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Kepler Again

A more formal example of the importance of the Law of Conservation of Angular Momentum under Central Forces is in its application to Celestial Mechanics, where the gravitational attraction of the Sun is certainly a classic central force. If we always use the Sun as our origin  O, neglecting the influence of other planets and moons, the orbits of the planets must obey Conservation of Angular Momentum about the Sun. Suppose we draw a radius  r  from the Sun to the planet in question, as in Fig. 11.6.

  

Figure: A diagram illustrating the areal velocity of an orbit. A planet (mass  m) orbits the Sun at a distance  r. the shaded area is equal to   ${1\over2} r \times r \, d\theta$  in the limit of infinitesimal intervals [ i.e. as   $d\theta \to 0$]. The areal velocity [ rate at which this area is swept out] is thus   ${1\over2} r^2 d\theta/dt \, = \, {1\over2} r^2 \omega$.

The rate at which this radius vector "sweeps out area" as the planet moves is   ${1\over2} r^2 \omega$, whereas the angular momentum about the Sun is   $m r^2 \omega$. The two quantities differ only by the constants  ${1\over2}$  and  m; therefore Kepler's empirical observation that the planetary orbits have constant "areal velocity" is equivalent to the requirement that the angular momentum about the Sun be a conserved quantity.