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Energy in elliptic orbits 
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#1
Oct205, 07:16 AM

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P: 481

How does one derive the total energy in an elliptic orbit:
[tex]E=  \frac{GMm}{2a}[/tex] where a is the semimajor axis? I did manage to get the result for the special case of circular orbit, as [tex]v = \sqrt{\frac{GM}{R}}[/tex] But the problem is that I can't figure out a way to express v in an elliptic orbit. If at all possible, give hints (that is: not a direct answer), as I'd rather try it myself first :). 


#2
Oct205, 08:11 AM

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P: 481

Not really, no.
I can't get the given equation: [tex]v=\sqrt{2\mu\left({1\over{r}}{1\over{2a}}\right)}[/tex] without assuming the result (total energy) I'm trying to get. EDIT: Either I am getting paranoid or someone replied, but deleted his/her message :). 


#3
Oct205, 08:16 AM

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P: 876

Would a formula for elliptical velocity be too much help ?



#4
Oct205, 08:18 AM

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P: 481

Energy in elliptic orbits
I can get it by assuming [tex]E=  \frac{GMm}{2a}[/tex] but that's the equation I want to prove. 


#5
Oct205, 08:22 AM

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#6
Oct405, 03:06 PM

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P: 481

Help, anyone?



#7
Oct405, 10:36 PM

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P: 6,671

[tex]E(r) = KE + PE = \frac{1}{2}mv^2  \frac{GMm}{r} = \frac{1}{2}mv_t^2 + \frac{1}{2}mv_r^2  \frac{GMm}{r}[/tex] Use the fact that the radial KE (middle term) is 0 when r is maximum or minimum (ie. when r = a or r=b) AM 


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