the ages of globular clusters

The Ages of Globular ClustersAuthor(s): D. H. McNamaraSource: Publications of the Astronomical Society of the Pacific, Vol. 113, No. 781 (March2001), pp. 335-343Published by: The University of Chicago Press on behalf of the Astronomical Society of the PacificStable URL: http://www.jstor.org/stable/10.1086/319332 .

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335

Publications of the Astronomical Society of the Pacific, 113:335–343, 2001 Marchq 2001. The Astronomical Society of the Pacific. All rights reserved. Printed in U.S.A.

The Ages of Globular Clusters

D. H. McNamara

Department of Physics and Astronomy, Brigham Young University, Provo, UT 84602; [email protected]

Received 2000 September 20; accepted 2000 November 20

ABSTRACT. We examine the luminosity levels of the main-sequence turnoffs, M , and horizontal branches,TOv

(HB), in 16 globular clusters. An entirely new approach of inferring the luminosity levels by utilizing high-Mv

amplitude d Scuti variables (HADS) is introduced. When the M values are compared with theoretical valuesTOv

inferred from models, we find all 16 clusters (metal-strong to metal-poor) are coeval with an average age of∼11.3 Gyr. A considerable scatter of (HB) values of clusters at similar [Fe/H] values is found. A trend forMv

clusters with blue horizontal branches to have brighter (HB) than clusters with blue-red horizontal branchesMv

is suggested by the data. The (HB) values appear to depend on another or other parameters in addition to theMv

[Fe/H] values. In spite of this problem, we derive an equation relating (HB) values of globular clusters toMv

their [Fe/H] values. We also derive an equation relating the M values of clusters to their [Fe/H] values. BothTOv

of these equations can be utilized to find cluster distances. The distance modulus of the LMC is found to be18.66 from the values of three LMC globular clusters; RR Lyrae stars in seven globular clusters yield 18.61,TOVand RR Lyrae stars in the LMC bar yield 18.64.

1. INTRODUCTION

It is generally recognized that globular clusters were amongthe first objects to be formed in the galaxy. Their ages provideus with information on the early formative stages of the galaxy.In fact, they can be utilized to provide a lower limit to the ageof the universe provided their ages can be ascertained. Asidefrom the difficulties involved in calculating isochrones orhorizontal-branch (HB) models, accurate distance determina-tions, which play a key role in age determinations, are difficultto achieve. Gratton et al. (1997) have pointed out that thereare at least three ways to derive distances to globular clusters:

1. The Baade-Wesselink (BW) method can be used to obtainvalues of RR Lyrae variables. The (V) values of RR LyraeMv

variables in the clusters can then be utilized to find the dis-tances. The zero point of the , [Fe/H] relation derived fromMv

the data is open to question because of uncertainties in thetemperature scale and p-value adopted to convert from radialvelocity to radial displacement. See papers by Walker (1992),Fernley (1994), Carney et al. (1995), and McNamara (1997a,1997b).

2. A second method employs the cooling sequence of whitedwarfs. A recent application of this method has been employedto derive the distance to NGC 6752 by Renzini et al. (1996)utilizing Hubble Space Telescope data.

3. The advent of accurate parallax measurements for a sig-nificant number of subdwarfs by the Hipparcos mission hasled to definitions of main-sequence templates as a function of[Fe/H] that can be utilized to infer the distances and ages of

globular clusters by main-sequence fitting. Both Reid (1997,1998) and Gratton et al. (1997) have employed this technique.

To the above three methods described in more detail byGratton et al. (1997), we may add the use of high-amplituded Scuti stars (HADS; McNamara 1997b) to infer distances andages of globular clusters. The results of this method are similarto those found by method 3, namely, larger distances andyounger ages of globular clusters. One weakness of this methodis the difficulty involved in determining the identification ofthe pulsation mode of the HADS, which can lead to uncer-tainties of ∼0.3 mag in the distance modulus. Since the pub-lication of the original paper describing the method, a consid-erable body of new data has become available, and the pulsationmode problem has largely been overcome (McNamara 2000).At least in principle distance moduli can be inferred to anaccuracy of ≤0.1 mag if photometry of several HADS in thecluster are available. This contribution deals with a reexami-nation of the luminosities of the “turnoff,” M , and HBs andTO

v

distances to globular clusters in light of new and more extensivedata.

2. THE OBSERVATIONAL DATA

The data utilized in this paper were obtained by severalinvestigators. They are available primarily through the re-markable efforts of Kaluzny and his collaborators. The dataare collected inTable 1 by cluster along with the primary ref-erence. More details about the observational data may be foundby referring to these references.



336 McNAMARA

2001 PASP, 113:335–343

TABLE 1Variable Stars in Clusters

Star(1)

log P(2)

log P0

(3)AVS(4)

Av

(5)Mode

(6)AB 2 VS

(7)Mv

(8)M (HB)v

(9)

TOMv

(10)Reference

(11)

NGC 104: 47 Tuc, , ,TOV p 17.65 V(HB) p 14.04 D log P p 0.111

V1 . . . . . . . . . . . . 21.199 21.087 15.59 0.15 0 0.32 2.12 0.57 4.18 1V2 . . . . . . . . . . . . 20.991 … 14.86 0.15 F 0.39 1.76 0.94 4.55V3 . . . . . . . . . . . . 21.254 … 15.93 0.08 NR 0.32 … … …V4 . . . . . . . . . . . . 21.33 … 16.46 0.04 F 0.34 3.02 0.60 4.21V5 . . . . . . . . . . . . 21.479 21.133 15.50 0.04 4, 0 0.14 2.29 0.83 4.44V6 . . . . . . . . . . . . 21.449 21.103 15.73 0.03 4, 0 0.25 2.18 0.49 4.10

NGC 288: , ,TOV p 19.00 V(HB) p 15.30 D log P p 0.109

V4 . . . . . . . . . . . . 21.102 … 17.24 0.30 F 0.28 2.17 0.23 3.93 2V5 . . . . . . . . . . . . 21.291 21.182 17.54 0.46 0 0.30 2.47 0.23 3.93V6 . . . . . . . . . . . . 21.172 … 17.28 0.41 F 0.29 2.43 0.45 4.15V7 . . . . . . . . . . . . 21.398 21.289 17.92 0.06 0 0.29 2.86 0.24 3.94V8 . . . . . . . . . . . . 21.332 21.233 17.78 0.06 0 0.28 2.62 0.14 3.84V9 . . . . . . . . . . . . 21.405 21.171 17.52 0.05 2, 0 0.16 2.43 0.21 3.91


V2 . . . . . . . . . . . . 21.389 21.302 18.30 0.32 0 0.73 2.92 20.08 3.42 3V6 . . . . . . . . . . . . 21.176 … 17.29 0.33 F 0.87 2.45 0.46 3.96V8 . . . . . . . . . . . . 21.193 … 17.76 0.64 F 0.67 2.51 0.05 3.55V14 . . . . . . . . . . . 21.249 … 18.18 0.28 F 0.91 2.72 20.16 3.34V15 . . . . . . . . . . . 21.333 21.247 18.08 0.26 0 0.75 2.71 20.07 3.43V18 . . . . . . . . . . . 21.322 21.236 17.48 0.32 0 0.66 2.67 0.49 3.99V25 . . . . . . . . . . . 21.224 … 17.62 0.32 F 0.72 2.63 0.31 3.81V29 . . . . . . . . . . . 21.335 21.249 17.75 0.43 0 0.92 2.72 0.27 3.77

NGC 4590: M68, ,TOV p 19.10 V(HB) p 15.64

V39 . . . . . . . . . . . 21.193 … 18.05 0.71 F 0.25 2.51 0.10 3.56 4V48 . . . . . . . . . . . 21.364 21.015 17.27 0.17 4, 0 0.27 1.85 0.22 3.68


NC 7 . . . . . . . . . 21.434 21.348 19.30 0.14 0 0.33 3.09 0.44 3.84 5NC 11 . . . . . . . . 21.456 21.370 19.50 0.15–0.25 0 0.32 3.17 0.32 3.72NC 13 . . . . . . . . 21.466 21.380 19.46 0.13 0 0.23 3.21 0.40 3.80NC 14 . . . . . . . . 21.406 21.320 19.41 0.08 0 0.26 2.98 0.22 3.62NC 15 . . . . . . . . 21.449 21.363 19.53 0.12 0 0.30 3.14 0.26 3.66

NGC 5272: M3, , ,TOV p 19.10 V(HB) p 15.65 D log P p 0.100

Anon . . . . . . . . . 21.509 21.409 18.36 0.08 0 0.20 3.32 0.61 4.06 6V237 . . . . . . . . . 21.397 21.297 18.00 0.10 0 0.15 2.90 0.55 4.00 7SE 174 . . . . . . . 21.398 21.298 17.98 0.07 0 0.26 2.90 0.57 4.02NW 449 . . . . . . 21.398 21.298 18.29 0.12 0 … 2.90 0.26 3.71NW 858 . . . . . . 21.432 21.322 18.28 0.09 0 0.16 3.03 0.40 3.85


NH 29 . . . . . . . . 21.398 21.298 18.90 0.12 0 0.19 2.90 0.62 3.97 8NH 49 . . . . . . . . 21.347 … 19.26 0.25 F 0.20 3.08 0.44 3.79NH 35 . . . . . . . . 21.302 … 18.93 0.51 F 0.26 2.92 0.61 3.96NH 39 . . . . . . . . 21.297 … 19.11 0.36 F 0.16 2.90 0.41 3.76NH 27 . . . . . . . . 21.295 … 18.78 0.12 F 0.23 2.89 0.73 4.08NH 28 . . . . . . . . 21.258 … 18.80 0.44 F 0.23 2.75 0.57 3.92


V1 . . . . . . . . . . . . 21.375 21.266 17.48 0.09 0 0.20 2.78 0.37 3.78 9V2 . . . . . . . . . . . . 21.384 21.275 16.76 0.02 2.0? 0.28 2.46 0.77 4.18V3 . . . . . . . . . . . . 21.380 21.271 17.04 0.10 0 0.19 2.80 0.83 4.24V9 . . . . . . . . . . . . 21.324 21.215 16.92 0.06 0 0.24 2.59 0.74 4.15V160 . . . . . . . . . 21.047 … 15.30 0.10 F? … … … …



AGES OF GLOBULAR CLUSTERS 337

2001 PASP, 113:335–343

TABLE1(Continued)

Star(1)

log P(2)

log P0

(3)AVS(4)

Av

(5)Mode

(6)AB 2 VS

(7)Mv

(8)M (HB)v

(9)

TOMv

(10)Reference

(11)


V38 . . . . . . . . . . 21.176 … 16.98 0.65 F … 2.45 0.75 3.97 10V48 . . . . . . . . . . 21.301 21.201 17.01 … … 0.36 2.54 … 4.25V47 . . . . . . . . . . 21.301 21.201 17.10 … … 0.33 2.54 … 4.16V46 . . . . . . . . . . 21.301 21.201 17.48 … … 0.37 2.54 … 3.78


31841 . . . . . . . . 21.379 21.279 17.06 0.03 0 … 2.83 0.22 3.72 734361 . . . . . . . . 21.086 … 16.11 0.17 F … 2.11 0.45 3.9534347 . . . . . . . . 21.404 … 16.74 0.07 … … … … …34461 . . . . . . . . 21.358 21.258 16.96 0.05 0 … 2.75 0.24 3.7431780 . . . . . . . . 21.312 21.212 16.57 0.07 0 … 2.58 0.46 3.9634261 . . . . . . . . 1.007 … 15.88 0.90 F … 1.82 0.39 3.8931835 . . . . . . . . 1.383 21.283 17.22 0.05 0 … 2.84 0.07 3.5734487 . . . . . . . . 21.387 21.287 17.09 0.03 0 … 2.86 0.22 3.7225078 . . . . . . . . 21.227 21.127 16.40 0.05 0 … 2.26 0.31 3.8119387 . . . . . . . . 21.479 … 17.04 0.10 … … … … …19220 . . . . . . . . 21.418 21.318 17.27 0.03 0 … 2.98 0.16 3.6619407 . . . . . . . . 21.382 21.282 16.92 0.10 0 … 2.84 0.37 3.8724966 . . . . . . . . 20.868 … 15.76 0.04 … … … … …19048 . . . . . . . . 21.418 … 16.69 0.04 … … … … …19480 . . . . . . . . 21.435 21.335 17.16 0.05 0 … 3.04 0.33 3.8319336 . . . . . . . . 21.411 21.311 17.23 0.04 0 … 2.95 0.17 3.6725204 . . . . . . . . 1.432 21.332 17.23 0.03 0 … 3.03 0.25 3.7515539 . . . . . . . . 21.432 21.332 17.20 0.03 0 … 3.03 0.28 3.7819176 . . . . . . . . 21.345 … 16.53 0.11 0? … 2.66 … …19164 . . . . . . . . 21.332 21.232 16.98 0.03 0 … 2.69 0.16 3.6625187 . . . . . . . . 21.341 21.241 16.81 0.34 0 … 2.81 0.45 3.956967 . . . . . . . . . . 21.273 … 16.94 0.02 F … 2.8 0.31 3.8110250 . . . . . . . . 21.384 21.284 17.18 0.05 0 … 2.85 0.12 3.6212618 . . . . . . . . 21.446 21.346 17.21 0.04 0 … 3.08 0.32 3.82


V10 . . . . . . . . . . 21.521 21.421 15.84 0.11v 0 0.35 3.36 0.39 3.87 11V11 . . . . . . . . . . 21.418 21.318 15.34 0.04 0 0.34 2.98 0.51 3.99


V7 . . . . . . . . . . . . 21.229 (21.129) 15.59 0.46 0 0.28 2.27 0.28 3.36 12V12 . . . . . . . . . . 21.388 21.288 16.12 0.035 0 0.28 2.86 0.34 4.02V13 . . . . . . . . . . 21.329 … 16.26 0.07 F 0.24 3.02 0.36 4.04


H1 . . . . . . . . . . . . 21.301 21.192 16.25 0.12 0 0.33 2.51 0.70 4.24 13

IC 4499: , ,TOV p 20.90 V(HB) p 17.65 D log P p 0.100

S4 . . . . . . . . . . . . 21.253 21.153 19.32 0.3v 0 … 2.36 0.69 3.94 14

Rup 106: , ,TOV p 21.05 V(HB) p 17.80 D log P p 0.108

V3 . . . . . . . . . . . . 21.403 21.295 20.05 0.25 0 0.36 2.92 0.67 3.92 15V4 . . . . . . . . . . . . 21.309 21.201 19.71 0.14 0 0.36 2.57 0.66 3.91V11 . . . . . . . . . . 21.321 … 20.25 0.18 F 0.35 2.99 0.54 3.79

References.—(1) Gilliland et al. 1998; (2) Kaluzny, Krzeminski, & Nalezyty 1997; (3) Kaluzny & Krzeminski 1993; (4) Walker1994; (5) Nemec et al. 1995; (6) Kaluzny et al. 1998; (7) Rodrıguez & Lopez-Gonzalez 2000;(8) Nemec, Nemec, & Lutz 1994; (9) Kaluzny et al. 1999; (10) Mazur, Kaluzny, & Krzeminski 1999; (11) Kaluzny 1997; (12)Thompson et al. 1999; (13) Hodder et al. 1992; (14) Walker & Nemec 1996; (15) Kaluzny, Krzeminski, & Mazur 1995.



338 McNAMARA

2001 PASP, 113:335–343

Fig. 1.—Six variable stars (HADS) of NGC 288 are plotted as open circlesin a , diagram. Lines indicate the regions of F pulsation and first-AVS log Povertone pulsation ( ). We have added , corresponding to1, 0 D log P p 0.100

to the three variables falling along the line. They are plottedP /P p 0.794 1, 01 0

(with this addition) as crosses. The star falling at ,AVS ∼ 17.5 log P ∼ 21.40is probably pulsating in the second overtone.

Table 1 is arranged according to the NGC number of thecluster. In addition to the NGC number, the row headings con-tain other well-known names, the V magnitude of the turnoff,

; the V magnitude (mean) of the horizontal branch, V(HB);TOVand , the change in required to convert a first-D log P log Povertone pulsator to a fundamental pulsator. The andTOV

(HB) values were determined from fiducial lines when avail-Mv

able. Otherwise, they were determined from the color-magni-tude diagrams. We list the HADS in each cluster (col. [1]) bystar numbers given in the reference; the observed valueslog Pand the fundamentalized period, , if a first-overtone var-log P0

iable by applying the corrections, are given columnsD log P(2) and (3) of the table. The mean V magnitude, ; total lightAV Samplitude, ; and pulsation mode are given in columnsAv

(4)–(6). The mean magnitude is generally an intensity mean,but since the amplitudes of the variables are small, any meanvalue should be similar.

The mean B2V color index, ; absolute magnitudeAB 2 V Sof the variable star, ; absolute magnitude of the horizontalMv

branch (HB); and M are given in columns (7)–(10). TheTOMv v

(HB) and M values are calculated with the aid of theTOM Mv v v

and values of each variable star and the V(HB) and TOAV S Vvalues given in the row identifier.

The mode assignment is based on three criteria: (1) lightamplitude, (2) light-curve asymmetry, and (3) position of thevariable in the , diagram. In general, the asymmetricAV S log Plight-amplitude variables are pulsating in the fundamentalmode, and symmetrical light-curve variables are pulsating inthe first overtone. In some rare cases this may be the secondor third overtone. Variables with small light amplitudes andextremely short periods are generally first-overtone stars. Ifseveral stars in different pulsation modes are present in a cluster,the mode identification can be resolved with a ,AV S log Pdiagram.

More details on the identification of pulsation modes maybe found in McNamara (2000). In addition, we illustrate themode identification in the cluster NGC 288.

NGC 288.—The variables V4 and V6 have large-amplitudeand asymmetrical light curves indicative of fundamental pul-sators. The variable V5 has a large amplitude but fairly sym-metric light curve indicative of a first-overtone pulsator. Theother stars, V7, V8, and V9, all have small amplitudes andvery short periods indicative of first-overtone or higher pul-sation modes. When the amplitudes are small ( mag),A ! 0.1v

it is hard to say anything meaningful regarding the symmetryproperties of the light curves. The variables are plotted as opencircles in a , diagram in Figure 1. At first glance theAV S log Pstars scatter over the diagram, but V4 and V6 are more lu-minous and have a longer period than the other stars. If

, corresponding to , is added toD log P p 0.100 P /P p 0.7941 0

the values of the first-overtone variables (crosses), welog Pfind they form a P-L relation along with the stars we haveassigned fundamental status. The star V9 (at andV ∼ 17.5

) is unique and evidently is pulsating in the secondlog P ∼ 21.4

overtone. The clusters where mode identifications are some-what in doubt are discussed in more detail below.

NGC 104 (47 Tuc).—We have adopted the periods and pul-sation modes for the stars in this cluster as determined byGilliland et al. (1998).

NGC 4590 (M68).—This cluster has two known HADS (Wal-ker 1994). V39 is a large-amplitude and very asymmetric light-curve variable. There is little doubt the star is an F pulsator.V38, on the other hand, is a brighter, smaller amplitude vari-able with a perfectly symmetrical light curve. If it isa member of the cluster, it must be pulsating in a highmode. In order to reconcile its apparent magnitude with V39, itmust be a pulsator with a fundamentalized P-value of4, 0

; Gilliland et al. 1998).log P p 21.015(P /P p 0.4484 0

NGC 5053.—This cluster has five short-period ( !log P21.40) variables of relatively small light amplitude. We as-signed pulsation mode to all five stars.1, 0

NGC 6362.—This cluster has one large-amplitude asym-metric light-curve variable and three shorter period small-amplitude variables with only approximate periods known. Weutilize only the longer period variable in our analysis.

NGC 6838 (M71).—Only one small-amplitude (A p 0.12v

mag) variable is known. The light curve is very symmetrical,and we assign status to the star.1, 0

IC 4499.—Again, only one variable is known. The amplitudeis large ( mag) but variable. We assign status toA p 0.3 1, 0v

the variable.In some cases, the amplitudes of the light curves are very

small. Many small-amplitude variables in the field are foundto be nonradial pulsators. For the most part, the small-amplitudestars fall on P-L relations. We have not attempted to analyzethe light curves to discern the nature of the pulsation. As longas the stars fit a P-L relation, we are confident that they canbe utilized to infer the luminosity levels of the “turn off” andthe horizontal branch. We suspect, however, that at least someof the small-amplitude variables are nonradial pulsators.

The mean magnitudes of the variable stars are given inAV S




2001 PASP, 113:335–343

column (4) ofTable 1, and the corresponding values of theMv

variables are listed in column (8). These values were calculatedfrom the P-L relation given by McNamara (1997b):

M p 23.725 log P 2 1.933. (1)v

The values used in the equation are the fundamentallog Pperiods or the fundamentalized periods in column (3) if thestar is pulsating in the first-overtone or higher mode. The

values range fromD log P D log P p 0.111(P /P p 0.775)1 0

to . The ratio is a functionD log P p 0.086(P /P p 0.820) P /P1 0 1 0

of [Fe/H] increasing as the metal abundance decreases (Petersen& Christensen-Dalsgaard 1996). Our choice of valuesD log PinTable 1 is based on this dependence and in some cases partlyon the observational data. An error of 0.01 in can leadD log Pto an error of 0.037 mag in the of the variable star.Mv

Petersen & Christian-Dalsgaard (1999) have discussed theP-L relation, suggesting that a more appropriate value of thezero point is 21.96. We have retained the larger number be-cause it predicts the values of the shorter period metal-poorMv

variables better. They conclude that the P-L relation shouldyield values to an accuracy of ∼0.1 mag.Mv

3. AGES AND HORIZONTAL BRANCHES

Our discussion in this section is based on the data presentedin Table 2. To the 15 clusters tabulated in Table 1 we haveadded q Cen (NGC 5139) (McNamara 2000), bringing the totalto 16 globular clusters containing HADS. We list the clustersby their NGC numbers in column (1) and other well-knownnames in column (2). The [Fe/H] values are given in columns(3) and (4). The first entry (col. [3]) is from the Harris (1996)catalog, which is the abundance scale established by Zinn &West (1984). The second entry (col. [4]) was taken from Rut-ledge, Hesser, & Stetson (1997), and the values in parentheseswere reduced to their system from the values given in column(3). Horizontal-branch morphologies are given in columns (5)and (6). Column (5) gives the HB ratio (B 2 R)/(B 1 V 1 R)introduced by Lee (1990), and column (6) gives the modifiedDickens (1972) HB type. In general, much of these data hasbeen extracted from the Harris catalog rather than going backto original sources.

The new key data in this investigation are given in column(7) (M ) and column (8) [ (HB)]. These values, absolute mag-TO Mv v

nitudes of the turnoff, M , and absolute magnitudes of the hor-TOv

izontal branches, have been inferred from the and mag-AV S Mv

nitudes of the HADS and the and V(HB) of the clustersTOVgiven inTable 1. Note that we bypass completely the HB as wellas the interstellar reddening in deriving the M values. TheTO

v

uncertainties (internal) listed in columns (7) and (8) reflect onlythe uncertainties due to the HADS values (see the scatter inAV STable 1). To these we must add the uncertainty due to inferringthe V values of the turnoffs and horizontal branches of the

clusters. A comparison of our values with thoseTOVof Rosenberg et al. (1999) yields a mean difference of

( ; 11 clusters). The standard deviation of0.00 5 0.02 j p 0.070.07 mag is similar to a typical 0.08 mag error of the quantity

used to derive by other investigators (see Chaboyer,TO TODV VHB

Demarque, & Sarajedini 1996). The error, 50.08 mag, is ob-tained from the average error given in Table 2 of Chaboyer etal. multiplied by the factor 0.61 derived in their Appendix.

Other possible sources of error include 50.037 mag for anerror in of 50.010 and photometric errors in the meanD log Pmagnitudes of the variables of 50.025 mag. The latter arealready incorporated in the scatter of the V magnitudes of var-iables about a cluster P-L relation and hence in the errorstabulated in columns (7) and (8) of Table 2. The most serioussource of error is misidentified modes, which can lead to errorsof 50.372 mag. Thus, if the modes are identified correctly,the three most serious errors are the following: 50.07 mag in

, 50.08 mag in the scatter of a single HADS about theTOVP-L relation, and 50.037 mag because of a poor choice of

. Thus, for a cluster with a single HADS, we shouldD log Pbe able to infer M to an accuracy of ∼0.11 mag. This assumes,TO

v

of course, there are no sources of systematic errors. We cannotrule out a systematic error of 50.10 mag in the zero point ofthe P-L relation (eq. [1]).

A plot of the M values against the [Fe/H] values in columnTOv

(4) (Zinn & West scale) is shown in Figure 2. The error (onlydue to HADS) is indicated by the vertical lines. We estimatethe errors in the [Fe/H] values as 50.15. The error of a typicaldata point is indicated by the error bars in the upper right-handcorner of the figure. The solid curves are the M curves forTO

v

10 and 12 Gyr of the VandenBerg et al. (2000) models, whilethe dot-dashed lines are the M curves calculated from theTO

v

equations and coefficients of Table 1 of Cassisi et al. (1999)again at 10 and 12 Gyr. The dashed line is a least-squares fitto the data:

TOM p 0.34(5.08)[Fe/H] 1 4.48(50.05),v

j p 50.10. (2)

We have allowed for errors in both coordinates in derivingequation (2). We assumed the error in M is 50.10 andTO

v

the error in [Fe/H] is 50.15. It is apparent that the scatter inFigure 2 is consistent with the errors. It is unlikely that we havemany modes misidentified, otherwise the scatter in Figure 2would be much larger. Since the scatter is consistent with theerrors, it suggests the globular clusters are coeval within a 1 Gyrtime frame. This result, little or no age spread, is in generalconsistent with the conclusion of Rosenberg et al. (1999), whodetected no age spread for clusters with [Fe/H . They] ! 21.2did find an age spread for the intermediate-metallicity group(2 Fe/H ) and for at least one cluster, Pal 12, in1.2 ≤ [ ] ≤ 20.9the metal-rich group ([Fe/H ). It is important to keep in] ≥ 20.9



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2001 PASP, 113:335–343

TABLE 2The Absolute Magnitude of “Turnoffs” and Horizontal Branches

Cluster(1)

Other(2)

[Fe/H]

HBR(5)

hbt(6)

TOMv

(7)(HB)Mv

(8)Harris 1996

(3)

Rutledgeet al. 1997

(4)a

NGC 104 . . . . . . . 47 Tuc 20.71 20.78 20.99 7 4.30 5 0.08 0.69 5 0.08NGC 288 . . . . . . . … 21.40 21.14 0.98 0 3.95 5 0.04 0.25 5 0.04NGC 4372 . . . . . . … 22.09 (21.87) 1.00 0 3.66 5 0.08 0.16 5 0.08NGC 4590 . . . . . . M68 22.09 22.00 0.17 3 3.62 5 0.04 0.16 5 0.04NGC 5053 . . . . . . … 22.41 21.98 0.52 3 3.73 5 0.04 0.33 5 0.04NGC 5139 . . . . . . q Cen 21.55 (21.25) … 2 3.88 5 0.03 0.14 5 0.02

… 21.28 (21.07) … … … 0.28 5 0.02NGC 5272 . . . . . . M3 21.66 21.33 0.08 4 3.93 5 0.06 0.48 5 0.06NGC 5466 . . . . . . … 22.22 22.13 0.58 2 3.89 5 0.04 0.56 5 0.04NGC 5904 . . . . . . M5 21.40 21.12 0.31 3 4.09 5 0.09 0.68 5 0.09NGC 6362 . . . . . . … 21.06 20.99 2.58 1/7 3.97 5 0.12 0.75 5 0.12NGC 6397 . . . . . . … 21.91 21.76 0.98 1 3.93 5 0.04 0.45 5 0.04NGC 6752 . . . . . . … 21.54 21.24 1.00 0 4.01 5 0.02 0.33 5 0.02NGC 6809 . . . . . . M55 21.81 21.54 0.48 1 3.78 5 0.03 0.28 5 0.03NGC 6838 . . . . . . M71 20.58 20.73 21.00 7 4.24 5 0.12 0.70 5 0.12

IC 4499 21.75 (21.45) 0.11 4 3.94 5 0.12 0.69 5 0.12Rup 106 21.69 (21.49) 20.82 6 3.87 5 0.03 0.62 5 0.03

a The values in parentheses were reduced to their system from the values given in col. (3).

Fig. 2.—Plot of M of globular clusters as inferred from HADS vs. theirTOv

[Fe/H] values (open circles). The vertical lines indicate the error due to HADSdata only. A more realistic indicator of the error bars is indicated by the crossin the upper right of the figure. The solid lines at 10 and 12 Gyr are basedon isochrones of VandenBerg et al. (2000). The dot-dashed lines are based onthe equations and coefficients given in Table 1 of Cassisi et al. (1999) for 10and 12 Gyr. The dashed line is a least-squares fit to the data.

mind that our conclusion regarding ages is based strictly on theclusters we have analyzed. We have not analyzed some of theunusual clusters that other investigators find to be younger thanthe typical cluster.

The VandenBerg et al. models yield an average age ofGyr for the 16 clusters, while the Cassisi et al.11.30 5 0.25

equation yields an average age of .11.0 Gyr. Note that in bothcases we find a slope in the M , [Fe/H] diagram in excellentTO

v

agreement with those predicted by the models.The ages may be compared with Gyr found by11.8 5 2.5

Gratton et al. (1997) for what they consider to be the olderclusters. Their age is based on five different isochrone sets, andtheir uncertainty reflects the differences between the isochronesets. Our error of 50.25 Gyr is internal for only one isochroneset. A more realistic error is at least 51 Gyr if other isochronesets are considered. For example, if we utilize equation (3) ofChaboyer et al. (1998) relating age to M and [Fe/H], we findTO

v

a mean age of 9.9 Gyr. It is evident that model isochronesdiffer. We shall adopt Gyr as a good estimate of the11.3 5 1age and uncertainty in the age of the clusters. Actually this agedoes compare favorably with the Gratton et al. (1997) valueabove and the Chaboyer et al. (1998) value of 11.5 Gyr forthe oldest clusters. We note that an uncertainty of 50.1 in thezero point of the P-L relation (eq. [1]) can introduce errors of50.8 Gyr in the ages of the clusters. If the P-L relation is inerror, it is more likely in the sense of overestimating the lu-minosities of the HADS. Thus, if a systematic error of 0.1 magis really present, we have underestimated the ages of the clus-ters by ∼0.8 Gyr.

It is important to emphasize that we have inferred the MTOv

values by a completely new observational approach. The gen-

eral overall agreement in the ages is actually quite remarkable.Our finding, of little or no age spread in the sample of Galacticglobular clusters investigated here, is contrary to the view heldby many astronomers who have suggested that later matterinfall and subsequent accretion played a key role in formingthe halo clusters over an extended period of time (Searle &Zinn 1978), giving rise to a considerable range in ages. Ourresults favor the concept of Eggen, Lynden-Bell, & Sandage(1962), who envisioned the halo forming rapidly during thecollapse of the proto-Galactic gas cloud which would lead toa common age for all clusters.




2001 PASP, 113:335–343

Fig. 3.—Relation between the (HB) values of globular clusters and theirMv

[Fe/H] values. The vertical lines represent only the internal errors of the HADSdata in inferring the (HB) values. Typical errors are 50.1 mag in andM Mv v

50.15 mag in [Fe/H]. The observational data are plotted with different symbolsdesignating different HB morphological types. Clusters with predominatelyblue HBs are designated with open circles (hbt 0–1), intermediate clusterswith blue to red HBs (hbt 2–6) are indicated with crosses, and clusters withpredominate red branches (hbt 6–7) are plotted as filled circles. The two opentriangles designate the two luminosity levels of RR Lyrae stars in q Cen. Theline is based on a least-squares solution. The scatter in the data points (j p

mag) is twice the expectation based on errors in the (HB) values. This0.19 Mv

suggests that the (HB) values depend on at least one additional parameterMv

other than [Fe/H]. Note that clusters with predominately blue HBs tend tohave brighter HBs than clusters with blue-red HBs.

We turn now to a discussion of the luminosity levels of thehorizontal branches of the clusters. It should be possible todetermine the (HB) values with greater precision than theMv

M values since we are dealing with a horizontal sequence ofTOv

stars rather than a vertical sequence.The V magnitudes of the horizontal branches of each cluster

are given in the row headings ofTable 1 as V(HB). These aremean values that I have determined from published color-mag-nitude diagrams. Where possible, I have adopted the mean Vmagnitudes of the RR Lyrae stars in each cluster, otherwise, themean V mag of the blue or red HB. The mean difference betweenthe values given inTable 1 and the V(HB) of Rosenberg et al.(1999) (12 clusters in common) is 2 mag, and the0.04 5 0.02V(HB) values of Chaboyer et al. (1998) (13 clusters in common)is 1 mag. The differences are in the sense of the0.01 5 0.02present values minus the others.

The absolute magnitudes of the cluster horizontal branches,(HB), have been derived from the V(HB), magnitudesM AV Sv

of the HADS and the values of the HADS. They are listedMv

in column (8) of Table 2 and plotted against the [Fe/H] valuesof the clusters in Figure 3. We list two sets of [Fe/H] valuesin Table 1. In column (2) are found the [Fe/H] values on theZinn & West (1984) scale, and in column (3) are [Fe/H] valueson the Rutledge et al. (1997) scale. In the latter, [Fe/H] valuesin parentheses in column (3) designate a value reduced to theirsystem.

A comparison of the (HB) values in Table 2, derived fromMv

the HADS, can be made with the (HB) values derived fromMv

cluster main-sequence fitting. For five clusters in common withthe study of Gratton et al. (1997) we find an average differenceof 2 mag; with seven clusters in common with0.05 5 0.08Reid (1997, 1998) the difference is 1 mag. The0.14 5 0.07differences are in the sense of (HB) present minus (HB)M Mv v

others. Thus, our values are slightly brighter than the Grattonet al. values and fainter than the Reid values, but the differencesare within the random and systematic errors of the variousinvestigations.

The most critical discussion of utilizing calibrating subdwarftemplates to find the luminosities of horizontal branches is thatof Chaboyer et al. (1998) of NGC 6752, M5, and M13. For twoclusters we can make direct comparisons. For NGC 6752 andM5 Chaboyer et al. find andM (HB) p 0.30 5 0.15v

. These values compare withM (HB) p 0.54 5 0.09v

(NGC 6752) andM (HB) p 0.33 5 0.07 M (HB) p 0.68 5v v

(M5) found from the HADS for these clusters. White dwarf0.11fitting to NGC 6752 (Renzini et al. 1996) yields M (RR) pv

. The agreement is very satisfactory: the scatter in0.45 5 0.14the data points is entirely consistent with the errors.

As indicated previously, the (HB) values inferred from theMv

HADS are plotted against the Zinn & West [Fe/H] values inFigure 3. The vertical lines represent only the internal errors ofthe HADS data in inferring the (HB) values. The typical errorsMv

are more nearly 50.11 mag in and 50.15 in [Fe/H]. TheMv

former is based on 50.08 mag error in the HADS data and

50.07 mag for the errors in V(HB) and 50.037 for a poorchoice of . The later is based on systematic and internalD log Perrors of [Fe/H] determinations.

Linear least-squares solutions to the data points yield

M (HB) p 0.30(50.16)[Fe/H]v

10.92(50.09), j p 50.19 (3)

for the data displayed in Figure 3 and

M (HB) p 0.34(50.17)[Fe/H]v

10.93(50.12), j p 50.20, (4)

for the Rutledge et al. (1997) [Fe/H] values. Allowance forerrors in both and [Fe/H] were taken into account. If oneMv

assumes all the error is in , we find smaller slopes (0.21 andMv

0.22).We note that Groenewegen & Salaris (1999) found

M p 0.18[Fe/H] 1 0.77(50.26) (5)v

(the slope 0.18 was assumed) from halo RR Lyrae stars in theHipparcos Catalogue by the “reduced parallax” method. At [Fe/H our equation (3) yields , compared with] p 20.6 M p 0.74v

given by the Groenewegen & Salaris equation. AtM p 0.66v

[Fe/H equation (3) gives , which compares] p 22.0 M p 0.32v

with given by the Groenewegen & Salaris expres-M p 0.41v



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2001 PASP, 113:335–343

Fig. 4.—Plot of the [Fe/H] values of HADs vs. their values. Openlog Pcircles are field variables and the crosses are the mean values of oflog Pvariables in clusters. First-overtone variables have been fundamentalized byadding corrections. The dashed line (drawn by hand) indicates theD log Pmean variation of [Fe/H] on .log P

TABLE 3LMC Clusters

Cluster(1)

TV(2)

Av

(3)

TOV0

(4)[Fe/H]

(5)

TOMv

(6)

TO TOV 2 M0 v

(7)

NGC 1466 . . . . . . 22.87 0.29 22.58 21.85 3.85 18.73NGC 2257 . . . . . . 22.45 0.13 22.32 21.85 3.85 18.47Hodge 11 . . . . . . . 22.80 0.25 22.55 22.05 3.78 18.77

sion. Thus over the typical range of [Fe/H] values of RR Lyraestars in globular clusters, the agreement between the two ex-pressions is less than 0.1 mag.

A very surprising result is the large errors in the (HB)Mv

values. The standard deviation, , is about twice the errorj p 0.19of the M values. We are forced to conclude that there is realTO

v

scatter in the (HB) values at similar [Fe/H] values. The as-Mv

sumption that ([Fe/H]) is clearly an oversimplifi-M (HB) p fv

cation and needs to be replaced by ([Fe/H], anotherM (HB) p fv

parameter, or other parameters). It appears that not only is themorphology of the clusters affected by a second parameter, butthe luminosity levels of the horizontal branches are as well. Notethat the observational data are plotted with different symbolsdesignating different HB morphological types. Blue horizontalbranches are designated with open circles (hbt 0–1), intermediateclusters with crosses (hbt 2–6), and clusters with predominatelyred branches with a filled circle (hbt 6–7). The two open trianglesdesignate the two luminosity levels of RR Lyrae stars in q Cen.There is a definite tendency of the blue horizontal-branch clustersto be the most luminous and the intermediate and red horizontal-branch clusters to be less luminous. The failure to take this intoaccount is no doubt responsible in part for the spread in ages ofglobular clusters described in many studies in the past.

The correlation between the F period of pulsation of the HADSand the abundance parameters [Fe/H] has been pointed outpreviously (McNamara 1997b). All of the new globular clusterdata now available make it worthwhile to examine this questionanew. We have plotted the average (F period) values oflog Pthe cluster variables versus the [Fe/H] values in Figure 4. Onlythe clusters with three or more HADS are displayed (crosses).The field variable star data (McNamara 1997b) are also plotted(open circles) in the figure. First-overtone pulsators have beenfundamentalized. It is evident that the shorter period variablesare very metal-poor and the solar-abundant variables are notfound with short periods. Only when variables have valueslog P∼20.90 and larger do we encounter solar-abundance variables.

The [Fe/H] value of a globular cluster with a large number ofHADS can be approximately inferred from the value.Alog PS

4. THE DISTANCE TO THE LARGEMAGELLANIC CLOUD

Johnson et al. (1999) have presented convincing evidencethat the oldest globular clusters in the LMC are the same ageas globular clusters in our galaxy. As such, we can apply equa-tion (2) relating M of the clusters to their [Fe/H] values toTO

v

infer distances.In Table 3, we list the three clusters studied by Johnson et

al. (1999). We have determined the values of the clusters,TOVlisted in column (2), from the clusters fiducial curves. We haveadopted the same interstellar extinctions as Johnson et al. whichare given in column (3) and corrected the “turnoffs” to VTO

0

values given in column (4). With the aid of the [Fe/H] valuesin column (5), the M are calculated and given in column (6).TO

v

The distance moduli are given in column (7). The mean valueis 18.6650.08, where the error is the uncertainty in the meanvalue.

We may utilize also the mean V magnitudes of RR Lyraevariables in LMC clusters to infer the distance to the LMC.We list in Table 4 Walker’s data for seven clusters. Clusternames, [Fe/H], , E(B2V), and values are found inAV S AV S0

columns (1)–(5), respectively. The values in column (6)Mv

were calculated with equation (3). Although we have suggestedthat (RR) depends on more than [Fe/H], the exact depen-Mv

dence is not known at the present time, so we will adopt thevalues given by equation (3). The distance moduli are givenin column (7). The mean value is , which com-18.61 5 0.04pares favorably with 18.66 found from the values of theTOV0

clusters.Carretta et al. (2000) suggest that the average dereddened

magnitude of 75 RR Lyrae variables in the LMC bar ismag. If the average metallicity of theseAV S p 19.11 5 0.020

stars is [Fe/H as they suggest, then ; this] p 21.5 M p 0.47v

yields a distance modulus of 18.64, in good agreement withthe other determinations.

Our results favor the long distance scale. We note that somerecent studies favor a shorter scale. For example, Guinan et al.(1998) find an LMC distance modulus of 18.30 from a carefulanalysis of an LMC eclipsing binary. We can also point to adistance modulus of 18.24 found from an analysis of red clumpgiants by Stanek et al. (2000). The issue of the long and short




2001 PASP, 113:335–343

TABLE 4RR Lyrae Stars in the LMC Cluster

Cluster(1)

[Fe/H](2)

AVS(3)

E(B2V)(4)

AVdS0

(5)Mv

(6)HB Type

(7)AVS02Mv

(8)

NGC 1466 . . . . . . 21.8 19.33 0.09 19.05 0.38 0.40 18.67NGC 1786 . . . . . . 22.3 19.27 0.07 19.05 0.23 … 18.82NGC 1835 . . . . . . 21.8 19.37 0.13 18.96 0.38 … 18.58NGC 1841 . . . . . . 22.2 19.31 0.18 18.75 0.26 0.72 18.49NGC 2210 . . . . . . 21.9 19.12 0.06 18.93 0.35 0.57 18.58NGC 2257 . . . . . . 21.8 19.03 0.04 18.90 0.38 0.49 18.52Reticulum . . . . . . . 21.9 19.07 0.03 18.98 0.35 20.04 18.63

distance scales has not been resolved to everyone’s satisfactionat the present time. Suffice it to say, it is difficult to understandhow our average LMC distance modulus of 18.64 can be inerror by much more than 0.1 mag.

5. SUMMARY AND CONCLUSIONS

We have utilized HADS in 16 globular clusters to fix theluminosity levels of the main-sequence “turnoffs,” M , andTO

v

the horizontal branches, (HB). The M values are found toTOMv v

agree with theoretical isochrones yielding an average age of

the globular cluster sample of Gyr. Both metal-poor11.3 5 1and metal-strong clusters have identical ages. An equation re-lating the (HB) values to the [Fe/H] values is derived forMv

the 16 clusters. The scatter in the (HB) values at similarMv

[Fe/H] values suggests that the (HB) value is a function ofMv

at least one additional parameter in addition to the [Fe/H] value.We apply some of the calibrations derived in this investi-

gation to find the distance to the LMC. We average the valuesof three different determinations and find the mean distancemodulus of the LMC to be 18.64.

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