Target Level: Middle School

Timetable: 35-40 minutes

Vocabulary: Kepler, orbital period, orbital resonance

Kepler's Third Law describes the motions of objects in different sized orbits. If you know how far away an object is from the body it orbits, you can calculate how long it takes for it to make a complete orbit. Although Kepler first described this fact for planets orbiting the Sun, it holds true for all objects in orbit around a central body. The mathematical equation is written d³ µ p², which means that the ratio of an object's relative distance from the central body cubed (usually given in Astronomical Units, or AU) is directly proportional to its relative orbital period squared. (usually given in Earth years). In this activity you will use Kepler's Third Law to show how the orbits of the first three Galilean moons are related.

First, we need to standardize the units to be used for distance and orbital period. For simplicity, we'll choose the moon Io, and say that 421,600 kilometers, which is the average distance from Jupiter to Io, is equal to one moon distance unit (M.U.): d = 1 M.U. Then we'll say that the time it takes for Io to go once completely around Jupiter is equal to one orbital period unit (Orb): p = 1 Orb. The answers you get for orbital period for each moon will be in terms of the actual values for Io.

Look again at Kepler's Third Law. We can now write it in this form: (M.U.)³ µ (Orb)²

For Io, we see that

d³ µ p²

(1 M.U.)³ µ (Orb)²

1 M.U. µ (Orb)²

Ö 1 M.U. µ Orb, or 1 µ Orb. This is the simplest case.

Now let's try Europa together. The average distance from Jupiter to Europa is 670,900 km. First, how many moon distance units (M.U.) is this?

Next, to determine Europa's orbital period (in Orbs) we have to cube the distance (in M.U.) and then take the square root.

From these calculations, we see that Europa's orbital period is twice as long as the orbital period of Io.

1) In terms of orbits, how often are Io and Europa aligned on the same side of Jupiter?

2) Determine the orbital period for Ganymede, whose average distance from Jupiter is 1,070,000 km.

3) How does the orbital period of Ganymede compare to that of Io and Europa?

4) Are Io, Europa, and Ganymede ever aligned on the same side of Jupiter? How often? What effect might this have on the orbit of Io, or on tidal forces acting on the moons?

5) If we know that it takes Io 1.77 Earth days to orbit Jupiter, how many Earth days does it take Europa and Ganymede to complete one orbit?

As a final step, consult a textbook on Jupiter and its moons to see how accurate your calculations were!

Teacher notes available here.

Click here for more information about Kepler's Laws.

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This module was written by Brian Exton (National Optical Astronomy Observatories, Tucson AZ).

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Galileo Solid State Imaging Team Leader: Dr. Michael J. S. Belton

The SSI Education and Public Outreach webpages were originally created and managed by Matthew Fishburn and Elizabeth Alvarez with significant assistance from Kelly Bender, Ross Beyer, Detrick Branston, Stephanie Lyons, Eileen Ryan, and Nalin Samarasinha.

Last updated: September 17, 1999, by Matthew Fishburn

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