# Derive angular momentum of planet with elliptical orbit

1. Oct 31, 2008

### psychicist

1. The problem statement, all variables and given/known data
A planet of mass m orbiting the Sun(mass=$$m_s$$ along an elliptical orbit, with aphelion r1 and perihelion r2. Find the angular momentum of the planet relative to the centre of the Sun.

2. Relevant equations
Angular momentum, $$M=m(R\times v)=mvR sin\theta$$
Distance from Sun to the planet, $$R=\frac{2{r_1}{r_2}}{(r_1+r_2)-(r_1-r_2)cos \theta}$$
while $$\theta$$ is the angle of $$\overrightangle{R}$$ from the semi-major axis, which varies with time.

3. The attempt at a solution
Known that the angular momtentum is conserved along its motion, with v, R, and \theta varies with time but not sure that whether it is useful to derive the value of R, and I don't know how to relate this 3 variables together. Can anyone help me? Thank you very much!

2. Nov 2, 2008

### Irid

Oh just use energy conservation to equate energies at perihelion and aphelion and you'll be able to get a velocity at either point. Plug it into the corresponding angular momentum formula!