Astron. Astrophys. 357, 515-519 (2000) 2. Evaluation of the distancesIn absence of interstellar extinction, the flux density of the radiation at a wavelength emitted by a circumstellar envelope of outer radius , is related to the flux density observed at a distance d by the well known equation: which can be used to evaluate the distance of the object. The basic idea of our approach is the determination of the distance of a carbon star by using the best-fit values of the emitted flux at 10.6 µm and of the outer radius of the circumstellar envelope. We have chosen to work at the wavelength = 10.6 µm, since this value lies in the wavelength range of the observed IRAS Low Resolution Spectra (LRS, IRAS Science Team 1986) and because at this wavelength the interstellar extinction is completely negligible. The best-fit parameters of the model discussed in Paper I are the optical thickness of the envelope at 0.55 µm, the star temperature T_{*}, the two ratios and (where is the inner radius of the envelope and R_{*} the stellar radius), and the percentages of the three dust components of the envelope (amorphous carbon, -SiC and -SiC). This means that, in order to derive R we have to know R_{*}. In Paper I the stellar radius has been derived, starting from the best-fit values of T_{*}, and using the semi-empirical formulas (Bergeat et al. 1978): However, when these values of R_{*}, together with the best-fit values of T_{*}, are used to obtain the stellar luminosity: ( is the Stephan-Boltzmann constant) we derive for many stars values of L_{*} which sensibly differ from the expected ones. To overcome this difficulty we have followed here a different approach. From the sample of carbon stars examined in Paper I it is possible to extract a subset of 10 Mira-like stars with well known period of variability. These stars constitute a very important group, since Feast et al. (1989), analysing a sample of 49 Mira variables in the Large Magellanic Cloud (LMC), found two distinct relations between the period and the bolometric luminosity of the star (P-L relation) which are valid for the 20 carbon-rich Miras and the 29 oxygen-rich Miras, separately. Since, as reported by Cohen & Hitchon (1996), oxygen-rich Miras in the LMC, in the solar neighborhood, and in globular clusters all fit the same P-L relation derived by Feast et al. (1989), it is reasonable to assume that the P-L relation obtained by Feast et al. (1989) for the 20 carbon-rich Miras in the LMC holds also for the galactic carbon-rich Miras. Even if the 10 galactic Mira variables of our sample have, on average, periods longer than those of the LMC C-Miras used to establish the P-L calibration, we do not expect large discrepancies from the values foreseen by the P-L relation. In fact the maximum period of our galactic Miras is only 30% greater than the maximum period of the LMC C-Miras and the two ranges overlap. We decided therefore, following Whitelock et al. (1994), to adopt the relation found by Feast et al. (1989) also for our sample of galactic Mira variables, deriving the average luminosity of these stars starting from their period. Except for one case, we have found that the luminosities evaluated in this way are in agreement within 17% with the mean value (L_{*} = 7050 ) observed for the LMC carbon stars (Frogel et al. 1980). On the contrary they disagree by a factor up to 7 with the luminosity obtained for the same stars if the starting assumption is the validity of Eqs. (2-3) and (4). This is not surprising, however, since the semi-empirical formulas (2) and (3) have been obtained for bright carbon stars (Bergeat et al. 1978) and, in principle, they could not be valid for evolved carbon stars which are usually quite faint in the visible. In the light of the previous discussion, for the 10 Miras of our sample we have adopted in this work the average luminosity derived from the P-L relation, while for all the remaining stars of the sample we have used the mean value L_{*} = 7050 . The use of the mean value L_{*}=7050 (strictly valid for the LMC) could not appear as an appropriate choice, since carbon stars in the Galaxy and in the LMC could be two different populations. In fact, observations show that the luminosity functions for carbon stars in the Large and Small Magellanic Clouds are different (see e.g. Groenewegen & de Jong 1993) and this, according to some theoretical models (Marigo et al. 1999), should depend on the different metallicity in the two galaxies. Therefore the mean luminosity of C-Miras in the LMC could be, in principle, different from that of the same kind of stars in our Galaxy. However, the approach of using a unique luminosity for a large sample of stars is the best we can do at the moment and it can be considered a reasonable approximation (Groenewegen et al. 1992), since we are taking into account an homogeneous sample of stars of the same type and belonging to the same spectral class. Moreover we note that our choice is supported by the satisfactory agreement between the luminosities of the 10 galactic C-Miras and the mean luminosity observed in the LMC (see above). In any case we think that the above choice should not affect dramatically our results; in fact, an indetermination of a factor of 2 on the star luminosity involves an indetermination of about 40% on the stellar distance. Starting from L_{*} and the best-fit value of T_{*}, it is possible to evaluate, by means of Eq. (4), the star radius and, from this, the outer radius of the envelope (using the best-fit values of the ratios R_{I}/R_{*} and R_{E}/R_{I}) and eventually from Eq. (1) the distance of the star. In Table 1 we have listed the values of the distances found in this work for all the 55 sources of our sample. In the same table we report, for comparison, also the distances of the same objects evaluated by other authors with different methods. Table 1. Star distances in parsec. The error in the determination of the distance is linked, in different extent, to the accuracy in the evaluation of the quantities T_{*}, R_{I}/R_{*}, and R_{E}/R_{I} (by means of the best-fit) and of L_{*} (by means of the above discussed assumptions). While the synthetic spectra are quite sensitive to variations in T_{*} and R_{I}/R_{*}, this is not the case for R_{E}/R_{I}. In fact, while variations of about in T_{*} and in R_{I}/R_{*} induce an appreciable variation in the calculated spectrum, the ratio R_{E}/R_{I} can be changed even by a factor 3, without producing any appreciable spectral change. Fortunately we have found that a large change of R_{E}/R_{I} does not produce a large variation of the distance. In fact a change by a factor of 10 in R_{E}/R_{I} produces a variation of only 0.5% in the stellar distance, if the other input parameters of the model are fixed. It is worthwhile to summarize here the main features of our method which allows, in our opinion, to obtain reliable values of the distances:
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