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

Multiband light curves of two RR Lyrae variables in Segue II and Ursa Major II ultrafaint dwarf galaxies were collected from near simultaneous observations using the Lulin One-meter Telescope in Vgri bands. Together with Gaia G-band light curves, we determined photometric metallicities using empirical relations involving pulsation period and Fourier parameter as dependent parameters. We demonstrated that the RR Lyrae photometric metallicity can be determined accurately when these empirical relations were employed at multiple wavelengths, which can potentially improve the distance determination based on RR Lyrae stars. The photometric metallicities based on our approach were found to be −2.27 ± 0.13 dex and −1.87 ± 0.16 dex for the RR Lyrae in Segue II and Ursa Major II UFD, respectively, with corresponding distance moduli of 17.69 ± 0.15 mag and 17.58 ± 0.15 mag, in agreement with previous literature determinations. This approach of photometric metallicity of RR Lyrae based on multiband optical light curves will be particularly relevant for distance measurements in the era of the Vera C Rubin’s Legacy Survey of Space and Time.

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 -band 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 multi-band 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), 1 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.
Note-This table is available in its entirety in machine-readable form, a portion is shown here for guidance on its content.
Section 2 presents multi-band light curves collected from LOT, together with archival Gaia light curves, for these two RR Lyrae.Based on these multi-band 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.

MULTI-BAND LIGHT CURVES
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 backilluminated CCD was mounted on LOT during these observations, providing a FOV of 13.2 ′ × 13.2 ′ (with a pixel scale of 0.386 ′′ /pixel) on the CCD images.A sequence of V gri exposures, with identical exposure time of 360 second 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-and dark-subtraction, followed by flat-fielding).Astrometric calibration on the reduced images were done using the SCAMP package (Bertin 2006).The PSF (point-spread function) 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. Table 1 and 2 provide multi-band 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. 2016Collaboration et al. , 2023) ) for these two RR Lyrae, with Gaia Note-This table is available in its entirety in machine-readable form, a portion is shown here for guidance on its content.
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 SIM-BAD/VizieR database), because the adopted periods are adequate to fold the light curves and there is no need to re-derive 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 multi-band 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: 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 φ 31 = φ 3 − 3φ 1 Fourier parameters which are needed to estimate the photometric metallicity for RR Lyrae (see next Section).The determined φ 31 from the multi-band light curves are provided in Table 3.

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 φ 31 (Jurcsik & Kovacs 1996).We collected the grGV I-band [Fe/H]-φ 31 -P relations that are available recently in the literature.These include the grV -band relations taken from Ngeow (2022, 2009) scale as determined in Mullen et al. (2021).We did not include Vband 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: Mullen et al. (2021) pointed out that a π is needed to be subtracted from φ 31 when using their relation, hence we included it in the above relation.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: For SEGUE II-V1 and UMaII-V1, the A 2 parameters in the I-band are 0.069 ± 0.003 and 0.114 ± 0.003, respectively.

The Derived Metallicity
The derived multi-band 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 [Fe/H] I S05 values are mainly due to the large error on 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 ∼ 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 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 Note-An error of 0.02 mag is adopted on these mean magnitudes.
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.

THE DISTANCES OF THE TWO RR LYRAE
Since the Wesenheit magnitude, denoted as W , is reddening-free by construction (see the Appendix 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 V Igr-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 multi-band light curves.The V Igrband mean magnitudes are listed in Table 4. Neeley et al. (2019) calibrated multi-band 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] ZW conversion.After applying the V I-band PWZ relation from Neeley et al. (2019), we determined the V I-band based distance moduli (µ V I ) to SEGUE II-V1 and UMaII-V1 as µ V I (SEGUE II-V1) = 17.67 ± 0.20 mag and µ V I (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, 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 grband 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 V Iband 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 V I and µ gr were determined from independent PWZ relations and in different filters, the averages of µ V I 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 µ = 17.68 +0.16  −0.17 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) and Marconi et al. (2022) were adopted instead of the empirical relations.

CONCLUSIONS
In this work, we obtained homogeneous grV I-band light curves using LOT for two RR Lyrae in SEGUE II and UMaII ultra-faint dwarf galaxies.Together with their Gaia G-band light curves, we derived the photometric metallicities for these two RR Lyrae using nine [Fe/H]-φ 31 -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 multi-band 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 multi-filter 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 Lyraebased 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 to remind that the gr-band and the V I-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 grV I-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.

Figure 1 .
Figure 1.The grV I-band light curves based on LOT observations and the G-band light curves retrieved from Gaia DR3 for SEGUE II-V1 (left panel) and UMaII-V1 (right panel).For a better visualization, these light curves, except the g-band, were offsetted.The black curves represent the best-fit light curves based on the Fourier-expansion.

Figure 2 .
Figure2.The derived photometric metallicities in various filters, as summarized in Table3, for SEGUE II-V1 (left panel) and UMaII-V1 (right panel).The orange horizontal lines represent the weighted means based on the nine adopted relations, and the orange shaded regions are the ±1σ dispersion of the weighted means.Horizontal lines in green and magenta colors are the published values, together with their ±1σ dispersion shown as dashed lines, based on spectroscopic observations of a number of bright member stars in these two dwarf galaxies.For a better visualization, we only plotted the values taken from two publications for these two RR Lyrae.theconstant term, ±0.832, in the S05 relation.Errors from other relations are around ∼ 0.3 to ∼ 0.4 dex.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 Table3giving 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 ∼ times smaller that those listed in Table3(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 case of SEGUE II-V1, our derived photometric metallicity is in good agreement with the values determined inBelokurov et al. (2009) andKirby et al. (2013), as shown in the left panel of Figure2, and almost identical to the value recalculated byBoettcher et al.  (2013, −2.26 ± 0.14 dex).For UMaII-V1, our photometric metallicity is close to the value given by

Figure 3 .
Figure3.The W 1-band light curves for SEGUE II-V1 and UMaII-V1 extracted from the ALLWISE (WISE Team 2019) database, which is composed of data collected from the WISE(Wright et al. 2010) and the NEOWISE(Mainzer et al. 2011) mission.The red triangles indicate data-points with "null" entries in the error bar.Both light curves were folded with periods adopted from Gaia DR3 (see Section 2).