Curved Stellar Paths and Proper Motion Determination
George H. Kaplan
U.S. Naval Observatory
Abstract The publication of the Hipparcos catalog has drawn new attention to the problem of the
determination of proper motions of stars that are components of binary or multiple systems. With
only three years of observational data, it was recognized that unmodeled binary motion could affect
the measured proper motions of many stars in the Hipparcos catalog. In fact, the problem occurs
on many time scales and undoubtedly affects many catalogs. This note presents some analytical
expressions for the effects of binary motion on proper motions when the orbital period is at least
several times the span of observations.
Key Words astrometry, proper motions, positional astronomy, binary stars, catalogs
1. Introduction
The publication of the Hipparcos catalog has drawn new attention to the problem of the determination
of proper motions of stars that are components of binary or multiple systems. With only about three years
of observational data, it was recognized that unmodeled binary motion could affect the proper motions of
many stars in the Hipparcos catalog. Most of the concern has been with binaries without known orbits that
may have periods of several years to several decades. Yet almost half of all known binary systems have
semimajor axes of 50 AU or more, implying, for solar-mass components, periods of a century or more. Such
systems can create problems when observational catalogs made years or decades apart are combined (as in
Tycho 2) in an effort to provide improved proper motions. Thus, the binary-motion problem occurs on many
time scales and may affect, to some degree, a significant fraction of the data in compiled catalogs.
In this paper we look at a particularly pernicious piece of this problem: stellar paths that are almost,
but not quite, linear. We provide a simplified model of astrometric observation analysis to quantify the
problem. We then use the results to estimate the range of orbital semimajor axes that the observations are
sensitive to, as a function of observational accuracy and time span, and the distance and mass of the system.
2. Simplified Development of Proper Motion Estimation
In this section we develop a simple model of how analysis of astrometric observations for stellar proper
motion determination can become contaminated by an unmodeled acceleration. The object is to provide
approximate expressions that will allow us to determine the order of magnitude of the effect as well as its
qualitative nature.
The equation of motion of a body can be expressed as a Taylor series in vector form as
P(t) = P
0
+ V
0
t +
1
2
Zt
2
+ · · ·
(1)
where P
0
and V
0
are the body's position and velocity at time t = 0, and Z is the acceleration. time
derivative. For a star where the acceleration is due to the gravitational attraction of a companion (either
seen or unseen), Z = (GM/R
2
) ^
R, where G is the constant of gravitation and M and R are the mass and
distance of the companion. The unit vector ^
R points toward the companion. If we assume a constant
acceleration -- i.e., truncate eq. (1) after the third term -- we are limited to considering a small segment
of the orbit. This is the problem we wish to investigate: the parabolic approximation constitutes a "weak
curvature" case.
Let the star's motion, projected onto the plane of the sky, be p(t). The plane of the sky is orthogonal
to the line of sight unit vector n, so we have
p(t) = P(t) - (P(t) · n)n
= (P
0
+ V
0
t +
1
2
Zt
2
) - ((P
0
+ V
0
t +
1
2
Zt
2
) · n)n
= P
0
+ V
0
t +
1
2
Zt
2
- (P
0
· n)n - (V
0
· n)tn -
1
2
(Z · n)t
2
n
= [P
0
- (P
0
· n)n] + [V
0
- (V
0
· n)n]t +
1
2
[Z - (Z · n)n]t
2
(2)
1
