Peremennye Zvezdy (Variable Stars) 45, No. 2, 2025 Received 3 January; accepted 17 January.	Article in PDF
	DOI: 10.24412/2221-0474-2025-45-7-31

Mid-infrared Period-luminosity Relations of RR Lyrae Variables via Color-transformed Data

A. K. Dambis, L. N. Berdnikov

Sternberg Astronomical Institute, Moscow State University, Universitetskij pr. 13, Moscow 119992, Russia; leonid.berdnikov@gmail.com

1. Introduction

RR Lyrae variables are old low-mass pulsating core-helium-burning stars occupying the instability strip. Most of them are fundamental-mode pulsators (RRab), a smaller fraction pulsates in the first overtone (RRc), and very few of them pulsate in both modes simultaneously (RRd). RR Lyraes serve as valuable distance indicators and kinematical tracers because they obey a rather tight photometric-band-dependent period-metallicity-luminosity relations of the form:

(Catelan et al. 2004). Here is the intensity-mean absolute magnitude in the photometric band and is the fundamental (or, in the case of overtone pulsators, fundamentalized) period, which is equal to the variability period for fundamental-mode pulsators (RRab) and /0.746 or log = log +0.127 (Iben 1974) for first-overtone pulsators (RRc), respectively. The -band absolute magnitude of RR Lyrae stars depends on metallicity and exhibits no appreciable dependence on period, whereas near- and mid-ifrared luminosities of these variables clearly depend both on period and metallicity (Marconi et al. 2015; Neeley et al. 2017). Period-metallicity-luminosity relations in the infrared are better suited for determining photometric distances to RR Lyraes for two major reasons. First, interstellar extinction is much smaller in the infrared (by a factor of in the band and in the WISE and Spitzer 3.5 m bands, respectively - see Yuan et al. 2013), resulting in much smaller (and practically negligible) effect of errors in the adopted extinction values. Second, amplitudes of light variations in the infrared are smaller than at shorter wavelengths, permitting the corresponding mean magnitudes to be determined with better precision.

There have been many empirical determinations of the parameters of RR Lyrae period-metallicity-luminosity relations in the near-infrared bands () using both the Baade-Wesselink method (for relatively bright field stars) and globular-cluster variables (Fernley et al. 1987; Liu & Janes 1990; Jones et al. 1992; Frolov & Samus 1999; Sollima et al. 2006; Prudil et al. 2024). The problem with the practical application of the near-IR period-metallicity-luminosity relations for actual population studies is that the three largest-area near-infrared surveys - DENIS (Epchtein et al. 1999), 2MASS (Skrutsie et al. 2006), and VISTA Hemisphere Survey, or VHS (McMahon et al. 2013) - provide only single-epoch data for this wavelength range. The multi-epoch near-infrared survey with the most extensive sky coverage is VVV (Minniti et al. 2024), but it focusses on the Galactic bulge and a part of the southern Milky-Way disk and thus covers only 500-plus square degrees.

The situation has improved substantially in the mid-infrared range with the advent of the Spitzer (Fazio et al. 2004) and WISE (Wright et al. 2010) photometric surveys where the latter provides all-sky multi-epoch coverage and therefore should be best suited for extensive RR-Lyrae-based Galactic structure and kinematics studies. However, WISE has two important limitations. First, the angular resolution, which is equal to 6 1 and 6 4 in the WISE and bands, respectively (Wright et al. 2010), and restricts the use of the survey data in crowded fields such as those of most of globular clusters. Second, the limited photometric sensitivity, which at the 5 level is equivalent to 16.6 and 15.6 Vega magnitudes for the and bands, respectively, with the light curves becoming quite noisy and scattered already appreciably above these limits: this factor prevents bona fide study of RR Lyrae stars in all but few of relatively nearby globular clusters. Such a study was carried out by Dambis et al. (2014), who calibrated the period-luminosity-metallicity relations in the WISE and bands based on WISE light curves for 360 and 275 RR Lyraes in 15 and 9 Galactic globular clusters, respectively. Spitzer survey performs better in both aspects, but it has limited sky coverage and its data have so far been used to study the period-luminosity relation in only two globular clusters: Reticulum, which is located close to the LMC (Muraveva et al. 2018), and the nearby cluster M4 (Neeley et al. 2015). Mullen et al. (2023) performed the most comprehensive calibration of RR Lyrae period-luminosity-metallicity and period-Wesenheit-metallicity relations based on WISE photometry and Gaia EDR3 parallaxes (Gaia Collaboration, 2021) of about 1000 mostly relatively bright field RR Lyraes, thus circumventing the two WISE survey limitations mentioned above. However, because their study relies of field-star parallaxes, the accuracy of the inferred period slope estimates is not as good as those obtained by Dambis et al. (2014) using WISE data for RR Lyraes in nearby globular clusters despite the rather large scatter of the period-magnitude relations in individual clusters due to crowding. Here we propose another way to circumvent the above two WISE-survey limitations using intensity-mean WISE and magnitude estimates computed by transforming the intensity-mean Gaia and magnitudes. This solution takes advantage of better resolution and deeper limiting magnitude of the Gaia survey compared to WISE and therefore makes it possible to study more distant globular clusters and RR Lyraes in more crowded cluster fields (e.g., stars located closer to cluster centers).

2. Calibration formulas

Formula (1) is linear in and [Fe/H], and hence all sorts of intensity-mean absolute-magnitude differences, which are distant independent and equal to the corresponding intensity-mean color indices (like, e.g., , ( , etc.), and linear combinations thereof also obey similar linear relations of the same form. As a consequence, the same is true for extinction-free  indices constructed from observed colors (with coefficients chosen so as to cancel out extinction). With the Cardelli et al. (1989) and O'Donnell (1994) extinction law and assuming =3.1, the extinction ratios are equal to:

With these ratios, the extinction-free and indices, which are equal to

and

respectively, should depend linearly on and [Fe/H]:

and

Given an appropriate calibrating sample of RR Lyraes with known , , , , , and [Fe/H], the coefficients , , , , , and can be estimated by solving sets of equations (5) and (6) via the least-squares method. Once these coefficients are computed, the mid-infrared intensity-mean magnitudes and of an RR Lyrae type variable can be estimated from its known intensity-mean Gaia DR3 magnitudes and , fundamental period, and [Fe/H]:

and

3. Calibration sample

Our calibrating sample is based on two RR Lyrae star lists with spectroscopic metallicity estimates. One is the subset of the catalog of Liu et al. (2020) including 5206 RR Lyraes with both metallicity and metallicity error estimates derived from spectra acquired within the framework of the LAMOST (Deng et al. 2012; Liu et al. 2014) and SDSS (SEGUE) (Yanni et al. 2009) surveys and assumed to be in the UVES high-resolution-spectroscopy-based scale (Carretta et al. 2009). The second list is the catalog of 850 RR Lyraes by Muhie et al. (2021) with metallicities based on (1) original [Fe/H] estimates determined from low-resolution spectra taken with the 11-m SALT (Southern African Large Telescope) (Buckley 2006; O'Donoghue et al. 2006) in the process of the Milky wAy Galaxy wIth SALT spectroscopy (MAGIC) project (Kniazev et al. 2019) and (2) published spectroscopic [Fe/H] determinations compiled and homogenized by Dambis et al. (2013). The metallicities in the catalog of Muhie et al. (2021) are in the old Zinn and West (1984) (ZW) scale. We use the formula from Carretta et al. (2009):

to transform them to the UVES scale adopted by Liu et al. (2020). The two lists have 139 stars in common, for which we adopt the [Fe/H] values from the more homogeneous catalog of Liu et al. (2020), with the combined list containing 5717 stars. We then cross-matched this combined list with the Clementini et al. (2023) calalog of stars classified as RR Lyrae variables based on an analysis of epoch photometry provided in Gaia DR3 (Gaia Collaboration 2023) and with ALLWISE Multiepoch Photometry and NEOWISE-R Single Exposure (L1b) Source Tables to extract the - and -band light curves, and derive the corresponding intensity-mean magnitudes and . We compute and by fitting Fourier series with up to seventh harmonic to the light curves containing at least 35 data points with available uncertainty estimates. We retain only stars for which all the following data are available: (1) spectroscopic [Fe/H] estimate from Liu et al. (2020) or Muhie et al. (2021); (2) both the intensity-mean Gaia magnitudes and together with variability periods and pulsation modes from Clementini et al. (2023), and (3) WISE -band intensity mean magnitude based on at least 35 individual photometric measurements accompanied with uncertainty estimates in the combined ALLWISE Multiepoch Photometry and NEOWISE-R Single Exposure (L1b) Source Tables. The resulting sample contains a total of 4845 RR Lyrae variables, with 4010 stars having also WISE -band intensity mean values based on at least 35 individual data points, accompanied with uncertainty estimates. Note, however, that, because of the photometric sensitivity limits, the quality of WISE photometry degrades already well above the limiting magnitude. This is apparent from both the dependence of the light-curve scatter (see Fig. 1) and, especially, from that of the magnitude difference (Fig. 2) on . To reduce the effect of photometry quality degradation, we further exclude stars fainter than  =14.0 from our final calibration sample, which thus contains a total of 1633 stars and all of them also have WISE -band intensity mean magnitudes based on at least 35 bona fide individual photometric measurements. With this final sample, the iteratively 3-clipped linear least square fits of Equations (5) and (6) for 1293 fundamental- and double-mode RR Lyraes (RRab and RRd) are:

and

The corresponding fits for 335 first-overtone RR Lyraes (RRc) are:

and

Equations (7) - (13) imply the following transformation formulas:

and

for RRab- and RRd-type variables and:

and

for RRc-type variables. We point out two important properties of the above formulas: (1) their rather small scatter (about 003-004) and (2) small (for RRab-type stars) or insignificant (for RRc-type stars) "metallicity slopes", permitting the intensity-mean infrared magnitudes to be rather accurately inferred even if metallicity is poorly known or just set equal to the average value for the population considered (e.g., for the Galactic halo; de Jong et al. 2010).

4. Calibration of the Period-Metallicity-Luminosity relation

4.1. The period slope

We start by determining the period slopes ( and ) of the - and -band Period-Metallicity-Luminosity relations (equation (1)) for RR Lyrae stars using Gaia photometry-based estimates of the and intensity-means of globular-cluster RR Lyraes computed via formulas (14-17). To this end, we first cross-match the catalog of Clementini et al. (2023) with the new version of the catalog of Milky-Way globular clusters by Harris (2010) within the clusters' tidal radii. To exclude likely fore- and background RR Lyraes, we retain only stars whose -band magnitudes are within 05 of the cluster horizontal-band magnutude given in the catalog. To compute the -band magnitudes of the stars of our sample from their and magnitudes from the catalog of Clementini et al. (2023), we use the following formula:

derived by fitting homogenized intensity-mean -band magnitudes from Dambis et al. (2013) combined with -band intensity means based on CCD observations of Muhie et al. (2021). We then exclude stars whose Gaia and fluxes are likely to be affected by contamination because of the strong crowding in central parts of globular-cluster fields. We do this by excluding all stars with the corrected and flux excess factor of Riello et al. (2021) greater than . The quantity is equal to the original and flux excess factor introduced by Evans et al. (2018) and defined as a simple ratio between the total flux in and bands, and the -band flux, i.e., , minus the function that takes into account the color dependence of . Whereas, due to the shape of the passbands, stars with uncontaminated and fluxes have values slightly greater than unity, their corrected factors are, by construction, close to zero. The and factors of stars that appreciably suffer from crowding should be significantly greater than unity and zero, respectively, because and fluxes, and , are not deblended and are calculated as the sum of the flux in a window of  arcsec, whereas the -band flux is computed from LSF or PSF fitting to a narrow image (with the pixel size of 58.9 and 176.8 mas, respectively, in the along- and across-the-scan directions).

To estimate the period slope or for a particular cluster, we use the procedure employed by Dambis et al. (2014) and just slightly modify the final step. The revised procedure goes as follows. Given that stars in a cluster are located practically at the same distance and (generally) have the same metallicity and the same amount of interstellar extinction (in any case, the latter is at least 17 times smaller in all WISE passbands compared to the  band and therefore its variations can therefore be ignored), equation (1) becomes:

where can be treated as a constant. We add the +0.25 term to in order to center the solution at , which is close to the average value of this parameter, so that the constant would be representative of the cluster distance modulus, in order to minimize the effect of differences in the inferred values between different clusters. To estimate the constant for some assumed slope , we proceed as follows. We compute the left-hand side of equation (18), , for each star. We then sort the values in the ascending order and try every subset containing consecutive values with (where N is the total number of RR Lyraes in the given cluster after all the above cuts performed) and seek the subset having the smallest dispersion of computed values, . We then try values from -1.0 to -5.0 in increments of 0.01 to find the one yielding the smallest . If the modal "core" of the distribution (i.e., the part of the distribution corresponding to stars whose data points outline the relation assumed to be linear) of values were normal, our subset would roughly consist at least of all stars with between and , where and are the mean and dispersion of values for the subset of stars defining the linear relation, respectively. The mean value averaged over the subset stars, , should then be close to the mean , and the (truncated) dispersion should be roughly equal to , implying . We therefore determine the final estimate of and by least squares solving the equation set

which is just a rewritten form of equation (18), for stars with values in the interval . In the original version, we set equal to 3 = 5.56 corresponding to the 0.9973 probability for a normal distribution. However, in real cases we have to deal with rather small RR Lyrae samples, and the 0.9973-probability interval has to be estimated based on the Student's - rather than normal distribution, and so in our revised procedure we set , where is the inverse Student's cumulative distribution. Hence, in our new version, stars are selected from a broader interval (whose halfwidth is equal to and for subsets consisting of 10 and 100 stars, respectively). Table 1 lists the solutions so obtained for 24 globular clusters with at least eight stars outlining the linear relation together with the cluster metalicities [Fe/H] and color excesses adopted from the Harris (2010) catalog.

Here, like in Dambis et al. (2014), we plot the scaled computed intensity-mean and magnitudes ( and ) for our calibrating clusters as a function of fundamentalized periods (Figs. (3-6). As it is evident from Table 1 and Figs. (3-6), the period-magnitude relation slopes are quite consistent among the clusters studied. Moreover, one can see from Figs. (7) and (8) that the slopes show only a marginal trend with metallicity: the corresponding linear weighted fits are:

and

The trends differ from zero by 1.4 , and, interestingly, they are in the sense opposite to that found by Dambis et al. (2014). We hereafter compute the combined solutions assuming the same slope for all clusters and using weights based on the scatter of individual-cluster PL solutions listed in Table 1. These combined solutions yield the slopes of and for the - and -band relations, respectively. Figs. (9) and (10) show the combined versus and versus plots for RR Lyraes passing the above filters in the 24 globular clusters of the final list. We list the and parameters obtained from these combined solutions in the last two columns of Table 1.

4.2. Intrinsic color calibration

To finalize our calibration of the mid-infrared period-metallicity-luminosity relations for RR Lyrae stars and to use it to compute distances to these variables, we need to be able to estimate interstellar extinction to individual stars. To this end, we now derive a calibration for the intrinsic colors of RR Lyrae variables based on the above final sample of globular-cluster stars. First, equations (2) imply the following formula for the color excess ratio:

Hence

or (in view of ):

and

We use formula (22) and color excesses from Harris (2010) to fit a linear function of and [Fe/H] to the so computed intrinsic colors of the cluster RR Lyraes of our final sample:

Given the extinction ratios (2), the total extinction in the and bands can then be computed as

and

respectively. The scatter of intrinsic color computed from formula (23) ( ) contributes only about 0.007 and 0.004 to the error of the computed and , respectively.

4.3. The metallicity slope and zero point

To estimate the two remaining parameters of the period-metallicity-luminosity relation (1), and , we cross-match our initial calibration sample of 5717 RR Lyraes having spectroscopic metallicity estimates with the Gaia DR3-based catalog of RR Lyrae-type variables by Clementini et al. (2023) and leave only stars for which the latter catalog provides both - and -band intensity-mean magnitudes. We use formulas (14)-(17) and (23)-(25) to compute the dereddened - and -band intensity-mean magnitudes of the stars considered. We make further astrometric and photometric quality cuts leaving only stars with RUWE , astrometric_excess_noise , corrected and flux excess factor . A total of 4658 stars pass these cuts.

Like Mullen et al. (2024), we determine and by fitting period-metallicity-luminosity relation using the Astrometric Based Luminosity (ABL) :

where is the Gaia DR3 trigonometric parallax in milliarcseconds with the parallax zero-point bias corrected as prescribed by Lindegren et al. (2021), is the absorption-corrected apparent magnitude ( or ), is the period slope estimate inferred above from globular-cluster RR Lyraes, and and are determined in the fit. Given the accurate globular-cluster-based estimates of the period slopes and inferred above, formula (26) for - and -band magnitudes can be rewritten as:

and

respectively. Ordinary 4-clipped weighted regression fits (27) and (28) with Gaia DR3 parallaxes bias-corrected as prescribed by Lindegren et al. (2021) and with errors assumed to be only in the dependent variable and computed based on quoted Gaia DR3 parallax errors and assuming the scatter of 005 both in and about the period-metallicity-luminosity relation yield the following results:

and

The standard unit-weight error is equal to 1.20 in both cases, indicating that the quoted Gaia DR3 parallax errors appear to be slightly underestimated.

However, in our case there are also significant errors in independent variables [Fe/H], and simple regression fits produce biased results. To estimate and correct the biases introduced, we generate two sets of 400 simulated 4658-star samples with the parallax of each star computed using formulas (27) and (28) based on observed , [Fe/H], and or values and with , , , and parameters inferred above, adding simulated random noise normally distributed with zero mean and errors computed based on quoted Gaia DR3 parallax errors and 005 scatter in or magnitudes and then multiplied by 1.2. The simulated metrallicities [Fe/H] are equal to the observed ones plus random noise normally distributed with zero mean and quoted metallicity errors. We found the bias and in the sense the estimated minus true value to be

and

implying the following bias-corrected values:

and

5. Results and validation

Thus, our final - and -band period-metallicity-luminosity relations for RR Lyrae variables are:

and

with standard errors about 004-005. These relations are very close to those inferred by Mullen et al. (2023) based on WISE - and -band data for about 1000 bright RR Lyraes (recentered to our adopted central and [Fe/H]=-1.6):

and

and differ by much higher accuracy of the period slopes and slightly steeper metallicity slopes. Note that our -band period-metallicity-luminosity relation is very close (both in terms of the slopes and slope errors) to the corresponding -band relation derived by Bhardwaj et al. (2023) based on accurate homogeneous near-IR photometry of 954 RR Lyraes in 11 globular clusters anchored using 346 Milky Way field RR Lyraes with Gaia EDR3 parallaxes:

(we recentered it to and [Fe/H]=-1.6 for consistency).

We use relations (31) and (32) combined with the intrinsic-color calibration (23) and extinction ratios (2) to compute the true distance moduli ((W1) and (W2)), the corresponding photometric parallaxes ( and ), and distances ( and ) for the 24 globular clusters of our final list, and compare them to the globular-cluster trigonometric parallaxes determined via Gaia EDR3 parallaxes and kinematic distances found from Gaia EDR3 or Hubble Space Telescope proper-motion dispersion profiles combined with radial-velocity dispersion profiles (Baumgardt & Vasiliev 2021). We summarize all these data in Table 2. Given that the errors of our globular-cluster photometric parallaxes and photometric distances are mush smaller than those of the cluster trigonometric parallaxes and kinematic distances reported by Baumgardt & Vasiliev (2021), we treat the former in the following weighted regression fits as independent variables with no errors and the latter as dependent variables with the corresponding quoted errors:

(24 clusters),

(12 clusters),

(5 clusters). As is evident from the above relations, the globular-cluster photometric distances based on our period-metallicity-luminosity relations are marginally consistent with the cluster parallaxes computed via Gaia EDR3 parallaxes of individual stars (in the sense that our distances are slightly longer, by a factor of 1.016-1.018 corresponding to +0.04 in terms of the true distance modulus), but are shorter (by up to 2.5 or a factor of 1.027-1.029 corresponding to -0.06 in terms of true distance modulus) than the kinematical distances computed from Gaia EDR3 proper-motion dispersion profiles. A comparison with the kinematical distances computed via Hubble Space Telescope proper-motion dispersion profiles yields the same discrepancy factor, but with a greater error corresponding to about 1 level.

Note that an analysis involving eventual bias in globular-cluster trigononetric parallax estimates reported by Baumgardt & Vasiliev (2021) yields:

and

(24 clusters), i.e. our photometric parallaxes of globular clusters appear to be totally consistent with estimates based on trigonometric parallaxes of individual cluster stars, and the unaccounted bias in globular-cluster parallaxes reported by Baumgardt & Vasiliev (2021) appears to be of about as in the sense that the said parallaxes are slightly overestimated.

We finally test our approach by applying to it to faint Gaia RR Lyrae variables. To this end, we cross-match the catalog of Clementini et al. (2023) with the list of Cusano et al. (2021) containing about 22000 RR Lyraes in the Large Magellanic Cloud having near-infrared photometry light curves from VISTA survey of the Magellanic Clouds system (VMC) and optical data from the Optical Gravitational Lensing Experiment (OGLE) IV survey and the Gaia Data Release 2 catalogue, which the authors consider to be LMC members. We leave only stars with corrected and flux excess factor with our final list containing a total of 11901 stars for which we compute the WISE - and -band intensity-mean magnitudes and via formulas (14)-(17) above adopting me same [Fe/H] equal to the median value for LMC fundamental-mode RR Lyraes (Skowron et al. 2016). We do not use individual-star metallicity estimates from Cusano et al. (2021) because they are unavailable for a large fraction of the list and those available have errors comparable to or greater than the scatter of the values themselves. We then deredden and by substracting =0.487 and =0.294 extinction values, respectively, where -band extinction estimates are adopted from Cusano et al. (2021) and the extinction ratios are computed based on the Cardelli et al. (1989) and O'Donnell (1994) extinction law and assuming =3.1. The 3- clipped regression fits of the period-magnitude relations and are:

and

Note that, despite the large scatter (which is to be expected for stars at the faint end of the Gaia catalog and located in a very crowded field), the slopes are remarkably close to the values inferred above from much brighter globular-cluster stars. Our relations (31) and (32), combined with the intrinsic-color calibration (23) and extinction ratios (2), yield the same true LMC distance modulus estimates for both passbands, and these values are rather close to the very accurate estimate (statistical) 0.026 (systematic) by Pietrzynski et al. (2019) based on observations of eclipsing binary systems composed of late-type stars and the calibration of the surface brightness-color relation. The statistical errors () are unrealistically small, and we consider a bona fide conservative error estimate to be 0.04-0.06 based on the largest discrepancy found in the case of the comparison of our photometric distances to globular clusters with the determinations based on Gaia EDR3 parallaxes of individual cluster stars and with the kinematical distances computed using Gaia EDR3 and Hubble Space Telescope proper motions - just enough to explain the departure from the very precise Pietrzynski et al. (2019) estimate.

6. Conclusions

We derived color transformations permitting the intensity-mean WISE - and -band magnitudes of RR Lyrae-type variables to be computed with errors from their known Gaia DR3 intensity-mean and magnitudes, fundamental periods, and [Fe/H]. We used our calibrations to study the mid-infrared fundamental period-magnitude relations in 24 Galactic globular clusters and find the average slopes to be and for the and bands, respectively, and the scatter of the relations to range from 0017 for M 107 to 01 for NGC 5824. These slope estimates, together with low-resolution spectroscopic [Fe/H] estimates for 5000 RR Lyraes adopted from the catalog of Liu et al. (2020) and the catalog of Muhie et al. (2021), combined with Gaia DR3 parallax and photometry data yield - and -band period-metallicity-luminosity relations with metallicity slopes =+0.1660.007 and =+0.1700.007, respectively. The photometric distance estimates for 24 Galactic globular clusters implied by these relations are totally consistent with average Gaia DR3 trigonometric parallaxes (with a possible parallax bias of as) and about a factor of 1.028 shorter than kinematical distances computed with Gaia DR3 or HST-based proper motions. We find the - and -band fundamental period-magnitude relations for LMC RR Lyraes to have the same slopes and as we find for Globular-cluster variables, but much greater scatter of . Both our - and -band period-metallicity-luminosity relations yield LMC RR Lyraes true distance modulus estimates with the mean and a scatter of (formal error of the mean is about 0.0015, but the actual error is determined by the zero point errors of the PML relations, transformation equations, and extinction values and may be as high as our conservative estimate of 0.06). This result is about greater than the currently best estimate  (statistical)  (systematic) obtained by Pietrzynski et al. (2019) based on LMC eclipsing variables.

7. Acknowledgments

The study was conducted under the state assignment of Lomonosov Moscow State University.

This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.

This publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration.

This publication also makes use of data products from NEOWISE, which is a project of the Jet Propulsion Laboratory/California Institute of Technology, funded by the Planetary Science Division of the National Aeronautics and Space Administration.

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Table 1. Parameters of the  =  + 0.25) + and  =  + 0.25) + fits for globular clusters with at least eight RR Lyraes outlining linear relation

Cluster	Alt.		[Fe/H]
name	name										(=	(=
											-2.418)	-2.449)
NGC 7089	M 2	0.06	-1.65	10	14.804	-2.431	0.048	14.824	-2.468	0.040	14.804	14.824
					0.017	0.149		0.017	0.152		0.015	0.016
NGC 5272	M 3	0.01	-1.50	93	14.485	-2.386	0.037	14.506	-2.422	0.038	14.484	14.505
					0.005	0.064		0.005	0.066		0.004	0.004
NGC 6121	M 4	0.35	-1.16	43	10.959	-2.412	0.029	10.953	-2.443	0.029	10.959	10.952
					0.006	0.054		0.006	0.054		0.005	0.005
NGC 5904	M 5	0.03	-1.29	44	13.822	-2.321	0.054	13.843	-2.344	0.056	13.817	13.838
					0.010	0.107		0.010	0.112		0.008	0.009
NGC 6402	M 14	0.60	-1.28	22	14.343	-2.502	0.041	14.325	-2.521	0.043	14.349	14.331
					0.012	0.087		0.012	0.091		0.009	0.009
NGC 7078	M 15	0.10	-2.37	34	14.469	-2.601	0.040	14.483	-2.668	0.041	14.473	14.488
					0.008	0.125		0.008	0.129		0.007	0.007
NGC 6656	M 22	0.34	-1.70	11	12.095	-2.313	0.037	12.093	-2.342	0.040	12.088	12.087
					0.015	0.144		0.016	0.152		0.011	0.012
NGC 5024	M 53	0.02	-2.10	30	15.766	-2.604	0.039	15.787	-2.665	0.041	15.768	15.789
					0.008	0.100		0.008	0.103		0.007	0.007
NGC 6266	M 62	0.47	-1.18	46	13.496	-2.439	0.060	13.484	-2.446	0.062	13.493	13.479
					0.011	0.092		0.011	0.096		0.009	0.009
NGC 4590	M 68	0.05	-2.23	26	14.414	-2.330	0.035	14.431	-2.388	0.036	14.412	14.430
					0.008	0.133		0.008	0.136		0.007	0.007
NGC 6981	M 72	0.05	-1.42	15	15.601	-2.267	0.031	15.621	-2.286	0.033	15.601	15.620
					0.009	0.236		0.009	0.247		0.008	0.009
NGC 6171	M 107	0.33	-1.02	12	13.258	-2.305	0.017	13.253	-2.320	0.018	13.250	13.244
					0.007	0.067		0.008	0.070		0.005	0.006
NGC 1851		0.02	-1.18	10	14.855	-2.325	0.050	14.876	-2.343	0.052	14.848	14.867
					0.022	0.180		0.023	0.186		0.016	0.017
NGC 3201		0.24	-1.59	67	12.834	-2.405	0.028	12.838	-2.451	0.030	12.834	12.837
					0.004	0.069		0.004	0.073		0.003	0.004
NGC 4833		0.32	-1.85	9	13.492	-2.127	0.053	13.492	-2.149	0.054	13.500	13.500
					0.021	0.241		0.021	0.246		0.018	0.018
NGC 5053		0.01	-2.27	9	15.567	-2.476	0.040	15.588	-2.532	0.041	15.566	15.587
					0.015	0.148		0.015	0.151		0.014	0.014
NGC 5466		0.00	-1.98	19	15.391	-2.339	0.054	15.412	-2.382	0.055	15.392	15.413
					0.014	0.144		0.014	0.148		0.012	0.013
NGC 5824		0.13	-1.91	24	16.991	-2.200	0.101	17.004	-2.241	0.104	16.984	16.997
					0.023	0.241		0.023	0.249		0.021	0.022
NGC 6101		0.05	-1.98	13	15.162	-2.396	0.042	15.177	-2.447	0.043	15.161	15.177
					0.013	0.144		0.013	0.146		0.012	0.012
NGC 6584		0.10	-1.50	25	15.089	-2.617	0.081	15.104	-2.651	0.082	15.097	15.112
					0.019	0.201		0.019	0.204		0.016	0.017
NGC 6638		0.41	-0.95	9	14.385	-2.983	0.092	14.377	-3.011	0.093	14.450	14.442
					0.065	0.482		0.066	0.485		0.031	0.031
NGC 6934		0.10	-1.47	17	15.426	-2.782	0.050	15.431	-2.805	0.069	15.425	15.431
					0.013	0.146		0.017	0.200		0.012	0.016
IC 4499		0.23	-1.53	72	15.888	-2.500	0.057	15.894	-2.537	0.058	15.889	15.895
					0.008	0.108		0.008	0.111		0.007	0.007
NGC 5139	Cen	0.12	-1.68	33	13.140	-2.371	0.061	13.158	-2.416	0.062	13.140	13.153
					0.011	0.135		0.008	0.137		0.011	0.011

Because of the metallicity spread among RR Lyrae stars in this cluster (Sollima et al. 2006 and reference therein), we left only the metal-poor stars ( corresponding to ).

Table 2. Comparison of our RR Lyrae-based photometric distance estimates to 24 globular clusters with Gaia DR3-based trogonometric parallax estimates and Gaia DR3- and HST-based kinematical distance estimates from Vasiliev & Baumgardt (2021)

Cluster	Alt.		,	,		,	,	,	(EDR3),	(HST),
name	name	(W1)	kpc	mas	(W2)	kpc	as	as	kpc	kpc
NGC 7089	M 2	15.288	11.416	87.64	15.290	11.429	87.50	82	11.940
		0.016	0.083	0.64	0.016	0.085	0.64	11	0.703
NGC 5272	M 3	14.951	9.778	102.27	14.951	9.778	102.27	110	10.116
		0.006	0.025	0.26	0.006	0.025	0.27	10	0.384
NGC 6121	M 4	11.310	1.828	547.04	11.304	1.823	548.57	556	1.878
		0.006	0.005	1.53	0.006	0.005	1.51	10	0.033
NGC 5904	M 5	14.246	7.067	141.50	14.246	7.065	141.53	141	7.467	7.456
		0.009	0.030	0.59	0.009	0.031	0.61	10	0.357	0.201
NGC 6402	M 14	14.676	8.616	116.06	14.677	8.617	116.06	129
		0.010	0.040	0.53	0.010	0.041	0.54	11
NGC 7078	M 15	15.069	10.322	96.88	15.072	10.337	96.73	97		10.375
		0.008	0.038	0.35	0.008	0.039	0.36	10		0.295
NGC 6656	M 22	12.531	3.207	311.78	12.532	3.209	311.65	306	3.181	3.161
		0.012	0.018	1.73	0.013	0.019	1.81	10	0.123	0.088
NGC 5024	M 53	16.333	18.475	54.13	16.336	18.504	54.04	67	17.313
		0.008	0.071	0.21	0.008	0.072	0.21	11	1.353
NGC 6266	M 62	13.826	5.824	171.70	13.822	5.812	172.04	185	6.395	6.502
		0.010	0.027	0.78	0.010	0.027	0.81	11	0.327	0.163
NGC 4590	M 68	14.994	9.970	100.30	14.995	9.979	100.02	113
		0.008	0.037	0.37	0.008	0.038	0.38	11
NGC 6981	M 72	16.048	16.203	61.72	16.048	16.207	61.70	84
		0.009	0.068	0.26	0.009	0.070	0.27	12
NGC 6171	M 107	13.581	5.203	192.20	13.574	5.187	192.79	194	6.017
		0.007	0.016	0.58	0.007	0.016	0.61	11	0.407
NGC 1851		15.260	11.273	88.70	15.258	11.260	88.81	88
		0.017	0.087	0.67	0.017	0.089	0.70	11
NGC 3201		13.276	4.520	221.24	13.274	4.517	221.37	222	4.745
		0.005	0.011	0.54	0.005	0.011	0.55	10	0.176
NGC 4833		13.971	6.226	160.62	13.972	6.230	160.52	164	5.822
		0.018	0.053	1.33	0.019	0.053	1.36	11	0.360
NGC 5053		16.161	17.071	58.58	16.163	17.088	58.52	50
		0.014	0.111	0.38	0.014	0.113	0.38	11
NGC 5466		15.941	15.427	64.82	15.942	15.430	64.81	57
		0.013	0.093	0.39	0.013	0.095	0.40	11
NGC 5824		17.499	31.604	31.64	17.500	31.629	31.62	57
		0.021	0.311	0.31	0.022	0.315	0.31	12
NGC 6101		15.702	13.814	72.39	15.700	13.805	72.44	84
		0.012	0.079	0.41	0.013	0.080	0.42	11
NGC 6584		15.548	12.872	77.68	15.548	12.872	77.69	77
		0.017	0.100	0.59	0.017	0.101	0.60	11
NGC 6638		14.756	8.936	111.90	14.752	8.920	112.10	115
		0.031	0.130	1.59	0.032	0.129	1.60	12
NGC 6934		15.872	14.944	66.92	15.862	14.872	67.24	78	16.718
		0.013	0.090	0.40	0.017	0.115	0.52	12	1.382
IC 4499		16.323	18.388	54.38	16.322	18.387	54.39	54
		0.008	0.067	0.20	0.008	0.068	0.20	11
NGC 5139	Cen	13.617	5.291	188.99	13.618	5.293	188.93	193	5.359	5.264
		0.011	0.028	0.99	0.012	0.028	1.00	9	0.141	0.121

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