Multi-Wavelength, Optical (VI) and Near-Infrared (JHK) Calibration of the Tip of the Red Giant Branch Method based on Milky Way Globular Clusters
William Cerny, Wendy L. Freedman, Barry F. Madore, Finian Ashmead, Taylor Hoyt, Elias Oakes, Nhat Quang Hoang Tran, Blake Moss
DDraft version December 18, 2020
Typeset using L A TEX twocolumn style in AASTeX62
Multi-Wavelength, Optical (VI) and Near-Infrared (JHK) Calibrationof theTip of the Red Giant Branch Methodbased onMilky Way Globular Clusters
William Cerny,
1, 2
Wendy L. Freedman,
1, 2
Barry F. Madore, Finian Ashmead, Taylor Hoyt, Elias Oakes, Nhat Quang Hoang Tran, and Blake Moss Department of Astronomy and Astrophysics, University of Chicago,5640 S. Ellis Ave., Chicago IL 60637, USA Kavli Institute for Cosmological Physics, University of Chicago,5640 S. Ellis Ave., Chicago, IL 60637, USA The Observatories, Carnegie Institution for Science813 Santa Barbara St., Pasadena CA 91101, USA Department of Astronomy, University of Texas at Austin, 2515 Speedway, Stop C1400Austin, Texas 78712-1205 , USA
ABSTRACTUsing high precision ground-based photometry for 46 low-reddening Galactic globular clusters, inconjunction with Gaia DR2 proper motions for member star selection, we have calibrated the zeropoint of the tip of the red giant branch (TRGB) method at two optical (
V I ) and three near-infrared(
JHK ) wavelengths. In doing so, we utilized the sharply-defined zero-age horizontal branch (ZAHB)of these clusters to relatively calibrate our cluster sample into a composite color-magnitude diagramspanning a wide range of metallicities, before setting the absolute zero point of this composite usingthe geometric detached eclipsing binary distance to the cluster ω Centauri. The I − band zero pointwe measure [ M I = − . ± .
02 (stat) ± .
10 (sys)] agrees to within one sigma of the two previouslypublished independent calibrations, using TRGB stars in the LMC [ M I = -4.047 mag; Freedman etal. 2019, 2020] and in the maser galaxy NGC 4258 [ M F W = -4.051 mag; Jang et al. 2020]. We alsofind close agreement for our J, H, K zero points to several literature studies.
Keywords: star clusters: general INTRODUCTIONFor over a century, the brightest stars in globularclusters have played a pivotal role in our understand-ing of the scale size of our universe. These includethe “Great Debate” (Shapley & Curtis 1921) over thesize (and the uniqueness) of our universe where HarlowShapley used the brightest giants for determining dis-tances to globular clusters. Several decades later, byusing newly-developed red-sensitive emulsions on pho-
Corresponding author: William [email protected] Shapley was on the wrong side of history in that application;he assumed that there were no corrections for dust, and that hisCepheid zero-point calibration was correct (But, of course, thereis plenty of dust in the plane of our Milky Way; and only later tographic plates, Walter Baade (Baade 1944) was ableto resolve tip of the Red Giant Branch (TRGB) starsin the Local Group galaxies M31, M32 and NGC 205,resulting in a major recalibration of the distance scale.Measuring the magnitude difference between tip starsin Galactic globular clusters and RR Lyrae variables inthe same clusters led Baade to revise Hubble’s measure-ment of the expansion rate by a factor of two. TheTRGB method also played an important role in end-ing the “factor-of-two” debate over the Hubble constantthat remained unresolved during the last three decadesof the 20th century: the HST Key Project on the extra-galactic distance scale used TRGB distances to galaxies was it found that there are two types of Cepheids, differing by afactor of 2 in luminosity). a r X i v : . [ a s t r o - ph . GA ] D ec Cerny et al. both as a consistency check on the zero point of theCepheid calibration, and to undertake a differential testfor metallicity effects on the Leavitt Law (e.g., see Freed-man et al. 2001, and references therein).The value of the Hubble constant has recently onceagain become a subject of debate (e.g. Freedman 2017;Efstathiou 2020). Planck measurements of the cosmicmicrowave background (CMB) temperature and polar-ization power spectra give a value of H o = 67.4 ± HST measurementsof the TRGB (Freedman et al. 2019, 2020) provide acalibration of the Type Ia supernovae distance scale,and find a value of H o = 69.6 ± ± V I ) and near-infrared (
JHK ) pho-tometry of red giant stars in a sample of 46 globularclusters in the Milky Way. These clusters have a rangeof metallicities that span those observed in the halos ofnearby galaxies for which TRGB distances can be mea-sured. The organization of this paper is as follows. InSection 2, we review previous calibrations of the TRGBusing Galactic globular clusters. In Section 3, we de-scribe the cluster data used for this work, and our proce-dure for identifying cluster member stars algorithmicallyusing proper motion data from
Gaia
DR2. In Section 4,we utilize the ZAHB to derive relatively calibrated clus-ter distances for all clusters in our sample, before finallyestablishing an absolute zero-point for these distancesusing the DEB distance to ω Cen . PREVIOUS CALIBRATIONS OF THE TRGBMETHODThe development of the TRGB as an accurate distanceindicator improved significantly with the availability ofCCD detectors. Obtaining
V I
CCD data for six MilkyWay globular clusters, Da Costa & Armandroff (1990)used distances based on theoretical horizontal branch models to calibrate the luminosities of RR Lyrae stars.The first application of this Galactic calibration to theextragalactic distance scale was undertaken by Lee et al.(1993). These authors compared the TRGB distancesto 10 nearby galaxies with those based on RR Lyraestars and Cepheids, and quantitatively demonstratedthe power of the TRGB method for measuring distancesto nearby galaxies, a method comparable in accuracy tothe Cepheid Leavitt law.Ferraro et al. (1999) updated the Milky Way TRGBcalibration, based on a larger sample of 60 globularclusters. These authors adopted the zero-age horizon-tal branch (ZAHB) as a standard candle upon which tobase their distances, avoiding altogether the use of RRLyrae variables. This method takes advantage of thesimplicity of the measurement of the horizontal branch,as compared to measurement of the period-luminosity-metallicity relation for variable RR Lyrae stars. As theseauthors noted, the mean RR Lyrae locus is not coinci-dent with the (fainter) position of the ZAHB. More re-cently, the development of a technique to measure accu-rate distances using detached eclipsing binaries (DEBs)provided a new and independent means of calibratingthe TRGB in Galactic globular clusters. Bellazzini et al.(2001, 2004) based their calibration of ω Cen and 47
T uc on the detached eclipsing binary distance to 47
T uc (Thompson et al. 2001). In a later study, Rizzi et al.(2007) used the horizontal branches of five Local Groupgalaxies (IC 1613, NGC 185, Fornax, Sculptor and M33)for their calibration of the TRGB.These calibrations of the TRGB have generally beenin good agreement, leading to an absolute magnitude forthe I-band TRGB in the range of -4.00 to -4.05 mag. Wenext describe our methodology for measuring the TRGBbased on a homogeneous sample of Milky Way globularclusters. CLUSTER DATA3.1.
Cluster Selection and Optical Data
We initially selected our sample of globular clustersfrom the catalogs presented in Stetson et al. (2019),hereafter “S19,” which included 48 clusters in total.These clusters were originally chosen in S19 based onthe availability of high-quality observations in all fiveof the optical UBVRI filter bands, where the criterionfor quality was based on the ability to derive a preciseand accurate absolute calibration for the photometry.As S19 describes, homogeneous reduction of the obser-vations was performed over more than 80,000 individ-ual CCD images using the DAOPHOT, ALLFRAME,and ALLSTAR suites of programs (Stetson 1987, 1992,1994). However, since we only sought to calibrate the ulti-Wavelength (VIJHK) TRGB Calibration from Globular Clusters Table 1.
Adopted Metallicities, Derived Distance Moduli& Reddenings for the 46 Selected Galactic Globular ClustersCluster [Fe/H]
H10 ( m − M ) E ( B − V )NGC 6362 -0.99 14.47 0.08NGC 6723 -1.10 14.69 0.06NGC 2808 -1.14 15.10 0.17NGC 1851 -1.18 15.42 0.03NGC 362 -1.26 14.76 0.04NGC 1261 -1.27 16.10 0.01NGC 6864 (M75) -1.29 16.67 0.14 NGC 5904 (M5) -1.29 14.41 0.03
NGC 288 -1.32 14.83 0.01NGC 6218 (M12) -1.37 13.56 0.17NGC 6981 (M72) -1.42 16.11 0.05NGC 6934 -1.47 16.01 0.10NGC 6229 -1.47 17.43 0.02NGC 6584 -1.50 15.69 0.10NGC 5272 (M3) -1.50 15.08 0.01NGC 7006 -1.52 18.05 0.07
NGC 5139 ( ω Cen) -1.53 13.678 0.12
NGC 6205 (M13) -1.53 14.30 0.02IC4499 -1.53 16.52 0.20NGC 6752 -1.54 13.04 0.05NGC 5986 -1.59 15.15 0.28NGC 3201 -1.59 13.49 0.23NGC 1904 (M79) -1.60 15.57 0.02NGC 7089 (M2) -1.65 15.41 0.04NGC 5286 -1.69 15.36 0.23NGC 6093 (M80) -1.75 15.01 0.18NGC 7492 -1.78 16.77 0.03NGC 4147 -1.80 16.37 0.02
NGC 6541 -1.81 14.40 0.12
NGC 5634 -1.88 17.02 0.05NGC 5897 -1.90 15.43 0.11NGC 5824 -1.91 17.52 0.14NGC 6809 (M55) -1.94 13.53 0.12NGC 5466 -1.98 15.98 0.01NGC 6779 (M56) -1.98 14.99 0.22NGC 5694 -1.98 17.75 0.09NGC 6101 -1.98 15.78 0.10NGC 6397 -2.02 11.90 0.17NGC 5024 (M53) -2.10 16.30 0.02NGC 2419 -2.15 19.61 0.08Terzan8 -2.16 17.10 0.12NGC 4590 (M68) -2.23 15.02 0.05NGC 7099 (M30) -2.27 14.52 0.04NGC 5053 -2.27 16.19 0.01
NGC 6341 (M92) -2.31 14.59 0.02
NGC 7078 (M15) -2.37 15.11 0.08
Cerny et al.
TRGB in the V and I bands, we opted to eliminate 11high-reddening ( E ( B − V ) > .
25) clusters from this ini-tial S19 sample, replacing them with nine other bright,low-reddening NGC globular clusters available from theCADC archive as of September 2020 – all processed inthe same homogeneous manner – to build our final sam-ple of 46 clusters. These 9 clusters were not in S19 be-cause data in all of the UBVRI bands were not availableor not yet reduced at the time of publication. Hereafter,we include these clusters when referring to S19 through-out this work. We list all clusters included in this studyin Table 1.As S19 emphasizes, the resulting photometry for theseGalactic globular clusters is well-suited to be utilizedin conjunction with the high-precision photometry andproper motion measurements provided by Gaia
DR2.To combine these catalogs, we performed a cross-matchbetween the S19 positional data and the complete
Gaia
DR2 database using a 0.5 arcsec matching radius em-ploying the CDS xMatch service. We then retained onlythe matched stars that also had proper motion measure-ments. We did not use any parallax information in thisstudy, anticipating superior determinations in the up-coming
Gaia
Early Data Release 3 (EDR3) release. Wethen further applied conservative quality cuts based onthe DAOPHOT parameters χ and sharp. In particular,we applied χ < | sharp | <
1, and after selectingmember stars for each cluster (described in the follow-ing subsection), we additionally removed all stars within min (core radius, 1 arcmin) of angular separation fromthe cluster centroids, utilizing core radii and centroidsfrom the Harris (2010) catalog of globular cluster prop-erties. Hereafter, we refer to this catalog, most recentlyupdated in 2010, as H10. These cuts are intentionallygenerous so as to be as complete as possible, while alsomaintaining a reasonably high level of purity.3.2.
Selecting Cluster Member Stars
In order to isolate cluster member stars from fore-ground and background stars, we utilized the scikit-learn (Pedregosa et al. 2012) implementation of the GaussianMixture Model clustering algorithm, inspired by themethods presented in Bustos Fierro & Calder´on (2019)and Vasiliev (2019). Briefly, a Gaussian Mixture Modelis a fast and robust supervised clustering algorithm ca-pable of fitting a multi-dimensional Gaussian to a fixednumber of classification components; applying the algo-rithm allows for the probabilistic labelling of all sourcesinto these distinct components. http://cdsxmatch.u-strasbg.fr/ The clustering algorithm we applied takes in four di-mensions as inputs: the two spatial coordinates (
X, Y )derived for each individual cluster in S19 and two propermotion vector components ( µ α cos ( δ ) , µ δ ) taken from the Gaia catalog. The (
X, Y ) coordinates are chosen forthe same reason presented in Bustos Fierro & Calder´on(2019), namely that they mitigate the potential for themixture model to treat clusters at varying declinationsdifferently due to spherical geometry; here, instead, theplanar (
X, Y ) coordinates were derived in S19 so as to fixa consistent radial scale centered at the position of eachcluster. For a given cluster, we begin by pre-processingthese measurements using scikit-learn ’s RobustScaler method over these four input features, which scales thefeature values to equal orders of magnitude based ontheir inter-quartile ranges in order to weight them equiv-alently in membership determinations. We do not uti-lize
Gaia parallaxes or parallax-derived quantities inthese models, as prior attempts in the literature to usethem for cluster member star determinations resultedin reduced machine learning model efficiency (eg, Bus-tos Fierro & Calder´on 2019), and since these measure-ments can be imprecise in densely populated regions ofclusters. We also note that
Gaia radial velocities remainunavailable for the vast majority of stars in most clus-ters, and thus we do not utilize these measurements forour mixture model classification.In using the Gaussian Mixture Model algorithm, wegenerally impose that there are two mixture model com-ponents for each cluster’s dataset: one representing sig-nal based on the cluster’s spatial distribution and propermotion vector, and the second “background.” After thealgorithm is run over each cluster, we visually inspectthe resulting spatial (coordinate) distribution, propermotion vector-point diagrams, and color-magnitude di-agrams. For a limited number of clusters where visualinspection elucidated high levels of contamination fromforeground/background stars, we adjusted the numberof expected components and re-run the algorithm in or-der to provide a better membership classification for allstars in the data. Such a change proved to be particu-larly important for cluster fields with other nearby as-tronomical objects; for example, in the case of the clus-ter NGC 362, the close proximity of the Small Magel-lanic Cloud (SMC) in projected (2D) coordinate sepa-ration and its subsequent contribution to the other twoclusters’ catalogs demanded that we increase the num-ber of expected mixture model components to three, re-flecting that of the SMC, the desired cluster, and fore-ground/background stars. One key advantage of usingthe Gaussian mixture model, however, is that we neednot specify the expected centroid or mean proper mo- ulti-Wavelength (VIJHK) TRGB Calibration from Globular Clusters
15 10 5 0 5 10 15 cos( ) X Y ( V I ) I Figure 1.
Diagnostic plots generated when applying our Gaussian Mixture Model membership to the cluster NGC 362 (acluster with two distinct signal components). (Left) 2D distribution of the
Gaia
DR2 proper motion measurements for all starscross-matched between the catalogs provided by S19 and
Gaia
DR2 within 1.25 tidal radii of the cluster centroid (H10;Vasiliev2019). The blue ellipsoid corresponds to stars identified as likely cluster members, while the red ellipsoid corresponds to starsthat are likely members of the SMC. Points drawn in grey are foreground or background stars that the model predicts do notbelong to either of the two “signals” in the S19 photometric catalogs. Grey stars that appear in the region dominated bythe blue ellipsoid are individual stars that featured proper motion measurements that are approximately consistent with thecluster’s systemic mean proper motion, but were excluded from being identified as likely cluster member stars due to large spatialseparation from the cluster centroid. (Center) Spatial distribution of NGC 362 in planar (
X, Y ) coordinates. Stars displayedin color are those identified as member stars by the Gaussian Mixture Model, while stars in grey correspond to those identifiedas either SMC or foreground/background stars. In general, stars are increasingly likely to be deemed members by the modelas their positions approach the cluster centroid; however, we note that the precision and accuracy of the
Gaia proper motionsdecreases significantly in the crowded cores of the majority of the clusters studied in this work. (Right) Color-magnitude diagram(CMD) of the identified member stars for NGC 362, with no distance, reddening, or extinction corrections applied. In order tomitigate crowding effects, we exclude stars within min (core radius, 1 arcminute) of the cluster centroid when constructing andanalyzing CMDs in this work. tion signal of each component – these are automaticallyderived, and our model only assumes that each compo-nent is well-represented by a multi-dimensional Gaus-sian over the input features. Additionally, in a limitednumber of additional cases where the proper motion sig-nal for the cluster is less prominent, we apply exceed-ingly conservative cuts on the proper motion compo-nents as priors; these priors are chosen so as to guide thealgorithm’s membership classifications without exclud-ing any potential member stars. We emphasize that bynot requiring parallaxes in our membership procedure,our catalogs are generally more complete than those pro-vided by Gaia Collaboration et al. (2018), featuring tensto thousands of additional member stars (typically atfainter magnitudes) without compromising the purity ofthe color-magnitude diagram for each cluster.In Figure 1, we visually depict diagnostic results ofthe cluster membership process described above for thecluster NGC 362; this cluster is chosen to demonstratea case when a three-component mixture model is usedto isolate the cluster signal from that of the SMC andthe foreground/background field of stars. In the left-hand panel, we plot the distribution of proper motionmeasurements ( µ α cos ( δ ) , µ δ ) for all stars cross-matchedbetween S19 and Gaia
DR2 within 1.25 tidal radii of the cluster centroid and that pass our initial qualitycuts. Points in blue are those sources identified aslikely cluster members, whereas points in red are likelySMC stars; light grey points correspond to foregroundor background stars, which belong to neither system.In the center panel, we plot the spatial distribution oflikely member stars in planar (
X, Y ) coordinates, col-ored by membership probability (normalized to unity).Field stars identified as non-members are plotted as greypoints. Lastly, in the right-hand panel, we plot the I vs.( V − I ) color–magnitude diagram for the likely memberstars for the cluster (uncorrected for reddening or ex-tinction). The quality of the S19 photometry and thepurity of the member-star sample is evident, with mini-mal contamination visible in the upper red giant branchand asymptotic giant branch. In Figure 2, we presentcomparable I vs. ( V − I ) color–magnitude diagrams forall clusters analyzed in this study.3.3. Near-Infrared (JHK) Data
In order to extend our TRGB measurement to thenear-infrared, we utilize the high-precision
J, H, K pho-tometry from the 2 Micron All-Sky Survey (2MASS;Skrutskie et al. 2006). To construct our stellar catalog,we perform a cross match between the full
Gaia
DR2
Cerny et al. M I NGC7078 [Fe/H]= 2.37
NGC6341 [Fe/H]= 2.31
NGC7099 [Fe/H]= 2.27
NGC5053 [Fe/H]= 2.27
NGC4590 [Fe/H]= 2.23
Terzan8 [Fe/H]= 2.16
NGC2419 [Fe/H]= 2.15
NGC5024 [Fe/H]= 2.10
NGC6397 [Fe/H]= 2.02 M I NGC6779 [Fe/H]= 1.98
NGC5694 [Fe/H]= 1.98
NGC6101 [Fe/H]= 1.98
NGC5466 [Fe/H]= 1.98
NGC6809 [Fe/H]= 1.94
NGC5824 [Fe/H]= 1.91
NGC5897 [Fe/H]= 1.90
NGC5634 [Fe/H]= 1.88
NGC6541 [Fe/H]= 1.81 M I NGC4147 [Fe/H]= 1.80
NGC7492 [Fe/H]= 1.78
NGC6093 [Fe/H]= 1.75
NGC5286 [Fe/H]= 1.69
NGC7089 [Fe/H]= 1.65
NGC1904 [Fe/H]= 1.60
NGC5986 [Fe/H]= 1.59
NGC3201 [Fe/H]= 1.59
NGC6752 [Fe/H]= 1.54 M I NGC5139 [Fe/H]= 1.53
NGC6205 [Fe/H]= 1.53
IC4499 [Fe/H]= 1.53
NGC7006 [Fe/H]= 1.52
NGC5272 [Fe/H]= 1.50
NGC6584 [Fe/H]= 1.50
NGC6229 [Fe/H]= 1.47
NGC6934 [Fe/H]= 1.47
NGC6981 [Fe/H]= 1.42 ( V I ) M I NGC6218 [Fe/H]= 1.37 ( V I )NGC288 [Fe/H]= 1.32 ( V I )NGC5904 [Fe/H]= 1.29 ( V I )NGC6864 [Fe/H]= 1.29 ( V I )NGC1261 [Fe/H]= 1.27 ( V I )NGC1851 [Fe/H]= 1.18 ( V I )NGC2808 [Fe/H]= 1.14 ( V I )NGC6723 [Fe/H]= 1.10 ( V I )NGC6362 [Fe/H]= 0.99
Figure 2.
46 Galactic Globular Cluster Optical CMDs used in this study, individually displayed and ordered by increasingmetallicity (shown in red) from the top-left to bottom-right. Adopted distance moduli and reddenings are given in Table 1. Weexclude NGC 362 in this plot, as an identical CMD is displayed in Figure 1. Clusters with “missing” main sequences correspondto some of the more distant globular clusters with stellar populations that extend to magnitudes fainter than the
Gaia limitingmagnitude of G ∼
21 mag (e.g., NGC 2419, NGC 5694 and NGC 7006). catalog and the 2MASS Point-Source Catalog using theCDS xMatch service, with a 0.5” matching radius. Wethen sub-select all sources within this cross-match thatwere also identified in the optical catalog (see previoussubsection), by selecting stars based on their
Gaia
DR2source IDs. CALIBRATION OF CLUSTER DISTANCES4.1.
Initial Reddening and Distance Corrections
In order to bring the individual 46 clusters onto a con-sistent relative calibration, we first begin by convertingall measurements to absolute magnitudes. To do so, weinitially adopted the apparent visual distance moduliand color excesses E ( B − V ) from the H10 catalog. Wethen converted the former to true distance moduli as-suming a Cardelli et al. (1989) reddening law with R V = 3.1 for all clusters. Next, we applied an extinction correction in each band by adopting the H10 redden-ing values and the following total-to-selective absorp-tion ratios for the Johnson-Cousins BV I filters: R B =4 . , R V = 3 . , R I = 1 .
485 from Cardelli et al. (1989).We later adopt R J = 0 . , R H = 0 . , R K = 0 . E ( B − V ) color excess values using theoptical data.4.2. Securing a Relative Cluster Calibration
With these corrections in place, we next sought tobring all the clusters onto a consistent relative cali-bration. To do so, we began by splitting the clustersample into four distinct metallicity bins based on the ulti-Wavelength (VIJHK) TRGB Calibration from Globular Clusters ( V I ) M I [Fe/H] < 2.0 , 9 clusters ( V I ) , 12 clusters ( V I ) , 15 clusters ( V I ) [Fe/H] > 1.4 , 10 clusters Figure 3.
Optical-wavelength ( I versus V − I ) composite color-magnitude diagrams for the four metallicity bins used duringthe relative alignment process in Section 4, corrected for distance and reddening/extinction, and brought onto the absolute zero-point set by the DEB distance to ω Cen . The clusters corresponding to each bin are listed in Table 1. These composite CMDsreveal the precise alignment of the cluster ZAHBs achieved by the process described in Section 4. While some contamination isvisible at fainter magnitudes, the four composites individually exhibit the rapid drop-off in density expected expected from theTRGB at I (cid:46) − . ω Cen . metallicity values reported in H10. These four binscorrespond to the metallicity ranges: [Fe / H] < − . − . ≤ [Fe / H] < − . − . ≤ [Fe / H] < − .
4, and[Fe / H] > − . V, I bands. With the derived bluehorizontal branch, we then applied a simple maximum-likelihood grid search technique in order to bring thelower envelopes of the remaining clusters into the samerelative calibration (based on the
V, I bands). Withina given bin, we initially assumed the calculated truedistance modulus and reddening values from H10, andstepped through potential distance modulus offsets in arange [-0.25 to 0.25 mag] in intervals of 0.01 mag in orderto find the distance modulus offset which best brought the horizontal branch of each cluster into alignment withthe fiducial HB. The likelihood function we sought tomaximize for this grid search was defined as the rawnumber of horizontal branch stars for each cluster thatfell between the fiducial lower envelope from the ref-erence cluster and a parallel branch displaced .05 magbrighter, factoring in photometric uncertainties for eachstar. For each value of the distance modulus shift inthe interval, we added the distance modulus offset to allstars’ I − band magnitudes, and re-calculated the num-ber of stars lying in the horizontal branch region definedabove. In some cases where the morphology of a givencluster varied to a non-negligible degree with respect tothe reference cluster for its bin (for example, when thehorizontal branch naturally truncates at redder photo-metric colors), we restricted the color range of the fidu-cial to match that of the given cluster when applyingthe likelihood procedure; this minimizes the possibilityof contaminant stars affecting the likelihood determina-tion. Additionally, in a number of clusters, we foundevidence for clear misalignment of the lower red giantbranch and subgiant branch of clusters compared to the(low reddening) reference cluster far beyond what mightbe expected due to differences in metallicity alone withinthe small ( ∼ . E ( B − V ) color excess of each cluster (recalculat-ing the extinction in both bands and the reddening) inintervals of 0.01 mag and took note of the pair of (red-dening, distance modulus offset) which resulted in boththe clearest alignment of the blue horizontal branch andthe maximization of the number of stars lying within thebounds constructed using the fiducial horizontal branch. Cerny et al.
In more metal-rich clusters where a reasonably-populated horizontal branch was not present, we insteadfit to the lower edge of the red clump; in our sample, themetallicity range in our most metal-rich bin is narrowenough that significant variation (metallicity dependenteffects) in the red clump is not expected across clusters.In particular, we utilized NGC 1261, which is low red-dening ( E ( B − V ) = 0 .
01 mag) and features both a wellpopulated red horizontal branch/clump and enough ofa blue horizontal branch to construct a fiducial ZAHBlower envelope. As a result of doing so, this clustercould be used as reference for clusters both with andwithout blue horizontal branches.We have excluded the bright, well-populated, cluster47 Tucanae from our analysis, as it is significantly moremetal-rich ([Fe / H] = − .
72; Harris 2010) than the otherclusters considered in this study, which are bounded bya metallicity of [Fe / H] = − .
99 (NGC 6362). Thismetallicity gap renders it nearly impossible to bring47 Tuc onto a relative calibration with all the otherclusters, as metallicity effects in the red clump appearto become significant. Additionally, while 47 Tuc alsohas a well-constrained geometric distance from detachedeclipsing binaries (Thompson et al. 2020), we did notuse this measurement to set the absolute zero-point inthis work due to both the aforementioned metallicitygap and the lack of a horizontal branch for this clus-ter. However, these issues will be resolved with accurateparallax-based distances to these clusters with the forth-coming
Gaia
Data Release 3, allowing for all clustersto be brought onto a self-consistent geometric distancescale (see Section 6).The final result of these procedures is a compositecolor-magnitude diagram for each of the four metallic-ity bins, shown in Figure 3, with the clusters in eachbin brought onto a uniform relative calibration. We es-timate the (statistical) uncertainty of cluster distancemoduli derived from this intra-bin cluster horizontalbranch alignment process as ± .
03 mag, correspondingto the typical full width of the maximum-likelihood re-sponse peak about the most probable value for a clus-ter’s vertical shift.4.3.
Absolute Calibration
We set the absolute zero-point for the relatively-calibrated composite of clusters constructed above toa DEB-based geometric distance to the cluster ω Cen .Thompson et al. (2001) measured an apparent distancemodulus to the system of ( m − M ) V = 14 . ± .
11 mag.In the previous subsection, we found a best-fit color ex- cess value for ω Cen of E ( B − V ) = 0 .
12 mag andthus we recalculate the true distance modulus to thesystem to be µ = 14 . − (3 . ∗ .
12) = 13 .
678 mag.Adopting this DEB-based distance to ω Cen , we re-zerothe third metallicity bin containing ω Cen . Then, wesimply applied our maximum-likelihood HB-fitting tech-nique three times more in total, once to each metallicitybin, in order to bring each of the remaining three bins’composite HBs onto the newly-set zero point providedby the ω Cen
DEB distance. In doing so, we found thatonly small (0.01 - 0.04 mag) shifts were required. Weestimate the error associated with this inter-bin align-ment to be ± .
02 mag, again based on the width of themaximum-likelihood response.We emphasize that our procedure for multi-clusteralignment is fully independent of RGB stars, insteadonly relying on the zero-age horizontal branch featurelocated 4 mag below the TRGB feature.We report our derived values of true distance modulusand color excess for all 46 clusters (calculated by firstassuming the H10 value, and then applying our deriveddistance modulus and/or reddening shifts, if applicable)in Table 1. For each true distance modulus, ( m − M ) ,we attribute a statistical error based on three sourcesof uncertainty: (1) a 0.015 mag statistical uncertaintyassociated with the alignment in color of a given clusteragainst ω Cen due to an empirical 0.01 mag uncertaintyin the E ( B − V ) color excess, propagated to the the I − band; (2) a 0.03 mag statistical uncertainty arisingfrom the intra-bin alignment described in the previoussubsection; and (3), a 0.02 mag statistical uncertaintyarising from the inter-bin alignment described above inthe current subsection. Adding the respective compo-nents of the statistical errors in quadrature, we assignan uncertainty to our reported true distance moduli of( m − M ) ± .
039 (stat) mag. We return to the issue ofsystematic errors in Section 5 below.We apply distance and extinction/reddening correc-tions using these values when generating the individ-ual cluster CMDs in Figure 2 and all composite CMDsthroughout this work. DETECTING THE TRGBIn the following sections, we utilize the composite clus-ter sample constructed above to measure the magni-tude/slope of the TRGB across five different photomet-ric bandpasses. 5.1.
Optical: I -Band For reference, Thompson et al. (2001) adopted a value E ( B − V ) = 0 .
13 mag, as derived from Schlegel et al. (1998) maps. ulti-Wavelength (VIJHK) TRGB Calibration from Globular Clusters Figure 4.
Composite I vs ( V − I ) CMD, luminosity function, and edge detector response for four bins of equal stellar counts,sorted by increasing metallicity from the top left to the bottom right. We note that these bins are distinct from those used inthe calibration procedure, which varied significantly in stellar counts. (Left Panels) Optical CMDs for clusters. We apply aloose color–magnitude selection, denoted by the dashed diagonal line, when constructing the CMD and luminosity function, andwhen measuring the edge response in the center and right-hand panels, respectively. (Center) GLOESS-smoothed luminosityfunction of RGB stars with M I > − . We first sought to measure the TRGB in the I -band,where the TRGB is expected to be roughly flat acrossa broad range of photometric color (1 . (cid:46) ( V − I ) (cid:46) . I − band RGB luminosity function (LF)into bins of 0.01 mag, and count the number of starsfalling in each bin. This LF is then smoothed usingGLOESS (Gaussian-windowed, Locally-Weighted Scat-terplot Smoothing). We then apply a [-1, 0, +1] Sobelfilter, which acts as an “edge detector” that respondsto discontinuities or steep derivatives in the LF, as ex-pected due to the rapid drop-off in stellar counts at themagnitude of the (flat) I − band TRGB.In Figure 4, we present plots depicting four individualmeasurements of the I − band TRGB absolute magni-tude using the method described above. The 46-clustersample is sorted here by increasing metallicity, splittingthe composite catalog into four bins of equal numbers ofstars. We note that these re-defined metallicity groups are distinct from those used in the calibration procedure(section 4), where the numbers of stars per bin was notconstant. The I − band TRGB absolute magnitude andits statistical uncertainty are shown in each of the in-dividual plots. The range of TRGB magnitudes is only0.020 mag, and the values for each individual metallicitygroup agree to within their measured uncertainties.In Figure 5, we present the same plots for the 46-cluster composite. For this full composite, we apply acolor selection, indicated by a dashed diagonal line inthe right-most panel, in order to remove contaminatingsources from our TRGB detection.5.1.1. Statistical Error on the Mean
In the I-band CMD (right panel) we include two (thin)horizontal lines to denote one and two sigma above andbelow the (thick) solid line at the level of our measuredTRGB. Contributing to the TRGB detection there are107 stars within two sigma of the tip. Our techniquefor aligning the horizontal branches of the clusters con-tributes to the observed ‘blurring’ of the TRGB, in theplot of the marginalized luminosity function in Figure 5.Applying a Sobel filter to the observed luminosity func-tion, we measure the tip at M T RGBI = − .
056 mag.0
Cerny et al.
Table 2. I − band TRGB detections based oncluster bins of equal counts ( ∼ ,
000 stars permetallicity bin across all magnitudes), and corre-sponding statistical uncertainties (before account-ing for all other uncertainties). These measure-ments originate from the detections presented inFigure 4.Bin Color Range at Tip M I TRGB . (cid:46) ( V − I ) (cid:46) . ± .
012 (stat)1 . (cid:46) ( V − I ) (cid:46) .
55 -4.050 ± .
013 (stat)1 . (cid:46) ( V − I ) (cid:46) . ± .
013 (stat)1 . (cid:46) ( V − I ) (cid:46) . ± .
013 (stat)
Full Composite: -4.056 ± .
012 (stat)
Using the statistical error of ± σ stat = ± . / (cid:112) ( N −
1) = ± Adopted Systematic Error
There are two sources of systematic uncertainty inthe TRGB calibration: (1) The systematic uncertaintyin the measured distance modulus to ω Cen , for whichwe have adopted the DEB measurement from Thomp-son et al. (2001), and (2) An additional 0.015 mag sys-tematic uncertainty in the DEB distance propagatedto the I − band apparent modulus measurement from a0.01 mag uncertainty in the ω Cen E ( B − V ) color ex-cess.Recent work by Braga et al. (2018) using near infraredobservations of RR Lyrae variables in ω Cen have con-firmed the DEB distance, but in addition, their high pre-cision suggests that the quoted systematic uncertaintyon the DEB distance from Thompson et al. (2001) mayhave been overestimated. Using a theoretical calibra-tion of the RR Lyrae variables, Braga et al. find truedistance moduli for ω Cen of 13.674 ± ± ± ± ± ω Cen over time, and over methods, (seeFigure 23 of Braga et al. 2018) we very conservatively adopt ± .
10 mag as the systematic error on the truedistance to ω Cen . This, of course, will soon be su-perseded with the release of Gaia DR3 (see Section 6).Our final adopted TRGB calibration using ω Cen is M T RGBI = − . ± .
02 (stat) ± .
10 (sys) mag. Wesummarize the quoted errors for this measurement inTable 3.
Table 3.
Measurement Error BudgetSource of Uncertainty σ stat σ sys Reddening Alignment 0 .
015 ...Intra-Bin ZAHB Alignment 0 .
03 ...Inter-Bin ZAHB Alignment 0 .
02 ... ω Cen
Distance ... 0 . ω Cen Extinction ... 0 . .
039 0 . N = 4 ref. bins) 0 .
022 ...
Total Uncertainty (mag.) 0 .
022 0 . Note —TRGB measurement uncertainties due to red-dening were propagated from 0 .
01 mag uncertainty in E ( B − V ) to the I -band extinction adopting R I = 1 . Optical: V − Band
Because the same stars that define the tip in the I -band also define the TRGB across all other wavelengths– from the optical to the near-infrared – the procedurefor detecting the TRGB in other filter bands is funda-mentally simple. In the V bands, theory predicts (andobservations have confirmed) that the TRGB exhibits apositive slope with increasing (redder) stellar color. (e.g.Freedman et al. 2020).As described in detail in Madore & Freedman (2020),one only needs knowledge of the calibrated run of theTRGB with photometric color and the color-color re-lations between two bandpasses (for a limited run ofspectral types) in order to predict the slope and thezero-point for any other bandpass combination. Theapplication of this type of relation is particularly sim-ple in the case of transforming between the run of theTRGB in I vs. ( V − I ) to V vs. ( V − I ): by assum-ing zero slope for the I − band TRGB with respect to V − I color, the V − band slope is uniquely determined as V / ( V − I ) = 1 .
00 . The V − band zero point can then beeasily determined by adopting a fiducial intrinsic colorat which to measure the zero point, and simply addingit to the I − band zero point, as V = I + ( V − I ). Adopt-ing a fiducial measurement color of ( V − I ) = 1 . V − band zero point of M V = − .
26 mag. ulti-Wavelength (VIJHK) TRGB Calibration from Globular Clusters Figure 5.
Full 46-cluster composite CMDS displaying the upper RGB/TRGB in B vs ( V − I ), V vs ( V − I ), and I vs( V − I ). We impose a color selection in the I − band, removing stars bluer than the dashed diagonal line represented by M I = − . V − I ) + 1 . To summarize, we find: M I = − .
06 + 0 . V − I ) − . M V = − .
26 + 1 . V − I ) − . B − band slope and zero point in this work,primarily due to increased scatter in the photometry forthis filter, we include an illustrative slope fit to highlightfor the reader the wavelength dependence of the TRGBacross the three panels.5.3. Near Infrared: J,H, and K Bands
In order to extend our analysis to measure the TRGBin the near-infrared, where the TRGB is expected tonegatively correlate (decrease in absolute magnitude)with redder intrinsic stellar colors (e.g. Madore et al.2018; Hoyt et al. 2018), we undertake a procedure simi-lar to the one described in the previous subsection. How-ever, in doing so, we emphasize two important distinc-tions from our analysis in the optical bands. First, wenote that the impact of line-of-sight reddening is greatlydiminished in the near-infrared compared to the visualbands, where, for example, the magnitude of extinctionin the K -band ( A K ) is nearly one tenth of the extinctionin the V − band ( A K = . A V ; Cardelli et al. 1989). Secondly, as introduced in Section 3, we now utilizehigh-precision JHK photometry from 2MASS for starscross-matched between S19,
Gaia
DR2, and S19’s cat-alogs that were identified as likely cluster members inSection 3.In order to derive a TRGB zero point for the
J, K bands, we assume the Madore et al. (2018) J − bandslope of J/ ( J − K ) = − .
85; the K − band slope thendiffers by 1.00, and thus, K/ ( J − K ) = − .
85 (seeprevious subsection and Madore & Freedman 2020).These slopes were measured based on TRGB stars in the(low-extinction) halo of the Local Group dwarf galaxy,IC1613. For completeness, the uncertainties on theslopes from that work are ± .
09 and ± .
19, for the J/ ( J − K ) and K/ ( J − K ) slopes, respectively. Usingthese slopes, we then “rectify” (i.e. flatten) the com-posite CMDs such that their TRGBs appear flat as afunction of intrinsic color (as is naturally the case inthe I − band). In doing so, the RGB luminosity func-tions show the greatest contrast in stellar counts atthe TRGB, allowing for the determination of the NIRTRGB zero points through applying the same Sobel fil-ter edge detection as employed in the previous subsec-tions. Similar to the procedure for the V − band, weadopt a fiducial color at which to measure the TRGB2 Cerny et al.
Figure 6.
Full 46-cluster composite CMDs for all three visual wavelengths. The dark line in each panel corresponds to thederived equations from Section 5. We note that the B − band photometry utilized in this work was found to have significantlylarger scatter than the V and I bands, and thus we did not seek to derive a zero point in this filter; however, we include anapproximate linear fit to the corresponding CMD in order to display the wavelength-dependent variation of the TRGB acrosseach of the panels. zero point, namely ( J − K ) = 1 .
00 mag, and measurethe J − band zero point of M J = − .
16 mag, which si-multaneously determines the K − band zero point to be(by definition) M K = M J − .
00 = − .
16 mag. Thesederived zero points above are based on a large sampleof 119 (254) stars within one (two) sigma of the NIRTRGB, reported based on the J − band edge detectoroutput.For the H − band, we re-fit the TRGB slope directlyvia simple linear regression over all stars within twosigma of the rectified J − band edge detection, deriv-ing a slope of H/ ( J − K ) = − .
72. Then, holding the H − band zero point to be the same as the J, K zeropoint at ( J − K ) = 0 .
00 mag, we run the Sobel filterover the rectified H − band data, deriving a zero pointof M H = − .
03 mag. Figure 7 depicts rectified CMDs,smoothed LFs, and Sobel filter responses for each of thethree NIR filter bands. In the left panels of each subplot,we include a CMD of the upper RGB and TRGB forthe 46-cluster composite, where the color-dependence ofthe TRGB has been flattened. The dashed, diagonalline represents a color cut applied before the LF is con-structed and the Sobel filter edge detector is run. This color is chosen as it corresponds to the same metallicityas the fiducial color chosen for the I − band, ( V − I ) = 1 . . In summary, we find: M J = − . − . J − K ) − . M H = − . − . J − K ) − . M K = − . − . J − K ) − . H − bandslope by assuming the J, K
TRGB slopes from (Madoreet al. 2018).In Figure 8, we present J vs ( J − K ), H vs. ( J − K ),and K vs ( J − K ) composite CMDs depicting the upperRGB and TRGB for the 46-cluster sample using the2MASS photometry. In each panel, we plot a diagonalline indicating the TRGB calibration described by eachof the above equations. DISCUSSION6.1.
Comparison with Prior Measurements
In Table 4 we summarize our measurements of theTRGB absolute magnitude and slope. We place theseresults in the context of calibrations derived from a vari-ety of physical systems, including other studies of Galac-tic globular clusters (Bellazzini et al. 2004, Freedmanet al. 2020), the LMC (Hoyt et al. 2018; Freedman et al.2020; G´orski et al. 2018), and the maser-hosting galaxyNGC 4258 (Jang et al. 2020). We find excellent agree-ment with the body of existing literature measurements,with our measurement lying within one sigma of nearly ulti-Wavelength (VIJHK) TRGB Calibration from Globular Clusters Figure 7.
Rectified (flattened) CMDs for the
J, H, K bands, smoothed LFs, and Sobel filter responses for each of the three NIRfilter bands. (Left Panels) NIR CMDs for the upper RGB/TRGB of the 46-cluster composite. We apply a color cut indicatedby the dashed diagonal line to remove bluer stars that likely belong to the asymptotic giant branch and other contaminantstars. The upper horizontal line in each plot at M = − . J − and K − band zero points differ by exactly 1.00 mag. (Right Panels) Sobel filter edge response all tabulated measurements across all wavelengths con-sidered in this work.6.2. Future Prospects with Gaia DR3
While our analysis did not utilize parallaxes to se-lect cluster member stars or to set our absolute zero-point, the upcoming
Gaia
Early Data Release 3 (EDR3;expected December 2020) and planned full Data Re-lease 3 (DR3; currently planned for 2022) will offerthe possibility of calibrating the TRGB based solely ongeometrically-defined distances to these clusters. Suchan advancement will be enabled both by corrections tothe underlying astrometric solution (currently known toexhibit a systematic parallax bias with respect to back-ground quasars; Sch¨onrich et al. 2019), and also signif-icantly improved astrometric systematics. Because theerror in our TRGB measurements is dominated by thesystematic error on the single Thompson et al. (2001)DEB distance, the use of parallax-based geometric dis- tances from EDR3 to each of the 46 clusters consid-ered here have the potential to improve the precisionand accuracy of this cluster-based calibration of theTRGB luminosity by an order of magnitude. Addi-tionally, EDR3’s expected factor-of-two improvementsin proper motion uncertainties, 20% improvements inparallax uncertainties, and significant increase in thenumber of stellar sources with full 5-parameter astro-metric measurements will make it possible to increasethe purity and completeness of cluster membership cat-alogs – for example, by utilizing parallaxes as a param-eter in mixture model classifications (Section 3), cap-italizing on more precise proper motion measurementsof stars located near the cores of clusters, and by reap-ing the benefits of increasingly confident probabilisticestimates of a star’s membership likelihood due to therefined proper motion measurements. CONCLUSION4
Cerny et al.
Figure 8.
Full 46-cluster composite CMDs based on the 2MASS
J, H, K photometry. The diagonal line corresponds in eachpanel represent the calibration equations presented in Section 5.3. Unfilled circles represent those excluded from when derivingthe zero point due a color cut.
Using
Gaia
DR2, we have applied Gaussian MixtureModel clustering to precisely distinguish member starsfor 46 Galactic globular clusters, and crossmatched withhomogeneous photometric datasets in the optical andnear-infrared bandpasses to build a multi-band catalogof cluster data spanning a wide range of metallicity. Us-ing the zero-age horizontal branch, we brought these 46clusters onto a consistent relative calibration, before re-zeroing the ensuing composite to a detached eclipsingbinary (geometric) distance measurement to ω Cen . Wethen utilized this uniformly-calibrated cluster compos-ite to measure the zero point of the TRGB across fiveoptical and near-infrared wavelengths, finding excellentagreement with previous calibrations.Our results provide an independent check on the ap-plication of the TRGB to nearby galaxies, reinforcingthe validity of TRGB-based rungs of the cosmologicaldistance ladder. The methods and calibrations outlinedin this paper will be greatly improved with the advent of
Gaia
DR3, which will bring reliable, self-consistent par-allax distances to these clusters, allowing for greatly di-minished systematic uncertainties associated with eachmeasurement and the overall absolute cluster zero point, along with improvements to the astrometry underlyingthe cluster member star selection process. ACKNOWLEDGMENTSWe thank the
Observatories of the Carnegie Institu-tion for Science and the
University of Chicago for theirsupport of our long-term research into the calibrationand determination of the expansion rate of the Universe.This work has made use of data from the Euro-pean Space Agency (ESA) mission
Gaia
Gaia
Gaia
Multilateral Agreement.This research made use of the cross-match serviceprovided by CDS, Strasbourg ([http://cdsxmatch.u-strasbg.fr/])
Facility:
Gaia . Software: astropy (AstropyCollaborationetal.2013;Price-Whelan et al. 2018),
Matplotlib (Hunter 2007), ulti-Wavelength (VIJHK) TRGB Calibration from Globular Clusters Table 4.
Comparison of TRGB calibrationsFilter Zeropoint Reference Calibration Technique I − . ± .
12 Bellazzini et al. (2004) DEB distances to 47 Tuc, ω Cen (averaged) I − . ± .
053 (stat) ± .
080 (sys) Freedman et al. (2020) 12 GCs with literature distances + DEB distance to 47 Tuc I − . ± .
022 (stat) ± .
039 (sys) Freedman et al. (2020) 18 DEBs in LMC F W − . ± .
027 (stat) ± .
045 (sys) Jang et al. (2020) Maser in NGC4258 I − . ± . (stat) ± . (sys) This Work 46 GCs - Cluster ZAHBs and DEB distance to ω Cen V − . ± .
022 (stat) ± .
039 (sys) Freedman et al. (2020) DEBs in LMC V − . ± .
053 (stat) ± .
080 (sys) Freedman et al. (2020) 12 GCs with literature distances + DEB distance to 47 Tuc V − . ± .
027 (stat) ± .
045 (sys) Jang et al. (2020) Maser in NGC4258 V − . ± . (stat) ± . (sys) This Work 46 GCs - Cluster ZAHBs and DEB distance to ω Cen J − . ± .
01 (stat) ± .
05 (sys) Hoyt et al. (2018) 8 DEBs in LMC J − . ± .
007 (stat) ± .
050 (sys) G´orski et al. (2018) 8 LMC DEBs; 4 SMC DEBs J − . ± .
022 (stat) ± .
039 (sys) Freedman et al. (2020) 18 DEBs in LMC J − . ± . (stat) ± . (sys) This Work 46 GCs - Cluster ZAHBs and DEB distance to ω Cen H − . ± .
01 (stat) ± .
05 (sys) Hoyt et al. (2018) 8 DEBs in LMC H − . ± .
008 (stat) ± .
048 (sys) G´orski et al. (2018) 8 LMC DEBs; 4 SMC DEBs H − . ± . (stat) ± . (sys) This Work 46 GCs - Cluster ZAHBs and DEB distance to ω Cen K − . ± .
01 (stat) ± .
05 (sys) Hoyt et al. (2018) 8 DEBs in LMC K − . ± .
008 (stat) ± .
048 (sys) G´orski et al. (2018) 8 LMC DEBs; 4 SMC DEBs K − . ± . (stat) ± . (sys) This Work 46 GCs - Cluster ZAHBs and DEB distance to ω Cen
Note —The Jang et al. (2020) V − band TRGB measurement listed here is converted from that work’s F W − band measurement followingthe same technique utilized in Section 4, marginalizing over the difference between the F W and Johnson-Cousins I filters (and thusshould be considered approximate). The Hoyt et al. (2018) zero-points are rescaled here to the reddening value determined in Freedmanet al. (2020). The statistical uncertainties for J, H, K from G´orski et al. (2018) are taken to be the errors on their fits to the TRGBmagnitude relations (their Table 3), and the systematic error is taken to be the quadrature sum of the uncertainty in their adopted LMCdistance modulus and the typical uncertainty in their absolute reddening zero-point. numpy (Van Der Walt et al. 2011), scikit-learn (Pe-dregosa et al. 2012)REFERENCES