Context. The method of distance determination of eclipsing binaries consists in combining the radii of both components determined from spectro-photometric observations with their respective angular diameters derived from the surface brightness-color relation (SBC). However, the largest limitation of the method comes from the uncertainty on the SBC relation: about 2% for late-type stars (or 0.04 magnitude) and more than 10% for early-type stars (or 0.2 mag). Aims. The aim of this work is to improve the SBC relation for early-type stars in the −1 ≤ V − K ≤ 0 color domain, using optical interferometry. Methods. Observations of eight B-and A-type stars were secured with the VEGA/CHARA instrument in the visible. The derived uniform disk angular diameters were converted into limb darkened angular diameters and included in a larger sample of 24 stars, already observed by interferometry, in order to derive a revised empirical relation for O, B, A spectral type stars with a V − K color index ranging from −1 to 0. We also took the opportunity to check the consistency of the SBC relation up to V − K 4 using 100 additional measurements. Results. We determined the uniform disk angular diameter for the eight following stars: γ Ori, ζ Per, 8 Cyg, ι Her, λ Aql, ζ Peg, γ Lyr, and δ Cyg with V − K color ranging from −0.70 to 0.02 and typical precision of about 1.5%. Using our total sample of 132 stars with V − K colors index ranging from about −1 to 4, we provide a revised SBC relation. For late-type stars (0 ≤ V − K ≤ 4), the results are consistent with previous studies. For early-type stars (−1 ≤ V − K ≤ 0), our new VEGA/CHARA measurements combined with a careful selection of the stars (rejecting stars with environment or stars with a strong variability), allows us to reach an unprecedented precision of about 0.16 magnitude or 7% in terms of angular diameter. Conclusions. We derive for the first time a SBC relation for stars between O9 and A3, which provides a new and reliable tool for the distance scale calibration.

Context. The B-W method is used to determine the distance of Cepheids and consists in combining the angular size variations of the star, as derived from infrared surface-brightness relations or interferometry, with its linear size variation, as deduced from visible spectroscopy using the projection factor. The underlying assumption is that the photospheres probed in the infrared and in the visible are located at the same layer in the star whatever the pulsation phase. While many Cepheids have been intensively observed by infrared beam combiners, only a few have been observed in the visible. Aims. This paper is part of a project to observe Cepheids in the visible with interferometry as a counterpart to infrared observations already in hand. Methods. Observations of δ Cep itself were secured with the VEGA/CHARA instrument over the full pulsation cycle of the star. Results. These visible interferometric data are consistent in first approximation with a quasi-hydrostatic model of pulsation surrounded by a static circumstellar environment (CSE) with a size of θCSE = 8.9 ± 3.0 mas and a relative flux contribution of fCSE = 0.07 ± 0.01. A model of visible nebula (a background source filling the field of view of the interferometer) with the same relative flux contribution is also consistent with our data at small spatial frequencies. However, in both cases, we find discrepancies in the squared visibilities at high spatial frequencies (maximum 2σ) with two different regimes over the pulsation cycle of the star, φ = 0.0 − 0.8 and φ = 0.8 − 1.0. We provide several hypotheses to explain these discrepancies, but more observations and theoretical investigations are necessary before a firm conclusion can be drawn. Conclusions. For the first time we have been able to detect in the visible domain a resolved structure around δ Cep. We have also shown that a simple model cannot explain the observations, and more work will be necessary in the future, both on observations and modelling.

The surface brightness colors (SBC) relation is a very important tool to derive the distance of extragalactic eclipsing binaries. However, for early-type stars, this SBC relation is critically affected by the stellar environment (wind, circumstellar disk, etc...) and/or by the fast rotation. We calculated 6 models based on the code of Domiciano de Souza et al. (2012) considering different inclinations and rotational velocities. Using these results, we quantify for the first time the impact of the rotation on the SBC relation.

Context. When a given observational quantity depends on several stellar physical parameters, it is generally very difficult to obtain observational constraints for each of them individually. Therefore, we studied under which conditions constraints for some individual parameters can be achieved for fast rotators, knowing that their geometry is modified by the rapid rotation which causes a non-uniform surface brightness distribution. Aims. We aim to study the sensitivity of interferometric observables on the position angle of the rotation axis (PA) of a rapidly rotating star, and whether other physical parameters can influence the determination of PA, and also the influence of the surface differential rotation on the determination of the β exponent in the gravity darkening law that enters the interpretation of interferometric observations, using α Cep as a test star. Methods. We used differential phases obtained from observations carried out in the Hα absorption line of α Cep with the VEGA/CHARA interferometer at high spectral resolution, R = 30 000 to study the kinematics in the atmosphere of the star. Results. We studied the influence of the gravity darkening effect (GDE) on the determination of the PA of the rotation axis of α Cep and determined its value, PA = −157 +17• −10• . We conclude that the GDE has a weak influence on the dispersed phases. We showed that the surface differential rotation can have a rather strong influence on the determination of the gravity darkening exponent. A new method of determining the inclination angle of the stellar rotational axis is suggested. We conclude that differential phases obtained with spectro-interferometry carried out on the Hα line can in principle lead to an estimate of the stellar inclination angle i. However, to determine both i and the differential rotation parameter α, lines free from the Stark effect and that have collision-dominated source functions are to be preferred.

Context. The eclipsing binary method for determining distance in the local group is based on the surface brightness-color relation (SBCR), and early-type stars are preferred targets because of their intrinsic brightness. However, this type of star exhibits wind, massloss, pulsation, and rotation, which may generate bias on the angular diameter determination. An accurate calibration of the SBCR relation thus requires careful analysis. Aims. In this paper we aim to quantify the impact of stellar rotation on the SBCR when the calibration of the relation is based on interferometric measurements of angular diameters. Methods. Six stars with V − K color indices ranging between -1 and 0.5 were modeled using the code for high angular resolution of rotating objects in nature (CHARRON) with various rotational velocities (0, 25, 50, 75, and 95% of the critical rotational velocity) and inclination (0, 25, 50, 75, and 90 degrees). All these models have their equatorial axis aligned in an east-west orientation in the sky. We then simulated interferometric observations of these theoretical stars using three representative sets of the CHARA baseline configurations. The simulated data were then interpreted as if the stars were non-rotating to determine an angular diameter and estimate the surface-brightness relation. The V − K color of the rotating star was calculated directly from the CHARRON code. This provides an estimate of the intrinsic dispersion of the SBCR relation when the rotation effects of flattening and gravity darkening are not considered in the analysis of interferometric data. Results. We find a clear relation between the rotational velocity and (1) the shift in zero point (∆a 0 ) of the SBCR (compared to the static relation) and (2) its dispersion (σ). When considering stars rotating at less than 50% of their critical velocity, ∆a 0 and σ have about 0.01 mag, while these quantities can reach 0.08 and 0.04 mag, respectively, when the rotation is larger than 75% of the critical velocity. Besides this, the inclination angle mostly has an impact on the V − K color: i < 50• (resp. i > 50 • ) makes the star redder (resp. bluer). When considering the 150 models, ∆a 0 and σ have 0.03 and 0.04 mag, respectively. These values are slightly but not significantly modified (about 0.03 and 0.01 mag in ∆a 0 and σ, respectively) when considering different CHARA configurations. Interestingly, these 150 models, regardless of the interferometric configuration, are consistent with the empirical SBCR, which is within its dispersion of 0.16 mag. In addition, if one only considers projected rotational velocity V rot sin i lower than 100 km s −1 , then ∆a 0 and σ have 0.02 and 0.03 mag, respectively. Conclusions. To calibrate the SBCR interferometrically at the 0.02 mag precision (or lower), one should consider (1) a baseline configuration covering all directions of the (u, v) plan; (2) a sample of stars with rotational velocity lower than 50% of their critical velocity or, alternatively, stars with V rot sin i lower than 100 km...

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Context. Rapid rotation is a common feature for massive stars, with important consequences on their physical structure, flux distribution and evolution. Fast-rotating stars are flattened and show gravity darkening (non-uniform surface intensity distribution). Another important and less studied impact of fast-rotation in early-type stars is its influence on the surface brightness colour relation (hereafter SBCR), which could be used to derive the distance of eclipsing binaries. Aims. The purpose of this paper is to determine the flattening of the fast-rotating B-type star δ Per using visible long-baseline interferometry. A second goal is to evaluate the impact of rotation and gravity darkening on the V − K colour and surface brightness of the star. Methods. The B-type star δ Per was observed with the VEGA/CHARA interferometer, which can measure spatial resolutions down to 0.3 mas and spectral resolving power of 5000 in the visible. We first used a toy model to derive the position angle of the rotation axis of the star in the plane of the sky. Then we used a code of stellar rotation, CHARRON, in order to derive the physical parameters of the star. Finally, by considering two cases, a static reference star and our best model of δ Per, we can quantify the impact of fast rotation on the surface brightness colour relation (SBCR). Results. We find a position angle of 23 ± 6 degrees. The polar axis angular diameter of δ Per is θ p = 0.544 ± 0.007 mas, and the derived flatness is r = 1.121 ± 0.013. We derive an inclination angle for the star of i = 85 +5−20 degrees and a projected rotation velocity V sin i = 175 +8 −11 km s −1 (or 57% of the critical velocity). We find also that the rotation and inclination angle of δ Per keeps the V − K colour unchanged while it decreasing its surface-brightness by about 0.05 mag. Conclusions. Correcting the impact of rotation on the SBCR of early-type stars appears feasible using visible interferometry and dedicated models.

The international Araucaria Project aims to provide an improved local calibration of the extragalactic distance scale out to distances of a few Megaparsecs. We present a brief, up-to-date review of different results and ongoing projects to (i) better understand the Baade–Wesselink projection factor used in distance determinations based on Cepheids, and (ii) improve the eclipsing-binary technique of distance determination using CHARA/VEGA interferometric observations.