Further Insights and Challenges
Exercises 23 and 24 prove Kepler’s Third Law. Figure 10 shows an elliptical orbit with polar equation
where p = J2/k. The origin of the polar coordinates is at F1. Let a and b be the semimajor and semiminor axes, respectively.
Question 13.350
This exercise shows that 
.
Show that CF1 = ae. Hint: rper = a(1 − e) by Exercise 13.
772
Show that
.Show that F1A + F2A = 2a. Conclude that F1B + F2B = 2a and hence F1B = F2B = a.
Use the Pythagorean Theorem to prove that
.
Question 13.351
The area A of the ellipse is A = πab
Prove, using Kepler’s First Law, that
, where T is the period of the orbit.Use Exercise 23 to show that
.Deduce Kepler’s Third Law:
.
Question 13.352
According to Eq. (7) the velocity vector of a planet as a function of the angle θ is
Use this to explain the following statement: As a planet revolves around the sun, its velocity vector traces out a circle of radius k/J with center at the terminal point of c (Figure 11). This beautiful but hidden property of orbits was discovered by William Rowan Hamilton in 1847.
