arxiv:1310.7218v1 [astro-ph.sr] 27 oct 2013 · (martins et al. 2008; crowther et al. 2010). they...

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3Astronomy amp Astrophysicsmanuscript no comp˙evolmod ccopy ESO 2013October 29 2013

A comparison of evolutionary tracks for single Galactic massivestars

F Martins1 and A Palacios1

LUPM Universite Montpellier 2 CNRS Place Eugene Bataillon F-34095 Montpellier Francee-mailfabricemartins AT univ-montp2fr

ABSTRACT

Context The evolution of massive stars is not fully understood The relation between different types of evolved massive stars is notclear and the role of factors such as binarity rotation or magnetism needs to be quantifiedAims Several groups make available the results of 1-D single stellar evolution calculations in the form of evolutionary tracks andisochrones They use different stellar evolution codes for which the input physics and its implementation varies In this paper weaim at comparing the currently available evolutionary tracks for massive stars We focus on calculations aiming at reproducing theevolution of Galactic stars Our main goal is to highlight the uncertainties on the predicted evolutionary pathsMethods We compute stellar evolution models with the codes MESA and STAREVOL We compare our results with those of fourpublished grids of massive stellar evolution models (Geneva STERN Padova and FRANEC codes) We first investigate the effects ofovershooting mass loss metallicity chemical composition We subsequently focus on rotation Finally we compare the predictionsof published evolutionary models with the observed properties of a large sample of Galactic starsResults We find that all models agree well for the main sequence evolution Large differences in luminosity and temperatures appearfor the post main sequence evolution especially in the coolpart of the Hertzsprung-Russell (HR) diagram Depending onthe physicalingredients tracks of different initial masses can overlap rendering any mass estimate doubtful For masses between 7 and 20 M⊙we find that the main sequence width is slightly too narrow in the Geneva models including rotation It is (much) too wide for the(STERN) FRANEC models This conclusion is reached from the investigation of the HR diagram and from the evolution of the surfacevelocity as a function of surface gravity An overshooting parameterα between 01 and 02 in models with rotation is preferred toreproduce the main sequence width Determinations of surface abundances of carbon and nitrogen are partly inconsistent and cannotbe used at present to discriminate between the predictions of published tracks For stars with initial masses larger than about 60M⊙ the FRANEC models with rotation can reproduce the observations of luminous O supergiants and WNh stars while the Genevamodels remain too hot

Key words Stars massive - Stars evolution

1 Introduction

Massive stars (Mgt 8 M⊙) are born as O and B stars on the zeroage main sequence (hereafter ZAMS) After a few million yearsthey evolve off the main sequence to become either (red) su-pergiants or Wolf-Rayet stars depending on their initial massThey may go through a phase during which they are seen asLuminous Blue Variables (LBV) blue or yellow supergiantsBeyond this qualitative scenario little is known about theevolu-tion of massive stars In particular the detailed relations betweenstars of different types is poorly constrained Very massive andluminous H-rich WN stars are probably core-H burning objects(Martins et al 2008 Crowther et al 2010) They may becomeLBVs when they evolve towards the red part of the Hertzsprung-Russell (HR) diagram (Smith amp Conti 2008) At lower masses(Msim50 M⊙) Crowther amp Bohannan (1997) and Martins et al(2007) provided relations between mid-O supergiants and sev-eral types of WN and WC stars These evolutionary sequencesremain partial and do not exist for the entire upper HR diagram

Stellar evolutionary models have been developed to explainand predict the physical properties of massive stars The effec-tive temperature and luminosity they predict are used to buildtracks followed by stars of different initial masses in the HR di-agram These predictions are confronted to observations totest

Send offprint requests to F Martins

the input physics Hamann et al (2006) studied the WN stars inthe Galaxy and concluded that the models of Meynet amp Maeder(2003) are able to account for the global properties of WN starsHowever some quantitative problems exist especially regard-ing the number of early and late WN stars Similarly the ra-tio of WC to WN stars provides a test of evolutionary modelsAccording to the classical scenario WC stars represent a moreadvanced state of evolution than WN stars simply because ofmass loss as evolution proceeds mass is removed by stellarwinds and deeper layers are unveiled It takes more time to reachdeeper layers and consequently these layers bear the imprintof the nucleosynthesis occurring at later stages of evolutionNeugent amp Massey (2011) showed that the ratio WCWN is cor-rectly reproduced by the models of Meynet amp Maeder (2003) atlow metallicity This is not the case at higher metallicity (see alsoNeugent et al 2012) where evolutionary models predict too fewWC stars Hunter et al (2008) studied the nitrogen content of Bstars in the Large Magellanic Cloud and found that single stan-dard stellar evolutionary models could account for the propertiesof roughly two thirds of the sample the remaining objects beingunexplained by current single star tracks with rotation The re-sults were confirmed by the calculations of Brott et al (2011b)Nonetheless Maeder et al (2009) cautioned that many parame-ters could affect the surface abundances and that they needed tobe considered

1

Martins amp Palacios Comparing evolutionary tracks for Galactic massive stars

Indeed evolutionary calculations rely on various prescrip-tions to describe the physical processes driving the evolutionand these prescriptions may vary from code to code The mostimportant ingredientsprocesses to be considered are convec-tion and its related properties (such as overshooting) mass loss(Chiosi amp Maeder 1986) chemical composition (and the rela-tive abundance of the various species considered in the mod-els) and of course initial mass Rotation is another key ingredi-ent since it affects the internal structure the physical properties(temperature luminosity) the surface chemical appearance andthe lifetimes of stars (Maeder amp Meynet 2000b) Several pre-scriptions usually exist to treat a given physical process in evo-lutionary codes As a consequence the outputs depend on theinput physics Since evolutionary calculations are a crucial toolto link the observed properties of stars to their physical state andevolution it is important to understand the limitations and un-certainties associated with evolutionary models

In this paper we present a comparison of various predictionsof the evolution of massive stars computed with different codesOur goal is to highlight the uncertainties in the outputs of evolu-tionary calculations especially concerning the HR diagram Wefocus on calculations for Galactic stars In Sect 2 we describethe different codes and models we have used in our comparisonsWe then present a study of the uncertainties associated withtheassumptions in the input physics (Sect 3) In the same sectionwe also compare the evolutionary tracks predicted by the differ-ent codes In Sect 4 we confront the predictions of publishedgrids of models to the observed properties of massive stars inthe Galaxy We highlight the limitations of each grid Finallywe summarize our main conclusions in Sect 5

2 Stellar evolution models

To achieve our goal of comparing stellar evolution tracks ofmassive stars we have used four databases of stellar evolutiontracks presented in Bertelli et al (2009) Brott et al (2011a)Ekstrom et al (2012) Chieffi amp Limongi (2013) and we com-puted models using the STAREVOL code (Decressin et al2009) and the MESA code (Paxton et al 2011 2013) We recallhere the main ingredients and physical parameters used in eachcase since they may differ largely and these differences appearto affect the evolutionary tracks Table 1 summarizes the maininputs for each code

21 STERN stellar evolution code (Brott et al 2011a)

We first make use of the grid of stellar evolution modelspublished by Brott et al (2011a) The computations have beenperformed with the code fully described in Heger et al (2000)In the following we will refer to this code as the STERN code

Solar reference chemical compositionBrott et al (2011a) adopt tailored reference chemical abun-dances for their models of LMC SMC and Galactic massivestars based on the solar abundances of Asplund et al (2005)with a modification of the C N O Mg Si and Fe abundancesThis results in unusual chemical mixtures described in Tables 1and 2 of their paper Their adopted values for the metal massfraction Z is 00088 00047 and 00021 for the Galaxy theLMC and the SMC respectively The Galactic metallicity isabout half the value used by the other codes (see below) Theinitial helium content is Y=0264 The OPAL radiative opacitiesof Iglesias amp Rogers (1996) are used in the calculations

ConvectionThey use the Ledoux criterion1 to determine the extension ofthe convective regions and model convection according to themixing length theory withαMLT = 15 The mixing length is thelength over which a displaced element conserves its propertiesand the mixing length parameterαMLT is the ratio of themixing length to the local pressure scale heightHP The zoneswhich are stable according to the Ledoux criterion but unstableaccording to the Schwarzschild criterion are considered tobesemi-convective Semi-convection is included as in Langeret al(1983) withαsc = 1 Finally Brott et al calibrate an additionalclassical overshooting parameter to adjust the evolution of therotation velocity as a function of surface gravity of a 16 M⊙model at LMC metallicity (see Sect 41) This parameteris applied to their entire grid and results in an extension ofthe convective cores beyond the limit defined by the Ledouxcriterion bydover = 0335Hp

Mass lossMass loss is implemented following a combination of prescrip-tions or recipes that are specific for each evolutionary phase ofmassive stars Brott et al (2011a) use Vink et al (2000 2001)(MV ) for winds of early O and B-type stars A switch toM byNieuwenhuijzen amp de Jager (1990) (MNdJ) is operated at Tefflt22000 K (bi-stability jump temperature) wheneverMV lt MNdJ Another switch is operated for the Wolf-Rayet phase andMby Hamann et al (1995) is adopted as soon asYs ge 07 (Ys isthe surface helium abundance) For intermediate values ofYsan interpolation between the Vink et al mass loss rates and theWolf-Rayet mass loss rates reduced by a factor 10 is performedBrott et al use a metallicity scaling of the mass loss by afactor (Fesur fFe⊙)085 based on the solar iron abundance fromGrevesse et al (1996) (ǫ(Fe) = 750) which is higher than thatof their models (ǫ(Fe) = 740) For the rotating models they

also apply the correction factor(

11minusVVcrit

)043to the mass loss

rate (Vcrit is the critical velocity)

Rotation and rotation-induced mixingThe effects of the centrifugal acceleration on the stellar struc-ture equations is considered according to Kippenhahn et al(1970) The transport of angular momentum and chemicalspecies is treated in a diffusive way following the formalismby Endal amp Sofia (1978) as described in Heger et al (2000)Eddington-Sweet circulation dynamical and secular shear andaxisymmetric (GSF) instabilities contribute to the transport ofboth angular momentum and chemical species The formalismthey use relies on two efficiency factors (free parameters) fc = 00228 which reduces the contribution to the rotation-induced hydrodynamical instabilities in the total diffusion coeffi-cient andfmicro = 01 which regulates the inhibiting effect of chem-ical gradients on the rotational mixing The values of thesepa-rameters are calibrated on observations They are fully describedin Eq 53 and 54 of Heger et al (2000)In addition to that the action of magnetic fields on thetransport of angular momentumonly is included through thehighly debated Tayler-Spruit dynamo (Spruit 2002) followingPetrovic et al (2005)

1 For a plasma described by a general equation of state theLedouxstability criterion is given bynablarad lt nablaad +

φ

δnablamicro with δ = minus lnρ

lnT andφ =lnρlnmicro micro being the mean molecular weight When there are no chemicalgradientsnablamicro = 0 and the Schwarzschild stability criterion is recoverednablarad lt nablaad

2


Table 1 Main ingredients of the evolutionary models

STERN1 Geneva2 FRANEC3 Padova4 MESA5 STAREVOL5

Initial metallicity (Z) 00088 00140 001345 00170 0014 00134Mixing length parameter (lHP) 15 16 10dagger 23 168 20 163Overshoot parameter (dHP) 0335 01 02 sim 05 f= 00001002Dagger 000102Rotation 0 - 550 km sminus1 ΩΩcrit = 04 300 km sminus1 0 0 200 km sminus1 0 220 km sminus1

Magnetic field Spruit-Taylor no no no no noSolar mixture AGS056 AGS05 AGSS097 GN938 GN93 AGSS09

with CNO with Ne enhancedMgSiFe modified (Cunha et al 2006)

NotesReferences 1 - Brott et al (2011a) 2 - Ekstrom et al (2012) 3 - Chieffi amp Limongi (2013) 4 - Bertelli et al (2009) 5 - this workHeavy elements solar mixture 6 - Asplund et al (2005) 7 - Asplund et al (2009) 8 - Grevesse et al (1993)dagger - For stars with initial masslt 40 M⊙ the mixing length parameter islHP = 16 For more massive stars it is defined with respect to the localdensity scale height andlHρ = 10Dagger - In MESA the overshooting is implemented as a decreasing exponential with parameterf (see text)

22 Geneva stellar evolution code (Ekstrom et al 2012)

We also use the grid published by Ekstrom et al (2012)and summarize briefly the main physical ingredients used tocompute the models provided in that paper

Solar reference chemical compositionIt is based on Asplund et al (2005) with a modification of theNe abundance according to Cunha et al (2006) Their adoptedmetals mass fraction isZ = 0014 resulting from a solarcalibration The initial helium content is Y=0266 The OPALradiative opacities of Iglesias amp Rogers (1996) are used

ConvectionThey use the Schwarzschild criterion to define the convectiveregions Convection is modelled following the mixing-lengthformalism with αMLT = 16 For models with Mgt 40 M⊙the mixing length parameter is computed using the densityheight scale instead of the pressure height scale withαMLT = 1following Maeder (1987)They also include classical overshoot at the convective coreedge with αover = 01 This parameter corresponds to theratio between the extension of the convective core beyond thevalue resulting from the Schwarzschild criterion to the localpressure scale heightdover = 01HP is calibrated to reproducethe width of the main sequence in the mass range 135 - 9 M⊙Semi-convection is not modelled

Mass lossMass loss is implemented following a combination of prescrip-tions or recipes chosen to best represent mass loss of massivestars along their evolution Ekstrom et al use the stellarwindsprescriptions from Vink et al (2000 2001) when log(Teff) gt 39A switch to de Jager et al (1988) mass loss formulae is operatedwhen the models reach log(Teff) lt 39 and then to Crowther(2000) as they evolve into the red supergiant phase For theWolf-Rayet phases (log(Teff) gt 4 andXS le 04 - XS being thesurface hydrogen abundance) they use Nugis amp Lamers (2000)or Grafener amp Hamann (2008)Mass loss rates are scaled according to Maeder amp Meynet(2000a) for the rotating modelsFor the most massive models (Mgt 15M⊙) in order to accountfor the supra-Eddington mass loss during the red supergiantphase they multiplyM by a factor of 3 whenever the luminosityin the envelope becomes 5 times larger than the Eddington

luminosity

Rotation and rotation-induced mixingThe modification of the stellar structure equations by thecentrifugal acceleration is taken into account followingMeynet amp Maeder (1997) The transport of angular momentumand of nuclides due to meridional circulation and turbulentshearis self-consistently included following the formalism by Zahn(1992) Maeder amp Zahn (1998) The prescriptions used for theturbulent diffusion coefficients are from Zahn (1992) for the hor-izontal component and from Maeder (1997) for the vertical com-ponentConvective regions are assumed to rotate as solid-bodiesNo additional transport due to the presence of magnetic fields isincluded

23 FRANEC stellar evolution code (Chieffi amp Limongi 2013)

Chieffi amp Limongi (2013) reported on the inclusion of rotationin the FRANEC code and have computed a grid of massivestellar evolution models with and without rotation

Solar reference chemical compositionThe models are computed using the heavy element solar mixturefrom Asplund et al (2009) The initial global metallicity andhelium content areZ = 001345 andY = 0265 The associatedopacities are from OPAL for radiative opacities

ConvectionConvection is treated following the MLT formalism and convec-tive limits are defined using the Schwarzschild criterion exceptfor the H burning shell appearing at the beginning of the coreHe burning phase for which the Ledoux criterion is applied(see Limongi et al 2003) The mixing length parameter adoptedis not given but should be ofα = ΛHp = 23 according toStraniero et al (1997) Classical overshooting is included witha value ofdover= 02Hp

Mass lossThey use the mass loss prescriptions of Vink et al (20002001) for the blue supergiant phase switching to de Jager etal(1988) when log(Teff) lt 39 and Nugis amp Lamers (2000) forthe WR phase The mass loss during the red supergiant phaseis enhanced according to van Loon et al (2005) Mass loss

3


of rotating models is also enhanced as in the STERN codefollowing Heger et al (2000)

Rotation and rotation-induced mixingThe modification of the stellar structure equations by thecentrifugal acceleration and the transport of angular momentumand of nuclides are the same as in the Geneva codeThe impact of the mean molecular weight gradients on thetransport of both angular momentum and nuclides is regulatedby the use of a free parameterfmicro defined bynablaadopted

micro = fmicro times nablamicroChieffi amp Limongi adopted fmicro=003 This value is calibratedto ensure that at solar metallicity the stars in the mass range15-20 M⊙ settling on the main sequence with an equatorialvelocity of 300 km sminus1 will increase their surface nitrogenabundance by a factor ofasymp 3 by the time they reach the TAMSConvective regions are assumed to rotate as solid-bodiesNo additional transport due to the presence of magnetic fields isincluded

24 Padova stellar evolutionary code (Bertelli et al 2009)

The evolutionary models for massive stars computed with thePadova code are described in Bertelli et al (2009) Additionalinformation regarding the code can be found in Bono et al(2000) and Pietrinferni et al (2004 2006) Rotation is notincluded in the grid of Bertelli et al (2009)

Solar reference chemical compositionThe metals mass fraction adopted for their solar-metallicitymodels is Z = 0017 We used the models with Y=026(several values are available) The OPAL radiative opacitiesof Iglesias amp Rogers (1996) are used A global scaling of therelative element mass fractions is made compared to the mixtureof Grevesse et al (1993) on which the OPAL tables are basedFor very high temperatures (logT gt 87) the opacities ofWeiss et al (1990) are used

ConvectionThey adopt the formalism of the mixing length theory withαMLT = 168 calibrated on solar models Stability is set by theSchwarzschild criterion Overshooting is taken into accountwith a free parameter corresponding to the ratio of the trueextent of the convection core to the convection core radiusdefined by the Schwarzschild criterion The notable differencein the Padova code is that this parameter expresses the extent ofovershootingacross (and not above) the border of the convectivecore (as set by the Schwarzschild criterion) A value of 05 forthe overshooting parameter is adopted Overshooting belowtheconvective envelopes is also accounted for with a parameterequal to 07Semi-convection is considered to be negligible for massivestarsmodels

Mass lossThe mass loss prescriptions of de Jager et al (1988) are usedfor all phases of evolution A metallicity scaling of radiativelydriven winds is taken into account according to Kudritzki etal(1989) (ieM prop Z05)

25 STAREVOL (Decressin et al 2009)

A detailed description of the STAREVOL code can be found inSiess et al (2000) Siess (2006) Decressin et al (2009) Themodels computed for the present study were obtained using theSTAREVOL v330 which includes a number of updates withrespect to previous descriptions of the code2 For the presentstudy we have adopted a setup close to that used in the Genevagrid

Solar reference chemical compositionWe use Asplund et al (2009) with OPAL tabulated opaci-ties modified accordingly At low temperature we use theFerguson et al (2005) opacities computed for the Asplund etal2009 solar composition The adopted solar metallicity is thusZ = 00134 The initial helium content isY = 0277 No furthermodification of the abundances is made When computingmodels for non-solar metallicities a simple proportionality isapplied

ConvectionThe Schwarzschild criterion is used to define the convectiveregions Convection is modelled following the mixing-lengthformalism as described in Kippenhahn amp Weigert (1990) Themixing length parameterαMLT = 164 is calibrated for the solarmodel In some models we have included classical overshootatthe edge of convective regions withαover = 01 or 02

Mass lossFor massive stars (Mgt 7 M⊙) with log(Teff) gt 39 we apply theprescriptions from Vink et al (2000 2001) which we changeto (a) de Jager et al (1988) when the models evolve to the redand have their temperature drop below log(Teff) = 39 and thento Crowther (2000) as they evolve into the red supergiant phase(b) to Reimers (1975) for models in the mass range 7-12 M⊙

when they evolve off the main sequence (c) to Nugis amp Lamers(2000) for those models that experience a Wolf-Rayet phase(eg with log(Teff) gt 4 andXS le 04)The mass loss is down-scaled by a factor (ZZ⊙)05 for non-solarmetallicity models We have included the correction to the massloss of rotating massive stars according to Maeder amp Meynet(2001)

Rotation and rotation-induced mixingThe modification of the stellar structure equations due tocentrifugal acceleration in rotating models is taken into accountfollowing Kippenhahn et al (1970) The expression for theeffective temperature following this formalism is implementedas described in Appendix A of Meynet amp Maeder (1997) Inaddition to this the transport of angular momentum and ofnuclides is as in the Geneva and FRANEC codes The prescrip-tions used for the turbulent diffusion coefficients are from Zahn(1992) for the horizontal component and from Talon amp Zahn(1997) for the vertical componentConvective regions are assumed to rotate as solid-bodiesNo additional transport due to the presence of magnetic fields isincluded

2 The updates concern opacities and reference solar abundances aswell as mass-loss prescriptions

4


26 MESA code (Paxton et al 2011)

MESA3 is a public distribution of modules for experiments instellar astrophysics The computation of evolutionary modelsis possible with the module ldquostarrdquo of the distribution Anexhaustive description of the code is available in Paxton etal(2011) and Paxton et al (2013) We have computed dedicatedevolutionary models for 7 9 15 20 25 40 and 60 M⊙ starsBoth non-rotating and rotating (initial equatorial velocity of 200km sminus1) models have been calculated

Solar reference chemical compositionWe have adopted a value ofY = 026 andZ = 0014 in our cal-culations MESA uses OPAL opacities from Iglesias amp Rogers(1996) The relative mass fraction of metals in the OPALcomposition is based on the solar composition of Grevesse etal(1993) Grevesse amp Sauval (1998) or Asplund et al (2009) Theuser can select any of these compositions A global scalingwith Z is made when non-solar metallicity models are computed

ConvectionThe standard mixing length formalism as defined by Cox (1968)is used to treat convection as a diffusive process in MESA Theonset of convection is ruled by the Schwarzschild criterion Inour calculations we usedαMLT = 20 Although it is availablewe did not include semi-convection in our models Convectiveovershooting is treated as a diffusive process following theformalism of Herwig (2000) The overshooting diffusion coef-ficient (Dov) is related to the MLT diffusion coefficient (Dconv)

throughDov = Dconveminus2z

f Hp where f is a free parameter Unlessstated otherwise we have adoptedf = 001 in our calculations

Mass lossA mixture of prescriptions is used to account for mass lossin the various phases of evolution The recipe of Vink et al(2001) is used forTeff gt 10000K andX(H) gt 04 For thesame temperature range but lower H content (X(H) lt 04) themass loss rates of Nugis amp Lamers (2000) are implemented ForTeff lt 10000K the values of de Jager et al (1988) are used Itis possible to scale these prescriptions by a constant factor Forhot star it is a way to take the metallicity dependence of massloss rates into account (see Sect 31)

Rotation and rotation-induced mixingThe geometrical effects of rotation are implemented followingthe formalism of Kippenhahn et al (1970) The transport of an-gular momentum and chemical species through meridional cir-culation and hydrodynamical instabilities turbulence is treatedas a purely diffusive process following the Endal amp Sofia (1978)formalism as in the STERN code The efficiency factor (see Sect21) have the following valuesfc = 130 similar to the theoret-ical value of Chaboyer amp Zahn (1992) andfmicro = 01 We did notinclude magnetism in our computation (although the formalismof Spruit (2002) is implemented in MESA and can be switchedon)

3 Code predictions and uncertainties

In this section we perform comparisons between the resultsofcalculations performed with the six codes described aboveWefocus on the evolutionary tracks in the Hertzsprung-Russell di-agram We first compare standard tracks (ie without rotation)

3 httpmesasourceforgenet

Fig 1 Effects of opacities (top) overshooting (middle) and massloss (bottom) on 20 M⊙ evolutionary tracks The computationshave been performed with the code MESA In the upper rightpanel different solar heavy elements mixtures are used forthe same initial metal fraction (Z = 0014) GN93 refers toGrevesse et al (1993) GS98 to Grevesse amp Sauval (1998) andAGS09 to Asplund et al (2009)

in order to test the various implementations of the basic physicsWe subsequently investigate the effects of rotation

31 Effects of physical ingredients on evolutionary tracks

In Fig 1 we illustrate the effect of modifying the opacities theovershooting and the mass loss on the evolution of aM = 20M⊙ model from the Zero Age Main Sequence (ZAMS) to the

5


Terminal Age Helium Main Sequence (TAHeMS) We have usedboth the MESA and STAREVOL codes to compute the evolu-tionary sequences We mainly discuss the MESA models in thissection (the results obtained using STAREVOL are very simi-lar)The top panel illustrates the effect of different heavy ele-ments solar mixtures on the opacities The solar compositionof Grevesse et al (1993) and Asplund et al (2009) are relativelysimilar the C N O and Ne abundances do not differ by morethansim005 dex The Grevesse amp Sauval (1998) abundances areon average 010-015 dex larger In Fig 1 we see that the maineffects of different solar composition on the opacities is reflectedin the post main sequence evolution On the main sequence theluminosity variations are negligible while beyond the terminalage main sequence (TAMS) the differences vary between 001and 002 dex depending on the temperature

The middle panel of Fig 1 shows the effect of overshootingThe calculations have been performed for a diffusive overshoot-ing with a parameterf equal to 0 001 and 002 as indicatedon the figure Overshooting is included only in the convectiveregions related to H burning Qualitatively the effect of an in-creasing overshooting is the lengthening of the main sequencephase As a result of the larger extension of the convective corea larger amount of hydrogen is available for helium production inthe core Quantitatively the main sequence duration is 860 Myrfor f=001 and 906 Myr forf=002 This corresponds to anincrease of 9 As a consequence of the longer main sequenceduration for larger overshooting the star exits the core H burn-ing phase at a lower effective temperature (by 2500 K) and at ahigher luminosity (increase of 005 dex) whenf increases from001 to 002 If the overshooting parameter is not constraineda degeneracy in the evolutionary status of a star located closeto the end of the main sequence can appear Depending on thetracks used and the amount of overshooting it can be identifiedas a core H burning object close to end of the main sequenceor as a post main sequence object Beyond the main sequencemodels with stronger overshooting evolve similarly but at higherluminosities

The bottom panel of Fig 1 illustrates the effect of mass lossrates on evolutionary paths In addition to the track with the stan-dard mass loss rate two additional tracks with mass loss ratesglobally scaled by a factor 033 and 010 are shown As expectedthe main sequence is barely affected The reason is the low val-ues of the mass loss rates during this phase for the initial massof 20 M⊙ considered here For the standard track (dashed blueline) the mass at the end of the main sequence is 1966 M⊙ cor-responding to a loss of only 17 of the initial mass over 860Myr The mass drops to 1826 M⊙ in the next Myr (time to reachthe bottom of the red giant branch) On average the mass lossrate is thus 35 times larger in the post-main sequence phase com-pared to the main sequence To first order the effect of mass losscan be understood as a simple shift to lower luminosity Sincethe luminosity is directly proportional to some power-law of themass (the exponent being around 10-20 depending on the masseg Kippenhahn amp Weigert 1990) a reduction of the mass im-mediately translates into a reduced luminosity This is what weobserve in Fig 1 Quantitatively a reduction by a factor close to3 (10) in the mass loss rates corresponds to a maximum increasein luminosity of sim001 (003) dex The changes are larger formore massive stars since mass loss rates are also higher

The prescriptions of mass loss rates for massive stars suf-fer from several uncertainties The presence of clumping inhotstars winds has lead to a reduction of the mass loss rates by afactor of roughly 3 (Puls et al 2008) But this value is stillde-

Fig 2 Effect of metallicity on a 20 M⊙model computed withMESA

bated reduction up to a factor of 10 being sometimes neces-sary to reproduce observational diagnostics (Bouret et al2005Fullerton et al 2006) For the cool part of the evolution of amas-sive star the very nature of the mass loss mechanism is stillnotclear Mauron amp Josselin (2011) have shown that the mass lossrates of de Jager et al (1988) are still valid But for a givenlumi-nosity the scatter in mass loss rates is large (up to a factor10)The uncertainties in the mass loss rates thus translate intouncer-tainties of the order of 002 dex in the luminosity of evolutionarytracks beyond the TAMS

Figure 2 highlights the well documented effects of metal-licity (eg Meynet amp Maeder 2003) We have computed mod-els for three different metallicities the solar value (Z = 0014)and the extreme values encountered in the Galaxy according tothe study of HII regions by Balser et al (2011) ndashZ = 15 Z⊙andZ = 125 Z⊙ No scaling of the mass loss rates was ap-plied in order to extract the effect of metallicity on the internalstructure and evolution A lower metal content correspondsto alower opacity which in turn translates into a higher luminosityOn average a reduction of the metal content by a factor of twotranslates into an increase in luminosity by 0005-0010 dex onthe main sequence and by 003-005 dex beyond

Assuming a typical uncertainty on the luminosity ofplusmn 002dex (opacity effect)plusmn004 dex (overshooting effect)plusmn001 dex(mass loss effect) plusmn003 dex (metallicity effect) and simplyadding quadratically the errors we obtain a global uncertaintyof aboutplusmn005 dex on the luminosity of a MESA track Thevalues we adopted are typical of the uncertainties at the endofthe main sequence and around Teff = 10000 K An uncertaintyof 005 dex on the luminosity is equivalent to an uncertaintyofabout 6 on the distance of the star On the main sequence theuncertainty on the luminosity is lower thanplusmn002 dex

We have computed a second set of these models using theSTAREVOL code and we also find that the choice of the over-shooting and of the metallicity are the ones affecting the most theluminosity The global uncertainty on the 20 M⊙ track computedwith STAREVOL is ofplusmn 006 dex around Teff = 10000 K of

6


the same order than that found for MESA models In Fig 3 wedisplay the envelope corresponding to the global intrinsicerrorfor the MESA and STAREVOL models of 20 M⊙ For both setsof models the shape of the envelope is similar The uncertaintyis maximum at temperatures around 10000 K (in the core Heburning phase see Table 2) The uncertainty on the luminosityat a given effective temperature is not symmetrical with respectto the reference track shown in dashed line These envelopesillustrate the intrinsic uncertainties of a given evolutionary track

We have performed the same type of calculations and com-parisons on aM = 7 M⊙ model The effect of mass loss is neg-ligible (changes in luminosity smaller then 001 dex) Differentchemical mixtures and their effect on opacities translates to un-certaintieslt 005 dex on the luminosity They are roughly simi-lar to the effects seen in theM = 20 M⊙ star A change in metal-licity from solar to 125 solar corresponds to an increase in lu-minosity by 01 dex Hotter temperatures are also obtainedTheeffect is larger than in the M=20 M⊙ model Finally the largesteffect on the evolutionary track of theM = 7 M⊙ model is due tochanges in the overshooting parameter Variations in luminosityby 010-015 dex are observed beyond the main sequence Theeffect of overshooting dominates the uncertainty on theM = 7M⊙ evolutionary track The global uncertainty on the luminosityof the M = 7M⊙ model amounts to 02 dex which equivalent toan error of 30 on the distance

32 Comparison between codes

In this section we make direct comparisons between the sixcodes presented in Sect 2 We focus on theM = 20 M⊙ trackcorresponding to a late O star on the main sequence Table 2gathers the effective temperature luminosity and age at four dif-ferent evolutionary phases for the six types of models The40M⊙ track is also briefly presented at the end of this section

Fig 4 (left) shows the evolutionary tracks of classical (noro-tation included) 20 M⊙ model at solar metallicity On the mainsequence there is an overall good agreement between the outputsof the different codes The STERN track is overluminous andbluer than the others which is expected from its lower metallic-ity (Z = 00088 vsZ = 0014minus 0017 for all the other tracks)The TAMS of the Geneva STAREVOL and MESA models islocated in the same region of the HR diagram as can also beverified from Table 2 However the age at the TAMS is quitedifferent for the Geneva Model which is younger byasymp 07 Myrcompared to the STAREVOL and MESA models A differencein age at the TAMS with no associated difference in Teff nor lu-minosity may indicate differences in the nuclear physicsThe characteristic hook at the end of the main sequence occurs atlower temperatures for the FRANEC Padova and STERN mod-els The larger amount of overshooting hence the size of theH core is responsible for these differences (see Table 1) FromTable 2 we also see that FRANEC and Padova models reach theTAMS later (bysim05 Myr) compared to MESA and STAREVOLmodels which is consistent with having a larger reservoir of fuelto be consumed during the hydrogen core burning phase On theother hand and surprisingly the STERN model is even youngerthan all the models except the Geneva one when it reaches theTAMS The lower metal content might lead to a higher core tem-perature and consequently to a faster hydrogen burning via theCNO reactions At the end of the main sequence an age spreadof 13 Myr (15) is observed between the six codes

Beyond the main sequence the differences between the vari-ous codes are larger The GenevaStarevolMESA tracks are still

Fig 5 Effective temperature as a function of central helium massfraction for aM = 20 M⊙ model computed with the six codesconsidered in this study

rather similar in the post main sequence evolution with differ-ences in luminosity usually lower than 005 dex The red super-giant part of the STAREVOL model is located redwards com-pared to that of the MESA and Geneva models The reason forthis behaviour is attributed to a combination of differences in themixing length parameter the opacities and the equation of stateThe Padova and FRANEC models reach the TAMS with largerluminosities due to the stronger overshoot and see their lumi-nosity subsequently decrease to a large amount (03 dex in thecase of the FRANEC model) This behaviour is also observed inthe STERN model The Padova STERN and FRANEC modelsreach logL

L⊙= 487 477 and 471 respectively at the bottom of

the red giant branch compared to the logLL⊙= 500ndash505 reached

by the MESA STAREVOL and Geneva models The decreaseof the total luminosity during the Hertzsprung gap results froma subtle balance between the core contraction the energy gen-eration by the H shell surrounding the core the mean molecularweight gradient profile and the opacity of the surface layers Forthe Padova STERN and FRANEC models the thermal instabil-ity of the envelope (triggered by the above conditions) seems tobe stronger leading to a larger overall reduction of the luminos-ity Given that the detailed structures associated with these tracksare not all available it is difficult to be more precise concerningthe different paths followed by the tracks presented hereThe beginning of the helium core burning phase (ZAHeMS) isdefined as the time at which the central helium mass fractionstarts to decrease from its maximum value reached after the cen-tral hydrogen burning phase The ZAHeMS starts at Teff sim 25000K for most models except the Padova (Teff = 15657 K) and theSTERN (Teff = 5388 K) ones We attribute the very differenttemperature of the STERN models to the large overshooting andthe inclusion of magnetism The temperatures at the TAHeMSdiffer by 350 K at most This is a large difference (10) af-fecting the interpretation of the properties of red supergiants(eg Levesque et al 2005 Davies et al 2013) The tracks fromFRANEC and Padova tracks present a blue hook similar to whatis observed during core helium burning for lower masses

7


Fig 3 Region occupied by the evolutionary tracks of 20 M⊙ models computed with MESA (left) and STAREVOL (right) withdifferent opacities metallicities mass loss rates and overshooting parameters The dashed red line shows the track of thestandardmodel with the parameters as indicated on the figure (the overshooting parameter - ov - is not defined in the same way in bothcodes hence the different values) The shaded envelope defines a rough global intrinsic uncertainty on 20 M⊙models

Fig 4 Evolutionary tracks forM = 20 M⊙ (left) andM = 40 M⊙ (right) without rotation For theM = 40 M⊙ case no Padovatrack exists

The large differences in the post main sequence evolution canalso be seen in Fig 5 where we show the evolution of effectivetemperature as a function of core helium mass fraction (YC) TheMESA and STAREVOL tracks spend most of the helium burn-ing phase at temperatures larger than 10000 K 40 of the he-lium burning phase takes place at hot temperatures in the Genevamodel The subsequent evolution takes place mainly at low Teff In the FRANEC model almost all the helium burning is donein the cool part of the HR diagram Finally the Padova model

features a blue loop so that helium burning is first done at lowTeff before finishing at Teff gt 10000 K

To summarize the evolution of theM = 20 M⊙ modelbecomes more uncertain as temperature decreases (ie as thestar evolves) with a wider spread in luminosity in the HRdiagram This type of differences also exists for lower massstars not analyzed here4 and reveals the uncertainty of theHe-burning phases understanding and modelling The lengthof

4 See for instance the results presented at the work-shop rdquo The Giant Branchesrdquo held in Leiden in May 2009 -

8


Fig 6 Envelopes of evolutionary paths forM =7 9 15 20 2540 and 60 M⊙ taking into account the predictions of the codesstudied in this paper Rotation is not included The envelopes forthe 7 and 9 M⊙ models do not include FRANEC since no tracksis available for these masses The Padova models do not existfor 40 and 60 M⊙ For the 40 and 60 M⊙ tracks the envelopeencompasses only the redward evolution (the Wolf-Rayet phasesare not included)

the main sequence depends on the treatment of overshootingasillustrated in Sect 31 Differences in luminosity up to 03 dex(a factor of 2) and in temperatures up to 10 are observed inthe coolest phases of evolution

Fig 4 (right) shows the evolutionary tracks for aM = 40M⊙ star Padova models for such a mass are not available (thegrid of Bertelli et al 2009 stops atM = 20 M⊙) The GenevaMESA and STAREVOL models are very similar during the Hand He burning phases The FRANEC track has the same pathas the GenevaMESASTAREVOL tracks on the main sequenceBeyond that its luminosity is about 005 to 008 dex lower untilit reaches the red part of the HR diagram where it drops byanother 012 dex before rising again by 025 dex and startingits blueward evolution towards the Wolf-Rayet phase Thebehaviour is very different from the other models since once inthe red part of the HR diagram the luminosityincreases insteadof decreasing This behaviour is tentatively attributed to the sizeof the H-rich envelope that develops in the FRANEC modelwhen the luminosity drops This envelope reaches 15 M⊙ atthe lowest effective temperature For comparison in the MESAmodel it represents at most about 10 M⊙ The STERN track ismore luminous than all other tracks on the main sequence Itdoes not show the characteristic hook revealing the end of theconvective core H burning phase This feature is very peculiar inthe STERN track and is attributed to the combination of the verylarge overshooting parameter and the inclusion of magnetism

httpwwwlorentzcenternllcweb2009324programphp3wsid=324

Fig 7 Uncertainty on the location of the evolutionary path fora 20 M⊙ stellar model with (dark grey envelope) and without(light grey delimited by dashes lines) rotation We have consid-ered tracks generated by five different codes (Geneva STERNFRANEC MESA and Starevol) with similar (yet not exactly thesame) initial rotation rates

(see Sect 41)

In Fig 6 we present a summary of the comparison betweenthe available standard tracks For each mass we present en-velopes encompassing the tracks produced by the six codesThese envelopes are defined from the ZAMS to temperaturesof about 3000 K For sake of clarity the subsequent evolution(back to the blue) of the most massive objects (Mgt40 M⊙) isnot taken into account to create these envelopes Such envelopesprovide a first guess of the uncertainty on the evolutionarytracks for specific masses The main sequence phase is relativelywell defined with the major uncertainties being encounteredat the exhaustion of central hydrogen when the track makesa hook in the HR diagram For the masses shown in Fig 6there is no overlap The same conclusion remains for the postmain sequence evolution above 40 M⊙ For the lower massesthere is a degeneracy on the mass at low temperatures wherethe envelopes defining the possible location for a specific massoverlap The overlap is the largest in the coolest phases (below5000 K) Severe degeneracies in the initial masses appearFor instance a star with log(Teff)=36 and logL

L⊙=42 can be

reproduced by tracks of stars with initial masses between 7 and15 M⊙ The corresponding stellar ages are thus very different(see also Martins et al 2012b)

The global intrinsic uncertainty within models computedwith the same stellar evolution code (005 dex at most for a20 M⊙ model) is much lower than the uncertainty coming fromthe use of tracks computed with different stellar evolution codes(04 dex at maximum forM = 20 M⊙ see Fig 6) The uncer-tainty is in both cases larger beyond the TAMS and it appearsthat no clear consensus exists on the position of lower end of

9


the red giant branch nor on the temperature of the red giantbranch itself making it very difficult to draw trustful conclu-sions when comparing effective temperatures and luminositiesobtained from spectroscopic analysis with predictions of evo-lutionary tracks for yellowred supergiants Age determinationsare also very uncertain

33 Rotation

The effects of rotation on evolutionary tracks of massive starshave been described in details in the literature We refer toMaeder amp Meynet (2000b) and Langer (2012) for reviews of themain effects To summarize the effects of rotation can be broadlydescribed as follows

bull geometrical effects rapid rotation tends to flatten stars theequatorial radius becoming larger than the polar radius Asa consequence the equatorial gravity is smaller than the po-lar gravity According to the Von Zeipel theorem von Zeipel(1924) the effective temperature at the pole is largerbull effects on transport processes rotation triggers various insta-

bilities and fluid motions in the stellar interior (eg merid-ional circulation shear instability) which transport angularmomentum and chemical species between the stellar coreand the surface Consequently the surface abundances andinternal distribution of species are strongly affected by rota-tionbull effect on mass loss rate rotation modifies the surface temper-

ature and gravity as described above Since radiatively drivenwinds are directly related to these quantities (Castor et al1975) mass loss rates are also affected Maeder amp Meynet(2000a) showed that on average they increase whith the ra-tio of rotational velocity to critical rotational velocity

A direct consequence of the mixing processes on the evolution ofa star is an increase of the duration of the core hydrogen burningphase Because of mixing material above the central convectivecore is brought to the center leading to a refuelling of the corehence to an extension of the hydrogen core burning phase dura-tion The effect is qualitatively the same as overshooting This issomewhat counterbalanced by the larger luminosity of rotatingtracks caused by a different radial profile of the mean molecu-lar weight A higher luminosity translates into a shorter nuclearburning timescale But this effect is smaller than the effect ofmixing so that on average rotation leads to longer nuclearburn-ing sequences (by 10-20 for the main sequence) Rotating starsthus end their main sequence with larger luminosities than nonrotating stars

In Fig 7 we show the envelopes of tracks forM = 20M⊙ The dark grey one corresponds to rotating models the lightgrey one to non rotating models The envelopes have been de-fined from the tracks computed with STERN FRANEC theGeneva code STAREVOL and MESA We have excluded thePadova tracks since they do not include rotation The MESA andSTAREVOL tracks have been computed assuming an initial sur-face equatorial velocity of 200 and 233 km sminus1 respectively TheSTERN and FRANEC tracks have an initial velocity of 300 kmsminus1 The Geneva tracks have been computed for a ratio of initialto critical rotation of about 04 corresponding with their defini-tion of the critical velocity to about 250 km sminus1 at the equator

The widening of the main sequence described above isvisible in Fig 7 the dark grey region extends over a widerluminosity range from the ZAMS to the TAMS Beyond themain sequence the envelope of the rotating models is wider

than that of the non-rotating models immediately after theTAMS and becomes subsequently narrower at temperaturesbelowsim 20000 K In the Hertzsprung gap (5000lt Teff lt 20000K) the less luminous of the rotating tracks are overluminous byabout 01 dex in logL

L⊙compared to the non rotating tracks The

width of the envelope remains large in the coolest phases 02dex at logTeff = 37 Since we are using models with differentrotational velocities (between 200 and 300 km sminus1) the widthof the corresponding envelope is most likely affected by thisdispersion of velocities and is probably an upper limit If allmodels had been computed with the same initial velocity thespread in luminosity would be slightly smaller

The global uncertainty associated with the choice of aspecific grid of stellar evolution models is reduced beyond theZAMS when considering models including rotation This isessentially due to the fact that the decrease in luminosity afterthe TAMS when the tracks crosses the Hertzsprung gap is muchless important in the rotating models generated with STERNand FRANEC codes We may attribute it to the larger mass lossand efficient mixing

4 Comparisons with observational results

After comparing results of calculation with different codes wenow turn to comparisons to observational data From now onwe use only the publicly available grids of tracks of Brott etal(2011a) Ekstrom et al (2012) and Chieffi amp Limongi (2013)All include rotational mixing

41 The main sequence width

In Fig 8 we show a HR diagram with OB stars and theevolutionary tracks of the three public grids The tracks aretruncated so that only the part corresponding to a hydrogenmass fraction higher than 060 is shown5 This is already ahigh value for O stars corresponding to HeH=035 by numberThe stellar parameters for the comparison stars result fromdetailed analysis with atmosphere models or from calibrationsaccording to spectral types In that case the calibrationsofMartins et al (2005a) have been used to assign an effective tem-perature A bolometric correction was subsequently computedfollowing Martins amp Plez (2006) Extinction was calculatedfrom BminusV The distance to the stars (taken from parallaxeswhen available or from cluster membership) finally lead tothe magnitudes and thus with the bolometric correction totheluminosity The data concerning the comparison stars have beentaken from the following studies Massey amp Johnson (1993)McErlean et al (1999) Vrancken et al (2000) Walborn et al(2002) Lyubimkov et al (2002) Levenhagen amp Leister(2004) Martins et al (2005b) Crowther et al (2006a)Crowther et al (2006b) Melena et al (2008) Searle et al(2008) Markova amp Puls (2008) Martins et al (2008)Hunter et al (2009) Crowther et al (2010) Lefever et al(2010) Liermann et al (2010) Negueruela et al (2010)Przybilla et al (2010) Martins et al (2012c) Martins et al(2012a) and Bouret et al (2012) In the following we will as-sume that the main sequence is populated by stars of luminosityclass V and IV This is a reasonable assumption for stars below sim40 M⊙ but certainly an oversimplification for stars above 40 M⊙

5 This value is chosen so that the end of the main sequence for the40M⊙ track is visible

10


Table 2 Properties of the 20 M⊙ model computed with different codes without rotation and evaluated at four different evolutionaryphases1

Phase STERN2 Geneva FRANEC Padova MESA STAREVOLXc = 05

Teff(K) 33522 33758 33709 32538 33011 33365log L

L⊙4764 4733 4736 4747 4745 4733

age (Myr) 3598 3745 4079 4279 4073 3963

TAMS3

Teff(K) 24630 29297 28094 25972 28733 27902log L

L⊙5096 5001 5036 5045 5007 4995

age (Myr) 8173 7819 9085 9100 8598 8535

ZAHeMS

Teff(K) 5388 24014 27382 15657 23200 24259log L

L⊙4987 5025 5012 5044 5042 5015

age (Myr) 8185 7828 9088 9111 8610 8541

TAHeMS3

Teff(K) -lowast 3753 3817 3550 3702 3462log L

L⊙-lowast 5050 5206 5090 5049 5027

age (Myr) -lowast 8713 9690 9820 9530 9748

Notes 1- when the central H mass fraction is 05 (Xc = 05) at the terminal age main sequence (TAMS) at the zero age helium burning mainsequence (ZAHeMS) at the terminal age helium burning main sequence (TAHeMS)2- for the STERN grid the data available stop at the beginning of the core He burning phase3- The TAMS and TAHeMS are defined as the points where the mass fraction of H (resp He) is lower than 10minus5

for which the spectroscopic luminosity class is not necessarilymatching the internal evolutionary status (a luminous starwitha strong mass loss can still be burning hydrogen in its coreand already have the appearance of a supergiant because of itswind)

The upper panel of Fig 8 shows the Geneva tracks with ini-tial rotation on the ZAMS between 180 and 270 km sminus1 depend-ing on the initial mass The width of the main sequence (betweenthe solid and short dashed - long dashed lines) corresponds wellto the position of main sequence stars belowsim 7 M⊙ The exten-sion of the main sequence might be slightly too small for massesbetween 7 and 25 M⊙ (compared with the location of red trian-gles and pentagons that is stars with luminosity classes ofIVand V ) The main sequence width is larger when rotation is in-cluded as expectedIn the middle panel of Fig 8 we show the HR diagram built us-ing the tracks of Brott et al (2011a) forV sini = 300 km sminus1 Themain sequence for models including rotation is wider than for theGeneva models For masses of 10ndash15 M⊙ the main sequence ex-tension corresponds to the area populated by luminosity class VIV and III objects Bright giants and supergiants are located be-yond the main sequence All giant stars (green squares) beingincluded in the main sequence width the core H-burning phasein the Brott et al models is too extended The models withoutro-tation have a narrower main sequence in better agreement withthe position of dwarfs and sub-giants belowsim 15 M⊙ The maindifference between the Geneva and STERN tracks is attributedto the overshooting parameter (α=0335 for STERN versus 01for Geneva) The larger overshooting in the Brott et al modelstranslates into a wider main sequence This is particularlytrue

abovesim 30 M⊙ where all blue supergiants are within the mainsequence width Even if some supergiant stars can in principlestill be main sequence objects at high luminosity it is unlikelythat all of them are core-H burning objects indicating thatanovershooting of 0335 is too large for stars with masses above20 M⊙Finally the bottom panel of Fig 8 shows the FRANEC mod-els of Chieffi amp Limongi (2013) Unfortunately this grid onlyincludes models for masses larger than 13 M⊙ so we focus onthe HR diagram above this mass Between 13 and 20 M⊙ themain sequence width appears to be a little wider than the exten-sion of the region where luminosity class V and IV stars are lo-cated Most giants (green squares) are within the predictedmainsequence band Non rotating models better account for the ob-served extension of the main sequence Beyond 20 M⊙ and upto 40 M⊙ the MS width remains roughly constant and includesmany supergiants It might thus be too wide As for the STERNmodels this may be due to the overshooting parameter (α=02for Chieffi amp Limongi) We note the peculiar behaviour of thenon rotating models above 40 M⊙ the core H burning phase ex-tends to cooler temperatures than the models including rotation

From the HR diagram we thus conclude that moderate val-ues of the overshooting parameter (α lt 02) in models with ini-tial rotational velocities of 250ndash300 km sminus1 correctly reproducethe main sequence width For models without rotation a largeramount of overshooting is required to compensate for the reduc-tion of the main sequence width

There are few determinations of the strength of overshootingin the literature and they provide conflicting results Ribas et al(2000) used eclipsing binaries to show thatα increases with

11


mass but Claret (2007) found thatα=02 could reproducecorrectly the properties of 3lt M lt 30 M⊙ stars Using aster-oseismology Briquet et al (2007) determinedα=044plusmn007for the B2IV star θ Oph and Briquet et al (2011) obtainedα=01plusmn005 for the O9V star HD 46202 One of the mainreasons for these differences is that it is usually the extent ofthe convective core that is constrained from observationsFromthis size the overshooting distance is determined by subtractionof the convective core predicted by models in absence ofovershooting This core size depends on the input physics andthus varies from model to model implying that the estimatesofthe overshooting distance also depend on the uncertaintiesofthe evolutionary models Although it relies on a physical effectovershooting can be partly viewed as an adjustment variabletocorrectly reproduce the extension of stellar convective cores thatare deduced from observations Keeping this limitation in mindwe can test the values of overshooting adopted in the variouscodes

The large overshooting parameter adopted by Brott et al(2011a) results from the comparison of the projected rotationalvelocity to the surface gravity of massive stars in LMC clustersHigh gravity objects show a wide range of rotational velocitieswhile low logg stars haveV sini lt 100 km sminus1 The transitionoccurs at about logg = 32 in the LMC stars and corresponds tothe rapid inflation of the star after the end of the core-H burningphase The larger radius immediately translates into a lower ve-locity Brott et al (2011a) calibrated the overshooting parameterof their 16 M⊙ model to reproduce this sudden drop

Fig 9 shows theV sini ndash logg diagram for the stars usedto build the HR diagram of Fig 8 Only the stars for whichboth the projected rotational velocity and surface gravityareavailable are included in Fig 9 As expected a clear transitionis seen at logg = 345 Above this value velocities span therange 0-400 km sminus1 while below only the lowest velocities (lt100 km sminus1) are observed Main sequence objects (red triangles)are all located on the left part of the transition while mostsupergiants (blue circles) are found on the right side Thisisqualitatively consistent with the results of Hunter et al (2009)and Brott et al (2011a) However the value we obtain for thetransition gravity at solar metallicity is larger than thatobtainedfor the LMC by these authors 345 versus 32 The overshootingused by Brott et al to reproduce the properties of LMC starsis too high for Galactic targets This is illustrated in the upperpanel of Fig 9 where we see that the Brott et alrsquos tracks havea terminal main sequence (identified by the loop in the tracks)at logg = 29ndash32 for masses between 5 and 20 M⊙ The largeovershooting increases the core-H burning phase resulting ina bigger star and thus a lower gravity at the hydrogen coreexhaustion The middle panel of Fig 9 shows the behaviour ofthe Geneva models In the 5ndash20 M⊙ range the TAMS is locatedat logg = 325ndash360 in better agreement with the observationaldata However the 5 and 9 M⊙ tracks have a TAMS at loggslightly too large compared to the position of luminosity classV stars A value ofα equal to 01 is a little too low to reproducethe bulk of main sequence stars The bottom panel presents thetracks from Chieffi amp Limongi (2013) in the mass range 13 M⊙to 60 M⊙ The overshoot parameter used for these models isdHp = 02 and we see an intermediate behaviour compared tothe two other families of tracks as expected for this intermediatevalue of overshooting The TAMS is located at logg = 30ndash32at gravities too low compared to the observationsLet us note that the Geneva and FRANEC tracks (Fig 9b andc) behave similarly beyond the TAMS (in particular the 40 M⊙

track) and only cover the slowest supergiants (blue circles) withpredicted surface velocities of less than 40 km sminus1for log g lt 30The models from the STERN grid on the other hand experiencea weaker braking and conserve a more rapid rotation beyondThey better reproduce the observed distribution of rotationvelocities of supergiants The treatment of rotation inducedmixing and angular momentum evolution in the STERN codeis different from that implemented in the Geneva and FRANECcodes which might explain these differences in the evolution ofthe surface equatorial velocity

In conclusion an overshooting parameter a little larger than01 is required to have a convective core extension able to ac-count for the observed width of the main sequence for GalacticOB stars

42 Surface carbon and nitrogen content

Rotation modifies the surface abundances because the of the in-duced mixing processes Surface abundances are thus impor-tant diagnostics to see how realistic is the treatment of rota-tion in evolutionary models In Fig 10 and 11 we compare theCH and NH ratios predicted by the models to those deter-mined for OB stars The symbols correspond to Galactic starsanalyzed by means of quantitative spectroscopic analysisTheevolutionary tracks of Brott et al (2011a) Ekstrom et al(2012)and Chieffi amp Limongi (2013) are shown They span a massrange between 15 and 60 M⊙ OB stars on the main sequencehave an initial carbon content between 50times10minus5 and 26times10minus4We note that values lower than 20times10minus4 are only obtained inthe study of Hunter et al (2009) The other studies of main se-quence stars (Przybilla et al 2010 Nieva amp Simon-Dıaz 2011Firnstein amp Przybilla 2012) all find CH larger than 20times10minus4consistent with the solar reference value of Grevesse et al(2010)

The Brott et al (2011a) tracks have an initial composi-tion roughly corresponding to the average of the B stars ofHunter et al (2009) The few B stars of Hunter et al withevidence for N enrichment are rather well accounted forby these tracks On the other hand the OB stars analyzedby Przybilla et al (2010) Nieva amp Simon-Dıaz (2011) andFirnstein amp Przybilla (2012) cannot be reproduced by the Brottet al tracks The Ekstrom et al (2012) and Chieffi amp Limongi(2013) predictions are better (although on average slightly toocarbon rich for the Geneva models) to account for the initialcomposition and the evolution of these stars One may wonderwhether the samples of Hunter et al (2009) and Przybilla et al(2010)Nieva amp Simon-Dıaz (2011)Firnstein amp Przybilla(2012) are taken from environments with different metal-licities Hunter et al (2009) focused on NGC 6611 towardsthe Galactic Center and NGC3293 NGC 4755 at Galacticlongitudes of 285o and 303o and distances close to 2 kpcThe stars of Przybilla et al (2010)Nieva amp Simon-Dıaz(2011)Firnstein amp Przybilla (2012) are mostly from the solarneighborhood We thus do not expect strong metallicity dif-ferences among the stars shown in Fig 10 The reason for thedifferences in C abundance at the beginning of the evolutionbetween the two sets of stars is thus not clear It might be relatedto the methods used to determine the stellar parameters and thesurface abundances

Turning to Fig 11 we see that evolved OB stars are wellreproduced by the Ekstrom et al and Chieffi amp Limongi modelsalthough the number of stars with good C and N abundances isstill too small to draw firm conclusions The Brott et al (2011a)

12


tracks have a too low carbon content at a given NH because ofthe initial offset discussed above

These comparisons indicate that accurate abundance deter-minations are necessary before the predictions of evolutionarymodels regarding surface composition can be tested at depthThe reason for the difference between the study of Hunter et alon the one hand and of Nieva Przybilla Firnstein and collabora-tors on the other hand needs to be understood before preferencecan be given to a set of tracks

43 The upper HR diagram

In this section we focus on the upper HR diagram ie at lumi-nosities larger than 3times 105 L⊙ The only publicly available tracksincluding rotation for stars with masses larger than 60 M⊙ areprovided by the Geneva group and Chieffi amp Limongi (2013) InFig 12 we plot the evolutionary tracks of Ekstrom et al (2012)and Chieffi amp Limongi in the upper and lower panels respec-tively Only the part of the tracks with a surface helium massfraction smaller than 075 is shown This is equivalent to a num-ber ratio HeH of about 08 This value is the largest determinedfor most WNh stars (Hamann et al 2006 Martins et al 2008Liermann et al 2010)

We first focus on the Geneva models The WNh stars and themost luminous supergiants cannot be accounted for by the evolu-tionary tracks with rotation of Ekstrom et al (2012) The 60 85and 120 M⊙ tracks turn to the blue almost immediately after thezero age main sequence (Fig 12 upper left panel) The 60 M⊙

track later comes back to the red part of the HRD but at thattime it is depleted in hydrogen while the comparison stars allstill contain a significant fraction of hydrogen The 85 and 120M⊙ tracks do not come back to the right of the ZAMS and cannotreproduce the position of the WNh stars We thus conclude thatthe solar metallicity tracks with rotation of Ekstrom et al (2012)do not account for the most massive stars and thus should notbe used for comparisons with normal stars with masses above60 M⊙ The Geneva tracks are computed forveq

vcritsim 04 where

veq andvcrit are the equatorial velocity and the break-up veloc-ity For stars with Mgt 60 M⊙ the initial velocity issim280 kmsminus1 Although not extreme and in spite of the rapid drop of therotational velocity with age (see Fig 10 of Ekstrom et al 2012)the fast initial rotation may explain the blueward evolution of themost massive models This behaviour is similar to that of quasi-chemically homogeneous tracks (Maeder 1987 Langer 1992)For comparison we show on the upper right panel of Fig 12 theGeneva non rotating tracks with the same cut in helium massfraction This time the 60 85 and 120 M⊙ models cross the re-gion occupied by the most luminous supergiants and the WNhstars With a low or moderate helium content they can thus re-produce the existence of these very luminous objects One canconclude that the current Geneva rotating tracks are probablycomputed with a too high initial rotational velocity to explain theexistence of most of the Galactic WNh and early O supergiantsThe tracks by Meynet amp Maeder (2003) include rotation with aninitial equatorial velocity of 300 km sminus1 and haveZ = 0020They are super-solar according to the current calibration of solarabundances (Z = 0014) They do not show the rapid bluewardevolution just after the ZAMS seen in the Ekstrom et al tracksThey better account for the very massive stars as exemplified bythe studies of Hamann et al (2006) Martins et al (2008) Sincethe initial rotational velocity in the Meynet amp Maeder (2003)tracks is larger than in the Ekstrom et al (2012) models weat-tribute the different behaviour to the metallicity difference

Levesque et al (2012) used the evolutionary tracks includingrotation of Ekstrom et al (2012) to compute population synthe-sis models They concluded that the inclusion of rotation lead toa significant increase of the ionizing flux in young stellar popu-lation The main reason is a shift towards higher luminosities ofthe tracks when rotation is included (see Sect 33) In the earliestphases the presence of stars with masses in excess of 60 M⊙ alsocontributes to the larger ionizing flux since as illustrated abovethe tracks predict much higher temperature when rotation isin-cluded As discussed by Levesque et al (2012) their predictionsare probably biased towards populations with larger than averagerotational velocities In view of the direct comparison betweenevolutionary tracks and position of Galactic massive starsin theHR diagram we confirm that the Ekstrom et al (2012) tracksshould be used with a good recognition that they are relevantforfast rotating objects

Coming back to the lower panels of Fig 12 we see that themodels of Chieffi amp Limongi do not suffer from the same prob-lems as the Geneva models The 60 and 80 M⊙ tracks includingrotation evolve classically towards the red part of the HR dia-gram They all have initial rotation of about 250 km sminus1 simi-lar to that of the 20M⊙ and 40 M⊙ models from Ekstrom et al(2012) as can be see from Fig 9 They can account for stars asluminous as 106 L⊙ The 120 M⊙track also evolves redwards butturns back to the blue at logTeff = 454 Many of the luminousWNh stars can be explained by this track Only three objects atlogTeff = 44 are not reproduced by the FRANEC tracks Butthe 120 M⊙ non-rotating track (lower right panel of Fig 12) ex-tends down to such temperatures Since the rotation of massivestars spans a range of values it is possible that some objects areslow rotators Unfortunately it is difficult to measure rotation ve-locities in WNh stars since their spectrum is dominated by linesformed above the photosphere in the windWe thus conclude that overall the Chieffiamp Limongi tracks betterreproduce the population of luminous O supergiants and WNhstars in the HR diagram For the most luminous stars it ap-pears that non-rotating tracks may give a better fit to the ob-servations than rotating ones when only considering the positionin the HR diagram The difference between the FRANEC andGeneva computations are 1) a larger overshooting and mixinglength parameter and 2) the inclusion of efficiency factors in thetreatment of rotation in the study of Chieffiamp Limongi The massloss rates in the temperature range of interest are the sameSincethe Geneva models are very different with and without rotationwe tend to attribute the differences with the FRANEC models tothe second point listed above The efficiency factors of Chieffiamp Limongi reduce the impact of rotation on the diffusion coef-ficients Qualitatively they could limit the strong effects of rota-tion seen in the grid of Ekstrom et al (2012)

5 Conclusion

We have performed a comparison of evolutionary modelsfor massive stars in the Galaxy The published grids ofBertelli et al (2009) Brott et al (2011a) Ekstrom et al(2012)and Chieffi amp Limongi (2013) have been used We havealso computed additional models with the codes STAREVOL(Decressin et al 2009) and MESA (Paxton et al 2013) Our goalwas to estimate and highlight the uncertainties in the output ofthese models Our conclusions are

bull Evolutionary tracks in the HR diagram are sensitive to theadopted solar composition mixture metallicity the amountof overshooting and the mass loss rate The extension of the

13


(a) Geneva - with rotation (b) Geneva - without rotation

(c) FRANEC - with rotation (d) FRANEC - without rotation

Fig 12 Same as Fig 8 but focusing on the upper part of the HR diagramThe orange squares correspond to WNh stars We haveused stars for which stellar parameters determined from spectroscopic analysis are available In the upper (lower) panels the Geneva(FRANEC) models are shown The tracks have been plotted for X(He)lt075 which corresponds to HeHsim08 by numberLefttracks with rotationRight non rotating tracks

convective core due to overshooting andor rotation-inducedmixing and the adopted initial metallicity are the majorsources of uncertainty on the determined luminosity onthe main sequence and during the core He burning phaseWithin one set of stellar evolution models we estimatea global intrinsic uncertainty on the luminosity of aboutplusmn005 dex for a star with an initial mass of 20 M⊙ Thisuncertainty is lower when studying main sequence starsThis uncertainty translates into an error of about 6 on thedistance and may rise up to an error of 30 on the distanceof lower mass red supergiants (typically stars with initial

mass of 7 M⊙)

bull The evolutionary tracks computed with the six differentcodes agree reasonably well for the main sequence evo-lution Beyond that large differences appear They arethe largest at low effective temperature In the red part ofthe HR diagram evolutionary tracks with different initialmasses and computed with different codes can overlap Thedifference on the luminosity of a star located near the redgiant branch can be as large as 04 dex depending on thestellar evolution models adopted which makes the estimate

14


of ages and initial masses for red supergiants extremelyuncertain

bull Comparison of the tracks of Brott et al (2011a)Ekstrom et al (2012) and Chieffi amp Limongi (2013) withthe properties (Teff luminosity) of a large number of OBand WNh stars in the Galaxy indicate that in the mass range7ndash20 M⊙ the Ekstrom et al tracks have a slightly too narrowmain sequence width while the Brott et al and Chieffi ampLimongi tracks have a too wide one This is due to theovershooting parameters used These results are confirmedby the analysis of the distribution of rotational velocities asa function of logg for Galactic stars with various spectraltypes A clear drop in rotational velocities is observed atlogg = 345 at larger gravity than that observed for LMCstars studied by Hunter et al (2008) An overshooting pa-rameter slightly above 01 is required to reproduce this trend

bull Measurements of surface abundances of carbon and nitrogenare currently too uncertain to help constrain evolutionarymodels

bull Stars with initial masses higher than about 60 M⊙ are not ac-counted for by the Ekstrom et al tracks with rotation Theybend to the blue part of the HR diagram quickly after leav-ing the zero age main sequence and do not reproduce theposition of the WNh and luminous blue supergiants Modelsprovided by Chieffi amp Limongi give a much better fit to theobservations Non rotating tracks of both grids can reproducethe position of the most luminous objects

Future analysis of surface abundances of large samples ofGalactic OB stars will provide critical constraints on evolution-ary models and the treatment of rotation Better prescriptions ofmass loss rates for a wide variety of stars are also required toimprove the predictions of evolutionary models

Acknowledgements We warmly thank the MESA team for making their codefreely available (httpmesasourceforgenet) We thank Marco Limongi andAlessandro Chieffi for kindly providing their models and associated informationWe aknowledge the comments and suggestions of an anonymous referee Wethank Georges Meynet Daniel Schaerer Sylvia Ekstrom Cyril Georgy Selmade Mink Hugues Sana for interesting discussions This study was supported bythe grant ANR-11-JS56-0007 (Agence Nationale de la Recherche)

ReferencesAsplund M Grevesse N amp Sauval A J 2005 in Astronomical Society of

the Pacific Conference Series Vol 336 Cosmic Abundances as Records ofStellar Evolution and Nucleosynthesis ed T G Barnes III amp F N Bash 25

Asplund M Grevesse N Sauval A J amp Scott P 2009 ARAampA 47 481Balser D S Rood R T Bania T M amp Anderson L D 2011 ApJ 738 27Bertelli G Nasi E Girardi L amp Marigo P 2009 AampA 508 355Bono G Caputo F Cassisi S et al 2000 ApJ 543 955Bouret J-C Hillier D J Lanz T amp Fullerton A W 2012 AampA 544 A67Bouret J-C Lanz T amp Hillier D J 2005 AampA 438 301Briquet M Aerts C Baglin A et al 2011 AampA 527 A112Briquet M Morel T Thoul A et al 2007 MNRAS 381 1482Brott I de Mink S E Cantiello M et al 2011a AampA 530 A115Brott I Evans C J Hunter I et al 2011b AampA 530 A116Castor J I Abbott D C amp Klein R I 1975 ApJ 195 157Chaboyer B amp Zahn J-P 1992 AampA 253 173Chieffi A amp Limongi M 2013 ApJ 764 21Chiosi C amp Maeder A 1986 ARAampA 24 329Claret A 2007 AampA 475 1019Cox J P 1968 Principles of stellar structure - Vol1 Physical principles Vol2

Applications to stars New York Gordon and BreachCrowther P A 2000 AampA 356 191Crowther P A amp Bohannan B 1997 AampA 317 532

Crowther P A Hadfield L J Clark J S Negueruela I amp Vacca W D2006a MNRAS 372 1407

Crowther P A Lennon D J amp Walborn N R 2006b AampA 446 279Crowther P A Schnurr O Hirschi R et al 2010 MNRAS 408 731Cunha K Hubeny I amp Lanz T 2006 ApJ 647 L143Davies B Kudritzki R-P Plez B et al 2013 ApJ 767 3de Jager C Nieuwenhuijzen H amp van der Hucht K A 1988AampAS 72 259Decressin T Mathis S Palacios A et al 2009 AampA 495 271Ekstrom S Georgy C Eggenberger P et al 2012 AampA537 A146Endal A S amp Sofia S 1978 ApJ 220 279Ferguson J W Alexander D R Allard F et al 2005 ApJ 623 585Firnstein M amp Przybilla N 2012 AampA 543 A80Fullerton A W Massa D L amp Prinja R K 2006 ApJ 6371025Grafener G amp Hamann W-R 2008 AampA 482 945Grevesse N Asplund M Sauval A J amp Scott P 2010 ApampSS 328 179Grevesse N Noels A amp Sauval A J 1993 AampA 271 587Grevesse N Noels A amp Sauval A J 1996 in Astronomical Society of the

Pacific Conference Series Vol 99 Cosmic Abundances ed S S Holt ampG Sonneborn 117

Grevesse N amp Sauval A J 1998 Space Sci Rev 85 161Hamann W-R Grafener G amp Liermann A 2006 AampA 4571015Hamann W-R Koesterke L amp Wessolowski U 1995 AampA 299 151Heger A Langer N amp Woosley S E 2000 ApJ 528 368Herwig F 2000 AampA 360 952Hunter I Brott I Langer N et al 2009 AampA 496 841Hunter I Brott I Lennon D J et al 2008 ApJ 676 L29Iglesias C A amp Rogers F J 1996 ApJ 464 943Kippenhahn R Meyer-Hofmeister E amp Thomas H C 1970AampA 5 155Kippenhahn R amp Weigert A 1990 Stellar Structure and EvolutionKudritzki R P Gabler A Gabler R Groth H G amp Pauldrach A W A

1989 in Astrophysics and Space Science Library Vol 157 IAU Colloq 113Physics of Luminous Blue Variables ed K Davidson A F JMoffat ampH J G L M Lamers 67ndash80

Langer N 1992 AampA 265 L17Langer N 2012 ARAampA 50 107Langer N Fricke K J amp Sugimoto D 1983 AampA 126 207Lefever K Puls J Morel T et al 2010 AampA 515 A74Levenhagen R S amp Leister N V 2004 AJ 127 1176Levesque E M Leitherer C Ekstrom S Meynet G amp Schaerer D 2012

ApJ 751 67Levesque E M Massey P Olsen K A G et al 2005 ApJ628 973Liermann A Hamann W-R Oskinova L M Todt H amp Butler K 2010

AampA 524 A82Limongi M Chieffi A amp Bonifacio P 2003 ApJ 594 L123Lyubimkov L S Rachkovskaya T M Rostopchin S I amp Lambert D L

2002 MNRAS 333 9Maeder A 1987 AampA 178 159Maeder A 1997 AampA 321 134Maeder A amp Meynet G 2000a AampA 361 159Maeder A amp Meynet G 2000b ARAampA 38 143Maeder A amp Meynet G 2001 AampA 373 555Maeder A Meynet G Ekstrom S amp Georgy C 2009 Communications in

Asteroseismology 158 72Maeder A amp Zahn J-P 1998 AampA 334 1000Markova N amp Puls J 2008 AampA 478 823Martins F Escolano C Wade G A et al 2012a AampA 538 A29Martins F Forster Schreiber N M Eisenhauer F amp Lutz D 2012b AampA

547 A17Martins F Genzel R Hillier D J et al 2007 AampA 468 233Martins F Hillier D J Paumard T et al 2008 AampA 478 219Martins F Mahy L Hillier D J amp Rauw G 2012c AampA 538 A39Martins F amp Plez B 2006 AampA 457 637Martins F Schaerer D Hillier D J et al 2005a AampA441 735Martins F Schaerer D Hillier D J et al 2005b AampA441 735Massey P amp Johnson J 1993 AJ 105 980Mauron N amp Josselin E 2011 AampA 526 A156McErlean N D Lennon D J amp Dufton P L 1999 AampA 349553Melena N W Massey P Morrell N I amp Zangari A M 2008 AJ 135 878Meynet G amp Maeder A 1997 AampA 321 465Meynet G amp Maeder A 2003 AampA 404 975Negueruela I Clark J S amp Ritchie B W 2010 AampA 516A78Neugent K F amp Massey P 2011 ApJ 733 123Neugent K F Massey P amp Georgy C 2012 ApJ 759 11Nieuwenhuijzen H amp de Jager C 1990 AampA 231 134Nieva M-F amp Simon-Dıaz S 2011 AampA 532 A2Nugis T amp Lamers H J G L M 2000 AampA 360 227Paxton B Bildsten L Dotter A et al 2011 ApJS 1923Paxton B Cantiello M Arras P et al 2013 ArXiv e-prints

15


Petrovic J Langer N Yoon S-C amp Heger A 2005 AampA435 247Pietrinferni A Cassisi S Salaris M amp Castelli F 2004 ApJ 612 168Pietrinferni A Cassisi S Salaris M amp Castelli F 2006 ApJ 642 797Przybilla N Firnstein M Nieva M F Meynet G amp Maeder A 2010 AampA

517 A38Puls J Vink J S amp Najarro F 2008 AampA Rev 16 209Reimers D 1975 Memoires of the Societe Royale des Sciences de Liege 8 369Ribas I Jordi C amp GimenezA 2000 MNRAS 318 L55Searle S C Prinja R K Massa D amp Ryans R 2008 AampA481 777Siess L 2006 AampA 448 717Siess L Dufour E amp Forestini M 2000 AampA 358 593Smith N amp Conti P S 2008 ApJ 679 1467Spruit H C 2002 AampA 381 923Straniero O Chieffi A amp Limongi M 1997 ApJ 490 425Talon S amp Zahn J-P 1997 AampA 317 749van Loon J T Cioni M-R L Zijlstra A A amp Loup C 2005 AampA 438

273Vink J S de Koter A amp Lamers H J G L M 2000 AampA 362 295Vink J S de Koter A amp Lamers H J G L M 2001 AampA 369 574von Zeipel H 1924 MNRAS 84 684Vrancken M Lennon D J Dufton P L amp Lambert D L 2000 AampA 358

639Walborn N R Howarth I D Lennon D J et al 2002 AJ123 2754Weiss A Keady J J amp Magee Jr N H 1990 Atomic Data and Nuclear Data

Tables 45 209Zahn J-P 1992 AampA 265 115

Fig 8 Comparison between evolutionary tracks (black lines)and the location of OB stars in the HR diagram The evolu-tionary tracks are from Ekstrom et al (2012) (top) Brott et al(2011a) (middle) and Chieffi amp Limongi (2013) (bottom) Theyare shown for hydrogen mass fraction lower than 060 for clar-ity The short dashed - long dashed line connects the cooler edgeof the main sequence for the different models and defines theTAMS The dotted lines is the same for the non-rotating mod-els Different symbols correspond to different luminosity classesFilled symbols correspond to stars for which the stellar param-eters have been determined through a tailored analysis whileopen symbols are for stars with parameters taken from calibra-tions according to their spectral type The data sources arelistedin the text sect 41

16


Fig 9 Projected rotational velocity as a function of surface grav-ity The symbols have the same meaning as in Fig 8Top evo-lutionary tracks from Ekstrom et al (2012)Middle evolution-ary tracks from Brott et al (2011a)Bottom evolutionary tracksfrom Chieffi amp Limongi (2013) The vertical dashed line is aguideline located at logg = 345

Fig 10 CH as a function of NH The symbols cor-respond to results from spectroscopic analysis fromMartins et al (2008) Hunter et al (2009) Przybilla et al(2010) Nieva amp Simon-Dıaz (2011) Firnstein amp Przybilla(2012) Bouret et al (2012) The evolutionary tracks for variousmasses between 15 and 60 M⊙ are taken from Ekstrom et al(2012) (black) Brott et al (2011a) (red) and Chieffi amp Limongi(2013) (magenta)

Fig 11 Same as Fig 10 on a wider scale to show abundances ofevolved objects The stars analyzed by Hunter et al (2009) havebeen omitted for clarity

17

1 Introduction


21 STERN stellar evolution code brott11a

22 Geneva stellar evolution code ek12

23 FRANEC stellar evolution code cl13

24 Padova stellar evolutionary code bert09

25 STAREVOL dmp09

26 MESA code pax11




33 Rotation





5 Conclusion


Indeed evolutionary calculations rely on various prescrip-tions to describe the physical processes driving the evolutionand these prescriptions may vary from code to code The mostimportant ingredientsprocesses to be considered are convec-tion and its related properties (such as overshooting) mass loss(Chiosi amp Maeder 1986) chemical composition (and the rela-tive abundance of the various species considered in the mod-els) and of course initial mass Rotation is another key ingredi-ent since it affects the internal structure the physical properties(temperature luminosity) the surface chemical appearance andthe lifetimes of stars (Maeder amp Meynet 2000b) Several pre-scriptions usually exist to treat a given physical process in evo-lutionary codes As a consequence the outputs depend on theinput physics Since evolutionary calculations are a crucial toolto link the observed properties of stars to their physical state andevolution it is important to understand the limitations and un-certainties associated with evolutionary models

In this paper we present a comparison of various predictionsof the evolution of massive stars computed with different codesOur goal is to highlight the uncertainties in the outputs of evolu-tionary calculations especially concerning the HR diagram Wefocus on calculations for Galactic stars In Sect 2 we describethe different codes and models we have used in our comparisonsWe then present a study of the uncertainties associated withtheassumptions in the input physics (Sect 3) In the same sectionwe also compare the evolutionary tracks predicted by the differ-ent codes In Sect 4 we confront the predictions of publishedgrids of models to the observed properties of massive stars inthe Galaxy We highlight the limitations of each grid Finallywe summarize our main conclusions in Sect 5


To achieve our goal of comparing stellar evolution tracks ofmassive stars we have used four databases of stellar evolutiontracks presented in Bertelli et al (2009) Brott et al (2011a)Ekstrom et al (2012) Chieffi amp Limongi (2013) and we com-puted models using the STAREVOL code (Decressin et al2009) and the MESA code (Paxton et al 2011 2013) We recallhere the main ingredients and physical parameters used in eachcase since they may differ largely and these differences appearto affect the evolutionary tracks Table 1 summarizes the maininputs for each code

21 STERN stellar evolution code (Brott et al 2011a)

We first make use of the grid of stellar evolution modelspublished by Brott et al (2011a) The computations have beenperformed with the code fully described in Heger et al (2000)In the following we will refer to this code as the STERN code

Solar reference chemical compositionBrott et al (2011a) adopt tailored reference chemical abun-dances for their models of LMC SMC and Galactic massivestars based on the solar abundances of Asplund et al (2005)with a modification of the C N O Mg Si and Fe abundancesThis results in unusual chemical mixtures described in Tables 1and 2 of their paper Their adopted values for the metal massfraction Z is 00088 00047 and 00021 for the Galaxy theLMC and the SMC respectively The Galactic metallicity isabout half the value used by the other codes (see below) Theinitial helium content is Y=0264 The OPAL radiative opacitiesof Iglesias amp Rogers (1996) are used in the calculations

ConvectionThey use the Ledoux criterion1 to determine the extension ofthe convective regions and model convection according to themixing length theory withαMLT = 15 The mixing length is thelength over which a displaced element conserves its propertiesand the mixing length parameterαMLT is the ratio of themixing length to the local pressure scale heightHP The zoneswhich are stable according to the Ledoux criterion but unstableaccording to the Schwarzschild criterion are considered tobesemi-convective Semi-convection is included as in Langeret al(1983) withαsc = 1 Finally Brott et al calibrate an additionalclassical overshooting parameter to adjust the evolution of therotation velocity as a function of surface gravity of a 16 M⊙model at LMC metallicity (see Sect 41) This parameteris applied to their entire grid and results in an extension ofthe convective cores beyond the limit defined by the Ledouxcriterion bydover = 0335Hp

Mass lossMass loss is implemented following a combination of prescrip-tions or recipes that are specific for each evolutionary phase ofmassive stars Brott et al (2011a) use Vink et al (2000 2001)(MV ) for winds of early O and B-type stars A switch toM byNieuwenhuijzen amp de Jager (1990) (MNdJ) is operated at Tefflt22000 K (bi-stability jump temperature) wheneverMV lt MNdJ Another switch is operated for the Wolf-Rayet phase andMby Hamann et al (1995) is adopted as soon asYs ge 07 (Ys isthe surface helium abundance) For intermediate values ofYsan interpolation between the Vink et al mass loss rates and theWolf-Rayet mass loss rates reduced by a factor 10 is performedBrott et al use a metallicity scaling of the mass loss by afactor (Fesur fFe⊙)085 based on the solar iron abundance fromGrevesse et al (1996) (ǫ(Fe) = 750) which is higher than thatof their models (ǫ(Fe) = 740) For the rotating models they

also apply the correction factor(

11minusVVcrit

)043to the mass loss

rate (Vcrit is the critical velocity)

Rotation and rotation-induced mixingThe effects of the centrifugal acceleration on the stellar struc-ture equations is considered according to Kippenhahn et al(1970) The transport of angular momentum and chemicalspecies is treated in a diffusive way following the formalismby Endal amp Sofia (1978) as described in Heger et al (2000)Eddington-Sweet circulation dynamical and secular shear andaxisymmetric (GSF) instabilities contribute to the transport ofboth angular momentum and chemical species The formalismthey use relies on two efficiency factors (free parameters) fc = 00228 which reduces the contribution to the rotation-induced hydrodynamical instabilities in the total diffusion coeffi-cient andfmicro = 01 which regulates the inhibiting effect of chem-ical gradients on the rotational mixing The values of thesepa-rameters are calibrated on observations They are fully describedin Eq 53 and 54 of Heger et al (2000)In addition to that the action of magnetic fields on thetransport of angular momentumonly is included through thehighly debated Tayler-Spruit dynamo (Spruit 2002) followingPetrovic et al (2005)

1 For a plasma described by a general equation of state theLedouxstability criterion is given bynablarad lt nablaad +

φ

δnablamicro with δ = minus lnρ

lnT andφ =lnρlnmicro micro being the mean molecular weight When there are no chemicalgradientsnablamicro = 0 and the Schwarzschild stability criterion is recoverednablarad lt nablaad

2













luminosity







3


















4

















5











6













7








8







(see Sect 41)


L⊙=42 can be



9



33 Rotation

















10




Teff(K) 33522 33758 33709 32538 33011 33365log L

L⊙4764 4733 4736 4747 4745 4733

age (Myr) 3598 3745 4079 4279 4073 3963

TAMS3

Teff(K) 24630 29297 28094 25972 28733 27902log L

L⊙5096 5001 5036 5045 5007 4995

age (Myr) 8173 7819 9085 9100 8598 8535

ZAHeMS

Teff(K) 5388 24014 27382 15657 23200 24259log L

L⊙4987 5025 5012 5044 5042 5015

age (Myr) 8185 7828 9088 9111 8610 8541

TAHeMS3


L⊙-lowast 5050 5206 5090 5049 5027

age (Myr) -lowast 8713 9690 9820 9530 9748







11











12








vcritsim 04 where




5 Conclusion



13






mass of 7 M⊙)


14























15








16





17

1 Introduction






25 STAREVOL dmp09

26 MESA code pax11




33 Rotation





5 Conclusion













luminosity







3


















4

















5











6













7








8







(see Sect 41)


L⊙=42 can be



9



33 Rotation

















10




Teff(K) 33522 33758 33709 32538 33011 33365log L

L⊙4764 4733 4736 4747 4745 4733

age (Myr) 3598 3745 4079 4279 4073 3963

TAMS3

Teff(K) 24630 29297 28094 25972 28733 27902log L

L⊙5096 5001 5036 5045 5007 4995

age (Myr) 8173 7819 9085 9100 8598 8535

ZAHeMS

Teff(K) 5388 24014 27382 15657 23200 24259log L

L⊙4987 5025 5012 5044 5042 5015

age (Myr) 8185 7828 9088 9111 8610 8541

TAHeMS3


L⊙-lowast 5050 5206 5090 5049 5027

age (Myr) -lowast 8713 9690 9820 9530 9748







11











12








vcritsim 04 where




5 Conclusion



13






mass of 7 M⊙)


14























15








16





17

1 Introduction






25 STAREVOL dmp09

26 MESA code pax11




33 Rotation





5 Conclusion


















4

















5











6













7








8







(see Sect 41)


L⊙=42 can be



9



33 Rotation

















10




Teff(K) 33522 33758 33709 32538 33011 33365log L

L⊙4764 4733 4736 4747 4745 4733

age (Myr) 3598 3745 4079 4279 4073 3963

TAMS3

Teff(K) 24630 29297 28094 25972 28733 27902log L

L⊙5096 5001 5036 5045 5007 4995

age (Myr) 8173 7819 9085 9100 8598 8535

ZAHeMS

Teff(K) 5388 24014 27382 15657 23200 24259log L

L⊙4987 5025 5012 5044 5042 5015

age (Myr) 8185 7828 9088 9111 8610 8541

TAHeMS3


L⊙-lowast 5050 5206 5090 5049 5027

age (Myr) -lowast 8713 9690 9820 9530 9748







11











12








vcritsim 04 where




5 Conclusion



13






mass of 7 M⊙)


14























15








16





17

1 Introduction






25 STAREVOL dmp09

26 MESA code pax11




33 Rotation





5 Conclusion

















5











6













7








8







(see Sect 41)


L⊙=42 can be



9



33 Rotation

















10




Teff(K) 33522 33758 33709 32538 33011 33365log L

L⊙4764 4733 4736 4747 4745 4733

age (Myr) 3598 3745 4079 4279 4073 3963

TAMS3

Teff(K) 24630 29297 28094 25972 28733 27902log L

L⊙5096 5001 5036 5045 5007 4995

age (Myr) 8173 7819 9085 9100 8598 8535

ZAHeMS

Teff(K) 5388 24014 27382 15657 23200 24259log L

L⊙4987 5025 5012 5044 5042 5015

age (Myr) 8185 7828 9088 9111 8610 8541

TAHeMS3


L⊙-lowast 5050 5206 5090 5049 5027

age (Myr) -lowast 8713 9690 9820 9530 9748







11











12








vcritsim 04 where




5 Conclusion



13






mass of 7 M⊙)


14























15








16





17

1 Introduction






25 STAREVOL dmp09

26 MESA code pax11




33 Rotation





5 Conclusion











6













7








8







(see Sect 41)


L⊙=42 can be



9



33 Rotation

















10




Teff(K) 33522 33758 33709 32538 33011 33365log L

L⊙4764 4733 4736 4747 4745 4733

age (Myr) 3598 3745 4079 4279 4073 3963

TAMS3

Teff(K) 24630 29297 28094 25972 28733 27902log L

L⊙5096 5001 5036 5045 5007 4995

age (Myr) 8173 7819 9085 9100 8598 8535

ZAHeMS

Teff(K) 5388 24014 27382 15657 23200 24259log L

L⊙4987 5025 5012 5044 5042 5015

age (Myr) 8185 7828 9088 9111 8610 8541

TAHeMS3


L⊙-lowast 5050 5206 5090 5049 5027

age (Myr) -lowast 8713 9690 9820 9530 9748







11











12








vcritsim 04 where




5 Conclusion



13






mass of 7 M⊙)


14























15








16





17

1 Introduction






25 STAREVOL dmp09

26 MESA code pax11




33 Rotation





5 Conclusion













7








8







(see Sect 41)


L⊙=42 can be



9



33 Rotation

















10




Teff(K) 33522 33758 33709 32538 33011 33365log L

L⊙4764 4733 4736 4747 4745 4733

age (Myr) 3598 3745 4079 4279 4073 3963

TAMS3

Teff(K) 24630 29297 28094 25972 28733 27902log L

L⊙5096 5001 5036 5045 5007 4995

age (Myr) 8173 7819 9085 9100 8598 8535

ZAHeMS

Teff(K) 5388 24014 27382 15657 23200 24259log L

L⊙4987 5025 5012 5044 5042 5015

age (Myr) 8185 7828 9088 9111 8610 8541

TAHeMS3


L⊙-lowast 5050 5206 5090 5049 5027

age (Myr) -lowast 8713 9690 9820 9530 9748







11











12








vcritsim 04 where




5 Conclusion



13






mass of 7 M⊙)


14























15








16





17

1 Introduction






25 STAREVOL dmp09

26 MESA code pax11




33 Rotation





5 Conclusion








8







(see Sect 41)


L⊙=42 can be



9



33 Rotation

















10




Teff(K) 33522 33758 33709 32538 33011 33365log L

L⊙4764 4733 4736 4747 4745 4733

age (Myr) 3598 3745 4079 4279 4073 3963

TAMS3

Teff(K) 24630 29297 28094 25972 28733 27902log L

L⊙5096 5001 5036 5045 5007 4995

age (Myr) 8173 7819 9085 9100 8598 8535

ZAHeMS

Teff(K) 5388 24014 27382 15657 23200 24259log L

L⊙4987 5025 5012 5044 5042 5015

age (Myr) 8185 7828 9088 9111 8610 8541

TAHeMS3


L⊙-lowast 5050 5206 5090 5049 5027

age (Myr) -lowast 8713 9690 9820 9530 9748







11











12








vcritsim 04 where




5 Conclusion



13






mass of 7 M⊙)


14























15








16





17

1 Introduction






25 STAREVOL dmp09

26 MESA code pax11




33 Rotation





5 Conclusion







(see Sect 41)


L⊙=42 can be



9



33 Rotation

















10




Teff(K) 33522 33758 33709 32538 33011 33365log L

L⊙4764 4733 4736 4747 4745 4733

age (Myr) 3598 3745 4079 4279 4073 3963

TAMS3

Teff(K) 24630 29297 28094 25972 28733 27902log L

L⊙5096 5001 5036 5045 5007 4995

age (Myr) 8173 7819 9085 9100 8598 8535

ZAHeMS

Teff(K) 5388 24014 27382 15657 23200 24259log L

L⊙4987 5025 5012 5044 5042 5015

age (Myr) 8185 7828 9088 9111 8610 8541

TAHeMS3


L⊙-lowast 5050 5206 5090 5049 5027

age (Myr) -lowast 8713 9690 9820 9530 9748







11











12








vcritsim 04 where




5 Conclusion



13






mass of 7 M⊙)


14























15








16





17

1 Introduction






25 STAREVOL dmp09

26 MESA code pax11




33 Rotation





5 Conclusion



33 Rotation

















10




Teff(K) 33522 33758 33709 32538 33011 33365log L

L⊙4764 4733 4736 4747 4745 4733

age (Myr) 3598 3745 4079 4279 4073 3963

TAMS3

Teff(K) 24630 29297 28094 25972 28733 27902log L

L⊙5096 5001 5036 5045 5007 4995

age (Myr) 8173 7819 9085 9100 8598 8535

ZAHeMS

Teff(K) 5388 24014 27382 15657 23200 24259log L

L⊙4987 5025 5012 5044 5042 5015

age (Myr) 8185 7828 9088 9111 8610 8541

TAHeMS3


L⊙-lowast 5050 5206 5090 5049 5027

age (Myr) -lowast 8713 9690 9820 9530 9748







11











12








vcritsim 04 where




5 Conclusion



13






mass of 7 M⊙)


14























15








16





17

1 Introduction






25 STAREVOL dmp09

26 MESA code pax11




33 Rotation





5 Conclusion




Teff(K) 33522 33758 33709 32538 33011 33365log L

L⊙4764 4733 4736 4747 4745 4733

age (Myr) 3598 3745 4079 4279 4073 3963

TAMS3

Teff(K) 24630 29297 28094 25972 28733 27902log L

L⊙5096 5001 5036 5045 5007 4995

age (Myr) 8173 7819 9085 9100 8598 8535

ZAHeMS

Teff(K) 5388 24014 27382 15657 23200 24259log L

L⊙4987 5025 5012 5044 5042 5015

age (Myr) 8185 7828 9088 9111 8610 8541

TAHeMS3


L⊙-lowast 5050 5206 5090 5049 5027

age (Myr) -lowast 8713 9690 9820 9530 9748







11











12








vcritsim 04 where




5 Conclusion



13






mass of 7 M⊙)


14























15








16





17

1 Introduction






25 STAREVOL dmp09

26 MESA code pax11




33 Rotation





5 Conclusion











12








vcritsim 04 where




5 Conclusion



13






mass of 7 M⊙)


14























15








16





17

1 Introduction






25 STAREVOL dmp09

26 MESA code pax11




33 Rotation





5 Conclusion








vcritsim 04 where




5 Conclusion



13






mass of 7 M⊙)


14























15








16





17

1 Introduction






25 STAREVOL dmp09

26 MESA code pax11




33 Rotation





5 Conclusion






mass of 7 M⊙)


14























15








16





17

1 Introduction






25 STAREVOL dmp09

26 MESA code pax11




33 Rotation





5 Conclusion























15








16





17

1 Introduction






25 STAREVOL dmp09

26 MESA code pax11




33 Rotation





5 Conclusion








16





17

1 Introduction






25 STAREVOL dmp09

26 MESA code pax11




33 Rotation





5 Conclusion





17

1 Introduction






25 STAREVOL dmp09

26 MESA code pax11




33 Rotation





5 Conclusion

arxiv:1310.7218v1 [astro-ph.sr] 27 oct 2013 · (martins et al. 2008; crowther et al. 2010). they...

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