Math 21a Supplement on
Planetary Motion
Suppose that an object (planet, asteroid, whatever) travels through space under the
gravitational influence of a star.
Newton’s laws allow one to describe trajectory of the
planet.
This supplement describes the situation.
a)
Newton’s Laws
There is a natural coordinate system to describe the motion.
This is the so called
center of mass coordinate system.
(See the final section, f.)
In this coordinate system, the
position vector
r
(t) for the object at time t (the vector from the origin to the planet’s
position) evolves in time according to a differential equation of the following form:
d
dt
2
2
r
= 
κ
r
/r
3
.
(1)
Here,
κ
is a positive constant which is determined by the masses of the object and the star.
Also, r = 
r
 is the distance of the object from the origin.
To simplify the typing and notation, allow me to introduce the following shorthand:
•
d
dt
r
≡
r
´ .
•
d
dt
2
2
r
≡
r
´´ .
(2)
For example, Equation (1) reads
r
´´ = 
κ
r
/r
3
.
(3)
b)
The energy
The standard strategy for solving (3) is to find so called “constants of the motion”.
These are quantities which are constant along each trajectory.
The energy,
E = 
r
´
2
/2 
κ
/r .
(4)
is one such constant.
To show that E is constant, consider its time derivative under the
assumption that
r
(t) obeys (3).
One finds that
E´ =
r
´·
r
´´ +
κ
r
´·
r
/r
3
.
(5)
The fact that the latter is zero follows by taking the dot product of both sides of (3) with the
vector
r
´.
The derivation of (5) uses the fact that r = (
r
·
r
)
1/2
and that the time derivative of a
dot product obeys
(
A
·
B
)´ =
A
´·
B
+
A
·
B
´
(6)
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when
A
and
B
are both vector valued functions of time.
(Equation (6) can be verified by
writing out
A
·
B
= a
1
b
1
+ a
2
b
2
+ a
3
b
3
and then differentiating.)
You should think about what it means if the energy E is constant along a trajectory.
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 Fall '11
 GregoryJ.Galloway
 Math, Derivative, Multivariable Calculus, Force, Trajectory, λ2/r2

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