Applying an accurate spherical model to gamma-ray burst afterglow observations
Konstantinos Leventis, Alexander J. van der Horst, Hendrik J. van Eerten, Ralph A.M.J. Wijers
aa r X i v : . [ a s t r o - ph . H E ] F e b Mon. Not. R. Astron. Soc. , 1–15 (2012) Printed 19 July 2018 (MN L A TEX style file v2.2)
Applying an accurate spherical model to gamma-ray burstafterglow observations
K. Leventis ⋆ , A. J. van der Horst , H. J. van Eerten , R.A.M.J. Wijers Astronomical Institute ‘Anton Pannekoek’, PO box 94248, 1090 SJ Amsterdam, the Netherlands Center for Cosmology and Particle Physics, Physics Department, New York University, New York, NY 10003, USA
Accepted ... Received ...; in original form ...
ABSTRACT
We present results of model fits to afterglow data sets of GRB 970508, GRB 980703and GRB 070125, characterized by long and broadband coverage. The model assumessynchrotron radiation (including self-absorption) from a spherical adiabatic blast waveand consists of analytic flux prescriptions based on numerical results. For the first timeit combines the accuracy of hydrodynamic simulations through different stages of theoutflow dynamics with the flexibility of simple heuristic formulas. The prescriptionsare especially geared towards accurate description of the dynamical transition of theoutflow from relativistic to Newtonian velocities in an arbitrary power-law densityenvironment. We show that the spherical model can accurately describe the data onlyin the case of GRB 970508, for which we find a circumburst medium density n ∝ r − .We investigate in detail the implied spectra and physical parameters of that burst. Forthe microphysics we show evidence for equipartition between the fraction of energydensity carried by relativistic electrons and magnetic field. We also find that for theblast wave to be adiabatic, the fraction of electrons accelerated at the shock has to besmaller than 1. We present best-fit parameters for the afterglows of all three bursts,including uncertainties in the parameters of GRB 970508, and compare the inferredvalues to those obtained by different authors. Key words: hydrodynamics radiation mechanisms: non-thermal methods: statisticalgamma-ray burst: general.
Afterglow observations of gamma-ray bursts (GRBs) haveprovided important insight into the nature of these events.Some of it has been direct, for example the measurementsof redshifts (Metzger et al. 1997), or the association of somebursts with supernova explosions (Hjorth et al. 2003). Onthe other hand, some has been indirect, accessible only oncethe available data are interpreted within the context of aphysical model. The commonly used fireball model (Rees &M´esz´aros 1992; Paczy´nski & Rhoads 1993), for instance, isfirmly supported by extensive modelling of afterglow obser-vations as synchrotron radiation originating from a decel-erating relativistic blast wave (Wijers et al. 1997; Waxman1997; Sari et al. 1998; Chevalier & Li 2000; Panaitescu &Kumar 2000).Despite the success of the aforementioned studies ininterpreting afterglow observations within a general frame-work, the values derived by independent groups for the phys-ical parameters of individual afterglows are often substan- ⋆ E-mail: [email protected] tially different. Such is the case for the well-studied after-glows of GRB 970508 and GRB 980703, for which large dif-ferences can be found in the derived values for blast-waveenergy, density of the circumburst medium (CBM) and mi-crophysics parameters from different authors (Wijers et al.1997; Granot & Sari 2002; Panaitescu & Kumar 2001, 2002;Frail et al. 2003). The CBM density seems to be especiallyunconstrained, as differences of many orders of magnitudecan be found in the literature. One of the most importantparameters of GRB outflows, that directly affects the in-ferred energetics and rate of these events, is the openingangle of the jet. Specifically, jetted instead of spherical out-flows would significantly aleviate the energy requirementsand boost the event rate of GRBs. The first strong infer-ence of their presence (Harrison et al. 1999) was perceivedas evidence for the ubiquitous role they play in the GRB phe-nomenon. Accumulating observations, however, have failedto fully confirm this picture, with many afterglows not show-ing any steepening in the light curves that can be attributedto a jet break (e.g. Racusin et al. 2009), rendering the in-fluence of collimation on GRB outflows for the most partambiguous. All these uncertainties on the inferred physical c (cid:13) Leventis et al. parameters of GRB blast-waves have called for refinementand greater precision in the methods that underlie afterglowmodelling.Theoretical afterglow calculations have been continu-ously improved to include more precise methods of calculat-ing the dynamics and spectra of the source (e.g. Kobayashiet al. 1999; Huang et al. 1999; Rhoads 1999; Granot & Sari2002; Granot & Piran 2012; Pe’er 2012). Many recent stud-ies (e.g. Meliani et al. 2007; Zhang & MacFadyen 2009; vanEerten et al. 2010a; Wygoda et al. 2011; De Colle et al.2012a; van Eerten et al. 2012) are based on high-resolutionrelativistic hydrodynamic (RHD) simulations which are es-sential to understand critical aspects of the outflow’s dy-namics, like lateral spreading of jets and the transition tothe non-relativistic phase. This allows in principle for accu-rate determination of spectra and light curves from simula-tion runs. However, this method is not suitable for iterativefitting of model parameters to observations due to the lim-itations posed by the necessary performance of numeroustime-consuming RHD simulations.Recently (van Eerten et al. 2012; Leventis et al. 2012;see also van Eerten & MacFadyen 2012b) a new approachhas been developed for the calculation of spectra and lightcurves that retains the accuracy of the numerical techniques,without requiring the long run times of simulations. Whilethe methods of these studies differ, they are common in howthey are based on sets of blast-wave simulations that spanthe parameter space. In the case of van Eerten et al. (2012)dynamical results of 2D simulations have been tabulatedallowing the user to perform a straightforward numericalcalculation of the afterglow radiation for any combinationof the physical parameters within the explored range. Evenso, this calculation can be lengthy and is best executed ona parallel computer network.The method of Leventis et al. (2012) is based on 1DRHD simulations that span the entire range of dynamics,from ultrarelativistic to Newtonian velocities. These sim-ulations, however, do not account for jet features as theyrely on the assumption of spherical symmetry. Several runshave been used to calibrate analytically derived scalings ofobserved synchrotron spectra. The resulting formulas havethe unique advantage of combining the accuracy of high-resolution trans-relativistic simulations with the versatilityof analytic equations. The fact that they cover a sequenceof dynamical phases has motivated us to use them in orderto fit model parameters to observational data for afterglowswith extensive monitoring. The bursts we are mainly con-cerned with in this paper are GRB 970508, GRB 980703 andGRB 070125, all monitored in several bands from radio toX-ray frequencies and covering observer times from hours toseveral months. The two former are among the most studiedafterglows with several groups publishing results they haveobtained through afterglow modelling.In this work we present fit results for the afterglows ofthese bursts and investigate the extent to which a spheri-cal outflow can provide an adequate description of the data.These results also serve as a basis for comparison to modelfits based on 2D simulations. Furthermore, the prescriptionsof Leventis et al. (2012) enable us to examine the densitystructure of the burster’s immediate environment, as a con-tinuous range of values for the slope of the CBM density isallowed. The resulting slope can then reveal unusual density distributions of the CBM, or confirm previous claims basedon models with only preset structures available, typicallyconstant density or a profile corresponding to a stellar-windenvironment ( ∝ r − ).The paper is organized as follows. A description of theobservational data that have been used during fitting is pre-sented in Section 2. In Section 3 we illustrate the main fea-tures of the physical model we have used and in Section 4we present our main results. In Section 5 we focus on theinferred parameters of GRB 970508 for which we obtain themost reliable results. In Section 6 we discuss the implicationsof this work on afterglow physics and modelling. Finally, inSection 7 we conclude by summarizing our main findings. In this study we focus on three sources: GRB 970508,GRB 980703 and GRB 070125. All three have well-sampledafterglows across the electromagnetic spectrum. In partic-ular they are among the few GRBs that have detectionsin multiple radio bands at hundreds of days after the initialgamma-ray trigger. This allows us to model the full evolutionof the GRB blast wave from the ultrarelativistic to the non-relativistic phase. Another burst with afterglow monitoringspanning almost a decade in the radio is GRB 030329. Wehave not fit that data set as it is clear from the light curvesthat a jetted model is needed to interpret the observations(see van der Horst et al. 2008 and references therein).Since the lauch of the
Swift satellite, it has become clearthat the early (10 − s) afterglow behaviour of manybursts cannot be explained by standard afterglow models(Nousek et al. 2006). Energy injection into the blast wavehas been proposed to explain the typically shallow decaythat the optical and X-ray light curves show (e.g. Gra-not & Kumar 2006; Nousek et al. 2006; Zhang et al. 2006;Panaitescu & Vestrand 2011). Other plausible explanationsare evolution of the shock microphysics parameters (Gra-not et al. 2006), or viewing angle effects (Eichler & Gra-not 2006). In our sample, GRB 970508 and GRB 070125 dis-play an atypical behaviour, lasting in both cases up to 1 . . . . .
43, 4 .
86 and 8 .
46 GHz (Galama et al. 1998b; Frail et al.2000). Near-infrared and optical data have been published at6 observing bands (Chary et al. 1998; Galama et al. 1998a;Sokolov et al. 1998; Sahu et al. 1997; Garcia et al. 1998).The magnitudes of the underlying host galaxy in the B , V , R c and I c bands have been presented in Zharikov & Sokolov(1999), while the observations in the K and U bands are suf-ficiently early that they are not affected by the host galaxybrightness. We have corrected the observed optical magni-tudes for galactic extinction, subtracted the host galaxy flux, c (cid:13) , 1–15 its of a spherical model to GRB afterglows and converted them to fluxes. The afterglow was observedin X rays with BeppoSAX (Piro et al. 1998), for which wehave converted the X-ray count rates to fluxes by assuminga spectral index of − . H , J , I , R , V and B bands (Bloom et al. 1998; Castro-Tirado et al.1999; Vreeswijk et al. 1999). We have corrected the observedmagnitudes for galactic extinction, but also for extinction inthe host galaxy with E ( B − V ) = 0 .
29 (Starling et al. 2007;Starling 2008). The host galaxy of GRB 980703 was bright,not only in the optical (Frail et al. 2003) but also at radiowavelengths (Berger et al. 2001), and we have subtractedthe host galaxy flux at all these wavelengths from our mea-sured fluxes. The afterglow has also been detected at X-rayenergies (Vreeswijk et al. 1999), for which we have used thesame conversion method as in the case of GRB 970508.For GRB 070125 we have used all the broadband datapresented in De Cia et al. (2011). Radio observations wereperformed at 4 .
86, 8 .
46, 15 and 22 . M = 0 .
27, Ω Λ = 0 .
73 and the Hubble-parameter H = 71 km s − Mpc − ; so for the GRB 970508redshift of z = 0 .
835 (Metzger et al. 1997) the luminos-ity distance is d L = 1 . · cm, for GRB 980703 theredshift z = 0 .
966 (Djorgovski et al. 1998) correspondsto d L = 1 . · cm, and for GRB 070125 the redshift z = 1 .
547 (Cenko et al. 2008) implies that d L = 3 . · cm.During the fit process, no data were excluded based onflux values. However, in the figures we present, data that arenot significant at the 2 σ level are depicted as upper limits,for display purposes. A noticeable feature of radio data is the high degree of scat-ter they show, especially compared to the size of the errorbars (see Section 4). In the case of GRB 970508, notable scat-ter is also present in near-infrared and optical frequencies.In these bands it is presumably caused by the use of datafrom various telescopes without the performance of cross-calibration analysis.In the radio, interstellar scintillation affects the flux lev-els (Goodman 1997). Its strength diminishes as the angularsize of the source grows and this has been used to infer theradius of GRB outflows (Frail et al. 1997; Taylor et al. 1997;Waxman et al. 1998; Frail et al. 2000; Yost et al. 2003).Various other groups have accounted for the effect of scin-tillation (Panaitescu & Kumar 2002; Chandra et al. 2008),especially in early data, by effectively increasing the size ofthe error bars, more so in early observer times. In our studywe have not included the effect of scintillation in the model.One reason is that it does not affect the best-fit values ofour results significantly, as the central values of the mea-surements are not perturbed. Another reason is the lack ofdetailed measurements for the amount of scattering materialoff the galactic plane, which makes the effect of scintillation in models of extragalactic sources uncertain (Chandra et al.2008).
The model we have used is a direct implementation of themethod presented in Leventis et al. (2012). In that pa-per we present simulation-calibrated flux prescriptions ofsynchrotron radiation, including self-absorption, through-out the entire dynamical evolution of GRB afterglows. Themodel assumes an initially ultrarelativistic spherical blastwave expanding adiabatically inside a medium with a den-sity profile described by a power law: n ( r ) ∝ r − k . The en-ergy distribution of the electrons accelerated at the forwardshock is also assumed to be a power law. The minimumLorentz factor of that distribution is calculated through theenergy density and mass density of the shocked gas. The syn-chrotron spectrum is then determined through the emissiv-ity and absorption coefficient of these relativistic electrons.In total there are seven free parameters. These are theblast-wave energy E in units of 10 erg, the number den-sity n at 10 cm (regardless of the density structure), theindex p of the electron power-law distribution, the index k ofthe density distribution of the matter surrounding the GRB,the fraction ξ of accelerated electrons, and ǫ e and ǫ B denot-ing the fractions of internal energy carried by the relativisticelectrons and magnetic field, respectively. In practice, due toa degeneracy of this model (Eichler & Waxman 2005) a valuefor one of these parameters has to be assumed in order touniquely determine the others. In this work we ‘break’ thedegeneracy by assuming ξ = 1 in all runs, unless otherwisestated.The flux prescriptions are based on analytic calculationsof flux scalings during the relativistic (Blandford & McKee1976) and Newtonian (Sedov 1959; Taylor 1950) phase ofthe blast-wave dynamics. In these two dynamical regimesthe flux at every power-law segment of the spectrum hasbeen calibrated in terms of p and k . Several hydrodynamicsimulations of the afterglow dynamics were run and subse-quently post-processed using a radiative-transfer code (vanEerten & Wijers 2009; van Eerten et al. 2010a). The cal-ibration was carried out by matching analytic expressionsfor the flux scalings to these numerical results. The sharp-ness of spectral breaks connecting different power laws ofthe spectrum is also expressed as a function of p and k . Thetransition from the relativistic to the Newtonian solution isnicely described as a temporal power-law break between theasymptotic behaviour of the critical parameters of the spec-trum, namely maximum flux F m , self-absorption frequency ν a and synchrotron characteristic frequency of the lowest-energy electrons ν m . It is worth noting that the characteris-tics (break time and sharpness) of those temporal breaks are,in general, unique for every parameter of the spectrum. Thisemphasizes the advantages of simulation-based flux prescrip-tions compared to simple analytic models for the transrela-tivistic behaviour of observed afterglows. c (cid:13) , 1–15 Leventis et al.
A feature of the synchrotron spectrum not covered in thetreatment of Leventis et al. (2012) is the cooling break, man-ifested as a fourth spectral parameter ν c . Its presence inthe spectrum, however, might be important, especially forobservations at optical wavelengths and X-ray energies. Forthat reason all the performed fits have been checked for con-sistency by calculating the value of ν c according to formu-las available in the literature (e.g. Granot & Sari 2002; vanEerten & Wijers 2009) and comparing it to the frequenciesof the observations. The results of the two aforementionedstudies are compatible. We have chosen to use those of vanEerten & Wijers (2009) due to the fact that a general valuefor k is allowed in their prescriptions. The consistency checkshave been performed throughout the range of observer timescovered by the data. A value of ν c greater than the observ-ing frequencies implies that cooling has not affected the fitsand the obtained values for the physical parameters are con-sistent with the underlying physical model. To the best ofour knowledge simulation-based analytic prescriptions for ν c beyond the relativistic phase do not exist in the literature.That being the case, we have used formulas applicable inthis phase throughout. This extrapolation provides a lowerlimit on the actual value of ν c because its temporal slopein the Newtonian phase is shallower than in the relativis-tic (van Eerten et al. 2010a), which is sufficient when ν c isfound not to interfere with the observing frequencies.On the other hand, when the value of ν c is found to belower than – or at about the same levels as – the observ-ing frequencies a different approach is necessary in order tofirmly constrain the influence of cooling on the data. Ourfitting code has been expanded to include a prescription forthe position of ν c as a function of time. We have made useof the formulas from van Eerten & Wijers (2009) by cal-culating ν c , of that paper. Formally this expression shouldonly apply in the case of slow cooling ( ν m < ν c ). However,it is easy to verify (see also Granot & Sari 2002) that theexpression for ν c in the case of fast cooling gives a similarresult within a factor of about 2. A few modifications inthe prescriptions are then required in order to account forthe influence of cooling in the broadband spectrum. When ν a < ν m < ν c or ν m < ν a < ν c the only modification is thatof appending another break in the spectrum at the coolingfrequency, beyond which the spectrum steepens by a half(Sari et al. 1998). The formula we have used is F ν ( ν obs ) = A "(cid:18) ν obs ν (cid:19) − a s + (cid:18) ν obs ν (cid:19) − a s − /s × " (cid:18) ν obs ν (cid:19) h ( a − a ) − /h × " (cid:18) ν obs ν (cid:19) r ( a − a ) − /r (1)The first line in eq. (1) describes the first break of the spec-trum at the lowest characteristic frequency, while each fac-tor on the second line stands for an extra break at pro-gressively higher frequencies. The parameters ν , ν , ν and s, h, r represent the values of the three critical frequenciesand the sharpness of the spectral breaks they correspondto, respectively, while a , a , a and a are the slopes of thefour power laws present in a spectrum with three breaks.Finally, A is the normalising factor of the spectrum derivedthrough modelling of the peak flux F m . When ν m , ν c < ν a the ordering of ν m and ν c doesnot play a role and one retrieves spectrum 3 of Granot &Sari (2002). In that case we have approximated the self-absorption frequency with the values applicable to the no-cooling case. Similarly, when ν a < ν c < ν m (spectrum 5 ofGranot & Sari 2002) we have approximated both ν m and ν a with their values in the absence of cooling, while the peakflux is attributed to ν c . Formally, when ν a < ν c < ν m theself-absorption break is split in two break frequencies withan extra power-law segment between them that has a slopeof 11 /
8. We have neglected that effect and used only one self-absorption frequency that has the value of ν a1 from Leventiset al. (2012). This frequency connects power laws of slope2 and 1 /
3. In reality, we have found that most of the timebest-fit values of the physical parameters imply that theseapproximations are not used since ν c > ν a , ν m . Howeverthere are instances when this is not the case and we addressthese in more detail in Section 5.A last issue that needs to be dealt with when coolinginfluences the fits is the application of the relativistic formu-las for ν c throughout the range of observer times. To assessthe validity of this application one needs to estimate theduration of the relativistic phase of the afterglow in the ob-server frame. In the absence of a detailed description for thetransrelativistic behaviour of the cooling frequency, the mostgeneral way to do that is by calculating the observer timewhich corresponds to the transition between the relativisticand Newtonian asymptotes, t NR (e.g. Piran 2004; Leventiset al. 2012). This calculation has been performed for all setsof best-fit parameters and is presented along with our mainresults in Section 4. The fitting method we have used is a χ -minimization algo-rithm following the downhill-simplex method combined withsimulated annealing, as explained in van Eerten et al. (2012).The errors for the best-fit parameters of GRB 970508 havebeen determined via a Monte Carlo process. In this anal-ysis the values of all data points are perturbed randomly,based on their error bars, and a new best-fit set of parame-ters is calculated for the synthetic data. For every physicalscenario (class and constraint) this has been repeated 1000times from which 683 best fits were drawn to determine therange of the parameters’ values at a 68 . σ , confi-dence level.The fitted parameters were allowed to vary within thefollowing ranges ( n in cgs units): 10 − < E < , 10 − 5, 10 − < ǫ B < . 0, 10 − < ǫ e < . − . < k < . For all afterglow data sets, we present three classes of mod-els. Each class corresponds to a different assumption (orthe lack thereof) for the value of k . We have run fits for k = 0 and 2, corresponding to constant density CBM (la-belled ISM ) and a constant-stellar-wind profile (labelled Wind ), respectively, and fits where k is a free parameter.For each class, a range of microphysics settings has beentested. Namely, we have either allowed for both ǫ e and ǫ B c (cid:13) , 1–15 its of a spherical model to GRB afterglows Figure 1. Afterglow of GRB 970508. Best-fit light curves for ISM (solid grey line) and Wind (dashed black line) classes. When k is a free parameter, the Wind scenario is retrieved with high precision. The three radio bands are on top and the rest followin order of increasing frequency, spanning near-infrared, optical, ultraviolet and X-ray energies. Data points before 1 . σ errors. Triangles depict upper limitsat the 2 σ level. to be free parameters, or connected them through a closurerelation that effectively reduces them to one free parameter.Two options for the closure relation have been explored. Onthe one hand we have imposed equipartition ( ǫ e = ǫ B ) andon the other the ‘Medvedev’ relation ( ǫ = ǫ B ; Medvedev2006). All other parameters have been kept free at all runs,apart from ξ which, for every run, has taken the value of 1. We have performed several fits both to the full data setand to different subsets (radio only, radio and optical only,radio, optical and X rays) of the afterglow observations ofGRB 970508. Radio data alone do not provide enough in-formation to determine simultaneously all the parameters.However, when k is frozen (either in the ISM or the Wind scenario) and a microphysics constraint is used, the resultsfrom fitting the radio only, are fairly similar to those fromfits to the full data set; all best-fit values of parameters areless than 50% off in the Wind class and less than a factor of c (cid:13) , 1–15 Leventis et al. Table 1. Best-fit parameters, with 1 σ errors, for GRB 970508 in all classes of models and for all microphysics settings. The fits havebeen performed to data including radio, near-infrared, optical, ultraviolet and X rays, but excluding observations made prior to 1 . erg. The fourth column represents n , the number density ata radius of 10 cm. When k = 2, n and A ∗ (Chevalier & Li 2000) are related by the formula n ≃ A ∗ . For example, the best-fitmodel (equipartition constraint) of the Wind class has A ∗ = 0 . 243 g cm − . The last column presents the value of χ divided by thedegrees of freedom (dof).Class Constraint E n p ǫ B ǫ e k χ / dof − . +0 . − . . +25 . − . . +0 . − . (4 . +11 . − . ) · − . +0 . − . ISM Equipartition 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . − . +0 . − . . +2 . − . . +0 . − . . +0 . − . . +0 . − . . Wind Equipartition 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . Medvedev 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . − . +0 . − . . +19 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . k free Equipartition 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . Medvedev 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . k free Equipartition 0 . +0 . − . . +14 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . aa Error bars of data points are rescaled by a factor of 5. ISM class. Including X-ray data has almost noinfluence on the inferred values of the physical parameters,as the fits are governed by the combination of radio and op-tical observations. Nevertheless, we present results and lightcurves from fits to all bands for completeness.In Fig. 1 we present light curves of best-fit models ap-plied to the full data set. We have found that the spheri-cal model can produce an adequate fit to the data, when k = 2. Results for the Wind scenario are almost identical tothose from fits where k is a free parameter. Models of the ISM class consistently overpredict late radio flux at 4 . 86 and8 . 46 GHz. On the other hand, Wind models provide a gooddescription at all observer times. In the optical and near-infrared bands, the ISM and Wind cases are practically in-distinguishable. One common feature of both is the system-atic, albeit minor, underprediction of early ( < 10 days) flux,especially in the R and V bands. This is less pronouncedin the surrounding K , I and B bands. It is worth notingthat the X-ray data cannot be fitted by any combination ofparameters. Along with the fact that the flux drops sharplyafter the first two data points, this hints towards a separateorigin of the early X-ray flux, for example, inverse Compton(e.g. Sari & Esin 2001). Alternatively, the high X-ray fluxat early times could be due to flaring activity, which is nottemporally resolved due to the poor coverage.In the Wind scenario, all critical frequencies lie belowthe near-infrared. On the other hand, both ν a and ν m passthrough the radio bands. This is in rough agreement withthe findings of Chevalier & Li (2000) and Panaitescu & Ku-mar (2002), although we do not confirm the expectationsof the former group regarding the passage of ν c from theoptical. Instead we find that ν c stays below 10 Hz duringthe observations. In the ISM case we find that ν m startsoff between the optical and radio and crosses ν a (5 · Hz) at ∼ 50 days. We also find that ν c remains between opti-cal and X-ray energies throughout, contrary to the resultsof Galama et al. (1998c) and Wijers & Galama (1999) whofind that ν c crosses the optical frequencies early on. Calcu-lation of t NR yields 145 and 180 days, in the best-fit modelsof the ISM and Wind class, respectively.In Table 1 we present best-fit parameters, with 1 σ er-rors, of runs to the full data set. A readily apparent featureis the value of k when it is a free parameter, which convergesto the Wind scenario. Actually, all best-fit values as well asthe χ of these two classes are almost identical, regardlessof the chosen microphysics. From the ISM class only therun with no constraints on the microphysics comes close interms of χ , but that model requires a low value for ǫ B andhigh value for ǫ e to work. The energy inferred in this case isan order of magnitude higher than the values correspondingto the Wind scenario.For all classes of models, the best-fit values of χ / dofare much higher than 1. This is mainly caused by the notablescatter that data in radio, near-infrared and optical bandsshow. The scatter (discussed in Section 2.1) is not reflectedin the size of the error bars. This is clearly demonstratedin the very small uncertainties that the inferred parame-ters have, when a microphysics constraint is used. To obtaina better measure for the uncertainties when scatter is ac-counted for, we have artificially increased the error bars ofall the data by a factor of 5 and re-calculated them for thebest-fit model of the k free class. The results are presentedin the bottom row of Table 1. The choice of the factor ismotivated by the value of χ / dof ≈ χ / dof, there are alsosystematic deviations from the data (for example in the R and V band during the first 20 days). Therefore, strictly c (cid:13) , 1–15 its of a spherical model to GRB afterglows Figure 2. Afterglow of GRB 980703. Best-fit light curves for ISM (solid grey line), Wind (dotted black line) and k free (dashedblack line) classes. Radio bands are shown in the top panel. The lower panel contains near-infrared, optical and X-ray bands. Alldata were taken into account for the light curves we present. In all bands, data points have 1 σ errors. Triangles depict upper limitsat the 2 σ level. speaking, the method of artificially increasing the error barsshould not be applied to the whole data set. Nevertheless,its application results in uncertainties that represent betterthe parameter range allowed by the data and is not used todraw any conclusions on the quality of the fits.X-ray data show a preference for the ISM class, buthardly influence the fit at all, due to the small number ofdata points. We have investigated the dependence of our re-sults (especially those for k ) on the relative importance ofX-ray data by increasing the error bars by a factor of 5 inall bands, apart from X rays, and recomputing the uncer-tainties in the values of the inferred parameters. The best-fit results are essentially identical to those of Table 1 for the k-free class. The 1 σ uncertainties, while larger than thosepresented in the lowest row of Table 1, exclude the ISM scenario.A discussion of the spectra, dynamics and inferred pa-rameters in the Wind scenario (that produces the best fits)is presented in Section 5. Another well-sampled afterglow that has been extensivelymodelled in the literature is that of GRB 980703. We have c (cid:13) , 1–15 Leventis et al. Table 2. Best-fit parameters of each class for GRB 980703. For column description see Table 1.Class Constraint E n p ǫ B ǫ e k χ / dof ISM − . 82 760 . . 538 10 − . . Wind − . 771 14 . 220 3 . 865 0 . . 133 2 9 . k free Equipartition 2 . 546 4 . 265 3 . 933 0 . 115 0 . 115 1 . performed fits to the full data set, from radio to X rays, andwe have found that no set of parameters can fit the data.The Wind model does better than the ISM , but the best fitis obtained for k ≈ . ISM model underperforms com-pared to the other classes. In the optical and near-infrarednone of the models seems to be able to reproduce the dataadequately, especially in the low-energy bands. X-ray data,on the other hand, can only be described within the ISM class. From this general picture we can conclude that thephysical scenario of synchrotron radiation from a sphericalblast wave is not realistic for this source.For every class of models, we have selected those withthe microphysics settings that produced the best fits andpresent them in Table 2. For the ISM and Wind class, themodel that performs better is the one with no constrainton the microphysics, whereas when k is free, equipartitionproduces the best χ / dof. Fitting the afterglow of this burstwe have allowed for p to range between 2 . . p < . k free classand the one from the Wind class have very high values for p ( > . χ / dof values are notably better than thoseof the ISM class. The values of t NR are 100, 1310 and 880days for the ISM , Wind and k free class, respectively. Dueto the overall-bad fits to the light curves and the extremebest-fit values of p , we consider the values we obtain unre-liable. For that reason we have not calculated any errors onthe derived parameters for this burst.It is worth noting the consensus over the outflow ge-ometry of GRB 980703. Several studies infer small open-ing angles and jet breaks in the timescale of days-weeks(Panaitescu & Kumar 2001; Yost et al. 2003; Frail et al.2003). In the spherical model, the very fast decays observedin the H , J and R bands, can only be explained by very largevalues of p , that result in steep light-curve profiles. However,a more natural explanation of the observed slopes would bethat the edge of the jet has become visible (Rhoads 1999;Panaitescu 2005). We therefore regard the results presentedin this paper implicit confirmation of the jet geometry inthe outflow of GRB 980703. The afterglow of the exceptionally luminous GRB 070125was observed in several bands, lasting more than ten daysin X rays and about a year in the radio. We find that Wind -like models provide the best description of the data, butwith noticeable outliers and with inferred parameters thatare fairly extreme ( E > erg, ǫ e = 1, ǫ B < − ). Weconsider the results indicative, but by no means conclusive,as additional physics (e.g. jets) may be needed to explain thedeviations and special conditions are required to account forthe physical parameters we obtain.In Fig. 3 we present light curves of the best-fit modelsfrom each class. Results for the Wind and k free classes aresimilar to each other and differ significantly from the ISM class in radio, millimetre and X-ray bands, where the formerperform better. However, late-time behaviour of the data at4 . , , 46 and 22 . Wind and k free models are able to describethe data.In agreement with De Cia et al. (2011) we find that ν c lies between optical and X-ray energies throughout the ob-servations. Chandra et al. (2008), on the other hand, findthat they can best explain the data when ν c lies below theoptical. Calculation of t NR yields 80 days in the ISM caseand ∼ 140 days in the other classes. This ensures that therelativistic formula for ν c is valid during near-infrared, op-tical, ultraviolet and X-ray observations, that last up to 10days after the gamma-ray trigger. In all classes, ν m startsoff bellow the optical and overtakes ν a at 30 − 80 days. Thedifferent temporal evolution of ν a makes for the deviationsin late radio light curves between the ISM class and theothers.In Table 3 we present the values of the inferred parame-ters for the best-fit models of each class. Deviations betweendifferent classes are moderate. Best χ / dof values are foundwhen no assumption for the microphysics is made. This isbecause to explain the data, all models require a high valuefor ǫ e and a very low one for ǫ B . Values of the parameterswhen k is free (model with the best χ / dof) are closer tothose from the Wind scenario, without, however, matchingthem. The inferred energies are high in all cases, as are thevalues for p . Given the imperfect fits and the extreme pa-rameters we infer, we have not calculated errors for theirvalues. c (cid:13) , 1–15 its of a spherical model to GRB afterglows Figure 3. Afterglow of GRB 070125. Best-fit light curves for ISM (solid grey line), Wind (dotted black line) and kfree (dashed black line) classes. Radio and millimetre bands are on top. The lower panel shows near-infrared, optical,ultraviolet and X-ray bands. Data points before 1 . σ errors.c (cid:13) , 1–15 Leventis et al. Table 3. Best-fit parameters of each class for GRB 070125. For column description see Table 1.Class Constraint E n p ǫ B ǫ e k χ / dof ISM − . 968 1 . · . 203 1 . · − . . Wind − . 85 1 . · . 717 1 . · − . . k free − . 32 3 . · . 831 5 . · − . . GRB 970508 is a unique burst in many ways. Its afterglowwas only the second ever observed and despite the multi-frequency monitoring, in some bands over the period of sev-eral months, the inferred physical parameters vary widelybetween difeerent authors (Wijers & Galama 1999; Chevalier& Li 2000; Frail et al. 2000; Panaitescu & Kumar 2002; Yostet al. 2003; Berger et al. 2004). From our sample, the fits toGRB 970508 are deemed the most reliable and the most suc-cessful, despite the higher values for χ / dof. There are twobasic reasons for this. The first is the overall behaviour ofmodel light curves that successfully reproduce the trends ofthe data at all well-probed wavelengths. The second reasonis the stability and convergence that the fits to GRB 970508show, especially in the Wind scenario, but also when k is afree parameter and a constraint on the microphysics is im-posed. If no constraint is placed on the microphysics and k is a free parameter, we cannot discern between the ISM and the Wind scenario. However, once either equipartitionor the Medvedev formula are used, the results clearly favoura wind-type CBM. In this Section we present an analysis ofthe physics implied by the best-fit parameters we obtain forGRB 970508 and compare those to values inferred by otherauthors. At first glance, the best-fit values presented in Table 1 re-veal an issue concerning the microphysics of the blast-wave,namely, the sum of ǫ B and ǫ e is greater than 1. In fact, inorder for the outflow to be adiabatic, as assumed by themodel, at least one of these parameters has to be muchsmaller than 1. A low value of ǫ e ensures that most of theenergy remains in the blast wave, even if the electrons ra-diate efficiently, while a low value of ǫ B moderates the en-ergy losses of the electron population. The degeneracy ofthe theoretical model (Eichler & Waxman 2005) which hasprompted us to freeze ξ = 1 during the fit process, can beused to solve this issue. The net effect of this degeneracy isthat a set of parameters E ′ = f − E , n ′ = f − n , ǫ ′ e = f ǫ e , ǫ ′ B = f ǫ B , ξ ′ = f ξ produce the same spectrum as theunprimed ones, regardless of the value of (the positive num-ber) f . Therefore, the inconsistency implied by the high val-ues of ǫ B and ǫ e may be seen as evidence that ξ < 1, whichmeans that not all electrons are accelerated at the shock .Consequently, the values for E and n presented in Table1 should be viewed as lower limits, whereas those for ǫ B and ǫ e as upper limits.Another notable feature of the results for the micro- physics in the Wind scenario is that we can not conclusivelydistinguish between the three possibilities (no constraint,equipartition, Medvedev relation). Equipartition settingsseem to be marginally favoured by the better χ / dof val-ues these models have, but the Medvedev relation cannotbe ruled out. The ambiguity of our results is mainly causedby the relatively high values that both ǫ B and ǫ e have. Wehave run fits where ξ was frozen at 0 . . 01 and moni-tored the behaviour of the two former quantities. They werefound to be approximately equal to each other and always(as did E and n ) followed the scalings implied by the de-generacy relations. This confirms energy equipartition be-tween power-law electrons and magnetic field, which is alsosuggested by χ / dof values. In the Wind scenario the synchrotron spectrum starts offat 1 . ν a =1 . · Hz, ν c = 1 . · Hz, ν m = 1 . · Hz. At 5days, ν m overtakes ν c , causing the wiggle in the radio lightcurves of the model (see Fig. 1). The flux at the highest-frequency power law of the spectrum (where near-infrared,optical, ultraviolet and X-ray data lie) is independent of theordering of critical frequencies and therefore no feature isobserved in those bands during the spectral transition. Af-ter 5 days the spectrum settles into the slow-cooling regime.During the fast-cooling phase (i.e. before 5 days), almost allavailable data lie above ν m and ν c ; there are hardly any sig-nificant radio observations during that time. Therefore, ourapproximations for ν a when ν a < ν c < ν m have a negligibleeffect on the fits. Moreover, given that the values of ν m and ν c are largely independent of their ordering in the spectrum(Granot & Sari 2002; van Eerten & Wijers 2009), the valid-ity of our approach towards optical data is ensured. From 5days onwards, no approximation is made for the value of anyof the critical frequencies and the model assumes its mostaccurate form.It is worth noting that the best-fit spectra naturallyexplain the spectral evolution (at ∼ 100 days) depicted inFig. 5 of Frail et al. (2000), due to the passage of ν m . Inthe ISM case ν m crosses the radio earlier, at around 45days, something excluded by the data. Frail et al. (2000)also find ν a = 3 GHz at seven days, whereas in our best fit ν a = 5 . . c (cid:13) , 1–15 its of a spherical model to GRB afterglows and 8 . 46 GHz varies between 0 . . ν c is observed to pass through thenear-infrared bands at ∼ 10 days. When all the availabledata from several different bands ( K, I, R, V ) are taken intoaccount, we find that the spectral index starts off (at ∼ Wind scenario, ν c lies be-low the optical bands throughout the duration of opticalobservations. Therefore, its exact value is important at allobserver times. As mentioned in Section 4.1, calculation of t NR yields ∼ 180 days. This implies that the values of ν c during late near-infrared and optical observations (extend-ing up to ∼ 200 days in the R band) should be mildly af-fected by the transition towards the Newtonian dynamicalphase. Since ν c is not included in the treatment of Leventiset al. (2012) we do not have a description of the transitionfor this critical frequency, at least not at the level of accu-racy that we do for the others. Assuming that the trans-relativistic behaviour of ν c is similar to those of the otherspectral parameters (smoothly broken power-law) and thatthe break is centered around t NR , we have explored vari-ous sharpnesses for that transition and found that the fitresults remain consistent. The only parameter that changesnoticeably is p which grows from 2 . 28 to about 2 . 34 whenthe transrelativistic evolution of ν c is taken into account.Having established that this evolution does not affect theinferred parameters, the results we present in Table 1 areobtained using the relativistic formula for ν c only. First noticed in van Eerten et al. (2010a) and subsequentlyquantified in Leventis et al. (2012), the duration of the trans-relativistic phase of a spherical outflow in the observer framecan be long (this also holds in the case of a jetted outflow;Zhang & MacFadyen 2009). The near-infrared and opticallight curves in Fig. 1 show strong deviations from the ultra-relativistic behaviour already at a few tens of days, in ob-server time. Their progressive steepening is caused entirelyby the dynamics slowly adjusting to the Sedov-Taylor solu-tion, as there is no critical frequency crossing these bands.The effect is similar in the radio, but less pronounced dueto the simultaneous spectral evolution.Deviations of the observed radio light curves from therelativistic scalings at timescales of several weeks promptedWaxman et al. (1998) to propose a jetted outflow forGRB 970508. In this paper we demonstrate how accuratemodelling of the transrelativistic phase can account for thedeviations from the ultrarelativistic scalings at observertimes ≪ t NR . This implies that a similar trend may holdfor at least some other GRB afterglows, the temporal evo-lution of which has been interpreted as evidence for a jetbreak.Differentiating between jet breaks and the transition tothe non-relativistic phase is important, as it directly affectsthe inferred geometry and energetics of GRB outflows. Thereare two main quantities that can serve as diagnostics for thisdifferentiation. The first is the change of the temporal indexof the flux. In the case of a jet break, a decrease in thevalue of the temporal index is expected, mainly due to the missing-flux effect (Panaitescu et al. 1998) that arises whenthe edges of the jet become visible. On the other hand, thechange in the temporal slope, as the outflow approaches thetransition to non-relativistic velocities, may be positive ornegative, is a function of k and (depending on the spectralregime) p , and is known from theory (e.g. van Eerten et al.2010a). A second diagnostic is the duration (smoothness) ofthe change in the temporal index. In the case of a jet breakthe transition lasts from factors of few (ISM environment)up to a decade (wind environment) in observer time (DeColle et al. 2012b; van Eerten & MacFadyen 2012a), whereasthe typical duration of the transrelativistic regime in thecase of spherical outflows is a few decades (Leventis et al.2012). The picture is slightly more complicated in the caseof a jet break observed off axis. In that case the jet-breaktransition is effectively stretched and postponed (in realityit splits in two). Good coverage is then critical to discernbetween the different interpretations. Several broadband fits to the afterglow of GRB 970508 havebeen performed and presented in the literature (Wijers& Galama 1999; Chevalier & Li 2000; Yost et al. 2003;Panaitescu & Kumar 2002). Others have fit only late-timeradio data (Frail et al. 2000; Berger et al. 2004), while Star-ling et al. (2008) have fit only the slopes of light curves andspectra to infer values for p and k . Most of these studiesassume or find that an ISM scenario fits the data better,apart from Chevalier & Li (2000) and Panaitescu & Ku-mar (2002) who favour the Wind case. In this study wehave presented a detailed investigation of both density struc-tures that clearly favours a stellar-wind CBM. In addition wedemonstrate how models with no assumption on the slopeof the CBM converge to the Wind scenario. Interestingly,Starling et al. (2008) find that, in their sample, four out offive afterglows with well-constrained values for k suggest thesame. In that study the density structure of GRB 970508 ispoorly constrained.There seems to be more agreement on the geometryof the outflow of GRB 970508. Most studies (also Rhoads1999) do not need to invoke a jet, while those that do in-fer a jet geometry, usually find large half-opening angles:18 ◦ (Panaitescu & Kumar 2002), 30 ◦ (Frail et al. 2000), 50 ◦ (Yost et al. 2003). We find that the spherical model pro-vides a good description of the data, capturing the trendsof the light curves at different wavelengths for more thantwo orders of magnitude in observer time. We argue thatGRB 970508 may have indeed originated from an almostspherical outflow. The energy of the prompt emission is es-timated around 5 · erg, if isotropic (Bloom et al. 2001).Although on the high side, this value is not unreasonable(e.g. Metzger et al. 2011).In terms of the whole set of fitted parameters, ourresults are similar to those of Chevalier & Li (2000) andPanaitescu & Kumar (2002). Given the uncertainties in thelast row of Table 1, their best-fit values are within, or justoutside the allowed range of our results. We find moderatelyhigher values for ǫ e and ǫ B than both studies, but these val-ues are effectively upper limits. Lowering ξ to 0 . E ≃ . A ∗ ≃ . 73 g cm − , ǫ e ≃ ǫ B ≃ . 19, while thevalue for p remains the same, 2 . 28. None of these parame- c (cid:13) , 1–15 Leventis et al. ters are more than a factor of three off compared to bothaforementioned studies (note, however, the inference of ajet from Panaitescu & Kumar 2002). It is worth mention-ing that the blast-wave energy inferred through modellingof the afterglow radiation is very similar to the radiativeoutput of the prompt emission. This result holds regardlessof the outflow geometry and implies a very high efficiencyof the gamma-ray radiation from the main burst. However,given the fast cooling at early times, adiabatic evolution ofthe blast wave demands ǫ e ≪ 1. For ξ < . 17, both ǫ e and ǫ B are smaller than 10%. The corresponding blast wave energybecomes > · erg, which reduces the efficiency of theprompt emission below 40%. In this Section we discuss the implications of our results forthe properties of GRB outflows and afterglow fitting. We have demonstrated how a spherical outflow can accountfor the observations of GRB 970508 and how it fails in thecase of GRB 980703. The former afterglow has often beensuccessfully modelled both with a spherical and a collimatedoutflow. For GRB 980703 a jet is invariably inferred and inthis research, similarly to Frail et al. (2003), we find that aspherical model cannot provide an adequate description tothe data under any combination of physical parameters.The degree of collimation in the case of GRB 070125is less clear. On the one hand, there are only a few studiesof the afterglow radiation and only one of them performsbroadband (radio to X rays) fitting (Chandra et al. 2008).On the other hand, our results provide a satisfactory de-scription of most of the broadband data, apart from latetime behaviour in the radio, when an additional componentis observed in the light curves. This component, however,cannot be explained in the jetted model of Chandra et al.(2008) either. Moreover, they propose that the X-ray fluxis dominated by inverse Compton, in order to explain whatseems to be a chromatic break in the optical and X-ray lightcurves ( at about 4 and 10 days, respectively). However, theclaim for a jet-break in the optical is based only on two datapoints (one in the I and one in the R band), while in the Xrays it is only based on one (see Fig. 3). We find that a spher-ical model offers a similar level of accuracy, without the needto invoke a jet or other radiation mechanisms beyond syn-chrotron. However, the parameters we obtain are extreme,both on the microphysics side, but also in the total energybudget they imply ( > erg). We therefore consider itlikely that the model we have used lacks some physics, whichat least in some bands and observer times ‘drives’ the radi-ated spectrum. That extra physics could be a jetted outflow,but the evidence from previous studies combined with ourfindings is not conclusive.In this study we cannot quantify the opening angle ofjets, in cases that one is inferred. We can, however, qualifyafterglows as spherical by successfully fitting their broad-band data set. This has been the case for GRB 970508 andwe consider this a clear demonstration of the diversity inthe geometry of GRB outflows. This is in accordance with searches for jet breaks in large samples of afterglow ob-servations that fail to clearly identify a jet break in morethan half of the sources (Kocevski & Butler 2008; Racusinet al. 2009). However, in the collapsar model (MacFadyen& Woosley 1999) for long GRBs, it is likely that outflowsare still collimated right after breaking out of the stellarenvelope (e.g. Morsony et al. 2007). If the opening angleof the jet is large, the light curves will exhibit deviationsfrom spherical symmetry during the transrelativistic phase.In such a quasi-spherical scenario, the observational signa-tures of decollimation might be weak and the expected dif-ferences from a perfectly spherical outflow have, to the bestof our knowledge, not been explored in the literature. In thecase of GRB 970508, t NR = 180 days in the Wind class, andthe decollimation should occur on similar timescales, closeto the end of data sampling. Therefore, if the outflow ofGRB 970508 had a very large opening angle, our fits will bedominated by observations of an almost conical flow. The in-ferred energy would then be the isotropic equivalent of thereal energy content of the blast wave, which is lower onlyby a factor of order unity. The best-fit values of p and k are inferred by the slopes of spectra and light curves which,at least for the best part of the observations, are not influ-enced by effects caused by a possible quasi-spherical geome-try. Therefore, we would not expect our general conclusionsconcerning the slope of the CBM to be significantly affectedby such a scenario.Quantifying the distribution of jet opening angles isnot an easy task, especially considering the inadequate (forbroadband modelling) coverage that a large fraction of after-glows have. On the observational side, Curran et al. (2008)have shown that jet breaks may be misidentified as sin-gle power laws, due to data-analysis effects. Moreover, vanEerten et al. (2010b) have shown that a moderately off-axisviewing angle (but smaller than the jet semi-opening an-gle) can ‘mask’ the appearance of a jet-break. If jets arepresent, observing them off axis should happen more oftenthan not. Therefore, this is an important effect that shouldbe taken into account in the model fits. Another issue thatneeds to be better understood is the early (10 − s) af-terglow behaviour which in a large fraction of bursts sug-gests some form of energy injection, continuous or irregular(Nousek et al. 2006; Panaitescu & Vestrand 2011). This mayaffect the overall dynamics of the outflow but also result inmisinterpreting a potentially coincident jet break (Racusinet al. 2009). Thus, connecting the dynamics of the early af-terglow with the more regular behaviour observed at largertimescales is essential to uncover evidence for jets that maynot be in the form of the canonical achromatic jet break. In this work we have treated the density structure of theCBM as a free parameter ( k ), assuming that a constantpower law applies. Out of the three data sets we studied,one (GRB 970508) showed convergence to a constant stellarwind, represented by k = 2. The best fit to GRB 980703 isobtained for k = 1 . k = 1 . 67, which is closer to that of a constant stel-lar wind than homogeneous CBM. For all data sets, Wind environments produce better fits than the ISM class. c (cid:13) , 1–15 its of a spherical model to GRB afterglows Similarly, however, to the discussion on the geometry ofthe outflows, GRB 970508 is the only one with reliable re-sults. In both GRB 980703 and GRB 070125 large values forboth p and k are needed to best describe the data within thespherical model, the applicability of which is at least doubt-ful in these cases. For GRB 970508, the value of A ∗ impliesthat the inferred density profile corresponds to a constantmass-loss rate of 2 . · − ξ − M ⊙ / yr, for a wind veloc-ity of 1 , 000 km s − . Interpreting the adiabatic condition as ξ < . 17, we find ˙ M > . · − M ⊙ / yr, which implies a rel-atively massive Wolf-Rayet star towards the end of its life(Chevalier & Li 1999).Several studies have fit individual bursts and found orassumed a homogeneous density structure for the CBM.When the fits are compared against those with stellar-windCBM the results are often ambiguous (e.g. Frail et al. 2003;Chandra et al. 2008), while in some cases the Wind scenarioseems to be favoured (Chevalier & Li 2000; Panaitescu &Kumar 2002). On the theoretical side, van Eerten & Mac-Fadyen (2012a) have shown that the majority of Swift postjet break slopes are not reconcilable with a constant den-sity CBM, if late energy injection and viewing angle do notsignificantly affect the observations. Instead, the observedslopes suggest a wind-type environment for the CBM. Star-ling et al. (2008) have studied a sample of 10 Beppo-SAX afterglows and found that the majority of the data sets thatwere sufficient to constrain the value of k implied a stellar-wind CBM. However, half of them have error bars that allowfor a wide range for k . Curran et al. (2009) have performeda similar study using Swift bursts and find a division in thesample between constant and wind-like profiles. It seems,therefore, likely that the density structure of the CBM inGRBs is diverse, similar to the geometric characteristics oftheir outflows. However, this does not necessarily translateto diversity of the progenitors as well, because in the col-lapsar model (Woosley 1993; MacFadyen & Woosley 1999)the CBM of a large fraction of long GRBs is modified bymultiple stellar winds from the neighbouring stars (Mimica& Giannios 2011). For all the afterglows we studied, we have found that a multi-frequency data set is more suitable for fitting all the parame-ters at once. This has led to the expansion of the model withthe inclusion of the cooling frequency of the synchrotronspectrum, ν c . However, even when radio to X-ray data arefitted and all details of the spectrum are taken into account,setting k a free parameter results in large uncertainties, ifno assumption for the microphysics is made. This is mani-fested in the large errors for the best-fit values of physicalparameters in the case of GRB 970508 (see row 7 of Table1). When k is free and no microphysics assumption is made,the number of fitted parameters is six, equal to the max-imum number of constraints we can have from the lightcurves – four from the positions of the critical frequenciesand the value of F m , plus two more from the slopes of spectraand light curves. However, our results imply that not all ofthese constraints are efficiently used during the fitting pro-cess. This means that the effects of two or more of the con-straints cannot be separated, leading to a case-specific de- generacy. In the case of GRB 970508, for the best-fit model,both ν m and ν c lie between radio and near-infrared bandsfor the best part of the observations ( ν m stays above theradio bands for about 100 days). Therefore, their positionsare not independently constrained by the data, leading toa wide range of possible values when all six parameters aresimultaneously fitted.An interesting feature of the prescriptions we have usedis the inclusion of ξ as a parameter. Due to the degeneracy ofthe model, the presence of ξ is not necessary per se. One canimagine a situation where a range in the allowed values for ξ is reflected in the adjustment of the ranges of the other pa-rameters. For example, by assuming that ξ = 1 and allowing ǫ B and ǫ e to obtain values > ξ being smaller than those two param-eters, while all of them are smaller than unity. However, theinclusion of ξ in the model demonstrates these situationsmore clearly. In the results we obtain for GRB 970508 it wasnot initially possible to discern between the Medvedev con-straint and the equipartition constraint for the microphysicsdue to the high values of both ǫ B and ǫ e , that, within theuncertainties, extend to the upper limit of the allowed range.By freezing ξ at values much lower than 1, we have excludedthe presence of better fits in which ǫ B > ξ and/or ǫ e > ξ ,and confirmed that energy equipartition between power-lawelectrons and magnetic field describes better the afterglowobservations of GRB 970508. We have performed broadband fits of three afterglow datasets using accurate analytic flux prescriptions applicable tospherical outflows. We have shown that GRB 970508 is suc-cessfully fit by a spherical model. The fits fail in the case ofGRB 980703 and GRB 070125 at varying degrees, implyingthat these sources may be indeed related to jetted outflows.This is supported by extensive modelling of the former andthe extremely high isotropic energy inferred for the latter.For GRB 970508 we find that the best-fit value for k is practically 2, strongly suggesting a stellar-wind environ-ment. Fits to GRB 970508 also show strong evidence for apopulation of electrons that is not accelerated at the forwardshock. The implied values for the microphysics parameters, ǫ e and ǫ B , suggest that they are close to equipartition.Modelling of GRB 970508 illustrates how an accuratespherical model accounts for the progressive deviations oflight curves from the ultrarelativistic scalings at t obs ≪ t NR .This feature had been previously interpreted as a jet breakin the context of simpler models, but emerges naturally fromprecise calculations of dynamics and spectra in the sphericalscenario. Therefore, we consider it possible that similar fea-tures in the data sets of other afterglows have been misinter-preted as jet breaks, in the absence of detailed calculationsfor the spherical case. We would like to thank A. De Cia for providing the data forGRB 070125. This research was supported by NOVA and inpart by NASA through grant NNX10AF62G issued through c (cid:13) , 1–15 Leventis et al. REFERENCES Berger E., Kulkarni S. R., Frail D. A., 2001, ApJ, 560, 652Berger E., Kulkarni S. R., Frail D. A., 2004, ApJ, 612, 966Blandford R. D., McKee C. F., 1976, Physics of Fluids, 19,1130Bloom J. S., Frail D. A., Kulkarni S. R., Djorgovski S. G.,Halpern J. P., Marzke R. O., Patton D. R., Oke J. B.,Horne K. D., Gomer R., Goodrich R., Campbell R.,Moriarty-Schieven G. H., Redman R. O., Feldman P. A.,Costa E., Masetti N., 1998, ApJ, 508, L21Bloom J. S., Frail D. A., Sari R., 2001, AJ, 121, 2879Castro-Tirado A. J., Zapatero-Osorio M. R., Gorosabel J.,Greiner J., Heidt J., Herranz D., Kemp S. N., Mart´ınez-Gonz´alez E., et al., 1999, ApJ, 511, L85Cenko S. B., Fox D. B., Penprase B. E., Cucchiara A., PriceP. A., Berger E., Kulkarni S. R., Harrison F. A., Gal-YamA., Ofek E. O., Rau A., Chandra P., Frail D. A., KasliwalM. M., Schmidt B. P., Soderberg A. M., Cameron P. B.,Roth K. C., 2008, ApJ, 677, 441Chandra P., Cenko S. B., Frail D. A., Chevalier R. A.,Macquart J.-P., Kulkarni S. R., Bock D. C.-J., BertoldiF., et al., 2008, ApJ, 683, 924Chary R., Neugebauer G., Morris M., Becklin E. E.,Matthews K., Kulkarni S. R., Lowrance P. J., ZuckermanB., Mastrodemos N., 1998, ApJ, 498, L9Chevalier R. A., Li Z.-Y., 1999, ApJ, 520, L29Chevalier R. A., Li Z.-Y., 2000, ApJ, 536, 195Curran P. A., Starling R. L. C., van der Horst A. J., WijersR. A. M. J., 2009, MNRAS, 395, 580Curran P. A., van der Horst A. J., Wijers R. A. M. J., 2008,MNRAS, 386, 859De Cia A., Starling R. L. C., Wiersema K., van der HorstA. J., Vreeswijk P. M., Bj¨ornsson G., de Ugarte PostigoA., Jakobsson P., Levan A. J., Rol E., Schulze S., TanvirN. R., 2011, MNRAS, 418, 129De Colle F., Granot J., L´opez-C´amara D., Ramirez-RuizE., 2012a, ApJ, 746, 122De Colle F., Ramirez-Ruiz E., Granot J., Lopez-CamaraD., 2012b, ApJ, 751, 57Djorgovski S. G., Kulkarni S. R., Bloom J. S., GoodrichR., Frail D. A., Piro L., Palazzi E., 1998, ApJ, 508, L17Eichler D., Granot J., 2006, ApJ, 641, L5Eichler D., Waxman E., 2005, ApJ, 627, 861Frail D. A., Kulkarni S. R., Nicastro L., Feroci M., TaylorG. B., 1997, Nature, 389, 261Frail D. A., Waxman E., Kulkarni S. R., 2000, ApJ, 537,191Frail D. A., Yost S. A., Berger E., Harrison F. A., Sari R.,Kulkarni S. R., Taylor G. B., Bloom J. S., Fox D. W.,Moriarty-Schieven G. H., Price P. A., 2003, ApJ, 590, 992Galama T. J., Groot P. J., van Paradijs J., KouveliotouC., Strom R. G., Wijers R. A. M. J., Tanvir N., BloomJ., Centurion M., Telting J., Rutten R. G. M., Smith P., Mackey C., Smartt S., Benn C., Heise J., in ’t Zand J.,1998a, ApJ, 497, L13Galama T. J., Wijers R. A. M. J., Bremer M., Groot P. J.,Strom R. G., de Bruyn A. G., Kouveliotou C., RobinsonC. R., van Paradijs J., 1998b, ApJ, 500, L101Galama T. J., Wijers R. A. M. J., Bremer M., Groot P. J.,Strom R. G., Kouveliotou C., van Paradijs J., 1998c, ApJ,500, L97Garcia M. R., Callanan P. J., Moraru D., McClintock J. E.,Tollestrup E., Willner S. P., Hergenrother C., RobinsonC. R., Kouveliotou C., van Paradijs J., 1998, ApJ, 500,L105Goodman J., 1997, New A, 2, 449Granot J., K¨onigl A., Piran T., 2006, MNRAS, 370, 1946Granot J., Kumar P., 2006, MNRAS, 366, L13Granot J., Nakar E., Piran T., 2003, Nature, 426, 138Granot J., Piran T., 2012, MNRAS, 421, 570Granot J., Sari R., 2002, ApJ, 568, 820Harrison F. A., Bloom J. S., Frail D. A., Sari R., KulkarniS. R., Djorgovski S. G., Axelrod T., Mould J., et al., 1999,ApJ, 523, L121Hjorth J., Sollerman J., Møller P., Fynbo J. P. U., WoosleyS. E., Kouveliotou C., Tanvir N. R., Greiner J., othersJ.2003, Nature, 423, 847Huang Y. F., Dai Z. G., Lu T., 1999, MNRAS, 309, 513Kobayashi S., Piran T., Sari R., 1999, ApJ, 513, 669Kocevski D., Butler N., 2008, ApJ, 680, 531Kumar P., Piran T., 2000, ApJ, 532, 286Leventis K., van Eerten H. J., Meliani Z., WijersR. A. M. J., 2012, MNRAS, 427, 1329MacFadyen A. I., Woosley S. E., 1999, ApJ, 524, 262Medvedev M. V., 2006, ApJ, 651, L9Meliani Z., Keppens R., Casse F., Giannios D., 2007, MN-RAS, 376, 1189Metzger B. D., Giannios D., Thompson T. A., BucciantiniN., Quataert E., 2011, MNRAS, 413, 2031Metzger M. R., Djorgovski S. G., Kulkarni S. R., SteidelC. C., Adelberger K. L., Frail D. A., Costa E., FronteraF., 1997, Nature, 387, 878Mimica P., Giannios D., 2011, MNRAS, 418, 583Morsony B. J., Lazzati D., Begelman M. C., 2007, ApJ,665, 569Nousek J. A., Kouveliotou C., Grupe D., Page K. L., Gra-not J., Ramirez-Ruiz E., Patel S. K., Burrows D. N., et al.,2006, ApJ, 642, 389Paczy´nski B., Rhoads J. E., 1993, ApJ, 418, L5Panaitescu A., 2005, MNRAS, 362, 921Panaitescu A., Kumar P., 2000, ApJ, 543, 66Panaitescu A., Kumar P., 2001, ApJ, 554, 667Panaitescu A., Kumar P., 2002, ApJ, 571, 779Panaitescu A., Meszaros P., Rees M. J., 1998, ApJ, 503,314Panaitescu A., Vestrand W. T., 2011, MNRAS, 414, 3537Pe’er A., 2012, ApJ, 752, L8Piran T., 2004, Reviews of Modern Physics, 76, 1143Piro L., Amati L., Antonelli L. A., Butler R. C., Costa E.,Cusumano G., Feroci M., Frontera F., Heise J., in ’t ZandJ. J. M., Molendi S., Muller J., Nicastro L., Orlandini M.,Owens A., Parmar A. N., Soffitta P., Tavani M., 1998,A&A, 331, L41Racusin J. L., Liang E. W., Burrows D. N., Falcone A.,Sakamoto T., Zhang B. B., Zhang B., Evans P., Osborne c (cid:13) , 1–15 its of a spherical model to GRB afterglows J., 2009, ApJ, 698, 43Rees M. J., M´esz´aros P., 1992, MNRAS, 258, 41PRhoads J. E., 1999, ApJ, 525, 737Sahu K. C., Livio M., Petro L., Bond H. E., MacchettoF. D., Galama T. J., Groot P. J., van Paradijs J., Kouve-liotou C., 1997, ApJ, 489, L127Sari R., Esin A. A., 2001, ApJ, 548, 787Sari R., Piran T., Narayan R., 1998, ApJ, 497, L17+Sedov L. I., 1959, Similarity and Dimensional Methods inMechanicsSokolov V. V., Kopylov A. I., Zharikov S. V., Feroci M.,Nicastro L., Palazzi E., 1998, A&A, 334, 117Starling R. L. C., 2008, A&A, 488, 915Starling R. L. C., van der Horst A. J., Rol E., WijersR. A. M. J., Kouveliotou C., Wiersema K., Curran P. A.,Weltevrede P., 2008, ApJ, 672, 433Starling R. L. C., Wijers R. A. M. J., Wiersema K., RolE., Curran P. A., Kouveliotou C., van der Horst A. J.,Heemskerk M. H. M., 2007, ApJ, 661, 787Taylor G., 1950, Royal Society of London Proceedings Se-ries A, 201, 159Taylor G. B., Frail D. A., Beasley A. J., Kulkarni S. R.,1997, Nature, 389, 263van der Horst A. J., Kamble A., Resmi L., WijersR. A. M. J., Bhattacharya D., Scheers B., Rol E., StromR., Kouveliotou C., Oosterloo T., Ishwara-Chandra C. H.,2008, A&A, 480, 35van Eerten H., MacFadyen A., 2012a, arXiv:1209.1985van Eerten H., van der Horst A., MacFadyen A., 2012, ApJ,749, 44van Eerten H. J., Leventis K., Meliani Z., WijersR. A. M. J., Keppens R., 2010a, MNRASvan Eerten H. J., MacFadyen A. I., 2012b, ApJ, 747, L30van Eerten H. J., Wijers R. A. M. J., 2009, MNRAS, 394,2164van Eerten H. J., Zhang W., MacFadyen A., 2010b, ApJVreeswijk P. M., Galama T. J., Owens A., Oosterbroek T.,Geballe T. R., van Paradijs J., Groot P. J., KouveliotouC., et al., 1999, ApJ, 523, 171Waxman E., 1997, ApJ, 489, L33Waxman E., Kulkarni S. R., Frail D. A., 1998, ApJ, 497,288Wijers R. A. M. J., Galama T. J., 1999, ApJ, 523, 177Wijers R. A. M. J., Rees M. J., Meszaros P., 1997, MNRAS,288, L51Woosley S. E., 1993, ApJ, 405, 273Wygoda N., Waxman E., Frail D. A., 2011, ApJ, 738, L23Yost S. A., Harrison F. A., Sari R., Frail D. A., 2003, ApJ,597, 459Zhang B., Fan Y. Z., Dyks J., Kobayashi S., M´esz´aros P.,Burrows D. N., Nousek J. A., Gehrels N., 2006, ApJ, 642,354Zhang W., MacFadyen A., 2009, ApJ, 698, 1261Zharikov S. V., Sokolov V. V., 1999, A&AS, 138, 485 c (cid:13)000