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3 August 2009 TAM University of San Francisco Physics 12_ Laboratory #_1 JUPITER AND ITS MOONS Kepler’s Third Law OBJECTIVES Learn how to gather and analyze data Determine the mass of Jupiter. Gain a deeper understanding of Kepler's third law INTRODUCTION AND BACKGROUND Astronomers cannot directly measure many of the things they study, such as the masses and distances of the planets and their moons. Nevertheless, we can deduce some properties of celestial bodies from their motions despite the fact that we cannot directly measure them. In 1543, Nicolaus Copernicus hypothesized that the planets revolve in circular orbits around the sun. Tycho Brahe (1546-1601) carefully observed the locations of the planets and 777 stars over a period of 20 years using a sextant and compass. These observations were used by Johannes Kepler, a student of Brahe’s, to deduce three empirical mathematical laws governing the orbit of one object around another. Kepler’s Third Law is the one that applies to this lab. For a moon orbiting a much more massive parent body, it states the following: a3 M 2 T where M is the mass of the parent body in units of the mass of the sun ( M ). a is the length of the semi-major axis in units of the mean Earth-Sun distance, 1 A.U. (astronomical unit). If the orbit is circular (as we assume in this lab), the semi-major 1 This laboratory exercise is based and draws extensively upon The Revolution of the Moons of Jupiter Student Manual produced by Department of Physics, Gettysburg College, Gettysburg, PA 17325 as part of the Contemporary Laboratory Experiences in Astronomy (CLEA) program sponsored by Gettysburg College and the National Science Foundation. Other input was taken from Kepler’s 3rd Law and the Orbits of the Moons of Jupiter a laboratory exercise produced by the Astronomy Department, University of Michigan, Ann Arbor, MI 48109. Modifications and additions are due to P.McNeff, T. A. Mulera and A. Venkatesan of USF’s Department of Physics. Thanks to K. R. Scanlon for “test driving” the exercise. Physics 12_ Lab _ – 1/9 3 August 2009 TAM axis is equal to the radius of the orbit. T is the period of the orbit in Earth years. The period is the amount of time required for the moon to orbit the parent body once. In 1609, the telescope was invented, allowing the observation of objects not visible to the naked eye. Galileo used a telescope to discover that Jupiter had four large moons orbiting it and made exhaustive studies of this system, which was especially remarkable because the Jupiter system is a miniature version of the solar system. Studying such a system could open a way to understand the motions of the solar system as a whole. Indeed, the Jupiter system provided clear evidence that Copernicus’ heliocentric model of the solar system was physically possible. Unfortunately for Galileo, the inquisition took issue with his findings; he was tried and forced to recant. We will observe the four moons of Jupiter that Galileo saw through his telescope, known today as the Gallilean moons. They are named Io, Europa, Ganymede and Callisto, in order of distance from Jupiter. You can remember the order by the mnemonic “I Eat Green Carrots”. If you looked at Jupiter through a small telescope, you might see the following: Figure 1. Jupiter and Moons through a Small Telescope The moons appear to be lined up because we are looking edge on at the orbital plane of the moons of Jupiter. If we watched, as Galileo did, over a succession of clear nights, we would see the moons shuttle back and forth, more or less in a line. While the moons actually move in roughly circular orbits, you can only see the perpendicular distance of the moon to the line of sight between Jupiter and Earth. If you could view Jupiter from “above” (see Fig. 2), you would see the moons traveling in apparent circles. VIEW FROM ABOVE THE PLANE OF ORBIT Rapparent shows the apparent distance between the moon and Jupiter that would be seen from earth. Rapparant Rmax sin t , the angular velocity, is constant Figure 2. View of Jupiter and one of its moons perpendicular to the orbital plane of the moon. Physics 12_ Lab _ – 2/9 3 August 2009 TAM As you can see from Fig. 3, the perpendicular distance of the moon should be a sinusoidal curve if you plot it versus time. By taking enough measurements of the position of a moon, you can fit a sine curve to the data and determine the radius of the orbit (the amplitude of the sine curve) and the period of the orbit (the period of the sine curve). Once you know the radius and period of the orbit of that moon and convert them into appropriate units, you can determine the mass of Jupiter by using Kepler’s Third Law. You will determine Jupiter’s mass using measurements of each of the four moons; there will be errors of measurement associated with each moon, and therefore your Jupiter masses may not be exactly the same. The apparent position of a moon varies sinusoidally with the changing angle from the line of sight, , as it orbits Jupiter. Here the apparent position is measured in units of the radius of the moon’s orbit, R, and the angle measured in degrees. Figure 3. Graph of the apparent position of a moon viwed edge on at the orbital plane of the moon. The CLEA program simulates the operation of an automatically controlled telescope with a charge coupled device (CCD) camera that provides a video image to a computer screen. It also allows convenient measurements to be made at a computer console, as well as adjustment of the telescope’s magnification. The computer simulation is realistic in all important ways, and using it will give you a good understanding of how astronomers collect data and control their telescopes. Instead of using a telescope and actually observing the moons for many days, the computer simulation shows the moons to you as they would appear if you were to look through a telescope at the specified time. EQUIPMENT AND MATERIALS 1. Windows computer or Macintosh Computer running a Windows emulator, e.g. Parallels. 2. The CLEA program Revolution of Jupiter Moons. Installation instructions on p.9. 3. Scientific calculator. PROCEDURE Activity I-Starting the program and familiarizing yourself with the Jupiter system 1. Start up the program by double clicking on the Revolution of Jupiter Moons icon. 2. Select Log in from the File menu. Enter your name(s) and table number and click OK. Physics 12_ Lab _ – 3/9 3 August 2009 TAM 3. The title page screen should appear. Select File>Run and click OK to accept the default values for Start Date and Time. The Observation Screen should appear as in Figure 4 below. 4. Jupiter appears in the center of the screen, while the small, point-like moons are on either side. Sometimes a moon is hidden behind Jupiter and sometimes it appears in front of the planet and is difficult to see. You can display the screen at four levels of magnification by clicking on the 100X, 200X, 300X, and 400X buttons. The screen also displays the date, Universal Time (the time at Greenwich, England), the Julian Date (a running count of days used by astronomers; the decimal is an expression of the time), and the interval between observations (or animation step interval if Animation is selected). 5. Select File>Features>Animation and then click the Cont. button on the Observation screen. Watch the moons zip back and forth as the time and date scroll by. 6. Select File>Features>Show top view and click OK. You can also select ID color to avoid confusing the moons. Figure 4 OBSERVATION SCREEN When you have familiarized yourself with the program and the motions of the Jovian moons, turn off the Animation and proceed to section II. Activity I - Set up observing sessions 1. Return to the Observation Screen. 2. Select File>Observation Date> Set Date/Time. Type the date and the time in the appropriate spaces. Universal Time = PST + 8 hours. Click OK. Physics 12_ Lab _ – 4/9 3 August 2009 TAM 3. Select File>Timing. Set the Observation Interval to 12.00 hrs. Click OK. Activity III - Collect Position Data for Moons 1. In order to measure the position of a moon, move the pointer to the moon and left-click the mouse. The lower right-hand corner of the screen will display the name of the moon (for example, II. Europa), the X and Y coordinates of its position in pixels on your screen, and its X coordinate expressed in diameters of Jupiter (Jup. Diam.) to the east or west of the planet’s center. To measure each moon’s position accurately, switch to the highest magnification that will keep the moon on the screen. 2. Click on the Record button. The Record/Edit Measurements dialog box will appear and show your measurement in Jupiter diameters East or West of center. Click OK. 3. Repeat 1 and 2 above until you have measured and recorded the positions of all visible moons. If a moon is eclipsed by Jupiter leave the measurement blank. 4. Click Next on the Observation Screen and repeat 1 through 3 above. Collect a total of at least eighteen sets of measurements. To safeguard your data in the event of a computer crash use File>Data>Save after each set of measurements. Save as a .CSV file. If the picture of Jupiter and the moons is suddenly replaced with a picture of a cloudy sky just click Next again and procede. Activity IV - Plot Graphs of Your Measurements 1. On the Observation Screen select File>Data>Analyze. A window titled “Jupiter Satellite Orbit Analysis” will appear. Select Data>Select Moon>Callisto (IV) in this window. Select Data>Plot>Plot Type>Connect Points and print using Data>Print>Current Display. 2. Repeat 1 above for the other three moons. 3. Attach plots to answer sheet Activity V - Determine the Period and Semi-Major Axis for the Orbits oEach Moon and Calculate the Mass of Jupiter 1. On each graph measure the period (time for a complete cycle) and semi-major axis (amplitude) in units of days and Jupiter diameters (J. D.). See Figures 2 and 3. Your graphs will probably not be this smooth but measure as well as you can. Enter your measurements in Table 1 on the answer sheet. 2. Convert these measurements to terrestrial years and astronomical units. 1 yr = 365 days and 1 A. U. = 1050 J. D. Enter in Table 1. 3. Apply Kepler’s 3rd Law to calculate the mass of Jupiter in units of solar masses. Enter the result for each moon in Table 2. Physics 12_ Lab _ – 5/9 3 August 2009 TAM University of San Francisco Physics 12_ Laboratory #_ Answer Sheet Lab section: ________________ (Day-Time) Lab Partners: Date: ___________________ (mm/dd/yyyy) _____________________________ _____________________________ Each member of the lab group should hand in an original lab report. Photocopies will not be accepted. _____________________________ _____________________________ Circle your name in the list at left. Attach your tables of values for the positions of the planets and your graphs to these answer sheets. Physics 12_ Lab _ – 6/9 3 August 2009 TAM Table 1 - Results from Graphs Moon a (J.D.) T (days) a (A.U.) T (years) Table 2 - Results of Mass Calculations Moon Mass of Jupiter M The average calculated mass is M J = ______________________ solar masses. From your text book, the mass of Jupiter is 9.54 x 10-4 solar masses. Your percentage error is: Physics 12_ Lab _ – 7/9 3 August 2009 TAM Percent error = M J 9.54 104 100 = ____________________% 9.54 104 Questions TEuropa Consider the three inner satellites. Form the ratios . What do TIo TIo you notice about the relative values of the three periods? To what phenomenon does this give rise? 2. There are moons beyond the orbit of Callisto. Will they have larger or smaller periods than Callisto? Why? 3. The orbit of Earth’s moon has a period of 27.3 days and a radius of 2.56 x 10-3 A. U. or 3.84 x 105 km. What is the Earth’s mass M in units of solar masses M ? Physics 12_ Lab _ – 8/9 and TGanymede 1. 3 August 2009 TAM Installation of Revolution of Jupiter Moons Software Go to CLEA website: http://ww3.gettysburg.edu/~marschal/clea/CLEAhome.html Click on Software on left side of page. Click on the Revolution of the Moons of Jupiter icon. “File Download – Security Warning” window will open. Click Save in this window. Set save destination to Desktop and procede. Save will take a good 5 or 6minutes and should leave an icon tiled JupLab on your desktop. Double click on JupLab. “File Download – Security Warning” window will open. Give permission and follow instructions to install Revolution of Jupiter Moons. An icon with this title should appear on your desktop. Return to p. 3 to run program. Physics 12_ Lab _ – 9/9