further evidence for a DPLH spectrum in GRB afterglows

# Modeling the Multi-band Afterglows of GRB 060614 and GRB 060908: Further Evidence for a Double Power-Law Hard Electron Energy Spectrum

Q. Zhang Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China S. L. Xiong Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China L. M. Song Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China
###### Abstract

Electrons accelerated in relativistic collisionless shocks are usually assumed to follow a power-law energy distribution with an index of . Observationally, although most gamma-ray bursts (GRBs) have afterglows that are consistent with , there are still a few GRBs suggestive of a hard () electron energy spectrum. Our previous work showed that GRB 091127 gave strong evidence for a double power-law hard electron energy (DPLH) spectrum with , and an “injection break” assumed as in the highly relativistic regime, where is the bulk Lorentz factor of the jet. In this paper, we show that GRB 060614 and GRB 060908 provide further evidence for such a DPLH spectrum. We interpret the multi-band afterglow of GRB 060614 with the DPLH model in an homogeneous interstellar medium by taking into account a continuous energy injection process, while for GRB 060908, a wind-like circumburst density profile is used. The two bursts, along with GRB 091127, suggest a similar behavior in the evolution of the injection break, with . Whether this represents a universal law of the injection break remains uncertain and more such afterglow observations are needed to test this conjecture.

acceleration of particles – gamma-ray burst: individual (GRB 060614, GRB 060908) – radiation mechanisms: non-thermal
journal: ApJ\correspondingauthor

Q. Zhang

## 1 Introduction

Gamma-Ray bursts (GRBs) are the most energetic stellar explosions in the universe. They produce a short prompt -ray emission followed by a long-lived afterglow phase. The afterglows of GRBs are believed to originate from the synchrotron emission of shock-accelerated electrons produced by the interaction between the outflow and the external medium (Rees & Mészáros, 1992; Mészáros & Rees, 1993, 1997; Sari et al., 1998; Chevalier & Li, 2000). Particle acceleration is usually attributed to the Fermi process (Fermi, 1954), which results in a power-law (PL) energy distribution , with a cutoff at high energies. Some analytical and numerical studies indicate a nearly universal spectral index of (e.g., Bednarz & Ostrowski, 1998; Kirk et al., 2000; Achterberg et al., 2001; Lemoine & Pelletier, 2003; Spitkovsky, 2008), though other studies suggest that there is a large range of possible values for of (Baring, 2004). The values of derived from the spectral analysis of the multi-band afterglow (e.g. Chevalier & Li, 2000; Panaitescu & Kumar, 2002; Starling et al., 2008; Curran et al., 2009; Fong et al., 2015; Li et al., 2015; Wang et al., 2015) or the X-ray data alone (e.g., Shen et al., 2006; Curran et al., 2010) show a rather wide distribution, but most of them are consistent with . Only a few GRBs, e.g., GRB 060908 (Covino et al., 2010), GRB 091127 (Filgas et al., 2011; Troja et al., 2012), GRB 110918A (Frederiski et al., 2013) and GRB 140515A (Melandri et al., 2015), show very flat spectra in the optical band and require a hard () electron energy spectrum.

To explain those afterglows that cannot be well modeled with a standard () electron energy spectrum, two types of electron energy distributions were proposed in literature: (1) a single PL electron energy distribution () with an exponential cutoff at a maximum electron Lorentz factor (Bhattacharya, 2001; Dai & Cheng, 2001); (2) a double PL electron energy distribution ( and ) with an “injection break” (Panaitescu & Kumar, 2001; Bhattacharya & Resmi, 2004; Resmi & Bhattacharya, 2008; Wang et al., 2012). A direct method to distinguish the two models is to see the passage of the injection break frequency (i.e., the synchrotron frequency corresponding to ) through a certain band, e.g, from the optical to the near-infrared (NIR) bands. Our previous work (Zhang et al., 2015, Paper I hereafter) showed that GRB 091127 was such a case and gave strong evidence for the double PL hard electron spectrum model (the so-called “DPLH model” in Paper I, ). The physical origin of is not clear. The DPLH model assumes in the highly relativistic regime, here is the bulk Lorentz factor of the jet. Paper I found by modeling the multi-band afterglow of GRB 091127. Does this imply a universal evolution of the injection break? More GRB 091127-like bursts are needed to test this conjecture.

The “smoking-gun” evidence for a DPLH spectrum requires high-quality and multi-wavelength afterglow observations to provide detailed spectral information, in order to identify the existence of and its evolution behavior. In this paper, we show that the multi-band afterglows of GRB 060614 and GRB 060908 can be well modeled by the DPLH model, thus providing further evidence for such a DPLH spectrum. Moreover, the two bursts, along with GRB 091127, seem to show a similar behavior in the evolution of the injection break.

Our paper is organized as follows. In Section 2, we summarize the observational results of GRB 060614 and GRB 060908. Based on the work of Resmi & Bhattacharya (2008, RB08 hereafter), the DPLH model for both a homogeneous interstellar medium (ISM) and a wind-like circumburst environment is described in Section 3. In this section we also extend the original model by taking into account a continuous energy injection process (Sari & Mészáros, 2000; Zhang & Mészáros, 2001) to explain the afterglow of GRB 060614. In Section 4, we constrain the model parameters and then compare our model with the multi-band afterglow data. Finally, we present our conclusion and make some discussions in Section 5. The convention is adopted throughout the paper, where is the spectral index and is the temporal decay index. We use the standard notation with being a generic quantity in cgs units and assume a concordance cosmology with , and (Jarosik et al., 2011). All the quoted errors are given at a confidence level (CL) unless stated otherwise.

## 2 Observational Results

### 2.1 Grb 060614

GRB 060614 triggered the Swift Burst Alert Telescope (BAT; Barthelmy et al., 2005) on 2006 June 14 at 12:43:48 UT (Parsons et al., 2006) and was also detected by Konus-Wind (Golenetskii et al., 2006). The light curve (LC) shows an initial hard, bright peak lasting  s followed by a long, somewhat softer extended emission, with a total duration of  s (Barthelmy et al., 2006). The spectrum of the initial pulse can be fitted in the 20 keV–2 MeV energy range by a PL with an exponential cutoff model, with the peak energy  keV, while the spectrum of the remaining part of the burst can be described by a simple PL with photon index (Golenetskii et al., 2006). The total fluence in the 20 keV–2 MeV energy range is , of which the initial intense pulse contributes a fraction of (Golenetskii et al., 2006). With a redshift of (Fugazza et al., 2006; Price et al., 2006), the isotropic equivalent energy was estimated as erg in the  keV rest-frame energy band (Mangano et al., 2007, M07 hereafter). In addition, GRB 060614 has null spectral lags, being consistent with typical short GRBs (Gehrels et al., 2006).

The X-ray Telescope (XRT; Burrows et al., 2005) began observing the field 91 s after the BAT trigger (Parsons et al., 2006). The X-ray afterglow of GRB 060614 exhibits a canonical LC which has been commonly observed in the Swift era (e.g., Nousek et al., 2006; Zhang et al., 2006; Evans et al., 2009). It begins with an initial fast exponential decay, followed by a plateau with slope ; at  ks, it steepens to a standard afterglow evolution with slope ; later on, the LC shows a further steepening to a slope at  ks ( M07, see Figure 1). The X-ray data observed in the photon counting (PC) mode show no significant spectral evolution, with the spectral index (M07).

The Swift Ultra-Violet/Optical Telescope (UVOT; Roming et al., 2005) commenced observations 101 s after the BAT trigger (Holland, 2006). Besides, the R-band afterglow was detected by several ground telescopes (e.g., Della Valle et al., 2006; French et al., 2006; Fynbo et al., 2006; Gal-Yam et al., 2006). M07 presented detailed spectral and temporal analysis of the optical/ultraviolet (UV) afterglow, below we summarize their main results. The optical/UV LCs show achromatic breaks with the X-ray afterglow, i.e.,  ks and  ks. The decay slopes after the two breaks are and , respectively. On the whole, the X-ray/UV/optical LCs have marginally consistent evolutions after  ks. What is puzzling is that the initial slope is dependent on wavelength: the UV LCs show nearly flat evolutions while the optical LCs rise slowly with slopes from to (see Figure 1). The spectral energy distributions (SEDs) of the afterglow from optical to X-rays show a spectral break passing through the optical/UV band between and  ks. The break frequency at 10 ks is around  Hz (see Figure 7 of M07, ). At this time, the optical/UV and X-ray afterglows have spectral indices and , respectively. At later times ( ks), the spectral index in the optical/UV band changes to be consistent with that of X-rays. Fits of the broad-band SEDs imply a weak host extinction (M07).

In addition, deep optical/NIR follow-ups of GRB 060614 show no evidence for an associated supernova down to very strict limits; the GRB host is a very faint star-forming galaxy with a specific star formation rate lower than most long GRB hosts; the GRB counterpart resides in the outskirts of the host (Della Valle et al., 2006; Fynbo et al., 2006; Gal-Yam et al., 2006). The recent discovery of a distinct NIR excess at about 13.6 days after the burst suggests a possible kilonova (or macronova) origin (Jin et al., 2015; Yang et al., 2015). Together with the vanishing time lags of the prompt emission, all these point towards a different origin from typical long GRBs; it is likely to be of a subclass of merger-type short GRBs (Gehrels et al., 2006; Zhang et al., 2007).

### 2.2 Grb 060908

GRB 060908 triggered the Swift/BAT on 2006 September 14 at 08:57:22.34 UT (Evans et al., 2006). Further analysis found the onset of the GRB occurs 12.96 s before the trigger time, i.e., (Covino et al., 2010). So the time used in this work is relative to . The BAT LC shows a multi-peaked structure with a total duration of  s (Palmer et al., 2006). The time-averaged spectrum is best fit by a simple PL and can be alternatively fit by a Band function (Band et al., 1993) with the high-energy photon index fixed. With a redshift of (Fynbo et al., 2009), the latter spectral model gives the rest-frame peak energy  keV and the isotropic equivalent energy  erg in the rest-frame  keV energy band (Covino et al., 2010).

The XRT began observing the field 72 s after the BAT trigger(Evans et al., 2006). The spectra were modeled with an absorbed power-law, which gave the spectral index and the host absorbing column density cm. The LC is characterised by a constant PL decay with index , while from to  s a complex flaring activity is superposed on the underlying decay (Covino et al., 2010, see Figure 2).

The UVOT commenced observations 80 s after the BAT trigger (Morgan et al., 2006). The optical/NIR afterglow was also monitored by several ground-based telescopes (e.g., Andreev et al., 2006; Antonelli et al., 2006; Nysewander et al., 2006; Wiersema et al., 2006). The LCs can be described by a broken PL with the initial decay index , the break time  s and the post-break decay index (Covino et al., 2010). There seems to be another break at  s, with the post-break decay slope of (see Figure 2). However, this break time cannot be well constrained by the data (Covino et al., 2010). In Subsection 4.2, we will show that such a late break is actually required by the afterglow modeling. The spectral analysis at 800 and 8000 s shows rather flat spectra with index and host dust extinction (Covino et al., 2010).

## 3 Model

Several clues should be considered before establishing the afterglow model for both GRBs. For GRB 060614: (i) The two achromatic breaks ( and ) shown in the multi-band LCs require a hydrodynamical origin. This canonical afterglow behavior was well described in Zhang et al. (2006). The first break is possibly an “energy-injection break”, implying the end of a continuous energy injection into the forward shock (Sari & Mészáros, 2000; Zhang & Mészáros, 2001), while the second break is most likely the so-called “jet break” (Rhoads, 1999; Sari et al., 1999); (ii) The early flat spectrum () in the optical/UV band definitely requires a hard electron energy spectrum; (iii) There should be a spectral break between the optical/UV and the X-ray bands, but neither the minimum synchrotron frequency nor the cooling frequency can accommodate the observations 111The reasons are as follows: (i) Although the passage of can produce a spectral evolution and slow-rising optical LCs (M07), this requires . The model predicted spectral index in this regime is which is inconsistent with the observed value (); (ii) If the observed break frequency is , it suggests a hard electron energy spectrum with . However, the single PL hard electron spectrum model of Dai & Cheng (2001) predicts the post-jet-break decay slope should be which is substantially lower than the observed value (; M07, ).. For GRB 060908, the SED analysis also requires a hard electron energy distribution and some kind of spectral break between the optical/NIR and the X-ray bands. The single PL hard electron spectrum model of Dai & Cheng (2001) with has difficulties in explaining the observations, since the model predicts which is obviously lower than the observed value; the predicted decay slopes are also inconsistent with the observations. Therefore, the DPLH model is a natural choice.

Using the derived spectral and temporal indices of RB08 (their Table 2), we found the afterglow properties of GRB 060614 can be well reproduced by the DPLH model for an ISM medium when an additional energy injection is invoked, while the properties of GRB 060908 can be well explained when a wind-like circumburst density profile is used. In this section, we give a basic description of the DPLH model and present relevant formulas which will be used in Section 4. We refer the reader to RB08 for more details.

The DPLH spectrum with indices and is represented as (RB08)

 N(γe)=Ce⎧⎪ ⎪⎨⎪ ⎪⎩(γeγb)−p1,γm⩽γe<γb,(γeγb)−p2,γe⩾γb, (1)

where is the normalization constant, is minimum electron Lorentz factors, and is the injection break. The physical origin of is not clear, RB08 assumed that it is a function of to accommodate the non-relativistic regime of expansion, i.e.,

 γb=ξ(βγ)q, (2)

where is a constant of proportionality, is the dimensionless bulk velocity, and is assumed to be a constant for simplicity.

For a a relativistic shock propagating through a cold medium with particle density , the post-shock particle density and energy density are and , respectively (Sari et al., 1998), from which one derives the minimum Lorentz factor (RB08)

 γm=(fpmpmeϵeξ2−p1)1p1−1β−q(2−p1)p1−1(γ−1)1p1−1γ−q(2−p1)p1−1, (3)

where and are the proton and electron rest mass, respectively; is the fraction of shock energy carried by electrons, and .

We calculate the break frequencies of synchrotron spectra , , and the peak flux according to the formulas given by Wijers & Galama (1999):

 νm = xp1+zqeB′πmecγγ2m, (4) νb,c = 0.2861+zqeB′πmecγγ2b,c, (5) Fν,max = √3ϕpNeq3e(1+z)4πd2Lmec2B′γ, (6)

where is the total number of swept-up electrons, is the electron charge, is the post-shock magnetic field density, is the fraction of shock energy carried by magnetic fields, is the luminosity distance corresponding to the redshift , is the cooling Lorentz factor of electrons. and represent the dimensionless peak frequency and the peak flux, respectively. Their dependence on can be obtained from Wijers & Galama (1999).

For the adiabatic self-similar evolution of a spherical blastwave, the radius and bulk Lorentz factor evolve as and in the ultra-relativistic regime (Blandford & McKee, 1976; Sari et al., 1998; Chevalier & Li, 2000; Gao et al., 2013). The above derivation used the density profile , for ISM and for wind medium. By substituting these expressions in Equations (4)–(6), one derives 222Different from RB08, we did not consider the effect of sideways expansion in the derivation of Equations (7)-(14). This can be seen as a reasonable approximate in the ultra-relativistic regime as long as the inverse Lorentz factor has not exceeded the initial jet opening angle (Rhoads, 1999). The coefficients in these equations are consistent with those of RB08 within a factor of a few that may be due to minor differences in the treatment of dynamics.

 νm = 8.2×106(1833fp)2p1−1(37.2)1−q(2−p1)p1−1xp11+z (7) ξ−2(2−p1)p1−1ϵ2p1−1eϵ1/2B,−2Ep1−q(2−p1)4(p1−1)52np1−2+q(2−p1)40 (td1+z)−3[p1−q(2−p1)]4(p1−1)Hz, νc = 1.5×1015ϵ−3/2B,−2E−1/252n−10[td(1+z)]−1/2Hz, (8) νb = 3.8×105(6.1)1+2q1+zξ2ϵ1/2B,−2E1+q452n1−q40 (9) (td1+z)−3(1+q)4Hz, Fν,max = 6.8×103ϕp1ϵ1/2B,−2E52n1/20d−2L,28(1+z)μJy, (10)

for the ISM case, and

 νm = 5.8×106(13.8)y(183.3fp)2p1−1xp11+zξ−2(2−p1)p1−1 (11) ϵ2p1−1e,−1ϵ1/2B,−2Ey/252A1−y2∗(td1+z)−2+y2Hz, νc = 1.6×1015(1+z)3ϵ−3/2B,−2E1/252A−2∗(td1+z)1/2Hz, (12) νb = 1.6×106(13.8)q1+zξ2ϵ1/2B,−2Eq/252A1−q2∗ (13) (td1+z)−2+q2Hz, Fν,max = 20.6ϕp1(1+z)ϵ1/2B,−2E1/252A∗d−2L,28 (14) (td1+z)−1/2mJy,

for the wind case333We note the exponent of in Equation (11) and the exponent of in Equation (13) are different from the results of RB08 (see their Equations (13) and (15)). Their expression of also missed out a factor of . We have carefully checked our derivations to make sure that our results are robust. Here is the time in days, and are related by  cm and .

The evolution of the synchrotron flux density at a given frequency () relies on the order of the three break frequencies and the regime in which resides. Below we give only some scaling laws444We refer the reader to RB08 for a complete reference of the scaling relationships for the spectral breaks and in various spectral regimes (but without an energy injection). for and that will be used in Section 4. Since the synchrotron self-absorption process is not relevant, we do not consider it in this work.

For GRB 060908, according to Equations (11)-(14), the relevant spectral regimes and flux densities are:

(i) ,

 Fν=Fν,max(ννm)−p1−12∝t14(2q−p1q−2p1−1). (15)

(ii) ,

 Fν = Fν,max(νbνm)−p1−12(ννb)−p2−12 (16) ∝ t14(2q−p2q−2p2−1).

(iii) ,

 Fν = Fν,max(νcνm)−p1−12(νbνc)−p12(ννb)−p22 (17) = Fν,max(νbνm)−p1−12(νcνb)−p2−12(ννc)−p22 ∝ t14(2q−p2q−2p2).

For GRB 060614, the situation is somewhat more complicated. Besides the adiabatic self-similar evolution phase, these should be a continuous energy injection process before  ks and a jet break at about 117 ks. The injected energy can be provided by a long-lived central engine (Dai & Lu, 1998; Zhang & Mészáros, 2001) or by slower material with significant energy which gradually piles up onto the decelerating ejecta and “refreshes” it (Ress & Mészáros, 1998; Sari & Mészáros, 2000). Here we do not consider a specific energy injection mechanism and generally assume that the isotropic equivalent blastwave energy evolves as

 E(t)=⎧⎨⎩Ef(ttf)1−e,ti⩽t

where is the time when the assumed PL energy injection ( begins, is the end time of the energy injection, is the final blastwave energy, and is required for an effective energy injection. When this energy injection is taken into account, the blastwave energy in Equations (7)-(10) should be replaced with Equation (18).

According to Equations (7)-(10), the relevant spectral regimes and flux densities are:

(i) ,

 Fν=Fν,max(ννm)−p1−12∝t[(1−e)−(2+e)(p1+p1q−2q)8]. (19)

(ii) ,

 Fν = Fν,max(νbνm)−p1−12(ννb)−p2−12 (20) ∝ t[(1−e)−(2+e)(p2+p2q−2q)8].

According to Equation (9), the injection break frequency scales as

 νb∝t−(2+e)(1+q)4. (21)

After the end of the energy injection (), the blastwave enters an adiabatic evolution phase and the corresponding scaling relationships can be easily obtained by setting in Equations (19)–(21).

We next discuss the physical origin of the jet break of GRB 060614. For a simplified conical jet with a half-opening angle , as it decelerates, the radiation beaming angle () would eventually exceed the jet half-opening angle, i.e., . At this time, a jet break may occur in the afterglow LC. Two effects could result in a jet break: the first is the pure jet-edge effect which steepens the LC by for an ISM medium (Mészáros & Rees, 1999); the second effect is caused by sideways expansion, which has important effects on the hydrodynamics when is satisfied and the post-jet-break flux decays as for a normal electron energy spectrum with index (Rhoads, 1999; Sari et al., 1999).

For GRB 060614, the jet break should be a result of significant sideways expansion rather than the jet-edge effect. The reasons are as follows: (i) The post-jet-break decay (in the optical/UV band) caused by the edge effect would have a slope , which is substantially lower than the observed value (); (ii) Using the expression (their Equation (11)) given by Wang et al. (2012) who considered the effect of sideways expansion in a similar DPLH model and the obtained parameter values ( and ) in Subsection 4.1, we estimate the post-jet-break slope to be which is excellently consistent with the observed value.

Based on the work of Wang et al. (2012), we give the scaling law for in the post-jet-break phase straightforwardly. For ,

 Fν∝t−q(p2−2)+(p2+2)2,   t>tj, (22)

where is the jet-break time.

## 4 Parameter Constraint and Afterglow Modeling

### 4.1 Grb 060614

Before constraining the free parameters (, , , , , , , and ), we first summarize the relevant observational results of GRB 060614: (i) , ; (ii) ; ; (iii)  ks,  ks; (iv)  Hz555Here and below we use to denote the observed break frequency in the SEDs, in order to distinguish with the injection break frequency in our model.; (v) , since the SED shows that the break frequency has just crossed the R-band at about 30 ks; (vi) the initial decay slope of the -band LC ; (vii) . In this section we use conditions (i)–(iv) to constrain the model parameters, and use (v)–(vii) for consistency checks.

Using condition (i), we get and . The values of and can be obtained from condition (ii) and Equation (20), i.e.,

 (2+e)(p2+p2q−2q)8−(1−e) = 0.11±0.03, (23) 3(p2+p2q−2q)8 = 1.11±0.03. (24)

Solving these equations gives and . With these values, we test our model predictions with conditions (v)–(vii). First, Equation (21) gives during the energy injection phase, then, with condition (iv) we have , which is excellently consistent with condition (v). Second, based on Equation (19), the predicted initial -band decay slope is that is consistent with the observational results (condition (vi)) within 1  errors. Finally, we estimate the post-jet-break decay slope from Equation (22) and the obtained value is , which is in perfect accord with that of the optical/UV afterglow, and marginally consistent with that of the X-ray afterglow. These exciting results encourage us to have a further check of our model by modeling the afterglow LCs. In the following calculations, we adopt , , and .

In the normal decay phase (), we have . Following Equations (7)–(10) and (20), one derives666 and were adopted in the derivations according to Wijers & Galama (1999) and our obtained . The same values were used to calculate and for GRB 060908 in Subsection 4.2.

 νm = 1.1×1010ξ−1.334ϵ3.33e,−1ϵ1/2B,−2E0.6f,52n−0.060t−1.8dHz, (25) νb = 1.0×1015ξ24ϵ1/2B,−2E0.35f,52n0.150t−1.06d Hz, (26) νc = 1.4×1015ϵ−3/2B,−2E−1/2f,52n−10t−1/2d Hz, (27) FνR = 7.5×103ξ0.684ϵe,−1ϵ0.92B,−2E1.37f,52n0.560t−1.11dμJy. (28)

To constrain the parameters, we require that (i) the -band flux at 52 ks is 777This value has been corrected for Galactic and host galaxy extinction with and , respectively, according to the results of M07., (ii)  Hz, and (iii) should well above 10 keV at the last measurement of the X-ray afterglow, i.e., . After a simple calculation, we get

 ϵB,−2n2/30 = 1.95×10−3ϵ−4/3e,−1E−5/3f,52, (29) ξ4 = 1.5ϵ−1/3e,−1E0.24f,52n0.090, (30) ϵe,−1 > 0.84E−1f,52. (31)

With only two equations, the model parameters (, , , and ) are strongly degenerate. Here we adopt a typical value of , which has been supported by recent large sample afterglow modelings (e.g., Nava et al., 2014; Santana et al., 2014; Beniamini & van der Horst, 2017). is the final blastwave energy after the energy injection, of which the mechanism was not specified above. Here we simply assume an equivalent prompt emission efficiency of and leave the discussion on the energy injection mechanism in Section 5. With  erg and , we obtain and Equation (31) is naturally satisfied. By substituting these values in Equation (29), we get . The values of and cannot be well constrained since both of them are highly uncertain parameters and vary over several orders of magnitude. By modeling the multi-band afterglows of 38 short GRBs, Fong et al. (2015) gave a median density of  cm , and found that 80%–95% of bursts have densities of  cm. For GRB 060614, if we take to , we get to . These values are well consistent with the recent results of Santana et al. (2014) and Barniol Duran (2014), who found the distribution of has a range of with a median value of a few . In the following calculations, we adopt and . Finally, we substitute the above values in Equation (30) and get . We note is weakly dependent on other parameters and can be well constrained; it is around , varying within a factor of two.

Since we interpret the achromatic break at as a jet break, we can estimate the half-opening angle of the jet according to (Rhoads, 1999; Sari et al., 1999). We thus have

 θj=9.4\arcdeg E−1/852n1/80(tb,2,d1+z)3/8=5.1\arcdeg. (32)

Using and (Rhoads, 1999), we have , which suggests a mildly relativistic jet even at the end of the X-ray observations. Therefore, our explanation of the entire afterglow of GRB 060614 in the highly relativistic regime is self-consistent.

Based on Equations (19), (20), (22), (25)–(28) and our obtained parameters, we can now compare our model with the multi-band afterglow LCs. As shown in Figure 1, the whole optical/UV and X-ray (except the last few data points) LCs can be well described with our model888The initial steep decay of the X-ray LC before about 500 s is likely the prompt emission tail (M07) which is not a concern of our model.. Especially in the optical/UV band, our model successfully explained the initial frequency-dependent decay feature and the corresponding spectral evolution. Besides the two achromatic breaks and , there is an chromatic break in the optical/UV LCs. It denotes the time that crosses an observational frequency . For , the corresponding breaks are  ks. The optical/UV LCs show a plateau between and with the same slope as the X-ray plateau; before , the LCs rise with a slope of (0.32). It should be noted that an exact calculation of afterglow radiation would give smooth spectral and temporal breaks (Granot & Sari, 2002), so such a chromatic break in the optical/UV LCs may not be clearly seen, especially when the data are sparsely sampled. Instead, the passage of through the optical/UV band may show an average effect in the LCs: slowly rising at low frequencies and flattening at higher energies, just like the afterglow of GRB 060614 (M07). We emphasize, however, that our simple analytic model perfectly described this feature and no need to employ complicated numerical calculations. For the X-ray afterglow, we note that the data points after  s obviously deviate from our modeling fit and suggest a late re-brightening or a flattening. M07 found that at the end the observations have small signal to noise ratios and approach the XRT sensitivity limit. We thus do not consider this inconsistency. There are also slight excesses between and  s, this is because in our modeling we used the central value of 2.48 for the post-jet-break slope. When the uncertainty of this parameter is considered, this problem would be alleviated.

We conclude this subsection by comparing the value of GRB 060614 with that of GRB 091127. For GRB 091127, (Paper I), while GRB 060614 gives . These results imply a similar evolution behavior of the injection break, with . However, at this stage it is premature to say that this represents a universal law of the injection break and more such events are needed to test this conjecture. We emphasize that the values of for both bursts are reliable, since the consistency checks have been performed with various afterglow observational constraints. Finally, we emphasize that this DPLH spectrum predicts an injection break frequency evolving as (for ), which is substantially faster than in a single PL hard electron spectrum model. Therefore, when this kind of spectral break along with flat spectra in the optical band is observed in afterglows, it provides strong support to the above conjecture.

### 4.2 Grb 060908

The parameters to be constrained are , , , , , , and . The observed spectral indices require , then we have and . The value of can in principle be determined by the observed X-ray decay index . According to Equation (28), we get . It is not strange that is badly constrained, since has a very weak dependence on and it is mainly determined by which has large uncertainties. Based on the results of Subsection 4.1, we assume for GRB 060809 and test whether it is consistent with other observational properties. With this value of and , we obtain . Given that and obtained from the spectral indices have relatively large uncertainties, for simplicity we adopt and in the following calculations.

To calculate the flux density , we should first determine the order between and . Below we give some arguments: (i) the spectral analysis of Covino et al. (2010) requires  Hz, with we have  keV; (ii) at the end of the X-ray observations, should not have crossed the X-ray band. We simply require  keV, with we get  Hz. That is, at the beginning of the observations ( s), should be near the low-end of the XRT band, while should be near the high-end of the ultraviolet band, i.e., . As decreases and increases, the spectrum transits to and eventually becomes .

Since we have throughout the observations, the corresponding electrons may suffer from significant inverse Compton losses, especially when has very small values. When the synchrotron self-Compton (SSC) effect is considered, the cooling frequency would be reduced by a factor of and the X-ray flux would be suppressed by , here is the Compton parameter (Sari & Esin, 2001). With the adopted parameters, we derive the break frequencies according to Equations (11)–(13) and replace with , i.e.

 νm = 3.4×107ξ−1.334ϵ3.33e,−1ϵ1/2B,−4E0.6752A−0.17∗t−1.67dHz, (33) νb = 7.8×1013ξ24ϵ1/2B,−4E0.2552A0.25∗t−1.25d Hz, (34) νc = 3.9×1016ϵ−3/2B,−4E1/252A−2∗t1/2d(1+Y)−2 Hz. (35)

The Compton parameter can be estimated as follows (RB08). For ,

 Y ≈ νICbFICνbνbFνb (36) = 2γbγcζ(γmγb)p1−1 = 670ϵe,−1ϵ−1B,−4(1+Y)−1,

where , and are the synchrotron and SSC flux at , respectively, and . We note that is only a simply function of and . As long as , we have . Small values of are required for GeV-detected bursts if the GeV emission arises from external shocks (e.g., Kumar & Barniol Duran, 2009, 2010; Beniamini et al., 2015), and are also supported by recent systematic studies using X-ray/optical (Santana et al., 2014) or radio (Barniol Duran, 2014) afterglow observations. Such small values of imply a large and significant SSC losses, which have important effects on the derived blastwave energy and thus the prompt emission efficiency (Beniamini et al., 2015, 2016). With this consideration, Equation (36) can be written as

 Y≈25.9 ϵ1/2e,−1ϵ−1/2B,−4. (37)

For ,

 Y ≈ νICcFICνcνcFνc (38) = 2γ2cζ(γmγb)p1−1(γbγc)p2−1 ∝ ζγ