sat - Physics 53 Satellite Motion You know it's at times...

Physics 53
Satellite Motion
—You know, it's at times likes this when I'm stuck in a Vogon airlock with a man from
Betelgeuse, about to die from asphyxiation in deep space, that I wish I had listened to what my
mother told me when I was young.
— Why, what did she say?
—I don't know. I didn't listen.
— Hitchhiker's Guide to the Galaxy
Radial and angular motion
We will be concerned here with the orbiting motion of a satellite, such as a planet
around the sun, or the moon around the earth. We neglect the effects of all other bodies.
We will assume that the satellite's mass ( m ) is much smaller than that ( M ) of the body it
moves around. Both bodies actually orbit around the CM of the system, but M moves
only a small amount because the CM is so close to it. We ignore this motion as an
approximation, and take the center of M to be the origin of our coordinate system.
For a system with two masses of comparable size, such as a double star, obviously this is a bad
approximation and one must analyze the motion of both bodies around the CM.
Because gravity (a central and conservative force) is the only force doing work:
The total mechanical energy E of the system is conserved.
The gravitational force acts along the line between the two bodies, so there is no torque
about the origin. This means that:
The total angular momentum L of the system about the origin is conserved.
This is still true if the CM is taken as origin even when we take account of the motion of both bodies.
We will find that the nature of the trajectory is determined by the values of these two
conserved quantities.
The velocity of the satellite has in general two components, one parallel and one
perpendicular to the position vector r of the satellite. We call these the radial and
tangential velocities and write
v = v
r
+ v
⊥
.
The kinetic energy then breaks into two parts:
PHY 53 1 Satellite Motion

K =
1
2
mv
r
2
+
1
2
mv
⊥
2
.