## 5. Mass determination## 5.1. StrategyThe first step is the determination of the total mass Unfortunately, the errors of the orbital elements are not
systematically published. Instead of estimating rather arbitrarily the
missing quantities, we have calculated for each mass an incomplete
error, and have indicated the nature of the missing element(s) in
Tables 7 - 8 by the flag noted In a second step we combine the mass ratio of the components with the total mass in order to obtain the individual masses: Two situations must be distinguished: - The mass ratio
*B*is the solution of the processing (Type I stars), so that Eq. 4is directly implemented and yields the two masses (100% Hipparcos results). This method is flagged 'A' in Table 7. - The only available ratio is the difference
(Type II stars). In that case, we use a ground-based estimate of the
magnitude difference in a
*V*band to calculate , and then*B*. This is flagged method 'B' in Table 7. The results obtained in this way are of course less accurate than the previous ones, because of the heterogeneity of the quantities involved. To improve the quality, the magnitude differences have been expressed, when possible, in the Hipparcos photometric system (see next section).
For solutions of the second type, the variance of the mass ratio
In any case, the errors on the individual masses are given by the classical expressions: ## 5.2. The photometric transformationTo transform the ground-based estimates of into the Hipparcos system, the knowledge of the spectral types of each component is required. They have been found for only 18 systems over the 38 binaries for which it was really needed, mainly by the use of the SIMBAD database. The result of this survey is summarised in Table 6, with the corresponding Johnson's colour indices.
The four steps of the photometric conversion are the following: - The Johnson's index is first deduced from the spectral type by means of the data compiled in Zombeck, 1990 .
- This index is then converted into the Cousin's system as,
- From we get the quantity
, the difference between the Hipparcos magnitude
and the Johnson
*V*magnitude, by using the well calibrated relations presented in Fig. 4, which depend on the spectral type and class of the components. - Then, the Hipparcos magnitude difference is
deduced from the Johnson's by the expression:
where and are related respectively to the primary and the secondary component.
When the spectral types of the two components are identical, the correction is equal to zero. For example, the correction for the well known system Algol reaches mag.
## 5.3. Results and commentsThe results are presented in Tables 7 - 8 respectively for 7 stars with solution of the first Type (direct determination of the mass ratio) and 36 stars of Type II. Even if the method 'A' is the most appropriate for the stars of Type I, the method 'B' has also been used for such systems for the sake of comparison. In all the cases, the individual masses obtained by either way are compatible. On the other hand, the method 'B' is the only one used for stars of Type II and then not mentioned in the table. When method 'A' is used, the magnitude difference is directly derived from the processing and thus does not need to be transformed into the Hipparcos system (there is a 'no' in the third column of Table 7). In all other cases, comes from ground-based measurements (see references in Table 9) and a correction was applied whenever possible. The standard deviation on has been taken equal to 0.15 mag whenever it was unknown; the resulting errors on the masses and must then be regarded with caution. The ground-based are of various origins: most of them come from the compilation files of the Observatory of the Côte d'Azur, some other from Worley, but whenever it was possible, we have chosen the most recently published result (see references in Table 9).
## 5.4. ComparisonsAn extensive bibliographical search has allowed to compare our
results with ground-based measurements for 17 systems out of the 46
considered. For six stars of Type I on a total of eight, the mass
ratios
Concerning the ratios, the general agreement between the two samples is excellent. For ten cases out of seventeen, the Hipparcos solution is more accurate than the ground-based one, or at worst simply identical. For five other cases it was not possible to conclude because of the absence of the standard deviation in the published data. When considering the masses of the components, the comparison reveals some disagreements related to the parallaxes. For such cases the error bars found in the literature are not of great help, as the uncertainties of the parallaxes are generally underestimated (especially for dynamical parallaxes). The good agreement seen in the two samples seriously strengthens the confidence in the 29 other results, for which no ground-based estimate of mass ratio has been found. Moreover, one must keep in mind the principles of the method used here, which is mainly dependent on the quality of the orbital elements and, for Type II binaries, on the quality of the magnitude difference used to derive the individual masses. The values presented in Table 9 are those used in this study to check the validity of our results. These are also the so called 'reference values' used as input of the algorithm to speed up the convergence of the process (see Sect. 3.2). © European Southern Observatory (ESO) 1998 Online publication: January 16, 1998 |