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Astron. Astrophys. 349, 243-252 (1999)

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5. Spectral energy distributions and synthetic colours

5.1. Frequency-dependent radiative transfer

The 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 distributions

To 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:

  • Jäger1000 : the spectrum has a lower flux in the short wavelength region (0.5 to 5 µm) and the maximum at longer wavelengths. The lower flux level in this region is due to the fact that [FORMULA] for the Jäger1000 data is higher than for the Rouleau data, therefore we have a higher total dust opacity which results in less flux coming out. The shift of the maximum is also due to the higher dust opacity in the Jäger1000 case.

  • Zubko : the spectrum has a lower flux level in the short wavelength region and the maximum is shifted to longer wavelengths, but not as far as the Jäger1000 spectrum. This is due to the fact that Jäger1000 is a more "graphite-like" amorphous carbon dust than the Zubko material.

  • Jäger400 : the spectrum has a comparable flux all over the spectrum and the maximum at slightly shorter wavelengths. The slightly higher flux level around the maximum results from the lower [FORMULA] of the Jäger400 data which is due to its more "diamond-like" nature compared to the Rouleau data. In the wavelength region where the maxima of the spectra lie, the two data sets are very similar, therefore the maxima of the SEDs do not differ much in wavelength.

Note that for the "inconsistent" SEDs the total flux may differ from the value of the consistent models.

[FIGURE] Fig. 3. Spectral energy distributions of a minimum phase model of DROR where the spectra have been calculated with different dust data: we compare Rouleau (consistent calculation) with Jäger1000 (upper), with Zubko (middle) and Jäger400 (lower). For details see text.

5.3. Spectral energy distributions including SiC

The 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 [FORMULA] for each wavelength is calculated from

[EQUATION]

where [FORMULA] are the fractional parts of amorphous carbon and SiC, respectively, and where [FORMULA] and [FORMULA] are the opacities of carbon and SiC. Fig. 4 shows how a mixture of dust grains consisting of amorphous carbon (Jäger1000 data) and SiC modifies the SED around 11.3 µm. Two different ratios of [FORMULA]: [FORMULA], 4:1 and 9:1 were adopted. The higher the amount of SiC, the stronger is the 11.3 µm feature (see inset of Fig. 4). Another choice of grain shape than spherical for the SiC particles, would result in a broader feature.

[FIGURE] Fig. 4. Spectral energy distributions of a minimum phase model of DJ1R where the spectra have been calculated with amorphous carbon alone (dotted), with a ratio [FORMULA] of 9:1 (dashed) and a ratio of [FORMULA] of 4:1 (solid).

5.4. Synthetic colours

For 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.

[FIGURE] Fig. 5. Synthetic colours: Upper panels: left: (J-H) vs. (H-L) for the maxima and minima of consistent models; crosses denote the Rouleau data (DROR), asterisks the Jäger1000 data (DJ1R), diamonds the Jäger400 data (DJ4R) and triangles the Zubko data (DZUR); right: (H-L) vs. (L-[12]) for the same models (same symbols). Lower panels: "inconsistent" colours, all based on model DROR, in comparison to the consistent Rouleau colours (crosses): circles represent spectra calculated with the Jäger400 data, squares denote the Jäger1000 data and x the Zubko dust data. This plot shows that the influence of the different dust data used in the radiative transfer calculation is much stronger than the effect of the model structures.

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).

[FIGURE] Fig. 6. Upper panel: (J-H) vs. (H-L) for maxima (asterisks) and minima (circles) of the DJ1R model. When connecting the points following each other one can see that the result is a spiral, the minima are always redder than the following maxima. The colours are strongly related to the formation of a new dust shell which takes place every 5 to 6 pulsation cycles. After this time scale the colours match very closely the ones during the formation of the prior dust shell. Lower panel: Same as above, only (H-L) vs. (L-[12]) is shown.

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]

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 [FORMULA] are very similar. The same applies for the Maron and the Rouleau data. Only the J flux differs between Maron and Rouleau , the reason being that the Maron data do not extend below 1 µm and therefore the corresponding contribution to the J filter is lacking.

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© European Southern Observatory (ESO) 1999

Online publication: August 25, 1999
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