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I'm trying to show that the time average of the potential energy of a 2-body system is equal to the instantaneous potential energy of that system when the two bodies are separated by a distance equal to the semi-major axis, a.

So I know that

[tex]U = \frac{-Gm_{1}m_{2}}{r}[/tex]

and the time average of a function [tex]f(t)[/tex] over a time interval [tex]\tau[/tex] is defined to be

[tex] <f(t)>=\frac{1}{\tau}\int_0^{\tau} f(t) dt [/tex]

and

[tex]r = \frac{a(1-e^{2})}{1+ecos \theta}[/tex]

Can anyone tell me what's wrong with my attempt? Here it is:

[tex]<U>=\frac{1}{P} \int_0^P \frac{-Gm_1m_2}{r} dt = \frac{-Gm_{1}m_{2}}{2\pi a(1-e^2)}\int_0^{2\pi}(1+ecos\theta) d\theta

=\frac{-Gm_1m_2}{a(1-e^2)}

[/tex],

but I'm supposed to be verifying that

[tex]<U>=\frac{-Gm_1m_2}{a}[/tex]

I can't seem to figure out how to get rid of that factor of (1-e^2) in the denominator. I suppose this can be simplified by just asking why/how

[tex] <\frac{1}{r}>=\frac{1}{a}[/tex]

and not

[tex]\frac{1}{a(1-e^2)}}[/tex]

Thank you for your time