Angular momentum in eliptical orbits

• arunma
In summary, the orbit is not conserved if you take the center of the ellipse as your reference point.

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.

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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.

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

Am I wrong? How could it be conserved?

Doc Al said:
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?

arunma said:
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.

Doc Al said:
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.

1. What is angular momentum in eliptical orbits?

Angular momentum is a measure of the rotational motion of an object around a fixed point. In the context of eliptical orbits, it refers to the momentum of a celestial body as it moves around a central point, such as a planet orbiting a star.

2. How is angular momentum related to the shape of an orbit?

The shape of an orbit, whether it is circular or eliptical, is directly related to the angular momentum of the orbiting body. The more elliptical the orbit, the greater the angular momentum will be.

3. Can angular momentum change in an eliptical orbit?

Yes, angular momentum can change in an eliptical orbit. This can occur due to external forces, such as gravitational pull from other objects, or internal forces, such as the release of gas or particles from the orbiting body.

4. What is the role of angular momentum in Kepler's laws of planetary motion?

Angular momentum plays a crucial role in Kepler's laws of planetary motion. The second law, also known as the law of equal areas, states that a line connecting a planet to the sun will sweep out equal areas in equal times. This is possible because the angular momentum of the planet remains constant as it moves around the sun.

5. How is angular momentum calculated in an eliptical orbit?

In an eliptical orbit, angular momentum is calculated by multiplying the mass of the orbiting body by its velocity and the distance from the central point. This is represented by the formula L = mvr, where L is angular momentum, m is mass, v is velocity, and r is the distance from the central point.