Law II: Law of Equal Areas
What does Kepler's second law tell us about the motion
of the planets?
From the Figure, we see that
area ~ 1/2 D x d so that
D'/d' = D/d
Now, a planet will move a distance d (or d') determined by how
fast it travels. In particular, d = speed x time = v x t. We then have
that
D' ~ v' x time and D ~ v x time
By assumption, the times are the same and so we have that
D'/v' = D/v
===> v' / v = D / D'
This shows that the speed of a planet in orbit varies depending upon
how far away the Sun the planet is. When a planet is close to the Sun,
it moves faster than when it is farther away from the Sun. For the
orbit of the Earth, we have the eccentricity of e = 0.017. This small
eccentricity means that the Earth travels at a speed which 3 % (or so)
greater when the Earth is closest to the Sun ( perihelion) as
compared to when it is farthest from the Sun ( aphelion).