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Astron. Astrophys. 330, 585-599 (1998)

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5. Mass determination

5.1. Strategy

The first step is the determination of the total mass M of the system with the Kepler's third law. The estimate of the parallax [FORMULA] is one of the six or seven unknowns solved in the processing (Martin et al., 1997 ), while the period P and the semi-major axis a of the relative orbit are taken from the literature. The references for the orbital elements are given in Tables 4 - 5. The combination of the uncertainty of the orbital period and that of the parallax is easily propagated in the standard deviation of the total mass as,

[EQUATION]

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 N. When [FORMULA], the three error estimates [FORMULA], [FORMULA] and [FORMULA] were known. On the other hand, [FORMULA] or [FORMULA] mean that only [FORMULA] or both [FORMULA] and [FORMULA] were unknown. Whereas the knowledge of [FORMULA] is not essential (the weight of this term is 2.3 times smaller than the two others, and P is generally the best known parameter for such short-period binaries), this is no longer true for the semi-major axis. The cases with [FORMULA] correspond to underestimating the variance of the total mass, and thus must be considered with caution.

In a second step we combine the mass ratio [FORMULA] of the components with the total mass in order to obtain the individual masses:

[EQUATION]

Two situations must be distinguished:

  1. 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.
  2. The only available ratio is the difference [FORMULA] (Type II stars). In that case, we use a ground-based estimate of the magnitude difference [FORMULA] in a V band to calculate [FORMULA], 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 B is computed from a combination of the variances of [FORMULA] and [FORMULA]:

[EQUATION]

In any case, the errors on the individual masses are given by the classical expressions:

[EQUATION]

5.2. The photometric transformation

To transform the ground-based estimates of [FORMULA] 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 [FORMULA] colour indices.


[TABLE]

Table 6. Spectral types and Johnson-Morgan's colour indices of the components of 18 double stars. The last column indicates the type of transformation to be made in order to get the Cousin's [FORMULA] index (see Fig. 4).



[TABLE]

Table 7. Masses of seven systems with solutions of first kind.



[TABLE]

Table 8. Astrometric binaries processing. Masses for 36 systems with solutions of second kind.


The four steps of the photometric conversion are the following:

  1. The Johnson's [FORMULA] index is first deduced from the spectral type by means of the data compiled in Zombeck, 1990 .
  2. This index is then converted into the Cousin's system as,


    [EQUATION]

  3. From [FORMULA] we get the quantity [FORMULA], 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.
  4. Then, the Hipparcos magnitude difference [FORMULA] is deduced from the Johnson's by the expression:


    [EQUATION]

    where [FORMULA] and [FORMULA] 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 [FORMULA] reaches [FORMULA] mag.

[FIGURE] Fig. 4. Relation between [FORMULA] and the [FORMULA] colour index ([FORMULA] and [FORMULA] are resp. the Hipparcos global magnitude and the Johnson V magnitude). The solid curve (A) concerns the O to [FORMULA] type stars of class V to II, and the red giants of types [FORMULA] to [FORMULA]. The dotted curve (B) holds for the G, K, and M dwarfs with [FORMULA].

5.3. Results and comments

The 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, [FORMULA] comes from ground-based measurements (see references in Table 9) and a correction was applied whenever possible. The standard deviation on [FORMULA] has been taken equal to 0.15 mag whenever it was unknown; the resulting errors on the masses [FORMULA] and [FORMULA] must then be regarded with caution. The ground-based [FORMULA] 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).


[TABLE]

Table 9. Reference values of component masses and physical ratios.


5.4. Comparisons

An 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 B have been directly compared, and for eleven stars of Type II for a total of thirty eight, the comparison refers to [FORMULA]. Results are shown in Fig. 5. It must be noted that the vertical error bars are not systematically present, as the standard deviations are not always available in the literature (the lack of error bar does not mean that the accuracy is outstanding!). In the same way, the Hipparcos and ground-based stellar masses have been compared for the same 17 systems. Results for each type are shown in Fig. 6.


[FIGURE] Fig. 5. Raw results of the processing: Comparison with ground-based measurements for 17 systems. For 11 double stars of Type II the difference [FORMULA] is plotted, while for 6 stars of type I the plotted value is the mass ratio B alone (upper part of the diagram). When available, the error bars are represented.

[FIGURE] Fig. 6. Stellar masses: Comparison with ground-based measurements for 17 systems. a 6 double stars of Type I; b 11 stars of type II (see text). When available, the error bars are represented.

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).

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© European Southern Observatory (ESO) 1998

Online publication: January 16, 1998
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