The magnitude of the acceleration projected onto the plane of the sky is related to the magnitude of
the true 3-D acceleration vector by z = (sin )|Z|/d, where is the angle between the direction of Z and the
line of sight, and d is the distance to the star. The parallax, p, is 1/d, so our h > condition becomes
1
24
(sin ) |Z| T
2
p >
(17)
The acceleration magnitude is related to the physical state of the binary system through |Z| = GM/R
2
,
where R is the instantaneous distance between components, and M is either (a) the mass of the companion,
if the star's motion is measured in an inertial system, or (b) the total mass of the system, if the motion is
measured with respect to the companion. If we use units of AU, years, and solar masses, the gravitational
constant G = 4
2
and the true acceleration |Z| is expressed in AU year
-2
. If d is in parsecs and p in
arcseconds, then z is in units of arcsec year
-2
and h is in arcsec.
The distance R is obviously closely related to the semimajor axis of the orbit, a. If we write R = a(R/a)
(for reasons that will become apparent shortly) our condition for detection of weak curvatures becomes
4
2
(sin ) M T
2
p
24 a
R
a
2
>
(18)
The strong curvature limit is defined by P 2T , where P is the orbital period, given by P = a
3/2
/
M ,
where M is the total mass of the system. For this limit we therefore have the condition a
3/2
/
M > 2T .
Rearranging the expressions for the weak and strong curvature limits, we obtain the limits on the semimajor
axis that define the fraction of stars of interest:
(2T )
2/3
M
1/3
< a <
T
R
a
M sin
6
1/2
p
1/2
(19)
Note that M on the left and right sides can have different meanings: on the left it is always the total mass
of the system; on the right it is the total mass of the system only if the motion of the star is measured with
respect to its companion. If the star's motion is measured with respect to an inertial frame then M on the
right is the mass of the companion. The expressions therefore can be used for investigating planet detection
by the astrometric method, in the long-period limit, simply by setting M on the right side to be the planet's
presumed mass.
We have chosen
= 2 e /
N , which is twice the 1 uncertainty of any angular variable derived from
the observations (assuming N ). Therefore, the quantity p/ appearing on the right side of eq. (19) can
be thought of as half the signal-to-noise ratio of the parallax: p/ =
1
2
p/
p
. (This holds only for the case
where the parallax of the star is determined using the same set of observations used for the proper motion
and acceleration determination.) If we fix the p/
p
ratio, eq. (19) provides the semimajor axis limits for stars
in a certain distance shell; for example, if we wish to consider stars with parallaxes good to 10% (p/
p
= 10)
when
p
10
-3
arcsec, we will obtain the limits on a for stars at a distance of approximately 100 pc. It
is interesting to note in this regard that if we have a parallax-limited sample of stars uniformly distributed
through space, the average parallax is only 1.5 times the minimum parallax. In such a case, and assuming
that the minimum parallax is
p
, then p/ < 1 for most stars in the sample, and the upper limit on a will
generally be quite restricted. However, most catalogs are magnitude-limited and their stellar distributions
are much more concentrated toward the Sun. In such catalogs, and in special samples of nearby stars, there
will be enough stars with p/ > 1 to provide a useful range of detectable semimajor axes.
To assess quantitatively how many stars fall between the limits defined by eq. (19), we need to know
something about the distributions of sin , a, M , and R/a. Two of the four distributions, for sin and R/a,
are easy to deal with. Although the projection factor sin is unknown for any specific binary system, we
can assume that the direction of the vector separating the two components is randomly distributed over 4
steradians. For such a distribution, the average projection factor onto any given plane is /4 0.79, the
median projection is 0.87, and 71% of such vectors have projection factors of 0.7 or more. Therefore we can
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