The Revolution of the Moons of Jupiter

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

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 0.999. 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

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

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 365.25 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

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

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. 15 10 5 0-5 0 5 10 15 20 25 30 35 40 45-10 -15 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

The Program: You should now be able to get started taking data. Double-click on the CLEA Jupiter icon with the mouse to start the program and login via the menu at the top of the screen. If you cannot find the appropriate icon or you have trouble logging in, please ask your instructor for assistance. Select START from the menu. This will bring up a window of parameters for your simulated observations. Enter the appropriate date and time. It asks for Universal time, which is the time in Greenwich, England. To convert Eastern Standard Time to Universal Time just add five hours (four if it is currently daylight savings time). The window also asks you to set the observation interval time. The default is 24 hours, but we do not want this. Set the interval to 12 hours. This will make it easier for you when it comes time to do your fits. Once you have entered everything, click OK. You should see an image of Jupiter and the 4 Galilean moons. At the bottom of the screen is the date, time, and buttons to change the magnification. Record the date, time, and day in your data table on the answer sheet. Your first observation should be day 1, second observation day 1.5, third observation day 2, and so on, because you are going by 12 hour intervals. Now measure the apparent positions of the moons and record them in your data table. To do this, hold down the left mouse button and move the cursor over the moons. When you are centered over a moon, the name of the moon and the apparent position is displayed in the low right side of the screen. You should see 4 numbers. The one you are concerned with is the X value on the bottom line next to the R value. Record that value for each moon. Remember if the moon is to the right of Jupiter, assign a positive sign to the apparent position and if it is to the left, assign a negative sign to it. Note: it is best to measure the apparent position on the moon using the largest magnification possible. Once you have recorded everything, move on to the next observation by clicking the NEXT button and repeat the procedure until you have 20 lines of data. Sometimes the moon is directly behind or in front of Jupiter. If it is behind, then you cannot get a measurement, so record the distance for that moon as zero. If it is in front of Jupiter, you can still make a measurement if you zoom in close enough. The program is also designed to randomly give you some cloudy days to make the simulation more real. For days were 7

you have clouds you cannot take measurement and will have a gap in your data (Remember this when you go to do you plots). Fill out the date, time, and day, and then indicate that it was a cloudy day. Once you have 20 lines of data, you can quit the program by clicking QUIT on the menu line. If you have any questions or something seems wrong with the program, please ask your instructor. Data Analysis: Now that you have all your data, you can begin the process of analyzing it. On each of the 4 graphs (one for each moon) provided at the back of this lab you want to plot the apparent position you measured (y-axis) versus the day you observed it (x-axis). Your plots should end up looking something like the one in Figure 4. Remember to skip days for which you have no measurements due to clouds. Once you have all the data plotted for each moon, you will want to go back and draw in a sine curve through your data that best fits the points. Remember the curve should have a regular period and have the same amplitude throughout. If you are having trouble getting a good curve, please ask your instructor for assistance. The curves are important for the rest of your analysis. From your curve you can now obtain the parameters you need to use Kepler s Third Law to determine the mass of Jupiter. The period of the orbit is going to be the period of the sine curve you fitted to the data. To measure this, just determine from your graph the time interval between one peak to the next peak or one trough to the next trough. If your observations allowed you to see the moon through many cycles (you see multiple peaks and troughs), then you can get a more accurate period by determining the time it takes the moon to complete, say 3 cycles and then divide that time by 3. If your graph does not have two peaks or troughs (this may happen for Ganymede and Callisto) then measure the time interval between a peak and the first trough after it or a trough and the next peak after it. This will give you half of the period, which you can then just multiple by two. Record your period for each moon in the space provided on the answer sheet. Remember your units. Now that you have the period, you need to determine the semi-major axis. Remember you assumed the orbits to be circular, therefore, the amplitude of your curves 8

will tell you how long the semi-major is. Measure the amplitude for each moon and record it in the space provided. Again remember your units. The last thing you need to do before you can determine the mass of Jupiter is convert your period and semi-major axis to the correct units. The period you measured was in days. Convert that to years by dividing by the number of days in a year, 365.25. The semi-major axis was measured in Jupiter diameters. There are 1050 JD in one AU, so divide your semi-major axis in JD by 1050 to convert it to AU. Record these values. Now you can compute the mass of Jupiter in solar masses by plugging your values into Kepler s Third Law. Calculation of Jupiter s Mass: Use Kepler s Third Law to calculate Jupiter s mass using the data for each moon and record the values in the spaces provided on your answer sheet. Once you have a value for Jupiter s mass from each moon, calculate the average value. M J = a 3 /P 2 Where M J = mass of Jupiter in solar masses a = semi-major axis of the orbit in AU P = period of the orbit in Earth years Remember to fill in all of the spaces on your answer sheet and answer all of the questions, which follow your calculations. 9

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

Graphs: Moon I Io Moon II Europa 11

Moon III Ganymede Moon IV Callisto 12

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

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

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