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# 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. 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)

as 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.   Next: Kepler's first law Up: Keplerian orbits Previous: Plane polar coordinates
Richard Fitzpatrick 2016-03-31