Probing high-mass stellar evolutionary models with binary stars

New windows on massive stars: asteroseismology, interferometry, and spectropolarimetry c 2014 International Astronomical Union Proceedings IAU Symposium No. 307, 2014

G. Meynet, C. Georgy, J.H. Groh & Ph. Stee, eds. DOI: 00.0000/X000000000000000X

Probing high-mass stellar evolutionary models with binary stars A. Tkachenko1 †

arXiv:1408.0949v1 [astro-ph.SR] 5 Aug 2014

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Instituut voor Sterrenkunde, KU Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium email: [email protected]

Abstract. Mass discrepancy is one of the problems that is pending a solution in (massive) binary star research field. The problem is often solved by introducing an additional near core mixing into evolutionary models, which brings theoretical masses of individual stellar components into an agreement with the dynamical ones. In the present study, we perform a detailed analysis of two massive binary systems, V380 Cyg and σ Sco, to provide an independent, asteroseismic measurement of the overshoot parameter, and to test state-of-the-art stellar evolution models. Keywords. asteroseismology, star: oscillations (including pulsations), line: profiles, methods: data analysis, techniques: photometric, techniques: spectroscopic, (stars:) binaries: spectroscopic, stars: fundamental parameters, stars: individual (V380 Cyg, σ Sco)

1. Introduction One of the major problems that is currently pending a solution in binary star research field is the so-called mass discrepancy problem. It stands for the difference between the component masses inferred from binary dynamics (hereafter, dynamical masses) and those obtained from spectral characteristics of stars and evolutionary models (hereafter, theoretical masses). The mass discrepancy problem observed in massive O- and B-type stars has been known for more than 20 years already and has been discussed in detail by Herrero et al. (1992). Hilditch (2004) pointed out that the discrepancy does not disappear when the effects of rotation are included into the models. A remarkable mass discrepancy has been reported by Guinan et al. (2000) for the primary components of the V380 Cyg system. The authors showed that large amount of core overshoot (αov = 0.6 pressure scale height) can account for the difference between dynamical and theoretical mass of the primary component. This large amount of overshoot contradicts the typical value of αov 0.35 is required to reproduce observed absolute dimensions of both components of the V578 Mon system. Thus, there seems to be a tendency of measuring larger core overshoot in binary stars than in single objects, within the same stellar mass range. The reason is that this parameter often effectively accounts for the above mentioned mass discrepancy, but we need to investigate the feasibility of using core overshoot alone to explain such a complex problem as discrepancy between dynamical and theoretical masses in binary stars. In this paper, we present a study of two massive binary star systems, V380 Cyg and σ Sco, and aim at measuring core overshoot parameter for pulsating components using asteroseismic methods. † Postdoctoral Fellow of the Fund for Scientific Research (FWO), Flanders, Belgium

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A. Tkachenko Table 1. Key orbital, physical, and atmospheric parameters of the V380 Cyg and σ Sco systems, as derived from our photometric and/or spectroscopic data. Parameter Period, Periastron passage time, Periastron long., eccentricity, RV semi-amplitude, Mass, Radius, Teff , log g, v sin i,

V380 Cyg Primary Secondary

σ Sco Primary Secondary

(day) (HJD) (degree)

12.425719 33.016±0.012 24 54 602.888±0.007 24 34 886.11±0.04 138.4±0.4 288.1±0.8 0.2261±0.0004 0.383±0.008 (km s−1 ) 93.54±0.07 152.71±0.22 30.14±0.35 47.01±0.98 (M ) 11.43±0.19 7.00±0.14 14.7±4.5 9.5±2.9 (R ) 15.71±0.13 3.819±0.048 9.2±1.9 4.2±1.0 (K) 21 700±300 22 700±1 200 25 200±1 500 25 000±2 400 (dex) 3.104±0.006 4.120±0.011 3.68±0.15 4.16±0.15 (km s−1 ) 98±2 38±2 31.5±4.5 43.0±4.5

2. V380 Cyg V380 Cyg is a bright (V = 5.68) double-lined spectroscopic binary (SB2, Hill & Batten 1984) consisting of two B-type stars residing in an eccentric 12.4 day orbit. The primary component is an evolved star, whereas the secondary just started its life on the mainsequence. Pavlovski et al. (2009) revisited the U, B, V light curves obtained by Guinan et al. (2000) and collected about 150 high-resolution ´echelle spectra using several telescopes. The authors presented a revised orbital solution, and used spectral disentangling technique (Simon & Sturm 1994) formulated in Fourier space (Hadrava 1995), as implemented in the fdbinary code (Iliji´c et al. 2004), to determine disentangled spectra of both binary components. Similar to the results of Guinan et al. (2000), a remarkable mass discrepancy was found for the primary component of V380 Cyg. Moreover, Pavlovski et al. (2009) came to the same conclusion as Hilditch (2004) did, namely that the effects of rotation included into evolutionary models could not fully account for the observed discrepancy. The discovery of seismic signal in the primary component of V380 Cyg (Tkachenko et al. 2012) opened up an opportunity of obtaining an independent measurement of the overshoot parameter for this binary system. We base our analysis on about 560 days long time-series of high-precision Kepler photometry, and about 400 high-resolution, high signal-to-noise ratio (S/N) spectra obtained with hermes spectrograph (Raskin et al. 2011) attached to 1.2-meter Mercator telescope (La Palma, Canary Islands). The effects of binarity in the Kepler light curve were modelled using the jktwd wrapper (Southworth et al. 2011) of the 2004 version of the Wilson-Devinney code (Wilson & Devinney 1971; Wilson 1979). The time-series of spectra were analysed with the fdbinary code to determine spectroscopic orbital solution and disentangled spectra of both stellar components. We found that light curve yields poor constraints on the shape of the orbit, because of strong correlation between eccentricity e and longitude of periastron ω. Since the quantities e cos ω and e sin ω are respectively well constrained from photometry and spectroscopy, we constrained the orbital shape by iterating between the two analyses: the light curve was used to determine best fit ω for a given e, and the spectral disentangling to find the best e for a given ω. Analysis of the disentangled spectra delivered accurate atmospheric parameters and individual abundances for both binary components. Table 1 lists some key orbital, physical, and atmospheric properties of this binary system; for more details, reader is referred to Tkachenko et al. (2014a). Photometric residuals obtained after the subtraction of our best fit model were sub-

log (Surface gravity [cm s−2 ]) [dex]

Cyg: orbital solution intrinsic variability of the primary3 Testing V380 evolutionary models withand binary stars 12 3.0

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Table 9. Evolutionary model parameters for both components of V380 Cyg.

A. Tkachenko et al.

Par

Unit

Primary Secondary Model1 Model2 Model3 Model1 Model2 Model3 one previously found by Cugier & Boratyn (1992). Th

tification for f5 presented in this study is done for the fi addition of the seismic constraints to the modelling im tic reduction in the uncertainties of the fundamenta 3.5 and provided an age estimate. Our best fitting evolut suggests a small decrease of 7.3×10−5 d−1 /century of t of the dominant radial mode; the effect cannot be dete data due to low frequency resolution. 4.0 This paper is the second one in a series devoted papers, we plan to take a closer look at another massive binary system, Spica, with the aim to see whether the above mentioned mass ysis of spectroscopic double-lined binary systems, discrepancy holds for this star too and whether asteroseismology two B-type stars of which at least one is a pulsating a fully consistent solution and agreement with stella can help us to constrain αov parameter in that case. 4.45 4.15 4.40 4.35 4.30 4.25 4.20 the primary component, after having taken into acco log (Effective temperature [K]) [dex] tional behaviour in the analysis of the data. This co in sharp contrast with the incompatibility between d Figure 8. Position of the primary (circle) Cyg and thein secondary (triangle) Figure 1. Left: Location of the primary and secondary components of V380 the Teff -log gof ACKNOWLEDGEMENTS Figure 14. Location of the primary and secondary components of V380 Cyg σ Sco system in the T eff -log g diagram, along with the  evolutionary els for the binary V380 Cyg, for which we found a di diagram. Solid, longand short-dashed tracks correspond to models 1, 2 and 3 in Table 2, in the T eff -log g diagram. Solid, long- and short-dashed tracks correspond tracks. The initial masses as well as the overshoot parameter fov are inThe research leading to these of results received funding from the Eu- ∼1.5 M between the dynamical and theoretical mas respectively. T and log g values are those from Table 1. Right: Position the primary (circle) to model 1, 2, and 3 ineff Table 9, respectively. dicated in the plot. The atmospheric parameters T eff and log g of the priropean Research Council under the European Community’s Sev- core overshoot was needed to explain the properties o and the secondary (triangle) of the σ Sco systemmary in and the Teff -log diagram, along and with the mesa secondary aregthose from asteroseismic spectroscopic analenth Framework Programme (FP7/2007–2013)/ERC grant agree- On the other hand, the fact that V380 Cyg is an eclip ysis, respectively. The isochrones corresponding to theindicated age of the system evolutionary tracks. The initial masses as well as the overshoot parameter fThis in ◦ ov are lined spectroscopic binary allowed us to measure the ment n 227224 (PROSPERITY). research is (partially) 12.1 Myr deduced from seismology of the primary are indicated by thefunded the plot. Teff andof log g are those from Table 3. The isochrones dividual component to precisions approaching 1%, wh analysis. The The results atmospheric of the photometricparameters and spectroscopic analyses by the Research Council of the KU Leuven under grant dashed lines. The error range in age is given by the dotted lines, which cor-agreeare consistent with each the of sense thatsystem the variability occursMyr corresponding to other the in age the of 12.1 and barsThe deduced from seismology ment GOA/2013/012. research leading these results has re- in masses of both components of the σ Sco system respond to theits besterror fit overshoot parameter of the primaryto fov = 0.0. at the orbital frequency andindicated its harmonicsby as well at the rotation ceived funding from the European Community’s Seventh Frame- ∼30% if we do not consider the seismic properties of of the primary are the asdashed lines. frequency of the primary. All other variability detected in the Kework Programme FP7-SPACE-2011-1, project number 312844 In this respect, the two systems are incomparable and sured dynamical masses of the σ Sco stellar componen pler data is of stochastic nature. We find that the characteristics of compute (SPACEINN). individual masses and radii of both components of σ Sco. Table 2. are Evolutionary model for both components of thethis V380 Cygwithin system. αov from drawing any firm conclusions with respect to th this variability not in contradiction withparameters the recent theoretical The values of the radii obtained way agree the estimated models. In our next paper, we will present a detailed a predictions and by Cantiello et al.for (2009) Shiode et al. (2013) for errors υ stand theand overshoot parameter andwith initial rate, respectively. those rotation computed from angular diameters presented by Spica system, based on the space-based photometric the g-mode oscillations excited both in the core and in a thin con- North et al. (2007). In order to verify the mass and radius of the priREFERENCESSecondary Parameter Primary mary component, based high-resolution spectroscopic data. vective sub-surface layer of massive stars. we performed the fitting of all three independent Model 1 the Model 2 Model 3 modes, Model 1  Model 2 –Model 3 A&A, Aerts, C., using de Pauw, M., & Waelkens, C.evolution 1992, We also explored the hypothesis of observing rotationally pulsation and stellar and 266, oscil-294 modulated signal in our data in more detail. We used the Doppler lation codes, Aerts,respectively. C., Thoul, A., ska, J.,theoretical Scuflaire,model R., Waelkens, OurDaszy´ best nfitting deliv- C., M,to study (M ) 11.43 12.00spectral12.90 7.00 7.42 Dupret, M. A., Niemczura, E.,7.42 & Noels,inA.good 2003, Science, 300, Imaging technique variability detected in several ers an effective temperature and surface gravity agreement ACKNOWLEDGEMENTS lines of doubly Z, ionized (dex) silicon. Two solutions provide nearly 0.014 that 0.012 0.012 0.014 values, 0.012and confirms 0.012the radius of the priwith the 1926 spectroscopic the same qualityαof fit but spot configurations Aerts, C., from Briquet, M., Degroote, Thoul, A.,the &system van Hoolst, T. The research leading to these results has received f , (H 0.2 0.6 on the0.3 0.2 0.2P.,The mary determined our0.6 orbital solution. age of is ovthe p ) different stellar surface have obtained. 534,0Table A98 10 summarizes the European Community’s Seventh Framework Prog estimated2011, to0beA&A, ∼12 Myr. the final fundamenυ, been(km s−1 )We0 favour the one 241that shows243 0 a lower abundance gradient and assumes two high- and one low- tal parameters Asplund, N.,ofSauval, A.J., & P., 2009, SPACE-2011-1, project number 312844 (SPACEINN of M., both Grevesse, components σ Sco, where theScott, mass and Age, (Myr) – 21.5 18 – 21.5 18 contrast spot located at (nearly) the same latitude but at different radius ofARA&A, 47, 481 the Fund for Scientific Research of Flanders (FWO), the secondary were computed from the theoretical mass of longitudes. Auvergne, M., Bodin, P.,mass Boisnard, al. 2009, A&A, 506, 411 der grant agreement G.0B69.13. Mode identification the primary and spectroscopic ratio, L., andetfrom the calculated Bagnulo, S., Landstreet, J. D., Fossati, L.,errors & Kochukhov, We used the MESA stellar evolution code to look at the mass and spectroscopic log g, respectively. The 1σ given in O., the software package  developed in the framewo A&A, 538, A129 model predictions for both components of V380 Cyg.majority For the sec- Tables European Coordination Action HELAS (http://www.h jected for frequency analysis. The of the frequencies are variable both in uncerap92012, and 10 do not take into consideration systematic A. H. 1962, Publications of the Dominion Astrophysical ondary, we found a discrepancy of ∼0.85 M between the dynam- tainties Batten, connected with themade choice of a given stellar evolution code pearance and amplitude, in agreement with the conclusions by Tkachenko et al. Observatory Victoria, 12, 91 ical and theoretically predicted masses assuming standard stellar and its input physics. Fig. 8 shows the position of both components (2012) about nature ofvalue theofsignal. variability consistent with the expected Bessell, M. S.,inCastelli, F., &g Plez, B. 1998, 333,evo231 evolutionary physics.stochastic However, a slightly larger the mass The of the σ Sco system the T eff -log diagram, alongA&A, with the REFERENCES Blomme, R., Mahy, L., top Catala, et al. binarity 2011, A&A, 533, A4 (within 3-σ from the dynamical and a component bulk metallicity of lutionary tracks. The erroron bars are those obtained from the and evolurotation period of the value) primary has been detected ofC.,the J., & Nordlund, Å.uncertainties 2006, A&A,for 450, 1077 Z = 0.012 dex can account for this difference. Two models that fit tionary Braithwaite, Aerts, C., de Pauw, M., & Waelkens, C. 1992, A&A models and 3σ spectroscopic the primary stochastic oscillation signals in the star. We speculate that this signal comes from271, ro-482 Breger, M., Stich, J., Garrido, R., et is al.an 1993, A&A, the position of the primary in the T eff -log g diagram were found and secondary, Aerts, C., & De Cat, P. 2003, SSRv, 105, 453 respectively. The primary evolved star near tational of spot-like abundance or structures on the and provide modulation significantly different age estimates chemical for the system: M.,temperature Morel, T.,whereas Thoul, A., Scuflaire, A., the end Briquet, of its main-sequence, the secondaryR., justMiglio, started its Mon- Aerts, C., Christensen-Dalsgaard, J., & Kurtz, D. W. 21.5±1.5 and (cf. Table component. 9). In both cases, the ap- main-sequence talb´an, evolution. J., Dupret,spectral M.-A., & Aerts, C.profiles 2007, MNRAS, 381, The position of the secondary in dia- 1482 oseismology, Springer, Heidelberg surface of18±1 theMyr primary Tosystem verify this hypothesis, line ofthethe pears to be much younger than 25.5±1.5 Myr, the age determined gram suggests Briquet,that M.,the Aerts, Baglin, A., et than al. 2011, A&A, starC., is more massive 8.7 M massA112 Asplund, M., Grevesse, N., Sauval, A.J., & Sco , the527, primary has been for spot signatures, after subtracting contribution of the by G2000 assuming αov =examined 0.6 for the primary component. Since we obtained Briquet, M., Neiner, C.,the Leroy, B., theoretical & P´apics, P. I. 2013, from binary dynamics and the mass of theA&A, ARA&A, 47, 481 companion star from composite spectra of V380 Cyg. We found a for remarkable variboth models assume extreme inputthe physics like a very high rotation primary. 557, L16discrepancy Beeckmans, F., & Burger, M. 1977, A&A, 61, 815 Similar was found the main-sequence secat the ZAMS a large amount of overshooting, we more concludemassive A. S., of Browning, K.,binary & Toomre, J. could 2005, ApJ, 629, ondary Brun, component the V380M. Cyg system (Tkachenko et 461 Breger, M., Stich, J., Garrido, R., et al. 1993, A&A, ability in and/or all observed silicon lines of binary component, which not that single star evolutionary models are not suitable for this partic- al. 2014). Bryson, S. T., Tenenbaum, P., Jenkins, J. M., et al. 2010, ApJ Let- Butler, K. 1984, PhD thesis, University of London, U be detected was found to be consistent ular binary system.in spectral lines of other chemical elements 713, and L97 analysis Cugier, H., & Boratyn, D. A. 1992, AcA, 42, 191 Ourters, asteroseismic of the primary of σ Sco delivered Butler, K. 1984, PhD thesis, University ofThis London, variability detected in the Kepler intrinsicinvers8 to the the identification withThethe rotational period of data theis star. (Piskunov Rise code was used Debosscher, J., Aerts, C., Tkachenko, A., et al. 2013 of all& three fitted1993) frequencies. way,UK two of Brott, I., et al. 2009, A&A, 499, primary component and of stochastic nature. The signal is variable them, fCantiello, A56 , M., are Langer, found toN., be fundamental and second over-279 1 and f5prominent to perform Doppler Imaging analysis based on several lines of doubly ionized Cantiello, Braithwaite, Brandenburg, Del Sordo, both in amplitude and in appearance on a short time-scale. This tone radial modes,M., respectively, andJ.,frequency f3 is A., identified as F.; Evans, D. S., McWilliam, A., Sandmann, W. H., silicon spectrum of the parameter primaryan component. The obtained suggest apyl¨a, P., &mode. Langer, N. is 2011, IAUresults Symposium, 272, 32 makes the found tuning ofin thethe convective core overshooting 1986, AJ, 92, 1210 l = 1K¨ non-radial This in perfect agreement with the Charbonneau, P. 1995, which ApJS, 309located αov from asteroseismic analysis impossible for this star. In surface future spectroscopic Fitch, W. S., 1967, ApJ, 148, 481 mode identification we101, obtained for f3 andeither f5 using the presence of two high-contrast stellar abundance spots are Giddings, J. R. 1981, PhD thesis, University of Lond the moment method. Identification for f1 is consistent with the maat the same latitude or longitude. c 2013 RAS, MNRAS 000, 1–18

Goossens, M., Lampens, P., De Maerschalck, D., & S jority of the previous studies (e.g., Cugier & Boratyn 1992; HeynFinally, we compare our revised fundamental parameters of both 1984, A&A, 140, 223 derickxstellar 1994) whereas the identification of f3 iscomponents different from the M Z αov v Age

M dex Hp km s−1 Myr

11.43 0.014 0.2 0 —

12.00 0.012 0.6 241 21.5

12.90 0.012 0.3 243 18

7.00 0.014 0.2 0 —

7.42 0.012 0.6 0 21.5

7.42 0.012 0.2 0 18

of the V380 Cyg system with the state-of-the-art evolutionary models computed with the mesa code (Paxton et al. 2011, 2013). Figure 1 (left) shows the location of both

c 2014 RAS, MNR

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A. Tkachenko

Table 3. Fundamental parameters of both components of the σ Sco, after seismic modelling of the primary. Parameters determined spectroscopically are highlighted in boldface. Parameter Mass, Radius, Luminosity (log (L)), Age of the system, Overshoot (fov ), Teff , log g,

Primary

Secondary

13.5+0.5 −1.4 8.95+0.43 −0.66 4.38+0.07 −0.15

(M ) 8.7+0.6 −1.2 (R ) 3.90+0.58 −0.36 (L ) 3.73+0.13 −0.15 (Myr) 12.1 (Hp ) 0.000+0.015 – +2 400 (K) 23 945+500 −990 25 000−2 400 +0.15 (dex) 3.67+0.01 4.16 −0.03 −0.15

components of V380 Cyg in the Teff -log g diagram along with the evolutionary tracks. The two models that fit the positions of both stars in the diagram, taking into account the error bars, are illustrated by long- and short-dashed lines (models 2 and 3 in Table 2, respectively). The dynamical mass models for both binary components are shown by solid lines (model 1 in Table 2). The mesa model predictions clearly point to mass discrepancy for the primary component, in agreement with the findings by Guinan et al. (2000) and Pavlovski et al. (2009). We conclude that present-day single-star evolutionary models are inadequate for this particular binary system, and lack a serious amount of near-core mixing.

3. σ Sco Sigma Scorpii is a double-lined spectroscopic binary in a quadruple system. Two components are early B-type stars, residing in an eccentric, 33 day period orbit. Though the star was a subject of numerous photometric and spectroscopic studies in the middle of 20th century, its double-lined nature was discovered by Mathias et al. (1991). So far, the studies by Mathias et al. (1991), Pigulski (1992), and North et al. (2007) have been the most extensive ones focusing on orbital and physical properties of the system. Besides the σ Sco system is a spectroscopic binary, its evolved primary component is known to be unstable to β Cep-type stellar pulsations. Moreover, according to Kubiak (1980), the amplitude of the dominant, radial pulsation mode of the primary is comparable to the orbital semi-amplitude K1 of this star. This fact was not taken into account in either of the previous studies focusing on orbital solution, but is taken into consideration in our study. Our analysis is based on some 1000 high-resolution spectra collected with the CORALIE spectrograph attached to the 1.2-meter Euler Swiss Telescope (La Silla, Chile). Orbital parameters of the system were initially derived based on an iterative approach, and further on refined using the method of spectral disentangling. The spectral disentangling was applied to a couple of dozen carefully selected spectra and corresponding to a zero pulsation phase (unperturbed profile), since the method assumes no variability intrinsic to stellar components forming a binary system. For more details on the procedure, reader is referred to Tkachenko et al. (2014b). The disentangled spectra were used to determine accurate atmospheric parameters and chemical composition of both stars. The final set of orbital parameters as well as the spectroscopically derived values of Teff and log g are given in Table 1. The masses and radii listed in this table were determined from our orbital parameters and interferometric value of the orbital inclination angle reported by North et al. (2007). We further made use of the fact that the primary component of σ Sco is a radial

Testing evolutionary models with binary stars

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mode pulsator, and performed asteroseismic analysis of this star. Evolutionary models were computed with the mesa code, while p- and g-mode eigenfrequencies for mode degrees l = 0 to 3 have been calculated in the adiabatic approximation with the gyre stellar oscillation code (Townsend & Teitler 2013). The addition of the seismic constraints to the modelling implied a drastic reduction in the uncertainties of the fundamental parameters, and provided an age estimate (see Table 3). Figure 1 (right) shows the position of both components of the σ Sco system in the Teff –log g diagram, along with the evolutionary tracks. The error bars are those obtained from the evolutionary models and 3σ spectroscopic uncertainties for the primary and secondary, respectively. Though we make an a priori assumption in our seismic modelling that the models are appropriate for the primary component, similar to the case of V380 Cyg, we still find mass discrepancy for the main-sequence secondary component of the σ Sco system. References Aerts, C. 2013, EAS Publications Series 64, 323 Aerts, C., Thoul, A., Daszy´ nska, J., et al. 2003, Science 300, 1926 Aerts, C., Briquet, M., Degroote, P., et al. 2011, A&A 534, A98 Briquet, M., Morel, T., Thoul, A., et al. 2007, MNRAS 381, 1482 Briquet, M., Aerts, C., Baglin, A., et al. 2011, A&A 527, A112 Garcia, E. V., Stassun, K. G., Pavlovski, K., et al. 2014, AJ 148, 39 Guinan, E. F., Ribas, I., Fitzpatrick, E. L., et al. 2000, ApJ 544, 409 Hadrava, P. 1995, A&AS 114, 393 Herrero, A., Kudritzki, R. P., Vilchez, J. M., et al. 1992, A&A 261, 209 Hilditch, R. W. 2004, ASP Conference Series 318, 198 Hill, G. & Batten, A. H. 2004, A&A 141, 39 Iliji´c, S., Hensberge, H., Pavlovski, K., & Freyhammer, L. M. 2004, ASP Conference Series 318, 111 Kubiak, M. 1980, AcA 30, 41 Mathias, P., Gillet, D., & Crowe, R. 1991, A&A 252, 245 North, J. R., Davis, J., Tuthill, P. G., et al. 2007, MNRAS 380, 1276 Pavlovski, K., Tamajo, E., Koubsk´ y, P., et al. 2009, MNRAS 400, 791 Paxton, B., Bildsten, L., Dotter, A., et al. 2011, ApJS 192, 3 Paxton, B., Cantiello, M., Arras, P., et al. 2013, ApJS 208, 4 Pigulski, A. 1992, A&A 261, 203 Piskunov, N. E. & Rice, J. B. 1993, PASP 105, 1415 Raskin, G., van Winckel, H., Hensberge, H., et al. 2011, A&A 526, A69 Simon, K. P. & Sturm, E. 2009, A&A 281, 286 Southworth, J., Zima, W., Aerts, C., et al. 2011, MNRAS 414, 2413 Tkachenko, A., Aerts, C., Pavlovski, K., et al. 2012, MNRAS 424, L21 Tkachenko, A., Degroote, P., Aerts, C., et al. 2014a, MNRAS 438, 3093 Tkachenko, A., Aerts, C., Pavlovski, K., et al. 2014b, MNRAS 442, 616 Townsend, R. H. D. & Teitler, S. A. 2013, MNRAS 435, 3406 Wilson, R. E. 1979, ApJ 234, 1054 Wilson, R. E. & Devinney, E. J. 1971, ApJ 166, 605