Finding Eccentricity of Orbit Given Masses, Positions, and Velocities

[jgoldst1]

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 = $\sqrt{1 + \frac{2 E L^2}{\mu k^2}}$
where
E = energy
L = $\mu r^2 \dot{\theta}$
μ = $\frac{m1m2}{m1+m2}$
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 $E = \frac{1}{2}μv^2- \frac{Gm1m2}{r}$ ?

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.

 

[faeriewhisper]

Hello!
One of my assignments for a discipline named planetary systems was to write a program and a paper about the orbit of Eris.
One of the tasks was to find the orbit's characteristics with only one arbitrary point of position and velocity.

Take a look ;)

My best regards, Iris.