AFTERGLOW OBSERVATIONS OF FERMI LARGE AREA TELESCOPE GAMMA-RAY BURSTS AND THE EMERGING CLASS OF HYPER-ENERGETIC EVENTS

GRB 090323 was detected by the Fermi GBM at 00:02:42.63 on 2009 March 23 (Ohno et al. 2009; UT dates are used throughout this work). In the 8 keV to 40 MeV bandpass of the GBM, the light curve was multipeaked with a duration¹⁶ of t₉₀ = 133.1 ± 1.4 s (Bissaldi 2010). GRB 090323 was also detected at MeV energies by several of the satellites comprising the Inter-Planetary Network (IPN; Hurley et al. 1999), including the Konus instrument on the Wind satellite (Hurley et al. 2009; Golenetskii et al. 2009c).

Bissaldi (2010) analyzed the time-averaged spectrum of GRB 090323 (see also van der Horst 2009a), finding that the spectrum is best fit with a Band function (Band et al. 1993) having α = −1.05 ± 0.02, β = −2.7^+0.2_{− 0.4}, and E_p = 591⁺³⁶_{− 33} keV. The resulting fluence in the 8–10³ keV (observer frame) bandpass is f_γ = (1.22 ± 0.01) × 10⁻⁴ erg cm⁻². The Konus-Wind instrument was able to measure the high-energy spectrum over the entire duration of the MeV emission. Fitting a Band model to their spectrum, Golenetskii et al. (2009c) find a reasonable fit with α = −0.96^+0.12_{− 0.09}, β = −2.09^+0.16_{− 0.22}, E_p = 416⁺⁷⁶_{− 73} keV, and f_γ = 2.02^+0.28_{− 0.25} × 10⁻⁴ erg cm⁻² (20–10⁴ keV observer-frame bandpass).

In order to facilitate direct comparisons between events, we must transform this high-energy fluence into a common (rest-frame) bandpass (i.e., a "k"-correction; Bloom et al. 2001). Here we adopt the rest-frame 1–10⁴ keV bandpass, as this encompasses the full range of peak energies observed from GRBs (Band et al. 1993) without (for most pre-Fermi satellites) requiring large extrapolations outside the observed bandpass. At z = 3.568 (Section 2.1.3), the GBM measurement corresponds to a prompt fluence of f_γ = (1.24 ± 0.05) × 10⁻⁴ erg cm⁻² in the 1–10⁴ keV rest-frame bandpass (correcting the Konus-Wind spectrum yields a consistent result).

In addition to the detection at MeV energies by the GBM, GRB 090323 was detected at GeV energies by the Fermi LAT. Like several previous GRBs observed at GeV energies (e.g., Hurley et al. 1994; González et al. 2003; Giuliani et al. 2008; Abdo et al. 2009b), the GeV emission began several seconds after the MeV emission was detected, and remained significant long (∼1000 s) after the MeV component had faded beyond detectability (Ohno et al. 2009). The highest energy photon from the direction of GRB 090323 had an energy of E ≈ 7.5 GeV (Piron et al. 2009). However, this photon arrived after the prompt MeV emission ended, and so its origin (i.e., prompt emission or afterglow, or, in physical terms, related to the internal or external shock) is somewhat uncertain. The highest energy photon detected by the LAT during the formal MeV t₉₀ measurement had an energy of E ≈ 500 keV (at t ≈ 90 s; Piron et al. 2009).

The Swift satellite began observations of the field of GRB 090323 with the onboard X-ray Telescope (XRT; Burrows et al. 2005b) at 19:27 on 2009 March 23 (∼19.4 hr after the burst; Kennea et al. 2009a). A candidate afterglow was promptly identified at $\alpha = 12^{\mathrm{h}} 42^{\mathrm{m}} 50\mbox{$.\!\!^{\mathrm s}$}26$, δ = +17°03'142, with a 90% containment radius of 27 (J2000.0). The XRT continued to monitor the evolution of the fading counterpart for the following two weeks. We have obtained the XRT light curve from the online compilation of N.R.B.;¹⁷ the resulting evolution is plotted in Figure 2.

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The optical afterglow of GRB 090323 was discovered shortly thereafter by GROND (Updike et al. 2009). The optical/near-infrared (NIR) spectral energy distribution, constructed from simultaneous imaging in the g'r'i'z'JHK filters, implied a spectral steepening around the observed g' band. Associating this break with absorption from Lyα in the GRB host galaxy, Updike et al. (2009) derive a photometric redshift of 4.0 ± 0.3 for GRB 090323. The full set of GROND observations is presented by McBreen et al. (2010); we include them in our modeling efforts, but for clarity they are omitted from Table 1.

We began observations of the afterglow of GRB 090323 with the automated Palomar 60 inch (1.5 m) telescope (P60; Cenko et al. 2006) beginning on 2009 March 24 (Cenko & Perley 2009). Images were obtained in the Sloan r' and i' filters and individual frames were automatically reduced using our custom IRAF¹⁸ software pipeline. To increase the signal-to-noise ratio (S/N), individual frames were astrometrically aligned using the Scamp software package and coadded using Swarp.¹⁹ We used aperture photometry to extract the flux of the afterglow from these coadded frames with the aperture radius roughly matched to the full width at half-maximum (FWHM) intensity of the point-spread function (PSF). Aperture magnitudes were then calibrated relative to field sources from the Sloan Digital Sky Survey (SDSS) Data Release 7 (Abazajian et al. 2009). Imaging continued on subsequent nights with the P60 until the afterglow was below our detection threshold. The results of this monitoring campaign, uncorrected for foreground Galactic extinction (E[B − V] = 0.025 mag; Schlegel et al. 1998), are presented in Table 1.

We obtained additional imaging of the field of GRB 090323 with the two Gemini Multi-Object Spectrographs (GMOS-N and GMOS-S; Hook et al. 2004). Images were taken at both Gemini North and Gemini South in the Sloan r' and i' filters, and they were reduced using the IRAF gemini package. Photometry was performed with the same methodology used for the P60 imaging of the field, and the resulting measurements are shown in Table 1.

Additionally, we have compiled optical and NIR measurements of the afterglow of GRB 090323 from the GCN²⁰ Circulars and included these in Table 1. The majority of the late-time measurements (of most interest to us for modeling purposes) were obtained in the R band. We note that most of these measurements were calibrated with respect to a single object (1070-0238439) from the USNO-B1 catalog (Monet et al. 2003), following Kann et al. (2009b). Using the SDSS observations of this field and the filter transformations of Jordi et al. (2006), we find an R-band magnitude for this source of R = 17.31 mag (compared to an assumed value of 17.15 mag from Kann et al. 2009b). We therefore offset all reported R-band magnitudes from the GCN Circulars by +0.16 mag.

Finally, we observed the afterglow of GRB 090323 with the Very Large Array²¹ (VLA) beginning a few days after the initial burst trigger. The flux-density scale was tied to 3C 286 or 3C 147 and the phase was measured by switching between the GRB and a nearby bright point-source calibrator. To maximize sensitivity, the full VLA continuum bandwidth (100 MHz) was recorded in two 50 MHz bands. Data reduction was carried out following standard practice in the AIPS software package.

Our initial VLA detection of GRB 090323 was reported by Harrison et al. (2009). GRB 090323 was also detected at radio wavelengths by the Westerbork Synthesis Radio Telescope (WSRT; van der Horst 2009b). The full set of VLA measurements is listed in Table 2. In order to improve the S/N and to reduce the modulation of the light curve caused by interstellar scintillation (ISS; e.g., Frail et al. 2000a), we binned the data from adjacent epochs. These binned points were used for our afterglow modeling (Section 3) and are plotted in Figure 2.

We began spectroscopic observations of the optical afterglow of GRB 090323 with GMOS-S on 2009 March 24 (∼29.93 hr after the GBM trigger; Chornock et al. 2009). We first obtained 2 × 600 s spectra with the B600 grating and a central wavelength of 6000 Å, providing coverage over the range ∼4500–7500 Å with a resolution of 2.5 Å. Immediately following these exposures, we obtained 2 × 600 s spectra with the R400 grating and a central wavelength of 8000 Å, providing a coverage of ∼6000–10000 Å with a resolution of 5.0 Å.

The spectra were reduced in a standard manner using routines from the IRAF gemini and specred packages (see, e.g., Cenko et al. 2008 and references therein for details). Wavelength calibration was performed relative to CuAr lamps and then adjusted based on measured night-sky emission lines. The resulting root-mean-square wavelength uncertainty was ≲ 0.25 Å for the spectra with the B600 grating and ≲ 0.40 Å for the spectra taken with the R400 grating. Telluric features were removed using the continuum from well-exposed spectrophotometric standard stars (e.g., Wade & Horne 1988; Matheson et al. 2000).

Flux calibration was performed relative to the spectrophotometric standard star LTT 7379 (Stone & Baldwin 1983; Baldwin & Stone 1984). We caution, however, that the standard-star observations were conducted on different nights from the GRB observations (2009 August 31 for the B600 grating; 2009 August 4 for the R400 grating), so the absolute flux calibration is somewhat uncertain (estimated to be ∼30%).

The resulting spectrum of the afterglow of GRB 090323 is shown in Figure 3. The broad absorption feature at λ ≈ 5570 Å is produced by Lyα at z ≈ 3.6. The spectrum blueward of this wavelength is dominated by absorption from the Lyα forest, while the lack of Lyα forest features redward of this transition indicates that it corresponds to the redshift of the GRB host galaxy. Unfortunately, the GMOS CCD chip gap over the range 5500–5525 Å precludes an accurate measurement of the Lyα profile and hence a determination of the H i column density.

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Redward of Lyα, we find a series of strong (rest-frame equivalent width W₀ ⩾ 1 Å) absorption features superposed on a relatively flat, featureless continuum. In particular, we identify Si ii λ1260, Si ii* λ1264, O i λ1302, Si ii λ1304, Si ii* λ1309, C ii λ1334, C ii* λ1335, Si iv λλ1393, 1402, Si ii λ1526, Si ii* λ1533, and C iv λλ1548, 1550 at z = 3.568 ± 0.004. Excited fine-structure lines are indicative of ultraviolet (UV) pumping and further suggest these features arise in the host galaxy of GRB 090323. The strongest features exhibit a complex velocity structure that is only marginally resolved with the B600 grating. We also detect intervening absorption systems of C iv λλ1548, 1550 at z = 3.379 and Mg ii λλ2796, 2803 at z = 2.101. A more thorough analysis of the host-galaxy properties of GRB 090323 will be presented in a subsequent work.

At z = 3.568, the isotropic prompt energy release from GRB 090323 in the rest-frame 1–10⁴ keV bandpass is E_{γ, iso} = (3.30 ± 0.13) × 10⁵⁴ erg. Using the formulation of Lithwick & Sari (2001), the lower limit on the outflow Lorentz factor (assuming a nonthermal spectrum up to E_obs ≈ 500 keV) is Γ₀ ≳ 600.

GRB 090328 triggered the Fermi GBM at 09:36:46 on 2009 March 28 (McEnery et al. 2009). In the GBM bandpass, the light curve is multipeaked with a duration of t₉₀ ≈ 57 ± 3 s (Bissaldi 2010). GRB 090328 was also detected at MeV energies by several IPN satellites (Golenetskii et al. 2009a), including Konus-Wind (Golenetskii et al. 2009d).

The time-averaged GBM spectrum is well fit by a power law having an exponential cutoff, with α = −1.07 ± 0.02 and E_p = 744⁺⁵⁰_{− 47} keV (Bissaldi 2010; see also Rau et al. 2009). The resulting fluence in the 8 keV to 1 MeV (observer frame) bandpass is f_γ = (5.09 ± 0.04) × 10⁻⁵ erg cm⁻². These results are all consistent with the analogous values derived by the Konus-Wind instrument. Given the wider bandpass of the GBM and the higher precision of its fluence measurement, we adopt this value for the remainder of this work. Using the observed redshift of 0.736 (Section 2.1.3), we find a prompt fluence of f_γ = (6.80 ± 0.7) × 10⁻⁵ erg cm⁻² in the 1–10⁴ keV rest-frame bandpass.

GRB 090328 was also detected at GeV energies by the Fermi-LAT. Much like GRB 090323, flux in the LAT band from GRB 090328 is detected out to ∼900 s after the GBM trigger (Cutini et al. 2009). Many photons with energies above 1 GeV are detected from the direction of GRB 090328; however, they are all detected well after the end of the MeV emission (Cutini et al. 2009; Piron et al. 2009). The highest energy photon detected from the direction of GRB 090328 was measured at ∼5 GeV (t₀ + 798 s), while the most energetic photon detected during the prompt emission had E ≈ 700 keV at t₀ + 60 s (Piron et al. 2009).

The Swift XRT began target-of-opportunity observations of GRB 090328 at 01:26 on 2009 March 29 (∼15.9 hr after the GBM trigger). A fading X-ray source at $\alpha = 6^{\mathrm{h}} 02^{\mathrm{m}} 39\mbox{$.\!\!^{\mathrm s}$}58$, δ = −41°52'575 (J2000.0; 20 containment radius) was promptly identified as the X-ray afterglow (Kennea 2009; Kennea et al. 2009b). The resulting X-ray light curve, showing the afterglow evolution over the following 10 days, is plotted in Figure 4.

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The Ultraviolet Optical Telescope (UVOT; Roming et al. 2005) onboard Swift began settled observations of the field of GRB 090328 approximately 16 hr after the Fermi trigger. The optical afterglow, detected in both the U-band and white filters, was identified shortly thereafter (Oates 2009). UVOT observations of the afterglow continued for ∼2 weeks, almost exclusively in these two filters. We have downloaded the UVOT U-band data from the HEASARC archive²² and conducted photometry with the images following the technique described by Li et al. (2006). Given the large uncertainty associated with flux calibration for the white filter, we have not included data taken in this band. The results of this analysis, not including a correction for foreground Galactic extinction (E[B − V] = 0.057 mag; Schlegel et al. 1998), are shown in Table 3.

We obtained a single i' image of the afterglow of GRB 090328 with GMOS-S on 2009 April 29. This image was reduced in an identical manner to our observations of GRB 090323, and the zero point was calculated using predetermined values from the Gemini Web site.²³ We have also included the simultaneous GROND optical and NIR measurements from McBreen et al. (2010) in our modeling, and therefore display them in Figure 4 as well (but not in Table 3).

Finally, we began observing the afterglow of GRB 090328 with the VLA on 2009 March 30 (Frail et al. 2009) and continued for nearly two months. The data were reduced in a manner identical to that described in Section 2.1.2. The results of our radio campaign are displayed in Table 4 (individual epochs) and plotted in Figure 4 (combined epochs).

We began spectroscopic observations of GRB 090328 with GMOS-S at 00:05 on 2009 April 30 (∼38.5 hr after the GBM trigger; Cenko et al. 2009). We obtained 2 × 1500 s spectra with the R400 grating, the first with a central wavelength of 6000 Å (providing a coverage of ∼4000–8000 Å) and the second with a central wavelength of 8000 Å (providing a coverage of ∼6000–10000 Å). The spectra were reduced in the manner described in Section 2.1.3. Flux calibration was performed relative to the standard star LTT 7379 taken with identical instrumental setups on 2009 August 4.

The resulting spectrum of GRB 090328 is shown in Figure 5. Superposed on a relatively flat continuum, we identify strong emission lines of oxygen ([O ii] λ3727, [O iii] λλ4959, 5007) as well as a series of narrow absorption features (Fe ii λ2586, Fe ii λ2600, Mg ii λλ2796, 2803, Mg i λ2852, Ca ii λλ 3934, 3969), all at z = 0.7357 ± 0.0004.

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At this redshift, the prompt isotropic gamma-ray energy release from GRB 090328 in the 1–10⁴ keV bandpass is E_{γ, iso} = (9.6 ± 1.0) × 10⁵² erg. Using the formulation of Lithwick & Sari (2001), the lower limit on the outflow Lorentz factor (assuming a nonthermal spectrum up to E_obs ≈ 700 keV) is Γ₀ ≳ 200.

At 11:05:08.31 on 2009 September 2, the Fermi-GBM triggered and located GRB 090902B (Bissaldi & Connaughton 2009). In the GBM bandpass, the light curve consisted of a bright, multipeaked pulse with a duration t₉₀ ≈ 21 s. GRB 090902B was also detected in a similar bandpass by Suzaku-WAM (Terada et al. 2009).

Furthermore, GRB 090902B was bright enough to be detected at GeV energies by the Fermi-LAT (de Palma et al. 2009). Like the other events in our sample, LAT emission was seen out to t₀ + 1000 s, at late times decaying like a power law with temporal index α_LAT ≈ 1.5 (Abdo et al. 2009a). GRB 090902B is responsible for the highest energy photon detected from a GRB to date, with E = 33.4^+3.7_{− 3.5} GeV at t₀ + 82 s. Within the MeV prompt emission phase, the highest energy photon was measured at E = 11.16^+1.48_{− 0.58} GeV (Abdo et al. 2009a).

Unlike the other events considered here, the prompt high-energy spectrum of GRB 090902B is not adequately described by a single Band function. While the 100–1000 keV bandpass is reasonably well fit by such a model, the spectrum exhibits excess emission at both low (≲ 100 keV) and high (≳ 10 MeV) energies. Abdo et al. (2009a) have suggested that the high-energy spectrum of GRB 090902B can be reproduced as the sum of two components: a Band function peaking at ∼700 keV, and a single power law (photon index Γ ≡ β + 1 = 1.93) extending over the entire GBM+LAT bandpass. In this model, the power-law component accounts for ∼24% of the total 10 keV to 10 GeV fluence. The physical mechanism responsible for this complex spectrum is still not entirely understood; possible explanations include a hadronic origin (either proton synchrotron radiation (Razzaque et al. 2010) or photohadronic interactions (Asano et al. 2009)) or thermal emission from the jet photosphere (Mészáros & Rees 2000; Ryde 2004; Ryde et al. 2010).

Using their two-component spectrum, Abdo et al. (2009a) report a total fluence of f_γ = (4.59 ± 0.05) × 10⁻⁴ erg cm⁻² in the 8 keV to 30 GeV (observer frame) bandpass. At a redshift of 1.8229 (Section 2.3.3), this corresponds to a prompt fluence of (3.83 ± 0.05) × 10⁻⁴ erg cm⁻² in the 1–10⁴ keV rest-frame bandpass.

The Swift XRT began target-of-opportunity observations of the field of GRB 090902B at 23:36 on 2009 September 2 (∼12.5 hr after the GBM trigger). A fading X-ray source at $\alpha = 17^{\mathrm{h}} 39^{\mathrm{m}} 45\mbox{$.\!\!^{\mathrm s}$}26$, δ = +27°19'281 (J2000.0; 21 containment radius) was promptly identified as the X-ray afterglow (Kennea & Stratta 2009; Evans 2009; Stratta et al. 2009). The resulting X-ray light curve is shown in Figure 6.

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The Swift UVOT began concurrently observing the field of GRB 090902B and first reported the detection of a candidate optical afterglow consistent with the X-ray position (Swenson & Siegel 2009). Subsequent observations revealed that the candidate had faded, confirming its association with GRB 090902B (Swenson & Stratta 2009). In Table 5 we present UVOT U-band observations of GRB 090902B, reduced in a manner identical to that described in Section 2.2.2.

We obtained a single r' image of the afterglow of GRB 090902B with GMOS-N on 2009 September 3. This image was reduced in the same manner as those of the other events, and the resulting photometry is presented in Table 5. We have included optical and NIR photometry of GRB 090902B from Pandey et al. (2010) and McBreen et al. (2010) in our modeling, and therefore display them in Figure 6 as well.

Finally, we began observing the afterglow of GRB 090902B with the VLA on 2009 September 3 (Chandra & Frail 2009) and continued for ∼5 months. The data were reduced as described in Section 2.2.2. The results of our radio campaign are displayed in Table 6 (individual epochs) and plotted in Figure 6 (combined epochs). We have also included the early radio detection at 4.8 GHz reported by the WSRT (van der Horst et al. 2009) in Table 6.

We began spectroscopic observations of GRB 090902B with GMOS-N at 06:29 on 2009 September 3 (∼19.4 hr after the GBM trigger; Cucchiara et al. 2009). We obtained 2 × 900 s exposures, both with the R400 grating and a central wavelength of 6000 Å, providing a coverage of ∼4000–8000 Å. The spectra were reduced as described in Sections 2.1.3 and 2.2.3. Flux calibration was performed relative to spectra of the standard star Feige 34 (Oke 1990) taken with the same instrumental setup on 2008 May 1.

The resulting spectrum of GRB 090902B is shown in Figure 7. Superimposed on a smooth power-law continuum, we identify strong absorption features corresponding to Mg i λ2852, Mg ii λλ2796, 2803, Mn ii λ2606, Fe ii λ2600, Mn ii λ2594, Fe ii λ2586, Mn ii λ2576, Fe i λ2484, Fe ii λ2382, Fe ii λ2344, Fe ii λ2260, Fe i λ2167, Cr ii λλ2056, 2062, and Mg i λ2026, all at a common redshift of 1.8229 ± 0.0004.

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The lack of evidence for Lyα absorption down to λ ≲ 4000 Å implies an upper bound on the redshift of the GRB host galaxy of z ≲ 2.3. Together with the strength of the above features (in particular Fe i λ2167), we consider it quite likely that the observed system at z = 1.8229 derives from the GRB host galaxy.

At this redshift, the prompt isotropic gamma-ray energy release in the 1–10⁴ keV rest-frame bandpass is E_{γ, iso} = (3.20 ± 0.04) × 10⁵⁴ erg. Assuming a nonthermal spectrum up to E_obs = 11 GeV, Abdo et al. (2009a) have determined a lower limit to the initial outflow Lorentz factor of Γ₀ ≳ 1000.

GRB 090926A triggered the GBM on Fermi at 04:20:26.99 on 2009 September 26 (Bissaldi 2009). The light curve consists of a single pulse with duration t₉₀ ≈ 20 s in the GBM bandpass. The prompt emission was sufficiently bright to trigger several additional satellite instruments, including Suzaku-WAM (Noda et al. 2009), Konus-Wind (Golenetskii et al. 2009b), and RT-2 on CORONAS-PHOTON (Chakrabarti et al. 2009).

Simultaneously fitting both the GBM and LAT (see below) data from t₀ to t₀ + 21 s, Bissaldi et al. (2009) report that the spectrum is reasonably well modeled by a Band function with α = −0.693 ± 0.009, β = −2.342 ± 0.011, and E_p = 268 ± 4 keV. The corresponding 10 keV to 10 GeV fluence (observer frame) is f_γ = (2.47 ± 0.03) × 10⁻⁴ erg cm⁻². These results differ only slightly from the values reported from other satellites. At z = 2.1062 (Malesani et al. 2009), the corresponding prompt fluence in the rest-frame 1–10⁴ keV bandpass is (1.73 ± 0.03) × 10⁻⁴ erg cm⁻².

GRB 090926A was also detected by the Fermi-LAT. The emission in the LAT bandpass lasted at least 400 s, with possible indications of significant flux out to ∼1000 s after the GBM trigger. Many photons with E > 1 GeV were detected from the position of GRB 090926A, with the highest energy measured of 20 GeV at t₀ + 26 s (Uehara et al. 2009).

Swift XRT began observing the field of GRB 090926A at 17:17 on 2009 September 26 (∼13 hr after the GBM trigger). A fading X-ray counterpart at $\alpha = 23^{\mathrm{h}} 33^{\mathrm{m}} 36\mbox{$.\!\!^{\mathrm s}$}18$, δ = −66°19'259 (J2000.0, 15 containment radius) was promptly identified in the XRT data (Vetere et al. 2009; Vetere 2009). The XRT observed GRB 090926A for the next 3 weeks, and we plot the X-ray light curve in Figures 8 and 9.

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Under the control of Skynet, four of the 16 inch diameter PROMPT telescopes (Reichart et al. 2005) at the Cerro Tololo Inter-American Observatory (CTIO) observed the Fermi-LAT localization of GRB 090926A beginning 19.0 hr after the GBM trigger in the B-, V-, R-, and I-band filters. Within the Swift-XRT localization, we identified an uncataloged and fading source as the optical afterglow of GRB 090926A (Haislip et al. 2009b, 2009a). Individual images were automatically reduced using our custom, IRAF-based reduction pipeline and then astrometrically aligned and stacked. We subsequently measured the afterglow flux with aperture photometry, where the inclusion radius was approximately matched to the FWHM of the PSF. PROMPT continued to observe the field for 10 more nights (Haislip et al. 2009e, 2009c, 2009d).

We also obtained three epochs of I- and J-band imaging of the afterglow of GRB 090926A using the ANDICAM (A Novel Dual Imaging CAMera) instrument mounted on the 1.3 m telescope at CTIO.²⁴ This telescope is operated as part of the Small and Moderate Aperture Research Telescope System (SMARTS) consortium.²⁵ Each epoch consisted of six individual 360 s I-band observations and 30 individual 60 s J-band observations. Between optical exposures, the telescope was slightly offset and the individual J-band exposures were additionally dithered via an internal tilting mirror system. Standard data reduction was performed on these images, including cosmic ray rejection, overscan bias subtraction, zero subtraction, flat fielding, and sky subtraction to correct for the NIR background and the I-band fringing. For each epoch, the individual images were then aligned and averaged to produce a single frame in each band with summed exposure times of 36 minutes in I and 30 minutes in J.

Relative aperture photometry was performed on the SMARTS data, in comparison with a number of nonvariable sources in the field of GRB 090926A. The I-band field was photometrically calibrated by comparison (on photometric nights) with Landolt standard stars in the field of T Phe (Landolt 1992). J-band photometric calibration was performed using Two Micron All Sky Survey (Skrutskie et al. 2006) field stars.

SMARTS BVRI observations of the field of GRB 090926A were also obtained on two photometric nights a few months after the GRB occurred (2009 December 16 and 18). For these observations, total summed exposure times amounted to 180 s in BRI and 120 s in V. The absolute photometry of the field was again established based on same-night observations of the T Phe Landolt standard stars. These observations were then used to provide absolute calibration for the PROMPT observations of GRB 090926A.

We acquired additional late-time images of the field of GRB 090926A on 2009 October 19 with GMOS-S on Gemini South. A total of 600 s of exposure time was obtained in the Sloan g'-, r'-, and i'-band filters. The data were reduced in the manner described in Section 2.1.2 and calibrated in the same way as the PROMPT and SMARTS observations.

The results of our optical and NIR monitoring campaign of the afterglow of GRB 090926A, uncorrected for the modest amount of Galactic extinction (E(B − V) = 0.024 mag; Schlegel et al. 1998), are shown in Table 7 and Figures 8 and 9. Also included in Figures 8 and 9 are the optical and NIR observations presented by Rau et al. (2010).

Finally, the field of GRB 090926A was observed in the radio (5.5 GHz) on 2009 October 1 with the Australia Telescope Compact Array. No source was detected at the afterglow location to a 2σ limit of f_ν < 1.5 mJy (Moin et al. 2009).

Malesani et al. (2009) obtained a spectrum of the afterglow of GRB 090926A with the X-Shooter instrument mounted on the 8 m Very Large Telescope (VLT) UT2. Based on the detection of a damped-Lyα system and many strong, narrow absorption features, these authors derive a redshift of 2.1062 for the host galaxy of GRB 090926A. Rau et al. (2010) find a consistent result based on a spectrum taken with FORS2 on the VLT.

At this redshift, the prompt isotropic gamma-ray energy release in the 1–10⁴ keV bandpass is E_{γ, iso} = (1.89 ± 0.03) × 10⁵⁴ erg. Assuming a nonthermal spectrum up to E_obs = 20 GeV, we infer a lower limit for the initial Lorentz factor of Γ₀ ≳ 700 (Lithwick & Sari 2001).

Before proceeding to a detailed model, we derive some initial constraints by looking at the spectral and temporal behavior of the afterglow. This type of "α − β" analysis is useful for comparing to other published work, but we do not use it in our detailed afterglow modeling.

Considering first the X-ray afterglow, we fit the flux density to a power-law decay and find a best-fit index of α_X = 1.5 ± 0.1 (χ² = 28.2 for 22 degrees of freedom (dof)). Performing an analogous power-law fit to the X-ray spectrum, we find β_X = 1.06^+0.34_{− 0.13} (χ² = 8.0 for 13 dof; see the online compilation of N.R.B. for details).

We perform a similar power-law fit in the optical/NIR (f_ν∝t^−αν^−β), forcing the decay index to be identical in all filters and ignoring the points at t > 10 days due to host-galaxy contamination (see below). Though the fit quality is somewhat worse (χ² = 153.3 for 99 dof), we find similar power-law indices as seen in the X-rays: α_O = 1.74 ± 0.05; β_O = 1.14 ± 0.06.

For typical values of the electron index p ≈ 2.5 (Shen et al. 2006; Starling et al. 2008; Curran et al. 2009), this steep temporal decay is difficult to reconcile with expansion into a constant-density medium, where α is typically ∼1.0–1.5 (Sari et al. 1998). However, expansion into a wind-like medium, even with relatively standard electron indices, can naturally accommodate such a temporal decay (Chevalier & Li 2000).

The temporal decay index we derive is slightly shallower than that inferred by McBreen et al. (2010; α_O = 1.90). These authors include an additional emission mechanism beyond the standard synchrotron emission, a "bump" at t ≈ 7 days that effectively removes flux from the power-law component at late times. Our derived spectral index is noticeably shallower than that found by McBreen et al. (2010; β_O = 0.65), predominantly because we have yet to include the effects of dust extinction in the GRB host galaxy. We shall correct for this in our full modeling in the following section.

The comparable spectral and temporal indices in the optical and X-rays suggest that both bandpasses fall in the same synchrotron spectral regime. In other words, the cooling frequency should fall either below the optical or above the X-ray bandpass for the duration of these observations (t ≈ 1–10 days). Examining the "closure" relations between α and β in different circumburst media and spectral regimes (e.g., Price et al. 2002), we find that we can rule out a low cooling frequency at large significance, as this would require α = (3β − 1)/2 ≈ 1.0 for both a wind-like and an ISM-like circumburst medium.

The relatively flat radio light curve (modulo effects from ISS) until t ≈ 20 days has two important implications. First, it suggests a wind-like circumburst medium, as the flux would be expected to rise in proportion to t^1/2 for ν_a < ν < ν_m in a constant-density environment. Because of the effects of ISS, however, the statistical significance is not very large and the full suite of broadband observations is required to compare the competing circumburst density profiles. Second, together with the lack of evidence for late-time steepening in the X-ray and optical light curves, this suggests that if a jet break is required, it should occur at t ≳ 20 days. (Another possibility for the decay could be the peak frequency ν_m passing through the radio bands.)

Finally, the relatively bright r'-band detection at t = 130 days is almost certainly dominated by flux from the host galaxy and not the afterglow of GRB 090323. If we assume this flux is due entirely to host light, the host will contribute a significant fraction of the flux in some of the late-time points of our optical light curve (∼40% at t = 14 days in r'). A similar conclusion is reached by McBreen et al. (2010) who also detect the host in the i' filter. In our detailed modeling, we assume a constant contribution from the host galaxy equal to the values from these measurements in the R/r' and i' filters.

With these constraints in hand, we have modeled the afterglow of GRB 090323 with the software described above. The resulting best-fit wind model is plotted in Figure 2 and the derived parameters are provided in Table 8. The overall fit quality is reasonable (χ² = 316.8 for 239 dof), with comparable residuals in the radio, optical, and X-ray bandpasses. The best-fit ISM model results in a dramatically worse fit (χ² = 585.0 for 239 dof), with poorer residuals in all bandpasses (i.e., not just the radio).

Table 8. GRB 090323 Afterglow Fit Parameters

Model	EKE, iso	A*	e	B	θ	p	AV(host)	χ2ν (dof)
	(1052 erg)	(g cm−1)	(%)	(%)	(°)		(mag)
Wind	116+13− 9	0.10 ± 0.01	7.0 ± 0.5	0.89+0.07− 0.18	2.8+0.4− 0.1	2.71 ± 0.02	0.12 ± 0.07	1.33 (239)
Wind-Isoa	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	90	⋅⋅⋅	⋅⋅⋅	1.35 (240)
ISM	623	0.11	33b	0.13	2.8	2.10	0.19	2.45 (239)

Notes. ^aFor the isotropic model, all parameters are identical to the jetted model, except for the opening angle, which is held fixed. ^bThe electron partition fraction was fixed to its equipartition value.

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As expected, the cooling frequency ν_c lies near or above the X-ray bandpass for the duration of our observations (t ≳ 1 day). Consequently, the ratio of E_{KE, iso} to A_* is required to be larger than inferred for previous GRBs (e.g., Panaitescu & Kumar 2001b; Yost et al. 2003). While the isotropic blast-wave kinetic energy is relatively large compared to that of previously modeled GRBs (E_{KE, iso} = 1.2 × 10⁵⁴ erg), it is in fact comparable to the prompt gamma-ray energy release, as would be expected for reasonable values of the gamma-ray efficiency η_γ. The inferred density is slightly lower than usual (A_* = 0.10 g cm⁻¹), though smaller values have been reported in the literature (e.g., GRB 020405; Chevalier et al. 2004).

The best-fit model results in a jet break time of t_j = 17.6^+11.2_{− 2.3} days. This occurs largely after the X-ray and optical observations have ceased (except for the host detections in the optical), and is therefore most directly constrained by the radio light curve. In Figure 2, we also plot the same model parameters but for an isotropic explosion, where the turnover in the radio at late times is due to the peak frequency ν_m passing through the radio bandpass (see also Table 8). For the entire data set, the fit quality for the isotropic model is only marginally worse (χ² = 324.3 for 240 dof); however, this masks the significant improvement in the fit achieved in the radio bandpass (χ² = 7.8 for the jet-like model compared to χ² = 13.5 for the isotropic model). Conservatively, the opening angle and beaming-corrected energetics derived here can at worst be treated as lower limits.

Though the upper bound on the jet break time is not extremely well constrained, due to the relatively weak dependence of θ on t_j in a wind-like circumburst medium (Equation (5)), the uncertainty in the jet opening angle is significantly smaller. Compared with previous GRB samples, the jet break occurs rather late in the observer frame (e.g., Zeh et al. 2006), and the inferred opening angle (θ = 28) is still relatively small compared with previous samples. Though the dependence is not strong (θ∝(E_{KE, iso}/A_*)^−1/4), the large ratio of E_{KE, iso} to A_* effectively lowers the opening angle for a given jet break time.

After applying the collimation correction, we find that the true energy release of GRB 090323 is E_γ = 4.0^+1.2_{− 0.4} × 10⁵¹ erg, E_KE = 1.4^+0.6_{− 0.2} × 10⁵¹ erg. We compare these results with a larger sample of events in Section 4.1.

Following the results of Section 3.1.1, we first consider the spectral and temporal indices of GRB 090328 in the observed bandpasses independently. The X-ray afterglow light curve is well fit by a single power-law decay with index α_X = 1.65 ± 0.09 (χ² = 24.1 for 26 dof). The spectral index, on the other hand, is less well constrained: β_X = 1.1^+0.4_{− 0.3} (χ² = 6.7 for 14 dof). We note that the X-ray spectral fit indicates the presence of an additional absorption component in the host galaxy of GRB 090328 (N_H = 4.9^+3.5_{− 2.8} × 10²¹ cm⁻² at z = 0.736).

Along the lines of our analysis of the optical light curve in Section 3.1.1, we fit our Swift U-band observations, together with the optical/NIR data from GROND (McBreen et al. 2010), to a power-law model of the form f_ν∝t^−αν^−β. After subtracting the host flux values derived in each filter from GROND photometry, we find a relatively poor fit (χ² = 99.7 for 41 dof) with α_O = 1.85 ± 0.13 and β_O = 1.40 ± 0.05. The resulting spectral index is comparable to that found by McBreen et al. (2010; before we account for host reddening; see below), though again we derive a somewhat shallower temporal index due to the fact that these authors added a bump to their light-curve model at late times.

In the case of GRB 090328, examining the α–β closure relations yields little insight. Because of the poor fit quality in the optical and the large uncertainty in the X-ray spectral index, we are unable to rule out much of the available parameter space. We note, however, that with the exception of the excess absorption, the X-ray and optical afterglow light curves and spectral energy distributions of GRB 090328 broadly resemble those of GRB 090323.

The results of our best-fit model for the afterglow of GRB 090328 are shown in Figure 4 and the parameters are provided in Table 9. The overall fit quality is quite good (χ² = 98.4 for 101 dof). The improvement over our simple power-law fits in the optical in the previous section results largely from incorporating the modest host-galaxy extinction (A_V = 0.33 mag). Only the early radio light curve, which suffers from strong scintillation at early times, differs systematically from the model predictions.

Table 9. GRB 090328 Afterglow Fit Parameters

Model	EKE, iso	A*	e	B	θ	p	AV(host)	χ2ν (dof)
	(1052 erg)	(g cm−1)	(%)	(%)	(°)		(mag)
Wind	82+28− 18	0.26+0.13− 0.07	11+6− 1	0.19+0.04− 0.08	4.2+1.3− 0.8	2.81+0.14− 0.07	0.33 ± 0.05	0.97 (101)
Wind-Isoa	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	90	⋅⋅⋅	⋅⋅⋅	1.63 (102)
ISM	87	82	19	0.12	7.6	2.26	0.30	1.11 (101)

Note. ^aFor the isotropic model, all parameters are identical to the jetted model, except for the opening angle, which is held fixed.

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Much like the case of GRB 090323, we find that only a wind-like circumburst medium is favored, though not nearly in as statistically significant a manner. The largest discrepancy between the wind-like and constant-density environments actually occurs in the X-ray bandpass, where the wind is strongly favored (χ²_{ν, wind} = 1.08 compared with χ²_{ν, ISM} = 1.48).

The steep decline in the radio light curve at late times offers relatively tight constraints on the presence of a jet break. The difference in model fits for the radio band is χ²_ν = 2.52 (beamed) compared with χ²_ν = 10.81 (isotropic), a fact clearly evidenced by the rising isotropic light curve in Figure 4. In fact, the break is consistent with (though not formally required by) the final few points in the X-ray light curve as well. For our best-fit model, we infer t_j = 9.0^+11.6_{− 6.0} days and θ = (4.2^+1.3_{− 0.8})°. The break is essentially impossible to detect in the optical due to the presence of the bright host galaxy.

Correcting for the effects of beaming, we find an energy release of E_γ = 2.6^+2.3_{− 1.1} × 10⁵⁰ erg and E_KE = 2.2^+2.9_{− 0.9} × 10⁵¹ erg. The total relativistic energy release, E_tot ≈ 2 × 10⁵¹ erg, is somewhat less than the value we derive for GRB 090323. We note that unlike most of the other events in our sample, the implied gamma-ray efficiency for GRB 090328 is relatively modest.

A single power-law model with α_X = 1.36 ± 0.03 provides a good fit to the X-ray light curve of GRB 090902B over the entire span of the XRT observations (χ² = 94.8 for 107 dof). Likewise, the X-ray spectrum is well fit by a power law with β_X = 0.90 ± 0.13 (χ² = 42.8 for 56 dof). The X-ray spectrum strongly favors the presence of absorption in excess of the Galactic value along the GRB line of sight. Assuming the dust arises in the GRB host galaxy at z = 1.82, the best-fit host column density is N_H = (6.3^+2.1_{− 1.7}) × 10²¹ cm⁻². These results are all consistent with a similar analysis of the X-ray afterglow performed by Pandey et al. (2010).

The decline inferred from the early-time ROTSE-III R-band observations is significantly steeper than the late-time (t ≳ 1 day) optical decay. Following Pandey et al. (2010), we attribute this emission largely to the presence of a reverse shock and do not include these data points in any subsequent analysis. Simultaneously fitting the remaining optical data to a power law of the form f_ν∝t^−αν^−β, we find α_O = 0.89 ± 0.03 and β_O = 0.82 ± 0.10 (χ² = 113.1 for 67 dof). Again, our results are consistent with those reported by Pandey et al. (2010) and McBreen et al. (2010).

The large discrepancy between the X-ray and optical temporal decay slopes suggests that the two bandpasses fall in different synchrotron spectral regimes (i.e., ν_O < ν_c < ν_X). Furthermore, the relatively shallow optical decline strongly rules out a wind-like circumburst medium. We find that both the X-ray and optical bandpasses are broadly consistent with expansion into a constant-density medium and an electron index p ≈ 2.2. The observed X-ray temporal decline is somewhat steeper than what is predicted, but this could be accounted for by synchrotron radiative losses at later times.

The radio light curve appears to decay from the earliest observations at t₀ + 1.5 days onward. This behavior is not expected in the simple forward-shock model where radio afterglows are typically observed to rise (for a constant-density medium) or remain flat (for a stratified environment) until either a jet break occurs or the peak frequency reaches the radio bandpass (typically weeks after the burst). While ISS can affect the radio afterglow dramatically, particularly at early times, such behavior nonetheless requires an alternate explanation.

Bright, early-time radio emission is actually rather common, having been detected in a quarter of all radio afterglows to date (Soderberg & Ramirez-Ruiz 2003; P. Chandra et al. 2011, in preparation). This prompt component is most commonly attributed to the reverse shock propagating in the shocked ejecta. The reverse-shock emission fades away quite quickly, and at late times (t ≳ 5 days) the afterglow will be dominated by the forward-shock emission. GRB 990123 was the first known example (Kulkarni et al. 1999), but there have been many more claimed detections since (e.g., Berger et al. 2003b; Panaitescu 2005; Chandra et al. 2010).

The radio light curve of GRB 090902B most closely resembles that of GRB 991216 in which the early power-law decline is due to the overlap of emission from the reverse and forward shocks (Frail et al. 2000b). Since we are concerned here with the modeling of the behavior of the forward shock, we will exclude the early radio point and postpone a full discussion to a subsequent paper dealing with a compilation of all known GRBs having prompt radio emission (P. Chandra et al. 2011, in preparation).

Using the constraints derived above, our best-fit model for GRB 090902B assuming a constant-density circumburst medium is shown in Figure 6, and the derived model parameters are presented in Table 10. The overall fit quality is quite reasonable (χ² = 114.8 for 102 dof), with comparable fit quality in all bandpasses.

Table 10. GRB 090902B Afterglow Fit Parameters

Model	EKE, iso	n	e	B	θ	p	AV(host)	χ2ν (dof)
	(1052 erg)	(cm−3)	(%)	(%)	(°)		(mag)
ISM	56+7− 3	(5.6+0.9− 0.7) × 10−4	13 ± 1	33a	3.9 ± 0.2	2.21 ± 0.02	0.05 ± 0.03	1.11 (103)
ISM-Isob	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	90	⋅⋅⋅	⋅⋅⋅	1.79 (104)
Wind	0.063	0.19	8.9	33a	28	1.54	0.04	1.54 (103)

Notes. ^aThe electron partition fraction was fixed to its equipartition value. ^bFor the isotropic model, all parameters are identical to the jetted model, except for the opening angle, which is held fixed.

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We stress, however, that we have not included the first two R-band data points and the first VLA detection in our fitting, as we believe the flux at these points is dominated by reverse-shock emission. These points are severely underpredicted by our models and, if included, could result in somewhat different fit parameters. Including the early-time radio emission at t ≈ 1.5 days would not dramatically affect our results, however, due mostly to the large effects of ISS.

We find that a wind-like environment is strongly disfavored by our models, with the best-fit wind model having χ² = 156.7 (102 dof). Furthermore, the parameters of the best-fit wind model are relatively unphysical, with an extremely low afterglow energy (E_{KE, iso} = 6.3 × 10⁵⁰ erg, more than four orders of magnitude less than the prompt gamma-ray energy) and an electron energy distribution index of p = 1.54 (p values less than 2 require a spectral cutoff to avoid an infinite energy).

Much like the other events considered here, our inferred parameters for GRB 090902B suggest a large afterglow kinetic energy (E_{KE, iso} = 5.6^+0.3_{− 0.7} × 10⁵³ erg; a factor of five less than the prompt gamma-ray energy release) and a low circumburst density (n = 5.6^+0.9_{− 0.7} × 10⁻⁴ cm⁻³). In fact, we are unable to find any acceptable solutions (χ²_ν < 2) with n > 10⁻² cm⁻³. The inferred circumburst density for many long-duration GRBs is lower than typical values observed in dense molecular clouds (where presumably their massive-star progenitors have formed). Yet the value we have derived for GRB 090902B is closer in fact to what one might find in the ambient ISM or even the intergalactic medium, as has been found for the few short-hard GRBs with sufficient afterglow data (e.g., Panaitescu 2006; Perley et al. 2009b). We return to this issue in more detail in Section 4.3.

Isotropic models are strongly disfavored as well (χ² = 182.8 for 104 dof), with evidence for a jet break from both the optical (in particular the late-time measurement from McBreen et al. 2010 at t ≈ 23 days) and the radio. Our models suggest that the peak synchrotron frequency, ν_m, will not fall to the radio bandpass until t ≳ 40 days. We find that the jet break time is relatively tightly constrained to be t_j = 18.6 ± 0.8 days. This is very similar to that found by McBreen et al. (2010). Again, because of the large ratio of E_{KE, iso} to n, this corresponds to a relatively narrow opening angle: θ = (3.9^+0.2_{− 0.2})°.

Correcting for collimation, we find the true energy release from GRB 090902B to be E_γ = (7.4 ± 0.8) × 10⁵¹ erg, E_KE = (1.3 ± 0.3) × 10⁵¹ erg.

Fitting a single power-law decay to the X-ray light curve, we find a reasonable quality fit (χ² = 120.8 for 97 dof) with α_X = 1.43 ± 0.03. The fit quality improves somewhat if we allow for a steeper decay at late times: α_{X, 1} = 1.35 ± 0.04, α_{X, 2} = 2.6^+0.9_{− 0.5}, t_b = 9.1^+2.1_{− 1.4} days, χ² = 111.9 (95 dof), although this in and of itself is not particularly statistically significant. The X-ray spectrum is well fit (χ² = 41.7 for 51 dof) by a single power law with index β_X = 1.12 ± 0.13 and does not require any absorption in addition to the Galactic component.

The behavior in the optical bandpass, however, is significantly more complex (Figure 8). The flux in the initial UVOT U- and V-band observations declines until t ≈ 0.8 days, at which point a rebrightening or plateau phase is evident in all five observed filters (UBVRI). The optical light curve of GRB 090926A is reminiscent of the rebrightening seen at t ≈ 1 day from GRB 970508 (Djorgovski et al. 1997) and GRB 071003 (Perley et al. 2008 and references therein). Such a phenomenon is difficult to reconcile with the standard afterglow paradigm. Possible explanations include a sharp change in the circumburst density (e.g., Lazzati et al. 2002; Tam et al. 2005; but see also Nakar & Granot 2007) or a smooth injection of energy into the forward shock from the central engine (e.g., Rees & Meszaros 1998). A more detailed analysis of the early-time behavior of GRB 090926A is beyond the scope of this work.

Simultaneously fitting all of the Bg'Vr'Ri'IJHK data in the range 1 day < t < 10 days, we find α_O = 1.38 ± 0.05, β_O = 0.88 ± 0.07 (χ² = 657.6 for 309 dof). Our derived spectral slope is in good agreement with that found by Rau et al. (2010). However, the temporal decay index we infer is significantly shallower than that reported by both Rau et al. (2010) and Swenson et al. (2010). Again, the discrepancy results largely from how these authors handled deviations from pure power-law behavior. Rau et al. (2010) introduced several bumps to their light curve, while Swenson et al. (2010) removed periods of flaring from their fits. We have chosen to instead adopt the simplest power-law models and accept the poorer fit residuals this encompasses. We return to the effect these differences introduce in the derived opening angles and energetics in Section 4.1.

Extrapolating the observed decay indices to our late-time g'r'i' Gemini imaging at t ≈ 23 days, we find that the flux at late times is significantly overpredicted (Figure 9), suggesting the presence of a break. To quantify this, we fit all of the available g', r', and i' data at t > 1 day first to a single power law. The resulting fit is relatively poor, with χ² = 412.7 (164 dof). Introducing a simultaneous break in all three filters, we find a significantly improved fit (χ² = 311.3 for 162 dof; t_b = 10.7 ± 1.3 days). The result strongly suggests the presence of a jet break, which we return to discuss in the following section.

Taken together, we find similar temporal (α_{X, 1} = 1.35, α_{O, 1} = 1.38) and spectral (β_X = 1.12, β_O = 1.03) indices in the X-ray and optical for t ≈ 1–9 days. Furthermore, the optical to X-ray spectral index, β_OX, is comparable to both the optical and X-ray spectral indices. These facts strongly suggest that both the X-ray and optical bandpasses fall in the same synchrotron spectral regime.

Considering the various synchrotron closure relations, the best fit appears to occur when both bandpasses fall above the cooling frequency, ν_c. The afterglow decay in this regime is independent of the circumburst medium. Such a low cooling frequency is somewhat unusual, though not unprecedented (e.g., GRB 050904; Frail et al. 2006), in GRB afterglows. The implied electron index in this case would be p ≈ 2.3.

Unfortunately, without a radio light curve we cannot uniquely solve for the physical parameters of the afterglow of GRB 090926A. We can, however, at the very least more robustly constrain the jet break time and opening angle of this event, as well as identify broad trends in the parameters associated with acceptable solutions. As discussed previously, we consider only data at t ⩾ 1 day due to the rebrightening observed in the optical before then. To simplify our models and not overly weight the optical bandpass, we have only included observations in the g', r', and i' filters (while this changes the details of the chosen models slightly, it does not alter any of our primary results).

As expected, we were able to find a number of acceptable solutions, both for constant-density and wind-like environments. The best achieved fits for both a wind-like and a constant-density medium are provided in Table 11, with the best-fit wind model plotted in Figure 9. The model provides a reasonable description of the data in all bandpasses, with χ² = 176.3.1 (202 dof). Without radio data, however, the solutions presented in Table 11 are not unique in a global sense, and therefore should only be taken as representative of a larger family of solutions.

Table 11. GRB 090926A Afterglow Fit Parameters

Model	EKE, iso	A*	e	B	θ	p	AV(host)	χ2ν (dof)
	(1052 erg)	(g cm−1)	(%)	(%)	(°)		(mag)
Wind	6.8 ± 0.2	0.56 ± 0.03	33a	8.1+1.3− 0.7	9.6 ± 0.2	2.11 ± 0.02	<0.05	0.87 (202)
Wind-Isob	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	90	⋅⋅⋅	⋅⋅⋅	2.56 (203)
ISM	185	2.5	33a	0.14	9.6	2.13	0	1.02 (202)

Notes. ^aThe electron partition fraction was fixed to its equipartition value. ^bFor the isotropic model, all parameters are identical to the jetted model, except for the opening angle, which is held fixed.

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Most importantly for our purposes, a jet break at t ≈ 10 ± 2 days is required to provide a reasonable fit to the late-time optical data (Figure 9). An analogous fit without a jet break results in a significantly worse quality (χ² = 519.7 for 203 dof). The precise translation into an opening angle is dependent on the circumburst density (both profile and absolute value) and the afterglow energy. Taking the values derived above into account, we find a typical value for a wind-like circumburst medium of θ ≈ 9°. Based on the observed spread of E_KE and A_* in our wind models, we adopt a conservative uncertainty in this value of θ ≈ (9⁺⁴_{− 2})°.

Correcting for collimation, we find a prompt energy release of E_γ = 2.3^+2.6_{− 0.9} × 10⁵² erg. For our best-fit wind model, we find E_KE = 8⁺¹⁰_{− 3} × 10⁵⁰ erg, although we note, despite the small statistical uncertainties, that this value is highly ambiguous due to the lack of radio data.