Young and Massive Binary Progenitors of Type Ia Supernovae and Their Circumstellar Matter
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YOUNG AND MASSIVE BINARY PROGENITORS OF TYPE Ia SUPERNOVAEAND THEIR CIRCUMSTELLAR MATTER I ZUMI H ACHISU
Department of Earth Science and Astronomy, College of Arts and Sciences, University of Tokyo, Komaba 3-8-1, Meguro-ku, Tokyo 153-8902, Japan M ARIKO K ATO
Department of Astronomy, Keio University, Hiyoshi 4-1-1, Kouhoku-ku, Yokohama 223-8521, Japan
AND K EN ’ ICHI N OMOTO
Department of Astronomy, University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, and Institute for the Physics and Mathematics of the Universe,University of Tokyo, Kashiwa, Chiba 277-8582, Japan to appear in the Astrophysical Journal
ABSTRACTWe present new evolutionary models for Type Ia supernova (SN Ia) progenitors, introducing mass-strippingeffect on a main-sequence (MS) or slightly evolved companion star by winds from a mass-accreting whitedwarf (WD). The mass-stripping attenuates the rate of mass transfer from the companion to the WD. As aresult, quite a massive MS companion can avoid forming a common envelope and increase the WD mass upto the SN Ia explosion. Including the mass-stripping effect, we follow binary evolutions of various WD + MSsystems and obtain the parameter region in the initial donor mass – orbital period plane where SNe Ia occur.The newly obtained SN Ia region extends to donor masses of 6 - M ⊙ , although its extension depends onthe efficiency of mass-stripping effect. The stripped matter would mainly be distributed on the orbital planeand form very massive circumstellar matter (CSM) around the SN Ia progenitor. It can explain massive CSMaround SNe Ia/IIn(IIa) 2002ic and 2005gj as well as tenuous CSM around normal SN Ia 2006X. Our newmodel suggests the presence of very young ( . yr) populations of SNe Ia, being consistent with recentobservational indications of young population SNe Ia. Subject headings: binaries: close — circumstellar matter — stars: winds, outflows — supernovae: individual(SN2002ic, SN 2005gj, SN 2006X) INTRODUCTION
The nature of Type Ia supernova (SN Ia) progenitors hasnot been clarified yet (e.g., Niemeyer & Hillebrandt 2004;Nomoto et al. 2000), although it has been commonly agreedthat the exploding star is a mass-accreting carbon-oxygenwhite dwarf (C+O WD). For the exploding WD itself, theobserved features of SNe Ia are better explained by theChandrasekhar mass model than the sub-Chandrasekhar massmodel (e.g., Livio 2000). However, there has been no clearobservational indication as to how the WD mass gets closeenough to the Chandrasekhar mass for carbon ignition; i.e.,whether the WD accretes H/He-rich matter from its binarycompanion [single degenerate (SD) scenario], or two C+OWDs merge [double degenerate (DD) scenario].Recently, the following two important findings have beenreported in relation to the SN Ia progenitors: (1) circumstellarmatter (CSM) around the progenitors, and (2) a very young( . yr) population of the progenitors. Circumstellar Matter:
In the SD scenario, H/He-richCSM is expected to exist around SNe Ia as a result ofmass transfer from the companion as well as the WD winds(e.g., Nomoto 1982; Hachisu, Kato, & Nomoto 1999a). Thussearching for H/He-rich CSM is one of the key observationsto identify the progenitors (e.g., Lundqvist et al. 2003). Re-cently detections of such CSM have been reported for several
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SNe Ia, i.e., observations of narrow H-emission lines in SNe2002ic (Hamuy et al. 2003) and 2005gj (Alderling et al.2006; Prieto et al. 2007) (Type Ia/IIn or IIa (Deng et al.2004)), thermal X-rays from 2005ke (Immler et al. 2006),and Na I D lines in 2006X (Patat et al. 2007a).The identification of SN 2002ic as an SN Ia has beenconfirmed by the recent spectral comparison between SN2005gj and SNe Ia (Prieto et al. 2007), being againstthe Type Ic suggestion by Benetti et al. (2006). Sev-eral CSM interaction models suggested a 1 - M ⊙ CSM(Chugai et al. 2004; Nomoto et al. 2005). The evolution-ary origin of such a massive CSM has been explored byLivio & Reiss (2003) based on a common envelope evolutionmodel, by Han & Podsiadlowski (2006) from the delayeddynamical instability model of binary mass transfer, and byWood-Vasey & Sokoloski (2006) based on a recurrent novamodel with a red giant companion.For normal SNe Ia, non-detection of radio has put the up-per limit of mass loss rate as ˙ M / v . - M ⊙ yr - , where v ≡ v /
10 km s - (Panagia et al. 2006). However, the opticalobservations of SN 2006X have detected variable Na I D linesfrom CSM, whose expansion velocity and mass have been es-timated to be v ∼
10 and ∼ - M ⊙ (Patat et al. 2007a).Patat et al. have suggested that the CSM in SN 2006X origi-nated from the red-giant companion because of relatively lowvelocities. Comparing the SN 2006X light curves with theother normal SNe Ia light curves, Wang et al. (2007a) sug-gested that the obvious deviation, the decline rate is slow-ing down in a later phase, can be explained by an interac- Hachisu et al.tion between ejecta and CSM or a light echo of circumstel-lar/interstellar matter (see also Wang et al. 2007b). Young Population:
According to Mannucci et al. (2006),the present observational data of SNe Ia are best matched by abimodal population of the progenitors, in which about 50 per-cent of SNe Ia explode soon after their stellar birth at the delaytime of t delay ∼ yr, while the remaining 50 percent have amuch wider distribution of the delay time of t delay ∼ t delay . yr) of SNe Iahave been suggested from the distribution of SNe Ia relativeto spiral arms (e.g., Bartunov et al. 1994; della Valle & Livio1994). Recently, Di Stefano & Kong (2003) reported, basedon the Chandra data from four external galaxies: an ellipticalgalaxy (NGC 4967), two face-on spiral galaxies (M101 andM83), and an interacting galaxy (M51), that in every galaxythere are at least several hundred luminous supersoft X-raysources (SSXSs) with a luminosity of & erg s - and that,in the spiral galaxies M101, M83, and M51, SSXSs appearto be associated with the spiral arms. The latter may indicatethat SSXSs are young systems, possibly younger than 10 yr,and has some close relation to the young population of SNeIa.The SD scenario has ever not predicted such young pop-ulations of t delay ∼ yr, corresponding to, at least, thezero-age main-sequence (ZAMS) stars at mass 5 - M ⊙ (see, e.g., Li & van den Heuvel 1997; Hachisu et al. 1999b;Langer et al. 2000; Han & Podsiadlowski 2004). In thepresent paper, we propose a scenario for such a young SNIa population by introducing mass-strippingeffectinto binaryevolutions. Mass-accreting WDs blow optically thick windswhen the mass transfer rate to the WD exceeds the criticalrate of ˙ M cr ∼ × - M ⊙ yr - (Hachisu et al. 1996). TheWD wind collides with the secondary’s surface and strips offmatter. When the mass-stripping effect is efficient enough, themass transfer rate to the WD is attenuated and the binary canavoid the formation of a common envelope even for a rathermassive secondary.The mass-stripping effect on a MS companion has beenfirst introduced by Hachisu & Kato (2003b,c), who ana-lyzed two quasi-periodic transient supersoft X-ray sources,RX J0513.9 - ∼ -
120 days)and low ( ∼
40 days) states with an amplitude of 1 mag(Alcock et al. 1996). RX J0513 is X-ray bright only duringthe optical low states (Reinsch et al. 2000). Hachisu & Kato(2003b) proposed a model that the mass transfer is modulatedby the WD wind because the WD wind collides with the com-panion and strips off its surface and attenuates the mass trans-fer rate. When the mass transfer rate decreases below the crit-ical rate ˙ M cr , the WD wind stops and supersoft X-ray turnson. This corresponds to an optical low sate. Then the mass-transfer rate recovers because of no attenuation by WD windsand the WD blows winds again. X-ray turns off and an opti-cal high state resumes and the binary starts the next cycle ofquasi-periodic oscillation. Such a self-sustained model natu-rally explains major characteristics of quasi-periodic high andlow states and this success encourages us to adopt the same idea in the evolution scenario of supersoft X-ray sources andSN Ia progenitors.In the present paper, we show that this mass-stripping effectderives (1) formations of circumstellar matter (CSM) aroundSNe Ia and (2) a very young population of SNe Ia. We sum-marize our basic treatments of mass-stripping effect and bi-nary evolutions in §2, and then show our numerical resultsand their relations to a very young population of SNe Ia in§3. In §4 we present the origin of CSM around SNe Ia basedon our results and show a relation between the very youngpopulation of SNe Ia and their massive CSM. Discussion andconcluding remarks follow in §§5 and 6. MASS-STRIPPING EFFECT AND BINARY EVOLUTION
Strong winds from a mass-accreting WD collide with thecompanion star and strip off its surface. This mass-strippingeffect plays an important role in binary evolutions (e.g.Hachisu et al. 1999a). Here we reformulate its treatment inour binary evolution calculation.
New Aspects of Binary Evolutions
First we briefly introduce a new binary evolutionary processthrough stages (a)-(d) below (also shown in Fig. 1a–d), where(c) and (d) are new stages introduced by mass-stripping.(a) The more massive (primary) component of a binaryevolves to a red giant star (with a helium core) or an AGBstar (with a C+O core) and fills its Roche lobe. Mass transferfrom the primary to the secondary begins and a common en-velope is formed. After the first common envelope evolution,the separation shrinks and the primary component becomes ahelium star or a C+O WD. The helium star evolves to a C+OWD after a large part of helium is exhausted by core-helium-burning. We eventually have a close pair of a C+O WD and amain-sequence (MS) star as shown in Figure 1a.(b) After the secondary evolves to fill its Roche lobe, themass transfer to the WD begins. This mass transfer occursin a thermal timescale because the secondary mass is moremassive than the WD. The mass transfer rate exceeds the crit-ical rate for the optically thick wind to blow from the WD(Hachisu et al. 1996, 1999a,b).(c) Optically thick winds from the WD collide withthe secondary surface and strips off its surface layer(Hachisu & Kato 2003a,b,c). This mass-stripping attenuatesthe rate of mass transfer from the secondary to the WD, thuspreventing the formation of a common envelope for a moremassive secondary in the case with than in the case withoutthis effect. Thus the mass-stripping effect widens the donormass range of SN Ia progenitors (see Fig. 3 below).(d) Such stripped-off matter forms a massive circumstellartorus on the orbital plane, which may be gradually expand-ing with an outward velocity of ∼ -
100 km s - (Fig. 1d),because the escape velocity from the secondary surface to L3point is v esc ∼ [( φ L3 - φ MS ) GM / a ] / ∼
100 km s - (see be-low). Subsequent interaction between the fast wind from theWD and the very slowly expanding circumbinary torus formsan hourglass structure (Fig. 1c–d). Formulation of Mass-stripping
Fast strong winds collide with the companion as illustratedin Figure 1. The companion’s surface gas is shock-heated andablated in the wind. We estimate the shock-heating by assum-ing that the velocity component normal to the companion’ssurface is dissipated by the shock and the kinetic energy isrogenitors of Type Ia Supernovae 3 (c)(b)(a) (d)WD MSwinds massstripping torus torus expandingtorus F IG . 1.— A schematic configuration of a binary evolution including mass-stripping effect. (a) Here we start a pair of a C+O WD and a more massivemain-sequence (MS) star with a separation of several to a few tens of solar radii. (b) When the secondary evolves to fill its Roche lobe, mass transfer ontothe WD begins. The mass transfer rate exceeds a critical rate for optically thick winds. Strong winds blow from the WD. (c) The hot wind from the WD hitsthe secondary and strips off its surface. (d) Such stripped-off material forms a massive circumstellar disk or torus and it gradually expands with an outwardvelocity of ∼ -
100 km s - . The interaction between the WD wind and the circumstellar torus forms an hourglass structure. The WD mass increases up to M Ia = 1 . M ⊙ and explodes as an SN Ia. When we observe the SN Ia from a high inclination angle such as denoted by “line of sight,” circumstellar matter(CSM) can be detected as absorption lines like in SN 2006X. converted into the thermal energy of the surface layer. Theheated surface layer expands to be ablated in the wind.To obtain the mass stripping rate, we use the same formula-tion proposed by Hachisu & Kato (2003b,c). We equate thestripping rate times the gravitational potential at the compan-ion surface to the net rate of energy dissipation by the shockas: GMa ( φ L3 - φ MS ) · ˙ M strip = 12 v · η eff · g ( q ) · ˙ M wind , (1)where M = M WD + M MS , M WD is the WD mass, M MS is themain-sequence companion mass, a is the separation of the bi-nary; φ MS and φ L3 denote the Roche potential (normalized by GM / a ) at the MS surface and the L3 point near the MS com-panion, respectively; v is the WD wind velocity, η eff is the ef-ficiency of conversion from kinetic energy to thermal energyby the shock, g ( q ) is the geometrical factor of the MS surfacehit by the wind including the inclination (oblique shock) ef- fect of the wind velocity against the companion’s surface (seeHachisu, Kato, & Nomoto 1999a, for more details on g ( q )),and q ≡ M / M = M MS / M WD is the mass ratio. Here we mod-ified equation (21) of Hachisu et al. (1999a) to include theeffect of Roche lobe overflow from the L3 point. Then thestripping rate is estimated as ˙ M strip = c ˙ M wind , (2)where c ≡ η eff · g ( q ) φ L3 - φ MS (cid:18) v a GM (cid:19) . (3)Here we assume that the WD wind is spherically symmetric.If the asphericity of the WD wind is not so large, having a lat-itudinal ( θ -angle) dependency like a broad angle jet, we havea different form of g ( q ) and its value may be much smallerthan that for the spherically symmetric WD winds. We alsoassume η eff = 1 in the present calculation. When the wind ve- Hachisu et al. F IG . 2.— SN Ia evolutions for two typical cases of WIND and CALM. (a) Case WIND: starting from M WD , = 1 . M ⊙ , M , = 5 . M ⊙ , and P = 2 .
15 dayswith c = 3, the WD reaches the SN Ia explosion in the wind phase at t = 6 . × yr. The WD mass ( M WD ), secondary mass ( M ), mass loss rate from thesecondary ( ˙ M ), WD wind mass loss rate ( ˙ M wind ), radius of the secondary ( R ), effective radius of the Roche lobe for the secondary ( R ∗ ), and orbital period( P orb ) are plotted. Only the orbital period is multiplied by four to easily see its change. (b) Case CALM: starting from M WD , = 1 . M ⊙ , M , = 5 . M ⊙ , and P = 6 .
79 days with c = 3, the WD reaches the SN Ia explosion but in an SSXS phase without winds at t = 6 . × yr. The WD wind stops at t = 5 . × yr.Even after that, the WD loses its mass due to weak helium shell flashes (Kato & Hachisu 1999). Here ˙ M wind includes an average mass loss rate by helium shellflashes and thus does not become zero after the optically thick wind of steady hydrogen shell burning stops. Values of the secondary radius ( R ) and the Rochelobe radius for the secondary ( R ∗ ) are divided by two to squeeze them into the figure. locity is as fast as 4,000 km s - , we have c ∼
10 as estimatedby Hachisu & Kato (2003b). Although there is a large ambi-guity in this kind of parameterization as c , Hachisu & Kato(2003b,c) found the best fit models with c = 1 . -
10 forRX J0513.9 - c = 7 - c = 1, 3, and 10 to examine the dependence on the mass-stripping effect, because the essential ambiguity of our for-mulation is included in the c parameter.When winds blow from the WD and strip off the compan-ion’s surface, the change of the separation, ˙ a , is calculatedfrom ˙ aa = ˙ M + ˙ M M + M - ˙ M M - ˙ M M + ˙ JJ = ˙ M + ˙ M M + M - ˙ M M - ˙ M M + M + M M M (cid:0) ℓ w ˙ M wind + ℓ s ˙ M strip (cid:1) , (4)where M = M WD , M = M MS , ℓ w and ℓ s are the specific an- gular momenta of the WD wind and the stripped-off matter,respectively, in units of a Ω orb with Ω orb being the orbital an-gular velocity. Since the WD wind is much faster than theorbital motion, the wind cannot get angular momentum fromthe orbital torque during its journey, so that the wind has thesame specific angular momentum as the WD, which is esti-mated as ℓ w = (cid:18) q + q (cid:19) . (5)The ablated gas from the companion is assumed to have theangular momentum at the companion’s surface. Then we havea numerical factor of ℓ s = h ( q ) g ( q ) , (6)which was given in Table 1 of Hachisu et al. (1999a) and israther small compared with ℓ w . (See Hachisu et al. 1999a,for more details of ℓ s .)rogenitors of Type Ia Supernovae 5 Modified Mass Transfer Rate
We have followed binary evolutions from the initial stateof ( M , , M , , P ), i.e., ( M WD , , M MS , , P ), where P is theinitial orbital period. Here, the subscript naught (0) denotesstage (a) in Figure 1, that is, before the mass transfer fromthe secondary starts. The radius, R ( M , t ), and luminosity, L ( M , t ), of stars which have slightly evolved off from thezero-age main-sequence (ZAMS), are calculated using the an-alytic form given by Tout et al. (1997).The mass transfer proceeds on a thermal time scale whenthe mass ratio M / M exceeds 0.79. We approximate the masstransfer rate as - ˙ M = M τ KH · max (cid:18) ζ RL - ζ MS ζ MS , (cid:19) , (7)where τ KH is the Kelvin-Helmholtz timescale given by τ KH ≈ × yr (cid:18) M M ⊙ (cid:19) (cid:18) R R ⊙ · L L ⊙ (cid:19) - (8)(e.g., Paczynski 1971), and ζ RL = d log R ∗ / d log M and ζ MS = d log R MS / d log M are the mass-radius exponents of the innercritical Roche lobe and the main sequence component, respec-tively (e.g., Hjellming & Webbink 1987). The effective ra-dius of the inner critical Roche lobe, R ∗ , is calculated fromEggleton’s (1983) empirical formula, i.e., R ∗ a = f ( q ) ≡ . q / . q / + ln(1 + q / ) , (9)where q = M / M .When the mass transfer rate to the WD exceeds a criticalvalue, which is given by ˙ M cr ≈ . × - (cid:18) M WD M ⊙ - . (cid:19) M ⊙ yr - , (10)for the solar composition (hydrogen content of X = 0 . Z = 0 . ˙ M wind ( < ˙ M cr is the sameas the critical rate for mass-accreting WDs to expand to a giantsize, i.e., ˙ M RG (see Nomoto et al. 2007, for the recent calcu-lation of ˙ M RG ). The mass loss from the WD also occurs duringthe hydrogen shell flashes when - ˙ M < ˙ M stable , where ˙ M stable is the lowest rate for steady hydrogen burning and given byequation ˙ M stable ≈ . × - (cid:18) M WD M ⊙ - . (cid:19) M ⊙ yr - (11)(Nomoto et al. 2007). When ˙ M stable < - ˙ M < ˙ M cr , we haveno mass loss associated with steady hydrogen shell-burningbut have mass loss by helium shell flashes. This massloss play some role in the binary evolution (Kato & Hachisu1999). Therefore, ˙ M wind is the summation of the opticallythick wind mass loss, hydrogen shell flashes, and helium shellflashes.We have the relation ˙ M + ˙ M = ˙ M wind + ˙ M strip , (12)from the total mass conservation, thus defining the net masstransfer rate to the WD as ˙ M transfer ≡ ˙ M strip - ˙ M = ˙ M - ˙ M wind , (13) where signs of ˙ M transfer > ˙ M strip ≤ ˙ M < ˙ M ≥
0, and ˙ M wind ≤ ˙ M is given, we have the netmass transfer rate of ˙ M transfer = (cid:26) ( c ˙ M cr - ˙ M ) / ( c + , for - ˙ M > ˙ M cr - ˙ M , for - ˙ M ≤ ˙ M cr , (14)where we use equations (2), (13), and a relation of - ˙ M wind = ˙ M transfer - ˙ M cr , (15)for - ˙ M > ˙ M cr . Other treatments for binary evolution are es-sentially the same as those in Hachisu et al. (1999b).Figure 2 shows two typical evolutionary sequences thatdemonstrate the effects by the modified mass transfer rate, ˙ M , in equation (7).(a) Starting from M WD , = 1 . M ⊙ , M , = 5 . M ⊙ , and P = 2 .
15 days with c = 3, the WD reaches the SN Ia ex-plosion in the wind phase (Case WIND) at t = 6 . × yrafter the secondary fills its Roche lobe. The WD increasesits mass ( M WD ) up to M Ia = 1 . M ⊙ to explode as an SN Ia.The secondary mass ( M ) decreases to 2 . M ⊙ at the explo-sion. Both the mass decreasing rate of the secondary (dashedline labeled ˙ M ) and the WD wind mass loss rate (dashed linelabeled ˙ M wind ) are also decreasing rapidly especially in theearly phase of t . × yr. This is because - ˙ M is largeand the mass transfer rate, ˙ M transfer , is large during this phase,and as a result, both the WD wind mass loss rate, ˙ M wind , andthe stripping rate, ˙ M strip , are also large. Shortly after this earlyphase, the Roche lobe’s mass-radius exponent, ζ RL , becomessmaller than the secondary’s mass-radius exponent, ζ MS , thatis, ζ RL - ζ MS <
0. This gives - ˙ M = M /τ KH from equation(7). We keep this mass transfer rate as long as the secondaryoverfills the Roche lobe, i.e., R > R ∗ . In Figure 2a, we plotthe secondary radius (the red line labeled R ) and the Rochelobe radius for the secondary component (the blue line labeled R ∗ ) to show the condition of R > R ∗ during the evolution.(b) Starting from M WD , = 1 . M ⊙ , M , = 5 . M ⊙ , and P = 6 .
79 days with c = 3, the WD reaches the SN Ia explo-sion but in a phase of no winds (Case CALM) at t = 6 . × yr after the secondary fills its Roche lobe. In this case the evo-lution of the mass transfer rate is different from Case WINDabove. With - ˙ M = M /τ KH for ζ RL < ζ MS in equation (7), thesecondary eventually underfills the Roche lobe, i.e., R < R ∗ .This can be seen in Figure 2b, where the line of R crossesthe line of R ∗ at t ∼ × yr. This is because the strippedmatter has rather low specific angular momentum (eq. [6]),so that the binary separation hardly shrinks or even increasesas seen from the temporal increase in the orbital period inFigure 2b. In realistic binary evolutions, the mass transferis tuned in a way that the secondary radius is always equalto the Roche lobe radius for the secondary, i.e., R = R ∗ .Therefore - ˙ M is drastically decreased after t ∼ × yr,as shown in Figure 2b. Thus, the optically thick WD windstops at t = 5 . × yr. In such a low mass transfer phase as ˙ M transfer ∼ × - M ⊙ yr - , weak helium shell flashes occurand play an important role as a mass loss mechanism. Thishelium flash wind also strips off the secondary surface, thusworking as a stripping effect. We introduce mass-strippingeffect by these helium shell flashes into our binary evolution.Very small but finite ˙ M wind in Figure 2b (after winds stop) rep-resents the mass loss from the WD at helium shell flashes and ˙ M includes the ensuing mass-stripping from the secondary. Hachisu et al. F IG . 3.— The initial parameter regions producing SNe Ia are plotted in the log P - M , (orbital period — donor mass) plane for the WD + MS systemswith various mass-stripping factors, c . Thick solid: c = 10. Medium solid: c = 3. Thin solid: c = 1. Dotted: c = 0. The (red) hatched region indicates aregion with a short delay time ( t delay ≤
100 Myr) for the case of c = 10. The region extends to the more massive donors for the larger c . Two supersoft X-raysources, RX J0513.9 - open circle ) and V Sge ( filled circle ), are plotted, masses of which are estimated to be 2 . M ⊙ (Hachisu & Kato 2003b) and 3 . M ⊙ (Hachisu & Kato 2003c), orbital periods of which are determined to be 0.76 days (Pakull et al. 1993) and 0.51 days (Herbig et al. 1965; Patterson et al. 1998),respectively. The position of V Sge suggests that c > IG . 4.— Dependence of the SN Ia parameter region on the initial WD mass, M WD , , for a mass-stripping factor of c = 3. From inside to outside, M WD , = 0 . thick solid line ), and 1 . M ⊙ . There is no region for M WD , = 0 . M ⊙ . The (red) sparse hatched region indicates the delay time of t delay ≤
100 Myrfor M WD , = 1 . M ⊙ but the (blue) dense hatched region for M WD , = 0 . M ⊙ . rogenitors of Type Ia Supernovae 7 F IG . 5.— The parameter region that produces SNe Ia is plotted in the log P - M d (orbital period — donor mass) plane for the WD + MS system. Here we assume M WD , = 1 . M ⊙ for the initial white dwarf mass. The initial WD + MS system inside the region encircled by the (red) thin solid line (labeled “initial”) willincrease its white dwarf mass up to the critical mass ( M Ia = 1 . M ⊙ ) for the SN Ia explosion to occur. The final state of the WD + MS system in the log P - M d plane just before the SN Ia explosion is encircled by the (blue) thick solid line (labeled “final”). The final state of the WD just before the SN Ia explosion isspecified by one of wind ( open circle ), steady H-burning ( filled triangle ), or recurrent nova ( open square ) phase. An hatched region indicates a region in whichthe progenitor explodes in a delay time of t delay ≤
100 Myr.
Dashed line : in a delay time of 200 Myr.
Dotted line : in a delay time of 400 Myr. Currently knownpositions of three recurrent novae are indicated by a star mark ( ⋆ ) for U Sco (e.g., Schaefer & Ringwald, 1995; Hachisu et al. 2000a,b), and by arrows for theother two recurrent novae, V394 CrA (Schaefer 1990) and CI Aql (Mennickent & Honeycutt 1995), of unknown companion masses. The WD masses of U Scoand V394 CrA were estimated to be 1 . M ⊙ (Hachisu et al. 2000a; Hachisu & Kato 2000) while that of CI Aql was 1 . M ⊙ (Hachisu & Kato 2003a).F IG . 6.— The final SN Ia region just before an SN Ia explosion. Each symbol has the same meaning as in Fig. 5. The upper black solid line and lower magentasolid line denote lines at - ˙ M = ˙ M cr and - ˙ M = ˙ M stable , respectively, just at the SN Ia explosion, where ˙ M is calculated from eq. (17) with R and L taken froma single star evolution given by Tout et al. (1997). Both the lines agree reasonably with the borders of WIND–CALM and CALM–RN, respectively. Hachisu et al. F IG . 7.— Same as Fig. 5, but for an initial WD mass of M WD , = 1 . M ⊙ .F IG . 8.— Same as Fig. 6, but for an initial white dwarf mass of M WD , = 1 . M ⊙ . Large difference in the border of WIND–CALM comes from the fact thatthe secondary considerably overfills the Roche lobe, i.e., R > R ∗ , at the SN Ia explosion in the Case WIND.TABLE 1T HREE TYPICAL CASES OF
SN I
A EXPLOSION case wind H burning CSM pre-SN history SN Ia delay time immediate radio/X-rayWIND wind steady massive: near WIND (V Sge type) IIa (02ic-like) young yesCALM no wind steady thin: far WIND → SSXS normal Ia young no ( ∼ -
100 yr)RN no wind flash very thin: many shells WIND → SSXS → RN normal Ia broad no ( ∼
100 – 1000 yr)or SSXS → RN rogenitors of Type Ia Supernovae 9 F IG . 9.— Same as Fig. 5, but for an initial white dwarf mass of M WD , = 0 . M ⊙ . There is no Case WIND (no open circles ).F IG . 10.— Same as Fig. 6, but for an initial white dwarf mass of M WD , = 0 . M ⊙ . YOUNG POPULATION TYPE IA SUPERNOVAE
Based on the binary evolution scenario proposed byHachisu et al. (1999a,b), we have followed binary evolutionsstarting from stage (b) in Figure 1, that is, just when the com-panion evolves to fill its Roche lobe. The main differencefrom the previous work cited above is the inclusion of mass-stripping effect. Our results are shown in Figures 3–10.Figure 3 shows the parameter regions that produce SNeIa (SN Ia region) in the log P - M , (the initial orbital pe-riod and the initial secondary mass) plane for the WD + MSsystem. Here the initial white dwarf mass is assumed to be M WD , = 1 . M ⊙ . The white dwarfs within these SN Ia re-gions will increase their mass, M WD , up to the critical mass( M Ia = 1 . M ⊙ ) for the SN Ia explosion to occur.The SN Ia region in the log P - M , plane is enclosed byfour boundaries. (1) The left boundary is given by the mass-radius relation for the zero-age main-sequence stars. (2) Thelower boundary is set by strong nova explosions, below which ˙ M transfer . × - M ⊙ yr - and the resultant nova explosionejects most of the accreted matter, thus preventing the WDmass from increasing. (3) The upper boundary is limited bythe formation of a common envelope. Here we assume thata common envelope is formed when ˙ M transfer & × - M ⊙ yr - because R , ph & a ∼ R ⊙ for such a high ˙ M transfer (seeHachisu et al. 1999b, for more details). (4) The right bound-ary corresponds to the end of central hydrogen burning of theMS companion: after that, it shrinks and underfills its Rochelobe.In Figure 3, the SN Ia regions for the various mass-strippingfactor c = 10, 3, and 1 are encircled by the thick, medium, andthin solid lines, respectively, and the no stripping case ( c = 0)by the dotted line. The position of the Galactic supersoft X-ray source V Sge is clearly outside the SN Ia region for c = 0,but inside the SN Ia region if c >
0. For larger c , the SNIa region extends to more massive M , , because the strongermass-stripping leads to the lower mass transfer rate, ˙ M transfer ,from the secondary to the WD (see eq.[14]), thus prevent-ing the formation of a common envelope for larger M , . Asshown in this figure quite massive secondaries produce SNe Ia(e.g., M , = 7 . M ⊙ for c = 10) for the strong mass-strippingcase of c & t delay ≤
100 Myr, by the red shadow inFigures 3, 4, 5, 7, and 9. Figure 4 shows the SN Ia regions fordifferent initial WD masses, M WD , = 0 .
7, 0.8, 0.9, 1.0, and1 . M ⊙ . The red (sparse) and blue (dense) hatched regions in-dicate the delay time of t delay ≤
100 Myr for M WD , = 1 . M ⊙ and 0 . M ⊙ , respectively.We apply the present result to equation (1) ofIben & Tutukov (1984), i.e., ν = 0 . · ∆ q · Z M u M l dMM . · ∆ log a yr - , (16)where ∆ q , ∆ log a , M l , and M u are the appropriate rangesof the mass ratio and the initial separation, and the lower andupper limits of the primary mass for SN Ia explosions in solarmass units, respectively. We then estimate the SN Ia birth ratein our Galaxy as ν WD + MS ∼ .
004 yr - , which is consistentwith the observation (Cappellaro et al. 1999).On the other hand, Hachisu et al. (1999a) proposed anotherchannel to SNe Ia, the symbiotic channel, binary of which consists of a white dwarf and a red giant (WD + RG), andestimated its birth rate to be ν WD + RG ∼ .
002 yr - .Assuming the initial distribution of binaries given byequation (16) at the burst of star formation (single event),we estimate the delay time distribution of SNe Ia for theWD + MS systems in Figure 11. The number ratio ofthese young populations is calculated for 10 bins of delaytime, (0 . , . . , . . , . . , . . , . . , . . , . . , . . , . . , . t delay ≤
100 Myr and t delay ≤
200 Myr are about 50% and 80%, respectively, of the to-tal SNe Ia coming from the WD + MS system, which isconsistent with the recent observational suggestions (e.g.,Mannucci et al. 2006; Aubourg et al. 2007).Short delay times ( t delay . yr) of some SNe Ia have beensuggested from the distribution of SNe Ia relative to spiralarms (e.g., Bartunov et al. 1994; della Valle & Livio 1994).Petrosian et al. (2005) reported that about 30–40% of SNe Iaare associated with spiral arms in their samples, being consist-ing with our results. Mannucci et al. (2006) have suggestedthat the delay time distribution function of SNe Ia has a bi-modality, one for young population ( t delay ∼
100 Myr) and theother with a broad distribution over ∼ t delay ≤
100 Myr fromthe WD + MS systems and a broad distribution from the WD+ RG systems (Hachisu et al. 1999a) as shown in Figure 12. FINAL STAGE OF BINARY EVOLUTION AND CIRCUMSTELLARMATTER
The final state of the WD depends mainly on the mass trans-fer rate ˙ M transfer from the donor star to the WD at the SN Ia ex-plosion (Nomoto 1982; Hachisu et al. 1999a; Nomoto et al.2007). As shown in Figure 2, ˙ M drops quickly in the earlystage and then slows down to almost a constant value. Atleast, in the early phase, the mass transfer proceeds on a ther-mal time scale, represented by the second term of equation(7), when the mass ratio M / M exceeds 0.79. So we approx-imate the mass transfer rate as - ˙ M ≈ M τ KH ∼ × - M ⊙ yr - (cid:18) R R ⊙ · L L ⊙ (cid:19) (cid:18) M M ⊙ (cid:19) - (17)By applying the approximate M - L relation of L ∝ M m ,where m ∼ . - M ⊙ zero-age main-sequence(ZAMS) stars or m ∼ . - M ⊙ ZAMS stars, - ˙ M ∝ R M m - . (18)Thus - ˙ M decrease as M decreases.Figures 5–10 show the SN Ia regions in the log P - M d (or-bital period — donor mass) plane for the initial WD + MSsystem (encircled by the red thin line and labeled “initial”)as well as the final state at the SN Ia explosion (encircledby the blue thick line and labeled “final”). Here we assume c = 3 and M WD , = 1 .
1, 1.0, and 0 . M ⊙ . In these figures, wedistinguish three final states just before the SN Ia explosion,i.e., optically thick WD wind phase (WIND: open circles),steady hydrogen burning phase without optically thick windsfrom WDs (CALM: filled triangles), and recurrent nova (RN)phase (RN: open squares). The characteristic properties forthese three progenitor stages are summarized in Table 1 andthe corresponding binary parameters are tabulated in Table 2. Case WIND rogenitors of Type Ia Supernovae 11
TABLE 2I
NITIAL PARAMETERS FOR T HREE
SN I
A EXPLOSIONS
WD mass secondary mass orbital period case pre-SN history SN Ia( M ⊙ ) ( M ⊙ ) (days)1 . - . - ∼ . - . - . - ∼ -
10 CALM WIND → SSXS normal Ia1 . - . . - ∼ . - → SSXS normal Ia1 . - . . - . ∼ . - → SSXS → RN normal Iaor SSXS → RN0 . . - ∼ . - → SSXS normal Ia0 . . - . ∼ . - → SSXS → RN normal Ia0 . - ∼ - → SSXS → RN normal Ia0 . - ∼ . - → SSXS normal Ia0 . . - ∼ . - → SSXS → RN normal Ia0 . - . ∼ . - → SSXS → RN normal Ia
When the mass transfer rate from the secondary continu-ously exceeds the critical rate of equation (10) until the fi-nal stage, the WDs explode during the wind phase (Fig. 2a).Therefore, we call this Case WIND. Case WIND is real-ized in the region of M , & M ⊙ and P , . M WD , = 1 . M ⊙ and 1 . M ⊙ (open circle), but no Case WINDexists for M WD , ≤ . M ⊙ as shown in Figures 5, 7, and 9.The stripped-off matter from the companion can easilyamount to ∆ M strip ∼ - M ⊙ and even reach 3 - M ⊙ as seenfrom the donor mass difference ∆ M between the “initial” andthe “final” in Figures 5, 7, and 9. More precisely, ∆ M con-sists of three parts, the stripped-off mass ∆ M strip , the accretedmass by the WD ∆ M , and the mass ejected by the WD wind ∆ M wind , i.e., ˙ M = ˙ M strip + ˙ M wind - ˙ M from equation (12). Thiscan be approximated as ˙ M ≈ ˙ M strip + ˙ M wind = (1 + / c ) ˙ M strip =4 / ˙ M strip because ˙ M ≪ - ˙ M , so that ∆ M strip ≈ / ∆ M for c = 3.The stripped-off material forms CSM very near the SN Ia.We expect that stripped-off matter did not go away from thesystem because the velocity of stripped-off matter may notexceed the escape velocity of the binary system. Then theSN Ia undergoes circumstellar interaction as observed in TypeIa/IIn (or IIa) SNe 2002ic and 2005gj.Alderling et al. (2006) suggested that the host galaxy ofSN 2005gj had a burst of star formation 200 ±
70 Myr ago. Ifthe progenitor of SN 2005gj was born at that time, its delaytime is consistent with our Case WIND as shown in Figure11.
Case CALM
When the mass transfer rate from the secondary is belowthe critical rate for optically thick winds but above the lowestrate of steady hydrogen burning, i.e., ˙ M stable < ˙ M transfer < ˙ M cr ,the WDs undergo steady H-burning at the time of SN Ia ex-plosion (filled triangles in Figs. 5–10). We call this CaseCALM because no optically thick winds occur. The WDs areobserved as supersoft X-ray sources (SSXSs) until the SN Iaexplosion. The stripped-off material forms CSM but it hasbeen dispersed too far to be detected immediately after theSN Ia explosion.The CALM case is realized in the region of M , & M ⊙ and P , & M WD , = 1 . M ⊙ and 1 . M ⊙ in Figures5, 7, and 9, where ˙ M transfer in the early phase is much largerthan that of P , . R and L are much larger than those for P , . ˙ M transfer is much larger, thus much more mass had been lostin the earlier phase. As a result, the wind phase finishes at an earlier time even for the same initial mass M , as seen inFigures 2a and 2b. Therefore, at the SN Ia explosion, no windoccurs.In the region of M , . M ⊙ for M WD , = 1 . M ⊙ and1 . M ⊙ in Figures 5–8 (filled triangles or open squares), M decreases to as small as the primary M WD , i.e., the mass ra-tio of q ∼
1, at the SN Ia explosion, which corresponds to alower part of the “final” region. Then the mass transfer ratedecreases down to ˙ M transfer < ˙ M cr or even ˙ M transfer < ˙ M stable be-cause L is smaller for the smaller M even if the mass transferitself is proceeding on a thermal time scale. The wind phasehas ended earlier than for M , & M ⊙ .The border between Case WIND and Case CALM canbe simply estimated from the condition ˙ M transfer = ˙ M cr and R = R ∗ at the SN Ia explosion. We have calculated ˙ M fromequation (17) with R and L being taken from Tout et al.(1997) (a single star evolution). This line agrees reasonablywith the border of WIND–CALM in Figure 6 but largely de-viates from it in Figure 8. This is because the secondary con-siderably overfills the Roche lobe, i.e., R > R ∗ , at the SN Iaexplosion for M WD , = 1 . M ⊙ .For M WD , . . M ⊙ (filled triangles or open squares), M decreases at the SN Ia explosion as shown in Figure 9 (“final”region) and ˙ M transfer decreases to be lower than ˙ M cr mainlybecause the time for the WD to reach M Ia = 1 . M ⊙ is longerand much more mass is lost during the evolution. The windphase has ended before the SN Ia explosion.For a typical case of c = 3, M WD , = 0 . M ⊙ , M , = 4 . M ⊙ ,and P = 1 . t = 9 × yr after the secondary fills its Roche lobe.The wind has already stopped 3 × yr ago (duration of theWIND phase, ∆ t wind = 6 × yr, and duration of the CALMphase, ∆ t calm = 3 × yr), so that the inner edge of stripped-off material has already gone to (10 -
100 km s - ) × (3 × yr) ∼ - cm from the SN Ia. Therefore, it takesabout 10 -
100 yr for the SN Ia ejecta to reach the inner edgeof stripped-off matter. We do not expect radio or X-ray until,at least, 10 -
100 yr after the explosion. Thus the resultantSNe Ia are mostly “normal.” The duration of CALM phaseis typically a third or fourth of the total evolution time to theSN Ia. These long durations of optically thick wind phasesmay reduce the statistical number of luminous supersoft X-ray sources because the photospheric temperature of the WDis lower than ∼
10 eV and not luminous in supersoft X-ray.The decline of SN 2006X light curves is slowing down ina later phase compared with the other normal SNe Ia lightcurves, suggesting an interaction between the ejecta and CSM2 Hachisu et al.in a later phase (Wang et al. 2007a) or a light echo of circum-stellar/interstellar matter (Wang et al. 2007b). This happensif SN 2006X is placed at the border between our Case WINDand Case CALM since the innermost part of slowly expand-ing circumstellar matter has not yet moved far away. A X-raydetected SN 2005ke may also belong to the same category(Immler et al. 2006).For the progenitor of SN 2006X, Patat et al. (2007a) sug-gested a WD + RG system like RS Oph from the circumstellarmatter (CSM) absorption lines. Here we suggest that a WD +MS system (like U Sco) may better explain a continuous ve-locity distribution (from ∼ -
30 to -
150 km s - ) of the CSMabsorption lines by the stripped matter with continuous veloc-ity distribution (see Fig. 1d). In this connection, very recentreport of the Na I D circumstellar lines of RS Oph during the2006 outburst is suggestive (Iijima 2007). These lines indi-cate no continuous distribution as observed in SN 2006X buta narrow velocity component of -
36 km s - against RS Ophthat is attributed to the red giant cool wind.Recently negative detections of time-variable Na I D lineshave been reported for two SNe Ia 2000cx (Patat et al.2007b) and 2007af (Simon et al. 2007). Patat et al. (2007b)and Simon et al. (2007) suggested a possibility that the distri-bution of CSM is torus/disk-like as illustrated in Figure 1. Insuch a case, variable Na I D lines would not be observed if theline of sight is perpendicular to or off the orbital plane. Sincethe hot WD winds have a large velocity of & - , theCSM formed by hot winds quickly diffuse away and is tootenuous to be detected.Recently, Badenes et al. (2007) reported that the fast WDwind of v &
200 km s - , which excavates its circumstellarmedium and forms a large cavity around an SN Ia, is incom-patible with the X-ray emission from the shocked ejecta in ourGalaxy (Kepler, Tycho, SN 1006), Large Magellanic Cloud(0509-67.5, 0519-69.0, N103B), and M31 (SN 1885). We canavoid this difficulty if the stripped-off matter has a velocity of10 -
100 km s - . Case RN
When the mass transfer rate from the secondary is below thelowest rate of steady hydrogen burning, i.e., ˙ M transfer < ˙ M stable ,hydrogen shell burning is unstable to flash and recur manytimes in a short period as a recurrent nova (RN) (the opensquares in Figs. 5–10). We call this Case RN. The recurrentnova U Sco, one of the candidates of SN Ia progenitors, is inthe middle of the “final” region (Hachisu et al. 2000a,b). Theresultant explosions are “normal” SNe Ia.A simple estimation gives the border between Case CALMand Case RN, ˙ M transfer = ˙ M stable and R = R ∗ at the SN Ia explo-sion. Here we calculate ˙ M from equation (17) with R and L being taken from Tout et al. (1997) (a single star evolution).These lines agree reasonably with the border of CALM–RNin Figures 6, 8, and 10.For a typical Case RN of M WD , = 1 . M ⊙ , M , = 2 . M ⊙ , P = 1 .
18 days with c = 3, the WD undergoes the SN Ia ex-plosion in the recurrent nova phase at t = 9 . × yr af-ter the secondary first fills its Roche lobe. The WD windstops at t = 4 × yr and the stable hydrogen burning endsat t = 8 . × yr. During the last 10 yr in the recur-rent nova phase, the secondary loses ∼ . M ⊙ , of whichthe WD accretes 0 . M ⊙ . Therefore the stripped-off mat-ter in the recurrent nova phase is very small. On the otherhand, the stripped-off matter in the early wind phase amounts ∆ M strip ≈ . M ⊙ , which has already been far from the SNat the SN Ia explosion, i.e., (10–100 km s - ) × (5 × yr)= (1 - × cm. It takes about 100-1000 yr for the SNejecta to reach the stripped-off matter. These features aresummarized in Table 1. DISCUSSION
Mass-stripping Effect and Modulated Mass TransferRate
As mentioned in §1, the existence of mass-stripping ef-fect has been demonstrated by Hachisu & Kato (2003b,c).They analyzed two quasi-periodic transient supersoft X-raysources, RX J0513 . - V ∼
11 and its duration of ∼
180 days) and low ( V ∼ ∼
120 days) states with total durations of ∼
300 days(see, e.g., Šimon & Mattei 1999, for the long-term behav-ior). (2) Very soft but very weak X-rays are detected only inthe long-term optical low state (e.g., Greiner & van Teeseling1998). (3) Radio observations indicate a wind mass-loss rateas large as ∼ - M ⊙ yr - (Lockley, Eyres, & Wood 1997;Lockley et al. 1999).Hachisu & Kato (2003c) explained these features based onthe mass-stripping effect; the mass transfer to the WD ismodulated by the WD wind because mass-stripping attenu-ates the mass transfer rate. This interaction leads to highand low states. The mass loss rate of the WD wind (with ahigh velocity of & - ) reaches as high as ˙ M wind ∼ × - M ⊙ yr - , being consistent with the radio observation.Thus, the mass transfer rate itself may not be constant butvary in time, thus being regarded as a time-averaged rate inthe present paper.From the light-curve fitting Hachisu & Kato also estimatedthe WD mass as M WD ∼ . M ⊙ and the secondary massto be M MS ∼ . M ⊙ , and concluded that V Sge will ex-plode as an SN Ia in a time scale of ∼ × yr. Since thepresent orbital period of V Sge is 0.51 days (Herbig et al.1965; Patterson et al. 1998), its position in the orbital periodvs. donor mass plane in Figure 3 indicates c >
0. Thus, wemay regard binaries in the wind phase as “V Sge type stars.”
Paucity of Progenitor Systems
The life time of V Sge type stars is typically a few to severaltimes 10 yr, mainly because the time-averaged mass strip-ping rate is as high as ˙ M MS ∼ - M ⊙ yr - . If this channel ofthe WD + MS system produces about four Type Ia supernovaeper millennium in our Galaxy (e.g., Cappellaro et al. 1999),we should have a chance to observe at least several hundredV Sge type stars in our Galaxy. Steiner & Diaz (1998) listedfour V Sge type stars in our Galaxy and discussed their similarproperties. Although the masses of the companion stars to theWDs are not yet clearly identified, their orbital periods fall inthe range of 0 . - . F IG . 11.— Upper blue thick histogram: delay time distribution for c = 3. Each bin is separated by the delay time (0 . , . . , . . , . . , .
8) Gyr. About 50% of SNe Ia coming from the WD + MS systems explode in within 0.1 Gyr. Lower red shadowed histogram: the ratio of SN 2002ictype (Case WIND) SNe Ia. About 7% of SNe Ia coming from the WD + MS systems explode in a wind phase.F IG . 12.— Same as Fig. 11, but for both the WD + MS (blue shadowed histogram) and WD + RG (red thick histogram) systems. Each bin is separated by thedelay time (0 . , . . , . . , . . , . . , . . , . . , . . , .
8) Gyr. The number ratio is normalized for each system.
Chandra data. They have estimated at least several hundred SSXSsin each galaxy, many of which are obscured by interstellarabsorption.
Angular Momentum Loss by Stripped Matter
The stripped matter is lost from the binary system withsome angular momentum. In our treatment, we assume thatthe specific angular momentum (angular momentum per unitmass) of the stripped matter is given by equation (6), that is,the ablated gas from the companion has the specific angu-lar momentum there just at the companion’s surface. Thisassumption may be too simplified because the stripped mat-ter may get some angular momentum from the binary motionduring its journey. Here we examine other two cases: one isthe same as the high velocity WD wind, i.e., ℓ s = (cid:18) + q (cid:19) , (19)the other is the slow velocity case, i.e., ℓ s = 1 , (20)where the stripped matter gets large angular momentum fromthe binary torque (see Jahanara et al. 2005, for recent three-dimensional hydrodynamic calculation).For the first case of equation (19), we have obtained es-sentially the same results as in equation (6). If we adopt thesecond case of equation (20), however, we have common en-velope formations in a hundred or thousand years for c = 3, M , = 5 . M ⊙ , and M WD , = 1 . M ⊙ in Figure 7 regardless of P . If we start the evolution with c = 3, M , = 4 . M ⊙ , and M WD , = 1 . M ⊙ , we obtain SN Ia explosions only for P = 2–5 days. These results hardly change even if we increase theefficiency of mass stripping effect to c = 10. This is becausetoo much angular momentum is removed from the binary forthe case of equation (20) and it makes the separation shrinkdrastically regardless of the c value. Evolutions with c = 3, M , = 3 . M ⊙ , and M WD , = 1 . M ⊙ result in the same finaloutcome as in equation (6).On the other hand, there exist four V Sge type stars withshort orbital periods of 0.2–0.5 days (Steiner & Diaz 1998).Therefore, we conclude that the angular momentum loss ismuch closer to equation (6) or (19) rather than equation (20)because these V Sge type stars cannot be realized with thelarge angular momentum loss like equation (20) that resultsin formation of a common envelope. Mass Transfer Rate of Simplified Treatment
Our treatment of thermal time scale mass transfer may betoo simplified compared with detailed mass transfer model studied by Langer et al. (2000) and Han & Podsiadlowski(2004). Han & Podsiadlowski compared our results based ona simplified model (Hachisu et al. 1999b) with their detailedmodel calculations, and pointed out that the difference is largefor lower mass WDs. Although we need detailed mass trans-fer model to obtain precise SN Ia regions, our treatment hasan advantage of easy and simple estimation for the SN Ia pa-rameter region. As pointed by Han & Podsiadlowski (2004),our SN Ia region thus calculated may deviate from the real-istic one for less massive WDs. However, our SN Ia regionis probably not so largely different from the realistic one formore massive WDs (compare with Fig. 12 of Hachisu et al.1999b and Figs. 3 and 5 of Han & Podsiadlowski 2004). CONCLUDING REMARKS
Both Cases WIND and CALM originate from the systemswith massive donors, i.e., young population. It would beimportant to make some comparisons with the observationaldata, such as frequency and population. The red hatched re-gions in Figures 5, 7, and 9 indicate a region in which the pro-genitor explodes at t delay ≤
100 Myr. Also the dashed line andthe dotted lines correspond to t delay = 200 Myr and 400 Myr,respectively. We see in Figure 11 that Case WIND and thusSNe Ia/IIn (IIa) are realized by the very young system with t delay . -
200 Myr.If M WD , . . M ⊙ , we have almost no region of CaseWIND, different from the cases of M WD , & . M ⊙ . Ifall the WD + MS system with M , & - M ⊙ ( c = 3), M WD , & . M ⊙ ( M , & . M ⊙ ), and P ∼ . - ∼
5% (including both the WD + MSand WD + RG systems with their total number ratio of 4:2).A group of Type IIn SNe such as SNe 1997cy and 1999Eshow a very similar spectroscopic and photometric features toSN 2002ic (Wang et al. 2004; Deng et al. 2004; Prieto et al.2007). If these are in fact all Type Ia/IIn (IIa) SNe, their fre-quency can be estimated to be ∼ + - % (Prieto et al. 2007),which is consistent with the above estimate.Type Ia supernovae play a key role in astrophysics, and thusour progenitor model has important implications. Our modeldepends essentially on the parameter of stripping effect, c ,which depends on the properties of WD winds, such as as-phericity, velocities, and the efficiency of energy conversion.Also we calculate the mass transfer rate using the simple ap-proximate binary models. In order to improve these param-eterization and approximations, we need multi-dimensionalhydrodynamical simulations, which are beyond the scope ofthe present study. In the present approach, we constrain the c parameter observationally, and estimate c ∼ - c ∼ . -
10 from the analysis of V Sge and RX J0513 . - ˙ M cr ∼ × - M ⊙ yr - . The WD wind collides with thesecondary’s surface and strips off its surface. If the mass-stripping effect is efficient enough, the mass transfer rate tothe WD is attenuated and the binary can avoid formation ofa common envelope even for a rather massive secondary. In-cluding this mass-stripping effect into our binary evolutionmodel of the WD + MS systems, we have found a new evo-lutionary scenario, in which a companion as massive as 6–rogenitors of Type Ia Supernovae 157 M ⊙ can produce an SN Ia for a reasonable strength of mass-stripping effect, say c ∼ P – M , (initial orbital pe-riod – initial donor mass) plane. The newly obtained SN Iaregion extends to massive donor masses up to M , ∼ - M ⊙ for P ∼ . -
10 days, although its extension depends on thestrength of mass-stripping effect, c , i.e., M , ∼ - M ⊙ for c = 10, M , ∼ - M ⊙ for c = 3, and M , ∼ M ⊙ for c = 1.(3) We have estimated that the SN Ia birth rate in our Galaxyis ν WD + MS ∼ .
004 yr - (for c = 3), which is consistent withthe observation. The rates of young populations, i.e., t delay ≤
100 Myr and t delay ≤
200 Myr, are about 50% and 80% ofthe total SN Ia rate of the WD + MS channel. These shortdelay times of SN Ia progenitors are consistent with the recentobservational suggestions that a half of SNe Ia belong to sucha very young population as the delay time of t delay ∼ yr.(4) Another channel of the WD + RG system shows a broaddistribution of the delay time over 2–3 Gyr (Hachisu et al.1999a), thus the two (WD + MS and WD + RG) channelsyield a bimodality of the delay time distribution.(5) The stripped-off material is probably distributed on theorbital plane and forms a massive circumbinary torus (or disk)around SNe Ia. Such circumstellar matter (CSM) may be con-sistent with the observed CSM feature in SN 2006X. WhenSN ejecta strongly interact with massive CSM, it can explainthe feature of Type Ia/IIn (IIa) SNe 2002ic and 2006gj. (6) Three different environments of SN Ia explosions canbe specified by three different states of WDs just at the SNIa explosion, i.e., the optically thick WD wind phase (CaseWIND), steady hydrogen burning phase without opticallythick winds from WDs (Case CALM), and recurrent novaphase (Case RN). In Case WIND, SN Ia ejecta strongly in-teract with massive CSM like SNe Ia/IIn (IIa) 2002ic and2005gj because CSM exists near the SN Ia. The estimatedrate of Case WIND is ∼
5% of the total SN Ia rate, being con-sistent with the observational estimate. In Cases CALM andRN, SNe show a normal SN Ia feature because the CSM isfar from the SN but the ejecta may interact with the CSM ina much later phase. SN 2006X may be on a border betweenCase WIND and Case CALM.We thank Massimo Della Valle and the anonymous ref-eree for their useful comments. I.H. and M.K. are gratefulto people at the Astronomical Observatory of Padova and atthe Department of Astronomy, University of Padova, Italy,for their warm hospitality and fruitful discussions, where wehave started and completed this work. This research has beensupported in part by the Grant-in-Aid for Scientific Research(16540211, 18104003, 18540231) of the Japan Society for thePromotion of Science, and by the NSF under grant PHY99-07949. We would like to thank stimulated discussion at theSanta Barbara workshop “Paths to Exploding Stars: Accre-tion and Explosion” (19-23 March 2007).
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