Radii and Mass-loss Rates of Type IIb Supernova Progenitors
aa r X i v : . [ a s t r o - ph . H E ] M a y Draft version May 11, 2017
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RADII AND MASS-LOSS RATES OF TYPE IIB SUPERNOVA PROGENITORS
Ryoma Ouchi and Keiichi Maeda Department of Astronomy, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan (Received March 17, 2017; Accepted April 19, 2017)
ABSTRACTSeveral Type IIb supernovae (SNe IIb) have been extensively studied, both in terms of the progenitor radius andthe mass-loss rate in the final centuries before the explosion. While the sample is still limited, evidence has beenaccumulating that the final mass-loss rate tends to be larger for a more extended progenitor, with the differenceexceeding an order of magnitude between the more and less extended progenitors. The high mass-loss rates inferredfor the more extended progenitors are not readily explained by a prescription commonly used for a single stellarwind. In this paper, we calculate a grid of binary evolution models. We show that the observational relation in theprogenitor radii and mass-loss rates may be a consequence of non-conservative mass transfer in the final phase ofprogenitor evolution without fine tuning. Further, we find a possible link between SNe IIb and SNe IIn. The binaryscenario for SNe IIb inevitably leads to a population of SN progenitors surrounded by dense circumstellar matter(CSM) due to extensive mass loss ( ˙ M & − M ⊙ yr − ) in the binary origin. About 4 % of all observed SNe IIn arepredicted to have dense CSM, produced by binary non-conservative mass transfer, whose observed characteristics aredistinguishable from SNe IIn from other scenarios. Indeed, such SNe may be observationally dominated by systemsexperiencing huge mass loss in the final 10 yr, leading to luminous SNe IIn or initially bright SNe IIP or IIL with acharacteristics of SNe IIn in their early spectra. Keywords: circumstellar matter – stars: mass-loss – supernovae: individual (SN 1993J, SN 2008ax,SN 2011dh, SN 2013df)
Corresponding author: Ryoma [email protected] INTRODUCTIONType IIb supernovae (SNe IIb) are characterizedby hydrogen lines in their early phase spectra, whichare gradually replaced by He lines at later phases(Filippenko 1997). The progenitor of SNe IIb is believedto be a massive star which retains only a small amountof hydrogen ( . M ⊙ ) in its outer layer at the time of theexplosion. For the removal of the hydrogen layer, twoscenarios have been considered. In one scenario, mas-sive single stars ( & M ⊙ ) eject their outer layer viatheir strong stellar wind (Georgy 2012; Gr¨afener & Vink2016). The other scenario the binary interaction, i.e., astar in a binary system transfers most of its hydrogen-rich layer to its companion by Roche lobe overflow(RLOF) (Stancliffe & Eldridge 2009). The question asto which is the dominant evolutionary scenario for theproduction of SNe IIb progenitors is still open, but ob-servations and theoretical models so far seem to favorthe binary scenario (Smith et al. 2011; Sana et al. 2012;Folatelli et al. 2014).Among SNe IIb, 1993J, 2008ax, 2011dh, and 2013dfhave been investigated in detail and have yielded abun-dant observational data, covering the long-term evolu-tion from early to late phases at various wavelengths.Their progenitors (or strong candidates) have also beenidentified (Maund et al. 2004, 2011; Van Dyk et al.2014; Folatelli et al. 2015). In Table 1, characteristicproperties of these progenitors are listed. Note the di-versity they show in the HR diagram. They cover alarge range in their radii, from ∼ R ⊙ (blue supergiant: BSG) to ∼ R ⊙ (yellow or red supergiant : YSGor RSG). Whether they represent two discrete groupsor a continuous distribution is still uncertain. Variousevolutionary models have been investigated for eachprogenitor that match both their location in the HR di-agram and classification of their SN type. Binary evolu-tion models seem to explain the observed features morenaturally than single-star evolution models for most ofthem (Woosley et al. 1994; Benvenuto et al. 2013). Ac-cording to these works, the progenitor masses for theseSNe are estimated to be in the range 12 ∼ M ⊙ .In addition to information on the progenitors, themass-loss rate just before the explosion contains im-portant information about their evolutionary paths.The mass-loss property is reflected in the density ofcircumstellar matter (CSM), which has been studiedby radio, X-ray, and optical observations in the latephase, through the signature of the SN-CSM interac-tion. Maeda et al. (2015) have found a correlation be-tween the progenitor radius and the average mass-lossrate shortly before the explosion thus derived, in whichmore extended progenitors ( ∼ R ⊙ ; e.g. 1993J, 2013df) have had a relatively large mass-loss rate be-fore the explosion at a rate of ˙ M ∼ × − M ⊙ yr − (Fransson et al. 1996), while less extended progenitors( ∼ R ⊙ ; e.g. SN 2011dh) have had a moderatemass-loss rate ( ˙ M ∼ × − M ⊙ yr − ) (Maeda et al.2014). This tendency is supported by a larger sample ofSNe without direct progenitor detection (Kamble et al.2016).For the less extended BSG or YSG progenitors likeSN 2008ax or SN 2011dh, a mass-loss rate from a singlestellar wind seems to be compatible with the observa-tionally derived mass-loss rates. However, in the caseof the more extended progenitors like SN 1993J or SN2013df, which are also YSGs, a mass-loss rate under thecommonly used prescription for a single stellar wind fallssignificantly short of the observationally derived values(de Jager et al. 1988). It may still be possible that theextensive mass loss for the more extended progenitorscan be explained solely by a single stellar wind, consid-ering that the high mass-loss rates reaching as high as˙ M ∼ − M ⊙ yr − from some RSGs have been reported(van Loon et al. 2005). At the same time, this unusu-ally extensive mass loss for the more extended progen-itors might indicate an additional mass-loss mechanismrelated to binary evolution. Solving this problem mayprovide us with a key to understanding the evolutionaryhistory of the progenitors of SNe IIb.In the context of the binary scenario, the binary in-teraction, especially the non-conservative mass trans-fer, may be the origin of this additional mass loss(van Rensbergen et al. 2011). Recently, Yoon et al.(2017) showed that the mass-loss rates of the binarymodels for SNe IIb are consistent with the observa-tionally derived values. They, however, did not dis-cuss whether the relation between the progenitor radiiand mass-loss rates are generally expected for differ-ent values of the initial mass ratios and mass accretionefficiency. Also, they did not discuss what kind of phys-ical processes are involved in determining the mass-lossrate. In this paper, we investigate whether the apparentrelation between progenitor size and mass-loss rate be-fore the explosion can be explained by binary evolutionmodels. We have found that this observed tendency canindeed be naturally explained by non-conservative masstransfer in the final phase. We also discuss the physicalmechanisms that produce the relation.We also suggest a possible link between binary evolu-tion (and SNe IIb) and some Type IIn SNe (SNe IIn),which are characterized by narrow hydrogen emissionlines in their spectra (Filippenko 1997). SNe IIn are be-lieved to have dense CSM in the vicinity of the progen-itors, which indicates extensive mass loss shortly beforethe explosion. The mass-loss rates have been estimatedfor many SNe IIn in various ways, and these values coverthe range 10 − –1 M ⊙ yr − , often assuming a wind ve-locity of ∼
100 km/s (Kiewe et al. 2012; Taddia et al.2013; Moriya et al. 2014). For a certain range of binaryparameters, our binary models can have dense CSMcomparable to these observationally derived values atthe time of the explosion, produced by non-conservativemass transfer shortly before it.This paper is structured as follows. In §
2, we describethe method used for the calculation of the binary evolu-tion models. In §
3, we show the results of these models,focusing on the property of the progenitor and its rela-tion to the mass-loss rate. In § § § METHODWe use MESA for the calculations of the binary evo-lution. MESA is a one-dimensional stellar evolutioncode that uses adaptive mesh refinement and adaptivetime stepping. See Paxton et al. (2011, 2013, 2015) fordetails. In this section, we briefly describe the key pa-rameters in our calculations.We assume a solar metallicity of Z = 0 .
02 for allthe models. Convection is modeled using the mix-ing length theory of Henyey et al. (1965), adopting theLedoux criterion. The mixing length parameter is setto be α = 2 .
0. Semiconvection is modeled follow-ing Langer et al. (1985) with an efficiency parameter α sc = 1 . f = 0 .
018 and f = 0 . ∼ . H P from the convectiveboundaries ( H P is the scale height near the convectiveboundaries). We adopt the overshooting for the convec-tive core during hydrogen-burning and the convectivehydrogen-burning shell, and also for the convective coreand shell where no significant burning takes place. Weuse the ‘Dutch’ scheme for the stellar wind, with a scal-ing factor of 1.0. The ‘Dutch’ wind scheme in MESAcombines results from several papers. Specifically, when T eff > K and the surface mass fraction of hydrogen isgreater than 0.4, we then use that of Vink et al. (2001),and when T eff > K and the surface mass fraction ofhydrogen is less than 0.4, we use that of Nugis & Lamers ’http://mesa.sourceforge.net/’ (2000). Then, in the case when T eff < K, we use thewind scheme of de Jager et al. (1988). When the pri-mary star fills its Roche lobe, we implicitly compute themass transfer rate following the scheme of Kolb & Ritter(1990).Some previous binary evolution calculations, includ-ing stellar rotation have shown that the mass accretionfrom the primary can bring the secondary close to thecritical rotation, which is then likely to enhance thewind. This enables the mass transfer to be highly non-conservative, especially for binaries with relatively largeorbital period ( &
10 days). This is usually expected forType IIb binary progenitor models (Langer et al. 2003;Petrovic et al. 2005). So, in this paper, we consider onlynon-conservative mass transfer and assume the efficiencyof mass accretion f to be 0.5 or 0.0, which is kept con-stant throughout the calculations. Here, the mass accre-tion efficiency f denotes the fraction of the mass trans-ferred by the primary which accretes onto the secondary.We assume that the matter ejected by non-conservativemass transfer has a specific angular momentum equal tothat of the accreting star.In some of our models, it happens that both stars filltheir own Roche lobes at the same time. In this case,it is likely that the system enters the common envelopephase, which MESA currently cannot deal with. In thiscase we stop the calculation. Furthermore, in severalmodels, after the major mass transfer by the primary hasoccurred, the secondary completes the main-sequencestage and expands to become a giant. This results inthe RLOF of the secondary, before the primary star ex-plodes. Also in this case we stop the calculation.In order to compare the model outcomes with the ob-servationally derived mass-loss rates before the explo-sion, we calculate the final mass-loss rate for each modelas follows. We pick up a model snapshot at about 1000yr before the end of the calculation, and compute thedifference between the total mass of the two stars atthis epoch and that at the end of the calculation. Thisis then divided by the time interval between the twophases. Throughout this paper, we denote this quantityas ˙ M .The initial primary star mass is fixed to be 16 M ⊙ , soas to be roughly consistent with the luminosity of thedetected progenitors. For the secondary star mass, weconsider different values of the initial mass ratio q = M /M , simulating the models with q = 0 . , . , . The subscripts 1 and 2 express the primary and the secondaryrespectively. The primary in this paper refers to the more massivestar at the beginning of the calculation. log( T eff (K)) log( L/L ⊙ ) Radius( R ⊙ ) ˙ M ( M ⊙ yr − )1993J 3.63 ± ± ∼
600 (2–6) × − ∼
50 6 . × − ± ∼
200 3 × − ± ∼
600 (5.4 ± × − Table 1.
Properties of the progenitors of Well-studied Type IIb SNe. For information on the HR diagram and the progenitorradius, the date are taken from Maund et al. (2004) for SN 1993J, Folatelli et al. (2015) for SN 2008ax, Maund et al. (2011) forSN 2011dh, and Van Dyk et al. (2014) for SN 2013df. The data of the mass-loss rate are taken from Fransson et al. (1996) forSN 1993J, Chevalier & Soderberg (2010) for SN 2008ax, Maeda et al. (2014) for SN 2011dh, and from Maeda et al. (2015) forSN 2013df. mass ratios implied for Galactic O stars appear to beuniformly distributed. Therefore, systems with q . . q & .
6) (Stancliffe & Eldridge 2009;Benvenuto et al. 2013; Folatelli et al. 2015), which mayindicate that these Type IIb progenitors mainly evolvefrom binaries with high initial mass ratio ( q & . P =5, 25, 50, 200, 600, 800,1200, 1600, 1800, 1950 and 2200 days. We follow theevolution of both stars, from the zero-age main sequenceuntil the mass fraction of carbon at the primary’s centerfalls below 1 . × − . This corresponds to the carbon-shell burning phase and it takes only a few years untilthe explosion from that point.We assume that models with final envelope mass morethan 1 M ⊙ explode as SNe IIP or SNe IIL, while thosewith envelope mass less than 0 . M ⊙ explode as SNe Ibor Ic. Models whose envelope mass is between 0 . M ⊙ and 1 M ⊙ are adopted as SN IIb progenitors. RESULTSTables 2 and 3 show the final properties of each model,including the average mass-loss rates before the explo-sion ( ˙ M ), for an accretion efficiency of f = 0 . P = 5 days lose all of their hydrogen-rich envelope andbecome SNe Ib/c. Following our criterion, the initialperiod of 1000 days separates SN IIP/IIL and SN IIbprogenitors. Thus the models with initial period in therange 10 days . P . E n v e l ope m a ss ( M ⊙ ) Initial orbital period (days)f=0.5q=0.6q=0.8q=0.95 0.01 0.1 1 10 1 10 100 1000 10000 E n v e l ope m a ss ( M ⊙ ) Initial orbital period (days)f=0.0q=0.6q=0.8q=0.95
Figure 1.
Final envelope mass of the progenitor as a func-tion of the initial orbital period P . The left and right panelsshow the models with the accretion efficiency of f = 0 . f = 0 .
0, respectively. The models with different initial massratios are shown by different symbols/colors.
Fig. 2 shows the final locations of the SN IIb pro-genitor models on the HR diagram (the SN II and SNIb/c progenitor models are not included in the figure).Note that our models cover the range of properties thatthe detected progenitors show in the HR diagram. Fromthis figure, together with Table 2 and Table 3, it is clearthat the models with a smaller initial period are locatedat the left side of the HR diagram, i.e., they are morecompact. Furthermore, the final locations on the HR di-agram are not sensitively dependent on the initial massratio and the accretion efficiency compared with the ini-tial period. l og ( L / L ⊙ ) log(T eff [K]) q=0.8, f=0.5q=0.95, f=0.5q=0.6, f=0.0q=0.8, f=0.0q=0.95, f=0.0 Figure 2.
Final locations of the SN IIb progenitor mod-els, together with the the observed values for the detectedprogenitors, on the HR diagram.
Next, we plot progenitor radius (hereafter R ) versus˙ M in Fig. 3, together with the observed values forthe well-studied progenitors. We find that the binarymodels do predict the relation between the progeni-tor radius and the mass loss, without any fine-tuning.Namely, the more extended progenitors have about anorder of magnitude higher mass-loss rates before the ex-plosion than the less extended ones. The outliers in therelation, i.e., those with R ∼ R ⊙ and ˙ M ∼ (5–7) × − M ⊙ yr − are the models with P = 2200 days. TheRoche lobe radii (hereafter R rl ) of these models are toolarge for the primary stars to start the mass transfer,and the final mass-loss rate is mostly determined by thestellar wind. They may explode as SNe IIP.Our models leading to the less extended progenitors(e.g., SN 2008ax, SN 2011dh) reproduce the absolutevalues of the mass-loss rates observed for these progeni-tors fairly well. However, for the more extended progen-itors like 1993J and 2013df, the values of the mass-lossrates found in our models are slightly lower than the ob-servations indicate. Nevertheless, given the intrinsic un-certainties in the stellar evolution calculations and the measurement of the mass-loss rates, further investiga-tion of this issue is beyond the scope of this paper. -6-5.5-5-4.5-4-3.5-3-2.5-2 0 100 200 300 400 500 600 700 800 900 10002011dh 2013df 1993J2008ax l og ( M · [ M ⊙ y r - ] ) Radius (R ⊙ ) q=0.6, f=0.5q=0.8, f=0.5q=0.95, f=0.5q=0.6, f=0.0q=0.8, f=0.0q=0.95, f=0.0 Figure 3.
Radius versus mass-loss rate for all the models,together with the observationally derived values.4.
DISCUSSION4.1.
The radius versus mass-loss relation
In all models with an initial period of P = 5 days, andmost of the models with an initial period of P = 25 days,there is no RLOF after He-burning, so the final mass-loss rate is determined by the stellar wind, which has avalue of ˙ M ∼ − M ⊙ yr − . The final mass-loss ratesof the models with P = 2200 days are also determinedby the stellar wind (Section 3). In all other models,the primary star experiences RLOF after exhaustion ofthe He fuel, and the RLOF continues until the explo-sion. For the less extended progenitors whose radius is R . R ⊙ , the mass loss via non-conservative masstransfer is a few × − M ⊙ yr − , which is comparableto or a little higher than the stellar wind (see Table 2,3). For the more extended progenitrs, the mass transferrates reach ∼ − M ⊙ yr − , or even larger, which farexceed the mass-loss rate predicted by the stellar wind.In this case, the mass-loss rate is mostly determined bythe RLOF mass transfer rate ( ˙ M tr ) in the final phase.In this subsection, we discuss why a more extended pro-genitor has a higher mass transfer rate in the final phasein our binary models.Omitting the models in which there is no, or a negli-gible amount of RLOF in the final phase, such as thosewith P =5 or 2200 days, the mass transfer history is di-vided into two classes, i.e. Case BB, and Case C (Yoon2015). For models with P . P & & (10 − –10 − ) M ⊙ yr − initially. Shortly after this phase, a mod-erate and stable mass transfer takes place as driven bythe expansion due to the core-evolution. For a certainrange of the initial period, this short intensive masstransfer phase occurs very shortly before the explosion,likely leading to a shell-like, dense CSM located near theprogenitor at the time of the explosion. We will discussthis issue further in § . × years (right) for a typical model in the caseBB mass transfer (No. 8 in Table 2) are shown. Therise in radius until about 1 . × yr corresponds tothe carbon-oxygen core contraction (He-shell burning)phase, and the carbon burning starts at the dip in theevolution of radius at ∼ . × yr. Carbon burningcontinues for ∼ × yr (until the next dip), and thisis then followed by carbon-shell burning, with a rapidincrease in radius.Note that, before the ignition of carbon burning, themass transfer is driven by the expansion due to thecarbon-oxygen core contraction, and the radius remainsalmost equal to the Roche lobe radius. Then, oncecarbon burning sets in, the radius changes rapidly ona short timescale. The rapid change in radius duringthis phase causes the mass transfer rate to fluctuate bysome factors around the value at the beginning of carbonburning. The final evolution in the last ∼ yr towardthe explosion corresponds to this fluctuation phase, andit is difficult to estimate the mass transfer rate duringthis phase by simple analysis.In the case of the stable mass transfer, the mass trans-fer rate does not change significantly after carbon igni-tion (Fig. 4). In Fig. 5, we compare ˙ M of themodels with f =0.0, which is approximately equal tothe average mass transfer rate in the last 1 × yr,with the mass transfer rate at 6 × yr before theexplosion, which we denote as ˙ M tr, . This epoch, RR rl l n ( R [ R ⊙ ] ) , l n ( R r l [ R ⊙ ] ) Age (yr)-8-7.5-7-6.5-6-5.5-5-4.5-4 1.21x10 l og ( M · t r [ M ⊙ y r - ] ) Age (yr)
Figure 4.
Left: time evolution of the stellar radius ( R )and Roche lobe radius ( R rl ) during the last 1 . × yearsof the primary star’s evolution. The purple thick line andgreen thin line represent the stellar radius and the Rochelobe radius, respectively. Right: time evolution of the masstransfer rate ( M tr ) during the same epoch as the left panel.Both are plotted for the model No. 8 in Table 2. × yr before the explosion, corresponds to thecarbon-ignition phase. These two values are closely con-nected, distributed tightly on the line ˙ M tr, = ˙ M .This supports the idea that the mass transfer ratedoes not change significantly after carbon ignition. Al-though some models with relatively low mass transferrate slightly deviate from the line ˙ M tr, = ˙ M ,this is because the contribution of the stellar wind tothe mass loss is not negligible at such a low rate. In ad-dition, there are some models with high mass-loss rates( & − M ⊙ yr − ) which deviate from the line signifi-cantly. This is because an unstable RLOF phase is in-volved during the last 6 × yrs. We discuss this issuein § yr, to compare with theobservations, by the mass transfer rate at the ignition -7-6-5-4-3-2 -7 -6 -5 -4 -3 -2 l og ( M · t r , [ M ⊙ y r - ] ) log( M · [M ⊙ yr -1 ]) q=0.6, f=0.0q=0.8, f=0.0q=0.95, f=0.0 Figure 5. ˙ M is compared to the mass transfer rate at6 × yrs before the explosion ( ˙ M tr, ), which correspondsto the ignition of carbon burning. Shown are the models with f = 0 . of carbon burning as analyzed by simple arguments. Inthis way, we analyze what mechanisms determine themass transfer rate at the ignition of carbon burning.During the stable mass transfer driven by carbon-oxygen core contraction, the mass transfer rate can beapproximated as follows (Soberman et al. 1997; Ivanova2015); − ˙ M tr = M env ζ eq , env − ζ L M env M (cid:16) ∂ ln R∂t − ∂ ln R rl ∂t (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ˙ M =0 . (1)Here, M env and M represent the hydrogen-rich enve-lope mass and the total mass of the donor star, respec-tively. This relation can be derived from d ln Rdt = ζ eq , env d ln M env dt + ∂ ln R∂t (cid:12)(cid:12)(cid:12)(cid:12) ˙ M =0 , and (2) d ln R rl dt = ζ L d ln Mdt + ∂ ln R rl ∂t (cid:12)(cid:12)(cid:12)(cid:12) ˙ M =0 , (3)by assuming that the equation R = R rl is fulfilledall the time, which is approximately correct during thephase we are focusing on here (Fig. 4). Here, ζ eq , env expresses the change in donor radius in response to thereduction of the envelope mass, assuming that this massloss occurs slowly enough to keep the donor in thermalequilibrium. ζ L is the change in the Roche lobe radius of the donor in response to the mass transfer. They aredefined as follows: ζ eq , env = (cid:16) ∂ ln R∂ ln M env (cid:17) eq , and (4) ζ L = ∂ ln R rl ∂ ln M . (5)The second term in brackets of equation (1) expressesthe angular momentum loss due to gravitational waveemission, and this is negligible compared to the firstterm in our binary models. Then, we can rewrite equa-tion (1) as follows: − ˙ M tr = M env ζ eq , env − ζ L M env M (cid:16) ∂ ln R∂t (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ˙ M =0 . (6)Physically, the term of M env × (cid:16) ∂ ln R∂t (cid:17)(cid:12)(cid:12)(cid:12) ˙ M =0 means thatas the donor expands considerably (∆ R ∼ R ) in atimescale of (cid:16) ∂ ln R∂t (cid:17)(cid:12)(cid:12)(cid:12) − M =0 , the whole envelope mass islost. The term of ζ eq , env represents the effect of changein the donor radius in the equilibrium state due tothe mass loss, while ζ L represents the change in theRoche lobe radius due to the mass transfer. In sum,five factors ( ζ eq , env , ζ L , M env , M, ∂ ln R∂t (cid:12)(cid:12) ˙ M =0 ) determinethe mass transfer rate. We consider these terms oneby one below.First, we consider ζ eq , env . In order to estimate thisvalue, we calculate a series of single-star models havingthe same physical condition as the primary star in ourbinary models, except that here we remove the envelopemass artificially at a high rate ( ˙ M ∼ − M ⊙ yr − )during He-burning and turn off the stellar wind of theprimary star until exhaustion of the carbon fuel. Thisprocedure allows us to construct a series of the primarymodels with different amounts of the hydrogen-rich en-velope under thermal equilibrium. By extracting the ra-dius and the envelope mass at the beginning of carbonburning of these single-star models, we create a relationbetween the envelope mass and the radius in its equilib-rium state. We then derive ζ eq , env by differentiating thecurve. Fig. 6 shows such a plot. Although this figureis constructed by evolving the single stars, the plot isapplicable to our primary star at the same evolutionarystage, because the physical conditions assumed are thesame.From Fig. 6, we see that the radius in the equilibriumstate increases rather quickly with increasing envelopemass as long as ln ( M env [ M ⊙ ]) . -2, while beyond thispoint, the radius is almost constant as a function of theenvelope mass. This sudden change of the shape of thecurve at this point is due to the low surface temperatureand thus the development of the convective layer in theenvelope, when ln ( M env [ M ⊙ ]) & -2. As shown in Fig.6, the radius in complete equilibrium increases with theincrease of the envelope mass when the envelope is ra-diative (ln ( M env [ M ⊙ ]) . -2). This behavior can be ex-plained approximately by an analytical argument, by ap-plying the approach of Cox & Salpeter (1961) to the sit-uation under consideration (see Appendix A for details).For an envelope mass beyond ln ( M env [ M ⊙ ]) ∼ -2 in theconvective regime, further increase of the radius is sup-pressed by the existence of the Hayashi line, which keepsthe radius almost constant. Therefore, ζ eq , env is close tozero for more extended progenitors ( M env & . M ⊙ ),while ζ eq , env & M env . M ⊙ ). l n ( R en v [ R ⊙ ] ) ln(M env [M ⊙ ]) Radius ζ eq, env Figure 6.
Equilibrium radius (purple) and ζ eq , env (green) asa function of the hydrogen-rich envelope mass at the ignitionof carbon burning. Next, we consider ζ L . After some calculations, usingthe approximation formula by Eggleton (1983), ζ L canbe written as follows (Soberman et al. 1997): ζ L = ∂ ln R rl ∂ ln M = ∂ ln a∂ ln M + ∂ ln( R rl /a ) ∂ ln q ∂ ln q∂ ln M = (cid:18) − βq (cid:19)(cid:18) − −
11 + q + 5 − β − β + q − q / . q / + q q / . q / ln(1 + q − / ) (cid:19) . (7)Here, we define β ≡ − f , as the fraction of the trans-ferred material that is lost from the system. We plot ζ L in Fig. 7 for three values of the mass accretion efficiency.For a large mass ratio ( q ≫ -2-1.5-1-0.5 0 0.5 1 1 2 3 4 5 6 7 ζ L Mass ratio (M /M ) β =0.0 β =0.5 β =1.0 Figure 7.
Value of ζ L as a function of the mass ratio, fordifferent value of β = M /M . ing, ζ L converges to ∼ − .
5. This behavior is explainedin an analytical way, and is described in Appendix B.Finally, in Fig. 8, we plot ∂ ln R∂t (cid:12)(cid:12) ˙ M =0 as a function ofthe envelope mass at the ignition of carbon burning. Incalculating this value, we use the same single-star mod-els as those used to create Fig. 6. We pick up two phases:6 × yr before the end of the calculation (which isaround the ignition of carbon burning) and 1 × yrbefore that point. We then calculate the difference in ln R between these two phases, which is then divided bythe time interval between them ( ∼ × yr). Fig. 8shows that stars with a small amount of the hydrogen-rich envelope (ln M env ( M ⊙ ) . −
2) have a large valueof ∂ ln R∂t (cid:12)(cid:12) ˙ M =0 ( ∼ × − yr − ), while those with alarge amount of the envelope (ln M env ( M ⊙ ) & − ∂ ln R∂t (cid:12)(cid:12) ˙ M =0 ( ∼ × − yr − ). Therapid decrease of the expansion rate ∂ ln R∂t (cid:12)(cid:12) ˙ M =0 beyondln M env ( M ⊙ ) ∼ − R . R ⊙ ).These have a hydrogen-rich envelope as small as M env ∼ . M ⊙ (Tables 2 and 3), which then gives ζ eq , env ∼ . ζ L ∼ − M env M ∼ − . Therefore, ζ eq , env − ζ L M env M ∼ ζ eq , env ∼ .
5. Recalling that for theless extended stars, ∂ ln R∂t (cid:12)(cid:12) ˙ M =0 ∼ × − yr − (Fig.8),the mass transfer rate at carbon ignition is estimated asfollows: ∂ l n ( R [ R ⊙ ] ) / ∂ t | M · = ln(M env [M ⊙ ]) Figure 8.
Expansion rate of the donor, assuming there is nomass loss, as a function of the envelope mass at the ignitionof carbon burning. − ˙ M RLOF = M env ζ eq , env − ζ L M env M (cid:16) ∂ ln R∂t (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ˙ M =0 ∼ . M ⊙ . × × − yr − ∼ × − M ⊙ yr − . (8)This value, taking the contribution from the stellar windinto account, is consistent with that extracted from thebinary evolution calculations (Table 3).Next, we consider a representative case among themore extended progenitors, with R ∼ R ⊙ having ahydrogen-rich envelope mass of M env ∼ M ⊙ (Table2and 3). From the value of M env and Fig. 6, we derive ζ eq , env ∼ .
1. For these progenitors, ζ L ∼ − M env M ∼ .
15. Therefore, ζ eq , env − ζ L M env M ∼ . ∂ ln R∂t (cid:12)(cid:12) ˙ M =0 ∼ × − yr − (Fig.8), the mass transfer rate at carbonignition is estimated as follows: − ˙ M RLOF = M env ζ eq , env − ζ L M env M (cid:16) ∂ ln R∂t (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ˙ M =0 ∼ M ⊙ . × × − yr − ∼ × − M ⊙ yr − . (9)This value is consistent with that extracted from thebinary evolution calculations (Table 3).In summary, the difference between the mass-lossrate of less extended progenitors and that of more ex-tended progenitors can be explained as follows. Firstly,the value of ∂ ln R∂t (cid:12)(cid:12) ˙ M =0 is an order of magnitude lowerfor more extended progenitors than less extended ones.This alone is contrary to the observed tendency. How-ever, the envelope mass M env is about an order of mag-nitude larger for more extended progenitors than less extended ones. Furthermore, the value of ζ eq , env − ζ L M env M is also about an order of magnitude larger for moreextended progenitors than less extended ones, mainlybecause ζ eq , env is an order of magnitude smaller formore extended ones. Therefore, these factors ( M env and ζ -terms) cause the mass transfer rate to be two or-ders of magnitude higher for more extended progenitorsthan less extended ones. Combined with the effect of ∂ ln R∂t (cid:12)(cid:12) ˙ M =0 , the mass transfer rate of more extended pro-genitors should be an order of magnitude higher.Thus, the simple intuitive estimation of mass transferrate by the M env × ∂ ln R∂t (cid:12)(cid:12) ˙ M =0 is not enough to recoverthe relation in Fig. 3, and the ζ terms (especially ζ eq , env )also play a key role in producing the relation.4.2. implications for Type IIn SNe A rate of Type IIn SNe through the binary evolution
As indicated in Fig. 3, some of our models show a finalmass-loss rate that is so high ( ≥ − M ⊙ yr − ) thatthey may be observed as SNe IIn ( § M ⊙ , and assume the distribution ofthe initial periods as f ( P ) ∝ P − (Kouwenhoven et al.2007).The range of the initial period for which SNe IIb areproduced is 10 days . P . § M ) as a function of the initial period. Assumingthat the models with the final mass-loss rates of ˙ M & − M ⊙ yr − explode as SNe IIn (Taddia et al. 2013;Moriya et al. 2014) , a primary star in a binary sys-tem with initial period in the range 1800 days . P . − ln1800) / (ln1000 − ln10) ∼ . Assuming that the velocity associated with the mass loss is ∼
10 km s − , this corresponds to ˙ M & − M ⊙ yr − if thevelocity is assumed to be the frequently adopted value of ∼ − for SNe IIn. ∼ . ∼ P .Thus, we conclude that a small but non-negligiblefraction of the observed SNe IIn is occupied by thosewhose surrounding CSM is produced by binary non-conservative mass transfer in the final phase of the pro-genitor evolution. While the expected rate is relativelysmall, it is highly interesting to identify these kinds ofSNe IIn observationally. Since this is a solid predictionfrom the binary evolution model for SNe IIb, identify-ing such a population in SNe IIn is important to clarifynot only the origin of SNe IIn but the role of the bi-nary evolution toward SNe IIb. Indeed, these types ofSNe IIn are expected to have two features which couldbe distinguishable from other scenarios, e.g., a luminousblue variable (LBV)-like progenitor. (1) As compared tothe popular LBV-like progenitor scenario, the velocityassociated with the mass loss will be smaller by a factorof a few (e.g. wind from the main sequence companion)or by more than an order of magnitude (e.g. from thegiant progenitor). This is an interesting target for high-dispersion spectroscopy of nearby and bright SNe IIn (orthey may look like SNe IIL if the narrow absorption cre-ated within the CSM is contaminated by an unrelatedbackground Kangas et al. 2016). (2) The CSM which isproduced in this way may well have characteristic struc-ture, because of its origin in the binary evolution, whichhas a specific axis as defined by the orbital plane. Un-covering the geometry of CSM around SNe IIn may allowus to confirm this idea (see, e.g., Katsuda et al. 2016).4.2.2. A population of SNe that interact with a shell-likeCSM
One of our models (No.28 in Table3) shows a veryhigh mass-loss rate, showing as much as ˙ M ∼ × − M ⊙ yr − . In this model, an intensive and unsta-ble mass transfer begins very shortly before the explo-sion. In general, when the primary begins to experi-ence RLOF for the first time, the mass transfer usuallyoccurs on thermal or dynamical timescales, dependingon whether their envelopes are radiative or convective.Both timescales are usually much shorter than the evolu-tionary timescales. Shortly after this rapid mass trans- -6-5.5-5-4.5-4-3.5-3-2.5-2 0 500 1000 1500 2000 2500 l og ( M · [ M ⊙ y r - ] ) Initial orbital period (days) q=0.6, f=0.5q=0.8, f=0.5q=0.95, f=0.5q=0.6, f=0.0q=0.8, f=0.0q=0.95, f=0.0
Figure 9.
Average mass-loss rate in the final 1 × yearsas a function of the initial period. fer phase, stable mass transfer takes place (Section 4.1).In most of the models, this first intensive mass transferoccurs well before the collapse. Therefore, the materiallost during this phase, if any, will have already gone toofar to interact with the SN ejecta during the observ-able time window after the progenitor explodes as anSN. However, in some of the case C models, this inten-sive mass transfer occurs so shortly before the explosion( . years), that we can observe the interaction ofthe SN ejecta with the CSM at the immediate vicinityof the progenitor as created by such an intensive massloss. In this case, the shape of the CSM will probablybe shell-like, unlike other models which have sustainedmass loss for a long period before the explosion.In Fig. 10, the mass transfer rate evolution in the final ∼ years is shown for two representative models withcase C mass transfer to illustrate this situation. The leftand right panels show the models with P = 1800 daysand P = 1950 days, respectively. The initial mass ratioand mass accretion efficiency are the same in both mod-els. The model with P = 1800 days has a smaller Rochelobe radius, therefore the mass transfer sets in earlier.The model with P = 1950 days, which corresponds toNo.28 in Table3, has a relatively larger Roche lobe ra-dius, and significant mass transfer sets in only when theprimary is about to explode; thus it may produce a shell-like CSM near the SN progenitor. Although the averagemass-loss rate during the final ∼ yrs of this model is˙ M ∼ × − M ⊙ yr − (Table3), the temporary mass-loss rate can reach as much as ˙ M & − M ⊙ yr − (Fig.10). Thus, exploring the probability of such an unstablemass transfer taking place shortly ( . yr) before thecollapse seems to be worthwhile.Let us estimate the rate of such events. For the SNejecta to interact with the shell-like CSM, intensive, un-stable mass transfer should begin for the first time some-1 -7-6-5-4-3-2-1 1.212x10 l og ( M · t r [ M ⊙ y r - ] ) Age (yr)-7-6-5-4-3-2-1 1.212x10 l og ( M · t r [ M ⊙ y r - ] ) Age (yr)
Figure 10.
Time evolution of the mass transfer rate ( M tr )for models No. 25 (left) and No, 28 (right) in Table 3 duringthe last 1 × yr. time during the last ∼ yr before the explosion. Fig.11 shows the time evolution of the radius of a single16 M ⊙ star in the last 5 × yrs of evolution, calcu-lated under the same physical conditions as the primarystars in the binary models. In the last ∼ yrs, theradius changes from 905 R ⊙ to 920 R ⊙ . In order for suchan event to occur, the progenitor’s Roche lobe radiusneed to be in this range.In MESA, the Roche lobe radius of the primary iscalculated as R rl = 0 . q − . q − + ln(1 + q − ) a . (10)Here, a is the binary separation expressed as a = n G ( M + M ) P π o . (11)This can be calculated as R rl R ⊙ = F ( q )( M M ⊙ ) ( P day ) . (12)Here, we define F ( q ) as F ( q ) = 5 . q − (1 + q ) . q − + ln(1 + q − ) . (13)
885 890 895 900 905 910 915 920 925 1.215x10 R ( R ⊙ ) Age (yr)
Figure 11.
Time evolution of the radius of the 16 M ⊙ single-star model in the last 5 × yrs of the evolution. For simplicity, we neglect the wind. In this case, untilthe beginning of intensive mass transfer, the mass ofeach star and orbital period remain constant. Assumingthat mass transfer begins as soon as R reaches R rl , thecondition for the initial period leading to a dense CSM inthe vicinity of the SN progenitor is described as follows: F ( q ) − × . P/ day . F ( q ) − × . (14)The range of the initial period to satisfy this conditioncorresponds to ∆ ln ( P /day) = ln (920) - ln (905) ∼ . F ( q ) is canceled out. Comparing thiswith the corresponding value for Type IIb SNe, i.e., ∆ln ( P /day) ∼ M & − M ⊙ yr − ) shortly before theexplosion is 5 × − times that of SNe IIb, or ∼ .
06 %of all the observed CCSNe. This covers roughly 0.65 %of all the observed SNe IIn.The ‘volumetric’ (intrinsic) rate as estimated abovemay sound like a prediction that would be impracti-cal to confirm, perhaps marginally being testable onlywith large future surveys like the Large Synoptic Sur-vey Telescope (LSST). However, we argue that this isnot the case. The luminosity of SNe IIn, or in generalSNe powered by SN-CSM interaction, is scaled to beroughly proportional to the CSM density (Moriya et al.2014). This is more complicated in the case of an opti-cally thick, dense CSM (Moriya & Maeda 2014), whilethe argument should in any case apply to the total en-ergy budget. The mass-loss rate of & − M ⊙ yr − inthe binary systems discussed here (Fig. 10) correspondsto the CSM density by at least one order of magnitudelarger than the less extreme SNe IIn with 10 − –10 − M ⊙ yr − ( § ∼ − M ⊙ yr − have been derived for the most luminous SNe IIn(Gal-Yam & Leonard 2009; Miller et al. 2010). Assum-ing that the luminosity of the SNe IIn in this class islarger than the less extreme case by two orders of mag-nitude (i.e., 5 mag), which is consistent with the lumi-nosity function of SNe IIn in observed samples (Li et al.2011), then the detectable volume of such luminous SNeIIn in the universe is three orders of magnitudes largerthan that of the less extreme SNe IIn. Note that deriv-ing the mass-loss rate may include an error at a levelof an order of magnitude, but the diversity discussedhere is well beyond such uncertainty. Therefore, despitethe intrinsically rare occurrence of these luminous SNeIIn, this population could indeed dominate, or at leastsignificantly contribute to, the observed SNe IIn fromthe binary evolution or even the observed SNe IIn as awhole. SUMMARYSeveral progenitors of Type IIb SNe have been identi-fied so far. Among these, four SNe have abundant obser-vational data sets already published, including their lo-cation in the HR diagram and the mass-loss rates shortlybefore the explosion. In addition to the diversity in theHR diagram, there is a tendency that their mass-lossrates increase by an order of magnitude with the increaseof the progenitor radii. In particular, the high mass-lossrates associated with the more extended progenitors arenot readily explained by a prescription commonly usedfor a single stellar wind.We have calculated a grid of binary evolution mod-els with various parameter sets. We have shown thatthe observational relation between the progenitor radiiand mass-loss rates can naturally be explained by non-conservative mass transfer in the final phase of progen-itor evolution without any fine tuning.We have also clarified that the mass transfer rate inthe final ∼ yr can be approximately estimated usingan analytical formula (eq (6)), which roughly claims thatthe progenitor loses envelope mass within the timescaleof the expansion of the radius. Using this formula, wecan explain why the mass transfer rate increases with theprogenitor radius (Fig. 3). This is mainly because lessextended progenitors have not only a smaller envelopemass to transfer but a larger value of ζ eq . The larger ζ eq means that the progenitor shrinks faster in response tothe mass loss.This is further support for the dominance of binaryevolution origin leading to Type IIb SNe. Therefore,further testing the relation between the size of the pro-genitor and the associated mass-loss rate with an in-creasing number of observed samples can provide a keyto clarifying the still-debated origin toward SNe IIb, andeventually also to SNe Ib/c.As a byproduct, we have also found a possible linkbetween the binary evolution scenario toward SNe IIband some SNe IIn. About 4 % of all observed SNeIIn should have CSM which is produced by binary non-conservative mass transfer in the final evolutionary stageof the progenitor, if the main path to SNe IIb is the bi-nary interaction. Such a population of SNe IIn will havecharacteristics of the velocity and geometry of the CSM,and therefore will be distinguishable from other SNe IInfrom different evolutionary scenarios. Identifying suchSNe IIn will provide a new test for the binary origin to-ward SNe IIb (and a fraction of SNe IIn, or even SNeIIL).Furthermore, about one tenth of SNe IIn related tobinary evolution are predicted to be associated with anextensively dense mass loss, reaching ˙ M & − M ⊙ yr − in the final ∼ yr, which likely produces a shell-likeCSM in the immediate vicinity of the SN progenitor.They may indeed be classified as either SNe IIP or IIL,but initially showing the spectroscopic features of SNeIIn. While the intrinsic (volume-limited) rate is pre-dicted to be small, these may dominate the observed(magnitude-limited) sample of SNe IIn through the bi-nary path (for which the CSM is produced by binarynon-conservative mass transfer), or even a large fractionof all the luminous SNe IIn.The authors thank Sung-Chul Yoon, Takashi Moriya,Ryosuke Hirai, Akihiro Suzuki and Takashi Nagao foruseful comments on this research and stimulating dis-cussions, and also thank the anonymous referee for theconstructive comments on this manuscript. The au-thors thank the Yukawa Institute for Theoretical Physicsat Kyoto University for useful discussions to completethis work during the YITP workshop YITP-T-16-05 on‘Transient Universe in the Big Survey Era: Understand-ing the Nature of Astrophysical Explosive Phenomena’.The work of K. M. has been supported by Japan Societyfor the Promotion of Science (JSPS) KAKENHI Grant26800100 and 17H02864 (K.M.).3APPENDIX A. THE RELATION BETWEEN THE HYDROGEN-RICH ENVELOPE MASS AND THE RADIUS IN THEEQUILIBRIUM STATEThe radius decreases with the decrease of the envelope mass when the envelope is radiative (ln ( M env [ M ⊙ ]) . -2)as shown in Fig. 6. In this Appendix, we show that this behavior can be described approximately in an analyticalway, following an argument similar to that presented by Cox & Salpeter (1961) in a different context. First, the basicequations determining the structure of the stellar envelope are d ( P gas + P rad ) dr = − GMr ρ , (A1) dP rad dr = − κL πcr ρ . (A2)Here, P gas and P rad are the pressure of the gas and radiation, respectively. For the opacity ( κ ), we assume that free-freeabsorption is the dominant source of the opacity, therefore κ = κ ρT − / . Here, we also assume that the mass andenergy generation in the envelope are negligible compared with those in the core and the surrounding shell. We furtherassume for simplicity that P gas = βP , with β being constant throughout the envelope. Using these relations, we cansolve Equations (A1) and (A2) analytically: ρ = C (cid:18) r − R (cid:19) / , and (A3) C = r πac κ L (cid:18) GM µHβ k B (cid:19) / , (A4)where µ is the mean molecular weight, while k B and H are Boltzmann’s constant and the reciprocal of Avogadro’snumber, respectively. From this solution, we can express the hydrogen-rich envelope mass as follows: M env = Z RR c dr πr ρ , (A5) M env /M ⊙ = 4 πC ′ I ( z ) , (A6)where C ′ = 4 πCM ⊙ R / , (A7) I ( z ) = z / (1 + z ) / Z du u / (1 + zu ) . (A8) R c and R are the radius of the helium core and the stellar radius, respectively. Here, we define a new variable z = R − R c R c . From (A6)–(A8), and substituting the typical values of the models (Table 4) for the corresponding physicalparameters in (A4) and (A7), we obtain approximately the radius as a function of the envelope mass. In Fig. 12,we compare this analytically derived relation with the equilibrium radius derived in Section 4. 1 (Fig. 6). Despitethe crude approximations (the opacity is dominated by the free-free absorption and the ratio of the gas pressure tothe radiation pressure is constant throughout the envelope), the analytical curve derived from (A6) reproduces thecurve obtained through the numerical evolution calculations fairly well. Thus, the radius in complete equilibriumincreases with increasing envelope mass when the envelope is radiative under the conditions we assumed, which is thensuppressed due to the development of convection for ln ( M env [ M ⊙ ]) & -2. B. ASYMPTOTIC BEHAVIOR OF ζ L For a large mass ratio ( q ≫ ζ L converges to ∼ − .
5. This is explained as follows. Instead of using the fitting formula of Eggleton(1983), here our argument is based on the Roche potential. If we set the origin of the coordinates at the center of the4primary (i.e. star 1), and set the y-axis and z-axis parallel to the line connecting the two stars and the orbital rotationaxis, respectively, the Roche potential φ is written as φ ( x, y, z ) = − GM ( x + y + z ) / − GM (( x − a ) + y + z ) / −
12 Ω [( x − µa ) + y ] . (B9) µ is defined as µ = M M + M , and a and Ω = q G ( M + M ) a are the binary separation and the angular velocity of theorbit, respectively. If we denote the position of the L1 point as ( x L ,0 ,0), then x L is derived from ∂φ ( x, , ∂x = 0 . (B10)Noting that 0 < x L < a , this leads to GM x − GM ( x − a ) − Ω ( x − µa ) = 0 . (B11)If q is sufficiently large, then we expect x L ≪ a . In this limit, Equation (19) is expanded in terms of ( x/a ) as follows:1 = (1 + 3 q ) (cid:16) xa (cid:17) + O (cid:18)(cid:16) xa (cid:17) (cid:19) (B12) ∼ q (cid:16) xa (cid:17) . (B13)Thus, we obtain x L a ∝ q − . (B14)Denoting the orbital angular momentum as J , a is expressed as a = J G M + M M M (B15)= J G M M ( q ≫ . (B16)When q is sufficiently large, M is practically constant, even with mass transfer from the primary. Moreover, becausewe assume that the specific angular momentum of the escaping material has a value identical to that of the accretingstar (secondary) in our calculations, the loss of angular momentum is negligible if q ≫
1. Then, from equations (B14)and (B16), we obtain the following: R rl, ∼ x L (B17) ∝ M − / (B18)Therefore, we finally reproduce the asymptotic behavior found in the binary evolution calculations: ζ L ∼ −
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Final properties of the models ( f = 0 . M env and M denote the hydrogen-rich envelope mass and star mass,respectively. ˙ M wind is the mass-loss rate due to the stellar wind at the end of the calculations. The subscript 1 and 2 referto the primary and the secondary, respectively, while the subscript f refers to the values at the end of the calculation. ’NONCONVERGENCE’ in the column of the final fate means the calculation has a convergence problem during the calclation. No. Initial period (P) Initial mass ratio (q) log T eff , log( L L ⊙ ) Radius R M env , M M ˙ M wind ˙ M Final fate(Days) ( R ⊙ ) ( M ⊙ ) ( M ⊙ ) (10 − M ⊙ yr − ) (10 − M ⊙ yr − ) of the primary1 5 0.6 - - - - - - - - CONTACT2 5 0.8 4.53 4.85 7.6 0.000 3.801 17.804 1.99 2.43 SN Ib3 5 0.95 4.54 4.85 7.5 0.000 3.816 19.937 2.07 2.51 SN Ib4 25 0.6 - - - - - - - - CONTACT5 25 0.8 4.24 4.94 32.11 0.059 4.533 17.864 1.57 1.93 SN IIb6 25 0.95 4.43 4.94 13.4 0.043 4.530 19.857 1.94 2.33 SN IIb7 50 0.6 - - - - - - - - CONTACT8 50 0.8 3.90 4.94 156.3 0.057 4.562 18.005 1.68 3.74 SN IIb9 50 0.95 3.87 4.95 178.5 0.057 4.607 20.030 2.53 4.88 SN IIb10 200 0.6 - - - - - - - - CONTACT11 200 0.8 3.72 4.95 364.0 0.065 4.611 17.957 2.50 4.32 SN IIb12 200 0.95 - - - - - - - - NON CONVERGENCE13 600 0.6 - - - - - - - - CONTACT14 600 0.8 3.61 4.99 626.4 0.151 4.962 17.585 3.18 7.58 SN IIb15 600 0.95 3.60 5.00 649.3 0.168 5.048 19.376 3.36 8.72 SN IIb16 800 0.6 - - - - - - - - CONTACT17 800 0.8 - - - - - - - - CONTACT18 800 0.95 3.59 5.00 688.4 0.212 5.156 18.942 3.61 10.51 SN IIb19 1200 0.6 - - - - - - - - CONTACT20 1200 0.8 - - - - - - - - CONTACT21 1200 0.95 3.55 5.01 843.4 0.714 5.746 18.369 4.94 22.83 SN IIb22 1600 0.6 - - - - - - - - CONTACT23 1600 0.8 3.53 5.02 948.2 2.040 7.127 15.488 5.53 46.10 SN IIP/IIL24 1600 0.95 3.53 5.02 960.0 2.768 7.875 16.752 6.99 67.13 SN IIP/IIL25 1800 0.6 - - - - - - - - CONTACT26 1800 0.8 3.52 5.020 965.07 2.714 7.802 15.145 5.58 67.27 SN IIP/IIL27 1800 0.95 3.53 5.02 955.9 3.725 8.832 16.274 6.86 100.30 SN IIn28 1950 0.6 - - - - - - - - CONTACT29 1950 0.8 3.52 5.02 965.2 3.408 8.498 14.799 5.65 146.32 SN IIn30 1950 0.95 3.53 5.02 936.7 5.030 10.14 15.625 6.90 334.51 SN IIn31 2200 0.6 3.54 5.02 890.3 7.553 12.664 9.548 5.09 5.94 SN IIP/IIL32 2200 0.8 3.54 5.02 887.9 7.669 12.757 12.670 5.30 6.40 SN IIP/IIL33 2200 0.95 3.54 5.02 890.8 7.549 12.657 14.385 6.52 7.30 SN IIP/IIL Table 3.
Final properties of the models ( f = 0 . No. Initial period (P) Initial mass ratio (q) log T eff , log( L L ⊙ ) Radius R M env , M M ˙ M wind ˙ M Final fate(Days) ( R ⊙ ) ( M ⊙ ) ( M ⊙ ) (10 − M ⊙ yr − ) (10 − M ⊙ yr − ) of the primary1 5 0.6 4.58 4.89 6.4 0.000 4.084 9.548 2.20 2.70 SN Ib2 5 0.8 4.54 4.85 7.4 0.000 3.832 12.659 2.03 2.48 SN IIb3 5 0.95 - - - - - - - - SECONDARY’S RLOF4 25 0.6 4.07 4.95 71.2 0.057 4.587 9.548 1.42 6.72 SN IIb5 25 0.8 4.21 4.94 37.0 0.062 4.538 12.670 1.58 1.92 SN IIb6 25 0.95 - - - - - - - - SECONDARY’S RLOF7 50 0.6 3.97 4.94 111.3 0.055 4.581 9.548 0.66 5.36 SN IIb8 50 0.8 3.88 4.94 168.4 0.056 4.565 12.670 0.99 4.98 SN IIb9 50 0.95 - - - - - - - - SECONDARY’S RLOF10 200 0.6 3.79 4.95 261.9 0.058 4.602 9.548 1.38 4.99 SN IIb11 200 0.8 3.7009 4.9481 394.0 0.068 4.620 12.670 2.00 5.35 SN IIb12 200 0.95 3.66 4.95 474.6 0.084 4.676 14.326 4.41 8.65 SN IIb13 600 0.6 3.65 4.96 507.0 0.092 4.714 9.548 2.49 7.32 SN IIb14 600 0.8 3.60 4.99 648.3 0.167 5.004 12.670 3.35 12.99 SN IIb15 600 0.95 3.59 5.00 698.9 0.231 5.152 14.396 4.83 18.49 SN IIb16 800 0.6 3.62 4.98 580.6 0.123 4.852 9.548 2.86 9.88 SN IIb17 800 0.8 3.59 5.00 695.4 0.222 5.125 12.670 3.57 16.50 SN IIb18 800 0.95 3.57 5.01 762.7 0.361 5.333 14.343 5.70 25.09 SN IIb19 1200 0.6 3.60 5.00 674.7 0.198 5.076 9.548 3.44 15.57 SN IIb20 1200 0.8 3.55 5.01 856.2 0.777 5.769 12.670 4.89 41.51 SN IIb21 1200 0.95 3.54 5.02 913.6 1.402 6.473 14.337 7.04 60.74 SN IIP/IIL22 1600 0.6 3.54 5.02 919.0 1.253 6.363 9.548 5.39 70.77 SN IIP/IIL23 1600 0.8 3.53 5.02 958.8 2.851 7.937 12.670 5.63 131.45 SN IIn24 1600 0.95 3.53 5.02 951.3 3.712 8.819 14.361 6.90 139.57 SN IIn25 1800 0.6 3.53 5.02 956.7 1.725 6.835 9.548 5.49 87.61 SN IIP/IIL26 1800 0.8 3.53 5.02 948.9 3.879 8.968 12.669 5.51 197.10 SN IIn27 1800 0.95 3.53 5.01 924.8 4.945 10.053 14.363 6.60 234.26 SN IIn28 1950 0.6 3.52 5.02 969.7 2.725 7.836 9.548 5.52 4739.21 SN IIn29 1950 0.8 3.53 5.02 931.9 5.179 10.267 12.669 5.49 506.14 SN IIn30 1950 0.95 3.54 5.02 912.4 6.386 11.493 14.359 6.86 443.59 SN IIn31 2200 0.6 3.54 5.02 890.4 7.553 12.664 9.548 5.17 5.95 SN IIP/IIL32 2200 0.8 3.54 5.02 888.7 7.669 12.757 12.670 5.31 6.81 SN IIP/IIL33 2200 0.95 3.54 5.02 891.22 7.549 12.657 14.385 6.53 7.39 SN IIP/IIL Table 4.
Typical values of the physical quantities used in Fig. 12.
M/M ⊙ L/L ⊙ R c /R ⊙ β µ κ (cm g − )4.6 1 . × . ×10