Coasting external shock in wind medium: an origin for the X-ray plateau decay component in Swift GRB afterglows
aa r X i v : . [ a s t r o - ph . H E ] S e p to appear in ApJ Preprint typeset using L A TEX style emulateapj v. 11/10/09
COASTING EXTERNAL SHOCK IN WIND MEDIUM: AN ORIGIN FOR THE X-RAY PLATEAU DECAYCOMPONENT IN
SWIFT
GRB AFTERGLOWS
Rongfeng Shen and Christopher D. Matzner
Department of Astronomy & Astrophysics, University of Toronto, M5S 3H4, Canada to appear in ApJ
ABSTRACTThe plateaus observed in about one half of the early X-ray afterglows are the most puzzling feature ingamma-ray bursts (GRBs) detected by
Swift . By analyzing the temporal and spectral indices of a largeX-ray plateau sample, we find that 55% can be explained by external, forward shock synchrotron emis-sion produced by a relativistic ejecta coasting in a ρ ∝ r − , wind-like medium; no energy injection intothe shock is needed. After the ejecta collects enough medium and transitions to the adiabatic, deceler-ating blastwave phase, it produces the post-plateau decay. For those bursts consistent with this model,we find an upper-limit for the initial Lorentz factor of the ejecta, Γ ≤
46 ( ǫ e / . − . ( ǫ B / . . ;the isotropic equivalent total ejecta energy is E iso ∼ ( ǫ e / . − . ( ǫ B / . − . ( t b / s) erg,where ǫ e and ǫ B are the fractions of the total energy at the shock downstream that are carried byelectrons and the magnetic field, respectively, and t b is the end of the plateau. Our finding supportsWolf–Rayet stars as the progenitor stars of some GRBs. It raises intriguing questions about the originof an intermediate-Γ ejecta, which we speculate is connected to the GRB jet emergence from its hoststar. For the remaining 45% of the sample, the post-plateau decline is too rapid to be explained inthe coasting-in-wind model, and energy injection appears to be required. Subject headings: gamma-ray burst: general – radiation mechanisms: non-thermal – relativistic pro-cesses – shock waves – X-rays: bursts INTRODUCTION
Among the new features in early X-ray afterglows re-cently discovered by
Swift , the X-ray plateau is the mosttroublesome to explain; see Zhang 2007 for a review.There is a long list of proposed explanations, including aprior emission model (Ioka et al. 2006; Yamazaki 2009),an evolving microphysical parameter model (Panaitescuet al. 2006; Ioka et al. 2006), a two-component (Granotet al. 2006) or multi-component (Toma et al. 2006) jetmodel, a slow energy transfer from the magnetized ejectato the ambient medium (Kobayashi & Zhang 2007), areverse-shock dominated afterglow model (Uhm & Be-loborodov 2007; Genet et al. 2007), a circumburst-dust-scattered echo of prompt X-rays (Shao & Dai 2007; butsee Shen et al. 2009), and a scattered internal shock(Shen, Barniol Duran & Kumar 2008) or external shockemission (Panaitescu 2008) model. Most fail to explainall the properties of the X-ray plateau. The leadingmodel invokes energy injection to refresh the externalshock. In this scenario the injected energy comes eitherfrom a long-lasting central engine activity (e.g., Dai & Lu1998a, b; Zhang & M´esz´aros 2001; Dai 2004; Yu & Dai2007) or from a slower portion of an outflow with a broadinitial Lorentz factor (LF) distribution (e.g., Granot &Kumar 2006). However, the standard refreshed shockmodel faces several serious issues. First, the flat slope( ∼ t − . ) and late ending time ( ∼ s) of the plateaufeature imply that the total energy carried by the later orthe slower ejecta is much larger than that of the prompt,fast component which gives rise to γ -rays. Second, the [email protected]@astro.utoronto.ca abrupt end to the plateau phase poses a serious theo-retical challenge to models for the origin of the injectedenergy. The late-activity scenario also suffers from thefact that prompt γ -rays, which are comparable in energyto the late-time injection, emerge so much earlier. Theimplied radiative efficiency can exceed 90% (e.g., Ioka etal. 2006; Zhang et al. 2007), much higher than the ∼ ± pairdominated outflow, such as a dissipated wind from a mil-lisecond magnetar (Dai 2004; Yu & Dai 2007; Yu, Liu &Dai 2007; Mao et al. 2010), the wind-decelerating rela-tivistic reverse shock radiates more efficiently than in thecase of a baryon-dominated injecting outflow and mighteven dominate over the forward shock emission; this al-leviates the energy budget and the required efficiency. Inthat model, the X-ray plateau ending time correspondsto the spin-down time of the magnetar.Some afterglow features under the fireball shock modelin a ρ ∝ r − k medium, where k is a constant, have beendescribed earlier by Dai & Lu (1998c) and M´esz´aros, Rees& Wijers (1998). Chevalier & Li (1999, 2000) have stud-ied specific behavior of afterglow due to the blastwaveexpanding into the wind of a Wolf–Rayet star, corre-sponding to k = 2. Panaitescu & Kumar (2000) stud-ied the afterglow light curves both in a constant den-sity ( k = 0) interstellar medium (ISM) and in a windmedium. Models of late-time ( t & s) afterglow data(e.g., Panaitescu & Kumar 2002; Schulze et al. 2011;Oates et al. 2011) have inferred that afterglows consis-tent with ISM are more numerous than those consistentwith a wind medium.In the external shock model for gamma-ray burst Shen & Matzner(GRB) afterglows, relativistic ejecta coasts freely beforeit collects 1 / Γ j times its rest mass from the circum-burstmedium and decelerates, where Γ j is its initial LF. Inthe past, afterglow studies have concentrated mainly onthe decelerating phase, in which the flatness of the X-rayplateau decay slope is difficult to explain. In this paperwe look at the early, coasting phase. If the medium den-sity is constant with radius, the light curve in this phaseshould rise as t − , simply because the total number ofemitting particles increases steeply with time. A featureconsistent with such a rise was only detected for a fewcases (e.g., Molinari et al. 2007). Such a rise may of-ten be difficult to detect if it finishes so early that it isburied below the prompt emission tail, which is dimmingbut still brighter than the rising afterglow.In a wind medium, however, the external shock lightcurve in the coasting phase is flat or slowly decaying, ifthe observing frequency is above both the synchrotroninjection frequency and the cooling frequency (see § Swift era. In this paperwe investigate systematically the scenario in which the
Swift -observed X-ray plateau is produced in the coastingphase of the ejecta plunging into a wind medium, andthe ‘normal’ decay following the plateau corresponds tothe subsequent decelerating phase of the ejecta in thesame wind medium; we refer to this as the “coasting-in-wind” model. We assume the synchrotron emission ofthe external forward shock is the dominating emissioncomponent.We start with reviewing in § § § § EMISSION FROM EXTERNAL FORWARD SHOCK IN AGENERALIZED MEDIUM
In this work we use the simplest self-consistent assump-tions and approximations, including assumptions that allelectrons are accelerated into a single power-law distri-bution, and that the magnetic and electron energy den-sities are fixed fractions of the post-shock energy density.Simulations of relativistic shock acceleration (Spitkovsky2008; Sironi & Spitkovsky 2009, 2011) show that only asmall fraction ( ∼ ∼
10% ofdownstream energy. This can be modeled using a fixed ǫ e (defined below), unless in reality either the numberfraction or the energy fraction of non-thermal electronschanges strongly with the shock LF (e.g., see § p . Thosenon-thermal electrons carry a small fraction, ǫ e , of theinternal energy downstream. The minimum electron en-ergy in this power law is γ m = f p ǫ e Γ µ e m p /m e , for 2 < p, (cid:16) f p ǫ e Γ γ p − M µ e m p /m e (cid:17) / ( p − , for 1 < p < , (1)where µ e is the number of nucleons per electron, f p = | p − | / ( p −
1) derives from integrating the number densityover the power law (e.g., Sari et al. 1998), and for thesecond case γ M = p e/ ( σ T B ) is the maximum electronenergy in the power law (Dai & Cheng 2001). Here σ T isthe Thomson cross section, e is the electron charge and B is the magnetic field, whose strength B = p πǫ B Γ ρc is determined by the fraction, ǫ B , of the downstream en-ergy which goes to the magnetic field. The acceleratedelectrons emit synchrotron radiation. Another criticalelectron energy is the cooling energy above which elec-trons lose significant energy in the dynamical time: γ c = 6 πm e c (1 + z )(1 + Y ) σ T Γ B t , (2)where z is the GRB host red shift and Y is the syn-chrotron self Compton (SSC) parameter; we neglect theSSC loss, i.e., Y = 0. Above γ c , cooling modifies thepower law slope of electrons distribution. Radiativelosses are necessarily important when p <
2, because inthis case the electron energy is concentrated at very highenergies above γ c ; conversely when p > γ c < γ m .For the dynamical model, we consider a relativisticejecta of isotropic equivalent energy E iso and LF Γ j plunging into the circum-burst medium. The mediumdensity profile is generally assumed to be a power law ρ = Ar − k , where for the steady stellar wind case, A k =2 = ˙ M w / (4 πv w ) = 5 × A ∗ g cm − (Chevalier& Li 1999); the fiducial value A ∗ = 1 would arise for awind of mass loss rate ˙ M w = 10 − M ⊙ yr − and speed v w = 10 km s − . In the case of a uniform density (‘ISM’)medium, A k =0 = ρ .A pair of shocks is produced, with the forward shock(LF Γ sh ) moving into the medium and the reverse shockinto the ejecta. The shocked fluid region comprisesthe forward-shocked medium and reverse-shocked ejecta,which have approximately the same LF Γ( t ) ≈ Γ sh / √ t ) isgiven by pressure balance at the discontinuity as Γ =Γ j / (1 + 2Γ j /a / ) / , where a = ρ ′ j /ρ ( r ) is the rest-frame density ratio between the ejecta and the ambientmedium (e.g., Panaitescu & Kumar 2004b). In the limitof a ≫ Γ j , Γ ≃ Γ j . It can be shown that, before the de-celeration, i.e., the reverse shock passage, of the ejecta,the dense shell condition a ≫ Γ j is satisfied. In thecoasting-in-wind model Γ j takes a single value rather The ambient density is ρ ( r ) = (3 − k ) M ( r ) / (4 πr ), and therest-frame jet density is ρ ′ j = M j / (4 πr ∆ ′ j ), where M j is theisotropic ejecta mass and ∆ ′ j is the ejecta width in the rest frame. oasting-in-Wind Model for GRB X-Ray Plateaus 3than a distribution, so that Γ( t ) has a definite initialvalue Γ ≃ Γ j .Right after the reverse shock passage, about half of theinitial ejecta total energy is transferred to the shockedmedium and ejecta. The combined shocked region entersthe Blandford & McKee (1976) self-similar solution, inwhich Γ( t ) declines. For convenience, we denote phase 1as the period in which the reverse shock is crossing theejecta, and phase 2 the later period of blastwave deceler-ation. In phase 1, since usually the reverse shock is tooweak to account for the X-ray emission, we consider theforward-shocked region only whenever radiation proper-ties are concerned.In phase 2 the shock compresses the density and en-hances the energy per unit mass both by a factor of ∼ Γ;energy conservation requires (Cohen et al. 1998; also seeWu et al. 2005 for similar treatment of radiative loss) E iso = 8 πA Γ r − k c − k + ǫ (cid:16) − ǫ (cid:17) (3)where, to account for synchrotron losses, ǫ ≃ (cid:26) , if p > γ m < γ c ( ǫ e + ǫ e ) , otherwise (4)is the fraction of post-shock energy, if any, radiated away;approximation (4) matches the more complicated exactsolution by Cohen et al (1998) to within 5%. If p > γ m < γ c so that E iso is conserved in phase 2, Γ ∝ r − (3 − k ) / ∝ t − (3 − k ) / (8 − k ) ; otherwise Γ ∝ r − m where(Cohen et al. 1998) m = (1 + ǫ ) + 3(1 + ǫ )(4 − k ) − − ǫ (5)which in the adiabatic limit becomes m ǫ =0 = 3 − k . Inany phase of continuous motion with Γ ∝ r − m , the ob-server’s time t is given by t = 1 + z m rc t Γ c −→ z − k rc t Γ c , (6)where the arrow evaluates m → (3 − k ) for a deceler-ating, adiabatic blastwave. For adiabatic blastwaves,we find Γ ∝ r − / ∝ t − / in a wind medium where t = (1 + z ) r/ (2 c t Γ c ), and Γ ∝ r − / ∝ t − / in a uni-form medium where t = (1 + z ) r/ (4 c t Γ c ). The prefactor c t = 4 if most of the observed emission is from a thin layerof shocked fluid right behind the shock front and on theline-of-sight axis (Sari 1997; Dai & Lu 1998c; Chevalier& Li 2000). Emission from off axis and from the fluidfurther downstream arrives somewhat later, so the effec-tive c t is somewhat smaller (Waxman 1997; Sari 1998;Panaitescu & Mesazaros 1998). We adopt c t = 2.The observed synchrotron frequency for electrons withenergy γ , averaged over an isotropic distribution ofthe electron’s pitch angle, is ν = 3Γ γ eB/ [16 m e c (1 + For a late deceleration time, which is appropriate for our coasting-in-wind model, the ejecta width is determined by radial spreading,so ∆ ′ j ∼ r/ Γ j . Thus a = ρ ′ j /ρ ( r ) ∼ Γ j M j /M ( r ). Prior to decel-eration, M j / Γ j ≫ M ( r ) (cf. Equation 3), so a ≫ Γ j ; however a ∼ Γ j at deceleration. z )]. The critical frequencies corresponding to γ m and γ c are ν m and ν c , respectively. The peak syn-chrotron specific power for a single electron is P ν, max =8 m e c σ T Γ B/ (9 πe ), independent of the electron energy.The peak flux density in the observed F ν spectrumis F ν, max = N P ν, max (1 + z ) / (4 πd L ), where N =4 πAr − k / [(3 − k ) m p ] is the total number of electrons inthe swept-up medium, and d L is the luminosity distance.Given the above formulae, we can find the followingtime evolution laws for the case of p > ν m ∝ t − k/ , ν c ∝ t k/ − , F ν, max ∝ t − k/ (7)for phase 1, and ν m ∝ t − k +4 m m +1) , ν c ∝ t k − m +1) , F ν, max ∝ t − k − m m +1) (8)for phase 2. For 1 < p <
2, the only difference is on ν m : ν m ∝ t − k p − in phase 1, and ν m ∝ t − m ( p +2)+ k m +1)( p − inphase 2. Note that for wind medium, ν m decreases and ν c increases monotonically with time in both phases.Following from the above evolution laws the observedflux density F ν evolves according to F ν ( t ) ∝ ν − β t − α (e.g., Sari, Piran & Narayan 1998). The spectral slope β is determined by p and the relative locations of ν , ν m and ν c . Therefore, a specific numerical relation between α and β , the so-called “closure relation”, exists depend-ing on which type of medium and which dynamical phasethe external shock is in. We refer the reader to Zhang& M´esz´aros (2004) for a summary of these relations inthe decelerating phase for both uniform and wind media.For demonstration purpose, we list in Table 1 the closurerelations for both the coasting phase and the deceleratingphase in a k = 2 wind medium. SAMPLE AND DATA ANALYSIS
We use the data from a sample of 53 X-ray plateauscompiled by Liang et al. (2007). This sample is providedwith fitted spectral and temporal slopes of the plateaucomponent, β and α , and of the normal decay compo-nent following the plateau, β and α , and the time of thetransition between the two components t b . We omit fourbursts whose phase 2 decay slopes are too steep ( α ≈ α ≈ α and β against the predicted closure relations for different dy-namical, medium and spectral models. It can be seenthat the coasting-in-wind model with max( ν m , ν c ) < ν X most successfully accommodates the majority of data.Within this model the post-plateau decay correspondsto the decelerating phase (phase 2) of the blast wave ina wind medium; the break cannot be due to a changeof spectral regime because there is no spectral evolution( β = β ) in all but one burst — GRB 061202 (e.g.,Liang et al. 2007; Shen et al. 2009). The lack of spectralevolution also signals a requirement for any model, that Shen & Matzner TABLE 1The spectral and temporal indices of the forward shock synchrotron emission ( F ν ∝ ν − β t − α ) in the coasting-in-wind modelfor the wind density index k = 2 . Spectral regime β α ( p > α ( p < ν m < ν c ) ν < ν m − − − p − − − p )12( p +1) ν m < ν < ν c p − p −
12 3 p −
14 12 p +88 Fast cooling ( ν c < ν m ) ν < ν c −
13 13 23 13 23 ν c < ν < ν m −
12 14 −
12 14 max( ν m , ν c ) < ν p p −
22 3 p − p +68 Note . — Phases 1 and 2 are the coasting and decelerating phases of the blast wave, respectively. β X,1 α X , k = . k = k = . Coasting in Wind max( ν m , ν c ) < ν X Coasting in Wind ν m < ν X < ν c Decelerating in Wind ν m < ν X < ν c Decelerating in ISM ν m < ν X < ν c Fig. 1.—
X-ray decay index vs. spectral index during the plateauphase. The lines are the closure relations predicted by variousmodels. The prediction from the coasting-in-wind model withmax( ν m , ν c ) < ν ( thick solid line) is the most capable one thataccounts the data. The predictions from the same model withslight variations in k ( thin solid lines) are also plotted. The datain open circles (magenta color) are those whose post-plateau de-cays are inconsistent with the decelerating phase (phase 2) of thecoasting-in-wind model (see Figure 2 and in the text). GRB 061202that shows a strong spectral evolution is plotted as a diamond. is, whichever spectral regime phase 1 is in, it remains inphase 2. We therefore expect a specific relation between α and β depending on the relative locations of ν X , ν m and ν c .Figure 2 plots the observed α vs. β with pre-dicted relations superimposed. For the deceleratingphase in a wind medium, we see that the spectral regimemax( ν m , ν c ) < ν X is the most consistent with the data,which also satisfies the requirement of zero spectral evo-lution. However, as in Figure 1, the data in Figure 2show broad scatter around the prediction of this model.While the scatter in the former figure could be attributedto a slight variation of the density index k from burstto burst, the scatter in the latter cannot — because α has no k -dependence in the max( ν m , ν c ) < ν X spectral β X,2 α X , k = . k = . k = k = GRB 061202
Decelerating in Wind max( ν m , ν c ) < ν X Decelerating in Wind ν m < ν X < ν c Decelerating in ISM ν m < ν X < ν c Fig. 2.—
X-ray temporal index vs. spectral index during thepost-plateau phase. The solid lines are for the decelerating phase ofthe coasting-in-wind model. Note that for a decelerating blastwavewith max( ν m , ν c ) < ν and p > α is independent of k . The datain open circles (magenta color) are for the inconsistent sub-sample.The thick dotted line (blue color) is the prediction for phase 2 of thecoasting-in-wind model considering the radiative loss of blastwaveenergy (requiring ν c < ν m when p >
2, and assuming ǫ e = 0 . α X, for a given β X, withinthis model because of the high ǫ e value assumed. The exceptionalburst that shows a spectral evolution is labeled. regime. This is a puzzle for all decelerating blast wavemodels that do not invoke late energy injection.In conclusion, we find that the coasting-in-wind modelfor the plateau, and its subsequent (adiabatic) decelerat-ing stage for the post-plateau decay, can account for 55%of the sample. We call these bursts the consistent sub-sample , and they are plotted as black data points in Fig-ures 1 and 2. GRB 061202 is consistent with the closurerelation predictions of the coasting-in-wind model bothduring and after the plateau phase, but its strong soft-ening evolution from β = 1 . ± .
07 to β = 2 . ± . ν m < ν X < ν c → max( ν m , ν c ) < ν X , nor by anyvariant of the coasting-in-wind model; we therefore ex-oasting-in-Wind Model for GRB X-Ray Plateaus 5clude it from the consistent sub-sample. The remainingof the sample, i.e., those located above the coasting-in-wind model prediction by at least 2 σ in Figure 2, aremarked as open circles (in magenta color) both in Fig-ure 2 and Figure 1. We call these the inconsistent sub-sample . We see in Figure 1 that the inconsistent sub-sample shows no significant difference from the consis-tent sub-sample in the α vs. β scatter plot. Figure 3shows the distribution of k derived from the consistentsub-sample in phase 1, which is centered around k = 2.At face value, this finding seems to contrast with pre-vious studies (e.g., Panaitescu & Kumar 2002; Schulzeet al. 2011;Oates et al. 2011) of late-time ( t & s)afterglow data, which concluded that GRBs with k ≃ k ≃
2. However, the dif-ference should not come as a surprise. Our consistentsub-sample includes only those bursts consistent with themax( ν m , ν c ) < ν X spectral regime — as required by ourmodeling of the plateau data and by the lack of spectralevolution. Since k cannot be determined from the modelprediction in phase 2 for this spectral regime (solid linein Figure 2), we infer k instead from plateau data. Ourinferred k values, based as they are on the assumptionof a coasting ejecta, are model dependent: we would ob-tain very different values within a model involving latecentral engine activity.As for the inconsistent sub-sample, interpreting theseunder the ν m < ν X < ν c spectral regime indicates thatmany are closer to k = 0 than k = 2. Two bursts,050801 and 060714, appear in both the Schulze et al.(2011) sample and our consistent sub-sample; Schulze etal. classify both as k = 0. Such overlap is expected,as we describe a burst as ‘consistent’, and derive its k from the coasting-in-wind model, if it is within 2 σ of themax( ν m , ν c ) < ν X locus in phase 2. Bursts at the upperperiphery of this sub-sample could also be interpretedassuming ν m < ν X < ν c , and the derived k would bevery different. Note also that our values for the twobursts’ post-plateau indices, from Liang et al. (2007),differ slightly from those used by Schulze et al., and thelatter authors treat a burst as consistent if it falls within3 σ of a model, whereas our threshold is 2 σ .Allowing for the radiative loss of blastwave energy inphase 2 can help to increase the size of the consistentsub-sample, but not by much. For instance, in the mostradiative case — ǫ e = 0 .
3, and ν c < ν m for p > α X, in Figure 2 is shifted upwardby ≈ .
2. That freedom could at most increase the con-sistent sub-sample size to about 2/3 of the total.To account for the inconsistent sub-sample, one mustinvoke the energy refreshment in the decelerating blastwave. That is the topic of the next subsection.
Refreshed shock models
There are two primary scenarios in which the blastwave can be refreshed . A third scenario was suggested by Kobayashi & Zhang (2007),in which the ejection is brief but is dominated by the Poyntingflux. The magnetic energy in the ejecta is not transferred to theshocked ambient medium until the ceasing of the reverse shock.The delayed transfer of the magnetic energy serves as a varied k N Fig. 3.—
Distribution of the derived density power-law index k for the sub-sample that are consistent with the coasting-in-windmodel in both its phase 1 and phase 2 – the consistent sub-sample.The top left horizontal bar shows the size of the typical error of k .
1. The central engine activity is long lasting (Zhang& M´esz´aros 2001) with an outflow luminosity history L ( t ) ∝ t − q . Since the LF of the outflow is always muchlarger than that of the blast wave, this is also the rate atwhich the total energy of the blast wave gets replenished,i.e., E iso ( t ) ∝ t − q . In order for the late energy injectionto be important, q < j such that outer shells have larger Γ j andmoreover the ejecta mass (or even energy) is dominatedby the inner, slower shells, i.e., the mass of ejecta is apower law in Γ j (Rees & M´esz´aros 1998; Panaitescu,M´esz´aros & Rees 1998; Sari & M´esz´aros 2000; Ramirez-Ruiz, Merloni & Rees 2001): M ( > Γ j ) ∝ Γ − sj . Thisextended ejecta distribution could be the outcome of aGRB jet breaking out of the progenitor star surface or theinternal shocks during the prompt emission phase. Thecoasting-in-wind model can be considered the s → ∞ limit of this class, in the sense that the ejecta have asingle Γ j .In both scenarios the blast wave energy is replenishedat some certain rate, so as to cause a flattened plateautemporal slope, and t b corresponds to the ending of en-ergy injection; the post-plateau decay is the standarddecelerating blast wave phase. For any given α anda medium type, there is a one-to-one correspondencerelation between values of q and s for which the for-ward shock evolution is identical (although the reverseshock is not), so it is not possible to distinguish betweenthe two scenarios on the basis of X-ray data alone. Wefind that the inconsistent sub-sample can be accommo-dated within these models in either a wind or a uniform version of the two scenarios mentioned here for the refreshed shockmodel. However, the process of the magnetic energy transfer inthe shocked region is poorly understood. Fan & Piran (2006) andMimica et al. (2009) claim that the time scale for the transfer is atmost several times the light crossing time of the ejecta, too shortto account for the plateau feature. Shen & Matzner −2 −1 0 112345678 q N ν m < ν X < ν c in ISM ν m < ν X < ν c in Wind −10 −5 0 5 1024681012 s N ν m < ν X < ν c in ISM ν m < ν X < ν c in Wind Fig. 4.—
Distribution of the derived q ( top ) and s ( bottom ) forthe inconsistent sub-sample that require a refreshed shock for in-terpretation, where q is the late central engine luminosity temporalindex and s is the catching-up ejecta mass distribution index (seesection 3.1). The spectral regime ν m < ν X < ν c is required by thedata in phase 2. For a refreshed shock in a wind medium, the datarequire unreasonable conditions, i.e., q < s <
0, for the twoshock refreshment scenarios, respectively. Therefore, ISM is theonly allowed medium type when a refreshed shock is considered. medium, so long as the observed band is below the cool-ing break ( ν m < ν < ν c ). The lack of spectral evolutionimplies that this is true in the plateau phase as well.With energy injection a new closure relation exists inphase 1 among α , β and q (or s ), within a given ambi-ent medium (Zhang et al. 2006). From this we can inferthe required q (or s ) value for each burst in the inconsis-tent sub-sample. The resultant distributions of q and s are shown in Figure 4.Figure 4 shows that within a wind medium the requiredparameters are unreasonable, in the sense that q < s < q -distribution peaks at q ∼ . s -distribution peaks at s ∼
3. Thismeans that in these two refreshed shock scenarios, re-spectively, the outflow total kinetic energy is dominatedeither by the late ejecta or by the slower, massive ejecta.
Optical data
A minority (13) of the original X-ray plateau sample inLiang et al. (2007) have simultaneous optical monitoringdata; six of them are within our consistent sub-sample.Comparison of the optical light curve with the X-ray lightcurve shows diversity. Three (GRB 050801, 060714, and060729) show achromatic breaks around t b , while in theother three the breaks are chromatic. This roughly equalratio of achromatic to chromatic bursts is about the samefor the total 13 X-ray plateaus with optical data in Lianget al. (2007). Among the latter three, GRB 060210shows an optical break much earlier than t b (there is nooptical coverage on t b or later), GRB 060526 shows anoptical break later than t b , and the optical light curve inGRB 050319 shows no break at all and is consistent witha single power law fit. Can our coasting-in-wind modelexplain the diverse optical behavior as well?The achromatic breaks seen in the former three burstsare consistent with the coasting-in-wind model, accord-ing to which the break at t b is of dynamical origin. Inthese three bursts the optical and X-ray temporal slopesare very similar, both before and after the break (seeFigure 2 and Table 2 of Liang et al. 2007). This impliesthe same physical origin for emission in the two wavebands, which fits well to the coasting-in-wind model.This consistency also applies to the sub-sample compiledby Panaitescu & Vestrand (2011) whose coupled opticaland X-ray light curves both show plateaus (their Figure4).An early optical break before t b , such as seen in GRB060210, can be interpreted as well in the coasting-in-windmodel by the passage of ν m or ν c across the optical bandduring phase 1; for instance, the spectral regime change ν o < ν m < ν c → ν m < ν o < ν c causes the slope steepenby ∆ α o = (3 p − / α o ≈ ν m decreases and ν c increasesin the model. This pure spectral change scenario couldalso explain those bursts in which the optical shows apeak at the beginning of or during the X-ray plateau(e.g., GRB 060607A and 061121; see Liang et al. 2010and Panaitescu & Vestrand 2011). Similarly, a ν m or ν c passage over ν o in phase 2 can explain the optical breaklater than t b , such as seen in GRB 060526.The real difficulty lies in explaining the lack of an opti-cal break around t b as seen in GRB 050319. If the end ofX-ray plateau is of a dynamical origin, as postulated inthe coasting-in-wind model, it is hard to avoid an achro-matic break at t b . Panaitescu et al. (2006) proposedthat one solution is to let ǫ e and ǫ B be functions of theblastwave LF Γ (also see Panaitescu & Kumar 2004a): ǫ e ∝ Γ − e , ǫ B ∝ Γ − b . (9)oasting-in-Wind Model for GRB X-Ray Plateaus 7We adopt this treatment here for the coasting-in-windmodel, which means in phase 1 ǫ e and ǫ B are still con-stant, but they are free to evolve with time in phase2. Note that Panaitescu et al.’s scenario uses energy in-jection for the X-ray plateau while the coasting-in-windmodel does not. In the following, we derive the conditionon e and b for the lack of optical break at t b .We first calculate the X-ray decay slope in phase 2 inthe presence of evolving ǫ e and ǫ B , since we know itsspectral regime is max( ν m , ν c ) < ν X : α X, = 14 + b
16 + ( p − (cid:18) − e − b (cid:19) . (10)Next we calculate the optical slope in phase 2. The ob-served slope α o in those single power-law optical lightcurves usually lies in the range of 0.5 – 0.8. In phase1, this slope is best consistent with the spectral regime ν m < ν o < ν c for which α o, = ( p − / ǫ e and ǫ B , α o, = 12 − b p − (cid:18) − e − b (cid:19) . (11)Then, the equality α o = α o, = α o, poses the followingcondition for the lack of an optical break: α o = 12 − b α o (cid:18) − e − b (cid:19) . (12)In the case of GRB 050319, α o ≈ .
5. The conditionbecomes 3 b + 4 e = 12 . (13)Together with the observed α X, ≈ e ≈ b ≈ ǫ e and ǫ B has to satisfy such relation as in Eq. (13). Thisis the difficulty of this scenario. Alternatively, the chro-maticity in these bursts suggests that the X-ray and op-tical afterglow emission may arise from different origins,for instance, from a two-component outflow in which alow E iso , high Γ j component decelerates very early andproduces the optical emission with a single power lawdecay (the LF of this jet component has dropped signif-icantly from its initial Γ j , which explains its dominancein the long-wavelength emission), and a high E iso , low Γ j component decelerates later at t b and is responsible forthe X-rays. CONSTRAINTS ON PARAMETERS OF THECOASTING-IN-WIND MODEL
In the previous section we examined a large sampleof X-ray plateaus and found the coasting-in-wind modelwith max( ν m , ν c ) < ν X to be consistent with the ma-jority of bursts. While this model is consistent with the data for all the sample in the plateau phase, it can ac-commodate only 55% of sample data in the post-plateauphase. In this section we aim to put constraints on themodel parameters based on the consistent sub-sample.For this model to work, the deceleration time t dec mustequal t b ; t dec is given by setting Γ( t dec ) = Γ in Eq.(3). Following from this, the blast wave isotropic en-ergy can be inferred. The synchrotron spectral regimemax( ν m , ν c ) < ν X must be justified for both phase 1 and2. Since with time ν m decreases and ν c increases mono-tonically during both phases (Eqs. 7 - 8), the spectralconstraints are: ν m ( t ) < ν X and ν c ( t ) < ν X , where t is the observed time of the earliest plateau data pointand t is that of the latest post-plateau decay data point.The last constraint is that the predicted flux density levelhas to match the observed one. Without losing general-ity, we choose to calculate the flux density at t b . Thoseconstraints are summarized as follows. • t dec = t b ∼ s. • ν m ( t = t ∼
500 s) ≤ ν X . • ν c ( t = t ∼ s) ≤ ν X . • F ν X ( t = t b ∼ s) ∼ µ Jy.The formulae to calculate the model predictions are E iso = 4 . × − k − k × − k A Γ − k (cid:18) t b z (cid:19) − k erg . (14) ν m ( t ) = × − k × − k/ f p ǫ e ǫ / B Γ − k A / t − k/ × (1 + z ) k/ − Hz , for 2 < p, . × × (4 . × − k × − k/ f p ǫ e ǫ / B A / ) / ( p − × Γ p +2 − kp − t − k p − (1 + z ) k p − − Hz , for 1 < p < , (15) ν c ( t ) = 0 . × [6(4 − k )] k/ × k − ǫ − / B A − / Γ k − × (cid:18) t z (cid:19) k/ − (1 + z ) − (cid:18) t t b (cid:19) − (3 − k )(3 k − − k ) Hz . (16) F ν X ( t b ) = . × × (3 . × ) p − × (6 × ) − ( p +2)4 k (3 − k )(1+ z ) p − D × f p − p ǫ p − e ǫ p − B A p +24 Γ ( p +2)(2 − k/ × (cid:16) t b z (cid:17) − ( p +2)4 k Jy , for 2 < p, . × × (1 . × ) p − × (6 × ) − k (3 − k )(1+ z ) p − D × f p ǫ e A Γ p − k (cid:16) t b z (cid:17) − k Jy , for 1 < p < , (17)where we have kept all model and observable parame-ters in the expressions. The flux density calculation hasincluded a correction factor ≈ . TABLE 2The sub-sample that is consistent with our coasting-in-wind model and their model parameter constraints.
GRB t t b t F ν X ( t b ) z ( ǫ e ǫ B ) max Γ , max A ∗ , min E iso (10 s) (10 s) (10 s) (10 − Jy) ( ǫ − e, − ǫ B, − ) (10 − ǫ − e, − ǫ − B, − ) (10 ǫ − e, − ǫ − B, − erg)050319 61.1 11.2 8.5 9.3 3.24 109.4 49.5 3.1 7.6050416A 2.5 1.7 26.2 23.8 0.65 3.5 45.1 7.7 5.1050713B 7.9 10.8 47.9 30.3 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · Note . — The subscript “max” means the upper limit and “min” the lower limit. In deriving those constraints, we assumed a commonvalue set p = 2 . k = 2. Observed red shifts are taken from Liang et al. (2007); for those with unknown red shift, z = 2 is assumed.A H = 71, Ω Λ = 0.73, Ω M = 0.27 universe is assumed. equal-arrival-time surface effects (Granot, Piran & Sari1999).In principle, one can calculate these predictions andwork out constraints using all the available parameters[ p , k , F ν X ( t b ), etc.]. However, the appearance of p and k in the exponents would then obscure the dependence onbasic parameters such as ǫ e and ǫ B , and make compar-isons very difficult. We choose instead to adopt the mostcommon set of values k = 2 and p = 2 . β X, = 1 . t , t , t b , F ν X ( t b ) and z .As an example for demonstration, in the following weuse all typical observable values, i.e., p = 2 . k = 2, t = 500 s, t b = 10 s, t = 10 s, ν X = 1 keV, z = 2, and F ν X ( t b ) = 1 µ Jy, to carry out the constraints. It turnsout the results using common values are consistent withthose listed in Table 2 using individual values. Adoptingthese values, the items to be constrained become E iso = 0 . × A ∗ Γ t b erg . (18) ν m ( t = 500s) = 5 . × ǫ e ǫ / B A / ∗ Γ Hz ≤ ν X , (19) ν c ( t = 10 s) = 1 . × ǫ − / B A − / ∗ Γ Hz ≤ ν X , (20) F ν X ( t b = 10 s) = 5 . × − ǫ / e ǫ / B A / ∗ Γ / Jy= 1 µ Jy , (21)Eq. (21) gives A ∗ = 7 . × ǫ − / e ǫ − / B Γ − . (22)Plugging the above into Eq. (19), we have ǫ / e ǫ / B ≤ . . (23)Various data modeling work and numerical experimentsgave various values of ǫ e and ǫ B . A fairly conservativerange is ǫ e ∼ . − .
5, and ǫ B ∼ . − .
1, respec-tively. Thus, the above constraint of ǫ e and ǫ B can beeasily met. Plugging Eq. (22) into Eq. (20) gives anupper limit on Γ :Γ ≤ ǫ − / e, − ǫ / B, − . (24)When plugged back into Eq. (22), the above implies A ∗ ≥ . × − ǫ − / e, − ǫ − / B, − . (25)Finally, plugging Eq. (22) into Eq. (18), we have E iso = 1 . × ǫ − / e, − ǫ − / B, − t b, erg . (26)oasting-in-Wind Model for GRB X-Ray Plateaus 9 E iso ( ε e,−1−14/11 ε B,−2−1/11 ergs) E γ , i s o ( e r g s ) Fig. 5.—
The total energy release in prompt γ -rays vs. the totalenergy of the ejecta inferred from X-ray data for our consistentsub-sample, both in isotropic equivalent values. Filled circles –bursts with measured z . Open circles – bursts with unknown z , forwhich z = 2 is assumed. The dotted line is E γ, iso = E iso . For properties of wind from a typical Wolf–Rayet star,the lower limit on A ∗ in Eq. (25) is easily met. Therefore,the major model parameter constraints from the aboveare Γ .
50 and E iso ∼ erg.We plot the individual E iso vs. E γ, iso — the isotropicenergy release in prompt γ -rays — for the consistentsub-sample in Figure 5. An almost linear correlationbetween the two can be seen, which is consistent withand likely derives from the observed correlation betweenthe plateau X-ray fluence and the prompt γ -ray fluence(Liang et al. 2007). It also shows that E γ, iso is compa-rable to or smaller than E iso , which alleviates the trou-blesome issue one faces in refreshed shock models of ex-tremely high γ -ray radiative efficiency ∼
90% (Zhang etal. 2007) .For all bursts in the sample, the post-plateau lightcurve does not show any further steepening break thatcan be identified as the jet break up to t . This fact setsa lower limit on the ejecta beaming angle θ j ≥ / Γ( t ) =Γ − ( t /t b ) (3 − k ) / [2(4 − k )] . For k = 2, t b ≈ s, t ≈ s and using Eq. (24), we obtain θ j ≥ . ǫ / e, − ǫ − / B, − radians — not a strong constraint. SUMMARY AND DISCUSSION
By analyzing the closure relations for a large sample ofX-ray afterglows with plateaus, we find that the plateaufeature and post-plateau decay can be explained by thecoasting-in-wind model in 55% of the sample. This sim-ple model is also able to explain the contemporaneouslyobserved optical afterglow emission when optical data areavailable; however, for a very few bursts for which the endof the plateau is chromatic, additional assumptions (suchas evolution of the microphysical parameters ǫ e and ǫ B ,or a two-component outflow) are needed to explain thelack of a break in the optical light curve. For the remain-ing 45% of the sample, the coasting-in-wind model canstill reproduce the plateau, but the post-plateau decay inthese bursts is too rapid to be explained in this model; a refreshed shock remains the most capable interpretation.Given the constraints derived from the consistent sub-sample, the coasting-in-wind model poses two physicalchallenges which must be addressed.First, is it reasonable for the wind medium to extend tothe large distances r = 4Γ c ( t b t ) / / (1 + z ) ≈ cm,implied by this model? In fact it is: for a wind speed10 v w, cm s − , the wind ram pressure at this distance is10 − . A ∗ v w, (10 cm /r ) dyne cm − , which exceeds thehydrostatic pressure ( P ≃ G Σ ) for any column den-sity Σ < . A / ∗ v w, (10 cm /r ) g cm − . For the low-est acceptable values of A ∗ we might therefore expectthe wind to have terminated, especially within the highpressures and column densities of a starburst environ-ment. However, a fiducial wind ( A ∗ ∼
1) is sufficient tocompete with starburst pressures (note that Dai & Wu2003 found A ∗ ∼ .
01 for a burst which lacked an X-ray plateau). In fact, it is quite possible that at theseradii the circum-burst medium is a merged wind from anentire star cluster (Chevalier & Clegg 1985), as massivestars are rarely found alone. The cluster mass function(McKee & Williams 1997) implies that the progenitorstar is equally likely to belong to a massive cluster as asmall one, and star clusters — especially massive ones— are very effective at clearing their immediate molec-ular environments with winds, radiation pressure, andphotoevaporation before any stellar cores collapse (e.g.,Krumholz & Matzner 2009).One must also consider the expansion of the wind-termination shock as the bubble expands (see Ramirez-Ruiz et al. 2005); this is > cm at 10 years if ei-ther n H < A ∗ v / w, cm − in a uniform medium, or ifthe bubble expands into a previous stellar wind which is < . A ∗ v / w, times denser (Koo & McKee 1992). How-ever, other cluster stars help to clear the ambient mediumand alleviate this constraint. The duration of phase 2 istherefore not a strong constraint on the model, except incases where there is independent evidence that the windis weak and the ambient pressure and density are veryhigh.The second and more puzzling challenge involves thearrangement of ejecta from the central engine. Our upperlimit Γ .
50 is well below the lower limit of 10 − de-rived from the prompt γ -rays using the pair opacity con-straint (Lithwick & Sari 2001) for a few Fermi
Large AreaTelescope (LAT) bursts, e.g., Γ &
600 – 900 for 080916C(Abdo et al. 2009a), Γ & for short-hard burst 090510(Ackermann et al. 2010), Γ & for 090902B (Abdo etal. 2009b), and Γ ≈
200 – 700 for 090926A (Ackermannet al. 2011). However, the above constraints assumethat both the MeV and GeV emission are produced inthe same region; considering two-zone models for thesetwo spectral components reduces the lower limit on Γ bya factor of ∼ Fermi -LAT bursts do not have early ( t < × t ∼
100 s and both X-rayand optical light curves show a shallow-to-steep transi-tion at t ∼ s which, however, is found to be con-sistent with a jet break (Kumar & Barniol Duran 2009).These four bursts are all among those most energetic ones( E γ, iso ∼ − ergs, even the short-hard GRB 090510has E γ, iso = 10 ergs). It is quite plausible that thesemost energetic bursts have much higher outflow LFs, sothat their plateau features are too short ( t b ∝ E iso Γ − ;see Eq. 14) to be caught or identified. Future accu-mulation of Fermi-LAT bursts with early X-ray / opti-cal coverage could either support or disprove this biaseffect. (2) The above constraints on Γ for each burstare derived from individual pulse(s) during the promptphase, while our work is for the afterglow phase. It isvery natural to have an ejecta bulk LF in the afterglowphase that is much lower than that of an ‘individual emit-ter’ during the prompt phase. This can be understoodin the frames of two major GRB prompt phase models.In the conventional internal shock model (e.g., Paczyn-ski & Xu 1994; Rees & M´esz´aros 1994), discrete shellswith large LF variation among them collide with eachother, with each collision corresponding to an individualprompt pulse. Approximately after the prompt phase, allshells merge together and external shocks develop. It ispossible that the prompt emission is produced only fromthe high LF portion of the outflow, while the outflow to-tal energy is dominated by the lower LF portion so thatexternal shock possesses a lower LF. In another categoryof models that involve ‘individual emitters’ due to, e.g.,magnetic turbulence inside a relativistic bulk flow (Lyu-tikov & Blandford 2003; Narayan & Kumar 2009), theprompt-emission-derived LF limit or value is actually forthe bulk flow LF times the ‘individual emitters’ LF. Af-ter the ‘individual emitters’ energy is released in form of γ -rays, the external shock LF is just that of the initialbulk flow.Alternatively the bulk LF may have been reduced bybaryon pollution during time between the prompt andafterglow phases. Such contamination cannot come fromthe wind, for this would imply that the fast ejecta de-celerate early and skip the observed plateau. Possiblestellar sources include: (1) the blowout of an inflated co-coon within the stellar envelope, which may have mixedwith an unknown amount of stellar envelope material,and (2) the cap of stellar envelope which is unable to es-cape the advancing jet head, and is cast forward by theprocess of runaway shock acceleration (Matzner 2003).Of these, it is unclear how the former would pose an ob-stacle to the high-Γ jet material after it has escaped thestar.The latter source, a trapped portion of the outer en-velope, is worthy of closer inspection. To evaluate this,we examine the approximate conditions for trapping ofmatter in front of the jet head (eqs. [26] and [27] ofMatzner 2003, subject to his eq. [6] ). When trappingoccurs while the head’s motion is nonrelativistic, the re- Note the typo (inverted expression) in the middle expressionof equation (6) in Matzner (2003). sult is simple: the jet traps material once it is within ∼ . θ j R ∗ of the stellar surface, where R ∗ is the star’sradius; in a polytropic region with index n , the trappedmass scales as θ nj (a very strong function of θ j ). Ifinstead the head is relativistic, the dependence weakensbecause a narrower, faster jet traps matter from deeperin the envelope; however it also depends on the intensityas well as the opening angle of the jet. For an example,consider the outer envelope profile for the SN 1998bwprogenitor by Woosley, Eastman, & Schmidt (1999) anddiscussed by Matzner (2003, § isotropic rest energy of trapped material ismax " . (cid:18) θ ◦ (cid:19) , . (cid:18) θ ◦ (cid:19) . (cid:18) L j, iso erg s − (cid:19) . × erg . (27)Although this material is swept into an accelerating for-ward shock (Matzner 2003), a negligible fraction attainsLFs higher than Γ j . Moreover, although it is small,the rest energy is well above the critical value Γ − j E iso required to decelerate the ejecta. Because it is about10 − (10 erg /E iso ) of the jet energy (for the fiducial casecited), the final LF after the interaction would be ∼ ,but strongly dependent on θ j ; our inferred Γ j ∼
40 isentirely plausible as a final value. We refer the readerto the discussion by Thompson (2006), who considers asimilar scenario.Whatever the origin of the lower-Γ j matter, it is clearthat it inherits effectively all of the GRB kinetic energyfrom the outflow powering the prompt phase. First, Fig-ure 5 shows that the afterglow energy budget is compat-ible with what is expected from the prompt phase, givenreasonable radiative efficiencies. Second, if any signif-icant fraction of the energy had proceeded beyond thecoasting shell at higher LF, it would have deceleratedearly and caused a noticeable departure from the plateauphase at early times. Indeed, since in the decelerationphase, when max( ν m , ν c ) < ν X (from Eqs. 14 and 17), E iso ∝ F ν X ( t b ) / ( p +2) t (3 p − / (2+ p ) b , (28)we infer that the persistence of the plateau from times t to t b rules out any early injection of fast ejecta withmore than a fraction ( t /t b ) (3 p − / (2+ p ) ∼ .