Kepler's Law

Time requirement: 50 minutes as an activity for older or mathematically advanced students.

Materials

• Calculator

Procedures

In order to make your calculations, you will use one of Kepler's Laws. In the early 1600's, Johannes Kepler published three laws that govern planetary motion based on observations of the planets by Tycho Brahe. The law that is of interest to us right now says that the square (2nd power) of the period of the orbit for any planet orbiting the Sun is proportional to the cube (3rd power) of the average distance between the planet and the Sun.

With respect to Earth, Kepler's law says that if T_p and R_p are the orbital period of a planet and the mean distance between the planet and the Sun, and T_e and R_e are the orbital period of the Earth and the mean distance between the Earth and the Sun, then:

T_p = T_e * (R_p/R_e)^3/2

Remember that the notation (R_p/R_e) ^3/2 means (R_p/R_e) to the power of 3/2.

The orbital period T_e of the Earth is, of course, one year, and the mean distance R_e between Earth and the Sun is one astronomical unit, or AU. An AU is equivalent to about 150 million kilometers.

As an example of how to use Kepler's Law to determine the orbital period of another planet we will calculate the orbital period for Mars. The average distance from between Mars and the Sun is 1.52 AU, so Mars is just slightly more than 50% farther from the Sun than is Earth. With this value, you can calculate the period of the Martian orbit by:

So, Kepler's Law (which can be derived easily for circular orbits from basic principles of Physics) tells us that the year on Mars is 87% longer than it is on Earth. The primary Mars Global Surveyor mission lasts for one Martian year, which is apparently nearly two years on Earth!

Make sure that you understand how the formula works and that you can correctly calculate the Martian orbital period. Once you are able to do that, calculate the orbital periods for the other planets in the Solar System using the table below to obtain the mean distances between each of the planets and the Sun. It is quite amazing what kind of things you can calculate using simple formulas which describe how the universe behaves!!!