Lesson Plan on Kepler's Laws of Planetary Motion

Objectives: Students will be able to

  1. state and understand Kepler's three laws of planetary motion.
  2. define the parts of an ellipse and construct an ellipse.
  3. calculate eccentricity of an ellipse.
  4. use a planet's eccentricity to construct its orbit.
  5. use Kepler's third law to calculate the period of revolution or the measure of the semimajor axis.

Grade Levels: 8-12

Background Information:

Kepler's first law states that the orbits of the planets and other celestial bodies around the Sun are ellipses. An ellipse is defined as a figure drawn around two points called the foci such that the distance from one focus to any point on the figure back to the other focus equals a constant.

[Diagram of an Ellipse]

This constant is the measure of the long diameter of the ellipse, the major axis. Half of this segment is called the semimajor axis. The short diameter, the minor axis, is a perpendicular bisector of the major axis. Half of the minor axis is called the semiminor axis. For planets, the Sun is at one focus, nothing is at the other.

The eccentricity of an ellipse is a measure of its flatness. Numerically, it is the distance between the foci divided by the length of the major axis. The following is a series of ellipses having the same major axis but different eccentricities:

[Eccentricity Diagram]

As the eccentricity approaches 1, the ellipse approaches a straight line. As the eccentricity approaches 0, the foci come closer together and the ellipse becomes more circular. A circle has an eccentricity of zero.

Kepler's second law states that a line from the planet to the Sun sweeps over equal areas in equal amounts of time. These areas in the ellipse are called sectors. In the following diagram, as the planet moves from point A to point B along its orbit, a long, skinny sector is created.

[Orbital Ellipse Diagram]

If we wanted to create a sector of equal area at points closer to the Sun (points C and D), the result is a short, fat sector. According to Kepler, the time it takes for the planet to get from A to B is equal to the time it takes the planet to get from C to D. This means that a planet orbits slower as it moves further from the Sun.

Kepler's third law deals with the length of time a planet takes to orbit the Sun, called the period of revolution. The law states that the square of the period of revolution is proportional to the cube of the planet's average distance to the sun:

P2=a3.
Because of the way a planet moves along its orbit, its average distance from the Sun is half of the long diameter of the elliptical orbit (the semimajor axis.) The period, P, is measured in years and the semimajor axis, a, is measured in astronomical units (AU), the average distance from the Earth to the Sun.

An example for using this formula would be to calculate how long it takes the near-Earth asteroid called Eros to orbit the Sun. The closest distance to the Sun that Eros orbits is 1.13 AU, and the farthest away from the Sun that it orbits is 1.78 AU. So, the average distance from Eros to the Sun, the semimajor axis, is (1.13 + 1.78)/2 = 1.46 AU. Substituting this in for a in the formula

P2=a3
and solving for P we see that it takes Eros about 1.76 years to orbit the Sun.

Activities

Optional Activity 1: Constructing Orbits

No Frames Table of Contents

[LMGFP home page] Contact Karen Krupinsky (kgurley@gsfc.nasa.gov) or
Tammy Seergae (tseergae@umd.edu) for further information.