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Kepler's 3rd Law: Orbital Period vs. Distance

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Author: Erin Bjornsson

There are 8 planets (and one dwarf planet) in orbit around the sun, hurtling around at tens of thousands or even hundreds of thousands of miles an hour. Because the planets are attracted to the sun and kept in orbit by its gravitational force, the distance they are away from the sun is directly related to the speed at which they travel.

Objective: Explore the relationship between orbital period distance.

What will have a larger orbital period, short strings or long strings?

Materials

  • Twine
  • Metal Washer
  • Scissors
  • Meter stick
  • Stopwatch
  • Notebook and pen or pencil

Procedure

  1. Cut several different lengths of twine and measure them. Record the measurements in your notebook.
  2. Tie a washer securely to one end of the twine.
  3. Start swinging the washer around your head so it barely keeps tension on the rope.
  4. Have a partner start the stopwatch and at the same time, start counting the number of revolutions of the washer.
  5. After 10 seconds, record how many revolutions the washer made in your notebook and calculate the orbital period:

Orbital Period

  1. Let’s say your washer orbited 8 times in ten seconds. We’d calculate the orbital period like this:

Orbital Period Example

  1. Repeat with different lengths of twine.
  2. Show your results in a chart. Can you find a mathematical relationship between distance and orbital period?

Results

Shorter lengths of twine will have faster speeds and shorter orbital periods. Longer lengths will have high orbital periods and slower speeds. Earth’s orbital period is 365 days/revolution.

Why?

Gravity is modeled by the tension in the rope, which keeps the washer from flying away on a tangent. For shorter lengths of twine, you will need to swing the system faster and harder than for longer lengths. This is due to centripetal force in the circle.

This is a decent model for a circular system. In reality, planetary orbits in our solar system are ellipses with two foci, with the sun being at one focus. Kepler’s 3rd Law of Planetary Motion states that the square of the orbital period of a body orbiting around a larger body is proportional to the cube of the semi-major axis of the body’s orbit, which is basically the body’s distance from the larger body. Here’s an easy way to think about it: the more distant a planet is from the sun, the greater its orbital period.

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