Jupiter's Mass and the Galilean Moons

Goals of the Lab

Determine the mass of Jupiter by observing the orbital characteristics of its largest moons.

Compare the mass of Jupiter to masses of other bodies in the solar system.

Required Equipment: Calculator, Position Data Table and Orbital Plot printouts, CLEA software

Background: Jupiter is regarded as the king of the planets and has numerous moons, some of which are only on the order of a couple miles in diameter. Most of the outer moons orbit Jupiter in a retrograde orbit, meaning they orbit Jupiter in the opposite direction that the planet rotates. This signifies that these moons are more than likely asteroids that were captured by Jupiter's immense gravitational field. Jupiter does, however, have several large moons, some of which are larger than our own moon. The four largest and most famous are known as the Galilean moons, a named attributed to their discovery by the famous Italian astronomer Galileo Galilei. These four moons, in ascending orbital size order, are Io, Europa, Ganymede, and Callisto. Read the background information of these bodies. Not only do they provide us a means to determine Jupiter's mass, but they give us a glimpse as to how strange extraterrestrial worlds can be.

The Galilean Moons to Scale

If we observe the motions of these moons, specifically how far they are from the planet and their orbital period, then we can determine the mass of Jupiter from Kepler's Third Law.The most general version of Kepler's third law is the form

where

p is the period of the orbiting body
a is the semimajor axis of the orbit
M is the mass of the body being orbited
m is the mass of the orbiting body
G is the gravitational constant where

In the case where the orbiting body's mass is very small compared to the body it orbits, the sum of the two masses is approximately equal to the orbited mass. The equation then simplifies to

Eq. 1

Rearranging terms, we can solve for the mass of the orbited body in terms of just the period and orbital radius of the orbiter:

Eq. 2

Part I: Collecting Your Data

To make our observations of the Jovian moons, we will employ CLEA (Contemporary Learning Experiences in Astronomy) software provided by Gettysburg College. Before starting the software, you will need to print a copy of the Position Data Table in which you will record your observations. Also go ahead and print out the Orbital Plot since you will use it later to determine the orbital periods and radii of the moons.

After starting the CLEA software:

Click on File -> Login. You do not have to enter any information, but you must click on "Login" to start the program. It is advisable to enter your name - if you print anything, your name will already be on the printout.
Click on File -> Run. The program will determine the time and date from the computer and simulate the current arrangement of the Jovian moons.
Since you are going to be observing three different moons that take three different amounts of time to orbit once, you will want to set observation intervals that are appropriate to the periods. Go to File -> Preferences -> Timing. Set the observation interval to 2 hours for Io. Later, when you finish observing Io, you will set the remaining time intervals to 3.5 hours for Europa and 7 hours for Ganymede. You do not need to change the animation step since we aren't concerned with that feature. You are now ready to begin your observation run. Note: The creators of the software tried to add a little realism to the program by letting your view be "clouded out." If an observation is clouded out, you do not record any measurement of the moons. There will be a gap in the period curve but this should not hinder the final result.
To make it easier to determine which moon is which and when a moon has gone through a full period, go to File -> Preferences -> Top View. A window will pop up next to the main program window displaying the system as viewed from the top. Also, click on File -> Preferences -> and ID Colors. This will color the moons to make it easier to keep track of them. Finally, set your magnification by clicking the "200X" button in the main window. The three moons of interest should always be in the field of view.
There are a number of ways you can record the times for the time columns in your table. Recording the Julian Date is not recommended because it is a little harder to calculate the period and it also takes more time to record the time. Therefore, you can either record the Universal Time (UT) or simply record the first observation time as zero and then subsequent measurements will look like (as in the case for Europa) 3.5, 7, 10.5, 13, etc. If you decide to do the latter, then it is advisable to record the UT at the first observation. That way if you are unsure if you accidently double click on "Next", you can work use the starting UT to determine if there is an error. Your TA can help you with this.
Starting with Io, click on the moon with the cursor. The name of the moon will appear in the bottom right of the window. Be sure that the name of the moon appears - if it does not, this means you do not have your cursor directly on your target and you will record an inaccurate measurement. Underneath the name is the distance of the moon from Jupiter (as viewed from Earth) in units of Jupiter diameters. Record the value, along with E or W, in the data table. Repeat this step until you have made 30 measurements for each moon. Do not record the X and Y coordinates. Note: If you click one too many times, you cannot got backwards. In this case, you will have another gap in your table. This is okay as long as you record the gap in the data!
Once you have finished observing Io, reset the observation interval before moving to the next moon. Do not record all three moon positions simultaneously or it will take you all night to observe Ganymede due to its longer period!

Part II: Plotting the Data

Once you have collected all thirty observations for all three of the moons, plot your data on the orbital plot you printed out earlier. You will need to label the axes with some sort of scale that makes it easy to read the data points. A recommended method is to label every third or so grid point.
After you have all of your points plotted, you should be able to see a very nice sinusoidal pattern (assuming you did everything correctly). It may make it easier to see if you connect all the points with a line. If you suspect that one or more of your points are outliers that don't seem to fit the sine curve, then you may ignore them when connecting your other points. This may occur if you accidently record the wrong moon.

Part III: Analysis

By examining the plot, we can see that the amplitude of the curve (how tall it is) is the orbital radius of the moon, and the wavelength (the distance between successive features such as crests) is the period. Determine the orbital radius (in Jupiter diameters) and period (in hours) of the three moons.
In order to use the above equation, we must convert our units - the orbital radius must be in terms of meters (Jupiter's radius = 71,500 km) and the period must be in terms of seconds. It should be noted that the constant in Eq. 2 simplifies to unity if our units are in terms of Earth years and astronomical units. After converting, substitute in the values for each moon to determine the mass of Jupiter in kilograms. Be sure to list your results! After doing this for all three moons, average your results to get a final mass.
Now determine how close your calculation is to the accepted mass of Jupiter (1.899*10^27 kg). Calculate your percent error with

How close were you to the accepted value?
Compare your result to the mass of the Sun (2*10^30 kg). Given that the Sun is about 333,000 times as massive as Earth, how much more massive is Jupiter compared to the Earth?

Astronomy 103, Spring 2007