Study of Globular Cluster M53: new variables, distance, metallicity

Study of Globular Cluster M53: new variables, distance, metallicity

Dékány Konkoly Observatory, PO Box 67, H-1525, Budapest, Hungary
   I. Konkoly Observatory, PO Box 67, H-1525, Budapest, Hungary
   Kovács Konkoly Observatory, PO Box 67, H-1525, Budapest, Hungary
   G. Konkoly Observatory, PO Box 67, H-1525, Budapest, Hungary
Received / Accepted
Key Words.:
globular clusters: individual: M53 – methods: data analysis – stars: variables: general – stars: variables: RR Lyr – stars: oscillations – stars: abundances


Aims:We study the variable star content of the globular cluster M53 to compute the physical parameters of the constituting stars and the distance of the cluster.

Methods:Covering two adjacent seasons in 2007 and 2008, new photometric data are gathered for 3048 objects in the field of M53. By using the OIS (Optimal Image Subtraction) method and subsequently TFA (Trend Filtering Algorithm), we search for variables in the full sample by using Discrete Fourier Transformation and Box-fitting Least Squares method. We select variables based on the statistics related to these methods combined with visual inspections.

Results:We identified 12 new variables (2 RR Lyrae stars, 7 short periodic stars – 3 of them are SX Phe stars – and 3 long-period variables). No eclipsing binaries were found in the present sample. Except for the 3 (hitherto unknown) Blazhko RR Lyrae (two RRab and an RRc) stars, no multiperiodic variables were found. We showed that after proper period shift, the PLC (period–luminosity–color) relation for the first overtone RR Lyrae sample tightly follows the one spanned by the fundamental stars. Furthermore, the slope is in agreement with the one derived from other clusters. Based on the earlier Baade-Wesselink calibration of the PLC relations, the derived reddening-free distance modulus of M53 is mag, corresponding to a distance modulus of  mag for the Large Magellanic Cloud. From the Fourier parameters of the RRab stars we obtained an average iron abundance of  (error of the mean). This is  dex higher than the overall abundance of the giants as given in the literature and derived in this paper from the three-color photometry of giants. We suspect that the source of this discrepancy (observable also in other, low-metallicity clusters) is the want of sufficient number of low-metallicity objects in the calibrating sample of the Fourier method.


1 Introduction

Among the oldest and most metal-poor Galactic globular clusters, the outer halo cluster M53 (NGC 5024, , ) is the second most abundant in variable stars after M15 (cvsgc, see also acs2009 for current age determinations). The search for its RR Lyrae content dates back to shapley1920, and yielded a total of 60 RR Lyrae stars until now. A thorough review of discoveries is given in the paper of kop2000, the only CCD time-series photometric study of M53 RR Lyrae stars prior to this work. This cluster is also very abundant in blue straggler stars (BSSs). Among the almost 200 BSSs, 8 were proven to be SX Phe variables (jeon2003). We also note that beccari2008 showed that the BSSs follow a bimodal distribution, implying a binary rate of , but no eclipsing binaries have been found so far. As for the most extensive color-magnitude study of M53, we refer to rey1998. The cluster is located at a very high Galactic latitude of . Therefore, its field contamination and interstellar reddening is expected to be negligible. This is confirmed by zinn1985, reporting E, and by schlegel, who give a reddening value of E. Despite its high variable content and its favorable position, no comprehensive wide-field CCD time-series photometric study has been published for this cluster. Among the two previous works of similar type, kop2000 observed only the central part of the cluster, while jeon2003 limited themselves to the investigation of SX Phe stars. The present study, based on two-color photometry covering most of the cluster, intends to improve the time and spatial coverage of this cluster.

To exploit the entirety of our photometric data-base and detect as many variables as possible, we employ the widely used method of Optimal Image Subtraction (OIS) developed by al98 and improved by alard2000, to obtain optimum relative light curves. Then, in a post-processing phase, we filter out temporal trends by using the Trend Filtering Algorithm (TFA) of kbn2005. This latter method has been extensively used during the past several years by the HATNet Project111Hungarian-made Automated Telescope Network, see
in search for transiting extrasolar planets and also employed successfully in general variable search (kob2008; szkw2009). The main purpose of this paper is to search for variables in the full dataset and not limit the investigation to the well-known regions of variability (i.e., the RR Lyrae strip, the blue straggler and the upper red giant branch regions of the Hertzspring-Russell diagram). With the aid of the reduction and post-processing methods mentioned above, we can increase photometric precision and attempt to reach the detection limit set by the photon statistics.

In Sect. 2 we give details on the observations and data processing, and show examples of the detections resulting from the methods applied. The next three sections (Sects. 3LABEL:lpv) are devoted to the more detailed analysis of the specific types of variables, most importantly to that of the RR Lyrae stars. Our attention will be focused mostly on the period–luminosity–color relation and the metallicity issue. A brief discussion and conclusions are given in Sect. LABEL:conclusions.

2 Observations, data reduction, method of analysis

2.1 Observations and data reduction

We performed time-series photometric observations of M53 through and filters of the Johnson and Kron-Cousins system, respectively, with the  cm Schmidt Camera of the Konkoly Observatory located at Piszkéstető. The telescope is equipped with a Photometrics AT 200 type CCD of 9- pixels in a array. The point spread function (PSF) is slightly undersampled with a scale of arcsec/pixel. The observations were carried out between March 25, 2007 and May 28, 2008 in two seasons, on a total number of 26 nights. The observations have a full time span of  days. Exposure times varied between and seconds in and and seconds in depending on atmospheric conditions. A total number of and frames have been collected in the and bands, respectively. While for the time series analysis we use only the -band data, for standard magnitude transformations, variable star classification, and for deriving cluster distance and metallicity, we also utilize the -band data.

The processing of the images (bias, dark, and flat-field corrections) and the alignment of all frames into a common pixel reference system were performed by standard iraf222IRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation packages. The joint field of view of all observations was somewhat truncated compared to the full detector area due to technical reasons. Only this common area (still covering a fairly large part of the cluster by spanning to tidal radius, see the catalog of harris1996, revised in February 2003), centered approximately on the cluster core, was subjected for further reduction.

We calibrated the astrometric transformations between pixels and celestial coordinates by using 30 bright stars from the second edition of the Guide Star Catalog (version 2.3.2, gsc2) uniformly distributed around the cluster center and located sufficiently well outside the cluster core. The standard deviations in the residuals of the coordinate mapping were and in right ascension and declination, respectively.

In order to deal with heavy crowding and to probe the regions close to the center of the cluster, we applied the OIS method of alard2000. For this method one has to prepare an accurate reference frame (RF) in each band which will be subtracted from all other frames after the appropriate scaling of the PSF of the stellar objects. The success of the method is highly sensitive to the quality of the RFs, therefore we carefully selected the best images with the lowest background, best seeing, and most regular PSFs and stacked them into one single image in order to increase their signal-to-noise ratio. Before the subtraction from each frame, the RF was convolved with a kernel function, which was determined by using the representative PSFs of 225 stars uniformly distributed on the image. The kernel functions were allowed to have second order spatial variations to account for a varying PSF across the image. The convolved RF, being the best approximation of the processed frame in a least-squares sense, was subtracted from the processed frame. The method yielded a set of residual images consisting of, in principle (but not in practice), the true light variations plus noise. For all procedures of image convolution and subtraction we used the isis 2.2 package of alard2000.

We applied the daophot PSF-fitting software package (daophot) on the RFs to identify stellar objects and to obtain their instrumental magnitudes. We found 3048 stars above the level. Linear transformation equations of these instrumental magnitudes into standard ones were established by using 110 photometric standard stars of standards around the cluster center.

We measured differential flux variations on the residual frames by aperture photometry at all stellar positions obtained from the RFs, using a custom IRAF routine. The sizes of the apertures were variable in order to conform to the FWHMs of the PSFs on the original frames, changing in time due to seeing variations. To diagnose if some faint variables might have been remained undetected on the RFs due to crowding but had light variations emerged on the residuals, we also employed the detection algorithms of isis 2.2 in two ways. First, all sources of significant variability were checked for the case if light variation comes from a yet undetected faint star falling within the same aperture. In these cases we performed its photometry using a new, accurately recentered aperture. Secondly, we checked if there were any remaining faint variables that had neither been detected on the RFs nor fell within any aperture (we note that we did not find any additional variables by this latter test). All light curves obtained in this way, expressed in linear flux units, were then further processed with the aid of the Trend Filtering Algorithm (TFA, see Sect. 2.2 for further details).

As the last step of the reduction, to convert differential fluxes of an object into magnitudes, one has to know the flux of the source on the RF in each band which has been subtracted from each processed frame (see e.g., woz2000). This additional information sets the zero point of the logarithmic transformation

between the differential magnitude and differential flux of the object (derived by the OIS method) on the -th frame. The value has been measured by PSF photometry for each object as described above. Clearly, a flux-to-magnitude transformation with an imprecise value yields an incorrect amplitude of the light variation and can significantly alter the magnitude averages as well, but its actual effect on these errors is specific to each object and depends on the brightness and the shape of the light variation. Therefore, the flux curves of variable stars were transformed into magnitudes only if their (already standardized) photometry on the RFs were reliable enough, which has been decided based on the formal errors of the PSF photometry. Generally, objects were transformed into magnitudes if the formal error of their was below  mag (see Sect. LABEL:plc for details on this cut in the case of RR Lyrae stars)333 Photometric time-series of the variables detected in this work and the Fourier decompositions of the RR Lyrae stars with magnitude-transformed light curves in color are available electronically online at CDS..

2.2 Post-processing with TFA

The time-series data-base obtained above is subjected to a subsequent post-processing phase, when we filter out various trends/systematics due to, e.g., imperfect reduction, lack of correction to position- and time-dependent extinction, anomalies in the convolution procedure prior to image subtraction, etc. The method is described in detail by kbn2005 and also summarized recently by bak2009 and szkw2009. Here we only briefly note that the method is based on the idea of correcting elements of the systematic variation in the target time series by using the light curves of many other objects, available in the CCD frame. In the course of signal search, we have no information on the temporal content of the time-series. Therefore, we assume that the target is trend- and noise-dominated. If it is the case, then the method suppresses the trend and gives rise to the signal to appear in some time-frequency transformation of the detrended signal. Here, on the expense of the suppression of the systematics, we also deform the true signal at some level. This effect is cured in the second step of filtering, when we reconstruct the signal by using a full time-series model that includes both the systematics and the signal with the period obtained in the first step. An extension of the method to multiperiodic signals is given in kob2008. In this method (adopted also in here) one fits the TFA template and the signal (represented by a Fourier series) simultaneously, thereby avoiding iteration by incomplete time-series model representations.

Figure 1: Histogram of the peak frequencies of the Fourier (DFT) spectra for the 3048 stars analyzed. Upper panel shows the frequency distribution of data obtained by aperture photometry on the residual frames of the OIS method. Lower panel shows the effect of TFA on these data.

Special attention is to be paid here to the effects related to the number of data points. In ideal case, the necessary number of TFA templates to handle all systematics is considerably lower than the number of data points constituting the time-series. Unfortunately, we do not have a good method to select a “representative” small set of templates, therefore we use a large set that includes both “useful” and “useless” (e.g., pure noise) template time-series. Attempts have been made to determine an optimum number of templates but it seems that it is hard to settle at a value lower than a few hundred (szkw2009). Since in the present case we have only data points per time-series, we have to limit the number of templates considerably (i.e., about half of the number of the data points) to avoid overfitting and sudden increase of the false alarm probability (FAP). This latter effect is thoroughly tested in each interesting case by using multiple template runs, which are capable of reducing FAP by several factors (kob2007). To avoid significant overfitting, we settled on the lowest TFA template numbers that yielded the most significant detection together with the lowest unbiased scatter around the signal. This optimum template number varied between and , the latter being an upper limit set by us due to the low number of available data points.

In general, we selected the TFA template sets from the brightest stars in the sample. The faintest stars in the template sets did not exceed  mag in band. The templates cover the frame in a quasi-uniform manner (see kbn2005, for more details on the distribution of templates).

The significance of the Fourier or BLS components (see kzm2002) was deduced by checking the SNR of the frequency spectra (the ratio of the amplitude of the highest peak to the standard deviation of the spectrum). Statistical tests (similar to the ones performed in our earlier papers, e.g., in nk2006) have led to the conclusion that for Gaussian signals with , the FAP due to uncorrelated random noise is less than . Considering that the processed signals always have some colored noise (either physical or instrumental) we selected variables well above this SNR value (usually with SNR exceeding ).

In demonstrating the ability of signal detection, first we compare the distributions of the peak frequency components obtained by OIS and by the subsequent application of TFA on the OIS time series. Figure 1 shows that already the OIS data are reasonably free of periodic systematics. Actually, when compared with earlier similar diagrams derived on data of aperture photometry (e.g., kbn2005; szkw2009) we see that the current data are much less dominated by the customary  d systematics. We see in the bottom panel that TFA has successfully filtered out the daily trends. In the final dataset the frequency distribution becomes nearly flat, with a slight surplus due to the RR Lyrae stars and the long-term irregular light variations of red giants.

In checking the effect of TFA in more detail, we plotted the DFT (Discrete Fourier Transformation) spectra and the folded light curves for the OIS and OISTFA data in the case of an RRc (V71, Fig. 2) and an SX Phe star (V78, Fig. 3). The effect of the signal reconstruction is large in both cases. The true variability of V71 would have been significantly more troublesome to decipher from the forest of high peaks in the original (OIS) data than in the clean, single-component spectrum of the TFA-filtered time-series. For V78 the situation is better because of the high frequency of the pulsation. Nevertheless, even in this case, the highest peak is near  d (and its aliases), therefore, an automatic search that only looks for the highest peak in the spectra, would not find this variable in the original (OIS) data. The improvement in the quality of the folded light curve derived from the reconstructed data is also substantial. We should note however that the small scatter seen is somewhat biased, because of the large number of TFA templates used. The effect of the bias in the residual scatter is common to all data regressions. The fit introduces correlation among the original array elements that leads to smaller observed scatter as it were expected if only the deterministic (model-predicted) part of the signal would have been fitted. In general, when least-squares fitting a data array of data points with parameters (larger than exact model representation required) the residual RMS will be -times of the value expected from the noise level in the original data (see also szkw2009 for further details on this subject). For this reason, the unbiased standard deviations of the reconstructed light curves are and higher due to the and templates used for V71 and V78, respectively. The different template numbers resulted from the multi-template tests mentioned earlier in this section, in the description of the method of analysis.

In a final example, in Fig. 4 we show the signal reconstruction capability of TFA. We used an th order Fourier series with TFA templates in the reconstruction (see kob2008). Inspection of the DFT spectrum of the OIS light curve of this star shows that it is not influenced by periodic systematics of type integer d. Therefore, the improvement introduced by TFA is attributed to filtering out some transient signals. We found transient systematics also in other variables, similarly to the ones reported by szkw2009. The unbiased standard deviation of the residuals around the best-fitting Fourier-sum dropped down by , which is a good improvement over the original data.

Figure 2: Detection of the RRc variable V71. Top left: folded light curve (with the period detected by TFA) of the OIS data; top right: DFT of the OIS data (light shade) and that of the TFAd data (dark shade); bottom left: folded light curve of the TFA-reconstructed data; bottom right: DFT of the TFA-reconstructed data. Star identification name and period (in days) are shown in the upper right corner. Please check note in text on the scatter of the TFA-filtered light curve.
Figure 3: Detection of the SX Phe star V78. Notation is the same as in Fig. 2.
Figure 4: Signal reconstruction in the case of the RRab star V45. Open circles denote the OIS fluxes while filled ones show the TFA-processed values. Please check note in text on the scatter of the filtered light curve.

3 RR Lyrae stars

Our prime interest in this paper is the study of the RR Lyrae stars, since they are in large number in the cluster and they are the ones that have fairly well established theoretical and observational understanding (e.g., variable and mode identification, computation of the physical parameters from observed ones, etc). We focus on the distance and on the metallicity (i.e., [Fe/H]) as the two most important parameters derivable from the available empirical relations.

3.1 General description

We identified altogether RR Lyrae stars in the cluster. Except for the two RRc stars V71 and V72, all others were known previously (see the Catalog of Variable Stars in Globular Clusters, hereafter CVSGC, cvsgc). We note that the variability status of V34, which had been questioned previously by vgend47 was clearly confirmed by our data. Due to the application of OIS and TFA, many of the variables ended up with light curves suitable for further analysis. We give the full list of variables together with their basic observed parameters in Table 3.1.1. Please note the very high SNR attached to each detection even for the variables sitting near the center of the cluster.

The accuracy of the periods of the RR Lyrae stars has been estimated in the following way. First we computed the Fourier decompositions from the TFA-reconstructed light curves. From these decompositions synthetic light curves were generated and Gaussian noise was added with standard deviations obtained from the unbiased estimates of the residual scatter between the synthetic light curves and the TFA reconstructions. The generated light curves were subjected to the standard Fourier frequency analysis as if they were real observed light curves. The periods obtained from each realization were stored and after completing 100 independent simulations for each object, the standard deviations of those period values were computed. The errors shown in Table 3.1.1 are these standard deviations.

The periods obtained in this study are generally in good agreement with previously determined values, except for the three RRc stars V44, V47, and V55, for which previous values are various aliases of the true periods. In the case of other first overtone stars the agreement between our and earlier published values are better than with a typical difference of about . As for RRab stars, deviations are generally below , with a few higher values not exceeding .

Figures LABEL:rrlyrlc and LABEL:rrlyrfc show the TFA-reconstructed and folded light curves. Standardizing the photometry for the highly crowded objects shown in Fig. LABEL:rrlyrfc was not reliable enough, therefore, we left the light curves in relative flux units. Unfortunately, these objects can only be partially utilized here due to the lack of empirical relations derived on fluxes and the absence of existing photometry with a higher resolution CCD (exceptions are V7, V8, V9, V33, and V37 that have good-quality light curves in magnitude from kop2000). We note that among the previously known RR Lyrae stars missing from our sample, V52 and V53 were merged with each other, V61 was merged with the long period variable V49, and the rest (V12, V13, V21, V30, and V48) were outside the field of view.

3.1.1 Blazhko-stars

We investigated possible multiperiodicity by performing successive prewhitening on all objects, including RR Lyrae stars. Surprisingly, from the whole sample we found only the three Blazhko stars given in Tables 3.1.1 and LABEL:blazhko, in spite of the relatively large scatter visible in several stars (see Fig. LABEL:rrlyrfc). We note that the detection of the Blazhko behavior in the three stars is firm. The modulation is clearly observable as side-lobe frequencies with uniform frequency separations after prewhitening the main frequency and its harmonics. For V11, significant side lobe frequencies could be identified for up to the 7th harmonic of at , except for , where we found . In the case of the RRc star V16 we found and . The phase modulations in both stars are very little, unlike to that of the third Blazhko-star, V57, whose modulation is very large both in phase and in amplitude. We detected and in the case of V57. The light curves of the Blazhko stars folded with the pulsation periods are shown in Fig. LABEL:rrlyrfc. The light curves were reconstructed by simultaneously fitting the template and the full modulated signal (see Sect. 2.2).

ID Type [hms] [dms] [″] [d] SNR
V1 RRab 13:12:56.3 18:07:13.8 176.8 0.6098298(9) 14.3 16.970 0.579
V2 RRc 13:12:50.3 18:07:00.8 202.0 0.386122(1) 22.2 16.879 0.402
V3 RRab 13:12:51.4 18:07:45.5 154.6 0.630605(1) 17.8 16.848 0.447
V4 RRc 13:12:43.9 18:07:26.4 230.3 0.385545(1) 26.0 16.800 0.383
V5 RRab 13:12:39.1 18:05:42.6 353.3 0.639426(1) 23.2 16.888 0.526
V6 RRab 13:13:03.9 18:10:19.8 123.7 0.664020(1) 14.4 16.819 0.522
V7 RRab 13:13:00.9 18:11:29.7 113.3 0.5448584(6) 13.5
V8 RRab 13:13:00.4 18:11:05.1 92.3 0.615528(1) 16.1
V9 RRab 13:13:00.1 18:09:25.0 81.9 0.6003690(7) 15.2
V10 RRab 13:12:45.7 18:10:55.6 143.3 0.6082612(7) 18.8 16.837 0.463
V11 RRabB 13:12:45.4 18:09:01.9 156.3 0.629940(5) 12.2
V14 RRab 13:13:20.5 18:06:42.8 415.6 0.5454625(7) 14.0 16.880 0.452
V15 RRc 13:13:12.4 18:13:55.0 332.5 0.3086646(9) 21.6 16.894 0.342
V16 RRcB 13:12:46.2 18:06:39.2 247.3 0.3031686(7) 18.7
V17 RRc 13:12:40.4 18:11:54.1 236.7 0.381282(1) 27.5 16.848 0.422
V18 RRc 13:12:48.7 18:10:12.9 93.9 0.336054(1) 22.2
V19 RRc 13:13:07.0 18:09:26.1 173.2 0.391377(1) 28.1 16.880 0.437
V20 RRc 13:13:09.0 18:04:16.2 404.7 0.384337(1) 22.5 16.875 0.407
V23 RRc 13:13:02.3 18:08:35.9 137.9 0.365804(1) 25.2 16.825 0.391
V24 RRab 13:12:47.2 18:09:33.0 120.4 0.763198(2) 20.3
V25 RRab 13:13:04.4 18:10:37.2 133.5 0.705162(1) 16.7 16.779 0.555
V26 RRc 13:12:35.7 18:05:20.5 401.7 0.391106(1) 24.1 16.845 0.395
V27 RRab 13:12:41.4 18:07:23.9 257.9 0.671071(1) 14.7 16.856 0.550
V28 RRab 13:12:42.1 18:16:37.9 430.9 0.6327804(7) 16.5 16.875 0.545
V29 RRab 13:13:04.3 18:08:46.9 152.9 0.823243(4) 26.6 16.782 0.588
V31 RRab 13:12:59.6 18:10:04.6 61.9 0.705665(1) 15.2
V32 RRc 13:12:47.7 18:08:35.9 142.6 0.390623(2) 20.4 16.712 0.416
V33 RRab 13:12:43.9 18:10:13.2 162.5 0.6245815(8) 16.9
V34 RRc 13:12:45.7 18:06:26.1 262.0 0.289611(1) 21.2 16.928 0.314
V35 RRc 13:13:02.4 18:12:37.5 179.8 0.372666(2) 22.7 16.943 0.442
V36 RRc 13:13:03.3 18:15:10.2 321.8 0.373242(1) 21.5 16.879 0.414
V37 RRab 13:12:52.3 18:11:05.1 69.6 0.717615(1) 21.0
V38 RRab 13:12:57.1 18:07:40.5 151.8 0.705792(2) 21.2 16.749 0.557
V40 RRc 13:12:55.9 18:11:54.7 105.3 0.3147939(9) 19.6
V41 RRab 13:12:56.8 18:11:09.3 63.6 0.614438(1) 19.2
V42 RRab 13:12:50.6 18:10:19.5 67.4 0.713717(2) 21.8
V43 RRab 13:12:53.1 18:10:55.7 55.5 0.712017(2) 24.2
V44 RRc 13:12:51.5 18:09:58.2 54.6 0.375099(2) 23.7
V45 RRab 13:12:55.2 18:09:27.4 42.4 0.654950(2) 16.2
V46 RRab 13:12:54.6 18:10:36.2 28.1 0.703655(3) 22.0
V47 RRc 13:12:50.4 18:12:24.6 151.4 0.335377(1) 21.1 16.791 0.372
V51 RRc 13:12:57.8 18:10:50.7 54.7 0.355203(2) 20.2
V54 RRc 13:12:54.4 18:10:31.4 25.0 0.315122(3) 14.2
V55 RRc 13:12:53.6 18:10:39.2 37.7 0.443386(2) 26.4
V56 RRc 13:12:53.7 18:09:29.1 46.3 0.328796(2) 15.1
V57 RRabB 13:12:55.5 18:09:58.0 12.1 0.568234(7) 11.8
V58 RRc 13:12:55.6 18:09:30.8 39.4 0.354954(3) 17.2
V59 RRc 13:12:56.7 18:09:20.4 53.4 0.303941(2) 19.7
V60 RRab 13:12:57.0 18:09:36.0 41.6 0.644755(2) 18.4
V62 RRc 13:12:54.0 18:10:30.1 27.0 0.359891(4) 13.8
V63 RRc 13:12:56.2 18:10:02.8 15.5 0.310476(4) 14.3
V64 RRc 13:12:52.6 18:10:12.5 38.5 0.319529(1) 19.2
V71* RRc 13:12:54.5 18:09:54.3 18.8 0.304242(5) 14.5
V72* RRc 13:12:55.8 18:09:50.6 20.7 0.254155(3) 13.5

Table 1: Properties of the RR Lyrae stars
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