2023 Spring - Kepler's Conundrum
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Colorado State University, Fort Collins *
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Course
101
Subject
Astronomy
Date
Apr 3, 2024
Type
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AA 101 – Astronomy Laboratory
Kepler’s Conundrum
Learning Objectives
●
Estimate the mass of a planet by measuring the orbital properties of its satellites.
●
Produce and analyze graphs that represent orbital motion.
Introduction
Johannes Kepler discovered in 1619 that
planets in our solar system orbit the Sun
with periods that are related to the semi-
major axis (or radius) of their orbits. The
relationship between period (P) and
semi-major axis (a) is not 1:1. What later
became known as Kepler’s Third Law of
Planetary Motion, or the Law of
Harmonies, is expressed as:
P
2
=
a
3
This
relationship
is
not
mere
coincidence. It depends upon the
gravitational force that keeps planets in orbit around the Sun. Today, you will use a more general
form of Kepler’s Third Law, given below, to determine the mass of a planet being orbited by a much
smaller moon.
P
2
=
(
4
π
2
GM
)
a
3
In this equation, P is the orbital period of the moon (in seconds), a is the semi-major axis of the
moon’s orbit (in meters), M is the mass of the planet (in kilograms), and G is the gravitational
constant.
G
=
6.67
×
10
−
11
m
3
kg∙s
2
(← This red text is just units.)
Before you start the main lab, check that this equation works using what you know about the Earth’s
orbit around the Sun. What do you calculate as the Sun’s mass based on Earth’s orbital period (1
year) and semi-major axis (1 AU = 150,000,000,000 m)? How does this compare to the Sun’s mass
according to Wikipedia?
Lab Activity
Page 1 of 5
Kepler’s Conundrum
You should spend roughly 2 hours completing this lab exercise. This includes completing tasks and
composing your lab report, so pace yourself. Jot down your answers to questions as you go along,
and make sure you reserve enough time at the end of class to gather your findings in a polished,
original lab report.
Groups are expected to measure the mass of one planet before class ends. If you finish early, collect
more data from additional satellites for more robust results. Instructions
Open Stellarium. Use the icons on the toolbars to select the following settings.
Icon
On/Off
Set Normal Time Rate
Off (stops the clock)
Ground
Off
Atmosphere
Off
Cardinal Points
Off
Planet Labels
On
Configuration Window
Information > Short
You will first use Saturn as a test subject to ensure that your Telescope Mount setting is correct.
Search for Saturn and use the toolbar icon to Center on Selected Object. Zoom in until the planet and
its moons fill your screen.
○
With a computer, you can do this with a scrolling mouse. ○
With a Windows machine, you can hold down the control button and push the
up/down arrow keys. ○
With a Mac machine, you can hold down the command button and push the
up/down arrow keys.
○
On a phone or tablet, you can zoom in/out with two fingers, and you can drag the
sky around with one finger.
Now open your Date/Time window, and scroll forward one hour at a time to watch what happens to
the planet. If it looks like the planet is twirling around, click on the Switch Between Equatorial and
Azimuthal Mount icon once, and try moving through time again. If you have the correct Mount setting,
Saturn should appear stable.
For your data collection, choose your favorite Jovian (Jupiter, Saturn, Uranus, or Neptune) and
Center on Selected Object. Zoom in until the planet and its moons fill your screen. Choose one of
your planet’s moons, and record its distance (in millimeters) from the planet’s center as you step
Page 2 of 5
Kepler’s Conundrum
forward in time (by days or hours). It is recommended that you tape your ruler to your screen before
taking measurements to improve consistency. Example
The distance between Jupiter and Io should be measured between the centers of both objects. In
this case, it is about 23 centimeters or 230 millimeters.
As you move forward in time, be careful not to change the zoom on your screen. Collect data for a
couple of orbits. Then create a plot of your data for your lab report. It should look something like the
example below.
Based on your plot, estimate the orbital period and semi-major axis of your moon’s orbit. Convert the
orbital period to seconds, and convert the semi-major axis to meters using the following equation.
Page 3 of 5
Orbital Distance on Screen
(in millimeters)
Time
(in days or hours)
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Kepler’s Conundrum
You can measure the planet’s diameter on your screen, and look up the real planet diameter using
Wikipedia. The “moon distance on screen” is the amplitude of the sine curve you just graphed. The
“real moon distance” is what you are solving for.
realmoon distance
(
¿
meters
)
real planet diameter
(
¿
meters
)
=
moon distanceonscreen
(
¿
millimeters
)
planet diameter onscreen
(
¿
millimeters
)
Example
In this picture, Jupiter’s diameter on the screen is about 80 millimeters. Wikipedia tells us Jupiter’s
actual diameter is about 140 million meters. At this moment, when Io is 230 millimeters from Jupiter
on the screen. . . x
140,000,000
meters
=
230
millimeters
80
millimeters
x
=
402,500,000
meters
Io is 402.5 million meters away from Jupiter.
Once you know the period and semi-major axis of your moon, you can calculate your planet’s mass
using the general form of Kepler’s Third Law provided above.
Lab Report
Summarize your findings from this exercise in an original lab report. Questions posed in the lab
instructions should be addressed in your report, but do not copy any text directly from the lab
instructions! When your report is complete, check that your lab instructor can access it via Google
Docs.
Reports will be evaluated based on the following criteria: scientific accuracy, quantitative analysis,
communication, and engagement.
Page 4 of 5
Kepler’s Conundrum
Your report should include your name and the date as well as the following sections:
Introduction
●
Describe the goal of today’s exercise.
Methods
●
Explain how you used Stellarium to collect data for this exercise. What planet/moon system did
you observe? (Include a picture of your setup.) How did you decide how far to zoom in/out on
the system? What time step did you use to collect your measurements, and why did you
choose this value? ●
Describe any challenges you encountered during this exercise and how you addressed them. Data
●
Include a graph that shows your moon’s orbital distance as a function of time. Show your work
for the calculations used to estimate your moon’s orbital period and semi-major axis.
●
Calculate the mass of your planet, and compare your result to the value provided by Wikipedia.
If you observed multiple moons for the same planet, be sure to include graphs for each moon
and your final results from each data set. What is the average value you calculated for the
planet’s mass?
●
Finally, quantify the percentage of “error” in your measurement compared to the Wikipedia
value.
Discussion
●
What were the largest sources of error or uncertainty in your calculations? How might you
change your procedure to get more accurate results?
Page 5 of 5