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Astron. Astrophys. 330, 1080-1090 (1998)

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5. Synthetic stellar spectra

The knowledge of the monochromatic absorption coefficient for the nano-diamonds allows construction of self-consistent models of the stellar environments in which the diamonds can have formed. We present here self-consistent photospheres and synthetic spectra of carbon stars with simplified diamond condensation. We assume that the diamonds form in the temperature range from 1600 K to 1500 K in such a way that the amount of 2 h2 which transforms into diamond grains increases linearly from 0% at 1600 K to a value at 1500 K which we have set to 0%, 5%, and 10%, respectively, in various models. Models where we induce considerably larger amounts of dust condensation have convergence problems, possibly due to conflicts with our assumption of hydrostatic equilibrium. For temperatures below 1500 K, the condensed fraction of 2 h2 is assumed to be constant (0%, 5%, and 10%, respectively). C2 H2 has been suggested as the primary species for diamond growth by Sharp & Wasserburg (1993) and Krüger et al. (1996), based on full chemical pathway calculations. The condensation temperatures we have adopted are based on the calculation by Sharp & Wasserburg (1993) and by Lodders & Fegley (1995); both of these groups have performed molecular equilibrium calculations to establish the stability fields of C(s), TiC(s) and SiC(s) and other high temperature phases under conditions of different pressures and C/O ratios.

Our computed model photospheres are based on an improved version (Jorgensen et al. 1992) of the MARCS code (Gustafsson et al. 1975). It assumes hydrostatic equilibrium and local thermodynamic equilibrium (LTE), but includes effects of sphericity and uses an opacity sampling (OS) treatment of molecular opacities from approximately 60 million spectral lines (Jorgensen 1994). Such models have proven to reproduce well the observed spectral features of carbon stars (Lambert et al. 1986; Jorgensen 1989; Jorgensen & Johnson 1991). Full line opacities of CO, C2, CN, CH, 2 h2, C3, HCN, and presolar diamonds were included. The model calculations were performed with the following parameters: C/O = 1.17, 1.35, 2.00; T [FORMULA] = 2600 K, 2700 K, 2800 K; log(g) = 0.0, Z = Z [FORMULA], and M=M [FORMULA].

Fig. 6 shows the temperature versus gas pressure structure of a model atmosphere of [FORMULA] = 2600 K, log(g) = 0, Z = Z [FORMULA], C/O = 1.17, without diamond formation, and with 5%, and with 10%, respectively, of the 2h2 transformed into diamond over a region of the atmosphere from 1600 K to 1500 K. The result of the inclusion of the diamond opacity into the model calculation, is a heating of the upper layers of the photosphere for a given value of the gas pressure. This is a consequence of the larger absorption coefficient of diamond than of 2 h2 in the wavelength region just long-ward of the Planck maximum (at 1 - 2 µm). As a result of this heating the model with diamond included produce a slight increase (compared to the model without diamond) in the spectral flux of the central part of the 3 µm (2 h2) band as well as of the spectral region of the diamond features themselves. The effect is, however, too small to make a clear identification of possible stellar extrasolar diamonds likely on this basis. It should also be mentioned that the effect of reducing the intensity of the 3 µm spectral feature due to the photospheric heating, is smaller than the effect caused by reducing the adopted gravity of the model atmosphere within observationally determined limits (Jorgensen 1989), so this indirect measure of the diamond absorption is not applicable either.

[FIGURE] Fig. 6. The gas temperature versus total gas pressure for three carbon star model atmospheres with [FORMULA] = 2600 K, log(g) = 0, C/O = 1.17, M = M [FORMULA], and Z = Z [FORMULA]. The full drawn line represent a model where no dust is allowed to form. The two other curves represent models where 2 h2 gradually (linearly in temperature) solidifies to diamonds in the temperature range from 1600 K to 1500 K. The maximum degrees of completeness (for T [FORMULA] 1500 K) of the 2h2 consumptions are 5% and 10%, for the dotted and the dashed curves, respectively.

Fig. 7 shows the diamond spectrum, normalised to the true continuum, for the model from Fig. 6 where 10% of the 2 h2 has transformed into diamonds. This spectrum is essentially the monochromatic absorption coefficient modified by the radiation field and the partial pressures through the atmosphere. The same features as in Fig. 3 are therefore seen here too. Some of these may be artifacts caused by the extraction procedure, as discussed above, but all the measured spectral features are shown here for completeness.

[FIGURE] Fig. 7. The absorption spectrum from 2 to 12 µm (5,000 to 800 cm-1) of presolar diamonds in a carbon star atmosphere (model as in Fig. 6, with 10% condensation), calculated based on the measured presolar diamond absorption coefficient shown in Fig. 5. Our assignments (from Table 1) are indicated along with the spectrum.

It is seen that the strongest spectral feature (at [FORMULA] 3 µm) from our measurements of the presolar diamonds gives rise to a reduction of the model spectral flux of 8%. In Fig. 8 is seen that the left over 2 h2 in the same model gives rise to a flux reduction of between 20% and 80% in the 3 µm spectral region. When also the other molecules in this region are taken into account, the outcome of synthetic spectrum computations based on a fixed model structure, is almost identical spectra independent of whether the diamonds are included in the spectra or not. In addition, the 3 µm diamond feature is likely to be an artifact of the chemical extraction procedure, as discussed above, so that the real intrinsic diamond spectral features may be even weaker. Also at 6 and 9 µm, where the total molecular absorption is weaker than at 3 µm, the diamond features are weak compared to the molecular spectral features.

[FIGURE] Fig. 8. Same as Fig. 7 (although limited to the spectral region 2.6 µm - 3.6 µm). In addition to the continuum and the diamond spectrum, also the 2 h2 spectrum, and the full self-consistent synthetic spectrum based on all included opacity sources, are shown.

Generally, grain nucleation (as opposed to grain growth) is the bottle neck in stellar dust formation. Polycyclic Aromatic Hydrocarbons (PAH) are often assumed to be the molecular nucleation seeds for dust growth in carbon stars. In hydrodynamic model atmospheres, however, the time scales for PAH formation are too long compared to the dynamical time scales, and in the corresponding hydrostatic photospheres the gas temperature is too high for PAH formation (Helling et al. 1996). The relatively modest opacity, and higher condensation temperature of the diamonds than that of the PAH molecules, may cause nucleation of diamond grains at relatively high atmospheric densities, where the velocity field is still negligible (hence, the hydrostatic approximation is acceptable here). If nucleation of diamond dust (as oppose to PAH) can act as seeds for grain growth in carbon stars, the long dynamical time scales in the region of diamond condensation may therefore contribute to the solution of the nucleation problem mentioned above.

If diamonds are the nucleation seeds of other dust grains, then we should of course expect that the presolar diamonds were part of agglomerates or part of heterogeneous larger grains when they left the parent stellar atmosphere, rather than being pure diamonds (but our considerations above about the photospheric spectra would still apply, since the small, high-temperature diamond seeds will form as individual grains in the pseudo-hydrostatic, denser photosphere before the agglomeration). Such larger grains may, however, have been destroyed in the interstellar space or in the solar nebula. If they survived all the way to being included in the carbonaceous chondrites, they would unfortunately most likely be dissolved during the present chemical extraction procedure, and the development of a more gentle extraction procedure could therefore add important new information about the origin of the presolar diamonds.

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© European Southern Observatory (ESO) 1998

Online publication: January 27, 1998