A COMPREHENSIVE STUDY OF GAMMA-RAY BURST OPTICAL EMISSION. I. FLARES AND EARLY SHALLOW-DECAY COMPONENT

Gamma-ray bursts (GRBs) and their broadband afterglows are the most luminous phenomena in the universe. According to the standard model, the broadband afterglow is from the external shock as the fireball is decelerated by the ambient medium (Mészáros & Rees 1997; Sari et al. 1998), whereas the prompt gamma-ray emission, is due to some internal dissipation processes within the relativistic ejecta, either due to internal shocks (Rees & Mészáros 1994; Kobayashi et al. 1997; Daigne & Mochkovitch 1998) or internal magnetic energy dissipation processes (e.g., Usov 1992; Thompson 1994; Drenkhahn & Spruit 2004; Giannios & Spruit 2006; Zhang & Pe'er 2009; Zhang & Yan 2011), either near or far above the photosphere. In the pre-Swift era, afterglow observations were mostly made in the optical bands. The data are well explained by the external shock model (e.g., Mészáros & Rees 1997; Sari et al. 1998; Panaitescu et al. 1998; Panaitescu & Kumar 2001; Huang et al. 2000; see Zhang & Mészáros 2004 for a review). The successful launch and operation of the Swift mission (Gehrels et al. 2004) have significantly improved our understanding of the physical origin of GRBs. In particular, early X-ray afterglow observations revealed erratic flares and early plateaus that are difficult to interpret within the standard theoretical framework (Zhang et al. 2006; Nousek et al. 2006). The flares are believed to be produced by late activity of the GRB central engine (Burrows et al. 2005; Fan & Wei 2005; Zhang et al. 2006; Dai et al. 2006; Proga & Zhang 2006; Perna et al. 2006), and the shallow-decay segment likely signals a long-lasting wind powered by the GRB central engine after the prompt gamma-ray phase (Zhang et al. 2006). These features indicate that the GRB central engine does not die quickly. One is obliged to accept a more complicated afterglow picture, namely, that the observed afterglow emission is a superposition of the traditional external shock afterglow and an afterglow related to the late central engine activity (Zhang 2011).

The prompt localization of GRBs with the X-Ray Telescope (XRT) on board Swift has significantly increased the number of GRBs with optical afterglow detection and redshift measurement. The simultaneous observations with XRT, UVOT, and ground-based optical telescopes in a much wider time window in the afterglow phase have revolutionized our knowledge of the GRB afterglow, and raised some critical problems with the conventional models (e.g., Zhang 2007, 2011; Liang 2010). For example, the multi-wavelength observations sometimes revealed chromatic behaviors between the optical and X-ray bands (e.g., Panaitescu et al. 2006; Liang et al. 2007), suggesting multiple emission components in the afterglow that dominate different energy bands at different epochs. The existence of engine-powered early X-ray emission also obscures the clear separation between long and short GRBs, causing confusion in GRB classification (see, e.g., Zhang et al. 2007; Zhang & Pe'er 2009; Lü et al. 2010 for a detailed discussion). In order to unveil the underlying physics, it is essential to decompose the light curve into different components that have distinct physical meanings.

There are two approaches in decomposing the light curves into different components: one through theoretical modeling and the other through empirical fitting. Theoretical modeling gives insights into the physical properties of the emission, including radiation mechanisms, micro-physical parameters, properties of the surrounding medium, etc. Intense modeling of optical afterglow data has been carried out in the pre-Swift era (e.g., Panaitescu et al. 1998; Panaitescu & Kumar 2001, 2002; Huang et al. 2000; Wu et al. 2005). This has become increasingly difficult in the Swift era. First, there are many more bursts with high-quality data to be modeled, and performing such modeling would be time consuming. More importantly, data suggest that it is essentially impossible to interpret all the data with the standard external shock model. The empirical fitting approach, on the other hand, makes use of some empirical functions to fit the data, and is suitable for handling a great amount of data. A morphological study can catch insights into various emission components, which can make statistical analyses of a large sample of data easier. This has been done by some authors (e.g., Liang & Zhang 2006; Panaitescu & Vestrand 2008, 2011; Kann et al. 2010, 2011). Some interesting features have been revealed. For example, Liang & Zhang (2006), Nardini et al. (2006), and Kann et al. (2006) found two universal tracks of the late optical luminosity light curves. Panaitescu & Vestrand (2008, 2011) showed some general features of the early optical bumps and plateaus in the optical light curves. Kann et al. (2010, 2011) compared the optical light curves of different types of GRBs.

Different from these previous statistical studies, we plan to perform empirical fitting to the observed optical light curves and to identify multiple emission components in a series of papers. After decomposing the light curves, we plan to perform statistical analyses of the parameters of various components, and discuss their physical implications. As the first paper in the series, here we present the sample (Section 2), the general results of the decomposition, and a "synthetic" light curve that shows eight possible components with distinct physical origins (Section 3). Since flares and the shallow-decay component may be directly related to the late central engine activity, in this paper we carry out a detailed study of these two components: flares in Section 4 and the shallow-decay component in Section 5. We discuss physical implications in Section 6, and summarize the results in Section 7.

We include all the GRBs that have optical afterglow detections before 2011 November (from 1997 February 28 to 2011 November) in our sample. A sample of 225 optical light curves are compiled from data reported in the literature. We extensively search for the optical data from published papers or from GCN Circulars if no published paper is available for some GRBs. Well-sampled light curves are available for 146 GRBs. In Table 1, we summarize the following information for each GRB: redshift, spectral properties of the optical afterglow and prompt gamma-ray emission, as well as time intervals of optical observations.⁶ We collect the optical spectral index β_O (defined with the convention $F_\nu \propto \nu ^{-\beta _O}$)⁷ and the extinction A_V of the host galaxy for each burst from the same literature to reduce the uncertainties introduced by different authors. Galactic extinction correction is performed by using a reddening map presented in Schlegel et al. (1998). Since the A_V values are available only for some GRBs and A_V is derived from the spectral fits using different extinction curves, we do not correct for the GRB host galaxy extinction. The k-correction in magnitude is calculated by k = −2.5(β_O − 1)log (1 + z). For the late epoch data (∼10⁶ s after the GRB trigger), possible flux contribution from the host galaxy is subtracted. The isotropic gamma-ray energy (E_{γ, iso}) is derived in the rest frame 1–10⁴ keV energy band using the spectral parameters.

The optical light curves are usually composed of one or more power-law segments along with some flares, humps, or rebrightening features. The mix of different components makes the diverse optical afterglow light curves. In order to decompose the rich features, we fit the light curves with a model of multiple components. The basic component of our model is either a power-law function

$$\begin{equation} F_1 = F_{01} t^{-\alpha } \end{equation} \tag{ 1 }$$

or a smooth broken power-law function

$$\begin{equation} F_2 = F_{02} \left[\left(\frac{t}{t_{\rm b,1}}\right)^{\alpha _1\omega } +\left(\frac{t}{t_{\rm b,1}}\right)^{\alpha _2\omega }\right]^{-1/\omega }, \end{equation} \tag{ 2 }$$

where α, α₁, α₂ are the temporal slopes, t_b is the break time, and ω measures the sharpness of the break. In some afterglow models, a double broken power-law light curve is expected. For example, it is theoretically expected that the afterglow light curve may have a shallow segment early on due to energy injection, and then transits to a normal decay segment when the energy injection is over, and finally steepens due to a jet break (e.g., in the canonical X-ray afterglow light curve; Zhang et al. 2006). We therefore also consider a smooth triple power-law function to fit some light curves. In this case, we extend Equation (2) to the following function (Liang et al. 2008):

$$\begin{equation} F_3=\big(F_{2}^{-\omega _2}+F_j^{-\omega _2}\big)^{-1/\omega _2}, \end{equation} \tag{ 3 }$$

where ω₂ is the sharpness factor of the jet break at t_{b, 2}, and

$$\begin{equation} F_j=F_2(t_{\rm b,2})\left(\frac{t}{t_{b,2}}\right)^{-\alpha _3}. \end{equation} \tag{ 4 }$$

We developed an IDL code to make best fits with a subroutine called MPFIT.⁸ The parameter ω is usually fixed to three or one in our fitting. The approach of our light curve fitting is as follows. Initially, we introduce a minimum number of components based on by-eye inspection of the global feature of the light curve. If the reduced χ²_r is much larger than one, then we continue to add more components and re-do the fit, until the reduced χ²_r becomes close to one. The reduced χ² of simple power-law fits to the light curves of some GRBs, such as GRBs 020813, 030328, 050416A, and 070110, are ∼1. However, a smooth broken power law can significantly improve the fit,⁹ which reduces the χ²_r value by >50%. We therefore adopt the smooth broken power-law fit for these GRBs. The χ²_r values for some light curves are much smaller than one, indicating that some model parameters are poorly constrained. For these cases, we hold some parameters constant to make the fits. The erratic fluctuations of some data points with small error bars in some GRBs, such as GRB 030329, make χ²_r much larger than one. We do not add additional components for these light curves, so that their χ²_r values remain much larger than one. The most challenging problem in our fit is extracting severely overlapping flares/bumps from the light curves. The slopes of these flares/bumps are usually quite uncertain. In our fitting, we first set all the parameters free to get the best fit to the global light curve, and then adjust the rising slopes to ensure that the fitting curve crosses the data point around the peak time of each flare/bump. Finally, we hold the rising slopes constant and perform the best fits again. As an example, Figure 1 shows six light curves with the best-fit multiple components decomposed.

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Even though individual optical light curves can differ significantly, after synthesizing many light curves, one can come up with a synthetic afterglow light curve, as shown in Figure 2. In contrast to the five-component canonical X-ray light curve (Zhang et al. 2006), the synthetic optical light curve includes eight components that may have distinct physical origins. These components are as follows. Ia: prompt optical flares; Ib: an early optical flare from the reverse shock; II: early shallow-decay segment; III: the standard afterglow component (an onset hump followed by a normal decay segment); IV: the post-jet-break phase; V: optical flares; VI: rebrightening humps; VII: late supernova (SN) bumps. Components II–V can find their counterparts in the canonical X-ray light curve. These components can be distinguished based on the parameters of our multi-component fits. For example, flares usually have rapid rise and fall. We define a flare if the absolute value of its rising and decaying slopes are steeper than two. If flares occur during the prompt emission phase, then they are grouped as Ia (prompt optical flares). A reverse shock flare (Ib) is a huge flare (or optical flash) that peaks slightly after the end of prompt emission (the decay slope can be somewhat shallower than two). All the optical flares (V) afterward are considered "late" (with respect to Ia and Ib), even though most of them actually happen in the early afterglow phase.

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We define different components based on theory. For example, flares are defined as features with both steep rising (α₁ < −2 with the convention F_ν∝t^−α) and decaying (α₂ > 2) slopes. A shallow-decay component, on the other hand, is defined as a segment whose decay is shallower than what is predicted in the constant energy afterglow model. As shown in Table 1, the spectral indices of the optical afterglows are usually smaller than one within error. This is consistent with the standard external shock synchrotron radiation model in the spectral regime of ν_m < ν_O < ν_c (with the electron power-law index p > 2), where ν_m and ν_c are the injection and cooling frequencies, respectively. In this spectral regime, one has the closure relation α_O = 3β_O/2. A shallow-decay component (II) is then defined by the condition α_O < 3β_O/2 (with the convention $F_\nu \propto t^{-\alpha _{\rm O}}$) within error. Similarly, the post-jet-break decay segment IV is defined as in the segment beyond a steepening break after the normal decay segment (with α_O = 3β_O/2).¹⁰ An afterglow onset feature is characterized by a smooth hump peaking at less than one hour post trigger, which is followed by a normal power-law decay component (III). A rebrightening hump (VI) is similar to the early afterglow onset hump but occurs much later. It differs from optical flares due to its much shallower rise and decay as well as a much smoother peak. The SN bump (VII) is a special late rebrightening peaking at around one to two weeks after GRB trigger, which usually shows a red color.

After decomposing the light curves, we are able to group all the identified components under one of these eight components. The early optical afterglow light curves (t < 10³ s) of about one-third of the GRBs show a smooth hump. Another one-third of the light curves start with a shallow-decay segment. In 19 GRBs, 24 optical flares are observed. Late rebrightening humps are observed in 30 GRBs. A jet-like break is detected in 10 GRBs. A clear SN bump is detected in 18 GRBs. The detected fraction of each component is marked in the synthetic light curve (Figure 2).

We will report our statistical results for the various components in a series of papers. As the first paper of the series, this paper focuses on the optical flares and the shallow-decay segment. The reason for discussing them together is because they are both likely related to the late central engine activity of GRBs (see Section 6 for more discussion). Note that the prompt optical flares and early reversed shock flares are not included in this paper, and we will discuss them separately. Throughout the paper, we mark the parameters of the flares and the shallow-decay segment with the superscripts "F" and "S," respectively.

We get 24 flares in 19 GRBs. A flare is clearly seen in 14 out of the 19 GRBs, as shown in Figure 3 along with our best-fit results. Some flares may be embedded in the light curves, as shown in Figure 1. Note that most of the well-sampled optical light curves are in the R band. For a few GRBs, the flares are well sampled in other bands. We correct these light curves to the R band using the optical spectral indices. The fitting parameters (the flux at peak time and the rising and decaying slopes) of the flares and the derived temporal properties, including the peak time (t^F_p), the width (w^F) measured at the full width at half-maximum, the rising timescale t^F_r, the decay timescale (t^F_d), the ratio of the t^F_r to t^F_d (R^F_rd), and the ratio of the t^F_r to t^F_p (R^F_rp) derived from the fitting parameters, are summarized in Table 2. With the fitting parameters, we calculate the isotropic peak luminosity (L^F_{R, iso}) and the total energy release (E^F_{R, iso}) in the R band. The E^F_{R, iso} is integrated from t^F_p/5 to 5t^F_p. Our results are reported in Table 2.

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We show the distributions of R^F_rp, R^F_rd, t^F_p, w^F, L^F_{R, iso} in Figure 4. The distribution of R^F_rp is clustered around 0.1–0.3, consistent with the expectation of an internal origin for the flares. The ratio R^F_rd is similar to that observed in GRB pulses, but the rising wing of some flares is even longer (in log scale) than the decaying wing. The t^F_p ranges from ∼tens of seconds to ∼10⁶ s. The w^F values are in the same range as t^F_p. The L^F_{R, iso} ranges from 10⁴³ to 10⁴⁹ erg s⁻¹, with a typical value of 10⁴⁶ erg s⁻¹.

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The w^F and L^F_{p, iso} of the flares as a function of t^F_p are shown in Figure 5. A tight correlation between w^F and t^F_p is found. The best fit gives log w^F = −0.32 + 1.01log t^F_p, i.e., w^F ∼ t^F_p/2. The L^F_{R, p} is anti-correlated with t^F_p in the burst frame, i.e., log L^F_{R, iso, 48} = (1.89 ± 0.52) − (1.15 ± 0.15)log [t^F_p/(1 + z)] with a Spearman correlation coefficient 0.85 and a chance probability p < 10⁻⁴. Therefore, later flares tend to be dimmer and wider than earlier flares.

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The possible relations between the flare properties and E_{γ, iso} are shown in Figure 6. Since more than one flare are detected in a few GRBs, we select only the brightest one for our analysis. As shown in Figure 6, E^F_{R, iso} is typically smaller than 1/100 of E_{γ, iso}. The flare R-band luminosity L^F_{R, iso} is correlated with the gamma-ray luminosity L_{γ, iso}, i.e., log L^F_{R, iso}/10⁴⁸ = (− 3.97 ± 0.60) + (1.14 ± 0.27)log L_{γ, iso}/10⁵⁰ with a Spearman correlation coefficient of r = 0.75 and a chance probability p ∼ 10⁻³. The flares in GRBs 050401, 060926, and 090726 are out of the 3σ region of the fit. Without considering the flares in these three GRBs, it is found that the $t^{^{\prime }\rm F}_{\rm p}$ is also tightly anti-correlated with E_{γ, iso}, i.e., $\log t^{^{\prime }\rm F}_{\rm p}=(5.38\pm 0.30)-(0.78\pm 0.09)\log E_{\rm \gamma , iso}/10^{50}$ (with r = 0.92). Similarly, a tight anti-correlation between L_{R, p} and t_p in the burst frame is found without considering the flares in the three GRBs, i.e., log [t^F_p/(1 + z)] = (7.57 ± 0.60) − (1.35 ± 0.17)log E_{γ, iso, 50}, with a Spearman correlation coefficient of 0.91. These results indicate that the optical flares of a GRB that has a larger E_{γ, iso} tend to peak earlier and brighter.

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It is interesting to study whether optical flares are associated with X-ray flares. Early flares are frequently seen in X-ray afterglow light curves (Burrows et al. 2005; O'Brien et al. 2006). However, as shown above, early optical flares are only observed in the light curves of GRBs 060210, 060926, 090618, and 090726. The fraction of GRBs with detected early optical flares is much lower than that for X-ray flares. Among the 19 GRBs with optical flare detections, 16 had early Swift/XRT observations. Their X-ray afterglow light curves are also shown in Figure 2. Simultaneous observations with XRT during the optical flares are available for GRBs 050401, 060206, 060210, 060607A, 060926, 070311, 071010A, 071031, 080506, 090618, and 100728B. An X-ray flare that may be associated with the optical flare is only observed in GRBs 060926, 070311, and 071010A. The optical flares of these three GRBs lagged behind the corresponding X-ray flares. Measuring the lags with the peak time of the flares, we get 196 s, 7.7 × 10⁴ s, and 2.45 × 10⁴ s for the flares in GRBs 060926, 070311, and 071010A, respectively. The lag is potentially proportional to the peak time of the flares with the three flares.

A shallow-decay segment is defined using the criterion that the initial decay slope of this segment is shallower than 3β_O/2 within error. Out of 146 GRBs, we get a sample of 39 that have such a shallow-decay segment. Some examples are shown in Figure 7. The data for the shallow decay segments are summarized in Table 3. Figure 8 shows the distributions of the decay slopes (α^S₁ and α^S₂), the break times (t^S_b), and the luminosity at the break (L^S_{b, iso}) of our sample. Thirty-one shallow-decay segments transit to a decay slope of 1 ∼ 2.5, and 5 have shallow-decay segment followed by a sharp drop with a decay slope steeper than 2.5. About half of the shallow-decay segments look like a plateau, with |α^S_{b, 1}| ⩽ 0.3. The break time ranges from tens of seconds to several days after the GRB trigger, with a typical t^S_p of ∼10⁴ s. The L^S_{R, b} typically varies from 10⁴³ to 10⁴⁷ erg s⁻¹, and even reaches ∼10⁴⁹ erg s⁻¹ in a few GRBs with an early break. The break luminosity L^S_{R, b} is anti-correlated with t^S_b, as shown in Figure 9. The best fit gives log L^S_{R, 48} = (1.75 ± 0.22) − (0.78 ± 0.08)log [t^S_b/(1 + z)], with a Spearman correlation coefficient of r = 0.86 and a chance probability of ρ < 10⁻⁴. No correlation between E^S_{R, iso} and t^S_b is observed.

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A shallow-decay segment is commonly seen in the well-sampled X-ray afterglow light curves detected by Swift/XRT (e.g., Liang et al. 2007; Evans et al. 2009), except for a few GRBs whose XRT light curves decay as a single power law (Liang et al. 2009). It was also reported that the X-ray luminosity at the break time is correlated with the break time (Dainotti et al. 2010). We overplot L_{b, iso} as a function of t_b in the burst frame in Figure 9. One can observe that optical data share a similar relation to the X-ray data. Since the X-ray luminosity is measured in the 0.3–10 keV energy band and the optical luminosity is measured in the R band, the X-ray luminosity lies significantly above the optical luminosity (the νF_ν peak is at ν_c for slow cooling, which may be close to the X-ray band). The observed photon spectral indices of the X-ray spectra are ∼2 (Liang et al. 2007). Therefore, the X-ray energy spectra are flat and the derived L_{b, iso}–t_b relation in the 1 keV band is roughly consistent with that derived from the entire X-ray band.

We examine the chromaticity of the shallow-decay segments in the X-ray and optical bands. The X-ray observations are available for 17 of the 34 GRBs. We extract the underlying afterglow components II and III (by removing flares) for the X-ray and optical samples, and compare the parameters α^S₁, α^S₂, and t^S_b of the two samples in Figure 10. It is found that the t^S_b data points are scattered around the equality line, and a tentative correlation between the break times of the optical and X-ray light curves is observed, with a chance probability of correlation of ∼0.15. These is no correlation between the decay slopes of the X-ray and optical light curves. The decay segment prior to the break times in the optical bands tends to be steeper than that in the X-ray band, but the post-break slopes are roughly consistent, except for α^S₂ > 2.5 in the optical bands.

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We integrate the total R-band energy release (E^S_{R, iso}) in the shallow-decay phase from 10 s to t^S_b post trigger. We show E^S_{R, iso} as a function of E_{γ, iso} in Figure 11. A rough proportionality between E^S_{R, iso} and E_{γ, iso} is observed. The Spearman correlation analysis shows that the chance probability of the correlation between the two quantities is ∼6 × 10⁻³. Our robust fit yields log E^S_{R, iso} = 0.40 + 0.47log E_{γ, iso}. The correlation between L^S_{b, iso} and E_{γ, iso} is much worse. We get log L^S_{R, b} = −5.57 + 1.13log E_{γ, iso}, with a chance probability of 0.16, as shown in Figure 11.

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Physically, there are two types of shallow-decay segment (plateau), as observed in the X-ray band (Liang et al. 2007). The majority of X-ray plateaus are followed by a normal decay with decay index typically around −1. These plateaus are likely of an external shock origin, with the shallow-decay segment caused by continuous energy injection into the blast wave (Rees & Mészáros 1998; Dai & Lu 1998; Sari & Mészáros 2000; Zhang & Mészáros 2001a). This scenario has been applied to interpret most X-ray plateaus discovered by Swift (Zhang et al. 2006; Nousek et al. 2006). A small fraction of plateaus, first found by Troja et al. (2007) in GRB 070110 and studied systematically by Liang et al. (2007), are followed by a much steeper decay (index steeper than −3), which cannot be interpreted within the external shock model. These plateaus are called "internal plateaus" by Liang et al. (2007), since they have to be powered by internal dissipation of a late outflow. Looking at our optical shallow-decay component sample, most light curves have a shallow-decay component followed by a normal decay segment. Nonetheless, we identify two possible internal plateaus in GRBs 060605 and 080413B, which show the superposition of a normal decay segment and a possible internal plateau with the sharp drop of the slope (see Figure 12).¹¹ As shown in Figure 12, the early optical light curves of GRBs 060605 and 080413B are a smooth bump and a normal decay segment, respectively, which are consistent with the standard afterglow model. Their late light curve rapidly decays with a slope of α > 2.5, and then flattens to a level consistent with the normal decay. This suggests that the plateau is likely internal and is superposed on the external component. We also revealed evidence of such a component from the light curves of GRB 970508 (ending at ∼10⁶ s with a slope 3.0) and 050319 (ending at ∼490 s and 3.3 × 10⁵ with slopes 3 and 2.5, respectively). The plateau end times of these GRBs range from tens of seconds to several days after the GRB trigger.

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6.1. From Prompt Gamma-Ray Pulses to Late Optical Flares: Global Evolution of an Erratic Central Engine

6.2. The Shallow-decay Segment as a Probe of Late Energy Injection

In the framework of the GRB fireball models, the shallow-decay segment followed by normal decay can be interpreted as a blast wave with continuous energy injection. There are two types of energy injection, one related to a long-lasting central engine (Dai & Lu 1998; Zhang & Mészáros 2001a) and another related to the distribution of the Lorentz factor in the promptly ejected outflow (Rees & Mészáros 1998). The two cases have effectively similar predictions and cannot be differentiated from the data. The existence of internal plateaus observed in some GRBs suggests that at least for some GRBs, a long-lasting central engine is indeed at work. This may be related to the spin-down of the central engine, either a rapidly spinning black hole or a rapidly spinning magnetar. Theoretical modeling also suggests that the energy of flares can pile up onto the blast wave and give rise to a shallow-decay segment (Maxham & Zhang 2009). Overlapping optical flares on the shallow-decay segment are observed in some GRBs, such as 970508 and 000301C. This is good evidence of two emission sites: an internal origin of the optical flares and the external shock origin of the shallow-decay segment. So an observed shallow-decay component can be a probe of the central engine activity and energy injection into the blast wave.

For a long-lasting central engine, one may parameterize the central engine luminosity history as L = L₀t^−q. The external shock closure relation between the decay slope and the spectral slope can be written as α = (q − 1) + (2 + q)β_O/2 in the case of ν_m < ν_O < ν_c (Zhang et al. 2006). With the observed α and β_O, we derive the q values for these GRBs in this spectral regime. The typical q value is 0.5, as shown in Figure 13. For a black hole–torus system, the wind may be driven by neutrino annihilation or the Blandford & Znajek (1977) mechanism. For a neutrino-driven wind, the annihilation luminosity can be estimated as $\log L_{\nu \bar{\nu }}=43.6+4.89\log (\dot{m}/0.01 \,M_\odot)+3.4 a_*$, where $\dot{m}$ is the accretion rate and, a_* is the spin parameter of the central black hole (W. H. Lei & B. Zhang 2012, in preparation). Assuming that the pre-collapse density profile is ρ∝r^τ, the mass enclosed within r increases with radius as $r^{\tau ^{^{\prime }}}$, where $\tau ^{^{\prime }}=3+\tau$ for τ > −3 and $\tau ^{^{\prime }}\sim 0$ for τ < −3. Then, the mass fall-back rate onto the disk is given as $\dot{m}_{\rm f}\propto t^{(2\tau +\tau ^{^{\prime }}+3)/(3-\tau)}$. Following Kumar et al. (2008), τ = −1.8, we get $\dot{m}_{\rm f}\propto t^{-0.3}$. If the fall-back mass is accreted onto the black hole, then we obtain $\log L_{\nu \bar{\nu }}\propto t^{-1.5}$. This is inconsistent with q values for most bursts in our sample. For the Blandford & Znajek (1977) mechanism, the maximum of the power can be estimated with $L_{\rm BZ}\sim \dot{m}c^2$. If $\dot{m}\propto t^{-0.3}$ as discussed above, then the decay slope is consistent with the q values for most GRBs in our sample. Alternatively, if the long-lasting wind is driven by a spinning-down magnetar, one has L(T) = L_{0, em}/(1 + T/T_em)², where T_em is the characteristic timescale for dipolar spin-down. The predicted q values are zero or two, not consistent with the typical value q = 0.5. On the other hand, if the shallow decay is caused by injection of flare energies into the blast wave (e.g., Maxham & Zhang 2009), then the predicted decay slope depends on the energetics and temporal distributions of the flares, and can be more flexible.

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One issue in explaining the shallow-decay segment with energy injection onto the blast wave is the chromatic breaks in the optical and X-ray bands (e.g., Panaitescu et al. 2006; Liang et al. 2007), as shown in Figures 7 and 10. This raises the concern regarding whether the X-ray and optical emission are from the same emission component (e.g., Liang et al. 2009; Racusin et al. 2008). Note that the decay slopes after the break in the optical and X-ray bands are usually consistent with the expectation of external shock models. This implies that the radiation from the two energy bands could share a similar origin. Introducing inverse Compton scattering in the X-ray band may cause chromatic behavior between the X-ray and optical bands (e.g., Panaitescu & Kumar 2000; Zhang & Mészáros 2001b). Alternatively, the X-rays may be emission from a long-lasting wind. A long-lasting reverse shock may introduce further complications (e.g., Uhm et al. 2012). In general, mixing of different emission components may be the reason for the complex chromatic behaviors observed in different energy bands.

We have systematically decomposed the optical afterglow light curves for 146 GRBs before 2011 November that have good quality optical data. By fitting the light curves with multiple components, we get a synthetic optical light curve that includes eight components with distinct physical origins. We plan to study these components in detail in a series of papers, and in this paper we focus on the optical flares and the shallow-decay segment. Our results can be summarized follows:

• 1.  
  We obtained 24 optical flare events in 19 GRBs. The t^F_p ranges from several tens of seconds to several days post-GRB trigger, and it is tightly correlated with the width and peak luminosity of the flares, i.e., w^F ∼ t^F_p/2 and log L^F_{R, iso, 48} = (1.89 ± 0.52) − (1.15 ± 0.15)log [t^F_p/(1 + z)], suggesting that flares peaking later tend to be dimmer and wider. The parameters t^F_p and E^F_{R, iso} are also corrected with E_{γ, iso}, suggesting that GRBs with larger E_{γ, iso} tend to have optical flares peaking earlier and be brighter.
• 2.  
  The fraction of GRBs with detected optical flares and the number of flares in a GRB are much smaller than in the case of X-ray flares. Among the 19 GRBs with detected optical flares, 16 have early Swift/XRT observations. Only four cases, i.e., GRBs 970508, 060926, 070311, and 071010A, show detection of associated X-ray flares. The optical flares in the three GRBs lag behind the corresponding X-ray flares, similar to the spectral lag observed in prompt gamma-ray emission, but the time lag is much longer than what is observed in the prompt phase.
• 3.  
  We get a sample of 42 shallow-decay segments from 39 GRBs. About half of the shallow-decay segments look like a plateau, with a decay slope α^S_b that is smaller than 0.3. Thirty-two out of the 39 shallow-decay segments transit to a decay with a slope of 1 ∼ 2.5, and 5 of them are followed by a sharp drop with a decay slope steeper than 2.5. The break times range from tens of seconds to several days after the GRB trigger, with a typical t^S_b ∼ 10⁴ s. No clear correlation between E^S_{R, iso} and E_{γ, iso} is found.
• 4.  
  The break times of the shallow-decay segment in the optical and X-ray bands are chromatic for most GRBs, but they are tentatively correlated. The decay prior to the break time in the X-ray band tends to be steeper than that in the optical band, and the decay slopes post-break in the two energy bands are roughly consistent with each other. L^S_{R, iso} is anti-correlated with t^S_b, which is similar to the case for X-ray plateaus.

We discussed the physical implications of the optical flares and the optical shallow-decay segment; both are related to late GRB central engine activities. The observations strengthen the theory that the GRB central engine dies out gradually with luminosity decreasing with time. The late central engine activity can be either erratic (for flares) or steady (for internal plateaus); both could add energy to the blast wave to make a shallow-decay segment in the light curve. The observed afterglow is a mix of various emission components of external and internal origins, and the variation of the strengths of different components leads to diverse chromatic afterglow behaviors.

We acknowledge the use of public data from the Swift data archive. We are grateful for helpful discussions with Zi-Gao Dai, Xue-Feng Wu, and Shuang-Nan Zhang. This work is supported by the National Natural Science Foundation of China (grant Nos. 11025313, 10873002, 11078008, 11063001, and 11163001), the "973" Program of China (grant 2009CB824800), Special Foundation for Distinguished Expert Program of Guangxi, the Guangxi SHI-BAI-QIAN project (grant 2007201), the Guangxi Natural Science Foundation (2010GXNSFA013112, 2011GXNSFB018063, and 2010GXNSFC013011), the special funding for national outstanding young scientist (contract No. 2011-135), and the Third Innovation Project of Guangxi University. B.Z. acknowledges support from NSF (AST-0908362).