adopt /4 as the average value of the projection factor sin , with some confidence that the average is also
typical.
The distribution of R/a values presents a similar situation. For circular orbits, R/a=1 at all times.
For stars in eccentric orbits, R/a varies between 1e and 1+e, where e is the orbital eccentricity. However,
binary stars spend more time near apastron than periastron, and over an orbital period the average value of
R/a is 1 +
1
2
e
2
. Since this ratio can only take on values between 1 and 1.5, we can adopt an average value
of R/a of 1.3 without much concern about the distribution of eccentricities.
Inserting the appropriate numerical quantities, eq. (19) becomes
1.59 T
2/3
M
1/3
< a < 0.88 T M
p
1/2
(20)
Since = 2
p
, the right side (upper limit to a) could also be expressed as 0.62T (M p/
p
)
1/2
. As we expect,
the range of applicable a increases with T . For M = 1 and p/ = 1, 7.4 a 8.7 if T = 10 and 34 a 87
for T = 100. These limits define the null set for T < 6.1 years for solar-mass binaries where the parallax is
at the limit of detection but do not preclude more massive or closer systems.
If we think about the astrometric method of detecting orbital motion in general, including both orbit
and acceleration solutions, the situation is this: at given distance, and for a given primary and companion
mass, the astrometric sensitivity increases with semimajor axis, provided at least one full period is observed.
However, once only a fraction of a period can be observed, the astrometric sensitivity decreases with in-
creasing semimajor axis, because the observed motion becomes more linear. As the distance increases, the
width of the regime of detectable semimajor axes narrows, since the observational scatter corresponds to an
increasing linear scale. Thus there is a reduced range of semimajor axes that are greater than the scatter
but less than that for which the orbital motion is statistically indistinguishable from a straight line.
7
