Opacity Build-up in Impulsive Relativistic Sources

Transcription

1 SLAC-PUB-769 arxiv: [astro-ph] Sptmbr 007 Opacity Build-up in Impulsiv Rlativistic Sourcs Jonathan Granot,,, Johann Cohn-Tanugi,, and Eduardo do Couto Silva ABSTRACT Opacity ffcts in rlativistic sourcs of high-nrgy gamma-rays, such as gamma-ray bursts (GRBs or Blazars, can prob th Lorntz factor of th outflow as wll as th distanc of th mission sit from th sourc, and thus hlp constrain th composition of th outflow (protons, pairs, magntic fild and th mission mchanism. Most prvious works considr th opacity in stady stat. Hr w study th ffcts of th tim dpndnc of th opacity to pair production (γγ + in an impulsiv rlativistic sourc, which may b rlvant for th prompt gamma-ray mission in GRBs or flars in Blazars. W prsnt a simpl, yt rich, smi-analytic modl for th tim and nrgy dpndnc of th optical dpth, τ γγ, in which a thin sphrical shll xpands ultra-rlativistically and mits isotropically in its own rst fram ovr a finit rang of radii, R 0 R R 0 + R. This is particularly rlvant for GRB intrnal shocks. W find that in an impulsiv sourc ( R R 0, whil th instantanous spctrum (which is typically hard to masur du to poor photon statistics has an xponntial cutoff abov th photon nrgy (T whr τ γγ ( =, th tim intgratd spctrum (which is asir to masur has a powr-law high-nrgy tail abov th photon nrgy ( T whr T is th duration of th mission pisod. Furthrmor, photons with nrgis > ar xpctd to arriv mainly nar th onst of th spik in th light curv or flar, which corrsponds to th short mission pisod. This ariss sinc in such impulsiv sourcs it taks tim to build-up th (targt photon fild, and thus th optical dpth τ γγ ( initially incrass with tim and (T corrspondingly dcrass with tim, so that photons of nrgy > ar abl to scap th sourc mainly vry arly on whil (T >. As th sourc approachs a quasi-stady stat ( R R 0, th tim intgratd spctrum dvlops an xponntial cutoff, whil th powr-law tail bcoms incrasingly supprssd. Kavli Institut for Particl Astrophysics and Cosmology, Stanford Linar Acclrator Cntr, Stanford Univrsity, P.O. Box 0450, MS 9, Stanford, CA Cntr for Astrophysics Rsarch, Univrsity of Hrtfordshir, Collg Lan, Hatfild, Hrts, AL 9AB, UK Th first two authors contributd qually to this work. Snd rprint rqusts to granot@slac.stanford.du, cohn@slac.stanford.du Submittd to Astrophysical Journal Work supportd in part by US Dpartmnt of Enrgy contract DE-AC0-76SF0055

2 Subjct hadings: gamma rays: thory rlativity mthods: analytical gamma rays: bursts galaxis: jts. Introduction and motivation Astrophysical sourcs of gamma-rays that ar both compact and vry luminous may b optically thick to pair production (γγ + within th sourc. Th corrsponding optical dpth, τ γγ, is usually an incrasing function of th photon nrgy, and thrfor a larg optical dpth would prvnt th scap of high-nrgy photons from th sourc, causing a high-nrgy cutoff in th obsrvd spctrum. For sufficintly high optical dpths, nough + pairs may b producd, so that th optical dpth of all photons (vn low nrgy photons that ar optically thin to pair production to scattring on ths lctrons/positrons would b much largr than unity, in which cas th photon nrgy spctrum would b thrmalizd. Th siz of th gamma-ray mitting rgion is usually hard to constrain dirctly from obsrvations, sinc th angular rsolution of gamma-ray tlscops is much poorr than thir countrparts in lowr nrgy photons (.g. X-rays, optical, or radio. Nvrthlss, th physical proprtis of th mitting rgion can b constraind using compactnss argumnts, and th obsrvd proprtis of th sourc. In particular, rapid flux variability of th sourc is oftn usd in ordr to st uppr limits on th siz of th mitting rgion, making highly variabl sourcs with significant non-thrmal high-nrgy mission a prim targt for such analysis. On of th bst xampls for such sourcs ar gamma-ray bursts (GRBs, and w shall focus on thm blow, although most of our analysis has a much broadr rang of applicability (similar opacity considrations hav also bn usd to constrain th proprtis of othr sourcs, such as Blazars,.g. Sikora, Bglman & Rs 994. It has bn ralizd arly on that, in GRBs, pair production within th sourc is xpctd to caus a high-nrgy cutoff in th obsrvd photon nrgy spctrum (s Piran 005, and rfrncs thrin. Naivly, if th sourc shows significant flux variability on an obsrvd tim scal of T, its siz is infrrd to b R c T/( + z whr z is its cosmological rdshift, and th optical dpth to pair production at a dimnsionlss photon nrgy E ph /m c is τ γγ ( σ T L /(+z /4πm c 3 R σ T L /(+z ( + z/4πm c 4 T 4 ( + z[l /ǫ(+z /( 5 rg s ][ T/( ms], whr L = F /(+z 4πd L ( + z and F ar th sourc isotropic quivalnt luminosity and obsrvd flux pr unit dimnsionlss photon nrgy, and d L is th luminosity distanc to th sourc. For GRBs th (obsrvd part of th F spctrum typically paks around, and bing at cosmological distancs thir isotropic quivalnt luminosity is typically in th rang of rg s. Furthrmor, thy oftn show significant variability down to milliscond timscals. This implis hug

3 3 valus of τ γγ, as high as 5, undr th abov naiv assumptions. Such hug optical dpths ar clarly inconsistnt with th non-thrmal GRB spctrum, which has a significant powr law high-nrgy tail. This is known as th compactnss problm (Rudrman 975. If th sourc is moving rlativistically toward us with a Lorntz factor Γ, thn in its own rst fram th photons hav smallr nrgis, ( + z/γ, whil in th lab fram (i.. th rst fram of th cntral sourc most of th photons propagat at angls /Γ rlativ to its dirction of motion. Th lattr implis that in th lab fram th typical angl btwn th dirctions of th intracting photons is θ /Γ, which has two ffcts. First, it incrass th thrshold for pair production, ( + z > /( cosθ, to ( + z Γ (compard to for th roughly isotropic distribution of angls btwn th dirctions of th intracting photons in th rst fram of th sourc, whr θ. This rducs τ γγ( by a factor of Γ ( α whr L L 0 α at high photon nrgis (corrsponding to dn ph /d α, i.. α is th high-nrgy photon indx, sinc L /(+z nds to b rplacd by L Γ /(+z = Γ ( α L /(+z. Scond, th xprssion for th optical dpth includs a factor of cosθ (that rprsnts th rat at which photons pass ach othr and hav an opportunity to intract which for a stationary sourc is, but for a rlativistic sourc moving toward us is Γ. Finally, th siz of th mitting rgion can b as larg as R Γ c T/( + z, which rducs τ γγ by an additional factor of Γ. altogthr, τ γγ ( is rducd by a factor of Γ (α+, and sinc typically α 3 this usually implis Γ in ordr to hav τ γγ < and ovrcom th compactnss problm. Using similar argumnts, th lack of such a high-nrgy cutoff du to pair production in th obsrvd spctrum of th prompt gamma-ray mission in GRBs has bn usd to plac lowr limits on th Lorntz factor of th outflow (Krolik & Pir 99; Fnimor, Epstin & Ho 993; Woods & Lob 995; Baring & Harding 997; Lithwick & Sari 00. W not, howvr, that τ γγ gnrally dpnds both on th radius of mission, R, and on th bulk Lorntz factor, Γ : τ γγ ( Γ α R L 0 α. Thrfor, on nds to assum a rlation btwn R and Γ in ordr to obtain a lowr limit on th lattr. Most works assum R Γ c T/(+z (.g., Lithwick & Sari 00, which givs τ γγ ( Γ (α+ ( T L 0 α, whil th lack of a high-nrgy cutoff up to som photon nrgy implis τ γγ ( <. This, in turn, provids a lowr limit on Γ sinc on can dirctly masur th variability tim T, th photon indx α, and L 0 4πd L ( + zα α F. Howvr, th rlation R Γ c T/( + z dos not hold for all modls of th prompt GRB mission. For xampl, this rlation dos not hold if th prompt GRB mission is gnratd by rlativistic magntic rconnction vnts, with angular scals /Γ, that crat local rlativistic motion with Lorntz factor γ rl 5 rlativ to th avrag bulk valu Γ of th mitting shll (Lyutikov & Blandford 00, 003. In this cas T/( + z R/cΓ and th infrrd valu of th Lorntz factor from standard opacity argumnts would b γ rl Γ rathr than th bulk Lorntz factor of th

4 4 shll, Γ. This allows th radius of th prompt mission to b as larg as R 6 7 cm, clos to th dclration radius whr most of th nrgy of th outflow is transfrrd to th swpt-up xtrnal mdium, and is much largr than th prompt mission radius that is xpctd in th intrnal shocks modl, R 3 4 cm. Thrfor, w adopt a mor modl-indpndnt approach and do not automatically mak this assumption. Instad, w driv most of our formulas without this assumption, as wll as driv xprssions for Γ undr this assumption, which could srv in ordr to tst its validity. Th Gamma-ray Larg Ara Spac Tlscop (GLAST mission (Ritz 007, to b launchd in arly 008, is xpctd to shd light on th high-nrgy mission from GRBs and othr impulsiv rlativistic sourcs. In particular, opacity ffcts du to th local photon fild within th sourc ar xpctd to b most rlvant in th GLAST Larg Ara Tlscop (LAT nrgy rang (0 MV to mor than 300 GV, s Rimr 007. Thus, it rprsnts a powrful tool for probing th physics of ths sourcs. GLAST is likly to dtct th high-nrgy cutoff du to pair production opacity which would actually dtrmin Γ α R, rathr than just provid a lowr limit for it. Furthrmor, in GRBs, th outflow Lorntz factor Γ may b constraind by th tim of th aftrglow onst (Panaitscu & Kumar 00; L, Ramirz-Ruiz & Granot 005; Molinari t al. 007, providd that th rvrs shock is not highly rlativistic, so that if GLAST dtcts th high-nrgy pair production opacity cutoff, th radius of mission R could b dirctly constraind, thus hlping to tst th diffrnt GRB modls. In particular, this could dirctly tst whthr th rlation R Γ c T/(+z that is xpctd in many modls indd holds, sinc both R and Γ could b dtrmind sparatly. This, howvr, rquirs a rliabl way of idntifying th obsrvd signaturs of opacity to pair production. This is on of th main motivations for this work. Th lading modl for th prompt mission in GRBs faturs intrnal shocks (Rs & Mészáros 994 du to collisions btwn shlls that ar jctd from th sourc at ultra-rlativistic spds (Γ 0. Th shlls ar typically quasi-sphrical, i.. thir proprtis do not vary a lot ovr angls a fw Γ around our lin of sight. Undr th typical physical conditions that ar xpctd in th shockd shlls, all lctrons cool on a tim scal much shortr than In th prsnt work, w will not considr opacity ffcts du to intraction of high nrgy photons with th xtra-galactic background light. Such an attnuation, intrsting in its own right, can b addd to th in sourc opacity in a straightforward way. Furthrmor, it is xpctd to bcom significant (i.. produc τ γγ > only at cosmological rdshifts (z and for vry high photon nrgis ( 56 0 GV at z = and 8 63 GV at z = 3; Knisk t al. 004, and is thrfor likly to significantly affct only th high nd of th GLAST nrgy rang, whr th photon statistic might b too poor to rliably masur this ffct. This sourc of opacity will b indpndnt of tim (and dpnds only on th rdshift of th sourc, and on th photon nrgy, which would hlp in disntangling it from th tim dpndnt opacity intrinsic to th sourc that w calculat in this work.

5 5 th dynamical tim (i.. th tim it taks th shock to cross th shll, and most of th radiation is mittd within a vry thin cooling layr just bhind th shock front. Thus, our modl which faturs an mitting sphrical thin shll that xpands outward ultra-rlativistically is appropriat for th intrnal shocks modl. As this mitting shll xpands outward to largr radii, it builds up a photon fild that can pair produc with high-nrgy photons from th sam mission componnt. This ffct has bn studid in th past (s spcially Baring 006, and rfrncs thrin, but th tmporal and spatial dpndncs of th photon fild hav bn avragd out, corrsponding ithr xplicitly or implicitly to a quasi-stady stat. Howvr, in impulsiv rlativistic sourcs, th tim scal for significant variations in th proprtis of th radiation fild within th sourc is comparabl to th total duration of th mission pisod, and thrfor th dpndnc of th opacity to pair production on spac and tim cannot b ignord, and may produc important ffcts that ar supprssd in th stady-stat limit. Thrfor, in th prsnt work w considr th full tmporal and spatial dpndnc of th opacity, in ordr to captur all th rsulting ffcts. W dvlop a simpl, yt rich, modl to invstigat quantitativly th intuitiv considration that in impulsiv sourcs it taks tim to build up th (targt photon fild, and thus th optical dpth initially incrass with tim, so that high nrgy photons might b abl to scap th sourc mainly at th vry arly part of th spik in th light curv. This rsults in a powr law tail for th tim-intgratd spctrum at high nrgis, whil th instantanous spctrum (which is hard to masur du to poor photon statistics has an xponntial cutoff. This ariss sinc th photon nrgy of th xponntial cutoff in th instantanous spctrum dcrass with tim, as th opacity incrass with tim at all nrgis. Thrfor, at sufficintly high photon nrgis, most of th photons scap during th short initial tim bfor th optical dpth incrass abov unity, i.. bfor th cutoff nrgy swps past thir nrgy. W prform dtaild smi-analytic calculations of th optical dpth to pair production, which improv on prvious works by first calculating th photon fild at ach point in spac and tim, and thn intgrating along th trajctory of ach photon. Th structur of th papr is as follows. In w introduc our modl and driv a gnral xprssion for th flux that rachs an obsrvr at infinity. This xprssion includs th optical dpth along th trajctory of ach photon that may rach th obsrvr, which is drivd in 3. Th calculation of th optical dpth rquirs th knowldg of th photon fild at ach point along th trajctory of ach (tst photon. This local photon fild is first xprssd in trms of th sourc missivity ( 3.. Nxt ( 3. it is convnintly rwrittn as th product of th typical optical dpth (that is approachd on a dynamical tim, and is similar to

6 6 that drivd in prvious works and dimnsionlss ordr unity xprssion (containing a fw intgrals which capturs th nw tim dpndnt ffcts that ar th focus of this work. In 4 xplicit xprssions ar drivd for th intgrands of ths dimnsionlss ordr unity intgrals. In 5 w driv th rlvant analytic scalings for th rsulting optical dpths and obsrvd flux, and in 6 w prsnt numrical rsults (i.. numrically valuat th smianalytic xprssions for th opacity, light curvs, and spctra (both th instantanous and tim-intgratd spctra ar addrssd in 5 and 6. Our conclusions ar discussd in 7.. Calculating th Obsrvd Flux.. Modl Assumptions W considr an ultra-rlativistic (with Lorntz factor Γ, thin (of width R/Γ in th lab fram sphrical xpanding shll, that mits ovr a finit rang of radii, R 0 R R 0 + R (i.. th mission turns on at R 0 and turns off at R 0 + R. This modl can b associatd with a singl puls or flar in th light curv. In th contxt of intrnal shocks within th outflow, R R 0 is typically xpctd (Rs & Mészáros 994; Piran 005, and rfrncs thrin. Th mission is assumd to b isotropic in th co-moving fram of th mitting shll (i.. th shll rst fram, and uniform ovr th sphrical shll. In this work primd quantitis ar always masurd in th co-moving fram, whil unprimd quantitis ar valuatd ithr in th lab fram, that is th rst fram of th cntral sourc, in which th shll is sphrical (.g. th Lorntz factor Γ, or in th obsrvr fram (.g. th obsrvd tim and photon nrgy which suffr cosmological tim dilation and rdshift, rspctivly, rlativ to th lab fram which is at th cosmological rdshift of th sourc. Th obsrvr is assumd to b locatd at a distanc from th sourc that is much largr than th sourc siz (so that th angl subtndd by th sourc, as sn by th obsrvr, is vry small, and th obsrvr can b considrd as bing at infinity. For convninc, w will us dimnsionlss photon nrgis,, in which th obsrvd photon nrgy, E ph, is normalizd by th lctron rst nrgy: E ph /m c. Whil gnral xprssions will b providd whn possibl, w also provid dtaild smi-analytical solutions to th modl by assuming that th luminosity in th shll rst fram has a powr-law dpndnc on rst fram photon nrgy and radius R, L ( α R b, and that th Lorntz factor scals as a powr law with radius, Γ R m. Th approximation that Γ and L scal as powr laws with radius is usually xpctd to hold rasonably wll. For intrnal shocks, th colliding shlls ar xpctd to b in th coasting stag nar th collision radius (R 0,

7 7 which corrsponds to m = 0 (s Piran 005; Mészáros 006, and rfrncs thrin. In th GRB aftrglow, both bfor and aftr th dclration radius, whr most of th nrgy is transfrrd from th jcta to th shockd xtrnal mdium, Γ (Blandford & McK 976 and L (.g., Sari 998; Granot 005 ar xpctd to scal as powr laws with radius. For GRB intrnal shocks, th scaling of L with radius R gnrally dpnds on th dtails of th colliding shlls. For uniform colliding shlls, whr th strngth of th shocks going into th shlls is constant with radius, abov th pak of th νf ν spctrum, pak, on xpcts 0.5 b 0. This may b undrstood as follows. In this cas th Lorntz factor in th shockd rgions of th colliding shlls is constant with radius, whil th magntic fild scals as B R. Thrfor, sinc th numbr of mitting lctrons scals linarly with radius, N R, thn L,max B N R 0. Th typical synchrotron photon nrgy scals as ǫ m B γm R sinc th typical Lorntz factor of th lctrons, γ m, is constant for a constant shock strngth. Th nrgy of a photon that cools on th dynamical tim (th tim sinc th start of th collision scals as ǫ c R. Thrfor, abov th pak of th νf ν spctrum, at > pak = max( c, m, w hav L = L,max( m/ c / ( / m p/ R ( p/, whr p is th powr law indx of th lctron distribution, dn /dγ γ p for γ > γ m. Sinc p 3 is typically infrrd for th GRB prompt mission, this corrsponds to 0.5 b 0. For fast cooling ( c < m blow pak = m, L = L,max ( / c / R /. For slow cooling ( c > m, howvr, blow pak = c, L,max( / m ( p/ R ( p/. Th simplifying assumption of a powr law mission spctrum [L ( α ], howvr, is not always valid (s,.g., Baring 006. For xampl, in GRB intrnal shocks it braks down for photons of nrgy Γ /(+z pak, i.. m c 5(+z (Γ/0 ( pak m c /0 kv GV. Indd, photons of such nrgy intract with photons blow th spctral brak nrgy brak which is th pak of th νf ν spctrum. A dtaild tratmnt of th cas of a mor ralistic spctrum for GRB intrnal shocks will b providd lswhr. Th xact shap of th spctrum at high nrgis is not wll constraind. Thus, w us a fiducial valu of α =, which corrsponds to a flat νf ν (i.. qual nrgy pr dcad in photon nrgy, in our dtaild illustrativ solutions, and also xplor th ffcts of varying th valu of α... Th Equal Arrival Tim Surfac of Photons to th Obsrvr (EATS-I Th obsrvd normalizd flux dnsity, F = (m c /hf ν, is calculatd as a function of tim and photon nrgy, closly following th drivation of Granot (005. For this purpos, th contributions to th obsrvd flux at any givn obsrvd tim T ar intgratd ovr th qual arrival tim surfac (EATS-I th locus of points from which photons that ar

8 8 mittd at th shll rach th obsrvr simultanously, at th obsrvd tim T. In th prsnt work, th ffcts of opacity to pair production will b addd at th nd of this calculation, as dtaild blow. W considr a photon initially mittd by th shll at a lab fram tim t 0 whn th radius of th shll is R t,0 R sh (t 0 and its Lorntz factor is Γ t,0, at an angl of θ t,0 from our lin of sight to th origin R = 0 (s Fig.. Du to th sphrical symmtry of our modl, thr is no dpndnc on th azimuthal angl. Th arrival tim T of th photon to a distant obsrvr is givn by th qual arrival tim formula: T ( + z = t 0 R t,0 cos θ t,0, ( c whr th lab fram tim t is rlatd to th shll radius at that tim, R sh (t, by t = Rsh (t 0 dr βc = R sh(t c c Rsh (t 0 dr Γ (R + O(Γ 4. ( In Eq. (, T = 0 is chosn to corrspond to a photon that is mittd at th origin at t 0 = 0. Eq. ( rlats t and R sh (t, so that th locus of points (R t,0, θ t,0 that kp T constant dfins th EATS-I at tim T. For a coasting shll (m = 0, it is a wll-known rsult that th EATS-I is an llips of smi-major to smi-minor axis ratio Γ (Rs 966. Th flux dnsity at th rscald nrgy is obtaind by intgrating ovr th luminosity in th shll rst fram, L, along th EATS-I (Granot 005: F (T = ( + z 4πd L δ 3 dl ( + z = 8πd L ymax dy dµ t,0 y min dy δ3 (yl (y, (3 whr δ ( + z/ is th Dopplr factor of th mittd photon (btwn th co-moving and lab frams, µ t,0 cosθ t,0 is th cosin of its angl of mission, and w dfind th normalizd radius y R t,0 /R L, whr R L = R L (T is th largst radius on th EATS-I at tim T. Th intgration is prformd along th EATS-I, and th boundaris for y ar y min (T = min [ ] R 0, R L (T, y max = min [, R ] 0 + R R L (T, (4 sinc th mission turns on at R 0 and turns off at R 0 + R. For th tims T rlvant to th problm, corrsponding to th arrival of photons to th obsrvr, R 0 /R L (T is always smallr than. It actually rprsnts an llipsoid, kping in mind th symmtry around th lin of sight to th cntr of th mitting sphrical shll, and th lack of dpndnc on th azimuthal angl.

9 9 In ordr to valuat th intgral abov, w now driv xprssions for th intgrand. Dfining Γ L Γ(R L, Γ R m can b rwrittn as Γ (RR m = Γ L Rm L = constant, and thus Γ = Γ L y m. Eq. ( now bcoms t 0 = R t,0 c + R L y m+ (m + Γ L c + O(Γ 4. (5 In th limit of small angls (θ t,0, which is rlvant for Γ, Eq. ( implis t 0 R t,0 /c = T/( + z R t,0 θ t,0/c, which togthr with Eq. (5 yilds T ( + z = R L y m+ (m + Γ L c + R t,0θt,0 c. (6 As can b sn in Fig., a photon that is mittd at R t,0 = R L (corrsponding to y = R t,0 /R L (T = rmains along th lin of sight (θ t = θ t,0 = 0, so that Eq. (6 yilds ( /(m+ R L (T = (m + Γ ct T L [T/( + z] ( + z = R 0, T 0 = ( + zr 0 T 0 (m + cγ 0, (7 whr Γ 0 Γ(R 0, and can b rwrittn as θ t,0 = y y m (m + Γ L. (8 W hav introducd th tim T 0 at which th first photons rach th obsrvr (corrsponding to a photon mittd at R 0 along th lin of sight, θ = 0: R L (T 0 R 0. Sinc µ t,0 θt,0 /, Eq. (8 implis dµ t,0 dy = y + my m. (9 (m + Γ L Finally, th Dopplr factor of th mittd lctron is givn by δ Γ( β cosθ t,0 Γ + (Γθ t,0 = (m + Γ Ly m/ m + y m, ( and its valu at R L (which corrsponds to y = is δ(r L = Γ L. Sinc whr = ( + z/δ, w obtain: [ ] L = α ( b Rt,0 L (+z/δ(r L (R L, ( (R L ( α ( α ( b m(α / δ δ L = L (+z/γ L (R L y b = L (+z/γ Γ 0 (R 0 y b RL. ( L Γ L R 0 R L

10 Th ffct of pair production opacity will b tratd in this work in a somwhat simplifid mannr, by assuming that photons which pair produc do not rach th obsrvr, and ignoring th additional opacity that is producd by th scondary pairs and th photons mittd by ths pairs. Undr ths simplifications, th ffcts of opacity to pair production can b includd by adding a trm xp( τ γγ into th intgrand in Eq. (3, whr τ γγ is a function of y,, R/R 0, and T/T 0. Thus, by combining qs. (9 with Eq. (3, w obtain: F (T = Γ L L ( + z (+z/γ L (R L 4πd L = Γ 0 L ( + z (+z/γ 0 (R 0 4πd L ymax ( m + dy m + y m y min ymax ( +α m + dy y b mα/ τγγ m + y m y min (b mα/[(m+] ( T T 0 +α y b mα/ τγγ, (3 whr Eq. (7 is usd to driv th scaling R L (T/R 0 = (T/T 0 /(m+, and ( τ γγ = τ γγ y,, R, T L 0, R 0 T 0 Γ α 0 R, (4 0 as is shown latr on, whr Γ 0 Γ(R 0, and L L 0 α is th obsrvd isotropic quivalnt luminosity. Unlss spcifid othrwis, th drivations throughout this work ar valid for a gnral valu of m. For a coasting shll (m = 0, which is a cas of spcial intrst (as it is xpctd,.g., for intrnal shocks, Eq. (3 simplifis to F (T = Γ 0 L ( + z (+z/γ 0 (R 0 4πd L ( b T ymax T 0 y min dy y α+b τγγ. W hav xprssd th obsrvd flux dnsity for our modl as a function of th obsrvd tim T, and w now nd to driv th xprssion of th optical dpth τ γγ. W gathr hr th dpndnc on y of two quantitis that will b ndd latr on: ˆR 0 R 0 = y min R t,0 y = R 0 R R R t,0 = y ( T T 0 /(m+, x (Γ t,0θ t,0 = y (m+ (m +. (5 In ordr to facilitat rading, w includ in Tabl th most common quantitis usd throughout this work. 3. Computation of th optical dpth As in th prvious sction, w considr a tst photon mittd by th shll at radius R t,0 and angl θ t,0 with rspct to th lin of sight (s Fig.. All th quantitis with a

11 subscript t will always rfr to such a tst photon. W wish to calculat its optical dpth to pair production with all th othr photons which ar mittd by th sam sourc and dnotd by a subscript i (for potntially intracting. Th diffrntial of th optical dpth to pair production is givn by (Wavr 976 dτ γγ = σ dn i [χ( t, i, µ ti ]( µ ti dω i d i ds. (6 dω i d i In this quation, ds is th diffrntial of th path lngth along th trajctory of th tst photon; n i, Ω i and E i i m c ar th numbr dnsity, solid angl, and photon nrgy of th photon fild along th path of th tst photon with which it might intract. 3 For convninc, t and i dnot th valus of th corrsponding dimnsionlss photon nrgis in th lab fram, rathr than in th obsrvr fram (as is th cas for, i.. without th cosmological rdshift, so that t = ( + z should vntually b usd in ordr to valuat th optical dpth at an obsrvd valu of. Th Lorntz invariant cross sction for pair production σ (χ is [ σ (χ = πr (χ 4 + χ ln(χ + χ χ χ( + χ ] χ, (7 6 t i ( µ ti χ =, (8 whr χ is th cntr of momntum nrgy (in units of m c, of ach particl ach of th two intracting photons, and th producd lctron and positron, and µ ti = ˆn t ˆn i is th cosin of th angl btwn th dirctions of motion of th tst photon (ˆn t and a potntially intracting photon (ˆn i. In ordr to valuat µ ti, w nd to spcify th gomtry for our modl: a sphrical mitting shll, whos mission dpnds only on its radius R sh (i.. at any givn radius its local mission dos not dpnd on th location within th shll and is isotropic in its own rst-fram. Undr ths assumptions, th radiation fild will dpnd only on th radius R and th (lab fram tim t, and at any givn plac and tim it will b symmtric around th radial dirction (s Fig.. Thrfor, at any point along th trajctory of th tst photon, w can us a local coordinat systm, S r, whos z-axis is alignd with th radial dirction (from th cntr of th shll to that point, ẑ r, and such that th dirction of motion of th tst photon is in th x-z plan. In this fram th polar 3 W do not add a factor of / du to doubl counting (as was don by,.g., Baring & Harding 997; Drmr & Schlickisr 994, as it should not appar in th xprssion for th optical dpth. W discuss this point in mor dtails in annx E.

12 angls ar dnotd by (θ r, φ r, and ˆn t = ˆx r sin θ t + ẑ r cosθ t, (9 ˆn i = ˆx r sin θ r cosφ r + ŷ r sin θ r sin φ r + ẑ r cosθ r, (0 µ ti = ˆn i ˆn t = sin θ t sin θ r cosφ r + cosθ t cosθ r. ( Not that θ t varis only with s. Th intgration ovr th solid angl in th lab fram in Eq. (6 can convnintly us th fram S r which is at rst in th lab fram, i.. dω i = dω r = dφ r dµ r. Th optical dpth of th tst photon is thn givn by: τ γγ ( t, θ t,0, R t,0 = ds d i dω r σ dn i [χ( t, i, µ ti ]( µ ti. ( dω r d i Nxt, w xprss th drivativ in th intgrand of Eq. (, which rprsnts th photon fild along th trajctory of th tst photon, in trms of th sourc missivity. In addition, w mak a sris of changs of variabl in ordr to simplify th xprssion for th optical dpth. 3.. Exprssing th photon fild in trms of th sourc missivity In., w xprssd th obsrvd flux as an intgral ovr th EATS-I of photons to th obsrvr at an obsrvd tim T. Ths photons travl along straight lin trajctoris that pass through th photon fild. As a rsult, w intgrat th contribution to th optical dpth at ach point along th path of ach photon, trating it as a tst photon. This is th intgration ovr ds in Eq. ( which, as w show blow, can b rplacd by an intgration ovr dr t. In th othr two innr intgrations R t is kpt fixd, and th photon fild, dn i /dω r d i, nds to b valuatd as a function of i, µ r and R t. For a givn tst photon that is mittd at (R t,0, µ t,0, th valu of R t also dtrmins th valu of th lab fram tim t t. W rmind th radr that R t and t t ar always computd in th lab fram, and that R t is in gnral diffrnt than R sh (t t, i.. at a gnral tim th position of th tst photon dos not coincid with that of th shll. W procd first to rlat th photon fild at (t t, R t to th missivity in th local fram of th mitting shll, which is asir to spcify, and simplr. Th Dopplr factor of th mittd photon is givn by δ i i = Γ( βµ i = Γ( + βµ i, (3 whr µ i cosθ i = ˆβ ˆn i and µ i cosθ i = ˆβ ˆn i ar th cosins of th angl btwn th bulk vlocity of th mitting fluid ( β and th dirction of th intracting photon in th lab

13 3 fram (ˆn i and in th comoving fram of th mitting fluid (ˆn i, rspctivly. Furthrmor, µ i = µ i β βµ i = dω i dω r = dω i dω i = dµ dµ = δ, (4 sinc dω i = dφ i dµ i and φ i = φ i. W ar intrstd in th diffrntial dnsity of photons of nrgy i and dirction of motion in th solid angl dω r around th dirction ˆn i, which is at an angl θ r from th radial dirction, at a radius R t and tim t t. This dnsity is rlatd to th spcific intnsity of th photon fild by: I i (ˆn i de = i m c 3 dn i (ˆn i, (5 dsdtd i dω i d i dω r whr th (normalizd spcific intnsity I i is th nrgy (de pr unit normal ara (ds whr d S/dS = ˆn, pr unit tim (dt, pr unit (normalizd photon nrgy (d i, pr solid angl (dω i = dω r around som dirction ˆn i of th (potntially intracting photons. Th diffrntial (normalizd spcific luminosity (in our cas, from a small part of th mitting shll is dfind as dl = de/d i dt, whil th isotropic quivalnt (normalizd spcific luminosity is dfind by: dl i,iso 4π dl i dω r. (6 Th contribution of an mitting lmnt with dl i,iso to th (normalizd flux dnsity df i de/dsdtd i and to th (normalizd spcific intnsity I i at a point locatd at a distanc r from it is df i = dl i,iso 4πr = I i (ˆndΩ r, (7 and is along th dirction ˆn from th mitting lmnt to that point (i.. hr ds is th diffrntial of th ara normal to ˆn, ds = ˆn d S. Finally, w can convnintly xprss dl i,iso in th comoving fram (i.. th local rst fram of th mitting shll, dl i,iso = 4π dl i de = 4π = δ 3 de 4π dω r d i dtdω i d dt dω i = δ 3 dl, (8 whr th last quality follows from th assumption that th mission is isotropic in th comoving fram. Bcaus th mission is assumd to b uniform throughout th shll, dl dpnds only on th radius of mission of th potntially intracting photon, R, and not on th location within th shll. Apart from th mission radius, R, th position of an mitting point on th shll is also spcifid by th polar angl, θ, which for convninc is masurd with rspct to th dirction from th cntr of th sphr to th location of th tst photon (at a radius R t whr th flux (or som othr proprty of th photon fild is calculatd

14 4 (s Fig.. As a rsult, w can writ dl = L (R dµ /, whr µ = cosθ and L (R has bn dfind and discussd in. W finally combin qs. (5, (7 and (8 to obtain th xprssions for th normalizd spcific intnsity, I = L (R 4π δ 3 dµ 4πr dµ r, (9 and th xprssion for th photon fild which appars in th intgrand in Eq. (, dn i = L (R δ 3 dµ d i dω r (4π m c 3 r dµ r. (30 Th drivativ in th last trm to th right of ths quations must b computd along th qual arrival tim surfac (EATS-II of photons to R t at t t, whr I or dn i /d i dω r ar to b calculatd. W can now rwrit Eq. ( as: τ γγ ( t, θ t,0, R t,0 = σ T (4π m c 3 ds d i σ [χ( t, i, µ ti ] L (R δ 3 i dω r ( µ ti dµ σ T i r dµ r. (3 W hav thus rplacd th photon fild by th spcific missivity in th xprssion for th optical dpth. Th boundaris of intgration will b spcifid xplicitly latr on. W now want to simplify this tripl intgration in ordr to mak it asir to valuat. 3.. Analytical rduction In th rmaindr of this work, w will mak us of various dimnsionlss radii, which ar gathrd in Tabl and gratly simplify th analysis. Furthrmor, it is much mor convnint to work with such quantitis of ordr unity insid th intgrand. W thus rscal R and R t by introducing R R /R t and ˆR t R t /R t,0. Furthrmor, th notations R R/R t and ˆR R/R t,0 will b usd for othr rscald dimnsionlss radii as wll. Whil clarly ˆR t <, th rang of R is much mor complx and will b xtnsivly discussd in 4. For now, w want to simplify Eq. (3 by changing intgration variabls. W giv hr th main rsults and lav th dtails of th drivations for Annx A. As has bn mntiond abov, th intgration ovr ds can b rplacd by an intgration ovr ˆR t. Undr th approximation of larg Lorntz factors (Γ, and thus small mission angls (θ t,0, on obtains ds = R t,0 d ˆR t (s th discussion following Eq. [A] for mor dtails. Bsids, sinc w intgrat ovr dω r = dφ r dµ r and th intgrand contains dµ /dµ r w can convnintly chang th intgration ovr µ r to an intgration ovr R. W show in

15 5 th Annx A that dµ dµ r dµ r = dµ d R d R, sinc dµ /d R > 0, whr th limit of intgration ovr R should b in incrasing ordr (i.. th intgration should b from small to larg valus of R. Th optical dpth now rads: τ γγ ( t, θ t,0, R t,0 σ T (4π m c 3 R t,0 d ˆR t ˆR t π d i dφ r d / t 0 R R 0 /R t σ [χ( t, i, µ ti ] L (R δ 3 dµ i ( µ ti σ T i r d R. (3 Nxt, w can follow th hind-sights of Stpny & Guilbrt (983 and Baring (994 in ordr to cast th intgrations ovr (dφ r, d i into a much mor practical form. In ordr to prform this chang of variabls, it is ncssary to spcializ th spcific luminosity to th dpndnc discussd in : L (R = L 0 h(r /R 0 ( i α Γ α 0 L 0 ( i α h( R ˆRt / ˆR 0, i whr h is a gnral function of R /R 0 that satisfis h( = (for dtails s Appndix A.3 and L 0 L = (R 0. Not that L 0 Γ α 0 L 0 is approximatly th obsrvd isotropic quivalnt luminosity at a photon nrgy of m c 5 kv, nar th pak of th spik in th light curv which corrsponds to th mission pisod that w modl for R R 0. For convninc, w rscal all th quantitis in th intgrand of Eq. (3 which ar not of ordr unity by th rlvant powr of th Lorntz factor at radius R t, Γ t = Γ(R t, so that th rscald quantitis (which ar dnotd by a bar will b of ordr unity. W rscal δ δ/γ t and d µ Γ t dµ, but do not rscal r which is alrady of ordr unity. Thus, δ +α r dµ d R = Γ α 0 ( mα/ +α ˆRt δ ˆR 0 r d µ d R, (33 and th xprssion for th optical dpth bcoms: τ γγ ( t, θ t,0, R t,0 = τ α t whr ˆR mα/ 0 d ˆR t ˆR mα/ t 0 L 0 ( τ Γ α 7σ T Γ α 0 R 0, α, L 0 = 48π 3 m c 3 α 5/3 R 0 = 0.40 ( α d R δ +α r d µ d R h ( ˆRt R ζ H ˆR α α (ζ, (34 0 5/3 4( α L 0,5 (Γ 0, α R 0,3, (35 and L 0,5 = L 0 /( 5 rg s, R 0,3 = R 0 /( 3 cm, Γ 0, = Γ 0 /0, ζ = (ζ + ζ /ζ, ζ + = [ cos (θ r + θ t ]/, ζ = [ cos (θ r θ t ]/, and H α (z is a function discussd in

16 6 Annx A.3. In Eq. (35, τ, th only quantity rquiring astrophysical input, is a constant of th ordr of th optical dpth to pair production at a photon nrgy of m c at R 0 in quasistady stat (nar th pak of th spik in th light curv for R R 0. Not that sinc both th photon indx α and L 0 (roughly th isotropic quivalnt luminosity ar obsrvabl quantitis (th lattr rquiring knowldg of th sourc rdshift, th obsrvation of a highnrgy spctral cutoff du to pair production opacity can nabl th dtrmination of Γ α 0 R 0. In th limit of small angls that is appropriat for larg Lorntz factors, ζ is of ordr Γ, so w dfin ζ Γ t ζ, whr Γ t = Γ(R t = Γ ˆR m/ 0 t. Thus, ζ α = Γ α 0 ( mα ˆRt α ( ζ. (36 ˆR 0 Undr th assumption that h is also a powr-law of indx b, h(r = (R /R 0 b = ( R ˆRt / ˆR 0 b, th xprssion for th optical dpth in our modl simplifis to: τ γγ ( t, θ t,0, R t,0 = τ 0 ( t, R t,0 F(x, (37 τ 0 ( t, R t,0 = τ α F(x = t ˆR b mα/ 0, (38 d R +α δ d µ r d R α Rb ζ H α (ζ. (39 d ˆR t ˆRb +mα/ t In ordr to procd furthr, w nd to obtain an xplicit xprssion for th innrmost intgrand of F, by a dtaild xamination of th gomtry of th photon fild. Th nxt sction will b dvotd to this analysis, which constituts th main novlty of this work. W will valuat th optical dpth (Eq. [37], taking into account that th photon fild is not homognous along th tst photon trajctory, but th contribution to th photon fild is actually built up in tim. 4. Calculating th Photon Fild 4.. Equal Arrival Tim Surfac of Photons to th Tst Photon (EATS-II In this sction w calculat th photon fild at a gnral radius R t and tim t t, along th trajctory of a tst photon. For this purpos w nd to considr th contribution from all photons that arriv at th instantanous location of th tst photon, (R t, t t, simultanously. Th locus of points whr all such photons ar mittd, taking into account that th mission occurs only in th shll, forms a two dimnsional surfac rfrrd to as th qual arrival tim surfac (EATS-II of photons to th instantanous location of th tst photon. Th local

17 7 photon fild at (R t, t t is calculatd by intgrating th contributions ovr this surfac. W strss that this surfac (EATS-II, is diffrnt from th qual arrival tim surfac of photons to th obsrvr at infinity (EATS-I. Fig. shows th basic configuration for our calculations and illustrats th rlation btwn th two diffrnt qual arrival tim surfacs (EATS of photons:. to th obsrvr at infinity (EATS-I,. to th instantanous location of a tst photon (EATS-II. It can b sn that th EATS-II grows with th lab fram tim t, and thrfor also with th radius of th tst photon R t. Furthrmor, ach EATS-II ncompasss all othr EATS-II corrsponding to smallr tims, and is ncompassd within all th EATS-II which corrspond to largr tims. In particular, all EATS-II ar within th EATS-I, which corrsponds to th limit of th EATS-II for an infinit tim (whn th tst photon rachs th obsrvr at infinity. All of th EATS-II and EATS-I pass through th mission point of th tst photon, and for cas and 3, also through th plac whr th photon crosss th shll (i.. its location in cas. Ths ar gnral proprtis of th EATS-II. W now procd to calculat th EATS-II and th xprssions for rlvant quantitis along this surfac, which ar ndd in ordr to calculat th local radiation fild. From th gomtry of our problm (s Fig., w can immdiatly driv th two following quations: r = + R R µ = ( R + R ( µ, (40 R = + r rµ r, (4 whr R R /R t and r r/r t. Th qual arrival tim surfac (EATS-II of photons to (R t, t t is dtrmind by th condition that r = c(t t t = c[t t t sh (R ], whr th photons ar mittd at a prvious tim t whn th shll is at a radius R = R sh (t. Th EATS-II quation is thus givn by r = c [t t t sh (R ] = ( R R + R ( µ, (4 t which rlats th radius (R and angl (θ = arccosµ of mission along this surfac. Th xprssion for t sh (R dpnds on our assumption about th xpansion of th shll. If th lattr occurs at constant spd, thn t sh (R = R /βc and in th limit of R t Eq. (4 rducs to β(ct t R t = R ( βµ, which is th usual polar quation of an llips (stting ct = ct t R t. In this simpl cas, using th short notation R sh = R sh (t t, w hav r = ( R sh R /β, and th EATS-II is givn by µ = β R [ ( Rsh R β ( R ], (43

18 8 whil th lowr and uppr limits for th rang of R valus along th EATS-II, which corrspond to µ = and µ =, rspctivly, ar givn by R,min = R sh β ( R sh + β/( + β Rsh,, R,max = (44 + β ( R sh β/( β Rsh. Not that w hav not assumd Γ = ( β /, so ths rsults ar valid for an arbitrary vlocity, as long as it is constant with radius. Combining qs. (40 and (4 w also obtain µ r = R µ ( R + R ( µ = R t c [ R ] µ, (45 t t t sh (R whr in th last quality w hav also usd Eq. (4, so that it is valid only along th EATS-II (whil th first quality is valid mor gnrally, as it is drivd dirctly from th gomtrical stup. 4.. Radial Dpndnc of Rlvant Angls, µ ( R and µ r ( R, along EATS-II Spcifying for Γ = Γ R t m, w can rwrit Eq. (5 as [ ] Thus, Eq. (4 implis t sh (R = R t c r = R sh R + r = ( Rm+ sh R m+ R + (m + Γ t (m + Γ t m+ R ( Rsh R + ( Rsh R ( Rm+ sh + O ( Γ 4 t. (46 + O ( Γ 4 t, (47 (m + Γ t m+ R + O ( Γ 4 t. (48 Not that R R sh, bcaus t t t (du to causality and R sh (t is an incrasing function of t. Th quality only holds whn R t = R sh (t t, i.. whn R sh = (cas blow. Thus, qs. (40 and (48 giv (to th ordr of Γ t, Γ t ( µ = (Γ t θ = R Γ t [ ( Rsh R ( R ] + ( Rsh R ( Rm+ sh (m + R m+. (49

19 9 Th two trms on th right hand sid of th quation ar typically of th sam ordr sinc R sh a fw Γ t, i.. Rsh = Rsh (t t = R t. This immdiatly implis [ dµ d R = ( Γ R Γ t t R sh + R ] ( sh Rm+ m+ m+ sh + m R R. (50 m + Now w turn to µ r. From qs. (4 and (48 w obtain ( Rsh R ( + R ( Rm+ m+ sh R µ r = ( Rsh R + R 4(m + Γ ( t Rsh R + O ( Γ 4 t, [ dµ r d R = d r R ] d R r R r r whr = R sh ( Rsh R + R m ( Rm+ sh + (m + Γ t 4Γ t R ( Rsh R ( m+ R Rsh R ( Rsh R 3 + O ( Γ 4 t d r d R = + R m Γ t = + (5, (5 Γ ( R = β( R. (53 This can asily b undrstood sinc r = c(t t t along th qual arrival tim surfac, so that dr = cdt and d r/d R = dr/dr = cdt /dr = c/(dr /dt = /β( R. Th maximal radius of mission, R,max, from which a photon rachs a point at radius R t at th tim t t is dtrmind by th photon that is mittd at θ = 0 (i.. µ =, along th lin conncting that point to th cntr of th sphr. Thus, r min = R,max = c R t [ = R sh R,max + t t t sh ( R ],max ( Rm+ sh R m+,max (m + Γ t and th problm naturally divids into thr cass. (54 + O ( Γ 4 t, (55

20 Proprtis of EATS-II According to Rlativ Location of Tst Photon and Shll Th proprtis of th EATS-II qualitativly chang according to th location of th tst photon rlativ to th shll at th sam lab fram tim, t t. Thus th problm naturally divids into thr cass, as illustratd in Figs. and 3. If th photon is mittd at an angl 4 θ t,0 > /Γ t,0, i.. x (Γ t,0 θ t,0 >, it initially lags bhind th shll (cas, sinc du to th abrration of light (also rfrrd to as rlativistic baming this corrsponds to an angl gratr than 90 from th radial dirction in th co-moving fram of th shll. Th photon vntually catchs-up with th shll and crosss it (cas, sinc th lattr is moving at a vlocity slight smallr than th spd of light. Aftr it crosss th shll, it rmains ahad of th shll (cas 3. A photon that is mittd at θ t,0 /Γ t,0, corrsponding to x, immdiatly gts ahad of th shll (cas 3. All photons ar always mittd at th shll, so th point of mission is considrd cas. Lik th latr shll crossing for photons with x >, cas corrsponds to a singl point along that trajctory of th tst photon, unlik cass corrsponds to a finit path along th trajctory, and cas 3 corrsponds to a (practically smi-infinit intrval (as far as th obsrvr is considrd to b at infinity ; th contribution to th opacity at larg distancs from th sourc, howvr, bcoms ngligibl. Th thr diffrnt cass ar discussd in dtail blow, and th rlvant xprssions for ach cas ar drivd. W start by dfining som usful quantitis for this purpos, which will b vry hlpful latr on. In th limit of small angls, Eq. (A yilds (Γ t θ t x ˆR m t, (56 whr x (Γ t,0 θ t,0 is th squar of th normalizd mission angl of th tst photon. Evaluating Eq. (46 at R sh = R sh (t t givs ct t = R R R m+ sh sh + t (m + Γ t + O ( Γ 4 t, (57 which can b rwrittn in trms of th quantity ( f m (m + Γ ctt t = (m + Γ t R ( R sh + t R m+ sh + O ( Γ t, (58 that plays a major rol in th following drivations. 4 Mor gnrally, th condition is cosθ t,0 < β, but for Γ t,0 and θ t,0 this rducs to θ t,0 > /Γ t,0.

21 For an mission pisod starting at R 0 = 0, th inquality ct t > R t is rquird in ordr to hav a non-vanishing radiation fild at th point (R t, t t. If th mission turns on at a non-zro radius R 0, this condition gnralizs to ct t R t R 0 (m + Γ (R 0 = Rm+ 0 (m + Γ t. (59 This implis that f m > 0 (for m >, which is assumd in this work, and is typically th cas for th astrophysical sourcs of intrst. W not that f m < for R sh < (whn th tst photon is travling in front of th shll, f m > for R sh > (whn th tst photon is travling bhind th shll, and f m = for R sh = (whn th tst photon is at th shll. It is convnint to xprss f m as a function of our primary variabls. Using R t = R + z = R t,0 sin θ t,0 + [R t,0 cosθ t,0 + c(t t t 0 ], (60 whr R is th distanc btwn th lin of sight to th origin and th trajctory of th tst photon (s Fig. and Eq. [A] and solving this scond ordr quation, on obtains c (t t t 0 ( ( = ˆRt + θ t,0 R t,0 ˆR + O(θt,0 4. (6 t Rcalling that ct 0 /R t,0 = + /(m + Γ t,0 + O(Γ 4 t,0, w finally obtain: ( ( f m ( ˆR t (m + Γ ctt + x(m + ˆR t t = R t ˆR m+ t + O ( Γ 4. (6 Fig. 4 shows th dpndnc of f m ( ˆR t on th paramtr x (Γ t,0 θ t,0. For ˆR t = w always hav f m = sinc th tst photon is mittd at th shll. For x > th photon initially lags bhind th shll (cas, and th quation f m = that can b xprssd as ˆR m+ t [ + (m + x] ˆR t + (m + x = 0 has an additional non-trivial solution, ˆR, which corrsponds th th point whr th photon crosss th shll. For m = 0 and m =, it is givn by ˆR = x and ˆR = ( + 8x /, rspctivly Cas : Tst Photon Bhind th Shll, R t < R sh (t t In this cas ( R t < R,max < R sh (t t R t + a fw Γ t < R,max < R sh (t t + a fw Γ t, (63

22 whr th last approximat inquality holds for mission angls (Γ t θ a fw, from which most of th contribution to th obsrvd flux ariss, and ar thrfor th ons of rlvanc. An xprssion for R sh (t t may radily b obtaind through (s Eq. [46] R sh (t t = ct t R t [ Rsh (t t ] m (m + Γ t + O ( Γ 4 t ct t = R t (m + Γ t + O ( Γ 4 t, (64 whil R,max is obtaind by quating th two xprssions for r, from Eq. (47 and Eq. (40 for µ =, ( Rm+ m+ r min = R,max = R sh R sh R,max,max + + O ( Γ 4 (m + Γ t = Rsh R,max + O ( Γ 4 t, t (65 which implis ( ( ( ctt R,max Rsh, (66 R,max = ( ctt + R t R t 4(m + Γ t + O ( Γ 4 t (m + Γ t Rsh +. (67 Whil θ is always small, (Γ t θ a fw, in th cas studid in this subsction θ r can rang from zro to π and it is not obvious a priori whthr it can b takn to b ithr larg, (Γ t θ r, or small, (Γ t θ r a fw. W argu that whn θ r is larg, th photons must b mittd at a larg angl rlativ to th dirction of motion of th mitting shll (θ i = θ r + θ, and ar thrfor significantly supprssd by rlativistic baming. This ffct wins ovr th incras in th raction rat du to th largr angl btwn th tst photon and th intracting photons, that is manifstd by th factor of ( µ ti in th intgrand for th optical dpth. Thrfor, th dominant contribution to th optical dpth occur from small θ r valus, and w can thrfor mak th approximations that ar appropriat for (Γ t θ r a fw. W xprss ths considrations mor quantitativly in Annx B. Thus, w

23 3 obtain: (Γ t θ = Γ t ( µ = = ( R (m + R [ f m ( ˆR t ( R,max R [ (m + R Rm+ m+,max R + 4(m + Γ t ] m+ R ( R,max ] + O ( Γ t + O ( Γ t, (68 [ ( R Rm+ m+ (Γ t θ r = Γ t ( µ sh R + (m + Γ t Rsh ] r = ( (m + Rsh R + O ( Γ t R [f m ( ˆR ] m+ t R = (m + ( R + O ( Γ t, (69 dµ d R = = (m + Γ t [ f m ( ˆR t dµ (m + r d R = [ m+ (m + R { R,max R R m+ + (m + [ ( Rm+ m+,max R + 4(m + Γ t R,max ] (m + Γ t R R m+ +(m + ( R ] ( m R R,max R }, (70 ( m+ R Rsh R R [ ( Rm+ m+ sh sh R + (m + Γ t Rsh ] ( (m + Γ t Rsh R ( R f m ( ˆR ] m+ t + R (m + Γ t ( R. (7 W not that, as xpctd, µ ( R,max =, sinc ( R ( = R,max R +O ( Γ t whil dµ /d R > 0. Th Dopplr factor is givn by (m+/ Γ (m + Γ t R δ + Γ (θ + θ r = ( R (m + R m+ ( R + f m ( ˆR t R m+, (7 whr w hav usd qs. (68 and (69 as wll as Γ = Γ t R m and θ + θ r. Finally,

24 4 r R, and thus δ α+ r dµ d R = Γ α t δ α+ r d µ d R (Γ t α (m + +α Rα+ m (+α ( R α ( R + f m ( ˆR ] +α. (73 t [ (m + R m+ R m Cas : Tst photon at th shll, R t = R sh (t t This is a limiting cas btwn cas and cas 3, whn th tst photon is locatd on th shll: t t = t sh ( R,max, r min = 0, and R,max =, i.. R,max = R sh (t t = R t, R,max = R sh (t t =. (74 This mans that th last mittd photons that still rach th point (R t, t t ar mittd at that sam point in spac and tim, i.. th qual arrival tim surfac nds at that point. Thrfor, (Γ t θ = Γ t ( µ = m+ R (m + R, (75 ( R ( (Γ t θ r = Γ t( µ r = m+ R ( R (m + ( R, (76 dµ d R = m ( dµ r d R = m+ R ( R ( + (m + Γ t R m+ R m+ R, (77 m+ (m + R ( R (m + Γ t ( R. (78 In th limit whr R (i.. R w hav: (Γ t θ ( R, (Γt θ r m + ( R, dµ d R R Γ t θ Γ t, dµ r + (m d R 4Γ t (79, In this limit r R, which implis [s Eq. (30] that I dµ /dµ r / r ( R, i.. th spcific intnsity divrgs at th angl θ r = θ r,max = /Γ t, and vanishs abov this

25 5 angl. This can b undrstood as follows. In this limit r, i.. r R t = R sh (t t and th curvatur of th shock front bcoms unimportant, so that in ordr for a photon to rach th point (R t, t t togthr with th shock front it must propagat along th shock front, which corrsponds locally to an angl of /Γ (or mor gnrally cosθ = β from th normal to th shock front, i.. th radial dirction in our cas Cas 3: Tst Photon Ahad of th Shll, R t > R sh (t t With th causality condition R,max < R sh (t t and Eq. (59, w now hav: As a rsult, Eq. (55 yilds: R m+ 0 R 0 R,max < R sh (t t < R t < ct t R t, (80 (m + Γ t R 0 R,max < R sh < ct t R 0 m+. (8 R t (m + Γ t R,max = R sh R,max + R m+ sh R m+,max ( Rm+ sh m+ R,max (m + Γ t m+ R sh R m+ 0 R sh = (m + Γ t (m + Γ t [ and R,max = R sh (m + Γ t ( R sh = [ (m + Γ t R m+ sh ( ctt R t ] /(m+ = + O ( Γ 4, (8 < ] /(m+ R m+ sh (m + Γ t <, (83 (m + Γ t [ f m ( ˆR t ] /(m+. (84 Taking ths rsults into account, w now driv th rlvant xprssions from qs. (49 5. Th lading trms for µ, µ r, and thir drivativs with rspct to R ar all of th ordr of O ( Γ t. Thus, w obtain:

26 6 (Γ t θ = Γ t ( µ = = (Γ t θ r = Γ t( µ r = = ( R ( Rm+,max m+ R (m + R + O ( Γ t ] ( R [ f m ( ˆR t R ( Rm+ R m+ (m + R + O ( Γ t,max m+ R ( R + O ( Γ t, (85 (m + R [f m ( ˆR ] m+ t R (m + ( R + O ( Γ t, (86 and for th drivativs dµ d R = (m + m+ R ( R [ + f m ( ˆR t (m + Γ t R ] m+ R + O ( Γ 4 t, (87 dµ r d R = (m + m+ R ( R (m + Γ t [ f m ( ˆR ] m+ t R ( R + O ( Γ 4 t. (88 Th Dopplr factor is givn by δ (m+/ Γ (m + Γ t R + Γ (θ + θ r = (m + R m+ ( R + R m ( R [ f m ( ˆR t ], (89 R m+ whr w hav usd qs. (85 and (86 as wll as Γ = Γ t and θ + θ r. Finally, r R, and thus [ δ α+ dµ δ r d R = Γ α α+ t d µ (Γ t α f m ( r d R ( R + ˆR ] (+α m+ t R +mα/ R (m + R m+ ( R (Γ t α +α (m + Rα+m(+α/ ( R α ( R + f m ( ˆR ] +α. (90 t [ (m + R m+ R m+

27 7 W not that th xprssions abov for cas 3 ar idntical to thos for cas Eq. (73. Whil dn i /d i dω r divrgs as R R (θ r,max θ r θ r,max / at θ r,max, dn i /d i = dωr (dn i /d i dω r π θ r dθ r (dn i /d i dω r rmains finit (i.. both th nrgy dnsity and th nrgy flux of th radiation fild rmain finit. This has bn noticd in th contxt of th divrging surfac brightnss of th aftrglow imag at its outr dg, whn th mission coms from an infinitly thin shll (Sari 998; Granot & Lob 00. In that contxt, it has also bn shown (Waxman 997; Granot, Piran & Sari 999a,b; Granot & Lob 00 that whn th mission coms from a shll of finit width, th surfac brightnss (i.. th spcific intnsity I dos not divrg Putting it all togthr Analytical xprssions for our modl hav now bn fully drivd, and ar rportd for convninc hr. Th scald spctral flux dnsity, Eq. (3 is rwrittn as: ( b mα F (T T (m+ y max ( +α ( m + = dy y b mα/ xp [ τ F,0 T 0 y min m + y m γγ y, t, R, T ], R 0 T 0 (9 whr t = ( + z, y min = min[, R 0 /R L (T] and y max = min[, (R 0 + R/R L (T], whil th flux normalization is givn by F,0 = Γ 0 L ( + z (+z/γ 0 (R 0 = α L 0 α ( + z α, (9 4πd L 4πd L F 0 α F,0 = (F,0 = = L ( α 0 + z (93 πd L ( α + z = L 0,5d L,8 rg cm s, whr d L = 8 d L,8 cm, and may b usd in ordr to infr th valu of L 0 from th obsrvd flux lvl. Th optical dpth in th intgrand abov is: τ γγ ( t, θ t,0, R t,0 = τ α t ˆR b mα/ 0 F(x, (94 whr ˆR 0 = y (T/T 0 /(m+ and x = (y (m+ /(m +. Th function F is th following doubl intgral: F(x = ˆR d ˆR t R, ˆR 0 / ˆR t d R I( ˆR t, R + R,3 d ˆR t d R I( ˆR t, R. (95 ˆR ˆR 0 / ˆR t