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An eclipsing-binary distance to the Large MagellanicCloud accurate to two per centG. Pietrzynski1,2, D. Graczyk1, W. Gieren1, I. B. Thompson3, B. Pilecki1,2, A. Udalski2, I. Soszynski2, S. Kozłowski2, P. Konorski2,K. Suchomska2, G. Bono4,5, P. G. Prada Moroni6,7, S. Villanova1, N. Nardetto8, F. Bresolin9, R. P. Kudritzki9, J. Storm10,A. Gallenne1, R. Smolec11, D. Minniti12,13, M. Kubiak2, M. K. Szymanski2, R. Poleski2,14, Ł. Wyrzykowski2, K. Ulaczyk2,P. Pietrukowicz2, M. Gorski2 & P. Karczmarek2

In the era of precision cosmology, it is essential to determine theHubble constant to an accuracy of three per cent or better1,2. Atpresent, its uncertainty is dominated by the uncertainty in thedistance to the Large Magellanic Cloud (LMC), which, being oursecond-closest galaxy, serves as the best anchor point for the cosmicdistance scale2,3. Observations of eclipsing binaries offer a uniqueopportunity to measure stellar parameters and distances preciselyand accurately4,5. The eclipsing-binary method was previouslyapplied to the LMC6,7, but the accuracy of the distance resultswas lessened by the need to model the bright, early-type systemsused in those studies. Here we report determinations of the dis-tances to eight long-period, late-type eclipsing systems in the LMC,composed of cool, giant stars. For these systems, we can accuratelymeasure both the linear and the angular sizes of their componentsand avoid the most important problems related to the hot, early-type systems. The LMC distance that we derive from these systems(49.97 6 0.19 (statistical) 6 1.11 (systematic) kiloparsecs) is accur-ate to 2.2 per cent and provides a firm base for a 3-per-cent deter-mination of the Hubble constant, with prospects for improvementto 2 per cent in the future.

The modelling of early-type eclipsing binary systems consisting ofhot stars is made difficult by the problem of obtaining accurate fluxcalibrations for early-type stars and by the degeneracy between thestellar effective temperatures and reddening8,9. As a result, the dis-tances determined from such systems are of limited (,5–10%) accu-racy. More-accurate distances can be obtained using binary systemscomposed of cool stars; however, among the frequent dwarf stars in theLMC such systems are too faint to be accurately analysed with present-day telescopes.

The Optical Gravitational Lensing Experiment (OGLE) has beenmonitoring around 35 million stars in the field of the LMC for morethan 16 years10. Using this unique data set, we have detected a dozenextremely scarce, very long-period (60–772-d) eclipsing binary systemscomposed of intermediate-mass, late-type giants located in a quietevolutionary phase on the helium-burning loop11 (SupplementaryTable 1). These well-detached systems provide an opportunity to usethe full potential of eclipsing binaries as precise and accurate distanceindicators, and to calibrate the zero point of the cosmic distance scalewith an accuracy of about 2% (refs 5, 12, 13).

To do so, we observed eight of these systems (Fig. 1 shows oneexample) over the past 8 yr, collecting high-resolution spectra with theMIKE spectrograph at the 6.5-m Magellan Clay telescope at the LasCampanas Observatory and with the HARPS spectrograph attachedto the 3.6-m telescope of the European Southern Observatory on La

Silla, together with near-infrared photometry obtained with the 3.5-mNew Technology Telescope located on La Silla.

The spectroscopic and OGLE V- and I-band photometric observa-tions of the binary systems were then analysed using the 2007 versionof the standard Wilson–Devinney code14,15, in the same way as in ourrecent work on a similar system in the Small Magellanic Cloud9.Realistic errors in the derived parameters of our systems were obtainedfrom extensive Monte Carlo simulations (Fig. 2). The astrophysicalparameters of all the observed eclipsing binaries were determined withan accuracy of a few per cent (Supplementary Tables 2–9).

For late-type stars, we can use the very accurately calibrated (2%)relation between their surface brightness and V2K colour to deter-mine their angular sizes from optical (V) and near-infrared (K) pho-tometry16. From this surface-brightness/colour relation (SBCR), wecan derive angular sizes of the components of our binary systemsdirectly from the definition of the surface brightness. Therefore, thedistance can be measured by combining the angular diameters of thebinary components derived in this way with their corresponding lineardimensions obtained from the analysis of the spectroscopic and pho-tometric data. The distances measured with this very simple but accur-ate one-step method are presented in Supplementary Table 12. Thestatistical errors in the distance determinations were calculated byadding quadratically the uncertainties in absolute dimensions, V2Kcolours, reddening and the adopted reddening law. The reddeninguncertainty contributes very little (0.4%) to the total error11,17. A sig-nificant change in the reddening law (from Rv 5 3.1 to 2.7, where Rv isthe ratio of total to selective absorption) causes an almost negligiblecontribution, at the level of 0.3%. The accuracy of the V2K colour forall components of our eight binary systems is better than 0.014 mag(0.7%). The resulting statistical errors in the distances are very close to1.5%, and are dominated by the uncertainty in the absolute dimen-sions. By calculating a weighted mean from the individual distances tothe eight target eclipsing binary systems, we obtain a mean LMCdistance of 49.88 6 0.13 kpc.

Our distance measurement might be affected by the geometry anddepth of the LMC. Fortunately, the geometry of the LMC is simple andwell studied18. Because nearly all the eclipsing systems are located veryclose to the centre of the LMC and to the line of nodes (Fig. 3), wefitted the distance to the centre of the LMC disk plane, assuming itsspatial orientation18. We obtained an LMC barycentre distance of49.97 6 0.19 kpc (Fig. 4), which is nearly identical to the simpleweighted mean value. This shows that the geometrical structureof the LMC has no significant influence on our present distancedetermination.

1Universidad de Concepcion, Departamento de Astronomıa, Casilla 160-C, Concepcion, Chile. 2Warsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland. 3Carnegie Observatories,813 Santa Barbara Street, Pasadena, California 91101-1292, USA. 4Dipartimento di Fisica Universita di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Rome, Italy. 5INAF-OsservatorioAstronomicodiRoma, Via Frascati 33,00040 MontePorzio Catone, Italy. 6Dipartimento di FisicaUniversita di Pisa, LargoB. Pontecorvo2, 56127Pisa, Italy. 7INFN,SezionediPisa, ViaE. Fermi2, 56127 Pisa,Italy. 8Laboratoire Lagrange, UMR7293, UNS/CNRS/OCA, 06300 Nice, France. 9Institute for Astronomy, 2680 Woodlawn Drive, Honolulu, Hawaii 96822, USA. 10Leibniz Institute for Astrophysics, An derSternwarte 16, 14482 Postdam, Germany. 11Nicolaus Copernicus Astronomical Centre, Bartycka 18, 00-716 Warszawa, Poland. 12Departamento de Astronomıa y Astrofısica, Pontificia UniversidadCatolica de Chile, Vicuna Mackenna 4860, Casilla 306, Santiago 22, Chile. 13Vatican Observatory, V00120 Vatican City, Italy. 14Ohio State University, 140 West 18th Avenue, Columbus, Ohio 43210, USA.

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The systematic uncertainty in our distance measurement comesfrom the calibration of the SBCR and the accuracy of the zero pointsin our photometry. The root mean squared scatter in the current SBCRis 0.03 mag (ref. 13), which translates to an accuracy of 2% in therespective angular diameters of the component stars. Because the sur-face brightness depends only very weakly on metallicity16,17, this effectcontributes to the total error budget at the level of only 0.3% (ref. 9).Both optical (V) and near-infrared (K) photometric zero points areaccurate to 0.01 mag (0.5%). Combining these contributions quadra-tically, we determine a total systematic error of 2.1% in our presentLMC distance determination.

The LMC contains significant numbers of different stellar distanceindicators, and, being the second-closest galaxy to our own, offers us aunique opportunity to study these indicators with the utmost pre-cision. For this reason, this galaxy has an impressive record of severalhundred distance measurements2,3,19. Unfortunately, almost all LMC

distance determinations are dominated by systematic errors, with eachmethod having its own sources of uncertainties. This prevents a cal-culation of the true LMC distance by simply taking the mean of the

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Figure 2 | Error estimation of the distance for one of our target binarysystems. The reduced x2 map (goodness of fit) for the system OGLE-LMC-ECL-15260 showing the dependence of the goodness of fit to the V-band andI-band light curves on the distance modulus of the primary component. Thismap was obtained from 110,000 models computed with the Wilson–Devinneycode14,15 within a broad range of the primary and secondary radii, R1 and R2, theorbital inclination, i, the phase shift, w, the secondary’s temperature, T2, andalbedo, A2. In each case the distance, d, was calculated from the V-band surface-brightness/colour (V2K) relation16 and translated into distance modulus usingm{M~5log(d){5, where m and M are the observed and absolutebrightnesses, respectively. The horizontal lines correspond to the standard-deviation limits of the derived distance modulus, of 18.509 mag (50.33 kpc): 1s,2s and 3s, from bottom to top.

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Figure 1 | Change of the brightness of the binary system OGLE-LMC-ECL-06575 and the orbital motion of its components. a, The main panel shows theorbital motion of the two binary components in the system OGLE-LMC-ECL-06575. Filled circles, primary component; open circles, secondary component.The top panel shows the residuals of the fit (see below): observed radial velocities(O) minus the computed radial velocities (C). b, The main panel shows theI-band light curve (1,200 epochs collected over 16 yr) of OGLE-LMC-ECL-06575 together with the solution obtained using the Wilson–Devinney code. P,orbital period. The top panel shows the residuals of the observed magnitudesfrom the computed orbital light curve. All individual radial velocities weredetermined by the cross-correlation method using appropriate template spectraand spectra from the Magellan Inamori Kyocera Echelle (MIKE) and HighAccuracy Radial Velocity Planet Searcher (HARPS) spectrographs, yielding inall cases velocity accuracies better than 200 m s21 (error bars smaller than theplotted symbols). The orbit (mass ratio, systemic velocity, velocity amplitudes,eccentricity and periastron passage) was fitted with a least-squares method to themeasured velocities. The resulting parameters are presented in SupplementaryTables 2–9. The spectroscopic orbits, light curves and solutions for theremaining systems are of similar quality.

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Figure 3 | Location of the observed eclipsing systems in the LMC. Most ofour eight systems (circles) are located quite close to the geometrical centre of theLMC and to the line of nodes (line), resulting in very small corrections to theindividual distances for the geometrical extension of this galaxy (in all casessmaller than the corresponding statistical error in the distance determination).The effect of the geometrical structure of the LMC on the mean LMC distancereported here is therefore negligible. The background image has a field of view of8u3 8u and is taken from the All Sky Automated Survey wide-field sky survey28.

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reported distances resulting from different techniques. Our presentLMC distance measurement, of 49.97 6 0.19 (statistical) 6 1.11 (sys-tematic) kpc (that is, a true distance modulus of 18.493 6 0.008 (statis-tical) 6 0.047 (systematic) mag), agrees well, within the combinederrors, with the most recent determinations of the distance to theLMC19,20,21. Our purely empirical method allows us to estimate bothstatistical and systematic errors in a very reliable way, which is normallynot the case, particularly in distance determinations relying in part ontheoretical predictions of stellar properties and their dependences onenvironment. In particular, our result provides a significant improve-ment over previous LMC distance determinations made using observa-tions of eclipsing binaries7,22. Those studies were based on early-typesystems for which no empirical SBCR is available, and so had to rely ontheoretical models to determine the effective temperature. Our presentdetermination is based on many (eight) binary systems and does notresort to model predictions.

The classical approach to deriving the Hubble constant (H0) consistsof deriving an absolute calibration of the Cepheid period–luminosityrelation (CPLR), which is then used to determine the distances tonearby galaxies containing type Ia supernovae23. These supernovaeare excellent standard candles that extend out to the region of unper-turbed Hubble flow once their peak brightnesses are calibrated thisway, and provide the most accurate determination of H0 (ref. 24). Analternative approach to calibrate the CPLR with Cepheids in the LMCis to calibrate it in our own Milky Way galaxy using Hubble SpaceTelescope (HST) parallax measurements of the Cepheids nearest to theSun25. However, the CPLR that results from that approach is lessaccurate for two reasons. First, the Cepheid sample with HST paral-laxes is very small (ten stars) relative to the Cepheid sample in the LMC(2,000 stars), which can be used to establish the CPLR once the LMCdistance is known. Second, the average accuracy of the HST Cepheidparallaxes is 8% (ref. 25) and suffers from systematic uncertainties thatare not completely understood, including Lutz–Kelker bias26,27.Therefore, at present the preferred route to determining the Hubbleconstant is that which uses the highly abundant LMC Cepheidpopulation whose mean distance is now known, with the result of thiswork, to 2.2%. This result reduces the uncertainty in H0 to a very firmlyestablished 3%.

We have good reason to believe that there is significant room toimprove on the high-accuracy LMC distance determination reportedhere, by improving the calibration of the SBCR for late-type stars12,16,which is the dominant source of systematic error in our present deter-mination. This work is in progress, and a determination of the distanceto the LMC accurate to 1% should be possible once the SBCR cal-ibration is refined. This will have a corresponding effect on furtherimproving the accuracy of H0. This is similar to the accuracy of thegeometrical distance to the LMC, which is to be delivered by Gaiamission in around 12 years from now. The eclipsing-binary techniquewill then probably provide the best opportunity to check on the futureGaia measurements for possible systematic errors.

Received 19 August; accepted 20 December 2012.

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Supplementary Information is available in the online version of the paper.

Acknowledgements We acknowledge financial support for this work from the BASALCentro de Astrofısica y Tecnologias Afines (CATA), the Polish Ministry of Science, the

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Figure 4 | Consistency among the distance determinations for the targetbinary systems. Distance offsets, Dd, between our particular eclipsing binarysystems and the best-fitted LMC disk plane, plotted against the angular distanceof the systems from the LMC centre, r. The identification of the systems is thesame as in Fig. 3. The error bars correspond to 1s errors. We assumed themodel of the LMC from ref. 18. We fitted one parameter, the distance to thecentre of the LMC (RA 5 5 h 25 min 06 s; dec. 5 269u 479 0099), using a fixedspatial orientation of the LMC disk (inclination, i 5 28u; position angle of thenodes, h 5 128u). The resulting distance to the LMC barycentre is49.97 6 0.19 kpc, with a reduced x2 value very close to unity.

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Foundation for Polish Science (FOCUS, TEAM), the Polish National Science Centre andthe GEMINI-CONICYT fund. The OGLE project has received funding from the EuropeanResearch Council ‘Advanced Grant’ Program. We thank the staff astronomers at LasCampanas andESO La Silla, who provided expert support in data acquisition. We thankJ. F. Gonzalez for making the IRAF scripts rvbina and spbina available to us. We alsothank O. Szewczyk and Z. Kołaczkowski for their help with some of the observations.

Author Contributions G.P.: photometric and spectroscopic observations andreductions. D.G.: spectroscopic observations, modelling and data analysis. W.G.:observations and data analysis. I.B.T.: observations, RV determination, data analysis.

B.P.: spectroscopic observations and reductions, RV measurements. A.U., I.S. and S. K.:optical observations and data reductions. P.K., K.S., M.K., M.K.S., R.P., Ł.W., K.U., P.P.,M.G. and P.K.: observations. G.B., P.G.P.M., N.N., F.B., R.P.K., J.S., A.G. and R.S.: dataanalysis. S.V.: analysis of the spectra. G.P. and W.G. worked jointly to draft themanuscript with all authors reviewing and contributing to its final form

Author Information Reprints and permissions information is available atwww.nature.com/reprints. The authors declare no competing financial interests.Readers are welcome to comment on the online version of the paper. Correspondenceand requests for materials should be addressed to G.P. (pietrzyn@astrouw.edu.pl).

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