Metallicity of Galactic RR Lyrae from Optical and Infrared Light Curves. II. Period–Fourier–Metallicity Relations for First Overtone RR Lyrae

The extraction of period and Fourier parameters directly follows the procedure outlined in Paper I, Section 3, and is summarized below. We first refine the period of each variable with the Lomb-Scargle method (Lomb 1976; Scargle 1982), applied to the large temporal baseline (>8 yr) of the ASAS-SN and WISE time-series data. A Gaussian locally weighted regression smoothing algorithm (GLOESS; Persson et al. 2004) is then applied to the phased data in order to smooth over unevenly sampled data and exclude outliers. A Fourier sine or cosine decomposition is finally executed on each GLOESS light curve by applying a weighted least-squares fit of the following equation (in the case of the sine decomposition):

$$\begin{eqnarray}&&m({\rm{\Phi }})={A}_{0}+\sum _{i=1}^{n}{A}_{i}\sin [2\pi i({\rm{\Phi }}+{{\rm{\Phi }}}_{0})+{\phi }_{i}],\end{eqnarray} \tag{ 1 }$$

where m(Φ) is the observed magnitude for either the ASAS-SN or WISE bands, A₀ is the mean magnitude, n is the order of the expansion, Φ is the phase from the GLOESS light curve varying from 0 to 1, Φ₀ is the phase that corresponds to the time of maximum light T₀, and A_i and ϕ_i are the ith order Fourier amplitude and phase coefficients, respectively. A similar equation can be written to represent the equivalent cosine decomposition. For RRc, n is often lower than five; however, we advise choosing the optimal number of terms for the data set with care (see Petersen 1986; Deb & Singh 2009) as the addition of higher-order terms may cause inconsistencies in the value of low-order Fourier parameters. For this work, we note that by using an intermediate GLOESS step to eliminate noise factors, the Fourier parameters are consistent between third-, fourth-, or fifth-order decompositions, with the exclusion of a small number of ≫3σ outliers (∼4% of total RRc for ASAS-SN and ∼8% for WISE). This small subset of outliers has been excluded in our processing to make our result invariant between different decomposition orders used. However, it is worth noting that with the robust fitting procedure outlined in Section 3.3, the results of this paper remain unaltered with the inclusion or exclusion of these sources. Figure 3 shows a typical V, W1, and W2 light curve from our RRc calibration stars, where the Fourier decomposition fit (solid black line) is plotted on top of the actual photometric data in red.

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As in Paper I, we explore the link between metallicity and Fourier coefficients through either the ratios of Fourier amplitudes R_ij = A_i /A_j , or the linear combinations of the phase coefficients ϕ_ij = j · ϕ_i − i · ϕ_j , where ϕ_ij is cyclic in nature and ranges from 0 to 2π. This is shown in Figure 4, where several low-order Fourier parameters of individual RRc stars are plotted against period and color-coded based on their [Fe/H]. In the case of the RRc V-band ϕ₃₁ values (top right), it is advantageous to represent the phase parameter as a product of a cosine decomposition rather than sine to avoid the rollover of the ϕ₃₁ parameter across the 2π boundary. This can be achieved by either adopting the cosine form of Equation (1) or using the simple transformation ${\phi }_{31}^{c}={\phi }_{31}^{s}-\pi$ between the sine (s superscript) and cosine (c superscript) form of this parameter.

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In all V-band panels, we can readily see a gradient in the [Fe/H] distribution of the stars. We found the most distinct separation to be in the ${\phi }_{31}^{c}$ versus period plot, confirming earlier studies, such as M07, which suggests that a relation between period and ${\phi }_{31}^{c}$ is a good indicator of metallicity for the V-band RRc. We similarly found that the ${\phi }_{31}^{s}$ versus period plot for WISE (bottom right panel) showed the most distinct trend with metallicity. However, the general metallicity trends in the WISE bands are less apparent than in the optical due to both the smaller sample size and the larger intrinsic dispersion of ϕ₃₁ values, covering almost the entire 2π range. As in the case of the RRab stars, we take advantage of the nearly identical shape of the light curves in the W1 and W2 bands (see, e.g., Figure 3) to improve the signal in the WISE Fourier parameters by averaging the ${\phi }_{31}^{s}(W1)$ and ${\phi }_{31}^{s}(W2)$ values for stars with both light curves available. See Paper I, Section 3.2 for a quantitative discussion demonstrating the validity of this approach for the RRab stars; we have verified that this result still holds for our RRc sample.

Upon initial creation of our ΔS metallicity sample, we found a significant population of stars that were previously marked as RRc in the literature, but were found on further analysis to be likely eclipsing contact binaries known as W Ursae Majoris (W UMa). These binaries have sinusoidal light curves similar to RRc variables, and can be confused with first-overtone RRL stars if they happen to have a similar period and amplitude. If the noise is high enough in a W UMa light curve, it can prevent detection of a characteristic secondary eclipse.

Indeed, we found that for our sample of W UMa, most combinations of Fourier parameters show some degree of degeneracy with regions typical of RRc. In particular, Figure 5 shows that some W UMa binaries largely overlap with RRc variables in the optical Bailey diagram, although their amplitude tends to be smaller. The overlap is partially mitigated in the infrared (W1, shown in the figure, or W2 diagrams), in which case the same W UMa tends to have larger amplitudes than RRc with a similar period. This opposite trend of amplitudes with wavelengths provided us with a simple criterion to automatically identify and reject the W UMa binaries from our sample in the case both bands are available (as in our joint sample): as shown in the left panel of Figure 5, the A_W1/A_V ratio provides a clear separation between RRc stars and the potential binaries.

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For all sources with only one available light curve (either in the optical or infrared), we resorted to less reliable criteria and manual inspection of the light curves. These criteria were developed for individual bands by analyzing an initial sample of the potential W UMa that have been identified with their amplitude ratio between the infrared and optical. If only the V-band photometry was available, we flagged as potential W UMa all sources with A_V ≲ 0.15 or ${\phi }_{21}^{s}(V)\gt 4.25$. For stars with infrared data alone, we instead flagged all stars with A_W > 0.15 (in either the W1 or W2 bands, depending on availability). We then visually examined the phased light curve in the one available band, searching for a secondary light curve minimum, which could have been missed in previous work based on less detailed photometry. We then rejected all sources showing evidence of such secondary eclipses.

The analysis presented in this paper (including Table 1) focuses solely on this sample, which has been cleaned of W UMa. We provide the final cautionary note that certain W UMa as shown can be easily misconstrued as RRc, and anyone using the relations published in this work should take extreme care in the validity of their RRc sample.

As discussed in Section 1, RRc photometric metallicities based on a Fourier decomposition of optical light curves are available for different literature relations (see, e.g., M07, N13, M14, and IB20). These relations are either linear in both period and ϕ₃₁ or use higher-order combinations of these two parameters. Our tests show that, for our sample, higher-order and nonlinear terms in the period, ϕ₃₁, or other Fourier parameters result in minimal benefits at the expense of complexity and decreased robustness in the fitting results. For these reasons, we decided to proceed using a simple linear relation in period and ϕ₃₁, similarly to what we did in Paper I for the RRab stars.

To validate this choice of a period-ϕ₃₁ plane fit, we followed the methodology described in Appendix B of Paper I. Principal component analysis showed that 91.30% of the variance in the V-band RRc data could be attributed to just two principal axes (i.e., a plane). This is comparable to the 92.9% value we found for RRab stars in Paper I, which we interpreted as suggesting that a third fit parameter was not needed. A similar analysis for the infrared RRc data set showed that 85.17% of the variance could be attributed to two dimensions (6.74% less than what we found for the RRab in the WISE bands). We attribute this larger variance to a larger relative scatter in the WISE data set itself (seen in the bottom right panel of Figure 4) due to the smaller amplitude of the pulsations, higher intrinsic photometric noise in the data, and smaller size of the sample. Search for a third observable parameter and/or higher-order fitting functions did not result in a better fit, leading us to adopt a period-ϕ₃₁ plane fit for the infrared data set as well.

We initially attempted to fit an RRc period-ϕ₃₁-[Fe/H] relation adopting the same procedure as described in Paper I. We quickly discovered that the smaller size of the RRc sample (by a factor of 3 and 10, compared to the RRab, in the ASAS-SN and WISE bands, respectively) resulted in a fit that was highly dependent upon the exact calibration sample used. Bootstrap analysis (Efron & Tibshirani 1986) highlighted that this instability in the fitting results was due to a combination of higher susceptibility to individual outliers and a stronger correlation between the period and ϕ₃₁ slopes. We found, however, that when different bootstrap samples were used, the range of period slopes we obtained for the best-fit RRc relation was consistent with the value we derived in Paper I for the RRab relation. Based on these considerations, we decided to freeze the period slope of the RRc period-ϕ₃₁-[Fe/H] relation to the RRab value, effectively reducing the RRc fit to a two-parameter fit in the metallicity zeropoint a and the ϕ₃₁ slope c,

$$\begin{eqnarray}&&[\mathrm{Fe}/{\rm{H}}]=a+{b}_{F}\cdot ({P}_{F}-{P}_{0})+c\cdot ({\phi }_{31}-{\phi }_{{31}_{0}}),\end{eqnarray} \tag{ 2 }$$

,where b_F is the period slope from the fit of RRab stars in Paper I, equal to −7.60 ± 0.24 dex/ day⁻¹ for the ASAS-SN V band, and −8.33 ± 0.34 dex/ day⁻¹ for the WISE bands. Note that for this choice of values to be appropriate, the period of the RRc variables needs first to be "fundamentalized", or transformed into its equivalent fundamental period by using the equation $\mathrm{log}{P}_{F}=\mathrm{log}{P}_{{FO}}+0.127$ (Iben & Huchra,J 1971; Rood 1973; Cox et al. 1983). Fundamentalizing the period of overtone pulsators is a common technique when deriving PLZ and PWZ relations of Cepheids and RRL variables (see, e.g., Groenewegen 2018; Marconi et al. 2015; Neeley et al. 2015) when their smaller number and narrower period range makes it prohibitive to fit them alone. It is worth noting that fundamentalization does have a minor dependence upon metallicity (Bragaglia et al. 2001; Soszyński et al. 2014); however, the effect is so small that regardless of the relation tried (e.g., Coppola et al. 2015), the results of this paper remain unchanged. Finally, as in Paper I, we have included the pivot offsets P₀ and ${\phi }_{{31}_{0}}$ to add further robustness to the fitting procedure; each value is close to the median of the period and [Fe/H] distribution for the RRc sample.

We fit the two remaining free parameters of Equation (2), and their uncertainties using an orthogonal distance regression (ODR) routine combined with bootstrap resampling. The analysis we presented in Paper I for RRab stars showed that ODR tends to minimize trends along all fitted dimensions as no differentiation is made between dependent and independent variables. The ODR fit was run multiple times using bootstrap resampling, where in each run, all the parameters of each individual calibrator star (period, ϕ₃₁, and [Fe/H]) are randomly replaced with all the corresponding parameters from any singular star in the same calibration data set. Namely, we are sampling with replacement our entire calibration data set before fitting our relation, and ensuring each resampled calibration data set is the same size as our initial sample. Note that all the stars in our calibration sample are fitted with equal weight, as the uncertainties in the fit are dominated by the intrinsic scatter we observe in the period-ϕ₃₁-[Fe/H] plane (see Figure 4), in comparison to the small uncertainties in period and the ϕ₃₁ parameter (∼ 10⁻⁶ days and ∼0.02 radians on average, respectively).

Figure 6 shows the cloud of best-fit values obtained with the process described above in the optical and infrared fit. The contours enclose the regions with 68%, 95%, and 99.7% of the ODR best-fit values obtained with 100,000 bootstrap resampled data sets. The contours provide a visual representation of the robustness of the fitting parameters with respect to outliers and are significantly larger than the corresponding 1, 2, and 3σ error ellipses of the best-fit parameters of individual ODR fits. For this reason, we adopted the mean values and standard deviations of the bootstrap resampled ODR fits as solutions and uncertainties for our RRc period-ϕ₃₁-[Fe/H] relations,

$$\begin{eqnarray}\begin{array}{rcl}{[\mathrm{Fe}/{\rm{H}}]}_{V} & = & (-1.62\pm 0.01)+(-7.60)\cdot ({P}_{F}-0.43)\\ & & +(0.30\pm 0.02)\cdot ({\phi }_{31}^{c}-3.20)\end{array}\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}\begin{array}{rcl}{[\mathrm{Fe}/{\rm{H}}]}_{W} & = & (-1.48\pm 0.05)+(-8.33)\cdot ({P}_{F}-0.43)\\ & & +(0.14\pm 0.05)\cdot ({\phi }_{31}^{s}-2.70)\end{array},\end{eqnarray} \tag{ 4 }$$

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where Equations (3) and (4) provide photometric metallicities for the ASAS-SN V band and WISE W1 and W2 infrared bands (averaged when possible; see Section 3.1). Note again how the V-band metallicities are expressed as a function of the cosine decomposition of the optical light curves (i.e., they depend on ${\phi }_{31}^{c}$), while the WISE formula is a function of the sine decomposition parameter ${\phi }_{31}^{s}$.

Figure 6 and the uncertainties quoted in Equations (3) and (4) show that the sampled coefficients occupy a much smaller region of the parameter space in the V band than in the W band. The larger uncertainties in the infrared coefficients are a reflection of the larger intrinsic scatter we noted in the W-band period-ϕ₃₁-[Fe/H] plane (Figure 4) as well as a consequence of the smaller size of the infrared calibrator sample. A detailed analysis of the performance of these relations is provided in Section 4. Table 2 lists the derived photometric properties for both the V-band and infrared data sets, including the period, ϕ₃₁ value, and photometric metallicity derived in each band with Equations (3) and (4).

Table 2. Derived Photometric Properties of the RRc Sample

Gaia ID	Period a	${\phi }_{31}^{c}$ (V)	${\sigma }_{{\phi }_{31}}$ (V)	[Fe/H]V	${\phi }_{31}^{s}$ (W)	${\sigma }_{{\phi }_{31}}$ (W)	[Fe/H]W
(DR3)	(day)	(radian)	(radian)	(dex)	(radian)	(radian)	(dex)
6914532141197318784	0.334879	5.002	0.017	−1.22
6913110953698726912	0.324371	3.180	0.008	−1.66
6910854717182648448	0.323513	4.150	0.013	−1.36
6897117354482002688	0.322392	3.418	0.007	−1.57
6731321171497007488	0.223019	2.917	0.005	−0.71
6688916306549500800	0.339563	3.588	0.004	−1.69	3.244	0.041	−1.61
6340460627660385920	0.285073	2.725	0.004	−1.39
6340096929829777152	0.384128	4.077	0.011	−2.00	2.619	0.007	−2.19
6307501113055775232	0.339337	3.149	0.009	−1.82
6299550445690958080	0.296671	3.944	0.004	−1.15

^a When both V-band and infrared (WISE) data are present, the period included was calculated from ASAS-SN (V-band) data as the period is usually more accurate due to the higher amplitude and steeper light curve. Period accuracies are quoted to an accuracy on the order of 10⁻⁶ days, corresponding to a readily detectable ∼1% shift in phase for a typical 0.32 -day period RRc star when phased over the large temporal (>8 yr) baseline of our data sets.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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In order to test the V-band relation obtained in Section 3.3 on an independent sample, we selected a list of 10 GCs with metallicity homogeneously spread between [Fe/H] = −1.0 and −2.3 dex. The clusters we sampled are the same as in Paper I, with two additional GCs: Reticulum (a Large Magellanic Cloud GC), and NGC 6171. Photometry of the clusters comes from Piersimoni et al. (2002; NGC 3201), Kuehn et al. (2013; Reticulum), M14 (NGC 6171), and the homogeneous data set of P. B. Stetson ¹⁵ (hereafter PBS) for the remaining majority.

The general properties of the clusters are listed in Table 3, which includes their spectroscopic metallicities (in the scale of C09) and the number of RRL with good-quality well-sampled light curves available in each cluster. The spectroscopic metallicity of the Galactic GCs are obtained from C09, while the metallicity of Reticulum is from Mackey & Gilmore (2004) and converted from the [Fe/H] scale of Zinn & West (1984, ZW84) to C09. A Fourier decomposition was performed on each light curve to obtain their ϕ₃₁ parameters, with the exception of the stars in NGC 6171 and NGC 3201, for which the Fourier parameters were taken directly from their respective photometric catalogs. The period-ϕ₃₁-[Fe/H] relations for RRc (Equation (3)) and RRab (Equation (6) from Paper I with [Fe/H] shifted to the scale of C09) were then applied to estimate the metallicity of each RRL star in the clusters. Note that no metallicity scale corrections are needed between that of C09 and the RRc relations presented in this work. The average [Fe/H] abundance of the RRc and RRab variables in each cluster, with their standard deviation, are listed in columns 4 and 6 of Table 3, respectively.

Figure 7 shows the spectroscopic [Fe/H] versus the mean photometric [Fe/H] values, calculated for each GC with our relations. Each RRab and RRc stellar photometric metallicity measurement contributes equal weight to the mean photometric [Fe/H] for a given GC. The figure, as well as the individual values listed in Table 3, demonstrates the good performance of our formulæ (including the new relations for RRc variables) to provide reliable [Fe/H] abundances. The combination of the photometric RRab metallicities (from Paper I) with the new photometric RRc metallicities is in good agreement with the spectroscopic metallicities of the clusters: overall ±0.08 dex, which is well within the respective uncertainties.

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In this section, we compare photometric metallicities derived using a variety of period-ϕ₃₁-[Fe/H] relations, including those found in this work (Equations (3) and (4)), as well as the fits provided by M07, M14, N13, and IB20. Figure 8 shows the photometric metallicities for all the RRc in our calibration sample, plotted as a function of their spectroscopic [Fe/H] abundances. The rms scatter of each relation, calculated with respect to the ideal one-to-one relation over the entire spectroscopic metallicity range within which each relation has been calibrated, is indicated in each case.

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The top row shows the analysis of our fits in the optical (ASAS-SN, left) and infrared (WISE, right) bands. Since we are directly comparing to the same sample used to derive these relations, the figure allows us to check for remaining trends in the residuals with respect to the ideal one-to-one relation to ensure that our relations provide a consistent estimate of each star's [Fe/H] over the entire metallicity range of our calibration sample. For the case of the V-band ASAS-SN data set, we show nearly symmetric residuals (with an rms ≈ 0.30 dex) over the entire range of metallicities with only a minor deviation at higher metallicities, still well within the error of our relation. The WISE band residuals, however, show some larger divergence at high metallicity ([Fe/H] ≳ −0.5), where Equation (4) systematically underpredicts the spectroscopic metallicities. The overall residual rms is also significantly larger (0.50 dex) than the one we obtained in the V band. Due to the small number of high-metallicity RRc stars, we cannot determine if this is due to nonlinearity of the period-ϕ₃₁-[Fe/H] in this metallicity regime or rather a reflection of less accurate spectroscopic metallicities for the stars in our calibration sample that approach solar [Fe/H] abundance. Indeed, Crestani et al. (2021b) found that the RRL that have been used to calibrate the ΔS method in Crestani et al. (2021a), which we in turn use as the basis for our own relation, show a broad dispersion in α-elements measured through high-resolution spectroscopy. Since the ΔS method is based on the strength of a Ca line, while our Fourier-ϕ₃₁-metallicity method aims to measure [Fe/H] metallicities, a spread in α-element abundances can potentially lead to the observed deviations at the ends of the metallicity scale. If we restrict our WISE bands analysis to [Fe/H] ≲ −0.5, we obtain a best-fit relation that is virtually indistinguishable from Equation (4) (because of the much larger number of low-metallicity RRc, and due to the robustness of our bootstrap fit), but with a residual rms equal to 0.29 dex, nearly identical to the one we found for the V band (calculated over the entire metallicity range). By directly comparing the V-band (Equation (3)) and infrared (Equation (4)) photometric metallicities for the RRc in the joint sample, we note that the two sets of metallicities are consistent with each other within their respective uncertainties.

The middle left panel shows the residuals found with the higher-order nonlinear period-ϕ₃₁-[Fe/H] relation of M07 (their Equation 4). Note that M07 published two equations: one based on the [Fe/H] scale of ZW84, and one based on that of Carretta & Gratton (1997). We choose to analyze the latter due to its smaller quoted dispersion and simpler functional form. Their relation was based on 106 RRc stars from 12 GCs with V-band Fourier parameters gathered from heterogeneous publications. In order to consistently compare with the HR+ΔS spectroscopic metallicities, we converted the metallicities derived from the M07 relation into that of C09 using the relation [Fe/H]_C09 = (1.137 ± 0.060)[Fe/H]_CG97 − 0.003 (provided by C09). A few years later, M14 published an updated version of their V-band M07 fit, using the same base of RRc extended to include a total of 163 stars gathered from 19 GCs, with a metallicity scale updated to that of C09. Although they found a slightly different nonlinear period-ϕ₃₁-[Fe/H] functional form, the metallicities predicted by M14 (center right panel in Figure 8) appear to be roughly the same as those provided by M07. Both relations do not appear to have a trend or bias with respect to the spectroscopic metallicities, but they do have a larger rms (0.36 and 0.39 dex, respectively) than our ASAS-SN relation.

N13 provided a nonlinear period-ϕ₃₁-[Fe/H] for the Kepler Space Telescope Kp band (their Equation (4)). Due to the small field of view (relative to large-area surveys) of Kepler's primary mission, this relation is calibrated by augmenting the sample of 3 RRc obtained in Kepler's field with the Fourier parameters of 98 GC RRc from M07, converted into the Kp system using the relation ϕ₃₁(V) = ϕ₃₁(Kp) − (0.151 ± 0.026) from Nemec et al. 2011. We use this same relation to convert the V-band ϕ₃₁ parameters of our spectroscopic sample into the Kp photometric system in order to generate the bottom left panel of Figure 8. The plot shows that the N13 largely predicts the [Fe/H] metallicities without noticeable trends even outside their [Fe/H] calibration range, albeit with a small negative bias (the average metallicity predicted by N13 is 0.26 dex lower than the spectroscopic values), and a larger scatter (RMS = 0.52 dex) than all other relations evaluated here.

Finally, IB20 used a sample of 50 GC RRc stars extracted from the Gaia DR2 database to derive a bilinear period-ϕ₃₁-[Fe/H] relation (Equation (4) in their paper) with the same functional form as our Equations (3) and (4). In order to apply this relation to our data set, we had to perform two transformations: (1) we converted the V-band ϕ₃₁ value of our ASAS-SN sample into the G-band system using the relation ϕ₃₁(G) = (0.104 ± 0.020) + (1.000 ± 0.008) ϕ₃₁(V) from Clementini et al. (2016), and (2) we transformed the metallicity provided by the IB20 relation (in the ZW84 scale) into the Carretta scale using the [Fe/H]_C09 = 1.105[Fe/H]_ZW84 + 0.160 relation from C09. The results are shown in the bottom right panel of Figure 8. Again, no obvious trend or offset is found, and the rms dispersion (0.32 dex) is among the smallest of the relations assessed in this section, comparable to the rms we measure in the V band with our Equation (3).

Overall, M07, M14, and N13 share the majority of their photometric calibration data sets, but differ in their functional form by adding slightly different combinations of nonlinear terms. Figure 8 shows that with respect to optical linear relations (IB20 and Equation (3) in this work), these nonlinear relations tend to have a larger residual dispersion for our large sample of spectroscopic metallicities. With the exception of the infrared WISE band (where we observe a possible departure at solar metallicities), our data set does not support the need for the addition of nonlinear terms in photometric metallicity relations based on the Fourier parameters of RRc optical light curves.

In this section, we test our V-band period-ϕ₃₁-[Fe/H] relations with RRL stars in the MW dSph satellite Sculptor. We compare our photometric Fourier metallicities to those derived by Martínez-Vázquez et al. (2016b; hereafter MV16b) by inverting the theoretical PLZ relation from Marconi et al. (2015). Being a relatively nearby Local Group galaxy (μ₀ = 19.62 mag; Martínez-Vázquez et al. 2015), Sculptor has been extensively studied as a probe for galaxy evolution, and detailed studies are available of its variable star content, with photometry stretching back over two decades (for a thorough review, see Martínez-Vázquez et al. 2016a, hereafter MV16a). We specifically chose Sculptor as a test case for our Fourier metallicity relations as this dwarf galaxy has been shown to have an early history of chemical enrichment, resulting in an older stellar population (including RRL stars) with a broad range of metallicity (≳1 dex; see Clementini et al. 2005; Martínez-Vázquez et al. 2015, 2016b). By studying Sculptor, we show that the relations provided in this work are widely applicable to complex stellar populations, beyond RRLs in the field or GCs.

Sculptor photometry is available from the PBS database in both the I band (used by MV16b in their work) and the V band. We refer to MV16a for the exact details of the observing runs, bands observed, instruments, and telescopes used to collect the Sculptor photometry in PBS. Out of 536 known RRLs (289 RRab, 197 RRc, and 50 RRd; MV16a), MV16b derived photometric metallicities for 276 RRab and 195 RRc stars. However, only 274 of these stars (126 RRab and 148 RRc) have good-quality V-band light curves (according to the criteria outlined in Paper I, Section 3). For these stars, we have extracted their V-band Fourier parameters following the procedures described in Section 3.1. We have then estimated their photometric [Fe/H] abundances using our Fourier metallicity relations for RRab (Equation 6 from Paper I with [Fe/H] shifted to the scale of C09) and RRc (Equation 3 from Section 3.3) stars, respectively.

The results of this analysis are shown in Figure 9. The top left panel shows the Fourier metallicity distributions of the Sculptor RRab and RRc stars. The two histograms have a similar mean (−1.88 dex and −1.83 dex for the RRab and RRc, respectively) and dispersion (σ_RRab = 0.48 dex and σ_RRc = 0.36 dex). The remaining panels of Figure 9 compare the metallicity distribution obtained with our Fourier method with the [Fe/H] abundance derived via PLZ inversion by MV16b. We find a remarkable agreement between the metallicities derived with these two methods. In the overall sample (RRab and RRc together, top right panel) we measured a mean metallicity [Fe/H] ≃ −1.85 dex, matching the value of [Fe/H] ≃ −1.90 dex found by MV16b. The dispersion and shape of the two [Fe/H] distributions is also very similar, with both methods suggesting a spread of metallicities in the Sculptor RRL population of ∼2 dex. Note that while the Fourier metallicities derived in this work suggest the presence of a small high-metallicity tail (with a few stars having [Fe/H] between −1.0 and −0.5 dex, which was not seen in MV16b), this excess may be an artifact originating in the calibration of our relation, as noted in Section 4.2. Comparison of the Fourier and PLZ metallicities derived for RRab and RRc separately (bottom row in Figure 9) leads to similar conclusions.

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These results are also consistent with the spectroscopic metallicities of Sculptor. Clementini et al. (2005) found a [Fe/H] peaking at ∼−1.8 dex via the ΔS spectroscopic method applied to a sample of 107 RRL. Achieving such a consistent mean metallicity and distribution validates both of these photometric approaches, especially because the two photometric relations are completely independent of one another both in methodology (shape of the light curve as opposed to inversion of the PLZ relation) and in the data set used (V band versus I band, and light curves with different sampling and coverage).