Issue 
A&A
Volume 569, September 2014



Article Number  A45  
Number of page(s)  14  
Section  Stellar structure and evolution  
DOI  https://doi.org/10.1051/00046361/201424185  
Published online  17 September 2014 
Beyond the diffraction limit of optical/IR interferometers^{⋆,}^{⋆⋆}
II. Stellar parameters of rotating stars from differential phases
^{1}
Laboratoire J.L. Lagrange − UMR 7293 – Observatoire de la Côte d’Azur
(OCA), Université de NiceSophia Antipolis (UNS), Centre National de la Recherche
Scientifique (CNRS), Campus
Valrose, 06108
Nice Cedex 2,
France
email:
Massinissa.Hadjara@oca.eu
^{2}
Centre de Recherche en Astronomie, Astrophysique et Géophysique
(CRAAG), Route de l’Observatoire, BP 63, Bouzareah, 16340
Alger,
Algérie
^{3}
Astronomical Observatory, Volgina 7, PO Box 74 11060
Belgrade,
Serbia
^{4}
UJFGrenoble 1/CNRSINSU, Institut de Planétologie et
d’Astrophysique de Grenoble (IPAG) UMR 5274, 38041
Grenoble,
France
^{5}
INAF − Osservatorio Astrofisico di Arcetri,
Largo E. Fermi, 5, 50125
Firenze,
Italy
^{6}
MaxPlanckInstitut für Radioastronomie,
Auf dem Hügel 69, 53121
Bonn,
Germany
Received:
12
May
2014
Accepted:
4
July
2014
Context. As previously demonstrated on Achernar, one can derive the angular radius, rotational velocity, axis tilt, and orientation of a fastrotating star from the differential phases obtained by spectrally resolved long baseline interferometry using earthrotation synthesis.
Aims. We applied this method on a small sample of stars for different spectral types and classes, in order to generalize the technique to other rotating stars across the HR diagram and determine their fundamental parameters.
Methods. We used differential phase data from the AMBER/VLTI instrument obtained prior to refurbishing its spectrometer in 2010. With the exception of Fomalhaut, which has been observed in the mediumresolution mode of AMBER (λ/δλ ≈ 1500), our three other targets, Achernar, Altair, and δ Aquilae offered highresolution (λ/δλ ≈ 12 000) spectrointerferometric data around the Brγ absorption line in K band. These data were used to constrain the input parameters of an analytical, still realistic model to interpret the observations with a systematic approach for the error budget analysis in order to robustly conclude on the physics of our 4 targets. We applied the super resolution provided by differential phases φ_{diff} to measure the size (equatorial radius R_{eq} and angular diameter ⌀_{eq}), the equatorial rotation velocity (V_{eq}), the inclination angle (i), and the rotation axis position angle (PA_{rot}) of 4 fastrotating stars: Achernar, Altair, δ Aquilae, and Fomalhaut. The stellar parameters of the targets were constrained using a semianalytical algorithm dedicated to fast rotators SCIROCCO.
Results. The derived parameters for each star were R_{eq} = 11.2 ± 0.5 R_{⊙}, V_{eq}sini = 290 ± 17 km s^{1}, PA_{rot} = 35.4° ± 1.4°, for Achernar; R_{eq} = 2.0 ± 0.2 R_{⊙}, V_{eq}sini = 226 ± 34 km s^{1}, PA_{rot} = −65.5° ± 5.5°, for Altair; R_{eq} = 2.2 ± 0.3 R_{⊙}, V_{eq}sini = 74 ± 35 km s^{1}, PA_{rot} = −101.2° ± 14°, for δ Aquilae; and R_{eq} = 1.8 ± 0.2 R_{⊙}, V_{eq}sini = 93 ± 16 km s^{1}, PA_{rot} = 65.6° ± 5°, for Fomalhaut. They were found to be compatible with previously published values from differential phase and visibility measurements, while we were able to determine, for the first time, the inclination angle i of Fomalhaut (i = 90° ± 9°) and δ Aquilae (i = 81° ± 13°), and the rotationaxis position angle PA_{rot} of δ Aquilae.
Conclusions. Beyond the theoretical diffraction limit of an interferometer (ratio of the wavelength to the baseline), spatial super resolution is well suited to systematically estimating the angular diameters of rotating stars and their fundamental parameters with a few sets of baselines and the Earthrotation synthesis provided a high enough spectral resolution.
Key words: stars: rotation / stars: general / methods: observational / methods: numerical / techniques: interferometric / techniques: high angular resolution
Based on observations performed at the European Southern Observatory, Chile, under ESO AMBERconsortium GTO program IDs 084.D0456 081.D0293 and 082.C0376.
Figure 5 is available in electronic form at http://www.aanda.org
© ESO, 2014
1. Introduction
The rotation of stars can reach more than 80% of their critical or breakup velocity in some specific cases. Here, G is the gravitational constant, M the mass of the star, and R_{c} the equatorial radius at this velocity. The Eddington factor can be ignored in the case of the stars studied in this paper presenting low luminosity. Fastrotating stars exhibit a number of peculiar characteristics, such as geometrical flattening, coupled with gravitational darkening (von Zeipel 1924), making the poles apparently hotter than the equator. Gravity darkening has a profound impact on the physics of the star, with important observational consequences. For example, the models from Collins & Sonneborn (1977) indicate a twocomponent spectral energy distribution (SED) for these stars, with an infrared (IR) excess due to gravity darkening. It is therefore not easy to place these stars in one single spectral classification, since the observed SED depends on their rotational velocity and inclination angle (Maeder & Peytremann 1972). In addition, rotationinduced mechanisms, such as meridional circulation or turbulence, may affect the internal structure of the star and its evolution (Meynet 2009). Thus, fast rotators can be considered as a physics laboratory for studying the impact of rotation on the stellar structure and evolution.
Opticallong baseline interferometry (OLBI) is a key observing technique for directly constraining the stellar diameter and the socalled von Zeipel effect (von Zeipel 1924; Domiciano de Souza et al. 2003; McAlister et al. 2005; Aufdenberg et al. 2006; Peterson et al. 2006; Monnier et al. 2007; Zhao et al. 2009; Che et al. 2011; van Belle 2012; Delaa et al. 2013). The study of rotation across the HR diagram may also give information on the internal structure of stars of different spectral types and luminosity classes and the physics of their atmosphere (Espinosa Lara & Rieutord 2011).
OLBI is classically based on measuring visibility amplitudes at different baselines to determine the angular size of the target being observed. However, when a rotating star is studied with spectrally resolved interferometry, the Doppler shift across a line, which implicitly contains spatial information, together with the change in the photocenter relative to this shift (Lagarde 1994), can measure the angular size of the visible hemisphere of the star.
In addition, spectrally resolved longbaseline interferometry – also called differential interferometry (DI; Petrov 1989; Lachaume 2003) once combined with Earthrotation synthesis and/or the use of simultaneous baselines from three or more telescopes, offers new possibilities for observing the details of rotationally flattened stars and directly checking whether the von Zeipel effect applies to different spectral types and luminosity classes (van Belle 2012). The application of DI in practice demands, however, special data analysis to access the socalled differential phase (Vakili et al. 1998), which in turn needs appropriate calibration procedures that go beyond the usual traditional angular diameter estimates versus visibility curves extrema and/or zeros.
This paper scrutinizes four stars of different spectral types, namely:

Achernar (α Eridani, B6Vep spectral type),

Altair (α Aquilae, A7V spectral type),

δ Aquilae (HD182640, F0IV spectral type)

Fomalhaut (α Piscis Austrini, A4V spectral type).
These stars were observed using DI with the AMBER focal instrument of the VLTI (Petrov et al. 2007). This comparative study focuses on the rotational characteristics of these stars. To interpret these data, we developed a dedicated analytical model called SCIROCCO (Simulation Code of Interferometricobservations for ROtators and CirCumstellar Objects, Hadjara et al. 2012, 2013) in order to derive the fundamental and geometrical parameters of our sample of stars.
The present paper is organized as follows: we first introduce the stars, their observations, and data reduction (Sect. 2). We recall the principles of differential phase measurements and present the SCIROCCO model (Sect. 3). In Sect. 4, we validate the SCIROCCO model, comparing it with previously reported measures of Achernar. We then determine the parameters of Altair, δ Aquilae, and Fomalhaut in Sect. 5. These results, together with future applications of SCIROCCO, are discussed for a broader study of fastrotating stars with AMBER/VLTI interferometry across the HR diagram (Sect. 6).
2. Observations and data reduction
2.1. The sample stars
The data of the four studied stars in this paper are available from ESO archive. Achernar (α Eri, HR 472, HD 10144) is a rotationnally flattened Be star with an oblateness ratio reported between 1.41 and 1.56 typically with 3% accuracy (Domiciano de Souza et al. 2003). Subsequent studies (Kervella & Domiciano de Souza 2006) considered a polar massloss component added to the photospheric disk, for a star with rotational parameter V_{eq}sini = 225 km s^{1} (from previous spectroscopic estimates by Slettebak 1982). Then a small residual disk (Carciofi et al. 2008) and/or the presence of a companion by Kervella et al. (2008) were invoked to explain the apparent strong flattening for Achernar from interferometric assymetric data. However, Kanaan et al. (2008) considered the circumstellar environment (CSE) contribution as important for explaining the strong flatness. Overall, Domiciano de Souza et al. (2012, Paper I), gave the recent parameters of Achernar as for its equatorial radius to be 11.6 R_{⊙}, its equatorial velocity 298 km s^{1}, the inclination 101° and the position angle 34.7°.
Altair (α Aql, HR 7557, HD 187642) is a bright (V = 0.77) A7IVV star (Johnson & Morgan 1953) that is known to be a rapid rotator (240 km s^{1} from Slettebak 1955). These characteristics, together with its location (close to celestial equator, therefore observable in both the northern and southern hemispheres), make Altair a prime target among the fast rotators. For example, Buzasi et al. (2005) show that it is a variable of the δ Scuti type by detecting several oscillating frequencies in this star. Several interferometric and spectroscopic observations indicate a V_{eq}sini value between 190 km s^{1} and 250 km s^{1} (Abt & Morrell 1995; van Belle et al. 2001; Royer et al. 2002; Reiners & Royer 2004b, among others). Domiciano de Souza et al. (2005) used the interferometer VINCI/VLTI and a model for fast rotators, including Roche approximation, limbdarkening, and von Zeipel gravitydarkening, to find the same value of V_{eq}sini = 227 km s^{1} as in Reiners & Royer (2004a), who used spectroscopy for determine the constraints on Altairs inclination angle and differential rotation from the global rotational broadening profile derived from about 650 spectral absorption lines. Most recently, Monnier et al. (2007) have reconstructed an image of the surface of Altair from CHARA observations, inferring several fundamental parameters: inclination, position angle, effective temperature, and radii of the pole and of the equator. They derived the values of V_{eq}sini = 240 km s^{1} and 241 km s^{1}.
Fig. 1 Left: baselines and the corresponding (u,v) coverage of VLTI/AMBER observations of Altair, δ Aquilae, and Fomalhaut. Earthrotation synthesis spanned over ~1.5 h/night for δ Aquilae and Altair and ~5 h/night for Fomalhaut. Right: same data for Achernar, where Earthrotation synthesis spanned over ~5 h/night. Notice the rather dense sampling of the Fourier space for the target stars. 

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The source δ Aquilae A (δ Aql, HR 7377, HD 182640) is a subgiant F0IV star (Cowley & Fraquelli 1974). Using a model atmosphere analysis of spectroscopic binaries and multiple star systems, Fuhrmann (2008) determined that the mass of this star is 1.65 M_{⊙}, and its radius is 2.04 R_{⊙}. Looking for signatures of differential rotation in line profiles, Reiners (2006) found an effective temperature of 7016 K. The star δ Aquilae A presents nonradial pulsations and is a Delta Scuti variable that in addition is spinning rapidly with a V_{eq}sini estimated around 89 km s^{1} (Mantegazza & Poretti 2005).
Fomalhaut (αPsA, HR 8728, HD 216956) is a young A4V star. As the nineteenth brightest star in the sky, it is one of the closest main sequence star surrounded by a spatially resolved debris disk with an inclined ring of about 140 AU in radius (Holland et al. 2003; Stapelfeldt et al. 2004). With HST/ACS imaging, under the assumption that the disk is intrinsically circular, Kalas et al. (2005) estimated the disk inclination and the position angle of PA_{disk} = 156° ± 0.3°. Kalas et al. (2008) confirm the presence of a planetary companion. Royer et al. (2007) estimated the equatorial velocity V_{eq} = 93 km s^{1} and Di Folco et al. (2004) find the equatorial radius R_{eq} = 1.1 mas(1.8 R_{⊙}). Recently, Le Bouquin et al. (2009) used the spectrally resolved nearIR long baseline interferometry to obtain spectroastrometric measurements across the Brγ absorption line, to find a position angle PA_{star} = 65° ± 0.3° for the stellar rotation axis, perpendicular to the literature measurement for the disk position angle (PA_{disk} = 156°.0 ± 0.3°), and they suggest backwardscattering properties for the circumstellar dust grains. However, they did not infer the star’s radius from their measurements, which we do in this article with the same dataset.
2.2. The observations
All the above stars were observed with the AMBER/VLTI instrument located at Cerro Paranal, Chile, with the Auxiliary Telescopes (ATs). They were observed using the high spectral resolution mode of AMBER (λ/ Δλ ≈ 12 000) with the exception of Fomalhaut, observed using the medium spectral resolution mode (λ/ Δλ ≈ 1500). Table 1 provides the observation log of Altair, δ Aquilae, and Fomalhaut. The UV coverage for all studied stars is provided in Fig. 1. The Achernar’s observations (that are not included in Table 1) are the same as in Paper I.
VLTI/AMBER observations of Altair, δ Aquilae, and Fomalhaut with details on the dates, times, and baseline triplets.
Fig. 2 Top: AMBER instrumental fringing effect on the differential phases as a function of wavelength. Left: Achernar’s differential phases showing highfrequency beating (black line), and the same filtered out (red line). Right: same as for Altair. Bottom: left: Achernar’s dynamical differential phases before removing the fringing effect, for the observing night of 2009 Nov. 01, for the E0H0 baseline. Center: after bias removal, the rotation effect around the Brackett γ line becomes more visible as a dark vertical strip next to a bright vertical strip. Right: by projecting the differential phases as a function of wavelength, one gets the typical sshaped effect as a function of wavelength. 

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2.3. Instrumental biais troubleshooting and data calibration
The differential phase φ_{diff} obtained from AMBER data reduction algorithm is related to the object’s Fourier phase φ_{obj} by (e.g., Millour et al. 2011, 2006): (1)where the spatial frequency coordinates u and v depend on the wavelength λ, the projected baseline length B_{proj} and the baseline position angle PA (from north to east; u = B_{proj}sin (PA) /λ and v = B_{proj}cos (PA) /λ). The parameters a and b correspond to an offset and a slope, given in appropriate units.
Data have been reduced using version 3.1 of the amdlib software (Chelli et al. 2009; Tatulli et al. 2007). We adopted a standard frame selection based on fringe signaltonoise ratio (Millour et al. 2007) and kept the 50% best frames calibrated later using appropriate software by Millour et al. (2008). The visibility amplitudes could not be reliably calibrated due to too chaotic a transfer function. Indeed, FINITO did not provide the additional data necessary at that time to properly calibrate the absolute visibilities. Our final dataset include the source spectrum, differential visibilities, differential phases, and closure phases.
Owing to the defective IRISfeeding dichroic plate in the VLTI optical train in front of AMBER, a highfrequency beating affected all our measurements (spectrum, visibilities, phases) and was also present in the calibrated data (Fig. 2). Owing to the polarizing prisms of AMBER, a lowfrequency beating (AMBER “socks”) was also present in our data. The highfrequency beating appears at a specific frequency in the wavenumber domain. We removed it by Fourier filtering at that specific frequency. The second beating was corrected by subtracting a sine wave fit to the continuum signal.
We performed a wavelength calibration by using the positions of telluric lines. We normalized the spectrum by adjusting and dividing a fourth order polynomial to the continuum. Figure 2 (top) shows an comparative example of the AMBER differential phases before and after processing, while Fig. 2 (bottom) shows dynamical differential phases before and after removing of the instrumental biases.
Our final data set consists of:

Altair: 6 (=2 × 3 baselines) φ_{diff}(λ) curves centered on Br γ,

δ Aquilae: 6 (=2 × 3 baselines) φ_{diff}(λ) curves centered on Br γ,

Fomalhaut: 18 (=6 × 3 baselines) φ_{diff}(λ) curves centered on Brγ,

Achernar: the same 84 (=28 × 3 baselines) φ_{diff}(λ) curves as in Paper I, used here for reference to our analytical model.
The high spectral resolution mode of AMBER (λ/ Δλ = 12000) leads to a velocity resolution of ≃25 km s^{1}, while the mediumresolution mode (λ/ Δλ = 1500) leads to the velocity resolution of ≃200 km s^{1}. Projected equatorial rotational velocities V_{eq}sini above ~150 km s^{1} would ensure that inside the Brγ line, the attained resolution is ~12 individual spectral canals in high resolution and two spectral canals in the middle resolution observational mode. In such cases, rotation effects need to be accounted for when modeling phase signatures in order to be consistent with the physics of the studied star.
Fig. 3 Left: 3D Brackett γ local line profile representation for a star with at the poles and at the equator from Kurucz/Synspec model. The polar line profiles (in the board) have less amplitude than the equatorial one in the middle. Right: 3 panels of 1D cuts of the same 3D figure at different latitudes; at the poles (red line), at the equator (blue) and the average (in green). 

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3. SCIROCCO code for modeling rotating stars
To interpret these spectrointerferometric observations, we developed a chromatic semianalytical model for fast rotators: Simulation Code of Interferometricobservations for ROtators and CirCumstellar Objects (SCIROCCO). The code, implemented in Matlab ^{1}, allows computing monochromatic intensity maps of uniformly rotating, flattened, and gravitydarkened stars from a semianalytical approach. SCIROCCO is described below, and in the next section it is used to investigate the dependence of observed interferometric differential phases of the four stars on the relevant input physical parameters.
3.1. Surface shape
The stellar surface is analytically described by an oblate ellipsoid of revolution with the semiminor axis R_{p} in the direction of the rotation axis and semimajor axis given by the equatorial radius R_{eq}. The polartoequatorial radii ratio is taken from the Roche model (Domiciano de Souza et al. 2002): (2)where G is the gravitational constant, V_{eq} the equatorial linear rotation velocity, and M the mass of the star.
The minortomajor axis ratio of the oblate spheroid projected onto the sky plane (apparent ellipse) depends on the polar inclination angle i and the orientation of the ellipse, which is defined by the position angle PA_{rot} of the minor (rotation) axis projected onto the sky (position angle of the minor axis).
3.2. Gravity darkening
Following the recent spectrointerferometric works on fast rotators, we consider a von Zeipellike (von Zeipel 1924) form for the gravity darkening effect with a β coefficient: (3)where T_{eff}(θ) and g_{eff}(θ) are the effective surface temperature and gravity, which depend on the colatitude θ; β is the gravity darkening coefficient; and σ is the StefanBoltzmann constant. The constant C is given by (4)where is the average effective temperature over the stellar surface (ellipsoidal surface in our model) and S_{⋆} the stellar surface. We adopt the Roche model expression for the effective gravity g_{eff}(θ), but considering the equation of the oblate ellipsoid of revolution for the dependence of the radius with colatitude θ. Given this definition of g_{eff}(θ), the gravity darkening is totally defined by the above equations with input parameters and β.
3.3. Continuum intensity map
Once the local effective temperature T_{eff}(θ) is defined, the specific intensity maps in the continuum I_{c}(θ,φ) (continuum specific intensity at each visible pixel of the stellar surface) are given by (5)where (θ,φ) are the (colatitude, longitude) coordinates on the stellar surface, I_{0} is approximated by the analytical (black body) Planck function, and μ(θ,φ) is the cosine of the angle between the normal to the surface at a given point and the observer’s direction. Limb darkening is analytically included following the fourparameter prescription from Claret (2000a), where the limbdarkening coefficients a_{k} are chosen from tabulated values for a given photometric band, local effective gravity, and temperature. In this paper we consider solar metallicity and turbulent velocity of 2 km s^{1} for the limbdarkening parameters.
Fig. 4 Topleft: velocity map of the same model (inclination 60°, orientation 0°). Then for the simulated rotation a rigid body is assumed, and a velocity around 90% of the critical velocity of the star. Topright: simulated intensity map in the continuum assuming von Zeipel gravity and limbdarkening effects where the poles are brighter than at the equator. Centerleft: corresponding 2D map of the visibility modulus, where the four interferometric baselines cited above are represented by circles: ([ 75 m,45° ] in red circle, [ 150 m,45° ] in green circle, [ 75 m,90° ] in blue circle and [ 150 m,90° ] in magenta circle). Centerright: same as for the 2D map of the visibility phase. ([ 75 m,45° ] in red circle, [ 150 m,45° ] in green circle, [ 75 m,90° ] in blue circle and [ 150 m,90° ] in magenta circle). Bottomleft: monochromatic intensity map for a given Doppler shift across the Brackett spectral line. Bottomright: top: photocenter along the Z axis. (Z axis represents the declination and Y axis the right ascension.) Middle: photocenter along Y (the photocenters are in radians). Bottom: normalized spectrum with the Brackett γ rotationally broadened line. 

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3.4. Intensity map in photospheric absorption lines
The specific intensity maps I(λ,θ,φ) across photospheric absorption lines are computed as the product of the continuum specific intensity I_{c}(λ,θ,φ) defined above and a local normalized intensity profile H, which can vary over the stellar surface both because of rotational Doppler shift, and gravity darkening I is thus given by (6)where V_{proj} is the surface rotational velocity projected onto the observer’s direction, c is the vacuum light velocity, and λ_{0} the central wavelength of the line profile.
The local profile H over the stellar surface is interpolated from synthetic spectra calculated with the SYNSPEC code based on stellar atmosphere models from the Kurucz ATLAS9 grid (Kurucz 1970; Hubeny & Lanz 1995), computed with the tabulated closest values [ T_{eff},log g ] according to the colatitude deducted by SCIROCCO. Figure 3 depicts examples of local profiles interpolated from the adopted spectral synthesis code.
Limbdarkening parameters adopted for Achernar.
Parameters and uncertainties estimated from a LevenbergMarquardt fit of our model to the VLTI/AMBER φ_{diff} observed on studied stars, and compared with what we found in the literature.
Once the specific intensity maps on the photospheric lines are computed, the spectrointerferometric observables are directly obtained as spectra and photocenters, and by Fourier transformations, which provide visibility amplitudes, phases, and closure phases (examples are given in the bottom row of Fig. 4). Thus we can give feedback on the input model parameters by comparing the synthesized observable quantities to spectrointerferometric measures within the error bars. The relevant input parameters of our model for fastrotating stars are

the equatorial radius R_{eq},

the stellar mass M,

the inclination angle i and the equatorial rotation velocity V_{eq} (or alternatively V_{eq}sini),

the mean effective temperature on the stellar surface ,

the gravitydarkening coefficient β,

the stellar distance d (to convert from linear to angular sizes).
For Fig. 4 we define a reference using the stellar model similar to Achernar (exactly as in our Paper I): R_{eq} = 11 R_{⊙}, d = 50 pc, V_{eq}sini = 250 km s^{1}, i = 60°, M = 6.1 M_{⊙}, K, PA_{rot} = 0° (north direction), β = 0.25 (theoretical value for radiative stellar envelopes; von Zeipel 1924), no differential rotation, with gravity darkening and limb darkening effects and Kurucz/Synspec line profile. These parameters correspond to V_{eq} equal to 90% of the critical velocity V_{crit}, R_{eq}/R_{p} = 1.4 and an equatorial angular diameter ⌀_{eq} = 2R_{eq}/d = 2 mas. To be compatible with the AMBER observations presented in Paper I, the simulations were performed around the hydrogen Br_{γ} line, B_{proj} = 75 m and 150 m, PA = 45° and 90°, and λ/ Δλ = 12 000.
Examples of maps of projected velocity and specific intensity in the continuum and in the line are shown in the top row of Fig. 4. The second row represents the 2D maps of the visibility modulus and visibility phase, photocenters (or centroid: the firstorder term of the phase according to Mac Lauren development, Jankov et al. 2001), and normalized spectra deduced from the monochromatic intensity maps.
4. Validation of SCIROCCO on VLTI/AMBER differential phases of Achernar
To test and validate our semianalytical code, we performed a χ^{2} minimization using the VLTI/AMBER differential phase data recorded on Achernar (cf. Sect. 2.1) and previously analyzed by Domiciano de Souza et al. (2012, Paper I).
The χ^{2} minimization was performed using a free Matlab implementation (from Matlab central file exchange) that we adapted for our needs using the generalized nonlinear nonanalytic chisquare fitting code, developed by N. Brahms^{2}. This code performs a fit to the measurements with known errors, and can use several Matlab library toolbox, such as the LevenbergMarquardt (LM) algorithm for faster convergence. We used the Monte Carlo method to confirm that the parameter uncertainties are normally distributed (Bevington & Robinson 2002) in order to fit the simulated differential phases to the observations and to constraint the free parameters and their uncertainties. The free parameters are R_{eq}, V_{eq}, i, and PA_{rot}. For our four stars cited in the introduction (Sect. 1), the VLTI/AMBER φ_{diff} observations were analyzed with the numerical model SCIROCCO presented in Sect. 3.
Limbdarkening parameters adopted, respectively, left to right, for Altair, δ Aquilae, and Fomalhaut.
Fig. 6 The 6 VLTI/AMBER φ_{diff} measured on Altair and δ Aquilae around Brγ at 2 different observing times (format YYYYMMDDTHH MM SS) and, for each time, three different projected baselines and baseline position angles, as indicated in the plots. The smooth curves superposed on the observations are the bestfit φ_{diff} obtained without a differential rotation. 

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The limbdarkening parameters are fixed assuming the Claret function (Claret 2000a), as cited in the Sect. 3 and summarized in the Table 2. The gravity darkening coefficient β has been fixed using the predicted theoretical relation between β and temperature from Claret (2000b, 2012), as well as recent observational results. Based on the similar β value found for studied rapid rotators (including β Cas, which has spectral type F2IIIIV, close to δ Aql), Che et al. (2011) recommend adopting a new standard β = 0.19 for future modeling of rapidly rotating stars with radiative envelopes. However, here (as well as in Paper I) we use the value β = 0.2, since this number of significant digits is consistent with our results, showing that φ_{diff} is not very sensitive to β and to the relative uncertainties ~10% in the derived parameters. The model parameters that were held fixed, following Paper I, are shown in Table 3. Following Espinosa Lara & Rieutord (2011), which established the relation between the gravity darkening coefficient and the flatness of the star (see Fig. 4 of this paper), we deduced β = 0.15 for Achernar, with which we have obtained similar results with β = 0.2.
Parameters and uncertainties estimated from a LevenbergMarquardt fit of our model to the VLTI/AMBER φ_{diff} observed on studied stars, and compared with values found in the literature.
Figure 5 shows the bestfit φ_{diff}, together with the corresponding observations of Achernar. The bestfit values of the free parameters found with the LM algorithm are given in Table 3. As shown in these table and figures, bestfit model parameters and differential phases agree with the results from Paper I, within the measured uncertainties.
Fig. 7 The 18 VLTI/AMBER φ_{diff} measured on Fomalhaut around Brγ at 7 different observing times (format YYYYMMDDTHH MM SS) and, for each time, three different projected baselines and baseline position angles, as indicated in the plots. The smooth curves superposed to the observations are the bestfit φ_{diff} obtained without a differential rotation (Domiciano de Souza et al. 2003), gravitydarkened Roche model, as described in Sect. 3. All the observed φ_{diff} points are shown here, knowing that the fit has been performed using all the wavelength points. 

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5. SCIROCCO fitting of Altair, δ Aquilae, and Fomalhaut
The free parameters adopted for Altair, δ Aquilae, and Fomalhaut modelfitting are R_{eq}, V_{eq}, i, and PA_{rot}. They were calculated from previous works cited in the Sect. 3. The limbdarkening parameters are fixed assuming the Claret function (Claret 2000a), as described in Sect. 3, and are summarized in Table 4. The remaining model parameters were held fixed, as shown in Table 5. Still following Espinosa Lara & Rieutord (2011), we deduced β = 0.19 for Altair (exactly what we and Monnier et al. 2007, used). For δ Aquilae, and Fomalhaut we computed β around 0.24, and for these values we also obtained similar results to β = 0.2 for these two last stars.
The bestfit values of the free parameters found with the LM algorithm are given in Table 5 as well. Figures 6 and 7 show the bestfit φ_{diff}, together with the corresponding observations of δ Aquilae, and Fomalhaut. The uncertainties of the parameters estimated by the LM algorithm are ≈1%.
6. Conclusions and discussion
In this work we applied the superresolution capabilities provided by differential interferometry to measuring size (equatorial radius and angular diameter), projected velocity, and position angle of rotation axis of four fastrotating stars: Achernar, Altair, δ Aquilae, and Fomalhaut. Our analysis was based on spectrointerferometric differential phases recorded with the VLTI/AMBER beam combiner, operating in its high and medium spectral resolution modes and centered on the Br γ line. All targets are only partially resolved by the interferometer in terms of fringe contrast (angular sizes ~2 −5 lower than the theoretical angular resolution λ/B_{max}).
The stellar parameters of the targets were constrained using a semianalytical model to interpret the observations: SCIROCCO. This simplified algorithm dedicated to fast rotators was validated by showing that it leads to results in agreement with previous values obtained from an independent analysis of the same differential phase observations of Achernar (Paper I).
The observations of Altair, δ Aquilae, and Fomalhaut were fitted with SCIROCCO to determine their R_{eq} (and ⌀_{eq}), V_{eq}sini, and PA_{rot} from the VLTI/AMBER differential phases alone. These parameters are in good agreement (within the uncertainties) with values found in the literature and derived from different observables, such as spectra, squared visibilities, and differential phases.
In this work, we were able to deduce the rotationaxis inclination angle i for δ Aquilae, and Fomalhaut (i = 81 ± 13° and 90 ± 9°, respectively) and the rotationaxis position angle PA_{rot} = −101.2 ± 14° for δ Aquilae, information that we have not found in the literature.
This work demonstrates that we can use the methodology shown in our Paper I for large specimens of fast rotators of different spectral types, and for several spectral interferometric observations.
The results from this work thus open new perspectives for using spectrointerferometry to study partially resolved fastrotating stars with present interferometers in order to simultaneously measure their sizes, projected velocities, and position angles.
Available at http://www.jmmc.fr/searchcal
Available at http://cdsweb.ustrasbg.fr/
Available at http://www.jmmc.fr/amberdrs
Acknowledgments
This research made use of the SIMBAD database, operated at the CDS, Strasbourg, France, and of the NASA Astrophysics Data System Abstract Service. The author, M. Hadjara, acknowledges the support from the CRAAG Institue (Algeria), the Lagrange and OCA financial grants to carry on the present work, as well as the grants from the Fizeau European interferometry initiative (I2E). This research made use of the JeanMarie Mariotti Center SearchCal service^{3} codeveloped by Lagrange and IPAG, and of the CDS Astronomical Databases SIMBAD and VIZIER^{4}. This research made use of the AMBER data reduction package of the JeanMarie Mariotti Center^{5}. Special thanks go to Romain Petrov for his precious advice.
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Online material
Fig. 5 The 84 VLTI/AMBER φ_{diff} measured on Achernar around Brγ at 28 different observing times (format YYYYMMDDTHH MM SS) and, for each time, three different projected baselines and baseline position angles, as indicated in the plots. The smooth curves superposed to the observations are the bestfit φ_{diff} obtained without a differential rotation (Domiciano de Souza et al. 2003), gravitydarkened Roche model, as described in Sect. 3. All the observed φ_{diff} points are shown here, knowing that the fit has been performed using all the wavelength points. 

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Fig. 5 continued. 

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Fig. 5 continued. 

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All Tables
VLTI/AMBER observations of Altair, δ Aquilae, and Fomalhaut with details on the dates, times, and baseline triplets.
Parameters and uncertainties estimated from a LevenbergMarquardt fit of our model to the VLTI/AMBER φ_{diff} observed on studied stars, and compared with what we found in the literature.
Limbdarkening parameters adopted, respectively, left to right, for Altair, δ Aquilae, and Fomalhaut.
Parameters and uncertainties estimated from a LevenbergMarquardt fit of our model to the VLTI/AMBER φ_{diff} observed on studied stars, and compared with values found in the literature.
All Figures
Fig. 1 Left: baselines and the corresponding (u,v) coverage of VLTI/AMBER observations of Altair, δ Aquilae, and Fomalhaut. Earthrotation synthesis spanned over ~1.5 h/night for δ Aquilae and Altair and ~5 h/night for Fomalhaut. Right: same data for Achernar, where Earthrotation synthesis spanned over ~5 h/night. Notice the rather dense sampling of the Fourier space for the target stars. 

Open with DEXTER  
In the text 
Fig. 2 Top: AMBER instrumental fringing effect on the differential phases as a function of wavelength. Left: Achernar’s differential phases showing highfrequency beating (black line), and the same filtered out (red line). Right: same as for Altair. Bottom: left: Achernar’s dynamical differential phases before removing the fringing effect, for the observing night of 2009 Nov. 01, for the E0H0 baseline. Center: after bias removal, the rotation effect around the Brackett γ line becomes more visible as a dark vertical strip next to a bright vertical strip. Right: by projecting the differential phases as a function of wavelength, one gets the typical sshaped effect as a function of wavelength. 

Open with DEXTER  
In the text 
Fig. 3 Left: 3D Brackett γ local line profile representation for a star with at the poles and at the equator from Kurucz/Synspec model. The polar line profiles (in the board) have less amplitude than the equatorial one in the middle. Right: 3 panels of 1D cuts of the same 3D figure at different latitudes; at the poles (red line), at the equator (blue) and the average (in green). 

Open with DEXTER  
In the text 
Fig. 4 Topleft: velocity map of the same model (inclination 60°, orientation 0°). Then for the simulated rotation a rigid body is assumed, and a velocity around 90% of the critical velocity of the star. Topright: simulated intensity map in the continuum assuming von Zeipel gravity and limbdarkening effects where the poles are brighter than at the equator. Centerleft: corresponding 2D map of the visibility modulus, where the four interferometric baselines cited above are represented by circles: ([ 75 m,45° ] in red circle, [ 150 m,45° ] in green circle, [ 75 m,90° ] in blue circle and [ 150 m,90° ] in magenta circle). Centerright: same as for the 2D map of the visibility phase. ([ 75 m,45° ] in red circle, [ 150 m,45° ] in green circle, [ 75 m,90° ] in blue circle and [ 150 m,90° ] in magenta circle). Bottomleft: monochromatic intensity map for a given Doppler shift across the Brackett spectral line. Bottomright: top: photocenter along the Z axis. (Z axis represents the declination and Y axis the right ascension.) Middle: photocenter along Y (the photocenters are in radians). Bottom: normalized spectrum with the Brackett γ rotationally broadened line. 

Open with DEXTER  
In the text 
Fig. 6 The 6 VLTI/AMBER φ_{diff} measured on Altair and δ Aquilae around Brγ at 2 different observing times (format YYYYMMDDTHH MM SS) and, for each time, three different projected baselines and baseline position angles, as indicated in the plots. The smooth curves superposed on the observations are the bestfit φ_{diff} obtained without a differential rotation. 

Open with DEXTER  
In the text 
Fig. 7 The 18 VLTI/AMBER φ_{diff} measured on Fomalhaut around Brγ at 7 different observing times (format YYYYMMDDTHH MM SS) and, for each time, three different projected baselines and baseline position angles, as indicated in the plots. The smooth curves superposed to the observations are the bestfit φ_{diff} obtained without a differential rotation (Domiciano de Souza et al. 2003), gravitydarkened Roche model, as described in Sect. 3. All the observed φ_{diff} points are shown here, knowing that the fit has been performed using all the wavelength points. 

Open with DEXTER  
In the text 
Fig. 5 The 84 VLTI/AMBER φ_{diff} measured on Achernar around Brγ at 28 different observing times (format YYYYMMDDTHH MM SS) and, for each time, three different projected baselines and baseline position angles, as indicated in the plots. The smooth curves superposed to the observations are the bestfit φ_{diff} obtained without a differential rotation (Domiciano de Souza et al. 2003), gravitydarkened Roche model, as described in Sect. 3. All the observed φ_{diff} points are shown here, knowing that the fit has been performed using all the wavelength points. 

Open with DEXTER  
In the text 
Fig. 5 continued. 

Open with DEXTER  
In the text 
Fig. 5 continued. 

Open with DEXTER  
In the text 
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