arXiv:1712.02556v2 [astro-ph.HE] 24 Jan 2018 rS-esln Rs(e.g., GRBs long SN-less or MNRAS sepce otigrsotGB n rdc Li-Paczy´nski produce and GRBs ( macronovae short trigger to expected ( is limit than larger is Chandrasekhar mass total their the may if supernovae mergers Ia WD Type produce Double phenomena. interesting different low quite are su- redshifts any their with ( du- though associated long even not their signature are of pernova they because However, GRBs time. long ration as classified be should 2006 ultra-compact ⋆ the explain to used been have systems WD 2014 ( favoured be in- may model binary the compact so Li-Paczy´nskivolved prob- macronova, which a 060614, from GRB arose of ably afterglow the in discovered was (BH) hole black white intermediate-mass massive ( an by a star and a of (NS) disruption star ( neutron includ- (WD) a 060614, dwarf of GRB merger explain the to ( ing proposed GRB were short models a luminosity eral like peak more and lag it temporal makes the example, an as 060614 2018 October 6 iZ og e-i Gu Wei-Min Dong, Yi-Ze for Model Association Supernova Binary without Compact Bursts Dwarf Gamma-ray White Long - Hole Black A rgnt rmteclas fmsiesasadaeaccom- are ( and stars supernovae massive with of (GRBs) panied collapse bursts the gamma-ray long from that originate believed generally is It INTRODUCTION 1 c eateto srnm,Xae nvriy imn Fujia Xiamen, University, Xiamen Astronomy, of Department u un,&Zag2008 Zhang & Huang, Lu, z -al [email protected] E-mail: ∼ < 00TeAuthors The 0000 opc iaysseshv enwdl ple to applied widely been have systems binary Compact .Temre fbnr S rta faB n nNS an and BH a of that or NS, binary of merger The ). 111005A and 060614 060505, GRBs Traditionally, ). 0 . ) uhGB r lonmda ogsotGRBs long-short as named also are GRBs Such 1). 000 , i&Paczy´nski 1998 & Li 1 ig lsn ais2007 Davies & Olsson, King, – 5 00)Pern coe 08Cmie sn NA L MNRAS using Compiled 2018 October 6 Preprint (0000) ose 1993 Woosley .Rcnl,ana-nrrdbump near-infrared a Recently, ). age l 2017 al. et Wang az anci Nelemans & Mannucci, Maoz, nacnatbnr ytmcnitn faselrms lc oeand evolution. hole binary words: black compact Key stellar-mass in a gap v important of extremely an consisting and fills system t which unstable path binary contact new the a a association, propose in supernova We origin. without 1110 star GRBs and non-massive classification a 060614 usual implies 060505, to supernova according GRBs convincingly events as these such classify m GRB association, of long supernova collapse low-redshift the some of by However, produced supernovae. the be with to in associated expected phenomena are violent GRBs and long luminous ditionally, are (GRBs) bursts Gamma-ray ABSTRACT us:gnrl–wiedwarfs white – general burst: ; erl ta.2006 al. et Gehrels eze ta.2010 al. et Metzger ; ⋆ ogLu n ufn Wang Junfeng and Liu, Tong , ose Bloom & Woosley age l 2015 al. et Yang ,adtetidal the and ), .Tkn GRB Taking ). crto,aceindss–sas lc oe iais ls gam close binaries: – holes black stars: – discs accretion accretion, .Sev- ). .NS- ). ). 605 China 361005, n e ( ses uhasse,otoscnpa nipratrole. important an play can outflows system, a such 2015 on studies those to system(e.g. similar momen- WD is angular double which the orbital away, the in carried the outflows is and strong tum system exist BH-WD there typical that show calculations Figure systems in binary presented compact is and the together of pieced picture be full may evolution- a this scenario, With long ary association. the explain supernova can without system GRBs BH-WD a in accretion unstable ( 121102) (FRB burst radio fast repeating icoie l 2017 al. et Fiacconi mass WD typical n bevtoshv hw ht uflw r signif- ( are case outflows super-Eddington fac- that, the a shown in have by icant observations super-Eddington, and extremely of is tor accretion the LIGO by detected was BHs originating double emission of wave ( merger gravitational the the from addition, In -a iais(CB)( (UCXBs) binaries X-ray rt h pnaglrmmnu fteaceigstar accreting the disc of accretion momentum the angular of con- spin momentum be ( the can angular to re- momentum the or also angular to degener- orbital may verted double of en- the mechanisms For shrink systems, momentum. away other the angular ate carries orbital addition, to the leads It In move and orbit. evolution. momentum the angular binary and the ergy during role uemn&Saa 1983 Shaham & Ruderman 2016a al. et Abbott h rvttoa aeeiso ly significant a plays emission wave gravitational The ; egrn ae,&Leet1992 Liebert & Saffer, Bergeron, it,Mdltn ain2016 Fabian & Middleton, Pinto, > 10 3 o tla-asB.Rcn simulations Recent BH. stellar-mass a for ; M , in,Soe ai 2017 Davis & Stone, Jiang, b WD .I hswr,w rps htthe that propose we work, this In ). eeas&Jne 2010 Jonker & Nelemans 0 = ; u aznk 1984 Paczynski & Hut a ebn 1999 Webbink & Han . 6 M ⊙ ,adtelc fthe of lack the and s, ; scoe o analy- for chosen is ¸ dwk Narayan S¸adowski & aen detection no have s ; elre l 2007 al. et Kepler 5.I shr to hard is It 05A. atne l 2016 al. et Walton A oetaccretion iolent T siesasand stars assive rdc long produce o ue l 2016 al. et Gu ht dwarf, white a E nvre Tra- universe. tl l v3.0 file style X .Tu,in Thus, ). n the and ) 1 ma-ray . .I a If ). .Our ). ), ). ; 2 Dong et al.
Figure 1. A full picture of compact binary systems. The figure shows different compact binary systems and potential corresponding observations.
We use an analytic method to investigate the dynamical respectively, a is the orbital separation, and M = M1 + M2 stability of the BH-WD system during mass transfer. For represents the total mass of two stars. A relatively accurate a typical WD donor, when the mass transfer occurs, the formula for the radius of WD is provided by Eggleton and accretion rate can be extremely super-Eddington and the is quoted by Verbunt & Rappaport (1988): accreted materials will be ejected from the system. Such 1 RWD M2 − 2 M2 2 2 strong outflows can carry away significant orbital angular = 0.0114 ( ) 3 − ( ) 3 momentum and trigger the unstable accretion of the system. R⊙ MCh MCh (2) − 2 The released energy from the accretion is sufficient to power 3 M2 − 2 M2 −1 GRBs. × 1 + 3.5( ) 3 +( ) , Mp Mp
where MCh = 1.44M⊙ and Mp = 0.00057M⊙ . This relation 2 MODEL AND ANALYSES offers a single relation that can be used in any mass range of WD (Marsh, Nelemans, & Steeghs 2004). For simplicity, In our model, the system consists of a stellar-mass BH and a we adopt a form to describe the Roche-lobe radius of the WD donor. In such a system, the mass transfer occurs when secondary M2 (WD)(Paczy´nski 1971): the WD fills the Roche lobe. If this process is dynamically R2 M2 1 unstable, the WD will be disrupted rapidly. The instabil- = 0.462( ) 3 . (3) a M ity of the system greatly depends on the relative expansion rate of the WD and the Roche lobe. Following the mass For the circular orbit, the variation of a due to the gravita- transfer, the radius of the WD will expand, and the orbital tional wave emission can be written as (Peters 1964) 3 separation also tends to increase if the orbital angular mo- da 64 G M1M2M = − . (4) mentum is conserved. If the WD expands more rapidly than dt 5 c5a3 the Roche lobe, i.e., the timescale for the expansion of the Based on Equations (1)-(4), we can derive the mass accretion Roche lobe is longer than that of the WD, the mass transfer rate for the system, will be dynamically unstable. In this scenario, the orbital 3 2 angular momentum loss can restrain the expansion of the 64G M1M2 M M˙ 2 = , Roche lobe, and therefore will have essential influence on 5 4 − 5 δ 5c a (2q 3 + 3 ) the binary evolution. We consider a compact binary system which contains M − 2 M 2 M M 1 2 3 2 3 p 7 p − ( M ) +( M ) 2 M [ ( M ) 3 + 1] a WD donor. The system is assumed to be tidally locked, δ = Ch Ch − 2 3 2 , M − 2 M 2 M − 2 M ( 2 ) 3 − ( 2 ) 3 1 + 3.5( 2 ) 3 +( 2 )−1 and the orbit is circular. The orbital angular momentum J MCh MCh Mp Mp of the binary system is given by where q = M2/M1. Ga 1 J = M1M2( ) 2 , (1) Then, we define a dimensionless parameterm ˙ = 2 M −M˙ 2/M˙ Edd, where M˙ Edd = LEdd/ηc is the Eddington ac- where M1 and M2 are the mass of the accretor and the WD, cretion rate, where LEdd = 4πGM1c/κes. We choose η = 0.1.
MNRAS 000, 1–5 (0000) BH-WD model for GRBs without SN association 3
108 0.3
6 10 0.2
4 10 0.1
102 0
100 -0.1
10-2 -0.2
10-4 -0.3 0 2 4 6 8 10 12 14 16 18 20 1 5 10 15 30 50
Figure 2. Accretion rates for the BH-WD system with mass Figure 3. Criterion for an unstable mass transfer. The red dashed transfer. The figure shows that the accretion rate is extremely and solid lines represent the results of King, Olsson, & Davies 4 super-Eddington by a factor larger than 10 for a 0.6M⊙ WD. (2007) and ours, respectively, where outflows are not considered. The critical mass of WD between the sub-Eddington and super- The green and blue lines correspond to our results including the Eddington cases is around 0.1M⊙. effects of outflows. The criterion for an unstable mass transfer is F > 0, as shown by Equation (9).
Thus, 2 2 angular velocity of the disc, b1 is the distance between the 4G M2 Mκes m˙ = , accreting star and the L1 point, and λ is a parameter prob- 25πc4a4(5/6 − q − δ/6) ably in the range 0 6 λ< 1. The parameter λ characterises 2 −1 where the opacity κes = 0.34 cm g . For M2 = the loss of orbital angular momentum through outflows. The 0.01M⊙, 0.1M⊙, 0.6M⊙, the variation of the stable accre- expression of b1 takes the form: tion rate with M1 is plotted in Figure 2. There exists dra- b1 = 0.5 − 0.227 log q . (8) matic increase for M1 < 2M⊙, which indicates that, for a relatively large mass ratio, the mass transfer is unstable for The instable mass transfer will occur if the radius the system consisting of a WD and an NS/WD. If M1 is of WD expands more rapidly than the Roche lobe, i.e., fixed,m ˙ is positively related to M2. The critical mass of ˙ ˙ WD between sub-Eddington and super-Eddington is around RWD/RWD > R2/R2. Based on Equations (1)-(3) and (5)- (8), we can derive the instability criterion for the mass trans- 0.1 M⊙. For M2 = 0.6M⊙, the accretion rate is extremely super-Eddington, so the outflows ought to be taken into con- fer: 2 sideration. Here we only considered the effects of gravita- b1 − a1 F(q,f,λ)= λ(1 + q) tional wave emission. Obviously, outflows can be another a (9) mechanism to take away the angular momentum, so the 5 − δ fq mass accretion rate should be even higher, which can re- − + q(1 − f)+ > 0 , 6 3(1 + q) sult in stronger outflows. We use a parameter f to describe the scale of mass loss in the mass transfer process due to where a1 is the distance between the BH and the mass centre outflows: of the system. We adopt f = 0.9 and f = 0.99 for calcula- tions, which means that 90% or 99% materials transferred dM1 + dM2 = fdM2 (0 6 f < 1) , (5) from the WD are lost the system due to strong outflows. The where dM2 is negative. In general, the stream of matter flow- material stream ejected from the system may form a com- ing from the inner Lagrange point can form a disc around mon envelope around the binary system (Han & Webbink the accreting star. If the disc is circular and nonviscous, its 1999). We assume λ > f since the orbital angular momen- radius can be written as (Verbunt & Rappaport 1988) tum can be carried away by the outflows. Moreover, the out- flows may possess larger angular momentum per unit mass Rh 2 = 0.0883 − 0.04858 log q + 0.11489 log q than the inflows thus can escape from the system. a (6) +0.020475 log3 q , −3 where 10
MNRAS 000, 1–5 (0000) 4 Dong et al. and λ = 0, is plotted by the red solid line, and the critical mass ratio is q = 0.52. Such a result is similar to previous studies (Ruderman & Shaham 1983; Hut & Paczynski 1984; Verbunt & Rappaport 1988). In this scenario, only when the accretor mass is smaller than about 1.15 M⊙, the accre- tion can be unstable, which means that the accretor cannot be a BH. The results of King, Olsson, & Davies (2007) are slightly different from ours (red dashed line), since an elabo- rated formula for the radius of WD is adopted in the present work.
The strong outflows during mass transfer can carry away angular momentum, which influences the stability of the system. In reality, the mass of the components and the separation of the binary also change due to the mass transfer during the evolution, so the system may not al- ways be unstable. We plot the critical condition (brown Figure 4. Variation of the critical BH mass with the WD mass line) in Figure 4, where f = 0.9 and λ = 0.99 are adopted for f = 0.9 and λ = 0.99. For the parameters beyond the critical for the super-Eddington accretion case. It should be noted BH mass (brown line), the mass transfer is dynamically unstable. that once the WD fills its Roche lobe within the unsta- The two blue dashed lines show the evolutionary track (from the ble region, the violent accretion will continue as long as filled circle to the arrow). The space is divided into two regions, the evolutionary track is still located in the unstable re- sub-Eddington (left) and super-Eddington (right), by the black gion. In Figure 4, the brown line has a maximum value solid line. The nine UCXBs are well located in the sub-Eddington MBH = 15.7M⊙ at MWD = 0.35M⊙. Thus, for a typical WD and stable region, where eight UCXBs are NS candidates and the with MWD = 0.6M⊙, the system can maintain the unstable other one is a BH candidate. mass transfer if the BH is sufficiently large MBH > 15.7M⊙, i.e., the accretion process will not stop until the accretion can be expressed as: becomes sub-Eddington, corresponding to a low-mass WD 2 around 0.1M⊙ (the nearly vertical gray line in Figure 4). Etot = Ma(1 − f)ηc , (10) The blue dashed line shows the mass changes of compo- where Ma is the material disrupted by BH. For MWD = nents along with the mass transfer. If the BH mass is lower, less material of the WD can be accreted by the BH. For ex- 0.6M⊙, MBH = 15.7M⊙, f = 0.9, and λ = 0.99, Ma = 0.25M⊙. We adopt η = 0.1, so the total energy ample, for MWD = 0.9M⊙ and MBH = 10M⊙, the critical 52 Etot is around 10 erg. For a typical half-opening angle mass is MWD = 0.63M⊙ for MBH = 10M⊙, and therefore ◦ ∆θ = 10 for GRBs, the isotropic energy is Etot/(1 − only around 0.27M⊙ will be accreted under the unstable cos θ) ≈ 6.6 × 1053erg, which is sufficient to power most phase. If we consider a BH with larger mass, such as 20 M⊙, GRBs. Obviously, it is impossible to detect a supernova for a 0.6 M⊙ WD, the accreted material is around 0.5 M⊙. Recently, the first BH UCXB candidate X9 has been dis- in such a event. The current merger rate for BH-WD × −6 −1 covered in globular cluster 47 Tucanae (Bahramian et al. binary is about 1.9 10 yr in the galactic disc (Nelemans, Yungelson, & Portegies Zwart 2001), with typ- 2017; Miller-Jones et al. 2015), and its highest accretion rate −10 −1 ical BH mass around 5 ∼ 7 M⊙, which is slightly less than is estimated as a few times of 10 M⊙ yr , which sug- gests that the BH mass can reach a few dozens of solar the typical required BH mass in our model, so the event rate for our system is lower. For a BH-WD binary system, mass. In this system, however, the WD mass is only around the initial system should contain two main-sequence stars, 0.02M⊙, and therefore such a system is under the stable type of mass transfer, as shown in Figure 4. Nine UCXB a large mass primary star (> 25M⊙) and a secondary star (1 ∼ 8M⊙). The primary star will evolve into a 3 ∼ 15 M⊙ candidates are plotted, including eight NS UCXB candi- dates, 4U 1626−67 (Takagi et al. 2017), 4U 1850−087, 4U BH via a supernova explosion (Woosley & Weaver 1995; 0513−40, M15 X−2 (Prodan & Murray 2015), XTE J1751- Zhang, Woosley, & Heger 2008), whereas the secondary star still stay at the main sequence. The system will experience 305 (Andersson, Jones, & Ho 2014; Gierli´nski & Poutanen 2005), XTE J1807−294 (Leahy, Morsink, & Chou 2011), a common envelope phase when the secondary star evolves off the main sequence, and finally become a WD. The initial 4U 1820−30 (Guver¨ et al. 2010) and 4U 1543−624 separation of the compact system should be comparable to (Wang & Chakrabarty 2004), and a BH UCXB candidate, 47 Tuc X9. It is shown that all the UCXB candidates are well the separation of the X-ray binary, so we can roughly es- located in the left “Stable” region. Such a location is quite timate the time required for the binary system to evolve into contact according to equation (4). For an X-ray bi- reasonable since the unstable mass transfer corresponds to extremely high accretion rates and therefore short timescale nary, the secondary star fills its Roche lobe, so the sepa- for existence, whereas the left stable region corresponds to ration of the X-ray binary can be estimated by Equation (3), while the radius of the main sequence star is given by sub-Eddington rates and therefore long timescale. 0.724 Demircan & Kahraman (1991): Rm = 1.01Mm (0.1M⊙ < 52 The isotropic energy for a GRB is around 10 erg, and M < 18.1M⊙), where Rm and Mm are the radius and the can be up to 1054 erg. It is difficult to calculate the accurate mass of the main sequence star, respectively. For a system energy released from the unstable accretion of the BH-WD contains a 5-15M⊙ BH and 1-8M⊙ main sequence star, af- system. Thus, a rough estimate of the total released energy ter the main sequence star become a WD, the time required
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ACKNOWLEDGEMENTS
We thank Taotao Fang for beneficial discussion, and thank the referee for helpful suggestions that improved the manuscript. This work was supported by the National Ba- sic Research Program of China (973 Program) under grants 2014CB845800, and the National Natural Science Founda- tion of China under grants 11573023, 11522323, 11473022, 11473021, and 11333004.
MNRAS 000, 1–5 (0000)