Physical parametres of GRB and GRB from their afterglow synchroton emission Wijers, R.A.M.J.; Galama, T.J.

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1 UvA-DARE (Digital Acadmic Rpository) Physical paramtrs of GRB and GRB from thir aftrglow synchroton mission Wijrs, R.A.M.J.; Galama, T.J. Publishd in: Astrophysical Journal DOI: / Link to publication Citation for publishd vrsion (APA): Wijrs, R. A. M. J., & Galama, T. J. (1999). Physical paramtrs of GRB and GRB from thir aftrglow synchroton mission. Astrophysical Journal, 523, DOI: / Gnral rights It is not prmittd to download or to forward/distribut th txt or part of it without th consnt of th author(s) and/or copyright holdr(s), othr than for strictly prsonal, individual us, unlss th work is undr an opn contnt licns (lik Crativ Commons). Disclaimr/Complaints rgulations If you bliv that digital publication of crtain matrial infrings any of your rights or (privacy) intrsts, plas lt th Library know, stating your rasons. In cas of a lgitimat complaint, th Library will mak th matrial inaccssibl and/or rmov it from th wbsit. Plas Ask th Library: or a lttr to: Library of th Univrsity of Amstrdam, Scrtariat, Singl 425, 1012 WP Amstrdam, Th Nthrlands. You will b contactd as soon as possibl. UvA-DARE is a srvic providd by th library of th Univrsity of Amstrdam ( Download dat: 15 Fb 2019

2 THE ASTROPHYSICAL JOURNAL, 523:177È186, 1999 Sptmbr 20 ( Th Amrican Astronomical Socity. All rights rsrvd. Printd in U.S.A. PHYSICAL PARAMETERS OF GRB AND GRB FROM THEIR AFTERGLOW SYNCHROTRON EMISSION R. A. M. J. WIJERS1 AND T. J. GALAMA2 Rcivd 1998 May 26; accptd 1999 April 29 ABSTRACT W hav calculatd synchrotron spctra of rlativistic blast wavs and Ðnd prdictd charactristic frquncis that ar mor than an ordr of magnitud di rnt from prvious calculations. For th cas of an adiabatically xpanding blast wav, which is applicabl to obsrvd gamma-ray burst (GRB) aftrglows at lat tims, w giv xprssions to infr th physical proprtis of th aftrglow from th masurd spctral faturs. W show that nough data xist for GRB to comput unambiguously th ambint dnsity, n \ 0.03 cm~3, and th blast wav nrgy pr unit solid angl, E \ 3 ] 1052 rgs/4n sr. W also comput th nrgy dnsity in lctrons and magntic Ðld. W Ðnd that thy ar 12% and 9%, rspctivly, of th nuclon nrgy dnsity and thus conðrm for th Ðrst tim that both ar clos to but blow quipartition. For GRB , w discuss th brak found in its spctrum by Ramaprakash t al. It can b intrprtd ithr as th pak frquncy or as th cooling frquncy; both intrprtations hav som problms, but on balanc th brak is mor likly to b th cooling frquncy. Evn whn w assum this, our ignoranc of th slf-absorption frquncy and prsnc or absnc of baming mak it impossibl to constrain th physical paramtrs of GRB vry wll. Subjct hadings: gamma rays: bursts È gamma rays: thory È radiation mchanisms: nonthrmal È shock wavs 1. INTRODUCTION Explosiv modls of gamma-ray bursts (GRBs), in which rlativistic jcta radiat away som of thir kintic nrgy as thy ar slowd down by swpt-up matrial, naturally lad to a gradual softning of th mission at lat tims. This lat-tim softr radiation has bn dubbd th aftrglow ÏÏ of th burst, and its strngth and tim dpndnc wr prdictd thortically (M sza ros & Rs 1997). Soon aftr this prdiction, th accurat location of GRB by th BppoSAX satllitïs Wid Fild Camras (Piro, Scarsi, & Butlr 1995; Jagr t al. 1995) nabld th dtction of th Ðrst X-ray and optical aftrglow (Costa t al. 1997b; van Paradijs t al. 1997). Its bhavior agrd wll with th simpl prdictions (Wijrs, Rs, & M za ros1997; Waxman 1997a; Richart 1997). Th basic modl is a point xplosion with an nrgy of ordr 1052 rgs, which xpands with high Lorntz factor into its surroundings. As th mass swpt up by th xplosion bgins to b signiðcant, it convrts its kintic nrgy to hat in a strong shock. Th hot, shockd mattr acquirs mbddd magntic Ðlds and acclratd lctrons, which thn produc th radiation w s via synchrotron mission. Th phnomnon is thus vry much th rlativistic analog of suprnova rmnant volution, playd out apparntly in sconds owing to th strong tim contractions rsulting from th high Lorntz factors involvd. Naturally, th Lorntz factor of th blast wav dcrass as mor mattr is swpt up, and consquntly th powr output and typical nrgy dcras with tim aftr th initial fw sconds of gamma-ray mission. This producs th X-ray aftrglows, which hav bn dtctd up to 10 days aftr th burst (Frontra t al. 1998), and th optical ons, which hav bn dtctd up to a yar aftr th burst (Fruchtr t al. 1997; Bloom t al. 1998a; Castro-Tirado t al. 1998). Th burst of 1997 May 8 was bright for a rlativly long tim and producd mission from gamma rays to radio. This nabld a dtaild analysis of th xpctd spctral faturs of a synchrotron spctrum, conðrming in grat dtail that w ar indd sing synchrotron mission and that th dynamical volution of th xpanding blast wav agrs with prdictions if th blast wav dynamics ar adiabatic (Galama t al. 1998a, 1998b). In principl, on can driv th blast wav proprtis from th obsrvd synchrotron spctral faturs. Th problm is that th charactristic synchrotron frquncis and Ñuxs ar takn from simpl dimnsional analysis in th publishd litratur, so thy ar not suitabl for dtaild data analysis. Sinc thr ar now nough data on th aftrglows of a fw GRBs to driv thir physical proprtis, w amnd this situation in 2, corrcting th cofficints in th quations for th brak frquncis by up to a factor 10. W thn us our thortical rsults to infr th physical proprtis of th aftrglows of GRB ( 3) and attmpt th sam for GRB ( 4). W conclud with a summary of rsults and discuss som prospcts for futur improvmnts in obsrvation and analysis ( 5). 2. RADIATION FROM AN ADIABATIC BLAST WAVE 2.1. Blast W av Dynamics W rdriv th quations for synchrotron mission from a blast wav in ordr to clan up som imprcisions in prvious vrsions. Sinc th dynamical volution of th blast wavs should b clos to adiabatic aftr th Ðrst hour or so, w spcializ 1 Institut of Astronomy, Madingly Road, Cambridg CB3 0HA, UK; and Dpartmnt of Physics and Astronomy, Stat Univrsity of Nw York, Stony Brook, NY Astronomical Institut Anton Pannkok,ÏÏ Univrsity of Amstrdam; and Cntr for High Enrgy Astrophysics, Kruislaan 403, 1098 SJ Amstrdam, Th Nthrlands. 177

3 178 WIJERS & GALAMA Vol. 523 to th cas of dynamically adiabatic volution. This mans that th radius r and Lorntz factor c volv with obsrvr tim as (Rs & M sza ros 1992; M sza ros & Rs 1997; Waxman 1997a; Wijrs t al. 1997) r(t) \ r (t/t )1@4 (1) dc dc c(t) \ g(t/t )~3@8. (2) dc Hr g 4 E/M c2 is th ratio of initial nrgy in th xplosion to th rst mass nrgy of th baryons (mass M ) ntraind in it, th dclration 0 radius r is th point whr th nrgy in th hot, swpt-up intrstllar matrial quals that 0 in th original xplosion, and t is th obsrvr dc tim at which th dclration radius is rachd. Dnoting th ambint particl numbr dnsity as n, in units dc of cm~3, w hav A r \ E B1@3 dc 4ng2nm c2 p \1.81 ] 1016 AE B1@3 52 g300 ~2@3 cm (3) n t \ r dc dc 2g2c \ 3.35AE B1@3 52 g300 ~8@3 s, (4) n with m th proton mass and c th spd of light, and w hav normalizd to typical valus: E \ E/1052 rgs and g \ p g/300. Strictly spaking, w hav dðnd n hr as n 4 o/m, whr o is th ambint rst mass dnsity. 52 Stting t \ t/1 day 300 w thn hav, for t [ t, p d dc r(t) \ 2.29 ] 1017(E /n)1@4t1@4 cm (5) 52 d c(t) \ 6.65(E /n)1@8t~3@8. (6) 52 d Not that nithr c nor r dpnd on g: onc th blast wav has ntrd its phas of slf-similar dclration, its initial conditions hav bn partly forgottn. Th nrgy E dnots th initial blast wav nrgy; it and th ambint dnsity do lav thir marks. It should also b notd that ths quations rmain valid in an anisotropic blast wav, whr th outñow is in cons of opning angl h around som axis of symmtry, as long as its proprtis ar uniform within th con and th opning angl is gratr than 1/c (Rhoads 1998). W should thn rplac E by th quivalnt nrgy pr unit solid angl E 4 E/). To xprss this quivalnc w shall writ th normalization for this cas as E \ E(4n/1052 rgs), so w can dirctly rplac E in all quations with E to convrt from th isotropic to th anisotropic cas. 52 Bfor w can calculat th synchrotron mission from th blast wav, w hav to comput th nrgis in lctrons and magntic Ðld (or rathr, summariz our ignoranc in a fw paramtrs). First, w assum that lctrons ar acclratd to a powr-law distribution of Lorntz factors, N(c ) P c~p, with som minimum Lorntz factor c. W ar ignorant of what p m should b, but it can in practic b dtrmind from th data. Th total nrgy in th lctrons is paramtrizd by th ratio, v, of nrgy in lctrons to nrgy in nuclons. This is oftn calld th lctron nrgy fraction, but that trm is only appropriat in th limit of small v. Th postshock nuclon thrmal nrgy is cm c2, and th ratio of nuclon to lctron p numbr dnsitis is th sam as th prshock valu, which w can paramtriz as 2/(1 ] X), whr X is th usual hydrogn mass fraction. In trms of ths w hav v 4 n SE T ncm c2 \ 1 ] X m p [ 1 c mc (7) 2 m p [ 2 p p c \ 2 m p [ 2 p m 1 ] X m p [ 1 v c. (8) Th strngth of th magntic Ðld in th comoving fram is paramtrizd by stting th Ðld nrgy dnsity, B@2/8n, toa constant fraction, v, of th postshock nuclon nrgy \ 4c2nm c2. (Primd quantitis ar masurd in th rst fram of th shockd, B swpt-up matrial; othrs ar masurd in th fram p of an obsrvr outsid th blast wav at rst rlativ to th xplosion cntr.) Consquntly, B@ \ ccj32nnm v p B \2.58v1@2E1@8n3@8t~3@8 G. (9) B 52 d From th abov rlations, w can xprss th volution of th synchrotron spctrum from th blast wav in trms of obsrvabl quantitis and six unknown paramtrs: E, n, X, p, v, and v. But Ðrst w nd to rlat th synchrotron spctrum to ths paramtrs. 52 B 2.2. Synchrotron Radiation W now driv th corrct synchrotron frquncis and Ñuxs. Ths ar strictly valid only for a uniform mdium moving with a constant Lorntz factor. Ral blast wavs dclrat, of cours, and hav a mor complicatd structur bhind th shock. Th blast wav dclration mans that surfacs of constant arrival tim ar no longr th idal llipss xpctd for a constant spd of th blast wav (Rs 1966), and at a givn tim w s contributions from gas with di rnt Lorntz factors.

4 No. 1, 1999 PHYSICAL PARAMETERS OF GRB AND GRB This ct has bn discussd thoroughly (Waxman 1997b; Panaitscu & M sza ros 1998; Sari 1998). Th uniformity bhind th shock is also a simpliðcation: in rality th Lorntz factor varis from just bhind th shock to th contact discontinuity. Th dnsity and othr paramtrs vary accordingly (Blandford & McK 1976). This ct has not yt bn tratd; it is xpctd to b comparabl in importanc to dclration. Sinc both ths cts ar rathr lss important than our corrctions to th synchrotron frquncis, w shall nglct both rathr than attmpt to apply only on of thm. Howvr, our improvd tratmnt of th synchrotron mission is purly local and can b incorporatd into any formalism that accounts for th varying local proprtis of th shockd mdium at a Ðxd obsrvr tim. W assum that th lctron population in any local volum has an isotropic distribution of angls rlativ to th magntic Ðld and that th magntic Ðld is sufficintly tangld that w may avrag th mission proprtis assuming a random mix of orintation angls btwn th Ðld and our lin of sight. Th radiatd powr pr lctron pr unit frquncy, intgratd ovr mission angls is A B A P@ l J3 3B@ sin a \ F l l (c ), a m c2 l sin M M ab rgs cm~2 s~1 Hz~1 lctron~1, (10) whr F is th standard synchrotron function (.g., Rybicki & Lightman 1979), and and m ar th lctron charg and mass. a is th angl btwn th lctron vlocity and th magntic Ðld, and l (c ) \ 3c 2B@ M 4nm c. (11) (i.., th traditional charactristic synchrotron frquncy quals l sin a in our notation.) Nxt, w dðn th isotropic synchrotron function F by avraging ovr an isotropic distribution M of a. Stting x (c ) \ l/l (c ), w gt iso M M J3 3B@ P@ (x ) \ F (x ) rgs cm~2 s~1 Hz~1 lctron~1 (12) iso M m c2 iso M F iso (x M ) \ P 0n@2 da sin2 af(xm /sin a) (13) (W hav mad us of th symmtry of sin a to absorb a factor 1/2 into conðning th intgral to th Ðrst quadrant. Th apparnt singularity at a \ 0 poss no problms bcaus F dcrass xponntially for larg valus of th argumnt.) Not that most calculations of blast wav spctra assum that th spctrum paks at frquncy c2 B@/m c. Owing to th nglct of th factor 3/4n and th fact that F(x) paks at x \ and F (x) at x \ , this stimat lads to quit rronous iso infrncs about blast wav proprtis. Finally, w must avrag th mission ovr a distribution of lctron nrgis. W assum a simpl powr-law probability distribution of lctrons btwn xtrm valus c and c : m t f (c ) \ f Ac B~p 0 c ¹ c ¹ c (14) c c n t m m Now lt x \ l/l (c ). M m f \ p [ [ (c /c )1~p. (15) t m Thn th avrag powr pr lctron bcoms J3 3B@ P@ (x) \ F (x) rgs cm~2 s~1 Hz~1 lctron~1 (16) PL m c2 PL P F (x) \ f x 0 x~(p~1)@2 du u(p~3)@2f (u), (17) PL 2 iso xcm2@ct2 in which w hav transformd th intgration variabl from c to u 4 xc2/c2. Th last quation shows th familiar rsult that for 1 [ x [ c2/c2 th spctrum from a powr law of lctrons is itslf a powr m law. Sinc this rgion is known to xtnd ovr many dcads t in m GRB and aftrglow spctra, w quot numrical rsults for th cas c? c, for which th quotd rsults ar indpndnt of c. Th most asily idntiðd point in th spctrum is its dimnsionlss t maximum, m x, and th dimnsionlss Ñux at this point, t F (x ) 4 / ; thir dpndnc on p is shown in Figur 1. Both now dpnd on th p lctron nrgy slop p. This dðns th Ðrst PL two p numbrs p that w can masur in th spctrum: l \ x p l M (c m ) \ 3x p 4n J3 3B@ P@ \ /. (19) lm p m c2 Th calculation of th brak frquncy l that sparats radiation from slowly and rapidly cooling lctrons (Sari, Piran, & Narayan 1998) is somwhat mor difficult c bcaus th cooling rat dpnds on both c and pitch angl a. Howvr, sinc th cooling and th mission ar both dominatd by a \ n/2, w may stimat th brak as th pak of F(x) for th valu of c c m 2 B@ m c (18)

5 180 WIJERS & GALAMA Vol. 523 FIG. 1.ÈDimnsionlss location x (solid lin) and dimnsionlss pak Ñux / (dashd lin) of a synchrotron spctrum from a powr law of lctrons, as a function of th powr-law indx, p, of th p lctron nrgy distribution. p whr th cooling tim for lctrons with a \ n/2 quals th xpansion tim, t: c \ 6nm c (20) c p cb@2t T l@ \ c2 B@ c c 4n m c. (21) In ordr to transform frquncy and powr from th rst fram of th mitting matrial to our fram, w not that th mission is isotropic in th rst fram by assumption. It is thn trivial to comput th angl-avrag Dopplr factors (s Rybicki & Lightman 1979, chap. 4). For th rcivd powr, w Ðnd P \ c2(1 ] b2/3)p@, which w shall simplify to P \ 4c2P@/3 in kping with th fact that our whol tratmnt is don in th ultrarlativistic limit, b ] 1. Similarly, th intnsitywightd man chang in any frquncy is l \ 4cl@/3. Consquntly, th appropriat man of a powr pr unit frquncy will transform as P \ cp@. Of cours, th spctrum also gts broadnd, but that will not a ct th locus of charactristic l l frquncis signiðcantly. Th synchrotron slf-absorption frquncy is usually st at th point whr q \ Using th comoving width of th l shock, *r@ \ r/4c, and th xprssion for th synchrotron absorption cofficint (Rybicki & Lightman 1979), w gt l \ 4cl a 3 \ 2.97 ] 108[(p ] 2)/p ] 2/3]3@5(p [ 1)8@5/(p [ 2)(1 ] X)8@5n3@5v ~1v B 1@5E 1@5(1 ] z)~1 Hz, (22) 52 whr w hav usd quations (5)È(9) for th blast wav dynamics to xprss l in trms of th unknowns w try to solv for, and hav addd th corrction for rdshift, i.., th quation in this form rlats a th obsrvd frquncy on Earth to th proprtis of th blast wav masurd by a local obsrvr at rst rlativ to th cntr of th xplosion. Not that th slf-absorption frquncy in this simplst form is tim-indpndnt. W now also translat th othr two frquncis into practical form: l \ 4cl@ /3 \ 5.73 ] 1016x [(p [ 2)/(p [ 1)]2v2 v1@2e1@2(1 ] X)~2(1 ] z)1@2t~3@2 Hz (23) m m p B 52 d l \ 4cl@/3 \ 1.12 ] 1012v~3@2E~1@2n~1(1 ] z)~1@2t~1@2 Hz. (24) c c B 52 d Not th nontrivial rdshift dpndnc of both, which stms from th fact that t is also masurd on Earth and thrfor rdshiftd. Th obsrvd Ñux at l can b obtaind by noting that our assumption d of uniformity of th shockd matrial mans that all swpt-up lctrons sinc m th start contribut th sam avrag powr pr unit frquncy at l (at any frquncy, in fact), which is givn by quation (19). Adding on factor of c to transform to th lab fram and accounting m for th rdshift, w hav F \ N (1 ] z) lm, (25) lm 4nd2 L whr N is th total numbr of swpt-up lctrons, rlatd to th blast wav paramtrs by N \ (4n/3)r3n(1 ] X)/2. Th luminosity distanc dpnds on cosmological paramtrs and for an ) \ 1, " \ 0 univrs, which w shall adopt hr, is

6 No. 1, 1999 PHYSICAL PARAMETERS OF GRB AND GRB givn by d \ 2c(1 ] z [ J1 ] z)/h. Consquntly, L 0 h2 F \ lm (J1 ] z [ 1)2 / p (1 ] X)E 52 n1@2v 1@2 mjy, (26) B whr h \ H /70 km s~1 Mpc~1. Equations 70 (22), 0 (23), (24), and (26) now ar four indpndnt rlations btwn th four paramtrs of intrst E, n, v, and v. This mans w can solv for all paramtrs of intrst if w hav masurd all thr brak frquncis (not ncssarily 52 at th B sam tim) and th pak Ñux of th aftrglow. In addition this rquirs us to know th rdshift of th burst, th lctron indx p, and th composition paramtr, X, of th ambint mdium. Not that multipl masurmnts of th sam brak at di rnt tims srv to tst th modl assumptions but do not provid xtra constraints on th paramtrs, sinc validity of th modl implis that any of th four ky quations is satisðd for all tim if it is satisðd onc. W thrfor dðn th constants C 4 l /l, C 4 l t3@2/l, C 4 l t1@2/l, and C \ F /F. Hr starrd symbols dnot th numrical cofficints in a ach a a* of th m four m dm quations, m* c and c tims dc c* dnot th F tim lm at lm* which th quantity in qustion was masurd. Rarranging th four quations thn yilds C E \ C~5@6C~5@12C1@4C(3@2)3@2x5@12/ ~3@2(p [ 1)1@2 p ] 2 D1@2 AJ1 ] z [ 1B3 (1 ] X)~1(1 ] z)~1@2 (27) 52 a m c F p p p ] 1/2 h 70 C(p [ 1)1@2DC p ] 2 D~1@2 AJ1 ] z [ 1B~1 v \ C5@6C11@12C1@4C~1@2x~11@12/ 1@2 (1 ] X)(1 ] z)1@2 (28) a m c F p p p [ 2 p ] 1/2 h 70 C v \ C~5@2C~5@4C~5@4C1@2x5@4/ ~1@2(p [ 1)3@2 p ] 2 D3@2 AJ1 ] z [ 1 (1 ] X)(1 ] z)~5@2 B (29) B a m c F p p p ] 3/2 h 70 C n \ C25@6C25@12C3@4C~3@2x~25@12/ 3@2(p [ 1)~5@2 p ] 2 D~5@2 AJ1 ] z [ 1B~3 (1 ] X)~1(1 ] z)7@2. (30) a m c F p p p [ 5/2 h 70 Th last factor in ach of ths stms from th spciðc cosmological modl adoptd, and has ntrd th solution only via quation (25). To gnraliz to any cosmology, all that is ndd is to rplac (J1 ] z [ 1)/h in th abov 70 quations by (d /8.57 Gpc)/J1 ] z. L 3. OBSERVED AND INFERRED PARAMETERS OF GRB GRB was a modratly bright gamma-ray burst (Costa t al. 1997b; Kouvliotou t al. 1997). It was dtctd on May UT with th Gamma-Ray Burst Monitor (GRBM; Frontra t al. 1991), and with th Wid Fild Camras (WFCs; Jagr t al. 1995) on board th Italian-Dutch X-ray obsrvatory BppoSAX (Piro t al. 1995). Optical obsrvations of th WFC rror box (His t al. 1997a), mad on May 9 and 10, rvald a variabl objct at R.A. \ 06h53m49s.2, dcl. \]79 16@19A(J2000), which showd an incras by D1 mag in th V band (Bond 1997). BppoSAX Narrow Fild Instrumnt obsrvations rvald an X-ray transint (Piro t al. 1997), th position of which is consistnt with that of th optical variabl, and Frail t al. (1997) found th Ðrst GRB radio aftrglow for GRB ; th radio sourc position coincids with that of th optical sourc (Bond 1997). Th spctrum of th optical variabl showd absorption lins at rdshifts 0.77 and 0.835, indicating that is th minimum rdshift of th aftrglow (Mtzgr t al. 1997a, 1997b). Subsquntly, an [O II] mission lin with z \ was also found in th hostïs spctrum (Bloom t al. 1998a), which is oftn associatd with star-forming rgions in galaxis. A faint undrlying galaxy or star-forming rgion is infrrd to indd xist from a lvlling o of th light curv aftr 6È11 months (Bloom t al. 1998a; Castro-Tirado t al. 1998). It must b vry compact, sinc th Hubbl Spac T lscop limits on an xtndd objct undrlying th GRB ar faintr than th magnitud infrrd from th light curv (Pian t al. 1998). It is thrfor almost crtain that th compact nbula is th sourc of th [O II] lin and thrfor also of th majority of th absorption lins. Givn its compactnss, a chanc location of th burst far bhind it is unlikly, and w shall assum that th burst occurrd in this nbula, i.., its rdshift is From th light-curv bhavior and broadband spctrum (Fig. 2) of GRB , Galama t al. (1998a, 1998b) dducd th othr proprtis of th burst rquird to calculat th physical paramtrs of th aftrglow. W summariz thm hr: at t \ 12.1 days aftr triggr, th brak frquncis ar l \ 2.5 ] 109 Hz, l \ 8.6 ] 1010 Hz, and l \ 1.6 ] 1014 Hz. Th pak Ñux is F \ 1.7 mjy and th lctron indx p \ 2.2. a Aftr th Ðrst 500 m s, lctrons no longr c coold fficintly, and th aftrglow lm must volv adiabatically. W shall st th cosmological paramtrs to b ) \ 1, " \ 0, H \ 70 km s~1 Mpc~1. As notd abov, thy ntr th solution only via th luminosity distanc, and altrnativs can thrfor 0 b incorporatd asily via th substitution givn following quation (30). Finally, w adopt X \ 0.7 for th composition of th ambint mdium. Thr ar no rasons in th modl to assum th ambint mdium would not hav normal cosmic abundanc. Whil th mtallicity Z is a strong function of rdshift, X is hardly rdshift-dpndnt, sinc th balanc btwn H and H in cosmic mattr has not bn changd vry much by nuclosynthsis. Using furthr that x \ 0.580, / \ 0.611, w Ðnd E \ 3.5, n \ 0.030, v \ 0.12, v \ (31) 52 B W do not onc mor our dlibrat us of E, th nrgy pr unit solid angl scald to that of an isotropic xplosion of rgs, instad of th total nrgy: E is truly constraind by th data, whras th total nrgy rquirs us to know th as yt 52 poorly constraind baming of bursts. Th rcnt Ðndings by Fruchtr t al. (1999a) suggst thr might b a brak in th lat

7 182 WIJERS & GALAMA Vol. 523 FIG. 2.ÈX-rayÈtoÈradio spctrum of GRB on May 21.0 UT (12.1 days aftr th vnt) from Galama t al. (1998b). Indicatd ar th infrrd valus of th brak frquncis l, l, and l for May 21.0 UT. a m c light curv (100È200 days aftr th burst), at which tim th Lorntz factor is 2 or lss. If this brak is du to baming, it would b vry modst baming, and th total nrgy would b only a factor of a fw lss than th isotropic stimat. Our valu of E 52 dos clarly rul out th vry high nrgy stimats by Brainrd (1998) from th radio data alon. W hav dmonstratd for th Ðrst tim that th lctron and magntic Ðld nrgy dnsitis ar indd clos to but somwhat blow quipartition. Th ambint dnsity is on th low sid of normal for a disk of a galaxy but dðnitly highr than xpctd for a halo, lnding furthr support to th notion that bursts occur in gas-rich nvironmnts. As an asid, w not that switching th valus of l m and l, which is allowd by th shap of th spctrum at 12.1 days, dos not giv a snsibl solution (.g., v \ 20). This c conðrms th choic of Galama t al. (1998b), who notd that this solution was not compatibl with th tmporal volution of th aftrglow. Th gamma-ray Ñunc of GRB was masurd with BATSE to b (3.1 ^ 0.2) ] 10~6 rgs cm~2. Using z \ and h \ 1, this implis E \ In othr words, th nrgy mittd in gamma rays is 18% of th total blast wav nrgy 70 52c (pr unit solid angl in our dirction). According to Galama t al. (1998b), th aftrglow was cooling fficintly until 500 s aftr triggr; this mans that during th gamma-ray phas all th nrgy givn to lctrons would b radiatd away quickly, and mostly in gamma rays. If this phas is not too long, th nrgy radiatd in gamma rays should b E \ v E, whr v is 52c c 52i c th valu of v during th arly, gamma-rayèmitting phas and E is th initial valu of E. Sinc th subsqunt phas will 52i 52 b adiabatic, th blast wav nrgy masurd at lat tims should b E \ (1 [ v )E. Eliminating th initial nrgy, w 52 c 52i conclud that v c \ E 52c. (32) 1 [ v E c 52 Thrfor th masurd ratio of gamma-ray Ñunc to lat-tim blast wav nrgy implis that v \ 0.15, or slightly gratr if c som of th initial nrgy output is at E \ 20 kv. Compard with v \ 0.12 at lat tims, this dmonstrats th nar- constancy of th fraction of nrgy that is givn to th lctrons. Sinc th infrncs about th initial gamma-ray Ñunc ar indpndnt of th whol machinry on blast wav synchrotron mission in th prvious sction, w may viw this agrmnt as modst vidnc that th cofficints drivd thr ar clos to corrct, dspit our simpliðcation of th dynamics. It is also intrsting to compar th proprtis at lat tims with thos drivd from radio obsrvations. Th scintillation siz aftr 1 month is about 1017 cm (Frail t al. 1997), whras our formula giv a transvrs diamtr of 5 ] 1017 cm; givn th statistical natur of th scintillation siz and our nglct of th gradints in proprtis in th transvrs dirction, to which this particular masurmnt is of cours snsitiv, this is not too bad. Th Lorntz factor at this tim is 3.4, so th volution is still just in th ultrarlativistic rgim. Th Ðld at this tim is B@ \ 0.07 G. Katz & Piran (1997) stimatd a siz of th aftrglow of GRB from a crud masurmnt of th slf-absorption frquncy. Thy found a siz of 1017 cm, and assuming an ambint dnsity of 1 cm~3, thy found that th Lorntz factor had alrady dcrasd to 2 and that most of th nrgy of th blast wav had bn lost, i.. it had volvd with radiativ dynamics. Th numbrs w driv from our full solution aftr 1 wk ar c \ 5.8, transvrs diamtr \2 ] 1017 cm. This mans th blast wav is still quit rlativistic, and with our low ambint dnsity thr is no nd for radiativ volution. 4. PROPERTIES OF GRB This burst occurrd on 1997 Dcmbr UT. With a Ñunc of 1.1 ] 10~5 rgs cm~2, it is a modratly bright burst (Kippn t al. 1997). Aftr localization by th BppoSAX Wid Fild Camra in X-rays (His t al. 1997b), th optical aftrglow of this burst was found by Halprn t al. (1998). It shows vidnc of strong rddning (Halprn t al. 1998; Ramaprakash t al. 1998). Onc th aftrglow had fadd, a host galaxy bcam visibl undrnath it, and its rdshift was masurd to b 3.42 (Kulkarni t al. 1998).

8 No. 1, 1999 PHYSICAL PARAMETERS OF GRB AND GRB On dðnit brak and anothr possibl on wr obsrvd in th spctrum. Th dðnit brak (hraftr optical brak ÏÏ) was found by Ramaprakash t al. (1998); thy notd a brak in th spctrum of th aftrglow at 3 ] 1014 Hz, 0.58 days aftr triggr, in th xtinction-corrctd V RIJK spctrum. Anothr possibl brak (hraftr IR brak ÏÏ) was found by Gorosabl t al. (1998) in K-band data 3È5 hr aftr triggr. In Figur 3 w show th K-band light curv. Th data ar not strongly inconsistnt with a pur powr-law Ðt (s2/dof \ 2.0) but ar suggstiv of a brak passing through K aftr about 5 hr. Th physical intrprtation of th aftrglow dpnds rathr strongly on whthr th optical brak is th pak frquncy l or th cooling frquncy l, so w shall discuss ths two cass with thir implications and problms in turn. m c 4.1. T h Optical Brak as l m Ramaprakash t al. (1998) intrprtd th optical brak as th pak frquncy, l. A complication with th data is that th spctral slop is much too stp to corrspond to any simpl Ðrball modl, which m can b intrprtd as du to rddning within th host galaxy (Ramaprakash t al. 1998; Halprn t al. 1998). Sinc rddning scals approximatly as 1/wavlngth, it cannot b dtrmind without knowing what th tru slop of th spctrum is. Assuming a blast wav modl, on can prdict this slop from th tmporal dcay rat of th Ñux and an intrprtation of what brak is sn. Following Ramaprakash t al., w now assum an adiabatic blast wav and intrprt th brak as l. Thn th Ñux abov th brak should dpnd on frquncy and tim as F \ F l~bt~d, whr b \ 2d/3. Thr ar svral m dtrminations of d, for th VRIÑuxs: 1.2 ^ 0.2 by Kulkarni t al. (1998) and ^ 0.2 by Halprn t al. (1998). W shall adopt th valu 1.3 ^ 0.2 in this papr, so b \ 0.87 ^ 0.13 for th cas undr considration. Th lft-hand panl of Figur 4 shows th rsulting drddnd spctrum at t \ 0.52 days (not that Ramaprakash t al. construct th spctrum at a slightly latr tim, 0.58 days). W Ðnd that l \ 4 ] 1014 Hz and F \ 30kJy. (As an asid, w not that xtinction Ðts with a man galactic xtinction curv ar wors, sinc m th rdshiftd 2200 m A bump falls within VRI.) Th amount of xtinction, 0.43 mag at a rst fram wavlngth of 5500 A, is vry modst and dos not imply a spcial location of th burst within th galaxy. A strong point of this Ðt is that th X-ray Ñux masurd at th sam tim, which was not includd in th Ðt, agrs nicly with it. Th rportd nondtctions in th radio at lvls of 10È50 kjy could b inconsistnt with th pak Ñux: th rddning-corrctd F is 30 kjy, so w may hav to m invok slf-absorption to supprss th Ñux at 8.46 GHz. A wak point is th fact that th Ñux at K, which is blow th pak frquncy at 0.58 days in this modl, would hav to ris with tim as t1@2 as th pak approachs. But th arly K data by Gorosabl t al. (1998) clarly indicat a signiðcant dclin of th Ñux in K from 0.21 to 0.58 days. In fact, th arly K Ñux vn xcds th supposd pak Ñux in th spctrum at 0.52 days, which mans th pak Ñux had to b dclining as wll. This rquirs a nonstandard blast wav modl (.g., a bamd on or a nonadiabatic on) and is thus inconsistnt with th blast wav modl usd in this intrprtation. Nonthlss, w shall briñy xplor th physical implications of this modl, sinc th intrprtation of th optical brak as a cooling brak is not fr of problms ithr. For th simpl adiabatic modl usd by Ramaprakash t al., w gt th blast wav nrgy from quation (23): E 52 \ ] z A v 0.2B~4A v B 0.1B~1. (33) Th cofficint is 60 tims largr than in quation (3) of Ramaprakash t al., almost solly du to our mor accurat calculation of th pak frquncy. W can us quation (26) to driv an indpndnt stimat of th blast wav nrgy from FIG. 3.ÈK Ñux of GRB as a function of tim. Extrapolations to arly tims of th Kck masurmnt aftr 14 hr (Kulkarni t al. 1998) ar shown for th sam thr valus of d usd in Fig. 4. Othr data ar by Gorosabl t al. (1998) and Garcia t al. (1997; th uppr limit). Th assumption hr is that K lis abov l and blow l (th situation in th right-hand panl of Fig. 4). Thrfor, th actual slop of ach curv is 0.25 lss than th valu of d (by which it is labld) m bcaus K lis c blow th cooling brak, and d is masurd in VRI, abov th cooling brak, whr th tmporal slop is stpr by 1/4 than blow it.

9 184 WIJERS & GALAMA Vol. 523 FIG. 4.ÈNar-infrard/opticalÈtoÈX-ray spctral Ñux distribution of GRB on Dcmbr UT (th poch of th I-band masurmnt). Opn symbols indicat th masurd valus (from Ramaprakash t al and His t al. 1999) xtrapolatd to Dcmbr UT. Not that th rror on th J-band data is much largr than usd by Ramaprakash t al. (1998), in agrmnt with th original rport (Tanvir t al. 1997). Filld symbols indicat th drddnd data. Th dottd lins indicat th 1 p rrors on th spctral slop as drivd from th tmporal slop. Lft: Rsult of assuming that th brak is th synchrotron pak. Th spctrum blow th pak follows th low-frquncy tail of th synchrotron spctrum, whr F P l1@3. Right: Rsult of assuming that th brak is th cooling brak. Th spctral slop changs by 0.5 across th brak. Th xtinction, A, drivd from th l Ðt, corrsponds to th rst-fram V band. V th pak Ñux of 30kJy: A E \ 0.09n~1@2 v B. (34) B~1@2 This valu is difficult to rconcil with th nrgy stimat from l, unlss w push th quipartition fractions vry clos to m unity and/or adopt a vry low ambint dnsity T h Optical Brak as l c In ordr to accommodat th dclin in th arly K-band Ñux, w now assum that l was alrady wll blow K band at m 0.52 days, and thrfor th optical brak is l. Thn th spctral slop in VRIshould b rlatd to th tmporal dcay at c thos frquncis by b \ (2d ] 1)/3, which for d \ 1.3 ^ 0.2 implis b \ 1.2 ^ 0.13 and p \ 2.4. This modl is xplord in th right-hand panl of Figur 4. Whil it solvs th K dclin problm, it is a wors Ðt to th K Ñux at this tim and dos not do vry wll in prdicting th X-ray Ñux, which is 2.3 p abov th xtrapolatd spctrum. Also, if w thn say that th IR brak is ral, w hav a pak Ñux of 60 kjy. This valu is gratr than th 8.46 GHz Ñux limits obtaind 0.8È20 days aftr th burst (Kulkarni t al. 1998). This would rquir that th slf-absorption frquncy in this aftrglow xcds 10 GHz. Altrnativly, th pak frquncy at 20 days could b at last a factor 200 abov 8.46 GHz, so that an xtrapolation from th pak Ñux to th radio using an optically thin synchrotron spctrum (F P l1@3) falls blow 10 kjy. Sinc w know in this cas that l was l m 1.4 ] 1014 Hz aftr 0.21 days, w can us l P t~3@2 to Ðnd that l \ 150 GHz at 20 days, too low to b compatibl with th radio uppr limits; w conclud that w must m rquir l Z 10 GHz, m as in th othr intrprtation of th brak, to supprss th radio Ñux. a Using l and F from th IR pak at t \ 5hr\ 0.21 days, w can again us quations (23) and (26) to gt two xprssions for th nrgy m in trms m of th othr unknowns: A E \ 3.0 v v B~ B~4A A B (35) E \ 1.1 n v B~1@2A B. (36) 0.089B~1@2 Hr w hav scald th unknowns to th valus found for GRB In this cas, th two indpndnt nrgy stimats ar quit compatibl. Now that w idntiðd all but th slf-absorption frquncy in th aftrglow of GRB , w may us quations (27)È(30) to gt all th paramtrs of th burst, laving thir dpndnc on th unknown l xplicit. Th cooling frquncy is l \ 4 ] 1014 Hz at 0.52 days. It follows that a c E \ 0.46l~5@6, n \ 0.60l25@6, v \ 0.26l5@6, v \ 0.027l~5@2. (37) 52 a,ghz a,ghz a,ghz B a,ghz Sinc w rquir l [ 10 GHz in ordr to satisfy th radio limits, w gt E \ 0.07, n [ 9 ] 104, v [ 1.8, and v \ 8 a 52 B ] 10~5. Ths valus ar rathr di rnt from thos of GRB , and v [ 1.8 is implausibl (but not impossibl). With such a low nrgy and high dnsity, th GRB would bcom nonrlativistic (and start a fastr dclin) within about 4 days.

10 No. 1, 1999 PHYSICAL PARAMETERS OF GRB AND GRB This mans that th lat-tim radio Ñux may b supprssd without invoking l [ 10 GHz, asing th constraints on th paramtrs somwhat. But with l \ 1 GHz w gt valus for which th GRB rmains a rlativistic for 400 days; this conñicts with th radio limits. W conclud a that l probably has to xcd 5 GHz for a consistnt solution, implying a fairly low nrgy and high ambint dnsity for GRB a With a high mittd gamma-ray nrgy pr unit solid angl, E \ 30, this mans that th ratio of mittd gamma-ray nrgy to rmaining nrgy in th blast wav is uncomfortably larg, 52c probably in xcss of 100. Eithr vry fficint radiation in th GRB phas of th burst is ndd to achiv this or a strong di rnc in th amount of baming btwn th gamma-ray and aftrglow mission. Baming of GRB to a dgr similar to that obsrvd in GRB (Fruchtr t al. 1999b; Kulkarni t al. 1999) would allow a mor standard solution for th paramtrs: all th troubl ariss from th fact that th radio limits forc a high l. If th burst was bamd so that it startd dclining mor sharply aftr a day or so, thn th radio Ñux at latr tims could b a naturally low, and w would b allowd to choos l D 1 GHz, which lads to paramtr valus for GRB that ar similar to thos of GRB It would vn opn th a possibility that th tmporal dcay is somwhat contaminatd by th stpr dcay aftr th baming brak, so w should hav usd a slightly shallowr tmporal and spctral slop in Ðtting th spctrum of GRB at 0.52 days with a cooling brak. Thn both th X-ray and K Ñux would agr much bttr with th Ðt. Howvr, this now is a littl too much spculation for th amount of data availabl, and it may wll b wrong to try to bring th paramtrs of GRB into agrmnt with thos of GRB W alrady know that aftrglows ar divrs. For xampl, GRB had a cooling brak in X rays aftr a day rathr than in th optical (Vrswijk t al. 1999; Bloom t al. 1998b). W simply nd mor wll-masurd bursts in ordr to stablish th allowd rangs of paramtrs such as v and v. In summary, nithr th idntiðcation of th optical brak as l nor as l is without B problms, so w should tak any drivd paramtrs for this burst with a grain of salt. But th l intrprtation m c is in our viw th last tnabl, sinc it prdicts that th Ñux at K should hav risn from th start until about m a day aftr th burst, whras th data clarly show a dclining K Ñux. Th problm of th l intrprtation is that th X-ray Ñux at 0.52 days is about 2.3 p highr than xpctd, and th c K-band Ñux lowr, which is somwhat uncomfortabl but prhaps tolrabl. 5. CONCLUSION W hav calculatd th synchrotron spctra from th blast wavs causing GRB aftrglows and driv improvd xprssions for th rlations btwn masurd brak frquncis and th intrinsic proprtis of th blast wav. Ths allow us to rlat th blast wav proprtis to obsrvabl quantitis mor accuratly. W corrct th xprssion for th blast wav nrgy by almost 2 ordrs of magnitud. Our xprssions ar xact for an undclratd, uniform mdium. Dclration and radial structur of th shock ar xpctd to chang th xprssions for th Ðnal paramtrs by anothr factor of a fw, much lss than th corrctions found hr but still of intrst. Combind with th uncrtaintis in th masurd valus of th spctral braks and Ñuxs, this mans that th blast wav paramtrs drivd hr ar still uncrtain by an ordr of magnitud (s th solution by Granot, Piran, & Sari 1998 as an illustration of th possibl di rncs). Thr ar nough data on GRB to comput all intrinsic paramtrs of th blast wav. Th nrgy in th blast wav is 3 ] 1052rgs/4n sr. Th ambint dnsity into which th blast wav xpands is 0.03 cm~3, on th low sid for a disk of a galaxy. Th fraction of postshock nrgy that gos into lctrons is 12%, and that into magntic Ðld, 9%. W also stimat th fraction of nrgy transfrrd to lctrons during th gamma-ray phas and Ðnd this to b 15%. Th agrmnt with th latr blast wav valu suggsts that th fraction of nrgy givn to lctrons is constant from 10 s to 106 s aftr th triggr. For GRB thr is ambiguity in th intrprtation of th brak sn in th optical a half-day aftr th burst. W argu that th brak is most likly to b th cooling brak, but th argumnt is not watrtight. Assuming it is th cooling brak, w still lack th slf-absorption frquncy, but radio limits constrain this to b in xcss of about 5 GHz. Th limits on paramtrs that follow from this indicat that th aftrglow proprtis of GRB ar di rnt from thos of GRB GRB must ithr hav had mor narrow baming in gamma rays than in optical or hav radiatd its initial nrgy with mor than 99% fficincy in th gamma-ray phas, according to th paramtrs w driv. Also its magntic Ðld was far blow quipartition, v [ 10~4. Howvr, baming or othr additions could as th constraints and allow paramtr valus similar to thos of GRB B , so th physical paramtrs of GRB ar vry poorly constraind. Our analysis mphasizs th importanc of arly masurmnts covring a wid rang of wavlngths. Th full idntiðcation of th cooling frquncy l in GRB hingd on abundant photomtry, including colors, bing availabl soon aftr th burst, sinc th brak passd c R aftr 1.5 days (Galama t al. 1998b). In H and K, th action lastd a wk (Galama t al. 1998b), and this is th gnral trnd: thr is mor tim in IR, sinc all braks pass latr thr. Howvr, our rvisd cofficint for th pak frquncy, l, shows that th pak can only b caught in th IR within hours of th triggr (or much latr in th radio). A cas in point ar m th vry arly K@-band masurmnts of GRB by Gorosabl t al. (1998), which provid an invaluabl constraint on this aftrglow as thy may hav caught th passag of l through K@. Thrfor, w ncourag Ðrst and formost arly long-wavlngth covrag, including sarchs for aftrglows in m IR, as a mthod of ctivly constraining aftrglow paramtrs. Two of th thr crucial brak frquncis in an aftrglow can pass th optical and IR within hours and days, rspctivly. Thr is no tim Ðrst to sarch and only thn to attmpt broad covrag. Instantanous alrts from HETE2 and SWIFT will thrfor gratly advanc our undrstanding of aftrglow physics. For HETE2, and to som xtnt for SWIFT, this will rquir an normous amount of work from a ntwork of ground-basd obsrvatoris with good covrag in longitud and latitud, so that always at last on obsrvatory is wll placd for immdiat rspons. R. A. M. J. Wijrs was supportd for part of th work by a Royal Socity URF grant. T. J. Galama is supportd through a grant from NFRA undr contract W thank Juls Halprn for alrting us to th X-ray data on GRB and pointing out that our original modl was inconsistnt with th X-ray Ñux.

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