## 2. The analysis of extinction and the pair method## 2.1. The classification schemeThe classification scheme of intrinsic (i.e. without selective dust
extinction) SEDs is described in Sect. 3 of Paper I. It was
derived from diagrams (colour-colour and galactic latitude Finally, the mean intrinsic SEDs were thus established for the CV1 to CV6-groups (Sect. 3.3 in Paper I). Sixteen mean unreddened indices: and dispersions were calculated, and being the unreddened magnitudes at (the reference wavelength) and at any used wavelengths. The former magnitudes were adapted from Baumert's (1972) data in a spectral band free of strong molecular features. The most remarkable feature is that the red-near infrared parts of the six SEDs nearly coincide while large differences are observed at shorter and longer wavelengths (see Fig. 3 in Paper I). The reader is referred to Sect. 2 of Paper II for the specific way in which the method was applied to the hot carbon stars (HC0 to HC5-groups). ## 2.2. The pair methodThe method fully described in Sect. 4 of Paper I is then applied to the whole sample, including presumably unreddened stars as well. It makes use of the differences between the observed magnitudes and, for a given group tentatively considered, the mean unreddened indices . If the latter are properly selected (i.e. if the appropriate CV-group is considered), a linear relation is thus expected between and the adopted extinction law , the extinction at = 1.25 being the slope and = - the intercept. Both quantities are simultaneously derived from the least square method. If the selected group and/or the adopted extinction law are wrong, the relation is no longer a linear one. The method is illustrated on the whole spectral range in Fig. 1 for C4121 = S Sct taken from Paper I: strong curvatures were observed when the indices of the group CV3 or CV5 were used instead of CV4 the right group for S Sct. Eqs. (2) and (3) should reduce to when there is no reddening: the observed points are then expected to scatter around an horizontal line (see Fig. 1 in Paper II). A computer code was thus written which studies the relation between and and the associated statistics as described in Sect. 2.3. At this stage, a few stars migrated from our "unreddened" list to the "slightly reddened" one and conversely.
## 2.3. The used statisticsAssuming a given group and extinction law were selected, the ordinate predicted is where and are the slope and intercept respectively, as deduced from the least squares method. The standard deviation on the slope is calculated from where , the reference to wavelength being omitted for clarity. The usual colour excess is for the diffuse law The quality of the linear fit may also be estimated by making use of the standard correlation coefficient. The differences were systematically analyzed for possible residual dependence on wavelength. Strong discrepancy of a given measurement may lead to reject it and start the analysis again. Consistent discrepancies on a large wavelength interval need
explanation. Most stars in this paper, especially carbon Miras, show
strong IR excesses starting usually from the H or K-band, less
frequently from the J or L-band. Those excesses are attributed to
thermal emission from ## 2.4. The k-factorOnce dereddening has been operated through Eq. (3), we should have ideally at every wavelength but some scatter is of course observed. Thus we calculate the mean value whose standard deviation is given by It may be used as a signal to noise ratio for the whole method. The k-coefficient of Eq. (7) is the ratio of the dereddened star fluxes to those of a reference star of the same group (CV1 to 6) which would have the magnitude zero at . Of course, the same ratio can directly be computed for unreddened stars. It will be confirmed that this is a squared angular diameter on a relative scale (Bergeat & Knapik, 1999). We emphasize here that the quantities already showed the correlation with true parallaxes as deduced from ESA, that are expected for stars populating a given range in linear diameters (see Fig. 3 and Sect. 6 of Knapik et al. 1998). © European Southern Observatory (ESO) 1999 Online publication: March 10, 1999 |