Red Giant Branch Bump Brightness and Number Counts in 72 Galactic Globular Clusters Observed with the Hubble Space Telescope
David M. Nataf, Andrew P. Gould, Marc H. Pinsonneault, Andrzej Udalski
RRed Giant Branch Bump Brightness and Number Counts in 72Galactic Globular Clusters Observed with the Hubble SpaceTelescope
David M. Nataf , Andrew P. Gould , Marc H. Pinsonneault , Andrzej Udalski [email protected] ABSTRACT
We present the broadest and most precise empirical investigation of red gi-ant branch bump (RGBB) brightness and number counts ever conducted. Weimplement a new method and use data from two
Hubble Space Telescope (HST) globular cluster (GC) surveys to measure the brightness and star counts of theRGBB in 72 GCs. The median measurement precision is 0.018 mag in the bright-ness and 31% in the number counts, respectively reaching peak precision valuesof 0.005 mag and 10%. The position of the main-sequence turnoff (MSTO) andthe number of horizontal branch (HB) stars are used as comparisons where ap-propriate. Several independent scientific conclusions are newly possible with ourparametrization of the RGBB. Both brightness and number counts are shown tohave second parameters in addition to their strong dependence on metallicity.The RGBBs are found to be anomalous in the GCs NGC 2808, 5286, 6388 and6441, likely due to the presence of multiple populations. Finally, we use our em-pirical calibration to predict the properties of the Galactic bulge RGBB. The up-dated RGBB properties for the bulge are shown to differ from the GC-calibratedprediction, with the former having lower number counts, a lower brightness dis-persion and a brighter peak luminosity than would be expected from the latter.This discrepancy is well explained by the Galactic bulge having a higher heliumabundance than expected from GCs, ∆Y ∼ +0.06 at the median metallicity. Subject headings:
Hertzspring-Russell and C-M diagrams – Galaxy:globular clus-ters: general – Galaxy: globular clusters: individual: (NGC 2808, NGC 5286,NGC 6388, NGC 6441) – Galaxy: Bulge Department of Astronomy, Ohio State University, 140 W. 18th Ave., Columbus, OH 43210 Warsaw University Observatory, Al. Ujazdowskie 4, 00-478 Warszawa,Poland a r X i v : . [ a s t r o - ph . GA ] M a r
1. Introduction
The red giant branch bump (RGBB) is a prominent feature of color-magnitude diagrams(CMD) along the red giant (RG) branch that was first theoretically described by Thomas(1967) and Iben (1968). It was first empirically confirmed nearly two decades later, by Kinget al. (1985), in their observations of the Galactic globular cluster (GC) 47 Tuc. During thefirst ascent of the RG branch when the hydrogen-burning shell moves outward, stars becometemporarily fainter before becoming brighter again due to a discontinuity in the chemicalabundance profile near the convective envelope (Sweigart et al. 1990). As the star thushas the same luminosity on three separate occasions, an excess in the luminosity functionat a characteristic magnitude becomes visible in the luminosity function of RG stars. Theproperties of this excess, such as its characteristic brightness and expected number counts,are a steeply sensitive function of a stellar population’s age, initial helium abundance, andmetallicity (Cassisi & Salaris 1997; Bono et al. 2001; Bjork & Chaboyer 2006; Di Cecco etal. 2010; Cassisi et al. 2011; Nataf et al. 2011a,b).Even though the RGBB occurs before more complex phases of stellar evolution suchas the helium flash, and should as such be well-constrained theoretically, there has been anongoing debate in the literature as to a discrepancy between the predicted and observedbrightnesses of the RGBB. In their pioneering study of the RGBB in 11 GCs, Fusi Pecci etal. (1990) found that a zero-point shift of several tenths of a magnitude was required to bringtheory into agreement with observations – the predicted luminosity of the RGBB was greaterthan that observed. Some disagreement followed. Cassisi & Salaris (1997) and Zoccali et al.(1999) argued that there was no significant disagreement between theory and observations ifone accounted for observational uncertainties in [ α /Fe] and [Fe/H] abundances. Moreover,as the the brightness of RGBB was measured relative to the HB, it was not clear how anydiscrepancy between models and data should be interpreted. 3 –Fig. 1.— TOP: The values of M V,RGBB and their statistical errors for all 72 Galactic GCsstudied in this work are shown as filled red circles. The error bars are the statistical errorsdue to the fit, and do not include the systematic error from the assumed extinction anddistance values. RGBB properties in the anomalous GCs NGC 5286, 2808, 6388 and 6441delineated by black x-shaped symbols. BOTTOM: Number counts on the RGBB normalizedwith respect to the number of HB stars as a function of [M/H]. The values of f HBRGBB for theGalactic spheroid systems M32 and the MW bulge, respectively measured by Monachesi etal. (2011) and Nataf et al. (2011a), are shown as the empty black squares in the lower panel.However, since those works the breadth of photometric catalogs, the range of availablediagnostic, and the accuracy of metallicity scales have evolved. More recently, three worksusing three distinct methods found that the brightness of the RGBB is 0.2-0.4 mag fainterthan predicted by models. Di Cecco et al. (2010) used a large sample of 15 GCs spanning2 dex in metallicity. They found that the difference in V -band between the RGBB and thezero-age HB, ∆ V RGBBZAHB = V RGBB − V ZAHB , was larger than predicted by models, i.e. that thebump was fainter in observations than in models. The discrepancy was found to be muchmore severe in metal-poor GCs. Cassisi et al. (2011) compared the RGBB brightness relative 4 –to the main sequence turnoff (MSTO) in the
Hubble Space Telescope (HST)
F606W band,∆ F W MST ORGBB = F W MST O − F W RGBB . They found that agreement with theorycould only be obtained if GCs are in fact ∼ ξ . Models predict this to be a very sensitivehelium diagnostic that only weakly depends on age. Unfortunately, these same models onlymatch the data if the helium abundance in the clusters is set to an unphysical value ofY=0.20 – lower than the primordial value Y=0.249 derived from observations of the cosmicmicrowave background and the standard model of particle physics (Steigman 2010). Threedifferent methods lead to one consistent result, that stellar models overestimate the predictedluminosity of the RGBB.A second observational challenge in RGBB astrophysics has also recently come on thescene. Nataf et al. (2011a) found that the RGBB in the Galactic bulge had deficient numbercounts relative to both stellar models and observations of GCs. They measured an RGBBto red clump (RC) ratio f RCRGBB = 0 . ± .
02 toward Baade’s window, compared to ∼
25% inthe most metal-rich GCs and stellar model predictions that the lifetime of the RGBB at theage and metallicity of the bulge should be ∼
25 Myr (Nataf et al. 2011a), which predicts a ∼
25% number fraction if one assumes a lifetime of ∼
100 Myr for the HB (Han et al. 2009).Meanwhile, in an analysis of
HST
CMDs toward the dwarf elliptical M32, Monachesi et al.(2011) measured f RCRGBB = 0 . ± . M V,RGBB = (0 . ± . . ± . / H] + 1 . , δ = 0 .
077 (1)∆ V MST ORGBB = V MST O − V RGBB = (3 . ± . − . ± . / H] + 1 . , δ = 0 . I RHBRGBB = I RGBB − I RHB = (0 . ± . . ± . / H] + 0 . , δ = 0 . EW RGBB = (0 . ± . . ± . / H] + 1 . , δ = 0 . f HBRGBB = N RGBB N HB = (0 . ± . . ± . / H] + 1 . , δ = 0 . , (4)where the five parameters denote the absolute magnitude of the RGBB peak in V , the bright-ness difference in V between the RGBB peak and the position of the MSTO, the brightnessdifference in I between the RGBB peak and the mean brightness of red horizontal branch(RHB) stars, the equivalent width EW RGBB of the RGBB feature in the RG luminosityfunction relative to the underlying RG continuum, and the relative number of RGBB starsto HB stars. The values of δ refer to the intrinsic scatter in these relations that is in addi-tion to the scatter due to statistical errors in the measurements and the errors in the inputspectroscopic metallicities. The significant values of δ demonstrate that all but one of thesevariables has at least one second parameter. The problematic GCs NGC 2808, 5286, 6388and 6441 are left out of the fit. The two key empirical trends probed by this work, quantita-tively summarized in the above equations and graphically summarized in Figure 1, are thedecreasing luminosity and increasing number counts of the RGBB as metallicity is increased.In this paper, we summarize the input photometric and spectroscopic data as well asnecessary calibrations used in Section 2. A fitting procedure for the different observationalparameters of the RGBB including star counts is outlined in Section 3. The methodologywith which we measure the relevant parameters for the MSTO and the HB are discussedin Sections 4 and 5. Results for the brightness, number counts and other parameters arerespectively discussed in Sections 6, 7, 8. A more detailed analysis of the anomalous RGBBsin the clusters NGC 2808, 5286, 6388 and 6441 is to be found in Section 9. We use themetallicity distribution of bulge stars to derive what the properties of the Galactic bulgeRGBB toward two distinct sightlines would be if bulge stars have the same input physicsas Galactic GCs stars in Section 11. Our conclusions are presented in Section 12. We useMonte Carlo methods to demonstrate the reliability of our approach in the Appendix. Tablessummarizing the measurements are to be found following the Appendix. 6 –
2. Data
In this study we make use of two different large-scale GC surveys conducted with
HST .We also use ground-based data for comparisons to the Galactic bulge.We use photometry obtained with
HST ’s Advanced Camera for Surveys (ACS) (Saraje-dini et al. 2007; Dotter et al. 2011), hereafter “the ACS GCs”. The data were taken as partof an
HST treasury program to obtain high signal-to-noise ratio photometry down to thelower main sequence for a large number of Galactic GCs. Artificial star tests demonstratethat the photometry is expected to be very precise and complete at the brightness of theRGBB (Anderson et al. 2008).We also use the
HST
F439W and F555W photometry obtained as part of an
HST
GC survey program with the WFPC2 camera (Piotto et al. 2002), hereafter “the WFPC2GCs”. We use the photometry from this database for 16 clusters with well-populated redgiant branches that were not observed within the ACS survey. Combining these two datasetsyields a richer sample with better completeness over the metallicity range of GCs. Zoccaliet al. (1999) and Riello et al. (2003) have already studied the RGBB in the WFPC2 GCsample, however we wish to study the two samples together, using a uniform methodology.The metallicities for these clusters are taken from the GC metallicity scale of Carrettaet al. (2009b), except for the three GCs studied in this work not listed there. The metallicityfor Lynga 7 is taken from Bonatto & Bica (2008), and those of NGC 6426 and Pyxis aretaken from Dotter et al. (2011). The remaining GC parameters are taken from the Harris(1996, 2010 edition) catalog of parameters for Milky Way GCs.OGLE-III optical photometry is used to compare the RGBB in GCs to that in theGalactic bulge. OGLE-III observations were obtained from the 1.3 meter Warsaw Telescope,located at the Las Campanas Observatory in Chile, and are complete to magnitude ∼ V and I . Detailed descriptions of the instrumentation, photometric reductions andastrometric calibrations are available in Szyma´nski et al. (2011) We make two adjustments to the input data motivated by the need for uniformity. Thefirst is to the definition of V used in the WFPC2 dataset, and the second relates to themetallicity scale.With respect to the photometric calibration, neither dataset obtained data in V . TheWFPC2 dataset obtained photometry in F W and F W , whereas the ACS dataset 7 –photometry is in F W and F W . Photometric values respectively transformed into the( V, B − V ) and ( I, V − I ) plane were given. For 13 of the WFPC2 clusters that had alsohad data in the ACS dataset, namely NGC 104, 362, 1851, 2808, 5904, 5927, 5986, 6304,6388, 6441, 6624, 6637, 7078, we compared photometric values obtained for the two datasetsat the level of the RGBB. We found that the V given in the WFPC2 dataset was ∼ V given in the ACS dataset. We adjusted the definition of V in theWFPC2 results, without adjusting the definition of ( B − V ).The second calibration pertains to the metallicity. The metallicity scale of Carretta etal. (2009b) has values of [Fe/H] and σ [Fe / H] for every cluster studied in this work. However,it does not have values of [ α /Fe] for all the clusters. We computed a linear fit to all clustersin the catalog that have both [Fe/H] and [ α /Fe] values, and are not associated to dwarfspheroidal galaxies (Arp 2, NGC 4147, NGC 6715, Palomar 12, Terzan 7, and Terzan 8).We obtained [ α /Fe] = 0.342 − α /Fe] value.Measurement errors in [Mg/Fe] and [Si/Fe] were derived for 17 GCs spectroscopicallyinvestigated in Carretta et al. (2009a). We take the mean of these errors as the estimate for σ [ α/ Fe] . The mean error is σ [ α/ Fe] =0.060. Since the scatter in the assumed relation of [ α /Fe]= 0.342 - 0.033([Fe/H]+1) is 0.085 dex, we assume an error of σ [ α/ Fe] = √ . − . =0.060 dex for all the remaining cluster for which we don’t have a reported value of [ α /Fe],and thus no measurement error. It is likely a coincidence that the mean measurement errorin [ α /Fe] is equal to the intrinsic scatter to the [Fe/H]-[ α /Fe]relation.We then compute the “total metallicity” of the GCs using the relation of Salaris et al.(1993): [M / H] = [Fe / H] + log(0 . ∗ [ α/ Fe] + 0 . [M / H] ≈ . N F e /N F e (cid:12) ) + 0 . N α /N α (cid:12) ) . (6)Finally, given the approximation that the errors to [ α /Fe] and [Fe/H] are uncorrelated,the error in the total metallicity is: σ [M / H] = (cid:115) σ [Fe / H]2 + (cid:20) . ∗ [ α/ Fe] . ∗ [ α/ Fe] + 0 . (cid:21) σ [ α/ Fe]2 (7) 8 –
3. Fitting for the RGBB
We jointly fit for the luminosity function of RG+RGBB stars in and near the RGBBusing the following parametrization: N ( I ) = A (cid:26) exp (cid:20) B ( I − I RGBB ) (cid:21) + EW RGBB √ πσ RGBB exp (cid:20) − ( I − I RGBB ) σ RGBB (cid:21)(cid:27) , (8)where A defines the total normalization of the population, B defines an exponential lumi-nosity function for the underlying RG branch, the equivalent width EW RGBB = N RGBB /A isthe ratio of the number of RGBB stars to the number density of RG stars at the brightnessof the RGBB, I RGBB is the mean brightness of the RGBB, and σ RGBB is the brightnessdispersion of the RGBB. This methodology has previously been used elsewhere (Nataf et al.2011a,b), but we will provide a stand-alone justification here.We fit in I because it is a more stable bandpass to blending and differential reddeningthan V , and because the absolute value of the derivative of the bolometric correction in I is much smaller than that in V for stars moving up to the RG branch, facilitating amore accurate comparison to models. We use I (cid:48) = V − B − V ) − eff –log g –[Fe/H] (An et al. 2007),calculated at the position of the RGBB. V RGBB is obtained for both datasets by measuringthe color of the RG branch at the position of the RGBB. CMDs, magnitude histograms andtheir corresponding best-fit
N(I) probability density functions are shown for 47 Tuc, NGC362, NGC 1261 and NGC 7078 in Figures 2, 3, 4, 5.Measurement of the best-fit values for the parameters and their associated errors aredone using a maximum-likelihood analysis to explore the parameter space via Markov ChainMonte Carlo (MCMC) – we do not bin the data before fitting the parameters. For eachvalue of the parameters tested by the MCMC, we compute the log-likelihood (cid:96) : (cid:96) = N obs (cid:88) i ln (cid:20) N (I i / A , B , EW RGBB , σ
RGBB , I RGBB ) (cid:21) − N obs (9)where N obs is the total number of stars included in the fit. In each element of the chain,the parameter A is selected such that the integral of the function N ( I ) over the magnituderange is equal to N obs . In other words, it is determined analytically for each combination ofthe other parameters rather than floated as a free parameter.We can thus use the statistical identities valid for distributions marginalized to a singleparameter, that parameter values with (cid:96) ≥ ( (cid:96) max − /
2) are within 1 σ of the best-fit, thosewith (cid:96) ≥ (cid:96) max − σ of the best-fit, and so on. This yields the reported 1- σ I , I +d I ) is equal to N ( I )d I , thestatistical noise is Poissonian, thereby relating (cid:96) to χ in the limit of a large number ofdatapoints: χ = − (cid:96) (10) 10 –Fig. 2.— LEFT: CMD of 47 Tuc in ACS data, with the color-magnitude selection contoursshown for the 2415 RG+RGBB stars and 545 RHB stars. The color of the RG branch at theRGBB is 1.03. The mean brightness of the RHB stars is I RHB = 13 .
09. RIGHT: Magnitudedistribution of the RG+RGBB stars, I RGBB = 13 . ± . EW RGBB = 0 . ± .
04. 11 –Fig. 3.— LEFT: CMD of NGC 362 in ACS data, with the color-magnitude selection contoursfor 1059 RG+RGBB stars and 296 RHB stars shown. The color of the RG branch at theRGBB is 0.985. The mean brightness of the RHB stars is I RHB = 14 .
67. RIGHT: Magnitudedistribution of the RG+RGBB stars, I RGBB = 14 . ± . EW RGBB = 0 . ± .
06. 12 –Fig. 4.— LEFT: CMD of NGC 1261 in ACS data, with the color-magnitude selectioncontours for 808 RG+RGBB stars. The color of the RG branch at the RGBB is 0.972.RIGHT: Magnitude distribution of the RG+RGBB stars, I RGBB = 15 . ± . EW RGBB =0 . ± .
06. The HB selection for this cluster can be found in Figure 8. 13 –Fig. 5.— LEFT: CMD of NGC 7078 in ACS data, with the color-magnitude selectioncontours for 1403 RG+RGBB stars shown. The color of the RG branch at the RGBB is0.972. RIGHT: Magnitude distribution of the RG+RGBB stars, I RGBB = 14 . ± . EW RGBB = 0 . ± . N RGBB ≥
10 and that were not known to be affected by severe patchy differential reddening. Thissample on its own would be at risk of overestimating the expected value for EW RGBB , sinceit has a minimum value of the normalization. We construct a “silver” sample as follows. On 14 –all the remaining clusters, we fit for the RGBB subject to the constraints: B = 0 . , (11)0 . ≤ σ RGBB ≤ . , (12) | V RGBB − V RGBB, predicted | ≤ . , (13)where the third constraint has a V RGBB, predicted derived from the metallicity of the clusterand the V -band apparent distance modulus to the cluster (Harris 1996, 2010 edition), andthe best-fit relation for those two parameters obtained by a linear fit to the gold sample.The first two priors are absolute – no parameter space is explored outside the specifiedrange. The third prior is slightly relaxed, other values of V RGBB are explored, but with steep∆ (cid:96) penalties outside the specified range. To be included in the silver sample, GC CMDshad to first met one of three conditions, N RGBB, measured ≥ . N RGBB, predicted ≥ .
0, or both N RGBB, measured ≥ . N RGBB, predicted ≥ . N RGBB, predicted is calculated from the fitto EW RGBB in the gold sample. We then also required that the error in the peak brightnessof the RGBB be less than 0.05 mag. Larger values occur when the MCMC jumps betweenoverdensities in the magnitude distribution – when it is not statistically clear which featureis the bump. We relaxed the σ RGBB constraint to a maximum value of 0.12 mag ratherthan 0.09 mag for NGC 6316 and NGC 6440 due to their moderate differential reddening.GCs with severe patchy differential reddening such as NGC 6266 are not included. As theWFPC2 dataset has a heavier bias toward Disk/Bulge clusters, severe differential reddeningproved to be a limiting criteria for a number of clusters. In total, there are 48 clusters in thegold sample and 24 clusters in the silver sample. 37 of the gold sample and 18 of the silversample come from the ACS dataset, and the remainder come from the WFPC2 dataset.While we have used the parametrization discussed in this section before (Nataf et al.2011a,b), we recognize that it is a break from the great majority of the literature pertainingto the RGBB. Our investigation demonstrates that the RG luminosity function, the bright-ness peak, and normalization of the RGBB can be degenerate parameters (i.e. the errors ofdifferent parameters are correlated), and thus must be fit for concurrently rather than se-quentially. In light of this significant development in methodology, we independently discussour parametrization for both the brightness and normalization of the RGBB. 15 –Fig. 6.— Correlation diagram for a representative sample of points showing the distributionof the values of the RG+RGBB parameter limits from the MCMC for 47 Tuc. Parametervalues within 4 σ of the best-fit value are shown in magenta, those within 3 σ in blue, thosewithin 2 σ in green, and yellow for trials within 1 σ of the best-fit. Best-fit parameter valuesand the 1 σ errors are shown as a legend in the histograms, with correlations between differentparameters shown as a legend in the scatter plots. Whereas the value of I RGBB is largelyindependent of the other parameters, that is not the case for EW RGBB . 16 –
Fusi Pecci et al. (1990) presented a method of measuring and interpreting the RGBBbrightness that has since been broadly used. Their method was to log-integrate the lumi-nosity functions of cluster RG stars from both sides of the RGBB, and to measure the pointat which the cumulative distribution function breaks with that of the two linear fits. Thebrightness is then compared to that of the zero-age horizontal branch (ZAHB) at the positionof the RR Lyrae instability strip in V-band – the ∆ V ZAHBRGBB = ( V RGBB − V ZAHB ) parameter.For this work, as well as our other recent works (Nataf et al. 2011a,b), we use a maximum-likelihood method to fit for the parameters that has the advantage of being independent ofbin size and of fitting all the parameters concurrently rather than sequentially.There are issues with the use of RR Lyrae stars as an anchor. First, ZAHB are muchless likely to lie on the RR Lyrae instability strip in both metal-poor and metal-rich stellarsystems. Additionally, in composite stellar populations such as dwarf galaxies, the Galacticbulge or indeed many massive GCs, the stars in the RR Lyrae instability strip may bebiased toward a different subset of stars than the stars populating the RGBB. For example,the Galactic bulge RR Lyrae stars have a metallicity peak near [Fe/H] ≈ − ≈ − V ZAHBRGBB = ( V RGBB − V ZAHB ) in complex stellar populations bymodelling their full star-formation history, which they applied to several Local Group dwarfgalaxies.We suggest the use of two different comparative anchors for the brightness peak of theRGBB. First, following Cassisi et al. (2011), we compute wherever possible the difference inbrightness between the RGBB and the main sequence turn-off, ∆ V MST ORGBB = ( V MST O − V RGBB ).This parameter is more theoretically robust than ∆ V ZAHBRGBB , as it does not require assump-tions concerning the theory of post-main-sequence stellar evolution, such as neutrino energyloss. Moreover, for composite stellar populations, both the MSTO and the RGBB shouldbe representative of the dominant population. We also compute the brightness parameter∆ I RHBRGBB = ( I RGBB − I RHB ), where the mean brightness of the RHB is used as a benchmark. I -band is preferred due to the reduced evolutionary effects for the RHB in that bandpass(Girardi & Salaris 2001; Pietrzy´nski et al. 2010). For heavily reddened systems such as 17 –some positions of the Galactic bulge (Nataf et al. 2011a), the RGBB may be measurableeven when the MSTO falls at or below the photometric detection limit, whereas ∆ I RHBRGBB compares two populations at similar locations on the CMD. Additionally, the RHB is notonly rigorously and thoroughly investigated in theory (Girardi & Salaris 2001), it is the onlyHipparcos-calibrated standard candle (Stanek & Garnavich 1998; Groenewegen 2008). Formetal-rich populations, the RHB (as well as the MSTO) should share the RGBB’s propertyof being representative of the numerically dominant population.
Many prior studies of the RGBB in the literature have attempted to investigate starcounts on and near the RGBB using the R
Bump parameter, which is the ratio of stars inthe RGBB region V RGBB − . ≤ V ≤ V RGBB + 0 . V RGBB +0 . ≤ V ≤ V RGBB +1 . Bump parameter is also quite sensitive tophotometric incompleteness. Both characteristics can be traced to its arbitrary integrationlimits.For typical values of EW RGBB and B , 0.3 mag and 0.72 mag − , the number of stars inthe numerator of the R Bump parameter, NR
Bump , will be proportional to:NR
Bump = 0 . (cid:90) . − . exp (cid:20) . − V RGBB ) (cid:21) dVNR Bump = 0 . .
81 = 1 . . (14)Meanwhile for the denominator:DR Bump = (cid:90) . . exp (cid:20) . − V RGBB ) (cid:21) dV = 2 . . (15)Only (0.3/1.11) ∼
27% of the stars in the numerator correspond to the excess lifetime spenton the RG branch due to the RGBB. Even with a weaker-than-predicted RGBB, there wouldstill be a significant number of stars in that region, contributing their own source of noise. Tosee the consequences of this, consider the example of a well-populated CMD that is expectedto have ∼
100 stars in its RG luminosity corresponding to the excess lifetime due to theRGBB, more than the number in 66 of our 72 clusters. Such a system would then also beexpected to have 270 additional stars in its numerator, and 700 stars in its denominator.The signal to noise ratio of the excess in the RG luminosity function would then only be ameager 100/sqrt(100+270+700) ∼ Bump value of 0.81/2.10 ∼ Bump for 47 Tuc as having a value of (0 . ± .
05) – a 4.8 σ detection. Bycontrast, using our parametrization, we measure N RGBB = (122 . ± .
2) for 47 Tuc – an8.6 σ detection.The situation will then be much worse for the vast majority of GCs that have less well-sampled CMDs, and/or for the metal-poor clusters that have a lower value of EW RGBB .Further, a signal of (cid:46) σ is just for the zeroth order existence of the RGBB. The signalwill be much lower if one investigates first-order effects such as gradients due to age, he-lium and metallicity. Fundamentally, R Bump is a composite parameter of the parameters B and EW RGBB , with a heavy bias toward B . We consider both these parameters to beindependently interesting, and argue they should be fit as distinct parameters.The two normalization parameters we introduced in Nataf et al. (2011a) and study infurther detail here mitigate this issue. The EW RGBB and f HBRGBB , are the excess in the RGluminosity function due to the RGBB respectively normalized by the number density of RGstars per magnitude at the brightness peak of the RGBB, and the number of HB stars.As the RGBB is observed as an excess over the continuum luminosity function of the RGbranch, its best-fit normalization will always be degenerate with the parameters A and B .We reduce the impact of this degeneracy by including as many stars in the fit as we can whilesimultaneously leaving out contamination from the SGB, the HB, the AGB, and foregrounddisk contamination where present. We do not discard the statistically meaningful stars thatare either brighter than the RGBB by more than 0.4 mag, between 0.4 and 0.5 mag fainterthan the RGBB, or fainter than I RGBB + 1 . Bump does not generalize well to composite stellar systems. In our investi-gation of the Galactic bulge RGBB (Nataf et al. 2011a), we did not integrate up to stars 1.5mag fainter than the
RGBB . That region of the CMD is heavily mixed with foreground diskstars and bulge SGB stars. The integration limits need to be flexible in order to account forthe diversity of stellar populations in which the RGBB is observable and will be observablein the future.Photometric incompleteness can also be a concern. It is true that both of the catalogsused in this work have artificial star tests confirming completeness on the RG branch, butthere is value in having a methodology that could generalize well to other kinds of catalogs.For any smooth photometric completeness function, the parameter EW RGBB has the ad-vantage of not incorporating photometric incompleteness as a systematic error, since RGBBstars will have the same detection probability as the RG stars at sufficiently similar bright- 19 –ness. The systematic effect will be limited to reducing the value of B , an unfortunate butcontained issue. The case is different for the R Bump parameter. Since the fainter stars inthe denominator will be less frequently detected, photometric incompleteness will end upmasquerading as a stronger normalization for the RGBB.We briefly comment on a potentially confusing issue of terminology. In this paper, weare measuring the excess in the luminosity function of the RG branch at the position of theRGBB. This does not exactly correspond to the stellar evolutionary processes involved increating the RGBB. The evolutionary process involves stars moving up the giant branch,briefly becoming fainter and moving down, and moving up again, thus crossing the sameluminosity interval three times. All of the stars in the luminosity interval as well as near theinterval, modulo any sources of noise, will be experiencing the stellar processes involved, butonly some of the stars contribute to the excess in the number counts, the observable we label N RGBB . The remaining stars are observationally equivalent (within our parameterization) tohaving an underlying distribution of stars continuously moving up through the RG branchat the luminosity of the RGBB, though that is not what happens structurally.
Fitting an exponential law to the number distribution of RG stars as a function ofmagnitude has the simple physical explanation that it corresponds to a power-law as afunction of luminosity. An exponential fit to the number counts also corresponds exactlyto a linear fit to the log of the number counts. Previously, the RG continuum distributionhas been modelled by fitting a linear relationship between the log of the cumulative numbercounts and magnitude (Fusi Pecci et al. 1990; Zoccali & Piotto 2000). We prefer to usethe number density functions rather than the cumulative density functions because it isstraightforward to calculate the errors for the former. Cumulative distribution functions donot have straightforward error calculations because adjacent bins have correlated numbercounts.
4. The Main-Sequence Turnoff
We fit for the MSTO in all of the ACS GCs for which we have a measurement of theRGBB. We do not use the MSTO measurements of Mar´ın-Franch et al. (2009) since thoseare reported in the ( F F W − F W ) absolute magnitude plane rather than in the( I , V − I ) apparent magnitude plane required by our investigation. 20 –To measure the MSTO of each cluster, we first fit 2nd and 3rd degree polynomials tothe upper main-sequence of each cluster, with color being a function of magnitude. Thefits typically cover a luminosity range between 0.3 mag brighter than the MSTO and 0.7mag fainter, the boundaries are selected to comfortably include the MSTO but to excluderegions of the CMD that would distort the fit, such as the subgiant branch. 3- σ outliers arerecursively removed from fits with the fits then recomputed, though we removed 2- σ outliersin NGC 6171 and NGC 6624 due to their thicker main-sequences. For both polynomials, wetake the bluest point on the best-fit curve as an estimate for the MSTO, and we report theaverage of the two values as our measurement.The values obtained by the 3rd and 4th order polynomials had an average discrepancyof 0.01 mag in V and 0.0002 mag in ( V − I ). Unfortunately, it is difficult to quantifythe errors in these fits due to the existence of a few systematics. For example, we expectsome contamination from binary stars, though the most egregious examples of those areleft out of the fit by our removing of outliers. We show the results of our method for tworepresentative GCs, NGC 104 (47 Tuc) and NGC 1261 in Figure 7. The summary of ourMSTO measurements in Table 5. 21 –Fig. 7.— TOP: CMD of the relatively metal-rich, disk/bulge cluster NGC 104 (47 Tuc)shown, with the measured position of the MSTO, ( V − I, V ) = (0 . , . V − I, V ) =(0 . , .
5. The Horizontal Branch Stars
We estimate the number of HB stars in all GCs for which we have a measurement of theRGBB. We also compute the mean brightness of the RHB for 31 of the clusters for which theRHB is observed to be a well-populated, visually distinct component of the CMD. The errorin the mean brightness, σ I RHB , is taken to be the standard error in the mean, σ/ (cid:112) ( N RHB ).Two examples of RHB selection cuts, NGC 1261 and NGC 7089 (M2), are shown in Figures8 and 9. Additionally, the examples of 47 Tuc and NGC 362 are shown in Figures 2 and 22 –3. The RHB is conservatively selected by drawing a box where a clear RC is visible. GCswithout clear RCs do not contribute to the ∆ I RHBRGBB statistic.Due to the fact that the HB populates a specific region of the CMD, it is relativelystraightforward to count up the number of HB stars. However, as is clearly discerniblein Figures 8 and 9, blue stragglers, background (or foreground) contamination stars, andAGB stars are sometimes photometrically indistinguishable from HB stars. Fortunately,the intersection of those populations with that of the HB on the CMD never totals morethan a few percent of the HB population. Since the uncertainty in the number of RGBBstars is typically ∼ f HBRGBB .Zoccali & Piotto (2000) counted the HB stars in 26 of the GCs observed in the WFPC2survey (Piotto et al. 2002). Our investigations have 8 CMDs in common. We measured 168HB stars in NGC 1904 to their 177, 145 HB stars in NGC 5634 to their 146, 529 HB starsin NGC 5824 to their 520, 302 HB stars in NGC 6139 to their 299, 34 stars in NGC 6235 totheir 35, 133 HB stars in NGC 6284 to their 133, and finally 365 HB stars in NGC 6356 totheir 370. The level of disagreement is thus of order 2%, miniscule compared to the typical, ∼
10% error in N RGBB , or even the typical ∼
7% Poisson error in N HB .We recognize that there may be a small bias in our measurement of f HBRGBB due to thefact these
HST data are taken toward the cores of GCs. Due to chemically-distinct multiplegenerations (D’Ercole et al. 2008) and dynamical relaxation (Leigh et al. 2011), differentphases of stellar evolution should have slightly different occupation ratios at differing coreradii. It is difficult to assess the impact of these effects at this time since they are bothrapidly evolving fields.The HB characterization for the clusters NGC 6388 and NGC 6441 are modified due totheir complex morphologies. These are discussed in Section 9.3. 23 –Fig. 8.— ACS CMD for the intermediate-metallicity GC NGC 1261. The 231 HB stars areshown by open circles. The mean brightness for the RHB stars is I RHB = 15 .
94. Some MSstars have been removed from the figure to reduce image size. 24 –Fig. 9.— ACS CMD for the low-metallicity GC NGC 7089 (M2). The 720 HB stars areshown by open circles. Some MS stars have been removed from the figure to reduce imagesize.
6. Results: The Brightness and Color of the RGBB
We have measured the brightness of the RGBB in 55 of the clusters from the ACSdataset and 17 from the WFPC2 dataset, for a total of 72 measurements. All of the ACSclusters have measured values of the MSTO positions and thus values of ∆ V RGBBMST O and∆( V − I ) RGBBMST O . The relative brightness between the RGBB and the RHB, ∆ I RHBRGBB , for 22of the ACS clusters and 9 of the WFPC2 clusters, for a total of 31 measurements. All themeasurements discussed and used in this section are listed in Tables 4 and 5. The GCs 25 –NGC 2808, NGC 5286, NGC 6388 and NGC 6441 are not included in the fits due to theiranomalous RGBB properties.The dominant empirical trend, previously observed by several investigations (Fusi Pecciet al. 1990; Zoccali et al. 1999; Riello et al. 2003; Rey et al. 2004; Di Cecco et al. 2010;Monelli et al. 2010; Nataf et al. 2011a), is the declining luminosity of the RGBB withincreasing metallicity. The M V,RGBB increases by ∼ ≈ − ≈ − ∼ I RHBRGBB = ( I RGBB − I RHB ), we find a variation of 1.1 magover a metallicity interval of ∼ M V,RGBB , a fit weighted by the statistical error in the brightness measurements has a χ =3785. The clear interpretation is that the dispersion due to errors in the input metallic-ities, apparent distance modulus in V , and undiagnosed second parameters are substantially larger than the statistical error in the measurement of the brightness. We adjust the errorsusing the following prescription: σM V,RGBB (cid:48) = σM V,RGBB + (cid:20) dM V,RGBB d [M / H] (cid:21) σ [M / H]2 + δσV RGBB , (16)where δσM V,RGBB is the noise added due to undiagnosed second parameters. The fit obtainedis M V,RGBB = (0.600 ± ± δσM V,RGBB = 0.077mag is needed to yield a fit with χ = 66 (68 measurements and 2 parameters). This scattercould also be due to errors in the values of the V -band apparent distance modulus summa-rized by Harris (1996, 2010 edition), as well as the fact that many of the distance estimatescome from different methods, rendering the list of distance moduli used heterogeneous. Thescatter in M V,RGBB could also be due to an additional scatter of ∼ (0 . / .
737 = 0 . − in the metallicity [M/H] above that which is assumed in this work. Another possibilityare variations in age or initial helium abundance (Cassisi & Salaris 1997).For the brightness relative to the MSTO, we obtain ∆ V RGBBMST O = (3.565 ± − ± χ =49 is δ ∆ V MST ORGBB =0.072 mag. Age could be the source of this extra scatter. Stellar evolution models predictthat older clusters should have larger values of ∆ V RGBBMST O , at a rate of ∼ V RGBBMST O can thus be entirely explained by anage scatter of ∼ δ ∆ V MST ORGBB =0.072 mag is almost as large as δσM
V,RGBB =0.077 mag, as the latter would be expected to far larger since it is directly dependent on 26 –estimates of total extinction and distance to globular clusters. The similarity of the two scat-ters suggests that the apparent distance modulus is precise to a level of √ . − . ≈ V − I ) color decreases with increasing metallicity. As metallicityincreases, both the MSTO and RGBB become redder, but the MSTO becomes redder faster. 27 –Fig. 10.— In all 4 panels, blue points are the gold sample measurements in the ACS clusters,red points are the gold sample measurements in the WFPC2 clusters, and green points comefrom the combined silver sample. TOP-LEFT: ∆ V MST ORGBB = ( V MST O − V RGBB ) for all theACS GCs. TOP-RIGHT: ∆( V − I ) MST ORGBB = ( V − I ) MST O − ( V − I ) RGBB for the ACSGCs. BOTTOM-LEFT: M
V,RGBB for all clusters with an RGBB measurement, using the V -band apparent distance modulus from Harris (1996, 2010 edition). BOTTOM-RIGHT:∆ I RHBRGBB = ( I RGBB − I RHB ) from 31 GCs for which we measured the RHB mean brightness.For ∆ I RHBRGBB , we obtain ∆ I RHBRGBB = (0.123 ± ± δ ∆ I RHBRGBB = 0.051 mag. δ ∆ I RHBRGBB is smaller in quadrature than δ ∆ V MST ORGBB by a value of 0.051 mag, so it may appear to be a more stable variable. However,it is calculated from a sample of 28 rather than 51 GCs, and over a smaller metallicityrange. Moreover, it has a hidden selection bias. Whereas all metal-rich clusters have a red 28 –component to their HB, only some of the intermediate-metallicity ([Fe/H] ∼ −
7. Results: The Number Density of RGBB Stars
We measure f HBRGBB and EW RGBB in 55 of the ACS GCs and 17 of the WFPC2 GCs for atotal of 72 measurements. The measurements are shown in Figure 11, and all measurementsdiscussed in this section are listed in Table 6. The GCs NGC 2808, NGC 5286, NGC 6388and NGC 6441 are not included in the fits due to their anomalous RGBB properties.The dominant empirical trend is the increasing number counts of the RGBB with in-creased metallicity. There are two factors involved. The first is that the RGBB gets moreprominent relative to the underlying RG branch at increased metallicity. As [M/H] is in-creased from − EW RGBB increases by a factor of ∼ − f HBRGBB increases by a factor of ∼ N RGBB , as well as expectedfluctuations due to hidden second parameters such as variations in age, CNO abundances,initial helium abundance, and other factors, since the strength of the RGBB will not be afunction of metallicity alone (Nataf et al. 2011a). For example, a fit for f HBRGBB with respectto metallicity weighted purely by the statistical error measurements yields χ = 96.8 for 68measurements and 2 parameters. It is clear that whereas metallicity is the first parameterof RGBB strength, it is not the only parameter.We add, in quadrature, a systematic noise to the error δσf HBRGBB with the followingprescription: σ (cid:48) f HBRGBB = σf HBRGBB + (cid:20) df HBRGBB d [M / H] (cid:21) σ [M / H]2 + δσf HBRGBB , (17)And we compute a weighted least squares using the combined error σ (cid:48) f HBRGBB , W i = 1 / ( σ (cid:48) f HBRGBB ) , (18)where we adjust the value of δσf HBRGBB until we obtain a fit with χ = 66. The analo-gous procedure is performed for EW RGBB . We thus measure f HBRGBB = (0.111 ± ± δσf HBRGBB = 0 . EW RGBB = (0.248 ± ± χ = 62.1 29 –for 68 measurements and 2 parameters, implying a perhaps surprising lack of evidence forhidden second parameters. Nataf et al. (2011a) found a predicted variation of dEW RGBB /dt = 0.008 mag Gyr − for a 10 Gyr old population with [M/H]=0, an effect which could likely betoo small to infer given the large statistical errors. More theoretical investigation is neededto ascertain whether the value of EW RGBB is slowly varying with age across the range ofmetallicities probed in this work. Moreover, our fits may have too great a degree of freedom.As we will show in the next section, the data is consistent with the RG luminosity parameter B being a constant of stellar evolution.Fig. 11.— For both panels, blue points are measurements in the gold sample of ACS clusters,and red points are measurements in the gold sample WFPC2 clusters. The points from thesilver sample are shown in green. The best-fit line is shown for both relations. TOP: EW RGBB for 63 GCs as a function of metallicity. BOTTOM: The fraction of RGBB starsto HB stars as a function of metallicity. 30 –
8. Results: Other Parameters
We briefly discuss the other measured stellar evolution parameters, the exponential slopeof the RG luminosity function B , and the magnitude dispersion of the RGBB σ RGBB . Theparameters discussed in this section are listed in Table 7. We do not include GCs from thesilver sample in our fits for these two parameters, as these GCs had these two parametersconstrained to match the distribution in the gold sample. Therefore, the fits in this sectionare done purely on the 44 measurements in the gold sample, with NGC 2808, 5286, 6388and 6441 removed as before.Fig. 12.— TOP: B , the exponential slope of the RG luminosity function, as a function of[M/H]. BOTTOM: The measured magnitude dispersion of the RGBB.We find that B has no significant dependence on metallicity. For a least squares weighted 31 –by the errors in the measurement of B and using only the measurements in the gold sample,we obtain B =(0.715 ± ± χ =47.2 for 44 measurementsand 2 parameters. The slope is detected at the ∼ σ level – it is not significant. A fit tothe weighted mean value B = 0 .
719 yields a χ =47.7. In light of the potential systematicspresent such as varying amounts of disk contamination in clusters, we argue that there is noconvincing evidence for a relation with metallicity. The mean of the measurements weightedby the errors in B is (0.715 ± B is predicted by stellar models, as a straightforward consequence ofthe relation between the total luminosity of a star on the RG phase and the mass of theHe-core (Paczynski 1984; Castellani et al. 1989). The prediction of Castellani et al. (1989),that B = 0 . ± .
04 across the age and metallicity range spanned by the Milky Way GCsystem is confirmed by our investigation.We also measure the relation for the magnitude dispersion of the RGBB. σ RGBB =(0.051 ± ± χ = 32 . σ RGBB may be useful in interpreting the bumps of specific GCs, one should be cautious ininterpreting the global relation.
9. Interesting Clusters: NGC 2808, 5286, 6388, and 6441
We comment on the interesting anomalies we measure in the RGBBs of the GCs NGC2808, 5286, 6388 and 6441.
NGC 2808 is known to have at least three main sequences, from which Piotto et al.(2007) estimates two helium-enhanced populations, each with ∼
15% of the cluster stars.Their inferred enhancements are ∆Y ∼ χ values are 1.50, 0.08, 3.49,0.66, and 0.53 respectively. Only NGC 2808 exhibits a strong detection of a skewed RGBB.Its parameters change from ( V RGBB , σ RGBB , EW RGBB ) = (16.235, 0.092, 0.347) to (16.219,0.112, 0.303) with a strongly negative skew of − . +0 . − . . The negative skew is exactlywhat one would expect if there were a relatively small number of brighter (helium-enhanced)RGBB stars. Moreover, it also contrasts to the expectation from models (Cassisi et al. 2002;Nataf et al. 2011b) that the RGBB of a single-metallicity, single-age population be positivelyskewed, i.e. with its mode at its bright end and a long tail to fainter luminosities (and thushigher values of magnitude). We show the CMD and magnitude histogram in Figure 13. 33 –Fig. 13.— LEFT: CMD of NGC 2808 from the ACS sample, zoomed in on the RG branchat the location of the RGBB. RIGHT: Magnitude histogram of 3308 RG+RGBB stars withfit. Unlike other well-populated clusters, the RGBB is much better fit by a skew-normaldistribution than by a standard normal distribution, consistent with findings that the clusterhas an extreme helium-enhancement subpopulation (Piotto et al. 2007).Consistent with the measurement of the third moment (skewness) of the RGBB’s mag-nitude distribution is that of the second moment (dispersion). At the cluster’s metallicity[M/H]= − σ RGBB = 0.047 ± σ RGBB = 0 . ± .
013 without a skew and 0.112 ± ∼ A I ∼ Y (cid:38) .
10, due to the magnitude separation between the RGBBs of the 34 –first and second generations.The second and third moments of the NGC 2808 RGBB are in a rather startling agree-ment with stellar evolution predictions and observations of other stellar populations of thecluster. A clear prediction is that the brighter RGBB stars should be oxygen-poor andsodium-rich relative to the fainter RGBB stars, since those are thought to be the mosthelium-enriched (Piotto et al. 2007; Valcarce & Catelan 2011). With a population of ∼ global brightness differenceof ∼ . ± .
042 mag difference in V for 1368 stars from 14 GCs characterized as eitherprimordial, or intermediate/extreme based on their [Na/Fe] abundances.
NGC 5286 may be displaying a split RGBB. We show the CMD and magnitude his-togram in Figure 14. There are two peaks, one at V RGBB = 16 .
287 and a second, smallerpeak ∼ M V,RGBB for the brighter, more populated peak is ∼ V MST ORGBB = 3.87, is larger than the expected 3.72by a similar amount. The number counts are also low. At [M/H]= − .
43, the predictedvalue of EW RGBB from the fit is 0.212 ± ± ± ∼ B − V , V ) CMD of NGC 5286 from ground-based datashown in Figure 8 of Zorotovic et al. (2009), where it is quantified as a ∼ σ effect. Theweighted mean of the brightnesses would yield the approximate expected values of ∆ V MST ORGBB and V RGBB , and the sum of their normalization would do likewise for EW RGBB and f HBRGBB .One solution is for this cluster to be an extreme member of the class formed by NGC1851 and NGC 6656 (Milone et al. 2011). Those two GCs do not show the behavior expectedfrom a spread in helium, but spectra demonstrate variations in heavy elements such as ironand yttrium. The two peaks are matched by theory if two thirds of the stars (the brighterRGBB) are in the first generation, and the remainder are in the second, [Fe/H]-enhancedsecond generation. Zorotovic et al. (2010) inferred the [Fe/H] of the RRc stars from their 35 –Fourier components. The [Fe/H] values are − − − − − − − − − − − − − − − − ± ∼ − − − ± We find a few peculiarities in the GCs NGC 6388 and NGC 6441. Their relative mea-surements are not consistent with their identical spectroscopic metallicities, the brightnessof the RGBB indicates the distance may be underestimated, and their low number countsare consistent with the presence of an extreme, helium-enhanced population.We first state the different HB calibration selection used for these two GCs. For theseGCs, a quantitatively significant portion of the red end of their RHB merges with the RGbranch. Due to the fact these are two of the most interesting GCs in the Galaxy (Yoon etal. 2008), it is critical to adapt our method to get these right. We first sum the numberof point sources toward the regions of the CMD that are clearly dominated by the HB. Wethen fit for a second
Gaussian in the combined RG+RGBB+RHB branch to measure thered end of the RHB. The total HB population is then the sum of the HB stars counted in therest of the CMD and the best-fit normalization value of the RHB component along the RGbranch. We take the weighted mean (by number counts) for the brightness of the RHB. It isnecessary to do a double-Gaussian fit if only to have a proper fit of the RG+RGBB+RHBbranch. Without doing this the parameters of the RGBB would be severely compromised.The procedure is visually summarized in Figure 16. 38 –Fig. 16.— LEFT: CMD of NGC 6441 in ACS data, with the color-magnitude selectioncontours for 5776 RG+RGBB+RHB stars and 1280 pure RHB stars shown. The color ofthe RG branch at the RGBB is 0.985. The weighted mean brightness of the 1280 RHBstars in the pure RHB box and the 433 mixed with the RG+RGBB stars is I RHB = 16 . I RGBB = 16 . ± . EW RGBB = 0 . ± . − [M/H] = 0.01 ± I RHBRGBB | − ∆ I RHBRGBB | = 0.109 ± ∼ M V,RGBB even as they have values of ∆ V MST ORGBB that agree with the global trends to within ∼ V RGBB value 0.32 mag fainter thanthat predicted by the fit, and the deviation is 0.41 mag for NGC 6441. These deviations arefar larger than could be reasonably attributed to variations in age or to errors in metallicity, 39 –so we conclude that a combination of the distance and reddening to the clusters listed inHarris (1996, 2010 edition) are underestimated. The apparent distance modulus is obtainedfrom RR Lyrae measurements (Pritzl et al. 2001, 2002). If the RR Lyrae stars are helium-enhanced, then they will be brighter than that predicted by the standard [Fe/H]-M V relationfor RR Lyraes (Caloi & D’Antona 2007), leading to a severe underestimate of the distance.We argue this to be the case here.Enhanced helium enrichment may play a role in the RGBB star counts for this cluster.The f HBRGBB derived from the relation to all the GCs at [M/H]= − ± ± ± ∼ σ and 3.3 σ respectively. Caloi & D’Antona (2007)argued that at 15% of the stellar content of NGC 6441 had to be extremely helium enhanced(∆Y ≥ ∼
30% were significantly brighter, andfell between the RGBB and the RHB or perhaps even in the RHB region of the CMD, theywould not be captured by our measurement. A similar scenario may be at work for NGC6388.Both GCs have values of σ RGBB that are larger than expectations from the fit to all theGCs. However, it is difficult to interpret these excesses due to the large uncertainty in thedifferential reddening toward these GCs (Yoon et al. 2008).Unfortunately, disk contamination could in principle corrupt the measurements in NGC6441, as can be clearly seen in Figure 16. We experimented with various selections for the RGbranch, shifting the limits for both color and magnitude, and our measured values of f HBRGBB did not change by more than 1%, nor was the brightness peak shifted. The measurementsalso remained the same when we removed the inner half of the GC stars, to test for effectsdue to photometric noise. As well as we can test with the available data, the measurementof a discrepancy appears robust. 40 –
10. Comparisons to the Milky Way Bulge and M32
The first measurement of the RGBB in the Milky Way bulge was discussed in Nataf etal. (2011a). The detection was later confirmed by Clarkson et al. (2011) and Gonzalez et al.(2011b). Subsequent analysis leads us to adopt the parameters f RCRGBB | Bulge = 0.201 ± I RCRGBB | Bulge =0.737 ± − σ RGBB = 0.220 ± σ RC = 0.241 ± HST , they report a value of N RC = 1422 .
8, and N RGBB = 219 ±
51, for f RCRGBB | M32 = 0.151 ± F W RCRGBB | M32 =0.56 ± I RCRGBB | M32 =0.56 ± f HBRGBB than thatexpected from the Galactic GC system, and that the bulge has a lower value than expectedfor ∆ I RCRGBB | Bulge once composite metallicity effects are taken into account. Because thesetwo measurements were obtained with different instrumentation, different methodology, andother different systematics such as crowding, their comparable deviation from the relationsfor Galactic GCs is independently derived, and may be due to a similar evolution.
11. Application: Empirically-Motivated Prediction of the Galactic BulgeRGBB Properties
The Galactic bulge is a complex stellar population for which different analyses lead todifferent results for parameters as fundamental as age (e.g. Bensby et al. 2011; Clarksonet al. 2011). In this section, we compute what the Galactic bulge RGBB should look likerelative to the RC toward two distinct sightlines given the assumption that the bulge RGBBpopulation will follow the same relations with metallicity as the Galactic GC system. Wewill show this to be a unique probe of the Galactic bulge stellar population. 41 –Our comparison between the measured and predicted values for sightlines close to theplane are summarized in Table 1, and that for the sightline ( l, b ) = (1 , −
6) in Table 2. Wespecify that while our calculation is done for a bulge MDF and compared to Galactic bulgeobservations, we expect the methodology to generalize well to future observations of theRGBB in kinematically-selected Gaia CMDs of the thin disk, thick disk and halo of the MWtaken with Gaia observations (Lindegren 2010).We use the [Fe/H] and [ α /Fe] of 204 bright RG stars toward Baade’s window respectivelymeasured by Zoccali et al. (2008) and Gonzalez et al. (2011a). The mean metallicity for thesebright red giants is [M/H]= +0.047. However, for a composite stellar population such as theGalactic bulge, the metallicity distribution of one population (e.g. bright red giants) willnot match the metallicity distribution of another population (e.g. RGBB stars), since therelative lifetimes are a function of metallicity. We correct for these two effects by using therelations from Renzini (1994):∆Log t HB ≈ +0 . . , (19)∆Log t RGB ≈ − . − . , (20)These relations predict that an HB star with [M/H]=0 and Y=0.27 will have a lifetimet HB ∼
17% longer than one with [M/H]= − reduction in lifetime of ∼ HB ≈ +0 . . , (21)∆Log t RGB ≈ − . − . . (22)The mean metallicity of RC stars has to be higher than that of the RG stars, since metal-rich RG stars have suppressed lifetimes whereas metal-rich HB stars have enhanced lifetimes,with the effect amplified by the monotonic relation between metallicity and initial heliumabundance. A weighted mean must be computed: W i,HB = exp (cid:20) (ln 10)(0 .
06 + 0 . / H] + (0 .
50 + 1 . (cid:21) , (23) W i,HB = exp (cid:20) . / H] + 0 . [M / H] − (cid:21) . (24)The estimated mean metallicity of bulge RC stars is thus [M/H]= +0.074, which is expectedto be a little higher than that of the bright red giants. 42 –We then compute the mean metallicity distribution of the RGBB distribution by weight-ing over their respective relative lifetimes. For each RGBB star, we compute: W i,RGBB = f HBRGBB | [M / H] ∗ exp (cid:20) . / H] + 0 . [M / H] − (cid:21) (25)where the first factor of W i,RGBB is the value of f HBRGBB | [M / H] at that metallicity, the first-orderapproximation to the relative lifetime. The second factor corrects for the fact that metal-richstars are numerically suppressed in the RG metallicity sample, as well as the fact that thenormalization N HB is enhanced at the high-metallicity end. The weighted-mean metallicityof the RGBB stars is thus [M/H] RGBB = +0.108. The predicted fraction is then: f RCRGBB | Bulge = f HBRGBB | [M / H]=0 . ∗ exp (cid:20) . . (0 . − . − (cid:21) ∗ (184 / . (26)The second factor corrects for the lower mean metallicity of the HB relative to that of theRGBB. The third factor, 184/180, accounts for the small number of BHB+RR+RHB starsthat won’t be included in a CMD selection box for the RC, estimated using HST propermotions toward the SWEEPS field (Clarkson et al. 2011). The term may be even largerif the bulge has an undiscovered extreme BHB population, a plausible outcome due to theUV-excess observed toward ellipticals and the bulges of disk galaxies (Terndrup & Walker1994). The derivation yields f HBRGBB | Bulge = (0.279 ± I at a rate of 0.2 mag dex − , a prediction confirmedin observations of both the local Hipparcos population (Udalski 2000) and extragalacticsystems (Pietrzy´nski et al. 2010).We take the weighted mean of the brightness distribution of RGBB∆ I RCRGBB | Bulge = W i,RGBB ∗ (∆ I RHBRGBB | i + 0 . / H] | i − . , (27)where the first term gives the first-order weighted sum over the predicted brightness differ-ences, and the second term corrects for the second-order effect of the RC becoming fainterwith increased metallicity, by bringing it back to its brightness at [M/H]= +0.074. Wethereby obtain a mean brightness of ∆ I RCRGBB | Bulge =0.767 ± I RCRGBB is shifted to 0.837mag, and f RCRGBB drops to 0.241. The predicted dispersion for the RGBB is 0.258 mag. Inthe second case (bottom panel of Figure 17), ∆ I RCRGBB is shifted to 0.844 mag, and f RCRGBB shifts to 0.259. The predicted dispersion for the RGBB is 0.280 mag. Adding an RC shiftsthe measured RGBB peak to a fainter luminosity and reduces its number counts. Adding anRG luminosity function then mostly restores the RGBB number counts. This is due to thefact that a symmetric Gaussian is being fit to a skewed RGBB. The Gaussian then “scoops”up some of the RG stars as a byproduct of enforcing it symmetric profile.These parameters are inconsistent with those measured in Nataf et al. (2011a) andrevised in Section 10. The brightness peak is brighter by 0.1 mag, the brightness dispersionis lower, and the number counts are smaller by 20%. This presents a strong case that theinput physics for the bulge stellar population (age, helium, etc) are different from those ofthe Galactic GC system. The measured and simulated luminosity function for the Galacticbulge, with their stark differences, are shown in Figure 17. 44 –Table 1: Observable parameters for the RGBB toward a triaxial ellipsoid sightline ofthe Galactic bulge. The second column lists the measured values. The third columnpredicts the values of the brightness, normalization and brightness dispersion for the RGBB.The fourth column predicts what values would be measured by the double-Gaussian method.Parameter Measured Predicted Predicted with RC+RG+RGBB LF∆ I RCRGBB ± ± ± f HBRGBB ± ± ± σ RGBB ± ± ± l, b ) = (1 , − I RCRGBB ± ± f HBRGBB ± ± σ RGBB ± ± I . Panel 3: Combined predicted brightness distribution for theRGBB and RC. Panel 4: Predicted brightness distribution with a RG luminosity function. 46 –Fig. 18.— TOP: Measured luminosity function for the Galactic bulge RC+RGBB+RGbranch. BOTTOM: Predicted luminosity function for the Galactic bulge RC+RGBB+RGbranch given the bulge metallicity distribution and the metallicity relations measured inGalactic GCs.We repeat the exercise for the metallicity distribution toward ( l, b ) = (1 , − − l, b ) = (1 , − I (Nataf & Udalski 2011). The meanbrightness of the RGBB stars is 0.658 mag fainter than the RC, with f RCRGBB = 0.265. After anRG+RC luminosity function is added, ∆ I RCRGBB = 0.745, σ RGBB = 0.234 mag, and f RCRGBB = 48 –0.256. If the Galactic bulge RGBB indeed has over 20% of the numbers counts of theGalactic bulge RC within the Milky Way’s X-wings, then it must be taken into accountwhen modelling those sightlines. In particular, it could explain why there is a low measureddifference in radial velocity distributions between the two RCs (De Propris et al. 2011):perhaps the brighter RGBB is heavily mixed with the fainter RC.
We run a few illustrative stellar models using the Yale Rotating Evolution Code (YREC)with diffusion (Sills et al. 2000; Delahaye et al. 2010) and empirically calibrated bolometriccorrections (Pinsonneault et al. 2004; An et al. 2007). The mixing length is set to α = 1.922to match current data of the solar radius, luminosity, and atmospheric metals to hydrogenratio (Grevesse & Sauval 1998). We test two hypotheses that might explain the increasedbrightness and decreased number counts of the Galactic bulge RGBB relative to the GalacticGC calibration. The first is that the Galactic bulge has enhanced helium enrichment withouta younger age, and the second is that the Galactic bulge has a younger age with standardhelium enrichment. The results are summarized in Table 3 and shown in Figure 20.Table 3: Observational discrepancy compared to predicted evolutionary effects of enhancedenrichment helium and a younger age.Parameter Observational Discrepancy ∆ Y = +0 .
06 ∆t = − I RCRGBB − ± − − . f HBRGBB − ± − − ∼ ± ∼
30% reduction in the lifetime of the RGBB, but theage modification yields a change in brightness 3 × that obtained by increasing the helium. 50 –Since there are three measurable parameters for the Galactic bulge RGBB, the prospectsof tightly constraining both the age-metallicity and helium-metallicity relations, to yield aGalactic bulge age-helium-metallicity relation, are decent. The prospects improve further ifthe RGBB properties are found to be measurable toward multiple sightlines of the Galacticbulge that differ in their metallicity distributions. However scientifically satisfactory it may be to have the GC relations with metallicityfor the RGBB-RHB pair with which to to construct a null hypothesis to test for the bulge,it must be pointed out that this anchor is itself imperfect, as it is not precisely known. Asignificant scientific concern is the underlying foundation of this approach – the assumptionthat there are pertinent mean relations for the GCs. These are diverse stellar systems withsignificant variations in self-enrichment profiles (Carretta et al. 2010; Caloi & D’Antona 2011;Valcarce & Catelan 2011), in age (Mar´ın-Franch et al. 2009; Dotter et al. 2011), and otherproperties. Milone et al. (2011) used observations of 47 Tuc in 9 bandpasses to estimatethat ∼
70% of the stars are helium and nitrogen enriched as well as oxygen deficient. If thisis found to be the norm for the Galactic GC system, it will be necessary to incorporatethe expectation that the RGBB in Galactic GCs should be a little brighter, and a littleless-populated, than from a population with canonical abundance patterns.The weights used are not a significant source of systematic error. We repeat the samecalculation with the nearly pure RC spectroscopic sample measured by Hill et al. (2011),which does not require weights derived from stellar evolution models as it is already anchoredat the core helium-burning phase. We find that ∆ I RCRGBB increases by 0.040 mag, and f RCRGBB increases by 0.006. The source of this difference is that the sample of Hill et al. (2011)report slightly higher metallicities, which they speculate in their Section 4.1.4 may be due toupgrades in their spectroscopic reduction procedure. In this paper we have used the combinedsample of Zoccali et al. (2008) and Gonzalez et al. (2011a) to have a uniform sample for thesightlines toward the triaxial ellipsoid component of the bulge and the X-shaped componentof the bulge.The predicted value of f RCRGBB | Bulge may be artificially decreased by systematic effects.Hill et al. (2011) argue that spectroscopy of bulge giants may systematically underestimatethe metallicities of the most metal-rich stars. If true, this would increase the predicted valuesof ∆ I RCRGBB and f RCRGBB , increasing the discrepancy with the observed values.We also point out that the metallicity distribution of the bulge is turning out to be 51 –more complex than previously assumed, as it is bimodal, and possibly more complex. Hillet al. (2011) found two peaks in their sample, one at [Fe/H]= − σ RGBB .Bensby et al. (2011), using observations of microlensed Galactic bulge dwarf and subgiantstars, also find two peaks, but that are more broadly separated than those measured fromRC stars, at [Fe/H]= −
12. Discussion and Conclusion
In this work, we have introduced and justified a scientifically robust parametrizationwith which to study the RGBB and the associated RG luminosity function. The relevant pa-rameters can be fit concurrently rather than sequentially, and the use of maximum-likelihoodestimation removes any concern that the size and position of bins could distort the outputparameters. By combining this parametrization with the photometry from the ACS GCsurvey (Sarajedini et al. 2007) and that of the WFPC2 GC survey (Piotto et al. 2002), we fitfor the brightness and color of the RGBB in 72 GCs, the brightness and color of the MSTOin 55 GCs, and the brightness of the RHB in 31 GCs. We also fit for the strength of theRGBB, EW RGBB and the number of HB stars for all 72 GCs. There are several empiricalachievements in this work.This is the most robust investigation of RGBB star counts in GCs. Measurements of N RGBB reach precisions of 10% in the most populous clusters, and the mean relations for EW RGBB and f HBRGBB are determined to 4% accuracy. Previous investigations had relied onthe R
Bump parameter, a composite measure of the RG luminosity function and the strengthof the RGBB, that has a lower signal to noise ratio. The RGBB in 47 Tuc, previouslydetected with ∼ σ -significance, is now detected with ∼ σ -significance due to this differentparametrization. Measurements of the strength of the RGBB feature are now on firm-enoughfooting that it is in itself a tool with which to precisely compare GC observations to stellarmodel predictions.We also compute predicted values of ∆ I RHBRGBB and f RCRGBB for two Galactic bulge sight-lines given the assumption that these parameters have the same functional dependence onmetallicity in the bulge as they do for GCs. The results are not consistent with those foundin Nataf et al. (2011a) and revised in this work – the predicted RGBB luminosity is fainter, 52 –broader in magnitude spread, and more significant in number counts. As discussed in Natafet al. (2011a), one path to resolve this discrepancy is to posit enhanced helium-enrichmentfor the Galactic bulge. A higher value of ∆Y/∆Z would make the RGBB stars brighter,thereby decreasing the size of the derivative of brightness with metallicity, and it would alsodecrease the lifetime of the RGBB. Enhanced helium for the bulge is also a prediction ofchemical evolution, due to the enhanced α -element abundance toward the bulge (Catelan2007). The question of exactly how much helium is needed will be tackled in a future paper,where we will compare these results to stellar model predictions.Our analysis of number counts may lead to a resolution of a longstanding issue in RGBBastrophysics, that the observed brightness of the RGBB in GCs is ∼ α -capture elements could adjust the predicted number counts of the RGBB tono more than σ R Bump ∼ ∼
3% uncertainty in the lifetime of the RGBB.These predictions of stellar evolution are now testable.In the next decade, large-scale surveys and improved-instrumentation will further thedepth of astrophysical research accessible with the RGBB. Observations of this galaxy bythe Gaia mission will allow detailed investigations of the RGBB in the disk of this galaxy.Meanwhile, if 30 meter telescopes are built, higher-quality CMDs of GCs will be availablethroughout the local group. Though we do not expect split main-sequences to be as observ-able as ω Cen is using
HST (Bedin et al. 2004) , a split RGBB should be observable towardthose kinds of systems in Andromeda or Triangulum if they exist. Indeed, the split RGBBof ω Cen can be viewed from ground-based, 1-meter telescopes without adaptive optics (Reyet al. 2004). As much as the astrophysics accessible with
HST observations of the RGBB inGCs is an upgrade over what was previously available, we forsee even more significant gainsin the coming decade due to observational efforts listed above.We thank the referee, Santi Cassisi, for his thorough analysis of the text, which has led
53 –to a substantial improvement in the manuscript. DMN and AG were partially supported bythe NSF grant AST-0757888. DMN was partially supported by the NSERC grant PGSD3-403304-2011. The OGLE project has received funding from the European Research Councilunder the European Community’s Seventh Framework Programme (FP7/2007-2013) / ERCgrant agreement no. 246678 to AU. We thank Aaron Dotter for helpful discussions. Finally,we thank the referee for his helpful suggestions.
A. Testing our Methodology with Monte Carlo Methods
We use Monte Carlo simulations to determine the statistical robustness of our method-ology. We compare the relative diagnostic power of the parameter EW RGBB and R
Bump , andwe investigate whether or not out maximum-likelihood approach gives reliable estimates ofthe luminosity function parameters and their errors.We sample from a population of stars drawn from the probability density functioncharacterized by Equation (8). We set B = 0 . EW RGBB = 0 . σ RGBB = 0 .
05, and m RGBB = 0, where we use the generalized notation m
RGBB for this section, and not I RGBB or V RGBB as used elsewhere in this paper, as the RGBB could in principle be investigatedwith other bandpasses. The stars are distributed within the range m RGBB − . ≤ m ≤ m RGBB + 7 .
0, with the total number of stars distributed log-uniformly between ∼ ∼ ∼
3% of the stars within the range m RGBB − . ≤ m ≤ m RGBB + 2 . N RGBB ≥ .
0. Over the 3,000 runs of the simulation,half have their MCMCs run without priors, and the other half have the same priors as thosewe imposed on the silver sample.The larger initial range is used so that bins in the selected range have uncorrelatedrather than anti-correlated number counts. In a real stellar population, the number of starsin a magnitude interval ( m , m + δm ) is independent of the number of stars in anotherinterval ( m , m + δm ) – both populations are distributed as their own independent Poissonrandom variables. This will not be true in a simulation with a fixed total number of stars,where 1 additional star in 1 bin means 1 less available star for the other bins. We areeffectively breaking this anti-correlation by sampling from a population reservoir that is ∼ A.1. The Relative Diagnostic Power of the EW RGBB and R Bump
Parameters
We find that the parameter EW RGBB averages approximately twice the statistical sig-nificance of the R
Bump parameter. The behavior of the two parameters is shown in Figure21. That the ratio of significances would be near 2 is not surprising. We demonstratedin Section 3.2 that for a typical RG+RGBB luminosity function, ∼
73% of the stars in theinterval V RGBB − . ≤ V ≤ V RGBB + 0 . ∼ Bump is in fact the expected outcome ofeffectively diluting the sample size by a factor of 4. Other sources of error, such as that ofthe denominator in the R
Bump parameter, are also present but they are not dominant.The behavior of R
Bump at low number counts is particularly devastating. The statisticalsignificance is frequently below zero, which would imply an unphysical negative normalizationfor the RGBB. The significance only reaches 1 in the 50th percentile when N RGBB surpasses10. 55 –Fig. 21.— The statistical significance of the RGBB given the choice of parametrization, asa function of the true number of RGBB stars in the model distribution. Filled black squaresdenote the median statistical significance for EW RGBB with the error bars denote 32nd and68th percentiles. Similarly for the empty black squares and R
Bump . A.2. The Inferred Population Parameters Versus the True PopulationParameters
We find that the maximum likelihood approach yields an unbiased estimator for EW RGBB at large number counts, but one which is biased toward higher inferred values of EW RGBB for N RGBB (cid:46)
10. The bias disappears if we impose the same priors that we imposed onthe silver sample, which was constructed out of GCs that had a measured best-fit value N RGBB ≤ m RGBB is found to be an unbiased estimator of thetrue value of m RGBB . However, the scatter increases for lower values of N RGBB . At lownumber counts, any methodology is at risk of fitting to other peaks in the distribution thatarise from statistical fluctuation. The lack of bias is due to the fact that these other peaksneed not be either fainter or brighter than the true peak of the RGBB. 56 –Fig. 22.— TOP: The distribution of differences between the value of EW RGBB inferred bythe MCMC, and the true value from which the histogram is constructed. Error bars denote32nd and 68th percentile. Empty circles denote the distribution of inferences without priors,filled circles with priors. BOTTOM: Same as top, but for the parameter m RGBB .Both parameter comparisons demonstrate the urgent need for a broad investigation ofthe RG+RGBB luminosity function over the full pertinent range of ages, metallicities andhelium enrichments, to ascertain the theoretical robustness of these priors.
A.3. The Inferred Errors in the Population Parameters Versus the TrueErrors in the Inferred Population Parameters
It is important to demonstrate not just that our parameter estimates are unbiasedin the mean, but that the errors in our parameter estimates are unbiased as well. Thestandard deviation of the differencce between the maximum-likelihood value and the truevalue of the parameters should be equal to mean of the errors reported. We find thatreliable determinations of the errors are obtained by our maximum-likelihood method for N RGBB (cid:38)
10, regardless of the use of priors. At low number counts, the errors in EW RGBB remain unbiased with the priors we used to construct our silver sample, but a small biasremains in the errors in the inferred brightness. The results are shown in Figure 23. 57 –Fig. 23.— TOP LEFT: The 32nd, 50th, and 68th percentile of the error in EW RGBB inferred,for MCMCs with priors (filled circles) and without (empty circles). BOTTOM LEFT: Thestandard deviation of the difference between the true EW RGBB and the value measured bythe maximum-likelihood method. TOP RIGHT: The 32nd, 50th, and 68th percentile of theerror in m RGBB inferred, for MCMCs with priors (filled circles) and without (empty circles).BOTTOM RIGHT: The standard deviation of the difference between the true m RGBB andthe value measured by the maximum-likelihood method.There is a simple explanation for this behavior. At low number counts, the MCMC risksjumping from the true RGBB brightness peak to other statistical fluctuations that may cropup. The measured errors will then be the errors in the position and normalization of thatpeak, rather than of the true peak.Whereas the best-fit values of EW RGBB and m RGBB are unbiased at low number counts, 58 –the error in m RGBB is likely underestimated by the MCMC for N RGBB (cid:46)
10. The errorreported is the error in the peak which is fit for, and not the difference between the locationof the true peak and that of the peak which is fit for.In practice, the relation for the brightness of the RGBB with metallicity will allowastronomers to rule out peaks that differ from the true peak by 1.0 mag or more. However,this is not possible in the less frequent case where a secondary peak shows up within 0.1mag or less. Additionally, other catastrophic failures of fitting at low number counts, suchas peaks with σ RGBB ≥ .
25, or EW RGBB ≥ .
0, will be selected against.
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This preprint was prepared with the AAS L A TEX macros v5.2.
63 –Table 4: Cluster metallicities, estimated V-band apparent distance modulus, V RGBB , aswell as ( V − I ) RGBB and ( B − V ) RGBB where measured. Clusters with an asterix in theirname are part of the silver sample. V -band magnitudes have for GCs from the WFPC2dataset have been made brighter by 0.0365 mag, in the manner described in Section 2 The( B − V ) measurements are not adjusted.Name [Fe/H] [M/H] (m-M) V V
RGBB ( V − I ) ( B − V )ARP0002* -1.74 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± V -band magnitudes have for GCs from the WFPC2 dataset have been shifted in the mannerdescribed in Section 2.Name V MST O ( V − I ) MST O ∆ V MST ORGBB ∆ I RHBRGBB
ARP0002* 21.633 0.700 3.693 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± EW RGBB N RGBB N HB f HBRGBB
ARP0002* 0 . ± .
209 6 . ± . . ± . . ± .
071 4 . ± . . ± . . ± .
104 30 . ± . . ± . . ± .
041 122 . ± . . ± . . ± .
063 22 . ± . . ± . . ± .
063 51 . ± . . ± . . ± .
069 10 . ± . . ± . . ± .
054 170 . ± . . ± . . ± .
183 14 . ± . . ± . . ± .
065 44 . ± . . ± . . ± .
128 9 . ± . . ± . . ± .
101 13 . ± . . ± . . ± .
040 14 . ± . . ± . . ± .
064 41 . ± . . ± . . ± .
036 23 . ± . . ± . . ± .
137 14 . ± . . ± . . ± .
065 41 . ± . . ± . . ± .
066 40 . ± . . ± . . ± .
101 72 . ± . . ± . . ± .
087 36 . ± . . ± . . ± .
051 26 . ± . . ± . . ± .
097 6 . ± . . ± . . ± .
079 34 . ± . . ± . . ± .
130 5 . ± . . ± . . ± .
248 22 . ± . . ± . . ± .
059 32 . ± . . ± . . ± .
108 11 . ± . . ± . . ± .
101 31 . ± . . ± . . ± .
163 6 . ± . . ± .
092 69 –NGC 6254 0 . ± .
103 17 . ± . . ± . . ± .
123 21 . ± . . ± . . ± .
084 44 . ± . . ± . . ± .
077 39 . ± . . ± . . ± .
062 14 . ± . . ± . . ± .
172 19 . ± . . ± . . ± .
063 78 . ± . . ± . . ± .
151 18 . ± . . ± . . ± .
188 7 . ± . . ± . . ± .
031 262 . ± . . ± . . ± .
284 7 . ± . . ± . . ± .
081 40 . ± . . ± . . ± .
117 6 . ± . . ± . . ± .
067 123 . ± . . ± . . ± .
023 286 . ± . . ± . . ± .
086 9 . ± . . ± . . ± .
120 37 . ± . . ± . . ± .
084 23 . ± . . ± . . ± .
109 46 . ± . . ± . . ± .
084 12 . ± . . ± . . ± .
091 56 . ± . . ± . . ± .
080 40 . ± . . ± . . ± .
101 19 . ± . . ± . . ± .
200 11 . ± . . ± . . ± .
185 20 . ± . . ± . . ± .
054 11 . ± . . ± . . ± .
103 16 . ± . . ± . . ± .
214 9 . ± . . ± . . ± .
060 18 . ± . . ± . . ± .
143 21 . ± . . ± . . ± .
109 46 . ± . . ± . . ± .
131 9 . ± . . ± . . ± .
173 13 . ± . . ± . . ± .
075 66 . ± . . ± . . ± .
086 14 . ± . . ± . . ± .
137 21 . ± . . ± .
060 70 –NGC 7006 0 . ± .
071 26 . ± . . ± . . ± .
056 27 . ± . . ± . . ± .
033 35 . ± . . ± . . ± .
086 5 . ± . . ± . . ± .
172 3 . ± . . ± . . ± .
218 7 . ± . . ± . . ± .
199 7 . ± . . ± .
105 71 –Table 7: Other Parameters for the RGBB. N is the total number of stars used in thefit, B is the exponential slope of the RG luminosity function, and σ RGBB is the brightnessdispersion of the RGBB in I -band. Only measurements for gold sample GCs are shown.Name N B σ
RGBB
LYNGA07 329 0.653 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±±