New near-infrared period-luminosity-metallicity relations for RR Lyrae stars and the outlook for Gaia
Tatiana Muraveva, Max Palmer, Gisella Clementini, Xavier Luri, Maria-Rosa L Cioni, Maria Ida Moretti, Marcella Marconi, Vincenzo Ripepi, Stefano Rubele
AAstrophysical Journal
New near-infrared period-luminosity-metallicity relations for RR Lyrae stars and theoutlook for Gaia T. Muraveva
INAF-Osservatorio Astronomico di Bologna, via Ranzani 1, 40127, Bologna, Italy [email protected]
M. Palmer
Dept. d’Astronomia i Meteorologia, Institut de Ciències del Cosmos, Universitat de Barcelona (IEEC-UB),Martí Franquès 1, E08028 Barcelona, Spain
G. Clementini
INAF-Osservatorio Astronomico di Bologna, via Ranzani 1, 40127, Bologna, Italy
X. Luri
Dept. d’Astronomia i Meteorologia, Institut de Ciències del Cosmos, Universitat de Barcelona (IEEC-UB),Martí Franquès 1, E08028 Barcelona, Spain
M.-R.L. Cioni , Universität Potsdam, Institut für Physik und Astronomie, Karl-Liebknecht-Str. 24 /
25, 14476 Potsdam,Germany
M. I. Moretti INAF-Osservatorio Astronomico di Bologna, via Ranzani 1, 40127, Bologna, Italy
M. Marconi
INAF-Osservatorio Astronomico di Capodimonte, Salita Moiariello 16, I-80131, Napoli, Italy
V. Ripepi
INAF-Osservatorio Astronomico di Capodimonte, Salita Moiariello 16, I-80131, Napoli, Italy Leibniz-Institut für Astrophysik Potsdam, An der Sternwarte 16, 14482 Potsdam, Germany University of Hertfordshire, Physics Astronomy and Mathematics, College Lane, Hatfeild AL10 9AB, United Kingdom Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, Italy a r X i v : . [ a s t r o - ph . S R ] M a y INAF-Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, 35122, Padova, Italy
ABSTRACT
We present results of the analysis of 70 RR Lyrae stars located in the bar of the Large Mag-ellanic Cloud (LMC). Combining spectroscopically determined metallicity of these stars fromthe literature with precise periods from the OGLE III catalogue and multi-epoch K s photometryfrom the VISTA survey of the Magellanic Clouds system (VMC) , we derive a new near-infraredperiod-luminosity-metallicity (PL K s Z) relation for RR Lyrae variables. In order to fit the rela-tion we use a fitting method developed specifically for this study. The zero-point of the relationis estimated in two di ff erent ways: by assuming the value of the distance to the LMC and byusing Hubble Space Telescope (HST) parallaxes of five RR Lyrae stars in the Milky Way (MW).The di ff erence in distance moduli derived by applying these two approaches is ∼ . K s Z relation based on 23 MW RR Lyrae starswhich had been analysed in Baade-Wesselink studies. We compared the derived PL K s Z rela-tions for RR Lyrae stars in the MW and LMC. Slopes and zero-points are di ff erent, but stillconsistent within the errors. The shallow slope of the metallicity term is confirmed by bothLMC and MW variables.The astrometric space mission Gaia is expected to provide a huge contribution to the de-termination of the RR Lyrae PL K s Z relation, however, calculating an absolute magnitude fromthe trigonometric parallax of each star and fitting a PL K s Z relation directly to period and ab-solute magnitude leads to biased results. We present a tool to achieve an unbiased solution bymodelling the data and inferring the slope and zero-point of the relation via statistical methods.
Subject headings: stars: variables: RR Lyrae - galaxies: Magellanic Clouds - (cosmology):distance scale - methods: data analysis - stars: statistics - astrometry
1. Introduction
RR Lyrae stars are radially pulsating variables connected with low-mass helium-burning stars on thehorizontal branch (HB) of the colour-magnitude diagram (CMD). These objects are Population II stars,which are abundant in globular clusters and in the halos of galaxies. RR Lyrae stars are a perfect tool forstudying the age, formation and structure of their parent stellar system. Moreover, they are widely used forthe determination of distances in the Milky Way (MW) and to Local Group galaxies. Based on observations made with VISTA at ESO under programme ID 179.B-2003. V − [Fe / H]) relation in the visual band and of period-luminosity-metallicity (PLZ) relations in the in-frared passbands. The near-infrared PL K s Z relation of RR Lyrae stars was originally discovered by Long-more et al. (1986), and later was the subject of study by many di ff erent authors (e.g., Bono et al. 2003,Catelan et al. 2004, Del Principe et al. 2006, Sollima et al. 2006, Sollima et al. 2008, Borissova et al. 2009,Coppola et al. 2011, Ripepi et al. 2012a). The near-infrared PL K s Z relation has many advantages in compar-ison with the visual M V − [Fe / H] relation. First of all, the luminosity of RR Lyrae stars in the K s passbandis less dependent on metallicity and interstellar extinction ( A K s ∼ . A V ). Furthermore, light curves ofRR Lyrae stars in the K s band have smaller amplitudes and are more symmetrical than optical light curves,making the determination of the mean K s magnitudes easier and more precise.In order to calibrate the PL K s Z relation a large sample of RR Lyrae stars is required, spanning a widerange of metallicities, for which accurate mean K s and [Fe / H] measurements are available. We have selected70 RR Lyrae variables in the Large Magellanic Cloud (LMC) with spectroscopically determined metallicitiesin the range of − . < [Fe / H] < − K s magnitudes, multi-epoch photometry is needed. Forthis reason we are using data from the near-infrared VISTA survey of the Magellanic Clouds System (VMC ,Cioni et al. 2011), which is performing K s -band observations of the whole Magellanic System in 12 (ormore) epochs, while in many previous studies only single-epoch photometry from the Two Micron All-SkySurvey (2MASS, Cutri et al. 2003) was used. To fit the PL K s Z relation we apply a fitting approach developedfor the current study. This method takes into account errors in two dimensions, the intrinsic dispersion ofthe data and the possibility of inaccuracy in the formal error estimates.One main issue in the determination of distances with the RR Lyrae PL K s Z relation is the calibrationof the zero-point. Trigonometric parallaxes remain the only direct method of determining distances toastronomical sources, free of any assumptions (such as, for instance, the distance to the LMC, etc.) andhence calibrating the PL K s Z zero-point. However, reasonably well estimated parallaxes exist, so far, onlyfor five RR Lyrae variables in the MW observed by Benedict et al. (2011) with the
Hubble Space Telescope
Fine Guidance Sensor (
HST / FGS). In this study we use both a global estimate of the LMC distance andthe
HST parallaxes in order to calibrate the zero-point of our PL K s Z relation based on LMC RR Lyraestars. Furthermore, to check whether the RR Lyrae PL K s Z relation is universal and could thus be appliedto measure distances in the MW and to other galaxies, we analyse a sample of 23 MW RR Lyrae stars, forwhich absolute magnitudes in the K and V passbands are available from the Baade-Wesselink studies (e.g.,Fernley et al. 1998b, and references therein). Based on these absolute magnitudes and applying our fittingapproach we fit the RR Lyrae PL K s Z relation. Then we compare the PL K s Z relations derived for RR Lyraestars in the MW and in the LMC.Gaia, the European Space Agency (ESA) cornerstone mission launched in December 2013, is expectedto provide a great contribution to the determination of the RR Lyrae PL K s Z relation and to the definition of itszero-point in particular. The satellite is designed to produce the most precise three-dimensional (3D) map ofthe MW to date (Perryman et al. 2001) by measuring parallaxes of over one billion stars during its five-year 4 –mission, among which are thousands of RR Lyrae variables. In the current study we present a method whichavoids the problems of the non-linear transformation of trigonometric parallaxes (and negative parallaxes)to absolute magnitudes, and apply this method to fit the PL K s Z relation of the 23 MW RR Lyrae stars, basedon simulated Gaia parallaxes.In Section 2 we provide information about the 70 RR Lyrae stars in the LMC that form the basis ofthe present study. In Section 3 we present our method and results of fitting the RR Lyrae PL K s Z relation inthe LMC and in the MW. In Section 4 we present the method to fit the PL K s Z relation with simulated Gaiaparallaxes and apply this method to the 23 MW RR Lyrae stars analysed in Section 3. Section 5 providesa summary of the results. In the Appendix sections we present a detailed description of the fitting methodwhich was developed for this study (Appendix A) and a compilation of metal abundances for the MW RRLyrae stars (Appendix B).
2. Data
Optical photometry for the LMC RR Lyrae stars discussed in this paper was obtained by Clementiniet al. (2003) and Di Fabrizio et al. (2005) using the Danish 1.54 meter telescope in La Silla, Chile. Twodi ff erent sky positions, hereafter called fields A and B were observed. Both are located close to the bar of theLMC (Clementini et al. 2003, Di Fabrizio et al. 2005). As a result, accurate B , V and I light curves tied to theJohnson-Cousins standard system and pulsation characteristics (period, epoch of maximum light, amplitudesand mean magnitudes) for 125 RR Lyrae stars were obtained (Di Fabrizio et al. 2005). Low-resolution spec-tra for 98 of these variables were collected by Gratton et al. (2004) using the FOcal Reducer / low dispersionSpectrograph (FORS1) instrument mounted at the ESO VLT. They were used to derive metal abundancesfor individual stars by comparing the strength of the Ca II K line with that of the H lines (Preston 1959).For the calibration of the method, four clusters with metallicity in the range [ − − (cid:48)(cid:48) , 3 (cid:48)(cid:48) and 7 (cid:48)(cid:48) . The 2 stars with a counterpart at more than 5 (cid:48)(cid:48) are OGLE-LMC-RRLYR-10345 and OGLE-LMC-RRLYR-10509; for these two objects we checked both the OGLE IIIfinding charts and Gratton et al. (2004) Figure 5 (field B1) in order to understand if they are a ff ected by anyproblem. Star OGLE-LMC-RRLYR-10345 is an isolated slightly elongated star without any clear blendingproblem, while star OGLE-LMC-RRLYR-10509 is very close to another source possibly making moredi ffi cult to locate accurately the star center. Considering that Gratton et al. (2004) and OGLE III periods forthese 2 stars agree within 0.5%, we kept these stars in our sample.We compared the periods of the 98 RR Lyrae stars provided by Di Fabrizio et al. (2005) and those in 5 –the OGLE III catalogue (Soszy´nski et al. 2009). For 96 objects the periods agree within ∼ ff er significantly. For star A6332 the di ff erence is of ∼
25% and for star A5148 it isof ∼
37% (star identifications are from Di Fabrizio et al. 2005). Moreover, star A5148 has been classifiedas a first-overtone RR Lyrae star (RRc) in the OGLE III catalogue, and as a fundamental-mode RR Lyrae(RRab) by Di Fabrizio et al. (2005). Since accurately estimated periods and classifications play a key rolein the current study, we discarded these two objects from the following analysis.Seven objects (B2811, B4008, B3625, B2517, A2623, A2119, A10360) from the sample are classifiedas RRc by Di Fabrizio et al. (2005) and as second-overtone RR Lyrae star (RRe) in the OGLE III catalogue.We removed them from our analysis because of the uncertain classification. Furthermore, since one of themain purposes of the current research is to study the PL K s Z relation of RR Lyrae stars of ab- and c-types wediscarded seven objects, which were classified as double-mode RR Lyrae stars (RRd) by Di Fabrizio et al.(2005): A7137, A8654, A3155, A4420, B7467, B6470 and B3347. This left us with a final sample of 61RRab and 21 RRc stars, which all have a counterpart in the OGLE III catalogue. The period search for theRR Lyrae stars in the OGLE III catalogue was performed using an algorithm based on the Fourier analysisof the light curves (Soszy´nski et al. 2009). The uncertainties in the OGLE III periods for the 82 RR Lyraestars in our sample are declared to be less than 5 × − days. Therefore we used the periods provided bythe OGLE III catalogue in order to fit the PL K s Z relation for our sample, and did not consider errors in theperiods since they are negligible in comparison to the other uncertainties.In order to derive mean K s magnitudes for the RR Lyrae stars in our sample we used data from the VMCsurvey (Cioni et al. 2011). Started in 2009, the VMC survey covers a total area of 116 deg in the LMCwith 68 contiguous tiles. The survey is obtaining YJK s photometry. The K s -band photometry is taken intime-series mode over 12 (or more) separate epochs and each single epoch reaches a limiting K s magnitude ∼ . / N ∼ K s ∼ . / N was enough to detectthe RR Lyrae stars. In the following analysis we used all 13 available epochs to fit the light curves of theRR Lyrae stars. PSF photometry of the time-series data for this tile was performed on the homogenisedepoch-tile images (Rubele et al. 2012) using the IRAF Daophot packages (Stetson et al. 1990). On eachepoch-tile image the PSF model was created using 2500 stars uniformly distributed, finally the DaophotALLSTAR routine was used to perform the PSF photometry on all epoch images and time-series catalogueswere correlated within a tolerance of one arcsec.We have cross-matched our sample of 82 RR Lyrae stars against the PSF photometry catalogue of theVMC tile LMC 5_5. VMC counterparts for 71 objects were found within a pairing radius of 1 (cid:48)(cid:48) . Amongthem, 70 sources have 13 epochs in the K s -band, while for one object (B4749) we have observations onlyin 6 epochs. Six data points are not enough for a reliable fit of the light curve and, consequently, forthe robust determination of the mean K s magnitude, hence, we discarded this source from the following 6 –analysis and proceeded with the 70 RR Lyrae stars, for which 13 epochs in the K s -band exist. We derivedthe mean K s magnitudes of these 70 RR Lyrae stars by Fourier fitting the light curves with the GRaphicalAnalyzer of TImes Series package (GRATIS, custom software developed at the Observatory of Bolognaby P. Montegri ff o, see e.g. Clementini et al. 2000). To fit the light curves we discarded obvious outliers.Nevertheless, after the σ -clipping procedure, each source still has 11 or more data points. Examples of the K s light curves are shown in Figure 1.After deriving K s mean magnitudes we performed the dereddening procedure. Clementini et al. (2003)estimated reddening values of E ( B − V ) = . ± .
017 and 0 . ± .
017 mag in LMC field A andB, respectively, using the method from Sturch (1966) and the colours of the edges of the instability stripdefined by the RR Lyrae variables. Applying the coe ffi cients from Cardelli et al. (1989) of A K / A V = R V = K s -bandas: A K s = . × E ( B − V ) (1)Table 1 summarizes the properties of the sample of 70 RR Lyrae stars which have a counterpart in theVMC catalogue. First and second columns give the identification of the stars in Di Fabrizio et al. (2005) andin the OGLE III catalogue, respectively. The table also shows coordinates and the classification of the starsfrom the OGLE III catalogue, metallicities with errors from Gratton et al. (2004) and dereddened mean K s magnitudes, determined with the GRATIS package, along with their errors. 7 –Fig. 1.— Examples of K s -band light curves for RR Lyrae stars in our sample. Identification numbers arefrom Di Fabrizio et al. (2005), periods are from the OGLE III catalogue (Soszy´nski et al. 2009) and aregiven in days. 8 –Table 1. Properties of the 70 RR Lyrae stars in the bar of the LMC analyzed in this paper Star OGLE ID RA DEC Type [Fe / H] σ [Fe / H] P (cid:104) K s , (cid:105) σ (cid:104) K s , (cid:105) (J2000) (J2000) (dex) (dex) (days) (mag) (mag)A28665 OGLE-LMC-RRLYR-12944 5:22:06.55 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − PL K s Z relation of RR Lyrae stars3.1. Method Using the dereddened mean K s magnitudes of the 70 RR Lyrae stars derived as described in Section 2,spectroscopically determined metallicities from Gratton et al. (2004) and accurately estimated periods fromthe OGLE III catalogue (with RRc stars "fundamentalized" by adding 0.127 to the logarithm of the period)we can now fit the PL K s Z relation. The fit is performed using a fitting approach developed specifically forthis work.Fitting a line to data is a common exercise in science. Most common approaches use Minimum-Least-Squares methods, however these are often based on assumptions which do not always hold for realobservational data. The most basic methods assume that data are drawn from a thin line with errors, whichare Gaussian, perfectly known, and exist in one axis only. These assumptions do not hold in the presentcase, as we have an unknown but potentially significant intrinsic dispersion, non-negligible errors in twodimensions ( K s and [Fe / H]), and the possibility of inaccuracy in the formal error estimates (e.g. in thedetermination of the precision metallicity estimates).We therefore follow the prescription of Hogg et al. (2010), who develop a method for fitting a lineto data which avoids the problems highlighted above by statistical modelling of the data. They present amethod for use in two dimensions, which has been extended to three dimensions in this paper.The method assumes that the data is drawn from a plane of the form L s( P , [Fe / H]) = A logP + B [Fe / H] + C (2)where A is the slope in the logP axis, B is the slope in the metallicity [Fe / H] axis, and C is the intercept.We assume a uniform Gaussian intrinsic dispersion around the luminosity axis, plus the scatter caused bythe Gaussian observational errors. The exact mathematical definition is given in Appendix A. The methodutilises adaptive Markov Chain Monte Carlo (MCMC) methods (Foreman-Mackey et al. 2013) to evaluatethe posterior probability density function (PDF) of each parameter, given an input dataset, and returns themaximum a posteriori probability (MAP) estimate of each parameter, the formal error estimate, and the fullposterior PDF. The formal error estimate is obtained from the 16% and 84% quartiles of the posterior PDF ofthe parameters, which give the 1 σ formal error estimate assuming that the posterior PDF is approximatelyNormal. The free fit parameters are: the slope in logP, the slope in metallicity, the zero-point, and theintrinsic dispersion perpendicular to the magnitude axis.By applying this method we found the following relation between period, metallicity and mean apparent K s magnitude: K s , = ( − . ± . P + (0 . ± . / H] + (17 . ± .
01) (3) 10 –Table 1—Continued
Star OGLE ID RA DEC Type [Fe / H] σ [Fe / H] P (cid:104) K s , (cid:105) σ (cid:104) K s , (cid:105) (J2000) (J2000) (dex) (dex) (days) (mag) (mag)B6798 OGLE-LMC-RRLYR-10044 5:17:11.37 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − K s magnitude from the VMC data, determined from the analysis of the light curve withGRATIS; 10) Error of the mean K s magnitude.
11 –The intrinsic dispersion of the relation is found to be 0.01 mag. The RMS deviation of the data aroundthe relation, neglecting the intrinsic dispersion, is 0.1 mag. Since the reddening in the K s band is negligiblewe suggest that the e ff ects of the LMC depth cause the intrinsic dispersion of the relation. The left panel ofFigure 2 presents the PL K s Z relation (Equation 3) of the 70 LMC RR Lyrae stars in the period-luminosity-metallicity space, whereas the right panels show the projection of the PL K s Z on the log(P) − K s (top-rightpanel) and K s − [Fe / H] (bottom-right panel) planes. The grey lines in the figure are lines of equal metallicity(top-right) or equal period (bottom-right). The method finds the relation (values of A, B, and C for therelation K s = A logP + B [Fe / H] + C) in the three dimensions (logP, K s , and [Fe / H]). Each of the grey lines inthe top-right plot are therefore K s = A logP + B [Fe / H] + C for the full range of periods, at the metallicity ofeach star (one line per star). Thus, by following the line up and down it is seen how K s changes with periodat some specific metallicity. The lines do not always cross the points on the diagram because the line is theresult of the fit, and the points are a ff ected by errors and intrinsic dispersion so may be above or below thefit. In the bottom-right plot the lines are K s = A logP + B [Fe / H] + C for the full range of metallicity with logPtaken from each star.It is worth noting that we find a very small dependence of the K s magnitude on metallicity. However,the metal abundance range spanned by the adopted sample does not reach the highest values (up to solar andsupersolar) observed in the MW bulge and disk RR Lyrae populations. In order to study the e ff ect of theadopted range of metallicities on the slope of the PL K s Z relation we derived this relation also for MW RRLyrae stars. We discuss the results in Section 3.3. PL K s Z relation in the LMC To use the derived PL K s Z relation for determining distances it is necessary to calibrate its zero-point.This can be done in a number of di ff erent ways. In this paper we follow two di ff erent approaches: thefirst one is based on adopting a value for the distance of the LMC; in the second one we use the absolutemagnitudes of Galactic RR Lyrae stars for which trigonometric parallaxes have been measured with the HST / FGS. Both approaches have their advantages and disadvantages, we discuss them in the followingsections.
The LMC is widely considered the first rung of the cosmic distance ladder as it contains a large numberof di ff erent distance indicators, such as Cepheids, RR Lyrae variables, eclipsing binaries (EBs), red giantbranch (RGB) stars, etc., allowing the galaxy distance to be determined with several independent techniques.Figure 8 of Benedict et al. (2002) shows an impressive summary of LMC distance moduli published duringthe ten years from 1992 to 2001. Values from 18.1 to 18.8 mag were reported in the literature, with thosesmaller than 18.5 mag supporting the so-called "short" scale, and those larger than 18.5 mag, the "long"one. In more recent years the dramatic progress in the calibration of the di ff erent distance indicators has 12 –led the dispersion in LMC distance moduli to shrink significantly. Extreme values such as those listed inBenedict et al. (2002) are not very often seen in the recent literature (Clementini 2008). Still a generalconsensus on the LMC distance has not been fully reached yet. Moreover, there have been significantconcerns about a possible "publication bias" a ff ecting the distance to the LMC (Schaefer 2008, Rubele et al.2012, Walker 2012). In particular, Schaefer (2008) claimed that from 2002 to 2007 June, 31 independentpapers reported new measurements of the distance of the LMC, and the new values clustered tightly aroundthe value ( m − M ) = . ± . HST
Key Project on the extragalactic distance scale(Freedman et al. 2001). Schaefer (2008) considered the e ff ects of the "publication bias" to be the most likelycause of the clustering of LMC distance measurements.A number of studies on the compilation of distances to the LMC as derived from di ff erent distance indi-cators can be found in the literature of the last 15 years (e.g., Gibson 2000; Benedict et al. 2002; Clementiniet al. 2003; Schaefer 2008; de Grijs et al. 2014). Clementini et al. (2003) analysed the distance to the LMCmeasured using Population I and Population II standard candles and showed that all distance determina-tions converge within 1 σ error on a distance modulus ( m − M ) = . ± .
085 mag. The most recentcompilation of LMC distance moduli is that of de Grijs et al. (2014) who compiled 233 separate distancedeterminations, published from 1990 March until 2013 December, and concluded that the canonical distancemodulus of ( m − M ) = . ± .
09 mag may be used for all practical purposes. The compilation of de Grijset al. (2014) includes the distance modulus of ( m − M ) = . ± .
03 estimated by Ripepi et al. (2012b)using LMC classical Cepheids observed by the VMC survey, and the recent determination of direct distancesto eight long-period EBs in the LMC by Pietrzy´nski et al. (2013), which is claimed to be accurate to within ∼ D LMC = ± ± m − M ) = ± ± ff erentclasses of pulsating stars in the LMC, also based on di ff erent samples and hydrodynamical codes, providesvalues consistent with 18.5 mag (see Bono et al. 2002; Marconi & Clementini 2005; Keller & Wood 2002,2006; McNamara et al. 2007).The RR Lyrae stars in our sample are located in a relatively small area close to the center of the LMCbar. Neglecting depth / projection e ff ects they can be considered as being all at the same distance from usand close to late-type EBs, which are all located relatively close to the barycenter of the LMC as derivedby Pietrzy´nski et al. (2013). Therefore, in the following we adopt for the distance modulus of the LMCthe value published by Pietrzy´nski et al. (2013). We subtracted this value from the dereddened mean K s apparent magnitudes of our 70 RR Lyrae stars and derived absolute magnitudes in the K s band ( M K s ).Then by applying the technique described in Section 3.1 we derived the relation between K s -band absolutemagnitudes, periods and metallicities, with the zero-point entirely based on the distance to the LMC byPietrzy´nski et al. (2013): M K s = ( − . ± . + (0 . ± . / H] − (1 . ± .
01) (4) 13 –
In order to obtain an estimate of the PL K s Z relation zero-point which is independent of the distance tothe LMC and, in turn, be able to measure the distance to this galaxy from the RR Lyrae PL K s Z relation, it isnecessary to know the absolute magnitude of the RR Lyrae stars with good accuracy. Trigonometric paral-laxes remain the only direct method to measure distances and hence derive absolute magnitudes. Benedict etal. (2011) derived absolute trigonometric parallaxes for five Galactic RR Lyrae stars (RZ Cep, XZ Cyg, SUDra, RR Lyr and UV Oct) with the
HST / FGS. With these parallaxes the authors estimated absolute magni-tudes in the K s and V passbands, corrected for interstellar extinction and Lutz-Kelker-Hanson bias (Lutz &Kelker 1973, Hanson 1979). Absolute magnitudes in the K s -band, periods and metallicities from Benedict etal. (2011), and the slopes of the relation derived in Equation 3 were used in order to determine a zero-pointfrom each of these five MW RR Lyrae stars. The metallicities in Benedict et al. (2011) are in the Zinn &West metallicity scale and were converted to the metallicity scale in Gratton et al. (2004) by adding 0.06dex (see Section 2). The logarithm of the period of the RRc star RZ Cep was "fundamentalized" by adding0.127. Then we calculated the weighted mean of the five zero-points, this corresponds to: − . ± .
08 mag.The relation between absolute magnitudes, periods and metallicities with the zero-point based on the fiveMW RR Lyrae stars from Benedict et al. (2011) is: M K s = ( − . ± . + (0 . ± . / H] − (1 . ± .
08) (5)A recent analysis (Monson 2015, private communication) shows that there is likely a typo in Benedictet al. (2011) parallax for the RR Lyrae star RZ Cep. Hence, we excluded this star from the sample andderived the zero-point based on parallaxes of remaining four RR Lyrae stars (XZ Cyg, UV Oct, SU Dra andRR Lyr): M K s = ( − . ± . + (0 . ± . / H] − (1 . ± .
06) (6)Situation improves, however there is still a di ff erence of ∼ . HST parallaxes of four RR Lyrae stars(Eq. 6). In fact, if we apply our PL K s Z relation with zero-point calibrated on Benedict et al. (2011) parallaxes(Eq. 6) to determine the absolute magnitudes of the 70 RR Lyrae stars in our sample, we obtain a distancemodulus for the LMC of ( m − M ) = . ± .
10 mag. This distance modulus is about 0.2 mag longer thanthe widely adopted value of ( m − M ) = . ff ects, also pointed out that Pietrzy´nski et al.(2013) distance to the LMC di ff ers significantly from the average distance inferred from four hot, early-typeEBs, D = ± m − M ) = . ± .
065 mag), published by Guinan et al. (1998), Fitzpatricket al. (2002, 2003), and Ribas et al. (2002). Furthermore, in using the late-type EBs to calibrate the RRLyrae PL K s Z relation we have implicitly assumed that RR Lyrae stars and EBs are at same distance from us.However, when pushing for distance comparisons at a few percent level the e ff ects of sample size, spatialdistribution, depth and geometric projection become important and properly accounting for the internalstructure of the LMC may become necessary. The RR Lyrae stars in our sample could be distributed alongthe whole depth of the LMC. Moreover, RR Lyrae stars and EBs from Pietrzy´nski et al. (2013) could reside indi ff erent sub-structures of the LMC, which could be the reason for the systematic error in the determinationof the zero-point (see e.g. Moretti et al. 2014 for di ff erent features of the LMC structure traced by classicalCepheids, RR Lyrae stars and hot EBs).On the other hand, when calibrating the zero-point by applying parallaxes of the four MW RR Lyraestars by Benedict et al. (2011) we implicitly assumed that the PL K s Z relation is the same in the MW andin the LMC, which may not be true (see Subsection 3.3). We may also wonder whether there might beunknown systematic errors a ff ecting Benedict et al.’s parallaxes. These come from HST fields, which providerelative and not absolute trigonometric parallaxes. Absolute parallaxes of the reference stars in each field areestimated via a complex procedure of fitting the spectral type and luminosity class of each star. A generalformal error of 0.5 mas is applied to the absolute parallax of the reference stars, equal for all stars in allfields, and without justification. This could result in miscalculated estimates of the precision of the finalabsolute parallax measurements of the four RR Lyrae stars. The Lutz-Kelker bias is corrected a posteriori.In this respect it is worth of notice that, according to van Leeuwen (2007), Hipparcos parallax of RR Lyraeitself, the only RR Lyrae variable for which the satellite measured the parallax with high precision ( ± K s Z relation is expected from the ESAastrometric satellite Gaia. We discuss this topic in Section 4. PL K s Z relation of RR Lyrae stars in the MW In spite of many studies in the literature, it remains still unsettled whether the RR Lyrae PL K s Z isa universal relation. To investigate this matter we have derived the PL K s Z relation for RR Lyrae stars inthe MW and compared it with the relation we have obtained in Section 3.2 for the LMC variables. Tothis end we selected 23 MW RR Lyrae stars which have their absolute magnitudes known from Baade-Wesselink (hereinafter B-W) studies based on near-infrared data (Jones et al. 1988, 1992; Fernley et al.1990; Liu & Janes 1990; Cacciari et al. 1992; Skillen et al. 1993; Fernley et al. 1994, and references therein)and metallicities determined from abundance analysis of high resolution spectra. Information about these23 RR Lyrae stars is presented in Table 2. Star’s coordinates in the table are from the SIMBAD database;periods, apparent V and K magnitudes and reddening E(B-V) are from Fernley et al. (1998a). The sample 15 –contains two first-overtone mode RR Lyrae stars (namely, T Sex and TV Boo). As done for the LMC RRcstars, their periods were fundamentalized by adding + M V and M K magnitudes in Columns 10 and 12 were taken from the compilations of B-W results in Table 11 ofCacciari et al. (1992) and from Table 16 of Skillen et al. (1993) for the variable stars: WY Ant, W Crt and RVOct. According to Cacciari et al. (1992) the K magnitudes of the stars analyzed with the B-W method are inthe Johnson photometric system. Following the discussion in Cacciari et al. (1992) and Skillen et al. (1993)we retained only 23 of the original lists of 29 stars field RR Lyrae stars analysed with the B-W method, asstars which are likely to be evolved (DX Del, SU Dra, SS Leo, BB Pup and W Tuc) were discarded. Wealso discarded DH Peg as there is suspect the star is a dwarf Cepheid (see Feast et al. 2008 and discussiontherein). Furthermore, following Fernley et al. (1994), original M V and M K values were revised (i) assumingfor the p factor used to convert the observed pulsation velocity to true pulsation velocity in B-W analysesthe value of p = ff erent spec-troscopic studies have targeted the stars in Table 2. In Appendix B we provide a summary of their results.The largest and most homogeneous samples are those by Clementini et al. (1995) and Lambert et al. (1996).These Authors measured [Fe / H] abundances from high resolution spectra for several of the stars in Table 2and provided recalibrations of the ∆ S index (Preston 1959), from which metal abundances can be derivedfor the stars which lack abundance analysis. For sake of homogeneity and ease of use in this paper weadopt metallicities and metallicities errors for the MW RR Lyrae stars as they are listed, ready for use, inTable 21 of Clementini et al. (1995). These [Fe / H] values are the average of the FeI and FeII abundances,adopting log (cid:15) (FeI) = (cid:15) (FeII) = (cid:15) (FeI) = / H] values and errors, are reported in columns2 and 3 of Table 4 in Appendix B. In Appendix B we also present the
PLZ relation obtained using Lambertet al. (1996)’s metallicities and our approach.By applying our fitting approach (Section 3.1) to the 23 MW RR Lyrae stars we derived the followingPL K Z relation: M K = ( − . ± . + (0 . ± . / H] − (0 . ± .
14) (7)The intrinsic dispersion of the relation is found to be 0.007 mag. The RMS deviation of the data aroundthe relation, neglecting the intrinsic dispersion, is 0.086 mag. It should be noted that the metallicities listedin Table 2 may di ff er slightly from the metal abundances used in the B-W analysis of these stars. However,this is not of great concern as the B-W based on near-infrared data is mildly a ff ected by small changesin metallicity and reddening. We also point out that the rather large error of the logP term in Equation 7 islargely driven by the large errors (0.15-0.25 mag) in the K -band absolute magnitudes from the B-W analyses.This is confirmed by the exercise with Gaia simulated parallaxes we present in Section 4. 16 –The slope in [Fe / H] in Equation 7 is higher than the slope obtained for the LMC RR Lyrae stars(Equations 4, 6), although they are still consistent within the respective errors. Equation 7 was derived overa wide range of metallicities [-2.5; 0.17] dex, nevertheless the slope of the metallicity term remains rathersmall. Thus, the relatively small metallicity range spanned by the LMC variables could be not responsiblefor the negligible dependence on metallicity of the RR Lyrae PL K s Z relation in the LMC.The distribution of the 23 MW RR Lyrae stars in the period-luminosity-metallicity space and the pro-jections of the PL K Z relation (Equation 7) on the log(P) − M K and M K − [Fe / H] planes are shown in Figure 3.Grey lines are the same as in Figure 2 and are described in Section 3.1.Some concern may arise since the K magnitudes of the 23 MW RR Lyrae stars used to derive Equation 7are in the Johnson photometric system (see, Cacciari et al. 1992), whereas for the LMC RR Lyrae variableswe have K s photometry in the VISTA system . To address this issue we have reported in column 10 ofTable 2 the average K s magnitudes in the 2MASS system of the 23 MW RR Lyrae stars as provided by Feastet al. (2008). The di ff erence with the Johnson K average magnitudes listed in column 9 is very small (of theorder of about 0.03 mag, on average) and definitely much smaller than individual errors in the B-W K -bandabsolute magnitudes of the MW variables (0.15-0.25 mag), or errors in the K s average apparent magnitudesof the LMC RR Lyrae stars (see column 10 of Table 1). Hence, we are confident that the di ff erence inphotometric system does not a ff ect significantly our comparison. The near-infrared PL K Z relation of the RR Lyrae stars has been studied by several authors both froma theoretical and an observational point of view. Longmore et al. (1986) pioneering work was followedby Liu & Janes (1990), Skillen et al. (1993) and Jones et al. (1996). A comprehensive analysis of the IRproperties of RR Lyrae stars was performed by Nemec et al. (1994).Some of the RR Lyrae PL K Z relations available in the literature are presented in Table 3. Bono et al.(2003) derived the semi-theoretical relation presented in the first row of Table 3. This theoretical relation hasbeen derived from an extended set of RR Lyrae nonlinear hydrodynamical models spanning a wide rangeof chemical compositions (Z from 0.0001 to 0.02, which approximately corresponds to [Fe / H] from -2.45to -0.15 dex). Catelan et al. (2004) presented a theoretical calibration of the RR Lyrae PL K Z relation basedon synthetic HB models computed for several di ff erent metallicities, fully taking into account evolutionarye ff ects besides the e ff ect of chemical composition. They derived the relation:M K = − .
353 log P + .
175 log Z − .
597 (8)By using Eqs. 9 and 10 in Catelan et al. (2004) and assuming [ α / Fe] ∼ The VISTA system is tied to the 2MASS photometry (Skrutskie et al. 2006), with the di ff erence in K s magnitude only mildlydepending on the J − K s colour, and being of the order of 3-4 mmag for the typical J − K s colour of RR Lyrae stars.
17 –Fig. 2.—
Left panel : PL K s Z relation of the 70 LMC RR Lyrae stars (Equation 3) analyzed in the paper, inthe period-luminosity-metallicity space.
Right panels : Projections of the PL K s Z relation (Equation 3) on thelog(P) − K s (top) and K s − [Fe / H] (bottom) planes. Grey lines represent lines of equal metallicity (top) andequal period (bottom). See text for the details. Uncertainties in the K s magnitude and [Fe / H] are omitted tosimplify the figure, but they are provided in Table 1.Fig. 3.—
Left panel : PM K Z relation of the 23 MW RR Lyrae stars (Equation 7) in the period-luminosity-metallicity space.
Right panel : Projections of the PM K Z relation (Equation 7) on the log(P) − M K (top panel)and M K − [Fe / H] (bottom panel) planes. Grey lines represent lines of equal metallicities (top panel) andperiods (bottom panel). See text for the details. Uncertainties in the M K magnitude and [Fe / H] are omittedto simplify the figure, but they are provided in Table 2. 18 –Table 2. Properties of 23 bright RR Lyrae stars in the Milky Way
Star RA DEC P [Fe / H] σ [Fe / H] E(B-V)
V K K s M V M K σ M V , M K (deg) (deg) (day) (dex) (dex) (mag) (mag) (mag) (mag) (mag) (mag) (mag)UU Cet 1.02135 − − − − − − − − − − − − − − − − − − − − − − − − − − − + − − − − − − − − − − − − − + − − − + + − − − + − − − V magnitude from Fernley et al. (1998a); 9) K magnitude in the Johnson system from Fernley et al.(1998a); 10) K s magnitude in the 2MASS system from Feast et al. (2008); 11) Absolute magnitude in the V passband from Fernley et al. (1994);12) Absolute magnitude in the K passband obtained from B-W analyses and corrected to p = V , K magnitudes.
19 –Dall’Ora et al. (2004) derived an empirical relation between apparent K magnitude and period for 21RRab and 9 RRc stars in the LMC globular cluster Reticulum. Del Principe et al. (2006) obtained therelation between apparent K s magnitude, metallicity and period from the analysis of RR Lyrae stars in theGalactic globular cluster ω Cen.Sollima et al. (2006) derived a PL K Z relation by analysing 538 RR Lyrae stars in 15 Galactic clustersand in the LMC globular cluster Reticulum. This relation spans the metallicity range − . < [Fe / H] < − . K magnitudes were estimated by combining Two-Micron-All-Sky-Survey (2MASS, Cutri et al.2003) photometry and literature data. The zero-point was calibrated on RR Lyrae itself, whose distancemodulus was derived using the HST trigonometric parallax measured for this star by Benedict et al. (2002).Sollima et al. (2008) presented
JKH time-series photometry of RR Lyrae and derived a new zero-point ofSollima et al. (2006)’s PL K Z relation.Borissova et al. (2009) presented near-infrared K s photometry and spectroscopically measured metal-licity for a sample of 50 field RR Lyrae stars in inner regions of the LMC. These authors had 5 measurementsin the K s passband for most of the stars in their sample and used templates from Jones et al. (1996) to fit thelight curves and derive the mean K s magnitudes. To improve statistics they added to their sample LMC RRLyrae stars from Szewczyk et al. (2008) dataset, and derived the PL K s Z relation based on the total sampleof 107 LMC variables. The zero-point was calculated using Sollima et al. (2008)’s mean K magnitude, thereddening and the trigonometric parallax of RR Lyrae.Benedict et al. (2011) recalibrated all the literature relations listed in Table 3, but Catelan et al. (2004)’sone, by fitting to equations in the form: a (logP + . + b ([Fe / H] + . + ZP , the Lutz-Kelker-Hanson-corrected absolute magnitudes of the five MW RR Lyrae stars for which HST parallaxes are available. Sincethere are concerns about Benedict et al. (2011) RR Lyrae star (namely RZ Cep) we transformed the PL K Zrelations of the LMC and MW RR Lyrae stars derived in this paper (Equations 4, 6) to the form adoptedby Benedict et al. (2011) and determined their zero-points on the basis of Benedict et al. (2011)’s
HST parallaxes but excluding RZ Cep. The zero-points based on Benedict et al. (2011) parallaxes are presentedin column 5 of Table 3, whereas the zero-point of our LMC PL K s Z relation calculated by assuming thedistance to the LMC (Subsection 3.2) and the zero-point of our MW PL K Z relation based on the B-Wstudies are (Section 3.3) are presented in Column 4.The slope in period of the RR Lyrae PL K Z relations (Column 2 of Table 3) di ff ers significantly inthe various studies. The slope we derived for the LMC RR Lyrae stars is in excellent agreement with thatderived by Del Principe et al. (2006), whereas the slope of the MW RR Lyrae PL K Z is in good agreementwith that derived by Sollima et al. (2006, 2008).The dependence on metallicity of the PL K Z relations (Column 3 of Table 3) also varies among di ff erentstudies and generally is larger in the theoretical and semi-theoretical relations. The comparison of the metal-licity dependence in the di ff erent empirical relations is complicated by the inhomogeneity of the metallicityscales adopted in these studies. Metallicities in Del Principe et al. (2006) are in the Zinn & West (1984)scale, whereas in Sollima et al. (2006, 2008) are in the Carretta & Gratton (1997) scale. In the current studyfor the LMC RR Lyrae stars we used the metallicity scale defined by Gratton et al. (2004) which is also the 20 –scale adopted by Borissova et al. (2009). As discussed in Gratton et al. (2004) this scale is systematically0.06 dex higher than Zinn & West scale. This di ff erence is small and systematic, hence should not a ff ectthe results of this comparison. Finally, for the MW RR Lyrae stars we used the metallicities measured byClementini et al. (1995). Because the spectroscopic [Fe / H] values in Clementini et al. (1995) are derivedfrom high dispersion spectra analyzed using standard reduction procedures, the derived metallicities are onthe scale of the high dispersion spectra (i.e., the Carretta et al. 2009 scale) and could be transformed to Zinn& West scale using the relations provided in Carretta et al. (2009).The slope in metallicity of the PL K Z relation based on the LMC RR Lyrae stars is the smallest amongthe various studies listed in Table 3 and it is close to Borissova et al. (2009)’s slope. This is consistent withthe two studies both involving LMC variables and using exactly the same metallicity scale. The slope inmetallicity we found for the MW RR Lyrae stars is larger than that of the LMC RR Lyrae stars and, in spiteof the di ff erence in metallicity scales, it is very close to the slope obtained by Sollima et al. (2006) for RRLyrae stars in globular clusters. However, taken at face value, the metallicity slopes of the empirical relationsin Table 3 appear to be all rather small and in agreement to each other within the relative uncertainties, thusgenerally suggesting a mild dependence the RR Lyrae PL K Z independently on the specific environment.
4. Gaia observation of RR Lyrae stars in the Milky Way
The Gaia astrometric satellite will revolutionise many fields of astronomy (Perryman et al. 2001). Ofparticular importance will be its catalogue of trigonometric parallaxes for more than one billion stars, withastrometric precision down to µ as level. Due to Gaia’s constant observation of the sky over the five yearnominal mission, Gaia will repeatedly observe all stars brighter than its limiting magnitude, with an averageof 70 observations per star. This will also make it possible for Gaia to discover and characterise many typesof variables, including RR Lyrae stars and Cepheids.Gaia is observing in the broad visual band G (Jordi et al. 2010) for its astrometric measurements, and istherefore not ideal for characterising the RR Lyrae PLZ relation, which exists only in the infrared passbands.However, since Gaia will provide accurate parallaxes for an expected tens of thousands of MW RR Lyraestars, it could serve as a perfect tool for the determination of the zero-point of the PL K s Z relation througha combination with external datasets. As it was discussed in Subsection 3.2.2, the current largest limitingfactor in zero-point calibration of PL K s Z and M V − [Fe / H] relations is the lack of a reliable and statisticallysignificant sample of parallax measurements. The current state of the art is the sample of five RR Lyraeparallaxes from Benedict et al. (2011) using the
HST . Gaia will improve this situation by several orders ofmagnitude in both precision and numbers of objects. Moreover, the distance to the LMC will be determinedthrough the combination of Gaia parallaxes for a large sample of LMC bright stars, hence, a zero-point ofthe PL K s Z relation based on the distance to the LMC will be obtained with a high precision. 21 –Table 3. PL K s Z relations from the literature
Relation a b ZP from the original relation ZP from Benedict et al. (2011) Theoretical or semi-theoretical relations
Bono et al. (2003) -2.101 0 . ± . − . ± . − . ± . Empirical relations
Dall’Ora et al. (2004) − . ± .
09 - - − . ± . , − . ± .
12 0 . ± .
04 - − . ± . − . ± .
04 0 . ± . − . ± . − . ± . Sollima et al. (2008) − . ± .
04 0 . ± . − . ± . − . ± . , − . ± .
17 0 . ± .
07 -1.05 − . ± . , − . ± .
25 0 . ± . − . ± . − . ± . This paper (MW) − . ± .
36 0 . ± . − . ± . − . ± . Zero-point of the original relation from the literature in the form: a logP + b [Fe / H] + ZP Zero-point of the relation in the form: a (logP + . + b ([Fe / H] + . + ZP , as recalibrated by Benedict et al. (2011) Near-infrared photometry in the K s band. Metallicity is on the Zinn & West (1984) metallicity scale Metallicity is on the Carretta & Gratton (1997) metallicity scale Zero-point was calibrated by Benedict et al. (2011) neglecting the metallicity term Metallicity on the scale adopted by Gratton et al. (2004), which is , on average, 0.06 dex higher than Zinn & West (1984) scale Metallicities from Clementini et al. (1995), they are on the scale of the high dispersion spectra (i.e., the Carretta et al. 2009) Zero-point was calibrated by us considering only four RR Lyrae stars (XZ Cyg, UV Oct, SU Dra and RR Lyr) from Benedictet al. (2011) and excluding RZ Cep, since there are concerns about the parallax of this star
22 –
Using parallax data for calibration of a PL relation is complicated by the presence of statistical biases(e.g. Lutz & Kelker 1973) and sample selection e ff ects (e.g. Malmquist 1936). Non-linear transformationson the parallax cause a highly asymmetric uncertainty on the absolute magnitude when calculated usingparallax and apparent magnitude information via the relation: m − M = − − (cid:36) ), where (cid:36) is theparallax. Additionally, stars with a negative parallax measurement can not be used to calculate an absolutemagnitude, though they do contain information. For these reasons, calculating an absolute magnitude foreach star and fitting a PL relation directly to period and absolute magnitude leads to a biased result.An unbiased solution can be achieved through modelling the data and inferring the slope and zero-pointof the relation via statistical methods. For a catalogue of N stars we can define x = ( x , x , ..., x N ) wherethe vector ( x i = m , l , b , P , (cid:36), A ) describes the observed data on each object. P is the period, m the apparentmagnitude, l and b the position, (cid:36) the parallax, and A the extinction). We can additionally define that thevector ( x = m , l , b , P , r , A ) gives the ‘true’ underlying object properties.We assume that the stars follow a PL relation of the form M = ρ log P + δ , although this can be changedto include other terms, such as metallicity, as needed. We can therefore model the true absolute magnitudesof the population as being Normally distributed around the PL relation, with the dispersion describing theintrinsic scatter on the relation: ϕ M ( x | ρ, δ, σ PL ) = σ PL √ π e − . (cid:18) M − ( ρ log P + δ ) σ PL (cid:19) (9)where σ PL is the intrinsic dispersion of the PL relation. The parameters ρ and δ are the slope and zero-pointof the PL relation, which are to be found.The true absolute magnitude is calculated through: M = m + (cid:36) ) + − A (10)The observations are Normally distributed around the true values with a standard deviation given bythe formal error on the measurement: E ( x | x ) = (cid:15) (cid:36) (cid:15) m (cid:15) A (2 π ) / e − . (cid:16) (cid:36) − (cid:36) (cid:15)(cid:36) (cid:17) e − . (cid:16) m − m (cid:15) m (cid:17) e − . (cid:18) A − A (cid:15) A (cid:19) δ ( l , b , P ) (11)Assuming negligible errors on the position and period, the observations are described by a delta function.The terms (cid:15) (cid:36) , (cid:15) m , (cid:15) A are the formal errors on the parallax, magnitude and extinction.With the above models defined, the joint probability density function for the observations is: P ( x i | ρ, δ, σ PL ) = S ( x ) (cid:90) ∀ x ϕ M ( x | ρ, δ, σ PL ) E ( x | x ) d x (12)the ‘true’ parameters x are never known and so these values are marginalised through integration. Theterm S ( x ) is the selection function, which takes the probability of observing a star into account, given the 23 –properties of the star and the instrument’s observational capabilities. To take the fact that Gaia is a magnitudelimited sample into account, a step function is used with S ( x ) = , if G < . , otherwise . (13)The Maximum Likelihood Estimation of the parameters are found by maximising equation 12 by vary-ing the parameters ( α , ρ , σ PL ). This formulation avoids non-linear transformations on error e ff ected data,and includes a selection function which avoids the Malmquist bias. In order to check the application of the method defined in Section 4.1 we have used the sample of 23RR Lyrae stars in the MW discussed in Section 3.3 (see also Table 2). In order to investigate the performanceof the Gaia satellite and the contents of the end-of-mission catalogue, Gaia’s Data Processing and AnalysisConsortium (DPAC) has a group working on the simulation of several aspects of the Gaia mission. Onemajor product of this work is the Gaia Object Generator (GOG; Luri et al 2014), designed to simulateboth individual Gaia observations and the full contents of the end-of-mission catalogue. GOG includes afull mathematical description of the nominal performance of the Gaia satellite, and is therefore capable ofdetermining the expected precision in astrometric, photometric and spectroscopic observations. In general,the precision depends on the apparent magnitude of the star, its colour, and its sky position, which a ff ectsthe number and type of observations made (due to the Gaia scanning law).To obtain a distance for each RR Lyrae star from the sample, we use: M K = − .
53 log P − .
95 (14)as determined in Equation 7 to obtain an absolute magnitude (neglecting the metallicity term for simplicity).We then determine a distance by combining this absolute magnitude with the apparent magnitude and ex-tinction as defined above. Colour information as (V-I) is obtained from the Hipparcos catalogue (Perrymanand ESA 1997) where available. The apparent magnitude, position, colour, and period data form the basisof a synthetic catalogue of RR Lyrae stars, along with the distance obtained from the PM K Z relation, and isused as the input catalogue of ‘true’ parameters for GOG.GOG then creates simulated Gaia observations for our sample. We take the PM K Z elation (Equation 14)as true, as a study of the possible precision in PM K Z calibration after the Gaia data will become available.Using the fitting method described in Section 4.1 to the data including the simulated parallax observa-tions and simulated errors applied to parallax and apparent magnitude, we find a PM K relation of: M K = ( − . ± . P + ( − . ± .
01) (15) 24 –Comparison of these results to the input relation shows very good agreement. It proves that the fittingprocedure given in Section 4.1 is accurate and unbiased. When Gaia parallaxes will become available forthe much larger sample of RR Lyrae stars, we will apply the described method to fit the PL K Z relation ofRR Lyrae variables in the MW. Moreover, precise distance to the LMC obtained from the combination ofGaia parallaxes for a large sample of the bright LMC stars, will allow us to calibrate zero-point of the PL K s Zrelation based on the LMC RR Lyrae stars. This provides a flavor of what will be possible to achieve withGaia parallaxes.
5. Summary
We studied a sample of 70 RR Lyrae stars in the LMC, for which multi-epoch K s photometry from theVMC survey, precise periods from the OGLE III catalogue and spectroscopically determined metallicities(Gratton et al. 2004), are available. There are 13 epoch data in the K s band for all stars in the sample, thatallowed us to determine mean K s magnitudes with a great accuracy.Specifically for this work we developed a fitting approach. This method has several advantages com-pared to the Minimum-Least-Squares fitting, such as taking into account potentially significant intrinsicdispersion of the data, non-negligible errors in two dimensions and the possibility of inaccuracies in theformal error estimates. We used this method to derive the PL K s Z relation of the 70 RR Lyrae stars in theLMC. Potentially the method could be used to fit any other sample of data.The zero-point of the derived PL K s Z relation was estimated in two di ff erent ways: (i) by assuming thedistance to the LMC determined by Pietrzy´nski et al. (2013); (ii) by applying HST parallaxes of four MWRR Lyrae stars by Benedict et al. (2011). The zero-point derived using the MW RR Lyrae stars is 0.2 maglarger and, consequently, gives a longer distance to the LMC: ( m − M ) = . ± .
10 mag. In future studieswe suggest to use the relation based on the precise distance to the LMC: M K s = ( − . ± . P + (0 . ± . / H] − (1 . ± .
01) (16)We found a negligible dependence of the M K s on metallicity, which could be caused by the relativelysmall range in metallicity covered by the LMC RR Lyrae stars. Thus, we applied the fitting approach to 23RR Lyrae stars in the MW, for which absolute M K and M V magnitudes are known from Baade-Wesselinkstudies. We derived the PL K s Z relation for MW RR Lyrae stars in the form: M K = ( − . ± . + (0 . ± . / H] − (0 . ± .
14) (17) 25 –Even though the metallicities of the MW RR Lyrae stars span a wide range [-2.5; 0.17] dex, the de-pendence on metallicity is relatively small and consistent, within the errors, with the slope in metallicityfound for the LMC RR Lyrae variables. We concluded that the small range of metallicities doesn’t cause thenegligible dependence of the M K on metallicity for the LMC RR Lyrae stars.To solve the problem of the PL K s Z zero-point, a large sample of RR Lyrae stars with precisely de-termined parallaxes is necessary. A great contribution to this field is expected by the Gaia satellite. Byusing GOG we simulated Gaia parallaxes of 23 MW RR Lyrae stars with observational errors. We present amethod for the calibration of the PL relation which avoids several of the problems which arise when usingparallax data. The method was tested by deriving the PL K s relation based on the simulated Gaia parallaxes.When combined with metallicity and photometry from other sources and a statistical tool such as the onedeveloped in the present study, the extraordinary large sample of Gaia parallaxes for RR Lyrae stars willallow us to estimate these relations with unprecedented precision. Acknowledgements
This work was supported by the Gaia Research for European Astronomy Training (GREAT-ITN) MarieCurie network, funded through the European Union Seventh Framework Programme [FP7 / ff ofor the development and maintenance of the GRATIS software. We thank the members of the OGLE teamfor making public their catalogues. A. Bayesian fitting approach
This method is based on the prescription of Hogg et al. (2010), extended into three dimensions andimplemented in Python using a Markov Chain Monte Carlo sampler to obtain parameter estimates alongwith their complete posterior PDF. Initially, we model the data as being drawn from a thin plane defined by: f ( x , y ) = A x + B y + C (A1)where A is the slope in the x axis, B is the slope in the y axis, and C is the intercept. In this initial model weassume that we have data in three axis, x , y and z , with errors only in the z axis.In this model, given an independent position ( x i , y i ), an uncertainty σ zi , slopes A and B , and an intercept C , the frequency distribution p ( z i | x i , y i , σ zi , A , B , C ) for z i is p ( z i | x i , y i , σ zi , A , B , C ) = (cid:113) π σ zi exp − [ z i − A x i − B y i − C ] σ zi (A2) 26 –Therefore the likelihood is defined as: L = N (cid:89) i = p ( z i | x i , y i , σ zi , A , B , C ) (A3)Taking the logarithm, ln L = K − N (cid:88) i = [ z i − A x i − B y i − C ] σ zi (A4)which is e ff ectively the least-squares solution. K is a normalisation coe ffi cient. Returning to Bayes rule it ispossible to define: p ( A , B , C |{ z i } Ni = , I ) = p ( { z i } Ni = | A , B , C , I ) p ( A , B , C | I ) p ( { z i } Ni = | I ) (A5) { z i } Ni = is all the data z i . I is all of the information of x and y , { x i , y i } Ni = , along with the formal errors { σ x i , σ y i , σ z i } Ni = , plus any other prior information which may be available. In our case we use uninformative(uniform) priors, making our inference method analogous to Maximum Likelihood Estimation. A.1. Multiple errors, no dispersion
As in this case there exist errors in more than one axis, they can be put together into a covariance tensor S i S i ≡ σ xi σ x y i σ xzi σ x y i σ y i σ y zi σ xzi σ y zi σ zi (A6)With errors in several dimensions, our observed data point ( x i , y i , z i ) could have been drawn from any truepoint along the plane ( x , y , z ). Making the probability of the data, given the model and the true position: p ( x i , y i , z i | S i , x , y, z ) = π √ det( S i ) exp (cid:32) −
12 [ Z i − Z ] (cid:62) S i − [ Z i − Z ] (cid:33) (A7)where we have implicitly made column vectors Z = x y z ; Z i = x i y i z i . (A8)In only two dimensions (e.g. y and z ), the slope (e.g. B ) can be described by a unit vector ˆ u orthogonal tothe line or linear relation (at any x ): ˆ u = √ + B − B = − sin θ cos θ (A9) 27 –where the angle θ = arctan B is made between the line and the y axis. The orthogonal displacement ∆ i ofeach data point ( y i , z i ) from the line is given by ∆ i = ˆ u (cid:62) y i z i − C cos θ (A10)Instead of extending fully into three dimensions, we will assume a negligible error in x (which will be theperiod, so has justifiably higher precision). The value of x can therefore be input directly into ∆ i withoutworrying about the interplay between the other parameters. ∆ i = ˆ u (cid:62) y i z i − ( C cos θ + A x i ) (A11)Assuming negligable errors in x also redefines the covariance matrix of the errors as: S i ≡ σ y i σ y zi σ y zi σ zi (A12)Similarly, each data point’s covariance matrix S i projects down to an orthogonal variance Σ i given by Σ i = ˆ u (cid:62) S i ˆ u (A13)and then the log likelihood for ( A , B , C ) or ( A , θ, C cos θ ) can be written asln L = K − N (cid:88) i = ∆ i Σ i (A14)where K is some constant. This likelihood can be maximized to find A , B and C . A.2. Dispersion
The final step is to introduce an intrinsic variance in the line, V, orthogonal to the line.According to Hogg et al. (2010), each data point can be treated as being drawn from a projecteddistribution function that is a convolution of the projected uncertainty Gaussian, of variance Σ i definedabove, with the Gaussian intrinsic scatter of variance V . Therefore the likelihood becomes:ln L = K − N (cid:88) i =
12 ln( Σ i + V ) − N (cid:88) i = ∆ i Σ i + V ] (A15)where again K is a constant, everything else is defined as above. We then solve for A , B , C , and V bymaximising the log likelihood. The optimisation is performed using the adaptive MCMC sampler EMCEEdeveloped by Foreman-Mackey et al. (2013). Any optimisation algorithm (e.g. Nelder-Mead, Powell, etc.)will find the maximum of the log likelihood. MCMC was chosen due to the evaluation of the full posteriorPDF of the parameters, which is useful for the determination of formal errors. 28 – B. Metallicities for the MW RR Lyrae stars analyzed with the B-W method
In this Appendix we provide a summary of spectroscopic metal abundances ([Fe / H]) derived for thefield RR Lyrae stars analyzed with the B-W technique, by studies mainly based on high resolution spectro-scopic material. Exceptions are the values with reference 1, 2 and 4 that come from compilations whichinclude metallicities measured from low resolution spectroscopic data and photometric indices (see discus-sion in Section 4.2 of Cacciari et al. 1992).By applying our fitting approach we derived the PM K Z relation for 23 MW RR Lyrae stars describedin Section 3.3 adopting as an alternative the metallicity values from Lambert et al. (1996): M K = ( − . ± . + (0 . ± . / H] − (1 . ± .
15) (B1)The intrinsic dispersion of the relation is found to be 0.007 mag. The RMS deviation of the data aroundthe relation, neglecting the intrinsic dispersion, is 0.090 mag. The slope in logP of the relation based onClementini et al. (1995)’s metallicities (Eq. 7) di ff ers from the slope obtained using Lambert et al. (1996)’smetallicities (Eq. B1), however the values are consistent within the errors. 29 –Table 4. Literature metallicities of the MW RR Lyrae starsanalyzed with the B-W method Star [Fe / H] σ [Fe / H] Reference(dex) (dex)UU Cet − − − ∆ S and ∆ s value adopted by (7) − ∆ S in (7) and ∆ s value adopted by (5) − − − − − − = − − − + − − − − − − − − − ∆ S and ∆ s value adopted by (7) − − − − − − − = − − − − − −
30 –Table 4—Continued
Star [Fe / H] σ [Fe / H] Reference(dex) (dex)AR Per − ∆ S and ∆ s value adopted in (5) − − − = − − − − − − ∆ S and ∆ S value adopted in (5) − ∆ S and ∆ S value adopted in (7) − − − ∆ S and ∆ S value adopted in (5) − ∆ S and ∆ S value adopted in (7) − − − ∆ S and ∆ S value adopted in (5) − − − = − − − − − − ∆ S and ∆ S in (5) − − − = − − − ∆ S and ∆ S value in (5)
31 –Table 4—Continued
Star [Fe / H] σ [Fe / H] Reference(dex) (dex) − − − = − − − − ∆ S and ∆ S value in (5) − − − + errors from us − ∆ S and ∆ S value in (7) − − − ∆ S and ∆ S value in (5) − ∆ S and ∆ S value in (7) − − − ∆ S and ∆ S value in (5) − − − = − − − − − − − ∆ S and ∆ S value in (5) − − − = − − −
32 –Table 4—Continued
Star [Fe / H] σ [Fe / H] Reference(dex) (dex)SW Dra − ∆ S and ∆ S value in (5) − − − ∆ S and ∆ S value in (7) − − − ∆ S and ∆ S value in (5) − − − − ∆ S in (7) and ∆ S value in (5) − − − ∆ S and ∆ S value in (5) − ∆ S in (7) and ∆ S value in (5) − − − ∆ S and ∆ S value in (5) − − − = − − − − − − − − = − − − − −
33 –Table 4—Continued
Star [Fe / H] σ [Fe / H] Reference(dex) (dex)V445 Oph + + − ∆ S in (7) and ∆ S value in (7) + − − − ∆ S and ∆ S value in (5) − ∆ S and ∆ S value in (7) − − − ∆ S and ∆ S value in (5) − − − = − − ∆ S and ∆ S value in (5) − ∆ S and ∆ S value in (7) − − / H] from Table 1 in Fernley et al. (1998a); (2) [Fe / H] values fromFeast et al. (2008); (3) [Fe / H] values from Nemec et al. (2013) using the values fromthe VWA analysis as they are listed in column (9) of Table 7 in that paper. They arethe average of the FeI and FeII abundances. (4) [Fe / H] values from Table 11 ofCacciari et al. (1992); (4*) [Fe / H] values from Table 16 of Skillen et al. (1993);(5) [Fe / H] values from abundance analysis of high resolution spectra performed byClementini et al. (1995). Values are the average of the FeI and FeII measurementsadopting for the solar abundance: log (cid:15) (FeI) = (cid:15) (FeII) = / H] values obtained fromClementini et al. (1995) re-calibration of the ∆ S index. (6) [Fe / H] values from Ta-ble 7 of Pancino et al. (2015). Abundances are from FeI averaging values fromdi ff erent spectra of the same star as described at the end of the paper Section 4.4.The error for the abundance of TU Uma is likely a typo, according to the paper Ta-ble 6 likely should be 0.14 dex; (7) Metal abundances from Lambert et al. (1996).
34 – [Fe / H] values were derived from the photometric determinations in the paper Ta-ble 3 adopting for the sun log (cid:15) (Fe) = / H] values obtained from Lambert et al. (1996) re-calibration of the ∆ S index (Equation 3 in that paper) which was derived by theseAuthors using the FeII abundances and ∆ S from Blanco (1992). Lambert et al.(1996) does not provide errors for the metallicities from ∆ S, hence we adopted anerror of 0.16 dex, as done by Clementini et al. (1995) for their metallicities from ∆ S;(8) [Fe / H] values from Table 10 of Haschke et al. (2012); (9) [Fe / H] values abun-dance analysis performed by Wallerstein & Huang (2010), no errors are provided;(10) [Fe / H] values from For et al. (2011) obtained as the average weighted by errorsof the values in the paper Table 5. Errors are the sum in quadrature of the individualerrors in Table 5 divided by the square root of N (with N number of measurements,i.e.: 11 for WY Ant and 17 for RV Oct). The average value from FeI for WY Antpublished in Table 11 of For et al. (2011). is − − / H] values from Fernley & Barnes(1996).They are from FeII with values taken from the paper Table 4b. Those fromFeI are taken from the paper Table 4a. Errors are estimated by the authors to be of ± / H].
35 –
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