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Astron. Astrophys. 330, 641-650 (1998)

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6. Conclusions and speculations

It has been shown that H-deficient model atmospheres may have radiative forces which exceed gravity, similar to the case of some luminous H-rich stars (Lamers & Fitzpatrick 1988; Gustafsson & Plez 1992). Such super-Eddington luminosities manifest themselves as [FORMULA] -inversions in hydrostatic atmospheres. The inversions are, however, not an artifact of this assumption but can also be present in dynamical atmospheres. The inversions are not removed unless a very high mass loss rate is present, and is neither necessarily removed by convection if the large radiative forces occur close to the surface. Thus, super-Eddington luminosities does not automatically initiate a stellar wind.

The location of the H-deficient R CrB stars in the immediate proximity of the computed opacity-modified Eddington limit (Fig. 2), is certainly suggestive of a connection between the enigmatic visual declines of the stars and the Eddington limit. It is therefore proposed that instabilities as the R CrB stars encounter the Eddington limit during their evolution towards higher [FORMULA] are the unknown trigger mechanism for their famous variability. The Eddington limit may thus be the "smoking gun" for the declines: gas is ejected due to the high radiative forces in the atmospheres, which then cools rapidly first radiatively and later from adiabatic expansion to reach sub-equilibrium temperatures and thus possibly enable dust condensation (cf. Woitke et al. 1996 for the similar case of shocks instead induced by pulsations). Such gas ejections may be observable as absorption components in strong lines, as observed by Rao & Lambert (1997) in R CrB at maximum light. If a connection indeed exists, it could explain why the EHe and HdC stars do not show R CrB-like variability as a result of being located on the stable side of the Eddington limit. It also points to an interesting similarity between the R CrB stars and the LBVs. Thus, the two types of stars known to be situated at the Eddington limit both show eruptive behaviour, which suggests a similar underlying physical explanation for their variability.

A search for radiative and dynamical instabilities in the atmospheres has been carried out for both H-rich and H-deficient late-type supergiants, but only partly successfully. Sound waves are found to be amplified by the large radiative forces but the linear stability analysis reveal only rather slow growth rates compared to for early-type stars despite the super-Eddington luminosities. Such radiation-modified sound waves may thus be partly responsible for the semi-regular pulsational variations of such supergiants, but it is doubtful whether the instability is efficient enough to eject gas clouds from the atmospheres as speculated above. The atmospheres are also found to be close to dynamically unstable, which might give rise to increased mass loss, as previously suggested for LBVs (Stothers & Chin 1993) and for the termination of the AGB phase (e.g. Paczyski & Ziól kowski 1968; Wagenhuber & Weiss 1994). This does not, however, explain in what way the R CrB stars are special compared to other similar stars. In fact, the H-rich models seem to be more vulnerable to such instabilities, which suggests that dynamical instabilities are not the answer to the declines of the R CrB stars. Another possibility is of course that the same dynamical instabilities occur in both types of stars but that the H-deficient and C-rich environment of the R CrB stars greatly favours dust condensation, and thus the observational manifestation, compared to for H-rich compositions.

One can also imagine that the Eddington limit is still responsible for the behaviour of the R CrB stars though in a more indirect way. Langer (1997a,b) has shown that the effect of rotation coupled with strong radiative forces may lead to increased mass loss in the equatorial plane, and it is possible that such a mechanism is at work in these supergiants, despite the presumably relatively small rotational velocities. The difference in observed variability between the R CrB stars and the HdC and EHe stars could then be the result of viewing angle. There are some observational evidence for such bipolarity (e.g. Rao & Lambert 1993; Clayton et al. 1997). Violent instabilities due to strange mode pulsations leading to dramatically increased mass loss is another possibility, which is partly related to the Eddington limit, since both have their origin in ionization zones, as do the dynamical instabilities discussed above. Whether any of these instabilities, or a combination thereof, could help explain the variability of the R CrB stars and the LBVs certainly deserves further investigation.

Finally, it should be remembered that the stability analyses presented here assume the unperturbed model to be a realistic description of the stellar atmosphere. At least for the R CrB stars, there are strong indications that this is in fact not the case (Gustafsson & Asplund 1996; Asplund et al. 1997a,b,c; Lambert et al. 1997). If the supergiant atmospheres are indeed distinctly different from the predictions of standard models, the efficiency of radiative instabilities may have been underestimated. One can suspect that models based on the normal assumptions, such as the mixing length theory for convection, near the Eddington limit may be very unsuitable. Clearly, radiative hydrodynamical simulations of such atmospheres would be valuable, also since they would shed further light on possible instabilities.

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© European Southern Observatory (ESO) 1998

Online publication: January 16, 1998