DraftversionFebruary1,2008 PreprinttypesetusingLATEXstyleemulateapjv.14/09/00 THE SPECTRAL ENERGY DISTRIBUTION AND MASS-LOSS RATE OF THE A-TYPE SUPERGIANT DENEB J. P. Aufdenberg SolarandStellarPhysicsDivision,Harvard-SmithsonianCenter forAstrophysics,60GardenStreet,Mail Stop15,Cambridge,MA02138 [email protected] P. H. Hauschildt DepartmentofPhysicsandAstronomy&CenterforSimulationalPhysics,UniversityofGeorgia,Athens,GA 30602-2451 [email protected] 2 E. Baron 0 DepartmentofPhysicsandAstronomy,UniversityofOklahoma,Norman,OK73019-0260 [email protected] 0 2 T. E. Nordgren DepartmentofPhysics,UniversityofRedlands,1200EastColtonAve,Redlands,CA92373 n tyler [email protected] a I. D. Howarth, A. W. Burnley J DepartmentofPhysicsandAstronomy,UniversityCollegeLondon,GowerStreet, LondonWC1E 6BT,UK 4 [email protected],[email protected] 1 and 1 K. D. Gordon, J. A. Stansberry v StewardObservatory,933NCherryAve,UniversityofArizona,Tucson,AZ85721 8 [email protected],[email protected] 1 Draft version February 1, 2008 2 1 ABSTRACT 0 A stellar wind module has been developed for the PHOENIX stellar atmosphere code for the purpose 2 ofcomputing non-LTE,line-blanketed,expandingatmosphericstructuresanddetailedsynthetic spectra 0 of hot luminous stars with winds. We apply the code to observations of Deneb, for which we report / h the first positive detections of mm and cm emission (obtained using the SCUBA and the VLA), as well p a strong upper limit on the 850µm flux (using the HHT). The slope of the radio spectrum shows that - the stellar wind is partially ionized. We report a uniform-disk angular diameter measurement, θ = o UD 2.40±0.06 mas, from the Navy Prototype Optical Interferometer (NPOI). The measured bolometric r t flux andcorrectedNPOIangulardiameteryieldaneffectivetemperatureof8600±500K. Least-squares s a comparisons of synthetic spectral energy distributions from 1220 ˚A to 3.6 cm with the observations : provideestimatesfortheeffectivetemperatureandthemass-lossrateof≃8400±100Kand8±3×10−7 v i M⊙ yr−1,respectively. Thisrangeofmass-lossratesisconsistentwiththatderivedfromhighdispersion X UV spectra when non-LTE metal-line blanketing is considered. We are unable achieve a reasonable fit r to a typical Hα P-Cygni profile with any model parameters over a reasonable range. This is troubling a because the Hα profile is the observational basis for Wind Momentum-Luminosity Relationship. Subject headings: methods: numerical — radio continuum: stars — stars: atmospheres — stars:individual (Deneb) — stars: winds, outflows — techniques: interferometric 1. introduction A-typesupergiantsarethebrighteststarsatvisualwavelengths(uptoM ≃−9)andarethereforeamongthebrightest V single stars visible in galaxies. For this reason, these stars have been of increasing interest in extragalactic astronomy where they show potential as independent distance indicators (Bresolinet al. 2001;Kudritzkiet al. 1999,hereafter K99). This potential lies in the use of the Wind Momentum-Luminosity Relationship (WMLR) which is derived from Sobolev radiation-driven stellar wind theory (K99 and references therein). Testing this relationship and the theory of radiation driven winds most critically requires an accurate determination of the stellar mass-loss rates. The WMLR states that M˙ v∞ ∝L1/α/R⋆0.5,where M˙ isthemass-lossrate, v∞ istheterminalvelocityofthewind,R⋆ isaphotosphericreference radius, and α is a parameter related to the distribution of atomic line strengths for the spectral lines which drive the wind. The application of the WMLR to nearby galaxies requires a calibration of the relationship using similar stars in the Milky Way. To determine mass-loss rates for A-type supergiants, K99 modeled the hydrogen Balmer lines Hα and Hγ for six stars in the Galaxy and M31 using a unified stellar atmosphere model. In their analysis, K99 synthesized the observedline profilesbyadjusting modelvaluesforthe mass-lossrate,the velocitylawexponentβ, andaline-broadening parameter, v , after first adopting parameters for the effective temperature, T , the radius, R , the gravity, logg, and t eff ⋆ the projected stellar rotational velocity. 1 2 Aufdenberg et al. TheuseoftheHαprofileasthesolediagnosticofA-typesupergiantwindmomentaisproblematicbecauseHαemission observed in stars of this type is known to be strongly variable (Rosendhal 1972; Kaufer et al. 1996). Nevertheless, the accuracy of mass-loss rate determinations for A-type supergiants in M31 is claimed to be at the level of 15% (McCarthy et al. 1997). However, there has been no consensus on the mass-loss rate for Deneb (α Cyg, HD 197345, spectral type A2Ia according to Morgan& Keenan (1973)), the brighest, nearest, and best-studied A-type supergiant, which we might expect to serve as an archetype and benchmark. Published estimates of its mass-loss rate range over more than three orders of magnitude, from methods which include modeling the line profiles Mg II λ2802 (Kunasz & Praderie 1981, 3.1×10−9−1.5×10−7 M⊙ yr−1) and Hα (Kunasz & Morrison 1982; Scuderi et al. 1992, 1.7±0.4×10−7 M⊙ yr−1; 3.7±0.8×10−6 M⊙ yr−1), analysing low-excitation Fe II lines in the ultraviolet (Hensberge et al. 1982, 1−5×10−9 M⊙ yr−1), modeling the IRexcess(Barlow& Cohen1977,6×10−7 M⊙ yr−1), and radio-fluxlimits (Abbott etal.1984, ≤2×10−7 M⊙ yr−1). It is not unreasonable to suggest that the degree to which we understand A-type supergiants in general may be tested by Deneb. Therefore, it is troubling that the published mass-loss estimates for Deneb vary so widely and this fact casts doubt on the what we can infer about the properties of more distant A-type supergiants. The proximity of Deneb allows one to more rigorously constrain its physical parameters and apply several different techniques to estimate its mass-loss rate. For these reasons, in this paper we investiage Deneb’s fundamental properties, and in particular its mass-loss rate. To do this we employ a new stellar wind-model atmosphere package developed by Aufdenberg (2000) for use with the PHOENIX1 general-purpose stellar and planetary atmosphere code (Hauschildt & Baron 1999, and references therein) and bring together new observations of Deneb at visible, infrared, millimeter, and radio wavelengths. Ourprincipalmotivationfortheconstructingunifiedwindmodelshasbeenthatmostanalysesofearly-typesupergiants still suffer from severe limitations present in the model atmospheres which are employed. Recent systematic analyses (Verdugo et al. 1999) of the atmospheric parameters of A-type supergiants have employed LTE, line-blanketed, plane- parallel,staticmodelatmospheres. Thesemodelsdonotincludetheeffects(onthetemperaturestructureandthesynthetic spectrum) of metal line-blanketing and a spherically extended expanding atmosphere. Furthermore, most analyses do not allow for departures from LTE, particularly for species heavier than H and He, in computation of the opacity and line formation, which are especially important in low-density extended atmospheres. Santolaya-Rey et al. (1997) have developed models, employed by K99, which address many of these limitations including the solution of the spherical expanding transfer equation in the co-moving frame and the non-LTE treatment of hydrogen and helium. Despite these important improvements, these models have had limited success in fitting spectral energy distributions (SEDs) due to the lack of metal line-blanketing in both the atmospheric structure and the synthetic spectrum. Fundamental stellar parameters derived from line-profile fitting of hydrogen and helium lines alone may not be consistent with the SED. Therefore, we believe that fitting both the line spectrum and the SED simultaneously should lead to more robust atmospheric parameters for A-type supergiants. In§2wedescribethespectrophotometric,spectroscopic,andinterferometicobservationsusedhere. In§3limb-darkening models and corrections to the angular diamter are discussed. The effective temperature, reddening and other parameters constrained by the observations are discussed in §4. In §5 we compare the spectrophotometic data from the UV to the radiowithoursyntheticspectralenergydistributions. ThecomparisonofourmodelstoportionsofDeneb’slinespectrum is discussed in §6. We summarize our results and conclusions in §7. Details on the model atmosphere construction and computation are left for Appendix A. 2. observations PreviousradioobservationsofDeneb(Drake&Linsky1989,andreferencestherein)haveyieldedonlyupperlimits. We report positive detections of radiation from Deneb at 1.35 mm with the Submillimetre Common-User Bolometer Array (SCUBA) located on the James Clerk Maxwell Telescope (JCMT) and at 3.6 cm with the Very Large Array (VLA). In addition, two of us (KDG & JAS) observed Deneb at 870 µm with the Heinrich Hertz Telescope (HHT) and we report an upper limit on the flux density at this wavelength. Futhermore, one of us (TEN) measuredDeneb’s angular diameter, previouslyuncertainatthe30%level,withtheNavyPrototypeOpticalInterferometer(NPOI)andwepresentanangular diameter measurement accurate to 3%. In addition to these new data, we have brought together observations of Deneb from Infrared Processing and Analysis Center (IPAC) data archive. The IR, millimeter and radio data are summarized in Table 1. 2.1. SCUBA Observation SCUBA was used to observeDeneb on 4 May 1998. Four runs weremade in the photometery mode using the 1350µm filter. Each run consisted of 50 integrations of 18 sec each. Each integration includes both the on-source and off-source time, the on-source time is half the value. Deneb was positively detected at 1.35 mm with flux density of 7.8± 1.9 mJy. 2.2. VLA Observation Deneb was observed by 22 antennas of the VLA in the D−→A array configuration on 29 January 1990. The central frequency of the observation was 8414 MHz with bandwidth of 50 MHz. The total observation time was 4800 sec, with 3450 sec on source. Deneb was positively detected at 3.6 cm with a flux density of 0.23±0.05 mJy. 1 Alsoseehttp://phoenix.physast.uga.edu The A-type Supergiant Deneb 3 Table 1 Infrared, Millimeter, and Radio Photometry of Deneb Wavelength Flux Density Observation 1.25 µm 609±19 Jy COBE/DIRBE band 1A a 2.2 µm 268±8 Jy COBE/DIRBE band 2A a 2.2 µm 282±28 Jy IRTFK band b 3.5 µm 125±4 Jy COBE/DIRBE band 3A a 3.5 µm 122±12 Jy IRTFL band b 4.29 µm 89.5±9.5 Jy MSX band B1 c 4.35 µm 84.5±8.0 Jy MSX band B2 c 4.8 µm 81±8 Jy IRTFM band b 4.9 µm 74.5±2.2Jy COBE/DIRBE band 4 a 7.76 µm 33.6±1.7 Jy MSX band A c 10.2 µm 22±2 Jy IRTFN band b 11.99 µm 15.6±0.5 Jy MSX band B c 14.55 µm 11.2±0.5 Jy MSX band C c 20.0 µm 6.7±0.7 Jy IRTFQ band b 20.68 µm 5.38±0.45 Jy MSX band D c 60 µm 957±287 mJy ISOPHOTd 870 µm <24 mJy HHTd 1.35 mm 7.8±1.9 mJy SCUBA d 2 cm <1.0 mJy VLA e 3.6 cm 0.23±0.05 mJy VLA d 6 cm <0.15 mJy VLA e ahttp://cobe.gsfc.nasa.gov/cio/ bAbbottet al. (1984) cPrice et al. (2001); Egan et al. (1999) dThis paper eDrake& Linsky (1989) 4 Aufdenberg et al. 2.3. HHT Observation Deneb was observed using the 10-m Heinrich Hertz Telescope (e.g., Baars et al. 1999) on Mt. Graham, Arizona on 22 December 2000. We used the MPIfR 19-element bolometer array which is sensitive at 870 µm = 345 GHz. The observations were done during good weather conditions (τ < 0.06). We used source-sky offsets of 100′′ compared 225GHz to the telescope beam of 22.4′′. The total on source integration time was 1600 seconds with individual integration times of 20 seconds. Mars was observed to provide calibration to physical units. While we did not detect Deneb, we were able to put a 3σ upper limit on this star of 24.0 mJy at 870 µm. 2.4. ISO Data Reduction, DIRBE and MSX Data Infrared Space Observatory(ISO) observed Deneb at 60 µm with the ISOPHOT instrument (Lemke et al. 1996). This datasetwasdownloadedfromtheISOarchiveandreducedusingthePIApackage(Gabrieletal.1997). Theobservations consist of three pointings bracketed by observations of the FCS1 calibration source. Each pointing was offset by almost the full array width from the last pointing and Deneb was observed in the middle pointing. Deneb was centered on the middle pixel of the array. The sky level at the position of Deneb was taken as the mean of the sky levels on the middle pixel in the two adjacent observations. The flux of Deneb was the difference between the observed flux and the sky level so computed. Including the correctionfor the flux falling outside the centralpixel, the 60 µmflux ofDeneb was 957mJy. The uncertainty of this measurement was estimated at 30% (Schulz et al. 1999). Infrared photometric data between 1 µm to 5 µm are from the Diffuse Infrared Background Experiment (DIRBE) aboard the Cosmic Background Explorer (COBE). These data were obtained from the Astrophysical Data Facility using the the DIRBE PointSource PhotometryResearchTool. Additional infraredphotometric data between 4 µm and20 µm from the Midcourse Space Experiment (MSX) (Price et al. 2001; Egan et al. 1999) were obtained through IPAC. 2.5. Ultraviolet Data Sources Spectrophotometric data of Deneb from the Skylab S-0109 Far-UV Objective Prism Spectrophotometer (Henize et al. 1979), the Orbiting Astronomical Observatory-2 (OAO-2) spectrometers (Code & Meade 1979), the S2/68 Ultraviolet Sky Survey Telescope aboard TD1 (Jamar et al. 1976), and the International Ultraviolet Explorer (IUE) are all in close agreement regarding the shape and absolute flux level despite being obtained at different epochs (see Table 2). Figure 1 shows these data over their common interval from 7 µm−1 to 4 µm−1 (1400 ˚A to 2500 ˚A). The OAO-2 data do show systematically higher fluxes relative to the other data sets between 7 µm−1 and 6 µm−1. This OAO-2 excess has been noted in studies on the UV spectrophotometry of B-type stars (Bohlin & Holm 1984; Aufdenberg et al. 1999) and is expected. There is good agreement between IUE and OAO-2 longward of 2500 ˚A until the end of the LWR camera beyond 3300 ˚A. The only absolute spectrophotometric data shortwardof 1400 ˚A are from IUE. Only two large aperture, low dispersion spectra of Deneb were obtained with IUE, one each in the Short Wavelength Primary (SWP) and Long Wavelength Redundant (LWR) cameras, exposures SWP 09133and LWR 07864,respectively. We use the IUE data set exclusively in our SED analysis because it has the greatest wavelength coverage, is consistent with the other UV data, has probably the best absolute (and relative) flux calibration, and both the long and short wavelength spectra were taken on the same date. For the high-dispersion analysis, we have chosen the well-exposed, large-aperture spectra SWP 10375 and LWR 07760. The IUE data were obtained from the ESA “IUE Newly Extracted Spectra” (INES) system (Rodr´iguez-Pascualet al. 1999;Gonz´alez-Riestra et al. 2000). 2.6. Additional Spectroscopic and Spectrophotometric Data Sources Table 2 Low-Resolution UV Spectrophotometry of Deneb Observatory Wavelength Range(˚A) Dates # of Scans OAO-2a 1840–3600 1969 June19 1 OAO-2 1400–1800 1970 May 11 1 1970 June21 4 TD1b 1360–2740 1972 March 19 3 1972 October31 – Skylab 1347–2290 1973 August 10 3 IUE 1150–2000 1980 May 26 1 IUE 1850–3300 1980 May 26 1 aOAO-2 observation dates from Meade (2001). bNoobservation datesavailable. Datesshown arethoseof theTD1mis- sion. The A-type Supergiant Deneb 5 Fig. 1.—Comparisonofultravioletabsolutespectrophotometry ofDenebfromSkylab,IUE,OAO-2,andTD1. Seetextfordetails. 6 Aufdenberg et al. Optical spectrophotometric data are from Glushneva et al. (1992) and are in good agreement with data from the compilation by Breger (1976) except at 3975 ˚A, where the datum from Glushneva et al. is ≃ 50% lower. Two high- dispersion (R ≃ 20000) optical spectra with wavelength coverage from 3500 ˚A to 5400 ˚A (“blue”) and 5775 ˚A to 8630 ˚A (“red”) were obtained on 1 August 1995 with the Heidelberg echelle spectrograph HEROS (Kaufer 1999). Fourier Transform Spectroscopy (R≃3000) in the K-band is from Wallace & Hinkle (1997). 2.7. NPOI Observations Each interferometric observation consists of a scan made over ten spectral channels spread in increments of equal wavenumber from 850 nm to 650 nm with a mean wavelength of 740 nm. Here we consider only the visibility data from the 38-meter East-West baseline, considered the most robust (Nordgren et al. 1999). Deneb was observed 74 times over 5 nights in 1997 December. Unfortunately, 62 of these observations have no suitable calibration data. Of the remaining data, 5 scans from 1997 December 5, 7 scans from 1997 December 6, and 1 scan from 1997 December 16 have suitable calibrationdatatakenwithinonehourofeachsciencescan. These dataarelistedinTable3andplottedinFigure2. The calibration star for each night and its estimated angular size (Nordgren et al. 1999) are noted in Table 3. For a detailed description of the calibration process used for these data see Nordgren et al. (1999). The A-type Supergiant Deneb 7 Table 3 NPOI Deneb Observations ObservationTime Cal. Timea Wavelength V2 SpatialFrequency (UT) (hours) (nm) (cycles/arcsecond) 1997December5b 01:40:05 0.305 849.4 0.5778± 0.0075 195.5441 01:40:05 0.305 820.9 0.5572 0.0074 202.3308 01:40:05 0.305 793.9 0.5155 0.0073 209.2131 01:40:05 0.305 768.3 0.5089 0.0080 216.1828 01:40:05 0.305 744.2 0.4803 0.0072 223.1807 01:40:05 0.305 722.9 0.4605 0.0073 229.7603 01:40:05 0.305 701.5 0.4388 0.0064 236.7691 01:40:05 0.305 683.1 0.4165 0.0068 243.1457 01:40:05 0.305 664.6 0.3947 0.0063 249.9149 01:40:05 0.305 648.9 0.3704 0.0066 255.9606 01:52:05 0.505 849.4 0.5914 0.0078 192.7878 01:52:05 0.505 820.9 0.5689 0.0079 199.4806 01:52:05 0.505 793.9 0.5279 0.0074 206.2675 01:52:05 0.505 768.3 0.5210 0.0083 213.1379 01:52:05 0.505 744.2 0.4962 0.0075 220.0392 01:52:05 0.505 722.9 0.4759 0.0074 226.5235 01:52:05 0.505 701.5 0.4544 0.0068 233.4355 01:52:05 0.505 683.1 0.4302 0.0076 239.7208 01:52:05 0.505 664.6 0.4045 0.0065 246.3961 01:52:05 0.505 648.9 0.3910 0.0068 252.3545 01:59:49 0.634 849.4 0.5929 0.0079 190.9612 01:59:49 0.634 820.9 0.5655 0.0081 197.5905 01:59:49 0.634 793.9 0.5261 0.0076 204.3120 01:59:49 0.634 768.3 0.5197 0.0084 211.1178 01:59:49 0.634 744.2 0.4928 0.0078 217.9575 01:59:49 0.634 722.9 0.4733 0.0078 224.3784 01:59:49 0.634 701.5 0.4519 0.0071 231.2249 01:59:49 0.634 683.1 0.4297 0.0078 237.4519 01:59:49 0.634 664.6 0.4079 0.0068 244.0599 01:59:49 0.634 648.9 0.3907 0.0074 249.9678 02:12:32 0.846 849.4 0.5999 0.0086 187.9221 02:12:32 0.846 820.9 0.5776 0.0082 194.4449 02:12:32 0.846 793.9 0.5388 0.0080 201.0607 02:12:32 0.846 768.3 0.5376 0.0089 207.7591 02:12:32 0.846 744.2 0.5136 0.0084 214.4886 02:12:32 0.846 722.9 0.4924 0.0080 220.8083 02:12:32 0.846 701.5 0.4722 0.0074 227.5415 02:12:32 0.846 683.1 0.4481 0.0085 233.6719 02:12:32 0.846 664.6 0.4306 0.0071 240.1741 02:12:32 0.846 648.9 0.4116 0.0076 245.9880 1997December6c 01:16:52 0.087 849.4 0.4991± 0.0100 199.7180 01:16:52 0.087 820.9 0.4534 0.0088 206.6525 01:16:52 0.087 793.9 0.4377 0.0090 213.6804 01:16:52 0.087 768.3 0.4091 0.0107 220.8014 01:16:52 0.087 744.2 0.3942 0.0095 227.9508 01:16:52 0.087 722.9 0.3724 0.0091 234.6640 01:16:52 0.087 701.5 0.3710 0.0088 241.8242 01:16:52 0.087 683.1 0.3338 0.0095 248.3401 01:16:52 0.087 664.6 0.3138 0.0088 255.2507 01:16:52 0.087 648.9 0.2957 0.0084 261.4261 01:44:35 0.074 849.4 0.6075 0.0103 193.6147 01:44:35 0.074 820.9 0.5023 0.0075 200.3382 01:44:35 0.074 793.9 0.4984 0.0077 207.1531 01:44:35 0.074 768.3 0.4752 0.0085 214.0514 01:44:35 0.074 744.2 0.4401 0.0081 220.9846 01:44:35 0.074 722.9 0.4189 0.0078 227.4956 01:44:35 0.074 701.5 0.4107 0.0077 234.4356 01:44:35 0.074 683.1 0.3948 0.0074 240.7516 01:44:35 0.074 664.6 0.3711 0.0076 247.4509 01:44:35 0.074 648.9 0.3683 0.0081 253.4416 02:14:50 0.048 849.4 0.5821 0.0094 187.9029 02:14:50 0.048 820.9 0.5490 0.0084 194.4257 02:14:50 0.048 793.9 0.5181 0.0086 201.0415 02:14:50 0.048 768.3 0.5153 0.0093 207.7393 02:14:50 0.048 744.2 0.4667 0.0085 214.4687 02:14:50 0.048 722.9 0.4568 0.0085 220.7885 02:14:50 0.048 701.5 0.4567 0.0092 227.5210 02:14:50 0.048 683.1 0.4371 0.0083 233.6514 02:14:50 0.048 664.6 0.4057 0.0083 240.1537 02:14:50 0.048 648.9 0.4228 0.0095 245.9638 02:27:25 0.054 849.4 0.6579 0.0140 183.3663 02:27:25 0.054 820.9 0.6186 0.0130 189.7327 02:27:25 0.054 793.9 0.5967 0.0127 196.1848 02:27:25 0.054 768.3 0.6022 0.0138 202.7225 8 Aufdenberg et al. A uniform-disk (UD) angular diameter θ was derived for each calibrated scan by fitting an Airy pattern UD 2 J (2kwα ) V2(kw)= 1 UD (1) kwα " UD # to the visibility data, where kw is the spatial frequency, k = 2π/λ is the wavenumber and w is the projected baseline, α = θ /2 is the angular radius, and J is the Bessel function of first order. Table 4 lists the least squares best fit UD UD 1 value of θ for each scan and the mean value of θ for each night. The model visibility curves corresponding to the UD UD mean and ±1σ values of θ for each night are shown in Figure 2. UD The scans of December 5th are very consistent with each other and with the single scan from December 16th which, whilenoisier,yieldsaconsistentangulardiameter. ThescansfromDecember6th,however,showconsiderablescatterand each scan is considerably noiser than the December 5th data, yielding higher χ2 values for each of the fits. A histogram of all the derived θ values is shown in Figure 3. Experience has shown that with well-observed stars the distribution UD of angular diameters follows a Gaussian distribution in the limit of many (∼100) scans. The weighted mean angular diameter for the 12 scans is θ =2.40±0.06 mas, (2) UD where the uncertainty represents our estimate for the error in the mean. This value is consistent with the published measurements by the CERGA group: θ = 2.04±0.45 mas (in the spectral band 600±25 nm) (Koechlin & Rabbia UD Table 3—Continued ObservationTime Cal. Timea Wavelength V2 SpatialFrequency (UT) (hours) (nm) (cycles/arcsecond) 02:27:25 0.054 744.2 0.5811 0.0133 209.2878 02:27:25 0.054 722.9 0.5517 0.0132 215.4553 02:27:25 0.054 701.5 0.5787 0.0142 222.0274 02:27:25 0.054 683.1 0.5387 0.0135 228.0098 02:27:25 0.054 664.6 0.5204 0.0142 234.3555 02:27:25 0.054 648.9 0.5041 0.0135 240.0260 02:51:50 0.060 849.4 0.6771 0.0187 177.5475 02:51:50 0.060 820.9 0.6405 0.0166 183.7125 02:51:50 0.060 793.9 0.6047 0.0155 189.9600 02:51:50 0.060 768.3 0.6327 0.0190 196.2900 02:51:50 0.060 744.2 0.5964 0.0165 202.6481 02:51:50 0.060 722.9 0.5696 0.0163 208.6200 02:51:50 0.060 701.5 0.6360 0.0245 214.9848 02:51:50 0.060 683.1 0.5725 0.0190 220.7742 02:51:50 0.060 664.6 0.5513 0.0199 226.9180 02:51:50 0.060 648.9 0.5760 0.0217 232.4076 03:18:22 0.065 849.4 0.5428 0.0099 171.7017 03:18:22 0.065 820.9 0.5794 0.0111 177.6646 03:18:22 0.065 793.9 0.5567 0.0108 183.7040 03:18:22 0.065 768.3 0.5434 0.0119 189.8256 03:18:22 0.065 744.2 0.5123 0.0103 195.9740 03:18:22 0.065 722.9 0.5012 0.0110 201.7472 03:18:22 0.065 701.5 0.5088 0.0111 207.9038 03:18:22 0.065 683.1 0.4848 0.0119 213.5034 03:18:22 0.065 664.6 0.4523 0.0111 219.4446 03:18:22 0.065 648.9 0.4825 0.0128 224.7564 03:44:02 0.095 849.4 0.6227 0.0121 166.8693 03:44:02 0.095 820.9 0.5874 0.0107 172.6637 03:44:02 0.095 793.9 0.5863 0.0124 178.5318 03:44:02 0.095 768.3 0.5590 0.0121 184.4802 03:44:02 0.095 744.2 0.5377 0.0115 190.4547 03:44:02 0.095 722.9 0.5383 0.0123 196.0689 03:44:02 0.095 701.5 0.5341 0.0125 202.0478 03:44:02 0.095 683.1 0.5186 0.0111 207.4907 03:44:02 0.095 664.6 0.4895 0.0127 213.2697 03:44:02 0.095 648.9 0.4918 0.0150 218.4261 1997December16d 01:03:29 0.214 849.4 0.5761± 0.0216 194.0322 01:03:29 0.214 820.9 0.5435 0.0134 200.7689 01:03:29 0.214 793.9 0.5133 0.0142 207.5969 01:03:29 0.214 768.3 0.5163 0.0155 214.5110 01:03:29 0.214 744.2 0.4526 0.0129 221.4574 01:03:29 0.214 722.9 0.4671 0.0129 227.9843 01:03:29 0.214 701.5 0.5164 0.0184 234.9374 01:03:29 0.214 683.1 0.4550 0.0148 241.2693 01:03:29 0.214 664.6 0.4434 0.0153 247.9844 01:03:29 0.214 648.9 0.4804 0.0193 253.9841 aTimebetweenthesciencesourcescanandthecalibrationscanusedtocalibrateit. bCalibration: γ Peg,θUD =0.421mas cCalibration: αLac,θUD =0.502mas dCalibration: ζ Cas,θUD =0.264mas The A-type Supergiant Deneb 9 Fig. 2.—NPOIvisibilitydata ofDeneb withuniform-disk(UD)fits. For eachdate, amodelvisibilitycurve(solidline)forthe meanUD diameter, θUD (see Table 4), is plotted along with the corresponding visibility curves for the ± 1σ error in the mean UD diameter (dotted lines). (a)5December1997: θUD=2.36±0.01mas(4scans);(b)6December1997: θUD=2.41±0.17mas(7scans);(c)16December1997: θUD =2.33±0.02mas(1scan). Table 4 Uniform-Disk Angular Diameters Observation Time θ χ2 UD (UT) (mas) 1997 December 5: θ = 2.364 ± 0.011 UD 01:40:05 2.364 ± 0.007 4.7 01:52:05 2.353 ± 0.007 5.0 01:59:49 2.379 ± 0.007 4.9 02:12:32 2.360 ± 0.008 7.1 1997 December 6: θ = 2.410 ± 0.168 UD 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 1997 December 16: θ = 2.325 ± 0.015 UD 01:03:29 2.325 ± 0.015 61 10 Aufdenberg et al. 1985)and θ =2.6±0.3mas(inthespectralrange500-650nm)(Bonneauetal.1981). HanburyBrown’sinterferometer UD at Narrabri could not observe Deneb due to its high declination. 3. angular diameter Ourmodelatmospheres(describedindetailinAppendixA)predictsubstantiallimbdarkeningfortheDenebparameters determined in the remainder ofthis paper. This motivates further analysisof the NPOI observationsin orderto obtain a more physically based estimate of the angular diameter. A limb-darkened angular diameter, θ , is normally employed LD in the calculation of the fundamental effective temperature 1/4 4F 0 T = . (3) eff σθ2 " LD# where F is the bolometric flux (corrected for interstellar extinction) and σ is the Stefan-Boltzmann constant. Unfortu- 0 nately, in extended atmospheres the effective temperature is not well defined or is ambiguous because the atmosphere is not sufficiently compact to have a unique radius (Baschek et al. 1991). Hence, a value for T cannot be separated from eff an adopted value for the radius and the corresponding angular size. For well resolved sources (e.g., Betelgeuse, Arcturus), observations can directly yield limb-darkening information from the shape of the visibility curve at spatial frequencies beyond the first null. Deneb, however, has yet to be observed at sufficiently long baselines (>100 m) for limb-darkening information to be obtained in this way. Therefore, the details of Deneb’s center-to-limb variation (CLV) must for now be supplied by theory. Limb-darkeningcoefficientsprovidedintheliterature(e.g.,Claret(2000))arederivedfromplane-parallelmodelsandare not appropriate for Deneb. For example, Figure 4a shows that the model CLV from an expanding atmosphere or “wind” model differ significantly from that of a hydrostatic model. Unlike the wind model CLV which falls very gradually from 40% of the central disk intensity to zero, the hydrostatic model CLV shows a fairly well defined edge. The unambiguous signature of the predicted wind model CLV lies in the height of the visibility function beyond the first null (see Figure 4b). Future measurements should be able to test this prediction. Fig. 3.— Histogram of the uniform-diskangular diameter values derived from each of the 12 scans of Deneb by the NPOI (see Table 4). Thedataarebinnedto0.05mas. Thedifferentcross-hatchedregionsspecifythedateofeachscan.