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

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5. Discussion

The main intention of this paper is to identify an instability which might initiate structure formation in dust forming media. We have considered a control loop of physical interactions which can be self-amplifying under certain circumstances. One remaining question in this context is, whether the chosen interactions depicted in Fig. 1 do in fact form a tightly coupled "closed" loop, where the internal perturbations have sufficient time to grow disregarding outer influences, or whether the chosen subset of feedbacks strongly interferes with other physical interactions not taken into account like the expansion of the system in the wind flow. In order to discuss the degree of coupling among the various interactions, we estimate some characteristic time scales in the following.

Radiative transfer time scale. The free travel time for photons crossing the dust shell is [FORMULA]hours for [FORMULA]. The photon diffusion time scale, which accounts for multiple absorption/reemission and scattering events, should not exceed this time by a large factor since the optical depths in the dust shell are of the order of unity. Hence, local changes of the opacity almost immediately affect the mean intensity structure in the entire part of the considered circumstellar envelope.

Radiative cooling time scale. This time scale determines how fast changes of J transform into changes of T. Assuming a thermally coupled, dusty gas in LTE, the relaxation time scale towards radiative equilibrium [FORMULA] is as short as [FORMULA], where [FORMULA] is the total (dust + gas) opacity per mass at 3 [FORMULA] and [FORMULA] is the heat capacity of the predominantly neutral, H2-rich gas. This short radiative cooling time scale enables us to assume radiative equilibrium in Sect. 3, even under the influence of the perturbations. However, thermal coupling between dust and gas is questionable. Realistic cooling time scales for the decoupled dust component may be as short as 0.01 sec (Woitke 2000). In contrast, the cooling time scale for the gas component may be as long as days, weeks or even months, mainly driven by molecular line cooling under non-LTE conditions (Woitke et al. 1996).

Chemical time scale. The formation of molecules accelerates the radiative cooling of the gas via the formation of new important coolants (e.g. CO, CS and HCN in the carbon-rich case) and provides the basic progenitor molecules for dust formation (e.g. C2H2). The characteristic time scale for the formation of these molecules is rather uncertain and may vary strongly among the molecules. Patzer et al. (1999) have shown that neutral-neutral reactions in the circumstellar envelopes of pulsating red giants already fail to maintain chemical equilibrium relatively close to the star ([FORMULA]K, [FORMULA]). At lower temperatures/densities, the chemical time scale exceeds the hydrodynamical expansion time scale and the chemistry is found to be frozen in. Neufeld & Hollenbach (1994) have investigated the time-dependent chemistry of a 50 km/s accretion shock wave propagating into a predominantly neutral [FORMULA] gas. The shock wave initially destroys virtually all molecules and compresses the gas up to a few [FORMULA]. Molecules like H2, CO, H2O and OH were found to be re-formed after about 5 days. It has been argued that the chemical time scale must be significantly smaller than the dust formation time scale, because the dust nucleation is a chemical process which requires many reaction steps (Gail & Sedlmayr 1988).

Dust formation time scale. The formation of dust usually introduces the longest internal time scale to astrophysical gases. According to the dust formation theory developed by Gail et al. (1984) and Gail & Sedlmayr (1988), typical dust formation time scales in stellar winds are found to be of the order of a few months in the carbon-rich case (Woitke 2000), depending on the gas density. This time scale usually exceeds the cooling and chemical time scales discussed above.

Hydrodynamical time scale. The expansion time scale [FORMULA] is found to be large, typically [FORMULA]years at [FORMULA].

Summarizing these estimates, the hydrodynamical processes, i.e. the gravitational forces and the acceleration of the dust/gas mixture by radiation pressure can be identified as the slowest processes, at least one order of magnitude slower than the radiative transfer, thermal, chemical and dust formation processes considered in this paper and sketched in Fig. 1. Consequently, the expanding wind seems to provide a slowly changing frame, wherein other (faster) processes have sufficient time to interact with each other and to amplify internal perturbations. Such an approximately "closed" control loop of internal feedbacks has been identified in Sect. 2 and discussed in the Sects. 3 and 4. Among the physical processes constituting this control loop, the dust formation is likely to be the slowest process and, hence, is expected to determine the growth time scale of the perturbations.

But of course, possible interrelations with hydrodynamical processes cannot be excluded. Interesting questions arise especially when asking for possible hydrodynamical consequences of a once existing cloudy dust structure. Some of these questions, which clearly go beyond the scope of this paper, are summarized below.

  • Does the radiative/thermal instability investigated in this paper interfere with other known dynamical instabilities (see e.g. Inogamov 1999)? What is, for example, the role of propagating shock waves caused by a pulsation of the central star which might induce Rayleigh-Taylor instabilities?

  • If the temperature in the shadowed regions behind the opaque dust clouds is in fact lower than in the in-between situated, illuminated regions (see Fig. 3), does the outer pressure compress the gas in these shadowed regions, causing a density contrast?

  • Are the opaque structures confined by radiation pressure? If the radiative flux preferentially escapes through the optically thin parts, the radial component of the flux is smaller in the shadowed regions as compared to the inner edges of the clouds. In this case, the dust cloud is expected to be pushed from the star side and the gas density inside the cloud might increase which could facilitate further dust formation.

  • Is the hydrodynamical process of dust cloud acceleration in an expanding wind flow dynamically stable?

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© European Southern Observatory (ESO) 2000

Online publication: June 8, 2000