Kepler's 3rd Law and the Orbits of the Moons of Jupiter
Objectives
- Determine the mass of Jupiter.
- Gain a deeper understanding of Kepler's third law
- Learn how to gather and analyze astronomical data
Background
In 1543, Copernicus formed his hypothesis about heliocenticity. Later Tycho Brache began to observe the planets and over 777 stars. These two men helped Kepler very much in the forming of his three mathematical laws governing the orbit of one body about another.
Kepler's third law for a moon orbiting a much larger body is
C = r^3/T^2
C is a constant.
r is the length of the semi-major axis of the elliptical orbit in units of the mean Earth-Sun distance.
T is the planetary orbit in Earth years.
A man named Newton used Kepler's Third Law, by using his Universal Law of Gravitation to solve for C and derived
r3/T2 = GM/4p2
G is the gravitational constant 6.67E-11 NM^2/kg^2.
M is the mass of the larger body.
In 1609, the telescope was invented, allowing the observations of objects not visible to the naked eye. A telescope aided Galileo to discover that Jupiter had four moons orbiting it and made exhaustive studies of this system. This newly visible system was especially important because it is a miniature version of the solar system which could be studied in order to understand the motions of the solar system in the Milky Way. The Jupiter system provided clear evidence that Copernicus' heliocentric model of the solar system was physically possible. Unfortunately, Galileo was tried and forced to take back the ideas of his findings.
The purpose of this lab is to observe the motion of the moons of Jupiter and then use Kepler's third law and the expansion to his law offered by Newton’s Universal Law of Gravitation to deduce the mass of Jupiter. The four Galilean moons, Io, Europa, Ganymede and Callisto are easily seen through a small telescope, and the CLEA software will simulate these observations.
Kepler's third law for a moon orbiting a much larger body is
C = r^3/T^2
C is a constant.
r is the length of the semi-major axis of the elliptical orbit in units of the mean Earth-Sun distance.
T is the planetary orbit in Earth years.
A man named Newton used Kepler's Third Law, by using his Universal Law of Gravitation to solve for C and derived
r3/T2 = GM/4p2
G is the gravitational constant 6.67E-11 NM^2/kg^2.
M is the mass of the larger body.
In 1609, the telescope was invented, allowing the observations of objects not visible to the naked eye. A telescope aided Galileo to discover that Jupiter had four moons orbiting it and made exhaustive studies of this system. This newly visible system was especially important because it is a miniature version of the solar system which could be studied in order to understand the motions of the solar system in the Milky Way. The Jupiter system provided clear evidence that Copernicus' heliocentric model of the solar system was physically possible. Unfortunately, Galileo was tried and forced to take back the ideas of his findings.
The purpose of this lab is to observe the motion of the moons of Jupiter and then use Kepler's third law and the expansion to his law offered by Newton’s Universal Law of Gravitation to deduce the mass of Jupiter. The four Galilean moons, Io, Europa, Ganymede and Callisto are easily seen through a small telescope, and the CLEA software will simulate these observations.
Directions
The CLEA software simulates a telescope allowing you to make your “observations” of these distant celestrial objects. The program has already been downloaded from the CLEA website onto your school laptop (a link is provided on the SHS AP Physics homepage and in the links section above if you'd like to download the software on your home computer). To begin the program, select CLEA_JUP and Login. After entering your names and lab table number, choose the Start option to set the starting date and time. Note that during the lab you may want to go back to this table to reset the “Interval Between Observations.”
Jupiter is in the center of the screen, while the small point-like moons are to either side. Sometimes a moon is behind Jupiter, so it cannot be seen. Even at high magnifications, they are very small compared to Jupiter. The current telescope magnification is displayed at the upper left hand corner of the screen. The date, UT (the time in Greenwich, England) and JD (Julian Date) are displayed in the lower left hand corner of the screen.
Click on each moon, using the highest magnification to get the best position measurement, R, which is recorded in number of “Jupiter Diameters”. R is the distance from the center of Jupiter to the center of the moon. Sometimes a moon will be behind Jupiter, and you will not be able to record data for that moon. To save measurements, simple click record.
You wish to give sufficient time coverage to all four orbits. Since Io's orbital period is significantly shorter than that of Callisto, you will have to change your observation interval to both get good time coverage and to make efficient use of your observing time. You need to cover a full period of the orbit from, for example, the moons most eastern position to its most western position and back.
You should collect approximately 20 days of data, then use the plot function to create r vs. t curves for each moon. The graph is a sine curve whose amplitude is orbital radius and wavelength is period. Now using the orbit of each Galilean moon, determine the quantities that you would have to graph in order to obtain a straight line whose slope will yield the mass of Jupiter. Create this plot manually to calculate the mass of Jupiter. You will have to convert Jupiter Diameters to meters and years to seconds. There are 1.43x10^8 meters in one Jupiter Diameter.
Jupiter is in the center of the screen, while the small point-like moons are to either side. Sometimes a moon is behind Jupiter, so it cannot be seen. Even at high magnifications, they are very small compared to Jupiter. The current telescope magnification is displayed at the upper left hand corner of the screen. The date, UT (the time in Greenwich, England) and JD (Julian Date) are displayed in the lower left hand corner of the screen.
Click on each moon, using the highest magnification to get the best position measurement, R, which is recorded in number of “Jupiter Diameters”. R is the distance from the center of Jupiter to the center of the moon. Sometimes a moon will be behind Jupiter, and you will not be able to record data for that moon. To save measurements, simple click record.
You wish to give sufficient time coverage to all four orbits. Since Io's orbital period is significantly shorter than that of Callisto, you will have to change your observation interval to both get good time coverage and to make efficient use of your observing time. You need to cover a full period of the orbit from, for example, the moons most eastern position to its most western position and back.
You should collect approximately 20 days of data, then use the plot function to create r vs. t curves for each moon. The graph is a sine curve whose amplitude is orbital radius and wavelength is period. Now using the orbit of each Galilean moon, determine the quantities that you would have to graph in order to obtain a straight line whose slope will yield the mass of Jupiter. Create this plot manually to calculate the mass of Jupiter. You will have to convert Jupiter Diameters to meters and years to seconds. There are 1.43x10^8 meters in one Jupiter Diameter.
Data, Graphs, and Data Analysis
| jupsatdata1.csv |
| position_vs_time_data_and_sine_wave_fits_for_moon_data.docx |
| jupiter_moons_data_analysis.xlsx |
Conclusion
Concluding Questions
1. Calculate the percentage error with the accepted mass of Jupiter (1.8986 × 1027 kg).
2. There are moons beyond the orbit of Callisto. Will they have larger or smaller periods than Callisto? Why?
3. Which do you think would cause the larger error in MJ: a ten percent error in "T" or a ten percent error in "r"? Why?
4. Why were Galileo's observations of the orbits of Jupiter's moons an important piece of evidence supporting the heliocentric model of the universe (or, how were they inst the contemporary and officially adopted Aristotelian/Roman Catholic, geocentric view)?
Answers:
1. Calculating the percentage error. This is shown on the excel spread sheet.
Error = -.25%
2. A moon that lies beyond Callisto would have a larger radius which would result in a larger period.
3. A ten percent error in r would result in a larger error in the mass because r is cubed and T is squared, so r would cause a greater change in the mass.
4. Galileo’s observations were especially important because he produced a miniature version of the solar system which was used to study our entire solar system, and it provided evidence that the heliocentric model was possible.
After the completion of the lab, I was able to figure out Jupiters mass and with very little error also. The lab itself was very helpful with the understanding of Kepler's 3rd Law and with the process of applying it. I also benefitted from ising Vernier Software and learned more about the system than ever before. This project was very interesting in my opinion, and I was very pleased with my data and analyzation.
1. Calculate the percentage error with the accepted mass of Jupiter (1.8986 × 1027 kg).
2. There are moons beyond the orbit of Callisto. Will they have larger or smaller periods than Callisto? Why?
3. Which do you think would cause the larger error in MJ: a ten percent error in "T" or a ten percent error in "r"? Why?
4. Why were Galileo's observations of the orbits of Jupiter's moons an important piece of evidence supporting the heliocentric model of the universe (or, how were they inst the contemporary and officially adopted Aristotelian/Roman Catholic, geocentric view)?
Answers:
1. Calculating the percentage error. This is shown on the excel spread sheet.
Error = -.25%
2. A moon that lies beyond Callisto would have a larger radius which would result in a larger period.
3. A ten percent error in r would result in a larger error in the mass because r is cubed and T is squared, so r would cause a greater change in the mass.
4. Galileo’s observations were especially important because he produced a miniature version of the solar system which was used to study our entire solar system, and it provided evidence that the heliocentric model was possible.
After the completion of the lab, I was able to figure out Jupiters mass and with very little error also. The lab itself was very helpful with the understanding of Kepler's 3rd Law and with the process of applying it. I also benefitted from ising Vernier Software and learned more about the system than ever before. This project was very interesting in my opinion, and I was very pleased with my data and analyzation.
