Effect of magnetic field on neutrino annihilation efficiency in gamma-ray bursts
aa r X i v : . [ a s t r o - ph . H E ] J u l Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 11 July 2019 (MN L A TEX style file v2.2)
Effect of magnetic field on neutrino annihilation efficiencyin gamma-ray bursts
Shuang Du , , ⋆ , Fang-Kun Peng , , , , Guang-Bo Long , Miao Li † Department of Physics and Astronomy, Sun Yat-Sen University, Zhuhai 519082, China Guizhou Provincial Key Laboratory of Radio Astronomy and Data Processing, Guizhou Normal University, Guiyang 550001, China Department of Physics And Electronics, Guizhou Normal University, Guiyang 550001, China GXU-NAOC Center for Astrophysics and Space Sciences, Department of Physics, Guangxi University, Nanning 530004, China Guangxi Key Laboratory for Relativistic Astrophysics, Nanning, Guangxi 530004, China
11 July 2019
ABSTRACT
Neutrino annihilation process on a hyperaccreting disk is one of the leading modelsto explain the generation of relativistic jets of gamma-ray bursts (GRBs). The neu-trino annihilation efficiency (NAE) has been widely studied in black hol-accretion disc(BHCdisc) system, and the published results are mutually corroborated. However,there is still uncertainty regarding the NAE of neutron star-accretion disc (NS-disc)system because of complicated microphysics processes and effects of strong magneticfield. In this paper, we investigate the latter NAE by assuming that the prompt jet ofGRB 070110 is driven by neutrino pair annihilation in the NS-disc system. Our calcu-lation shows η ν ¯ ν > . × − under the estimated accretion rate ˙ M ≃ . ⊙ · s − .Independent of the detailed accretion disc models, our result shows that the magneticfield may play an important role in the neutrino annihilation process on the hyperac-creting magnetized accretion disc. Compared with the theoretical value of η ν ¯ ν of thenon-magnetized BH-disc system, the NAE should increase significantly in the case ofthe NS-disc system if the GRB is powered by magnetar-disc system. Key words: accretion, accretion disk - star: gamma-ray burst - star: magnetar
Gamma-ray bursts (GRBs) are the brightest events in theuniverse after the Big Bang. In general, GRBs can be di-vided into four stages (Kumar & Zhang 2015): (i) a compactbinary merger or massive star collapse forms a compact star-Caccretion disc system (so-called central engine); (ii) inter-mittent ultrarelativistic jets are launched from the centralengine; (iii) energy dissipation in these relativistic jets gen-erates gamma-ray emission; (iv) the interaction between thejets and the surrounding medium results in multiband af-terglows. The central compact star can be a black hole (BH;e.g., Eichler et al. 1989; Narayan, Paczynski, & Piran 1992;Woosley 1993) or a neutron star (NS; e.g., Usov 1992; Dai& Lu 1998a,b; Zhang & Mszros 2001). The neutrino anni-hilation process (Popham, Woosley, & Fryer 1999) and theBlandford-Znajek (BZ) process (Blandford & Znajek 1977)are the two leading models used to explain the generationof these relativistic jets.One of these leading models, the neutrino annihilation ⋆ E-mail: dushuang [email protected] † E-mail: [email protected] process, has been widely studied theoretically (e.g., Popham,Woosley, & Fryer 1999; Gu, Liu, & Lu 2006; Lei et al. 2009)and using simulations (e.g., Harikae, Kotake, & Takiwaki2010; Zalamea & Beloborodov (2011); Just et al. 2016; seeLiu, Gu, & Zhang 2017 for review). According to the thepublished results, the values of neutrino annihilation effi-ciency (NAE) under the BH-disk system are mutually cor-roborated (see Section 4). However, there is still uncertaintyregarding NAE of NS-accretion disc (NS-disc) system be-cause of the complicated the microphysics processes and theeffects of strong magnetic fields (Xie, Huang, & Lei 2007; Xieet al. 2009; Lei et al. 2009; Zhang & Dai 2009, 2010). Untilnow, there have been no constraints on the NAE from GRBobservations as a result of the unknown launch mechanismof relativistic jets..The premise of solving this problem is to determine thetype of the central compact star. In some GRB afterglows,X-ray plateaus can be followed by a very steep decay (e.g. t − , the so-called ‘internal plateau’; see Figure 1). This fea-ture indicates that the GRB central engine remains activefor some time after the prompt emission is over, and thensuddenly shuts down. Therefore, the internal plateaus aredifficult to explain by using the scenario of BH central en- c (cid:13) Du et al. gines. It is wildly believed that supramassive strongly mag-netized NSs (also called magnetars), which are the centralengines of these GRBs, need to be invoked (Fan & Xu 2006;Gao & Fan 2006). The spin down radiation of the supra-massive NS powers the X-ray internal plateau. The tran-sition from the supramassive NS to the BH through thegravitational collapse after losing rotation energy naturallyaccounts for the steep decay.Usually, the magnetar model is incompatible with theBZ mechanism . Therefore, if it is true that the magne-tar is the central engine of a GRB, we may calculate theNAE by using the observation data of this GRB. In thispaper, we find that GRB 070110 is a potential candidatewhich can be used to address this interesting question. InSection 2, we present the properties of GRB 070110. Wethen calculate the NAE η ν ¯ ν of this NS-disk system in Sec-tion 3. We compare the NAE of the NS-disc and BH-discsystems in Section 4. We interpret the results of the com-parison in Section 5. Finally, we give a summary in Section6. Throughout this paper, a concordance cosmology with pa-rameters H = 70kms − Mpc − , Ω M = 0 .
30 and Ω Λ = 0 . The
Swift
Burst Alert Telescope (BAT) detected GRB070110 on 2007 January 10 (Troja et al. 2007). Its du-ration time T (15 keV ,
150 keV) is ∼
89s and redshift z is 2 . ,
150 keV) is ∼ . × − erg · cm − . Then one can obtain the isotropicprompt emission energy E γ, iso ≃ . × erg (Du et al.2016).In Figure 1, it is clear that a near flat plateau is fol-lowed by a steep decay (red line). As mentioned in the intro-duction, this internal plateau implies that a hyperaccretingstrong magnetized NS central engine is required. Besides, inthe X-ray afterglow of GRB 070110, a bump correspondingto the fall-back BH accretion after the central supramassiveNS collapsing to the BH was proposed by Chen et al. (2017).For these two reasons, we believe that the central object ofGRB 070110 is a supramassive magnetar (hereafter Mag07).The break time t b (1 + z ) of the internal plateau is ∼ . × s in observer frame. After considering the Galacticextinction, the flux of X-ray plateau in 0 .
310 keV is F x , pla ∼ . × − erg cm − s − . So the isotropic energy of theX-ray plateau in source frame is E X , iso , pla = 4 πD F x , pla t b ≃ . × erg , (1)where D L is the the luminosity distance. Without jet breakfeature, the jet opening angle only can be constrained as θ j > . ◦ according to the last observed point ( t j ∼
25 d) ofthe X-ray afterglow (Du et al. 2016). Considering the cor-rection of the jet opening angle, the total energy of prompt Some authors also point out that the NS with a stiff equationof state (EoS), which has an ergosphere, can power jets via theBZ mechanism (Ruiz et al. 2012). But since the EoS may be soft(Margalit & Metzger 2017), and plasma is coupled to magneticfield in NSs, we believe that this scenario need to be further stud-ied. emission is E γ = E γ, iso (1 − cos θ j ) > . × erg . (2)In principle, following the equations and method of Yost etal. (2003), we can obtain the total kinetic energy of GRB jets E K , jet . However, when this method is used to fit the data,some of the parameters are degenerate, we should choose aset of seemingly reasonable parameter values by hand. Notethat 0 . − . .
1, since the NAE is inversely proportionalto the prompt emission efficiency. So the kinetic energy ofthe jet is E K , jet = E γ / . . × erg , (3)and the total jet power iis L jet = E K , jet (1 + z ) /T = 9 . × erg · s − . (4)Based on the above results, we discuss how to constrainthe properties of Mag07. Unlike the prompt emission, theenergy injection of a magnetar is approximately isotropic.So the spin-down luminosity L sd can be expressed as η X , pla L sd = E X , iso , pla /t b ≃ . × erg · s − , (5)where η X , pla is the X-ray radiation efficiency of the spin-down power.The collapse of a magnetar into a black hole shouldoccur when a considerable amount of rotation energy is lost.So the spin-down timescale τ should be close to the breaktime t b . We assume τ = t b , and we have2 π IP = L sd t b , (6)where I and P are the moment of inertia and rotation pe-riod of Mag07, respectively. Theoretically, when the massof Mag07 M NS equals the maximum mass M max that thestar can support, the collapse will occur. The critical mass M max depends on the equation of state (EoS) and rota-tional period P of NSs. According to the observation ofthe NS binary merger (GW170817/GRB 170817A/kilonovaAT2017gfo), the upper limit on rest mass of NSs is con-strained as M res , max ≤ . ⊙ (Margalit & Metzger 2017) . Here, we adopt the parameters in the EoS APR4 (Readet al. 2009) that M res , max = 2 . ⊙ , I = 2 . × g · cm and star radius R = 11 . L sd = 8 π B R c P , (7)where B eff is the effective dipole magnetic field strength ofMag07, and c is the speed of light.In combination with equations (5) and (6), one has P = (cid:18) η X , pla . × erg · s − · π It b (cid:19) / . (8) A stiff EoS still has the possibility of existence (Yu, Liu, & Dai2018). c (cid:13) , 000–000 ffect of magnetic field on NAE -14 -13 -12 -11 -10 -9 -8 XRT X -r a y F l ux ( e r g c m - s - ) Time since BAT trigger (s) ~t -9 Figure 1.
The X-ray (black squares) lightcurve of GRB 070110.The solid line is the empirical smooth broken power law functionfitting.
Combinating equations (5), (6), and (7) gives B eff = (cid:18) η X , pla . × erg · s − · c I R t (cid:19) / . (9)The dependence of period P and magnetic field B eff on theradiation efficiency η X , pla is also shown in Figure 2. As wecan see in the upper panel of Figure 2, in addition to hav-ing to be less than 1, since there is a breakup spin period0 .
96 ms (dash line) for NSs (Lattimer & Prakash 2004), η X , pla has a lower limit 0 .
01. In the lower panel of Figure2, corresponding to η X , pla ∈ (0 . , B eff is(6 . × Gs , . × Gs).
Observations show that the mass distribution of the NSs inthe Milky way is usually very homogeneously in the rangeof 1 . − . ⊙ . The average mass of these NSs is closeto 1 . ⊙ (Zhang et al. 2011). So we assume the mass ofthe protomagnetar of GRB 070110 is M pro = 1 . ⊙ . Af-ter accretion, the magnetar mass increases from 1 . ⊙ to M NS = M res , max +∆ m , where ∆ m is the mass correction af-ter considering the centrifugal force. The gravitational forceof the extra mass should be balanced by the centrifugal force,so we have G ∆ mR = 4 πRP , (10)where G is Newtonian gravitational constant. For P > .
96 ms, there is ∆ m < . ⊙ . This result is also con-sistent with the conclusion of Breu & Rezzolla (2016) thatthe maximum critical mass of a uniformly rotating NScan increase at most by 20 percent of the upper limit onrest mass. Accordingly, the total mass of accretion disk is M dis = M NS − M pro < . ⊙ , and the accretion rate is˙ M = M dis /T ≃ .
04 M ⊙ · s − . The neutrino annihilation A more accurate method can be referred to Lyford, Baumgarte,& Shapiro (2003). P ( m s ) X pla P- X pla
P=0.96 ms B e ff ( G s ) X pla B eff - X pla
X pla =0.01
Figure 2.
The dependence of period P and magnetic field B eff on the radiation efficiency η X , pla (solid lines). In the upper panel,there is a breakup spin period (dash line) for NSs (Lattimer &Prakash 2004) that the lower limit of the spin period is 0 .
96 ms.Therefore, the radiation efficiency η X , pla has a lower limit 0 . η X , pla is in (0 . , B eff is (6 . × Gs , . × Gs). efficiency η ν ¯ ν can be expressed as η ν ¯ ν = L jet T (1 + z ) M dis c ≥ . × − . (11) To see what η ν ¯ ν ≥ . × − means, we compare this valueto that of the BH-disk system. In order to exclude unnec-essary interferences, the angular momentum, mass and ac-cretion rate of the BH are the same as that of Mag07. Since P > .
96 ms, the dimensionless spin of the BH is a ∗ ≃ πIcGP M , max < . . (12)As mentioned in the introduction, the neutrino annihi-lation luminosity under BH-disk system is wildly discussed.Some analytic results are shown as follows.(i) By fitting the results of Popham et al. (1998), Fryeret al. (1999) obtain an approximate formula, i.e.log L ν ¯ ν (erg · s − ) ≃ . . a ∗ + 4 .
89 log ˙ m c (cid:13) , 000–000 Du et al. < . , (13)where ˙ m = ˙ M/ (1 M ⊙ · s − ) is the dimensionless accretionrate. The second line of equation (11) uses a ∗ < .
03 and˙ m = 0 .
04, which is the same below.(ii) Similarly, by fitting results in Xue et al. (2013),there is (Liu, Gu, & Zhang 2017)log L ν ¯ ν (erg · s − ) ≃ . . a ∗ + 2 .
17 log ˙ m< . . (14)(iii) When taking the effect of BH mass into considera-tion, the annihilation luminosity is (Liu et al. 2016)log L ν ¯ ν (erg · s − ) ≃ .
98 + 3 . a ∗ − .
55 log m NS +5 . m < . , (15)where m NS = M NS / M ⊙ .(iv) Lei et al. (2017) also develop a formula to calculatethe luminosity of neutrino annihilation, i.e. L ν ¯ ν = L ν ¯ ν, ign "(cid:18) ˙ m ˙ m ign (cid:19) − α ν ¯ ν + (cid:18) ˙ m ˙ m ign (cid:19) − β ν ¯ ν − × " (cid:18) ˙ m ˙ m ign (cid:19) β ν ¯ ν − γ ν ¯ ν − , (16)where ( L ν ¯ ν, ign = 10 . . a ∗ (cid:0) m NS (cid:1) log ( ˙ m/ ˙ m ign ) − . erg s − α ν ¯ ν = 4 . , β ν ¯ ν = 2 . , γ ν ¯ ν = 0 . m ign = 0 . − . a ∗ , ˙ m trap = 6 . − . a ∗ , (17)and ˙ m ign and ˙ m trap are the dimensionless igniting and trap-ping accretion rates, respectively. According to equations(16) and (17), one can obtain L ν ¯ ν < . × erg · s − under a ∗ < .
03 and ˙ m = 0 . L ν ¯ ν ∼ erg · s − . If one believes that this value is right, thenit means the NAE on the non-magnetized accretion discaround the BH is at least two orders of magnitude lowerthan that of the magnetar case under ˙ M ≃ .
04 M ⊙ · s − . It is certain that the differences between the NS-disc systemand the BH-disc system lead to the different NAEs. In bothsystems, mass falls by the way of neutrino dominated ac-cretion flow (Narayan, Paczynski, & Piran 1992). The maindifferences should be: (1) NSs have solid surfaces, but BHsare not; (2) Mag07 have strong magnetic field, so the ac-cretion disc is magnetized, but situations under the BH-discsystems considered in section 4 are just the opposite.For the first difference, two extra channels that the an-nihilation of neutrinos emitted from the NS surface and an-nihilation between the neutrinos emitted from the accretiondisc and the NS surface will make contributions to the totalNAE. This enhancement depends on the accretion geometry(Zhang & Dai 2010). Considering the stability of the accre-tion disc, magnetic pressure should not be greater than theram pressure of the accretion flow (also can see Figure 11of Zhang & Dai (2010)). So here, the effect of funnel accre-tion flow is ignored. These two extra channels may increase the NAE by one order of magnitude (Zhang & Dai 2009).The problem of excessive NAE under the NS-disc system isunsolved.For the second difference, the energy deposition through ν → ν + e + + e − and ¯ ν → ¯ ν + e + + e − in the strong magneticfield will also contribute to the jet power. But, since themagnetic field of Mag07 is ∼ Gs (see Figure 2), and themass accretion rate is ˙ M = 0 .
04 M ⊙ · s − , this enhancementshould be smaller than one order of magnitude (see Figure 7of Zalamea & Beloborodov (2011)). In this sense, the NAEwe have calculated previously should be called “equivalentneutrino annihilation efficiency”. But this effect is still notenough to improve the NAE.It is necessary to re-examine the energy transfer in ac-cretion process. In general, to convert gravitational poten-tial energy into thermal energy in an accretion disc, a largeviscosity coefficient is needed. This large viscosity may beinduced by the magneto-rotational instability of small-scalemagnetic field. However, the viscosity will lead to the de-crease of the surface density of the accretion disc. That’s tosay, the magneto-rotational instability will, in turn, inhibitthe generation of thermal energy. Therefore, the thermal lu-minosity is not sensitive to the viscosity coefficient (Kato,Fukue, & Mineshige 1998). When there is a large-scale mag-netic field, this situation may be relieved. According to theflux conservation, the magnetic field on the accretion discsatisfies B ∝ r − , where r is the radius of the disk. Alarge-scale magnetic field whose magnetic-pressure gradienttoward the outside of the accretion disc may prevent thefalling motion of disc matter, as will as the reduction of thesurface density of the disc. So, the thermalization of the ac-cretion disc will become stronger under this situation. Leiet al. (2009) show a similar result that the strong magneticfield will increase the density of the accretion disc and theNAE from numerical aspect.Therefore, we get the following conclusion: the large-scale magnetic field in the hyperaccreting accretion disccan effectively enhance the neutrino annihilation luminos-ity. This enhancement should be about several tens timeslarger than that of non-magnetized accretion disc, such thatwhen the first two enhancements are also taken into consid-eration, the higher NAE is acceptable. The range of mass accretion rate in GRBs is believed to be0 .
01 M ⊙ · s − −
10 M ⊙ · s − . In this paper, we only calculatethe NAE under the accretion rate ˙ M ≃ .
04 M ⊙ · s − througha case study: GRB 070110. In order to get a complete η ν ¯ ν − ˙ m − B eff correlation, more samples like GRB 070110are required. On the observation, the internal plateaus areobserved both in the afterglows of short and long GRBs. Itis possible to achieve this goal in the foreseeable future. Inour case, η ν ¯ ν > . × − is two order of magnitude largerthan that of BH central engine. The different accretion ge-ometry between the NS-disc system and the BH-disc systemmay be not enough to explain the high NAE of GRB 070110.Another effect that the large-scale magnetic field in the ac-cretion disc will obviously improve the NAE can supplementthe deficiency of the former.It is worth reminding that, usually, to power a jet c (cid:13) , 000–000 ffect of magnetic field on NAE through the BZ mechanism, there must be a strong large-scale magnetic field in the accretion disc. Therefore, ourresult indicates that BZ mechanism and neutrino annihi-lation process are both important under the hyperaccret-ing BH-disc system. The structure of the jet from this BH-disc system may be very different from that of the jet onlylaunched by the BZ mechanism or only produced by the neu-trino annihilation process. This structural difference may bevery important for explaining some special GRBs, e.g., GRB170817A (Abbott et al. 2017). We thank the anonymous referee for his/her useful com-ments. We acknowledge the use of the public data from theSwift data archive, and the UK Swift Science Data Cen-ter. This work is supported by the National Natural Sci-ence Foundation of China (Grant No. 11275247, and GrantNo. 11335012) and a 985 grant at Sun Yat-Sen Univer-sity. F. K. Peng acknowledges support from the DoctoralStarting up Foundation of Guizhou Normal University 2017(GZNUD[2017] 33).
REFERENCES
Abbott B. P., et al., 2017, ApJ, 848, L13Blandford R. D., Znajek R. L., 1977, MNRAS, 179, 433Breu C., Rezzolla L., 2016, MNRAS, 459, 646Chen W., Xie W., Lei W.-H., Zou Y.-C., L¨u H.-J., Liang E.-W.,Gao H., Wang D.-X., 2017, ApJ, 849, 119Dai Z. G., Lu T., 1998a, A&A, 333, L87Dai Z. G., Lu T., 1998b, PhRvL, 81, 4301Du S., L¨u H.-J., Zhong S.-Q., Liang E.-W., 2016, MNRAS, 462,2990Eichler D., Livio M., Piran T., Schramm D. N., 1989, Nature,340, 126Fan Y.-Z., Xu D., 2006, MNRAS, 372, L19Fryer C. L., Woosley S. E., Herant M., Davies M. B., 1999, ApJ,520, 650Gao W.-H., Fan Y.-Z., 2006, ChJAA, 6, 513Gu W.-M., Liu T., Lu J.-F., 2006, ApJ, 643, L87Harikae S., Kotake K., Takiwaki T., 2010, ApJ, 713, 304Just O., Obergaulinger M., Janka H.-T., Bauswein A., SchwarzN., 2016, ApJ, 816, L30Kato S., Fukue J., Mineshige S., 1998, bhad.confKumar P., Zhang B., 2015, PhR, 561, 1Lattimer J. M., Prakash M., 2004, Sci, 304, 536Lei W.-H., Zhang B., Wu X.-F., Liang E.-W., 2017, ApJ, 849, 47Lei W. H., Wang D. X., Zhang L., Gan Z. M., Zou Y. C., Xie Y.,2009, ApJ, 700, 1970Liu T., Gu W.-M., Zhang B., 2017, NewAR, 79, 1Liu T., Xue L., Zhao X.-H., Zhang F.-W., Zhang B., 2016, ApJ,821, 132Lyford N. D., Baumgarte T. W., Shapiro S. L., 2003, ApJ, 583,410Margalit B., Metzger B. D., 2017, ApJ, 850, L19Narayan R., Paczynski B., Piran T., 1992, ApJ, 395, L83Popham R., Woosley S. E., Fryer C., 1999, ApJ, 518, 356Read J. S., Lackey B. D., Owen B. J., Friedman J. L., 2009,PhRvD, 79, 124032Ruiz M., Palenzuela C., Galeazzi F., Bona C., 2012, MNRAS,423, 1300Troja E., et al., 2007, ApJ, 665, 599Usov V. V., 1992, Nature, 357, 472 Woosley S. E., 1993, ApJ, 405, 273Xie Y., Huang C.-Y., Lei W.-H., 2007, ChJAA, 7, 685Xie Y., Huang Z.-Y., Jia X.-F., Fan S.-J., Liu F.-F., 2009, MN-RAS, 398, 583Xue L., Liu T., Gu W.-M., Lu J.-F., 2013, ApJS, 207, 23Yost S. A., Harrison F. A., Sari R., Frail D. A., 2003, ApJ, 597,459Yu Y.-W., Liu L.-D., Dai Z.-G., 2018, ApJ, 861, 114Zalamea I., Beloborodov A. M., 2011, MNRAS, 410, 2302Zhang B., M´esz´aros P., 2001, ApJ, 552, L35Zhang C. M., et al., 2011, A&A, 527, A83Zhang D., Dai Z. G., 2010, ApJ, 718, 841Zhang D., Dai Z. G., 2009, ApJ, 703, 461c (cid:13)000