Activity 7A: Determining the Mass of Jupiter

Many years ago Sir Isaac Newton continued the work of Johannes Kepler and discovered that the mass of any planet could be calculated by knowing the period and orbital radius of any of its moons. The result of this work** is contained in the following equation:

M  =             4p²r³/T²G

where M is the mass of the planet, T is the period of one of its moons and r is the orbital radius of the same moon. The letter G represents the universal gravitation constant which has the value 6.67 x 10^-11 N-m²/kg².

The units used with this equation are of utmost importance. The period, T, must be in seconds. The orbital radius, r, must be in meters. The resulting mass, M, will be in kilograms.

**If you are interested in how Newton arrived at this equation please refer to the *Library in the EDO - Orbits nexus community. The document is called "Derivation of the formula for determining the Mass of Jupiter".

In Activity 6 you were able to determine the period and orbital radius of the moon Io. Possibly you were able to do this for other moons. You will now be able to use this information to perform a relatively simple calculation and determine the mass of Jupiter.

Do some math:

1. Go back to Activity 6 and locate the period and orbital radius you found for Io.
2. Convert the period from hours into seconds.
3. Convert the radius from kilometers to meters.
4. Now insert these values into the formula above to calculate the mass of Jupiter in kilograms.

You will want to use a calculator for this purpose. Be careful with the exponents on the powers of ten.

Locate a source for the published value of the Jupiter's mass. Compare the result of your calculation above to the published value.

Activity 7B: Determining the mass of the Earth.

You now have had the practice of using Newton's equation to find the mass of Jupiter. Recall that you needed to know only the period and orbital radius of one of its moons, Io.

You can use this same process to determine the mass of the Earth by obtaining the same orbital information about the earth's moon.

Find a source of information that will give you the moon's period and orbital radius. Once you have found these values remember to convert them into the units you used in Activity 7A. Then insert these numbers into the mass equation in Activity 7A to calculate the mass of the Earth.

Now find a source that will give you the published value for the mass of the Earth. Compare your calculation to this accepted value. If you find that your calculated value was not close to the published value, go back and check your work, especially the conversion of units.

Activity 7C: Measurement of the speed of light.

Interestingly enough, the moons of Jupiter can be used to determine the speed of light. In the late 1600's Ole Roemer made a well known set of measurements to give us an early value of light's speed. Although the value that Roemer determined was quite different from the value we know today, it gave other scientists an idea of how large it was and suggested other methods that could be used in future measurements.

In this activity you are directed to some internet websites to learn more about the measurement of the speed of light and how Ole Roemer used the moons of Jupiter to make this measurement.

Use the link above to go to the EDO Project Links cybrary. Here you will find three websites that are very useful for understanding how the speed of light was first calculated.

The first website listed will give you some background on some early attempts at this measurement, and then it gives information about how Roemer went about his measurement. In contains a diagram that is helpful in visualizing Roemer's thought process. Please note that there is an important error on this website. Roemer did not get 186,000 miles/sec for his calculation. That is the currently accepted value. He got a value that was somewhat higher.

The second website is a quick summary of his experiment, but also contains a little more detail of the actual measurements and the numbers Roemer used.

The final website contains a biography of Roemer's life, including his work on the speed of light.

After you have read some of these references, particularly the first two, you might want to attempt a repeat of Roemer's measurements.

First, try to go through the numbers that Roemer actually used to calculate the speed of light. The only equation you really need is: speed = distance/time. The distance involved here is related to the radius of the earth's orbit around the sun which is called an Astronomical Unit. You can find this number in the third reference. The time you need is the delay of the light coming from Io as it appears around Jupiter. You'll find that number in the first and second references. Remember to use the correct units, usually meters and seconds, to get the correct units for speed, usually meters/second.

After this you could try to find your own values for the time delay of Io by looking for published values of Io's transit behind Jupiter at two different times of the year. The times must be 6 months apart: one time when the earth is moving toward Jupiter; the other time when it is moving away from Jupiter.

Good luck!