That is, the star's motion in the plane of the sky can be represented as
p(t) = p
0
+ v
0
t +
1
2
zt
2
(3)
where p
0
is P
0
projected onto the sky = P
0
- (P
0
· n)n
v
0
is V
0
projected onto the sky = V
0
- (V
0
· n)n
z is Z projected onto the sky = Z - (Z · n)n
The vectors p(t), p
0
, v
0
, and z have no component along the line of sight n and are therefore 2-vectors in a
coordinate system on the plane of the sky.
In standard current practice, a star's motion is expressed as p (t) = p
0
+ v t. Here, v represents
the star's proper motion as conventionally defined -- assumed constant, hence without subscript. (In this
development we are ignoring terms for the curvature of the sky and are assuming that aberration and parallax
have already been removed from the data.) It is tempting to think of eq. (3) as a simple extension of the
conventional expression, carried to higher order. But v
0
in eq. (3) does not correspond to proper motion. For
a gravitationally-bound binary, proper motion properly refers to the projection on the sky of the (constant)
space velocity of the center of mass of the system. In eq. (3), v
0
is the instantaneous linear component of the
star's apparent motion, which is the sum of the proper motion of system plus the projected orbital velocity
of the star (at t=0) around the center of mass. Determination of the proper motion of the system would
require observations of the star spanning nearly an orbital period, or observations of both the star and its
companion over a shorter period together with an estimate of their mass ratio.
But here we wish to consider a series of observations where the orbital motion of the star is not obvious.
Consider an observation of the star's position, p
i
, taken at time t
i
, with a measurement error e
i
(all vectors
are now in the plane of the sky). Then p
i
= p
0
+ v
0
t
i
+
1
2
zt
2
i
+ e
i
. But if we have no knowledge of the
acceleration, we will model the star's motion in the conventional way as p (t) = p
0
+ v t. The difference
between the observation and this incomplete model of the star's motion at time t
i
is then
e
i
= p
0
+ v
0
t
i
+
1
2
zt
2
i
+ e
i
- p
0
- v t
i
(4)
Compared to our incomplete model, the observation will appear to be in error by an amount e
i
. This error
is the sum of the random error of observation and the systematic error resulting from the use of the incorrect
model. Despite the fact that it is not a purely stochastic quantity, it provides a basis for investigating what
would happen if the incomplete model were fit to a set of N two-dimensional observations using least squares.
Let us take the common situation where the measurement errors e
i
are assumed to all have approximately
the same magnitude e and the observations are therefore all given unit weight. In such a case, the quantity
that would be minimized is the sum of e
2
i
over all observations. The method would determine the position
p
0
(at time t=0) and proper motion v that minimize
e
2
i
. The quantities p
0
and v are not particularly
interesting in themselves but the differences between these quantities and the corresponding ones from eq. (3)
for this case are key to further analysis. That is, we are interested in p = p
0
- p
0
and v = v
0
- v . These
quantities measure in some sense the "error" in the position-at-epoch and proper motion derived from the
linear fit.
As noted above, the quantity to be minimized is
e
2
i
, the sum of the squares of the post-linear-fit
2
