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' = Dd

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'time = Dvtime

By assumption, the times are the same and so we have that
D'v' = Dv ===> v' / v = D / D'

This implies 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). Comment--for planets with small eccentricity, e pretty close to zero, we have that

v(peri)/v(ap) ~ 1 + 2e