IS ETA CARINAE A FAST ROTATOR, AND HOW MUCH DOES THE COMPANION INFLUENCE THE INNER WIND STRUCTURE?^*

We compute two-dimensional latitude-dependent wind models for η_A using an updated version of the two-dimensional radiative transfer code of Busche & Hillier (2005, hereafter BH05) and via a methodology similar to that described in Groh et al. (2008). In the following, we briefly describe only the essential aspects of the code. We refer the reader to BH05, Groh et al. (2006, 2008, 2009a), and Driebe et al. (2009) for further details on the code's applications.

The two-dimensional models use as input several quantities (e.g., energy-level populations, ionization structure, and radiation field) from the spherically symmetric model of η_A (H01, H06) computed using the non-local thermal equilibrium, fully line blanketed radiative transfer code CMFGEN (Hillier & Miller 1998). We assume the same parameters derived by H01: stellar temperature $\mathit {T}_{\star }$= 35,310 K (at Rosseland optical depth, τ_Ross = 150), effective temperature $\mathit {T}_{\rm eff}$= 9210 K (at τ_Ross = 2/3), luminosity $\mathit {L}_{\star }= 5 \times 10^6 \, \mathit {L}_{\odot }$, $\dot{M}= 10^{-3} \, \mathit {M}_{\odot }{\rm \ yr}^{-1}$, v_∞ = 500 km s^-1, a clumping volume-filling factor f = 0.1, and distance d = 2.3 kpc. The atomic model and abundances are described by H01 and H06.

The code allows for the specification of any arbitrary latitude-dependent variation of the wind density ρ and v_∞. For the latitudinal variation of ρ, we adopt the predictions from Owocki et al. (1998) for gravity-darkened (GD) line-driven prolate winds,

$$\begin{equation} \mathrm{GD, prolate}: \frac{\rho (\theta)}{\rho _{0}} \propto \sqrt{1 - W^2 \sin ^{2}\theta }, \end{equation} \tag{ 1 }$$

and the non-GD scaling for oblate winds,

$$\begin{equation} \mathrm{\mbox{non-GD}, oblate}: \frac{\rho (\theta)}{\rho _{0}} \propto \frac{1}{1 - W^2 \sin ^{2}\theta }, \end{equation} \tag{ 2 }$$

where θ is the colatitude angle (0°= pole, 90°= equator), ρ₀ is the density at the pole, and W is the ratio of the rotational velocity of η_A ($v_{\rm {rot}}$) to its critical velocity ($v_{\rm {crit}}$). Note that the oblate-wind density scaling assumes a standard CAK (Castor et al. 1975) α parameter value of 2/3. The inclination angle i is the tilt angle between the rotation axis of the star and the line of sight (i = 0° = pole-on view, i = 90° = equator-on view). Emissivities and opacities are scaled according to the modified density, but the code assumes a spherically symmetric ionization structure. Gravity darkening and distortion of the shape of the hydrostatic core of the star are not included at this time; however, these are not expected to significantly affect the K-band emitting region since the source function in the infrared continuum depends only weakly on the temperature. For consistency, we also assume the Owocki et al. (1998) predictions for v_∞(θ).

We calculated a grid of prolate and oblate models with i ranging from 0° to 90° in steps of 1°, and W ranging from 0 to 0.99 in steps of 0.01, totaling 18,200 models. For a desired position angle (P.A.) orientation on the sky,⁵ the K-band image, visibilities, and closure phases (CPs) are computed.

Since the spherical CMFGEN model of H01 does not fit the geometry of the K-band continuum (VB03, W07), we first attempted prolate-wind models (Figure 1) to reproduce the suggested elongation of the K-band continuum observed with VINCI in 2003 January–February. We find the model parameters to be degenerate, as models with P.A. = 108°–142°, W= 0.80–0.99, and i = 30°–90° fit the VINCI data equally well. Figure 2(a) presents a color-coded plot of the reduced χ² values of the fit to the VINCI data, with the whitish regions corresponding to the best-fit models. In general, models with relatively large W require lower i, while those with relatively lower W need larger i to fit the data.

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Additional constraints to W and i can be obtained by analyzing the CP, which measures the asymmetry of the brightness distribution. W07 measured CP=0° ± 3° in the K-band continuum in 2005 February, and a color-coded plot of the model CPs, computed using the same telescope array configuration used by W07, is shown in Figure 2(b). We find that prolate models with high W (0.82–0.99) and intermediate inclination angles (i ≃ 30°–60°) yield a K-band image which is noticeably not point-symmetric (Figure 1(h)), and thus are ruled out since they have much larger CPs (7°–19°) than observed. To fit simultaneously the VLTI/VINCI visibilities and the VLTI/AMBER CPs, W = 0.77–0.92 and i = 60°–90° are needed (Figure 2(c)). Strikingly, this inclination is significantly higher than that of the Homunculus (i_Hom = 41°; Smith 2006), which has been assumed so far to align with the current rotation axis of η_A. We find that, based on the available interferometric data, if η_A has a prolate wind, its rotation axis is not necessarily aligned with the Homunculus polar axis.

The image asymmetries from models with i = 30°–60° (red spot in Figure 1(h)) arise from the latitude-dependent K-band photosphere associated with the wind density variations. Rays from the northwest hemisphere cross the low-density equator, and so accumulate a lower optical depth than those from the southeast hemisphere at the same impact parameter. The deeper penetration of northwest versus southeast rays thus leads, for the nearly spherical source function which is larger for smaller radii, to a stronger emerging radiation field, and thus a higher surface brightness in the northwest hemisphere.

One may ask if a prolate-wind model is unique in being able to fit the interferometric data. Therefore, we also analyze the visibility as a function of telescope P.A. and CP for oblate-wind models. We find that the oblate model fits the visibilities as well as the prolate model, provided that P.A. = 210°–230° (Figure 3), i.e., the stellar rotation axis projected on the plane of the sky is rotated by 80°–100° from the Homunculus symmetry axis. Oblate models with W = 0.73–0.90 and i = 80°–90° are able to fit both the observed visibilities and CP (Figures 2(d)–(f)). Therefore, for oblate-wind models, the rotation axis of η_A is not aligned with the Homunculus polar axis.

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Our models are based on the spherically symmetric models of η_A (H01, H06), with the same stellar parameters as in Section 2.1, but using the two-dimensional code of BH05 to create a low-density cavity and dense interaction-region walls. Further details are given in J. H. Groh et al. (2010, in preparation), and here we outline only the main characteristics of our implementation.

We approximate the cavity as a conical surface with half-opening angle α and interior density 0.0016 times lower than that of the spherical wind model of η_A. We include cone walls of angular thickness δα and, assuming mass conservation, a density contrast in the wall of f_α = [1 − cos(α)]/[sin(α)δα] (Gull et al. 2009) times higher than the wind density of the spherical model of η_A at a given radius. The conical shape is justified since the interferometric observations were taken at orbital phases sufficiently before periastron when such a cavity has an approximately two-dimensional axisymmetric conical form (Okazaki et al. 2008). Based on the expected location of the cone apex during these phases (Canto et al. 1996; Okazaki et al. 2008), we place the cavity at a distance d_apex from the primary star. We assume that the material inside the cavity and along the walls has the same ionization structure as the wind of η_A, implying that the shocked wind of the primary star cools radiatively.

The amount of influence of the wind–wind interaction on the observables depends on two factors: how close the cavity gets to the K-band emitting region of η_A ("bore hole effect"; Madura & Owocki 2010), and how much free–free radiation is emitted by the dense walls of the shock cone ("wall effect"). We find that the interferometric observables in the K band are insensitive to the bore hole effect if d_apex ≳ 8AU. This is because most of the observed K-band emission comes from a region with a characteristic 50% encircled-energy radius of 4.8 AU (W07), and a halo with significant emission exists up to ∼8 AU. For the standard assumed orbital parameters of Eta Car (see, e.g., Okazaki et al. 2008), d_apex ≳ 8AU during most of the orbit (0.055 < ϕ < 0.945), which includes the epochs when the VINCI (ϕ_vb03 = 0.92–0.93) and AMBER (ϕ_w07 = 0.27–0.30) observations were taken. The VINCI observations were obtained when d_apex ≃ 10AU, thus a bore hole effect alone is not able to explain the VINCI observations.

According to our models, the main effect from the wind–wind collision zone at the orbital phases analyzed is extended free–free emission from the compressed walls. Such an effect may influence the geometry of the K-band emitting region even at orbital phases far from periastron, depending on the observer's location, geometry of the cavity, i, α, and f_α. A model with α = 54°, f_α = 9.7 (i.e., δα = 3°; Gull et al. 2009), and i = 41° shows significant wall emission and is able to produce a significant elongation of the K-band continuum (Figure 4). To explain the VINCI observations, the symmetry axis of the cavity has to be oriented along P.A. ≃35°–45° (i.e., southwest–northeast axis; Figure 4), which is roughly consistent with that expected from a longitude of periastron of ω = 243° (e.g., Okazaki et al. 2008; Parkin et al. 2009; Gull et al. 2009; Groh et al. 2010) and a counterclockwise motion of η_B on the sky (Figure 4(b)). The $\dot{M}$ of η_A needs to be slightly reduced to $8\times 10^{-4} \, \mathit {M}_{\odot }{\rm \ yr}^{-1}$ to compensate for the extra extension in the K-band emitting region caused by the wall effect. Comparing Figures 1 and 4, the cavity and walls have an effect on the available interferometric observables that is as large as that due to a latitude-dependent wind caused by rapid rotation, although the morphology of the K-band images is noticeably different.

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The cavity model assumed here is very idealized, since it does not take into account the distortion of the shock cone (Okazaki et al. 2008), or the instabilities thought to arise in the wind–wind collision zone (Parkin et al. 2009). Nevertheless, our conclusion is that, using a two-dimensional model, noticeable elongation of the K-band region can occur if the apex of the cone penetrates a significant distance into the K-band emitting region, that cone is open toward the observer, and/or the compressed walls are dense enough and emit free–free radiation. Three-dimensional radiative transfer models, when available, would be desirable to test the conclusions found here.