Further critical tests of stellar evolution by means of double-lined eclipsing binaries

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1 Mon. Not. R. Astron. Soc. 289, (997) Further critical tests of stellar evolution by means of double-lined eclipsing binaries Onno R. Pols,* Christopher A. Tout,,2 Klaus-Peter Schr5der,3, Peter P. Eggleton and James Manners 4 nstitute of Astronomy, Madingley Road, Cambridge CB3 OHA 2 Konkoly Observatory, Hungarian Academy of Sciences, H-525 Budapest, P. O.B. 67, Hungary 3 nstitut for Astronomie und Astrophysik, Hardenbergstrasse 36, D-J0623 Berlin, Germany 4Royal Greenwich Observatory, Madingley Road, Cambridge CB3 OEZ Accepted 997 April 2. Received 997 March 7; in original fonn 996 December 9 NTRODUCTON Binary stars can provide a fundamental and stringent test of stellar evolution theory, because the masses, radii and effective temperatures or luminosities of the stars can be determined, in principle with high accuracy. The most accurate data of stellar masses and radii from detached, double-lined eclipsing binary stars have been reviewed most recently by Andersen (99). These data provide one of the most critical tests of evolution models of single stars within and beyond the main sequence. The stars in this sample are relatively unevolved, mostly on the main sequence or slightly beyond. n any case they can be assumed to have evolved as single stars and thus can be compared with single-star evolution models. These and similar data have been regarded as evidence for nonstandard evolution during the main sequence, in particular for convective overshooting or some other form of mixing beyond the formal convective core boundary. ndeed, Andersen alludes to *Present address: nstituto de Astrofisica de Canarias, Via Lactea, La Laguna, Tenerife, Spain. ABSTRACT The most accurately measured stellar masses and radii come from detached, double-lined eclipsing binaries, as compiled by Andersen. We present a detailed quantitative comparison of these fundamental data with evolution models for single stars computed with our evolution code, both with and without the effects of enhanced mixing or overshooting beyond the convective cores. We use the same prescription for overshooting that Schroder, Pols & Eggleton found to reproduce the properties of r Aurigae binaries. For about 80 per cent of the 49 binary systems in the sample, both sets of models provide a good fit to both stars at a single age and metallicity within the observational uncertainties. We discuss possible causes for the discrepancies in the other systems. For only one system, A Hya, do the enhanced-mixing models provide a significantly better fit to the data. For two others (WX Cep and TZ For) the fit to the enhanced-mixing models is also better. None of the other systems can individually distinguish between the models with and without enhanced mixing. However, the number of systems in a post-main-sequence phase is in much better agreement with the enhanced-mixing models. This test provides supportive evidence for extended mixing in main-sequence stars in the range 2-3M 0. Key words: binaries: eclipsing - binaries: spectroscopic - stars: evolution - stars: fundamental parameters. the fact that standard models are inconsistent, first with the data for several individual systems, and furthermore with the distribution of stars across the main-sequence band. n particular, the number of stars near or beyond the 'blue hook' seems to be much larger than standard models predict. n this paper we attempt to quantify this evidence by comparing these data quantitatively with stellar models computed with the latest version of our evolution code (Pols et al. 995, Paper ). n a previous paper, Schroder, Pols & Eggleton (997, Paper T) performed a similar test by comparing these models with the properties of a different class of eclipsing binaries, the r Aur systems. These are wide binaries containing a late-type giant or supergiant and a hotter, main-sequence companion. As pointed out in that paper and by other authors, the effects of enhanced mixing during the main sequence become much more pronounced during later evolutionary stages. n particular the luminosity of the helium burning loop, where the giant components are most likely to be, depends sensitively on the amount of enhanced mixing, so these systems allow a very sensitive test of this quantity. From the analysis of three r Aur binaries and a few other giants with well-determined luminosities, Downloaded from by guest on 05 January RAS

2 870 O. R. Pols et al. Table. Results ofisochrone fitting. The observational data are from Andersen (99), unless otherwise indicated at the bottom of the table. An asterisk (*) after the Z value indicates that the minimum i was obtained at the boundary of the range , and therefore that the best fit lies outside this range. no. star compo EM Car 2 YCyg 3 V478 Cyg 4 CWCep 5 QXCar 6 V539 Ara 7 CVVel 8 AGPer 9 UOph 0 DHer V760 Sco 2 GGLup 3 rphe 4 i Hya 5 Q Per 6 PV Cas 7 V450ph 8 WXCep 9 TZMen 20 ARAur 2 V030ri 22 {J Aur 23 SZCen 24 YZCas 25 V624 Her 26 GZCMa Observational data R log Teff models () (K) SD: SD: SD: SD: SD: SD: SD: SD: t (:<:: 0") (yr) 5.45e6 (-.07,+0.4) 5.70e6 (-0.94, +) 3.26e6 (-0.25, +0.2) 3.58e6 (-0.26, +0.25) 7.66e6 (-.42, +0.23) 8.20e6 (-3., +0.27) 7.e6 (-4.0, +.48) 7.8e6 (-3.87, +2.3) 8.26e6 (-5.29, +2.78) 7.40e6 (-4.09,+3.72) 4.05e7 (-0.32, +0.30) 4.37e7 (-0.34, +0.33) 3.5ge7 (-6, +3) 3.65e7 (-0.82, +) 2.6ge7 (-0.98, +0.38) 3.04e7 (-.26,+7) 4.2ge7 (-.7,+0.95) 4.5e7 (-.6,+.0) 0.00e7 (-0.00, +6) 0.00e7 (-0.00, +0.6) 3.60e7 (-.,+0.77) 3.35e7 (-.0, +.30).52e7 (-9, +0.6).66e7 (-3, +0.74) 8.93e7 (-6, +5) 9.95e7 (-8, +7).90e8 (-0.08, +0.08) 2.0e8 (-0.0, +0.) 0.63e8 (-0.0, +0. 0) 0.7e8 (-0., +0.) l.le8 (-0., +0.2).26e8 (-0.2, +0.3) 2.46e8 (-0.29, +0.28) 2.68e8 (-0.32, +0.33) 5.0e8 (-0.7,+0.8) 5.47 e8 (-0.26, +0.23).68e8 (-0.20, +0.8).87e8 (-0.22, +0.23) 0.84e8 (-0.84, +0.88) 0.90e8 (-0.90, +0.90) 5.52e8 (-0.,+0.2) 6.ge8 (-0.,+0.2) 4.7e8 (-0.,+0.09) 5.ge8 (-0.3,+0.2) 6.68e8 (-0.34, +0.27) 7.4e8 (-, +0.9) 4.56e8 (-0.20, +0.8) 5.0e8 (-0.2, +0.2) 6.82e8 (-0., +0.) 6.40e8 (-, +) 5.28e8 (-0.35, +0.25) 5.73e8 (-0.39, +0.30) Model parameters Z (:<::0-) 0.009* (-0.000, +0.00) 0.009* (-0.000, ) 2 Xmin * (-0.000,+0.00) * (-0.000, +0.00) * (-0.000, ) * (-0.000, ) (-0.003, +0.02) (-0.007, +0.09) (-0.005, +0.06) (-0.008, +0.0) 0.03 (-0.003, ) (-0.004, ) (-0.005, ) (-0.006,+0.04) 0.009* (-0.000, ) * (-0.000, ) (-0.006, +0.00) (-0.006, +0.00) (-0.005, ) (-0.005, ) (-0.003, ) (-0.004, ) (-0.002, ) (-0.002, ) (-0.00, +0.00) (-0.00,+0.00) (-0.002, ) (-0.002, ) (-0.002, ) (-0.002, ) * (-0.00, ) * (-0.00, ) (-0.004, ) (-0.005, ) (-0.004, ) (-0.004, ) (-0.00, ) 0.06 (-0.00, ) 0.05 (-0.005, ) 0.06 (-0.005, ) (-0.007, ) (-0.007, +0.00) (-0.002, ) (-0.002, ) (-0.009, +0.05) (-0.00, ) (-0.003, ) (-0.004, ) (-0.003, ) * (-0.002, ) (-0.004, ) (-0.005, ) 0.89 Downloaded from by guest on 05 January RAS, MNRAS 289,869-88

3 Table - continued no. star 27 V647 Sgr 28 EEPeg 29 AHya 30 VVPyx 3 TZ For 32 WWAur 33 KWHya 34 RS Cha 35 MYCyg 36 PVPup 37 V442 Cyg 38 RZCha 39 BWAqr 40 DMVr 4 V43 Cyg 42 UXMen 43 APhe 44 RTAnd 45 FLLyr 46 EW Ori 47 CGCyg 48 HSAur 49 YYGem compo M (Mo) Observational data R log Teff models () (K) SD: SD: SD: SD: SD: SD: SD: SD: SD: SD: SD: SD: SD: SD: SD: SD: SD: SD: SD: Further critical tests of stellar evolution 87 t (± 0') (yr) 2.06e8 (-0.60, +9) 2.8e8 (-7, +0.63) 2.94e8 (-0.67, +0.6) 3.23e8 (-0.28, +0) 9.2ge8 (-0.23, +0.24) 9.7ge8 (-0.20, +0.2) 5.22e8 (-0.2,+0.2) 5.87e8 (-0.9, +0.8) O.7ge8 (-0.37, +0.38) 2.42e8 (-5,+4) 3.38e8 (-2, +0.75) 3.85e8 (-8, +0.73) 6.24e8 (-0, +8) 6.84e8 (-2, +7) 8.8ge8 (-0.30, +0.25) 9.73e8 (-0.39, +) 8.9ge8 (-6, +5) 0.00e8 (-2, +2) 0.00e9 (-0.00, +0.7) 0.00e9 (-0.00, +0.7).52e9 (-0.09, +0.09).65e9 (-0.0, +0.0) 2.24e9 (-0.08, +0.09) 2.4e9 (-0.09,+0.0) 2.5e9 (-0.4, +0.) 2.2e9 (-0.2, +0.4) l.64e9 (-0.7, +0.6).66e9 (-, +0.7) 0.00e9 (-0.00, +0.2) 0.00e9 (-0.00, +0.2) 2.3e9 (-0.20, +0.2) 2.32e9 (-0.20, +0.2) 4.60e9 (-0.27, +0.33) 5.0e9 (-0.30, +0.9) 2.3ge9 (-0.68, +.03) 2.75e9 (-.73, +0.73) 2.37e9 (-2, +3) 2.47e9 (-7,+4) 0.7ge9 (-2, +0.39) 0.83e9 (-5, +0) 6.68e9 (-.24, +.22) 6.82e9 (-.45, +.08) 2.68e9 (-.40, +.65) 2.68e9 (-.38, +.65) 0.00e9 (-0.00, +.63) 0.00e9 (-0.00, +.68) Model parameters Z (± 0') 0.06 (-0.003, ) 0.07 (-0.003, ) (-0.007, +0.00) 0.033* (-0.006, ) 0.Q28 (-0.004, ) 0.033* (-0.002, ) 0.009* (-0.000, +0.00) (-0.00, ) (-0.005, ) 0.02 (-0.002, ) 0.033* (-0.005, ) 0.033* (-0.005, ) 0.06 (-0.002, ) 0.06 (-0.002, ) 0.02 (-0.002, ) 0.05 (-0.003, ) 0.033* (-0.005, ) 0.033* (-0.002, ) (-0.005, ) (-0.005, ) 0.04 (-0.002, ) 0.06 (-0.002, ) (-0.006, ) 0.06 (-0.003, ) 0.08 (-0.002, ) (-0.003, ) (-0.006, ) 0.033* (-0.006, ) (-0.005, ) (-0.005, ) (-0.004, ) (-0.004, ) 0.02 (-0.002, ) 0.04 (-0.003, ) 0.033* (-0.000, ) 0.033* (-0.000, ) 0.03 (-0.006, ) 0.03 (-0.005, ) 0.033* (-0.003, ) 0.033* (-0.003, ) 0.033* (-0.00, ) 0.033* (-0.00, ) (-0.005, ) (-0.005, ) 0.033* (-0.00, ) 0.033* (-0.00, ) Downloaded from by guest on 05 January 209 References: [2] Simon, Sturm & Fiedler (994), [6] Clausen (996), [8] Gimenez & Clausen (994), [2] Andersen, Clausen & Gimenez (993), [20] Nordstrom & Johansen (994a), [22] Nordstrom & Johansen (994b), [44,47] Popper (994), [46] Popper et al. (986). we found strong evidence for enhanced mixing in stars with mainsequence (MS) masses between 2.5 and 6.5 Mo, and we were able to constrain the amount to an equivalent overshooting length of about pressure scaleheights. Although less evolved and therefore less sensitive to the effects of enhanced mixing, the binaries in Andersen's sample have much more accurately determined masses and radii than the t Aur binaries. The purpose of the present paper is to test whether our 997 RAS, MNRAS 289,869-88

4 872 O. R. Pols et al. stellar models are consistent with these very accurate data, and whether those data provide evidence for the same amount of enhanced mixing. We expect the best test of enhanced mixing to come from the several quite evolved systems that are near or beyond the terminal-age main sequence (TAMS). To this end we compare with two sets of models, one without enhanced mixing and one with the same amount that was found to best reproduce the data in Paper. These models are described in Section 3. The high quality of the data and the large number of systems warrant the use of a more quantitative, least-squares fitting, method of comparison than was possible in Paper (see Section 4). The results of this fitting procedure are presented in Section 5. We perform a statistical test of the number of stars in post-ms phases in Section 6. 2 OBSERVATONAL DATA Our sample comprises 49 detached, double-lined eclipsing binary systems. The data we use for the masses, radii, and effective temperatures are listed in Table. Hereafter, numbers in square brackets will refer to system numbers in Table. For the greater part (40 binaries) the data come from the review by Andersen (99). For a few systems (V539 Ara [6], GG Lup [2] and (3 Aur [22]) we use more accurate, recent determinations as indicated in Table. For EW Ori [46], we use the original data of Popper et al. (986) which Andersen (99) quotes as the source but, for an unknown reason, the radius of the secondary has been increased in Andersen's review. Furthermore, we have added recent measurements for five systems not included in Andersen's review: Y Cyg [2], AG Per [8], AR Aur [20], RT And [44] and CG Gyg [47]. We exclude from Andersen's sample the binary EK Cep because the secondary of this system is known to be in a pre-main-sequence phase (Popper 987) and our stellar models do not include this phase. The errors in masses and radii of the stars in Andersen's (99) sample (not given in Table ) are generally less than 2 per cent. The effective temperatures are usually less accurate and, in addition, may be subject to systematic errors. They are in most cases determined from the photometry and the conversion to temperature is often uncertain. However, the temperature difference between the components of a binary is usually reliable and unaffected by systematic errors. As was also pointed out by Andersen, a critical test of the models is possible only if all available data are taken into account. This includes the metallicity, a quantity which is in most cases undetermined, although efforts are made to measure them for many systems (J. V. Clausen, private communication). 3 STELLAR MODELS We have evolved a series of stellar models with masses M between and 40Mo, spaced by approximately 0. in 0gM, and with metallicities Z = 0.0, 0.02 and We have used the Eggleton stellar evolution code (Eggleton 97, 972), with updated input physics as described by Pols et al. (995, Paper ). The latter paper describes the most recent version of the code, and a more detailed description of the grid of models computed will be given in a future paper (Pols, SchrOder & Eggleton 997, Paper V in preparation). The hydrogen and helium abundance are assumed to be functions of the metallicity, as follows: X = Z and Y = Z. The mixing length is assumed to be a = 2.0. We neglect stellarwind mass loss. Apart from this set of models which we label the standard (STD) models, we have computed a similar set of models with a moderate amount of enhanced mixing beyond the Schwarzschild boundary of the convective regions. We label this set the overshooting (OVS) models, although we emphasize that we do not believe that such enhanced mixing is necessarily a result of convective overshooting alone. A detailed description of the way in which we model this enhanced mixing is given in Paper ; see also Paper V. We allow mixing to occur into the region where Vr > Va - Dov( r + 6r 2 ), where r is the ratio of radiation pressure to gas pressure and Dov is a free parameter. n Paper we found that the best correspondence with the properties of r Aurigae binaries occurs for a value of Dov = 0.2. We therefore adopted Dov = 0.2 in our set of overshooting models. For both sets of models, we construct isochrones as a function of age t and metallicity Z. The construction of these isochrones is not trivial, and we describe our procedure in detail in Paper V. We allow a small extrapolation of 0 per cent from the Z grid, i.e. between and METHOD For each component of a binary, the observational data give the mass M*i, radius R*i and effective temperature T*i and their error estimates, which we adopt as the standard deviations of a normal distribution: am", ar" and at". Although luminosity L and effective gravity g are also given, and often used in comparisons with stellar models, we do not use these data as they are derived quantities (L is determined from R and Teff and g from M and R) and the errors accumulate. The stellar-evolution models yield the observable properties of a star (radius R and effective temperature Teff) as a function of mass M, age t and chemical composition (X and Z, which are assumed to be related). For binary stars the age is completely unknown, as is the metallicity in most cases. However, a constraint is still present if we make the reasonable assumption that both stars have the same age and initial composition: i.e. fall on to a single isochrone. This leaves us with four input parameters for modelling a given binary: M J, M 2, t and Z, where M J and M2 are constrained within their observational errors. We thus define l as a function of these four parameters: 2 2 {(M. _ M ") 2 X (MJ,M2,t,Z) = L 'a *. r=l M., + [R(Mi, tz) - R*i] 2+ [Teff(Mi,;,Z) - T*i] 2}. R. T., n practice we use the logarithms of M, R and T but, because the errors a are small, this is essentially the same. We obtain a bestfitting model by minimizing l with respect to the input parameters. This minimization is done in two steps. First, the goodness-of-fit of a certain isochrone l (t, Z) is obtained by minimizing equation () with respect to M J and M 2. Since we represent our isochrones by a series of straight-line segments, this minimization step is done analytically. We transform the coordinate system so that the error bars have unit length, and then the minimum X2 is the sum of the minimum distances squared of each data point to the closest isochrone line segment. Secondly, we minimize X2 with respect to t and Z using the routine E04JAF from Numerical Algorithms Limited, Oxford (NAG). We have six data values and four model parameters, so we expect Xn to follow a l distribution with two degrees offreedom (d.o.f.), and a typical X of about 2. () Downloaded from by guest on 05 January 209

5 :8 z "- z STD (N tot = 37) OVS (N tot = 34) Figure. Cumulative distribution of Xin values for the STD and OVS models, compared to the X2 distribution with two degrees of freedom (dotted line). Systems with a best-fitting Z outside the allowed range are excluded, since the in found for those are not true minima. Note that the abscissa is linear in x, not in -, in order to resolve small X2 values. Once we have found the best-fitting model, characterized by t and Z, we can estimate the lu error intervals on these quantities by determining the upper and lower bounds of the contour in (t,z) space where i = Xin +. We find Uz by minimizing i(t,z) with respect to t at fixed Z using NAG routine E04ABF, and solving min t i(t, Z) = Xin + for Z using NAG routines COSAVF and COSAZF. With the same procedure we find Ut. Of course, Ut and Uz only have meaning if a good fit can be obtained. The least-squares fitting procedure described above does not take into account the speed of evolution. f the best-fitting models for a particular system imply that a star should be in a fast evolution phase (e.g. crossing the Hertzsprung gap to the giant branch), this is a less likely situation (i.e. in some sense a 'worse fit') than if the models imply a slow phase, such as the main sequence. On the other hand, in a sample of 49 systems we expect a certain (small) number of systems in a fast evolution phase, according to the fraction of time spent in that phase and possible selection effects. This provides an additional test for the models, which we will address in Section 6. 5 RESULTS OF MODEL FTTNG 5. Overall results of the isochrone fitting The results of the fitting procedure are presented in the right-hand columns of Table. For each binary system the best-fitting age and metallicity are given for both the STD and the OVS models, together with lu error intervals and the minimum X2 values. The choice of a goodness-of-fit criterion is somewhat arbitrary, particularly because the model is non-linear in its input parameters. n Fig. we compare the minimum X2 values from Table with the i distribution with 2 d.o.f. For both sets of models the distributions have about the same width (Le. median i =.5) as the i distribution, but with an excess of systems with both small and large i. Many systems with i < have components that are very similar to each other (in some cases almost identical), so that the data become correlated and there are effectively fewer degrees of freedom in the fitting. However, for most other systems there do seem to be two d.o.f., and a X2 :5 4 is certainly acceptable (corresponding to about 3 per cent in confidence level). This is confirmed by eye inspection of the best-fitting isochrones. n fact, as can be seen from Table and Fig., there are very Further critical tests of stellar evolution 873 few systems with i = 4-5, and the group of acceptable fits is quite distinct from the doubtful or bad fits. Of the 49 systems analysed, nine are bad fits (i > 8) for both sets of models. This is quite a high proportion: on the basis of the i distribution with two d.o.f. we expect only one system with i > 8. An additional three systems for both sets have 5 < i < 8 and are also quite bad fits, but this number is in accordance with the i distribution and may be a result of statistical fluctuation. We will discuss each of the 2 systems with i > 5 and possible causes for discrepancy in Section 5.2. Of the remaining 37 systems, 34 are acceptable fits (i < 4) to a single isochrone for both the SD and OVS sets of models, for metallicities within the range considered ( ). A few of these 34 indicate that a somewhat better fit could be obtained with a Z slightly outside this range (see Table ). The average i of this group of 34 acceptable fits is about the same for both sets of models: (i) =.0 for the SD models and (i) =.00 for the OVS models. The majority of systems in the sample cannot therefore distinguish between SD and OVS models on the basis of isochrone fitting alone, without taking into account the difference in evolutionary phase implied by the models. Fig. also shows that the OVS models give, on the whole, marginally smaller X2 values than the SD models. n most cases the OVS models yield ages that are about 0 per cent larger than with the SD models, and also slightly larger metallicities. As an example of a well-matched system, Fig. 2 shows the bestfitting isochrones for YZ Cas [24] for both sets of models. Systems like YZ Cas, with quite different component masses, in principle provide the best test of the models because they probe the isochrones over a large range of mass. However, since the system is rather unevolved (the primary has only completed about two thirds of its MS lifetime) it does not distinguish between STD and OVS models. There are three systems (WX Cep [8], A Hya [29] and TZ For [3]) for which only the OVS models yield a fit with i < 4. For two of these systems, however, the STD models are in fact still marginally acceptable (i = 4.8) and for only one (A Hya) do the OVS models provide a greatly improved fit. On the other hand, there are no systems for which the STD models give a significantly better fit than the OVS models (V624 Her [25] and KW Hya [33] both have smaller i with STD but the fit is still quite bad, see Section 5.2). Perhaps not surprisingly, all three systems that favour the OVS models are quite evolved, with at least one component near or beyond the TAMS. Besides providing a better fit, the OVS models also predict a different evolutionary state for one of the components. There are a further four systems that fit well with either set of models in a formal sense but for which a different evolutionary state is predicted too. These differences are discussed in more detail in Section 5.3 below. 5.2 Discussion of badly fitting systems 5.2. Y Cyg and PV Cas n both these binaries [2, 6] the components are nearly identical and the radii indicate unevolved systems, not very far from the zeroage main sequence (ZAMS). However, the effective temperatures are incompatible with the models for any metallicity in the range considered. n the case ofpv Cas [6], the model TeffS are too high even for Z = 0.033, which could indicate a very high metallicity. The opposite is true for the O-type binary Y Cyg [2], for which the model Teffs are too low even for Z = t is possible that Y Cyg has a very low metallicity (about a tenth of the solar value) but the Downloaded from by guest on 05 January 209

6 874 O. R. Pols et al. temperature scale for 0 stars is uncertain and other authors give much lower Teffs (Hill & Holmgren 995). These would correspond to more or less solar metallicity Phe, Q Per and KW Hya The components in these systems [3, 5,33] are quite unequal in mass (mass ratios between and 0.75) and their masses and radii can be fitted well to a single isochrone in each case. The best-fitting metallicities are well constrained by the radii of the secondaries, which are quite unevolved since they have a much longer nuclear time-scale than the primaries. f we only fit to the masses and radii, we find a perfect fit (i = 0) at the following metallicities for both sets of models: 2 = 0.04 ::!:: for. Phe [3], 0.D8 ::!:: for Q Per [5] and 0.03 ::!:: for KW Hya [33], at essentially the same ages as given in Table. However, the models at these metallicities do not match the observed temperatures (see Fig. 3). n the case of. Phe the primary is reasonably matched but the model Teff for the secondary is too low. This may be attributed to difficulties in determining this parameter because of its low luminosity (Andersen 99). For Q Per and KW Hya the model Teffs for both components are about 5 per cent too high. Such a discrepancy could be explained by an anomalous helium content, i.e. much lower than given by our assumed Y -2 relation (Section 3). However, in the absence of other indications of such anomalies, we prefer not to introduce another free parameter (Y) in our models. t is also possible that the Teffs for these systems are systematically in error V624 Her, WW Aur and RS Cha n these systems, it is not possible to fit the masses and radii of both stars at the same age, for any metallicity. n all three cases (and also in some low-mass systems, see Section 5.2.4) the radius of the secondary is too large to fit the models at the same age as the primary. This is most clear in RS Cha [34] where the less-massive secondary is significantly larger and cooler than the primary: see Fig. 4. Andersen (975) suggested the components are at opposite ends of the hook in the main sequence, the primary being at the end of the post-ms contraction phase. However, the present models attain much larger radii at the TAMS and do not permit such a solution. n V624 Her [25], the STD models put the primary in the post-ms contraction phase and thus provide a better fit to the masses and radii, although the temperatures are better matched by the OVS models: see Fig. 5. Owing to the much larger error bars on Teff the overall fit to the STD models is better, but it is still very unsatisfactory. For WW Aur [32] the discrepancy may be only statistical. The radii are assumed to be equal because the photometry is inadequate to determine them individually (Popper 980), but Cester et al. (978) find the secondary to be 7 per cent smaller than the primary, which would adequately match the mass ratio of about 0.9. For RS Cha and V 624 Her a satisfactory fit to the data can be obtained if we allow the components different ages at a single metallicity. n both systems the secondary should then be more than 0 8 yr older than the primary, which is very unlikely. A perhaps more probable explanation is that some interaction-or mass loss has occurred in these system's.'.. llt RT And, FL Lyr, CG Cyg and YY Gem These four systems are among the eight lowest-mass binaries in the sample, with M, <.25 Mo. The primary components of RT And [44], FL Lyr [45] and CG Cyg [47] can be fitted with the models at somewhat greater than solar metallicity and at ages of a few Gyr (.4::!:: 0.7, 2. ::!:: 0.6 and 4.8 ::!::.6 x 0 9 yr, respectively). However, their secondary components are too big and too cool to be matched by the same models and appear to be very much older and at much larger 2 than the primaries: see Figs 6 and 7. Any compromise best fit to both components is always at very high 2. Similarly, the models fail to fit both (identical) components of YY Gem [49] at any metallicity or age less than the Hubble time (Fig. 7). All four systems are short-period active binaries, the first three showing starspot activity, while YY Gem is a flare star. We notice that the greatest discrepancies occur for the most active binaries RT And, CG Cyg and YY Gem, which all have orbital periods P < d, while for FL Lyr at 2.2 d the fit is somewhat better. The other four low-mass binaries have P > 4 d and are good fits to the models. This may indicate a past interaction phase, as was suggested in the case ofrt And by Arevalo, Lazaro & Claret (995). Alternatively, these systems may be pre-ms binaries with the secondary or both components still in contraction. This has been suggested by Chabrier & Baraffe (995) for YY Gem, who found agreement with their stellar models at t yr. Finally we note that high 2 values are implied for all low-mass binaries, including EW Ori [46] and HS Aur [48], which are well fitted. This is worrying in the case ofhs Aur, which has an implied age older than the Galactic disc. We thus find anomalies for all stars with masses below Mo, which may indicate that our lowmass models are systematically too hot. However, the temperature calibration for these low-mass systems, which mainly fixes the model metallicity, may also have systematic errors (Popper et al. 986; Popper 994). On the other hand, the 'twin' systems UX Men and A Phe are successfully fitted by the models at metallicities that are in agreement with the measured values (see Section 5.3.6). 5.3 Differences between overshooting and standard models The differences between standard and overshooting models become more pronounced towards and beyond the TAMS, and we therefore expect the most sensitive tests to come from the most evolved binaries. There are eight systems in the sample in which the STD models predict (at least) one component to have evolved beyond the TAMS. Another system, V624 Her, could be in post-ms contraction with the STD models but this is a very bad fit, see Section 5.2.3, so we exclude this system from the discussion here. This applies to the primary (A) components of x 2 Hya, WX Cep, V03 Ori and A Hya, and to both components of SZ Cen, RZ Cha, TZ For and A Phe. n the last two systems the primary star is already on the giant branch. On the other hand, the OVS models put nine of these 2 stars (in seven out of eight systems) within the main-sequence band, all except for the primary (giant) component of TZ For and both stars in the A Phe system. The three systems for which the OVS models yield a noticeably better fit (WX Cep, A Hya and TZ For) are in this group of most evolved binaries in the sample. Also for three of the other evolved systems the OVS models fit slightly better but not significantly so. We discuss each of these syste"fus individually..".,. ';',,0.,; '... :,:,.:..".;-,,:( :"'l'_' 5.3. A Hya The masses and radii of A Hya [29] are well fitted by either set of models, but the effective temperatures clearly favour the OVS models (see Fig. 8). The primary is cooler than the secondary, as Downloaded from by guest on 05 January 209

7 et:: - STD log t = 8.66 Z = 0.D OVS log t = 8.70 Z = ZAMS Z = 0.05 C yz Cas Further critical tests of stellar evolution _ "... "-' log Tefl Figure 2. Comparison of the best-fitting isochrones to the observational data for YZ Cas [24), in case of the STD (solid lines) and OVS models (dashed lines). The left panel shows the mass-radius diagram, and the right panel log Teff versus log R, which is equivalent to the HR diagram (lines of constant luminosity running diagonally). The ZAMS is shown as a dash-dotted line. et:: C " - STD log t = 7.90 Z = OVS log t = 7.95 Z = STD log t = 8.79 Z = OVS log t = 8.84 Z = ZAMS Z = Phe J KW Hya.J;'.,OJ.,;,,,.j" ".:'.l.',.. ;',..i!." Figure 3. Same as Fig. 2 for r Phe [3) and KW Hya (33). The isochrones shown are those that fit best to the masses and radii, to show the discrepancy in T eff. '.,,.. -STD log t 8.95 Z 0.04, OVS log t = 8.99 Z = 0.04 _._.- ZAMS Z = 0.04, RS Cha log Teff, -' Downloaded from by guest on 05 January 209 et:: C '- ---' : log T eff Figure 4. Same as Fig. 2 for RS Cha [34). 997 RAS, MNRAS 289,869-88

8 876 O. R. Pols et al STD log = 8.83 Z = OVS log = 8.8 Z = ZAMS Z = V624 Her ", 0:: c>..q o ZAMS Z = STD log = 9.25 Z = STD log = 9.85 Z = RT And o FL Lyr,.,.,.,.,. f'..,' ",' -' ".'..-::.-;...., o'!: '-'- '-' '-' Figure 5. Same as Fig_ 2 for V624 Her [25]. i i i, o 0.05,. t: ogtef,.,- i,,,.. i...-- ',. -'- '- ' log T elf Figure 6. ZAMS and STD isochrones fot RT And [44] and FL Lyr [45]. We do not show the OVS isochrones since they are indistinguishable from the SD ones at the relevant masses and radii. Note the discrepancy in radii and temperatures of the secondary components. 0. o - ZAMS Z = STD log=9.78 Z= STD log = Z = CG Cyg o YY Gem! i.... " --, '. Downloaded from by guest on 05 January 209 0::-0. c>..q log Telf Figure 7. Same as Fig. 6 for CG Cyg [47] and YY Gem [49]. CG Cyg B and both components of YY Gem are too big and cool to fit the models even at 4 Gyr. 997 RAS, MNRAS 289,869-88

9 tt: Cl.Q 0.8 -STD log t = B.95 Z = OVS log t = B.99 Z = A Hyo Further critical tests of stellar evolution 877 " , predicted by the OVS models which put the primary inside the main-sequence band. The STD models, however, put the primary beyond the TAMS at a slightly higher Teff than the secondary. Because the error bars on Teff are very small for this system this is a significantly worse fit (see Table ). These temperatures are low which indicates a high metallicity: with the OVS models the best fit within our Z-grid is obtained at the upper boundary, Z = This indicates that a better fit can be obtained at an even higher Z, about twice the solar value. The photometric metallicity index also indicates such a high Z (Jflrgensen & Grfllnbech 978) l Hya, WX Cep and V03 Ori These systems, at somewhat higher mass, are similar to A Hya in that the STD models put the primary beyond the TAMS, while it is comfortably within the MS band with the OVS models.l Hya [4] is the most massive system with an evolved primary (3.6Mo). A good fit to the data is obtained at subsolar metallicity for both sets of models. The STD models then put the primary just beyond the TAMS, although a somewhat worse but still acceptable fit can be obtained in which the primary is just at the red edge of the MS, at slightly higher Z and smaller age. We also note that the primary 3.9 Figure 8. Same as Fig. 2 for A Hya [29] log Teff might be slightly tidally deformed because its radius is 80 per cent of the Roche radius. The WX Cep system [8] is very similar to A Hya: the masses and radii are a good fit to both models but the temperature difference between the components favours the OVS models (see Fig. 9). The best fit to both models is obtained at solar metallicity. However, this case is not quite as significant as A Hya because the error bars on Teff are larger. The same applies to V03 Ori [2], which is almost identical to WX Cep. n this case the masses are very accurately known, to less than per cent, and masses and radii indicate a somewhat larger than solar metailicity. However, the temperatures are not accurate enough to indicate a significant difference between the model sets SZ Cen The components of SZ Cen [23] have nearly equal masses and are both clearly beyond the TAMS with the STD models. The OVS models put both within the MS band, with the primary just on the TAMS (Fig. 0). Also here, the difference in Teff favours the OVS models but the STD models (in this case predicting a larger difference) are also consistent. The near equality of the masses, Downloaded from by guest on 05 January 209

10 R. Pols et al. (t:: '" STD log t = B.B2 Z = OVS log t = B.B7 Z = SZ Cen 0.7..Q STD log t = 9.03 Z = OVS log t = 9.09 Z = " TZ For " 0.9 a:: 0.8 '" Figure 0. Same as Fig. 2 for SZ Cen [23]. " Figure. Same as Fig. 2 for TZ For [3] log Tefl rr- 3.8 log Tefl _ , 3.7 Downloaded from by guest on 05 January log Tefl Figure 2. Same as Fig. 2 for RZ Cha [38].

11 0.6 cr C..Q - STO log t = 9.67 Z = OVS log t = 9.69 Z = 0.03 A Phe Further critical tests of stellar evolution however, is somewhat better fitted with the STD models, making the overall goodness-of-fit nearly the same in both cases. Note that, with an overshooting parameter {jov slightly less than 0.2, the primary would be beyond the TAMS. This puts it in a very fast evolution phase (see Section 6) and makes the overall fit worse Z For This binary [3] is much wider than any other system in the sample, and the primary is on the giant branch. The secondary is also beyond the TAMS with the STD models, while the OVS models put it just within the MS band (Fig. ). n both cases the masses should be nearly equal, which is possible within the error bars. The temperature difference is somewhat more comfortably fitted with the OVS models, giving a lower overall i. As is the case with SZ Cen A, the secondary would be beyond the TAMS with {jov < 0.2, in this case making the fit much worse. The best-fitting metallicity for both models is in good agreement with the spectroscopically determined Z = ± (Andersen et al. 99, assuming Zo = 0.088). Although both models give the best fit with the primary on the first giant branch (FGB), the OVS moqels suggest another interpretation where it is in a core-helium burning loop after the FGB (see Fig. ). The minimum radius reached in the loop is 8.9 Ro, which is only slightly but significantly larger than the radius of TZ For A. Nevertheless, this interpretation is attractive since - in addition to putting the star in a longer-lived phase - it provides an explanation for the circular orbit of the binary (Andersen et al. 99; Claret & Gimenez 995). ts maximum radius on the FGB would have been about 25 Ro, a significant fraction of its present Roche radius of 46 Ro, and the tidal circularization time-scale could have been short enough to circularize the orbit. The circular orbit is difficult to understand if the primary is now ascending the FGB because the present radius is only 0.8 times the Roche radius. For the STD models, a post-fgb loop interpretation is ruled out because the maximum FGB radius is > 00 Ro and the system would have undergone Roche-lobe overflow RZ Cha This system [38] has almost identical components of.5 Mo and consequently an almost perfect fit with i = 0 can be obtained because there is effectively only one star. The OVS models put the components well within the MS band while the SD models put them Figure 3. Same as Fig. 2 for A Phe [43). log Teft just beyond the TAMS (see Fig. 2). These different interpretations require quite different metallicities to achieve a good fit to the data, Z = with SD and 0.06 with OVS. Measurement of a supersolar or subsolar metallicity could therefore distinguish between the models in this case. However, a marginal fit to the SD models is also possible with the stars just on the TAMS. So if a subsolar metallicity is confirmed, this would not strongly constrain the amount of enhanced mixing (i.e. the value of {joy) in this mass range A Phe Both stars of A Phe [43] are beyond the TAMS according to either set of models, although only just for the OVS models (see Fig. 3). The best-fitting metallicities are also consistent, within the error bars, with the spectroscopically determined Z = 0.04 ± (Andersen et al. 988, corrected for Zo = 0.088). To fit the secondary in the MS band with the OVS models would require a lower metallicity, Z < 0.0, inconsistent with the observed Z. This puts an upper limit to the amount of enhanced mixing {jov :S 0.2 at about.2mo. UX Men [42] has very similar masses to A Phe but is less evolved. The measured metallicity of Z = 0.02 ± (Andersen, Clausen & Magain 989) is also consistent with the models in this case. 6 A STATSTCAL TEST Unfortunately, the sample of systems considered here is strongly biased by selection effects because the only reason for a binary to end up in the sample is the high quality of the data. This favours equal masses and luminosities (to observe both spectral lines with high resolution) and short orbital periods (to obtain high-quality lightcurves, etc.). Therefore we have to be extremely careful to make statistical inferences from the sample, as Andersen (99) has warned. However, we may suppose that any given system, characterized by its known masses and orbital parameters, is fairly unbiased by evolutionary age effects. That is, it would not have been much more likely to end up in the sample if it were at any particular age. We might expect a certain bias towards late evolutionary phases because stars become bigger and brighter as they age, thus increasing the likelihood of eclipses and the volume of space for which accurate photometry and spectra can be obtained. However, the luminosity difference also increases which would Downloaded from by guest on 05 January 209

12 R. Pols et al. Table 2. Number statistics. For both sets of models we give the observed and expected number of systems with the primary in the Hertzsprung gap ( HG), the primary on the giant branch ( GB) and the secondary in the Hertzsprung gap (2 HG), and the probability P(2:nobs) if nobs > n exp, or P(05nobs) if nobs < n exp. This is performed for the whole sample and for two subsamples with primary masses in the given ranges. STD models OVS models sample state obs. expo P(%) obs. expo P(%) all HG (40 syst.) GB HG Mo HG (6 syst.) GB HG <.9Mo HG ( syst.) GB HG make the secondary harder to detect and therefore has the opposite effect. Since we are dealing with detached binaries which (presumably) have not interacted, their lifetime is limited to the time when the most massive component fills its Roche lobe. For each system, we can compute this maximum lifetime tmax on the basis of the stellar models for their known masses and their orbital period. The models give the radii as a function of time, while the Roche lobe radii follow from the masses and orbital period using Kepler's law and the formula for the fractional Roche radius of Eggleton (983). Some of the binaries have eccentric orbits and for these we assume that the orbit will be circularized before the stars fill their Roche lobe. This gives a small correction to the maximum radii and therefore to tmax. We take the orbital periods and eccentricities from the catalogue of Batten, Fletcher & MacCarthy (989) or from the original sources if not available there. The stellar models also give the main-sequence lifetimes for both components, tms,] and tms,2' These we define as the age at the blue end of the hook, i.e. the end of the contraction phase, or as the age when the central hydrogen mass fraction is down to 0. per cent for low-mass stars. We also define tgb,] as the age at the base of the giant branch. The fraction of time for which a given system can be observed with the primary on the giant branch is fgb,] = max(o, - tgb,]tmax ) and the fraction of time with the primary in the Hertzsprung gap (HG) is A g,] = max (0, - fgb,] - tms,]tmax). Similarly, the fraction of time for which the secondary of that system can be observed in the HG is fhg,2 = max(o, - f gb,2 - tms,2tmax), etc. The latter will be zero unless the masses are nearly equal. All time-scales involved, and therefore all the fractionsfi, depend on which set of models is used. These fractions will be smaller for the OVS models than for the STD models, because () the post-ms phases evolve much faster because the helium cores formed after central H-burning are larger, and (2) the radii at the end of the MS are larger and the stars are therefore already closer to their Roche lobes. n the absence of any selection effects for age, the fi also define the probabilities, Pi' that a given system is caught in a particular evolution phase of one of its components. The sum of these probabilities over all systems then gives the expected number of systems in the sample with a primary or secondary in the HG, etc. These numbers can be compared to the actual numbers implied to be in these phases by the models. We can also compute the probability P(n) that a certain number n is observed with, say, the primary in the HG, from the individual Phg,]S of all systems (see Appendix). We have done this for all systems for which a reasonable fit could be obtained, with x 2 < 8. We thus exclude nine of the 49 binaries, as discussed in Section 5.2. The results are given in Table 2. Considering the sample of 40 systems as a whole, the observed numbers of stars in the Hertzsprung gap are within the expectations for the OVS models, but for the STD models the observed numbers are much larger than expected. The probabilities of finding (at least) the observed number of primaries and secondaries in the HG are only 2.8 and 4.6 per cent, respectively. The number of giants is greater than expected for both sets of models. The discrepancy becomes clearer if we divide the sample in different mass ranges (see Table 2). Most of the systems for which the STD models imply one or two components in the HG have primary masses between about 2 and 3 Mo. However, most of the expected systems with post-ms components should have masses less than.9 Mo. The probabilities of finding the observed number of primaries and secondaries in the HG in the mass range Mo are both less than per cent. For the OVS models the fact that none is observed in this mass range is expected because the probability of finding any stars in the Hertzsprung gap is very small. We notice that if we had used a smaller amount of overshooting, say oov = 0.0, we would have found two stars (SZ Cen A and TZ For B) in the HG, which would have been very unlikely. This is an indication that oov ;?; 0.2 is appropriate in the mass range 2-3Mo For masses> 3 Mo or <.9 Mo we find that the number of HG stars is in accordance with both sets of models. Hence we have no indication of enhanced mixing outside the mass range 2-3 Mo from the present sample. On the basis of the STD models we expect 0. primaries in the HG with masses> 3 Mo. Since i Hya A mayor may not be in the HG (Section 5.3), this provides no conclusive evidence. 7 SUMMARY AND CONCLUSONS For 37 out of 49 binaries, we find a good correspondence between our stellar models and the masses, radii and effective temperatures of both components at the same age and metallicity. For six of the remaining 2 systems the discrepancy is likely to be owing to statistical fluctuations, or systematic errors in the effective temperatures or a composition outside the considered range. The discrepancy in the other six binaries is more serious because the masses and radii of both stars cannot be fitted by models of the same age. Some of these systems may have components still contracting to the main sequence, while others have possibly interacted or lost mass in the past. Four of these systems are low-mass binaries with strong chromo spheric activity, but our low-mass models may be incorrect. Since the metallicities of most of the systems are unknown we cannot yet apply the most stringent of tests. However, for the few systems with measured metallicities, the data correspond to the best-fitting metallicities implied by the stellar models. For other systems our predicted metallicities may be tested in the future if more values are spectroscopically determined. For the moment, we conclude that our models of main-sequence stars seem to be consistent with the most accurate available data of fundamental stellar properties, at least for masses;?; Mo. The model fits imply a large spread in metallicity among the systems, independent of age. Downloaded from by guest on 05 January 209