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Open Access
Issue
A&A
Volume 639, July 2020
Article Number A67
Number of page(s) 8
Section Stellar atmospheres
DOI https://doi.org/10.1051/0004-6361/202037679
Published online 09 July 2020

© N. Nardetto et al. 2020

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1 Introduction

In the era of precision cosmology, it is essential to determine the Hubble constant to an accuracy of 2% or better (Komatsu et al. 2011; Freedman & Madore 2010). Recently, Riess et al. (2019) achieved a 1.9% precision by relying on the distance to the Large Magellanic Cloud (LMC), which is the best anchor point for the cosmic distance scale (Schaefer 2008; Walker et al. 2012; Riess et al. 2016). Late-type eclipsing-binary systems provide the opportunity of measuring accurate distances by combining the linear diameter (derived from the light curve and velocimetry) and angular diameters (derived from a surface-brightness color relation, SBCR) of their components.

Based on the observation of some 35 million stars in the LMC for more than 20 yr during the Optical Gravitational Lensing Experiment (OGLE; Udalski et al. 2008), a few dozen extremely scarce long-period eclipsing systems composed of late-type clump giants were cataloged (Graczyk et al. 2011; Pawlak et al. 2016). In the course of the Araucaria project (Gieren et al. 2005), Pietrzyński et al. (2013) studied 8 systems and derived a distance to the LMC at the 2.2% level. This distance relied on the SBCR derived by Di Benedetto (2005) and thus was completely dominated by the error of 2% (or 0.03 magnitude) on this relation. Then, Laney et al. (2012) secured uniform and precise K-band photometry (precision of about ≃0.01 mag) for a large sample of nearby very bright red giants located in a very well defined and quiet evolutionary phase of the core-helium burning (red clump). In addition, Gallenne et al. (2018) secured observations of 48 of these red-clump stars in the H band with the Precision Integrated Optics Near-infrared Imaging ExpeRiment (PIONIER, Le Bouquin et al. 2011) installed at the Very Large Telescope Interferometer. These measurements allowed deriving an SBCR with an rms of 0.018 magnitude. This relation was used to derive the individual angular diameters of the 20 eclipsing binaries at the 0.8% precision level, which finally led to a distance of the LMC that is accurate to 1% (Pietrzyński et al. 2019).

The purpose of this paper is to calibrate the SBCR in the visible domain in order to compare it to the one derived in the H band. For this, we observed a subsample of eight stars in the visible domain with interferometry, using the Visible spEctroGraph and polArimeter (VEGA) beam combiner (Mourard et al. 2009, 2011) operating at the focus of The Center for High Angular Resolution Astronomy (CHARA) array (ten Brummelaar et al. 2005), which is located at the Mount Wilson Observatory. In Sect. 2 we describe the VEGA/CHARA interferometric observations and provide the limb-darkened angular diameters of the eight stars in our sample. Section 3 is dedicated to the calibration of the SBCR. The results are then discussed in Sect. 4, and we conclude in Sect. 5.

Table 1

Summary of the observing log.

2 VEGA/CHARA observations of eight late-type stars

We selected eight late-type stars with existing infrared photometry by Laney et al. (2012), with a (V−K)0 color index ranging from 2.12 to 2.43, which corresponds to the index that was used to derive the distance to the LMC (2.05–2.75). They are red giants stars (δ between −9° to 12°) with spectral types ranging from G8 to K2. They have a visual magnitude mV ranging from 5.8 to 6.2, which is well below the limiting magnitude of VEGA. They are also bright in the K band (with mK <3.9), which allows us to track the group delay simultaneously with the infrared CLIMB combiner (Sturmann et al. 2010). We observed our sample from 2013 July to 2014 October using different suitable telescopes available on the CHARA array. A summary of the observationsis given in Table 1.

For these observations, we used the medium spectral resolution mode of VEGA (R ≃ 5000) and the standard VEGA pipeline in order to calibrate the squared visibilities in spectral bands of 15 or 20nm (Mourard et al. 2009, 2011; Ligi et al. 2013). The integration time of our observations is of 500 s. For each calibrated visibility, the statistical and systematic calibration errors are given separately in Table B.1. The systematic uncertainty stems from the uncertainty on the calibrator diameter, which is given in Table 2. To perform the model fitting, we used a JMMC1 tool, LITpro (Tallon-Bosc et al. 2008). The systematic and statistical errors are considered separately in the fitting procedure and are propagated until the final error on the surface brightness. In Figs. A.1 and A.2, the calibrated visibilities are plotted as a function of the spatial frequency together with the corresponding (u,v) coverage for each star. We fit these calibrated visibilities by a uniform disk, where the so-called uniform-disk angular diameter (θUD) is the only parameter. The results are given in Table 3. We then converted these θUD diameters into limb-darkened (θLD) angular diameters. A common and convenient approach is to use a linear law for the continuum-intensity profile of the star, defined by I(cos(α)) = 1 − u + ucos(α), where u is the linear limb-darkening coefficient of the star in the interferometric wavelength band of observation (Claret & Bloemen 2011). α is the angle between the normal of the star and the line of sight. The limb-darkened angular diameter is then calculated using θLD=θUD[(1−u3)(1−7u15)]12$\theta_{\mathrm{LD}}=\theta_{\mathrm{UD}}\left[\frac{(1-\frac{u}{3})}{(1-\frac{7u}{15})}\right]^{\frac{1}{2}} $ (Hanbury Brown et al. 1974). In the Claret tables, u is given as a function of the effective temperature Teff, the surface gravity logg, the metallicity Z and the microturbulence velocity Vt, in several photometric bands, including the R band (λeff = 670 nm) and the I band (λeff = 856nm). The fundamental parameters of the stars are given in Table 4 together with the rounded values used as an input in the Claret tables. After they are extracted from these tables, the limb-darkened coefficients in the R and I bands were interpolated at the typical wavelength band of our observations, that is, 710 mn, in order to derive the corresponding linear limb-darkening coefficient (u[710]). The resulting u[710] parameters are given in Table 3 together with the corresponding limb-darkened angular diameters. The reduced χ2 ranges from 0.4 to 2.9.

Changing the effective temperature by 500 K and/or the surface gravity by 0.5 have an effect on the limb-darkening angular diameter of 1.3%. The grid in effective temperature has a step of 250 K, thus the largest error we can make on the temperature is 125 K, which corresponds to an additional uncertainty on the limb-darkened diameter of about 0.3%. However, the fact that the derived diameters are weakly sensitive to the input parameters of the limb-darkening law does not exclude that they might be sensitive to the limb-darkening law itself or to the method. We also tested the method by fitting the limb-darkened angular diameter with the limb_linear function of the LITpro tool, that is, by fixing the limb-darkening coefficient to its value indicated in Table 4. The derived limb-darkened angular diameters are 0.4 to 0.8% larger (except for HD 23526, for which we find 1.2%) than those indicated in Table 4 based on the approximate analytic conversion law. This bias is larger than what has been stated by Hanbury Brown et al. (1974), which is an approximation of < 0.4% if uR ranges from 0.5 to 1.0. The uncertainties are strictly the same, however. For the limb-darkening law itself, we also used the dedicated SATLAS models by Neilson & Lester (2013a,b) to derive the limb-darkened diameters using the same input physical parameters as those indicated in Table 4. With this approach, we find diameters that are about 0.4% larger than the values presented in Table 3. These tests are interesting and should be considered with caution when a limb-darkened angular diameter is derived at the sub-percent level, but our statistical precision here is about 2.4% and the common analytical approach leads to a bias on our SBCR calibration at a far lower level than our statistical precision.

Our sample includes four stars that are in common with samples of Gallenne et al. (2018) and Pietrzyński et al. (2019): HD13468, HD 23526, HD 360, and HD 40020. Their PIONIER/VLTI angular diameters are listed in Table 3. This is an interesting opportunity to compare the angular diameters derived from CHARA/VEGA and VLTI/PIONIER (see Fig. 1). We find that the CHARA/VEGA angular diameters are consistent within their uncertainties with those from VLTI/PIONIER. Visible (this work) and infrared (Gallenne et al. 2018) interferometric observations are thus in agreement, which means that the conversions from UD into LD angular diameters are at least consistent in the two wavelength domains (R and H) at the sub-percent level. Moreover, it indicates that there is negligible systematics on the instrumental point of view in the determination of the angular diameters. Both instruments might be biased in exactly the same way (for the four stars), but this is highly unlikely. The calibrators used in the VLTI/PIONIER analysis are giants with spectral types ranging from G8 to K2, which is significantly colder and more resolved than the calibrators we used in our analysis in the visible domain (Table 2). We therefore expect that the systematics on the squared visibilities (Table 2) that stem from the uncertainties on the calibrators are significantly lower in the visible than in the infrared.

Table 2

Reference stars and their parameters, including the spectral type, the visual magnitude (mV), and the predicted uniform-disk angular diameter (in mas) together with its corresponding uncertainty derived from the JMMC SearchCal software (Bonneau et al. 2006; Lafrasse et al. 2010).

Table 3

Angular diameters obtained with VEGA/CHARA and the corresponding surface brightnesses.

Table 4

Fundamental parameters of the stars in our sample based on spectroscopy and combined with photometry for stars in Luck & Heiter (2007).

thumbnail Fig. 1

Angular diameters derived from VEGA/CHARA compared to those from PIONIER/VLTI. The blue line marks the 1:1 relation.

3 Calibrating the surface brightness relation

The surfacebrightness of a star is linked to its visual intrinsic dereddened magnitude mV0$m_{\mathrm{V_{{0}}}}$ and its limb-darkened angular diameter θLD by the following relation: SV=mV0+5logθLD$S_{\mathrm{V}}=m_{\mathrm{V_{{0}}}}&#x002B;5\log \theta_{\textrm{LD}}$. In order to derive mV0$m_{\mathrm{V_{{0}}}}$, we first selected the apparent mV magnitude from the main HIPPARCOS and Tycho catalog (ESA 1997) expressed in the Johnson system (Johnson et al. 1966). For the dereddening of these magnitudes, we used mV0=mV−AV$ m_{\mathrm{V_{{0}}}}=m_{\mathrm{V}}-A_{\textrm{V}}$, where AV is either takendirectly from Pietrzyński et al. (2019) for the four targets we have in common or based on Stilism2 (Lallement et al. 2014; Capitanio et al. 2017). Pietrzyński et al. (2019) used the 2D reddening maps of Schlegel et al. (1998) and Schlafly & Finkbeiner (2011). These maps give a total extinction in a given line of sight, while the reddening to a given star isonly a small fraction of it. To calculate this fraction, they used a simple exponential model of dust distribution in the Milky Way, the HIPPARCOS parallaxes, and the local extinction-free bubble radius of 40 pc (Suchomska et al. 2015). Stilism is a 3D maps of the Galactic interstellar medium based on a combination of various methods and databases. The extinction we used for each star in our sample is indicated in Table 3, and the effect of these extinctions on the SBCR is discussed in the next section. Then the color follows with (V−K)0=(V−K)−AV−AK$(V-K)_{0}=(V-K)-A_{\textrm{V}}-A_{\textrm{K}}$, where AK = 0.089 AV (Nishiyama et al. 2009). The mK magnitudes are homogeneously derived fromLaney et al. (2012). The SV values are given in Table 3.

In order to fit our SBCR, we used the same formalism as Pietrzyński et al. (2019), SV=α[(V−K)0−2.405]+β,\begin{equation*}S_{\mathrm{V}} =\alpha [(V-K)_{\mathrm{0}} - 2.405] &#x002B; \beta ,\end{equation*}(1)

where α and β are the slope and zero-point of the relation, respectively. The value 2.405 is the average color of the stars that were used to built the SBCR in Pietrzyński et al. (2019). We kept this value in our fit to allow a direct comparison. We first fit both parameters and then assumed the slope from the SBCR of Pietrzyński et al. (2019). The statistical uncertainty on the magnitudes mV and mK, on the extinction AV, and on the surface brightness Sv were taking into account in the fitting process. Results are summarized in Table 5. We obtain very similar results when we fit one or two parameters (1.7% difference on the slope and 0.05% on the zero-point), as shown by Fig. 2. The SBCRs from Kervella et al. (2004a), Di Benedetto (2005), and Challouf et al. (2014) are shown for comparison.

4 Discussion

Our relation has an rms of 0.04 magnitude (Table 5) and is consistent with all other SBCRs presented in Fig. 2, except for that of Challouf et al. (2014), in particular for a (V − K) color larger than 2.3 magnitude. This is explained by the fact that the relation by Challouf et al. (2014) is nonlinear, was based on 132 starsof all classes, and was fit over a very wide range of color from early to late types. We note that this relation was dedicated toO, B, A stars.

Conversely, the relation by Pietrzyński et al. (2019), which has an rms of 0.018 magnitude, is consistent with all SBCRs (except for that of Challouf et al. (2014)), including the SBCR in this work. These relations are indeed consistent (within 1%), even thoughdifferent methods were used. The relation by Pietrzyński et al. (2019) is based on the observation of 48 giant stars in the H band with the PIONIER/VLTI (Gallenne et al. 2018), while the relation by Kervella et al. (2004a) is based on a set of angular diameters of dwarfs and subgiant stars derived from different optical and infrared interferometers: NPOI, NII (Narrabri Intensity Interferometer, Hanbury Brown et al. 1967), Mark III (Shao et al. 1988), PTI (Palomar Testbed Interferometer, Colavita et al. 1999), and VINCI (Kervella et al. 2004b). The relation by Di Benedetto 2005 is based on the NPOI (Navy Prototype Optical Interferometer, Armstrong et al. 1998) observations by Nordgren et al. (2001) of 24 giants and three dwarfs obtained in the optical domains (wavelength of reference of 740 nm). Thus, following Salsi et al. (in prep.), who found a different SBCR depending on the spectral type and class of the stars, we can consistently compare therelations by Pietrzyński et al. (2019), Di Benedetto (2005) and ours, which are all based on giants. On the color range considered, the relation in this work (rms of 0.039 mag) is consistent with the relation by Di Benedetto (2005) (rms of 0.04 mag) and with the relation by Pietrzyński et al. (2019) (rms of 0.018 mag).

It is well known that the extinction has a negligible effect on the SBCR because the color and surface brightness are sensitiveto it in almost the same way. The only difference comes from the extinction in K, which is about 10% of the extinction in V. Adding an extinction of 0.1 magnitude on a measurement, for instance, results in a brighter color of the star of about 0.054 magnitude, and similarly, in a brighter surface-brightness of 0.05 magnitude. In order to quantify the effect of extinction on the SBCR, we arbitrarily changed the absorption (AV) of all stars in our sample. We find that a higher extinction of 0.1 magnitude results in an increase in zero-point of the SBCR by 0.045 magnitude, which corresponds to the rms of our SBCR. This offset becomes 0.048 magnitude when we use the older relationship AK = 0.114 AV from Cardelli et al. (1989) instead of the relation we considered in Sect. 3 (AK = 0.089 AV) from Nishiyama et al. (2009). This means that even if the AV values of the star in the sample were all under-estimated by 0.1 magnitude (which is highly unlikely), our SBCR would still be consistent with the SBCR by Pietrzyński et al. (2019).

5 Conclusion

We observed eight red clump stars (late-type giants) with the VEGA/CHARA instrument and derived their angular diameters with an average precision of 2.4%. By combining these diameters with a homogeneous set of infrared photometry (Laney et al. 2012), we derived the slope and zero-point of the SBCR with a residual dispersion of 0.04 magnitude (or 1.8% precision in terms of angular diameter). The zero-points of the SBCR derived in the H band and in the visible domain are consistent at the 0.4% level (or 0.5σ). The angular diameters of four stars in our sample are consistent with those found by VLTI/PIONIER, which shows that the systematics of both instruments are under control It also indicates that it is possible to combine interferometric measurements in different bands when the SBCR is calibrated. All these results confirm the recent claim of a 1% precision and accuracy on the LMC distance by Pietrzyński et al. (2019).

thumbnail Fig. 2

Surface brightness – color relation derived from VEGA/CHARA data of eight late-type stars with a homogenous set of infrared photometry (Laney et al. 2012). The black arrow shows the effect of an additional extinction of 0.1 magnitude on a measurement. The gray zone corresponds to the rms.
Table 5

Fit of the SBCR. Slope and zero-point of the SBCR were first fit using Eq. (1).

Acknowledgements

We acknowledge M. Schultheis for fruitful discussions about the extinction of stars. This research has made use of the SIMBAD and VIZIER (http://cdsweb.u-strasbg.fr/) databases at CDS, Strasbourg (France) and of electronic bibliography maintained by the NASA/ADS system. This research has made use of the Jean-Marie Mariotti Center LITpro service co-developped by CRAL, IPAG and Lagrange (LITpro software available at http://www.jmmc.fr/litpro). This research has made use of the Jean-Marie Mariotti Center Aspro service (http://www.jmmc.fr/aspro). This work was supported by the “Programme National de Physique Stellaire (PNPS) of CNRS/INSU co-funded by CEA and CNES”. This work was supported by the “Action Spécifique pour la Haute Résolution Angulaire (ASHRA) of CNRS/INSU co-funded by CNES”. The CHARA Array is funded by the National Science Foundation through NSF grants AST-0606958 and AST-0908253 and by Georgia State University through the College of Arts and Sciences, as well as the W. M. Keck Foundation. R.L. has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement n. 664931. W.G. and G.P. gratefully acknowledge financial support for this work from the BASAL Centro de Astrofisica y Tecnologias Afines (CATA) AFB-170002. We acknowledge financial support for this work from ECOS-CONICYT grant C13U01. The authors acknowledge the support of the FrenchAgence Nationale de la Recherche (ANR), under grant ANR-15-CE31-0012-01 (project UnlockCepheids). Support from the Polish National Science Centre grants MAESTRO UMO-2017/26/A/ST9/00446 and from the IdPII 2015 0002 64 grant of the Polish Ministry of Science and Higher Education is also acknowl-edged. The research leading to these results has received funding from the European Re-search Council (ERC) under the European Unions Horizon 2020 research and innovation program (grant agreement No 695099).

Appendix A Plot of the CHARA/VEGA squared visibility measurements

thumbnail Fig. A.1

Squared visibility vs. spatial frequency for all stars in our sample with their corresponding statistical uncertainties (red dots). The solid blue lines indicate the best uniform-disk model obtained from the LITpro fitting software.
thumbnail Fig. A.2

Squared visibility vs. spatial frequency for all stars in our sample with their corresponding statistical uncertainties (red dots). The solid blue lines indicate the best uniform-disk model obtained from the LITpro fitting software.

Appendix B Table of the observing log

Table B.1

Observing log.

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All Tables

Table 1

Summary of the observing log.

In the text
Table 2

Reference stars and their parameters, including the spectral type, the visual magnitude (mV), and the predicted uniform-disk angular diameter (in mas) together with its corresponding uncertainty derived from the JMMC SearchCal software (Bonneau et al. 2006; Lafrasse et al. 2010).

In the text
Table 3

Angular diameters obtained with VEGA/CHARA and the corresponding surface brightnesses.

In the text
Table 4

Fundamental parameters of the stars in our sample based on spectroscopy and combined with photometry for stars in Luck & Heiter (2007).

In the text
Table 5

Fit of the SBCR. Slope and zero-point of the SBCR were first fit using Eq. (1).

In the text
Table B.1

Observing log.

In the text

All Figures

thumbnail Fig. 1

Angular diameters derived from VEGA/CHARA compared to those from PIONIER/VLTI. The blue line marks the 1:1 relation.
In the text
thumbnail Fig. 2

Surface brightness – color relation derived from VEGA/CHARA data of eight late-type stars with a homogenous set of infrared photometry (Laney et al. 2012). The black arrow shows the effect of an additional extinction of 0.1 magnitude on a measurement. The gray zone corresponds to the rms.
In the text
thumbnail Fig. A.1

Squared visibility vs. spatial frequency for all stars in our sample with their corresponding statistical uncertainties (red dots). The solid blue lines indicate the best uniform-disk model obtained from the LITpro fitting software.
In the text
thumbnail Fig. A.2

Squared visibility vs. spatial frequency for all stars in our sample with their corresponding statistical uncertainties (red dots). The solid blue lines indicate the best uniform-disk model obtained from the LITpro fitting software.
In the text

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