The Moons of Jupiter A CLEA Program

Purpose: In this lab, you will measure the orbital properties of Jupiter's moons and analyze their motions using Kepler's Third Law in order to obtain the mass of Jupiter.

Equipment: Scientific calculator, computer, and the Contemporary Laboratory Experiences in Astronomy (CLEA) computer program Moons of Jupiter.

Requirements: This lab is to be performed in groups of no more than two. If you work in pairs, take turns manipulating the computer and taking the data. You should switch off from time to time so that each of you gets a chance to use the computer. Each computer will use different initial data. Although you may use the computer and the program with your partner to collect data, all calculations, graphing, and any narratives in your lab report must be your own original work! Be sure to include all of the data you collect tonight.

Introduction and Historical Background: We can deduce some properties of celestial bodies from their motions despite the fact that we cannot directly measure them. In 1543, Nicolas 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. In 1609, Galileo Galilei heard of the invention of a new optical instrument by a Dutch spectacle maker, Hans Lippershey. By using two lenses, one convex and one concave, Lippershey found that distant objects could be made to look nearer. This instrument was called a telescope. Without even having seen an assembled telescope, Galileo was able to construct his own telescope with a magnification of about three. He soon perfected the construction of the telescope, and his best telescopes had a magnification of about thirty. Galileo immediately began observing celestial objects with his crude instruments. He was a careful observer and in 1610, published a small book of his remarkable discoveries called the Sidereal Messenger. One can imagine the excitement these new discoveries caused in the scientific community. Suddenly, a whole new world was opened! Galileo found

sunspots on the Sun and craters on the Moon. He found that Venus had phases, much as the Moon has phases, and he was able to tell that the Milky Way was a myriad of individual stars. He could see that there was something strange about Saturn, but his small telescope was not able to resolve its rings. One of the most important discoveries was that Jupiter had four moons revolving around it. Galileo made such exhaustive studies of these moons that they have come to be known as the Galilean satellites. This "miniature solar system" was clear evidence that the Copernican theory of a Sun-centered solar system was physically possible. Because he was developing a world view which was not easily reconciled with the religious dogma of his period, Galileo was compelled by the Inquisition to neither "hold nor defend" the Copernican hypothesis. Nevertheless, in 1632 he published his Dialogue on the Great World Systems, which was a thinly disguised defense of the Copernican system. This led to his trial, his forced denunciation of the theory, and confinement to his home for the rest of his life. In this lab, you are going to repeat Galileo's observations (without threat of government or religious condemnation!). Today, however, we also know the size of Jupiter. Jupiter's diameter is 11 times Earth's diameter, or about 1.5 × 105 kilometers. The data collected in this experiment and this information allow us to determine the mass of Jupiter. KEPLER'S THIRD LAW: When one body such as a moon orbits around a much more massive parent body, Kepler's Third Law is:

a3 m= 2 P where: m is the mass of the parent body, in units of the mass of the Sun. a is the length of the semi-major axis of the elliptical orbit in units of the mean Earth-Sun distance, 1 AU (Astronomical Unit). If the orbit is circular (as will be assumed in this lab), the semi-major axis is the same as the radius of the orbit. p is the period of the orbit in Earth years. The period is the amount of time required for a moon to orbit the parent body once. This law applies to planets orbiting about the Sun (check: for the Earth orbiting around the Sun: a = 1 AU and P = 1 year, and we obtain for the mass of the Sun: m = 1 solar mass) or to any moon orbiting around its planet. You will be determining a and P for the Galilean moons of Jupiter and then mJ, the mass of Jupiter, in solar masses.

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In this computer simulation, you will first determine a in units of Jupiter's diameter (JD) and the period P will be in Earth-days. You will convert these units to AU and years at the end of the lab.

Observing Jupiter's Moons: This lab can in principle be done by anyone with a set of binoculars or a small telescope. The computer simulation Moons of Jupiter replaces actual observing sessions at the observatory using the telescope. The computer simulation is based on the real orbital data for each satellite. As a matter of fact, if you were to set the simulation for today's date and time, you could verify the position of the Jovian moons by direct observation through a telescope at an observatory. You will note that the computer also provides you some of the pitfalls of actual live telescopic observations such as occasional cloudy nights! You will obtain data from 18 clear observing sessions making observations twice per evening, spaced 12 hours apart. We could do this lab for any one moon of Jupiter. If we did the experiment very accurately, the answer for Jupiter's mass should be the same no matter which moon we use. There will be errors, however, and we shall use data collected on all four Galilean moons of Jupiter. They are named Io, Europa, Ganymede, and Callisto, in order of increasing actual distance from Jupiter. You can remember the order by the mnemonic "I Eat Green Carrots". We also refer to them in this exercise as moons I, II, III, and IV. If you looked through any small telescope, the picture might look like what is seen in Figure 1:

Figure 1. Moons of Jupiter The moons appear to be lined up because we are looking edge-on to the orbital plane of the moons around Jupiter. As time goes by, the moons will move about Jupiter. Thus, while the moons move in roughly circular orbits, we generally see only the apparent distance of each moon from Jupiter's center as projected in the east-west direction which is perpendicular to the line-of-sight between Jupiter and Earth. On the computer screen, Jupiter and its moons will look much like Galileo's original sketches. Remember that west is to the right (+) and east is to the left (-) on the screen. This is the way the sky looks through a telescope. It will be necessary to record the apparent (east-west) distance of a moon from Jupiter's center in units of Jupiter diameters (JD). Lucky for you, the measurement mode of the computer program provides a direct readout in JD. The computer simulation will be presenting data on the moons as they would be seen every 12 hours.

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Such observations are possible only in the winter time when the nights are long. The observations are complete when you have obtained a total of at least 18 actual observations – NOT counting cloudy days. For each moon and for each of the 18 observing sessions, you are to measure the apparent distance of the moon from Jupiter. The data you will collect will be placed on the Data Table.

Procedure: Open the Moons of Jupiter program by clicking on START → Programs → Academic Department → CLEA Exercises → Jupiter Moons. The Moons of Jupiter 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 is a sophisticated computer program that allows convenient measurements to be made and the telescope's magnification to be adjusted. The computer simulation is realistic in all important ways, and using it will give you a good feel for how modern astronomers actually collect data and control their telescopes. Instead of using a telescope and actually observing the moons for many nights, the Moons of Jupiter computer simulation shows the moons to you as they would appear if you were to look through a telescope at the specified time. Click on File → Log In. A dialog box appears. Enter the names of each student working at your computer. When all the information had been entered to your satisfaction, click OK to continue. Initial Setup Click on File → Run. The next dialog box to appear is called Start Date & Time. Startup values are needed by the computer to establish your initial observation session, and students at each computer will perform and analyze a different set of observing sessions. The Start Dates for each computer will be one partner’s birthday, including year. Keep the Time = 0, 0, 0.00. The Observation Interval should be set to 12.00 hours. You can adjust this time interval by clicking on File → Preferences → Timing. When all of the information has been entered to your satisfaction, click OK to continue. The Main Telescope Screen After you have entered all of this information into the computer, it will display a screen similar to that shown in Figure 2. If you wish to return to the original Start Date & Time screen at any time, go to the menu bar and select File → Run. You control the observing session from this Main Telescope Screen. Notice that Jupiter is displayed in the center of your computer screen. To either side are the small point-like Galilean satellites. Sometimes a moon is behind Jupiter, so it cannot be seen. Even at high magnifications, the moons are very small compared to Jupiter. The current telescope magnification is shown in the upper left corner. The date, the UT (Universal Time – the time in Greenwich, England), and the Julian Day (JD) are all displayed in the lower left-

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Figure 2. CLEA Moons of Jupiter main telescope screen. hand corner of the screen. You may select Help from the menu to receive on-line help at any time while you are in the Main Telescope Screen. Within the Help menu are three choices that you can select: Help → How to Start advises you on how to set up the program before taking data, Help → How to Use Jupiter Lab explains how to manipulate the program in order to take data, and Help → About Help Windows explains how to use the Help windows. To close the Help windows, click on the small close box located in the upper left corner of the window. You may leave the Help windows open while working if you prefer, in which case you will want to re-size the window. Position the cursor on the box located in the lower right corner of the window. Click and hold down the mouse button while re-sizing the window. You can display the screen at four scales of magnification by clicking on the 100X, 200X, 300X, and 400X buttons at the bottom of the screen. Try them now. To improve the accuracy of your measurement of a moon, you should always use the largest possible magnification which leaves the moon visible on the screen. Be careful NOT to click the button marked Next just yet! 5

Now, position the mouse cursor over the screen and hold down the button. The measurement system turns on and displays the apparent perpendicular distance X (in JDs) that the cursor is away from the center of Jupiter. Notice that each edge of Jupiter is X = 0.5 JD. Recording Data In order to measure the perpendicular distance of each moon from Jupiter, move the cursor until the moon is centered in the cross hairs and then hold down the mouse button. When the satellite is carefully centered, release the mouse button and information about the moon will appear at the lower right corner of the screen. This information includes the name of the selected moon, the X and Y pixel location on the screen, and the perpendicular distance X (in units of Jupiter's Diameter) from the Earth-Jupiter line-of-sight for the selected moon, as well as E or W to indicate whether it is east or west of Jupiter. If the moon's name does not appear, you did not center the moon in the cross hairs exactly and you should try again! To measure a moon, always switch to the highest magnification that still leaves the moon on the screen. It is important to use the highest magnification possible for each moon for best accuracy. If a moon is behind Jupiter, record the apparent distance for that moon as ±0.5 since you don't know its location any better than this. Now, begin the data collection process by recording the data for the first observations on the attached Data Table: Column 1: Date Column 2: Universal Time Column 3: Day - number of the day (like 1.0, 1.5, 2.0). Remember to keep incrementing the Day number even on cloudy days Columns 4 – 7: Record each moon's position under the column for that moon. Count positions to the left (east) of Jupiter as negative and those to the right (west) as positive. If Europa were selected and had X = 2.75 west of Jupiter, you would enter +2.75 in column 5. You can also enter all this data electronically into the computer by choosing the Record Measurements button. A box in which you can input each moon’s data appears. Here, you record “E” or “W”, rather than “-“ or “+”. Click Ok when you are finished recording data for each observation interval. Please note that the computer does not recognize ±0.5 when it creates the plots, so do not enter this information when a moon can’t be seen. You should still record it on your data sheet, though, because a good astronomer always keeps a hand-written record of the data as well. When you have recorded the Universal Time and perpendicular distance for each moon, click the Next button and the image will advance by the amount of time you specified in the Observation Interval, 12 hours. Note that a certain percentage of observing sessions will be "cloudy". 6

If you encounter bad weather, just enter "Cloudy Day" in the space provided after you write in the Day number. Do NOT remeasure any of the Cloudy Days – you will just have to allow for any gaps in your data due to these days. Again, when it comes time to plot this data, leave these lines blank. The observations are complete when you have obtained a total of 18 actual observations. When you have completely finished filling out the Data Table, select File → Exit → OK. Note: Once you exit the program altogether, you cannot continue where you left off! Data Reduction You will use the data from the Data Table to obtain a graph that looks similar to what is shown in Figure 3. The data are for an imaginary moon, Clea. Alternatively, if you recorded the data on the computer, you can analyze them by clicking File → Data → Analyze and Select a moon from the menu. All of that moon’s data are plotted and you can now determine its orbital period and its distance from Jupiter. Use the Plot → Fit Sine Curve → Set Initial Parameters to help you do this. You may want to use the Help button for assistance. Be sure to continue reading here to learn more about what you are doing and why.

Figure 3. Orbital period for Moon Clea. P = 14 days a = 3.0 J.D. P = 0.0383 years a = 0.00286 AU

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Each dot in the figure is one observation of Moon Clea. Note the irregular spacing of dots, due to clouds or poor weather and other observing problems on some nights. The curve drawn through the points is the smooth curve that would be made by Clea if you had enough observations. The shape of the curve is called a sine curve. You will need to determine the sine curve that best fits your data in order to determine the orbital properties of each moon. Here are a few hints: (i) the orbits of the moons are regular, that is, they don't speed up or slow down from one period to the next, and (ii) the actual radius of each orbit does NOT change from one period to the next. The sine curve that you draw through your data points should therefore also be REGULAR and SMOOTH. It should go through all of the points, and NOT be higher at the maxima in some places than others. It should also NOT be wider in some places than others. Please pay attention to how you draw your graph as this will be used to determine the orbital parameters for each moon. Using the data from Moon Clea shown in Figure 3, it is possible to determine both the radius, a, and the period, P, of the orbit for each moon. The period is the time it takes to get back to the same point in the orbit. Thus: (i) the time between two consecutive maxima (or minima) is the period, (ii) the time between two consecutive crossings at 0 JD is equal to 1/2 the period because this is the time it takes to get from the front of Jupiter to the back of Jupiter (or vise versa) which is 1/2 way around in its orbit, and (iii) the time between a crossing at 0 JD and the nearest maximum or minimum is equal to 1/4 the period. For some moons, you may not get enough observations for a full period, so these points may be of use to you in determining the period, even though the moon has not gone through a complete orbit. On the other hand, if you have enough observations for several cycles, you can find a more accurate period by taking the time it takes for a moon to complete, say, 4 cycles, and then dividing it by 4. Once you find P, it must then be converted to units of years by dividing by 365.25 days. Remember, it is important to stay in the correct set of units if you are going to use Kepler's Third Law. You can determine the semi-major axis (the radius), a, for each moon by measuring the maxima and minima in your smooth sine curves. When a moon is at the maximum position eastward or westward, it is at the largest apparent distance from the planet. Remember that the orbits of the moons are nearly circular, but since we see the orbits edge on, we can only determine the actual radius of the orbit when the moon is at its maximum position eastward or westward from Jupiter.

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Repeat Steps 1–4 below for each of the four Jovian moons that you measured: Step 1. Enter the data for each of the four moons into an Excel spreadsheet. The first column should contain the observing interval (i.e. 1, 1.5, 2, 2.5); the second column should contain the distance. Create a graph by highlighting both columns and choosing the XY(Scatter) option. Along the horizontal scale, should be the day numbers, starting on the left with the number of the first day for which you have data. Calibrate the horizontal scale of each graph to make your data as stretched out as possible. (The graph for Io, in particular, will benefit from this stretching). The vertical scale should be divided by a zero (0) line, with positive and negative numbers on each side of it. Each day's measurement of a moon's apparent separation from Jupiter should give you one dot on the graph for that moon. Since each observing day has two sessions, be sure that each point on the graph on the horizontal axis represents only one session. Remember NOT to plot any points for Cloudy Days since you don't know where any of the moons are on those days. Also remember that the computer does not recognize ±0.5 when it creates the plots, so do not enter this information either. You will have to make a best guess as to the moon’s location based on the other observations. Step 2. For each moon, print out the plot and draw a smooth sine curve through the points. Mark all maxima and minima on the curve by crosses. They need not fall on one of the grid lines. The curve should be symmetric about the horizontal line corresponding to zero apparent separation. The maxima and minima should have the SAME values, except for their sign. Step 3. Read off the period, P, and the semi-major axis, a, from your figure, in the manner shown in the earlier example for Moon Clea on page 7 (Figure 3). These units will be days for P and JD for a. Step 4. To obtain P in years, divide your result in days by 365.25 since there are 365.25 days in a year. To obtain a in AU, divide your result in JD by 1050, since there are 1050 Jupiter diameters (JD) in one AU. Include your converted values on each of the graphs you’ve created for each moon. You now have all the information you need from each of the four moons to use Kepler's Third Law to determine the mass of Jupiter, mJ, as:

a3 mJ = 2 P where: P has units of years, a has units of AU, and mJ has units of solar masses.

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Data Table Collect 18 nights of data, not counting cloudy nights. Mark cloudy sessions as in the sample below. Remember to enter “E” or “W” as “-“ or “+”. Example: (1) Date

(2) Time

(3) Day

(4) Io

(5) Europa

7/24

0.0

1.0

2.95W

7/24

12.0

1.5

CLOUDY- ------------ -----------

(1) Date

(2) Time

(3) Day

2.75W

(6) Ganymede

(4) Io

(5) Europa

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7.43E

(6) Ganymede

(7) Calisto

13.15W ----------(7) Callisto

Use the space below to show your four separate calculations of Jupiter's mass based on your observations of each Galilean moon. Don't forget to include the appropriate units for the mass of Jupiter.

Summarize your calculations for the mass for Jupiter in each of the four cases below: From moon I, Io:

mJ = ___________________ solar masses

From moon II, Europa:

mJ = ___________________ solar masses

From moon III, Ganymede:

mJ = ___________________ solar masses

From moon IV, Callisto:

mJ = ___________________ solar masses

Hint: If one of the values is very different from the other three, look for a source of error. Perhaps the data are not adequate for a better result, in which case leave the value as you obtained it. Average mJ = ______________________ solar masses.

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Additional Questions and Discussion 1. Convert your average value for mJ into kilograms by multiplying your previous result (in solar masses) times the mass of the Sun in kilograms (this can be found in your textbook). What is the percent error between your average value for mJ and the accepted value (also given in your textbook) for the mass of Jupiter in kilograms?

2. To express the mass of Jupiter in units of the mass of the Earth, divide your result from Question 1 above by 5.97 × 1027 grams, which is the mass of the Earth. SHOW YOUR WORK BELOW. Think about your answer to make sure you’ve done the correct calculation.

mJ = ______________________ Earth masses 3. Which of the four Galilean moons was the most difficult to fit a smooth sine curve to? Why?

4. How would you change the time interval between observations for this "difficult" moon? (Remember that since you are observing from the surface of the Earth, you are still stuck with a certain amount of time each day when Jupiter is not visible!)

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5. When a moon was not visible because it was behind Jupiter, you entered ±0.5 JD for its apparent distance from Jupiter. However, all you really know from your measurement is that its apparent distance was somewhere between the edges of Jupiter at +0.5 JD and at – 0.5 JD. Rather than just ignore the data points, how would you account for this uncertainty when drawing your smooth sine curve through all of your "0.5 JD" points?

6. Suppose that you were only able to get telescope time to observe Jupiter's moons once every 4 days instead of once every 12 hours as in this lab. Which of Jupiter's four moons would you still be able to fit accurately to a sine curve, and which ones would be very difficult to fit? Why?

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