Angular momentum in eliptical orbits

[arunma]

OK, I'm sure I must be doing something wrong here, because I've run into a seeming contradiction. Please refer to the image I've attached below: an ellipse with the center and a focus marked. If there is a massive body at point P, then an object can orbit in the ellipse shown. If I know the velocity of the object at point B, and want to compute the velocity at point A, I can use conservation of angular momentum,

$$mv_{A}r_{PA} = mv_{B}r_{PB}$$

$$v_{A} = v_{B} \dfrac{r_{PB}}{r_{PA}}$$

Here, $$r_{PA}$$ and $$r_{PB}$$ are the distances from the focus to points A and B respectively.

However, if I switch coordinate systems and measure angular momentum with respect to the center of the ellipse, then I can write,

$$mv_{A}a = mv_{B}a$$

$$v_{A} = v_{B}$$

Where a is the semimajor axis of the ellipse.

OK, clearly I can't do this, since switching coordinate systems gives me a different result. Can someone explain why the latter is an invalid means to solve this problem? Thanks.

 

[Doc Al]

The reason why taking the focus (P) as your reference point works so easily is that the gravitational force on the orbiting body never exerts a torque about that point. (The force is radial towards P.) That's why angular momentum about point P is conserved. But if you take the center of the ellipse as your reference point, angular momentum is no longer conserved.

 

[cesiumfrog]

But if you take the center of the ellipse as your reference point, angular momentum is no longer conserved.

You're sure about that?

 

[Doc Al]

Am I wrong? How could it be conserved?

 

[arunma]

The reason why taking the focus (P) as your reference point works so easily is that the gravitational force on the orbiting body never exerts a torque about that point. (The force is radial towards P.) That's why angular momentum about point P is conserved. But if you take the center of the ellipse as your reference point, angular momentum is no longer conserved.

Thanks, this makes qualitative sense. But one point still confuses me. If we take O as the origin, then the net torque on the orbiting body is equal to the cross product of the radius vector and the force. And since the force is still antiparallel to the radius vector, the net torque on the body should still be zero.

Am I still missing something here?

 

[Doc Al]

And since the force is still antiparallel to the radius vector, the net torque on the body should still be zero.

They are only antiparallel at those two points in the orbit. In general, they are not. As the body goes from A to B, gravity exerts a torque about point O.

 

[arunma]

They are only antiparallel at those two points in the orbit. In general, they are not. As the body goes from A to B, gravity exerts a torque about point O.

Oh, I understand now. Thank you.