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.

The velocity vector traces out a circle as the planet travels along its orbit.