MNRAS 505, 5957–5977 (2021) https://doi.org/10.1093/mnras/stab1474 Advance Access publication 2021 May 24

Accurate distances to Galactic globular clusters through a combination of EDR3, HST, and literature data

H. Baumgardt 1‹ and E. Vasiliev 2,3 1School of Mathematics and Physics, The University of Queensland, St. Lucia, QLD 4072, Australia 2Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK 3Lebedev Physical Institute, Leninsky prospekt 53, 119991 Moscow, Russia Downloaded from https://academic.oup.com/mnras/article/505/4/5957/6283731 by UQ Library user on 30 July 2021 Accepted 2021 May 18. Received 2021 May 17; in original form 2021 March 17

ABSTRACT We have derived accurate distances to Galactic globular clusters by combining data from the Gaia Early Data Release 3 (EDR3) with distances based on (HST) data and literature-based distances. We determine distances either directly from the Gaia EDR3 parallaxes, or kinematically by combining line-of-sight velocity dispersion profiles with Gaia EDR3 and HST-based proper motion velocity dispersion profiles. We furthermore calculate cluster distances from fitting nearby subdwarfs, whose absolute luminosities we determine from their Gaia EDR3 parallaxes, to main sequences. We finally use HST-based stellar number counts to determine distances. We find good agreement in the average distances derived from the different methods down to a level of about 2 per cent. Combining all available data, we are able to derive distances to 162 Galactic globular clusters, with the distances to about 20 nearby globular clusters determined with an accuracy of 1 per cent or better. We finally discuss the implications of our distances for the value of the local Hubble constant. Key words: : distances – globular clusters: general.

using for example theoretical models (e.g. Catelan, Pritzl & Smith 1 INTRODUCTION 2004)orHipparcos or Gaia parallaxes (Fernley et al. 1998; Neeley Galactic globular clusters constitute an important rung on the et al. 2019;Ripepietal.2019). Accurate distances have also been extragalactic distance ladder. They are nearby enough to be resolved obtained for globular clusters by using eclipsing binaries (Kaluzny into individual stars, making them ideal objects to calibrate the et al. 2007; Thompson et al. 2020), however the faintness and scarcity brightnesses of physically interesting stars like RR Lyrae or Type II of suitable binaries means that observations have so far been limited Cepheids. In addition, they are also massive enough to contain statis- to a few globular clusters. Finally, it is possible to determine distances tically significant samples of these stars. Hence, globular clusters are by comparing the magnitudes of main-sequence stars with stars of useful to determine the slopes and zero-points of RR Lyrae period– similar in the solar neighbourhood, the so-called subdwarf luminosity (P/L) relations (e.g. Bono, Caputo & Di Criscienzo 2007; method (e.g. Reid & Gizis 1998; Cohen & Sarajedini 2012)or Dambis et al. 2014) as well as a possible metallicity dependence kinematically by comparing line-of-sight and proper motion velocity of the zero-points (e.g. Sollima, Cacciari & Valenti 2006). Globular dispersion profiles in globular clusters (e.g. McNamara, Harrison & clusters can also be used as calibrators for other distance methods, Baumgardt 2004;vandeVenetal.2006; Watkins et al. 2015a). like the tip of the red giant branch distance method (TRGB), which The latter method has the advantage that the derived distance is not allows to determine distances to external without having influenced by the reddening of the cluster. to rely on P/L relations of variable stars (e.g. Cerny et al. 2020; Typical uncertainties in the zero-points of the P/L relations of Freedman et al. 2020; Soltis, Casertano & Riess 2021). RR Lyrae stars are thought to be of the order of 0.05 mag (Bhardwaj Distances to globular clusters are determined either through 2020) and other methods like CMD fitting have similar uncertainties. fits of their colour–magnitude diagrams (CMDs) with theoretical In addition, the reddening of many clusters presents an additional isochrones (e.g. Ferraro et al. 1999; Dotter et al. 2010; Gontcharov, challenge for the determination of accurate distances since it can be Mosenkov & Khovritchev 2019; Valcin et al. 2020), or by using variable even across small, arcminute size fields (Bonatto, Campos variable stars that follow known relations between their periods and & Kepler 2013; Pallanca et al. 2019). Furthermore, there is evidence absolute luminosities like RR Lyrae stars (e.g. Bono et al. 2007; for non-standard reddening laws in the directions of several globular Hernitschek et al. 2019), Type II Cepheids (Matsunaga et al. 2006), clusters like for example M4 (Dixon & Longmore 1993; Hendricks or Mira type variables (Feast, Whitelock & Menzies 2002). In order et al. 2012). to avoid circularity, the absolute luminosities of these stars need It is therefore important to use independent methods to verify dis- to be determined independently from globular cluster distances by tances to globular clusters, especially methods that are not affected by the reddening of stars. In this paper, we use data from the Gaia EDR3 catalogue (Gaia Collaboration 2016, 2020) to determine distances E-mail: [email protected] to Galactic globular clusters. In addition to using the Gaia EDR3

C 2021 The Author(s) Published by Oxford University Press on behalf of Royal Astronomical Society 5958 H. Baumgardt and E. Vasiliev parallaxes directly, we also determine moving group distances and The small-scale correlated errors were already taken into account kinematic distances derived by comparing proper motion velocity by the parallax averaging procedure of VB21. We list the median dispersion profiles with line-of-sight velocity dispersion ones. We distances D = 1/ for clusters with /σ > 10 in Table 2 to allow finally use the Gaia EDR3 parallaxes of nearby subdwarfs as well as a comparison of the Gaia EDR3 parallax distances with the other Hubble Space Telescope (HST) counts together with kinematic distances that we derive. When calculating mean cluster distances information to determine distances. we use the data for all clusters and the parallax values directly as will Our paper is organized as follows: In Section 2, we present our be described further below. derivation of cluster distances using the methods mentioned above. In Section 3, we describe our survey of literature distances. We compare the distances that we derive from the different methods and derive 2.3 Kinematic distances the average distance to each globular cluster in Section 4. We finally draw our conclusions in Section 5. For clusters with accurately measured line-of-sight and proper mo- Downloaded from https://academic.oup.com/mnras/article/505/4/5957/6283731 by UQ Library user on 30 July 2021 tion velocity dispersion profiles one can determine cluster distances kinematically by varying the cluster distance until the velocity 2ANALYSIS dispersions are the same or a best match to a theoretical model of the cluster is achieved (e.g. Henault-Brunet´ et al. 2019). The 2.1 Cluster sample and selection of member stars advantage of such kinematic distances is that, like parallaxes, the We take our target list of globular clusters from Baumgardt et al. kinematic distance method is a direct method which does not rely (2019b). To this sample, we add six additional globular on other, more nearby methods in the distance ladder. Similar to clusters that have been found in recent years: VVV-CL001 (Minniti parallax distances, kinematic distances are also not affected by cluster et al. 2011;Fernandez-Trincado´ et al. 2021), BH 140 and FSR 1758 reddening, allowing in principle to derive accurate distances for (Cantat-Gaudin et al. 2018), Sagittarius II (Laevens et al. 2015; highly reddened clusters. In order to calculate kinematic distances, Mutlu-Pakdil et al. 2018), RLGC 1 and RLGC 2 (Ryu & Lee 2018), one either needs to know the anisotropy profile of the internal and Laevens 3 (Longeard et al. 2019). The status of Sagittarius II velocity dispersion of a cluster, or assume that the cluster is isotropic. is still debated, while Mutlu-Pakdil et al. (2018) argue for it to be a The latter is most often assumed since observed velocity dispersion based on its location in a size versus luminosity plane, profiles usually show that globular clusters are isotropic, at least in Longeard et al. (2020) argue, based on a small spread in [Fe/H] their inner parts (van Leeuwen et al. 2000; Watkins et al. 2015b;Raso that they find among the cluster stars, for it to be a dwarf . We et al. 2020;Cohenetal.2021), N-body simulations of star clusters tentatively include the object in our list of clusters. The other systems also show that most globular clusters should have isotropic velocity are most likely globular clusters based on their CMDs, kinematics, dispersion profiles (Baumgardt & Makino 2003;Lutzgendorf¨ et al. and location in the Milky Way. Together with the 158 GCs studied in 2011; Tiongco, Vesperini & Varri 2016). Baumgardt et al. (2019b), we therefore have a sample of 162 globular We take the line-of-sight velocity dispersion profiles for the clusters. kinematic distance fitting from Baumgardt & Hilker (2018)and Baumgardt et al. (2019a, b). Baumgardt & Hilker (2018) calculated line-of-sight velocity dispersion profiles by combining literature 2.2 Cluster parallax distances velocities with line-of-sight velocities derived from archival spectra Determining distances to globular clusters via the parallaxes of from the ESO and Keck data archives. Baumgardt et al. (2019a, individual stars has the advantage that the distances are determined b) added line-of-sight velocities from Gaia DR2 and the Anglo- directly, without having to rely on another distance method as an Australian Observatory data archive to these profiles. In addition, we intermediate step in the distance ladder. In addition, parallaxes are use the line-of-sight velocity dispersion profiles published by Ka- not influenced by the reddening of the clusters. In this paper, we take mann et al. (2018) based on MUSE data. We take the proper motion the cluster parallaxes from Vasiliev & Baumgardt (2021, hereafter dispersion profiles either from VB21 based on Gaia EDR3 data, or VB21). VB21 determined mean parallaxes for 170 globular and outer from published profiles based on multi-epoch HST measurements. halo star clusters by averaging the cluster parallaxes of individual Table 1 lists the HST proper motion velocity dispersion profiles that member stars from the Gaia EDR3 catalogue. Cluster members were we use in this paper together with the distances that we derive from selected based on the Gaia EDR3 proper motions and parallaxes. these profiles. Unfortunately, we could not use the recently published VB21 also applied multi-Gaussian mixture modelling to classify proper motion dispersion profiles for nine inner Milky Way globular stars into cluster members and background stars and determined the clusters by Cohen et al. (2021) since no accurate line-of-sight velocity mean cluster parameters and their errors over multiple realizations dispersion profiles are available for these clusters. of statistically created member catalogues. In order to account for We determine a kinematic distance independently for each proper possible magnitude, colour, and position-dependent biases in the motion velocity dispersion profile that we have from either Gaia Gaia catalogue, VB21 applied the parallax corrections of Lindegren EDR3 or HST data. The best-fitting distance is determined by fitting et al. (2020) to the individual stellar parallaxes. However, when each cluster with the grid of N-body models calculated by Baumgardt testing the derived cluster parallaxes against the literature distances (2017) and Baumgardt & Hilker (2018). Their grid contains several that we derive further below, VB21 found that even after applying the thousand N-body models of star clusters, varying the size, initial Lindegren et al. (2020) corrections, there is still a mean offset in the density profile, metallicity, and stellar mass function of each cluster. Gaia parallaxes. In particular, they found that the parallax corrections We interpolate within this grid and determine for each cluster the N- suggested by Lindegren et al. (2020) might have been overcorrecting body model that simultaneously provides the best fit to the internal the Gaia EDR3 parallaxes by ∼ 0.007 mas. VB21 also found mass function, velocity dispersion, and surface density profile. More evidence for spatially correlated small scale systematic errors of details about the N-body models can be found in Baumgardt (2017) order ∼ 0.01 mas. We therefore subtract = 0.007 mas and Baumgardt & Hilker (2018) and we refer the reader to these from the derived mean parallaxes to correct for the parallax bias. papers for a full description of the fitting procedure.

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Table 1. Kinematic distances based on HST proper motion velocity disper- 2.4 Subdwarf distances sion profiles. The second last column gives the kinematic distance derived by the authors of the original HST data set, the last column gives our distances The main idea of our subdwarf distances is to make use of the for the same data set. excellent accuracy of Gaia EDR3 parallaxes for nearby stars. As discussed in Section 2.2, Gaia EDR3 parallaxes have small-scale, Cluster HST data set Lit. dist. Our dist. systematic uncertainties of ∼0.01 mas that cannot be removed (kpc) (kpc) through calibration of the parallaxes against objects with known parallaxes. This means that even the distances to the nearest globular NGC 104 Watkins et al. (2015b)4.45± 0.50 4.545 ± 0.047 clusters cannot be determined directly with an accuracy better than Heyl et al. (2017)4.29± 0.47 4.347 ± 0.236 NGC 288 Watkins et al. (2015b)9.98± 0.37 9.098 ± 0.291 a few per cent. The accuracy of parallax distances also quickly NGC 362 Watkins et al. (2015b)9.37± 0.18 9.202 ± 0.280 deteriorates with increasing distance and drops below the accuracy of NGC 1261 Raso et al. (2020) – 16.775 ± 0.824 CMD fitting distances beyond distances of about 5 kpc. In contrast,

NGC 1851 Watkins et al. (2015b) 11.41 ± 0.20 11.440 ± 0.254 the distance to a nearby star at d = 100 pc has a relative uncertainty Downloaded from https://academic.oup.com/mnras/article/505/4/5957/6283731 by UQ Library user on 30 July 2021 NGC 2808 Watkins et al. (2015b) 10.18 ± 0.12 9.837 ± 0.122 of only 0.1 per cent using Gaia EDR3 parallaxes. NGC 5139 Watkins et al. (2015b)5.22± 0.05 5.264 ± 0.082 Our basic approach for deriving subdwarf distances is the same NGC 5904 Watkins et al. (2015b)8.77± 0.15 7.456 ± 0.146 as the one used by Cohen & Sarajedini (2012),exceptthatwe ± ± NGC 6266 Watkins et al. (2015b)5.670.07 6.354 0.128 replace the Hipparcos parallaxes by Gaia EDR3 ones. We start using ± McNamara et al. (2012) – 6.945 0.264 the compilation of precise Johnson–Cousins UBVRI photometry NGC 6341 Watkins et al. (2015b)8.43± 0.34 8.231 ± 0.347 of nearby subdwarfs by Casagrande et al. (2010) and identify the NGC 6352 Libralato et al. (2019) – 6.260 ± 0.762 NGC 6362 Watkins et al. (2015b)7.34± 0.24 7.720 ± 0.315 counter-parts of these stars in the Gaia EDR3 catalogue. We then NGC 6388 Watkins et al. (2015b) 10.44 ± 0.12 10.894 ± 0.137 apply the Lindegren et al. (2020) parallax corrections to the stars as NGC 6397 Watkins et al. (2015b)2.54± 0.05 2.332 ± 0.057 well as the additional shift of = 0.007 mas found by VB21. Heyl et al. (2012)2.20± 0.60 2.416 ± 0.062 Using only stars that have published V-andI-band photometry by NGC 6441 Watkins et al. (2015b) 11.77 ± 0.20 12.059 ± 0.172 Casagrande et al. (2010), a reddening E(B − V) < 0.02 and Gaia ± ± Haberle¨ et al. (2021) 12.74 0.16 12.364 0.226 EDR3 parallaxes with a relative precision / > 20 leaves us with NGC 6624 Watkins et al. (2015b)6.69± 0.36 7.972 ± 0.277 a sample of 206 stars. Despite our more stringent limit on the parallax ± ± NGC 6656 Watkins et al. (2015b)3.180.07 3.161 0.070 accuracy, our sample is almost a factor of 10 larger than the one used ± ± NGC 6681 Watkins et al. (2015b)9.330.14 9.260 0.165 by Cohen & Sarajedini (2012). Thanks to the superior accuracy of the NGC 6715 Watkins et al. (2015b) 23.79 ± 0.33 25.019 ± 0.646 ± ± Gaia EDR3 parallaxes, our sample also contains about 20 stars with NGC 6752 Watkins et al. (2015b)4.280.03 4.005 0.130 − NGC 7078 Watkins et al. (2015b) 10.23 ± 0.13 10.375 ± 0.237 [Fe/H] < 2.0, while no accurate Hipparcos parallaxes were available for any of these stars. We finally use the SIMBAD astronomical data base to exclude known binaries from this sample, leaving us with a sample of 185 stars. Our source for the globular cluster photometry is the compilation We restrict the radial extent of the line-of-sight velocity dispersion of HST photometry published by Sarajedini et al. (2007) based on profiles to the same range for which we have proper motion observations made with the Advanced Camera for Surveys (ACS) on- dispersions in order to avoid a possible bias due to the N-body board HST. Their survey contains deep photometry in the ACS/WFC models not providing a match to the full profile. In the N-body F606W and F814W bands for 67 globular clusters. In addition to the models, we measure the line-of-sight velocity dispersion profile photometry for 67 globular clusters from Sarajedini et al. (2007), we for stars brighter than the main-sequence turn-off. The magnitude also use the ACS/WFC photometry published by Dotter, Sarajedini limit for the proper motion velocity dispersion profiles was varied & Anderson (2011) for six globular clusters as well as the photometry for each data set independently so we match the magnitude limit for NGC 6528 from Lagioia et al. (2014) that was kindly provided of the observed data. Hence, except for the nearest clusters like to us by the authors. We finally use the photometry from Baumgardt NGC 6121, we chose a magnitude limit equal to the main-sequence et al. (2019b) for an additional six globular clusters (NGC 6325, turn-off for fits to the Gaia EDR3 proper motions, while we choose NGC 6342, NGC 6355, NGC 6380, NGC 6401, and NGC 6558). We deeper limits for the HST data. The left-hand and middle panels of skipped Pal 2 due to significant differential reddening and E 3 since Fig. 1 show the resulting fits to two clusters that have both Gaia the cluster has such a low mass that the location of its main sequence and HST data and Table 1 gives the derived distances from the HST cannot be accurately determined. This leaves us with a sample of 79 data together with the kinematic distances derived by the authors globular clusters with accurate, deep ACS/WFC photometry in the of the original papers. There is usually good agreement between the F606W and F814W bands. distances derived by us and the original kinematic distances. The only We dereddened the photometry of each globular cluster using exceptions are NGC 5904 and NGC 6266. At least for NGC 5904, one the reddening values given by Dotter et al. (2010) together with the reason for different distances could be the lack of a radial overlap the extinction coefficients from Sirianni et al. (2005). We then between the line-of-sight and proper motion velocity dispersion transformed the subdwarf photometry, from the Johnson–Cousins profiles used by Watkins et al. (2015b), which makes the derived UBVRI system to the ACS/WFC filter system using equation 12 of distance strongly dependent on the dynamical model used for the Sirianni et al. (2005) together with the transformation coefficients cluster. given in table 18 of their paper. We then fitted isochrones We give the full set of distances including the Gaia ED3 kinematic (Bressan et al. 2012) to the CMD of each cluster, and used the distances in the appendix. The right-hand panel of Fig. 1 shows the isochrone to select the cluster members. We then calculated the error weighted distribution of differences in the kinematic distances average colour of the cluster stars as a function of magnitude and for the Gaia EDR3 and HST data. It can be seen that they roughly define a main-sequence ridge line by cubic spline interpolation follow a normal distribution, indicating good agreement between the between the data points. HST and Gaia distances.

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Figure 1. Illustration of the kinematic distance determination for the globular clusters NGC 6624 (left-hand panel) and NGC 6656 (middle panel). Shown are the χ 2 values from the fit of our N-body models to the velocity dispersion profiles as a function of distance for the HST proper motions (red circles) and Gaia EDR3 proper motions (blue triangles). Solid lines show a quadratic fit to the χ 2 values of each data set and the dashed lines and shaded areas mark the best-fitting distances and their 1σ error. The literature distances are D = 8020 pc (NGC 6624) and D = 3330 pc (NGC 6656) and are in agreement with the kinematic ones. The right-hand panel compare the error weighted distance differences from Gaia and HST data for clusters that have proper motion velocity dispersion profiles from both satellites. The resulting distribution is compatible with a normal distribution (shown by a blue solid line).

In order to compare the subdwarf photometry with the glob- velocities, the total cluster mass changes linearly with distance, while ular cluster photometry and derive the cluster distances, we se- for a cluster with a velocity dispersion profile based only on proper lect for each globular cluster all subdwarfs with metallicities motions, the mass changes with the distance to the third power. For a |[Fe/H]SD − [Fe/H]GC| < 0.2. We took the metallicities of the cluster with measured line-of-sight and proper motion velocities, the subdwarfs from Casagrande et al. (2010) and the metallicities of scaling will be in between these limits. Since the predicted number the globular clusters from Carretta et al. (2009). In order to correct of main-sequence stars changes linearly with the cluster mass (for the colours of the subdwarfs to the colour they would have at the a given mass function), one can use the observed number of cluster metallicity of the globular cluster, we created a set of PARSEC stars in a given magnitude interval and a given field, together with the isochrones equally spaced in metallicity by [Fe/H] = 0.5 and measured cluster kinematics and a theoretical model for the cluster, calculate the colours of zero-age main-sequence stars as a function of to determine the cluster distance. their absolute V-band luminosity. We then use this grid of isochrones We use as source for the globular cluster photometry the com- and the absolute magnitudes of the subdwarfs which we calculate pilation of ACS/WFC F606W/F814W photometry published by using their Gaia EDR3 parallaxes to correct the subdwarf colours Sarajedini et al. (2007)aswelltheHST photometry that was derived for the difference in metallicity. Due to the proximity of the subdwarf in Baumgardt et al. (2019b), Baumgardt, Sollima & Hilker (2020), metallicities to the metallicity of the clusters, the resulting shifts are and Ebrahimi et al. (2020). We also use the photometry published always below 0.02 mag. by Kerber et al. (2018) for NGC 6626 as input photometry. We After transforming the subdwarf photometry to the ACS/WFC exclude NGC 5139 from this list due to the significant metallicity system and correcting the subdwarf metallicities, we fitted the cluster spread among the cluster stars. For each HST observation, we again main sequences in a F814W0 versus (F606W–F814W)0 CMD with fit PARSEC isochrones (Bressan et al. 2012) to the cluster CMDs. To the subdwarfs. To this end, we vary the assumed cluster distance create the isochrones, we use the cluster ages derived by VandenBerg modulus until the error-weighted difference between the subdwarfs et al. (2013), or, for clusters not studied by them, from available and the previously determined main-sequence ridgeline are minimal. literature sources, and take the cluster metallicities and reddenings Due to the steepness of the cluster ridgelines and the accurate Gaia from Harris (1996). From the isochrones we then derive individual parallax distances, this essentially results in minimizing the error stellar masses for the main-sequence stars in the clusters. We fit weighted colour differences between the subdwarfs and the main- the N-body models of Baumgardt (2017) and Baumgardt & Hilker sequence ridge lines. We exclude subdwarfs with M814 < 4 from (2018) to the observed surface density, velocity dispersion profiles, the fits since these could already have evolved away from the main and the observed mass function of main-sequence stars in the HST sequence. We also exclude subdwarfs with discrepant photometry fields and determine the best-fitting N-body model from the fit. For since these could be undetected binaries. Fig. 2 shows examples each HST observation, we fit a power-law mass function N(m) ∼ mα of our subdwarf fits for four globular clusters. The clusters shown to the observed stellar mass distribution as well as the masses of stars are the same as the ones depicted in fig. 2 of Cohen & Sarajedini in the N-body models in the same field. We use the mass function (2012). We will derive an error bar for the subdwarf distance moduli slopes α as fit parameters but not the absolute number of stars. Fig. 3 in Section 3 when we compare the subdwarf distance moduli against shows as an example the distribution of HST fields with measured literature data. photometry (left-hand panel), the derived CMD (middle panel), and the measured stellar mass functions at different distances from the centre (right-hand panel) for the globular cluster NGC 104. 2.5 Star count distances Varying the cluster distance, we then determine the distance that For a given velocity dispersion profile, the derived mass of a cluster gives the best match between the number of main-sequence stars increases (decreases) with increasing (decreasing) cluster distance. in the best-fitting N-body model and the observed number of main- If the velocity dispersion profile is based entirely on line-of-sight sequence stars in the different HST fields. We restrict ourselves to

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Figure 2. Illustration of the subdwarf distance determination for four globular clusters with a range of metallicity and reddening values. Grey points mark stars considered to be background stars, and black points are cluster members. The green line shows our fitted ridge line for each cluster. Subdwarfs used in the distance fitting are shown by red circles, subdwarfs not used are shown by blue circles. Photometric error bars are also shown. Note that the error in due to the Gaia EDR3 parallax error is shown but usually too small to be seen. well-observed clusters in which the errors of the total mass (based binaries. Since the average mass of binaries is larger than that of on the cluster kinematics) are less than 5 per cent. single stars, real globular clusters contain fewer stellar systems than Fig. 4 depicts the ratio between the best-fitting cluster distances models with only single stars for the same total mass, leading to based on the star count method against the mean cluster distances an underprediction of the cluster distances with our method. The that we derive from the other methods. We split the cluster sample size of this underprediction is also in rough agreement with the into two groups, clusters with 10 or more distance determinations observed binary fractions, which are around 10 per cent (Sollima from the other methods and clusters with less than 10 measurements. et al. 2007; Milone et al. 2012; Giesers et al. 2019). Fig. 4 also We expect that the distances of clusters in the first group are indicates that the star count distances contain additional errors that well determined so that any deviation is mainly due to errors in are not accounted for by the uncertainties in the cluster kinematics, the star count distances. It can be seen that the average distance since even when applying a constant shift, the deviations between the ratio is well below unity. The reason is probably that the N-body star count distances and the mean distances of the other methods are models that we use to fit the clusters do not contain primordial larger than what can be accounted for by the errors in either method.

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Figure 3. Illustration of the fit of our N-body models to the globular cluster NGC 104. The left-hand panel shows all fields in NGC 104 with available HST photometry. The different fields are labelled by the name of the Principal Investigator of the HST observations and the white circle marks the observed half-light radius of NGC 104. The middle panel shows the fit of a PARSEC isochrone (shown in red) to the combined photometry of the four HST/WFPC2 fields of Rhoads (HST Proposal ID: 9634) from the left-hand panel. The parameters of the isochrone are given in the figure. Grey circles mark all stars in the field and black stars are the stars used for the determination of the stellar mass function of NGC 104. The group of stars visible in the lower left part of the middle panel is theSMC. The right-hand panel compares the derived mass functions as a function of radius against the mass function of the best-fitting N-body model (shown by solid lines). There is a clear change in the derived mass function slope from increasing towards higher masses in the centre to increasing towards lower masses in the outermost parts, indicating that NGC 104 is mass segregated. This trend is reproduced by the best-fitting N-body model.

2.6 Moving group distances Stars in star clusters move on very similar trajectories through the Milky Way since the internal motion of stars in the cluster is usually two orders of magnitudes smaller than the velocity with which the cluster moves around the Galactic centre. As the distance and viewing angle to the stars changes along the orbit, the proper motion and line- of-sight velocity of the stars will change as well and one can use this variation to measure the distance to the cluster. This is the so-called moving group or convergent point method (Brown 1950; de Bruijne 1999). To first order, the change in proper motion μ due to these perspective effects can be approximated by v α μ =− Los mas yr−1, (1) DA −1 where vLos is the line-of-sight velocity of the cluster in km s , α the distance from the cluster centre in the direction of motion in radians, D the distance of the cluster from the Sun in kpc, and A = 4.74 (Brown 1950;vandeVenetal.2006). Using the above equation, VB21 determined moving group distances to about 40 globular clusters based on Gaia EDR3 proper motions and the average line-of-sight velocities of the clusters from Baumgardt & Hilker (2018). Here, we use the two most precisely determined distances D = 4.62 ± 0.28 kpc Figure 4. Ratio of the best-fitting distance based on star counts to the (NGC 3201) and D = 5.34 ± 0.55 kpc (NGC 5139). For all other average distance from the other methods as a function of the global mass clusters, the derived distances have too large error bars to be useful. function slope α of the clusters. The cluster sample is split into clusters with 10 or more distance measurements (blue circles) and those with less than 10 individual measurements (red triangles). The dashed line shows the 3 LITERATURE SURVEY OF GLOBULAR average DStarCount/DMean ratio for the former group which has more secure CLUSTER DISTANCES distance determinations. It can be seen that the average ratio is well below unity. In addition to our own distance determinations, we also performed a literature search of globular cluster distances. Limiting our search to the last 20–25 yr, we were able to find about 1300 distance These additional errors could arise due to for example mismatches determinations in total, with each globular cluster having between between the chosen mass functions of the N-body models and those 1 and 35 individual measurements. The distances used are given of the real clusters or errors in the conversion from luminosity to in Table B1 in the Appendix (available online in the supporting mass from the isochrones. We will fit for an additional shift in the information). Whenever a paper gave multiple measurements for the star count distances as well as additional errors in Section 4 when same cluster, we averaged these and took the scatter between the we compare all distances. individual values as an indication of the uncertainty. When we had

MNRAS 505, 5957–5977 (2021) Accurate distances to Galactic globular clusters 5963 multiple distance determinations from the same research group which in the data. We use only clusters with E(B − V) < 0.4 mag were based on the same method and similar input data, we included for the comparison since more strongly reddened clusters show only the newest value since we expect the different distances to not larger deviations between the individual measurements, and would be independent from each other and the newest measurement to be therefore require a separate treatment. We compare all distance the most accurate one. When a paper gave both a distance and a moduli against the CMD fitting ones since these are available for distance modulus to a cluster, we used the distance modulus since the largest number of clusters. we expect the errors to be Gaussian only in distance modulus. The It can be seen that the RR Lyrae distance moduli are smaller by resulting list of distance moduli and distances can be found in the RRL =−0.021 mag compared to the CMD fitting ones, which is appendix. significant at the 2σ level. Our subdwarf distance moduli are also In order to calculate a mean distance modulus for each cluster, smaller, while the other methods show no statistically significant we de-reddened distance moduli using the reddening values given in offset to the CMD distances. Since without independent information, the papers, or, if no value was available, given in the 2010 edition it is not possible to determine which of these distance scales is closer Downloaded from https://academic.oup.com/mnras/article/505/4/5957/6283731 by UQ Library user on 30 July 2021 of Harris (1996). We also transformed linear distances into distance to the true distances, we do not apply shifts to any individual method moduli for those papers which did not give distance moduli. For and perform a straight averaging of the individual measurements distance moduli without an error bar, we assumed an error equal instead. Our MCMC sampling also finds evidence for additional to the mean error of the other distance determinations for the same systematic errors of ≈ 0.023 in the CMD fitting and RR Lyrae cluster, with a minimum of 0.10 mag. A comparison of distance distance moduli and additional systematic errors of EB ≈ 0.038 mag moduli for a few well-observed clusters shows that even in clusters in the eclipsing binary/IR RR Lyrae distance moduli. The MCMC with low reddening, individual distance moduli show a scatter of sampling finally shows that our subdwarf distance moduli have errors around 0.07–0.10 mag around the mean. This scatter is independent of about SD = 0.077 mag, which we use when calculating the mean of the chosen method, with only eclipsing binary and RR Lyrae distance moduli. distances based on infrared P/L relations showing smaller scatter. We Adding the above systematic errors in quadrature to the individual therefore set the minimum error of any measurement except for EB errors for the CMD, RR Lyrae, and eclipsing binary distances, we or infrared RR Lyrae distances, for which we use the quoted errors, to then calculate a mean literature distance modulus and its formal error 0.07 mag. We neglected distances that deviate strongly from the other for each cluster from the mean distance moduli of each individual 2 measurements. In total we had to exclude about 25 measurements out method. We also calculated a reduced χr value of all distance of the 1300 measurements we found this way. Most of the excluded determinations according to measurements were for highly reddened clusters with E(B − V) > 1.0  2 1 2 2 where the individual measurements also often show a larger scatter. χ = (μLit −μi ) /σ , (2) r N − 1 i This scatter probably reflects an uncertainty in the exact amount i of reddening and the proper selection of cluster and background where μLit and μi are the average distance modulus and the stars. individual distance moduli respectively and σ i is the error of the 2 We also found that the subdwarf distance moduli of Cohen individual measurements. For clusters with χr > 1, i.e. a larger than & Sarajedini (2012) were systematically smaller than the other expected scatter of the individual measurements, we increase the − 2 literature values for low-metallicity clusters with [Fe/H] < 1.2. final distance error by the square root of χr .Table2 gives the The average difference increased with decreasing metallicity and average distance and its 1σ upper and lower error that we obtain reached about 0.30 mag for the lowest-metallicity clusters. In order after averaging the literature distances for each cluster in this way. to correct this bias, we fitted a linear relation between the difference For easier comparison with the other methods, we give the literature in distance modulus and metallicity to the data and shifted the data in linear distances, however when calculating a final value for Cohen & Sarajedini (2012) distance moduli upwards to bring them the distance we use the distance modulus as explained below. Plots into agreement with the other literature data. We finally shifted the depicting the individual measurements for each cluster can be found distances that Recio-Blanco et al. (2005) derived from CMD fitting in the appendix. down by 0.10 mag since we found a systematic bias of this size when comparing their distances with other literature distances. 4 RESULTS In order to assess the accuracy of the derived distances and the reliability of the quoted errors, we compare in Fig. 5 the mean 4.1 Calculation of final distances distance moduli calculated from various methods (isochrone fitting of CMDs, visible light RR Lyrae P/L relations, eclipsing binaries Fig. 6 compares the distances that we derive from the Gaia EDR3 and infrared RR Lyrae P/L relations and a combination of all data (separated into parallax and kinematic distances), the HST other literature methods) plus the distance moduli we derived from kinematic distances and the star count distances with the mean subdwarfs with each other. For each cluster we take a weighted literature distances determined in the previous section. We again average of all individual measurements of a given type. We split the use MCMC sampling and determine offsets between the distances RR Lyrae distances into distances derived from visible or infrared as well as additional unaccounted for errors. We restrict ourselves P/L relations, since infrared P/L relations are usually thought to be again to clusters with E(B − V) < 0.4 mag for which the distances more accurate (e.g. Bono et al. 2016). We also group the infrared should be least influenced by reddening uncertainties. P/L distances together with the EB distances to have a larger We find a constant shift of the Gaia parallax distances of = sample of clusters with very accurate distances. For each cluster, 0.007 mas, confirming the results of VB21. There is no evidence for we calculate average distance moduli for each method separately a significant shift of the CMD distances against the other methods. and then use Monte Carlo Markov chain (MCMC) sampling and However, given that the error bar on CMD is about 0.04, the maximum likelihood optimization to determine possible shifts RR Lyrae distances would also be in statistical agreement with between the CMD distance moduli and the other distance moduli, the distances that we determined from the Gaia EDR3 data. The as well as additional systematic errors that could still be present MCMC sampling finds DGaia =−0.016 ± 0.020, implying that the

MNRAS 505, 5957–5977 (2021) 5964 H. Baumgardt and E. Vasiliev Downloaded from https://academic.oup.com/mnras/article/505/4/5957/6283731 by UQ Library user on 30 July 2021

Figure 5. Comparison of the distance moduli derived from isochrone fitting of CMDs (CMD), optical RR Lyrae P/L relations (RRL), eclipsing binaries and infrared RR Lyrae P/L relations (EB), and a combination of all other literature methods (OTH) and our subdwarf distance moduli (SD). For each of the methods shown, we use MCMC sampling to determine a maximum likelihood solution, solving for a constant shift in distance modulus of each method against the CMD isochrone fitting distances as well as additional systematic errors of each method. The RR Lyrae distance moduli are on average about RRL = 0.021 mag smaller than the CMD fitting and our subdwarf distance moduli are smaller by about 0.025 mag. The MCMC sampling also indicates additional errors of order 0.02 mag for the CMD and RR Lyrae distance moduli and 0.04 mag for the eclipsing binary ones. It finally finds an error of 0.08 mag for our subdwarf distance moduli. The insets compare the distance moduli for individual clusters. Clusters with E(B − V) < 0.40 that are used for the comparison are shown in blue, clusters with E(B − V) > 0.40 are shown in grey.

Gaia kinematic distances are in statistical agreement with the other the HST and Gaia kinematic distances to be in agreement with each methods. However the HST kinematic distances are smaller by about other, meaning that the error might well be with the CMD distances. 3.0 per cent. There is also some evidence for significant, additional The MCMC analysis finally shows that the star count distances need unaccounted for errors in the HST kinematic distances. We therefore to be increased by a factor 1/0.93 = 1.075 and that their uncertainties add an error equal to 0.017 times the distance in quadrature to the have an additional systematic contribution of 4.4 per cent of the formal distance error of the HST distances for each cluster. We do not cluster distances. apply a shift to the HST kinematic distances since it is first not clear After increasing the star count distances and adding additional which method gives the correct distance, and also because we found errors to the HST kinematic and star count distances, we determine

MNRAS 505, 5957–5977 (2021) Accurate distances to Galactic globular clusters 5965

Table 2. Derived distances of the studied globular clusters. The final column lists the number of independent distance determinations used to calculate the mean distance. The Gaia EDR3 parallax distances give the mean and 1σ uncertainty of the distance. We have applied shifts to the star count distances and Gaia parallaxes to bring all distances to a common scale and also have added systematic errors as described in the text.

Name GEDR3 parallax dist. GEDR3 kin. dist. HST kin. dist. Subdwarf dist. Star count dist. Lit. dist. Mean distance NTot (kpc) (kpc) (kpc) (kpc) (kpc) (kpc) (kpc)

+0.682 +0.682 2MASS-GC01 – – – – – 3.373−0.567 3.373−0.567 1 +0.456 +0.456 2MASS-GC02 – – – – – 5.503−0.421 5.503−0.421 2 +3.444 +3.444 AM 1 – – – – – 118.905−3.347 118.905−3.347 2 +0.951 +0.951 AM 4 – – – – – 29.013−0.920 29.013−0.920 2 +1.058 +0.346 +0.346 Arp 2 – – – 28.184−1.019 –28.721−0.342 28.726−0.342 11 + + 0.258 0.258 Downloaded from https://academic.oup.com/mnras/article/505/4/5957/6283731 by UQ Library user on 30 July 2021 BH 140 4.808−0.233 –––––4.808−0.233 1 +0.279 +0.265 BH 261 – – – – – 6.183−0.267 6.115−0.253 4 +4.334 +4.334 Crater – – – – – 147.231−4.210 147.231−4.210 3 +0.676 +0.671 Djor 1 – – – – – 9.854−0.632 9.879−0.628 4 +0.179 +0.178 Djor 2 – – – – – 8.758−0.176 8.764−0.174 5 +0.765 +0.268 +0.254 E3 7.326−0.633 ––––7.940−0.259 7.876−0.245 3 +0.656 +0.656 ESO 280 – – – – – 20.941−0.636 20.945−0.636 3 +0.208 +0.202 ESO 452 – – – – – 7.440−0.203 7.389−0.196 4 +2.937 +2.936 Eridanus – – – – – 84.723−2.839 84.684−2.838 1 +0.275 +0.270 FSR 1716 – – – – – 7.321−0.265 7.431−0.260 4 +0.543 +0.543 FSR 1735 – – – – – 9.082−0.512 9.082−0.512 4 +0.832 +0.764 FSR 1758 – – – – – 11.482−0.776 11.085−0.710 2 +0.145 +0.144 HP 1 – – – – – 6.944−0.142 6.995−0.141 10 +1.474 +1.469 IC 1257 – – – – – 26.632−1.397 26.587−1.392 2 +0.491 +0.345 +0.263 IC 1276 5.023−0.411 ––––4.159−0.318 4.554−0.243 3 +0.652 +0.254 +0.253 IC 4499 – – – 17.378−0.629 –18.880−0.250 18.891−0.250 11 +1.672 +1.672 Laevens 3 – – – – – 61.773−1.628 61.767−1.628 3 +0.353 +0.353 Liller 1 – – – – – 8.061−0.338 8.061−0.338 5 +0.368 +0.165 +0.163 Lynga 7 – – – 7.798−0.351 –7.867−0.161 7.899−0.159 8 +0.533 +0.523 Mercer 5 – – – – – 5.495−0.486 5.466−0.476 1 +0.189 ± ± +0.165 ± +0.040 +0.031 NGC 104 4.367−0.174 4.533 0.067 4.522 0.085 4.406−0.159 4.779 0.197 4.512−0.039 4.521−0.031 33 +0.731 ± ± +0.350 ± +0.096 +0.089 NGC 288 7.692−0.615 9.814 0.576 9.098 0.348 9.333−0.338 8.798 0.446 8.983−0.095 8.988−0.088 23 +0.317 ± ± +0.330 ± +0.114 +0.096 NGC 362 9.099−0.296 8.267 0.646 9.202 0.321 8.790−0.318 8.707 0.374 8.770−0.112 8.829−0.096 21 ± +0.611 ± +0.205 +0.192 NGC 1261 – – 16.775 0.872 16.293−0.589 16.680 0.726 16.353−0.202 16.400−0.190 15 ± +0.462 ± +0.156 +0.134 NGC 1851 – – 11.440 0.320 12.303−0.445 12.376 0.528 12.017−0.154 11.951−0.133 18 +0.182 +0.181 NGC 1904 – – – – – 13.080−0.179 13.078−0.179 13 +0.471 ± +0.175 +0.170 NGC 2298 – – – 10.000−0.450 7.471 1.208 9.890−0.172 9.828−0.167 10 +2.437 +2.437 NGC 2419 – – – – – 88.471−2.371 88.471−2.371 6 ± ± +0.435 ± +0.141 +0.112 NGC 2808 – 9.688 0.428 9.837 0.233 10.280−0.417 10.603 0.494 10.139−0.139 10.060−0.111 20 +0.239 ± +0.172 ± +0.046 +0.043 NGC 3201 4.819−0.217 4.745 0.176 – 4.571−0.165 4.416 0.202 4.749−0.046 4.737−0.042 29 +0.674 +0.214 +0.214 NGC 4147 – – – 17.947−0.649 –18.510−0.212 18.535−0.212 16 +0.425 ± ± +0.385 +0.214 NGC 4372 5.688−0.370 5.933 0.423 – – 4.593 0.614 5.959−0.362 5.713−0.207 5 +0.388 ± +0.101 +0.100 NGC 4590 – – – 10.328−0.374 10.396 0.841 10.409−0.100 10.404−0.099 24 +0.490 ± +0.246 ± +0.091 +0.084 NGC 4833 6.557−0.426 5.822 0.360 – 6.546−0.237 6.453 0.395 6.519−0.089 6.480−0.083 12 ± +0.696 ± +0.189 +0.185 NGC 5024 – 17.313 1.353 – 18.535−0.670 18.803 1.446 18.518−0.187 18.498−0.183 20 +0.677 +0.235 +0.235 NGC 5053 – – – 18.030−0.652 –17.515−0.232 17.537−0.232 20 +0.302 ± ± +0.056 +0.047 NGC 5139 5.485−0.272 5.359 0.141 5.264 0.121 – – 5.468−0.055 5.426−0.047 31 ± +0.362 ± +0.085 +0.082 NGC 5272 – 10.116 0.384 – 9.638−0.349 10.770 0.610 10.167−0.084 10.175−0.081 38 ± +0.408 +0.149 +0.145 NGC 5286 – 11.181 0.689 – 10.864−0.393 –11.071−0.147 11.096−0.143 13 +0.592 +0.164 +0.164 NGC 5466 – – – 15.776−0.571 –16.106−0.162 16.120−0.162 21 +0.630 +0.628 NGC 5634 – – – – – 25.990−0.615 25.959−0.613 4 +0.746 +0.745 NGC 5694 – – – – – 34.834−0.730 34.840−0.730 6 +0.605 +0.604 NGC 5824 – – – – – 31.754−0.594 31.713−0.593 5 +0.464 ± +0.246 +0.238 NGC 5897 – – – 12.359−0.447 12.252 1.141 12.607−0.241 12.549−0.233 8

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Table 2 – continued

Name GEDR3 parallax dist. GEDR3 kin. dist. HST kin. dist. Subdwarf dist. Star count dist. Lit. dist. Mean distance NTot (kpc) (kpc) (kpc) (kpc) (kpc) (kpc) (kpc)

+0.677 ± ± +0.283 ± +0.066 +0.060 NGC 5904 7.651−0.575 7.467 0.357 7.456 0.201 7.551−0.273 7.721 0.350 7.471−0.065 7.479−0.060 38 +0.900 ± +0.318 ± +0.118 +0.111 NGC 5927 8.606−0.744 8.209 0.647 – 8.472−0.306 8.757 0.371 8.215−0.116 8.270−0.109 14 +0.589 +0.529 NGC 5946 – – – – – 9.554−0.555 9.642−0.496 4 ± +0.393 ± +0.137 +0.132 NGC 5986 – 10.269 0.668 – 10.471−0.379 10.885 0.837 10.549−0.135 10.540−0.130 14 +0.389 ± +0.120 +0.116 NGC 6093 – – – 10.375−0.375 10.193 0.489 10.342−0.118 10.339−0.115 20 +0.537 +0.188 +0.187 NGC 6101 – – – 14.322−0.518 –14.454−0.185 14.449−0.184 13 +0.035 ± +0.071 ± +0.020 +0.015 NGC 6121 1.830−0.034 1.878 0.033 – 1.888−0.068 2.016 0.090 1.839−0.020 1.851−0.016 23 +0.488 +0.463 Downloaded from https://academic.oup.com/mnras/article/505/4/5957/6283731 by UQ Library user on 30 July 2021 NGC 6139 – – – – – 10.046−0.465 10.035−0.441 6 +0.866 +0.319 +0.129 +0.127 NGC 6144 8.292−0.716 ––8.511−0.308 –8.147−0.127 8.151−0.125 8 +0.411 ± +0.254 ± +0.081 +0.077 NGC 6171 5.464−0.357 6.017 0.407 – 5.998−0.244 5.479 0.407 5.629−0.080 5.631−0.076 18 +0.822 ± +0.277 +0.079 +0.076 NGC 6205 8.621−0.690 6.906 0.336 – 7.379−0.267 –7.427−0.078 7.419−0.075 25 +0.282 ± +0.200 ± +0.052 +0.049 NGC 6218 5.068−0.254 4.632 0.194 – 5.321−0.192 5.352 0.265 5.136−0.052 5.109−0.048 24 +0.475 +0.475 NGC 6229 – – – – – 30.102−0.468 30.106−0.467 7 +0.395 +0.387 NGC 6235 – – – – – 11.896−0.383 11.937−0.374 5 +0.412 ± +0.235 ± +0.070 +0.064 NGC 6254 5.394−0.358 4.976 0.199 – 4.989−0.225 5.346 0.310 5.051−0.069 5.067−0.063 19 +0.650 ± +0.326 +0.299 NGC 6256 – – – 7.516−0.598 7.680 0.804 7.204−0.312 7.242−0.288 9 +0.426 ± ± ± +0.172 +0.105 NGC 6266 5.764−0.371 6.395 0.327 6.502 0.163 – 6.956 0.325 6.249−0.167 6.412−0.104 13 +0.683 ± ± +0.203 +0.165 NGC 6273 7.657−0.580 7.658 0.424 – – 8.463 0.536 8.527−0.198 8.343−0.163 10 +0.435 +0.427 NGC 6284 – – – – – 14.309−0.422 14.208−0.415 6 +0.721 +0.427 +0.376 NGC 6287 7.184−0.601 ––––8.143−0.406 7.929−0.356 6 ± +0.297 +0.282 NGC 6293 – – – – 9.273 1.131 9.209−0.288 9.192−0.274 7 +0.504 ± +0.214 +0.161 +0.150 NGC 6304 6.373−0.435 7.304 0.684 – 5.702−0.206 –6.059−0.157 6.152−0.146 13 ± ± +0.499 +0.393 NGC 6316 – 9.631 0.901 – – 9.131 1.785 11.792−0.479 11.152−0.382 7 +0.646 +0.795 +0.369 +0.330 NGC 6325 6.627−0.540 ––8.241−0.725 –7.755−0.352 7.533−0.314 7 +0.775 +0.147 +0.144 NGC 6333 8.052−0.650 ––––8.310−0.144 8.300−0.142 8 ± ± +0.333 ± +0.075 +0.071 NGC 6341 – 8.627 0.473 8.231 0.374 8.872−0.321 8.188 0.345 8.511−0.074 8.501−0.070 35 +0.787 +0.302 +0.244 +0.233 NGC 6342 7.716−0.654 ––8.054−0.291 –8.043−0.237 8.013−0.226 6 +0.378 ± +0.214 ± +0.077 +0.073 NGC 6352 5.583−0.333 – 6.260 0.816 5.702−0.206 5.564 0.295 5.534−0.076 5.543−0.072 16 +0.707 +0.318 ± +0.259 +0.224 NGC 6355 7.236−0.591 ––8.472−0.306 9.444 0.449 8.527−0.251 8.655−0.220 8 +0.973 +0.946 NGC 6356 – – – – – 15.900−0.917 15.656−0.890 4 +0.731 ± ± +0.287 ± +0.071 +0.067 NGC 6362 8.026−0.618 7.717 0.440 7.720 0.341 7.656−0.277 6.849 0.326 7.677−0.070 7.649−0.066 28 +0.181 ± +0.167 ± +0.055 +0.051 NGC 6366 3.643−0.165 3.404 0.256 – 3.548−0.160 3.861 0.297 3.407−0.054 3.444−0.050 14 +0.358 +0.312 +0.306 NGC 6380 – – – 9.550−0.345 –9.532−0.302 9.607−0.296 4 ± +0.404 ± +0.280 +0.162 NGC 6388 – – 10.894 0.230 10.765−0.389 11.196 0.466 11.545−0.273 11.171−0.161 15 +0.061 ± ± +0.090 ± +0.024 +0.019 NGC 6397 2.458−0.058 2.467 0.056 2.371 0.051 2.410−0.087 2.531 0.107 2.512−0.024 2.482−0.019 29 +0.781 +0.308 ± +0.316 +0.238 NGC 6401 7.283−0.643 ––8.204−0.297 8.448 0.376 7.910−0.304 8.064−0.234 7 +0.847 +0.267 +0.255 NGC 6402 8.482−0.706 ––––9.204−0.259 9.144−0.248 6 +1.022 +0.356 +0.355 NGC 6426 – – – 21.677−0.976 –20.701−0.350 20.710−0.349 8 +0.742 ± +0.281 +0.248 NGC 6440 7.283−0.617 7.394 0.776 – – – 8.461−0.272 8.248−0.241 7 ± ± +0.245 +0.163 NGC 6441 – – 12.190 0.244 – 13.095 0.674 13.176−0.240 12.728−0.162 16 ± +0.238 +0.220 NGC 6453 – – – – 11.149 0.523 9.804−0.232 10.070−0.216 8 ± +0.345 ± +0.161 +0.153 NGC 6496 – 9.823 0.612 – 9.204−0.333 9.326 0.988 9.647−0.159 9.641−0.151 11 ± +0.717 +0.578 NGC 6517 – 10.202 1.704 – – – 8.913−0.664 9.227−0.538 4 ± +0.221 +0.211 NGC 6522 – 8.120 0.929 – – – 7.158−0.214 7.295−0.205 8 +0.244 +0.239 NGC 6528 – – – – – 7.791−0.237 7.829−0.232 8 +0.669 +0.274 +0.127 +0.124 NGC 6535 6.570−0.556 ––6.486−0.263 –6.353−0.125 6.363−0.122 8 +0.866 ± ± +0.609 +0.395 NGC 6539 8.292−0.716 8.686 0.746 – – 8.694 1.140 7.565−0.563 8.165−0.379 5 +0.301 +0.279 NGC 6540 – – – – – 5.897−0.286 5.909−0.265 4 +0.648 ± +0.274 ± +0.113 +0.102 NGC 6541 7.429−0.552 7.056 0.324 – 7.311−0.264 8.493 0.416 7.617−0.111 7.609−0.101 11 +0.074 ± +0.113 +0.059 NGC 6544 2.585−0.070 2.516 0.239 – – – 2.589−0.109 2.582−0.057 6 +0.370 ± +0.141 +0.128 NGC 6553 5.488−0.326 6.211 0.457 – – – 5.219−0.138 5.332−0.125 12

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Table 2 – continued

Name GEDR3 parallax dist. GEDR3 kin. dist. HST kin. dist. Subdwarf dist. Star count dist. Lit. dist. Mean distance NTot (kpc) (kpc) (kpc) (kpc) (kpc) (kpc) (kpc)

+0.298 +0.306 +0.294 NGC 6558 – – – 7.943−0.287 –7.495−0.294 7.474−0.282 6 ± +0.275 +0.261 NGC 6569 – – – – 10.961 1.031 10.529−0.268 10.534−0.255 9 +0.475 +0.177 +0.176 NGC 6584 – – – 12.647−0.457 –13.602−0.174 13.611−0.174 13 ± ± +0.286 +0.119 +0.108 NGC 6624 – 8.027 0.530 7.972 0.308 7.621−0.276 –8.017−0.117 8.019−0.107 15 +0.325 ± ± +0.114 +0.099 NGC 6626 5.297−0.289 5.734 0.306 – – 5.571 0.403 5.311−0.111 5.368−0.098 10 ± +0.345 ± +0.107 +0.106 NGC 6637 – 9.960 1.149 – 9.204−0.333 9.139 1.054 8.880−0.106 8.900−0.104 17 +0.363 +0.347 NGC 6638 – – – – – 9.786− 9.775−0.334 5

0.350 Downloaded from https://academic.oup.com/mnras/article/505/4/5957/6283731 by UQ Library user on 30 July 2021 +0.302 +0.209 +0.204 NGC 6642 – – – 8.054−0.291 –8.009−0.204 8.049−0.198 7 +0.357 +0.141 +0.139 NGC 6652 – – – 9.506−0.344 –9.467−0.138 9.464−0.137 13 +0.119 ± ± +0.127 ± +0.049 +0.037 NGC 6656 3.368−0.111 3.181 0.123 3.161 0.088 3.373−0.122 3.556 0.149 3.327−0.049 3.303−0.037 14 ± ± +0.342 ± +0.130 +0.107 NGC 6681 – 9.722 0.578 9.260 0.228 9.120−0.330 9.634 0.414 9.354−0.128 9.362−0.106 15 +0.676 ± +0.294 +0.240 NGC 6712 7.402−0.571 7.463 0.546 – – – 7.355−0.282 7.382−0.233 9 ± ± +0.383 +0.328 NGC 6715 – – 25.019 0.829 – 25.386 1.185 26.644−0.378 26.283−0.325 17 +0.135 +0.133 NGC 6717 – – – – – 7.464−0.133 7.524−0.131 10 +0.829 ± +0.318 ± +0.103 +0.100 NGC 6723 8.237−0.690 8.708 0.586 – 8.472−0.306 8.691 0.601 8.241−0.102 8.267−0.099 18 +0.779 +0.230 +0.218 NGC 6749 7.849−0.650 ––––7.561−0.223 7.591−0.212 5 +0.171 ± ± +0.149 ± +0.050 +0.041 NGC 6752 4.092−0.158 4.036 0.114 4.005 0.147 3.981−0.144 3.917 0.167 4.176−0.050 4.125−0.041 18 +0.839 ± +0.600 +0.441 NGC 6760 8.244−0.697 9.322 0.900 – – – 8.110−0.559 8.411−0.418 4 +0.381 +0.145 +0.144 NGC 6779 – – – 10.139−0.367 –10.404−0.143 10.430−0.142 12 +0.285 ± +0.195 ± +0.055 +0.052 NGC 6809 5.020−0.256 5.201 0.246 – 5.200−0.188 5.223 0.303 5.370−0.054 5.348−0.051 27 +0.190 ± +0.151 +0.053 +0.050 NGC 6838 4.160−0.174 4.154 0.237 – 4.036−0.146 –3.977−0.053 4.001−0.050 15 +0.459 +0.457 NGC 6864 – – – – – 20.549−0.449 20.517−0.448 5 ± +0.555 +0.175 +0.173 NGC 6934 – 16.718 1.382 – 14.791−0.535 –15.704−0.173 15.716−0.171 19 +0.603 +0.185 +0.185 NGC 6981 – – – 16.069−0.581 –16.672−0.183 16.661−0.183 15 +1.474 +0.565 +0.565 NGC 7006 – – – 39.264−1.420 –39.319−0.557 39.318−0.557 9 ± +0.398 ± +0.104 +0.096 NGC 7078 – – 10.375 0.295 10.617−0.384 10.769 0.452 10.740−0.103 10.709−0.095 28 ± +0.419 ± +0.119 +0.115 NGC 7089 – 11.940 0.703 – 11.169−0.404 12.267 0.543 11.647−0.117 11.693−0.114 21 +0.753 ± +0.324 ± +0.094 +0.090 NGC 7099 7.981−0.633 8.682 0.496 – 8.630−0.312 7.959 0.435 8.480−0.093 8.458−0.089 19 +0.581 +0.579 NGC 7492 – – – – – 24.434−0.567 24.390−0.566 4 +0.547 +0.329 +0.328 Pal 1 – – – 10.520−0.520 –11.189−0.320 11.176−0.319 4 +1.321 +1.317 Pal 2 – – – – – 26.158−1.258 26.174−1.253 2 +3.288 +3.288 Pal 3 – – – – – 94.842−3.178 94.842−3.178 2 +2.601 +2.601 Pal 4 – – – – – 101.391−2.536 101.391−2.536 2 +0.521 +0.520 Pal 5 – – – – – 21.928−0.509 21.941−0.508 6 +0.472 +0.463 Pal 6 – – – – – 6.887−0.442 7.047−0.433 6 +0.722 +0.652 Pal 8 – – – – – 11.695−0.680 11.316−0.611 4 +1.621 +1.292 Pal 10 – – – – – 8.402−1.359 8.944−1.072 3 +0.523 +0.517 Pal 11 – – – – – 14.125−0.505 14.024−0.498 3 +0.686 +0.300 +0.300 Pal 12 – – – 18.281−0.661 –18.484−0.296 18.494−0.295 8 +0.403 +0.403 Pal 13 – – – – – 23.475−0.397 23.475−0.397 6 +1.645 +1.645 Pal 14 – – – – – 73.587−1.609 73.579−1.609 4 +2.227 +1.152 +1.152 Pal 15 – – – 42.855−2.117 –44.096−1.123 44.096−1.123 4 +1.414 +0.662 +0.662 Pyxis – – – 37.670−1.363 –36.526−0.650 36.526−0.650 6 +4.288 +4.288 RLGC 1 – – – – – 28.840−3.733 28.840−3.733 1 +2.356 +2.356 RLGC 2 – – – – – 15.849−2.051 15.849−2.051 1 +0.735 +0.366 +0.365 Rup 106 – – – 19.588−0.709 –20.730−0.360 20.711−0.359 6 +2.514 +1.581 +1.581 Sgr II – – – 66.988−2.423 –66.527−1.544 66.527−1.544 5 +0.175 +0.175 Ter 1 – – – – – 5.673−0.170 5.673−0.170 6 +0.336 +0.332 Ter 2 – – – – – 7.681−0.322 7.753−0.318 4 +0.338 +0.318 Ter 3 – – – – – 7.568−0.324 7.644−0.304 3 +0.316 +0.315 Ter 4 – – – – – 7.558−0.304 7.591−0.302 5 ± ± +0.224 +0.150 Ter 5 – 6.780 0.220 – – 6.346 0.503 6.462−0.217 6.617−0.148 10

MNRAS 505, 5957–5977 (2021) 5968 H. Baumgardt and E. Vasiliev

Table 2 – continued

Name GEDR3 parallax dist. GEDR3 kin. dist. HST kin. dist. Subdwarf dist. Star count dist. Lit. dist. Mean distance NTot (kpc) (kpc) (kpc) (kpc) (kpc) (kpc) (kpc)

+0.360 +0.360 Ter 6 – – – – – 7.271−0.343 7.271−0.343 4 +0.497 +0.496 Ter 7 – – – – – 24.277−0.487 24.278−0.487 5 +1.038 +0.422 +0.421 Ter 8 – – – 27.669−1.001 –27.542−0.415 27.535−0.415 8 ± +0.434 +0.356 Ter 9 – 5.030 1.037 – – – 5.605−0.403 5.770−0.333 5 +0.414 +0.412 Ter 10 – – – – – 10.247−0.398 10.212−0.396 3 ± +0.444 +0.400 Ter 12 – 5.002 1.851 – – – 5.107−0.408 5.166−0.369 4 +0.360 +0.344 Ton 2 – – – – – 6.934−0.343 6.987−0.326 3 Downloaded from https://academic.oup.com/mnras/article/505/4/5957/6283731 by UQ Library user on 30 July 2021 +0.570 +0.570 UKS 1 – – – – – 15.581−0.550 15.581−0.550 2 +1.664 +1.622 VVV-CL001 – – – – – 8.204−1.383 8.082−1.343 1 +1.192 +1.192 Whiting 1 – – – – – 30.591−1.147 30.591−1.147 2 +0.472 +0.463 Pal 6 – – – – – 6.887−0.442 7.047−0.433 6

the final distance for each cluster by minimizing the combined D = 4.53 ± 0.06 kpc that Ma´ız Apellaniz,´ Pantaleoni Gonzalez´ & likelihood of all measurements according to Barba(´ 2021) determined from the Gaia EDR3 parallaxes, correcting   2 for the small scale angular covariances in the Gaia data by comparing 1 G −  2 the parallaxes of 47 Tuc stars with those of background SMC stars. It 1 D 1 (Di −D) ln L =− − also agrees very well with the distance of D = 4.45 ± 0.12 kpc that 2 2 2 σ 2 i i Chen et al. (2018) derived with the same method from Gaia DR2 data  2 1 (μj − 5logD+5) as well as the distance of D = 4.53 ± 0.05 kpc that Thompson et al. − . (3) 2 σ 2 (2020) found from the analysis of two detached eclipsing binaries in j j the cluster. The distance to 47 Tuc therefore seems to be established Here, G and are the cluster parallaxes and their 1σ errors, with an accuracy of better than 1 per cent. Di and σ i are the cluster distances and associated 1σ errors of the kinematic, star count, and moving group distances. μj and σ j are the distance moduli and error bars that we obtain from the averaging 4.2.2 NGC 362 of the literature data and our subdwarf fits. D is the mean cluster distance that we want to determine. We use the Gaia parallaxes here We correct the Gaia EDR3 parallaxes in the same way done by instead of the distances that can be calculated from them since errors Ma´ız Apellaniz´ et al. (2021) for NGC 104 and Chen et al. (2018)for on the parallax distance are Gaussian only in parallax space. For the NGC 104 and NGC 362. We again use SMC field stars in the direction = ± same reason, we use distance moduli for the literature and subdwarf of NGC 362 as reference, assuming a distance of D 62.1 1.9 kpc distances. We calculate the mean distance from the point where as distance to the SMC (Cioni et al. 2000; Graczyk et al. 2014). The = ± ln L has a maximum and calculate the associated upper and lower resulting cluster parallax is 109.9 3.7 μas, corresponding . +0.32 distance errors from the condition that ln L = 1.0. The resulting to a distance of 9 10−0.30 kpc. This distance is in agreement with = ± mean distances along with the individual values from the various the earlier value of Chen et al. (2018)(D 8.54 0.48 kpc) as methods are given in Table 2. All data have been transformed to linear well as the distance we derive from an averaging of literature values ± distances to make the different measurements easier comparable with (8.80 0.11 kpc) and our own kinematic distance determinations each other. (see Fig. A1). Fig. 7 depicts the relative distance accuracy that we achieve by combining all data as a function of the cluster reddening. It can be seen that for clusters with low reddening values E(B − V) < 0.3 we 4.2.3 ω Cen achieve relative accuracies of about 1 per cent. For larger reddening There is also good agreement in our distance determinations for values E(B − V) > 0.3, there is a continuous rise of the distance ω Cen. The literature distances lead to an average distance of D errors and we achieve only 20 per cent accuracy for the most highly = 5.47 ± 0.06 kpc (see Fig. A1). The Gaia kinematic distance is reddened clusters. This rise could partly be driven by the fact that in good agreement with the literature distance, however the HST heavily reddened clusters have often only few individual distance kinematic distance (5.26 ± 0.12 kpc) is about 200 pc shorter. One determinations. possible reason for the mismatch could be the intrinsic flattening of ω Cen. Mackey & Gilmore (2004) find an ellipticity of = 0.17 for ω Cen based on its projected isophotes. The true flattening is 4.2 Individual clusters likely even larger, with the exact value depending on the cluster inclination. van de Ven et al. (2006) found through axisymmetric 4.2.1 47 Tuc Schwarzschild modeling that the resulting flattening of the velocity We derive a distance of D = 4.52 ± 0.03 kpc to 47 Tuc (NGC 104). ellipsoid can decrease the kinematic cluster distance to ω Cen by For 47 Tuc, both literature distances and kinematic distances have several hundred pc if the flattening is not taken into account in small formal errors and are fully consistent with each other (see the modeling. Since the N-body models that we use to derive the Fig. A1). Our mean distance also agrees very well with the value of kinematic distances are spherically symmetric, there is a chance that

MNRAS 505, 5957–5977 (2021) Accurate distances to Galactic globular clusters 5969 Downloaded from https://academic.oup.com/mnras/article/505/4/5957/6283731 by UQ Library user on 30 July 2021

Figure 6. Comparison of the CMD fitting distances with the Gaia parallax and kinematic distances and the HST kinematic distances. We again use MCMC sampling and solve for constant scale factors D in the distance scales of the various methods as well as additional systematic errors . DGaia and DHST are relative shifts in the kinematic distances of the Gaia and HST kinematic distances and is a shift in the Gaia parallaxes. We obtain a significant shift in the Gaia parallaxes of ≈ 0.007 mas, in agreement with VB21.TheHST kinematic distances are on average also smaller than the literature distances by about 3.0 per cent, while the Gaia kinematic distance shift DGaia is compatible with zero. The HST kinematic distances also show evidence of additional relative distance errors of about 1.7 per cent. The inset compares the kinematic and star count distances with the distances from isochrone fitting of the clusterCMDs. our kinematic distances underestimate the true distance to ω Cen. 4.2.4 NGC 6121 This effect is much less likely for most other clusters that have much We derive a distance of D = 1.851 ± 0.015 kpc to M4 smaller relaxation times and should therefore be closer to being (NGC 6121). This distance is slightly larger than the distance of spherically symmetric. This is indeed also seen in observations (e.g. D = 1.82 ± 0.04 kpc found by Kaluzny et al. (2013) based on three White & Shawl 1987). We therefore regard the literature distance as eclipsing binary stars as well as recent results from RR Lyrae stars more reliable for ω Cen. The Gaia EDR3 parallaxes after application (e.g. Braga et al. 2015; Neeley et al. 2019;Bhardwajetal.2020b) of the systematic parallax offset also favour a longer distance, (see Fig. A1). Nevertheless, their distances are compatible with our however the error bar is too large to be decide between the two distance within the error bars. The true distance to NGC 6121 is possibilities.

MNRAS 505, 5957–5977 (2021) 5970 H. Baumgardt and E. Vasiliev

−4.05 ± 0.06, using eclipsing binaries in 47 Tuc and the SMC to calibrate the LMC distance. The ω Cen distance of Soltis et al. (2021) was based on a direct averaging of the Gaia EDR3 parallaxes of individual member stars, applying only the Lindegren et al. (2020) parallax corrections. Their parallax value did not take systematic biases in the Gaia parallaxes or small-scale correlated errors into account. Accounting for both, we derived a parallax of = 182.3 ± 9.5 μas for ω Cen (Vasiliev & Baumgardt 2021), corresponding to a distance D = 5.48 ± 0.24 kpc, slightly larger than Soltis et al. (2021) and also with a larger formal error than found by them. Combining this data with the other distances, we then derive a distance of D = 5.43 ± 0.05 to ω Cen. Downloaded from https://academic.oup.com/mnras/article/505/4/5957/6283731 by UQ Library user on 30 July 2021 A similar analysis as the one done by Soltis et al. (2021)leadstoan absolute TRGB magnitude of MI,TRGB =−4.06 ± 0.06, confirming the value found by Freedman et al. (2020). In addition, our results also confirm their adopted distance to NGC 104. The resulting value −1 of the local Hubble constant of H0 = 69.4 ± 2.0 km s would be in better agreement with the expected value based on the Planck data of 67.4 ± 0.5 km s−1 (Verde, Treu & Riess 2019).

5 CONCLUSIONS Figure 7. Final distance error σ D/D as a function of the cluster reddening E(B − V). Clusters with reddening values E(B − V) < 0.20 have an average We have derived mean distances to globular clusters using a com- distance error of about 1 per cent. More reddened clusters show a continuous bination of our own measurements based on Gaia EDR3 data and increase of the distance error with reddening. The location of three of the published literature distances. We derived our own distances using clusters discussed in the text is marked in the plot. six different methods: Gaia ED3 parallaxes, kinematic distances using proper motion velocity dispersion profiles based on Gaia therefore likely somewhere in the range 1.83–1.85 kpc, making M4 EDR3 proper motions, fitting nearby subdwarfs to the globular another cluster whose distance is determined with an accuracy of cluster main sequences where the distances and absolute luminosities about 1 per cent. of the subdwarfs are calculated from their Gaia parallaxes and moving group distances based on Gaia ED3 proper motions. In addition, we also derive kinematic distances from HST proper motion 4.2.5 NGC 6397 and NGC 6656 velocity dispersion profiles and distances using HST star counts in combination with the cluster kinematics. NGC 6397 and NGC 6656 (M 22) are among the closest globular Apart from a bias in the Gaia EDR3 parallaxes, we find good clusters to the Sun. Their small distances together with their low internal agreement among the distances calculated with the different reddening make them ideal targets for globular cluster studies, so methods and also with published distances in the literature down a large amount of observational data exists for both clusters. For to a level of about 2 per cent. There is a possibility that our HST NGC 6397, we find a total of 27 individual distance determinations kinematic distances could slightly underestimate the distances to in the literature, from which we derive a mean cluster distance of globular clusters, however the difference that we see could also ± 2.521 0.025 kpc. Despite a relative distance error of only about signify a problem in the isochrone fitting distances. We also find 1 per centf or the literature distance and similarly small errors for some evidence that isochrone based distance moduli are larger than most of the other methods, most distances are in good statistical RR Lyrae based ones by about 0.02 mag. agreement with each other, with the exception of the HST kinematic Averaging over the various distance determinations, we are able to distances that are too short by about 100 pc. Taking the average determine distances to a number of nearby globular clusters with an ± over all methods, we derive a distance of 2.488 0.019 kpc for accuracy of 1 per cent, making these clusters valuable objects for the NGC 6397, making this cluster the second closest cluster to the establishment of a . In particular, our results Sun. There is a similar good agreement in the individual distances for the distances to 47 Tuc and ω Cen argue for an absolute TRGB = values for NGC 6656, for which we derive a mean distance of D magnitude of M =−4.05 leading to a value of the local Hubble ± I,TRGB 3.307 0.037 kpc. constant that is in agreement with the expected value based on Planck data. They however do not remove the tension between the Cepheid based value of H and the Planck data, which is currently significant 4.3 Absolute luminosity of the TRGB and the value of H 0 0 at about the 4σ level (Riess et al. 2019). Given that we find that Globular clusters can be used to determine the absolute luminosity the different literature distances to globular clusters are consistent of the tip of the red giant branch (TRGB), which in turn can down to a level of about 1–2 per cent, it is also unlikely that our be used to measure the distance to nearby galaxies. Using Gaia results would lead to a significant revision of the distance modulus to EDR3 parallaxes, Soltis et al. (2021) recently derived a distance nearby galaxies like the LMC, and therefore a reduction of the tension of 5.24 ± 0.11 kpc to ω Cen. Using this distance, they derived an in H0. absolute I-band magnitude of the TRGB of MI,TRGB =−3.97 ± 0.06, Future releases of the Gaia catalogue will further increase the sam- from which they deduced a value of the local Hubble constant of H0 ple of globular clusters with highly accurate distances by reducing the = 71.2 ± 2.0 km s−1.Incontrast,Freedmanetal.(2020)usedstars random and systematic errors in the parallaxes and proper motions in the LMC to derive an absolute TRGB magnitude of MI,TRGB = of stars. There is currently also still a substantial uncertainty in the

MNRAS 505, 5957–5977 (2021) Accurate distances to Galactic globular clusters 5971 distances of highly reddened clusters, where individual distances Arellano Ferro A., Ahumada J. A., Calderon´ J. H., Kains N., 2014, Rev. Mex. can deviate from each other by up to a factor of 2. At least for the Astron. Astrofis., 50, 307 more nearby clusters, Gaia parallaxes and proper motion velocity Arellano Ferro A., Mancera Pina˜ P. E., Bramich D. M., Giridhar S., Ahumada dispersion profiles based on future Gaia data releases should bring a J. A., Kains N., Kuppuswamy K., 2015, MNRAS, 452, 727 significant improvement to the distances to these clusters. Arellano Ferro A., Luna A., Bramich D. M., Giridhar S., Ahumada J. A., Muneer S., 2016, Ap&SS, 361, 175 Arellano Ferro A., Rojas Galindo F. C., Muneer S., Giridhar S., 2018, Rev. ACKNOWLEDGEMENTS Mex. Astron. Astrofis., 54, 325 Arellano Ferro A., Bustos Fierro I. H., Calderon´ J. H., Ahumada J. A., 2019, EV acknowledges support from STFC via the consolidated grant Rev. Mex. Astron. Astrofis., 55, 337 to the Institute of Astronomy. We thank Domenico Nardiello for Arellano Ferro A., Yepez M. A., Muneer S., Bustos Fierro I. H., Schroder¨ K. providing us with the HST photometry for NGC 6626. This work P., Giridhar S., Calderon´ J. H., 2020, MNRAS, 499, 4026 presents results from the European Space Agency (ESA) space Armandroff T. E., 1988, AJ, 96, 588 Downloaded from https://academic.oup.com/mnras/article/505/4/5957/6283731 by UQ Library user on 30 July 2021 mission Gaia. Gaia data are being processed by the Gaia Data Balbinot E., Santiago B. X., Bica E., Bonatto C., 2009, MNRAS, 396, 1596 Processing and Analysis Consortium (DPAC). Funding for the DPAC Barba´ R. H., Minniti D., Geisler D., Alonso-Garc´ıa J., Hempel M., Monachesi is provided by national institutions, in particular the institutions A., Arias J. I., Gomez´ F. 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Same as Fig. A1 for Arp 2, Djor 1, Djor 2, and E 3, ESO 115, 1476 452-SC11, and FSR 1716. Testa V., Chieffi A., Limongi M., Andreuzzi G., Marconi G., 2004, A&A, 421, 603 Figure A3. Same as Fig. A1 for HP 1, IC 1257, IC 4499, Liller 1, Thompson I. B., Kaluzny J., Pych W., Burley G., Krzeminski W., Paczynski´ Lynga 7, and NGC 288. B., Persson S. E., Preston G. W., 2001, AJ, 121, 3089 Figure A4. Same as Fig. A1 for NGC 1261, NGC 1851, NGC 1904, Thompson I. B. et al., 2020, MNRAS, 492, 4254 NGC 2298, NGC 2419, and NGC 2808. Tiongco M. A., Vesperini E., Varri A. L., 2016, MNRAS, 455, 3693 Figure A5. Same as Fig. A1 for NGC 3201, NGC 4147, NGC 4372, Tsapras Y. et al., 2017, MNRAS, 465, 2489 NGC 4590, NGC 4833, and NGC 5024.

MNRAS 505, 5957–5977 (2021) 5976 H. Baumgardt and E. Vasiliev

Figure A6. Same as Fig. A1 for NGC 5053, NGC 5272, NGC 5286, Figure A16. Same as Fig. A1 for NGC 6624, NGC 66226, NGC NGC 5466, NGC 5694, and NGC 5824. 6637, NGC 6638, NGC 6642, and NGC 6652. Figure A7. Same as Fig. A1 for NGC 5897, NGC 5904, NGC 5927, Figure A17. Same as Fig. A1 for NGC 6656, NGC 6681, NGC 6712, NGC 5946, NGC 5986, and NGC 6093. NGC 6715, NGC 6717, and NGC 6723. Figure A8. Same as Fig. A1 for NGC 6101, NGC 6139, NGC 6144, Figure A18. Same as Fig. A1 for NGC 6749, NGC 6752, NGC 6760, NGC 6171, NGC 6205, and NGC 6218. NGC 6779, NGC 6809, and NGC 6838. Figure A9. Same as Fig. A1 for NGC 6229, NGC 6235, NGC 6254, Figure A19. Same as Fig. A1 for NGC 6864, NGC 69346, NGC NGC 6256, NGC 6266, and NGC 6273. 6981, NGC 7006, NGC 7078, and NGC 7089. Figure A10. Same as Fig. A1 for NGC 6284, NGC 6287, NGC 6293, Figure A20. Same as Fig. A1 for NGC 7099, NGC 7492, Pal 1, Pal NGC 6304, NGC 6316, and NGC 6325. 5, Pal 6, and Pal 12. Figure A11. Same as Fig. A1 for NGC 6333, NGC 6341, NGC 6342, Figure A21. Same as Fig. A1 for Pal 13, Pal 14, Pal 15, Pyxis, Rup NGC 6352, NGC 6355, and NGC 6356. 106, and Ter 1. Downloaded from https://academic.oup.com/mnras/article/505/4/5957/6283731 by UQ Library user on 30 July 2021 Figure A12. Same as Fig. A1 for NGC 6362, NGC 6366, NGC 6380, Figure A22. Same as Fig. A1 for Ter 4, Ter 5, Ter 6, Ter 7, Ter 8, NGC 6388, NGC 6397, and NGC 6401. and Ter 9. Figure A13. Same as Fig. A1 for NGC 6402, NGC 6426, NGC 6440, APPENDIX B: LITERATURE DISTANCES USED IN THIS NGC 6441, NGC 6453, and NGC 6496. WORK Figure A14. Same as Fig. A1 for NGC 6517, NGC 6522, NGC 6528, Please note: Oxford University Press is not responsible for the content NGC 6535, NGC 6539, and NGC 6540. or functionality of any supporting materials supplied by the authors. Figure A15. Same as Fig. A1 for NGC 6541, NGC 6544, NGC 6553, Any queries (other than missing material) should be directed to the NGC 6558, NGC 6569, and NGC 6584. corresponding author for the article.

MNRAS 505, 5957–5977 (2021) Accurate distances to Galactic globular clusters 5977

APPENDIX A: COMPARISON OF DISTANCE DETERMINATIONS FOR GLOBULAR CLUSTERS Figs A1 and A2–A22 (available online in the supporting information) depict the individual distances determinations to globular clusters used in this work. Only clusters with more than five measurements are shown. Downloaded from https://academic.oup.com/mnras/article/505/4/5957/6283731 by UQ Library user on 30 July 2021

Figure A1. Comparison of recent literature distance determinations for the globular clusters NGC 104, NGC 362, NGC 5139, and NGC 6121 with our data. Black symbols depict distance determinations taken from the literature, while red symbols show our own measurements based on Gaia EDR3 data. Red triangles show Gaia EDR3 kinematic distances, red circles subdwarf distances, red squares moving group distances, and red sexagons show Gaia EDR3 parallax distances. Blue triangles show HST kinematic distances and dark green squares with error bars star count distances. The bright green square is the distance given by Harris (1996) based on an averaging of literature values up to 2010. The distances are sorted according to publication year, with newest measurements at the top of each panel and the first and last year indicated. Distance determinations shown in grey have been neglected when calculating the mean cluster distance.For literature distances without an error, the dashed error bars show our adopted error. Solid error bars are literature measurements with error bars. The red vertical line and shaded region marks the average distance and its 1σ error bar. The average distances are also given at the top of each panel.

This paper has been typeset from a TEX/LATEX file prepared by the author.

MNRAS 505, 5957–5977 (2021)