Solar Limb Darkening and the Overshooting Approximation 848 F

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Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 Interferometry Collaborators S. T. Ridgway (NOAO, LESIA, Observatorie de Paris-Meduon) P. Kervella, A. Mérand, V. Coudé du Foresto (LESIA, Observatorie de Paris-Meduon) D. Mozurkewich (Seabrook Engineering) CHARA Team (Georgia State University)

Model Atmosphere Collaborators H.-G. Ludwig (Lund Observatory) R. Kurucz (Smithsonian Astrophysical Observatory) P. Hauschildt, A. Schweitzer (Hamburger Sternwarte) program phoenix F. Allard (CRAL-ENS, Lyon) c ------c-- E. Baron (University of Oklahoma) c-- used modules: c-- use phoenix_variables T. Barman (Wichita State University) use phx_interfaces, dummy_takeband => takeband implicit none C. I. Short (St. Mary’s University) ************************************************************************ * the main program for phoenix * version 13.0.0 of 30/Jan/2002 by Peter H. Hauschildt et al. ************************************************************************

Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 What’s to Come....

Intro: All about limb darkening

Part I: Procyon, Convection, and Limb Darkening

Part II: Deneb, Stellar Winds, and Limb Darkening

Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 Limb Darkening Basics

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μ= Cosθ Gray (1992)

Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 Early Multi-Wavelength Limb Darkening

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Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 Early Model Limb Darkening Models

Schwarzschild (1906) Models vs. Contemporary Data

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�� 1906 - K. Schwarzschild Derived a center-to-limb proﬁle for the Sun with a radiative equilibrium temperature structure. He showed this to be consistent with observations, ruling out an adiabatic equilibrium temperature structure.

Assumptions: mass absorption coefﬁcent

is both wavelength and depth independent. Adapted from K. Schwarzschild (1906) “Über das Gleichgewicht der Sonnenatmosphäre” Angular-dependent intensity is replaced by Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen. Math.-phys. Kalsse, 295, 41 its mean. Translation in D. H. Menzel, Ed., Selected Papers on the Transfer of Radiation (1966) NY: Dover

Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 New Models, New Physics, Hydrogen and the Origins of Convective Instability

1921 - E. A. Milne Replaced Schwarzchildʼs mean intensity by an angular average producing a radiative equilibrium temperature structure with better ﬂux conservation, yielding a limb darkening coefﬁcent of in better agreement with observations.

1925 - C. Payne and H. N. Russell Established hydrogen as the principal component of the solar atmosphere.

1930 - A. Unsöld Investigated the effects of hydrogen ionization on the stability of radiative equilibrium against convection. 1939 - R. Wildt Recognizes the importance of _ wavelength dependent H bound-free and free-free opacity. This opacity causes the solar atmosphere to be unstable to convection.

Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 Wavelength-Dependent Opacity of the Solar Atmosphere

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Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 1976ApJS...30....1V

Reconstructing the Sun’s Temperature Structure by Inverting the Planck Function

1976ApJS...30....1V

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Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 18 CHAPTER 2. BASIC RADIATIVE TRANSFER

18 CHAPTER 2. BASIC RADIATIVE TRANSFER Eddington-Barbier approximation. The emergent intensity at the stellar surface (τν = 0, µ > 0) is given by: Eddington-Barbier approximation. The emergent intensity at the stellar surface (τν = 0, µ > 0) is given by: + ∞ tν /µ I (τ =0, µ) = S (t ) e− dt /µ. (2.43) ν ν + ∞ ν ν tν /µ ν Iν (τν =0, µ) =0 Sν(tν) e− dtν/µ. (2.43) � �0 Substitution of Substitution of

∞ n 2 n 18 Sν(τν) = anτνCHAPTER= a0 + a1τν + a2τ 2.ν + ... BASIC+ anτν RADIATIVE TRANSFER ∞ n=0n 2 n Sν(τν) = anτ�ν = a0 + a1τν + a2τν + ... + anτν n and use of 0∞ x exp( x) dx = n! gives n�=0− � I+(τ =0, µ) = a + a µ + 2a µ2 + ... + n! a µn. n ν ν o 1 2 n andEddington-Barbier use of ∞ xLinkingexp( Intensityx approximation.) dx = ton Depth:! gives The Eddington-BarbierThe emergent Approximation intensity at the stellar surface 0 Truncation− of both expansions after the first two terms produces the important Eddington- (τν = 0, µ > 0)Barbier is given approximation by: + 2 n � I+(τ =0, µ) S (τ = µ) (2.44) Iν (τν =0, µ) = aoν +ν a1µ +≈ 2aν 2νµ + ... + n! anµ . which is exact when+ Sν varies linearly with∞τν. Likewise fort theν /µ emergent flux: Iν (τν =0, µ) = Sν(tν) e− dtν/µ. (2.43) Truncation of both expansions after the+ first0 two terms produces the important Eddington- ν (0) � πSν(τν = 2/3). (2.45) Barbier approximation F ≈ Substitution of A formal derivation is given+ on page 85, a simple one in Exercise 2 on page 225. Figure 2.3 illustrates the Eddington-BarbierIν (τν =0 approximation, µ) S simplistically,ν(τν = µ) Figure 2.4 its application (2.44) to solar limb darkening, Figure 2.5 its application≈ to line formation at increasing sophis- tication. ∞ n 2 n which is exact whenSSν(ντνvaries) = linearlyanτν with= a0 τ+ν.a1 Likewiseτν + a2τν for+ the... emergent+ anτν flux: n=0 � + θ ν (0) πSν(τν = 2/3). Iν (2.45) n ∞ −Fτν ≈ 0 and use of 0 x exp( x) dx e= n! givesSν A formal derivation is given− on page 85, a simple one in Exercise 2 on page 225. Figure 2.3 � + −τν 2 n I (τν =0, µ) = aSoν+e a1µ + 2a2µ + ... + n! anµ . illustrates the Eddington-Barbierν approximation simplistically,1 Figure 2.4 its application to solar limb darkening, Figure0 2.5 its application to line formation at increasing sophis- Truncation of both expansions0 1 after 2 the 3 first 4 twoτ termsν produces the important Eddington- tication. τ ν Barbier approximation 2 + I (τν =0, µ) Sν(τν = µ) (2.44) Figure 2.3: The Eddington-Barbierν approximation. Left: the integrandFrom Rob Rutten’sSν exp(excellentτ lectureν ) measures notes: the contri- ≈ http://www.phys.uu.nl/~rutten/Astronomy_lecture.html− bution to the radially emergent intensity Iν (τν =0, µ=1) from layers with different optical depth τν . The Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitáθ di Pisa 13 April 2005 which is exact whenvalue of SνSatντν varies= 1 is a good linearly estimator of the with area underτν. the Likewise integrand curve, fori.e., the total emergent contribution.I flux: Right: for a slanted beam the characteristic Eddington-Barbier depth is shallower than for a radial beam;ν it lies at τν =−µτ. ν e Sν +(0) πS (τ =0 2/3). (2.45) Fν ≈ ν ν −τν A formal derivation2.3 is Line given transitions on pageSν e 85, a simple one in Exercise 2 on page 225. Figure 2.3 illustrates the Eddington-BarbierBound-bound transitions between approximation the lower l and upper simplistically,1 u energy levels Figure of a discrete 2.4 its application to solar limb0 darkening,electromagnetic Figure energy-storing 2.5 system its application such as an atom, to ion line or molecule formation may occur at as: increasing sophis- tication. 0– radiative 1 excitation; 2 3 4 τν τ ν 2 θ Iν Figure 2.3: The Eddington-Barbier approximation. Left: the integrand Sν exp( τν ) measures the contri- −τν 0 − bution to the radially emergente intensitySIνν (τν =0, µ=1) from layers with different optical depth τν . The value of Sν at τν = 1 is a good estimator of the area under the integrand curve, i.e., the total contribution. Right: for a slanted beam the characteristic Eddington-Barbier−τν depth is shallower than for a radial beam; Sν e it lies at τ = µ. ν 1 0 0 1 2 3 4 τν τ ν 2.3 Line transitions 2

Bound-bound transitions between the lower l and upper u energy levels of a discrete Figure 2.3: The Eddington-Barbier approximation. Left: the integrand Sν exp( τν ) measures the contri- − electromagneticbution to the radially energy-storing emergent intensity systemIν ( suchτν =0, as µ=1) an atom, from layers ion withor molecule diﬀerent optical may occur depth as:τν . The value– radiative of Sν at τν excitation;= 1 is a good estimator of the area under the integrand curve, i.e., the total contribution. Right: for a slanted beam the characteristic Eddington-Barbier depth is shallower than for a radial beam; it lies at τν = µ.

2.3 Line transitions

Bound-bound transitions between the lower l and upper u energy levels of a discrete electromagnetic energy-storing system such as an atom, ion or molecule may occur as: – radiative excitation; Solar Limb Darkening and the Overshooting Approximation 848 F. Castelli et al.: Notes on the convection in the ATLAS9 model atmospheres Castelli, Gratton & Kurucz (1997) A&A 318, 841

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 WITH OVERSHOOTING Models at τRoss=1 APPROXIMATION [M/H]=0 [M/H]= 3.0 −

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 Intensity Convective Flux / Total Flux Total Convective Flux /

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 different structure depending whether the “overshooting” option is switched on or off. In this section we investigate the effect of the Fig. 4a and b. Comparison between ob- different model structure on the V K, B V, and b y colour − − − served (points) and computed (full line) so- indices, which are often adopted for ﬁxing effective tempera- lar limb-darkening curves I (cosθ)/I (0). tures for cool stars; furthermore, we will try to state whether λ λ the color indices from the “overshooting” models (COLK95) or Observations are from Pierce & Slaughter Overshooting those from the “no-overshooting” models (COLNOVER) give NO OVERSHOOTING (1977) and Pierce, Slaughter & Weinberger Teff closer to the values derived from the infrared ﬂux method APPROXIMATION (1977). Computed limb-darkening curves (IRFM), which is almost model independent. are from models which differ only for the

Temperature (K) Temperature “overshooting” option. a it is “on” (SUNK94 7.1. The dependence of the synthetic colour indices on the model) b it is “off” (SUNNOVERC125 convection model). The different curves correspond to differentLog values (Rosseland of cosθ Optical Depth) We computed grids of synthetic colours UBV, uvby, and RIJKL Fig. 11. Upper plot: the ratio Fconv/Ftot as a function of log τRoss in the from models having the “overshooting” option switched off, mi- Wavelength (Å) Sun for: (a) the SUNK94 model (full line), (b) the standard ML theory 1 croturbulent velocity ξ=2 km s− , and metallicities [M/H]=0.0 Table 2. Observed and computed color indices for the Sun without any modification (crosses), (c) the SUNNOVERC125 model and [M/H]= 3.0. For each gravity, we derived T by inter- This implies that the continuum(no “overshooting”)(dashed windows line). selected Lower plot: by the Pierce T-log τ relation eff Ross polating in the− COLNOVER grids for the (V K), (B V), Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar& Slaughter Atmosphere (1977) Models.... andfor thePierce Dipartimento (a) and et (c) casesal. (1977)di Fisica may Universitá not always di Pisa 13 April 2005 and (b y) color indices of the COLK95 grids.− In this− way, Color Observed Computed Computed correspond to the real continuum at several wavelengths. we may− estimate the effect of the convection on the effective indices SUNK94 SUNNOVERC125 temperatures derived from the color indices. The temperature differences ∆Teff =Teff over-Teff nover as function of Teff for the (U B) 0.1951 0.17 0.18 (V K), (B V), and (b y) indices are shown for different 4.3. Color indices − − − − 0.17 0.012 and decreasing electron pressure which cause a growth of the gravities and solar metallicity in the upper panels of Fig. 13-15. (B V) 0.650±1 0.67 0.66 Table 2 compares (U hydrogenB), (B ionizationV), and zone. (b y) observed color The largest ∆Teff differences are about 60 K, 100 K, and − 2 − − − 170 K for the (V K), (B V), and (b y) indices respectively. 0.68 0.005 indices with those derivedTable from 3 shows the whichSUNK94 models and are affected the SUN- by convection They occur for T− and log− g between− 7500-8000 K and 4.0- ± 3 eff 0.656 0.005 NOVERC125 models.for Owing gravities to ranging the uncertaintyfrom log g=5.0 to in log theg=1.0. solar Furthermore, 4.5 for (V K), 6750-7250 K and 3.0-4.0 for (B V), 6500- ± 4 − − (b y) 0.406 0.004 0.41 0.397 colors and to the smallfor differences the metallicities between [M/H]=0 the and SUNK94 [M/H]= 3, it and lists the mo- 7000 K and 3.0-4.0 for b y. Temperatures from the color − ± 3 − − 0.414 0.003 dels which show the largest difference, at τross=1, between the indices computed from the “no overshooting” models are lower ± the SUNNOVERC125F colorsconv/Ftot wecomputed cannot with state the “overshooting”which model option has switched than those from color indices computed from the “overshooting 1 Neckel (1994) to be preferred. on and off respectively. The maximum effective temperature of models”. For all the three indices the value of ∆Teff weakly 2 Schmidt-Kaler (1982) models affected by the different convection options decreases depends on gravity for Teff <6500 K. For Teff > 6500 K the with decreasing log g. effect of the convection increases with increasing gravity. 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 Multi-Wavelength Diameters from Speckle Interferometry (@Palomar) Alpha Ori Alpha Lyr ~50 milli-arcseconds ~3 milli-arcseconds 1977ApJ...214L..79B

blue

green

yellow

red Gezari et al. (1972)

ApJ, 173,1972ApJ...173L...1G L1 Blazit et al. (1977) ApJ, 214, L79 Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 1982AJ.....87.1044R

Limb-Darkening Measurements from Lunar Occultation 1982AJ.....87.1044R Ridgway et al. (1982) AJ 87, 104

Worth Hill Observatory Wavelength

Delaware Valley Amateur Astronomers

Angular Diameter

Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 F. Abe et al.: Probing the atmosphere of a solar-like star by galactic microlensing at high magniﬁcation L495

Table 1. Microlensing parameters of MOA 2002–BLG–33. Here rE is thus expected to rotate through 1◦–2◦. Light curves were denotes the Einstein radius of the lens, as determined by its total mass, computed for rotating lenses with various periods of rotation, + M1 M2, DOL and DOS the distances to the lens and source stars. The but with the plane of rotation held fixed and perpendicular to errors are 2σ limits calculated as in Avni (1976). the lens-source axis, to maximise the effect. It was found that periods of rotation <150 days yielded light curves that were in- Parameter Value consistent with the data at a level greater than 2σ. Limb darkening coefficients, and stellar profiles, were then computed, in Lens mass ratio M /M 0.54 0.20 1 2 ± the static approximation, for a simulated dataset with a rotation Lens separation a 0.11 0.01 r ± E period of 150 days. It was found that the computed stellar pro- Einstein time tE 50.7 1.0 day files differed from the inputted profiles by approximately 2 ± × Position angle φ 112◦ 2◦ the 2σ statistical uncertainty of the profiles computed without ± 3 Source radius r (2.25 0.05) 10− r D /D rotation. The effect of rotation was therefore allowed for, in the s ± × E OS OL 4 Impact parameter u (4.29 0.09) 10− r present work, by doubling the errors in the stellar profiles cal- 0 ± × E Centre time t HJD 2 452 460.496 0.004 culated in the static approximation. 0 ± Limb darkening c 0.41 0.04 Limb DarkeningI ± Measurements from Microlensing . . dI 0 08 0 65 − ± 3. The surface of the source star c 0.49 0.02 VE ± d 1.15 0.56 1318 FIELDS ET AL. Vol. 596 VE ± We used colour measurementsFields to identifyet al. (2003) the source ApJ star 596, type 1305 Abe et al. (2003) A&A 411, L493 in MOA 2003–BLG–33. The baseline (i.e. unmagnified) flux of the source star was determined by from the fluxes measured by the MDM and Wise telescopes at high magnification. The instrumental magnitudes were calibrated using the catalogue of Udalski et al. (2002). The magnitudes thus determined in the V and I bands (Landolt system) were m = 19.94 0.05 V ± and m = 17.96 0.07. Assuming a distance to the source star I ± of 7 2 kpc, and absorption and reddening due to galactic dust ± with A = 1.18 0.05 mag and E(V I) = 1.24 0.05 (Sumi I ± − ± et al. 2003; Udalski 2003), we derive its absolute magnitude M = 2.6 0.6 and colour index V I = 0.74 0.10.These val- I ± − ± ues identify the source star as an F8–G2 main-sequence turn- off star with age approximately 3 10 Gy and temperature in − the range 6200 5800 K (Girardi et al. 2002), i.e. a solar-like − star that has just commenced evolving away from the main sequence, or is just about to do so. Limb-darkening coefficients have been calculated for the ATLAS and PHOENIX models of stellar atmospheres for a wide variety of stars (Claret 2000). The limb darkening coefficients, (cλ , dλ ), for the ATLAS model of a star with L494 F. Abe et al.: Probing the atmosphere of a solar-like star by galacticTe microlensingff = 6000 K, at high surface magnification gravity log g = 4.0, and metallicity [M/H] = 0.3 are (0.363, 0.459) for the I-band and (0.461, Fig. 2. The passage of the source star of MOA 2002–BLG–33 through − 2 the lens-caustic. The magnification map of the lens is represented in 0.394) for the VE-band . These compare favourably with the blue, with regions of higher magnification represented by lighter shad- measured values in in Table 1 that were derivedfrom our obser- ing. The caustic is the diamond shaped line of very high magnification. vations without including the effect of rotation described above. The red lines indicate the track of the source star relative to the caus- Figure 3 shows the theoretical intensity profiles for the range tic. The position angle of the lens binary system with respect to the of possible star types identified for MOA 2003–BLG–33, and φ source trajectory is denoted by . The angular diameter of the source the intensityFig. profiles 1. Observed derived light from curve, the best measurements fitting with the µ star would be 2–4 as. effect ofFig. rotation8.—Brightnessmodel, included. and profile residuals in theIt isI band. from apparent The the black model that line andfor the shading ATLAS are EROS and BLG-2000-5 and its associated 3 � error envelope. The red lines give 2 Next Gen models at log g 2:5 and M=H 0:3 for Teff between 4000 and 4600 K, in 100 K increments with the lowest-temperature model the most limb PHOENIXdarkened profiles (i.e.,the the peak curve are of that¼ confirmed MOA is the brightest½ 2002–BLG–33.�at¼ in þ the the center 95% and The dimmestconfidence at the limb). level The blue line corresponds to the best-fit Next2Gen model at T 4600 K, log g 2:5, and M=H 0:25. Also given is the I-band curve of the Sun in green. As can be seen, there is a point that all the model curves share with¼ the Sun to within¼ abouttotal 4%.½ dataset The� ¼ � precision consists of of 473 the measure- present measurements at approximately (r, SI ) = (0.73, 1.02). This fixed point is the same as the one that the ATLAS models share (but the microlensing-determined profiles do not) we obtained. Only the above close binary model gave an ac- resultedand from is not apparentments the fact in the the that (c�I, dband� the) formalism. magnification from Boyden 1.5 was m high through- ceptable fit to the data. out the caustictelescope crossing, (South and Africa), that it could 186 points therefore be mon- Unfortunately,neither the rotation period of the binary lens, itoredexpecting accuratelyin theVE wrong everywhere,from type EROS of star. in 1m particular In principle, (La Silla), at differential the end points 5. CONCLUSION extinction260 across points the in fieldI from could MDM redden 2.4 the m star (Kitt more than The observational limb darkening found from microlens- nor the plane of rotation, was able to be determined from the whichthe are clump most stars sensitive against which to limb it is darkening calibrated (see and Fig. reddening, 10 of as Peak), 480 points in I from MOA 0.6 m ing formally disagrees with the limb darkening derived from data. We estimated the contribution of binary rotation to our An et al. 2002), thus affecting our photometric estimate of atmospheric models by many standard deviations. We have its intrinsic(New color. Zealand), However, and the 21 spectroscopic points in I analysis measurements of limb darkening as follows. For typical values 2 argued that this difference is unlikely to be the result of the Foryields the V aEand-band, source 10 the temperature points average in valuesV similarfrom given to Wise the by one 1 Claret m we (2000) find inobservations, the but it is more likely due to something related of the lens parameters, the rotation period lies in the range 100– V and photometrically.R bands were used, to allow for the extended red transmissionto the atmospheric models. It is possible that these models One should(Israel). note The that inset reddening shows has the the separation effect of shifting 200 days. During the caustic crossing shown in Fig. 2, the lens of the EROS V filter, and also the pre-filtering effect of galactic dust.include physics that is not applicable in all surface gravity the observedof the bands. light curves We find, in however, the VE and thatI bands the measured regimes. It is a testament to the theoretical models that they extinctioncaused for EROS by limb BLG-2000-5 reddening. [E(V I ) = 1.3] shifts the � ˚ approximate reality in several bands without any previous mean wavelength of the I bandpass by only +100 A. This physical data. We hope that now that giant stars have a cali- shift is about 28% of the difference between the mean wave- bration point, much like dwarfs have from the Sun and lengths of I and IE and only 4% of the difference between I supergiants have from interferometry, stellar models can and V. Comparing Figures 7 and 1, one sees that it cannot continue to improve in all stellar regimes. Jason2. Light P. curveAufdenberg of MOA Where 2002–BLG–33 did Half of the Sun’s Oxygenon a clusterGo? Testing computeraccount Stellar (Rattenbury for differences Atmosphere et between al. 2002). the Models.... observations The inverse andDipartimento the di Fisica Universitá di Pisa 13 April 2005 ray tracing techniqueatmospheric naturally models. accommodates This is partly because the finite the magnitude size The event MOA 2002–BLG–33 was first alerted by the MOA of the effect is too small and partly because it causes a shift We would like to thank Joachim Wambsganss for his of the source starin theand wrong the direction.limb darkening profiles across its comments on this manuscript. We would also like to thank collaboration on 2002 June 18 in its ongoing microlensing alert face (Wambsganss 1997). To find the best fitting model light program (Bond et al. 2001). About 48 hours before its time of curve, we implemented a χ2 minimisation technique using peak amplification on 2002 July 4, MOA identified this event a Metropolis Hastings Markov chain Monte Carlo procedure as a high magnification event in progress and a second alert (Ford 2003). This avoids the problem of traditional minimisa- was issued. Subsequently, an intensive campaign of follow-up tion techniques which can be fooled by false minima due to observations was carried out using a number of telescopes cir- numerical noise in the simulated light curves. cling the globe (Fig. 1). Photometry of the event was derived The best fitting theoretical curve is shown in Fig. 1. A total using difference imaging analysis applied to the CCD images. of 11 physical parameters plus 6 flux normalisation parameters This procedure was carried out separately on each passband of (one for each passband-telescope set) are required to describe each telescope resulting in six sets of time series photometry. the light curve1. The best fitting physical parameters are given These are shown in Fig. 1 where they have been normalised to in Table 1. The fitting procedure yielded correlated values for the microlensing light curve using the best fitting model (de- the limb darkening coefficients. In Table 1, we report the coef- scribed below). ficients (c , d ), which are related to the coefficients in Eq. (1) The M-shaped feature on the light curve results from the λ λ throughthe phase space rotation c = cλ cos 33◦+dλ sin 33◦ and source star entering and exiting the central caustic curve pro- λ d = dλ cos 33◦ cλ sin 33◦. In this representation, the d term duced by a binary lens. The detailed shape of the light curve λ − λ contributes a small correction to Iλ(θ)/Iλ(0). The limb darken- depends on the characteristics of the lens, the trajectory of the ing parameters allow a colour map of the face of the source star source star, and the radius and limb darkening profile of the to be drawn. This is depicted in Fig. 2 along with its passage source star. The parameters associated with the binary lens are through the caustic. the lens mass ratio M /M , the separation a of the two lenses, 1 2 Our best best fitting model light curve and the associated the Einstein radius crossing time t , the source trajectory im- E caustic correspond to a close binary lensing system. Dominik pact parameter, u , with respect to the centre-of-mass of the 0 (1999) found that qualitatively similar caustic curves can be lens, the time t when the source is closest to the centre of mass 0 produced by lensing configurations that differ greatly. Caustics of the lens, and the angle, φ, between the source trajectory and similar in shape to the one shown in Fig. 2 can be produced by the lens components. lenses comprised of wide binary systems. Our best fitting wide The source star is parametrized by its radius r and an ap- s binary solution for our observations of MOA 2003–BLG–33 propriate limb darkening model. For this we used the square- yielded a mass ratio corresponding to that of a giant planet or- root law (Claret 2000) biting a stellar mass object. While this configuration qualita- Iλ(θ) tively produced a short duration M-shaped amplification pro- = 1 cλ(1 cos θ) dλ(1 √cos θ) (1) Iλ(0) − − − − file, it could not reproduce the observed light curve profile at a level demanded by the precise, densely sampled photometry where cλ and dλ are the wavelength dependent limb darkening coefficients. 1 Blending parameters were not required because difference imag- Model binary microlensing light curves were generated nu- ing analysis removes the effects of blending due to nearby neighbours merically using an inverse ray tracing technique, implemented and the lens itself. VLTI Chile

SUSI Australia CHARA California

GI2T France Keck Hawaii

MIRA Japan

COAST England

IOTA Arizona ISI California

NPOI Arizona NPOI

Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 Schematic Two-Telescope Interferometer

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Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 the BIG picture Stellar Interferometry: An Application of Second-Order Coherence Theory (Mandel & Wolf 1995; Optical Coherence & Quantum Optics)

The Observable:The observable: Visibility Visibility

Imax - Imin V = ______Imax + Imin Imax

Imin

Thinking in the (U,V) Plane and Thinking in Intensities

on the sky what the interferometer measures

Testing Model Stellar Atmospheres Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005

Fluxes & Intensities

F (r, ν, t) = ∫ I(r, n, ν, t) n dω

Interferometry tests Spectroscopy the intensity tests the flux 1996A&A...312..160Q

“Direct” Limb darkening measurements from interferometry 1974MNRAS.167..475B �� 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 �� M4 III �� 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 678 G. Perrin et al.: Interferometric observations of α Orionis and α Herculis with FLUOR at IOTA 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, 466 (dashed line) plane-parallel ATLAS 9 model, all with correspondingOHISHI, model parameters NORDGREN, and best fitting θLD. As a reference & for the HUTTER strength of the Vol. 612 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 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 K 1.5 III 2 ∼ − corresponding reduced χν values, and finally the Rosseland angular differences are caused by systematic effects of our data calibra- diameter θRoss. The error σ(θLD) is estimated to be uniformly 0.2 mas M1 Iab ± tion or by the model structures. The squared visibility ampli- A1 V for all model fits (see text). tudes derived from the siderostat data show a systematic off- 2 Teff log g MCRoss/LD θLD χν θRoss set of about 1 σ towards larger values with respect to our best fitting curves. This is likely caused by possible small system- Spherical PHOENIX models: Fig. 3. K band visibilitiesatic for α calibrationOrionis. The χ2 is computed effects, with since first lobe thevisibilities calibration only. of our high squared visibility amplitudes in the range 0.8 0.95 derived from the 986 M. Wittkowski et al.: Limb-darkening measurements3550 0.7 with NPOI 1.3 0.9388 8.664 1.80 8.13measurements. The maxima of the averaged visibility func- frequency ∼30 and− 40 cycles/arcsec for α Orionis and tion are smaller thansiderostat the maxima of adata monochromatic was di visibilityfficultα (seeHerculis Sect. respectively)≤ 4.3). and for all data points. The results are 3500 0.7 1.3 0.9394 8.663 1.81 8.14model thus mimicking a limbThe darkening best effect. fitting Besides the Rosseland ze- the following angular for all epochs: diameters are well HR 7635, scan no. 1 HR 7635, scan no. 1 HR 7635, scan no. 1 roes of the monochromatic visibility function are replaced by 0.40 3600 0.7 1.3 0.9381 8.665 1.79 8.13minima of a few percent.constrained It is therefore important by our to take measurem aver-A7α VOrionisent. The measurements in the 0.030 π 2 0.02 aging into account. ØUD, 1st = 43.26 0.04 mas χ = 30.68 3550 0.5 1.3 0.9218 8.814 1.79 8.12 ± 2 0.30 M0 III 2nd lobe of the visibility functionØUD, all = 43 also.33 0.04 constrain mas χ the= 21. positions45 3550 1.0 1.3 0.9577 8.504 1.79 8.14 ± 0.01 3.2. Model fitting of the 1st minimum and 2ndα Herculis maximum of the visibility func- 0.020 π/2 �� 2 3550 0.7 1.0 0.9302 8.739 1.80 8.13 ØUD, 1996, 1st = 31.66 0.08 mas χ = 0.004 0.20 The visibility data have been fitted by a wide band uniform disk ± 2 tion, which is a constraintØ ofUD, 1996 the, all diameter.= 31.64 0.08 mas Thisχ constraint= 3.37 is diameter visibility curve by minimizing the function: ± 2 Plane-parallel PHOENIX model: ØUD, 1997 = 31.24 0.07 mas χ = 0.85. Triple Amplitude 0.010 independent of possible small systematic± calibration uncertain- 0.10 Closure Phase [rad] N 2 2 200 250 0 1 V M(ØUD; S i) 2 2 2 ��i The data are presented in Figs. 3 and 4, with the uniform disk χ = ties− of the V values. The(1) reduced χ values for the different 3550 0.7 / 1 8.168 1.72 8.17 N 1 σ ν Squared Visibility Amplitude 0.00 0.000 − i=1  i  | | modelfits. The agreementof the first lobe data of α Herculis to �  consideredmodels do not showthe model significant is virtually “perfect”, differences, setting the upper which level of is accu- 100 150 200 250 850 800 750 700 850 800 750 Plane-parallel700 ATLAS 12 model:   2 where σi is the estimated error on Vi and: racy of the FLUOR data to 0.2%, and confirming that the mea- uv-Radius [cycles/arcsec] Wavelength [nm] Wavelength [nm] consistent with the very similar predicted shapes of the model 2 surement error bars are reasonable. This conclusion has also 3550 0.7 / 1 8.191 1.78 8.19 ��2J1 (πφUDBik) M(Ø ; S ) = w(k)dk (2) been verified on other bright stars. Despite the correctness of UD i visibilityπφ B k functions. These similarities are expected since our band UD i the error bar estimates, the accuracy of the diameter measure- HR 7635, scan no. 2 HR 7635, scan no. 2 HR 7635, scan no. 2 � � � 0.40 Plane-parallel ATLAS 9 models: is the wide band uniform�measurement disk model� with isk dominatedthe wavenum- byments continuum may be questionable. photons, As a matter and of thefact the con- diameter Jason P. Aufdenberg Where0.030 did Half of the Sun’s Oxygenπ Go? Testing Stellar Atmosphere Models.... Dipartimento� di Fisica� Universitá di Pisa 13 April 2005 0.02 3500 0.5 / 1 8.244 1.74 8.24ber, S i = Bikeff thetinuum effective spatial forming frequency, regionBi the pro- ofestimates the atmosphere depend on the estimated is almost effective compact wavelength of the 0.30 jected baseline and ØUD the uniform disk diameter, w(k) being instrument and are directly proportional to it. The direct mea- 0.01 3500 1.0 / 1 8.243 1.73 8.24a weighting function.(see Fig. 5). surement of the effective wavelength is quite difficult and de- 0.020 π/2 If the model is a perfect representation of the source and if pends on a large number of parameters. We have assessed the 0.20 3750 0.5 / 1 8.227 1.71 8.23the error bars are well estimated then the mean of χ2 is equal quality of our diameter estimates by comparing them to some 3750 1.0 / 1 8.228 1.71 8.23to 1. The value of the χ2 can therefore be used as a criterion to independent measurements in the paragraph 4.5.3 of Perrin

Triple Amplitude 0.010 assess the validity of the error bar estimates. et al. (1998). The diameters of α Boo and α Tau measured 0.10 Closure Phase [rad] 200 250 0 The K band visibility6. Discussion data of α Orionis have and been gath- conclusionswith an accuracy of 1 part in 200 with FLUOR were statis- ered and fitted by a single visibility function. The 1996 and tically compatible with measurements obtained at the same Squared Visibility Amplitude �� 0.00 0.000 otherwise) and FDD (I = µ) model visibility functions1997 K band are dataSpherical for α Herculis are PHOENIX fitted separately. models We or differentWe wavelengths have constructed with other interferometers a spher- or by the 100 150 200 250 850 800 750 700 850 800 750 700 present the results of the fits for the first lobe data only (spatial lunar occultation technique. In the case of α Orionis and uv-Radius [cycles/arcsec] Wavelength [nm] Wavelengthshown, [nm] 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 0.40 HR 7635, scan no. 3 HR 7635, scan no. 3 HRnificantly 7635, scan no. from 3 UD and FDD models, confirming the limb- CLV by comparing it with our VLTI/VINCI measurement 0.030 π 0.02 darkening effect. All considered PHOENIX and ATLAS model of the visibility function in the second lobe. We find that 0.30 �� 0.01 CLV predictions lead to very similar model visibility functions the model predicted shape of the visibility function is con- 0.020 π/2 0.20 up to the 2nd lobe, which are all consistent with our data. sistent with our VLTI/VINCI measurements. Simultaneously,

Triple Amplitude 0.010 Hence, our data conﬁrm the model-predicted strength of the the Rosseland angular diameter derived from the model and

0.10 Closure Phase [rad] 200 250 0

Squared Visibility Amplitude 0.00 0.000 100 150 200 250 850 800 750 700 850 800 750 700 uv-Radius [cycles/arcsec] Wavelength [nm] Wavelength [nm]

0.40 HR 7635, scan no. 4 HR 7635, scan no. 4 HR 7635, scan no. 4 π 0.02 0.030 0.30 0.01 0.020 π/2 0.20

Triple Amplitude 0.010

0.10 Closure Phase [rad] 200 250 0

Squared Visibility Amplitude 0.00 0.000 100 150 200 250 850 800 750 700 850 800 750 700 uv-Radius [cycles/arcsec] Wavelength [nm] Wavelength [nm]

0.40 HR 7635, scan no. 5 HR 7635, scan no. 5 HR 7635, scan no. 5 π 0.02 0.030 0.30 0.01 0.020 π/2 0.20

Triple Amplitude 0.010

0.10 Closure Phase [rad] 200 250 0

Squared Visibility Amplitude 0.00 0.000 100 150 200 250 850 800 750 700 850 800 750 700 uv-Radius [cycles/arcsec] Wavelength [nm] Wavelength [nm]

0.40 HR 7635, scan no. 6 HR 7635, scan no. 6 HR 7635, scan no. 6 π 0.02 0.030 0.30 0.01 0.020 π/2 0.20

Triple Amplitude 0.010

0.10 Closure Phase [rad] 200 250 0

Squared Visibility Amplitude 0.00 0.000 100 150 200 250 850 800 750 700 850 800 750 700 uv-Radius [cycles/arcsec] Wavelength [nm] Wavelength [nm] Fig. 3. As Fig. 2, but for HR 7635. Fig. 1.—From left to right, squared visibility amplitudes on OB2, OB1, and OB3, triple amplitudes, and closure phases of Vega measured on 2001 May 25. Six scans were obtained. Dashed lines: Uniform-disk model with a diameter of 3.11 mas. Solid lines: Limb-darkened–disk model with a diameter of 3.22 mas. We used linear limb-darkening coefﬁcients calculated by Van Hamme (1993), with TeA 9500 K and log g 4:0. The limb-darkened–disk model reproduces the measured squared visibility amplitudes and the triple amplitudes better than the uniform-disk¼ model. ¼

visibility of the uniform disk with an angular radius of rUD The triple amplitude is the absolute value of the triple product, is written as follows:

VTP V1 V2 V3 ; 9 ¼ ð Þ 2J1 kBprUD j j j jj jj j V r : 6 UD UD kB r ð Þ ¼ �p UD � ð Þ and the closure phase is the phase of the triple product,

The visibility of the limb-darkened disk with an angular radius �c �1 �2 �3: 10 of rLD and a linear limb-darkening coefﬁcient of u(k) is written ¼ þ þ ð Þ as follows: As we can see from equation (5), if the brightness distribution

6 2J1 kBprLD of the source projected to a baseline, I(x), is symmetric with VLD rLD 1 u k x, the imaginary part of the visibility is zero, and the phase ð Þ ¼ 3 u k ½� ð Þ � kBprLD � ð Þ ( � � 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 i�1 V2 exp i�2 V3 exp i�3 : 8 the source. ¼ j j ð� Þj j ð� Þj j ð� Þ ð Þ �� New Direct Limb Darkening Measurements �� B- and A-type Supergiants & F-type stars ��

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Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 Part I Procyon, Convection, and Limb Darkening

Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 A&A 417, 751–768 (2004) Astronomy DOI: 10.1051/0004-6361:20034328 & c ESO 2004 � Astrophysics

A&A 417, 751–768 (2004) Astronomy DOI: 10.1051/0004-6361:20034328 & c ESO 2004 Line formation� in solar granulation Astrophysics

IV. [O I],O I and OH lines and the photospheric O abundance

M. Asplund1, N. Grevesse2,3, A. J. Sauval4, C. Allende PrietoLine5, and formation D. Kiselman6 in solar granulation

1 Research School of Astronomy and Astrophysics, Mt. Stromlo Observatory, Cotter Rd., Weston, ACT 2611, Australia 2 Centre Spatial de Liège, Université de Liège, avenue Pré Aily, 4031IV. Angleur-Li` [O I],Oege, BelgiumI and OH lines and the photospheric O abundance 3 Institut d’Astrophysique et de Géophysique, Université de Liège, Allée du 6 août, 17, B5C, 4000 Liège, Belgium 4 Observatoire Royal de Belgique, avenue circulaire, 3, 1180 Bruxelles, Belgium 1 2,3 4 5 6 5 McDonald Observatory and Department of Astronomy, University of Texas,M. Asplund Austin, TX, N. 78712-1083, Grevesse USA, A. J. Sauval , C. Allende Prieto , and D. Kiselman 6 The Institute for Solar Physics of the Royal Swedish Academy of Sciences, AlbaNova University Centre, 106 91 Stockholm, Sweden 1 Research School of Astronomy and Astrophysics, Mt. Stromlo Observatory, Cotter Rd., Weston, ACT 2611, Australia A&A 417, 751–768 (2004) Astronomy DOI: 10.1051/0004-6361:20034328 2 & c ESO 2004 � CentreAstrophysics Spatial de Liège, Université de Liège, avenue Pré Aily, 4031 Angleur-Liège, Belgium Asplund et al. (2004) A&A3 Institut 417, d’Astrophysique 751 Recent et de Géophysique, 3-D radiative Universit´ transfere de Li` ege,models Allée du 6 août, 17, B5C, 4000 Liège, Belgium Received 17 September 2003 / Accepted 2 December 20034 Line formation in solar granulationObservatoire Royal de Belgique, avenue circulaire, 3, 1180 Bruxelles, Belgium 5 have reduced the solar oxygen IV. [O I], O I and OH lines and the photosphericMcDonald O abundance Observatory and Department of Astronomy, University of Texas, Austin, TX 78712-1083, USA 6 abundance by nearly a factor of 2! Abstract. The solar photospheric oxygenM. Asplund1, N. abundance Grevesse2,3, A. J. Sauval4, has C. Allende been PrietoThe5, and determined D. InstituteKiselman6 for from Solar [O  Physics], O , OH of the vibration-rotation Royal Swedish and Academ OH purey of Sciences, AlbaNova University Centre, 106 91 Stockholm, rotation lines by means of a realistic1 Research School oftime-dependent, Astronomy and Astrophysics, Mt. Stromlo Observatory, 3D, Cotter hydrodynamical Rd.,Sweden Weston, ACT 2611, Australia model of the solar atmosphere. In the case of the 2 Centre Spatial de Liège, Université de Liège, avenue Pré Aily, 4031 Angleur-Liège, Belgium 3 Institut d’Astrophysique et de Géophysique, Université de Liège, Allée du 6 août, 17, B5C, 4000 Liège, Belgium O  lines, 3D non-LTE calculations4 Observatoire have Royal de Belgique, been avenue performed, circulaire, 3, 1180 Bruxelles, Belgium revealing significant departures from LTE as a result of photon losses 5 McDonald Observatory and Department of Astronomy, University of Texas, Austin, TX 78712-1083, USA in the lines. We derive a solar6 The oxygen Institute for Solar Physics abundance of the Royal Swedish Academ ofy of Sciences, log AlbaNova� University= 8 Centre,.66 106 91 Stockholm,0.05. All oxygen diagnostics yield highly consistent Sweden ReceivedO 17± September 2003 / Accepted 2 December 2003 abundances, in sharp contrast withReceived 17 September the 2003 results/ Accepted 2 December of 2003classical 1D model atmospheres. This low value is in good agreement with

Abstract. The solar photospheric oxygen abundance has been determined from [O ], O , OH vibration-rotation and OH pure measurements of the local interstellarrotation lines by means ofmedium a realistic time-dependent, and 3D, hydrodynamical nearby model of theB solar stars. atmosphere. In theThis case of the low abundance is also supported by the excellent O  lines, 3D non-LTE calculations have been performed, revealing significant departures from LTE as a result of photon losses in the lines. We derive a solar oxygen abundance of log � = 8.66 0.05. All oxygen diagnostics yield highly consistent correspondence between lines of very different line formationO ± sensitivities, and between the observed and predicted line shapes   abundances, in sharp contrast with the results of classical 1D model atmospheres.Abstract. This low value is in good agreementThe with solar photospheric oxygen abundance has been determined from [O ], O , OH vibration-rotation and OH pure measurements of the local interstellar medium and nearby B stars. This low abundance is also supported by the excellent and center-to-limb variations. Togethercorrespondence between lines with of very different the line formation corresponding sensitivities, and between the observed down-ward and predicted line shapes revisions of the solar carbon, nitrogen and neon and center-to-limb variations. Together with the corresponding down-ward revisionsrotation of the solar carbon, lines nitrogen andby neon means of a realistic time-dependent, 3D, hydrodynamical model of the solar atmosphere. In the case of the abundances, the resulting significant decrease in solar metal mass fraction to Z = 0.0126 can, however, potentially spoil the abundances, the resulting significantimpressive agreement decrease between predicted and observedin solar sound speed in metal the solar interiorO  determined masslines, from fractionhelioseismology. 3D non-LTE to Z = calculations0.0126 can, have however, been performed, potentially revealing spoil the significant departures from LTE as a result of photon losses impressive agreement betweenKey predicted words. convection – line: and formation observed – Sun: abundances – Sun: granulation sound –in Sun: photosphere speed the lines. in the We solar derive interior a solar determined oxygen abundancefrom helioseismology. of log � = 8.66 0.05. All oxygen diagnostics yield highly consistent O ± 1. Introduction abundances usingabundances, classical 1D stellar model atmospheres. in sharp In contrast with the results of classical 1D model atmospheres. This low value is in good agreement with particular in the solar case, the freedom of parameters to ob- Oxygen is the most abundant element in the Universe with a tain consistency is very restricted yet the discrepancy is present Key words. convection – line:non-Big formation Bang nucleosynthesis origin. – As Sun: a consequence, abundances oxy- in full. measurements – Sun: granulation of the – local Sun: interstellar photosphere medium and nearby B stars. This low abundance is also supported by the excellent gen plays a central role in many different fields of astrophysics Until recently the commonly adopted solar oxygen abun- ranging from supernova physics and galaxy evolution to dat- 1 dance was log �O = 8.93 0.04 (Anders & Grevesse 1989). ing stars and production of the light elements through cosmic This historicallycorrespondence high abundance± was suggested by analyses between lines of very different line formation sensitivities, and between the observed and predicted line shapes ray spallation. Yet it appears that in many crucial objects for of the forbidden [O ] 630.0 nm line (Lambert 1978) as well which accurate knowledge of the oxygen abundances is neces- as OH vibration-rotationand center-to-limb and pure rotation lines in the in- variations. Together with the corresponding down-ward revisions of the solar carbon, nitrogen and neon sary the oxygen content is hotly debated. Recent disputes re- frared (Grevesse et al. 1984; Sauval et al. 1984) using the volve around the overabundance of oxygen in metal-poor halo 1D hydrostatic Holweger-Müller (1974) semi-empirical model stars (see Asplund & Garc´ıa Pérez 2001; Nissen et al. 2002, of the solar atmosphereabundances, and LTE line formation. the On the other resulting significant decrease in solar metal mass fraction to Z = 0.0126 can, however, potentially spoil the and references therein), the Galactic radial abundance gradient hand, a much lower abundance is indicated by the permit- 1. Introduction (Rolleston et al. 2000; Cunha & Daflon 2003), and, astonish- ted high-excitation O  lines,abundances most noteworthy the IR triplet using classical 1D stellar model atmospheres. In ingly, the solar oxygen abundance. Partly these disagreements at 777 nm, whenimpressive employing the same model agreement atmosphere with between predicted and observed sound speed in the solar interior determined from helioseismology. stem from differences in the adopted input data (e.g. g f -values, non-LTE line formation. This discrepancy of about 0.2 dex be- effective temperatures Teff, surface gravities log g)butmoreim- tween different abundance indicatorsparticular have often been blamed in the solar case, the freedom of parameters to ob- Oxygen is the most abundant elementportantly they reflect in the choice the of spectral Universe lines to derive the on with over-estimated a departures from local thermodynamic Send offprint requests to: M. Asplund, 1 On the customaryKey abundance words.tain scale defined consistency as �convection(X) = 1012 is – veryline: formation restricted – yet Sun: the abundances discrepancy – Sun: is present granulation – Sun: photosphere × non-Big Bang nucleosynthesis origin.e-mail: [email protected] As a consequence,N(X)/N(H). oxy- in full. gen plays a central role in many different fields of astrophysics Until recently the commonly adopted solar oxygen abun- ranging from supernova physics and galaxy evolution to dat- dance was log � = 8.93 0.041 (Anders & Grevesse 1989). 1. Introduction O abundances using classical 1D stellar model atmospheres. In ing stars and production of the light elements through cosmic This historically high abundance± was suggested by analyses particular in the solar case, the freedom of parameters to ob- ray spallation. Yet it appears that in many crucial objects for of the forbidden [O ] 630.0 nm line (Lambert 1978) as well Oxygen is the most abundant element in the Universe with a tain consistency is very restricted yet the discrepancy is present which accurate knowledge of theJason oxygen P. Aufdenberg abundances Where did Half is neces-of the Sun’s Oxygenas OH Go? Testing vibration-rotation Stellar Atmosphere Models.... and pureDipartimento rotation di Fisica linesUniversitá in di Pisa the 13 in-April 2005 sary the oxygen content is hotly debated. Recent disputesnon-Big re- Bang nucleosynthesis origin. As a consequence, oxy- in full. gen plays a centralfrared role(Grevesse in many et di al.fferent 1984; fields Sauval of astrophysics et al. 1984) using the volve around the overabundance of oxygen in metal-poor halo Until recently the commonly adopted solar oxygen abun- ranging from1D supernova hydrostatic physics Holweger-M¨ and galaxyuller (1974)evolution semi-empirical to dat- model stars (see Asplund & Garc´ıa Pérez 2001; Nissen et al. 2002, dance was log � = 8.93 0.041 (Anders & Grevesse 1989). of the solar atmosphere and LTE line formation. On the other O ± and references therein), the Galactic radial abundanceing gradient stars and production of the light elements through cosmic This historically high abundance was suggested by analyses ray spallation.hand, Yet a it much appears lower that abundance in many crucial is indicated objects byfor the permit- (Rolleston et al. 2000; Cunha & Daflon 2003), and, astonish- of the forbidden [O ] 630.0 nm line (Lambert 1978) as well which accurateted knowledge high-excitation of the O oxygen lines, abundances most noteworthy is neces- the IR triplet ingly, the solar oxygen abundance. Partly these disagreements as OH vibration-rotation and pure rotation lines in the in- sary the oxygenat 777 content nm, when is hotly employing debated. the Recent same disputes model atmosphere re- with stem from differences in the adopted input data (e.g. g f -values, non-LTE line formation. This discrepancy of about 0.2frared dex be- (Grevesse et al. 1984; Sauval et al. 1984) using the volve around the overabundance of oxygen in metal-poor halo 1D hydrostatic Holweger-Müller (1974) semi-empirical model effective temperatures Teff, surface gravities log g)butmoreim- tween different abundance indicators have often been blamed stars (see Asplund & Garc´ıa Pérez 2001; Nissen et al. 2002, of the solar atmosphere and LTE line formation. On the other portantly they reflect the choice of spectral lines to derive the on over-estimated departures from local thermodynamic and references therein), the Galactic radial abundance gradient hand, a much lower abundance is indicated by the permit- (Rolleston et al. 2000; Cunha & Daflon 2003), and, astonish- 1 ted high-excitation12 O  lines, most noteworthy the IR triplet Send offprint requests to: M. Asplund, ingly, the solar oxygenOn the customary abundance. abundance Partly these scale disagreements defined as �(X) = 10 e-mail: [email protected] N(X)/N(H). at 777 nm,× when employing the same model atmosphere with stem from differences in the adopted input data (e.g. g f -values, non-LTE line formation. This discrepancy of about 0.2 dex be- effective temperatures Teff, surface gravities log g)butmoreim- tween different abundance indicators have often been blamed portantly they reflect the choice of spectral lines to derive the on over-estimated departures from local thermodynamic

Send oﬀprint requests to: M. Asplund, 1 On the customary abundance scale deﬁned as �(X) = 1012 e-mail: [email protected] N(X)/N(H). × 2430 GIRARD ET AL. Vol. 119

event, the two solutions yielded orbital elements that agreed al. (1992). The Irwin et al. values are those of their combined to within their estimated errors. A large number of high- astrometric and radial velocity solution, extracted from residual outliers were discarded from the all-plateÈseries their Table 6. The errors listed in the Ðrst column are solution, leaving 599 exposures for the orbit determination. formal errors from the least-squares Ðt but are to be viewed Figure 1 shows the Ðnal astrometric orbit Ðt to the plate with caution because of the large amount of data trimming 2432 GIRARD ET AL. measuresVol. 119 with parallax and proper motion subtracted. performed. For several of the elements, our values di†er Procyon: The Visual BinaryTable (P 2 lists = the derived40.82 elements of theyr) astrometric orbit, signiÐcantly from those of Irwin et al. (1992), notably in along with those obtained by Strand (1951) and by Irwin et semimajor axis, eccentricity, and period, in which there is Girard et al (2000) ApJ 119, 2428 actually better agreement with Strand (1951). @ 3.5 pc, the 13th closest As described above, the Ðnal orbital elements were then applied to the individual plate measures, and the parallax star system to the Sun was determined Ðrst with all of the plate series and Ðnally http://www.chara.gsu.edu/~thenry/ with just the USNO plates. These relative parallaxes are RECONS/TOP100.htm transformed to absolute by applying a correction based on the mean magnitude of the reference stars (see van Altena, Lee, & Hoffleit 1995 for details). The resulting absolute parallax estimates are presented in Table 3, along with that of the USNO result (Harrington et al. 1993) for their own plates and the value obtained by the Hipparcos mission A B (ESA 1997). Our value is consistent with that of USNO (within the combined uncertainties), with a formal improve- ment in accuracy to 1.5 milliarcseconds. This can be attributed to the use of the PDS to measure these excellent plates. The Hipparcos parallax di†ers from our value by approximately 1.6 times the combined uncertainties and, in fact, is in closer agreement with the USNO value. In both the USNO and Hipparcos investigations, the observation baseline was short enough that an orbit was assumed, instead of being solved for. In the case of the Hipparcos reductions, the orbital elements assumed were those of Irwin et al. (1992). What would be the e†ect on the parallax derived from the Hipparcos data had one assumed the revised orbital elements presented here? To answer this question, we rederived the Hipparcos astrometric solution from the inter- mediate astrometric data that accompany the Hipparcos catalog CD version and that are also available on the Hip- parcos Web site.10 The revised parallax solution was performed using the procedures and software described by Pourbaix & Jorissen (2000). The e†ect of adopting the new orbital parameters was minimal, decreasing the derived FIG. 1.ÈAstrometric-orbit determination for Procyon A. The points parallax by a mere 0.1 mas. As one might expect, the e†ect Massindicate = PDS 1.497 measures ± of individual0.037 exposures M⊙ and the curve shows the least-squaresA solution for the orbit. The plate measures have been trans- Massformed = to a0.602 common reference ± 0.015 system. Only M those⊙ measures which contrib- utedB to the orbit solution are included. The x- and y-coordinates are ÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈ nominally aligned with right ascension and declination. 10 At http://astro.estec.esa.nl/Hipparcos.

TABLE 2 Procyon A (F5 IV): Fundamental ParametersASTROMETRIC ORBIT ELEMENTS OF PROCYON FIG. 3.ÈWFPC2 PCGirard image ofet Procyon al (2000) A-B taken ApJ 1995.18 119, with2428 the HST . This 160 s exposure was with the F218W Ðlter. The inner 256 ] 256 portion of ± the PC frame is shown. The position of the primary was determined from a 0.11 s exposure taken moments earlier. Angular diameter = 5.45 Element0.05 mas (KervellaStrand (1951) et al Irwin. 2003) et al. (1992) Present Study

Parallax = 285.93±Semimajor 0.88 mas axis, a (arcsec)(Hipparcos:...... Perryman1.217 et al.)1.179 1.232 HST/WFPC2 PC image (160 s) A of the frame, well separated from the occulted primary. We redetermination of its visual orbit based on all available ± ^0.002 ^0.011 ^0.008 have calculated theF218W A-B separation ﬁlter based on this frame, as measures contained in the Washington DoubleRadius Star = (WDS) 2.05 0.02Eccentricity, R⊙ e ...... 0.40 0.365 0.407 well as another similar exposure from the same night. The Catalog (Worley & Douglass 1996), supplemented by an ... ^0.008 ^0.005 11 Log(g) = 3.95± 0.02Inclination, cgs (deg) ...... 35.7 31.9 31.1 position of the primary was determined by modeling the additional speckle interferometric measure made in 1996 i ± -9 -2 ^0.2 ^0.9 ^0.6 di†raction spikes beyond the apodized portion of the mask. February as part of the Yale/San Juan DoubleBolometric Star Project ﬂux = 17.8Angle of node,0.)9(deg) x 10...... W m 104.3 104.8 97.3 The secondaryÏs position was determined by Ðtting an ellip- (Horch et al. 1996). This Ðnal observation strengthens the ± ^0.3 ^1.5 ^0.3 tical paraboloid to its image, Ðtting only those pixels above relevant ephemerides separation and positionEffective angle (P.A.)TemperatureLongitude = 6516 of periastron, 87u (deg) K...... 89.8 88.8 92.2 an appropriate threshold. The mean separation from the of Vir at 1995.249, making it a true interpolation. Our new ^0.3 ^2.0 ^0.3 c Period, P (yr) ...... 40.65 40.38 40.82 two frames, at epoch 1995.09, was found to be 90.4 ^ 0.9 orbit solution for c Vir leads to a CoCo-scale estimate of ... ^0.15 ^0.06 pixels. ThisJason includes P. Aufdenberg a correction of Where]0.3 pixels did toHalf o†set of the Sun’s0A.0566 Oxygen^ 0A.0006 Go? pixel Testing~1. Details Stellar of A thistmosphere calculation, Models.... includ- DipartimentoPeriastron passage, di FisicaT (yr) ...... Universitá 1927.6di Pisa 13 April1967.86 2005 1967.97 bias from the sloping background at the position of the ing the new visual orbit solution of c Vir, are described in (1968.3) ^0.16 ^0.05 secondary image, caused by the wings of the primary image. the Appendix. With the above value for the instrument The 0.9 pixel uncertainty is dominated by the Ðt to the scale, the CoCo separation of Procyon A-B becomes 5A.12 overexposed di†raction spikes of the primary. It was esti- ^ 0A.07(1995.09; P.A. \ 41¡.0 ^ 1¡.0). mated from the scatter in the Ðtted positions as derived using various Ðtting techniques and is consistent with the 4.2. HST W FPC2 Observations di†erence in separation values, 0.6 pixels, derived from the In 1995 March, a series of observations were made of two separate frames. Procyon using the WFPC2 Planetary Camera (PC) of the The pixel scale of the CoCo-NSFCAM combination is HST with the intention of accurately measuring the A-B not well known, and thus we have deduced a nominal scale separation. One such PC frame, a 160 s exposure taken with based on four short exposures of c Vir (STF 1670) observed the F218W Ðlter, is shown in Figure 3. In it, the secondary is in 1995 March with the same instrument. Unfortunately, readily measurable and well separated from the heavily the orbit of this bright (V \ V \ 3.4) binary is also not A B well known, having been last updated (coincidentally) by ÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈ Strand (1937) in 1937. Thus, we have undertaken a 11 Electronic version available at http://aries.usno.navy.mil/ad/wds. letters to nature

...... There have been at least seven independent reports of p-mode detections in Procyon since 1986, based on high-resolution ground- No stellar p-mode oscillations in based spectroscopy of the star2–8. The longest of these data sets8 space-based photometry of Procyon spanned 13 nights, covering 131 h (or about 42%) of the total timespan. All these observers found excess power in the Fourier Jaymie M. Matthews1, Rainer Kusching1, David B. Guenther2, spectra of their data in a frequency range of about 0.5–1.5 mHz 1 3 4 (periods from about 33 to 11 min), with peak radial velocity (RV) Gordon A. H. Walker , Anthony F.J. Moffat , Slavek M. Rucinski , 21 Dimitar Sasselov5 & Werner W. Weiss6 amplitudes around 1 m s . Large-amplitude p-modes had also been predicted for Procyon by theorists11 as early as 1983. More 1Department of Physics and Astronomy, University of British Columbia, 6224 recent calculations of stochastic excitation of p-modes by turbulent Agricultural Road, Vancouver V6T 1Z1, Canada convection in the star12,13 suggest velocity amplitudes of about 2 Institute for Computational Astrophysics, Department of Astronomy and Physics, 1.3 m s21 and luminosity amplitudes of about 60–70p.p.m. St Mary’s University, Halifax, Nova Scotia B3H 3C3, Canada 3 (ref. 12). Even the most conservative estimate based on observed De´partement de physique, Universite´ de Montreál, CP 6128, Succ. Centre-Vile, RVamplitudes and theoretical light-to-RVamplitude ratios predicts Montreál H3C 3J7, Canada 4Department of Astronomy and Astrophysics, University of Toronto, 60 St George a luminosity amplitude of 20 p.p.m. (ref. 14). Therefore, this star Street, Toronto M5S 3H8, Canada qualified as a prime target for the MOSTmission, whose oscillation 5Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, detection limit for Procyon would be a few p.p.m. Massachusetts 02138, USA We present in Fig. 2a the MOST photometry of Procyon collected 6Institut fu¨r Astronomie, Universita¨t Wien, Tu¨rkenschanzstrasse 17, A-1180 Wien, from 8 January to 9 February 2004. Exposures of 0.9 s each were Austria obtained at an average rate of 6 times per minute with only 7 h of ...... gaps during 32 days — an unprecedented duty cycle of 99%. The Pressure-driven (p-mode) oscillations at the surface of the Sun, best ground-based global campaigns to monitor stars other than the resulting from sound waves travelling through the solar interior, Sun, such as the Whole Earth Telescope, have never achieved duty are a powerful probe of solar structure, just as seismology can cycles of more than 35% in one month. The MOST photometry is reveal details about the interior of the Earth. Astronomers have 1 contaminated by scattered Earthlight entering the instrument focal hoped to exploit p-mode asteroseismology in Sun-like stars to plane, but this contributes less than 1% of the stellar signal from test detailed models of stellar structure and evolution, but the Procyon. This stray light is modulated by the known orbital period observations are extremely difficult. The bright star Procyon has of the satellite (P 101.413 min) so it is easy to distinguish from been considered one of the best candidates for asteroseismology, orb ¼ 2–8 other sources of variability. The data presented here have undergone on the basis of models and previous reports of p-modes only a preliminary reduction which is more than sufficient to detected in ground-based spectroscopy. Here we present a search support our conclusions. for p-modes in 32 days of nearly continuous photometric satellite- based observations of Procyon. If there are p-modes in Procyon, they must have lifetimes less than 2–3 days and/or peak amplitudes <15 parts per million, which defy expectations from the Sun’s oscillations and previous theoretical predictions. Target selection for future planned asteroseismology space missions may need to be reconsidered, as will the theory of stellar oscillations. The MOST (Microvariability and Oscillations of Stars) micro- satellite9 is Canada’s first orbiting space telescope and its first scientific research satellite to be launched in over 30 years. It houses a telescope of aperture 15 cm feeding a CCD photometer through a broad-band optical filter (350–700 nm). From the vantage point of its polar Sun-synchronous orbit (altitude, 820 km), MOST can monitor stars in a band of sky about 548 wide for up to 8 weeks without interruption. Starlight from each Primary Science Target (brighter than V-band magnitude V < 6) is projected onto the CCD as a fixed pupil image of the telescope covering about 1,500 pixels for high stability. The satellite was launched on 30 June 2003, and letters to naturecommissioned in the latter half of 2003. MOST is at present the only instrument capable of reaching photometric precision of a few parts Figure 2b presents the MOSTper million data binned (p.p.m.) into in averages stars other of the thansignificant the Sun. peaks with the characteristicFigure 1 The roughly location equal of Procyon spacings in a plot of of luminosity versus effective temperature. The Principal goals of the MOST mission include1 the detection of basic physical parameters of Procyon are well specified: distance d 3.50 ^ 0.01 pc orbital period to better show the variability of the star on timescales p-modes . Our detection thresholds for coherent oscillations are ¼ of hours and days. To gaugep-modes instrumental in Sun-like drifts, we starsshow inand Fig. detection 2c about of 3 reflected p.p.m. at frequencies light from abovebased 2 mHz, on a Hipparcos and 7 p.p.m. parallax, between mass M 0.5 1:50 ^ 0:04 M ( from measurements of the 10 21 ¼ 22 the light curve of a fainterexoplanets Secondary Science. MOST’s Target first (HD Primary 61199, Scienceand 2.0 Target mHz. was This Procyon, is the best a photometricbinary orbit precision, radius R so2 far:05 reported^ 0:03 R( determined by optical interferometry , and letters to nature 15–17 ¼ 23 V 8.2) in the Procyon field,visual binned binary in system the same containing way. We an have F5 subgiantin the literature primary (Vby about0.38) antemperature order of magnitude.T eff 6530 ^ 50 K from model fits to its spectrum .(M (; R( and L( ¼ ¼ ¼ discovered this star to be a multiperiodicand a white dwarf variable secondary, of the d Scuti 5 0 away type, and moreIn the than Sun, 10 individual mag fainter. p-modeare oscillations respectively the have solar finite mass, lifetimesradius and luminosity.) Three different stellar evolutionary whoseFigure main 2b presents pulsation the MOSTperiod(Here data is 57 binned wemin, refer withinto averages to an amplitudethe of primary the ofsignificant only Procyontypically peaks A with simply of the days characteristic as to Procyon.)weeks18 roughlyand dotracks equalnot remain24 spacings, all with coherent. primordial of However, compositions in of (X, Z) (0.71, 0.02) (where X and Z are the 1 19 ¼ 1.5orbital millimagnitudes period to better (1,500show the p.p.m.).Procyon variability It occupiesalso of the exhibits star on an timescales a interesting 3.9-day period-p-modes position.extended Our in detection the standard photometry thresholds diagramof for the coherent Sunmass oscillations from fractions space of are hydrogen, a characteristic and all elements heavier than helium, respectively) all pass of hours and days. To gauge instrumental drifts, we show in Fig. 2c about 3 p.p.m. at frequencies above 2 mHz, and 7 p.p.m. between 0.5 icity which we attribute to rotation or binarity. The slow trend p-mode pattern of roughly equal spacing clearly25 rises above the the light curve of a fainter Secondaryof luminosity Science Target versus (HD surface 61199, temperature.and 2.0 mHz. This In is Fig. the best1 we photometric show that precisionthrough so far the reported observed position of Procyon in the luminosity–temperature plane. The commonV 8.2) in to the Fig. Procyon 2b and field, cevolutionary binnedin the in last the third same tracks of way. the for We datathree have has differentin been the literature modelsnoise15–17 with ofby solar aboutamplitudes composition an order up of to magnitude.,6models p.p.m. are To part test of how an extensive sensitive grid is of the models being computed to study the oscillation removed¼ in the subsequent analysis. The variations seen in Fig. 2b at complete MOST data set to oscillations of finite26 lifetimes, we 9 discovered this star to be a multiperiodicall pass variable through of the thed Scuti observed type, positionIn the Sun, of individual Procyon. p-mode All are oscillations viable haveproperties finite lifetimesof stars . The 1.45 M ( model is 2.47 10 yr 2.47 gigayear (Gyr) old and timescales less than a few days appear intrinsic to Procyon. A shorter embedded in18 the raw data modulated sine waves of known fre- £ ¼ whose main pulsation period ismodels, 57 min, with yet an each amplitude is in a of distinct only typically evolutionary of days to phase weeks withand do potentially not remain coherent.has exhausted However, its in core of hydrogen. The 1.50 M model is 2.20 Gyr old, and still has segment1.5 millimagnitudes of the Procyon (1,500 p.p.m.). light curve It also spanning exhibits a 3.9-dayonly 0.3 period- day is plottedextended photometryquencies, whose of the amplitudes Sun from space were19, allowed a characteristic to grow and decay with a ( measurable differences in the eigenspectra or oscillation driving and a mass fraction of hydrogen of X 0.005 in its core. The 1.56 M model is 1.74 Gyr inicity Fig. which 2d, towe indicate attribute the to rotation point-to-point or binarity. scatter. The slow trend p-mode patternrange of of roughly lifetimes, equal and spacing whose clearly phases rises above were the random. Wecore then¼ ( common to Fig. 2b and c in thedamping last third characteristics. of the data has been It wasnoise hoped with amplitudes that observing up to ,6 p.p.m. p-mode To test howold, sensitive with X core is the0.138. The depth of the convective envelope in Procyon severely tests Figure 3a is the Fourier amplitude spectrum of the entire processed these data in the same way as¼ before. As expected, removed in the subsequentProcyon: analysis.frequencies The variations would seen enableAstroseismology in Fig. 2b us at to distinguishcomplete MOST between data set the to possibilities. oscillations Target of finitethe model lifetimes, physics we because it is extremely shallow. unbinnedtimescales less data than set. a few The days peaks appear at intrinsic very low to Procyon. frequencies A shorter are dueembedded to oscillations in the raw data which modulated have largersine waves amplitudes of known for fre- longer times are the variations seen in Fig. 2bNATURE (except | VOL for 430 the | 1 JULY trend 2004 which | www.nature.com/nature has been moreletters easily recovered to nature in the Fourier spectrum of the data (see 51 segmentletters of the to Procyon nature light curve spanning only 0.3 day is plotted quencies, whose amplitudes© were 2004 allowedNatur toe P growublis andhing decayGrou withp a removed).in...... Fig. 2d, to The indicate notches...... the point-to-point in the spectrum...... scatter. correspond to the frequen-range of lifetimes,Supplementary and whose Information). phases were We random. show We in Fig. then 3b a conservative ...... There have been at least sevencies independentFigure of the 3a orbit reports is the and of Fourier p-mode its harmonics, amplitude where spectrum power of duetheThere entire to modulated haveprocessed been at leastcase these seven of independentdata modes in the with reports same lifetimes of way p-mode as ofbefore. only As 3 days expected, and maximum input detections in Procyon since 1986, based on high-resolution ground- detections in Procyon since 1986,strayunbinnedNo based stellar on light high-resolution data has p-modeset. been The ground- peaks removed. oscillations at very Figure low frequencies in 3a does are not due reveal to oscillations any amplitudes which2–8 have between larger amplitudes 15 and8 30 for p.p.m., longer timeswhere are the frequencies are No stellar p-mode oscillations in based spectroscopy of the star2–8. The longest of these data sets8 based spectroscopy of the star . The longest of these data sets the variations seen in Fig. 2b (except for the trend which hasspanned been 13 nights,more covering easilythose recovered 131 hreported (or about in the in 42%) Fourier the of most the spectrum total extensive of the RV data studies (see6,8. The presence of space-based photometry of Procyon spanned 13 nights, covering 131removed).space-based h (or about The 42%) notches of photometry the in total the spectrum of correspond Procyon to the frequen- Supplementary Information). We show in Fig. 3b a conservative timespan. All these observers found excess power in the Fourier timespan. All these observersp-modes found excess would power be in recognisable the Fourier in our data even by eye for modes cies of the orbit and1 its harmonics,1 where power2 due to modulatedspectra of theircase data of in modes a frequency with range lifetimes of about of only 0.5–1.5 3 mHzdays and maximum input 1 1 2 spectra of their data in a frequencyJaymie range M. Matthews of about, Rainer 0.5–1.5 Kusching mHz , David B. Guenther , Jaymie M. Matthews , Rainer Kusching , David B. Guenther , stray light has1 been removed.3 Figure 3a does4 not reveal(periods any fromamplitudes about 33 towith 11 between min), lifetimes with 15 peak and as radial 30 short p.p.m., velocity as (RV)this. where the frequencies are 1 3 4 (periods from about 33 to 11 min),Gordon with A. peak H. Walker radial, Anthony velocity F.J. (RV) Moffat , Slavek M. Rucinski , 21 Gordon A. H. Walker , Anthony F.J. Moffat , Slavek M. Rucinski , 21 5 6 6,8 5 6 Dimitar Sasselov & Werner W. Weiss amplitudes around 1 m s .Perhaps Large-amplitude the average p-modes hadmode also lifetimes in Procyon are indeed much Dimitar Sasselov & Werner W. Weiss amplitudes around 1 m s . Large-amplitude p-modes had also those reported in the most11 extensive RV studies . The presence of 11 been predicted for Procyon by theorists as early as 1983. More been predicted for Procyon by1 theorists asa early as 1983. More p-modes wouldshorter be recognisable than in the in Sun? our data To explore even by eye this, for we modes divided the data into 1 Department of Physics and Astronomy, University of British Columbia, 6224 recent calculations of stochastic excitation of p-modes by turbulent Department of Physics and Astronomy, University of British Columbia, 6224 recent calculations of stochastic excitation4 of p-modes by turbulent Agricultural Road, Vancouver V6T 1Z1, Canada with lifetimes12,13 as short as this. Agricultural Road, Vancouver V6T 1Z1, Canada 12,13 convection in the star varioussuggest velocity subsets amplitudes to look of for about recurring frequencies and/or an overall convection in the star suggest2Institute velocity for Computational amplitudes Astrophysics, of about Department of Astronomy and Physics, 21 2Institute for Computational Astrophysics, Department of Astronomy and Physics, 21 0 1.3 m s and luminosityPerhaps the amplitudes average mode of about lifetimes 60–70p.p.m. in Procyon are indeed much 1.3 m s and luminosity amplitudesSt Mary’s University, of about Halifax, 60–70p.p.m. Nova Scotia B3H 3C3, Canada pattern formed by intermittent frequencies seen in short data St Mary’s University, Halifax, Nova Scotia B3H 3C3, Canada a (ref. 12). Evenshorter the most than conservative in the Sun? estimate To based explore on observedthis, we divided the data into (ref. 12). Even the most conservative3De´partement estimate de physique, based on Universite observed´ de Montreál, CP 6128, Succ. Centre-Vile, 3De´partement de physique, Universite´ de Montreál, CP 6128, Succ. Centre-Vile, –44 RVamplitudes and theoreticalsegments. light-to-RVamplitude We repeated ratios thepredicts experiments with simulations (sampled RVamplitudes and theoretical light-to-RVamplitudeMontreál H3C 3J7, Canada ratios predicts various1. 32 subsetsdays, to99% look fortemporal recurring frequenciescoverage and/or an overall Montreál H3C 3J7, Canada 4 a luminosity amplitude ofidentically 20 p.p.m. (ref. 14). to Therefore, the MOST this star data) containing pink noise, whose 4 Department0 of Astronomy and Astrophysics, University of Toronto, 60 St George Department of Astronomy and Astrophysics, University of Toronto, 60 St George a luminosity amplitude of 20 p.p.m. (ref.–8 14). Therefore,0.2 % this star pattern formed by intermittent frequencies seen in short data Street, Toronto M5S 3H8, Canada qualified as a prime target for the MOSTmission, whose oscillation Street, Toronto M5S 3H8, Canada qualified as a prime target for the MOSTmission, whose oscillation segments.2. p-mode Weamplitude repeated peak amplitude thescales experiments inversely < with 15 with simulationsparts frequency. per (sampled million This is roughlyor what is 5Harvard-Smithsonian–4 Center for Astrophysics, 60 Garden Street, Cambridge, detection limit for Procyon would be a few p.p.m. 5Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, detection limit for Procyon would be a few p.p.m.b identicallyexpected to the MOST in the data) light containing variations pink due noise,to granulation, whose the pattern of Massachusetts 02138,0.2 % USA We present in Fig. lifetimes 2a the MOST photometryless than of Procyon2-3 days collected Massachusetts 02138, USA We present in Fig. 2a the MOST6 photometry–84 of Procyon collected 6 Institut fu¨r Astronomie, Universita¨t Wien, Tu¨rkenschanzstrasse 17, A-1180 Wien, from 8 January to 9 February 2004. Exposures of 0.9 s each were Institut fu¨r Astronomie, Universita¨t Wien, Tu¨rkenschanzstrasse 17, A-1180 Wien, from 8 January to 9 February 2004. Exposures of 0.9 s each were amplitude scales inversely with frequency. This is roughly what is Austria Austria b obtained at anexpected average rate in of the 6 times light per variations minute with due only to 7granulation, h of the pattern of obtained at an average rate of 6...... times per0 minute with only 7 h of ...... 4 gaps during 32 days — an unprecedented duty cycle of 99%. The gaps during 32 days — an unprecedentedPressure-driven duty cycle (p-mode) of 99%. oscillations The at the surface of the Sun, Pressure-driven (p-mode) oscillations at the surface of the Sun, best ground-based global campaigns to monitor–4 stars other than the best ground-based global campaigns to monitor stars other than the resulting from sound waves travelling through the solar interior, resulting0 from sound waves travelling through the solar interior, Sun, such as the Whole Earth Telescope, have never achieved duty Sun, such as the Whole Earth Telescope,are a powerful have never probe achieved of solar duty structure, just as seismology can are a powerful probe of solar structure, just as seismology can cycles of more than 35% in one month.–4–8 The MOST photometry is cycles of more than 35% in one month. The MOST photometry is reveal details about the interior of the Earth. Astronomers have reveal details about the interior of the Earth. Astronomers have contaminated by scattered Earthlight entering the instrument focal contaminated by scattered Earthlight entering the instrument focal 1 hoped to exploit p-mode asteroseismology1 in Sun-like stars to hoped–8 to exploit p-mode asteroseismology in Sun-like stars to plane, but this contributes less than 1% of the stellar signal from plane, but this contributes less thantest detailed 1% of thec models stellar of signal stellar from structure and evolution, but the test detailed models of stellar structure and evolution, but the Relative signal (1,000 × p.p.m.) 4 Procyon. This stray light is modulated by the known orbital period Procyon. This stray light is modulatedobservations by thec known are extremely orbital period difficult. The bright star Procyon has observations are extremely difficult. The bright star Procyon has of the satellite (Porb 101.413 min) so it is easy to distinguish from of the satellite (P 101.413 min) soRelative signal (1,000 × p.p.m.) it4 is easy to distinguish from been considered one of the best candidates for asteroseismology, orb ¼ been considered0 one of the best candidates for asteroseismology, other sources of variability.¼ The data presented here have undergone other sources of variability. The data presented here have undergone 2–8 on the basis of models and previous reports2–8 of p-modes on the0 basis of models and previous reports of p-modes only a preliminary reduction which is more than sufficient to only a preliminary reduction whichdetected–4 is in more ground-based than sufficient spectroscopy. to Here we present a search detected in ground-based spectroscopy. Here we present a search support our conclusions. support our conclusions. for p-modes in 32 days of nearly continuous photometric satellite- for p-modes–4 in 32 days of nearly continuous photometric satellite- based observations–8 of Procyon. If there are p-modes in Procyon, based observations of Procyon. If there are p-modes in Procyon, –8 they must have lifetimes less than 2–3 days and/or peak ampli- they must have lifetimes less than 2–3 days and/or peak ampli- < 14701470 1475 1475 1480 1480 1485 1485 1490 1490 1495 14951500 1500 tudes <15 parts per million, which defy expectations from the tudes 15 parts per million, which defy expectations from the Sun’s oscillations and previous theoretical predictions. Target Sun’s oscillations and previous theoretical predictions. Target selection for future planned asteroseismology space missions selection for future planned asteroseismology space missions d may need tod be reconsidered, as will the theory of stellar may need to be reconsidered, as will the theory of stellar 11 oscillations. oscillations. The MOST (Microvariability and Oscillations of Stars) micro- The MOST (Microvariability and Oscillations of Stars) micro- 0 90 satellite9 is Canada’s first orbiting space telescope and its first satellite is Canada’s first orbiting space telescope and its first scientific research satellite to be launched in over 30 years. It houses scientific–1 research satellite to be launched in over 30 years. It houses a telescope of aperture 15 cm feeding a CCD photometer through a a telescope–1 of aperture 15 cm feeding a CCD photometer through a broad-band optical filter (350–700 nm). From the vantage point of broad-band optical filter (350–700 nm). From the vantage point of 1482.55 1482.60 1482.65 1482.70 1482.75 its polar Sun-synchronous orbit (altitude, 820 km), MOST can its polar Sun-synchronous orbit (altitude, 820 km), MOST can 8 monitor stars in a band of sky about 548 wide for up to 8 weeks monitor stars in a1482.55 band of sky 1482.60 aboutJD–JD2000 54 wide 1482.65 (d) for up to 1482.70 8 weeks 1482.75 Figure 3 The Fourier amplitude spectrum of Procyon and the detection sensitivity of the without interruption. Starlight from each Primary Science Target without interruption. Starlight from each Primary Science Target (brighter than V-band magnitude V 6)JD–JD2000 is projected onto (d) the CCD (brighter than V-band magnitude V < 6) is projected onto the CCD Figure 2 The light curve of Procyon. a,< All 231,524 0.9-s exposures spanning 32 days. MOST data. aFigure, The spectrum 3 The ofFourier the photometry amplitude presented spectrum in Fig. of Procyon 2a; the maximum and the detection sensitivity of the as a fixed pupil image of the telescope covering about 1,500 pixels Foras a the fixed first pupil21 days, image exposures of the were telescope obtained covering 4 times aboutper minute; 1,500 for pixels the remaining time, frequency corresponds to a period of 2.8 min. (There is no excess power above the noise Figurefor high 2 stability.The light The curve satellite of Procyon. was launcheda, All 231,524 on 30 June 0.9-s 2003, exposures and spanning 32 days. MOST data. a, The spectrum of the photometry presented in Fig. 2a; the maximum for high stability. The satellite was launched on 30 June 2003, and Jasonat twice P. this Aufdenberg rate. The mean signal Where level perdid exposure Half of is 1.34the Sun’s107 ADU Oxygen (analogue-to- Go? Testingout to the Stellar Nyquist frequency Atmosphere of about 50 Models.... mHz.) The dashed Dipartimento vertical lines mark di the Fisica range inUniversitá di Pisa 13 April 2005 Forcommissioned the first 21 indays, the latterexposures half of were 2003. obtained MOST is 4 at times present per£ the minute; only for the remaining time, frequency corresponds to a period of 2.8 min. (There is no excess power above the noise commissioned in the latter half of 2003. MOST is at present the only digital units). The gaps in this light curve total only 7 hours owing to brief interruptions of which p-modes were reported by previous observers. The inset is an expanded view of instrument capable of reaching photometric precision of a few parts atinstrument twice this capable rate. The of reaching mean signal photometric level per precision exposure of a is few 1.34 parts 107 ADU (analogue-to- out to the Nyquist frequency of about 50 mHz.) The dashed vertical lines mark the range in satelliteper million pointing. (p.p.m.)b, The in same stars data, other binned than the at intervals Sun. of the satellite orbitalFigure period 1 The of locationthat of frequency Procyon in a range. plot of luminosity The notches versus in effective the noise temperature. level of theThe spectra are where power has per million (p.p.m.) in stars other than the Sun. Figure 1 The location of Procyon in a plot of luminosity versus effective temperature. The £ digital101.413Principal units). min. goalsWe The estimate gaps of the in the MOST this r.m.s. light missionpoint-to-point curve include total uncertainty only the 7 detectionhours in the owing unbinned of tobasicbrief data physical to interruptions be parametersbeen filtered of of Procyon atwhich the are satellite well p-modes specified: orbital distancewere frequency reportedd 3.50 and^ by its0.01 harmonicsprevious pc observers. to remove known The inset stray is an expanded view of Principal goals of the MOST mission include the detection of basic physical parameters of Procyon are well specified: distance d 3.50 ^ 0.01 pc ¼ p-modes in Sun-like¼ stars and detection of reflected light from based on a Hipparcos parallax, mass M 1:50 ^ 0:04 M from measurements of the based on a Hipparcos parallax, mass M satelliteabout1 50 ^ 5000 pointing.04 p.p.m.;M from inbthe measurements, The binned same data, of thedata, 75 p.p.m., binned so most at intervals of the structure of the seen satellite in Fig. orbital 1b is periodlight artefacts. of thatb, The frequency spectrum range. of( the photometry The notches in which in the artificial noise level signals of were the spectra are where power has p-modes in Sun-like stars and detection of reflected light from : : 10 ( 21 ¼ 22 10 ¼exoplanets . MOST’s first Primary Science Target was Procyon, a binary orbit , radius R 2:05 ^ 0:03 R determined by optical interferometry , and exoplanets . MOST’s first Primary Science Target was Procyon, a binary orbit21, radius R 2:05 ^ 0:03 R determined by optical interferometry22, and ( 101.413signal,( not min. noise. We (We estimate are not yet the at r.m.s. the limit point-to-point of Poisson statistics, uncertainty for which in the the unbinned r.m.s. dataembedded to¼ be priorbeen to data filtered processing. at the The satellite23 signals orbital simulate frequency stochastically and excited its harmonics p-modes to remove known stray ¼ visual binary system23 containing an F5 subgiant primary (V 0.38) temperature T eff 6530 ^ 50 K from model fits to its spectrum .(M (; R( and L( visual binary system containing an F5 subgiant primary (V 0.38) temperature T eff 6530 ^ 50 K from modelvalues fits are to its about spectrum 275 and.(M ( 11; R p.p.m.,( and L respectively.)( c, The light curve¼ of a fainter star in the ¼with a lifetime of only 3 days and peak amplitudes between 15 and 30 p.p.m. The ¼ ¼ aboutand a white500 p.p.m.; dwarf secondary, in the binned 5 0 away data, and 75 more p.p.m., than so 10 most mag of fainter. the structureare respectively seen in the Fig. solar 1b mass, is radiuslight and artefacts. luminosity.) Threeb, The different spectrum stellar evolutionary of the photometry in which artificial signals were and a white dwarf secondary, 5 0 away and more than 10 mag fainter. are respectively the solar mass, radius and luminosity.) Three different stellar evolutionary 6,8 Procyon(Here we field, refer binned to in the the primary same way Procyon as b. These A simplydata show as an Procyon.) intrinsic 3.9-daytracks period24, all with primordialfrequencies compositions embedded of (X, areZ) from(0.71, previously 0.02) (where reportedX and Z detectionsare the . The inset is at the same (Here we refer to the primary Procyon A simply as Procyon.) tracks24, all with primordial compositions ofsignal, (X, Z) not(0.71, noise. 0.02) (where (We areX and notZ are yet the at the limit of Poisson statistics, for which the r.m.s. embedded¼ prior to data processing. The signals simulate stochastically excited p-modes Procyon¼ occupies an interesting position in the standard diagram mass fractions of hydrogen and all elements heavier than helium, respectively) all pass Procyon occupies an interesting position in the standard diagram mass fractions of hydrogen and all elementsvisible heavier in the than light helium, curve, respectively) and a main all pass pulsation period of 57 min, which is responsible for scale as in a. The short horizontal bars represent the input amplitudes, with vertical lines valuesof luminosity are about versus 275 surfaceand 11 p.p.m.,temperature. respectively.) In Fig. 1c we, The show light that curvethrough of a fainter the observed star in25 position the ofwith Procyon a in lifetime the luminosity–temperature of only 3 days plane. and The peak amplitudes between 15 and 30 p.p.m. The of luminosity versus surface temperature. In Fig. 1 we show that through the observed25 position of Procyonmuch in the of luminosity–temperature the apparent scatter. plane.d, An The expanded view of 0.3 day of the data in Fig. 3a and b. extending above to make the input frequency spacings more obvious. Enough of the input models are part of an extensive grid of models being computed to study the oscillation 6,8 models are part of an extensive grid of modelsProcyonevolutionary being field, computed tracks binned to study for in three the the oscillation same different way models as b. These of solar data composition show an intrinsic 3.9-day period frequencies embedded are from previously reported detections . The inset is at the same evolutionary tracks for three different models of solar composition The small open squares are the unbinned photometry; the large filled squares connected 26mode frequencies – even with9 both lifetimes and amplitudes lower than expected from 26 all pass through9 the observed position of Procyon. All are viable properties of stars . The 1.45 M ( model is 2.47 10 yr 2.47 gigayear (Gyr) old and all pass through the observed position of Procyon. All are viable properties of stars . The 1.45 M ( modelvisible isby 2.47 straight in10 the linesyr light2.47 are curve,thegigayear data (Gyr) and binned old a and main at the pulsation orbital period. period JD, of Julian 57 min, day. which is responsibletheory for – are recoveredscale as to in£ distinguisha. The¼ short the horizontal simulation from bars the represent real data the in a input. amplitudes, with vertical lines models,£ yet each¼ is in a distinct evolutionary phase with potentially has exhausted its core of hydrogen. The 1.50 M ( model is 2.20 Gyr old, and still has models, yet each is in a distinct evolutionary phase with potentially has exhausted its core of hydrogen. The 1.50 M ( model is 2.20 Gyr old, and still has muchmeasurable of the differences apparent scatter. in the eigenspectrad, An expanded or oscillation view of driving0.3 day and of thea data mass in fraction Fig. of 3a hydrogen and b. of X coreextending0.005 in its above core. The to 1.56 makeM ( themodel input is 1.74 frequency Gyr spacings more obvious. Enough of the input measurable differences in the eigenspectra or oscillation driving and a mass fraction of hydrogen of X core 0.00552 in its core. The 1.56 M ( model is 1.74 Gyr ¼ NATURE | VOL 430 | 1 JULY 2004 | www.nature.com/nature ¼ damping characteristics. It was hoped that observing p-mode© 200old,4 withNatXucorere Pu0.138.blishi Theng depthGrou ofp the convective envelope in Procyon severely tests damping characteristics. It was hoped that observing p-mode old, with X core 0.138. The depth of theThe convective small envelope open insquares Procyon severely are the tests unbinned photometry; the large filled squares connected¼ mode frequencies – even with both lifetimes and amplitudes lower than expected from ¼ frequencies would enable us to distinguish between the possibilities. the model physics because it is extremely shallow. frequencies would enable us to distinguish between the possibilities. the model physics because it is extremelyby shallow. straight lines are the data binned at the orbital period. JD, Julian day. theory – are recovered to distinguish the simulation from the real data in a. NATURE | VOL 430 | 1 JULY 2004 | www.nature.com/nature 51 NATURE | VOL 430 | 1 JULY 2004 | www.nature.com/nature 51 © 2004 Nature Publishing Group © 2004 Nature Publishing Group 52 NATURE | VOL 430 | 1 JULY 2004 | www.nature.com/nature © 2004 Nature Publishing Group 3-D Hydrodynamical Simulations of Procyon

Allende Prieto et al (2002) ApJ 567, 544 No. 1, 2002 CONVECTION IN SPECTRUM OF PROCYON 551

τ 3-D versus 1-D = 0.3 No. 1, 2002 Center-to-LimbCONVECTION Proﬁles IN SPECTRUM OF PROCYON 561

applies to the solar case. Other stars and elements may 3-D show larger di†erences. 3. Procyon is marginally deÐcient in iron compared to the Sun by about 0.05 dex.

7. CENTER-TO-LIMB VARIATION τ= 1.0 Prediction:One of the applications of model atmospheres is to derive 1-D (MARCS) limb-darkening laws. These are required, for example, to 1.6correct % interferometric measurements of stellar angular angulardiameters size and a†ect directly the otherwise fully empirical calibrations of e†ective temperature against color indices difference(see, e.g., Mozurkewich et al 1991). Here we compare the center-to-limb variation predicted betweenby homogeneous models and the three-dimensional simulations. Taking the emerging intensity for di†erent angles at τ= 3.0 1 micronall di†erent spatial locations, we produce a spatially and 3-D time-averaged intensity for di†erent ray inclinations. Figure and19 shows the predicted limb darkening for the one- dimensional and three-dimensional models. Two wave- 450lengths nm were selected, 4500 and 10000A� . Strong di†erences are obvious in the plot, the limb darkening being markedly nonlinear for the inhomogeneous model. We have Ðtted the data to third-order polynomials by regular least-squares Ðts τ (see eq. [6] in Hanbury Brown, Davis, & Allen 1974), and = 10 1-D (MARCS) the results correspond to the solid (three-dimensional) and dashed (one-dimensional) lines in Figure 19. Lower order polynomials were inappropriate for the three-dimensional model atmosphere. Hanbury Brown et al. (1974) calculated the interferometric correlation factor associated with this particular limb-darkening law. At 4500A� , their determined FIG. 19.ÈLimb darkening for the three-dimensional (rhombi) and one- dimensional (asterisks) models. Third-order polynomials have been Ðtted angular diameter for a uniformly emitting disk should be by regular least-squares Ðts to the data. corrected by factors of 1.081 and 1.064 for the one- dimensional and three-dimensional cases, respectively. The one-dimensional correction obtained from an older version FIG. 7.ÈTemperature on surfaces of equal optical depths in the convection simulations of Procyon (left panels) and the Sun (right panels). The surfaces dimensional models need to be improved, there are prob- of MARCS (Gustafsson et al. 1975) used by Mozurkewich correspond to continuum optical depths at 5000A� of 0.3, 1, 3, and 10 from top to bottom. It should be noted that these iso-tau surfaces are highly corrugated ably other important error sources in our calculated line et al. (1991) was 1.07 at 4500A� . The radius and the e†ective and therefore the temperature contrast is much greater across surfaces of equal geometrical depths. All images share the same maximum and minimum proÐles. Departures from LTE are a likely candidate. We do temperatureJason cuts, which P. highlights Aufdenberg the signiÐcantly largerWhere temperature did contrastHalf inof Procyon. the ForSun’s each iso-tauOxygen surface, theGo? temperature Testing contrast Stellar(T /ST T) isAtmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13temperature April 2005 of Procyon we derived in ° 4.1 should therefore given. rms not observe a larger scatter or a systematic deviation for be slightly corrected from 2.071 to 2.059R_ and from 6512 low-excitation Fe I lines that one could expect from the to 6530 K, respectively. The correction between three- three-dimensional NLTE calculations of Shchukina & Tru- dimensional and one-dimensional models, which amounts jillo Bueno (2001), but we notice a signiÐcantly larger computed with the same spectral synthesis code as the to roughly 1.6%, implies a correction to the e†ective tem- three-dimensional proÐles. Without the convective Doppler 4.4. L ine Data scatter for the iron abundance determined from Fe I lines in perature of roughly 50 K for a star like Procyon A, showing shifts, additional ad hoc broadening in the form of the Iron is the best represented element in the spectrum of Procyon than in the Sun, and departures from LTE are the importance of detailed model atmospheres for estab- micro- and macroturbulence must be invoked in one late-type stars. Neutral iron has been the subject of a expected to grow for warmer stars. Because of the shown lishing a truly empiricalT scale from measurements of dimension in order to obtain correct line widths, with the number of Ðne laboratory works to derive radiative tran- imperfections in the three-dimensional models, the pre- angular diameters (see, e.g.,eff di Benedetto 1998). former a†ecting the line strengths and the latter a†ecting sition probabilities at Oxford (e.g., Blackwell et al. 1986). ferred iron abundance for Procyon is that derived from the only the line shapes. In both cases, Gaussian distributions The number of lines measured with high accuracy and in a three-dimensional analysis of weak lines: log v(Fe)_ \ 7.36 8. ABSOLUTE RADIAL VELOCITY are assumed. homogeneous manner has been enlarged by the work of 0.03 ( 0.15) dex. ^ p \ When discussing measurements of stellar radial veloci- The main conclusions of our study of the iron abundance ties, it is commonly assumed that an observed shift of the are as follows: spectral features from their wavelengths at rest can be 1. The abundances derived from Fe I and Fe II lines are directly translated to a velocity projected along the line of consistent for both one-dimensional and three-dimensional sight. Lindegren, Dravins, & Madsen (1999) have recently analyses, and with a higher coherence for the three- discussed the dangers in doing so, especially when it comes dimensional model. This implies that departures from LTE, to precise measurements, pointing out several e†ects that which should be minimal for Fe II, are likely small. Never- may need to be considered. theless, they could be responsible for a signiÐcant fraction of The gravitational shift is proportional to the gravita- the scatter between the abundances retrieved from di†erent tional Ðeld, Vg \ GM/(Rc), and for late-type dwarfs lines, providing a plausible explanation for the larger scatter amounts to about 0.5 km s~1. In the case of Procyon A, we observed for Procyon than for the Sun. have a special advantage, as its mass, radius, and proper 2. The di†erences between the iron abundances derived motion are well determined. The gravitational redshift, from one-dimensional and three-dimensional analyses including the blueshift induced by Earth, is Vg \ 0.436 (setting aside lines stronger than 50mA� in the case of ^ 0.019 km s~1, where we have used the corrected stellar Procyon) are small([0.05 dex), and the same conclusion radius R \ 2.06 ^ 0.02R_ (see ° 7). The transverse Doppler A Tail of Three Atmosphere Codes

1) 1-D PHOENIX: A general-purpose state-of-the-art stellar and planetary atmosphere code B. Freytag et al.: Spots on the surface of Betelgeuse 215 + 17 other PHOENIX developers deep convection zone. It is important to note that these models enclose the transition from the optically thick convective See: Hauschildt & Baron (1999) layers to the optically thin photosphere, where the plasma J. Comp. and App. Math, 109, 41 ascending from the solar interior looses its energy by efﬁ- 1-D Spherical Symmetry, and cient radiative coolingwithin a very narrow thermal boundary non-LTE line blanketing, layer. The cool downdrafts generated at the surface by radia- Aufdenberg et al. (2002) ApJ 570, 344 tive cooling actually drive the ﬂow. In fact, the properties of Peter Hauschildt France Allard Eddie Baron MLT convection, the entire convection zone are largely determined by the radiation hydrodynamicsof the surface layers. For this reason, lo- Expanding Atmospheres cal simulations with realistic input physics turned out to be a very successful tool to study the solar granulation (e.g. Nord- 2) 3-D CO5BOLD: The ``COservative COde for COmputation of COmpressible COnvection in a BOx of L Dimensions’’ lund 1982; Steffen et al. 1989; Asplund et al. 2000; Skartlien et al. 2000; Stein & Nordlund 2000) and stellar surface convection (e.g. Ludwig et al. 1994, 1999; Freytag et al. 1996; 3-D hydrodynamics, Freytag & Salaris 1999), even though they are much smaller than the whole convection zone. A few examples of such lo- plane parallel cal box models computed recently with the CO BOLD code are displayed in Fig. 2. non-grey opacities Among other things, the local box simulations of stellar surface convection have been used to investigate the size of See: Freytag, Steffan & Dorch (2002) stellar granules, , as a function of stellar parameters. It turns out that, to a good approximation, scales with Hans-Günther Ludwig Astronomische Nachrichten, 323, 213. the pressure scale height at the surface. Roughly, (Freytag et al. 1997; is the universal gas constant), which is equivalent to (2) 3) 1-D ATLAS 12: Latest version of the Kurucz atmosphere and spectrum synthesis code

2.2. Global models: Supergiant convection

Substituting the stellar parameters of Betelgeuse into Eq (2), 1-D Plane-Parallel we obtain , implying the presence of a few LTE line-blanketing hundred convection cells at the surface of this star. Under these conditions, the star-in-a-box approach becomes feasi- MLT convection with “Overshooting Approximation” See: http://kurucz.harvard.edu/ ble. and Fig. 2. Snapshots from 3D local box simulations of stellar surface Castelli, Gratton & Kurucz (1997) A&A 318, 841 3. State-of-the-art results convection performed with the CO BOLD code. Top: Solar Gran- ulation ( K, , M/H =0, dimension: 5.6 x 5.6 x 2.25 Mm, grid: 140 x 140 x 150), Middle: Metal-poor F- 3.1. Toy supergiant: mini-Sun dwarf ( K, , M/H =-2, dimension: 28.6 x In order to verify the star-in-a-box approach and to pro- 28.6 x 7.0 Mm, grid: 140 x 140 x 122), Bottom: Metal-poor G giant (Jason P. AufdenbergK, , WhereM/H =-2, did dimension: Half of 126 the x 126 Sun’s x 47 Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 duce first test models, we have started with a scaled-down Mm, grid: 140 x 140 x 122). The panels show emergent intensity at supergiant. The idea is to keep the solar surface parame- A˚ (top face) and specific entropy (side faces of simulation ters ( K, ) which are known to volume). be numerically well tractable, but to reduce the stellar radius to km, such that by design. The central potential of this toy model is characterized ‘granules’ which become visible as bright spots at the sur- by M and , the specific en- face. As time proceeds, the small cells quickly merge to form tropy enforced in the central core is erg/g/K. The larger convection cells. Finally, the typical solar granulation initial model is a hydrostatic sphere with an adiabatic interior pattern has formed, consisting of bright granules and a net- and an atmosphere in radiative equilibrium, perturbed by tiny work of dark intergranular lanes, wrapped around a moon- random velocity fluctuations. sized object. By design, only a few convection cells occupy Fig. 3 shows the resulting time sequence from the very the surface. start. Initially, the luminosity of the model drops as the en- In the later stages of this simulation, a new (unwanted) velope cools because the convective energy supply from the convective mode begins to dominate the convective flow, with deep interior is not yet established. After about 400 s, the a jet penetrating through the center from one hemisphere to convective instability has led to the formation of many small the other, leading to a more irregular surface structure. As A CO5BOLD model for Procyon

15 y [Mm] y [Mm]

0 0 5 10 15 20 25 x [Mm]

d3gt65g40n3.10-26-3 d3gt65g40n3.10-26-3 10 4•104 ]

2•104 5

0•104

0•103

-5 0•103 Velocity (km/s) Velocity Group averaged vertical velocity [km s Temperature (K) Temperature

0•103 -10

4 2 0 -2 -4 4 2 0 -2 -4 log10 �Ross log10 �Ross Log (Optical Depth) Log (Optical Depth)

Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 7

A grid of forty-two 1-D, spherical, hydrostatic models from the PHOENIX general-purpose stellar and planetary atmosphere code (version 13.07, for a description see Hauschildt et al. 1999; Hauschildt & Baron 1999) have been constructed with the following parameters: effective temperatures, Teff = 6420 K to 6620 K, 10 K steps; log(g) = 3.95, radius, 1 1 R = 1.46 10 1 cm; depth-independent micro-turbulence, ξ = 2.0 km s− ; mixing-length to pressure scale ratio, α = 0.5 and 1.24;× solar elemental abundance (Anders & Grevesse 1989); LTE atomic and molecular line blanketing, typically 106 atomic lines and 4 105 molecular lines dynamically selected at run time; non-LTE line blanketing of H I (30 levels, 435 bound-bound transitions),× He I (19 levels, 171 bound-bound transitions), and He II (10 levels, 45 bound-bound 2 transitions); and boundary conditions: outer pressure, Pgas = 10−4 dynes cm , extinction optical depth at 1.2 µm: out- 10 2 side 10− , inside 10 outside. For this grid of models we use 100 radial shells (depths) with 100 tanget rays and 14 core-intersecting rays. The intersection of the rays with the outermost radial shell describes a center-to-limb intensity profile with 114 angular points. The mixing-length theory (MLT) formulation for convection in PHOENIX is the Mihalas formulation (Ludwig et al. (2002); Mihalas (1978)) with no overshooting. [convergence details?]

5 3.3. CO BOLD + PHOENIX Models 7 [More COBOLD details?] A grid[Convergence of forty-two details?] 1-D, spherical, hydrostatic models from the PHOENIX general-purpose stellar and planetary atmosphere code (version 13.07, for a description see Hauschildt et al. 1999; Hauschildt & Baron 1999) have been constructed These two codes have used together previously to study surface convection in a late-M dwarf (Ludwig et al. 2002). The with the following parameters: effective temperatures, Teff = 6420 K to 6620 K, 10 K steps; log(g) = 3.95, radius, R =1-D 1.46PHOENIX1011 cm;code depth-independent provides direct micro-turbulence, opacity samplingξ at= 2.0high km spectral s 1; mixing-length resolution for to pressure calculation scale of ratio, the radiationα = field based − 5 0.5 andthe 1.24; temporally× solar elemental and spatially abundance averaged (Anders 3-D & Grevesse structure1989); provided LTE atomic by CO andBOLD molecular. The 1-Dline blanketing, temperature, typically pressure, and depth 5 5 106 arraysatomic (91 lines depth and 4 points)105 molecular from CO linesBOLD dynamicallyare read by selectedPHOENIX at run. Physical time; non-LTE depths provided line blanketingCO BOLD of HareI (30 relative lev- to a reference els,depth, 435 bound-bound but PHOENIX transitions),× requires He absoluteI (19 levels, radii 171 for constructionbound-bound transitions), of the 1-D spherical and He II radiation(10 levels, field. 45 bound-bound The value 1.46 1011 cm, 2 5 × transitions);corresponding and boundary to the conditions:physical radius outer of pressure, Procyon,Pgas is= therefore 10−4 dynes added cm , to extinction the CO opticalBOLD depths. depth at The 1.2 µ accuracym: out- of this value is side 10 10, inside 102 outside. For this grid of models we use 100 radial shells (depths) with 100 tanget rays and 14 4 not− critical because the width of the line-forming region, extending to a Rosseland optical dpeth τRoss = 10− , is 0.1% of core-intersectingthe stellar radius. rays. The intersection of the rays with the outermost radial shell describes a center-to-limb intensity profile with 114 angular points. The mixing-length theory (MLT) formulation for convection in PHOENIX is the Mihalas formulation (Ludwig et al. (2002); Mihalas (1978)) with no overshooting. 7 [convergence details?] 3.4. ATLAS 12 Models We also use the ATLAS 12 code ofA Kurucz grid of forty-two (1992), 1-D, which spherical, includes hydrostatic an “overshooting” models from the inPHOENIX the prescriptiongeneral-purpose for stellar convective and planetary atmosphere code5 (version 13.07, for a description see Hauschildt et al. 1999; Hauschildt & Baron 1999) have been constructed flux transport in the mixing-length theory3.3. CO (CastelliBOLD + PHOENIX et al. 1997),Models for comparison with 1-D and 3-D models just discussed. with the following parameters: effective temperatures, Teff = 6420 K to 6620 K, 10 K steps; log(g) = 3.95, radius,7 1 1 [MoreIn the COBOLD approximate details?] overshooting (AO)R = 1. formulation,46 10 1 cm; depth-independent the convective micro-turbulence, flux extends toξ = lower 2.0 km optical s− ; mixing-length depths, reducing to pressure the scale ratio, α = [Convergencetemperature details?] gradient relative to a0.5 models andA 1.24; with× grid solar of no forty-two elemental overshooting. 1-D, abundance spherical,ATLAS9 hydrostatic (Andersmodels & models Grevesse with from1989); the overshootingPHOENIX LTE atomicgeneral-purpose and have molecular been stellar shownline and blanketing, planetary to atmo- typically Thesebetter two fit codes solar have LD used curves together than previously models106 atomic withsphere to study no lines code overshooting surface (versionand 4 13.07, convection105 molecular for (Castelli a description in linesa late-M et dynamically see al. Hauschildt dwarf 1997). (Ludwig selected et al. 1999; at et run al. Hauschildt 2002). time; non-LTE The& Baron line1999) blanketing have been constructed of H I (30 lev- with the following× parameters: effective temperatures, Teff = 6420 K to 6620 K, 10 K steps; log(g) = 3.95, radius, 1-D PHOENIX code provides direct opacity samplingels, 435 bound-bound at high spectral1 transitions), resolution He I for(19 calculation levels, 171 bound-bound of the radiation transitions), field1 based and He II (10 levels, 45 bound-bound Three ATLAS 12 models have been constructedR = 1.46 10 with1 cm; (1) depth-independent no5 overshooting, micro-turbulence, (2) 50% overshooting,ξ = 2.0 km s− ; and mixing-length2 (3) “full” to pressure overshoot- scale ratio, α = the temporally and spatially averaged 3-Dtransitions); structure0.5 and provided and 1.24;× boundary solar by elementalCO conditions:BOLD abundance. The outer (Anders1-D pressure, temperature, & GrevessePgas1 =1989); 10 pressure,−4 dynes LTE atomic andcm , depth extinctionand molecular opticalline blanketing, depth at 1.2 typicallyµm: out- ing with the following parameters:5 sideTeff 10=610 6530, inside K; 10 log(2 outside.g) =5 3 For.95; thisξ = grid 0.0 of models km5 s− we; α use= 100 1.25; radial solar shells elemental (depths) with abundance 100 tanget rays and 14 arrays (91 depth points) from CO BOLD are read by10− PHOENIXatomic lines. Physical and 4 10 depthsmolecular provided lines dynamicallyCO BOLD are selected relative at run to time;a reference non-LTE line blanketing of2 H I (30 lev- depth,(Anders but PHOENIX & Grevesserequires 1989); absolute LTE radii atomic forcore-intersecting constructionels, and 435 bound-bound molecular rays. of the The 1-D transitions),× lin intersectio sphericale blanketing; Hen radiation ofI (19 the levels, rays and field. with 171 boundary bound-bound The the outermost value conditions: 1 transitions),.46 radial101 shel1 and cm,l inner describes He II τ(10Ross a levels, center-to-limb= 45 10 bound-bound intensity 2 [outer?]. These models have 72 depthprofile pointstransitions); with 114 with angular and 17 boundary angular points. conditions: The “µ” mixing-length5 points outer describing pressure, theoryPgas (MLT) the= 10− center-4 formulation dynesto-limb× cm , extinction for convection intensity optical in profile. depthPHOENIX at 1.2isµ them: Mihalasout- corresponding to the physical radius of Procyon, is therefore10 added2 to the CO BOLD depths. The accuracy of this value is formulationside 10− (Ludwig, inside et 10 al.outside. (2002); Mihalas For this (1978)) grid of models with no we overshooting. use 100 radial4 shells (depths) with 100 tanget rays and 14 not critical[The becausePHOENIX themodels width of have the line-forming 83% less N region, and 87% extending less O to (by a Rosseland number) optical relative dpeth to theτ ATLAS= 10− 12, ismodels. 0.1% of The (Anders & [convergencecore-intersecting details?] rays. The intersection of the rays with theRoss outermost radial shell describes a center-to-limb intensity theGrevesse stellar radius. 1989) oxygen abundance is 186%profile larger with 114 found angular by points. (Asplund The mixing-length et al. 2004). theory Does (MLT) this formulation have any for relevance convection in herePHOENIX otheris the Mihalas formulation (Ludwig et al. (2002); Mihalas (1978)) with no overshooting. than completeness?] 3.3. CO5BOLD + PHOENIX Models 3.4.[convergenceATLAS details?] 12 Models [More COBOLD details?] 5 We also use the ATLAS 12 code of Kurucz[Convergence (1992),3.5. Synthetic which details?] includes Visibilities an “overshooting” and Fit3.3. Procedure inCO theBOLD prescription+ PHOENIX Models for convective flux transport in the mixing-length theory (Castelli et al. 1997), for comparison with 1-D and 3-D models just discussed. For the computation of synthetic visibilitiesThese[More two codes COBOLD from have the details?] used mode togetherl radiation previously fields to study we follow surface (Wittkowskiconvection in a late-M et al. dwarf 2004) (Ludwig in order et al. 2002). The In the approximate overshooting (AO) formulation,1-D PHOENIX[Convergence thecode convective provides details?] direct flux opacity extends sampling to lower at optical high spectral depths, resolution reducing for calculation the of the radiation field based temperatureto take into gradient account relative the to effect a models of “bandwidth withthe temporally noThese overshooting. two smearing” and codes spatially haveATLAS9 in used averaged the togethermodels broadband 3-D previously with structure overshooting VINCI to study provided surface filter. by have convectionCO At5 beenBOLD a projected. shown in The a late-M 1-D to temperature, dwarf baseline (LudwigB pressure, etand al. 2002). and The depth mean wavelength 1-D PHOENIX code provides direct5 opacity sampling at high spectral resolution for calculation5 of the radiation field based better fit solar LD curves than models witharrays no overshooting (91 depth points) (Castelli from etCO al.BOLD 1997).are read by PHOENIX. Physical5 depths provided CO BOLD are relative to a reference the temporally and spatially averaged 3-D structure provided by CO BOLD. The 1-D temperature, pressure, and depth1 Three ATLAS 12 models have been constructeddepth, with but PHOENIX (1) no overshooting,requires∞ λ absolute S(λ) (2)5F radii 50%(λ) for overshooting,dλ construction andof the (3) 1-D “full” spherical overshoot- radiation5 field. The value 1.46 10 1 cm, arrays (91 depth points)0 from CO BOLD are read by PHOENIX. Physical depths provided5 CO BOLD are relative to a reference× ing with the following parameters: T = 6530corresponding K; log(g to)λ =0 the= 3. physical95; ξ = radius 0.0 km of s Procyon,1; α = is 1.25; therefore solar added elemental to the abundanceCO BOLD depths. The accuracy(2) of this1 value is eff depth, but PHOENIX requires∞ absolute radii− for construction of the 1-D spherical radiation field. The value 1.46 10 1 cm, S(λ) F (λ) dλ 5 2 × 4 (Anders & Grevesse 1989); LTE atomic andnot critical molecularcorresponding because line to the blanketing; the� width0 physical of the radius and line-forming of boundary Procyon, region, is conditions: therefore extending added inner to to a the RosselandτRossCO BOLD= 10depths. optical The dpeth accuracyτRoss = of 10 this− , value is 0.1% is of 4 [outer?]. These models have 72 depth pointsthe with stellarnot 17 critical radius. angular because “µ” the points width describing of the line-forming the center- region,to-limb extending intensity to a Rosseland profile. optical dpeth τRoss = 10− , is 0.1% of where Sλ is the instrument sensitivity curve, � [The PHOENIX models have 83% less N and 87%the less stellar O radius. (by number) relative to the ATLAS 12 models. The (Anders & 1 3.4. ATLAS 12 Models Grevesse 1989) oxygen abundance is 186% larger found by (Asplund et al. 2004). Does this3.4. haveATLAS any 12 relevanceModels here other than completeness?] We also use theFATLASλ = 12 Icode(µ, ofλ) Kuruczµ dµ, (1992), which includes an “overshooting” in the prescription(3) for convective flux transportWe also in use the the mixing-lengthATLAS0 12 code theory of Kurucz (Castelli (1992), et which al. 1997), includes for comparison an “overshooting” with 1-D in the and prescription 3-D models for just convective discussed. flux transport in the� mixing-length theory (Castelli et al. 1997), for comparison with 1-D and 3-D models just discussed. In the approximate overshooting (AO) formulation, the convective flux extends to lower optical depths, reducing the is the flux, and I(µ, λ) is the model3.5. Synthetic radiationIn the Visibilities approximate field at angle and overshooting‹ Fitµ Procedureand (AO) wavelength formulation,λ the, the convective synthetic flux extends squared to lower visibility optical is depths, reducing the temperatureModeltemperature intensities gradient gradient relative relative t oM a toodel models a models visibilities with with no no overshooting. overshooting. ATLAS9ATLAS9 modelsmodels with with overshooting overshooting have have been been shown shown to to For the computation of synthetic visibilities from the model radiation∞ fields we follow2 (Wittkowski et al. 2004) in order betterbetter fit solar fit solar LD curves LD2 curves than than0 modelsV models(B, with λ with) nodλ no overshooting overshooting (Castelli (Castelli et al. al. 1997). 1997). The visibility V is the HankelV (B, transform λ ) = of a set of intensities I, angles μ, and angular diameterθ (4) to take into account the effect of “bandwidthThree smearing”ThreeATLASATLAS 12 in0models the 12 models broadband have have been been VINCI constructed2 constructed filter.2 with with At (1) (1) a no projected no overshooting, overshooting, baselineLD (2) (2) 50% 50%B overshooting,and overshooting, and and (3) (3) “full” “full” overshoot- overshoot- ∞ S(λ) F (λ) , dλ 11 mean wavelength ing withing thewith following the following parameters: parameters:0 � TeffTeff== 6530 6530 K; K; log( log(g)g) = = 3 3.95;.95; ξξ == 0.0 km km s s−−; ;αα== 1.25; 1.25; solar solar elemental elemental abundance abundance 2 2 (Anders(Anders & Grevesse∞ & Grevesse 1989); 1989); LTE LTE atomic atomic and and molecular molecular lin linee blanketing; blanketing; and and boundary boundary conditions: conditions: inner innerτRossτ = 10= 10 where 0 λ S(λ) F (λ) dλ Ross [outer?].λ[outer?].0 = These1 These models models have� have 72 depth 72 depth points points with with 17 17 angular angular “ “µµ”” points points describing describing the(2) the center- center-to-limbto-limb intensity intensity profile. profile. ∞ S(λ) F (λ) dλ [The PHOENIX[The� PHOENIX0 modelsmodels have have 83% 83% less less N andN and 87% 87% less2 less1 O/ O2 (by (by number) number) relative to to the theATLASATLAS 12 12models.models. The The (Anders (Anders & & V (B, λ)Grevesse = 1989)S(λ) oxygenI(µ, λ abundance) πθLD( isB/λ 186%)(1 largerµ found) by (Asplundµ dµ et al. 2004). Does this have any relevance(5) here other Grevesse 1989) oxygen abundance is 186% larger− found by (Asplund et al. 2004). Does this have any relevance here other where Sλ is the instrument sensitivity curve, than completeness?]0� than completeness?]� � � is the Hankel transform of the center-to-limb inten1 sity profile of a given “limb-darkened” angular size, θ . 3.5. Synthetic Visibilities and Fit Procedure LD Fλ = I(µ, λ) µ dµ, (3) 3.5.λ Synthetic Visibilities andλ Fit Procedure Squared visibility forFor a thebaseline�0 computation B and mean of syntheticwavelength visibilities 0 for filter from transmission the model S( radiation): fields we follow (Wittkowski et al. 2004) in order is the flux, and I(µ, λ) is the model radiationFor fieldto the take at computation angle into accountµ and of the wavelength synthetic effect of visibilities “bandwidthλ, the syntheticfrom smearing” the mode squared in thel radiation broadband visibility fields VINCI is we follow filter. (Wittkowski At a projected et baseline al. 2004)B inand order to takemean into wavelength account the effect of “bandwidth smearing” in the broadband VINCI filter. At a projected baseline B and ∞ 2 ∞ λ S(λ) F (λ) dλ mean wavelength2 0 V (B, λ) dλ 0 V (B, λ ) = λ0 = (4) (2) 0 2 2 ∞ ∞ S(λ) F (λ) dλ ∞ S(λ) F (λ) , dλ 0� 0λ S(λ) F (λ) dλ 0 � λ0 = (2) where Sλ is the instrument sensitivity curve, ∞ S(λ) F (λ) dλ where � 0� 1 � 1 where Sλ is the instrument sensitivity2 curve,1/2 F =� I(µ, λ) µ dµ, (3) V (B, λ) = S(λ)I(µ, λ) πθLD(B/λ)(1 µ ) µ dµ λ (5) − 10 �0 � � � is the flux, and I(µ, λ) is the model radiationFlux: fieldFλ = at angleI(µ,µ and λ) µ wavelength dµ, λ, the synthetic squared visibility is (3) is the Hankel transform of the center-to-limb intensity profile of a given “limb-darkened” angular0 size, θLD. ∞ 2 2 � 0 V (B, λ) dλ is the flux, and I(µ, λ) is the model radiationV field(B, λ at0) angle= µ and wavelength λ, the synthetic squared visibility is(4) ∞ 2 2 0 �S(λ) F (λ) , dλ ∞ 2 where 2 0 V (B, λ) dλ V (B,1 λ0) = � (4) ∞ S(λ)2 F (λ)2 , dλ2 1/2 Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar AtmosphereV (B, Models.... λ) = DipartimentoS(λ)I di( Fisicaµ, λ0 Universitá)� πθLD di (PisaB/λ 13 April)(1 2005µ ) µ dµ (5) 0 − where � � � � is the Hankel transform of the center-to-limb1 intensity profile of a given “limb-darkened” angular size, θLD. V (B, λ) = S(λ)I(µ, λ) πθ (B/λ)(1 µ2)1/2 µ dµ (5) LD − �0 � � is the Hankel transform of the center-to-limb intensity profile of a given “limb-darkened” angular size, θLD. Mark III and VLTI: Optical and Near-IR Interferometry Mark III Very Large Telescope Interferometer Mt. Wilson, California Paranal, Chile

20 m

16 m m λ = 500 nm, 800 nm 63 23 m CHARA Array Mt. Wilson, California λ = 2.2 µm 152 m

λ = 2.2 µm Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 3-D CO5BOLD Model 5.392 mas

�� vs. �� Mark III 500 nm, 800nm VLTI/VINCI K-band �� and �� CHARA/FLUOR K-band ��

��

Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 Angular Diameters from 1-D & 3-D Model Fits 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

Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 B: PHOENIX “B” A: PHOENIX “A” Comparing D: ATLAS “D” F: ATLAS “F” 1-D and 3-D Model E: ATLAS “E” C: CO5BOLD + PHOENIX Temperature Structures

A D PHOENIX F ATLAS E

C CO5BOLD

Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 Procyon’s 2-D Surface: Many Temperatures, Many Colors

Model Intensity Map Intensity Historgram

8•10-11 25

6•10-11 20

15 4•10-11 PDF y [Mm] y [Mm]

10 2•10-11

0 0 1•1010 2•1010 3•1010 4•1010 0 5 10 15 20 25 2 x [Mm] Emergent continuum intensity [erg/s/cm /srad/band]

Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 Procyon’s Spectral Energy Distribution vs. Composite Model Spectrum

IUE SWP IUE LWP

STIS GHRS

12 component model 1 component model

u v b y Glushneva et al. (1992)

Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 Synthetic vs. Observed Strömgren Photometry

(b-y) c = (u-v) - (v-b)

Model

ATLAS 12 “E” overshooting ATLAS 12 “F” } ATLAS 12 “D”

PHOENIX “A” NO overshooting

PHOENIX “B” }

Composite SED A

Composite SED B

Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 Part II Deneb & Rigel, Stellar Winds, and Limb Darkening

Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 Future Targets for Hot FromStar F5 IVInterferometry to A2 Ia & B8 Ia...

A Theoretician’s HR Diagram

Deneb

Rigel Angular Diameters from Hanbury Brown et al. (1974) Aufdenberg et al. (2002)

Distances from HIPPARCOS

Tracks from Schaller et al. (1992) and Girardi et al. (2000)

Procyon

Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 Plane Parallel vs. Spherical Limb Proﬁles

a) plane-parallel case b) spherical case

Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 Plane Parallel Limb Proﬁles

��cos-1 �

Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 Spherical Limb Proﬁles

��cos-1 � z

r0 r1

r2 rN r3

Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 Deneb (α Cygni): the prototypical A-type supergiant

New!

Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 356 AUFDENBERG ET AL. Vol. 570

TABLE 7

Least-Squares Results for Parameters R , log g, �, and �t � Model Fit Results

log g(R ) R � � v2 P(F-test) � E(B V ) t � (1) � (2)� (3) (4) (5) (6) (7)� (8)

1.3...... 1.50E13 3.0 15 1.15E3 5.00E 1 2.080 0.060 � 1.3...... 1.20E13 3.0 15 1.17E3 4.32E 1 2.080 0.060 1.3...... 9.00E12 3.0 15 1.26E3 2.25E �1 2.080 0.060 � 1.3...... 1.20E13 3.0 25 1.31E3 1.36E 1 2.060 0.050 � 1.5...... 1.20E13 3.0 15 1.46E3 2.66E 2 2.060 0.050 � 1.3...... 1.20E13 2.0 15 1.69E3 8.17E 4 2.260 0.120 1.7...... 1.20E13 3.0 15 1.87E3 3.63E �5 2.040 0.050 � 1.3...... 1.20E13 3.0 5 2.13E3 2.98E 7 2.120 0.090 � 1.3...... 1.20E13 4.0 15 2.23E3 5.96E 8 2.020 0.050 � Note.—Columns (1)–(4): the models listed here have the fixed values T 9000 K and _ 6 1 � ¼ M 10� M yr� . Columns (5)–(8) are as described in Table 6. ¼ � tinuum opacity wind model produces 10 times more flux sphere (classic surface cooling), especially at depths where than our most sophisticated non-LTE line-blanketed� wind the radio continuum forms. In LTE, lower electron temper- model, and the LTE line-blanketed wind model produces atures mean lower ionization, and therefore less thermal 1 2 as much flux. The LTE continuum-only model spectrum radio emission is expected in the presence of line blanketing �shows a turnover at 30 cm due to truncation at the outer relative to the continuum-only opacity case. Second, photo- boundary of the model,� 200 R . A similar model with an ionization is enhanced in non-LTE. Non-LTE photoioniza- other boundary of 400 R does� not show a turnover. The tion affects the ionization state of the outer layers because line-blanketed wind models,� which are not fully ionized to they are radiatively coupled to the hotter radiation field at the outer boundary, are not sensitive to this truncation. depth. In non-LTE, the ionization fraction in the outermost There are two important factors at play here. First, the layers is enhanced relative to the LTE line-blanketed case, incorporation of metal lineDeneb blanketing’s inStellar the radiative Wind: equi- Ultravioletwhich leads to to Radio more thermal Constraints radio emission. librium solution cools the outermost layers of the atmo- The sensitivity of the radio continuum to the degree of ionization in the extended stellar envelope makes it clear 360 AUFDENBERGthat the ET application AL. of a fully ionized, uniform, sphericallyVol. 570 Mg II h&k double P-Cygni lines symmetric mass flow model commonly employed in mass- loss ratebroad estimates emission for wings O- andnormally B-type attributed supergiants to electron (Scuderi scat- et tering. Despite being the prototype of the class, Deneb’s H � al. 1998;profile Drake has only & Linsky been modeled 1989) will in detail not twicework to for our A-type knowl- supergiants.edge. Furthermore, in such models the radio spec- 2=3 trum followsReasonable the power fits to severallaw ofS Deneb’s� � H, but Deneb� profiles exhibits have a models spectralbeen slope achieved between by Kunasz 1.35 &mm Morrison and/ 3.6 (1982) cm consistent and Scuderi with et data al. (1992). Unfortunately, the model parameters and assumptions usedIR, in thesemm, works radio are inconsistent continuum with the present observational data. In the former work, the best fits were achieved by adopting terminal velocities of 120 and 1 140 km s � , which are inconsistent with the blue edges of the Mg ii h and k profiles (see Fig. 19). In the latter work, the critical assumption of a fully ionized wind is made in addi- tion to the assumption of LTE and the Sobolev approximation. As shown above ( 5.3; Fig. 11), a fully ionized wind is inconsistent with the observedx slope of the millimeter-radio continuum. models We have attempted to fit the H � profile from the HEROS Fig. 19.—IUE spectrum LWR 07760 in a region showing the Mg ii h and 1 k resonance lines. Shown for comparison are synthetic spectra from two red spectrum by fixing v at 225 km s � and computing the 1 models: LTE Mg ii and non-LTE Mg ii. With the exception of the non- synthetic profiles in the comoving framedata from the set of radi- Fig. 9.—SyntheLTE tic metalUV continua line blanketing, from three these models sets (from have top identical to bottom) parameters. of The ative equilibrium expanding atmosphere models computed ˚ three models: surfacezero gravity point ofthe log velocityg scale1:3 is ( thesolid lineh),line, 1.5 (�airdotted2795 line:523), andA shifted to ¼ 1 ¼ for the SED analysis. We are unable to improve on earlier 1.7 ( dashed line );radial the microturbul velocity of Deneb, ent velocity RV �4t:5 km5 s ( �dotted(Evans line 1967).), 15 The ( solid interstellar 1 ¼ � attempts to fit the observed H � profile by adjusting any of line), and 25 km s compo1 (dashed nents at line70); km and s � thehave velocity not been¼ law modeled. exponent � 2:0 � � ¼ the model parameters (see Fig. 20). The most significant and (dotted line ), 3.0 ( solid line ), and 4.0 ( dashed line ). These synthetic spectra persistent discrepancy between the synthetic and observed have been binned to 20 A ˚ resolution to show the gross features of the spectrum. All models have the following parameter6.2. Visual-IR s unless Line otherwis Spectrum e noted: profiles is the depth of the absorption component, which is _ 6 1 significantly weaker in the observed spectrum (residual T 9000 K, log g R 1:3, M 106.2.1.� M Theyr�H,��Line3:0, and �t 15 � ¼ 1 ð �Þ ¼ ¼ � ¼ ¼ intensity 0.6). Observed H � profiles from 1991 (Kaufer et km s � . All models have been scaled by � 2:08 mas. Constants of ’ 9 At optical9 wavelengths,1 2 ˚ the1� most¼ conspicuous indicator al. 1996) have a broader absorption component than shown 1:25 10� and 2 :5 Aufdenberg10� ergs s � cm et� al.A� (2002)have been addedApJ to570, the top 344 two sets� of models forof mass-loss clarity.� The in effects early-type of log supergiantsg can be is seen the on P Cygni both sides charac- in Figure 20, but they are not deeper. Furthermore, while 1 of the Balmer discontinuityter of the H . The effects� line. of Like varying all BA-type�t from supergiants, 5 to 25 km Deneb’ss � Fig.the10. velocity—Infrared of the , millimeter, absorption and component radio photom minimum etry of Deneb is quite from Jason P. Aufdenberg˚ Where1 did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2001 5 are apparent belowH 3000� profile A (>3.5 is variablelm� (Kaufer), where the et al. flux 1996). drops Deneb’s by roughly H a � pro-Table 1.variable, Synthetic it continua is rarely, forif ever, models shifted listed blueward in Table 6 withof 50T km s 9000� , K factor of 2. The effectsfile ofdistinguishes varying � itselffrom from 2.0 to most 4.0 are supergiants significant below in that the it lacks and six different20% of mass-lossthe terminal rates velocity. are shown Models for comparison. with mass-loss The� inset rates� shows¼ Balmer jump, where the flux drops by roughly 50%. a close� up of the COBE/DIRBE and MSX photom etry.

Fig. 20.—HEROS red spectrum ( thick solid line ) at the H � line shown with syntheti c profiles from a grid of model atmosphere s with different parameteriza - 6 1 1 tions about the base model [ T 9000 K, log g R 1:3, M_ 10� M yr� , � 3:0, �t 15 km s � ]. The syntheti c profile for zero mass loss is from the � ¼ ð �Þ ¼ ¼ 1 � ¼ ¼ hydrostatic model T 8500 K, log g R 1:3, and �t 15 km s � . Shown are the effects on the synthetic profile due to the ( a) mass-loss rate, ( b) velocity law exponent, ( c) microturbul� ¼ ence parameterð �Þ ¼ , and ( d ) the input¼ radius. Positions of telluric lines are mark by circled cross symbols. All syntheti c profiles have been shifted from vacuum to air wavelen gths and rotationally broadened for v sin i 25 km s 1. ¼ � NPOI and CHARA: More Optical and Near-IR Interferometry

Navy Prototype Optical Interferometer Center for High Angular Resolution Astronomy Array Flagstaff, Arizona Mt. Wilson, California

38m λ = 500 nm, 800 nm 251 m 331 m 152 m

λ = 650 - 850 nm

λ = 2.2 µm

Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 348 AUFDENBERG ET AL. Vol. 570 Deneb: NPOI 1st lobe data and 2nd lobe model predictions

1st Lobe

No. 1, 2002 A-TYPE SUPERGIANT DENEB 349

Center-to-Limb Models

2nd lobe Predictions

Aufdenberg et al. (2002) ApJ 570, 344 Fig. 2.—NPOI visibility data of Deneb withFig. UD4.—( fits.a) Comparison For each of date, the I-band a model CLVs from visibility two model curve atmospheres (solid line and a) foruniform the disk: mean uniform-disk UD diameter model (dotted� line(see), spherical Table hydrostatic 4) is plotted model (dashed line), and spherical hydrostatic plus expanding wind model (solid line). The three CLVs have been scaled so that theirUD visibility curves match the 1 along with the corresponding visibility curvesNPOI for data the from 19971 � Decembererror in 5 at the 220 mean arcsec� UD.(b) Different diameter CLVs ( candotted be most lines easily). discriminated (a) 1997 December beyond the first 5: null;�UD the uniform-disk2:36 squared0:01 mas visibil- (four Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá1 di Pisa 13 April 2005 ities are predicted� to be 60% larger than the wind model squared visibilities at a spatial frequency of 700 arcsec� in the I band. The¼ spherical,� expanding atmo- scans); (b) 1997 December 6: �UD 2:41 0:17 mas (seven scans); (c) 1997 December 16:13�UD 2:33 0:02 mas (one scan). 2 ¼ � sphere model is parameterized by T 9000 K at R 1:2 10 cm and¼ a surface� gravity, log g R 1:3, equivalent to Teff 8543 K at R �Ross 3 and � ¼ _ � ¼ �7 1 ð �Þ ¼ 1 ¼ ¼ log g R 1:21 (see eq. [9]); a mass-loss rate M 5:0 10� M yr� ; a terminal wind velocity v 225 km s� ; and a velocity law exponent � 3:0 (see ð Þ ¼ ¼ � � 1 ¼13 �¼ � Appendix A2). The spherical, hydrostatic model is parameterized by T 8500 K at R 1:2 10 cm and log g R 1:3, equivalent to Teff 8746 K at R � x 2 and log g R 1:35. � ¼ � ¼ � ð �Þ ¼ ¼ Ross ¼ 3 ð Þ ¼ � � effective temperature: Unfortunately, in extended atmospheres the effective temperature is not well defined or is ambiguous because the at- from1=4 the shape of the visibility curve at spatial frequencies would have an immeasurable effect on the observed visibil- 4F0 mosphere is not sufficiently compact to have a unique radius Teff beyond; the first null. Deneb,3 however, has yet to be ities. This is defined quantitatively by the following proce- 2 observed at sufficiently long baselines (>100(Baschek, m) for limb- Scholz,dure. & Wehrse 1991). Hence, a value for T ¼ ��LD ð Þ eff � �darkening information to be obtained in thiscannot way. There- be separatedThe intensity from of an the stellaradopted disk as value a function for of the the angu- radius fore, the details of Deneb’s center-to-limb variation (CLV) lar radius I(�) is extracted from the model atmosphere. The where F0 is the bolometric flux (correctedmust for now be for supplied interstellar by theory. and the correspondingcoordinate system angular for the size. solution of the spherical model at- extinction) and � is the Stefan-BoltzmannLimb-darkening coefficients constant. provided in theFor literature well-resolvedmosphere sourcesproblem is specified (e.g., by Betelgeuse, a set of parallel Arcturus), rays tan- (see, e.g., Claret 2000) are derived from plane-parallel mod- gent to nested spherical shells (see, e.g., Fig. 1 in Hummer & els and are not appropriate for Deneb.observations For example, canRybicki directly 1971; Mihalas, yield Kunasz, limb-darkening & Hummer 1975). information The rays Figure 4a shows that the model CLV from an expanding at- are parameterized by the perpendicular distance p from the mosphere or ‘‘ wind ’’ model differs significantly from that line of symmetry though the center of the star. A solution to TABLE 4of a hydrostatic model. Unlike the wind model CLV, which the model atmosphere problem provides the specific inten- Uniform-Disk Angularfalls Diameters very gradually from 40% of the central disk intensity to sity I� at wavelength � at depths z along each ray. The zero, the hydrostatic model CLV shows a fairly well-defined observed intensity for a given ray is taken to be the intensity edge. The unambiguous signature of the predicted wind at the intersection of that ray with the outermost shell. The Observation Time modelhUD CLV lies in the height of the visibility function relation between the directional cosine of the emergent radi- (UT) (mas)beyond the firstv null2 (see Fig. 4b). Future measurements ation l cos � and the angular displacement on the sky � is should be able to test this prediction. ¼ The gradual decline of the wind model CLV presents a � 1 l2 1=2 : 4 1997 Dec 5: �UD = 2.364 0.011 ¼ ð � Þ ð Þ problem� for deriving hLD. As noted by Baschek et al. (1991), a flat intensity decrease toward the outer disk leaves an The spectral response of the NPOI is approximated by 01:40:05 ...... 2.364 0.007 4.7 the mean of the R and I bands (Nordgren et al. 1999). ‘‘ intensity� radius,’’ defined by an arbitrary cutoff of the 01:52:05 ...... 2.353CLV,0.007 e.g., at 1%, 5.0 poorly defined. Such a profile prohibits a Hence, after calculating I�(�) on a fine-wavelength grid, a � passband-weighted CLV for each filter, 01:59:49 ...... 2.379simple0.007 relation between 4.9 an ‘‘ optical depth radius,’’ which corresponds� to reference optical depth along a ray normal bi 02:12:32 ...... 2.360 0.008 7.1 I� � Si � d� to the atmosphere, and an ‘‘ intensity radius,’’ which corre- ai � �Ii � ð Þ ð Þ ; 5 sponds to the impact parameter of a ray at the angle of the ð Þ ¼ bi ð Þ R a Si � d� 1997 Dec 6: �UD = 2.410 0.168 i ð Þ intensity� cutoff. We choose to define the limb-darkened edge to be the angle beyond which the model intensity profile is computed, where Si(�)R is the response function of the 01:16:52 ...... 2.578 0.009 17 � 01:44:35 ...... 2.470 0.008 57 � 02:14:50 ...... 2.432 0.009 38 02:27:25 ...... 2.180 � 0.014 20 � 02:51:50 ...... 2.169 0.020 30 � 03:18:22 ...... 2.541 0.012 90 � 03:44:02 ...... 2.501 0.014 32 � Fig. 3.—Histogram of the uniform-disk angular diameter values derived 1997 Dec 16: �UD = 2.325 0.015 � from each of the 12 scans of Deneb by the NPOI (see Table 4). The data are 01:03:29 ...... 2.325 0.015 61 binned to 0.05 mas. The different crosshatched regions specify the date of � each scan. 2004 CHARA/FLUOR Observations of Deneb: Wind Detected!

Deneb: Peak of 2nd Lobe Consistent with Mass Loss

Uniform Disk Model Hydrostatic Model Wind Model

Deneb: Position Angle Coverage Suggests Wind is Asymmetric

Wind Model Position Angles +20o to +85o; Diameter = 2.39 mas Position Angles --90o to +20o; Diameter = 2.46 mas

Jason P. Aufdenberg Where didb) Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 A Surprise! Deneb Appears Asymmetric

DENEB DENEB Θ(WBα=0.28) = 2.455±0.002 mas Θ(WBα=0.28) = 2.391±0.001 mas

2 2 1.0 χ r=2.791 1.0 χ r=19.493 200 200

0 v 0 v

−200 −200 0.1 u 0.1 u −200 0 200 −200 0 200 2 2 V V

0.01 0.01

0.001 0.001

100 200 300 100 200 300 20 2 2

V 10 V σ σ /

/ 0 0 2 2 −10 −50 −20 errV errV 100 200 300 100 200 300

Base (m) Base (m)

Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 ��

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Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005 Summary

1. Multi-wavelength (optical + near-IR) visibility measurements are providing very-high precision angular diameters: ≈ 0.1%. Such measurements can tightly constrain limb-darkening.

2. For Procyon, such measurements test models with different temperature gradients due to different convection treatments. 3-D predictions are conﬁrmed: Procyonʼs limb less darkened than most 1-D models predict.

3. 1-D models can be made to ﬁt observed limb darkening, but canʼt to match full SED. For Procyon, the mean temperature structure appears to be partially decoupled from its mean photometric colors. What do precise colors tell us about Teff and convection?

4. Visibility measurements of Deneb yield a mass-loss rate consistent with UV and radio diagnostics, but uncertainties are still large (>> 15%), but now within an order of magnitude.

5. Denebʼs stellar disk appears to asymmetric at ~3% level, asymmetric outﬂow?

6. More insights into stellar atmospheres and stellar interiors will be coming from interferometry. Stay tuned!

Jason P. Aufdenberg Where did Half of the Sun’s Oxygen Go? Testing Stellar Atmosphere Models.... Dipartimento di Fisica Universitá di Pisa 13 April 2005