Here, the e
i
are the lengths of the 2D error vectors, that is, e
2
i
= e
i
·e
i
. In the large-N case we are considering,
N = T /t. But e is defined such that
N
i=1
e
2
i
= N e
2
, so eq. (13) becomes
X
2
= N
1
720
z
2
T
4
+ e
2
(14)
and the fractional increase in X
2
(the "extra sum of squares") due to the modeling error from the linear
approximation (and assuming no other modeling errors) would be
X
2
X
2
=
1
720
z
2
T
4
e
2
=
1
26.8
zT
2
e
2
h
e
2
(15)
where z and e are expressed in the same angular units, and h is the previously defined amplitude of the
modeling error, zT
2
/24. Suppose we set h = g e /
N , where g is a factor to be determined; then the
fractional increase in X
2
is g
2
/N . The ratio F that is subject to the F-test is the fractional increase in X
2
(= fractional increase in
2
, since all observations have a an uncertainty of e ) times the number of degrees
of freedom, , in the quadratic fit:
F =
X
2
X
2
/
=
g
2
N
(16)
If N is sufficiently large that N , then F g
2
. For an F-test probability of 95%, F would have to be
about 4 for a wide range of degrees of freedom (F =4.17 for =30, 3.92 for =120, 3.84 for =). Thus
we require g = 2, i.e., h = 2 e /
N . This result is not surprising, since a "two sigma" value for a model
parameter determined from Gaussian-distributed data has a 95% chance of being significantly different from
zero. If we wish a more stringent test, we can always set g=3 (probability > 99%); the value of g can be
adjusted according to the acceptable ratio of false positives / false negatives in the results.
3. Estimating the Magnitude of the Effect
This section provides an assessment of the range of orbital semimajor axes that astrometric measure-
ments of a given accuracy and time span are sensitive to. This result would form the basis for any estimate
of the fraction of stars whose observations would be affected by accelerated motion. The development given
above treats the statistics of stellar loci at the weak curvature limit -- those representing a very small part
(a few percent) of a binary orbital period. The strong curvature limit might be plausibly defined by the point
at which a reliable orbital solution becomes feasible. That point will be somewhat arbitrarily defined here
as being half an orbital period, although preliminary orbital solutions are often formed from observations
spanning much less time.
To assess the practical effect of unmodeled accelerations, a reasonable approach is to compare, for each
candidate binary system, the amplitude of the linear-track modeling error, h, to some detection criterion
that we are free to choose. The results of the previous section show that a reasonable choice for is 2 e /
N ,
where e is the mean error of a single observation and N is the number of observations. (More correctly,
the denominator should be
, where is the number of degrees of freedom in the quadratic solution, but
we are assuming that N is sufficiently large that N .)
The value of h is defined for a specific time period of duration T , and we will use the expression
h = zT
2
/24 derived in the previous section. Although this expression somewhat underestimates the effect
of orbital motion on the data for stars that traverse a significant fraction of their orbits in time T , it is the
appropriate expression to use for the weak curvature limit.
Our task, then, is to determine over what range of conditions h > . Since h is proportional to the
star's projected acceleration on the sky, z, we must first relate z to the magnitude of the true acceleration
of the star in 3-space, |Z|. The true acceleration can be expressed in terms of the physical parameters of the
binary system, and we can then use what is known about the distribution of these physical parameters to
estimate the frequency of significant acceleration effects on astrometric data.
5
