The Astrophysical Journal, 725:2209–2224, 2010 December 20 C 2010.
doi:10.1088/0004-637X/725/2/2209
The American Astronomical Society. All rights reserved. Printed in the U.S.A.
CONSTRAINING GAMMA-RAY BURST INITIAL LORENTZ FACTOR WITH THE AFTERGLOW ONSET FEATURE AND DISCOVERY OF A TIGHT Γ0 –Eγ ,iso CORRELATION En-Wei Liang1,2,3 , Shuang-Xi Yi1,3 , Jin Zhang3,4 , Hou-Jun Lu¨ 1,3 , Bin-Bin Zhang2 , and Bing Zhang2 1 Department of Physics, Guangxi University, Nanning 530004, China;
[email protected] Department of Physics and Astronomy, University of Nevada, Las Vegas, NV 89154, USA;
[email protected] 3 GXU-NAOC Center for Astrophysics and Space Sciences, Nanning, Guangxi 530004, China 4 College of Physics and Electronic Engineering, Guangxi Teachers Education University, Nanning, Guangxi 530001, China Received 2009 December 24; accepted 2010 October 16; published 2010 December 6 2
ABSTRACT The onset of gamma-ray burst (GRB) afterglow is characterized by a smooth bump in the early afterglow light curve as the GRB fireball is decelerated by the circumburst medium. We extensively search for GRBs with such an onset feature in their optical and X-ray light curves from the literature and from the catalog established with the Swift/XRT. Twenty optically selected GRBs and 12 X-ray-selected GRBs are obtained, among which 17 optically selected and 2 X-ray-selected GRBs have redshift measurements. We fit these light curves with a smooth broken power law and measure the width (w), rising timescale (tr ), and decaying timescale (td ) at full width at half-maximum. Strong mutual correlations among these timescales and with the peak time (tp ) are found. The ratio tr /td is almost universal among bursts, but the ratio tr /tp varies from 0.3 to ∼1. The optical peak luminosity in the R band (LR,p ) is anti-correlated with tp and w in the burst frame, indicating a dimmer and broader bump peaking at a later time. The isotropic prompt gamma-ray energy (Eγ ,iso ) is also tightly correlated with LR,p and tp in the burst frame. Assuming that the bumps signal the deceleration of the GRB fireballs in a constant density medium, we calculate the initial Lorentz factor (Γ0 ) and the deceleration radius (Rd ) of the GRBs with redshift measurements. The derived Γ0 is typically a few hundreds, and the deceleration radius is Rdec ∼ 2 × 1017 cm. More intriguingly, a tight correlation between Γ0 and Eγ ,iso is found, namely Γ0 182(Eγ ,iso /1052 erg)0.25 . This correlation also applies to the small sample of GRBs which show the signature of the afterglow onset in their X-ray afterglow, and to two bursts (GRBs 990123 and 080319B) whose early optical emission is dominated by a reverse shock. The lower limits of Γ0 derived from a sample of optical afterglow light curves showing a decaying feature from the beginning of the observation are also generally consistent with such a correlation. The tight lower limits of Γ0 of GRBs 080916C and 090902B derived from the opacity constraints with Fermi/LAT observations are also consistent with the correlation at the 2σ confidence level, but the short GRB 090510 is a clear outlier of this relation. This correlation may give insight to GRB physics and could serve as an indicator of Γ0 for long GRBs without early afterglow detections. A comparison of the early X-ray and optical afterglow light curves shows that the early bright X-ray emission is usually dominated by a non-forward-shock component, but occasionally (for one case) the forward shock emission is observable, and an achromatic deceleration feature is observed. The superposition of the internal and external components in X-rays causes the diversity of the observed X-ray light curves. Key words: gamma-ray burst: general – radiation mechanisms: non-thermal Online-only material: color figures the coasting phase is called the initial Lorentz factor (Γ0 ), which is a crucial parameter to understand GRB physics, but is poorly known for most GRBs. Three methods have been proposed to estimate Γ0 of a GRB fireball. The first one is based on the “compactness” argument with the high-energy cutoff of the prompt gamma-ray spectrum (Fenimore et al. 1993; Woods & Loeb 1995; Baring & Harding 1997; Lithwick & Sari 2001). However, this method suffers great uncertainties. Theoretically, the cutoff energy depends on both Γ0 and the emission radius Rγ (Gupta & Zhang 2008), which is Rγ = Γ02 cδt in the internal shock model. Such an assumption is not necessarily correct, and the minimum variability timescale δt is subject to large uncertainty because the GRB light curves are chaotic without a characteristic timescale. Observationally, so far no clear cutoff feature is observed in the GRB spectrum for most GRBs (Abdo et al. 2009a; Zhang et al. 2010), and a distinct high-energy component in the GeV range, which may come from a different emission region, is observed in a few GRBs, such as 090510 (Abdo et al. 2009b) and GRB 090902B (Abdo et al. 2009c). Therefore, constraints on Γ0 with the cutoff energy or the detected highest photon energy may lead
1. INTRODUCTION The fireball model is the most popular one for the gamma-ray burst (GRB) phenomenon (M´esz´aros 2002; Zhang & M´esz´aros 2004; Piran 2004), in which the observed prompt gamma-ray emission is explained by synchrotron (or inverse Compton) emission from the internal shocks in an erratic, unsteady, relativistic fireball (M´esz´aros & Rees 1993; Rees & M´esz´aros 1994) and broadband afterglow emission is attributed to synchrotron emission from the external shock when the fireball is decelerated by a circumburst medium (M´esz´aros & Rees 1997; Sari et al. 1998). To avoid the “compactness problem” of highenergy non-thermal photons detected from GRBs, the fireball is required to move relativistically toward Earth. After an initial radiation-dominated acceleration phase, the fireball enters a matter-dominated “coasting” phase, keeping an approximate constant Lorentz factor (Γ) until it sweeps up a considerable amount of mass from the ambient medium at the so-called deceleration radius (Rd ), after which Γ decreases significantly and approaches a self-similar solution characterized by a power-law decay with R and the observer time t. The Lorentz factor during 2209
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to erroneous conclusions. The second approach to estimate Γ0 is using the blackbody component detected in some GRB spectra (Pe’er et al. 2007). However, observations with Fermi show that no thermal emission component is reliably identified from the broadband spectra for the Fermi GRBs, except for GRB 090902B (Ryde et al. 2010; Zhang et al. 2010). The third, more commonly adopted method is using the early afterglow light curves that show the signal of fireball deceleration. A smooth onset bump peaking at a time when roughly half of the fireball energy is transferred to the medium in the early afterglow light curve is predicted in the fireball model (Sari & Piran 1999; Kobayashi & Zhang 2007). The most relevant case is the thin shell regime, defined when the thickness of the fireball −8/3 shell satisfies Δ < (E/nmp c2 )1/3 Γ0 , where E is the kinetic energy of the fireball, n is the circumburst medium density, mp is the mass of proton, and c is the speed of light (Sari & Piran 1999; Kobayashi 2000). Within this regime, the deceleration −8/3 time (the peak time at the light curve bump), tp ∝ Γ0 (E/n)1/3 (M´esz´aros & Rees 1993), sensitively depends on the initial Lorentz factor but is rather insensitive to other parameters. The detection of tp can be then used to infer Γ0 . In the optical band, early emission may be contaminated by the emission from the reverse shock (M´esz´aros & Rees 1997; Sari & Piran 1999; Kobayashi 2000; Zhang et al. 2003). However, under certain conditions (either a Poynting flux dominated flow, Zhang & Kobayashi 2005; or a relatively low typical synchrotron frequency in the reverse shock, Jin & Fan 2007), the reverse shock component would not show up in the optical band. In these bursts, a smooth onset bump can be detected, which signals the deceleration feature of the fireball, and hence, can be used to constrain the initial Lorentz factor and the deceleration radius (Sari & Piran 1999; Zhang et al. 2003; Molinari et al. 2007; Xue et al. 2009; Zou et al. 2009). In this paper, we constrain Γ0 with the early GRB afterglows that show the deceleration signature and investigate the possible correlations among deceleration parameters (including Γ0 ) as well as the prompt gamma-ray emission properties. We extensively search for the onset of afterglow signature in the optical and X-ray light curves. Our sample selection criteria are presented in Section 2. The temporal characteristics and their correlations are presented in Section 3. The relation between the prompt gamma-ray properties and the deceleration properties is investigated in Section 4. In particular, we constrain Γ0 and the deceleration radius of the fireball for the z-known sample, and discover a tight correlation between Γ0 and Eγ ,iso . Discussion and conclusions are presented in Sections 5 and 6, respectively. A concordance cosmology with parameters H0 = 71 km s−1 Mpc−1 , ΩM = 0.30, and ΩΛ = 0.70 is adopted. The notation Qn denotes Q/10n in cgs units throughout the paper. 2. SAMPLE SELECTION AND LIGHT CURVE FITTING We extensively search for the smooth “bump” feature at the onset of the GRB afterglows. Two criteria are employed. (1) Since flares and some X-ray plateau features followed by steep drop near the end (internal plateaus; Liang et al. 2007; Troja et al. 2007; Lyons et al. 2010) are related to internal emission of late central engine activities (Zhang et al. 2006), our first criterion is to search for smooth bumps without the superposition of significant flare-like components. (2) In the optical band, early emission is sometimes contaminated by the emission from the reverse shock or internal shocks, which are characterized by a decay slope of −2 or steeper. Our second criterion is therefore
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to require the decay slope after the bump feature to be shallower than −2. Through the literature search, we obtain 20 optically selected GRBs that show the afterglow onset feature. We also go through the Swift/XRT light curve archive that has been processed by our group in the past (from 2005 January to 2009 September), and identify 12 X-ray selected sample with afterglow onset signature. The observational results of these bursts are summarized in Tables 1 and 2. The details of XRT data reduction have been presented in a series of papers published by our group (Zhang et al. 2007, Paper I; Liang et al. 2007, 2008, 2009; Papers II, III, IV). The prompt gamma-ray properties of the bursts are taken from the published papers or GCN reports. The afterglow light curves of the optically selected and X-ray-selected samples are presented in Figures 1 and 2, respectively. The early X-ray afterglow light curves for the optically selected sample and the optical afterglow light curves for the X-ray selected sample are also presented, if they are available. In the X-ray selected sample, only a few cases have simultaneous optical observations. Nineteen out of 20 GRBs in the optically selected sample have simultaneous X-ray observations. We find only one case, GRB 080319C, that tentatively shows an achromatic onset bump in both the optical and X-ray bands, although the X-ray peak has a large error. This suggests that the external shock emission indeed contributes to both the optical and the X-ray band for this burst. For most optically selected GRBs, on the other hand, either the early X-ray light curves show erratic X-ray flares or internal plateaus that are believed to be powered by the GRB central engine, or the X-ray observations started only after the optical bump peak. Inspecting the two samples shown in Figures 1 and 2, we find that the onset bumps in the optically selected sample are usually smoother than those observed in the X-ray-selected sample. We fit the light curves with an empirical model proposed by Kocevski & Liang (2001), F (t) = Fp
t + t0 tp + t 0
r
r d + r +d r +d
t + t0 tp + t 0
r+1 − r+d r+1 ,
(1) where Fp is the maximum flux at tp , t0 is a reference time, and r and d are the rising and decaying power-law indices, respectively. An IDL routine named mpfitfun.pro is employed for our fitting. This routine performs Levenberg–Marquardt least-square fit to the data for a given model. It optimizes the model parameters so that the sum of the squares of the deviations between the data and the model becomes minimal. The time interval and the fitting curve for each GRB are shown in Figures 1 and 2, and the fitting parameters are summarized in Tables 1 and 2.5 Note that the light curves of some GRBs, such as 050820A, 060607A, 070411, 071031, and 080330, show significant energy injection or re-brightening features after the deceleration bump. We make our fits only around the bump. We take the full width at half-maximum (FWHM) of a fitting curve as a characteristic width (w) of the bump and measure the rising and decaying timescales (tr and td ) at FWHM. We also derive the ratios of tr /tp and tr /td . The results are reported in Tables 1 and 2. Seventeen out of 20 GRBs in the optically selected sample and 2 out of 12 GRBs in the X-ray-selected sample have 5
The reduced χ 2 of our fits for some optical light curves are large. This is due to the fluctuations in the light curves and small observational errors in the optical data.
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Table 1 Fitting Results of the Optical Light Curves Fm a
GRB (Band) 030418(V) 050730(V) 050820A(R) 060418(H) 060605(R) 060607A(H) 060904B(V) 061007(R) 070318(V) 070411(R) 070419A(R) 070420(R) 071010A(R) 071031(R) 080319C(N) 080330(R) 080710(R) 080810(R) 081126(R) 081203A(U)
2.51 4.00 21.10 88.70 9.59 28.60 4.91 1820.0 14.20 3.74 0.46 12.80 3.90 0.71 2.57 1.43 3.31 107.00 12.70 110.00
± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±
0.06 0.45 1.25 1.71 0.20 0.49 0.29 12.0 0.43 2.60 0.01 1.10 0.26 0.01 0.03 0.02 0.03 1.70 0.20 0.20
tp b 1344.5 ± 590.7 ± 391.0 ± 153.3 ± 399.1 ± 180.9 ± 467.9 ± 78.3 ± 301.0 ± 450.1 ± 587.0 ± 213.2 ± 368.2 ± 1018.6 ± 338.3 ± 621.9 ± 2200.9 ± 117.6 ± 201.3 ± 367.1 ±
78.6 131.5 16.7 3.3 13.0 2.4 48.4 0.4 21.3 5.0 20.9 18.7 24.4 1.6 5.6 17.0 4.1 1.1 1.2 0.8
r
d
χ 2 /dof
wa
tr a
td a
tr /td
tr /tp
Refs.
1.17 ± 0.20 1.36 ± 0.43 4.45 ± 0.76 2.70 ± 0.22 0.90 ± 0.09 4.15 ± 0.22 1.56 ± 0.43 2.17 ± 0.04 1.05 ± 0.14 0.76 ± 0.03 2.20 ± 0.24 2.59 ± 1.07 2.36 ± 0.43 1.11 ± 0.01 2.00 (fixed) 0.34 ± 0.03 1.34 ± 0.01 1.34 ± 0.04 2.04 ± 0.06 2.09 ± 0.01
0.74 ± 0.08 1.02 ± 0.15 1.04 ± 0.01 1.27 ± 0.02 1.17 ± 0.05 1.32 ± 0.04 0.85 ± 0.22 1.71 ± 0.01 1.12 ± 0.05 1.54 ± 0.03 1.27 ± 0.04 1.10 ± 0.33 0.74 ± 0.01 0.92 ± 0.01 1.50 (fixed) 1.77 ± 0.12 0.97 ± 0.01 1.21 ± 0.01 0.44 ± 0.01 1.49 ± 0.01
26/9 25/7 47/7 16/8 74/50 46/23 15/13 811/79 9/6 754/17 102/43 4/4 12/20 9819/22 22/4 30/36 2511/61 854/62 244/5 3176/32
5816 1836 711 298 1313 259 1524 142 1090 1603 1387 433 1234 4009 628 3552 6754 344 1330 794
953 339 120 65 281 57 281 31 233 359 239 94 202 657 137 632 1245 85 134 202
4863 1497 592 233 1032 202 1243 111 857 1243 1148 339 1032 3352 491 2920 5508 259 1197 592
0.20 0.23 0.20 0.28 0.27 0.28 0.23 0.28 0.27 0.29 0.21 0.28 0.20 0.20 0.28 0.22 0.23 0.33 0.11 0.34
0.71 0.57 0.31 0.42 0.70 0.31 0.60 0.39 0.77 0.80 0.41 0.44 0.55 0.65 0.40 1.02 0.57 0.72 0.66 0.55
(1) (2) (3) (4) (5) (4) (6) (5) (7) (8) (9) (6) (10) (11) (12) (13) (14) (15) (16) (17)
Notes. a In units of 10−12 erg cm−2 s−1 . b In units of seconds. References. (1) Rykoff et al. 2004; (2) Pandey et al. 2006; (3) Cenko et al. 2006; (4) Molinari et al. 2007; (5) Rykoff et al. 2009; (6) Klotz et al. 2008; (7) Roming et al. 2009; (8) Ferrero et al. 2008; (9) Melandri et al. 2009; (10) Covino et al. 2008; (11) Kr¨uhler et al. 2009a; (12) Li & Filippenko 2008; (13) Guidorzi et al. 2009; (14) Kr¨uhler et al. 2009b; (15) Page et al. 2009; (16) Klotz et al. 2009; (17) Kuin et al. 2009. Table 2 Fitting Results of the XRT Light Curves GRB 060319 060801 060804 070103 070208 070714A 080307 080319C 080409 090429B 090607
Fm a 8.46 20.20 7.60 1.60 1.80 1.38 67.10 58.40 0.78 1.52 40.40
± ± ± ± ± ± ± ± ± ± ±
0.98 3.50 0.66 0.13 0.29 0.24 1.50 3.50 0.08 0.14 2.50
tp b 267.0 ± 20.2 114.3 ± 9.5 418.9 ± 176.1 685.5 ± 64.3 968.1 ± 72.9 234.0 ± 36.9 210.5 ± 3.12 432.8 ± 29.1 395.9 ± 100.0 540.2 ± 51.6 118.9 ± 3.48
r 5.46 6.69 0.85 0.76 1.09 2.25 2.22 1.55 0.50 1.57 4.86
± ± ± ± ± ± ± ± ± ± ±
d 2.05 4.42 0.67 0.16 0.27 1.46 0.16 0.41 0.00 0.38 1.13
1.13 1.77 1.22 1.47 1.29 0.86 2.05 1.41 1.11 1.34 2.79
± ± ± ± ± ± ± ± ± ± ±
0.04 0.27 0.07 0.08 0.06 0.12 0.04 0.03 0.11 0.08 0.39
χ 2 (dof)
ω
tr b
td b
tr /td
tr /tp
43/30 3/6 25/23 22/24 39/30 16/8 141/153 72/52 13/10 15/13 25/20
490 134 1603 2325 3274 685 338 994 2007 1487 134
82 39 281 521 657 94 94 281 336 339 39
408 94 1322 1803 2616 591 244 713 1671 1148 94
0.20 0.42 0.21 0.29 0.25 0.16 0.39 0.39 0.20 0.29 0.42
0.31 0.34 0.67 0.76 0.68 0.40 0.45 0.65 0.85 0.63 0.33
Notes. a In units of 10−11 erg cm−2 s−1 . b In units of seconds.
redshift measurements. In the following, we mostly only use the z-known optically selected sample in our analysis, but use the z-known X-ray-selected sample to confirm the findings. Most of these GRBs are detected with Swift/BAT. It is well known that the GRB spectrum is well fit with the Band function (Band et al. 1993). Due to the narrowness of the BAT energy band (15–150 keV), the true spectral parameters and the bolometric energy of the prompt gamma-rays (Eγ ,iso ; defined in the 1–104 keV band in the rest frame of the burst) of these GRBs are poorly constrained. The spectra observed with BAT are usually adequately fit with a single power law. Seven out of the 17 z-known GRBs in the optically selected sample, including GRBs 050820A, 060418, 060904B, 061007, 070318, 080319C, and 080810, were simultaneously observed with Konus/Wind, Suzaku, or Fermi/GBM. Their spectral parameters and Eγ ,iso
were derived from the joint spectral fits of the observations by these instruments (Krimm et al. 2009; Cenko et al. 2006; Page et al. 2009). For other GRBs, the spectral parameters and Eiso in the 1–104 keV band are taken from Butler et al. (2007, 2010), in which a complete and extensive spectral analysis for the Swift GRBs is presented using a unified methodology, although the spectral parameters and Eγ ,iso have great uncertainties. The derived Eiso is reported in Table 3 We collect the optical spectral index (βO ) and host galaxy extinction (AV ) of each GRB from the literature. They are reported in Table 3. The spectral index βO is taken as 0.75 if it is not available. Most optically selected light curves in our sample are measured in the R band. We therefore correct the observed light curves in the other bands to the R band using the spectral index. Since the βO value of GRB 060605 is unreasonably
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large,6 we take its βO as 0.75. All data have been corrected for extinction in the Milky Way and the GRB host galaxy assuming an LMC extinction curve. Finally, we derive the rest-frame R-band flux using the k-correction FRc = FRobs (1 + z)βO −1 . We then calculate the rest-frame R-band peak luminosity (LR,p ) based on the burst redshift, and calculate the isotropic restframe R-band energy (ER,iso ) by integrating the luminosity from 10 to 105 s after the GRB trigger. We do not consider the error on ER,iso since they are calculated using integration of the fitted light curves. The results are also reported in Table 3.
GRB 060605 is at z = 3.773. The spectral index is affected by Lyα and Lyman-limit blanketing.
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3. CHARACTERISTICS OF THE ONSET BUMP AND THEIR CORRELATIONS The following statistics applies to the optically selected sample. The distributions of r, d, tp ,tr , td , Fp , w, tr /tp , and tr /td are shown in Figure 3. It is found that the rising index r of most bursts is in the range of 1–2, with three exceptional cases, i.e., GRBs 080330 (r = 0.34 ± 0.03), 060607A (r = 4.15 ± 0.22), and 050820A (4.45 ± 0.76). The optical light curve of the afterglow of GRB 080330 rises slowly, keeping almost constant in 300–1000 s post-GRB trigger. This feature is similar to that observed in GRB 060614. For GRBs 060607A and 050820A, their optical light curves rapidly rise and decay normally after the peak as predicted by the forward shock models. Considering the first data point of GRB 061007, the rising index is also very steep (r ∼ 4.90), but our fitting model cannot yield
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104
105
10-12
103
104 Time (s)
Time (s) 10-10
105
10-12
10-16 102
106
10-9
070714A
χ2/dof=118/76
10-14
10-14 102
106
106
10-10
10-13
102
105
070521
χ /dof=39/30
Flux (erg cm-2 s-1)
Flux (erg cm-2 s-1)
Flux (erg cm-2 s-1)
10-13
104 Time (s)
2
10-11
10-12
103
10-8
070208
2
10-11
10-14 101
10-14 102
1000
103
104 Time (s)
105
106
10-9
080307
χ /dof=16/8 2
080319C
χ /dof=141/153 2
10-10
10-10
10-11
10-11
χ2/dof=78/78
10-12
Flux (erg cm-2 s-1)
Flux (erg cm-2 s-1)
Flux (erg cm-2 s-1)
10-11
10-12
10-12
10-13
10-13
10-14
10-14
10-13
10-14 101
102
103
104
105
10-15 102
106
103
Time (s)
106
10-15 101
107
10-12
10-13
102
103
104 Time (s)
105
106
10-14 102
105
106
χ2/dof=25/20
10-10
Flux (erg cm-2 s-1)
Flux (erg cm-2 s-1)
10-13
104
090607
χ2/dof=15/13
10-11
10-12
103
10-9
090429B
χ2/dof=13/10
10-11
102
Time (s)
10-10
080409
Flux (erg cm-2 s-1)
105 Time (s)
10-10
10-14 101
104
10-11
10-12
103
104 Time (s)
105
106
10-13 10
100 Time (s)
Figure 2. Same as Figure 1, but for the X-ray-selected sample. The optical data points are presented for GRBs 080307 and 080319C. (A color version of this figure is available in the online journal.)
1000
GRB INITIAL LORENTZ FACTOR AND Γ0 –Eγ ,iso RELATION
No. 2, 2010
2215
Table 3 Derived Intrinsic Properties of the Bursts with Redshift Measure in the Optically Selected Sample GRBa
zref
050730
3.97(1)
050820A
2.615(2)
060418
1.49(3)
060605
3.8(4)
060607A
3.082(5)
060904B
0.703(6)
061007
1.262(7)
070318
0.84(8)
070411
2.954(9)
070419A
0.97(10)
071010A
0.98(11)
071031
2.692(12)
080319C
1.95(13)
080330
1.51(14)
080710
0.845(15)
080810
3.35(16)
081203A
2.1(17)
ER,iso d
Γ0
Rd e
4.33 ± 0.58
120
0.065 ± 0.008
11.13 ± 0.78
2.26+0.82 −0.68
162
∼0.1
12.57 ± 0.77
111
···
8.82 ± 0.82
185
0.41+0.14 −0.3
11.42 ± 0.83
65
0.44 ± 0.05
0.09 ± 0.01
6
0.66 ± 0.02
155.75 ± 1.21
738
···
0.39 ± 0.04
17
···
1.97 ± 0.27
59
0.37 ± 0.19
0.02 ± 0.01
1
0.62 ± 0.15
0.16 ± 0.03
7
0.5
0.30 ± 0.05
24
0.67 ± 0.06
0.65 ± 0.01
10
···
0.15 ± 0.01
15
289+41 −28 332+42 −21 379+33 −10 283+44 −9 426+41 −12 155+14 −14 627+5 −5 206+10 −10 299+30 −8 131+16 −4 145+34 −4 191+25 −4 327+7 −7 150+43 −3 90+11 −6 588+49 −49 315+30 −9
Eγ ,iso a
tp,z b
βO c
AR c
LR,p d
9+8 −3 97+31 −14 10+7 −2 2.5+3.1 −0.6 9+7 −2 0.72+0.43 −0.43 104.65+6.94 −6.94 1.45+0.38 −0.38 10+8 −2 0.24+0.23 −0.05 0.13+0.24 −0.01 3.9+4.1 −0.6 22.55+3.35 −3.35 0.41+0.94 −0.06 0.8+0.8 −0.4 30+20 −20 17+13 −4
120.09 ± 27.76
0.82+0.04 −0.04 0.96+0.03 −0.03 0.65+0.06 −0.06 4.64+0.58 −0.58 0.56+0.03 −0.05 0.75+0.1 −0.1 0.9+0.005 −0.005 0.75+0.1 −0.1 0.75+0.1 −0.1 0.82+0.16 −0.07 0.76+0.23 −0.26 0.78+0.03 −0.03 0.75+0.1 −0.1 0.61+0.03 −0.03 1+0.02 −0.02 0.51+0.22 −0.22 0.9+0.01 −0.01
0.100 ± 0.015
108.17 ± 4.62 60.73 ± 0.82 83.14 ± 2.70 42.89 ± 0.62 271.91 ± 33.75 34.62 ± 0.18 162.09 ± 15.26 113.83 ± 1.27 297.98 ± 10.62 185.95 ± 12.31 275.88 ± 0.42 117.38 ± 3.22 247.77 ± 6.79 1192.91 ± 2.24 27.02 ± 0.26 118.09 ± 0.46
···
0.11 ± 0.01
32
···
52.55 ± 17.02
306
0.08
29.91 ± 0.36
557
2.69+0.69 −0.37 1.96+0.34 −0.10 1.50+0.47 −0.11 1.75+0.34 −0.10 1.48+0.32 −0.32 3.06+0.05 −0.05 1.54+0.21 −0.21 2.29+0.46 −0.12 1.15+0.28 −0.08 0.87+0.41 −0.07 2.26+0.59 −0.09 2.83+0.14 −0.14 1.25+0.72 −0.06 2.19+0.55 −0.27 2.10+0.35 −0.35 2.64+0.50 −0.16
Notes. a GRBs 050820A, 060418, 060904B, 061007, 070318, 080319C, and 080810 were simultaneously observed with BAT and Konus/Wind, or Sazaku, or Fermi/GBM. Their spectral parameters and the Eγ ,iso (in units of erg) in 1–104 keV band in the burst frame were taken from Krimm et al. (2009), Cenko et al. (2006), and Page et al. (2009). The spectral parameters and the Eγ ,iso of other GRBs are taken from Butler et al. (2007, 2010). b In units of seconds. c The observed optical spectral index and A of the GRB host galaxy. The references of the optical data are the same as that in Table 1. R d The R-band peak luminosity (in units of 1047 erg cm−2 s−1 ) and isotropic energy (in units of 1048 erg) in 101 –104 s post the GRB trigger. e In units of 1017 cm. References. (1) Rol et al. 2005; (2) Ledoux et al. 2005; (3) Prochaska et al. 2006; (4) Ferrero et al. 2009; (5) Ledoux et al. 2006; (6) Fugazza et al. 2006; (7) Jakobsson et al. 2007b; (8) Chen et al. 2007; (9) Jakobsson et al. 2007a; (10) Cenko et al. 2007b; (11) Prochaska et al. 2007; (12) Ledoux et al. 2007; (13) Wiersema et al. 2008; (14) Cucchiara 2008; (15) Perley et al. 2008a; (16) Prochaska et al. 2008; (17) Landsman et al. 2008.
an acceptable fit around the bump. The decaying index d is distributed in the range of 0.44–1.77, with an average of 1.16 ± 0.34. Except for GRBs 080330 and 061007, the decay indices are well consistent with the isotropic forward shock models in a constant density medium. The decay indices of the two exceptions are ∼1.7, slightly steeper than the normal decay slope predicted by the isotropic forward shock models, but may be accommodated within the forward shock model if the jet carries an angular structure and the line of sight is close to the jet axis (M´esz´aros et al. 1998; Dai & Gou 2001; Schady et al. 2007). The peak time tp is in the range of 102 –103 s with a median value of ∼380 s. The distribution of Fp is in the range of 10−13 –10−8 erg cm−2 s−1 , with a mean 7.25 × 10−12 erg cm−2 s−1 . The width w is distributed in 102 –103 s. The distributions of tr and td peak around 102 s and 103 s, respectively. The ratio tr /td is narrowly distributed in the range of 0.1–0.3. The distribution of the ratio tr /tp is however much wider in the range of 0.3–1. We show various correlations among the characteristics of the optically selected light curves in Figure 4 and summarize the linear correlation coefficients from the Spearman pair correlation analyses in Table 4. Tight correlations are found among tr , td , tp , and w, with the linear correlation coefficients being larger than 0.93. These correlations read log td = (0.48 ± 0.13) + (1.06 ± 0.06) log tr ,
(2)
log td = (−0.09 ± 0.29) + (1.17 ± 0.11) log tp ,
(3)
Table 4 Spearman Pair-correlation Coefficients of the Characteristics of the Optically Selected Sample Lp,O LR,p tp r d tr td tr /td tr /tp
tp
r
d
−0.90
X X
X X X
tr −0.89 0.95 X X
tr /td
tr /tp
w
−0.88 0.93 X X
X X 0.90 X
X X 0.90 X
−0.88 0.94 X X
0.98
X X
X X X
0.98 ∼1 X X
log tr = (−0.54 ± 0.22) + (1.11 ± 0.08) log tp ,
(4)
log w = (0.05 ± 0.27) + (1.16 ± 0.10) log tp ,
(5)
log w = (0.61 ± 0.11) + (1.05 ± 0.05) log tr ,
(6)
log w = (0.15 ± 0.02) + (0.98 ± 0.01) log td .
(7)
These tight correlations suggest that the structures of the bumps among these bursts are similar, indicating a universal physical origin. An interesting characteristic is that a wider bump tends to peak at a later time. This is consistent with the expectation of the external shock model, since a later deceleration time corresponds to a smaller Lorentz factor, and hence, a longer
2216
LIANG ET AL.
Vol. 725
Figure 3. Distributions of the fitting parameters of the optically selected sample.
angular spreading time R/Γ2 c. No correlation between the decay index d and other parameters is found. This is also consistent with the expectations from the fireball model, where the decay slope is dictated by the density profile and the electron spectral index p but is independent on the details of the afterglow onset. On the other hand, the rising index r is tightly anti-correlated with both the ratio tr /td and tr /tp , although it is not correlated with tr and td . These correlations read (8) log r = (−0.21 ± 0.06) − (1.68 ± 0.19) log tr /tp , log r = (−0.15 ± 0.02) − (0.48 ± 0.05) log tr /td .
(9)
In addition, both w and tp in the burst frame (w z and tp,z ) are anti-correlated with LR,p , i.e., log LR,p,47 = (5.61 ± 0.83) − (2.49 ± 0.39) log tp,z , (10) log LR,p,47 = (5.43 ± 0.84) − (2.00 ± 0.32) log wz . (11) These results suggest that a dimmer bump tends to peak at a later time with a longer duration. This is again consistent with the expectation of the external shock model, since a later deceleration time corresponds to a smaller Lorentz factor, and hence, a weaker forward shock with fainter emission.
No. 2, 2010
GRB INITIAL LORENTZ FACTOR AND Γ0 –Eγ ,iso RELATION
2217
Figure 4. Pair correlations among the fitting parameters of the optically selected sample. Lines are the best fits.
4. INITIAL LORENTZ FACTOR CONSTRAINTS AND THE γ0 –Eγ ,iso CORRELATION The observational properties of the early optical bumps seem to be due to the onset of the external shock afterglow in the thin shell regime. To further test this hypothesis, we show the correlations of LR,p , ER,iso , and tp,z with Eγ ,iso in Figure 5, and report their linear coefficients and chance probabilities in Table 5. It is found that they are correlated, i.e., log LR,O,47 = (0.83 ± 0.15) + (1.40 ± 0.08) log Eγ ,iso,52 , (12) log ER,iso,48 = (1.30 ± 0.14) + (0.76 ± 0.14) log Eγ ,iso,52 (13) log tp,z = (2.35 ± 0.09) − (0.40 ± 0.07) log Eγ ,iso,52 .
(14)
These correlations indicate that a GRB with a larger Eγ ,iso tends to have a brighter optical afterglow peaking at an earlier time, being consistent with the afterglow onset theory.
Table 5 Spearman Pair-correlation Coefficients (rs ) and Chance Probability (ps ) Between the Afterglow Onset and Prompt Gamma-rays for the Optically Selected Sample
r p
tp,z –Eγ ,iso
LR,p –Eγ ,iso
ER,iso –Eγ ,iso
LR,p –Γ0
tp,z –Γ0
Eγ ,iso –Γ0
−0.69 0.002
0.87
