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ASP Conference Series, Vol. 145, 1998

Editors: R. Albrecht, R. N. Hook and H. A. Bushouse

**E. Grocheva and A. Kiselev**

Photographic Astrometry Department, Pulkovo Observatory,
S.Petersburg, 196140, Russia, Email: gl@spb.iomail.lek.ru

*
We shall present methods which allow the identification of physical binary stars in a
probabilistic sense. The application of probability theory provides a more complete picture of the
frequency of stellar binarity than simple methods based only on proximity or proper motion. We will
also present preliminary results from the application of these methods to Pulkovo's observation
program of binary star systems, and outline how such methods might be applied to present and future
high precision astrometric catalogues.
*

We propose to use the real distribution of proper motions for estimation of **P**. Let the probability of
finding the primary component in the area equal 1 as a probability of a reliable
event. The probability **P** of finding *n* components in a small area may be represented as:

(1) |

where

- probability of finding n stars in small area limited with angular distance for the case of a random distribution, (Deutsch,1962):

(2) |

*n* - is multiplicity of a star system;

- is area where *N* stars are randomly disposed;

- probability of the proximity of proper motions and ;

- error of determination of proper motions.

Let be a random vector, , where , . If takes on a value from the cell , then we say that random vector takes the value . Hence, vector may take the following values:

Let the probabilities that takes one or another value be equal correspondingly to
*p _{11}*,

(3) |

where is the Borel's set, where the tips of vectors are situated at:

(4) |

Now we will describe a procedure for the solution of this problem without a concrete realisation.

- Using any astrometric catalogue with stellar positions and proper motions we obtain the
differential law of distribution (density of probability) of proper motions. We construct the matrix M
whose elements will be
*n*_{ij}(3). Let us consider the stars whose and satisfy certain specified conditions, for example, . Properly speaking, selection conditions can be very different, but for the present we solve this problem from an astrometric point of view only. For every star we calculate the position of corresponding element in matrix M. We add 1 to the element thus found. The elements of resulting matrix will yield the probability density of proper motion distribution. - Let us consider the binary star whose components have proper motions
, and angular spacing . The required probability is equal to
- We choose assumed binaries from some catalogue. The sample must be restricted to pairs whose are limited by some quantity. For example it is possible to use the Aitken's criterion(Aitken,1932). Then we determine the probability of random distribution for every pair. The sample must include few known optical and physical pairs to obtain a definite criterion for the identification.

The distribution of proper motions was derived from the PPM catalogue for stars of the North-polar
area. The parameters of the distributions are in Table 1. The predominance of negative motions, especially of , is readily observable. This is due to the location of the North-polar area relative to the Solar apex.
We chose 76 double stars from Pulkovo's observation program and
calculated the probabilities of a random distribution for these pairs. There are 8 physical and 12 optical
systems among them (Grocheva,1996 & Catalogue of relative positions
and motions of 200 visual double stars,1988).
The proper motions of these pairs was obtained by using the catalogue
*``Carte du Ciel''* and modern observations with the 26''refractor. The precision of is 0''.005
/yr.

Mean | Variance | max | min | ||

-0''.0006 | 0.047 | 1''.545 | -2''.97 | ||

-0''.0059 | 0.041 | 0''.818 | -1''.74 | ||

Analysing the resulting probabilities we conclude that only the
probability of random
proximity of proper motions can be used to identify true physical binaries. Multiplication
by corrupts the probability pattern. Figure 1 shows
probabilities on a logarithmic scale (we numbered the pairs of this sample
from 1 to 76 and used these numbers as *x*-coordinates). We see that probabilities for physical pairs are less than
0.01, whilst those for optical ones are large. Hence, the quantity
may be used to identify
physical binaries and the limit of probability of proper motions proximity *S*/*N* is 0.01 for
physical pairs. It turned out that only 27 physical binaries were among the sample of 76
double stars.

The technique presented identifies physical binaries. This method has a simple algorithm and can be used for the automatic treatment of large stellar catalogues. We are pleased to also note that this method requires only minimal data such as positions and proper motions. This method was used to correct the Pulkovo's program of binary star observations.

Deutsch, A. N., 1962, ``The visual double stars''. in The course of astrophysics and stellar astronomy, p.60, Moscow, (in Russian).

Catalogue of relative positions and motions of 200 visual double stars.,1988, Saint-Petersburg, (in Russian).

Grocheva, E. A., 1996, ``Physical and optical double stars...'', Workshop The Visual Double Stars.., Spain.

Aitken, R. G., 1932, New General Catalogue of Double Stars, Edinburgh

© Copyright 1998 Astronomical Society of the Pacific, 390 Ashton Avenue, San Francisco, California 94112, USA

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Eclipsing Binaries** Up:** Computational Astrophysics** Previous:** Modelling Spectro-photometric Characteristics of Nonradially Pulsating Stars

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