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Astronomical Applications Department, U.S. Naval Observatory - Precession Memo (Page 7)

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Astronomical Applications Department, U.S. Naval Observatory - Precession Memo
(31)
K
q
K
g
K
a
=
0
2
o
f
f
+
S
r %
(1
-
A
C
)
o
q
(1
-
A
C
)
o
g
(1
-
A
C
)
o
a
-
2 A
C
cos
x
$ cos x $ q sin a dq dg
where is the vector from the center of mass to a point on the cone. Using eqs. (16) and (18), we
r
have
(32)
r
=
`(a , g)
-
1
x
y
z
=
`(a , g)
-
1
q sin a cos g
q sin a sin g
h
+
a
tan
a
-
q cos a
=
-
h cos
a
-
a
cos
2
a
sin
a
+
q
0
h sin
a
+
a cos
a
Substituting eq. (32) into eq. (31), performing the integral in from
to
, and sim-
q
f
=
a
sin
a
f
+
S
=
b
sin
a
plifying, we find the result
(33)
K
q
K
g
K
a
=
P (b
-
a)
0
2
o
-
B
2
o
g
B
2
o
q
+
B
1
Q
o
a
cos
x
-
2B
1
A
C
cos
x
B
2
(1
-
A
C
) P
g
cos
x
cos
x dg
where
(34)
B
1
h
1
2
1
sin
2
a
U (a
+
b) cos
a
-
2
3
(a
2
+
a b
+
b
2
)
B
2
h
1
2
1
sin
a
Q U (a
+
b)
and
(35)
Q h 1
-
A
C
U h h sin
a
+
a cos
a
Now make use of eq. (18) to transform back to the Cartesian body frame.
(36)
K
x
K
y
K
z
=
P (b
-
a)
0
2
o
`(a , g)
-
B
2
o
g
B
2
o
q
+
B
1
Q
o
a
cos
x
-
2B
1
A
C
cos
x
B
2
(1
-
A
C
) P
g
cos
x
cos
x dg
Finally, performing the remaining integral and simplifying, we arrive at the result
(37)
K
x
K
y
K
z
=
V
o
X
(cos
y cos h sin v
+
sin
h cos v)
+
o
Y
(sin
h sin v
-
cos
y cos h cos v)
-
o
Z
cos
h sin y
o
X
(cos
h cos v
-
cos
y sin h sin v)
+
o
Y
(cos
h sin v
+
cos
y sin h cos v)
+
o
Z
sin
h sin y
0
where
(38)
V h
oP (b
-
a) (
-
o
X
sin
y sin v
+
o
Y
sin
y cos v
-
o
Z
cos
y)
$ [B
1
(3
+
A
C
) cos
a sin a
-
B
2
(2 sin
2
a
-
cos
2
a)]
3.4 Force and Torque Components Due to Radiation Pressure on the "Flattop" Surface
Now we will calculate the torque due to an incident pressure on the top of the frustum, the "flattop".
Letting
in eqs. (37), (38), (34), and (35), we find
a
d 0, b d a, and a d
o
2
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