1. A Brief Panorama Of The M_V-[Fe/H] Relation

The relevance of RR Lyrae (RRL) stars as distance indicators has been well known since the early 20th century. ^{Shapley (1917)} recognized that “the median magnitude of the short-period variables [RR Lyrae stars] apparently has a rigorously constant value in each globular cluster” and “the observed differences in the mean values then become sensitive criteria of distance, and the relative parallaxes of these remote systems can be known with an accuracy...”, a fact that was used later by Shapley himself to describe the Galactic distribution of globular clusters (^{Shapley 1918}). This apparently constant value of the mean magnitude of the RRL can now be interpreted as the luminosity level of the horizontal branch (HB) being constant in all globular clusters. The fact that this is not exactly the case, but instead that metallicity plays a role in determining the luminosity level of the HB, has been demonstrated from theoretical and observational grounds. ^{Sandage (1981)}; ^{Lee et al. (1990)}; ^{Sandage (1990)} provided a calibration of the M_V -[Fe/H] relation. ^{Lee et al. (1990)} discussed its dependence on helium abundance. Other empirical calibrations followed in the works of ^{Walker (1992)}), Carney et al. (1992), Sandage(1993) and ^{Benedict et al. (2011)}. Complete summaries on the calibration of the M_V -[Fe/H] relation can be found in the works of ^{Chaboyer (1999)}, ^{Cacciari & Clementini (2003)} and ^{Sandage & Tammann (2006)}.

While the relation has been considered to be linear in most empirical works, a non-linear nature is advocated by theoretical approaches, e.g. ^{Cassisi et al. (1999)} and ^{VandenBerg et al. (2000)}. A linear relation of the form M_V = a + b[Fe/H] has been broadly accepted in the literature and the slope, resulting from a variety of independent calibrations, ranges a wide span (0.13 0.30) as different strategies have been adopted, mainly towards the calculation of M_V. The relevance of the slope and zero point of this relation on the relative and absolute ages of the globular clusters has been amply discussed by ^{Chaboyer et al. (1996}, ¹⁹⁹⁸⁾ and ^{Demarque et al. (2000)}. That a linear relation may be an over simplification (^{Catelan & Smith 2015}) becomes very clear from the theoretical analysis of ^{Demarque et al. (2000)}, that clearly demonstrates the role of the HB structure and that the slope itself is a function of metallicity. The higher complexity of the HB luminosity and metallicity interconnection has however defied clear empirical demonstrations and calibrations, for which, a very extensive and homogeneous endeavor is required.

Our approach to the determination of mean M_V and [Fe/H] of RRL stars has been the Fourier light curve decomposition of both the fundamental mode and first overtone pulsators RRab and RRc respectively. This followed by the employment of solidly established semi-empirical calibrations between the Fourier parameters and the physical quantities. As early as 2002, our group started studying individual clusters in great detail from CCD time-series imaging through the Johnson-Kron-Cousins VI bands. Each cluster in the sample has been the subject of a dedicated study, and the discussions include several key aspects of the nature of the globular clusters, such as distinction of likely cluster members from field stars, structure of the colour-magnitude diagram (CMD), pulsating mode distribution on the HB, and theoretical approaches to the mass loss events in the red giant branch and the subsequent mass distribution at the stage of core He-burning (zero age horizontal branch or ZAHB) and post ZAHB evolution.

In 2017, ^{Arellano Ferro et al. (2017)} (hereinafter ABG17) summarized the results of a 15-year old program dedicated to study the variable star populations in globular clusters. The program was mainly aimed to the determination of the mean distance and metallicity of the clusters in a homogeneous way, via the Fourier decomposition of the light curves of their RRL stars and the use of well tested semi-empirical calibrations of the Fourier parameters in terms of luminosity and [Fe/H]. The program is based on the Johnson-Kron-Cousins VI CCD time-series imaging, and their subsequent scrutiny via the difference images analysis (DIA), that produces accurate photometry even in the crowded central regions of the globular clusters. In the process numerous variables of virtually all types typically present in globular clusters were discovered, of which ABG17 gave a detailed account.

Presently, five years after ABG17 paper, our group has systematically continued enlarging the sample of studied clusters which has increased from 25 to 34. This should enable a better sustained discussion of the M_V -[Fe/H] relation, i.e. the metallicity dependence of the luminosity of the horizontal branch (HB), and the influence of the cluster Oostherhoff type and the HB structure (^{Demarque et al. 2000}). In the present paper we update our discussion of the nature and calibration of the M_V -[Fe/H] relation, and introduce the role of the HB structure parameter £ which is shown to be of obvious relevance. We also perform a census of variable stars per cluster per variable type and reinforce the resulting cluster distance scale from the Fourier approach via the comparison with independent distances recently obtained from Gaia and HST accurate data.

We shall mention at this point that the parameters listed in the tables below for specific clusters and number of variables may occasionally differ slightly from the equivalent tables in ABG17, as a result of a critical evaluation of the original samples. The present tables supersede the previous ones.

2. Observations and Image Reductions

3. Calculation of M_V and [Fe/H]

Our approach to the calculation of mean M_V and [Fe/H] for each GC in the sample has been through the RRL light curve Fourier decomposition, and the application of ad hoc, well tested, semi empirical calibrations. The Fourier decomposition of the RRL light curves is performed by fitting the observed light curve in V -band with a Fourier series model of the form:

where m(t) is the magnitude at time t, P is the period, and E is the epoch. A linear minimization routine is used to derive the best-fit values of the amplitudes A_κ and phases Φ_κ of the sinusoidal components. From the amplitudes and phases of the harmonics in equation 1, the Fourier parameters, defined as Φ_ij = jΦ_i - iΦ_j, and R_ij = A_i / A_j, are computed.

Subsequently, the low-order Fourier parameters can be used in combination with semi-empirical calibrations to calculate [Fe/H] and M_V for each RRL, and hence the mean values of the metallicity and absolute magnitude for the RRL population in the host cluster.

The specific calibrations and zero points used for RRab and RRc stars for this purpose are described in § 3.1. Numerous Fourier decompositions of RRL light curves can be found in the literature. However, over the years, each author has used different calibrations and zero points to estimate MV and [Fe/H]. Our group has also used slightly different equations in the earlier papers but in the work by ^{Arellano Ferro et al. (2010)} zero points of the M_V calibrations (see their § 4.2), were discussed and adopted, and we have used them subsequently. In the present paper we have recalculated M_V and [Fe/H] for all clusters in the sample, using the calibrations described in the following section.

The final values found for M_V and [Fe/H], the later expressed in the three different scales defined in § 3.1, are listed in Table 1, which is organized by Oosterhoff types; Oosterhoff (1939, 1944) realized that the periods of fundamental-mode RRL, or RRab stars in a given cluster, group around two values; 0.55 d (Oosteroff Type I or OoI) and 0.65 d (Oosteroff Type II or OoII). OoI clusters are systematically more metal-rich than OoII clusters. A third Oosterhoff class (OoIII) (^{Pritzl et al. 2000}), which presently contains only two GCs, NGC 6388 and NGC 6441, is represented by very metal-rich systems, where the periods of their RRab stars average about 0.75 d. A few clusters have been classified as of the intermediate type or OoInt, since the average periods of their RRab stars and their mean [Fe/H] fall between those of Type I and Type II clusters; it has been argued that OoInt clusters may be associated to an extragalactic origin (^{Catelan 2009}) due to their similarity to dSph galaxies, satellites of the Milky Way, and their respective clusters. In Table 1 we include of 16 OoI, 14 OoII, 2 OoIII and 2 OoInt clusters. The calculations have been performed independently for RRab and RRc stars. For clusters with differential reddening, i.e. NGC 1904, NGC 3201, NGC 6333 and NGC 6401, care has been taken in calculating the individual reddening for each RRL. The interested reader is referred to the original papers for a detailed discussion on that subject.

4. Vindication of the Photometric Approach to the Metallicities

There is absolutely no doubt that the most precise approach to the determination of metallicities of heavenly bodies is via high-resolution spectroscopy. The practical limitations to that technique are several however; to reach deep in magnitude, typical of the HB in most globular clusters, long exposure times with large telescopes are required, making it unaffordable. The spectroscopic values for 19 clusters listed by ^{Carretta et al. (2009)} were obtained in numerous previous papers cited by these authors. The analyses were carried on luminous red giants of V ≈ 14 16 mag, i.e. 2-3 magnitudes brighter than RRL stars at the HB, and after an enormous compromise of observational and computational resources. While this situation may change with the advent of in-orbit high resolution spectrographs, the competition for access to the instrumentation will likely remain tough. The photometric approach to the metallicity and luminosity calculation in RRL stars was envisaged in the 1980’s (^{Simon & Teays 1982}), and became a popular alternative since it reaches as low as V ≈ 20 mag with sufficient accuracy with very short exposure times on 1-2 m-class telescopes, enabling the access to larger samples of clusters. The Fourier decomposition approach was further developed to produce the calibrations in § 3 employed in this paper.

Our goal in this section is to compare the photometric values reported in this work with the spectroscopic values of ^{Carretta et al. (2009)}. Figure 1 shows the photometric based values of [Fe/H] in the UV and Nemec scales (Table 1) for the RRab and RRc stars, plotted versus the spectroscopic values given by Carretta et al. (2009). In Panels (a) and (b) it is clear that the comparison is satisfactory for the case of [Fe/H]_UV. However, for [Fe/H]_N the RRab calibration seems to systematically overestimate the metallicity relative to the spectroscopic values. The iron values from Nemec’s calibration for RRc stars also compare well with the spectroscopic values. There is a mild suggestion in both panels (b) and (d) that the calibrations for the RRc stars of equations 3 and 5, which are in fact based on the same set of calibrators, may require a small adjustment of about −0.2 dex for iron values smaller than −2.0.

Fig. 1 Comparison of the Fourier based [Fe/H] (Table 1), with those from high-resolution spectroscopy based values from ^{Carretta et al. (2009)}. Filled and empty circles represent OoII and OoI clusters respectively. Open triangles and squares represent OoInt and OoIII clusters. See § 4 for a discussion. The color figure can be viewed online. 

5. The M_V -[Fe/H] correlation

It has been argued that equation 2 overestimates [Fe/H] for metal-poor clusters. This problem has been addressed by ^{Jurcsik & Kovács (1996)}, ^{Schwarzenberg-Czerny & Kaluzny (1998)}, ^{Kovács (2002)}, ^{Nemec (2004)} and ^{Arellano Ferro et al. (2010)}. It is difficult to quantify a correction to be applied, and it is likely also a function of the metallicity. However, empirical estimations in the above papers point to a value between -0.2 and -0.3 dex on the scale of equation 2. We have adopted -0.3 dex, which on the ZW scale is equivalent to - 0.21 dex. Equally difficult is to define a value of [Fe/H]_ZW below which the corrections should be applied. Guided by the metallicity values of globular clusters in the spectroscopic scale of ^{Carretta et al. (2009)}, we estimated that a reasonable limit would be [Fe/H]_ZW < 1.7 Therefore, the values listed in Table 1 for clusters with [Fe/H]_ZW < -1.7 dex were obtained by adding -0.21 dex to the value of [Fe/H]_ZW found via equation 2. As a consequence the [Fe/H]_UV values for these clusters are also affected by this correction. Note that the good comparison between the photometric [Fe/H]_UV and the spectroscopic values displayed in Figure 1 (a) was obtained after the application of the correction above.

In Figure 2 we show the distribution of clusters in the M_V -[Fe/H] plane obtained from the RRab stars (left panel) and the RRc stars (right panel). In each of these panels we display the resulting distributions for the three involved scales [Fe/H]_ZW, [Fe/H]_UV and [Fe/H]_N. In the middle and bottom boxes, for the spectroscopic scales [Fe/H]_UV and [Fe/H]_N, we include as reference, in gray colour, two theoretical, non-linear, versions of the M_V -[Fe/H] relation of ^{Cassisi et al. (1999)} and ^{VandenBerg et al. (2000)}.

Fig. 2 The [Fe/H] versus MV correlations for RRab and RRc stars. The involved metallicity scale is, from top to bottom panels, [Fe/H]ZW, [Fe/H]UV and [Fe/H]N. Filled and open circles represent OoII and OoI clusters respectively. Open triangles and squares represent OoInt and OoIII clusters. All fits have been weighted by the number of stars included in each cluster. In the left panel, the gray curves are the theoretical predictions of ^{Cassisi et al. (1999)} (long dash) and ^{VandenBerg et al. (2000)} (short dash), which are strikingly similar to the photometric solution in panel (b). The color figure can be viewed online. 

We shall discuss these correlations separately for the RRab and RRc stars.

5.3 The Role of the HB Structure Parameter

^{Demarque et al. (2000)}, have argued on theoretical grounds that the overall structure of the HB plays a relevant role and may be interconnected with the HB luminosity and the metallicity of the parental globular cluster. Here we explore the role of the HB type parameter, or Zinn-Lee parameter, defined as ₤ = (B - R)/(B + V + R) (^{Zinn 1986}; ^{Lee 1990}), from empirical arguments. B, V and R represent the number of stars to the blue of the instability strip (IS), the number of RRL stars, and to the red of the IS respectively. For a better calculation of ₤, it is convenient to include, as far as possible, only cluster member stars in the counting. Since the list of ₤ values for a large number of clusters presented by ^{Torelli et al. (2019)} is the result of membership considerations, we have adopted them for the subsequent analysis. When a cluster is not included in the above list we used the value reported by ^{Catelan (2009)}. The one exception is NGC 7079 (M2). For this cluster, the reported value is ₤ =+0.96. The analysis of the projected positions and proper motions available in the Gaia-eDR3 data base, and the application of the method of ^{Bustos Fierro & Calderón (2019)}, kindly performed by Dr. Bustos Fierro, renders a CMD of very likely cluster member stars, with a substantial population of red HB stars for ₤ =+0.34, which shall be adopted. For a few other clusters where similar analyses have been carried out, ₤ values close to those of ^{Torelli et al. (2019)} were found.

Figure 3 illustrates the correlation between ₤ and M_V, the latter as obtained from the Fourier decomposition for RRab stars (top panel) and RRc stars (bottom panel). In the case of the RRab stars the correlation clearly shows a quadratic trend, once the two Oo-Int clusters, NGC 6388 and NGC 6441, were excluded. In the case of the RRc stars the correlation appears linear with a very mild slope. The corresponding fits in Figure 3 are of the form:

MV=0.620±0.011-0.029±0.018L-0.135±0.036L2 (12)

with an rms = 0.058, and

MV=0.558±0.006-0.019±0.013L, (13)

with an rms = 0.026.

Considering the trends in Figure 2 for the RRab and RRc, both for the [Fe/H]_UV values (middle panels), and in Figure 3, the dependence of M_V on the metallicity [Fe/H] and HB structure parameter ₤, can be expressed, for the RRab and RRc respectively as:

MV=A+BFeHUV+CFeHUV2+DL+EL2 (14)

with A=+1.096(±0.141), B=+0.519(±0.172), C=+0.119(±0.050), D=+0.006(±0.014), E=−0.111(±0.029), and rms = 0.053 mag.

MV=+0.609±0.016+0.032±0.009FeHUV+0.015±0.011L (15)

with rms = 0.024 mag.

The equivalent calibrations in terms of the metallicity in the scale of ^{Nemec et al. (2013)}, [Fe/H]_N are:

MV=A+BFeHN+CFeHN2+DL+EL2 (16)

with A=+0.720 (±0.082), B=+0.130(±0.098), C=+0.033(±0.029), D=­0.043(±0.020), E=-0.145(±0.030), and rms = 0.055 mag.

Mv=+0.655±0.019+0.063±0.013FeHN+0.012±0.009L (17)

with rms = 0.019 mag.

For the calibration of equations 14 and 16 from the RRab solutions, the terms involving ℒ are small but significant. On the contrary, equations 15 and 17 from the RRc solutions, the last term is insignificant, and in fact, for instance equations 10 and 15 in the UV scale, or equations 11 and 17 in the Nemec’s scale, are, within the uncertainties, indistinguishable, confirming the linearity and sufficiency of an M_V -[Fe/H] relation for the RRc stars.

Fig. 3 The HB structure parameter ℒ vs. mean M_V for a family of clusters coded as in Figure 1. Two open squares for the OoIII clusters were not considered in the weighted fits. 

Therefore, the empirical M _V -[Fe/H] relation as worked out from the Fourier decomposition of RRab stars light curves, has turned out to be much more complex, with the metallicity and HB structure playing a measurable role, and equations 14 and 16 are a good representation.

For the RRc stars, with simpler light curves and generally being more confined near the ZAHB, the M_V -[Fe/H] relation remains linear and simple, and the structure of the HB does not seem to play any pertinent role; equation 10 and 11, or for any purpose equations 15 and 17, are good empirical calibrations, with a well established slope around 0.06.

6. Globular Cluster Distances

Once the mean absolute magnitude M_V for the HB is obtained for a given cluster, its distance can be estimated for an assumed value of E(B - V). Using the values listed in Table 1 the distances were calculated and are listed in Table 2 as they were obtained either for the RRab or RRc stars from equations 6 and 7 respectively. We perform a comparison with accurate mean distances recently estimated by ^{Baumgardt & Vasiliev (2021)} (hereinafter BV21), calculated for a large sample of globular clusters using the data from Gaia-eDR3, HST and selected literature distances.

It should be obvious from Table 2 that the distances derived from our Fourier approach (Columns 2 and 3) agree with those of BV21. Figure 4 is a graphical comparison and shows how the distance differences do not correlate either with the metallicity or with the HB structure parameter. The Fourier and the BV21 distances differences are all smaller than 1.7 kpc and display a standard deviation of 0.7 kpc.

Fig. 4 Comparison of distances obtained from the RRL Fourier decomposition and those of BV21. Blue and green symbols stand for distances derived from RRab and RRc Fourier light curve treatment, respectively. The color figure can be viewed online. 

The agreement is remarkably good considering that the distance determinations come from completely independent approaches. We note that the distances obtained from the M_V calibrations for RRab and RRc stars, are also independent, as they come from different and independent calibrations. This gives further support to the M_V calibrations of equations 6 and 7 and to their zero points (^{Arellano Ferro et al. 2010}).

The accurate distances of individual globular clusters obtained from the RRL stars listed in Table 4, can serve as a frame of comparison of other independent methods to calculate cluster distances. Given that SX Phe stars are common in globular clusters, they can be used as secondary distance indicators through their well established P-L relation, of which, however, different calibrations are found in the literature. We shall explore the consistency of the results. We considered the calibrations of ^{Arellano Ferro et al. (2011)} (AF11), and ^{Cohen & Sarajedini (2012)} (CS12). The resulting distances from these two calibrations of 13 clusters with well observed SX Phe stars are listed in Columns 4 and 6 of Table 2, Column 5 indicates the number of member SX Phe stars available in each cluster. We should emphasize that the two calibrations lead to distances agreeing within 1.3 kpc, except for NGC 6934 where the distance differences is ≈ 2.0 kpc for the CS12 calibration. The SX Phe distances agree with the RRL results well within 1 kpc, i.e. the average RRL and SX Phe distance match in average by ≈ 4% of the corresponding distance.

7. Variable Stars in Our Sample of Globular Clusters

Once a CCD time-series photometry is performed on a given cluster, a by-product of the exercise is the discovery of previously undetected variables. In the work carried by our group, we have systematically searched for variables via a variety of approaches described in the individual papers, e.g. ^{Arellano Ferro et al. (2013b)}. In Table 3 we summarize the number of variables, and their types, known in the globular clusters of our sample, noting the ones found by our work. We have found 326 new variables in the field of the clusters, 23 of them are either considered field stars or have not been classified. The most numerous families are in order RRab, RRc, SR, SX Phe, eclipsing binaries, CW and double mode RRd stars. The total number of variables detected in these clusters is 2047 but only 1886 are likely to be truly cluster members. Thus, about 16% of the variables in this sample of clusters has been found by our VI CCD time-series imaging program.

8. Conclusions

A homogeneous approach towards the determination of mean M_V and [Fe/H] from the Fourier decomposition of the cluster member RRL light curves, enables a new empirical exploration of the nature of the M_V -[Fe/H] relation, which describes the dependence of the luminosity of the HB on the metallicity. Although numerous efforts, from assorted strategies, have been performed to establish the zero point and slope of the relation, universal values have been elusive. We found that, if the RRL stars are to be used as indicators of the form of the relation, or if this is to be employed as a distance indicator instrument, it should be treated independently for RRab and RRc stars. The reason is that the relation displays a different nature; for the RRab stars it is non linear with considerable scatter, while for the RRc it is tight, linear and the slope is mild.

Following the suggestion of theoretical works (^{Demarque et al. 2000}), the inclusion of the HB structural parameter ℒ demonstrates that M_V is also correlated with ℒ, in a nonlinear fashion for the RRab analysis. For the RRc the role of ℒ is negligible. We offer a calibration M_V -[Fe/H], with [Fe/H] in the spectroscopic scale of ^{Carretta et al. (2009)} (equation 14) or in the ^{Nemec et al. (2013)} scale (equation 16) valid for RRab stars, and linear calibrations M_V -[Fe/H] in the above two scales (equation 10 or equation 11) valid for RRc stars.

We find pertinent at this point to recall a well established result: that globular clusters harbour more that one generation of stars (e.g. ^{Bedin et al. 2004}; ^{Piotto et al. 2005}, ²⁰⁰⁷⁾, and that each generation has a measurable different chemical abundance; particularly, the He content increases in later generations (^{Milone et al. 2018}). As a result of different evolutionary sequences, the objects on the HB have a large range of He abundance causing the observed HB structure, particularly the color range, and the breadth; the higher the value of Y, the more luminous the corresponding ZAHB would be. Recently it has been shown that small variations in the He-burning core mass would also contribute to the observed breadth of the HB (^{Yepez et al. 2022}). Hence, the values of the ℒ parameter employed in the present investigation may be responding to these effects, which in turn may be responsible, at least partially, of the scatter observed in the correlations. Therefore, while it is helpful to identify the cluster star members, as we have done, the identification of stars belonging to the different generations in each cluster might offer an important improvement on the calibration of the M_V -[Fe/H]-ℒ correlation.

To give support to our results, we compared the Fourier determinations of [Fe/H] with the spectroscopic values in the scale of ^{Carretta et al. (2009)}) and found them to be in excellent agreement. The distances obtained from the mean Fourier M_V calibrations, and their zero points, have proven to match within 1.7 kpc and to display a difference dispersion with an rms of 0.7 kpc, with the independent and also homogeneous distances determined by ^{Baumgardt & Vasiliev (2021)} from Gaia-eDR3 and HST data.

The CCD time-series photometric study of globular clusters, in combination with difference image analysis, has been be very fruitful in the discovery of new variables. Following the latest version of the Catalogue of Variable Stars in Globular Clusters (^{Clement et al. 2001}), we updated in Table 3 the number of know variables per type and per cluster and indicated the numbers of variables found and classified by our program over the years, for a total of 303 out of the 1886 variables, likely to be cluster members, presently known in the family of the 35 clusters considered.