aa r X i v : . [ a s t r o - ph ] J a n submitted to ApJ Cepheid Mass-loss and the Pulsation – Evolutionary MassDiscrepancy
Stefan C. Keller
RSAA, Australian National University, Canberra A.C.T. 2600, Australia
ABSTRACT
I investigate the discrepancy between the evolution and pulsation masses forCepheid variables. A number of recent works have proposed that non-canonicalmass-loss can account for the mass discrepancy. This mass-loss would be suchthat a 5 M ⊙ star loses approximately 20% of its mass by arriving at the Cepheidinstability strip; a 14 M ⊙ star, none. Such findings would pose a serious challengeto our understanding of mass-loss. I revisit these results in light of the Padovastellar evolutionary models and find evolutionary masses are (17 ± < M/M ⊙ <
14. I find thatmild internal mixing in the main-sequence progenitor of the Cepheid are able toaccount for this mass discrepancy.
Subject headings:
Cepheids:pulsation stellar:evolution
1. Introduction δ Cepheid variables are an essential step in our determination of extragalactic distances.Apart from their use as distance indicators, the regularity of Cepheid pulsation providesa well-defined set of observational parameters with which to probe the course of stellarevolution for intermediate mass stars.Over the preceeding decades much effort has been devoted to reconciling mass determi-nations for Cepheids from the various methods at our disposal (see Cox (1980) for a review).The longest standing of these, the Cepheid pulsation mass discrepancy was first revealedby Stobie (1969) who showed that pulsation masses for bump Cepheids were significantly Bump Cepheids are distinguished by a secondary local maximum of the light curve seen in Cepheidswith periods in the range of 6 to 16 days (Bono et al. 2000b; Hertzsprung 1926). . × K < T < × K bya factor of 2.5 it was possible to remove the discrepancy. More detailed modelling of opac-ities by the OPAL (Rogers & Iglesias 1992) and the Opacity Project (Seaton et al. 1994)confirmed an increase in this temperature range due to metal opacity.The implementation of new opacities largely resolved the bump Cepheid mass discrep-ancy (Moskalik et al. 1992). However, despite convergence, a number of subsequent studieshave shown that the discrepancy remains significant and requires explanation. Studies ofGalactic (Natale et al. 2007; Caputo et al. 2005), LMC (Wood et al. 1997; Keller & Wood2002, 2006; Bono et al. 2002) and SMC (Keller & Wood 2006) Cepheids have shown thatthe masses determined via pulsation modelling are ∼ ∼
15% larger.From an evolutionary perspective, Cepheids are understood to be post-red giant starscrossing the instability strip on so-called blue loops following the initiation of core-He burn-ing. To ascribe an evolutionary mass one takes the Cepheid’s luminosity and, using a mass-luminosity (M-L) relation that is derived from evolutionary models, derives the mass of theCepheid. The M-L relation can be modified substantially by the treatment of internal mixingand mass-loss. Both processes feature complex hydrodynamical and radiative mechanismsfor which we, at present, only possess empirical approximations.The treatment of internal mixing modifies the size of the helium core established duringthe star’s main-sequence (MS) evolution. Overshoot at the edge of the convective core of theCepheid progenitor mixes additional hydrogen into the core and hence increases the heliumcore mass. As a consequence the post-MS evolution occurs at a higher luminosity. Massloss, in an ad-hoc manner at least, offers a mechanism to modify the M-L relation by directlyreducing the mass of a Cepheid.The properties of pulsation, on the other hand, are dependent on the structure of theatmosphere of the Cepheid. Keller & Wood (2006) show that the morphology of a bumpCepheid light curve is highly sensitive to the mass, luminosity, effective temperature andmetallicity. Hence modelling of the light curve can be used to determine a pulsation Cepheidmass that is entirely independent of stellar evolution calculations.In the work of Bono et al. (2002) and Caputo et al. (2005) it is proposed that mass- 3 –loss can account for the mass discrepancy between pulsation and evolutionary masses.Caputo et al. (2005) also conclude that models that incorporate additional internal mix-ing in the vicinity of the convective core are not able to explain the mass discrepancy. Inthis paper we revisit these conclusions and present a scenario for resolution of the massdiscrepancy.
2. The Cepheid Pulsation – Evolutionary Mass Discrepancy
A useful way of expressing the mass discrepancy is to form the quantity ∆
M/M E inwhich ∆ M is the difference between the pulsation mass, M P , and the canonical (i.e. notincluding CCO) evolutionary mass, M E , as shown in Figure 1. In Figure 1 we also show theeffects of the inclusion of mild (Λ c =0.5) and moderate (Λ c =1.0) CCO. The effect of CCO isto raise the luminosity of the Cepheid of a given mass as discussed above.Caputo et al. (2005) utilise the BV IJ K absolute magnitudes derived by Storm et al.(2004) using distances from the near-infrared surface brightness (IRSB) method. Caputo et al.(2005) have used the mass dependent period-luminosity-color relation to determine the pulsa-tional Cepheid mass using the M-L relation of Bono et al. (2000a). They find that ∆
M/M E ranges from 20% at M ∼ M ⊙ to ∼
0% at M ∼ M ⊙ (see Figure 1). Such a trend can not beexplained by a uniform level of core overshoot. Monte-Carlo simulation using the individualquoted uncertainties of the data shown in Figure 1 reveals a gradient of (3 . ± . M − ⊙ .A caveat to IRSB analysis that underlies the results of Storm et al. (2004) and Caputo et al.(2005) is the necessary introduction of the poorly understood projection factor, p , that em-bodies the effects of limb-darkening (Nardetto et al. 2006), and is dependent on the pulsationvelocity and the species under study (Nardetto et al. 2007). p is approximated as a functionof period, and it has been speculated that this could introduce a period dependency in thederived distances. However, the work of Fouque et al. (2007) shows that direct HST paral-laxes for a sample of Cepheids agree within uncertainties with the IRSB measures albeit foronly the five stars with distance determinations from both techniques.To explain the mass discrepancy found at lower masses, Caputo et al. (2005), proposemass-loss from the Cepheid progenitor before, or during the central He-burning phase. Theimplied total mass-loss declines from ∼
20% at M ∼ M ⊙ and vanishes by M ∼ M ⊙ . Suchmass-loss is seemingly at odds with empirical estimates of mass-loss rates which show thatmass-loss increases with stellar luminosity and radius (Reimers 1975; de Jager et al. 1988;Schroeder & Cuntz 2007). Furthermore, Caputo et al. conclude that CCO can not accountfor this trend in ∆ M/M E with mass since this would lead to unphysical ∆ M/M E, Λ < M/M E, Λ is the mass at a given Λ c ) for higher mass Cepheids.The Cepheid M-L relation implemented in analysis is critical. In their study, Caputo et al.(2005) have chosen to utilise a linear M-L relation derived from Bono et al. (2000a) evolu-tionary models ( Z =0.02, Y =0.28 and Λ c =0) that do not incorporate mass loss. This M-Lrelation is shown as the dashed line in Figure 2. However, as shown by the evolutionarymodels of Bono et al. (2000a) there are significant departures from this linear relationshipfor M ≥ M ⊙ (see the dotted line in Figure 2).In the analysis to follow we incorporate the non-linear nature of the Cepheid M-L intoour analysis. The models of (Bono et al. 2000a, Z =0.02, Y =0.27 and Λ c = 0) extend to M =12 M ⊙ , however the derived pulsation masses reach to 14.9 M ⊙ . Let us consider thestellar evolutionary sequences of Bressan et al. (1993) that extend to higher masses. Thesemodels ( Z =0.02, Y =0.28 and Λ c =0) implement mass-loss according to the Reimers (1975)and de Jager et al. (1988) formulation and are shown in Figure 2 as the solid line. Formasses less than 8 M ⊙ there is clearly a different gradient between the Bono et al. (2000a) andBressan et al. (1993) M-L, a feature discussed by Beaulieu et al. (2001). What is commonto both models is a marked departure from a linear M-L in the mass range 9-10 M ⊙ . Thisdeparture results in evolutionary masses that are greater for a given luminosity comparedwith those derived from the linear M-L for masses greater than 9-10 M ⊙ .The use of the Bressan et al. (1993) M-L results in Figure 3. Figure 3 reveals no sig-nificant trend in ∆ M/M E as a function of M E (Monte-Carlo simulation of the data andassociated uncertainties shown in Figure 3 reveals a gradient of (0 . ± . M − ⊙ ). Thisdemonstrates that a consideration of the non-linear Cepheid M-L accounts for the decrease in∆ M/M E seen in the work of Caputo et al. The absence of a significant gradient also negatesthe argument by Caputo et al. (2005) that CCO can not be the source of the Cepheid massdiscrepancy.Reanalysis of the results of Caputo et al. (2005) shows that the evolutionary Cepheidmass is (17 ±
3. The Source of the Cepheid Pulsation – Evolutionary Mass Discrepancy
The hiding place of the source of discrepancy between pulsation and evolutionaryCepheid masses has shrunk dramatically over the last three decades. To account for thediscrepancy we have three key options. Firstly, it is possible that some input physics in pul-sation calculations are not sufficiently described. In this regard, the only input that could 5 –affect pulsation to this magnitude is the description of radiative opacity. Second, that theCepheid mass is smaller than their main-sequence progenitors due to mass-loss. And finally,that evolutionary calculations underestimate the mass of the He core in intermediate massstars and so underestimate their luminosity. We now discuss each of these possible sourcesin detail.
The pulsation properties of Cepheids are critically dependent on the so-called Z -bumpopacity arising from the dense spectrum of transitions originating from highly ionized Fe.The inclusion of these transitions in the works of OPAL (Rogers & Iglesias 1992) and OP(Seaton et al. 1994) resulted in a substantial increase in opacity at log T ≈ .
2. The OpacityProject (Badnell et al. 2005) has included further details of atomic structure (in particular,the treatment of atomic inner shell processes) in their calculation of opacity. The newopacities do show an increase over the 1992 OP and OPAL values of opacity in the Z -bump,but at a level of only 5-10% (Badnell et al. 2005). To account for the mass discrepancy theopacity would need to be raised by 40-50%, equivalent to the increase between the early LosAlamos opacities (Cox & Tabor 1976) and OP and OPAL opacities. Hence the uncertaintyin radiative opacity is an unlikely resolution to the Cepheid mass discrepancy. The studies of Bono et al. (2002, 2006) and Caputo et al. (2005) propose that mass-losscan account for the mass discrepancy. Candidate mass-loss phases include the red giantbranch phase, subsequent blue loop evolution, or possibly from the action of pulsation itself(Bono et al. 2006). The removal of mass from the Cepheid is a straightforward, albeit ad-hoc, way to bring the evolutionary mass in line with that derived from pulsation. The timingof, and the changes in stellar structure brought about by significant mass-loss are importantto the net change in the Cepheid M-L. For instance, significant mass-loss on the MS causes areduction in overall mass and hence helium core mass, resulting in a reduction in luminosityin the instability strip (de Loore 1988). Enhanced mass-loss on the red giant branch reducesenvelop mass, and the material available to the hydrogen-burning shell within the Cepheid,again leading to a reduction in luminosity relative to a star without mass-loss (Yong et al.2000). One of the difficulties with the proposal for mass-loss to solve the Cepheid massdiscrepancy is that it would require the rather artificial bulk removal of material withoutconsequent modifications to stellar structure and energy production. 6 –Furthermore, standard mass-loss can account for at most a few percent reduction inCepheid mass and not the 15%-20% required. Mass loss is usually treated the semi-empiricalrelation of Reimers (1975, ‘Reimers’ law’). While not providing any physical reasoning onwhy the mass-loss is generated, ‘Reimers’ law’ provides an adequate match to observedmass-loss rates over a broad range of stellar parameters (Schroeder & Cuntz 2007).Major mass-loss is expected during the red giant branch (RGB) evolution. The modelsof Girardi et al. (2000) and Bressan et al. (1993) use a parameterised, empirical fit dM/dt = − × − ηL/gR (Reimers 1975). The value of η is set by a consideration of the massesof stars on the horizontal branch (HB) of globular clusters. Determination of η is madedifficult due to the variety of HB morphology exhibited by globular clusters. If we considerthe distribution of effective temperatures for HB stars is entirely due to variable mass-lossthen η must range from 0 to somewhat more than 0.4. Using the canonical value of η = 0 . M ⊙ star looses ∼ M ⊙ during the RGB phase. To accomodate the mass discrepancyseen in Figure 1 this would have to be increased to 0.8 M ⊙ corresponding to a 20-30 foldincrease in η which is not plausible. Therefore mass-loss on the RGB does not resolve themass discrepancy.Caputo et al. (2005) and Bono et al. (2006) suggest that pulsation may give rise toan enhancement of mass-loss. Attempts to measure the mass-loss rates for Cepheids havebeen made using IRAS infrared excesses (McAlary & Welch 1986) and IUE UV line profiles(Deasy 1988) and in the radio (Welch & Duric 1988). Deasy (1988) found mass-loss rates forthe majority of Cepheids of the order of a few × − M ⊙ yr − . The study of Welch & Duric(1988) places upper limits on the mass-loss rate of < − M ⊙ yr − . Welch & Duric (1988)and Deasy (1988) conclude that mass-loss during the Cepheid phase is insufficent to explainthe observed mass discrepancy. In the case of a 5 M ⊙ Cepheid, mass loss can account forthe mass discrepancy if either the lifetime in the IS is ten times longer than derived byBono et al. (2000a) or mass loss is thirty times greater than found by Deasy (1988). Massloss as an explanation for the mass discrepancy in a 12 M ⊙ Cepheid is more challenging. Themass loss found by Deasy over the entire lifetime of the star can not account for the massdiscrepancy and a 600-fold increase in the mass loss rate would be required.Recently, M´erand et al. (2007) found using near-infrared interferometry that, from asample of four Cepheids all show the presence of some circumstellar material. α Persei, a non-variable supergiant residing in the instability strip, does not show evidence for circumstellarmaterial, suggesting that pulsation does have a role in enhancing mass-loss. The conversionfrom historical and ongoing mass loss that lead to the circumstellar material and the rateof mass loss is beyond the scope of the study of M´erand et al. (2007) however. On thecurrent evidence I conclude that mass loss does not present a solution to the Cepheid mass 7 –discrepancy.
Cepheid luminosity is critically dependent on the He core mass. The mass of the He coreis determined by the extent of the convective core during core H burning. Classical modelsdefine the limit to convection via the Schwarzschild criterion. This places the boundary toconvection at the radius at which the buoyant force acting on a hot clump of material risingfrom the convective core drops to zero. However, the temperature and density regime in thevicinity of the convective boundary of the main sequence Cepheid progenitor are such thatrestorative forces in the region formally stable to convection are mild, and some significantlevel of overshoot of the classical boundary is expected (Zahn 1991; Deng & Xiong 2007).The description of convection is the weakest point in our understanding of the physicsof intermediate mass – massive stars. Numerical modeling of core convection requires adescription of the turbulence field at all scales. Three-dimensional hydrodynamical calcu-lations capable of adequate resolution have only recently become a possibility and are intheir infancy (Meakin & Arnett 2007; Dearborn et al. 2006; Eggleton et al. 2003, 2007). Inthe absence of a general theory of CCO, which would enable us to calculate the amount ofcore overshoot for a star of a given mass and chemical composition, a semiempirical phe-nomenological approach must be used in calculations of stellar evolution. Several computa-tional schemes of various degrees of sophistication in treating the physics of overshoot havebeen discussed in the literature (Maeder & Meynet 1988; Bertelli et al. 1990; Girardi et al.2000; Demarque et al. 2007; Straka et al. 2005). The most common parameterization of coreovershoot utilizes the mixing-length formulation, where gas packets progress a distance ofΛ c pressure scale heights into the classically stable region. Λ c offers a convenient way toparametrize the extension of the convective core it does not constrain its physical origin. Inaddition to the CCO mechanism outlined above, rotationally induced mixing can similarlybe invoked to bring about a similar range of internal mixing. As shown by Heger & Langer(2000) and Meynet & Maeder (2000) mixing in the sheer layer formed at the interface be-tween the convective and radiative regions can lead to larger He core masses for massivestars.At present we must rely on observation for constraint of Λ c . Observational determi- We quantify core overshoot using the formalism of Bressan et al. (1981). Note that Λ c is a factor of2.0 times the overshoot parameter d over /H p in the formalism of the Geneva group (Schaller et al. 1992;Demarque et al. 2004) M ⊙ ), where thesignature of CCO is expected to be most clearly seen. The studies of Mermilliod & Maeder(1986) and Chiosi et al. (1992) of Galactic open clusters converge on the necessity for mildcore overshoot; Λ c ≈ .
5. A number of subsequent studies have presented evidence both for(Barmina et al. 2002) and against (Testa et al. 1999) core overshoot. From studies of theyoung populous cluster systems of the Magellanic Clouds, Keller et al. (2001) found evidencefor Λ c = 0 . ± .
11, while Cordier et al. (2002) examined the Magellanic Cloud field pop-ulation and found the necessity for a level of overshoot between 0.2 and 0.8 pressure scaleheights. Measurements with the lowest associated uncertainties are derived from the pulsa-tion modelling of Cepheid light curves by Keller & Wood (2002, 2006); Bono et al. (2002).Keller & Wood (2006) finds evidence for a weak dependence of Λ c with metallicity with Λ c rising from 0.688 ± ± ± . c = 0 . ± .
17. This degree of overshoot is within the range of previous studies discussedabove. The scenario I have outlined offers a straightforward explanation for the findings ofCaputo et al. (2005), one that uses canonical mass-loss and mild convective core overshoot.
4. Conclusions
In this paper I revisit the conclusions of Caputo et al. (2005) regarding the cause of theobserved discrepancy between the evolution and pulsation masses for Cepheids. Caputo et al.(2005) find that Cepheids of 5 M ⊙ have 20% less mass, as determined from pulsation analysis,than expected from evolutionary calculations, while the evolution and pulsation masses forCepheids of 14 M ⊙ agree. In order to explain this finding Caputo et al. propose a scenarioof non-standard mass-loss:- one that sees increased mass-loss at lower masses and drops tonegligible mass-loss at M ≥ M ⊙ . This scheme of mass-loss would be counter to the obser-vationally grounded evidence that accumulated mass-loss by the epoch of core-He burningincreases with increasing stellar mass. Furthermore, Caputo et al. claim that convectivecore overshoot is unable to provide a solution as it can not account for the trend of massdiscrepancy with Cepheid mass.The findings of Caputo et al. are based on a Cepheid mass-luminosity relationship thatproves erroneous when extrapolated to higher masses. In this paper I show that including afull description of the mass-luminosity relation results in a mass discrepancy in which evolu- 9 –tionary masses are (17 ± c = 0 . ± .
17, a value which agrees with previousdeterminations from a range of techniques.I thank A. Bressan et al. for providing us with unpublished evolutionary models forΛ c = 1 .
0. I would also like to thank Peter Wood for discussions during the preparation ofthis paper and Guiseppe Bono for his comments on a draft of this paper.
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This preprint was prepared with the AAS L A TEX macros v5.2.
13 –Fig. 1.— Here we show the mass discrepancy as a function of mass. ∆
M/M E expresses themass discrepancy and is equivalent to the difference between the pulsation mass, M P , andclassical evolutionary mass, M E (i.e. M E does not incorporate convective core overshoot),normalised by M E . Here the extrapolated mass-luminosity relation of Bono et al. (2000a)was used (the dashed line in Figure 2, see text for details). Note the mass discrepancyvanishes at higher masses. Overlaid are the locii of models that incorporate mild (Λ = 0 . .
0; Bressan (2001)) convective core overshoot. 14 –Fig. 2.— The Cepheid mass-luminosity relation used by Caputo et al. (2005) consists of thedashed line (from Caputo et al. Equation 2). The models of Bono et al. (2000a, Z =0.02, Y =0.27 and Λ c = 0) without mass-loss are shown by the dotted line. The Bressan et al.(1993) mass-luminosity relationship for Cepheids is shown as the solid line. Note the signif-icant departures from the linear relation of relation of Caputo et al. 15 –Fig. 3.— As in Figure 1 except here the mass-luminosity relation is due to Bressan et al.(1993). The mass discrepancy shows no significant dependence on mass, but rather a uniformoffset of (17 ±±