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# 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. 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.

1. If we estimate the orbits of Earth and Jupiter as circles (a good approximation), Jupiter is closest to Earth when it is at opposition.  How far away from Earth is it?

### Part 1: Observing Jupiter

First, make sure you understand how the software works and what you are looking at. 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?

2. 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)?

3. 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.
If you loose the lock on Jupiter, you can go to the Find tab, click the arrow icon next to Jupiter, and choose "Centre".
Hold the cursor over one of the moons to show the HUD, which will tell you which moon you are looking at as well as other information.

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.
 Date Time Hours since start Angular Separation Io Europa Ganymede Callisto 0 2 4 6 8

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

## Part 3: generating the orbit

1. Your GSI will give you a data table with the angular separation in both arc minutes and seconds (like what you measured from Starry Night) and in radians.  There are also 4 graphs of angular separation in radians vs time in hours, one for each of the moons.
2. 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.
 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. Below is a sketch of the positions of Earth, Jupiter (solid circle) and one of its moons at the maximum apparent distance as seen from Earth. Label the Earth-Jupiter distance with the distance you found in the preview questions, and the semi-major axis of the moon's orbit with "a".

5. The right-triangle small angle formula is where the opposite and adjacent legs must be in the same units. Use this formula and the figure above to convert the semi major axis for each moon from radians to the unit in column 4 of table 3.  Show a sample calculation here.

6. 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.

 Moon Mass of Jupiter (__________) Io Europa Ganymede Callisto
1. Average the 4 values of M to get the mass of Jupiter  MJ = ______________________ solar masses.

## Concluding Questions

1. The currently accepted mass of Jupiter is 9.54x10-4 solar masses. Is this very close to your answer? Explain your answer.

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

4. If Mars had a moon with the same period as Io, will it have the same semi-major axis? Explain.

5. Starry Night was able to tell you what the angular seperation was. How do you think Galileo figured out the angular seperation (give one possibility)?