572
CHAPTER 14
CALCULUS OF VECTORVALUED FUNCTIONS
(ET CHAPTER 13)
SOLUTION
Kepler’s Third Law states that the period
T
of the orbit is given by:
T
2
=
Ã
4
π
2
GM
!
a
3
or
T
=
2
√
a
3
/
2
If
a
is increased fourfold the period becomes:
2
√
(
4
a
)
3
/
2
=
8
·
2
√
a
3
/
2
That is, the period is increased eightfold.
Exercises
1.
Kepler’s Third Law states that
T
2
/
a
3
has the same value for each planetary orbit. Do the data in the following table
support this conclusion? Estimate the length of Jupiter’s period, assuming that
a
=
77
.
8
×
10
10
m.
Planet
Mercury
Venus
Earth
Mars
a
(10
10
m)
5.79
10.8
15.0
22.8
T
(years)
0.241
0.615
1.00
1.88
Using the given data we obtain the following values of
T
2
/
a
3
,whe
re
a
, as always, is measured not in
meters but in 10
10
m:
Planet
Mercury
Venus
Earth
Mars
T
2
/
a
3
2
.
99
·
10
−
4
3
·
10
−
4
2
.
96
·
10
−
4
2
.
98
·
10
−
4
The data on the planets supports Kepler’s prediction. We estimate Jupiter’s period (using the given
a
)a
s
T
≈
√
a
3
·
3
·
10
−
4
≈
11
.
9 years.
2.
A satellite has initial position
r
=
h
1
,
000
,
2
,
000
,
0
i
and initial velocity
r
0
=
h
1
,
2
,
2
i
(units of kilometers and
seconds). Find the equation of the plane containing the satellite’s orbit.
Hint:
This plane is orthogonal to
J
.
The vectors
r
(
t
)
and
r
0
(
t
)
lie in the plane containing the satellite’s orbit, in particular the initial position
r
= h
1000
,
2000
,
0
i
and the initial velocity
r
0
= h
1
,
2
,
2
i
. Therefore, the cross product
J
=
r
×
r
0
is perpendicular to the
plane. We compute
J
:
J
=
r
×
r
0
=
¯
¯
¯
¯
¯
¯
ij
k
1
,
000
2
,
000
0
12
2
¯
¯
¯
¯
¯
¯
=
¯
¯
¯
¯
2
,
000
0
22
¯
¯
¯
¯
i
−
¯
¯
¯
¯
1
,
000
0
¯
¯
¯
¯
j
+
¯
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 Winter '08
 GANGliu
 Astronomical unit, Kepler's laws of planetary motion, Celestial mechanics, Semimajor axis

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