JET BREAKS AND ENERGETICS OF Swift GAMMA-RAY BURST X-RAY AFTERGLOWS

We assume that all X-ray light curves in our sample inherently follow the canonical form (Figure 1) described by Zhang et al. (2006) and Nousek et al. (2006). These four segments and additional component are as follows. I: the initial steep decay often attributed to high-latitude emission or the curvature effect (Kumar & Panaitescu 2000; Qin et al. 2004; Liang et al. 2006; Zhang et al. 2007a); II: the plateau, which is frequently attributed to continuous energy injection from the central engine (Rees & Mészáros 1998; Dai & Lu 1998; Sari & Mészáros 2000; Zhang & Mészáros 2001; Granot & Kumar 2006; Zhang et al. 2006; Liang et al. 2007); III: the normal decay due to the deceleration of an adiabatic fireball (Mészáros 2002; Zhang et al. 2006); IV: the post-jet break phase (Rhoads 1999; Sari et al. 1999; Mészáros 2002; Piran 2005); V: flares, which are seen in ∼1/3 of all Swift GRB X-ray afterglows during any phase (I–IV) and are believed to be caused by sporadic emission from the central engine (Burrows et al. 2005; Zhang et al. 2006; Chincarini et al. 2007; Falcone et al. 2007). We classify the data into these segments based upon the criteria described below. Only 25 cases contain all four light curve segments, with 14 cases also containing flares.

This analysis does not address the phenomenon of X-ray flares, but rather excludes them from the spectral and temporal analysis. See Chincarini et al. (2007) and Falcone et al. (2007) for detailed studies of X-ray flares and analysis on this data set. These studies have shown that significant spectral evolution occurs throughout the flares, therefore in order to constrain the properties of the underlying afterglows, we remove the time intervals of significant flaring from subsequent temporal and spectral analysis. Flaring was determined by visual inspection of the light curves and hardness ratios. Only the most apparent flares were removed, with no attempt to constrain small-scale or micro-flaring. Large flares that overlap and significantly exceed the level of the underlying afterglow can also mask whole segments, making it impossible to determine the underlying temporal and spectral properties. The flaring in these cases usually occurs at the beginning of light curves, overwhelming segment I and leading to light curves with apparent segments II–III afterwards. Rather than guessing the properties or presence of these specific flaring segments, we remove those time intervals and proceed as if they were not part of the rest of the light curves.

For those X-ray afterglows that were also observed by Chandra at late times, we include those data points in our temporal but not spectral fits. These bursts include GRB 051221A (Burrows et al. 2006), GRB 050724 (Grupe et al. 2006), and GRB 060729 (D. Grupe et al. 2009).

2.1.1. Light Curve Construction

2.1.2. Light Curve Fitting

We have developed tools to fit single power laws, broken power laws, double broken power laws, and triple broken power laws with the following functional forms to the XRT light curves: single power law:

$$\begin{equation} F(t)=N\,t^{-\alpha _0}, \end{equation} \tag{ 1 }$$

broken power law:

$$\begin{equation} F(t)=N \left\lbrace \begin{array}{@{}l@{\quad }l} t^{-\alpha _1} & t<t_{b_1} \\ t_{b_1}^{(\alpha _2-\alpha _1)}\,t^{-\alpha _2} & t>t_{b_1}, \\ \end{array}\right. \end{equation} \tag{ 2 }$$

double broken power law:

$$\begin{equation} F(t)=N \left\lbrace \begin{array}{@{}l@{\quad }l} t^{-\alpha _1} & t<t_{b_1} \\ t_{b_1}^{(\alpha _2-\alpha _1)}\,t^{-\alpha _2} & t_{b_1}<t<t_{b_2} \\ t_{b_1}^{(\alpha _2-\alpha _1)}\,t_{b_2}^{(\alpha _3-\alpha _2)}\,t^{-\alpha _3} & t>t_{b_2}, \\ \end{array}\right. \end{equation} \tag{ 3 }$$

triple broken power law:

$$\begin{equation} F(t)=N \left\lbrace \begin{array}{@{}l@{\quad }l} t^{-\alpha _1} & t<t_{b_1} \\ t_{b_1}^{(\alpha _2-\alpha _1)}\,t^{-\alpha _2} & t_{b_1}<t<t_{b_2} \\ t_{b_1}^{(\alpha _2-\alpha _1)}\,t_{b_2}^{(\alpha _3-\alpha _2)}\,t^{-\alpha _3} & t_{b_2}<t<t_{b_3} \\ t_{b_1}^{(\alpha _2-\alpha _1)}\,t_{b_2}^{(\alpha _3-\alpha _2)}\,t_{b_3}^{(\alpha _4-\alpha _3)}\,t^{-\alpha _4} & t>t_{b_3}, \\ \end{array}\right. \end{equation} \tag{ 4 }$$

where N is the normalization, t is the time since the burst trigger, F(t) is the count rate over the soft X-ray band (0.3–10 keV), $t_{b_1,b_2,b_3}$ are the times of breaks in the light curves, and α_(0,1,2,3,4) are the temporal indices of the power-law fits.

Our software, written in IDL, requires user input for initial guesses of the location and number of breaks (and therefore power-law model). Based upon visual inspection, the user first eliminates all obvious time intervals with significant flaring.⁶ The user then makes initial guesses for light curve break times, and a least-squares fitting routine is used to fit each model. When the addition or removal of light curve segments from the initial model also provides an adequate fit, we perform an F-test and if the fit is improved at a 99% confidence level, then the new model is retained.

In order to accurately measure the light curve model parameter errors without overestimating them, we tested both Δχ² confidence interval mapping and Monte Carlo simulation methods. In the Monte Carlo method, we created 10,000 simulated light curves for each GRB light curve jiggling the data points by an amount drawn randomly from the Poisson distributions derived from the source and background counts. Each of these light curves was fit with the same method as the real light curves. The 90% and 2σ confidence intervals were taken from the distributions of each fit parameter from the simulations. The broken power-law fits were not well behaved in Δχ² contour space due to data binning, light curve gaps, and logarithmic fits, resulting in larger error estimates compared to the Monte Carlo method. The latter method is more free from assumptions and biases, therefore we chose to use the Monte Carlo light curve parameter error estimates for the following analysis.

We classify each segment of the light curves in terms of the canonical form (Figure 1) using the following criteria. If the light curve is a triple broken power law then identification of its segments is unambiguous, and it is designated as a type I–II–III–IV. If the light curve is best fit by a double broken power law, we apply the criteria that if α₁ > α₂ then it is designated as segments I–II–III or if α₁ < α₂ then it is designated as segments II–III–IV. If the light curve is best fit by a singly broken power law, we apply the criteria that if α₁ > α₂ then it is designated as segments I–II, while if α₁ < α₂ then it can be interpreted as either segments II–III or III–IV. If the light curve is best fit by a single power law then any segment (I, II, III, IV) is possible.

The distributions of these temporal fits are given in the left panel of Figure 2. Although the α distributions are broad, the different segments are clearly separated by our classification criteria described above. Kolmogorov–Smirnov (K–S) tests show that the segments I through IV are different at >99.9% level. The single power law distribution also differs from segments I, II, and IV at >99.9% level. However, the single power law and segment III distributions are more similar (4% probability of begin drawn from same inherent distribution), suggesting that some of the single power laws are segment III, with others drawn from segments I, II, and IV.

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This scheme has no implicit constraints on the range of temporal indices, but does leave some ambiguity in the case of a broken power law with α₁ < α₂. This case is equivocal between segments II–III and III–IV. We generally assume that they are cases of II–III when looking at sample distributions. However, we still fit the post-jet break closure relations, allowing for the possibility of III–IV. This ambiguity and distinguishing criteria are further addressed in Section 4.3.

We have constructed spectra for each segment of each light curve distinguished using the temporal fits defined in Section2.1.2. These spectra were extracted using XSELECT with 20 pixel radius source extraction region, and a 40 pixel radius background region. All analysis used the version 011 (release date 2008 May 14) response matrices from the Swift CALDB, and Ancillary Response Files (ARFs) were made using the xrtmkarf task. The spectra were grouped using the grppha task with 20 counts per bin and were fit using χ² statistics unless there were fewer than 150 counts, in which case they were grouped with 10 counts per bin and were fit using Cash statistics. We fit these spectra in XSPEC to an absorbed power law with two absorption components, one fixed to the Galactic value (Kalberla et al. 2005), and another freely varying using the measured redshift if available.

The photon indices (Γ) of the spectral fits are used to measure β, the energy spectral index, where β = Γ − 1. The distributions of these spectral fits are shown in Figure 2. Segments II through IV are statistically similar (as tested with a K–S test). Of those light curves with segment IV, ∼90% are consistent with minimal or no spectral evolution through segments II, III, and IV (90% confidence errors). The K–S test shows that segment I differs from segments II and III at >98% level, with the distribution peaking at a slightly lower β, consistent with the possibility of a different physical origin of this phase (Zhang et al. 2006, 2007a, 2009; Liang et al. 2006). The distribution of βs for the single power-law light curves is statistically consistent with the other individual segments. These spectral properties are in agreement with the suggestion from the temporal distributions that this sample is a mixture of the other segments.

The steep decline of segment I in the canonical X-ray light curve is probably due to the tail end of the prompt emission and is governed by the curvature effect, for which emission from different viewing angles reaches the observer with different delays due to light propagation effects (Kumar & Panaitescu 2000; Zhang et al. 2006). The relationship between the temporal and spectral slopes of the high latitude emission is

$$\begin{equation} \alpha =2+\beta \end{equation} \tag{ 8 }$$

or in our formulation

$$\begin{equation} \Psi =\alpha -2-\beta \end{equation} \tag{ 9 }$$

and is independent of any of the environmental or other parameters that affect the closure relations for the external shock. Therefore, we only use this relation to help discriminate segment I from other segments and we do not use this relation to constrain any of the burst properties explored throughout the rest of this paper. We find that ∼41% of the segment I in our sample are consistent with this relation, with the remainder either steeper (∼26%) or shallower (∼34%) than this relation.

Zhang et al. (2007a, 2009) explore steep declines in the XRT afterglow data set and discuss several physical explanations, finding that steep decays with and without significant spectral evolution can be explained by the curvature effect. They explore the subset with distinct spectral evolution and no clear and obvious contaminating flaring to conclude that the data are best characterized by an apparent evolution of a cutoff power-law (CPL) spectrum. Those that are contaminated with flaring mix spectral evolution during the flares with possible spectral evolution of the underlying afterglow, leading to difficulty in characterizing the mean spectral and temporal properties. Zhang et al. (2009) can interpret the apparent spectral evolution during the steep decay phase using a curvature effect model that invokes a non-power-law spectrum at the end of the prompt emission phase. Those afterglows with α steeper than 2 + β could also be caused by only seeing the tail of a flare and interpreting it as the steep decay. This is further complicated by the choice of t₀ for determining the slope of the temporal decay that affects segment I much more strongly than the other segments (Zhang et al. 2006; Liang et al. 2006). Zhang et al. (2007a) fit t₀ in their multicomponent spectral evolution models to best characterize the temporal decay, rather than using the BAT trigger time. This method could help explain some of the deviations. However, these efforts are beyond the scope of this study and have little consequence for the analysis of segments II–IV and the study of jet breaks, so are not repeated here.

The closure relations governing external shocks depend on the local environment, range of electron spectral index, cooling regime, energy injection, peak frequency and cooling frequency. The pre-jet break "isotropic" relations and the post-jet break relations are given in Table 1. The framework for the closure relations for the slow and fast cooling cases is explored by Sari et al. (1998) and expanded upon for the collimated case by Sari et al. (1999). Dai & Cheng (2001) include the 1 < p < 2 cases, with additional cases given in Zhang & Mészáros (2004). The addition of the energy injection mechanism to explain the plateau portion of X-ray light curves brought additional parallel closure relation sets for p > 2 given in Zhang et al. (2006). We extract the additional jet break relations from information provided in Panaitescu (2005) and Panaitescu et al. (2006). We choose not to include the cases of energy injection for 1 < p < 2 because this scenario is unduly complicated, unlikely, and often not analytically solvable.

In segments II–III, the relationship between the electron slope (p) and the measured X-ray spectral slope (Γ) is derived in (Sari et al. 1998) as

$$\begin{equation} \Gamma -1 \equiv \beta = \left\lbrace \begin{array}{@{}l@{\quad }l} 1/2 & \nu _c<\nu <\nu _m \\ (p-1)/2 & \nu _m<\nu <\nu _c \\ p/2 & \nu _m,\nu _c<\nu, \\ \end{array}\right. \end{equation} \tag{ 10 }$$

where ν_m and ν_c are the synchrotron and cooling frequencies, respectively.

Other large-scale studies (Liang et al. 2008; Panaitescu 2007; Evans et al. 2008) of jet break closure relations address only the simplest cases of the uniform jet (Zhang & Mészáros 2004). We also include the nonspreading uniform jet with energy injection, the laterally spreading uniform jet with and without energy injection and the simplest form of structured jets with power-law angular distribution of energy outflow from Panaitescu et al. (2006). We apply all of these cases for both ISM and wind environments. We choose not to apply any of the 1 < p < 2 closure relations that include energy injection and occur post-jet break, as well as anything more complex than the most simple p > 2 structured jet relations due to their complexity and impracticability. Closure relations for all jet models are listed in Table 1. All of these relations are for on-beam geometry. In particular, the power-law structured jet model requires that the line of sight is within the central cone beam.

The power-law structured jet relations are valid for a particular α and β provided the index, k, of the angular energy distribution is less than $\tilde{k}$, where

$$\begin{equation} \tilde{k}= \left\lbrace \begin{array}{@{}l@{\quad }l} \displaystyle\frac{8}{2\beta +5} \ (\hbox{ISM}\ \nu <\nu _c) \\ \displaystyle\frac{8}{2\beta +3} \ (\hbox{ISM}\ \nu >\nu _c) \\ \displaystyle\frac{8}{2\beta +4} \ (\hbox{Wind}). \\ \end{array}\right. \end{equation} \tag{ 11 }$$

The case of $k>\tilde{k}$ reduces to the nonspreading uniform jet relations because the core is dominant (Panaitescu 2005).

The models in Table 1 can be used in succession throughout an individual X-ray afterglow light curve, however several models often fit equally well for any given light curve segment. After fitting each closure relation to each set of α and β, we combine them to form a coherent physical model for each afterglow. While we often cannot distinguish a unique set of closure relations, we can use information from one segment to exclude inconsistent relations from other segments. We assume there is no perceptible change in the circum-GRB environment probed throughout an individual afterglow, and therefore if either an ISM or Wind environment can be excluded in any segment, we exclude it for the other segments. We do not allow transitions from slow cooling to fast cooling within any light curve. Since we do not see evidence of a change of spectral index between segments in most individual afterglows, we require the spectral regime to remain constant throughout a light curve (i.e., either ν < ν_c or ν>ν_c for slow cooling, or ν_c < ν < ν_m or ν>ν_m for fast cooling), therefore excluding light curve breaks due to transitions of the cooling frequency through the X-ray band. Theoretically, we do not expect the electron spectral index, p, to change during an afterglow light curve. However, because we do not include the full suite of 1 < p < 2 relations with energy injection, we are careful not to exclude a 1 < p < 2 model just because only p > 2 models are available to fit other segments. We eliminate models only on the the basis of their p values being inconsistent between segments, which is especially important for those segments with p ∼ 2 and large error bars. We require a segment consistent with a particular isotropic model to also be consistent with a corresponding jet model in the following segment, and conversely a jet model must be consistent with a corresponding previous isotropic model in terms of environment and spectral frequency regime. Similarly, any energy injection model in segment II must be consistent with the models for the following segment (if present) in terms of environment and frequency regime. When a post-jet break relation with energy injection is applied, we require q to be consistent between pre- and post-jet break segments. We also only allow jet breaks to directly precede potential segments III or IV. Ambiguous single power laws have no restrictions on available sets of closure relations.

Using these criteria, we eliminate some models to better constrain a coherent picture for each afterglow. Unfortunately, a unique set of models is often still unattainable, as several equally likely relations remain. In an attempt to extract any available information from the remaining closure relations, we look at constraints from families of closure relations to determine specific properties. For example, if several closure relations are consistent with a particular segment or whole afterglow but they are all ISM relations, then we can conclude that this afterglow is consistent with the expectations of an ISM environment. This also works for families of relations with a common spectral frequency regime, environment, and pre/post-jet break regime.

Our sample is broken up into four groups (as described in Section 2.1.2) based on their temporal properties. These groups include the I–II–III–IV/II–III–IV sample which have distinct segment IV, segments I–II–III, ambiguous segments II–III/III–IV, and single power laws. The latter three samples may contain jet breaks and still fit within the canonical light curve picture. The presence of observing biases can explain the missing contextual clues that would make jet break distinctions more clear. In order to avoid imposing biases on the results, we have not attempted to distinguish jet breaks based on a priori assumptions for decay indices or break times.

The fact that segments II–III/III–IV light curves do not look like the canonical light curve is likely a result of two different observing biases: a late start or early end. The behavior prior to segment II was either not observed at all due to a late observation start, or was complicated by flaring behavior making the underlying afterglow shape unclear, both leading to an unknown initial steep decline. If the observations began even later, segment I and II would be missed, leaving only segments III–IV. Similarly, an early observation end would also lead to the observer missing the jet break. This end could be due to either a manual end of the observations because of observing constraints, or the afterglow becoming too faint for XRT detection. These two-segment ambiguous light curves may contain jet breaks as either the observed break, or with the inclusion of energy injection, the jet break would have occurred sometime prior to the start of the first segment. In this latter case, the actual jet break time is impossible to determine because there were no observations prior to the start of the apparent segment II. Considering the wide range of decay indices, observing times, and redshifts, these limitations are likely to have influenced the data.

Apparent segments I–II–III show some suggestions of deviation from the canonical form, which can also be explained by observing biases. In some cases, the temporal decay of these segments III is substantially steeper than typical segments III, as if they transitioned directly from segment II to segment IV. Alternatively, it is possible that the segment III is simply missing or buried in the data due to a short duration, large error bars, or gaps in the light curve with segment IV appearing to follow directly after segment II. Willingale et al. (2007) proposed an empirical form of the canonical light curve that is made up of two falling exponential plus power-law functions that can explain the structure of most X-ray afterglow light curves. They suggest that missing phases, like that seen in the apparent II–IV transitions, are a result of one of the two components being particularly bright, weak, or short lived.

Other types of light curves exist in the sample for which we do not observe jet breaks, and would not expect them to be hidden or ambiguous. These include light curves showing only segments I–II, where the later segments presumably occurred after observations ended. Another type and possible jet break sample contaminant is the so-called "naked bursts" which show only the initial steep decline from the prompt emission without the subsequent segments (II–IV) attributed to the afterglow. This is thought to occur because the surrounding medium is not dense enough to produce the external shocks that power GRB afterglows (Kumar & Panaitescu 2000). These GRBs can appear as either single power-law decays or broken power laws where the prompt emission and high latitude emission masquerade as segment II–III light curves in the absence of comparison to the γ-ray prompt emission. Three examples of possible naked GRBs that have been investigated extensively are the long GRBs 050421 (Godet et al. 2006) and 050412 (Mineo et al. 2007), and the short hard burst GRB 051210 (La Parola et al. 2006). To filter out these bursts we search for any light curves that appear to be segments II–III, have break times <1000 s, and are consistent with the high latitude closure relation (Equation (8)); these should not be considered candidates for jet breaks. Using these criteria, we identify GRB 060801 as another possible naked burst in addition those identified in the literature that is also a short hard GRB. There may be more naked bursts in the single power-law sample like that of GRB 050412, but we have no clean way of distinguishing them, therefore we leave them for later discrimination. It is possible that additional naked bursts are present in the 29 GRBs excluded from the final sample due to their faintness and limited observations which made temporal and spectral analysis not possible.

In order to learn more about these ambiguous afterglows and distinguish jet breaks, we apply the closure relations. The following section describes how we use the closure relations and the temporal behavior to identify additional jet break candidates.

We define the Prominent jet break class as those light curves for which we can clearly distinguish a break between segments III and IV, with segment IV being consistent with post-jet break closure relations, consistent with the canonical morphology (Figure 3). This conservative classification criterion requires that the light curve is composed of either segments I–II–III–IV or II–III–IV, and the final segment is consistent with post-jet break closure relations. We find 30 such afterglows, 28 of which are consistent within 2σ of at least one post-jet break closure relation in their segment IV after internal consistency checks.

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In the two inconsistent cases (GRBs 061121 and 070508), all models were eliminated in the process of internal consistency checks with the 2σ criteria. The unusually bright GRB 061121 (Page et al. 2007) was triggered by a precursor, causing the choice of T₀ to affect the slopes, which may account for some of the deviations from the models. Page et al. (2007) also suggest the presence of a Comptonized component. Therefore, these outlier cases may not be well represented by the canonical form, have breaks due to other mechanisms such as the cooling frequency moving through the X-ray band, or are not valid with the suite of closure relations used here, and are ignored in the following analysis.

Based on the canonical light curve form, we assume that segment IV is post-jet break. Therefore, the post-jet break closure relations are the only models allowed in segment IV. In contrast, many closure relations are allowed by the canonical form for segments II and III. Figure 4 shows an example of a burst in the Prominent jet break category with its light curve and the closure relations that are consistent with the data. All four segments of this burst can be adequately characterized by the closure relations: segment I is consistent with the high latitude relation; segment II requires p > 2 and energy injection, but is consistent with either slow or fast cooling and either ISM or wind environment; segment III is consistent with either isotropic or jet-with-energy-injection relations; and segment IV is consistent with either a spreading or nonspreading jet with an ISM or wind environment. The resulting fits imply that the jet break time cannot be unambiguously established; if the II–III break is due to the cessation of energy injection then the III–IV break is a jet break, but it is also possible that the II–III break is the jet break and the III–IV break is due to the end of energy injection (see also Ferrero et al. 2008 for a similar analysis of this GRB). We list the properties of the Prominent jet break bursts in Table 2. The distributions of these properties will be discussed in Section 4.4.

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We wish to assess the deficit of jet breaks in the XRT afterglow sample, therefore we must make a reasonable estimate of the fraction of our sample with jet breaks, accounting for a variety of observing biases. Due to various observing constraints and light curve profiles, not all burst observations began with an immediate slew nor were they all observed out to a time at which the jet break is expected to have occurred. Therefore, to calculate an accurate jet break fraction, we reduce our sample to only those GRBs for which the observations span a time frame where we would expect a jet break. Previous studies of optical jet breaks (Frail et al. 2001; Bloom et al. 2003; Zeh et al. 2006) showed them to occur several days after the GRB trigger. Instead of making a priori assumptions about achromaticity or assuming similar behavior, we determine the time frame during which we would expect a jet break by studying the Prominent jet break sample.

The Prominent jet break sample consists of 28 X-ray afterglows with t_start ranging from a few minutes for those light curves that start with segment I to an hour for those that start during segment II, and t_stop ranging between 2 days and five months (excluding late-time Chandra observations). The important measurement from the Prominent sample is the jet break time (t_b) which ranges between 0.02 and 26.2 days (excluding earlier breaks that suggest post-jet-breaks-with-energy-injection). Excluding the extremely late jet break case of GRB 060729, and the extremely early jet break case of GRB070328, the latest light curve break in the whole sample is ∼12 days, with 90% occurring within 10 days. Therefore, we define our "complete" sample (those GRBs for which observations sufficiently cover a time range where a jet break could have been measured), to begin before 0.1 days and end after 10 days. Of the 230 GRBs in our sample, 82 fit these completeness criteria. The distributions of observation start, stop, break, last detection, and the jet break lower limit (described in Section 5) are shown in Figure 5.

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The Prominent jet break category constitutes only ∼12% of the total sample. We examined the remaining light curves for evidence of "hidden" jet breaks, identified by their closure relations rather than the light curve morphology. Our Hidden jet break category includes those light curves with ambiguous final segments. We consider three ambiguous cases. As noted in Section 2.1.2, broken power laws with α₂ > α₁ cannot be distinguished a priori between segments II–III and III–IV. If the final segment is only consistent with post-jet break closure relations, then we designate it as a III–IV case with a Hidden jet break. The second ambiguous case involves three segment light curves initially classified as I–II–III in which the final segment is steeper than typical segment III and consequently only consistent with post-jet break closure relations. Therefore, this type appears more like a I–II–IV or a jet-break-with-energy-injection. The third ambiguous case involves single power-law light curves which are only consistent with post-jet break closure relations even though the jet break itself is not observed.

We find an additional 12 light curves that fit these criteria, of which three are from the two-segment ambiguous sample, nine are segment III from the I–II–III sample, and none are single power laws. Figure 6 shows an example that was classified as an ambiguous II–III/III–IV until we found that its final segment is only consistent with post-jet break closure relations. We list the properties of the Hidden jet breaks in Table 3.

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Our classification of XRF 060218 as a Hidden jet break illustrates one limitation of our methodology. Using the closure relations in Table 1, we find a post-jet break decay to be the only possible outcome. In doing so we assume that the emission is due to a purely forward shock origin. Individual studies on XRF/GRB 060218 (Campana et al. 2006; Ghisellini et al. 2007a) show a strong early thermal component related to the associated SN 2006aj and the observed shock break out, followed by a possible Compton component. Our analysis reveals an unusually steep late-time spectral slope, which may suggest that our models are not applicable in this individual case. Therefore, this break may not be due to a jet break at all. We choose to treat all GRBs in our sample in the same way, and our methodology did not account for possible non-power-law spectral components. As a result, a small fraction of the sample could be misclassified. We note that only ∼1% of GRB afterglows show evidence for thermal spectral components.

We now examine the remaining light curves to search for additional jet breaks in the data set that have not been previously identified as Prominent or Hidden. Of the remaining 190 light curves, there are 36 that are consistent with only pre-jet break relations in their last segment, and therefore classified as nonjet breaks. The 3 naked bursts are also classified as nonjet breaks, and 6 light curves that are inconsistent with all closure relations after consistency checks are discussed below. The remaining 145 light curves are ambiguous because they are consistent with both pre- and post-jet break closure relations. This sample includes those ambiguous segments II–III/III–IV, single power laws, and segments I–II–III in which segment III is potentially post-jet break. We use the properties of the Prominent jet break sample to identify possible jet breaks in the ambiguous sample.

The samples of ambiguous segments II–III/III–IV, segments I–II–III, and the Prominent sample have the common feature of apparent segments II–III with a break in between, which is a shared distinct component that can be used to compare them. We use the Prominent jet break sample as a "control group" for comparisons with the other categories. Figure 7 shows the distributions of temporal decay indices and break times for segments II and III for all three groups. The temporal indices split relatively cleanly into several distinct distributions, especially in the Prominent sample. There is a small amount of overlap between segments III and IV in the Prominent jet break sample, but this is not surprising considering the multitude of possible model scenarios employed to explain these light curves. The distributions for the ambiguous segments II–III/III–IV are plotted assuming that they are all segments II–III. The broader and steeper distributions in this sample are consistent with some contamination by actual segments III–IV. The steeper than expected segments II and III from the sample of segments I–II–III are also consistent with the hypothesis of segment III confusion and post-jet-break-with-energy-injection. The break times between segments II and III, as plotted in the right-hand side of Figure 7, are also suggestive of these findings. The distributions are not nearly as narrow as those suggested by the canonical light curve form (Nousek et al. 2006; Zhang et al. 2006), but do suggest a spread to larger break times in the contaminated ambiguous II–III/III–IV sample and the segments I–II–III sample. The break times are also dispersed due to redshift effects whose amplitude is unknown for ∼60% of the GRBs. When looking only at those GRBs with known redshifts, these same distributions are narrower and more cleanly separated.

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To further distinguish those potential jet breaks that are in this contaminated ambiguous group, we look at the correlations between the temporal decay slopes from different segments in the same light curves. In the left side of Figure 8 we plot α_III versus α_II, α_IV versus α_III, and α_IV versus α_II for the Prominent sample as our "control group" and see a reasonably clean distinction between the light curve transitions in this α–α parameter space, which can be used to classify the remaining ambiguous segments. We use the "control group" to determine the area of α–α space for each cluster of specific segment combinations. The mean values for each cluster from the control group are indicated by the black crosses. For each ambiguous light curve, we calculate the distance in α–α space to each cluster mean weighted by σ_α, and categorize the ambiguous segment transitions based upon their proximity to these mean values (right side, Figure 8).

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Applying this technique to the 98 light curves with segments II–III/III–IV or segments I–II–III from the remaining 145 ambiguous light curves, we identify 21 bursts that are similar in alpha–alpha space to the Prominent jet break sample for the segments II–IV transition, and 26 bursts that are similar to the segments II–IV. Their consistency with post-jet break closure relations and temporal slopes suggest that there are indications of jet breaks in those light curves. These are categorized as Possible jet breaks (Table 4). The remaining 51 ambiguous light curve transitions appear to be segments II–III and therefore probably pre-jet break. However, they could possibly be segment II with a post-jet-break-with-energy-injection segment III. We are unable to distinguish these cases, therefore we put these remaining afterglows into a new category called Unlikely jet breaks. (These light curves are called Unlikely jet breaks because their slopes are too shallow compared to the Prominent jet break αs to be post-jet break.) We show example light curves for each of these newly segregated groups in Figures 9–11.

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Finally, we examine single power laws that are consistent with both pre- and post-jet break closure relations. The remaining ambiguous light curves consist of 47 single power-laws. These single power-law light curves do not easily fit into the canonical picture unless they are either a short snapshot of one segment or are the post-jet break component and therefore relevant to this study. Often these light curves are plagued by a low signal-to-noise ratio and few counts, which leads to minimal information to be extracted. Other light curves in this group are dominated by large flaring during part or all of their light curves, which therefore makes determination of the underlying afterglow shape impossible during that time interval. We fit only the portion of the light curves that clearly returns to the underlying nonflaring level. However, there are a few examples of outliers that do not have flares and do have strong counting statistics. The most notable and best sampled X-ray afterglow in this outlier group is that of GRB 061007 which displayed a very bright exceptionally smooth single power law. Schady et al. (2007) showed that this must be either due to a very late-time jet break requiring enormous kinetic energy or an exceptionally early jet break (t_b < 80 s) with highly collimated outflow from a jet that includes continuous energy injection throughout. We do not exclude these isotropic models with extreme energy requirements from our global study, but these extreme energy requirements are a valid concern addressed in Section 4.5.

We attempt to filter the single power laws that are a result of a short pre-jet break segment from those that represent a post-jet break decay. In the context of the canonical X-ray afterglow, we should be able to look at the relationship between observation start and stop times, and the temporal decay to distinguish pre- and post-jet break. Figure 12 shows the relationship between the time that the observations begin and the time of last detection as a function of α. Unfortunately, there are only 13 redshifts measured of the 47 total GRBs in this category that are consistent with at least one post-jet break closure relation. Therefore, to include as many potential jet breaks as possible, the times used in the following distinctions are in the observed frame and not the rest frame. There appears to be a general trend that suggests that those single power laws for which observations began late (greater than 10⁴ s) tend to be steeper (α>1.2) than those that start early and continue for a long observation. We make a cut in this parameter space and deem those afterglows that start within or traverse the time frame for which we would expect jet breaks (from the Prominent sample) and have steep (α>1.5) decays as Possible jet breaks and add them to that sample. These six bursts are indicated in Figure 12. The remaining single power-law afterglows that appear to be pre-jet break are put into the Unlikely jet break category.

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We present the complete sample of Possible jet breaks in Table 4. The probable jet break time for each light curve is identified as t_b. However, some ambiguity still remains of whether this is the time of the jet break. Therefore, we also list the t_start and t_lastdet (the time of last detection, if relevant) because they provide limits on the jet break time if the jet break is not t_b. Using the completeness criteria described in Section 4.1, we find that at least 23% of the Possible jet breaks and 35% of the Unlikely jet breaks were observed sufficiently long enough that we would expect to have seen a jet break during their observations.

The six afterglows that are inconsistent with all closure relations after internal consistency checks, but have temporal behavior of their final segments similar to Prominent segments III–IV or II–IV are also included in the Unlikely jet break sample. The breaks in these cases may be due to origins other than those in the canonical model such as transition of the cooling frequency through the X-ray band or Compton processes. They may also be contaminated by small-scale flaring that is not removed by our methods. They are denoted in Table 4 by "none" in the requirements field.

The complete sample of Unlikely jet breaks are listed in Table 5, and the secure Nonjet breaks are listed in Table 6. The criteria, inputs, and final memberships of the Prominent, Hidden, Possible, Unlikely, and Nonjet break categories are summarized in Table 7. The nonjet break category includes all remaining bursts, most of which are segment I–II transitions.