Workshop on Jupiter's moons and Dark Matter

This workshop is designed to be done primarily in small discussion groups. There are 5 parts; spend about 40 minutes each on Goals 1-4, and the last few minutes on Goal 5.

Goals:

(1) To become more comfortable and skilled at interpreting and learning from graphical representations of information, and at expressing your understanding with graphs.

(2) To develop an understanding of the relationship between the size and period of a gravitationally bound circular orbit.

(3) To develop an understanding of some of Newton's and Kepler's laws

(4) To extend your understanding of planetary orbits to galactic rotation, and use it to predict distribution of dark matter.

(5) To do effective collaborative work with small and large groups of people with diverse skills and backgrounds.

Assumptions: Jupiter is much more massive than any of its moons. The Galilean satellites have nearly circular orbits.

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GOAL (1): To become more comfortable and skilled at interpreting and learning from graphical representations of information, and at expressing your understanding with graphs.

Graphing: How can oscillations be represented graphically? First you will try your hand at sketching how the position of an oscillator varies in time, (a) qualitatively and (b) quantitatively. (c) Then you will interpret a graph of Jupiter's moons.

(a) Consider a swinging pendulum. At rest, it hangs at equilibrium. Call this position zero. When it's displaced from equilibrium, it oscillates about this zero point. How does the angular displacement (or height) change with time?

Everyone in the group should sketch their own representation of the pendulum's displacement versus time in your notebook. Don't worry about assigning scales to your graph's axes yet.

Then compare and discuss your graphs. Graphs may be oriented in different ways, yet be substantially the same. What are the main features shared by the graphs which your group agrees on?

Sketch your team's graph on the board to compare with other groups.

(b) Quantitatively: Now measure the period for your pendulum. Your estimate can be more accurate if you average over several periods. The period is the time it takes to complete a full swing - say, to fall from maximum displacement, swing out, and swing back to the starting point again.

Post your pendulum data on the chart on the board.

Use your data to label time and distance scales on your graph. Focus on key points such as the zero point and the turnaround points. Be sure to include units on your axes.

Compare your labeled graph with those of teammates. When you agree, go label the axes on your graph on the board. Tabulate the results of the whole class in your notebook

Finally, plot the period versus length data for all these pendula, together on one graph. Do you see a trend? Are any data suspect? In your notebook, summarize in one sentence how the period of a pendulum appears to depend on its length.

(c) Obtain a graph of the motions of Jupiter's moons. Analyze the graph qualitatively first, to get a general overview and to practice your basic graph-interpreting skills. Answer each question on your own, then compare your answers with your teammates, as usual.

Which moon has the fastest orbit? The slowest?

Which moon orbits furthest from Jupiter? which orbits closest?

Discuss your results with teammates until you reach a consensus.

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GOAL (2): To develop an understanding of the relationship between the size and period of a gravitationally bound circular orbit.

(a) Choose one of Jupiter's four moons to analyze carefully. Talk with all the tables around you to make sure they are analyzing different moons, because you will need their results on those other moons.

(b) Before you do any calculating, first qualitatively predict how your moon's orbit will compare with the others, in your notebook. Second shortest period? Third largest orbit radius?

(c) Quantitative analysis of your moon:

Check the time and space axes on your graph to make sure you understand their units.

Measure the period of your moon as carefully as possible, with a clear plastic ruler. Your result will be more accurate if you measure over several periods and divide by the number of periods. How far off could your result be? You probably will have no more than three significant figures in your answer ( 1.53 seconds, +/- 0.02 sec) Don't waste time on meaningless over-precision (such as 1.5386149 seconds).

Measure the orbit radius of your moon as carefully as possible, with a clear plastic ruler. What is the uncertainty in your measurement? (R +/- dR)

Post your results on the chart on the front board, and tabulate the class results in your notebook.

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GOAL (3) : To develop an understanding of some of Newton's and Kepler's laws.

(a) Using the class data on the board, plot (period T) versus (orbit radius R) for Jupiter's four moons. Is this a straight line? Do you expect it to be?

(b) Recall the derivation of Kepler's 3d law from Astronomy class on Tuesday. Start with F=ma (Newton's 2d law) and derive an algebraic relationship between T and R for a circular orbit, in your notebook. Astronomy students should work closely with Stars, Sky, and Culture students so everyone feels comfortable with the derivation.

(c) What power of T should you plot against what power of R to get a straight line graph? Do it. Compare to your graph in (a) above.

(d) You can also use your measurements of your moon's period and orbit radius to find the mass of Jupiter! How uncertain is your answer? (e.g. 3.56 +/- 0.3 kg) Check your Jupiter mass (M_J) against the results of other groups, with each group using only their own moon's R and T.

Data: G=6.67 x 10^-11 m³/kg.s²

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GOAL (4): To extend your understanding of orbits to galactic rotation, and use it to predict distribution of dark matter.

(a) The "velocity profile" or "rotation curve" of a system of orbiting masses tells you how the system's mass is distributed, whether you can see all this mass shining or not. Graph speed versus R for Jupiter's moons, where speed v=distance/time = 2*Pi*R/T and R is the moon's distance to Jupiter and T is the orbit period, as usual.

(b) Where is most of the mass in the system consisting of Jupiter and its moons? Your graph is typical of a system dominated by a central "point" mass.

(c) Vera Rubin found that galaxies have a different rotation profile than our solar system: v is nearly constant with respect to R. Sketch that relation on a v versus R graph in your notebook. Do you expect that most of a galaxy's mass is located at a central point? Galactic light is concentrated in the center.

(d) "The Sun orbits the center of the Milky Way galaxy with an orbital speed of about 250 km/sec. The distance to the center of the galaxy is ... about 9.1 kpc. ... [For stars nearer the edge, at] a distance of 15kpc from the center of the Milky way, the rotational speed is still 250 km/sec. ... What percentage of the mass of the Milky Way lies between 9.1 and 15 kpc? ... What can one say about the mass gravitationally revealed in the outer regions of the Milky Way," where there is little mass visible in the form of stars? (from Ferguson's Introductory Astronomy Exercises, p.160)

Data: one parsec = 1 pc = 2.06 x 10⁵ AU, (k = kilo = 1000)

AU = astronomical unit = Earth's mean distance from Sun = 1.5 x 10⁸ km)

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GOAL (5): To do effective collaborative work with small and large groups of people with diverse skills and backgrounds.

(a) What did you learn about working in your small group? Discuss with teammates what went well and what you'd like to do differently next time.

(b) What did you learn from sharing information with nearby table and the rest of the class? If you have suggestions for changes, be sure to indicate them on the blue worksheet your group turns in.

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What other questions or ideas occured to you in the course of this workshop?

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