# Angular momentum and eccentricity

1. Jul 11, 2012

### Alexrey

I'm trying to find the relationship between angular momentum and orbital eccentricity but so far I haven't really found anything. I did find an indirect relationship, though, which looked like it should come out to,$$L=\sqrt{\frac{a(1-e^{2})}{m_{1}+m_{2}}},$$ but I may be completely wrong. Anyone know the correct answer?

2. Jul 11, 2012

### Staff: Mentor

Even if you remove as many variables as possible, orbits always have two degrees of freedom, they can be written as semi-major axis and eccentricity. The angular momentum will depend on both, together with the masses and the gravitational constant.

I found this formula at wikipedia:

The energy is $E = \frac{-G(M+m)}{2a}$

Therefore, $e^2 = 1-\frac{c}{a} L^2$ with $c=\frac{M+m}{GM^2m^3}$ and $L=\sqrt{\frac{1-e^2}{ca}}$ where c just depends on the masses.

3. Jul 16, 2012

### Alexrey

Thanks very much.