ at 1.5 d, the decay rate would be α ≈ −1.5 in the ISM model or α ≈ −2.0 in the wind model, whereas in the latter we would expect α ≈ −1 in both ISM and wind scenarios. The observed UV/NIR/X-ray common decay rate after the re-brightening of α ≈ −1 indicates that ${\nu }_{{\rm{m}}},{\nu }_{{\rm{c}}}\lt {\nu }_{\mathrm{NIR}}\lt {\nu }_{{\rm{X}}}$ at 1.5 d and $p\approx 2$.

Zoom In Zoom Out Reset image size

The 8.46 GHz light curve drops as ${t}^{-2.5}$ between the ATCA observation at 2 d and the VLA observation at 3 d, while the SMA 345 GHz light curve is flat over this period (Figure 9). At ≈2 d, the radio spectral index is $\approx 0.7$ from 4.9 to 8.46 GHz, and $\approx 0.3$ from 8.46 to 345 GHz. The rapid decline in the light curve at a single waveband can not be explained in the standard synchrotron model by forward shock emission. We consider an alternative scenario, in which the radio to millimeter emission at 2 d is dominated by a RS. In that case, the peak frequency of the RS must pass through 8.46 GHz before 2 d with a peak flux density of at least $\approx 1.3$ mJy. Propagating the RS spectrum back in time assuming the peak frequency and peak flux evolve as ${t}^{-1.5}$ and ${t}^{-1}$, respectively (Kobayashi & Sari 2000), we would expect a minimum R-band flux density of $\approx 2$ Jy at 120 s. This is more than five orders of magnitude brighter than Swift White-band observations at this time. Thus the rapid decline and re-brightening of the 8.46 GHz light curve between 2 and 20 d cannot be explained in the standard synchrotron framework. It is possible that these data suffer from strong ISS, and we therefore exclude these data from our analysis.

Zoom In Zoom Out Reset image size

The optical and X-ray light curves after 0.4 d are insensitive to the circumburst density profile since ${\nu }_{{\rm{m}}},{\nu }_{{\rm{c}}}\lt {\nu }_{\mathrm{NIR}}\lt {\nu }_{{\rm{X}}}$ after the re-brightening. The best-sampled radio light curve is the composite 8.46 GHz formed from VLA, VLBA, and ATCA observations. However, this light curve exhibits significant scatter about a smooth power-law evolution, possibly due to either ISS or inter-calibration issues across the three telescopes.¹⁶ Thus we are unable to constrain the circumburst density profile from the afterglow data. For the remainder of this section we focus on the ISM model, discussing the possibility of a wind environment in Appendix C.

Since this event occurred at a relatively low redshift, z = 0.6235 (Antonelli et al. 2010), the host galaxy is detected in the Sloane Digital Sky Survey (SDSS) as SDSS J170527.10+112742.5. We use the SDSS photometry ($u=24.55\pm 1.14$, $g=22.92\pm 0.17$, $r=22.45\pm 0.17$, $i=21.95\pm 0.17$, $z=22.54\pm 0.83$) as fixed a priori measurements of the host galaxy flux in the griz bands. We compute the host flux density at B- and V-band using $B=g+0.33(g-r)+0.2$ and $V=g-0.58(g-r)-0.01$ (Jester et al. 2005) and hold these values fixed for our MCMC analysis. Since there is no a priori measurement of the host flux density in the UVOT/White-band and much of the UVOT data was taken in White-band, we keep the host flux density in the White-band as a free parameter, and integrate over the GRB SED using the White-band filter function.¹⁷

Using our MCMC analysis as described in Section 4, we fit the data after 0.4 d, and our highest-likelihood model (Figure 9) has the parameters $p\approx 2.1$, ${\epsilon }_{{\rm{e}}}\approx 0.12$, ${\epsilon }_{B}\approx 1.1\times {10}^{-2}$, ${n}_{0}\approx 1.4$ ${\mathrm{cm}}^{-3}$, ${E}_{{\rm{K}},\mathrm{iso}}\approx 3.4\times {10}^{52}$ erg, ${t}_{\mathrm{jet}}\approx 17$ d, ${A}_{{\rm{V}}}\lesssim 0.1$ mag, and ${F}_{\nu ,\mathrm{host},\mathrm{White}}\approx 1.6\;\mu$Jy, with a Compton y-parameter of $\approx 2.9$, indicating IC cooling is important. The blastwave Lorentz factor is ${\rm{\Gamma }}=4.9{(t/1\;{\rm{d}})}^{-3/8}$ and the jet opening angle is ${\theta }_{\mathrm{jet}}\approx 20^\circ$. The beaming-corrected kinetic energy is ${E}_{{\rm{K}}}\approx 2.1\times {10}^{51}\;\mathrm{erg}$, while the beaming corrected γ-ray energy is ${E}_{\gamma }\approx 6\times {10}^{49}\;\mathrm{erg}$ (1–10⁴ keV; rest frame), corresponding to an extremely low radiative efficiency of ${\eta }_{\mathrm{rad}}\equiv {E}_{\gamma }/({E}_{{\rm{K}}}+{E}_{\gamma })\approx 3\%$. These results are summarized in Table 5. We present histograms of the marginalized posterior density for all parameters in Figure 10 and contour plots of correlations between the physical parameters in Figure 11.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

View extended figure (37 panels)

In concordance with the basic analysis outlined above, we find that the break frequencies at 1 d for this model are located at ${\nu }_{{\rm{a}}}\approx 9.0$ GHz, ${\nu }_{{\rm{m}}}\approx 3.4\times {10}^{11}$ Hz, and ${\nu }_{{\rm{c}}}\approx 2.3\times {10}^{14}$ Hz, while the peak flux density is about 12 mJy at ${\nu }_{{\rm{m}}}$. The spectrum transitions from fast to slow cooling at about 230 s after the burst, and ${\nu }_{{\rm{m}}}$ drops below ${\nu }_{{\rm{a}}}$ at about 25 d. The high value of ${\nu }_{{\rm{a}}}$ is required to suppress the flux density at 8.46 GHz and 5 GHz (VLA), relative to the mm-band (PdBI and SMA). The ordering of the break frequencies relative to the observing bands, ${\nu }_{{\rm{m}}},{\nu }_{{\rm{c}}}\lt {\nu }_{\mathrm{NIR}}$ at 1 d, ensures that the optical and X-ray are on the same part of the synchrotron spectrum at this time. The jet break at $\approx 17$ d is largely driven by the millimeter and radio data. The blastwave becomes non-relativistic at ${t}_{\mathrm{NR}}\approx 40$ d, resulting in a subsequent slow decline (${F}_{\nu }\sim {t}^{0.6+3\beta }\sim {t}^{-1}$, with $\beta =(1-p)/2\approx -0.5$; Frail et al. 2000) in the radio light curves after the peak.

Moin et al. (2013) use a VLBA limit on the size of the afterglow at 65 d to estimate the average ejecta Lorentz factor. However, we note that the apparent physical size of the afterglow is given by ${R}_{\perp }={\left[{2}^{12-3k}{(4-k)}^{5-k}/{(5-k)}^{5-k}\right]}^{1/(8-2k)}{\rm{\Gamma }}{{ct}}_{{\rm{z}}}\approx 2.3{\rm{\Gamma }}{{ct}}_{{\rm{z}}}$, where ${\gamma }_{l}$ is the ejecta Lorentz factor of the ejecta at time, ${t}_{{\rm{z}}}={t}_{\mathrm{obs}}/(1+z)$, and k = 0 for the ISM model (Granot & Sari 2002); therefore the VLBI measurement actually represents an upper limit to the ejecta Lorentz factor at the time of the measurement, such that ${\rm{\Gamma }}({t}_{\mathrm{obs}}=65\;{\rm{d}})\lt 5.8$. Since the blastwave becomes non-relativistic at around 40 days, our model does not violate the VLBA limit on the angular size of the afterglow.

Taking the forward shock model described in Section 6.1.2 as a starting point, we find that the X-ray and UV/optical data before the re-brightening can be explained by two successive periods of energy injection,

$$\begin{eqnarray}&&{E}_{{\rm{K}},\mathrm{iso}}(t)\\ \quad =\ \left\{\begin{array}{ll}{E}_{{\rm{K}},\mathrm{iso},{\rm{f}}}, & t\gt {t}_{0}=0.45\;{\rm{d}}\\ {E}_{{\rm{K}},\mathrm{iso},{\rm{f}}}{\left(\displaystyle \frac{t}{{t}_{0}}\right)}^{1.55}, & {t}_{1}=0.05\;{\rm{d}}\lt t\lt {t}_{0}\\ {E}_{{\rm{K}},\mathrm{iso},{\rm{f}}}{\left(\displaystyle \frac{{t}_{1}}{{t}_{0}}\right)}^{1.55}{\left(\displaystyle \frac{t}{{t}_{1}}\right)}^{0.7}, & {t}_{2}=1.7\times {10}^{-3}\;{\rm{d}}\lt t\lt {t}_{1}\\ {E}_{{\rm{K}},\mathrm{iso},{\rm{f}}}{\left(\displaystyle \frac{{t}_{1}}{{t}_{0}}\right)}^{1.55}{\left(\displaystyle \frac{{t}_{2}}{{t}_{1}}\right)}^{0.7}, & t\lt {t}_{2}.\end{array}\right.\end{eqnarray} \tag{ 4 }$$

In this model, the energy increases by a factor of $\approx 11$ from ${E}_{{\rm{K}},\mathrm{iso},{\rm{i}}}\approx 1.0\times {10}^{50}$ erg at $1.7\times {10}^{-3}$ d to ${E}_{{\rm{K}},\mathrm{iso}}\approx 1.1\times {10}^{51}$ erg at 0.05 d followed by another increase by a factor of $\approx 30$ to ${E}_{{\rm{K}},\mathrm{iso},{\rm{f}}}\approx 3.4\times {10}^{52}$ erg at 0.45 d. The overall increase in energy is a factor of $\approx 340$, corresponding to an injected energy fraction of $\approx 99.7\%$ over this period. In comparison, ${E}_{\gamma ,\mathrm{iso}}={9.9}_{-3.4}^{+6.3}\times {10}^{50}$ (Marshall et al. 2011). The blastwave Lorentz factor decreases from ${\rm{\Gamma }}\approx 38$ at the start of energy injection at $1.7\times {10}^{-3}$ to ${\rm{\Gamma }}\approx 11$ at 0.05 d, and then to ${\rm{\Gamma }}\approx 7$ at the end of energy injection at 0.45 d. The values of m derived above correspond to an ejecta distribution with $E(\gt {\rm{\Gamma }})\propto {{\rm{\Gamma }}}^{1-s}$ with $s\approx 9.6$ for $7\lesssim {\rm{\Gamma }}\lesssim 11$ and $s\approx 3$ for $11\lesssim {\rm{\Gamma }}\lesssim 38$.

GRB 100901A was detected and localized by the Swift BAT on 2010 September 01 at 13:34:10 UT (Immler et al. 2010). The burst duration is ${T}_{90}=(439\pm 33)$ s, with a fluence of ${F}_{\gamma }=(2.1\pm 0.3)\times {10}^{-6}$ erg cm⁻² (15–150 keV observer frame; Immler et al. 2010). The afterglow was detected in the X-rays and optical bands by XRT (Page & Immler 2010), UVOT (Pritchard & Immler 2010), and by multiple ground-based observatories (e.g., de Cia et al. 2010; Hentunen et al. 2010; Gorbovskoy et al. 2010; Kopac et al. 2010; Kuroda et al. 2010c; Pandey & Zheng 2010; Sahu et al. 2010b; Updike et al. 2010b), as well as in the radio by the WSRT and the VLA (Chandra & Frail 2010a; van der Horst et al. 2010b). Spectroscopic observations with Magellan yielded a redshift of z = 1.408 (Chornock et al. 2010). Using the BAT fluence, Gorbovskoy et al. (2012) determined the isotropic equivalent γ-ray energy of this burst to be ${E}_{\gamma ,\mathrm{iso}}=6.3\times {10}^{52}$ erg (1–10⁴ keV; rest frame). However, the γ-ray spectrum does not exhibit a turnover within the BAT energy range, indicating that the peak of the γ-ray spectrum, ${E}_{\mathrm{peak}}$, is outside the BAT energy range. Margutti et al. (2013) obtain a correlation between ${E}_{\mathrm{BAT},\mathrm{obs}}$ (15–150 keV) and ${E}_{\gamma ,\mathrm{iso}}$: $\frac{{E}_{\gamma ,\mathrm{iso}}}{\mathrm{erg}}={10}^{-3.7}{\left[\frac{{E}_{\mathrm{BAT}}^{15-150}}{\mathrm{erg}}\right]}^{1.08\pm 0.01}$ with a scatter of $\sigma =0.24$ (68%), which we use to perform a K-correction and obtain ${E}_{\gamma ,\mathrm{iso}}=(8\pm 1)\times {10}^{52}$ erg. Note that the error bars do not include the uncertainty in the correlation itself.

XRT began observing the GRB during the γ-ray emission interval. The X-ray spectra from 1 ks to the end of XRT observations are well fit by an absorbed single power-law model with ${N}_{{\rm{H}},\mathrm{Gal}}=7.1\times {10}^{20}\;{\mathrm{cm}}^{-2}$ (Kalberla et al. 2005), intrinsic hydrogen column ${N}_{{\rm{H}},\mathrm{int}}=(3.1\pm 0.7)\times {10}^{21}\;{\mathrm{cm}}^{-2}$, and photon index, ${{\rm{\Gamma }}}_{{\rm{X}}}=2.15\pm 0.06$, with no evidence for spectral evolution during this period. We convert the count rate light curve published by Margutti et al. (2013) to a flux density light curve at 1 keV using their time-resolved spectra. We also analyze all UVOT photometry for this burst and report our results in Table 7, and a compilation of all photometry listed in GCN circulars¹⁸ or published elsewhere for this event in Table 8.

Table 7.  Swift/UVOT Observations of GRB 100901A

$t-{t}_{0}$	Filter	Frequency	Flux Densitya	Uncertainty	Detection?
(days)		(Hz)	(μJy)	(μJy)	(1 = Yes)
0.00258	White	8.64e+14	27.4	5.65	1
0.00498	u	8.56e+14	146	11.4	1
0.00661	b	6.92e+14	230	75.6	0
0.00689	White	8.64e+14	58.6	18.9	1
0.00717	uvw2	1.48e+15	67.9	12.1	0

Note.

^aIn cases of non-detections, we report the formal flux measurement at the position of the afterglow.

Machine-readable versions of the table is available.

Download table as:  DataTypeset image

Table 8.  Optical Observations of GRB 100901A

$t-{t}_{0}$	Observatory	Telescope/	Filter	Frequency	Flux Densitya	Uncertainty	Detection?	Reference
(days)		Instrument		(Hz)	(μJy)	(μJy)	(1 = Yes)
0.00131	MASTER	⋯	CR	4.4e+14	82.1	436	0	Gorbovskoy et al. (2012)
0.00178	MASTER	⋯	CR	4.4e+14	156	170	0	Gorbovskoy et al. (2012)
0.00246	MASTER	⋯	CR	4.4e+14	131	143	0	Gorbovskoy et al. (2012)
0.00396	MASTER	⋯	CR	4.4e+14	119	107	0	Gorbovskoy et al. (2012)
0.00494	MASTER	⋯	CR	4.4e+14	304	78.6	1	Gorbovskoy et al. (2012)

Note.

^aIn cases of non-detections, we report the formal flux measurement at the position of the afterglow.

Machine-readable versions of the table is available.

Download table as:  DataTypeset image

The X-ray data before the first orbital gap at around 10⁻² d exhibits rapid flaring, ending in a steep decay. The earliest UVOT observations during this period detect a counterpart at multiple wavelengths. Owing to the contemporaneous γ-ray emission, indicating on-going central engine activity during this period, we do not consider data before the first orbital gap for our afterglow analysis. The X-ray re-brightening and subsequent fading between 0.08 d and 20 d can be well fit with a broken power-law model (Equation (5)), with ${t}_{{\rm{b}}}=0.40\pm 0.02{\rm{d}},$ ${F}_{\nu ,{\rm{X}}}({t}_{{\rm{b}}})=2.0\pm 0.1\;\mu \mathrm{Jy}$, ${\alpha }_{1}=0.77\pm 0.15$, ${\alpha }_{2}=-1.55\pm 0.06$, and $y=2.3\pm 1.1$ (Figure 12).

Zoom In Zoom Out Reset image size

A broken power-law fit to the UVOT U-band data between 0.02 d and 20 d yields ${t}_{{\rm{b}}}=0.40\pm 0.03$ d, ${F}_{\nu ,{\rm{U}}}({t}_{{\rm{b}}})=144\pm 7\;\mu \mathrm{Jy}$, ${\alpha }_{1}=0.45\pm 0.06$, and ${\alpha }_{2}=-1.63\pm 0.12$, with y = 2.5, similar to the results from fitting the X-ray re-brightening. Over the same time range, a broken power-law fit to the UVOT B-band data yields ${t}_{{\rm{b}}}=0.39\pm 0.08$ d, ${F}_{\nu ,{\rm{B}}}({t}_{{\rm{b}}})=201\pm 24\;\mu \mathrm{Jy}$, ${\alpha }_{1}=0.40\pm 0.06$, and ${\alpha }_{2}=-1.46\pm 0.21$, similar to the X-ray and U-band fits. Finally, we fit the X-ray, UVOT U-band, and UVOT B-band data jointly, where we constrain the model light curves at the two frequencies to have the same rise and decay rate and time of peak, and use independent normalizations in the three bands. Using a Markov Chain Monte Carlo simulation using emcee (Foreman-Mackey et al. 2013), we find ${t}_{{\rm{b}}}=0.36\pm 0.02$, ${\alpha }_{1}=0.57\pm 0.09$, and ${\alpha }_{2}=-1.47\pm 0.07$, with a fixed value of y = 2.5.

The spectral index between the UKIRT K-band observation at 2.88 d (Im et al. 2010) and the X-rays is ${\beta }_{\mathrm{NIR},\ {\rm{X}}}=-0.78\pm 0.08$. The spectral index between the NIR K-band and the Swift/U-band is slightly steeper, ${\beta }_{\mathrm{NIR},\mathrm{UV}}=-1.0\pm 0.2$, indicating that some extinction may be present. The spectral slope in the X-rays following the re-brightening is ${\beta }_{{\rm{X}}}=-1.15\pm 0.06$, leading to ${\beta }_{\mathrm{NIR},\ {\rm{X}}}-{\beta }_{{\rm{X}}}=0.4\pm 0.1$ suggesting that ${\nu }_{\mathrm{opt}}\lt {\nu }_{{\rm{c}}}\lt {\nu }_{{\rm{X}}}$ at 2.88 d (Figure 13).

Zoom In Zoom Out Reset image size

If the X-rays indeed lie above ${\nu }_{{\rm{c}}}$, we would have $p=-2{\beta }_{{\rm{X}}}=2.3\pm 0.12$. The expected light curve decay rate is $\alpha =(2-3p)/4=-1.2\pm 0.1$ above ${\nu }_{{\rm{c}}}$ and $\alpha =3(1-p)/4=-1.0\pm 0.1$ (ISM) or $\alpha =-1.5\pm 0.1$ (wind) below ${\nu }_{{\rm{c}}}$. The U-band light curve after the re-brightening can be fit either with a single power law with α = −1.5 or with a break from α ≈ −0.9 to α ≈ −1.8 at ≈1 d and α = −1.0 ± 0.1 between 0.2 d and 1.0 d. The former is consistent with a wind-like environment, while the latter suggests an ISM model and a jet break at ≈1 d. We return to this point later.

There are multiple epochs of observations with the VLA (PI: Kulkarni) and with WSRT (PI: van der Horst) in the NRAO and ASTRON archives. We downloaded and analyzed these observations with CASA using techniques described in Section 2.5. We list our derived radio photometry for the afterglow of GRB 100901A in Table 9, and additionally include the SMA observation of the afterglow at 345 GHz reported by de Ugarte Postigo et al. (2007) in our analysis.

Table 9.  Millimeter and Radio Observations of GRB 100901A

Date	$t-{t}_{0}$	Observatory	Frequency	Integration time	Flux density	Uncertainty	Detection?
(UT)	(days)		(GHz)	(min)	(mJy)	(mJy)	($1=$ Yes)
2010 Sep 06.49	4.92	VLA	4.495	31.3	289	43	1
2010 Sep 06.49	4.92	VLA	7.915	25.7	488	37	1
2010 Sep 08.42	6.85	VLA	4.495	26.7	247	40	1
2010 Sep 08.42	6.85	VLA	7.915	24.3	349	54	1
2010 Sep 13.46	11.90	VLA	4.495	33.4	245	38	1
2010 Sep 13.46	11.90	VLA	7.915	28.2	419	46	1
2010 Sep 13.46	11.90	VLA	33.559	22.3	270	90	0
2010 Sep 21.44	19.87	VLA	4.495	32.7	138	32	1
2010 Sep 21.44	19.87	VLA	7.915	32.7	209	46	1
2010 Sep 21.44	19.79	VLA	33.559	14.7	288	96	0
2010 Oct 15.18	43.61	VLA	4.495	19.1	126	42	0
2010 Oct 15.18	43.61	VLA	7.915	19.4	213	71	0
2010 Oct 15.18	43.86	VLA	33.559	17.6	246	82	0
2010 Sep 04.10	2.54	WSRT	4.90	567.6	189	46	1
2010 Sep 07.10	5.53	WSRT	4.90	531.9	275	61	1
2010 Sep 12.08	10.52	WSRT	4.90	551.9	311	47	1
2010 Sep 21.05	19.48	WSRT	4.90	525.7	252	84	0
2010 Sep 42.00	40.44	WSRT	4.90	550.8	123	41	0
2010 Sep 03.44	1.875	SMA	345.0	...	2250	750	0

VLA and WSRT data are from this work. SMA data are from de Ugarte Postigo et al. (2007). Dates in column 1 refer to the mid-point of each observation.

Download table as:  ASCIITypeset image

Extrapolating the Lightbuckets R-band data at 1.7 d to the time of the SMA 345 GHz upper limit at 1.8 days using the fit to the optical U-band light curve described above, we find a spectral index of ${\beta }_{\mathrm{mm}-{\rm{R}}}$ ≳ −0.6 from the millimeter to the optical at this time. Combined with the steeper spectrum of ${\beta }_{\mathrm{NIR},{\rm{X}}}=-0.78\pm 0.08$ between the optical and X-rays, this suggests that the spectrum turns over above the millimeter band, indicating that ${\nu }_{{\rm{m}}}\gtrsim 345$ GHz at 1.8 d.

The spectral index between the VLA 4.5 GHz and 7.9 GHz bands at 4.92 d is β = 0.9 ± 0.3. However, the light curve at 4.5 GHz declines as α = −0.17 ± 0.12 between 3 d and 12 d, and as α = −1.1 ± 0.5 between 12 d and 20 d. The rising radio spectrum coupled with the declining light curve implies that the jet break has occurred before 5 d. The steepening of the 4.5 GHz light curve at ≈12 d implies ${\nu }_{{\rm{m}}}\ \approx 4.5$ GHz at this time. Given that ${\nu }_{{\rm{m}}}\gt 345$ GHz at 1.8 d, this would imply an evolution of $\sim {\nu }_{{\rm{m}}}\propto {t}^{-2.3}$, consistent with the expected evolution of ${\nu }_{{\rm{m}}}\propto {t}^{-2}$ following the jet break. Together, this implies that the jet break must have occurred before 1.8 d, which agrees with the results from the X-ray and optical analysis above and indicates an ISM-like environment. We therefore focus on the ISM model for the remainder of this section, and discuss the wind model for completeness in Appendix D.

To summarize, the NIR to X-ray SED exhibits mild evidence for extinction and suggests that ${\nu }_{\mathrm{NIR}}\lt {\nu }_{{\rm{c}}}\lt {\nu }_{{\rm{X}}}$ at $\approx 3\;$ d, with $p=2.3\pm 0.1$. The radio observations indicate ${\nu }_{{\rm{m}}}\approx 5\;\mathrm{GHz}$ at $\approx 12\;$ d. Together, the X-ray, NIR and radio data suggest ${t}_{\mathrm{jet}}\approx 1$ d and an ISM-like environment. All radio observations took place after this time, and are therefore insensitive to the density profile of the circumburst environment.

We employ our MCMC analysis to fit the multi-band data for GRB 100901A after $t\approx 0.25$ d. At the redshift of z = 1.408, the UVOT White-, uvw1-, uvw2-, and uvm2-band data are affected by IGM absorption, and we do not include these bands in our analysis. The highest-likelihood ISM model (Figure 14) the parameters $p\approx 2.03$, ${\epsilon }_{{\rm{e}}}\approx 0.33$, ${\epsilon }_{B}\approx 0.32$, ${n}_{0}\approx 3.2\times {10}^{-3}$ ${\mathrm{cm}}^{-3}$, ${E}_{{\rm{K}},\mathrm{iso}}\approx 3.0\times {10}^{53}$ erg, ${t}_{\mathrm{jet}}\approx 0.96$ d, ${A}_{{\rm{V}}}\approx 0.09$ mag, and ${F}_{\nu ,\mathrm{host},{{\rm{r}}}^{\prime }}\approx 4.1\;\mu \mathrm{Jy}$, with a Compton y-parameter of $\approx 0.6$. The blastwave Lorentz factor is ${\rm{\Gamma }}=15.9{(t/1\;{\rm{d}})}^{-3/8}$ and the jet opening angle is ${\theta }_{\mathrm{jet}}\approx 2\buildrel{\circ}\over{.} 1$. The beaming-corrected kinetic energy is ${E}_{{\rm{K}}}\approx 2.1\times {10}^{50}\;\mathrm{erg}$, while the beaming corrected γ-ray energy is ${E}_{\gamma }\approx 5\times {10}^{49}$ erg (1–10⁴ keV; rest frame). We plot histograms of the measured parameters in Figure 15 and correlation contours between the physical parameters in Figure 16, providing summary statistics from our MCMC analysis in Table 5.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

View extended figure (11 panels)

Zoom In Zoom Out Reset image size

View extended figure (46 panels)

In this model, the break frequencies at 1 d are located at ${\nu }_{{\rm{a}}}\approx 0.7$ GHz, ${\nu }_{{\rm{m}}}\approx 2\times {10}^{12}$ Hz, and ${\nu }_{{\rm{c}}}\approx 9\times {10}^{14}$ Hz, while the peak flux density is about 4.2 mJy at ${\nu }_{{\rm{m}}}$. The spectrum transitions from fast to slow cooling at about 350 s after the burst. The cooling frequency is located between the optical and X-rays and the jet break is before 1 d, both of which are expected from the basic analysis outlined above. The low value of p is driven by the shallow measured decline in the X-rays following the jet break, and the resulting spectrum remains consistent with the optical-to-X-ray SED (Figure 13).

Taking the forward shock model described in Section 6.2.2 as a starting point, we find that the X-ray and UV/optical data before the re-brightening can be explained by a single period of energy injection between 0.105 d and ≈0.26 d:

$$\begin{eqnarray}{E}_{{\rm{K}},\mathrm{iso}}(t)=\left\{\begin{array}{ll}{E}_{{\rm{K}},\mathrm{iso},{\rm{f}}}, & t\gt {t}_{0}=0.26\;{\rm{d}}\\ {E}_{{\rm{K}},\mathrm{iso},{\rm{f}}}{\left(\displaystyle \frac{t}{{t}_{0}}\right)}^{1.88}, & {t}_{1}=0.105\;{\rm{d}}\lt t\lt {t}_{0}\\ {E}_{{\rm{K}},\mathrm{iso},{\rm{f}}}{\left(\displaystyle \frac{{t}_{1}}{{t}_{0}}\right)}^{1.88}, & t\lt {t}_{1}.\end{array}\right.\end{eqnarray} \tag{ 5 }$$

In this model, the energy increases by a factor of ≈5.5 from ${E}_{{\rm{K}},\mathrm{iso},{\rm{i}}}\approx 5.4\times {10}^{52}$ erg at 0.105 d to ${E}_{{\rm{K}},\mathrm{iso},{\rm{f}}}\approx 3.0\times {10}^{53}$ erg at 0.26 d, corresponding to an injected energy fraction of $\approx 85\%$, while the Lorentz factor decreases from ${\rm{\Gamma }}\approx 30$ to ${\rm{\Gamma }}\approx 26$ over this period. In comparison, ${E}_{\gamma ,\mathrm{iso}}\approx 8\times {10}^{52}$ erg (Section 6.2.1). The value of $m\approx 1.88$ corresponds to $s\approx 14.4$ for $26\lesssim {\rm{\Gamma }}\lesssim 30$.

GRB 120404A was detected and localized by the Swift BAT on 2012 April 04 at 12:51:02 UT (Stratta et al. 2012). The burst duration was ${T}_{90}=38.7\pm 4.1$ s, with a fluence of ${F}_{\gamma }=(1.6\pm 0.1)\times {10}^{-6}$ erg cm⁻² (15–150 keV observer frame, Ukwatta et al. 2012). Swift and ground-based observatories detected an afterglow in the X-rays and UV/optical (Breeveld & Stratta 2012; Gorbovskoy et al. 2012; Guidorzi et al. 2012; Xin et al. 2012b), as well as in the radio (Zauderer et al. 2012). Spectroscopic observations with Gemini-north yielded a redshift of z=2.876 (Cucchiara & Tanvir 2012). The isotropic-equivalent γ-ray energy for this event is ${E}_{\gamma ,\mathrm{iso}}=(9\pm 4)\times {10}^{52}$ erg (1–10⁴ keV, rest frame; Guidorzi et al. 2014).

This burst has been previously studied in detail by Guidorzi et al. (2014), who interpret the optical re-brightening starting around 800 s in their well-sampled, multi-band optical light curves as due to the passage of the characteristic synchrotron frequency, ${\nu }_{{\rm{m}}}$. They additionally invoke RS emission to explain the flat (${t}^{0.0\pm 0.1}$) portion of the optical light curve before the onset of the re-brightening. In the following, we propose an alternate model for the multi-band radio through X-ray light curves in the context of energy injection.

The X-ray light curve before 700 s can be modeled as a power-law decay with ${\alpha }_{{\rm{X}}}\approx -2$. This light curve phase is likely part of the high latitude emission, and we ignore the data before 0.008 d in our analysis. The X-ray photon index at 0.12–0.24 d, ${{\rm{\Gamma }}}_{X}=2.3\pm 0.3$ (Guidorzi et al. 2014), implying a spectrum, ${F}_{\nu }\propto {\nu }^{-1.3\pm 0.3}$, is consistent with the spectral slope between the optical i'-band and the X-rays at 0.07 d, ${\beta }_{\mathrm{opt}-{\rm{X}}}=-0.91\pm 0.04$, suggesting that the optical and X-ray bands are on the same segment of the afterglow SED, although the large uncertainty in ${\beta }_{{\rm{X}}}$ leaves open the possibility that ${\nu }_{{\rm{c}}}$ lies between the optical and X-rays. Additionally, the spectral slope within the optical (B- to i'-band) is ${\beta }_{\mathrm{opt}}=-1.3\pm 0.2$ at 0.07 d, indicating that extinction is present. The spectral index between the 19.2 and 24.5 GHz observations at 0.75 d is ${\beta }_{\mathrm{radio}}\approx 2$, which indicates that ${\nu }_{{\rm{a}}}$ is located above 24.5 GHz at this time (Figure 17).

Zoom In Zoom Out Reset image size

The optical R-band light curve declines as ${t}^{-1.9\pm 0.02}$ after 0.13 d, consistent with the X-ray decline rate of ${t}^{-1.8\pm 0.1}$ after 0.05 d. The steep decline of α ≈ −2 is indicative of a jet break before $\approx 0.1$ d. A broken power-law fit to the B-band light curve results in the parameters ${t}_{{\rm{b}}}=(2.8\pm 0.4)\times {10}^{-2}$ d, ${F}_{\nu ,{\rm{B}}}({t}_{{\rm{b}}})=252\pm 10\;\mu \mathrm{Jy}$, ${\alpha }_{1}=1.74\pm 0.58$, ${\alpha }_{2}=-1.71\pm 0.17$, and $y=0.78\pm 0.43$, making this the earliest re-brightening episode of the four events studied in this paper.

We interpret the optical light curve peaks at around 2500 s as the end of a period of energy injection, after which the afterglow evolves according to the standard framework with a fixed energy, ${E}_{{\rm{K}},\mathrm{iso},{\rm{f}}}$. We use the data after 0.04 d for estimating the parameters of the blastwave shock and employ our MCMC tools described in Section 4 to model the afterglow after this time.

The parameters of our highest likelihood ISM model are $p\approx 2.06$, ${\epsilon }_{{\rm{e}}}\approx 0.27$, ${\epsilon }_{B}\approx 0.16$, ${n}_{0}\approx 2.8\times {10}^{2}$ ${\mathrm{cm}}^{-3}$, ${E}_{{\rm{K}},\mathrm{iso}}\approx 1.2\times {10}^{53}$ erg, ${t}_{\mathrm{jet}}\approx 6.6\times {10}^{-2}$ d, and ${A}_{{\rm{V}}}\approx 0.13$ mag. The Compton y-parameter for this model is ≈0.9. The blastwave Lorentz factor is ${\rm{\Gamma }}=4.1{(t/1\;{\rm{d}})}^{-3/8}$ and the jet opening angle is ${\theta }_{\mathrm{jet}}\approx 3\buildrel{\circ}\over{.} 1$. The beaming-corrected kinetic energy is ${E}_{{\rm{K}}}\approx 1.7\times {10}^{50}\;\mathrm{erg}$, while the beaming corrected γ-ray energy is ${E}_{\gamma }\approx 1.2\times {10}^{50}$ erg (1–10⁴ keV; rest frame). The MCMC analysis yields an isotropic equivalent kinetic energy of ${E}_{{\rm{K}},\mathrm{iso}}$ $={1.3}_{-0.2}^{+0.4}\times {10}^{53}$ erg, which is similar to the value of ${1.9}_{-0.1}^{+0.7}\times {10}^{53}$ erg derived by Guidorzi et al. (2014). The high circumburst density of $\mathrm{log}({n}_{0})={2.5}_{-0.3}^{0.4}$ is also consistent with the value of $\mathrm{log}({n}_{0})={2.4}_{-0.2}^{+0.02}$ determined by Guidorzi et al. 2014), and is driven by the low flux density and steep spectrum in the radio. However, unlike Guidorzi et al. (2014), our low value of $p\approx 2.1$ allows us to match the NIR to optical SED and the normalization of the X-ray light curve. We note that the high circumburst density also results in a low cooling frequency¹⁹ : in our highest likelihood model ${\nu }_{{\rm{c}}}\ \lt {\nu }_{{\rm{a}}}$ and the spectrum remains in the fast cooling regime through the duration of the observations (spectrum 4 of Granot & Sari 2002), with ${\nu }_{\mathrm{ac}}\approx 5.9\times {10}^{10}$ Hz, ${\nu }_{\mathrm{sa}}\approx 2.0\times {10}^{11}$ Hz, ${\nu }_{{\rm{m}}}\approx 9.1\times {10}^{11}$ Hz, and ${\nu }_{{\rm{c}}}\approx 6.3\times {10}^{10}$ Hz at 1 d.

We plot the posterior density functions for the all parameters in Figure 18 and correlation contours in Figure 19; we list our measured values for the physical parameters in Table 5. In the wind and ISM models, light curves at all frequencies become indistinguishable following a jet break. Since the jet break occurs early, soon after the start of the data used for deriving the parameters of the blastwave, we expect a viable wind model to exist as well. We discuss this model in Appendix E.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

View extended figure (37 panels)

We model the light curves before ${t}_{0}=0.04$ d by injection of energy into the blastwave shock. We use the afterglow parameters for the highest-likelihood model (Section 6.3.2) as the final parameters following the end of energy injection, with ${E}_{{\rm{K}},\mathrm{iso},{\rm{f}}}=1.2\times {10}^{53}$ erg. We find that the optical and X-ray light curves can be modeled well by two subsequent periods of energy injection, beginning at ${t}_{2}=7\times {10}^{-3}$ d (Figure 20):

$$\begin{eqnarray}&&{E}_{{\rm{K}},\mathrm{iso}}(t)\\ \quad =\;\left\{\begin{array}{ll}{E}_{{\rm{K}},\mathrm{iso},{\rm{f}}}, & t\gt {t}_{0}=0.04\;{\rm{d}}\\ {E}_{{\rm{K}},\mathrm{iso},{\rm{f}}}{\left(\displaystyle \frac{t}{{t}_{0}}\right)}^{0.75}, & {t}_{1}=2.2\times {10}^{-2}\;{\rm{d}}\lt t\lt {t}_{0}\\ {E}_{{\rm{K}},\mathrm{iso},{\rm{f}}}{\left(\displaystyle \frac{{t}_{1}}{{t}_{0}}\right)}^{0.75}{\left(\displaystyle \frac{t}{{t}_{1}}\right)}^{2.5}, & {t}_{2}=7.0\times {10}^{-3}\;{\rm{d}}\lt t\lt {t}_{1}\\ {E}_{{\rm{K}},\mathrm{iso},{\rm{f}}}{\left(\displaystyle \frac{{t}_{1}}{{t}_{0}}\right)}^{0.75}{\left(\displaystyle \frac{{t}_{2}}{{t}_{1}}\right)}^{2.5}, & t\lt {t}_{2}.\end{array}\right.\end{eqnarray} \tag{ 6 }$$

In this model, ${E}_{{\rm{K}},\mathrm{iso}}$ increases by a factor of ≈18 from $4.4\times {10}^{51}$ erg to $7.8\times {10}^{52}$ erg between $7.0\times {10}^{-3}$ d and $2.2\times {10}^{-2}$ d, and then by ≈50% to ${E}_{{\rm{K}},\mathrm{iso},{\rm{f}}}\approx 1.2\times {10}^{53}$ erg between $2.2\times {10}^{-2}$ d and 0.04 d. Thus $\approx 95\%$ of the final kinetic energy is injected into the blastwave between $7.0\times {10}^{-3}$ d and 0.04 d. In comparison, ${E}_{\gamma ,\mathrm{iso}}\approx {10}^{53}$ erg is similar to the final kinetic energy. The blastwave Lorentz factor decreases from ${\rm{\Gamma }}\approx 17.4$ at $7.3\times {10}^{-3}$ to ${\rm{\Gamma }}\approx 16.2$ at $2.2\times {10}^{-2}$ d, and then to ${\rm{\Gamma }}\approx 13.7$ at the end of energy injection at 0.04 d. The value of m derived above corresponds to $s\approx 3.7$ for $13.7\lesssim {\rm{\Gamma }}\lesssim 16.2$ and $s\approx 40$ for $16.2\lesssim {\rm{\Gamma }}\lesssim 17.4$.

Zoom In Zoom Out Reset image size