Fundamental Parameters of Stellar (Photospheric) Physics

Home , Beta Sagittae, Limb darkening

RADIATIVE TRANSFER IN STELLAR ATMOSPHERES

Jason Aufdenberg (ERAU)

illustration: R. J. Rutten (2003)

R.J. Rutten Sterrekundig Instituut Utrecht Institute of Theoretical Astrophysics Oslo May 8, 2003

ESMN What’s to Come.... 90 Years of stellar angular diameters

Diameters + Distances  Radii

Quick intro to temperature structure of the Sun

Quick introduction to limb darkening

How an interferometer sees a star

Limb darkening observed on 23 stars

Giants, geometry, and limb darkening

Convection, 3D models, and limb darkening

Warm supergiants and limb darkening

Rapid Rotators: observations and models 1921ApJ....53..249M Michelson andPeaseMichelson (1921): Angular Size ofBetelgeuse Astrophysical 53, Journal 249-259

1921ApJ....53..249M 1921ApJ....53..249M Interferometric Fringes from Creekside Observatory, Daytona Beach, Florida, USA α Lyrae 29 September 2011

Wavelength: R-band (~600 nm) Apertures: 12.1 cm, Baseline: 37.5 cm Scale = 0.08 arcseconds/pixel 0.05 sec exposures, 5 frames per second movie Radius from Angular Size and Distance

2R θ

θd R = 2 θ 1 radian Parallax error > 1% for 4811 of the brightest (V < 6) stars

∆d • Seven stars =0.461 ∆M (V < 6) have d π < 0.05 mas: HR 8154 (O8e) 32 Eri A (G8 III), V399 Car (A9Ia), θ Mus (WC+...), HR 5223 (B2IIne), HR 5680 (O8IIIp). “Famous” stars have radius precision limited by parallax Fundamental EffectiveEffective Temperature Temperature from Angular Size and Flux

Bolometric Flux (corrected Angular for extinction) Diameter (corrected for limb darkening) follows from:

luminosity distance radius Effective Temperature radiates same integrated flux as the star diameter correction extinction correction Effective temperature of black body radiates same integrated flux

4 Total 3

H- 1.6 µm 2 bound-free

1 H- free-free Continuous Opacity Continuous 0 400 800 1200 1600 2000 2400 Wavelength [nm] 1976ApJS...30....1V Absolute Intensities Reconstructing Temperature Structure: Spatially Resolved Vernazza, Avrett, &Loeser (1976)ApJS 30, 1 Orbiting Solar Observatory 6 Observatory Orbiting Solar 1976ApJS...30....1V 1976ApJS...30....1V Limb Darkening Reveals Temperature Structure

(a) Deeper, hotter layers are visible near the disk center

(b) Shallower, cooler layers are visible near the disk limb

isothermal atmospheres do not exhibit limb darkening

Solar and Heliospheric Observatory (SOHO), Michelson Doppler Imager (MDI) Very narrow band Ni 6768 Å (sohowww.nascom.nasa.gov) Limb Darkening Depends on Wavelength

Credit: undergraduate students Edward Muller and Keily Grubaugh Limb Darkening Depends on Wavelength

@I 4 λ @T /⇠

Credit: undergraduate students Edward Muller and Keily Grubaugh Stellar Limb Darkening: Geometry of a plane-parallel atmosphere

θ = cos-1 µ

0.1% of stellar z radius { –10–

The intensity vectors Iλ(µ) are linearly interpolated (in the log) at µ(x, y) around the grid square,

00 Iλ = Iλ(Tj,gl,µ(x, y)) 10 Iλ = Iλ(Tj+1,gl,µ(x, y)) 11 Iλ = Iλ(Tj+1,gl+1,µ(x, y)) 01 Iλ = Iλ(Tj,gl+1,µ(x, y)).

Next, the intensity is bilinearly interpolated at the local Teﬀ and log(g) for each (x, y) position in the map:

Iλ(x, y)=Iλ[Teff (x, y),g(x, y),µ(x, y)] = (1 a)(1 b) I00 + a(1 b) I10 − − λ − λ + ab I11 + (1 a)bI01 (17) λ − λ where a =(T (x, y) T )/(T T ) eff − j j+1 − j b =(g(x, y) g )(g g ) − l l+1 − l Finally, a Delaunay triangulation is computed (using the IDL routine TRIGRID) to regrid the intensity map Iλ(x, y), originally gridded in ϑ and ϕ, onto a regular 512x512 grid of points in x and y. The coordinates x and y have the units of milliarcseconds and correspond to offsets in right ascension and declination on the sky (∆α,∆δ) relative to the origin, the subsolar point.

Angular Size from Interferometry 4.2. Synthetic Squared Visibility Computation

Due to the lack of symmetry in the synthetic intensity maps, we evaluate a set of discrete 2-D Fourier transformsOn the insky order to generate aOn set the of uv-plane synthetic squared visibilities comparable to the CHARA/FLUORintensity observations. The ﬁrst step is to compute the discrete V2 Fourier transform for each wavelength at each of the spatial frequency coordinates (u, v) corresponding to the projected baseline and orientation of each data point (see Table 1). The mean (u, v) coordinates for each data point, in units of meters, are converted to the corresponding spatial frequency coordinates (uk,vk) in units of cycles per arcsecond for each y v wavelength λk. The Fourier transform,x u 2 2 ∞ ∞ i2π(ux+vy) Vλ (u, v)= SλIλ(x, y)e dx dy (18) −∞ −∞ 2 2 2J1(πθB/λ) V (B,λ, θ)= (B/λ)2 = u2 + v2 πθB/λ How Does an Interferometer See a Single Star? Bird’s Eye View of Fourier Map On the Sky Intensity Map

Intensity

Visibility: Double-beam Interference

Dec [milliarcsec]

RA [milliarcsec]

Fourier Map high visibility, Visibility: highDouble-beam contrast Interference

Fringe Contrast Model Curve, 3.356±0.006 milliarcsecond diameter function.

low contrast

Visibility: Double-beamhigh visibility, Interference high contrast low visibility, spatial frequency low contrast [1/arcsec]

(from Hecht: Optics) spatial frequency high contrast [1/arcsec ] very low contrast

high visibility, high contrast low visibility, low contrast

(from Hecht: Optics)

20 m] [M y 15 low visibility, low contrast 10

5 (from Hecht: Optics)

x [Mm] 15

25 Corrections to Uniform-Disk Diameters

Uniform disk: 3.228±0.003 mas ( ∆θ χν= 37) ∆θ Single Effective Temperature 0.01 0.01 Model Atmosphere: 3.263±0.002 mas θ θ ( ∆θ χν= 20) ∆θ 0.04 α 0.04 Analytic Model, I = μ θ θ 3.356±0.006 mas, α = 0.31±0.01 ( χν= 6)

Vega (α Lyr), K-Band, CHARA/FLUOR (Aufdenberg et al. 2006, ApJ, 645, 664) X. Haubois et al.: Imaging the spotty surface of Betelgeuse in the H band 5

Substituting the intensity equation in the visibility expression, the Mellin transform appears. For the general case of a power law of index δ, I(µ) = µδ, this can be written as follows (Hestroﬀer, 1997):

466 OHISHI, NORDGREN, & HUTTER Vol.1 612 2 ν 1 Jν(x) Vν(x) = 2ν (1 µ ) − J0(xµ)µdµ =Γ(ν + 1) | | (4) − × (x/2)ν 0 where J ν is the Bessel function of the first kind of index ν, defined as ν = δ/2 + 1. The expression for the visibility of a quadratic limb- darkening is thus finally:

Exemplary second-lobe measurements from interferometryJ1(z) π J3/2(z) J2(z) A. Domiciano de Souza et al.: Diameter and photospheric structures of Canopus fromα AMBER| /VLTI| + interferometryβ | L7 | + 2γ | | Ohishi et al. (2004) Domiciano de Souza et al.(2008)986 Wittkowskiz etM. Wittkowskial.2 (2001)z et3/ al.:2 Limb-darkeningz2 measurements with NPOI Vλ(z) = (5) 3 α β γ 0.40 HR 7635, scan no. 1 HR 7635, scan no. 1 HR 7635, scan no. 1 d) π ra 0.030 + + A7 V F0 Ib ( 0.02 0.30 2 M0 III

ase 2 3 4

Altair Canopus h 0.01 0.020 π/2 0.20 γ Sge 1 B osure p α = β = + γ = = πφ ⊥ Triple Amplitude

Cl 0.010

with 1 A B, A 2B,Closure Phase [rad] B and z LD . 0.10 200 250 λ Fig. 3. Amount of beam integrated ﬂux and dust shell intensity com- 0 − − 0 −

Squared Visibility Amplitude 0.00This model and the0.000 linear one have been squared and fit to the 4. 5. 6. 7. pared to the fiber profile. Black line: beam integrated flux, normalized 100 150 200 250 850 10+7800 750 700 850 800 750 700 datauv-Radius (Fig. [cycles/arcsec] 4). The resultsWavelength of this [nm] fitting upWavelength to the [nm] second lobe are to the photospheric flux, as a function of beam radius, either in units of Sp. Freq largest base (Bproj/λ) Fig. 3.summarizedAMBER/VLTI closure phases in Table (absolute values) 2. of Canopus as dust shell inner radius (bottom axis) or in arcseconds assuming a 13 R 0.40 HR 7635, scan no. 2 HR 7635, scan no. 2 HR 7635, scan no. 2 Che et al. (2011) a function of the spatial frequency calculated0.030 on the longest baseline. π dust shell inner radius (top axis). Grey line: the intensity profile of theValues indicateXaubois that0.02 Canopus does et not have al. a completely (2010) centrally sym- metric0.30 intensity distribution. For example the absolute values of the clo- 0.01 7 dust shell, normalized to the central photosphericMérand et intensi al.ty. (2010) TheA&A dashed 517, A64 (2010)sure phases at low spatial frequencies ( 0.0205.5 10 cycles/rad) are of π/2 A. Mérand et al.: CSE around Polaris and δ Cep. II. 157the order0.20 of 0.1 0.2 rad. The triangles indicate≤ × the theoretical closure Table 2. Calibrated squared− visibility measurements of η Ser. line represents the Gaussian fiber profile.0.03 200 phases obtained from the fit of a single convective cell model to the Triple Amplitude 0.010 F2 IV 0.10 Closure Phase [rad] 0 visibility amplitudes alone200 (see250 text for details). 0 Table 2. Best fit model parameters for Polaris and its CSE. θ� is the lobe. For this purpose we chose a single parameterv (m) CLD law, theMJD B PA V2 σ L1

α K0 III Squared Visibility Amplitude ± stellar angular diameter (mas), α the CLDβ Cas coefficient, θs the shell an- power law: I(µ)/I(1) = µ (Michelson & Pease−200 1921; Hestroffer 0.00 (m) (◦) (%) 0.000 −200 0 200 850 800 750 700 gular diameter (mas), w the shell width (mas) and F /F the relative 0.02 53Table 930.377 2. Parameters 214.68100 1507 and.77200 uncertainties 0.00715250 0.00041 estimated850 from800 a least-square750 700 fit 850 800 750 700 s � 1997). The hydrostatic model computed Ser from Claretu (m) coefficients − ± 2 η 53of 930.400 a LLD 219.44 diskuv-Radius model13 [cycles/arcsec].90 to the 0.00781 observed0.00020 visibilities. TheWavelength reduced [nm]χ2 of Wavelength [nm] brightness (Fig. 3). Last column4. tabulates Limb-darkening the reduced χ . Only param- and physical diameter − ± for Polaris corresponds to α = 0.16. Even using a single pa-53the 936.364 fit is χ 171.462 = 7.0,38 suggesting.16 0.00924 that0.00053 the LLD disk cannot completely eters with error bars (lower scripts) have been fitted. The first line is the V² 53 937.302 157.44red −32.36 0.02710 ± 0.00114 • rameter CLD model compared to the 4-parameters Claret’s law,explain the observations.− Physical± parameters• of Canopus derived from hydrostatic model; the second line is the adjusted CLD; the model offf 53 937.3200.40 162.43 HR34 7635,.71 scan 0.01978 no. 30.00093 HR 7635, scan no. 3 HR 7635, scan no. 3 The limb darkening e ect is a gradual0.01 2 decrease of surface3 bright-53the 937.338 measured 166.67/ are−36 also.46 listed. 0.01272 ± 0.00065 • π corresponding V only differ at most by 10− (relative) in the 0.02 0.030 the last line includes a shell. Fig. 2. Top: AMBER/VLTI visibility amplitudes and uncertainties in the � − ± • first two lobes. We therefore prefer to use a single parameterNotes. MJD: modified0.30 Julian date of the observation; B: projected base- 564 S. Lacourness et al.: from The limb-darkened the center Arcturus: to imaging theH (green) stellar with theand IOTAK limbbands/IONIC (black). caused interferometer The model by visibilities the (red decreas squares)line length;e PA:Fitted azimuth parameters of the0.01 projected baseline (countedDerived positively parameters CLD law (the power law), for the sake of simplicity. 0.020 2 were calculated with a linear limb-darkened disk model fitted to thefrom North to East); V2: squared visibility; L1: “ ” means measure- π/2 θ� αθs w Fs/F� χ 0.20 a +00.00 / = 6.93 0.15 mas • R/R = 71.4 4.0 100 of temperature with altitude:data (Table10 the2). We optical also show the depth theoretical ofvisibilities 1 is (dots reach for H andmentsed obtained� in the first± lobe of the visibility. � b ± c 3.152 0.003 0.16 – – –H (broad band 4.5 filter) The best fit, adjusting α as a free parameter, leads to θ� = �H = 0.04 0.01 Teff = 7284 107 K or 7582 252 K

± dashes for K) 2 expected 160 for Canopus 180 from a LLD200 disk with 220 parame- ± M1 Iab±Triple Amplitude 0.010 d ± 11, 12 & 13 May 2 �0.10= 0.07 0.01 log g = 1.7 0.1 Closure Phase [rad] 3.189 0.005 0.26 0.01 for– lower – – altitudes 2.5 at the3.189ters center determined0.005masV² of by and Claret theα (=2000 stellar0.26). Clearly0.UD01; theseDiameter: disk the limb-darkenedreduced 20.304+/−0.011 thanχ is at models then tTablehe 2.5 3. ResultsK of the fits to the data200 for diff250erent models. 0 ± ± 14 & 16 May 0 2 − ± ± 3.123 0.008 0.16 7.5 0.2 0.5 1.5 0.4% 1.4 ± ± Reduced2 χ = 31 a (Tabledo not 2, account second−1 for line). the observations, Based on the especiallyχ , the after fit is the significantly first minimum. bet- FromSquared Visibility Amplitude the Hipparcos distance d = 95.Ori9 4.9 pc (Perryman et al. 1997). ± ± ± ff 10 −2 1.53 μm 0.00 α ± 0.000 Mérand etedge. al. Taking (2006) this e ect intoBottom account: fit residualsΔ V²/ σ in isσ units mandatory showing2 a relatively for important an accu- depar-ModelThe Lutz-KelkerθLD correctionα is negligibleFirst lobe forCanopus All (Jerzykiewicz & 50 ter: the hydrostatic−4 1.55 CLD μm V led to χ of 4.5. The CLD is stronger 100(mas) 150 200 250 χ2 P χ2 850P 800 750 700 850 800 750 700 ture from zero, as1.57 well μm as residual trends indicating that additional ef- Molenda-Zakowiczuv-Radius2000 [cycles/arcsec]). Wavelength [nm] Wavelength [nm] rateS1−S2 measurementE1−E2 of thethan diameter predicted by of1.59 hydrostatic 160 theμm photosphere. models180 and 200 the corresponding To 220 do soUniform an-,b disk 2.855 0.007 0.000 1.89 1 17.1 0

1.0 2 From the bolometric flux measured by Code et al. (1976): f = 1.00 fects should10−2 be included in the modeling of Canopus. ± 0.65 gular diameter is thus1.61 μ larger,m asBaseline expected. (m) However, before tryingATLAS9 2.936 0.0076 0.1762 1 1.59 1 2.2 0 S1−S2 ff (45.0 1.8) ±10− erg cm− s− . 1.63 μm MARCS-ppc ±0.40 2.940× 0.007HR 7635, 0.183 scan no. 1.58 4 1 2.1 0 HR 7635, scan no. 4 HR 7635, scan no. 4 Transmission [%] we used parametric models (see Claret, 2004; Hestro er, 1997) Obtained by± using values from Smiljanic et al. (2006) and references 564 S. Lacour et al.: The limb-darkened0.60 Arcturus:to imaging interpret withFig. this the 1. Squared IOTA result/ visibilitiesIONIC in terms interferometerand adjusted of photospheric two-parameter limb characteristics, darkened PHOENIX 2.944 0.007 0.196 1.57 1 1.9 0 0.95 1.65 μm M. Wittkowski et al.: VLTI/VINCI limb-darkening measurements of0.030ψ Phe π 721 0 disk visibility1.67 modelm of η Ser (with bandwidth smearing). The dotted MARCS-sphtherein (visual 2.945 ± 0 apparent.0070.02 0.193 magnitude 1.56V = 10 1.9.72, extinction 0 A = 0.08, one should−3 noticeLacour thatμ this modelet stillal. fails to(2008) fit the mid-first Wittkowski et al. (2004)V 1.5 1.6and 1.7 a physical 1.8 model 1.9 derived4. ModelVisibility 10 from fitting a and MARCS physical parameters simulation of the ± − 0.55 line+0 is the1.69 monochromatic μm visibility curve for λ = 2.16 microns. The adjustedvisual LD0.30 bolometric 2.958 0.027 correction 0.23 0.05BC 1.52= 0 0.0) 2.1 and from 2 Drilling & Arlo 100 lobe (see E1-E210 panel1.0 in Fig. 2, dotted line, which actually over- 0.020± ± V 18.8 19.0 19.2 H59.2 (broad band 59.4 filter) lower panel1.71 shows μm the normalized residuals, as well as the difference 0.01 The observed visibilities (Fig. 2) and closure2 phases (Fig. 3) in-Notes.(θ2000: CLD) (solar corrected luminosity diameter; α and: equivalent bolometric power law magnitude).0.020 parame- The uncertainty Betelgeuse atmosphere.11, 12 & 13 May between the1.74 monochromatic μm visibility profile and the one using band- LD Fig. 4. SquaredVLTI/VINCI visibility dataPHOENIX fittedsph. by theπ/2 best limb darkening model λ [μm] laps with the solid line). The measuredUD Diameter:V data 20.304+/−0.011 are lower than2 0.20 2 0.035 W2−E2 width smearing1.75 μ (dottedm line, with a typical error bar of 5%). The smaller ter; χ inandTPe:ff reducedwas estimatedχ and number from of parametersσ / and considering fitted (if P = 0, a the conservative cumula- 14 & 16 May dicate that,−4 as a zero-order approximation, Canopus2 can be con- ψPhe (M4� III) PHOENIX pl. computed10 forpanel a in limbthe1.78 upper μm darkened right corner shows disk. theReducedu-v Aplane. change χ = 31 in CLD affχ2ectsis only computed). “first lobe” referes to data marked with a “ ” in E1−E2 10−1 0.8 tive error of 0.1 in V, AV , and BCV. Fig. 3. Relative photometry between the different spectral channels as sidered to be a star1.53 having μm a centrally-symmetric intensity dis-Table 2d. (dotted line). To improveTriple Amplitude 0.010 • theATLAS modeling, 12 pl. we should reproduce the excess 1.80 μm Closure Phase [rad] primarily the second lobe (higher spatial frequencies), and only Assuming0.10 0.015 a mass of M200/M =2509 2. a function 50 of the wavelength (arbitrary vertical units). The wavelength tribution (see also1.55 discussion μm in Sect. 5). We have thus fitted a � ± ATLAS 9 pl. 0 0.030 1.57 μm 3 of contrast at high frequencies. The squared visibility of a UD is drawn the scale ofTable the 1. firstCalibrators lobe. used In for the order observations to change of η Ser. the shape of the Squared Visibility Amplitude UD/FDD was determined by measuring the frequency of the fringes as shown in linearly limb-darkened1.59 μm disk (LLD) model to both H and K band3. Angular0.00 diameter 0.000 2 −2 0 5 10 15 20 V² 10 0.6 Fig.0.52. F84.1. Ib Parametric modelsfirstobserved lobe, one visibilities. has1.61 to invokeμ Them free something parameters larger of the that model Polaris are the itself in100 full150 line200 for250 comparison.850 800 750 700 850 800 750 700 145.0 145.5 146.0 20 Star 1.63 mmV mKs Spect. θUD(K) γ 3.1. Limb darkeninguv model-Radius comparison [cycles/arcsec] Wavelength [nm] Wavelength [nm] 0.014 μ AMBER, the uncertainty in R is dominated by the uncertainty in

Transmission [%] to disturbLLD angular the lower diameter spatial/ , frequencies. and the H and Thus,K band we think LLD coef-that this 1.65 m (mas) (◦) 0.010 Polaris μ � In orderthe to distance,estimate the contrarily angular diameter to the from previous the measured measurements visi- of / . 0 ffstrongficients CLD�H isandIRC not+00339�1.67K realistic.. The μm 5.5 estimated 2.6 K2III parameters 1.402 0 and.018 correspond- 6.8 0.013 10−3 � 1.5 1.6 1.7 1.8 1.9 Visibility 10 0.4 ± In order to compare di erent observations,HR 66501.69 μm 6.2 3.2 K1III-IV one needs 1.123 0.015 standard 16.7 bilities (TableAs2) indicated it is necessaryHR in 7635, to Table know scan the 2no., intensity the5 two distribution values ofHRT e7635,ff were scan ob-no. 5 HR 7635, scan no. 5 ing uncertainties obtained from a Levenberg-Marquardt± (L-M) 0.40 A second limitation is the field of view of the interferometer. HR 78091.71 μm 6.1 3.6 K1III 1.055 0.015 31.1 of the light on the stellar disk, that is the limb darkening0.030 (LD). π 0.012 least-squares fit are given in Table 2. The model± visibilities and tained from different0.02 approaches, corresponding to distinct es- It is delimited by the maximummodelsλ distance [μm] of between the twostellar objects intensity 0 distribution.1.74 μm TheK simplest1.5 III onesIn this Section, we present the results of two approaches based Notes. mv, mKs: magnitudes in V and K bands; UD(K): uniform disk timates0.30 of0.005 the bolometric fluxes. One can find in the lit- W1−E2 fit residuals−4 are shown1.75 μm in Fig. 2. on an assumed intensity distribution:

Squared visibility amplitude The third and fourth visibility lobes of Betelgeuse clearly 0.011 10 0.2 Squared visibility amplitude whose fringesHydrostatic overlap LD on the detector. To be rigorous, one should 3.4. Companionangular diameter; and1.78 μ pulsationγm: angular distance to η Ser. erature values compatible0.01 with bothM4 estimates. III For example, 246.0 246.5 The−10 uncertainty in / (σ / = 2.2% / ) isBoo dominated by the 2 0.020 Fig. 3. Relative photometry betweenare the the diff linearerent spectral and channels quadratic as limbResiduals [ σ ] darkening1.80 μm laws.α The visibility1. adjust a simple uniform disk model to the V measurements. π/2 take into accountFitted LD parameters like the mode of recombinaison, the � � � Desikachary0.20 & Hearnshaw (1982) and references therein give a function of the wavelength (arbitrary vertical units). The wavelength uncertainty in the instrumental spectral calibration (cf. Sect. 3). The uniformcontain disk (UD) model, high although spatial unphysical,Phe is a use-frequency information more complex than Hydrostatic LD + Shell ψ stroke of the piezo (in the caseof of a IONIC), point-symmetric and even the spectral objectWhenEven isseeking with−20 a this Hankel limitation, possible0.0 AMBER transform explanations/VLTI enabled for (whose the us to departure derive kerful-T tooleff for interferometry0.0007300 7400 when K, the while stars are Kovtyukh marginally re- (2007), Smiljanic was determined by measuring the frequencyW2−E2 of the fringesW1−E2 as shown in 0 5 10 15 20 Triple Amplitude 0.010 � − Closure Phase [rad] 2 0 5 10 15 20 solvedet al. since (0.10just2006 it is) a and single limb references parameter darkening model; therein give T and,7500 up7600 to K. this point, have been excluded energyFig.0.02. distribution of the target. However, to establish a simple aroundthe mostV precise50%, angular0 two diameterobvious50 100 of possibilities Canopus150 known must200 today. be con-250 140 160200 250180 200 e220ff 240 260 280 0 nel is a Bessel function) normalizedKervella 20 et byal. 2004a the; Mérand total et al. spatial2006a). For the int reportedensity2. assume a limb darkened profile from existing atmo-� − This interferometrically∼ measuredSpatial angular frequency diameter B/λ (1/arcsec) is compati- FigureSquared Visibility Amplitude 2 shows thatSpatial the near-IR frequency intensity B/λ (1/arcsec) distribution of ffi = relation, we will only take into account the spectral bandwidth sidered beforeobservations, invoking we used a CSE:Baseline the CHARA Polaris len baselinesgth is [M a S2-W2λ pulsating] 0 and E2-S1, star andsphere a models.0.00from Here the we use fits. the stellar The parameters0.000 linear de-0 limb darkening coe cient A 0.43 for 0 100 200 300 ble with several previous measurements within the uncertainties, rivedCanopus recently is100 for notη Ser entirely150 by200 Hekker compatible250 & Meléndez with (a2007 simple850): 800 limb-darkened750 700 850 800 750 700 of a spectral channel (Δλ), asdistribution. well as the distance For between the two monochromaticspectroscopic-astrometricFig. 4.Fig.Overview 6.with 10Our ground measured of thelengths case, dataset: squared of binary 177 the and visibility visibility 279 as visibility m, well respectively. square amplitudes (Wielen measurements The i calibra-etofs ψ al.Phe 2000). (“x” as symbols a with error bars) together with the (solid black line) spherical PHOENIX baseline (m) for example,tor stars/ = were6.6 chosen0.8 from mas the (Hanbury-Brown catalogue compiled et by al. Mérand1974; in- Temodel.ff = 4955 In K, particularuv log-Radiusg = [cycles/arcsec]3 the.20, height[Fe/H] = of0 the.21, secondVturb = lobeWavelength is quite [nm] im- Wavelength [nm] telescopesA second (B): limitation is the field of view of the interferometer. We shall now� show that± neither of these two hypotheses can3.52 km sa1. For diameter the computation of of the 44.28> intensity− profile<0.15 mas can be compared with the esti- functiontensitymodel of interferometry),et theprediction al. ( baseline2005), using with length./ criteria model= 7.00 The defined parameters wavelength0.41 by these masT authors(Heraseff is, log color (Table etg, al. and coded12002). mass; as as inportant, derived− corresponding from spectrophotometry to �H 0 and and model�K± evolutionary0 (i.e., a slight tracks (Sect. 3.2), explain theThey discrepancy 0 were observed in� immediately the first± lobe. before and/or after η Ser in or- of the star, three models are used: the∼ classical ATLAS9∼ = Fig.It 2.isResults delimited of fit by for the di2fferent maximum models. distance Squared visibilitybetween with two respect objects Fig.Bspectral1and. The best fit); solid fitting additional curveθLD correspondvalue. values Shown are to listedthe are byvisibility also Heras the (dashed-dottedet cu al.rve (2002 of a uniform). line)limb-brightening). plane-parallelmationHRPHOENIX 7635, These by scan Perrin measuredmodel, no. 6 (dotted values et al. line) do not (2004) plane-parallelHR agree 7635, with scan in theno.ATLAS 6 K 12bandmodel, whoHR 7635, scan found no. 6 A 0.09 for λ der to monitor the interferometric transfer function of the instru- and PHOENIX0.40 models, approximated using Claret’s (2000) towhose baseline. fringes Solid lineoverlap is the on hydrostatic the detector. CLD To from be Claretrigorous,∞ (2000),,λ one dotted should πstellar⊥ disk of angular diameter 20.30 mas (not accounting for band-theoretical predictions of linear limb-darkening0.030 for Canopus. To π FOVinterferometer = I(r )J0(2)(2 (dashedInr Table)rdr−10ment. line)2 we plane-parallel give the linearATLAS radius2 9 Rmodel,and eff allective with tempera- correspondingfour-parameter model parameterslaws in0.02 the K andband, best and the fitting MARCSθLD. model As a reference for the strength of the If theResiduals [ σ ] departure detected at V is believed to be due to the line is a fitted powerΔ lawλ B CLD while the dashed line0 is the hydrostatic width smearing). The bottom panel presents the residual of that(Gustafsson fit,illustratea et smaller this al. 2008 point)1, using we diameter show intensities in Fig. computed2 the of using theoretical43.6 visibility mas. Those measurements obtained take into account parameters· like the mode of recombinaison, the tureλlimb-darkening,Teff derivedAll our from theresults our gray angular were lines obtaineddiameter denote using corre and visibilitysponding other previously models UD (upper line)0.30 and FDD (lower line) model visibility functions. The left panel shows companion, it should vary with the position angle angle of(2) theTURBOSPECTRUM (Alvarez & Plez 1998), with spheri- model surrounded by the shell (see Tableλ 2 for= the models parameters). showingmeasured the−20that parameters.clear take inconsistency into account Thanks bandwith ofto theprecision second smearing, lobe. in as/ describedattained in with curves for linear0.01 limb-darkening coefficients from∼ Claret (2000): stroke of the piezo (in the caseV of(r IONIC),) and even the spectral the full range of the visibility function while the right panel is an enlargement of the lowff squared visibility0.020 amplitudes in the second lobe. All Note that the interferometric field of view is baseline dependent. projected baseline. 0 Our 5 sampled 10 range 15 in projection� 20 anglecal is� (“-sph”)= in0.28 and two andplane-parallel� di= 0 geometries.24.erent These (“-pp”). values bands correspond do not to T correspondand π/2 to the same extension Noteenergy that solid distribution and dotted of line the overlap target. However,in the main to panel, establish the S1-S2∞ a simple and,λ consideredAufdenberg PHOENIX et al. (2006and).ATLAS Since themodel optical predictions filter for FLUOR result is in a veryH similar0.20 shapeK of the visibility function in the 2nd lobe.eff Our measurements E1-E2It will small be panels.larger for shorter baselines, and smaller for longerI(r )quiterdr3 I large(μ)/I(1)quite and= 1 wide, densely�(1 it isμ extremely), populated where importantI is the for specific to E1-E2 takeband intensity, (Fig. width and1). smear-μ Howeveris theThe resultslog g offrom the fits Table are presented2, and in to Table the3 metallicity. In each case, and the microturbulence ff relation, we will only take into account the spectral bandwidth0 Baseline length [M ] 2 are significantlying in account,− di−ff especiallyerent from if one uniform collects datadiskλ near and a fully-darkened visibility angular disk diameter models,of of the the and starphotosphere. is consistent calculated by with fitting, allTriple Amplitude using considered0.010 aThatchi PHOENIX couldand explainATLAS models. the di erence in diame- baselines. Moreover, this field limitation is valid only in the cos of the angle between the line of sight and the emergent intensity. velocity0.10 from Smiljanic et al. (2006). Closure Phase [rad] theofsu departurefficient contrast does not being change enough significantly to track withthe fringes respect on to all pro- the 2 200 250 of a spectral channel (Δλ), as well as the distance between two null where the visibility varies substantially inside the wave-2 minimization, only the four V measurements obtained in the 0 direction along the baseline. Perpendicular to the baseline, the jectionFig. 4. angleOverviewband. of We the of shall baselines. the not dataset: fully describe As visibility seen this in method Fig.square in 2: thismeasurements the paper,V asrecorded as a = baselines. Squared Visibility Amplitude ter. On the theoretical side, Manduca (1979) measured A 0.52 telescopes (B): where B is the projected baseline of the interferometer, r thefirst lobe of the0.00 visibility function, where our measurements0.000 are functionTable ofit 5. istheResults very baseline specific for to length.θLD theobtained instrument The wavelength and by filterfits to details. our is color Theseinterferometric de- codedinsensitive as indata. to limblimb-darkening darkening. We then computeeffect. the Our reduced measuredχ2 values in the second lobe measuresbandpass the does flux not as cause a number any of field photons); limitation.⊥T is It the is chromaticdifficult to usingA E1-E2 few things are consistent are striking: within first, their the errors.object isBecause relatively our achro- typi- 100 150 200 250 850 800 750 700 850 800 750 700 r 2 tails are available, however, directly from the authors. For visual using all the V2 measurementsµ which gives us a metric of the Fig. Listed1.,λ The are solid the curve model correspond input parameters to the visibilityTeff, log cu2grve, and of amass uniformM, theatof 1.2uv the-Radius visibility [cycles/arcsec]m by function fitting seem a to simulated lieWavelength systematically [nm] intensity above theWavelength profile [nm] of Betelgeuse instrumentalestablish the transmission, field ofλ2 view whic ofdistance anh has interferometer been to measured the as star a internally whole. center, A calmatic.VI(rprecision Thiscomparison,) can the isbe of spatialseen we the plot on order in the Fig. intensityof residuals1 the 3%, monochromatic the of companion the distributionV visibility. Secondly, must for be theas stellarmodel-specific disk of angular correction diameter factors 20.30C mas (not, the accounting fit results for for1 θ band-http://www.marcs.astro.uu.se/, their Fig. 3. As Fig. 2, but for HR 7635. (aconservativeFOV standardinterferometer atmospheric way= to do transmission so is to consider model the is maximum also applied). baseline(2) faintsecond as 1.5% lobeλ = of of2.16 the the microns. data main is star not flux, well or fittedRoss less/LD by (in a the uniformK band), disk. in ThisLD or- model predictions by 0.5 1σ. It is not yet clearT if these= small = Δλ B · at the observing wavelengthwidthλ smearing).and J The0 isbottom the2 panel firstpresents Bessel the residual function of that fit, with the following∼ − parameters: ef f 3750 K, log g 1.5 and a length for a given direction. der tocorresponding remain undetected reduced byχ CHARAν values,/ andFLUOR. finally Moreover, the Rosseland based angular The only parameter adjusted in the fit is the angular diameter isshowing due to the thePage clear 2 presence of 4 inconsistency of limb of darkening.the second lobe. Thirdly, the closure differences are caused by systematic effects of our data calibra- In the North-East/South-West direction, using the spectral on nondiameter detectionθRoss in. The UV error andσ X-ray,(θLD) is Evans estimated et al. to be (2002) uniformly estimate0.2 massolar composition. More recently van Hamme (1993) predicted Fig. 1.—From left to right, squared visibility amplitudes on OB2,ofNote the OB1, star, that the which and interferometric is found OB3, toof befield triple theθ� of= view first3.152 amplitudes, is baseline kind.0.003 mas.dependent. We The present andphases closure are now close the to phases zero calculation or π. This of means Vega of the the measured object visibi is likely±l- to on 2001tion or by May the model 25. structures. Six The squared visibility ampli- dispersion mode of IOTA2 (D = 45 cm, Δ±λ = 40 nm and that thefor companion all model fits mass (see text). is between 1.7 and 1.4 solar masses. = = correspondingIt will be larger reduced for shorterχ isity 4.5 baselines, (Table for a 2). and quadratically Note smaller that for we longer take limbbe point darkened symmetric. disk. For an achromatic Atudes0.49 derived for from a limb the siderostat darkening data show in a systematic J band o withff- Tef f 3500 K, log scans were obtained. Dashed lines: Uniform-disk model with a diameterintoBbaselines.= account35 m),of Moreover, the the 3.11 correlations field mas. ofthis viewfield betweenSolid is limitation not error limited lines bars isvalidby of di: theff Limb-darkened–disk onlyerent telescope in data the Thus,of su thisfficient starT contrastis mostlog likelyg being modelMC a enough main sequence with to trackθ star thea diameter offringesχ similar2 onθ spec- all the of 3.22set of about mas. 1 σ Wetowards used larger values with respect to our best (FOV = 750 mas), but by the bandwidth. The field tral type (bute lowerff luminosity) toRoss Polaris./LD WielenLD etν al. (2000),Ross = points.directiontelescopes These along correlations the baseline. comequadratic Perpendicular from the multiple limb to the use baseline, ofdarkening, a single the baselines. the spatial intensity distribution is a g fitting0.5 curves. and This a is solar likely caused composition. by possible small Given system- that limb darkening is linear limb-darkening coefficients calculated by Van Hamme (1993),of with view isT 350eA mas at9500 1.55 μm, K and and 480 maslog atg 1.80 μ4m.:0.in3. The their Comparison study limb-darkened–disk of the with astrometric atmosphere orbit, conclude models/ model that the di reproducesffer- the measured calibratorbandpass in does the dataset. not cause They any ar fielde properly limitation. treatedµ It according is difficult to to A few things are striking: first, the object is relatively achro- probablyatic calibration smaller effects, since in the calibration H band of our than high insquared the J band, the agreement Inestablish the North-West the field¼/South-East of viewfunction of andirection, interferometer ofthe shorter, as the a baselines whole.¼ cosine A ence ofprescriptions in the magnitudeSpherical angle between between PHOENIX the models:two the components radius2 is ∆ vectorV = 6.5 squared visibility amplitudes and the triple amplitudes better thanthe the formalism uniform-disk developed by Perrin model. (2003). matic. This can be seen on the residuals of the V . Secondly, the visibility amplitudes in the range 0.8 0.95 derived from the (conservativeB = 15 m) allow way to a largerdo so is interferometric to consider the field maximum of view, baseline hence fromsecond which lobe3550 we of deduce, the 0.7 data because is 1.3 not well of 0.9388 th fittede similarity by8.664 a uniform in spectral1.80 disk.8.13 type, This with our results is quite∼ good.− In Fig. 2, we display theand data pointsthe line and the of models. sight. The It can3.1. be Fitting written limb-darkening as follows: prescriptions 3 siderostat data was difficult (see Sect. 4.3). solidalength 750 line mas for corresponds field a given limitation direction. to the due hydrostatic to the telescope CLD model size. for Polaris ∆Kis due6.5. to This the presence corresponds of limb to aflux darkening. ratio of Thirdly, 2.5 10 the− which closure ≈ 3500 0.7 1.3 0.93942 8.663× 1.81 8.14 The best fitting Rosseland angular diameters are well fromWeIn Claret thewill (2000).North-East consider It appearsin/South-West the following that the direction, model a 400 fails using750 to masthe reproduce field spectral of translatesSincephases limb are into close darkening an interferometric to zero is or apparent,π. ThisV meansmodulation a logical the object firsttwice step is as likely large, is to to 1 µ = × µ 3µ 2 viewdispersion for IOTA mode. of IOTAI (D( =) 451 cm, ΔAλ (1= 40 nm) andB5(110− or)3600 half a percent. 0.7 1.3 0.9381 8.665 1.79 8.13(3) constrained by our measurement. The measurements in the the data in the second lobe (see W1-E1 baseline)− and marginally− − fitbe× a point− model symmetric. for the brightness distribution of the photosphere. intermediateB = 35 m), baselines the field (E1-E2), of view where is notV limited2 50%. by the telescope Numerous3550 types of 0.5 limb-darkening 1.3 0.9218 (LD)8.814 prescriptions 1.79 exist8.12 in 4.2.2nd lobeThe of MARCSthe visibility function modeling also constrain of the the positions photosphere According3550 to the 1.0 latest radial1.3 0.9577 velocity surveys,8.504 the 1.79 radial8.14 pul- (FOVtelescopes = 750 mas), but by the∼ bandwidth. The field the literature. We used two of them, which we supposed achro- of the 1st minimum and 2nd maximum of the visibility func- visibility of the uniform disk with an angular radius2.5. of The datasetrUD The triple amplitude2r 2sation of Polaris is3550 the is 0.7 of the absolute order1.0 0.9302of 0.4% in8.739 valuediameter 1.80 (Moskalik of8.13 the triple product, of view is 350 mas at 1.55withμm,µ and= 480 mas1 at 1.80( φ μm.) matic.3.where Comparison A powerφLD law withis (Hestro the atmosphereffer limb 1997): darkened models/ disk tion, which is a constraint of the diameter. This constraint is 3.3. Adjusted center-to-limb variation LD & Gorynya 2005). In the case of FLUOR, for which the rela- To model the photosphere of this supergiant, we use the TheIn the dataset North-West consists/ ofSouth-East 924 visibility direction, measurements the shorter and− baselines308 clo- prescriptionsPlane-parallel PHOENIX2 model: independent of possible small systematic calibration uncertain- is written as follows: 2 tive error in squaredα visibility (σV2 /V ) is almost constant, the sure(B = phases.15 m) The allowV aare larger plotteddiameter. interferometric as a function With field of of the view, B baseline=0, hence oneI(μ) gets/I(1) = theμ , linear limb darkening case.(3) SOSMARCSties of the V 2 values. code, The reduced versionχ2 values May for the 1998, different described in Verhoelst Because the second lobe is not well reproduced by the hydro- most3.1. e Fittingffective3550 limb-darkening baseline 0.7 to/ search prescriptions1 for diameter8.168 variations 1.72 8.17 maxi- ν lengtha 750 inmas the field upper limitation panel of due Fig. to4 the. The telescope CP are plotted size. in the up- considered| models| do not show significant differences, which is static model, a simple way to improve the model is to adjust mizesand a the quadratic followingPlane-parallel law criterion: (Manduca ATLAS for et a al. given 12 1977 model: baseline) were used:b and angular theper strength panelWe will of of Fig. consider the5. CLD. The in frequency theIndeed, following the plane CLD a coverage 400 profile750 was changes mas presented field the of diameterSince limbθ, a diameterdarkening increase is apparent, of δθ should a logical lead first to the step max- is to consistent with the very similar predicted shapes of the model inview Fig. for1. TheIOTA solid1. curve corresponds to the best× fit of a uniform VTP V1 V2 V3 ; 9 scale of the first lobe (not its shape) and the height of the second imumIfit(μ) a/I relativemodel(1) =35501 for increasea the(1 0.7 brightnessμ in) / squaredb(1 1 distributionμ) visibility2, 8.191δ ofV2 the/V 1.782 photosphere.. Thus,8.19 the(4) visibility functions. These similarities are expected since our disk. The residuals are plotted on the lower panels. It is inter- NumerousjjPlane-parallel types− of− limb-darkening¼− ATLASjj− 9jj models:(LD) prescriptionsjj exist in ð Þ 2J1 kBprUD esting to note that fringes have been observed with a contrast measurement is dominated by continuum photons, and the con- wherethe literature.μ = 1 We(2 usedr/θ two)2), ofr them,being whichthe angular we supposed distance achro- from below2.5. The 1%. dataset Such a low contrast exists thanks to the dispersive 3500 0.5 LD/ 1 8.244 1.74 8.24 tinuum forming region of the atmosphere is almost compact VUD rUD : 6 thematic. star A center, power and− lawθ (Hestrothe angularffer 1997 diameter): of the photosphere. mode, which allows a deep first null. If the full H band was ob- 3500 1.0LD / 1 8.243 1.73 8.24 (see Fig. 5). kB r The dataset consists of 924 visibility measurements and 308 clo- In terms of complex visibilities, the power law limb darkening ð Þ¼ p UD served, the eðffectÞ of bandwidth2 smearing would have limited the 3750α 0.5 / 1 8.227 1.71 8.23 �� sure phases. The V are plottedand as a function the of closure the baseline prescription phaseI(μ)/I(1) = yields:μ is, the phase of the triple(3) product, depthlength of in the the null upper to panel several of percents Fig. 4. The (Perrin CP are & plottedRidgway in 2005 the up-). 3750 1.0 / 1 8.228 1.71 8.23 Probingper panel the of null Fig. was5. The possible frequency with planebootstrapping, coverage two was baselines presented and a quadratic law (Manduca et al. 1977) were2 used:k 6. Discussion and conclusions Γ(α/2 + 2) (πvrθLD) in Fig. 1. The solid curve corresponds to the best fit of a uniform V(vr) = − , (5) 1 I(μ)/otherwise)I(1) =Γ1(α/ anda2(1+ k FDD+μ)2)Γ (bI((1k=+ 1)µμ)2 model, 4 visibility functions(4) are Spherical PHOENIX models We have constructed a spher- disk.IOTA’s The field residuals of view are decreases plotted when on usingthe lower large bandpanels. filters. It is inter- k 0 shown,≥ with− diameters− − θ − and θ corresponding to our ical hydrostatic PHOENIX model atmosphere for ψ Phe in The visibility of the limb-darkened disk with an angularesting radius to note that fringes have been observed with a contrast UD FDD where μ = 1 (2r/θ )2), r being the angular distance from below 1%. Such a low contrast exists thanks to the dispersive favoriteÈcPHOENIXÈLDmodel1 fit.È Our2 measurementsÈ3: differ sig- Sect. 3.2. Here, we confront10 this model’s prediction for the the star center, and− θ the angular diameter of the photosphere. mode, which allows a deep first null. If the full H band was ob- nificantly fromLD UD and FDD models, confirming the limb- CLV by comparing it with our VLTI/VINCI measurement of rLD and a linear limb-darkening coefficient of u(k) is written In terms of complex¼ visibilities,þ the powerþ law limb darkening ð Þ served, the effect of bandwidth smearing would have limited the darkening effect. All considered PHOENIX and ATLAS model of the visibility function in the second lobe. We find that prescription yields: as follows: depth of the null to several percents (Perrin & Ridgway 2005). CLV predictions lead to very similar model visibility functions the model predicted shape of the visibility function is con- Probing the null was possible with bootstrapping, two baselines up to the 2nd lobe, which are all consistent2 k with our data. sistent with our VLTI/VINCI measurements. Simultaneously, Γ(α/2 + 2) (πvrθLD) As we can see fromV(vrHence,) = equation our data confirm (5), the model-predicted− if the, brightnessstrength of(5) the the Rosseland distribution angular diameter derived from the model and 1 Γ(α/2 + k + 2)Γ(k + 1) 4 IOTA’s field of view decreases when using large band filters. k 0 ≥ 6 2J1 kBprLD of the source projected to a baseline, I(x), is symmetric with VLD rLD 1 uk x, the imaginary part of the visibility is zero, and the phase ð Þ¼3 uk ½�� ðÞ kBprLD � ðÞ( �� of the visibility is 0� or 180�. If the brightness distribution of the source is asymmetric, the imaginary part of the visibility � J3=2 kBprLD u(k) : 7 becomes nonzero, and the phase is neither 0� nor 180�. Gen- þ 2 3=2 ð Þ rffiffiffiffi kB��prLD ) erally, it is not easy to measure the phase of visibility with a ground interferometer because of atmospheric turbulence. (Quirrenbach et al. 1996). �� However, the closure phase cancels the effect of atmospheric The triple product is the product of the visibilities on three turbulence, and so source information is obtainable. Conse- baselines that form a triangle: quently, the closure phase is a useful interferometric observable for discussing the asymmetry of the brightness distribution of VTP V1 exp iÈ1 V2 exp iÈ2 V3 exp iÈ3 : 8 the source. ¼ jj ðÞ� jj ðÞ� jj ðÞ� ð Þ Astronomy & Astrophysics manuscript no. 16279 c ESO 2011 � April 7, 2011

Imaging the dynamical atmosphere of the red supergiant ⋆ Betelgeuse in the CO first overtone lines with VLTI/AMBER , , K. Ohnaka1, G. Weigelt1, F. Millour1 2, K.-H. Hofmann1, T. Driebe1 3, D. Schertl1, A. Chelli4, F. Massi5, R. Petrov2, and Ph. Stee2 Ohnaka et al.: Imaging the dynamical outer atmosphere of Betelgeuse in the CO first overtone lines 9 1 Max-Planck-Institut fürRadioastronomie, Auf dem Hügel 69, 53121 Bonn, Germany e-mail: [email protected] 2 Observatoire de la Côte d’Azur, Departement FIZEAU, Boulevard de l’Observatoire, B.P. 4229, 06304 Nice Cedex 4, France 3 Deutsches Zentrum fürLuft- und Raumfahrt e.V., Königswinterer Str. 522-524, 53227 Bonn, Germany 4 Institut de Planétologie et d’Astrophysique de Grenoble, BP 53, 38041 Grenoble, Cédex 9, France 5 INAF-Osservatorio Astrofisico di Arcetri, Instituto Nazionale di Astrofisica, Largo E. Fermi 5, 50125 Firenze, Italy Received / Accepted

ABSTRACT

Aims. We present one-dimensional aperture synthesis imaging of the red supergiant Betelgeuse (α Ori) with VLTI/AMBER. We reconstructed for the first time one-dimensional images in the individual CO first overtone lines. Our aim is to probe the dynamics of the inhomogeneous atmosphere and its time variation. Methods. Betelgeuse was observed between 2.28 and 2.31 µm with VLTI/AMBER using the 16-32-48 m telescope configuration with a spectral resolution up to 12000 and an angular resolution of 9.8 mas. The good nearly one-dimensional uv coverage allows us to reconstruct one-dimensional projection images (i.e., one-dimensional projections of the object’s two-dimensional intensity distri- butions). Results. The reconstructed one-dimensional projection images reveal that the star appears differently in the blue wing, line center, and red wing of the individual CO lines. The one-dimensional projection images in the blue wing and line center show a pronounced, asymmetrically extended component up to 1.3 R⋆, while those in the red wing do not show such a component. The observed one- ∼ dimensional projection images in the lines can be reasonably explained by a model in which the CO gas within a region more than 1 half as large as the stellar size is moving slightly outward with 0–5 km s− , while the gas in the remaining region is infalling fast 1 with 20–30 km s− . A comparison between the CO line AMBER data taken in 2008 and 2009 shows a significant time variation in the dynamics of the CO line-forming region in the photosphere and the outer atmosphere. In contrast to the line data, the reconstructed one-dimensional projection images in the continuum show only a slight deviation from a uniform disk or limb-darkened disk. We derive a uniform-disk diameter of 42.05 0.05 mas and a power-law-type limb-darkened disk diameter of 42.49 0.06 mas and a ± 2 ± limb-darkening parameter of (9.7 0.5) 10− . This latter angular diameter leads to an effective temperature of 3690 54 K for the Fig. 2. Continuum visibilities of± Betelgeuse× averaged over the continuum spectral channels between 2.28 and 2.293 µm. The insets ± continuum-formingshow enlarged layer. views of These the second, diameters third, and confirm fourth/fifth that lobes. the The near-IR solid and size dashed of Betelgeuse lines represent was the visibil nearlyities constant for a uniform over the last 18 years, in marked contrastdisk with ato diameter the recently of 42.05 reported mas and for noticeable a limb-darkened decrea disksewith in a the diameter mid-IR of 42.49 size. mas The andcontinuum a limb-darkening data parametertaken in of 2008 and 2009 reveal no or only0.097 marginal (power-law-type time variations, limb-darkened much disk smaller of Hestroff thaner 1997), the marespectively.ximum The variation dotted lines predicted represent by the the maxi currentmum range three-dimensiona of the l convection Ohnakavariations et al. in the (2011) 2.22 µm visibility A&A, due to time-dependent 520, 163 inhomogeneous [More surface on structures CO predicted in by the three-dimensionalOri @ 11:30!] simulations. 1 convection simulation of Chiavassa et al. (2009), who presents the model prediction up to 70 arcsec− . α Our two-epoch AMBER observations show that the outer atmosphere extending to 1.3–1.4 R⋆ is asymmetric and its Conclusions. ∼ dynamics is dominated by vigorous, inhomogeneous large-scale motions, whose overall nature changes drastically within one year. This is likely linked to the wind-driving mechanism in red supergiants. Key words. infrared: stars – techniques: interferometric – stars: supergiants – stars: late-type – stars: atmospheres – stars: individual: Betelgeuse

arXiv:1104.0958v1 [astro-ph.SR] 5 Apr 2011 1. Introduction no satisfactory theories for the RSG mass loss at the moment, as stressed by Harper (2010). Red supergiants (RSGs) experience slow, intensive mass loss up 4 1 to 10− M yr− , which is very important for understanding the ⊙ Studies of the outer atmosphere, where the winds are accel- final fate of massive stars. For example, our poor understanding erated, are a key to tackling this problem. The outer atmosphere of the RSG mass loss makes it difficult to estimate the main- of RSGs has complicated structures. The UV observations of sequence mass range of the progenitors of the Type IIP super- the well-studied RSG Betelgeuse (α Ori, M1-2Ia-Ibe) with the novae, which are the most common type of core-collapse super- Hubble Space Telescope reveal that the hot ( 6000–8000 K) ∼ novae. The mass loss also plays a significant role in the chem- chromospheric plasma is more than twice as extended as the ical enrichment of galaxies. Despite this importance, there are photosphere (Gilliland & Dupree 1996). However, radio continuum observations with the Very Large Array show that much ff cooler ( 1000–3000 K) gas extends to several stellar radii (Lim Send o print requests to: K. Ohnaka ∼ ⋆ Based on AMBER observations made with the Very Large et al. 1998), suggesting that the hot chromospheric plasma and Telescope Interferometer of the European Southern Observatory. cooler gas coexist. IR spectroscopic and interferometric studies Program ID: 082.D-0280 (AMBER Guaranteed Time Observation) of a few bright RSGs also show the presence of dense H2O gas in Second-lobe measurements from interferometry to date

M5 II α Her (Perrin et al. 2004) M4.5 III BY Boo (Wittkowski et al. 2001) M4 III V416 Lac (Wittkowski et al. 2001) M4 III ψ Phe (Wittkowski et al. 2001; Wittkowski et al. 2004 (VLTI)) M1.5 III α Cet (Wittkowski et al. 2006) (VLTI) M1Iab α Ori (Burns et al. 1997; Perrin et al. 2004; Xaubois et al. 2010, Ohnaka et al. 2011 (VLTI)) M0 III γ Sge (Wittkowski et al. 2001; Wittkowski et al. 2006 (VLTI)) K3 II γ Aql (Nordgen et al. 1999) K2 III α Ari (Hajian et al. 1998) K2 III 11 Lac (Baines et al. 2010) K1.5 III α Boo (Quirrenbach et al. 1996, Lacour et al. 2008) K1 V α Cen B (Bigot et al. 2006) (VLTI) K1 IV γ Cep (Baines et al. 2009) K0 III α Cas (Hajian et al. 1998) K0 III η Ser (Mérand et al. 2010) F8 Ib α UMi (Mérand et al. 2006) F5 Ib α Per (Mérand et al. 2007) F5 IV-V α CMi (Aufdenberg et al. 2005) F2 IV β Cas (Che et al. 2011) F0 Ib α Car (Domiciano de Souza et al. 2008) (VLTI) A7 V α Aql (Ohishi et al. 2004; Monnier et al. 2007) A1 V α CMa (Hanbury Brown et al. 1974) A0 V α Lyr (Peterson et al. 2006; Aufdenberg et al. 2006) A&A 413, 711–723 (2004) Astronomy DOI: 10.1051/0004-6361:20034149 & c ESO 2003 � Astrophysics

Tests of stellar model atmospheres by optical interferometry

VLTI/VINCI limb-darkening measurements of the M4 giant ψ Phe�

1 2 3 M. WittkowskiM. Wittkowski et al.: VLTI/,VINCI J. P. Aufdenberg limb-darkening, and measurements P. Kervella of ψ Phe 721

1.0 0.020 1 European Southern Observatory, Karl-Schwarzschild-Str. 2, 85748 Garching bei M¨unchen, Germany VLTI/VINCI PHOENIX sph. e-mail: [email protected] ψPhe (M4 III) PHOENIX pl. 2 0.8Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Mail Stop 15, Cambridge, MA 02138, USAATLAS 12 pl. e-mail: [email protected] 0.015 ATLAS 9 pl. 3 European Southern Observatory, Casilla 19001, Santiago 19, Chile UD/FDD 0.6e-mail: [email protected] 0.010 Received0.4 1 August 2003 / Accepted 17 September 2003

Abstract. We present K-band interferometric measurements of0.005 the limb-darkened (LD) intensity profile of the M 4 giant Squared visibility amplitude star0.2ψ Phoenicis obtained with the Very Large Telescope InterferometerSquared visibility amplitude (VLTI) and its commissioning instrument VINCI. High- precision squared visibility amplitudes in the second lobe of the visibility function were obtained employing two 8.2 m Unit Telescopes (UTs). This took place one month after light from UTs was first combined for interferometric fringes. In addition, 0.0 0.000 we sampled the visibility function at small spatial frequencies using the 40 cm test siderostats. Our measurement constrains 0 50 100 150 200 250 140 160 180 200 220 240 260 280 the diameter ofSpatial the star frequency as well B/ asλ its (1/arcsec) center-to-limb intensity variation (CLV). WeSpatial construct frequency a spherical B/λ (1/arcsec) hydrostatic PHOENIX model atmosphere based on spectrophotomet0 ric data from the literature and compare its CLV prediction0 with our interferometric Fig. 6. Our measuredmeasurement. squared We visibility compare as amplitudes well CLV of predictionsψ Phe (“x” by symbols plane-parallel with error hydrostatic bars) togetherPHOENIX with, ATLAS the 9 (solid, and ATLAS black line) 12 models. spherical We PHOENIX model predictionfind that with the model Rosseland parameters angularT diametereff, log g as, and predicted mass by as comparison derived from of thespectrophotometry spherical PHOENIX andmodel model with evolutionary spectrophotometry tracks is(Sect. 3.2), and best fittingin goodθLD value. agreement Shown with are our also interferometric the (dashed-dotted diameter line) measuremen plane-parallelt. ThePHOENIX shape of ourmodel, measured (dotted visibility line) plane-parallel function in theATLAS second 12 model, (dashed line)lobe plane-parallel is consistentATLAS with all 9 model, considered all withPHOENIX correspondingand ATLAS modelmodel parameterspredictions, and and isbest significantly fitting θLD di. Asfferent a reference to uniform for disk the (UD) strength of the limb-darkening,Wittkowski,and the fully gray darkened lines Aufdenberg denote disk (FDD) corresponding models. & Kervella We UD derive (upper (2004) high-precision line) and A&A, FDD fundamental 413, (lower 711 line) parameters model visibility for ψ Phe, functions namely. a The Rosseland left panel shows the full rangeangular of the visibilitydiameter of function 8.13 while0.2 mas, the with right the panel Hipparcos is an enlargement parallax corresponding of the low squaredto a Rosseland visibility linear amplitudes radius R inof the 86 second3 R , lobe. All ± ± � considered PHOENIXand an effandectiveATLAS temperaturemodel predictions of 3550 50 result K, with in aR verycorresponding similar shape to a of luminosity the visibility of log functionL/L = in3.02 the 2nd0.06. lobe. Together Our measurements with ± � ± are significantlyevolutionary different models, from uniform these values disk and are consistentfully-darkened with a disk mass models, of 1.3 and0.2 consistentM , and a withsurface all gravity considered of logPHOENIXg = 0.68 and0.11.ATLAS models. ± � ± Key words. techniques: interferometric – stars: atmospheres – stars: fundamental parameters – stars: late-type Table 5. Results for θLD obtained by fits to our interferometric data. limb-darkening effect. Our measured values in the second lobe Listed are the model input parameters Teff, log g, and mass M, the of the visibility function seem to lie systematically above the model-specific correction factors C , the fit results for θ , their Ross/LD LD modelimportant predictions parameters by 0.5 for modeling1σ. It is not stellar yet clear atmospheres if these smalland corresponding1. Introduction reduced χ2 values, and finally the Rosseland angular ∼ − ν distellarfferences evolution. are caused by systematic effects of our data calibra- diameter θRoss. The error σ(θLD) is estimated to be uniformly 0.2 mas tion or by the model structures. The squared visibility ampli- Stellar atmosphere models predict the spectrum emerging± from Further tests of stellar atmosphere models by interferomet- for all model fits (see text). tudes derived from the siderostat data show a systematic off- every point of a stellar disk. However, model atmospheres are ric observations help to improve the reliability of results in all T log g MC θ χ2 θ usuallyeff only constrainedRoss by/LD comparisonLD toν integratedRoss stellar setareas of about of astrophysics 1 σ towards where larger such values models with are respect used. Cool to our giants best spectra. Optical interferometry has proven its capability to go fittingand AGB curves. stars This are is of likely interest caused for atmosphere by possible modeling small system- since beyondSpherical this principal PHOENIX test models: of the predicted flux, and to probe the aticthey calibration allow the e studyffects, of since extended the calibration stellar atmospheres of our high and squared the wavelength-dependent center-to-limb intensity variation (CLV) visibility amplitudes in the range 0.8 0.95 derived from the 3550 0.7 1.3 0.9388 8.664 1.80 8.13 stellar mass-loss process. Red giants∼ are− also used as probes of across the stellar disk. In addition, the measurement of the stel- siderostatthe chemical data enrichment was difficult history (see of Sect. nearby 4.3). galaxies through de- lar3500 angular 0.7 diameter 1.3 together 0.9394 with8.663 the bolometric 1.81 8.14 flux is the tailedThe abundance best fitting measurements Rosseland of angular the calcium diameters infrared are triplet well primary3600 measure 0.7 1.3 of the 0.9381 effective temperature,8.665 1.79 one8.13 of the most constrainedwhich rely on by model our measurem atmospheres.ent. The measurements in the 3550 0.5 1.3 0.9218 8.814 1.79 8.12 2nd lobeHowever, of the the visibility required function direct measurements also constrain of the stellar positions in- 3550 1.0 1.3 0.9577 8.504 1.79 8.14 oftensity the 1st profiles minimum are among and 2nd the maximum most challenging of the visibility programs func- in Send3550 offprint 0.7 requests 1.0 to: 0.9302 M. Wittkowski,8.739 1.80 8.13 e-mail: [email protected] tion,modern which optical is a constraint interferometry. of the Since diameter. more than This one constraint resolu- is Plane-parallel� Based on public PHOENIX data released model: from the European Southern independenttion element of across possible the stellar small systematic disk is needed calibration to determine uncertain- sur- 2 2 Observatory VLTI obtained from the ESO/ST-ECF Science Archive tiesface of structure the V parametersvalues. The beyond reduced diameters,χ values the for long the baselines different 3550 0.7 / 1 8.168 1.72 8.17 | | ν Facility. The VLTI was operated with the commissioning instrument consideredneeded to models obtain this do not resolution show significant also produce differences, very low which visi- is Plane-parallel ATLAS 12 model: VINCI and the MONA beam combiner. consistentbility amplitudes with the corresponding very similar predicted to vanishing shapes fringe of contrasts. the model 3550 0.7 / 1 8.191 1.78 8.19 visibility functions. These similarities are expected since our Plane-parallel ATLAS 9 models: measurement is dominated by continuum photons, and the con- 3500 0.5 / 1 8.244 1.74 8.24 tinuum forming region of the atmosphere is almost compact Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20034149 3500 1.0 / 1 8.243 1.73 8.24 (see Fig. 5). 3750 0.5 / 1 8.227 1.71 8.23 3750 1.0 / 1 8.228 1.71 8.23 6. Discussion and conclusions otherwise) and FDD (I = µ) model visibility functions are Spherical PHOENIX models We have constructed a spher- shown, with diameters θUD and θFDD corresponding to our ical hydrostatic PHOENIX model atmosphere for ψ Phe in favorite PHOENIX model fit. Our measurements differ sig- Sect. 3.2. Here, we confront this model’s prediction for the nificantly from UD and FDD models, confirming the limb- CLV by comparing it with our VLTI/VINCI measurement darkening effect. All considered PHOENIX and ATLAS model of the visibility function in the second lobe. We find that CLV predictions lead to very similar model visibility functions the model predicted shape of the visibility function is con- up to the 2nd lobe, which are all consistent with our data. sistent with our VLTI/VINCI measurements. Simultaneously, Hence, our data confirm the model-predicted strength of the the Rosseland angular diameter derived from the model and M. Wittkowski et al.: VLTI/VINCI limb-darkening measurements of ψ Phe 721

1.0 0.020 VLTI/VINCI PHOENIX sph. ψPhe (M4 III) PHOENIX pl. 0.8 ATLAS 12 pl. 0.015 ATLAS 9 pl. UD/FDD 0.6 0.010 0.4

0.005 Squared visibility amplitude 0.2 Squared visibility amplitude

0.0 0.000 0 50 100 150 200 250 140 160 180 200 220 240 260 280 Spatial frequency B/λ (1/arcsec) Spatial frequency B/ (1/arcsec) 0 λ0 Fig. 6. Our measured squared visibility amplitudes of ψ Phe (“x” symbols with error bars) together with the (solid black line) spherical PHOENIX model prediction with model parameters Teff, log g, and mass as derived from spectrophotometry and model evolutionary tracks (Sect. 3.2), and best fitting θLD value. Shown are also the (dashed-dotted line) plane-parallel PHOENIX model, (dotted line) plane-parallel ATLAS 12 model, (dashed line) plane-parallel ATLAS 9 model, all with corresponding model parameters and best fitting θLD. As a reference for the strength of the limb-darkening, the gray lines denote corresponding UD (upper line) and FDD (lower line) model visibility functions. The left panel shows the full range of the visibility function while the right panel is an enlargement of the low squared visibility amplitudes in the second lobe. All considered PHOENIX and ATLAS model predictions result in a very similar shape of the visibility function in the 2nd lobe. Our measurements are significantly different from uniform disk and fully-darkened disk models, and consistent with all considered PHOENIX and ATLAS models.

Table 5. Results for θLD obtained by fits to our interferometric data. limb-darkening effect. Our measured values in the second lobe Listed are the model input parameters Teff, log g, and mass M, the of the visibility function seem to lie systematically above the Extractingmodel-specific correction a factorsDiameter:CRoss/LD, the fit results for θLD, their model predictions by 0.5 1σ. It is not yet clear if these small 2 ∼ − corresponding reduced χν values, and finally the Rosseland angular differences are caused by systematic effects of our data calibra- Modeldiameter θRoss .Geometry The error σ(θLD) is estimated Matters to be uniformly at the0.2 mas 1-2%tion or by level the model in structures. K-band The squared visibility ampli- for all model fits (see text). ± tudes derived from VLTI/VINCI theM. siderostat Wittkowski K-band data et al.: show VLTI/VINCI a systematic limb-darkening off- measurements of ψ Phe 721 2 Teff log g MCRoss/LD θLD χν θRoss set of about1.0 1 σ towards larger values with respect to0.020 our best fitting curves. This is likely caused by possible small system-VLTI/VINCI PHOENIX sph. ψPhe (M4 III) PHOENIX pl. atic calibration0.8 effects, since the calibration of our high squared ATLAS 12 pl. Spherical PHOENIX models: 0.015 visibility amplitudes in the range 0.8 0.95 derived from the ATLAS 9 pl. 3550 0.7 1.3 0.9388 8.664 1.80 8.13 ∼ − UD/FDD siderostat0.6 data was difficult (see Sect. 4.3). 3500 0.7 1.3 0.9394 8.663 1.81 8.14 The best fitting Rosseland angular diameters0.010 are well 3600 0.7 1.3 0.9381 8.665 1.79 8.13 constrained0.4 by our measurement. The measurements in the 3550 0.5 1.3 0.9218 8.814 1.79 8.12 2nd lobe of the visibility function also constrain the positions0.005 Squared visibility amplitude 3550 1.0 1.3 0.9577 8.504 1.79 8.14 of the 1st0.2 minimum and 2nd maximum of the visibilitySquared visibility amplitude func- 3550 0.7 1.0 0.9302 8.739 1.80 8.13 tion, which is a constraint of the diameter. This constraint is Plane-parallel PHOENIX model: 0.0 0.000 independent0 of possible50 100 small systematic150 200 calibration250 uncertain-140 160 180 200 220 240 260 280 2 Spatial frequency B/λ (1/arcsec)2 Spatial frequency B/λ (1/arcsec) ties of the V values. The reduced0 χ values for the different 0 3550 0.7 / 1 8.168 1.72 8.17 | | ν Fig.considered 6. Our measured models squared do visibility not show amplitudes significant of ψ Phe di (“x”fferences, symbols with which error isbars) together with the (solid black line) spherical PHOENIX Plane-parallel ATLAS 12 model: modelconsistent prediction with with modelthe very parameters similarTeff predicted, log g, and mass shapes as derived of the from model spectrophotometry and model evolutionary tracks (Sect. 3.2), 3550 0.7 / 1 8.191 1.78 8.19 andvisibility best fitting functions.θLD value. Shown These are also similarities the (dashed-dotted are expected line) plane-parallel sincePHOENIX our model, (dotted line) plane-parallel ATLAS 12 model, (dashed line)∆ plane-parallelθ ATLAS 9 model, all with corresponding model parameters and best fitting θLD. As a reference for the strength of the Plane-parallel ATLAS 9 models: limb-darkening,measurement the grayis dominated lines0 denote.015 by corre continuumsponding UD photons, (upper line) and and the FDD con- (lower line) model visibility functions. The left panel shows 3500 0.5 / 1 8.244 1.74 8.24 thetinuum full range forming of the visibility region function of the while atmosphere the right panel is an almost enlargement compact of the low squared visibility amplitudes in the second lobe. All considered(see Fig.PHOENIXθ 5). and ATLAS model predictions result in a very similar shape of the visibility function in the 2nd lobe. Our measurements 3500 1.0 / 1 8.243 1.73 8.24 are significantly different from uniform disk and fully-darkened disk models, and consistent with all considered PHOENIX and ATLAS models. 3750 0.5 / 1 8.227 1.71 8.23

3750 1.0 / 1 8.228 1.71 8.23 Table 5. Results for θLD obtained by fits to our interferometric data. limb-darkening effect. Our measured values in the second lobe Listed6. Discussion are the model input and parameters conclusionsTeff, log g, and mass M, the of the visibility function seem to lie systematically above the model-specific correction factors CRoss/LD, the fit results for θLD, their model predictions by 0.5 1σ. It is not yet clear if these small 2 ∼ − otherwise) and FDD (I = µ) model visibility functions are correspondingSpherical reduced PHOENIXχν values, models and finallyWe the have Rosseland constructed angular di afferences spher- are caused by systematic effects of our data calibra- Phe: Wittkowski, Aufdenberg & Kervella diameter(2004)θ .A&A, The error σ413,(θ ) is estimated711 to be uniformly 0.2 mas shown, with diameters θUD and θFDD corresponding to our ical hydrostaticRoss PHOENIXLD model atmosphere for tionψ Phe or by in the model structures. The squared visibility ampli- ψ for all model fits (see text). ± favorite PHOENIX model fit. Our measurements differ sig- Sect. 3.2. Here, we confront this model’s predictiontudes for derived the from the siderostat data show a systematic off- T log g MC θ χ2 θ set of about 1 σ towards larger values with respect to our best nificantly from UD and FDD models, confirming the limb- CLVeff by comparing itRoss/ withLD LD our VLTIν /VINCIRoss measurement fitting curves. This is likely caused by possible small system- darkening effect. All considered PHOENIX and ATLAS model of the visibility function in the second lobe. We find that Spherical PHOENIX models: atic calibration effects, since the calibration of our high squared CLV predictions lead to very similar model visibility functions the model predicted shape of the visibility functionvisibility is con- amplitudes in the range 0.8 0.95 derived from the 3550 0.7 1.3 0.9388 8.664 1.80 8.13 up to the 2nd lobe, which are all consistent with our data. sistent with our VLTI/VINCI measurements. Simultaneously,siderostat data was difficult (see Sect.∼ 4.3).− Hence, our data confirm the model-predicted strength of the the3500 Rosseland 0.7 angular 1.3 0.9394 diameter8.663 derived 1.81 8.14 from the modelThe and best fitting Rosseland angular diameters are well 3600 0.7 1.3 0.9381 8.665 1.79 8.13 constrained by our measurement. The measurements in the 3550 0.5 1.3 0.9218 8.814 1.79 8.12 2nd lobe of the visibility function also constrain the positions 3550 1.0 1.3 0.9577 8.504 1.79 8.14 of the 1st minimum and 2nd maximum of the visibility func- 3550 0.7 1.0 0.9302 8.739 1.80 8.13 tion, which is a constraint of the diameter. This constraint is Plane-parallel PHOENIX model: independent of possible small systematic calibration uncertain- 2 2 3550 0.7 / 1 8.168 1.72 8.17 ties of the V values. The reduced χν values for the different considered| models| do not show significant differences, which is Plane-parallel ATLAS 12 model: consistent with the very similar predicted shapes of the model 3550 0.7 / 1 8.191 1.78 8.19 visibility functions. These similarities are expected since our Plane-parallel ATLAS 9 models: measurement is dominated by continuum photons, and the con- 3500 0.5 / 1 8.244 1.74 8.24 tinuum forming region of the atmosphere is almost compact 3500 1.0 / 1 8.243 1.73 8.24 (see Fig. 5). 3750 0.5 / 1 8.227 1.71 8.23 3750 1.0 / 1 8.228 1.71 8.23 6. Discussion and conclusions otherwise) and FDD (I = µ) model visibility functions are Spherical PHOENIX models We have constructed a spher- shown, with diameters θUD and θFDD corresponding to our ical hydrostatic PHOENIX model atmosphere for ψ Phe in favorite PHOENIX model fit. Our measurements differ sig- Sect. 3.2. Here, we confront this model’s prediction for the nificantly from UD and FDD models, confirming the limb- CLV by comparing it with our VLTI/VINCI measurement darkening effect. All considered PHOENIX and ATLAS model of the visibility function in the second lobe. We find that CLV predictions lead to very similar model visibility functions the model predicted shape of the visibility function is con- up to the 2nd lobe, which are all consistent with our data. sistent with our VLTI/VINCI measurements. Simultaneously, Hence, our data confirm the model-predicted strength of the the Rosseland angular diameter derived from the model and Spherical Atmospheres: same gravity, different mass

Two stars: same T and log(g), different Mass eff a) plane-parallel case

θ = cos-1 µ

z Spherical Geometry and Atmospheric Extension

b) spherical case

θ = cos-1 µ z

r0 r1

r2 rN r3

p 6 M. Wittkowski et al.: NPOI & VLTI Interferometry of γ Sagittae

0.4 0.4 0.4 γ Sge γ Sge γ Sge NPOI NPOI NPOI 0.3 0.3 0.3 ATLAS 9 ATLAS 9 ATLAS 9 PHOENIX pp PHOENIX pp PHOENIX pp PHOENIX sph PHOENIX sph PHOENIX sph 0.2 0.2 0.2

0.1 0.1 0.1

Squared visibility amplitude (CE) 0.0 Squared visibility amplitude (CE) 0.0 Squared visibility amplitude (CE) 0.0 A&A 460, 843–853500 (2006)600 700 800 900 500 600 700 800Astronomy900 500 600 700 800 900 DOI: 10.1051/0004-6361:20065853Wavelength (nm) Wavelength (nm) & Wavelength (nm) c ESO 20060.4 � 0.4 Astrophysics γ Sge γ Sge 0.4 γ Sge NPOI NPOI NPOI 0.3 0.3 ATLAS 9 ATLAS 9 0.3 ATLAS 9 PHOENIX pp PHOENIX pp PHOENIX pp Tests0.2 of stellar modelPHOENIX atmospheres sph 0.2 by optical interferometryPHOENIX sph PHOENIX sph 0.2 III. NPOI and VINCI interferometry of the M0 giant γ Sagittae 0.1 covering 0.5–2.20.1 µm�,�� 0.1

Squared visibility amplitude (CE) 0.0 M. Wittkowski1, C. A. Hummel2, J. P.Squared visibility amplitude (CE) Aufdenberg0.0 3, and V. Roccatagliata4 Squared visibility amplitude (CE) 0.0 500 600 700 800 900 500 600 700 800 900 500 600 700 800 900 Wavelength (nm) Wavelength (nm) Wavelength (nm) 1 European Southern Observatory, Karl-Schwarzschild-Str. 2, 85748 Garching bei München, Germany Fige-mail:. 5. [email protected]. 3, but showing the squared visibility amplitudes on the NPOI CE baseline. 2 European8 Southern Observatory, CasillaM. Wittk19001,owski Santiagoet al.: NPOI 19,& ChileVLTI Interferometry of γ Sagittae 3 National Optical Astronomy Observatory, 950 North Cherry Avenue, Tucson, AZ 85719, USA 4 Max-Planck-InstitutTable 3. Details of our fürVL Astronomie,TI/VINCI observ Königsstuhlations of γ 17,Sagittae 69117(date Heidelberg,0.4 Germany 0.015 and0.04time of observation, spatial frequency, azimuth angle of the pro- γ γ Sge 0.015 Sge γ Sge γ Sge Receivedjected baseline 17 June(E 2006of N)),/ Acceptedtogether with 15 Septemberthe measured 2006squared visibil- NPOI VLTI/VINCINPOI NPOI ity amplitudes and their errors. The last column denotes the number 0.3 ATLAS 9 of successfully processed interferograms for each series. The effective 0.03 PHOENIXATLAS 9 pp 0.010 ATLAS 9 wavelength for our γ Sagittae observations is ∼ 2.19ATLASµm. F9or eachABSTRACTdate 0.010 PHOENIX pp PHOENIX pp of observation, we list the equivalent uniform discPHOENIX(UD) diameter pp ob- PHOENIX sph PHOENIX sph PHOENIX sph Context.tained0.02fromOpticalonly interferometrythe data of the specific allowsnight. a measurementUsingPHOENIXall data ofsphtogether the intensity, profile0.2 across a stellar disc, leading to a direct test and calibra- tionwe ofobtain theoreticalan equi modelvalent UD atmospheresdiameter of asΘUD well= 5 as.93 to± 0 a.02 precisemas, or determinationan of fundamental stellar parameters. 0.005 Aims.equiWevalent presentFDD diameter a comparisonof ΘFDD of= the6.69 visual± 0.02 andmas. near-infrared intensity0.005 profile of the M0 giant γ Sagittae to plane-parallel ATLAS 9 Triple amplitude Triple amplitude as well as to plane-parallel and spherical PHOENIX model atmospheres. Triple amplitude 0.01 0.1 Methods. We use previously described2 visual interferometric data obtained with the Navy Prototype Optical Interferometer (NPOI) UT Sp. freq az V σV2 # in July 2000. We[1 apply/ ] the[de recentlyg] developed technique of coherent0.000 integrationSquared visibility amplitude , and thereby obtain visibility data of more spectral0.000 channels0.00 (526–85228 nm)June and2002, withΘUD higher= 5.91 preci± 0.03sionmas than before. In addition, we employ new measurements of the near-infrared K-band ( 220005:16:57 nm) diameter131.14 of γ136.63Sagittae1.825e-01 obtained with8.074e-03 the instrument161 VINCI at0.0 the ESO VLT Interferometer (VLTI) in 2002. ∼ Results.05:22:40500The spherical130.08600PHOENIX136.65 model7001.828e-01 leads to8008.852e-03 a precise definition900152 of the500 Rosseland40 600 a50ngular diameter60700 and70 a cons800istent80 high-precision90090 500 600 700 800 900 diameter05:32:34 value for128.15 our NPOI136.73Wavelength and VLTI2.077e-01 /(nm)VINCI7.578e-03 data sets of172Θ = 6.06 0.02 mas, withWavelengthProjected the Hipparcos baseline (nm) parallax (m) corresponding to Wavelength (nm) Ross ± RRoss 06:44:22= 55 4 R111.69, and with139.42 the bolometric3.112e-01 flux1.079e-02 corresponding80 to an effective temperature Teff = 3805 55 K. Our visual visibility 0.012 � Fig. 8. Measured γ Sagittae squared visibility amplitudes obtained data close to± the first8 July minimum2002, Θ and= in5.89 the± second0.04γmas Sge lobe constrain the limb-darkening effect and are generally± consistentγ Sge with the model γ Sge UD with VLTI/VINCI in June to September 2002, together with the syn- atmosphere05:12:10 predictions.124.21 The137.05 visual2.241e-01 closure phasesNPOI7.199e-03 exhibit a397 smooth transition between 0 and π. NPOI NPOI 0.010 thetic visibility curves of the best fitting models. Conclusions.05:20:13 The122.44 agreement137.27 between2.426e-01 the NPOI7.846e-03 and VINCI418 diameter0.010 values increases the confidence in the model atmosphere pre-0.010 ATLAS 9 ATLAS 9 ATLAS 9 dictions0.008 from optical11 July to2002, near-infraredΘUD = 5.98 wavelengths± 0.04 mas as well as in the calibration and accuracy of both interferometric facilities. The PHOENIX pp consistentWittkowski03:06:45 night-by-night142.16 et al. diameter138.28 (2006) values1.064e-01 A&A, of VINCI1.256e-02 460, give 843 additional55 confidence in the given uncertainties. ThePHOENIX closure pp phases suggest PHOENIX pp 0.006 PHOENIX sph creasing wavelength. A detailed comparisonPHOENIXof oursph data to PHOENIX sph a slight05:29:49 deviation117.47 from circular138.08 symmetry,2.495e-01 which8.775e-03 may be due69 to surfacemodel features,atmospheres an asymmetricfollows belo extendedw in Sect. layer,4. or a faint unknown companion.05:34:22 116.38 138.31 2.660e-01 9.929e-03 207 0.005 0.005 0.00405:40:41 114.83 138.64 2.878e-01 8.524e-03 297

KeyTriple amplitude words. techniques: interferometric – stars: late-type – stars:Triple amplitude AGB and post-AGB – stars: fundamental parameters – Triple amplitude 05:47:01 113.25 139.02 2.977e-01 1.228e-02 198 Analysis of calibration uncertainties In order to test and ver- stars:0.002 atmospheres – stars: individual: γ Sagittae 15 July 2002, ΘUD = 5.94 ± 0.05 mas ify the calibration uncertainties that are used in our analysis, 0.00004:22:40 128.76 136.70 1.926e-01 6.824e-03 90 0.000we investigate the night-to-night variation of the obtained di-0.000 04:41:27 124.90 136.98 2.223e-01 1.315e-02 139 ameter values. All derived single nights’ diameter values and -0.002 8 August 2002, Θ = 5.92 ± 0.03 mas UD uncertainties are consistent within 1.3σ with the value obtained 1. Introduction03:05:49500 125.17600 136.95 7002.179e-01 8007.678e-03 900469 500Theoretical600 atmosphere700 models predict800 in general900 the spec-500 600 700 800 900 from all data together (5.93 mas) as well as with the weighted 03:12:20 123.77 137.10Wavelength2.244e-01 (nm) 9.005e-03 436 trum emerging fromWavelength every point (nm) of a stellar disc. Optical in- Wavelength (nm) Cool giants03:18:34 on the red122.39 giant137.27 branch (RGB)2.309e-01 and7.979e-03 asymptotic468 gi- meanterferometryof the single providesnights’ thevalues strongest(5.92 mas). observational constraint of ant branchFig. 6.03:24:34 (AGB)As Fig. are121.033, verybut luminous137.47showing2.471e-01 andthe extended,NPOI1.901e-02 havetriple a178 lowamplitudes.thisThis predictionconfirms bythat resolvingour diameter the stellarvalue is disc.reliable In addition,and that the con- our estimate of uncertainties is realistic. The obtained high- surface temperature,04:13:56 108.94 and their140.21 atmospheres3.376e-01 can1.212e-02 thus be rich423 in straints on the intensity profiles allow us to find meaningful def- 12 September 2002, Θ = 5.99 ± 0.11 mas precision (0.3%) UD and FDD diameter values of Θ = molecules. Cool giants are the most importantUD source of dust for- initions of the stellar radius and its precise measurement.UD 01:26:22 116.44 138.29 2.607e-01 1.547e-02 190 5.93 ± 0.02 mas and Θ = 6.69 ± 0.02 mas can thus be used mation and its delivery to the interstellar medium. The detailed For regular coolFDD non-pulsating giants, the centre-to-limb structureTable of01:47:171. theirFit extended111.24results atmospheres,139.54of our3.407e-01NPOI includingdata4.631e-02 theto models eff105ects withoutof a uniformfurther uncertainties.disc Interferometer (VLTI, Glindemann et al. 2003), the instrument 18 September 2002, Θ = 5.88 ± 0.05 mas variation (CLV) is mainly characterised by the limb-darkening from(UD) circumstellarand a molecularfully dark andened dustUD(FDD) layers, aredisc. stillThe a matterformal errorsAdditionalofpossiblethe di-systematicVINCIerrors that(Kervare constantella etoval.er 2003), and the two VLTI test siderostats 00:32:53 123.38 137.15 2.347e-01 1.081e-02 428 timeeffect,scales whichlarger isthan anco effvectered ofby theour verticalanalysis, temperaturei.e. about 2 profile of investigation00:40:53 and121.59 debate∼ (cf.,137.39. e.g.2.393e-01 Scholz 1985,1.177e-02 1998, 2001;407 ameter values are 0 01 mas, additional calibrationmonths,ofuncertainties the stellarcan not atmosphere.be ruledare out.onSuch TheJunesystematic strength28, oferrorsJuly thecould limb-darkening8, Julyin 11, July 15, August 8, September 12, Perrin et al.00:47:14 2004; Ohnaka120.14 2004a;137.61 Ireland2.667e-01 & Scholz1.751e-02 2006).210 ∼ 0.06 mas, total errors thus ∼ 0.06 mas. principlecan be probedbe related byto opticalthe calibration interferometryof the interferometric in two waysar- (cf., e.g. Hanbury Brown et al. 1974;and QuirrenbachSeptember et al.18, 1996;2002. BurnsThese data are public commission- � Based on data obtained with the Navy Prototype Optical ray and the instrument, such as the calibration of the baseline 1 Table 4. Fit results of our VINCI data to UD and FDD models. lengthet al.or 1997;the eff Hajianective w etav al.elength. 1998;Such Wittkowskiuncertainties et al.are 2001,not 2004; Interferometer (NPOI). The NPOI is a joint project of the Naval 2 ing data released from the VLTI . The VLTI stations E0 and Model Diameter Parameter αeAufdenbergxpectedχtoνrepresent et al. 2005):a considerable (1) by measuringsource of error variations. of an equiv- Research Laboratory and the United States Naval Observatoryχ2 in coop- G1 forming a ground baseline length of 66 m were used for Model Diameter ν alent uniform disc diameter (i.e. the uniform disc that has the eration with LowellUD Observatory,Θ Θ and=UD is= funded±5.64 bymas the Office of Navalα = 0 11.0 UD UD 5.93 0.02 mas 0.63 same integral flux as the true intensity profile) as a function of Research and the OceanographerΘ of= the Navy.± all our observations. The observations were repeated during FDDFDD ΘFDD 6.69= 60..5902 masmas0.63 α = 1 5.6 �� Based on public commissioningFDD data released by the Paranal 4.wavelength,Comparison andto (2)predictions by directly7 diffb constrainingyerentmodelnights the star’sspread intensityover more than 2 months in order to Observatory, Chile, and received via the ESO/ST-ECF Science Archive profileatmospheres in the second and higher lobes of the visibility function Facility. at one or several bandpasses.compute the night-to-night variation of the obtained diameter 3. parametersVLTI/VINCIlisted in Tmeasurementsable 4. Since our VINCI data cover only 4.1. Employed model atmospheres one bandpass and only data of the first lobe of the visibility We compare our measured visibilityand datatherebyto predictionsto estimateby the- the calibration uncertainty caused by function, it is -contrary to our NPOI data- not feasible to con- oretical model atmospheres indiorderfferentto calibrateatmosphericand test these and possibly instrumental conditions. All 3.1.strainVLArticletheTI/VINCIlimb-dark publishedening byobser EDPeffect Sciencessolelyvationsbased and availableon these atVINCI http://www.aanda.orgmodels, and toorderihttp://dx.doi.org/10.1051/0004-6361:20065853ve fundamental stellar parameters of γ Sge. 2 data. This is also reflected by equal χν values obtained for UD We use plane-parallel ATLAS 9 (Kurucz 1993) as well as TheandnearFDD-infraredmodels as wellKas-bandby the virtuallyinterferometricidentical modeldataplane-parallelof γ Sagittaeand spherical PHOENIX (Hauschildt et al. 1999) 1 http://www.eso.org/projects/vlti/instru/vinci/vinci data sets.html werevisibilityobtainedcurves in Fig.with8. the ESO Very Larmodelge atmospheresTelescopeto calculate synthetic visibility data, as The increased equivalent UD diameter with respect to the done in Papers I&II. We refer to the descriptions in Papers shorter NPOI wavelengths is consistent with the general trend I&II for more details on the employed model atmosphere files of decreasing strength of the limb-darkening effect with in- and their use. Differences between ATLAS 9 and PHOENIX mod- Spherical Model Yields Consistent Angular Diameter Across Optical/Near-IR for γ Sge (M0 III) 10 M. Wittkowski et al.: NPOI & VLTI Interferometry of γ Sagittae

Table 5. Results for the ﬁt of ATLAS 9 and PHOENIX model atmospheres to our interferometric VLTI/VINCI and NPOI data sets of γ Sagittae.

Model atmosphere NPOI (526 nm to 852 nm) VLTI/VINCI (2190 nm) ATLAS 9, plane-parallel, Teﬀ = 3750 K, log g =1.0 ΘLD = 6.18 ± 0.06 mas ΘLD = 6.05 ± 0.02 mas 2 2 χν = 2.2 χν = 0.6

PHOENIX, plane-parallel, Teﬀ = 3750 K, log g = 1.0 ΘLD = 6.11 ± 0.06 mas ΘLD = 6.05 ± 0.02 mas 2 2 χν = 2.3 χν = 0.6

PHOENIX, spherical, Teff = 3750 K, log g = 1.0, M = 1.3 M ΘLD = 6.30 ± 0.06 mas ΘLD = 6.30 ± 0.02 mas 2 2 χν = 2.4 χν = 0.6 ΘRoss = 6.02 ± 0.06 mas ΘRoss = 6.02 ± 0.02 mas All atmosphere models have: Teff = 3750 K, log(g) = 1.0 ter. The shallobut differentwer temperature geometriesgradient of the plane-parallel matching spatial structure of the layers where TiO molecules PHOENIX model leads to a better agreement between the NPOI reside. and VLTI/VINCI diameters ( 1 σ difference). The obtained diameter values in Tab. 5 are not affected by ≈ Finally, the spherical PHOENIX model leads to a Rosseland these observed discrepancies of the details of measured and model-predicted visibility function since the best-fitting diam- angular diameter of ΘRoss = 6.02 mas for both the NPOI and VLTI/VINCI data. The different and larger best-fit diameters eter for any given model file is mostly constrained by the posi- for the two datasets for plane-parallel PHOENIX model ap- tion of the first minimum and the global shape of the visibility pears to be due to model geometry. The agreement of NPOI curve. and VLTI/VINCIWittkowskidata sets etgi al.ves (2006)confidence A&A,in both,460, the843atmo- It is remarkable that the ATLAS 9 and PHOENIX models with sphere models and the accuracy of the results from NPOI and parameters listed in Tab. 5 lead to measurable differences of the VLTI/VINCI. predicted visibility values for the visual spectral channels of NPOI (Figs. 3-6). Similarly different models lead to identical visibility predictions for the (broad) K-band even in the second Shape of the visibility function The measured and model- lobe as shown in Paper II. This illustrates that the use of nar- predicted visibility functions are generally consistent. rower spectral channels and the use of the visual wavelength 2 ff However, the obtained reduced χν values for the NPOI data range provide stronger constraints on the limb-darkening e ect between 2.2 and 2.4 are above unity, as would be expected than observations in the broad K-band. for a perfect match. This indicates differences at the 2σ level between observed visibility data and the model predictions. Model atmosphere fluxes Fig. 9 shows the measured flux of These differences are most evident in (1) a lower second γ Sge from Alekseeva et al. 1997 in the wavelength range from maximum of the measured visibility function with respect to 0.4-1.0 µm, i.e. covering the NPOI range used in this paper. the model prediction on the EW baseline (Fig. 3), and (2) a Also shown are the predictions by the model atmospheres with flattened measured visibility function with respect to the model parameters listed in Tab. 5. The limb-darkened 0% diameter predictions at the blue end on the CW baseline (Fig. 4). It is values ΘLD derived from the fit to the interferometric data were not yet clear if and by how far these deviations of measured used to scale the model SEDs. The spectral resolution of the and synthetic visibility functions indicate different details of model SEDs is convolved to the resolution of the data used, the limb-darkening effect at visual spectral channels, or if they i.e. to 10 nm. The model-predicted flux curves based on the are caused by additional calibration uncertainties that are not three considered models are well consistent with the general included in the error bars. In particular the flattening of the shape of the measured flux, while the detailed description of the measured visibility function at the bluest spectral channels on spectral bands and features differs between the different models the CW baseline can most likely be explained by additional and the measured values. These differences can most likely be calibration uncertainties of our NPOI data, as the instrumental explained by the treatment of TiO absorption lines which are transfer function for these data exhibits a drop which may not important for the visual wavelength range and difficult to model be fully compensated. (Allard et al. 2000), as mentioned in the paragraph above. At optical wavelengths including all our NPOI spectral channels, TiO absorption bands are very important for the mod- Deviations from circular symmetry Differences between data elling of atmospheres of cool giants. It has been shown that the and models are also observed for the closure phases (Fig. 7). use of different line list combinations of TiO and H2O leads The observed smooth variation of the closure phases from 0 to significantly different model structures and spectra, in par- to π instead of the expected instantaneous flip may indicate ticular in the optical where TiO bands are important (Allard a small deviation from spherical symmetry as already men- et al. 2000). A possible explanation for differences between tioned in Paper I. As our maximum spatial resolution reaches our visibility data and the model predictions could thus also 2.7 mas, and the stellar disc has a size of ΘRoss = 6.02 mas, be mismatching opacity tables for the TiO bands and/or a mis- the stellar disc is well resolved with 2.2 resolution elements A&A 446, 635–641 (2006) Astronomy DOI: 10.1051/0004-6361:20053187 & c ESO 2006 � Astrophysics 638 L. Bigot et al.: The limb darkening of α Cen B

Table 4. limb-darkening I(λ, µ) for various wavelengths over the JHK range. λ (µm) / µ 0.0 0.1The 0.2 limb 0.3 darkening 0.4 0.5 of 0.6α Centauri 0.7 0.8 B 0.9 1.0 1.050 0.4434 0.5745 0.6453 0.7069 0.7605 0.8087 0.8527 0.8932 0.9311 0.9667 1.0000 1.270 0.4646 0.6017 0.6738 0.7347 0.7860 0.8310 0.8711 0.9074 0.9406 0.9715 1.0000 1.650Matching 0.4838 3D 0.6752 hydrodynamical 0.7487 0.8039 0.8462 models 0.8812with 0.9110 interferometric 0.9369 0.9601 measurements 0.9811 1.0000 2.000 0.5442 0.7063 0.7707 0.8202 0.8585 0.8905 0.9178 0.9417 0.9630 0.9825 1.0000 2.200 0.5729 0.7220 0.7817 0.8283 0.8646 0.8950 0.9211 0.9439 0.9645 0.9831 1.0000 1 2 1 3 2.400 0.5968 0.7353L. 0.7912 Bigot 0.8352, P. Kervella 0.8698, F. Thévenin 0.8988 0.9239, and D. 0.9458 Ségransan 0.9656 0.9836 1.0000 L. Bigot et al.: The limb darkening of α Cen B 641 1 Département Cassiopée, UMR 6202, Observatoire de la Côte d’Azur, BP 4229, 06304 Nice Cedex 4, France e-mail: [email protected] for the same stellar fundamental parameters. It appears that 2 LESIA, UMR 8109, Observatoire de Paris-Meudon, 5 place Jules3D Janssen, RHD 92195 produces Meudon a lessCedex, significan France t center-to-limb variation 3 Observatoire de Genève, 1290 Sauverny, Switzerland than a 1D static model. The departureUD from a 1D model increases with decreasing wavelengths.3D Such behavior was also Received 4 April 2005 / Accepted 12 September 2005 found by Allende Prieto et al. (2002) for Procyon. However, in the case of α Cen B, the departure1D from 1D to 3D limb- ABSTRACTdarkening is smaller, as a consequence of a less efficient convection in K dwarfs as compared to F stars. For the nearby dwarf star α Cen B (K1 V), we present limb-darkening predictions from a 3D hydrodynamical radiative transfer model of its atmosphere. We first compared the results of this model to a standardThe Kurucz’s reason atmosphere. the emergent Then intensity we used both differs predictions between to 1D fit the and new interferometric visibility measurements of α Cen B obtained with the3D VINCI models instrument is that theof the properties VLT Interferometer. of the superadiabatic Part of these new and visibility sur- measurements were obtained in the second lobe of the visibility function,face convective which is sensitive layers cannot to stellar be limb-darkening. described well The by best the agreement mixing is found for the 3D atmosphere limb-darkening model and a limb-darkenedlength formalism, angular diameter whatever of θ3D parameter= 6.000 we0.021 choose. mas, corresponding The temper- to a linear radius of 0.863 0.003 R (assuming π = 747.1 1.2 mas). Our new linear radius agrees well with the± asteroseismic value predicted ± � ± ature inhomogeneities (granulation) together with the strong by Thévenin et al. (2002, A&A, 392, L9). In view of future observationssensitivity of this of the star opacitywith the VLTI (H )/AMBER to the temperature instrument, we make also present the limb-darkening predictions in the J, H, and K bands. − warm ascending flows more opaque than they would be for a Key words. instrumentation: interferometers – stars: atmospheres homogeneous 1D model. This purely 3D effect, added to the Fig. 5. Close-up views of the squared visibilities of α Cen B in the lower partcontribution of the first of lobe the (left turbulent panel) and pressure, the second pushes lobe the (right location panel of). The continuous line represents the broadband, limb darkened disk visibilitythe model surface derived to lower from densities. the 3D The RHD temperature with θ3D = gradient6.000 mas. in these The dashed linesFig. 2. correspondSnapshot of3D to the results disk-centerRHD obtained Surface (µ = from1) intensity the 1D ATLAS emerging model at the with θ = 6.017 mas. The upper dotted curve is a UD model with 1. Introduction regions1D(Kurucz is steeper 1992) than or Phoenix in the 1D (Hauschildt case (see 1999).Nordlund However, & Dravins in spite θ = 5stellar.881 mas. surface at a representative time. UD 1990).of Since the detailed the continuum physics is included formed in in these these codes,layers, their the emer- 1D na- Limb-darkening (hereafter LD) is a well-known effect in stel-gentture intensity is a limitation is different. for deriving realistic emergent intensities. lar physics.Bigot Itset manifestational. (2006) isA&A, a non-uniform 446, 635 brightness of the The correction due to 3D simulations (a few percents) comparedchecked with for VINCI. solar line There formations will be (e.g. two Asplund major advantages et al. 2000b,c,Asplund,Indeed, M., Grevesse, these codes N., Sauval, contain A. free J., parameters, Allende Prieto, like the C., well- & disk, whose edges appear fainter than the center. This effect oc-is smallknown but mixing not negligible length parameter, with respect which to are the injected precision artificially ob- with AMBER:2004), helioseismology (e.g. Rosenthal et al. 1999), and also Blomme, R. 2005, A&A, 431, 693 curs because of the decrease in the source function outwardstained in in by order the to new reproduce generation the properties of interferometric of the turbulent instruments convec- for stellar line formations (e.g. Allende-Prieto et al. 2002). Bordé, P., Coudé du Foresto, V., Chagnon, G., & Perrin, G. 2002, – It will providethe atmosphere. a wavelength The disk dependence center then of shows the deeper visibility and warmerlikeA&A, VINCItion 393, at 183 theor AMBER. stellar surface. This improvement As a consequence, is essential the comparison for de- The adopted atmospheric parameters are those of Morel ([1.9 2.4]layers,µm) whereas therefore the allowing edges show di higherfferential and cooler studies material. of Bouchy, Thisrivingbetween F., an & accurate Carrier, these F. angular1D 2001, models A&A, diam andeter 374, observations of L5 the star. depends We report on the our in- et al.means (2000), that i.e. analysisT of= the5260 intensity K,I log(µg) at= different4.51 latitudinal and limb-darkening− as a functioneff of wavelength.λ Bouchy,limb-darkeningput F., & parameters, Carrier, predictions F. 2002, which A&A, thereby in Table 390, creates 205 4 for an a seriesimportant of contin- source of [Fe/H] =+0.2. The simulation was run for a few hours of stel- angles µ = cos θ provides information on the temperatureCarrier, vari-uum F.,uncertainties. wavelengths & Bourban, between Moreover, G. 2003, 1.0 A&A, convec and 406,2.4tionµ 23m, is correspondingby nature a 3D to process. the – AMBERlar time canation that simultaneously with covered depth several in the combine external convective layers the turn-over light of the from star. times. three This The is there- telescopes and therefore measure the closure phase. This Claret,JHK A.Itsrange 2000, manifestation accessible A&A, 363, tois 1081 the AMBER presence ofinstrument. bright granules and dark result isfore a 3D, an excellenttime-dependent constraint box for representing testing atmospheric the stellar models, sur- to intergranular lanes. Reducing it to a 1D process, i.e. ignoring gives an advantage to determining the angular size of the Cohen, M., Walker, R. G., Carter, B., et al. 1999, AJ, 117, 1864 face. Avalidate snapshot or of invalidate the disk-center assumptions surface used intensity to derive is theserepre-Eggenberger, mod- horizontal P., Charbonnel, flows and C., temperature Talon, S., inhomogeneities, et al. 2004, A&A, changes 417, 235 the star whenels observing (like NLTE in the/LTE), minima and to of provide the visibility suggestions function. for improving sented in Fig. 2. The structure of our model is similar to the oneGustafsson,4. Visibilitypressure B., Bell, scale model R. A., height, Eriksson, and the angular location K., & Nordlund, diameter of the surface, Å. 1975, and A&A, there- obtainedthe by input Nordlund physics & (equation-of-state Dravins (1990) but and is/or even opacities more real- in partic- These improvements will lead to better constrained angular di- 42,of 407foreα Cen also B the emergent intensity (see e.g. Allende-Prieto et al. istic, sinceular). the The present center-to-limb version of variation the code of solves the Sun compressible has been known 2004; and Asplund et al. 2000a, for a comparison of multi- ameters of α Cen A and B and, therefore, to high precision mea- Hauschildt, P. H., Allard, F., & Baron, E. 1999, ApJ, 512, 377 equationsfor of many hydrodynamics years and been and measured uses more for grid-points, numerous µ whichand λHestro(e.g.Inff thiser,dimensional D., section, & Magnant, we simulations). describe C. 1998, the A&A, application 333, 338 of our 3D limb- surement of the ratio of the linear radii of A and B, independent allowsPierce a better & treatment Slaughter of 1977; the turbulence. Neckel & Labs 1994; HestroffKervella,erdarkening & P., Coudé models du to Foresto, the interpretation V., Glindemann, of the A., VINCI & Hofmann, measure- R. of the parallax.Magnan 1998) leading to a plethora of theoretical works that The precise measurements of the center-to-limb variation ments2000,achieved SPIE, of α Cen 4006, nowadays B. 31 require realistic stellar atmospheric models have improved our knowledge of the external layers of theKervella, Sun. P., Gitton, Ph., Ségransan, D., et al. 2003a, SPIE, 4838, 858 Acknowledgements. We thank Vincent Coudé du Foresto for impor- that take all the complexity of the stellar surface into account, 3.2. 3D limb-darkeningTraditionally, analysis of solar and stellar LD is madeKervella, by P., Thévenin, F., Ségransan, D., et al. 2003b, A&A, 404, tant remarks that led to improvements at an early stage of this pa- and motivates the use of the new generation of 3D radiative adopting an approximated law for Iλ(µ), generally a polyno-4.1.1087 Limb-darkened (Paper I) disk visibility model per. WeThe thank monochromatic the anonymous surfacereferee for intensity constructive was computed remarks. These for var- hydrodynamical (hereafter RHD) simulations. mial expansion in µ that are either linear or non-linearKervella, (see P., Ségransan, D., & Coudé du Foresto, V. 2004, A&A, 425, interferometricious latitudinal measurementsµ and were longitudinal obtainedϕ usingdirections the VLTI at the (ESO stellar In the simple case of a centro-symmetric star such as α Cen B, e.g. Claret 2000, for recent developments) and with coeffi1161- In this paper we propose a study of α Cen B (HD128621), Paranal,surface. Chile), andThe were limb-darkening retrieved fromIλ(µ the) was ESO obtained/ST-ECF by Archive horizon- the visibility function measured using a broadband inter- cients determined from 1D atmospheric models, like ATLASKurucz, R.a nearby L. 1992, K1V The Stellar dwarf Populations star. It is part of Galaxies, of a visual IAU triple Symp., star (Garching,talGermany). (x,y), longitudinal LB thanks and the CNEStime averages for financial of the support surface and in- ferometric instrument such as VINCI is defined by three 149, 225 Å. Nordlundtensity. for Forproviding the time his RHD average, code. we We considered also thank Claude a sequence Van’t of wavelength-dependent functions: Ludwig, H., Allard, F., & Hauschildt, P. H. 2002, A&A, 395, 99 Veer for2 providing hours of the stellar ATLAS time. model. The results are plotted in Fig. 3 for Morel, P., Provost, J., Lebreton, Y., Thévenin, F., & Berthomieu, G. the twoArticle extreme published wavelengths by EDP Sciences of our spectraland available domain, at http://www.edpsciences.org/aa 1.0 and 1. The spectralor http://dx.doi.org/10.1051/0004-6361:20053187 energy distribution S (λ) of the star, ex- 2000, A&A, 363, 675 2.4 µm. For comparison, we overplot limb-darkening obtained pressed in terms of photons (VINCI uses a photon counting Neckel, H., & Labs, D. 1994, SoPh, 153, 91 Referencesfrom a 1D ATLAS9 model for the same wavelengths and detector). Nordlund, Å. 1982, A&A, 107, 1 Allende Prieto, C., Asplund, M., Garcia Lòpez, R. J., & Lambert, D. L. Nordlund, Å., & Dravins, D. 1990, A&A, 228, 155 2002, ApJ, 567, 544 Petrov, R. G., Malbet, F., Richichi, A., et al. 2000, SPIE, 4006, 68 Allende Prieto, C., Asplund, M., & Fabiani Bendicho, P. 2004, A&A, Pierce, A. K., & Slaughter, C. D. 1977, SoPh, 51, 25 423, 1109 Pourbaix, D., Nidever, D., McCarthy, C., et al. 2002, A&A, 386, 280 Asplund, M., Nordlund, Å., Trampedach, R., & Stein, R. F. 1999, Robbe-Dubois, S., Petrov, R. G., Lagarde, S., et al. 2004, SPIE, 5491, A&A, 346L, 17 1089 Asplund, M., Ludwig, H.-G., Nordlund, Å., & Stein, R. F. 2000a, Rosenthal, C. S., Christensen-Dalsgaard, J., Nordlund, Å., Stein, R. F., A&A, 359, 669 & Trampedach, R. 1999, A&A, 351, 689 Asplund, M., Nordlund, Å., Trampedach, R., Allende Prieto, C., & Stein, R. F., & Nordlund, Å. 1989, ApJ, 342, L95 Stein, R. F. 2000b, A&A, 359, 729 Stein, R. F., & Nordlund, Å. 1998, ApJ, 499, 914 Asplund, M., Nordlund, Å., Trampedach, R., & Stein, R. F. 2000c, Söderhjelm, S. 1999, A&A, 341, 121 A&A, 359, 743 Thévenin, F., Provost, J., Morel, P., et al. 2002, A&A, 392, L9 Asplund, M., Grevesse, N., Sauval, A. J., Allende Prieto, C., & Thoul, A., Scuflaire, R., Noels, A., et al. 2003, A&A, 402, 293 Kiselman, D. 2004, A&A, 417, 751 3-D Radiative Hydrodynamic vs. 1-D Center-to-Limb Variations

L. Bigot et al.: The limb darkening of α Cen B 639

Fig. 3. Normalized monochromatic center-to-limb variation Iλ(µ)/Iλ(1) of the surface intensity obtained by 3D RHD simulations of α Cen B as a function of µ for two different wavelengths: 1 µm(left panel) and 2.4 µm(right panel), which correspond to the extreme limits of the spectral• domain Limb we havedarkening considered in this weaker paper. The solid at lineslonger represent wavelengths the 3D RHD limb-darkening, whereas the dashed lines represent limb darkening derived from 1D ATLAS atmospheric models. In both case, the symbols represent the values extracted from both 1D (white) and 3D (black) simulations. � • 3D RHD models less limb darkened than (standard) 1-D models 2. The wavelength-dependent intensity profile of the star: To obtain the visibility function observed by VINCI in broad- I(λ, µ)/I(λ, 1). band, we have to integrate this function taking the transmission 3. The spectral transmission T(λ) of the instrument, including of the instrument and the spectral energy distribution of the star the atmosphere, all optical elements and the detector quan- into account: tum efficiency. 2 2 K [Vλ(B,θ) T(λ) S (λ)] λ dλ VK(B,θ) = (3) Bigot et al. (2006) A&A, 446, 635 � 2 2 · Out of these three functions, T(λ) is known from the concep- K [T(λ) S (λ)] λ dλ tion of the instrument, as well as from calibrations obtained on � 2 the sky (see Kervella et al. 2003b, for details). The spectral en- Note the λ term that is necessary, as the actual integration ergy distribution of the star S (λ) can be measured directly using of the squared visibility by VINCI over the K band is done a spectrograph or taken from atmospheric numerical models. in the Fourier conjugate space of the optical path difference From the 3D RHD simulations presented in Sect. 3, we ob- (expressed in meters), and is therefore done in wavenumber tained intensity profiles for ten distinct wavelengths over the σ = 1/λ. This corrective term ensures that the integration of K band (chosen in the continuum). For each of these pro- the star’s spectrum is done precisely in the same way as in the files, ten values of µ were computed. The resulting 10 10 instrument. element 2D table I(λ, µ)/I(λ, 1) was then interpolated× to a This formulation is very general, as it does not make any larger 60 50 element table, with a 10 nm step in wavelength particular assumption about the spectrum of the star or about (over the× 1.90 2.50µm range) and a 0.02 step in µ. This in- the wavelength dependence of its intensity profile I(λ, µ). terpolation preserves− the smooth shape of the intensity profile function well. This procedure was also used to build the 4.2. Fit of the interferometric data and angular I(λ, µ)/I(λ, 1) table based on the 1D Kurucz model. The origi- diameter of α Cen B nal sample (10 20) was interpolated to the same final grid as the 3D model. × Considering the model discussed in Sect. 4.1, we now derive the limb-darkened angular diameter θ of α CenB. It is ob- We can derive the monochromatic visibility law Vλ(B,θ) LD 2 from the monochromatic intensity profile I(λ, µ) using the tained by a standard χ analysis of the data. We define the re- 2 Hankel integral: duced χ of our fit by N 2 2 2 1 1 Vi V 1 π B θLD 2 model 2 χred = − , (4) Vλ(B,θ) = I(λ, µ)J0 1 µ µ dµ, (1) N n ⎛ σi ⎞ A �0 � λ � − � − �i=1 ⎜ ⎟ ⎝⎜ ⎠⎟ where B is the baseline (in meters), θ the limb darkened an- where n is the number of variables (n = 1 for our fit), N the total number of measurements, i the index of a particular mea- gular diameter (in radians), J0 the zeroth order of the Bessel function, λ the wavelength (in meters), µ = cos θ the cosine of surement, and σi the standard deviation of the measurement with index i. the azimuth of a surface element of the star, and A the normal- 2 ization factor: The χ minimization was computed for three center-to-limb models: uniform disk (UD), 1D ATLAS, and 3D RHD. In each 1 2 case, the broadband square visibility curve VK(B,θ) is shown in A = I(λ, µ) µ dµ. (2) Figs. 4 and 5. In addition to the purely statistical error, we must �0 The Astrophysical Journal, 633:424–439, 2005 November 1 # 2005. The American Astronomical Society. All rights reserved. Printed in U.S.A.

ON THE LIMB DARKENING, SPECTRAL ENERGY DISTRIBUTION, AND TEMPERATURE STRUCTURE OF PROCYON J. P. Aufdenberg1 National Optical Astronomy Observatory, 950 North Cherry Avenue, Tucson, AZ 85719 H.-G. Ludwig Lund Observatory, Lund University, Box 43, 22100 Lund, Sweden and P. Kervella LESIA, UMR 8109, Observatoire de Paris-Meudon, 5 place Jules Janssen, 92195 Meudon Cedex, France Received 2005 May 23; accepted 2005 July 13 a) Model “C” 5.395 mas Procyon = α CMa (F5 IV-V) ABSTRACT Mark III 0.5 µm • Different UD angular We have fit synthetic visibilities from three-dimensional (CO5BOLD+PHOENIX) and one-dimensional (PHOENIX, ATLAS 12) model stellar atmospheres of Procyon (F5 IV) to high-precision interferometricdiameters data at from500 the nm, VLT 800nm, Interferometer (K band) and from the Mark III interferometer (500 and 800 nm). These data sets provide a test of theoretical wavelength-dependent limb-darkening predictions. The work of Allende2.2 Prietoμm etreveal al. has shown limb that darkening the temperature structure from a spatially and temporally averaged three-dimensional hydrodynamic model produces significantly less limb darkening at 500 nm relative to the temperature structure of a one-dimensional MARCS model atmosphere with a standard mixing-length approximation forVL convection.TI/VINCI 2.2 µm Our direct• LImb fits to darkening the interferometric most data sensi- confirm this prediction. A one-dimensionalMark III 0.8 µm ATLAS 12 model with ‘‘approximatetive overshooting’’ to temperature provides the structure required temperature gradient. We show, however, that one-dimensional models cannot reproduce the ultraviolet spectrophotometry below 160 nm with effective temperatures in the range constrainedat by shortest the measured wavelength bolometric flux and angular diameter. We find that a good match to the full spectral energy distribution can be obtained with a com- posite model consistingMark of III a 0.5 weighted µm average of 12 one-dimensional model atmospheres based on the surface intensity distribution of a three-dimensional granulation simulation. We emphasize that one-dimensional models with overshooting may realistically represent the mean temperature structure of F-type stars such as Procyon, but the same models will predict redder colors than observed because they lack the multicomponent temperature distribution b) expected for the Aufdenberg, surfaces of these Ludwig stars. & Kervella (2005) ApJ, 633, 424 Subject headinggs: convection — methods: numerical — stars: atmospheres — stars: fundamental parameters (colors, temperatures) — stars: individual (Procyon) — techniques: interferometric

1. INTRODUCTION adiabatic equilibrium temperature structure. Milne (1921) improved on Schwarzschild’s mean intensity approximation and The connection between the transport of energy in stellar at- found that a radiative equilibrium temperature structure with mospheres and the limb darkening (LD) of stellar photospheres improved flux conservation yielded an LD coefficient of � 3, has been under investigation for nearly 100 years in the case of ¼ 5 the Sun. Since the work of Schwarzschild (1906), the darkening in better agreement with observations. of the solar limb has been investigated to understand the roles of The studies of Schwarzschild and Milne were completed be- convection and radiation in the transport of energy through the fore hydrogen was recognized as the principal constituent of the Sun’s atmosphere. Assuming that the mass absorption coefficient Sun’s atmosphere by Payne (1925), before the work of Unso¨ld of the solar atmosphere was both wavelength and depth indepen- (1930) concerning the effects of hydrogen ionization on the sta- dent and that the angle-dependent intensity could be replaced by bility of radiative equilibrium against convection, and before the the mean intensity at each depth, Schwarzschild derived a center- importance of the bound-free and free-free opacity of the nega- to-limb intensity variation based on a purely radiative equilib- tive ion of hydrogen was recognized by Wildt (1939). Plaskett rium temperature structure: (1936) inverted solar LD observations to find a temperature structure inconsistent with radiative equilibrium, suggesting that con- I(� 0; cos �) 2 vection has a significant effect on LD. Keenan (1938) showed ¼ 1 �(1 cos �);� ; 1 that this conclusion was acceptable only if the adiabatic gradient I(0; 1) ¼ � � ¼ 3 ð Þ began at optical depths considerably less than� ¯ 2, contrary to � where � is the optical depth, � is the angle between the line of Unso¨ld’s calculations. The reasoning was that convection cur- sight and the emergent intensity, and � is the linear LD coeffi- rents should transport little energy above the zone of instability cient. Schwarzschild showed this LD law to be more consistent and that radiative equilibrium should prevail at smaller optical with contemporary observations than an LD law based on an depths significant to LD. Plaskett’s conclusion was further cri- tiqued by Woolley (1941), who showed that it necessitates convective velocities large enough to blur the Fraunhofer lines. 1 Michelson Postdoctoral Fellow. Woolley attributed the differences between the observations and 424 Procyon: Convection Effects on Limb Darkening Models without overshooting B B: PHOENIX “B” A A: PHOENIX “A” D: ATLAS “D” D B F: ATLAS “F” A B A { F D D F C C: CO5BOLD + PHOENIX C F E C E E: ATLAS “E” Models with E overshooting{ U: Uniform Disk All atmosphere U models have:

Teff = 6530 K log(g) = 3.95

U but different convection treatments

Aufdenberg, Ludwig & Kervella (2005) ApJ, 633, 424 848 Limb DarkeningF. Castelli et al.: Notes on theand convection Convection in the ATLAS9 model atmospheres in the Sun

854 F. Castelli et al.: Notes on the convection in the ATLAS9 model atmospheres

Table 3. The parameters of models affected by convection (columns 1 and 2) and the parameters of models which show the largest difference at log τRoss=0 between Fconv/Ftot computed for the “overshooting” option switched on and off respectively. Columns 1 and 3 are for [M/H]=0 and columns 1,4 are for [M/H]= 3 − Convective Max[ F(conv) - F(conv) ] FtotOVER FtotNOVER Models at τRoss=1

[M/H]=0 [M/H]= 3.0 WITH OVERSHOOTING −

APPROXIMATION log g Teff (K) Teff (K) Teff (K)

No Overshooting 5.0 8500-3500 8000-7500 8500-7000 Overshooting 4.0 8000-3500 7500-7000 8000-6000 3.0 7500-3500 7000-6500 7000-5500 2.0 7000-3500 6500-6000 6500-4500 1.0 6500-3500 6000-5500 6000-4000 Convective Flux / Total Flux Total / Flux Convective

7. Teff from colour indices (V K), (B V), and (b y) − − − No Overshooting In the previous section we showed that some models have a dif- Intensity RatioIntensity ferent structure depending whether the “overshooting” option is switched on or off. In this section we investigate the effect of the different model structure on the V K, B V, and b y colour indices, which are often adopted− for fixing− effective− tempera- Fig. 4a and b. Comparison between ob- tures for cool stars; furthermore, we will try to state whether served (points) and computed (full line) so- the color indices from the “overshooting” models (COLK95) or lar limb-darkening curves Iλ(cosθOvershooting)/Iλ(0). those from the “no-overshooting” models (COLNOVER) give Observations are from Pierce & Slaughter Teff closer to the values derived from the infrared flux method (IRFM), which is almost model independent. NO OVERSHOOTING (1977) and Pierce, Slaughter & Weinberger APPROXIMATION (1977). Computed limb-darkening curves Temperature (K) Temperature are from models which differ only for the 7.1. The dependence of the synthetic colour indices on the “overshooting” option. a it is “on” (SUNK94 convection model) b it is “off” (SUNNOVERC125 Log (Rosseland Optical Depth) We computed grids of synthetic colours UBV, uvby, and RIJKL model).Fig. The 11. Upper different plot: the curves ratio Fcon correspondv/Ftot as a function to of log τRoss in the from models having the “overshooting” option switched off, mi- Sun for: (a) the SUNK94 model (full line), (b) the standard ML theory 1 different values of cosθ croturbulent velocity ξ=2 km s− , and metallicities [M/H]=0.0 without any modification (crosses), (c) the SUNNOVERC125 model and [M/H]= 3.0. For each gravity, we derived T by inter- Wavelength (Å) (no “overshooting”)(dashed line). Lower plot: the T-log τ relation eff Ross polating in the− COLNOVER grids for the (V K), (B V), Castelli, Gratton & Kuruczfor the (a) and (c)(1997) cases A&A 318, 841 Table 2. Observed and computed color indices for the Sun This implies that the continuum windows selected by Pierce and (b y) color indices of the COLK95 grids.− In this− way, & Slaughter (1977) and Pierce et al. (1977) may not always we may− estimate the effect of the convection on the effective Color Observed Computed Computed correspond to the real continuum at several wavelengths. temperatures derived from the color indices. The temperature indices SUNK94 SUNNOVERC125 differences ∆Teff =Teff over-Teff nover as function of Teff for the (V K), (B V), and (b y) indices are shown for different − − − and decreasing electron pressure which cause a growth of the gravities and solar metallicity in the upper panels of Fig. 13-15. (U B) 0.1951 0.17 0.18 4.3. Color indices hydrogen ionization zone. The largest ∆Teff differences are about 60 K, 100 K, and − 2 0.17 0.01 170 K for the (V K), (B V), and (b y) indices respectively. ±1 − − − (B V) 0.650 0.67 0.66 Table 2 compares (U B), (B TableV), and3 shows (b whichy) observed models are color affected by convection They occur for Teff and log g between 7500-8000 K and 4.0- − 2 − for− gravities ranging− from log g=5.0 to log g=1.0. Furthermore, 4.5 for (V K), 6750-7250 K and 3.0-4.0 for (B V), 6500- 0.68 0.005 indices with those derived from the SUNK94 and the SUN- − − ± 3 for the metallicities [M/H]=0 and [M/H]= 3, it lists the mo- 7000 K and 3.0-4.0 for b y. Temperatures from the color 0.656 0.005 − − NOVERC125 models. Owingdels to which the show uncertainty the largest difference, in the solar at τross=1, between the indices computed from the “no overshooting” models are lower (b y) 0.406±0.0044 0.41 0.397 colors and to the small differencesFconv/Ftot betweencomputed with the the SUNK94 “overshooting” and option switched than those from color indices computed from the “overshooting − ± 3 0.414 0.003 the SUNNOVERC125 colorson we and cannot off respectively. state which The maximum model effective has temperature of models”. For all the three indices the value of ∆Teff weakly ± models affected by the different convection options decreases depends on gravity for Teff <6500 K. For Teff > 6500 K the 1 Neckel (1994) to be preferred. with decreasing log g. effect of the convection increases with increasing gravity. 2 Schmidt-Kaler (1982) 3 Gray (1992) 4 Edvardsson et al. (1993) 4.4. The Balmer profiles Comparison of Balmer profiles from BALMER9 with those from SYNTHE has shown that the metallic lines do not affect when SUNNOVERC125 or SUNNOVERC20 models are used the shape of the wings of the Hα and Hβ profiles. The violet wing to compute the solar intensities Iλ(cosθ) (Fig. 4b). of Hγ predicted by the synthetic spectrum is a little bit broader A fact to be considered when observed limb-darkening than that predicted by BALMER9, owing to the presence of a curves are compared with the computations is that the line opac- strong Fe I line at 432.576 nm. ity is negligible only for few wavelengths, as we can infer from Observed high-resolution spectra were taken from the Solar the analysis of high-resolution spectra (Kurucz et al, 1984). Flux Atlas of Kurucz et al. (1984). We lowered the continuum Temperature Structure Constrained by Interferometric Diameters B: PHOENIX “B” A: PHOENIX “A” D: ATLAS “D” F: ATLAS “F” E: ATLAS “E” C: CO5BOLD + PHOENIX

A D PHOENIX F ATLAS • Steepness of temperature E gradient determines limb darkening C CO5BOLD

Aufdenberg, Ludwig & Kervella (2005) ApJ, 633, 424 Stars up close, More Than One Temperature Structure

d3gt65g40n3.10-26-3

4•104 25

2•104 20

0•104

15 3 y [Mm] 0•10

10 0•103 Temperature (K) Temperature

5 0•103

4 2 0 -2 -4 log τ 0 10 Ross 0 5 10 15 20 25 Log (Optical Depth) x [Mm]

Hotter, rising granules have a warmer temperature structure than cooler, descending dark lanes.

Procyon: Aufdenberg, Ludwig & Kervella (2005) ApJ, 633, 424 Spectral Energy Distribution from 3-D Stellar Surfaces

Procyon = α CMa (F5 IV-V)

IUE SWP IUE LWP

STIS GHRS

12 component model 1 component model

• Multi-temperature component photosphere matches SED in the short ultraviolet, 1D model with overshooting does12 component not model 1 component model

Aufdenberg, Ludwiguv & Kervella (2005)by ApJ, 633, 424 Glushneva et al. (1992) Solar spectral line strengthsPereira as a et function al.: Oxygen lines of in limb solar granulation. angle II. 11

tures in this spectral region, most notably the feature at around Observations 615.793 nm. 0.65 1D MARCS 1D HM The presence of these blends depresses the continuum level i 3D Model around the O lines, so that it becomes a significant source of un- 0.60 certainty. Our analysis, either in computing the equivalent widths from the observations or fitting the line profiles, suffers from the same problems. For example, if one assumed this oxygen line to 0.55 have no blends and took the continuum level of the FTS disk- centre intensity atlas, then the equivalent width could be up to 33% higher than our estimate for the oxygen-only contribution.

Equivalent width [pm] 0.50 This effect is amplified as one looks at spectra closer to the limb because the different blends vary differently in strength over the 0.45 solar disk, which makes a precise determination of the continuum level difficult. Unlike the stronger O i 777 nm triplet the 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 predicted NLTE effect is rather small for the O i 615.8 nm lines: µ Asplund et al. (2004) predict it to be 0.03 dex when neglecting H i collisions. − Fig. 9. Equivalent width vs. µ for the [O i]+Ni i 630.03 nm lines blend, for our observations and different models at different positions in the solar disk. 4.3.2. Wavelength calibration Pereira, Asplund & Kiselman (2009) A&A, 508, 1403 A similar approach to the wavelength calibration of [O i] 4.2.6. Comparison with previous work 630.03 nm was used. Because a precise wavelength calibration for this line is not as crucial as [O i], we derive the wavelength An easy comparison to be made is with ALA01. Some things calibration using only the nearby Fe i 615.77 nm line. It is im- have been done differently in the present study. We compute a portant to make the wavelengths of the O i consistent with this proper blend of the two transitions (adding line opacities), in- Fe i line because they lie on its red wing, and a different wave- stead of computing separate line profiles for each line and co- length difference between them would imply a lower or higher adding them. We also perform a more precise wavelength cali- blend influence. bration usinging Fe i lines. And finally, we employ a more recent 3D Model and a higher Ni abundance. Fitting the flux profiles Using the reference laboratory wavelengths from the NIST i i against the FTS flux atlas and using the same fitting range in database for the Fe 615.77 nm and O 615.81 nm lines and i i ALA01, we obtain an oxygen abundance 0.07 dex lower. The calibrating using the waveshift of Fe lines, the synthetic O reason for this difference is mainly due to a higher Ni abun- profiles are shifted from the observations. This can be seen in dance used (6.22 vs. 6.05), which translates to a 0.07 dex the left panels of Fig. 10. The 3D synthetic profile is shifted to difference in the derived oxygen abundance. The proper≈− blend- the blue in regards to the observations, while for the 1D mod- ing of the line instead of co-adding profiles causes a difference els the opposite happens. None of them matches the observations perfectly. For the 3D case, the shift to the observations is of 0.01 dex in oxygen. The proper wavelength calibration us- 1 ing− Fe i lines instead of only correcting for gravitational redshift about 0.89km s− , or 1.8 pm. This could mean that there is a sig- causes a difference of +0.01 dex. nificant error in the laboratory wavelengths; the predicted line shifts from the 3D models are inconsistent for the Fe i and O i Fitting the line profiles with log Ni = 6.22 instead of 6.05 leads to a better agreement with the observed disk-centre profile lines; or there are blending features causing a shift in the profile. (χ2 0.53χ2 ), but a worse agreement for the observed flux Including known molecular and weak atomic lines is not the ex- 6.22 ≈ 6.05 planation. In the top middle panel of Fig. 10 we have artificially profile (χ2 1.51χ2 ). 6.22 ≈ 6.05 shifted the O i lines, so that the synthetic profile matches the FTS intensity atlas. For the rest of the analysis we will be using these 4.3. O I 615.81 nm artificially shifted profiles (both for 3D and 1D models), but the derived abundance from fitting the disk-centre profiles remains 4.3.1. Context essentially the same if the profiles are not shifted (but as seen The O i 615.81 nm line is a triplet, with components at 615.8149, in Fig. 10, the agreement with the observations is worse). The ff 615.8172 and 615.8187 nm, according to the NIST database. results for the equivalent widths are not a ected by this shift. The latter is the strongest component. The components are so close that they are unresolved in the solar spectrum. Its position 4.3.3. Molecular blends and comparison with observations in the solar spectrum is far from optimal. The line is close to a strong ( 4.8 pm) Fe i line at 615.77 nm and there are multiple We have included several CN and C2 lines in the 615.8 nm re- weak blends≈ around. The VALD database lists several atomic gion (J. Sauval, 2008, private communication). The line data are lines in this region, but besides the strong Fe i line only the given in Table 5. These data were obtained from from J. Sauval Nd ii 615.782 nm, Ni i 615.800 nm and Fe i 615.803 nm lines (2008, private communication). The molecular line data is un- have any measurable effect, though still just barely noticeable. certain, especially wavelengths for CN lines. With this uncer- Spectral synthesis using only these lines is still not enough to tainty in mind, we have adjusted some of the wavelengths and explain the solar spectrum. However, there seem to be several log gf of the CN lines from Table 5. The two (9, 4) band lines molecular lines in this region. These are mostly CN lines, with at 615.8522 and 615.8796 nm have been red shifted by 7.2 and afewC2 lines. The molecular lines are important for our anal- 11.6 pm, respectively, so that they match two features in the so- ysis and their effect is discussed below. It should be noted that lar disk-centre spectrum. Furthermore, the log gf values of these even accounting for the molecules there are still unidentified fea- two lines have been increased so that they fit reasonably the ob- A&A 489, L5–L8 (2008) Astronomy DOI: 10.1051/0004-6361:200810450 & c ESO 2008 � Astrophysics Letter to the Editor

Diameter and photospheric structures of Canopus from AMBER/VLTI interferometry�,��

A. Domiciano de Souza1, P. Bendjoya1, F. Vakili1, F. Millour2, and R. G. Petrov1

1 Lab. H.A. Fizeau, Domiciano CNRSA. Domiciano de UMR Souza 6525,de et Souzaal.: Univ.Diameter et al.: de Diameter Nice-Sophia and photospheric and photospheric Antipolis, structures structures Obs. of deCanopus of la Canopus Côte from d’Azur, from AMBER AMBER Parc/VLTI Valrose,/VLTI interferometry interferometry 06108 Nice, France L7 L7 e-mail: [email protected] = α Car (F0Ib): H- and K -bands Non-Zero Closure Phase 2 Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, 53121 Bonn, Germany H K Model Fit 3 Single 3 d) Received 23 June 2008 / Accepted 25 July 2008 d) ra ra ( Standard Models ( Convective 2 2 ase ase

ABSTRACT h

h Cell Model

1 Context. Direct measurements of fundamental parameters and photospheric 1 structures of post-main-sequence intermediate-mass stars osure p osure p are required for a deeper understanding of their evolution. Cl Cl 0 Aims. Based on near-IR long-baseline interferometry we aim to resolve the0 stellar surface of the F0 supergiant star Canopus, and to precisely measure its angular diameter and related physical parameters. 4. 5. 6. 7. 4. 5. 6. 10+7 7. Methods. We used the AMBER/VLTI instrument to record interferometric data on Canopus: visibilities and cl10osure+7 phases in the Sp. Freq largest base (Bproj/λ) H and K bands with a spectral resolution of 35. The available baselines ( 60 110 m)Sp. and Freq the largest high base quality (Bproj of/λ) the AMBER/VLTI observations allowed us to measure fringe visibilities as far as in the thirdFig. visibility 3.�AMBER− lobe./VLTI closure phases (absolute values) of Canopus as Results. We determined an angular diameter of / = 6.93 0.15Fig. mas 3.aAMBER by function adopting/ ofVLTI the a spatialclosure linearly frequency phases limb-darkened (absolute calculated values)disk on the model. of longest Canopus From baseline. this as angular diameter and Hipparcos distance we derived� a stellar± radiusa functionRValues= 71 of. indicate4 the4 spatial.0 thatR . Canopus frequency Depending does calculatednot on havebolometric a on completely the longestfluxes centrally existing baseline. sym- in the literature, the measured / provides two estimates of the effectiveValues temperature:metric indicate intensity± thatT Canopus distribution.�= 7284 does For107 not example have K and a thecompletelyT absolute= 7582 valuescentrally252 of the K.sym- clo- metricsure intensity phases distribution. at loweff spatial For frequencies example the ( absolute5.5eff 107 valuescycles of/rad) the are clo- of Conclusions. In addition to� providing the most precise angular diameter obtained to date, the± AMBER interferometric± data point • Canopus less limb darkened thatsure expected phasesthe order at oflow 0. 1 spatialfrom0.2 rad. frequencies standard The triangles ( 5 indicate.≤5 models10× 7 thecycles theoretical /rad) are closure of − towards additional photospheric structures on Canopus beyondthe orderlimb-darkenedphases of 0 obtained.1 0.2 model rad. from The the alone. triangles fit of A a promising single indicate≤ convective× the explanation theoretical cell model forclosure to such the surface structures• Evidence is the presence for of presence convection cells. for We convective checkedphases suchvisibility aobtained hypothesiscell amplitudes− from using the alone fit first of (see aorder single text star-cell for convective details). models cell and model concluded to the that the AMBER observations are compatible with the presencevisibility of surface amplitudes convective alone structures. (see text This for details). direct detection of convective cells on Canopus from interferometry can provide strong constraints toTable radiation-hydrodynamics 2. Parameters and uncertainties models estimated of photospheres from a least-square of F-type fit supergiants.Domiciano de Souza et al. (2008) A&A, 489,of a LLDL8 disk model to the observed visibilities. The reduced χ2 of Tablethe 2. Parameters fit is χ2 = and7.0, uncertainties suggesting that estimated the LLD from disk a cannot least-square completely fit Key words. stars: fundamental parameters – stars: individual: Canopusof a LLD – supergiants disk modelred – to methods: the observed observational visibilities. – The reduced χ2 of explain2 the observations. Physical parameters of Canopus derived from techniques: high angular resolution – techniques: interferometricthe fitthe is χ measuredred = 7.0,/ suggestingare also listed. that the LLD disk cannot completely Fig. 2. Top: AMBER/VLTI visibility amplitudes and uncertainties inexplain the the observations.� Physical parameters of Canopus derived from the measured / are also listed. H (green) and K bands (black). The model visibilities (red squares) Fitted� parameters Derived parameters Fig. 2. Top:were AMBER calculated/VLTI with visibility a linear amplitudes limb-darkened and uncertainties disk model in fitted the to the / = 6.93 0.15 mas R/R = 71.4 4.0a 1.H Introduction(green) anddata (TableK bands2). We(black). also show The the model theoretical visibilities visibilities (red squares) (dots for H andperformed a detailed spectroscopic study� of Canopus, and other Fitted�� parameters= 0.04± 0.01 T = Derived7284 107 parameters Kb or± 7582 252 Kc dashes for K) expected for Canopus from a LLD disk with parame- H eff were calculated with a linear limb-darkened disk model fitted to the intermediate-mass� = 0.07± 0.01 stars, in orderlog to± g investigate= 1.7 0.a1d the± influence of The evolvedters star determined Canopus by Claret (α Carinae, (2000). Clearly HD these 45348) limb-darkened is the sec- models / = 6.93K 0.15 mas R/R = 71.4 4.0 data (Table 2). We also show the theoretical visibilities (dots for H and rotation� on± − the internal± mixing of� elements.b ±± They concludec that do not account for the observations, especially after the first minimum. �aHFrom= 0. the04 Hipparcos0.01 distanceTeff = d7284= 95.91074.9 K pcor (Perryman 7582 252 et al. K1997). onddashes brightest for K) expected star (V for= Canopus0.72) from in a the LLD night disk with sky, parame- just af- ± ± ± d ± the�TheK abundance= Lutz-Kelker0.07 0. of01 correction many stars is negligible oflog theirg = for sample,1 Canopus.7 0.1 including (Jerzykiewicz Canopus, & terters Sirius. determinedBottom Although by: fit Claret residuals Canopus (2000 in−).σ ClearlyV isunits classified showing these limb-darkened a relativelyas a F0II important models star depar- in − ± ± do not accountture forfrom the zero, observations, as well as residual especially trends after indicating the first thatminimum. additionala ef-FromareMolenda-Zakowicz the better Hipparcos explained distance2000 by).d models= 95.9 including4.9 pc (Perryman rotation et al. e1997ffects). in the the SIMBADfects should database, be included several in the works modeling (e.g. of Canopus. Achmad et al. b From the bolometric flux measured± by Code et al. (1976): f = Bottom: fit residuals in σV units showing a relatively important depar- Theevolutionary Lutz-Kelker correction models.6 is negligible2 1 for Canopus (Jerzykiewicz & 1991; Desikachary & Hearnshaw 1982) present it as a F0 su-Molenda-Zakowicz(45.0 1.8) 102000− erg). cm− s− . ture from zero, as well as residual trends indicating that additional ef- c ObtainedTo± constrain by× using evolutionary values from Smiljanic models et al. it ( is2006 crucial) and references to have pre- pergiantfects should (F0Ib), be included which in theis moremodeling adapted of Canopus. to the measured lu- b Fromcise the measurements bolometric flux of measured fundamental by Code stellar et al. parameters, (1976): f = such as therein (visual6 apparent2 magnitude1 V = 0.72, extinction AV = 0.08, minosity of4. ModelL fitting13 000 and15 000physicalL (e.g. parameters Jerzykiewicz &(45.0 1.8) 10− erg cm− s− . − thevisual effective bolometric temperature correction TBCV, luminosity= 0.0) and fromL, radius DrillingR &. If Arlo the dis- � − � c Obtained± by× using values from Smiljaniceff et al. (2006) and references Molenda-ZakowiczThe observed2000 visibilities; Smiljanic (Fig. et2) al. and2006 closure). phases (Fig. 3) in- (2000) (solar luminosity and bolometric magnitude). The uncertainty thereintance (visuald and apparent the angular magnitude diameterV = /0.72,are extinction known, thenA = a0 quite.08, direct 4.During Modeldicate their fitting evolution, that, and as a physical zero-order intermediate-mass approximation, parameters stars Canopus like Canopus can be con- in Teff was estimated from σ / and considering� a conservativeV cumula- visualmethod bolometric to estimate correctionR isBC to� apply= 0.0)− the and simple from Drilling relation &R Arlo= 0.5 / d. (M 10 Msidered) leave to be the a star red having giant brancha centrally-symmetric (RGB) and enter intensity the dis- tive error of 0.1 in V, AV , andVBCV. � The� observed� visibilities (Fig. 2) and closure phases (Fig. 3) in- (2000The)d (solarAssuming most luminosity direct a mass and of andM precise/M bolometric= 9 measurements2. magnitude). of The/ uncertaintycan be obtained blue-giant region,tribution significantly (see also discussion increasing in Sect. their5). e Weffective have temper-thus fitted a � ± � 3 in Tbyeff was modern estimated long from baselineσ / and interferometers. considering a conservative cumula- aturedicate and that, performinglinearly as a zero-order limb-darkened the so-called approximation, disk blue-loop. (LLD) model Canopus Evento both can if most beH and con- starsK band � sidered to be a star having a centrally-symmetric intensity dis- tive errorIn of this 0.1work in V, A weV , and presentBCV. a precise determination of the angu- found in theobserved blue-giant visibilities. region The of free the parameters HR diagram of the are model experi- are thed Assuming a mass of M/M = 9 2. tribution (see also discussion in Sect. 5). We have thus fitted a larAMBER, diameter the (and uncertainty other� derived in R is dominated physical parameters)by the uncertainty of Canopus in encing theirLLD blue-loop angular phase, diameter there/ , and are the stillH puzzlingand K band questions LLD coef- ± linearly limb-darkenedficients � and disk� . (LLD) The� estimated model3 to parameters both H and andK correspond-band fromthe distance,interferometric contrarily observations to the previous performed measurements with of the/ . AMBER concerning this phaseH of stellarK evolution (e.g. Xu & Li 2004). � observed visibilities.ing uncertainties The freeobtained parameters from a Levenberg-Marquardt of the model are the (L-M)focalAs instrument indicated (Petrovin Table 2 et, the al. two2007 values), installed of Teff were at ESO-VLTI ob- LLDAnother angularleast-squares unresolved diameter fit issue/ are, and given in the the inH Table evolutionand2.K Theband of model intermediate- LLD visibilities coef- andAMBER,(Glindemanntained the from uncertainty di etfferent al. 2004 in approaches,R)is located dominated corresponding at Cerro by the uncertaintyParanal, to distinct Chile. in es- The massficients stars� fit concernsand residuals� . The some are� shownestimated discrepancies in Fig. parameters2. between and correspond- observed andthe distance,timates contrarily of the bolometric to the previous fluxes. measurements One can find of in/ the. lit- H K observationserature values and compatible data reduction with both are described estimates. For in Sects.� example,2 and 3. predicteding uncertainties abundances.The obtained uncertainty Additional from in / a( Levenberg-Marquardt internalσ / = 2. mixing2% / ) is processes dominated (L-M) are by the As indicated in Table 2, the two values of Teff were ob- � � � The results and conclusions are outlined in Sects. 4 and 5. invokedleast-squares touncertainty explain fit are given the in the observations. in instrumental Table 2. The spectral Smiljanic model calibration visibilities et al. (cf. ( and2006 Sect.)tained3). Desikachary from different & Hearnshaw approaches, (1982) corresponding and references to distinct therein es- give T 7300 7400 K, while Kovtyukh (2007), Smiljanic fit residualsEven are with shown this in limitation, Fig. 2. AMBER/VLTI enabled us to derivetimateseff of� the bolometric− fluxes. One can find in the lit- � Based onthe observations most precise performed angular diameter at the of European Canopus known Southern today. et al. (2006) and references therein give Teff 7500 7600 K. The uncertainty in / (σ / = 2.2% / ) is dominated by the erature2. Observations values compatible with both estimates.� For− example, Observatory,This Chile interferometrically under ESO� Program� measured 079.D-0507.� angular diameter is compati-DesikacharyFigure &2 Hearnshawshows that (1982) the near-IR and references intensity distribution therein give of uncertainty in the instrumental spectral calibration (cf. Sect. 3). Canopus is not entirely compatible with a simple limb-darkened �� Table 1 isble only with available several in previous electronic measurements form at within the uncertainties,T The AMBER7300 7400 observations K, while were Kovtyukh obtained (2007 at), low Smiljanic spectral reso- Even withfor this example, limitation,/ = 6 AMBER.6 0.8 mas/VLTI (Hanbury-Brown enabled us to et derive al. 1974; in-eff model.� In− particular the height of the second lobe is quite im- http://www.aanda.orgthe most precise angular� diameter± of Canopus known today. et al.lution (2006R)s and= λ/ referencesΔλ 35 therein in the> giveH andTeff K 7500bands< 7600 (LR-HK K. mode), tensity interferometry), / = 7.00 0.41 mas (Heras et al. 2002; portant, corresponding� to �H 0 and ��K 0− (i.e., a slight This interferometricallyspectral fit); additional measured values� angular are± listed diameter by Heras is compati- et al. (2002). Figurelimb-brightening).2 shows that These the measured near-IR∼ values intensity do∼ not distribution agree with of the ble with severalIn previousTable 2 we measurements give the linear within radius theR and uncertainties, effective tempera-Canopustheoretical is not entirely predictions compatible of linear limb-darkeningwith a simple limb-darkened for Canopus. To for example,ture/T = derived6.6 0.8 from mas our (Hanbury-Brown angular diameterArticle et and al. other1974 published; previously in- model. by EDPillustrate In particularSciences this point the we height show of in the Fig. second2 the theoretical lobe is quite visibility im- � eff ± tensity interferometry),measured parameters./ = 7.00 Thanks0.41 to the mas precision (Heras etin al./ attained2002; withportant,curves corresponding for linear limb-darkening to �H > 0 coe andfficients�K < from0 (i.e., Claret a (slight2000): spectral fit); additional values� are± listed by Heras et al. (�2002). limb-brightening).�H = 0.28 and These�K = measured0.24.∼ These values values do correspond∼ not agree to withTeff theand In Table3 I2(μwe)/I(1) give= 1 the� linear(1 μ), radius where IRisand the specific effective intensity, tempera- and μ istheoretical the log g from predictions Table 2 of, and linear to the limb-darkening metallicity and for microturbulence Canopus. To cos of the angle between− − the line of sight and the emergent intensity. velocity from Smiljanic et al. (2006). ture Teff derived from our angular diameter and other previously illustrate this point we show in Fig. 2 the theoretical visibility measured parameters. Thanks to the precision in / attained with curves for linear limb-darkening coefficients from Claret (2000): � �H = 0.28 and �K = 0.24. These values correspond to Teff and 3 I(μ)/I(1) = 1 �(1 μ), where I is the specific intensity, and μ is the log g from Table 2, and to the metallicity and microturbulence cos of the angle between− − the line of sight and the emergent intensity. velocity from Smiljanic et al. (2006). The Astrophysical Journal, 664:1093Y1101, 2007 August 1 # 2007. The American Astronomical Society. All rights reserved. Printed in U.S.A.

EXTENDED ENVELOPES AROUND GALACTIC CEPHEIDS. III. Y OPHIUCHI AND � PERSEI FROM NEAR-INFRARED INTERFEROMETRY WITH CHARA/FLUOR Antoine Me´rand Center for High Angular Resolution Astronomy, Georgia State University, PO Box 3965, Atlanta, GA 30302-3965; [email protected] Jason P. Aufdenberg Per (F5Ib): K-bandPhysical UV Sciences coverage Department, Embry-RiddleNo. 2, 2007 Aeronautical University, Daytona Beach,CSE FL 32114 AROUND GALACTIC CEPHEIDS: Y OPH AND � PER 1099 1098 ME´ RAND ETα AL. Vol. 664 α Per (F5Ib): K-band Residuals to hydrostatic Model Pierre Kervella and Vincent Coude´ du Foresto TABLE 4 Journal of Observations: � Per LESIA, UMR 8109, Observatoire de Paris, 5 place Jules Janssen, 92195 Meudon, France TABLE 6

B P.A. and � Per PHOENIX Models MJD 54,000 (m) (deg) V 2 � Theo A. ten Brummelaar, Harold A. McAlister, Laszlo Sturmann, Judit Sturmann, and Nils H. Turner 46.281...... 147.82 79.37 0.02350 0.00075 � 46.321...... 153.87 69.04 0.01368 0.00067 Center for High Angular Resolution Astronomy, Georgia State University, PO Box 3965, Atlanta, GA 30302-3965 TeA � 46.347...... 155.79 62.17 0.01093 0.00060 � Received 2007 February 23; accepted 2007 April 2 (K) log g 1:4 log g 1:7 46.372...... 156.27 55.30 0.01134 0.00039 � ¼ ¼ 46.398...... 155.65 48.10 0.01197 0.00051 � 46.421...... 154.41 41.32 0.01367 0.00052 � 6150...... 0.137 0.136 46.442...... 152.94 34.84 0.01581 0.00045 ABSTRACT � 46.466...... 151.13 27.03 0.01824 0.00064 � 6270...... 0.135 0.134 46.488...... 149.58 19.40 0.02167 0.00071 Unbiased angular diameter measurements are required for accurate distances to Cepheids using the interferometric � 46.510...... 148.49 12.06 0.02250 0.00058 6390...... 0.133 0.132 � 46.539...... 147.77 1.53 0.02370 0.00070 Baade-Wesselink method (IBWM). The precision of this technique is currently limited by interferometric measure- � 47.225...... 96.66 44.51 0.27974 0.00282 � � ments at the 1.5% level. At this level, the center-to-limb darkening (CLD) and the presence of circumstellar envelopes 47.233...... 97.75 47.34 0.27321 0.00274 � � Notes.—Models tabulated for different effective tempera- 47.252...... 100.37 53.97 0.24858 0.00258 (CSE) seem to be the two main sources of bias. The observations we performed aim at improving our knowledge � � 47.274...... 103.03 60.91 0.23529 0.00242 tures and surface gravities. The K-band CLD is condensed into � � of the interferometric visibility profile of Cepheids. In particular, we assess the systematic presence of CSE around 47.327...... 249.17 81.65 0.01363 0.00032 � � a power-law coefficient : I( ) . 47.352...... 251.22 74.84 0.01340 0.00032 Cepheids in order determine accurate distances with the IBWM free from CSE biased angular diameters. We ob- � � � � 47.374...... 251.04 68.91 0.01330 0.00035 / � served a Cepheid (Y Oph) for which the pulsation is well resolved and a nonpulsating yellow supergiant (� Per) using 47.380...... 250.67 67.10 0.01316 0.00035 � long-baseline near-infrared interferometry. We interpreted these data using a simple CSE model we previously de- Note.—Date of observation (Modified Julian Day), telescope projected sep- Fig. 6.—� Per. Projected baselines are in meters. North is up, and east is right. aration, baseline projection angle, and squared visibility. The shaded areaveloped. corresponds We to the found squared visibility’s that our second observations lobe. The base- of � Per do not provide evidence for a CSE. The measured CLD is explained describes a center-to-limb intensity profile with 64 angular lines are W1-W2,by E2-W2, an hydrostatic and E2-W1, from photospheric the shortest to the longest. model. Our observationsFig. 7.— of� YPer Oph, squared when visibility compared models: to smaller UD, baseline adjusted measurements, CLD, PHOENIX, and points. larger angular diameter; the surface brightness effect wins over the suggest that it is surrounded by a CSE with characteristics similar to CSEs found previously around other Cepheids. We total luminosity. Based on their angular diameter, � 1:45 mas, it two different CSE models are plotted here as the residuals to the PHOENIX CLD Power-law fits to the hydrostatic model visibilities range from � small baselinehave where determinedB�/k 1/3. As the demonstrated distance by to our Y Polaris Oph to be d 491 18 pc. Additional evidence points toward the conclusion that appears that Barnes et al. (2005) overestimated Y Oph’s angu- measurements,• Per the presence also� of CSE less is expected limb to weaken darkened the with¼ respect than� to baseline. spherical The common hyrdostatic point at B 175 m is models the first minimum of lar size. Among Cepheids brighter than mV 6:0, Y Oph has α most Cepheids are surrounded by faint CSEs, detected by near-infrared interferometry: after observing� four Cepheids, � 0:132 to 0.137 (Table 6), which correspond to 0.8Y1.1 � ¼ second lobe of visibility curve, mimicking stronger CLD. the visibility function. The top of the second lobe is reached for B 230 m. See the strongest B V color excess, E(B V ) 0:645 (Fernie et al. FLUORall operates show in evidence the near-infrared for aK CSE.band with Our a mean CSE nondetection around a nonpulsating supergiant in the instability strip, Per, ¼ 1995a) and one� of the highest fraction� of¼ polarized light, p 2 �� above the value we measured. Our measured CLD is below that wavelength of k0 2:13 �m. This sets the small, first-lobe and Table 7 for parameters and reduced � of each model. 1:34% 0:12% (Heiles 2000). Indeed, Y Oph is within the Ga-¼ provides� confidence in the detection technique and suggests a pulsation driven mass-loss mechanism for the Cepheids. � second-lobe baselines at approximately 50, 150, and 220 m, predicted by the models, or in other words the predicted dark- lactic plane: this means that it has a probably a large extinction which are wellSubject matched headin to CHARAgs:g baselinesCepheids W1-W2 — ( 100 circumstellar m), matter — stars: individual (Y Ophiuchi, � Persei) — due to the interstellar medium. E2-W2 ( 150m),andE2-W1( 250 m). See Figure� 6 for a ening is slightly overestimated. graphical� representation of the baseline� techniques: coverage. interferometricthe angle between the stellar surface and the line of sight. The 3. THE NONPULSATING YELLOW SUPERGIANT � PER The data reduction was performed using the same pipeline as 3.3. Possible Presence of CSE � Per (HR 1017, HD 20902, F5Ib) is among the most luminous forMérand Y Oph. Squared visibilities et al. were (2007) calibrated using ApJ, a similar 664,uniform 1093 disk [More diameter about case corresponds Cephieds to @� 10:50!]0:0, whereas an nonpulsating stars inside the Cepheids’ instability strip. The Dop- strategy we adopted for Y Oph. We used two sets of calibrators: ¼ pler velocity variability has been found to be very weak, of the one for the shorter baselines,1. W1-W2 INTRODUCTION and E2-W2, and one for increasing valueAmong of current� corresponds optical interferometers, to a stronger CHARA/FLUOR CLD. (ten Firstly, we employ the CSE model used for Cepheids (Papers I 1 the longest baseline, E2-W1 (Table 5). order of 100 m s� in amplitude (Butler 1998). This amplitude is Brummelaar et al. 2005; Me´rand et al. 2006a) is one of the few 10 times less than what is observed for the very low amplitude All visibility models for this work have been computed taking and II). This model is purely geometrical, and it consists of a limb- In our two previous3.2. Simple papers, Model (Kervella et al. 2006; Me´rand capable of such a measurement for Cepheids. The only Cepheid Cepheid Polaris (Hatzes & Cochran 2000). into account instrumental bandwidth smearing (Aufdenberg et al. � Per’s apparent angular size, approximately 3 mas (Nordgren et al. 2006b, hereafter3.2.1. Limb-darkened Paper I andDisk Paper II), we reported the dis- accessible to CHARA/FLUOR, i.e., large enough in angular di- darkened disk surrounded by a faint CSE, whose two-dimensional et al. 2001), makes it a perfect candidate for direct center-to-limb coveryTo probe of the faint shape circumstellar of the measured visibility envelops profile, (CSEs) we first around2006). Galactic Fromameter, this two-parameter for such a measurement fit, we is Polaris deduce (� UMi), a stellar which diam- we ob- geometric projection (on the sky) is modeled by a ring. The ring darkening detection with CHARA/FLUOR. Following the ap- used an analytical model, which includes the stellar diameter and proach we used for Polaris (Paper II), we observed � Per using classicala CLD law. Cepheids. Because of its Interestingly,versatility, we adopt all here the a power Cepheids law weeter observed of �� 3served:130 and0 found:007 mas to have and a aCLD power-law compatible coefficient with hydrostatic� parameters consist of the angular diameter, width, and flux ratio three different baselines, including one sampling the second lobe, (‘(HestrofferCar in Paper 1997): I( I,�)/�I(0)UMi�� andwith �� Cepcos (� in), where Paper� is II) were0: found119 to 0¼:016.photospheric The� fit models, yields although a reduced surrounded� 2 by1: a0. CSE There (Paper is II).¼ a in order to measure its CLD strength, but also in order to be able to harbor CSEs with similar¼ characteristics:¼ a CSE 3Y4 times larger Polaris, however, is a very low-amplitude pulsation Cepheid 0.4% with respect to the star (Fs/F?). We adopt the same geometry assess the possible presence of a CSE around this star (Table 4). strong� correlation between the diameter and CLD¼ coefficient, 0.9. than the star, which accountsTABLE 5 for a few percent of the total flux in in diameter, compared to 15%Y20% for type I Cepheids (Moskalik found for Cepheids: a ring with a diameter equal to 2.6 times 3.1. Interferometric Observations the infrared K band.� Per The Calibrators presence of CSEs was discoveredThis in is our a well known& Gorynya effect, 2005); that thus, a thechange agreement in CLD is not induces necessarily a change expected If the star is purely a CLD disk, then only two baselines are the stellar diameter (Paper II) with no dependency with respect required to measure the angular diameter and the CLD strength. attempt to improve our knowledgeUD Diameter of Cepheids in thein context apparent of diameter.for large amplitude Cepheids, whose photospheres are more out of Name Spectral Type (mas) Baselines to the width (fixed to a small fraction of the CSE angular diameter, Observations must take place in the first lobe of the squared the distance determination via the interferometric Baade-Wesselink equilibrium. The direct measurement of CLD of a2 high-amplitude visibility profile, in order to set the star’s angular diameter �. The HD 18970 ...... G9.5 III 1.551 0.021 W1-W2, E2-W2 This fit is entirely satisfactory, as the reduced � is of order say 1/5). method (IBWM). Part of the� method requires the measurement Cepheid during its pulsation phase remains to be performed. first lobe is defined by B�UD/k < 1:22. Additional observations HD 20762 ...... K0 IIYIII 0.881 0.012 E2-W1 � unity, and the residuals of the fit do not show any trend (Fig. 7). should be taken in the second lobe, in particular near the local HD 22427 ...... K2 IIIYIV 0.913 0.013 E2-W1 This model restriction is justified because testing the agreement of the angular diameter variation� of the star during its pulsation. Hydrodynamic simulations (Marengo et al. 2003) suggest that HD 31579 ...... K3 III 1.517 0.021 W1-W2, E2-W2 maximum (B�/k 3/2), because the strength of the CLD is di- � � TheHD 214995 determination ...... K0 ofIII the 0.947 angular0.013 diameters W1-W2 from sparseWe inter-note thatthe this CLD is notvariations the case during for the Polaris pulsation (Paper do not account II). Polaris, for more between a generic CSE model and interferometric data requires a rectly linked to the height of the second lobe. To address the pres- � ence of a CSE, observations should be made at a small baseline. ferometric measurements is not straightforward, because optical than a 0.2% bias in distance determination in the near-infrared Notes.—Uniform disk diameters, given in mas, are only intended for com- with u-v coverage very similar to � Per, does not follow a simple complete angular resolution coverage, from very short to very Because the CSEs that were found around Cepheids are roughly interferometersputing the expected squared gather visibility high in the K angularband. These stars resolution come from our data onlyLD at a disk few model,using because the IBWM, it is where surrounded most of the by IBWM a faint observing CSE. work has 3 times larger that the star itself (Papers I and II), we chose a baselinescatalog of interferometric at a time; calibrators thus (Me good´rand et al. phase 2005a). and angular resolution cov- been done in recent years: the best formal distance determina- large baselines. Although our � Per data set is diverse regarding the baseline coverage, we lack a very short baseline (B 50 m), erage cannot be achieved in a short time. For Cepheids, the main tion to date3.2.2. usingHydrostatic the IBWM is of Models the order of 1.5% (Me´rand et al. uncertainty in the IBWM was thought to be the center-to-limb 2005b). which was not available at that time at the CHARA Array.� darkening (CLD), which biases the interferometric angularWe di- computedWhereas a small the grid near-infrared of models, IBWM which seems consists to be relatively of six one- im- The simplest fit consists in adjusting the geometrical scale; the ameter measurements (Marengo et al. 2004). dimensional,mune spherical, to bias hydrostaticfrom CLD, the models recent discovery using the of CSEs PHOENIX raises the angular size ratio between the CSE and the star is fixed. This The direct measurement of CLD is possible using an optical issue of possible bias in angular diameter measurements, and 2 interferometer, given sufficient angular resolution andgeneral-purpose precision. hence stellar bias in distance and planetary estimations at atmosphere the 10% level (Papercode II).ver- It yields a reduced � of 3, than at 1.1 for a hydrostatic LD and 1.0 sion 13.11; for a description see Hauschildt et al. (1999) and 1093 for an adjusted CLD law (Table 7). Hauschildt & Baron (1999). The range of effective temperatures We can force the data to fit a CSE model by relaxing the CLD and surface gravities is based on a summary of � Per’s parameters of the star or the CSE flux ratio. A fit of the size of the star and the by Evans et al. (1996): brightness of the CSE leads to a reduced �2 of 1.4 and results in a

1. Effective temperatures, TeA 6150, 6270, 6390 K. 2. log (g) 1:4; 1:7. ¼ TABLE 7 12 3. Radius,¼R 3:92 ; 10 cm. Models for � Per 4. Depth-independent¼ microturbulence, � 4:0kms 1. ¼ � 5. Mixing length to pressure scale ratio, 1.5. �? Fs/F? 2 6. Solar elemental abundance LTE atomic and molecular line Model (mas) � (%) �r 6 5 blanketing, typically 10 atomic lines and 3 ; 10 molecular lines UD...... 3.080 0.004 0.000 ... 5.9 � dynamically selected at run time. PHOENIX CLD...... 3.137 0.004 0.135 ... 1.1 � 7. Non-LTE line blanketing of H i (30 levels, 435 bound- adjusted CLD...... 3.130 0.007 0.119 0.016 ... 1.0 � � bound transitions), He i (19 levels, 171 bound-bound transitions), CSE ‘‘Polaris’’ ...... 3.086 0.007 0.135 1.5 3.0 � and He ii (10 levels, 45 bound-bound transitions). CSE (1) ...... 3.048 0.007 0.066 0.004 1.5 1.4 4 2 � � 8. Boundary conditions: outer pressure 10� dynes cm� , ex- CSE (2) ...... 3.095 0.010 0.135 0.06 0.26 1.4 10 2 � � tinction optical depth at 1.2 �m: outer was 10� , inner was 10 . Notes.—The parameters are: �? is the stellar angular diameter, the CLD power- For this grid of models the atmospheric structure is computed at law coefficient is �, and if relevant, the brightness ratio between the CSE and the 50 radial shells (depths), and the radiative transfer is computed star is Fs/F?. The first line is the uniform disk diameter, the second one is expected CLD from the PHOENIX model, and the third one is the adjusted CLD. The fourth along 50 rays tangent to these shells and 14 additional rays that line is a scaled Polaris CSE model (Paper II). The last two lines are attempts to force impact the innermost shell, the so-called core-intersecting rays. a CSE model to the data. Lower script are uncertainties, the absence of lower script The intersection of the rays with the outermost radial shell means that the parameter is fixed. Astronomy & Astrophysics manuscript no. alpha˙Ori˙v9 c ESO 2010 March 9, 2010

Radiative hydrodynamics simulations of red supergiant stars: II. simulations of convection on Betelgeuse match interferometric observations , , A. Chiavassa1 2, X. Haubois3, J. S. Young4, B. Plez2, E. Josselin2, G. Perrin3, and B. Freytag5 6

1 Max-Planck-Institut fürAstrophysik, Karl-Schwarzschild-Str. 1, Postfach 1317, D85741 Garching b. München, Germany [email protected] e-mail: 2 2 GRAAL,IOTA Universitéde data image Montpellier reconstruction II - IPM, CNRS, Place Eugéne3D Bataillon Simulation/Best 34095 Montpellier Cedex match 05, France to V 3 Observatoire de Paris, LESIA, UMR 8109, 92190 Meudon, France and Closure phase data 6 4 Astrophysics Group, CavendishA. Chiavassa Laboratory, etJJ Thomson al.: Convection Avenue, Cambridge on Betelgeuse CB3 0HE 5 Centre de Recherche Astrophysique de Lyon, UMR 5574: CNRS, Universitéde Lyon, Ecole´ Normale Supérieure de Lyon, 46 allée d’Italie, F-69364 Lyon Cedex 07, France 6 Department of Physics and Astronomy, Division of Astronomy and Space Physics, Uppsala University, Box 515, S-751 20 Uppsala, Sweden Received; accepted

ABSTRACT

Context. The red supergiant (RSG) Betelgeuse is an irregular variable star. Convection may play an important role in understanding this variability. Interferometric observations can be interpreted using sophisticated simulations of stellar convection. Aims. We compare the visibility curves and closure phases obtained from our 3D simulation of RSG convection with CO5BOLD to various interferometric observations of Betelgeuse from the optical to the H band in order to characterize and measure the convection pattern on this star. Methods. We use 3D radiative-hydrodynamics (RHD) simulation to compute intensity maps in different filters and we thus derive interferometric observables using the post-processing radiative transfer code OPTIM3D. The synthetic visibility curves and closure phases are compared to observations. Results. We provide a robust detection of the granulation pattern on the surface of Betelgeuse in the optical and in the H band based on excellent fits to the observed visibility points and closure phases. Moreover, we determine that the Betelgeuse surface in the H band is covered by small to medium scale (5–15 mas) convection-related surface structures and a large ( 30 mas) convective cell. In ≈ this spectral region, H2O molecules are the main absorbers and contribute to the small structures and to the position of the first null of the visibility curve (i.e. the apparent stellar radius). Key words. stars: betelgeuse – stars: atmospheres – hydrodynamics – radiative transfer – techniques: interferometric Fig. 4. Left panel: recontructed image from Haubois et al. (2009). Right panel: our best fitting 3D simulation snapshot of Fig. 3 convolved with a 6 6 mas PSF derived by fitting the central peak of the interferometric dirty beam. The intensities in both panels × 1. Introduction The position of Betelgeuse on the H-R diagram is highly are normalized to the range [0, 1] and some contour lines are indicated (0.2,uncertain, 0.5, due 0.7, to 0.8, uncertainty 0.9 of the in its peak effective brightness). temperature. Betelgeuse is a red supergiant star (Betelgeuse, HD 39801, M1– Levesque et al. (2005) used one-dimensional MARCS models Chiavassa2Ia–Ibe) and is oneet ofal. the (2010) brightest stars A&A in the optical515, and12 near [more(Gustafsson 3D etModels al. 2003, 1975) @ 11:10!] to fit the incredibly rich TiO infrared. This star exhibits variations in integrated brightness, molecular bands in the optical region of the spectrum for sev- surface features, and the depths, shapes, and Doppler shifts of eral RSGs. They found an effective temperature of 3650 K for its spectral lines. There is a backlog of visual-wavelength obser- Betelgeuse. Despite the fact that they obtained a good agree- vations of its brightness which covers almost a hundred years. ment with the evolutionary tracks, problems remain. There

arXiv:1003.1407v1 [astro-ph.SR] 6 Mar 2010 The irregular fluctuations of its light curve are clearly aperiodic is a mismatch in the IR colors that could be due to atmo- and rather resemble a series of outbursts. Kiss et al. (2006) stud- spheric temperature inhomogeneities characteristic of convec- ied the variability of different red supergiant (RSG) stars includ- tion (Levesque et al. 2006). Also the distance of Betelgeuse has ing Betelgeuse and they found a strong noise component in the large uncertainities because of errors related to the positional photometric variability, probably caused by the large convection movement of the stellar photocenter. Harper et al. (2008) derived cells. In addition to this, the spectral line variations have been a distance of (197 45 pc) using high spatial resolution, mul- ± analyzed by several authors, who inferred the presence of large tiwavelength, VLA radio positions combined with Hipparcos granules and high convective velocities (Josselin & Plez 2007; Catalogue Intermediate Astrometric Data. Gray 2008). Gray also found line bisectors that predominantly have reversed C-shapes, and line shape variations occurring at Betelgeuse is one of the best studied RSGs in term of multi- 1 the 1 km s− level that have no obvious connection to their shifts wavelength imaging because of its large luminosity and an- in wavelength. gular diameter. The existence of hot spots on its surface has been proposed to explain numerous interferometric observations with WHT and COAST (Buscher et al. 1990; Wilson et al. Send offprint requests to: A. Chiavassa 1992; Tuthill et al. 1997; Wilson et al. 1997; Young et al. 2000,

Fig. 5. Top row: maps of the linear intensity in the IONIC filter of the molecular species (Fig. 1). The simulation snapshot and intensity range is the same as in Fig. 3. Bottom row: the best matching visibility curve (solid line) of Fig. 2 is compared to the visibility curve obtained from the intensity maps above: CO (dotted line), CN (dashed line), and H2O (dash dot line). Red data points are IOTA observations. pears blurry and seems to contain only the information about the lobe also carries information on the presence of large convective stellar radius. However, the top right panel displays the signal cells. related to all the frequencies larger than the first lobe: in this Fig. 6 (bottom row) shows the second lobe with convection- image we clearly miss the central convective cell of 30 mas related structures of 10-15 mas, (30% of the stellar radius), and ≈ ≈ size (60% of the stellar radius) visible in Fig. 3. Thus, the first the third and fourth lobes with structures smaller than 10 mas. We conclude that we can estimate the presence of convection- A&A 407, L47–L50 (2003) DOI: 10.1051/0004-6361:20030786 Astronomy c ESO 2003 & Astrophysics

The spinning-top Be star Achernar from VLTI-VINCI

A. Domiciano de Souza1, P. Kervella2, S. Jankov3, L. Abe1, F. Vakili1,3, E. di Folco4, and F. Paresce4 L48 A. Domiciano de Souza et al.: The spinning-top Be star Achernar from VLTI-VINCI L48 A. Domiciano de Souza et al.: The spinning-top Be star Achernar from VLTI-VINCI 1 Laboratoire Univ. d’Astroph. de Nice (LUAN), CNRS UMR 6525, Parc Valrose, 06108 Nice Cedex 02, France 2 European Southern Observatory (ESO), Alonso de Cordova 3107, Casilla 19001, Vitacura, Santiago 19, Chile 3 Observatoire de la Cˆote d’Azur, D´epartement FRESNEL, CNRS UMR 6528, Boulevard de l’Observatoire, BP 4229, 06304 Nice, France 4 European Southern Observatory (ESO), Karl-Schwarzschild str. 2, 85748 Garching, Germany

Received 5 May 2003 / Accepted 22 May 2003

Abstract. We report here the first observations of a rapidly rotating Be star, α Eridani, using Earth-rotation synthesis on the Very Large Telescope (VLT) Interferometer. Our measures correspond to a 2a/2b = 1.56 0.05 apparent oblate star, 2a and 2b ± being the equivalent uniform disc angular diameters in the equatorial and polar direction. Considering the presence of a circumstellar envelope (CSE) we argue that our measurement corresponds to a truly distorted star since α Eridani exhibited negligible Hα emission during the interferometric observations. In this framework we conclude that the commonly adopted Roche approx- α imationFig. (uniform 1. VLTI ground rotation baselines and centrally for Achernar condensed observations mass) and should their not apply to Eridani. This result opens new perspectives to basiccorresponding astrophysical projections problems, onto such the as sky rotationally at different observing enhanced times. mass loss and internal angular momentum distribution. In addition to itsLeft intimate: Aerial relation view of VLTI with ground magnetism baselines and for the pulsation, two pairs ofrapid 40 cm rotation thus provides a key to the Be phenomenon: one of the outstandingsiderostats non-resolved used for Achernar problems observations. in stellar Color physics. magenta represents the 66 m (E0-G1; azimuth 147◦, counted from North to East) and green the 140 m (B3-M0; 58◦). Right: Corresponding baseline pro- Key words.jections ontotechniques: the sky (B highproj) as angular seen from resolution the star. Note – techniques: the very effi- interferometric – stars: rotation – stars: emission-line, Be – stars:cient individual: Earth-rotation Achernar synthesis resulting in a nearly complete coverage Fig. 2.*DiskFit of an of ellipse Achernar over the observed (B3 Vpe) squared resolved visibilities V 2astrans- in azimuth angles. lated to equivalent uniform disc angular diameters. Each V2 is plotted togetherellipsoid with its symmetrical by VLTI value(Axial in azimuth. ratio: Magenta1.56±0.05 points are for the 66 m baseline and green points are for the 140 m baseline. 1. Introductiondetection of stellar asymmetries. Moreover, Earth-rotation has 2.The Observations fitted ellipse results in and major data axis 2a processing= 2.53 0.06 milliarcsec, Fig. 1. VLTI ground baselines for Achernar observations and their ± produced an efficient baseline synthesis effect (Fig. 1, right). minor axis 2b = 1.62 0.01 milliarcsec, and minor axis orientation corresponding projections onto the sky at different observing± times. A total of more than 20 000 interferograms were recorded on Dedicatedα0 = 39◦ 1◦ (from observations North to East). of The Achernarpoints distribution have reveals been an carried α ± The southern star Achernar ( Eridani, HD 10144, spectral extremely oblate shape with a ratio 2a/2b = 1.56 0.05. DomicianoAchernar,Left: Aerial and de approximately view Souza of VLTI aset many al. ground on(2003) its calibrators, baselines A&A, cor- for 407,out the during two L47 pairs the[More of ESO 40 rotating cm period 70, stars from± 11 @ September14:00!] to type B3Vpe)responding is the brightest to more Be than star 20 inhours the of sky. integration. A Be star From is de- these 12 November 2002, with quasi-uniform time coverage, on the siderostats used for Achernar2 observations. Color magenta represents fined as a non-supergiantdata, we obtained B type 60 individual star thatV hasestimates, presented at anepisodic effective VLT Interferometer (VLTI, Glindemann et al. 2003) equipped wavelengththe 66 m of λ (E0-G1;ff = 2.175 azimuth0.003 µm. 147◦, counted from North to East) and Balmer lines in emissione (Jaschek± et al. 1981), whose origin is withoblateness the VINCI only in a beam first-order combiner UD approximation. (Kervella To et interpret al. 2003a). This our data in terms of physical parameters of Achernar, a consis- attributed togreen a CSE the ejected 140 by m the (B3-M0; star itself. 58 Physical◦). Right mech-: Correspondinginstrument recombines baseline pro- the light from two telescopes in the tent scenario must be tailoredffi from its basic known properties, anisms like3. non-radialjections Results onto pulsations, the sky magnetic (Bproj) activity, as seen or from bina- theastronomical star.so that Note we can the safelyK band, very establish e which the- conditions is centered where at a 2.2 coherentµm and cov- rity are invokedThecient determination to Earth-rotation explain of the the CSE shape synthesis formation of Achernar resulting of from Be our stars set in of in aV2 nearlyersmodel 0.4 complete canµm. be The built observable and coverage discussed. measuredFig. 2. byFit VINCI of an is ellipse the squared over the observed squared visibilities V2 trans- conjunctionis within not azimuth atheir straightforward fundamental angles. task property so that some of rapid prior assumptions rotation. coherence factor µ2 of the star light. It is derived from the raw 2 need to be made in order to construct an initial solution for lated to equivalent uniform disc angular diameters. Each V is plotted Theoretically, rotation has several consequences on the star interferograms4. Discussion after photometric calibration using a wavelet structure (Cassinelliour observations. 1987). A The convenient most obvious first approximation is the geometri- is to de- together with its symmetrical value in azimuth. Magenta points are rive from each V2 an equivalent uniform disc (UD) angu- basedAchernar’s method pronounced (Ségransan apparent et al. asymmetry 1999). The obtained reduction in this procedure cal deformation that results in a larger radius at2 the equator2 than for the 66 m baseline and green points are for the 140 m baseline. lar diameter UD from the relation V = 2J1(z)/z . Here, isfirst detailed approximation, by Kervella together et al.with (2003b) the fact thatand it has is a successfully Be star, been 1 ff | | at the poles.zdetection Another= π ( wellα) ofB established stellar(α) λ− , J asymmetries.is e theect, Bessel known function as Moreover, grav- of the raises Earth-rotation the question of whether has we observeThe fitted the stellar ellipse photo- results in major axis 2a = 2.53 0.06 milliarcsec, UD proj eff 1 applied to dwarf stars observations with the VLTI (Ségransan ff α 2 ± ity darkeningfirstproduced or kind the and von of Zeipel an first order, effi e cientect and foris baseline hot the azimuthstars (von angle synthesis Zeipel of Bproj et esphereff al.ect 2003). with (Fig. or without The 1, instrumental right). an additional contributionminor value of axisµ from 2isb a then CSE.= 1 calibrated.62 0.01 milliarcsec, and minor axis orientation 1924), is thatat di bothfferent surface observing gravity times anddue to emitted Earth-rotation. flux decrease The appli- For example, a flattened envelope in the equatorial plane ± A total of more than 20 000 interferogramsthrough were the recordedobservation onof stableα0 stars= 39 with◦ 1 known◦ (from angular North di- to East). The points distribution reveals an from the polescation to the of this equator. simple Although procedure well reveals studied the extremely in the liter- oblate would increase the apparent oblateness of the star± if it were shape of Achernar from the distribution of (α)onanel- ameters.to introduce The a calibrators significant infrared chosen (IR) for excess Achernar with are respect presented in / = . . ature, the effectsAchernar, of rotation and have approximately rarely been testedUD againstas many ac- on its calibrators, cor- extremely oblate shape with a ratio 2a 2b 1 56 0 05. lipse (Fig. 2). Since α, Bproj(α), and λeff are known much bet- Tableto the photospheric 1. The final continuum. product Theoretical of the processing models (Poeckert is the squared ± curate enoughterresponding observations than 1%, the measured (Reiners to more errors & Schmitt than in V2 are 20 2003; associated hours van Belle only of to integration.visibility& MarlboroughV2 Fromof 1978) the predictobject these a for rather each low baseline CSE contribution projected on the et al. 2001),the a gap uncertainties bridged in by our. We interferometric performed a non-linear observations2 regres- in the K band2 especially for a star tilted at higher inclina- data, we obtainedUD 60 individual V estimates,sky (Bproj at). V anis edirectlyffective related to the Fourier transform of the of Achernar.sion fit using the equation of an ellipse in polar coordinates. brightnesstions, which distribution should be our of case the as object discussed via the below. Zernike-Van Indeed, Cittert Althoughwavelength this equation of λ caneff be= linearized2.175 in0 Cartesian.003 µ coor-m. Yudin (2001) reported a near IR excessoblateness (difference only between in a first-order UD approximation. To interpret ± theorem. For these observations, two interferometric baselines dinates, such a procedure was preferred to preserve the orig- observed and standard color indicesour in visible data andin termsL band of physical parameters of Achernar, a consis- inal, and supposedly Gaussian, residuals distribution as well werecentered used, at 3.6 66µ mm) (E0-G1; to be E(V azimuthL) = 0 147m. 2,◦ with, counted the same from North − as to correctly determine the parameters and their expected tolevel East) of uncertainty. and 140 Moreover, m (B3-M0; thistent author 58◦), scenario reports equipped a zero must with in- be40 tailoredcm from its basic known properties, errors. We find a major axis 2a = 2.53 0.06 milliarcsec trinsic polarization (p ). These values are significantly smaller Send offprint requests3. Results to: A. Domiciano de Souza, ± siderostats (Fig. 1∗ left). Their orientations are almost perpen- (mas), a minor axis 2b = 1.62 0.01 mas, and a minor- than mean values for Be stars earlierso than that B3 we (E(V canL) safely> establish the conditions where a coherent e-mail: [email protected]± dicularm to each other giving an excellent configuration− for the axis orientation α0 = 39◦ 1◦. Note that the correspond- 0. 5 and p > 0.6%), meaning2 that the Achernar’s CSE is The determination± of the shape of Achernar from∗ our set of V model can be built and discussed. ing ratio 2a/2b = 1.56 0.05 determines the equivalent star weaker than in other known Be stars. Further, an intermediate is not a straightforward± task so that some prior assumptions need to be made in order to construct an initial solution for Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:200307864. Discussion our observations. A convenient first approximation is to derive from each V2 an equivalent uniform disc (UD) angu- Achernar’s pronounced apparent asymmetry obtained in this 2 2 lar diameter UD from the relation V = 2J1(z)/z . Here, first approximation, together with the fact that it is a Be star, 1 | | z = π (α) B (α) λ− , J is the Bessel function of the raises the question of whether we observe the stellar photo- UD proj eff 1 first kind and of first order, and α is the azimuth angle of Bproj sphere with or without an additional contribution from a CSE. at different observing times due to Earth-rotation. The appli- For example, a flattened envelope in the equatorial plane cation of this simple procedure reveals the extremely oblate would increase the apparent oblateness of the star if it were shape of Achernar from the distribution of (α)onanel- to introduce a significant infrared (IR) excess with respect UD lipse (Fig. 2). Since α, Bproj(α), and λeff are known much bet- to the photospheric continuum. Theoretical models (Poeckert ter than 1%, the measured errors in V2 are associated only to & Marlborough 1978) predict a rather low CSE contribution the uncertainties in . We performed a non-linear regres- in the K band especially for a star tilted at higher inclina- UD sion fit using the equation of an ellipse in polar coordinates. tions, which should be our case as discussed below. Indeed, Although this equation can be linearized in Cartesian coor- Yudin (2001) reported a near IR excess (difference between dinates, such a procedure was preferred to preserve the orig- observed and standard color indices in visible and L band inal, and supposedly Gaussian, residuals distribution as well centered at 3.6 µm) to be E(V L) = 0m. 2, with the same − as to correctly determine the parameters and their expected level of uncertainty. Moreover, this author reports a zero in- errors. We find a major axis 2a = 2.53 0.06 milliarcsec trinsic polarization (p ). These values are significantly smaller ± ∗ (mas), a minor axis 2b = 1.62 0.01 mas, and a minor- than mean values for Be stars earlier than B3 (E(V L) > ± m − axis orientation α0 = 39◦ 1◦. Note that the correspond- 0. 5 and p > 0.6%), meaning that the Achernar’s CSE is ± ∗ ing ratio 2a/2b = 1.56 0.05 determines the equivalent star weaker than in other known Be stars. Further, an intermediate ± Theory of Rotating Stars Revitalized by Observations

1198 JACKSON, MacGREGOR, &Models SKUMANICH for Achernar with Differential RotationVol. 606 for stars of Achernar’s given spectral type are about M � 6 M and TeA 19; 000 K, respectively. Although the value � � of TeA for our model of a nonrotating, 6 M star is in good agreement with the mean derived from observations� (Table 1), the radius must be increased by more than a factor of 4 to conform with the interferometrically inferred value of Req. Uniformly rotating models cannot account for such an en- hancement of the equatorial dimension. The polar radius of a 6 M model rigidly rotating at the breakup rate (i.e., � 0:0, � ¼ � 1:0) is Rpol 2:84 R , while the equatorial radius has the ¼ ¼ � value Req 4:28 R 1:51 Rpol, characteristic of a Roche- type model.¼ We note� that¼ the polar dimension of a rotating star is always reduced relative to a nonrotating model for the same mass, although rotation may also lead to significant departures from spheroidal shape (see JMS04). To account for the Req and Rpol of Achernar using a fully consistent, uniformly rotating stellar model would require that the star have a radius Fig. 1.—Cross-sectional representations of the photospheric shapes of 8 R in the absence of rotation, implying a mass k20 M Jackson,. models MacGregor, for the indicated Skumanich masses, where(2004) (� ApJ,, �) is 606, (a) (3.0, 1196 8.4), (b) (2.0, 4.9), � The� properties of nonuniformly rotating, 6 M models are� (c) (2.0, 4.4), and (d) (2.5, 5.4). In each panel, the rotation axis and equatorial given in Table 1. These models have been chosen� on the basis diameter are shown, and the arrow points in the direction of a distant observer. The apparent polar radius is the perpendicular distance between the horizontal of having Req 12:0 0:5 R , in rough agreement with the ¼ � � dotted and dashed lines and for the inclination angles given here is equal to estimate from observations. The models are listed in order of 7.7 R for each model. increasing degree of internal differential rotation, i.e., in order � of decreasing $1=2, which varies from 1:6 R0 for � 2:5 to ¼ values 1.6, 1.3, and 1.0 R , respectively. While TeA for these 0:93 R0 for � 4:5, where R0 Req( 0 0). In each case, 0 h i the axial angular¼ velocity exceeds� that¼ of the equator, models is now more representative of stars with mid-B spec- 2 tral types, the combination of higher surface temperatures and 0= eq (1 � ). As is evident from these results, rotation at a rate¼ and scaleþ sufficient to distend the equatorial radius by larger surface areas leads to increased luminosities, with the the required amount produces highly flattened, disklike con- consequence that the positions of these stars in a theoretical figurations. To the extent that any of these models reflect H-R diagram are well above the main sequence. To illustrate the degree to which the shape of a differentially Achernar’s photospheric shape, their tabulated values of Rpol rotating star with Achernar’s inferred dimensions may deviate and Zmax imply that the estimated polar radius of 7:7 R cannot represent the true polar dimension of the star. Instead,� from sphericity, we depict cross-sectional views of the pho- the stellar rotation axis must be inclined by considerably less tospheres of four models in Figure 1. In each panel, we show than 90� with respect to the line of sight, making the derived the surface contour in the plane containing the rotation axis and the line of sight. The models have been inclined by the Rpol an apparent rather than actual radius. The polar and equatorial effective temperatures given in indicated amounts, so that the apparent dimension perpen- 1=4 dicular to the line of sight corresponds to the estimated polar Table 1 are in accord with the von Zeipel law, TeA geA , where is the sum of the surface gravitational and centrif-/ radius, Rpol 7:7 R . The way in which the shape of the geA � � ugal accelerations. In addition, modified central thermody- stellar envelope is determined by the internal balance of forces namic conditions, resulting from centrifugal support of the is examined in detail in JMS04. Here, we note that by virtue of inner stellar core, are manifest in reduced luminosities for equation (1), when �>1 the rotation speed V $ attains a 2 ¼ maximum Vmax (1 � )=(2�) Veq at $ Req=�, imply- increasingly rapid, differential rotation. Together with the in- ¼½ þ � ¼ creased surface area that accompanies rotational distortion of ing that Veq is not the highest value of V. Thus, the 6 M model 1 � the stellar shape, this implies that the photospherically aver- in Figure 1a has Veq sin i 225 km s� , but Vmax sin i 375 km s 1. All of the models� for which the rate and degree of� aged effective temperature TeA should decrease relative to � h i differential rotation are large enough that Req k12 R have a the value for a nonrotating star of the same mass. Inspection of � Table 1 reveals that rotation rapid enough to produce a 6 M portion of the stellar surface that rotates at speeds between � 300 and 480 km s 1. star with Req 12 R has a profound effect on TeA , reduc- � ing it from 18,900� K� (� � 0) to values lessh thani 9900 K 4. SUMMARY AND DISCUSSION for models with � 2:50.¼ ¼ � If TeA is an accurate indicator of the average thermal Taking Achernar’s interferometrically determined size and conditionsh i in the stellar photosphere, a value below 104 K shape at face value, it is possible to construct differentially would probably preclude assignment of the spectral type B3 to rotating stellar models with dimensions that are consistent Achernar. However, a higher TeA would be obtained if the with those derived from the observations. For the putative stellar mass were larger than theh nominali value of 6 M . The massof6M implied by its B3 V spectral type, these con- remaining entries in Table 1 pertain to differentially rotating� figurations are� severely distorted from a spherical shape: with models with masses 9, 12, and 15 M . The results suggest that Rpol=Req 0:2, their appearances are more disklike than for sufficiently rapid rotation, the� equatorial radii of these spheroidal.� The agreement of our computational results with more massive objects can be similarly enlarged, from the non- observations is only partial, however, since there are a number rotating values R0 3:68; 4:33, and 4.91 R , respectively, to of discrepancies between the models and other stellar prop- sizes that compare¼ favorably with the measurements� of erties. The reduction in average effective temperature brought Achernar. We find that the scale of the required differential on by rapid rotation is of a sufficient magnitude to suggest that rotation is quite similar to that for M 3 with $ having the the spectral types of these models would be later than B3 V. ¼ 1=2 A&A 453, 1059–1066 (2006) Astronomy 1064 DOI: 10.1051/0004-6361:20054771P. Kervella and A. Domiciano de Souza: The polar wind of Achernar& c ESO 2006 � Astrophysics Table 4. Best fit parameters (and corresponding uncertainties) of our CSE model, and the residuals of the fit present satisfactory sta- simple model consisting in a Gaussian elliptical envelope superimposed tistical properties (see Sect. 3.7). on a uniform ellipse representing the central star. The fit was computed An instrumental origin for these outliers cannot be formally on our complete H and K band data set. excluded, especially as these data points were obtained on the The polar wind of the fast rotatingvery first Be night star of VLTI Achernar operations of the UT1-UT4 baseline. θeq 2.13 0.05 mas stellar equatorial angular size However, no particular technical problem was reported, and the θ 1.51VINCI/VLTI± 0.02 mas stellar interferometric polar angular size observations of an elongated polar envelope pol ± other stars observed on this night showed consistent results. As α1 131.6 1.4 deg azimuth of the stellar equator ± 1 we could not2 distinguish,3 these measurements from the rest of our ρeq 2.7 1.3 mas envelope FWHMP. along Kervella stellarand equator A. Domiciano de Souza ρ 17.6± 4.9 mas envelope FWHM along stellar pole data, we chose to publish them all together for the sake of homo- pol ± f 14LESIA,.7 0.3% CNRS UMR relative 8109, (envelopeObservatoire to de star) Paris-Meudon, near-IR flux 5 place Julesgeneity. Janssen, 92195A possible Meudon astrophysical Cedex, France cause for these low visibilities e-mail:± [email protected] would be a stellar eruption that could have suddenly increased 2 Lab. Univ. d’AstrophysiqueP. de Kervella Nice (LUAN), and A. Domiciano CNRS UMR de 6525,Souza: UNSA, Thethe polar Parc CSE wind Valrose, brightness of Achernar 06108 Nice, and France/or angular extension.1061 3 Observatoire de la Côte d’Azur, CNRS UMR 6203, Département GEMINI, BP 4229, 06304 Nice Cedex 4, France Table 2. Relevant parameters of the calibrators used for VINCI observations ofReceived Achernar 23 (continued December from 2005 Table/ Accepted 1). The 20 references February 2006 are the same as in Table 1. 3.7. Comparison with other models and residuals of the fit ABSTRACT Name � Ind A α Cet υ Cet In order to assess the level of adequation of our star+CSE model HDContext. numberBe stars HD show 209100 evidence HD of 18884 mass loss HD and 12274 circumstellar envelopesto (CSE) the from data, UV we resonance also tried lines, near-IR to fit them excesses, with and twothe simpler models: mV presence of episodic4.7 hydrogen emi 2.5ssion lines. The 4.0 geometry of these envelopesa circular is still uniform uncertain, disk although and it a is uniform often assumed ellipse. that The residuals for mK they are formed by2.2 a disk around –1.7 the stellar equator 0.0 and a hot polar wind. Aims. We probe the close environment of the fast rotating Be star Achernareach at angular of the scales three of models a few milliarcsec are presentedonds (mas) in in the Fig. 4 as a function Sp. type K4.5V M1.5IIIa K5/M0III of the projected baseline azimuth angle. We obtained in the first Teff infrared,(K) in order to4500 constrain the 3730 geometry of a possible – polar CSE. log gMethods. We obtained4.5 long-baseline– interferometric– observations of Achernarcase awith uniform the VINCI disk/VLTI angular beam combiner diameter in the ofHθandUD = 1.78 mas, with K bands,1 using various telescope configurations and baseline lengths with a wide azimuthal2 coverage. v sin i (km s− )1 – – the large χ of 4.9 characteristic of a bad fit. Fitting a uniform Results.a The observed visibility measurements along the polar direction are significantlyred lower than the visibility function of the θLD (mas) 1.89 0.02 – – ellipse results in the following best-fit values: θ = 2.31 mas, θ photosphere(mas)b of1. the84 ± star0.02 alone, 11 in.6 particular0.40 at 5.3 low0 spatial.5 frequencies. This points to the presence of an asymmetric diffuse CSE eq UD ± ± ± 2 a elongated along the polar direction of the star. To our data, we fit a simpleθpol model= 1.68 consisting mas, and of twoα1 components:= 135.7◦. a Again, 2D elliptical the χ of 3.2 shows The angularGaussian diameter superimposed of � Ind on a A uniform was measured ellipse representin by Kervellag the distorted photosphere of the fast rotating star. red b poor agreement of this model to our data. It thus appears that et al. (2004b).Conclusions.The uniformWe clearly disk detected angular sizesa CSE of elongatedα Cet and alongυ Cet the polar axis of the star, as well as rotational flattening of the stellar were taken from Dyck et al. (1998) and Richichi & Percheron (2005), photosphere. For the uniform-ellipse photosphere we derive a major axisour of starθeq +=Gaussian2.13 0.05 CSE mas andmodel a minor is a axis much of θ betterpol = fit to our data set respectively.1.51 0.02 mas. The relative near-IR flux measured for the CSE compared(χ2 to the= stellar0.79) photosphere than± the models is f = 4.7 without0.3%. Its CSE. angular ± red ± dimensions are loosely constrained by the available data at ρeq = 2.7 1.3 masAs and shownρpol = 17 in.6 Fig.4.9 mas. 4 (bottom), This CSE could the residuals be linked of our star+CSE to free-free emission from the radiative pressure driven wind originating± from the hot polar caps± of the star. The resulting calibrated squared visibilities are listed in fit appear to be homogeneous with respect to azimuth angle. Tables 3a–d.Key For words. eachtechniques: measuremen hight, angular the calibrator resolution is – listed. techniques: No interferometricSimilarly, – stars: emission-line, we do not Be detect – stars: any mass-loss significant – residual either with stars: rotation – stars: individual: Achernar Fig.systematic 3. Graphical deviation representation of the visibility of the was best-fit observed model for intensity any of our distribu-Fig. 1. Coveragerespect of to the projected (u,v) plane baseline for the VINCI length observations or with time of (Fig. 5). The calibrators at a 1σ level. Achernar. The K band observations (MONA beam combiner) are repre- tion of Achernar. The relative flux contributions from the star andsented the usingH circlesand K (openband for siderostat data sets observations, do not solid show for any the Unit systematic devia- envelope are not to scale. This illustration should not be consideredTelescopes), as tion, and which the H band justifies observations a posteriori are represented our combined using crosses. treatment of these a true image1. Introduction of the star. The scalesRapidtwo arein data rotation units sets. of andB/λ Due, expressedgravity to the darkening in relatively cycles/arcsec. seem small to be number impor- of measure- 3. Model fitting tant keys to explaining the two-component circumstellar en- The southern star Achernar (α Eridani, HD 10144) is the bright- vironmentments (CSE) in the ofH Beband stars: and (1) their a dense lower (particle accuracy densities compared to the 3.1. PolarKervella and equatorial & Domiciano visibilities de Souza (2006) A&A,11 453,12 3 1059 8 est of all Be stars (V = 0.46 mag). Depending on the author (and3.2.N Star-envelopeK10band,10 theircm model− influence), descriptionhigh mass-loss on the (best-fit10− M parameters/yr) and low is very limited. � − 1 � � In orderthe technique to define used) a plausible the spectral model typefor the of light Achernar2 distribution ranges of from radialHowever, velocity ( they10 100 are km overall s− ) equatorial in excellent envelope agreement and (2) a with the best- visibility function is a two-dimensional V (u,v) map. FigureAs 2 discussed in Sect.� − 3.1,9 it3 appears that a di2ffuse envelope,10 Achernar,B3-B4IIIe we toexamine B4Ve (e.g., here the Slettebak2 shape 1982; of the Balona polar and et al. equa- 1987). rarefiedfit model, (N with10 acm specific− ), low reduced mass-lossχ of ( 10 only− 0.2.M /yr) Considering the shows the cuts of this best-fit V (u,v) map along the stellar poleconfined to the direction� of1 its polar axis, is present� around� torialThe visibility estimated functions. projected The rotation orientation velocity ofv thesin i minorranges axis from of 220 and fast ( 1000 km s− ) polar wind (e.g. Damineli Neto & 1 Achernar.limited In order� amount to study of theH fluxband contribution data, we of currentlythis envelope, cannot investigate andAchernar theto equator 270 on km the s− (solid planeand thecurves). of the effective sky Note relative temperature that to the North rapidTeff from was visibility 15ob- 000 de-de Freitas Pacheco 1982; Waters et al. 1987, and references we needthe to definewavelength a simple dependence model to fit of the the observed CSE properties, visibility but additional creasetainedto observed 20 by 000 D03 K using at(e.g., low a Viniciussubset spatial of et the frequencies al. data 2006; discussed Rivinius, in the in the priv.polar present comm.; directiontherein). This picture of a two-component CSE is based on many data. Forobservations this purpose, we with considered the AMBER the following instrument components: of the VLTI in the J is reproducedpaper.Chauville Using well et a al.simplified by 2001). the The presence analysis difficulty of of the in the deriving dependence elongated these of polar parame- the enve-observations of Be stars performed in the past few decades. For ters more precisely is a direct consequence of the rapid rotation example,and opticalH bands/IR will data soonhave shown allow emission such studies. lines and It should IR ex- be noted that lopeequivalent in the model. uniform disk angular diameter with the azimuth of the – the stellar photosphere is represented by a uniform ellipse. of Achernar. Such rapid rotation ( 80% of the critical velocity) cessesinterferometric that essentially probe observations the denser of regions the bright of the CSE B0IVpe (e.g. star γ Cas in projected baseline, they obtained an orientation≥ of the minor axis The parameters are the equatorial and polar angular sizes θ Ainduces graphical mainly representation two effects on of the the star star structure: and its a rotational polar envelope flat- Waters 1986; Dougherty et al. 1994), while UV resonance lineseq of Achernar (assumed to be the polar axis) of α0 = 39 1◦ east and θtheand visible the azimuth have shown orientation that ofthe the apparent equatorial size axis of on this star can vary basedtening on the and best-fit a gravity parameters darkening, is which presented can be indescribed Fig.± 3. by We the em-of highlypol ionized species can probe regions of lower density (e.g. of North. the skyconsiderablyα ; with wavelength (Stee et al. 1998). The scatter ap- phasizevon that Zeipel this eff figureect (von is Zeipel not a 1924). true image of the star, but onlySnow 1981;1 Peters 1982). Gehrz et al. (1974) showed that the To visualize the polar visibility function of Achernar, we ex- – the diffuse envelope is represented by a bidimensional ellip- near-IRpears excess to be measured slightly in larger Be stars along is due the to free-free polar direction radiation. than along the thetracted representationDomiciano the interferometric of de the Souza best-fit m eteasurements al. light(2003, distribution hereafter with azimuth D03) with anglesmeasured the a pri-tical Gaussian. It is parametrized by its full widths at half the apparent rotational flattening of Achernar using the Very Theequator disk-like of the shape star of (Fig. the dense 5, top). equatorial This could CSE has be been caused by devia- oribetween hypothesis 10◦ and that 70 the◦, i.e. star can30◦ from be described the sky-projected by a uniform polar el-maximum (FWHM) along the polar and equatorial axes of axisLarge of the Telescope star. As shown Interferometer in�± Fig. 2 (left), (VLTI). it appears They showed that the that dis- the directlytions measured from our by interferometric simple star-CSE observations model on in the small radio angular scales. lipse surrounded by an elliptical Gaussian envelope aligned withthe starForρ instance,pol and ρeq the, and presence its integrated of clumps flux relative in the CSE to the could create this itstribution principalflattening of axes. the ratio measured This measured intensity visibilities on this distribution star does cannot not follow be reproduces explained that of in a the the ob-andstellar optical flux/IRf . We (e.g. make Dougherty the assumption & Taylor that 1992; its principal Stee et al. axes 1995; Roche approximation, especially when taking the von Zeipel Quirrenbachapparent et al.instability 1997). In of a recent the visibility work, Tycner function. et al. (2005) However, our data serveduniform visibilities disk, and well, there isbut a clearseveral deficit others of visibility could also at low fit. spa- In partic-are aligned with the principal axes of the stellar photosphere. tiale frequencies.ffect into account. FittingRecently, a simple uniform this effect disk was model revealed (through in two exploreset the is relationshipstill too limited between to the constrain angular size their of properties the Hα emit- significantly. ular,a classical weother cannot rapidly least-square determine rotating minimization) stars if the thanks envelope to to interferometric these is symmetric data leads observa- to relativeUsingting a region simple (measured uniform ellipse by interferometry) model is naturally and the a net very Hα sim-emis- 2 to theθUDtions: star,= 1.78 Altairdue mas, to (A7V,Ohishi the but baselinethe reduced et al.orientationχ 2004;of 6.2 Domiciano is ambiguity characteristic de Souza of 180 of et al.◦.plifiedsion approximation measured spectroscopically of the photospheric for seven light Be stars. distribution They find of an a poor2005) fit. and The Regulus deficit of (B7V, visibility McAlister at low et al.spatial 2005). frequencies is Achernar.interesting4. In Nature reality, correlation the of rapid the between rotation CSE the of of two Achernarthe quantities, star causes which signif- they typical of the presence of an extended, incoherent source that icant brightening of the polar caps of the star as a consequence 3.6.is alreadyExcluded resolved data by points the interferometer on the short baselines. of its flattening.4.1. Total Though extension numerical models can accurately pre- In other words, a diffuse and extended envelope appears to be dict the distribution of light on the photosphere of uniformly Article published by EDP Sciences and available at http://www.edpsciences.org/aa or http://dx.doi.org/10.1051/0004-6361:20054771 Inpresent the fitting along process, the polar we axis chose of the star. to exclude the four data pointsrotatingAs stars listed (see e.g. in Domiciano Table 4, thede Souza angular et al. sizes 2002), of the the axes of the obtainedThe on equatorial the UT1-UT4 visibility baseline function (see can be Table evaluated 3b). With by re- a posi-underlyingphotosphere Roche approximation ellipse (θ ieqs notand necessarilyθpol) are verified well for constrained, as is tionstricting angle ourof sample46 deg to for the the visibility projected measurements baseline, obtained they correspond in Achernar.the In flux particular, ratio Jacksonf = et4 al..7 (2004)0.3% show between that stellar the star and the to athe measurement azimuth≈ range thatα1 = is129 almost30◦ aligned. As shown with in the Fig. pole 1, we of do themodels star polar of Achernar envelope. including However, differential± the internal angular rotation dimensions re- of the en- not have as many measurements± at high spatial frequencies along sult in better agreement with the interferometric profile obtained (α = 41.6 1.4 deg). Although they satisfy the data quality cri- velope itself are poorly constrained. In particular, as visible 0this range± of azimuth, due to the limitations in the available VLTI by D03. Though there are good prospects for differential rota- teriabaselines that we during applied commissioning. to the other Figure data 2 (right) points, shows they the are dis- locatedtion beingin constrained Fig. 2, we observationally lack very short by spectro-interferometry baseline measurements to esti- 6 tribution7σ away of from squared the visibilities best-fit observed model. It in shouldthis azimuth be noted range as that(Domiciano the mate de the Souza total et extensional. 2004), the of current the envelope uncertainties in the on polar direction. 114−a function remaining of the data spatial points frequency. are in In excellent this case, the agreement fit of a simple withthe our lightConsidering distribution of ourthe photosphere data, it could lead beus to much prefer more the sim- extended than the uniform disk model with θUD = 2.38 mas produces satisfactory ple approach of a uniform ellipse. We also make the hypothesis results with a reduced χ2 of only 0.6. In this case, we conclude that the axes of the envelope are aligned with the principal axes that we do not detect any significant diffuse envelope along the of the stellar photosphere. Due to the limited coverage of our equatorial plane of the star. data set in terms of azimuth angle at intermediate baselines, we arXiv:0812.2531v1 [astro-ph] 13 Dec 2008 [email protected] (E). : to and Correspondence (D) 078.D-0295(C), programs under Observatory l 03,wt ao n io xso respectively of axes minor and major with 2003), Souz was al. de photosphere (Domiciano interferometry distorted long-baseline strongly by solved its since interest ( Achernar much star Be southern The Introduction 1. asinmdladcmaete oSMC oe predic- model simple SIMECA a 3). to using (Sect. them Achernar tions compare of and CSE model polar the Gaussian t of and scale contribution angular the derive ical we 2), (Sect. measurements i infr our new thermal report VLTI the the we using in Letter, domain, Achernar present of the A1V-A3V observations In type terferometric 2008). spectral al. of et Achernar, discovered (Kervella of (2007) companion Souza close-in de a Domiciano & Kervella envelop CSE), circumstellar this after to addition In photosphere. the uea h oe rae atplrwn htwsdtce i 4 detected reaches was flux that its wind where K06, polar pres by fast radiative infrared a high near creates The the poles Ka08). the hereafter at 2008, sure al. et Kanaan to compared e polar the e oz 06 eefe 0) u oiseteeyfs rotati fast extremely its to Due ( K06). hereafter 2006, Souza 0 hw h uso hsbest-fit this of cuts the shows otesa,det h aeieoinainabgiyo 180 of ambiguity orientation baseline relative the symmetric to is due envelope star, the the to if determine partic- In cannot fit. we also ob- could ular, others the several el- reproduces but well, distribution uniform visibilities intensity served a This by axes. with aligned described principal envelope its Gaussian be elliptical can an only by pri- star but surrounded a star, lipse the the the with that of distribution hypothesis image light ori best-fit true em- the We a of 3. not representation Fig. is the in figure presented this is that parameters phasize best-fit the on based enve- polar elongated the model. the of in presence direction lope the polar by de- the well visibility in reproduced frequencies rapid is spatial the low that at Note observed curves). crease (solid equator the and oamaueetta sams lge ihtepl ftestar the of pole the with aligned almost ( is that measurement posi- points a a to data With four 3b). Table the of (see angle exclude tion baseline to UT1-UT4 chose the on we obtained process, fitting the points In data Excluded 3.6. two-dimensional a is function visibility the star. the and of star illu image the This true from a scale. to contributions not flux are relative envelope The Achernar. of tion 3. Fig. computed was fit The star. complete central our the on representing ellipse uniform superimposed envelope a elliptical on Gaussian a in consisting model simple 4. Table 1064 ei htw ple oteohrdt ons hyaelocated are they points, data other the to 6 applied we that teria 1 eann aapit r necletareetwt our with agreement excellent in are points data remaining 114 edoff Send v α ff . srnm Astrophysics & Astronomy ⋆ eebr1,2008 15, December − 5ad1. and 05 sin c vnZie 94 assteplrcp ob overheated: be to caps polar the causes 1924) Zeipel (von ect 0 7 θ α θ ρ ρ f ae nosrain aewt S eecpsa Paranal at Telescopes ESO with made observations on Based rpia ersnaino h tradisplrenvelope polar its and star the of representation graphical A pol eq = σ pol eq 1 nthe in 3 2 1 e words. Key Conclusions. Results. Methods. Aims. Context. nteplrdrcino cenr h interferometricall The Achernar. of direction polar the in eevd;Accepted ; Received jce ytehtplrcp ftestar. the of caps polar hot the 2.2 by at ejected detected Already model. SIMECA existing nyetbiha pe ii f51 o h qaoilenv equatorial the for % 5-10 of spect limit the upper fitting an by measured establish 10-20% only of excess IR photometric xsigmdlo hssa optdwt h IEAcode. SIMECA the with computed star this of model existing i Kervella etal.Kervella (2009) A&A, 493, 53 41. rpia ersnaino h etfi oe nest distribu- intensity model best-fit the of representation Graphical wyfo h etfi oe.I hudb oe htthe that noted be should It model. best-fit the from away ≈ .Kervella P. : to requests print 32 on Germany Bonn, 53121 69, Hugel dem Auf für Radioastronomie, Max-Planck-Intitut a.H ieu NSUR62,Ui.d ieSpi Antipoli Nice-Sophia de Univ. 6525, UMR CNRS Fizeau, H. Lab. EI,Osraor ePrs NSUR80,UM,Ui.Par Univ. UPMC, 8109, UMR CNRS Paris, de Observatoire LESIA, etfi aaees(n orsodn netite)o our of uncertainties) corresponding (and parameters fit Best 5 km.s 250 1 2 131. 6 17. ff 51 . . 2 esac o h intr fcruselreiso tdis at emission circumstellar of signature the for search We N 51 13 ± 4 . cietmeauecudb ihrta 000K, 000 20 than higher be could temperature ective 7 . 6 ad ihafl it thl aiu f9. of maximum half at width full a with band, try. interferome near-IR by detected recently envelope, an by ed surround is Achernar stars, Be other several of case the is As ± 7 ≈ 1 6 inadflxcnrbto fthe of contribution flux and extension angular the derive we visibilities, measured the From ± ± ± eotie nefrmti bevtoso he LIbas VLTI three on observations interferometric obtained We h niomn ftefs oaigsa Achernar star rotating fast the of environment The .Kervella P. ± . 6dgfrtepoetdbsln,te correspond they baseline, projected the for deg 46 ± 0 ± e) lhuhte aif h aaqaiycri- quality data the satisfy they Although deg). 4 H 000Ka h qao Jcsne l 2004; al. et (Jackson equator the at K 000 10 0 0 1 tr:idvda:Ahra;Sas msinln,B;Me Be; emission-line, Stars: Achernar; individual: Stars: . 4 0 2mlirscns(evla&Dmcaode Domiciano & (Kervella milliarcseconds 02 . . . 1 − 2msselrplraglrsize angular polar stellar size angular equatorial stellar mas 02 mas 05 a neoeFH ln tla equator stellar along FWHM envelope mas 3 . and . a ieto fAhra swl erdcdb the by reproduced well is Achernar of direction polar the along IR thermal the in emission extended observed The a neoeFH ln tla pole stellar along FWHM envelope mas 9 . 1 %rltv evlp osa)na-Rflux near-IR star) to (envelope relative 3% e zmt fteselrequator stellar the of azimuth deg 4 n osqetflteig h o Zeipel von the flattening, consequent and ) K / IIisrmn.Atradsrpinof description a After instrument. MIDI addt set. data band I hra nrrditreoer ihVLTI/MIDI with interferometry infrared Thermal II. 1 .Dmcaod Souza de Domiciano A. , aucitn.ms0980hal no. manuscript V α tainsol o ecniee as considered be not should stration 2 r,H 04)hsreceived has 10144) HD Eri, ( u ,v .Krel n .Dmcaod oz:Teplrwn fAchernar of wind polar The Souza: de Domiciano A. and Kervella P. a ln h tla pole stellar the along map ) V 2 ( u ,v aueo h atwind fast the of nature sig observational an probably most is envelope polar this µm, . 7 a.Fgr 2 Figure map. ) θ ± L = te othe to etter 0 2 (here- e 9 . eovdplrevlp otiue 13. contributes envelope polar resolved y .Kanaan S. , of % 3 2 ± . 13 ABSTRACT ared 2 et a yp- re- on . n- a ( mas 3 ± lp.W opr h bevdpoete fteevlp wi envelope the of properties observed the compare We elope. n a nrydsrbto.Det u iie zmt coverag azimuth limited our to Due distribution. energy ral - acso e tla ai rmAhra,i h hra IR thermal the in Achernar, from radii stellar few a of tances ◦ . bevtr ( Observatory and für Astronomie used Planck-Institut of we processing, min di 5 data approximately two raw by the For followed calibrations. inter min, photometric 3 recording target of on fringes time ometric a to corresponds observation o length each angle baseline of position projected dates corresponding average the The with vation, visibilities. acc more the a of allowing simultane- calibration signals, recorded interferometric is the telescope with each ously of photometry the and mode, UT1-UT2, (UT1-UT4, the baselines and us- telescope UT3-UT4), 2007, m and 8.2 the 2006 three sample in properly ing observed to was Hz Achernar 2 background. of sky frequency Telesco pes a Unit with two chopping the were of mirrors secondary the observations, fAhra,w sdapimwt pcrlrslto of resolution spectral a with prism a astr used R the we in Achernar, working of interferometer Telescope Michelson nomical Large classical Very a instrument This is the 2004). al. of et Glindemann (VLTI; Interferometer combiner mid-infr the is beam processing 2007) al. et two-telescope data Ratzka 2003; al. and et (Leinert MIDI setup Instrumental 2.1. .Observations 2. airtdsurdvisibilities squared calibrated ,Osraor el ˆt Au,018Nc ee ,France 2, Cedex Nice 06108 Côte dAzur, la de Observatoire s, sDdrt lc ue ase,915Muo ee,Franc Cedex, Meudon 92195 Janssen, Jules place 5 Diderot, is hd:osrainl ehius interferometric Techniques: observational; thods: ≈ 1 lnsi the in elines = E 2 ino cenr for Achernar, of tion eomlzd(e et . o details). for 2.2 Sect. (see renormalized ere.Teistdarm hwec aeiei h ( the in baseline each show diagrams giv inset are baseline The projected degrees. each of angles position The ware. ttsia iprino the of dispersion statistical 1. Fig. i.4.Tedte iei h pe ltsostepolar the shows plot upper the in line dotted The baseli 4). each Fig. for of simultaneously range sampled broad is relatively frequencies a tial MIDI, Thanks of coverage line). spectral (dashed the Achernar to of direction polar the and plane, ino cenr for Achernar, of tion eomlzd(e et . o details). for 2.2 Sect. (see renormalized ere.Teistdarm hwec aeiei h ( the in baseline each show diagrams giv inset are baseline The projected degrees. each of angles position The ware. ttsia iprino the of dispersion statistical 1. Fig. i.4.Tedte iei h pe ltsostepolar the shows plot upper the in line dotted The baseli 4). each Fig. for of simultaneously range sampled broad is relatively frequencies a tial MIDI, Thanks of coverage line). spectral (dashed the Achernar to of direction polar the and plane, http: .Meilland A. , 6 ditor λ/ R oee,te r vrl necletareetwt h best- the reduced specific with a agreement with model, excellent fit in overall are they measure- However, of number small relatively the the in to ments Due sets. these of data treatment combined The two our posteriori 5). a (Fig. justifies which time tion, with with or angle. either length residual azimuth baseline significant projected to any to respect detect respect with not do homogeneous we be Similarly, to appear fit aho h he oesaepeetdi i.4a function of a diameter angular as first the disk 4 in uniform Fig. obtained We a in angle. case presented azimuth for baseline models: are projected residuals simpler models the The two of three ellipse. with the uniform them a of fit and each to disk uniform tried circular also a we data, the to ( lis eut ntefloigbs-tvalues: best-fit following the in results ellipse S oe,adtersdaso h tpeetstsatr sta- satisfactory present 3.7). fit Sect. the (see of properties residuals tistical the and model, CSE slse nTbe4 h nua ie fteae fthe of axes the of sizes angular the 4, ( Table ellipse photosphere in listed As extension Total 4.1. Achernar of CSE the of Nature 4. fit the of residuals star and our of adequation of models level the assess other to order In with Comparison 3.7. h u ratio flux the nefrmti bevtoso h rgtBIp star B0IVpe bright the of observations the interferometric in VLTI the of and instrument AMBER the additional with but observations properties, CSE the of dependence wavelength the of amount limited orareeto hsmdlt u aa ttu per that appears thus It data. our to model star our this of agreement poor θ h large the ol easelreuto htcudhv udnyincreased suddenly have could and/ brightness that CSE eruption the stellar visibilities a low be these homo- for would of cause sake the astrophysical for possible together A all geneity. them publish our to of chose rest we As the data, from results. measurements consistent these distinguish showed not night could we this baseline. the on UT1-UT4 observed and reported, stars the was other of problem technical operations particular VLTI no However, of night first d very these as especially excluded, osdrn u aa tcudb uhmr xeddta the than extended esti- direction. more polar to much the be measurements could in baseline it envelope data, the short visible our of Considering very extension as lack total particular, the we en- In mate 2, the constrained. Fig. of poorly in dimensions are angular itself the velope However, envelope. polar e ssiltolmtdt osri hi rprissignificantly. properties their constrain to limited data too our However, still function. is set visibility this create the could of CSE scales. instability the angular in apparent devia- clumps small of by on presence caused the model instance, be star-CSE For could simple This our top). from 5, tions (Fig. the star along the than direction of polar equator the ap- along scatter larger slightly The be 1998). vary to al. can pears star et this (Stee of wavelength size with apparent considerably the that shown have visible the K H χ ff pol ∆ ⋆ //www.strw.leidenuniv.nl red 2 .Ti u otiuini ngo gemn ihthe with agreement good in is contribution flux This ). ad hi nuneo h etfi aaeesi eylimited. very is parameters best-fit the on influence their band, rn otaepackages: software erent λ and N ssoni i.4(otm,tersdaso u star our of residuals the (bottom), 4 Fig. in shown As nisrmna rgnfrt for origin instrumental An qae iiiiiso cenr h ro asaethe are bars error The Achernar. of visibilities Squared H = qae iiiiiso cenr h ro asaethe are bars error The Achernar. of visibilities Squared ≃ = ad(7.6–13.3 band 0t bansetal ipre rne.Drn the During fringes. dispersed spectrally obtain to 30 ad ilso lo uhsuis tsol entdthat noted be should It studies. such allow soon will bands 1 N 0 + MIA+EWS K . 8ms and mas, 68 . PA asinCEmdli uhbte tt u aaset data our to fit better much a is model CSE Gaussian χ ad(8-13 band 9 hntemdl ihu CSE. without models the than 79) red 2 addt esd o hwaysseai devia- systematic any show not do sets data band 3 r ie nTbe1 ahsa rcalibrator or star Each 1. Table in given are .Spang A. , H SCIPHOT f49caatrsi fabdfi.Ftigauniform a Fitting fit. bad a of characteristic 4.9 of nrwt VLTI with Achernar of interferometry infrared Thermal al.: et Kervella P. nrwt VLTI with Achernar of interferometry infrared Thermal al.: et Kervella P. adadterlwracrc oprdt the to compared accuracy lower their and band f B 1 B eso ..)i re oetatthe extract to order in 1.5.2) version , = = ) sn h IIinstrument. MIDI the using µm), H = 4 3 .TeUT1-UT2 The m. 130 µ 3 .TeUT1-UT2 The m. 130 V α addt,w urnl antinvestigate cannot currently we data, band ± V ) o h eotdobservations reported the For m). θ V / 2 1 oeo II nti observing this In MIDI. of mode eq 4 2 N ∼ 2 2 2 raglrextension. angular or esrmnsfo the from measurements . . esrmnsfo the from measurements nevec n h Stee Ph. and , ( = ftepoopei flux photospheric the of % 5 7 adcruselremission circumstellar band Cena 07.W found We 2007). 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9500K 10500K -0.5 10000K -0.5 -0.5 -0.5 6100K 6100K North (milliarcseconds) North (milliarcseconds) North (milliarcseconds) -1.0 North (milliarcseconds) -1.0 -1.0 -1.0 -1.5 -1.5 1.0 0.5 0.0 -0.5 -1.0 1.0 0.51.5 0.01.0 0.5-0.50.0 -1.0-0.5 -1.0 -1.5 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 East (milliarcseconds) East (milliarcseconds)East (milliarcseconds) East (milliarcseconds) *Disk of Regulus (B7 V) resolved *Disk of Caph (F2 IV) resolved as *Pole of Vega (A0 V) resolved by as ellipsoid by CHARA (McAlister ellipsoid by CHARA/MIRC (Che et CHARA (Aufdenberg et al. 2006). et al. 2005, Che et al. 2011). al. 2011). Fig. 5.— Images of α Leo . The notations are all the sameFig. as 4.—those Imagesin images of ofβ βCasCas . (see The left one shows the surface intensity distribution of β 2 Fig. 4). The angular resolution is 0.55 mas. The reducedCasχ fromof the MACIM, images overplotted fromVe MACIMga Model with latitudes (i = 90 degrees) and longitudes from the model. The angular and model fitting are 0.78 and 1.32. resolution is 0.57 mas(milli-arcsecond). The dashed contours represent the surface brightness temperatures of the image. The right one shows the image from model fitting, overplotted with brightness temperature contour from the model. The reduced χ2 of the images from MACIM and model fitting are 1.20 and 1.36.

*Disk of Altair (A7 V) resolved by *Disk of Alderamin (A7 V) resolved by *Disk of Rasalhague (A5 III) resolved CHARA (J. Monnier et al. 2007). CHARA (M. Zhao et al. 2009). by CHARA (M. Zhao et al. 2009). Astronomy & Astrophysics manuscript no. espinosa c ESO 2011 � September 15, 2011

Gravity darkeningF. Espinosa in rotating Lara and M. stars Rieutord: Gravity darkening in rotating stars 5 1,2 1,2 F. Espinosa Lara and M. Rieutord 4. Conclusions 1 Universitéde Toulouse; UPS-OMP; IRAP; Toulouse, France 2 CNRS; IRAP; 14, avenue Edouard Belin, F-31400 Toulouse, France Observing that the energy radiated by a star is produced almost entirely in the stellar core and that most stars may be considered September 15, 2011 • Observationsclose to a steady state, we have noted that the energy flux is es- ABSTRACT by CHARA/sentially a divergence-free vector field. We also observed from Context. Interpretation of interferometric observations of rapidly rotating stars requires a good modelMIRC of their show surface effective tem- β full two-dimensional models of rotating stars that the direction perature. Until now, laws of the form T ff g have been used, but they are only valid for slowly rotating stars. e ∝ eff gravity ff Aims. We propose a simple model that can describe the latitudinal variations in the flux of rotating stars at any rotationof this rate. vector is always very close to that of the e ective grav- Methods. This model assumes that the energy flux is a divergence-free vector that is antiparallel to thedarkening effective gravity. ff ity, so it only depends on the mass distribution. Following this Results. When mass distribution can be described by a Roche model, the latitudinal variations in theweaker e ective temperature than only depend on a single parameter, namely the ratio of the equatorial velocity to the Keplerian velocity. We validate this modelpicture,by comparing the physical conditions in the outer layers can only af- its predictions to those of the most realistic two-dimensional models of rotating stars issued from theβ ESTER=0.25 code. The agreement is very good, as it is with the observations of two rapidly rotating stars, α Aql and α Leo. fect the flux radiated outside the star very slightly as long asa Conclusions. We suggest that as long+/ as a gray 2-D atmosphere Star Model can be accepted, the inversion of data on flux distributiongray coming atmosphere from can be assumed. Thus, we proposed that the interferometric observations of rotating stars uses such a model, which has just one free parameter. Analytic Model energy flux in the envelope of a rotating star be approximated by Key words. stars: atmospheres - stars: rotation L Von Zeipel β = 0.25 F = Fω(r,θ)g ff. − 4πGM e We have shown how the non-dimensional function Fω(r,θ) 1. Introduction In Sect. 2 we explain the derivation of the new model of gravity darkening and the associated assumptions.can be In evaluated the next by assuming that mass distribution is repre- The recent development of long-baseline optical/infrared intersection, results are compared to the valuessented of two-dimensio by thenal Roche model. We then demonstrated that the lati- ferometry has allowed direct observation of gravity darkening at models and to observations. Conclusions follow. ff the surface ofEspinosa some rapidly Lara rotating & Rieutord stars. This phenomenon (2011) A&A,has 533, 43 tudinal variation of the e ective temperature only depends on a been known since the work of von Zeipel (1924), who noticed ff single parameter ω = Ω2R3/GM. Such a model is very appro- Fig. 3. Ratio between equatorialff andff polar2. The e modelective temperature e that the radiative flux is proportional to the e ective gravity ge in a barotropicas a function star (i.e. when of the flattening pressure onlyǫ depends= 1 onR tphe/ReBefore. Solid presenting line: our The model, analyti- we briefly recallpriate the to origin interpreting of the the interferometric observations of rotating density). Thus, the effective temperature Teff varies with latitude,− 1/4 von Zeipel law and Lucy’s exponent. Instars a radiative since, region unlike the von Zeipel’s or Lucy’s laws, it is valid for followingcalT ff modelg . This presented is known as in the this von Zeipel paper. law. Dashed line: Von Zeipel’s law.ff e ∝ eff =flux is essentially carried by the di usionhigh of photons rotation and thus rates (up to breakup) and depends only on a single Barotropicity,Crosses however,: ESTER is a strong models hypothesis, of set and is 1 actu (X-c correctlyX). Triangles described by Fourier’s: ESTER law, = . parameter (the β-exponent is removed). Adjustment of the ob- ally incompatiblemodels with of a set radiative 2 (X zonec in0 solid5X). body rotation. F = χ T, (1) This problem was noticed by Eddington (1925) and has led to − r∇ served surface flux would thus only require variations in ω and χ many discussions over subsequent decades (see Rieutord 2006, where r is the radiative heat conductivity.i If the star is assumed for a review). Now, the recent interferometric observations of to be barotropic, density and temperature (and, the thus inclinationχr) are only of the rotation axis on the line of sight. some nearby rapidly rotating stars by McAlister et al. (2005), functions of the total potential Φ (gravitational plusAs centrifugal). previously mentioned, this model fits the fully 2D mod- Aufdenberg et al. (2006), van Belle et al. (2006), and Zhao et al. Hence, (2009) have shown that gravity darkening is not well represented els well using a gray atmosphere with a rigidly rotating surface F = χ (Φ) T(Φ) = χ (Φ)T ′(Φ) Φ=χ (Φ)T ′(Φ)g ff. (2) by von Zeipel’s law, which seems to overestimate the tempera- − r ∇ − r ∇ (butr withe interior differential rotation). This dynamical feature ture difference between pole and equator. This conclusion is usu- β If we define the surface of the star asof a the place models of constant may not be very realistic, so we tested a surface arXiv:1109.3038v1 [astro-ph.SR] 14 Sep 2011 ally expressed using a power law, T ff g , with an exponent e ∝ eff given optical depth, we see that this model implies latitudinal 2 less than 1/4. This is traditionally explained by the existence of Ω θ =Ω . θ variations in the flux that strictly followrotation those of the eff(ective) eq(1 0 1 cos ) inspired by observations a thin convective layer at the surface of the star, in reference to 1/4 − ff ff ff the work of Lucy (1967), who found β 0.08 for stars with a gravity. Using the e ective temperature, we(Collier recover Te Camerongeff . 2007). The di erence is hardly perceptible, ∼ ∝ convective envelope. Lucy’s approach is based on a first-ordertherefore development the of proposed model of gravity darkening looks quite the dependence between Teff and geff in a convective enve- Actually, the weaker variation in the flux with latitude, com- lope. Since the convective flux is almostrobust. orthogonal Future to isen- improvement of two-dimensional models will of pared to von Zeipel’s law has also been noticed by Lovekin et al. tropic surfaces, log Teff and log geff are relatedcourse on such be a surface. used to confirm this robustness. (2006), Espinosa Lara & Rieutord (2007), and Espinosa Lara Deep enough where the entropy is almost the same everywhere, (2010) in two-dimensional numerical models of rotating stars. s(log Teff, log geff) = s0, where s0 is the specific entropy in the β This prompted us to reexamine this question using these new bulk of the convection zone. Assuming thatAcknowledgements.T ff g at first The authors acknowledge the support of the French Agence e ∝ eff models. It leads us to a new approach that avoids the strong as- order, one finds that Nationale de la Recherche (ANR), under grant ESTER (ANR-09-BLAN- sumption of von Zeipel and that is presumably able to better 0140). This work was also supported by the Centre National de la Recherche represent the gravity darkening of fast rotating stars. This paper ∂ ln s/∂ ln geff β = Scientifique (C.N.R.S.,(3) UMR 5572), through the Programme National de aims at presenting this new model. −∂ ln s/∂ ln T ff e Physique Stellaire (P.N.P.S.). The numerical calculations were carried out on the CalMip machine of the ‘Centre Interuniversitaire de Calcul de Toulouse’ (CICT), which is gratefully acknowledged.

Fig. 4. Gravity darkening coefficient β as a function of flat- References tening. The values were calculated using a least-square fit to Aufdenberg, J. P., Mérand, A., Coudédu Foresto, V., et al. 2006, ApJ, 645, 664 the profile of effective temperature. Solid line: Model presented Che, X., Monnier, J. D., Zhao, M., et al. 2011, ApJ, 732, 68 in this paper. Crosses: ESTER models of set 1 (Xc = X). Collier Cameron, A. 2007, Astronomische Nachrichten, 328, 1030 Domiciano de Souza, A., Kervella, P., Jankov, S., et al. 2005, A&A, 442, 567 Triangles: ESTER models of set 2 (Xc = 0.5X). The correspond- β Eddington, A. S. 1925, The Observatory, 48, 73 ing value of for von Zeipel’s law is 0.25. The boxes repre- Espinosa Lara, F. 2010, Ap&SS, 328, 291 sent the values of β, with their corresponding errors, obtained by Espinosa Lara, F. & Rieutord, M. 2007, A&A, 470, 1013 interferometry for several rotating stars: Altair (Monnier et al. Iglesias, C. A. & Rogers, F. J. 1996, ApJ, 464, 943 2007), α Cephei (Zhao et al. 2009), β Cassiopeiae, and α Leonis Lovekin, C. C., Deupree, R. G., & Short, C. I. 2006, ApJ, 643, 460 Lucy, L. B. 1967, ZAp, 65, 89 (Che et al. 2011). McAlister, H. A., ten Brummelaar, T. A., Gies, D. R., et al. 2005, ApJ, 628, 439 Monnier, J. D., Zhao, M., Pedretti, E., et al. 2007, Science, 317, 342 Peterson, D., Hummel, C., Pauls, T., et al. 2006a, ApJ, 636, 1087 surements of the gravity darkening exponent β derived by inter- Rieutord, M. 2006, in EAS publications, Vol. 21, Stellar Fluid dynamics and ferometry for several rapidly rotating stars, Altair (Monnier et al. numerical simulations: From the Sun to Neutron Stars, ed. M. Rieutord & 2007), α Cephei (Zhao et al. 2009), β Cassiopeiae, and α Leonis B. Dubrulle, 275–295 α Rieutord, M. & Espinosa Lara, F. 2009, Communications in Asteroseismology, (Che et al. 2011). We can see that there is good (Altair, Leo) 158, 99 orfair(α Cep) agreement between the observed values and the Rogers, F. J., Swenson, F. J., & Iglesias, C. A. 1996, ApJ, 456, 902 model. The discrepancy for β Cas may come from its small incli- van Belle, G. T., Ciardi, D. R., ten Brummelaar, T., et al. 2006, ApJ, 637, 494 nation angle ( 20 deg), so we see the star near pole-on, which von Zeipel, H. 1924, MNRAS, 84, 665 ∼ makes determining of its flattening more difficult and makes the Zhao, M., Monnier, J. D., Pedretti, E., et al. 2009, ApJ, 701, 209 results depend more on the model used for gravity darkening and limb darkening. Limb and Gravity Darkening for Rotating Stars

Observed SED Vega Model (i = 5 degrees) Bohlin and Gilliland (2004)

Pole-on view

Equator-on view

Vega Model (i = 90 degrees)

Aufdenberg et al. (2006) ApJ, 645, 664 Self-Consistant Field (SCF) Model Aufdenberg & MacGregor, Limb and Gravity Darkening Movie in preparation SCF Models: SEDs and Line Spectra Movie

Aufdenberg & MacGregor, in preparation Interferometry at visible wavelengths: higher angular resolution --> O & B diameters & darkening limb darkening more sensitive to temperature gradients Thank you for your attention!