The effect of massive binaries on stellar populations and supernova progenitors
aa r X i v : . [ a s t r o - ph ] N ov Mon. Not. R. Astron. Soc. , 1–11 (2005) Printed 1 November 2018 (MN L A TEX style file v2.2)
The effect of massive binaries on stellar populations andsupernova progenitors
John J. Eldridge , ⋆ , Robert G. Izzard & Christopher A. Tout Astrophysics Research Centre, Physics Building, Queen’s University, Belfast, County Antrim, BT7 1NN Sterrekundig Instituut Utrecht, Postbus 80000, 3508 TA Utrecht, The Netherlands Institute of Astronomy, The Observatories, University of Cambridge, Madingley Road, Cambridge, CB3 0HA
ABSTRACT
We compare our latest single and binary stellar model results from the CambridgeSTARS code to several sets of observations. We examine four stellar population ra-tios, the number of blue to red supergiants, the number of Wolf-Rayet stars to Osupergiants, the number of red supergiants to Wolf-Rayet stars and the relative num-ber of Wolf-Rayet subtypes, WC to WN stars. These four ratios provide a quantitativemeasure of nuclear burning lifetimes and the importance of mass loss during variousstages of the stars’ lifetimes. In addition we compare our models to the relative rateof type Ib/c to type II supernovae to measure the amount of mass lost over the entirelives of all stars. We find reasonable agreement between the observationally inferredvalues and our predicted values by mixing single and binary star populations. Howeverthere is evidence that extra mass loss is required to improve the agreement further, toreduce the number of red supergiants and increase the number of Wolf-Rayet stars.
Key words: stars: evolution – binaries: general – supernovae: general – stars: Wolf-Rayet – stars: supergiants
There are many problems outstanding in the field of stel-lar evolution. The most massive stars, those that end theirlives as a supernova (SN), the main questions are ‘At whatrate do stars lose mass?’, ‘Which stars give rise to whichSNe?’, ‘What physical processes occur in binary systems?’and ‘How important are rotation and magnetic fields?’ Oneway to gain insight into these questions is to observe pop-ulations of stars and SNe. Massive stars are classified in arange of possible spectroscopic stellar types. The main cate-gories are blue supergiants (BSGs), red supergiants (RSGs)and Wolf-Rayet (WR) stars. If we count the numbers of suchstars in a single system (e.g. a stellar cluster or galaxy) wecan test the accuracy of our stellar models by comparing tomodel predictions.When such comparisons are performed the agree-ment can be poor. For example the ratio of the BSG toRSG populations has been an outstanding problem forsome time (Langer & Maeder 1995; Maeder & Meynet 2001;Massey & Olsen 2003). The ratio of the number of RSGs tothe number of WR stars is also not correctly predicted bystellar models (Massey 2003). However the ratios of WR star ⋆ E-mail: [email protected] subtypes WN and WC stars are predicted by modern stellarevolution (Meynet & Maeder 2005; Eldridge & Vink 2006).Massive single stars end their lives as core-collapse SNebut the relation between the star and SN type is not straightforward. A list of core-collapse SN types includes IIP, IIL,IIb, IIn, IIpec, Ib and Ic (Turatto, Benetti & Pastorello2007). The difference between type II and type I SNe is thepresence or absence of hydrogen in the SN spectrum. It iseasy to identify which stellar models contain hydrogen or notbut the subtype classifications are fairly arbitrary (Popov1993; Dray & Tout 2003; Heger et al. 2003; Eldridge & Tout2004b; Young, Smith & Johnson 2005). The most directmethod to link supernovae to stellar end point models isto search for the progenitor stars of SNe in pre-explosionimages (Van Dyk et al. 2003; Smartt el al. 2004).If we only split the SN types into II and Ib/c anothercomparison can be made between models and observations.The relative rates of these two SN types and how theyvary with metallicity have been estimated observationally(Prantzos & Boissier 2003). Meynet & Maeder (2005) haveshown that standard non-rotating single star models do not Type Ia SNe are thought to be thermonuclear events fromexplosive carbon burning in a degenerate carbon-oxygen whitedwarf and are not considered in this work.c (cid:13)
John J. Eldridge, Robert G. Izzard & Christopher A. Tout agree with these observed rates and that mass loss must beenhanced, in their case by rotation, to find an agreementbetween predicted SN ratios and those inferred from obser-vations.The studies mentioned so far only compare singlestar models to the observations. This ignores the observedfact that many stars are in binaries. Binary evolution hasbeen studied widely, (e.g. Podsiadlowski, Joss & Hsu 1992;Wellstein & Langer 1999; Belczynski, Kalogera & Bulik2002; Izzard, Ramirez-Ruiz & Tout 2004;Vanbeveren, Van Bever & Belkus 2007; Vzquez et al. 2007;Dray & Tout 2007). The methods employed to estimatethe effect of binaries is based either on rapid populationsynthesis or a small number of detailed models. Populationsynthesis employs formulae or tables based on detailedmodels to follow the evolution, (e.g. Hurley, Pols & Tout2000 and the evolution of a star can be calculated in afraction of a second rather than many minutes taken bya detailed code. The speed of calculations means thatthe full range of parameters that govern the outcome ofbinary evolution can be studied and their relative effectsevaluated. The disadvantage is that in the complex phasesof evolution, such as when the envelope is close to beingstripped, the results of population synthesis can be quitespurious.We have constructed a set of binary stellar models cov-ering a wide range of binary parameter space using a detailedstellar evolution code (Eldridge & Tout 2004a). With thesemodels we have investigated the effect binary evolution hason the lifetimes of the various phases of stellar evolutionand therefore the effect it has on the relative populationsof massive stars and the relative numbers of different typesof SNe. While the binary parameter-space resolution is lowcompared to rapid population synthesis studies it is still oneof the highest resolution studies undertaken with a detailedevolution code.We note that we do not consider here the effect of rota-tion or magnetic fields. The effect of rotation on the struc-ture and evolution of stars is a complicated process andhas been studied by Heger, Langer & Woosley (2000) andHeger & Langer (2000) and the extensive study beginningin Meynet & Maeder (1997) and continuing up to most re-cently in Meynet & Maeder (2007). The effect of magneticfields on stellar structure and evolution has also been in-vestigated by Meynet & Maeder (2003), Meynet & Maeder(2004), Heger, Woosley & Spruit (2005). Our code does notinclude rotation therefore our study reveals the importanceof duplicity alone. We discuss how rotation may affect ourresults in Section 6.In this paper we start by providing details of our stel-lar model construction. We then specify our definitions ofthe different phases of stellar evolution and discuss how bi-nary evolution affects the relative lifetimes of stars in variousphases of stellar evolution. Using these lifetimes we predicthow the BSG to RSG, the WR to O supergiant, the RSGto WR and WC to WN ratios vary with metallicity. Wethen predict how the relative rates of type Ib/c and type IISNe vary with metallicity. With each of these predictions wecompare the values to the ratios inferred from observations.Finally we discuss the implications of our results.
We use a detailed stellar evolution code to construct setsof single and binary star models rather than a rapid pop-ulation synthesis code, because we require accurate treat-ment of complex stages of evolution, such as when the en-velope is close to being stripped, to obtain more accuratepre-SN models. We model the binary parameters with areasonable resolution owing to the great amount of com-puter power that is now available. We have compared our bi-nary models to the predictions of a rapid population synthe-sis code (Izzard & Tout 2003; Izzard, Ramirez-Ruiz & Tout2004; Izzard & Tout 2005) and found reasonable agreementfor most of our predictions. However difficulty in assigningWR types to population synthesis helium stars meant theagreement was poor for the WC/WN ratio.
Our detailed stellar models were calculated with the Cam-bridge stellar evolution code, STARS, originally developedby Eggleton (1971) and updated most recently by Pols et al.(1995) and Eldridge & Tout (2004a). Further details canbe found at the code’s web pages . Our models are avail-able from the same location for download without restric-tion. They are similar to those described by Eldridge & Tout(2004b) but we use 46 rather than 21 zero-age main-sequencemodels. Every integer mass from 5 to 40M ⊙ is modelledalong with 45, 50, 55, 60, 70, 80, 100 and 120M ⊙ . Theinitially uniform composition is X = 0 . − . Z and Y = 0 .
25 + 1 . Z , where X is the mass fraction of hydro-gen, Y that of helium and Z the initial metallicity takingthe values 0.001, 0.004, 0.008, 0.02 and 0.04. The compo-sition mixture is scaled solar and Z ⊙ = 0 .
02. Our modelsend after core carbon-burning or at the onset of core neonignition. During these burning stages, the envelope is onlyaffected to a small degree because t nuclear ≪ t thermal . Fur-thermore the late stages of evolution are rapid and have anegligible effect on the final observable state of the progen-itor (Woosley, Heger & Weaver 2002).Our opacity tables include the latest low temperaturemolecular opacities of Ferguson et al. (2005). This updateleads to tiny changes in the radii and surface temperaturesof red supergiant models because the altered opacity valuesonly slightly modify the surface boundary conditions.All the models employ our standard mass-loss pre-scription because it agrees best with observations ofSN progenitors (Eldridge & Tout 2004b). The prescrip-tion is based upon that of Dray & Tout (2003) but mod-ified in several ways. For pre-WR mass loss we use therates of de Jager, Nieuwenhuijzen & van der Hucht (1988)except for OB stars for which we use the theoreticalrates of Vink, de Koter & Lamers (2001). When the starbecomes a WR star ( X surface < .
4, log( T eff / K) > .
0) we use the rates of Nugis & Lamers (2000). Wescale all rates with the standard factor ( Z/ Z ⊙ ) . (Kudritzki, Pauldrach & Puls 1987; Heger et al. 2003), ex- ∼ stars c (cid:13) , 1–11 he effect of massive binaries on stellar populations and supernova progenitors cept for the rates of Vink, de Koter & Lamers (2001) whichinclude their own metallicity scaling.The most important change is that we scale theWR mass-loss rates with the by metallicity, even thoughthis is not normally included. Vanbeveren (2001) first in-cluded the scaling in a population synthesis model whileCrowther et al. (2002) first suggested that the scaling shouldbe included from observations of WR stars in the LMCand in the Galaxy. Recent theoretical predictions also sug-gest the scaling should be included (Vink & de Koter 2005).Inclusion of this scaling results in a greater agreementwith the change of WR population ratios with metallicity(Eldridge & Vink 2006). There is some uncertainty in theexact magnitude of the exponent we should use. To remainconsistent we use the same exponent as for the non-WRstars.We synthesise a single star population by assuming aconstant star formation rate and the initial mass function(IMF) of Kroupa, Tout and Gilmore (1993). Binary stars experience different evolution from that of sin-gle stars if their components interact. In a wide enough orbita star is not affected by a companion. Duplicity allows thepossibility of enhanced mass loss, mass transfer and otherbinary specific interactions (e.g. irradiation, colliding winds,surface contact, gravitational distortion) and hence a greaterscope for interesting evolutionary scenarios.We have modified our stellar evolution code to modelbinary evolution. The details of our binary interaction algo-rithm are relatively simple compared to the scheme outlinedin Hurley, Tout & Pols (2002). We use their scheme as a ba-sis but we change some details which cannot be directlyapplied to our detailed stellar evolution calculation. We alsomake a number of assumptions in producing our code tokeep it relatively simple. Our aim is to investigate the effectof enhanced mass-loss due to binary interactions on stellarlifetimes and populations, therefore we concentrate on thisrather than every possible physical process which would addmore uncertainty to our model. We also make assumptionsin calculating our synthetic population to avoid calculatinga large number of models. For example we do not model theaccretion on to the secondary in the detailed code. We takethe final mass of the secondary at the end of the primarycode as the initial mass of the secondary when we create ourdetailed secondary model. This avoids calculating ten timesmore secondary models than primary models.
In our binary models we always treat the primary as the ini-tially more massive star when and we only evolve one star ata time with our detailed code. When we evolve the primaryin detail it has a shorter evolutionary timescale than thesecondary which remains on the main sequence until afterthe primary completes its evolution and so we can determinethe state of the secondary using the single stellar evolutionequations of Hurley, Pols & Tout (2000). When we evolvethe secondary in detail we assume its companion is the com-pact remnant of the primary (a white dwarf, neutron star or black hole) and treat this as a point mass. We describe howwe combine these models to synthesise a population below.To model a binary we begin by specifying a primarymass, M , a secondary mass, M , and an initial separa-tion, a . We assume the orbit is circular and conserve an-gular momentum unless mass is lost from the system. Wefollow the rotation of the stars, considering them to ro-tate as solid bodies but we do not include processes suchas rotationally-enhanced mass loss or mixing. We give thestars initial rotation rates linked to their initial masses asin Hurley, Pols & Tout (2000). When the stars lose massin their stellar wind angular momentum is lost both fromtheir spin and orbit. We ignore tides so stellar rotation andorbital rotation evolve independently until Roche-lobe over-flow (RLOF) when we force the stars into synchronous ro-tation with the orbit, transferring angular momentum fromthe spin to the orbit or vice versa. In a binary system the equipotential surfaces are not spher-ical but are distorted by the gravity of the companion starand by the rotation of the system. If the stars are small com-pared to the size of the orbital separation these surfaces arenearly spherical but as the star grows relative to the sepa-ration the shape is distorted and becomes more ellipsoidal.A star’s surface eventually reaches the L1 Lagrange pointwhere the gravity of both stars cancels exactly. If a starexpands beyond this point then material begins to flow to-wards the other star. To include this in our models we definea Roche lobe radius, R L1 , such that the sphere of this radiushas the same volume as the material within the Roche Lobedefined by the equipotential surface passing through the L1point. Therefore when the star has a radius greater than theRoche lobe radius it transfers material on to the other star.We use the Roche lobe radius given by Eggleton (1983), R L1 a = 0 . q / . q / + ln(1 + q / ) , (1)where q = M /M and a is the orbital separation. It isaccurate to within 2% for the range 0 < q < ∞ . When R > R L1 we calculate the rate at which mass is lost fromthe primary according to,˙ M = F ( M )[ln( R /R L1 )] M ⊙ yr − , (2)where F ( M ) = 3 × − [min( M , . , (3)chosen by experiment to ensure mass transfer is steady(Hurley, Tout & Pols 2002).Mass lost from the primary is transferred to the sec-ondary but not all is necessarily accreted. Accretion causesthe star to expand owing to increased total mass and there-fore an increased energy production rate if ˙ M > M /τ KH where τ KH is the thermal, or Kelvin-Helmholtz, timescale.We assume that the star’s maximum accretion rate is de-termined by its current mass and its thermal timescale.We define a maximum accretion rate for a star such that˙ M M /τ KH . If the accretion rate is greater than thisthen any additional mass and its orbital angular momentumare lost from the system. In general stars with lower masses c (cid:13) , 1–11 John J. Eldridge, Robert G. Izzard & Christopher A. Tout have longer thermal timescales than more massive stars. Ef-ficient transfer is only possible if the two stars are of nearlyequal mass so the thermal timescales are similar. This is anapproximate treatment but provides a similar result to themore complex model of Petrovic, Langer & van der Hucht(2005) who included rotation and found it led to inef-ficient mass-transfer. For compact companions we derivethe maximum accretion rate from the Eddington limit(Cameron & Mock 1967).
If RLOF occurs but does not arrest the expansion of themass losing star, growth continues until the secondary isengulfed in the envelope of the primary. This is common en-velope evolution (CEE). During CEE it is thought that theenvelope is ejected by some dynamical process and that theprincipal source of energy for the envelope ejection is theorbital energy (Paczynski 1976; Livio & Soker 1984). Theorbital separation decreases and there is a chance that thetwo stars may coalesce before the envelope is ejected. Alter-natively the two stars are left in a close orbit, commonly onehelium star for a massive progenitor and one main-sequencestar. If the secondary then evolves to a helium giant a sec-ond CEE phase can occur possibly leading to a merger or avery close binary.CEE is modelled in population synthesis by first cal-culating the energy required for the ejection of the enve-lope, the binding energy, E , bind = GM M , env / ( λR , where M , env is the mass of the envelope, R is the radius of thestar and λ is a constant to reflect the structure of the en-velope. This is then compared to the initial orbital energy, E orb ,i = − GM c, M / (2 a i ), where M c, is the mass of thecore and a i is the initial separation at the onset of CEE. Ifthere is not enough orbital energy to eject the envelope andleave a stable binary then the two stars merge to form onestar. Otherwise the envelope is removed leaving a stable bi-nary system (e.g. Hurley, Tout & Pols 2002). The new orbitis calculated by the equation E , bind = α CE ( E orb ,i − E orb ,f ),where E orb ,f is the final orbital energy from which the finalseparation is calculated. There is some uncertainty in theconstant α CE as there are other sources of energy availablesuch as the energy from nuclear reactions and re-ionisationenergy of the hydrogen in the envelope so it can be greaterthan one.In our models when CEE occurs we derive the mass-loss rate from the above equation for RLOF, however welimit the mass-loss rates to a maximum of 10 − M ⊙ yr − be-cause more rapid mass loss causes the evolution code tobreak down. We assume that the secondary accretes nomass because the CEE occurs on the thermal timescaleof the primary which is normally shorter than the ther-mal timescale of the less massive secondary. The accre-tion rate for a compact remnant is restricted to the Ed-dington limit (Cameron & Mock 1967). To calculate thechange in the orbital separation we calculate the bindingenergy of the material lost in the CEE wind ( δE binding = − ( M + M ) δM /R ) and remove it from the orbital energy( E orbit = − GM M / (2 a )). We find, δa = a R M + M M δM M , (4) where δa is the change in orbital separation and δM is themass lost in the time period considered, one model timestep.In this formalism δM is negative so the orbit shrinks. Bydetermining the mass-loss rate from the RLOF equation wefind that the CEE ends naturally with the stellar radiusshrinking once most of the envelope has been removed. In the most compact binaries the stars can merge duringCEE to form a single star whose mass is the sum of its par-ents. We find that some of our binary models enter CEEand the binary separation tends to zero. When this occurswe use a different binary model. The primary is evolved inthe detailed code and once its radius is equal to the binaryseparation all the mass of the secondary is accreted on tothe surface of the primary star and the evolution contin-ues as a more massive single star. This is an extremely sim-ple model. Other models such as Podsiadlowski, Joss & Hsu(1992) show that a more detailed treatment may be requiredbut the efficiency of the merger is always uncertain. Ourmodels provide an estimate of the effects of stellar merg-ers and are an upper limit as we assume all the secon-daries mass is accreted. Furthermore the outcome dependson the evolutionary state of the primary, it is possible thatfresh hydrogen could be mixed into the helium burning zoneand therefore cause the envelope to be explosively removed(Podsiadlowski, Morris & Ivanova 2006).
We calculate three subsets of binary models which we thencombine to simulate a population of stars. The first subsetconsists of our primary models where we evolve the primarystar in the detailed code and evolve the secondary accordingto the single-star evolution equation of Hurley, Pols & Tout(2000). We choose our grid to be M =5, 6, 7, 8, 9, 10, 11, 12,13, 15, 20, 25, 30, 40, 60, 80, 100 and 120; q = M M = 0 .
1, 0 . .
5, 0 . .
9; log( a/ R ⊙ ) =1.0, 1.25, 1.5, 1.75, 2, 2.25, 2.5,2.75, 3, 3.25, 3.25, 3.5, 3.75 and 4;. We do not model lowermasses because these stars are never luminous enough toaffect our population ratios and do not lead to core-collapseSNe.The second subset consists of our secondary modelswhere we evolve the secondary in the detailed code after theend of the primary evolution and replace the primary byits remnant. We chose three masses for the compact com-panion to model a white dwarf, neutron star or black hole, M , post − SN = 0 .
6, 1 . ⊙ respectively. We choose thegrid to use the same separation and mass grid as for our pri-mary model subset. The final subset consists of our mergermodels and we choose this grid to have the same M , a and q distribution as our primary model. We only calculatemodels with initial orbital separation smaller than the max-imum stellar radius for each initial primary mass. These areonly used if the orbital separation reaches zero in a primarymodel. c (cid:13) , 1–11 he effect of massive binaries on stellar populations and supernova progenitors The binary population is synthesised by first assumingthe primary masses are distributed with the IMF ofKroupa, Tout and Gilmore (1993). Then we assume flat dis-tribution over 1 log ( a/ R ⊙ ) . q .
9. Wecalculate the primary stars’ lifetimes and SN types using theprimary models. If a primary model during evolution entersa CEE and the binary separation shrinks to zero we replacethe primary model with a merger model.The end point of the primary model is taken to be atthe onset of neon burning, or the onset of thermal pulses inan AGB star. First we determine the remnant the primarywill form. A white dwarf is selected unless the primary un-derwent a SN if there is an oxygen-neon core and the carbon-oxygen core mass is greater than 1.3 M ⊙ when a neutron staris selected. A black hole remnant replaces this if the heliumcore mass is greater than 8 M ⊙ . In the case of a white dwarfthe system remains bound. When a neutron star is formedwe consider the effect of its natal kick on the binary orbit. Weuse the prescription of Brandt & Podsiadlowski (1995) andthe kick distribution of Hansen & Phinney (1997) to deter-mine whether the binary remains bound or not. In the blackhole case we assume the kick velocity is zero and estimatethe remnant mass and ejected mass as in Eldridge & Tout(2004b).If the system is unbound we calculate the contributionof the secondary to the stellar lifetimes and SN type fromour single star models. The contribution from binaries thatremain bound are taken from our secondary models. The ini-tial mass taken for the secondary star is the initial secondarymass unless the star accreted material from the primary inwhich case we use the new mass after mass-transfer.Combining the model subsets creates our synthetic pop-ulation of binary stars based on detailed models from whichwe can estimate the relative numbers of different stellartypes and the relative rates of different types of SNe. Wecalculate these populations at five different metallicities, Z = 0 . Uncertainties arise from assumptions we have made in thephysical model for binary systems. The greatest uncertaintyis that we do not model the primary and secondary in detailsimultaneously. This means we are not able to model sys-tems where q is close to one. In these mass transfer can bequite efficient with the secondary accreting a large fractionof the mass of the primary (Cantiello et al. 2007). Our ap-proximation is reasonable because for primaries with initialmasses . M ⊙ , the secondary stars with q . . > M ⊙ with companions that evolve off the mainsequence before the primary experiences a SN also rarelyinteract as both stars lose the hydrogen envelopes beforebecoming RSGs. The error introduced by this approxima-tion is dwarfed by the uncertainty in our assumed initialparameter distribution for the binaries. If we only considerbinaries closer than log( a/R ⊙ ) < Table 1.
Our definitions of stellar types.Name log( T eff / K) log( L/ L ⊙ ) X ( x C + x O ) y Blue supergiants > . > . > . . > . > .
001 -O supergiants > .
48 - > . > . > . . > . > . . > . > . . . > . > . . > . Figure 1.
Hertzsprung-Russel diagram indicating regions thatwe include as BSGs, RSGs and WR stars with single star tracksto indicate the mass ranges they sample. The numbers indicatethe initial stellar mass in M ⊙ for each track. The initial metallic-ity was Z=0.02. The BSG and WR regions overlap in the lightershaded region adjacent to the BSG region. The distinction be-tween WR and BSGs is made by surface hydrogen abundance. Stellar types may be defined by observable characteristicssuch as colour and luminosity. However for a complete andaccurate classification spectra are required. For examplesome objects may initially appear to be BSGs but theirspectra indicate they are quite different and are in fact WRstars. WR spectra contain broad emission lines and weak orno hydrogen lines indicative of fast and dense stellar winds.In this work we use the theoretical stellar definitions thatare summarised in Table 1.First, we only consider massive BSGs, RSGs and WRstars that have log( L/ L ⊙ ) > .
9, equivalent to a luminositycut-off of M bol > − .
5. If we compare our models to sin-gle clusters with well defined ages the limit would not becritical. However, because we are considering a less specificpopulation with an assumed constant star-formation rate,lowering the limit slightly to log( L/ L ⊙ ) > . c (cid:13) , 1–11 John J. Eldridge, Robert G. Izzard & Christopher A. Tout log( T eff / K) > .
9. This includes O, B and A stars. RSGs arecooler with log( T eff / K) .
66. This includes K and M stars.We show how these regions fit the tracks of our single starevolution models in Figure 1. RSGs are stars more massivethan 12M ⊙ while BSGs are stars more massive than 20 M ⊙ .Our luminosity limit reduces contamination from AGB stars(Massey & Olsen 2003) in the RSG sample. The limit is alsogreater than the luminosity of stars that experience a blueloop during helium burning. This feature is very sensitiveto the details of convection included in the stellar modelsand so best excluded (Langer & Maeder 1995). Our mod-els with blue loops lie outside of our RSG definition so ourpredicted ratios are less sensitive to the complications intro-duced by the details of convection and mixing employed inour models.We also consider the O supergiants (OSGs) which are asubset of the BSGsfor the WR/O ratio. OSGs are the hottesthydrogen-rich stars with log( T eff / K) > .
48 and a surfacehydrogen mass fraction,
X > .
4. There is no luminositylimit for these stars as in Meynet & Maeder (2005).The other stellar types we consider are WR stars. Wedefine a WR star to be any star that has log( T eff / K) > . X < .
4. We alsorequire that log( L/ L ⊙ ) > .
9. Further to this we subdivideWR stars into WNL, WNE and WC stars. In each case theelement nitrogen or carbon is dominant in their emissionspectrum. The sequence is due to the exposure of nuclearburning products on the surface of the WR stars. We assumea WR star is initially a WNL star. When
X < .
001 itbecomes a WNE star. The star becomes a WC star when
X < .
001 and ( x C + x O ) /y > .
03 where x C , x O and y arethe surface number fractions of carbon, oxygen and helium. In this section we compare the stellar lifetimes of our sin-gle and primary models at two different metallicities. Wecalculate a mean lifetime from our primary models only torepresent the effect on the lifetimes of RLOF and CEE. Themean lifetime for primary models with the same initial massis calculated assuming a flat distribution in q and log a .The primary model BSG mean lifetimes are slightly in-creased relative to the single star BSG lifetime. The total av-erage lifetime for BSGs increases by 4 to 8 per cent and thereis no trend with metallicity. This is because close merger sys-tems extend the burning lifetimes of the primary star. Alsostars initially too low in mass to be counted as BSGs accreteenough mass to rise above our luminosity limit.The RSG lifetimes of our models are shown by the dot-ted lines in Figures 2 and 3. The single star lifetimes at Z = 0 .
004 are longer than those at Z = 0 .
02. This is dueto the reduction in mass-loss rates at the lower metallicityso more time is required for the hydrogen envelope to be re-moved. Our primary models have RSG lifetimes two to threetimes shorter. RLOF and CEE greatly reduce the numberof RSGs by removing their hydrogen envelopes to form WRstars.The WR lifetimes of our single stars are identical tothose of Eldridge & Vink (2006). Mass loss affects the life-times. Firstly, it determines the total
WR lifespan by af- fecting the total stellar mass and thus the vigour of thenuclear reactions in the core. Secondly, stronger mass lossstrips mass more quickly from the stellar surface, which leadsto the exposure of hydrogen and then helium burning prod-ucts and results in the appearance of different WR subtypes.The WR lifetimes are longer at higher metallicities asthe WR phase is entered at an earlier stage and there isa longer period of core helium burning during the WRphase because of larger pre-WR mass loss. The kink seenin the Z=0.02 plot (Fig. 2) is due the fact that stars withmasses below it undergo a red supergiant phase when massis stripped and the surface hydrogen abundance drops belowthe limit set for WNL stars ( X surface . K). Stars more massivethan the mass at the kink spend most of their evolution attemperatures hotter than this limit and become WR starsas soon as the surface abundance requirement is reached.Our primary model WR lifetimes in Figures 2 and 3have relatively minor differences as far as the total WR life-times are concerned. The main difference is that the min-imum mass for a WR star is decrease to around 15 M ⊙ atboth metallicities. These stars will produce lower mass WRstars than are created by single stars.The lifetimes of the WR subtypes at Z = 0 .
02 indicatethat the primary models have longer WNE lifetimes thansingle stars and shorter WC lifetimes. The WR stars belowthe minimum mass limit for single WR stars do not have aWC phase and exist only as WN stars. At Z = 0 .
004 we seethe same trend but many of the lower mass WR stars onlyhave WNL phases.
Counting and comparing the number of stars in differentstellar populations is a useful method for investigating stel-lar evolution. A comparison between observed populationsand those predicted by a stellar evolution code is only worth-while if in the count stars are not missed or misclassified.Many older observations relied on groups of stars deter-mined by photometry. The use of spectrometry has im-proved the accuracy and completeness of observational sur-veys (Massey & Olsen 2003).The metallicity mass fraction of the observed stellarpopulations was calculated from the log[O / H] + 12 valuesand comparing them to the values from our sets of models.Because this process is ambiguous and the position of solarmetallicity for the mass-loss scaling is indistinct, we haveassumed the metallicities are uncertain by 25 per cent.
The ratio of the number of BSGs to the number of RSGshas been a problem in stellar astrophysics for some time(Langer & Maeder 1995). Observations tend to indicate thatthe BSG/RSG ratio should decrease with metallicity whilemost stellar evolution codes predict a constant or increas-ing ratio. Eggenberger, Meynet & Maeder (2002) suggestedthat the situation could be improved by using spectroscopyto confirm the identity of supergiants. If we compare ourmodels to their observed trend we find good agreement c (cid:13) , 1–11 he effect of massive binaries on stellar populations and supernova progenitors Figure 2.
RSG, WNL, WNE and WC lifetimes at Z=0.02. Upperpanel, our single star models. Lower panel, our primary models.The lifetimes of the binary stars are mean lifetimes assuming aflat q and log a distribution. Each region is labelled but progressesfrom bottom to top by WNL, WNE and WC phase. Table 2.
Observed numbers and population ratio of BSGs andRSGs. Values are taken from Massey & Olsen (2003).System log[O / H] + 12 N BSG N RSG N BSG /N RSG
SMC 8.13 1484 90 16 ± ± at solar metallicity but no agreement at SMC metallicity.However Chiosi et al. (1995) indicate that the SMC clus-ter Eggenberger, Meynet & Maeder (2002) observed has aspread of ages similar to the age of the cluster itself, there-fore the ratio they observe is not due to a single populationof stars with the same age.Massey & Olsen (2003) provide a detailed observationalstudy of RSGs in the Magellanic clouds with rigid definitionsfor BSGs and RSGs (see Table 1). They found from observ-ing the entire SMC and LMC that the ratios of BSGs toRSGs were 16 and 14 respectively (see Table 2).Comparing the observations to predicted values fromour models in Figure 4 we find that single stars predict toolow a value because either there are too many RSGs, too Figure 3.
As for Figure 2 but Z=0.004.
60 40 20 10 8 6 4 2 0.04 0.02 0.008 0.004 0.001 N BS G / N R S G Metallicity SingleBinaryObservations
Figure 4.
Ratio of the numbers of BSGs to RSGs versus metal-licity. Observations are taken from Massey & Olsen (2003). Thesolid line is from our single star models while the dashed line isfrom our binary models.c (cid:13) , 1–11
John J. Eldridge, Robert G. Izzard & Christopher A. Tout few BSGs or a combination of these two. Extra BSGs canbe made via stellar mergers. Such stars are seen in globularclusters when the mass of the merged star is greater thanthe mass of the cluster turn-off, these are blue stragglers(Sills et al. 2000). The second binary effect is that stars be-low the minimum initial single star mass for a WR star willlose their hydrogen envelopes to become WR stars becauseof RLOF or CEE. These are stars that would have beenRSGs and therefore the total number is reduced. In com-bination these processes increase the predicted BSG/RSGratio as our binary models show.The binary model ratio is dependent on how we choosethe distribution of binaries in q and log ( a/ R ⊙ ). In our bi-nary population we include very wide binaries, log ( a/ R ⊙ ) >
3, which evolve as single stars so our binary populationis a mix of interacting binaries and single stars. Approx-imately one third of our primary models evolve as singlestars. Therefore, to reproduce observations, two thirds of allstars must be in interacting binaries. If our binary modelsare not completely correct something extra, such as rotation,may decrease the required fraction of interacting binaries.Maeder & Meynet (2001) find that rotation has a strong ef-fect on the BSG to RSG ratio but the trend they found wasthat the BSG to RSG ratio decreased rather than increased.
The WR to OSG ratio has been studied for sometime(Maeder & Meynet 1994). If binaries leave the RSG pop-ulation to become WR stars therefore this ratio should in-crease if the BSG to RSG ratio decreases. The observed ratiois less certain than the BSG to RSG ratio because there isgreater uncertainty in the completeness of the observations(Massey 2003). However the trend that the ratio decreaseswith metallicity is well established (Crowther 2007).Figure 5 shows that our single star models underpredictthe ratio while our binary models are in better agreement,as are the Geneva rotating models. This is because in ourmodels more stars are stripped of their envelopes and be-come WR stars. There is also a contribution to the OSGpopulation from stellar mergers and secondary accretion asfor the BSG/RSG ratio above. The trend our binary modelspredict with metallicity is a little too shallow but within theuncertainty of the observed ratios. This again could indicatesomething extra may need to be included in our models.
The RSG to WR ratio compares the relative populationsof the two stellar types that have completed core hydrogenburning and so measures the influence of mass loss.In Table 3 we list the observed populations. We plot theratio in Figure 6, it decreases dramatically with increasingmetallicity. Such a change would require a stronger scalingof mass loss with metallicity than we currently employ.Our single star models agree with the observations atSMC metallicity while our binary models agree with the ob-servations around LMC metallicity. One way to match theobserved trend between these metallicities is to have a metal-licity dependent binary fraction. However something extra isstill required around solar metallicity to get an exact agree-ment. At the higher metallicities the ratios are based on a N W R / N O MetallicitySingleBinaryGeneva ModelsObservations
Figure 5.
Ratio of the numbers of Wolf-Rayet stars to O-supergiants versus metallicity. Observations are taken fromMaeder & Meynet (1994). The solid line is our single star modelswhile the dashed line is our binary models. The dashed-dotted linefor the Geneva models is taken from Meynet & Maeder (2005).The y-axis error bars are an assumed error of 50 percent of thevalues given by Maeder & Meynet (1994).
Table 3.
Observed numbers and population ratios of RSG andWR stars. The values are taken from Massey (2003).System log[O / H] + 12 N RSG N WR N RSG /N WR SMC 8.13 90 12 7 . ± . . ± . . ± . . ± . . ± . . ± . . ± . small number of observed stars (see Table 3) and thereforedo not sample the full binary parameter space. Furthermore,small numbers mean our assumptions of constant star for-mation and IMF become invalid. For example, if all the WRstars are in very close binaries the RSG/WR ratio would beeven smaller than we estimate. A large number of stars mustbe observed to calculate the population ratio to ensure thefull range of possible binary systems is probed. Table 4.
Observed numbers and population ratios of WC andWN stars. The values are taken from Meynet & Maeder (2005).System log[O / H] + 12 N WC N WN N WC /N WN SMC 8.13 1 11 0 . ± . . ± . . ± . . ± . . ± . < . ± . . ± . . ± . (cid:13) , 1–11 he effect of massive binaries on stellar populations and supernova progenitors
20 10 6 4 2 1 0.6 0.4 0.2 0.1 0.06 0.04 0.02 0.01 0.04 0.02 0.008 0.004 0.001 N R S G / N W R MetallicitySingleBinaryObservations
Figure 6.
Ratio of the number of RSG to WR stars versus metal-licity. The observations are taken from Massey (2003). The solidline is from our single star models while the dashed line is fromour binary models. N W C / N W N Metallicity SingleBinaryGeneva modelsObservations
Figure 7.
Ratio of the number of WC to WN stars versus metal-licity. The observations and the dashed-dotted line for the Genevamodels are taken from Meynet & Maeder (2005). The solid lineis from our single star models while the dashed line is from ourbinary models.
Finally, we consider a ratio which provides a measure of therelative lifetimes of the two main WR star types: WN andWC (we include WO with WC). The lifetimes of the twotypes are determined by the mass-loss rates of WR stars.We plot the observations and predicted ratios in Fig-ure 7 it shows our predicted ratios and the most recentGeneva group rotating models (Meynet & Maeder 2005) forcomparison. The trend of the observations is for the ratioto decrease with metallicity. In agreement with the resultsof Eldridge & Vink (2006) we find that the single-star mod-els that scale the mass-loss rate of WR stars with initialmetallicity agree with the observed trend.The binary models in this case give a lower WC/WNratio than the single star models. This is because, as can beseen in Figures 2 and 3, the lifetimes of WN stars increase bya greater factor than the lifetimes of WC stars in binaries.If we combine a population of single and binary stars theresulting ratio is too low and requires an increase in the WC population relative to the WN population to regain animproved agreement. The result is similar to that found byVanbeveren, Van Bever & Belkus (2007).
There are many studies which investigate the con-nection between SNe and massive stars. Some stud-ies consider single-star evolution and predict the ini-tial parameter space and the relative rate of dif-ferent SN types (Heger et al. 2003; Eldridge & Tout2004b; Hirschi, Meynet & Maeder 2004), while other stud-ies are concerned with the evolution of binary-stars(Podsiadlowski, Joss & Hsu 1992; de Donder & Vanbeveren2003; Izzard, Ramirez-Ruiz & Tout 2004). In this section weuse our single- and binary-star models to predict the rela-tive SN rates and determine how they vary with metallicity.Firstly, we link our models to each SN type then, secondly,we predict how the relative SN rates vary with metallicity.Core-collapse SNe are classified according to theirlightcurve shapes and spectra. Matching stellar models toobserved SNe is difficult and example schemes can be foundin Heger et al. (2003) and Eldridge & Tout (2004b). Herewe check the amount of hydrogen in the progenitor model:if there is more than 0.001M ⊙ of hydrogen left in the stellarenvelope the SN is of type II, otherwise type Ib/c.There are many subtypes of SNe. The main distinguish-ing criterion is the mass of the hydrogen or helium enve-lope at the time of explosion but sometimes the circum-stellar environment must also be considered (Heger et al.2003; Eldridge & Tout 2004b). In this paper we group SNII sub-types (e.g. P, L) together as type II and SN Ib andIc together as type Ib/c. In our single-star models the ini-tial mass range of Ib/c progenitors is restricted to the mostmassive stars (Eldridge & Tout 2004b). In our binary-starmodels the full range of masses can lead to type Ib/c SNe.SN rates in different galaxy types have beenmeasured for some time (Cappellaro et al. 1997;Cappellaro, Evans & Turatto 1999) and, more recently, SNeobservations have been used to determine how the relativerate of type Ib/c to type II SNe varies with metallicity(Prantzos & Boissier 2003). The errors especially in theabsolute rates of the searches are considerable owing to thesmall sample size and the uncertainties in the completeness.The relative rates are less uncertain as the selection effectsare of similar magnitude and cancel.We plot our predicted SN rate ratios against the ob-served ratios in Fig. 8. The observations indicate a generaltrend of a decreasing rate of type Ib/c SNe relative to typeII as metallicity decreases. This is as expected owing to thedecreasing strength of stellar winds with reduced metallic-ity meaning that fewer stars lose all the hydrogen beforecore-collapse.We find that our theoretical predictions agree withthe trend indicated by the observations. However the sin-gle star models predict a lower relative rate for type Ib/cSNe than the observations and the binary star models pre-dict a value that agrees with the observations. We also plotthe SN rate ratio predicted by the Geneva rotating mod-els (Meynet & Maeder 2004) which agree with observations.The conclusion we draw is that we must consider rotating c (cid:13) , 1–11 John J. Eldridge, Robert G. Izzard & Christopher A. Tout N I b / c / N II MetallicitySingleBinaryGeneva ModelsObservations
Figure 8.
The observed and predicted ratios of the type Ib/cSN rate to the type II SN rate. Observations are taken fromPrantzos & Boissier (2003). The Geneva model predictions aretaken from Meynet & Maeder (2005), the upper line is for theirrotating models while the lower line is from their non-rotatingmodels. models and/or binary star models to exactly simulate theobservations.
We have compared our stellar models of massive stars to sev-eral observations and find that by including binary stars weachieve better agreement. This is because duplicity increasesthe chance of a star losing its hydrogen envelope which leadsto fewer RSGs, more WR stars and more type Ib/c SNe thansingle star models predict as required by observations.Problems remain and the agreement is not perfect. TheBSG/RSG ratio and RSG/WR ratio indicate that we stillpredict too many RSGs or not enough WR stars. Also thebinary models predict to many WN stars and too few WCstars. Physics we have not included in our models mightresolve these problems. The main culprit is rotation. Ithas two effects on our predictions. First it introduces mix-ing that extends the time stars spend on the main se-quence and therefore increases the number of BSGs. Sec-ond, if rotation is rapid it enhances mass loss by reduc-ing the depth of the potential from which mass must es-cape and therefore increases the population of WR starsat the expense of the RSG population. Rotationally en-hanced mass loss affects the WC/WN ratio by shorteningthe WN lifetimes. However the effect of rotation can bemuch more complex then these simple effects. For exam-ple Maeder & Meynet (2001) find that the BSG to RSG ra-tio is decreased rather than increased by rotation owing toa change of the internal helium gradient at the hydrogen-burning shell. In addition tides in a binary may boost theimportance of rotation during important phases of evolutionand complicate the situation further (Petrovic et al. 2005;Petrovic, Langer & van der Hucht 2005).Another limitation of our models is that we do notinclude mass losing eruptions that are common in theluminous-blue variable (LBV) stars. These most massivestars experience giant outbursts, losing large amounts of mass in one short event. This has dramatic implications forthe evolution of the object (Smith & Owocki 2006). There isgrowing evidence that SN progenitors can experience theseoutbursts prior to the SN explosion (Kotak & Vink 2006;Pastorello et al. 2007). Such outbursts would have a similareffect on stellar populations as rotation. They reduce thetime stars spend as RSGs and could reduce the time takento remove the helium envelopes for WN stars. Therefore abetter understanding of these outbursts may also improveagreement between stellar models and observations.In summary, binary stars must be considered when com-paring stellar evolution models to observations. However,binary evolution alone cannot explain all the observations.Fine tuning of stellar models is still required and the effect ofenhanced mass loss owing to rotation or LBV-like eruptionsmust be further considered.
The authors would like to thank the referee Andre Maederfor his helpful comments that improved the paper. JJE con-ducted part of this work during his time at the IAP in Franceas a CRNS post-doc. The remainder of the work was carriedout as part of the award “Understanding the lives of massivestars from birth to supernovae” made under the EuropeanHeads of Research Councils and European Science Founda-tion EURYI Awards scheme and was supported by fundsfrom the Participating Organisations of EURYI and the ECSixth Framework Programme. CAT thanks Churchill Col-lege, Cambridge for his Fellowship. RGI thanks the NWO forhis current fellowship in Utrecht. JJE also thanks StephenSmartt, Norbert Langer and Onno Pols for useful discussion.
REFERENCES
Asplund M., Grevesse N., Sauval A.J., 2005, ASPC, 336,25Belczynski K., Kalogera V., Bulik T., 2002, ApJ, 572, 407BBrandt N., Podsiadlowski P., 1995, MNRAS, 274, 461BCameron A.G.W., Mock M., 1967, Natur, 215, 464CCantiello M., Yoon S.-C., Langer N., Livio M., 2007, A&A,465L, 29CCappellaro E., Turatto M., Tsvetkov D.Yu., Bartunov O.S.,Pollas C., Evans R., Hamuy M., 1997, A&A, 322, 431CCappellaro E., Evans R., Turatto M., 1999, A&A, 351,459CChiosi C., Vallenari A., Bressan A., Deng L., Ortolani S.,1995, A&A, 293, 710Crowther P.A., Dessart L., Hillier D.J., Abbott J.B., Fuller-ton A.W., 2002, A&A, 392, 653Crowther P.A., 2007, ARA&A, 45, 177Cde Donder E., Vanbeveren D., 2003, NewA, 8, 817Dde Donder E., Vanbeveren D., 2004, NewAR, 48, 861Deupree R.G., 2001, ApJ, 552, 268DDray L.M., Tout C.A., 2003, MNRAS, 341, 299Dray L.M., Tout C.A., 2007, MNRAS, 376, 61DEggleton P.P., 1971, MNRAS, 151, 351Eggleton P.P., 1983, ApJ, 268, 368EDearborn D.S.P., Lattanzio J.C., Eggleton P.P., 2006, ApJ,639, 405D c (cid:13) , 1–11 he effect of massive binaries on stellar populations and supernova progenitors Eldridge J.J., Tout C.A., 2004a, MNRAS, 348, 201Eldridge J.J., Tout C.A., 2004b, MNRAS, 353, 87Eldridge J.J., Vink J.S., 2006, A&A, 452, 295Eggenberger P., Meynet G., Maeder A., 2002, A&A, 386,576EJason W. Ferguson, David R. Alexander, France Al-lard, Travis Barman, Julia G. Bodnarik, Peter H.Hauschildt, Amanda Heffner-Wong, Akemi Tamanai, 2005astro-ph/0502045Hansen B.M.S., Phinney E.S., 1997, MNRAS, 291, 569HHeger A., Langer N., Woosley S.E., 2000, ApJ, 528, 368HHeger A., Langer N., 2000, ApJ, 544, 1016HHeger A., Fryer C.L., Woosley S.E., Langer N., HartmannD.H., 2003, ApJ, 591, 288Heger A., Woosley S.E., Spruit H.C., 2005, ApJ, 626, 350Hirschi R., Meynet G., Maeder A., 2004, A&A, 425, 649Hurley J.R., Pols C.A., Tout O.R., 2000, MNRAS, 315,543HHurley J.R., Tout C.A., Pols O.R. 2002, MNRAS, 329, 897Izzard R.G., Tout C.A., 2003, PASA, 20, 345Izzard R.G., Dray L.M., Karakas A.I., Lugaro M., ToutC.A., 2006, A&A, 460, 565IIzzard R.G., Ramirez-Ruiz E., Tout C.A., 2004, MNRAS,348, 1215Ide Jager C., Nieuwenhuijzen H., van der Hucht K.A., 1998,A&ASS, 72, 259Kotak R., Vink J.S., 2006, A&A, 460L, 5KKroupa P., Tout C.A., Gilmore G., 1993, MNRAS, 262, 545Kudritzki R.P., Pauldrach A., Puls J., 1987, A&A, 173, 293Langer N., Maeder A., 1995, A&A, 295, 685Livio M., Soker N., 1984, MNRAS, 208, 763LMaeder A., Meynet G., 1994, A&A, 287, 803Maeder A., Meynet G., 2001, A&A, 373, 555Meynet G., Maeder A., 1997, A&A, 321, 465Meynet G., Maeder A., 2003, A&A, 411, 543Meynet G., Maeder A., 2004, A&A, 422, 225Meynet G., Maeder A., 2005, A&A, 429, 581Meynet G., Maeder A., 2007, A&A, 464L, 11MMassey P., Olsen K.A.G., 2003, AJ, 126, 2867Massey P., 2003, ARA&A, 41, 15Nugis T., Lamers H.J.G.L.M., 2000, A&A, 360, 227Paczynski B., 1976, IAUS, 73, 75PPastorello A., 2007, Nature, in press.Petrovic J., Langer N., Yoon S.-C., Heger A., 2005, A&A,435, 247PPetrovic J., Langer N., van der Hucht K.A., 2005, A&A,435, 1013PPodsiadlowski Ph., Joss P.C., Hsu J.J.L., 1992, ApJ, 391,246Podsiadlowski P., Langer N., Poelarends A.J.T, RappaportS., Heger A., Pfahl E., 2004, ApJ, 612, 1044Podsiadlowski Ph., Morris T.S., Ivanova N., 2006, ASPC,355, 259PPols O.R., Tout C.A., Eggleton P.P., Han Z., 1995, MN-RAS, 274, 964Popov D.V., 1993, ApJ, 414, 712Prantzos N., Boissier S.,2003, A&A, 406, 259Sills A., Bailyn C.D., Edmonds P.D., Gilliland R.L., 2000,ApJ, 535, 298SSmartt S.J., Gilmore G.F., Tout C.A., Hodgkin S.T., 2002,ApJ, 565, 1089Smartt S.J., Maund J.R., Hendry M.A., Tout C.A., Gilmore G.F., Mattila S., Benn C.R., 2004, Science, 303,499Smith N., Owocki S.P., 2006, ApJ, 645L, 45STuratto M., Benetti S., Pastorello A., 2007, astro-ph/0706.1086Vanbeveren D., 2001, in
The influence of binaries on stel-lar population studies , Dordrecht: Kluwer Academic Pub-lishers, 2001, xix, 582 p. Astrophysics and space sciencelibrary (ASSL), Vol. 264. ISBN 0792371046, p.249Vanbeveren D., Van Bever J., Belkus H., 2007,astro-ph/0703796Van Dyk S.D., Li, W., Filippenko A.V., 2003, PASP, 115,1289Vzquez G.A., Leitherer C., Schaerer D., Meynet G., MaederA., 2007, ApJ, 663, 995VVink J.S., de Koter A., Lamers H.J.G.L.M., 2001, A&A,369, 574Vink J.S., de Koter A., 2005Wellstein S., Langer N., 1999, A&A, 350, 148Woosley S.E., Heger A., Weaver T.A., 2002, RvMP, 74,1015Young T.R., Smith D., Johnson T.A., 2005, ApJ, 625L, 87This paper has been typeset from a TEX/ L A TEX file preparedby the author. c (cid:13)000