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arXiv:1312.5637v2 [astro-ph.HE] 23 Nov 2014 stemi ups fti work. this of purpose main the is 20)adKaee l 21)sdsoeyb uncov- (hereafter 40 by than discovery bursts more (2012)’s radio al. all et FRB) millisecond-duration Lorimer al. four confirmed et ering Keane have (2013) and (2007) al. et Thornton i,adtehs aayaditraatcmedium redshifts intergalactic cosmological ori- and have terrestrial they than that ( host suggest rather the models celestial and favor gin, data rent siaeo osbecnrbto fFBcnrlengines ( central The ultra-high FRB producing accelerated. of in contribution be possible FRBs, may of of rays estimate engine cosmic central cir- energy the the ultra-high by into released medium was interstellar energy of cum amount huge If 2013). 21)adi eodtesoeo hswr.Tert of al. rate The et Kulkarni work. to this in up of high found scope is the be FRBs beyond disadvan- can is and and models al. advantages (2014) et flar- these the (Loeb of and of Galaxy the tages discussion 2013), in The instead bi- are al. 2014). of FRBs et mergers if 2014)), (Kashiyama ing dwarfs Lasky al. & et Ravi (Bannister nary 2014; FRBs Zhang and be- a 2012; connection Bursts 2014), possible Gamma-ray the tween Rezzolla is & hypothesis (Falcke (i.e., relevant model highly holes of collapses black delayed so-called to 2007), the (SMNS) stars al. neutron et magne- al. supramassive pro- (Popov energetic et flares Totani 2014), been radio 2001; tar Pruzhinskaya have & Lyutikov black Lipunov models & binary 2013; stellar- of (Hansen few mergers a stars the a including neutron or FRBs, quite interpret to Currently posed neutron engine a central hole. either that likely suggests is millisecond-duration The z 1 [email protected] (YZF) [email protected] f05t n itne fu to up of distances and 1 to 0.5 of ) e aoaoyo akMte n pc srnm,Prl Mo Purple , Space and Dark of Laboratory Key rpittpstuigL using 2018 typeset 2, Preprint April version Draft narcn uvyfrplasadfs transients, fast and for survey recent a In OE-EEDN SIAEO H ONCINBTENFS RA FAST BETWEEN CONNECTION THE ON ESTIMATE MODEL-DEPENDENT nago rcino egr n h egrrt is rate merger the and supramas mergers if th of if ray fraction even cosmic good stars, EeV a neutron producing in double in significan of role no for mergers non-ignorable flares, including a the radio models that FRB accelerate suggest energetic other the also the en In FRBs, and huge of dwarfs ones. the rate white observed stars, high the neutron rather to binary the significantly merging with of together model engine the and model nmiuul etdi h r fgaiainlwv astronomy. wave gravitational of headings: era Subject the in tested unambiguously eetyadqieafwmdl aebe rpsd nti okwe work this ( In energy proposed. ultra-high been and have sources FRB models the few between a quite and recently h xsec ffs ai uss(Rs,anwtp fextraga of type new a (FRBs), bursts radio fast of existence The 2 rdaeUiest fCieeAaeyo cecs Yuquan Sciences, of Academy Chinese of University Graduate 1. ∼ in Li Xiang INTRODUCTION 10 A ◦ T 4 E > rmteGlci ln.Cur- plane. Galactic the from sky tl mltajv 5/2/11 v. emulateapj style X ai otnu:gnrl ceeaino atce—omcrays particles—cosmic of acceleration general— continuum: radio 10 − 1,2 18 1 e Zhou Bei , day V nrycsi rays cosmic energy eV) − 1 LR-IHEEG OMCRAYS COSMIC ENERGY ULTRA-HIGH Tono tal. et (Thornton 1,2 a-igHe Hao-Ning , ∼ rf eso pi ,2018 2, April version Draft Gpc. 3 ABSTRACT hn.and China. > 1 10 nanOsraoy hns cdm fSine ajn,2 Nanjing, Science, of Academy Chinese Observatory, untain iZogFan Yi-Zhong , h oeso antrgatrdoflares binary in radio of released giant energy magnetar merger dou- total of the of models merging because the model is the This and dwarfs. model, model, white blitzar star neutron the ble particular in and h ai nfr oaincnehnetemaximum 1986). the of factor enhance a al. can by et mass rotation (Friedman gravitational uniform NS that rapid than non-rotating larger The a is collapse SMNS for not of to allowed is mass quickly very gravitational NS rotate the normal to since Rezzolla and has & former SMNS (Falcke the between that distances difference magneto- observ- The cosmological is the 2014). that to burst of out radio massive fraction able a produce significant and a sphere the magnetic dissipate vio- traveling of snap the shock result will from a lines As Accelerated magnetic-field lently. the holes. SMNS, black of to collapse SMNSs the of collapse cosmic energy account ultra-high are then of 2014) and fraction rays. accelerate non-ignorable to al. a sufficient et for be (Loeb to stars small flaring too Galactic and 2007) tffeog eg,Dvse l 94 a u1998a; possible be to Lu suggested & been had Dai limit breakup 1994; is centrifugal material NS al. even the et or of Davis state SMNS (e.g., of enough some produce equation of stiff the also merger if can NS the normal stars (2014), neutron Rezzolla pro- binary & Falcke (ccSNe) supernovae 1986). in core-collapse posed the al. to et addition (Friedman stable In SMNS the make help thus Vcsi a ore Aos20) tpeeti sunclear is it present burst At not. radio or fast quick giant 2003). so the rotated generating (Arons tially sources the ray whether cosmic eV 18 & 1 V omcry.W hwta nteblitzar the in that show We rays. cosmic eV) h el-onmgeaswt nta pnrtscoet th to close rates spin initial with magnetars newly-born The 10 nti eto efcso h omlgclmodels cosmological the on focus we section this In ntebizrmdl h Rswr rgee ythe by triggered were FRBs the model, blitzar the In CEEAIN OE-EEDN ESTIMATE MODEL-DEPENDENT ACCELERATION: 3 2. yr − OSBEHG-NRYCSI RAY COSMIC HIGH-ENERGY POSSIBLE od1,Biig 009 China. 100049, Beijing, 19, Road yaeirlvn oFB,myplay may FRBs, to irrelevant are ey 1 ai rnins a enestablished been has transients, latic e omcry a contribute may rays cosmic EeV d Gpc e omcryi xetd We expected. is ray cosmic EeV t 1 ryrlaeo ahFBcentral FRB each of release ergy n aMn Wei Da-Ming and , 2.1. ics h osbeconnection possible the discuss ienurnsaswr formed were stars neutron sive − xml h egro double of merger the example 3 uhapsiiiywl be will possibility a Such . h lta model blitzar The I USSAND BURSTS DIO ∼ 1 0 . 05( P 1 0 Ppve al. et (Popov / ms) 1 − 10008, ∼ 2 ini- s 10 and 20 e 2

Dai et al. 2006; Shibata et al. 2000; Baumgarte et al. of the massive star, Chevalier et al. (2004) assumed 2000; Gao & Fan 2006; Metzger et al. 2008; Zhang that the surrounding medium has a typical of 2013). Usually the nascent NSs formed in both ccSNe the hot, low-density phase of a starburst galaxy (i.e., and merger of double NSs are differentially rotating and n ∼ 0.2 cm−3). If it is the case, the contribution to the differential rotation is suggested to be more efficient the DM by the surrounding medium can be ignored. to keep the SN stable. However, the differential rotation However, the ccSN may be born in 2 −3 is expected to be terminated by the magnetorotational with a typical number density ncloud ∼ 10 cm 2 1/3 instability as well as magnetic braking (Cook et al. 1992, and a size of Rcloud ∼ (3Mcloud/4πncloudmpc ) ∼ 1994; Hotokezaka et al. 2013) very quickly. That is why 4 1/3 2 −3 −1/3 10 pc (M /10 M⊙) (n /10 cm ) , we only consider the effect of uniform rotation in stabi- cloud cloud which can give rise to an observed DM ≈ 103 (1 + lizing the SMNS. Here we concentrate on ccSN-formed −1 4 1/3 2 −3 2/3 −3 SMNS and will discuss the merger-formed SMNS in some z) (Mcloud/10 M⊙) (ncloud/10 cm ) pc cm . Therefore the surrounding molecular cloud should not detail in section 2.2. 4 Before discussing the high-energy cosmic ray be very massive/dense (i.e., Mcloud < 10 M⊙ for 2 −3 acceleration, we examine whether the blitzar model ncloud ∼ 10 cm ) otherwise the central ccSNe could can account for the observed dispersion measures not be viable progenitors of the observed FRBs. Such or not. The SN outflow likely has a total mass a request is denoted as Request-III. All these three Msh ∼ a few M⊙, and the observed dispersion requests impose some challenges on the ccSN scenario in measures DM ∼ 1021 cm−2 (Thornton et al. 2013; the blitzar model. Nevertheless, the blitzar model can Lorimer et al. 2007) require that the SN outflow should explain some aspects of FRBs (e.g., Falcke & Rezzolla 1/2 2014) and is still widely adopted in the literature. reach a radius Rsh > (Msh/4πmp(1 + z)DM) ∼ 18 1/2 −1/2 Below we discuss the possible high energy cosmic ray 10 cm (Msh/10 M⊙) (1 + z) , where the term acceleration in such a scenario. (1 + z) is introduced to address the effects of both the In view of the sensitive dependence of stabilization cosmological time dilation and the frequency shift on the on P0, SMNS is unlikely to exist for P0 > 2 ms unless measured DM. Clearly, at such a huge radius the SN shell there is the fine tuning that the mass of SMNS is just is also transparent for the 1.3 GHz FRB. The velocity of 1/2 tiny above that allowed by the non-rotating NS. The the SN shell can be estimated as Vsh ∼ (Einj/Msh) ∼ rotational of SMNSs is quite large, i.e., 9 −1 −1/2 52 1/2 52 45 2 −2 10 cm s (Msh/10 M⊙) (Einj/10 erg) . The ESN,r ≈ 3 × 10 erg (I/1.5 × 10 g cm )(P0/1 ms) , life of the SMNS should satisfy tlife > Rsh/Vsh ∼ where I is the moment of inertia. Before collapsing 9 −1/2 52 −1/2 10 s (1 + z) (Msh/10 M⊙) (Einj/10 erg) , into a , the SMNS should have lost its ro- suggesting a dipole magnetic field strength tational energy mainly via magnetic dipole radiation 12 1/4 B⊥ < 1.3 × 10 G (1 + z) (I/1.5 × and possibly also radiation (Usov 45 2 3/4 6 −3 1/2 −1/2 1992; Duncan & Thompson 1992; Fan et al. 2013a). 10 g cm ) (R /10 cm) (P /1 ms) (M /10 M⊙) , s 0 sh The amount of energy injected into the surrounding where B⊥ = Bs sin α, Bs is the surface magnetic field 52 strength at the pole, R is the radius of the NS and medium is Einj ∼ ESN,r/2 ∼ 1.5 × 10 erg (I/1.5 × s 45 2 −2 α is the angle between the rotational and dipole 10 g cm )(P0/1 ms) , if the gravitational wave radi- axes. Hence the SMNS should not be significantly ation is not dominant. The frequency of the electromag- 3 magnetized otherwise the FRB can not be accounted netic wave is ∼ 10 Hz, which is much lower than the 4 1/2 for (see also Falcke & Rezzolla (2014)). We denote surrounding ’s frequency ωp ∼ 5.6 × 10 Hz ne , such a request as Request-I. Another highly-related where ne is the number density of the free electrons in request is that the gravitational wave radiation should the plasma (i.e., the SN outflow) and can be estimated as 4 −3 −1 18 −3 be weak enough to not dominate over the dipole ∼ 10 cm (Msh/10 M⊙)(Rsh/0.1) (Rsh/10 cm) , radiation of the (i.e., Request-II). In terms where the width of the outflowing material shell is taken of the ellipticity (ǫ) of the pulsar, we need ǫ < 5 × to be a fraction Rsh ∼ 0.1 of the Rsh. Therefore the −6 45 2 −1/2 2 9 1/2 10 (I/1.5 × 10 g cm ) (P0/1 ms) (tlife/10 s) electromagnetic wave will be “trapped” by the plasma. (Shapiro & Teukolsky 1983). Alternatively, for a The pressure of the electromagnetic wave outflow is so rapidly-rotating pulsar, some instabilities, for exam- high that can work on the surrounding plasma and then ple the Chandrasekhar-Friedman-Schutz instability the magnetic wind energy will be transferred into the (Chandrasekhar 1970; Friedman & Schutz 1978), kinetic energy of the outflow, as indicated by the shal- may occur when the ratio R = T /|W | of the rota- low decay of some GRB afterglows (Dai & Lu 1998a; tional kinetic energy T to the gravitational binding Dai & Lu 1998b; Zhang et al. 2006) and the light curves energy |W | is sufficiently large. In the Newto- of some superluminous supernovae (Kasen & Bildsten nian limit, the l = m = 2 f−mode, which has 2010; Woosley 2010; Inserra et al. 2013; Nicholl et al. the shortest growth time of all fluid modes 2013). Energetic forward shocks will be driven, and then −6 6 4 −5 (i.e., τGW ∼ 5 × 10 s (Rs/10 cm) (R − Rsec) , accelerate protons and other charged particles to ultra- see Lai & Sharpiro 1995), becomes unstable when high energies 2. Hence in the blitzar model FRBs may −3 R & Rsec ≈ 0.135. Hence R − Rsec . 10 is needed be promising sources of EeV cosmic ray protons. to satisfy τ > t ∼ 109 s otherwise the SMNSs GW life 2 The above estimate is for the assumption that at a radius 18 collapsed too early to get an sufficient small DM. Rsh ≥ 10 cm the SMNS wind is still Poynting-flux dominated. If Finally, the contribution to the DM by the circum-burst instead the SMNS wind at such a large distance is /positron medium is less clear. In the estimate of the wind pair dominated and the Poynting-flux is a tiny amount (for example < 0.1) of the total (e.g., Kennel & Coronitti 1984), most of the bubble structure at the end of the Wolf-Rayet stage wind energy will be converted into radiation and the acceleration 3

52 So far, no reliable γ−ray/X-ray/optical counterparts SMNS is Einj ∼ 10 erg, implying that the total energy of FRBs has been identified and the of the cen- input by FRB sources is comparable to the input by all tral engine can not be pinned down. A possible test of other ccSNs, and thus FRBs should be one kind of the the blitzar model is to observe the very early radio emis- main sources of cosmic rays. In particular, as found in sion of GRBs since at least for some GRBs the central eq.(1), the most energetic cosmic ray protons can reach engine may be SMNSs (see Zhang (2014) and the refer- the energy of 1018 eV, and might be the dominant com- ences therein). Intriguingly, tentative association of two ponent at such energies. The injected cosmic-ray density single dispersed millisecond radio pulses with two GRBs 47 Einj RFRB −1 −3 has been reported and these two single dispersed mil- ǫ˙CR & 10 ηω( )( ) erg yr Mpc lisecond radio pulses were detected in the few minutes 1052 erg 104 yr−1 Gpc−3 following two GRBs and the arrival times of both pulses (2) are found to coincide with breaks in the GRB X-ray light per energy decade with η as the cosmic ray acceleration curves, which likely label the phase transition of the cen- efficiency, and ω =1/ ln(εmax/εmin) ≃ 0.1 as the fraction tral engine, i.e., the collapse of the SMNS to stellar black of the total cosmic ray energy at each energy decade. hole (Bannister et al. 2012). Such a correlation between Then the corresponding observed cosmic ray flux at ∼ FRBs and GRBs, if confirmed in the future, will be in 1018 eV is support of the blitzar model.3 −28 ω Einj Now we discuss the cosmic ray particle generation FEeV−CR ∼ 10 η( )( ) 0.1 1052 erg by the SMNS wind-accelerated outflow of the ccSNe. R The progenitor star of a ccSN would experience signif- ( FRB ) m−2 s−1 sr−1 eV−1.(3) icant mass loss stage and then the surrounding medium 104 yr−1 Gpc−3 is usually not a simple free-wind structure or a con- 4 −1 −3 stant density structure. For simplicity, here we adopt Roughly, a rate of RFRB ∼ 10 yr Gpc the structure shown in Fig.1 of Chevalier et al. (2004) would result in the observed flux of cosmic-ray as 10−28η m−2 s−1 sr−1 eV−1, which can meet the observed to estimate the maximum energy of the protons acceler- 18 −3.2±0.05 cosmic ray flux Fobs(ε) = C(ε/6.3 × 10 eV) ated at the shock front of the SMNS wind-driven outflow. −33 −2 −1 −1 −1 In such a scenario, significant particle acceleration takes with C = (9.23 ± 0.065) × 10 m s sr eV place at R ∼ 5 × 1018 cm, where is the supergiant shell (Nagano & Watson 2000), for the EeV cosmic ray ac- with a number density n ∼ 102 − 103 cm−3. Following celeration efficiency η ∼ 0.03. Bell & Lucek (2001), the maximum energy of the pro- EeV cosmic rays in principle can produce PeV neu- tons accelerated by the forward shock can be estimated trinos via interacting with the , but as we do not expect significant PeV emission since the > 1017 eV protons can not be effectively confined ccSN 18 V 2 n 1/2 ǫB 1/2 and the energy loss via pion production is ignorable un- εp,M ∼ 10 eV ( ) ( ) ( ) , 109 cm s−1 102 cm−3 10−2 less the FRBs were born in the starburst and in (1) particular the so-called ultra-luminous infrared galaxies where ǫB is the fraction of shock energy given to (He et al. 2013). magnetic field and the velocity of the SN shell 1/2 has been estimated to be Vsh ∼ (Einj/Msh) ∼ 2.2. The model of merging double Neutron Stars 9 −1 −1/2 52 1/2 10 cm s (Msh/10 M⊙) (Einj/10 erg) . In the model of merging double neutron stars for The charged particles reach energies larger by FRBs, the radiation mechanism may be coherent ra- a factor of Z, the charge number. The mag- dio emission, like radio pulsars, by magnetic brak- netic field generated by the shock is B ∼ ing when magnetic fields of neutron stars are synchro- −3 −2 1/2 2 −3 1/2 9 −1 10 (ǫB/10 ) (n/10 cm ) (Vsh/10 cm s ), nized to binary rotation at the time of coalescence which is too low to effectively cool the accelerating EeV (Totani et al. 2013). In addition to FRBs, the merg- cosmic rays. ers of binary neutron stars may give rise to short GRBs −3 −1 The rate of FRBs is RFRB ∼ 10 year per galaxy or other kinds of violent explosions with possible cen- (Thornton et al. 2013), which is about one order of mag- tral engines of magnetized millisecond neutron stars −2 −1 nitude lower than the ccSN rate RccSN ∼ 10 year per (e.g., Duncan & Thompson 1992; Davis et al. 1994; galaxy. If FRBs are indeed the cry of the dying SMNSs, Dai & Lu 1998a; Baumgarte et al. 2000; Shibata et al. the energy released into the surrounding material of each 2000; Duez et al. 2006; Price & Rosswog 2006; Rosswog 2007; Giacomazzo & Perna 2013). The latest numeri- of the high energy cosmic rays by the outflow is less significant. cal simulations suggest that SMNS can be formed in 3 As already mentioned in above paragraph, to be a valid source the merger of a NS binary with Mtot ∼ 2.6 M⊙ (note of the observed FRBs, the SMNSe formed in the normal ccSNe that among the ten NS binaries identified so far, five 12 should have a typical dipole magnetic field strength B⊥ . 10 systems have such a total gravitational mass (Lattimer Gauss while the SMNS candidates found in both long and short 2012)) for reasonably stiff equation of states that are GRBs usually have B⊥ > 1014 Gauss. The energy available for generating FRBs of the GRB SMNSs is about 4 orders of magni- favored by current rest mass measurements of pulsars tude larger than that of normal ccSN SMNSs. Some differences (see Hotokezaka et al. (2013) and Fan et al. (2013b) between the FRBs from these two different groups of central en- and the references therein). There are growing, though gines may be expected. On the other hand, the dipole radiation timescale of such FRB pulsars are so long that at early times inconclusive, observational evidences for forming SMNS the associated supernovae are still “normal” with a typical energy or even stable NS in NS-NS mergers. The most-widely 52 ≪ Einj ∼ a few × 10 erg, which do not belong to the so-called discussed one in the literature is the X-ray plateau fol- hypernovae. lowed by an abrupt cease in the afterglow light-curve 4 of many short GRBs (Gao & Fan 2006; Metzger et al. 1 2008; Rowlinson et al. 2010, 2013). In addition, the ob- servations on short bursts such as GRB 051221A and 0.8 GRB 130603B indicate an energy injection from a highly- magnetized SMNS and the injected energy is as high as ) 0.6 1051 − 1052erg (Fan & Xu 2006; Rowlinson et al. 2010; max

M P = 0.5 Dall’Osso et al. 2011; Fan et al. 2013a). Moreover, as < M pointed out firstly by Fan & Xu (2006), usually the ma- ( 0.4 terial ejected during the binary merger is P not expected to be more than ∼ 0.01 M⊙ and could be 0.2 accelerated to a mildly-relativistic velocity by the wind of P0 >> 1 ms P0 = 1 ms SMNS and then produce X-ray/optical/radio afterglow 0 emission (see Gao et al. (2013a) for detailed numerical 2.2 2.3 2.4 2.5 2.6 calculation of the lightcurves), which can well account for Mmax [M⊙] the cosmological relativistic fading source PTF11agg, a remarkable event not associated with a high energy coun- Fig. 1.— The possibility distribution, as a function of Mmax, of terpart (Wang & Dai 2013; Wu et al. 2014). It thus forming supramassive (P0 = 1 ms) neutron stars or stable neutron seems reasonable to assume that SMNSs, which likely stars (P0 ≫ 1 ms) in the mergers of binary neutron stars. collapsed at t ∼ 102 − 104 s (Rowlinson et al. 2013), 2 2 c dNNS/dM ∝ exp[−(M − M0) /2σ ] with these parame- were formed in a good fraction of NS-NS mergers. ters are adopted. The possibility distribution of the grav- The prospect of forming SMNS in the mergers can itational of supermassive remnants (i.e., M ≈ also be roughly estimated as the following. On the one 2 2 hand, the gravitational mass (M) of the isolated neu- [−1+ p1+4α[M1 + M2 + α(M1 + M2 ) − mloss]/2α) formed in the simulated double neutron star mergers is tron star is related to the baryonic mass (Mb) as Mb ≈ 2 −1 presented in Fig.1. We find for Mmax ≥ 2.36 M⊙, about M + αM , where α ≈ 0.08 M⊙ (Lattimer & Yahil half of the mergers will produce SMNSs with P0 = 1 ms. 1989; Timmes et al. 1996). On the other hand, in Observationally the pulsar PSR J0348+0432 has an the numerical simulations of mergers of binary neu- accurately measured gravitational mass 2.01 ± 0.04 M⊙ tron stars performed in full incorpo- (Antoniadis et al. 2013) and J1748-2021B has a gravi- rating the finite-temperature effect and neutrino cooling, tational mass ≈ 2.74±0.21 M⊙ (Lattimer 2012). Hence Sekiguchi et al. (2011) found that the effect of the ther- Mmax ∼ 2.36 M⊙ is still possible, with which a sizeable mal energy is significant and can increase the maximal fraction of NS-NS mergers may produce SMNSs. gravitational mass Mmax by a factor of 20% − 30% for a We have briefly mentioned in section 2.1 that in high-temperature state with T ≥ 20 MeV. Since they are the blitzar model a small fraction of FRBs may be rel- not supported by differential rotation, the supermassive evant to the mergers of double neutron stars that pro- remnants were predicted to be stable until neutrino cool- 53 −1 duce SMNS remnants. In such a scenario some FRBs ing, with of & 3 × 10 erg s , has removed are expected to be detected in the afterglow emission the pressure support in a few seconds (Sekiguchi et al. phase of short GRBs. In the model of merging dou- 2011). After the neutrino cooling, the supermassive rem- ble neutron stars, FRBs are generated at the time of nant is still stable if coalescence of double NSs and are expected to precede 2 2 2 short GRBs or other kinds of violent explosions (i.e., αM +Mr,max −(M1 +M2)−α(M +M )+mloss > 0, r,max 1 2 no FRB is expected to occur in the afterglow phase of where mloss is the baryonic mass loss of the system during short GRBs) and the three requests outlined in Sect. −2 the merger, Mr,max ≈ [1 + 0.05(P0/1 ms) ]Mmax, and 2.1 do not apply. The follow-up observations of FRBs M1 and M2 are the gravitational masses of the binary and short GRBs would be crucial to distinguish between −3 −2 neutron stars, respectively. mloss ∼ 10 − 10 M⊙ the blitzar model and the merging neutron star model. has been inferred in the numerical simulation The origin of merging double neutron stars for FRBs, (Rosswog et al. 1999). In the modeling of the infrared if confirmed in the future, has interesting implication −2 bump of short GRB 130603B, mloss ∼ a few × 10 M⊙ on the sources of EeV cosmic ray protons. This is be- is needed (Tanvir et al. 2013; Berger et al. 2013; cause there are some tentative evidence for the forma- Fan et al. 2013a). As a conservative estimate we tion of SMNSs in a sizeable fraction of NS-NS merg- take mloss = 0.01 M⊙. In order to estimate the mass ers and such a kind of long-lived central engine can distribution of the neutron stars in the NS-NS binary accelerate the materials ejected during the merger to systems, Ozel¨ et al. (2012) divided the sample into one very high velocities (Fan & Xu 2006; Gao et al. 2013a; of pulsars and one of the companions (For the double Wu et al. 2014). The physical reason for converting the pulsar system J0737-3039A, they assigned the faster SMNS wind energy into the kinetic energy of the for- ward shock is the same as that in the case of ccSNe. pulsar to the “pulsar” and the slower to the “compan- 14 4 ion” categories). Repeating the above inference for these Within a radius Rsh . ctc ∼ 3 × 10 cm (tc/10 s), two subgroups individually, Ozel¨ et al. (2012) found the SMNS wind is likely Poynting-flux dominated rather that the most likely parameters of the mass distribution than electron/positron pair dominated (Vlahakis 2004). The frequency of the electromagnetic wave of the SMNS for the pulsars are M = 1.35M⊙ and σ = 0.05 M⊙, 0 (∼ 103 Hz) is much smaller than the surrounding whereas for the companions they are M0 = 1.32M⊙ 4 1/2 and σ = 0.05M⊙. Hence in our simulation, the dis- plasma’s frequency ωp ∼ 5.6 × 10 Hz ne , where 11 −3 −1 tributions of gravitational masses of neutron stars as ne ∼ 3 × 10 cm (Mej/0.01 M⊙)(Rsh/0.1) (Rsh/3 × 5

1014 cm)−3 is the number density of the free electrons in rect than in the blitzar model 4. Possible byproducts are the merger outflow with a rest mass Mej ∼ 0.01 M⊙. As high energy if there are dense/energetic seed a result, the electromagnetic wave is “trapped” by the photons. Even with very optimistic assumptions, the re- plasma. The high magnetic pressure works on and hence sulting PeV neutrinos are likely too weak to give rise accelerate the surrounding plasma (see Yu et al. 2013 to significant detection for IceCube like detectors (see and the references therein). The SMNS-driven outflow, Gao et al. (2013b) for a relevant estimate). almost isotropic, will generate energetic forward shocks and then accelerate ultra-high energy cosmic rays. 2.3. The model of merger of binary white dwarfs The maximum energy of the cosmic rays accelerated by the wide outflow driven by the SMNS can be esti- In the model of merger of binary degenerate mated as white dwarfs, the FRBs were produced by coher- ent emission from the polar region of a rapidly ro- NS−NS tating, magnetized massive formed af- εp,M ∼ βZeBdRd ter the merger (Kashiyamaet al. 2013). The en- β Einj 1 ergy budget of the nascent massive white dwarf ∼ 1.5 × 1018 eV Z( )2( ) 3 2 52 can be estimated as Ebug ∼ GMWD /RWD ∼ 3 × 0.5 10 erg 50 2 9 −1 n 1 ǫ 1 1 10 erg (MWD /1 M⊙) (RWD /10 cm) . Magnetic ac- 6 B 2 3 ( −2 −3 ) ( −2 ) Γ , (4) tivity of the post-merger object has been demonstrated 10 cm 10 by recent numerical simulations (Ji et al. 2013), in which the magnetic energy of the remnant at its peak where β is the velocity of the ejecta in units of the is found to exceed 1048 erg (the corresponding volume- c, e is the electron’s charge, Rd ∼ 1.2 × ¯ 11 18 52 1/3 −2 −3 −1/3 −2/3 averaged magnetic field strength is B ∼ 10 Gauss) and 10 cm (Einj/10 erg) (n/10 cm ) (Γ/5) −3 about Mej,WD ∼ 10 M⊙ mass is ejected from the sys- is the deceleration radius, and Bd = 4 −3 1/2 1/2 tem over the run time of the simulations, i.e., t =2×10 0.02 Gauss β(Γ/5)(n/0.01 cm ) (ǫB/0.01) is s. With a spin-down luminosity of the magnetized mas- the magnetic field strength at Rd. The β has been 38 9 2 sive white dwarf Ldip,WD ∼ 2 × 10 (B⊥,WD /10 G) , normalized to 0.5 since in the presence of a highly the ejected material as well as the swept circum medium magnetized SMNS, the material ejected during the with a density n ∼ 0.01 cm−3 can not be accelerated to NS-NS merger is expected to be accelerated to a a velocity larger than ∼ 109 cm s−1, where we have nor- trans-relativistic or even mildly-relativistic velocity 9 malize B⊥ to a value of 10 G, the surface magnetic (Fan & Xu 2006). Different from the ccSN scenario, ,WD −2 −3 field strength of the highly magnetized white dwarfs ob- we normalize n to the value of 10 cm to address served so far. With eq.(4) we find that the cosmic ray the fact that some mergers of NSs are expected to take protons more energetic than ∼ 1016 eV can not be ac- place in low density medium. Again the cooling of the celerated, which are not of our interest. If some FRBs accelerating EeV cosmic rays do not suffer significant are associated with type Ia supernovae, as suggested in energy loss via synchrotron radiation due to the low Bd. Kashiyama et al. (2013), the supernova outflow can ac- How large is Einj? The answer is somewhat uncertain celerate cosmic rays too. However, it is widely known since the SMNS formed in double neutron star mergers that SNe Ia outflow can at most accelerate protons to might suffer significant energy loss in addition to the reg- the so-called “knee” of the cosmic ray spectrum (i.e., ular dipole magnetic radiation. For GRB 130603B dis- 51 ∼ 3 PeV). Therefore in the model of merger of binary playing a SMNS signature, Einj & 2 × 10 erg is needed white dwarfs, EeV cosmic rays are not expected. to account for the multi-wavelength afterglow data. In a good fraction of short GRBs with distinguished X-ray af- 3. DISCUSSION terglow plateau, as reported in Rowlinson et al. (2013), the energy release by the central SMNS is found to be Since the origin of FRBs is still to be pinned down, 52 51 Einj < 10 erg. The inferred Einj ∼ a few × 10 erg in this work we have carried out model-dependent esti- is also favored by the weak radio afterglow emission of mate on the possible role of FRB sources in accelerating most short GRBs and has been suggested to be the sig- EeV cosmic rays. In the models of magnetar giant radio nature of the significant gravitational wave radiation of flares, the merger of binary white dwarfs, and Galactic the SMNS (Fan et al. 2013a,b), which may be possi- flaring stars, significant EeV cosmic ray acceleration is ble if the interior toroidal magnetic field was high up not expected. While in the blitzar model, the cosmo- to ∼ 1017 Gauss that can give rise to a sizeable de- logical FRB sources are very promising EeV cosmic-ray formation ǫ ∼ 0.01 of the magnetar or the secure in- accelerators thanks to the huge energy release into the stability occurred with a R ∼ Rsec + 0.03 ∼ 0.165. surrounding medium by each supramassive neutron star While in the modeling of GRB 051221A and PTF11agg, (see Sec. 2.1). In the model of merging neutron stars, 52 Einj ∼ 10 erg is needed. On average Einj is likely FRBs may still be promising ultra-high energy cosmic ∼ quite a few × 1051 erg. With Eq.(3), it is straight- ray sources if supramassive neutron stars are formed in a 51 −1 considerable fraction of binary neutron star mergers (see forward to show that η ∼ 0.1 (Einj/3 × 10 erg) is needed otherwise the accelerated ultra-high energy cos- Sec. 2.2). We also suggest that in the blitzar model the mic rays can not account for a sizeable fraction of the 4 The additional assumptions are either SMNSs are formed in observed EeV ones. We then suggest that in the model a considerable fraction of binary neutron star mergers or alterna- of merging double NSs (Totani et al. 2013), the FRBs tively the ejection of material with velocities > 0.3c during the may still have significant connection with ultra-high en- merger is very important, as hinted by current short GRB obser- ergy cosmic ray sources though the argument is less di- vations and by latest numerical simulation. 6

GRB-related FRBs may show some difference from the binary neutron star mergers may produce only ∼ 10% normal ccSN-related FRBs. of the observed EeV cosmic rays. Nevertheless, the im- The neutron star mergers, if irrelevant to FRBs, portance of NS-NS mergers in producing > 1018 eV cos- are expected to have a rate lower than RFRB . Even mic rays can be reliably estimated in the foreseeable fu- so, their role in producing EeV cosmic rays may be ture, since (i) The gravitational wave observations by non-ignorable. We are aware that the role of NS-NS- advanced LIGO/VIRGO will pin down or impose a tight merger outflow in accelerating ≤ 1017 eV cosmic ray constraint on the merger rate of binary neutron stars; protons has been discussed in (Takami et al. 2013), (ii) The dedicated electromagnetic counterpart searches where energy injection and then acceleration of the out- of the merger events will help us to tightly constrain the flow caused by the SMNS central engine have not been total energy injected into the surrounding medium. taken into account. However, as summarized in Sec. In view of the fact that all the Active Galactic 2.2, there are growing evidence for the formation of Nuclei (Ginzburg & Syrovatskii 1964; Hillas 1984), SMNS in plausibly a non-ignorable fraction of binary bright Gamma-ray Bursts (Waxman 1995; Vietri neutron star mergers, which help accelerate EeV cos- 1995) and low-luminosity GRBs (Murase et al. 2006; mic rays. For example, inserting the physical parame- Liu et al. 2011), Type Ic supernovae in particular the ters inferred from the modeling of GRB 130603B with so-called hypernovae associated with GRBs (Dermer a SMNS central engine (Fan et al. 2013a) into eq.(4), 2001; Wang et al. 2007; Budnik et al. 2008; Fan 2008; NS−NS 18 Chakraborti et al. 2011; Liu & Wang 2012) and clusters we have εp,M ∼ 2 × 10 eV. Based on extrapo- lations from observed binary pulsars in the Galaxy, a of galaxies (Murase et al. 2008) can also accelerate cos- −4 −1 mic rays to the energies ≥ 1018 eV, we suggest that the likely coalescence rate of binary NSs is ∼ 10 yr 18 per Milky-Way Equivalent Galaxy (Abadie et al. 2010), > 10 eV cosmic rays consist of multiple components which is about one order of lower than the from different astrophysical sources, possibly including observed rate of FRBs (The very optimistic estimate that producing FRBs. of the NS-NS merger rate could be comparable to the rate of FRBs). The beaming-corrected estimates of ACKNOWLEDGMENTS short GRB rate can be as high as ∼ 103 yr−1 Gpc−3 (Coward et al. 2012), and the merger rate of binary NSs We thank the anonymous referee for helpful com- is expected to be higher. These two independent esti- ments. This work was supported in part by 973 mates are consistent with each other. If each merger Programme of China under grants 2013CB837000 and injects energy of ∼ 1052 erg into the surrounding mate- 2014CB845800, National Natural Science of China rial (i.e., the rotational energy of SMNS is mainly lost under grants 11273063, 11303098 and 11361140349, and via magnetic dipole radiation), and then drives trans- the Foundation for Distinguished Young Scholars of relativistic shocks (β > 0.5), the flux of cosmic rays Jiangsu Province, China (No. BK2012047). YZF is also with the energy of ∼ 1018 eV detectable on the supported by the 100 Talents programme of Chinese −29 52 −2 −1 −1 −1 Academy of Sciences. DMW is partly supported by the would be ∼ 10 η(Einj/10 erg) m s sr eV , which can account for ∼ 1/3 of the observed flux ∼ Strategic Priority Research Program “The Emergence 3 × 10−30 m−2 s−1 sr−1 eV−1, for a reasonable EeV cos- of Cosmological Structures” of the Chinese Academy 52 −1 of Sciences, Grant No. XDB09000000. HNH is also mic ray acceleration efficiency η ∼ 0.1(Einj/10 erg) . As summarized in Sec. 2.2, if instead E is only a few supported by China Postdoctoral science foundation inj under grants 2012M521137 and 2013T60569. Bethe (i.e., 1051 erg), as found in GRB 130603B, the

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