A CORRELATED STUDY OF OPTICAL AND X-RAY AFTERGLOWS OF GRBs

While the prompt, gamma-ray emission of gamma-ray bursts (GRBs) generally is interpreted as being due to energy dissipation processes inside the relativistic outflow (e.g., Usov 1992; Rees & Mészáros 1994; Thompson 1994; Kobayashi et al. 1997; Daigne & Mochkovitch 1998; Drenkhahn & Spruit 2004; Giannios & Spruit 2006; Zhang & Pe'er 2009; Zhang 2011; Kumar & Zhang 2015), the afterglow emission is interpreted as synchrotron emission generated in a forward shock as the blast wave interacts with the circumburst medium (e.g., Mészáros & Rees 1997; Sari et al. 1998; Panaitescu et al. 1998; Huang et al. 2000; Panaitescu & Kumar 2001; see Zhang & Mészáros 2004 for a review; Kumar & Zhang 2015). Sari et al. (1998) calculated the broadband spectrum and corresponding light curves and reported quantitative relationships (the standard afterglow model) with a constant interstellar medium (ISM), which can generally describe the X-ray and optical afterglow emission as being from the same synchrotron emission process. Prior to the launch of the Swift satellite, observational studies largely confirmed the standard model. By comparing the light curve decays at different wavelengths, Wijers et al. (1997) showed that the standard model fitted the afterglow data for GRB 970228 and 970402. Likewise, Panaitescu (2005a, 2005b) concluded that the X-ray and optical afterglow emissions of a sample of 10 bursts were consistent with a single component origin. Furthermore, Nardini et al. (2006) studied the spectral energy distribution over the optical and X-ray bands of 24 bursts and concluded that the majority of the bursts were consistent with a single emission component. Furthermore, L. Li et al. (2015, in preparation) recently studied the evolution of color indices for a large GRB sample that have good multiband, optical afterglow observations and also find that most of these GRBs can be explained by the standard synchrotron afterglow model.

Although the standard afterglow model is successful in explaining the overall features of the observed afterglow behavior, the increasing amount of observational data, as well as the gain in detail, clearly call for additional assumptions to be made, e.g., on the ambient density profile, prolonged energy injection, jet, accretion around the central engine, and viewing angle effects (e.g., Mészáros et al. 1998; Dai & Lu 1998, 2002; Rees & Mészáros 1998; Rhoads 1999; Sari et al. 1999; Chevalier & Li 1999, 2000; Huang et al. 2000; Zhang & Mészáros 2001; Pisani et al. 2013; Zhang 2014; Ruffini et al. 2014). Findings based on these assumptions indicate that the afterglow emission is more complicated than that predicted by the simplest standard afterglow model, and further developments studying "poststandard effects" were made by various authors. For example, Chevalier & Li (1999) took a wind-like medium into account, that is, a circumburst medium density that decreases as r⁻², where r is the distance from the center, due to the wind from the progenitor. This gave rise to new, quantitative "closure relations" of the external-shock model. Detailed modeling of early and late afterglow properties in a wind medium has been carried out by Panaitescu & Kumar (2000), Wu et al. (2003), Kobayashi & Zhang (2003), and Kobayashi et al. (2004). The energy-injection models have been studied by Dai & Lu (1998), Rees & Mészáros 1998, Sari & Mészáros (2000), and Zhang & Mészáros (2001). The jet models are studied in detail by Rhoads (1999), Sari et al. (1999), Huang et al. (2000), and Wu et al. (2004), and more recently, simulations have been carried out by Van Eerten & MacFadyen (2013) and Duffell & MacFadyen (2014). Zhang et al. (2006) derived the "closure relations" of relevant external-shock models and collected them in their Table 2. A more complete review of closure relations can be found in Gao et al. (2013).

The Swift satellite (Gehrels et al. 2004) has provided an unprecedented sample of a long time series of broad energy band observations of bursts. Simultaneous observations with the Swift instruments BAT, XRT, and UVOT and ground-based optical telescopes have revolutionized our understanding but have also revealed some critical problems with the standard model (e.g., Zhang 2007, 2011). In contrast to the expected results from the standard model, many bursts showed a chromatic behavior between that of the X-ray and optical bands, that is, showing different behaviors at different frequencies. For instance, Panaitescu et al. (2006) studied six Swift bursts that exhibited distinctly different behaviors of the X-ray and optical light curves, prompting them to argue for a different origin of the emission in the two bands, at least for the bursts in the study. Moreover, Oates et al. (2009) studied a sample of 26 Swift bursts and showed that the XRT X-ray and UVOT optical light curves have similar statistical properties. However, they found that they are remarkably different during the first 500 s after the BAT trigger. The late-time light curves have a better resemblence. Zhang et al. (2007) and Liang et al. (2007, 2008, 2009), who studied X-ray afterglow and their consistency with the external-shock model in a series of papers, also suggest that the X-ray and optical bands have different origins within the emission. Systematic studies of Swift X-ray afterglows and their consistency with external-shock models have also been carried out by O'Brien et al. (2006), Willingale et al. (2007), Evans et al. (2007, 2009), and Zaninoni et al. (2013). In Ruffini et al. (2014), GRBs originating from a binary system are investigated, and one family of GRBs is classified as being from a standard black hole formed by a neutron star accreting to the critical mass of gravitational collapse; this family holds typical scaling laws and overlapping features in the X-ray afterglow (Pisani et al. 2013).

In this paper, we analyze a sample of 87 bursts, focusing on the clean afterglow emission after omitting the components that are related to internal shock origin, afterglow reverse shock emission, and the late-time associated supernova component. These bursts are observed by XRT and simultaneously by UVOT or ground-based telescopes and are able to make a large statistical study. We examine whether, and how often, the X-ray and optical afterglow light curves are consistent with the external-shock model. This paper is organized as follows. We define the data sample, and the fitting results are presented and analyzed in Section 2. We carry out a detailed statistical analysis in Section 3. In Section 4, we perform several tests to the external-shock models for all of our afterglow samples. A more detailed investigation regarding how well the models can interpret the data can be found in Wang et al. (2015). We discuss the properties of afterglow break behaviors in Section 5, and our conclusions and discussion are in Section 6.

We compiled a database of 270 GRBs with optical afterglow detections made from 1997 February to 2013 December that have been presented in GCN Circulars and in the literature. We included observations made by Swift/UVOT, as well as other ground-based telescopes, such as the Gamma-ray burst optical/near-infrared detector (Greiner et al. 2007, 2008), ROTSE-III (Akerlof et al. 2003), TAROT (Klotz et al. 2009), RAPTOR (Vestrand et al. 2004), and REM (Zerbi et al. 2001). For all of these bursts we also compile the full sample of XRT data from the Swift data archive.

In order to study the afterglow emission expected from the external-shock model, we need to define a subsample of these bursts because multiple emission components can exist. For instance, Zhang et al. (2006) summarized the X-ray afterglow behavior with a synthetic cartoon of an X-ray light curve based on the observational data from the Swift archive, showing that the X-ray afterglow included five emission components that have distinctly different physical origins or processes. Likewise, Li et al. (2012) summarized the optical afterglow behavior with a synthetic cartoon of optical light curves, based on an extensive analysis of the optical data statistics, morphology, and theory. In contrast to the X-ray afterglow, they identified eight clear emission components in the optical afterglow light curves, which may have different physical origins or processes (Li et al. 2012). To extract the cleanest possible signal of the afterglow component, we therefore apply the following four rules:

Rule 1: We exclude the following components: (1) the steep decay in early X-ray afterglow light curves, which is involved in the tail of prompt emission related to internal energy dissipation (Zhang et al. 2006); (2) the flare behavior that may be related to the late-time activity of the central engine (Burrows et al. 2005; Fan & Wei 2005; Dai et al. 2006; Perna et al. 2006; Proga & Zhang 2006; Zhang et al. 2006); (3) using the relation $\alpha \geqslant 2+\beta$, we exclude the "internal plateau" samples (Troja et al. 2007; Liang et al. 2007); (4) the early optical reverse shock emission; and (5) the GRBs-SNe bumps that are typically detected in the optical afterglow light curves at late times.

Rule 2: We restrict our sample to include only bursts for which there are good, simultaneous observations in both the X-ray and optical bands because we want to compare the optical and X-ray light curve behaviors.

Rule 3: All of the optical light curves correspond to R-band observations. For bursts that only have, or have better, observations in other bands, we corrected the observations into the R band by assuming that a spectrum ${{F}_{\nu }}={{F}_{\nu ,0}}{{\nu }^{-{{\beta }_{{\rm o}}}}},{{\beta }_{{\rm o}}}$ is the optical spectral index. This is particularly the case when a longer time series exists in other bands.

Rule 4: We fit the broadest-possible time interval with either a single power law (SPL) or a broken power law (BKPL). Such a fit will ignore wiggles on short timescales. For instance, GRB 051109A is fitted with a BKPL, which ignores the flare behavior (maybe a late rebrightening bump behavior) that is embedded in the light curve in the narrow time range [10⁴, 10⁵ s].

These rules give us a total of 87 bursts, which we classify into four samples depending on their light curves in the X-ray and optical bands. We denote these samples as Group I, Group II, Group III, and Group IV, which are defined in the following way:

• 1.  
  Group I: all of the bursts for which there is a clear light-curve break detected in both the X-ray and optical afterglows, as shown in the left-hand panel in the cartoon of light curves in Figure 1. Of 87 GRBs, our full sample consists of 24 such bursts, and the BKPL fits to the light curves are shown in Figure 3.
• 2.  
  Group II: all bursts for which there is a break detected only in the X-rays, while the optical band generally shows an SPL decay. Note that a few bursts have a detected afterglow onset that is followed by the characteristic decay of the sample. A cartoon of a typical light curve in this sample is shown in the center-left panel in Figure 1. Of 87 GRBs, our full sample consists of 24 such bursts, and the fits to the light curves are shown in Figure 4. We note that Group II GRBs do not always point toward a chromatic behavior. This is because in many GRBs there is no optical data around the X-ray break time, so the existence of an X-ray break around the same time is not ruled out. This issue is fully addressed in Wang et al. (2015).
• 3.  
  Group III: all bursts for which the break is detected only in the optical band, while the X-ray light curve in general is described by an SPL. Note that a few bursts have a detected afterglow onset. A cartoon example is shown in the center-right panel in Figure 1. Of 87 GRBs, this sample comprises nine bursts, and the fitted afterglows are depicted in Figure 5. Again, some GRBs in this group do not have X-ray data around the optical break time, so they are not necessarily chromatic (Wang et al. 2015).
• 4.  
  Group IV: all bursts that do not have any obvious breaks, either in the X-ray or in the optical bands. They are fitted with SPL functions. A few of these bursts have afterglow onsets. Of 87 GRBs, this sample includes 30 bursts, and the best-fit light curves are shown in Figure 6.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

One example of the light curve fittings is given in Figure 2.

These light curves were fitted by either an SPL

$$\begin{eqnarray}&&F={{F}_{0}}\;{{t}^{-\alpha }},\end{eqnarray} \tag{ 1 }$$

or a smooth BKPL

$$\begin{eqnarray}&&F={{F}_{0}}\;{{[{{(t/{{t}_{b}})}^{{{\alpha }_{1}}\omega }}+{{(t/{{t}_{b}})}^{{{\alpha }_{2}}\omega }}]}^{-1/\omega }}.\end{eqnarray} \tag{ 2 }$$

Here α, α₁, and α₂ are the temporal slopes, t_b is the break time, ${{F}_{b}}={{F}_{0}}\;{{2}^{-1/\omega }}$ is the flux of the break time, ω describes the sharpness of a break: the smaller the ω, the smoother the break. We often fixed this parameter to 3 (for a few cases which have a smoother break we fixed it to 1, which improves the fit). Equation (2) is used both for light curves that have a break and light curves that exhibit the afterglow onset. These functions are fitted to the data with a nonlinear, least-squares fitting using the Levenberg–Marquardt algorithm. We use an IDL routing called mpfitfun.pro¹⁷ (Markwardt 2009). Errors in the parameters are calculated either through error propagation or through the bootstrap method, for which we produce 1,000 realizations that give a Gaussian distribution, whose standard deviation is an estimate of the parameter error.

The fitting results of the X-ray afterglow, including the break time (T_x,b), the power-law decay slopes before (α_x,1) and after (α_x,2) the break¹⁸ , and time intervals during which the model fittings are performed¹⁹ , [T_x,1,T_x,2], are summarized in Table 1. Based on the fitting parameters, we also calculated the fluence of the X-ray band (S_x), which is the integral of the observed flux over the time intervals. We also calculate the properties in the rest frame for bursts that have a measured redshift (80 of 87 bursts), including the isotropic luminosity at the break (L_x,b) and the total isotropic energy release in the X-ray band (E_x,iso). The photon spectral indices before (Γ_x,1) and after (Γ_x,2) the break time are also given and were retrieved from the Swift website.²⁰ Most of the spectral indices were derived using the PC mode. Our results are reported in Table 1.

Similarly, for the X-rays, in Table 2, we report the fitting results of the optical afterglow over the time interval [T_o,1,T_o,2], the break time, T_o,b, the power-law slope before (after) the break, α_o,1 (α_o,2), the fluence in the optical band, S_o, the break luminosity, L_o,b, and the isotropic energy release in the optical band, E_R,iso. We also include in the table the optical afterglow spectral index, β_o, and the redshift, z, found in the literature. We converted the observed magnitudes to energy fluxes and calculated the magnitude of the K correction by $k=-2.5({{\beta }_{{\rm o}}}-1){\rm log} (1+z)$ and the Galactic extinction correlation was calculated by a reddening map presented in Schlegel et al. (1998). Any flux contribution in the very late epoch (∼10⁶ s after the GRB trigger) that possibly comes from the host galaxy is also subtracted.

A comparison of the relation properties between the X-ray and the optical afterglow is summarized in Table 3. We compare the X-ray and optical data set and, in particular, the fit results of the light curves. We present the ratio of break times ($\frac{{{T}_{{\rm x},b}}}{{{T}_{{\rm o}}},b}$). Next, we calculate the differences in power-law slopes in different temporal segments or different energy bands, as well as differences in the spectral slopes in the two bands. We define

$$\begin{eqnarray}&&{\Delta }\alpha \equiv {{\alpha }_{{\rm x}}}-{{\alpha }_{{\rm o}}},\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{\Delta }{{\alpha }_{1}}\equiv {{\alpha }_{{\rm x},1}}-{{\alpha }_{{\rm o},1}},\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{\Delta }{{\alpha }_{2}}\equiv {{\alpha }_{{\rm x},2}}-{{\alpha }_{{\rm o},2}},\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{\Delta }{{\alpha }_{{\rm x}}}\equiv {{\alpha }_{{\rm x},2}}-{{\alpha }_{{\rm x},1}},\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{\Delta }{{\alpha }_{{\rm o}}}\equiv {{\alpha }_{{\rm o},2}}-{{\alpha }_{{\rm o},1}},\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{\Delta }\beta \equiv {{\beta }_{{\rm x}}}-{{\beta }_{{\rm o}}},\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&{\Delta }{{\beta }_{{\rm x}}}\equiv {{\beta }_{{\rm x},2}}-{{\beta }_{{\rm x},1}},\end{eqnarray} \tag{ 9 }$$

and present their values in Table 3. We also calculate the X-ray-to-optical ratio of the break luminosity ($\frac{{{L}_{{\rm X},{\rm b}}}}{{{L}_{{\rm R},{\rm b}}}}$), isotropic energy release ($\frac{{{E}_{{\rm X},{\rm iso}}}}{{{E}_{{\rm R},{\rm iso}}}}$), as well as the flux at 1 hr ($\frac{{{F}_{{\rm X}}}}{{{F}_{{\rm o}}}}$).

In total, 833 Swift GRBs were detected by the Swift satellite up to the end of 2013 (from 2004 December 18 to 2013 December 31), of which only one-tenth (87/833) had good, simultaneously observed X-ray and optical afterglow data. This fraction is far smaller than the total 691 of 833 Swift GRBs that had good XRT data observations.

The distributions of statistical properties comparing the X-ray with the optical bands are shown in Figure 7. We compare the distribution of the decay slopes of the X-ray and the optical bands in Figure 7(a). We gather all of the information on the X-ray and optical decay slopes. In total, we had 135 decay slope indices for the X-ray (α_x) and 120 for the optical bands (α_o).²¹ The distribution of α_x ranges within [−0.01, 2.89] and of α_o within [−0.08, 2.59]. The X-ray emission usually decays faster than the optical emission.²² This is consistent with previous findings (e.g., Oates et al. 2011; Zaninoni et al. 2013) and is consistent with the theoretical predictions of the afterglow model (e.g., Sari et al. 1998; Gao et al. 2013). In Figures 7(b)–(e), we show the distributions of the break times and the break luminosity for the bursts that have a break behavior in their afterglow light curves (t_b, L_b; X-ray: Group I and Group II; optical: Group I and Group III), of the fluences (S_x, S_o), and finally the distribution of the isotropic energy (E_X,iso, E_R,iso). The break time ranges from ∼100 to ∼10⁶ s with a typical value ∼10⁴ s, and the X-ray and the optical distributions are similar. The peak distribution of X-ray fluence is ∼10⁻⁶ erg cm⁻², and in the optical band it is ∼10⁻⁸ erg cm⁻². The isotropic break luminosity (L_X,b or L_R,b), with a wider distribution, ranges from ∼10⁴³ erg s⁻¹ to ∼10⁴⁹ erg s⁻¹ for the X-rays with a typical value of 10⁴⁷ erg s⁻¹ and from ∼10⁴² erg s⁻¹ to ∼10⁴⁷ erg s⁻¹ for the optical band with a typical value of 10⁴⁶ erg s⁻¹. The isotropic energy release in the X-ray band (E_X,iso) generally ranges from ∼10⁴⁹ to ∼10⁵³ erg, whereas for the optical band (E_R,iso) it ranges from ∼10⁴⁸ to ∼10⁵¹ erg. The peak distribution value of the X-ray band is ∼10⁵² erg and of the optical band ∼10⁵⁰ erg.

Zoom In Zoom Out Reset image size

Figure 8 shows the correlation between the X-ray and the optical properties for Group I. We examine the pair correlations for Δα₁−Δα₂ and Δα_x−Δα_o and obtain r = 0.40 with p = 0.42 for Δα₁−Δα₂ and r = 0.40 with p = 0.55 for Δα_x−Δα_o, which indicates that these are only weak correlations. Notice that the peak value of the distribution for T_x,b/T_o,b is around zero in the logarithmic space. This indicates that a good fraction of Group I GRBs have the same break time in both the X-ray and the optical bands. This is generally related to the achromatic break behavior. A detailed analysis of the chromaticity of afterglow behavior is presented in Wang et al. (2015).

Zoom In Zoom Out Reset image size

We compare the start (T_x,1, T_o,1) and end times (T_x,2, T_o,2) of the X-ray and optical band observations in Figure 9. The T_x,1−T_o,1 panel (Figure 9(a)) shows that the data points generally cluster around the equality line in the logarithmic space, and the X-rays range from ∼10² to ∼10⁵ s and the optical band ranges from ∼10 to ∼10⁵ s. Most of the data points are below the equality line in the logarithmic space for the T_x,2−T_o,2 panel (Figure 9(b)), and the distribution is from ∼10⁴ to ∼10⁷ s for the X-ray band and from ∼10² to ∼10⁷ s for the optical band. This indicates that the clear afterglow component was observed for the optical band earlier than for the X-ray band (it is likely that the early X-ray afterglow is usually polluted by the steep decay) and also for the end time of the observation.

Zoom In Zoom Out Reset image size

Our discussion concerns the slow cooling case because most GRBs in our sample have a starting time (T_x,1) hundreds of seconds after the trigger. If the cooling frequency (${{\nu }_{c}}$) lies between that of the X-ray (${{\nu }_{{\rm x}}}$) and the optical (${{\nu }_{{\rm o}}}$) bands (ν_m < ν_o < ν_c < ν_x), then the differences in the temporal indices between the X-ray and the optical bands are the following: (1) for the cases with no energy injection (q = 1), one has Δα = $\frac{1}{4}$ for the ISM scenario (Sari et al. 1998) and Δα = $-\frac{1}{4}$ for the wind scenario (Chevaler & Li 2000); (2) for the energy-injection case with q = 0 due to pulsar spin-down (Dai & Lu 1998; Zhang & Mészáros 2001), one has ${\Delta }\alpha =\frac{1}{2}$ for the ISM case and ${\Delta }\alpha =-\frac{1}{2}$ for the wind case, which have the maximum difference between the X-ray and optical bands (e.g., Oates et al. 2009); (3) the difference in the spectral indices between the X-ray and the optical bands is at most ${\Delta }\beta =\frac{1}{2}$ for both the ISM and the wind models (the cooling break). If the cooling frequency is above the X-ray band (ν_m < ν_o < ν_x < ν_c), one has the same spectral and temporal indices in both the X-ray and optical bands, regardless of the medium type and whether or not there is energy injection. The ranges of relations in the temporal and spectral indices are summarized in Table 5 (see also Table 2 of Zhang et al. 2006). The expressions of Δα and Δβ can be summarized in the following two formulae.

$$\begin{eqnarray}{\Delta }\alpha =\left\{ \begin{array}{ccccccccccccccc} \frac{1}{2}, & {{\nu }_{m}}\lt {{\nu }_{{\rm o}}}\lt {{\nu }_{c}}\lt {{\nu }_{{\rm x}}}, \\ {} & {\rm injection}(q=0),({\rm ISM}) \\ \frac{1}{4}, & {{\nu }_{m}}\lt {{\nu }_{{\rm o}}}\lt {{\nu }_{c}}\lt {{\nu }_{{\rm x}}}, \\ {} & {\rm no}\;\;{\rm injection}(q=1),({\rm ISM}) \\ 0, & {{\nu }_{m}}\lt {{\nu }_{{\rm o}}}\lt {{\nu }_{{\rm x}}}\lt {{\nu }_{c}}, \\ {} & {\rm injection}(q=0){\rm and}\;\;{\rm no}\;\;{\rm injection}(q=1), \\ {} & ({\rm ISM}\;\;{\rm and}\;\;{\rm Wind}) \\ -\frac{1}{4}, & {{\nu }_{m}}\lt {{\nu }_{{\rm o}}}\lt {{\nu }_{c}}\lt {{\nu }_{{\rm x}}}, \\ {} & {\rm no}\;\;{\rm injection}(q=1),({\rm wind}) \\ -\frac{1}{2}, & {{\nu }_{m}}\lt {{\nu }_{{\rm o}}}\lt {{\nu }_{c}}\lt {{\nu }_{{\rm x}}}, \\ {} & {\rm injection}(q=0),({\rm wind}) \\ \end{array} \right.\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}{\Delta }\beta =\left\{ \begin{array}{ccccccccccccccc} 0, & {{\nu }_{m}}\lt {{\nu }_{{\rm o}}}\lt {{\nu }_{{\rm x}}}\lt {{\nu }_{c}}, \\ {} & {\rm injection}(q=0){\rm and}\;\;{\rm no}\;\;{\rm injection}(q=1), \\ {} & ({\rm ISM}\;\;{\rm and}\;\;{\rm Wind}) \\ \frac{1}{2}, & {{\nu }_{m}}\lt {{\nu }_{{\rm o}}}\lt {{\nu }_{c}}\lt {{\nu }_{{\rm x}}}, \\ {} & {\rm injection}(q=0){\rm and}\;\;{\rm no}\;\;{\rm injection}(q=1), \\ {} & ({\rm ISM}\;\;{\rm and}\;\;{\rm Wind}) \\ \end{array} \right.\end{eqnarray} \tag{ 11 }$$

4.1. Temporal and Spectral Indices for the Four Groups

In Figure 10, we present the correlations between the temporal decay indices in the optical band versus those in the X-ray band (α_x−α_o panel). We present the correlation separately for our four groups. In the figure we mark the range deviating by −0.25 and 0.25 from the equality line, which is the range expected from the standard forward-shock model for different circumstellar media (ISM and wind-like), as well as the range of −0.5, 0.5 (dashed lines), which is related to the energy-injection model for different circumstellar media (ISM and wind-like). Note that if the points lying above the equality line decay more quickly in the optical than in the X-ray band, then the data are consistent with the wind-like model; if the points below the equality line decay more quickly in the X-ray than in the optical band, then the data are consistent with the ISM model. In Figure 11, we similarly present the correlation between the spectral indices in the optical band versus those in the X-ray band for each of the four groups. We indicate the line corresponding to β_x−β_o lying within the range [0.0, 0.5], which is expected for different spectral regimes of the external-shock model.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The differences in temporal indices between the X-ray and the optical bands are compared to the empirical relations for various external-shock models (Urata et al. 2007) and are summarized in Table 4. For Group I (Figure 10(a)), 18 of 24 and 16 of 24 data points fall within the external-shock model ranges (−0.5, 0.5) for α_x,1−α_o,1 and α_x,2−α_o,2, respectively. Out of these, 13 of 24 and 10 of 24 data points fall within the standard afterglow model ranges (−0.25, 0.25). Moving to Group IV (Figure 10(d)), there are 27 of 30 data points that fall within (−0.5, 0.5), which includes 13 of 30 data points within (−0.25, 0.25) for α_x,2−α_o,2. This indicates that the data for Group I and Group IV satisfy the external-shock model for the global behavior. However, for Group II, although there are 17 of 24 data points that fall within [−0.5, 0.5] and it includes five of 24 points within [−0.25, 0.25] for α_x,1−α_o,2, only eight of 24 data points fall within (−0.5, 0.5) and only rare points (only one case) fall within the ranges of (−0.25, 0.25) for α_x,2−α_o,2. Similar to Group II, for Group III, three of nine data points fall within [−0.5, 0.5] and no point within [−0.25, 0.25] for α_x,2−α_o,1. However, there are eight of nine data points that fall within [−0.5, 0.5] and six of nine points within [−0.25, 0.25] for α_x,2−α_o,2. These observations indicate that the data are generally inconsistent with the external-shock model for the global behavior for Group II and Group III. Because these two groups include GRBs that potentially have a chromatic behavior (which is not predicted in the simplest afterglow model), this result is understandable: the GRBs that do not comply with the models may require two emission components to interpret the X-ray and optical emissions, respectively. One of the two components should still come from the external shock. The data indeed show that for a certain section of the afterglow the data are still consistent with the external-shock model. A full investigation of this issue is presented in Wang et al. (2015).

Table 4.  Test the Temporal Indices for Various External-shock Models

	Totala	Chromatic?b	Injectionc	Standardd	Standardd	Injectionc	Chromatic?b	Externale	Standarde
		Wind	Wind	Wind	ISM	ISM	ISM	ISM and Wind	ISM and Wind
		<−0.5	(−0.5,−0.25)	(−0.25, 0.0)	(0.0, 0.25)	(0.25,0.5)	>0.5	(−0.5, 0.5)	(−0.25, 0.25)
Group Samples
Group I
Δα(αx,1−αo,1)	24	1	1	8	5	4	5	18(75%)/23(96%)f	13(54%)/16(67%)
Δα(αx,2−αo,2)	24	3	3	5	5	3	5	16(67%)/21(88%)	10(42%)/16(67%)
Group II
Δα(αx,1−αo,2)	24	7	11	3	2	1	0	17(71%)/20(83%)	5(21%)/10(42%)
Δα(αx,2−αo,2)	24	0	0	0	1	7	16	8(33%)/14(58%)	1(4%)/7(29%)
Group III
Δα(αx,2−αo,1)	9	0	0	0	0	3	6	3(33%)/6(67%)	0(0)/1(11%)
Δα(αx,2−αo,2)	9	1	2	4	2	0	0	8(89%)/9(100%)	6(67%)/8(89%)
Group IV
Δα(αx,2−αo,2)	30	1	0	6	7	14	2	27(90%)/29(97%)	13(43%)/20(67%)
Epoch Samples
epoch 1
Δα(αx−αo)	50	6	7	9	7	10	11	33(66%)/42(84%)	16(32%)/23(46%)
epoch 2
Δα(αx−αo)	97	8	13	15	15	22	24	65(67%)/76(78%)	30(31%)/45(46%)
epoch 3
Δα(αx−αo)	80	12	6	13	13	15	21	47(59%)/60(75%)	26(33%)/37(46%)
epoch 4
Δα(αx−αo)	35	4	2	5	7	12	5	26(74%)/28(80%)	12(34%)/19(54%)

^aTotal number of data points for each Group or epoch sample. ^bNumber of data points for which Δα < −0.5 or Δα > −0.5, which do not fall into our external-shock model ranges. ^cNumber of data points for the energy-injection model, ISM and wind-like medium. ^dNumber of data points for the standard afterglow model, ISM and wind-like medium. ^eNumber of data points (brackets correspond to the fraction) satisfied with the external-shock and standard afterglow model ranges for each Group or epoch sample. ^fNot considering error bar/considering error bar.

Download table as:  ASCIITypeset image

Based on the statistics of the α_x−α_o relation with the definitions of the four Group samples, within the error, in total we find 54 of 87 (62%) GRBs that are consistent with the standard afterglow model (Δα = [−0.25, 0.25]). However, when more advanced modeling is invoked (e.g., long-lasting central engine emission, reverse shock emission, or structured jets), 79 of 87 (91%) GRBs may also satisfy the external-shock models (Δα = [−0.5, 0.5]). We also find that 53 of 87 (61%) GRBs correspond to the ISM model, and 34 of 87 (39%) GRBs correspond to the wind model.

The spectral indices for the GRBs in each Group are shown in the β_x−β_o panels of Figure 11; within errors, the data generally fall well within the range of [0, 0.5], which is expected for the external-shock models (the data of Group II have a little scatter around the external-shock model ranges). Notice that we have collected the optical spectral indices from the literature. No time-resolved optical spectral analysis was available for most GRBs. A detailed time-resolved joint analysis of the afterglow power density spectrum using X-ray and optical data will be presented in a separate paper (Li et al. 2015, in preparation).

In order to find the properties of the mean values of the temporal and spectral indices, in Figure 12 we show the distributions of the pre- and postbreak decay slopes and the spectral indices for the X-ray and the optical bands and compare them.²³ All of the distributions can be reasonably fitted with a Gaussian function in order to find the mean values. The decay slopes of the prebreak slopes are 0.46 ± 0.01 (optical) and 0.68 ± 0.03 (X-ray). This indicates that a shallow decay segment is commonly detected in both the X-ray and optical bands at the early time. The corresponding values of the postbreak slopes are 1.23 ± 0.03 (optical) and 1.48 ± 0.02 (X-ray), for which the value of α_x,2−α_o,2 is 0.25, and they fit very well with the case of an ISM medium and slow cooling within the spectrum regime (ν_m < ν_o < ν_c < ν_x). The spectral indices are 0.99 ± 0.23 (X-ray) and 0.69 ± 0.25 (optical), also very consistent with the external-shock model.

Zoom In Zoom Out Reset image size

4.2. Temporal Indices for Different Epochs

In order to study the evolution in the relations, we compare the X-ray and the optical temporal indices at different epochs and divide the data into four epochs and then test the data against the external-shock model.

We study a sample of GRBs that is selected according to the following: (1) We select the GRBs with redshift measurements (80 out of 87 bursts) so that their observed times can be corrected to the burst rest frames (Melandri et al. 2014). (2) We chose four epochs in the rest frames: epoch 1: <10³ s, epoch 2: 10³−10⁴ s, epoch 3: 10⁴−10⁵ s, and epoch 4: >10⁵ s. The reason to choose these different epochs is to investigate how the joint X-ray/optical emission properties are consistent or not consistent with each other in different epochs. (3) We selected the cases for which there was at least one epoch during which both X-ray and optical data are available. (4) If a light curve break time (X-ray or optical) lies within one epoch, we include both the pre- and postdecay indices in this epoch. Using these selection criteria, we finally get 50, 97, 80, and 35 cases in the four epochs, respectively. For the different samples, the number of cases are (19, 45, 41, 11) for Group I, (14, 26, 15, 7) for Group II, (5, 7, 5, 2) for Group III, and (12, 19, 19, 15) for Group IV.

As shown in Figure 13, in the α_x−α_o panels for our epoch samples, considering the error, 42 of 50 (84%), 76 of 97 (78%), 60 of 80 (75%), and 28 of 35 (80%) data points for Δα fall within our external-shock model ranges (−0.5, 0.5) for epochs 1, 2, 3, and 4, respectively. Out of these, 23 of 50 (46%), 45 of 97 (46%), 37 of 80 (46%), and 19 of 35 (54%) data points are within the standard afterglow model ranges (−0.25, 0.25). This indicates that the data are generally consistent with the external-shock models and there is no significant evolutionary effect in our epoch samples. The results can also be seen in Table 3.

Zoom In Zoom Out Reset image size

In order to consolidate these results, and in particular to show that the data are not randomly distributed, we compare the observational data with a simulation using the Monte Carlo (MC) method. We simulated a random set of decay slopes with a Gaussian distribution in both the X-ray and optical bands, lying within the observed range of [0, 3]. For each epoch, the simulated set contains the same number of points as the observed set. The simulated points (red open circles) are also shown in Figure 13. We evaluate the consistency between the simulated and observed sets by a Kolmogorov–Smirnov (K–S) test and use the probability of the K–S test (P_K−S) to measure the consistency. A larger value of P_K−S indicates a better consistency. If a value of P_K−S is less than 0.1, then there is no consistency. Our results are shown in Table 5, which suggests that the simulated and observed data are generally inconsistent with each other. We conclude that the observed data sets are not randomly distributed.

Table 5.  Probability of the K–S Test for Our Epoch Samples

epochs	PK−S(αx,obs,αx,Sim)a	PK−S(αo,obs,αo,Sim)b
epoch 1	0.02	1.32 × 10−10
epoch 2	8.36 × 10−3	2.28 × 10−9
epoch 3	0.24	7.32 × 10−7
epoch 4	0.03	0.49

^aThe probability of the K–S test for observed decay slope and simulated decay slope for the X-ray band. ^bThe probability of the K–S test for observed decay slope and simulated decay slope for the optical band.

Download table as:  ASCIITypeset image

The canonical X-ray light curve (Zhang et al. 2006) can be described as an initially steep decay phase I (with a temporal power-law index, α_x ∼−3 and ${{F}_{({\rm t})}}\propto {{t}^{\alpha }}$) that changes to a shallow decay segment (or plateau) II (−1.0 < α_x < 0). This phase is then followed by a normal decay segment III (−1.5 < α_x < 1.0), and finally the light curve changes into a post-jet-break decay IV. Compared to the X-rays, the synthetic optical light curves can generally be summarized as eight emission components with distinct physical origins²⁴ , and the segments II–IV are usually observed as the counterparts in the optical afterglow light curves (see Li et al. 2012, Liang et al. 2013). Thus, break behaviors in afterglow light curves are commonly observed, the shallow-to-normal breaks at early times and the normal-to-jet-break behavior at late times. Another possibility is a spectral break that is due to the cooling frequency crossing the X-ray or optical band. We summarized the properties of afterglow break behaviors, comparing the X-ray with optical bands and their statistical behaviors, in Table 6.

Table 6.  The Closure Relation of Afterglows in Gamma-ray Bursts

	α(q = 1)	α(0 ≤ q < 1)	α	β	α(β)(q = 1)	α(β)(0 ≤ q < 1)	α(β)
	No Injection	Injection	Jet		No Injection	Injection	Jet
	ISM, Slow cooling
ν < υm	$\frac{1}{2}$	$\frac{8-5q}{6}$	$\frac{1}{2}$	$\frac{1}{3}$	$\alpha =\frac{3\beta }{2}$	$\alpha =(q-1)+\frac{(2+q)\beta }{2}$	$\alpha =\frac{3\beta }{2}$
${{\upsilon }_{m}}\lt \nu \lt {{\upsilon }_{c}}$	$\frac{3(1-p)}{4}$	$\frac{(6-2p)-(p+3)q}{4}$	$\sim -p$	$\frac{1-p}{2}$	$\alpha =\frac{3\beta }{2}$	$\alpha =(q-1)+\frac{(2+q)\beta }{2}$	$\alpha =2\beta +1$
$\nu \gt {{\upsilon }_{c}}$	$\frac{2-3p}{4}$	$\frac{(4-2p)-(p+2)q}{4}$	$\sim -p$	$-\frac{p}{2}$	$\alpha =\frac{3\beta -1}{2}$	$\alpha =\frac{q-2}{2}+\frac{(2+q)\beta }{2}$	$\alpha =2\beta$
	ISM, Fast cooling
$\nu \lt {{\upsilon }_{c}}$	$\frac{1}{6}$	$\frac{8-7q}{6}$	⋯	$\frac{1}{3}$	$\alpha =\frac{\beta }{2}$	$\alpha =(q-1)+\frac{(2-q)\beta }{2}$	⋯
${{\upsilon }_{c}}\lt \nu \lt {{\upsilon }_{m}}$	$-\frac{1}{4}$	$\frac{2-3q}{4}$	⋯	$-\frac{1}{2}$	$\alpha =\frac{\beta }{2}$	$\alpha =(q-1)+\frac{(2-q)\beta }{2}$	⋯
$\nu \gt {{\upsilon }_{m}}$	$\frac{2-3p}{4}$	$\frac{(4-2p)-(p+2)q}{4}$	⋯	$-\frac{p}{2}$	$\alpha =\frac{3\beta -1}{2}$	$\alpha =\frac{q-2}{2}+\frac{(2+q)\beta }{2}$	⋯
	Wind, Slow cooling
$\nu \lt {{\upsilon }_{m}}$	0	$\frac{(1-q)}{3}$	$\frac{1}{2}$	$\frac{(q-1)}{3}$	$\frac{1}{2}$	$\alpha =\frac{q}{2}+\frac{(2+q)\beta }{2}$	$\alpha =\frac{3\beta }{2}$
${{\upsilon }_{m}}\lt \nu \lt {{\upsilon }_{c}}$	$\frac{1-3p}{4}$	$\frac{(2-2p)-(p+1)q}{4}$	$\sim -p$	$\frac{1-p}{2}$	$\alpha =\frac{3\beta +1}{2}$	$\alpha =\frac{q}{2}+\frac{(2+q)\beta }{2}$	$\alpha =2\beta +1$
$\nu \gt {{\upsilon }_{c}}$	$\frac{2-3p}{4}$	$\frac{(4-2p)-(p+2)q}{4}$	$\sim -p$	$-\frac{p}{2}$	$\alpha =\frac{3\beta -1}{2}$	$\alpha =\frac{q-2}{2}+\frac{(2+q)\beta }{2}$	$\alpha =2\beta$
	Wind, Fast cooling
$\nu \lt {{\upsilon }_{c}}$	$-\frac{2}{3}$	$-\frac{(1+q)}{3}$	$...$	$\frac{1}{3}$	$\alpha =\frac{1-\beta }{2}$	$\alpha =\frac{q}{2}-\frac{(2-q)\beta }{2}$	⋯
${{\upsilon }_{c}}\lt \nu \lt {{\upsilon }_{m}}$	$-\frac{1}{4}$	$\frac{2-3q}{4}$	$...$	$-\frac{1}{2}$	$\alpha =\frac{1-\beta }{2}$	$\alpha =\frac{q}{2}-\frac{(2-q)\beta }{2}$	⋯
$\nu \gt {{\upsilon }_{m}}$	$\frac{2-3p}{4}$	$\frac{(4-2p)-(p+2)q}{4}$	⋯	$-\frac{p}{2}$	$\alpha =\frac{3\beta -1}{2}$	$\alpha =\frac{q-2}{2}+\frac{(2+q)\beta }{2}$	⋯
	$p=2.5(p=2,p=3),q=0$
	ISM, Slow cooling
$\nu \lt {{\upsilon }_{m}}$	0.5(0.5, 0.5)	1.3(1.3, 1.3)	0.5(0.5, 0.5)	0.3(0.3, 0.3)	⋯	⋯	⋯
${{\upsilon }_{m}}\lt \nu \lt {{\upsilon }_{c}}$	−1.1(−0.75, −1.5)	0.25(0.5, 0.0)	−2.5(−2.0, −3.0)	−0.75(−0.75, −0.75)	⋯	⋯	⋯
$\nu \gt {{\upsilon }_{c}}$	−1.4(−1.0, −1.75)	−0.25(0.0, −0.5)	−2.5(−2.0, −3.0)	−1.25(−1.25, −1.25)	⋯	⋯	⋯
	ISM, Fast cooling
$\nu \lt {{\upsilon }_{c}}$	0.2(0.2, 0.2)	1.3(1.3, 1.3)	⋯	0.3(0.3, 0.3)	⋯	⋯	⋯
${{\upsilon }_{c}}\lt \nu \lt {{\upsilon }_{m}}$	−0.25(−0.25, −0.25)	0.5(0.5, 0.5)	⋯	−0.5(−0.5, −0.5)	⋯	⋯	⋯
$\nu \gt {{\upsilon }_{m}}$	−1.4(−1.0, −1.75)	−0.25(0.0, −0.5)	⋯	−1.25(−1.25, −1.25)	⋯	⋯	⋯
	Wind, Slow cooling
$\nu \lt {{\upsilon }_{m}}$	0.0(0.0, 0.0)	0.3(0.3, 0.3)	0.5(0.5, 0.5)	0.3(0.3, 0.3)	⋯	⋯	⋯
${{\upsilon }_{m}}\lt \nu \lt {{\upsilon }_{c}}$	−1.6(−1.25,−2.0)	−0.75(−0.5, −1.0)	−2.5(−2.0, −3.0)	−0.75(−0.75, −0.75)	⋯	⋯	⋯
$\nu \gt {{\upsilon }_{c}}$	−1.4(−1.0, −1.75)	−0.25(0.0, −0.5)	−2.5(−2.0, −3.0)	−1.25(−1.25, −1.25)	⋯	⋯	⋯
	Wind, Fast cooling
$\nu \lt {{\upsilon }_{c}}$	−0.7(−0.7, −0.7)	−0.3(−0.3, −0.3)	⋯	0.3(0.3, 0.3)	⋯	⋯	⋯
${{\upsilon }_{c}}\lt \nu \lt {{\upsilon }_{m}}$	−0.25(−0.25, −0.25)	−0.5(−0.2, −0.5)	⋯	−0.5(−0.5, −0.5)	⋯	⋯	⋯
$\nu \gt {{\upsilon }_{m}}$	−1.4(−1.0, −1.75)	−0.25(0.0, −0.5)	⋯	−1.25(−1.25, −1.25)	⋯	⋯	⋯

Notes. This table provides the ranges of relations in the temporal index and the spectral index that are expected from synchrotron emission with or without energy injection and the post-jet-break case (Zhang & Mészáros 2004; Zhang et al. 2006; Sari et al. 1999; Panaitescu et al. 2006). Here p is the electron index and q is the luminosity index. When q = 1, noninjection is the case, and when q = 0, energy injection is the case; and calculated numerical values for each case with p = 2.5, p = 2.0, p = 3.0, q = 1, and q = 0.

Download table as:  ASCIITypeset images: 1 2

The external forward-shock model provides well-predicted spectral and temporal properties and allows for a wide variety of behaviors; it has been successful in explaining the broadband afterglow emission (see, e.g., Rees & Mészáros 1992; Mészáros & Rees 1997; Paczynski & Rhoads 1993; Sari et al. 1998). The relations between the spectral and temporal properties are denoted "closure relations," which have been widely used to interpret the rich multiwavelength afterglow observations. The temporal index α and spectral index β in various afterglow models and the typical numerical values of each case are summarized in Table 5 (see also Table 2 in Zhang et al. 2006; Sari et al. 1998, Sari et al. 1999; Chevalier & Li 1999).

The closure relations (α−β) of the various external-shock models in different regimes for all of the cases are shown in Figure 14, together with the closure relations for the ISM and wind-like models (Liang et al. 2008; Lü & Zhang 2014). The X-ray data are shown in Figure 14(a) for the ISM model and 14(b) for the wind-like model. For X-ray break samples (Group I + Group II), the prebreak segments (the black solid) are shallower than the model predictions according to the standard afterglow model, generally due to the energy injection causing a shallow decay segment. The postbreak segments (the blue triangles) are very consistent with the standard afterglow model or fall well within the jet-break model region. We find that the data for the X-ray nonbreak samples (the pink half-solid points, Group III + Group IV) are very consistent with the standard afterglow model. The optical data are shown in Figure 14(c) and (d) for the ISM model and the wind-like model.²⁵ It can be seen that the optical data in the jet-break regions (which are clustered in our optical break samples, Group I and Group III) are fewer in number than the X-ray data. By contrast, the fraction of GRBs that fall in the jet-break region for optical nonbreak samples is significantly larger than in the X-ray cases. A few GRBs fall in the jet-break regions with spectral regime I (${{\nu }_{{\rm opt}}}\gt {\rm max} ({{\nu }_{m}},{{\nu }_{c}})$).

Zoom In Zoom Out Reset image size

5.1. Energy-injection Break

The shallow decay segments were commonly detected in the early afterglow light curves and are generally believed to be caused by the GRB outflows catching up with the external forward shock and with a long-lasting energy injection into the external shock that refreshes the afterglow emission. It can be defined by the afterglow theory, by which the temporal decay is shallower than what is predicted in the external-shock model.

We defined the shallow decay segment by the closure relation ${{\alpha }_{1}}\lt \frac{3{{\beta }_{1}}-1}{2}$ for the spectral regime I (ν > max(ν_m,ν_c)) for both the ISM and the wind-like models, ${{\alpha }_{1}}\lt \frac{3{{\beta }_{1}}}{2}$ for the ISM model, and ${{\alpha }_{1}}\lt \frac{3{{\beta }_{1}}+1}{2}$ for the wind-like model for spectral regime II (ν_m < ν < ν_c), respectively. We find that generally a very high fraction of ${{\alpha }_{1}}$ are in a shallow decay segment for both the X-ray and the optical break samples. As summarized in Table 7, 42 of 48 GRBs have a shallow decay segment for X-ray break samples (including 21 GRBs for Group I and Group II, respectively). Twenty-seven of 30 GRBs have a shallow decay segment for optical break samples (including 18 of 21 GRBs for Group I and 9 of 9 GRBs for Group III). No significant spectral evolution is observed for the shallow decay segment to the following phase for our 42 X-ray, shallow decay samples (see also in Table 7); only four of 42 GRBs showed significant spectral evolution, indicating that the shallow decay should have an external-shock origin (see also Liang et al. 2007).

The energy-injection behavior can be described as $L(t)\propto {{t}^{-q}}$ (see, e.g., Zhang & Mészáros 2001); here q is the temporal index. The difference between the decay slopes before and after the time of the break depends on the spectral regime observed and the type of ambient medium, which can be summarized as in Table 5 (see also Zhang et al. 2006 in Table 2). This model is called the "energy-injection model" (Zhang et al. 2006) and produces a break in the light curves, which are denoted "injection breaks" (Zhang et al. 2004 for review). The prebreak slope α₁ should give q = 0 for a constant luminosity:

$$\begin{eqnarray}&&{{\alpha }_{1}}=\left\{ \begin{array}{ccccccccccccccc} (q-1)+\frac{(2+q)\beta }{2}=\frac{(2p-6)+(p+3)q}{4}, \\ \quad {{\nu }_{m}}\lt {{\nu }_{{}}}\lt {{\nu }_{c}}{\rm (ISM)} \\ \frac{q}{2}+\frac{(2+q)\beta }{2}=\frac{(2p-2)+(p+3)q}{4}, \\ \quad {{\nu }_{m}}\lt {{\nu }_{{}}}\lt {{\nu }_{c}}{\rm (wind)} \\ \frac{(q-2)}{2}+\frac{(2+q)\beta }{2}=\frac{(2p-4)+(p+3)q}{4}, \\ \quad {{\nu }_{{}}}\gt {{\nu }_{c}}{\rm (ISM}\;{\rm and}\;{\rm wind)}, \\ \end{array} \right.\end{eqnarray} \tag{ 12 }$$

and the postbreak slope α₂ should correspond to q = 1 (for a constant-energy fireball, the scaling law is the same as q = 1; Zhang & Mészáros 2001). Then, according to Zhang et al. (2006), one should have

$$\begin{eqnarray}&&{{\alpha }_{2}}=\left\{ \begin{array}{ccccccccccccccc} \frac{3\beta }{2}=\frac{3(p-1)}{4}=\frac{3{{\alpha }_{1}}+3}{2}, \\ \quad {{\nu }_{m}}\lt {{\nu }_{{}}}\lt {{\nu }_{c}}{\rm (ISM)}, \\ \frac{3\beta +1}{2}=\frac{3p-1}{4}=\frac{3{{\alpha }_{1}}+1}{2}, \\ \quad {{\nu }_{m}}\lt {{\nu }_{{}}}\lt {{\nu }_{c}}{\rm (wind)}, \\ \frac{3\beta -1}{2}=\frac{3p-2}{4}=\frac{3{{\alpha }_{1}}+2}{2}, \\ \quad {{\nu }_{{}}}\gt {{\nu }_{c}}{\rm (ISM}\;{\rm and}\;{\rm wind)}. \\ \end{array} \right.\end{eqnarray} \tag{ 13 }$$

The temporal and spectral properties of the afterglow after the break (the normal decay phase) should satisfy the "closure relation" of the external-shock model (e.g., Zhang & Mészáros 2004), i.e., as described in Table 5. The relation of the α₁−α₂ panel for all of our break samples (Group I + II for the X-ray emission and Group I + III for the optical emission) is shown in Figure 15.

Zoom In Zoom Out Reset image size

Because the calculation of the q and p indices depends on which afterglow model is used, our first step is to use the observed quantity of the temporal index α and spectral index β to determine which spectral regime the burst belongs to. Spectral regime I (ν_obs > max(ν_m,ν_c)) corresponds to the observed band lying above the cooling frequency, and spectral regime II (${{\nu }_{m}}\lt {{\nu }_{{\rm obs}}}\lt {{\nu }_{c}}$) corresponds the observed band lying below the cooling frequency. Because different spectral regimes give different predicted decay indices through the "closure relations" of the external-shock model (e.g., Table 2 of Zhang et al. 2006; Gao et al. 2013), we can combine the α and β information to determine the spectral regime of a temporal segment in each GRB. Because the normal decay phase is not contaminated by energy injection (Zhang 2007), we use its temporal index α to perform the analysis. We first use the temporal index α to calculate the theoretically expressed spectral index (β^') for the different spectral regimes:

$$\begin{eqnarray}\beta _{\alpha }^{\prime }=\left\{ \begin{array}{ccccccccccccccc} \frac{2\alpha +1}{3}, & \nu \gt {\rm max} ({{\nu }_{m}},{{\nu }_{c}}){\rm (ISM}\;{\rm and}\;{\rm wind)} \\ \frac{2\alpha }{3}, & {{\nu }_{m}}\lt \nu \lt {{\nu }_{c}}{\rm (ISM)} \\ \frac{2\alpha -1}{3}, & {{\nu }_{m}}\lt \nu \lt {{\nu }_{c}}{\rm (wind)}. \\ \end{array} \right.\end{eqnarray} \tag{ 14 }$$

Second, we compare this index ($\beta _{\alpha }^{\prime }$) to the observed spectral index (β_obs): ${{\chi }^{2}}={{(\beta _{\alpha }^{\prime }-{{\beta }_{{\rm obs}}})}^{2}}/(\sigma _{\beta (\alpha )}^{2}+\sigma _{\beta ({\rm obs})}^{2})$. We thereby can identify which spectral regime the observation belongs to. We find that for our X-ray observations, 19 of 87 GRBs belong to spectral regime I, and 68 of 87 GRBs belong to spectral regime II. For optical emission the following was found: 9 of 80 GRBs belong to spectral regime I, and 71 of 80 GRBs belong to spectral regime II. Note that the optical spectral index of seven GRBs is not available, and therefore we did not judge the spectral regime of the optical band for those GRBs.

After determining the spectral regimes for each individual burst, the electron spectral index p can be determined through

$$\begin{eqnarray}p=\left\{ \begin{array}{ccccccccccccccc} \frac{4{{\alpha }_{2}}+2}{3}, & \nu \gt {\rm max} ({{\nu }_{m}},{{\nu }_{c}}){\rm (ISM}\;{\rm and}\;{\rm wind)} \\ \frac{4{{\alpha }_{2}}+3}{3}, & {{\nu }_{m}}\lt \nu \lt {{\nu }_{c}}{\rm (ISM)} \\ \frac{4{{\alpha }_{2}}+1}{3}, & {{\nu }_{m}}\lt \nu \lt {{\nu }_{c}}{\rm (wind)} \\ \end{array} \right..\end{eqnarray} \tag{ 15 }$$

Similarly, the luminosity injection index q (for energy-injection cases, calculated by shallow decay segments) can be determined from the temporal index α and the spectral index β:

$$\begin{eqnarray}q=\left\{ \begin{array}{ccccccccccccccc} \frac{2{{\alpha }_{1}}-2{{\beta }_{1}}+2}{1+{{\beta }_{1}}}, & \nu \gt {\rm max} ({{\nu }_{m}},{{\nu }_{c}}){\rm (ISM}\;{\rm and}\;{\rm wind)} \\ \frac{2{{\alpha }_{1}}-2{{\beta }_{1}}+2}{2+{{\beta }_{1}}}, & {{\nu }_{m}}\lt \nu \lt {{\nu }_{c}}{\rm (ISM)} \\ \frac{2{{\alpha }_{1}}-2{{\beta }_{1}}}{1+{{\beta }_{1}}}, & {{\nu }_{m}}\lt \nu \lt {{\nu }_{c}}{\rm (wind)}. \\ \end{array} \right.\end{eqnarray} \tag{ 16 }$$

The distributions of the p and q indices for both the X-ray and optical bands are shown in Figures 16(a) and (b). We calculated the q index for our break samples with the temporal index of the prebreak in a shallow decay segment. The q value ranges within [−0.90, 0.88] for the X-ray band and within [−0.79, 0.90] for the optical band, and the best Gaussian function fittings have center values q_x(c) = 0.46 ± 0.03 and q_o(c) = 0.13 ± 0.11 for the X-ray and the optical bands, respectively. The p index is derived from the observed spectral index of the postbreak decay slope for all of the cases. The p value ranges within [1.78, 4.18], and six of 87 cases had p < 2.0 for the X-ray band and range within [1.28, 4.42], and 17 of 87 cases have p < 2.0 for the optical band. The best Gaussian fittings have center values p_x(c) = 2.63 ± 0.02 and p_o(c) = 2.58 ± 0.04 for the X-ray and the optical bands, respectively.

Zoom In Zoom Out Reset image size

The Δα_(x,o) for each case is summarized in Table 3. The functional dependence of Δα_(x,o)−p, q for the different external-shock model cases can be summarized as follows:

$$\begin{eqnarray}{\Delta }{{\alpha }_{({\rm x},{\rm o})}}=\left\{ \begin{array}{ccccccccccccccc} \frac{1}{4}(p+2)(1-q), & {{\nu }_{{}}}\gt {{\nu }_{c}}{\rm (ISM}\;{\rm and}\;{\rm wind)}, \\ \frac{1}{4}(p+3)(1-q), & {{\nu }_{m}}\lt {{\nu }_{{}}}\lt {{\nu }_{c}}{\rm (ISM)}, \\ \frac{1}{4}(p+1)(1-q), & {{\nu }_{m}}\lt {{\nu }_{{}}}\lt {{\nu }_{c}}{\rm (wind)}. \\ \end{array} \right.\end{eqnarray} \tag{ 17 }$$

The two-dimensional panels between the observed quantity Δα_(x,o) and the value of the model parameters p and q are shown in Figure 16(c). For the Δα_(x,o)−p panel, the various external-shock model lines are well covered by all of the data points when q = −0.5, 0, and 0.5, respectively. For the Δα_(x,o)−q panel, we also find that the external-shock model lines are well covered by the data when we have p = 2.0, 3.0, respectively. Compared with the optical band, we find that the X-ray data are gathered in the regions that have higher p values.

5.2. Jet Break

5.3. Spectral Break

Another possibility that can cause a break in the X-ray (or optical) light curve is a spectral evolution where the spectral peak of the synchrotron emission moves through the X-ray band. However, because in general there is a lack of significant spectral evolution, this possibility is largely excluded (De Pasquale et al. 2006; Nousek et al. 2006). If the cooling frequency (ν_c) crosses the X-ray band, the change of the X-ray spectrum should be reflected by ${\Delta }{{\beta }_{{\rm x}}}=\frac{1}{2}$ for the ISM scenario and ${\Delta }{{\beta }_{{\rm x}}}=-\frac{1}{2}$ for the wind-like circumburst medium scenario (Sari et al. 1999; Chevalier & Li 1999). The change in the light curve should then correspond to ${\Delta }{{\alpha }_{{\rm x}}}=\frac{1}{4}$ for both the ISM and wind-like scenarios.

We show the distributions of Δα_(x,o) for all of our cases with breaks in Figure 17. We have 48 X-ray break cases (Group I + II), and the distribution of Δα_x is in the range [0.44, 2.27], which is larger than Δα_x = 0.25 as expected by the cooling frequency crossing the X-ray band model. This is similar for the 33 cases with optical afterglow breaks (Group I + III). The distribution of Δα_o is in the range [0.47, 1.84]. All bursts have Δα₀ values that are larger than the value expected by the cooling frequency crossing the optical band (Δα_o = 0.25).

Zoom In Zoom Out Reset image size

On the other hand, we derived the photon spectral indices of the prebreak and postbreak portions of the X-ray afterglow for our break samples (Group I + II) and then calculated the spectral indices using β_x = Γ_x−1. The distribution of the spectral indices of the prebreak and postbreak portions are shown in Figure 17(b), and the center values with fitting by Gaussian functions to the distributions are ${{\beta }_{{\rm x},1}}=0.95\pm 0.03$ and β_x,2 = 0.98 ± 0.01. The β_x,2−β_x,1 panel shows that all of the bursts (to within the error) are in the vicinity of the equality line. This indicates that the prebreak and postbreak portions of the light curves lack the significant spectral evolution in most of the cases. Because no optical time resolution was available for analysis of all of the optical break sample, we are not able to examine the change of the spectral index for the transition from prebreak to postbreak.

Both the X-ray and optical afterglow break behavior can thus not be interpreted by the cooling frequency crossing the X-ray or optical band. Indeed, there is also a lack of any significant spectral evolution from prebreak to postbreak.

We extracted the cleanest possible signals of the afterglow emission components, and we include the observations not only by Swift/UVOT, but also by other ground-based telescopes. We obtain a sample of, in total, 87 bursts that have good quality and simultaneous detections in both the X-ray and optical bands. We divided the sample into four groups depending on the existence of breaks in their afterglow light curves, and we examine their behavior in light of various external-shock model empirical relations. We find that the data generally satisfy the various external-shock empirical relations. Our main results can be summarized as follows:

• 1.  
  About 10% of Swift GRBs (up to the end of 2013) had good simultaneous observational data in both the X-ray and optical afterglow. Studying this sample in detail, we find that the X-ray and optical afterglow light curves show similar statistical properties.
• 2.  
  Characterizing the light curves in four different groups (Figure 1), we investigate how well GRBs in each group satisfy the external-shock model predictions. We find that in groups I and IV, a majority of GRBs follow the predictions of the external-shock model. The bursts in Groups II and III have, per definition, chromatic light curves, so one cannot interpret the full evolution directly with the external-shock model. However, a large fraction of these bursts have individual segments in the light curves that do satisfy the relations of the external-shock model. This is consistent with the following picture: for these bursts, one cannot interpret both the X-ray and optical data simultaneously in the external-shock model. Only one band may be of the external-shock origin. Part of the other band may arise from a different emission component (e.g., long-lasting central engine emission, reverse shock emission, or structured jets) causing the chromatic behavior; see further discussion in Wang et al. (2015).
• 3.  
  Overall, we find 62% of GRBs that are consistent with the standard afterglow model (Δα = [−0.25, 0.25]). When more advanced modeling is invoked, up to 91% of the bursts in our sample may be consistent with the external-shock model (Δα = [−0.5, 0.5]). Of these, 61% correspond to an ISM medium, and 39% of GRBs correspond to a wind medium.
• 4.  
  The fraction of the bursts that are consistent with the external-shock model is independent of the observational epoch (measured in the rest frame of GRBs).
• 5.  
  Approximately half of all bursts have light curves for which the break time of the shallow-to-normal decay is similar to that of the normal-to-jet decay, making an immediate change from the plateau phase to the jet phase in both the X-ray and the optical. We suggest that the normal decay segment for these bursts is short and therefore not detected.
• 6.  
  The average values of the postbreak decay slopes, in the X-ray and optical bands, are consistent with the case of an ISM medium and slow-cooling electrons (${{\nu }_{m}}\lt {{\nu }_{{\rm o}}}\lt {{\nu }_{{\rm x}}}\lt {{\nu }_{c}}$).
• 7.  
  We do not detect one single case in our sample with a temporal break that can be interpreted as the cooling frequency crossing the X-ray or optical bands.

The models we have tested are the simplest analytical external-shock models of GRBs. More sophisticated afterglow modeling has been carried out, which gives somewhat different predictions than the simplest models. For example, the recent numerical simulations of Duffell & MacFadyen (2014) simulated hydrodynamical evolution of a jet starting from a collapsar engine to the late afterglow phase. The change of decay slope in their simulated light curve from the plateau to the following power-law decay is Δα_x = 9/8. We investigated 48 GRBs in X-ray and 33 GRBs in optical that show such a break, and we find that 23 of 48 (48%) GRBs for the X-ray and 17 of 33 (52%) for the optical are consistent with having ${\Delta }\alpha \approx 9/8$, which is consistent with their simulation results. More detailed analyses are needed to fully confront this and other more sophisticated modeling with the observational data.

One important question is whether the X-ray and optical emissions come from the same emission component, as predicted by the standard afterglow model. One interesting fact is that we have identified nine of 87 GRBs whose afterglow light curves show a genuine afterglow behavior, namely an SPL decay sometimes with an early afterglow onset behavior, for both the X-ray and the optical emission. These GRBs are fully consistent with the simplest standard fireball model. There are no flares, no shallow decay segments, no rebrightening at late time (without extra energy injection), and no post-jet-breaks. The nine candidates are GRBs 050603, 061007, 080804, 090323, 090328, 090902B, 090926A, 120815A, and 130427A.

Other GRBs have more complicated features. Testing whether these GRBs are consistent with the external-shock models requires more detailed modeling. In this paper, we performed a series of consistency checks, including the α−β closure relations in individual segments and epochs and Δα and Δβ predictions. Our results show that in general most of them are still consistent with the external-shock model predictions. Nonetheless, in order to address whether these GRBs are fully consistent with the afterglow models, much more detailed analysis and modeling, including the achromaticity and global consistency with the closure relations, are needed. This has been carried out in a separate work (Wang et al. 2015). That work also suggests that most GRBs are consistent with the external-shock model of GRBs, which is fully consistent with the findings of this paper.

We acknowledge the use of public data from the Swift data archive. We appreciate valuable comments and suggestions by the referee and thank Prof. Xiang-Yu Wang for helpful discussions. This work is supported by the Swedish National Space Board, the Swedish Research Council, the National Basic Research Program of China (973 Program, Grant No. 2014CB845800), and the National Natural Science Foundation of China (Grant No. 11322328). Liang Li is supported by the Erasmus Mundus Joint Doctorate Program by Grant Number 2013-1471 from the EACEA of the European Commission. X.F.W. acknowledges support by the One-Hundred-Talent Program, the Youth Innovation Promotion Association, and the Strategic Priority Research Program "The Emergence of Cosmological Structures" (Grant No. XDB09000000) of the Chinese Academy of Sciences. Y.F.H. was supported by the National Natural Science Foundation of China with Grant Nos. 11473012 and 11033002. X.G.W. acknowledges the National Natural Science Foundation of China (Grants 11303005, u1331202). B.Z. acknowledges NASA NNX14AF85G for support.