Astronomy c ESO 2014 Astrophysics - oca.eu · A&A 570, A104 (2014) DOI: 10.1051/0004-6361/201423772 c ESO 2014 Astronomy & Astrophysics Improving the surface brightness-color relation

AampA 570 A104 (2014)DOI 1010510004-6361201423772ccopy ESO 2014

Astronomyamp

Astrophysics

Improving the surface brightness-color relation for early-type starsusing optical interferometry

M Challouf12 N Nardetto1 D Mourard1 D Graczyk3 H Aroui2 O Chesneau1 O Delaa1 G Pietrzynski34W Gieren3 R Ligi1 A Meilland1 K Perraut56 I Tallon-Bosc78 H McAlister910 T ten Brummelaar10

J Sturmann10 L Sturmann10 N Turner10 C Farrington10 N Vargas10 and N Scott10

1 Laboratoire Lagrange UMR 7293 UNSCNRSOCA 06300 Nice Francee-mail mounirchalloufocaeu

2 Laboratoire Dynamique Moleacuteculaire et Mateacuteriaux Photoniques UR11ES03 Universiteacute de TunisESSTT Tunisie3 Departamento de Astronomiacutea Universidad de Concepcioacuten Casilla 160-C Concepcioacuten Paraguay4 Warsaw University Observatory AL Ujazdowskie 4 00-478 Warsaw Poland5 Univ Grenoble Alpes IPAG 38000 Grenoble France6 CNRS IPAG 38000 Grenoble France7 Universiteacute Lyon 1 Observatoire de Lyon 9 avenue Charles Andreacute 69230 Saint Genis Laval France8 CNRSUMR 5574 Centre de Recherche Astroph de Lyon Eacutecole Normale Supeacuterieure 69007 Lyon France9 Georgia State University PO Box 3969 Atlanta GA 30302-3969 USA

10 CHARA Array Mount Wilson Observatory 91023 Mount Wilson CA USA

Received 7 March 2014 Accepted 2 September 2014

ABSTRACT

Context The method of distance determination of eclipsing binaries consists in combining the radii of both components determinedfrom 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 004 magnitude) and more than 10 for early-type stars (or 02 mag)Aims The aim of this work is to improve the SBC relation for early-type stars in the minus1 le V minus K le 0 color domain using opticalinterferometryMethods Observations of eight B- and A-type stars were secured with the VEGACHARA instrument in the visible The deriveduniform disk angular diameters were converted into limb darkened angular diameters and included in a larger sample of 24 starsalready observed by interferometry in order to derive a revised empirical relation for O B A spectral type stars with a V minus K colorindex ranging from minus1 to 0 We also took the opportunity to check the consistency of the SBC relation up to V minus K 4 using100 additional measurementsResults 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 Lyrand δ Cyg with V minus K color ranging from minus070 to 002 and typical precision of about 15 Using our total sample of 132 stars withV minus K colors index ranging from about minus1 to 4 we provide a revised SBC relation For late-type stars (0 le V minus K le 4) the resultsare consistent with previous studies For early-type stars (minus1 le V minus K le 0) our new VEGACHARA measurements combined with acareful selection of the stars (rejecting stars with environment or stars with a strong variability) allows us to reach an unprecedentedprecision of about 016 magnitude or 7 in terms of angular diameterConclusions We derive for the first time a SBC relation for stars between O9 and A3 which provides a new and reliable tool for thedistance scale calibration

Key words stars early-type ndash techniques interferometric ndash stars distances ndash binaries eclipsing ndash methods observational ndashstars atmospheres

1 Introduction

The distance measurements to extragalactic targets in the lastcentury revolutionized our understanding of the distance scaleof the universe The distance to the Large Magellanic Cloudis a critical rung on the cosmic distance ladder and numerousindependent methods involving for instance RR Lyrae stars(Feast 1997 Szewczyk et al 2008 Pietrzynski et al 2008)Cepheids (Bohm-Vitense 1985 Evans 1991 1992 Freedmanamp Madore 1996 Freedman et al 2008) or red clump stars

Partly based on VEGACHARA observations Appendix A is available in electronic form athttpwwwaandaorg

(Udalski et al 1998ba Pietrzynski amp Gieren 2002 Laney et al2012) have been used to derive its distance

The main goal of the long term program called the Araucariaproject is to significantly improve the calibration of the cos-mic distance scale based on observations of several distance in-dicators in nearby galaxies (Gieren et al 2005) Eclipsing bi-nary systems are particularly important to provide the zero pointof the extragalactic distances and study in detail populationaldependence on other distance indicators like RR Lyrae starsCepheids red clump stars etc Thirteen long period systemscomposed of late-type giants were analyzed in the MagellanicClouds so far eight in the Large Magellanic Cloud (Pietrzynskiet al 2009 2013) and five in the Small Magellanic Cloud(Graczyk et al 2012 2014) For such systems the linear dimen-

Article published by EDP Sciences A104 page 1 of 13

AampA 570 A104 (2014)

sion of both components can be measured with a precision upto of 1 from the analysis of high-quality spectroscopic andphotometric data (eg Torres et al 2010) The distance to aneclipsing binary follows from the dimensions determined in thisway plus the angular diameters derived from the absolute sur-face brightness which is very well calibrated for late-type stars(Di Benedetto 2005) This conceptually very simple techniquevery weakly depends on reddening and metallicity and providesthe most accurate tool for measuring distances to nearby galax-ies (Pietrzynski et al 2013 Graczyk et al 2014)

However the heart of this method the surface brightness-color (SBC) relation is very well calibrated only for late-typestars which significantly limits its usage The late-type systemscomposed of main-sequence stars are usually faint while thosecomposed of giants have very long periods (several hundreddays) that makes them very difficult to find As a result onlyabout 45 late-type systems well suited to precise distance deter-mination have been discovered so far in the Magellanic Cloudsby the Optical Gravitational Microlensing Experiment (OGLEPawlak et al 2013 Graczyk et al 2011) On the other handmany more relatively bright systems are known in nearby galax-ies (Massey et al 2013 Graczyk et al 2011 Wyrzykowski et al2003 2004 Bonanos et al 2006 Macri et al 2001 Mochejskaet al 2001 Vilardell et al 2006 Pawlak et al 2013) Thereforein order to derive the distance to nearby galaxies and to study thegeometry of the Magellanic Clouds it is imperative to calibrateSBC relation for early-type stars

The purpose of this paper is to improve the SBC relation forearly-type stars by using the resolving power of the Visible spEc-troGraph and polArimeter (VEGA) beam combiner (Mourardet al 2009) operating at the focus of the Center for High AngularResolution Astronomy (CHARA) Array (ten Brummelaar et al2005) located at Mount Wilson Observatory (California USA)The CHARA array consists of six telescopes of 1 m in diameterconfigured in a Y shape which offers 15 different baselines from34 m to 331 m These baselines can achieve a spatial resolutionup to 03 mas in the visible which is necessary in order to resolveearly-type stars Early-type stars are very small in terms of angu-lar diameter and can be affected by several physical phenomenalike fast rotation winds and environment which can potentiallybias the interferometric measurements

This paper is structured as follows Section 2 is devoted toa description of the stars in our sample In Sect 3 we presentthe data reduction process and the method used to derive theangular diameters Section 4 is dedicated to the calibration ofthe SBC relation and we discuss our results in Sect 5 We drawconclusions in Sect 6

2 VEGACHARA observations of eight early-typestars

We carefully selected eight early-type stars with a (V minus K) colorindex ranging from minus070 to 002 They are north hemispheremain-sequence subgiant and giant stars (δ gt 4) with spectraltypes ranging from B1 to A1 They are much brighter (with avisual magnitude mV ranging from 16 to 47) than the limitingmagnitude of VEGA (about mV = 7 in medium spectral reso-lution) They are also bright in the K band (with a mK magni-tude lower than 51) which makes it possible to track the fringessimultaneously with the infrared CLIMB combiner (Sturmannet al 2010) All the apparent magnitudes in V and K bands thatwe have collected from the literature are in the Johnson system(Johnson et al 1966 see also Mermilliod et al 1997) The ac-curacy of their parallaxes π spans from 15 to 15 The color

excess E(V minus K) the visual absorption AV the effective tem-perature Teff the mass M the radius R the luminosity L thesurface gravity log g and the metallicity index [FeH] are listedin Table 1 We emphasize that for our purpose (limb-darkeningestimates see end of Sect 31) we do not need very precise es-timates of the fundamental parameters of the stars in our samplewhich explains why we do not provide any uncertainty on theseparameters in the second part of Table 1

Among the eight early-type stars in our sample there are sixlow rotators (λAql γ Ori γ Lyr ιHer 8 Cyg and ζ Per) and twofast rotating stars (δ Cyg and ζ Peg) In the following we definefast rotators as stars with vrot sin i gt 75 km sminus1 A theoreticalstudy which aims at quantifying the impact of fast rotation onthe SBC relation for early-type stars is currently in progress andwill be published in a forthcoming paper

We observed our sample stars from July 23 2011 toAugust 29 2013 using different suitable triplets available onthe CHARA array A summary of the observations is given inTable 2

3 The limb-darkened angular diameters

In this section we describe how we derive the limb-darkenedangular diameter for all the stars in our sample

31 Data reduction and methodology

The first step is to calibrate the visibility measurements of ourtargets using observations of reference stars These calibrators(Table 3) were selected using the SearchCal1 software devel-oped by the Jean-Marie Mariotti Center (JMMC Bonneau et al2006) The way this calibration is done is shown in Fig 1 in thecase of γ Lyr (data obtained on June 21 2012 with the E1E2W2three-telescope configuration) We used the standard sequenceC6-S-C6 in which S is the target and C6 is the reference starThe light blue dots are the raw visibilities obtained on the sciencestar for the three corresponding baselines E2E1 (upper panel)E2W2 (middle panel) and E1W2 (bottom panel) Our VEGAmeasurements are typically divided into 30 blocks of observa-tions and each block contains 1000 images with an exposuretime of 15 millisec For each block the raw squared visibility iscalculated using the auto-correlation mode (Mourard et al 20092011) The red dots in the figure represent the transfer functionobtained by comparing the expected visibility of the referencestar to the one that has been measured This transfer functionis then used to calibrate the visibilities obtained on the sciencetarget (blue dots) A cross-check of the quality of the transferfunction is usually done for several bandwidths and over thewhole night Under good seeing conditions the transfer func-tion of VEGACHARA is generally stable at the level of 2for more than one hour The squared calibrated visibilities V2

targetobtained from our VEGA observations are listed in Tables A1The systematic uncertainties that stem from the uncertainty onthe reference stars are negligible compared to the statistical un-certainties and are neglected in the rest of this study

The calibrated visibility curves obtained for each star inour sample (Fig 2) are then used to constrain a model of uni-form disk that contains only one parameter the so-called uni-form disk angular diameter (θUD) This is performed using theLITpro2 software developed by the JMMC (Tallon-Bosc et al2008) The following formula of Hanbury Brown et al (1974b)

1 Available at httpwwwjmmcfrsearchcal2 Available at httpwwwjmmcfrlitpro

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M Challouf et al Improving the surface brightness-color relation for early-type stars

2prime57primeprime+

1prime22primeprime+

0prime58primeprime+

7prime10primeprime+

0prime22primeprime+

3prime01primeprime+

9prime52primeprime

7prime50primeprime

7plusmn06

26plusmn02

2plusmn05

79plusmn01

17plusmn01

34plusmn01

6plusmn01

7plusmn04

003plusmn00

12plusmn00

00plusmn00

15plusmn01

04plusmn00

87plusmn00

05plusmn00

04plusmn00

(VminusK

03plusmn00

11plusmn00

00plusmn00

12plusmn01

03plusmn00

77plusmn00

04plusmn00

minus00

4ominus00

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Table 2 Summary of the observing log

Name 3 telescope configurations N Reference stars

γ Lyr E2E1W2 23 C6 C7γ Ori E2E1W2 W2W1S2 E2E1W2 8 C1 C2 C38 Cyg W2W1E1 8 C10ι Her W2W1E1 8 C4λ Aql S2S1W2 45 C5 C9ζ Per E2E1W2 6 C12δ Cyg E2E1W2 22 C8 C10ζ Peg E2E1W2 12 C11

Notes All the details are given in Table A1 N corresponds to the num-ber of visibility measurements for each star The reference stars usedare also indicated (cf Table 3)

Table 3 Reference stars and their parameters including the spectraltype the visual magnitude (mV) and the predicted uniform disk an-gular diameter (in mas) derived from the JMMC SearchCal software(Bonneau et al 2006)

No Reference SType mV θUD[R]stars [mas]

C1 HD 34989 B1V 57 0130 plusmn 0009C2 HD 37320 B8III 58 0153 plusmn 0011C3 HD 38899 B9IV 48 0265 plusmn 0019C4 HD 167965 B7IV 55 0150 plusmn 0011C5 HD 170296 A1IVV 46 0429 plusmn 0031C6 HD 174602 A3V 52 0330 plusmn 0024C7 HD 178233 F0III 55 0399 plusmn 0029C8 HD 184875 A2V 53 0295 plusmn 0021C9 HD 184930 B5III 43 0317 plusmn 0022C10 HD 185872 B9III 54 0200 plusmn 0014C11 HD 216735 A1V 49 0310 plusmn 0022C12 HD 22780 B7Vn 55 0167 plusmn 0012

provides an analytical way to convert the equivalent uniform diskangular diameter θUD into the limb-darkened disk θLD

θLD(λ) = θUD(λ)

⎡⎢⎢⎢⎢⎢⎣(1 minus Uλ

(1 minus 7Uλ15 )

⎤⎥⎥⎥⎥⎥⎦12

middot (1)

For each star the limb-darkening coefficient Uλ is derived fromthe numerical tables of Claret amp Bloemen (2011) These ta-bles are based on the ATLAS (Kurucz 1970) and PHOENIX(Hauschildt et al 1997) atmosphere models The input param-eters of these tables are the effective temperature (Teff) themetallicity ([FeH]) the surface gravity (log g) and the micro-turbulence velocity The steps for these quantities are 250 K 0505 and 2 km sminus1 respectively The three first parameters aregiven in Table 1 and were rounded for each star to the closestvalue found in the table of Claret The micro-turbulence veloc-ity has almost no impact on the derived limb-darkened diame-ter (fifth decimal) We took arbitrarily 8 km sminus1 for stars withTeft gt 15 000 K and 4 km sminus1for stars with Teft lt 15 000 KWe also consider the limb-darkening coefficient applicable to theR band of VEGA (UR in the following)

32 Results

The uniform disk angular diameter (θUD) the limb-darkening co-efficients (UR) and the derived limb-darkened angular diameters

Fig 1 Time sequences of raw visibilities of the science observations(light blue dots) calibrated (blue dots) using the transfer function (reddots)

(θLD) are listed in Table 4 for each star in our sample The valueof θLD ranges from 031 mas to 079 mas with a relative preci-sion from 05 to 35 (average of 15) The reduced χ2

red isfrom 04 to 29 depending on the dispersion of the calibrated vis-ibilities For γ Lyr our result (θUD = 0742 plusmn 0010 mas) agreesat the 1σ level with the measurements from the PAVOCHARAinstrument (θUD = 0729 plusmn 0008 mas Maestro et al 2013) Forγ Ori our angular diameter (θLD = 0715 plusmn 0005 mas) is con-sistent with the value derived from the Narrabri Stellar IntensityInterferometer (NSII) (θLD = 072 plusmn 004 mas Hanbury Brownet al 1974a) For other stars with angular diameters lower than06 mas (ιHer λ Aql 8 Cyg and ζ Per) and for the two fast rota-tors (ζ Peg and δ Cyg) there are no interferometric observationsavailable to our knowledge

For the two rotators we derive the apparent oblateness us-ing the approximate relation provided by van Belle et al (2006)their Eq (A1) Rb

Ra 1minus (v sin i)2 Rb

2GM where Rb Ra M and G arethe major and minor apparent radius of the star its mass and thegravitational constant We find Rb

Ra= 107 for ζ Peg considering

Rb R = 403 R and M = 322 M where R is the mean ra-dius (see Table 1) while the rotational projected velocity v sin iis set to 140 km sminus1(Abt et al 2002) For δ Cyg we find similarlyRbRa= 106 considering v sin i = 140 km sminus1(Slettebak et al 1975

Gray 1980 Carpenter et al 1984 Abt amp Morrell 1995 Abt et al2002 van Belle 2012) Consequently our data might be sensi-tive to the expected gravity darkening intensity distribution andthe flatness of the star However this also depends on the base-line orientation For both stars the three telescopes (Table 2) arealigned Thus even if our reduced χ2

red are rather low (17 forζ Peg and 12 for δ Cyg) we cannot exclude a bias on our de-

A104 page 4 of 13

Fig 2 Squared visibility versus spatial frequency for all stars in our sample with their corresponding statistical uncertainties The red solid linesindicate the best uniform disk model obtained from the LITpro fitting software

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Table 4 Angular diameters obtained with VEGACHARA and the corresponding surface brightness

Star (V minus K)0 θUD [mas] χ2 UR θLD [mas] S v[mag]

λ Aql minus0265 plusmn 0055 0529 plusmn 0003 10 0301 0544 plusmn 0003 2079 plusmn 0030γ Lyr minus0102 plusmn 0072 0742 plusmn 0010 29 0402 0766 plusmn 0010 2544 plusmn 0059γ Ori minus0703 plusmn 0097 0701 plusmn 0005 04 0269 0715 plusmn 0005 0909 plusmn 00818 Cyg minus0492 plusmn 0147 0229 plusmn 0011 13 0299 0234 plusmn 0011 1456 plusmn 0177ι Her minus0459 plusmn 0076 0304 plusmn 0010 12 0280 0310 plusmn 0010 1225 plusmn 0082ζ Per minus0592 plusmn 0092 0531 plusmn 0007 12 0343 0542 plusmn 0007 0652 plusmn 0081ζ Peg minus0204 plusmn 0055 0539 plusmn 0009 17 0442 0555 plusmn 0009 2076 plusmn 0152δ Cyg +0021 plusmn 0055 0766 plusmn 0004 13 0408 0791 plusmn 0004 2318 plusmn 0129

Notes The systematical uncertainties for the two fast rotating stars ζ Peg and δ Cyg are of 0039 mas and 0047 mas respectively (see Sect 32)

rived limb-darkened angular diameters In order to get a roughestimate of this bias we only consider in first approximation theoblateness of the star while the gravity darkening is set to benegligible As a consequence if the orientation of the baselineis aligned with the polar or equatorial axis we can estimate amaximum systematic error of about 0039 mas (6) for ζ Pegwhile we find 0047 mas (7) for δ Cyg We translate these un-certainties in terms of S v magnitude in Sect 4

4 The calibration of the surface brightness relation

41 Methodology

As already mentioned in the introduction the SBC relation is avery robust tool for the distance scale calibration The surfacebrightness SV of a star is linked to its visual intrinsic dereddenedmagnitude mV0 and its limb-darkened angular diameter θLD bythe following relation

S V = mV0 + 5 log θLDmiddot (2)

Instead of SV the surface brightness parameter FV = 42207 minus01S V is often adopted in the literature to determine the stel-lar angular diameters (Barnes amp Evans 1976) In order to derivemV0 we first selected the apparent mV magnitudes for all thestars in our sample (Mermilliod et al 1997) These magnitudesare expressed in the Johnson system (Johnson et al 1966) andtheir typical uncertainty is of about 0015 mag In order to cor-rect these magnitudes from the reddening we then use the fol-lowing formulae mV0 = mV minus AV where AV is the extinctionin the V band Determining the extinction is a difficult task Weadopt the following strategy For stars lying closer than 75 pc weuse the simple relation

AV =08π (3)

where π is the parallax of the stars [in mas] This equation isstandard in the literature (Blackwell et al 1990 Di Benedetto1998 2005) The corresponding uncertainty is set to 001 mag

For distant stars we derive the absorption using the (B minus V)extinction (Laney amp Stobie 1993)

AV = 31E(Bminus V)middot (4)

The difficulty is then to derive E(B minus V) We have several pos-sibilities First we use the so-called Q method with Q = (U minusB) minus 072(B minus V) which was originally proposed by Johnson amp

Morgan (1953) The value of Q is derived for each star using ob-served UBV magnitudes from (Ducati 2002) Then a relation be-tween (BminusV)0 and Q can be found in Pecaut amp Mamajek (2013)and E(BminusV) is finally derived using E(BminusV) = (BminusV)minus(BminusV)0

Second from the spectral type of the stars in our sample wecan derive their intrinsic colors in different bands using Table 5of Wegner (1994) We thus obtain (B minus V)0 (V minus R)0 (V minus I)0(V minus J)0 (V minus H)0 and (V minus K)0 Once compared with the ob-served colors from Ducati (2002) we derive E(BminusV) E(V minusR)E(V minus I) E(V minus J) E(V minusH) and E(V minus K) and we finally useTable 2 (Col 4) from Fitzpatrick (1999) and assume total to se-lective extinction ratio in B-band AB

E(BminusV) = 41447 (Table 3 fromCardelli et al 1989) to perform a conversion into E(BminusV) usingthe following equations

E(B minus V) =E(V minus R)(AB minus AV )

(AV minus AR)= 12820E(V minus R) (5)

E(B minus V) =E(V minus I)(AB minus AV )

(AV minus AI)= 06536E(V minus I) (6)

E(B minus V) =E(V minus J)(AB minus AV )

(AV minus AJ)= 04464E(V minus J) (7)

E(B minus V) =E(V minus H)(AB minus AV )

(AV minus AH)= 03891E(V minus H) (8)

E(B minus V) =E(V minus K)(AB minus AV )

(AV minus AK)= 03650E(V minus K)middot (9)

We finally obtain seven values of the extinction (Q method andsix values derived from Table 5 of Wegner 1994) These quan-tities are averaged and their statistical dispersion provides a re-alistic uncertainty (indicated in Table 1 for the VEGA sample)However the Q method is applicable only for stars of class IVand V while Table 5 of Wegner (1994) can be used only for spec-tral types O and B We thus have in some cases fewer than sevenvalues And even in the case of γ Lyr for instance (which isan A1III star standing at a distance greater than 75 pc) we usedother E(BminusV) estimates available in the literature (see Table 1)The uncertainty on S V is finally derived from the uncertainty onmV (typically 0015) the angular diameter (see Table 4) and AV

In order to mitigate the effects from a somewhat erroneouscalibration of the intrinsic colors we recalculate (V minus K)0 fromthe derived E(B minus V) value First we calculate E(B minus V) fromaveraging via Eqs (5)minus(9) Then using this value and Eq (9)we derive E(V minus K) (given in Table 1) From E(V minus K) mV andmK we obtain (V minus K)0 The uncertainty on (V minus K)0 is derivedassuming an uncertainty of 0015 for mV 003 for mK (follow-ing Di Benedetto 2005) and the uncertainty on E(V minus K) itselfderived from the uncertainty obtained on E(B minus V) The mV and

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Fig 3 Relation between visual surface brightness S V as a function ofthe color index (V minus K)0 The black light blue green brown and redmeasurements are from Di Benedetto (2005) Boyajian et al (2012)Hanbury Brown et al (1974a) Maestro et al (2013) and VEGA (thiswork) respectively The red line corresponds to our fit when consider-ing all stars The rms of the difference between the surface brightnesscomputed from our fit and measured surface brightness is presented inthe lower panels (see the text for more detail)

mK magnitudes for the stars in our sample are given in Table 1together with π E(V minus K) and AV The derived values of thesurface brightness for each star are given in Table 4

In order to calibrate the SBC relation we also need to com-bine the eight limb-darkened angular diameters derived fromthe VEGA observations with different sets of diameters alreadyavailable in the literature

42 A revised SBC relation for late- and early-type stars

Historically the SBC was first derived from interferometric ob-servations of 18 stars by Wesselink (1969) using the (B minus V) in-

Fig 4 Same as Fig3 but with the names of the VEGACHARA starsin our sample (in red)

dex Five years later the apparent angular diameters of 32 starsin the spectral range O5 to F8 have been measured using the NSII(Hanbury Brown et al 1974a) Based on this sample Barneset al (1976) and Barnes amp Evans (1976) calibrated the SBC forlate-type and early-type stars respectively but this was not donewith the VminusK color index In order to constrain the SBC relationas a function of V minus K we therefore use these 32 angular diame-ters (but 6 are rejected see below) This is the first set of data wehave used We emphasize that the (V minus K)0 color index is usu-ally used to calibrated the SBC relation because it provides thelowest rms and it is mostly parallel to the reddening vector onthe S vminus (V minusK) diagram Moreover for all the datasets we haveconsidered we have recalculated the (V minus K)0 and AV values ina similar way as for the VEGA objects (see Sect 41)

More than ten years later Di Benedetto (1998) made a care-ful compilation of 22 stars (with A F G K spectral types) forwhich angular diameters were available in the literature and cal-ibrated the SBC relation Moreover the direct application of theSBC relation to Cepheids was done by Fouque amp Gieren (1997)and Di Benedetto (1998) Later 27 stars were measured by NPOIand Mark III optical interferometers and the derived high preci-sion angular diameters were published by Nordgren et al (2001)and Mozurkewich et al (2003) respectively Finally using acompilation of 29 dwarfs and subgiant (including the sun) inthe 00 le (V minus K)0 le 60 color range Kervella et al (2004)calibrated for the first time a linear SBC relation with an intrin-sic dispersion of 002 mag or 1 in terms of angular diameterA short time later Di Benedetto (2005) made the same kind ofcompilation but with 45 stars in the minus01 le (V minusK)0 le 37 colorrange (accuracy of 004 mag or 2 in terms of angular diame-ter) We use this larger second data set for our analysis

One year later Bonneau et al (2006) provided a SBC rela-tion (as function of V minus K color magnitude) based on interfero-metric measurements lunar occultation and eclipsing binariesWe compare our results with those of Bonneau et al (2006) andalso Di Benedetto (2005) in Sect 5

Recently Boyajian et al (2012) enlarged the sample to44 main-sequence A- F- and G-type stars using CHARAarray measurements In addition ten stars with spectral typesfrom B2 to F6 were observed using the astronomical visi-ble observations (PAVO) beam combiner at the CHARA array

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(Maestro et al 2013) and these recent CHARA measurementshave been incorporated in our analysis

However in order to derive a SBC relation accurate enoughfor distance determination one has to perform a consistent selec-tion Our strategy is the following we consider all stars in mul-tiple systems (as soon as the companion is far and faint enoughnot to contaminate interferometric measurements) fast rotatorsand single stars Fast rotating stars should be included as theyimprove the statistics of the relation (in particular for early-typeobjects) even if a slight bias is not excluded as discussed inSect 32 (see also next section) Conversely we exclude starswith environments (like Be stars with strong wind) or stars witha strong variability Following these criteria we found sevenstars to reject The first one is Zeta Orionis (ζ Ori) Its angulardiameter measured by Hanbury Brown et al (1974a) is mostprobably biased by a companion which was discovered laterand with a separation of 40 mas (Hummel et al 2000 2013)Kappa Orionis (κ Ori) shows a P Cygni profile in Hα caused bya stellar wind (Searle et al 2008 Stalio et al 1981 Cassinelliet al 1983) Delta Scorpii (δ Sco) is an active binary star ex-hibiting the Be phenomenon (Meilland et al 2013) Gamma2Velorum (γ2 Vel) is a binary system with a large spectral con-tribution from the Wolf-Rayet star (Millour et al 2007) ZetaOphiuchi (ζ Oph) is a magnetic star of Oe-type (Hubrig et al2011) Alpha Virginis (α Vir) is a double-lined spectroscopicbinary (B1V+B4V) with an ellipsoidal variation of 003 magdue to tidal distortion (Harrington et al 2009) The last oneZeta Cassiopeiae (ζ Cas) is in the PAVO sample (Maestro et al2013) It stands at 7σ from the relation It is a β Cepheid andthe photometric contamination by a surrounding environmentandor a close companion is not excluded (Sadsaoud et al 1994Nardetto et al 2011)

We finally end with 26 stars from Hanbury Brown et al(1974a) 44 stars from Boyajian et al (2012) 9 stars fromMaestro et al (2013) and 45 values of S v from Di Benedetto(2005) to which we can add our eight angular diameters ob-tained with VEGACHARA The total sample is composedof 132 stars (with minus0876 lt V minus K lt 369) including 32 early-type stars with minus1 lt V minus K lt 0 Using this sample of 132 starswe find the relation

S v =n=5sum

Cn(V minus K)n0 (10)

with C0 = 2624 plusmn 0009 C1 = 1798 plusmn 0020 C2 = minus0776 plusmn0034 C3 = 0517 plusmn 0036 C4 = minus0150 plusmn 0015 and C5 =0015 plusmn 0002 Uncertainties on coefficients of the SBC relationdo not take into account the X-axis uncertainties on (V minus K)0This relation can be used consistently in the range minus09 leV minus K le 37 with σS v = 010 mag This corresponds to a rel-ative precision on the angular diameter of σθ

θ= 461σS V 46

derived from Eq (5) of Di Benedetto (2005) For stars earlierthan A3 (minus09 lt V minus K lt 00) we successfully reached amagnitude precision of σ = 016 or 73 in terms of angulardiameter

5 Discussion

Figure 3a shows the resulting SB relation as a function of the(V minus K)0 color index for the five different data sets we haveconsidered The VEGA data appear in red in the figure Theresidual OminusCv which is the difference obtained between themeasured surface brightness (O) and the relation provided by

Eq (10) (Cv) is shown in Fig 3b In the following we defineσ+ and σminus as the positive and negative standard deviation Weobtain σ+ = 007 and σminus = 009 for 0 lt V minus K lt 37 (late-type stars dot-dashed line in the figure) and σ+ = 013 mag andσminus = 018 mag for minus09 lt V minus K lt 0 (early-type stars dottedline in the figure) In Fig 3c we derive the residual comparedto the Di Benedetto (2005) relation (Eq (2)) which is applicableonly in the minus01 lt V minusK lt 4 color domain We obtain a residual(OminusCd) which are similar σ+ = 008 mag and σminus = 007 magThis basically means that improving the statistics does not im-prove the thinnest of the relations For this purpose a homoge-neous set of V and K photometry is probably required

We also compare our results with those of Bonneau et al(2006) which is to our knowledge the only SBC relation ver-sus V minusK provided for early-type stars in the literature (actuallythe relation is set from minus11 to 7 their Table 2) but instead ofusing Eq (2) they considered another quantity θ

930610minusV5

therefore made a conversion to compare with the S V quantityThe residual (OminusCb) is shown in Fig 3d We findσ+ = 010 magand σminus = 011 mag for 0 lt V minus K lt 4 (or late-type stars) andσ+ = 023 mag and σminus = minus023 mag for minus1 lt V minus K lt 0(or early-type stars) These residuals are significantly larger thenthe ones obtained when using our Eq (10) or Eq (2) fromDi Benedetto (2005)

In Fig 3a we also have indicated the fast rotating stars andbinaries In Fig 4 we provide a zoom of the SBC relation overthe minus1 lt V minus K lt 025 color range In this zoom we havealso indicated the uncertainties and the names of the stars in ourVEGA sample We find that the OminusCv residual in the (V minus K)color range minus1 to 0 is σ = 006 σ = 017 and σ = 018 forstars in binary systems (6) for fast rotating stars (8) and forsingle stars (18) We note the following points

First we want to emphasize that a careful selection (by re-jected stars with environment and stars with companions in con-tact) in particular in the range of minus1 lt V minus K lt 0 can sig-nificantly improve the precision on the SBC relation We obtainσ 04 otherwise

Second the dispersion of the OminusC residual for stars in binarysystems is significantly lower (006) than to the one obtainedwith the whole sample (about 016) which indicates that inter-ferometric and photometric measurements are not contaminatedby the binarity

Third we obtain a large dispersion (σ = 017) for fast ro-tating stars Five stars are beyond 1σ while three are withinincluding ζ Peg in our VEGA sample (see Fig 3b) Delta Cygni(δ Cyg) is in particular at 2σ In Sect 32 we estimated theimpact of the fast rotation on the angular diameters of ζ Pegand δ Cyg to be 0039 mas and 0047 mas respectively UsingEq (2) it translates into a magnitude effect of plusmn0150 mag andplusmn0127 mag respectively As already said fast rotation mod-ifies several stellar properties such as the shape of the photo-sphere (Collins 1963 Collins amp Harrington 1966) and its bright-ness distribution (von Zeipel 1924ab) which should be takeninto consideration However studying these effects requires ded-icated modeling and this will be done in a forthcoming paperFinally in our VEGA sample four stars are beyond 1σ from therelation but when also considering the uncertainty in VminusK theyremain consistent with the relation (see Fig 4)

A104 page 8 of 13

Fig 5 Relation between the visual surface brightness S V and the colorindex (V minus K)0 for luminosity class I () luminosity class II (lowast) lumi-nosity class III (times) luminosity class IV () and luminosity class V ()

Fourth we calculate the SBC relation for luminosity classes Iand II III IV and V (Fig 5) We obtain the following results

minus 088 le (V minus K)0 le 321

S v = 2291 + 2151(V minus K)0 minus 0461(V minus K)20 + 0073(V minus K)3

0[σS v = 008 magσθ 35 12 stars Class I + II

0[σS v = 007 magσθ 34 41 stars Class III

0[σS v = 010 magσθ 48 79 stars Class IV + V

] middot (13)

We find a slight difference in the zero-points of these relationsTheir dispersion is however similar about 009 mag which isslightly lower than the global dispersion of 016 mag that weobtain when considering the whole sample

6 Conclusions

Taking advantage of the unique VEGACHARA capabilities interms of spatial resolution we determined the angular diametersof eight bright early-type stars in the visible with a precision ofabout 15 By combining these data with previous angular di-ameter determinations we provide for the very first time a SBCrelation for early-type stars with a precision of about 016 magwhich means that this SBC relation can be used to derive theangular diameter of early-type stars with a precision of 73This relation is a powerful tool for the distance scale calibrationas it can be used to derive the individual angular diameters ofdetached early-type and thus bright eclipsing binary systemsIt will be used in the course of the Araucaria Project (Gierenet al 2005) to derive the distance of different galaxies in theLocal Group for exemple M33 As the eclipsing binary methodis independent of the metallicity of the star it can be used as areference to test the impact of the metallicity on several other

distance indicators in particular the Cepheids In the course ofthe Araucaria project we also aim to test the method consistentlyon galactic early-type eclipsing binaries using photometry spec-troscopy and interferometry

Acknowledgements This research has made use of the SIMBAD and VIZIERdatabases at the CDS (httpcdswebu-strasbgfr) Strasbourg (France)of the Jean-Marie Mariotti Center Aspro service (httpwwwjmmcfraspro) and of the electronic bibliography maintained by the NASAADSsystem The research leading to these results has received funding fromthe European Communityrsquos Seventh Framework Programme under GrantAgreement 312430 and financial support from the Ministry of Higher Educationand Scientific Research (MHESR) minus Tunisia The CHARA Array is fundedby the National Science Foundation through NSF grants AST-0606958 andAST-0908253 and by Georgia State University through the College of Artsand Sciences as well as the W M Keck Foundation WG gratefully ac-knowledges financial support for this work from the BASAL Centro deAstrofisica y Tecnologias Afines (CATA) PFB-062007 and from the MilleniumInstitute of Astrophysics (MAS) of the Iniciativa Cientifica Milenio delMinisterio de Economia Fomento y Turismo de Chile project IC120009We acknowledge financial support for this work from ECOS-CONICYT grantC13U01 Support from the Polish National Science Center grant MAESTRO201206AST900269 is also acknowledged We also wish to thank the refereeDr Puls for his numerous and precise suggestions for improving the photomet-ric aspects of the paper This was an enormous help in refining our results Thisresearch has largely benefited from the support suggestions advice of our col-league Olivier Chesneau who passed away this spring The whole team wish topay homage to him

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1990 AampA 232 396Bohm-Vitense E 1985 ApJ 296 169Bonanos A Z Stanek K Z Kudritzki R P et al 2006 ApJ 652 313Bonneau D Clausse J-M Delfosse X et al 2006 AampA 456 789Boyajian T S McAlister H A van Belle G et al 2012 ApJ 746 101Cardelli J A Clayton G C amp Mathis J S 1989 ApJ 345 245Carpenter K G Slettebak A amp Sonneborn G 1984 ApJ 286 741Cassinelli J P Myers R V Hartmann L Dupree A K amp Sanders W T

1983 ApJ 268 205Claret A amp Bloemen S 2011 AampA 529 A75Collins II G W 1963 ApJ 138 1134Collins II G W amp Harrington J P 1966 ApJ 146 152Di Benedetto G P 1998 AampA 339 858Di Benedetto G P 2005 MNRAS 357 174Ducati J R 2002 VizieR Online Data Catalog II237Evans N R 1991 ApJ 372 597Evans N R 1992 ApJ 389 657Feast M W 1997 MNRAS 284 761Fitzpatrick E L 1999 PASP 111 63Fitzpatrick E L amp Massa D 2005 AJ 129 1642Fouque P amp Gieren W P 1997 AampA 320 799Freedman W L amp Madore B F 1996 in Clusters Lensing and the Future of

the Universe eds V Trimble amp A Reisenegger ASP Conf Ser 88 9Freedman W L Madore B F Rigby J Persson S E amp Sturch L 2008

ApJ 679 71Gieren W Pietrzynski G Soszynski I et al 2005 ApJ 628 695Gies D R amp Lambert D L 1992 ApJ 387 673Graczyk D Soszynski I Poleski R et al 2011 Acta Astron 61 103Graczyk D Pietrzynski G Thompson I B et al 2012 ApJ 750 144Graczyk D Pietrzynski G Thompson I B et al 2014 ApJ 780 59Gray D F 1980 PASP 92 771Hanbury Brown R Davis J amp Allen L R 1974a MNRAS 167 121Hanbury Brown R Davis J Lake R J W amp Thompson R J 1974b

MNRAS 167 475Harrington D Koenigsberger G Moreno E amp Kuhn J 2009 ApJ 704 813Hauschildt P H Baron E amp Allard F 1997 ApJ 483 390Hohle M M Neuhaumluser R amp Schutz B F 2010 Astron Nachr 331 349Hubrig S Oskinova L M amp Schoumlller M 2011 Astron Nachr 332 147Hummel C A White N M Elias II N M Hajian A R amp Nordgren T E

2000 ApJ 540 L91

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Hummel C A Rivinius T Nieva M-F et al 2013 AampA 554 A52Johnson H L amp Morgan W W 1953 ApJ 117 313Johnson H L Mitchell R I Iriarte B amp Wisniewski W Z 1966

Communications of the Lunar and Planetary Laboratory 4 99Kervella P Theacutevenin F Di Folco E amp Seacutegransan D 2004 AampA 426 297Kurucz R L 1970 SAO Special Report 309Laney C D amp Stobie R S 1993 MNRAS 263 921Laney C D Joner M D amp Pietrzynski G 2012 MNRAS 419 1637Lyubimkov L S Rachkovskaya T M Rostopchin S I amp Lambert D L

2002 MNRAS 333 9Macri L M Stanek K Z Sasselov D D Krockenberger M amp Kaluzny J

2001 AJ 121 870Maestro V Che X Huber D et al 2013 MNRAS 434 1321Massey P Neugent K F Hillier D J amp Puls J 2013 ApJ 768 6McDonald I Zijlstra A A amp Boyer M L 2012 MNRAS 427 343Meilland A Stee P Spang A et al 2013 AampA 550 L5Mermilliod J-C Mermilliod M amp Hauck B 1997 AampAS 124 349Millour F Petrov R G Chesneau O et al 2007 AampA 464 107Mochejska B J Kaluzny J Stanek K Z amp Sasselov D D 2001 AJ 122

1383Mourard D Clausse J M Marcotto A et al 2009 AampA 508 1073Mourard D Beacuterio P Perraut K et al 2011 AampA 531 A110Mozurkewich D Armstrong J T Hindsley R B et al 2003 AJ 126 2502Nardetto N Mourard D Tallon-Bosc I et al 2011 AampA 525 A67Nordgren T E Sudol J J amp Mozurkewich D 2001 AJ 122 2707Pasinetti Fracassini L E Pastori L Covino S amp Pozzi A 2001 AampA 367

521Pawlak M Graczyk D Soszynski I et al 2013 Acta Astron 63 323Pecaut M J amp Mamajek E E 2013 ApJS 208 9Pietrzynski G amp Gieren W 2002 AJ 124 2633Pietrzynski G Gieren W Szewczyk O et al 2008 AJ 135 1993Pietrzynski G Thompson I B Graczyk D et al 2009 ApJ 697 862

Pietrzynski G Graczyk D Gieren W et al 2013 Nature 495 76Sadsaoud H Le Contel J M Chapellier E Le Contel D amp

Gonzalez-Bedolla S 1994 AampA 287 509Searle S C Prinja R K Massa D amp Ryans R 2008 AampA 481 777Slettebak A Collins II G W Parkinson T D Boyce P B amp White N M

1975 ApJS 29 137Stalio R Sedmak G amp Rusconi L 1981 AampA 101 168Sturmann J ten Brummelaar T Sturmann L amp McAlister H A 2010 in

SPIE Conf Ser 7734 45Szewczyk O Pietrzynski G Gieren W et al 2008 AJ 136 272Tallon-Bosc I Tallon M Thieacutebaut E et al 2008 SPIE 7013ten Brummelaar T A McAlister H A Ridgway S T et al 2005 ApJ 628

453Tetzlaff N Neuhaumluser R amp Hohle M M 2011 MNRAS 410 190Torres G Andersen J amp Gimeacutenez A 2010 AampARv 18 67Udalski A Pietrzynski G Wozniak P et al 1998a ApJ 509 L25Udalski A Szymanski M Kubiak M et al 1998b Acta Astron 48 1van Belle G T 2012 AampARv 20 51van Belle G T Ciardi D R Ten Brummelaar T et al 2006 ApJ 637

494van Leeuwen F 2007 AampA 474 653Vilardell F Ribas I amp Jordi C 2006 AampA 459 321von Zeipel H 1924a MNRAS 84 665von Zeipel H 1924b MNRAS 84 684Wegner W 1994 MNRAS 270 229Wesselink A J 1969 MNRAS 144 297Wu Y Singh H P Prugniel P Gupta R amp Koleva M 2011 AampA 525

A71Wyrzykowski L Udalski A Kubiak M et al 2003 Acta Astron 53 1Wyrzykowski L Udalski A Kubiak M et al 2004 Acta Astron 54 1Zorec J amp Royer F 2012 AampA 537 A120Zorec J Cidale L Arias M L et al 2009 AampA 501 297

Pages 11 to 13 are available in the electronic edition of the journal at httpwwwaandaorg

A104 page 10 of 13

Appendix A

Table A1 Journal of the observations

Star Date obs TU HA MJD λ Base Arg V2plusmnstat plusmnsyst

[yyyy-mm-dd] [h] [h] [days] [nm] [m] [deg]

γ Ori 2011-10-12 10206 minus1707 55 8455 710 65574 minus114974 0830 plusmn 0059 plusmn 00012011-10-12 10209 minus1704 55 8455 7315 65578 minus114977 0817 plusmn 0064 plusmn 00002011-10-12 10194 minus1719 55 8455 7315 153578 minus110390 0228 plusmn 0019 plusmn 00012011-10-13 9654 minus2194 55 8465 710 64439 minus114627 0828 plusmn 0078 plusmn 00002011-10-13 9708 minus2141 55 8465 710 149824 minus110214 0222 plusmn 0036 plusmn 00012011-10-13 9693 minus2155 55 8465 7315 149659 minus110214 0244 plusmn 0024 plusmn 00012011-11-22 10851 1633 55 8865 730 104340 92583 0554 plusmn 0038 plusmn 00012011-12-10 7745 minus02984 55 9045 7085 154258 minus112675 0220 plusmn 0012 plusmn 0006

γ Lyr 2011-07-27 5633 minus0916 55 7685 715 654495 minus116028 0809 plusmn 0060 plusmn 00082011-07-27 5633 minus0916 55 7685 715 154448 minus109563 0168 plusmn 0020 plusmn 00102011-07-27 5633 minus0916 55 7685 735 654495 minus116028 0907 plusmn 0050 plusmn 00092011-07-27 5633 minus0916 55 7685 735 154448 minus109563 0188 plusmn 0014 plusmn 00122011-07-27 5633 minus0916 55 7685 735 219605 minus111486 0002 plusmn 0066 plusmn 00002011-09-01 5244 1060 55 8045 715 63897 minus133244 0699 plusmn 0056 plusmn 00072011-09-01 5244 1060 55 8045 715 152073 minus126300 0190 plusmn 0026 plusmn 00122011-09-01 5244 1060 55 8045 715 215640 minus128353 0016 plusmn 0011 plusmn 00022011-09-01 5244 1060 55 8045 735 63897 minus133244 0784 plusmn 0044 plusmn 00072011-09-01 5244 1060 55 8045 735 152073 minus126300 0222 plusmn 0020 plusmn 00132011-09-01 5244 1060 55 8045 735 215640 minus128353 0005 plusmn 0012 plusmn 00002012-06-21 7755 minus1103 56 0985 707 65125 minus114643 0659 plusmn 0025 plusmn 00052012-06-21 7755 minus1103 56 0985 707 153401 minus108177 0239 plusmn 0040 plusmn 00102012-06-21 7755 minus1103 56 0985 707 218236 minus110103 0026 plusmn 0050 plusmn 00022012-06-21 7755 minus1103 56 0985 7305 65125 minus114643 0684 plusmn 0024 plusmn 00042012-06-21 7755 minus1103 56 0985 7305 153401 minus108177 0178 plusmn 0017 plusmn 00072012-06-21 7755 minus1103 56 0985 7305 218236 minus110103 0046 plusmn 0044 plusmn 00042012-06-21 9712 0858 56 0985 7075 64388 minus131216 0719 plusmn 0044 plusmn 00052012-06-21 9712 0858 56 0985 7075 153274 minus124351 0167 plusmn 0018 plusmn 00072012-06-21 9712 0858 56 0985 7075 217337 minus126380 0003 plusmn 0025 plusmn 00002012-06-21 9712 0858 56 0985 7305 64388 minus131216 0672 plusmn 0046 plusmn 00042012-06-21 9712 0858 56 0985 7305 153274 minus124351 0153 plusmn 0020 plusmn 00062012-06-21 9712 0858 56 0985 7305 217337 minus126380 0043 plusmn 0045 plusmn 0003

λ Aql 2013-07-24 8216 1388 56 4965 5325 29264 minus23993 1003 plusmn 0095 plusmn 00022013-07-24 8216 1388 56 4965 5325 160270 minus35208 0205 plusmn 0020 plusmn 00232013-07-24 8216 1388 56 4965 5325 189061 minus33483 0162 plusmn 0088 plusmn 00192013-07-24 8216 1388 56 4965 5475 29264 minus23993 1025 plusmn 0131 plusmn 00052013-07-24 8216 1388 56 4965 5475 160270 minus35208 0201 plusmn 0015 plusmn 00292013-07-24 8216 1388 56 4965 5475 189061 minus33483 0096 plusmn 0024 plusmn 00212013-07-24 8925 2100 56 4965 5325 30595 minus28275 0950 plusmn 0107 plusmn 00022013-07-24 8925 2100 56 4965 5325 167700 minus37803 0142 plusmn 0010 plusmn 00122013-07-24 8925 2100 56 4965 5325 197938 minus36337 0042 plusmn 0061 plusmn 00052013-07-24 8925 2100 56 4965 5475 30595 minus28275 0911 plusmn 0041 plusmn 00022013-07-24 8925 2100 56 4965 5475 167700 minus37803 0169 plusmn 0012 plusmn 00142013-07-24 8925 2100 56 4965 5475 197938 minus36337 0082 plusmn 0021 plusmn 00102013-07-24 9319 2494 56 4965 5325 31311 minus30138 0924 plusmn 0071 plusmn 000262013-07-24 9319 2494 56 4965 5325 171056 minus38738 0094 plusmn 0011 plusmn 00082013-07-24 9319 2494 56 4965 5325 202070 minus37410 0005 plusmn 0042 plusmn 00002013-07-24 9319 2494 56 4965 5475 31311 minus30138 0928 plusmn 0050 plusmn 00022013-07-24 9319 2494 56 4965 5475 171056 minus38738 0170 plusmn 0009 plusmn 00142013-07-24 9319 2494 56 4965 5475 202070 minus37410 0045 plusmn 0021 plusmn 00052013-07-24 9802 2979 56 4965 5325 32121 minus31950 0928 plusmn 0105 plusmn 00022013-07-24 9802 2979 56 4965 5325 174274 minus39432 0116 plusmn 0013 plusmn 00112013-07-24 9802 2979 56 4965 5325 206164 minus38270 0028 plusmn 0062 plusmn 00042013-07-24 9802 2979 56 4965 5475 32121 minus31950 0998 plusmn 0090 plusmn 00022013-07-24 9802 2979 56 4965 5475 174274 minus39432 0105 plusmn 0012 plusmn 00092013-07-24 9802 2979 56 4965 5475 206164 minus38270 0068 plusmn 0043 plusmn 00092013-07-24 8215 1388 56 4965 7030 29263 minus23990 0972 plusmn 0235 plusmn 00012013-07-24 8215 1388 56 4965 7030 160265 minus35206 0488 plusmn 0155 plusmn 0037

A104 page 11 of 13

AampA 570 A104 (2014)

Table A1 continued

2013-07-24 8215 1388 56 4965 7340 29263 minus23990 0918 plusmn 0118 plusmn 00022013-07-24 8215 1388 56 4965 7340 160265 minus35206 0467 plusmn 0021 plusmn 00352013-07-24 8215 1388 56 4965 7340 189055 minus33480 0236 plusmn 0068 plusmn 00262013-07-24 8925 2099 56 4965 7030 30594 minus28272 0935 plusmn 0124 plusmn 00072013-07-24 8925 2099 56 4965 7030 167695 minus37802 0293 plusmn 0119 plusmn 00242013-07-24 8925 2099 56 4965 7340 30594 minus28272 0895 plusmn 0075 plusmn 00012013-07-24 8925 2099 56 4965 7340 167695 minus37802 0399 plusmn 0015 plusmn 00172013-07-24 8925 2099 56 4965 7340 197932 minus36335 0434 plusmn 0146 plusmn 00252013-07-24 9318 2494 56 4965 7030 31309 minus30134 0910 plusmn 0269 plusmn 00022013-07-24 9318 2494 56 4965 7030 171050 minus38736 0431 plusmn 0115 plusmn 00202013-07-24 9318 2494 56 4965 7340 31309 minus30134 0895 plusmn 0075 plusmn 00012013-07-24 9318 2494 56 4965 7340 171050 minus38736 0358 plusmn 0015 plusmn 00162013-07-24 9318 2494 56 4965 7340 202062 minus37408 0550 plusmn 0279 plusmn 00622013-07-24 9801 2979 56 4965 7030 32120 minus31948 0936 plusmn 0090 plusmn 00022013-07-24 9801 2979 56 4965 7030 174269 minus39431 0369 plusmn 0189 plusmn 00182013-07-24 9801 2979 56 4965 7030 206158 minus38269 0885 plusmn 0394 plusmn 00542013-07-24 9801 2979 56 4965 7340 32120 minus31948 1122 plusmn 0100 plusmn 00022013-07-24 9801 2979 56 4965 7340 174269 minus39431 0367 plusmn 0015 plusmn 00172013-07-24 9801 2979 56 4965 7340 206158 minus38269 0318 plusmn 0089 plusmn 0036

ι Her 2013-08-29 4351 1325 56 5325 5385 106359 minus94209 0875 plusmn 0048 plusmn 00062013-08-29 4351 1325 56 5325 5385 310123 minus120677 0081 plusmn 0056 plusmn 00042013-08-29 4351 1325 56 5325 5535 106359 minus94209 0930 plusmn 0052 plusmn 00062013-08-29 4351 1325 56 5325 5535 310123 minus120677 0118 plusmn 0024 plusmn 00072013-08-29 4350 1324 56 5325 7075 106361 minus94203 0933 plusmn 0122 plusmn 00042013-08-29 4350 1324 56 5325 7075 310127 minus120670 0430 plusmn 0164 plusmn 00152013-08-29 4350 1324 56 5325 7385 106361 minus94203 0904 plusmn 0035 plusmn 00042013-08-29 4350 1324 56 5325 7385 310127 minus120670 0315 plusmn 0054 plusmn 0010

8 Cyg 2013-08-28 6088 1130 56 5315 5385 216020 minus129391 0630 plusmn 0053 plusmn 00352013-08-28 6088 1130 56 5315 5385 306627 minus116749 0407 plusmn 0108 plusmn 00492013-08-28 6088 1130 56 5315 5535 216020 minus129391 0617 plusmn 0046 plusmn 00332013-08-28 6088 1130 56 5315 5535 306627 minus116749 0412 plusmn 0079 plusmn 00472013-08-28 6088 1129 56 5315 7075 216025 minus129386 0776 plusmn 0053 plusmn 00242013-08-28 6088 1129 56 5315 7075 306634 minus116743 0442 plusmn 0080 plusmn 00292013-08-28 6088 1129 56 5315 7400 216025 minus129386 0973 plusmn 0151 plusmn 00282013-08-28 6088 1129 56 5315 7400 306634 minus116743 0431 plusmn 0074 plusmn 0025

ζ Per 2011-10-13 8507 minus1828 55 8465 7150 63035 minus109781 0965 plusmn 0051 plusmn 00012011-10-13 8507 minus1828 55 8465 7150 147169 minus103253 0471 plusmn 0028 plusmn 00042011-10-13 8484 minus1851 55 8465 7150 209574 minus105057 0181 plusmn 0014 plusmn 00032011-10-13 8507 minus1828 55 8465 7345 63035 minus109781 0904 plusmn 0032 plusmn 00012011-10-13 8507 minus1828 55 8465 7345 147169 minus103253 0494 plusmn 0016 plusmn 00042011-10-13 8500 minus1835 55 8465 7345 209811 minus105163 0228 plusmn 0012 plusmn 0004

A104 page 12 of 13

Table A1 continued

δ Cyg 2011-07-23 7421 minus0153 55 7645 715 65642 minus121782 0772 plusmn 0041 plusmn 00022011-07-23 7425 minus0149 55 7645 715 155612 minus115029 0166 plusmn 0011 plusmn 00022011-07-23 7421 minus0153 55 7645 715 65642 minus121782 0772 plusmn 0040 plusmn 00022011-07-23 7425 minus0149 55 7645 715 155612 minus115029 0166 plusmn 0010 plusmn 00022011-07-23 7421 minus0153 55 7645 735 65642 minus121782 0698 plusmn 0035 plusmn 00012011-07-23 7419 minus0155 55 7645 735 155596 minus114963 0147 plusmn 0008 plusmn 00022011-07-23 9054 1484 55 7645 715 65146 minus139701 0708 plusmn 0059 plusmn 00012011-07-23 9050 1480 55 7645 715 154249 minus132820 0127 plusmn 0012 plusmn 00012011-07-23 9054 1484 55 7645 715 219055 minus134898 0011 plusmn 0019 plusmn 00032011-07-23 9054 1484 55 7645 735 65146 minus139701 0684 plusmn 0040 plusmn 00012011-07-23 9048 1478 55 7645 735 154256 minus132797 0160 plusmn 0009 plusmn 00022011-07-23 9054 1484 55 7645 735 219055 minus134898 0016 plusmn 0010 plusmn 00002011-07-27 8124 0815 55 7685 715 65744 minus132043 0726 plusmn 0040 plusmn 00012011-07-27 8124 0815 55 7685 715 155931 minus125241 0128 plusmn 0013 plusmn 00022011-07-27 8124 0815 55 7685 735 657443 minus132043 0788 plusmn 0049 plusmn 00022011-07-27 8124 0815 55 7685 735 155931 minus125241 0157 plusmn 0014 plusmn 00022011-07-27 8124 0815 55 7685 735 221349 minus127257 0028 plusmn 0019 plusmn 00002011-07-27 8917 1610 55 7685 715 65007 minus141195 0770 plusmn 0073 plusmn 00012011-07-27 8917 1610 55 7685 715 153819 minus134361 0163 plusmn 0015 plusmn 00022011-07-27 8917 1610 55 7685 715 218501 minus136390 0018 plusmn 0020 plusmn 00002011-07-27 8917 1610 55 7685 735 65007 minus141195 0797 plusmn 0053 plusmn 00012011-07-27 8917 1610 55 7685 735 153819 minus134361 0160 plusmn 0012 plusmn 0002

ζ Peg 2011-07-24 8586 minus1860 55 7655 715 65002 minus114225 0867 plusmn 0039 plusmn 00052011-07-24 8583 minus1863 55 7655 715 151533 minus109301 0428 plusmn 0021 plusmn 00152011-07-24 8576 minus1870 55 7655 715 216288 minus110765 0263 plusmn 0028 plusmn 00222011-07-24 8586 minus1860 55 7655 735 65002 minus114225 0839 plusmn 0034 plusmn 00052011-07-24 8583 minus1863 55 7655 735 151533 minus109301 0414 plusmn 0015 plusmn 00142011-07-24 8589 minus1857 55 7655 735 216431 minus110787 0222 plusmn 0017 plusmn 00152011-07-28 8236 minus1948 55 7695 715 64774 minus114068 0891 plusmn 0059 plusmn 00542011-07-28 8236 minus1948 55 7695 715 150724 minus109167 0442 plusmn 0027 plusmn 00142011-07-28 8213 minus1971 55 7695 715 215043 minus110606 0187 plusmn 0028 plusmn 00132011-07-28 8240 minus1944 55 7695 735 64784 minus114074 0882 plusmn 0055 plusmn 00052011-07-28 8240 minus1944 55 7695 735 150760 minus109173 0440 plusmn 0030 plusmn 00132011-07-28 8243 minus1941 55 7695 735 215419 minus110650 0188 plusmn 0045 plusmn 0012

A104 page 13 of 13

Introduction

VEGACHARA observations of eight early-type stars

The limb-darkened angular diameters

Data reduction and methodology

Results

The calibration of the surface brightness relation

Methodology

A revised SBC relation for late- and early-type stars

Discussion

Conclusions

References

Appendix A

AampA 570 A104 (2014)

sion of both components can be measured with a precision upto of 1 from the analysis of high-quality spectroscopic andphotometric data (eg Torres et al 2010) The distance to aneclipsing binary follows from the dimensions determined in thisway plus the angular diameters derived from the absolute sur-face brightness which is very well calibrated for late-type stars(Di Benedetto 2005) This conceptually very simple techniquevery weakly depends on reddening and metallicity and providesthe most accurate tool for measuring distances to nearby galax-ies (Pietrzynski et al 2013 Graczyk et al 2014)

However the heart of this method the surface brightness-color (SBC) relation is very well calibrated only for late-typestars which significantly limits its usage The late-type systemscomposed of main-sequence stars are usually faint while thosecomposed of giants have very long periods (several hundreddays) that makes them very difficult to find As a result onlyabout 45 late-type systems well suited to precise distance deter-mination have been discovered so far in the Magellanic Cloudsby the Optical Gravitational Microlensing Experiment (OGLEPawlak et al 2013 Graczyk et al 2011) On the other handmany more relatively bright systems are known in nearby galax-ies (Massey et al 2013 Graczyk et al 2011 Wyrzykowski et al2003 2004 Bonanos et al 2006 Macri et al 2001 Mochejskaet al 2001 Vilardell et al 2006 Pawlak et al 2013) Thereforein order to derive the distance to nearby galaxies and to study thegeometry of the Magellanic Clouds it is imperative to calibrateSBC relation for early-type stars

The purpose of this paper is to improve the SBC relation forearly-type stars by using the resolving power of the Visible spEc-troGraph and polArimeter (VEGA) beam combiner (Mourardet al 2009) operating at the focus of the Center for High AngularResolution Astronomy (CHARA) Array (ten Brummelaar et al2005) located at Mount Wilson Observatory (California USA)The CHARA array consists of six telescopes of 1 m in diameterconfigured in a Y shape which offers 15 different baselines from34 m to 331 m These baselines can achieve a spatial resolutionup to 03 mas in the visible which is necessary in order to resolveearly-type stars Early-type stars are very small in terms of angu-lar diameter and can be affected by several physical phenomenalike fast rotation winds and environment which can potentiallybias the interferometric measurements

This paper is structured as follows Section 2 is devoted toa description of the stars in our sample In Sect 3 we presentthe data reduction process and the method used to derive theangular diameters Section 4 is dedicated to the calibration ofthe SBC relation and we discuss our results in Sect 5 We drawconclusions in Sect 6

2 VEGACHARA observations of eight early-typestars

We carefully selected eight early-type stars with a (V minus K) colorindex ranging from minus070 to 002 They are north hemispheremain-sequence subgiant and giant stars (δ gt 4) with spectraltypes ranging from B1 to A1 They are much brighter (with avisual magnitude mV ranging from 16 to 47) than the limitingmagnitude of VEGA (about mV = 7 in medium spectral reso-lution) They are also bright in the K band (with a mK magni-tude lower than 51) which makes it possible to track the fringessimultaneously with the infrared CLIMB combiner (Sturmannet al 2010) All the apparent magnitudes in V and K bands thatwe have collected from the literature are in the Johnson system(Johnson et al 1966 see also Mermilliod et al 1997) The ac-curacy of their parallaxes π spans from 15 to 15 The color

excess E(V minus K) the visual absorption AV the effective tem-perature Teff the mass M the radius R the luminosity L thesurface gravity log g and the metallicity index [FeH] are listedin Table 1 We emphasize that for our purpose (limb-darkeningestimates see end of Sect 31) we do not need very precise es-timates of the fundamental parameters of the stars in our samplewhich explains why we do not provide any uncertainty on theseparameters in the second part of Table 1

Among the eight early-type stars in our sample there are sixlow rotators (λAql γ Ori γ Lyr ιHer 8 Cyg and ζ Per) and twofast rotating stars (δ Cyg and ζ Peg) In the following we definefast rotators as stars with vrot sin i gt 75 km sminus1 A theoreticalstudy which aims at quantifying the impact of fast rotation onthe SBC relation for early-type stars is currently in progress andwill be published in a forthcoming paper

We observed our sample stars from July 23 2011 toAugust 29 2013 using different suitable triplets available onthe CHARA array A summary of the observations is given inTable 2

3 The limb-darkened angular diameters

In this section we describe how we derive the limb-darkenedangular diameter for all the stars in our sample

31 Data reduction and methodology

The first step is to calibrate the visibility measurements of ourtargets using observations of reference stars These calibrators(Table 3) were selected using the SearchCal1 software devel-oped by the Jean-Marie Mariotti Center (JMMC Bonneau et al2006) The way this calibration is done is shown in Fig 1 in thecase of γ Lyr (data obtained on June 21 2012 with the E1E2W2three-telescope configuration) We used the standard sequenceC6-S-C6 in which S is the target and C6 is the reference starThe light blue dots are the raw visibilities obtained on the sciencestar for the three corresponding baselines E2E1 (upper panel)E2W2 (middle panel) and E1W2 (bottom panel) Our VEGAmeasurements are typically divided into 30 blocks of observa-tions and each block contains 1000 images with an exposuretime of 15 millisec For each block the raw squared visibility iscalculated using the auto-correlation mode (Mourard et al 20092011) The red dots in the figure represent the transfer functionobtained by comparing the expected visibility of the referencestar to the one that has been measured This transfer functionis then used to calibrate the visibilities obtained on the sciencetarget (blue dots) A cross-check of the quality of the transferfunction is usually done for several bandwidths and over thewhole night Under good seeing conditions the transfer func-tion of VEGACHARA is generally stable at the level of 2for more than one hour The squared calibrated visibilities V2

targetobtained from our VEGA observations are listed in Tables A1The systematic uncertainties that stem from the uncertainty onthe reference stars are negligible compared to the statistical un-certainties and are neglected in the rest of this study

The calibrated visibility curves obtained for each star inour sample (Fig 2) are then used to constrain a model of uni-form disk that contains only one parameter the so-called uni-form disk angular diameter (θUD) This is performed using theLITpro2 software developed by the JMMC (Tallon-Bosc et al2008) The following formula of Hanbury Brown et al (1974b)

1 Available at httpwwwjmmcfrsearchcal2 Available at httpwwwjmmcfrlitpro

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2prime57primeprime+

1prime22primeprime+

0prime58primeprime+

7prime10primeprime+

0prime22primeprime+

3prime01primeprime+

9prime52primeprime

7prime50primeprime

7plusmn06

26plusmn02

2plusmn05

79plusmn01

17plusmn01

34plusmn01

6plusmn01

7plusmn04

003plusmn00

12plusmn00

00plusmn00

15plusmn01

04plusmn00

87plusmn00

05plusmn00

04plusmn00

(VminusK

03plusmn00

11plusmn00

00plusmn00

12plusmn01

03plusmn00

77plusmn00

04plusmn00

minus00

4ominus00

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θLD(λ) = θUD(λ)

⎡⎢⎢⎢⎢⎢⎣(1 minus Uλ

(1 minus 7Uλ15 )

⎤⎥⎥⎥⎥⎥⎦12

middot (1)

32 Results

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41 Methodology

AV =08π (3)

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6 Conclusions

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2prime57primeprime+

1prime22primeprime+

0prime58primeprime+

7prime10primeprime+

0prime22primeprime+

3prime01primeprime+

9prime52primeprime

7prime50primeprime

7plusmn06

26plusmn02

2plusmn05

79plusmn01

17plusmn01

34plusmn01

6plusmn01

7plusmn04

003plusmn00

12plusmn00

00plusmn00

15plusmn01

04plusmn00

87plusmn00

05plusmn00

04plusmn00

(VminusK

03plusmn00

11plusmn00

00plusmn00

12plusmn01

03plusmn00

77plusmn00

04plusmn00

minus00

4ominus00

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θLD(λ) = θUD(λ)

⎡⎢⎢⎢⎢⎢⎣(1 minus Uλ

(1 minus 7Uλ15 )

⎤⎥⎥⎥⎥⎥⎦12

middot (1)

32 Results

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41 Methodology

AV =08π (3)

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6 Conclusions

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θLD(λ) = θUD(λ)

⎡⎢⎢⎢⎢⎢⎣(1 minus Uλ

(1 minus 7Uλ15 )

⎤⎥⎥⎥⎥⎥⎦12

middot (1)

32 Results

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AV =08π (3)

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6 Conclusions

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Documents

Astronomy c ESO 2014 Astrophysics - oca.eu · A&A 570, A104 (2014) DOI: 10.1051/0004-6361/201423772 c ESO 2014 Astronomy & Astrophysics Improving the surface brightness-color relation