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

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4. Dynamical instabilities

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

[EQUATION]

is less than [FORMULA] in a region effectively isolated from the rest of the star. An ideal gas has [FORMULA] while pure radiation has [FORMULA], but the combined effect of non-negligible radiation pressure and ionization may push [FORMULA] below the limiting value. For dynamical instability to occur, the pressure weighted volumetric mean value of [FORMULA] between the depth r and the stellar surface [FORMULA]

[EQUATION]

must be reduced below [FORMULA] 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 [FORMULA] and [FORMULA] as functions of optical depth are shown for two model atmospheres corresponding to [FORMULA] and log [FORMULA] for a solar and a H-deficient composition. In particular, [FORMULA] is reduced much below [FORMULA] for the model with solar abundances in the H I and He I ionization zones. At the surface [FORMULA] dominates the total pressure and hence [FORMULA] is very close to [FORMULA]. 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 [FORMULA]. The higher temperatures at depth in the H-deficient model make He I ionization occur at smaller [FORMULA]. Also seen is the minor effect of the C I ionization zone at [FORMULA]. These models only extend down to [FORMULA] and hence it is not possibly to tell how [FORMULA] varies further in, though the He II ionization zone will also reduce [FORMULA].

[FIGURE] Fig. 3. a The generalized first adiabatic exponent [FORMULA] as a function of optical depth in model atmospheres with solar (solid) and R CrB (dashed) compositions. b The pressure weighted first adiabatic exponent [FORMULA] for the same models as in a. In both panels the parameters are [FORMULA] and log [FORMULA]. Values below the dotted lines at [FORMULA] correspond formally to dynamical instability

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.

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© European Southern Observatory (ESO) 1998

Online publication: January 16, 1998
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