# A black hole–white dwarf compact binary model for long gamma-ray bursts without supernova association

A black hole–white dwarf compact binary model for long gamma-ray bursts without supernova... Abstract Gamma-ray bursts (GRBs) are luminous and violent phenomena in the Universe. Traditionally, long GRBs are expected to be produced by the collapse of massive stars and associated with supernovae. However, some low-redshift long GRBs have no detection of supernova association, such as GRBs 060505, 060614, and 111005A. It is hard to classify these events convincingly according to usual classifications, and the lack of the supernova implies a non-massive star origin. We propose a new path to produce long GRBs without supernova association, the unstable and extremely violent accretion in a contact binary system consisting of a stellar-mass black hole and a white dwarf, which fills an important gap in compact binary evolution. accretion, accretion discs, gamma-ray burst: general, white dwarfs 1 INTRODUCTION It is generally believed that long gamma-ray bursts (GRBs) originate from the collapse of massive stars and are accompanied with supernovae (Woosley 1993; Woosley & Bloom 2006). Traditionally, GRBs 060505, 060614, and 111005A should be classified as long GRBs because of their long duration time. However, they are not associated with any supernova signature even though their redshifts are quite low (z ≲ 0.1). Such GRBs are also named as long-short GRBs or SN-less long GRBs (e.g. Wang et al. 2017). Taking GRB 060614 as an example, the temporal lag and peak luminosity makes it more like a short GRB (Gehrels et al. 2006). Several models were proposed to explain GRB 060614, including the merger of a neutron star (NS) and a massive white dwarf (WD) (King, Olsson & Davies 2007), and the tidal disruption of a star by an intermediate-mass black hole (BH) (Lu, Huang & Zhang 2008). Recently, a near-infrared bump was discovered in the afterglow of GRB 060614, which probably arose from a Li–Paczyński macronova, so the model involved compact binary may be favoured (Yang et al. 2015). Compact binary systems have been widely applied to different interesting phenomena. Double WD mergers may produce Type Ia supernovae if their total mass is larger than the Chandrasekhar limit (Maoz, Mannucci & Nelemans 2014). The merger of binary NS, or that of a BH and an NS is expected to trigger short GRBs and produce Li–Paczyński macronovae (Li & Paczyński 1998; Metzger et al. 2010). NS–WD systems have been used to explain the ultracompact X-ray binaries (UCXBs) (Nelemans & Jonker 2010) and the repeating fast radio burst (FRB 121102) (Gu et al. 2016). In addition, the gravitational wave emission originating from the merger of double BHs was detected by LIGO (Abbott et al. 2016a,b). In this work, we propose that the unstable accretion in a BH–WD system can explain the long GRBs without supernova association. With this evolutionary scenario, a full picture of the compact binary systems may be pieced together and is presented in Fig. 1. Figure 1. View largeDownload slide A full picture of compact binary systems. The figure shows different compact binary systems and potential corresponding observations. Figure 1. View largeDownload slide A full picture of compact binary systems. The figure shows different compact binary systems and potential corresponding observations. The gravitational wave emission plays a significant role during the binary evolution. It carries away energy and angular momentum and leads to the shrink of the orbit. In addition, other mechanisms may also remove the orbital angular momentum. For double degenerate systems, the orbital angular momentum can be converted to the angular momentum of the accretion disc or to the spin angular momentum of the accreting star (Ruderman & Shaham 1983; Hut & Paczynski 1984). Our calculations show that there exist strong outflows in the typical BH–WD system and the orbital angular momentum is carried away, which is similar to those studies on the double WD system (e.g. Han & Webbink 1999). If a typical WD mass MWD = 0.6 M⊙ is chosen for analyses (Bergeron, Saffer & Liebert 1992; Kepler et al. 2007), the accretion is extremely super-Eddington, by a factor of >103 for a stellar-mass BH. Recent simulations and observations have shown that, outflows are significant in the super-Eddington case (Sa̧dowski & Narayan 2015; Pinto, Middleton & Fabian 2016; Walton et al. 2016; Fiacconi et al. 2017; Jiang, Stone & Davis 2017). Thus, in such a system, outflows can play an important role. We use an analytic method to investigate the dynamical stability of the BH–WD system during mass transfer. For a typical WD donor, when the mass transfer occurs, the accretion rate can be extremely super-Eddington and the accreted materials will be ejected from the system. Such strong outflows can carry away significant orbital angular momentum and trigger the unstable accretion of the system. The released energy from the accretion is sufficient to power GRBs. 2 MODEL AND ANALYSES In our model, the system consists of a stellar-mass BH and a WD donor. In such a system, the mass transfer occurs when the WD fills the Roche lobe. If this process is dynamically unstable, the WD will be disrupted rapidly. The instability of the system greatly depends on the relative expansion rate of the WD and the Roche lobe. Following the mass transfer, the radius of the WD will expand, and the orbital separation also tends to increase if the orbital angular momentum is conserved. If the WD expands more rapidly than the Roche lobe, i.e. the time-scale for the expansion of the Roche lobe is longer than that of the WD, the mass transfer will be dynamically unstable. In this scenario, the orbital angular momentum loss can restrain the expansion of the Roche lobe, and therefore will have essential influence on the binary evolution. We consider a compact binary system which contains a WD donor. The system is assumed to be tidally locked, and the orbit is circular. The orbital angular momentum J of the binary system is given by   \begin{eqnarray} J=M_{1}M_{2} \left(\frac{Ga}{M} \right)^\frac{1}{2} \ , \end{eqnarray} (1)where M1 and M2 are the mass of the accretor and the WD, respectively, a is the orbital separation, and M = M1 + M2 represents the total mass of two stars. A relatively accurate formula for the radius of WD is provided by Eggleton and is quoted by Verbunt & Rappaport (1988):   \begin{eqnarray} \frac{R_{\rm WD}}{R_{{\odot }}}=0.0114\left[ \left(\frac{M_2}{M_{\rm Ch}} \right) ^{-\frac{2}{3}}- \left(\frac{M_2}{M_{\rm Ch}} \right)^\frac{2}{3}\right] ^\frac{1}{2} \end{eqnarray} (2)where MCh = 1.44 M⊙ and Mp = 0.00057 M⊙. This relation offers a single relation that can be used in any mass range of WD (Marsh, Nelemans & Steeghs 2004). For simplicity, we adopt a form to describe the Roche lobe radius of the secondary M2 (WD; Paczyński 1971):   \begin{eqnarray} \frac{R_{2}}{a}=0.462 \left(\frac{M_{2}}{M} \right)^\frac{1}{3} . \end{eqnarray} (3)For the circular orbit, the variation of a due to the gravitational wave emission can be written as (Peters 1964)   \begin{eqnarray} \frac{{\rm d}a}{{\rm d}t} = - \frac{64}{5} \frac{G^{3} M_1 M_2 M}{c^5 a^{3}} . \end{eqnarray} (4)Based on equations (1)– (4), we can derive the mass accretion rate for the system,   \begin{equation*} \dot{M_2}=\frac{64G^3M_1M_2^2M}{5c^5a^4 \left(2q-\frac{5}{3}+\frac{\delta }{3} \right)} \ , \end{equation*}   \begin{equation*} \delta =\frac{ \left(\frac{M_2}{M_{\rm Ch}} \right)^{-\frac{2}{3}} + \left(\frac{M_2}{M_{\rm Ch}}\right)^\frac{2}{3}}{ \left(\frac{M_2}{M_{\rm Ch}} \right)^{-\frac{2}{3}}- \left(\frac{M_2}{M_{\rm Ch}} \right) ^\frac{2}{3}}-\frac{2\frac{M_{\rm p}}{M_2} \left[\frac{7}{3} \left(\frac{M_{\rm p}}{M_2} \right)^{-\frac{1}{3}}+1 \right]}{1+3.5 \left(\frac{M_2}{M_{\rm p}} \right)^{-\frac{2}{3}}+ \left(\frac{M_2}{M_{\rm p}} \right)^{-1}} \ , \end{equation*} where q = M2/M1. Then, we define a dimensionless parameter $$\dot{m}=-\dot{M_2}/ \dot{M}_{\rm Edd}$$, where $$\dot{M}_{\rm Edd}=L_{\rm Edd}/{\eta c^2}$$ is the Eddington accretion rate, where LEdd = 4πGM1c/κes. We choose η = 0.1. Thus,   \begin{equation*} \dot{m} = {\frac{4 G^2 M^2_2 M \kappa_{\rm es}}{25\pi c^4 a^4 \left(5/6-q-\delta/6 \right)}} \,, \end{equation*} where the opacity κes = 0.34 cm2 g−1. For M2 = 0.01, 0.1, 0.6 M⊙, the variation of the stable accretion rate with M1 is plotted in Fig. 2. There exists dramatic increase for M1 < 2 M⊙, which indicates that, for relatively large mass ratio, the mass transfer is unstable for the system consisting of a WD and an NS/WD. If M1 is fixed, $$\dot{m}$$ is positively related to M2. The critical mass of WD between sub-Eddington and super-Eddington is around 0.1 M⊙. For M2 = 0.6 M⊙, the accretion rate is extremely super-Eddington, so the outflows ought to be taken into consideration. Here, we only considered the effects of gravitational wave emission. Obviously, outflows can be another mechanism to take away the angular momentum, so the mass accretion rate should be even higher, which can result in stronger outflows. We use a parameter f to describe the scale of mass loss in the mass transfer process due to outflows:   \begin{eqnarray} {\rm d}M_1 + {\rm d}M_2 = f {\rm d}M_2\ (0 \leqslant f < 1) \ , \end{eqnarray} (5)where dM2 is negative. In general, the stream of matter flowing from the inner Lagrange point can form a disc around the accreting star. If the disc is circular and non-viscous, its radius can be written as (Verbunt & Rappaport 1988)   \begin{eqnarray} \frac{R_{\rm h}}{a} &=& 0.088\,3 - 0.048\,58\log q +0.114\,89\log ^2 q \nonumber\\ &&+\,0.020\,475\log ^3 q, \end{eqnarray} (6)where 10−3 < q < 1. A fraction of the orbital angular momentum of the stream comes from the spin angular momentum of the WD, so this part should be eliminated. Thus, the orbital angular momentum carried away by the outflows can be expressed as   \begin{eqnarray} {\rm d}J = - \lambda [({\rm d} M_{2}\Omega _1R_{\rm h}^2-{\rm d} M_2\Omega (a-b_1)^2], \end{eqnarray} (7)where Ω is the orbital angular velocity, Ω1 is the orbital angular velocity of the disc, b1 is the distance between the accreting star and the L1 point, and λ is a parameter probably in the range 0 ≤ λ < 1. The parameter λ characterises the loss of orbital angular momentum through outflows. The expression of b1 takes the form:   \begin{eqnarray} \frac{b_1}{a}=0.5-0.227 \log q . \end{eqnarray} (8) Figure 2. View largeDownload slide Accretion rates for the BH–WD system with mass transfer. The figure shows that the accretion rate is extremely super-Eddington by a factor larger than 104 for a 0.6 M⊙ WD. The critical mass of WD between the sub-Eddington and super-Eddington cases is around 0.1 M⊙. Figure 2. View largeDownload slide Accretion rates for the BH–WD system with mass transfer. The figure shows that the accretion rate is extremely super-Eddington by a factor larger than 104 for a 0.6 M⊙ WD. The critical mass of WD between the sub-Eddington and super-Eddington cases is around 0.1 M⊙. The instable mass transfer will occur if the radius of WD expands more rapidly than the Roche lobe, i.e. $$\dot{R}_{\rm WD}/R_{\rm WD}>\dot{R_2}/R_2$$. Based on equations (1)–(3) and (5)–(8), we can derive the instability criterion for the mass transfer:   \begin{eqnarray} {\mathcal {F}} (q,f,\lambda ) &=& \lambda (1+q) \left( \frac{b_1-a_1}{a} \right)^2 \nonumber\\ &&-\frac{5-\delta }{6} + q(1-f) + \frac{fq}{3(1+q)} > 0, \end{eqnarray} (9)where a1 is the distance between the BH and the mass centre of the system. We adopt f = 0.9 and f = 0.99 for calculations, which means that 90 per cent or 99 per cent materials transferred from the WD are lost in the system due to strong outflows. The material stream ejected from the system may form a common envelope around the binary system (Han & Webbink 1999). We assume λ ≥ f since the orbital angular momentum can be carried away by the outflows. Moreover, the outflows may possess larger angular momentum per unit mass than the inflows thus can escape from the system. 3 RESULTS The results for M2 = 0.6 M⊙ are shown in Fig. 3. An unstable region exists for M1 > 15.97 M⊙ under f = 0.5 and λ = 0.9 (green line), and for M1 > 11.09 M⊙ under f = 0.9 and λ = 0.99 (blue line). The required mass of M1 for the unstable accretion corresponds to a BH system. For comparison, the results under the conservative condition, i.e. f = 0 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 accretion 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 elaborated formula for the radius of WD is adopted in this work. Figure 3. View largeDownload slide Criterion for an unstable mass transfer. The red dashed and solid lines represent the results of King, Olsson & Davies (2007) and ours, respectively, where outflows are not considered. The green and blue lines correspond to our results including the effects of outflows. The criterion for an unstable mass transfer is $$\mathcal {F} >0$$, as shown by equation (9). Figure 3. View largeDownload slide Criterion for an unstable mass transfer. The red dashed and solid lines represent the results of King, Olsson & Davies (2007) and ours, respectively, where outflows are not considered. The green and blue lines correspond to our results including the effects of outflows. The criterion for an unstable mass transfer is $$\mathcal {F} >0$$, as shown by equation (9). 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 always be unstable. We plot the critical condition (brown line) in Fig. 4, where f = 0.9 and λ = 0.99 are adopted for the super-Eddington accretion case. It should be noted that once the WD fills its Roche lobe within the unstable region, the violent accretion will continue as long as the evolutionary track is still located in the unstable region. In Fig. 4, the brown line has a maximum value MBH = 15.7 M⊙ at MWD = 0.35 M⊙. Thus, for a typical WD with MWD = 0.6 M⊙, the system can maintain the unstable mass transfer if the BH is sufficiently large MBH > 15.7 M⊙, i.e. the accretion process will not stop until the accretion becomes sub-Eddington, corresponding to a low-mass WD around 0.1 M⊙ (the nearly vertical grey line in Fig. 4). The blue dashed line shows the mass changes of components along with the mass transfer. If the BH mass is lower, less material of the WD can be accreted by the BH. For example, for MWD = 0.9 M⊙ and MBH = 10 M⊙, the critical mass is MWD = 0.63 M⊙ for MBH = 10 M⊙, and therefore only around 0.27 M⊙ will be accreted under the unstable phase. If we consider a BH with larger mass, such as 20 M⊙, for a 0.6 M⊙ WD, the accreted material is around 0.5 M⊙. Recently, the first BH UCXB candidate X9 has been discovered in globular cluster 47 Tucanae (Miller-Jones et al. 2015; Bahramian et al. 2017), and its highest accretion rate is estimated as a few times of 10− 10M⊙yr− 1, which suggests that the BH mass can reach a few dozens of solar mass. In this system, however, the WD mass is only around 0.02 M⊙, and therefore such a system is under the stable type of mass transfer, as shown in Fig. 4. Nine UCXB candidates are plotted, including eight NS UCXB candidates, 4U 1626−67 (Takagi et al. 2016), 4U 1850−087, 4U 0513−40, M15 X−2 (Prodan & Murray 2015), XTE J1751-305 (Andersson, Jones & Ho 2014; Gierliński & Poutanen 2005), XTE J1807−294 (Leahy, Morsink & Chou 2011), 4U 1820−30 (Güver et al. 2010), and 4U 1543−624 (Wang & Chakrabarty 2004), and a BH UCXB candidate, 47 Tuc X9. It is shown that all the UCXB candidates are well located in the left ‘stable’ region. Such a location is quite reasonable since the unstable mass transfer corresponds to extremely high accretion rates and therefore short time-scale for existence, whereas the left stable region corresponds to sub-Eddington rates and therefore long time-scale. Figure 4. View largeDownload slide Variation of the critical BH mass with the WD mass for f = 0.9 and λ = 0.99. For the parameters beyond the critical BH mass (brown line), the mass transfer is dynamically unstable. The two blue dashed lines show the evolutionary track (from the filled circle to the arrow). The space is divided into two regions, sub-Eddington (left) and super-Eddington (right), by the black solid line. The nine UCXBs are well located in the sub-Eddington and stable region, where eight UCXBs are NS candidates and the other one is a BH candidate. Figure 4. View largeDownload slide Variation of the critical BH mass with the WD mass for f = 0.9 and λ = 0.99. For the parameters beyond the critical BH mass (brown line), the mass transfer is dynamically unstable. The two blue dashed lines show the evolutionary track (from the filled circle to the arrow). The space is divided into two regions, sub-Eddington (left) and super-Eddington (right), by the black solid line. The nine UCXBs are well located in the sub-Eddington and stable region, where eight UCXBs are NS candidates and the other one is a BH candidate. The isotropic energy for a GRB is around 1052 erg, and can be up to 1054 erg. It is difficult to calculate the accurate energy released from the unstable accretion of the BH–WD system. Thus, a rough estimate of the total released energy can be expressed as:   \begin{eqnarray} E_{\rm tot}=M_{\rm a}(1-f)\eta c^2 , \end{eqnarray} (10)where Ma is the material disrupted by BH. For MWD = 0.6 M⊙, MBH = 15.7 M⊙, f = 0.9, and λ = 0.99, Ma = 0.25 M⊙. We adopt η = 0.1, so the total energy Etot is around 1052 erg. For a typical half-opening angle Δθ = 10° for GRBs, the isotropic energy is Etot/(1 − cos θ) ≈ 6.6 × 1053erg, which is sufficient to power most GRBs. Obviously, it is impossible to detect a supernova in such a event. The current merger rate for BH–WD binary is about 1.9 × 10− 6yr− 1 in the Galactic disc (Nelemans, Yungelson & Portegies Zwart 2001), with typical BH mass around 5–7 M⊙, which is slightly less than the typical required BH mass in our model, so the event rate for our system is lower. For a BH–WD binary system, the initial system should contain two main-sequence stars, a large mass primary star (>25 M⊙) and a secondary star (1–8 M⊙). The primary star will evolve into a 3–15 M⊙ BH via a supernova explosion (Woosley & Weaver 1995; Zhang, Woosley & Heger 2008), whereas the secondary star still stay at the main sequence. The system will experience a common envelope phase when the secondary star evolves off the main sequence, and finally become a WD. The initial separation of the compact system should be comparable to the separation of the X-ray binary, so we can roughly estimate the time required for the binary system to evolve into contact according to equation (4). For an X-ray binary, the secondary star fills its Roche lobe, so the separation of the X-ray binary can be estimated by equation (3), while the radius of the main-sequence star is given by Demircan & Kahraman (1991): $$R_{\rm m} = 1.01 M_{\rm m}^{0.724} (0.1 {\rm M}_{{\odot }} < M < 18.1 {\rm M}_{{\odot }})$$, where Rm and Mm are the radius and the mass of the main-sequence star, respectively. For a system contains a 5–15 M⊙ BH and 1–8 M⊙ main-sequence star, after the main-sequence star become a WD, the time required for the WD to fill its Roche lobe is about 109 yr, which is shorter than a Hubble time. Bahramian et al. (2017) reported that they detected a contact binary system with a BH and a WD. The accretion in this system ought to be stable due to the low-mass WD. It implies that more undetected BH–WD systems may exist. For the unstable mass transfer, the WD can be disrupted within a few orbital periods (Fryer et al. 1999). The orbital period can be simply estimated as P ≈ 46(M⊙/M2) s. For a typical WD M2 = 0.6 M⊙, we have P = 77 s. Thus, the accretion time-scale is roughly in agreement with that of the long GRBs. For GRB 060614, the burst lasts about 100 s, and such a time-scale is reasonable in our model. 4 DISCUSSION We have shown that outflows can play a significant role in the stability of a contact BH–WD binary system, and the WD can be nearly disrupted by the BH in a short time-scale. In our binary system, the gamma-ray burst will take place along the rotational axis, and the outflows will spread over the orbital plane. Thus, the environment should not be baryon-rich, and the corresponding spectral signature may be quite weak even if it exists. It should be noted that the dynamical region for conserved mass transfer (q > 0.52) disappears when outflows are considered. For the extreme super-Eddington case, strong outflows are inevitable. Thus, our model including the effects of outflows may be more reasonable. Fryer et al. (1999) conducted numerical simulations of the mergers of WDs and BHs and proposed that these mergers can explain long GRBs. They assumed that no mass is ejected from the system, so the orbital angular momentum in their simulations is carried away by accretion discs instead of outflows. The studies about NS–WD or double WD mergers give a similar critical unstable mass ratio q < 0.002–0.005 (Ruderman & Shaham 1983; Hut & Paczynski 1984; Bonsema & van den Heuvel 1985; Ruderman & Shaham 1985). However, in their models, the angular momentum loss can occur only when the mass ratio is less than the critical value before the mass transfer starts. 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# A black hole–white dwarf compact binary model for long gamma-ray bursts without supernova association

, Volume 475 (1) – Mar 1, 2018
5 pages

/lp/ou_press/a-black-hole-white-dwarf-compact-binary-model-for-long-gamma-ray-zAYaacQEWW
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journal_eissn:11745-3933
ISSN
1745-3925
eISSN
1745-3933
D.O.I.
10.1093/mnrasl/sly014
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### Abstract

Abstract Gamma-ray bursts (GRBs) are luminous and violent phenomena in the Universe. Traditionally, long GRBs are expected to be produced by the collapse of massive stars and associated with supernovae. However, some low-redshift long GRBs have no detection of supernova association, such as GRBs 060505, 060614, and 111005A. It is hard to classify these events convincingly according to usual classifications, and the lack of the supernova implies a non-massive star origin. We propose a new path to produce long GRBs without supernova association, the unstable and extremely violent accretion in a contact binary system consisting of a stellar-mass black hole and a white dwarf, which fills an important gap in compact binary evolution. accretion, accretion discs, gamma-ray burst: general, white dwarfs 1 INTRODUCTION It is generally believed that long gamma-ray bursts (GRBs) originate from the collapse of massive stars and are accompanied with supernovae (Woosley 1993; Woosley & Bloom 2006). Traditionally, GRBs 060505, 060614, and 111005A should be classified as long GRBs because of their long duration time. However, they are not associated with any supernova signature even though their redshifts are quite low (z ≲ 0.1). Such GRBs are also named as long-short GRBs or SN-less long GRBs (e.g. Wang et al. 2017). Taking GRB 060614 as an example, the temporal lag and peak luminosity makes it more like a short GRB (Gehrels et al. 2006). Several models were proposed to explain GRB 060614, including the merger of a neutron star (NS) and a massive white dwarf (WD) (King, Olsson & Davies 2007), and the tidal disruption of a star by an intermediate-mass black hole (BH) (Lu, Huang & Zhang 2008). Recently, a near-infrared bump was discovered in the afterglow of GRB 060614, which probably arose from a Li–Paczyński macronova, so the model involved compact binary may be favoured (Yang et al. 2015). Compact binary systems have been widely applied to different interesting phenomena. Double WD mergers may produce Type Ia supernovae if their total mass is larger than the Chandrasekhar limit (Maoz, Mannucci & Nelemans 2014). The merger of binary NS, or that of a BH and an NS is expected to trigger short GRBs and produce Li–Paczyński macronovae (Li & Paczyński 1998; Metzger et al. 2010). NS–WD systems have been used to explain the ultracompact X-ray binaries (UCXBs) (Nelemans & Jonker 2010) and the repeating fast radio burst (FRB 121102) (Gu et al. 2016). In addition, the gravitational wave emission originating from the merger of double BHs was detected by LIGO (Abbott et al. 2016a,b). In this work, we propose that the unstable accretion in a BH–WD system can explain the long GRBs without supernova association. With this evolutionary scenario, a full picture of the compact binary systems may be pieced together and is presented in Fig. 1. Figure 1. View largeDownload slide A full picture of compact binary systems. The figure shows different compact binary systems and potential corresponding observations. Figure 1. View largeDownload slide A full picture of compact binary systems. The figure shows different compact binary systems and potential corresponding observations. The gravitational wave emission plays a significant role during the binary evolution. It carries away energy and angular momentum and leads to the shrink of the orbit. In addition, other mechanisms may also remove the orbital angular momentum. For double degenerate systems, the orbital angular momentum can be converted to the angular momentum of the accretion disc or to the spin angular momentum of the accreting star (Ruderman & Shaham 1983; Hut & Paczynski 1984). Our calculations show that there exist strong outflows in the typical BH–WD system and the orbital angular momentum is carried away, which is similar to those studies on the double WD system (e.g. Han & Webbink 1999). If a typical WD mass MWD = 0.6 M⊙ is chosen for analyses (Bergeron, Saffer & Liebert 1992; Kepler et al. 2007), the accretion is extremely super-Eddington, by a factor of >103 for a stellar-mass BH. Recent simulations and observations have shown that, outflows are significant in the super-Eddington case (Sa̧dowski & Narayan 2015; Pinto, Middleton & Fabian 2016; Walton et al. 2016; Fiacconi et al. 2017; Jiang, Stone & Davis 2017). Thus, in such a system, outflows can play an important role. We use an analytic method to investigate the dynamical stability of the BH–WD system during mass transfer. For a typical WD donor, when the mass transfer occurs, the accretion rate can be extremely super-Eddington and the accreted materials will be ejected from the system. Such strong outflows can carry away significant orbital angular momentum and trigger the unstable accretion of the system. The released energy from the accretion is sufficient to power GRBs. 2 MODEL AND ANALYSES In our model, the system consists of a stellar-mass BH and a WD donor. In such a system, the mass transfer occurs when the WD fills the Roche lobe. If this process is dynamically unstable, the WD will be disrupted rapidly. The instability of the system greatly depends on the relative expansion rate of the WD and the Roche lobe. Following the mass transfer, the radius of the WD will expand, and the orbital separation also tends to increase if the orbital angular momentum is conserved. If the WD expands more rapidly than the Roche lobe, i.e. the time-scale for the expansion of the Roche lobe is longer than that of the WD, the mass transfer will be dynamically unstable. In this scenario, the orbital angular momentum loss can restrain the expansion of the Roche lobe, and therefore will have essential influence on the binary evolution. We consider a compact binary system which contains a WD donor. The system is assumed to be tidally locked, and the orbit is circular. The orbital angular momentum J of the binary system is given by   \begin{eqnarray} J=M_{1}M_{2} \left(\frac{Ga}{M} \right)^\frac{1}{2} \ , \end{eqnarray} (1)where M1 and M2 are the mass of the accretor and the WD, respectively, a is the orbital separation, and M = M1 + M2 represents the total mass of two stars. A relatively accurate formula for the radius of WD is provided by Eggleton and is quoted by Verbunt & Rappaport (1988):   \begin{eqnarray} \frac{R_{\rm WD}}{R_{{\odot }}}=0.0114\left[ \left(\frac{M_2}{M_{\rm Ch}} \right) ^{-\frac{2}{3}}- \left(\frac{M_2}{M_{\rm Ch}} \right)^\frac{2}{3}\right] ^\frac{1}{2} \end{eqnarray} (2)where MCh = 1.44 M⊙ and Mp = 0.00057 M⊙. This relation offers a single relation that can be used in any mass range of WD (Marsh, Nelemans & Steeghs 2004). For simplicity, we adopt a form to describe the Roche lobe radius of the secondary M2 (WD; Paczyński 1971):   \begin{eqnarray} \frac{R_{2}}{a}=0.462 \left(\frac{M_{2}}{M} \right)^\frac{1}{3} . \end{eqnarray} (3)For the circular orbit, the variation of a due to the gravitational wave emission can be written as (Peters 1964)   \begin{eqnarray} \frac{{\rm d}a}{{\rm d}t} = - \frac{64}{5} \frac{G^{3} M_1 M_2 M}{c^5 a^{3}} . \end{eqnarray} (4)Based on equations (1)– (4), we can derive the mass accretion rate for the system,   \begin{equation*} \dot{M_2}=\frac{64G^3M_1M_2^2M}{5c^5a^4 \left(2q-\frac{5}{3}+\frac{\delta }{3} \right)} \ , \end{equation*}   \begin{equation*} \delta =\frac{ \left(\frac{M_2}{M_{\rm Ch}} \right)^{-\frac{2}{3}} + \left(\frac{M_2}{M_{\rm Ch}}\right)^\frac{2}{3}}{ \left(\frac{M_2}{M_{\rm Ch}} \right)^{-\frac{2}{3}}- \left(\frac{M_2}{M_{\rm Ch}} \right) ^\frac{2}{3}}-\frac{2\frac{M_{\rm p}}{M_2} \left[\frac{7}{3} \left(\frac{M_{\rm p}}{M_2} \right)^{-\frac{1}{3}}+1 \right]}{1+3.5 \left(\frac{M_2}{M_{\rm p}} \right)^{-\frac{2}{3}}+ \left(\frac{M_2}{M_{\rm p}} \right)^{-1}} \ , \end{equation*} where q = M2/M1. Then, we define a dimensionless parameter $$\dot{m}=-\dot{M_2}/ \dot{M}_{\rm Edd}$$, where $$\dot{M}_{\rm Edd}=L_{\rm Edd}/{\eta c^2}$$ is the Eddington accretion rate, where LEdd = 4πGM1c/κes. We choose η = 0.1. Thus,   \begin{equation*} \dot{m} = {\frac{4 G^2 M^2_2 M \kappa_{\rm es}}{25\pi c^4 a^4 \left(5/6-q-\delta/6 \right)}} \,, \end{equation*} where the opacity κes = 0.34 cm2 g−1. For M2 = 0.01, 0.1, 0.6 M⊙, the variation of the stable accretion rate with M1 is plotted in Fig. 2. There exists dramatic increase for M1 < 2 M⊙, which indicates that, for relatively large mass ratio, the mass transfer is unstable for the system consisting of a WD and an NS/WD. If M1 is fixed, $$\dot{m}$$ is positively related to M2. The critical mass of WD between sub-Eddington and super-Eddington is around 0.1 M⊙. For M2 = 0.6 M⊙, the accretion rate is extremely super-Eddington, so the outflows ought to be taken into consideration. Here, we only considered the effects of gravitational wave emission. Obviously, outflows can be another mechanism to take away the angular momentum, so the mass accretion rate should be even higher, which can result in stronger outflows. We use a parameter f to describe the scale of mass loss in the mass transfer process due to outflows:   \begin{eqnarray} {\rm d}M_1 + {\rm d}M_2 = f {\rm d}M_2\ (0 \leqslant f < 1) \ , \end{eqnarray} (5)where dM2 is negative. In general, the stream of matter flowing from the inner Lagrange point can form a disc around the accreting star. If the disc is circular and non-viscous, its radius can be written as (Verbunt & Rappaport 1988)   \begin{eqnarray} \frac{R_{\rm h}}{a} &=& 0.088\,3 - 0.048\,58\log q +0.114\,89\log ^2 q \nonumber\\ &&+\,0.020\,475\log ^3 q, \end{eqnarray} (6)where 10−3 < q < 1. A fraction of the orbital angular momentum of the stream comes from the spin angular momentum of the WD, so this part should be eliminated. Thus, the orbital angular momentum carried away by the outflows can be expressed as   \begin{eqnarray} {\rm d}J = - \lambda [({\rm d} M_{2}\Omega _1R_{\rm h}^2-{\rm d} M_2\Omega (a-b_1)^2], \end{eqnarray} (7)where Ω is the orbital angular velocity, Ω1 is the orbital angular velocity of the disc, b1 is the distance between the accreting star and the L1 point, and λ is a parameter probably in the range 0 ≤ λ < 1. The parameter λ characterises the loss of orbital angular momentum through outflows. The expression of b1 takes the form:   \begin{eqnarray} \frac{b_1}{a}=0.5-0.227 \log q . \end{eqnarray} (8) Figure 2. View largeDownload slide Accretion rates for the BH–WD system with mass transfer. The figure shows that the accretion rate is extremely super-Eddington by a factor larger than 104 for a 0.6 M⊙ WD. The critical mass of WD between the sub-Eddington and super-Eddington cases is around 0.1 M⊙. Figure 2. View largeDownload slide Accretion rates for the BH–WD system with mass transfer. The figure shows that the accretion rate is extremely super-Eddington by a factor larger than 104 for a 0.6 M⊙ WD. The critical mass of WD between the sub-Eddington and super-Eddington cases is around 0.1 M⊙. The instable mass transfer will occur if the radius of WD expands more rapidly than the Roche lobe, i.e. $$\dot{R}_{\rm WD}/R_{\rm WD}>\dot{R_2}/R_2$$. Based on equations (1)–(3) and (5)–(8), we can derive the instability criterion for the mass transfer:   \begin{eqnarray} {\mathcal {F}} (q,f,\lambda ) &=& \lambda (1+q) \left( \frac{b_1-a_1}{a} \right)^2 \nonumber\\ &&-\frac{5-\delta }{6} + q(1-f) + \frac{fq}{3(1+q)} > 0, \end{eqnarray} (9)where a1 is the distance between the BH and the mass centre of the system. We adopt f = 0.9 and f = 0.99 for calculations, which means that 90 per cent or 99 per cent materials transferred from the WD are lost in the system due to strong outflows. The material stream ejected from the system may form a common envelope around the binary system (Han & Webbink 1999). We assume λ ≥ f since the orbital angular momentum can be carried away by the outflows. Moreover, the outflows may possess larger angular momentum per unit mass than the inflows thus can escape from the system. 3 RESULTS The results for M2 = 0.6 M⊙ are shown in Fig. 3. An unstable region exists for M1 > 15.97 M⊙ under f = 0.5 and λ = 0.9 (green line), and for M1 > 11.09 M⊙ under f = 0.9 and λ = 0.99 (blue line). The required mass of M1 for the unstable accretion corresponds to a BH system. For comparison, the results under the conservative condition, i.e. f = 0 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 accretion 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 elaborated formula for the radius of WD is adopted in this work. Figure 3. View largeDownload slide Criterion for an unstable mass transfer. The red dashed and solid lines represent the results of King, Olsson & Davies (2007) and ours, respectively, where outflows are not considered. The green and blue lines correspond to our results including the effects of outflows. The criterion for an unstable mass transfer is $$\mathcal {F} >0$$, as shown by equation (9). Figure 3. View largeDownload slide Criterion for an unstable mass transfer. The red dashed and solid lines represent the results of King, Olsson & Davies (2007) and ours, respectively, where outflows are not considered. The green and blue lines correspond to our results including the effects of outflows. The criterion for an unstable mass transfer is $$\mathcal {F} >0$$, as shown by equation (9). 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 always be unstable. We plot the critical condition (brown line) in Fig. 4, where f = 0.9 and λ = 0.99 are adopted for the super-Eddington accretion case. It should be noted that once the WD fills its Roche lobe within the unstable region, the violent accretion will continue as long as the evolutionary track is still located in the unstable region. In Fig. 4, the brown line has a maximum value MBH = 15.7 M⊙ at MWD = 0.35 M⊙. Thus, for a typical WD with MWD = 0.6 M⊙, the system can maintain the unstable mass transfer if the BH is sufficiently large MBH > 15.7 M⊙, i.e. the accretion process will not stop until the accretion becomes sub-Eddington, corresponding to a low-mass WD around 0.1 M⊙ (the nearly vertical grey line in Fig. 4). The blue dashed line shows the mass changes of components along with the mass transfer. If the BH mass is lower, less material of the WD can be accreted by the BH. For example, for MWD = 0.9 M⊙ and MBH = 10 M⊙, the critical mass is MWD = 0.63 M⊙ for MBH = 10 M⊙, and therefore only around 0.27 M⊙ will be accreted under the unstable phase. If we consider a BH with larger mass, such as 20 M⊙, for a 0.6 M⊙ WD, the accreted material is around 0.5 M⊙. Recently, the first BH UCXB candidate X9 has been discovered in globular cluster 47 Tucanae (Miller-Jones et al. 2015; Bahramian et al. 2017), and its highest accretion rate is estimated as a few times of 10− 10M⊙yr− 1, which suggests that the BH mass can reach a few dozens of solar mass. In this system, however, the WD mass is only around 0.02 M⊙, and therefore such a system is under the stable type of mass transfer, as shown in Fig. 4. Nine UCXB candidates are plotted, including eight NS UCXB candidates, 4U 1626−67 (Takagi et al. 2016), 4U 1850−087, 4U 0513−40, M15 X−2 (Prodan & Murray 2015), XTE J1751-305 (Andersson, Jones & Ho 2014; Gierliński & Poutanen 2005), XTE J1807−294 (Leahy, Morsink & Chou 2011), 4U 1820−30 (Güver et al. 2010), and 4U 1543−624 (Wang & Chakrabarty 2004), and a BH UCXB candidate, 47 Tuc X9. It is shown that all the UCXB candidates are well located in the left ‘stable’ region. Such a location is quite reasonable since the unstable mass transfer corresponds to extremely high accretion rates and therefore short time-scale for existence, whereas the left stable region corresponds to sub-Eddington rates and therefore long time-scale. Figure 4. View largeDownload slide Variation of the critical BH mass with the WD mass for f = 0.9 and λ = 0.99. For the parameters beyond the critical BH mass (brown line), the mass transfer is dynamically unstable. The two blue dashed lines show the evolutionary track (from the filled circle to the arrow). The space is divided into two regions, sub-Eddington (left) and super-Eddington (right), by the black solid line. The nine UCXBs are well located in the sub-Eddington and stable region, where eight UCXBs are NS candidates and the other one is a BH candidate. Figure 4. View largeDownload slide Variation of the critical BH mass with the WD mass for f = 0.9 and λ = 0.99. For the parameters beyond the critical BH mass (brown line), the mass transfer is dynamically unstable. The two blue dashed lines show the evolutionary track (from the filled circle to the arrow). The space is divided into two regions, sub-Eddington (left) and super-Eddington (right), by the black solid line. The nine UCXBs are well located in the sub-Eddington and stable region, where eight UCXBs are NS candidates and the other one is a BH candidate. The isotropic energy for a GRB is around 1052 erg, and can be up to 1054 erg. It is difficult to calculate the accurate energy released from the unstable accretion of the BH–WD system. Thus, a rough estimate of the total released energy can be expressed as:   \begin{eqnarray} E_{\rm tot}=M_{\rm a}(1-f)\eta c^2 , \end{eqnarray} (10)where Ma is the material disrupted by BH. For MWD = 0.6 M⊙, MBH = 15.7 M⊙, f = 0.9, and λ = 0.99, Ma = 0.25 M⊙. We adopt η = 0.1, so the total energy Etot is around 1052 erg. For a typical half-opening angle Δθ = 10° for GRBs, the isotropic energy is Etot/(1 − cos θ) ≈ 6.6 × 1053erg, which is sufficient to power most GRBs. Obviously, it is impossible to detect a supernova in such a event. The current merger rate for BH–WD binary is about 1.9 × 10− 6yr− 1 in the Galactic disc (Nelemans, Yungelson & Portegies Zwart 2001), with typical BH mass around 5–7 M⊙, which is slightly less than the typical required BH mass in our model, so the event rate for our system is lower. For a BH–WD binary system, the initial system should contain two main-sequence stars, a large mass primary star (>25 M⊙) and a secondary star (1–8 M⊙). The primary star will evolve into a 3–15 M⊙ BH via a supernova explosion (Woosley & Weaver 1995; Zhang, Woosley & Heger 2008), whereas the secondary star still stay at the main sequence. The system will experience a common envelope phase when the secondary star evolves off the main sequence, and finally become a WD. The initial separation of the compact system should be comparable to the separation of the X-ray binary, so we can roughly estimate the time required for the binary system to evolve into contact according to equation (4). For an X-ray binary, the secondary star fills its Roche lobe, so the separation of the X-ray binary can be estimated by equation (3), while the radius of the main-sequence star is given by Demircan & Kahraman (1991): $$R_{\rm m} = 1.01 M_{\rm m}^{0.724} (0.1 {\rm M}_{{\odot }} < M < 18.1 {\rm M}_{{\odot }})$$, where Rm and Mm are the radius and the mass of the main-sequence star, respectively. For a system contains a 5–15 M⊙ BH and 1–8 M⊙ main-sequence star, after the main-sequence star become a WD, the time required for the WD to fill its Roche lobe is about 109 yr, which is shorter than a Hubble time. Bahramian et al. (2017) reported that they detected a contact binary system with a BH and a WD. The accretion in this system ought to be stable due to the low-mass WD. It implies that more undetected BH–WD systems may exist. For the unstable mass transfer, the WD can be disrupted within a few orbital periods (Fryer et al. 1999). The orbital period can be simply estimated as P ≈ 46(M⊙/M2) s. For a typical WD M2 = 0.6 M⊙, we have P = 77 s. Thus, the accretion time-scale is roughly in agreement with that of the long GRBs. For GRB 060614, the burst lasts about 100 s, and such a time-scale is reasonable in our model. 4 DISCUSSION We have shown that outflows can play a significant role in the stability of a contact BH–WD binary system, and the WD can be nearly disrupted by the BH in a short time-scale. In our binary system, the gamma-ray burst will take place along the rotational axis, and the outflows will spread over the orbital plane. Thus, the environment should not be baryon-rich, and the corresponding spectral signature may be quite weak even if it exists. It should be noted that the dynamical region for conserved mass transfer (q > 0.52) disappears when outflows are considered. For the extreme super-Eddington case, strong outflows are inevitable. Thus, our model including the effects of outflows may be more reasonable. Fryer et al. (1999) conducted numerical simulations of the mergers of WDs and BHs and proposed that these mergers can explain long GRBs. They assumed that no mass is ejected from the system, so the orbital angular momentum in their simulations is carried away by accretion discs instead of outflows. The studies about NS–WD or double WD mergers give a similar critical unstable mass ratio q < 0.002–0.005 (Ruderman & Shaham 1983; Hut & Paczynski 1984; Bonsema & van den Heuvel 1985; Ruderman & Shaham 1985). However, in their models, the angular momentum loss can occur only when the mass ratio is less than the critical value before the mass transfer starts. 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### Journal

Monthly Notices of the Royal Astronomical Society: LettersOxford University Press

Published: Mar 1, 2018

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