Good morning. Bobby, please read Kepler’s Second Law of Planetary Motion.
♪ Flipping Physics ♪ Each planet moves such that an imaginary
line drawn between the sun and the planet sweeps out equal areas in equal time periods.”
Thank you Bobby. In order to prove Kepler’s Second Law that states that all the areas
in the illustration are equal, we need to start by proving that the angular
momentum of a planet about the axis of rotation of the Sun is conserved. We
have actually already done this before, so Bo, could you please walk us through that?
Uhhhhh… It starts with
net external torque equals the derivative of angular momentum with respect to time.
Yeah… So, the net external torque
acting on the planet with the axis of rotation of the sun equals just torque caused by the
force of gravity acting on the planet because that is the only force acting on the planet.
Sure And the torque caused by the force of
gravity equals the cross product of the r position vector and the force of gravity which equals zero
because the area of the parallelogram created by those two vectors equals zero.
And therefore, the derivative of the angular momentum of the planet with respect to time
equals zero which means the angular momentum of the planet about the axis of rotation of the Sun
is constant; angular momentum is conserved. Thank you Bo.
(Yeah. Uh huh. Right.) Bobby, please determine the angular momentum
of the planet, which we now know is constant. Well, angular momentum equals the cross product
of the r position vector and linear momentum. Linear momentum equals mass times velocity. So,
I guess the angular momentum of a planet with the axis of rotation at the center of the Sun
equals the mass of the planet times the cross product of the planet’s r position and velocity.
Correct Bobby. And we can rearrange that to get that the cross product of the r position
vector and the velocity of the planet equals the angular momentum of the planet divided by the mass
of the planet. We already showed that the angular momentum of the planet is constant,
and we know the mass of the planet is constant, therefore, we now know that the cross product
of the r position vector and velocity of the planet is constant. This will be helpful a bit
later in this derivation. Now let’s discuss the concept of “equal areas in equal time periods”.
Over an infinitesimally small time period dt, the planet has moved an infinitesimally small
displacement dr. And, because velocity equals the derivative of position with respect to time,
we know displacement dr equals velocity times time period dt. Area dA is the area swept out
over the time period dt and the area of dA equals half the area of the cross product of
the r position vector and displacement dr. We can substitute velocity times time period dt
in for displacement dr. Time period dt is a scalar and we can remove that from the cross product
and divide the whole equation by time period dt. Therefore, we have just shown that the derivative,
with respect to time, of the area swept out by the planet equals one half the cross product of the r
position vector and the velocity of the planet. And, earlier we proved that to be constant,
therefore we know this whole equation is constant. Which means the area swept out by a planet will
be the same in equal time periods. We have just proved Kepler’s Second Law of Planetary Motion!
The derivative, with respect to time, of the area swept out by a planet
is constant. That means the rate of change of the area is constant which means,
for equal time periods, the area swept out by the planet will be the same. Okay. That makes sense.
Why is the shape of area dA different than the shapes in the original
illustration showing Kepler’s Second Law? Uh, he must have exaggerated the length of
displacement dr so we could see it in the illustration. Displacement dr is actually
infinitesimally small, which means area dA is also infinitesimally small. Each of the areas in
the original illustration is the summation of an infinite number of infinitesimally small areas dA
and therefore those areas in the illustration will not have the same shape as area dA.
Sure, that makes sense. Thanks Bobby. In our next lesson, we will learn
about Kepler’s Third Law of Planetary Motion. (That still seems logical.)
Thank you very much for learning with me today, I enjoy learning with you.