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3. Radial velocities

Relative radial velocities of both the programme and the standard stars were calculated from cross-correlating (using the cross-correlation programme available in the FIGARO package) each programme star with a set of appropriate model spectra. The stellar atmosphere models were produced using the MARCS code (Gustafsson et al. 1975), and the synthetic spectra generated using the SSG code (Bell & Gustafsson 1978).

The models had a range in tex2html_wrap_inline1065 between 3700 K and 4500 K, and log g between 0 and 0.7, following roughly the corresponding ranges in the model atmospheres for carbon stars of Querci et al. (1974). We also assumed metal abundance of tex2html_wrap_inline1067, carbon abundances [C/A] in the range between 0.0 and 2.0, and tex2html_wrap_inline1069 in the range between 3 and 30. The radial velocity corresponding to the highest cross-correlation peak (from a series of peaks obtained from cross-correlating the programme star with the set of model spectra) was adopted in each case. The corresponding model can be considered as being closest to the particular stellar spectrum.

Zero-point of the velocity system: The absolute zero-point of the velocity system was determined from the velocities of the standard stars. During the first observing run, the zero-point was determined with an accuracy of tex2html_wrap_inline1071 km/s (which becomes tex2html_wrap_inline1073 km/s, if we exclude one discrepant point). In the second observing run, a larger number of observations of standards was secured, and the zero-point was determined with an accuracy of tex2html_wrap_inline1075 km/s. This very good agreement between the standard (carbon) stars confirms in an indirect way the suitability of the models used. However, two stars (HD 75021 and HD 223392) showed a systematically different velocities from the bibliographic values (of tex2html_wrap_inline1077 km/s in both cases). They are suspected variables, and they were excluded from the zero-point determination. Finally, it should be noted that the velocities of the five programme stars that were observed in both observing runs (Table 1) showed a mean difference of tex2html_wrap_inline1079 km/s, which is well within the combined zero-point error from the two observing runs.

The final heliocentric radial velocities of the programme stars are given in Table 1. All of the programme stars given in Tables 1a and b yielded cross-correlation peaks (ccp) higher than 30%. Whenever two consecutive observations were available, the two spectra were co-added before deriving the final velocity appearing in Table 1.

Random errors in the velocity measurements:
1. From repeated observations: To estimate the random errors in the velocities, we compared the velocities derived from the individual observations of the same star for the cases for which all repeated spectra yielded ccp higher than 30%. For the first observing run, there were 30 such cases, yielding a rms of the velocity difference of 4.7 km/s for the whole sample, or 3.4 km/s, if we remove two discrepant stars from the sample (with velocity differences between the two measurements of 13.0, and 14.5 km/s respectively). For the second observing run, there were 25 such cases yielding a rms of the velocity difference of 1.7 km/s. It should be noted that for the majority of the repeats in the second observing run both measurements gave ccp higher than 60%, while in the first observing run, most ccp were less than 60%, hence the higher absolute velocity difference in the latter case. The final velocities of Table 1 come from the co-addition of the repeated observations, therefore they should be more accurate than the individual measurements. All these repeats were generally done consecutively. There is one case where a programme star was observed on two different nights of the same observing run (star MH 1185). The velocity difference between the two velocities was 0.4 km/s. As mentioned in the discussion of the zero-point above, there were also five programme stars that were observed in both observing runs (see Table 1a). The rms of the velocity difference for the five stars was 6.1 km/s, or 2.4 km/s excluding one discrepant point (MH 1153). It should be noted that this particular star was observed twice in each of the observing runs. In the first run the velocity difference between the two measurements was 3 km/s, and in the second run, 2 km/s. Therefore, it is unlikely that the 10 km/s difference between the two runs is just due to random errors. It is possible that the star (MH 1153) is a binary (see below).

2. From simulation: Finally, we conducted the following experiment: we artificially added different amounts of noise to a high signal-to noise spectrum of a standard star, and cross-correlated the resulting spectrum with the series of models, as was done for the real data. This was repeated several times in each case. We found that, the rms of the velocity difference was 3.6 km/s (with a maximum of 6.7 km/s) for a ccp of 30%. This rms decreased to 1.1 km/s (max of 2.7 km/s) for a ccp of 55%, and to 0.3 km/s (max of 0.6 km/s) for a ccp of 70%.

Using all of the above, we estimate that the individual random errors in the velocities of Table 1 range from about 5 km/s for the lowest ccp (i.e. <0.40) to 1 km/s for the best observed stars. Most of the stars have velocity determinations of intermediate accuracy, i.e. of 2-4 km/s.

The effect of the choice of models on the derived velocities:
The choice of models used as cross-correlation templates may have an influence on the derived radial velocities. We examined the cases for which more than one exposure was available, and for which all individual spectra gave acceptable ccp (this exercise could only be performed for the red spectra). It was found that for more than half the cases, the models that gave the highest ccp were identical for the different exposures of the same star. In the rest of the cases, the models picked were similar, and most importantly the effect on the radial velocities was well within the error margins previously mentioned (i.e. <6 km/s). In 70% of the stars, the overall range of velocities yielded by the full range of models used was less than 6 km/s. The maximum range observed was tex2html_wrap_inline1087 km/s (just for one star). However in this latter case the large range was caused by models that gave low ccp (well below 30%). From the above analysis it becomes clear that the use of models is necessary for the proper derivation of the velocity shifts, as the correct matching of the template to the programme star can be crucial. Therefore, just the use of a selection of a few galactic carbon stars as templates for the cross-correlation, which is the usual practice, may not be good enough.

Comparison between the Red and Blue spectra: The above analysis was based on the velocities derived from the red spectra. Generally, the "red'' spectra (8335-8879 Å) had much higher signal-to-noise ratios than the "blue'' spectra (4656-5785 Å), mostly due to the fact that carbon stars are much brighter in the red. This, coupled with the smaller pixel size (by a factor of 3 in km/s) of the red spectra and with the fact that in the red spectra there is a forest of strong but unsaturated lines, led to much more accurate velocities from the red spectra.

It is important, however, to compare the radial velocities derived from the two different wavelength regions. The results for the first run are discussed here. There is a small zero-point difference between the blue and red spectra velocities amounting to tex2html_wrap_inline1093 km/s, corresponding to less than 0.10 of the combined pixel size. The rms of the velocity difference between the red and blue spectra of the standard stars is 8.7 km/s, corresponding to 0.13 of the combined pixel size (the maximum difference being 17.4 km/s, corresponding to 0.26 of the combined pixel size). For the programme stars, the rms of the velocity difference between the two sets of velocities, for ccp higher than 30%, was 20.5 km/s corresponding to 0.31 of the combined pixel size, which is expected given the lower signal-to-noise of the blue spectra. This comparison between the velocities yielded by the red and blue spectra provides a check against gross errors caused by mis-identification of the ccp. No such instances were noted. In Table 1, we have adopted the velocities derived from the "red'' spectra.

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