arXiv:astro-ph/0605445v2 12 Jul 2006 o.Nt .Ato.Soc. Astron. R. Not. Mon. iZogFan Yi-Zhong magnetar? 051221A: millisecond GRB a short from in injection segment Energy flat afterglow X-ray The cetd... eevd...;i rgnlfr ...... form original in ...... ; Received ...... Accepted α
rmtyosre yboth by observed promptly idctffpwrlwseta tig nte2-00KeV 20-2000 3 the fluence a in shows fitting, band, Konus- spectral The power-law 2005). al. cutoff et Wind (Cummings band KeV 15-150 the in ergy ntuet h wf bevtosrva hsi short a is this with reveal burst, observations hard Swift The instrument. Telescope Alert Burst the the by onboard localized (BAT) was 051221A GRB INTRODUCTION 1 stoi rmteiso nryis energy emission prompt isotropic n h CMcnodnemdlo Ω of model concordance ΛCDM the ing index photon redshift a ( us sdsigihdb nXryflteigat one flattening X-ray only Th available. an band by is distinguished limits radio is upper the burst several by in followed while point data detected, well been have and ⋆ 2006), al. et super (Soderberg event bright current the the of in detection component of nova short/hard lack the the ac by for widely supported model the burst, merger the In star clear. However, neutron not 2006). double is cepted al. injection energy et that injection Burrows of energy 2006; nature significant al. et a (Soderberg suggests strongly which day, c 4 3 2 1 ∼ akCsooyCnr,NesBh nttt,Uiest of University Institute, Bohr Niels S Centre, of Cosmology Academy Dark Chinese Observatories, Astronomical National h aa nt fPyis erwUiest,Jrslm91 Jerusalem University, Hebrew Physics, of Inst. Racah The ayDvsFlo,Emi:[email protected] E-mail: Fellow, Davis Lady = upeMuti bevtr,CieeAaeyo cec,N Science, of Academy Chinese Observatory, Mountain Purple 00RAS 0000 10 h ohteXry( X-ray the Both − E 4 1 0 = peak − . 39 4 . 71. ± z × 402 = 0 0 = 10 . α 6 n ene1 fleunce a and 06, 5 = . 49(oebr ta.20) hsburst’s this 2006), al. et (Soderberg 5459 )atrlwlgtcre fGRB051221A of curves light afterglow s) − +93 T − 72 90 Swift 1 1 . , e Glntkie l 05.With 2005). al. et (Golenetskii KeV ∼ 1 = 08 2 . 2 , +0 − 3 000 10 ± aelt Prose l 05 and 2005) al. et (Parsons satellite 1 ⋆ . . Swift . 4 2 n ogXu Dong and , 1 7 0 0–0 00)Pitd7Fbur 00(NL (MN 2020 February 7 Printed (0000) 000–000 , . × ± − 4 n nosre eken- peak observed an and 14, 10 0 2 BTadteKonus-Wind the and /BAT . h a emn lasting segment flat The ABSTRACT R fego.I hswr,w hwta ilscn usrwt dipo with pulsar exte millisecond the a in that show injection we energy work, field strong this of In case afterglow. GRB clear first the resents n uflw–aito ehnss nonthermal mechanisms: outflows–radiation and words: this Key of engine central the that magnetar. suggests millisecond thus segment flat X-ray . ,ahr htnindex photon hard a s, 2 − 16 × 6 E ± 10 r cm erg γ 0 M ∼ 6 . ∼ 04 )adteoptical the and s) 0 = 10 2 × . − 4 14 10 . 2 × 7 Ω 27, low-energy a , t − as ol elacutfrta nryijcin h odqualit good The injection. energy that for account well could Gauss 10 ∼ 6 am as bursts Rays: Gamma 51 rscm ergs Λ 4 0 . r,us- erg, 03 0 = − . 0 73, − oehgn uin aisVj3,20 oehgn Denmar Copenhagen, 2100 30, Vej Maries Juliane Copenhagen, . ine,Biig101,China 100012, Beijing ciences, is 0,Israel 904, 2 2 - - ∼ nig200,China 210008, anjing 10 tar, where at h ioerdainlmnst famgea a be L can magnetar a of by luminosity estimated radiation impor dipole not is the Similarly, radiation tant, wave 2006). gravitational al. the et detail that M´esz´aros Zhang provided & some 2004; Zhang in 2001; Ramirez-Ruiz 2001; Dai 2004; discussed & Dai Wang been 1998; has Lu caused & injection pulsar (Dai energy millisecond the a scenario, by GRB long/soft the In INVESTIGATION ANALYTICAL 2 short this magnetar. of millisecond a engine be central flat may the X-ray burst that quality suggests good thus The segment data. multi-wavelength the for neeg neto rmamlieodpla ihadipole a with pulsar millisecond field a magnetic from injection energy an the in GRBs work not long/soft does case. of current 2001), scenario Heger collapsar & fall- en- Woosley the significant (MacFadyen, the to in rise So injection give correction. may ergy which beaming model, accretion moderate back with even erg, h aeileetdi h egris merger the in ejected material the eti o ieyt eal opm nryu to up energy pump ob- to able post-merger be central to the likely onto not material is that ject of part of Rswge l 99 uet&Jna20) ie nen- an Given 2001). Janka efficiency & conversion Ruffert ergy 1999; al. et (Rosswog 4 dip eod nteXryatrlwo R012Arep- GRB051221A of afterglow X-ray the in seconds R ( nti etr el hwta h fego undergoing afterglow the that show we’ll Letter, this In t s b B ) − sterdu ftemgea,Ωi h nta angular initial the is Ω magnetar, the of radius the is ⊥ ≃ Rs niiul(R 051221A) (GRB individual GRBs: stedpl antcfil tegho h magne- the of strength field magnetic dipole the is 2 . 6 × ∼ A 10 T 10 E 48 tl l v2.2) file style X 14 r s erg as ie,amgea)wl accounts well magnetar) a (i.e., Gauss − ∼ 1 B 0 . ⊥ 2 0,tefl-akaccretion fall-back the 001, , 14 R hr us a ea be may burst short nlsoko short a of shock rnal s, 6 6 ∼ Ω 4 4 (10 1+ (1 − 4 t emagnetic le − − b /T S:jets ISM: 10 k o ) − ∼ − 2 2 ) 10 , M (1) 52 ⊙ y - 2
frequency of radiation, the subscript “b” represents the time stitute p and f into equation (7), we find Ek ∼ 7.6 × 4 −2 −2 −6 51 2 measured in the burst frame, To = 1.6×10 B⊥,14Ω4 I45Rs,6 10 I45Ω4 erg is well consistent with the observational result 51 s is the initial spin-down timescale of the magnetar, I ∼ Ek ∼ 2Eγ ∼ 4.8 × 10 erg when taking typical I45 ∼ 1.5, 45 2 10 g cm is the typical moment of inertia of the magnetar Ω4 ∼ 0.65 (i.e. 1 millisecond period), and the GRB effi- (Pacini 1967; Gunn & Ostriker 1969). Here and throughout ciency η = Ek/(Ek + Eγ ) ∼ 30%. Furthermore, the mea- x this text, the convention Qx = Q/10 has been adopted in surements of tc, tf , f, and p are well consistent with the (p+2)/4 cgs units. constraint relation f ≃ (tf /tc) . So we conclude that The energy emitted at tb will be injected into the pre- the central engine may be a magnetar. Its physical parame- −1 14 vious GRB ejecta at time Tb satisfying ters are (Ω, Rs, B⊥,I) ∼ (6500 s , 13km, 10 Gauss, 1.5 × 45 2 Tb tb 10 g cm ) according to the above constraint relations.
[1 − β(τb)]cdτb = c β(τb)dτb ≈ ctb, (2) An additional constraint is on the ellipticity ǫ of the Ztb Z0 magnetar. As shown in Shapiro & Teukolsky (1983), the where β, in units of the speed of light c, is the velocity of the spin-down timescale due to the gravitational wave radiation ∼ × −3 −2 −1 −4 ∼ × 4 ejecta moving toward us. The corresponding observer time is τgw 3 10 ǫ I45 Ω4 s. Now τgw >tf 1.5 10 s, −4 −1/2 −2 is we have ǫ< 5 × 10 I45 Ω4 .
Tb t ≈ (1 + z) [1 − β(τ )]dτ . (3) Z b b 0 3 NUMERICAL FIT TO THE AFTERGLOWS t Equations (2) and (3) yield t ≈ (1 + z)t +(1+ z) b [1 − OF GRB 051221A b 0 β(τb)]dτb ≈ (1 + z)tb. R We interpret the apparent flattening in the X-ray light curve At time Tb, the energy injected into the ejecta satisfies being caused by an energy injection from the central magne- dE/dTb = [1 − β(Tb)]Ldip(tb). With the relation dt = (1+ tar. Yet the flattening episode is unapparent in the optical z)[1 − β(Tb)]dTb, we have and radio bands because of limited observations. The code dE 1 t here to fit the multi-band lightcurves has been used in Fan ≈ Ldip( ) dt 1+ z 1+ z & Piran (2006) and Zhang et al. (2006), and has been con- 48 2.6 × 10 −1 2 6 4 t −2 firmed by J. Dyks independently (Dyks, Zhang & Fan 2006). = erg s B⊥,14Rs,6Ω4[1 + ] (4). (1 + z) (1 + z)To The dynamical evolution of the outflow is calculated using the formulae in Huang et al. (2000), which are able to de- So the energy injection rate dE/dt ∼ const for t ≪ (1+z)T o scribe the dynamical evolution of the outflow for both the and dE/dt ∝ t−2 for t ≫ (1 + z)T . o relativistic and the non-relativistic phases. One modification A general energy injection form can be written as is that now we have taken into account the energy injection dE/dt = A(1 + z)−1(t/t )−q for t
c 0000 RAS, MNRAS 000, 000–000 3
2 ≤ 10 γm(t×) 20, where γm(t×) is the minim random Lorentz fac- tor of the electrons accelerated in the reverse shock front. 1 ≡ − − − 1 Such a small γm(t×) 3ǫe(p 2)(Γrsh 1)mp/[4(p 1)me] 10 requires that the fraction of shock energy given to the elec- Jy) trons ǫe is in order of me/mp because it is very likely
0
10 that now Γw ≫ Γ(t×), where me and mp are the rest mass of the electron and the proton, repsectively. Taking
-1 r'-band( flux 10 ǫe ∼ me/mp and p ∼ 2.3, we thus have Γrsh ≤ 115 and Γw ≃ 2ΓrshΓ(t×) ≤ 4600. After the reverse shock phase,
-9 -2
10 10 the magnetic energy may be translated to the forward shock mainly by the magnetic pressure working on the initial GRB )
-10 -2 10 ejecta and the shocked medium. However, the details are far cm
-11 -1
10 from clear. At t ∼ 0.91 day, the 8.46 GHz emission flux
-12 10 14 is 0.155 mJy. Therefore the optical (4.6 × 10 Hz) 2 -13
10 flux at t× ∼ 0.2 day is ≤ 0.155mJy(0.91/0.2) (4.6 × 1014Hz/8.46GHz)−0.65 = 2.7 µJy. It is about one
-14 10 order lower than the forward shock optical emission.
Flux (0.3-10keV,Flux ergs 15 -15 10 As for the X-ray, now νc(t×) ≈ 2 × 10 Hz,
1 2 3 4 5 6 7
10 10 10 10 10 10 10 the X-ray (at 2 keV) flux at t× ∼ 0.2 day is t (second) ≤ 0.155mJy(0.91/0.2)2 (2 × 1015Hz/8.46GHz)−0.65(4.84 × Figure 1. Modeling the XRT and R-band afterglow light curves 1017/2 × 1015)−1.15 ∼ 10−6 mJy, is also significantly lower of GRB 051221A with energy injection from an underlying mil- than the forward shock X-ray emission. Our estimates made lisecond magnetar. The optical and X-ray data are taken from Soderberg et al. (2006) and Burrows et al. (2006), respectively. here are independent of the poorly known magnetized re- The r′−band extinction of our Galaxy ∼ 0.19 mag and that of verse shock dynamics. We thus conclude that the reverse the host galaxy ∼ 0.2 mag (Verkhodanov et al. 2000) have been shock X-ray/optical emission is unimportant and can be ig- taken into account. The solid lines are our numerical results. Fit- nored. ted parameters of the GRB jet undergoing energy injection are One caveat is that the dipole radiation of the magnetar presented in Section 3. is almost isotropic. The medium surrounding the magne- tar but out of the GRB ejecta cone would be accelerated by the energetic wind. Because the energy injected into the emission has been calculated in Dai (2004), assuming that GRB ejecta is larger than that contained in the initial ejecta, the outflow is electron/positron pairs dominated and the re- the outflow contributing to the afterglow emission would be verse shock parameters are similar to, or even larger than close to be an isotropic fireball rather than a highly jetted that of the forward shock. The resulting reverse shock emis- ejecta. So the magnetar energy injection model is somewhat sion is so bright that long-time flat bump should be evi- challenged by the late jet break detected in GRB 051221A. dent in multi-band afterglow lightcurves. This prediction is However, this puzzle could be resolved if in other directions not confirmed by current observations. This puzzle, however, a large amount of baryons (∼ 0.01 M⊙) ejected in the dou- could be resolved if the fraction of the reverse shock energy ble neutron star merger have existed there, as found in the given to the electrons is very small, as shown below. previous numerical simulations (Rosswog et al. 1999; Ruffert In our numerical code, the reverse shock dynam- & Janka 2001). The ejected material would be accelerated ics/emission has not been taken into account. It is inves- by the magnetar wind to a bulk Lorentz factor ∼ a few, 52 tigated analytically instead. We assume that the reverse provided that Einj ∼ 10 erg. This wide but only mild- 6 7 shock emission accounting for the radio data is powered relativistic outflow will give rise to a very late (∼ 10 − 10 by the energy injection. Following Sari & Piran (1999), af- s after the burst) multi-wavelength re-brightening. However, ter the reverse shock crosses the ejecta at t× (noted that for typical parameters taken in this Letter (the isotropic en- 52 at t ∼ t× ∼ 0.2 day the central magnetar collapses), the ergy is 1.5 × 10 erg and the initial Lorentz factor is 5.0, −2 observed reverse shock emission flux Fν⊕ ∝ (t/t×) for i.e., assuming the material ejected from the merger is about −3 νm < ν⊕ < νc, where νm is the typical synchrotron emis- 1.5 × 10 M⊙), the flux is not bright enough to be de- 6 sion frequency of the shocked electrons and νc is the cool- tectable. The emission peaks at t ∼ 3 × 10 s. The 0.3-10 −1/2 ∼ × −15 −1 −2 ing frequency. On the other hand, Fν⊕ ∝ (t/t×) for keV flux is 1 10 erg s cm which is marginal for ν⊕ < νm < νc. The current radio observation thus requires the detection of Chandra, and the 8.46 GHz flux is ∼ 0.01 a νm(t×) ≤ 8.46 GHz. At t× ∼ 0.2 day, the ejecta is at a mJy. Both are consistent with the extrapolation of current radius R ∼ 7 × 1017 cm and moves with a bulk Lorentz fac- observations (Soderberg et al. 2006; Burrows et al. 2006). tor Γ(t×) ∼ 20 (obtained in our numerical calculation), thus If an event like GRB 051221A takes place much closer, for the comoving toroidal magnetic field of the magnetar wind is B ∼ [2L /(R2Γ2 c)]1/2 ∼ 20/Γ Gauss, where Γ is the w dip w w w 1 The numerical coefficient 3/4 is adopted because in the ideal bulk Lorentz factor of the magnetar wind. In the reverse MHD shock jump condition, the random Lorentz factor of the shock phase, the toroidal magnetic field will be amplified post-shock protons is proportional to 3(Γrsh − 1)/4 rather than by a factor ∼ Γ ≈ Γ /2Γ(t×) for Γ ≫ Γ(t×) (Kennel & rsh w w Γrsh − 1 (see eq. (17) of Fan, Wei & Zhang 2004b; see also Fan, Coronitti 1984). On the other hand, the constraint νm(t×) ≈ Wei & Wang 2004a and Zhang & Kobayashi 2005 for numerical 6 2 × × ≤ 2.8 10 Hz γm(t×)Γ(t )ΓrshBw/(1 + z) 8.46GHz yields verification).
c 0000 RAS, MNRAS 000, 000–000 4 example, at z ∼ 0.1, such very late multi-wavelength re- ACKNOWLEDGMENTS brightening may be detectable for Swift XRT or Chandra We thank Tsvi Piran for an important remark and Jens and other radio telescopes. This predication could be tested Hjorth for valuable comments that improved this paper. We in the coming months or years. also thank the referee and X. W. Liu for constructive sug- gestions. YZF is supported by the National Natural Science Foundation (grants 10225314 and 10233010) of China, the National 973 Project on Fundamental Researches of China (NKBRSF G19990754), and US-Israel BSF. DX is at the Dark Cosmology Centre funded by The Danish National Re- search Foundation.
4 DISCUSSION AND SUMMARY REFERENCES Short hard GRBs may be powered by the merger of double neutron stars (e.g. Eichler et al. 1989). For GRB 051221A, Burrows D. N., et al., 2006, ApJ, submitted (astro-ph/0604320) ruling out of bright supernova component in the late opti- Dai Z. G., 2004, ApJ, 606, 1000 cal afterglow lightcurve suggests that the progenitor prob- Dai Z. G., Lu T., 1998, A&A, 333, L87 ably is not a massive star and instead is consistent with Dai Z. G., Wang X. Y., Wu X. F., Zhang B. 2006, Science, 311, the double neutron star merger model. After the energetic 1127 merger, a black hole (Eichler et al. 1989), or a differentially Duez M. D., Liu Y. T., Shapiro S. L., Shibata M., Stephens B. C., 2006, Phys. Rev. D., in press (astro-ph/0605331) rotating neutron star (Klu´zniak & Ruderman 1998; Dai et Dyks J., Zhang B., Fan Y. Z., 2006, ApJ, submitted (astro- al. 2006), or a magnetar may be formed. 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c 0000 RAS, MNRAS 000, 000–000