Name: ___________________________
Class: ___________________________
Date: ____________________________
Assignment #Y – Using Kepler to get
Jupiter’s Mass
!!
Kepler’s Third Law is an amazing equation! It tells us that the equation, ! = 𝐶, is the same for any
!
planet or comet or asteroid or spacecraft that orbits the Sun. In other words, if we know a planet’s distance from
the Sun, we can use Kepler’s Third Law to give us its period, or vice versa! Kepler’s laws apply not just to
objects circling the Sun, of course, but to any object orbiting any other object due to the force of gravity. The
constant C may change for different orbital systems, but Kepler’s laws are true anywhere! Kepler’s Third Law
is used over and over again in astronomy. Astronomers use it to measure the mass of the central object around
which an object orbits. In this assignment we’ll put it to the test! Previously, we measured the planets’ orbital
periods and radii, and see if Kepler’s Third Law holds true for the sun. Now we are going to link this set of
observations and relationship with the observations of Galileo.
Galileo is famous for a number of observations. For example, Galileo made observations of sunspots,
the phases of the planet Venus, the non-linearity of the moon’s terminator, and the moons of Jupiter. He may
not have been the first to make these observations, but he made the observations and published them in a timely
fashion. As an aside, some people can observe the moons of Jupiter with the naked eye! What we are going to
do is recreate the observations that Galileo did with respect to Jupiter and its moon. Most likely Galileo used a
20x telescope, which had a field-of-view (FOV) of 15 arc minutes. In Stellarium, we will use this value to
observe Jupiter. Remember 15’ is 0.25°. Use Scientific Notation to one decimal place, unless
otherwise stated.
PART A
The first task is to verify that the angle measuring tools works. Start Stellarium.
Now move your cursor to the lower part of the screen. This should bring up some menu items/tool
buttons. Just to the right of middle portion of these tools you should see 3 items: an angle, a circle, and what
looks like a belt buckle. The Angle is the angle-measuring tool.
If the Angle doesn’t appear, then you need to enable the tool. Move the cursor to the left portion of the
screen, and select the Configuration Menu (the icon is the wrench and star), then select the PlugIn menu, then
the Angle Measuring Tool. Now check the button at the bottom of the page, which will enable the tool on
startup. Finally, restart Stellarium.
Set the date to November 8, 2011, and the time to 2200 (10PM). Stop the progression of time. Set the
Field-of-view (FOV) to 153o. You should see Orion rising in the east.
1) What is the altitude of Rigel? ___________
2) What is the altitude of Betelgeuse? __________
3) What is the altitude of Jupiter? ___________
To utilize the Angle Measuring Tool…
a.
b.
c.
d.
Enable the tool by clicking the tool-bar button, or by pressing control-A. A message will appear at the bottom of
the screen to tell you that the tool is active.
Drag a line from the first point to the second point using the left mouse button
To clear the measurement, click the right mouse button
To deactivate the angle measure tool, press the tool-bar button again, or press control-A on the keyboard.
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With the angle-measuring tool active, we are going to do some measurements. Select Aldebaran and
center it in the screen. Make the FOV about 67o. You should see Rigel, Betelgeuse, and Jupiter. Move the
cursor over Rigel and click on it. Keeping your “finger down”, move the cursor over to Betelgeuse, and release
your figure. You should see a line between the two stars and a set of numbers like “8d 5m 3.5s”. This means 8
degrees, 5 minutes, and 3.5 seconds.
1) What is the angular separation between Rigel and Betelgeuse? _________.
o
Now click on Betelgeuse and measure the angular “distance” to Rigel; what is it? __________
o
Now click on Rigel again and measure the angular distance to Betelgeuse; what is it? ________
2) Now measure the angular separation between Rigel and Jupiter. _________.
o
Now click on Jupiter and measure the distance to Rigel; what is it? _________
o
Now click on Rigel and remeasure the distance to Jupiter; what is it? __________
3) There should be a spread in the numbers in 1 and 2, and they should be about the same. This is your
measurement error. What is your measurement error? _________
Advance the time by 2 weeks to November 29, 2011.
4) What is the angular separation between Rigel and Betelgeuse? _________.
5) What is the angular separation between Rigel and Jupiter? __________.
Compare your two sets of measurements between Rigel, Betelgeuse, and Jupiter. Remember that during
your first set of measurements, you couldn’t tell the difference to arbitrary accuracy. Your ruler is limited to
the size of your measurement error. It is basically wrong to talk about changes that are the size of your
ruler. For example, if you report a separation of 6 and your error is 5, then it is just as likely that there was
no change at all!
6) Did the angular separation between Rigel and Betelgeuse change over the 2 weeks? ________
o
If so, then how much? ________
o
Now divide the Rigel/Betelgeuse change in separation by your measurement error: _________
7) Did the angular separation between Rigel and Jupiter change over the 2 weeks? _________
o
If it did, then by how much? ________
o
Now divide the Rigel/Jupiter change in separation by your measurement error: _________
8) Explain the observations that you’ve just made with Rigel, Betelgeuse, and Jupiter.
PART B
The second task is to use the angle-measuring tool. We are going to do this by measuring the angular size of
Jupiter. Start Stellarium. If they are on, turn off the Ground, the Atmosphere and the Fog by pressing G, A
and F. Set the date and time to 24 October 2011, at 8PM (20:00). Open the Search window and search for
Jupiter. Press Enter. Set the field-of-view (FOV) to 0.25o, by “paging up or down”(PC) or “command
up/down arrow”(Mac).
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Select the angle-measuring tool. Move the cursor to the left side of Jupiter and press down. Continue the
pressure as you move the cursor to the right size of the planet and then release (in other words, click and drag).
1. What is the value of the angle? ________ Call this A1. The units will be in terms of degrees (d), minutes
(m), and seconds (s).
From the information presented in the upper left portion of the screen,
2. What is the distance in AU from Jupiter to Earth (DEJ)? __________. Convert to kilometers________.
3. What is the apparent angular diameter? ____________. Call this A2. The units will be in degrees (o),
minutes (‘), and seconds (“).
4. Now calculate the size of your “measurement” error. Subtract A1 from A2. Divide the result by A2 and
multiple by 100%. What is the value? ____________. Discuss the difference between A1 and A2.
1. Now let’s do the same for Io. First use the angle-measuring tool to determine the angular diameter of IO.
Call this ID1. __________
2. Select Io either by using the search button, or by using the cursor. What does the information bar (upper left
corner) “say” about the apparent diameter of Io (call this ID2)? ________
3. Now use your Astronomy book or some other source and find the diameter (in km) of Io (call this DI)
___________. What was your source? ____________
4. Given the diameter of Io and its distance from us, we can calculate the apparent diameter. Take DI and
divide it by the distance from “us” to Jupiter (DEJ) in kilometers (Call this ID3). _________
5. ID3 will be in radians, convert it to the units of ID1 and ID2, call this ID4. Remember that radians and
degrees/minutes/seconds are two ways to measure angles. Also remember that, π radians is equivalent to
180 degrees. ID1: _______ ID2: _________ ID4: ________
6. Compare the values ID1, ID2, and ID4. Which ones agree? Which ones don’t? Which one do you believe?
Why?
PART C
Start Stellarium. If they are on, turn off the Ground, the Atmosphere and the Fog by pressing G, A
and F. Set the date and time to 24 October 2011, at 8PM (20:00). Open the Search window and search for
Jupiter. Press Enter. Change the mount from equatorial to azimuthal. That way Jupiter and its moons will not
rotate so much on the computer screen. Finally, change the field-of-view until it reads 0.25°.
Now we are ready. Use the angle-measuring tool to measure the angle from the center of Jupiter to the
center of each of its moons. Do this for each day from 24 October 2011 to 7 November 2011. Remember that if
you use the “-“ and “=” keys you advance and retard the date by one day. Remember that these angles will have
both positive and negative values. So when the moon is to the right of the planet, label it as positive, when it is
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on the left side, label it as negative. Also, record the angle to the nearest arc second. So each entry would
most likely be in arc minutes and arc seconds.
Table 1
Date
24 October 2011
25 October 2011
26 October 2011
27 October 2011
28 October 2011
29 October 2011
30 October 2011
31 October 2011
1 November 2011
2 November 2011
3 November 2011
4 November 2011
5 November 2011
6 November 2011
7 November 2011
Io angle
Europa angle
Ganymede angle
Calisto angle
Now convert the above Table 1 into decimal arc-minutes and put the result into Table 2. KEEP ONLY
ONE DECIMAL PLACES FOR EACH MEASUREMENT! For example 1min 30 sec equals 1.5 min. Note
that if there was a negative angle in Table 1, there must be a negative number in Table 2.
Table 2
Date
24 October 2011
25 October 2011
26 October 2011
27 October 2011
28 October 2011
29 October 2011
30 October 2011
31 October 2011
1 November 2011
2 November 2011
3 November 2011
4 November 2011
5 November 2011
6 November 2011
7 November 2011
Io angle
Europa angle
Ganymede angle
Calisto angle
Probably what made Galileo someone to remember is exemplified by his recognition that these 4 objects
(Io, Europa, Ganymede, and Calisto) were revolving about Jupiter. He noticed that their motions were periodic.
Let us theorize that we are viewing the orbit from the side, and not from above. Now let’s explore this
periodicity. Plot the moon angles on the following chart. Plot the days from 24 October (so 0, 1, 2….) along
the horizontal and the moon angles vertically. Put all 4 sets of “moon” data on the plot. I’d presume that
Callisto has the biggest range, so choose your vertical axis to make sure that the Callisto data fits.
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Figure 1 - showing y =sin(x) and y = cosine(x)
Figure 1 shows how the sine and cosine behave. In our case we really have data that fits
!
𝑎𝑛𝑔𝑙𝑒 = 𝑎𝑛𝑔𝑙𝑒!"# 𝑠𝑖𝑛 2𝜋 + 𝜙 , where t is the time in days, T is the period in days, ϕ is the phase, and
!
anglemax is the maximum angle associated with the observation. We are not interested in the phase of the orbit,
just that it exists. So when you plot your data notice when the value of angle is at its largest, smallest, and when
it is zero. Those are special cases for the sine. When x=0 or π or 2π or 3π, the sine is zero. When x= π/2 the
sine is maximized. When x= 3π/2 the sine is minimized. It is in this way that we’ll find the period of the orbit.
Plot 1
From the data in Table 2 or the preceding chart, you should be able to estimate the two parameters of
each moons orbit. Namely, find the maximum angular distance from Jupiter to each moon, and the period.
Remember the period goes around the entire planet. So a half period is when the moon goes from one side of
the planet to the other. Sometimes it is easier to determine the half-period and multiply by two in order to
estimate the period. Put these estimates into Table 3. The values that go into Table 3, must be
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representative to your measurements from Plot 1. If they are not, then you haven’t done the exercise. To get
the distance from the angle review Part B.
Table 3
Moon
Maximum Angle
(arc-minutes)
Maximum Angle
(radians)
Maximum distance from
Jupiter (AU)
Period (days)
Io
Europa
Ganymede
Calisto
You may have noticed that it is very difficult to make these estimates for Io and Europa. This is
because your measuring frequency is not frequent enough (relative to the oscillation that you are observing).
This effect is known as aliasing. Aliasing is well known in communications theory, and must be accounted for.
But this is easily done. Basically, we just make more frequent observations. So reset the date and time to 8PM
on the 24th of October 2011, and fill out the following table for Io and Europa. I’m sure that Galileo would have
redone the measurements, and instead of observing and recording the date once per evening. That once we saw
that Io and Europa revolved about Jupiter quickly, that we’d make several observations per evening. As before,
keep track of the minutes and seconds. Once you have the angle in minutes and seconds, convert the seconds to
fractions of a minute. Utilizing the “command” and “-“ and “command” “=” might be useful here.
Table 4
Date
Time
24 October 2011
24 October 2011
25 October 2011
25 October 2011
25 October 2011
25 October 2011
25 October 2011
26 October 2011
26 October 2011
26 October 2011
26 October 2011
26 October 2011
27 October 2011
27 October 2011
27 October 2011
8:01 PM
10:01 PM
Midnight + 1min
2:01AM
4:01AM
8:01PM
10:01PM
Midnight + 1min
2:01AM
4:01AM
8:01PM
10:01PM
Midnight + 1min
2:01AM
4:01AM
Io angle
(min & sec)
Io angle
(decimal min)
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Europa angle
(min & sec)
Europa angle
(decimal min)
Plot 2
Now that you’ve measured Io and Europa more frequently, plot the data in Plot 2 (just like you did for Plot 1).
Now that you’ve redone the measurements for Io and Europa, determine the period and maximum angle
for them. Also pull the data from Table 3 for Ganymede and Calisto and reproduce it below (in Table 5). In
particular, look for the zero-crossings. That is when the angle between the moon and Jupiter goes to zero.
Table 5
Moon
Maximum Angle
(arc-minutes)
Maximum distance
from Jupiter (AU)
Period (days)
Period (years)
Io
Europa
Ganymede
Calisto
Now that you’ve got the maximum angle in arc-minutes, convert that into Astronomical Units (AU) by
dividing by 866. We’re converting from arc-minutes to degrees to radians, and we are using the small-angle
approximation ( opposite_side = adjacent_side * angle_in_radians ). Also convert the period from days to
years.
PART D
Now, according to Kepler,
R3
= C for any small object(s) orbiting something massive (notice all the
T2
caveats). If we use AUs for R and years for T, then C is the mass of the object that is the center of the orbit (in
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terms of solar mass). In the case of the sun and the planets C=1, that is one solar mass (of course). So let’s
calculate C for Jupiter and its moons. So now we have if we measure R in astronomical units and T in years.
Let’s see if this is true. Use the data from Table 5 to fill in Table 6 below. For each moon, find R3 and T2.
Next divide its value of R3 by its value of T2 to get the value of C for each moon. Keep three decimal places
in each calculated value of C.
Table 6
Moon
R
3
T2
C (R3 / T2)
Io
Europa
Ganymede
Calisto
If Kepler was right, all the values of R3/T2 should be the same – they should all equal the same constant
“C,” which in the case for the planets is equal to 1. For this case, the central body is Jupiter, and not the sun.
Furthermore, the orbiting bodies are satellites/moons of Jupiter and not the planets. The constant C will be the
mass of Jupiter in units of solar mass!
PART E
Often in science we do experiments over and over again and measure some quantity many times, and
then take the average value, hoping to average out any random errors. In Table 3 we’ve calculated C eight
different times. Unless you made perfect observations, your values of C will not all be exactly the same.
What we can do is calculate a precise number that tells us how close to each other all our values of C
are. This number is called the standard deviation. The lower the standard deviation, the closer all your
answers are to each other. For example, if all our values of C were exactly the same, the standard deviation
would be zero. That would mean we were very careful experimenters! If the values of C were all the same and
were all equal to 1, that would also be a very good indication that Kepler was right!
To calculate the standard deviation, first we need to calculate your average value of C. Add up all the
values of C in Table 6, and divide the result by 4(since we measured four moons). This is the average value of
C. We’ll call it <C>.
What is your value of <C>? _______________________________.
Now we need to calculate how far each value of C is from <C>. For each value of C, we call this
number “ΔC”. Calculate this number for each planet and fill in Table 7.
Table 7
Planet
ΔCi = C - <C>
Io
Europa
Ganymede
Calisto
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(ΔCi)2= ΔCi *ΔCi
Now we’re ready to calculate the standard deviation. The standard deviation is given by
𝜎=
1
𝑁
Δ𝐶 !
!
Where N is the number of times we made the measurement of C – in this case, four. What this equation
says is that we SQUARE each ΔC for each moon, then ADD UP all those squared numbers (the Σ means “add
up”), then divide by our value of N, which is 4, and then take the square root of the result.
What is your standard deviation,
σ?
________________________________.
Remember, the closer the standard deviation is to zero, the more all your values are C are nearly the
same. In other words, the standard deviation is our first consideration at determining the error of our
measurements. Typically we report the measurement and standard deviation as C = <C> ± σ .
PART F
Now consider, we have just measured <C> for Jupiter. We’ve measured the mass of Jupiter as a ratio of
its mass to that of the sun. So what is our expected value for Ce? Look up the mass of Jupiter and the mass of
the sun in your Astronomy book. Also, look up the value on the Internet – do not use Wikipedia as your
primary source. Fill out Table 8, calculate the ratio of the Mass of Jupiter to the Mass of the Sun. Also, report
your Internet source here __________________________________.
Table 8
Mass
Jupiter
Sun
Ratio
Ce = MJupiter/Msun
Astronomy Textbook
Internet Source
Compare the two values of Ce, the expected value of C. Are they the same? ________. If not, then
why do think they are different? _____________________________________________________________
_______________________________________________________________________________________
Now compare your value of <C> to that from the textbook and your Internet source. For this
comparison, evaluate the following equation for each source (textbook and the Internet).
𝐴=
𝐶𝑒−< 𝐶 >
𝜎
What was your value of A from the textbook? _____________
What was your value of A from your Internet source? ___________
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PART F
By using Kepler’s 3rd Law, we have determined the ratio of Jupiter’s Mass to that of the Sun. Now
determine the mass of Jupiter in terms of kilograms? __________________________
_______________________________________________________________________________________
Write a brief conclusion summarizing your results and what you learned in this assignment. Use the
back of this page if you need more room.
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