Time average of the potential energy of a planet

  • Thread starter Dustgil
  • Start date
  • #1
42
0

Homework Statement


(a) Prove that the time average of the potential energy of a planet in an elliptical orbit about the sun is -k/a.
(b) Calculate the time average of the kinetic energy of the planet.

Homework Equations



[tex]F = \frac {-dV} {dr} = - \frac {k} {r}[/tex]

The Attempt at a Solution


[/B]
So the first part is what's giving me trouble. k obviously doesn't vary, yet r does. So if we find the average radius of orbit we can therefore easily find the potential energy. I know that a is a length of the semimajor axis of the ellipse and that it makes sense that is would be the average radius, as it lies between the aphelion and perihelion. I see that if a is the semimajor axis and Ea is half the length of the distance between the foci then the aphelion is a + Ea and the perihelion is a - Ea, where E is the eccentricity. This is where I get stuck. I'm not sure how to directly relate this to the average distance. Anyone have a hint?

Part b is pretty easy really, since I've already shown the total energy E = -k / 2a. Just use conservation of energy and

[tex]
K - \frac {k} {a} = - \frac {k} {2a}[/tex]

[tex]K = \frac {k} {2a}[/tex]
 

Answers and Replies

Related Threads on Time average of the potential energy of a planet

  • Last Post
Replies
18
Views
2K
Replies
4
Views
6K
Replies
8
Views
5K
Replies
2
Views
2K
Replies
1
Views
2K
  • Last Post
Replies
16
Views
293
Replies
2
Views
4K
Replies
5
Views
2K
Top