# Derive Angular Momentum in Elliptical Orbit

1. Oct 31, 2008

### psychicist

1. The problem statement, all variables and given/known data
A planet with mass m is orbiting about the Sun (mass m(s)) with elliptical orbit such that the maximum and minimum distance is r1 and r2 respectively. What is its angular momentum, L, of the planet relative to the centre of the Sun?

2. Relevant equations
$$\vec{L}=\vec{r}\times\vec{p}=m(\vec{r}\times \vec{v})=mrv sin\theta$$
Semi-major axis, $$a=(r_1+r_2)/2$$
Semi-minor axis, $$c=(r_1-r_2)/2$$
Distance of the planet from the Sun, $$R=\frac{a(1-e^2)}{1-e cos\theta}$$
$$e=c/a$$

3. The attempt at a solution
Known that the angular momentum is conserved along the motion, but I am confused about how to relate the variables, i.e. r and v.

I have tried to derive R and tried to put into the equation of angular momentum but the formula seems make it more complicated as it introduce one more variable theta inside.

Can someone give me some idea? Thank you very much.

Last edited: Oct 31, 2008
2. Oct 31, 2008

### nasu

Try to use conservation of energy (kinetic plus potential) between the two points.
Write the speed in terms of the angular momentum:
L=m*v1*r1=m*v2*r2
Then solve for L.

3. Oct 31, 2008

### psychicist

Yes! I got it! thank you very much!

4. Oct 31, 2008

### nasu

You are welcome.