Astron. Astrophys. 357, 572-580 (2000)

4. Results

A system of coupled, stiff, first-order differential equations describing the chemistry has been solved between the inner radius of the dust shell R_inner and R_inner + 5 , assuming the constant gas temperature of Table 1 and the wind chemical composition of Table 2. We assume that no molecules are initially present at the inner radius of the dust shell. In order to test the chemistry we vary the conditions at R_inner and study the effect on the nucleation of dust precursors. We consider gas number densities at the inner radius ranging from to cm^-3, allow the gas temperature to vary from 1000 to 6000 K and introduce an arbitrary wind opacity in the UV ( 1100 Å) for a gas number density of cm^-3.

The results after integration between R_inner and R_inner + 5 and for various gas number densities at are shown in Figs. 2-4. A constant gas temperature of 4000K and no wind opacity are assumed. It is clear from inspection of these figures that the chemistry is highly dependent on the gas number density. At the low density given by the spherically symmetric wind model (see Table 1), the dominant species in the gas are atomic ions while as the gas number density increases, the recombination of ions takes place and the gas composition is governed by neutral-phase chemistry, that is, the dominant species are neutral atoms and molecules although electrons and some ions are still present in relatively large amounts (for example, C⁺, O⁺ and He⁺).

Fig. 2. Abundances of atoms and their ions as a function of gas number density at the inner dust shell radius. The gas temperature is that of Table 1.

Fig. 3. Molecular abundances as a function of gas number density at the inner dust shell radius (The gas temperature is that of Table 1) - Left: carbon chains - Right: C and O-bearing species.

Fig. 4. Molecular abundances as a function of gas number density at the inner dust shell radius (The gas temperature is that of Table 1) - Left: Silicon species - Right: Sulphur species.

We can identify three density regimes over which the dominant formation and destruction processes vary: the low density range (n_gas= cm^-3), the intermediate density window (n_gas= cm^-3), and finally the high density range (n_gas= cm^-3). The processes at play for these three density windows are summarized in Table 5 and we discuss in turn results on carbon and silicon species below.

Table 5. Dominant formation (top line) and destruction (bottom line) processes in the chemical model. Formation - RA: radiative association; AD: e^- associative detachment; RXXX: reaction No. XXX in Appendix A; CE: charge exchange - Destruction - PP: photo-processes (dissociation, ionisation); R: recombination; He⁺: He⁺ attack; C⁺: C⁺ attack; O: O attack; RXXX: reaction No. XXX in Appendix A

4.1. Carbon-bearing species

As illustrated in Fig. 3, small carbon chains are extremely sensitive to the gas density as all the chain abundances vary by 20 orders of magnitude over the density range considered. This variation is due to the shift in the chemistry as the gas density increases from ionic to neutral, and the changes in formation and destruction processes shown in Table 5.

At low gas densities, the dominant formation processes for C₂, the carbon chains and the dominant C-bearing molecular species (CO, CS) are radiative association (RA) reactions, such as

Destruction of the chains, CO and CS results from photo-processes (dissociation and ionisation) at low densities while He⁺ attack as

starts to play a role as the density increases.

In the intermediate density range, a new type of formation channel appears, that is, associative detachment (AD) reactions such as

where C^- is formed from radiative recombination reactions. However, radiative association reactions still represent the dominant formation mechanism for the chains and the dominant carbon-bearing species. Destruction is now essentially governed by He⁺ attack and C⁺ attack such as

while photo-processes become less important.

For still larger densities, formation occurs mainly via radiative association reactions and to a less extent, associative detachment processes. As for the destruction of molecules, a new destructive channel becomes dominant and involves the attack by atomic oxygen, i.e.,

while He⁺ and C⁺ attacks become less important. This result illustrates the shift in the chemistry as the gas density increases and the fact that ions recombine more efficiently at higher densities. Once the neutral gas phase dominates, the destruction processes for molecules and their ions arise from atomic oxygen attack. Because we have ignored CO self-shielding in the treatment of our photo-processes, more atomic oxygen not locked up in CO is available in the flow, leading to more effective destruction of carbon chains.

Apart from C₂, the final abunbances of small carbon clusters and their ions are never large even at high gas number densities, due to the fact that the formation processes involved are slow radiative association and associative detachment reactions. Carbon monoxide is the dominant molecular species to form at high densities, but unlike late-type, carbon stars, it does not lock up the entire oxygen in this carbon-rich environment. The wind is still rich in free He, C and O atoms and their ions, and the observational evidence for the absence of other types of dust, such as oxides, is puzzling. We shall discuss this point in Sect. 5.

4.2. Silicon and sulphur species

Several silicon and sulphur-bearing species are considered, in particular molecules involved in the process of SiC cluster condensation. The formation and destruction processes for silicon and sulphur species are summarised in Table 5 and are similar to those involved in the formation of carbon species, except for a few specific reactions at play in the formation of CS, SiO or SO. Results on chemical abundances are presented in Fig. 4 and indicate that the dominant species are CS, SiO, SO and SiC and that the trend displayed by carbon-bearing molecules, that is, increasing abundances with increasing gas number densities, is also followed by these molecules. However, the final abundances are quite low at large densities, especially for the species which could act as intermediates in the nucleation of silicon carbide (e.g., SiC₂, SiC). As previously mentioned, SiC has not been observed at 11.3 µm in the wind of WC stars, but this can be understood in the light of the present results. Again, the difficulty in building SiC dust molecular precursors may reside in the slow chemical reactions governing the formation mechanism.

4.3. Temperature and opacity variations

Variation of the gas temperature at the inner dust shell was also considered for the various density regimes to check if the chemistry was very sensitive to temperature. Results are presented in Fig. 5 for the intermediate density range and the dominant chemical species in the wind. Apart from C₂, CS and SO, the atomic and molecular abundances do not show much variation with temperature. This is mainly due to the fact that the destruction and formation processes for these species are similar at low and high gas temperatures and that their chemical rates do not depend strongly on temperature. The situation is different for C₂, CS and SO which show higher abundances at low temperatures. For C₂, the destruction processes remain the same over the temperature range considered, but an extra formation channel (associative detachment) adds up to radiative association reactions at low temperatures and increases the total net formation rate. For CS and SO, the formation processes are very different at high and low temperatures. At high temperatures, RA reactions dominate while neutral-neutral reactions are more important at low T. As the latter channels are usually characterized by faster chemical rates than RA processes, this results in a greater net formation rate at low temperatures. These results hold at lower and higher gas densities than cm^-3 and imply that the chemistry of dust precursors is quite temperature independent.

Fig. 5. Left: Molecular abundances as a function of gas temperature at the inner dust shell radius. The gas number density is cm-3 and . Right: Molecular abundances as a function of gas optical depth at the inner dust shell radius. T K, n(gas) cm-3.

Results induced by considering an arbitrary wind opacity in the UV are also presented in Fig. 5. All molecular species show increasing abundances with increasing wind opacity and this is understood quite simply by the fact that the He⁺ abundance drops of two orders of magnitude as the opacity becomes larger. In the intermediate density region, the main destruction processes are He⁺ and C⁺ attacks (see Table 5) and photo-processes, and a higher wind opacity results in less ionisation of helium and carbon and lower photo-rates. Therefore, molecular destruction is hindered by the opacity of the wind.

© European Southern Observatory (ESO) 2000

Online publication: June 5, 2000
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