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Find eccentricity of orbiting bodies given their position and velocity components

  1. Jul 3, 2012 #1
    1. The problem statement, all variables and given/known data
    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.

    2. Relevant 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

    3. 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.
     
  2. jcsd
  3. May 16, 2013 #2
    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.
     

    Attached Files:

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