A NEW COLOR MAGNITUDE DIAGRAM FOR 47 TUCANAE: A STATISTICAL ANALYSIS

Transcription

1 The Astronomical Journal, 138: , 2009 November C The American Astronomical Society. All rights reserved. Printed in the U.S.A. doi: / /138/5/1455 A NEW COLOR MAGNITUDE DIAGRAM FOR 47 TUCANAE: A STATISTICAL ANALYSIS P. A. Bergbusch 1,3 and P. B. Stetson 2 1 Department of Physics, University of Regina, Regina, Saskatchewan S4S 0A2, Canada; pbergbusch@accesscomm.ca 2 DAO-HIA, NRC, 5071 W. Saanich Rd., Victoria, BC V9E 2E7, Canada Received 2009 June 29; accepted 2009 August 22; published 2009 October 6 ABSTRACT We present a statistical analysis of color magnitude diagrams of 47 Tuc derived from original and archival BVI photometry that produces the most probable locus for single stars. After adopting E(B V ) = 0.04, we derive an apparent distance modulus (m M) V = and achieve good matches to the most probable locus in the [B V, V], [V I, I], and [B I, I] planes with 12 Gyr, [α/fe] = +0.3, [Fe/H] = 0.83 isochrones from the Victoria Regina models. This metallicity is generally lower than recent spectroscopically derived estimates for the cluster, but it is reinforced by the main-sequence match with a sample of subdwarfs. Key words: globular clusters: general globular clusters: individual (47 Tuc) 1. INTRODUCTION More than half a century has passed since globular cluster color magnitude diagrams (CMDs) calibrated with photoelectric measurements were published (e.g., Arp et al. (1952, 1953) in the Stebbins Whitford (P,V) photometric system; Sandage & Walker(1955), Arp & Johnson (1955), Johnson & Sandage (1956) in the Johnson UBV photometric system). Those early CMDs were obtained from the iris photometry of photographic plates calibrated against photoelectric sequences of relatively isolated stars in the cluster fields. They were relatively sparse and the tabulations of magnitudes and color indices did not include estimates of the photometric errors for each star. Consequently, their CMDs consisted of points plotted to correspond with the magnitude and color measurements, and this has remained as the common practice. Although the Stebbins Whitford photometric system has long been abandoned, the Johnson system (with some modifications) has continued to be popular because of tremendous efforts to define standard stars (e.g., Landolt 1992) for photometry and because of continuing improvements to the color temperature transformations and bolometric corrections over a wide range of metallicities that permit direct comparisons between theory and observation. Now that it is possible to derive both magnitudes and error estimates via profile fitting photometry for tens of thousands of stars within a given globular cluster, it makes sense to revisit the way in which a CMD is constructed. In this paper, we present CMDs in which we plot the probability of a star having a particular magnitude and color index in the globular cluster 47 Tuc. Such CMDs give a direct visualization of the most probable loci for the evolutionary sequences while simultaneously revealing the significant evolutionary phases and the rates of evolution through them. In globular cluster studies, it has been customary to compare isochrones and/or theoretical horizontal branch (HB) calculations with observed CMDs to constrain cluster ages, distances, Based in part on observations made with the European Southern Observatory (ESO) telescopes and obtained from the ESO/ST-ECF Science Archive facility. 3 Visiting Astronomer, Cerro Tololo Inter-American Observatory (CTIO). CTIO is operated by AURA, Inc. under contract to the National Science Foundation. reddenings, metallicities, abundance mixtures, and mixing scenarios. Evolutionary rates derived from the theoretical calculations have been compared to the observed luminosity functions to probe the interior evolution of the stars, usually to constrain some aspect of the physics employed in the modeling process Tuc 47 Tuc has derived considerable attention over the years because it is relatively near, it is one of the more massive (and hence populous) clusters, and it is relatively metal rich. Estimates of its age help to constrain scenarios for the formation of stellar populations within the Galaxy since it is a member of the thick disk population. It has also been a useful laboratory in which to test a wide variety of distance determinations for consistency, ranging from white dwarf fitting at the faint end (Zoccali et al. 2001) to red giant branch (RGB) tip magnitudes (Bono et al. 2008) at the bright end, and every other indicator in between. Integrated light studies extend its significance beyond the Galaxy, (e.g., McWilliam & Bernstein 2008), by helping to constrain the ages and metallicities of unresolved extragalactic systems. The most significant CCD-based study of the cluster dates back to Hesser et al. (1987), but over the past decade photometric studies for substantial numbers of 47 Tuc stars in BVI (Kaluzny et al. 1998), in uvby (Grundahl et al. 2002), and in the near-infrared (Salaris et al. 2007) have appeared. A recent Hubble Space Telescope study (Anderson et al. 2009) reveals structure in the subgiant branch (SGB) suggestive of two distinct populations and a main sequence that has an intrinsic width greater than that expected from the photometric errors Metallicity and Reddening Dating back at least to Zinn & West (1984), metal abundance estimates for 47 Tuc have typically centered near [Fe/H] = 0.7 ± 0.1. For example, Carretta et al. (2004) usedhigh dispersion spectroscopy of subgiant and main-sequence stars to find [Fe/H] = 0.67, but a more recent estimate favors a slightly more metal-poor value, [Fe/H] = 0.76 (Koch & McWilliam 2008). Estimates of the reddening to 47 Tuc (Grundahl et al. 2002, and references therein) typically cover the range 0.03 < E(B V ) < 0.055, but the most robust (and the preferred) estimate comes from the IRAS/COBE/DIRBE dust map by 1455

2 1456 BERGBUSCH & STETSON Vol. 138 Schlegel et al. (1998). The mean reddening for a 2 deg 2 field centered on 47 Tuc derived from the dust map is E(B V ) = , where the quoted limits reflect the maximum and minimum values over the field. However, anticipating the results of our analysis, we adopt E(B V ) = 0.04 ± 0.01 given the range of published values and their associated uncertainties, and we acknowledge that all subsequent interpretations are uncertain at the 0.01 mag level. From the studies on interstellar extinction by He et al. (1995) and McCall (2004), we adopt the reddening relations E(V I) = 1.35 E(B V ) = and E(B I) = 2.35 E(B V ) = (Cousins I band), with R V = A V /E(B V ) = 3.08 ± OBSERVATIONS The B, V, and I data 4 considered in this investigation come from a database of original and archival observations (Stetson 2000), calibrated on the Landolt (1992) photometric system. The total body of data consists of 856 CCD images from 436 individual exposures obtained during 18 observing runs. In particular, 60 exposures producing 480 images were obtained from the eight-ccd Wide-Field Imager of the European Southern Observatory (ESO)/MP 2.2 m telescope. Observations that span from near the tip of the RGB to well below the main-sequence turnoff point made by P. Bergbusch with the 1.5 m telescope at the Cerro Tololo Inter-American Observatory in 1998 November account for another 200 images that provide wide spatial coverage of the cluster and serve to unify the other archival observations. Not all the images were centered at the same location, so any individual star appears in a maximum of 91 B-, 96 V-, and 97 I-band exposures. Some U-band observations do exist but we have not used them because of difficulties in calibration. We have selected the data for this analysis from a total of 210,250 objects with measurements in both B and V (204,280 in both V and I, and 203,083 in both B and I) that survived the reduction process in a field approximately 35 41, centered on the cluster Selection Criteria Imposed on the Data As is common practice, we have restricted our analysis to those objects that pass selection criteria based on spatial distribution as well as the photometric errors and profile fitting statistics derived via DAOPHOT (Stetson 1987) and ALLFRAME (Stetson 1994). Figure 1 shows the [B V, V] CMD for the entire unrestricted data set, color coded with respect to distance from the cluster center. The photometric scatter induced by the blending of unresolved pairs or groups of stars, as evidenced by the population of objects brighter than the turnoff and blueward of the giant branch and those fainter than the turnoff and redward of the main sequence (but not members of the Small Magellanic Cloud), is substantially reduced at a distance of 200 and almost completely absent from the sample outside 400. These spatial restrictions are further reinforced by the plots of magnitude versus radial distance in Figure 2, color coded with respect to σ B V. The lower envelope of the HB population, evident as a band of stars spanning the width of the diagram near V = 14.15, clearly shows evidence of a brightward bias induced by blending within 200 of the cluster center. The transition from the turnoff 4 The calibrated photometry will be made available upon publication from ormore simply Figure 1. CMD for 47 Tuc. Stars within 50 of the cluster center are plotted as gold dots, those between 50 and 100 are plotted in magenta, those between 100 and 200 are cyan, those between 200 and 400 are red, and those outside 400 are plotted in black. The order in which the regions are listed is the order in which the colors were plotted. to the base of the RGB, evident near V = 17.2, shows the onset of a similar bias around 400, and this spatial limit appears to be robust down to V 21. Excluding stars within 200 of the cluster center reduces the number of objects with measurements in both B and V to 172,582, in both V and I to 166,611, and in both B and I to 165,415. To excise objects that are probably non-stellar from the list of candidate objects, we used the mean shape statistic SHARP and the mean goodness of fit statistic χ derived from the ALLFRAME reductions. The mismatch χ, between an actual stellar profile and the point-spread function (PSF) used to model it, is the ratio of the observed pixel-to-pixel flux residuals to the scatter expected from readout noise and photon statistics. It tends to unity for faint objects because any actual profile mismatch becomes small relative to the noise sources. SHARP is a measure of the difference between an object s perceived radius and the effective radius of the PSF. At the faint end of the data, it becomes poorly defined and therefore shows increasing scatter. The correlation between χ and SHARP evident at intermediate magnitudes, shown in Figure 3, suggests that both indices are doing a good job of distinguishing point sources from extended objects in this regime. The value of both of these indices is seriously degraded for the faintest stars. Based on the evidence of Figure 3, we selected data via the following recipe:

3 No. 5, 2009 THE COLOR MAGNITUDE DIAGRAM OF 47 TUC Figure 2. Magnitudes are plotted as a function of distance from the cluster center. In the upper two panels, the color code is with respect to the uncertainties in B V : black indicates σb V > 0.1, green 0.05 < σ 0.10, magenta 0.03 < σb V 0.05, cyan 0.01 < σb V 0.02, and red σb V The colors were plotted in the order listed. In the lower panel, the coding is with respect to σb I. The lower envelope to the horizontal branch distribution is plotted as a dashed line at B = 14.95, V = 14.15, and I = The vertical dashed line marks the 200 limit inside of which the lower envelope of the HB shows a brightward bias S 1.0, (1) χ (V 14). (2) Brighter than V = 15.5, our prescription for χ accepts virtually all of the data, but at V = 15.5, the limiting value is χ = 3.81, while at V = 21.5, it is χ = In Figure 4, we plot the photometric errors as a function of magnitude. Although there are artifacts in this diagram that arise from different subsets of the archival data, for example, the population of objects in the range 16 V 20, they do not stand out in the CMD plotted in Figure 1. Given that poorly measured stars will smear out in a probability density analysis, as will be discussed in the following section, there is no need to delete any objects from the sample. However, for computational reasons (it takes a disproportionate amount of CPU time to analyze poorly measured stars on PAB s SUN Ultra 10 workstation) we have imposed a σb V 0.2. This generous limit gives access to the better measured data at the faint end of the sample. The CMDs plotted in Figure 5 show the results of selecting data with the exclusion limit set at 100 and the sigma limit 1457 Figure 3. DAOPHOT profile fitting statistic χ plotted as a function of the shape parameter SHARP at 1 mag intervals. The points are color coded with respect to σb V with the same limits as for the upper two panels of Figure 2 and are plotted in the same order. at 0.2 in the color index. The total number of stellar objects in the plotted samples is 179,181 for the B and V passbands, to 177,938 for V and I, and to 172,319 for B and I Probability Density Calculations Assuming that the stellar sample is complete, the probability of observing a star at a particular magnitude and color index in a CMD simply depends on the number density of stars at that magnitude convolved with the uncertainties in the photometry. To avoid the coupling of the photometric errors, for example, in V and B V, it is advantageous to compute the probability on the [B, V ] plane. Then, the probability of observing a star at a particular [B, V ] location is calculated by accumulating the total probability in a parallelogram-shaped cell with a base δb wide and an equal height δv. On transformation to the [B V, V ] plane, which skews the cell by 45, the cell becomes square with a height δv and an equal width δ(b V ). We treated each star as a two-dimensional Gaussian profile with σv and σb from the photometry, and computed the probability contribution in

4 1458 BERGBUSCH & STETSON Vol. 138 Figure 5. CMDs of 47 Tuc in the [V, B V ], [I, V I ], and [I, B I ] planes are color coded with respect to the uncertainty in the color index, as indicated on the plot. The points were plotted in the order of decreasing σ, the σ 0.01 sample being plotted last. The samples of data plotted are restricted to the region outside of 100 of the cluster core. The subsequent probability distribution analysis is derived from these samples. Figure 4. Photometric errors in the color indices are plotted as a function of magnitude. The location of the HB is evident near V = 14 or I = 13 while the RGB bump is evident near V = 14.5 or I = The population of objects with greater than typical sigmas most obvious between 17 < V < 20.5 is derived from one field of the archival ESO data near the edge of the region covered by this study. There are no more than 11 V images and 9 B images for this field and few local standard stars this far out. The upper red dashed line at σ = 0.2 in each panel indicates the limit we imposed on the data. cells out to a distance of 5σ from the locus of the star in the [B, V ] plane. The probability of finding a star at a particular location in the CMD, derived from the data plotted in Figure 5, is plotted as a Hess diagram in Figure 6, with square bins mag high in magnitude and mag wide in color. For the illustration, the probabilities are sorted into seven equally spaced logarithmic levels, with the lower threshold set at 1% of the maximum probability in the sample. (For example, in the [B V, V] pane, plotted in the left-hand panel, the maximum cumulative probability in any one cell is Consequently, only cells containing a probability in excess of 0.16 are shown. Pmax=15.75 implies that that one cell contains the equivalent of stars.) The main-sequence locus, as well as the transition from the turnoff to the base of the RGB and the location of the RGB bump are clearly evident. It is also apparent that the RGB locus shifts slightly blueward just at the location of the bump in all three of the color indices, most obviously in B I. Figure 6. Probability of finding a star at any location of the CMD, derived from the data plotted in Figure 5 reveals the detailed morphology of the single star sequences. The RGB bump, clearly evident approximately half a magnitude below the level of the horizontal branch, appears to demarcate a slight blueward jog in the upper RGB locus. The values of Pmax in each panel give the equivalent number of stars within the one mag cell that contains the maximum cumulative probability. For comparisons with isochrones, we derive the locus for the main sequence, the turnoff, the SGB and the RGB up to the tip by identifying the most probable (modal) point at each magnitude level (the bins are mag high, so each level corresponds to a mag step) of the probability distribution. The most probable locus in the region of the turnoff and lower RGB is plotted in Figure 7 for data outside the regions limited by 100, 200, and 600, respectively. To avoid contamination of the sample by objects that clearly do not belong to the single star locus on the RGB of 47 Tuc, we constructed the red and blue edges by eye, as indicated by the dashed lines. Remarkably, there is no obvious bias in the selection of the most probable points even in the sample outside 100, which indicates that blended images of unresolved single stars do not significantly affect the analysis this close to the cluster center. The locus defined by the

5 No. 5, 2009 THE COLOR MAGNITUDE DIAGRAM OF 47 TUC 1459 Figure 7. Most probable locus of single stars in the turnoff region, derived from the probability distributions (like the ones shown in Figure 6) based on the samples of data restricted stars outside a 100 radius (dark blue squares), outside a 200 radius (green squares), and outside a 600 radius (red squares). The only significant difference among the three samples is the increased scatter in the 600 sample; even in the 100 sample blended images of unresolved single stars do not significantly affect the analysis. 600 sample is not as tightly defined as the others, but this is just a statistical artifact. The robustness of this method for identifying the most probable locus for single stars relies on a populous sample of well-measured stars at each magnitude level. Obviously, this cannot be achieved on the upper portions of the RGB and it eventually fails at the faint end of the main sequence (see Figures 8 and 10, near V = 21, I = 20) where the mag square bins contain the equivalent of only one or two stars, but we have chosen to let the observations speak for themselves. Given the small measuring errors on the upper RGB, the observed distribution is indicative of real star-to-star differences in physical properties, and any attempt to impose a fiducial line through these data presumes that one actually knows what the differences are. The question that remains to be addressed is whether the most probable locus actually represents the locus for single stars. True binaries along the main sequence up to the turnoff region will occupy the region redward and up to 0.75 mag brightward of the single star locus and thus could cause a redward shift in the most probable locus. The numerical simulation described in the Appendix to this paper demonstrates the validity of our calculations and the robustness of the technique against binary star contamination. It suggests that any shift in color has to be no more than or mag in B V since the probabilities have been summed in bins mag wide. Figure Gyr VR isochrones and ZAHBs are superimposed on the most probable locus (plotted as small cyan squares) for single stars derived from the 100 sample together with the HB and AGB probability distributions derived from the sample outside a 200 radius. We have adopted E(B V ) = for the [Fe/H] = 0.61 isochrone even though it is not completely consistent with E(B V ) = 0.04 for the [Fe/H] = 0.83 isochrone, since negative reddenings are not physical. 3. ISOCHRONE FITS The locus for single stars derived from the data outside of 100 of the cluster center together with the probability density plot of the HB and asymptotic giant branch (AGB) from data outside of 200 is plotted in Figure 8. The 12 Gyr, [α/fe] = +0.3 isochrones from the Victoria Regina models (hereafter referred to as the VR models; VandenBerg et al. 2006) superimposed on them encompass the range of metallicities attributed to 47 Tuc in the literature. (The dashed line represents the [Fe/H] = 0.83 isochrone, the dotted line is for [Fe/H] = 0.61.) Regardless of the metallicity, its clear that the simultaneous match of the turnoff region and the level of the HB implies that 12 Gyr is a reasonable age estimate. However, if E(B V ) = 0.04 is the appropriate choice for the reddening and if the zero point of color calibration of the VR models is correct, the preferred metallicity is [Fe/H] = 0.83, which is slightly lower 5 than any of the recent spectroscopically determined metallicities (e.g., Koch & McWilliam 2008) for the cluster. 5 A lower value for the reddening, as the Schlegel et al. (1998) dust maps suggest, could imply an age slightly greater than 12 Gyr, or a metallicity slightly lower than [Fe/H] = 0.83.

6 1460 BERGBUSCH & STETSON Vol. 138 We note that the RGB locus is situated redward of both the [Fe/H] = 0.71 and the 0.83 isochrones, but that the level of the RGB tip is only matched by the [Fe/H] = 0.83 isochrone. These features will be discussed in more detail in Section 3.3 of the paper Subdwarfs We have identified a sample of 56 subdwarfs with metallicities in the range 0.45 [Fe/H] 0.95 from the study by Karatas & Schuster (2006). Four of them with spectroscopic orbits in the SB9 catalog (Pourbaix et al. 2004) are not used in our analysis. The parallaxes quoted by Karatas & Schuster have been updated by van Leeuwen (2007), and we have used them to recalculate the absolute magnitudes (with Lutz Kelker corrections), and have limited the sample to stars with M V > 4.6. The updated parallaxes and absolute magnitudes for the sample of subdwarfs used in this study are presented in Table 1. Chosen this way, the sample only contains stars for which σ π /π Each star in the sample was corrected in color to the adopted metallicity in the following way. The B V color on 10 Gyr isochrones spanning the range of metallicities 1.14 [Fe/H] 0.30 (a total of eight isochrones) was interpolated at the absolute magnitude of the subdwarf. Then, an Akima spline 6 with B V as a function of [Fe/H] was constructed through these isochrone points. The theoretical color of the star at its observed metallicity was then interpolated via the spline. The difference between its theoretical color and the color on the [Fe/H] = 0.83 isochrone at the same absolute magnitude was applied as the shift in color to [Fe/H] = Experiments employing isochrones ranging from 5 to 14 Gyr show that such differential color corrections are insensitive to age. Figure 9 shows the subdwarfs corrected for color superimposed on the most probable locus of the 47 Tuc CMD, for which a distance modulus (m M) = and a reddening E(B V ) = 0.04 have been adopted. (We have followed the caveat that Karatas & Schuster applied with respect to the reddening estimates for the subdwarfs, so only reddenings greater than 0.02 mag in B V have been applied.) The vertical error bars were derived directly from the parallax errors. (Judging from the VR isochrones, horizontal error bars amounting to about 0.02 mag are consistent with uncertainties of 0.1 dex in the metallicity.) The 5 and 10 Gyr isochrones suggest that evolutionary effects are potentially relevant for the subdwarfs brighter than V 18.5 on the diagram. The overall impression is that there remain a number of unresolved potential binaries in the sample. Given that small amounts of reddening have been ignored for the subdwarfs and that the reddening of 47 Tuc is uncertain by no more than ±0.01 mag, the overall quality of the match between the subdwarfs and the locus of most probable points for 47 Tuc cannot be improved by invoking a slightly higher metallicity. For example, based on the VR models, correcting the subdwarf colors to [Fe/H] = 0.71 would shift them approximately 0.02 mag redward from their locations in Figure 9, and this differential adjustment is independent of the zero point in the color temperature relations. On the other hand, if we had applied E(B V ) = 0.032, the reddening favored by the dust 6 The Akima spline is particularly useful when the data exhibit non-polynomic behavior: it avoids overshooting between data points. The result is a good drawn-by-eye look to the interpolant. Akima splines have been used extensively in the interpolations performed by the suite of software incorporated into the VR models. Table 1 Subdwarf Data HIP V B V E(B V) [Fe/H] M V π σ π SB9 (mag) (mag) (mag) (mag) (mas) (mas) maps of Schlegel et al. (1998), the agreement at [Fe/H] = 0.83 would be slightly better Mixing Length Parameter, Bolometric Corrections, and Color T eff Relations In Figure 10, 12 Gyr, [Fe/H] = 0.83 isochrones are superimposed on the most probable loci for single stars in the [B V, V], [V I, I], and [B I, I] planes. It should be

7 No. 5, 2009 THE COLOR MAGNITUDE DIAGRAM OF 47 TUC 1461 Figure Gyr, [Fe/H] = 0.83 VR isochrones, as well as the corresponding ZAHBs, are superimposed on the most probable locus for single stars (the small cyan squares) on the three CMD planes of this study. The small circles on the SGB mark the location of the local minimum on the isochrone population function predicted by the VR models; those on the RGB mark the location of the RGB bump. The HB and AGB probability distributions, plotted as the color-coded small squares, are derived from the sample outside a 200 radius. The limits for the HB and AGB distributions are taken to imply non-overlapping ranges, e.g., the red symbols are plotted for the range 0.6 <P 1.1. Figure 9. Sample of subdwarfs, color corrected to [Fe/H] = 0.83, superimposed on the most probable locus (plotted as small cyan squares) for single stars. Stars within the metallicity range 0.45 [Fe/H] > 0.65 are plotted as large red squares, those within 0.65 [Fe/H] < 0.85 are plotted as dark blue circles, and those within 0.85 [Fe/H] < 0.95 are plotted as green triangles. As discussed in the text, this fit implies that the metallicity of 47 Tuc is even lower than the most recent spectroscopic studies have claimed. emphasized that the isochrones have been plotted without resort to any arbitrary color shifts to force agreement at the turnoff: we have simply applied the reddenings as described in Section The apparent distance modulus applied to the I-band photometry is (m M) I = , which enforces consistency at the level of both the HB and the SGB transition. It is well known that both the turnoff color and the level of the SGB transition are affected by the mixing length parameter, α MLT, but experiments with [Fe/H] = 1.01 isochrones computed for α MLT = 1.60 and 2.20 (unpublished, kindly provided by D. A. VandenBerg) show that the effects are rather small: ΔM V /Δα MLT 0.13 and ΔM I /Δα MLT 0.06 at the level of the SGB transition. Since the effects with respect to absolute magnitude work in the same sense in both V and I, givena fixed value for α MLT, the differential effect between the V and I magnitudes amounts to only 0.07 mag per unit mixing length parameter. The same differential effect with respect to B V and B I is obtained for mixing length induced color shifts at the level of the turnoff. While the mixing length parameter does affect the shape of an isochrone, the fact that we achieve good simultaneous agreement among the ZAHB level, the SGB transition level, and the turnoff colors together with consistency in the quality of the fits in all three CMDs plotted in Figure 10 argues that it cannot be a significant cause of the discrepancies in color evident along the RGB nor in the I-band level of the RGB tip. The isochrone experiments show that if agreement in B V is forced at the turnoff and in the level of the SGB transition, increasing α MLT results in a bluer RGB. Over the range of mixing length parameters considered, the RGB tip varies in brightness by more than 1 mag in the sense that decreasing α MLT decreases V RGBT, but the color at the tip varies only by about 0.05 mag. On the other hand, while variations in α MLT induce similar behavior along the lower RGB in B I, I RGBT varies by less than 0.3 mag and (B I) RGBT varies by more than 1 mag. Returning to Figure 10, the mixing length parameter cannot be responsible for the discrepancies between the isochrones and the observations that occur on the RGB because discrepancies inb V must be accompanied by even greater discrepancies in V RGBT. Moreover, the sense of the discrepancies must remain the same, so the isochrone should remain either too blue or too red in V I and B I, and I RGBT should not be significantly affected. It is more likely that some revisions both to the color T eff relations and to the bolometric corrections for the I-band photometry are in order Evolutionary Phases: the SGB Transition, the RGB Bump, and Tip It has been suggested (Bergbusch & VandenBerg 1997) that the distribution in color of stars along the SGB transition from the turnoff to the base of the RGB could provide a useful constraint on cluster ages. However, subsequent (unpublished) studies showed that at least 50% of the sensitivity of their method was not due to age, but rather to the morphology of the transition. The problem arises from the difficulty in accurately defining the cluster turnoff point: while the turnoff color can be determined with good precision, the turnoff magnitude in an observed CMD is difficult to nail down more precisely than ±0.1 mag. The method applied by Bergbusch and VandenBerg

8 1462 BERGBUSCH & STETSON Vol. 138 nothing to distinguish the sample in the annulus between 50 and 100 (green) from the sample outside 400 (black), but the latter one is less likely to contain observations of blended images. While the neither the HB nor the AGB can be distinguished from the RGB locus in this diagram, the red end of the distribution converges to the dashed line defined by B V = 1.7, which is the appropriate color at the tip, consistent with the observations and in reasonable agreement (the observed B V color of the RGB tip is somewhat redder than the isochrone but, as can be seen in Figure 10, the observed V I and B I colors are slightly bluer) with the RGB tip the color of the 12 Gyr, [Fe/H] = 0.83 isochrone. The confusion evident on the [B V, V] plane derives mainly from the intrinsic scatter in the V magnitudes. Figure 11. Two-color diagram for stars brighter than the level of the ZAHB. Stars within the annulus between 50 and 100 of the cluster center are plotted in green, those between 100 and 200 are in red, those between 200 and 400 are in cyan, and those outside 400 are in black. The dashed line represents the locus for stars with B V = 1.7. attempted to circumvent this problem by identifying a point on the SGB transition 0.05 mag redward of the turnoff, but to make isochrone comparisons based on this offset assumes that the isochrone morphology is correct. The circle plotted on the SGB transition portion of the isochrones in Figure 10 marks the location of the local minimum on the isochrone population function (IPF; VandenBerg et al. 2006) predicted by the VR models. On the [B V, V] plane, it occurs at B V = and V = , which coincides quite accurately with the gap in the SGB distribution centered at B V = 0.7 in Figure 7, further reinforcing significance of the isochrone match. The circle plotted on the RGB portion of the isochrone near the level of the HB corresponds to the location of the evolutionary pause on the RGB that produces a local maximum in the IPF (aka the RGB bump) predicted by the VR models, at V = and I = The observed location of the bump in Figure 6 is slightly fainter than V = 14.5 and slightly brighter than I = 13.5, both occurring slightly earlier in evolution than the VR models suggest. This is not unexpected, as the VR isochrones are consistent with the Bergbusch & VandenBerg (2001) models, which also failed to predict the luminosity of the bump by a similar amount. Such a discrepancy has also been suggested by Salaris et al. (2007), who concluded that the observed location of the RGB bump in the near-infrared is possibly fainter than their models predict. The RGB tip is quite well delineated by the observations in the I band, as illustrated in Figures 5, 6, and 10, but its morphology in the V band is difficult to ascertain. Some of the confusion arises from the merging of the AGB with the RGB near the tip. Some of it may derive from the presence of intrinsic variables, whichwouldbemoreobviousinthev band than in the I band. Additional considerations are (1) there could be unrecognized artifacts in the photometry arising from incipient nonlinearity in the CCDs and (2) the brightest stars are measured in fewer images than most of the fainter stars because they are saturated in some exposures. Some of the confusion can be removed by constructing a twocolor plot, as shown in Figure 11, in which the observations are color coded with respect to their spatial distribution. There is 4. CONCLUDING REMARKS The high-quality observations of 47 Tuc, extending from at least 3 mag in the I band and 4 mag in the V band below the main-sequence turnoff right up to the RGB tip, presented in this paper constitute one of the most robust sets of photometric data for a globular cluster published to date. We have combined the photometric information with the numbers of stars at each location in the CMD to perform a probability distribution analysis. This kind of analysis provides for a direct visualization of evolutionary rates on the CMD and allows for the extraction of the most probable locus for single stars. The technique is well suited to populous samples and works particularly well for the main sequence and subgiant transition regions of the CMD. It is less successful on the upper portions of the RGB because of the intrinsic scatter in the observed sample and its reduced size. Consequently, the analysis is dominated by the well-measured stars on the upper RGB. Comparisons between the most probable loci for single stars and VR isochrones and ZAHBs on the [B V, V], [V I, I], and [B I, I] planes suggest that the cluster age is 12 Gyr for metallicities in the range 0.61 [Fe/H] We obtain a good fit to the color of the turnoff and the level of the SGB transition when E(B V ) = 0.04, without invoking an arbitrary zero-point adjustment to the isochrone colors, with [Fe/H] = 0.83 and an apparent distance modulus (m M) V = We achieve good consistency with the models on all three observational planes. Although recent high-resolution spectroscopy suggests that the cluster metallicity is [Fe/H] = 0.76, our result is further reinforced by a main-sequence fit to local subdwarfs. If anything, our subdwarf analysis implies that 47 Tuc may be even more metal poor since we did not apply small reddening corrections to the subdwarf sample and the adjustments in color to correct the subdwarf sample to [Fe/H] = 0.83 still left them generally redward of the single star locus for 47 Tuc. In light of the claim by Anderson et al. (2009) that the SGB of 47 Tuc exhibits an intrinsic spread in luminosity of about 0.12 mag in m F 435W, suggestive of at least two distinct populations, it is worth comparing the most probable locus for single stars as we have derived it with the evidence presented in their Figure 1. The interpretation of their histograms is, to some extent, influenced by the way they partitioned the data by luminosity into three groups. Without these partitions, a global histogram of the entire SGB population could be interpreted as a normal distribution with an extended faint tail. From our analysis, the most probable locus for single stars would correspond to the peak of such a distribution; as defined by their data, we

9 No. 5, 2009 THE COLOR MAGNITUDE DIAGRAM OF 47 TUC 1463 would identify points near m F 475W = at several values of the color indices across the SGB. We do not observe as large an intrinsic spread in SGB luminosity in a sample of our V data for stars with σ B V taken from the region outside 400. Anderson et al. (2009) also report that the main sequence of 47 Tuc is broader than can be accounted for by observational errors and suggest a dispersion of 0.1 dex in [Fe/H] would be sufficient to produce the effect. The distribution in color of the observations appears to be symmetric, so if our method of locating the most probable locus for single stars were applied to their data, it would have produce a ridge line consistent with the mean metallicity of the cluster. The numerical simulation presented in the Appendix to this paper shows that our method is insensitive to the potential bias introduced by a reasonable binary star population. This work was supported by the Natural Sciences and Engineering Research Council of Canada through an operating grant to P. A. Bergbusch. It is a pleasure to thank D. A. VandenBerg for providing the unpublished evolutionary tracks used in the discussion regarding the mixing length parameter, for input regarding color temperature relations and bolometric corrections, and for useful discussions regarding the subdwarf sample. P. A. Bergbusch also thanks a former student, Mr. Keith Nakonechky, for his unsung efforts to perform the initial analysis of my 47 Tuc observations from CTIO, superseded by the results presented in this paper. APPENDIX NUMERICAL SIMULATION To test the function of the code we developed for the probability distribution analysis and to evaluate the potential bias due to the presence of binary stars in the location of the most probable (modal) locus for single stars, we constructed a synthetic population of cluster stars on which to perform a numerical simulation. The potential bias is inherent in the smoothing effect produced by convolving the photometric errors with the star counts, since all true binaries will populate the region redward of the single star main sequence. We limited the simulation to 16 <V <20.5 for which the cluster sample outside 100 contains approximately 124,500 stars. A 12 Gyr, [Fe/H] = 0.83 IPF (VandenBerg et al. 2006), with x = 0.5asthe mass spectrum exponent [in Φ(M) M (1+x) ] was used as the seed for the synthetic stellar population. Since the model IPF only reached down to M V = (M = M ), we had to extrapolate the main sequence down to M V = 13 (M = 0.12 M ) to ensure that a significant fraction of the binary stars lay close enough to the single star locus. The B and V magnitudes were derived using different seeds for the random number generator. We assigned 10% of the sample to the binary star component. After each of the synthetic stars had been assigned its B and V magnitudes and its mass, those that had been designated as binaries were assigned a lower mass companion from the same seed IPF. The synthetic CMD is shown in the left panel of Figure 12. We modeled the error estimates for each star by constructing plots of σ B and σ V versus V, similar to those shown in Figure 4, and used independent random number generators to derive them assuming uniform distributions for each one. The adopted limits on σ V and σ B were such that σ V and σ B , or σ B V at V = 18. This recipe ensures that the contribution to the total Figure 12. Synthetic sample of stars for the numerical simulation to test the correctness of the code used in our probability density calculations and to evaluate the potential bias introduced by binaries is plotted in the left panel. The center panel shows the results of the probability distribution analysis with the threshold and scaling set to discriminate against the outlying binaries. The right panel shows the most probable locus in red, with the seed isochrone superimposed in black. probability within a cell by stars lying near the single star locus is not diluted significantly: binaries that lie far from the single star locus do not contribute much even if they have large sigmas. The center panel of Figure 12 shows the probability distribution for the synthetic sample. The probabilities are sorted into seven equally spaced logarithmic levels, and the lower threshold has been adjusted to discriminate against the binary components brightward of the transition from the turnoff to the RGB and redward of the single star locus. A comparison between this simulated distribution and the left panel of Figure 6, which shows the actual distribution for the cluster, suggests that x = 0.5is too low, since the simulation produces the highest probability density brightward of the turnoff point. The right panel shows the most probable locus (modal points, in red) derived from the distribution in the center panel. The isochrone superimposed on the most probable locus demonstrates that no significant bias has been introduced by the small amount of smoothing from the convolution of the photometric errors with the numbers of stars. The bluest point on the seed IPF is at B V = 0.506, exactly the same as the bluest point on the recovered locus. Thus, if there is a bias it can only be of the order of mag in B V. REFERENCES Anderson, J., Piotto, G., King, I. R., Bedin, L. R., & Guhathakurta, P. 2009, ApJ, 697, L58 Arp, H. C., Baum, W. A., & Sandage, A. R. 1952, AJ, 57, 4 Arp, H. C., Baum, W. A., & Sandage, A. R. 1953, AJ, 58, 4 Arp, H. C., & Johnson, H. L. 1955, ApJ, 122, 171 Bergbusch, P. A., & VandenBerg, D. A. 1997, AJ, 114, 2604 Bergbusch, P. A., & VandenBerg, D. A. 2001, ApJ, 556, 322 Bono, G., Stetson, P. B., & Sanna, N., et al. 2008, ApJ, 686, L87 Carretta, E., Gratton, R. G., Bragaglia, A., Bonifacio, P., & Pasquini, L. 2004, A&A, 416, 925 Grundahl, F., Stetson, P. B., & Anderson, M. I. 2002, A&A, 395, 481 He, Lida, Whittet, D. C. B., Kilkenny, D., & Spencer Jones, J. H. 1995, ApJS, 101, 335 Hesser, J. E., Harris, W. E., VandenBerg, D. A., Allwright, J. W. B., Shott, P., & Stetson, P. B. 1987, PASP, 99, 739 Johnson, H. L., & Sandage, A. R. 1956, ApJ, 124, 379

10 1464 BERGBUSCH & STETSON Vol. 138 Kaluzny, J., Wysocka, A., Stanek, K. Z., & Krzeminski, W. 1998, Acta Astron., 48, 439 Karatas, Y., & Schuster, W. J. 2006, MNRAS, 371, 1793 Koch, A., & McWilliam, A. 2008, AJ, 135, 1551 Landolt, A. U. 1992, AJ, 104, 340 McCall, M. L. 2004, AJ, 128, 2144 McWilliam, A., & Bernstein, R. A. 2008, ApJ, 684, 326 Pourbaix, D., Tokovinin, A. A., Batten, A. A., & Fekel, F. C. 2004, A&A, 424, 727 Salaris, M., Held, E. V., Ortolani, S., Gullieuszik, M., & Momany, Y. 2007, A&A, 476, 243 Sandage, A. R., & Walker, M. F. 1955, AJ, 60, 230 Schlegel, D. J., Finkbeiner, D. P., & Davis, M. 1998, ApJ, 500, 525 Stetson, P. B. 1987, PASP, 99, 191 Stetson, P. B. 1994, PASP, 106, 250 Stetson, P. B. 2000, PASP, 112, 925 VandenBerg, D. A., Bergbusch, P. A., & Dowler, P. D. 2006, ApJS, 162, 375 van Leeuwen, F. 2007, in Astrophysics and Space Science Library, Vol. 350, Hipparcos, The New Reduction of the Raw Data (Dordrecht: Springer) Zinn, R., & West, M. J. 1984, ApJS, 55, 45 Zoccali, M., et al. 2001, ApJ, 553, 733