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Astron. Astrophys. 344, 263-276 (1999)

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2. The analysis of extinction and the pair method

2.1. The classification scheme

The 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 vs. colour index). Provisional lists of reddened and presumably unreddened stars were thus made, which were slightly revised later on in the process. Then, intrinsic loci for unreddened stars were split into consistent boxes or groups in the colour-colour diagrams. The number of boxes was chosen so as to provide regular and significant intervals by taking into account the expected accuracies of observations and standard deviations on the mean values of colour indices. The classification scheme was achieved through a trial and error approach. No marked gap was observed which could have helped.

Finally, the mean intrinsic SEDs were thus established for the CV1 to CV6-groups (Sect. 3.3 in Paper I). Sixteen mean unreddened indices:

[EQUATION]

and dispersions were calculated, [FORMULA] and [FORMULA] being the unreddened magnitudes at [FORMULA] (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 method

The 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

[EQUATION]

between the observed magnitudes

[EQUATION]

and, for a given group tentatively considered, the mean unreddened indices [FORMULA]. If the latter are properly selected (i.e. if the appropriate CV-group is considered), a linear relation is thus expected between [FORMULA] and the adopted extinction law [FORMULA], the extinction [FORMULA] at [FORMULA] = 1.25 [FORMULA] being the slope and [FORMULA] = [FORMULA] - [FORMULA] 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 [FORMULA] 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 [FORMULA] and [FORMULA] 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.

[FIGURE] Fig. 1. The plot of the quantity [FORMULA] vs. [FORMULA] illustrating Eqs. (2), (3) and (4) in the case of C 4121 (S Sct). The abscissae are extinctions normalized at [FORMULA]. The ordinates are the colour excesses plus unreddened magnitude at [FORMULA] once a linear fit is obtained for a given box (here CV4). The intercept is thus [FORMULA] since the colour excess vanishes with extinction. The straight line was derived from the least square method (see text for details). Its slope is an evaluation of the extinction at [FORMULA], namely [FORMULA] (internal error).

2.3. The used statistics

Assuming a given group and extinction law were selected, the ordinate predicted is

[EQUATION]

where [FORMULA] and [FORMULA] are the slope and intercept respectively, as deduced from the least squares method. The standard deviation on the slope [FORMULA] is calculated from

[EQUATION]

where [FORMULA], the reference to wavelength being omitted for clarity. The usual colour excess is for the diffuse law

[EQUATION]

The quality of the linear fit may also be estimated by making use of the standard correlation coefficient. The differences [FORMULA] 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 CS grains. The data points at shorter wavelengths can be fitted by a linear relation with good accuracy and increasing differences are then noticed in the IR (see e.g. Figs. 2, 5, 6 and 7). The extinction law of the diffuse interstellar medium taken from Mathis (1990) was successfully used as can be seen in the following sections and diagrams. The consequences and the interpretations we place on those diagrams are discussed hereafter, separately for each category of studied objects.

2.4. The k-factor

Once dereddening has been operated through Eq. (3), we should have ideally

[EQUATION]

at every wavelength but some scatter is of course observed. Thus we calculate the mean value [FORMULA] whose standard deviation is given by

[EQUATION]

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 [FORMULA]. 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 [FORMULA] 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).

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© European Southern Observatory (ESO) 1999

Online publication: March 10, 1999
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