and similarly for X
2
/p
y
, X
2
/v
x
, and X
2
/v
y
. If we expand the dot products in eq. (8), (for
example, p · v = p
x
v
x
+ p
y
v
y
), perform the indicated partial differentiations, and multiply by
t, we obtain
2p
x
T + v
x
T
2
+
1
3
a
x
T
3
= 0
(9a)
2p
y
T + v
y
T
2
+
1
3
a
y
T
3
= 0
(9b)
p
x
T
2
+
2
3
v
x
T
3
+
1
4
a
x
T
4
= 0
(9c)
p
y
T
2
+
2
3
v
y
T
3
+
1
4
a
y
T
4
= 0
(9d)
Equations (9a) and (9c) can be solved for p
x
and v
x
:
p
x
=
1
12
a
x
T
2
and
v
x
= -
1
2
a
x
T
Equations (9b) and (9d) are identical to (9a) and (9c), respectively, except that the y components of the
vectors have replaced the x components, so it must be true that
p
y
=
1
12
a
y
T
2
and
v
y
= -
1
2
a
y
T
Obviously these results can be expressed as
p =
1
12
zT
2
and
v = -
1
2
zT
(10)
Since the vectors p, v, and z are collinear, any dot products among them would simply equal the
arithmetic product of their respective magnitudes. Also e
2
i
= e
2
i
. Therefore we can dispense with the vector
notation at this point and use use the symbols p, v, and z for the vector magnitudes. Equation (10) then
becomes
p =
1
12
zT
2
and
v = -
1
2
zT
(11)
The difference between the actual motion of the star and the linear model as a function of time t is in
the direction of z (+ or ). The acceleration z is always toward the companion, and for z to be considered
essentially constant, the companion must be sufficiently far away that neither its direction nor distance
change significantly over the short orbital arc we are considering. The magnitude of the difference between
the actual motion and the linear model is
= p + vt +
1
2
zt
2
=
1
12
zT
2
-
1
2
zT t +
1
2
zt
2
(12)
where z = |z|. The function has an extremum (d/dt=0) at t=T /2, in the middle of the span of observations,
where = -zT
2
/24. At the beginning and end of the span of observations (t=0 and t=T ), = zT
2
/12. The
total range of over the time interval of interest is thus zT
2
/8. The locus of actual motion of the star on
the sky crosses the least-squares-determined straight where the function has zeros, at t = T /2 ±
T /12 =
0.211T and 0.789T .
We will refer to zT
2
/24, which is the absolute value of at t = T /2, as the amplitude of the modelling
error from the linear approximation, designated by h. The total range of over the time span of interest
is 3h. In the following developments we will use h as the metric for determining the sensitivity of the N
observations to the acceleration. Intuitively, it would seem that if h is a few times e /
N , where e is the
mean error of a single observation (unit weight), then the observations should be at least marginally sensitive
to the acceleration. More precisely, if we compute the ratio of the post-fit sum-of-squares X
2
for the linear
and quadratic models, expressed as a function of h, we can use the F-test to determine the significance of
the acceleration term.
To do this, we revisit eq. (7) for X
2
, substituting in it the expressions for p and v from eq. (10).
We obtain the post-fit sum-of-squares for the case of the linear fit to accelerated motion:
X
2
=
1
t
(
1
144
z
2
T
5
-
1
24
z
2
T
5
+
1
36
z
2
T
5
+
1
12
z
2
T
5
-
1
8
z
2
T
5
+
1
20
z
2
T
5
) +
N
i=1
e
2
i
=
1
t
(
1
720
z
2
T
5
) +
N
i=1
e
2
i
(13)
4
