# Averaging the cube of semimajor axis to position ratio wrt to time

antythingyani
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Summary:: Averaging (a power of) semimajor axis to position ratio wrt to time - celestial mechanics

I evaluated it this far, but i don't know how to change the dt to d theta ... the final solution is

supposedly (1-e^2)^-(3/2) . Any help will be appreciated.

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It's not very easy to read your post or to understand precisely what question you are asking.

antythingyani
It's not very easy to read your post or to understand precisely what question you are asking.
The question is basically : derive the average of (a/r)^3 taking time as an independent variable. (where a is the semi major axis and r is the distance in an elliptical keplerian orbit.

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The question is basically : derive the average of (a/r)^3 taking time as an independent variable. (where a is the semi major axis and r is the distance in an elliptical keplerian orbit.
Given that we cannot easily express ##r## as a function of time, what is your strategy?

antythingyani
we can express r as a function of semimajor axis (a) and the true anomaly (theta). In that case maybe finding a way to turn dt into dtheta would be handy... or trying to express theta as a function of t.

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we can express r as a function of semimajor axis (a) and the true anomaly (theta). In that case maybe finding a way to turn dt into dtheta would be handy... or trying to express theta as a function of t.
I don't think you can get ##\theta## as a function of ##t## either. We have: $$\frac{d\theta}{dt} = \frac{L}{mr^2}$$But I don't know that solves the problem. There might be some trick using Kepler's law.

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... we also have $$\frac{a(1 - e^2)}{r} = 1 + e\cos \theta$$ Perhaps that does the trick?

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... which it does.

antythingyani
antythingyani
... which it does.
The 1-e^2 terms cancel and then again we will be left with a/r isn't it?!?

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The 1-e^2 terms cancel and then again we will be left with a/r isn't it?!?
I'm not sure what you mean by that. I used my notes on the derivation of elliptical orbits to find the relevant equations. This looks like a tricky problem where you'll need to do the same. I've given you the two equations to get you started.

At this level, I think you need to learn a little Latex:

https://www.physicsforums.com/help/latexhelp/

If you reply to my posts you'll see what I've typed to render the mathematics.

berkeman