3. IRAS two-colour diagram: dynamical models and observations
3.1. Spectral energy distributions
Fig. 1 shows a typical spectral energy distribution (SED) based on pure amorphous carbon grains where the full line represents the extended version of model R10C18 and the dotted line depicts the original dynamical calculation (cf. Table 1 for the model parameters). The comparison clearly illustrates how the infrared excess caused by the dust envelope leads to a significant shift in the IRAS two colour diagram and demonstrates the influence of the size of the circumstellar dust shell on the far-IR emission.
Moreover, the analysis of mid-IR carbon star spectra indicates that -SiC is the best candidate to reproduce the astronomical observations around the m region (Groenewegen 1995, Speck et al. 1996). Therefore it is plausible to consider -SiC as an additional dust component although the formation of SiC is not included in the self-consistent calculations because little is known about the condensation process. While Gauger et al. (1996) find that carbon condenses before SiC, Kozasa et al. (1996) arrive at the opposite conclusion. The effect of -SiC as dust component is described here in a qualitative manner, i.e. the dust opacity is calculated from
where are the fractional parts of carbon and SiC, respectively, and where are the opacities of carbon and SiC in the case of pure grains. To illustrate the influence of -SiC on the SEDs we have adopted a ratio of (Lorenz-Martins & Lefèvre 1994, Groenewegen 1995, Ivezi & Elitzur 1995). This mixture of dust grains consisting of amorphous carbon and -SiC modifies the SED around m as shown in Fig. 2.
3.2. Synthetic IRAS colours
We obtain the IRAS colours corresponding to the calculated SEDs by convolving with the IRAS filter transmission curves and by transforming to monochromatic fluxes at , and (cf. Joint IRAS Science Work Group, 1986). Fig. 3 depicts self-consistent models at various phases together with black body fluxes (dotted line) and the observations of carbon stars (Guglielmo et al. 1993 : small triangles, Kerschbaum & Hron 1994 : small diamonds) in the IRAS two-colour diagram with boundaries of certain regions according to van der Veen & Habing (1988). The physical properties of all our theoretical models used in this study are summarized in Table 1. The displacements within a group of symbols are caused by the variations during a pulsation cycle and are associated with the time-dependent dust formation and growth processes. Note that based on the measured laboratory data of amorphous carbon (Maron 1990), these IRAS colours can be directly calculated from the extended dynamical models without further assumptions like optical depths, density and temperature structures of the circumstellar envelope, particle size distributions and condensation radii.
The locations of these models in the IRAS two-colour diagram basically reveal a mass loss sequence and are almost independent of the details how the increasing mass loss is generated, i.e. by a different pulsation strength, a lower stellar mass, a higher luminosity or by an increased carbon-to-oxygen ratio. However, the actual position of a model in the two-colour diagram is phase-dependent and can vary by about mag in colour as well as about mag in the colour. Although we have considered typically ten instants of time for each model covering all pulsational phases the number of symbols which can be distinguished in Fig. 3 depend on specific properties of the model. Some models show similar colours and phases. The maximum light correlates with smallest and colours, i.e. both colours increase monotonically with increasing phase. These variations occur within a pulsation and/or dust formation period and are not due to a long term evolution on the AGB like dredge-up phases and He-shell flashes. Therefore, inspecting the mean mass loss rates of the corresponding dynamical models we can relate an observed IRAS source to an estimated mass loss rate but such an estimate is only accurate within a factor of three. Again, we want to emphasize the important influence of the circumstellar envelope on the observed IRAS colours since the mean black body colours of all our initial models constructed with (cf. Table 1) are given for the colour by mag and for the colour by mag.
3.3. Influence of SiC and detached shells
In Fig. 4 we compare the positions of our original models containing amorphous carbon grains (filled symbols) to models that have been modified by the inclusion of additional -SiC grains (open symbols) and/or of detached shells (which manifest themselves in higher fluxes as discussed below). The symbols plotted represent mean values (averaged over different phases), except for the stationary model VS.
First, the SiC-feature around results in a shift in both colours which is typically given by mag in and mag in . Due to the continuous absorption by SiC this displacement occurs also for models with large mass loss rates () where the SiC-feature is not visible in the SED anymore. Secondly, the colour shift caused by the inclusion of the SiC dust grains is almost parallel to the model sequence and mimics smaller mass loss rates of models with no SiC dust. An accurate determination of the mass loss rate of a particular object requires therefore the knowledge of a large portion of the SED for several pulsational phases to disentangle the time-dependence of the stellar outflow from the details of the dust forming processes. With IRAS data alone, the effect due to SiC cannot be easily separated from the effects of variations during a pulsational cycle.
We have also simulated the existence of detached shells as found by observations (Olofsson et al. 1990). Without knowing the history of the mass loss and without trying to make 'realistic' models we have simply increased the gas and dust densities at a certain radius by a factor of 10 and 100, respectively and have re-done the SED calculations. This radius corresponds to a temperature of typically K which requires a typical flow time of more than years or a few hundred pulsational cycles. The resulting IRAS colours (marked by a star for model VS and by a triangle for model R7C18) are shifted into the regions VIa and VII above the main locations of all our models. The redder colours correspond to the smaller increase of the mass loss rate, i.e. a factor of 10 in the dust and gas density. Such a detached shell having a temperature of K yields the largest effects on the m flux. Note that both original models are found within region I near in the IRAS colours.
We regard these simulations of detached shells as numerical experiments to see where an increased mass loss at earlier epochs would shift our dynamical models. We can conclude from varying the parameters of the detached shells that only a rather narrow temperature range around K as well as a mass loss rate increased at least by a factor of 10 is necessary to identify such sources as detached shells in the IRAS database.
3.4. Comparison with observations and previous works
The overall agreement between our synthetic colours and those of observed carbon stars (e.g. Guglielmo et al. 1993 ) is quite good except that our colours seem to be too blue by about mag. This could be due to specific dust properties like the chemical composition or the opacity or due to the small particle limit adopted for this study. Kerschbaum et al. (1996 ) have made a systematic comparison of the observational properties of carbon-rich Mira, semiregular (SRa and SRb) and irregular (Lb) variables finding that C-Lbs and C-SRabs populate the same areas in the IRAS two colour diagram whereas Miras are separated from them. While the first two groups are mainly found at higher values in the upper part of region VII and in region VIa, indicating significant amounts of cold dust, the later are located mainly in the lower part of region VII. All variable groups include a small fraction of optically thin objects in the "photospheric" region I extending close to the Rayleigh-Jeans point. Finally, for objects having period information (SRab, Mira) Kerschbaum et al. (1996 ) have found a positive correlation with the colour for their samples. When compared with these results, the models without detached shells resemble Mira properties best while the detached shell models are located in regions where mostly semiregular and irregular variables are found. The large fraction of SR and Lb variables among carbon stars with possibly detached shells has already been noted by Willems & de Jong (1988).
Our synthetic colours are similar to the tracks presented by Ivezi & Elitzur (1995) where a parameterized circumstellar envelope is used to model the SEDs of AGB stars. An observable difference is the aforementioned small colour shift. Comparing the mass loss rates, there is an agreement that redder colours are caused by higher mass loss rates, a result which has also been obtained in earlier investigations (e.g. Chan & Kwok 1990 ). However, at a given colour our mass loss rates are significantly larger (factors 2 to 5). The reason for this difference is most likely the effect of pulsation on the density structure of the atmosphere and the complexity of the dust formation process since both aspects are not covered by earlier models. Although it is evident that such kind of parameterized models can reproduce the observed IRAS colours it is very difficult to link the model parameters to actual mass loss rates and other stellar properties.
Using stationary models, Winters et al. (1994a) have presented both synthetic IRAS and near-IR colours. While near IR colours are very useful tools for studying AGB stars (e.g. Epchtein et al. 1990 , Kerschbaum & Hron 1994 ) and well suited to highlight the differences between stationary and dynamical models all modelling requires a proper inclusion of the photospheric radiation. A black body at the inner boundary is apparently not sufficient as can be seen from the systematic differences between observed and synthetic colours found by Winters et al. (1994a) and the difference between blackbody temperatures and effective temperatures (Kerschbaum & Hron 1996 ). We have therefore concentrated on the IRAS colours while synthetic near IR colours will be the subject of a future paper.
In Fig. 5 we have plotted the mass loss rate against the flux at m for the dynamical models of Table 1. The horizontal bars give the luminosity and m flux variations during a pulsational cycle. In order to compare our theoretical mass loss with empirical formulae we adopt Jura's estimate (Jura 1987) in the version of van der Veen & Olofsson (1989)
where µ is the gas-to-dust ratio, the outflow velocity in units of , D the source distance in kpc, the luminosity in , the IRAS flux at m in Jy, the mean wavelength at which the circumstellar envelope emits in units of m and the dust absorption coefficient at m in units of . Since all properties needed to evaluate this formula result from our models we have plotted the 'Jura' mass loss rates as crosses (joined by a dotted line) for the models using our calculated values for and . We have assumed a source distance of . The mean emission occurs in our models around , the mean dust absorption coefficient is at and the gas-to-dust ratios vary between 277 and 1000 as given in Table 1.
While the general correlation between the mass loss rate and the m flux contribution is similar for our models and for Jura's estimate, there are significant systematic differences. These differences can however easily be explained by the fact that for his assumed values for µ and of 220 and 150, respectively he has found good agreement with observed gas mass loss rates. Moreover, his formula is based on several simplifying assumptions like a stationary outflow and small optical depths. This comparison again demonstrates that far-IR observations combined with simple models for the circumstellar envelope can yield realistic mass loss rates but that the properties of the circumstellar envelope may indeed be quite different from the model assumptions.
© European Southern Observatory (ESO) 1997
Online publication: May 26, 1998