Gas Distribution, Kinematics, and Excitation Structure in the Disks around the Classical Be Stars Beta Canis Minoris and Zeta Tauri
Stefan Kraus, John D. Monnier, Xiao Che, Gail H. Schaefer, Yamina Touhami, Douglas R. Gies, Jason Aufdenberg, Fabien Baron, Nathalie Thureau, Theo A. Brummelaar, Harold McAlister, Nils H. Turner, Judit Sturmann, Laszlo Sturmann
aa r X i v : . [ a s t r o - ph . S R ] S e p Submitted to The Astrophysical Journal on 2011, June 6; accepted for publication on 2011, September 11.
Preprint typeset using L A TEX style emulateapj v. 11/10/09
GAS DISTRIBUTION, KINEMATICS, AND EXCITATION STRUCTURE IN THE DISKS AROUND THECLASSICAL BE STARS β CANIS MINORIS AND ζ TAURI S. Kraus , J.D. Monnier , X. Che , G. Schaefer , Y. Touhami , D.R. Gies , J.P. Aufdenberg , F. Baron ,N. Thureau , T.A. ten Brummelaar , H.A. McAlister , N.H. Turner , J. Sturmann , L. Sturmann Department of Astronomy, University of Michigan, 918 Dennison Building, Ann Arbor, MI 48109-1090, USA The CHARA Array, Georgia State University, P.O. Box 3965, Atlanta, GA 30302-3965, USA Center for High Angular Resolution Astronomy and Department of Physics and Astronomy, Georgia State University, P.O. Box 4106,Atlanta, GA 30302-4106, USA Department of Physical Sciences, Embry-Riddle Aeronautical University, 600 S. Clyde Morris Blvd., Daytona Beach FL 32114, USA Department of Physics and Astronomy, University of St. Andrews, Scotland, UK
Submitted to The Astrophysical Journal on 2011, June 6; accepted for publication on 2011, September 11.
ABSTRACTUsing CHARA and VLTI near-infrared spectro-interferometry with hectometric baseline lengths (upto 330 m) and with high spectral resolution (up to λ/ ∆ λ = 12 000), we studied the gas distributionand kinematics around two classical Be stars. The combination of high spatial and spectral resolutionachieved allows us to constrain the gas velocity field on scales of a few stellar radii and to obtain, forthe first time in optical interferometry, a dynamical mass estimate using the position-velocity analysistechnique known from radio astronomy. For our first target star, β Canis Minoris, we model theH+K-band continuum and Br γ -line geometry with a near-critical rotating stellar photosphere anda geometrically thin equatorial disk. Testing different disk rotation laws, we find that the disk isin Keplerian rotation ( v ( r ) ∝ r − . ± . ) and derive the disk position angle (140 ± . ◦ ) inclination(38 . ± ◦ ), and the mass of the central star (3 . ± . M ⊙ ). As a second target star, we observed theprototypical Be star ζ Tauri and spatially resolved the Br γ emission as well as nine transitions fromthe hydrogen Pfund series (Pf 14-22). Comparing the spatial origin of the different line transitions,we find that the Brackett (Br γ ), Pfund (Pf 14-17), and Balmer (H α ) lines originate from differentstellocentric radii ( R cont < R Pf < R Br γ ∼ R H α ), which we can reproduce with an LTE line radiativetransfer computation. Discussing different disk-formation scenarios, we conclude that our constraintsare inconsistent with wind compression models predicting a strong outflowing velocity component, butsupport viscous decretion disk models, where the Keplerian-rotating disk is replenished with materialfrom the near-critical rotating star. Subject headings: circumstellar matter – stars: emission-line, Be – stars: individual ( β CMi, ζ Tau) –stars: fundamental parameters – techniques: interferometric INTRODUCTIONClassical Be stars are main-sequence (or near main-sequence) B-type stars associated with hydrogen lineemission, indicating the presence of ionized circumstel-lar gas, which is believed to be arranged in an equato-rial disk-like structure. Optical/infrared spectroscopicobservations have revealed non-radial pulsations (e.g.Rivinius et al. 1998), which might provide a way to feedmaterial from the photosphere to the inner disk. Po-larimetric studies (e.g. Draper et al. 2011) can constrainthe disk density structure. Further unique insights intothe structure and physics of these disks can also be ob-tained with interferometry at visual and infrared wave-lengths, allowing one to unravel the inner disk structureon scales of a few stellar radii directly. For instance,interferometric studies have allowed associating quasi-cyclic variations in the ratio between the blue- and red-shifted wing of the H α -line emission ( V /R variability)with global oscillations in the circumstellar disk, likelyin the form of a one-armed spiral density pattern (e.g.Vakili et al. 1998; ˇStefl et al. 2009; Carciofi et al. 2009; [email protected] Based on observations made with ESO telescopes at theParanal Observatory under programme IDs 084.C-0848(A) and085.C-0911(A) and with the CHARA array.
Schaefer et al. 2010). Besides studies on the disk con-tinuum geometry, interferometric observations in spec-tral lines have provided the first direct constraints onthe gas kinematics, in particular for the hydrogen spec-tral lines of the Balmer (e.g. Quirrenbach et al. 1994;Vakili et al. 1998; Tycner et al. 2005; Delaa et al. 2011),Brackett (Meilland et al. 2007, 2011), and Pfund series(Pott et al. 2010). These studies have provided growingevidence that the disks around classical Be stars exhibita near-Keplerian rotation profile, which might allow toeffectively rule out several disk-formation scenarios (e.g.see review by Carciofi 2010). However, most earlier stud-ies using spectro-interferometry were limited in terms ofspectral resolution or baseline position angle (PA) cover-age, leaving significant uncertainties about the detailedgas velocity field and the evidence to distinguish betweena purely rotational versus an expanding velocity com-ponent in the disk. Obtaining such evidence is essen-tial in order to decide between different scenarios whichhave been proposed to explain the disk formation mecha-nism, including radiatively driven winds, ram pressure ormagnetically induced wind compression, and viscous de-cretion (Porter & Rivinius 2003). Furthermore, recentlythere has been a controversy about the appearance ofa phase inversion in spectro-interferometric observationsof several classical Be stars, which triggered speculationsabout secondary dynamical effects or the need for anadditional kinematical component beyond the canonicalstar+disk paradigm (Stefl et al. 2011).Here, we present near-infrared spectro-interferometricobservations on the classical Be star β CMi with a highspectral resolution of R = 12 ,
000 in the hydrogen Br γ -line, enabling us to constrain the rotation profile directly.In addition, we observed the classical Be star ζ Tau forthe first time in multiple hydrogen line transitions (Br γ and Pfund lines), providing direct information about theexcitation structure within the disk.In the following, we present our CHARA and VLTIinterferometric observations (Sect. 2). The observationsin spectral lines are then first interpreted using a model-independent photocenter analysis approach (Sect. 3). InSect. 4, we present our continuum and kinematical mod-eling on β CMi, followed by our discussion of the resultson ζ Tau (Sect. 5). Finally, we summarize our findingsin Sect. 6. OBSERVATIONS
Fig. 1.— uv -coverage achieved with our CHARA/MIRC ( H -band), CHARA/CLIMB ( K -band), and VLTI/AMBER ( K -band)interferometric observations on β CMi.
Our near-infrared H -band continuum observationson β CMi were obtained using the CHARA array(ten Brummelaar et al. 2005), which is operated byGeorgia State University. The MIRC beam combiner(Monnier et al. 2006b) allowed us to combine the lightfrom four of the six CHARA 1 m-telescopes simultane-ously, yielding baseline lengths of up to 330 m, with goodbaseline coverage (Fig. 1). The MIRC data cover the H -band with low spectral dispersion ( R = 35) and wasreduced using the University of Michigan MIRC data re-duction pipeline (Monnier et al. 2007).In order to investigate the K -band disk geometry of β CMi, we employed the CHARA/CLIMB 3-telescopebeam combiner (Sturmann et al. 2010). Visibilities andclosure phases were derived using the “redclimb” and“reduceir” software. Besides the statistical errors, wealso add a calibration uncertainty of 0.05 for the derived T A B L E O b s e r va t i o n l o g o f o u r C HA R A an d V L T I o b s e r va t i o n s . T a r g e t D a t e I n s t r u m e n t Sp ec t r a l N P D I TT e l e s c o p e C a li b r a t o r ( s )( U T ) m o d e [ s ] c o nfi g u r a t i o n β C M i - - A M B E R H R - K . U T - U T - U T H D - - A M B E R H R - K . K - G - A H D - - M I R C H - S - E - W - W H D , H D , H D - - M I R C H - S - E - W - W H D - - M I R C H - S - E - W - W H D - - M I R C H - S - E - W - W H D - - M I R C H - S - E - W - W H D , H D - - C L I M B K - S - W - W H D - - C L I M B K - S - E - W H D - - C L I M B K - S - E - W H D ζ T a u - - A M B E R M R - K . . U T - U T - U T H D N o t e . — C o l u m n ( N P ) d e n o t e s t h e nu m b e r o f p o i n t i n g s o b t a i n e d o n t h e s c i e n ce s t a r . F o r t h ec a li b r a t o r s , w e a ss u m e t h e f o ll o w i n g un i f o r m d i s k ( U D ) d i a m e t e r s : F o r H D ( . ± . m a s ) , H D ( . ± . m a s ) , H D ( . ± . m a s ) , a nd H D ( . ± . m a s ) w e u s e t h e d i a m e t e r e s t i m a t e s f r o m s e a r c h C a l ( B o nn e a u e t a l. ) . T h e U D d i a m e t e r s o f H D ( . ± . m a s ) a nd H D ( . ± . m a s ) h a v e b ee n e s t i m a t e db y a v e r ag i n g t h ee s t i m a t e s f r o m t h r ee i nd e p e nd e n t ph o t o m e t r i c m e t h o d s ( B a r n e s e t a l. ; B o nn e a u e t a l. ; K e r v e ll a & F o u q u ´ e ) . as distribution, kinematics, and excitation structure in the disks around β CMi and ζ Tau 3
Fig. 2.—
Closure phases measured with the MIRC beam com-biner on β CMi (blue data points), overplotted with the modelpredictions from our best-fit H -band model (red data points;Sect. 4.2). visibilities, which represents an empirical value for thetypical scatter in the instrument transfer function.Spectro-interferometric observations with medium(MR mode, R = 1500) and high spectral dispersion (HRmode, R = 12 000) were obtained with the Very LargeTelescope Interferometer (VLTI) of the European South-ern Observatory and the AMBER 3-telescope beam com-biner instrument (Petrov et al. 2007). The AMBER ob-servations on β CMi were recorded using three 8.2 munit telescopes (2009-12-31) and three 1.8 m auxiliarytelescopes (2010-04-23), respectively. The atmosphericpiston was stabilized using the FINITO fringe tracker(Le Bouquin et al. 2008), which allowed us to use longdetector integration times (DITs) of 1 s and 6 s and torecord data with high spectral dispersion around thehydrogen Br γ -line ( λ vacuumBr γ = 2 . µ m). Due tothe presence of residual phase jitter, the absolute vis-ibility calibration of the AMBER data is not reliable,while the important wavelength-differential observablesare not affected. Spectra and wavelength-differential vis-ibilities and phases (DPs, Fig. 3) were derived from theAMBER data using the amdlib (V3.0) data reductionsoftware (Tatulli et al. 2007; Chelli et al. 2009). Thewavelength calibration was done using atmospheric tel-luric features close to the Br γ -line (yielding an accu-racy of approximately 1 spectral channel) and by ap-plying a heliocentric-barycentric system correction us-ing heliocentric velocities of +5 .
97 km s − (2009-12-31)and − .
61 km s − (2010-04-23), respectively. For thesystemic velocity we assume +22 . − (Duflot et al.1995).From all CHARA and VLTI interferometric observa-tions, we also derived closure phases (CPs). Both the H - and K -band continuum closure phases are consistentwith zero on a 2 σ -level, which leads us to conclude thatthe brightness distribution does not show significant indi-cations for deviations from centro-symmetry. The mostconstraining CPs have been recorded with MIRC in the H -band, which are shown in Fig. 2. Our AMBER HRmeasurements from 2009-12-31 provide us also with a CPmeasurement in the Br γ -line of β CMi, while the derivedCPs from the 2010-04-23 dataset are rather noisy andare therefore not included for our model fits. ζ Tau was observed on 2010-01-01 using AMBER’s MR-mode covering the upper K -band (2.12 to 2.46 µ m).The assumed systemic velocity for ζ Tau is +21 . − (Duflot et al. 1995). In the ζ Tau data, we detected notonly the Br γ transition (7-4), but also hydrogen Pfundtransitions (Fig. 4, 1st row), including clear detectionof Pf14 (19-5, 2.4477 µ m) to Pf22 (27-5, 2.3591 µ m).Higher Pfund transitions are also present in the spec-trum, but cannot be clearly separated due to the wide,double-peaked profile of the individual lines. For ζ Tau,the derived CPs are too noisy to provide additional in-formation and are therefore not included in our furtheranalysis.Details about the observational setup for all interfero-metric observations are listed in Tab. 1. Each observationon a science star was accompanied by observations on in-terferometric calibrators, allowing us to monitor and cor-rect for the atmospheric and instrumental transfer func-tion. PHOTOCENTER ANALYSISDifferential phases provide unique information aboutsmall-scale (sub-mas) photocenter displacements be-tween the blue- and red-shifted line wings. These dis-placements provide a very sensitive measure of the gaskinematics on scales of a few stellar radii. In a first anal-ysis step, we reconstruct the on-sky 2-D photocenter dis-placement from the measured DPs by solving the systemof linear equations ~p = − φ i π · λ~B i , (1)where φ i is the differential phase measured on baseline i , ~B i is the corresponding baseline vector, and λ is thecentral wavelength (Le Bouquin et al. 2009). The de-rived photocenter plots for β CMi and ζ Tau are shown inFigs. 3 and 4 ( middle panel ), respectively, and clearly re-veal rotation-dominated velocity fields for both objects,as indicated by the linear alignment of the photocentervectors for the different gas velocities and the oppositesign of the photocenter displacement for the blue- andred-shifted emission. In order to associate the DP withthe on-sky orientation, it is necessary to calibrate thesign of the DP measurements. For this purpose, we re-processed the ζ Tau VLTI/AMBER data set presentedby ˇStefl et al. (2009) and calibrated our DP sign in orderto match the published on-sky orientation.For β CMi, we determine the position angles θ for thedisk rotation plane at the two epochs to be 138 . ± . ◦ (2009-12-31) and 141 . ± . ◦ (2010-04-22). The photo-center vectors corresponding to our highest-SNR obser-vation (2009-12-31) show an interesting arc-like structurein the red-shifted line wing (Fig. 3, bottom left ), wherethe photocenter vectors corresponding to low gas veloci-ties are above the derived disk plane, while the high ve-locities are displaced in the opposite direction. Althoughthe significance of this pattern is still only marginal inour data, we speculate that this pattern might resultfrom opacity effects, with the more distant parts of thedisk appearing fainter than the disk parts facing the ob-server. Such an obscuration screen would displace thephotocenter perpendicular to the disk plane, where theamplitude of the displacement is stronger for lower gasvelocities, since the low-velocity emission is distributed Fig. 3.—
Upper panel:
VLTI/AMBER spectra (1st row) , visibilities (2nd row) , and DPs (3rd row) measured on β CMi for the epochs2009-12-31 (left) and 2010-04-22 (right) . Middle panel:
From the measured DPs, we derive for each spectral channel the 2-D photocenterdisplacement vector (including continuum and line contributions; East is plotted left and North is up). The flux and DP data points aswell as the derived photocenter vectors have been color-coded based on the Doppler velocity (see the upper panel to relate each color to awavelength). The DPs corresponding to the astrometric solutions are shown in the middle panel as solid black lines.
Bottom panel:
Usingthe procedure outlined in Sect. 3, we corrected for the continuum contributions in the line spectral channels, revealing the photocenterdisplacement corresponding to the line emission only. Besides the determined photocenter offsets, for comparison we also show the size ofthe H -band continuum-emitting disk (black ellipse, FWHM Gaussian, Sect. 4.2) and of the stellar photosphere (grey ellipse, Sect. 4.1). as distribution, kinematics, and excitation structure in the disks around β CMi and ζ Tau 5
Fig. 4.—
Upper panel:
VLTI/AMBER spectra (1st row) , visibilities (2nd row) , and DPs (3rd row) measured on ζ Tau for the epoch2010-01-01 in the Br γ (left) and Pf14-22 transitions (right) . Middle/bottom panel:
Photocenter displacement vectors derived from themeasured DPs (including line and continuum emission, middle panel ) and the continuum-corrected DPs (tracing the line emission only, bottom panel ). For comparison, we also show the size of the H -band continuum-emitting disk, as determined by Schaefer et al. (2010) forepoch 2009-11-10 (black ellipse, FWHM Gaussian) and of the stellar photosphere (grey ellipse). over a more extended region. Accordingly, the displace-ment would be strongest at zero velocities, and then sym-metrically decrease towards higher velocities. The super-position of this weak displacement (perpendicular to thedisk plane) with the displacement due to Keplerian ro-tation (parallel to the disk plane) might result in theobserved arc-shaped structure, which would also providea unique tool to determine the orientation of the disk inspace and the disk rotation sense. In the case of β CMi,this implies that the north-eastern part of the disk is fac-ing towards the observer (based on the displacement ofthe low-velocity channels in this direction) and that thedisk is in clockwise rotation (based on the location ofthe red-shifted photocenter displacements in the north-western quadrant).Another intriguing feature in the β CMi data is the W -shaped profile, which we observe in the wavelength-dependent visibilities and DP on our longest interfero-metric baselines (Fig. 3, and ). Likely, thisprofile indicates that the visibility function of the line-emitting region passes through a visibility null and tran-sits from the first to the second visibility lobe. Sincethis effect would reverse the direction of the photocen-ter vector, we manually correct the phase sign in thesecorresponding spectral channels close to the line center.For ζ Tau, we can determine the rotation axis for theBr γ (123 . ± . ◦ ) and the nine Pf14-Pf22 transitions(127 . ± . ◦ ) separately and find that the line-emittinggas rotates in the same disk plane within the observa-tional uncertainties of ∼ ◦ .Using the aforementioned procedure, it is possible toreliably measure the direction of the photocenter dis-placement, while the length of the displacement vectoris biased by the contributions from the underlying con-tinuum emission. In order to remove these contributionsfrom the measured observables ( F , V , φ ), we apply themethod outlined by Weigelt et al. (2007) and interpolatethe continuum flux ( F c ) and continuum visibility ( F c )from the adjacent continuum. The visibility and DP ofthe pure line emitting-region ( V l , φ l ) are then given by | F l V l | = | F V | + | F c V c | − · F V · F c V c · cos φ (2)sin φ l = sin φ | F V || F l V l | , (3)where F l = F − F c denotes the flux contribution fromthe spectral line. The continuum-corrected DPs are thenused to derive the photocenter displacement of the pureline-emitting region (Figs. 3 and 4, bottom panel ). Ap-plying this correction will provide the real centroid off-set of the line emission, but also introduce noise fromthe visibility and flux measurements, resulting in an in-creased scatter in the position angle distribution. There-fore, we decided to measure the position angle of the ro-tation axis from the uncorrected line+continuum photo-center displacements (Figs. 3 and 4, middle panel ), whilethe continuum-corrected photocenter displacements ( bot-tom panel ) will be used later on to construct a position-velocity diagram and to compare the spatial origin indifferent line tracers. DISCUSSION ON β CANIS MINORIS β CMi (HR 2845) is a relatively quiet B8V-type clas-sical Be star located at a distance of 52 . +2 . − . pc (Tycner et al. 2005). Ground-based photometric moni-toring has provided no clear indications for significantvariability (Pavlovski et al. 1997), while there is somemarginal evidence for long-term variations in the H α -profile (Pollmann 2002; Hesselbach 2009).In the following, we present a refined model for thephotospheric emission of β CMi taking the near-criticalstellar rotation into account (Sect. 4.1), followed by ourmodeling of the interferometric data in continuum emis-sion (Sect. 4.2). Given that the disk rotation signaturesin the Br γ emission line are overlayed on the rotationsignatures of the star (in Br γ absorption), we investigatethe influence of the stellar rotation on our measurements(Sect. 4.3). In the following, we construct a position-velocity diagram (Sect. 4.4) and present a full kinemati-cal modeling in Sect. 4.5.4.1. Constraining the stellar parameters
In order to obtain a model for the photospheric emis-sion of β CMi, we searched for a consistent set of stellarparameters, taking the evidence for near-critical rota-tion into account (Saio et al. 2007). For this purpose, weemployed our rapid rotator code (Monnier et al. 2007;Che et al. 2011), which simulates the stellar oblatenessand surface temperature distribution using the modi-fied von Zeipel theorem (gravity darkening coefficient =0.188), and computes model images in the continuumand in the photospheric Br γ absorption line.In order to constrain the stellar parameters, we com-puted a small parameter grid, in which we systemati-cally varied the inclination angle i and the fractionalangular velocity ω/ω crit (where ω crit is the critical an-gular velocity). For a given inclination angle and frac-tional angular velocity, we start our iterative processby assuming a stellar mass M ⋆ , from which we com-pute the polar radius R pole using the rotation velocity v sin i = 244 ± − (Yudin 2001). The polar tem-perature T pole is derived from the V -band magnitude( V = 2 . Fig. 5.—
Spectrum of the He I II β CMi (solid line). The dashed line shows a photospheric modelwithout disk contributions, while the dotted line corresponds to amodel where 20% disk continuum contributions have been added. as distribution, kinematics, and excitation structure in the disks around β CMi and ζ Tau 7
TABLE 2Grid of stellar parameters, obtained for β CMi using our fast rotator model (Sect. 4.1).
Model i ω/ω crit M ⋆ R pole R eq T pole P L app T appeff [ ◦ ] [ M ⊙ ] [ R ⊙ ] [ R ⊙ ] [K] [days] [ L ⊙ ] [K]A 45 0.99 3.70 3.32 4.61 13200 0.676 267 11558B 45 0.96 4.11 3.03 3.94 14600 0.577 343 13085C 45 0.93 4.47 2.87 3.59 15650 0.527 413 14235D 40 0.99 3.98 2.95 4.10 14550 0.546 337 12871E 40 0.96 4.49 2.74 3.56 16050 0.475 435 14505F 40 0.93 4.91 2.60 3.25 17300 0.433 532 15847G 35 0.99 4.43 2.62 3.64 16200 0.433 437 14447H 35 0.96 5.03 2.44 3.17 18100 0.377 590 16474 β CMi obtained by Grundstrom (2007). Fig. 5 showsthe observed profiles ( solid line ) of He I II R = 12 , dashed line ) from thegrid of LTE models by Rodr´ıguez-Merino et al. (2005).We adopted average parameters for the visible hemi-sphere of the star of T eff = 11 ,
800 K, log g = 3 .
8, and V sin i = 230 km s − (Fr´emat et al. 2005), and the modelspectrum was convolved with a simple rotational broad-ening function for a linear limb darkening coefficient of ǫ = 0 .
42 (Wade & Rucinski 1985). Note that this ap-proach assumes a spherical star, ignores gravity darken-ing, and neglects changes in the local intensity spectrumwith orientation, but these simplifications are acceptablefor our purpose of checking for systematic line depth dif-ferences. We see that the match of the Mg II λ I λ dot-ted line ), which appears to be much too weak compared Fig. 6.—
Hertzsprung-Russell diagram, including the solutionsfrom our small rapid rotator grid for β CMi (Tab. 2). The num-bers on the left side give the stellar mass of the correspondingevolution track, while the numbers at the bottom gives the evolu-tionary age corresponding to the shown isochrones Yi et al. (2003);Demarque et al. (2004). As outlined in Sect. 4.1, only models A, B,and D satisfy the observational constraints on the stellar effectivetemperature. Furthermore, model D satisfies best our inclinationobtained with interferometry (38 . ± ◦ ). with the observed spectrum. This comparison suggeststhat the disk continuum emission in the blue range isnegligible, and thus we will ignore any disk contributionto the optical flux in the following discussion.With the derived polar radius and bolometric lumi-nosity, we are able to locate the stellar position on theHR diagram, after correcting for the rotational effect(Che et al. 2011). The stellar mass from the HR diagramis used as initial value for the next iteration step, untilconvergence between the assumed mass and the mass es-timated from the HR diagram is reached.For each parameter combination, our model providesthe rotation period, apparent luminosity L app , and ap-parent effective temperature T appeff , as listed in Tab. 2.Comparing the model effective temperature and luminos-ity with the observational constraints for β CMi ( T eff =12 ,
050 K; L = 195 ± L ⊙ ; Saio et al. 2007), we findgood agreement for models A, B, and D. Due to its con-sistency with the inclination and effective temperature(Sects. 4.2 and 4.5), we favor model D, suggesting i =40 ◦ , ω/ω crit = 0 . T pole = 14 , v sin i = 244 km s − , T appeff = 12 871 K, and a polar and equatorial radius of R pole = 2 . R ⊙ = 0 .
26 mas and R eq = 4 . R ⊙ =0 .
36 mas, respectively. We plot the values correspondingto our model grid in the HR diagram shown in Fig. 6and compare it to the evolutionary tracks from Yi et al.(2003) and Demarque et al. (2004), yielding an evolu-tionary age of ∼ . − .
15 Gyr.Obviously, an important input parameter for ourmodeling procedure is the rotation velocity v sin i , forwhich we use an average value from the literature.Townsend et al. (2004) argued that most v sin i measure-ments on Be stars might systematically underestimatethe true projected rotation value due to the effect ofgravity darkening. In order to test this scenario, we haveartificially increased the measured v sin i -value by 10%and repeated our modeling procedure. We find that theincrease in v sin i results in a significant increase in thestellar mass, apparent effective temperature, and appar-ent luminosity, making the model prediction much lessconsistent with the observed values. Therefore, we sug-gest that in the case of β CMi, the v sin i -measurementsare not significantly biased by gravity darkening, whichis likely a result of the intermediate inclination angleof β CMi (the effect would be stronger for an equator-on viewing angle) and the use of our moderate gravitydarkening law ( T eff ∝ g . , Che et al. 2011), which islikely more relatistic than the classical von Zeipel darken-ing coefficient ( T eff ∝ g . ) adopted by Townsend et al.(2004). Clearly, the accurate determination of v sin i re-mains a fundamental problem for constraining the stellarparameters of near-critical rotating stars. Therefore, weare currently working on incorporating the computationof model spectra in our modelling procedure, which willthen be fitted to observed spectra together with the pho-tometric and interferometric constraints (Che et al., inprep.).The modeling provides spectra as well as model imagesin the continuum emission and in hydrogen photosphericabsorption lines, which we will use as a representationof the stellar brightness distribution for our modeling ofthe disk in the following sections.4.2. Continuum disk geometry
With projected baseline lengths up to B = 314 m,our CHARA/MIRC interferometric observations allowus to constrain the disk geometry in eight spectralchannels in the H -band with an effective resolutionof λ/ B = 0 . r < R ⋆ , and therefore wrongly attributed to thedisk instead of the stellar flux. Accordingly, these modelswill systematically overestimate the disk emission withrespect to the stellar emission ( F disk /F ⋆ ), resulting in in-consistencies with SED fitting results. In order to avoidthese biases, we employ an elliptical Gaussian model,where the radial intensity profile at r > R ⋆ is given bya half-Gaussian. resulting in a proper estimation of the F disk /F ⋆ ratio. As alternative model, we considered aninclined ring model, where the radial intensity profile isgiven by a Gaussian centered at radius a from the star,with a fixed fractional width of 25% (see Monnier et al.2006a for details). Based on radiative transfer simul-tations of the brightness distribution in classical Be stardisks (e.g. Waters 1986), we consider the Gaussian modela better representation of the expected emission profile ina Be disk, while the ring model is more suited for compar-ison with results from the literature (e.g. Meilland et al.2006).For all models, the stellar photosphere is representedby the aforementioned rapid rotator model (Sect. 4.1)and the position angle θ of the stellar equator and thedisk major axis are aligned. Besides the disk major axis a (ring major radius or Gaussian half-width half max-imum), the PA θ , and the inclination i , we also treatthe flux ratio between the stellar and disk component( F disk /F ⋆ ) H as a free parameter. The model was fitted tothe MIRC squared visibilities, closure phases, and tripleamplitudes using a least-square fitting procedure, result-ing in the best-fit model shown in Fig. 7. The best-fitmodel parameters are displayed in Tab. 3, including 1 σ -errors, which we have determined using a boot strappingprocedure. We also tested whether the agreement canbe improved using a skewed ring model (Monnier et al.2006a), but find that introducing any disk asymmetryonly marginally improves the fit. Our model includesminor asymmetries due to the brightened polar regionin our rapid rotator model, which results in small clo- T A B L E M o d e l f i tt i n g r e s u l t s f o r t h e d i s ko f β C M i . C o n t i nuu m C o n t i nuu m L i n e H - b a nd K - b a nd B r a c k e tt γ ( R i n g m o d e l )( G a u ss i a n m o d e l )( G a u ss i a n m o d e l )( K e p l e r i a n r o t . )( F r ee r o t a t i o n l a w ) F l u x r a t i o F d i s k / F ⋆ . + . − . . + . − . . + . − . –– M a j o r a x i ss i ze a [ m a s ] . + . − . ( b ) . + . − . ( a ) . ( a ) , ( c ) –– P o s i t i o n a n g l e ( e ) θ [ ◦ ] . + . − . . + . − . . ( c ) . ( c ) . ( c ) I n c li n a t i o n i [ ◦ ] . + . − . . + . − . . ( c ) . ± . ( c ) O u t e r d i s k r a d i u s R o u t [ m a s ] –––5 . ± . . ( c ) S t e ll a r m a ss M ⋆ [ M ⊙ ] –––3 . ± . . ( c ) R o t a t i o n l a w i nd e x β ––– − . ( c ) − . ± . R o t a t i o n v e l o c i t y f k e p ( AU ) –––1 . ( c ) . ± . R a d i a li n t e n s i t y i nd e x q ––– − . ± . − . ( c ) χ r . . . . . N o t e . — ( a ) F o r t h e G a u ss i a n m o d e l, w e g i v e t h e h a l f - w i d t hh a l f m a x i m u m ( H W H M ) o f t h e h a l f - G a u ss i a n i n t e n s i t y p r o fi l e , m e a s u r e d a l o n g t h e m a j o r a x i s . ( b ) F o r t h e r i n g m o d e l s , w e g i v e t h e r i n g m a j o r a x i s d i a m e t e r . ( c ) T h i s p a r a m e t e r h a s b ee n k e p t fi x e d t h r o u g h o u tt h e m o d e li n g p r o ce du r e ( s ee S ec t s . . nd . f o r d e t a il s ) . ( d ) A n i n c li n a t i o n a n g l e o f i = d e n o t e s a f a ce - o nd i s k o r i e n t a t i o n . ( e ) T h r o u g h o u tt h i s p a p e r , a ll p o s i t i o n a n g l e s a r e m e a s u r e d a l o n g t h e d i s k m a j o r a x i s a nd E a s t o f N o r t h . as distribution, kinematics, and excitation structure in the disks around β CMi and ζ Tau 9
Fig. 7.—
Left:
Continuum visibilities measured on β CMi with CHARA/MIRC in the H -band (red points) and CHARA/CLIMB in the K -band (blue points). In order to show the position-angle dependence of the visibility function, we have binned the data by the projectedbaseline PA (number in lower left corner of each panel). For the H -band, the best-fit ring model (red line) and Gaussian model (greendotted line) are shown, with both models yielding nearly indistinguishable visibility profiles. For K -band, we show the best-fit Gaussianmodel (blue dashed line; see Sect. 4.2). Right:
Intensity distribution corresponding to our best-fit ring (top) and Gaussian model (bottom) . sure phases ( . . ◦ ), consistent with the measurement(Fig. 2).The determined disk position angles of 139 . ◦ (ringmodel) and 139 . ◦ (Gaussian model) are in excellentagreement with the gas disk rotation axis determinedwith our VLTI/AMBER photocenter analysis (140 . ± . ◦ ).Using the detailed information about the H -band ge-ometry obtained with our extensive MIRC data set, wethen fitted our Gaussian model to the CLIMB datato determine the K -band continuum geometry. Giventhe lower amount of observational constraints for the K -band, we treated only the disk-to-star flux ratio( F disk /F ⋆ ) K as a free parameters and kept the remainingparameters fixed. The resulting best-fit visibility curvesare shown in Fig. 7. With ( F disk /F ⋆ ) K = 0 . +0 . − . , wedo not find any significant deviations between the K -band and H -band flux ratio.4.3. Investigating the differential phase signatures ofthe stellar rotation in the Br γ absorption line With the currently achievable accuracy, DP measure-ments can already reveal photocenter displacements twoto three orders smaller than the formal angular resolution( λ/ B ). For instance, our 2009-12-31 AMBER observa-tions exhibit a DP accuracy of ∼ . ◦ (standard devia-tion over all continuum spectral channels), correspondingto a photocenter displacement of ∼ ∼
660 micro-arcsecond), it is important to investigatewhether the measured DP signatures might also containcontributions from the photospheric Br γ absorption line, which is tracing the stellar rotation and is underlying theBr γ -line emission.In order to simulate the effect of stellar rotation on ourinterferometric observables, we employ our rapid rotatorcode and compute the stellar surface brightness distribu-tion around the Br γ -line with a similar resolution as ourAMBER observations (Fig. 8, top ). For this, we alignthe stellar rotation axis with the measured rotation axis( θ = 227 . ◦ , Sect. 3). Br γ absorption on one side ofthe photosphere will shift the photocenter towards theopposite direction. Taking this into account, we adjustthe orientation of our model photosphere, so that theBr γ absorption in the blue-shifted line wing matches themeasured direction of the photocenter displacement atred-shifted wavelengths. In addition to the photosphericemission, we include disk emission in our model, assum-ing ( F disk /F ⋆ ) K = 0 .
25, as determined in Sect. 4.2.From the model, we compute the corresponding DPsand find signatures φ . . ◦ on all baselines (Fig. 8, bot-tom ). This result reflects the fact that the fraction of thephotospheric absorption to the total K -band emission israther small ( < Constraints on the disk kinematics from theposition-velocity diagram Fig. 8.—
Top:
Model images from our rapid rotator model for β CMi for some representative wavelengths (Sect. 4.3).
Bottom:
Modeldifferential phases (including the stellar photosphere only) computed for the baselines employed by our β CMi VLTI measurements from2009-12-31 (left) and 2010-04-23 (right) . Position-velocity ( p − v ) diagrams provide a powerfultool for the interpretation of a disk velocity field and arecommonly employed in radio astronomy to derive therotation profile of circumnuclear galactic disks or proto-stellar disks using, for instance, maser (e.g. Miyoshi et al.1995; Pestalozzi et al. 2009) or molecular emission trac-ers (e.g. Sofue & Rubin 2001; Isella et al. 2007). Usingour continuum-corrected photocenter displacement mea-surements, we can construct an equivalent diagram fromour VLTI interferometric data by measuring the length ofthe continuum-corrected photocenter displacement vec-tors projected on the disk plane θ = 140 . ◦ . Performingthis projection on the disk plane also allows us to avoida potential bias due to opacity effects, since these effectswould move the photocenter only perpendicular to thedisk plane (Sect. 3). The resulting p − v diagram (Fig. 9)shows a symmetric rotation curve, which we interpretusing the simple model of a thin Keplerian-rotating disk(Weintroub et al. 2008). The disk extends from an in-ner ( R in ) to an outer ( R out ) radius and the line-of-sight(LOS) velocity v of the emission element located at ra-dius r and at angle ϑ in the disk plane is given by v kep ( r, ϑ ) = r GM ⋆ r sin ϑ · cos i, (4)where ϑ is measured against the LOS and G is thegravitational constant. Considering only the emissionfrom the outer-most disk annulus (i.e. at R out and ϑ = − π... + π ) will result in a straight line in the p − v diagram(line A-B in Fig. 9). Adding the emission from the re-maining disk annuli ( R in to R out ) will result in a charac-teristic “bowtie”-shaped filled region in the p-v diagram,such as commonly observed in CO imaging observations(Sofue & Rubin 2001; Isella et al. 2007). However, astro-metric observations (such as radio masers or photocenter displacements) trace the light barycenter of each annu-lus, corresponding to the B-C and
A-D curves in Fig. 9(for more details see Weintroub et al. 2008). We fit thissimple model to the β CMi p − v diagram assuming afixed inclination of i = 40 ◦ (as determined in Sect. 4.2),which yields best agreement for M ⋆ = 3 . ± . M ⊙ , R out = 2 . ± . R in < . Detailed modeling of the disk kinematics
By reducing the information content to purely astro-metric data, the p − v diagram analysis method presentedin the last section provides a very intuitive and power-ful method to constrain the gas velocity field in the ob-served line tracer. In this section, we will use a moresophisticated model in order to make full use of the richinformation contained in our spectro-interferometric ob-servations, including spectra, wavelength-differential vis-ibilities, DPs, and CPs. The Br γ -emitting gas is assumedto be located in a thin disk plane, which is justified dueto the expected small opening angle of Be star disks (e.g.Bjorkman & Cassinelli 1993) and the intermediate incli-nation angle under which β CMi is observed (Sect. 4.2).The gas kinematics is parameterized with a rotationprofile | ~v ( r ) | = f kep ( R ref ) · ( r/R ref ) β , where β = − β = − . β = 0 for constant rotation,and β = +1 for solid body rotation (e.g. Stee 1996). f kep = | ~v ( R ref ) | / | ~v kep ( R ref ) | is the orbital velocity at thereference radius R ref = 1 AU, expressed as fraction ofthe Keplerian velocity | ~v kep ( r ) | = ( GM ⋆ /r ) − / . In themodel, the line emission extends from the stellar surfaceto an outer radius R out with a radial power-law inten-sity profile I l ( r ) ∝ r q . To include thermal line broad-ening in our kinematic model, we adopt a constant ra-dial gas temperature of 0 . · T eff (=7860 K for β CMi),as distribution, kinematics, and excitation structure in the disks around β CMi and ζ Tau 11
Fig. 9.—
Left:
Position-velocity diagram, derived from the continuum-corrected photocenter displacement vectors in Fig. 3 ( bottom, left ).The color of the data points matches the color-coding of the wavelength channels in Fig. 3. The model curves show the Keplerian rotationprofile for three stellar masses with R in = 0 .
68 mas, and R out = 2 . Right:
In order to illustrate the interpretation of the p − v -diagram, we show the Br γ -line model images from our best-fit kinematical model (Sect. 4.5, Fig. 10) for some representative velocitiesand mark the corresponding photocenter displacement (corresponding to the position offset plotted on the abscissa in the diagram) ineach channel map with a blue cross. The yellow vertical line marks the position of the star, which corresponds to the photocenter of thecontinuum emission and the zero position in the p − v diagram. as suggested by the radiative transfer modeling fromCarciofi & Bjorkman (2006).We include both line and continuum emission in ourmodel and compute the interferometric observables forour given VLTI array configurations and the coveredwavelength channels. The wavelength-dependent modelvisibilities and phases are then fitted to the VLTI inter-ferometric data using a reduced χ r goodness-of-fit esti-mator (see Kraus et al. 2009 for a definition), includingour flux, visibility, DP, and CP constraints.For the continuum emission, we assume the geometrydetermined with our CHARA observations (Sect. 4.2)and use this model to renormalize the AMBER contin-uum visibilities. In order to incorporate the underly-ing photospheric Br γ absorption, we include the photo-sphere model discussed in Sect. 4.3, although, as dis-cussed above, the influence on the differential phase( . . ◦ on all baselines) is still within the measurementuncertainities.The disk position angle is fixed to θ = 140 . ◦ , as de-termined by our model-independent photocenter analysis(Sect. 3). The remaining six parameters in our modelingare the outer disk radius R out , the inclination i , the stel-lar mass M ⋆ , the radial intensity power-law index q , thedisk rotation index β , and the velocity at the referenceradius R ref (expressed as fraction of the Keplerian veloc-ity, f kep ). In a first step, we fix the velocity profile toKeplerian rotation ( β = − . f kep = 1) and vary the re-maining four parameters systematically on a grid, yield-ing the best-fit model shown in Fig. 10. The correspond-ing parameters and uncertainties are listed in column 4of Tab. 3. In a second step, we test also non-Keplerianvelocity fields, yielding the best-fit values listed in col-umn 5 (Fig. 12). The resulting χ -surfaces are shown in Fig. 11 and the uncertainties have been derived using thebootstrapping technique.Our model fits show that the intriging phase inversionobserved at the line center on our longest VLTI base-lines (Fig. 10, ) can be explained with the phasejumps in the Fourier phase crossing a visibility null.These phase jumps appear in the same spectral chan-nels where we measure visibility minima in the W -shapedvisibility profile, indicating that the visibility function ofthe pure line-emitting geometry transits here from thefirst to the second visibility lobe (Fig. 10, and ). At the same time, the continuum emission is onlymarginally resolved, which results in the measured com-posite line+continuum visibility with a rather high con-trast of ∼ .
7. The phase jumps and visibility mini-mums are a basic property of the Fourier transform ofstrongly resolved objects and are reproduced very natu-rally and without finetuning from our kinematical mod-eling. Therefore, our results do not support the ideaoutlined by Stefl et al. (2011) that these features mightindicate secondary dynamical effects or polar mass out-flows.As best-fit value for the radial intensity index, we yield q = − . ± .
2. Isothermal viscous decretion disk models,such as discussed in Bjorkman & Carciofi (2005), pre-dict a radial disk surface density law Σ( r ) ∝ r − , cor-responding to q = − r ) ∝ r − ... − ) due to a steeptemperature drop in the inner few stellar radii of thedisk Carciofi & Bjorkman (2008). Therefore, we con-clude that our measured intensity profile is well consis-tent with these models.The most significant deviation of our simple kinemat-2 Fig. 10.—
Upper panel:
Comparison of the VLTI/AMBER β CMi spectra (1st row) , visibilities (2nd row) , DPs (3rd row) , and CPs (4th row) with our Keplerian disk model (solid lines). The three different colors correspond to the three baselines, where the associatedprojected baseline lengths and PAs are labeled in the 2nd row.
Lower panel:
To illustrate our kinematical model, we show intensity channelmaps for five representative wavelengths. as distribution, kinematics, and excitation structure in the disks around β CMi and ζ Tau 13
Fig. 11.— χ -surface derived from our β CMi model grid (Sect. 4.5) around the best-fit solution assuming a Keplerian velocity field(Tab. 3).
Fig. 12.— χ -surface for the rotation law index β derived fromour β CMi model grid including also non-Keplerian velocity fields(Sect. 4.5). ical model from the data is in the precise shape of theBr γ -spectrum. For instance, the measured spectra showa weak emission component in the line center (Fig. 10),which is not reproduced by the model and which might berelated either to the presence of a uniformly distributedlow-velocity gas halo or, more likely, to radiative transfereffects.Another deviation between the model and the observa-tion concerns the weak V /R -asymmetry which can be ob-served in our VLTI/AMBER Br γ spectra at both epochs(Fig. 3, top row ) and which is not reproduced by our ax-ialsymmetric kinematical model. Within the 113 dayscovered by our AMBER observations, no changes inthe V /R asymmetry could be observed, which is con-sistent with the conclusion by Tycner et al. (2005) andJones et al. (2011) that the H α profile does not showsignificant long-term variability. It is interesting to com-pare the average photocenter displacement between theblue- and red-shifted line emission (2.5 mas) with thecharacteristic size of the H α -emitting region (2.13 mas,Tycner et al. 2005), which suggests that Br γ emergesfrom a similar spatial region as H α and a much more ex-tended region than the H - and K -band continuum emis-sion ( R Br γ ∼ R H α > R cont , Fig. 3), in agreement withthe prediction from Carciofi (2010). However, a detailedcomparison is difficult due to the non-simultaneity of thedifferent observations. DISCUSSION ON ζ TAURI ζ Tau is a particularly well-studied classical Be starwhich shows a cyclic variability in the flux ratio ofthe violet- and red-shifted wing of the double-peaked H α emission line (Rivinius et al. 2006; ˇStefl et al. 2009).These V /R variations exhibit a period of ∼ γ -line of some ear-lier AMBER MR observations on ζ Tau was presentedby ˇStefl et al. (2009) and Carciofi et al. (2009) andsuggested a Keplerian rotation profile. Our spectro-interferometric observations on ζ Tau provide a Br γ mea-surement at a new epoch in the H α V /R -phase and cover,for the first time, also the wavelength region around thehydrogen Pfund lines. Besides our Br γ and Pfund seriesdata, we include H α -sizes from the literature, namely theGI2T photocenter measurement by Vakili et al. (1998, ∼ R ⋆ or ∼ .
33 mas) and the Mark III and NPOImeasurements by Quirrenbach et al. (1994, 1997) andTycner et al. (2004). It is important to note that theMark III and NPOI results were based on visibility am-plitudes instead of differential phases. Also, a direct com-parison with these earlier observations is complicated bythe known V/R variability of ζ Tau, although we notethat our measurement at phase 0.484 is reasonably closeto the H α -measurement at phase 0.577, presented byTycner et al. (2004). We convert the Gaussian FWHMderived by these studies to photocenter displacementsby computing the centroid of the corresponding Gaus-sian brightness distributions. We find that Br γ emerges Fig. 13.— ζ Tau disk position angle, plotted as function of theH α V/R phase, including data presented in Schaefer et al. (2010)and in this study. Our new phase measurement (phase 0.484)is in good agreement with the disk precession law proposed bySchaefer et al. α , but a more ex-tended region than the near-infrared continuum emis-sion, as already found for β CMi (Sect. 4.5). The Pfundemission originates from intermediate stellocentric radii( R H α ∼ R Br γ > R Pf > R cont ; Fig. 4).5.1. Signatures of the known one-armed oscillation
Both in Br γ and in the Pf14-Pf22 lines, we detect adouble-peaked line profile and clear rotation signaturesin the differential phases. All line transitions exhibit aphotocenter displacement with a stronger amplitude inthe south-eastern (red-shifted) than in the north-western(blue-shifted) lobe (Fig. 4, middle panel ). Such an asym-metric displacement is consistent with the presence ofa one-armed oscillation in the disk (ˇStefl et al. 2009;Carciofi et al. 2009). CHARA/MIRC observations bySchaefer et al. (2010) showed that the one-armed oscilla-tion pattern can also be observed as an asymmetry in the H -band continuum emission. Using multi-epoch data,they find that the position angle of the asymmetry is cor-related with the spectroscopic variability and precessesaround the star with the H α V /R period. Our observa-tion (2010-01-01) adds an additional epoch for this analy-sis at the V /R phase 0.484 (assuming maximum phase atJD=2454505.0 and a period of 1429 days, Schaefer et al.2010). We find that the spiral density maximum is lo-cated towards the south-east of the central star and themeasured PA is consistent with the sinusoidal PA mod-ulation proposed by Schaefer et al. (2010), as shown inFig. 13.The reported polarization angle for ζ Tau (32 . − . ◦ ,Quirrenbach et al. 1997; 35 ± ◦ , Ghosh et al. 1999; 32 − . ◦ , McDavid 1999) is in excellent agreement with therotation axis position angle (35 . ± .
0) found by ourspectro-interferometric observations.5.2.
Probing the disk excitation structure usingmulti-transition spectro-interferometry
In order to test whether the measured differences inthe stellocentric emitting radius of the Br γ and Pfundtransitions are consistent with the expected excitationstructure in the disk, we construct a simple radiativetransfer model assuming local thermodynamic equilib-rium (LTE). We assume that the line-emitting mate- Fig. 14.—
Radial intensity profile, as computed with our LTEmodel for different line transitions (Sect. 5.2). The intensity hasbeen weighted by the emitting area and normalized to the peakintensity. rial is located in an equatorial disk which extends out-wards from the stellar radius with a constant verticaldensity per unit volume and a half-opening angle ofΘ = 5 ◦ . Then, we integrate for each radius r the op-tical depth τ ν in vertical direction, assuming hydrogenunder LTE conditions and an isothermal temperaturedistributions. The number density for the different ex-citation levels and ionization stages is computed usingthe Saha and Boltzmann equation assuming a Gaussianline profile with thermal line broadening (Wilson et al.2009). The Einstein coefficients for the different hy-drogen transitions are estimated using the series expan-sion published by Omidvar & McAllister (1995). The ra-dial temperature and surface density profile are param-eterized with T ( r ) = T eff ( r/R ⋆ ) − . (Stee & de Araujo1994) and Σ( r ) = Σ ( r/R ⋆ ) − with Σ = 2 . − (Carciofi et al. 2009). For the stellar temperature, equa-torial stellar radius, and distance, we assume T eff =19 370 K, R ⋆ = 7 . R ⊙ , and d = 126 pc, respectively(Carciofi et al. 2009).Using the radiative transfer equation for LTE condi-tions ( I ν ( r ) = B ν ( T )(1 − e − τ ν ( r ) )), we compute the emit-ted intensity per unit area as function of radius r inthe disk, where B ν is the Planck spectrum for temper-ature T at frequency ν . From the radial intensity pro-files (Fig. 14) we compute the centroid of the brightnessdistribution, which is then compared to the stellocentricemission radius measured in the different line transitions.Given that our model assumes a simplified vertical den-sity structure and does not include inclination effects,we do not aim to match the absolute sizes of the emit-ting region in all line transitions, but focus instead onthe relative sizes. For this step, we normalize both themeasured and the model photocenter offsets relative toBr γ . The comparison between the H α , Br γ , and Pf14-17relative sizes and our LTE model is shown in Fig. 15.We find that we can reproduce important observationalfeatures, in particular that Fig. 15.—
Origin of the Br γ and Pfund line emission from ζ Tau,as measured by the length of the photocenter displacement vector,averaged over all velocity channels for a given line transition (theerror bars represent the standard deviation in the different veloc-ity channels). In addition, we include the H α -line photocenterdisplacement by Vakili et al. (1998, black data point) and the H α size estimates from Quirrenbach et al. (1994), Quirrenbach et al.(1997), Tycner et al. (2004), which we converted from GaussianFWHM to the corresponding photocenters (grey data point). Thedashed line shows the prediction from our LTE excitation model(Sect. 5.2). as distribution, kinematics, and excitation structure in the disks around β CMi and ζ Tau 15 a) Br γ originates from a similar spatial region in thedisk as H α ( R Br γ ∼ R H α ). This result is also inagreement with the predictions from more sophis-ticated non-LTE radiative transfer computations,such as made by Carciofi (2010). In order tobetter characterize the differences between theBr γ and H α -emitting region, contemporaneousobservations in these two wavelength bands willbe required. b) Br γ originates from a more extended region thanthe Pfund lines ( R Br γ > R Pf ). Computing theweighted average of the measurements in the indi-vidual transitions, we yield R Pf /R Br γ = 0 . ± . R Br γ > R Pf on the classi-cal Be star 48 Lib, and consolidates their suggestion thatthe measured size differences can already be explainedwith the expected optical depth differences between theseline transitions. We encourage theoreticans and model-ers to employ their more sophisticated non-LTE radiativetransfer codes in order to test the influence of inclina-tion and non-LTE effects, although, based on the resultsfrom Iwamatsu & Hirata (2008, e.g. Fig. 3), we expectno significant departure from LTE for the inner disk re-gions and the high transitions traced by our observations.Future multi-transition spectro-interferometric observa-tions with improved uv -coverage might also measure theprecise radial intensity profile in a model-independentfashion. Together with sophisticated radiative transfersimulations, these observations will reveal the excitationstructure of the disk and constrain parameters such asthe temperature profile and the vertical disk structure,which are currently difficult to access. CONCLUSIONSUsing CHARA and VLTI interferometry, our studycombined high angular resolution (with baseline lengthsup to 330 m) with kinematical information obtained athigh spectral dispersion, yielding direct observationalconstraints on the gas distribution, excitation structure,and kinematics of the disks around two classical Be stars.Using a model-independent photocenter analysis methodwe derived the disk rotation axis for the prototypical ob-jects β CMi and ζ Tau and spatially and spectrally re-solved the disk rotation profile on scales of a few stellarradii. For both objects, we find that the determinedgas rotation plane agrees well with the orientation ofthe continuum-emitting disk as resolved by CHARA, al-though there is also clear evidence for substructure inthe disk around ζ Tau, revealing a one-armed oscillation,as indicated by different displacement amplitudes in theblue- and red-shifted line wing.Using our data set on β CMi we constructed a position-velocity diagram, which can be interpreted using thewell-established procedures from radio interferometry,but probes the milli-arcsecond scale position displace-ments resulting from the rotating disk around this Bestar. From our kinematical constraints, we derive thedynamical mass of the central star to 3 . ± .
2, which is in excellent agreement with earlier spectroscopic stud-ies (Saio et al. 2007). The inclination of the system is38 . ± ◦ , as determined with our CHARA continuumand VLTI line observations. As shown with our de-tailed kinematical modeling, the rotation law is Keple-rian ( β = − . ± .
1) and we do not have to includean expanding velocity component in order to explain ourdata. Furthermore, our kinematical model allowed us toidentify the origin of the phase inversion, which has nowbeen observed in the differential phases in five out of eightBe stars. These phase jumps correspond to the transitionfrom the first to the second visibility lobe, removing thenecessity for speculations beyond the canonical star+diskparadigm (Stefl et al. 2011).For ζ Tau, we obtained spectro-interferometric obser-vations covering simultaneously the Br γ and at least ninetransitions from the Pfund line series. For all transi-tions, we detect a significantly stronger photocenter dis-placement in the red-shifted line wing than in the blue-shifted line wing, tracing the one-armed oscillation whichhas been deduced for the ζ Tau disk before. Comparingthe photocenter displacement in the different line tran-sitions, we find that the Pfund, Brackett, and Balmerlines originate from different stellocentric emitting re-gions ( R cont < R Pf < R Br γ ∼ R H α ), which we can re-produce qualitatively with a simple LTE line radiativetransfer model. More work, including non-LTE radia-tive transfer modeling, will be required in order to derivequantitative constraints.By detecting a purely Keplerian velocity field, our ob-servations are inconsistent with disk-formation mecha-nisms incorporating a strong outflowing velocity com-ponent, such as the wind compression scenario, whichpredicts a strong radial velocity component compara-ble to the escape velocity (Bjorkman & Cassinelli 1993).On the other hand, our kinematical constraints, as wellas the measured hydrogen line intensity profiles (with aradial power law index q = − . ± .
2) are consistentwith the predictions from Keplerian viscous decretiondisk models (Lee et al. 1991). As shown by Kato (1983)and discussed in various reviews (e.g. Carciofi 2010), vis-cous Keplerian disks are also able to produce one-armeddensity oscillations, such as detected for ζ Tau.Considering that our kinematic constraints have beenobtained using a very limited number of individual mea-surements (2 pointings on β CMi, 1 pointing on ζ Tau),our study also illustrates the high effectiveness achievablewith spectro-interferometry, in particular if a very highspectral resolution is employed or several line transitionsare observed.We thank A. Carciofi for helpful discussions on polar-ization effects in the infrared, A. M´erand for validatingour spectral line imaging code, N. Morrison for providingRitter spectra, and C. Jones for providing us informationabout the photometric variability of β CMi. This workwas done in part under contract with the California Insti-tute of Technology (Caltech), funded by NASA throughthe Sagan Fellowship Program (SK is a Sagan fellow).JDM and GHS acknowledge support for this work pro-vided by the National Science Foundation under grantsAST-0707927 and AST-1009080. The MIRC beam com-biner was developed with funding from the University of6Michigan. The CHARA Array is funded by the Geor-gia State University, by the National Science Foundationthrough grant AST-0908253, by the W.M. Keck Founda- tion, by the NASA Exoplanet Science Institute, and theDavid and Lucile Packard Institute.
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