On the Old Open Clusters M67 and NGC 188: Convective Core Overshooting, Color‐Temperature Relations, Distances, and Ages

Open cluster studies conducted during the past decade have shown that stellar models must allow for convective core overshooting if they are to provide satisfactory matches to observed color‐magnitude diagrams (CMDs); see, e.g., Daniel et al. (1994), Demarque et al. (1994), Nordström et al. (1997), Rosvick & VandenBerg (1998), Gim et al. (1998), and Kalirai et al. (2001). This finding, coupled (in particular) with the supporting evidence from binary stars (see Andersen 1991; Schröder et al. 1997), has provided a compelling case against models that neglect overshooting. It is, in fact, for this reason that most modern grids of evolutionary tracks and isochrones (including those by, e.g., Meynet et al. 1994; Claret 1995; Girardi et al. 2000; Yi et al. 2001) assume that convective cores are enlarged by some fraction of a pressure scale height, H_P (usually 0.2H_P–0.25H_P or an equivalent amount if a different formalism is employed).

New sets of overshooting stellar models are currently being computed by D. A. VandenBerg, P. A. Bergbusch, & P. D. Dowler (2004, in preparation). Unlike the case with existing grids, the sizes of convective cores are determined using a modified version of the Roxburgh (1978) criterion. As discussed subsequently by Roxburgh (1989), convective cores should not extend beyond the radius r_max where

(L and L_rad represent the total luminosity produced by nuclear reactions and the radiative luminosity, respectively, while the other symbols have their usual meanings.) This equation gives the maximum size of a convective core, because it does not allow for the viscous dissipation of energy, which requires the evaluation of a second, much more complicated integral. According to the most general form of Roxburgh's criterion, the radius for which the difference between these two integrals vanishes defines the actual boundary of a central convective zone. Because this second integral is so intractable, we have opted to represent it as a fraction of the first integral.

To be more specific, at smaller radii than the classical core boundary r₀ (where ∇_ad = ∇_rad), L_rad<L, and the integrand in equation (1) is positive. In this part of the star, we can allow for dissipative processes by multiplying equation (1) by a factor (1 - F_diss), where 0.0≤F_diss≤1. At larger radii (i.e., in the overshooting zone), L_rad>L, and given that the dissipation term is always positive, the multiplicative factor must be changed to (1 + F_diss). In fact, we prefer to use F_over = 1 - F_diss as the overshooting parameter, because F_over = 0 implies that there is no overshooting beyond the boundary given by the Schwarzschild criterion (i.e., r_cc = r₀), whereas F_over = 1 implies that the core is enlarged by the maximum possible amount of overshooting (i.e., r_cc = r_max). With this change of variable, the integral equation that must be solved for the convective core boundary r_cc can be written as

Observational constraints must then be used to calibrate the parameter F_over as a function of mass and chemical composition. As long as this quantity does not vary appreciably with evolutionary state, this approach should yield results that are faithful to the generalized Roxburgh criterion, which was derived from the relevant equations of fluid dynamics.

The main purpose of the present investigation is to determine the value of F_over that applies to turnoff stars in M67, which is arguably the oldest known open cluster with accurately determined parameters (metallicity, distance) that has a well‐defined gap at the main‐sequence turnoff (MS TO). It provides an especially important constraint on the variation of F_over with stellar mass, in the case of near‐solar abundance stars, because its ≈1.3 M_⊙ TO stars are only 0.15–0.2 M_⊙ more massive than the lowest mass MS star that is expected to retain a convective core until central H exhaustion. Earlier studies have revealed that F_over ≈ 0.5 in both NGC 6819 (Rosvick & VandenBerg 1998) and NGC 7789 (Gim et al. 1998), which were found to have TO masses near 1.55 and 1.8 M_⊙ and ages of ∼2.5 and ∼1.6 Gyr, respectively. As shown in § 2, a much smaller value of F_over must be assumed in models that are fitted to the ∼4 Gyr TO stars of M67. (As shown in a forthcoming study by P. D. Dowler & D. A. VandenBerg [see also Dowler 1994], the amount of overshooting that is calculated using eq. [2] with F_over = 0.5 is approximately equivalent to the net effect over the MS lifetime of extending the convective core by Λ = 0.2H_P–0.25H_P. The two parameters [F_over and Λ] are not simply related; measured as a fraction of a pressure scale height, the extent of overshooting derived from the Roxburgh criterion is initially small at the zero‐age MS, rising to ≳0.35H_P just prior to central H exhaustion.)

In what follows, we also explore the implications of different assumptions concerning the helium and metal abundances of M67 for F_over, and also for the cluster distance/age and the quality of the isochrone fit to the observed CMD (by Montgomery et al. 1993, hereafter MMJ93).¹ Most of our results are based on BV data, given current uncertainties in both the V−I transformations for stars with [Fe/H]≳0.0 (see VandenBerg & Clem 2003; hereafter VC03) and the available VI photometry (Sandquist 2004, hereafter S04). Further discussion of the difficulties with current VI photometry for M67 is presented at the end of § 2.

Fortunately, the quite well defined CMDs and color‐color diagrams for NGC 188 that can be constructed from the BV(RI)_C data given by Stetson et al. (2004)² provide important clues about the color‐temperature relations that are obeyed by near‐solar abundance stars. As described in § 3, in order to satisfy the constraints afforded by both NGC 188 and Landolt (1992) photometric standard stars, it appears to be necessary to make modest revisions to the semiempirical (V−R)‐T_eff and (V−I)‐T_eff relations published by VC03, primarily at log g≲3.5 and T_eff≲5000 K. A derivation of the dependence of the inferred distance and age of NGC 188 on the assumed helium and metal abundances, along with our best estimate of the cluster age, is also contained in § 3. The main results of this study are then summarized in § 4.

The reddening and metallicity of M67 appear to be particularly well established. The E(B−V) estimates derived from Strömgren photometry (Nissen et al. 1987), the two‐color diagram (Sarajedini et al. 1999), and the latest dust maps (Schlegel et al. 1998) are within ±0.005 mag of the mean value (0.036 mag) from these three sources. Throughout this study, we will adopt E(B−V) = 0.038 mag, as given by the Schlegel et al. reddening maps. High‐resolution spectroscopic studies have found the cluster stars to be slightly metal poor relative to the Sun: Tautvaišienė et al. (2000) have recently obtained [Fe/H] = -0.03 ± 0.03, in excellent agreement with the iron abundance ([Fe/H] = -0.04 ± 0.12) derived previously by Hobbs & Thorburn (1991). Very similar values (between −0.05 and −0.09) have been found using lower resolution spectroscopy (Friel & Janes 1993), calibrations of metallicity sensitive DDO indices (Janes & Smith 1984), and calibrations of the ultraviolet excess at (B−V)₀ = 0.6 as a function of [Fe/H] (Montgomery et al. 1993).

VC03 have already shown that if E(B−V) = 0.038 and [Fe/H] = -0.04 are adopted for M67, then a MS fit of the MMJ93 CMD to isochrones for the adopted metallicity yields (m−M)_V ≈ 9.65. This should be quite an accurate estimate of the apparent distance modulus, because the temperature scale of the models (for solar abundances) satisfies the solar constraint and because the color‐T_eff relations that were used agree exceedingly well with the empirical transformations derived by Sekiguchi & Fukugita (2000) from a sample of solar neighborhood stars with well‐determined parameters. Encouragingly, the best‐fitting isochrone (for 4 Gyr) to the cluster subgiants also matches the red giants very well, and as shown by VC03, the predicted temperatures for the latter are in excellent agreement with those inferred from empirical (V−K) versus T_eff relations. There are no obvious discrepancies between theory and observations, although the morphology of the TO can be matched by nondiffusive models only if a small amount of convective core overshooting is assumed, as demonstrated in the next section. (Note that the isochrone used by VC03 was taken from the present investigation.)

2.1. Overshooting in M67 Turnoff Stars

2.2. On the (V−I)‐T_eff Relations Applicable to M67

The reddening and metallicity of NGC 188 appear to be quite well determined. In this investigation, we have chosen to adopt E(B−V) = 0.087 mag, as derived from the Schlegel et al. (1998) dust maps. This agrees very well with the latest estimate of the reddening from the two‐color diagram by Sarajedini et al. (1999), who obtained 0.09 ± 0.02 mag. As far as the cluster metal abundance is concerned, most determinations during the past decade or so have tended to fall within the range −0.12≲[Fe/H]≲0.0 (e.g., Hobbs et al. 1990; Friel & Janes 1993; Friel et al. 2002; Randich et al. 2003), which is very similar to what has generally been found for M67. Consequently, the same grids of isochrones that were used to fit the M67 CMD are probably applicable to NGC 188 as well. By considering the same models, one can determine (1) which values of Y and Z result in the best matches of isochrones to the NGC 188 CMD, and (2) whether or not the inferred values of these chemical composition parameters are similar to those obtained for M67.

Figure 7 contains the answers to these questions. Each panel gives the result of performing a MS fit of those stars from the Stetson et al. (2004) study that are deemed to have ≥50% membership probabilities to isochrones for the indicated parameters. (In order to be able to distinguish the isochrones from the stellar data, only half of the cluster main‐sequence stars satisfying the adopted membership criterion have been plotted. To be more specific, the probable members were first sorted in magnitude, and then every second star in the resulting list was plotted. Moreover, to avoid too much clutter in each panel, only that isochrone that does the best job of reproducing the cluster subgiants is shown; this isochrone provides our best estimate of the age of NGC 188 in each of the cases considered.)

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Recall from Figure 3 that the Y, Z values assumed in the panels along the diagonal running from the upper left to the lower right resulted in the best agreement between the isochrones and the M67 CMD (for ages ranging from 4.3 to 3.9 Gyr, respectively). Essentially the same result is found for NGC 188, although slightly larger Y values at a given Z seem to be favored, and the inferred ages are larger by 2.6–3.0 Gyr. (Very similar age differences were found by Sarajedini et al. 1999.) Interestingly, the variation in the morphology of the isochrones with Y and Z is such that if the metallicity of either cluster is less than Z = 0.015, it will not be possible to obtain a simultaneous match of both their MS and lower giant branch populations unless the assumed helium abundance is less than Y = 0.25. This seems rather unlikely, given that the primordial He abundance from the concordance between Wilkinson Microwave Anisotropy Probe (WMAP) observations (Spergel et al. 2003) and big bang nucleosynthesis calculations is very close to Y = 0.248 (Cyburt et al. 2003; Coc et al. 2004)—which in turn argues against a low metal abundance for either cluster.

To be sure, it would be risky to place too much reliance on such a conclusion, given that (as noted in § 2) many factors can affect the predicted age and metallicity dependence of the location of the giant branch relative to the TO and MS. On the other hand, both the model temperature scale and the adopted (B−V)‐T_eff relations are fully consistent with the best available empirical constraints (see VC03), and the isochrones are able to reproduce the observed CMDs quite well on the assumption of [Fe/H] values that are in the middle of the ranges found from spectroscopy and helium abundances that are close to the solar helium content. Although the agreement is not perfect (e.g., the isochrones tend to lie redward of the mean locus through the brightest TO stars in NGC 188), small discrepancies could easily be due to the effects of binaries, problems with the color transformations, differences between the actual and assumed heavy‐element mixtures, etc. In view of such uncertainties, minor problems with the fits of isochrones to observed CMDs can hardly be avoided.

While Figure 7 contains several interesting results, our main motivation for studying NGC 188 is to find out whether our isochrones yield the same interpretation of its CMD, irrespective of whatever color is used. In fact, as illustrated in Figure 8, such consistency is not found. In this figure, the three plots in both the upper and lower rows correspond, in turn, to the cases presented in Figs. 7a, 7e, and 7i: the former involve V−R and V−I, respectively, whereas the latter are concerned with B−V colors. To get comparably good fits to the VR and VI photometry for the TO and MS stars as those found on the ([B−V]₀, M_V) diagram, it is necessary to apply small zero‐point shifts to the isochrone colors (as noted in the individual panels of Fig. 6). However, after such adjustments are made, the predicted giant branch segments are all appreciably bluer than the mean locus through the cluster giants, while in the case of the BV data, they either agree well with the observations or are slightly too red. For consistency, the differences between the isochrones and observations on the various color planes should be in the same sense and of a comparable size.

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As discussed by Stetson et al. (2004), small zero‐point differences (≲0.02 mag) are easily within calibration uncertainties and thus are not a serious concern. Indeed, some fairly compelling evidence presented in that investigation indicates that the V−R colors determined for the cluster stars are probably too red by about 0.02 mag. This was found from comparisons of the NGC 188 fiducial sequence with (1) Landolt standards on various color‐color diagrams, and (2) the CMDs for Gliese (1969) catalog stars having M_V values from Hipparcos and BV(RI)_C photometry from Bessell (1990). (If such an offset were applied to the V−R data, then essentially no zero‐point correction to the isochrone V−R colors would be necessary.) Importantly, in a systematic sense, the variations of B−V, V−R, and V−I colors with V magnitude along the lower MS of NGC 188 agreed very well with those indicated by the Gliese stars.

One way of reconciling the fits of the isochrones to the cluster observations in Figure 8 with their counterparts in Figure 7 is to adopt somewhat redder (V−I)‐T_eff and (V−R)‐T_eff relations for stars of lower gravity (log g≲4.0) than those by VC03. It is evident from the plots presented by VC03 (see their Figs. 8 and 15) that the color transformations advocated by Castelli (1999) have the desired behavior; i.e., they agree well with those adopted by VC03 at gravities appropriate to MS stars, but they become systematically redder as the gravity is reduced. The solid curve in Figure 9 represents the 6.8 Gyr isochrone for Y = 0.2774 and Z = 0.0173 (that plotted in Figs. 5e, 6b, and 6e) when the colors are derived from Castelli V−I and VC03 B−V transformations. The predicted color‐color diagram for the isochrone so obtained is clearly in good agreement with the same sample of Landolt standard stars considered in Figure 4 (the large filled circles and crosses represent class V and class III standards, respectively) and with the fiducial sequence for NGC 188 (small open circles). [The latter was dereddened assuming E(B−V) = 0.087 and the E(V−I)/E(B−V) relation given by Dean et al. 1978.]

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The same consistency was not found in our examination of the BVI photometry for M67 by MMJ93 (see Fig. 4). In this case, a representative isochrone produced a fine match to the color‐color diagram of M67 (when VC03 color‐T_eff relations were used), but the predicted giant branch segments of the two loci were offset from the location of Landolt class III standards. The dashed curve in Figure 9 shows where the giant branch segment of the solid curve would be located if the V−I colors were derived from the VC03 transformations, instead of those by Castelli (1999). It is quite clear that the solid curve provides a superior fit to the data. This implies that (1) the Castelli (V−I)‐T_eff relations are more realistic than those by VC03 (at least for [Fe/H] values close to solar), and (2) the MMJ93 V−I colors for giants in M67 are too blue by ∼0.02 mag. (Whereas the BV CMDs of M67 and NGC 188 appear to be consistent with one another, judging from the isochrone fits presented in Figs. 3 and 7, the same cannot be said for the cluster VI data. The comparisons that we have made of the cluster fiducials with the Landolt standards suggests that the problem must rest mainly with the M67 photometry.)

Figure 10 presents an analogous comparison of the different data sets on the V−R versus B−V diagram. [E(V−R)/E(B−V) = 0.59 was assumed; see Bessell et al. 1998.] Note that in order for the NGC 188 locus to overlie the same Landolt standards in the color range 0.6≲B−V≲0.9 as in the previous figure, it was necessary to subtract 0.018 mag from the fiducial V−R colors. (As discussed by Stetson et al. 2004, the zero‐point of the cluster R‐band photometry appears to be slightly too bright.) Here, too, it was necessary to adopt the Castelli (1999) V−R transformations instead of those published by VC03 in order to obtain the good agreement of the isochrone with the cluster sequence and the Landolt standards that is evident in this figure.

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Thus, one of the main conclusions reached in this investigation is that the Castelli (1999) transformations to V−R and V−I (for [Fe/H] ≈ 0.0) appear to be preferred over those by VC03. [As shown by Sekiguchi & Fukugita 2000 and VC03, the Castelli (B−V)‐T_eff relations are not consistent with empirical constraints.] If we then adopt her color‐temperature relations and repeat the comparisons between theory and observations given in Figure 8, we obtain the results shown in Figure 11. The quality of the isochrone fits to the observed data is now much improved, and importantly, they are now much more consistent with those on the [(B−V)₀, M_V] plane (see Fig. 7). That is, consistent explanations of the data on the three color planes can been achieved by using VC03's transformations to B−V, coupled with those by Castelli for V−R and V−I. We believe that our analysis has given us an improved understanding of the color‐temperature relations appropriate to stars having near‐solar abundances of the heavy elements.

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For the reasons discussed above, the zero‐point offset of 0.018 mag that was applied to the isochrones shown in Figures 11a–11c should probably be applied the NGC 188 observations instead. It is not clear whether the corrections to the predicted V−I colors in Figs. 11d–11f are due to a small problem with the color‐T_eff relations or with the photometric zero point.

The two most important results of this investigation are the following. First, when using the parameterized Roxburgh criterion described in § 1 to treat convective core overshooting, the value of the parameter F_over that should be assumed in computing nondiffusive stellar models for ≈1.3 M_⊙ near‐solar metallicity stars is ≈0.07. With this choice, isochrones are able to provide a good match to the CMD of M67 in the vicinity of the turnoff (and elsewhere). Second, the color‐color relations defined by Landolt standards in both the main‐sequence and giant phases offer considerable support for the VandenBerg & Clem (2003) (B−V)‐T_eff relations, together with the Castelli (1999) transformations to V−R and V−I. If these color‐temperature relations are adopted, then stellar models having a well‐constrained T_eff scale provide consistent (and quite good) fits to the various CMDs that can be constructed from the BVRI photometry for NGC 188 by Stetson et al. (2004), and they are able to reproduce the Landolt color‐color diagrams very well.

In carrying out our analyses of the M67 and NGC 188 CMDs, we explored only a small range in [Fe/H] (from −0.12 to +0.01, which nevertheless encompasses most spectroscopic determinations for these open clusters) in addition to a modest range in helium abundance (0.25≤Y≤0.30). Depending on the assumed chemistry, the derived ages of M67 and NGC 188 varied from 3.6 to 4.6 Gyr (see Fig. 3) and from 5.9 to 8.1 Gyr (see Fig. 7), respectively. Our best estimates of their ages, from stellar models that do not take diffusive processes into account, are 4.0 and 6.8 Gyr (treating such processes will lead to a reduction in these age estimates by 5%–7%; see Michaud et al. 2004). The resulting age determinations should be accurate to within ±10%, even if some allowance is made for the effects of reddening and calibration uncertainties.

The comparisons of isochrones with the M67 CMD presented in Figure 3 also showed that the value of F_over that is assumed in the models has to be a fairly small number, irrespective of the assumed chemical abundances. This indicates that core overshooting is much less important in the turnoff stars of M67 than in those of NGC 6819 (Rosvick & VandenBerg 1998) and NGC 7789 (Gim et al. 1998). Thus, based on the studies carried out to date, we have learned that F_over increases from a value near zero in ≈1.3 M_⊙ stars to about 0.5 in ≈1.55 M_⊙ stars, and the indications are that F_over remains fairly constant thereafter. (Many more constraints on the mass dependence of this parameter will be reported in a forthcoming study by P. D. Dowler & D. A. VandenBerg 2004, in preparation).

Finally, although we have argued from the Landolt standards and the NGC 188 photometry by Stetson et al. (2004) that Castelli (1999) transformations to V−R and V−I (for giant stars) are more realistic than those by VandenBerg & Clem (2003), the reliability of this result is difficult to assess. This is mainly because the (B−V)‐(V−I) diagram for giants in M67, based on CCD photometry either by Montgomery et al. (1993) or Richer et al. (1998), disagrees with that for giants in NGC 188 (Stetson et al.) at the level of 0.02–0.03 mag in V−I at a given B−V value (or vice versa). The Stetson et al. fiducial is definitely in better agreement with the mean locus for the Landolt standards, but the latter are relatively few in number, and they do not themselves define a tight sequence. Further work needs to be undertaken to understand the small discrepancies between the M67 and NGC 188 color‐color diagrams. Further study of a similar (≈0.02 mag) offset between the (B−V)‐(V−I) relation for main‐sequence stars derived from Landolt standards and that for Hyades main‐sequence stars would also be worthwhile.