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Astron. Astrophys. 358, 665-670 (2000)

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2. The nature of the instability

We consider radiative/thermal instabilities in a dust forming system as illustrated in Fig. 1. The following assumptions on the important interactions in the system are made:

  • 1. A decrease/increase of the local mean intensity [FORMULA] causes a decrease/increase of the local temperature in the dust forming medium T (positive feedback), as it is always true in radiative equilibrium.

  • 2. A decrease of the temperature T improves the conditions for dust formation and, thereby, leads to an increase of the degree of condensation [FORMULA] (negative feedback).

  • 3. An increase of the degree of condensation [FORMULA] causes an increase of the opacity [FORMULA] (positive feedback).

[FIGURE] Fig. 1. Control loop of interactions in a dust forming system. Positive feedbacks are marked by [FORMULA], negative feedbacks by [FORMULA]. The sign of the feedback marked by the dashed arrow (non-local effect of opacity changes on the mean intensity) determines whether the control loop is self-amplifying, or whether it damps initial perturbations.

As will be discussed in Sect. 3, an increase of the opacity [FORMULA] may either cause a decrease (negative feedback) or an increase (positive feedback) of the mean intensity [FORMULA] via radiative transfer effects, depending on the circumstances. In any case, a control loop is constituted (see Fig. 1) which acts on relatively short time scales, where the slowest process inside the loop likely is the dust formation (see Sect. 5). If the latter influence is negative, the overall feedback in the control loop is positive ([FORMULA]). In such a case the control loop is self-amplifying, i.e. unstable against small perturbations which may arise e.g. from fluctuations. This kind of instability possibly causes a spatial structuring of the dust forming medium. In the opposite case, initial perturbations are damped by the dust forming system. According to the assumptions outlined, the stability of dust forming regions is hence controlled by the non-local effect of opacity variations on the mean intensity, which is investigated below.

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© European Southern Observatory (ESO) 2000

Online publication: June 8, 2000