Astronomical Applications Department, U.S. Naval Observatory Sensitivity Integral Page 22
=
(
)
S , , , ,
,
r s g T
1
2
d
1
2
integrand
1
x
(
)
S , , , ,
,
r s g T
1
2
=
d
1
2
(
)
+
+
+
+
a
0
a
1
x
a
2
x
2
a
3
x
3
a
4
x
4
(
)
sin s x
2
(
)
-
(
)
sin r x
(
)
sin r g x
2
x
becomes, to fourth order in combinations of s and r,
=
(
)
S
4
, , , ,
,
r s g T
1
2
d
1
2
(
)
factor
(
)
expansion
,
,
integrand
1
[
]
,
s r
4
x
=
(
)
S
4
, , , ,
,
r s g T
1
2
d
1
2
(
)
+
+
+
+
a
0
a
1
x
a
2
x
2
a
3
x
3
a
4
x
4
x
4
(
)
- +
1
g
2
r
2
s
2
x
(Indeed, this fourth-order term is the first nonzero one to appear.) Hence we have
(
)
factor
(
)
value %
(
)
S
4
, , , ,
,
r s g T
1
2
1
2520
s
2
r
2
(
)
- +
1
g
2
(
)
- +
1
2
504
2
2
1
2
a
0
315
2
7
a
3
+
=
360
2
6
a
2
420
2
5
a
1
504
2
4
a
0
280 a
4
2
8
360 a
2
1
6
315 a
3
1
7
+
+
+
+
+
+
280 a
4
1
8
504
1
4
a
0
420 a
1
1
5
315
2
6
1
a
3
280
2
7
1
a
4
+
+
+
+
+
280
2
5
1
3
a
4
315
2
5
1
2
a
3
360
2
5
1
a
2
280
2
6
1
2
a
4
315
2
2
a
3
1
5
+
+
+
+
+
360
2
2
a
2
1
4
420
2
2
a
1
1
3
280
2
3
a
4
1
5
315
2
3
a
3
1
4
360
2
3
a
2
1
3
+
+
+
+
+
420
2
3
a
1
1
2
504
2
3
1
a
0
280
2
4
a
4
1
4
315
2
4
a
3
1
3
360
2
4
a
2
1
2
+
+
+
+
+
420
2
4
a
1
1
280
2
a
4
1
7
315
2
a
3
1
6
360
2
a
2
1
5
420
2
a
1
1
4
+
+
+
+
+
504
2
1
3
a
0
280
2
2
a
4
1
6
+
+
Page 22
