A systematic study of carbon-oxygen white dwarf mergers: mass combinations for Type Ia supernovae
Yushi Sato, Naohito Nakasato, Ataru Tanikawa, Ken'ichi Nomoto, Keiichi Maeda, Izumi Hachisu
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A SYSTEMATIC STUDY OF CARBON–OXYGEN WHITE DWARF MERGERS: MASS COMBINATIONS FOR TYPE IaSUPERNOVAE Y USHI S ATO , , N AOHITO N AKASATO , A TARU T ANIKAWA , , K EN ’ ICHI N OMOTO , , K EIICHI M AEDA , , I ZUMI H ACHISU Department of Astronomy, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan Department of Earth Science and Astronomy, College of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8902, Japan ;[email protected] Department of Computer Science and Engineering, University of Aizu, Tsuruga Ikki-machi Aizu-Wakamatsu, Fukushima 965-8580, Japan RIKEN Advanced Institute for Computational Science, 7-1-26, Minatojima-minami-machi, Chuo-ku, Kobe, Hyogo 650-0047, Japan Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8583, Japan Department of Astronomy, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan
Received 2015 Jannuary 29; accepted 2015 May 4; published 2015 MM DD
ABSTRACTMergers of two carbon–oxygen (CO) WDs have been considered as progenitors of Type Ia supernovae (SNeIa). Based on smoothed particle hydrodynamics (SPH) simulations, previous studies claimed that mergersof CO WDs lead to an SN Ia explosion either in the dynamical merger phase or stationary rotating mergerremnant phase. However, the mass range of CO WDs that lead to an SN Ia has not been clearly identifiedyet. In the present work, we perform systematic SPH merger simulations for the WD masses ranging from0 . M ⊙ to 1 . M ⊙ with higher resolutions than the previous systematic surveys and examine whether or notcarbon burning occurs dynamically or quiescently in each phase. We further study the possibility of SN Iaexplosion and estimate the mass range of CO WDs that lead to an SN Ia. We found that when the both WDs aremassive, i.e., in the mass range of 0 . M ⊙ ≤ M , ≤ . M ⊙ , they can explode as an SN Ia in the merger phase.On the other hand, when the more massive WD is in the range of 0 . M ⊙ ≤ M ≤ . M ⊙ and the total massexceeds 1 . M ⊙ , they can finally explode in the stationary rotating merger remnant phase. We estimate thecontribution of CO WD mergers to the entire SN Ia rate in our galaxy to be of < ∼ Subject headings: binaries: close — galaxies: evolution — supernovae: general — white dwarfs — hydrody-namics INTRODUCTION
SNe Ia play the important roles in the determination ofcosmological parameters as luminous standard candles (e.g.,Riess et al. 1998; Perlmutter et al. 1999) and in the chemi-cal evolution of galaxies as major sources of iron group el-ements (e.g., Kobayashi et al. 1998). However, their pro-genitors are still not identified yet (e.g., Maoz et al. 2014).They are considered as a thermonuclear explosion of a COWDs in a binary system, in which the WD accretes massfrom its companion and the WD mass approaches the Chan-drasekhar mass (M Ch ∼ . M ⊙ ). But, it is still controversialwhether its companion is a non-degenerate star, i.e., singledegenerate (SD) model, or a degenerate star, i.e., double de-generate (DD) model. In the SD model, a CO WD accreteshydrogen/helium-rich gas from the companion and increasesits mass upto M Ch . Finally, carbon burning starts at the cen-ter of the CO WD and explodes as an SN Ia (Whelan & Iben1973; Nomoto 1982; Hachisu et al. 1996, 1999a,b). On theother hand, in the DD model, both the components are COWDs. Because binaries lose their orbital angular momentumby emitting gravitational waves, they will eventually merge.If their total mass exceeds M Ch , the binary finally explodes asan SN Ia (Iben & Tutukov 1984; Webbink 1984).Some clues on the progenitors of SNe Ia were found in re-cent observations. In particular, neither surviving compan-ions nor signatures of them were detected in some SNe Ia(e.g., Schaefer & Pagnotta 2012). This fact supports the DDsystem, although no detection of companions can also be ex- Hamamatsu Professor plained by the SD model (Di stefano et al. 2011; Justhum2011; Hachisu et al. 2012). In some SNe Ia, signatures ofcircumstellar matter (CSM) were detected (e.g., Patat et al.2007; Blondin et al. 2009; Simon et al. 2009; Sternberg et al.2011). This supports the SD model (e.g., Maoz et al. 2014,for a recent review), although this can be explained by someDD models (Raskin & Kasen 2013; Shen et al. 2013; Sokeret al. 2013). Observations do still not clarify which model isthe main progenitors of SNe Ia, i.e., SD or DD (or other type)system.The DD model has a theoretical difficulty as a progenitormodel. Some theoretical studies indicated that CO WD merg-ers can not become an SN Ia, but collapse to a neutron star(e.g., Saio & Nomoto 1985, 2004; Nomoto & Kondo 1991).These studies calculated only the evolution of merger rem-nants after DD systems merged. Their calculations were one-dimensional (1D) spherically symmetric and assumed station-ary state. However, the merger of a DD system is a three-dimensional (3D) and dynamical event, so multi-dimensionalhydrodynamical simulations are necessary to reach a definiteconclusion. Benz et al. (1990) used 3D smoothed particle hy-drodynamics (SPH) code and simulated mergers of two COWDs. Following this work, there were several studies onmergers of DD systems (Rasio & Shapiro 1995; Segretain &Chabrier 1997; Guerrero et al. 2004; Yoon et al. 2007; Lorén-Aguilar et al. 2009; Fryer et al. 2010; Pakmor et al. 2010; Danet al. 2011; Raskin et al. 2012, 2014; Moll et al. 2014). Thesestudies concluded that some DD pairs can explode as SNeIa. Such successful models can be divided by the dynamicalphase when the SN Ia explosion occurs. Sato et al.Pakmor et al. (2010) simulated mergers of massive COWDs ( ∼ . M ⊙ ) and found that carbon detonation initiatesduring the dynamical merger phase because of the compres-sional heating by the disrupted secondary which violently ac-cretes onto the primary. The binary system finally explodesas a subluminous SN Ia. Pakmor et al. (2012a) also simulatedmergers of more massive CO WDs (1 . + . M ⊙ ) and foundthat the system leads to a normal SN Ia.If carbon detonation does not initiate in the dynamicalmerger phase, the remnant undergoes three phases. The firstphase is the early remnant phase (e.g., Shen et al. 2012;Kashyap et al. 2015), 100–1000 s after the secondary is com-pletely disrupted. In this phase, the merger remnant doesnot reach a quasi-stationary state yet and still has small non-axisymmetric structures. The second phase is the viscous evo-lution phase (Schwab et al. 2012; Shen et al. 2012; Ji et al.2013), 10 –10 s after merging. In this phase, the remnantreaches a quasi-stationary, axisymmetric state and it evolvesin a viscous timescale. The third is the thermal evolutionphase (Saio & Nomoto 1985, 2004; Yoon et al. 2007; Shenet al. 2012), > ∼ years. If off-center carbon burning occursin these three phases before the rotating core of the remnantreaches M Ch , it likely converts the CO WD to an oxygen–neon–magnesium (ONeMg) WD. The ONeMg WD finallycollapses to a neutron star when its core mass reaches M Ch .Yoon et al. (2007) simulated the merger of CO WDs withmasses of 0 . + . M ⊙ and followed the evolution of themerger remnant. They found that the remnant can avoid off-center carbon burning and explodes as an SN Ia in the thermalevolution phase, if it satisfies several conditions.Thus, these works concluded that some DD systems can be-come SNe Ia. However, the mass range of CO WDs that leadto SNe Ia is not clarified yet. In this work, we simulate merg-ers of CO WDs with a mass range of 0 . . M ⊙ with ourSPH code until the end of the early remnant phase, and iden-tify the mass range of CO WDs that lead to SNe Ia. Using theobtained mass range of CO WDs, we estimate the contribu-tion of CO WD mergers to the entire SNe Ia in our galaxy.Similar parameter surveys have already been done by Danet al. (2012, 2014) and Zhu et al. (2013). Zhu et al. (2013)mainly focused on the (early) remnant phase and the resolu-tions of their simulations are lower than our study. AlthoughDan et al. (2012, 2014) covered the merger phase and remnantphase, their numerical resolution is much lower than ours. Weexpect that SN Ia explosions occur not only in the early rem-nant phase but also in the dynamical merger phase. In thepresent work, we adopt higher resolution than two times thoseof the previous works (e.g., Zhu et al. 2013). This is becausethe numerical resolution is one of the most important param-eters to identify the initiation of detonation in the dynamicalmerger phase (Pakmor et al. 2012b). Therefore, we adopt fourdifferent resolutions and check the numerical convergence.To investigate the possibility that carbon burning leads to anSN Ia, we check the density and temperature of SPH particlesand identify carbon burning by the condition of τ CC < τ cool ,where τ CC is the timescale of carbon burning and τ cool isthe cooling timescale. In the dynamical merger phase, τ cool is the dynamical timescale of adiabatic expansion. On theother hand, in the (early) remnant phase, τ cool is the timescaleof neutrino cooling. These are necessary but not sufficientconditions for a thermonuclear explosion. In this sense, ourresults would not be conclusive but are sufficiently sugges-tive. Since the temperature has numerical noise in our SPH simulation, we have to treat the temperature carefully. Thisnoise comes from fluctuations of density and internal energy,which arise due to numerical resolution, finite neighbor par-ticles, and the accuracy of time-integration. We use both theraw and smoothed temperatures to estimate the noise in ourSPH simulation (Dan et al. 2012). The raw temperature isthe temperature that our SPH code originally generates. Thesmoothed temperature is an averaged temperature calculatedfrom neighbor particles, which could avoid large numericalnoises (defined in Section 3.1). In the present version of ourSPH simulation, we do not include nuclear reactions becausethey are so sensitive to temperature noises and possibly en-hance them erroneously.This paper is organized as follows. Section 2 briefly de-scribes our numerical methods. In section 3, we show ourresults and then estimate the rate of SNe Ia coming from theDD merger systems and their contribution to the entire SNeIa in our galaxy. In section 4, we compare our results withprevious works and discuss the dependence of our numeri-cal results on the resolution. Finally we conclude the presentwork in Section 5. METHODS
Here, we present a brief summary of our numerical method.The details of the numerical method are already described inNakasato et al. (2012) and Tanikawa et al. (2015).
Numerical Code
SPH is the Lagrangian mesh-free particle method devel-oped for the study of astrophysical fluid phenomena (Lucyet al. 1977; Gingold & Monaghan 1982). Nowadays, it isapplied to more various subjects, e.g., engineering, meteorol-ogy, and Hollywood movies. Recent good reviews about SPHcodes are available, e.g., in Monaghan (2005) and Rosswog(2009). Our formulation is the one called "vanilla ice" SPHformulation in Rosswog (2009). Our basic SPH equationsconsist of the equation of continuity, equation of motion, andenergy equation for self-gravitating fluid in Lagrangian for-mulation.We use "OcTree On OpenCL"(OTOO) code for our 3Dsimulations of CO WD mergers, which is developed for var-ious particle simulations of astrophysical fluid phenomena(Nakasato et al. 2012). This code implements the octreemethod (Barnes & Hut 1986) to calculate gravity and neigh-bor particles for SPH. It is optimized for multiple CPUs andGPUs in heterogeneous computational resources.In this code, we adopt the 3rd-order spline kernel and itsderivative is modified in the same way as that proposed byThomas & Couchman (1992) to avoid the pairing instability.The smoothing length is determined to keep the average num-ber of neighbors being about 75 in every step (Thacker et al.2000). The formulation of the artificial viscosity is basicallythe same as Monaghan (1992), but the viscosity coefficientis time-dependent (Morris & Monaghan 1997). More detailexplanation can be seen in Rosswog et al. (2000) and Mon-aghan (2005). We also introduce the Balsara switch to shutoff the artificial viscosity in no shock regions (Balsara 1995).We adopt a leap-frog scheme for time integration.We use the Helmholtz equation of state (EOS) (Timmes &Swesty 2000) and assume a uniform chemical composition of50% carbon and 50% oxygen. This chemical composition isnot changed throughout the merger simulation because we donot include nuclear reactions.erger of White Dwarfs 3
Initial Setup
The initial setup of our simulations is similar to those ofRasio & Shapiro (1995) and Dan et al. (2011). We separatelygenerate each CO WD from spherically symmetric densityprofiles of a perfectly degenerate star, and set the tempera-ture at 10 K everywhere. To reduce numerical noise in themapping from the 1D spherically symmetric density profile toour 3D SPH density distribution, we relax the SPH particlesfor 20 physical seconds with velocity damping force but with-out the evolution of internal energy. For the damping force,we adopt d v i dt = - v i C damp dt , (1)where v i is the velocity of i th particle, C damp is the inverse ofrelaxation timescale and we fix C damp = 128 . i th) particle of the secondaryapproaches closely enough the L1 point, i.e., k r sec , i - r L1 k < . R , (2)we stop the calculation. Here r sec , i is the position of i th par-ticle of the secondary, r L1 is that of L1 point, and R is theeffective radius of the secondary.Finally, we transfer the SPH particles from the co-rotatingframe to a rest frame. RESULTS
We summarize the results of our simulations in Table A1.Because we aim to identify the mass range of CO WDs thatlead to an SN Ia, our initial models cover the entire mass rangeof CO WDs, i.e., from M WD = 0 . . M ⊙ . We prepare bi-nary models of 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, and 1 . M ⊙ , andperform the simulations of mergers for all 28 mass combina-tions. Since we also investigate the dependence of our resultson the numerical resolution, we perform the same simulationswith different resolutions, which are 10 k , k , k , k SPH particles per one solar mass (here k ≡ , . + . M ⊙ and resolution is 500 k M - ⊙ . The merger phasecovers the first ( t = 25 s) to fifth ( t = 135 s) panels, andthe early remnant phase corresponds to the sixth ( t = 240 s)panel. Their morphological structures are consistent with theprevious works (e.g., Pakmor et al. 2012a).First we check whether dynamical carbon burning starts ornot in the merger phase for all simulations because it is a nec-essary condition for an SN Ia explosion. Then, we checksteady carbon burning in the early remnant phase. If it oc-curs in the early remnant phase, carbon burning converts a CO WD into an ONeMg WD and the merger remnant will notbecome an SN Ia. Although we perform simulations with fourdifferent resolutions, we first focus on the highest resolutions(= 500 k M - ⊙ ). Merger Phase
Pakmor et al. (2010) first suggested that CO WD mergerslead to an SN Ia in the merger phase. They called such amodel the (carbon-ignited) violent merger (VM) scenario. Inthis scenario, matter of the secondary violently accretes ontothe primary and such violent accretion causes dynamical car-bon burning. As a result, detonation wave would be formedand propagate into the primary with converting its carbon–oxygen into iron group elements. Finally, the system explodesas an SN Ia. Because our simulations cannot directly resolvethe initiation of detonation, we try to judge the occurrence ofdynamical carbon burning in the merger phase. This conditionis a necessary condition for an SN Ia explosion.For this purpose, we extract the highest temperature particlein the merger phase for all simulations. The condition fordynamical carbon burning is τ CC < τ dyn , (3)where τ CC is a carbon burning timescale defined by τ CC = C P T ǫ CC , (4)and τ dyn is a dynamical timescale (Nomoto 1982) defined by τ dyn = 1 √ π G ρ , (5)where C P is the specific heat at constant pressure, ǫ CC is theenergy generation rate of carbon burning. We calculate theboth timescales for each particle in the merger phase and ex-amine whether the particles satisfy Equation (3). In this work, C P is derived from the Helmholtz EOS of Timmes & Swesty(2000) and ǫ CC is the same as that of Dan et al. (2014), origi-nally proposed by Blinnikov & Khokhlov (1987). Its formu-lation is ǫ CC = ρ q C A T Y exp( - Q / T / + f CC ) , (6)where q C = 4 . × erg mol - (Blinnikov & Khokhlov1987), A T = 8 . × T / T - / s - mol - cm , T ≡ T / K, T ≡ T / (1 + . T ), Q = 84 .
165 (Fowleret al. 1975). The carbon abundance is calculated as Y C = n C / ( ρ N a ) = 0 .
033 mol g - , where n C is the number den-sity of carbon and N a is the Avogadro number. A screeningfactor f CC is ignored here, because we focus on the start ofdynamical carbon burning and the factor of self-accelerationfor nuclear burning described in Frank-Kamenetskii (1967) isnot applied to the initiation of carbon burning.Figure 2 shows the density and temperature of a particlewith the highest temperature in the merger phase for all masscombinations (their resolutions are 500 k M - ⊙ ), similar to Fig-ure 12 of Dan et al. (2014). The shapes and colors of sym-bols indicate the primary’s mass and the total mass, respec-tively. Solid lines indicate τ CC = τ dyn , and dashed lines do τ CC = 0 . τ dyn . For the mass combinations above the solid line,dynamical carbon burning occurs in the merger phase. So, themerger of CO WDs would lead to an SN Ia explosion.In our SPH simulation, physical raw temperature of eachparticle has numerical noise. So we adopt another definition Sato et al. t=25s0.01R sun t=75s0.01R sun t=120s0.01R sun t=125s0.01R sun t=135s0.01R sun l og R HO [ g c m - ] t=240s0.01R sun F IG . 1.— Density profiles in the equatorial plane for the dynamical evolution of our merger simulation. The mass combination is 1 . + . M ⊙ , and theresolution is 500 k M - ⊙ . Colors indicate density in a logarithmic scale. of temperature to reduce the effect of noise, i.e., smoothedtemperature. It is defined by T s , i = X j m j ρ j T j W ( r i j , h i j ) , (7)where m j , ρ j , T j are the mass, density, temperature of j th par-ticle, respectively. r i j = k r i - r j k , and h i j = ( h i + h j ) / i th and j th particles.Figure 2(a) and 2(b) show the results for raw temperatureand smoothed temperature thus defined, respectively. It isclear that all symbols in Figure 2(a) move to a lower placein Figure 2(b). As a result, the number of mass combina-tions above the solid line of τ CC = τ dyn decreases. Figure 2(b)of smoothed temperature shows that the mass combinationsabove the solid line would certainly trigger dynamical carbonburning in the merger phase. Remnant Phase
If dynamical carbon burning does not occur in the mergerphase, the merged object goes into a stationary phase, i.e.,the (early) remnant phase. Figure 3(a) and 3(b) show thedensity and temperature, respectively, of the merger remnantwhose mass combination and resolution are 1 . + . M ⊙ and500 k M - ⊙ . The remnant consists of three components, i.e., acold core, hot envelope, and outer disk. The structure of sucha remnant has been studied in the several works as alreadymentioned in Section 1. Although there are small differencesamong these previous works, their results are almost consis-tent with each other. Our results are also consistent with theseprevious results.It has long been discussed that such a hot envelope gradu-ally accrete onto a cold core because of angular momentumloss by some mechanisms (e.g. viscosity or magnetic field).If off-center carbon burning occurs quiescently during suchan accretion phase, carbon deflagration waves propagate intothe core. At last, the whole core is converted into an ONeMgWD (e.g., Saio & Nomoto 1985, 1998, 2004). In such a case, it can not explode as an SN Ia, even if its total mass exceeds M Ch . Instead, it collapses to a neutron star (Nomoto & Kondo1991). On the other hand, if off-center carbon burning doesnot occur, the core remains unchanged as a CO WD and sur-rounding matter continues to accrete onto the core. Whenthe core mass exceeds M Ch , it explodes as an SN Ia. It iscritically important to examine whether carbon burning startsquiescently off-center in the remnant phase.We examine, in the same way as that of dynamical carbonburning in the merger phase, whether or not off-center carbonburning occurs in the remnant phase. Carbon burning quies-cently occurs near the boundary between the cold core and hotenvelope, if the condition for carbon burning, i.e., τ CC < τ ν , (8)is satisfied in the remnant phase. Here, τ ν is a timescaleof neutrino cooling and we use the description of Itoh et al.(1996) for calculating it from the density and temperature ofSPH particles. We find the highest temperature particle inthe remnant phase for all mass combinations, and examineEquation (8). If the highest temperature particle satisfies thecondition, we regard that off-center carbon burning starts andconverts the CO core into an ONeMg core. Then, the systemfinally collapses to a neutron star if the total mass exceeds M Ch . On the other hand, if there are no particles that satisfythe condition and the total mass of the remnant exceeds M Ch ,we consider that the remnant becomes an SN Ia.Figure 4 is the same plot as Figure 2, but for the remnantphase. Magenta solid lines indicate τ CC = τ ν . Figure 4(a)and 4(b) show results of the raw and smoothed temperatures,respectively. Off-center carbon burning occurs in the remnantphase for the mass combination models above the magentasolid line. They would finally collapse to a neutron star. Onthe other hand, the models below the line would become anSN Ia if the total mass exceeds the Chandrasekhar mass, i.e., M + M > M Ch .We have to follow the viscous and thermal evolution phaseserger of White Dwarfs 5 l og T [ K ] t CC = t d y n t CC = . t d y n (a) M t o t [ M s un ] l og T s [ K ] log r [gcm -3 ] t CC = t d y n t CC = . t d y n (b) M t o t [ M s un ] F IG . 2.— Density and temperature of the highest temperature particle inthe merger phase for all mass combinations. Numerical resolutions of thesemodels are 500 k M - ⊙ . Colors of symbols indicate the total mass of the sys-tem as indicated in the rightside of the figures, and shapes of symbols do themass of the primary. Filled squares are the 1 . M ⊙ primary, filled circles1 . M ⊙ , filled triangles 0 . M ⊙ , filled inverted triangles 0 . M ⊙ , filled di-amonds 0 . M ⊙ , open squares 0 . M ⊙ , open circles 0 . M ⊙ . Solid linesindicate τ CC = τ dyn , and dashed lines do τ CC = 0 . τ dyn . (a) Raw temperatureof SPH particles. (b) Smoothed temperature, T s , of SPH particles defined byEquation (7). for obtaining definite conclusion on the off-center carbonburning. However, we stop our SPH calculation at the endof the early remnant phase because our SPH code includesno physical viscosities (e.g., magnetic viscosity). In the vis-cous and thermal evolution phases, the hot envelope furtheraccretes onto the cold core and compress itself on the coldcore. The temperature and density might further increase. Asa result, carbon ignites off-center even for the cases of no off-center burning in the early remnant phase. In fact, Shen etal. (2012) and Schwab et al. (2012) followed the evolution ofmerger remnants and showed that off-center carbon ignitionstarts in the viscous and thermal evolution phases for somecases. Yoon et al. (2007) performed SPH simulation of COWD merger whose mass combination is 0 . + . M ⊙ andfurther followed the evolution of the merger remnant with a1D stellar evolution code. They found that off-center car-bon burning can be avoided when the highest temperatureis lower than the threshold for carbon ignition, angular mo-mentum loss occurs with a timescale longer than that of neu-trino cooling, and the mass accretion rate is ˙ M ≤ × - to 10 - M ⊙ yr - . Our present condition for off-center carbonburning is posed only for the (early) remnant phase but not ap-plied yet for the viscous and thermal accretion phases. In thissense, our results only for the early remnant phase are not def- F IG . 3.— Structure of a merger remnant at about 200 s ( ∼ . P orb , init ) af-ter the secondary is disrupted completely. Here, P orb , init is the initial orbitalperiod. Its mass combination is 1 . + . M ⊙ and the numerical resolution is500 k M - ⊙ , which is the same model as in Figure 1. (a) Density profile (in alogarithmic scale of g cm - ) in the x - - z plane. (b) Temperature structure (ina linear scale of 10 K) in the x - - z plane. The central part becomes slightlyhot. This is caused by numerical noise and artificial viscosity, so it has nophysical meaning. inite answers. We leave such viscous and thermal evolutionsof the merger remnants in our future works. White Dwarf Mass Combinations For SNe Ia
Now, we have obtained the mass range of CO WDs whichpossibly lead to an SN Ia. Using this mass range of COWDs, we estimate their contribution to the entire SNe Ia inour galaxy. We consider four paths that CO WD mergerswould follow. The first one is the VM path, the condition ofwhich is that dynamical carbon burning occurs in the mergerphase. The systems satisfying this condition would explodeas an SN Ia immediately after merging (Pakmor et al. 2010,2011, 2012a). When dynamical carbon burning does not oc-cur in the merger phase, the system enters the remnant phaseand the disrupted secondary surrounds the primary. If its totalmass exceeds M Ch and off-center carbon burning does not oc-cur during the accretion phase, carbon burning occurs at thecenter of the CO WD and it would finally explode as an SNIa. We regard this evolutionary path as the accretion inducedexplosion (AIE) path. On the other hand, if off-center car-bon burning occurs, the core of the remnant will be convertedinto an ONeMg WD and then collapse to a neutron star whenthe core mass exceeds M Ch . This is the accretion induced col-lapse (AIC) path (Saio & Nomoto 1985, 1998, 2004; Nomoto& Kondo 1991). When dynamical carbon burning does notoccur in the merger phase and the total mass of the systemdoes not exceed M Ch , the system would form a massive COWD. We call this evolutionary path as the massive white dwarf(MWD) path.Figure 5 shows all mass combinations of our simulationswith 500 k M - ⊙ particles and identifies which path they take.Colors of symbols indicate the four paths, i.e., the VM (red),AIE (green), AIC (blue), and MWD (magenta) paths. Amongthese four paths, the VM and AIE paths are possible paths toSNe Ia. Sato et al. l og T [ K ] t CC = t n (a) M t o t [ M s un ] l og T s [ K ] log r [gcm -3 ] t CC = t n (b) M t o t [ M s un ] F IG . 4.— Same as Figure 2, but for the remnant phase. Magenta solid linesindicate τ CC = τ ν . SN Ia Rate
We estimate the rate of SNe Ia which originate from COWD mergers and their contribution to the entire SNe Ia in ourgalaxy. Here, we assume that all mergers of CO WDs sat-isfying the VM or AIE condition can explode as an SN Ia.Strictly speaking, for the VM path, dynamical carbon burningis a necessary, but not a sufficient condition for carbon deto-nation. So it is not trivial if those mergers explode as an SN Ia.On the other hand, we have to follow the evolution of mergerremnants for a much longer time (e.g., Yoon et al. 2007) inorder to identify their final fates, the AIE (exploding as an SNIa) or AIC (collapsing to a neutron star) path. In this sense,our estimate is just an upper limit.The SN Ia rate depends on the Hubble type and stellarmass of a galaxy. According to Li et al. (2011), in SBctype galaxies with similar stellar mass to our galaxy, the SNIa rate is about 1 . × - yr - M - ⊙ . Badenes et al. (2012)estimated the merger rate of binary WDs in our galaxy as1 . × - yr - M - ⊙ . The mass distribution of binary WDs inour galaxy is still uncertain because there are small sampleseven in our neighborhood. We assume that both the primaryand the secondary follow the mass distribution of single DAWDs in our galaxy derived from SDSS-DR7 (see, e.g., Figure10 of Kleinman et al. 2013). Then we can calculate the rate ofCO WD mergers that satisfy the condition of each scenario.The merger rate is about 1 . × - yr - M - ⊙ (0 . × - yr - M - ⊙ ) for the VM, 5 . × - yr - M - ⊙ (8 . × - yr - M - ⊙ ) for the AIE path, and6 . × - yr - M - ⊙ (9 . × - yr - M - ⊙ ) for the both M [ M s un ] M = - M + . M s un M = M VMAIEAICMWD 0.5 0.6 0.7 0.8 0.9 1 1.1 0.5 0.6 0.7 0.8 0.9 1 1.1 M [ M s un ] M [M sun ] M = - M + . M s un M = M F IG . 5.— The outcome of our merger simulations for the (a) raw and (b)smoothed temperatures. Red symbols mean the VM path, green symbolsthe AIE path, blue the AIC, and magenta the MWD path. Two black linesindicate that the primary and secondary have the same mass and that the totalmass equals M + M = M Ch ∼ . M ⊙ . Mass combinations of red and greensymbols result in an SN Ia. paths, if we adopt the case of raw temperature (smoothedtemperature). This is only < ∼
9% of the entire galactic SNe Ia.Therefore, at least in our galaxy, DD merger systems mightnot dominate the progenitors of SNe Ia. DISCUSSION
Comparison With Previous Studies
We compare the present results with previous studies. Inparticular, we mainly focus on two mass combinations. Oneis 1 . + . M ⊙ and the other is 0 . + . M ⊙ , because thesetwo combinations were well studied in the previous works. . + . M ⊙ First we compare our result of 1 . + . M ⊙ and 100 k M - ⊙ (i.e. the total number of SPH particles is about 2 × ) withthat of Pakmor et al. (2012b). Figure 6(a) shows the timeevolutions of the orbital separation and Figure 6(b) shows thenumber of particles having temperature higher than 2 × K.Our Figure 6 should be compared with Figure 4 and Table 1of Pakmor et al. (2012b).It should be noted that the WDs in our model merge morequickly (about 100 s) than those in Pakmor et al. (2012b)(about 600 s) although we start our simulation with the ini-tial condition similar to theirs. This is because our initial sep-aration ( ∼ . × cm) is less than Pakmor et al. (2012b)( ∼ . × cm). Indeed, Figure 6(a) and 6(b) resemble thecase of their smaller initial separation. We suppose that theerger of White Dwarfs 7 F IG . 6.— Time evolution of (a) the orbital separation and (b) the number ofhigh temperature ( ≥ . × K) particles. The mass combination of WDs is1 . + . M ⊙ and the numerical resolution is 100 k M - ⊙ . main reason of the difference is the relaxation method of asingle WD. As mentioned in Section 2.2, we relax a singleWD with a velocity-dependent damping force but without theevolution of internal energy. On the other hand, Pakmor et al.(2012b) changed the damping timescale (see their Equation14). After the relaxation method of Pakmor et al. (2012b) isapplied to our models, the radii of the relaxed WDs are a fewpercent larger than our original ones. Therefore, their separa-tions when the RLOF starts are larger than ours and it resultsin a longer merging time.If the initial separation is smaller when the RLOF starts, themass transfer tends to occur more violently and the secondaryis completely disrupted in a few orbital periods (Dan et al.2011). As a result, accreted matter is more strongly shock-heated and dynamical carbon burning easily occurs. AlthoughPakmor et al. (2012b) concluded that the initial condition isnot so important for dynamical carbon burning, we shouldnote that our highest temperatures in the merger phase couldbe overestimated. . + . M ⊙ We also compare our result of 0 . + . M ⊙ and 100 k M - ⊙ with the previous studies of Yoon et al. (2007), Dan et al.(2011), and Zhu et al. (2013). Since these studies have simi-lar resolution (a few × SPH particles in all), it is suitablefor comparison. Figure 7 shows the time evolution of highesttemperature in each density zone. Since no dynamical car-bon burning occurs (see Figure 5), we examine whether or notquiescent carbon burning occurs in the remnant phase, i.e., at t ∼
300 s in Figure 7. The temperature is about 6 × K andthis is consistent with the above three previous studies.We compare our Figure 7 with Figure 4 in Yoon et al.(2007). For the highest temperature in the merger phase (at t ∼
100 s in Figure 7), their results of 1 . × K is higher thanours (1 . × K), although dynamical carbon burning doesnot occur in the merger phase for both ours and theirs. Wesuppose that the difference in the highest temperature comesfrom the difference in the initial condition. Dan et al. (2011) F IG . 7.— Evolution of highest temperature in each density range,log ρ (g cm - ) < . . < log ρ < . . < log ρ < . . < log ρ < . . + . M ⊙ and theresolution of 100 k M - ⊙ . reported that the morphology of merger remnant could be af-fected by the initial condition. They found that the remnantwhose initial separation is larger has longer trailing arm thanthe one whose initial separation is smaller (see Figure 10 ofDan et al. 2011). On the other hand, we find that the high-est temperature in the remnant phase is barely affected by theinitial condition (see Figure 7). Similar discussion appears inTanikawa et al. (2015).Other factors might affect the results of our simulations. Forexample, Zhu et al. (2013) and Dan et al. (2014) performedmerger simulations of non-spinning WDs and found that thestructures of such merger remnants are different from those inmergers of synchronously spinning WDs, especially for thecase of nearly equal masses. In non-spinning cases, the hightemperature region is formed near the center of the mergerremnant, and carbon burning might occur in that region. Sinceit is still uncertain whether binary WDs maintain synchroniza-tion until their merger, our results for the AIE path might bechanged. The topic of synchronization is out of the scope inthis paper, so we leave this in our future works. Numerical Resolution
Pakmor et al. (2012b) concluded that the numerical reso-lution of simulation is one of the most important factors forcarbon burning in the merger phase because very high reso-lution is required to identify very small hot spots. Therefore,we examine the dependence of our results on the numericalresolution. We perform the same simulations with four differ-ent resolutions, i.e., 10 k , 50 k , 100 k , and 500 k per one solarmass. The highest temperature is critical both for dynamicalcarbon burning and quiescent carbon burning, so we focus onthe dependence of the highest temperature on the resolution.Figure 8(a) and 8(b) show the dependence of the highesttemperature in the merger phase on the numerical resolutionfor the (a) raw and (b) smoothed temperatures, respectively.The highest temperature increases with the number of SPHparticles. In other words, our simulation does not convergeyet and the final fates of some models could be changed from Sato et al. l og T m a x [ K ] (a) sun sun sun sun sun sun sun sun sun sun sun sun l og T s , m a x [ K ] log N [M sun-1 ](b) F IG . 8.— Dependence of highest temperature on the numerical resolutionin the merger phase for the (a) raw and (b) smoothed temperatures. Thehorizontal axis is the number of SPH particles per a solar mass. The verticalaxis is the highest temperature. Shapes and colors of symbols have the samemeaning as those in Figure 2. The highest temperature tends to increase withthe numerical resolution. the AIC path to the VM path. Such a tendency was also re-ported in Pakmor et al. (2012b). In this sense, we must furtherincrease the numerical resolution to definitely identify the fateof merger products, at least, more than 500 k M - ⊙ . Figure 9(a)and 9(b) show the dependence of the minimum τ CC /τ dyn ra-tio in the merger phase on the numerical resolution for the(a) raw and (b) smoothed temperatures, respectively. The ma-genta dashed line indicates τ CC /τ dyn = 1 .
0. It tends to decreaseas the numerical resolution increases. This trend is consistentwith that of the highest temperature. For smoothed temper-ature, only very massive pairs (both masses > ∼ . M ⊙ ) canignite carbon dynamically.On the other hand, Figure 10(a) and 10(b) show the high-est temperature in the remnant phase for the (a) raw and (b)smoothed temperatures, respectively. Comparing with the re-sults in the merger phase (Figure 8), the highest temperaturedepends barely on the resolution. Especially, for the smoothedtemperature in Figure 10(b), it converges for almost all themass combinations. This tendency of weak dependence onthe resolution was already reported in the previous studies(Raskin et al. 2012; Dan et al. 2014). CONCLUSIONS
We have performed SPH simulations of CO WD merg-ers for the mass combinations of 0 . . M ⊙ from the startof the RLOF to the formation of a quasi-stationary mergerremnant, and examined whether carbon burning occurs ei-ther in the merger phase or remnant phase. Using the results -50510152025 l og t CC / t d y n (a) 1.1+1.0M sun sun sun sun sun sun sun sun sun sun sun sun -50510152025 4 5 6 l og t CC / t d y n log N [M sun-1 ](b) F IG . 9.— Dependence of minimum τ CC / τ dyn ratio on the numerical resolu-tion in the merger phase for the (a) raw and (b) smoothed temperatures. Thehorizontal magenta dashed lines indicate τ CC / τ dyn = 1 . of SPH simulations, we have investigated the mass range ofCO WDs that possibly lead to an SN Ia in the merger phaseor remnant phase. We have obtained the mass range as fol-lows. When the primary and secondary are as massive as0 . M ⊙ ≤ M , ≤ . M ⊙ , the binary results in an SN Ia inthe merger phase. On the other hand, when the primary is0 . M ⊙ ≤ M ≤ . M ⊙ and the total mass of the binary ex-ceeds 1 . M ⊙ , they lead to an SN Ia in the remnant phase.From the obtained mass range, we have estimated the rateof SNe Ia coming from CO WD mergers in our galaxy. It is6 . × - yr - M - ⊙ if we use the results of our raw tempera-ture calculations, while it is 9 . × - yr - M - ⊙ if we use thatof the smoothed temperature. These are only less than 9% ofthe entire SN Ia rate. Therefore, it is unlikely that the mergersof CO WDs are the main progenitors of SNe Ia.Of course, the above estimate is not conclusive becauseof several uncertainties in our calculation. We have checkedthe dependence of the highest temperature on the numericalresolution both in the merger and remnant phases in order toexamine the numerical convergence of our simulations. Wehave found that the highest temperature in the merger phasedepends on the numerical resolution. It tends to increase withthe resolution as already reported in Pakmor et al. (2012b).On the other hand, in the remnant phase, the highest tempera-ture depends barely on the numerical resolution. Therefore, itis necessary to increase the number of SPH particles, at least,up to ≥ k M - ⊙ , for definite conclusion. Additionally, ourcalculations for SNe Ia in the remnant phase is not sufficient.In order to obtain the decisive conclusion, we have to followerger of White Dwarfs 9 l og T m a x [ K ] (a) sun sun sun sun sun sun sun sun sun sun sun sun l og T s , m a x [ K ] log N [M sun-1 ](b) F IG . 10.— Same as Figure 8, but for the highest temperature in the remnantphase. The highest temperature seems to converge. the further evolution of the merger remnant, like Yoon et al.(2007). This is one of our future works.We thank the anonymous referee for many detailed com-ments that help to improve the paper. These simulationswere performed by using computational resources of KavliInstitute for the Physics and Mathematics of the Universe(IPMU), and HA-PACS at the Center for Computational Sci-ences in University of Tsukuba under Interdisciplinary Com-putational Science Program. This research has been supportedin part by Grants-in-Aid for Scientific Research (23224004,23540262, 23740141, 24540227, 26400222, and 26800100)from the Japan Society for the Promotion of Science andby the World Premier International Research Center Initia-tive, MEXT, Japan. This work is partly supported by MEXTprogram for the Development and Improvement for the NextGeneration Ultra High-Speed Computer System under itsSubsidies for Operating the Specific Advanced Large Re-search Facilities. APPENDIXSUMMARY OF CO WD MERGER SIMULATIONS
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TABLE 1S
UMMARY OF ALL CALCULATED MODELS M M a init T max ρ ( T max ) T s , max ρ ( T s , max ) T max , rem ρ ( T max , rem ) T s , max , rem ρ ( T s , max , rem )( M ⊙ ) ( M ⊙ ) (10 cm) (10 K) (10 g cm - ) (10 K) (10 g cm - ) (10 K) (10 g cm - ) (10 K) (10 g cm - )Resolution = 10 k M ⊙ - k M ⊙ - TABLE 2S
UMMARY OF ALL CALCULATED MODELS M M a init T max ρ ( T max ) T s , max ρ ( T s , max ) T max , rem ρ ( T max , rem ) T s , max , rem ρ ( T s , max , rem )( M ⊙ ) ( M ⊙ ) (10 cm) (10 K) (10 g cm - ) (10 K) (10 g cm - ) (10 K) (10 g cm - ) (10 K) (10 g cm - )Resolution = 100 k M ⊙ - k M ⊙ -1