Chapter 8.2, Problem 72E

Solution

a.

To define: How Kepler discovers his three laws of planetary motion.

Explanation:

Johannes Kepler was able to observe the planets' orbits, which adhere to three laws with reference to the planet Mars, while working as Tycho Brahe's rich assistant.

The first two laws were found by Kepler after he realized that an imaginary line traced from a planet to the Sun covered the same amount of space in the same amount of time (2nd law), and that in order for this to be true, a planet's orbit had to be elliptical (1st law).

Also demonstrated was the connection between a planet's distance from the Sun and how long it takes for it to complete one orbit of the Sun (3rd law).

b.

To define: The Kepler’s second law.

Kepler's second law states that a ray from the sun to a planet sweeps out equal areas in equal times.

Explanation:

Kepler's second law states that a ray from the sun to a planet sweeps out equal areas in equal times.

  

From the above diagram, at same time $T$

  $A_1=A_2$