# I'm confused about 1st Kepler's law and the 2 body problem

1. Dec 13, 2008

### fluidistic

Hi PF,
According to Kepler's first Law any planet in orbit of the Sun describes an elliptical orbit.
I don't know why I can't go on wikipedia since days. But if I remember well I've seen that when there are 2 bodies like 2 stars for example then the center of mass will remain quiet (since there's no external force) while the 2 bodies would rotate around it.
So I'm confused and don't understand why the motion of a satellite should be an ellipse. Probably due to conservation of angular momentum. But then I don't understand why the 2 bodies are rotating around their barycenter.
Or maybe it's a combination of the 2 situation. I'm all confused. Could you please explain to me what's happening?

2. Dec 13, 2008

### covariance

Any two-body attractive potential problem is equivalent to that of a single body in a central potential $$V(r)$$ (i.e. independent of angle $$\theta$$). Since the potential is spherically symmetric, angular momentum is conserved. Since the original potential is a function of $$\frac{1}{r_c^2}$$ (where $$r_c$$ denotes the distance between the centers of mass of the two bodies), all radially bounded trajectories are ellipses. The analysis is quite lengthy and can be found in Goldstein's Classical Mechanics. I think even Landau's Mechanics has a derivation.

EDIT: Ah crap, I messed up my TeX. I'll fix it asap.

Last edited: Dec 13, 2008
3. Dec 13, 2008

### fluidistic

Thank you covariance. I didn't study this stuff yet and I'll do that in my 3rd year.
If I understood well the 2 bodies are rotating around their center of mass and it's equivalent to say that one of them has an elliptical orbit around the other body.
EDIT : I could enter in wikipedia. Now I visualize it, it's true! I mean if one chose as origin a body then the other body seems to orbit elliptically. While in fact both bodies are rotating around the barycenter of the 2 bodies. Very nice indeed.

Last edited: Dec 13, 2008