# Time average of the potential energy of a planet

## 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

$$F = \frac {-dV} {dr} = - \frac {k} {r}$$

## The Attempt at a Solution

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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

$$K - \frac {k} {a} = - \frac {k} {2a}$$

$$K = \frac {k} {2a}$$