Institute of Astronomy, 48 Pyatnitskaya St., Moscow 109017, Russia

Astrophysikalisches Institut Potsdam, An der Sternwarte 16, D-14482 Potsdam, Germany

The available
observational data on masses and/or luminosities of binary components
and the current models for different
chemical compositions of Pop I stars
were used to
improve the mass-luminosity relation for stars of moderate masses.
We have developed a technique of simultaneous
minimization of the discrepancies between the predicted and
observed masses/radii, assuming equal ages
and chemical compositions of binary system components.
The method of descending coordinates was used to
optimize the observed and theoretical parameters.

This technique was also applied to decompose the low-mass components of several stars suspected to be unresolved binaries. In order to explain the offsets of these stars from the ``normal'' location in the mass-luminosity plane, we computed the most probable masses and luminosities of their hidden components.

This paper presents a mathematical approach which can be used in investigations of different astrophysical problems. The method is based on the minimum determination technique applied to functions of several variables covering a limited range of values. Generally, the method can be described as follows.

Let *p*_{1}, *p*_{2}, ..., *p*_{n} be *n* independently observational parameters
(IOP) for a given kind of objects. We assume *p*^{0}_{1}, *p*^{0}_{2}, ..., *p*^{0}_{n}
and
to be their corresponding observed quantities
and observational errors, respectively.
Further, we define a set of non-directly observable parameters (NDP)
*q*_{1}, *q*_{2}, ..., *q*_{m} for the objects under study.
In addition, we consider a set of astrophysical conditions
relevant for the investigated objects.
There are two different kinds of
these conditions.
First, the strong mathematical relations between
parameters
*p*_{i} = *p*_{i}(*q*_{1}, ..., *q*_{m}, *p*_{j}, ..., *p*_{k})
(e.g., the mass of a binary is equal to the sum of the component masses);
second, the empirical relations
between two parameters
*Q*_{i} = *Q*_{i}(*q*_{j}) (e.g., mass-luminosity relation).

We define an initial set of parameters (ISP hereafter)
selected from NDP and IOP.
The remaining parameters
from IOP
can be computed in accordance with the relations given above. Treating
the observational data *p*^{0}_{i} and their computed values *p*_{i}, we can find
a new set of ISP which optimally satisfy both the observations
and the adopted conditions.

In other words, we have to minimize a function

(1) |

Minimizing the function *D*, we can derive optimal sets of IOP and
NDP and, consequently, determine more suitable values
within their error boxes for observable and
non-directly observable parameters. The function *D* is dimensionless and
can be used as a criterion for the accuracy of the solution.
To find the minimum of the function *D*, the method of descending coordinates
(or other appropriate numerical algorithms) can be used.
An application of the method to a given astrophysical problem
assumes a careful construction of the suitable function *D* and definition
of reasonable limits for the included parameters.

Below we present two examples of possible applications.

The main objective of the project was
to improve the
mass-luminosity relation (MLR) for stars of moderate masses
by optimal using
the available data
on binaries components.
For this purpose we considered the data (IOP) on masses
(
) and radii
(
for the components of
eclipsing binary stars from the Catalogue of
Astrophysical Parameters of
Binary Stars
(Malkov 1993). The corresponding
set of NDP includes the age *a* (
) and abundance *z*
of the components.
Assuming a common origin for the components, we may consider their ages
and chemical compositions to be equal.
As additional conditions, we used theoretical stellar
models of Schaller et al. (1992)
for different chemical compositions of Pop I stars. The values of
and
derived from these models were used to determine the parameters of the
relations
*r*_{i} = *R*(*m*_{i}, a, z) and the corresponding accuracy .

The function *D* to be minimized can be written as

(2) |

As the result, the optimal values of masses and radii
as well as their chemical
composition (*z*) and evolutionary stage (*a*) were obtained for
12 double star systems.
For one of the considered binaries (AS Cam), Fig. 1
shows how these parameters correlate with the original observational
data.

About 30 other double star systems are included in the project. Extending the investigations to a larger number of binaries will improve the database on optimal masses and luminosities. This will allow us to derive the mass-luminosity relation for stars of moderate masses with higher accuracy.

Here we considered stars located in the area of underluminous objects of the MLR. Using the approach described above, we tested whether at least some of them may be considered as photometrically unresolved binaries.

In this case the corresponding IOP are mass *m* = *m*_{1} + *m*_{2} and luminosity
of an
underluminous star supposed to be a binary.
The initial set of observational
data
,
for the low mass
stars were taken from Malkov, Piskunov, & Shpil'kina (1997).
As additional condition, we used the theoretical mass-luminosity
relation for the low mass region represented by a function .
The width of MLR was assumed to be
.
In this case we have to minimize the function

(3) |

The parameters of components were determined for four recently resolved binaries and for two other stars which have been suspected to be multiple (Kovaleva & Malkov 1999). The distribution of probable values of masses and luminosities is plotted in Fig. 2 for one of the resolved stars (GJ 508 A).

Applying our method to the suspected binaries, we derived the parameters of
the components which may describe the situation realistically.
Nevertheless, we
do not claim that these results present the * final proof* for the duplicity
or multiplicity of these two stars examined here. The offsets of the stars from
the usually adopted MLR may be also explained both by large observational
errors and by insufficient knowledge of the actual MLR. The results presented
here merely show that the possibility for these stars to be unresolved
binaries cannot be rejected.

Kovaleva D. A. & Malkov O. Y. 1999, Astronomy Reports, in press

Malkov, O. Y. 1993, Bull. Inf. CDS, 42, 27

Malkov, O. Y., Piskunov, A. E., & Shpil'kina, D. A. 1997, A&A, 320, 79

Schaller G., Schaerer D., Meynet G., Maeder A. 1992 A&AS, 96, 269

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