arXiv:astro-ph/0509236v1 9 Sep 2005 eno,VA20171 Herndon, [email protected] idlynlnv.i,he [email protected], email: [email protected], 20375 [email protected] DC Washington, [email protected], SW, Ave. ahntn C 09-40eal [email protected] email: 20392-5420 DC, Washington, 00 mi:[email protected] email: 20706 nvriy tn ro,N 19-80eal dpeter- email: [email protected] email: 11794-3800 Chile 19, ago NY Brook, sig- [email protected] Stony that mean University, to indica- taken these is of types absence spectral The earlier al. at 1995). et tors al. (Simon et chromosphere Walter a of 1994; indicators as taken sion, bevtr d,Fasa,A 60-51eal jben- email: 86001-8521 AZ Flagstaff, [email protected] [email protected], Rd., Observatory r h w ots tr hwn Ly showing stars hottest two the are eoiy siae er40k s breakup km 400 the near of estimated fraction velocity, rotational significant true a the already for are limit velocity, lower a velocity, projected its s km 200 s km and 217 1982), Fukuda & (Uesugi u bv hto h u.Seicly larand Altair near Specifically, masses Sun. at the stars of that sequence above main but of atmospheres the oainlvlct aiul siae t22k s km 242 at estimated projected variously large a velocity to Unlike due spectrum be rotational diffuse to stars. rather recognized early a A was shows which stars notable “Sum- Altair the brightest Deneb, other and in two the Vega membership with of sharing Triangle” one sky, mer is Northern VI-V) the A7 in type spectral of eie rf fJn 1 2018 21, June of draft Revised 4 7 3 2 6 5 1 larhsbcm infiatojc nunderstanding in object significant a become has Altair lar(variously Altair umte othe to Submitted rpittpstuigL using typeset Preprint .M Peterson M. D. uoenSuhr bevtr EO,Csla101 Santi 19001, Casilla (ESO), Observatory Southern European nefrmtis n. 35 urs alyDie Suite Drive, Valley Sunrise 13454 Inc., Interferometrics, ..NvlOsraoy lgtffSain 09 .Naval W. 10391 Station, Flagstaff Observatory, Naval U.S. ebokEgneig 30DbryR. ebok MD Seabrook, Rd., Dubarry 9310 Engineering, Seabrook ..NvlOsraoy 40MsahstsAe NW, Ave. Massachusetts 3450 Observatory, Naval U.S. aa eerhLbrtr,Cd 25 55Overlook 4555 7215, Code Laboratory, Research Naval eateto hsc n srnm,SoyBrook Stony Astronomy, and Physics of Department odeapeo h iclisrtto a nrdc nteinterpr the in introduce o headings: can barely Subject rotation only difficulties is the Altair of diagram. VI-V, A7 example classification, good its a by indicated as evolved n hwta h reoltns ssgicnl agroigt nin an to owing larger significantly o is apparent oblateness the true with of the consistent that are show results and Our velocity. (angular) cuayo hsfi,wihnvrhls niae htAti srota is appear Altair star that magnitude indicates first nevertheless this which of fit, flux this visible detect of high such Hutte accuracy the first & the to with Nordgren, rotation, due made Ohishi, to effects Altair by due distribution of claim intensity observations the metric to confirm We law, darkening Interferometer. gravity Zeipel von erpr ucsflfitn faRcemdl ihasraetemp surface a with model, Roche a of fitting successful report We ∼ − 64 1 umte othe to Submitted At&Mre 95.Teeetmtsof estimates These 1995). Morrel & (Abt ◦ Hindsley oteln fsgt fpriua neet ecnld htinstead that conclude we interest, particular Of sight. of line the to .INTRODUCTION 1. srpyia Journal Astrophysical α 1 .A Hummel A. C. , A q,5 q,H 57 D187642, HD 7557, HR Aql, 53 Aql, T E tl mltajv 6/22/04 v. emulateapj style X tr:rtto—tr:iaigsas niiul( individual imaging—stars: rotation—stars: stars: 3 metric .J Hutter J. D. , EOVN H FET FRTTO NALTAIR IN ROTATION OF EFFECTS THE RESOLVING srpyia Journal Astrophysical − − IHLN-AEIEINTERFEROMETRY LONG-BASELINE WITH 1 . 1 2,4 Ryre l 2002), al. et (Royer α ue9 2005 9, June .A Pauls A. T. , n C and 5 .J Johnston J. K. , Rcie 05Jn 9) June 2005 (Received II ue9 05Rvsddato ue2,2018 21, June of draft Revised 2005 9, June il α emis- 3 Cep ABSTRACT 240, .T Armstrong T. J. , − n- 1 - 4 .Mozurkewich D. , eerdt s“rvt aknn” ihsffiin rota- sufficient With darkening”. “gravity as to referred ae oteeetv rvt as re- gravity is effective temperature the de- centrifugal effective to one for local if lated accounting the Specifically, gravity then effective acceleration, equator. local the rotational a display at the fines cooler would on and disks hotter axes stellar temperatures, rotation surface variable moderate for that the on vector sky. that the of angular of the projection plane of the angle vector, position momentum oblateness the theory, yield by provid- predicted do flattening Besides measurements the determine retrograde. of test or one a pro- can ing rotation, Nor of separately. sense two the the not but well, n 8 ms km 180 s km and 196 1982), Fukuda & (Uesugi up- the that note on We point sequence. this main at per disappears convection nificant oainvlct:oedtrie h quantity the determines between apparent one the degeneracy as velocity: (inclination) same rotation tilt and the velocity displays equatorial through Except theory, interpreted incomplete. oblateness, Roche is apparent it epochal, breakup, of near short nothing is tion rotation. projected observed con- flattening the completely significant was with 1924) flattening sistent Zeipel a the (von showed that classical models Roche to (2001) Comparison 1999) Palo- al. detected. al. et was the et Belle it with van Colavita However, observations (PTI; by near-IR Interferometer Inter- Testbed the Intensity 1974). mar until the rota- al. not with its et was Brown starting of (Hanbury years, effects ferometer the geometrical over the tion measure to tempts rjce oainvlct ihetmtso 4 ms km 246 of estimates with velocity rotation projected nadto oflteig o epl(94 predicted (1924) Zeipel von flattening, to addition In lhuhti gemn ewe hoyadobserva- and theory between agreement this Although at- prompted has rate rotation high known Altair’s 3 .A Benson A. J. , o na sltdsa.Instrumental star. isolated an in ion − ltns on yvnBlee al. et Belle van by found blateness 1 iga 0 at ting At&Mre 95 listed. 1995) Morrel & (Abt htAti ipasa asym- an displays Altair that r α ob h iiigfco nthe in factor limiting the be to lnto fterttoa axis rotational the of clination rtr rdetfloigthe following gradient erature tto fti ato h HR the of part this of etation 6 q) ehius interfero- techniques: Aql)— h ayPooyeOptical Prototype Navy the .Schmitt H. & , h ASadrepresents and ZAMS the ff . 90 5 fbigsubstantially being of ± .G Gilbreath G. C. , 0 . 2o t breakup its of 02 T − α eff 3,7 4 1 e lohsahigh a has also Cep Ryre l 2002), al. et (Royer ∝ g eff 3 .B. R. , hc is which , v eq sin − 1 i 2 Peterson et al. tion and at intermediate inclinations, gravity darkening 2. OBSERVATIONS predicts that stellar disks will display asymmetric inten- Altair was observed on four nights, 25–27 May and sity distributions. 1 June of 2001, with the NPOI. These are the same ob- As we describe below, this prediction is of great inter- servations used by Ohishi et al. (2003) and Ohishi et al. est in the field of optical interferometry. Asymmetric in- (2004); we refer the reader to those papers for a journal tensity distributions produce significant imaginary com- of observations and a description of the observing details, ponents in the visibilities, usually represented as a non- but we briefly reprise them here. We have focused here trivial visibility phase. Recently developed techniques on the data set obtained May 25, 2001. This is by far the for recovering a closely related quantity, “closure phase” largest set of data, while the other data do not increase (Baldwin et al. 1996; Benson et al. 1997), are now be- the range of hour angles observed in the first night. ing applied to the first round of stellar objects, (e.g., The observations used the Astrometric West (AW), As- Wittkowski et al. 2001). trometric East (AE), and West 7 (W7) stations, form- Although originally proposed as a follow-up on the ing a triangle of interferometric baselines with lengths oblateness observations, Altair was observed at the of 37.5 m (AW–AE), 29.5 m (W7–AW), and 64.4 m (AE– Navy Prototype Optical Interferometer (hereafter NPOI, W7). The backend combined these three input beams to Armstrong et al. 1998) while the three beam combiner produce three output beams, with one baseline on each. was in operation, allowing measurement of closure phase The output beams were dispersed into 32 spectral chan- around one complete triangle. Examination of the data nels covering λλ443 − 852 nm, although the bluest four immediately revealed the intermediate phase angles, un- channels (λλ443 − 460 nm) of the W7–AW output were ambigiously signaling the presence of an asymmetric in- not functioning. tensity profile (Ohishi et al. 2003). The Altair observations were interleaved with obser- Using a model consisting of a limb-darkened disk and a vations of a calibrator, ζ Aql (A0 V), about 12◦ away bright spot, Ohishi et al. (2004) demonstrated both the on the sky. We initially estimated its diameter to be previously discovered oblateness and the necessity of in- around 0.85 mas, with which it would have acted as cluding asymmetries in the intensity distribution. They a quite acceptable calibrator. However, as noted by argued that the probable interpretation was that of ro- Ohishi et al. (2004), ζ Aql is a rapid rotator with values tational flattening and gravity darkening. − − of 345kms 1 (Uesugi & Fukuda 1982) and 317kms 1 In the meantime we have become aware of some lim- −1 itations in those data due to inadequate corrections for (Royer et al. 2002) reported. We adopted 325kms , “deadtimes” in the avalanche photodiode detectors which raising the question of aspect dependent corrections to affect the high signal levels from objects as bright as Al- the squared visibilities and phases. This possibility was tair. We therefore reconsider a subset of these data that discussed by Ohishi et al. (2004) who concluded that the is relatively immune to the detector problems, using a effects were not important at the level of the analysis they full implementation of von Zeipel’s theory (von Zeipel conducted, but who cautioned that the problem needed 1924) for the model fitting, and redoing the reductions to be reconsidered if a detailed analysis of Altair was in a way that dramatically reduces noise in the bluest attempted with these observations. channels. We find that a Roche model rotating at 90% We have found a number of occasions in 2004 when ◦ ζ Aql was observed with a second calibrator, γ Lyr of the breakup angular velocity and inclined ∼ 64 from −1 pole-on fits the observations with high fidelity. (B9 III). γ Lyr is a relatively slow rotator (∼ 70kms , We show that the parameter that sets the overall tem- Royer et al. 2002) and a bit fainter (V ∼ 3.24) which, perature scaling for the model, the effective temperature coupled with slightly higher temperature leads to the ex- at the poles, Tp, is close to 8700 K for this model, and pectation of a symmetric, nearly unresolved calibrator the polar surface gravity is correspondingly fairly high. for ζ Aql. This turned out to be correct, and to our This suggests that Altair is less evolved than one might surprise, we found that we were able to deduce a mean- naively expect from its spectral type and luminosity clas- ingful fit of a Roche model to ζ Aql using the phase and sifications. visibility amplitude data. Since such results are rare, In this model the equator is 1850K cooler than the our report on Altair here is the first, we have decided pole. Given that the model includes both polar bright- to present a detailed discussion of the case in a separate ening and a long equatorial swath of low intensity, this is communication (Peterson et al. 2006). a complex intensity distribution, and the agreement with We summarize in columns 6 and 7 of Table 1 a pre- the observations is a strong endorsement for the simple liminary set of the relevant Roche parameters for ζ Aql. von Zeipel (1924) theory. Given the small angular diameter, the quoted errors pro- Below, we describe the new reductions, give a brief duce uncertainties in the calibration of Altair that are review of Roche theory and then present the fits. We undetectable compared to other error sources. One prop- note that the existence of large amounts of surface at erty of note from these parameters is the near equator-on near-solar temperatures suggests that the role of Altair orientation of the spheroid. At most we would have ex- (and probably α Cep) in defining the high temperature pected a few degrees uncertainty in the triple phases in- end of convection on the main sequence may need to duced by the range of possible inclinations. However, the be reconsidered. We also note the recent announcement observations calibrated by γ Lyr show that these phases are truly small, indistinguishable from zero at the ±1◦ that Altair is a low amplitude (δ Sct) pulsating star ◦ (Buzasi et al. 2005), which may give hope that astero- level. The star is apparently seen at nearly i ∼ 90 . seismology will be able to put useful limits on any gra- Ohishi et al. (2004) used contemporaneous observa- dient of the angular velocity in the outer envelope. tions of Vega as a check star - one that should show a circular outline. Although we agree those observations Resolving Altair 3 do seem to imply that Vega is circular, we are not in- quantities were then averaged (averaging real and imag- clined to place much weight on the result. Vega is twice inary part of the complex triple products separately) for as bright as Altair in this wavelength range. Detector a total integration time per data point of 2 s. nonlinearities, already serious in the Altair observations The alignment of the raw visibility phasors neccessary as we describe in §3, are overwhelming here. We sim- before integration in order to avoid detrimental coher- ply do not know how to interpret the observed squared ence losses was performed as follows. Two steps are nec- visibilities. essary to rotate the phasors onto a common fringe in Finally, we note that since the NPOI records phases order to enable a phase tracking algorithm. and visibilities over a range of wavelengths, there is little First, average power spectra of the channeled visibility possibility of contamination by unknown companions, at as a function of delay were computed for 10 ms intervals. least those within the 0 ′′. 5 field of view of the siderostats. Their maxima, corresponding to the group delays of the If such existed there would be clear, strongly modulated fringe packets, are not zero but have a typical RMS on phases and visibilities. ζ Aql is a known member of a the order of one micron as the NPOI group delay fringe wide multiple system, ADS12026, with companions no tracker tries to center the fringe, but does not lock onto closer than 5 arcseconds and magnitude differences no its phase. less than 8. Second, from the deviation in position of the fringe 3. DATA REDUCTION from the estimated geometrical value, which is non-zero and has a typical RMS on the order of 10 µm due to 3.1. Incoherent Integration atmospheric refractive index fluctuations, we estimated The NPOI observes interference fringes by modulating the differential amount of air and thus the phase shift be- the optical path on the delay line for each array element, tween the peak of the envelope of the fringe packet and using a triangle-wave pattern at a frequency of 500 Hz. the nearest fringe peak. In other words, this phase is the The resulting modulation of the intensity is detected in phase of the complex Fourier transform of the visibility 8 bins evenly spaced over one fringe in each channel by as a function of wavenumber. The modulus of this trans- Avalanche Photo Diodes (APD). The phase of the inten- form peaks at the value of the group delay. The phase sity modulation changes on time scales of milliseconds of the transform at this delay is called the group delay since the fringe tracker employed by NPOI tracks the phase. envelope of the (bandwidth limited) fringe packet rather We converted the group delay phase to a delay using than the fringe phase. the mean wavelength of the white-light fringe, and added The data in the delay bins were processed to produce it to the group delay. Rotation of the visibility phasors the complex visibility and squared visibility modulus V 2 of different channels by an angle corresponding to the for each baseline at each wavelength. From these the ratio of this delay value and the wavelength of the spec- triple product V123 exp iφcl can be calculated, where the trometer channel will align them on the same fringe. At triple amplitude V123 = |V1||V2||V3| is the product of the this point, the algorithm implements a photon-noise lim- amplitudes of the complex visibilities of the individual ited off-line fringe phase tracker enabling the use of much baselines and the closure phase φcl = φ1 + φ2 + φ3 is longer coherent integration times. the sum of the individual phases. Although the base- line phases themselves are affected by atmospheric tur- 3.3. Baseline Bootstrapping bulence, those effects cancel in the sum of three phases We used an important modification of the above around a closed triangle, so the closure phase preserves procedure by applying the baseline bootstrapping information about the source structure. These data method, a design feature of the NPOI interferometer products are produced for each 2ms cycle of delay mod- (Armstrong et al. 1998). It exploits the fact that the ulation. In the standard incoherent integration as de- sum of the fringe delays along a closed loop of baselines scribed by Hummel et al. (1998), the squared visibilities is zero (if the same fringe is identified on each baseline). and complex triple products are summed to provide av- Therefore, if a long baseline in a multi-telescope array erage values in one second intervals. sees a low contrast fringe due to, e.g., object extension, 3.2. Coherent Integration and this baseline involves two telescopes which are at the same time involved with other telescopes of the ar- We employed a new algorithm for the coherent inte- ray on much shorter baselines seeing much higher fringe gration of the complex visibilities of the NPOI first pre- contrast, the fringe delay of the long baseline can be sented by Hummel et al. (2003). Compared to the in- computed from the fringe delays on the shorter baselines coherent integration of the squared visiblities, coherent which are “bootstrapping” the long one. In the simple integration achieves a higher SNR of the averages due to case of the observations described on Altair, the fringe the larger number of photons detected in a coherent sam- delay on the long 64 m W7–AE baseline is just the dif- ple of the fringe. We have exploited this fact to recover ference between the fringe delays on the shorter AE–AW meaningful results from all NPOI spectrometer channels, and W7–AW baselines. while the channels on the blue side of about 560 nm had usually been discarded in incoherent reductions due to the insufficient number of photons detected during a 2 ms 3.4. Averaging and Editing instrumental integration time. For the coherent integra- The 2 s data points are edited for outliers as described tion time we selected 200 ms, and the resulting complex by Hummel et al. (1998). The final averaging is done visibilities were both combined to form complex triple over the full length of a pointing (called a scan at NPOI) products and transformed individually into squared am- which lasts typically 90 s. The computation of the for- plitudes of the modulus. Every ten samples of these mal errors also follows Hummel et al. (1998), except that 4 Peterson et al. we have implemented a different approach for the com- where N¯0 = QI0. A DFT at the dither frequency now plex triple products based on a suggestion by D. Buscher extracts an amplitude different than the nominal N¯0V (priv. comm.). Under simple assumptions, the error of (and after division by the nominal mean signal produces a complex triple product is described by an error ellipse an estimate that can differ significantly from the true which has one axis aligned with the triple product pha- visibility amplitude). However, the phase comes through sor. Assuming this, we compute the error of the triple the process unaffected, as do the frequencies of the min- product as the error of the imaginary and real parts of ima in the visibility amplitudes. By floating the overall the mean after applying a rotation of all phasors by the amplitudes in the data reduction (see below) we retain mean triple product phasor. The errors of amplitude and the important spatial scale information contained in the phase of the triple product are then equal to the error of minima. But the most important conclusion is that the the real part and the error of the imaginary part divided phases may be assumed to be essentially free of detector by the respective amplitudes. induced biases.

3.5. Detector Non-linearity 3.6. Visibility Calibration A source of systematic error comes from deadtime in The degradation of the measured visibilities due to at- the pulse counting electronics controlling the avalanche mospheric and instrumental effects is measured, as with photodiode detectors. These systems saturate at about all interferometers, by observing calibrator stars with di- 1 MHz and display significant non-linearity in the appar- ameters as small as possible to reduce uncertainties in the ent count rates as they near this limit. The nominal de- visibility estimates for them. As shown by Hummel et al. sign of the detector systems included a τ = 200ns dead- (1998), the NPOI visibility amplitudes sometimes show a time, but we have subsequently found that not only do negative correlation with the RMS of the delay line mo- those time constants vary significantly, channel to chan- tion which is related to the seeing. But at other times, nel, but also that in a given channel they depend on the instrumental effects which correlate with time or other mean signal level because of the effects of heating. We systematic effects which correlate with position on the believe that it will be possible to model and remove these sky (e.g., hour angle) can dominate the visibility vari- effects, but some effort is involved, which we will report ations. Therefore, formal photon-noise based visibility on in the future. Unfortunately, these problems, which errors usually require the addition (in quadrature) of a do not affect the fainter objects usually observed by the calibration error which is derived from the residual vis- instrument, were not recognized at the time the Altair ibility variations of the calibrator after calibration. For observations were made. the amplitude calibrations, we smoothed the calibrator However, we believe that through a rather unique set visibilities with a 20 min Gaussian kernel in hour angle, of circumstances the phase and some of the amplitude and obtained calibration errors ranging from about 4% at measurements acquired during the 2001 observations are the red end to 15% at the blue end of the spectrometers. to first order free of the effects of these non-linearities. For the closure phase, we used the same smoothing tech- One reason was that during these observations only three nique but applied to the calibrator phases as a function stations were in use and the three spectrographs recorded of time. single baseline data. During model fitting, however, this standard procedure The other reason was that the amplitude and phase yielded inconsistent fits. One can already see this from measurements from each channel were accomplished with the results of Ohishi et al. (2004) (their Fig. 6), where a simple Discrete Fourier Transform. By dithering the amplitudes can be systematically high or low with the optical delay at a frequency of ω, the signal was modu- important characteristic that the deviation is very con- lated according to sistently independent of wavelength. The effect is exac- erbated by the fact that due to the brightness of Altair, I(t)= I0 [1 + V cos(ωt + φ)] (1) the formal amplitude errors are quite small. The reason where V is the (instrumental) amplitude of the visibil- for the scan-to-scan variations is most likely the same as ity and φ the instantaneous phase. The detector system for the residual variations of the calibrator after calibra- responded to the modulated signal according to tion, except that there is no perfect correlation due to target and calibrator not being at the same location in QI(t) the sky. (Past experience has shown that visibilities do N(t)= ∼ N (t) − τN 2(t)+ ··· . (2) 1+ τQI(t) 0 0 correlate quite well if the calibrator is very near the tar- get.) Therefore, we allowed the calibration for each scan Where Q is the quantuum efficiency, N(t) is the apparent and baseline to float by applying “achromatic” calibra- pulse rate and N0(t) = QI(t) is the true photon detec- tion factors to improve the fit between data and model. tion rate. In the linearized form we assumed τN0 << 1, We will discuss the implications for the model fitting in which for a 300kHz count rate, typical of the wider red Section 4. Finally we note that an error in the calibrator channels on the higher visibility baselines, is adequate star diameter produces chromatic errors across the NPOI to 1% or better. Substituting equation 1 into 2, clear- spectrometer as do uncompensated deadtime corrections, ing the quadratic cosine using the half angle formula and neither of which can be removed by the calibration fac- collecting terms, this becomes tors. ¯ ¯ 2 N(t) ∼ N0 1 − τN0 1+ V /2
 4. MODELING

+N¯0V (1 − 2τN¯0) cos(ωt + φ) 4.1. Roche Spheroids 2 τN¯ V 2 The theory for the equilibrium shapes and surface − 0 cos(2ωt +2φ) (3) 2 properties of rotating stars was first presented 80 years Resolving Altair 5 ago (von Zeipel 1924) assuming solid body rotation and ignore all these effects here, except to acknowledge the a point source gravitational potential. This model has limitations inherent in this theory. proved quite successful in describing the figures of stars in close binary systems, where tidal effects to first order 4.2. Roche Models produce the same distortions as rotation (Collins 1989). We use that model here, but note its limitations. First, We have constructed a suite of programs to evalu- there is no a priori reason that stars should rotate as ate the run of specific intensity across the surface of a solid bodies, and the surface layers of the Sun have long Roche spheroid. The definitions of the various angles been known to rotate differentially. However, among the are from Collins (1963). Otherwise, we follow the pre- early results of helioseismology was the discovery that the scription for the surface figure and the notation given transition from the outer convection zone to the inner by Hardorp & Strittmatter (1968) with one exception: radiative layers coincided with an abrupt transition to following the discussion by Hardorp & Scholz (1971) we solid body rotation (Spiegel & Zahn 1992). Since early- take the polar radius, Rp, as a fixed parameter. Specifi- type stars have radiative envelopes and relatively small cally we do not allow it to be a function of the fractional (in radius) convective cores, one might expect solid body rotation. rotation to be a good approximation for the external lay- The modeling requires that we specify six quantities: ers of early type stars. the ratio of the angular velocity to that of breakup, There has been some observational support for this ω = Ω/ΩB, the inclination (or tilt) of the rotational expectation. Reiners & Royer (2004) have analyzed the axis, i, defined such that i = 0 is pole-on, the posi- rotational profiles of a large number of A stars looking tion angle, P A, of the pole on the sky (measured North for evidence of differential rotation following a solar-type through East), the angular diameter of the polar axis, latitudinal dependence. In the 78 stars for which the de- θp, the effective temperature at the pole, Tp, and the termination could be made, they found 4 objects where surface gravity, or more commonly the logarithm of the pecularities were seen which might be from differential surface gravity (cgs), at the pole, log gp. From the re- rotation (or other causes). However, 95% of the line lations in the cited references it is then possible to cal- shapes were fully consistent with solid body rotation. culate the radius of the star for a given stellar latitude, In addition there is the long known consistency θ, and hence the surface gravity geff and effective tem- 4 4 between the largest rotational velocities measured in perature (T (θ) = Tp (geff /gp)) at that latitude (see the early-type stars and the predicted maximum ro- Hardorp & Strittmatter 1968, for the definition of geff ). tation velocities associated with “equatorial breakup” Note that for stars with accurate parallaxes like Altair, (Fr´emat et al. 2005). In recent decades (e.g., Tassoul specifying the polar surface gravity and angular diame- 1978) it has been demonstrated that rotation laws other ter is equivalent to specifying the mass and linear (polar) than rigid rotation do not generally impose maximum radius. These in turn fix the breakup angular velocity, 2 3 rotation velocities. ΩB = (8/27)GM/Rp, along with the equatorial and pro- The second limitation, one which we will spend some jected velocities (when the inclination is specified). Fi- time on, involves the exponent in the Teff − geff re- nally, it is useful to recall the relation between the polar lation (geff , the effective gravity, includes centrifugal radius and equatorial radius at breakup angular velocity: terms). In the original work, von Zeipel (1924) consid- Re,B = 3Rp/2 (Hardorp & Strittmatter 1968). Accord- ered the case of a fully radiative envelope, deriving the ing to this first order theory the maximum rotational 0.25 well known “gravity darkening” relation Teff ∝ geff . flattening is 2/3. Lucy (1967) reconsidered the problem in fully convec- The model definition is completed by specifying, for tive stars, deriving a much reduced gravity dependence, each wavelength and surface point, the specific intensity 0.08 Teff ∝ geff . Other approximations lead to other expo- at the angle of the line of sight from the local normal. nents (see Reiners 2003, for references). In our nominal As noted by the early authors, plane-parallel model at- calculations we adopt the original von Zeipel (1924) pre- mospheres are entirely adequate in the context of Roche scription. As we shall show, the Altair observations bear models for stars on and near the main sequence. The only significantly on this issue. exception is that these models develop a cusp at the equa- Even in the limit of rigid rotation, the von Zeipel tor at critical rotation velocity. However, the cusp does (1924) theory is only first order in rotation rate. Distor- not appear until fractional rotation velocities of ω =0.99 tions of the interior figure, allowing some gravitational or larger and, as we have indicated, the breakdown of the quadrapole contribution, can be expected as rotation plane-parallel approximation is only one of several prob- rates approach breakup. That is even more likely if there lems with the model in this limit. A number of auxillary are significant deviations from solid body rotation, even quantities are also calculated, both integrated over the if confined to the inner convective regions (e.g., Tassoul spheroid and as seen from that particular direction, in 1978). addition to the predicted complex visibilities. An exam- Probably even more relevant are the effects of radia- ple of the former is the integrated luminosity, calculated tion pressure, which is treated simplistically in the the- as described in Collins (1963). Examples of the latter ory, and stellar winds. Significant envelope extension due include the magnitudes in various bandpasses. to radiation can be expected in the low effective surface In practice we need to solve the inverse problem: given gravity regions of rapidly rotating stars. And we would a point on the sky, (α, δ), determine whether the point certainly expect a dramatic increase in mass loss at the is on the stellar disk and if so what the corresponding equator, both due to enhanced convection as gas temper- latitude and radius are. We have solved this problem atures decrease into the solar and sub-solar regime. We explicitly using simple iteration, and first and second or- der versions of Newton-Raphson iteration. The routines, 6 Peterson et al. written in C, are quite flexible, reasonably fast and freely However, this gives very small estimated errors which available (from the first author). we feel do not properly reflect the influence of the (small) The properties of Roche models for isolated rotat- remaining biases in the triple amplitudes. We have there- ing early-type stars have recently been reviewed by fore decided to take a conservative approach to our error Domiciano de Souza et al. (2002), who use a slightly estimates. We have run a separate reduction using the different but completely equivalent parameterization. squared visibilities for the individual baselines in place Those authors summarize some of the results from their of the triple amplitudes. This solution, which has a sub- models for massive stars, which have provided a useful stantially larger reduced χ2 reflecting the larger residuals check on our own routines. in the amplitudes of the individual baselines, is summa- rized in column 4 of Table 1. Our adopted errors, shown 4.3. Implementation of Roche code in column 3, are the difference between these two solu- The Roche code consists of a library of functions writ- tions. ten in C, with a main function enabling its use as stan- Figure 3 shows how Altair appears projected on the dalone software and a wrapper enabling it to be called sky. The intensity distribution at 500 nm, as would be from within the NPOI standard data reduction software seen for example by an interferometer, is color encoded: OYSTER. The Roche spheroid parameters are part of blue for high intensity, red for low intensity. Except for the standard hierarchical model format of OYSTER and limb-darkening, this is also a temperature encoding. The are passed to the Roche code along with pointers to range in intensities, a factor of 18, is about a factor of the extensive tables of linear, logarithmic, and square- 2.5 more than would be expected due to limb-darkening root law monochromatic limb darkening coefficients for alone in a non-rotating star of this spectral type. a grid of Kurucz model atmospheres as published by Included in Table 1 is the integrated B − V as calcu- Van Hamme (1993). The Roche code, with the addi- lated for the models. Although we force all models to re- tional input of the (u, v) coordinates, computes the vis- produce the observed V magnitudes, as described above, ibilities for a grid of wavelengths supplied by OYSTER, this constraint does not automatically mean B − V will which are subsequently integrated over the NPOI band- be reproduced. The fact that the color does agree with passes. the measurements is a significant consistency check on The fitting of the model parameters (except for the the models, particularly given that the range of temper- gravities) utilizes the Marquardt-Levenberg algorithm atures across the surface could produce a wide range in (Press et al. 1992) implemented in OYSTER, with the B − V . A number of ancillary parameters derived from derivatives computed numerically. In addition to the the adopted fit are given in Table 2 and are discussed visibility and phase measurements the reductions were further in the §6. also constrained to reproduce the observed V magnitude. This provided particularly strong constraints on the po- 5.2. Comparison with Previous Results lar effective temperature. As noted, Altair has been the subject of a number of in- 5. DISCUSSION terferometric measurements over the years, most notably by the PTI array (van Belle et al. 2001). The diameter 5.1. Model Fitting determinations are not simply intercomparable, even if The analysis proceded with few complications. In par- we consider only the projected major and minor axes of ticular, a close examination of the χ2 surface indicated our model, since the PTI observations were fit to the no unusual morphology, and indeed the iterations con- physical dimensions of a Roche model, which included verged to the same final solution independent of our limb-darkening but not gravity darkening. Still, our de- starting guess, whether from larger or smaller values of termination of an equatorial radius of 1.988 R⊙ seems in the parameters. Our final solution, given in column 2 of reasonable accord with their quoted 1.88 R⊙. Table 1, is based on the triple phases and triple ampli- More problematic are the reported position angles of tudes only, the latter with an overall floating multiplier the rotational pole. First, we note that in a preliminary for each scan as described in §3.6. The reduced χ2 for report of this work (Peterson et al. 2004) the pole is off this solution is still a bit large, which we attribute to by 180◦ due to a sign error. More complicated is the some residual non-gray problems of uncertain origin in disagreement between the position angle from the PTI the amplitudes, as shown in Figure 1. The fit to the clo- measurements, −25◦ ± 9 compared to our 123◦.2 ± 2◦.8. sure phases, Figure 2, on the other hand is remarkable, Communication with van Belle (2003, private commu- showing no trends with wavelength or hour angle. nication) indicated that the (u, v) coordintates of each The χ2/ν for our best model is well in excess of unity. baseline were inadvertantly exchanged in their analysis. This is the result of both the remaining residuals in the Fitting a simple uniform ellipse model to the PTI data, triple amplitudes evident in Fig. 1 and the very small corrected for the component swap, yields P A = 122◦.2, formal errors in both the amplitudes and phases owing in excellant aggrement with our result. to the high signal levels. If the large χ2 had resulted Recently Reiners & Royer (2004) have reported the from just the latter, we could have taken the usual expe- determination of Altair’s equatorial rotational velocity, −1 dient and scaled the errors by a common multiplier which veq ≤ 245kms . They analyzed the star’s rotational would have normalized away the excess χ2. Estimates of broadening profile to determine the first two zeros of its the uncertainties in the resulting parameters would then Fourier transform, the ratio of which has been shown have been obtained through the usual process of varying by Reiners (2003) to depend on the equatorial velocity each parameter until χ2 had increased by unity, thereby rather than the usual v sin i. This is a new approach to mapping out out formal errors and the correlation ma- measuring total velocities in stars, and it is difficult to trix. know how much weight it should be given. One notable Resolving Altair 7

Fig. 1.— Triple amplitudes as fit by best Roche model. Error bars of ±σ are shown. The analytic fits (solid line) include a constant multiplicative renormalization.

TABLE 1 Roche Model Fits for Altair and ζ Aql

Altair ζ Aql a 2 b 2 b Parameter V123 & φcl Errors Vi & φcl β = 0.09 Vi & φcl Errors

ω =Ω/Ωc 0.90 ±0.02 0.88 0.978 0.990 ±0.005 θp (mas) 2.96 0.04 3.00 3.04 0.815 0.005 Tp (K) 8740 140 8600 7980 11750 ··· i (deg) 63.9 1.7 62.2 65.6 90 +0, -5 PA (deg) 123.2 2.8 120.4 97.4 45 ±5 χ2/DOF 3.8 ··· 9.5 13.4 V (obs: 0.77) 0.765 ··· 0.765 0.76 B-V (obs: 0.22) 0.215 ··· 0.22 0.26 −1 Veq sin i (km s ) 245 ··· 231 295

aModel fit to the triple amplitude and closure phase data. bModels fit to the closure phase data and the squared visibilities of the three baselines. aspect of that analysis was the adoption of an exponent valid for these models, the situation is not so clear with for the gravity darkening law (β ∼ 0.09) that was about regard to the exponent on the gravity darkening law, 1/3 that of the von Zeipel (1924) value. We discuss this β Teff ∝ geff . The classical work of von Zeipel (1924) aspect of calculating rotationally distorted stars next. would seem to apply to an A star, even one somewhat evolved, yielding β = 0.25. However, convection does 5.3. Gravity Darkening occur in late A stars, and particularly in the photosphere While we have several lines of reasoning, described in where it competes with radiation in carrying the flux. It §4.1, that lead us to believe that solid body rotation is 8 Peterson et al.

Fig. 2.— Triple phases as fit by the best Roche model. No renormalization has been applied. Small circles are used for the observations since on the red side of the plots the error bars tend to be smaller than (and fall within) the width of the line showing the analytic fit. is then important to note that Lucy (1967) has shown plicitly points out that the derivation is valid only for that for small distortions the appropriate coefficient in small changes in the effective gravity. Of course, the fully convective envelopes is closer to β ≃ 0.08. effective gravity changes by orders of magnitude as rota- This leaves matters in a somewhat uncertain state. On tion approaches critical, and it is not clear whether the the one hand, there are stars in the transition region exponent derived by Lucy can be used to describe grav- between having fully convective and fully radiative en- ity darkening for anything but the most modest rotation. velopes, Altair arguably one of them, and there is no Again, this is in contrast to the modest contribution of an obvious guidance in choosing an appropriate value for induced gravitational quadrapole, even for stars rotating this parameter. On the other hand, even where the en- at breakup. velopes are unambigiously in either one of those states Even so, in a recent series of papers, Claret (2004, and or the other, the classical results apply under rather dif- references therein) has attempted to deal with the is- ferent circumstances, which we next discuss. sue of a smooth interpolation between these two extreme Both the von Zeipel (1924) and Lucy (1967) results cases. He has noted that as stars evolve off the main treat rotation as a perturbation. However, in the ra- sequence and toward the red giant branch, their interior diative case where uniform rotation is adequate (and is- structures trace out approximately straight line loci in sues like mass loss, etc. can be ignored), the quantity a (log Teff , log g) diagram. On the main sequence for treated as a perturbation is the size of the quadrapole massive (mostly radiative) stars the slope of this line is moment of the gravitational field. Since stars are cen- about 0.25, and for intermediate mass stars (∼ 1 M⊙, trally condensed, even for velocities approaching critical mostly convective) the slope is about 0.06, the two val- the distortions in the core are modest, and one can ex- ues being remarkably close to the radiative and convec- pect that the analysis given by von Zeipel (1924) will be tive exponents cited above. Working with the interiors reasonably accurate. This has been found to be true in codes, Claret (2004) is able to evaluate this exponent at practice (Sackmann 1970). each point in the evolutionary paths of models covering For the convective case, the gravity darkening expo- 40 ≥ M⊙ ≥ 0.08, offering the results as appropriate ex- nent is obtained by analyzing the adiabats found in the ponents to use in rotating stars and stars in close binary envelopes of representative stars. Lucy (1967) quite ex- systems over their entire evolutionary lifetimes. Resolving Altair 9

Fig. 4.— One of the observed V2’s for the AE–AW baseline. The effect of the strong Hβ feature in the λ 468.3 channel is clearly evident and well matched by the model.

cate that Altair has hardly evolved from the zero age main sequence. The Claret (1998, 2004) tables give Fig. 3.— A false-color rendering of Altair’s visible surface. In- values for β in essential agreement with the von Zeipel tensity at 500 nm increases from red to blue. Except for the effects (1924) result during this phase of evolution. Thus, Al- of limb-darkening, this is also a map of temperature, which varies from 8740 K at the pole to 6890 K at the equator. tair does not provide a test of those tables, but does highlight a difficulty in applying the technique proposed by Reiners & Royer (2004) for finding total velocities, namely the difficulty of obtaining a priori reliable esti- This is a constructive suggestion for the thorny prob- mates for β. lem of choosing an appropriate gravity darkening law. However, we are not fully convinced of the leap of going 5.4. Hβ from deriving a quantity based on evolutionary changes In Figure 4 we show the blue squared visibilities of to using it to describe the effects of rotational distor- the AE-AW baseline for the HA = 0h. 83 observation. tion. One might make the case for small amounts of ro- The notable feature at 486 nm is Hβ; the agreement with tation or if it could be shown that rotational distortions the calculations shown in this scan, and the others not and evolutionary effects were close to being homologous shown, is striking. This feature is nearly centered in the transformations from one to the other. But rotational 486.3 nm channel. In contrast, Hα, which has a smaller distortions are not homologous to evolutionary changes equivalent width and is split between channels at λ 665.4 and it is not at all clear how well these “interpolations” and λ 648.7, is much less noticable. The main reason for work. the feature being reflected in the amplitudes is the reduc- The observations reported here bear somewhat on this tion in the limb-darkening coefficient, the star appears to problem. We have tried converging our Roche models be more like a uniform disk at this wavelength, and thus using the value β = 0.09 adopted by Reiners & Royer the visibility is reduced. This close agreement is a nice (2004). The results are shown in column 5 of Table 1. confirmation of the details of the model fits. The V 2 data set was used so the comparison is with column 4. To achieve the degree of asymmetry found 6. CONCLUSIONS in the triple phase data with this low exponent value, the rotation parameter is forced to near critical rotation, 6.1. Imaging Altair ω ∼ 0.978. In turn the predicted projected rotational The primary result of these observations is, we be- velocity, v sin i ∼ 295kms−1 conflicts with the observed lieve, the first detection of asymmetric surface intensi- value, and the predicted color is significantly redder than ties on the surface of a star induced by rotation. Al- observed. Further, the reduced χ2 is significantly worse though we have imposed a model on the data and fit the for this fit. model parameters, the simple conclusion, first reported We feel it is premature to use these observations to by Ohishi et al. (2004), is that the surface of Altair dis- derive a “best” value of the gravity darkening parameter plays an extremely asymmetric intensity distribution and until the remaining visibility residuals are better under- that the asymmetry is consistent with that expected from stood. Since it is the phase measurements, which we the known high rotation and with the previously reported trust, which are sensitive to asymmetries in brightness oblateness (van Belle et al. 2001). We have, in effect, im- across the disk and not the visibility amplitudes, it does aged the surface of an A star. appear that the von Zeipel (1924) value for that param- In Table 2 we summarize various physical parameters eter is superior to the value adopted by Reiners & Royer for the adopted model — column 2 of Table 1. Most (2004). quantities should be self-explanatory. Subscript “B” Moreover, as we will describe in §6.2, our results indi- refers to the model if it were rotating at breakup. The 10 Peterson et al. angular diameter θmin is for the projected minor axis TABLE 2 while θMax is for the projected major axis, i.e., the an- a gular diameter of the equator. Altair Physical Parameters 6.2. The Status of Altair Quantity Unit Value Error Knowing now the rotational state of Altair, we can better answer fundamental questions such as its evolu- Rotating Model Parameters tionary status. Over the years Altair has been classi- −1 Veq km s 273 13 fied as A7 IV-V (e.g., Johnson & Morgan 1953), the lu- b −1 Veq,B km s 374 3 minosity class usually indicating an object slightly past Ω day−1 2.71 0.11 the end of its main sequence evolution while the spectral b −1 ΩB day 3.01 0.06 type is that of a star having an effective temperature Tp K 8740 140 in the vicinity of 7800K (e.g., Erspamer & North 2003; Teq K 6890 60 van Belle et al. 2001). In the context of analyzing Al- Rp R⊙ 1.636 0.022 Req R⊙ 1.988 0.009 tair’s pulsations, it is important to know in detail its θmin mas 3.056 0.047 evolutionary state, mainly the extent of its core, and to θMax mas 3.598 0.017 remove any biases that might be introduced by rotation. log L L⊙ 1.027 0.011 Fortunately, in the context of rigid rotation, this is not log gp cgs 4.266 0.012 log g cgs 3.851 0.035 so difficult. Early results (e.g., Sackmann 1970) showed eq that two quantities were relatively insensitive to the ef- Non-rotating Parametersc fects of rotation: polar radius and total luminosity. The Roche model fits give polar radius directly while it is a M M⊙ 1.791 0.018 straightforward matter to calculate the total luminosity R R⊙ 1.652 0.022 (e.g., Collins 1963). log g cgs 4.256 0.002 Teff K 8200 98 These two quantities are not perfectly conserved. In Xc – 0.607 0.019 the range of interest, 3 M⊙ ≥ M ≥ 1.4 M⊙, both quan- tities decrease with increasing rotation, approximately aUncertainties due to the parallax 2 in proportion to ω , reaching a maximum correction of have not been included in the errors. about 6% in luminosity and 1.5% in radius (Sackmann bRotating at breakup but with the same mass and polar radius. 1970). For stars in the neighborhood of 1.8 M⊙, rotating c 2 The parameters of a non- with ω ∼ 0.8 we find from the Sackmann (1970) cal- rotating star from the Geneva grid culations that the non-rotating star would be 4% more (Schaller et al. 1992) which would luminous and 1% larger than our deduced polar radius. reproduce the (corrected) luminosity To estimate the parameters of the appropriate non- and polar radius – see text. rotating star we have used the evolutionary tables by the Geneva group (Schaller et al. 1992). These models were calculated with modest convective overshoot (0.2Hp). We have used the grid for a composition of X = 0.68, As we also indicated in §1, rather than just being Y =0.30, Z =0.02. another rotating A star, Altair may prove to be a The quantities given in Table 2 as “Non-rotating” are valuable laboratory for examining the internal rotation those estimated from the Schaller et al. (1992) models state of a star with a predominantly radiative enve- and are quite striking. The last entry in Table 2 is the lope. Buzasi et al. (2005) have announced the discovery mass fraction of hydrogen remaining in the core. This is of δ Scuti pulsations in Altair and have identified several to be compared to a starting value of Xc =0.68. Altair of the periods with frequencies mostly in the range 15 – is almost on the ZAMS. 18 day−1. Interestingly, two of the frequencies reported by Buzasi et al. (2005), 3.526day−1 and 2.57 day−1 were 6.3. Chromospheric indicators significantly lower, the latter being quite close to the ro- As mentioned in §1, Altair is one of two A7 objects, tational frequency we have derived, e.g.,Table 2. the other being α Cep, in which certain ultraviolet emis- This was immediately recognized as providing a poten- sion lines, taken as indicators of chromospheric temper- tial probe of the interior structure and particularly the ature inversions, are seen. No objects of earlier spectral rotation law, and an attempt has been made to iden- type are known to show these features, and it is usually tify and model the modes (Su´arez et al. 2005). Unfortu- argued that these therefore represent the hottest photo- nately, not knowing the rotational state of the star and spheres where convection is still capable of creating such making the assumption that equator-on was the most temperature profiles. The model for Altair adopted here likely orientation, Su´arez et al. (2005) adopted a total calls that conclusion into question. As shown in Table rotation significantly lower than now seems likely. Other 2 Altair has a broad swath of 6900K gas at its equa- effects of this choice included identifying the evolution- tor, which is the likely source of the strong convection. ary state as being substantially more advanced than we We also note the recent announcement (van Belle et al. believe is the case. As is clear from their results, rota- 2005) that substantial oblateness and gravity darkening tional velocities above the 180–240 km s−1 range they in- have been found in α Cep, suggesting significant amounts vestigated lead to large changes in the oscillation modes, of cool, convective gas in that object as well. making mode identification difficult. It will not be an easy task to tap the information being provided by Al- 6.4. A δ Scuti star tair. Resolving Altair 11

The referee, Gerard van Belle, asked a number of pen- US Naval Observatory in association with Lowell Obser- etrating questions that significantly improved this work. vatory, and was funded by the Office of Naval Research Notably, he wondered about the effects of rotation on our and the Oceanographer of the Navy. This research has calibrators, leading to the discovery that we could mea- made use of the SIMBAD literature database, operated sure those effects in ζ Aql. The NPOI facility is a collab- at CDS, Strasbourg, France, and of NASA’s Astrophysics oration between the Naval Research Laboratory and the Data System.

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