Angular momentum in two body systems?

Question

Considering one body to be very heavy with respect to the other, the barycentre lies almost at te centre of the heavy body. So angular momentum being conserved, we can apply kepler's second law, can't we?
If we can, then r^2 * w = constant, so w is a function of r
Now this means that if the trajectory of the small body is eliptical, its angular velocity will change with time < as r will change>. Now for change in angular velocity we need angular acceleration. Angular acceleration will be caused due to some torque acting on the body.
How can gravitational force, being radial, produce a torque?
Where am I going wrong?

Answer 1

You're going wrong here: "Now for change in angular velocity we need angular acceleration. Angular acceleration will be caused due to some torque acting on the body."

It's only angular momentum that needs a torque to change it, not angular velocity. The gravitational force in this situation can't produce a torque, and in the absence of torque, angular momentum is constant. But that doesn't mean that angular velocity is constant: if the moment of inertia changes, the angular velocity changes so as to keep angular momentum constant.

One reason it's counterintuitive is that the corresponding change for linear momentum can't happen: mass is conserved. So you can't have a change of velocity in an object without a force acting on it and hence with constant momentum.

It may help to consider a zero force. As the object approaches the center, its angular velocity increases, without any acceleration at all.

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Edit: heh, an hour went by and then you beat me to the submit button by 19 seconds. Good answer, mine's a little verbose.

Answer 2

If you consider that in an elliptical orbit, the distance between the orbiting body and the central body is always changing, this is where gravitation increases and decreases tangential or instantaneous speed.

When an orbiting body is moving toward the perihelion/perigee point, it is actually falling toward the center of mass or barycenter. This gives you your gravitational component that increases velocity. It isn't a torque as that term is normally defined, that is a force or component of force applied at a distance from the center of mass. While it is true that gravitational force is always radial and directed toward the center of mass, this does not necessisarily correspond to the direction of travel of the object being acted upon. If you apply a force through the center of mass to a moving body, you introduce an acceleration that changes straight-line motion into a curve. There is your ellipse.

Draw an elliptical orbit, and at several points draw in these vectors for the orbiting body (free body diagram):
a) gravitational force
b) instantaneous velocity
c) radial and tangential acceleration

That should make it clearer. And yes, Kepler's law does apply, but keep in mind that this is just a quantitative description of elliptical / planetary motion. Kepler knew nothing about gravitation, or the causes of the elliptical motion we now know as orbits.

Answer 3

Actually, the angular momentum does not change; Kepler's second law implies as much. It states that an imaginary line drawn from the center of the sun to the center of the planet will sweep out equal areas in equal intervals of time. For small angles, the area approaches 1/2 r·(r·w·Δt); which is to say wr^2 (angular momentum) is constant. There is no change in angular momentum, and no applied torque.

Answer 4

A torque is a change in angular momentum, and not angular velocity. And since angular momentum in this case is constant (L=mwr^2, with m being the constant mass), it's time derivative (it's rate of change with time) is zero, and thus the torque is zero.

The angular velocity can and does change as you said, but is compensated for by the change in radial distance to produce the constant angular momentum.

Answer 5

In a two body system of rotation assuming that their motion is not relative to any other body are really in local space. The Rotation energy of the larger mass relative to the barry center must equal to the rotational energy of the Smaller body in order for orbital motion to continue.
The forces of each body relative to the barry center are not equal.(newton's third law in this case applies to energy only).

Torques refers to the product of a distance and a moving force.
Actually torque is equal to rotational motional energy. So for the Sun and Planet to balance each other in orbit their torque must be the same for each body.
Now in more then two body motion such as the nine Planets + other small celestial debries orbiting the Sun ,the following scenario occurs;
Each Planet has its own barry center relative to the Sun and all Planets are orbiting the sun all at the same time.
For all torques to balance at the same time for all Planets, the barry center becomes a dynamic moving frame of reference ,following a three dimensional trajectory.
Therefore whether its a three body or more ,the Gravitational Energy between them must balance al all time.

So the old idea that the Sun was revolving around the Earth was not such a far fetched idea. The Sun actually rotates around the barry center of each individual Planets all at the same time.
So the Sun actually orbits all the Planets not only the Earth. Never the less the Planets form a larger orbit.

The orbit of the Planets are eliptical because the Sun's mass changes continually ,as it orbit the Galaxy's barry Center.
This is what Keppler never expained to us.Why is the Earth following an eliptical orbit?He only expained to us that its an eliptical orbit.
Note; I had to elaborate a long time to figure this out.