A&A 622, A35 (2019) Astronomy https://doi.org/10.1051/0004-6361/201833010 & c ESO 2019 Astrophysics

The progenitors of type-Ia supernovae in semidetached binaries with donors D. Liu1,2,3,4 , B. Wang1,2,3,4 , H. Ge1,2,3,4 , X. Chen1,2,3,4 , and Z. Han1,2,3,4

1 Yunnan Observatories, Chinese Academy of Sciences, Kunming 650216, PR China e-mail: [email protected], [email protected] 2 Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese Academy of Sciences, Kunming 650216, PR China 3 University of Chinese Academy of Sciences, Beijing 100049, PR China 4 Center for Astronomical Mega-Science, Chinese Academy of Sciences, Beijing 100012, PR China Received 13 March 2018 / Accepted 25 November 2018

ABSTRACT

Context. The companions of the exploding carbon-oxygen white dwarfs (CO WDs) that produce type-Ia supernovae (SNe Ia) have still not been conclusively identified. A red-giant (RG) can fill this role as the mass donor of the exploding WD − this channel for producing SNe Ia has been named the symbiotic channel. However, previous studies on this channel have given a relatively low rate of SNe Ia. Aims. We aim to systematically investigate the parameter space, Galactic rates, and delay time distributions of SNe Ia arising from the symbiotic channel under a revised mass-transfer prescription. Methods. We adopted an integrated mass-transfer prescription to calculate the mass-transfer process from a RG star onto the WD. In this prescription, the mass-transfer rate varies with the local material states. First, we obtain the parameter space that leads to SNe Ia by evolving a large number of semidetached WD+RG systems with the Eggleton stellar-evolution code. Second, we investigate the Galactic rates and delay-time distributions of SNe Ia using a binary population synthesis method. Results. The parameter space of WD+RG systems that can produce SNe Ia is enlarged significantly judging by our calculations. This channel could produce SNe Ia with intermediate and old ages, contributing up to 5% of all SNe Ia in the Galaxy. Our model increases the SN Ia rate from this channel by a factor of five. We suggest that the symbiotic systems RS Oph and T CrB are strong candidates for the progenitors of SNe Ia. Key words. binaries: close – : evolution – supernovae: general – white dwarfs

1. Introduction hole model, and so on (for recent reviews see Wang 2018; Soker 2018; Livio & Mazzali 2018). Type-Ia supernovae (SNe Ia) are good distance indicators in cos- In the single-degenerate model, the primary WDs can accrete mology. They revealed the accelerating expansion of the uni- H-rich matter from RG stars and form SNe Ia when they grow verse and led to the discovery of dark energy (e.g., Howell 2011; close to MCh in terms of mass. This formation channel is known Meng et al. 2015). It is generally believed that SNe Ia result from as the symbiotic channel (e.g., Whelan & Iben 1973; Kenyon the thermonuclear explosions of carbon-oxygen white dwarfs 1986; Kenyon et al. 1993; Munari & Renzini 1992; Hachisu et al. (CO WDs) in binaries (e.g., Hoyel & Fowler 1960; Nomoto et al. 1996; Li & van den Heuvel 1997; Yungelson & Livio 1998; King 1984). However, the identity of the mass donor for the exploding et al. 2003; Lü et al. 2006, 2009; Chen et al. 2011). Although the CO WD is still not fully confirmed (e.g., Podsiadlowski et al. actual number of symbiotic stars in the Galaxy is still unknown 2008; Wang & Han 2012; Maoz et al. 2014; Ruiz-Lapuente (e.g., Mikołajewska 2012; Rodríguez-Flores et al. 2014), many 2014). The mass donor could be a (MS) star, a symbiotic systems have been observed (e.g., Belczynski´ et al. red giant (RG) star, or a helium (He) star in the single-degenerate 2000; Miszalski & Mikołajewska 2014; Li et al. 2015). In these (SD) model, in which the CO WD that accretes H-/He-rich mat- systems, the symbiotic novae T CrB and RS Oph are possi- ter may produce an SN Ia when its mass approaches the Chan- ble progenitor candidates for SNe Ia (Kraft 1958; Brandi et al. drasekhar limit (MCh; e.g., Whelan & Iben 1973; Nomoto et al. 2009). Patat et al.(2007) detected Na I absorption lines with low 1984; Li & van den Heuvel 1997; Langer et al. 2000; Han & expansion velocities in SN 2006X, and speculated that the com- Podsiadlowski 2004; Chen & Li 2007; Wang et al. 2009). The panion of the exploding WD may be an early RG star, although mass donor may also be another CO WD in the double- Chugai(2008) argued that the absorption lines detected in SN degenerate (DD) model, in which the merger of the double 2006X cannot be formed in the RG wind. Voss & Nelemans WDs may produce SNe Ia (e.g., Iben & Tutukov 1984; Webbink (2008) suggested that the progenitor of SN 2007on may be a 1984; Han 1998; Nelemans et al. 2001; Toonen et al. 2012). In WD+RG system after studying the pre-explosion X-ray images addition, some other progenitor models have been proposed to at the same position. In addition, the surviving companions of explain the observed diversity of SNe Ia, such as for example the SNe Ia from the symbiotic channel may be related to the forma- double-detonation model, the core-degenerate model, the colli- tion of single low-mass He WDs in observations (e.g., Justham sional WD model, the single star model, the WDs near black et al. 2009; Wang & Han 2010).

Article published by EDP Sciences A35, page 1 of8 A&A 622, A35 (2019)

However, previous studies argued that the rate of SNe Ia investigate the semidetached symbiotic channel for producing from the symbiotic channel is relatively low (e.g., Li & van den SNe Ia. In Sect.2, we show the methods for detailed binary evo- Heuvel 1997; Yungelson & Livio 1998; Han & Podsiadlowski lution computations and the corresponding results. The method 2004). These studies usually adopted a surface boundary condi- for and results of synthesizing a binary population are provided tion to calculate the process of Roche-lobe overflow (RLOF), as in Sect.3. We present a discussion in Sect.4 and finally a sum- follows: mary in Sect.5.  !3 ˙  rstar  M2 = −Cmax 0, − 1  , (1) 2. Detailed binary evolution computations  rlobe  2.1. Methods where M˙ 2 is the mass-transfer rate, rstar is the radius of the lobe- We use the Eggleton stellar evolution code (Eggleton 1973) to filling star, rlobe is the radius of its Roche lobe and C is a con- stant (see Han et al. 2000). The constant C is usually set to follow the binary evolution of semidetached WD+RG systems. −1 be 1000 M yr . According to this prescription, the exceeding The typical Pop I composition is adopted for the initial MS mod- mass of the donor star would be transferred onto the accretor els with H fraction X = 0.7, He fraction Y = 0.28, and metallic- immediately as soon as the donor star exceeds its Roche-lobe, ity Z = 0.02. In this work, we do not calculate the structure of the WD and consider it as a point mass. When the WD grows in since (rstar/rlobe − 1) is always less than 0.001 once Eq. (1) is used. In this case, the mass-transfer rate in WD+RG systems is mass to MCh (set to be 1.378 M ), an SN Ia explosion is assumed usually relatively high, resulting in two cases that prevent the for- to occur. In this work, we adopted the integrated mass-transfer mation of SNe Ia: (1) A common envelope (CE) may be formed prescription presented in the Appendix of Ge et al.(2010) to if the mass-transfer is dynamically unstable (e.g., Ivanova et al. calculate the RLOF in semidetached WD+RG systems (see also 2013). The binary is likely to merge after the formation of a CE, Kolb & Ritter 1990). The mass-transfer rate which prevents the formation of SNe Ia. If the CE can be ejected, 3 Z φs 2πRL 1/2 2 Γ+1 1/2 the binary may evolve into a CO WD+He WD system or a CO M˙ 2 = − f (q) Γ ( ) 2(Γ−1) (ρP) dφ, (2) WD+He-burning star system (e.g., Han et al. 2000). The CO GM2 φL Γ + 1 WD+He-burning star system may evolve to double CO WDs and then produce SNe Ia via the DD model (Ruiter et al. 2013; Liu where RL is the effective Roche-lobe radius, G is the gravita- et al. 2016, 2018). (2) The driven by the WD radia- tional constant, M2 is the donor mass, Γ is the adiabatic index, ρ tion may blow away too much accreted matter from the surface is the local gas density, and P is the local gas pressure. The inte- gration is from the Roche-lobe potential energy (φL) to the stellar of the WD, preventing the WD from growing in mass to MCh. The local gas density and sound velocity in the region around surface potential energy (φs). The potential energy is written as the inner Lagrange point L1 of a RG star are significantly dφ = GM R−2 dR, (3) lower than those of a MS star. Furthermore, hydrodynamic esti- 2 mates of the dimensionless parameter C are of the order of in which R is the donor radius. The coefficient f (q) is a slowly M2/Porb (Paczynski´ & Sienkiewicz 1972; Eggleton 2006) and varying function of the mass ratio q: therefore the parameter C may also be smaller in a binary sys- tem containing a RG star rather than a MS star due to the q 1 f q ≡ , ( ) 3 1/2 (4) larger orbital period. Previous studies usually employed the same r (1 + q) [a2(a2 − 1)] mass-transfer prescription presented in Eq. (1) for WD+RG sys- L tems and WD+MS systems. Thus, the constant C in Eq. (1) is in which a2 is defined as too large for semidetached WD+RG systems in the previous ˙ q 1 simulations, which may overestimate M2 when the RG star fills a = + , (5) 2 3 q − x 3 its Roche-lobe. Lubow & Shu(1975) proposed that the mass- xL(1 + q) (1 + )(1 L) transfer process can be investigated by integrating the local gas density and sound velocity over the plane that is perpendicu- where xL is simply but accurately approximated as lar to the line of centres connecting the two stars, and passing ( (0.7 − 0.2q1/3)q1/3, q ≤ 1 through the inner Lagrangian point. By assuming that the equa- xL = −1/3 −1/3 (6) tion of state of stars obeys an adiabatic power law, and that the 1 − (0.7 − 0.2q )q , q > 1. mass outflow is laminar and occurs along equipotential surfaces, The accreted H-rich matter from RG stars first burns into He. Ge et al.(2010) obtained an approximated prescription for the ˙ The He experiences He flashes and burns into C and O, which is mass-transfer process. In this prescription, M2 is a function of then accumulated onto the surface of the WD. The mass growth local material states, which is equivalent to a variable C in rate of the WDs (M˙ ) is defined as Eq. (1). In this case, the exceeding mass will be transferred at WD a limited rate (C is around M/Porb) when the RG star fills its M˙ WD = ηHeηH M˙ 2, (7) Roche-lobe. Thus, M˙ 2 is lower than that of previous models, leading to a lower mass-loss rate and more matter being accumu- where ηH is the mass accumulation efficiency for H-shell burning lated onto the primary WD. Ge et al.(2015) also found that the from Wang et al.(2010), and ηHe is the mass accumulation effi- critical mass ratio for the formation of a CE becomes larger com- ciency for He-shell flashes from Kato & Hachisu(2004) 1. When pared with the results given by Hjellming & Webbink(1987), implying that more interacting binaries would experience stable 1 We note that the values of η from different studies are quite dif- mass-transfer processes (see also Passy et al. 2012; Pavlovskii & He ferent, for instance, the value of ηHe can range from ∼0.4 to ∼0.9 for −6 −1 Ivanova 2015). MWD = 1.3 M and M˙ 2 = 10 M yr (e.g., Yoon et al. 2004; Kato In this work, we adopted the integrated mass-transfer pre- & Hachisu 2004; Piersanti et al. 2014; Wang et al. 2015; Brooks et al. scription described in the Appendix of Ge et al.(2010) to 2016; Wu et al. 2017).

A35, page 2 of8 D. Liu et al.: The progenitors of SNe Ia in the semidetached binaries with RG donors

Fig. 1. A typical binary evolution for producing SN Ia via the symbiotic channel. In panel a, the solid curve represents the evolutionary track of the mass donor in the Hertzsprung–Russell diagram, and the dash-dotted curve shows the evolution of the orbital periods. The crosses in this panel represent the positions where mass transfer starts. In panel b, the evolution of M˙ 2, M˙ WD, and MWD as a function of time are shown as solid, dashed, and dash-dotted curves, respectively. The position of t = 0 corresponds to the moment when the RG fills its Roche-lobe. The mass-transfer process can be divided into four phases: (1) the strong H-shell flash phase; (2) the stable H-shell burning or the weak H-shell flash phase; (3) the optically thick wind phase; (4) the stable H-shell burning or the weak H-shell flash phase. The asterisks in both panels indicate the position where an SN Ia explosion occurs.

Fig. 2. Similar to Fig.1, but for a comparison using the traditional method to calculate the mass-transfer rate.

M˙ 2 is larger than the critical mass-transfer rate M˙ cr described in 2.2. Results Nomoto(1982), we assume that the accreted H stably burns into Figure1 shows a typical example of binary-evolution computa- He at the rate of M˙ cr, and the rest of the H-rich matter would be blown away in the form of the optically thick wind (e.g., tions, in which the WD accretes H-rich material from a RG star and explodes as an SN Ia when its mass reaches MCh. Here, we Hachisu et al. 1996). The critical mass accretion rate is written i set the initial WD mass MWD to 1.0 M , the initial mass of donor as i i star M2 to 1.2 M , and the orbital period to log (P /day) = 2.4. −7 1.7 − X −1 When the mass donor evolves to the RG stage, it expands quickly M˙ = 5.3 × 10 (M − 0.4) M yr , (8) cr X WD and fills its Roche-lobe. The crosses in panel a indicate the posi- where X is the mass fraction of H and M is the mass of the tion where the donor begins to fill its Roche-lobe, which also WD corresponds to the position of t = 0 in panel b. Subsequently, primary WD in units of M . When M˙ is lower than M˙ , the fol- 2 cr the binary will experience four phases shown in panel b before lowing assumptions are adopted (e.g., Wang et al. 2010): (1) the an SN Ia is produced: (1) At the beginning (t = 0), M˙ 2 is lower H-shell burning is stable and no matter is lost from the binary ˙ ˙ ˙ ˙ than 1/8Mcr, during which the H-shell flash is relatively strong when 1/2Mcr < M2 < Mcr; (2) a very weak H-shell flash occurs and no matter is accumulated onto the surface of the WD. (2) At ˙ ˙ and no matter is lost from the binary when 1/8Mcr < M2 < about t = 2.4 × 105 yr, M˙ increases to be larger than 1/8M˙ ˙ 2 cr 1/2Mcr; (3) the H-shell flash becomes so strong that no matter is but still smaller than M˙ , during which the H-shell burns sta- ˙ ˙ cr accumulated onto the WD when M2 < 1/8Mcr. We also assume bly or flashes very weakly, and all of the accreted H-rich matter that the mass loss depletes the specific orbital angular momen- is assumed to be burned into He. However, some of the matter tum of the primary WD. In our calculations, we do not consider would be lost due to the He flashes. (3) At about t = 3.8×105 yr, the effect of the mass stripping from the RG caused by the opti- M˙ 2 becomes larger than M˙ cr, and the binary enters the optically cally thick wind (see Hachisu et al. 1999). thick wind stage, during which the transferred H-rich matter

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Fig. 3. Regions of WD+RG systems at their formation moment in the Fig. 4. Initial and final regions of WD+RG systems for producing SNe i i i initial orbital period−initial secondary mass (log P −M2) plane for pro- Ia with various MWD. The filled triangle and asterisk represent two sym- i ducing SNe Ia with MWD = 1.2 M . The red solid contour shows the biotic systems RS Oph and T CrB, respectively. parameter space from this work, and the blue dashed contour is from Li & van den Heuvel(1997) for a comparison. The filled circles repre- sent binaries that can evolve to SNe Ia via the semidetached symbiotic undergo a rapid mass-transfer process as a result of the rapid channel, whereas the crosses indicate binaries that cannot. expansion of the RG stars, leading to the mass-loss of too much material in the form of the optically thick wind. We note that the mass donor with the shortest orbital period for the case of ˙ i stably burns into He at the rate of Mcr, while the rest of the H- MWD = 1.2 M would fill its Roche lobe at the bottom of its RG rich matter is assumed to be blown away via the optically thick branch. For binaries below the lower boundaries, the primary 5 wind. (4) At about t = 8.3 × 10 yr, the binary enters the H- WDs can grow in mass but cannot reach MCh due to the small shell burning phase again, and subsequently the very weak H- masses of the donors. The upper boundaries are constrained by shell flash phase. The primary WD grows in mass to MCh and the high mass-transfer rate owing to a large mass ratio, leading explodes as an SN Ia at about t = 1.42 × 105 yr. At this moment, to the formation of a CE. The wind-accreting channel of symbi- SN the mass of the RG star is M2 = 0.5365 M with a 0.42 M He otic stars may also slightly contribute to the formation of SNe Ia, core, and the orbital period is log (PSN/day) = 2.8178. In Fig.2, which is not considered in this work as its contribution is almost we present the evolution of the same binary using the traditional negligible (e.g., Yungelson & Livio 1998). mass-transfer prescription shown in Eq. (1) for a comparison. From this figure, we can see that the mass-transfer rate increases 3. Binary population synthesis rapidly after the RG fills its Roche-lobe, which may lead to the formation of a CE. 3.1. Methods We evolved more than 460 WD+RG systems, for which the initial masses of the WDs are in the range of 1.0−1.3 M , the ini- By employing the Hurley rapid binary evolution code (see Hur- ley et al. 2002), we conduct a series of Monte Carlo simula- tial masses of the donors range from 0.7 to 2.0 M and the initial orbital periods are ∼1−630 days. Thus, we obtained the initial tions in a binary population synthesis (BPS) approach to cal- parameter space for the production of SNe Ia via the symbiotic culate the rates and delay times of SNe Ia. In each simulation, × 7 channel. Figure3 shows the parameter space of WD +RG sys- we evolve a sample of 4 10 primordial binaries until the formation of WD+RG systems. The metallicity in our simula- tems for producing SNe Ia with an initial WD mass of 1.2 M . For a comparison, we show the results of Li & van den Heuvel tions is set to be 0.02. We assume that an SN Ia will be pro- 1997). This figure shows that the grid from the present work has duced if the parameters of the formed lobe-filling WD+RG sys- larger initial donor masses and shorter initial orbital periods. In tem are located in the initial regions for producing SNe Ia in Fig.4. these regions, M˙ 2 would be so high that the binary enters a CE process or a strong, optically thick wind process in the model The following initial parameters and basic assumptions of of Li & van den Heuvel(1997), preventing them from forming the Monte Carlo BPS computations are adopted: SNe Ia. (1) All stars are assumed to be in binary systems, and their orbits Figure4 presents the initial parameter space for the produc- are assumed to be circular. tion of SNe Ia, and the final regions of WD+RG systems at the (2) The initial mass function of Kroupa(2001) is adopted for the moment of SN Ia explosions in the orbital period–secondary primordial primaries, in which the masses of the primordial − mass (log P−M ) plane, in which Mi = 1.0, 1.1 and 1.2 M . primaries (M1) are in the range of 0.01 100 M . 2 WD (3) For the primordial secondary mass (M ), we simply adopt We found that Mi = 1.0 M is the minimum initial WD mass 2 WD a constant mass ratio (q0 = M /M ) distribution, that is, for producing SNe Ia as its region almost vanishes. We also 2 1 n(q0) = 1, in which 0 < q0 ≤ 1. found that the symbiotic systems RS Oph and T CrB could (4) The initial distribution of separations a is supposed to be con- form SNe Ia via the symbiotic channel (for more details, see stant in log a for wide binaries, and fall off smoothly for close Sect.4). The WDs in the binaries beyond these initial contours binaries (see Eggleton et al. 1989), as follows: cannot grow in mass to M : binaries beyond the left boundaries Ch ( will experience strong H-shell flashes that will blow away too αset, a0 < a < a1 a · n(a) = m , (9) much material, while binaries beyond the right boundaries will αset(a/a0) , a ≤ a0

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Fig. 5. Evolution of SN Ia rates in the Galaxy as a function of time based Fig. 7. Density distribution of WD+RG systems producing SNe Ia. 7 on the symbiotic channel. The solid, dashed, and dash-dotted curves Here, αCEλ is set to be 1.5 and 4 × 10 primordial samples are included. represent the cases with αCEλ = 0.5, 1.0 and 1.5, respectively. The solid contour is the initial parameter space of WD+RG systems for i producing SNe Ia with MWD = 1.2 M .

3.2. Results

Figure5 presents the evolution of SN Ia rates from the sym- biotic channel with different values of αCEλ, in which a con- −1 stant star formation rate of 5 M yr is adopted. This figure shows that the Galactic rates of SNe Ia are in the range of ∼0.5−1.3 × 10−4 yr−1. The Galactic SN Ia birthrate in obser- vations is about 3−4 × 10−3 yr−1 (e.g., Cappellaro et al. 1997). Thus, the semidetached symbiotic channel may contribute up to 5% of all SNe Ia in the Galaxy. We note that the SN Ia rate increases with αCEλ. This is because the orbital period of the binaries evolving from CE ejections would be larger for a larger value of αCEλ, resulting in more WD+RG systems located in the SN Ia production region in Fig.3. For comparison, the tra- ditional method for calculating the mass-transfer rate yields a Fig. 6. Delay time distributions of SNe Ia from the symbiotic channel, contribution of only 1% under the same star formation rate (i.e., in which a star burst of 1010 M in stars is adopted. The open circles −1 5 M yr ; e.g., Han & Podsiadlowski 2004; Wang et al. 2010). are from Totani et al.(2008), and the filled triangles, the filled circles, Obviously, our model increases the rate of SNe Ia from the sym- the filled squares, and the open square represent, respectively, observed biotic channel by a factor of five. However, the rate of SNe Ia results from Maoz et al.(2010, 2011, 2012) and Graur & Maoz(2013), all of which have been rescaled by Maoz & Graur(2017). arising from the semidetached symbiotic channel is still low, as most WD+RG systems lie outside of the initial parameter space for producing SNe Ia. Further investigation is required if in which we assume αset ≈ 0.07, m ≈ 1.2, a0 = 10 R and the symbiotic channel is the primary source of SNe Ia. Other- 6 a1 = 5.75 × 10 R . (5) The standard energy prescription from wise, some other explosion mechanisms or formation channels Webbink(1984) is employed to describe the CE ejection pro- are required (e.g., Tout 2005; Wang & Han 2012). cess, in which the uncertain parameters αCE and λ are com- Figure6 shows the theoretical delay time distributions of 10 bined as a single parameter and set to be αCEλ = 0.5, 1.0, SNe Ia from the symbiotic channel. A single starburst of 10 M and 1.5 for comparison. (6) The star formation rate is adopted in stars is assumed here. According to the symbiotic channel, the −1 to be constant (5 M yr ) for the Galaxy over the past 15 Gyr SN Ia delay times range from about 400 Myr–10 Gyr, indicating (see Yungelson & Livio 1998; Willems & Kolb 2004; Han & that this channel mainly contributes to the observed SNe Ia in Podsiadlowski 2004), or alternatively, modeled as a delta func- the intermediate and old populations. We note that Wang et al. 10 tion (a single star burst of 10 M in stars). (2010) suggested that the symbiotic channel only contributes to The formation channel of WD+RG systems is similar to that SN Ia rates in old populations. The present work extends the in Wang et al.(2010). The primordial primary fills its Roche- contribution of the symbiotic channel to the SN Ia rates in the lobe when it evolves to the thermal pulsing asymptotic giant intermediate populations. branch. In this case, the mass transfer is dynamically unstable, Figure7 presents the density distribution of WD +RG sys- i i leading to the formation of a CE. If the CE can be ejected, the tems that produce SNe Ia in the log P −M2 plane. For a compar- primordial primary becomes a CO WD. After that, a WD+RG ison, we also show the initial contour for producing SNe Ia with i system will be formed when the primordial secondary evolves MWD = 1.2 M . From this figure, we can see that the orbital to its RG phase. The parameters of primordial binaries for pro- periods of the majority of WD+RG systems are mainly dis- ducing SNe Ia via the symbiotic channel are M1,i ∼ 5.0−6.5 M , tributed from about 15 d to the right boundary of the initial i 0.15 < M2,i/M1,i < 0.5, and P ∼ 600−5000 days. contour, and the donor masses lie in the range from 1.0 M to

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1.8 M . The contribution of the remaining space to the SN Ia that the relative rates increase due to the new mass-transfer rates is negligible. Here, αCEλ is adopted to be 1.5. We note that prescription. the absolute number of WD+RG systems will decrease and the There are some other alternative paths for producing SNe relative number will not change significantly if we adopt a lower Ia via the symbiotic channel, such as for example the mass- value of αCEλ. stripping model, the aspherical stellar wind model, the tidally enhanced stellar wind model, and so on. (1) Hachisu et al.(1999) proposed a mass-stripping model to 4. Discussion stabilize the mass-transfer process and avoid the formation In the present work, we adopted an improved method to calcu- of a CE, in which the stellar wind from the WD collides late the mass-transfer rate, and found that the parameter space with the RG surface and strips some of the mass from the for producing SNe Ia via the semidetached symbiotic channel RG. We note that this model can reproduce the basic features is significantly enlarged. The mass-transfer prescription adopted of the observed light curves of two supersoft X-ray sources here is based on a power-law adiabatic assumption, which is still (RX J0513.9−6951 and V Sagittae), although the predicted under debate. Woods & Ivanova(2011) argued that the enve- light curves may be influenced by a coefficient defined as the lope of the RG will not expand adiabatically, when the mass-loss ratio of the mass-stripping rate to the optically thick wind- timescale is comparable with the local thermal timescale of the mass-loss rate (see Hachisu & Kato 2003a,b). superadiabatic outer surface layer of the envelope. In this case, (2) Lü et al. 2009 assumed an aspherical stellar wind with an our model might underestimate M˙ 2, and thus we might obtain an equatorial disk from a RG to investigate the symbiotic chan- upper limit of the parameter space for producing SNe Ia. nel of SNe Ia, leading to a higher SN Ia rate. However, the The Galactic SN Ia rates calculated by the revised mass- results of Lü et al. 2009 are strongly affected by the mass- transfer prescription are five times larger than that when assum- loss rate and the outflow velocity of the equatorial disk. ing the traditional mass-transfer prescription. However, the (3) Chen et al. 2011 adopted the tidally enhanced stellar wind present work only provides an upper limit for the SN Ia rate from assumption presented by Tout & Eggleton(1988) to study the symbiotic channel. There are still some uncertainties in our the symbiotic channel, and found that the parameter space BPS calculations: for producing SNe Ia was enlarged. Chen et al.(2011) esti- (1) From Fig.5, we can see that di fferent CE models may result mated that the SN Ia rates from the symbiotic channel would − − in a difference in the SN rate that may modify it by a factor increase to be 6.9×10 3 yr 1 using Eq. (1) of Iben & Tutukov of three. Some recent studies indicate that the value of αCEλ (1984). We note that Eq. (1) of Iben & Tutukov(1984) may for WD+MS systems may be much lower than 1 (∼0.25; overestimate the rate since some parameter space for produc- e.g., Zorotovic et al. 2010, 2014; Toonen & Nelemans 2013; ing SNe Ia may not contribute to the SN Ia rate. Importantly, Camacho et al. 2014). The rates of SNe Ia would decrease to the rate obtained by Chen et al.(2011) strongly depends on a −4 −1 ∼0.4 × 10 yr if αCEλ were adopted to be 0.25. tidal wind enhancement parameter Bw, which is still poorly (2) We adopted a constant initial-mass ratio distribution in our understood and may critically influence the predicted SN Ia calculations and note that the predicted rates of SNe Ia will rate; the parameter space for producing SNe Ia would shrink decrease sharply when an extreme uncorrelated mass ratio dramatically if a lower value of Bw were adopted (see Chen distribution is adopted (e.g., Wang et al. 2010). et al. 2011). (3) In Fig.5, we assumed a constant star formation rate of In the symbiotic channel, some of the matter would be blown −1 5 M yr for the Galaxy over the past 15 Gyr, which has away from the system before the SN explosion, mainly due to the been calibrated by assuming that a binary with its primary optically thick wind when M˙ 2 > M˙ cr. This blown-away matter mass larger than 0.8 M is formed each year (see Iben & would remain as circumstellar matter. We found that the circum- Tutukov 1984; Han et al. 1995; Hurley et al. 2002). How- stellar matter in SN Ia explosions has masses of up to ∼1.0 M . ever, recent studies suggested that the current Galactic star Some direct evidence has been found for the existence of cir- −1 formation rate is about 1.9 ± 0.4 M yr (see Chomiuk & cumstellar matter in the normal SN Ia SN 2006X (e.g., Patat Povich 2011). If the Galactic star formation rate is adopted et al. 2007). The symbiotic channel may be responsible for SN −1 to be 2 M yr , the Galactic rates of SNe Ia from the symbi- 2006X-like events (see also Patat et al. 2007; Voss & Nelemans otic channel decrease to ∼0.2−0.6 × 10−4 yr−1, contributing 2008). to at most 2% of all SNe Ia in the Galaxy. In the observations, there are many symbiotic novae that (4) We note that the binary fraction may vary with the primor- are progenitor candidates of SNe Ia. For instance, RS Oph and dial primary mass (e.g., van Haaften et al. 2013). In the T CrB are two symbiotic systems that both consist of a mas- present work, we simply assume that about 50% of binary sive WD and a lobe-filling RG star. RS Oph has a 1.2−1.4 M systems have orbital periods shorter than 100 yr, and that WD and a 0.68−0.80 M RG star with an orbital period of there is an equal binary number per logarithmic interval for 454.1 ± 0.41 days (Brandi et al. 2009), and T CrB has a ∼1.2 M wide systems with orbital periods longer than 100 yr. If the WD and a ∼0.7 M RG star with an orbital period of ∼227.6 days binary fraction is less than 50%, the SN Ia rate may also be (Kraft 1958; Belczynski´ & Mikołajewska 1998). Recently, overestimated. Mikołajewska & Shara(2017) suggested that the WD in RS Oph (5) The age of the Galactic disk is around 10 Gyr, which is less is a CO WD according to their analysis of its spectra, which than the Hubble timescale adopted in Fig.5. However, the strongly supports the idea that RS Oph will form an SN Ia. value of this timescale has almost no influence on the final The binary parameters of RS Oph and T CrB are located in the results as the Galactic rates of SNe Ia are almost constant regions for producing SNe Ia in Fig.3, which indicates that they once t is larger than 10 Gyr. can form SNe Ia in their future evolution via the semidetached Generally, if the different assumptions listed above are adopted symbiotic channel. in our BPS calculations, the Galactic SN Ia rate may also Some published observations support the existence of sin- decrease by a factor of five, similar to that due to the differ- gle low-mass He WDs with masses lower than 0.45 M (e.g., ent mass-transfer prescription. However, the argument still holds Marsh et al. 1995; Kilic et al. 2007), which may correspond to

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