Calorimetry of GRB 030329: Simultaneous Fitting to the Broadband Radio Afterglow and the Observed Image Expansion Rate
aa r X i v : . [ a s t r o - ph . H E ] J u l Draft version September 4, 2018
Preprint typeset using L A TEX style emulateapj v. 5/2/11
CALORIMETRY OF GRB 030329: SIMULTANEOUS MODEL FITTING TO THE BROADBAND RADIOAFTERGLOW AND THE OBSERVED IMAGE EXPANSION RATE
Robert A. Mesler
Department of Physics and Astronomy, University of New Mexico MSC07 4220, Albuquerque, NM 87131 andYlva M. Pihlstr¨om Department of Physics and Astronomy, University of New Mexico, MSC07 4220, Albuquerque, NM 87131
Draft version September 4, 2018
ABSTRACTWe perform calorimetry on the bright gamma ray burst (GRB) 030329 by fitting simultaneouslythe broadband radio afterglow and the observed afterglow image size to a semi-analytic magneto-hydrodynamical (MHD) and afterglow emission model. Our semi-analytic method is valid in boththe relativistic and non-relativistic regimes, and incorporates a model of the interstellar scintillationthat substantially effects the broadband afterglow below 10 GHz. The model is fitted to archivalmeasurements of the afterglow flux from 1 day to 8.3 years after the burst. Values for the initial burstparameters are determined and the nature of the circumburst medium is explored. Additionally, directmeasurements of the lateral expansion rate of the radio afterglow image size allow us to estimate theinitial Lorentz factor of the jet.
Subject headings: gamma rays: bursts INTRODUCTION
A Gamma ray burst (GRB) afterglow is produced bythe interaction of the GRB jet with the circumburstmedium in which it is immersed. An afterglow spec-trum is composed of a series of power laws separatedby breaks (Sari et al. 1998), and is produced when elec-trons in the circumburst medium spiral around the tan-gled and compressed magnetic field lines present at theshock boundary between the jet and the circumburstmedium. In this manner, the kinetic energy of the jetis gradually converted into radiation and particle energy(Meszaros & Rees 1993; Katz 1994). The properties ofthe jet- including its initial kinetic energy, mass, andhalf-opening angle- only partially determine the tem-poral evolution of the afterglow. Because the after-glow luminosity is dependent upon the density of thecircumburst medium, the nature of that medium (i.e.,whether it is wind-like or a uniform density ISM) playsa crucial role in determining the temporal evolution aswell (Sari et al. 1998; Huang et al. 2000; Chevalier & Li2000; Granot & Sari 2002; Mesler et al. 2012b). Ana-lytical models of relativistic jets in power law mediums( n ( r ) ∝ r − k , with k = 0 for a uniform density and k = 2for a wind) have yielded light curves that are in rea-sonable agreement with observations (Berger et al. 2000;Yost et al. 2003; Curran et al. 2011; Price et al. 2002).Semi-analytic models developed by Berger et al.(2003), Frail et al. (2005), and van der Horst et al.(2005) have been used to estimate parameters of theGRB 030329 radio afterglow, including the isotropicequivalent energy E k , the electron spectral index p , theelectron and magnetic field energy fractions ( ǫ e and ǫ B ,respectively), and the jet half-opening angle θ j , usingbroadband radio observations over the course of the first [email protected] Ylva Pihlstr¨om is also an Adjunct Astronomer at the Na-tional Radio Astronomy Observatory year. van der Horst et al. (2008) extend this to the first ∼ .
84 and 250 GHz as well asarchival VLA observations at 1.4, 4.9, and 8.5 GHz takenbetween 1.7 and 8.3 years after the burst to perform burstcalorimetry deep into the non-relativistic regime. Thebroadband afterglow is modelled using a method devel-oped in Mesler et al. (2012b). GRB 030329
At a redshift of z = 0 . CDM cosmol-ogy with H = 71 km s − Mpc − , Ω M = 0 .
27, andΩ Λ = 0 .
73, GRB 030329 is located at an angular dis-tance of d A = 587 Mpc, with 1.0 mas corresponding to2.85 pc. The burst was first detected on March 29th,2003 at 11:37 UTC by the High Energy Transient Ex-plore 2 ( HETE-2 ) satellite, and was subsequently local-ized in the optical by Peterson & Price (2003). The GRB030329 radio afterglow was (and still is) the most lumi-nous afterglow to ever have been observed, achieving amaximum flux density of 55 mJy at 43 GHz one weekafter the burst. The relative proximity of the burst tothe Earth, coupled with its extremely high luminosity,made it possible for the radio afterglow to be directlyresolved by the Very Long Baseline Array (VLBA) andthe Global VLBI Array at 5 GHz (Taylor et al. 2004,2005; Pihlstr¨om et al. 2007; Mesler et al. 2012a). Addi-tionally, the 5 GHz radio afterglow was detected by theVLA for 8.3 years. GRB 030329 has therefore provideda unique opportunity to study the evolution of both theluminosity and the physical size of a GRB afterglow ARCHIVAL DATA
We compare our model to observations of the GRB030329 radio afterglow luminosity. In addition to pre-viously published data, we include late-period 1 .
4, 4 .
9, Mesler and Pihlstr¨omand 8 . . . . . MODELING THE JET EXPANSION
In this section, we will discuss the semi-analyticmethod that we have developed for modelling GRB after-glow emission. This model builds upon work appearingin Mesler et al. (2012b). It will be outlined below withemphasis on improvements that have been made since itspublication in Mesler et al. (2012b).Gamma ray bursts are modelled as initially relativistic,double-sided jets that propagate outward into an ambientcircumburst medium. In the following discussion, we willuse primed quantities to refer to the reference frame thatis comoving with the jet, we will use un-primed quan-tities with no subscript to refer to the reference framein which the ISM is at rest, and we will use quantitieswith the subscript ⊕ to refer to the reference frame of anEarthbound observer. Jet Hydrodynamics
An expression for the evolution of the jet Lorentz fac-tor Γ can be found by invoking the requirement for theconservation of energy. The jet will sweep up materialfrom the circumburst medium as it propagates, forcingthe jet to decelerate. Pe’er (2012) shows that d Γ dm = − ˆ γ (cid:0) Γ − (cid:1) − (ˆ γ −
1) Γ β M ej + ǫm + (1 − ǫ ) m [2ˆ γ Γ − (ˆ γ −
1) (1 + Γ − )] , (1)where Γ is the Lorentz factor of the jet, M ej is the initialmass of the jet ejecta, m is the total mass that has beenswept up by the jet, β = (cid:0) − Γ − (cid:1) / is the normalizedbulk velocity, ˆ γ ≃ (4Γ + 1) / (3Γ) is the adiabatic index(Huang et al. 2000), and ǫ = ǫ e t ′− t ′− + t ′ ex ) − (2)is the radiative efficiency of the jet (Dai et al. 1999).The expansion time t ′ ex = t/ ( β Γ c ) and the synchrotroncooling time t ′ syn = 6 πc/σ T ǫ e Γ m p B ′ (Dai & Lu 1998), where c is the speed of light, σ T is the Thompson scat-tering cross-section, ǫ e is the fraction of the burst energystored in the electrons, m p is the proton mass, and B ′ isthe magnitude of the comoving frame magnetic field.Our semi-analytic method is capable of producing lightcurves in arbitrary density profiles, but we limit ourselvesfor the sake of simplicity to a uniform density typical ofan interstellar medium (ISM) and an n ( r ) ∝ r − stellarwind. Density profiles are modelled as a series of uniformgrid points of width 10 − pc. The model is fed an initialkinetic energy E K , ejecta mass M ej , and jet half-openingangle θ . The total mass swept up by the jet ( m ), thejet’s Lorentz factor (Γ), the width of the leading edgeof the jet perpendicular to the observer’s line of sight( a ), and the isotropic frame time ( t ) are then solved forsimultaneously as a function of the radius ( r ) of the jet.The evolution of the jet is identical to the case wherethe GRB outflow is isotropic until t = t jet , when thecenter of the jet comes into causal contact with its edge.For a jet with half-opening angle θ j = arctan a/r , thisoccurs when Γ ≃ /θ j . At t jet , the jet experiences rapidlateral expansion, leading to an increase in the amount ofcircumburst material being swept up by the jet with time.The increase in swept-mass leads to a faster decelerationand a decrease in the jet luminosity. Eventually, the jetis decelerated to the point that it is no longer relativistic,and it transitions to spherical expansion. The transitiontime to non-relativistic expansion is denoted t NR . Determining the Initial Lorentz Factor
GRB 030329 is unique in that its afterglow has been di-rectly resolved using VLBI at 5 GHz (Taylor et al. 2004,2005; Pihlstr¨om et al. 2007; Mesler et al. 2012a). Withmultiple epochs of observation, the expansion history ofthe burst can be determined directly, allowing us to placeconstraints on the initial Lorentz factor of the burst.Initially, the mass being swept up by the GRB jet willbe negligible as compared to the initial jet ejecta mass M ej . The jet will coast at nearly constant speed in thisregime assuming that ǫ ≪ ≪ M ej /m . It is only whenthe sum of the two terms in the denominator of equa-tion 1 that are dependent upon the swept mass m is ofthe same order as the initial ejecta mass M ej that thejet begins to decelerate appreciably. We will define theisotropic frame time t = t dec as the time at which M ej = ǫm + (1 − ǫ ) m (cid:2) γ Γ − (ˆ γ − (cid:0) − (cid:1)(cid:3) . (3)The afterglow linear size evolves according to the co-moving sound speed (Huang et al. 2000) c ′ s = s ˆ γ (ˆ γ −
1) (Γ − γ (Γ − c. (4)In the regime where t < t dec , the Lorentz factor is nearlyconstant, meaning that the afterglow expands laterallyat a nearly constant rate β ⊥ . Using the relationship be-tween the isotropic frame time and the Earth frame time dt = Γ dt ′ = Γ (cid:16) Γ + p Γ − (cid:17) dt ⊕ , (5)we find that the relationship between the expansion ratealorimetry of GRB 030329 3 Date ∆t Frequency Flux Density VLA Project Code(days) (GHz) ( µ Jy)2004 Dec 09 621.2 1.4 650 ±
70 AF4142004 Dec 09 621.2 8.5 250 ±
30 AF4142004 Dec 23 635.1 1.4 590 ±
70 AF4142004 Dec 23 635.1 4.9 370 ±
60 AF4142004 Dec 23 635.1 8.5 300 ±
40 AF4142005 Jan 23 665.9 1.4 420 ±
80 AK5832005 Jan 30 672.9 1.4 480 ±
70 AK5832005 Jan 31 673.7 1.4 510 ±
80 AS7962005 Mar 31 732.7 8.5 260 ±
40 AK5832005 Apr 07 739.7 1.4 650 ±
40 AF4142005 Apr 07 739.7 8.5 300 ±
30 AF4142005 Jun 08 801.5 1.4 380 ±
10 AK5832005 Oct 21 937.0 8.5 100 ±
30 AK5832005 Oct 29 944.5 8.5 120 ±
30 AK5832005 Dec 14 990.5 8.5 180 ±
20 AK5832005 Dec 26 1003.1 8.5 140 ±
30 AK5832006 Mar 22 1088.7 1.4 630 ±
30 AS8642006 Mar 22 1088.7 8.5 170 ±
20 AS8642006 Apr 15 1266.2 8.5 80 ±
30 AS8642006 Apr 29 1126.5 4.9 080 ±
20 AS9332006 Sep 15 1266.2 4.9 170 ±
50 AS8642011 Jul 03 3018.2 4.9 32 ±
10 10C-203
TABLE 1Observations of the GRB 030329 radio afterglow at . , . , and . GHz taken from the NRAO data archive and notappearing in a previous publication. β ⊥ of the afterglow and the initial Lorentz factor is β ⊥ ≃ √ (cid:18) Γ + q Γ − (cid:19) s + Γ − − (4Γ −
1) (6)for t < t dec .The average apparent expansion rate of the afterglow(in units of the speed of light) is defined as h β app i = (1 + z ) R ⊥ ct ⊕ , (7)where R ⊥ is the physical radius of the image, z is thesource’s cosmological redshift, t is the Earth-frame timeof observation, and c is the speed of light. The observedaverage lateral expansion rate of the GRB 030329 radioafterglow is shown in Fig. 1. At early times ( t < t dec ),the lateral expansion rate is nearly constant, and, there-fore, β ⊥ ≃ h β app i . After t = t dec , however, the jet beginsto decelerate and both β ⊥ and h β app i begin to decrease.The time t dec will therefore show up in the plot of h β app i vs. t as a break where d h β app i dt begins to decrease from itsinitial value of ∼ − .
42, as in Mesler et al. (2012a). In this interpretation,the time t dec occurs before the time of the first VLBIobservation at day 15, and we do not see the coastingphase during which the jet is moving at a nearly con-stant velocity. In the second model, the jet coasts at aconstant rate until day t dec = 83 days and then beginsto decelerate. Because both models fit the data equallywell, it is not possible to distinguish between them. Mod-els of the average apparent expansion rate which assume t dec >
83 days, however, produce steadily worse fits as t dec is increased, so we argue that t dec .
83 days. Thelower bound on the possible values of t dec can be foundby turning to our MHD models. An early t dec implies ahigh initial Lorentz factor and a longer transition time between the coasting phase and the decelerating phase.In order for t dec to have occurred before the date of thefirst observation at day 15, the jet must have had a lowenough initial Lorentz factor for its average apparent ex-pansion rate to be adequately modelled as a single powerlaw from day 15 onward. Our MHD models produce jetswith average apparent expansion rates that can be mod-elled as single power laws after day 15 only if t dec & t dec of 1 . t dec .
83 days, we obtain4 . . h β app i . .
0. Using Eqn 6, we then obtain anestimate for the initial Lorentz factor of 4 . Γ . θ ≃ ◦ and was responsible for thehigh energy emission (optical and higher frequencies),while the other component was only mildly relativisticwith a larger jet half-opening angle θ ≃ ◦ , and wasresponsible for emission at frequencies in the optical andbelow. Our value of 4 . Γ . M ej and the kinetic energy E k via E k = (Γ − M ej c . (8)Determination of the initial Lorentz factor can thereforeprovide an important constraint on the ratio between theinitial burst kinetic energy and the ejecta mass. Fitting the χ r MHD Models
Using the observations of the burst linear size (Fig.1), we produce a best-fit to our MHD models by vary-ing the kinetic energy E k , the ejecta mass M ej , and theinitial jet half-opening angle θ . Best-fits to a varietyof wind and uniform density profiles are shown in Fig.2. We find that the afterglow size evolution can be suc-cessfully fitted to either a wind or a uniform density, but Mesler and Pihlstr¨om −1 t (in days) 〈 β app 〉 t dec < 15 dayst dec = 83 days Fig. 1.—
The observed transverse expansion rate h β app i . Thered line corresponds to a single power law with index − .
42 as inMesler et al. (2012a). The blue line is a piecewise function consist-ing of a constant value before a time t dec = 83 days and a singlepower law thereafter. that lower values of χ r = χ /N , where N is the num-ber of degrees of freedom present in the model, can beobtained for fits of uniform media to the afterglow sizeevolution than for stellar winds ( χ r , uniform & . χ r , wind & . . Γ .
10 over the entire range in n and A ∗ thatwe searched (10 − < n < cm − and 0 . < A ∗ < − ). MODELLING THE LIGHT CURVE
While the linear size evolution of the afterglow cangive us insight into the initial Lorentz factor of the jet,it can only be used to determine the ratio between theinitial kinetic energy and the ejecta mass if the natureof the circumburst medium is unknown. In order to de-termine the values of the kinetic energy and the ejectamass, as well as the values of the other burst parameters,we must expand our model to include the properties ofthe afterglow emission. We build upon the model de-tailed in section 5 of Mesler et al. (2012b) to model theGRB 030329 synchrotron emission by fitting to broad-band radio afterglow observations between 840 MHz and250 GHz taken between 1 and 3018 days after the burst.
Spherical Emission and Beaming
The leading edge of a GRB jet is not a flat planethat propagates directly toward an Earthbound observer.Rather, it is a section of a spherical surface, meaning thatmaterial at different positions along the leading edge ofthe jet has different velocity components toward the ob-server. Relativistic beaming will cause radiation that isemitted by material with the largest velocity componentin the direction of the observer to be beamed toward theobserver more than radiation that is emitted by materialwith a slightly smaller velocity component. Additionally,jet material at different locations in the jet is not all ata uniform distance from the observer, leading to differ-ences in light travel time throughout the jet. The overalleffect of the relativistic beaming on the afterglow is anincrease in the total observed flux at early times when the jet is still relativistic. The difference in arrival timesof photons emitted from different regions of the jet makesthe light curve broader and more smooth than it wouldbe otherwise.To incorporate the effects of spherical emission andbeaming into our model, equations 27-32 of Mesler et al.(2012b) must be modified to account for the fact thatradiation emitted by different parts of the jet at the sametime t emit will not all reach the observer at the same time t ⊕ , nor will it all be beamed at the observer to the samedegree.The arrival time t ⊕ at which radiation emitted by thejet reaches the observer will be dependent upon the angle θ at which it was emitted with respect to a line connect-ing the center of the GRB progenitor to the observer.Material at angle θ = 0 will arrive at the observer first,while material emitted at θ = θ j will arrive last. Theequation for the equal arrival time surface is t ⊕ = Z (1 − β cos θ ) cβ dr = const . (9)Numerical integration of equation 9 yields the radiusat which emission at some angle θ was emitted in orderto reach the observer at time t ⊕ . To account for the timeand angular dependence of the beaming of radiation to-ward the observer, we integrate up the luminosities of thematerial at each location [r, θ , φ ] which emit radiationthat arrives at the observer at time t ⊕ to find the totalafterglow flux density: F ν, ⊕ = 14 πD Z Z Ω j L ′ ν ′ [ r ( θ )] D Ω j d cos θ dφ, (10)where D is the Doppler factor D = 1 / Γ(1 − β cos θ )and Ω j = 2 π (1 − cos θ j ) is the solid angle occupied bythe jet (Moderski et al. 2000). The quantity L ′ ν ′ is thecomoving frame luminosity of the afterglow, which can bedetermined via equations 27-32 of Mesler et al. (2012b). BURST CALORIMETRY FROM THE BROADBANDAFTERGLOW
The MHD and emission models detailed above wereused to perform calorimetry on the broadband radio af-terglow of GRB 030329. Seven parameters were fit si-multaneously: the kinetic energy ( E k ), the ejecta mass( M ej ), the jet half-opening angle ( θ j ), the fraction of thetotal burst energy stored in magnetic fields ( ǫ B ) and inelectrons ( ǫ e ), the electron power law index ( p ), and themedium density (n) in the case of a uniform mediumor the wind density scaling factor ( A ∗ ) in the case of awind. The factor A ∗ is set such that the medium density ρ ( r ) = 5 × A ∗ r − g cm − for a stellar wind.The values that are obtained for the GRB 030329 burstparameters appear in Table 2. Synthetic light curves pro-duced using the best fit uniform medium and wind mod-els are shown in Fig. 3. We incorporate the Goodman(1997) model of interstellar scintillation to account forthe time-dependent scatter of the low-frequency ( . &
10 GHz) ex-hibits mild departures from the expected smooth be-haviour of the afterglow emission, possibly due to slightclumping of the circumburst medium. The large valuesalorimetry of GRB 030329 5 −1 t (in days) R ⊥ ( c m ) n = 0.1 cm −3 n = 1.0 cm −3 n = 10 cm −3 n = 100 cm −3 A * = 0.1 cm −1 A * = 1.0 cm −1 A * = 10 cm −1 A * = 100 cm −1 Fig. 2.—
Fits of the GRB 030329 radio afterglow linear size to MHD models. Solid lines refer to models in which a uniform medium wasassumed, while dashed lines indicate a stellar wind. of χ r are due to these mild departures from smooth be-haviour. Fitting only the data at or below 4 . χ r = 1 . χ r = 24for the best-fit wind.The best-fit uniform density and wind models are bothcapable of reproducing the broadband afterglow for thefirst ∼
300 days. The transition to non-relativistic ex-pansion begins at approximately 42 days. The fluxwill decay more quickly in the case of a stellar windthan in the case of a uniform medium after the non-relativistic transition time t NR . The scatter of the datadue to refractive scintillation and large-scale diffractiveeffects means that the shallower decay due to a uni-form medium is not obvious until several hundred daysafter the burst (Fig. 3). It is clear that a uniformmedium is preferred over a stellar wind, however, bylooking at the data that was obtained after ∼ A ∗ >
1. In doing so, however, the ini-tial Lorentz factor must also be increased in order tokeep the jet from becoming non-relativistic too early. Alarger value of A ∗ , then, will provide a better fit to thebroadband afterglow at the expense of a poorer fit tothe afterglow expansion rate if E k , M ej , and θ j are heldconstant.Our fitted burst parameters are broadly consistentwith previous work (Berger et al. 2003; Frail et al. 2005;van der Horst et al. 2005; van der Horst et al. 2008)(Table 3). We find that the initial burst kinetic energy is E k = 1 . × erg. Radiative losses and energy lossesdo to adiabatic expansion reduce this to E k = 0 . × ergs by t jet = 13 days. The ejecta mass is found to be M ej = 2 . × − M ⊙ , yielding Γ = 4 .
5. We also findthat θ = 24 ◦ , ǫ B = 0 . ǫ e = 0 .
33, and p = 2 .
2. The value we obtain for the medium density of n = 6 . − is somewhat larger than has been found by previous au-thors. The wide range in densities obtained for the GRB030329 circumburst medium is probably due to differ-ences in the individual models employed by each author.Uncertainties in the structure of the jet magnetic fieldsand the time dependence of ǫ B and ǫ e limit the accu-racy of any gamma ray burst emission model. Given theimperfect nature of the current understanding of the jetphysics, we estimate that the values we obtain for theburst parameters are accurate to within approximately afactor of 1 . χ = 1 . χ = 3 . = 4 . = 10 . . Γ .
6, while the wind modelinitial Lorentz factor does not. CONCLUSIONS
We have presented the first simultaneous fit of agamma-ray burst afterglow model to a GRB’s broadbandlight curve and its observed afterglow expansion rate.More than eight years of afterglow observations at radiofrequencies between 840 MHz and 250 GHz were em-ployed to perform accurate calorimetry on the gamma-ray burst GRB 030329 deep into the non-relativisticphase. Values for the burst parameters were determined,and the nature of the circumburst medium was explored.By noting that a GRB jet will coast at a nearlyconstant velocity until a time t dec where it has be-gun to sweep up a significant amount of circumburstmaterial, we derive a relationship between the aver- Mesler and Pihlstr¨om Medium Type E k M ej θ ǫ B ǫ e p A χ r (10 erg) ( M ⊙ ) ( ◦ )uniform (ISM) 1 . . × −
24 0 .
046 0 .
33 2 . . . . . × −
37 0 .
15 0 .
33 2 . . . TABLE 2Best fits to the GRB 030329 burst parameters assuming either a uniform density or a stellar wind circumburst medium.The medium density scaling parameter A = n for a uniform medium and A = A ∗ for a wind. Note that the values of E k listed here correspond to the kinetic energy of the jet at the time of the burst. −2 −2 −2 F l u x D en s i t y ( m Jy )
15 GHz −2
23 GHz 44 GHz −2
100 GHz t (in days)
250 GHz
Fig. 3.—
Best fits to the GRB 030329 broadband radio afterglow assuming a uniform medium (red line) and a stellar wind (blue line). age apparent afterglow lateral expansion rate, h β app i ,and the initial Lorentz factor Γ . The initial Lorentzfactor for GRB 030329 is found to be 4 . Γ .
6. Berger et al. (2003) found that the GRB 030329radio afterglow was best modelled as the product ofa jet component that was initially mildly-relativistic.This interpretation has been subsequently supportedby other authors (Frail et al. 2005; van der Horst et al.2005; van der Horst et al. 2008), and is in agreementwith our estimate of Γ .The flux falls off more slowly in the case of a uniformmedium than in the case of a wind once the jet becomesnon-relativistic, so late-time observations of a GRB af-terglow provide important insight into the nature of thecircumburst medium. Refractive scintillation and large- scale diffractive effects produce a significant scatter in thedata at .
10 GHz, meaning that observations of the fluxat very late times, when the predicted afterglow evolu-tion is significantly different for a wind than for a uniformmedium, are extremely valuable in determining the na-ture of the circumburst medium. Our light curve modelsincorporate data out to 3018 days after the burst, allow-ing us to better determine the burst parameters and thenature of the circumburst medium than has been previ-ously possible for GRB 030329.We find that the best fit of a uniform density modelto the broadband radio afterglow predicts values for theburst parameters that are similar to the values thatare obtained when a stellar wind density profile is as-sumed. The goodness of fit χ r of the model that as-alorimetry of GRB 030329 7 Model t j t NR E k θ ǫ B ǫ e p n (days) (days) (10 erg) ( ◦ )Relativistic
10 N/A 0 .
67 26 0 .
042 0 .
19 2 . .
14 48 0 .
90 26 0 .
074 0 .
17 2 . . N/A 50 0 .
78 N/A 0 .
13 0 .
06 2 . .
10 N/A 0 .
24 42 0 .
43 0 .
28 2 . . N/A 80 0 .
34 N/A 0 .
49 0 .
25 2 . . .
59 24 0 .
045 0 .
33 2 . . TABLE 3Comparison of best-fit parameters for the various models that have been produced for the GRB 030329 radio afterglow.These models were first published in Berger et al. (2003) , Frail et al. (2005) , van der Horst et al. (2005) , andvan der Horst et al. (2008) . The bottommost model is the best-fit uniform density model presented in this work. Notethat the kinetic energies listed here are valid at t = t jet for models valid in the relativistic regime and at t = t NR formodels that are valid solely in the non-relativistic regime. −1 t (in days) R ⊥ ( c m ) uniformwind Fig. 4.—
Fits of the uniform and wind density models from Table2 to the observed apparent lateral afterglow expansion rate h β app i . sumes a uniform density medium, however, is better bya factor of ∼ . . Γ .
6, while the wind model does not,again suggesting that the circumburst medium is morecharacteristic of a uniform ISM than a stellar wind.The ability to determine the initial Lorentz factor of aGRB jet provides a powerful constraint on models of theburst evolution. The initial Lorentz factor of the burstdetermines the time t dec when the jet transitions to non-relativistic expansion as well as the jet’s initial lateralexpansion rate. By fitting the afterglow expansion his-tory to determine Γ , we limit the parameter space fromwhich we can build a model of the broadband afterglow.In the case of GRB 030329, this means that no stellarwind environment remains that can be used as a suitablemodel of the circumburst medium. In order to deter-mine the initial Lorentz factor, however, we must haveenough observations of the afterglow image size to deter-mine t dec . The GRB 030329 afterglow is currently theonly gamma ray burst afterglow to have been directly re-solved. In the future, more luminous, low-redshift burstswill need to be imaged with VLBI so that combined af-terglow and lateral expansion evolution fitting can beapplied beyond the case of GRB 030329.The National Radio Astronomy Observatory is a facil-ity of the National Science Foundation operated undercooperative agreement by Associated Universities, Inc. REFERENCESBerger, E., Kulkarni, S. R., Pooley, G., Frail, D. A., McIntyre, V.,Wark, R. M., Sari, R., Soderberg, A. M., Fox, D. W., Yost, S.,& Price, P. A. 2003, Nature, 426, 154Berger, E., Sari, R., Frail, D. A., Kulkarni, S. R., Bertoldi, F.,Peck, A. B., Menten, K. M., Shepherd, D. S.,Moriarty-Schieven, G. H., Pooley, G., Bloom, J. S., Diercks, A.,Galama, T. J., & Hurley, K. 2000, ApJ, 545, 56Chevalier, R. A. & Li, Z.-Y. 2000, ApJ, 536, 195Curran, P. A., Starling, R. L. C., van der Horst, A. J., Wijers,R. A. M. J., de Pasquale, M., & Page, M. 2011, Advances inSpace Research, 47, 1362Dai, Z. G., Huang, Y. F., & Lu, T. 1999, ApJ, 520, 634Dai, Z. G. & Lu, T. 1998, MNRAS, 298, 87Fenimore, E. E., Madras, C. D., & Nayakshin, S. 1996, ApJ, 473,998Frail, D. A., Soderberg, A. M., Kulkarni, S. R., Berger, E., Yost,S., Fox, D. W., & Harrison, F. A. 2005, ApJ, 619, 994Goodman, J. 1997, New Astronomy, 2, 449Granot, J. 2007, in Revista Mexicana de Astronomia y AstrofisicaConference Series, Vol. 27, Revista Mexicana de Astronomia yAstrofisica, vol. 27, 140–165Granot, J. & Sari, R. 2002, ApJ, 568, 820 Greiner, J., Peimbert, M., Estaban, C., Kaufer, A., Jaunsen, A.,Smoke, J., Klose, S., & Reimer, O. 2003, GRB CoordinatesNetwork, 2020, 1Huang, Y. F., Gou, L. J., Dai, Z. G., & Lu, T. 2000, ApJ, 543, 90Katz, J. I. 1994, ApJ, 432, L107Mesler, R. A., Pihlstr¨om, Y. M., Taylor, G. B., & Granot, J.2012a, ApJ, 759, 4Mesler, R. A., Whalen, D. J., Lloyd-Ronning, N. M., Fryer, C. L.,& Pihlstr¨om, Y. M. 2012b, ApJ, 757, 117M´esz´aros, P. 2002, ARA&A, 40, 137Meszaros, P. & Rees, M. J. 1993, ApJ, 405, 278Moderski, R., Sikora, M., & Bulik, T. 2000, ApJ, 529, 151Pe’er, A. 2012, ApJ, 752, L8Peterson, B. A. & Price, P. A. 2003, GRB Coordinates Network,1985, 1Pihlstr¨om, Y. M., Taylor, G. B., Granot, J., & Doeleman, S.2007, ApJ, 664, 411