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jgoldst1
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Homework Statement
Find the eccentricity of an orbit given the masses, cartesian position components, and cartesian velocity components for particles 1 and 2. The case is reduced to the xy plane.
Homework Equations
I am attempting this problem using reduced mass from the center of mass frame.
ε = [itex]\sqrt{1 + \frac{2 E L^2}{\mu k^2}}[/itex]
where
E = energy
L = [itex]\mu r^2 \dot{\theta}[/itex]
μ = [itex]\frac{m1m2}{m1+m2}[/itex]
k = Gm1m2
r = distance between the two particles
The Attempt at a Solution
I have two general questions. 1) Is the method below correct? If no, I would appreciate guidance to correct the method. 2) If there a better method?
If I knew the velocity, energy, and angular momentum of the reduced mass "particle", I could input the information into the relevant equation.
Is the velocity v of the reduced mass "particle" the difference between the velocities of particles 1 and 2? Similarly, is the position r of the "particle the difference between the positions of particles 1 and 2?
Given the velocity, would the energy of the "particle" be [itex] E = \frac{1}{2}μv^2- \frac{Gm1m2}{r} [/itex] ?
Would the angular momentum L of the "particle" be μ* r x v? Where I would take the cross product of the "particle's" position and velocity components then find the square L^2?
Thank you.