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Astron. Astrophys. 349, 243-252 (1999) 5. Spectral energy distributions and synthetic colours5.1. Frequency-dependent radiative transferThe dynamical calculation yields the structure of the atmosphere and circumstellar envelope (density, temperature, degree of condensation, etc.) as a function of time. The time-independent radiative transfer equation is solved for each frequency separately along parallel rays to obtain spectral energy distributions (Windsteig et al. 1997and references therein). The grey gas opacity (Planck mean) is taken directly from the dynamical models. The dust opacity is calculated from the optical properties of amorphous carbon and in some cases SiC (see Sect. 3 for the description of the different dust data). 5.2. Spectral energy distributionsTo investigate how the different dust data influence the resulting spectral energy distributions (SEDs), spectra were calculated using the optical constants of amorphous carbon of Rouleau , Zubko , Jäger400 and Jäger1000 on top of a fixed atmospheric structure. Two different kinds of SEDs were calculated; (1) fully consistent ones where the same amorphous carbon data were used in the dynamical model and in the SED calculations and (2) "inconsistent" ones where we used different dust opacity data for the detailed radiative transfer on top of the same dynamical model structure (fixed spatial distribution of density, temperature, degree of condensation). The latter spectra enable us to distinguish between the effect of the various dust data in the radiative transfer calculation and the effect on the model structure. Fig. 3 shows the result for the SEDs based on a minimum phase model of the DROR model sequence. The full line always denotes the SED of the consistent model where the Rouleau data were used for the underlying dynamical model as well as for the calculation of the spectrum. In the upper panel the (inconsistent) Jäger1000 spectrum (dotted) calculated on top of the same Rouleau model is shown in addition to the consistent Rouleau spectrum. The middle panel shows the same for Zubko and the lower panel for Jäger400 . The effects of the different dust data for a given structure compared to a consistent model using the Rouleau data can be summarised as follows:
Note that for the "inconsistent" SEDs the total flux may differ from the value of the consistent models.
5.3. Spectral energy distributions including SiCThe analysis of mid-IR carbon star spectra indicates that SiC is the best candidate to reproduce the observations around the 11 µm region. We have therefore considered SiC as an additional dust component (see Sect. 3.3 for details). The formation of SiC is not included in the self-consistent model calculations because (1) little is known about the condensation process and (2) because we do not expect that SiC will have a significant influence on the model structures. We use either Planck or Rosseland means for the model computations and SiC will contribute only in a very narrow wavelength region with small amounts to these mean opacities compared to amorphous carbon. The effect of SiC as dust component is described in a qualitative
manner. The dust opacity where
5.4. Synthetic coloursFor a comparison of the consistent spectra (model structure and spectra computed with the same dust data) we have calculated synthetic J, H and L colours as well as the IRAS 12 µm colour. In a (J-H) versus (H-L) diagram (Fig. 5a) the models based on different amorphous dust data fall into distinct regions.
The models calculated with the Jäger1000 dust data have the reddest colours in (H-L). The Jäger400 colours are the bluest, while Rouleau and Zubko lie in between. In (J-H) models with the Zubko data are reddest and the others do not differ much. The reason for the different slopes of Rouleau and Zubko compared to both of the Jäger data sets is that in these cases the maxima of the SEDs are changing between the J and the H filter depending on the phase. The maxima of the SEDs resulting from the Jäger1000 model structures are always at longer wavelengths and the ones of the Jäger400 structures lie mainly in one filter. From Fig. 5b, which shows the "inconsistent" colours based on model DROR (structure was calculated with the Rouleau data and the spectra with other dust data) in addition to the consistent Rouleau colours, it is clear that the influence of the different dust data used in the radiative transfer calculation is much stronger than the effect of the underlying hydrodynamic model structure. Note that in Fig. 5 only maximum and minimum phases are shown. Other phases would fill in the gaps between successive extremes. The colours are strongly related to the formation of a new dust shell which takes place every 5 to 6 pulsation cycles (see Sect. 4.3). After this time scale the colours match very closely the ones of the preceding dust formation cycle as shown in Fig. 6. When connecting the succeeding points it can be seen that they form a spiral. The minima (circles) are always redder than the following maxima (asterisks).
To investigate also the mid-IR properties of the models we calculated the 12 µm colour. A (L-[12]) vs. (H-L) diagram (Fig. 5c) shows, that again the consistent colours fall into distinct regions. The sequence in (L-[12]) (Zubko - Rouleau - Jäger1000 - Jäger400 ) is a sequence of decreasing optical depths. Table 3 lists the mean dust optical depths for a few selected wavelengths. In Fig. 5d the inconsistent colours based on the model structure of DROR are shown for comparison. Table 3. Mean dust optical depths at a few selected wavelengths From Fig. 5 we can infer that the influence of the different amorphous carbon dust data used in the radiative transfer calculation is much stronger than the effect of the model structures. The colours resulting from the same dust data fall approximately into the same region of a two-colour-diagram, whether they are calculated on top of the corresponding model structure (upper panel) or a fixed model sequence (lower panel). This applies for all data sets. In addition we computed synthetic colours for the Maron and
the Preibisch data, but they are not shown in Fig. 5 to avoid
overlaps in the plot. The Preibisch colours fall into the same
region as the Zubko colours as one would expect because the
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