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Astronomical Data Analysis Software and Systems VII
ASP Conference Series, Vol. 145, 1998
R. Albrecht, R. N. Hook and H. A. Bushouse, e
Ö Copyright 1998 Astronomical Society of the Pacific. All rights reserved.
ds.
Identification and Analysis of Binary Star Systems using
Probability Theory
E. Grocheva and A. Kiselev
Photographic Astrometry Department, Pulkovo Observatory,
S.Petersburg, 196140, Russia, Email: gl@spb.iomail.lek.ru
Abstract. Accurate identification of physical binary star systems has
become of increasing interest in recent years. The frequency of occurrence
of binary and multiple star systems remains uncertain at the present time,
and is a stumbling block to our understanding of the formation of stars,
stellar systems, and planetary systems. Identification of physical binary
star systems, with an estimated reliability of such identifications based on
the theory of probability, is also important for current and future e#orts
to find binary star systems that are candidates for observations with the
aim of determining their orbits and component masses.
We shall present methods which allow the identification of physical
binary stars in a probabilistic sense. The application of probability the­
ory 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 meth­
ods 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.
1. Introduction
It is usually di#cult to identify physical binary star systems without long­term
observations. A simple comparison of proper motions does not provide certain
verification of the connection between the components. In many cases we do
not know the distances to stars or stellar space velocities. Therefore two stars
seen in the same direction and having similar proper motions may have di#erent
distances and space velocities. Estimating the component masses is interesting
from the point of view of searching for hidden masses. It is possible to make
this identification on the basis of probability theory. We propose to estimate
the probability P of the random disposition of two or more stars, having similar
proper motions, on small angular distance #. True physical systems will have
the lowest such probabilities.
2. Estimation of Probability.
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 # = ## 2 equal
15

16 Grocheva and Kiselev
1 as a probability of a reliable event. The probability P of finding n components
in a small area # may be represented as:
P = P n-1 (#)P µ (µ A , µB , #) (1)
where
P n (#) ­ probability of finding n stars in small area limited with angular
distance # for the case of a random distribution, (Deutsch,1962):
P n (#) = 1
n!
# #N
# # e - #N
# (2)
n ­ is multiplicity of a star system;
# ­ is area where N stars are randomly disposed;
P µ (µ A , µB , #) ­ probability of the proximity of proper motions µA and µB ;
# ­ error of determination of proper motions.
Let #µ be a random vector, µ = (µ x , µ y ), where µ x = 15µ # cos #, µ y = µ # . If
#µ takes on a value from the cell g ij = (µ xj , µ xj +#µ x ) â (µ yi , µ yi +#µ y ), then
we say that random vector #µ takes the value µ ij . Hence, vector µ may take the
following values:
µ 11 , µ 12 , .., µ nn .
Let the probabilities that µ takes one or another value be equal correspondingly
to
p 11 , p 12 , .., p nn .
and n
# ij
p ij = 1
i.e., the density of probability of the random vector is known. In the case of sta­
tistical estimation of probabilities the quantities p ij are the relative frequencies
n ij
N
, then P µ (µ A , µB , #) is equal to:
P µ = #
µ ij #G #
n ij
N
= S
N
(3)
where G # is the Borel's set, where the tips of vectors are situated at:
G # = (µ xmin - #, µ xmax + #) â (µ y min - #, µ y max
+ #) (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 di#erential 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, # > 70. Properly speaking, selection conditions
can be very di#erent, but for the present we solve this problem from an
astrometric point of view only. For every star we calculate the position of

Identification and Analysis of Binary Star Systems 17
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
µA ,µ B and angular spacing #. The required probability is equal to
P = #N
#
S
N
= ## 2 S
#
according to (1) ­(3).
. 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.
3. Results of Analysis of Stars from Pulkovo's Observation Program.
The distribution of proper motions was derived from the PPM catalogue for stars
of the North­polar area. The parameters of the distributions are in Table 1. The
predominance of negative motions, especially of µ y , is readily observable. This
is due to the location of the North­polar 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 µ is 0''.005 /yr.
Table 1. Parameters of the proper motions distribution
Mean Variance max min
µ x ­0''.0006 0.047 1''.545 ­2''.97
µ y ­0''.0059 0.041 0''.818 ­1''.74
Analysing the resulting probabilities we conclude that only the probability
of random proximity of proper motions P µ = S/N can be used to identify true
physical binaries. Multiplication P µ by #N/# corrupts the probability pattern.
Figure 1 shows probabilities P µ on a logarithmic scale (we numbered the pairs
of this sample from 1 to 76 and used these numbers as x­coordinates). We see
that probabilities for physical pairs are less than 0.01, whilst those for optical
ones are large. Hence, the quantity P µ 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.

18 Grocheva and Kiselev
0.0001
0.001
0.01
0.1
1
S/N
physical optical other
Figure 1. The pattern of probabilities P µ = S/N shown on a loga­
rithmic scale.
4. Conclusion.
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.
References
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,
Saint­Petersburg, (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