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The most precise and accurate method to know the mass of a planet in our solar system is to use Kepler's second law. If a planet has a moon, measure the period and semimajor axis of the moon's orbit and calculate the planet's mass.
This method also works for very massive stars that have low-mass companion stars. Measure the period and semimajor axis of the companion star and calculate the mass of the star (i.e. the high-mass star). We'll apply Kepler's second law to stars later in a later topic.
In this experiment, you will use the CLEA Moons of Jupiter software to measure the period and radius of the orbits of Jupiter's largest moons (called the Galilean Moons): Io, Europa, Ganymede, and Callisto.
Juipter and its moons are in the same orbital plane (called the ecliptic plane) around Sun as Earth. As a result, we view Jupiter's moons from a side view. We do not see Jupiter's moons travel in a circle as we would if we were looking at a top view of the orbits. But rather, we see Jupiter's moons move back and forth from one side of Jupiter to the other.
For example, consider the simulation of the orbit of Io shown below. Use the Play button to play or pause the simulation. Click the Reset button to start it at the beginning.
The simulation shows Io's orbit from a top view. This is the easiest way to visualize its orbit. You'll notice that Io's orbit is nearly a perfect circle (very low eccentricity ε).
Period is the time elapsed for one complete orbit.The clock shows time in Earth days. How many Earth days is the period of Io?
Because Earth is in the same plane as Jupiter and its moons, we see Io's orbit from the side when we look at Jupiter and Io through a telescope. Click the "Show Side View" button to view Io and Jupiter as we see it through a telescope.
Let's use the variable x to represent the the horizontal distance of Io from Jupiter with positive values of x to the right of Jupiter and negative values of x to the left of Jupiter. In the side view, Io moves from right to left during the first half of its orbit. Io's radius is approximately 3 times the diameter of Jupiter. We will use "Jupiter diameters" as our unit for the position of Io. Thus, x goes from the value +3 to the value -3 in about 0.88 seconds. Then, it goes from -3 to 3 from Time=0.88 seconds to Time=1.77 seconds. It then repeats this motion.
If we measure Io's position relative to Jupiter at various clock readings and graph the horizontal position of Io as a function of time, we will see the graph shown in the following animation. To view the graph, click the Play button and watch the motion for at least 10 s.
This mathematical function is called a sine curve. The height of a peak of the sine curve is called the amplitude and gives us the radius of the orbit. The time elapsed between successive peaks is called the period. Click the "Show Period and Amplitude" button in order to see these defined on the graph. Be sure to view the animation until it automatically stops in order to most clearly understand how amplitude and period are defined.
The period of the sine curve is the period of Io's orbit. The amplitude of the sine curve is the radius of Io's orbit. Knowing, it's period and radius, we can calculate the mass of Jupiter.
Download the two files: (1) Jupiter.doc and (2) data-analysis.xls.
The Word doc describes how to use the CLEA Moons of Jupiter software. The Excel spreadsheet does the calculations for you. The following videos will help you see how to collect data and fit a sine curve to the data.
Moons of Jupiter Part 1. This video describes how to start the CLEA simulation and collect data for the positions of the moons of Jupiter.
Moons of Jupiter Part 2. This video describes how to fit a sine curve to the position of a moon (as a function of time) and how to analyze the sine curve to get the period and radius of the orbit.
You do not have to answer all questions in the Word doc nor do you have to calculate the mass of Jupiter by hand. The spreadsheet does all calculations for you.
You merely have to collect data for the position of Jupiter's moons on various days. Then, analyze the sine curves to get the period and radius of each moon's orbit. Then, insert the period and radius of each moon's orbit into the appropriate cells in the spreadsheet.
WebAssign -- Lab: Moons of Jupiter. Go to WebAssign and answer questions about this lab.
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