1 A brief critical evaluation of the existing models of Be stars

1.1 The Be phenomenon

More that one tenth of all B stars exhibit time-variable emissions in their Balmer (and some other) line profiles. These stars, called Be stars, represent perhaps the most variable objects among massive stars. Their continuum and line spectra vary on several time scales, ranging from minutes to decades or more. Actually, the longest time scale of their variability is not known and it is conceivable that every B star may become a Be star for an interval during its evolution. The phenomenon also partly extends to hotter Oe stars and cooler Ae stars.

In spite of all the effort of several generations of stellar astronomers, neither the nature of the Be phenomenon nor the physical causes of variations are well understood at present. There is general agreement that the emission lines are formed in extended gaseous envelopes around these stars. Such envelopes have dimensions of one or even two orders of magnitude larger than the stars themselves and re-radiate the stellar radiation in all directions. Evidence has been accumulated to confirm Struve's (1931) original suggestion that the Be envelopes are rotationally flattened disks. An ultimate proof of this statement came from spectro-interferometric and spectro-polarimetric observations of several Be stars which allowed a partial spatial resolution of their envelopes - see Quirrenbach et al. (1993,1997).

However - as far as the understanding the very complex Be phenomenon is concerned - the current situation with many partial models, explaining often only one aspect of it, is not very satisfactory.

1.2 Attempts to explain the Be phenomenon

Concerning the very origin of the Be envelopes, the number of hypotheses has been growing steadily and there is no general agreement on a single one. These hypotheses are as follows:

1.

Rotational hypothesis by Struve (1931) explains the formation of Be envelopes by rotational instability of underlying, rapidly rotating stars. Its main argument is the observed correlation between the width of the observed Balmer emission lines and v sin iof the respective stars (which has been confirmed by several later studies - cf., e.g., Hanuschik et al. 1988). Struve himself was aware of the fact that the rotational model does not explain variability, namely the long-term V/R variations, observed for a number Be stars at certain epochs. He ingeniously suggested a slow spatial revolution of temporarily elongated envelopes as an explanation. Critics of the rotational model argue that one observes a number of similarly rapid rotators, called Bn stars, which were never observed to have any emission lines (e.g. Baade 1992). Many of the investigators are also convinced that the Be stars are rotating below their break-up speeds at the equator (cf., e.g., Porter 1996). Boyarchuk (1958) pointed out that the rapid rotation itself cannot account for the Be phenomenon and all later quantitative models led to the same conclusion (see, e.g., Limber & Marlborough 1968). Note also that the rotational hypothesis itself does not offer a clue to the observed variations on several time scales.

2.

Outflow models Several attempts were made to explain the origin of Be envelopes as a spheroidal outlow of gaseous material from the underlying star: the stellar-wind model by Gerasimovic (1934,1935), the variable mass flux model by Doazan & Thomas (1987) and Doazan (1987), the "bi-stable'' or "axi-symmetric'' radiation-driven wind model by Lamers & Pauldrach (1991), Araújo et al. (1994) and Stee & Araújo (1994) and the wind-compressed disk model by Bjorkman & Cassinelli (1993) (see also the review by Bjorkman 2000). The latest one was considered as a very promising idea several years ago but the radiation-hydrodynamics simulations by Owocki et al. (1996) showed that the small non-radial components in line forces inhibit formation of an equatorial disk by the wind-compressed model. Also, this model could have problems providing strong enough stellar winds in Be stars of later spectral subclasses. More generally - outflow models do not offer an explanation for the variability of Be stars.

3.

Non-radial pulsations of Be stars were first suggested to explain the observed rapid light and line-profile variations (Baade 1979,1982; Bolton 1982) but the idea has been developed in a complex attempt to explain the Be phenomenon by Baade and his students. In particular, Rivinius et al. (1998a) argued that constructive interference of several pulsational modes can lead to release of a new Be envelope and claimed agreement with the observed emission-line episodes for $\mu$ Cen. They did not present any energy balance considerations of whether such a mechanism could really work, however. It is certainly true that the series of line profiles obtained for a few Be stars are remarkably similar to theoretical line profiles, based on non-radial pulsation modelling. It is conceivable, however, that the description of line-profile variations in terms of spherical harmonic functions would also work to model line-profile variations due to co-rotating structures, located slightly above the stellar surface, which has been advocated, for instance, by Harmanec (1989), Smith et al. (1998) or Balona (2000). Note also that the attempts to derive the macroscopic stellar properties from the modelling of line-profile changes, interpreted as non-radial pulsations, led to too small values of stellar radii, contradicting the estimates based on Hipparcos parallaxes (compare the Maintz et al. 2002 radius of 3.4 $R_\odot$ of the B2e star $\mu$ Cen from the line-profile modelling to Harmanec's 2000 estimate of 5.2-5.4 $R_\odot$ from the V-band brightness outside emission-line episodes and the Hipparcos parallax). Aerts (2000) also warned that the interpretation of Be stars in terms of non-radial pulsations needs further careful tests. Harmanec (2001) pointed out that the formal description of radial-velocity variations of $\mu$ Cen by six non-radial modes did not lead to the gradual decrease of the rms error of the fit, as would be expected for real multiperiodicity. It is fair to say, however, that until such a modelling as was carried out for the non-radial pulsation model will also be carried out for alternative models, the model of non-radial pulsations represents the best available description of the observed rapid line-profile changes of Be stars.

4.

The helmet-type magnetic field model was presented by Underhill (1983,1987) which assumes the presence of organized magnetic fields in Be stars and interprets their envelopes as helmet-type co-rotating structures. This model has been largely ignored by the community (perhaps due to the fact that there was no observational technique allowing the detection of the putative magnetic fields). Given the observational evidence in favour of co-rotating structures slightly above the stellar photospheres (e.g. Harmanec 1989,1999; Harmanec & Tarasov 1990; Smith et al. 1998 or Balona 2000), and the improving sensitivity to detect even weaker magnetic fields in hot stars, this hypothesis probably deserves further critical examination.

5.

Binary models The last class of models are models that assume that the Be phenomenon is somehow related to the duplicity of the Be stars. They are discussed in detail in the next section.