The Age Of Globular Clusters In Light Of Hipparcos Resolving

a r

X

i

v

:a

s t

r o

-

p

h

/

9

7

6

1

28

v

3

4

A

u

g

1

9

9

7

CERN-TH/97-121CWRU-P4-97astro-ph/9706128June 1997(revised July 97)The Age Of Globular Clusters In Light Of Hipparcos:Resolving the Age Problem?Brian Chaboyer 1,2,P.Demarque 3,Peter J.Kernan 5and Lawrence M.Krauss 4,5,6ABSTRACT We review ?ve independent techniques which are used to set the distance scale to globular clusters,including subdwarf main sequence ?tting utilizing the recent Hippar-cos parallax catalogue.These data together all indicate that globular clusters are farther away than previously believed,implying a reduction in age estimates.We now adopt a best ?t value M v (RR)=0.39±0.08(stat )at [Fe /H]=?1.9with an additional uniform systematic uncertainty of +0.13?0.18.This new distance scale estimate is combined with a detailed numerical Monte Carlo study (previously reported by Chaboyer et al.1996a)designed to assess the uncertainty associated with the theoretical age-turno?luminosity relationship in order to estimate both the absolute age and uncertainty in age of the oldest globular clusters.Our best estimate for the mean age of the oldest globular clusters is now 11.5±1.3Gyr,with a one-sided,95%con?dence level lower limit of 9.5Gyr.This represents a systematic shift of over 2σcompared to our earlier estimate,due completely to the new distance scale—which we emphasize is not just due to the Hipparcos data.This now provides a lower limit on the age of the universe which is consistent with

either an open universe,or a ?at,matter dominated universe (the latter requiring H 0≤67km s ?1Mpc ?1).Our new study also explicitly quanti?es how remaining uncertainties in the distance scale and stellar evolution models translate into uncertainties in the

–2–

derived globular cluster ages.Simple formulae are provided which can be used to update

our age estimate as improved determinations for various quantities become available.

Formulae are also provided which can be used to derive the age and its uncertainty for

a globular cluster,given the absolute magnitude of the turn-o?,or the point on the

subgiant branch0.05mag redder than the turn-o?.

Subject headings:stars:interiors–stars:evolution–stars:Population II–globular

clusters:general–cosmology:theory–distance scale

1.Introduction

The absolute age of the oldest Galactic globular clusters(GCs)currently provides the most stringent lower limit to the age of the universe,and as such,provides a fundamental constraint on cosmological models.In particular,for some time the best GC age estimates have been in direct contradiction with the maximum Hubble age for the preferred cosmological model,a?at matter dominated universe.The most recent comprehensive analyses suggested a lower limit of approx-imately12Gyr for the oldest GC’s in our galaxy(e.g.Chaboyer,Demarque,Kernan&Krauss 1996,hereafter Paper I),which,for a?at matter dominated model,implies H0≤53km s?1Mpc?1, a value which is low compared to almost all observational estimates.

Because of this apparent discrepancy,it remains critically important to continue to re-evaluate the errors associated with the GC age determination process itself.GC age estimates are obtained by comparing the results of theoretical stellar evolution calculations to observed color magnitude diagrams.The absolute magnitude of the main-sequence turn-o?(M v(TO))has small theoretical errors,and is the preferred method for obtaining the absolute ages of GCs(e.g.Renzini1991).Age determination methods which utilize the color of the models,or post main-sequence evolutionary models are subject to much larger theoretical uncertainties,and do not lead to stringent age limits.

In recent years,a number of authors have examined the question of the absolute age GCs (e.g.Chaboyer&Kim1995,Mazzitelli et al.1995,Salaris et al.1997)using di?erent assumptions for the best available input physics.Chaboyer1995presented a table of absolute GC ages based on a variety of assumptions for the input physics needed to construct the theoretical age-M v(TO) relationship.VandenBerg et al.1996have presented a review of the absolute ages of the GCs,and by comparing results from di?erent authors,include an discussion on how various uncertainties in the age dating process e?ect the?nal age estimate.

In order to obtain both a best estimate,and a well-de?ned lower limit to the absolute age of the oldest GCs,we earlier adopted a direct approach of running a Monte Carlo simulation.In our Monte Carlo,the various inputs into the stellar evolution codes were varied within their inferred uncertainties,utilizing1000sets of isochrones,and the construction of over4million stellar models. From these theoretical isochrones,the age-M v(TO)relationship was determined,and combined

–3–

with an empirical calibration of M v(RR)7in order to calibrate age as a function of the di?erence

).This calibration in magnitude between the main sequence turn-o?and horizontal branch(?V TO

HB

was used to derive the mean age of17old,metal-poor GCs using?V TO

.The principal result of

HB

this work was an estimate for the age of the oldest GCs of14.6±1.7Gyr,with the one-sided95% C.L.lower bound of12.1Gyr(Paper I)mentioned above.Another important result was an explicit demonstration that the uncertainty in M v(RR)overwhelmingly dominated the uncertainty in the GC age determination.We chose a Gaussian distribution for the uncertainty in M v(RR)because the data,while scattered,appeared to be appropriately distributed about the mean value,which we then determined to be M v(RR)=0.60at[Fe/H]=?1.9,with an uncertainty of approximately 0.16at the95%con?dence level.

Since this work was completed,the Hipparcos satellite has provided improved parallaxes for a number of nearby subdwarfs(metal-poor stars)(Perryman et al.1997),the distance to a GC has been estimated using white dwarf sequence?tting(Renzini et al.1996),a number of new astrometric distances to GCs have been published(Rees1996),and improved theoretical horizontal branch models have become available(Demarque et al.1997).This has lead us to critically re-evaluate the globular cluster distance scale(and hence,the M v(RR)calibration),and update our estimate for the absolute age of the oldest GCs.We?nd,using the full Hipparcos catalogue along with the other independent distance estimators that all the data suggests that this distance scale, and hence the GC age estimate have shifted by a signi?cant amount,suggesting that the dominant uncertainty in M v(RR)was,and still is,not statistical but rather systematic in character.

A detailed discussion of the globular cluster distance scale is presented in§2.The input parameters and distributions in the Monte Carlo are presented in§3.The principal results of this paper are presented in§4,which includes simple formulae which can be used to update the absolute age of the oldest GCs when improved estimates for the various input parameters become available. Finally,§5contains a brief summary of our results,and a brief discussion of their cosmological implications.

2.The Globular Cluster Distance Scale

It is currently impossible to directly determine distances to GCs using trigometric parallaxes. While such distance estimates may be available in the future from micoarcsecond space astrome-try missions(Lindegren&Perryman1996,Unwin,Boden&Shao1996),at present a variety of secondary distance estimates are all that is available for GC’s.The di?erent techniques rely upon di?erent data and assumptions.As such,we have elected to review a number of these techniques, and present a GC distance scale which is based on combining5independent estimates.To facilitate

–4–

this,we have reduced the various distance estimates to a calibration of M v(RR).This allows us

method(see§4).As we are interested in absolute ages,we have to derive GC ages via the?V TO

HB

focused our attention on those techniques which rely upon the minimum number of assumptions and thus hopefully should provide a priori the most reliable absolute distances.

2.1.Astrometric Distances

A comparison of the proper motion and radial velocity dispersions within a cluster allows for a direct determination of GC distances,independent of reddening(Cudworth1979).Although this method requires that a dynamical model of a cluster be constructed,it is the only method considered here which directly measures the distance to a GC without the use of a‘standard’candle. The chief disadvantage of this technique is its relatively low precision.This problem is avoided by averaging together the astrometric distances to a number of di?erent GCs.Rees(1996)presents new astrometric distances to eight GCs,along with two previous determinations.As pointed out by Rees,there are possibly large systematic errors in the dynamical modeling of M15,NGC6397 and47Tuc.As such,these clusters will be excluded in our analysis.In addition Rees(private communication)cautions that the distance to M2will be revised soon to due to a new reduction of the M2proper motions.Excluding this cluster from the analysis results in six clusters whose distances have been estimated astrometrically.Table1tabulates the astrometric distances from Rees(1996).Unless otherwise noted,the numbers are those given by Rees(1996)..For the[Fe/H] values,we have given preference to the high dispersion results of Kraft,Sneden and collaborators. Taking the weighted average of the M v(RR)values listed in Table1results in M v(RR)=0.59±0.11 at<[Fe/H]>=?1.59,where the average[Fe/H]value has been calculated using the same weights as in the M v(RR)average.

2.2.White Dwarf Sequence Fitting

Renzini et al.(1996)have utilized deep HST WFPC2observations of NGC6752to obtain accurate photometry of the cluster white dwarfs.They have combined this with similar observations of local white dwarfs with known parallaxes and masses(close to those in the cluster)to derive the distance to NGC6752using a procedure similar to main sequence?tting.The derived distance modulus is(m?M)O=13.05±0.10assuming E(B?V)=0.04.This reddening estimate is from Zinn1985,and is identical to those found by Burnstein&Heiles1982and Carney1979. NGC6752is a moderately metal-poor([Fe/H]=?1.54from Zinn&West1984)cluster with V(HB)=13.63(tabulated by Chaboyer et al.1996c).NGC6752has an extremely blue HB, thus,an estimate of V(HB)relies upon an extrapolation of the observed photometry.As such the determination of V(HB)in NGC6752is rather uncertain,and so have elected to take a rather generous error bar in the determination of V(HB)of±bd40081fc281e53a5802ff11bining the above quantities yields M v(RR)=0.45±0.14.

–5–

2.3.Subdwarf Main Sequence Fitting

Using parallaxes of nearby?eld stars,it is possible to de?ne the position of the ZAMS,and via a comparison to deep GC color magnitude diagrams obtain a rather direct estimate of the distance to a cluster.Unfortunately,the position of the ZAMS is a rather sensitive function of metallicity, and there are few nearby subdwarfs.Hence,there are few metal-poor stars with well determined absolute magnitudes.

The release of the Hipparcos data(Perryman et al.1997)has improved this situation some-what,providing a large database of high quality parallax measurements.The Hipparcos catalogue contains over100,000stars,of which nearly21,000stars have parallax errors less than10%.The Hipparcos catalogue has been searched for stars which are suitable for GC main sequence?tting. When selecting stars for use in main sequence?tting,it is important to avoid potential biases due to unresolved binaries and stars which are evolved o?the ZAMS.Known or suspected binaries which are not resolved photometrically should be avoided as both magnitudes and colors may be signi?cantly altered by the presence of a companion.The use of stars which have evolved o?the ZAMS may lead to systematic errors in the derived distance moduli,as it is not clear if GCs and metal-poor?eld stars are exactly of the same age.For example,a2Gyr age di?erence between a calibrating subdwarf at M V=5and a GC would lead to a systematic error of0.14mag in the distance modulus(based on our standard isochrones).To be safe,we will only consider stars with M V～>5.5.Fainter than this,the stars are essentially unevolved.

Many of the stars in the Hipparcos catalogue have large relative parallax errors and are not useful for main sequence?tting.For these reason,we have elected to only consider stars with σπ/π<0.10.This stringent selection criterion was selected to minimize potential Lutz-Kelker type biases(Lutz&Kelker1973,Brown et al.1997).The Hipparcos catalogue was searched for stars which(a)haveσπ/π<0.10,(b)are fainter than M V?5.5,and(c)are not known or suspected Hipparcos binaries or variables.This resulted in a list of2618stars,of which the great majority have near solar metallicity.As we are interested in the most metal-poor globular clusters, we require stars with[Fe/H]～5.5,[Fe/H]～

–6–

the Hipparcos parallax observations.Another indication that possible Lutz-Kelker type corrections are small for our sample is that the the maximumσπ/πvalue is0.08,well below our threshold of 0.10.This makes it extremely unlikely that stars whose true parallax are systematically smaller than the observed parallax are preferentially included in our sample(the source of Lutz-Kelker type biases).

Our subdwarf sample only has two stars in common with the sample of Pont et al.1997.This is because the Pont et al.1997sample only includes?ve stars on the ZAMS(M V～>5.5).As discussed above,even as small as a2Gyr age di?erence between evolved subdwarfs and the GC will lead to systematic errors in the distance modulus of～0.14mag.Of the unevolved stars in the Pont et al.1997sample,three are known binaries,which we do not use.Pont et al.1997apply an average binary correction of+0.375mag to the6binaries in their total sample.The Poisson(root N)noise in this correction is±0.15mag.If one has a large sample of binaries(～>30),then the approach taken by Pont et al.1997to include average binary corrections is sound.However,given that the small number statistics in the present sample results in a very large error in the binary correction,we believe it is best not to use the binaries.

Theoretical models predict that the location of the ZAMS is a sensitive function of metallicity. Even with the Hipparcos data,the current observations are not accurate(or numerous)enough to empirically derive the ZAMS location as a function of metallicity.There are only four subdwarfs whose absolute magnitudes are known to within±0.1mag.Unfortunately,the colors predicted by the models are still rather uncertain,and so we do not have a reliable calibration of how the location of the ZAMS changes as a function of metallicity.Thus one should ensure that the mean metallicity of subdwarf sample used in the main sequence?tting should be as close as possible to the metallicity of the GC.This requires accurate metallicity determinations.

We have searched the literature for abundance analyses,based upon high dispersion,high signal to noise spectrum.King(1997)has performed a detailed abundance analysis of HD134439and HD 134440(a common proper motion pair).Rather surprisingly,King?nds that the abundances of theα-capture elements are consistently some～0.3dex below the vast majority of metal-poor?eld stars,and those observed in GC giant stars.Due to their relatively high abundance,theoretical models predict that theα-capture elements play an important role in determining the position of a star in the color magnitude diagram.Given the peculiar abundances in these two stars,we have elected not to use them in main sequence?tting.The calibrating subdwarf data for the remaining 8stars is presented in Table2.Most of the data in Table2has been taken from the Hipparcos catalogue.The reddening estimates are from Carney et al.1994when available,or Pont et al.1997. The[Fe/H]values are discussed in detail below.

The[Fe/H]abundance of HD193901has been determined by a number of groups.Recent values of[Fe/H]are?1.00(Carretta&Gratton1997),?0.98(Axer et al.1994)and?1.22(Tomkin et al.1992).The higher abundances derived by Axer et al.1994and Gratton et al.1997may be largely due to the di?erent e?ective temperature scales adopted by these authors compared to

–7–

Tomkin et al.1992.This is still a matter of active debate,so we have elected to simple average the above[Fe/H]values.In our main sequence?tting analysis(§2.3.1and§2.3.2),we explore the consequences of the various[Fe/H]values.HD145417has not been extensively studied,and the only spectroscopic metallicity determination is[Fe/H]=?1.15±0.13(Gratton et al.1997).

Balachandran&Carney(1996)have presented a detailed abundance analysis of HD103095 (Groombridge1830),the subdwarf with the best determined absolute magnitude.They found [Fe/H]=?1.22±0.04.Other abundances appeared to be typical of metal-poor stars.This is very similar to the value obtained by Gratton et al.1997([Fe/H]=?1.24±0.07).The Balachandran& Carney(1996)value is adopted in this work.The abundance of HD120559has been determined to be[Fe/H]=?1.23±0.07(Axer et al.1994).Tomkin et al.1992have found[Fe/H]=?1.45for HD126681.

The metallicity of BD+592407is[Fe/H]=?1.60±0.16(Gratton et al.1997)based on high signal to nose data.Carney et al.1994found[Fe/H]=?1.91in their low signal to noise data.An examination of the Gratton et al.1997and Carney et al.1994abundances indicates that the former are systematically more metal-rich than the later.Gratton et al.(1997)found that on average, their[Fe/H]values were+0.34dex more metal rich than Tomkin et al.1992.Once again,part of this di?erence is attributable to the di?erent e?ective temperature scales.We have again elected to adopt the average value[Fe/H]=?1.75,and will discuss the di?erent[Fe/H]scales in§2.3.1 and§2.3.2.

A detailed abundance analysis of HD25329has been presented by Beveridge&Sneden(1994). They found[Fe/H]=?1.84±0.05,while Gratton et al.1997report[Fe/H]=?1.69±0.07. Beveridge&Sneden(1994)note that HD25329is a N-enhanced star;only～3%of observed halo dwarfs are N-enhanced.However,they?nd that the relative abundances ofα-capture and iron peak elements are normal for metal-poor stars.Nitrogen comprises only～3%of the mass fraction of the heavy elements in a star.Thus,the fact that HD25329is N-enhanced is unlikely to a?ect its position on the color magnitude diagram,and so it will be used in the main sequence?tting. Taking a simple average of the above two[Fe/H]determinations results in[Fe/H]=?1.76.

Finally,CPD-80349has[Fe/H]=?2.26±0.2(Pont et al.1997).This is based on low signal to noise spectra,and is on the same system as Carney et al.1994.This is apparently the most metal-poor star in the Hipparcos catalogue(see Figure1),and an improved abundance determination would be of great bene?t.This star has E(B?V)=0.02(Pont et al.1997).Given the poor quality of the abundance determination,and the fact that it is di?cult to determine the reddening for a single star,we have elected not to use CPD-80349in our main sequence?ts.

Figure1presents the HR diagram for the calibrating subdwarf data,along with a comparison with our standard isochrones.Most of the stars have metallicities in the range?1.1to?1.5and provide a nice calibration of the ZAMS in this[Fe/H]range.The position of HD25329([Fe/H]=?1.76)is somewhat surprising,as it lies along the same isochrone as HD103095([Fe/H]=?1.22). Both of these stars have very well determined metallicities and parallaxes.Note that BD+59

–8–

2407([Fe/H]=?1.75)does not lie along the same isochrone as HD25329.This could be due to an error in the reddening or parallax of BD+592407.Alternatively,it suggests that HD25329 is anomalously bright for its metallicity and color.Clearly more data is needed to di?erentiate between these hypothesis.Unfortunately,an inspection of the Hipparcos catalogue reveals there are no candidate metal-poor,unevolved single stars withσπ/π<0.10which are likely to have [Fe/H]

Given the[Fe/H]values of the calibrating subdwarfs,accurate GC distances using main se-quence?tting can be obtained for GC with?1.8～<[Fe/H]～

2.3.1.NGC6752

High resolution spectra of three giants yields[Fe/H]=?1.58(Minniti et al.1993),while the six giants studied by Norris&Da Costa1995yield[Fe/H]=?1.52.Carretta&Gratton1997obtained data for4other giants,and re-analyzed the above data to obtain[Fe/H]=?1.42.Averaging these three abundance determinations,we adopt[Fe/H]=?1.51.The reddening is E(B?V)=0.04,as discussed in§2.2.Subdwarfs with?1.23≤[Fe/H]≤?1.76were used in the weighted,least squares ?t.The mean abundance of these subdwarfs(using the same weighted as in the least squares?t to the NGC6752?ducial)is[Fe/H]=?1.55,very similar to our adopted[Fe/H]abundance of NGC 6752.

The distance to NGC6752was determined using a weighted least squares?t to the deep photometry of this cluster as presented by Penny&Dickens1986.The weights for the?t were the

,presented in Table2).These absolute errors in the absolute magnitudes of the subdwarfs(σM

V

magnitude errors only include the parallax errors.To allow for errors in the photometry,an error of±0.02mag was added in quadrature with theσM

tabulated in Table2when performing the

V

?t.The resultant distance modulus is(m?M)V=13.33±0.04mag,where this error represents the error associated with the weighted least squares?t of the NGC6752?ducial to the subdwarf

–9–

data.To this error,one must add in errors associated with the reddening,and allow for possible metallicity errors.An uncertainty in the reddening of±0.01translates into an error in the derived distance modulus of±0.05.

Due to the possible systematic uncertainties in the metallicity abundances of the subdwarfs and NGC6752we have examined various possibilities,in order to determine how a possible mis-match between the metallicity of NGC6752and the calibrating subdwarfs might a?ect the distance modulus estimates.

1.Adopting the Carretta&Gratton1997abundance for NGC6752([Fe/H]=?1.42),and using

subdwarfs with?1.22≤[Fe/H]≤?1.76.The weight of HD103095in the?t was decreased by

error to±0.045,ensuring that the mean weighted mean metallicity of the increasing itsσM

V

5calibrating subdwarfs was[Fe/H]=?1.42.The resultant distance modulus is(m?M)V=

13.31±0.03mag.

2.Adopting the Carretta&Gratton1997abundance for NGC6752along with the Gratton

et al.1997and Axer et al.1994abundances for the subdwarfs.This was done as the Axer et al.1994and Gratton et al.1997subdwarf abundances are systematically more metal-rich than other determinations.HD126681does not have an abundance determinations by Gratton et al.1997or Axer et al.1994and was removed from the list.The3stars with?1.24≤[Fe/H]≤?1.69were used in the?t(HD103095,25329and BD+592407).

error to Once again,the weight of HD103095in the?t was decreased by increasing itsσM

V ±0.038,ensuring that the mean weighted mean metallicity of the calibrating subdwarfs was [Fe/H]=?1.42.The resultant distance modulus is(m?M)V=13.25±0.03mag.

3.Assuming that a systematic zero-point error(+0.20dex)exists between the subdwarf[Fe/H]

determinations and NGC6752implying that NGC6752has a metallicity of[Fe/H]=?1.71 (in the subdwarf[Fe/H]system).This results in the use of3stars with?1.45≤[Fe/H]≤?1.76,and decreasing the weight of HD25329in the?t by increasing itsσM

error to

V ±0.073mag.The derived distance modulus is(m?M)V=13.24±0.06mag.

4.Assuming that a systematic zero-point error(?0.20dex)exists between the subdwarf[Fe/H]

determinations and NGC6752implying that NGC6752has a metallicity of[Fe/H]=?1.31 (in the subdwarf[Fe/H]system).The three subdwarfs with?1.22≤[Fe/H]≤?1.45with equal weighting(implying a mean[Fe/H]=?1.30)were used in the?t.The resultant distance modulus is(m?M)V=13.42±0.09mag.

5.Assuming the anomalous position of HD25329in the Figure1is due to an incorrect abundance

determination,and so removing HD25329from the?t.The three subdwarfs with?1.23≤[Fe/H]≤?1.75were used in the?t,and theσM

error in HD120559was increased to±0.22

V

so that the weighted mean subdwarf[Fe/H]was?1.51.The derived distance modulus is (m?M)V=13.30±0.11mag.

–10–

6.Assuming that the[Fe/H]determination of BD+592407is in error,and removing it from

the?t.This results in the use of the three subdwarfs with?1.23≤[Fe/H]≤?1.76and a distance modulus of(m?M)V=13.34±0.04mag.

The maximum change in the derived distance modulus is±0.09mag,which we take to be the1-σerror in the distance modulus due to possible metallicity errors.Adding the metallicity,reddening and?tting errors together in quadrature yields a distance modulus of(m?M)V=13.33±0.11. As discussed in§2.2,V(HB)=13.63±0.1,and so M v(RR)=0.30±0.15from the subdwarf distance modulus.This subdwarf visual distance modulus corresponds to(m?M)O=13.20±0.11 (with A V=3.2),which is within1-σof the distance obtained from the white dwarfs(m?M)O= 13.05±0.10(Renzini et al.1996).

2.3.2.M5

High dispersion spectroscopic analysis indicates that this cluster has[Fe/H]=?1.17(Sneden et al.1992).The reddening is E(B?V)=0.03,as summarized by Reid1997.A deep color magnitude diagram for this cluster has been presented by Sandquist et al.1996.Subdwarfs with ?1.07≤[Fe/H]≤?1.23were used in the weighted,least squares?t.The weighted mean abundance of these subdwarfs[Fe/H]=?1.19,very similar to our adopted[Fe/H]abundance of NGC6752. Subdwarf?tting yields a distance modulus of(m?M)V=14.51±0.02.To this error,one must add in errors associated with the reddening,and allow for possible metallicity errors.An uncertainty in the reddening of±0.01translates into an error in the derived distance modulus of±0.05.

Due to the possible systematic uncertainties in the metallicity abundances of the subdwarfs and M5,we have once again examined examined the e?ects that various scenarios for metallicity errors have on the derived distance modulus.

1.Adopting the Gratton et al.1997and Axer et al.1994abundances for the subdwarfs,and

using the4subdwarfs with?0.99≤[Fe/H]≤?1.23(HD103095,120559,145417and193901) resulting in a mean metallicity of the subdwarfs of[Fe/H]=?1.20and(m?M)V=14.51±

0.03.

2.Assuming that a systematic zero-point error(+0.20dex)exists between the subdwarf[Fe/H]

determinations and M5implying that M5has a metallicity of[Fe/H]=?1.37(in the subdwarf [Fe/H]system).This results in the use of5stars with?1.22≤[Fe/H]≤?1.76,and decreasing

error to±0.032mag.The derived the weight of HD103095in the?t by increasing itsσM

V

distance modulus is(m?M)V=14.49±0.02mag.

3.Assuming that the[Fe/H]value for HD103095is in error,and so removing it from the?t.This

results in the use of three subdwarfs(?1.07≤[Fe/H]≤?1.23and(m?M)V=14.58±0.03 mag.

–11–

4.Removing HD193901from the?t(leaving HD103095,120559and145417with a mean

metallicity of[Fe/H]=?1.20).The resultant distance modulus is(m?M)V=14.51±0.02 mag.

5.Removing HD145417from the?t,and giving equal weight to the remaining three stars(HD

103095,120559and193901)to ensure a mean[Fe/H]=?1.17.The derived distance modulus is(m?M)V=14.58±0.07mag.

6.Only using HD145417([Fe/H]=?1.15)in the?t,resulting in(m?M)V=14.56±0.03mag.

The maximum change in the derived distance modulus is±0.07mag,which we take to be the1-σerror in the distance modulus due to possible metallicity errors.Adding the metallicity,reddening and?tting errors together in quadrature yields a distance modulus of(m?M)V=14.51±0.09. Utilizing V(RR)=15.05±0.02(Reid1996),results in M v(RR)=0.54±0.09.

2.3.3.M13

High dispersion spectroscopic analysis indicates that this cluster has[Fe/H]=?1.58(Kraft et al.1997).The reddening is E(B?V)=0.02(Zinn&West1984).A deep color magnitude diagram and?ducial has been obtained by Richer&Fahlman1986.This is a somewhat di?cult metallicity to deal with,as none of the calibrating subdwarfs has a metallicity near[Fe/H]=?1.58.We have explored a number of possible options for a subdwarf sample selection.

1.Utilizing the subdwarfs with?1.23≤[Fe/H]≤?1.76,results in a weighted mean[Fe/H]=

?1.55for the subdwarfs and(m?M)V=14.54±0.04is obtained from a weighted?t to the ?ducial.

2.The subdwarf[Fe/H]range is restricted to?1.45≤[Fe/H]≤?1.76and the stars are

equally weighted(resulting in a mean[Fe/H]=?1.65)then the derived distance modulus is (m?M)V=14.46±0.10

3.A systematic o?set error of?0.2dex is assumed between the subdwarfs and M13,implying

that M13has[Fe/H]=?1.78in the subdwarf system.In this case,only the two stars with [Fe/H]=?1.75and?1.76are used,resulting in(m?M)V=14.39±0.05

4.A systematic o?set error of+0.2dex is assumed between the subdwarfs and M13,implying

that M13has[Fe/H]=?1.38in the subdwarf system.In this case subdwarfs with?1.22≤[Fe/H]≤?1.76are used,and the weight of HD103095in the?t is decreased by increasing itsσM

error to±0.034(to ensure a subdwarf mean[Fe/H]=?1.38).The derived distance V

modulus is(m?M)V=14.51±0.02

–12–

The derived distance moduli vary from14.39–14.54.We have elected to adopt the mid-point as our best value,and utilize a generous1-σerror of±0.09,hence(m?M)V=14.47±0.09.Adding in quadrature the error due to reddening,a total distance modulus error of±0.10is adopted.M13 has a very blue HB,so the determination of V(HB)is di?cult.We adopt V HB=14.83±0.10 (Chaboyer et al.1996c),implying M v(RR)=0.36±0.14.

2.4.Calibration of M v(RR)via the LMC

Walker(1992)determined the mean magnitudes of a number of RR Lyr stars in several LMC clusters.Adopting a distance modulus to the LMC ofμLMC=18.50±0.10,he found M v(RR)= 0.44±0.10.This distance modulus was based upon the traditional calibration of bd40081fc281e53a5802ff11ing Hipparcos based parallaxes,Feast&Catchpole(1997)derivedμLMC=18.70±0.10.This distance relied upon a period-color relation,and parallaxes of rather low quality(σπ/π～>0.3).An analysis of the Hipparcos Cepheid data by Madore&Freedman1997yieldsμLMC=18.57±0.11who noted that“other e?ects on the Cepheid PL relation(e.g.reddenning,metallicity,statistical errors)are as signi?cant as this reassessment of its zero point”.The distance to the LMC may be estimated independent of the Cepheid or RR Lyr8from geometric considerations using the‘light echo’times to the ring around SN1987A(Panagia et al.1991,Gould1995).Using the same data set,but independent analysis,the SN1987A ring distance has been re-calculated by a few groups.Sonneborn et al.1997foundμLMC=18.43±0.10,while Gould&Uza(1997)determinedμLMC<18.44±0.05. Recently,Lundquist&Sonneborn1997reported a lower limit ofμLMC<18.67±0.08.In light of these contradictory results,we have elected to follow the conclusion of Madore&Freedman1997 and adopt a distance modulus of18.50mag for the LMC,and assume an uncertainty±0.14to fully encompass the range of recently published values.Adopting this distance modulus,along with the photometry of Walker(1992)yields M v(RR)=0.44±0.14at[Fe/H]=?1.9.

2.5.Theoretical HB models

Continued advances in our understanding in the basic physics which governs stellar evolution have lead to ever more reliable theoretical HB models.Recently,Demarque et al.1997have constructed synthetic HB models for various clusters,based upon new evolutionary models for HB stars.Assuming a primordial helium abundance of0.23,these models predict M v(RR)=0.34 for M92([Fe/H]=?2.25)and M v(RR)=0.42for M15([Fe/H]=?2.15).These two clusters are among the17old clusters whose mean age is determined in§4.Taking a simple average of the above

–13–

two numbers yields M v(RR)=0.38at[Fe/H]=?2.20.Also,since the primordial helium value utilized in this analysis was on the low side,we have adjusted this mean value to0.36to account for a mean primordial helium value of0.235.We also adopt an error of0.10mag on M v(RR),to allow for possible errors in the models and in the primordial helium abundance estimate.

bd40081fc281e53a5802ff11bining the Distance Estimates

The inpidual determinations of M v(RR)at the various metallicities are summarized in Table 3.While it is not evident from this table,there is considerable evidence from other observations and theoretical modeling that M v(RR)is a function of[Fe/H]:

M v(RR)=μ([Fe/H]+1.9)+γ.(1) We have chosen distance calibrations which yield reliable absolute numbers with the minimum possible systematic uncertainties9Hence,they are useful in deriving the value ofγ.However,these M v(RR)determinations do not provide reliable information on the M v(RR)-[Fe/H]slopeμ.For this,one needs to utilize techniques which yield reliable relative M v(RR)values as a function of [Fe/H].Theoretical HB models,as well as Baade-Wesselink studies of?eld RR Lyr stars provide the best estimate of the M v(RR)-[Fe/H]relationship10The semi-empirical Baade-Wesselink method has been applied by Jones et al.1992and Skillen et al.1993.Reanalysis of these data suggest that μ=0.22±0.05(Sarajedini et al.1997).The latest theoretical models of blue HB clusters yield slopes ofμ=0.25±0.07(Demarque et al.1997).A weighted mean value ofμ=0.23±0.04 was adopted.The third column in Table3lists the various values ofγimplied by the inpidual determinations of M v(RR).There is a considerable spread in these values(0.21to0.52),reinforcing the notion that the dominant uncertainty remains systematic and that our previous procedure of assigning a Gaussian uncertainty to this quantity was ill-advised.We have thus now chosen to utilize a uniform top-hat uniform distribution which evenly weights all values in the range0.21to

–14–

0.52.However to give some emphasis to the mean value of the measured data,we have added to this distribution a Gaussian distribution centered on the weighted mean ofγ=0.39with an uncertainty of0.08,doubling the calculated error in the mean to account for the average deviation from the mean.Note that this new mean value is0.21mag(more than2?σ)below the value adopted in Paper I.This will lead to a considerably downward revision in our GC age estimates,which we believe will also now have a distribution which is more appropriate to the systematic nature of the existing uncertainties.(We emphasize once again that while the Hipparcos data provided a motivation for re-examining this value,all of the other distance estimators we have examined apear now to be consistent,within the systematic uncertainties quoted,with a much lower value than we previously adopted.)

Spectroscopic studies of blue horizontal-branch(BHB)stars provide further support for the longer GC distance scale adopted here.From both the continuous spectrum and absorption line pro?les,it is possible,with the help of model stellar atmospheres,to derive the e?ective temperature and surface gravities of these stars.This combined information yields the mass-to-light ratio M/L of the star,and if its distance is also known,its mass.This method has in the past yielded masses incompatible with the standard HB evolution theory(masses lower than evolutionary models)(de Boer et al.1995;Moehler et al.1995,1997).A recent attempt to rederive the distances of some?eld BHB stars using Hipparcos parallaxes,could not be given much weight in view of the smallness of the parallaxes(de Boer et al.1997).Using Reid’s(1997)reanalysis of the distances to some globular clusters based on larger Hipparcos subdwarf parallaxes,Heber et al.1997have reconsidered this problem,and concluded that the higher luminosities for BHB stars now yield masses in better agreement with the evolutionary masses.This important result provides independent support, based on physical modeling,for revising upward the distance scale to globular clusters,as suggested by several lines of reasoning,including the Hipparcos parallax data.

Finally,we should point out that the new distance scale yields a poor?t to calculated isochrones near the cluster turno?.This suggests that the stellar model radii may need revision(better stellar atmospheres and convection modeling).Improvements in atmosphere models may lead to revisions of the T eff to color transformations,particularly for the most metal poor stars.Since the?V TO

HB method is little a?ected by surface e?ects,it further justi?es our preference for this approach over the?(B?V)approach,and the approach of?tting to the shapes of theoretical isochrone turno?s, both of which are sensitively a?ected by atmosphere and outer envelope physics.

3.The Monte Carlo Variables

In order to access the range of error associated with stellar evolution calculations and age determinations,the various inputs into the stellar evolution codes were varied within their uncer-tainties.In this Monte Carlo analysis,the input parameters were selected randomly from a given distribution.The distributions are based on a careful analysis of the recent literature,as summa-rized in Paper I.As ages will be derived using M v(TO)most attention was paid to parameters

–15–

which could e?ect the age-M v(TO)relationship.Table4provides an outline of the various input parameters and their distribution.If the distribution is given as statistical(stat.),then the pa-rameter in question was drawn from a Gaussian with the statedσ.If the error was determined to be a possible systematic(syst.)one,then the parameter was drawn from a top-hat(uniform) distribution.In total,1000independent sets of isochrones were calculated.Each set of isochrones consisted of three di?erent metallicities([Fe/H]=?2.5,?2.0and?1.5)at15di?erent ages(8?22 Gyr)(see Paper I for further details).

4.Results

4.1.The Technique

The absolute magnitude of the main sequence turn-o?is the favored age determination tech-nique when absolute stellar ages are of interest(see discussion in Paper I).Turn-o?luminosity ages can be determined independent of reddening by using the di?erence in magnitude between the main sequence turn-o?and the HB,?V TO

.Each set of Monte Carlo isochrones provides an

HB

independent calibration of M v(TO)as a function of age.This was combined with the M v(RR)

values as a function of age and calibration discussed in§2to determine a grid of predicted?V TO

HB

[Fe/H]which is then?t to an equation of the form

t9=β0+β1?V+β2?V2+β3[Fe/H]+β4[Fe/H]2+β5?V[Fe/H],(2)

and[Fe/H],along with their corresponding where t9is the age in Gyr.The observed values of?V TO

HB

errors,are input in(2)to determine the age and its error for each GC in our sample.

The age determination for any inpidual globular cluster has a large uncertainty,due to the large observational errors in V(TO).This error is minimized by determining the mean age of a number of globular clusters.However,there is a signi?cant age range among the globular clusters (e.g.Sarajedini&Demarque1990,VandenBerg et al.1990,Buonanno et al.1994b,Chaboyer et al. 1996c).This problem was avoided by selecting a sample of globular clusters which are well observed, metal-poor([Fe/H]≤?1.6),and which are not known to be young(based on HB morphology and/or the di?erence in color between the turn-o?and giant branch).In the tabulation of Chaboyer et al.1996c,17GCs satisfy the above criteria:NGC1904,2298,5024,5053,5466,5897,6101,6205, 6254,6341,6397,6535,6809,7078,7099,7492,and Terzan8.The observational data for each cluster was taken from Chaboyer et al.1996c.The mean(and median)metallicity of this sample is[Fe/H]=?1.9.

4.2.A Likelihood Distribution for the Age of the Oldest Globular Clusters

To derive our best estimate for the age,and uncertainty in the age of the oldest GCs,a mean age and1σuncertainty in the mean was determined for each set of isochrones,and a given value of

–16–

M v(RR).The value of M v(RR)was taken to be a random variable,weighted as described earlier (a top hat distribution between0.21and0.52superimposed on a Gaussian distribution with mean and uncertainty M v(RR)=0.39±0.08),and the sets of isochrones were sampled with replacement 12,000times.For each sample,we recorded a random age drawn from a Gaussian distribution with the mean age and variance for that isochrone set at that value of M v(RR).

The age data were sorted and binned,to produce the histogram shown in Figure2.The median and mean age is11.5Gyr,with a standard deviation of1.3Gyr.The1-sided,95%lower con?dence limit is9.5Gyr,and is believed to represent a robust lower limit to the age of the GCs,and more properly takes into account the residual systematic uncertainties in M v(RR),which largely determine the width of the derived age distribution.We are fully aware that due to our revision of the M v(RR)zero-point,these ages are considerably reduced compared to the ages given in Paper I.Indeed,our new mean age is below our previous claimed95%lower limit which was based on the assumption of Gaussian uncertainty in M v(RR).In any case,our new results considerably alter the constraints one can derive on cosmological models(see§5).

Even though we have considered four independent distance determinations in addition to the Hipparcos parallaxes,our age estimate is in good agreement with two recent works which relied solely on Hipparcos parallaxes to determine the distances(and hence,ages)to a number of GCs (Reid1997;Gratton et al.1997).Pont et al.1997have determined an age of14Gyr for M92,which is in disagreement with our work.Pont et al.1997made a new?t of the CMD of M92to theoretical isochrones,based on the Hipparcos subdwarf data.This paper represents a comprehensive analysis of the available data,and attempts the di?cult task of correcting for selection e?ects which are more relevant in this case than the classical Lutz&Kelker1973corrections.However,our own analysis suggests that they have overestimated the corrections to the Hipparcos parallaxes due to biases (see Appendix).Their corrections due to the presence of binaries is very uncertain(±0.15mag);

a fact which was not considered by Pont et al.1997in their analysis.A better procedure,which is not to include the suspected binaries in the?t,yields a larger distance modulus for M92.This approach,as pointed out by Pont et al.1997yields(m?M)V=14.74±0.08mag.With this distance modulus,and the photometry of Stetson&Harris1988(the same photometry used by Pont et al.1997),we calculate that the absolute magnitude of the point on the subgiant branch which is0.05mag redder than the turn-o?is V(BTO)=3.39±0.08mag.This point is an excellant diagnostic of the absolute age of M92(Chaboyer et al.1996b),and using our isochrones(as outlined in§4.5)results in an age for M92of12.1±1.3Gyr.This is in good agreement with our estimate for the mean age of the oldest GCs(which includes M92)given above.

In the?nal analysis,the Pont et al.1997paper puts most of the weight of their?t on the agreement between the shapes of the theoretical isochrones and the data near the turno?.However, this optimistic assessment of the models does not seem warranted in view of the well-known uncer-tainties associated with the treatment of convection,and the neglect of di?usion in the isochrones used(helioseismoly has taught us that di?usion must be taken into account in the Sun(Basu et al. 1996,Guenther&Demarque1997).The need to apply an arbitrary color shift to the VandenBerg

–17–

et al.1997isochrones to reproduce the observed colors of M92,is another indication of the uncer-

method in dating globular tainties involved,and lends further support to the choice of the?V TO

HB

clusters.We conclude that taking into account the di?erences in adopted distance moduli,and the neglect of di?usion by VandenBerg et al.1997,our age estimate for M92,which is11.5±1.3Gyr, is in good agreement with the Hipparcos data presented by Pont et al.1997.

4.3.E?ect of M v(RR)on the age estimate

As was emphasized in Paper I(and by other authors),the principal uncertainty in absolute GC age determinations is the distance scale.With the?V TO

age determination technique,this

HB

translates into the uncertainty in M v(RR).We explicitly display this e?ect in Figure3,where the GC ages are plotted as a function of M v(RR).In order to quantify this uncertainty,median and ±1σpoints were determined as a function of M v(RR).These were obtained by sorting the data based on M v(RR),and then binning the ages as a function of M v(RR).Sixty bins(corresponding to200ages per bin)were used,and in each bin the median age,and±1σ(68%range)ages were determined.An inspection of these points revealed that a simple linear relationship existed when one used the log of the age.A linear function of the form log(t9)=a+b M v(RR)was?tted to this data,and the coe?cients of this?t are given in the?gure caption.

The median and±1σ?ts are extremely useful summaries of our result.For example,at M v(RR)=0.40,the median?t yields11.7Gyr,identical to that given by the entire distribution (Fig.2).The±1σ?ts yield ages of12.6and11.0Gyr.Thus,if M v(RR)was known to be exactly0.40,then the error in the age of the oldest GCs would be±0.8Gyr,due solely to the residual theoretical uncertainties in the stellar evolution calculations.The median and±1σ?ts we present here may be used to update our age estimate as further data are obtained.For example, if M v(RR)=0.50±0.05then from the?ts,the median age would be13.03Gyr,with an error of ±0.94Gyr due to the theoretical uncertainties aside from those associated with M v(RR).Next,from the median?t,the median age at M v(RR)=0.45and0.55may be determined(corresponding to ±0.05mag)in order to estimate that the uncertainty in age associated with the M v(RR)uncertainty is±bd40081fc281e53a5802ff11bining these two error estimates in quadrature(±0.94and±0.68)would result in a best estimate of13.1±1.2Gyr for M v(RR)=0.50±0.05.To verify this result,we have re-run the Monte Carlo analysis with the above choice of M v(RR)and found identical results to those obtained from the M v(RR)median and±1σ?ts above.

4.4.E?ect of the stellar evolution parameters on the age estimate

In order to examine how the inpidual stellar evolution parameters(given in Table4)a?ect the estimated age,the mean age of the17GCs was determined for each of the1000Monte Carlo isochrones assuming?xed value M v(RR)=0.40.In a procedure analogous to that used for the

–18–

M v(RR)?ts,median and±1σ?ts were determined for each of the13continuous variables listed in Table4.As only1000points were available,only20bins were used.In addition,it was found that(due to the reduced age range),a linear?t provided as good a description as a log?t.Thus, the median and±1σ?ts for each parameter x,were of the form t9=a+b x.

This procedure revealed that several of the input parameters had a negligible e?ect on the derived ages of the globular clusters.In order of importance,the following parameters were found to impact the GC age estimate:α/[Fe],mixing length,helium abundance,14N+p?→15O+γreaction rate,helium di?usion coe?cient,and the low temperature opacities.The plots of age as a function of these important parameters are shown in Figures4—9.The?gure captions give the coe?cients of the median and±1σ?ts for each of the variables.These?ts can be used to update our best estimate for the age of the oldest globular clusters(in a manner analogous to that described for the M v(RR)?ts),as improved determinations of the above quantities become available.

In addition to the13continuous variables,we considered two binary variables(surface bound-ary condition,and color table,see Table4).To examine the e?ect these parameters have on the derived ages,the ages were pided into2groups depending on which surface boundary condition (color table)was used in the stellar evolution codes.Histograms were constructed for each group, and compared.Not surprisingly,we found that the choice of the surface boundary condition had a negligible impact on the derived ages.However,the choice of the color table was important,and the two histograms are plotted in Figure10.The choice of the color table changes the median age by0.7Gyr.

4.5.Calibration of the M v(TO)and M v(BTO)age relations

If the distance modulus to some cluster is known,then an accurate absolute age may be determined using M v(TO)or alternatively using M v(BTO)(Chaboyer et al.1996b).This later point is de?ned to be the point on the subgiant branch which is0.05mag redder(in B—V)than the turn-o?.As we have discussed,this point is easy to measure on an observed color magnitude diagram,yet has similar theoretical uncertainties to M v(TO)(Chaboyer et al.1996b).As a result, the precision in age estimation for inpidual clusters is better using M v(BTO).The Monte Carlo isochrones may be used to quantify the error associated with an age determined via either method. To facilitate such error estimates,we have calculated the median and±1σM v(BTO)(M v(TO)) points as a function of age(in a manner similar to that described in the previous subsection)for four values of[Fe/H]:?2.5?2.0?1.5and?1.0.For ages between8and17Gyr,these points were then?t to a function of the form

log(t9)=β1+β2M V+β3[Fe/H]+β4[Fe/H]2+β5[Fe/H]M V(3) where M V was chosen to be either M v(BTO)or M v(TO).The coe?cients of the median and±1σ?ts,for both M v(BTO)and M v(TO)are given in Table5.

–19–

The use of these?ts for determining ages via M v(BTO)is illustrated for NGC6752.Averaging the white dwarf distance modulus(§2.2)and the subdwarf main sequence?tting modulus(§2.3.1) results in(m?M)O=13.12±0.07,or(m?M)V=13.25±bd40081fc281e53a5802ff11ing the photometry of Penny &Dickens1986,we?nd V(BTO)=16.83±0.04,so that M V(BTO)=3.58±0.08.Recall that [Fe/H]=?1.51±0.08(§2.3.1).Using the coe?cients of the?ts in Table1,this corresponds to an age of11.15±0.8Gyr if M v(BTO)and[Fe/H]were known exactly.The e?ects of the M v(BTO)and[Fe/H]errors may be taken into account by using the median?t,and calculating ages for the±1σvalues for M v(BTO)and[Fe/H].This procedure results in estimated errors of ±0.9Gyr due to the M v(BTO)error(±0.08mag)and±0.4Gyr due to the[Fe/H]error of±0.08dex. Adding all three errors together in quadrature yields an age of11.2±1.3Gyr for NGC6752.This intermediate metallicity cluster has an age quite similar to the mean age of the17metal-poor clusters(11.5±1.3Gyr)determined in§4.2.More important,note that the uncertainty on the age of NGC6752determined in this way is comparable to the uncertainty in the mean of the set of17 old globular clusters,illustrating the potential power of the method based on M v(BTO).

Similarly,for M5,we calculate V(BTO)=18.03±0.02using the photometry of Sandquist et al.1996.With(m?M)V=14.51±0.09(§2.3.2),this results in M v(BTO)=3.52±0.09.Assuming [Fe/H]=?1.17±0.08(Sneden et al.1992),and using the technique outlined for NGC6752,an age of8.9±1.1Gyr is derived.Finally,for M13with[Fe/H]=?1.58and(m?M)V=14.47±0.09 (§2.3.3)we?nd V(BTO)=18.00±0.04using the photometry11of VandenBerg et al.1990,resulting in M v(BTO)=3.53±0.10and an age of10.9±1.4Gyr.Our results for the distances and ages of these three clusters are summarized in Table6.

5.Summary

Our new work has two primary results.First,we have updated the absolute age estimate, and quanti?ed the uncertainty in this estimate for the oldest globular cluster mean age.This update is primarily due to a reanalysis of estimates for the quantity which dominates the age uncertainty:M v(RR)(the distance scale to GCs).We have concentrated on exploring in detail di?erent estimates in order to account for the mean value,and the distribution in the uncertainty of this quantity.We?nd that all the data,not merely the recent Hipparcos parallax measurements, suggests a large systematic shift in M v(RR)of approximately0.2magnitudes compared to earlier estimates.This has the e?ect of reducing the mean age of the oldest globular clusters by almost 3Gyr.At the same time,this new data makes it clearer that M v(RR)residual uncertainties are primarily systematic,reminding us that even apparently gaussianly distributed measurements in astrophysics may be subject to large systematic shifts.As a result,we now incorporate a large

–20–

systematic uncertainty in the claimed mean value of M v(RR)in our estimates.

Next,we provide a formalism which may be used by other researchers to update the estimates given here as new data emerges.In particular,we have presented an explicit discussion of the e?ect of other input parameter uncertainties from stellar evolution theory on the inferred GC ages estimates.We have displayed these e?ects in Figures2-9,and provided analytical?ts for both median ages,and uncertainties in age as a function of these parameters,and also as a function of M v(RR).

We have also explicitly provided the?t for inpidual globular cluster ages and uncertainties as a function of metallicity and turn-o?magnitude,using both the M v(TO)and M v(BTO)schemes. This should allow one to derive the age,and uncertainty in age for any GC with?2.5≤[Fe/H]≤?1.0.We have illustrated this scheme,for the M v(BTO)method for NGC6752,using the average distance modulus from white dwarf sequence?tting,and subdwarf main sequence?tting,yielding an age of11.2±1.2Gyr,illustrating that the M v(BTO)method in principle allows an age precision on inpidual GC age determinations comparable to the M v(TO)method applied to the ensemble of17old Globular clusters used in our analysis.

Finally,we brie?y comment here on the cosmological implications of our central result that the mean of17old,metal-poor GC is11.5±1.3Gyr,with a one-sided,95%con?dence level lower bound of9.5Gyr(see Krauss(1997)for further details).First and foremost,this results suggests that the long-standing con?ict between the Hubble age,and GC age estimates for a?at matter dominated universe is now resolved for a realistic range of Hubble constants.A?at universe has an age which exceeds our lower limit on the GC ages for a Hubble constant H0≤67km s?1Mpc?1,which is well within the range of current measured values.Thus,it now appears that the“age problem”is now no longer the primary motivation for considering a non-zero cosmological constant in the universe (i.e.Krauss and Turner1995),and requires an alteration in the arguments associated with the debate between an open,?at matter dominated,and?at cosmological constant cosmologies(Krauss 1997).

If measurements of the Hubble constant continue to converge on the range60?70km s?1Mpc?1, as suggested by the most recent analyses,cosmological concordance,at least as far as age is con-cerned,will perhaps for the?rst time be possible in all three scenarios.

We would like to thank Bill van Altena,Sidney van den Bergh and the anonymous referee whose comments have signi?cantly improved the?nal product as presented here.This research has made use of the SIMBAD database,operated at CDS,Strasbourg,France and data obtained from the ESA Hipparcos Astrometry Satellite.BC was supported for this work by NASA through Hubble Fellowship grant number HF–01080.01–96A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy,Inc.,for NASA under contract NAS5–26555.LMK was supported in part by funds from CERN,CWRU,and a grant from the DOE.

–21–

A.Potential Biases in the Subdwarf Sample

The parallaxes and absolute magnitudes for the stars listed in Table2do not include any statistical correction for possible biases in the sample.There are a few sources of potential biases in the sample.The classical Lutz&Kelker1973correction is a statistical correction which takes into account systematic e?ects due to the fact that(a)stars with parallaxes measured too high have a higher probability of being included in the sample than those with parallaxes measured too low (due to ourσπ/π<0.1selection criterion),and(b)more weight is given to stars with parallaxes that are overestimated rather than to stars with underestimated parallaxes(due to our use of a weighted least squares?t).In addition to this,Pont et al.1997point out that the since metal-poor stars are far less numerous than more metal-rich stars,there may be an average underestimation of [Fe/H]in the sample.The importance of these biases will depend on the selection criterion which are used to select the subdwarfs used in the main sequence?tting.The3papers which have used Hipparcos subdwarf parallaxes to determine GC distances have all had di?erent selection criterion, and have determined di?erent bias corrections.In their study,Pont et al.1997determined that the unevolved subdwarfs had a mean bias of+0.64mag.In contrast,Gratton et al.1997determined a bias correction of?0.004mag.Reid1997whose subdwarf sample consisted of high proper motion stars,elected to use inpidual Lutz-Kelker corrections,whose magnitude depended on the uncertainty in the parallax.In general,the corrections used by Reid1997were small,and in the opposite sense to those employed by Pont et al.1997.

Our subdwarf study di?ers from the those of Reid1997,Gratton et al.1997and Pont et al. 1997in that we have access to the entire Hipparcos catalogue.Stars were selected for inclusion in the Hipparcos input catalogue based on a variety of considerations,and so there is no well de?ned selection criterion for the entire Hipparcos catalogue.Thus,it is di?cult to assess the importance of the various biases a priori.For this reason,we have elected to use a stringent selection criterion σπ/π<0.1which minimizes the importance of the Lutz&Kelker1973type bias(Brown et al. 1997).As it turns out,the?nal sample only contains stars withσπ/π<0.08,strongly suggesting that the stars whose true parallax are systematically smaller than the observed parallax are not preferentially included in our sample.

To study the possible biases which remain in our subdwarf sample,we have constructed a Monte Carlo simulation to generate synthetic data whose properties are known,and compared to ‘observed’properties which are calculated in the Monte Carlo.This is similar in spirit to the bias studies of Gratton et al.1997and Pont et al.1997.We have attempted to construct a subdwarf data set whose properties and selection biases closely match those in our actual data set.In particular, our subdwarf sample consists of stars with[Fe/H]5.5,andσπ/π<0.1and these facts are incorporated in the Monte Carlo.The Monte Carlo was constructed in the following steps:

1.An intrinsic[Fe/H]value(below[Fe/H]=?1.0)was drawn from one of two probability

functions.The?rst function is that given by the observed[Fe/H]distribution in the Carney

本文来源：https://www.bwwdw.com/article/gdrl.html