## 4. Dynamical instabilitiesDynamical 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 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 |