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

[antythingyani]

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.

[Image re-inserted with correct orientation by Mentor]

 

[PeroK]

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.

 

[PeroK]

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.

 

[PeroK]

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.

 

[PeroK]

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

 

[PeroK]

... which it does.

 

[antythingyani]

... which it does.

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

 

[PeroK]

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.