A strong blast wave is fully speciðed by two parameters: the blast-wave energy E \ 1052E ergs, and the ISM

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1 THE ASTROPHYSICAL JOURNAL, 511:852È861, 1999 Fbruary 1 ( Th Amrican Astronomical Socity. All rights rsrvd. Printd in U.S.A. GAMMA-RAY BURST AFTERGLOW: POLARIZATION AND ANALYTIC LIGHT CURVES ANDREI GRUZINOV AND ELI WAXMAN Institut for Advancd Study, School of Natural Scincs, Princton, NJ 854 Rcivd 1998 July 1; accptd 1998 Sptmbr 4 ABSTRACT Gamma-ray burst aftrglow polarization is discussd. W Ðnd an obsrvabl, up to D1%, polarization, if th magntic Ðld cohrnc lngth grows at about th spd of light aftr th Ðld is gnratd at th shock front. Dtction of a polarizd aftrglow would show that collisionlss ultrarlativistic shocks can gnrat strong larg-scal magntic Ðlds and conðrm th synchrotron aftrglow modl. Nondtction, at th D1% lvl, would imply that ithr th synchrotron mission modl is incorrct or that strong magntic Ðlds, aftr thy ar gnratd in th shock, somhow manag to stay undissipatd at microscopic, ÏÏ skin dpth, scals. Analytic light curvs of synchrotron mission from an ultrarlativistic slf-similar blast wav ar obtaind for an arbitrary lctron distribution function, taking into account th cts of synchrotron cooling. Th pak synchrotron Ñux and th Ñux at frquncis much smallr than th pak frquncy ar insnsitiv to th dtails of th lctron distribution function; hnc, thir obsrvational dtrmination would provid strong constraints on blast-wav paramtrs. Subjct hadings: gamma rays: bursts È polarization È radiation mchanisms: nonthrmal È shock wavs È stars: magntic Ðlds 1. INTRODUCTION X-ray, optical, and radio missions following gamma-ray bursts (GRBs) ar in broad agrmnt with modls basd on rlativistic blast wavs at cosmological distancs (Waxman 1997a; Wijrs, Rs, & M sza ros 1997; Vitri 1997b; Richart 1997; Katz & Piran 1997; Sari, Piran, & Narayan 1998). In ths modls th nrgy rlasd by an xplosion, D152 rgs, is convrtd to kintic nrgy of a thin baryon shll xpanding at ultrarlativistic spd. Aftr producing th GRB, th shll impacts on surrounding gas, driving an ultrarlativistic shock into th ambint mdium. In what follows, w rfr to th surrounding gas as intrstllar mdium (ISM) gas, although th gas nd not ncssarily b intrstllar. Th xpanding shock continuously hats frsh gas and acclrats rlativistic lctrons, which produc th obsrvd aftrglow radiation through synchrotron mission (Paczyn ski & Rhoads 1993; Katz 1994; M sza ros & Rs 1997; Vitri 1997a). To match th obsrvations, th magntic Ðld bhind th shock has to b D1% of quipartition with th shockhatd, comprssd ISM. What is th origin of this Ðld? Th shock-comprssd ISM Ðld is many ordrs of magnitud smallr than ndd. Th magntic Ðld frozn into th initial GRB Ðrball loss strngth by th tim th aftrglow stag bgins, and it is in a wrong plac anyway. During th aftrglow, th dcomprssd GRB Ðld is locatd far bhind th shock, whil most of th nrgy is in th rcntly shockd ISM. Thrfor, th magntic Ðld most likly must b gnratd in and by th blast wav. If th cohrnc lngth of th gnratd Ðld is comparabl to th thicknss of th blast wav, th radiation will b polarizd. A D1% dgr of polarization is xpctd. This is signiðcantly smallr than th maximal synchrotron polarization, D7%, bcaus th mitting rgion is thin and broad; it must b covrd by D1 mutually incohrnt patchs of magntic Ðld. In a papr on microlnsing of GRB aftrglows, Lob & Prna (1998) hav mntiond th possibility that th aftrglows ar polarizd. Hr w stimat th dgr of polarization ( 4). This papr also provids ( 3) xact analytic aftrglow light curvs for an arbitrary lctron distribution function, including th cts of lctron cooling. In 2 w dscrib th undrlying modl assumptions. W discuss th implications of our rsults to aftrglow obsrvations in 5. Most of th dtails of our drivations ar givn in Appndics AÈE. 2. THE BLAST-WAVE MODEL A strong blast wav is fully spciðd by two paramtrs: th blast-wav nrgy E \ 152E rgs, and th ISM 52 dnsity n \ 1n cm~3. With sufficint accuracy, th i 1 unshockd ISM may b takn to b cold unmagntizd hydrogn. To calculat th synchrotron mission w nd to know th fraction of nrgy in magntic Ðlds m and in B lctrons m and th shap of th lctron distribution func- tion [a function f (z) with Ðrst two momnts qual to 1]. W includ m, m, and f (z) in th list of indpndnt param- B trs. In principl, ths ar dtrmind by th blast-wav nrgy and th ISM dnsity, but a thory of strong collisionlss shocks is not availabl (Sagdv 1966; Krall 1997). Th plasma Ñow in th shockd ISM is assumd to b dscribd by th Blandford-McK (Blandford & McK 1976) slf-similar solution. Th Lorntz factor of th shock wav!, th Lorntz factor of th Ñow c, th propr nrgy dnsity, and th propr numbr dnsity n for all spactim points in th shockd plasma ar givn in Appndix A. W assum that magntic Ðlds and lctrons ar dscribd by simpl scalings. Th magntic Ðld is B2/8n \ m, and m is th sam in all spactim points. Th lctron distribution B B function in th local rst fram has th sam shap in all spactim points on th shock front; aftr th shock passag it volvs by adiabatic and synchrotron cooling. At th shock front, th man nrgy of an lctron in th local rst fram is c m c2\m /n and m is constant. Th shap of th lctron distribution function is not spci- Ðd at this point. W includ f (z) in th dðnition of th synchrotron mission function F. Th synchrotron powr pr unit frquncy pr lctron mittd in th local rst 852

2 GRUZINOV & WAXMAN 853 fram is P(u) \ J3 2n 3B m c2 FA u u c B, (1) u \ 3c 2 B c 2m c, (2) but F is not th standard dimnsionlss synchrotron mission function givn by Rybicki & Lightman (1979). F dpnds on th shap of th lctron distribution function shown in quation (B7). Givn th st of blast-wav paramtrs, w will masur tim, frquncy, and spctral luminosity (nrgy pr tim pr frquncy) in units of T \ c~1 A17 E AE B1@3 \ 5.5 ] s, (3) 8n n m c2b1@3 n i p 1 u 4 3Jn Am p m B5@2 c r m B 1@2 m 2(n i r 3)1@2 A \ 3.9 ] 11 m m.1b2a B n1 1@2 s~1, (4).1B1@2 L 4 17 Am B1@2 mb 1@2(n r3)1@2 2J6n m i p A E \ 2.2 ] 131 m B n1 1@2E rgs. (5).1B1@2 52 Th formal origin of ths units is xplaind in Appndics A and B. Thir physical maning is illustratd by th following ordr-of-magnitud statmnt. At obsrvd tim t /T \ o 1 th blast wav slows down to Lorntz factor 2; it radiats at frquncy u/u \ 1, with spctral luminosity L /L \ 1. Our analysis is rstrictd to th ultrarlativistic stag, that is, to dimnsionlss obsrvd tims t > 1. o 3. LIGHT CURVES W Ðrst calculat in 3.1 synchrotron mission of th blast wav nglcting radiativ cooling of lctrons, i.., assuming that th shap f (z) is th sam in th ntir shockd plasma. In 3.2 w rlax this assumption: f (z) is dtrmind at th shock front and volvs by synchrotron and adiabatic cooling thraftr. Our analytic light curvs ar xact undr th following assumptions: (1) Th blast-wav hydrodynamics is dscribd by th Blandford-McK slf-similar solution (Blandford & McK 1976). (2) Th magntic Ðld nrgy dnsity is a Ðxd fraction of th total nrgy dnsity, indpndnt of spac and tim. (3) Th lctron distribution function is dtrmind at th shock front and volvs aftrward only through adiabatic and synchrotron cooling. Granot, Piran, & Sari (1998) numrically drivd xact light curvs for powr-law lctron distribution, undr th assumptions dscribd abov and nglcting lctron cooling. Sinc a thory of strong collisionlss shocks is not availabl at prsnt, non of th abov assumptions can b justiðd. Thus, th numrical valus (.g., of th pak Ñux and pak frquncy as function of tim) drivd hr undr ths assumptions ar not ncssarily mor accurat than thos obtaind by ordr-of-magnitud stimats (.g., Waxman 1997b; Sari t al. 1998; Wijrs & Galama 1998). Th xact analytic light curvs ar usful bcaus thy allow us to dtrmin which aftrglow charactristics ar strongly dpndnt on th dtails of th lctron distribution function and which ar insnsitiv to ths dtails and dpnd mainly on th global blast-wav paramtrs (i.., th blast-wav nrgy, th ambint mdium dnsity, and th nrgy fractions carrid by lctrons and magntic Ðlds) Adiabatic L ight Curv In Appndix B, w driv an xprssion for th spctral luminosity, nglcting synchrotron cooling of lctrons. At obsrvd tim t aftr th gamma burst, at frquncy u, a distant obsrvr o (with ngligibl rdshift) infrs a slf-similar narrowband luminosity of th blast wav L (t ) \ L (ut3@2), whr u o A o L A (u) \ 48 P 1 da a3(1 ] 7a2)~2F[ 2a(1 ] 7a2)u ]. (6) W show th light curvs in Figur 1. Th thr curvs corrspond to di rnt doubly normalizd lctron distribution functions: 1. A powr law with indx p \ 2.4: f \ f \ 31.2z2(1 P ] 122zp`2)~1. 2. Maxwllian: f \ f \ 13.5z2~3z. M 3. Mixd: f \.7f ].3f. M P Not, that our powr-law ÏÏ distribution, for which f P z~p for z? 1, includs a low-nrgy tail, f D z2 for z > 1. W bliv th inclusion of such a thrmal ÏÏ low-nrgy tail is mor ralistic than assuming a sharp cuto of th lctron distribution blow a crtain minimum z-valu L ight Curv with Synchrotron Cooling Synchrotron cooling of th lctron distribution function has a noticabl ct on th light curv. W calculat th nonadiabatic light curv, L (t ) \ L (u, t ) in Appndix C, u o NA o FIG. 1.ÈAdiabatic light curvs q. (6) for di rnt lctron distribution functions ( 3): powr law (P), Maxwllian (M), and mixd (MP). This graph can b intrprtd as luminosity at a givn frquncy as a function of tim, or as th spctrum at a givn tim. P MP M

3 854 GAMMA-RAY BURST AFTERGLOW Vol. 511 nglcting cts of P cooling on th blast-wav propagation: 1 L (u, t ) \ 192 dy y3 P 1 da a3(1 ] 7a2)~2 P dz f (z ) NA o ] F [2a(1 ] 7a2)ut3@2 z~2], (7) o whr z~1 \ z~1 ] A(8t )~1@2a~1y~2(1 [ y19@6), (8) o and A \ 8 Am B2 p pt ct n m m 19 m i B A \ 1.6 ] 1~2 m m.1ba.1b B E52 1@3 n2@3. (9) 1 Scald spctra at di rnt obsrvd tims t ar shown in Figur 2 for th powr-law (and in Fig. 3 o for th mixd) lctron distribution function of 3.1 and A \.1. At high frquncis, quations (7) and (8) prdict a powr-law luminosity L P u~p@2 for an lctron distribution function with a powr-law tail of indx p Obsrvabls From quations (3)È(5) and Figurs 2 and 3, an aftrglow at rdshift z, obsrvd t days aftr th gamma burst, will b day show maximal Ñux at a frquncy A l D 3 ] 112 J1 ] z m m b m J2.1B2A B E52 1@2 t~3@2 Hz..1B1@2 day (1) Th maximal Ñux dos not dpnd on th tim of obsr FIG. 2.ÈNonadiabatic light curvs q. (7) for A \.1, for di rnt obsrvd tims, and for a powr-law lctron distribution function ( 3). Th adiabatic light curv is shown for comparison. Nonadiabatic curvs ar markd by th Lorntz factors of th shock front at th tim whn obsrvd photons wr mittd from th shock front from h \. Obsrvd tim in days is t day \ 8(E 52 /n 1 )1@3!~8@ FIG. 3.ÈSam as Fig. 2, but for th mixd (powr-law plus Maxwllian) lctron distribution function ( 3). vation. In a Ñat univrs with Hubbl constant H \ 75 km s~1 Mps~1, B~2A F \ 4 AJ1 ] z b [ 1 m B n1 1@2E mjy. (11) lm J2 [ 1.1B1@2 52 As sn in Figurs 1, 2, and 3, th pak Ñux F is robust, lm i.., it is indpndnt of th dtails of th lctron distribution function. Th Ñux blow th pak is also robust, and for l > l it is givn by m B~2A1 ] z B~2@3 b F l \.6 AJ1 ] z b [ 1 J2 [ 1 2 B~1@6A m.1 A ] m B n1 1@2 E5@6 t1@2 l1@3 mjy. (12).1B1@3 52 day GHz Th pak frquncy l is modl dpndnt and may di r by an ordr of magnitud m btwn di rnt lctron distribution functions with similar m. Th spctral shap (quivalntly th tim proðl) abov th pak is also strongly dpndnt on th dtails of th lctron distribution function. 4. POLARIZATION Synchrotron radiation is highly polarizd (Ginzburg 1989), but for this polarization to b masurabl in an unrsolvd sourc, th magntic Ðld cohrnc lngth should b comparabl to th sourc siz. Hr w show that GRB aftrglows might b polarizd. A D1% polarization sms to b an uppr bound, corrsponding to a cohrnc lngth that grows at about th spd of light aftr th Ðld is gnratd at th shock front. Qualitativly, our polarization analysis can b summarizd as follows. Suppos that th magntic Ðld cohrnc lngth in th local rst fram is l D cq, whr q is th propr tim aftr th shock. Th xtnsion of th mitting

4 No. 2, 1999 GRUZINOV & WAXMAN 855 rgion transvrs to th lin of sit is D5cq. Thr ar D5 cohrnt patchs. Th dgr of polarization is D6%/ J5 D 1%. If th cohrnc lngth is smallr than th propr tim, l D vcq, v\1, th dgr of polarization is dcrasd to % D 1v3@2%. (13) Th dgr and dirction of polarization should dpnd on tim, and th polarization cohrnc tim should b Dvt o Magntic Fild Gnratd by a Rlativistic Collisionlss Shock As far as w know, magntic Ðld gnration in collisionlss shocks is not undrstood. It sms possibl that, at th shock front, Wibl instability gnrats nar-quipartition (m D.1) small-scal (Dskin dpth d, hr d \ c/u, u2 D n2/c B m [ n2/c m ) magntic Ðlds. By magntic momnt p p consrvation, p p lctrons ar acclratd to narquipartition nrgis (rlativistic vrsion of Sagdv 1966; Kazimura t al. 1998). Th ultimat fat of th Ðld many skin dpths bhind th shock front is not clar. What happns to th magntic Ðld cohrnc lngth l and to th magnitud B at a distanc *? d bhind th shock front? Thr scnarios sm to mak sns: 1. Th gnratd Ðld dis out aftr Ðnishing its job of isotropizing th plasma and bringing it to a stat givn by th shock jump conditions. 2. Th magnitud stays at about quipartition, and th cohrnc lngth stays at about th skin dpth. 3. Th magnitud stays at about quipartition, and th cohrnc lngth grows as l D *. Scnario 1 is not consistnt with th synchrotron mission modl for th aftrglow, bcaus too littl synchrotron radiation is producd by a skin-dp shll with strong magntic Ðlds. Scnario 2 mans unpolarizd radiation. W will valuat th dgr of polarization for scnario Cohrnt Patch Assum that two vnts in th shockd ISM blong to th sam cohrnt patch if th di rnc in thir propr tims lapsd aftr th shock passag dq and thir spatial sparation transvrs to th lin of sit dh is small: dq \ v q q, (14) dh \v h cq, (15) whr v,v \ 1 and q is th avragd propr tim sinc th shock. By q h th propr tim of an vnt in th shockd ISM w man th propr tim aftr th shock of a Ñuid particl at this vnt Dgr of Polarization As shown in Appndix D, for th mission vnt with obsrvd tim t, th propr tim q (in units of T ) and th transvrs distanc o (in units of ct ) ar q \ 1.15t o 5@8a5@4y1@4(1 [ y9@4), (16) h \ 1.83t o 5@8a1@4(1 [ a2)1@2y1@4. (17) Hr th maning of dimnsionlss variabls a, y is unimportant, what mattrs is that th luminosity is givn by th intgral quation P (B2) ovr a and y: 1 L (u) \ 192 dy y3 P 1 da a3(1 ] 7a2)~2F[2a(1 ] 7a2)u]. (18) Now w can sparat th full luminosity of quation (18) into cohrnt parts according to th critria in quations (14) and (15). This givs, approximatly, th dgr of polarization. By numrical simulations (Appndix E), % D 1v v1@2 %. (19) h q 5. SUMMARY OF RESULTS 5.1. L ight Curvs W hav drivd simpl, xact analytic aftrglow light curvs for an arbitrary lctron distribution function, including th cts of lctron synchrotron cooling, quation (7), and in th limit whr synchrotron cooling is ngligibl, quation (6). Our light curvs ar xact undr th following assumptions: (1) Th blast-wav hydrodynamics is dscribd by th Blandford-McK slf-similar solution (Blandford & McK 1976). (2) Th magntic Ðld nrgy dnsity is a Ðxd fraction of th total nrgy dnsity, indpndnt of spac and tim. (3) Th lctron distribution function is dtrmind at th shock front and volvs aftrward only through adiabatic and synchrotron cooling. W hav shown (s Figs. 1, 2, and 3) that th pak synchrotron Ñux, quation (11), and th synchrotron Ñux at frquncis wll blow th pak Ñux, quation (12), ar insnsitiv to th dtails of th lctron distribution function. Sinc th pak Ñux is also insnsitiv to th dtails of th blast-wav hydrodynamic proðls (Fig. 1 of Waxman 1997c), th pak Ñux and th Ñux at frquncis wll blow th pak dpnd mainly on th global blast-wav paramtrs: blast-wav nrgy, ambint mdium dnsity, magntic Ðld, and lctron nrgy fractions (cf. qs. [11] and [12]). Obsrvational dtrmination of ths Ñuxs would thrfor provid strong constraints on blast-wav paramtrs. Th numrical valu of th pak Ñux drivd hr, quation (11), is similar to that drivd by Granot t al. (1998) and Wijrs & Galama (1998), within a factor of D3of th valus givn by Waxman (1997b) and a factor D1 smallr than thos givn by Sari t al. (1998). Th discrpancy with Sari t al. (1998) is mainly du to th fact that it is assumd by ths authors that th spctral width of th obsrvd spctrum at Ðxd tim is dtrmind mainly by th intrinsic spctral width of synchrotron mission, whil th actual width is signiðcantly largr and dominatd by contribution to th obsrvd spctrum at a givn tim from di rnt spactim points with di rnt plasma conditions. Th pak frquncy and spctral shap at frquncis abov th pak ar strongly dpndnt on dtails of th lctron distribution function (s Figs. 1, 2, and 3). Furthrmor, th pak frquncy is also strongly dpndnt on th dtails of th blast-wav hydrodynamic proðls (Fig. 1 of Waxman 1997c). This, and th fact that th spctral pak is Ñat, implis that obsrvational dtrmination of th pak frquncy and of spctral faturs abov th pak at a givn tim cannot b usd dirctly to constrain global blast-wav paramtrs. Ths faturs would mostly provid information on th lctron distribution function. Th numrical valu of th pak frquncy drivd hr, quation (1), is within a factor of D3 of th valus givn by Sari t al.

5 856 GAMMA-RAY BURST AFTERGLOW Vol. 511 (1998), Granot t al. (1998), and Wijrs & Galama 1998 and a factor of D1 smallr than thos givn by Waxman (1997b). Th discrpancy with Waxman (1997b) is du mainly to th fact that it is assumd by Waxman (1997b) that th spctral pak is clos to th synchrotron frquncy of lctrons with avrag Lorntz factor, whil th actual pak is closr to th synchrotron frquncy of lctrons nar th pak of th lctron distribution function. Th brak frquncy [th frquncy whr th highfrquncy spctrum changs th slop from (p [ 1)/2 to p/2 bcaus of synchrotron cooling] is not prominnt. Th transition to th p/2 slop occurs in a mannr that strongly dpnds on th dtails of th lctron distribution function Polarization If th obsrvd aftrglows ar indd synchrotron mission from ultrarlativistic blast wavs propagating into th ISM, th magntic Ðld ndd to account for th mission must b gnratd by th blast wav. If th cohrnc lngth of th gnratd Ðld grows at about th spd of light aftr th Ðld was gnratd at th shock front, aftrglows should b noticably polarizd. W thank John Bahcall for a discussion that initiatd this study. Our work was supportd by NSF PHY E. W. was also supportd by th W. M. Kck Foundation. APPENDIX A ULTRARELATIVISTIC BLAST WAVE Th Blandford & McK (1976) solution can b dscribd as follows: Lt (t, r, h) b th spactim coordinats in th blast fram and h b th polar angl, which is assumd to b small, with h \ in th obsrvr dirction. Lt E b th nrgy of th blast wav and n th unshockd ISM numbr dnsity, c \ 1. Th shock front propagats into th ISM with a Lorntz factor i! that dcrass with tim according to DÐn a similarity variabl!2t3\ 17 E 4 T 3. 8n n m i p (A1) s 4 8!2 A 1 [ r tb. (A2) Th shockd rgion is s[1, and th Ñuid Ñow in th shockd rgion is givn by c2\ 1!2s~1, (A3) 2 \ 2!2s~17@12n i m p, (A4) n \ 2J2!s~5@4n i. Hr c is th Lorntz factor of th Ñow, is th propr nrgy dnsity, and n is th propr numbr dnsity. In Appndix B w us (!, c, t ) as indpndnt variabls instad of (t, r, h). Hr t is th tim at which a photon mittd at (t, r, h) is obsrvd; with sufficint obs accuracy, obs t \ t [ r ] r h2 obs 2 \ t [ r ] t h2 2. (A6) Th coordinat transformation is (old coordinats in trms of nw coordinats) r \ A 1 [ 1 16c2 t \ T!~2@3, B t \ A 1 [ 1 (A5) (A7) 16c2B T!~2@3, (A8) From quation (A3) h2 2 \ t obs t [ 1 16c2 \ t obs T!2@3 [ 1 16c2. s \!2 2c2. (A9) (A1)

6 No. 2, 1999 GRUZINOV & WAXMAN 857 With s from quation (A1), quations (A4) and (A5) giv th dpndnt quantitis, nrgy dnsity, and dnsity in trms of nw indpndnt variabls. In nw coordinats, th spactim domain of th shockd Ñuid (t [, s[1, h[) is givn by O [ t [, (A11) obs O [![ A8t B~3@8 obs, T! J2 [c[ 1 4 At B~1@2 obs!~1@3. T (A12) (A13) APPENDIX B LIGHT CURVE Hr w calculat th light curv of synchrotron mission from an ultrarlativistic blast wav. Th physical assumptions of th modl ar discussd in th main txt. It is convnint to start from th following xprssion for th total mittd nrgy: E r \ P r2dr P 2nh dh P dtn P du@p(u@, c, B) u u@ Th factors in quation (B1) ar 1. Total numbr of mitting lctrons at a givn tim P r2 dr2nh dhcn. d)@ d). (B1) (B2) 2. Total nrgy mittd by on lctron P dt P du@p(u@, c, B) u c u@, (B3) whr u is th photon frquncy in th burst fram, and u@ is th frquncy in th local rst fram, u@ \ 1 ] c2h2 u. (B4) 2c P(u@, c,b) is th synchrotron radiation spctral powr in th local rst fram mittd at th frquncy u@ by on lctron from a distribution with a man Lorntz factor c in th magntic Ðld B. It is givn by (Rybicki & Lightman 1979) B P(u@, c, B) \ J33 2nm c2 BFAu@, (B5) u c u (c,b) \ 3 c 2m c Bc 2. (B6) Th dpndnc of mission on th man Lorntz factor of lctrons is shown xplicitly. Th dpndnc on th pitch angl is nglctd, but this might b partially accountd for by rdðning B. Th dpndnc on th dtaild distribution function of lctrons is hiddn in th dðnition of th synchrotron mission function F(x). Namly, w dðn with F (x) bing th standard synchrotron mission function P A x F(x) \ dzf (z)f z2b, (B7) F (x) 4 x P x= dmk5@3 (m). (B8) In quation (B7), th normalizd lctron distribution function f is dðnd by th following xprssion for th probability for th lctron to hav a Lorntz factor c l dprobability \ 1 Ac f l B. (B9) dc c c l

7 858 GAMMA-RAY BURST AFTERGLOW Vol Th last factor in quation (B1) is th ratio of inðnitsimal solid angls in th local rst and blast frams: d) \ 4c2 (1 ] c2h2)2. (B1) W assum that magntic Ðlds and lctrons tak up a Ðxd fraction of th propr nrgy dnsity (r, t): B2 8n \ m B, c nm c2\m. (B11) (B12) W also assum that th normalizd lctron distribution function f (z) in th shockd ISM is Ðxd. Ths assumptions might b approximatly corrct whn synchrotron cooling bcoms unimportant at latr stags of th aftrglow. W us th Blandford & McK (1976) slf-similar solution for th Lorntz factor c, dnsity n, and nrgy dnsity and chang th indpndnt variabls in quation (B1) from (t, r, h, u@)to(!, c, t, u). Using Appndix A, w gt obs E \ 17 E P = du P = dtobs P P =!@S2 d!!~3 dcc~3nd2p(d~1u, c, B). (B13) r 48 n m c2 i p (8tobs@T) ~3@8 (tobs@t) ~1@2! Hr D is th Dopplr factor,ïï D \ 2c 1 ] c2h2 \ 2cA7 8 ] 2 t obs T!2@3c2B~1, (B14) and T is th charactristic tim of th blast wav introducd in Appndix A. Th spctral light curv is dðnd as luminosity pr unit frquncy: From quation (B13), P L (t ) 4 de r u obs dt du. obs P L (t ) \ 17 E =!@S2 d!!~3 dcc~3nd2p(d~1u, c, B). (B16) u obs 48 n m c2 i p (8tobs@T)~3@8 (tobs@t)~1@2!~1@3@4 Now w us quation (B5) for th synchrotron powr P, quations (B11) and (B12) for B and c, and quations (A4), (A5), and (A1) for and n. Also, from now on w will dnot by t th obsrvd tim masurd in units of T. W also dðn th o frquncy and spctral luminosity units, quations (4) and (5). Ths ar dvisd to gt rid of constant factors in th rsulting xprssion for th luminosity. W dnot th frquncy u in units of u by u and th spctral luminosityl (t ) in units ofe u obs by L (t ). Thn quation (B16) taks th following form: u o P P A =!@S2 L (t ) \ d!!~1 dcc~3!2 B~47@24 D2F C D~1u!~3 A!2 u o (8to) ~3@8 to~1@2! ~1@3 2c2 2c2B25@24D. DÐn nw intgration variabls x and y: c 4 (!/J2)y,! 4 (8t )~3@8x~3@4. W obtain a slf-similar spctral light curv o (B15) L u (t o ) \ L A (ut o 3@2). (B18) Hr P 1 L (u) \ 192 dxx~1 P 1 dyy35@12 A y2b~2 7 ] F C 2x3y~37@12 A y2 7 ] A x2 x2b u D. (B19) x With only a D2% rror in th rsulting luminosity, w can rplac th indics 35/12 and 37/12 by 3 and gt a simplr xprssion P 1 L (u) \ 192 dyy3 P 1 daa3(1 ] 7a2)~2F[2a(1 ] 7a2)u], (B2) A whr a 4 x/y. From quation (B2), th adiabatic light curv is L A (u) \ 48 P 1 daa3(1 ] 7a2)~2F[2a(1 ] 7a2)u]. (B21)

8 No. 2, 1999 GRUZINOV & WAXMAN 859 APPENDIX C SYNCHROTRON COOLING Synchrotron plus adiabatic cooling of an lctron with Lorntz factor c is dscribd by l dc l dq \ 1 c dn l 3 n dq [ 4 3 p T c B2 8nm c2 c 2, (C1) l whr q is th propr tim of th Ñuid lmnt at th lctronïs location. W hav dq \ dt/c. From quation (A5), d ln n \ d ln! [ (5/4)d ln s. From quation (D8), d ln s \ 4d ln t. Thn dc l dn \ 1 c l 3 n [ 4 3 p T c B2 8nm c2 c 2 1 l cn Ad ln! [ 5. (C2) dt tb~1 Using quations (A1) and (B11), dc l dn \ 1 c l 3 n ] 8 p T 39 m c m B c 2 t l cn. (C3) To intgrat, w nd to xprss t,, and c in trms of n. Lt c, n, and b th Lorntz factor, propr dnsity, and propr nrgy dnsity at th shock passag tim t. From quations (A1) and (A3)È(A5), c \ 1 J2!, (C4) n \ 2J2! n, (C5) i \ 2! 2 n m c2, (C6) i p whr! 2\T 3/t3. From quations (D8), (A1), and (A3)È(A5), c \ c (t/t )~7@2, (C7) n \ n (t/t )~13@2, (C8) \ (t/t )~26@3. (C9) Now quation (C3) can b writtn as dc l dn \ 1 c l 3 n ] 8 p T 39 m c m t A nb~14@39 cl 2, (C1) B c n n and intgratd 1 z \ 1 ] 4 p T z 19 m2 c3 m B m t 2 (1 [ y19@6). (C11) c n Hr z 4 c /c, with c dðnd by quation (B12); y is dðnd by quation (D1). Plug in quations (C4)È(C6), and xprss! l in trms of a, y, and t. W gt o z~1 \ z~1 ] A(8t )~1@2a~1y~2(1 [ y19@6), (C12) o whr A is dðnd by quation (9) and a is dðnd by quation (D2). With th synchrotron cooling givn by quation (C12), th spctral luminosity is P 1 L (t) \ 192 dyy3 P 1 daa3(1 ] 7a2)~2 P dz f (z )F [2a(1 ] 7a2)ut3@2/z2]. (C13) u o APPENDIX D TRANSVERSE DISTANCES AND PROPER TIMES Adiabatic light curv of quation (B2) is an intgral ovr dimnsionlss variabls y and a: y \ J2c!, (D1) ay \ (8t )~1@2!~4@3. o (D2) To calculat polarization using quation (B2), w hav to xprss th distanc from th obsrvr, burst cntr lin h, and th propr tim sinc th shock passag q, in trms of y and a. This is don hr.

9 86 GAMMA-RAY BURST AFTERGLOW Vol. 511 Th transvrs distanc is h \ rh, and from quations (A8) and (A9) and from quations (D1) and (D2) h \ J2T!~2@3 A t o!2@3 [ 1 16c2B1@2, (D3) h \ 1 2 T (8t )5@8(ay)1@4(1 [ a2)1@2. o (D4) Now w calculat th propr tim. Th quation of motion of a shockd particl is dr dt \ 1 [ 1 2c2 \ 8 r t [ 7. (D5) Hr t is th burst fram coordinat tim. Intgration givs r \ t [ Ct8. Sinc th shock front is at R \ t [ t4 8T 3, (D6) w gt r \ t [ t8 8T 3t4, (D7) whr t is th burst coordinat tim at which th particl was shockd. Th similarity variabl at th particl is s \ t4/t4, (D8) and Using quations (D8), (A1), and (A7), w gt q \ P dt c \ J2 P dt s1@2! \ 2J2 9T 3@2t 2 (t9@2 [ t 9@2). q \ 2J2 9 T (8t o )5@8a5@4y1@4(1 [ y9@4). APPENDIX E DEGREE OF POLARIZATION To stimat th dgr of polarization % w us th adiabatic light curv and assum F(u) D u~s in quation (B2). Th lattr simpliðcation should not lad to a larg rror, bcaus % turns out to b approximatly th sam in th rlvant rang 1 [ s [ [1. With ths assumptions, th dgr of polarization can b stimatd as 3 % \ s ] 1 s ] 5/3 (D9) (D1) C1@2 LL. (E1) L Hr th Ðrst factor is polarization of a powr-law mission from on patch, L is th total unpolarizd luminosity, and C is th polarizd luminosity corrlator. Up to an irrlvant factor, LL P daa3~s(1 ] 7a2)~2~s, L \ 1 4 (E2) P A C \ da dy da dy (y y )3(a a )3~s[(1 ] 7a2)(1 ] 7a2)]~2~sC min 1, v h q LL nhB. (E3) Hr th min trm coms from th azimuthal angl intgral, q \ (q ] q )/2, h \ (h ] h )/2. C is th normalizd magntic Ðld corrlator, for which w tak a simpl form corrsponding to quations 1 2 (14) and 1 (15): 2 12 C \ h( o q [ q o [ v q)h( o h [ h o [ v q). (E4) q 1 2 h W calculatd quation (E1) numrically for di rnt valus of s, v, and v. h q Blandford, R. D., & McK, C. F. 1976, Phys. Fluids, 19, 113 Ginzburg, V. L. 1989, Applications of Elctrodynamics in Thortical Physics and Astrophysics (Nw York: Gordon & Brach) Granot, J., Piran, T., & Sari, R. 1998, prprint (astro-ph/986192) Katz, J. I. 1994, ApJ, 432, L17 Katz, J. I., & Piran, T. 1997, ApJ, 49, 772 REFERENCES Kazimura, Y., Sakai, J. I., Nubrt, T., & Bulanov, S. V. 1998, ApJ, 498, L183 Krall, N. A. 1997, Adv. Spac Rs., 2, 75 Lob, A., & Prna, R. 1998, ApJ 495, L597 M sza ros, P., & Rs, M. 1997, ApJ, 476, 232 Paczyn ski, B., & Rhoads, J. 1993, ApJ, 418, L5

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