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Astron. Astrophys. 317, 121-124 (1997)

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2. The criterion for the occurrence of dynamical instability in the convective envelope

The dynamical instability occurs in a region of the convective envelope, if there the sum of the outwards accelerations due to radiation and turbulent pressure exceeds the inward gravitational acceleration. This can be expressed as

[EQUATION]

where the inward gravitational acceleration is

[EQUATION]

the outward acceleration by radiation is

[EQUATION]

[FIGURE]Fig. 1. The changes in the density [FORMULA] and the value of [FORMULA] in the envelope of an AGB star of 7 [FORMULA]. The solid and the dotted curves indicate the cases with and without the consideration of turbulent pressure respectively.

and the acceleration by turbulent pressure is

[EQUATION]

[FORMULA] and [FORMULA] are the radiation pressure and the turbulent pressure, respectively. The motion of the gas element in the dynamical instable region conforms to the equation of

[EQUATION]

where [FORMULA] is the gas pressure, and v is the velocity of the gas element.

In the stellar model calculations one assumes, however, that all the regions in the star are in the hydrostatic equilibrium and conform to the equation of

[EQUATION]

This means that in the dynamical instable region, in which both Eqs. (5) and (6) and the condition (1) exist, the time depending term in Eq. (5) is neglected artifitially. Therefore, from condition (1) and Eq. (6) one obtains

[EQUATION]

[FIGURE]Fig. 2. The changes in the density [FORMULA] and the value of [FORMULA] in the envelope of a red giant with 7 [FORMULA]. The solid and the dotted curves have the same meaning as in Fig. 1.

Owing to the expression of the gas pressure [FORMULA] and the relation [FORMULA] in the region of radiation eqilibrium, one obtains from Eq. (7)

[EQUATION]

From Eq. (8) we know that the density gradient must reverse in the dynamical instable region.

According to the mentioned-above discussion a criterion for the occurrence of dynamical instability in the convective envelope is obtained as follows: The dynamical instability occurs when the function [FORMULA], and the density gragient reverses [FORMULA].

[FIGURE]Fig. 3. The changes in the density [FORMULA] and the value of [FORMULA] in the envelope of an AGB star of 2.8 [FORMULA]. The solid and the dotted curves have the same meaning as in Fig. 1.

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