PSC 102 Introduction to Astronomy Lab
The Revolutions of the Moons of Jupiter
Lab manual based on the Physical Sciences 21 Laboratory by
Fred Ringwald & Steve White California State University
Fresno 2003 and the CLEA student laboratory manual.
PURPOSE: To measure the orbits of Jupiter's moons, and to use Kepler's Third Law to
find the mass of Jupiter.
Why is this important? This lab gives a taste of just what it is that observational
astronomers (and most experimental scientists of all kinds) do in their day-to-day work.
They measure from their observations (or experiments), look at the numbers carefully,
and use patterns in these numbers to find new knowledge about the Universe.
REQUIREMENTS:
This lab is to be done in groups of two. You should switch off so both get a chance to
use the computer. This lab is a computer simulation, which is based on the real orbits for
each moon: if you were to set the simulation for today's date and time, you could verify
the positions of Jupiter’s moons by observing them with a telescope.
INTRODUCTION AND HISTORICAL BACKGROUND:
In 1609, Galileo Galilei heard of an invention by Hans Lippershey, a Dutch spectacle
maker. By using two lenses, Lippershey found that distant objects could be made to look
nearer. This instrument was called a spyglass. Without even having seen an assembled
spyglass, Galileo was able to construct his own telescope, and soon became famous as the
builder of the world's best telescopes.
Galileo immediately began to observe objects in the sky. He was a careful observer, and
in 1610 published a small book, "The Starry Messenger." His discoveries caused great
excitement. Suddenly, whole new worlds were opened! Galileo found spots on the Sun,
and craters on the Moon. He found that Venus went through a series of phases, much as
the Moon has phases. He was able to tell that the Milky Way was composed of
"innumerable stars". He could see there was something strange about the shape of Saturn,
but his small telescopes could not quite resolve its rings.
Galileo also discovered the four largest moons of Jupiter, and recorded their positions
night after night. Jupiter and its moons form a "miniature Solar System," with the moons
moving around, or orbiting, Jupiter. These four moons are now known as the Galilean
satellites. Their names are Io, Europa, Ganymede, and Callisto, in order of distance from
Jupiter. One can remember the moons in order with the memory aid "I Eat Green
Carrots."
In 1543, Nicolaus Copernicus had published a theory that all the planets, including the
Earth, move in circular orbits around the Sun, and that only the Earth’s Moon orbited the
Earth. The Galilean satellites provided the first clear evidence that the Copernican theory
was physically possible. Prior to Galileo’s discovery, an objection to the Copernican
theory was that a moving Earth would leave its Moon behind. The motion of the satellites
around Jupiter proved that a moving planet would not lose its moons.
Figure 1. Galileo’s drawings of Jupiter and its four largest moons, during 1620.
Many people were upset by the idea that the Earth was not the center of the Universe, and
Galileo was ordered in 1616 neither to "hold nor defend" the Copernican theory.
Nevertheless, in 1632 he published his Dialogue on the Two Great World Systems, which
was a thinly disguised defense of the Copernican system. This led to his forced retraction,
under the threat of torture, and confinement to his home for the rest of his life.
KEPLER'S THIRD LAW:
Over 20 years up to 1601, the Danish astronomer Tycho Brahe carefully used a sextant
and compass to observe the locations of the planets and stars. These observations were
used by his assistant, Johannes Kepler, to deduce three mathematical laws governing the
orbit of one object around another. Kepler's Third Law (1619), for a moon with a circular
orbit around a much more massive parent body (here, the planet Jupiter), is:
P2  a3
a is the semi-major axis of the orbit, measured in Astronomical Units, or A.U. One A.U.
is the average distance from the Earth to the Sun.
P is the period of the orbit, measured in years. The period is the time required for a moon
to complete one orbit around its parent body (here, the planet Jupiter).
Kepler’s finding was empirical: he could show from the observations that this law works,
but he didn’t know why it worked. In 1687, Isaac Newton showed why: a moon’s orbit
depends on how strong its parent body’s gravity is. Gravity depends on mass, so that if:
M is the mass of the parent body, in solar masses (The Sun has 1 solar mass.) then
a3
M  2
P
We will use computers to simulate observing one of the four Galilean moons of Jupiter.
The moons appear to be lined up because we are looking edge-on to the Jupiter system.
As time goes by, the moons move around Jupiter, each one following its own circular
orbit. From Earth, we will see the moons traveling in a back-and-forth motion.
Jupiter and one of its
moons, as seen from Earth.
Again, Jupiter and one of its moons,
seen from a point above Jupiter.
orbit
Jupiter
a
x
moon
To Earth
Figure 2.
From this figure, we see that x is the moon’s apparent distance from Jupiter, as we see it
from Earth. Therefore, a plot of x versus time should give a curve that repeats: such a
curve is called a sinusoid, or a sine wave. One repetition of this curve is called a cycle.
By taking enough measurements of the position of a moon, one can fit this curve to the
data and determine the radius of the orbit, also called the amplitude of the cycle. One can
also find the period of the orbit, which is the time it takes to repeat. Once one knows the
semi-major axis and period of the orbit of that moon, and converts them into appropriate
units, one can find the mass of Jupiter by using Kepler's Third Law. We will determine
Jupiter's mass using the data from one of the four moons. It is really quite amazing that
we can use careful observation and a law of nature to determine the mass of Jupiter!
Using the computer controlled telescope to take data
1. You must first select Log In from the menu before taking data. You don’t need to
enter your names: just hit “Return” to get by this.
2. Choose one of the four moons. Important: Before taking any data, set the
corresponding observation interval:
Io
Europa
Ganymede
Callisto
2 hrs
4 hrs
8 hrs
16 hrs.
Under File > Preferences > Timing… you will find the Observation Interval. Set it to
the value listed above based on whatever moon you have chosen.
IMPORTANT: Ask your lab instructor to check this, before going any further.
(Note to lab instructors: make sure the students get this right, since the data will be
useless if they use the default value of 24 hours, since they are too sparsely sampled for
all the moons except Callisto.)
3. Once you have set this interval begin the exercise by selecting File > Run…. You will
see another window appear with today’s date and the time, just hit return to begin. You
can display the screen at four scales of magnification by clicking on the 100X, 200X,
300X and 400X buttons at the bottom of the screen. In order to improve the accuracy of
your measurement of a moon, you should use the largest possible magnification that
leaves the moon on the screen.
4. In order to measure the apparent distance of a moon from Jupiter, move the pointer
until the tip of the arrow is centered on the moon and click the mouse. Information about
the moon will appear at the lower right corner of the screen. This includes the name of
the selected moon, the x location on the screen, and the apparent distance (in units of
Jupiter diameters) of the moon from Jupiter. Also included is an E or W, to signify
whether the moon is east or west of Jupiter. If the moon's name does not appear, center
the arrow more exactly on the moon: it should show up. Select Record Measurements…
at the top of the window. Record the moon's position in the data table. For example, if
Europa were selected and it had an x = 2.75W, you would enter 2.75W next to Europa
[II]: in the data table.
5. When you have recorded the position (that is the apparent distance from Jupiter) for
your moon in the data table, you may make the next observation by clicking on the Next
button. Note that under the Preferences menu you can also choose ID Color to avoid
confusing the four moons.
6. In total you should take between 18 and 20 measurements, which should be enough
data points for the moon to return to the approximate position it was in when you began
taking measurements. Remember to let your lab partner take some of the measurements.
7. In order to make the simulation more realistic, every now and then, you get a blank
screen and a message about "cloudy weather." This is to stimulate what astronomers
actually face sometimes when they try to make observations. Go on to the next reading.
Plotting the Data
1. The software will now plot your measurements and draw a sine curve from which you
can determine the period and semi-major axis. Under File > Data > Analyze… you will
find the plotting and curve-fitting tool. You will first be asked to select the moon to have
plotted which you can choose from the menu at the top of the window. Each
measurement of the moon's apparent separation from Jupiter will show up as one dot on
the graph for that moon. In order to find the period and the semi-major axis of the orbit of
the moon we have to fit a curve to the data points.
By plotting position versus time, you will use the data to draw a graph similar to this:
X
P
o
s
i
t
i
o
n
J.
D.
P = 1 Period
4
3
2
1
0
-1
a
-2
-3
-4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Time (hours)
Each dot in the figure is one observation of the moon you chose. Note the irregular
spacing of dots, due to poor weather and other problems on some nights. The curve
drawn through the points is the smooth curve that would be observed if you decreased the
observation interval enough. The shape of the curve is called a sinusoid, or sine wave.
Using the data from the moon you choose, you can find the semi-major axis and period of
the orbit. The period is the time it takes to go all the way around the orbit. Thus the time
between two peaks is the period, P. The semi-major axis of the orbit, a, is the greatest
apparent distance your moon is from Jupiter. This is also the greatest distance (in Jupiter
diameters, or J.D.) the peak is from zero. You can read both a and P directly from your
plot, as you can in the one above.
2. When the graph of data points appears, try to find a pattern by eye, and take note of
any data points that seem to be out of place. To better see the pattern, select Plot > Plot
Type > Connect Points. This will display a jagged, “connect-the-dots” version of your
graph. Click on a point at which the line connecting the points crosses from negative to
positive. (If this happens several times, choose a point toward the left of the screen.) A
date and a value very close to zero should appear in the box marked Cursor Value.
Select Plot > Fit Sine Curve > Set Initial Parameters and enter the date you just found
as T-zero. You will also be asked for two other values, the Period and the Amplitude.
You will need to make an educated guess to their values based on the data points in order
to find a curve which best fits the data. To find the period you need to find the time
difference between two peaks or two valleys in your data. The amplitude is the distance
the peak is above zero. Enter these values and hit return.
3. You should now see a blue sine curve over you data points. It should be a rough fit for
all your data points. If not, you may need to repeat the Set Initial Parameters step. Now
you will begin adjusting the fit with the three scroll bars that have appeared. First adjust
the T-Zero point. As you adjust this value, the entire curve will slide to the left or right.
Try to achieve the best fit for the data points closest to the T-zero value you selected
above. At this stage, pay no attention to the number labeled RMS Residual; just use your
eyes. After you’ve adjusted the T-zero value, you may notice that the points farther away
from your T-zero point no longer fit as well. Now use the Period scroll bar to stretch or
shrink the curve and achieve a better fit. If at any time you cannot scroll far enough to
get a good fit, select Plot > Reset Scrollbars > Normal Sensitivity to re-center the bars.
Next adjust the Amplitude bar to better fit the points near the peaks and valleys. You
may wish to return to the other scroll bars for further adjustment, but the goal at this point
is a good, though not necessarily perfect, fit.
Now it is time to fine-tune the curve. Return to adjusting the T-zero scroll bar, this time
observing the RMS Residual. The smaller the value of this number is (note that it’s
expressed in scientific notation), the better the fit. Move the scroll bar with the arrows
until this number is smallest, that is, when one click in either direction would make it
larger. Continue this process with the Period and Amplitude bars. Read off the value of
the period and amplitude from the values to the right of each scroll bar. Enter these
values and the name of the moon on your worksheet. You can now compute the mass of
Jupiter!
Name: _____________________________
Lab Partner: _______________________
Date: ________________
The Revolutions of the Moons of Jupiter
Plot the data you measured, and read off the period, P and the orbital semi-major axis, a
from your graph for whichever one of the four moons you chose. These values will have
units of days for the period P, and Jupiter Diameters (J.D.) for a. Enter your results here:
P (period) = _________ days
a (orbital semi-major axis) = ________ J.D.
In order to use Kepler's Third Law, you need to convert the period from days into years
by dividing by 365. The semi-major axis must be converted to A.U. You can use the
relationship that 1050 J.D. = 1 A.U. Divide the semi-major axis by 1050 and enter your
converted values here:
P (period) = _________ years
a (orbital semi-major axis) = ________ A.U.
You now have all the information you need to find the mass of Jupiter with Kepler's
Third Law, in which:
a3
M  2
P
where:
M is the mass of Jupiter in solar masses. (The Sun = 1 solar mass.)
a is the semi-major axis of the orbit in A.U.
P is the period in years.
Calculate the mass of Jupiter, using the results from the motion of your moon.
Moon: _________________
MassJupiter = __________________ solar masses
QUESTIONS
1: The Earth's Moon has an orbital period of 27.3 days and an elliptical orbit with a
semi-major axis of 2.56 x 10-3 A.U. What is the mass of the Earth? In what units? Show
your work.
2: Express the mass of Jupiter in Earth units by dividing it by 3.00 x 10-6, which is the
mass of Earth in solar mass units.
MassJupiter =___________________________________earth masses