- #1
tomwilliam2
- 117
- 2
Homework Statement
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##
Homework 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}##
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...