The puzzle of the CNO abundances of α Cygni variables resolved by the Ledoux criterion?
MMon. Not. R. Astron. Soc. , 000–000 (0000) Printed 5 November 2018 (MN L A TEX style file v2.2)
The puzzle of the CNO abundances of α Cygni variablesresolved by the Ledoux criterion?
Cyril Georgy (cid:63) , Hideyuki Saio , Georges Meynet Astrophysics, Lennard-Jones Laboratories, EPSAM, Keele University, Staffordshire, ST5 5BG, UK Astronomical Institute, Graduate School of Science, Tohoku University, Sendai, Japan Geneva Observatory, University of Geneva, Maillettes 51, 1290 Versoix, Switzerland
ABSTRACT
Recent stellar evolution computations show that the blue supergiant (BSG) stars couldcome from two distinct populations: a first group arising from massive stars that justleft the main sequence (MS) and are crossing the Hertzsprung-Russell diagram (HRD)towards the red supergiant (RSG) branch, and a second group coming from stars thathave lost considerable amount of mass during the RSG stage and are crossing theHRD for a second time towards the blue region. Due to very different luminosity-to-mass ratio, only stars from the second group are expected to have excited pulsationsobservable at the surface. In a previous work, we have shown that our models wereable to reproduce the pulsational properties of BSGs. However, these models failedto reproduce the surface chemical composition of stars evolving back from a RSGphase. In this paper, we show how the use of the Ledoux criterion instead of theSchwarzschild one for convection allows to significantly improve the agreement withthe observed chemical composition, while keeping the agreement with the pulsationperiods. This gives some support to the Ledoux criterion.
Key words: stars: abundances – stars: early-type – stars: evolution – stars: mass-loss– stars: oscillations – stars: rotation
The evolution of massive stars after the main sequence (MS)remains mostly unknown, despite numerous improvementsof stellar evolution codes during the last twenty years. Ac-tually, it strongly depends on several physical processes thatare suspected to take place in stellar interiors, such as in-ternal mixing, rotation, magnetic fields, of which the exactimplementation in evolution codes is still mostly uncertain.Mass loss also plays a key role, particularly during the redsupergiant (RSG) phase, but is also a major source of un-certainties (van Loon et al. 2005; Vanbeveren et al. 1998a,b;Georgy 2012).On the observational side, the determination of themaximal luminosity of Galactic RSGs (Levesque et al. 2005)at around log(
L/L (cid:12) ) = 5 . ∼− .
2) and recent stellar evolution models (Ekstr¨om et al.2012) indicate that the most luminous RSGs should origi-nate from stars up to ∼ − M (cid:12) . On the other hand,Smartt et al. (2009) determined that the maximal mass fora type IIP supernova (SN) progenitor is around 17 M (cid:12) . AsRSGs are expected to explode as type IIP SNe, there is here (cid:63) Email: [email protected] an indication that at least the most massive RSGs do notend their life as RSGs, but evolve further to become starsof other types.The recent release by the Geneva group of a new set ofrotating models at solar metallicity (Ekstr¨om et al. 2012)shows that these massive RSGs can cross the Hertzsprung-Russell diagram (HRD) a second time towards the blue side,if the mass-loss rates used during the RSG phase are in-creased by a factor of 3 . Stars with an initial mass above20 M (cid:12) explode as luminous blue variables or Wolf-Rayetstars (Georgy et al. 2012; Groh et al. 2013a,b).In this framework, two distinct populations of blue su-pergiant (BSG) stars are expected: the first one (group 1hereafter) composed by the stars leaving the MS and goingto the RSG branch (first crossing of the HRD), and the sec-ond one (group 2 hereafter) by the stars that were on theRSG branch, and that cross a second time the HRD for anyreason. In a previous paper (Saio et al. 2013), we showed thatthese two populations have very distinct pulsational prop-erties, allowing to distinguish them. Group 1 stars exhibitfew excited pulsation modes at the surface, due to their low These increased rates are still compatible with the rates inferredby van Loon et al. (2005).c (cid:13) a r X i v : . [ a s t r o - ph . S R ] N ov C. Georgy et al.
L/M ratio. On the contrary, stars of the group 2 have a lot ofexcited modes, that are compatible with the observed pulsa-tion period of α -Cyg type variable stars. However, at leasttwo stars (Rigel and Deneb) for which the period is com-patible with our predicted period for group 2 have chemicalsurface abundances (Przybilla et al. 2010) that are compat-ible with our stellar evolution models for group 1, but notfor group 2.In this paper, we show how to tackle this discrepancyby using another prescription for the convection in our stel-lar models. In Section 2, we describe the evolution of ournew models, and explain how they solve the problem of thechemical abundances. In Section 3, we show that these newmodels are still compatible with the pulsational properties ofobserved BSG stars. Finally, our conclusions are presentedin Section 4. In this Section, we discuss the main processes affecting theevolution of the chemical abundances at the surface of therotating 25 M (cid:12) model we showed in Saio et al. (2013). Thiswill serve as a basis for the discussion of the new modelpresented below. The physical ingredients used to computethis model are exactly the same as in Ekstr¨om et al. (2012).Particularly, the Schwarzschild criterion for convection wasused. In addition, an overshoot distance equal to 0 . H P be-yond the Schwarzschild boundary was considered, where H P is the pressure scale height.After the MS, the characteristic evolution timescale be-comes short enough to neglect the rotational mixing in thediscussion . The chemical structure of the star is thus mostlyaffected by the various nuclear-burning episodes (in the cen-tre or in shells), and by the development of convective zones.The surface abundances are also affected by the mass loss.As shown in Fig. 1 (left panel), two phenomena con-tributes to the change of the surface chemical compositionof the surface after the MS: the development of a convec-tive zone appearing when hydrogen burning migrates fromthe centre to a shell (at t = 7 .
96 Myr in the figure), andthe development of an external convective zone when thestar reaches the coldest part of the HRD (log( T eff ) (cid:46) . t = 7 .
975 Myr in the figure). The intermediate convectivezone plays an important role, as it homogenises a region thatwill be later uncovered by the strong mass loss that thesestars encounter during the RSG phase. The total mass ofthe star when it reaches log( T eff ) = 4 . At least at solar metallicity and for the masses considered here.The situations is different, for example, at low metallicity (Meynet& Maeder 2002). chemical gradient progressively bringing the abundances toalmost the ZAMS values (in this rotating model, the rota-tional mixing has slightly changed the surface compositiondue to the diffusion of the chemical species).Just after the MS and before central He ignition, an in-termediate H-burning convective shell appears immediatelyon top of the core (the shaded area in the top-right panelof Fig. 2). The effect of this deep convective zone is the cre-ation of a quite extended region strongly depleted in C andO, and enriched in N, between roughly 10 and 15 M (cid:12) .At the arrival on the RSG branch (bottom left panel),the situation is almost unchanged, except for the appearanceof a quite small external convective zone below the surface,that homogenises the chemical composition. During the laterevolution of the star, the huge mass loss encountered duringthe RSG phase progressively uncovers the layer below thesurface. When the star reaches log( T eff ) = 4 . Trying to reconcile the pulsational properties of our modelsduring the second HRD crossing (that are compatible withthe observations of variable BSGs) with the observed sur-face chemical composition (that are compatible with modelsduring the first crossing), the following options are available: • Changing the mass-loss rates. • Changing the internal mixing processes.Keeping exactly the same physical properties, but vary-ing the mass-loss rates during the RSG phase does not allowto significantly change the surface chemical composition ofBSGs during the second crossing. It changes the time spenton the RSG branch before the second crossing occurs, butthe surface chemical composition when the star reaches theBSG region for the second time is roughly independent ofthe mass-loss rates used. Actually, for the star to evolve bluewards from the RSG stage, a fixed amount of mass has tobe lost. However, the timescale for this loss does not reallymatter. Indeed, what determines the surface composition ofthe BSG in group 2 is the composition of the layers thatare exposed to the surface due to this mass removal and thecomposition of these layers is already fixed (at least in thetwo models shown in Figs 2 and 3) at the entrance into theRSG stage.To change the chemical structure of the star at the firstentrance into the RSG stage, we have to modify the way theelements are mixed in the interior. On the one hand, we canchange the mixing in the radiative zones, using other pre-scription for the rotational mixing (see Meynet et al. 2013).Some tests were performed in that direction without obtain-ing significant changes in the surface abundances of BSGs inthe group 2 with respect to the computation detailed in Sec-tion 2.1. This can be understood since rotational mixing isthe most efficient during the MS (due to the long timescaleavailable), and as the chemical abundances during the sec-ond crossing depends on phenomena that occur after theMS. c (cid:13) , 000–000 he puzzle of the CNO abundances of α Cygni variables L3 Figure 1.
Kippenhahn diagram of a rotating 25 M (cid:12) , from the very end of the MS up to the appearance of the external convective zone.The red line indicates the position of the surface, and the grey zones represents the convective regions. The blue (respectively green) linesindicate the position where H- (respectively He-) burning energy generation rate is maximal (solid line), and 100 erg · s − · g − (dottedlines). The dashed black line indicates the layer that is uncovered (due to mass loss) when the star has log( T eff ) = 4 . Left panel:
Modelcomputed with the Schwarzschild criterion.
Right panel:
Model computed with the Ledoux criterion.
Figure 2.
Chemical structure of the rotating 25 M (cid:12) computedwith the Schwarzschild criterion for convection. The logarithmsof the abundances of H (black), He (grey), C (red), N (green),and O (blue, all in mass fraction) is plotted as a function of theLagrangian mass coordinate (the star’s centre is on the left, thesurface on the right). The shaded areas represent the convectiveregions. Top left panel:
End of the MS; top right panel:
He-coreburning ignition and H-shell burning; bottom left panel:
First ar-rival on the RSG branch; bottom right panel:
Second HRD cross-ing when the star reaches log( T eff ) = 4. On the other hand, we can change the treatment of con-vection in our models. In our recent works (Ekstr¨om et al.2012; Georgy et al. 2013b,a), we used the Schwarzschild cri-terion to determine the convective region. However, anothercriterion commonly used in the literature is the Ledoux one(e.g. Brott et al. 2011). To examine the impact of changingthis criterion in our model, we computed a model strictly
Figure 3.
Same as Fig. 2, but for the model computed with theLedoux criterion. equivalent to the one presented in Section 2.1, only changingthe criterion for convection. In particular, note the following:1) We kept the same overshooting as in the Schwarzschildcase, in order that our rotating model is still able to repro-duce the width of the MS for lower mass stars (see Ekstr¨omet al. 2012). In addition, with the instantaneous mixing ap-proximation for convective mixing done in the Geneva code,this prevents any differences to develop between the convec-tive core sizes computed with the Ledoux and Schwarzschildcriteria. 2) No semi-convective mixing is applied in our mod-els. As semi-convection develops in region that are Ledouxstable, but Schwarzschild unstable, any model accountingfor this effect should behaves between the two set of modelspresented in this paper: a very inefficient semi-convection c (cid:13)000
Same as Fig. 2, but for the model computed with theLedoux criterion. equivalent to the one presented in Section 2.1, only changingthe criterion for convection. In particular, note the following:1) We kept the same overshooting as in the Schwarzschildcase, in order that our rotating model is still able to repro-duce the width of the MS for lower mass stars (see Ekstr¨omet al. 2012). In addition, with the instantaneous mixing ap-proximation for convective mixing done in the Geneva code,this prevents any differences to develop between the convec-tive core sizes computed with the Ledoux and Schwarzschildcriteria. 2) No semi-convective mixing is applied in our mod-els. As semi-convection develops in region that are Ledouxstable, but Schwarzschild unstable, any model accountingfor this effect should behaves between the two set of modelspresented in this paper: a very inefficient semi-convection c (cid:13)000 , 000–000 C. Georgy et al. corresponding to our “Ledoux” model, and a very efficientsemi-convection to our “Schwarzschild” model.Figure 1 (right panel) shows the evolution of the stellarstructure at the very end of the MS, up to the reaching ofthe RSG branch for the “Ledoux” model. Comparing withleft panel, we notice the following trends: • There is no obvious difference between the size of thecore in both models. • In the “Schwarzschild” model, the intermediate convec-tive zone is linked with the development of the H-burningshell. In the “Ledoux” model, the convective zone is inhib-ited in the deepest region due to the strong µ -gradient atthe edge of the core. Since the H-shell burning is not linkedwith a convective zone, it is weaker than the H-burning shellof the “Schwarzschild” model, and disappears quickly. Thismakes the star to cross the HRD towards the RSG branchmuch more rapidly in the “Ledoux” case, associated with adecrease of the luminosity. This allows the appearance of adeep external convective zone.In Fig. 3 we show the internal chemical structure of thestar, at the same evolutionary stages as in Fig. 2. At theend of the MS, the structure between both models are thesame. Divergences in the behaviour appears immediately af-ter central H exhaustion, when the H-burning migrates ina shell. In the “Schwarzschild” case, it is associated with adeep intermediate convective region, producing a quite ex-tended zone in the star strongly affected by the CNO cycle(see Sect. 2.1). Since the external convective zone is rela-tively small, the region in the star where the chemical abun-dances are compatible with the measurements at the surfaceof Deneb and Rigel (see Table 1) is close to the surface (be-tween a Lagrangian mass coordinate M r ∼ −
24, see top-right panel of Fig. 2). In order to evolve toward the blue, thestar has to uncover much deeper layers (deeper than about6 M (cid:12) ), which makes the surface composition incompatiblewith the one observed at the surface of Rigel and Deneb.On the contrary, in the case of the “Ledoux” model, theintermediate convective zone is less deep, keeping the zonewith a strong C and O depletion deeper in the star comparedto the “Schwarzschild” case (top-right panel of Fig. 3). More-over, the development of a deep external convective zone onthe RSG branch (bottom-left panel) will strongly dilute theCNO products. These two factors imply that the BSGs ingroup 2 will present much weaker N/C and N/O ratios ascan be seen in Table 1. Adopting the Ledoux criterion in-stead of the Schwarzschild one strongly decreases the N/Cand N/O ratios for the BSGs in the group 2. Here we didnot try to obtain a perfect match with the observed valuesfor Rigel and Deneb. However, adopting a lower initial ro-tation would likely allow to obtain a still better agreement.This is not the case when the Schwarzschild criterion is usedbecause the structure at the entrance in the RSG phase isfundamentally different whatever the rotation rate. Fig. 4 shows evolutionary tracks (upper panel) and the evo-lution of the total mass (lower panel) for models with theSchwarzschild criterion (red) and the Ledoux criterion (blue)for convection. Evolutionary tracks in the MS are nearly in-
Table 1.
Surface chemical ratio in the “Schwarzschild” and“Ledoux” models of a rotating 25 M (cid:12) when the surfacelog( T eff ) = 4 . / C N / O X He “Schwarzschild” (model) 57 .
86 4 .
17 0 . .
97 1 .
61 0 . . .
46 0 . . .
65 0 . Figure 4.
Top panel:
HRD of the 25 M (cid:12) computed with theLedoux (blue) and the Schwarzschild (red) criteria for convection.The solid line indicates models in which radial pulsations areexcited, while dashed line indicates models for which pulsationstability is examined but no radial modes are excited. Middlepanel:
Logarithm of the pulsation period as a function of theeffective temperature for the model with Ledoux criterion, as wellas the observed period range for the same stars as in top panel.The periods (or period ranges) are shown for some α Cyg variablesin the Milky Way (crosses) and in NGC 300 (pluses) that haveluminosities within the considered range. The references of theobservational data and the names of the stars are given in Saioet al. (2013).
Bottom panel:
Total mass of as a function of T eff . dependent of the convection criterions, while the post-MSmodel with the Ledoux criterion is generally less luminous.Pulsation properties are insensitive to the convectioncriterions as long as the models lie at the same positionin the HRD and have the same mass-luminosity ratio. Inboth cases, during the first crossing after the MS, no radialpulsations are excited in the range 3 . (cid:46) log( T eff ) (cid:46) . α Cygni variables are located, while radialpulsations are excited during the second crossing, due to thelarger
L/M ratio produced by the strong mass loss duringthe RSG phase (see bottom panel of Fig. 4).We compare observed periods with those predicted for25 M (cid:12) models with the Ledoux criterion in the second cross-ing in Fig. 4. Many radial and non-radial pulsations are ex-cited in these models; the excitation mechanisms and theamplitude distributions in the stellar interior are discussedin detail in Saio et al. (2013). c (cid:13) , 000–000 he puzzle of the CNO abundances of α Cygni variables L5 Also shown are observational properties of some α Cygni variables whose luminosities are comparable to theevolutionary track. Most of the α Cygni type variations arequasi periodic, which indicates that several periods are si-multaneously excited. Such period ranges are represented byvertical lines in the middle panel of Fig. 4. The variable A10and D12 in NGC 300 are rare exceptions; Bresolin et al.(2004) obtained regular light curves with periods of 96 . . α Cygni variables are more or less explained bythe pulsations excited in the second crossing models. Onlythe shorter period range of Deneb cannot be explained byour models. This difficulty is also present for the modelswith the Schwarzschild criterion as discussed in Saio et al.(2013).We found in this section that the pulsational proper-ties of the 25 M (cid:12) models with the Ledoux criterion in thesecond crossing are similar to those with the Schwarzschildcriterion, and they largely agree with the properties of α Cygni variables having comparable luminosities. This con-firms that models computed with the Ledoux criterion pro-vides in our case better agreement with the observationsthan the Schwarzschild criterion.
In this paper, we have studied how the choice of the crite-rion for convection changes the behaviour of the post-MSevolution of a massive star, with an emphasis on the surfacechemical composition and the pulsational properties of ourmodels. In Saio et al. (2013), we showed that models of BSGscomputed with the Schwarzschild criterion well reproducedthe pulsational properties of BSGs only after a strong massloss during the RSG phase (group 2). However, our modelsfailed to reproduce the observed surface abundances of suchstars. Using the Ledoux criterion for convection, the agree-ment between our results and the observed surface abun-dances is strongly improved, while the predicted pulsationperiods are still in good agreement with the observed one.The discussion on whether one has to adopt the Ledouxor the Schwarzschild criterion is a long lasting problem instellar evolution and of course we cannot pretend to closethe discussion here by concluding that Ledoux has to bedefinitely preferred. First, as usual, more observational datahave to be collected (here the conclusions are based on onlytwo stars). Second, we cannot be completely sure at the mo-ment that the solution proposed here is unique, in the sensethat some other combinations of overshoot, semi-convectionor treatment of convective mixing may also provide a rea-sonable solution while keeping the Schwarzschild criterion.However, an interesting outcome of the present work is toillustrate through an original problem (i.e. how to make thepulsational and surface abundances properties of BSGs inthe group 2 compatible), the strong impact of the way thelimit of the convective zones are computed. Another way todiscriminate between both criteria is the ratio of red- to blue-supergiants. This ratio is well reproduced by our standardmodels (Ekstr¨om et al. 2013). Using the Ledoux criterionincreases the duration of the RSG stage by a factor of about 2, and changes only marginally the BSG duration. However,this ratio is highly sensitive to the mass-loss prescription,and it is yet not possible, on the basis of the two models pre-sented here, to use it as a strong observational constraint.We are confident that the accumulations of observationalconstraints will narrow the set of possible solutions and willallow to chose the criterion that has to be adopted.
ACKNOWLEDGMENTS
The authors would like to warmly thank Arlette Noelswho suggested us to investigate the impact of using theLedoux criterion instead of the Schwarzschild one, as wellas the anonymous referee for precious comments on thiswork. CG acknowledges support from EU-FP7-ERC-2012-St Grant 306901.
REFERENCES
Bresolin, F., Pietrzy´nski, G., Gieren, W., et al. 2004, ApJ,600, 182Brott, I., de Mink, S. E., Cantiello, M., et al. 2011, A&A,530, A115Ekstr¨om, S., Georgy, C., Eggenberger, P., et al. 2012, A&A,537, A146Ekstr¨om, S., Georgy, C., Meynet, G., Groh, J., & Granada,A. 2013, in EAS Publications Series, Vol. 60, EAS Publi-cations Series, ed. P. Kervella, T. Le Bertre, & G. Perrin,31–41Georgy, C. 2012, A&A, 538, L8Georgy, C., Ekstr¨om, S., Eggenberger, P., et al. 2013a,A&A, 558, A103Georgy, C., Ekstr¨om, S., Granada, A., et al. 2013b, A&A,553, A24Georgy, C., Ekstr¨om, S., Meynet, G., et al. 2012, A&A,542, A29Groh, J. H., Meynet, G., & Ekstr¨om, S. 2013a, A&A, 550,L7Groh, J. H., Meynet, G., Georgy, C., & Ekstr¨om, S. 2013b,A&A, 558, A131Levesque, E. M., Massey, P., Olsen, K. A. G., et al. 2005,ApJ, 628, 973Meynet, G., Ekstr¨om, S., Maeder, A., et al. 2013, in Lec-ture Notes in Physics, Berlin Springer Verlag, Vol. 865,Studying Stellar Rotation and Convection, ed. M. Goupil,K. Belkacem, C. Neiner, F. Ligni`eres, & J. J. Green, 3–642Meynet, G. & Maeder, A. 2002, A&A, 390, 561Przybilla, N., Firnstein, M., Nieva, M. F., Meynet, G., &Maeder, A. 2010, A&A, 517, A38+Saio, H., Georgy, C., & Meynet, G. 2013, MNRAS, 433,1246Smartt, S. J., Eldridge, J. J., Crockett, R. M., & Maund,J. R. 2009, MNRAS, 395, 1409van Loon, J. T., Cioni, M.-R. L., Zijlstra, A. A., & Loup,C. 2005, A&A, 438, 273Vanbeveren, D., De Donder, E., van Bever, J., van Rens-bergen, W., & De Loore, C. 1998a, New A, 3, 443Vanbeveren, D., De Loore, C., & Van Rensbergen, W.1998b, A&A Rev., 9, 63 c (cid:13)000