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Gamma-Ray Burst Afterglows and Evolution of Postburst Fireballs With A&A manuscript no. (will be inserted by hand later) ASTRONOMY AND Your thesaurus codes are: ASTROPHYSICS (08.14.1; 08.16.6; 13.07.1) 15.1.2018 Gamma-ray burst afterglows and evolution of postburst fireballs with energy injection from strongly magnetic millisecond pulsars Z. G. Dai1 and T. Lu2,1,3 1Department of Astronomy, Nanjing University, Nanjing 210093, P.R. China 2CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, P.R. China 3LCRHEA, IHEP, CAS, Beijing, P.R. China Received ???; accepted ??? Abstract. Millisecond pulsars with strong magnetic Most of the models proposed to explain the occur- fields may be formed through several processes, e.g. rence of cosmological GRBs are related with compact ob- accretion-induced collapse of magnetized white dwarfs, jects from which a large amount of energy available for merger of two neutron stars, and accretion-induced phase an extremely relativistic fireball is extracted through neu- transitions of neutron stars. During the birth of such a trino emission and/or dissipation of electromagnetic en- pulsar, an initial fireball available for a gamma-ray burst ergy. One such possibility is newborn millisecond pulsars (GRB) may occur. We here study evolution of a post- with strong magnetic fields, which may be formed by sev- burst relativistic fireball with energy injection from the eral models, e.g. (1) accretion-induced collapse of magne- pulsar through magnetic dipole radiation, and find that tized white dwarfs, (2) merger of two neutron stars, and the magnitude of the optical afterglow from this fireball (3) accretion-induced phase transitions of neutron stars. first decreases with time, subsequently flattens, and fi- In the first model, an accreting white dwarf, when its mass nally declines again. This may provide a natural expla- increases up to the Chandrasekhar limit, may collapse to nation for the behavior of the lightcurve of the afterglow a rapidly rotating neutron star, and the magnetic field of of GRB970228 if this burst resulted from the birth of a the neutron star may become very strong due to magnetic- strongly magnetic millisecond pulsar. flux conservation (Usov 1992) or efficient dynamo action (Duncan & Thompson 1992). During the birth of the neu- Key words: stars: neutron – pulsars: general – gamma- tron star, an initial fireball may occur through neutrino rays: bursts processes (Dar, et al. 1992) and/or electromagnetic pro- cesses (Usov 1992; Yi & Blackman 1997). Second, it is usually thought that a neutron-star bi- nary will eventually merge into a black hole due to gravi- arXiv:astro-ph/9810402v1 25 Oct 1998 tational radiation. However, since the equations of state at high densities are likely stiff for several observational and 1. Introduction theoretical facts summarized by Cheng & Dai (1997), Dai The observational results from the BATSE detector on & Cheng (1997), and Cheng & Dai (1998), neutron-star CGRO (Fishman & Meegan 1995) strongly suggested that binaries like the Hulse-Taylor system might merge into the sources of gamma-ray bursts (GRBs) are at cosmolog- massive neutron stars which have both millisecond peri- ical distances or in the galactic halo, leading to a great ods and very strong magnetic fields generated by some debate on the origin of GRBs. The recent discovery of physical processes, e.g. dynamo action powered by tidal fading sources at X-ray (Costa, et al. 1997a, b; Yoshida, heating (Vietri 1996). During the coalescence and merging et al. 1997) and optical wavebands (Groot, et al. 1997; of two neutron stars, an initial fireball may be produced van Paradijs, et al. 1997; Sahu, et al. 1997a, b, c), and the through neutrino-antineutrino annihilation (Eichler, et al. detection of the redshift of the counterpart to GRB970508 1989; Mathews, et al. 1997) and/or efficient energy trans- (Metzger, et al. 1997), clearly demonstrate that GRBs are fer from the orbital energy (Vietri 1996). Third, it has at cosmological distances. been proposed by Cheng & Dai (1996) that some neutron stars in low-mass X-ray binaries accrete sufficient mass to Send offprint requests to: Z. G. Dai undergo phase transitions to become strange stars, and af- 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<T , L can be thought as a a relativistic fireball through magnetic dipole radiation. In constant; but for t ≫ T , L decays as ∝ t−2. this Letter we study evolution of the radiation from such The power of the pulsar is radiated away mainly a postburst fireball. through electromagnetic waves with frequency of ω = 2π/P . Once the electromagnetic waves propagate in the 2. The model shocked ISM, they will be absorbed if ωp > 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 (Paczy´nski 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- & M´esz´aros 1994; Paczy´nski & 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 & M´esz´aros 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 (M´esz´aros, 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 (Paczy´nski & Rhoads t 2 2 3 E 1993; Katz 1994; Vietri 1997b; M´esz´aros & 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.
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