# Two body problem, deriving energy expression

1. Aug 31, 2016

### tomwilliam2

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. Aug 31, 2016

### drvrm

i think you should substitute for h in terms of mu , a and e and try to simplify

3. Sep 1, 2016

Thanks.