HOW BAD OR GOOD ARE THE EXTERNAL FORWARD SHOCK AFTERGLOW MODELS OF GAMMA-RAY BURSTS?

The standard external shock models of GRB afterglows have clear theoretical predictions that can be verified or falsified by the observational data. These models attribute the multiwavelength afterglow emission to synchrotron radiation of electrons accelerated in the shock front as the fireball jet interacts with the circumburst medium. The models largely do not depend on the details of the central engine activities, so the afterglow behaviors only depend on a limited number of parameters. In the convention of ${F}_{\nu }\propto {t}^{-\alpha }{\nu }^{-\beta }$, where α and β are the temporal and spectral indices of the afterglows that can be measured directly from observations, the models predict certain relationships between the α and β values, which are called the "closure relations" of the models (e.g., Zhang & Mészáros 2004; Zhang et al. 2006; Gao et al. 2013). Technically there are many submodels (e.g., interstellar medium (ISM) versus wind, adiabatic versus radiative, whether or not there is energy injection), physical regimes (reverse shock crossing phase, self-similar phase, post-jet-break phase, Newtonian phase), and spectral regimes (different orders among the observed frequency (ν) and several characteristic frequencies (${\nu }_{{\rm{m}}}$, ν_c, the self-absorption frequency ν_a)). We refer to a comprehensive review of Gao et al. (2013) and references therein for the details of various models.

For the time frame of our interest (hours to weeks after the trigger), the reverse shock crossing phase is usually over, and the blast wave is still in the relativistic phase. This greatly reduces the number of relevant models. In Table 2, we summarize the α and β predictions of various models studied in this paper following Zhang et al. (2006) and Gao et al. (2013). This includes the ISM and wind models for adiabatic blast waves,¹³ for both pre- and post-jet-break temporal phases, with and without continuous energy injection, and for two spectral regimes (I: $\nu \gt {\nu }_{{\rm{c}}}$ and II: ${\nu }_{{\rm{m}}}\lt \nu \lt {\nu }_{{\rm{c}}}$) in the slow cooling (${\nu }_{{\rm{m}}}\lt {\nu }_{{\rm{c}}}$) regime. By doing so, we have assumed that ${\nu }_{{\rm{a}}}\lt \mathrm{min}({\nu }_{{\rm{m}}},{\nu }_{{\rm{c}}})$ and $\mathrm{min}({\nu }_{{\rm{X}}},{\nu }_{{\rm{O}}})\gt {\nu }_{{\rm{m}}}$, which is usually satisfied for optical and X-ray afterglow emission for typical GRB parameters.

Table 2.  Temporal Decay Index α and Spectral Index β in Different Afterglow Models

CMB	Spectral Regime	$\beta (p)$	$\alpha (p)/\alpha (p,q)$	$\alpha (\beta )/\alpha (\beta ,q)$	$\alpha (p)/\alpha (p,q)$	$\alpha (\beta )/\alpha (\beta ,q)$
			$p\gt 2$		$1\lt p\lt 2$
	Adiabatic deceleration without energy injection
ISM	${\nu }_{{\rm{m}}}\lt \nu \lt {\nu }_{{\rm{c}}}$	$\frac{p-1}{2}$	$\frac{3(p-1)}{4}$	$\alpha =\frac{3\beta }{2}$	$\frac{3(p+2)}{16}$	$\alpha =\frac{6\beta +9}{16}$
	$\nu \gt {\nu }_{{\rm{c}}}$	$\frac{p}{2}$	$\frac{3p-2}{4}$	$\alpha =\frac{3\beta -1}{2}$	$\frac{3p+10}{16}$	$\alpha =\frac{3\beta +5}{8}$
Wind	${\nu }_{{\rm{m}}}\lt \nu \lt {\nu }_{{\rm{c}}}$	$\frac{p-1}{2}$	$\frac{3p-1}{4}$	$\alpha =\frac{3\beta +1}{2}$	$\frac{p+8}{8}$	$\alpha =\frac{2\beta +9}{8}$
	$\nu \gt {\nu }_{{\rm{c}}}$	$\frac{p}{2}$	$\frac{3p-2}{4}$	$\alpha =\frac{3\beta -1}{2}$	$\frac{p+6}{8}$	$\alpha =\frac{2\beta +6}{8}$
	Adiabatic deceleration with energy injection
ISM	${\nu }_{{\rm{m}}}\lt \nu \lt {\nu }_{{\rm{c}}}$	$\frac{p-1}{2}$	$\frac{(2p-6)+(p+3)q}{4}$	$\alpha =(q-1)+\frac{(2+q)\beta }{2}$	$-\frac{12-18q-p(q+2)}{16}$	$\alpha =\frac{19q-10}{16}+\frac{(2+q)\beta }{8}$
	$\nu \gt {\nu }_{{\rm{c}}}$	$\frac{p}{2}$	$\frac{(2p-4)+(p+2)q}{4}$	$\alpha =\frac{q-2}{2}+\frac{(2+q)\beta }{2}$	$\frac{14q+p(q+2)-4}{16}$	$\alpha =\frac{7q-2}{8}+\frac{(2+q)\beta }{8}$
Wind	${\nu }_{{\rm{m}}}\lt \nu \lt {\nu }_{{\rm{c}}}$	$\frac{p-1}{2}$	$\frac{(2p-2)+(p+1)q}{4}$	$\alpha =\frac{q}{2}+\frac{(2+q)\beta }{2}$	$\frac{4+(p+4)q}{8}$	$\alpha =\frac{5q+4}{8}+\frac{\beta q}{4}$
	$\nu \gt {\nu }_{{\rm{c}}}$	$\frac{p}{2}$	$\frac{(2p-4)+(p+2)q}{4}$	$\alpha =\frac{q-2}{2}+\frac{(2+q)\beta }{2}$	$\frac{(6+p)q}{8}$	$\alpha =\frac{(\beta +3)q}{4}$
	Post-jet-break phase without energy injection
ISM	${\nu }_{{\rm{m}}}\lt \nu \lt {\nu }_{{\rm{c}}}$	$\frac{p-1}{2}$	$\frac{3p}{4}$	$\alpha =\frac{6\beta +3}{4}$	$\frac{3(p+6)}{16}$	$\alpha =\frac{3(2\beta +7)}{16}$
	$\nu \gt {\nu }_{{\rm{c}}}$	$\frac{p}{2}$	$\frac{3p+1}{4}$	$\alpha =\frac{6\beta +1}{4}$	$\frac{3p+22}{16}$	$\alpha =\frac{3\beta +11}{8}$
Wind	${\nu }_{{\rm{m}}}\lt \nu \lt {\nu }_{{\rm{c}}}$	$\frac{p-1}{2}$	$\frac{3p+1}{4}$	$\alpha =\frac{3\beta +2}{2}$	$\frac{p+12}{8}$	$\alpha =\frac{2\beta +13}{8}$
	$\nu \gt {\nu }_{{\rm{c}}}$	$\frac{p}{2}$	$\frac{3p}{4}$	$\alpha =\frac{3\beta }{2}$	$\frac{p+10}{8}$	$\alpha =\frac{\beta +5}{4}$
	Post-jet-break phase with energy injection
ISM	${\nu }_{{\rm{m}}}\lt \nu \lt {\nu }_{{\rm{c}}}$	$\frac{p-1}{2}$	$\frac{p(q+2)-4(1-q)}{4}$	$\alpha =\frac{5q-2}{4}+\frac{(2+q)\beta }{2}$	$\frac{22q-4+p(q+2)}{16}$	$\alpha =\frac{11q-2}{8}+\frac{(2+q)\beta }{8}$
	$\nu \gt {\nu }_{{\rm{c}}}$	$\frac{p}{2}$	$\frac{3q-2+p(q+2)}{4}$	$\alpha =\frac{3q-2+2\beta (q+2)}{4}$	$\frac{18q+4+p(q+2)}{16}$	$\alpha =\frac{9q+2+\beta (q+2)}{8}$
Wind	${\nu }_{{\rm{m}}}\lt \nu \lt {\nu }_{{\rm{c}}}$	$\frac{p-1}{2}$	$\frac{3q-2+p(q+2)}{4}$	$\alpha =q+\frac{(2+q)\beta }{2}$	$\frac{{pq}+8q+4}{8}$	$\alpha =\frac{1}{2}+\frac{(2\beta +9)q}{8}$
	$\nu \gt {\nu }_{{\rm{c}}}$	$\frac{p}{2}$	$\frac{p(q+2)-4(1-q)}{4}$	$\alpha =\frac{\beta (q+2)-2(1-q)}{2}$	$\frac{(p+10)q}{8}$	$\alpha =\frac{(\beta +5)q}{4}$

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The energy injection model invokes either a long-lasting central engine (Dai & Lu 1998; Zhang & Mészáros 2001) or Lorentz-factor-stratified ejecta (Rees & Mészáros 1998; Sari & Mészáros 2000; Uhm et al. 2012). The two scenarios are equivalent with each other in terms of light curve behaviors given a relationship between the central engine parameter q and the stratification parameter k (Zhang et al. 2006). We adopt the description of a long-lasting central engine with a power-law luminosity history $L(t)={L}_{0}{\left(\frac{t}{{t}_{0}}\right)}^{-q}$ (Zhang & Mészáros 2001), so the injected energy is ${E}_{\mathrm{inj}}=\frac{{L}_{0}{t}_{0}^{q}}{1-q}{t}^{1-q}$. The prescription applies when $q\lt 1$. The relevant closure relations are presented in Table 2.

Many observations suggest that GRB outflows are collimated. Assuming a conical jet with opening angle ${\theta }_{j}$, a steepening in the afterglow light curve is predicted when $1/{\rm{\Gamma }}\gt {\theta }_{j}$ (Γ is the bulk Lorentz factor of the blast wave). The main reason for this steepening is the so-called "edge effect" (e.g., Panaitescu et al. 1998):¹⁴ the $1/{\rm{\Gamma }}$ cone is no longer filled with emission beyond the jet break time (when $1/{\rm{\Gamma }}\gt {\theta }_{j}$). There is a reduction factor in flux ${\theta }_{j}^{2}/{(1/{\rm{\Gamma }})}^{2}={{\rm{\Gamma }}}^{2}{\theta }_{j}^{2}$. The relevant closure relations are also presented in Table 2.

It is possible that in some GRBs the energy injection phase lasts longer than the jet break time, so a jet break with energy injection in both the pre- and postbreak phases can be observed. The relevant closure relations of such models were derived in Gao et al. (2013) and are also presented in Table 2.

For the time domain we are interested in and for the optical and X-ray bands, there are four types of light curves (Figure 6).

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(1) BPL light curves with an energy injection break: in reference to the canonical X-ray light curve (Zhang et al. 2006), as reproduced in Figure 6(a), the energy injection break connects the shallow decay phase (segment II) to the normal decay phase (segment III), and a typical light curve is shown in Figure 6(b). Before and after the break, the adiabatic deceleration α(β) relations with and without energy injection (as listed in Table 2, Zhang et al. 2006; Gao et al. 2013) are used to check whether the data are consistent with model predictions.

(2) BPL light curves with a jet break: this corresponds to the transition from segment III to IV in the canonical light curve, and a typical light curve is shown in Figure 6(c), upper curve. Light curves in such a category should satisfy the constant-energy, isotropic closure relations before the break and the edge-effect post-jet-break closure relations after the break, with no energy injection effect either before or after the break (Table 2). The postbreak decay index is required to be steeper than 1.5 for this model.

(3) BPL light curves with a jet break with energy injection: This model allows the energy injection to extend longer than the jet break. The temporal break is still defined by the edge effect of a canonical jet, but the decay slopes before and after the break are shallower than the previous case (lower curve in Figure 6(c)), so a q parameter is introduced for both the pre- and postbreak phases.

(4) SPL decay: for some GRBs, a SPL function is adequate to describe the afterglow data (Figure 6(d)) after the deceleration phase. In the X-ray band, there might be a steeper decay phase before this SPL phase, which is due to the tail emission from the prompt emission (Barthelmy et al. 2005; Tagliaferri et al. 2005; Zhang et al. 2006). We ignore the steep decay phase and treat it as an SPL decay (upper curve of Figure 6(d)). Similarly, in the optical band, some GRBs show an early rising phase, which is a signature of the onset of afterglow at the deceleration radius (peak of the light curve, lower curve of Figure 6(d)). We also treat these light curves as SPL decay ones.

For all of the types, sometimes there are X-ray flares overlapping the power-law decay segments. We do not include the flares in our data fitting because they originate from a different emission component due to late central engine activities (e.g., Zhang et al. 2006; Maxham & Zhang 2009).

One important task is to perform a self-consistency check between the optical and X-ray bands. If a GRB is consistent with the external forward shock model, we demand that the GRB satisfy the following criteria.

• 1.  
  The X-ray and optical light curves are consistent with having an achromatic break, if any.
• 2.  
  Both the X-ray and optical light curves should satisfy the closure relations of the same circumburst medium type (ISM or wind) in both the pre- and postbreak temporal segments.
• 3.  
  Either both bands belong to the same spectral regime, or the two bands are separated by a cooling break ν_c, with the X-ray band above the break and the optical band below the break (with allowance for a gray zone; see more discussion below).
• 4.  
  The inferred electron spectral index p from both bands and from both pre- and postbreak segments should be consistent with each other within errors.
• 5.  
  For energy injection models, the energy injection parameter q values derived from the X-ray and optical bands should be consistent with each other.

Technically, we check the consistency between the closure relations for an individual temporal segment in an individual energy band. To ensure the same p value is derived for different bands, we also check the consistency between the data and models in the $\triangle {\beta }_{{\rm{X}},{\rm{O}}}-\triangle {\alpha }_{{\rm{X}},{\rm{O}}}$ plane. Here $\triangle {\alpha }_{{\rm{X}},{\rm{O}}}={\alpha }_{{\rm{X}}}-{\alpha }_{{\rm{O}}}$ is the difference between the decay indices in the X-ray and optical bands, respectively, in the same temporal segment, and $\triangle {\beta }_{{\rm{X}},{\rm{O}}}={\beta }_{{\rm{X}}}-{\beta }_{{\rm{O}}}$ is the difference between the spectral indices in the X-ray and optical bands, respectively. Based on the closure relations (Table 2), one can derive the $\triangle {\beta }_{{\rm{X}},{\rm{O}}}-\triangle {\alpha }_{{\rm{X}},{\rm{O}}}$ relations of all of the models (Tables 3 and 4). One can see that even though the α and β values can be very different in different models, the ${\rm{\Delta }}{\alpha }_{{\rm{X}},{\rm{O}}}$ and ${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}$ values have several well-predicted values. In particular, for the SPL and jet break models, both pre- and postbreak values are well-defined constants. For the energy injection breaks, the postbreak segment does not depend on the free parameter q. As a result, if one focuses on the second component only, all of the models can be expressed as several representative coordinate values in the $\triangle {\beta }_{{\rm{X}},{\rm{O}}}-\triangle {\alpha }_{{\rm{X}},{\rm{O}}}$ plane. Considering the possible gray zones (see below for details), these points define several straight lines in the $\triangle {\beta }_{{\rm{X}},{\rm{O}}}-\triangle {\alpha }_{{\rm{X}},{\rm{O}}}$ space (Figure 7 for details). If the observed data intersect with these model lines (within errors), one can regard them as being consistent with the model predictions.

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Table 3.  The ${\rm{\Delta }}{\alpha }_{{\rm{X}},{\rm{O}}}$ and ${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}$ Values in Different Afterglow Models, $p\gt 2$

		$p\gt 2$
Same Regimea	Different Regimesb		Gray Zonec
ISM,Wind	ISM	Wind	ISM	Wind
		SPL
${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}=0$	${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}=\frac{1}{2}$	${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}=\frac{1}{2}$	${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}$ = (0, $\frac{1}{2}$)	${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}$ = (0, $\frac{1}{2}$)
${\rm{\Delta }}{\alpha }_{{\rm{X}},{\rm{O}}}=0$	${\rm{\Delta }}{\alpha }_{{\rm{X}},{\rm{O}}}=\frac{1}{4}$	${\rm{\Delta }}{\alpha }_{{\rm{X}},{\rm{O}}}=-\frac{1}{4}$	${\rm{\Delta }}{\alpha }_{{\rm{X}},{\rm{O}}}$ = (0, $\frac{1}{4}$)	${\rm{\Delta }}{\alpha }_{{\rm{X}},{\rm{O}}}$ = (−$\frac{1}{4}$, 0)
		Energy injection break
${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}=0$	${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}=\frac{1}{2}$	${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}=\frac{1}{2}$	${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}$ = (0, $\frac{1}{2}$)	${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}$ = (0, $\frac{1}{2}$)
${\rm{\Delta }}{\alpha }_{1,{\rm{X}},{\rm{O}}}=0$	${\rm{\Delta }}{\alpha }_{1,{\rm{X}},{\rm{O}}}=\frac{2-q}{4}$	${\rm{\Delta }}{\alpha }_{1,{\rm{X}},{\rm{O}}}=\frac{-2+q}{4}$	${\rm{\Delta }}{\alpha }_{1,{\rm{X}},{\rm{O}}}=(0,\frac{2-q}{4})$	${\rm{\Delta }}{\alpha }_{1,{\rm{X}},{\rm{O}}}=(\frac{-2+q}{4},0)$
${\rm{\Delta }}{\alpha }_{2,{\rm{X}},{\rm{O}}}=0$	${\rm{\Delta }}{\alpha }_{2,{\rm{X}},{\rm{O}}}=\frac{1}{4}$	${\rm{\Delta }}{\alpha }_{2,{\rm{X}},{\rm{O}}}=-\frac{1}{4}$	${\rm{\Delta }}{\alpha }_{2,{\rm{X}},{\rm{O}}}$ = (0, $\frac{1}{4}$)	${\rm{\Delta }}{\alpha }_{2,{\rm{X}},{\rm{O}}}$ = (−$\frac{1}{4}$, 0)
		Jet break
${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}=0$	${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}=\frac{1}{2}$	${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}=\frac{1}{2}$	${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}$ = (0, $\frac{1}{2}$)	${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}$ = (0, $\frac{1}{2}$)
${\rm{\Delta }}{\alpha }_{1,{\rm{X}},{\rm{O}}}=0$	${\rm{\Delta }}{\alpha }_{1,{\rm{X}},{\rm{O}}}=\frac{1}{4}$	${\rm{\Delta }}{\alpha }_{1,{\rm{X}},{\rm{O}}}=-\frac{1}{4}$	${\rm{\Delta }}{\alpha }_{1,{\rm{X}},{\rm{O}}}$ = (0, $\frac{1}{4}$)	${\rm{\Delta }}{\alpha }_{1,{\rm{X}},{\rm{O}}}$ = (−$\frac{1}{4}$, 0)
${\rm{\Delta }}{\alpha }_{2,{\rm{X}},{\rm{O}}}=0$	${\rm{\Delta }}{\alpha }_{2,{\rm{X}},{\rm{O}}}=\frac{1}{4}$	${\rm{\Delta }}{\alpha }_{2,{\rm{X}},{\rm{O}}}=-\frac{1}{4}$	${\rm{\Delta }}{\alpha }_{2,{\rm{X}},{\rm{O}}}$ = (0, $\frac{1}{4}$)	${\rm{\Delta }}{\alpha }_{2,{\rm{X}},{\rm{O}}}$ = (−$\frac{1}{4}$, 0)
		Energy injection with jet break
${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}=0$	${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}=\frac{1}{2}$	${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}=\frac{1}{2}$	${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}$ = (0, $\frac{1}{2}$)	${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}$ = (0, $\frac{1}{2}$)
${\rm{\Delta }}{\alpha }_{1,{\rm{X}},{\rm{O}}}=0$	${\rm{\Delta }}{\alpha }_{1,{\rm{X}},{\rm{O}}}=\frac{2-q}{4}$	${\rm{\Delta }}{\alpha }_{1,{\rm{X}},{\rm{O}}}=\frac{-2+q}{4}$	${\rm{\Delta }}{\alpha }_{1,{\rm{X}},{\rm{O}}}=(0,\frac{2-q}{4})$	${\rm{\Delta }}{\alpha }_{1,{\rm{X}},{\rm{O}}}=(0,\frac{2-q}{4})$
${\rm{\Delta }}{\alpha }_{2,{\rm{X}},{\rm{O}}}=0$	${\rm{\Delta }}{\alpha }_{2,{\rm{X}},{\rm{O}}}=\frac{2-q}{4}$	${\rm{\Delta }}{\alpha }_{2,{\rm{X}},{\rm{O}}}=\frac{-2+q}{4}$	${\rm{\Delta }}{\alpha }_{2,{\rm{X}},{\rm{O}}}=(0,\frac{2-q}{4})$	${\rm{\Delta }}{\alpha }_{2,{\rm{X}},{\rm{O}}}=(0,\frac{2-q}{4})$

Notes.

^aSame regime: X-ray and optical bands are in the same spectral regime, regime I ($\nu \gt {\nu }_{{\rm{c}}}$) or regime II (${\nu }_{{\rm{m}}}\lt \nu \lt {\nu }_{{\rm{c}}}$). In this table, $\triangle {\beta }_{{\rm{X}},{\rm{O}}}={\beta }_{{\rm{X}}}-{\beta }_{{\rm{O}}}$ and $\triangle {\alpha }_{{\rm{X}},{\rm{O}}}={\alpha }_{{\rm{X}}}-{\alpha }_{{\rm{O}}}$. The subscripts "1" and "2" denote the pre- and postbreak segments for the BPL light curves, respectively. ^bDifferent regimes: X-ray band in regime I ($\nu \gt {\nu }_{{\rm{c}}}$), and optical band in regime II (${\nu }_{{\rm{m}}}\lt \nu \lt {\nu }_{{\rm{c}}}$). ^cGray zone: one band or two bands in the gray zone regime I–II.

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Table 4.  The ${\rm{\Delta }}{\alpha }_{{\rm{X}},{\rm{O}}}$ and ${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}$ Values in Different Afterglow Models, $1\lt p\lt 2$

		$1\lt p\lt 2$
Same Regimea	Different Regimesb		Gray Zonec
ISM,Wind	ISM	Wind	ISM	Wind
		SPL
${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}=0$	${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}=\frac{1}{2}$	${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}=\frac{1}{2}$	${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}=(0,\frac{1}{2})$	${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}=(0,\frac{1}{2})$
${\rm{\Delta }}{\alpha }_{{\rm{X}},{\rm{O}}}=0$	${\rm{\Delta }}{\alpha }_{{\rm{X}},{\rm{O}}}=\frac{1}{2}$	${\rm{\Delta }}{\alpha }_{{\rm{X}},{\rm{O}}}=\frac{1}{4}$	${\rm{\Delta }}{\alpha }_{{\rm{X}},{\rm{O}}}=(0,\frac{1}{2})$	${\rm{\Delta }}{\alpha }_{{\rm{X}},{\rm{O}}}=(0,\frac{1}{4})$
		Energy injection break
${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}=0$	${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}=\frac{1}{2}$	${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}=\frac{1}{2}$	${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}=(0,\frac{1}{2})$	${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}=(0,\frac{1}{2})$
${\rm{\Delta }}{\alpha }_{1,{\rm{X}},{\rm{O}}}=0$	${\rm{\Delta }}{\alpha }_{1,{\rm{X}},{\rm{O}}}=\frac{2-q}{4}$	${\rm{\Delta }}{\alpha }_{1,{\rm{X}},{\rm{O}}}=\frac{-2+q}{4}$	${\rm{\Delta }}{\alpha }_{1,{\rm{X}},{\rm{O}}}=(0,\frac{2-q}{4})$	${\rm{\Delta }}{\alpha }_{1,{\rm{X}},{\rm{O}}}=(\frac{-2+q}{4},0)$
${\rm{\Delta }}{\alpha }_{2,{\rm{X}},{\rm{O}}}=0$	${\rm{\Delta }}{\alpha }_{2,{\rm{X}},{\rm{O}}}=\frac{1}{2}$	${\rm{\Delta }}{\alpha }_{2,{\rm{X}},{\rm{O}}}=\frac{1}{4}$	${\rm{\Delta }}{\alpha }_{2,{\rm{X}},{\rm{O}}}=(0,\frac{1}{2})$	${\rm{\Delta }}{\alpha }_{2,{\rm{X}},{\rm{O}}}=(0,\frac{1}{4})$
		Jet break
${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}=0$	${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}=\frac{1}{2}$	${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}=\frac{1}{2}$	${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}=(0,\frac{1}{2})$	${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}=(0,\frac{1}{2})$
${\rm{\Delta }}{\alpha }_{1,{\rm{X}},{\rm{O}}}=0$	${\rm{\Delta }}{\alpha }_{1,{\rm{X}},{\rm{O}}}=\frac{1}{2}$	${\rm{\Delta }}{\alpha }_{1,{\rm{X}},{\rm{O}}}=\frac{1}{4}$	${\rm{\Delta }}{\alpha }_{1,{\rm{X}},{\rm{O}}}=(0,\frac{1}{2})$	${\rm{\Delta }}{\alpha }_{1,{\rm{X}},{\rm{O}}}=(0,\frac{1}{4})$
${\rm{\Delta }}{\alpha }_{2,{\rm{X}},{\rm{O}}}=0$	${\rm{\Delta }}{\alpha }_{2,{\rm{X}},{\rm{O}}}=\frac{1}{4}$	${\rm{\Delta }}{\alpha }_{2,{\rm{X}},{\rm{O}}}=-\frac{1}{4}$	${\rm{\Delta }}{\alpha }_{2,{\rm{X}},{\rm{O}}}=(0,\frac{1}{4})$	${\rm{\Delta }}{\alpha }_{2,{\rm{X}},{\rm{O}}}=(-\frac{1}{4},0)$
		Energy injection with jet break
${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}=0$	${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}=\frac{1}{2}$	${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}=\frac{1}{2}$	${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}=(0,\frac{1}{2})$	${\rm{\Delta }}{\beta }_{{\rm{X}},{\rm{O}}}=(0,\frac{1}{2})$
${\rm{\Delta }}{\alpha }_{1,{\rm{X}},{\rm{O}}}=0$	${\rm{\Delta }}{\alpha }_{1,{\rm{X}},{\rm{O}}}=\frac{2-q}{4}$	${\rm{\Delta }}{\alpha }_{1,{\rm{X}},{\rm{O}}}=\frac{-2+q}{4}$	${\rm{\Delta }}{\alpha }_{1,{\rm{X}},{\rm{O}}}=(0,\frac{2-q}{4})$	${\rm{\Delta }}{\alpha }_{1,{\rm{X}},{\rm{O}}}=(\frac{-2+q}{4},0)$
${\rm{\Delta }}{\alpha }_{2,{\rm{X}},{\rm{O}}}=0$	${\rm{\Delta }}{\alpha }_{2,{\rm{X}},{\rm{O}}}=\frac{2-q}{4}$	${\rm{\Delta }}{\alpha }_{2,{\rm{X}},{\rm{O}}}=\frac{-2+q}{4}$	${\rm{\Delta }}{\alpha }_{2,{\rm{X}},{\rm{O}}}=(0,\frac{2-q}{4})$	${\rm{\Delta }}{\alpha }_{2,{\rm{X}},{\rm{O}}}=(\frac{-2+q}{4},0)$

Notes.

^aSame regime: X-ray and optical bands are in the same spectral regime, regime I ($\nu \gt {\nu }_{{\rm{c}}}$) or regime II (${\nu }_{{\rm{m}}}\lt \nu \lt {\nu }_{{\rm{c}}}$). In this table, $\triangle {\beta }_{{\rm{X}},{\rm{O}}}={\beta }_{{\rm{X}}}-{\beta }_{{\rm{O}}}$ and $\triangle {\alpha }_{{\rm{X}},{\rm{O}}}={\alpha }_{{\rm{X}}}-{\alpha }_{{\rm{O}}}$. The subscripts "1" and "2" denote the pre- and postbreak segments for the BPL light curves, respectively. ^bDifferent regimes: X-ray band in regime I ($\nu \gt {\nu }_{{\rm{c}}}$), and optical band in regime II (${\nu }_{{\rm{m}}}\lt \nu \lt {\nu }_{{\rm{c}}}$). ^cGray zone: one band or two bands in the gray zone regime I–II.

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Taking the energy injection break as an example, our analysis uses the following procedure. (1) Use the observed spectral indices β_O and β_X to predict the postbreak temporal indices ${\alpha }_{{\rm{O}},2}$ and ${\alpha }_{{\rm{X}},2}$ in two possible spectral regimes. Then compare these theoretical predictions with the observational values. If the theoretical values are consistent with the fitting results within errors, then go to the next step. Otherwise, it indicates that this GRB does not fall into this light curve type. (2) Use the identified spectral regime to calculate the electron spectral index p from the spectral index β, i.e., $p=2\beta +1$ for ${\nu }_{{\rm{m}}}\lt \nu \lt {\nu }_{{\rm{c}}}$, or $p=2\beta$ for $\nu \gt {\nu }_{{\rm{c}}}$. Compare the p values derived from the optical and X-ray data, respectively. If ${p}_{{\rm{O}}}={p}_{{\rm{X}}}$ within errors, then move to the next step. Otherwise, this GRB does not fall into such a light curve type. (3) Use the inferred p value and spectral regimes to infer the energy injection parameter q using the temporal index before the break (${\alpha }_{{\rm{O}},1}$ and ${\alpha }_{{\rm{X}},1}$). Compare the derived q values from the optical and X-ray bands, respectively. If ${q}_{{\rm{O}}}={q}_{{\rm{X}}}$ within errors, then move to the next step. Otherwise, this GRB does not fall into such a light curve type. (4) Using the $\triangle {\beta }_{{\rm{X}},{\rm{O}}}-\triangle {\alpha }_{{\rm{X}},{\rm{O}}}$ relation to double-check the data, if the data fall into the predicted region in the $\triangle {\beta }_{{\rm{X}},{\rm{O}}}-\triangle {\alpha }_{{\rm{X}},{\rm{O}}}$ plane, then this burst can be fully interpreted by such a model. Otherwise, the burst does not fall into this category.

The simplest analytical model (Sari et al. 1998) predicts $\beta =p/2$ for Regime I ($\nu \gt {\nu }_{{\rm{c}}}$) and $\beta =(p-1)/2$ for Regime II (${\nu }_{{\rm{m}}}\lt \nu \lt {\nu }_{{\rm{c}}}$). Detailed numerical calculations (Uhm & Zhang 2014a) showed that the transition between the two regimes may take several orders of magnitude in observer time. As a result, some "gray zones" with $(p-1)/2\lt \beta \lt p/2$ are allowed by the model. Therefore, the parameter space between the two closure relation lines defined by the two spectral regimes in the $\alpha -\beta$ plane is allowed by the theory. Data points falling into this gray zone should be regarded as consistent with the model. There are three possibilities: (1) the optical band is in Regime II, while the X-ray band is in the gray zone; (2) the X-ray band is in Regime I, while the optical band is in the gray zone; and (3) both bands are in the gray zone.

For the cases where both the optical and X-ray bands are in the same spectral regime, we demand that the three spectral indices be the same within errors, i.e., ${\beta }_{{\rm{O}}}={\beta }_{\mathrm{OX}}={\beta }_{{\rm{X}}}$, where ${\beta }_{\mathrm{OX}}$ is the spectral index between the optical and X-ray band in the joint spectral energy distribution (SED).¹⁵ If the two bands are in different spectral regimes, we demand ${\beta }_{{\rm{O}}}\lt {\beta }_{\mathrm{OX}}\leqslant {\beta }_{{\rm{X}}}$ or ${\beta }_{{\rm{O}}}\leqslant {\beta }_{\mathrm{OX}}\lt {\beta }_{{\rm{X}}}$.

With the above preparation, everything is in place for us to systematically compare the broadband data with the external forward shock afterglow models. Based on how badly the data violate the models, we define the following five grades (see also Table 5).

• 1.  
  Grade I: Both X-ray and optical bands have SPL light curves or BPL light curves with an acceptable achromatic break. Both bands satisfy closure relations and are self-consistent (same medium type, p and q values). These are the best examples where the GRB afterglow data abide by the external shock model predictions.
• 2.  
  Grade II: Some GRBs have a clear break at t_b in one band (e.g., X-rays), but do not have a break in another band (e.g., optical). The missing break is likely due to incomplete observational coverage before or after the break. The data are consistent with the hypothesis of an achromatic break, and both bands satisfy closure relations self-consistently. These GRBs are almost as good as Grade I in terms of abiding by the external shock models.
• 3.  
  Grade III: Both X-ray and optical bands have SPL light curves or BPL light curves with an acceptable achromatic break. However, at least one temporal segment in one band does not satisfy the closure relations in a self-consistent manner with respect to other segments or band.
• 4.  
  Grade IV: This is the Grade II equivalent for Grade III. One band does not have a break, but the data are consistent with the hypothesis of having an achromatic break. At least one temporal segment in one band does not satisfy the closure relations in a self-consistent manner with respect to other segments or band.
• 5.  
  Grade V: There is clear evidence of chromatic breaks and violation of closure relations. These GRBs cannot be interpreted within the one-component external shock models.¹⁶

With these five grades, we define three samples.

• 1.  
  Gold Sample: The GRBs in Grade I and II are defined as the Gold sample GRBs because no observed information violates any predictions of the external shock models.
• 2.  
  Silver Sample: The GRBs in Grade III and IV are included in this sample. Even though at least one segment or band does not satisfy the closure relations self-consistently, the basic requirement of achromaticity is not violated. We note that the closure relations are the predictions of the simplest analytical external forward shock models. More complicated models invoking, for example, a structured jet (Rossi et al. 2002; Zhang & Mészáros 2002; Granot & Kumar 2003; Kumar & Granot 2003) or a circumburst density medium with an arbitrary k value (at least for a certain distance range) predict light curve behaviors that may not fully abide by the simple closure relations. Furthermore, if the GRB engine is long-lived and a long-lasting reverse shock outshines the forward shock, a variety of rich light curve behaviors can be generated, which do not follow the simple closure relations (e.g., Uhm et al. 2012; Uhm & Zhang 2014b). So it is possible that the GRBs in the Silver sample are still consistent with the external shock models.
• 3.  
  Bad Sample: The GRBs in Grade V violate the basic achromaticity principle of the external shock models and do not abide by the closure relations and therefore cannot be interpreted within the framework of the external shock models.

The 85 well-sampled GRBs in our sample are graded based on the above-defined grading criteria. The GRBs in the five grades are presented in Figures 1–5, respectively. The relevant data of different grades are presented in Tables 1 and 6.

• 1.  
  Grade I: As can be seen from Tables 1 and 6 and Figure 1, within errors, 43 of 85 GRBs satisfy the Grade I criteria. Out of 43 GRBs, 13, 8, and 22 GRBs are constrained to have an energy injection break, jet break, and SPL decay, respectively.
• 2.  
  Grade II: Within errors, two of 85 GRBs fall into this grade (Figure 2).
• 3.  
  Grade III: There are 34 of 85 GRBs that fall into this grade (Figure 3). Within the sample, 15 and 19 GRBs have SPL and BPL light curves, respectively. GRBs 060906, 080319B, and 100219A have two breaks at different times.
• 4.  
  Grade IV: Three of 85 GRBs fall into this grade (Figure 4).
• 5.  
  Grade V: Three of 85 GRBs fall into this grade (Figure 5). Two of them (GRBs 060607A and 070208) show clear chromatic breaks with good temporal coverage in both bands at the break times. One GRB (GRB 070420) shows a chromatic behavior based on the available data and simple model fitting, even though no observational data are available in the optical band at the break time of the X-ray band, so the existence of a break in the optical band (even though very contrived in shape) at the same epoch is not completely ruled out.

Table 6.  Gold Sample GRBs and Their Derived Parameters

GRB	z	Opticala	X-raya	p	q	${\epsilon }_{B}$ b	${E}_{\gamma ,\mathrm{iso}}$ c	${E}_{{\rm{K}},\mathrm{end}}$ c	${E}_{{\rm{K}},\mathrm{in}}$ c	${E}_{{\rm{K}},\mathrm{dec}}$ c	${\eta }_{\gamma ,\mathrm{dec}}$ d	${\eta }_{\gamma ,\mathrm{end}}$ d	Typee
050408	1.24	windII	windI-windII	2.12 ± 0.12	0.25 ± 0.17	...	11.76	3410.39 ± 76.12	3386.60 ± 75.06	23.80 ± 6.84	33 ± 10	0.3 ± 0.0	1
050801	1.56	ISMII	ISMII	2.78 ± 0.28	0.22 ± 0.15	2.7E-05	0.52 ± 0.11	124.89 ± 17.39	81.83 ± 20.10	43.06 ± 17.65	1 ± 1	0.4 ± 0.1	1
051109A	2.35	windI-windII	windI-windII	2.4 ± 0.1	0.22 ± 0.06	...	9.15 ± 6.83	14675.86 ± 763.28	14060.42 ± 730.61	615.44 ± 38.42	1 ± 1	0.1 ± 0.1	1
060206	4.05	windI-windII	windI	2.46 ± 0.1	0.41 ± 0.06	...	4.78 ± 20.79	386.76 ± 93.02	344.28 ± 83.18	42.47 ± 15.11	10 ± 44	1 ± 5	1
060714	2.71	windI-windII	windI-windII	1.89 ± 0.07	0.19 ± 0.05	...	18.22 ± 2.53	250.46 ± 248.11	239.83 ± 237.24	10.62 ± 11.25	63 ± 67	7 ± 7	1
060729	0.54	windI-windII	windI-windII	2.04 ± 0.08	0.06 ± 0.01	...	0.75 ± 0.07	2312.14 ± 33.54	2304.60 ± 33.43	7.54 ± 0.12	9 ± 1	0.03 ± 0.01	1
070411	2.95	windI-windII	windI-windII	2.48 ± 0.02	0.86 ± 0.09	...	17.06 ± 1.84	24456.57 ± 256.42	14460.59 ± 794.78	9995.98 ± 782.91	0.2 ± 0.1	0.1 ± 0.1	1
070518	1.16	windI-windII	windI-windII	2.4 ± 0.2	0.24 ± 0.08	...	0.27 ± 0.13	917.47 ± 22.99	911.31 ± 22.81	6.16 ± 0.52	4 ± 2	0.03 ± 0.01	1
080710	0.85	ISMII	ISMII	2.78	0.32	9.6E-06	1.34 ± 0.32	424.26 ± 40.68	393.99 ± 37.78	30.27 ± 2.90	4 ± 1	0.3 ± 0.1	1
090102	1.55	windI-windII	windI-windII	1.58 ± 0.22	0.33 ± 0.05	...	22.74 ± 2.12	1750.25 ± 1149.59	1490.68 ± 1009.33	259.57 ± 141.45	8 ± 4	1 ± 1	1
090426	2.61	windI-windII	windI	2.06 ± 0.3	0.13 ± 0.02	...	0.83 ± 0.28	2.77 ± 3.33	1.80 ± 2.17	0.97 ± 1.17	46 ± 58	23 ± 29	1
100418A	0.62	ISMII	ISMII	2.96 ± 0.18	0.05 ± 0.02	2.9E-06	0.14 ± 0.02	1754.24 ± 403.22	1754.06 ± 403.18	0.18 ± 0.05	43 ± 14	0.01 ± 0.01	1
101024A	...	ISMII	ISMII	2.64 ± 0.26	0.06 ± 0.11	...	...	...	...	...	...	...	1
050820A	2.61	ISMI-ISMII	ISMI-ISMII	1.78 ± 0.1	...	...	114.67 ± 33.59	...	...	4117.25 ± 2462.50	3 ± 2	...	2
050922C	2.20	ISMII	ISMI	2.06 ± 0.05	...	...	9.93 ± 1.06	...	...	118.25 ± 17.29	8 ± 1	...	2
060111B	...	ISMI	ISMI	1.57 ± 0.03	...	...	11.00 ± 5.00	...	...	...	...	...	2
081008	1.97	ISMII	ISMI	1.96 ± 0.18	...	...	9.30 ± 1.91	...	...	253.98 ± 93.28	4 ± 3	...	2
081203A	2.10	ISMI-ISMII	ISMI-ISMII	2.2 ±	...	...	34.71 ± 17.12	...	...	594.07 ± 14.99	6 ± 3	...	2
090618	0.54	ISMII	ISMI	1.92 ± 0.02	...	...	25.30 ± 0.00	...	...	89.10 ± 14.81	22 ± 4	...	2
091127	0.49	ISMII	ISMI	1.36 ±	...	...	1.52 ± 0.08	...	...	263.04 ± 22.94	1 ± 0.1	...	2
130427A	0.34	ISMII	ISMII	2.38 ± 0.02	...	9.1E-06	81.00	...	...	1665.96 ± 62.14	5 ± 0.2	...	2
051028	3.70	ISMI-SIMII	ISMI-ISMII	1.9 ± 0.1	...	...	11.94 ± 1.87	...	...	2515.02 ± 1557.52	0.5 ± 0.3	...	4
060418	1.49	ISMII	ISMI-ISMII	2.58 ± 0.18	...	...	12.45 ± 4.44	...	...	386.85 ± 42.95	3 ± 1	...	4
060512	0.44	ISMI-SIMII	ISMI-ISMII	1.94 ± 0.06	...	...	0.02 ± 0.01	...	...	5.70 ± 0.12	0.3 ± 0.2	...	4
060904B	0.70	ISMI-SIMII	ISMI-ISMII	3.25 ± 0.17	...	...	0.79 ± 0.56	...	...	750.73 ± 388.34	0.1 ± 0.1	...	4
060912A	0.94	ISMII	ISMII	2.2 ± 0.3	...	8.7E-05	1.12 ± 0.08	...	...	46.23 ± 1.16	2 ± 0.2	...	4
060927	5.46	windII	windI-windII	2.22 ± 0.1	...	...	10.37 ± 3.03	...	...	37039.49 ± 2373.94	0.03 ± 0.01	...	4
061007	1.26	ISMII	ISMII	2.04 ± 0.1	...	7.3E-05	99.81 ± 7.22	...	...	1085.73 ± 18.15	8 ± 1	...	4
061126	1.16	ISMII	ISMII	2.64 ± 0.18	...	1.0E-05	13.26 ± 1.90	...	...	1282.64 ± 177.08	1 ± 0.1	...	4
070318	0.84	ISMII-ISMII	ISMI-ISMII	2.74 ± 0.02	...	...	1.39 ± 0.35	...	...	277.28 ± 22.56	0.5 ± 0.1	...	4
071025	1.55	ISMII	ISMII	3.07 ± 0.13	...	9.2E-06	85.42 ± 12.42	...	...	1394.60 ± 851.47	6 ± 4	...	4
071031	2.69	ISMI	ISMI	2.01 ± 0.27	...	...	4.73 ± 9.26	...	...	68.53 ± 38.27	6 ± 5	...	4
080319C	1.95	windI-II	windI-windII	2.22 ± 0.2	...	...	23.54 ± 1.47	...	...	12314.17 ± 3183.23	0.2 ± 0.1	...	4
080413A	2.43	windII	windI-windII	2.3 ± 0.48	...	...	14.09 ± 7.30	...	...	249.98 ± 71.42	5 ± 3	...	4
080603A	1.68	ISMI	ISMI	2.96 ± 0.08	...	...	2.20 ± 0.80	...	...	3577.17 ± 272.70	0.1 ± 0.1	...	4
080804	2.20	windI-windII	windI-windII	1.86	...	...	24.61 ± 4.79	...	...	143.09 ± 16.57	15 ± 3	...	4
080913	6.70	ISMI-SIMII	ISMI-ISMII	2.79 ± 0.27	...	...	8.44 ± 1.55	...	...	354.60 ± 174.38	2 ± 1	...	4
080928	1.69	ISMII	ISMII	2.44 ± 0.04	...	5.7E-05	6.30 ± 0.75	...	...	548.00 ± 26.66	1 ± 0.1	...	4
090323	3.57	windII	windII	2.65 ± 0.13	...	9.5E-04	372.38 ± 16.86	...	...	209526.73 ± 18842.90	0.2 ± 0.1	...	4
090328	0.74	ISMII	ISMII	3.19 ± 0.21	...	3.4E-06	19.03	...	...	3861.76 ± 2039.23	0.5 ± 0.3	...	4
090926A	2.11	ISMII	ISMII	2.29 ± 0.19	...	...	185.13 ± 9.10	...	...	1859.10 ± 111.85	9 ± 1	...	4
100901A	1.41	windII	windI-windII	2.1 ± 0.1	...	...	2.95 ± 0.63	...	...	27847.74 ± 871.45	0.01 ± 0.01	...	4
120326A	1.80	windII	windII	2.51 ± 0.09	...	5.2E-04	3.18 ± 0.40	...	...	41757.87 ± 1069.46	0.01 ± 0.01	...	4
Grade II
051111	1.55	windII	windI-windII	2.52 ± 0.14	...	...	10.79 ± 3.07	...	...	3793.13 ± 273.16	0.3 ± 0.1	...	4
090313	3.38	windII	windI-windII	2.54 ± 0.72	...	5.0E-06	13.02 ± 2.94	...	...	173028.58 ± 189747.00	0.01 ± 0.01	...	4

Notes.

^aISMI: The ISM model in the spectral regime I ($\nu \gt {\nu }_{{\rm{c}}}$); ISMII: the ISM model in the spectral regime II (${\nu }_{{\rm{m}}}\lt \nu \lt {\nu }_{{\rm{c}}}$); windI: the wind model in the spectral regime I; windII: the wind model in the spectral regime II; ISMI-ISMII: the ISM model in the gray zone (between $\nu \gt {\nu }_{{\rm{c}}}$ and ${\nu }_{{\rm{m}}}\lt \nu \lt {\nu }_{{\rm{c}}}$); windI-windII: the wind model in the gray zone. ^bUpper limit, calculated with the X-ray data in the spectral regime II (${\nu }_{{\rm{m}}}\lt \nu \lt {\nu }_{{\rm{c}}}$). ^cIn units of ${10}^{52}\;\mathrm{erg}$; ${E}_{{\rm{K}},\mathrm{end}}$ is the kinetic energy at the end of energy injection (${t}_{\mathrm{end}}$); ${E}_{{\rm{K}},\mathrm{in}}$ is the injected kinetic energy during the energy injection phase; and ${E}_{{\rm{K}},\mathrm{dec}}$ is the kinetic energy at the fireball deceleration time (${t}_{\mathrm{dec}}$). ^dIn units of $\%$; ${\eta }_{\gamma ,\mathrm{dec}}$ is the radiative efficiency calculated using ${E}_{{\rm{K}},\mathrm{dec}}$; ${\eta }_{\gamma ,\mathrm{dec}}$ is the radiative efficiency calculated using ${E}_{{\rm{K}},\mathrm{end}}$. ^e1: Energy injection break; 2: jet break; 3: jet break with energy injection; 4: SPL decay.

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Consequently, we get three samples.

• 1.  
  Gold sample: This sample has 45 of 85 GRBs, including 13/45, 8/45, and 24/45 GRBs satisfying the energy injection, jet break, jet break with energy injection, and SPL decay models, respectively. Among them, 27/45 and 18/45 are consistent with the ISM and wind models, respectively; 17/45, 4/45, and 24/45 GRBs are consistent with being in the  same spectral regime, different spectral regimes (X-ray band in regime I and optical band in regime II), and gray zone, respectively. Among the 17 GRBs with the same spectral regime, 15 and two GRBs are consistent with being in the ISM II and wind II spectral regimes, respectively. For the four GRBs with a different spectral regime, all of them are consistent with having an ISM medium.
• 2.  
  Silver sample: This sample has 37 of 85 GRBs, which may (or may not) be interpreted within the more complicated numerical external shock models.
• 3.  
  Bad sample: Only three of 85 GRBs definitely violate the basic achromaticity principle of the external shock models and therefore belong to the bad sample.

Figure 7(a) shows $\triangle {\beta }_{{\rm{X}},{\rm{O}}}-\triangle {\alpha }_{{\rm{X}},{\rm{O}}}$ distributions for the Gold sample. For the energy injection sample, we only used the postbreak segment to remove the q dependence. These are the GRBs that also satisfy the closure relations in all temporal segments. We do not show the closure relation $\alpha -\beta$ plots since the energy injection models have an extra q dependence on the α values. To show the details of how each burst may fall into the model predictions of each model (gray zone included), in Figures 7(b)–(e) we show the $\triangle {\beta }_{{\rm{X}},{\rm{O}}}-\triangle {\alpha }_{{\rm{X}},{\rm{O}}}$ distributions of those Gold sample GRBs that satisfy the ISM and wind medium models with $p\gt 2$ and $1\lt p\lt 2$, respectively. The Silver sample GRBs are collected in Figure 7(f). About half of them fall outside the predicted region (red box) defined by the models. Even though some fall into the box, they do not satisfy the closure relations in all of the temporal segments in all energy bands.

Figure 8 shows the distributions of the temporal indices α in different energy bands and different temporal segments. They are all well fitted with Gaussian distributions for each band or temporal segment. For the GRBs having a BPL light curve, the typical α values are ${\alpha }_{{\rm{O}},1}=0.49\pm 0.45$, ${\alpha }_{{\rm{O}},2}=1.44\pm 0.39$, ${\alpha }_{{\rm{X}},1}=0.58\pm 0.63$, and ${\alpha }_{{\rm{X}},2}=1.50\pm 0.27$, respectively (Figure 8(a)). For the GRBs with an SPL light curve, one has ${\alpha }_{{\rm{O}}}=1.26\pm 0.38$ and ${\alpha }_{{\rm{X}}}=1.39\pm 0.26$ (Figure 8(b)). For the BPL sample, we also separate it into the energy injection sample and the jet break sample and perform the statistics. For the energy injection breaks, one has ${\alpha }_{{\rm{O}},1}=0.25\pm 0.12$, ${\alpha }_{{\rm{O}},2}=1.26\pm 0.26$, ${\alpha }_{{\rm{X}},1}=0.30\pm 0.27$, and ${\alpha }_{{\rm{X}},2}\;=1.35\;\pm 0.24$, respectively (Figure 8(c)). For the jet breaks, one has ${\alpha }_{{\rm{O}},1}=0.77\pm 0.18$, ${\alpha }_{{\rm{O}},2}=1.66\pm 0.16$, ${\alpha }_{{\rm{X}},1}=0.95\pm 0.16$, and ${\alpha }_{{\rm{X}},2}=1.70\pm 0.19$, respectively (Figure 8(d)). Both the prebreak and the postbreak α values in the energy injection sample are systematically shallower than those in the jet break sample. On average, the X-ray light curves are steeper than the optical light curves, consistent with the expectations of the theoretical models (i.e., the X-ray band is more likely above ν_c while the optical band is more likely below ν_c).

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Another self-consistency check is to compare the observed change of decay slope, ${\rm{\Delta }}\alpha ={\alpha }_{2}-{\alpha }_{1}$, with the model predictions. From the closure relations (Table 2), one can derive these relations:

• 1.  
  for energy injection breaks:

  $$\begin{eqnarray}&&{\rm{\Delta }}\alpha \\ =\left\{\begin{array}{ll}\displaystyle \frac{(1-q)(2+\beta )}{2}, & \mathrm{ISM}\;\mathrm{II}\;(p\gt 2)\\ \displaystyle \frac{(1-q)(19+2\beta )}{16}, & \mathrm{ISM}\;\mathrm{II}\;(1\lt p\lt 2),\\ \displaystyle \frac{(1-q)(1+\beta )}{2}, & \mathrm{ISM}\;{\rm{I}}\;(p\gt 2),\\ & \;\mathrm{wind}\;\mathrm{II}\;(p\gt 2),\mathrm{wind}\;{\rm{I}}\;(p\gt 2)\\ \displaystyle \frac{(1-q)(7+\beta )}{8}, & \mathrm{ISM}\;{\rm{I}}\;(1\lt p\lt 2)\\ \displaystyle \frac{(1-q)(5+2\beta )}{8}, & \mathrm{wind}\;\mathrm{II}\;(1\lt p\lt 2)\\ \displaystyle \frac{(1-q)(3+\beta )}{4}, & \mathrm{wind}\;{\rm{I}}\;(1\lt p\lt 2);\end{array}\right.\end{eqnarray} \tag{ 5 }$$
• 2.  
  for jet breaks:

  $$\begin{eqnarray}{\rm{\Delta }}\alpha =\left\{\begin{array}{ll}\displaystyle \frac{3}{4}, & \mathrm{ISM}\;{\rm{I}}\;\mathrm{and}\;\mathrm{II}\;(p\gt 2\;\mathrm{and}\;1\lt p\lt 2)\\ \displaystyle \frac{1}{2}, & \mathrm{wind}\;{\rm{I}}\;\mathrm{and}\;\mathrm{II}\;(p\gt 2\;\mathrm{and}\;1\lt p\lt 2);\end{array}\right.\end{eqnarray} \tag{ 6 }$$
• 3.  
  for jet breaks with energy injection:

  $$\begin{eqnarray}{\rm{\Delta }}\alpha =\left\{\begin{array}{ll}\displaystyle \frac{(q+2)}{4}, & \mathrm{ISM}\;\mathrm{II}\;(p\gt 2),\\ & \quad \mathrm{ISM}\;{\rm{I}}\;(p\gt 2\;\mathrm{and}\;1\lt p\lt 2)\\ \displaystyle \frac{(3q+6)}{16}, & \mathrm{ISM}\;\mathrm{II}\;(1\lt p\lt 2),\\ \displaystyle \frac{q}{2}, & \mathrm{wind}\;\mathrm{II}\;\mathrm{and}\;\mathrm{wind}\;{\rm{I}}\\ & \quad (p\gt 2\;\mathrm{and}\;1\lt p\lt 2).\end{array}\right.\end{eqnarray} \tag{ 7 }$$

Figure 9 shows a comparison between the observed ${\rm{\Delta }}{\alpha }_{\mathrm{obs}}$ and the theoretically predicted ${\rm{\Delta }}{\alpha }_{\mathrm{th}}$ for each GRB derived from the measured β and q values using the corresponding closure relations. One can see that the two are consistent with each other.

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Figure 10 displays the observed Δα distributions of various samples. For the Gold sample, the Δα distributions of optical and X-ray data are consistent with each other, i.e., ${\rm{\Delta }}{\alpha }_{{\rm{O}}}=0.94\pm 0.23$ and ${\rm{\Delta }}{\alpha }_{{\rm{X}}}=0.88\pm 0.28$ (Figure 10(a)). Furthermore, the ${\rm{\Delta }}\alpha$ values of both bands in subgroups (energy injection breaks and jet breaks) are also consistent with each other: ${\rm{\Delta }}{\alpha }_{{\rm{O}}}=1.05\pm 0.17$ and ${\rm{\Delta }}{\alpha }_{{\rm{X}}}=1.10\pm 0.21$ for the energy injection breaks, and ${\rm{\Delta }}{\alpha }_{{\rm{O}}}=0.75\pm 0.22$ and ${\rm{\Delta }}{\alpha }_{{\rm{X}}}=0.75\pm 0.22$ for the jet breaks (Figure 10(b)). The Silver sample, on the other hand, shows a poorer statistical behavior (Figures 10(c) and (d)).

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Figure 11 shows the spectral index distributions for the Gold sample. In general, the distributions can be fitted with Gaussian functions. For the global sample, one has ${\beta }_{{\rm{O}}}=0.70\pm 0.15$ and ${\beta }_{{\rm{X}}}=0.98\pm 0.15$ (Figure 11(a)). In the Gold sample, 17 of 45 GRBs have both the optical and X-ray bands in the same spectral regime. One has ${\beta }_{{\rm{O}}}=0.77\pm 0.19$ and ${\beta }_{{\rm{X}}}=0.89\pm 0.15$, which are consistent with each other (Figure 11(b)). The remaining 28 of 45 GRBs are identified to have X-ray and optical bands separated by a cooling break. The results show ${\beta }_{{\rm{O}}}=0.68\pm 0.18$, ${\beta }_{{\rm{X}}}=1.01\pm 0.14$, with ${\rm{\Delta }}\beta ={\beta }_{{\rm{X}}}-{\beta }_{{\rm{O}}}=0.37\pm 0.18$, which is consistent with the theoretically expected value $0\lt {\rm{\Delta }}\beta \leqslant 0.5$ (Figure 11(c)).

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We investigate the β distributions in different types of light curves. For the energy injection sample, one has ${\beta }_{{\rm{O}}}=0.78\pm 0.12$ and ${\beta }_{{\rm{X}}}=1.01\pm 0.13$ (Figure 11(d)); for the jet break sample, one has ${\beta }_{{\rm{O}}}=0.59\pm 0.11$ and ${\beta }_{{\rm{X}}}=0.97\pm 0.08$ (Figure 11(e)); and for the SPL sample, one has ${\beta }_{{\rm{O}}}=0.74\pm 0.24$ and ${\beta }_{{\rm{X}}}=0.95\pm 0.19$ (Figure 11(f)).

We also investigate the β distributions in different ambient medium types. For the ISM model (27/45 GRBs), one has ${\beta }_{{\rm{O}}}=0.72\pm 0.21$ and ${\beta }_{{\rm{X}}}=0.98\pm 0.10$ (Figure 11(g)); for the wind model (18/45 GRBs), one has ${\beta }_{{\rm{O}}}=0.70\pm 0.10$ and ${\beta }_{{\rm{X}}}=1.00\pm 0.20$ (Figure 11(h)). The ISM model is more favored than the wind model, which is consistent with the previous results (e.g., Panaitescu & Kumar 2002; Yost et al. 2003; Zhang et al. 2006; Schulze et al. 2011).

Figure 12 shows the distributions of the electron spectral index p of the Gold sample. It has a Gaussian distribution with p = 2.33 ± 0.48 (Figure 12(a)), which is very consistent with the typical value of p for relativistic shocks due to first-order Fermi acceleration (e.g., Achterberg et al. 2001; Ellison & Double 2002). It also has a wide distribution, which is consistent with previous studies (e.g., Shen et al. 2006; Liang et al. 2007, 2008; Curran et al. 2010).

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The p distributions in different subsamples are also generally consistent with each other. Within the Gold sample, those GRBs with optical and X-ray bands in the same spectral regime have p = 2.58 ± 0.39, whereas those with optical and X-ray bands in different spectral regimes have p = 2.17 ± 0.44 (Figure 12(a)). For the three light curve subsamples, one has p = 2.34 ± 0.38 for the energy injection break sample, p = 1.91 ± 0.37 for the jet break sample, and p = 2.48 ± 0.47 for the SPL sample, respectively (Figure 12(b)). For the two medium type models, one has p = 2.43 ± 0.57 for the ISM model and p = 2.28 ± 0.33 for the wind model, respectively (Figure 12(c)).

Figure 13 shows the distributions of the observed achromatic break times, t_b. The global distribution in the Gold sample gives $\mathrm{log}({t}_{{\rm{b}}}/\mathrm{ks})=(3.8\pm 0.9)$. Separating the energy injection sample and jet break sample, one has $\mathrm{log}({t}_{{\rm{b}}}/\mathrm{ks})=(3.6\pm 1.9)$ for the energy injection break sample and $\mathrm{log}({t}_{{\rm{b}}}/\mathrm{ks})=(3.9\pm 0.7)$ for the jet break sample. The energy injection end (which depends on central engine) is on average earlier than the jet break time (which depends on the geometry of the jet). The distribution of the energy injection break time is wider than the jet break time distribution.

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Within the Gold sample, 13 of 45 GRBs show an energy injection type break, either an energy injection break or a jet break with energy injection. Among them, 4/13 and 9/13 GRBs satisfy the ISM and wind model, respectively. The distributions of the energy injection parameter q of various samples are shown in Figure 14. The global sample has q = 0.22 ± 0.11. The ISM and wind models have q = 0.20 ± 0.12 and q = 0.23 ± 0.13, respectively, which are consistent with each other.

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Among the derived shock parameters, the magnetic field equipartition factor _B is of special interest. If the shock simply compresses the upstream magnetic field, then the expected _B is low, of the order of ${10}^{-6}-{10}^{-7}$. If, however, various plasma instabilities are playing a role to amplify the magnetic fields (e.g., Medvedev & Loeb 1999; Nishikawa et al. 2009), one would expect a relatively large _B, as high as 0.1. Early afterglow modeling (e.g., Wijers & Galama 1999; Panaitescu & Kumar 2001, 2002; Yost et al. 2003) derived a relatively large _B, with a typical value of ∼0.01. On the other hand, modeling of gigaelectronvolt emission in several Fermi/LAT-detected GRBs led to the suggestion that _B should be relatively low, at least for some GRBs (Kumar & Barniol Duran 2009, 2010). Santana et al. (2014) derived _B for a large sample of GRBs and derived a medium value of 10⁻⁵.

With our Gold sample, we can constrain _B independently. Even though in most GRBs _B cannot be constrained due to the degeneracy of the data, one can still place interesting upper limits on _B based on the medium type and spectral regime of the GRBs. For example, in the ISM model, ν_c decreases with time. For a regime II (${\nu }_{{\rm{m}}}\lt \nu \lt {\nu }_{{\rm{c}}}$) GRB, the last data point in the light curve would set a lower limit on ν_c at that epoch and, hence, an upper limit on _B. Similarly, in the wind model, ν_c increases with time. For a regime II GRB, the first data point in the light curve would set a lower limit on ν_c at that point and, hence, an upper limit on _B.

In the appendix, we present expressions of ${\nu }_{{\rm{m}}}$, ν_c, and ${F}_{\nu ,\mathrm{max}}$ and the kinetic energy of the afterglow, ${E}_{{\rm{k}},\mathrm{iso}}$. For the $p\gt 2$ cases, we adopt the formalism of previous works (Zhang et al. 2007a; Gao et al. 2013; Lü & Zhang 2014). New expressions for the $1\lt p\lt 2$ regime are also presented following Gao et al. (2013). We then derive the expressions of _B in various models, Equations (18), (21), (32), and (35), which are used to constrain _B.

The derived upper limits of _B are presented in Figure 15, with other parameters fixed as ${\epsilon }_{e}=0.1$, n = 1, or ${A}_{*}=1$. One can see that in general these upper limits point toward a relatively low _B value. In some cases for the ISM model, the upper limits are even lower than 10⁻⁵. These results are consistent with the findings of Kumar & Barniol Duran (2009, 2010), Barniol Duran (2014), and Santana et al. (2014).

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The isotropic γ-ray energy ${E}_{\gamma ,\mathrm{iso}}$ is calculated as

$$\begin{eqnarray}&&{E}_{\gamma ,\mathrm{iso}}=\displaystyle \frac{4\pi {D}_{{\rm{L}}}^{2}{S}_{\gamma }k}{1+z},\end{eqnarray} \tag{ 8 }$$

where ${S}_{\gamma }$ is the gamma-ray fluence in the BAT band, ${D}_{{\rm{L}}}$ is the luminosity distance of the source at redshift z, and the parameter k is a factor to correct the observed γ-ray energy in a given bandpass to a broad band (e.g., 1–10⁴ keV in the rest frame) with the observed GRB spectra (Bloom et al. 2001). It is well known that a typical GRB spectrum is well fitted with the so-called Band function (Band et al. 1993). If the Band parameters are measured for a burst, these parameters are used to derive the k parameter. However, owing to the narrowness of the Swift/BAT band, the spectra of many Swift GRBs in our sample are adequately fitted with an SPL, $N\propto {E}^{-{\rm{\Gamma }}}$, so the Band parameters are not well constrained. For these GRBs, we use an empirical relation between E_p and the BAT-band photon index Γ (Zhang et al. 2007b; Sakamoto et al. 2009; Virgili et al. 2012) to estimate E_p. Taking typical values of the photon indices α = −1.1 and β = −2.2 (Preece et al. 2000; Kaneko et al. 2006), we can derive the ${E}_{\gamma ,\mathrm{iso}}$ values of the GRBs with redshift measurements in our Gold sample, which range from 10⁵¹ to 10⁵⁵ erg, with a typical value of $\mathrm{log}({E}_{\gamma ,\mathrm{iso}}/\mathrm{erg})=53.15\pm 0.69$ (Figure 16(a)).

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The isotropic kinetic energy of the afterglow ${E}_{{\rm{K}},\mathrm{iso}}$ can be derived from the afterglow data. In general, broadband modeling is needed to precisely measure ${E}_{{\rm{K}},\mathrm{iso}}$ (Panaitescu & Kumar 2001, 2002). Most GRBs do not have adequate data to perform such an analysis. More conveniently, one may use the X-ray data only to constrain ${E}_{{\rm{K}},\mathrm{iso}}$ because the X-ray band is usually above ν_c, so the X-ray flux does not depend on the ambient density and only weakly depends on ${\epsilon }_{e}$ (Kumar 1999; Freedman & Waxman 2001; Berger et al. 2003; Lloyd-Ronning & Zhang 2004; Zhang et al. 2007a); see appendix for detailed derivations. For the cases with energy injection, ${E}_{{\rm{K}},\mathrm{iso}}$ is a function of time during the energy injection phase. Following Zhang et al. (2007a), we calculate ${E}_{{\rm{K}},\mathrm{iso}}$ at two different epochs, one at the break time t_b, when energy injection is over, and another at a putative deceleration time ${t}_{\mathrm{dec}}\sim \mathrm{max}(60\;{\rm{s}},{T}_{90})$. We use the X-ray flux at t_b to derive ${E}_{{\rm{K}},\mathrm{end}}$ and then derive ${E}_{{\rm{K}},\mathrm{dec}}={E}_{{\rm{K}},\mathrm{end}}{({t}_{\mathrm{dec}}/{t}_{{\rm{b}}})}^{1-q}$. The total injected energy is calculated as ${E}_{{\rm{K}},\mathrm{inj}}={E}_{{\rm{K}},\mathrm{end}}-{E}_{{\rm{K}},\mathrm{dec}}$. Based on our constraints on _B, we take ${\epsilon }_{B}={10}^{-5}$ for all of the GRBs in our calculations. Other parameters are taken as typical values: ${\epsilon }_{e}=0.1$, n = 1 or ${A}_{*}=1$, and Y = 1.

In Figure 16, we present several statistical results of the ${E}_{{\rm{K}},\mathrm{iso}}$ calculations. For the GRBs with energy injection, the distributions are log (${E}_{{\rm{K}},\mathrm{end}}/\mathrm{erg})=54.99\pm 0.86$ (Figure 16(b)), log (${E}_{{\rm{K}},\mathrm{inj}}/\mathrm{erg})=54.95\pm 0.61$ (Figure 16(c)), and log (${E}_{{\rm{K}},\mathrm{dec}}/\mathrm{erg})=53.29\pm 0.45$ (Figure 16(d)). For the entire Gold sample, one has log (${E}_{{\rm{K}},\mathrm{dec}}/\mathrm{erg})=54.66\pm 1.18$ (Figure 16(d)). It is interesting to see that the energetics of the energy injection sample reach a level similar to the no-energy-injection sample after the energy injection is over. Clear correlations are found among different energy components: ${E}_{{\rm{K}},\mathrm{end}}-{E}_{{\rm{K}},\mathrm{inj}}$ relation ${E}_{{\rm{K}},\mathrm{inj},52}=0.69{E}_{{\rm{K}},\mathrm{end},52}^{1.02\pm 0.02}$ (Figure 17(a)), ${E}_{{\rm{K}},\mathrm{dec}}-{E}_{{\rm{K}},\mathrm{inj}}$ relation ${E}_{{\rm{K}},\mathrm{inj},52}=41.7{E}_{{\rm{K}},\mathrm{dec},52}^{0.76\pm 0.20}$ (Figure 17(b)), ${E}_{\gamma ,\mathrm{iso}}-{E}_{{\rm{K}},\mathrm{end}}$ relation ${E}_{{\rm{K}},\mathrm{end},52}=476.3{E}_{\gamma ,\mathrm{iso},52}^{0.53\pm 0.23}$ (Figure 17(c)), ${E}_{\gamma ,\mathrm{iso}}-{E}_{{\rm{K}},\mathrm{dec}}$ relations in the entire Gold sample, ${E}_{{\rm{K}},\mathrm{dec},52}=56.2{E}_{\gamma ,\mathrm{iso},52}^{0.93\pm 0.19}$ (Figure 17(d)), in the energy injection sample, ${E}_{{\rm{K}},\mathrm{dec},52}=8.9{E}_{\gamma ,\mathrm{iso},52}^{1.10\pm 0.29}$ (Figure 17(e)), and in the no-energy-injection sample, ${E}_{{\rm{K}},\mathrm{dec},52}=316.2{E}_{\gamma ,\mathrm{iso},52}^{0.55\pm 0.21}$ (Figure 17(f)), respectively. In particular, $\mathrm{the}\;{E}_{{\rm{K}},\mathrm{inj},52}=41.7{E}_{{\rm{K}},\mathrm{dec},52}^{0.76\pm 0.20}$ correlation suggests a substantial energy injection during the shallow decay phase for most GRBs.

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The GRB radiative efficiency, defined as (Lloyd-Ronning & Zhang 2004)

$$\begin{eqnarray}&&{\eta }_{\gamma }=\displaystyle \frac{{E}_{\gamma ,\mathrm{iso}}}{{E}_{\gamma ,\mathrm{iso}}+{E}_{{\rm{K}},\mathrm{iso}}},\end{eqnarray} \tag{ 9 }$$

is an essential parameter to probe how efficiently a burst converts its global energy into prompt γ-ray emission. As mentioned above, if there is a continuous energy injection, the kinetic energy of the afterglow ${E}_{{\rm{K}},\mathrm{iso}}$ takes different values if one chooses different epochs. In principle, η_γ can be defined for two different epochs, ${t}_{\mathrm{dec}}$ and t_b, which have different physical meanings (see a discussion in Zhang et al. 2007a).

Figure 18 shows the radiative efficiencies calculated at ${t}_{\mathrm{dec}}$ and t_b as a function of ${E}_{\gamma ,\mathrm{iso}}$ along with their histograms. No significant correlation between η_γ and ${E}_{\gamma ,\mathrm{iso}}$ is found. The fireball internal shock model predicts a relatively small efficiency of a few percent (e.g., Kumar 1999; Panaitescu et al. 1999; Maxham & Zhang 2009; Gao & Mészáros 2015). Previous constraints on GRB radiative efficiencies give relatively large values, as large as above 90% (e.g., Lloyd-Ronning & Zhang 2004; Zhang et al. 2007a; Racusin et al. 2011) for some GRBs. This challenges the internal shock models and favors alternative prompt emission models, such as dissipation of magnetic fields (Zhang & Yan 2011) or photospheric emission (Lazzati et al. 2013).

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The derived efficiencies can be fit with rough log-normal distributions (Figures 18(b), (d), (f)). For the entire sample, one has $\mathrm{log}({\eta }_{\gamma ,\mathrm{dec}}/\%)=0.75\pm 0.86$. For the subsample of GRBs without energy injection, the radiative efficiency is lower, with $\mathrm{log}({\eta }_{\gamma ,\mathrm{dec}}/\%)=0.37\pm 0.61$. For the subsample of GRBs with energy injection, the radiative efficiencies read as $\mathrm{log}({\eta }_{\gamma ,\mathrm{dec}}/\%)=0.92\pm 1.25$ for ${t}_{\mathrm{dec}}$, and $\mathrm{log}({\eta }_{\gamma ,\mathrm{end}}/\%)=-0.89\pm 0.97$ for t_b, respectively.

The derived efficiencies are somewhat smaller than the values derived in previous work (e.g., Zhang et al. 2007a). The main reason is the adoption of a smaller value of ${\epsilon }_{B}\sim {10}^{-5}$, so the derived ${E}_{{\rm{K}},\mathrm{iso}}$ are systematically larger. This greatly alleviates the low-efficiency problem of the internal shock models. Nonetheless, some GRBs still have tens of percent efficiency, which demands a contrived setup for the internal shock models (e.g., Beloborodov 2000; Kobayashi & Sari 2001). If ${t}_{\mathrm{dec}}$ is adopted, which is more natural for most prompt emission models to calculate efficiency (see Zhang et al. 2007a for a detailed discussion), η_γ is still typically too large for the internal shock model. This is, on the other hand, consistent with the suggestion that internal collision-induced magnetic reconnection and turbulence (ICMART) is the dominant process to power GRB prompt emission in the majority of GRBs, which can typically gives tens of percent radiative efficiency (Zhang & Yan 2011; Deng et al. 2015). This conclusion is also consistent with independent studies of modeling the GRB prompt emission spectrum (Uhm & Zhang 2014c) and quasithermal photosphere emission component (Gao & Zhang 2015).

In the Gold sample, 8 of 45 GRBs show a jet break. These include five GRBs without energy injection and one more with energy injection. The ambient medium type of all six GRBs is ISM. Five out of these six GRBs have redshift information (Table 7).

Table 7.  Parameters of the Jet Break Sample

GRB	${\theta }_{j}^{o}$	$\mathrm{log}({E}_{\gamma }/\mathrm{erg})$	$\mathrm{log}({E}_{{\rm{K}}}/\mathrm{erg})$
050820A	4.5 ± 3.0	${51.54}_{-0.57}^{+0.24}$	${53.05}_{-0.99}^{+0.28}$
050922C	1.8 ± 0.3	${49.69}_{-0.10}^{+0.08}$	${50.13}_{-0.12}^{+0.09}$
060111Ba	⋯	⋯	⋯
081008	1.3 ± 0.4	${49.41}_{-0.21}^{+0.14}$	${50.54}_{-0.29}^{+0.17}$
081203A	1.0 ± 0.6	${49.76}_{-0.56}^{+0.24}$	${50.72}_{-0.33}^{+0.19}$
090618	3.5 ± 1.3	${50.68}_{-0.21}^{+0.14}$	${50.84}_{-0.23}^{+0.15}$
091127	2.7 ± 0.4	${49.22}_{-0.08}^{+0.07}$	${50.73}_{-0.09}^{+0.07}$
130427A	3.8 ± 0.3	${51.25}_{-0.04}^{+0.04}$	${51.54}_{-0.04}^{+0.04}$

Note.

^aNo redshift z available.

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Under the assumption of a conical jet, one can derive the jet opening angle based on observational data (Rhoads 1999; Sari et al. 1999; Frail et al. 2001):

$$\begin{eqnarray}{\theta }_{j} & = & 0.070\;\mathrm{rad}\;{\left(\displaystyle \frac{{t}_{{\rm{b}}}}{1\;\mathrm{day}}\right)}^{3/8}{\left(\displaystyle \frac{1+z}{2}\right)}^{-3/8}\\ & & \times {\left(\displaystyle \frac{{E}_{{\rm{K}},\mathrm{iso}}}{{10}^{53}\;\mathrm{erg}}\right)}^{-1/8}{\left(\displaystyle \frac{n}{0.1\;{\mathrm{cm}}^{-3}}\right)}^{1/8}.\end{eqnarray} \tag{ 10 }$$

We then calculate the geometrically corrected γ-ray energy

$$\begin{eqnarray}&&{E}_{\gamma }=(1-\mathrm{cos}{\theta }_{j}){E}_{\gamma ,\mathrm{iso}},\end{eqnarray} \tag{ 11 }$$

and kinetic energy

$$\begin{eqnarray}&&{E}_{{\rm{K}}}=(1-\mathrm{cos}{\theta }_{j}){E}_{{\rm{K}},\mathrm{iso}}.\end{eqnarray} \tag{ 12 }$$

Here ${E}_{{\rm{K}},\mathrm{iso}}$ is taken as ${E}_{{\rm{K}},\mathrm{end}}$ for the energy injection sample. The medium density is taken as n = 1 cm⁻³. The results are presented in Table 7 and Figure 19. The best-fitting results give ${\theta }_{j}=(2\buildrel{\circ}\over{.} 8\pm 1\buildrel{\circ}\over{.} 5)$, $\mathrm{log}({E}_{\gamma }/\mathrm{erg})=49.86\pm 0.65$, and $\mathrm{log}({E}_{{\rm{K}}}/\mathrm{erg})=50.89\pm 0.54$.

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For the $p\gt 2$ case, the forward shock emission can be characterized as (Yost et al. 2003; Zhang et al. 2007a)

$$\begin{eqnarray}{\nu }_{{\rm{m}}} & = & 3.3\times {10}^{12}\;\mathrm{Hz}\;{\left(\displaystyle \frac{p-2}{p-1}\right)}^{2}\\ & & \times {(1+z)}^{1/2}{\epsilon }_{B,-2}^{1/2}{\epsilon }_{e,-1}^{2}{E}_{{\rm{K}},\mathrm{iso},52}^{1/2}{t}_{d}^{-3/2},\\ {\nu }_{{\rm{c}}} & = & 6.3\times {10}^{15}\;\mathrm{Hz}\;{(1+z)}^{-1/2}\\ & & \times {(1+Y)}^{-2}{\epsilon }_{B,-2}^{-3/2}{E}_{{\rm{K}},\mathrm{iso},52}^{-1/2}{n}^{-1}{t}_{d}^{-1/2},\\ {F}_{\nu ,\mathrm{max}} & = & 1.6\;\mathrm{mJy}\;(1+z){D}_{28}^{-2}{\epsilon }_{B,-2}^{1/2}{E}_{{\rm{K}},\mathrm{iso},52}{n}^{1/2},\end{eqnarray} \tag{ 13 }$$

where ${E}_{{\rm{K}},\mathrm{iso},52}$ is the isotropic kinetic energy (in units of 10⁵² erg) in the blast wave, t_d is the time since trigger (in units of days), n is the density of the constant ambient medium, ${D}_{L}={10}^{28}\;\mathrm{cm}\;{D}_{28}$ is the luminosity distance, and

$$\begin{eqnarray}&&Y=[-1+{(1+4{\eta }_{1}{\eta }_{2}{\epsilon }_{e}/{\epsilon }_{B})}^{1/2}]/2\end{eqnarray} \tag{ 14 }$$

is the inverse Compton parameter, with ${\eta }_{1}=\mathrm{min}[1,{({\nu }_{{\rm{c}}}/{\nu }_{{\rm{m}}})}^{(2-p)/2}]$ (Sari & Esin 2001), and ${\eta }_{2}\leqslant 1$ is a correction factor introduced by the Klein–Nishina correction.

For $p\gt 2$ and in the $\nu \gt \mathrm{max}({\nu }_{{\rm{m}}},{\nu }_{{\rm{c}}})$ regime, one has (Zhang et al. 2007a)

$$\begin{eqnarray}\nu {F}_{\nu }(\nu ={10}^{18}\;\mathrm{Hz}) & = & {F}_{\nu ,\mathrm{max}}{\nu }_{{\rm{c}}}^{1/2}{\nu }_{{\rm{m}}}^{(p-1)/2}{\nu }_{{{\rm{X}}}^{(2-p)/2}}\\ & = & 5.2\times {10}^{-14}\;\mathrm{erg}\;{{\rm{s}}}^{-1}\;{\mathrm{cm}}^{-2}\;{D}_{28}^{-2}\\ & & \times {(1+z)}^{(p+2)/4}{(1+Y)}^{-1}\\ & & \times {f}_{p1}{\epsilon }_{B,-2}^{(p-2)/4}{\epsilon }_{e,-1}^{p-1}\\ & & \times {E}_{{\rm{K}},\mathrm{iso},52}^{(p+2)/4}{t}_{d}^{(2-3p)/4}{\nu }_{18}^{(2-p)/2},\end{eqnarray} \tag{ 15 }$$

where $\nu {f}_{\nu }(\nu ={10}^{18}\;\mathrm{Hz})$ is the energy flux at 10¹⁸ Hz (in units of ergs s⁻¹ cm⁻¹), and

$$\begin{eqnarray}&&{f}_{p1}=6.73{\left(\displaystyle \frac{p-2}{p-1}\right)}^{p-1}{(3.3\times {10}^{-6})}^{(p-2.3)/2}\end{eqnarray} \tag{ 16 }$$

is a function of electron spectral index p (Zhang et al. 2007a). One can then derive

$$\begin{eqnarray}{E}_{{\rm{K}},\mathrm{iso},52} & = & {\left(\displaystyle \frac{\nu {F}_{\nu }(\nu ={10}^{18}\;\mathrm{Hz})}{5.2\times {10}^{-14}\;\mathrm{erg}\;{{\rm{s}}}^{-1}\;{\mathrm{cm}}^{-2}}\right)}^{4/(p+2)}\\ & & \times {D}_{28}^{8/(p+2)}{(1+z)}^{-1}{(1+Y)}^{4/(p+2)}\\ & & \times {f}_{p1}^{-4/(p+2)}{\epsilon }_{B,-2}^{(2-p)/(p+2)}\\ & & \times {\epsilon }_{e,-1}^{4(1-p)/(p+2)}{t}_{d}^{(3p-2)/(p+2)}{\nu }_{18}^{2(p-2)/(p+2)},\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}{\epsilon }_{B,-2} & = & {\left(\displaystyle \frac{6.3\times {10}^{15}\;\mathrm{Hz}}{{\nu }_{{\rm{c}}}}\right)}^{(p+2)/(p+4)}\\ & & \times {\left(\displaystyle \frac{\nu {F}_{\nu }(\nu ={10}^{18}\;\mathrm{Hz})}{5.2\times {10}^{-14}\mathrm{erg}\;{{\rm{s}}}^{-1}\;{\mathrm{cm}}^{-2}}\right)}^{-2/(p+4)}{D}_{28}^{-4/(p+4)}\\ & & \times {(1+Y)}^{-2(p+3)/(p+4)}{n}^{-(p+2)/(p+4)}\\ & & \times {f}_{p1}^{2/(p+4)}{\epsilon }_{e,-1}^{2(p-1)/(p+4)}{t}_{d}^{-2p/(p+4)}{\nu }_{18}^{(2-p)/(p+4)}.\end{eqnarray} \tag{ 18 }$$

For $p\gt 2$ and in the ${\nu }_{{\rm{m}}}\lt \nu \lt {\nu }_{{\rm{c}}}$ regime, one has

$$\begin{eqnarray}\nu {F}_{\nu }(\nu ={10}^{18}\;\mathrm{Hz}) & = & {F}_{\nu ,\mathrm{max}}{\nu }_{{\rm{m}}}^{(p-1)/2}{\nu }_{{\rm{X}}}^{(3-p)/2}\\ & = & 6.5\times {10}^{-13}\;\mathrm{erg}\;{{\rm{s}}}^{-1}\;{\mathrm{cm}}^{-2}\\ & & \times {D}_{28}^{-2}{(1+z)}^{(p+3)/4}{f}_{p1}{\epsilon }_{B,-2}^{(p+1)/4}\\ & & \times {\epsilon }_{e,-1}^{p-1}{E}_{{\rm{K}},\mathrm{iso},52}^{(p+3)/4}\\ & & \times {n}^{1/2}{t}_{d}^{(3-3p)/4}{\nu }_{18}^{(3-p)/2},\end{eqnarray} \tag{ 19 }$$

so

$$\begin{eqnarray}{E}_{{\rm{K}},\mathrm{iso},52} & = & {\left(\displaystyle \frac{\nu {F}_{\nu }(\nu ={10}^{18}\;\mathrm{Hz})}{6.5\times {10}^{-13}\mathrm{erg}\;{{\rm{s}}}^{-1}\;{\mathrm{cm}}^{-2}}\right)}^{4/(p+3)}\\ & & \times {D}_{28}^{8/(p+3)}{(1+z)}^{-1}{f}_{p1}^{-4/(p+3)}{\epsilon }_{B,-2}^{-(p+1)/(p+3)}\\ & & \times {\epsilon }_{e,-1}^{4(1-p)/(p+3)}{n}^{-2/(p+3)}{t}_{d}^{(3p-3)/(p+3)}{\nu }_{18}^{2(p-3)/(p+3)},\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}{\epsilon }_{B,-2} & = & {\left(\displaystyle \frac{6.3\times {10}^{15}\;\mathrm{Hz}}{{\nu }_{{\rm{c}}}}\right)}^{(p+3)/(p+4)}\\ & & \times {\left(\displaystyle \frac{\nu {F}_{\nu }(\nu ={10}^{18}\;\mathrm{Hz})}{6.5\times {10}^{-13}\;\mathrm{erg}\;{{\rm{s}}}^{-1}\;{\mathrm{cm}}^{-2}}\right)}^{-2/(p+4)}{D}_{28}^{-4/(p+4)}\\ & & \times {(1+Y)}^{-2(p+3)/(p+4)}{n}^{-(p+2)/(p+4)}\\ & & \times {f}_{p1}^{2/(p+4)}{\epsilon }_{e,-1}^{2(p-1)/(p+4)}{t}_{d}^{-2p/(p+4)}{\nu }_{18}^{(3-p)/(p+4)}.\end{eqnarray} \tag{ 21 }$$

Following Gao et al. (2013), below we derive the expressions in the $p\lt 2$ case.

In the $\nu \gt \mathrm{max}({\nu }_{{\rm{m}}},{\nu }_{{\rm{c}}})$ regime, one has

$$\begin{eqnarray}{E}_{{\rm{K}},\mathrm{iso}} & = & {(\nu {F}_{\nu }(\nu ={10}^{18}\;\mathrm{Hz}))}^{16/(p+14)}{D}^{32/(p+14)}\\ & & \times {f}_{p2}^{-16/(p+14)}{n}^{(p-2)/(p+14)}{t}^{(3p+10)/(p+14)}\\ & & \times {(1+Y)}^{16/(p+14)}\\ & & \times {(1+z)}^{-12/(p+14)}{\epsilon }_{e}^{-16/(p+14)}{\nu }_{18}^{8(p-2)/(p+14)},\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&{f}_{p2}=0.00529\times {e}^{0.767p}\left(\displaystyle \frac{2-p}{p-1}\right),\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}{\epsilon }_{B} & = & 9.44\times {10}^{27}{\nu }_{{\rm{c}}}^{-2/3}{(\nu {F}_{\nu }(\nu ={10}^{18}\;\mathrm{Hz}))}^{-16/3(p+14)}\\ & & \times {D}^{-32/3(p+14)}{f}_{p2}^{16/3(p+14)}{n}^{-(3p+26)/(3p+42)}\\ & & \times {t}^{-4(p+6)/3(p+14)}{(1+Y)}^{-4(p+18)/3(p+14)}\\ & & \times {(1+z)}^{-(p+2)/3(p+14)}{\epsilon }_{e}^{16/3(p+14)}{\nu }_{18}^{-8(p-2)/3(p+14)}.\end{eqnarray} \tag{ 24 }$$

In the ${\nu }_{{\rm{m}}}\lt \nu \lt {\nu }_{{\rm{c}}}$ regime, one has

$$\begin{eqnarray}{E}_{{\rm{K}},\mathrm{iso}} & = & {(\nu {F}_{\nu }(\nu ={10}^{18}\;\mathrm{Hz}))}^{16/(p+18)}{D}^{32/(p+18)}\\ & & \times {f}_{p3}^{-16/(p+18)}{n}^{(p-10)/(p+18)}{t}^{3(p+2)/(p+18)}\\ & & \times {(1+z)}^{-16/(p+18)}{\epsilon }_{e}^{-16/(p+18)}\\ & & \times {\epsilon }_{B}^{-12/(p+18)}{\nu }_{18}^{8(p-3)/(p+18)},\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}{\epsilon }_{B} & = & {e}^{(1159.5+64.4p)/(p+14)}{\nu }_{{\rm{c}}}^{-2(p+18)/3(p+14)}\\ & & \times {(\nu {F}_{\nu }(\nu ={10}^{18}\;\mathrm{Hz}))}^{-16/3(p+14)}{D}^{-32/3(p+14)}{f}_{p3}^{16/3(p+14)}\\ & & \times {n}^{-(3p+26)/(3p+42)}{t}^{-4(p+6)/3(p+14)}\\ & & \times {(1+Y)}^{-4(p+18)/3(p+14)}{(1+z)}^{-(p+2)/3(p+14)}\\ & & \times {\epsilon }_{e}^{16/3(p+14)}{\nu }_{18}^{-8(p-3)/3(p+14)},\end{eqnarray} \tag{ 26 }$$

and

$$\begin{eqnarray}&&{f}_{p3}=5.53\times {10}^{-15}{e}^{0.767p}\left(\displaystyle \frac{2-p}{p-1}\right).\end{eqnarray} \tag{ 27 }$$

The following derivations follow Chevalier & Li (2000), Gao et al. (2013), and Lü & Zhang (2014).

For the $p\gt 2$ case, the forward shock emission can be characterized as

$$\begin{eqnarray}{\nu }_{{\rm{m}}} & = & 5.2\times {10}^{11}\;\mathrm{Hz}\;{\left(\displaystyle \frac{p-2}{p-1}\right)}^{2}\\ & & \times {(1+z)}^{1/2}{\epsilon }_{B,-2}^{1/2}{\epsilon }_{e,-1}^{2}{E}_{{\rm{K}},\mathrm{iso},52}^{1/2}{t}_{d}^{-3/2},\\ {\nu }_{{\rm{c}}} & = & 1.7\times {10}^{18}\;\mathrm{Hz}\;{(1+z)}^{-3/2}\\ & & \times {(1+Y)}^{-2}{\epsilon }_{B,-2}^{-3/2}{E}_{{\rm{K}},\mathrm{iso},52}^{1/2}{A}_{*,-1}^{-2}{t}_{d}^{1/2},\\ {F}_{\nu ,\mathrm{max}} & = & 1.6\;\mathrm{mJy}\;{(1+z)}^{3/2}{D}_{28}^{-2}{\epsilon }_{B,-2}^{1/2}{E}_{{\rm{K}},\mathrm{iso},52}^{1/2}{A}_{*,-1}{t}_{d}^{-1/2},\end{eqnarray} \tag{ 28 }$$

where A_* is the density parameter of the stellar wind medium.

For $p\gt 2$ and in the $\nu \gt \mathrm{max}({\nu }_{{\rm{m}}},{\nu }_{{\rm{c}}})$ regime, one has

$$\begin{eqnarray}\nu {F}_{\nu }(\nu ={10}^{18}\;\mathrm{Hz}) & = & {F}_{\nu ,\mathrm{max}}{\nu }_{{\rm{c}}}^{1/2}{\nu }_{{\rm{m}}}^{(p-1)/2}{\nu }_{{\rm{X}}}^{(2-p)/2}\\ & = & 2.6\times {10}^{-13}\;\mathrm{erg}\;{{\rm{s}}}^{-1}\;{\mathrm{cm}}^{-2}\;{D}_{28}^{-2}\\ & & \times {(1+z)}^{(p+2)/4}{(1+Y)}^{-1}\\ & & \times {f}_{p4}{\epsilon }_{B,-2}^{(p-2)/4}{\epsilon }_{e,-1}^{p-1}\\ & & \times {E}_{{\rm{K}},\mathrm{iso},52}^{(p+2)/4}{t}_{d}^{(2-3p)/4}{\nu }_{18}^{(2-p)/2},\end{eqnarray} \tag{ 29 }$$

$$\begin{eqnarray}&&{f}_{p4}=6.73{\left(\displaystyle \frac{p-2}{p-1}\right)}^{p-1}{(5.2\times {10}^{-7})}^{(p-2.3)/2},\end{eqnarray} \tag{ 30 }$$

$$\begin{eqnarray}{E}_{{\rm{K}},\mathrm{iso},52} & = & {\left(\displaystyle \frac{\nu {F}_{\nu }(\nu ={10}^{18}\;\mathrm{Hz})}{2.6\times {10}^{-13}\;\mathrm{erg}\;{{\rm{s}}}^{-1}\;{\mathrm{cm}}^{-2}}\right)}^{4/(p+2)}\\ & & \times {D}_{28}^{8/(p+2)}{(1+z)}^{-1}{(1+Y)}^{4/(p+2)}\\ & & \times {f}_{p4}^{-4/(p+2)}{\epsilon }_{B,-2}^{(2-p)/(p+2)}\\ & & \times {\epsilon }_{e,-1}^{4(1-p)/(p+2)}{t}_{d}^{(3p-2)/(p+2)}{\nu }_{18}^{2(p-2)/(p+2)},\end{eqnarray} \tag{ 31 }$$

$$\begin{eqnarray}{\epsilon }_{B,-2} & = & {\left(\displaystyle \frac{1.7\times {10}^{18}\;\mathrm{Hz}}{{\nu }_{{\rm{c}}}}\right)}^{(p+2)/(2p+2)}\\ & & \times {\left(\displaystyle \frac{\nu {F}_{\nu }(\nu ={10}^{18}\;\mathrm{Hz})}{2.6\times {10}^{-13}\;\mathrm{erg}\;{{\rm{s}}}^{-1}\;{\mathrm{cm}}^{-2}}\right)}^{1/(p+1)}\\ & & \times {D}_{28}^{2/(p+1)}{(1+z)}^{-(p+2)/(p+1)}\\ & & \times {(1+Y)}^{-1}{f}_{p4}^{-1/(p+1)}{\epsilon }_{e,-1}^{(1-p)/(p+1)}\\ & & \times {A}_{*,-1}^{-(p+2)/(p+1)}{t}_{d}^{p/(p+1)}{\nu }_{18}^{(p-2)/2(p+1)}.\end{eqnarray} \tag{ 32 }$$

For $p\gt 2$ and in the ${\nu }_{{\rm{m}}}\lt \nu \lt {\nu }_{{\rm{c}}}$ regime, one has

$$\begin{eqnarray}\nu {F}_{\nu }(\nu ={10}^{18}\;\mathrm{Hz}) & = & {F}_{\nu ,\mathrm{max}}{\nu }_{{\rm{m}}}^{(p-1)/2}{\nu }_{{\rm{X}}}^{(3-p)/2}\\ & = & 2.0\times {10}^{-13}\;\mathrm{erg}\;{{\rm{s}}}^{-1}\;{\mathrm{cm}}^{-2}{D}_{28}^{-2}\\ & & \times {(1+z)}^{(p+5)/4}{f}_{p4}{\epsilon }_{B,-2}^{(p+1)/4}{\epsilon }_{e,-1}^{p-1}{E}_{{\rm{K}},\mathrm{iso},52}^{(p+1)/4}\\ & & \times {A}_{*,-1}{t}_{d}^{(1-3p)/4}{\nu }_{18}^{(3-p)/2},\end{eqnarray} \tag{ 33 }$$

$$\begin{eqnarray}{E}_{{\rm{K}},\mathrm{iso},52} & = & {\left(\displaystyle \frac{\nu {F}_{\nu }(\nu ={10}^{18}\;\mathrm{Hz})}{2.0\times {10}^{-13}\;\mathrm{erg}\;{{\rm{s}}}^{-1}\;{\mathrm{cm}}^{-2}}\right)}^{4/(p+1)}\\ & & \times {D}_{28}^{8/(p+1)}{(1+z)}^{-(p+5)/(p+1)}{f}_{p4}^{-4/(p+1)}{\epsilon }_{B,-2}^{-1}\\ & & \times {\epsilon }_{e,-1}^{4(1-p)/(p+1)}{A}_{*,-1}^{-4/(p+1)}{t}_{d}^{(3p-1)/(p+1)}{\nu }_{18}^{2(p-3)/(p+1)},\end{eqnarray} \tag{ 34 }$$

$$\begin{eqnarray}{\epsilon }_{B,-2} & = & {\left(\displaystyle \frac{1.7\times {10}^{18}\;\mathrm{Hz}}{{\nu }_{{\rm{c}}}}\right)}^{1/2}\\ & & \times {\left(\displaystyle \frac{\nu {F}_{\nu }(\nu ={10}^{18}\;\mathrm{Hz})}{2.0\times {10}^{-13}\;\mathrm{erg}\;{{\rm{s}}}^{-1}\;{\mathrm{cm}}^{-2}}\right)}^{1/(p+1)}\\ & & \times {D}_{28}^{2/(p+1)}{(1+z)}^{-(p+2)/(p+1)}\\ & & \times {(1+Y)}^{-1}{A}_{*,-1}^{-(p+2)/(p+1)}{f}_{p4}^{-1/(p+1)}\\ & & \times {\epsilon }_{e,-1}^{(1-p)/(p+1)}{t}_{d}^{p/(p+1)}{\nu }_{18}^{(p-3)/2(p+1)}.\end{eqnarray} \tag{ 35 }$$

For $p\lt 2$ and in the $\nu \gt \mathrm{max}({\nu }_{{\rm{m}}},{\nu }_{{\rm{c}}})$ regime, one has

$$\begin{eqnarray}{E}_{{\rm{K}},\mathrm{iso}} & = & {(\nu {F}_{\nu }(\nu ={10}^{18}\;\mathrm{Hz}))}^{8/(p+6)}{D}^{16/(p+6)}\\ & & \times {f}_{p5}^{-8/(p+6)}{A}^{(p-2)/(p+6)}t\\ & & \times {(1+Y)}^{8/(p+6)}\times {(1+z)}^{-6/(p+6)}\\ & & \times {\epsilon }_{e}^{-8/(p+6)}{\nu }_{18}^{4(p-2)/(p+6)},\end{eqnarray} \tag{ 36 }$$

$$\begin{eqnarray}&&{f}_{p5}=405854\times {3}^{3p/8}{5}^{-p/4}{e}^{-7p}{\pi }^{(4-3p)/8}\left(\displaystyle \frac{2-p}{p-1}\right),\end{eqnarray} \tag{ 37 }$$

$$\begin{eqnarray}{\epsilon }_{B} & = & 6.87\times {10}^{-11}{\nu }_{{\rm{c}}}^{-2/3}{(\nu {F}_{\nu }(\nu ={10}^{18}\;\mathrm{Hz}))}^{8/3(p+6)}\\ & & \times {D}^{16/3(p+6)}{f}_{p5}^{-8/3(p+6)}{A}^{-(3p+26)/(3p+18)}{t}^{2/3}\\ & & \times {(1+Y)}^{-4(p+4)/3(p+6)}{(1+z)}^{-(p+8)/(p+6)}\\ & & \times {\epsilon }_{e}^{-8/3(p+6)}{\nu }_{18}^{4(p-2)/3(p+6)}.\end{eqnarray} \tag{ 38 }$$

For $p\lt 2$ and in the ${\nu }_{{\rm{m}}}\lt \nu \lt {\nu }_{{\rm{c}}}$ regime, one has

$$\begin{eqnarray}{E}_{{\rm{K}},\mathrm{iso}} & = & {(\nu {F}_{\nu }(\nu ={10}^{18}\;\mathrm{Hz}))}^{8/(p+4)}{D}^{16/(p+4)}\\ & & \times {f}_{p6}^{-8/(p+4)}{A}^{(p-10)/(p+4)}{t}^{(p+8)/(p+4)}\\ & & \times {(1+z)}^{-12/(p+4)}{\epsilon }_{e}^{-8/(p+4)}{\epsilon }_{B}^{-6/(p+4)}{\nu }_{18}^{4(p-3)/(p+4)},\end{eqnarray} \tag{ 39 }$$

$$\begin{eqnarray}{f}_{p6} & = & 1.72\times {10}^{22}\times {2}^{-17p/2}{3}^{3p/8}{5}^{-37p/4}\\ & & \times {e}^{13.38p}{\pi }^{(4-3p)/8}\left(\displaystyle \frac{2-p}{p-1}\right),\end{eqnarray} \tag{ 40 }$$

and

$$\begin{eqnarray}{\epsilon }_{B} & = & {(5.69\times {10}^{-16})}^{2(p+4)/3(p+6)}{\nu }_{{\rm{c}}}^{-2(p+4)/3(p+6)}\\ & & \times {(\nu {F}_{\nu }(\nu ={10}^{18}\;\mathrm{Hz}))}^{8/3(p+6)}{D}^{16/3(p+6)}{f}_{p6}^{-8/3(p+6)}\\ & & \times {A}^{-(3p+26)/(3p+18)}{t}^{2/3}{(1+Y)}^{-4(p+4)/3(p+6)}\\ & & \times {(1+z)}^{-(p+8)/(p+6)}{\epsilon }_{e}^{-8/3(p+6)}{\nu }_{18}^{4(p-3)/3(p+6)}.\end{eqnarray} \tag{ 41 }$$