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.

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