R.G. Melton
AERSP 309
HW #12
Fall 2005
(Practice Problems only – not to hand in)
1.
A satellite in low Earth orbit is oriented as shown.
The s/c is oriented such that its
2
1
ˆ
ˆ
b
b
−
plane
coincides with the orbital (
θ
i
i
r
ˆ
ˆ
−
) plane.
a.
Develop an expression for the gravity gradient torques in terms of the angle
.
b.
For the case of small librations (oscillations) about only the 3axis and
I
3
>I
2
>I
1
, and
assuming that
remains small, show that
0
2
3
=
+
k
I
±
±
, where
k
is a constant.
c.
The equation above describes simple harmonic motion about the 3axis.
Find an
expression for the period of this librational motion, and evaluate it for the case of
R
= 7000
km,
I
1
=
3000,
I
2
= 5000,
I
3
= 7500 kg
⋅
m
2
.
d.
Suppose that this satellite were placed into an elliptical orbit with period equal to that of
the librational motion.
What is the consequence?
(Hint:
don’t ever do this)
2.
If the specific impulse of a given propellant combination is
I
sp
= 450 s. and the s/c has initial mass
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 Fall '05
 MELTON
 Planet, Spacecraft propulsion, propellant combination, radiator fin

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