Age Estimates of Globular Clusters in the Milky S Way: Constraints on Cosmology ECTION

G LOBULAR C LUSTERS S

REVIEW PECIAL Age Estimates of Globular Clusters in the Milky S Way: Constraints on Cosmology ECTION

Lawrence M. Krauss1* and Brian Chaboyer2*

Recent observations of stellar globular clusters in the Milky Way Galaxy, combined evolution. Thus, an accurate determination of with revised ranges of parameters in stellar evolution codes and new estimates of the the age of the oldest clusters can yield one of earliest epoch of globular cluster formation, result in a 95% conﬁdence level lower the most stringent lower limits on the age of limit on the age of the Universe of 11.2 billion years. This age is inconsistent with the our galaxy, and thus the Universe. expansion age for a ﬂat Universe for the currently allowed range of the Hubble Globular cluster age estimates in the constant, unless the cosmic equation of state is dominated by a component that 1980s fell in the range of 16 to 20 Ga (1–3), violates the strong energy condition. This means that the three fundamental observ- producing a new apparent incompatibility ables in cosmology—the age of the Universe, the distance-redshift relation, and the with the Hubble age, then estimated to be 10 geometry of the Universe—now independently support the case for a dark energy– to 15 Ga on the basis of an estimated lower

dominated Universe. limit on the Hubble constant H0 of 50 to 75 km sϪ1 MpcϪ1. This provided one of the Hubble’s first measurement of the expansion such clusters. This abundance can be less earliest motivations for reintroducing a cos- of the Universe in 1929 also resulted in an than one-hundredth of that measured in the mological constant into astrophysics. Such a embarrassing contradiction: Working back- Sun, which suggests that the gas from which term in Einstein’s equations results in an ward, on the basis of the expansion rate he these objects coalesced had not previously increased Hubble age because it allows for a measured, and assuming that the expansion experienced significant star formation and cosmic acceleration, implying a slower ex- on April 3, 2011 has been decelerating since the Big Bang—as one would expect given the attractive nature of gravity—allowed one to put an upper limit on the age of the Universe since the Big Bang of 1.5 billion years ago (Ga). Even in 1929 this age was grossly inconsistent with well- accepted lower limits on the age of Earth. Although this contradiction evaporated as further measurements of the expansion rate of the Universe yielded a value that was up to an www.sciencemag.org order of magnitude less than Hubble’s estimate, much of the subsequent history of 20th- century cosmology has involved a continued tension between the so-called Hubble age— derived on the basis of the Hubble expansion—and the age of individual objects with-

in our own galaxy. Downloaded from Of special interest in this regard are per- haps the oldest objects in our galaxy, called globular clusters. Compact groups of 100,000 to 1 million stars with dynamical collapse times of less than 1 million years, many of these objects are thought to have coalesced out of the primordial gas cloud that only later collapsed, dissipating its energy and settling into the disk of our Milky Way Galaxy (see Fig. 1). Those globular clusters that still populate the halo of our galaxy are thus among the oldest visible objects within it, a fact confirmed by measuring the abundance of heavy elements such as iron in stars within

1Departments of Physics and Astronomy, Case West- ern Reserve University, 10900 Euclid Avenue, Cleve- land, OH 44106, USA. 2Department of Physics and Astronomy, Dartmouth College, 6127 Wilder Labora- Fig. 1. The oldest globular clusters (compact collections of stars shown as bright dots in the ﬁgure) tory, Hanover, NH 03755, USA. are thought to have coalesced early on from small-scale density ﬂuctuations in the primordial gas *To whom correspondence should be addressed. cloud, which itself later coherently collapsed, dissipating its energy and settling into the disk of our E-mail: [email protected], chaboyer@heather. Milky Way Galaxy. As a result, these objects populate a roughly spherical halo in our galaxy today. dartmouth.edu This sequence of events is shown schematically in four stages, from upper left to lower right.

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pansion rate at earlier times. Thus, galaxies been tentatively detected in the metal-poor certainties in age estimates, while also located a certain distance away from us today star BDϩ17° 3248 (9). On the basis of the searching out other techniques whose inde-

ECTION would have taken longer to achieve this sep- thorium and uranium abundance measure- pendent uncertainties may be under better

S aration since the Big Bang than they other- ments and a comparison of these abundances control if globular cluster ages are to be used wise would have. However, as more careful to the abundance of other r-process elements, to constrain the equation of state for a flat examinations of uncertainties associated with an age of 13.8 Ϯ 4 Ga is found for this star Universe. stellar evolution were performed, as well as (9). Although these ages are in agreement, the

PECIAL refined estimates of the parameters that gov- uncertainty is large and depends critically on Main Sequence Turnoff Ages S ern stellar evolution, the lower limit on glob- the theoretical calculation of the initial pro- Numerical stellar evolution models provide ular cluster ages progressively decreased, so duction ratio of thorium and uranium. Im- estimates of the surface temperature and lu- that a wide range of cosmological models proving these age estimates will require fur- minosity of stars as a function of time. These produced Hubble ages consistent with this ther measurements of uranium and thorium in can be compared to observations of stellar lower limit (4–7). metal-poor stars, along with progress in un- colors and magnitudes in order to determine, In the interim, other classic cosmological derstanding the r-process in general and the in principle, the age of an ensemble of stars in tests [including cosmic microwave back- production ratio of thorium and uranium in a globular cluster. As stars evolve, their lo- ground (CMB) measurements and measure- particular. cation on a temperature-luminosity plot ments of redshift versus distance for distant White dwarf cooling. White dwarfs are the changes. Thus, the distribution on this plot supernovae], when combined with mass esti- end stage of the evolution of stars with initial for a system of stars of different masses (such mates from large-scale structure observa- masses less than about eight solar masses. No as one finds in a globular cluster) matches the tions, seem to point toward the need for nuclear energy generation is occurring in theoretical distribution at only one time. Al- something like a cosmological constant. It is these white dwarfs, which are supported by though probing the entire distribution of stars thus timely to reexamine stellar age estimates electron degeneracy pressure. The white would, in principle, provide the best con- in light of new data, in order to explore how dwarf radiates energy into space, slowly straint on the age of the system, in practice this might complement these other cosmolog- cooling and becoming less luminous over stellar models are best determined for main- ical constraints. time. As a result, the luminosity of the faint- sequence stars. The most robust prediction of There are three independent ways to reli- est white dwarfs in a cluster of stars can be the theoretical models is the time it takes a on April 3, 2011 ably infer the age of the oldest stars in our used to estimate the age of the cluster by star to exhaust the supply of hydrogen in its galaxy: radioactive dating, white dwarf cool- comparing the observed luminosities to the- core (4, 14), at which point the star leaves the ing, and the main sequence turnoff time scale. oretical models of how white dwarfs cool main sequence (Fig. 2). Thus, absolute glob- Radioactive dating. Recent observations (11). However, calculation of the models for ular cluster age determinations with the of thorium and uranium in metal-poor stars very old white dwarfs is complicated by com- smallest theoretical uncertainties are those (8, 9) now make it possible to determine plexities in the equation of state, uncertainties that are based on the luminosity at this “main ages of individual stars using the known in the core composition of white dwarfs, and sequence turnoff ” point. radioactive half-lives of these elements the cool temperatures at their surfaces, which The theoretical stellar isochrones used for (232Th has a half-life of 14 billion years, require detailed radiative transfer calculations comparison with observations are dependent whereas 238U has a half-life of 4.5 billion in order to estimate the flux emitted by these on a variety of parameters that are used as www.sciencemag.org years), if one can estimate the initial abun- stars in the observed wavelength regions. inputs for the numerical codes used to model dance ratio of these two elements. There An initial attempt to observe the faintest stars and compare them to observations. are two principal difficulties with radioac- white dwarfs in a globular cluster was made These input parameters are subject to various tive dating: (i) Thorium and uranium have in 1995 with the Hubble Space Telescope theoretical and experimental uncertainties. weak spectral lines in stars, and accurate (HST). These observations found white The one method that explicitly incorporates abundance measurements of these elements dwarfs at the limit of the photometry, which all of these in a self-consistent fashion uses

is only possible if the stars have enhanced implied that the globular cluster M4 had a Monte Carlo techniques (4). Input parameters Downloaded from thorium and uranium abundances relative minimum age of 9 Ga (12). A much deeper can be randomly chosen from distributions fit to lighter elements like carbon and nitro- exposure of M4 with HST (approximate ex- to the inferred uncertainties from observa- gen; and (ii) one must know how thorium posure time 8 days) was obtained in 2001 tions, allowing one to derive output distribu- and uranium are produced in order to cal- (13), allowing observation of fainter white tions for stars to be compared to observed culate their initial abundance ratio. The dwarfs. This study resulted in an estimated stellar distributions for low-metallicity glob- first difficulty is overcome by surveying age for this cluster of 12.7 Ϯ 0.7 Ga (13). ular clusters. It has been found that the un- large numbers of metal-poor stars to find However, the quoted error in this age only certainty in translating magnitudes to lumi- rare stars with enhanced abundances. The takes into account the observational uncer- nosities (i.e., the uncertainties in deriving second difficulty is more problematic, be- tainties and does not include the possible distances to globular clusters) is the chief cause thorium and uranium are produced by error introduced by uncertainties in the white source of uncertainty in globular cluster age rapid neutron capture (called the r-process) dwarf cooling calculations. It is particularly estimates. far from nuclear stability. This makes it difficult to estimate the complex equation of difficult to estimate the systematic uncer- state of these stars, so that the systematic Globular Cluster Distance Estimates tainties associated with theoretical predic- uncertainties are likely to be at least as large There are five main methods that can be used to tions of the initial abundance ratio of tho- as, if not larger than, the quoted statistical estimate the distance to globular clusters. All of rium and uranium. errors. these methods rely on various assumptions or At present, uranium has been definitively Age estimates from both these techniques prior calibration steps. The most common tech- detected in only a single star: CS 31082-001 thus suggest a galactic age in excess of 10 Ga, nique involves what are called “standard can- (8, 10). The computed production ratios give which can marginally be accommodated in a dles.” This method assumes that some class of a quoted age for CS 31082-001 of 14.0 Ϯ 2.4 flat, matter-dominated Universe, given cur- stars have the same intrinsic luminosity, regard- Ga (10). Uranium and a large number of other rent uncertainties in the Hubble constant. It is less of their location in the Galaxy. One then r-process elements, including thorium, have thus important to attempt to reduce the un- observes the flux (apparent luminosity on the

66 3 JANUARY 2003 VOL 299 SCIENCE www.sciencemag.org G LOBULAR C LUSTERS S PECIAL sky) of the same class of stars in ϭ –1.9, as this is the mean of a globular cluster and determines the globular clusters whose av- the distance to the globular clus- erage age will be determined.

ter by comparing the observed Because the different distance S flux to the assumed intrinsic lu- estimates span a relatively ECTION minosity of the star. Classes of modest range in [Fe/H] (0.54 stars used to determine the dis- dex), the exact value of the tances to globular cluster stars in- Mv(RR)–[Fe/H] slope has only clude white dwarfs (15), main- a minor effect in our resultant sequence stars (16), horizontal distance scale. The weighted branch (HB) stars (17), and RR mean value of the absolute Lyrae stars (a subclass of HB magnitude of the RR Lyrae stars) (18). Details regarding stars at [Fe/H] ϭ –1.9 is ϭ these different types of stars are Mv(RR) 0.46 mag. found in Fig. 2 and its caption. The statistical parallax tech-

The key to the success of this nique yields values for Mv(RR) approach involves the accuracy in that are larger (i.e., fainter) than determining the intrinsic lumi- the other distance techniques. nosity of the standard candle. When statistical parallax results This can be done by using theo- are included in the weighted retical models, or by directly mean, the standard deviation measuring the distance to a near- about the mean is 0.13 mag. by standard candle star using trig- When the statistical parallax re- onometric parallax. This distance sults are not included in the anal- estimation can be combined with ysis, the mean becomes ϭ a measurement of the star’s flux Mv(RR) 0.44 and the standard on April 3, 2011 to determine the intrinsic lumi- deviation about the mean drops to nosity of that standard candle. 0.07 mag. In earlier analyses, the Studies of the internal dy- statistical parallax data were not namics of the stars within a included (5), because there were globular cluster provide an in- suggestions that some systematic dependent method for determin- Fig. 2. A schematic color-magnitude diagram for a typical globular cluster differences might exist between ing the distance to a globular (33) showing the location of the principal stellar evolutionary sequences. RR Lyrae stars in the field and cluster. The dynamical distance This diagram plots the visible luminosity of the star (measured in magni- those in globular clusters. How- estimate compares the relative tudes) as a function of the surface color of the star (measured in B-V ever, subsequent investigations motion of globular cluster stars magnitude). Hydrogen-burning stars on the main sequence eventually ex- have shown that this is not the www.sciencemag.org in the plane of the sky (proper haust the hydrogen in their cores (main sequence turnoff ). After this, stars case (21). generate energy through hydrogen fusion in a shell surrounding an inert Ϯ motion) to their motions along hydrogen core. The surface of the star expands and cools (red giant branch). Using the 0.13 mag stan- the line of sight to the star (ra- Eventually the helium core becomes so hot and dense that the star ignites dard deviation results in a long dial velocities). The measured helium fusion in its core (horizontal branch). A subclass is unstable to radial tail at low values of Mv(RR). It radial velocities are independent pulsations (RR Lyrae). When a typical globular cluster star exhausts its is inappropriate to include this of distance, whereas the mea- supply of helium, and fusion processes cease, it evolves to become a white spurious low tail when quoting dwarf.

sured proper motions are small- an allowed range. An asymmet- Downloaded from er for more distant objects; ric Gaussian distribution ϭ ϩ0.13 hence, a comparison of the two observations visual (V) magnitude of an RR Lyrae star, Mv(RR) 0.46–0.09 mag has a low range allows one to estimate the distance to a glob- Mv(RR). The results are summarized in consistent with that derived when the statis- ular cluster. Table 1 for the different distance indica- tical parallax result is not included, but has a As these discussions make clear, the dis- tors. There are three new features of this mean and high range equivalent to the value tance determination for globular clusters is compilation, as compared to those associ- derived by including the statistical parallax subject to many uncertainties. However, our ated with previous analyses: (i) Hipparcos result in a straightforward way. This is the knowledge of the distance scale is evolving parallaxes for metal-poor, blue HB stars in distribution that will be used to derive the rapidly. Many of the distance indicators to the field to calibrate the globular cluster allowed distance scale for metal-poor globu- globular clusters make use of HB stars. Our distance scale are included (17); (ii) the lar clusters. knowledge of the evolution of HB stars con- statistical parallax results on field RR tinues to advance through the use of increas- Lyrae stars are included; (iii) a new HST Stellar Evolution Input Parameters ingly more realistic stellar models. Studies of parallax for the star RR Lyrae itself is Seven critical parameters used in the com- the evolution of HB stars and their use as included, which is considerably more accu- putation of stellar evolution models have distance indicators have shown that the lumi- rate than the Hipparcos parallax (18); and been identified whose estimated uncertain- nosity of the HB stars depends not only on (iv) only distance estimates for systems ty can significantly affect derived globular metallicity, but also on the evolutionary sta- with [Fe/H] Ͻ –1.4 are included. cluster age estimates (5). In order of impor- tus of the stars on the HB in a given globular To compare the different distance esti- tance, they are (i) oxygen abundance cluster (19, 20). mates, these Mv(RR) values must be translat- [O/Fe] (22), (ii) treatment of convection To compare different distance indica- ed to a common [Fe/H] value. For this, an within stars, (iii) helium abundance, (iv) Ϯ 14 ϩ 3 15 ϩ␥ tors, it is convenient to parameterize the Mv(RR)–[Fe/H] slope of 0.23 0.06 is used, N p O reaction rate, (v) distance estimate by what it implies for the as suggested by models (19). We used [Fe/H] helium diffusion, (vi) transformations from

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lower bound is This can be seen as follows. For a flat ⍀ more stringent Universe with fraction 0 in matter density ⍀ ECTION than previous esti- and x in radiation at the present time, the

S Ј mates because of a age tzЈ at redshift z is given by confluence of fac- H0tzЈ tors; the new ϱ M (RR) distance v dz PECIAL estimates contrib- ϭ ͵ ͑ ϩ ͒ ͑ ϩ ͓͒⍀ ͑ ϩ ͒3 ϩ ⍀ ͑ ϩ ͒3 1 w ͔1/ 2 S 1 z 1 z 1 z uted 60% of the in- 0 x crease relative to zЈ previous estimates. (1) The best-fit age has where w, the ratio of pressure to energy den- also increased and sity for the dark energy, represents the equa- is now 12.6 Ga. tion of state for the dark energy (assumed Moreover, we also here for simplicity to be constant). For w Ͻ find the 95% confi- –0.3, as required in order to produce an ⍀ Ϸ dence upper limit accelerating Universe, and for 0 0.3 and ⍀ Ϸ on the age to be 16 x 0.7 as suggested by observations (27– ⍀ Ga. To use these 29), the x term is negligible compared to the Fig. 3. Histogram representing results of Monte Carlo presenting 10,000 ﬁts ⍀ of predicted isochrones for differing input parameters to observed iso- results to constrain 0 term for all redshifts greater than about 4. chrones to determine the age of the oldest globular clusters. cosmological pa- This relation implies that the age of the rameters, one Universe at a redshift z ϭ 6 was greater than theoretical temperatures and luminosities to needs to add to these ages a time that corre- about 0.8 Ga, independent of the cosmologi- observed colors and magnitudes, and (vii) sponds to the time between the Big Bang and cal model. Using this relation and the esti- opacities below 104 K. Table 2 details the the formation of globular clusters in our galaxy. mates given above, we find, on the basis of range of these parameters used in a Monte Observations of large-scale structure, com- main sequence turnoff estimates of the age of on April 3, 2011 Carlo simulation to evaluate the errors in bined with numerical simulations and CMB the oldest globular clusters in our galaxy, a the globular cluster age estimates. The measurements of the primordial power spec- 95% confidence level lower limit on the age ranges for the oxygen abundance, the pri- trum of density perturbations, have now estab- of the Universe of 11.2 Ga, and a best fit age mordial helium abundance, and the helium lished that structure formation occurred hierar- of 13.4 Ga. diffusion coefficients have changed as a chically in the Universe, with galaxies forming These limits can be compared with the result of recent observations. before clusters. Moreover, observations of gal- inferred Hubble age of the Universe, given by axies at high redshift definitively imply that Eq. 1 for zЈϭ0, for different values of w, and The Age and Formation Time of gravitational collapse into galaxy-size halos oc- for different values of H0 today. CMB deter- Globular Clusters and Constraints on curred at probably less than 1.5 Ga, and defi- minations of the curvature of the Universe the Cosmic Equation of State and nitely less than 5 Ga, after the Big Bang. Al- suggest that we live in a flat Universe [i.e., www.sciencemag.org Mass Density though this puts a firm upper limit of 21 Ga on (30)]. When we combine the CMB result Using the estimates for the parameter ranges for the age of the Universe, the task of putting a with these age limits on the oldest stars, some the variables associated with stellar evolution lower limit on the time in which galaxies like form of dark energy is required. The Hubble

(4, 5), we have determined the age of the oldest ours formed is somewhat more subtle. Key Project (31) estimated range for H0 is Galactic globular clusters with a Monte Carlo Fortunately, this is an area in which our 72 Ϯ 8kmsϪ1 MpcϪ1 (note that this estimate

simulation (Fig. 3). Taking a one-sided 95% observational knowledge has increased in re- is based on an estimate of Mv(RR) that is

range, one finds a lower bound of 10.4 Ga. This cent years. In particular, recent studies using consistent with our estimates). Figure 4 dis- Downloaded from observations of globular clusters in plays the allowed range of w versus matter ϭ Ϫ1 nearby galaxies and measurements of energy density for the case H0 72 km s ␣ Ϫ1 ⍀ Ͼ high-redshift Lyman objects all in- Mpc . In this case, for 0 0.25, as sug- dependently put a limit z Ͻ 6 for the gested by large-scale structure data, w Ͻ –0.7 maximum redshift of structure forma- at the 68% confidence level, and w Ͻ –0.45 tion on the scale of globular clusters at the 95% confidence level. (23–26). Although the precise limits and allowed To convert the redshifts into times, it parameter range are sensitive to the assumed is generally necessary to know the equa- value of the Hubble constant, the lower limit tion of state of the dominant energy den- on globular cluster ages presented here defin- sity in order to solve Einstein’s equations itively rules out a flat, matter-dominated (i.e., for an expanding Universe. However, the w ϭ 0) Universe at the 95% confidence level

age of the Universe as a function of for the entire range of H0 determined by the redshift is insensitive to the cosmic equa- Key project. Interestingly, for the best fit tion of state today for redshifts greater value of the Hubble constant, globular cluster than about 3 to 4. This is because for age limits also put strong limits on the total these early times the matter energy den- matter density of the Universe. In order to Ͼ ⍀ sity would have exceeded the dark ener- achieve consistency, if w –1, 0 cannot gy density because such energy, by vio- exceed 35% of the critical density at the 68% Fig. 4. Range of allowed values for the dark energy equation of state versus the matter density, assuming a lating the strong energy condition, de- confidence level and 50% of the critical den- ﬂat Universe, for the lower limit derived in the text for creases far more slowly as the Universe sity at the 95% confidence level. ϭ Ϫ1 Ϫ1 the age of the Universe, and for H0 72 km s Mpc . expands than does matter. One might wonder whether the upper lim-

68 3 JANUARY 2003 VOL 299 SCIENCE www.sciencemag.org G LOBULAR C LUSTERS S PECIAL 11. B. M. S. Hansen, Astrophys. J. 520, 680 (1999).

Table 1. Estimates of Mv(RR). 12. H. B. Richer et al., Astrophys. J. 484, 741 (1997). Method Objects [Fe/H] M (RR) M (RR) at [Fe/H] ϭ –1.90 13. B. M. S. Hansen, Astrophys. J., in press. v v 14. A. Renzini, in Observational Tests of Cosmological

Theoretical HB models Metal-poor blue HB –1.90 0.35 Ϯ 0.10 0.35 Ϯ 0.10 (19) Inflation, T. Shanks, A. J. Banday, R. S. Ellis, Eds. S clusters (Kluwer, Dordrecht, Netherlands, 1991), pp. 131– ECTION Main sequence fitting M13, M68, M92, –1.83 0.35 Ϯ 0.10 0.36 Ϯ 0.10 (16) 141. N6752 15. A. Renzini et al., Astrophys. J. 465, L23 (1996). White dwarf fitting N6397 –1.42 0.52 Ϯ 0.15 0.41 Ϯ 0.15 (15) 16. B. Chaboyer, in Globular Cluster Distance Determina- LMC RR Lyr RR Lyr in the LMC –1.90 0.44 Ϯ 0.10 0.44 Ϯ 0.10 (34) tions in Post-Hipparcos Cosmic Candle, A. Heck, F. Trigonometric parallax RR Lyrae –1.39 0.61 Ϯ 0.11 0.49 Ϯ 0.15* (21) Caputo, Eds. (Kluwer, Dordrecht, Netherlands, 1998), Trigonometric parallax Field halo HB stars –1.51 0.60 Ϯ 0.12 0.51 Ϯ 0.16* (17) pp. 111–124. Dynamical M2, M13, M22, –1.74 0.59 Ϯ 0.15 0.55 Ϯ 0.15 (16, 35) 17. R. G. Gratton, Mon. Not. R. Astron. Soc. 296, 739 (1998). M92 Ϯ Ϯ 18. G. F. Benedict et al., Astron. J. 123, 373 (2002). Statistical parallax Field RR Lyr –1.60 0.77 0.13 0.71 0.16* (36) 19. P. Demarque, R. Zinn, Y. Lee, S. Yi, Astron. J. 119, *Includes additional 0.10 mag uncertainty due to uncertain evolutionary status. 1398 (2000). 20. F. Caputo, S. degl’Innocenti, Astron. Astrophys. 298, 833 (1995). 21. M. Catelan, Astrophys. J. 495, L81 (1998). Table 2. Monte Carlo input parameters. 22. [O/Fe] is the oxygen abundance in a star relative to the iron abundance, in solar units, and is defined by Parameter Distribution References ϭ the equation [O/Fe] log(O/Fe)star – log(O/Fe)Sun. [O/Fe] 0.45 Ϯ 0.25 (uniform) (37–41) For example, a star with the same relative oxygen- ϭ Convection: Mixing length 1.85 Ϯ 0.25 (uniform) (5) to-iron ratio as the sun would have [O/Fe] 0. Helium abundance 0.2475 Ϯ 0.0025 (uniform) (30, 42, 43) 23. H. A. Kobulnicky, D. C. Koo, Astrophys. J. 545, 712 (2000). 14N ϩ p 3 15O ϩ␥reaction rate 1.00 Ϯ 0.12 (uniform) (5) Ϯ 24. D. Burgarella, M. Kissler-Patig, V. Buat, Astron. J. 121, Helium diffusion 0.50 0.30 (uniform) (44–47) 2647 (2001). Color transformation Binary (48, 49) 25. O. Y. Gnedin, O. Lahav, M. J. Rees, available at http:// Opacities below 104 K 1.0 Ϯ 0.3 (uniform) (5) arXiv.org/abs/astro-ph/0108034. 26. S. van den Bergh, Astrophys. J. 559, L113 (2001).

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