Download Jupiter and its Moons Fromm

3 August 2009
TAM
University of San Francisco
Physics 12_ Laboratory #_1
JUPITER AND ITS MOONS
Kepler’s Third Law
OBJECTIVES



Learn how to gather and analyze data
Determine the mass of Jupiter.
Gain a deeper understanding of Kepler's third law
INTRODUCTION AND BACKGROUND
Astronomers cannot directly measure many of the things they study, such as the
masses and distances of the planets and their moons. Nevertheless, we can deduce some
properties of celestial bodies from their motions despite the fact that we cannot directly
measure them. In 1543, Nicolaus 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. Kepler’s
Third Law is the one that applies to this lab. For a moon orbiting a much more massive
parent body, it states the following:
a3
M 2
T
where M is the mass of the parent body in units of the mass of the sun ( M ). a is the
length of the semi-major axis in units of the mean Earth-Sun distance, 1 A.U.
(astronomical unit). If the orbit is circular (as we assume in this lab), the semi-major
1
This laboratory exercise is based and draws extensively upon The Revolution of the Moons of Jupiter
Student Manual produced by Department of Physics, Gettysburg College, Gettysburg, PA 17325 as part of
the Contemporary Laboratory Experiences in Astronomy (CLEA) program sponsored by Gettysburg
College and the National Science Foundation. Other input was taken from Kepler’s 3rd Law and the Orbits
of the Moons of Jupiter a laboratory exercise produced by the Astronomy Department, University of
Michigan, Ann Arbor, MI 48109. Modifications and additions are due to P.McNeff, T. A. Mulera and A.
Venkatesan of USF’s Department of Physics. Thanks to K. R. Scanlon for “test driving” the exercise.
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axis is equal to the radius of the orbit. T is the period of the orbit in Earth years. The
period is the amount of time required for the moon to orbit the parent body once.
In 1609, the telescope was invented, allowing the observation of objects not
visible to the naked eye. Galileo used a telescope to discover that Jupiter had four large
moons orbiting it and made exhaustive studies of this system, which was especially
remarkable because the Jupiter system is a miniature version of the solar system.
Studying such a system could open a way to understand the motions of the solar system
as a whole. Indeed, the Jupiter system provided clear evidence that Copernicus’
heliocentric model of the solar system was physically possible. Unfortunately for
Galileo, the inquisition took issue with his findings; he was tried and forced to recant.
We will observe the four moons of Jupiter that Galileo saw through his telescope,
known today as the Gallilean moons. They are named Io, Europa, Ganymede and
Callisto, in order of distance from Jupiter. You can remember the order by the mnemonic
“I Eat Green Carrots”.
If you looked at Jupiter through a small telescope, you might see the following:
Figure 1. Jupiter and Moons through a Small Telescope
The moons appear to be lined up because we are looking edge on at the orbital plane of
the moons of Jupiter. If we watched, as Galileo did, over a succession of clear nights, we
would see the moons shuttle back and forth, more or less in a line. While the moons
actually move in roughly circular orbits, you can only see the perpendicular distance of
the moon to the line of sight between Jupiter and Earth. If you could view Jupiter from
“above” (see Fig. 2), you would see the moons traveling in apparent circles.
VIEW FROM ABOVE THE PLANE OF ORBIT
Rapparent shows the apparent distance
between the moon and Jupiter that would
be seen from earth.
Rapparant  Rmax sin t
 , the angular velocity, is constant
Figure 2. View of Jupiter and one of its moons perpendicular to the orbital plane of the moon.
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As you can see from Fig. 3, the perpendicular distance of the moon should be a
sinusoidal curve if you plot it versus time. By taking enough measurements of the
position of a moon, you can fit a sine curve to the data and determine the radius of the
orbit (the amplitude of the sine curve) and the period of the orbit (the period of the sine
curve). Once you know the radius and period of the orbit of that moon and convert them
into appropriate units, you can determine the mass of Jupiter by using Kepler’s Third
Law. You will determine Jupiter’s mass using measurements of each of the four moons;
there will be errors of measurement associated with each moon, and therefore your
Jupiter masses may not be exactly the same.
The apparent position of a moon
varies sinusoidally with the changing
angle from the line of sight, , as it
orbits Jupiter. Here the apparent
position is measured in units of the
radius of the moon’s orbit, R, and the
angle measured in degrees.
Figure 3. Graph of the apparent position of a moon viwed edge on at the orbital plane of the moon.
The CLEA 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 also allows convenient measurements to be made at a computer
console, as well as adjustment of the telescope’s magnification. The computer simulation
is realistic in all important ways, and using it will give you a good understanding of how
astronomers collect data and control their telescopes. Instead of using a telescope and
actually observing the moons for many days, the computer simulation shows the moons
to you as they would appear if you were to look through a telescope at the specified time.
EQUIPMENT AND MATERIALS
1. Windows computer or Macintosh Computer running a Windows emulator, e.g.
Parallels.
2. The CLEA program Revolution of Jupiter Moons. Installation instructions on p.9.
3. Scientific calculator.
PROCEDURE
Activity I-Starting the program and familiarizing yourself with the Jupiter system
1. Start up the program by double clicking on the Revolution of Jupiter Moons icon.
2. Select Log in from the File menu. Enter your name(s) and table number and click
OK.
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3. The title page screen should appear. Select File>Run and click OK to accept the
default values for Start Date and Time. The Observation Screen should appear as in
Figure 4 below.
4. Jupiter appears in the center of the screen, while the small, point-like moons are on
either side. Sometimes a moon is hidden behind Jupiter and sometimes it appears in
front of the planet and is difficult to see. You can display the screen at four levels of
magnification by clicking on the 100X, 200X, 300X, and 400X buttons. The screen
also displays the date, Universal Time (the time at Greenwich, England), the Julian
Date (a running count of days used by astronomers; the decimal is an expression of
the time), and the interval between observations (or animation step interval if
Animation is selected).
5. Select File>Features>Animation and then click the Cont. button on the Observation
screen. Watch the moons zip back and forth as the time and date scroll by.
6. Select File>Features>Show top view and click OK. You can also select ID color to
avoid confusing the moons.
Figure 4
OBSERVATION SCREEN
When you have familiarized yourself with the program and the motions of the
Jovian moons, turn off the Animation and proceed to section II.
Activity I - Set up observing sessions
1. Return to the Observation Screen.
2. Select File>Observation Date> Set Date/Time. Type the date and the time in
the appropriate spaces. Universal Time = PST + 8 hours. Click OK.
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3. Select File>Timing. Set the Observation Interval to 12.00 hrs. Click OK.
Activity III - Collect Position Data for Moons
1. In order to measure the position of a moon, move the pointer to the moon and
left-click the mouse. The lower right-hand corner of the screen will display the
name of the moon (for example, II. Europa), the X and Y coordinates of its
position in pixels on your screen, and its X coordinate expressed in diameters of
Jupiter (Jup. Diam.) to the east or west of the planet’s center. To measure each
moon’s position accurately, switch to the highest magnification that will keep
the moon on the screen.
2. Click on the Record button. The Record/Edit Measurements dialog box will
appear and show your measurement in Jupiter diameters East or West of center.
Click OK.
3. Repeat 1 and 2 above until you have measured and recorded the positions of all
visible moons. If a moon is eclipsed by Jupiter leave the measurement blank.
4. Click Next on the Observation Screen and repeat 1 through 3 above. Collect a
total of at least eighteen sets of measurements. To safeguard your data in the
event of a computer crash use File>Data>Save after each set of measurements.
Save as a .CSV file.
If the picture of Jupiter and the moons is suddenly replaced with a picture of a
cloudy sky just click Next again and procede.
Activity IV - Plot Graphs of Your Measurements
1. On the Observation Screen select File>Data>Analyze. A window titled “Jupiter
Satellite Orbit Analysis” will appear. Select Data>Select Moon>Callisto (IV)
in this window. Select Data>Plot>Plot Type>Connect Points and print using
Data>Print>Current Display.
2. Repeat 1 above for the other three moons.
3. Attach plots to answer sheet
Activity V - Determine the Period and Semi-Major Axis for the Orbits oEach
Moon and Calculate the Mass of Jupiter
1. On each graph measure the period (time for a complete cycle) and semi-major
axis (amplitude) in units of days and Jupiter diameters (J. D.). See Figures 2 and
3. Your graphs will probably not be this smooth but measure as well as you can.
Enter your measurements in Table 1 on the answer sheet.
2. Convert these measurements to terrestrial years and astronomical units. 1 yr =
365 days and 1 A. U. = 1050 J. D. Enter in Table 1.
3. Apply Kepler’s 3rd Law to calculate the mass of Jupiter in units of solar masses.
Enter the result for each moon in Table 2.
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University of San Francisco
Physics 12_ Laboratory #_
Answer Sheet
Lab section: ________________
(Day-Time)
Lab Partners:
Date: ___________________
(mm/dd/yyyy)
_____________________________
_____________________________
Each member of the lab group
should hand in an original lab
report. Photocopies will not
be accepted.
_____________________________
_____________________________
Circle your name in the list at
left.
Attach your tables of values for the positions of the planets and your graphs to these
answer sheets.
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Table 1 - Results from Graphs
Moon
a (J.D.)
T (days)
a (A.U.)
T
(years)
Table 2 - Results of Mass Calculations
Moon
Mass of
Jupiter
M 
The average calculated mass is M J = ______________________ solar masses.
From your text book, the mass of Jupiter is 9.54 x 10-4 solar masses. Your percentage
error is:
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Percent error =
M J  9.54 104
100 = ____________________%
9.54 104
Questions
TEuropa
Consider the three inner satellites. Form the ratios
. What do
TIo
TIo
you notice about the relative values of the three periods? To what phenomenon does
this give rise?
2.
There are moons beyond the orbit of Callisto. Will they have larger or smaller
periods than Callisto? Why?
3.
The orbit of Earth’s moon has a period of 27.3 days and a radius of 2.56 x 10-3 A. U.
or 3.84 x 105 km. What is the Earth’s mass  M   in units of solar masses  M  ?
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and
TGanymede
1.
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Installation of Revolution of Jupiter Moons Software
Go to CLEA website: http://ww3.gettysburg.edu/~marschal/clea/CLEAhome.html
Click on Software on left side of page.
Click on the Revolution of the Moons of Jupiter icon.
“File Download – Security Warning” window will open.
Click Save in this window. Set save destination to Desktop and procede. Save will take a
good 5 or 6minutes and should leave an icon tiled JupLab on your desktop.
Double click on JupLab. “File Download – Security Warning” window will open. Give
permission and follow instructions to install Revolution of Jupiter Moons. An icon with this
title should appear on your desktop.
Return to p. 3 to run program.
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