Discovery of RR Lyrae in the Ultra-faint-dwarf Galaxy Virgo III

Virgo III is a newly discovered ultra-faint-dwarf (UFD) candidate, and finding RR Lyrae associated with this galaxy is important to constrain its distance. In this work, we present a search of RR Lyrae in the vicinity of Virgo III based on the time-series r-band images taken from the Lulin One-meter Telescope (LOT). We have identified three RR Lyrae from our LOT data, including two fundamental mode (ab-type) and a first-overtone (c-type) RR Lyrae. Assuming these three RR Lyrae are members of Virgo III, we derived the distance to this UFD as 154 ± 25 kpc, fully consistent with the independent measurements given in the literature. We have also revisited the relation between absolute V-band magnitude (M V ) and the number of RR Lyrae (of all types, N RRL) found in local galaxies, demonstrating that the M V -N RRL relation is better described with the specific RR Lyrae frequency.


INTRODUCTION
Finding RR Lyrae in dwarf galaxies, especially the ultra-faint dwarfs (UFD; for a recent review, see Simon 2019), is particularly interesting (Sesar et al. 2014;Baker & Willman 2015).This is because RR Lyrae are well-known standard candles, therefore distances measured from RR Lyrae can be used to constrain the properties of their host UFD.Recently, Homma et al. (2023) reported the discovery of Virgo III as a candidate UFD.Based on the empirical relation derived in Martínez-Vázquez et al. (2019), and using the integrated V -band absolute magnitude (M V ) given in Homma et al. (2023, with M V = −2.69+0.45  −0.56 mag), the "expected" number of RR Lyrae (N RRL ) in Virgo III is 1 ± 1. Boötes II and Willman 1 have M V = −2.9mag and −2.5 mag, respectively, bracketing Virgo III.Yet Boötes II has one RR Lyrae and Willman 1 has none (Tau et al. 2024).Therefore, Virgo III could have (at least) one RR Lyrae or none, and it is useful to search and identify these variables in Virgo III.
In this work, we present our search for potential RR Lyrae in the vicinity of Virgo III using the Lulin Onemeter Telescope (LOT), located at the central Taiwan.We first describe our time-series observations carried out at LOT, as well as the image reduction and photometric calibration, in Section 2. We then create a set of simulated light-curves to evaluate the feasibility of detecting RR Lyrae based on the characteristics of our LOT observations, and search for potential RR Lyrae using the calibrated light curves in Section 3 and 4, respectively.In Section 5 we present our detected RR Lyrae, and revisit the M V -N RRL relation in Section 6.We concluded our work in Section 7.

LOT OBSERVATIONS, REDUCTION, AND CALIBRATION
LOT is a F/8 Cassegrain reflector, and it was equipped with the Andor iKon-L 936 CCD imager during our queued observations.As a result, the LOT images have a pixel scale of 0.345 ′′ pixel −1 and a field-ofview (FOV) of 11.8 ′ × 11.8 ′ .Note that the half-light radius for Virgo III is r h = 1 ′ (Homma et al. 2023), therefore the FOV of LOT can cover the entire galaxy.Given the expected faintness of the RR Lyrae (r ∼ 21.5 mag), we only observed Virgo III using the r-band filter commercially available from Astrodon, with exposure time of 1200 s (except the first two nights in 2023, when the exposure time was set to 900 s).Log of our time-series observations is given in Table 1.
All of the collected images were bias-subtracted and dark-subtracted using the master-bias and master-dark frames acquired from the same night, followed by flatfielding using either dome flat or twilight flat images.Astrometric calibration on the reduced images were done using the SCAMP (Bertin 2006) software.For photometric calibration, we selected ∼ 20 reference stars from the Pan-STARRS1 (PS1) photometric catalog (Chambers et al. 2016;Flewelling et al. 2020).Criteria for selecting the PS1 reference stars (when- b Averaged full-width at half-maximum (FWHM) of the point sources in the image in unit of arc-second.
c The 5σ limiting magnitude was adopted to represent the depth of each image.
ever applicable) were same as in Ngeow (2022) and Ngeow & Bhardwaj (2024), and hence will not be repeated here.The r-band magnitudes and colors of the PS1 reference stars, r P S1 and (g − r) P S1 respectively, were then used to iteratively fit the regression in the following form: The instrumental magnitudes of the reference stars, r instr on each images, were based on the point-spreadfunction (PSF) photometry measured from using the Source-Extractor (Bertin & Arnouts 1996) and PSFEx (Bertin 2011) package.After solving equation ( 1), the detected sources in each images were calibrated to the PS1 AB magnitude system.We then fitted a low-order polynomial to the calibrated r vs. σ r plot, and estimated the 5σ limiting magnitude from the fitted polynomial (these fitted polynomials would be used in the light-curve simulations as described in the next Section).They are listed in the last column of Table 1, and most of the images can reach to a nominal depth of r ∼ 23 mag.
We have also estimated the expected photometric error at r = 21.5 mag, which has a median of ∼ 0.05 mag.

LIGHT CURVES SIMULATIONS
Given the small number of epochs (∼ 20) collected from LOT, and our targeted RR Lyrae are faint and close to the detection limit, we ran light-curve simulations to evaluate the feasibility of detecting RR Lyrae using our LOT data.

Constant Stars
We first simulated light-curve for 1000 constant stars based on the epochs as listed in the third column of Table 1 (∆t).The r-band magnitudes were uniformly drawn from interval between 20.5 mag and 23.0 mag.Since we expect the RR Lyrae will have r ∼ 21.5 mag, we set the upper limit to be 1 mag brighter than this.The lower limit of 23.0 mag was set by the nominal depth of our LOT images, as we won't be able to detect any stars fainter than this limit.For each drawn magnitudes at a given epoch, we added a Gaussian uncertainty based on the fitted low-order polynomial mentioned in Section 2, to the simulated magnitudes.We discarded the simulated magnitudes if such magnitudes were fainter than the depth at a given epoch as listed in the last column of Table 1.Therefore, some simulated light-curves would have less data-points than others.
We calculated the following two quantities on the simulated light-curves with more than 10 data-points: r and M AD = median(|r i − r|), where r and r are the weighted means and medians for r i , respectively.The product of these two quantities appeared to be a good metric to identify (large-amplitude) RR Lyrae against constant stars (Ngeow et al. 2020).The gray points in the left panel of Figure 1 show the distribution of χ 2 × M AD as a function of magnitudes for the simulated constant stars.As expected, χ 2 ×M AD become larger at fainter magnitudes due to the increasing of photometric uncertainties.Nevertheless, values of χ 2 × M AD did not exceed 0.3 for the simulated constant stars.

RR Lyrae
Light-curve simulations for RR Lyrae are similar to the constant stars, except an additional step of adopting the r-band template light-curves (T r ) available from Sesar et al. (2010).In brief, light-curve for a RR Lyrae was constructed using the following expression: (2) where AM P r , t 0 , and P are the r-band light-curve amplitude, epoch at the maximum light, and pulsation period, respectively.For each RR Lyrae, we generated a uniformly distributed random number for r, P , t 0 , and AM P r .The range for r is same as in the cases of constant stars (20.5 mag to 23.0 mag), while for t 0 we set its range to be −P and 0 (days).Finally, a Gaussian uncertainty, based on the polynomial fits for each of the epochal photometry, was added to r(∆t i ) in equation ( 2).Same as in the cases of constant stars, we discarded r(∆t i ) if it is fainter than the depth on each epoch.
RR Lyrae can pulsates either in fundamental or firstovertone mode, known as ab-type and c-type, respectively.2Both types of RR Lyrae follow a different distribution of P and AM P r .The adopted ranges for P (in days) are P ab = [0.45,0.80] and P c = [0.25,0.45], while for the r-band amplitudes, the adopted ranges are AM P ab = [0.15,1.40] and AM P c = [0.13,0.55] (Ngeow et al. 2022).For template light-curves, 20 and 2 r-band templates were available from Sesar et al. (2010) for the ab-and c-type, respectively.They were randomly selected when constructing the simulated lightcurves via equation (2).
We simulated light-curves for 200 ab-type and 100 ctype RR Lyrae.Values of χ 2 × M AD for those lightcurves with more than 10 data-points were over-plotted in the left panel of Figure 1 alongside with the constant stars.As can be seen from this plot, the (large amplitude) RR Lyrae and constant stars can be well separated using χ 2 × M AD, except for some low-amplitude RR Lyrae toward the faint end (due to larger photometric errors that are comparable to the light-curve amplitudes).The red line shown in the inset figure is a good compromise to separate the RR Lyrae and constant stars, and there were 90% and 84% of the simulated ab-type and c-type RR Lyrae, respectively, located above the  Our simulated RR Lyrae light-curves can also be used to evaluate the period recovery rate.We employed a combination of Lomb-Scargle based and template lightcurve period search methods, both implemented in the gatspy (VanderPlas & Ivezić 2015) package, to search for the periods on our simulated light-curves.We emphasized that these period-search approaches are same as in the search of RR Lyrae in Virgo III using the real LOT data.Figure 2 presents the result on the periodsearch, which shows the recovery of 66.1% of the input periods (indicated as dashed lines in Figure 2).Other periods tend to lie along the tracks for different aliasing periods.

SEARCHING FOR RR LYRAE
Since majority of our LOT images can reach to a depth similar to, or slightly deeper than, the Sloan Digital Sky Survey (SDSS) Data Release 16 (DR16) catalog (Ahumada et al. 2020, at r ∼ 22.7 mag) 3 , and sources in SDSS DR16 have been classified into either stars or galaxies, we adopted SDSS DR16 catalog as our master catalog.There are 226 stellar sources in SDSS D16 located within the footprint of our LOT images, these stellar sources were used to construct a master stars list.We then cross-matched the detected and calibrated sources in each LOT images with this master stars list to create light curves for all of the 226 stellar sources.
We searched for potential RR Lyrae among 194 light curves that have more than 10 data-points.Values of χ 2 × M AD for them as a function of mean magnitudes are shown in the right panel of Figure 1.We visually inspected the light curves and ran a preliminary Lomb-Scargle periodogram analysis for stars above the red line drawn in the right panel of Figure 1.We identified three RR Lyrae because their periods, amplitudes and folded light-curves resembling a typical RR Lyrae.Two of them, V1 (SDSS objID = 1237654879650775304) and V2 (SDSS objID = 1237654879650841005), are ab-type RR Lyrae, and V3 (SDSS objID = 1237654879650775856) is a c-type RR Lyrae.LOT r-band light-curves for them are presented in Table 2.
Figure 3 shows the location of the three detected RR Lyrae with respect to Virgo III.The foreground reddening returned from the Bayerstar2019 3D reddening map (Green et al. 2019)   (Green 2018) 5 package, toward Virgo III is E = 0.002 ± 0.002 mag.This translates to an r-band extinction of A r = 2.617E = 0.005 ± 0.005 mag.

PROPERTIES OF THE DETECTED RR LYRAE
To improve the period determination on the detected RR Lyrae, we added single-epoch SDSS DR16 r-band PSF photometry (see Table 3) to the LOT light curves, after converting the SDSS photometry to the PS1 photometric system using the transformation provided in Tonry et al. (2012).We further employed the template light-curve based period search algorithm, available in the gatspy package (VanderPlas & Ivezić 2015), for periods refinements.The improved and final adopted periods for the three RR Lyrae are listed in Table 3. Fig-  a Errors on r , σ r , were estimated using the empirical relation between σ r and number of data-points on light-curves (Ngeow et al. 2022).
ure 4 and 5 present the folded light-curves for the abtype and c-type RR Lyrae, respectively.The dashed curves in these two figures are the best-fitted template light-curves found by gatspy, and subsequently used to determine the r-band amplitudes and intensity mean magnitudes r .The determined values are listed in Table 3. Figure 6 compares the r-band amplitudes and the extinction-corrected absolute r-band magnitude (M r , by adopting the distance modulus, µ, given in Homma et al. 2023) for the three detected RR Lyrae with the counterparts in the globular clusters (Ngeow et al. 2022).All of the three RR Lyrae are located within the distributions of known RR Lyrae, strongly supporting their identification and membership to Virgo III.Photometric metallicity, [Fe/H], for the two ab-type RR Lyrae can be estimated using the empirical re-  2023), who adopted an isochrone filter at metallicity of −2.2 dex to fit the Virgo III colormagnitude diagram (CMD).We did not use the r-band relation between [Fe/H], P , and Fourier parameter φ 31 (Ngeow 2022) to estimate their photometric metallicity because the low-order Fourier expansion failed to provide reasonable fit to the observed light-curves.Using both of the fundamental mode and firstovertone mode r-band period-luminosity-metallicity relations derived in Ngeow et al. (2022), where the metallicity is in the D16 scale, and the adopted [Fe/H], we calculated the distance moduli for these three RR Lyrae.The results are summarized in the last row of Table 3.By taking a weighted average and assum- For the three RR Lyrae, we have also applied approximate color corrections to bring the single-epoch SDSS photometry close to the mean values.We assumed such corrections carry an error of ∼ 0.1 mag.The mean r-band magnitudes for them were taken from Table 3.The green curves are three evolutionary tracks for horizontal branch (HB) models with mass (M ) of 0.58, 0.64, and 0.70 M⊙, taken from the BaSTI library.These HB models are α-enhanced (as found in many UFD; for a review, see Simon 2019), with [Fe/H]= −1.9 dex, and vertically shifted to the absolute magnitudes using the distance modulus derived in our work.
ing all three RR Lyrae are genuine member of Virgo III, the distance modulus for Virgo III was found to be µ = 20.937±0.355mag (statistical error only from small number statistics), or a linear distance of 154 ± 25 kpc.Our distance modulus is fully consistent with the values given in Homma et al. (2023), 20.9 ± 0.2 mag and 20.91 ± 0.04 mag, which were derived using isochrone fitting and the blue horizontal-branch (BHB) stars, respectively.
In Figure 7, we presented the extinction-corrected CMD for SDSS stars located within the 4 × r h ellipse of Virgo III (see Figure 3).Since the SDSS photometry was based on the single-epoch observations, we applied additional color corrections such that the SDSS photometry are close to the mean (g − r) colors for these RR Lyrae.These approximate corrections were estimated based on the template light curves (Sesar et al. 2010) or a (g − r) template color curve (Ngeow 2022).Overplotted on the CMD are the evolutionary tracks for the horizontal branch (HB) models at three representative masses.These evolutionary tracks were taken from the BaSTI (a Bag of Stellar Tracks and Isochrones, Pietrinferni et al. 2021) stellar isochrones library.As can be seen from the CMD, the locations of three detected RR Lyrae are consistent with the HB evolution-  ary tracks corresponding to our adopted distance modulus to Virgo III, strengthening their identification as RR Lyrae and the membership to the galaxy.
6. THE M V -N RRL RELATION In this Section, we revisited the M V -N RRL (of all types) relation presented in Martínez-Vázquez et al. (2019).We began with 63 galaxies listed in the appendix of Martínez-Vázquez et al. (2019), and added new local galaxies from Monelli & Fiorentino (2022) and Tau et al. (2024), supplemented with several additional galaxies (such as Virgo III from this work) not included in these compilations.We have also updated the number of RR Lyrae in a few dwarf galaxies based on the latest publications (in particular, for Fornax and Draco).The updated list of local galaxies, with their M V and N RRL , is provided in Table 4, alongside with the references for M V and N RRL .In total, there are 57867 RR Lyrae found in 94 galaxies, but ∼ 78.5% of them come from the Magellanic Clouds.
Figure 8 presents the M V -N RRL relation for the galaxies listed in Table 4.The (green) solid line is the empirical relation derived in Martínez-Vázquez et al. (2019), which describes the trend of the data well, therefore we did not re-derive the relation.Instead, the relation shows a large scatter.Therefore, we over-plotted the in-verted specific RR Lyrae frequency (S RR , Suntzeff et al. 1991;Mackey & Gilmore 2003) as: which is normalized to M V = −7.5 mag. Figure 8 shows the curves of equation ( 3) for several representative S RR , and most of the galaxies are confined between the curves for S RR = 1 and S RR = 100.Furthermore, the local galaxies seems to follow various tracks at given S RR (for examples, LMC, SMC, and NGC 185 are located at the S RR = 1 track, and some of the M V > −10 mag galaxies are located along the S RR = 100 track).Clearly, there is no single value of S RR to fit the majority of the galaxies (also, see Baker & Willman 2015).
Tucana III is the only faint UFD with S RR > 1000.However, all of the six detected RR Lyrae are extratidal stars of Tucana III (Vivas et al. 2020).Furthermore, Tau et al. (2024) only recovered one of them.It is possible that Tucana III might only have one extratidal RR Lyrae, which reduced its S RR to ∼ 302.On the other hand, there are several galaxies below the curve of S RR = 1, implying the detection of RR Lyrae on these galaxies are not yet completed.This is especially true for Triangulum, as the known RR Lyrae in this galaxy were detected based on the several narrow "pencil-beam" fields around Triangulum (Tanakul et al. 2017).

CONCLUSION
In this work, we searched for RR Lyrae in Virgo III using the time-series LOT observations.We have also ran light-curve simulations by taking the characteristics of LOT observations (such as photometric errors and depths on each images) into account, and demonstrated that RR Lyrae can be detected using the LOT data and our searching method (i.e. the χ 2 × M AD metric).We identified two ab-type and one c-type RR Lyrae with periods and amplitudes consistent with the known RR Lyrae in the globular clusters.Given that they are located within the 4×r h ellipse of Virgo III, and have similar distance modulus as Virgo III, we assume they are true members of Virgo III.Based on the three detected RR Lyrae in Virgo III, together with the latest findings in the literature, we have also revisited the M V -N RRL relation for local galaxies, showing their relations are better described using the specific RR Lyrae frequency.
It is worth to point out that both Virgo III and the three RR Lyrae are fainter than the detection limit of Gaia, hence there is no proper-motion information to verify the status of their membership.Confirmation of their membership has to wait for the future radial-velocity measurements.We have also estimated the metallicity of Virgo III to be [Fe/H] = −1.93dex based on the two ab-type RR Lyrae.Nevertheless, assuming the three detected RR Lyrae are members of Virgo III, the RR Lyrae-based distance modulus is fully consistent with the independent measurements given in Homma et al. (2023).

Figure 1 .Figure 2 .
Figure 1.Products of χ 2 and M AD as a function of mean r-band magnitudes.The left panel is for simulated light-curves, including constant stars (gray points), ab-type RR Lyrae (circles) and c-type RR Lyrae (squares).The color bar represents the input amplitudes for the simulated light-curves.The inset figure is the zoomed-in version to highlight constant stars.The red line, given as χ 2 × M AD = 0.045 r − 0.875, is adopted to separate RR Lyrae and constant stars.The right panel is for those stars, classified in SDSS DR16, with LOT light-curves.The red line is same as the line shown in the inset figure on the left panel.We identified three RR Lyrae labeled as V1, V2, and V3.Light-curve for the star with χ 2 × M AD ∼ 4.5 was affected by outliers, and hence it was rejected as a RR Lyrae candidate.

3Figure 3 .
Figure 3.A deep-stack image created by median-combined all of the LOT images using SWARP (Bertin et al. 2002).The cyan ellipses, centered at (α, δ) Virgo III J 2000 , have semi-major axes of {1, 2, 3, 4} × r h , a projected ellipticity ǫ = 1 − a/r h (where a is the projected semi-minor axis, and r h is the halflight radius in arc-minute), and with a position angle of θ.Values for (α, δ) Virgo III J 2000 , r h , ǫ, and θ are all adopted from Homma et al. (2023).Locations of the three detected RR Lyrae were also shown in this deep-stack image.

Figure 5 .
Figure 5. Same as Figure 4, but for the c-type RR Lyrae V3.

Figure 6 .
Figure 6.Comparison of r-band amplitudes (upper panel) and absolute magnitudes (lower panel) of the three detected RR Lyrae in Virgo III to the RR Lyrae in the globular clusters (orange and cyan symbols, adopted from Ngeow et al. 2022).

Figure 7 .
Figure7.The color-magnitude diagram for stars located within the 4 × r h ellipse of Virgo III, including the three detected RR Lyrae, which have been corrected for extinction.For the three RR Lyrae, we have also applied approximate color corrections to bring the single-epoch SDSS photometry close to the mean values.We assumed such corrections carry an error of ∼ 0.1 mag.The mean r-band magnitudes for them were taken from Table3.The green curves are three evolutionary tracks for horizontal branch (HB) models with mass (M ) of 0.58, 0.64, and 0.70 M⊙, taken from the BaSTI library.These HB models are α-enhanced (as found in many UFD; for a review, see Simon 2019), with [Fe/H]= −1.9 dex, and vertically shifted to the absolute magnitudes using the distance modulus derived in our work.

Figure 8 .
Figure 8. Number of RR Lyrae (NRRL) as a function of MV for the 94 galaxies listed in Table4.To deal with the "log of zero" problem on the y-axis (in log-scale), we added +1 to the NRRL.The green solid curve represents the empirical relation derived inMartínez-Vázquez et al. (2019).The dashed curve are for the different selected values of specific RR Lyrae frequency SRR.Several "outliers" are also marked in this plot.The inset figure is the zoomed-in version for UFD galaxies with NRRL < 7, where Virgo III is marked as a magenta star.For clarity, error bars and the curves for SRR < 1 are omitted in the inset figure.Note that the y-axis is in linear scale in the inset figure.

Table 1 .
Log of LOT observations.
a Time-difference, in days, between a given image and the first image.
Sesar et al. (2010)-curves for the two ab-type RR Lyrae V1 and V2.The dashed curves are the best-fit template light-curves returned from the gatspy's periodic.RRLyraeTemplateModeler module.The template light-curves were adopted fromSesar et al. (2010).

Table 3 .
Basic properties of the detected RR Lyrae.

Table 4 .
Martínez-Vázquez et al. (2019) problem on the y-axis (in log-scale), we added +1 to the NRRL.The green solid curve represents the empirical relation derived inMartínez-Vázquez et al. (2019).The dashed curve are for the different selected values of specific RR Lyrae frequency SRR.Several "outliers" are also marked in this plot.The inset figure is the zoomed-in version for UFD galaxies with NRRL < 7, where Virgo III is marked as a magenta star.For clarity, error bars and the curves for SRR < 1 are omitted in the inset figure.Note that the y-axis is in linear scale in the inset figure.