Kepler's laws

Essential to proving Kepler's second law (and further laws) is the concept of torque. A torque is a tendency to change something's state of rotation; it is the rotational analogue of force. For instance, if I apply torque to a wheel, I'm providing a tendency to rotate that wheel. Torque is in rotational mechanics what force is in linear mechanics.

Torque, tau can be employed as

tau = r cross F equation 1
where F is the impressed force and r is the lever arm over which it acting; that is, the vector that begins at the axis of rotation and ends at the point where the impressed force is acting. Note that torque is a vector quantity; this means that it has a direction. The direction of the torque indicates in which direction the body tends to rotate. That doesn't seem very directly related to celestial mechanics, does it? But while torque is usually applied to rigid bodies, such as wheels and levers, it does not have to be. The concept of torque can be applied to any body with respect to a fixed point in space. The vector between this fixed point and the body then becomes the lever arm, although it is by no means a solid one.

We shall apply this notion of torque to a planet orbiting the Sun. Here, however, the impressed force will be gravity. Our fixed reference point will be the Sun itself. We know that r = r r and F = (-G m M/r2) r so we can get

tau = r cross F equation 2
tau = (r r) cross [(-G m M/r2) r] equation 3
tau = (-G m M/r) (r cross r) equation 4
tau = o. equation 5
We know that any vector crossed with itself is the zero vector, o, so the Sun never impresses a torque on a planet. This makes perfect sense: if you can only pull radially on bucket (as the Sun can only pull radially on a planet), you won't be giving the bucket rotation about an axis a tendency to speed up or slow down in its rotation.
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