17.3.3 Kepler’s Third Law
Kepler’s third law involves the time it takes for the planet to orbit the sun. From 17.13
you can complete the square and obtain
and this yields
Now note this is the equation of an ellipse and that the diameter of this ellipse
This follows because
Now let T denote the time it takes for the planet to make one revolution about the sun.
It is left as an exercise for you to show that the area of an ellipse whose long axis is 2a
and whose short axis is 2b is πab. This is an exercise in trig. substitutions and is a little
tedious but routine. Using this formula, and 17.3 the following equation must
Now using 17.12, recalling that k = GM, and 17.15,
Written more memorably, this has shown
This relationship is known as Kepler’s third law.