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## Homework Statement

Given the parametric equations for a satellite in orbit around a spherical mass find angular momentum L in terms of ε, a, k, m, where k=GMm.

Also, find the energy E in the same terms.

Lastly, I can only use the equations provided and "fundamental definitions."

## Homework Equations

Equations provided:

y=a[itex]\sqrt{1-ε^2}[/itex]sinψ

x=a(cosψ-ε)

t=([itex]\frac{T}{2π}[/itex])(ψ-εsinψ)

[itex]\frac{T^2}{a^3}[/itex]=[itex]\frac{4π^2}{GM}[/itex]

What I think is necessary and otherwise fundamental:

L=

**r**x

**p**

Not yet sure about the equations for energy that would be considered fundamental...

## The Attempt at a Solution

I believe the first part is relatively straight forward even if it wouldn't otherwise be the most elegant way to go about the problem.

L=

**r**x

**p**

I can find

**r**using a

^{2}+b

^{2}=c

^{2}:

**r**

^{2}=x

^{2}+y

^{2}

**r**=[itex]\sqrt{x^2+y^2}[/itex]

**r**=[itex]\sqrt{a^2(cosψ-ε)^2+a^2(1-ε^2)sin^2ψ}[/itex]

v=[itex]\frac{2π}{t}[/itex] (edit: need correction here; this is average velocity... not sure how to do this without using theta; is the nasty relationship between theta and ψ fundamental?)

I'm given t so:

v=T(ψ-εsinψ)

**p**=m

**v**

**p**=mT(ψ-εsinψ)

I can solve for T from the given equation that contains it:

T=sqrt([itex]\frac{4π^2}{GMa^3}[/itex])

**p**=msqrt([itex]\frac{4π^2}{GMa^3}[/itex])(ψ-εsinψ)

Now I would just have to solve for ψ and replace it and get rid of the GM factor using k. Assuming I do my algebra and final cross product correctly does that seem like a sound solution to the angular momentum?

With regard to the energy portion I am a little weary about how to proceed given the requirement to use only fundamental definitions. I have come across a number of formulas for E of elliptical orbits that utilize kinetic energy and effective potential. Could it be as simple as:

E=[itex]\frac{1}{2}[/itex]mv

^{2}+[itex]\frac{GMm}{r}[/itex] and plug in the values for v and r that I already solved for?

Thank you for your help!

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