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

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2. The Eddington limit and the R CrB stars

2.1. The Eddington limit in the [FORMULA] -log g diagram

The location of the Eddington limit in a [FORMULA] -log g diagram can be estimated by model atmosphere calculations (Lamers & Fitzpatrick 1988; Gustafsson & Plez 1992). It should be emphasized that these models consistently include the radiative acceleration in the equation of hydrostatic equilibrium. Therefore, the location of the limit is not obtained by extrapolation towards lower gravities nor does it imply zero density, as has sometimes been claimed in the literature (e.g. Humphreys & Davidson 1994; Nieuwenhuijzen & de Jager 1995). An interesting conclusion from the above studies is that the theoretical opacity-modified Eddington limit coincides with the upper luminosity limit for stars (Lamers & Fitzpatrick 1988). It is therefore possible that the Eddington limit, possibly in connection with rotation (Langer 1997a,b; Owocki & Gayley 1997), prevents the evolution of stars into the super-Eddington regime by the existence of radiative instabilities. These may give rise to the eruptions of the Luminous Blue Variables (LBVs) (cf. Humphreys & Davidson 1994).

Here a similar investigation but for the H-deficient R CrB and related stars will be carried out; a preliminary report can be found in Asplund & Gustafsson (1996). The presence of [FORMULA] -inversions in such model atmospheres was noted by Asplund et al. (1997a), who speculated that it may be the trigger mechanism for the enigmatic visual declines of the R CrB stars. The variability of these stars is likely due to the formation of an obscuring dust cloud close to the stellar surface in the line-of-sight, but how the dust condensation can proceed despite the relatively high equilibrium temperatures has long been an enigma (cf. Clayton 1996). The presence of shocks seems to be a very promising mechanism for the gas to cool sufficiently and thereby trigger dust formation (Woitke et al. 1996).

The models used here are recently constructed line-blanketed model photospheres with typical abundances of R CrB stars (Asplund et al. 1997a). In Fig. 1 the variation of [FORMULA] with optical depth is displayed for a sequence of models corresponding to constant luminosity and mass (e.g. [FORMULA] L [FORMULA] and [FORMULA] M [FORMULA]). The super-Eddington luminosities only occur in the deep atmospheric layers where He I ionizes and, as expected, where [FORMULA] -inversions occur. Since the R CrB stars presumably evolve towards higher [FORMULA] (Kilkenny 1982), the stars will move from the sub-Eddington regime into having super-Eddington luminosities, as is seen in Fig. 2 where the theoretical Eddington limit in the [FORMULA] -log g diagram is shown. Compared with solar abundances the limit is shifted towards higher [FORMULA] due to the higher ionization potential of He compared with H, but also extends to slightly larger log g. The location of the Eddington limit for higher [FORMULA] has not yet been explored, since a self-consistent hydrodynamical treatment is needed for [FORMULA] K when [FORMULA] occurs for [FORMULA] and line opacity may determine [FORMULA].

[FIGURE] Fig. 1. [FORMULA] as a function of optical depth for a sequence of models with R CrB abundances at constant luminosity and mass. All models have C/He = 1%. The [FORMULA] K model is only partly affected by convection in the inner layers, while the hottest model is not at all. Increased molecular absorption causes the slight increase in [FORMULA] towards the surface for the coolest model

[FIGURE] Fig. 2. The location of the opacity-modified Eddington limit (solid) in relation to the observed values of R CrB stars (Lambert et al. 1997; Asplund et al. 1997c). Also shown is the classical Eddington limit (dashed) and the hot R CrB and EHe stars with determined parameters (Jeffery 1996). The cross in the upper left corner illustrates the typical uncertainties in the estimated parameters of the R CrB stars

The exact location of the computed Eddington limit will naturally depend on the detailed properties of the model atmospheres. For [FORMULA] K convection in the deeper layers of the He I ionization zone restricts the limit to lower log g, since the radiative flux diminishes in the presence of a significant convective flux. Also, the lower temperatures decrease [FORMULA] and [FORMULA] further. Including turbulent pressure would therefore bring the limit towards greater gravities by typically 0.2 dex, due to less efficient convection with lower densities. A C/He ratio of 1% has been assumed on the basis of the measured ratio in extreme helium (EHe) stars (Jeffery 1996), which presumably the R CrB stars are related to (Lambert et al. 1997). A higher carbon abundance pushes the Eddington limit towards higher gravities the same way as turbulent pressure does. A lower overall metallicity by 0.5 dex would decrease the limit by about 0.2 dex by lowering the temperatures in the inner layers with a diminished line-blanketing.

2.2. Possible consequences for H-deficient stars

In Fig. 2 the estimated parameters of the R CrB stars (Lambert et al. 1997; Asplund et al. 1997c) are also shown. Obviously, rather than being randomly located in the diagram as one a priori might have guessed, the R CrB stars fall close to the limit, which suggests that a connection between the declines of R CrB stars and the Eddington limit exists. Thus, there is an alluring resemblance between the R CrB stars and the Luminous Blue Variables in their proximity to the Eddington limits and their eruptive behaviours (Asplund & Gustafsson 1996). If the R CrB stars are evolving towards higher [FORMULA] (Kilkenny 1982) and the EHe domain at constant luminosity, their immediate progenitors should be the H-deficient carbon (HdC) stars (Lambert et al. 1997). The EHe and HdC stars experience no visual fadings despite being otherwise similar to the R CrB stars. It is therefore plausible that the reason for the non-variability is that the HdC stars have not yet reached the Eddington limit while the EHe stars are located on the stable side of the limit at higher [FORMULA].

If the declines of the R CrB stars are indeed triggered by the super-Eddington luminosities, one could expect a correspondence between the maximum value of [FORMULA] in the atmosphere and frequency of declines. In the pulsation-induced dust condensation model for the R CrB stars (Woitke et al. 1996), in which the dust formation is triggered by propagating shock waves due to ordinary pulsations, one would sooner expect a correlation with pulsation strength. Intriguingly, V854 Cen, which is the R CrB star most frequent in decline, has very weak photospheric pulsations (Lawson & Cottrell 1997) but by a wide margin the largest [FORMULA] found of the 18 R CrB stars analysed sofar. RY Sgr, on the other hand, has by far the strongest photospheric pulsations but only average amount of declines, which might reflect the fact that [FORMULA] is not unusually high. Pulsation may, however, push some stars across the limit by periodically shifting them towards higher [FORMULA], e.g. [FORMULA] K for R CrB itself (Rao & Lambert 1997).

The proposed connection between the Eddington limit and the declines of the R CrB stars is certainly not without problems, which deserve to be addressed further. A few stars seem not to behave as expected if the proposal is correct. In particular this is true for some of the coolest EHe stars and the three hot R CrB stars.

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Online publication: January 16, 1998