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Astron. Astrophys. 348, 831-842 (1999)

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6. Keplerian rotation of disks around Be stars

The question whether disks of Be stars are rotationally supported is still considered an open issue by many. Truly mysterious appears the mechanism which would supply the necessary angular momentum transfer to the disk, since even the most rapidly rotating Be stars do not seem to reach much more than [FORMULA]70% of the break-up velocity (e.g., Porter 1996). Two reasons may explain why this matter has for a long time perhaps not been given the emphasis that it deserves. One is the realisation that Struve's concept of a purely rotational instability is not supported by the actual distribution of equatorial velocities (cf. Porter 1996). The other one is the discovery that Be stars do lose mass in a high-speed wind (cf. Prinja 1989) that is more prominent at higher stellar latitude.

However, a fair amount of at least circumstantial observational evidence that disks of Be stars are rotating has been mounting during the past couple of years, often as a by-product of other work:

  • Polarisation and geometrical flatness: The very presence of a disk-like, as opposed to spheroidal, geometry is with the least number of additional assumptions attributable to rotational flattening. This has for long been inferred from the intrinsic polarisation of Be stars (Poeckert et al. 1979). The constancy of the polarisation angle through all phases of disk transformation (e.g., Hayes & Guinan 1984) shows that the plane of the disk around single Be stars is constant in space and probably corresponds to the one of the equator. The final proof of the disk geometry has come from direct interferometric imaging (Quirrenbach et al. 1994).

  • Stellarv sin iand width of emission lines:The clear correlation between the stellar [FORMULA] and the width of circumstellar emission lines (Slettebak 1976, Hanuschik 1989) is also most easily reconciled with an equatorial rotating disk model.

  • Emission line profiles: Symmetric H[FORMULA] emission lines of relatively low equivalent width often show a V-shaped central absorption. Hummel & Vrancken (1999) have modelled the absorption by the circumstellar shell, taking into account the velocity shear in the shell, obscuration of the shell by the star, and the finite size of the stellar disk. They conclude that the depth of the central absorption (and consistency with interferometrically measured shell radii) requires that the parameter j in the shell velocity law [FORMULA] (where r is the distance from the center of the star and [FORMULA] denotes the rotational velocity at the stellar surface) is on average less than 0.65 in their small sample of stars (incl. [FORMULA] Cen). The value [FORMULA] corresponds to a Keplerian disk.

  • Outbursts: Rivinius et al. (1998a, 1998b) have constructed a detailed temporal profile of the line emission outbursts of the Be star µ Cen, which are events of mass ejection and related to the beating of nonradial pulsation modes (Rivinius 1999, Baade 1999). It shows that matter is ejected at super-equatorial velocity which, after allowance for the relatively weakly constrained inclination angle and for plausible values of stellar mass and radius, comes at least close to the critical velocity. Kroll and Hanuschik (1997) studied somewhat less complete observations of outbursts of the same star. They find that inclusion of viscosity in the simulation of the orbital evolution of ballistically ejected matter (by some arbitrary mechanism) leads to the formation of a Keplerian disk from some fraction of the ejecta.

  • [FORMULA] variability and disk oscillations:Numerous Be stars undergo cyclic variations of the ratio in strength, [FORMULA], of the violet and red components of their emission lines with cycle lengths of the order of a few years (cf. the compilation in Okazaki 1997). Okazaki (1996, 1997) and Savonije and Heemskerk (1993) have modelled this general behaviour in terms of, respectively, retro- and pro-grade global one-armed oscillations of the disk. Actual line profile variations based on such dynamics were calculated by Hummel & Hanuschik (1997). The prograde-mode version was recently given strong observational support by interferometric observations of [FORMULA] Tau (Vakili et al. 1998) and [FORMULA] Cas (Berio et al. 1999) at different [FORMULA] phases. This matter is of relevance in the context of this paper as the disk oscillation models by necessity require (quasi-)Keplerian rotation.

  • Tilted/warped disks: Hummel (1998) recently suggested that particular peculiarities in the long-term emission-line variability of [FORMULA] Cas and 59 Cyg at certain epochs may be caused by a temporary tilt or warping of a precessing disk. This explanation, too, would require the disk to be rotating.

It would be premature and, given the observed spectrum of variabilities and the importance also of non-gravitational forces, probably even wrong to conclude from the above enumeration that the azimuthal velocity law in Be star disks is strictly Keplerian. But there can be no doubt that the disks (i) are rotating and (ii) do so sufficiently much in a Kepler-like way that Hanuschik's model is based on a reasonable approximation. Accordingly, this model does provide the correct qualitative explanation of CQE's: they are due to the line transfer in a rotating gaseous envelope and the finite size of the disk of the central star.

The inverse reasoning is also valid: Since Hanuschik's model explains CQE's, it re-inforces independent conclusions that rotation plays an important role in supporting disks of Be stars against gravitational collaps. With respect to CQE's alone, this is, more strictly speaking, correct only for weakly developed disks. But, e.g., the explanation of long-term V/R variations by disk oscillations is not subject to such a limitation. Because the issue of rotation in disks of Be stars has prior to the work of Hanuschik, Hummel, and co-workers for quite some time not been very explicitly addressed, the velocity law may in some areas have been used almost as a free parameter. This may have possible repercussions. An important topic, that is worthwhile to re-visit in this connection, is the formation of disks around Be stars. This is the subject of the following section (for a recent review see also Bjorkman 1999).

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© European Southern Observatory (ESO) 1999

Online publication: August 13, 199
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