A CHARA ARRAY SURVEY OF CIRCUMSTELLAR DISKS AROUND NEARBY Be-TYPE STARS

Our measurements of the sizes and orientations of Be star disks are based upon the observed decline in visibility caused by the extended angular distribution of disk flux. However, a drop in visibility can also occur if a stellar companion is within the effective field of view of the interferometer, and if the binary component is ignored, then the disk dimensions will be overestimated (in the extreme case apparently implying the presence of a large disk where none is present). Binary systems are relatively common among B-stars in general and Be stars in particular (Abt 1987; Mason et al. 1997; Gies 2001), so it is important to investigate what role they play in the interpretation of our interferometric measurements.

Fortunately, the signatures of binary companions in interferometric observations are well understood (Herbison-Evans et al. 1971; Armstrong et al. 1992; Dyck et al. 1995; Boden 2000), and given the projected separation and magnitude difference we can determine how the companion will affect our measurement. According to the van Cittert–Zernike theorem, the complex visibility is related to the Fourier transform of the angular spatial distribution in the sky, and the measured fringe amplitude is proportional to the real part of the visibility (Dyck et al. 1995; Boden 2000). In the ideal case of a binary star consisting of two point sources, the visibility varies according to a cosinusoidal term with a frequency that depends on the projected separation and the baseline and wavelength of observation. The interferometric fringe observed will display an amplitude that depends on the real part of the visibility according to the projected separation and binary flux ratio. For observations like ours that record the flux over a wide filter band, the fringe pattern is only seen over a range in optical path delay that is related to the coherence length. Binaries with separations smaller than the coherence length will display the full amplitude variation expected from the visibility dependence on binary separation and flux ratio (Boden 2000; Raghavan et al. 2009), while those with separations larger than the coherence length will appear as separated fringe packets (Dyck et al. 1995; O'Brien et al. 2011; Raghavan et al. 2012).

Thus, there are several separation ranges that are key to this discussion. First, we may ignore those binary companions that are outside of the field of view of the interferometer (0.8 × 0.8 arcsec; Section 3.6) and separated by more than atmospheric seeing disk. Second, there are binary companions that are effectively within the field of view, but whose projected separations are large enough that the fringe packets of the components do not overlap (≳ 9 mas for an observation with a 300 m baseline). This is by far the most common case for our observations, and indeed the fringe packet of the companion is usually located far beyond the recorded scan. In this situation the flux of the companion will act to dilute the measured visibility of the fringe but will not change its morphology (Section 3.3). Third, there are binaries that have such small projected separations (≲ 9 mas for an observation with a 300 m baseline) that their fringe packets overlap and create an oscillatory pattern in the observed visibility due to the interference between fringe packets (Section 3.3). This probably occurred in only a few cases among our observations (Section 3.7).

Ideally, we should model the fringe visibility in terms of the binary projected separation and flux ratio together with a parameterization of the Be disk properties (Section 5.1). However, because this is a survey program, our observational results are generally too sparse in coverage of baseline and position angle range to attempt such a general solution. For example, most of the known companions have angular separations that are relatively large, and the fringe packet of a companion would have only been recorded in short baseline observations. However, such short baseline data would not resolve the Be disks, which is our primary scientific goal. There are a few cases where the projected separations are small and the binary creates oscillations in visibility with projected baseline, but again our baseline coverage is generally too sparse to measure these together with the disk properties. Consequently, we made the decision to rely solely on published data on the binary companions of our targets (Section 3.2) in order to perform the corrections to the measured visibility where it was necessary to do so (i.e., the known companion was sufficiently bright and close enough to influence our visibility measurements).

The visibility correction procedure we adopted is outlined in the following subsections. The method is built upon the scheme described by Dyck et al. (1995) and considers how a binary companion influences the appearance of the combined fringe packets. We use estimates of the flux ratio (Sections 3.5 and 3.6) and the orbital projected separation (Section 3.2) for each observation to build a numerical relation between the Be star visibility and net observed visibility. Then we interpolate within these relations to determine the Be star visibility alone (Section 3.3). We also extend this approach to correct the visibilities for two multiple star systems (Section 3.4).

Many Be stars in our sample are known binaries or multiple systems. We checked for evidence of the presence of companions through a literature search with frequent consultation of the Washington Double Star (WDS) Catalog (Mason et al. 2001), the Fourth Catalog of Interferometric Measurements of Binary Stars (Hartkopf et al. 2001), and the Third Photometric Magnitude Difference Catalog.⁶ We only considered those companions close enough to influence the interferometry results (i.e., those with separations less than a few arcsec). We show the binary search results in Table 4, which includes visual binaries (typical periods >1 yr) and spectroscopic binaries (typical periods <1 yr). Columns of Table 4 list the star name, number of components, reference code for speckle interferometric observations, and then in each row, the component designation, orbital period, angular semimajor axis, the estimated K-band magnitude difference between the components ▵K, a Y or N for whether or not a correction for the flux of companions was applied to the data, and a reference code for investigations on each system. Entries appended with a semicolon in Table 4 indicate parameter values with large uncertainties. These include the single-lined spectroscopic binaries, where we simply assumed a primary mass from the spectral classification and 1 M_☉ for the companion to derive the semimajor axis a, which was transformed to an angular semimajor axis using the distance from Hipparcos (van Leeuwen 2007).

We find that 10 of the 24 Be stars in our sample have no known companion, and five others have companions that are too faint (all single-lined spectroscopic binaries) to influence the interferometric measurements. Thus, no corrected visibilities are listed for these 15 targets in Table 3. However, the companions were bright enough (▵K < 3.2) and close enough (separation <1'') for the remaining nine targets that we had to implement the corrections method outlined below.

We need to determine the companion's separation and position angle at the time of each observation in order to find the angular separation projected along the baseline we used. This was done by calculating the binary relative separation for the time of observation from the astrometric orbital parameters using the method outlined by Raghavan et al. (2009). We show in Table 5 the adopted orbital parameters for those binaries where we made visibility corrections. Note that no entry is present for HD 166014, where only one published measurement exists (see the Appendix), and we simply assumed that the projected separation was larger than the recorded scan length. There are four cases where we present new orbital elements that are based upon published astrometric measurements, and we caution that these are preliminary and used only to estimate the separations and position angles at the times of the CHARA Array observations. Details about these preliminary fits are given in the Appendix.

The changes in visibility caused by a binary companion can be described equivalently in terms of the real part of the complex visibility or the amplitude of interfering fringe packets (Dyck et al. 1995; Boden 2000). Here we develop a fringe packet approach to the problem that simulates the binary changes and that is directly applicable to the fringe amplitudes that we measure. We begin by considering how the fringe patterns of binaries overlap in order to assess the changes in visibility caused by a binary consisting of two unresolved stars, and then we extend the analysis to the situation where one star (Be plus disk) is partially resolved. The fringe packet for star i observed in an interferometric scan of changing optical path length has the form

$$\begin{eqnarray} &&F_{i} = {{\sin x} \over x} \cos \left(2\pi \frac{a}{\lambda } + \phi \right), \end{eqnarray} \tag{ 1 }$$

where x = πa/∧_coh, a is the scan position relative to the center of the fringe, ∧_coh is the coherence length given by λ²/δλ (equal to 13 μm for the CHARA Classic K' filter), and ϕ is a phase shift introduced by atmospheric fluctuations (O'Brien et al. 2011). If two stars are present in the field of view (Section 3.6) with a projected separation along the scan vector of x₂, then their fringe patterns may overlap and change the composite appearance of the fringe according to (Boden 2000)

$$\begin{equation} F_{{\rm tot}} = \frac{1}{1 + f_2/f_1} F_1 + \frac{f_2/f_1}{1 + f_2/f_1} F_2, \end{equation} \tag{ 2 }$$

where f₁ and f₂ are the monochromatic fluxes of the stars. Then the fringe visibility is calculated by

$$\begin{equation} V=\frac{\max (F_{{\rm tot}})-\min (F_{{\rm tot}})}{2 + \max (F_{{\rm tot}}) + \min (F_{{\rm tot}})}. \end{equation} \tag{ 3 }$$

In this discussion we will ignore the very small decline in fringe amplitude related to the angular diameters of the stars themselves, because their angular diameters are all very small (see Table 6).

We show a series of such combined fringe patterns in the panels of Figure 2 for an assumed flux ratio of f₂/f₁ = 0.5. In the top panel, the projected separation is zero, and the two patterns add to make the fringe pattern of a single unresolved star. The associated visibility that we would measure equals one in this case. However, in the second panel from the top, we show how a projected separation of 1 μm results in a much lower visibility because the peaks associated with star 1 are largely eliminated by the troughs associated with star 2. In the third panel from the top, the separation is just large enough (comparable to the coherence length) that the fringe pattern of the companion emerges from the blend, and the lower panel shows a separated fringe packet in which both fringe patterns are clearly visible. Note that the relationship between the projected angular separation (mas = milli-arcsecond) and the separation of fringe centers (μm) is given by

$$\begin{equation} \rho _{\rm mas} = \frac{206.265 \rho _{\rm \mu m}}{B}, \end{equation} \tag{ 4 }$$

where B is the projected baseline in meters. For instance, using a 330 m baseline, the longest scan length is 150 μm, which corresponds to a separation of ∼94 mas.

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We show in Figure 3 the net visibility that would be measured as a function of projected separation x₂. This shows that in general the observed visibility will be less than that of a single star. As expected, for very close separations the visibility varies cosinusoidally with x₂ with a frequency of 2π/λ. On the other hand, for projected separations larger than the coherence length ∧_coh, the visibility approaches the value 1/(1 + f₂/f₁) equal to the semiamplitude of the flux diluted fringe pattern of the target.

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Now suppose that star 1 is a Be star with a disk that is partially resolved, so that if it were observed alone, it would show a visibility V = V_c < 1. Consequently, its fringe pattern would have an semiamplitude given by V_c/(1 + f₂/f₁). We show a selection of model binary fringe patterns in Figure 4 again for f₂/f₁ = 0.5 and a specific separation of x₂ = 10 μm. The panels show from top to bottom the progressive appearance of the combined fringe patterns as V_c drops from 1 to 0.25. Now the visibility drops in tandem until V_c = 0.50 where the maximum and minimum are set by the fringe pattern of the companion. We show the relationship between the Be star visibility V_c and the net observed visibility V_o in Figure 5 (solid line for x₂ = 10 μm). At this separation, there is some slight destructive interference between the fringe patterns that decreases the maximum amplitude for star 2, and as V_c declines to zero, the net visibility attains the amplitude of star 2 alone (f₂/f₁)/(1 + f₂/f₁). Figure 5 also shows the (V_c, V_o) relationship for two other separations. The dotted line shows the case of zero separation for maximum constructive interference, and here the visibility declines linearly to (f₂/f₁)/(1 + f₂/f₁) as V_c tends to zero. Finally, the dashed line shows the case for a very large separation in which the fringe pattern of star 2 falls beyond the recorded portion of the scan. Here the visibility starts at its diluted value of 1/(1 + f₂/f₁) at V_c = 1 and declines to near zero at V_c = 0.

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Thus, to correct the observed visibilities for the presence of a companion, we need a diagram like Figure 5 for each observation of a target. We calculated the projected separation of the stars at the time of the observation from the dot product of the relative position vector (from the angular orbital elements given in Table 5) and the (u, v) spatial frequencies for the baseline used. Then we created an associated (V_c, V_o) diagram for each observation based upon the projected separation and effective flux ratio. The corrected visibility V_c was then found by interpolating in the relationship at the observed V_o value. In some rare circumstances, we encountered a double-valued (V_o, V_c) relation, so no correction was attempted because of this ambiguity.

There are six systems in our sample with two or more close companions. Both companions of HD 5394 (γ Cas) are faint, so no correction was made. In the cases of HD 23862 (Pleione) and HD 200120 (59 Cyg), the inner companion is very faint, so corrections were made for only the outer, brighter companion. The spectroscopic pair that comprises the B component of HD 4180 (o Cas) has such a small angular semimajor axis that it was treated as a single object, and thus this system was also corrected as a binary star (Section 3.3). This left two systems, HD 198183 (the triple λ Cyg) and HD 217675 (the quadruple o And), that required corrections for the flux of additional components. The very close B pair of o And was regarded as a single object (see the Appendix), so both λ Cyg and o And were treated as triple star systems.

We made visibility corrections for these two systems in much the same way as for the binaries, except in this case the fringe normalizations were assigned by

$$\begin{eqnarray} F_{{\rm tot}} &=& \frac{F_1}{1 + f_2/f_1 + f_3/f_1}+ \frac{(f_2/f_1)F_2}{1 + f_2/f_1 + f_3/f_1} \nonumber\\ && +\frac{(f_3/f_1)F_3}{1 + f_2/f_1 + f_3/f_1}. \end{eqnarray} \tag{ 5 }$$

Again, we formed model visibilities from the coaddition of the fringe patterns, determined the (V_c, V_o) relationships for the time and baseline configuration of each observation, and then used the inverted relation (V_o, V_c) to determine the corrected visibility.

Unfortunately, there are significant uncertainties surrounding both the magnitude differences and orbital elements for the companions of HD 198183 and HD 217675, and these introduce corresponding uncertainties in the amounts of visibility correction. Our results on these two systems must therefore be regarded as representative visibility solutions rather than definitive ones. However, the corrected interferometry visibilities are all close to one for these two Be stars, which suggests that their disks are only marginally resolved if at all. On the other hand, the much lower uncorrected visibilities of these two show that the influence of the companions is clearly present. Both targets will be important subjects for future, multiple baseline observations with the CHARA Array to determine their orbital properties.

Our visibility correction procedure requires a knowledge of both the projected separation of the stars (Section 3.2) and their monochromatic flux ratio. Unfortunately, the magnitude differences between the Be star primary and the companion are generally available only in the V-band, and we need to estimate the magnitude differences in the K-band. We must consider the color difference between the components and how much brighter the Be star plus disk appears in the K-band compared to the V-band. The predicted magnitude difference is given by

$$\begin{eqnarray} \triangle K_{\rm obs} &=& {-}2.5 \log _{10} \frac{F_2}{{F_1+F_d}} =-2.5 \log _{10} \frac{F_2}{F_1}\nonumber\\ && + 2.5 \log _{10} \left(1 + \frac{F_d}{F_1}\right), \end{eqnarray} \tag{ 6 }$$

where F₂, F₁, andF_d are the monochromatic K-band fluxes for the Be companion, the Be star, and the Be disk, respectively. We can estimate the first term from the color differences of the Be star and companion using

$$\begin{equation} \triangle K_{\rm bin} = -2.5 \log _{10} {F_2\over F_1} =\triangle V_{\rm bin} + (V-K)_{1} - (V-K)_{2} \end{equation} \tag{ 7 }$$

where we will assume that the disk contribution is negligible in the V-band so that ▵V_bin = ▵V_obs. In the absence of other information, we estimated the color differences (V − K) by assuming that both the Be star and companion are main-sequence objects, and we used the relationship between (V − K) and magnitude difference from a primary star of effective temperature T_eff(Be) (Frémat et al. 2005) for main-sequence stars from Lejeune & Schaerer (2001) to find (V − K) for both stars.

We determined the infrared flux excess term (1 + F_d/F₁) from our observed estimate of E^⋆(V^⋆ − K) (Touhami et al. 2011), which are related by

$$\begin{eqnarray} E^\star (V^\star - K) &=& 2.5 \log {{F_{\rm tot}^K} \over {F_{\rm tot}^V (F_1^K/F_1^V)}} \nonumber\\ &=& 2.5 \log _{10}{{[1 + F_d/F_1 + F_2/F_1]^K} \over {[1 + F_d/F_1 + F_2/F_1]^V}}, \end{eqnarray} \tag{ 8 }$$

where the superscripts indicate the filter band. If we assume that the disk contributes no flux in the V-band, then we can rearrange this equation to find the K-band flux excess relative to that of the Be star alone. Then we can combine the results from the two equations above to predict the observed K-band magnitude difference

$$\begin{eqnarray} \triangle K_{\rm obs} &=& \triangle K_{\rm {bin}}+E^\star (V^\star -K) + 2.5 \log _{10} \nonumber\\ && \times (1 + 10^{-0.4 \triangle V_{\rm {bin}}} - 10^{-0.4 (\triangle K_{\rm {bin}}+ E^\star (V^\star - K))}). \end{eqnarray} \tag{ 9 }$$

Our estimates for ▵K_obs are listed in Tables 4 and 5. Note that there are several instances where we give a range for the magnitude difference; these are Be stars that are single-lined spectroscopic binaries with a companion of an unknown type. We consider two hypothetical cases. First, we assume that the companion is a main-sequence star of one solar mass, and we use the Lejeune & Schaerer (2001) main-sequence relation to obtain the magnitude and color differences of the companion. The second case is to assume that the companion is a hot subdwarf (similar to the case of ϕ Per; see Gies et al. 1998) with a typical effective temperature of 30 kK and a stellar radius of 1 R_☉. We then estimate ▵K_obs by adopting the main-sequence radius for the Be star according to its effective temperature and by using the Planck function for both stars in order to estimate the monochromatic K-band flux ratio. Table 4 lists those cases with a hyphen in the ▵K column giving the magnitude difference range between that for a hot subdwarf (smaller) and for a solar-type companion (larger). However, we made no visibility corrections in most of these cases because the nature of the companion is so uncertain (with the exception of ϕ Per where the companion's spectrum was detected and characterized by Gies et al. 1998).

The CHARA Classic observations were recorded on a single pixel of the Near Infrared Observer camera (NIRO), and the physical size of the pixel corresponds to a square of dimensions 0.8 × 0.8 arcsec on the sky. The flux of companions with separations small compared to 0.8 arcsec will be more or less completely included in the observations, but the flux of companions with larger separations from the central Be star may be only partially recorded according to the separation and seeing conditions at the time of observation. Therefore, we need to calculate the effective flux ratio of companion to target based upon the relative amounts of flux as recorded by this one pixel.

Seeing is usually computed in real time according to the CHARA tip-tilt measurements of the V-band flux of the targets. By assuming that the K-band seeing varies with wavelength by λ^−1/5 (Young 1974), the K-band seeing disk is about ∼0.76 times that in V. The effective flux ratio is given by

$$\begin{equation} f_2 / f_1 = I_2 / I_1, \end{equation} \tag{ 10 }$$

where I₁ and I₂ are the net intensity contributions of the primary and secondary component, respectively, recorded by the pixel. We assume a Gaussian profile for the point-spread function as projected on the detector,

$$\begin{equation} I(x, x_0, y, y_0) = \frac{1}{2\pi \sigma ^2} \exp \left[ -{{1}\over {2}} \frac{(x-x_0)^2 + (y-y_0)^2}{\sigma ^2} \right], \end{equation} \tag{ 11 }$$

where (x₀, y₀) are the coordinates of the central position of the star on the detector chip, and σ is related to the seeing (σ = 2.355⁻¹θ_seeing). The intensity distributions of the primary and the secondary components integrated over one pixel on the detector are given by

$$\begin{eqnarray} &&I_1 = Q \int \!\!\int I(x, 0, y, 0) dx dy \end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray} &&I_2 = \int \!\!\int I(x, 0, y, \rho) dx dy \end{eqnarray} \tag{ 13 }$$

where ρ is the separation of the binary, and Q is the actual ratio of primary to secondary flux, which is derived from the magnitude difference of the two components in the K-band,

$$\begin{eqnarray} &&Q = 10^{ 0.4 \triangle K_{\rm obs}}. \end{eqnarray} \tag{ 14 }$$

We used the visibility correction scheme described in Sections 3.3 and 3.4 with the predicted projected angular separation (from the orbital elements in Table 5 and the observed (u, v) spatial frequencies in Table 3) and the effective flux ratio (Sections 3.5 and 3.6) to derive estimates of the Be star visibility alone. The corrected visibilities and their corresponding uncertainties are listed in the last two columns of Table 3 for the nine cases with significant companions. These uncertainties do not include the contributions to the error budget from uncertainties in projected separation and flux ratio. In most cases the projected separations are much larger than the coherence length (typically 9 to 27 mas for 300 m to 100 m baselines), which corresponds to the separated fringe packet case (see lower panel of Figure 2). Thus, the correction depends mainly on the flux dilution term, 1/(1 + f₂/f₁), and typical uncertainties of 0.2 mag for ▵K_obs only amount to correction uncertainties of ≈0.02 in visibility, i.e., generally smaller than the measurement errors. On the other hand, binaries with projected separations less than the coherence length (mainly observations of o Cas and ϕ Per; see Table 5) have corrections that reflect the fringe amplitude oscillation caused by beating between the fringe patterns of the components (Figure 3). These corrections may amount to (f₂/f₁)/(1 + f₂/f₁) in visibility, ≈0.10 for the relatively faint companions considered here. Figure 3 shows that the full range of the correction varies over a projected separation difference of ▵x₂ = λ/2, which corresponds to a projected angular separation difference of approximately 0.7 to 2.2 mas for 300 m and 100 m baselines, respectively. The predictive accuracy of the orbital separation probably has comparable uncertainties for the cases of o Cas and ϕ Per, so it is possible that the uncertainties in the corrections may be similar to the corrections themselves in these close separation instances. Nevertheless, we found that implementing the corrections even in these close cases tended to reduce the scatter in the fit of the Be disk visibilities (Section 5.1), so we will adopt the corrected visibilities in the subsequent analysis of all nine targets with significant companions.

Here we show how we can use the interferometric visibility measurements to estimate the stellar and disk flux contributions and to determine the spatial properties of the disk emission component. We used a two-component geometrical disk model to fit the CHARA Classic observations and to measure the characteristic sizes of the circumstellar disks. The model consists of a small uniform disk representing the central Be star and an elliptical Gaussian component representing the circumstellar disk (Quirrenbach et al. 1997; Tycner et al. 2004). Because the Fourier transform function is additive, the total visibility of the system is the sum of the visibility function of the central star and the disk,

$$\begin{equation} V_{{\rm tot}} = c_p V_s + (1-c_p) V_d, \end{equation} \tag{ 17 }$$

where V_tot, V_s, and V_d are the total, stellar, and disk visibilities, respectively, and c_p is the ratio of the photospheric flux contribution to the total flux of the system. The visibility for a uniform disk star of angular diameter θ_s is

$$\begin{equation} V_s = 2 J_1(\pi x \theta _s) / (\pi x \theta _s), \end{equation} \tag{ 18 }$$

where J₁ is the first-order Bessel function of the first kind and x is the spatial frequency of the interferometric observation, $x = \sqrt{u^2 + v^2}$. The central Be star is usually mostly unresolved even at the longest baseline of the interferometer, which brings its visibility close to unity, V_s ≃ 1. The disk visibility is given by a Gaussian elliptical distribution

$$\begin{equation} V_d = \exp \left[ -\frac{(\pi \theta _{\rm maj} s)^2}{4 \ln 2} \right], \end{equation} \tag{ 19 }$$

where θ_maj is the full width at half-maximum (FWHM) of the spatial Gaussian distribution along the major axis and s is given by

$$\begin{equation} s = \sqrt{r^2(u \cos {\rm P.A.} - v \sin {\rm P.A.})^2 + (u \sin {\rm P.A.} + v \cos {\rm P.A.})^2}, \end{equation} \tag{ 20 }$$

where r is the axial ratio and P.A. is the position angle of the disk major axis. Thus, the Gaussian elliptical model has four free parameters: the photospheric contribution c_p, the axial ratio r, the position angle P.A., and the disk angular size θ_maj.

The brightness distribution of the disk projected onto the sky is a function of the inclination and position angle of the major axis. Interferometric observations with a given baseline B_p will sample the disk elliptical distribution according to the angle between the projection of the baseline on the sky and the position angle of the disk major axis. It is helpful to consider rescaling the baseline to account for the changes in the disk size with direction. For a flat disk inclined by an angle i and oriented with the major axis at a position angle P.A. (measured from north to east), we can rescale using the effective baseline B_eff (Tannirkulam et al. 2008)

$$\begin{equation} B_{\rm eff} = B_p \sqrt{\cos ^2 (\phi _{\rm obs}-{\rm P.A.}) + \cos ^2 i \sin ^2 (\phi _{\rm obs}-{\rm P.A.})}, \end{equation} \tag{ 21 }$$

where ϕ_obs is the baseline position angle at the time of the observations. This new quantity, the effective baseline, takes into consideration the decrease in the interferometric resolution due to the inclination of the disk in the sky, and thus for the purposes of analysis, it transforms the projected brightness distribution of the disk into a nearly circularly symmetric brightness distribution. Thus, the disk part of the visibility can be considered as a function of B_eff alone, and below we will use this parameter to present the interferometric results. However, if there is also a stellar flux contribution, than its projection and visibility will be a function of the projected baseline B_p only (if the star appears spherical in the sky). Consequently, models with both stellar and disk contributions should be presented for both the major and minor axis directions in order to show the range in visibility with baseline direction in a single plot. Along the minor axis, for example, ϕ_obs − P.A. = 90°, so that the relation becomes B_eff = B_pcos i, and consequently for edge-on systems with i ∼ 90° a given B_eff may correspond to a B_p far larger than available with the Array. Such observations (if possible) would start to resolve the stellar disk and lead to low net visibility (see the dotted line in Figure 8.3).

The simple geometrical representation of the Be star system may in some cases present a solution family or degeneracy that exists between two fundamental parameters of the Gaussian elliptical model: the Gaussian FWHM θ_maj and the stellar photospheric contribution c_p. In order to explain the ambiguity in the model, we consider the case of a locus of (c_p, θ_maj) that produces the same visibility measurement at a particular baseline. We illustrate this relation in Figure 6, which shows an example of a series of (c_p, θ_maj) Gaussian elliptical visibility curves that all produce a visibility point V = 0.8 at a projected baseline of 200 m (for λ = 2.1329 μm and a Be star angular diameter θ_UD = 0.3 mas). The plot shows models for (c_p, θ_maj) = (0.156, 0.6 mas) (solid line), (c_p, θ_maj) = (0.719, 1.2 mas) (dotted line), and (c_p, θ_maj) = (0.816, 2.4 mas) (dashed line). Figure 7 shows the relationship between the Gaussian elliptical FWHM and the stellar photospheric contribution for the family of curves that go through the observed point V = 0.8 at a 200 m baseline. Larger circumstellar disks are associated with larger stellar flux contributions, and vice versa, which demonstrates that a single interferometric measurement does not discriminate between a bright small disk and a large faint one. Additional measurements at different baselines are necessary to resolve this ambiguity.

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Note that if the interferometric observations are all located on one projected baseline in the (u, v) plane, then the disk properties are defined in only one dimension. Thus, in such circumstances it is not possible to estimate the axial ratio r or the position angle of the disk major axis P.A. Only a lower limit for θ_maj can be set in such cases.

The circumstellar disk is modeled with a Gaussian elliptical flux distribution centered on the Be star. This disk model has four independent parameters (r, P.A., c_p, θ_maj), and one assumed parameter, the stellar diameter θ_s (Table 6). The fitting procedure consists of solving for the model parameters using the IDL nonlinear least-squares curve fitting routine MPFIT (Markwardt 2009), which provides a robust way to perform multi-parameter surface fitting. Model parameters can be fixed or free depending on the (u, v) distribution of the observations.

For the cases with limited coverage in the (u, v) plane, setting some model parameters to fixed values was necessary in order to successfully fit the data. We adopted additional constraints on some model parameters that are well defined from studies such as inclination estimates from Frémat et al. (2005) and intrinsic polarization angles from McDavid (1999) and Yudin (2001). Also, we occasionally used values of the IR flux excess derived from the SEDs of Be stars to estimate the stellar photospheric contribution c_p when needed.

Our fitting results are summarized in Table 7, where the cases with a fixed parameter are identified by a zero value assigned to its corresponding uncertainty. Column 1 of Table 7 lists the HD number of the star, Columns 2 and 3 list the axial ratio r and its uncertainty, Columns 4 and 5 list the disk position angle along the major axis P.A. and its uncertainty, Columns 6 and 7 list the photospheric contribution c_p and its uncertainty, Columns 8 and 9 list the angular FWHM of the disk major axis θ_maj and its uncertainty, Column 10 lists the reduced $\chi _{\nu }^2$ of the fit, Columns 11 and 12 list the corrected photospheric contribution c_p(corr) and its uncertainty (see Section 5.5), respectively, Columns 13 and 14 list the disk-to-star radius ratio R_d/R_s and its uncertainty, and finally, Column 15 indicates the cases of fully resolved disks (Y), marginally resolved disks (M), and unresolved disks (N) in our Be star sample.

Table 7. Gaussian Elliptical Fits of the Interferometric Visibilities

HD	r	δra	P.A.	δP.A.a	cp	δcpa	θmaj	δθmaj	$\chi _{\nu }^2$	cp(corr)	δcp(corr)	Rd/Rs	δRd/Rs	Res.
Number			(°)	(°)			(mas)	(mas)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)	(13)	(14)	(15)
HD 004180	0.583	0.101	101.4	13.8	0.500	0.000	1.027	0.173	3.03	0.558	0.019	3.243	0.493	Y
HD 005394	0.722	0.038	38.2	5.0	0.082	0.036	1.236	0.063	15.63	0.190	0.032	2.946	0.134	Y
HD 010516	0.100	0.000	135.5	3.5	0.682	0.017	2.441	0.198	8.18	0.700	0.016	10.437	0.838	Y
HD 022192	0.251	0.562	136.8	4.5	0.518	0.000	1.030	0.264	1.50	0.612	0.090	3.502	0.898	Y
HD 023630	1.000	0.000	00.0	0.0	0.448	0.297	0.091	0.000	5.85	1.000	0.000	1.010	0.000	N
HD 023862	0.438	0.000	159.0	0.0	0.500	0.000	0.364	0.220	3.33	0.715	0.057	1.879	2.315	M
HD 025940	0.866	0.000	108.0	0.0	0.412	0.000	0.597	0.245	2.55	0.539	0.009	2.072	0.676	Y
HD 037202	0.148	0.027	125.4	1.3	0.422	0.016	1.790	0.073	4.30	0.534	0.013	4.146	0.159	Y
HD 058715	0.783	0.000	139.5	0.0	0.717	0.000	0.777	0.181	2.77	0.851	0.132	1.539	0.194	Y
HD 109387	1.000	0.000	0.0	0.0	0.803	0.025	3.214	0.749	6.36	0.805	0.025	8.407	1.932	Y
HD 138749	0.200	0.000	177.0	0.0	0.500	0.000	0.261	0.177	2.06	0.907	0.056	1.332	0.470	N
HD 142926	0.270	0.082	98.4	5.7	0.822	0.000	1.329	0.714	0.51	0.830	0.005	7.330	3.824	M
HD 142983	0.405	0.000	50.0	0.0	0.242	0.000	0.836	0.164	0.75	0.294	0.005	4.963	0.936	Y
HD 148184	0.947	0.000	20.0	0.0	0.123	0.000	0.858	0.142	1.59	0.157	0.004	4.385	0.690	Y
HD 164284	0.685	0.000	18.0	0.0	0.728	0.000	0.892	0.187	2.92	0.739	0.005	5.082	1.025	M
HD 166014	0.435	0.279	67.8	32.5	0.250	0.000	0.337	0.105	1.88	0.941	0.069	1.191	0.108	M
HD 198183	0.826	0.000	30.0	0.0	0.133	0.000	0.163	0.198	4.91	0.749	0.177	1.292	0.779	N
HD 200120	0.310	0.000	95.0	0.0	0.143	0.000	0.554	0.190	4.67	0.264	0.023	3.852	1.219	M
HD 202904	0.258	0.129	108.8	2.9	0.541	0.370	1.211	0.786	1.75	0.604	0.319	4.225	2.544	Y
HD 203467	0.799	0.000	76.0	0.0	0.343	0.000	0.528	0.087	1.62	0.418	0.006	2.861	0.415	Y
HD 209409	0.249	0.059	107.5	2.2	0.617	0.000	1.525	0.642	1.80	0.647	0.006	5.776	2.374	Y
HD 212076	0.955	0.000	148.0	0.0	0.308	0.000	0.295	0.044	1.61	0.487	0.032	1.852	0.202	M
HD 217675	0.022	0.000	25.0	0.0	0.340	0.195	0.056	0.000	2.23	1.000	0.000	1.010	0.000	N
HD 217891	0.702	0.150	30.6	17.1	0.692	0.309	0.804	0.593	1.81	0.722	0.279	3.261	2.098	Y

Note. ^aUncertainty of zero indicates cases where the model parameter was fixed in advance of the fit.

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Plots of the best-fit solutions showing the visibility curves of the system disk-plus-star as a function of the effective baseline in meters along with our interferometric data are presented in Figure 8. The solid lines in Figure 8 represent the best-fit visibility model of the disk along the major axis, the dotted lines represent the best-fit visibility model of the disk along the minor axis, and the star signs represent the interferometric data.

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We find that the circumstellar disks of the four Be stars, HD 23630, HD 138749, HD 198183, and HD 217675 were unresolved at the time of our CHARA Array observations, while the circumstellar disks of HD 23862, HD 142926, HD 164284, HD 166014, HD 200120, and HD 212076 were only marginally resolved. Those are the cases where we had to fix r, P.A., and/or c_p in order to make Gaussian elliptical model fits to the data. On the other hand, we successfully resolved the circumstellar disks around the other 14 Be stars and were able to perform four-parameter Gaussian elliptical fits for most of them as listed in Table 7.

Modeling a circumstellar disk with an elliptical Gaussian intensity distribution is convenient but not completely realistic. The flux distribution in the model assumes that light components from the circumstellar disk and the central star are summed and that no mutual obscuration occurs. It is important to note that in the case of small disks most of the model disk flux is spatially coincident with the photosphere of the star, so the assignment of the flux components becomes biased.

To illustrate this effect, we show in Figure 9 the model components for a case with a faint disk. The dotted line in Figure 9 shows the assumed form of the intensity of the uniform disk of the star (angular diameter of 0.68 mas), the dashed line shows the Gaussian distribution of the circumstellar disk along the projected major axis (FWHM = 0.55 mas), and the solid line shows the sum of the two intensity components. In the case where the FWHM of the circumstellar emission is similar to or smaller than the stellar diameter, most of the disk flux occurs over the stellar photosphere where the sum produces a distribution similar to that of a limb-darkened star.

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The interpretation of the results obtained from the Gaussian elliptical fits must be regarded with caution in situations where the derived disk radius is smaller than the star's radius and a significant fraction of the disk flux is spatially coincident with that of the star. Consequently, we decided to correct the Gaussian elliptical fitting results in two ways. First, the disk radius was set based upon the relative intensity decline from the stellar radius, and we adopted the disk radius to be that distance where the Gaussian light distribution along the major axis has declined to half its value at the stellar equator. The resulting ratio of disk radius to star radius is then given by

$$\begin{equation} {{R_d}\over {R_s}} = (1 + ({\theta _{\rm maj}} / \theta _s)^2)^{1/2}, \end{equation} \tag{ 22 }$$

where θ_maj is the Gaussian FWHM along the major axis derived from the fits, and θ_s is the angular diameter of the central star. Second, the model intensity over the photosphere of the star from both the stellar and disk components was assigned to the flux from the star in an optically thin approximation. The fraction of the disk flux that falls on top of the star is f(1 − c_p), where f is found by integrating over the stellar disk the Gaussian spatial distribution given by (Tycner et al. 2004)

$$\begin{equation} I_{\rm env}(x,y) = \frac{4 \ln 2}{\pi r \theta _{\rm maj}^2} \exp \left[ -\frac{(x^2/r^2 + y^2)}{\theta _{\rm maj}^2/ 4 \ln 2} \right], \end{equation} \tag{ 23 }$$

where r is the axial ratio and (x, y) are the sky coordinates in the direction of the minor and major axes. Consequently, this fraction of the model disk flux should be reassigned to the star and removed from the disk contribution. Then the revised ratio of total disk to stellar flux is

$$\begin{equation} {{F_d}\over {F_\star }} = {{(1-c_p)(1-f)}\over {c_p + (1- c_p) f}} = {{1-c_p({\rm corr})}\over {c_p({\rm corr})}} \end{equation} \tag{ 24 }$$

which is lower than the simple estimate of (1 − c_p)/c_p. We list in Columns 11–14 of Table 7 the revised values of the stellar flux contribution c_p and the Be disk radius R_d/R_s (along with the corresponding uncertainties) obtained by applying this correction using the stellar angular diameters from Table 6.

It is important to validate our results against other published data wherever possible. However, since this is the first large-scale study of the K-band emission of northern Be stars, it is difficult to make direct comparisons. We list in Table 8 the available measurements of Gaussian elliptical model parameters of Be star disks for the stars in our sample (excluding the work of Gies et al. 2007 that is included in our analysis). There is only one other K-band measurement available by Pott et al. (2010), but even in this case a direct comparison of θ_maj is difficult because of the different assumptions made about the remaining parameters. Measurements of the disk emission in other bands will likely yield different diameter estimates (Section 6.2). Nevertheless, we can compare our results on the geometry of the disks with previous work, and we can consider the parameter estimates resulting from other kinds of observations.

We begin by comparing the stellar and disk flux contributions that we derived from the Gaussian elliptical fits (Table 7) with those estimated from the IR-excess in the SEDs (Table 6). Among the 14 stars in our sample with reliable detections of disk emission, we fit for the stellar flux contribution in six cases. The corrected parameter c_p(corr) = F_s/(F_s + F_d) is plotted together with the SED estimate of this ratio in Figure 10. The uncertainties are significant, but these two estimates of stellar flux in the K-band appear to be consistent (perhaps surprisingly so, because of the known temporal flux variations and large time span between the 2MASS photometry and our interferometric observations).

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Next we consider the disk axial ratio r that is related approximately to the disk normal inclination i by (Grundstrom & Gies 2006)

$$\begin{equation} r \approx \cos i + 0.022 \sqrt{\frac{R_d}{R_s}} \sin i, \end{equation} \tag{ 25 }$$

where the second term accounts for the increase in the minor axis size caused by the increase in disk thickness with radius. We expect that the disk inclination will be the same as the stellar spin inclination because the disk is probably fed by equatorial mass loss. Frémat et al. (2005) estimated the stellar inclinations for most of our targets by comparing the projected rotational velocity Vsin i with the critical rotational velocity V_crit, for which the equatorial centripetal and gravitational accelerations balance, and by assuming that Be stars as a class share a common ratio of angular rotational to critical velocity. The axial ratios derived from their estimates of spin inclination are given in the final column of Table 1. There are seven targets with fitted estimates of r among our 14 reliable detections, and we plot these values together with r(i) from the inclinations from Frémat et al. (2005) in Figure 11 (indicated by square symbols). Figure 11 includes other r estimates from previous interferometry (Table 8). With two exceptions ($\upsilon$ Cyg = HD 202904 above and γ Cas = HD 5394 below), the estimates from interferometry and rotational line broadening are in broad agreement. Three of our targets have prior interferometric estimates of r, and our results agree within the uncertainties. Note that the increase in disk thickness with radius implies that r will appear larger in bands where the disk emission extends to larger radius (Hα), and in two cases (ψ Per = HD 22192 and ζ Tau = HD 37202) r(Hα) is larger than r(K-band), while they are the same in the third case (γ Cas = HD 5394).

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The disk normal is expected to have the same position angle in the sky as that for the intrinsic polarization (from scattering in the inner disk; Quirrenbach et al. 1997). We show in Figure 12 a comparison of the position angles derived from interferometry with those from polarimetric studies (McDavid 1999; Yudin 2001). These generally agree within the uncertainties, but there are some exceptions ($\upsilon$ Cyg = HD 202904 and several cases with large axial ratio r where it is difficult to determine position angle). Our results are fully consistent for four targets with previous interferometric estimates of P.A. All these comparisons lend support to our strategy of fixing the c_p, r, and P.A. parameters in those cases where there were ambiguities with the full four-parameter, Gaussian elliptical fit.

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We clearly detected the visibility decline due to the disk in 14 stars in our sample. The ratio of disk to stellar radius (Table 7, Column 13) varies from 1.5 to 10 with a mean value of 4.4 among this subsample, but our detections are probably biased toward those cases with larger and brighter disks. We compare these K-band diameters with those measured for Hα (Table 8) in Figure 13, and this demonstrates that in most cases the Hα disk diameters are much larger. Gies et al. (2007) attributed this difference to the larger opacity of Hα compared to the free–free and bound–free opacities that dominate the disk emission in the near-infrared. There are three targets that fall below the trend: $\upsilon$ Cyg = HD 202904 (but with an uncertainty that may be consistent with the trend), ϕ Per = HD 10516, and κ Dra = HD 109387. It is curious that the latter two are both short-period binaries, and it may be that the usually larger Hα emitting regions are truncated by tidal forces, so that their K-band and Hα disk sizes are comparable. After setting aside these three discrepant cases, the unweighted slope of the relation starting from an origin at (R_d(K-band)/R_s, R_d(Hα)/R_s) = (1, 1) is ▵R_d(Hα)/▵R_d(K-band) = 4.0 ± 0.5 (s.d. of the mean), and this slope is indicated by a dotted line in Figure 13. Such a correlation is helpful in predicting the size of the circumstellar disk in the K-band from Hα observations, and vice versa, especially since Be star disks are generally highly variable and simultaneous, multi-wavelength observations are usually difficult to obtain.

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Next, we consider the connection between disk size and brightness. If the disk gas near the central Be star has a temperature that is some fraction of the stellar temperature, then we might expect to find a correlation between the surface intensity I of the inner disk and star in the limit of high disk optical depth,

$$\begin{equation} I_{\rm env}(x=0,y=\theta _s/2) = b I_s \end{equation} \tag{ 26 }$$

for some constant b. We can explore this relationship by considering the Gaussian elliptical model prediction for the disk brightness at the stellar equator (Tycner et al. 2004),

$$\begin{equation} I_{\rm env}(x=0,y=\theta _s/2) = (1 - c_p) {{4 \ln 2}\over {\pi r \theta _{\rm maj}^2}} 2^{-(\theta _s / \theta _{\rm maj})^2}. \end{equation} \tag{ 27 }$$

The mean stellar intensity in the model is

$$\begin{equation} I_s = {{c_p(\rm corr)} \over {\pi (\theta _s / 2)^2}}. \end{equation} \tag{ 28 }$$

Then we can equate these with proportionality constant b to obtain

$$\begin{equation} {{1 - c_p}\over {c_p({\rm corr})}} = b r (\theta _{\rm maj}/\theta _s)^2 2^{(\theta _{\rm maj}/\theta _s)^{-2}}. \end{equation} \tag{ 29 }$$

This relation predicts that the disk to star flux ratio (left-hand side) is approximately proportional to the ratio of disk to star projected area (right-hand side).

The Gaussian elliptical parameters from Table 7 and the stellar angular diameters from Table 6 were used to calculate the terms on both sides of the equation, and they are plotted together in Figure 14 (where we assumed a minimum uncertainty of 5% for those cases where the formal uncertainties may be underestimated). We see that the Be stars with the largest disk to stellar flux ratio are often those with a large ratio of circumstellar to stellar angular diameter. However, a counter example is the case of κ Dra = HD 109387 that is plotted with a large but faint disk (lower right location in Figure 14). The fit of the Gaussian elliptical model parameters in this case is constrained by a set of 100 m baseline measurements from Gies et al. (2007) (see Figure 8.10), and if a fit was made from the new measurements alone, then the disk size would be much smaller (as also suggested by the smaller Hα disk size shown in Figure 13). Given these uncertainties, our measurement for κ Dra may be set aside from consideration here. An unweighted, nonparametric, correlation test for the remaining 13 Be stars yields a Kendall's statistic of τ = 0.205, which has null rejection probability of 37%, i.e., a value which is not small enough to reject with confidence the null hypothesis of no correlation. We suspect that the poor correlation results from a break down among the faint disk stars of our simplifying assumption that the disks are optically thick, which may only be applicable to the brightest disk cases. We show for completeness in Figure 14 a linear fit of the sample of 13 stars that has a constant of proportionality of b = 0.18 ± 0.04 (s.d. of the mean). We caution again that such a relation may only hold for dense, optically thick disks and that our results may be biased against detection of fainter disks in general. We are planning to investigate further the question of disk surface intensity in another paper that will apply physical models and radiative transfer calculations to fit the observed visibilities.

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We found that the criterion for a confident detection of the circumstellar disk is usually a decline in visibility below V = 0.8 at the longest baselines available. We can use this criterion to estimate the limitations on disk sizes that we can detect for Be stars at different distances. The visibility measured along baselines aligned with the projected major axis of the disk is a function of the photospheric flux fraction c_p, the ratio of disk to stellar radius R_d/R_s, and the stellar angular diameter θ_s (Equations (17)– (19)). Thus, given R_d/R_s and θ_s we can find θ_maj, and the remaining parameter to estimate is c_p. In practice this could be estimated from a simultaneous analysis of the SED of the Be star. However, for our purpose here, we adopted the relationship between the disk to star flux ratio and projected surface areas (Equation (29) and Figure 14) to estimate c_p from r and R_d/R_s. This relationship is poorly defined for fainter disks ((1 − c_p)/c_p < 1), where the disks become optically thin and the ratio of areas argument no longer applies.

The stellar angular diameter is found from the assumed stellar radius and distance, and we made estimates for two cases, B0 V and B8 V types for the Be star, and for three distances corresponding to visual magnitudes 3, 5, and 7. We used stellar radii and absolute magnitudes for these classifications from the compilation of Gray (2005), and we neglected any extinction in the calculation of distance from the magnitude difference. Figure 15 shows the resulting predicted visibilities for a K-band measurement with a projected baseline of 300 m as a function of R_d/R_s for these different cases. Each plot shows how the visibility at this baseline declines as the disk size increases, and we can use these to estimate the smallest disk detectable. For example, we see in Figure 15(a) for a B0 V star of apparent magnitude 5 that the curve dips below V = 0.8 at R_d/R_s = 3.4 from which we would infer that only disks larger than this would be detected with the CHARA Array. As expected, we can detect smaller disks in nearer (brighter) Be stars. Figure 15(b) shows the case for a later B8 V type that is somewhat more favorable for detection of smaller disks. Note that at small disk radii we simply assume that all the flux is stellar, so the limiting visibility near R_d/R_s = 1 corresponds to the stellar visibility.

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Be stars are generally fast rotators, but it is difficult to determine the actual equatorial rotational velocity V_rot from the observed projected rotational velocity Vsin i (measured from the rotational Doppler broadening of the lines; Townsend et al. 2004) because of the unknown stellar inclination i. If we assume that the stellar spin and disk rotational axes are coaligned, then we can use the disk axial ratio to set the stellar inclination (Equation (25)). We collected the seven r values derived from our fits (Table 7) together with estimates from previous work (Table 8) to find mean sin i factors for 12 Be stars in our program. We adopted the projected photospheric rotational velocities Vsin i and the critical rotational velocities V_crit from Frémat et al. (2005), who derived these values after making corrections for the effects of gravity darkening. Then we divided Vsin i by sin i to find the equatorial rotational velocity V_rot. Our results appear in Table 9 that lists the HD number of the star, Vsin i and V_crit (Frémat et al. 2005), V_rot, and two ratios, V_rot/V_crit and Ω_rot/Ω_crit. The ratio of angular rotational velocities is given for the Roche approximation using expressions from Ekström et al. (2008),

$$\begin{equation} \frac{\Omega _{\rm rot}}{\Omega _{\rm crit}} = {3 \over 2} \frac{V_{\rm rot}}{V_{\rm crit}} \left[1 - {1\over 3}\left(\frac{V_{\rm rot}}{V_{\rm crit}}\right)^2\right]. \end{equation} \tag{ 30 }$$

We find that most of the stars in this subsample are rotating very quickly, and two targets may have attained the critical rate (γ Cas = HD 5394 and 48 Lib = HD 142983). However, there are two targets with much more moderate rotational velocities, β Psc = HD 217891 and $\upsilon$ Cyg = HD 202904. The former has a rather large axial ratio r = 0.70 ± 0.15, and it is possible that the uncertainties would allow a small inclination and hence large rotational velocity. However, the case of $\upsilon$ Cyg is more difficult to understand. Neiner et al. (2005) argue on the basis of the narrow lines in the spectrum that the axial ratio should be close to r ≈ 0.9 if it is actually a rapid rotator. This is much larger than the value we derive from the Gaussian elliptical fits, r = 0.26 ± 0.13. Additional interferometric observations are needed to confirm or remove this discrepancy.

We find a mean angular velocity ratio of Ω_rot/Ω_crit = 0.88 ± 0.17 (s.d.) for the sample of 12 Be stars or 0.95 ± 0.04 (s.d.) if we remove β Psc and $\upsilon$ Cyg from the sample. These results support earlier Be star angular velocity ratio estimates of 0.88 (Frémat et al. 2005), 0.7–1.0 (Cranmer 2005), 0.8–1.0 (Huang et al. 2010), and 0.95 ± 0.02 (Meilland et al. 2012). The consistency of these results suggests that rapid rotation in Be stars plays a key role in the Be phenomenon.