In the era of precision cosmology, it is essential to determine the Hubble constant to an accuracy of three per cent or better. At present, its uncertainty is dominated by the uncertainty in the distance to the Large Magellanic Cloud (LMC), which, being our second-closest galaxy, serves as the best anchor point for the cosmic distance scale. Observations of eclipsing binaries offer a unique opportunity to measure stellar parameters and distances precisely and accurately. The eclipsing-binary method was previously applied to the LMC, but the accuracy of the distance results was lessened by the need to model the bright, early-type systems used in those studies. Here we report determinations of the distances to eight long-period, late-type eclipsing systems in the LMC, composed of cool, giant stars. For these systems, we can accurately measure both the linear and the angular sizes of their components and avoid the most important problems related to the hot, early-type systems. The LMC distance that we derive from these systems (49.97 ± 0.19 (statistical) ± 1.11 (systematic) kiloparsecs) is accurate to 2.2 per cent and provides a firm base for a 3-per-cent determination of the Hubble constant, with prospects for improvement to 2 per cent in the future.

In the era of precision cosmology, it is essential to empirically determine the Hubble constant with an accuracy of one per cent or better 1 . At present, the uncertainty on this constant is dominated by the uncertainty in the calibration of the Cepheid period -luminosity relationship 2, 3 (also known as Leavitt Law). The Large Magellanic Cloud has traditionally served as the best galaxy with which to calibrate Cepheid period-luminosity relations, and as a result has become the best anchor point for the cosmic distance scale 4,5 . Eclipsing binary systems composed of late-type stars offer the most precise and accurate way to measure the distance to the Large Magellanic Cloud. Currently the limit of the precision attainable with this technique is about two per cent, and is set by the precision of the existing calibrations of the surface brightness -colour relation 5,6 . Here we report the calibration of the surface brightness-colour relation with a precision of 0.8 per cent. We use this calibration to determine the geometrical distance to the Large Magellanic Cloud that is precise to 1 per cent based on 20 eclipsing binary systems. The final distane is 49.59 ± 0.09 (statistical) ± 0.54 (systematic) kiloparsecs.All data are available upon request from G.P. Extended DataFig.1. Comparison of our relation with the relation of Di Benedetto obtained for giant stars 6 . Top panel, comparison of relations: data points show our results, with the fitted line shown in blue. The blue shaded area represents our obtained r.m.s. scatter of 0.018 mag. The green line is from ref. 6 . Very good agreement is demonstrated. Both S V and (V − K) 0 are in magnitudes. S V physically corresponds to the V band magnitude of a red giant star whose angular diameter is 1 mas. The error bars correspond to 1σ errors. Bottom panel, observed minus calculated values. Extended Data Fig.2. Observed minus calculated surface brightness versus metallicity 6 , [Fe/H]. In a relatively large range of metallicities (about 1 dex) no correlation is found. A formal linear fit gives O − C = 0.0009[Fe/H] -0.002 dex with coefficient of determination R 2 = 0.0001. Fig.3. Example of Monte Carlo simulations for one of our objects, ECL-12669. We computed 10,000 models with the JKTEBOP code 77 from which we obtained statistical uncertainties on the radii R 1 and R 2 , the orbital inclination i, the phase shift φ, the surface brightness ratio j 21 , radial velocity semi-amplitudes K 1 and K 2 , and the systemic velocities γ 1 and γ 2 . For every model we computed the distance modulus converting j 21 into temperature ratio T 2 /T 1 by using Popper's calibration 78 and our original solution with the Wilson-Devinney code 79 . We plot the number of calculated models versus distance modulus (m − M). The dashed line is the best fitted Gaussian and the blue line is the distance determined for this object. The intrinsic (V − K) 0 colours used to estimate the angular diameters of the components were computed using a temperature-colour calibration 28 . Extended DataExtended Data...

Context. The universality of the Cepheid period-luminosity (PL) relations has been under discussion since metallicity effects were assumed to play a role in the value of the intercept and, more recently, of the slope of these relations. Aims. The goal of the present study is to calibrate the Galactic PL relations in various photometric bands (from B to K) and to compare the results to the well-established PL relations in the LMC. Methods. We use a set of 59 calibrating stars, the distances of which are measured using five different distance indicators: Hubble Space Telescope and revised Hipparcos parallaxes, infrared surface brightness and interferometric Baade-Wesselink parallaxes, and classical Zero-Age-Main-Sequence-fitting parallaxes for Cepheids belonging to open clusters or OB stars associations. A detailed discussion of absorption corrections and projection factor to be used is given. Results. We find no significant difference in the slopes of the PL relations between LMC and our Galaxy. Conclusions. We conclude that the Cepheid PL relations have universal slopes in all photometric bands, not depending on the galaxy under study (at least for LMC and Milky Way). The possible zero-point variation with metal content is not discussed in the present work, but an upper limit of 18.50 for the LMC distance modulus can be deduced from our data.

Aims. We determine period-luminosity relations for Milky Way Cepheids in the optical and near-IR bands. These relations can be used directly as reference for extra-galactic distance determination to Cepheid populations with solar metallicity, and they form the basis for a direct comparison with relations obtained in exactly the same manner for stars in the Magellanic Clouds, presented in an accompanying paper. In that paper we show that the metallicity effect is very small and consistent with a null effect, particularly in the near-IR bands, and we combine here all 111 Cepheids from the Milky Way, the LMC and SMC to form a best relation. Methods. We employ the near-IR surface brightness (IRSB) method to determine direct distances to the individual Cepheids after we have recalibrated the projection factor using the recent parallax measurements to ten Galactic Cepheids and the constraint that Cepheid distances to the LMC should be independent of pulsation period. Results. We confirm our earlier finding that the projection factor for converting radial velocity to pulsational velocity depends quite steeply on pulsation period, p = 1.550− 0.186 log(P) in disagrement with recent theoretical predictions. We find PL relations based on 70 Milky Way fundamental mode Cepheids of M K = −3.33(±0.09)(log(P) − 1.0) − 5.66(±0.03), W VI = −3.26(±0.11)(log(P) − 1.0) − 5.96(±0.04). Combining the 70 Cepheids presented here with the results for 41 Magellanic Cloud Cepheids which are presented in an accompanying paper, we find M K = −3.30(±0.06)(log(P) − 1.0) − 5.65(±0.02), W VI = −3.32(±0.08)(log(P) − 1.0) − 5.92(±0.03). Conclusions. We delineate the Cepheid PL relation using 111 Cepheids with direct distances from the IRSB analysis. The relations are by construction in agreement with the recent HST parallax distances to Cepheids and slopes are in excellent agreement with the slopes of apparent magnitudes versus period observed in the LMC.

Context. Accretion and outflow processes are of fundamental importance for our understanding of the formation of stars and planetary systems. To trace these processes, diagnostic spectral lines such as the Brγ 2.166 μm line are widely used, although due to a lack of spatial resolution, the origin of the line emission is still unclear. Aims. Employing the AU-scale spatial resolution which can be achieved with infrared long-baseline interferometry, we aim to distinguish between theoretical models which associate the Brγ line emission with mass infall (magnetospheric accretion, gaseous inner disks) or mass outflow processes (stellar winds, X-winds, or disk winds). Methods. Using the VLTI/AMBER instrument, we spatially and spectrally (λ/Δλ = 1500) resolved the inner ( < ∼ 5 AU) environment of five Herbig Ae/Be stars (HD 163296, HD 104237, HD 98922, MWC 297, V921 Sco) in the Brγ emission line as well as in the adjacent continuum. From the measured wavelength-dependent visibilities, we derive the characteristic size of the continuum and Brγ lineemitting region. Additional information is provided by the closure phase, which we could measure both in the continuum wavelength regime (for four objects) as well as in the spectrally resolved Brγ emission line (for one object). The spectro-interferometric data is supplemented by archival and new VLT/ISAAC spectroscopy. Results. For all objects (except MWC 297), we measure an increase of visibility within the Brγ emission line, indicating that the Brγ-emitting region in these objects is more compact than the dust sublimation radius. For HD 98922, our quantitative analysis reveals that the line-emitting region is compact enough to be consistent with the magnetospheric accretion scenario. For HD 163296, HD 104237, MWC 297, and V921 Sco we identify an extended stellar wind or a disk wind as the most likely line-emitting mechanism. Since the stars in our sample cover a wide range of stellar parameters, we also search for general trends and find that the size of the Brγ-emitting region does not seem to depend on the basic stellar parameters (such as the stellar luminosity), but correlates with spectroscopic properties, in particular with the Hα line profile shape. Conclusions. By performing the first high-resolution spectro-interferometric survey on Herbig Ae/Be stars, we find evidence for at least two distinct Brγ line-formation mechanisms. Most significant, stars with a P-Cygni Hα line profile and a high mass-accretion rate seem to show particularly compact Brγ-emitting regions (R Brγ /R cont < 0.2), while stars with a double-peaked or single-peaked Hα-line profile show a significantly more extended Brγ-emitting region (0.6 < ∼ R Brγ /R cont < ∼ 1.4), possibly tracing a stellar wind or a disk wind.

Context. The ratio of pulsation to radial velocity (the projection factor) is currently limiting the accuracy of the Baade-Wesselink method, and in particular of its interferometric version recently applied to several nearby Cepheids. Aims. This work aims at establishing a link between the line asymmetry evolution over the Cepheids' pulsation cycles and their projection factor, with the final objective to improve the accuracy of the Baade-Wesselink method for distance determinations. Methods. We present HARPS high spectral resolution observations (R = 120 000) of nine galactic Cepheids: R Tra, S Cru, Y Sgr, β Dor, ζ Gem, Y Oph, RZ Vel, Car and RS Pup, having a good period sampling (P = 3.39d to P = 41.52d). We fit spectral line profiles by an asymmetric bi-Gaussian to derive radial velocity, Full-Width at Half-Maximum in the line (FWHM) and line asymmetry for all stars. We then extract correlations curves between radial velocity and asymmetry. A geometric model providing synthetic spectral lines, including limb-darkening, a constant FWHM (hereafter σ C ) and the rotation velocity is used to interpret these correlations curves. Results. For all stars, comparison between observations and modelling is satisfactory, and we were able to determine the projected rotation velocities and σ C for all stars. We also find a correlation between the rotation velocity (V rot sin i) and the period of the star: V rot sin i = (−11.5 ± 0.9) log (P) + (19.8 ± 1.0) [km s−1 ]. Moreover, we observe a systematic shift in observational asymmetry curves (noted γ O ), related to the period of the star, which is not explained by our static model: γ O = (−10.7 ± 0.1) log (P) + (9.7 ± 0.2) [in %]. For long-period Cepheids, in which velocity gradients, compression or shock waves seem to be large compared to short-or mediumperiod Cepheids we observe indeed a greater systematic shift in asymmetry curves. Conclusions. This new way of studying line asymmetry seems to be very promising for a better understanding of Cepheids atmosphere and to determine, for each star, a dynamic projection factor.

Context. The projection factor is a key quantity for the interferometric Baade-Wesselink (hereafter IBW) and surface-brightness (hereafter SB) methods of determining the distance of Cepheids. Indeed, it allows a consistent combination of angular and linear diameters of the star. Aims. We aim to determine consistent projection factors that include the dynamical structure of the Cepheids' atmosphere.Methods. Hydrodynamical models of δ Cep and Car have been used to validate a spectroscopic method of determining the projection factor. This method, based on the amplitude of the radial velocity curve, is applied to eight stars observed with the HARPS spectrometer. The projection factor is divided into three sub-concepts : (1) a geometrical effect, (2) the velocity gradient within the atmosphere, and (3) the relative motion of the "optical" pulsating photosphere compared to the corresponding mass elements (hereafter f o−g ). Both, (1) and (3) are deduced from geometrical and hydrodynamical models, respectively, while (2) is derived directly from observations.Results. The Fe i 4896.439 Å line is found to be the best one to use in the context of IBW and SB methods. A coherent and consistent period-projection factor relation (hereafter Pp relation) is derived for this specific spectral line: p = [−0.064 ± 0.020] log P + [1.376 ± 0.023]. This procedure is then extended to derive dynamic projection factors for any spectral line of any Cepheid. Conclusions. This Pp relation is an important tool for removing bias in the calibration of the period-luminosity relation of Cepheids. Moreover, it reveals a new physical quantity f o−g to investigate in the near future.

Abstract. The recent VINCI/VLTI observations presented in Paper I have nearly doubled the total number of available angular diameter measurements of Cepheids. Taking advantage of the significantly larger color range covered by these observations, we derive in the present paper high precision calibrations of the surface brightness-color relations using exclusively Cepheid observations. These empirical laws make it possible to determine the distance to Cepheids through a BaadeWesselink type technique. The least dispersed relations are based on visible-infrared colors, for instance F V (V − K) = −0.1336 ±0.0008 (V − K) + 3.9530 ±0.0006 . The convergence of the Cepheid (this work) and dwarf star (Kervella et al. 2004c) visible-infrared surface brightness-color relations is strikingly good. The astrophysical dispersion of these relations appears to be very small, and below the present detection sensitivity.