Time average of the potential energy of a planet

In summary, the conversation discusses how to prove that the time average of the potential energy of a planet in an elliptical orbit around the sun is -k/a and how to calculate the time average of the kinetic energy of the planet. The conversation also mentions using the semimajor axis and eccentricity to find the average distance and the use of the virial theorem to solve for the kinetic energy.
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Dustgil
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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]
 
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Related to Time average of the potential energy of a planet

1. What is the time average of the potential energy of a planet?

The time average of the potential energy of a planet is the average value of the potential energy over a certain period of time. It takes into account the changes in potential energy due to the planet's motion and the influence of external forces.

2. How is the time average of the potential energy of a planet calculated?

The time average of the potential energy of a planet can be calculated by taking the integral of the potential energy function over a specific time interval and dividing it by the length of the interval.

3. Why is it important to consider the time average of the potential energy of a planet?

The time average of the potential energy of a planet allows us to account for the variations in potential energy that occur due to the planet's motion. This is important in understanding the overall energy balance of a planet and its behavior over time.

4. Does the time average of the potential energy of a planet change over time?

Yes, the time average of the potential energy of a planet can change over time as the planet's motion and external forces change. This is why it is necessary to calculate the average over a specific time interval in order to get an accurate representation of the planet's potential energy.

5. How is the time average of the potential energy of a planet related to its kinetic energy?

The time average of the potential energy of a planet and its kinetic energy are both components of the total energy of the planet. As the potential energy of a planet changes, so does its kinetic energy, and vice versa. The time average of both energies together gives us a better understanding of the overall energy balance of the planet.

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