Gamma-Ray Burst Afterglows and Evolution of Postburst Fireballs With

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arXiv:astro-ph/9810402v1 25 Oct 1998 edopitrqet to requests offprint Send are GRBs distances. that cosmological demonstrate at clearly the GRB970508 1997), and to al. 1997; counterpart c), et the b, al. (Metzger, of 1997a, redshift et al. the et (Groot, of Sahu, detection wavebands 1997; Yoshida, al. of b; optical et 1997a, Paradijs, discovery and van al. recent 1997) great et The al. (Costa, a GRBs. et X-ray to at of leading sources origin halo, fading the galactic the on in debate cosmolog- or at are distances (GRBs) ical bursts gamma-ray on of detector sources the BATSE the from CGRO results observational The Introduction 1. Dai magnetic G. Z. strongly from pulsars injection postburst millisecond energy of evolution with and fireballs afterglows burst Gamma-ray as bursts rays: a words: of Key birth the from pulsar. resulted afterglow millisecond burst the magnetic of this strongly expla- lightcurve if natural the GRB970228 of a of fi- behavior provide the and may for This flattens, nation again. fireball subsequently declines this time, from nally with afterglow that decreases optical find the first and of the post- radiation, magnitude from a dipole the injection of magnetic energy evolution through with pulsar study fireball here relativistic a burst We such gamma-ray burst occur. a of may for birth available (GRB) fireball the initial During an stars. pulsar, e.g. phase neutron accretion-induced processes, of dwarfs, and stars, transitions several white neutron two magnetized through of merger of formed collapse be accretion-induced may fields Abstract. ??? accepted ???; Received 3 2 1 0.41 81.;13.07.1) 08.16.6; (08.14.1; are: codes thesaurus later) Your hand by inserted be (will no. manuscript A&A CHA HP A,Biig ..China P. P.R. 100080, Beijing, Beijing CAS, 8730, IHEP, Box LCRHEA, P.O. Laboratory), 2100 (World Nanjing University, CCAST Nanjing Astronomy, of Department Fsmn&Mea 95 togysgetdthat suggested strongly 1995) Meegan & (Fishman 1 ilscn usr ihsrn magnetic strong with pulsars Millisecond n .Lu T. and tr:nurn–plas eea gamma- – general pulsars: – neutron stars: .G Dai G. Z. : 2 , 1 , 3 3 ..China P.R. 93, neg hs rniin obcm tag tr,adaf- and stars, strange to become mass to has sufficient transitions neutron accrete phase it some undergo binaries that Third, X-ray (1996) low-mass 1996). Dai in & stars (Vietri Cheng energy by proposed orbital trans- been energy the efficient from and/or 1997) fer produced al. et be Mathews, may al. 1989; et fireball (Eichler, initial annihilation an neutrino-antineutrino through stars, neutron merging tidal two and by some coalescence of the powered by During 1996). action generated (Vietri dynamo heating fields e.g. magnetic processes, peri- strong physical millisecond very into both and have merge ods which might neutron-star stars system (1998), neutron Hulse-Taylor Dai massive the & like Cheng and binaries Dai (1997), (1997), Dai Cheng & and & Cheng observational by several summarized for facts stiff theoretical likely gravi- at are state to densities of due high equations hole the black since However, a radiation. into tational merge eventually will nary 1997). pro- Blackman & electromagnetic Yi neutrino and/or 1992; through 1992) (Usov occur al. cesses may et fireball (Dar, neu- initial processes the an of birth star, the action tron During dynamo 1992). efficient Thompson & or (Duncan 1992) magnetic- to (Usov of due conservation field strong very flux magnetic become the may and star to neutron star, collapse the neutron may rotating limit, rapidly Chandrasekhar mass a its the when to dwarf, up white stars. accreting increases an and neutron model, stars, first of the neutron transitions In two phase of accretion-induced merger (3) magne- (2) of dwarfs, collapse white sev- accretion-induced by tized (1) formed e.g. be models, may pulsars which eral fields, millisecond magnetic newborn strong en- is with electromagnetic possibility such of One for dissipation ergy. available and/or neu- energy through emission extracted of trino is amount fireball relativistic large extremely ob- a an compact which with from related are jects GRBs cosmological of rence .China R. eod ti sal huh htanurnsa bi- neutron-star a that thought usually is it Second, occur- the explain to proposed models the of Most ASTROPHYSICS ASTRONOMY 15.1.2018 AND 2 Z.G. Dai & T. Lu: Gamma-ray burst afterglows and evolution of fireballs with pulsars ter the phase transitions fireballs of GRBs naturally occur moment and rotation axis, Pms is the period in units of 1 6 from the quark surfaces of the hot strange stars (Cheng ms, and R6 is the stellar radius in units of 10 cm. This & Dai 1996; Usov 1998) because of very-low-mass crusts. power varies with time as The resulting strange stars may have short periods due −2 to accretion-induced spin-up prior to the phase transi- L(t) ∝ (1 + t/T ) , (2) tions, and they could have strong magnetic fields (Cheng & Dai 1997). Therefore, it can be seen that two common where t is one measure of time in the burster’s rest frame, products of these three energy-source models are newborn and T is the initial spin-down timescale defined by millisecond pulsars with strong magnetic fields and initial P 8 −2 2 −6 fireballs available for GRBs. Furthermore, Klu´zniak & Ru- T ≡ =5 × 10 s B⊥ P I45R , (3) 2P˙ ,12 ms 6 derman (1997) proposed that strongly magnetized neutron 0 stars differentially rotating at millisecond periods may be where P˙ is the rate of period increase due to magnetic the central engine of gamma-ray bursts. dipole radiation and I45 is the stellar moment of inertia It is natural to expect that if a GRB results from the in units of 1045 g cm2. (Note that the stellar spin-down birth of a strongly magnetic millisecond pulsar, then after timescale in the burster’s rest frame is equal to that in the main GRB the pulsar continuously supplies energy to the observer’s frame.) For t ω, where ω is the plasma frequency of the shocked ISM. Since We assume, as a pulsar is born by the models summarized p the electron number density of the shocked ISM in the in the introduction, an amount of energy E comparable to burster’s rest frame is n = 4γ2n (Blandford & McKee that observed in gamma rays, E ∼ 1051 ergs, is released e 1976), where γ is the Lorentz factor of the shocked ISM over a time interval less than 100 seconds. This initial and n is the electron number density of the unshocked fireball will expand and accelerate to relativistic velocity ISM, for ω ∼ 104 s−1 the inequality ω >ω is always valid because of the huge optical depth in the source (Paczyński p if n > 0.01γ−2 cm−3. This shows that the electromag- 1986; Goodman 1986; Shemi & Piran 1990). Perhaps, in- netic waves cannot propagate through the shocked ISM ternal shocks are formed due to collisions between the and by this means energy can be continuously pumped shells with different Lorentz factors in this period (Rees from the pulsar into the shocked ISM. This idea is sim- & Mészáros 1994; Paczyński & Xu 1994). During the ac- ilar to that of Pacini (1967), who first proposed that a celeration, the initial radiation energy will be converted pulsar can continuously supply energy to its surrounding to bulk kinetic energy of the outflow. Subsequently, the supernova remnant through magnetic dipole radiation. expansion of the fireball starts to be significantly influ- In the following we assume that the expansion of the enced by the swept-up interstellar medium (ISM) and two postburst blastwave in uniform ISM is relativistic and adi- shocks will be formed: a forward blastwave and a reverse abatic. At a time t, the shocked ISM energy is given by shock (Rees & Mészáros 1992; Katz 1994). A GRB will E =4πr2(r/4γ2)γ2e′ (Waxman 1997a), where r ≈ ct is be produced once the kinetic energy is dissipated and ra- sh the blastwave radius and e′ = 4γ2nm c2 is the shocked diated as gamma rays through synchrotron or possibly p ISM energy density in the comoving frame (Blandford & inverse-Compton emission from the accelerated electrons McKee 1976). This energy should be equal to the sum of in the shocks (Mészáros, et al. 1994; Sari, et al. 1996). The E/2 and the energy which the fireball has obtained from postburst fireball will be decelerated continuously, and an the pulsar based on energy conservation: X-ray, optical and/or radio afterglow will be formed, as predicted early by many authors (Paczyński & Rhoads t 2 2 3 E 1993; Katz 1994; Vietri 1997b; Mészáros & Rees 1997). 4πnmpc γ r = + (1 − β)L(t − r/c)dt , (4) 2 Z At the center of the fireball, the pulsar loses its ro- 0 tational energy through magnetic dipole radiation, whose where β = (1−1/γ2)1/2 and the factor 1/2 arises from the power is given by fact that about half of the total energy of the initial fire-

4 ball has been radiated away during the GRB phase (Sari 2 2π 6 2 2 & Piran 1995). The term (1−β) accounts for the fact that L = R Bs sin θ 3c3 P the fireball moves away from the incoming photons, and 43 −1 2 −4 6 − = 4 × 10 ergss B⊥,12Pms R6 , (1) the power L(t r/c) shows the fact that the radiation ab- sorbed by the fireball at time t was emitted at the retarded 12 where B⊥,12 = Bs sin θ/10 G, Bs is the surface dipole time t − r/c. According to Eq. (2), this power should de- −2 −2 field strength, θ is the angle between the magnetic dipole crease as L(t − r/c) ∝ [1 + (t − r/c)/T ] = (1+ t⊕/T ) , Z.G. Dai & T. Lu: Gamma-ray burst afterglows and evolution of fireballs with pulsars 3 where t⊕ is the observed time. Because the Lorentz fac- Third, for t⊕ ≫ T , the power of the pulsar due to mag- −2 tor of the fireball at the initial evolution stage decays as netic dipole radiation rapidly decreases as L ∝ t⊕ , and γ ∝ t−3/2, the timescale in which the fireball has ob- thus the fireball is hardly influenced by the stellar radia- tained energy ∼ E/2 from the pulsar can be estimated tion and the optical flux of the afterglow will significantly by 4γ2E/L, which is measured in the burster’s rest frame. decline with time again. The corresponding observer-frame timescale (τ) is equal to this timescale divided by 2γ2 (Waxman 1997b), viz., 3. Constraints on stellar parameters τ = 2E/L. We require τ ∼ 6 days and T ≫ τ for the optical afterglow of GRB970228. The analysis of the lightcurve in the last section can We now analyze evolution of the afterglow from a be directly applied to discussion on the afterglow of postburst fireball with energy injection from the pulsar GRB970228. As pointed out by Galama et al. (1997), the through magnetic dipole radiation. First, at the initial initial decrease of the optical afterglow can be easily ex- stage of the afterglow, viz., t⊕ ≪ τ, the first term on plained by the popular fireball model (Mészáros & Rees the right hand of Eq. (4) is much larger than the sec- 1997; Wijers, et al. 1997; Waxman 1997a; Vietri 1997a; ond term. In this case, one expects that the fireball is not Tavani 1997; Reichart 1997; Dai & Lu 1997), but the significantly influenced by the stellar radiation, and thus subsequent flattening is not consistent with this simple 1/8 −1/8 −3/8 its expansion evolves based on γ = 324E51 n1 t⊕ , model. Further considering the result observed by HST 51 −3 where E51 = E/10 ergs, n1 = n/1 cm , and t⊕ is in on 4 September (Fruchter, et al. 1997), therefore, we can units of 1 second. According to Dai & Lu (1997), therefore, see that the magnitude of the R-band brightness of the af- the synchrotron flux (X-ray) integrated between 2 and 10 terglow of GRB970228 first rapidly decreased with time, −αX keV decays as FX ∝ t⊕ , where αX =3/2 − 3(3 − p)/16, subsequently flattened, and finally declined again. The and the synchrotron flux density at some optical band de- lightcurve analyzed in the last section is well consistent −αopt clines as Sopt ∝ t⊕ , where αopt = 3(p − 1)/4. Here p is with these observational results. the index of the power-law distribution of the accelerated If the afterglow of GRB970228 was radiated from electrons in the shocked ISM. The observations of the af- a relativistic fireball with energy injection from a pul- terglow of GRB970228 have given αX =1.4 ± 0.2 (Costa, sar through magnetic dipole radiation, we now give con- +0.3 et al. 1997a, b; Yoshida, et al. 1997) and αopt =2.1−0.5 in straints on some parameters of this pulsar. Inserting Eq. R band (Galama, et al. 1997). In order to fit these values, (1) into τ = 2E/L, we get the timescale at which the we require p ∼ 3. fireball has obtained energy ∼ E/2: Second, for T >t⊕ ≫ τ, according to Eq. (4), the 7 −2 4 −6 energy which the fireball has obtained from the pulsar τ ≈ 5 × 10 s E51B⊥,12PmsR6 . (6) is much larger than E/2, and the expansion of the fireball is significantly affected by the pulsar’s radiation. In This equation can be further changed into the following the case with steady energy supply, the Lorentz factor form: −1/2 −1/4 of the fireball decays as γ ∝ r ∝ t⊕ (Blandford ≈ 1/2 −1/2 2 −3 & McKee 1976). This scaling law can also be found by B⊥,12 10E51 (τ/6 d) PmsR6 . (7) neglecting the term E/2 in Eq. (4). Here we have as- sumed that the power radiated from the pulsar doesn’t The R-band magnitude of the optical afterglow dur- vary with time. The comoving-frame equipartition mag- ing one month following 6 March after GRB970228 didn’t ′ −1/4 vary with time (Galama, et al. 1997), which means that netic field decays as B ∝ γ ∝ t , and the synchrotron ⊕ during this period the stellar radiation significantly af- break frequency drops in time as ν ∝ γ3B′ ∝ t−1. At the m ⊕ fected the evolution of the fireball. This in fact requires same time, since the comoving electron number density is one condition: T ≥ 36 days. On the other hand, the V- ′ ∝ ∝ −1/4 ne γ t⊕ and the comoving width of the emission band magnitude of the afterglow on 4 September dropped ′ 3/4 region ∆r ∼ r/γ ∝ t⊕ , according to Mészáros & Rees to V = 28.0 ± 0.25 which corresponds to the R-band mag- (1997a) and Wijers et al. (1997), the comoving intensity nitude R ≈ 27.0 (Fruchter, et al. 1997), showing that for ′ ′ ′ ′ 1/4 Iν ∝ neB ∆r ∝ t⊕ . So the observed peak flux density t⊕ ≫ 1 month the stellar radiation played no role in the 2 5 ′ increases in time based on Sνm ∝ t⊕γ Iνm ∝ t⊕. Thus, for expansion of the fireball. This requires another condition: ν ≫ νm, the observed optical flux density as a function of T ≪ 188 days. Furthermore, after comparing the observed observed time is data with the decay law of the optical flux at the early −3/2 −(p−1)/2 (3−p)/2 stage or at the late stage (Sopt ∝ t⊕ ), we can infer that S = S m (ν/ν ) ∝ t . (5) ν ν m ⊕ the flattening phase of the optical flux must have ended Therefore, the requirement of p ∼ 3 inferred from the around 60 days. This means T ≤ 60 days. From these early-time behavior of the afterglow leads to the result conditions and Eqs. (3) and (7), we obtain that the observed R-band flux at the subsequent stage 1/2 −1/2 −1/2 may keep relatively constant. 1.0 ≤ PmsE51 (τ/6 d) I45 ≤ 1.2 . (8) 4 Z.G. Dai & T. Lu: Gamma-ray burst afterglows and evolution of fireballs with pulsars

The equations of state at high densities are likely stiff and Blandford R.D., McKee C.F., 1976, Phys. Fluids 19, 1130 thus the moments of inertia of rapidly rotating neutron Cheng K.S., Dai Z.G., 1996, Phys. Rev. Lett. 77, 1210 stars with mass ≥ 1.4M⊙ are I45 ∼ 2 (Datta 1988; Weber Cheng K.S., Dai Z.G., 1997, ApJ 476, L39 Cheng K.S., Dai Z.G., 1998, ApJ 492, 281 & Glendenning 1993). Further adopting R6 ∼ 1, E51 ∼ 1 Costa E., et al., 1997a, IAU Circ. No. 6576 and τ ∼ 6 days, we can find 1.4 ≤ Pms ≤ 1.7 and 20 ≤ Costa E., et al., 1997b, Nature 387, 783 B⊥ ≤ 30. Therefore, we suggest that the central engine ,12 Dai Z.G., Cheng K.S., 1997, Phys. Lett. B 401, 219 of GRB970228 could be a strongly magnetic millisecond Dai Z.G., Lu T., 1997, MNRAS, accepted pulsar. Dar A., Kozlovsky B.Z., Nussinov S., Ramaty R., 1992, ApJ 388, 164 4. Discussion Datta B., 1988, Fund. Cosmic Phys. 12, 151 Duncan R.C., Thompson C., 1992, ApJ 392, L9 We have studied evolution of a postburst fireball with Eichler D., Livio M., Piran T., Schramm D., 1989, Nature 340, energy injection from a millisecond pulsar with a strong 126 magnetic field through magnetic dipole radiation if the Fishman G.J., Meegan C.A., 1995, ARA&A 33, 415 initial fireball of a GRB has been produced during the Fruchter A., et al., 1997, IAU Circ. No. 6747 birth of the pulsar. In the case of τ T 1/2 −1/2 Rees M.J., Mészáros P., 1994, ApJ 430, L93 or Pms > 3.2 msI45 E51 , the millisecond pulsar as the Reichart R.E., 1997, ApJ 485, L57 central engine cannot supply energy to the postburst fire- Sahu K., et al., 1997a, IAU Circ. No. 6606 ball effectively enough for the flattening phase to occur. It Sahu K., et al., 1997b, IAU Circ No. 6619 is interesting to note that this feature doesn’t depend on Sahu K., et al., 1997c, Nature 387, 476 magnetic field strength, which influences τ and T in the Sari R., Narayan R., Piran T., 1996, ApJ 473, 204 same way. Sari R., Piran T., 1995, ApJ 455, L143 We further assume that some cosmological GRBs re- Shemi A., Piran T., 1990, ApJ 365, L55 sult from the birth of strongly magnetic millisecond pul- Tavani M., 1997, ApJ 483, L87 Usov V.V., 1992, Nature 357, 452 sars. If the effective dipole magnetic-field strength of a Usov V.V., 1998, Phys. Rev. Lett. 80, 230 newborn pulsar is extremely high (e.g., B⊥ = Bs sin θ ∼ 15 van Paradijs J., et al., 1997, Nature 386, 686 10 G), the stellar rotational energy is dissipated due to Vietri M., 1996, ApJ 471, L91 2 magnetic dipole radiation in observed time T < 10 s. In Vietri M., 1997a, ApJ 488, L105 other words, the fireball of a GRB can obtain a large Vietri M., 1997b, ApJ 478, L9 amount of energy from the pulsar only in such a short Waxman E., 1997a, ApJ 485, L5 timescale. For t⊕ ≫ T , thus, the expansion of the post- Waxman E., 1997b, preprint: astro-ph/9709190 burst fireball is hardly affected by the magnetic dipole Weber F., Glendenning N.K., 1993, ApJ 390, 541 radiation of the pulsar. In the superstrong-magnetic-field Wijers R., Rees M.J., Mészáros P., 1997, MNRAS 288, L51 case, the flattening behavior might not be observable in Yi I., Blackman E.G., 1997, ApJ 482, 383 the optical counterparts of GRBs. Yoshida A., et al., 1997, IAU Circ. No. 6593

Acknowledgements. We would like to thank the referee for very valuable suggestions, and Drs. J. M. Wang and D. M. Wei for many discussions. This work was supported by the National Natural Science Foundation of China. This article was processed by the author using Springer-Verlag References a LTEX A&A style ﬁle L-AA version 3.