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Astron. Astrophys. 330, 585-599 (1998)
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
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]](img50.gif)
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
, the three error estimates
, and
were known. On the other hand,
or mean that only
or both and
were unknown. Whereas the knowledge of
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
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
of the components with the total mass in order to obtain the
individual masses:
![[EQUATION]](img57.gif)
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
B is computed from a combination of the variances of
and :
![[EQUATION]](img60.gif)
In any case, the errors on the individual masses are given by the
classical expressions:
![[EQUATION]](img61.gif)
5.2. The photometric transformation
To 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.
![[TABLE]](img64.gif)
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 index (see Fig. 4).
![[TABLE]](img59.gif)
Table 7. Masses of seven systems with solutions of first kind.
![[TABLE]](img117.gif)
Table 8. Astrometric binaries processing. Masses for 36 systems with solutions of second kind.
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.
![[FIGURE]](img78.gif) |
Fig. 4. Relation between and the colour index ( and are resp. the Hipparcos global magnitude and the Johnson V magnitude). The solid curve (A) concerns the O to type stars of class V to II, and the red giants of types to . The dotted curve (B) holds for the G, K, and M dwarfs with .
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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,
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).
![[TABLE]](img20.gif)
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
. 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]](img84.gif) |
Fig. 5. Raw results of the processing: Comparison with ground-based measurements for 17 systems. For 11 double stars of Type II the difference 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.
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![[FIGURE]](img86.gif) |
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.
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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
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