PART 1:
We will be using actual orbital data from a satellite orbiting the Moon to verify Kepler’s
laws and find the mass of the Moon.
1) Using the data provided and a sheet of graph paper plot the orbit of the satellite. Start by
plotting the Moon. Make the Moon’s diameter an even number of boxes across. What works
well is just making it 4 boxes across. Then if the diameter is 4 boxes the radius on the Moon
is 2 boxes. You can either print the graph paper, edit the pdf or you can try using this online
digital graph paper https://fillable-grid-online.pdffiller.com/
The plot points are in terms of x and y coordinates starting from the center of the Moon.
The value is in terms of the radius of the Moon. So +1 x means 1 Moon radius in the positive
x direction. -3 y means 3 Moon radii in the negative y direction. As you plot the points you
will see the elliptical shape of the orbit appear.
2) Find the length of the major axis, minor axis and the distance between the foci in
kilometers. Label these axes and the foci. To do this you need to figure out the scale of your
plot. First measure the Moon’s radius on your plot. Then look up the actual radius of the
Moon (It’s 1737 km). Now divide that by the measured Moon radius on your plot to get a
scale for your plot. For example, if the Moon’s radius on my plot is 3 cm then I divide 1737
by 3 and get 579 kilometers per cm. Now I can multiply any measurement from my plot in
cm by the scaling factor and get the actual distance in km.
Major Axis:
Minor Axis:
Distance between foci:
3) Find the eccentricity of your orbit. Remember the equation is e (eccentricity) = (distance
from the center to one focus)/(The length of the semi-major axis)
Eccentricity:
PART 2:
Let’s confirm that this is indeed an elliptical orbit. Kepler’s First Law.
4) Pick any point on the orbit. Measure the distance from that point to each focus. Let’s call
these D1 and D2. Pick another point on your ellipse and again measure the distances to each
focus. Call these D3 and D4. Does D1 + D2 = D3 + D4? Is there a reason it’s not exact?
D1: D2: D1 + D2:
D3: D4: D3 + D4:
Let’s confirm that over the same time the same area is swept out. Kepler’s Second Law.
5) Pick two points on the ellipse that are 2 hours apart in time. Draw lines from those points
to the center of the Moon. Count up as best as you can the number of boxes in-between the
two lines. Do this for another set of two points that have the same time difference. Do these
two values match?
Number of boxes in area 1:
Number of boxes in area 2:
Let’s confirm the orbital period and distance relationship. Kepler’s Third Law.
6) What is the orbital period of the satellite? Convert this answer to seconds for our
formula.
Orbital Period (seconds):
7) Using the formula from the lecture slide calculate the mass of the Moon. As an estimate
set the mass of the satellite to zero.
Mass of the Moon:
8) Find what percentage your value differs from the true mass of the Moon. [(Your
measured mass - The actual mass)/(The actual mass)]*100%
Percent Difference:
9) What reasons can you think of make this value more than 0%?
Sample Solution