1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Two body problem, deriving energy expression

  1. Aug 31, 2016 #1
    1. The problem statement, all variables and given/known data
    This problem relates to the two body problem of two rotating point masses, where one is much larger than the other. Equate the orbital energy per unit mass ##\epsilon## with the moments of momenta at the apses to get:
    ##\epsilon = -\mu/2a##

    2. Relevant equations
    The orbital energy per unit mass is ##\epsilon = \frac{1}{2}V^2 - \frac{\mu}{r}## where I think ##\mu = GM##
    Moment of momentum in massless form is, I think, ##\mathbf{h}=\mathbf{r \times}\mathbf{v}##

    3. The attempt at a solution
    The apses are the periapsis ##r_p = a(1-e)## and the apoapsis ##r_a = a(1+e)##. I can insert these into the energy equation above to get:
    ##\epsilon = \frac{1}{2}V_p^2 - \frac{\mu}{a(1-e)}=\frac{1}{2}V_a^2 - \frac{\mu}{a(1+e)}## (1)

    I know that the position and velocity vectors are always perpendicular, so
    ##h = rv##
    ##h^2 = r^2v^2##
    I can use the values of ##r## at the apses (and conservation of h) to get:
    ##\frac{h^2}{a^2(1-e)^2}=V_p^2##
    ##\frac{h^2}{a^2(1+e)^2}=V_a^2##
    Now inserting these values into the energy equation (1) above, I get:
    ##\epsilon = \frac{h^2}{2a^2(1-e)^2} - \frac{\mu}{a(1-e)}=\frac{h^2}{2a^2(1+e)^2} - \frac{\mu}{a(1+e)}##

    Now I'm a bit stuck...
     
  2. jcsd
  3. Aug 31, 2016 #2
    i think you should substitute for h in terms of mu , a and e and try to simplify
     
  4. Sep 1, 2016 #3
    Thanks.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Two body problem, deriving energy expression
Loading...