A strange dwarf scenario for the formation of the peculiar double white dwarf binary SDSS J125733.63+542850.5

A strange dwarf scenario for the formation of the peculiar double white dwarf binary SDSS... Abstract The Hubble Space Telescope observation of the double white dwarf (WD) binary SDSS J125733.63+542850.5 reveals that the massive WD has a surface gravity log g1 ∼ 8.7 (which implies a mass of M1 ∼ 1.06 M⊙) and an effective temperature T1 ∼ 13 000 K, while the effective temperature of the low-mass WD (M2 < 0.24 M⊙) is T2 ∼ 6400K. Therefore, the massive and the low-mass WDs have a cooling age τ1 ∼ 1 Gyr and τ2 ≥ 5 Gyr, respectively. This is in contradiction with traditional binary evolution theory. In this paper, we propose a strange dwarf (SD) scenario to explain the formation of this double WD binary. We assume that the massive WD is a SD originating from a phase transition (PT) in a ∼1.1 M⊙ WD, which has experienced accretion and spin-down processes. Its high effective temperature could arise from the heating process during the PT. Our simulations suggest that the progenitor of SDSS J125733.63+542850.5 can be a binary system consisting of a 0.65 M⊙ WD and a 1.5 M⊙ main-sequence star in a 1.492 d orbit. Especially, the secondary star (i.e. the progenitor of the low-mass WD) is likely to have an ultra-low metallicity of Z = 0.0001. binaries: close, stars: evolution, stars: individual: SDSS J125733.63+542850.5, white dwarfs 1 INTRODUCTION As the end products of binary evolution, double white dwarf (WD) binaries are good probes for testing stellar and binary evolutionary theory (Marsh, Dhillon & Duck 1995; Toonen et al. 2014). Especially, they are thought to be progenitors of Type Ia supernovae (Iben & Tutukov 1984; Webbink 1984), AM CVn systems (Camilo et al. 1996; Kilic et al. 2014), and R CrB stars (Webbink 1984). Furthermore, close double WDs are believed to be the main Galactic gravitational sources in the frequency range of 10−4 to 0.1 Hz, which will be detected by the Laser Interferometer Space Antenna detector (Hils, Bender & Webbink 1990; Nelemans, Yungelson & Portegies Zwart 2001; Hermes et al. 2012). Based on the Sloan Digital Sky Survey (SDSS; York et al. 2000; Eisenstein et al. 2006) subspectra, the double WD binary SDSS J125733.63+542850.5 (hereafter J1257) was first discovered by the Sloan WD Radial velocity Mining Survey (Badenes et al. 2009). Its radial velocity variations with a semi-amplitude of 323 km s − 1 were interpreted to originate from a 0.9 M⊙ WD while the companion was suggested to be a neutron star (NS) or black hole (Badenes et al. 2009). The two distinct components were revealed in the spectra of B- and R-band spectroscopy, and the Balmer lines with deep radial velocity variable were identified to come from a cool, extremely low-mass WD with mass less than 0.3 M⊙ (Kulkarni & van Kerkwijk 2010; Marsh et al. 2011). Recently, Bours et al. (2015) fit both the Hubble Space Telescope Cosmic Origins Spectrograph and the Space Telescope Imaging Spectrograph spectra, and the SDSS ugriz flux with a Markov Chain Monte Carlo approach. Their results indicate that the massive component has a surface gravity log g1 ∼ 8.73 ± 0.05, and an effective temperature T1 ∼ 13 030 ± 70 K. Detailed evolutionary models reveal the mass to be M1 = 1.06 ± 0.05 M⊙, and a corresponding cooling age of τ1 = 1.0 or 1.2 Gyr for carbon/oxygen and oxygen/neon WD models, respectively (Kowalski & Saumon 2006; Althaus et al. 2007; Tremblay, Bergeron & Gianninas 2011). However, the low-mass WD with M2 < 0.24 M⊙ has a temperature of T2 ∼ 6400 K and a cooling age of τ2 ≥ 5 Gyr (Marsh et al. 2011; Bours et al. 2015). The two cooling ages are in contradiction with binary stellar evolutionary theory. The progenitor of the low-mass He WD probably had a mass of 1–2 M⊙ (Istrate, Tauris & Langer 2014), and should have evolved much more slowly than 5–6 M⊙ progenitor of the more massive WD. After the formation of the low-mass WD, CNO flashes could cause it to fill its Roche lobe, and accretion heating would alter the thermal structure of the massive WD with a duration of ∼ 106 yr (Bildsten et al. 2006). However, this mass transfer time-scale ( ∼ 100 yr) is too short to influence the cooling history of the massive WD. In this paper, we propose that the difference in the cooling ages originate from the heating process during the formation of a strange dwarf (SD). We describe the SD scenario in Section 2. In Section 3, we discuss the changes of the orbital parameters during phase transition and the possible spin-down process of the massive WD. Employing the mesa code, we simulate the evolutionary history of J1257 in Section 4. A brief summary and discussion are presented in Section 5. 2 STRANGE DWARF SCENARIO Based on the hypothesis that the strange quark matter may be the most stable state of matter, Witten (1984) proposed that the pulsars may be strange quark stars (SSs) rather than NSs. Following this idea, the concept of SDs was introduced by Glendenning et al. (1995a) and Glendenning, Kettner & Weber (1995b) as strange counterpart of WDs. They pointed out that the inner density of ordinary stable WDs is always below critical density ρc ∼ 109 g cm3, which is the central density of a maximum-mass (∼1 M⊙) WD (Baym, Pethick & Sutherland 1971). The structure and thermal evolution of SDs with different masses have been studied by Benvenuto & Althaus (1996) in detail. Their computation indicates that the thermal evolution of SD with mass larger than 1 M⊙ is similar to that of a WD with the same mass. Following these studies, we assume that if the central density of a WD exceeds a critical density, a strange quark core will emerge in the centre region and the star evolves to be a SD. During this phase transition (PT, hereafter) process the mass of star will decrease slightly (about several percent). Part of them will translate to the binding energy of the more compact SD, and the other is lost from the star. The energy transportation during the process may heat the star and result in a hotter SD. Under the assumptions mentioned above, we outline the evolutionary stages of J1257 as follows, which is also illustrated in Fig. 1. We start from a compact binary (with an orbital period Pi ∼ 1.5  d), which consists of a WD with M1 ∼ 0.6–0.8 M⊙ and a main-sequence star with M2 ∼ 1.5 M⊙. Roche lobe overflow. The WD accretes around 0.3–0.5 M⊙ material from the donor star, and spins up to nearly breakup rotation. The spin-up process would also significantly broaden the hydrogen line profiles, as reported by Kulkarni & van Kerkwijk (2010). Formation of the second WD. The mass transfer terminates and the secondary star evolves into a low-mass WD. The central density of the primary WD, which is centrifugally diluted by the fast spinning due to accretion, starts to spin down. Meanwhile the second WD gradually cools. PT. After several Gyr, the massive WD’s spin-down causes its density to be above the critical density, thus PT takes place in its core and a nascent SD is heated up to 108K. Cooling of the massive SD. According to Benvenuto & Althaus (1996), the cooling process of the SD is similar to a normal WD with the same mass. Figure 1. View largeDownload slide Illustration of our SD evolution from the compact WD binary to the present observed system J1257. Figure 1. View largeDownload slide Illustration of our SD evolution from the compact WD binary to the present observed system J1257. The temperature of the nascent SD can be estimated as follows. Assuming that the difference between the gravitational masses of a WD and a SD with the same quark number is ΔM, and that a fraction α (<1) of the rest energy of the mass-loss during PT was used to heat the nascent SD, the thermal energy received by the SD is   \begin{eqnarray} Q=\alpha \Delta Mc^2. \end{eqnarray} (1) The internal energy of the SD with a temperature T can be estimated as   \begin{eqnarray} U_{\rm SD}=\frac{3}{2}{k_{\rm B}}NT, \end{eqnarray} (2)where kB is Boltzmann’s constant and N = MSD/Amu is the total number of quarks and electrons. Here, MSD is the mass of the SD, mu is the atomic mass unit, A is the relative particle mass. Following Alcock, Farhi & Olinto (1986), we assume that the SD is composed by roughly equal numbers of up, down, and strange quarks, and a small number (ignorable) electrons. Considering that the mass of SD decreased a little during PT while the quark number keeps constant, we take A ∼ 0.3. Since the effective temperature of the WD before PT was much lower than the nascent SD, the internal energy of WD UWD (≪USD) is ignorable. Because Q = USD − UWD ≈ USD, the temperature of the nascent SD can be written as   \begin{eqnarray} T=\frac{2}{3}\frac{\alpha \Delta Mc^2}{k_{\rm B}N}=\frac{2}{3}\frac{\alpha \Delta Mc^2Am_{\rm u}}{k_{\rm B}M_{\rm SD}}. \end{eqnarray} (3) PT between NS and SS has been extensively investigated for different equations of state. Several works suggested that the difference between the gravitational masses a NS and an SS with the same baryon number is roughly MNS − MSS ≈ 0.15 M⊙ for a NS with mass ∼1.5 M⊙ (e.g. Bombaci & Datta 2000; Drago, Lavagno & Parenti 2007; Marquez & Menezes 2017).1 Considering the difference in the mass and compactness between a WD and a NS, in this work we take ΔM ∼ 0.05 M⊙. Considering that most of the energy liberated during PT is assumed to be taken away by neutrinos (and antineutrinos), similar as in supernova explosions (Kippenhahn, Weigert & Weiss 2012), and only a small fraction is used to heat the nascent SD, we set α ∼ 0.001 as lower limit. Taking A ∼ 0.3, and MSD = 1.05 M⊙, the nascent SD had an initial effective temperature T ≃ 108 K. In comparison, the temperature of a 1.0 M⊙ WD is < 107K. 3 ORBITAL CHANGE DURING PT AND THE SPIN-DOWN OF PROGENITOR WD 3.1 Orbital change during PT We first discuss the influence of PT on the eccentricity of the binary. Considering that the PT in the core of WD took place quickly, and a kick velocity Vk was imparted to the new born SD, one can solve the orbital parameters during PT following Shao & Li (2016). Due to the mass transfer with a long duration, the orbit of the binary before PT can be thought to be circular. Setting ϕ to be the positional angle of Vk with respect to the pre-PT orbital plane and θ the angle between Vk and the pre-PT orbital velocity V0 (=(2πGM0/Porb, 0)1/3), the ratio between the semimajor axes before and after PT is (Hills 1983; Dewi & Pols 2003)   \begin{eqnarray} \frac{a_0}{a}=2-\frac{M_{0}}{M_{0}-\Delta {M}}(1+\nu +2\nu {\rm cos} \theta ), \end{eqnarray} (4)where ν = Vk/V0, and M0 and Porb, 0 are the total mass of the binary and orbital period of the binary before PT, respectively. Under the influence of mass-loss and kick, the eccentricity after PT can be written as (Hills 1983; Dewi & Pols 2003)   \begin{eqnarray} 1-e^2=\frac{a_{\rm 0}M_{\rm 0}}{a({M_{0}-\Delta {M})}}[1+ 2\nu {\rm cos} \theta + \nu ^2({\rm cos}^2\theta + {\rm sin}^2\theta {\rm sin}^2\phi )].\nonumber\\ \end{eqnarray} (5) Taking M0 = M2 + M1 = 1.3 M⊙, ΔM = 0.05 M⊙, Porb, 0 = 0.22 d, we simulated the possibility of small eccentricities (e < 0.01) after PT for different Vk in the range of 0–50 km s−1. For each Vk, we set 107 independent random values for cosθ and ϕ of uniform distribution in the interval of −1 to 1 and 0 to π, respectively. According to the observations of Badenes et al. (2009) and Marsh et al. (2011), the current orbit of J1257 is circular and a WD binary with e < 0.01 could evolve into a circular orbit on a time-scale of ∼1 Gyr. Fig. 2 shows the possibility distribution for the eccentricity less than 0.01 with different kick velocities. When Vk ≤ 5 km s−1 and Vk ≥ 50 km s−1, the possibilities with e < 0.01 are less than 0.1 per cent and 0.2 per cent, respectively. However, the relevant possibility is ≥ 1 per cent when 6 ≤ Vk ≤ 20 km s−1. Especially, the relevant possibility is as high as 10 per cent for a kick velocity range of 8–9 km s − 1. Similar to accretion induced collapse of NS (Hurley et al. 2010), the nascent SD should obtain a low kick velocity. Therefore, PT process has a relatively large possibility to result in a nearly circular orbit. Figure 2. View largeDownload slide Possibility distribution for the eccentricity less than 0.01 under different values of the kick velocity. Figure 2. View largeDownload slide Possibility distribution for the eccentricity less than 0.01 under different values of the kick velocity. According to the relation between the pre-PT and the post-PT orbital separation a0/(1 + e) ≤ a ≤ a0/(1 − e) (Flannery & van den Heuvel 1975), one can derive that the change of orbital period is ≤ 2 per cent when e < 0.01. Since the changes are relatively small, we ignore the orbital change of the binary during PT in our simulation. 3.2 Spin-down of massive WD Similar to pulsars, we consider the spin-down of WD with an angular velocity Ω = 2π/Ps is dominated by magnetic dipole radiation,2 and the energy loss rate is   \begin{eqnarray} \dot{E}_{\rm d}=-\frac{2}{3c^{3}}\mu ^{2}\Omega ^{4}, \end{eqnarray} (6)where $$\mu =BR^3=B_{7}R_{9}^{3}\times 10^{34}\,{\rm G \,cm^3}$$ is the magnetic dipole moment, and B7 is the surface magnetic field in units of 107 G and R9 is the radius in units of 109 cm of the WD. The rotational energy of WD changes at a rate   \begin{eqnarray} \dot{E}_{\rm s}=I\Omega \dot{\Omega }, \end{eqnarray} (7)where $$I\sim MR^2\approx 10^{51}R_{9}^{2}\,{\rm g \,cm}^{2}$$ is the moment of inertia of WD. If we assume that the braking torque of the WD fully originate from the magnetic dipole radiation, the spin period of the WD changes at a rate   \begin{eqnarray} \dot{P}_{\rm s}=\frac{8\pi ^2}{3c^3}\frac{\mu ^2}{IP_{\rm s}}=K/{P_{\rm s}}, \end{eqnarray} (8)where $$K=8\pi ^2\mu ^2/3c^3I\sim B_{7}^{2}R_{9}^{4}\times {10^{-13}}\,{\rm s}$$. With simple integration, one can get the spin-down time-scale of the WD from the initial spin period Ps, 0 to the spin period of PT Ps:   \begin{eqnarray} \tau _{\rm SD}=\frac{P_{\rm s}^{2}-P_{\rm s,0}^{2}}{2K}\approx \frac{P_{\rm s}^{2}-P_{\rm s,0}^{2}}{6B_{7}^{2}R_{9}^{4}}{\rm Myr}. \end{eqnarray} (9) Based on the theory of accretion disc–magnetic field interaction developed by Ghosh & Lamb (1979), Kulkarni & van Kerkwijk (2010) inferred that the magnetic field of J1527 was ∼105 G when it spins up to its current spin period of ∼60 s. However, Cumming (2002) showed that rapid accretion could reduce the field strength at the surface of the accreting WD because the field is advected into the interior by the accretion flow. Therefore, many non-magnetic WDs (B ≲ 105 G) may have submerged magnetic fields when they were accreting at rates greater than the critical rate $$\dot{M}_{\rm cr}=(1\text{--}5)\times 10^{-10}\,\rm M_{\odot }\,yr^{-1}$$, and the magnetic field would re-emerge when the mass transfer terminates, and the re-emergence time-scale of the field is   \begin{eqnarray} \tau _{\rm re}\simeq 300\times \left(\frac{\Delta M_{\rm acc}}{0.1\,{\rm M_{\odot }}}\right)^{7/5}{\rm Myr}, \end{eqnarray} (10)where ΔMacc is the accreted mass of the WD. Based on the model of Cumming (2002) for magnetic field evolution, we propose that the surface field of the more massive WD in J1257 decreased from ∼107 to ∼105 G due to rapid accretion. According to the simulation in the next section, the accreted mass of the massive WD is ΔMacc ≈ 0.45 M⊙, so the re-emergence time-scale is τre ≈ 2.5 Gyr. Taking Ps, 0 = 10 s (close to breakup rotation), Ps = 60 s, B7 = 1, and R9 = 0.6, one can derive that the spin-down time-scale of the massive WD is about 4.5 Gyr. Considering the re-emergence time-scale of the magnetic field τre ≈ 2.5 Gyr, the total duration of 7 Gyr before the PT for the massive WD is approximately in agreement with the cooling age of the low-mass WD. 4 NUMERICAL SIMULATION Using the mesa module (Paxton et al. 2015), we have simulated the evolution of the compact WD binary consisting of a WD and a main-sequence star to test whether it is possible to reproduce the characteristics of J1257. According to the estimation in the previous section, the orbital change during PT can be neglected while the mass growth of the WD is considered. For an accreting WD, hydrogen and helium shell flashes always trigger nova outbursts, which blow off the accreted matter and even result in convective dredge-up. Therefore, the mass accumulation efficiency for accreting hydrogen should be less than 1. During the mass transfer, the mass growth rate of the accreting WD is described as follows:   \begin{eqnarray} \dot{M}_{\rm 1}=\eta _{\rm He}\eta _{\rm H}|\dot{M}_{2}|, \end{eqnarray} (11)where $$\dot{M}_{2}$$ is the mass transfer rate of the donor star, ηH and ηHe are the accumulation efficiencies during hydrogen burning and helium burning, respectively. For the accumulation efficiency of hydrogen, a prescription given by Hachisu et al. (1999) and Han & Podsiadlowski (2004) was adopted, i.e.   \begin{eqnarray} \eta _{\rm H}= \left\lbrace \begin{array}{@{}l@{\quad }l@{}}\dot{M}_{\rm cr}/|\dot{M}_{2}| &\text{$|\dot{M}_2|>\dot{M}_{\rm cr}$}, \\ 1 &\text{$\dot{M}_{\rm cr}>|\dot{M}_2|>0.125\dot{M}_{\rm cr}$},\\ 0 &\text{$|\dot{M}_2|<0.125\dot{M}_{\rm cr}$}. \end{array}\right. \end{eqnarray} (12)In equation (12), $$\dot{M}_{\rm cr}$$ is a critical mass-accretion rate:   \begin{eqnarray} \dot{M}_{\rm cr}=5.3\times 10^{-7}\frac{1.7-X}{X}(M_{1}-0.4)\,{\rm M}_{{\odot }}\,{\rm yr}^{-1}, \end{eqnarray} (13)where X is the mass abundance of hydrogen in the accreted matter. For the accumulation efficiency ηHe during helium burning, the prescriptions given by Kato & Hachisu (2004) was adopted. The mass-loss $$(1-\eta _{\rm H}\eta _{\rm He})\dot{M}_{2}$$ during hydrogen and helium burning is assumed to be ejected in the vicinity of the WD in the form of isotropic winds, carrying away the specific angular momentum of the WD (Hachisu, Kato & Nomoto 1996; Soberman, Phinney & van den Heuvel 1997). In addition, we also consider angular momentum loss caused by gravitational radiation and magnetic braking (Verbunt & Zwaan 1981 with γ = 3.0; Rappaport, Verbunt & Joss 1983; Paxton et al. 2015). To study the progenitor properties of J1257, we simulated the evolution of a large number of WD binaries. The relevant binaries would be thought to be the progenitor candidates of J1257 if the following three conditions are satisfied: (1) the current orbital period is 4.6 h when the age of the system is within Hubble time; (2) the binary evolves into a detached system when the accreting WD’s mass is about 1.05–1.15 M⊙ (because the WD must experience a long-term spin-down process, hence the donor star should not overflow its Roche lobe when the WD mass increases to 1.05–1.15 M⊙); and (3) the effective temperature of the donor star is near 6400 K. Figs 2–4 show an example of evolution in which the initial donor mass and the initial WD mass are 1.5 and 0.65 M⊙, respectively. We change the initial metallicity in order to fit the observed parameters of J1257. As shown in Fig. 3, because the material is transferred from the more massive donor star to the less massive WD, the orbital period first decreases. With the reversal of the mass ratio, the orbital period increases until the binary evolve into a detached system. Subsequently, the donor star gradually evolves into a WD and enters the cooling stage, and magnetic braking and gravitational radiation induce a compact double WD binary. Once the mass transfer ceases, the massive WD spins down due to magnetic dipole radiation, then triggers PT during the cooling of the low-mass WD. The heating process during PT results in the formation of a hot SD. Similar to Fig. 4, the simulated effective temperatures of the low-mass WD are always higher than the observation in the Hubble time except for z = 0.0001, in which the donor star orbits a WD with an initial orbital period of 1.492 d. Fig. 5 shows the evolutionary tracks of the donor-star mass and the accreting WD’s mass. It is clear that our simulated donor-star masses are consistent with the observed data for three different metallicities. In calculation, the donor star with higher metallicity would produce systems with lower mass secondary WDs. These differences should arise from the metallicity dependence of the stellar wind mass-loss, which tend to reduce the stellar mass of donor star. Figure 3. View largeDownload slide Evolution of orbital period for WD binaries including a 1.5 M⊙ donor star and a 0.65 M⊙ WD. The solid, dashed, and dotted curves represent the metallicity z = 0.0001, 0.0005, and 0.001, respectively. Numbers beside the curves denote the initial orbital periods. Figure 3. View largeDownload slide Evolution of orbital period for WD binaries including a 1.5 M⊙ donor star and a 0.65 M⊙ WD. The solid, dashed, and dotted curves represent the metallicity z = 0.0001, 0.0005, and 0.001, respectively. Numbers beside the curves denote the initial orbital periods. Figure 4. View largeDownload slide Evolutionary tracks of the effective temperature of donor star for WD binaries including a 1.5 M⊙ donor star and a 0.65 M⊙ WD. Three cases are same to Fig. 3. Figure 4. View largeDownload slide Evolutionary tracks of the effective temperature of donor star for WD binaries including a 1.5 M⊙ donor star and a 0.65 M⊙ WD. Three cases are same to Fig. 3. Figure 5. View largeDownload slide Evolution of the donor-star mass (top panel) and the WD mass (bottom panel) for WD binaries including a 1.5 M⊙ donor star and a 0.65 M⊙ WD. Three cases are same to Fig. 3. Figure 5. View largeDownload slide Evolution of the donor-star mass (top panel) and the WD mass (bottom panel) for WD binaries including a 1.5 M⊙ donor star and a 0.65 M⊙ WD. Three cases are same to Fig. 3. 5 SUMMARY AND DISCUSSION Assuming both WD and SD are different stages of stellar evolution, in this work we propose a SD scenario to interpret the puzzle of the cooling age of two WD in J1257. The massive WD is thought to be a SD originating from the PT of the 1.05–1.15 M⊙ WD, thus its higher effective temperature can be interpreted as a result of heating during PT. A simple estimation indicates that a mass-loss ΔM ∼ 0.05 M⊙ during PT can heated the nascent SD up to 108 K. Based on these assumptions, we use the mesa code to simulate the evolution of a large number of WD binaries consisting of a 0.65 M⊙ WD and a 1.5 M⊙ main-sequence star for different initial orbital period and metallicities. Our simulation indicates that metallicities have important influence on the effective temperature of the donor star. When z = 0.0001, the calculated orbital period, the donor-star mass, and the effective temperature of the donor star are consistent with the observed data. Therefore, we propose that the PT of a massive WD may be responsible for the puzzling cooling age of two WDs. We expect further detailed multiwaveband observations for this source to obtain more precise constraints. ACKNOWLEDGEMENTS This work was supported by the National Natural Science Foundation of China under grant numbers 11573016, 11733009, 11773015, 11333004, U1731103, 11463004 and 11605110, the National Key Research and Development Program of China (2016YFA0400803), the Scientific Research Fund of Hunan Provincial Education Department (No. 16B250 & 16C1531), and the Program for Innovative Research Team (in Science and Technology) at the University of Henan Province. Footnotes 1 Lower value is also possible, for example, the gravitational mass between NS and hyperon star given by Schaffner-Bielich et al. (2002) is ∼ 0.03 M⊙. 2 Gravitational wave radiation might be an efficient mechanism extracting angular momentum from fast rotating WDs due to r-mode instability in a short time-scale ( < 108 yr, Yoon & Langer 2004). REFERENCES Alcock C., Farhi E., Olinto A., 1986, ApJ , 310, 261 https://doi.org/10.1086/164679 CrossRef Search ADS   Althaus L. G., García-Berro E., Isern J., Córsico A. 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A strange dwarf scenario for the formation of the peculiar double white dwarf binary SDSS J125733.63+542850.5

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Abstract

Abstract The Hubble Space Telescope observation of the double white dwarf (WD) binary SDSS J125733.63+542850.5 reveals that the massive WD has a surface gravity log g1 ∼ 8.7 (which implies a mass of M1 ∼ 1.06 M⊙) and an effective temperature T1 ∼ 13 000 K, while the effective temperature of the low-mass WD (M2 < 0.24 M⊙) is T2 ∼ 6400K. Therefore, the massive and the low-mass WDs have a cooling age τ1 ∼ 1 Gyr and τ2 ≥ 5 Gyr, respectively. This is in contradiction with traditional binary evolution theory. In this paper, we propose a strange dwarf (SD) scenario to explain the formation of this double WD binary. We assume that the massive WD is a SD originating from a phase transition (PT) in a ∼1.1 M⊙ WD, which has experienced accretion and spin-down processes. Its high effective temperature could arise from the heating process during the PT. Our simulations suggest that the progenitor of SDSS J125733.63+542850.5 can be a binary system consisting of a 0.65 M⊙ WD and a 1.5 M⊙ main-sequence star in a 1.492 d orbit. Especially, the secondary star (i.e. the progenitor of the low-mass WD) is likely to have an ultra-low metallicity of Z = 0.0001. binaries: close, stars: evolution, stars: individual: SDSS J125733.63+542850.5, white dwarfs 1 INTRODUCTION As the end products of binary evolution, double white dwarf (WD) binaries are good probes for testing stellar and binary evolutionary theory (Marsh, Dhillon & Duck 1995; Toonen et al. 2014). Especially, they are thought to be progenitors of Type Ia supernovae (Iben & Tutukov 1984; Webbink 1984), AM CVn systems (Camilo et al. 1996; Kilic et al. 2014), and R CrB stars (Webbink 1984). Furthermore, close double WDs are believed to be the main Galactic gravitational sources in the frequency range of 10−4 to 0.1 Hz, which will be detected by the Laser Interferometer Space Antenna detector (Hils, Bender & Webbink 1990; Nelemans, Yungelson & Portegies Zwart 2001; Hermes et al. 2012). Based on the Sloan Digital Sky Survey (SDSS; York et al. 2000; Eisenstein et al. 2006) subspectra, the double WD binary SDSS J125733.63+542850.5 (hereafter J1257) was first discovered by the Sloan WD Radial velocity Mining Survey (Badenes et al. 2009). Its radial velocity variations with a semi-amplitude of 323 km s − 1 were interpreted to originate from a 0.9 M⊙ WD while the companion was suggested to be a neutron star (NS) or black hole (Badenes et al. 2009). The two distinct components were revealed in the spectra of B- and R-band spectroscopy, and the Balmer lines with deep radial velocity variable were identified to come from a cool, extremely low-mass WD with mass less than 0.3 M⊙ (Kulkarni & van Kerkwijk 2010; Marsh et al. 2011). Recently, Bours et al. (2015) fit both the Hubble Space Telescope Cosmic Origins Spectrograph and the Space Telescope Imaging Spectrograph spectra, and the SDSS ugriz flux with a Markov Chain Monte Carlo approach. Their results indicate that the massive component has a surface gravity log g1 ∼ 8.73 ± 0.05, and an effective temperature T1 ∼ 13 030 ± 70 K. Detailed evolutionary models reveal the mass to be M1 = 1.06 ± 0.05 M⊙, and a corresponding cooling age of τ1 = 1.0 or 1.2 Gyr for carbon/oxygen and oxygen/neon WD models, respectively (Kowalski & Saumon 2006; Althaus et al. 2007; Tremblay, Bergeron & Gianninas 2011). However, the low-mass WD with M2 < 0.24 M⊙ has a temperature of T2 ∼ 6400 K and a cooling age of τ2 ≥ 5 Gyr (Marsh et al. 2011; Bours et al. 2015). The two cooling ages are in contradiction with binary stellar evolutionary theory. The progenitor of the low-mass He WD probably had a mass of 1–2 M⊙ (Istrate, Tauris & Langer 2014), and should have evolved much more slowly than 5–6 M⊙ progenitor of the more massive WD. After the formation of the low-mass WD, CNO flashes could cause it to fill its Roche lobe, and accretion heating would alter the thermal structure of the massive WD with a duration of ∼ 106 yr (Bildsten et al. 2006). However, this mass transfer time-scale ( ∼ 100 yr) is too short to influence the cooling history of the massive WD. In this paper, we propose that the difference in the cooling ages originate from the heating process during the formation of a strange dwarf (SD). We describe the SD scenario in Section 2. In Section 3, we discuss the changes of the orbital parameters during phase transition and the possible spin-down process of the massive WD. Employing the mesa code, we simulate the evolutionary history of J1257 in Section 4. A brief summary and discussion are presented in Section 5. 2 STRANGE DWARF SCENARIO Based on the hypothesis that the strange quark matter may be the most stable state of matter, Witten (1984) proposed that the pulsars may be strange quark stars (SSs) rather than NSs. Following this idea, the concept of SDs was introduced by Glendenning et al. (1995a) and Glendenning, Kettner & Weber (1995b) as strange counterpart of WDs. They pointed out that the inner density of ordinary stable WDs is always below critical density ρc ∼ 109 g cm3, which is the central density of a maximum-mass (∼1 M⊙) WD (Baym, Pethick & Sutherland 1971). The structure and thermal evolution of SDs with different masses have been studied by Benvenuto & Althaus (1996) in detail. Their computation indicates that the thermal evolution of SD with mass larger than 1 M⊙ is similar to that of a WD with the same mass. Following these studies, we assume that if the central density of a WD exceeds a critical density, a strange quark core will emerge in the centre region and the star evolves to be a SD. During this phase transition (PT, hereafter) process the mass of star will decrease slightly (about several percent). Part of them will translate to the binding energy of the more compact SD, and the other is lost from the star. The energy transportation during the process may heat the star and result in a hotter SD. Under the assumptions mentioned above, we outline the evolutionary stages of J1257 as follows, which is also illustrated in Fig. 1. We start from a compact binary (with an orbital period Pi ∼ 1.5  d), which consists of a WD with M1 ∼ 0.6–0.8 M⊙ and a main-sequence star with M2 ∼ 1.5 M⊙. Roche lobe overflow. The WD accretes around 0.3–0.5 M⊙ material from the donor star, and spins up to nearly breakup rotation. The spin-up process would also significantly broaden the hydrogen line profiles, as reported by Kulkarni & van Kerkwijk (2010). Formation of the second WD. The mass transfer terminates and the secondary star evolves into a low-mass WD. The central density of the primary WD, which is centrifugally diluted by the fast spinning due to accretion, starts to spin down. Meanwhile the second WD gradually cools. PT. After several Gyr, the massive WD’s spin-down causes its density to be above the critical density, thus PT takes place in its core and a nascent SD is heated up to 108K. Cooling of the massive SD. According to Benvenuto & Althaus (1996), the cooling process of the SD is similar to a normal WD with the same mass. Figure 1. View largeDownload slide Illustration of our SD evolution from the compact WD binary to the present observed system J1257. Figure 1. View largeDownload slide Illustration of our SD evolution from the compact WD binary to the present observed system J1257. The temperature of the nascent SD can be estimated as follows. Assuming that the difference between the gravitational masses of a WD and a SD with the same quark number is ΔM, and that a fraction α (<1) of the rest energy of the mass-loss during PT was used to heat the nascent SD, the thermal energy received by the SD is   \begin{eqnarray} Q=\alpha \Delta Mc^2. \end{eqnarray} (1) The internal energy of the SD with a temperature T can be estimated as   \begin{eqnarray} U_{\rm SD}=\frac{3}{2}{k_{\rm B}}NT, \end{eqnarray} (2)where kB is Boltzmann’s constant and N = MSD/Amu is the total number of quarks and electrons. Here, MSD is the mass of the SD, mu is the atomic mass unit, A is the relative particle mass. Following Alcock, Farhi & Olinto (1986), we assume that the SD is composed by roughly equal numbers of up, down, and strange quarks, and a small number (ignorable) electrons. Considering that the mass of SD decreased a little during PT while the quark number keeps constant, we take A ∼ 0.3. Since the effective temperature of the WD before PT was much lower than the nascent SD, the internal energy of WD UWD (≪USD) is ignorable. Because Q = USD − UWD ≈ USD, the temperature of the nascent SD can be written as   \begin{eqnarray} T=\frac{2}{3}\frac{\alpha \Delta Mc^2}{k_{\rm B}N}=\frac{2}{3}\frac{\alpha \Delta Mc^2Am_{\rm u}}{k_{\rm B}M_{\rm SD}}. \end{eqnarray} (3) PT between NS and SS has been extensively investigated for different equations of state. Several works suggested that the difference between the gravitational masses a NS and an SS with the same baryon number is roughly MNS − MSS ≈ 0.15 M⊙ for a NS with mass ∼1.5 M⊙ (e.g. Bombaci & Datta 2000; Drago, Lavagno & Parenti 2007; Marquez & Menezes 2017).1 Considering the difference in the mass and compactness between a WD and a NS, in this work we take ΔM ∼ 0.05 M⊙. Considering that most of the energy liberated during PT is assumed to be taken away by neutrinos (and antineutrinos), similar as in supernova explosions (Kippenhahn, Weigert & Weiss 2012), and only a small fraction is used to heat the nascent SD, we set α ∼ 0.001 as lower limit. Taking A ∼ 0.3, and MSD = 1.05 M⊙, the nascent SD had an initial effective temperature T ≃ 108 K. In comparison, the temperature of a 1.0 M⊙ WD is < 107K. 3 ORBITAL CHANGE DURING PT AND THE SPIN-DOWN OF PROGENITOR WD 3.1 Orbital change during PT We first discuss the influence of PT on the eccentricity of the binary. Considering that the PT in the core of WD took place quickly, and a kick velocity Vk was imparted to the new born SD, one can solve the orbital parameters during PT following Shao & Li (2016). Due to the mass transfer with a long duration, the orbit of the binary before PT can be thought to be circular. Setting ϕ to be the positional angle of Vk with respect to the pre-PT orbital plane and θ the angle between Vk and the pre-PT orbital velocity V0 (=(2πGM0/Porb, 0)1/3), the ratio between the semimajor axes before and after PT is (Hills 1983; Dewi & Pols 2003)   \begin{eqnarray} \frac{a_0}{a}=2-\frac{M_{0}}{M_{0}-\Delta {M}}(1+\nu +2\nu {\rm cos} \theta ), \end{eqnarray} (4)where ν = Vk/V0, and M0 and Porb, 0 are the total mass of the binary and orbital period of the binary before PT, respectively. Under the influence of mass-loss and kick, the eccentricity after PT can be written as (Hills 1983; Dewi & Pols 2003)   \begin{eqnarray} 1-e^2=\frac{a_{\rm 0}M_{\rm 0}}{a({M_{0}-\Delta {M})}}[1+ 2\nu {\rm cos} \theta + \nu ^2({\rm cos}^2\theta + {\rm sin}^2\theta {\rm sin}^2\phi )].\nonumber\\ \end{eqnarray} (5) Taking M0 = M2 + M1 = 1.3 M⊙, ΔM = 0.05 M⊙, Porb, 0 = 0.22 d, we simulated the possibility of small eccentricities (e < 0.01) after PT for different Vk in the range of 0–50 km s−1. For each Vk, we set 107 independent random values for cosθ and ϕ of uniform distribution in the interval of −1 to 1 and 0 to π, respectively. According to the observations of Badenes et al. (2009) and Marsh et al. (2011), the current orbit of J1257 is circular and a WD binary with e < 0.01 could evolve into a circular orbit on a time-scale of ∼1 Gyr. Fig. 2 shows the possibility distribution for the eccentricity less than 0.01 with different kick velocities. When Vk ≤ 5 km s−1 and Vk ≥ 50 km s−1, the possibilities with e < 0.01 are less than 0.1 per cent and 0.2 per cent, respectively. However, the relevant possibility is ≥ 1 per cent when 6 ≤ Vk ≤ 20 km s−1. Especially, the relevant possibility is as high as 10 per cent for a kick velocity range of 8–9 km s − 1. Similar to accretion induced collapse of NS (Hurley et al. 2010), the nascent SD should obtain a low kick velocity. Therefore, PT process has a relatively large possibility to result in a nearly circular orbit. Figure 2. View largeDownload slide Possibility distribution for the eccentricity less than 0.01 under different values of the kick velocity. Figure 2. View largeDownload slide Possibility distribution for the eccentricity less than 0.01 under different values of the kick velocity. According to the relation between the pre-PT and the post-PT orbital separation a0/(1 + e) ≤ a ≤ a0/(1 − e) (Flannery & van den Heuvel 1975), one can derive that the change of orbital period is ≤ 2 per cent when e < 0.01. Since the changes are relatively small, we ignore the orbital change of the binary during PT in our simulation. 3.2 Spin-down of massive WD Similar to pulsars, we consider the spin-down of WD with an angular velocity Ω = 2π/Ps is dominated by magnetic dipole radiation,2 and the energy loss rate is   \begin{eqnarray} \dot{E}_{\rm d}=-\frac{2}{3c^{3}}\mu ^{2}\Omega ^{4}, \end{eqnarray} (6)where $$\mu =BR^3=B_{7}R_{9}^{3}\times 10^{34}\,{\rm G \,cm^3}$$ is the magnetic dipole moment, and B7 is the surface magnetic field in units of 107 G and R9 is the radius in units of 109 cm of the WD. The rotational energy of WD changes at a rate   \begin{eqnarray} \dot{E}_{\rm s}=I\Omega \dot{\Omega }, \end{eqnarray} (7)where $$I\sim MR^2\approx 10^{51}R_{9}^{2}\,{\rm g \,cm}^{2}$$ is the moment of inertia of WD. If we assume that the braking torque of the WD fully originate from the magnetic dipole radiation, the spin period of the WD changes at a rate   \begin{eqnarray} \dot{P}_{\rm s}=\frac{8\pi ^2}{3c^3}\frac{\mu ^2}{IP_{\rm s}}=K/{P_{\rm s}}, \end{eqnarray} (8)where $$K=8\pi ^2\mu ^2/3c^3I\sim B_{7}^{2}R_{9}^{4}\times {10^{-13}}\,{\rm s}$$. With simple integration, one can get the spin-down time-scale of the WD from the initial spin period Ps, 0 to the spin period of PT Ps:   \begin{eqnarray} \tau _{\rm SD}=\frac{P_{\rm s}^{2}-P_{\rm s,0}^{2}}{2K}\approx \frac{P_{\rm s}^{2}-P_{\rm s,0}^{2}}{6B_{7}^{2}R_{9}^{4}}{\rm Myr}. \end{eqnarray} (9) Based on the theory of accretion disc–magnetic field interaction developed by Ghosh & Lamb (1979), Kulkarni & van Kerkwijk (2010) inferred that the magnetic field of J1527 was ∼105 G when it spins up to its current spin period of ∼60 s. However, Cumming (2002) showed that rapid accretion could reduce the field strength at the surface of the accreting WD because the field is advected into the interior by the accretion flow. Therefore, many non-magnetic WDs (B ≲ 105 G) may have submerged magnetic fields when they were accreting at rates greater than the critical rate $$\dot{M}_{\rm cr}=(1\text{--}5)\times 10^{-10}\,\rm M_{\odot }\,yr^{-1}$$, and the magnetic field would re-emerge when the mass transfer terminates, and the re-emergence time-scale of the field is   \begin{eqnarray} \tau _{\rm re}\simeq 300\times \left(\frac{\Delta M_{\rm acc}}{0.1\,{\rm M_{\odot }}}\right)^{7/5}{\rm Myr}, \end{eqnarray} (10)where ΔMacc is the accreted mass of the WD. Based on the model of Cumming (2002) for magnetic field evolution, we propose that the surface field of the more massive WD in J1257 decreased from ∼107 to ∼105 G due to rapid accretion. According to the simulation in the next section, the accreted mass of the massive WD is ΔMacc ≈ 0.45 M⊙, so the re-emergence time-scale is τre ≈ 2.5 Gyr. Taking Ps, 0 = 10 s (close to breakup rotation), Ps = 60 s, B7 = 1, and R9 = 0.6, one can derive that the spin-down time-scale of the massive WD is about 4.5 Gyr. Considering the re-emergence time-scale of the magnetic field τre ≈ 2.5 Gyr, the total duration of 7 Gyr before the PT for the massive WD is approximately in agreement with the cooling age of the low-mass WD. 4 NUMERICAL SIMULATION Using the mesa module (Paxton et al. 2015), we have simulated the evolution of the compact WD binary consisting of a WD and a main-sequence star to test whether it is possible to reproduce the characteristics of J1257. According to the estimation in the previous section, the orbital change during PT can be neglected while the mass growth of the WD is considered. For an accreting WD, hydrogen and helium shell flashes always trigger nova outbursts, which blow off the accreted matter and even result in convective dredge-up. Therefore, the mass accumulation efficiency for accreting hydrogen should be less than 1. During the mass transfer, the mass growth rate of the accreting WD is described as follows:   \begin{eqnarray} \dot{M}_{\rm 1}=\eta _{\rm He}\eta _{\rm H}|\dot{M}_{2}|, \end{eqnarray} (11)where $$\dot{M}_{2}$$ is the mass transfer rate of the donor star, ηH and ηHe are the accumulation efficiencies during hydrogen burning and helium burning, respectively. For the accumulation efficiency of hydrogen, a prescription given by Hachisu et al. (1999) and Han & Podsiadlowski (2004) was adopted, i.e.   \begin{eqnarray} \eta _{\rm H}= \left\lbrace \begin{array}{@{}l@{\quad }l@{}}\dot{M}_{\rm cr}/|\dot{M}_{2}| &\text{$|\dot{M}_2|>\dot{M}_{\rm cr}$}, \\ 1 &\text{$\dot{M}_{\rm cr}>|\dot{M}_2|>0.125\dot{M}_{\rm cr}$},\\ 0 &\text{$|\dot{M}_2|<0.125\dot{M}_{\rm cr}$}. \end{array}\right. \end{eqnarray} (12)In equation (12), $$\dot{M}_{\rm cr}$$ is a critical mass-accretion rate:   \begin{eqnarray} \dot{M}_{\rm cr}=5.3\times 10^{-7}\frac{1.7-X}{X}(M_{1}-0.4)\,{\rm M}_{{\odot }}\,{\rm yr}^{-1}, \end{eqnarray} (13)where X is the mass abundance of hydrogen in the accreted matter. For the accumulation efficiency ηHe during helium burning, the prescriptions given by Kato & Hachisu (2004) was adopted. The mass-loss $$(1-\eta _{\rm H}\eta _{\rm He})\dot{M}_{2}$$ during hydrogen and helium burning is assumed to be ejected in the vicinity of the WD in the form of isotropic winds, carrying away the specific angular momentum of the WD (Hachisu, Kato & Nomoto 1996; Soberman, Phinney & van den Heuvel 1997). In addition, we also consider angular momentum loss caused by gravitational radiation and magnetic braking (Verbunt & Zwaan 1981 with γ = 3.0; Rappaport, Verbunt & Joss 1983; Paxton et al. 2015). To study the progenitor properties of J1257, we simulated the evolution of a large number of WD binaries. The relevant binaries would be thought to be the progenitor candidates of J1257 if the following three conditions are satisfied: (1) the current orbital period is 4.6 h when the age of the system is within Hubble time; (2) the binary evolves into a detached system when the accreting WD’s mass is about 1.05–1.15 M⊙ (because the WD must experience a long-term spin-down process, hence the donor star should not overflow its Roche lobe when the WD mass increases to 1.05–1.15 M⊙); and (3) the effective temperature of the donor star is near 6400 K. Figs 2–4 show an example of evolution in which the initial donor mass and the initial WD mass are 1.5 and 0.65 M⊙, respectively. We change the initial metallicity in order to fit the observed parameters of J1257. As shown in Fig. 3, because the material is transferred from the more massive donor star to the less massive WD, the orbital period first decreases. With the reversal of the mass ratio, the orbital period increases until the binary evolve into a detached system. Subsequently, the donor star gradually evolves into a WD and enters the cooling stage, and magnetic braking and gravitational radiation induce a compact double WD binary. Once the mass transfer ceases, the massive WD spins down due to magnetic dipole radiation, then triggers PT during the cooling of the low-mass WD. The heating process during PT results in the formation of a hot SD. Similar to Fig. 4, the simulated effective temperatures of the low-mass WD are always higher than the observation in the Hubble time except for z = 0.0001, in which the donor star orbits a WD with an initial orbital period of 1.492 d. Fig. 5 shows the evolutionary tracks of the donor-star mass and the accreting WD’s mass. It is clear that our simulated donor-star masses are consistent with the observed data for three different metallicities. In calculation, the donor star with higher metallicity would produce systems with lower mass secondary WDs. These differences should arise from the metallicity dependence of the stellar wind mass-loss, which tend to reduce the stellar mass of donor star. Figure 3. View largeDownload slide Evolution of orbital period for WD binaries including a 1.5 M⊙ donor star and a 0.65 M⊙ WD. The solid, dashed, and dotted curves represent the metallicity z = 0.0001, 0.0005, and 0.001, respectively. Numbers beside the curves denote the initial orbital periods. Figure 3. View largeDownload slide Evolution of orbital period for WD binaries including a 1.5 M⊙ donor star and a 0.65 M⊙ WD. The solid, dashed, and dotted curves represent the metallicity z = 0.0001, 0.0005, and 0.001, respectively. Numbers beside the curves denote the initial orbital periods. Figure 4. View largeDownload slide Evolutionary tracks of the effective temperature of donor star for WD binaries including a 1.5 M⊙ donor star and a 0.65 M⊙ WD. Three cases are same to Fig. 3. Figure 4. View largeDownload slide Evolutionary tracks of the effective temperature of donor star for WD binaries including a 1.5 M⊙ donor star and a 0.65 M⊙ WD. Three cases are same to Fig. 3. Figure 5. View largeDownload slide Evolution of the donor-star mass (top panel) and the WD mass (bottom panel) for WD binaries including a 1.5 M⊙ donor star and a 0.65 M⊙ WD. Three cases are same to Fig. 3. Figure 5. View largeDownload slide Evolution of the donor-star mass (top panel) and the WD mass (bottom panel) for WD binaries including a 1.5 M⊙ donor star and a 0.65 M⊙ WD. Three cases are same to Fig. 3. 5 SUMMARY AND DISCUSSION Assuming both WD and SD are different stages of stellar evolution, in this work we propose a SD scenario to interpret the puzzle of the cooling age of two WD in J1257. The massive WD is thought to be a SD originating from the PT of the 1.05–1.15 M⊙ WD, thus its higher effective temperature can be interpreted as a result of heating during PT. A simple estimation indicates that a mass-loss ΔM ∼ 0.05 M⊙ during PT can heated the nascent SD up to 108 K. Based on these assumptions, we use the mesa code to simulate the evolution of a large number of WD binaries consisting of a 0.65 M⊙ WD and a 1.5 M⊙ main-sequence star for different initial orbital period and metallicities. Our simulation indicates that metallicities have important influence on the effective temperature of the donor star. When z = 0.0001, the calculated orbital period, the donor-star mass, and the effective temperature of the donor star are consistent with the observed data. Therefore, we propose that the PT of a massive WD may be responsible for the puzzling cooling age of two WDs. We expect further detailed multiwaveband observations for this source to obtain more precise constraints. 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Monthly Notices of the Royal Astronomical SocietyOxford University Press

Published: May 1, 2018

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