Photometric Metallicity and Distance for the Two RR Lyrae in Segue II and Ursa Major II Dwarf Galaxies Based on Multiband Light Curves

The old-population pulsating stars RR Lyrae are well-known distance indicators because they exhibit an absolute V-band magnitude-metallicity relation and well-defined period–luminosity(-metallicity) relations in other filters (for a review, see Bhardwaj 2020). In most cases, the metallicity of RR Lyrae, parameterized as [Fe/H], needs to be known a priori to apply these relations. The best [Fe/H] measurements are based on (high-resolution) spectroscopic observations. However, such observations could be time consuming and are only limited to relatively bright RR Lyrae in the milky Way. Alternatively, [Fe/H] can be obtained using the light-curve information, as shown in the seminar paper by Jurcsik & Kovacs (1996). The [Fe/H] measured using this approach, known as photometric metallicity, is expected to be widely applied to the distant RR Lyrae discovered from the multiband time-domain sky surveys, especially the Vera C. Rubin Observatory's Legacy Survey of Space and Time (LSST; Ivezić et al. 2019), because only few spectrographs on large-aperture telescopes, or perhaps none, can be used to collect high-resolution spectra of very distant RR Lyrae stars.

In the past, RR Lyrae photometric metallicities were typically obtained using light-curve structure information in a single filter. On the other hand, empirical relations that use the light-curve information to estimate [Fe/H] have been derived in several filters, ranging from optical V filter to infrared WISE filters (some examples can be found in Section 2). Higher-precision photometric metallicity can be achieved by averaging out the [Fe/H] obtained from several such relations in the same or different filters mitigating possible systematic effects. In this work, we demonstrate this is indeed the case based on the multiband observations of two ab-type (i.e., fundamental-mode) RR Lyrae discovered in the ultrafaint dwarf galaxies SEGUE II (Boettcher et al. 2013) and Ursa Major II (hereafter UMaII; Dall'Ora et al. 2012). These near-simultaneous time-series observations were carried out using the Lulin One-meter Telescope (LOT), ³ a general-purpose Cassegrain reflector located at the Lulin Observatory in central Taiwan. The collected data allow us to obtain the photometric metallicities for these two RR Lyrae based on the homogeneous light-curve data at multiple wavelengths.

Section 2 presents multiband light curves collected from LOT, together with archival Gaia light curves, for these two RR Lyrae. Based on these multiband light curves, we obtained their photometric metallicity in Section 3. A by-product of our work is the derivation of distance moduli to these dwarf galaxies, which will be discussed in Section 4. The conclusions of our work is presented in Section 5. Throughout this work, these two RR Lyrae will be referred as SEGUE II-V1 and UMaII-V1, respectively.

Time-series observations of SEGUE II-V1 and UMaII-V1 were carried out with LOT in 12 and 18 nights, respectively, between 2019 November 14 and 2020 April 15. The Princeton Instruments SOPHIA 2048B back-illuminated CCD was mounted on LOT during these observations, providing a FOV of $13\buildrel{\,\prime}\over{.} 2\times 13\buildrel{\prime\prime}\over{.} 2$ (with a pixel scale of 0386 pixel⁻¹) on the CCD images. A sequence of Vgri exposures, with identical exposure time of 360 s in all filters, were taken multiple times for each RR Lyrae in a given night when they were visible from the Lulin Observatory. All of the collected images were reduced in a standard manner (including bias subtraction and dark subtraction, followed by flat fielding). Astrometric calibration on the reduced images were done using the SCAMP package (Bertin 2006). The point-spread-function (PSF) photometry of RR Lyrae in the reduced images were measured using the combined PSFEx (Bertin 2011) and Source-Extractor (Bertin & Arnouts 1996) package, at which the details on the photometric calibration were presented in the Appendix A. Tables 1 and 2 provide multiband light curves collected from the LOT observations for SEGUE II-V1 and UMaII-V1, respectively.

In addition to the new LOT observations, we have also downloaded G-band light curves released from the Gaia Data Release 3 (DR3; Gaia Collaboration et al. 2016, 2023) for these two RR Lyrae, with the Gaia DR3 source ID for SEGUE II-V1 and UMaII-V1 as 87207528534363264 and 1043841876592990208, respectively. We have also adopted their periods as derived in Gaia DR3 (Clementini et al. 2023; Eyer et al. 2023; from the I/358/vrrlyr table available in the SIMBAD/VizieR database), because the adopted periods are adequate to fold the light curves and there is no need to rederive their periods. The adopted periods are P(SEGUE II-V1) = 0.7494938 ± 0.0000164 days and P(UMaII-V1) =0.5651226 ± 0.0000037 days.

Figure 1 presents multiband phase folded light curves with the adopted periods. These light curves were then fitted with a low-order Fourier expansion (see Ngeow 2022, and reference therein). In brief, we fitted a Fourier series of the following form:

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

to the phased light curves, where Φ is pulsational phase from 0 to 1, and Fourier order n varied from 4 to 9. The best order of fit was determined based on the method as described in Ngeow et al. (2023). From the best-fitted Fourier expansions, we determined the ϕ₃₁ = ϕ₃ −3ϕ₁ Fourier parameters which are needed to estimate the photometric metallicity for RR Lyrae (see next Section). The determined ϕ₃₁ from the multiband light curves are provided in Table 3.

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3.1. The Adopted Relations

It is well-known that [Fe/H] of a RR Lyrae can be obtained from its pulsation period P and its light-curve parameter, primarily the Fourier parameter ϕ₃₁ (Jurcsik & Kovacs 1996). We collected the grGVI-band [Fe/H]-ϕ₃₁–P relations that are available recently in the literature. These include the grV-band relations taken from Ngeow (2022; hereafter N22):

$$\begin{eqnarray*}\begin{array}{rcl}[\mathrm{Fe}/{\rm{H}}{]}_{{\rm{N}}22}^{g} & = & -1.01[\pm 0.06]-7.43[\pm 0.80](P-0.58)\\ & & +\,1.69[\pm 0.16]({\phi }_{31}-5.25),\ \sigma =0.24,\end{array}\end{eqnarray*}$$

$$\begin{eqnarray*}\begin{array}{rcl}[\mathrm{Fe}/{\rm{H}}{]}_{{\rm{N}}22}^{r} & = & -1.54[\pm 0.07]-8.28[\pm 1.04](P-0.58)\\ & & +\,1.30[\pm 0.15]({\phi }_{31}-5.25),\ \sigma =0.30,\end{array}\end{eqnarray*}$$

$$\begin{eqnarray*}\begin{array}{rcl}[\mathrm{Fe}/{\rm{H}}{]}_{{\rm{N}}22}^{V} & = & -1.21[\pm 0.05]-7.67[\pm 0.78](P-0.58)\\ & & +\,1.50[\pm 0.14]({\phi }_{31}-5.25),\ \sigma =0.24.\end{array}\end{eqnarray*}$$

These relations were derived based on a sample of 30 RR Lyrae located in the Kepler Field, and their spectroscopic [Fe/H] were derived from high-resolution spectra (Nemec et al. 2013). The metallicity scale for N22, [Fe/H]_N22, is in the Carretta et al. (2009) scale. We have also adopted V-band [Fe/H]-ϕ₃₁–P relation from Mullen et al. (2021; hereafter M21), derived from 1980 field RR Lyrae, as:

$$\begin{eqnarray*}\begin{array}{rcl}[\mathrm{Fe}/{\rm{H}}{]}_{{\rm{M}}21}^{V} & = & -1.22[\pm 0.01]-7.60[\pm 0.24](P-0.58)\\ & & +\,1.42[\pm 0.05]({\phi }_{31}-5.25),\ \sigma =0.41.\end{array}\end{eqnarray*}$$

The M21 metallicity is in Crestani et al. (2021) scale, therefore we applied an offset of −0.08 dex to convert $[\mathrm{Fe}/{\rm{H}}{]}_{{\rm{M}}21}^{V}$ to the Carretta et al. (2009) scale as determined in Mullen et al. (2021). We did not include V-band relation from Jurcsik & Kovacs (1996) because the qualities for some of the light curves used in their work (as commented in Mullen et al. 2021; Zong et al. 2023) are not as good as those studied either in M21 or N22.

In the Gaia G band, Iorio & Belokurov (2021; hereafter I21) derived the relation using a set of 86 field RR Lyrae, as given below:

$$\begin{eqnarray*}\begin{array}{rcl}[\mathrm{Fe}/{\rm{H}}{]}_{{\rm{I}}21}^{G} & = & -1.68[\pm 0.05]-5.08[\pm 0.5](P-0.60)\\ & & +\,0.68[\pm 0.11]({\phi }_{31}-2.00-\pi ),\ \sigma =0.31.\end{array}\end{eqnarray*}$$

Mullen et al. (2021) pointed out that a π is needed to be subtracted from ϕ₃₁ when using their relation, hence we included it in the above relation. The metallicity scale for the I21 relation is in Zinn & West (1984) scale, therefore we converted the Zinn & West (1984; ZW84) scale to the Carretta et al. (2009; C09) scale using the relation given in Carretta et al. (2009): [Fe/H]_C09 = 1.105[Fe/H]_ZW84 + 0.160. Independently, Jurcsik & Hajdu (2023; hereafter J23) derived the G-band relation using RR Lyrae-hosted globular clusters. Since it is well known that globular clusters can be divided into Oosterhoff-I (OoI; with higher metallicity) and Oosterhoff-II (OoII; with lower metallicity) types, Jurcsik & Hajdu (2023) derived three sets of G-band relation: one for each Oosterhoff types and a relation for the combined sample. Since both SEGUE II-V1 and UMaII-V1 are OoII RR Lyrae (Dall'Ora et al. 2012; Boettcher et al. 2013), we adopted the OoII relation:

$$\begin{eqnarray*}\begin{array}{rcl}[\mathrm{Fe}/{\rm{H}}{]}_{{\rm{J}}23}^{G} & = & -4.324[\pm 0.157]-7.212[\pm 0.248]P\\ & & +\,1.353[\pm 0.048]{\phi }_{31},\ \sigma =0.17.\end{array}\end{eqnarray*}$$

As Jurcsik & Hajdu (2023) adopted a slightly different Solar compositions, and offset of −0.04 dex was added to $[\mathrm{Fe}/{\rm{H}}{]}_{{\rm{J}}23}^{G}$ to bring the metallicity to the Carretta et al. (2009) scale. Besides these two G-band relations, another relation is available from Li et al. (2023; hereafter L23), which includes an additional R₂₁ = A₂/A₁ Fourier parameter:

$$\begin{eqnarray*}\begin{array}{rcl}[\mathrm{Fe}/{\rm{H}}{]}_{{\rm{L}}23}^{G} & = & -1.888[\pm 0.002]-5.772[\pm 0.026](P-0.6)\\ & & +\,1.090[\pm 0.005]({\phi }_{31}-2.0-\pi )\\ & & +\,1.065[\pm 0.030]({R}_{21}-0.45),\ \sigma =0.24.\end{array}\end{eqnarray*}$$

The R₂₁ values from the G-band light curve are 0.495 ± 0.022 and 0.423 ± 0.006 for SEGUE II-V1 and UMaII-V1, respectively. Similar to the I21 relation, a π is needed to be subtracted from the ϕ₃₁ value. The above relation was derived from a sample of ∼2000 RR Lyrae with spectroscopic [Fe/H] measured from Liu et al. (2020). According to Liu et al. (2020), the difference between their [Fe/H] and published [Fe/H], in the Carretta et al. (2009) scale, is negligible. Hence, we assume $[\mathrm{Fe}/{\rm{H}}{]}_{{\rm{L}}23}^{G}$ is in the Carretta et al. (2009) scale.

Finally, the I-band [Fe/H]-ϕ₃₁–P relations were published in Smolec (2005; hereafter S05) and Dékány et al. (2021; hereafter D21). S05 provides a two-parameter and a three-parameter relation. However, Hajdu et al. (2018) demonstrated that the two-parameter relation suffers from a systematic bias, therefore we adopted the three-parameter relation (which also has a smaller dispersion):

$$\begin{eqnarray*}\begin{array}{rcl}[\mathrm{Fe}/{\rm{H}}{]}_{{\rm{S}}05}^{I} & = & -6.125[\pm 0.832]-4.795[\pm 0.285]P+1.181\\ & & [\pm 0.113]{\phi }_{31}+7.876[\pm 1.776]{A}_{2},\sigma =0.14.\end{array}\end{eqnarray*}$$

The metallicity scale for S05 relation is in the Jurcsik (1995; J95) scale, and we converted it to the Carretta et al. (2009) scale using [Fe/H]_C09 = 1.001[Fe/H]_J95 −0.112 (Kapakos et al. 2011). Dékány et al. (2021) performed a feature selection and found that the optimal relation could also be obtained with three parameters, and subsequently applied a Bayesian linear regression on a sample of ∼80 RR Lyrae with good quality I-band light curves to derive the following relation, which is similar to the S05 relation:

$$\begin{eqnarray*}\begin{array}{rcl}[\mathrm{Fe}/{\rm{H}}{]}_{{\rm{D}}21}^{I} & = & -5.819[\pm 0.149]-6.350[\pm 0.067]P+1.248\\ & & [\pm 0.020]{\phi }_{31}+5.785[\pm 0.320]{A}_{2},\sigma =0.20.\end{array}\end{eqnarray*}$$

According to Dékány et al. (2021), the D21 metallicity scale is in, or has been shifted to, the Crestani et al. (2021) scale. Hence, we applied the same offset of −0.08 dex as in the case of M21 relation to bring the D21 metallicity scale to the Carretta et al. (2009) scale. For SEGUE II-V1 and UMaII-V1, the A₂ parameters in the I band are 0.069 ± 0.003 and 0.114 ± 0.003, respectively.

3.2. The Derived Metallicity

The derived multiband photometric metallicities, using the nine relations listed in the previous subsection, are listed in Table 3 and graphically presented in Figure 2. As mentioned, all of these photometric metallicities are in the Carretta et al. (2009) scale. For both RR Lyrae, all of the derived photometric metallicities are mutually consistent with each others from the nine relations adopted in the previous subsection. The errors on the derived metallicities are the quadrature sum of the propagated errors of each parameters (except we omitted the errors on period as they are negligible) and the dispersion σ for each relations. The large errors on the $[\mathrm{Fe}/{\rm{H}}{]}_{{\rm{S}}05}^{I}$ values are mainly due to the large error on the constant term, ±0.832, in the S05 relation. Errors from other relations are around ∼0.3 to ∼0.4 dex.

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We assume that the adopted nine relations in the previous subsection are equally well calibrated, although each relation may have their own systematic uncertainties (Jurcsik & Hajdu 2023). The uncertainties on the coefficients of these relations were propagated to the derived photometric metallicities. Therefore, we took the weighted means for the photometric metallicities listed in Table 3 giving smaller weights to more uncertain measurements. When taking a weighted mean, we obtained [Fe/H] = −2.27 ± 0.11 dex and −1.87 ± 0.10 dex for SEGUE II-V1 and UMaII-V1, respectively. Dispersions around these mean values were 0.13 dex and 0.16 dex, respectively, and will be adopted as the errors on these weighted means, which are ∼2 to ∼3 times smaller that those listed in Table 3 (except for the S05 relation). We have also tried to eliminate certain data points, such as excluding those based on S05 relation or with the largest deviations, the resulted weight means do not differ by more than ±0.03 dex from our preferred values. Small statistics of two RR Lyrae precludes us from quantifing the optimal number of relations or the preferred relation for accurate photometric metallicity determinations.

In the case of SEGUE II-V1, our derived photometric metallicity is in good agreement with the values determined in Belokurov et al. (2009) and Kirby et al. (2013), as shown in the left panel of Figure 2, and almost identical to the value recalculated by Boettcher et al. (2013; −2.26 ± 0.14 dex). For UMaII-V1, our photometric metallicity is close to the value given by Simon & Geha (2007), but more metal-rich than those determined in Kirby et al. (2013) or Willman & Strader (2012; −2.36 dex with σ = 0.56). However, dwarf galaxies are known to exhibit a wide spread in metallicity (for examples, see Simon & Geha 2007; Willman & Strader 2012; Kirby et al. 2013; and reference therein). The published metallicities are mostly measured from the spectroscopic observations of red-giant-branch stars and/or blue-horizontal-branch stars, and hence they may not necessarily represent the photometric metallicities for RR Lyrae.

Since the Wesenheit magnitude, denoted as W, is reddening-free by construction (see the Appendices in Madore & Freedman 1991, 2023), we adopted the recent period–Wesenheit–metallicity (PWZ) relations available in the literature to derive the distances to these two RR Lyrae with our photometric metallicities determined in the previous Section. For our VIgr-band light curves, the number of data points per light curves is more than 100, hence according to Ngeow et al. (2022) the errors on the mean magnitudes derived from these light curves are negligible. Nevertheless, we adopted a conservative error of 0.02 mag on the intensity-averaged mean magnitudes derived from the fitted Fourier expansion (see Figure 1) on our multiband light curves. The VIgr-band mean magnitudes are listed in Table 4.

Neeley et al. (2019) calibrated multiband PWZ relations based on the 55 Milky Way RR Lyrae together with their Gaia DR2 parallaxes. According to Neeley et al. (2019), metallicities for these 55 RR Lyrae are assumed to be in the Zinn & West (1984) scale. Therefore, we converted our derived metallicities to the Zinn & West (1984) scale by inverting the [Fe/H]_C09-[Fe/H]_ZW84 conversion. After applying the VI-band PWZ relation from Neeley et al. (2019), we determined the VI-band based distance moduli (μ_VI) to SEGUE II-V1 and UMaII-V1 as μ_VI(SEGUE II-V1) =17.67 ± 0.20 mag and μ_VI(UMaII-V1) = 17.56 ± 0.19 mag.

As an independent check, we have also obtained the distance moduli for the two RR Lyrae using the gr-band PWZ relation as derived in Ngeow et al. (2022; using the RR Lyrae in globular clusters), where the metallicity scale is in the Dias et al. (2016; hereafter D16) scale. The conversion of such metallicity scale to the Carretta et al. (2009) scale is given as [Fe/H]_C09=0.99[Fe/H]_D16 −0.05 (Dias et al. 2016). After we inverted our derived metallicity to the Dias et al. (2016) scale, we obtained gr-band based distance moduli (μ_gr ) as μ_gr (SEGUE II-V1) = 17.71 ± 0.24 mag and μ_gr (UMaII-V1) =17.61 ± 0.24 mag. The derived distance moduli based on the VI-band and the gr-band PWZ relations are summarized in Table 5.

It is worth to mention that the impact on the determined distance moduli is negligible (±0.01 mag at most) if we did not convert the metallicity to the Zinn & West (1984) or Dias et al. (2016) scale. This is mainly due to the small metallicity terms (0.13 and 0.05 dex/mag, respectively) in the adopted PWZ relations. Since both of the μ_VI and μ_gr were determined from independent PWZ relations and in different filters, the averages of μ_VI and μ_gr were adopted as the final distance moduli for these two RR Lyrae. For SEGUE II-V1, we obtained 〈μ〉 = 17.69 ± 0.15 mag, which is almost identical to the value found in Belokurov et al. (2009; 17.7 ± 0.1 mag) and Boettcher et al. (2013; D = 34.4 ± 2.6 kpc or $\mu ={17.68}_{-0.17}^{+0.16}$ mag) measured from the blue-horizontal-branch stars. Similarly, we obtained 〈μ〉 = 17.58 ± 0.15 mag for UMaII-V1, and it is in good agreement with the estimated value given by Zucker et al. (2006; ∼17.5 ± 0.3 mag), based on the isochrone fitting, and is almost the same as the one determined in Vivas et al. (2020; 17.60 ± 0.20 mag) when including additional RR Lyrae for UMaII. We also found statistically consistent distance measurements to these two dwarf galaxies if independent theoretical PWZ relations from Marconi et al. (2015, 2022) were adopted instead of the empirical relations.

In this work, we obtained homogeneous grVI-band light curves using LOT for two RR Lyrae in SEGUE II and UMaII ultrafaint dwarf galaxies. Together with their Gaia G-band light curves, we derived the photometric metallicities for these two RR Lyrae using nine [Fe/H]-ϕ₃₁–P relations (some of them include an additional Fourier parameter) recently published in the literature. Hence, the weighted means of the photometric metallicities for SEGUE II-V1 and UMaII-V1 were found to be [Fe/H] = −2.27 ± 0.13 dex and −1.87 ± 0.16 dex, respectively. Using these photometric metallicities, together with empirical PWZ relations, we determined the distance moduli to these two RR Lyrae as μ(SEGUE II-V1) = 17.69 ± 0.15 mag and μ(UMaII-V1) = 17.58 ± 0.15 mag. These distance and photometric metallicity measurements are in good agreement with previous determinations in the literature.

As mentioned in the Introduction, the motivation of this work is to demonstrate that a higher precision on photometric metallicity for RR Lyrae can be acheived when averaging the values derived from multiple metallicity–Fourier parameter(s)–period relations. This is indeed the case for the two RR Lyrae, SEGUE II-V1 and UMaII-V1, investigated in this work. Our proposed approach will be particularly important for various ongoing and upcoming multiband time-domain sky surveys providing well-sampled RR Lyrae light curves in the Milky Way. This is especially true for LSST, which will provide unprecedentedly large samples of distant RR Lyrae beyond our galaxy and particularly in the LSST-discovered new dwarf galaxies. The spectroscopic follow-up observations of these variables will not be feasible either because they are too far/faint or due to lack of rather competitive observational time. In this scenario, photometric metallities for RR Lyrae can be determined using the proposed approach, since LSST will provide multifilter light curves of RR Lyrae variables. However, it is critically important to adopt a homogeneous metallicity scale or homogenize metallicity measurements based on different empirical relation for better constrain on the photometric metallicities, as shown in detail in the manuscript. These photometric metallicities can then be used to derive the RR Lyrae-based distances, as well as to compare and/or constraint the inferred metallicity based on isochrone fitting to the color–magnitude diagrams of the newly discovered dwarf galaxies. The large number of expected RR Lyrae in dwarf galaxies with LSST will also allow to derive photometric metallicity maps and gain insight into chemical and morphological structure of these stellar systems.

Finally, it is worth reminding that the gr-band and the VI-band metallicity–Fourier parameter(s)–period relations used in Section 3 are in the Pan-STARRS1 and the Johnson–Kron–Cousins photometric systems, respectively. Therefore, the gri-band light curves based on the LSST observations need be photometrically transformed to the grVI band, as well as the Gaia G band, by properly taking the color-terms into account (as discussed in Ngeow 2022). We expect such photometric transformations would be available after the first-light of LSST, and hence it will be straight forward to apply our proposed approach to the LSST light-curve data for RR Lyrae.

We thank the useful discussions and comments from an anonymous referee to improve the manuscript. C.C.N. is thankful for funding from the National Science and Technology Council (NSTC; Taiwan) under the contract 109-2112-M-008-014-MY3. A.B. acknowledges funding from the European Union's Horizon 2020 research and innovation program under the Marie Skodowska-Curie grant agreement No. 886298. This publication has made use of data collected at Lulin Observatory, partly supported by NSTC grant 109-2112-M-008-001. This research has made use of the SIMBAD database and the VizieR catalog access tool, operated at CDS, Strasbourg, France. This research made use of Astropy, ⁴ a community-developed core Python package for Astronomy (Astropy Collaboration et al. 2013, 2018, 2022).

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.

The Pan-STARRS1 Surveys (PS1) and the PS1 public science archive have been made possible through contributions by the Institute for Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the Max-Planck Society and its participating institutes, the Max Planck Institute for Astronomy, Heidelberg and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins University, Durham University, the University of Edinburgh, the Queen's University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las Cumbres Observatory Global Telescope Network Incorporated, the National Central University of Taiwan, the Space Telescope Science Institute, the National Aeronautics and Space Administration under grant No. NNX08AR22G issued through the Planetary Science Division of the NASA Science Mission Directorate, the National Science Foundation grant No. AST-1238877, the University of Maryland, Eotvos Lorand University (ELTE), the Los Alamos National Laboratory, and the Gordon and Betty Moore Foundation.

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, and NEOWISE, which is a project of the Jet Propulsion Laboratory/California Institute of Technology. WISE and NEOWISE are funded by the National Aeronautics and Space Administration.

Facilities: LO:1m - , Gaia - , PS1 - Panoramic Survey Telescope and Rapid Response System Telescope #1 (Pan-STARRS).

Software: astropy (Astropy Collaboration et al. 2013, 2018, 2022), PSFEx (Bertin 2011), SCAMP (Bertin 2006), Source-Extractor (Bertin & Arnouts 1996).

Like in Ngeow (2022), photometric calibrations on the LOT reduced images were done using the Pan-STARRS1 (PS1) Data Release 1 photometric data (Chambers et al. 2016; Flewelling et al. 2020) as reference catalogs. The selection criteria of the PS1 reference stars are identical to Ngeow (2022; whenever applicable), and will not be repeated here. In the gr-band, the photometric calibration was performed via the following equations:

$$\begin{eqnarray}&&{g}^{\mathrm{PS}1}-{g}^{\mathrm{instr}}={\mathrm{ZP}}_{g}+{C}_{g}({g}^{\mathrm{PS}1}-{r}^{\mathrm{PS}1}),\end{eqnarray} \tag{ A1 }$$

$$\begin{eqnarray}&&{r}^{\mathrm{PS}1}-{r}^{\mathrm{instr}}={\mathrm{ZP}}_{r}+{C}_{r}({g}^{\mathrm{PS}1}-{r}^{\mathrm{PS}1}).\end{eqnarray} \tag{ A2 }$$

These equations also allow the (g^PS1 −r^PS1) colors to be measured via

$$\begin{eqnarray}&&({g}^{\mathrm{PS}1}-{r}^{\mathrm{PS}1})=\displaystyle \frac{{\mathrm{ZP}}_{g}-{\mathrm{ZP}}_{r}+({g}^{\mathrm{instr}}-{r}^{\mathrm{instr}})}{1-{C}_{g}+{C}_{r}}.\end{eqnarray} \tag{ A3 }$$

For the V band, we first transformed the gr-band photometry in the PS1 reference stars catalog to the V-band using the transformation equation provided by Tonry et al. (2012), followed by fitting the reference stars with the following equation:

$$\begin{eqnarray}&&V-{v}^{\mathrm{instr}}={\mathrm{ZP}}_{V}+{C}_{V}({g}^{\mathrm{PS}1}-{r}^{\mathrm{PS}1}).\end{eqnarray} \tag{ A4 }$$

Calibration for the i-band data required a further treatment, this is because we would like to calibrate the i-band photometry to the Johnson-Cousin I-band magnitudes. Combining ${i}^{\mathrm{PS}1}-{i}^{\mathrm{instr}}={\mathrm{ZP}}_{i}+{C}_{i}({g}^{\mathrm{PS}1}-{r}^{\mathrm{PS}1})$ with the Tonry et al. (2012) transformation, the i-band data can be calibrated using the following equation:

$$\begin{eqnarray}&&I-{i}^{\mathrm{instr}}=({\mathrm{ZP}}_{i}-0.367)+({C}_{i}-0.149)\times ({g}^{\mathrm{PS}1}-{r}^{\mathrm{PS}1}).\end{eqnarray} \tag{ A5 }$$

In Equations (A1)–(A5), $\{g,r,v,i\}{}^{\mathrm{instr}}$ are instrumental magnitudes measured with MAG_PSF implemented in the Source-Extractor package (together with the PSF models derived from the PSFEx package; Bertin 2011), while g^PS1 and r^PS1 are magnitudes taken from the PS1 reference catalog. After solving the coefficients ZP and C in these equations, we then apply them to the RR Lyrae by first obtaining the (g^PS1 −r^PS1) colors using Equation (A3), and subsequently calibrate the grVI-band photometry from the same set of Vgri images.

Besides the V-band relation, Mullen et al. (2021) have also derived a similar relation in the infrared WISE filter. However, the W1-band light curves for both RR Lyrae, extracted from the ALLWISE database, do not show a RR Lyrae-like light curve in the W1-band, rather the extracted light curves are quite scatter with large error bars (see Figure3). As a result, we omitted the W1-band [Fe/H]–ϕ₃₁–P relation in this work.

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