Dynamical stability has been extensively studied, in particular in connection with the termination of the AGB (e.g. Paczyski & Ziólkowski 1968; Wagenhuber & Weiss 1994). Radial adiabatic oscillations will grow exponentially in time for dynamical instability, which is formally encountered if the first generalized adiabatic exponent
is less than in a region effectively isolated from the rest of the star. An ideal gas has while pure radiation has , but the combined effect of non-negligible radiation pressure and ionization may push below the limiting value. For dynamical instability to occur, the pressure weighted volumetric mean value of between the depth r and the stellar surface
must be reduced below in a significant fraction of the star. Stars with a pronounced core-envelope structure and high luminosity-to-mass ratios, like the supergiants investigated here, may show dynamical instabilities in ionization zones (e.g. Stothers & Chin 1993; Wagenhuber & Weiss 1994).
In Fig. 3 the run of and as functions of optical depth are shown for two model atmospheres corresponding to and log for a solar and a H-deficient composition. In particular, is reduced much below for the model with solar abundances in the H I and He I ionization zones. At the surface dominates the total pressure and hence is very close to . A more complete study for H-rich model atmospheres has been presented by Lobel et al. (1992), who arrive at the same conclusions. In the R CrB model only the He I ionization zone exists, which explains the quite different depth variation of . The higher temperatures at depth in the H-deficient model make He I ionization occur at smaller . Also seen is the minor effect of the C I ionization zone at . These models only extend down to and hence it is not possibly to tell how varies further in, though the He II ionization zone will also reduce .
It seems that atmospheres of late-type supergiants are close to being dynamically unstable, or may even be so for sufficiently high luminosity-to-mass ratios. Violent instabilities due to this might lead to ejection of material, as found for the termination of the AGB (Wagenhuber & Weiss 1994), as well as for the LBVs (Stothers & Chin 1993). Also R CrB stellar models seem to suffer from a similar dynamical instability as the stars on the tip of the AGB (A. Weiss, private communication). It should be remembered though that the above-mentioned evolutionary studies lack a proper hydrodynamical treatment, and therefore interpreting the instabilities found in the models as actually occurring in the stars must still be made with some caution.
© European Southern Observatory (ESO) 1998
Online publication: January 16, 1998