Kepler's second law
Multiplying our planet's tangential equation of motion, Equation (4.19),
by , we obtain

(4.20) 
However, this equation can be also written

(4.21) 
which implies that

(4.22) 
is constant in time. It is easily demonstrated that is the magnitude
of the vector defined in Equation (4.6). Thus, the fact that
is constant in time is equivalent to the statement that the angular
momentum of our planet is a constant of its motion. As we have already mentioned, this is the case
because gravity is a central force.
Figure 4.2:
Kepler's second law.

Suppose that the radius vector connecting our planet to the origin (i.e., the Sun) rotates through an angle
between times and
. See Figure 4.2. The approximately triangular region swept out by the radius vector has the area

(4.23) 
because the area of a triangle is half its base (
) times its
height (). Hence, the rate at which the radius vector sweeps out area
is

(4.24) 
Thus, the radius vector sweeps out area at a constant rate (because is
constant in time); this is Kepler's second law of planetary motion. We conclude that Kepler's
second law is a direct consequence of angular
momentum conservation.