University of Michigan - Department of Astronomy

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Version: Lab

Kepler's 3rd Law and the Moons of Jupiter

Worksheets

Instructions for the software, a brief history and background on Kepler's 3rd law are in the introduction.

Preview Questions

  1. Write down Kepler's three laws of planetary motion.



  2. If you write down Kepler's third law as a3=P2 for planets in the solar system, what are the units of length, time and mass used? Can we use this formula as it stands for the satellites orbiting Jupiter? Why not?




The sketch below shows the orbits of Earth and Jupiter around the Sun approximated as circles.  The position of Earth is indicated by the circle with the + sign.  The average Sun-planet distance is also shown.Diagram of the orbits of Earth and Jupiter 

  1. Indicate the position of Jupiter at opposition and label this point “O”. Indicate Jupiter’s position at conjunction and label it point “C”.
  2. Indicate Jupiter’s position when it is closest to Earth.  Label it “M” (for minimum distance).
  3. Indicate Jupiter’s position when it is farthest from Earth and label it “F”

 


  1. Based on your labels, what position is Jupiter at when it is closest to Earth?  How far away is it?  Show your work.

 

 

  1. Based on your labels, what position is Jupiter at when it is farthest from Earth?  How far away is it?  Show your work.

 

 

 

Part 1: Observing Jupiter

See the introduction for instructions on using the software. Use the step time forward button to step forward 1 hour at a time.  Watch Jupiter and the other objects on the screen.  Stop when Jupiter sets.  It may be useful to step backward and forward again, or to zoom in and out while answering the following questions.

  1. Does Jupiter remain in a fixed orientation?  Why would it change?

 

  1. There are both stars and moons in the field of view.  You can tell what the objects are by holding the pointer over them to show the heads-up display.  How would you tell them apart just by looking through the telescope (hint, watch Io)?

 

 

  1. If the moons are on the left side of the planet in the evening (7PM), are they east or west of the planet?  What about at midnight, and in the morning? You may want to zoom out for the cardinal points.

 

 

Part 2: measuring the orbits

Go back to 6 PM on Jan 7 1610.
Change the time step to 2 hours.
Go to the options tab along the side.  Find the “Stars” section and un-check the box.  This will turn off the stars so the only things visible should be solar system objects.  Click the options tab again to collapse it.
Hold the cursor over one of the moons to show the Heads-Up Display of information about it. 

  1. Sketch Jupiter and the Galilean moons.  Label the four moons.



  2. Switch to the “angular separation” cursor tool. Hold the cursor over Jupiter – the information HUD should appear and say you are on Jupiter (sometimes the little moons get in the way). 
  3. Click and drag to Io.  The angular separation between Io and Jupiter will show up.  Record the angular separation in table 1.  If Io is to the west, record the angular separation as a negative number.  If Io is in front or behind Jupiter, record its distance as zero.  Also record the date and time of the observation.  The hours since the start are recorded for you. Note if the HUD gets in the way, you can start with the mouse over the moon, and click and drag to Jupiter. Again, make sure the HUD says you're over Jupiter, not one of the smaller moons.
  4. Return to Jupiter.  Click and drag to Europa. Record the angular separation in table 1, again making it negative if it is to the west.
  5. Repeat for Ganymede and Callisto.
  6. Step forward 2 hours.
  7. Record the date and time of the observation in table 1.
  8. Measure the angular separation for each of the moons again and record the numbers in table 1.  Don’t forget to make the numbers negative if they are west of Jupiter, or zero if they are aligned with the planet.
  9. Continue stepping forward at 2 hour intervals until Jupiter sets.  Table 1 should be filled in when you are done.
Table 1: Observations from Jan 7 1610

Date

Time

Hours since start

Angular Separation

Io

Europa

Ganymede

Callisto

 

 

0

 

 

 

 

 

 

2

 

 

 

 

 

 

4

 

 

 

 

 

 

6

 

 

 

 

 

 

8

 

 

 

 

  1. Make another sketch of Jupiter with its moons labeled.







  2. Which moon moved the most?  Which moon moved the least?  What does that tell you about these moons? 

  3. How will you know when the moon has completed a full orbit?  Check your answer with your GSI.







  4. Create a new table, similar to table 1, but leave yourself a lot more room.  Call it table 2. Information on using excel for creating the tables and graphs can be found at https://dept.astro.lsa.umich.edu/ugactivities/Labs/Comp/index.html.
  5. Select the Options tab.  Turn off Daylight and Local Horizon under Local View. You should be able to see Jupiter and it’s moons again.  Change the time interval to 4 hours.
  6. Continue stepping forward by 4 hour intervals until you have a full orbit of Io.  Record the data for the other moons as well. Make sure you record the distances as negative or positive depending on what side of Jupiter they are on.  You’ll have to watch what’s happening since you can’t get the direction with the horizon turned off.
  7. Change the time interval to 8 hours. Record the date, time, hours since start and angular separation for the three outer moons.  Continue until you have a full orbit for Europa.
  8. Change the time interval to 16 hours. Record the data for just Ganymede and Callisto
  9. Finally, change the interval to 24 hours and collect the data for Callisto.

Part 3: generating the orbit

  1. Convert you angular separations to radians.  Record the results in 4 new columns added to table 2, or create a new table with 4 columns for the angular separation in radians for each moon, plus a column for the hours since start. (reminder, there are 60” in 1’, 60’ in 1º and 180º=pi radians.)
  2. Plot angular separation in radians vs hours since start for each moon (4 graphs).  Your graphs should look like sine curves.
  3. Determine the period and semi-major axis from each graph.  Record these in table 3.  Reminder, the period is the time it takes the moon to go around the planet, which is also the time it takes for the graph to repeat.  The semi-major axis is the average distance from the planet, which is also the average of the maximum eastward and maximum westward distances.
Table 3: results from graphs

Moon

a (rad)

P (hours)

a (_______)

P (_______)

Io

 

 

 

 

Europa

 

 

 

 

Ganymede

 

 

 

 

Callisto

 

 

 

 

  1. In the pre-lab questions, you figured out the units you need for P and a.  Record those units in the top of table 4
  2. Convert P into the right units and record them in table 4.
  3. Jupiter was just past opposition during these observations.  Based on the pre-lab questions, how far away is Jupiter during these observations?  Explain



  4. Use the right-triangle small angle formula, small angle formula where the opposite and adjacent legs must be in the same units, to convert a from radians to AU and record the results in table 3.  Sketch the right triangle and show a sample calculation here.





  5. Calculate M for each of the 4 moons using the equation for Kepler’s 3rd law was given in the introduction.  Record the unit and your results in table 4, and show a sample calculation here.


Table 4 - Results of Mass Calculations

Moon

Mass of Jupiter (__________)

Io

 

Europa

 

Ganymede

 

Callisto

 

  1. Average the 4 values of M to get the mass of Jupiter  MJ = ______________________ solar masses.
  1. The currently accepted mass of Jupiter is 9.54x10-4 solar masses. Your percentage error is:

Percent error = (MJ - 9.54x10-4)/(9.54x10-4) x 100% = ______________________%

Concluding Questions

  1. There are moons beyond the orbit of Callisto. Will they have larger or smaller periods than Callisto? Why?





  2. Which do you think would cause the larger error in MJ:  a ten percent error in P or a ten percent error in a? Why?





  3. Convert your mass of Jupiter into kg.  Show your work.





  4. The mass of Earth is 5.997x10^24 kg.  How many times bigger is Jupiter?


  5. Why were Galileo's observations of the orbits of Jupiter's moons an important piece of evidence supporting the heliocentric model of the universe (or, how were they evidence against the contemporary and officially adopted Aristotelian/Roman Catholic, geocentric view)?



Last modified: 2/25/08 by SAM