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Astronomical Applications Department, U.S. Naval Observatory - Precession Memo (Page 9)

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Astronomical Applications Department, U.S. Naval Observatory - Precession Memo
Notice that, even if
initially, the torques will drive nutation and precession anyway.
y
=
w
=
y
.
=
w
.
=
0
Equations (41)-(44) are the final form for our symmetric, conically shielded spinning spacecraft.
They consist of terms describing force-free motion (the terms containing
), with the addition of per-
turbative terms due to pressures on the top of the spacecraft and on the protective conical shield.
These equations have been implemented in the numerical program, SymTop.
5 Precession
In this section, let us assume a fast-spinning top, so that
. Further, assume the pressure
W
h
>>
W
w
,
W
y
terms are small. Take the last three equations of eqs. (41), differente them, substitute eqs. (41) for the
first-order derivatives in the resulting equations. Finally, drop terms beyond first order in the small
quantities. We find the resulting second-order system of ODEs,
(45)
sin
y d
2
dt
2
W
w
= -
(1
-
b)
2
W
h
2
W
w
sin
y
+
(1
-
b) W
h
K
2
(
a, b, h,
a , A
C
, A
T
,
w, y
)
d
2
dt
2
W
y
= -
(1
-
b)
2
W
h
2
W
y
-
(1
-
b) W
h
K
1
(
a, b, h,
a , A
C
, A
T
,
w, y
)
sin
y d
2
dt
2
W
h
=
(1
-
b)
2
W
h
2
W
w
sin
y
-
(1
-
b) W
h
K
2
(
a, b, h,
a , A
C
, A
T
,
w, y
)
cos
y
Notice that the value of
merely scales the time. Since
is large, we can assume it is slowly vary-
W
h
ing compared to
and
. Hence, we may set
. There are two consequences of this
W
w
W
y
d
2
dt
2
W
h
l 0
from eqs. (45). First, the second equation implies simple harmonic mostion for
. Letting
W
y
, we have as solution
W
h
d const
(46)
W
y
l A cos((1
-
b) W
h
t)
+
B sin((1
-
b) W
h
t)
-
K
1
(
a,b,h,
a ,A
C
,A
T
,
w,y
)
(1
-
b) W
h
Notice the superimposed constant. This implies a small, monotonic drift in the inclination angle .
y
The second consequence is that we can solve for the precession rate. For the third equation of eqs.
(45) to hold, we require
(47)
W
w
l
K
2
(
a, b, h,
a , A
C
, A
T
,
w, y
)
(1
-
b) W
h
sin
y
If we further assume that the pressure is mainly along the fixed-frame Z axis,
, then eq.
o
Z
>>
o
X
,
o
Y
(47) becomes
(48)
W
w
l
o
Z
(1
-
b) W
h
o
Z
cos
y
-
(
o
X
sin
w
-
o
Y
cos
w)
cos
2
y
-
sin
2
y
sin
y
G
(
a, b, h,
a , A
C
, A
T
)
This equation becomes more clear by further letting
and
.
o
X
=
o
Y
=
0,
o
Z
=
1, A
C
=
A
T
h A
a
=
o
2
Then we have
(49)
W
w
l
(1
-
A)
o b
2
h
(1
-
b) I
xy
W
h
P cos
y
Recall that
is the inclination of the symmetry axis to the fixed-frame Z axis.
page 9 of 10

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