1. LETTER doi:10.1038/nature11878
An eclipsing-binary distance to the Large Magellanic
Cloud accurate to two per cent
G. 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 the Silla, together with near-infrared photometry obtained with the 3.5-m
Hubble constant to an accuracy of three per cent or better1,2. At New Technology Telescope located on La Silla.
present, its uncertainty is dominated by the uncertainty in the The spectroscopic and OGLE V- and I-band photometric observa-
distance to the Large Magellanic Cloud (LMC), which, being our tions of the binary systems were then analysed using the 2007 version
second-closest galaxy, serves as the best anchor point for the cosmic of the standard Wilson–Devinney code14,15, in the same way as in our
distance scale2,3. Observations of eclipsing binaries offer a unique recent work on a similar system in the Small Magellanic Cloud9.
opportunity to measure stellar parameters and distances precisely Realistic errors in the derived parameters of our systems were obtained
and accurately4,5. The eclipsing-binary method was previously from extensive Monte Carlo simulations (Fig. 2). The astrophysical
applied to the LMC6,7, but the accuracy of the distance results parameters of all the observed eclipsing binaries were determined with
was lessened by the need to model the bright, early-type systems an accuracy of a few per cent (Supplementary Tables 2–9).
used in those studies. Here we report determinations of the dis- For late-type stars, we can use the very accurately calibrated (2%)
tances to eight long-period, late-type eclipsing systems in the LMC, relation between their surface brightness and V2K colour to deter-
composed of cool, giant stars. For these systems, we can accurately mine their angular sizes from optical (V) and near-infrared (K) pho-
measure both the linear and the angular sizes of their components tometry16. From this surface-brightness/colour relation (SBCR), we
and avoid the most important problems related to the hot, early- can derive angular sizes of the components of our binary systems
type systems. The LMC distance that we derive from these systems directly from the definition of the surface brightness. Therefore, the
(49.97 6 0.19 (statistical) 6 1.11 (systematic) kiloparsecs) is accur- distance can be measured by combining the angular diameters of the
ate to 2.2 per cent and provides a firm base for a 3-per-cent deter- binary components derived in this way with their corresponding linear
mination of the Hubble constant, with prospects for improvement dimensions obtained from the analysis of the spectroscopic and pho-
to 2 per cent in the future. tometric data. The distances measured with this very simple but accur-
The modelling of early-type eclipsing binary systems consisting of ate one-step method are presented in Supplementary Table 12. The
hot stars is made difficult by the problem of obtaining accurate flux statistical errors in the distance determinations were calculated by
calibrations for early-type stars and by the degeneracy between the adding quadratically the uncertainties in absolute dimensions, V2K
stellar effective temperatures and reddening8,9. As a result, the dis- colours, reddening and the adopted reddening law. The reddening
tances determined from such systems are of limited (,5–10%) accu- uncertainty contributes very little (0.4%) to the total error11,17. A sig-
racy. More-accurate distances can be obtained using binary systems nificant change in the reddening law (from Rv 5 3.1 to 2.7, where Rv is
composed of cool stars; however, among the frequent dwarf stars in the the ratio of total to selective absorption) causes an almost negligible
LMC such systems are too faint to be accurately analysed with present- contribution, at the level of 0.3%. The accuracy of the V2K colour for
day telescopes. all components of our eight binary systems is better than 0.014 mag
The Optical Gravitational Lensing Experiment (OGLE) has been (0.7%). The resulting statistical errors in the distances are very close to
monitoring around 35 million stars in the field of the LMC for more 1.5%, and are dominated by the uncertainty in the absolute dimen-
than 16 years10. Using this unique data set, we have detected a dozen sions. By calculating a weighted mean from the individual distances to
extremely scarce, very long-period (60–772-d) eclipsing binary systems the eight target eclipsing binary systems, we obtain a mean LMC
composed of intermediate-mass, late-type giants located in a quiet distance of 49.88 6 0.13 kpc.
evolutionary phase on the helium-burning loop11 (Supplementary Our distance measurement might be affected by the geometry and
Table 1). These well-detached systems provide an opportunity to use depth of the LMC. Fortunately, the geometry of the LMC is simple and
the full potential of eclipsing binaries as precise and accurate distance well studied18. Because nearly all the eclipsing systems are located very
indicators, and to calibrate the zero point of the cosmic distance scale close to the centre of the LMC and to the line of nodes (Fig. 3), we
with an accuracy of about 2% (refs 5, 12, 13). fitted the distance to the centre of the LMC disk plane, assuming its
To do so, we observed eight of these systems (Fig. 1 shows one spatial orientation18. We obtained an LMC barycentre distance of
example) over the past 8 yr, collecting high-resolution spectra with the 49.97 6 0.19 kpc (Fig. 4), which is nearly identical to the simple
MIKE spectrograph at the 6.5-m Magellan Clay telescope at the Las weighted mean value. This shows that the geometrical structure
Campanas Observatory and with the HARPS spectrograph attached of the LMC has no significant influence on our present distance
to the 3.6-m telescope of the European Southern Observatory on La determination.
1
Universidad 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-Osservatorio
`
Astronomico di Roma, Via Frascati 33, 00040 Monte Porzio Catone, Italy. 6Dipartimento di Fisica Universita di Pisa, Largo B. Pontecorvo 2, 56127 Pisa, Italy. 7INFN, Sezione di Pisa, Via E. Fermi 2, 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 der
Sternwarte 16, 14482 Postdam, Germany. 11Nicolaus Copernicus Astronomical Centre, Bartycka 18, 00-716 Warszawa, Poland. 12Departamento de Astronomıa y Astrofısica, Pontificia Universidad
´ ´
Catolica 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.
´ ˜
7 6 | N AT U R E | VO L 4 9 5 | 7 M A R C H 2 0 1 3
©2013 Macmillan Publishers Limited. All rights reserved

2. LETTER RESEARCH
a 3.5
0 0.2 0.4 0.6 0.8 1
O – C (km s–1)
1
0
3.0
–1
310
Radial velocity (km s–1)
χr2
300
2.5
290
280
270
260 2.0
250
240
230 18.35 18.40 18.45 18.50 18.55 18.60 18.65 18.70
0 0.2 0.4 0.6 0.8 1 m – M (mag)
b 0 0.2 0.4 0.6 0.8 1
Figure 2 | Error estimation of the distance for one of our target binary
–0.02
O – C (mag)
systems. The reduced x2 map (goodness of fit) for the system OGLE-LMC-
0 ECL-15260 showing the dependence of the goodness of fit to the V-band and
0.02 I-band light curves on the distance modulus of the primary component. This
14.6 map was obtained from 110,000 models computed with the Wilson–Devinney
code14,15 within a broad range of the primary and secondary radii, R1 and R2, the
14.7 orbital inclination, i, the phase shift, w, the secondary’s temperature, T2, and
albedo, A2. In each case the distance, d, was calculated from the V-band surface-
14.8 brightness/colour (V2K) relation16 and translated into distance modulus using
I (mag)
m{M~5log(d){5, where m and M are the observed and absolute
14.9 brightnesses, respectively. The horizontal lines correspond to the standard-
deviation limits of the derived distance modulus, of 18.509 mag (50.33 kpc): 1s,
15 P = 189.995 d 2s and 3s, from bottom to top.
15.1 distance determinations are dominated by systematic errors, with each
0 0.2 0.4 0.6 0.8 1
Orbital phase
method having its own sources of uncertainties. This prevents a cal-
culation of the true LMC distance by simply taking the mean of the
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 the
orbital 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 the
I-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, 09660
orbital period. The top panel shows the residuals of the observed magnitudes
from the computed orbital light curve. All individual radial velocities were
determined by the cross-correlation method using appropriate template spectra
and spectra from the Magellan Inamori Kyocera Echelle (MIKE) and High
Accuracy Radial Velocity Planet Searcher (HARPS) spectrographs, yielding in
01866
all cases velocity accuracies better than 200 m s21 (error bars smaller than the
10567
plotted symbols). The orbit (mass ratio, systemic velocity, velocity amplitudes,
eccentricity and periastron passage) was fitted with a least-squares method to the 09114
measured velocities. The resulting parameters are presented in Supplementary 26122
06575 03160
Tables 2–9. The spectroscopic orbits, light curves and solutions for the 15260
remaining systems are of similar quality.
The systematic uncertainty in our distance measurement comes
from the calibration of the SBCR and the accuracy of the zero points
in our photometry. The root mean squared scatter in the current SBCR
is 0.03 mag (ref. 13), which translates to an accuracy of 2% in the
respective angular diameters of the component stars. Because the sur-
face brightness depends only very weakly on metallicity16,17, this effect
contributes to the total error budget at the level of only 0.3% (ref. 9).
Both optical (V) and near-infrared (K) photometric zero points are
accurate to 0.01 mag (0.5%). Combining these contributions quadra-
tically, we determine a total systematic error of 2.1% in our present Figure 3 | Location of the observed eclipsing systems in the LMC. Most of
LMC distance determination. our eight systems (circles) are located quite close to the geometrical centre of the
LMC and to the line of nodes (line), resulting in very small corrections to the
The LMC contains significant numbers of different stellar distance individual distances for the geometrical extension of this galaxy (in all cases
indicators, and, being the second-closest galaxy to our own, offers us a smaller than the corresponding statistical error in the distance determination).
unique opportunity to study these indicators with the utmost pre- The effect of the geometrical structure of the LMC on the mean LMC distance
cision. For this reason, this galaxy has an impressive record of several reported here is therefore negligible. The background image has a field of view of
hundred distance measurements2,3,19. Unfortunately, almost all LMC 8u 3 8u and is taken from the All Sky Automated Survey wide-field sky survey28.
7 M A R C H 2 0 1 3 | VO L 4 9 5 | N AT U R E | 7 7
©2013 Macmillan Publishers Limited. All rights reserved

3. RESEARCH LETTER
2 We have good reason to believe that there is significant room to
improve on the high-accuracy LMC distance determination reported
03160
1.5 D = 49.97 ± 0.19 kpc here, by improving the calibration of the SBCR for late-type stars12,16,
χr2= 0.992 which is the dominant source of systematic error in our present deter-
01866
1 06575 mination. This work is in progress, and a determination of the distance
to the LMC accurate to 1% should be possible once the SBCR cal-
10567 ibration is refined. This will have a corresponding effect on further
0.5
Δd (kpc)
15260 26122
improving the accuracy of H0. This is similar to the accuracy of the
0 geometrical distance to the LMC, which is to be delivered by Gaia
mission in around 12 years from now. The eclipsing-binary technique
–0.5
will then probably provide the best opportunity to check on the future
Gaia measurements for possible systematic errors.
–1 09114 09660 Received 19 August; accepted 20 December 2012.
–1.5 1. Komatsu, E. et al. Seven-year Microwave Anisotropy Probe (WMAP) observations:
0 0.5 1 1.5 2 2.5 3 3.5 cosmological interpretation. Astrophys. J. Suppl. Ser. 192, 18–65 (2011).
ρ (°) 2. Freedman, W. L. & Madore, B. F. The Hubble constant. Annu. Rev. Astron. Astrophys.
48, 673–710 (2010).
Figure 4 | Consistency among the distance determinations for the target 3. Schaefer, B. E. A problem with the clustering of recent measures of the distance to
binary systems. Distance offsets, Dd, between our particular eclipsing binary the Large Magellanic Cloud. Astron. J. 135, 112–119 (2008).
systems and the best-fitted LMC disk plane, plotted against the angular distance 4. Lacy, C. H. Distances to eclipsing binaries: an application of the Barnes-Evans
relation. Astrophys. J. 213, 458–463 (1977).
of the systems from the LMC centre, r. The identification of the systems is the
5. ´
Paczynski, B. in The Extragalactic Distance Scale (eds Livio, M., Donahue, M. & Panagia,
same as in Fig. 3. The error bars correspond to 1s errors. We assumed the N.) 273–280 (Space Telescope Sci. Inst. Ser., Cambridge Univ. Press, 1997).
model of the LMC from ref. 18. We fitted one parameter, the distance to the 6. Guinan, E. F. The distance to the Large Magellanic Cloud from the eclipsing binary
centre of the LMC (RA 5 5 h 25 min 06 s; dec. 5 269u 479 0099), using a fixed HV 2274. Astrophys. J. 509, L21–L24 (1998).
spatial orientation of the LMC disk (inclination, i 5 28u; position angle of the 7. Fitzpatrick, E. L., Ribas, I., Guinan, E. F., Maloney, F. P. & Claret, A. Fundamental
nodes, h 5 128u). The resulting distance to the LMC barycentre is properties and distances of Large Magellanic Cloud eclipsing binaries. IV. HV
5936. Astrophys. J. 587, 685–700 (2003).
49.97 6 0.19 kpc, with a reduced x2 value very close to unity.
8. Groenewegen, M. A. T. & Salaris, M. The LMC eclipsing binary HV 2274 revisited.
Astrophys. J. 366, 752–764 (2001).
reported distances resulting from different techniques. Our present 9. Graczyk, D. et al. The Araucaria project: an accurate distance to the late-type
double-lined eclipsing binary OGLE SMC113.3 4007 in the Small Magellanic
LMC distance measurement, of 49.97 6 0.19 (statistical) 6 1.11 (sys- Cloud. Astrophys. J. 750, 144–156 (2012).
tematic) kpc (that is, a true distance modulus of 18.493 6 0.008 (statis- 10. Udalski, A. et al. The Optical Gravitational Lensing Experiment: OGLE-III
tical) 6 0.047 (systematic) mag), agrees well, within the combined photometric maps of the Large Magellanic Cloud. Acta Astron. 58, 89–102 (2008).
11. Graczyk, D. et al. The Optical Gravitational Lensing Experiment: the OGLE-III
errors, with the most recent determinations of the distance to the catalog of variable stars. XII. Eclipsing binary stars in the Large Magellanic Cloud.
LMC19,20,21. Our purely empirical method allows us to estimate both Acta Astron. 61, 103–122 (2011).
statistical and systematic errors in a very reliable way, which is normally 12. Kruszewski, A. & Semeniuk, I. Nearby Hipparcos eclipsing binaries for color-
surface brightness calibration. Acta Astron. 49, 561–575 (1999).
not the case, particularly in distance determinations relying in part on 13. ´
Pietrzynski, G. et al. The Araucaria project: determination of the Large Magellanic
theoretical predictions of stellar properties and their dependences on Cloud distance from late-type eclipsing binary systems. I. OGLE051019.64–
environment. In particular, our result provides a significant improve- 685812.3. Astrophys. J. 697, 862–866 (2009).
14. Wilson, R. E. & Devinney, E. J. Realization of accurate close-binary light curves:
ment over previous LMC distance determinations made using observa- application to MR Cygni. Astrophys. J. 166, 605–620 (1971).
tions of eclipsing binaries7,22. Those studies were based on early-type 15. Van Hamme, W. & Wilson, R. E. Third-body parameters from whole light and
systems for which no empirical SBCR is available, and so had to rely on velocity curves. Astrophys. J. 661, 1129–1151 (2007).
16. Di Benedetto, G. P. Predicting accurate stellar angular diameters by the near-infrared
theoretical models to determine the effective temperature. Our present surface brightness technique. Mon. Not. R. Astron. Soc. 357, 174–190 (2005).
determination is based on many (eight) binary systems and does not 17. Thompson, I. B. et al. Cluster AgeS Experiment. The age and distance of the
resort to model predictions. globular cluster v Centauri determined from observations of the eclipsing binary
OGLEGC 17. Astron. J. 121, 3089–3099 (2001).
The classical approach to deriving the Hubble constant (H0) consists 18. van der Marel, R. P., Alves, D. R., Hardy, E. & Suntzeff, N. B. New understanding of
of deriving an absolute calibration of the Cepheid period–luminosity Large Magellanic Cloud structure, dynamics, and orbit from carbon star
relation (CPLR), which is then used to determine the distances to kinematics. Astron. J. 124, 2639–2663 (2002).
19. Mazzarella, J. M. NED for a new era. Astron. Soc. Pacif. Conf. 376, 153–162 (2007).
nearby galaxies containing type Ia supernovae23. These supernovae 20. Walker, A. R. The Large Magellanic Cloud and the distance scale. Astrophys. Space
are excellent standard candles that extend out to the region of unper- Sci. 341, 43–49 (2012).
turbed Hubble flow once their peak brightnesses are calibrated this 21. Monson, A. J. et al. The Carnegie Hubble Program: The Leavitt Law at 3.6 and 4.5
mm in the Milky Way. Astrophys. J. 759, 146–165 (2012).
way, and provide the most accurate determination of H0 (ref. 24). An 22. Bonanos, A. Z., Castro, N., Macri, L. M. & Kudritzki, R. P. The distance to the massive
alternative approach to calibrate the CPLR with Cepheids in the LMC eclipsing binary LMC-SC1–105 in the Large Magellanic Cloud. Astrophys. J. 729,
is to calibrate it in our own Milky Way galaxy using Hubble Space L9–L15 (2011).
23. Freedman, W. L. et al. Final results from the Hubble Space Telescope key project to
Telescope (HST) parallax measurements of the Cepheids nearest to the measure the Hubble constant. Astrophys. J. 553, 47–72 (2001).
Sun25. However, the CPLR that results from that approach is less 24. Riess, A. G. et al. A 3% solution: determination of the Hubble constant with the
accurate for two reasons. First, the Cepheid sample with HST paral- Hubble Space Telescope and Wide Field Camera 3. Astrophys. J. 730, 119–137
(2011).
laxes is very small (ten stars) relative to the Cepheid sample in the LMC 25. Benedict, G. F. et al. Hubble Space Telescope fine guidance sensor parallaxes of
(2,000 stars), which can be used to establish the CPLR once the LMC galactic Cepheid variable stars: period-luminosity relations. Astron. J. 133,
distance is known. Second, the average accuracy of the HST Cepheid 1810–1827 (2007).
26. Lutz, T. E. & Kelker, D. H. On the use of trigonometric parallaxes for the calibration of
parallaxes is 8% (ref. 25) and suffers from systematic uncertainties that luminosity systems: theory. Publ. Astron. Soc. Pacif. 85, 573–578 (1973).
are not completely understood, including Lutz–Kelker bias26,27. 27. van Leeuwen, F., Feast, M. W., Whitelock, P. A. & Laney, C. D. Cepheid parallaxes and
Therefore, at present the preferred route to determining the Hubble the Hubble constant. Mon. Not. R. Astron. Soc. 379, 723–737 (2007).
28. ´
Pojmanski, G. The All Sky Automated Survey. Acta Astron. 47, 467–481 (1997).
constant is that which uses the highly abundant LMC Cepheid
population whose mean distance is now known, with the result of this Supplementary Information is available in the online version of the paper.
work, to 2.2%. This result reduces the uncertainty in H0 to a very firmly Acknowledgements We acknowledge financial support for this work from the BASAL
established 3%. ´
Centro de Astrofısica y Tecnologias Afines (CATA), the Polish Ministry of Science, the
7 8 | N AT U R E | VO L 4 9 5 | 7 M A R C H 2 0 1 3
©2013 Macmillan Publishers Limited. All rights reserved

4. LETTER RESEARCH
Foundation for Polish Science (FOCUS, TEAM), the Polish National Science Centre and B.P.: spectroscopic observations and reductions, RV measurements. A.U., I.S. and S. K.:
the GEMINI-CONICYT fund. The OGLE project has received funding from the European optical observations and data reductions. P.K., K.S., M.K., M.K.S., R.P., Ł.W., K.U., P.P.,
Research Council ‘Advanced Grant’ Program. We thank the staff astronomers at Las 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.: data
Campanas and ESO La Silla, who provided expert support in data acquisition. We thank analysis. S.V.: analysis of the spectra. G.P. and W.G. worked jointly to draft the
J. F. Gonzalez for making the IRAF scripts rvbina and spbina available to us. We also manuscript with all authors reviewing and contributing to its final form
thank O. Szewczyk and Z. Kołaczkowski for their help with some of the observations.
Author Information Reprints and permissions information is available at
Author Contributions G.P.: photometric and spectroscopic observations and www.nature.com/reprints. The authors declare no competing financial interests.
reductions. D.G.: spectroscopic observations, modelling and data analysis. W.G.: Readers are welcome to comment on the online version of the paper. Correspondence
observations and data analysis. I.B.T.: observations, RV determination, data analysis. and requests for materials should be addressed to G.P. (pietrzyn@astrouw.edu.pl).
7 M A R C H 2 0 1 3 | VO L 4 9 5 | N AT U R E | 7 9
©2013 Macmillan Publishers Limited. All rights reserved