# The Revolution of the Moons of Jupiter

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1 The Revolution of the Moons of Jupiter Overview: During this lab session you will make use of a CLEA (Contemporary Laboratory Experiences in Astronomy) computer program generously developed and supplied by the Department of Physics at Gettysburg College. This program will simulate observations of the same four moons that Galileo saw through his telescope when he observed Jupiter back in the 1600 s. Using these observations and Kepler s Third Law, you will go about determining the mass of Jupiter. Hopefully, by the time you are finished, you will have learned something about how planets (and moons) behave in their orbits and how Kepler s Third Law can be used to gain valuable information. Background: You know that the Moon orbits the Earth and that the Earth orbits the Sun. Likewise you probably know that the other planets of our solar system orbit the Sun and have moons that orbit them. But, do you know what the geometry is of the orbits or any of the other orbital properties? Well, by the time you are finished with this lab, you should have a much better understanding of orbits. In the 1500 s, Nicholaus Copernicus hypothesized that the planets orbit in circles about the Sun. Later, in the early 1600 s, using observations of the planets and stars made by his mentor, Tycho Brahe, Johannes Kepler found that the orbits were really ellipses (see Figure 1). He then went on to deduce three mathematical laws concerning planetary orbits. These laws can also be applied to any one body orbiting another body. The first of Kepler s three laws of planetary motion concerns the overall geometry of an orbit. Put simply it states: The orbit of a planet about the Sun is an ellipse with the Sun located at one focus. Now what does that really mean? First it would be good to understand what an ellipse is. This is best explained with the help of a diagram (Figure 1). 1

2 Minor Axis F a F b c a Major Axis Figure 1 An ellipse is a closed curve where the sum of the distances of every point on the curve from the two foci (points F a and F b ) is the same. The dashed horizontal line that runs through the foci and the center is called the major axis. Half this distance is termed the semi-major axis and is labeled with the letter a. The semi-major axis will be important to you later. The dashed vertical line that runs through the center is called the minor axis and likewise half this distance is the semi-minor axis. Another important quantity you should be familiar with is the eccentricity (denoted with the letter e). This is a measure of how elongated an ellipse is. Eccentricity is determined by dividing the value c in Figure 1 (the distance from the center to the focus of the ellipse) by the semi-major axis. As an ellipse becomes more circular the foci and the center get closer together. Therefore the distance c becomes smaller. As c becomes smaller, so does the eccentricity, which remember is e = c/a. From this you should be able to see that eccentricity ranges from 0 (a perfect circle) to It can never reach 1, however. Okay so now you can start to understand what Kepler s First Law is saying. Each planet s orbit is an ellipse, that is; it follows an elliptical path around the Sun. The orbit, 2

3 since it is elliptical, has two foci and the Sun is located at one of them while nothing is at the other focus. This tells you how the orbit is positioned relative to the Sun (Figure 2). Kepler s Second Law deals with the speed at which the planets orbit. He noticed that as the planets neared the Sun their speed increased and conversely when planets are farther away from the Sun they moved slower. At a planet s closest approach to the Sun, called perihelion, it moves fastest. When a planet is at its farthest distance from the Sun, called aphelion, it moves slowest. This led to Kepler s Second Law, which states: A line joining a planet and the Sun sweeps out equal areas in equal time intervals. Another diagram at this point will be helpful. Figure 2 is another drawing of an ellipse illustrating Kepler s Second Law. Figure 2 Imagine it takes a planet 5 days to go from point A to point B along the orbit. A line joining the Sun and the planet will sweep out a somewhat triangular area between A and B (shaded region). If it also takes 5 days to travel from point C to point D, then the other shaded triangular region swept out will have the same area as the region swept out from A to B. This has come to be known as the law of equal areas. 3

4 The last of Kepler s laws relates the size of the planet s orbit to the time it takes the planet to go around the Sun once or its period. This is the law that you will be most concerned with in this lab. Simply put it states: The squares of the sidereal periods of the planets are proportional to the cubes of their semi-major axes. It sounds mathematical, but it s not too difficult to understand. You ve already learned that the semi-major axis is half of the major axis, which was defined earlier. See Figure 1 if you have forgotten. The sidereal period is just the time it takes for one body to orbit another with respect to the stars. For example the Earth s sidereal period is days. Proportional just means that the two quantities are related via some constant. The full-blown form of Kepler s Third Law is somewhat complicated, so you will be using this much simpler form: P 2 = a 3 /M J In this case the proportionality constant is just 1 over the mass of Jupiter in units of the solar masses. This form of Kepler s Third Law does require, however, that you make measurements of the period in Earth years and measurements of the semi-major axis in astronomical units. One astronomical unit (AU) is the average distance between the Earth and the Sun. What you are going to do now is make simulated observations of the 4 Galilean moons of Jupiter. You will record data from the computer, which will allow you to deduce the period and semi-major axis for each moon. Then using the form of Kepler s Third Law above, you will calculate the mass of Jupiter. Introduction: The CLEA computer program provided for this lab will simulate optical telescope observations of Jupiter and the 4 moons Galileo observed. In order of increasing distance from Jupiter they are Io, Europa, Ganymede, and Callisto. 4

5 The images displayed on the screen were actually taken by the Voyager spacecraft. The moons will appear lined up horizontally because the view is edge-on to the plane of their orbit. The orbits themselves have fairly small eccentricities; that is, they are roughly circular. Therefore for your purposes in this lab you will consider them to be circular, which means that the semi-major axis will be the same as the radius. Notice that, because your view is edge-on, you see only the apparent distance (R app ) from Jupiter and not the true distance or radius (R). Look at Figure 3, which is a top view of the system. Moon R θ Jupiter R app = Rsinθ To Observer Figure 3 From the computer you will be able to take measurements of R app over time. Because the motion in circular, the apparent position should form a sine curve if plotted versus time. You will measure the apparent position of each moon (in Jupiter diameters) for a number of time intervals and plot the data. Once you have the data plotted you will fit a sine curve to the data points from which you will ultimately deduce the period and semi-major axis. A sine curve is a smooth oscillating curve with a regular period and 5

6 amplitude. The period of your fitted sine curve will tell you the period of the orbit and the amplitude will tell you the radius of the orbit. (Remember you made the assumption that the orbit is circular, therefore, the radius is equal to the semi-major axis.) Hopefully Figure 4 will clarify this for you. If you are still having trouble, ask your instructor for help Days Figure 4 This is basically what your plots should look like. You then can draw a smooth sine curve through these points, which represents how the apparent position changes over time. Note that the negative values indicate that the moon is on the left side of Jupiter as we look at it, while positive values indicates its position to the right of Jupiter. For this example the amplitude (maximum height of the peaks) is roughly 11 Jupiter diameters (JD), which tells you that the radius, and thus, the semi-major axis is 11 JD. The period for this curve is roughly 20 days. The period is determined by measuring the time between one peak to the next on the graph. These values can now be converted to the appropriate units so you can use Kepler s Third Law to calculate the mass of Jupiter. 6

10 Name: Date: Session: Answer Sheet: The Revolution of the Moons of Jupiter Data Table: Date Time Day Io Europa Ganymede Callisto 10

11 Graphs: Moon I Io Moon II Europa 11

12 Moon III Ganymede Moon IV Callisto 12

13 Graph Measurements: Io: Period = days Period = years Europa: Period = days Period = years Ganymede: Period = days Period = years Callisto: Period = days Period = years a (semi-major axis) = JD a (semi-major axis) = AU a (semi-major axis) = JD a (semi-major axis) = AU a (semi-major axis) = JD a (semi-major axis) = AU a (semi-major axis) = JD a (semi-major axis) = AU Mass Calculations: From Io From Europa From Ganymede From Callisto M J = solar masses M J = solar masses M J = solar masses M J = solar masses Average M J = solar masses Questions and Discussion: 1. Are any of the values for the mass of Jupiter from each case significantly different? If yes, what might be some sources of error? Hint: Think about the data you took. 13

14 2. Express the mass of Jupiter in Earth masses. The Earth is 3.0 x 10-6 solar masses. 3. Jupiter has more moons beyond the orbit of Callisto. Do these moons have larger or smaller periods than Callisto? Why? 4. Which of the following do you think would cause a larger error in your calculation of Jupiter s mass, a ten percent error in the period or a ten percent error in the semi-major axis? Explain. 14

15 5. The orbit of the earth s moon has a period of 27.3 days and a semi-major axis of 2.56 x 10-3 AU. What is the mass of the Earth? Show all your work for credit and remember the units. 15

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