1. The Eddington limit and the development of gas pressure inversions
The Eddington limit corresponds to a situation where the radiative acceleration outwards equals the gravitational acceleration inwards. Eddington (1926) originally only considered electron scattering for the opacity (the classical Eddington limit), but in an ionization zone of an important element the true opacity can be much higher. Therefore, the opacity-modified Eddington limit in stellar atmospheres can occur at significantly greater gravities than the classical Eddington limit, as is evident for example from the calculations of Lamers & Fitzpatrick (1988), Gustafsson & Plez (1992) and Asplund & Gustafsson (1996).
If hydrostatic equilibrium is required, a stellar luminosity which locally exceeds the Eddington luminosity automatically forces the development of a gas pressure () inversion, as seen from the equation of hydrostatic equilibrium:
From rearranging the above expression one obtains
with the radiative acceleration defined by:
(e.g Mihalas 1978). Here denotes the physical flux and the total mass extinction coefficient (with dimension cm2 g-1). Obviously a positive -gradient must occur when the Eddington limit is encountered if hydrostatic equilibrium is valid.
It should be emphasized that a density inversion does not necessarily imply that the Eddington limit is exceeded. A density inversion predominantly occurs due to a changing molecular weight in the ionization zone of a dominant species, while a -inversion reflects a super-Eddington luminosity due to high opacity (Asplund et al. 1997a). Since both are related to ionization they may often occur in the same atmospheric layers, which has probably caused the confusion found in the literature (e.g. Maeder 1989).
It should be noted that -inversions only seem possible to develop in optically thick conditions. If the radiative acceleration exceeds gravity when lines dominate the opacity, the radiative force is highly unstable to perturbations induced by velocity gradients, since the spectral lines may then absorb unattenuated continuum flux. Therefore, rather than the development of a -inversion, a stellar wind will be initiated, which is the case for radiatively driven winds of hot stars. The aim of the present study is to investigate whether -inversions in late-type stars, where the opacity instead is dominated by continuous opacity, are subject to a similar instability which would prevent their existence. Before considering possible instabilities, in Sect. 2the derived Eddington limit for H-deficient stars is presented and a possible connection between the limit and the declines of the R Coronae Borealis (R CrB) stars is discussed.
© European Southern Observatory (ESO) 1998
Online publication: January 16, 1998