A DECADE OF SHORT-DURATION GAMMA-RAY BURST BROADBAND AFTERGLOWS: ENERGETICS, CIRCUMBURST DENSITIES, AND JET OPENING ANGLES

We present afterglow observations for 103 short GRBs discovered by the Swift satellite (Gehrels et al. 2004), Fermi satellite (Atwood et al. 2009; Meegan et al. 2009), Konus-Wind (Aptekar et al. 1995), High Energy Transient Explorer 2 (Ricker et al. 2003), or the Interplanetary Network (IPN; Hurley et al. 2010) between 2004 November and 2015 March. We restrict our sample to all bursts with T₉₀ ≲ 2 s and follow-up observations in any of the X-ray, optical, near-infrared (NIR) or radio bands on timescales of δt ≲ few days (where δt corresponds to the time after the γ-ray trigger). We also include three events (GRBs 050724A, 090607 and 100213A) which have T₉₀ ≈ 2.5–3 s but which exhibit the spectral hardness and negligible spectral lags typical of short GRBs (Marshall et al. 2009; Grupe et al. 2010). For bursts with optical/NIR follow-up, we only include data from bursts with afterglow detections or meaningful limits of ≳20 mag at δt ≲ 1 day. We note that included in our sample are three bursts with T₉₀ ≲ 2 s but which do not exhibit the same γ-ray properties as canonical short-hard GRBs, either due to their large spectral lags and soft γ-ray spectra (GRBs 090426; Ukwatta et al. 2009 and 131004A Stamatikos et al. 2013), or due to where they fall on the lag-luminosity relation of GRBs (GRB 060121; Kann et al. 2011). We also notably exclude GRBs 060614 and 100816A, both of which had durations of T₉₀ > 2 s but which otherwise exhibit some properties similar to traditional short-hard GRBs (e.g., Zhang et al. 2007; Jin et al. 2015; Yang et al. 2015). However, for complete uniformity and simplicity, we only employ a T₉₀ cut in our sample selection so as not to introduce any additional priors on the distributions. Basic information for the sample of 103 events, including durations, redshifts, and the available follow-up in each observing band, is presented in Table 1.

Of the 103 bursts in our sample, 96 (93%) have X-ray follow-up observations, 87 (84%) have optical/NIR observations, and 60 (58%) have radio observations. These observations have uncovered 71 X-ray afterglows, 30 optical/NIR afterglows, and 4 radio afterglows, leading to detection fractions of 74%, 34%, and 7%, respectively (Table 1). The observations for these bursts are cataloged in the Appendix (Tables 6–8).

Thirty-one bursts in this sample have redshifts determined from their host galaxies (Table 1), while two events have spectroscopic redshifts from absorption in their afterglows, GRB 090426A: (Antonelli et al. 2009; Levesque et al. 2010) and GRB 130603B (Cucchiara et al. 2013; de Ugarte Postigo et al. 2014). Furthermore, we place upper limits on the redshift from the detection of the optical afterglow, and therefore the lack of suppression blueward of the Lyman limit (λ₀ = 912 Å) or Lyα line (λ₀ = 1216 Å), for 11 bursts (Table 1). We also place a lower limit of z > 1.4 for GRB 051210 from the lack of detection of emission or absorption features in the spectrum of the host galaxy (Berger et al. 2007). In addition to the broadband afterglow observations that have been published thus far, we present new optical/NIR observations for 11 bursts (Table 7), and new radio observations for 25 events (Table 8).

2.1.1. Observing Constraints

We gather all available X-ray afterglow observations from the Swift light curve repository⁵ (Evans et al. 2007a, 2009), the GRB Coordinates Network (GCN) Circulars, and the literature (Table 6). The data were taken with the XRT on board Swift, the Chandra X-ray Observatory, and the X-ray Multi-Mirror Mission (XMM-Newton). Ten bursts have Chandra observations, while three bursts have XMM-Newton observations (Table 6). We use unabsorbed fluxes and uncertainties in the 0.3–10 keV energy band when they are available; otherwise, we use the count-rate light curves in the same energy range and convert to fluxes (described later in this section). For bursts with multiple upper limits, we only include those which help to constrain the temporal behavior of the X-ray light curve. Of the 96 short GRBs with X-ray observations, four events have no reported measurements or upper limits.⁶ Therefore, the resulting late-time X-ray afterglow catalog is comprised of 92 events (Table 6).

When applicable, we convert the count rate light curves to unabsorbed fluxes using the count-rate-to-unabsorbed-flux conversion factors provided by the Swift light curve repository. These factors are derived from the automatic spectral fitting routine (Evans et al. 2009). This routine fits the X-ray spectrum for each burst to an absorbed power law model characterized by photon index, Γ, and the intrinsic neutral hydrogen absorption column, N_H,int, in excess of the Galactic column density in the direction of the burst (Kalberla et al. 2005; Wakker et al. 2011; Willingale et al. 2013). We use spectral parameters extracted in the Photon Counting mode when available; otherwise, we use parameters from the Windowed Timing mode. In ten cases, the value of N_H,int is highly uncertain, but consistent with zero. Therefore, utilizing the median value of N_H,int may result in an over-estimate of the true unabsorbed flux. Instead of using the given conversion factors for these bursts, we calculate the unabsorbed fluxes using WebPIMMS,⁷ setting N_H,int = 0. For 16 events, no count-rate-to-unabsorbed-flux conversion factor is available, so we employ a fiducial value of 1 × 10⁻¹¹ erg cm⁻² s⁻¹ ct⁻¹ set by the median value for all of the events in our sample. Applying these conversion factors to each of the count-rate light curves from Swift/XRT, Chandra and XMM-Newton, we obtain the unabsorbed fluxes, 1σ uncertainties, and 3σ upper limits for each burst (Table 6).

To enable comparison of the X-ray light curves to the optical and radio data, we convert the X-ray fluxes to flux densities, F_ν,X, at a fiducial energy of 1 keV (${F}_{\nu ,{\rm{X}}}\propto {\nu }^{{\beta }_{{\rm{X}}}}$ where ${\beta }_{{\rm{X}}}\equiv 1-{\rm{\Gamma }}$). When no spectral information is available, we use a fiducial spectral index of β_X,med = −1.0, set by the median value of the events in our sample. The flux densities, 1σ uncertainties, and 3σ upper limits are listed in Table 6, and the resulting light curves are shown in Figure 1.

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Finally, we present Chandra observations for three bursts that have not been published in the literature thus far: GRBs 120804A (PI: Burrows), 140930B (PI: Fong) and 150101B (PIs: Troja, Levan). We retrieve the pre-processed Level 2 data from the Chandra archive. We use the CIAO data reduction package to extract a count-rate within a 25-radius source aperture centered on the X-ray afterglow position, and utilize source-free regions on the same chip to estimate the background. For GRB 120804A, we obtain spectral parameters from earlier Chandra epochs of the same burst (Berger et al. 2013). For GRBs 140930B and 150101B, we use CIAO/specextract to extract a spectrum and obtain the spectral parameters. We then apply these parameters to the count rate to convert to flux density as described above (Table 6).

For each burst we gather all available optical and NIR afterglow observations from the literature and GCN Circulars (see Table 7 for references). When there are multiple upper limits for a given burst, we include only the deepest limits at δt ≲ 1 day. We convert all magnitudes to the AB system using instrument-specific conversion factors when available, or the standard conversions following Blanton & Roweis (2007). We correct all observations for Galactic extinction in the direction of each burst (Schlegel et al. 1998; Schlafly & Finkbeiner 2011), and convert AB magnitudes to flux densities, F_ν,opt. A log of observations for 87 events with optical/NIR follow-up is provided in Table 7, and the light curves and upper limits are shown in Figure 1. Bursts with no detected optical afterglow are further classified by the detection of an X-ray afterglow (Figure 1).

We also present optical/NIR observations of 11 short GRBs that have not been published in the literature thus far: GRBs 070724A, 100628A, 110420B, 120229A, 130716A, 140402A, 140606A, 140619B, 140930B, 150101B, and 150120A. These observations were enabled by target-of-opportunity programs on the twin 6.5-m Magellan telescopes (PI: Berger), the twin 8 m Gemini telescopes (PIs: Berger, Cucchiara, Fox, Tanvir), and the 6.5-m MMT (PIs: Berger, Fong). We use standard tasks in the IRAF/ccdred package to process the Magellan and MMT data, and the IRAF/gemini package to process the Gemini data. For each of these bursts, we obtained the first epoch of observations at δt ≈ 1–20 hr and at least one additional set of observations at δt ≳ 24 hr to provide a template (Table 7). To assess any fading between the two epochs, we perform digital image subtraction for each burst and filter using the ISIS software (Alard 2000). With the exception of GRBs 070724A, 140930B, and 150101B, we find no significant emission in any of the subtracted images. We therefore employ aperture photometry using standard tasks in IRAF to place 3σ upper limits on the optical/NIR afterglow brightness. To measure the afterglow brightness for GRBs 070724A and 140903B, we employ aperture photometry on the detected point source in the subtracted image. For GRB 150101B, the afterglow position in the subtracted image is contaminated by residual emission from its bright host galaxy; thus, we use PSF photometry to measure the afterglow brightness. Details of the photometry will be outlined in an upcoming paper (W. Fong et al. 2015, in preparation) The observational details, afterglow brightness and 3σ upper limits for these bursts are presented in Figure 1 and listed in Table 7.

We gather all available radio afterglow data taken with the Karl G. Jansky Very Large Array (VLA), Westerbork Synthesis Radio Telescope (WSRT), Australia Telescope Compact Array (ATCA), and Combined Array for Research in Millimeter-wave Astronomy (CARMA). The resulting radio afterglow catalog is comprised of 60 short GRBs (Table 8). The large majority of events, 53 (88%), were observed with the VLA (Table 8), 28 of which were observed with the upgraded VLA, which has a tenfold improvement in sensitivity (Perley et al. 2011). We present new observations enabled by target-of-opportunity programs with the upgraded VLA (PI: Berger) and CARMA (PI: Zauderer) for 25 bursts (Table 8). In 11 cases, we obtained multiple sets of observations to probe the radio emission on timescales spanning δt ≈ 1–10 days.

For data calibration and analysis of VLA observations, we follow standard procedures in the Astronomical Image Processing System (AIPS; Greisen 2003). For CARMA observations, we use the Multichannel Image Reconstruction, Image Analysis and Display (MIRIAD) software package (Sault et al. 1995). For the majority of cases, we do not find any uncataloged radio sources in or around the X-ray or γ-ray positions. To calculate the 3σ upper limits on the radio afterglow brightness, we measured the rms noise in the map using a large source-free central region utilizing AIPS/IMSTAT for VLA data and MIRIAD/IMSTAT for CARMA data. The radio afterglow detections and upper limits for 60 short GRBs with radio observations are listed in Table 8 and displayed in Figure 1. Flux densities reported here supercede those reported in GCN Circulars.

Four bursts have detected radio afterglows, and all discoveries were made with the VLA (GRB 050724A: Berger et al. 2005; Panaitescu 2006; GRB 051221A: Soderberg et al. 2006; GRB 130603B: Fong et al. 2014; GRB 140903A: this work). In particular, we present the discovery of the radio afterglow of GRB 140903A, which is detected at three frequencies: 1.4, 6.0, and 9.8 GHz (Figure 1 and Table 8), and will be further discussed in an upcoming work.

Thanks to Swift, X-ray and optical follow-up typically commences within ≈60 s after the GRB is detected. While Swift/XRT is responsible for nearly all of the X-ray afterglow detections, only three short GRBs have detected optical afterglows with Swift/UVOT (Table 7). Thus, the detection of short GRB afterglows, or placement of meaningful upper limits, in the optical and radio bands relies on ground-based target-of-opportunity programs. For the 30 short GRBs with detected optical afterglows, the median optical afterglow brightness is ≈23.0 mag at δt ≈ 7.0 hr. The median 3σ limit placed on bursts with an X-ray afterglow and no detected optical afterglow is ≳23.8 mag with a median response time of δt ≈ 7.4 hr, while the limit placed on bursts with no detected X-ray or optical afterglow is less constraining and more delayed, ≳22.7 mag at δt ≈ 12.2 hr (Figure 1). This is likely due to the more limited number of facilities with instruments that can cover the comparatively larger γ-ray positional error circles. However, since we only include bursts with optical limits of ≳20 mag, we are excluding a fraction of bursts with very shallow follow-up; thus the limits here for bursts with only γ-ray localizations are an optimistic representation of the entire population. In the radio band, the median 3σ upper limit for all observations is ≲74.1 μJy, and the median response time to the first observation is δt ≈ 24.7 hr. In more recent cases, we set unprecedented 3σ limits of 15–20 μJy using the upgraded VLA on timescales of δt ≈ 1–10 days.

3.1.1. X-Rays

To investigate the temporal behavior of the X-ray afterglows, we utilize χ²-minimization to fit a single power law model to each light curve in the form ${F}_{\nu ,{\rm{X}}}\propto {t}^{{\alpha }_{{\rm{X}}}},$ with temporal index α_X as the single free parameter and the best-fit flux normalization C₀ given by

$$\begin{eqnarray}&&{C}_{0}=\displaystyle \frac{{\displaystyle \sum }_{i=1}^{N}\displaystyle \frac{{F}_{{\rm{model}},i}\times {F}_{{\rm{X}},i}}{{\sigma }_{{\rm{X}},i}^{2}}}{{\displaystyle \sum }_{i=1}^{N}\displaystyle \frac{{F}_{{\rm{model}},i}^{2}}{{\sigma }_{{\rm{X}},i}^{2}}},\end{eqnarray} \tag{ 1 }$$

where F_model,i are the un-normalized model fluxes, F_X,i and σ_X,i are the observed fluxes and uncertainties, respectively, and N is the number of data points. Since early-time X-ray afterglow light curves are often subject to steep decays, plateaus, or flares which may contaminate the afterglow emission (Nousek et al. 2006; Zhang et al. 2006; Margutti et al. 2011, 2013), we only utilize X-ray data at δt ≳ 1000 s, when bursts have typically settled into the power-law afterglow phase. For the X-ray light curves, we initially include all of the available data at δt ≳ 1000 s in the fit. In a few cases, there are light curve features beyond δt ≈ 1000 s which significantly affect the fit: flares (GRBs 050724A and 111121A), plateaus (GRB 051221A), or steepenings (GRBs 051221A and 111020A). For these bursts, we exclude the time intervals that contain such features in the fits. In 10 cases, there only exists a single detection and an upper limit at δt ≳ 1000 s, so we can only extract an upper limit for α_X. The resulting best-fit values for α_X, along with 1σ uncertainties, are listed in Table 2. Also listed are the X-ray spectral indices, β_X, from the relation ${\beta }_{{\rm{X}}}\equiv 1-{\rm{\Gamma }}.$ The X-ray afterglows have weighted mean values and 1σ uncertainties of $\langle {\alpha }_{{\rm{X}}}\rangle =-{1.07}_{-0.37}^{+0.31}$ and $\langle {\beta }_{{\rm{X}}}\rangle =-{0.89}_{-0.78}^{+0.38},$ and median values of α_X,med ≈ −1.16 and β_X,med ≈ −1.06.

Table 2.  Short GRB Spectral and Temporal Power-law Indices

GRB	αX	βX	αopt	βopt
050509B	−1.10 ± 0.25	−0.88 ± 0.34	⋯	⋯
050709	−1.23 ± 0.10	−1.24 ± 0.35	−1.42 ± 0.08	⋯
050724A	−0.93 ± 0.08	−0.81 ± 0.15	−1.74 ± 0.11a	−0.82 ± 0.03a
050813	<−0.004	$-{1.3}_{-1.3}^{+2.1}$	⋯	⋯
051210	⋯	−2.1 ± 0.5	⋯	⋯
051221A	−1.08 ± 0.12	−1.0 ± 0.2	−0.97 ± 0.06	⋯
060121	−1.23 ± 0.20	−1.07 ± 0.16	−0.60 ± 0.24	⋯
060313	−1.47 ± 0.39	−0.96 ± 0.09	−0.70 ± 0.19	−1.35 ± 0.19
060502B	⋯	$-{2.07}_{-0.54}^{+1.50}$	⋯	⋯
060801	⋯	−0.68 ± 0.12	⋯	⋯
061006	−0.85 ± 0.30	−0.90 ± 0.25	−0.43 ± 0.08	⋯
061201	−1.96 ± 1.18	−0.66 ± 0.12	⋯	⋯
061210	−1.71 ± 1.15	$-{1.86}_{-0.61}^{+1.26}$	⋯	⋯
070429B	<−1.12	−2.0 ± 0.63	⋯	⋯
070707	<−0.65	⋯	−2.55 ± 0.22	⋯
070714B	−1.96 ± 0.69	−1.07 ± 0.19	−0.81 ± 0.11	⋯
070724A	−0.95 ± 0.33	−0.60 ± 0.25	<−0.15	−0.58 ± 0.02
070729	⋯	$-{0.5}_{-0.25}^{+0.44}$	⋯	⋯
070809	−1.09 ± 1.10	$-{0.37}_{-0.07}^{+0.13}$	−0.73 ± 0.33	⋯
071227	−0.97 ± 0.27	−0.90 ± 0.31	<−0.24	⋯
080123	−0.77 ± 0.19	−1.6 ± 0.4	⋯	⋯
080426	−1.54 ± 0.33	−1.03 ± 0.16	⋯	⋯
080503	⋯	$-{1.53}_{-0.12}^{+0.26}$	⋯	⋯
080702A	<−0.38	−1.00 ± 0.43	⋯	⋯
080905A	⋯	−0.53 ± 0.18	−0.39 ± 1.36	⋯
080919	−1.23 ± 0.69	$-{1.9}_{-0.38}^{+0.63}$	⋯	⋯
081024A	⋯	−0.70 ± 0.38	⋯	⋯
081226A	⋯	$-{2.27}_{-0.32}^{+1.24}$	−0.95 ± 0.30	−1.67 ± 0.67b
090305A	⋯	⋯	−0.74 ± 0.09	−0.71 ± 0.27
090426A	−1.15 ± 0.16	−1.04 ± 0.09	−0.58 ± 0.16	−0.94 ± 0.06b
090510	−1.79 ± 0.63	−0.75 ± 0.08	−2.37 ± 0.29	−0.85 ± 0.05
090515	⋯	$-{1.53}_{-0.28}^{+0.78}$	<−0.12	⋯
090607	<−0.72	−1.2 ± 0.4	⋯	⋯
090621B	−1.48 ± 0.54	$-{2.7}_{-1.0}^{+0.68}$	⋯	⋯
091109B	−0.83 ± 0.28	−1.1 ± 0.3	−0.49 ± 0.45	⋯
100117A	⋯	−1.6 ± 0.3	−1.60 ± 0.33	⋯
100206A	⋯	$-{2.05}_{-0.53}^{+1.07}$	⋯	⋯
100213	⋯	$-{2.0}_{-1.0}^{+1.2}$	⋯	⋯
100625A	<0.37	−1.5 ± 0.2	⋯	⋯
100702A	⋯	−1.7 ± 0.3	⋯	⋯
101219A	−1.37 ± 0.13	−0.8 ± 0.1	⋯	⋯
101224A	⋯	$-{2.4}_{-0.6}^{+1.8}$	⋯	⋯
110112A	−1.10 ± 0.05	−1.2 ± 0.2	<−0.32	⋯
111020A	−0.78 ± 0.05	−1.04 ± 0.16	⋯	⋯
111117A	−1.21 ± 0.05	−1.0 ± 0.2	⋯	⋯
111121A	−1.55 ± 0.30	−0.90 ± 0.12	⋯	⋯
111222A	<−0.30	$-{0.7}_{-0.4}^{+1.4}$	⋯	⋯
120305A	⋯	$-{2.2}_{-0.4}^{+0.6}$	⋯	⋯
120521A	⋯	−0.8 ± 0.25	⋯	⋯
120630A	⋯	−0.8 ± 0.3	⋯	⋯
120804A	−1.02 ± 0.10	−1.1 ± 0.1	⋯	⋯
121226A	−1.12 ± 0.28	−1.5 ± 0.25	⋯	⋯
130313A	⋯	$-{1.6}_{-2.5}^{+2.1}$	⋯	⋯
130515A	⋯	−0.7 ± 0.31	⋯	⋯
130603B	−1.88 ± 0.15	−1.2 ± 0.1	−1.26 ± 0.05	−2.0 ± 0.1b
130716A	<−0.52	$-{1.1}_{-0.38}^{+0.56}$	⋯	⋯
130822A	⋯	$-{0.89}_{-0.34}^{+1.1}$	⋯	⋯
130912A	−1.33 ± 0.18	−0.57 ± 0.13	<−0.42	⋯
131004A	−1.1 ± 0.61	−0.8 ± 0.3	−1.94 ± 0.25	−1.52 ± 0.13b
140129B	−1.6 ± 0.26	−0.97 ± 0.11	−1.54c	⋯
140320A	<−0.23	$-{3.8}_{-0.88}^{+2.6}$	⋯	⋯
140516A	−0.42 ± 0.14	−0.70 ± 0.44	⋯	⋯
140622A	<−0.16	$-{0.55}_{-0.18}^{+0.42}$	⋯	⋯
140903A	−1.05 ± 0.20	−0.60 ± 0.10	⋯	⋯
140930B	−1.27 ± 0.15	−0.73 ± 0.28	⋯	⋯
141212A	⋯	$-{1.7}_{-0.81}^{+2.75}$	⋯	⋯
150101A	<−0.12	−0.40 ± 0.56	⋯	⋯
150101B	−1.07 ± 0.15	−0.67 ± 0.17	−1.01 ± 0.62	⋯
150120A	⋯	−1.0 ± 0.31	⋯	⋯
150301A	⋯	−0.7 ± 0.13	⋯	⋯

Notes. Error bars correspond to 1σ confidence. When applicable, α_X and α_opt represent pre-jet break values. Values of β_opt are observed values and are uncorrected for intrinsic rest-frame extinction, A_V.

^aThese values are computed over the same time interval as the X-ray flare that is superimposed on the underlying afterglow power-law decay. No optical detections exist for the underlying afterglow. ^bThis value corresponds to the spectral behavior after the jet break. ^cNo uncertainties are given for the optical light curve.

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3.1.2. Optical

3.1.3. Rest-frame Extinction

In four cases where there are contemporaneous observations in multiple optical/NIR filters, a single power law model provides a poor fit to the broadband photometry (${\chi }_{\nu }^{2}\gtrsim 3$). For these bursts, we include the line-of-sight rest-frame extinction from the host galaxy (${A}_{V}^{{\rm{host}}}$) as a second free parameter in the fit. We constrain the extinction using the Milky Way extinction curve (Cardelli et al. 1989), and employ the burst redshifts (Table 1) to obtain the rest-frame extinction. For bursts with no determined spectroscopic redshift, we assume z = 0.5 set by the median of the short GRB population (Berger 2014). We find non-zero extinction values for GRBs 060121, 070724A, 081226A, and 130603B (Table 3). The resulting values for the optical spectral indices, uncertainties, and ${A}_{V}^{{\rm{host}}}$ are listed in Table 3.

Table 3.  Inferred Properties

GRB	${A}_{V}^{{\rm{host}}}$	νc < νX ?	p	B	Eγ,iso,52	$\langle {E}_{{\rm{K,iso,52}}}\rangle$	ηγ	$\langle {\text{}}n\rangle$
	(mag)				(1052 erg)	(1052 erg)		(cm−3)
050709	0	Y	2.31 ± 0.13	0.1	0.09	${2.6}_{-0.5}^{+0.8}\times {10}^{-3}$	0.97	${1.0}_{-0.4}^{+0.5}$
	0	Y	2.31 ± 0.13	0.01	0.09	${6.2}_{-0.6}^{+0.4}\times {10}^{-3}$	0.93	${1.6}_{-0.2}^{+0.2}$
050724A	0	N	2.29 ± 0.10	10−4	0.24	${0.18}_{-0.05}^{+0.08}$	0.58	${0.89}_{-0.49}^{+0.58}$
051221A	0	Y	2.24 ± 0.07	0.1	1.3	${0.16}_{-0.01}^{+0.01}$	0.89	${0.03}_{-0.005}^{+0.006}$
	0	Y	2.24 ± 0.07	0.01	1.3	${0.27}_{-0.03}^{+0.03}$	0.83	${0.14}_{-0.04}^{+0.05}$
060121	1.6	Y	2.24 ± 0.20	0.1	4.5	${0.20}_{-0.04}^{+0.05}$	0.96	${5.4}_{-2.2}^{+5.6}\times {10}^{-3}$
	1.6	Y	2.24 ± 0.20	0.01	4.5	${0.23}_{-0.04}^{+0.05}$	0.95	${0.16}_{-0.06}^{+0.17}$
060313a,b	0	Y	2.03 ± 0.20	0.1	2.9	${0.45}_{-0.05}^{+0.05}$	0.87	${3.3}_{-0.5}^{+1.0}\times {10}^{-3}$
061006	0	N	2.39 ± 0.31	0.1	1.1	${0.64}_{-0.37}^{+0.85}$	0.63	${2.2}_{-1.9}^{+16}\times {10}^{-5}$
	0	N	2.39 ± 0.31	0.01	1.1	${1.1}_{-0.7}^{+2.1}$	0.50	${1.2}_{-1.1}^{+29}\times {10}^{-4}$
061201	0	N	2.35 ± 0.24	0.1	0.05	${0.05}_{-0.03}^{+0.10}$	0.47	${5.0}_{-4.6}^{+66}\times {10}^{-5}$
	0	N	2.35 ± 0.24	0.01	0.05	${0.1}_{-0.09}^{+0.4}$	0.29	${2.7}_{-2.6}^{+120}\times {10}^{-4}$
070714Bb,c	0.5	Y	2.30 ± 0.35	0.1	1.7	${0.1}_{-0.02}^{+0.02}$	0.94	${0.056}_{-0.011}^{+0.024}$
070724A	1.5	N	2.24 ± 0.33	0.1	0.03	${0.35}_{-0.20}^{+0.49}$	0.07	${1.9}_{-1.6}^{+12}\times {10}^{-5}$
	2.0	N	2.24 ± 0.33	0.01	0.03	${1.1}_{-0.8}^{+3.0}$	0.02	${9.3}_{-9.2}^{+210}\times {10}^{-5}$
070809	0	N	2.12 ± 1.47d	0.1	0.09	${0.5}_{-0.3}^{+0.7}$	0.14	${2.2}_{-1.9}^{+15}\times {10}^{-5}$
	0	N	2.12 ± 1.47d	0.01	0.09	${1.1}_{-0.8}^{+2.7}$	0.07	${1.2}_{-1.1}^{+30}\times {10}^{-4}$
071227	0	Y	1.92 ± 0.31	0.1	0.14	${8.4}_{-1.9}^{+2.5}\times {10}^{-3}$	0.94	${1.9}_{-1.1}^{+2.4}$
	0	Y	1.92 ± 0.31	0.01	0.14	${8.9}_{-1.8}^{+2.6}\times {10}^{-3}$	0.94	${60}_{-33}^{+75}$
080426	0	Y	2.29 ± 0.26	0.1	0.35	${0.05}_{-0.01}^{+0.01}$	0.87	${0.04}_{-0.02}^{+0.04}$
	0	Y	2.29 ± 0.26	0.01	0.35	${0.06}_{-0.01}^{+0.01}$	0.85	${1.2}_{-0.5}^{+1.2}$
080905A	0	N	2.06 ± 0.36	0.1	0.02	${0.04}_{-0.03}^{+0.12}$	0.34	${1.3}_{-1.2}^{+33}\times {10}^{-4}$
	0	N	2.06 ± 0.36	0.01	0.02	${0.08}_{-0.07}^{+0.44}$	0.21	${7.1}_{-7.1}^{+610}\times {10}^{-4}$
080919	≳6	Y	2.97 ± 0.68	0.1	0.07	${0.019}_{-0.004}^{+0.005}$	0.78	${0.20}_{-0.07}^{+0.19}$
	≳6	Y	2.97 ± 0.68	0.01	0.07	${0.029}_{-0.006}^{+0.007}$	0.70	${5.1}_{-1.7}^{+4.5}$
081024A	0	N	2.40 ± 0.76	0.1	0.11	${0.11}_{-0.07}^{+0.22}$	0.51	${8.1}_{-7.7}^{+150}\times {10}^{-5}$
	0	N	2.40 ± 0.76	0.01	0.11	${0.25}_{-0.20}^{+0.92}$	0.31	${4.3}_{-4.2}^{260}\times {10}^{-4}$
081226A	1.0	N	2.27 ± 0.39	0.1	0.09	${0.20}_{-0.13}^{+0.36}$	0.32	${3.2}_{-2.9}^{+29}\times {10}^{-5}$
	1.0	N	2.27 ± 0.39	0.01	0.09	${0.43}_{-0.33}^{+1.5}$	0.18	${1.7}_{-1.6}^{+54}\times {10}^{-4}$
090426A	0	Y	2.13 ± 0.14	0.1	2.0	${1.4}_{-0.3}^{+0.4}$	0.59	${0.04}_{-0.02}^{+0.04}$
	0	Y	2.13 ± 0.14	0.01	2.0	${1.5}_{-0.3}^{+0.4}$	0.57	${1.2}_{-0.6}^{+1.4}$
090510	0	N	2.65 ± 0.08	0.1	0.77	${0.77}_{-0.33}^{+0.57}$	0.50	${1.2}_{-1.0}^{+5.5}\times {10}^{-5}$
	0	N	2.65 ± 0.08	0.01	0.77	${1.9}_{-1.2}^{+3.0}$	0.29	${6.4}_{-6.0}^{+100}\times {10}^{-5}$
090607	0	Y	2.40 ± 0.76	0.1	0.10	${2.1}_{-0.3}^{+0.4}\times {10}^{-3}$	0.98	${0.84}_{-0.22}^{+0.52}$
	0	Y	2.40 ± 0.76	0.01	0.10	${2.6}_{-0.4}^{+0.5}\times {10}^{-3}$	0.98	${24}_{-6.3}^{+15}$
090621Bc	2.5	Y	2.64 ± 0.72	0.1	0.07	${0.023}_{-0.006}^{+0.007}$	0.75	${0.05}_{-0.02}^{+0.06}$
	2.5	Y	2.64 ± 0.72	0.01	0.07	${0.031}_{-0.007}^{+0.009}$	0.68	${1.0}_{-0.27}^{+0.52}$
091109B	0	N	2.40 ± 0.32	0.1	0.18	${0.25}_{-0.16}^{+0.18}$	0.42	${2.8}_{-2.5}^{+24}\times {10}^{-5}$
	0	N	2.40 ± 0.32	0.01	0.18	${0.8}_{-0.6}^{+2.2}$	0.18	${9.4}_{-9.0}^{200}\times {10}^{-5}$
100117A	0	Y	2.36 ± 0.30	0.1	0.22	${0.019}_{-0.003}^{+0.003}$	0.92	${0.04}_{-0.01}^{+0.03}$
	0	Y	2.36 ± 0.30	0.01	0.22	${0.023}_{-0.004}^{+0.004}$	0.90	${1.2}_{-0.3}^{+0.9}$
101219A	≳2.5	N	2.73 ± 0.13	0.1	0.74	${0.3}_{-0.2}^{+0.5}$	0.68	${4.6}_{-4.3}^{+59}\times {10}^{-5}$
	≳2.5	N	2.73 ± 0.13	0.01	0.74	${0.87}_{-0.64}^{+2.3}$	0.46	${2.4}_{-2.3}^{+97}\times {10}^{-4}$
110112Ab	0	Y	2.49 ± 0.07	0.1	0.03	${0.064}_{-0.005}^{+0.008}$	0.31	${2.4}_{-0.4}^{+0.4}\times {10}^{-2}$
111020Aa b c	0.5	Y	2.08 ± 0.32	0.1	0.17	${0.48}_{-0.08}^{+0.09}$	0.26	${4.5}_{-3.8}^{+6.0}\times {10}^{-3}$
111117A	0	Y	2.27 ± 0.07	0.1	0.55	${0.06}_{-0.01}^{+0.01}$	0.90	${8.3}_{-2.3}^{+5.9}\times {10}^{-3}$
	0	Y	2.27 ± 0.07	0.01	0.55	${0.07}_{-0.01}^{+0.02}$	0.89	${0.25}_{-0.07}^{+0.16}$
111121A	N/A	N	2.87 ± 0.21	0.1	2.1	${3.1}_{-1.2}^{+1.9}$	0.40	${8.2}_{-6.3}^{+27}\times {10}^{-6}$
	N/A	N	2.87 ± 0.21	0.01	2.1	${8.3}_{-4.8}^{+11.1}$	0.20	${4.2}_{-3.8}^{+48}\times {10}^{-5}$
120804Ac	2.5	Y	2.08 ± 0.11	0.1	3.4	${1.1}_{-0.2}^{+0.3}$	0.76	${3.2}_{-1.5}^{+3.1}\times {10}^{-3}$
	2.5	Y	2.08 ± 0.11	0.01	3.4	${2.3}_{-0.2}^{+0.2}$	0.60	${0.014}_{-0.001}^{+0.002}$
121226Aa b	1	Y	2.50 ± 0.37	0.1	0.37	${0.6}_{-0.09}^{+0.08}$	0.37	${4.0}_{-0.6}^{+1.0}\times {10}^{-3}$
130603B	1.2	Y	2.70 ± 0.06	0.1	0.37	${0.11}_{-0.01}^{+0.02}$	0.77	${0.09}_{-0.03}^{+0.04}$
	0.3	Y	2.70 ± 0.06	0.01	0.37	${0.15}_{-0.02}^{+0.02}$	0.72	${0.31}_{-0.04}^{+0.08}$
130912Ac	1.3	N	2.49 ± 0.17	0.1	0.16	${0.50}_{-0.20}^{+0.37}$	0.25	${1.7}_{-1.5}^{+9.9}\times {10}^{-5}$
	1.3	N	2.49 ± 0.17	0.01	0.16	${1.4}_{-0.8}^{+0.2}$	0.11	${5.2}_{-4.9}^{+7.6}\times {10}^{-5}$
131004A	0	N	2.57 ± 0.48	0.1	0.45	${1.2}_{-0.36}^{+0.56}$	0.28	${1.2}_{-1.0}^{+5.7}\times {10}^{-5}$
	0	N	2.57 ± 0.48	0.01	0.45	${2.8}_{-1.5}^{+3.4}$	0.45	${6.5}_{-6.2}^{+11}\times {10}^{-4}$
140129B	0	N	3.00 ± 0.19	0.1	0.07	${0.98}_{-0.53}^{+1.14}$	0.06	${3.1}_{-2.8}^{+29.4}\times {10}^{-5}$
	0	N	3.00 ± 0.19	0.01	0.07	${3.8}_{-2.9}^{+12.0}$	0.02	${1.6}_{-1.5}^{+47}\times {10}^{-4}$
140516A	0	N	2.40 ± 0.88	0.1	0.02	${0.02}_{-0.01}^{+0.04}$	0.54	${1.0}_{-0.99}^{+24}\times {10}^{-4}$
	0	N	2.40 ± 0.88	0.01	0.02	${0.04}_{-0.03}^{+0.18}$	0.34	${5.5}_{-5.4}^{+41}\times {10}^{-4}$
140622A	0	N	2.10 ± 0.60	0.1	0.07	${0.12}_{-0.08}^{+0.26}$	0.36	${5.8}_{-5.4}^{+8.8}\times {10}^{-5}$
	0	N	2.10 ± 0.60	0.01	0.07	${0.25}_{-0.20}^{+0.97}$	0.21	${3.2}_{-3.2}^{+16.2}\times {10}^{-4}$
140903A	0	N	2.27 ± 0.16	10−3	0.08	${2.9}_{-0.74}^{+0.92}$	0.03	${3.4}_{-1.6}^{+2.9}\times {10}^{-3}$
140930B	0	N	2.67 ± 0.19	0.1	0.40	${0.28}_{-0.17}^{+0.42}$	0.58	${4.6}_{-4.3}^{+58}\times {10}^{-5}$
	0	N	2.67 ± 0.19	0.01	0.40	${1.8}_{-0.9}^{+1.7}$	0.18	${1.8}_{-1.5}^{11}\times {10}^{-5}$
150101Bc	0.5	N	2.40 ± 0.17	0.1	4.0 $\;\times \;{10}^{-3}$	${0.61}_{-0.25}^{+0.45}$	6.6 $\;\times \;{10}^{-3}$	${8.0}_{-6.1}^{+24}\times {10}^{-6}$
	0.5	N	2.40 ± 0.17	0.1	4.0 $\;\times \;{10}^{-3}$	${3.0}_{-1.1}^{+1.7}$	1.3 $\;\times \;{10}^{-3}$	${5.3}_{-3.6}^{+11}\times {10}^{-6}$

Notes. Quoted uncertainties are 1σ. All solutions presented here are for a fixed _e = 0.1 and assume a lower density bound of n_min = 10⁻⁶ cm⁻³.

^aWe assume a redshift of z = 1 for this burst. ^bNo valid solution is found for _B = 0.01. ^cValue of ${A}_{V}^{{\rm{host}}}$ is determined from a comparison of the optical and X-ray bands, and not directly from the optical/NIR SED. ^dDetermined from α_X alone.

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For bursts where we do not have enough information from the optical/NIR bands alone to constrain the spectral behavior of the afterglow, we initially assume that there is no rest-frame extinction, ${A}_{V}^{{\rm{host}}}=0.$ However, in eight cases, the difference in slope between the X-ray and optical bands is shallower than expected (e.g., $| {\beta }_{\mathrm{OX}}| \lesssim | {\beta }_{{\rm{X}}}| -0.5,$ where β_OX is the spectral index between the X-ray and optical bands), suggesting that either there is intrinsic extinction, or that the X-rays do not originate from the forward shock. Assuming the former explanation, we include the minimum amount of extinction required until the X-ray and optical solutions agree to within the 1σ uncertainties. In two cases, GRBs 080919 and 101219A, there is only an upper limit on the optical afterglow brightness; thus we can only determine a lower limit on the rest-frame extinction. We note that GRB 080919 has the highest value of rest-frame extinction, with ${A}_{V}^{{\rm{host}}}\gtrsim 6$ mag. However, this burst has a sightline close to the Galactic plane and therefore has a highly uncertain Galactic extinction, which likely affects the inferred value for ${A}_{V}^{{\rm{host}}}.$ For the 12 events with rest-frame extinction, we find values of ${A}_{V}^{{\rm{host}}}$ ≈ 0.3–1.5 mag. These ${A}_{V}^{{\rm{host}}}$ values are also listed in Table 3.

To determine the electron power law index p, we use α_X and β_X to constrain the location of the cooling frequency, ν_c, with respect to the X-ray band. We use the relations given by Granot & Sari (2002) which relate α_X and β_X to the value of p for the scenarios ${\nu }_{{\rm{m}}}\lt {\nu }_{{\rm{X}}}\lt {\nu }_{{\rm{c}}}$ and ν_c < ν_X by

$$\begin{eqnarray}&&p=\left\{\begin{array}{c}1-2{\beta }_{{\rm{X}}}\\ 1-\displaystyle \frac{4}{3}{\alpha }_{{\rm{X}}}\end{array}\right\}\qquad {\nu }_{{\rm{m}}}\lt {\nu }_{{\rm{X}}}\lt {\nu }_{{\rm{c}}},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&p=\left\{\begin{array}{c}-2{\beta }_{{\rm{X}}}\\ \displaystyle \frac{2-4{\alpha }_{{\rm{X}}}}{3}\end{array}\right\}\qquad {\nu }_{{\rm{c}}}\lt {\nu }_{{\rm{X}}}.\end{eqnarray} \tag{ 3 }$$

For a given burst, we calculate the values of p and 1σ uncertainties using Equations (2) and (3) and standard propagation of errors. We select the valid scenario under the condition that the values of p independently determined from α_X and β_X for a given scenario agree within the 1σ uncertainties. Following this condition, we can constrain the location of ν_c with respect to the X-ray band for 38 bursts (Table 3). Using Equations (2) and (3), we calculate the weighted mean for the value of p in the valid scenario; the resulting values and uncertainties are listed in Table 3 and displayed in Figure 2. We note that in the case of GRB 071227, the condition is satisfied for ν_c < ν_X, but has a median of p = 1.92 ± 0.31, which yields a divergent total integrated energy. Thus for this burst, we employ p = 2.05 in our subsequent analysis (Table 3).

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Under the reasonable assumption that the optical band lies between the peak frequency, ν_m, and the cooling frequency, ν_c (i.e., ν_m < ν_opt < ν_c), we use the available values for α_opt and β_opt and Equation (2) to independently determine the value of p. We then include this in our weighted average of p for each burst (Figure 2). The weighted mean for the sample of 38 bursts is $\left\langle p\right\rangle ={2.43}_{-0.28}^{+0.36}$ (1σ; Figure 2).

For the remaining bursts in the afterglow catalog, there is not enough information to determine the location of the cooling frequency with respect to the X-ray band, and therefore the correct value of p. We thus concentrate on the subset of 38 bursts with determined values of p for our subsequent analysis. We find ν_c < ν_X for 18 cases, while ν_c > ν_X for 20 cases.

In the standard synchrotron model from Granot & Sari (2002), the dependencies on the isotropic-equivalent kinetic energy, circumburst density, and the microphysical parameters are as follows for a given flux density, F_ν,i and observing band, ν_i:

$$\begin{eqnarray}{F}_{\nu ,i}\propto \left\{\begin{array}{ll}{n}_{0}^{1/2}{E}_{{\rm{K,iso,52}}}^{5/6}{\epsilon }_{e,-1}^{-2/3}{\epsilon }_{B,-1}^{1/3} & {\nu }_{a}\lt {\nu }_{i}\lt {\nu }_{{\rm{m}}}\\ {n}_{0}^{1/2}{E}_{{\rm{K,iso,52}}}^{\frac{3+p}{4}}{\epsilon }_{e,-1}^{p-1}{\epsilon }_{B,-1}^{\frac{1+p}{4}} & {\nu }_{{\rm{m}}}\lt {\nu }_{i}\lt {\nu }_{{\rm{c}}}\\ {E}_{{\rm{K,iso,52}}}^{\frac{2+p}{4}}{\epsilon }_{e,-1}^{p-1}{\epsilon }_{B,-1}^{\frac{p-2}{4}} & {\nu }_{i}\gt {\nu }_{{\rm{c}}}\end{array}\right.,\end{eqnarray} \tag{ 4 }$$

where n₀ is in units of cm⁻³, E_K,iso,52 is in units of 10⁵² erg, and _e,−1 and _B,−1 are in units of 0.1. In addition to these four parameters, F_ν,i is also dependent on the redshift, luminosity distance, ν_i, time after the burst, δt, and the value of p; the exact dependencies are provided in Granot & Sari (2002). We note that for ν_i > ν_c, the flux density is independent of circumburst density. For bursts with no spectroscopic redshift, we assume z = 0.5. In all cases, we cannot independently constrain _e and _B since this requires knowledge of the locations of the three break frequencies (ν_a, ν_m, and ν_c), which generally necessitates well-sampled light curves and spectra in multiple bands. Thus, in order to determine ranges for E_K,iso and n, we fix the values of the microphysical parameters. We first consider the fiducial case that _e = 0.1 and _B = 0.1.

For each burst, we determine the constraints on E_K,iso and n by computing individual probability distributions for each observation. We then assign the probabilities to a grid of values, and use a joint probability analysis to calculate the distributions in each parameter. For the grid, the ranges of the density and isotropic-equivalent kinetic energy are n_i = 10⁻⁶–10³ cm⁻³ and E_i = 10⁴⁶–10⁵⁴ erg, with 1000 logarithmically, uniformly spaced steps in each parameter. We choose the lower bound of the density range, n_min = 10⁻⁶ cm⁻³ to match the typical density of the intergalactic medium (IGM).

3.3.1. Detections

To calculate the individual probability distributions for afterglow detections, we apply Equation (4) to the observations using the relevant regime for each observing band, ν_i. For the X-ray band, we use the location of ν_c as determined in Section 3.1.1 to determine which branch of Equation (4) to use. Since the value of p is primarily determined from the X-ray band, the ${E}_{{\rm{K,iso}}}\mbox{--}n$ relation remains unchanged when using different X-ray observations that follow the same temporal decline; thus, we only use one X-ray observation per burst. For the optical band, we make the reasonable assumption that ${\nu }_{{\rm{m}}}\lt {\nu }_{{\rm{opt}}}\lt {\nu }_{{\rm{c}}}$ and utilize the second branch of Equation (4). In some cases, individual optical/NIR observations lead to ${E}_{{\rm{K,}}\;{\rm{iso}}}\mbox{--}n$ relations that do not overlap within their 1σ uncertainties. In these cases, we use the weighted mean and standard deviation of these relations (i.e., systematic uncertainty) as the optical/NIR solution. For the radio band, we assume that ${\nu }_{a}\lt {\nu }_{{\rm{radio}}}\lt {\nu }_{{\rm{m}}},$ (first branch of Equation (4)) to calculate the ${E}_{{\rm{K,iso}}}\mbox{--}n$ relations. Since the radio band lies on a different segment of the synchrotron spectrum than the other bands, it contributes a ${E}_{{\rm{K,iso}}}\mbox{--}n$ relation with a different slope, and thus enables tighter constraints on the physical parameters (Figure 3).

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After calculating the unique probability distribution from each of the radio, optical/NIR and X-ray bands, we normalize the area under each of the distributions to unity. We assume that the uncertainties in the flux densities are Gaussian, and thus each band contributes a unique, log-normal distribution. These distributions are shown for the four bursts with radio afterglow detections (Figure 3), and 34 bursts with radio afterglow non-detections: 16 bursts with ν_c < ν_X (Figure 4) and 18 bursts with ν_c > ν_X (Figure 5).

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3.3.2. Upper Limits

3.3.3. Cooling Frequency Constraint

We utilize the relative location of the cooling frequency as a final constraint, since it depends on a combination of energy and density, by

$$\begin{eqnarray}&&{\nu }_{{\rm{c}}}\propto {n}_{0}^{-1}{E}_{{\rm{K,iso,52}}}^{-1/2}{\epsilon }_{B,-1}^{-3/2},\end{eqnarray} \tag{ 5 }$$

with additional dependencies on δt and redshift (Granot & Sari 2002). For the cases in which the X-ray band is located above the cooling frequency (ν_c < ν_X), we employ a maximum value at the lower edge of the X-ray band, ν_c,max = 7.3 × 10¹⁶ Hz (0.3 keV), to obtain a lower limit on the combination of energy and density. The corresponding probability distribution has zero value in the ${E}_{{\rm{K,iso}}}\mbox{--}n$ parameter space below the relation, and a constant value above the relation, where the area in the allowed parameter space is normalized to unity. The lower limits are shown as as blue dotted–dashed lines for GRBs 051221A and 130603B in Figure 3 and for all bursts in Figure 4. In cases where the X-ray band is below the cooling frequency (${\nu }_{{\rm{m}}}\lt {\nu }_{{\rm{X}}}\lt {\nu }_{{\rm{c}}}$), we set the cooling break to a minimum value, ν_c,min = 2.4 × 10¹⁸ Hz (10 keV) at the upper edge of the X-ray band, and determine the ${E}_{{\rm{K,iso}}}\mbox{--}n$ relation for each burst using Equation (5). This constraint sets an upper limit on the combination of energy and density. We form the probability distribution in the same manner as for afterglow upper limits. The limits for each burst set by the cooling frequency are shown as blue dashed lines for GRBs 050724A and 140903A in Figure 3 and all bursts in Figure 5.

Since each of the observing bands, as well as the location of the cooling frequency, contribute an independent probability distribution, we calculate the joint probability from a product of these distributions for each burst. To obtain one-dimensional probability distributions in E_K,iso and n, we integrate over each of the parameters. Finally, we normalize the area under each one-dimensional distribution to unity. The resulting distributions, P(n) and P(E_K,iso), for 34 bursts are shown in Figures 4–5 for the fiducial microphysical parameters, _e = _B = 0.1. The median values and 1σ uncertainties in isotropic-equivalent kinetic energy and circumburst density are also shown in these figures and listed in Table 3.

Figure 3 shows the probability distributions for the four events with radio afterglow detections. In three of the four cases, we can use the available data to place additional constraints on _B, fixing _e = 0.1. For GRBs 050724A and 140903A, the afterglow data require that _B ≲ 10⁻⁴ and ≲10⁻³, respectively. For larger values of _B, the constraint from the cooling frequency becomes more stringent and conflicts with the solutions obtained from the afterglow observations on the grid of allowed values. For GRB 051221A, we find a significantly better fit at _B = 0.01. In all cases, the addition of the radio band enables tighter constraints on both the energy and the density.

We compare the inferred isotropic-equivalent kinetic energy to the γ-ray energy by computing the isotropic-equivalent γ-ray energy, E_γ,iso, to represent the energy range ≈1–10⁴ keV (to match the widest energy ranges for current GRB detection satellites),

$$\begin{eqnarray}&&{E}_{\gamma ,{\rm{iso}}}={k}_{{\rm{bol}}}\times \displaystyle \frac{4\pi {d}_{L}^{2}}{1+z}{f}_{\gamma }\;{\rm{erg}}\end{eqnarray} \tag{ 6 }$$

where k_bol is the bolometric correction factor to convert the fluence to an energy range of ≈1–10⁴ keV, d_L is the luminosity distance in cm, and f_γ is the fluence in units of erg cm⁻². For cases in which the fluence is calculated over the 15–150 keV Swift energy range, we use k_bol = 5. If a burst is detected by other γ-ray satellites which cover a wider energy range of ≈10–1000 keV (e.g., Fermi, Konus-Wind, Suzaku), we utilize the measured fluences from these satellites and k_bol = 1 to calculate the γ-ray energy. Individual values of E_γ,iso are shown in Figure 3–5 and listed in Table 3, while the distribution for 38 bursts is shown in Figure 7. We find that the range in isotropic-equivalent γ-ray energy is ≈(0.04–45) × 10⁵¹ erg, with a median value of $\langle {E}_{\gamma ,{\rm{iso}}}\rangle \approx 1.8\times {10}^{51}$ erg, similar to the ranges and median values of the isotropic-equivalent kinetic energy (Figure 7).

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We also calculate the γ-ray efficiency,

$$\begin{eqnarray}&&{\eta }_{\gamma }=\displaystyle \frac{{E}_{\gamma ,{\rm{iso}}}}{{E}_{\gamma ,{\rm{iso}}}+{E}_{K,{\rm{iso}}}},\end{eqnarray} \tag{ 7 }$$

as well as the 1σ uncertainties in η_γ for each burst, following standard propagation of errors from the 1σ uncertainties in E_K,iso. The resulting values of η_γ are listed in Table 3 and the cumulative distributions, after incorporating the 1σ uncertainties, are shown in Figure 7. We find a wide range in η_γ, ≈10⁻³ − 1, and note that the lower bound is set by the single outlier, GRB 150101B (Table 3). Excluding this burst, the lower bound is ≈0.03 (GRB 140622A).

The distribution for _B = 0.1 has a median of $\langle {\eta }_{\gamma }\rangle ={0.56}_{-0.37}^{+0.36}$ (1σ uncertainties). The isotropic-equivalent kinetic energy scale for the _B = 0.01 case is comparatively high (c.f., Figure 6), and thus the median value for the γ-ray efficiency is relatively low, $\langle {\eta }_{\gamma }\rangle ={0.40}_{-0.35}^{+0.49}$ (1σ uncertainties; Figure 7).

We have thus far made assumptions about the values of the microphysical parameters (_e = _B = 0.1), the redshifts (z = 0.5 unless otherwise determined), the minimum allowed density (n_min = 10⁻⁶ cm⁻³), and the cooling frequency (0.3 keV for ν_c < ν_X; 10 keV for ν_c > ν_X). To explore the impact of these assumptions on our resulting distributions for the kinetic energy and circumburst density, we consider alternative values for these parameters.

4.2.1. The Value of _B

4.2.2. Redshift

4.2.3. The Value of n_min

4.2.4. The Value of ν_c

4.2.5. Trends

The X-ray, optical, and radio afterglow catalogs (Tables 6–8) allow us to analyze the observational afterglow properties of short GRBs as a population. By fitting the afterglow light curves, we find a weighted mean pre-jet break decline rate of $\langle {\alpha }_{{\rm{X}}}\rangle \approx -1.07$ at δt ≳ 1000 s for bursts with measured temporal indices, similar to the pre-jet break declines measured from long GRB light curves (Nysewander et al. 2009; Racusin et al. 2009; Kann et al. 2010; Zaninoni et al. 2013), and slightly shallower than the value of α_X ≈ −1.2 found for 11 short GRBs in an earlier study (Nysewander et al. 2009). We measure the optical decline rates and find the same weighted mean decline rate of $\langle {\alpha }_{{\rm{opt}}}\rangle \approx -1.07$ from 19 well-sampled bursts with measured indices.

From spectral fitting of the optical, near-IR, and X-ray data, we find 12 short GRBs which require rest-frame extinction, with measured values of ${A}_{V}^{{\rm{host}}}$ ≈ 0.3–1.5 mag, and two bursts with lower limits of ${A}_{V}^{{\rm{host}}}$ ≳ 2.5–6 mag (Table 3). We note that GRB 080919 has the highest value of rest-frame extinction; however, the sightline to this burst is close to the Galactic plane and therefore has a highly uncertain Galactic extinction which likely affects the measurement of ${A}_{V}^{{\rm{host}}}.$ Prior to this study, evidence for ≳0.5 mag of extinction has only been reported in three short GRBs: GRB 070724A (Berger et al. 2009; Kocevski et al. 2010), GRB 111020A (Fong et al. 2012b), and GRB 120804A (Berger et al. 2013). Afterglow modeling in this work results in the same conclusions for those three events, and includes nine additional events with rest-frame extinction. We note that most of the events with non-zero extinction and robust host associations are in star-forming host galaxies; the single exception is GRB 150101B which has an inferred value of ${A}_{V}^{{\rm{host}}}\approx 0.5$ mag and is located on the outskirts of an early-type galaxy (Fong et al., in prep). In comparison, ≈15%–20% of Swift long GRBs have optically sub-luminous afterglows that have been attributed to dust extinction. For long GRBs, inferred values of ${A}_{V}^{{\rm{host}}}\approx 0.5$ mag are common, with a substantial fraction of events with ${A}_{V}^{{\rm{host}}}$ ≳ 1–2.5 mag (Cenko et al. 2009; Perley et al. 2009a, 2013).

While rest-frame extinction can be explained by dust in star-forming regions in the local environments of long GRBs, substantial extinction in short GRBs cannot be easily explained in the context of the double compact object merger progenitor. It is possible that these events are "impostors" which in actuality have massive star progenitors; in that case, we might expect them to be distinct in their γ-ray properties with longer durations and softer γ-ray spectra. However, there does not appear to be a correlation between short GRBs with extinction and their durations as they span the full range, with T₉₀ ≈ 0.02–2 s, and only three events with non-zero extinction have T₉₀ ≳ 1 s (GRBs 060121, 070714B, and 121226A). Furthermore, a study conducted by Bromberg et al. (2013) assigned six of these events probabilities of having non-collapsar progenitors based on their γ-ray properties. According to this study, 4/6 events have ≳60% probabilities that they do not originate from collapsars. Thus, it is unlikely that the majority of these events are in fact "impostors." Instead, this suggests that some short GRBs may originate in star-forming regions, or have progenitor systems that can produce dust.

Most well-sampled short GRBs exhibit a single afterglow decline rate in the X-ray (at δt ≳ 1000 sec) and optical bands. However, there are four short GRBs (GRBs 051221A, 090426A, 111020A, and 130603B) which have temporal steepenings on timescales of δt ≈ 2–5 days, attributed to jet breaks (Table 5; Soderberg et al. 2006; Nicuesa Guelbenzu et al. 2011; Fong et al. 2012b, 2014). Jet breaks are achromatic features, and in principle can be detected in the X-ray through radio bands. The measurement of jet break time, in conjunction with the energy, density, and redshift, can be converted to a jet opening angle, θ_j, using (Sari et al. 1999; Frail et al. 2001)

$$\begin{eqnarray}&&{\theta }_{j}=9.5\;{t}_{j,{\rm{d}}}^{3/8}{(1+z)}^{-3/8}{E}_{{\rm{K,iso,52}}}^{-1/8}{n}_{0}^{1/8}\ \mathrm{deg},\end{eqnarray} \tag{ 8 }$$

where t_j,d is in days. The opening angle measurements for the four short GRBs with jet break detections are listed in Table 5. Using these four short GRB opening angle measurements alone, taking into account the published range of angles for individual bursts, the median is $\langle {\theta }_{j}\rangle =6^\circ \pm 1^\circ$ (Figure 8).

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However, the majority of short GRBs do not have detected jet breaks and instead exhibit a single power-law decline as long as they are detected. In these cases, the time of the last observation (δt_last) can be used to place lower limits on the opening angles. The inclusion of these bursts is essential in understanding the true opening angle distribution. In most cases, Swift/XRT observes short GRBs until they fade below the detection threshhold at δt ≲ 1 day, and enables relatively shallow lower limits of θ_j ≳ 2°–5° (Coward et al. 2012; Fong et al. 2012b). The inclusion of such limits will not have a significant effect on the opening angle distribution, as they virtually span the entire range of allowable angles.

Therefore, in order to have a more complete understanding of the opening angle distribution, we collect all existing published lower limits of θ_j ≳ 5°, and calculate lower limits for GRBs 101219A, 120804A, 140903A, and 140903B using the observations and physical parameters presented in this work. The inferred lower limits, the band in which the jet break was detected, and the value of δt_last used to compute the limits, are listed in Table 5. These seven events demonstrate that multi-wavelength afterglow observations to δt_last ≈ 3–25 days enable more meaningful lower limits on the opening angles of ≳5°–25° (Table 5).

To calculate the opening angle distribution, we give each of the 11 events equal weighting, where measurements are given Gaussian probability distributions to represent their allowed range of angles (Table 5), while lower limits are given probability that is evenly distributed between the lower limit and the maximum possible opening angle, θ_j,max = 90°. The resulting cumulative distribution for 11 short GRBs, including measurements and lower limits, is shown in Figure 8. Assuming an upper bound of θ_j,max = 90°, the short GRB population median is $\langle {\theta }_{j}\rangle ={33}_{-27}^{+38}$ deg (1σ). Motivated by simulations of post-merger black hole accretion predict jets with θ_j ∼ 5°–30° (Ruffert & Janka 1999b; Rosswog & Ramirez-Ruiz 2003; Aloy et al. 2005; Rosswog 2005; Rezzolla et al. 2011), we also calculate the cumulative distribution employing a more realistic maximum value of θ_j,max = 30°, and find a median of $\langle {\theta }_{j}\rangle =16\pm 10$ deg (1σ).

To compare these distributions to those for long GRBs, we collect opening angle measurements for 265 long GRBs, including 17 events with limits (Frail et al. 2001; Berger et al. 2003; Bloom et al. 2003; Ghirlanda et al. 2004; Friedman & Bloom 2005; Racusin et al. 2009; Cenko et al. 2010, 2011; Filgas et al. 2011; Goldstein et al. 2011; Ryan et al. 2015) and calculate the cumulative distributions in the same manner (Figure 8). We find a median value for the 248 long GRBs with measurements of $\langle {\theta }_{{\rm{j}}}\rangle ={13}_{-9}^{+5}$ deg. Including the 17 events with limits (θ_j,max = 90°), the median becomes ${14}_{-10}^{+9}$ deg.

The opening angle distribution of short GRBs impacts the true energy scale, as the true energy is lower than the isotropic-equivalent value by the beaming factor, f_b, where ${f}_{b}\equiv 1-{\rm{cos}}({\theta }_{j})$ and therefore E_true = f_bE_iso. To calculate the cumulative beaming factor distribution, we use the individual opening angle probability distributions for each burst to convert to individual distributions in beaming factor. We then sum the individual distributions in a cumulative sense and calculate the median and 1σ uncertainties about the median. Including all short GRBs with opening angle measurements and limits and assuming the more realistic scenario of θ_j,max = 30°, the median beaming factor is ${f}_{b}={0.04}_{-0.03}^{+0.07}.$ The beaming correction is less substantial if we assume θ_j,max = 90°, ${f}_{b}={0.17}_{-0.16}^{+0.52},$ and is much more substantial if we only include short GRBs with measurements, f_b = 0.005 ± 0.002.

We find median isotropic-equivalent γ-ray and kinetic energy scales of E_γ,iso ≈ 2 × 10⁵¹ erg and E_K,iso ≈ (1–3) × 10⁵¹ erg. Applying the beaming correction for the most realistic scenario gives median beaming-corrected γ-ray and kinetic energy scales of $\langle {E}_{\gamma }\rangle ={0.8}_{-0.6}^{+1.4}\times {10}^{50}$ erg and $\langle {E}_{K}\rangle ={0.8}_{-0.7}^{+2.5}\times {10}^{50}$ erg, resulting in a total beaming-corrected energy release of $\langle {E}_{{\rm{tot}}}\rangle ={1.6}_{-1.3}^{+3.9}\times {10}^{50}$ erg. The inferred energy scales can be used to constrain the mechanism of energy extraction to power the relativistic jet: the thermal energy release from $\nu \bar{\nu }$ annihilation in a baryonic outflow (Jaroszynski 1993; Mochkovitch et al. 1993) and magnetohydrodynamic (MHD) processes in the black hole's accretion remnant (e.g., Blandford & Znajek 1977; Rosswog et al. 2003). The general consensus is that $\nu \bar{\nu }$ annihilation can only produce beaming-corrected total energy releases of 10⁴⁸–10⁴⁹ erg, while MHD processes can more easily produce energy releases in excess of 10⁴⁹ erg (Popham et al. 1999; Ruffert & Janka 1999a, 1999b; Rosswog 2005; Birkl et al. 2007; Lee & Ramirez-Ruiz 2007). Thus, if the majority of short GRBs have wider opening angles than the four short GRBs with measurements, and thus have a smaller overall correction to the isotropic-equivalent energy scale, it will be necessary to invoke MHD processes to explain the observed energy releases.

The opening angles also impact the event rate, as the true event rate is elevated compared to the observed rate by a factor of ${f}_{b}^{-1},$ so ${{\mathfrak{R}}}_{{\rm{true}}}={f}_{b}^{-1}{{\mathfrak{R}}}_{{\rm{obs}}}.$ The current estimated observed short GRB volumetric rate is ${{\mathfrak{R}}}_{{\rm{obs}}}\approx 10$ Gpc⁻³ yr⁻¹ (Nakar et al. 2006). Using ${f}_{b}^{-1}={27}_{-18}^{+158},$ which corresponds to all short GRBs with opening angle measurements and limits (θ_j,max = 30°), we find a true event rate of ${{\mathfrak{R}}}_{{\rm{true}}}\approx {270}_{-180}^{+1580}$ Gpc⁻³ yr⁻¹. The observed all-sky event rate of ≈0.3 yr⁻¹ within 200 Mpc (Guetta & Piran 2005) then becomes ${8}_{-5}^{+47}$ yr⁻¹. We note that this rate is conservative compared to previously reported rates based on short GRB observations (Coward et al. 2012; Fong et al. 2012b, 2014), as it properly incorporates opening angle lower limits, with the only assumption of an upper bound on the opening angle of 30°. This range is fully consistent with the expected detection rates of neutron star mergers within a volume of 200 Mpc by Advanced LIGO/VIRGO of ≈0.2–200 yr⁻¹ (LIGO Scientific Collaboration et al. 2013). Since there are a limited number of short GRBs with meaningful information on the opening angles, any additional measurements will greatly help to elucidate the true opening angle distribution, and therefore true energy scale and event rate.

We connect the afterglow properties of short GRBs to their larger-scale, galactic environments. In particular, we investigate trends between the circumburst densities, which probe the explosion environment on sub-parsec-scales, to the predicted distributions for NS–NS mergers, host galaxy type, and locations within the host galaxies.

We find a wide range of inferred circumburst densities for the 38 bursts that we have studied in detail, and can compare the properties of the sub-parsec-scale environment with the global host galaxy environment. Separating the bursts by host galaxy type according to Fong et al. (2013) and additional data collected since, we find that bursts in both star-forming and elliptical host galaxies span a wide range of densities, 10⁻⁶–5 cm⁻³ (Figure 9). We compare the inferred densities to predictions for NS–NS mergers from population synthesis for varying Galactic potentials, which have input distributions for merger timescales and kick velocities (Belczynski et al. 2006). Considering only the four bursts with elliptical hosts, we find particularly good agreement with the distributions for the large elliptical galaxy model (M_* = 10¹¹M_⊙, M_halo = 10¹²M_⊙) which has probability peaks at 10⁻⁵ and 1 cm⁻³, and poor agreement with the small elliptical galaxy model (M_* = 10⁸ M_⊙), which is dominated by very low densities of ≲10⁻⁶ cm⁻³ (Belczynski et al. 2006). This conclusion is commensurate with the known stellar masses of short GRB elliptical hosts, which have a median of ${M}_{*}\approx {10}^{11.0}\;{M}_{\odot }$ (Leibler & Berger 2010; Berger 2014). There are no predicted density distributions for star-forming galaxies; however, we would expect short GRBs which originate in elliptical galaxies to have lower inferred densities due to the lower average ISM densities in elliptical galaxies (e.g., Fukazawa et al. 2006). Based on the small number of events, we do not find any significant difference between the inferred circumburst densities of short GRBs in star-forming versus elliptical hosts.

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We next investigate the relationship between the inferred circumburst densities and burst offsets from their host galaxies. If short GRBs trace the large-scale distribution of the ISM, we expect the inferred circumburst densities to decrease as a function of offset. In this vein, we gather all available projected physical offsets, δR, between the afterglow location and host galaxy center, derived from ground-based (Margutti et al. 2012; Berger et al. 2013; Sakamoto et al. 2013) and Hubble Space Telescope (HST) observations (Fong et al. 2010; Fong & Berger 2013). For bursts with no spectroscopic redshift, we calculate the physical offsets at z = 0.5 to be consistent with this work. However, for GRBs 060313 and 111020A, afterglow modeling implies that z > 0.5 (c.f. Section 3.3; Roming et al. 2006; Fong et al. 2012b) so we keep the original fiducial value of z = 1 for these bursts for complete uniformity. The distribution of circumburst densities with respect to projected physical offset for 22 bursts is shown in Figure 9. We find that four bursts with δR ≳ 10 kpc have very low densities of ≲10⁻³ cm⁻³, while bursts with δR ≲ 10 kpc have a wider range of densities, spanning ≈6 orders of magnitude (Figure 9). Overall, we find that for δR ≈ 1–15 kpc, there is no obvious trend between circumburst density and projected physical offset. We also find no obvious trends when considering only bursts with relatively well-measured densities (ν_c < ν_X).

To analyze the relationship with offsets in a more uniform manner, we utilize offsets that have been normalized by the sizes of their host galaxies (δR/r_e where r_e is the galaxy half-light radius). The sample of bursts with host-normalized offsets is smaller since precise galaxy size measurements require the resolution of HST; thus the sample comprises 16 events (Fong et al. 2010; Fong & Berger 2013). The circumburst densities as a function of projected host-normalized offset is provided in Figure 9. Our analysis suggests that for ≲5 r_e, the inferred densities are largely independent of host-normalized offset. We discuss a couple of possible contributing factors. First, since we can only measure projected offsets, we are not sensitive to the distance component along our line of sight, which could contribute a significant amount to the absolute distance. This may explain the case of GRB 061006, which has a small projected offset ≈0.4r_e but has a very low density of ≈2 × 10⁻⁵ cm⁻³ (Table 3 and Figure 9). Second, the afterglow only probes the sub-parsec circumburst environment and to a certain extent will be more sensitive to small-scale fluctuations in the ISM rather than the average ISM density on kiloparsec scales.

However, bursts that appear to have no coincident host galaxy to deep optical/NIR limits of ≳26 mag and are located ≈30–75 kpc from the nearest most probable host galaxy ("host-less" bursts; Berger 2010; Fong & Berger 2013; Tunnicliffe et al. 2013) are expected to have low inferred densities. Indeed, the three bursts located at offsets of ≳10r_e have low densities of ≲10⁻⁴ cm⁻³, as expected if these bursts occur in the IGM or outer halos of their hosts.