DraftversionFebruary2,2008 PreprinttypesetusingLATEXstyleemulateapjv.04/03/99 RELATIVISTIC WIND BUBBLES AND AFTERGLOW SIGNATURES Z. G. Dai DepartmentofAstronomy,NanjingUniversity,Nanjing210093, China;dzg@nju.edu.cn Draft version February 2, 2008 ABSTRACT Highly magnetized, rapidly rotating compact objects are widely argued as central energy sources of γ-raybursts(GRBs). AftertheGRB,suchamagnetar-likeobjectmaydirectlyloseitsrotationalenergy 4 throughsomemagnetically-drivenprocesses,whichproduceanultrarelativisticwinddominatedpossibly 0 by the energyflux ofelectron-positronpairs. The interactionofsucha windwith anoutward-expanding 0 fireballleadstoarelativisticwindbubble,beingregardedasarelativisticversionofthewell-studiedCrab 2 Nebula. We here explore the dynamics of this wind bubble and its emission signatures. We find that n when the injection energy significantly exceeds the initial energy of the fireball, the bulk Lorentz factor a of the wind bubble decays more slowly than before, and more importantly, the reverse-shock emission J could dominate the afterglow emission, which yields a bump in afterglow light curves. In addition, high 8 polarization of the bump emission would be expected if a toroidal magnetic field in the shocked wind 1 dominates over the random component. Subject headings: gamma-rays: bursts — relativity — shock waves — stars: winds, outflows 3 v 8 1. INTRODUCTION dominated, because the initial explosion should have left 6 a clean passage with very few baryoncontamination for a The recent observation of high linear polarization dur- 4 subsequent outflow. The interaction of this outflow with ing the prompt γ-ray emission of GRB 021206 (Coburn 8 an outward-expanding fireball implies a continuous injec- 0 & Boggs 2003) suggests that GRBs be driven by highly tionofthestellarrotationalenergyintothefireball. Dai& 3 magnetized, rapidly rotating compact objects. Two pop- Lu(1998,2000),Zhang&Mésza´ros(2001)andChang,Lee 0 ular scenarios for their birth are the merger of a compact &Yi(2002)discussedtheevolutionofarelativisticfireball / binary or the collapse of a massive star (for a recent re- h by assuming a pure electromagnetic-wave energy outflow, view see Mésza´ros 2002). In both scenarios, a rapidly ro- p while Rees & Mésza´ros (1998), Sari & Mésza´ros (2000), tating black hole surrounded by an accretion disk seems - Zhang & Mésza´ros (2002), and Granot, Nakar & Piran o to be a common remnant (Narayan, Paczyn´ski & Piran (2003)tookintoaccountavariableandbaryon-dominated r 1992; Woosley 1993; Mésza´ros & Rees 1997a; Paczyn´ski t injection. s 1998). However, a millisecond magnetar has also been a However, based on the successful models of the well- argued as an alternative interesting product (Usov 1992; : observed Crab Nebula (Rees & Gunn 1974; Kennel & v Duncan & Thompson 1992; Klu´zniak & Ruderman 1998; Coroniti1984;Begelman&Li1992;Chevalier2000),are- i Dai & Lu 1998; Spruit 1999; Ruderman, Tao & Klu´zniak X alistic,continuousoutflowduringtheafterglowisexpected 2000;Wheeleretal. 2000). Toexplainthecomplextempo- r to be ultra-relativistic and dominated by the energy flux ralfeature,theburstitself,insomeoftheseenergymodels, a of electron-positron pairs. As in the Crab Nebula, even is understood to arise from a series of explosive reconnec- if an outflow from the pulsar is Poynting-flux-dominated tion events in a rising, amplified magnetic field because at small radii, the fluctuating component of the magnetic of the Parker instability. This in fact dissipates the dif- field in this outflow can be dissipated by magnetic recon- ferentially rotational energy and magnetic energy of the nection and used to accelerate the outflow, which is even- newborn magnetar or accretion disk. tually dominated by the energy flux of e+e− pairs within After the GRB, the remaining object is reasonably as- a larger radius 1017 cm (Coroniti 1990; Michel 1994; sumed to be a millisecond magnetar or a rapidly rotating ∼ Kirk & Skjæraasan 2003). In the case of an afterglow, black hole surrounded by an accretion disk. For the lat- therefore, it is natural to expect that the central object ter object, the magnetic field in the disk could have been still produces an ultra-relativistic e+e−-pair wind, whose amplified initially by differential rotation to a magnetar- interaction with the fireball leads to a relativistic wind like strength of 1015 G, and particularly, within the ∼ bubble. This can be regarded as a relativistic version of framework of the collapsar/hypernova model, such a field the Crab Nebula. could be kept, due to longevity (with days or longer) of In this paper, we explore the dynamics of such a wind the disk maintained by fallback of the ejecta. During the bubble and its emission signatures. In 2 we present ex- afterglow,the object at the center will directly lose its ro- § pressions of the luminosity of a relativistic wind from a tational energy by the magnetic dipole radiation or the highly magnetized, rapidly rotating object. In 3 and Blandford-Znajek mechanism. §§ 4 we discuss evolution of the wind bubble and temporal An energy outflow driven magnetically includes three features of the radiation respectively. In the final section components: low-frequencyelectromagneticwaves,a rela- we summarize our findings and give a brief discussion on tivisticwind,andatoroidalmagneticfieldassociatedwith implications of the bubble. the wind. The wind energy flux is unlikely to be baryon- 1 2 2. THELUMINOSITYOFARELATIVISTIC WIND where TBH,0 = 1.0BB−H2,15(MBH/3M⊙)−1(a0/0.2)−2 days is the “initial” spin-down time of the black hole and a is We assume that a burst itself arises from a series of 0 the “initial” rotation parameter. One can easily see that explosive reconnection events. After the GRB, we are left equations (2) and (4) are similar, which implies that our withahighlymagnetized,rapidlyrotatingcompactobject. discussion of a relativistic wind bubble in the remaining Let’s first assume that it is a millisecond magnetar with text for magnetars should be valid for black holes. period P, surface magnetic field strength B , moment of s inertia I, radius R , and angle between the rotation axis M 3. THEBUBBLEDYNAMICSINTHETHIN-SHELL and magnetic dipole moment θ. Since such a pulsar loses APPROXIMATION its rotational energy through the magnetic dipole torque, Similarly to the Crab Nebula, a rotating magnetar at the luminosity of a resulting relativistic wind is given by the center of an afterglow generates a highly relativistic L 4 1047B2 R6 P−4ergs−1, (1) wind dominated by the energy flux of e+e− pairs, with w ≃ × ⊥,14 M,6 ms bulk Lorentz factor of γ 104 107. Atoyan (1999) w where B⊥,14 = Bssinθ/1014G, RM,6 = RM/106cm, and taerrgpureedttthhaetmtheaesCurreadbrpaudlisoars∼pinecittiraul−mly ohfadthγewC∼rab10N4etbouilna-. P =P/1ms. Because of spin-down, this luminosity will ms We adopt γ =104 as a fiducial value in our calculations. evolve with time as w Because γ is much larger than the Lorentz factor of the w const., if tTMM,,00, (2) a shock front and decelerates to match the expansion ve- ∝ locity of the swept-up medium. Therefore, a relativistic where t is the observer time in units of day, T = wind bubble, as a result of interaction of the wind with M,0 0.58B−2 I R−6 P2 days is the “initial” spin-down the medium, should include two shocks: a reverse shock ⊥,14 45 M,6 0,ms that propagates into the cold wind and a forward shock timescale of the magnetar at the onset of the afterglow, thatpropagatesintotheambientmedium. Thus,thereare P =P 1ms is the rotation period at this time, and I0 = I0,/m1s0×45gcm2. Usov (1992) also assumed that the four regions separated in the bubble by these shocks: (1) 45 the unshocked medium, (2) the forward-shockedmedium, early spin-down could be due to gravitational wave radi- (3) the reverse-shocked wind gas, and (4) the unshocked ation besides magnetic dipole radiation. However, the lu- cold wind, where regions 2 and 3 are separated by a con- minosity for gravitational wave radiation depends on the tactdiscontinuity. Forsimplicity,wehereassumethattwo stellar ellipticity which is poorly known, and so we ne- initially-formingforwardshocksduring interactions ofthe glected the effect of this mechanism on spin-down. fireball both with the medium and with the wind have Wenowdiscussanothercaseinwhichthecentralobject eventually merged to one forward shock, and also neglect isarapidlyrotatingblackholesurroundedbyanaccretion effectsofthebaryonloadingwhosemassismuchlessthan disk. If the amplified magnetic field in the disk does not theswept-upmass,whentheobservertimefarexceedsthe evolvesignificantlywithtimeduringfallbackoftheejecta, initial deceleration timescale. the rotational energy of this black hole will be gradually We denote n andP′ asthe baryonnumberdensity and extracted by the Blandford-Znajek mechanism, whose lu- i i pressure of region “i” in its own rest frame respectively, minosity is approximated by and γ is the Lorentz factor of region “i” measured in the i Lw =1.5×1051BB2H,15(MBH/3M⊙)2a2f(a)ergs−1 lfoaccatolrmoefdiruemgi’osnre3stmfreaamsuer.eWd eindetrhieverethstefrrealamteiveofLorergeniotnz ≃3×1047BB2H,15(MBH/3M⊙)2(a/0.2)4ergs−1,(3) 4 as γ34 ≃ (1/2)(γw/γ3 + γ3/γw) ≃ γw/(2γ3) ≫ 1 for γ γ 1, implying a relativistic reverse shock. Be- w 3 where f(a) = 1 [(1 + √1 a2)/2]1/2 a2/8 for the caus≫ethee≫lectronnumberdensityintherestframecomov- rotation paramete−r a 1, M− is the b≃lack-hole mass, ing with the unshocked wind is n = L /(4πr2γ2m c3) and B is the disk≪field stBrHength in units of 1015 G (where r is the radius of region 34and mw is the ewlecteron BH,15 e (Lee,Wijers&Brown2000). We notethatatypicalvalue mass), according to the jump conditions for a relativistic ( 1047ergs−1) of the wind luminosity in equations (1) shock (Blandford & McKee 1976), the pressure of region ∼ and (3) has been invoked by Rees & Mésza´ros (2000) to 3 is calculated by explain the observed iron lines from some GRBs within 4 L the framework of the collapsar/hypernova model. Con- P′ = γ2 n m c2 w . (5) sidering the rotational energy of the black hole E = 3 3 34 4 e ≃ 12πr2γ2c BH 3 f(a)M c2 (a2/8)M c2 for a 1 (Lee et al. 2000), BH ≃ BH ≪ Neglecting the presence of the reverse shock and the and assuming E˙ = L , we find that the Blandford- BH w radiative energy loss of region 2, and assuming an ambi- Znajekmechanismyieldsspin-down,similarlytothemag- ent interstellar medium with constant density of n , the 1 netarcase,ifthe accretedangularmomentumisneglected properties of the shocked medium in region 2 should sat- because its rate seems to be below the torque driven by isfytheBlandford-McKeeadiabaticself-similaritysolution theBlandford-Znajekmechanismattimeofdaysafterthe with the similarity variable at any radius r, burst for typical parameters in the collapsar/hypernova model. Thus,weobtainacrudeevolutionlawoftheabove r χ=(1+16γ2) 1 , (6) luminosity, 2 (cid:18) − ct (cid:19) l Lw ∝(1+t/TBH,0)−2(cid:26) ∼ct−on2,st., iiff tt<>TTBBHH,,00,, (4) mwhederiuemγ2ju≡stγb2e(hRi)ndistthheefoLrowreanrdtzsfhaocctkorwohfotsheerashdoiucskeids ∝ 3 denoted as R, and t = (R/c)[1+1/(16γ2)] is the time and the Lorentz factor of region 3, l 2 measured in the local medium’s rest frame. The radius r can thus be expressed as function of χ by (4L )12/17(17E )5/17 1/8 w 0 γ = 1 χ 3 (cid:20) 1024πn1mpc5t39/17 (cid:21) r =R 1+ 1 , (7) (cid:18) 16γ22(cid:19)(cid:18) − 1+16γ22(cid:19) =5.3L3w/,3447E552/136n1−1/8t−39/136 (17) which implies r R as long as χ 16γ2 for an ultrarela- ≃ ≪ 2 where L = L /1047ergs−1, E = E /1052ergs, and tivistic forwardshock. This justifies thethin-shell approx- w,47 w 52 0 n and t are in units of 1cm−3 and 1 day respectively. imation, in which the width of region 2 is insignificant as 1 Letting χ=1, we define a critical time compared to the shock radius R. According to Blandford & McKee (1976), the pressure t =4.9E L−1 days. (18) and Lorentz factor of the shocked medium at radius r are cr 52 w,47 given by At this time, the injection energy to the fireball signifi- 4 cantlyexceeds its initialenergy. For t 1. It should be emphasized that the dy- γ (r)=γ χ−1/2, (9) namics denoted by equation (17) is simply calculated by 2 2 equating the pressures of the two-sided shocked fluids at where m is the proton mass. Along the contact disconti- the contact discontinuity. In this derivation, we have ne- p nuity, γ =γ (r) and P′ =P′(r), which yield glectedanyworkdoneonregion2byregion3becausethe 3 2 3 2 pressure of region 3 is much less than P′(R) at t T (viz., E > 0.46I P−2 ), 2 cr M,0 52 45 0,ms thewindbubbleshouldevolvebasedonequations(15)and Usingequations(12)and(14),the Lorentzfactorofthe shocked medium (i.e., region 2) just behind the forward (17); for tcr < TM,0 (viz., E52 < 0.46I45P0−,m2s), however, shock is found to evolve with time as the Lorentz factors of the wind bubble decay initially as γ t−3/8 and γ t−39/136 at t < t (stage I), subse- 2 3 cr 17E 1/8 que∝ntly as γ = γ ∝ t−1/4 at t (t ,T ) (stage II), γ = 0 . (15) 2 3 ∝ ∈ cr M,0 2 (cid:18)1024πn1mpc5t3(cid:19) and finally again as γ2 ∝ t−3/8 at t > TM,0 (stage III). It should be pointed out that this discussion assumes a From equations (10)-(13), we have the similarity variable negligible radiative loss of region 3. However, no matter at the location of the contact discontinuity, whether region3 is at stage I or II, and once it enters the fastcoolingregimeatsometime,itspressurewillbeginto −12/17 −12/17 4Lwt Lw,47t become much smaller than that of region 2 and then the χ= =3.1 , (16) (cid:18)17E (cid:19) (cid:18) E (cid:19) Lorentz factor of region 2 will decay as equation (15). 0 52 4 4. SYNCHROTRONRADIATIONANDLIGHTCURVES νI =8.5 1013ξ−3/2E−1/2n−1t−1/2Hz, (27) c,2 × B,−1 52 1 In this section we discuss light curves of the emission FI =35ξ1/2 E n1/2D−2 mJy, (28) from a relativistic wind bubble, assuming t T is assumed. If t TM,0 (stageIII), becauseofadiabaticexpansion source in units of 1028 cm. We here considered of region3, the break frequencies and the peak flux decay only the characteristic frequency and the cooling fre- as νIII t−73/48, νIII t−73/48 and FIII t−47/48 quency for the optical to X-ray emission discussed in m,3 ∝ cut,3 ∝ ν,max,3 ∝ (Sari & Piran 1999). this paper. In our derivation, we used equations (8), According to the derived scaling laws of the break fre- (10) and (17) to obtain the energy density of region quencies and peak flux with time at three stages, we can 3, e′3 = 4n1mpc2γ22χ−17/12 = 4n1mpc2γ32χ−5/12 = obtainthelightcurveindicesfordifferentfrequencybands 0.11L8/17E−15/68n3/4t−19/68ergcm−3, and the mag- in the case of t < T (see Table 1). As an example, w,47 52 1 cr M,0 netic field strength, B′ = (8πǫ e′)1/2 = Figure 1 presents R-band light curves for typical values 3 B 3 0.52ǫ1/2 L4/17E−15/136n3/8t−19/136 G. We define a cool- of the model parameters: I45 = 3, P0,ms = 1, γw,4 = 1, B,−1 w,47 52 1 L = 1, p = q = 2.5, ǫ = 0.9, ǫ = ξ = ξ = 0.1, w,47 e B e B ing time tI0,3 through νmI ,3 =νcI,3 as E52 =1,n1 =1cm−3,andDL,28 =1. Wecanseethatthe emission flux from region 2 decays rapidly at time < t , tI0,3=14(gpǫeǫB,−1γw,4)−68/49 subsequently fades more slowly at time (tcr,TM,0), ancrd ∈ L−32/49E15/49n−51/49days. (25) finally declines based on the initial evolution law (Dai & × w,47 52 1 Lu 1998). More importantly, the emission from region 3 From equations (22) and (23), we find that region 3 is in dominates the afterglow emission, which leads to a bump the slow-coolingregime for t tI , in contrast to the standard afterglow 0,3 5. DISCUSSION ANDCONCLUSIONS model(Sarietal. 1998). Alargervalueofγ wouldimply w fast cooling in region 3 earlier on, in which case region 2 Based on the successful models of the Crab Nebula, we could still stay atstage I. For region2, the break frequen- discuss the dynamics of a relativistic wind bubble and its cies and peak flux are given by emission signatures. Such a wind bubble is naturally expected when an ultrarelativistic e+e−-pair wind from a νI =1.6 1013g2ξ2 ξ1/2 E1/2t−3/2Hz, (26) highlymagnetized,rapidlyrotatingobjectatthecenterof m,2 × q e,−1 B,−1 52 5 stage I stage II (t T tII T 0,3 M,0 0,3 M,0 0,3 M,0 0,3 M,0 0,3 M,0 3... ν <ν 107 1 17 1 5 17 p −204 −2 −12 −2 3 36 ν <ν <ν 5p+34 1 1 1 p+3 73p+21 p 0 − 68 −2 4 −2 2 96 ν >ν 5(p−2) 0 p−2 0 p+4 p+3 0 − 68 − 4 2 2 2... ν <ν 1 4 1 4 1 1 ν <ν ν 3q−2 q−2 3q−2 q−2 3q−2 3q−2 0 4 2 4 2 4 4 Table 1 The light curve index α as function of p or q (Fν ∝t−α). Definition: νp =min(νm,νc) and ν0 =max(νm,νc) anafterglowinteractswithanoutward-expandingfireball, afterglow in our model could be as high as Π 70% syn ∼ regardlessofwhetherthisobjectisamillisecondmagnetar for p 2 when the reverse shock emission dominates the ∼ or a Kerr black hole. We find that when the injection en- afterglow emission during an obvious bump. Even if the ergy significantly exceeds the initial energy of the fireball, total afterglow flux is dominated by the forward shock the bulk Lorentz factor of the wind bubble declines more emission, and the reversely-shocked wind provides only a slowlythanbefore. Inaddition,thereverse-shockemission small fraction of the total flux, ζ, then the shocked pair could dominate the afterglow emission, which leads to a wind could still dominate the polarized flux with linear bump in afterglow light curves. In this paper, we discuss polarization of Π ζΠ = 7%(ζ/0.1)(Π /0.7). This syn syn ∼ the case of t < T . However, even if t > T , a value is larger than the currently observed data of GRBs cr M,0 cr M,0 bump in Figure 1 still appears for some parameter space 021004and 030329( 2 3%, Rol et al. 2003;Lazzati et ∼ − as we move the late-time light curve of region 3 to early al. 2003; Greiner et al. 2003). If, on the other hand, the times. random-fieldstrengthexceedsthetoroidalcomponent,the Bump features have been detected in some events (e.g., calculateddegreeoflinear polarizationis significantly less GRBs 970508, 000301C, 021004, and 030329). To in- than that estimated above. terpret these features, some other models invoked the We have considered a spherical wind bubble in our pa- microlensing event (Garnavich, Loeb & Stanek 2000), per. However, there is evidence that GRBs are colli- density-jumpmedium(Dai&Lu2002;Lazzatietal. 2002; mated into narrow jets, whose kinetic energy is clustered Dai & Wu 2003), pure Poynting-flux injection (Dai & atE 1050 1051 ergs(Frail et al. 2001;Panaitescu& jet ∼ − Lu 1998; Zhang & Mésza´ros 2001), baryon-dominated in- Kumar 2002; Berger, Kulkarni & Frail 2003; Bloom, Frail jection (Rees & Mésza´ros 1998; Sari & Mésza´ros 2000; & Kulkarni 2003). On the other hand, relativistic winds Zhang & Mésza´ros 2002; Granot, Nakar & Piran 2003), frommillisecondmagnetarsareroughlyisotropicandtheir and two-component jet (Berger et al. 2003; Huang et al. total energy is 1053 ergs. Beaming correction gives an 2003). The magnetic field in the reversely-shockedregion injectionenergy∼of 1051ergswithintheinitialsolidangle ∼ of the wind bubble seems to consist of two components: ofajet. ThisenergyisofthesameorderasE . Actually, jet a toroidal field and a random field. The latter may be the jet cangetmore energyfromits centralmagnetardue naturally generated by the relativistic two-stream insta- to sideways expansion. This would favor our model. bility (Medvedev&Loeb1999). Ifthe toroidalfielddomi- nates overthe randomcomponent, one wouldexpect high polarization of the bump emission, which could be used The author thanks the referee and B. Zhang for valu- to distinguish between our wind-bubble model and other able comments and suggestions, and Y. F. Huang, X. Y. explanations. It is well known that the degree of linear Wang, D. M. Wei and X. F. Wu for useful discussions. polarization of synchrotron radiation from the reversely- This workwassupportedby the NationalNaturalScience shockedwindisaboutΠ =(p+1)/(p+7/3)foralarge- FoundationofChina (grants10233010and10221001)and syn scale toroidal magnetic field. Thus, the polarization of an the National 973 Project (NKBRSF G19990754). REFERENCES Atoyan,A.M.1999,A&A,346,L49 Frail,D.A.etal.2001,ApJ,562,L55 Begelman,M.C.,&Li,Z.Y.1992,ApJ,397,187 Garnavich, P.,Loeb,A.,&Stanek, K.2000,ApJ,544,L11 Berger,E.etal.2003, Nature,426,154 Granot,J.,Nakar,E.,&Piran,T.2003,Nature,426,138 Berger,E.,Kulkarni,S.R.,&Frail,D.A.2003,ApJ,590,379 Greiner,J.etal.2003,Nature,426,157 Blandford,R.,&McKee,C.1976,Phys.Fluids,19,1130 Huang, Y. F., Wu, X. F., Dai, Z. G., Ma, H. 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Thesolidlinecorrespondstothetotalflux.

Relativistic Wind Bubbles and Afterglow Signatures

Similar Relativistic Wind Bubbles and Afterglow Signatures