Spring 2003
 Orbits reveal central mass Astronomy & Cosmologies

This workshop is designed to be done in groups of 2-3. There are 5 parts; spend about 40  minutes each on parts 1-3, and finish by spending about 10 minutes on part 5 (we will return to part 4 later).  Bring your work to the next class, so we can compare results and build on your understanding.

GOALS:

(1) To become more comfortable and skilled at interpreting graphical representations of information; to learn to use graphs to express your understanding of quantitative or qualitative relationships.

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

(3) To better understand Newton's 2d and Kepler's 3d laws, and use them to find Jupiter's mass.

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

(5)  To develop skills collaborating with small and large groups, sharing data and discussing ideas.

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

EQUIPMENT:
various oscillators
plot of orbits of Jupiter's moons
clear plastic rulers
ACTIVITIES:
 1.a. Oscillations:  qualitatively 2.  Orbits:  size and period (plot) 4.  Dark matter in the Galaxy 1.b. quantitatively 3.  Laws -> Mass (derivation) 5.  Collaborations  (feedback)

Data: universal gravitational constant G=6.67 x 10-11 m3/kg.s2
one parsec = 1 pc = 2 x 105 AU (k = kilo = 1000, so kpc = 1000 pc)
AU = astronomical unit = Earth's mean distance from Sun = 1.5 x 108 km)

1.  How can you represent oscillations graphically?
First you will try your hand at sketching qualitatively how the position of an oscillator varies in time,
then you will make measurements to quantify your graph.   This will prepare you for interpreting the orbit plots in part 2.

(a) Qualitatively (without numbers):  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.  In your own words, how does the angular displacement (or height) change with time?
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Everyone in the group should sketch your own representation of the pendulum's displacement versus time. Don't worry about assigning scales to your graph's axes yet.
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Then compare and discuss your graphs. What are the main features shared by the graphs which your group agrees on?
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Sketch your team's graph on the board to compare with other groups.
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(b) OPTIONAL Quantitatively (with numbers): Now measure the period for your pendulum.  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. (Hint:  you can get more accurate measurements by timing, say, 10 periods together and then dividing by ten.  Check each other's counting to make sure you get ten, not nine, periods).

Post your pendulum data on the chart on the board.

Use your data to label time and distance scales on your graph in part (a) above. Where is the displacement zero?  Where are the points of maximum displacement?  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  Group initials Length L(cm) 10 cm 20 cm 30 cm 40 cm 50 cm 60 cm 70 cm 80 cm 90 cm 100 cm Period T(sec) +/- DT

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.

(2):  Investigate relationships between the size and period of a gravitationally bound circular orbit.

(a) 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. First answer each question on your own.

Which moon has the fastest orbit? The slowest?  How can you tell?
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Which moon orbits furthest from Jupiter? which orbits closest?  How can you tell?
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Discuss your results with teammates until you reach a consensus.  Predict your moon's rank, in orbit speed and distance from Jupiter.
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(b) 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.  Which is your moon?
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(c) Before you do any calculating, first qualitatively predict how your moon's orbit will compare with the others. Second shortest period? Third largest orbit radius?  Explain your reasoning.
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(d) Quantitative analysis of your moon's orbit: Check the time and space axes on your graph to make sure you understand their units.

What changes as you go across the page, left to right?  What is the circle in the middle?
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What changes as you go down the page?  What are the units?
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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).
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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)
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Post your results on the chart on the front board, and tabulate the class results in your notebook.  INCLUDE YOUR UNITS!  Group initials Moon Io Io Europa Europa Ganymede Ganymede Callisto Callisto Orbit Radius Period Speed (pt.4)

(3) : Use Newton's and Kepler's laws to find Jupiter's M from your moon's T and r.

(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?
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(b) Recall the derivation of Kepler's 3d law from class.  Start with F=ma (Newton's 2d law) and derive an algebraic relationship between T and r for a circular orbit.
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Where is the central mass represented, e.g. Jupiter?  Where is the mass of the moon, or the body orbiting at a distance r?
How does the moon's mass affect its orbit?
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(c) What power of T should you plot against what power of r to get a straight line graph?

Do it with the class data on Jupiter's moons. Compare to your graph in (a) above.  Explain the curve in your own words.
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(d) You can also use your measurements of your moon's period and orbit radius to find the mass of Jupiter!
First, solve your equation for M, the central mass about which the moons are orbiting.
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Now plug in your moon's data to get a number for Jupiter's mass, M.  Do your units work out?
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How uncertain is your answer?  If your radius could be off 4% and your period could be off 5%, then since M is proportional to r3 and T2, M could be off 3*4% + 2*5% = 22% (this is probably an overestimate - errors could either cancel or add).
 % uncertainty in r your estimate of M % uncertainty in T times % uncertainty in M % uncertainty in M DM = uncertainty in M

Check your Jupiter mass against the results of other groups, with each group using only their own moon's r and T. Don't look it up in the textbook!
If any values are terribly different, discuss with that group to try to resolve the difference.
When every group is confident of their calculations, then average the results.  As a class, what do you find for Jupiter's mass M +/- DM?
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(4): Extend your understanding to stars orbiting in the galaxy,
and find the distribution of dark matter in the Milky Way!

(a) The "velocity profile" or "rotation curve" of a system of orbiting masses tells you how the system's mass is distributed, whether you the mass emits light or not. Graph speed v 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.
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(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.
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(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. Do you expect that most of a galaxy's mass is located at a central point? Galactic light is concentrated in the center.
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(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)
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(5): Reflect on your collaborative work.

(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.  Tell me about it on your blue workshop feedback form.

(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.

Maintained by: E.J. Zita