Objectives:
Background:
Johannes Kepler studied the motion of planets orbiting stars as well as moons orbiting planets. Through this he discovered that there is a certain constant that applies to all orbiting celestial bodies, C. This C is equal to the radius of the semi-major axis cubed divided by the period of the orbit squared. Later Newton expanded on this discovery and deduced an addendum to the equation, discovering what C is equal to:
R^3/T^2=GM/(4 π^2)
With this equation we can find the mass of any body that is orbited by another body. WE will use this equation to discover the mass of Jupiter through the movement of four of its moons: Io, Callisto, Ganymede, and Europa.
Procedure:
The CLEA software simulates a telescope allowing you to make your “observations” of these distant celestial objects. The program has already been downloaded from the CLEA website onto your school laptop. 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.
- To calculate the mass of Jupiter through the orbits of its moons
Background:
Johannes Kepler studied the motion of planets orbiting stars as well as moons orbiting planets. Through this he discovered that there is a certain constant that applies to all orbiting celestial bodies, C. This C is equal to the radius of the semi-major axis cubed divided by the period of the orbit squared. Later Newton expanded on this discovery and deduced an addendum to the equation, discovering what C is equal to:
R^3/T^2=GM/(4 π^2)
With this equation we can find the mass of any body that is orbited by another body. WE will use this equation to discover the mass of Jupiter through the movement of four of its moons: Io, Callisto, Ganymede, and Europa.
Procedure:
The CLEA software simulates a telescope allowing you to make your “observations” of these distant celestial objects. The program has already been downloaded from the CLEA website onto your school laptop. 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.
DATA
This is the raw data acquired from the CLEA software for each of the 4 moons.
This is the raw data acquired from the CLEA software for each of the 4 moons.
| Raw Data |
Data Analysis
Percent Error Calculation:
Error=(1.77e27-1.90e27)/(1.90e27)*100
Error= 6.84%
Error=(1.77e27-1.90e27)/(1.90e27)*100
Error= 6.84%
Conclusion:
1. There are moons beyond the orbit of Callisto. Will they have larger or smaller periods than Callisto? Why?
They will have a larger period. According to the ratio r^3/T^2=C, if r gets bigger than T, the period, will have to get bigger in order to keep C a constant number, which it is.
2. Which do you think would cause the larger error in the mass of Jupiter calculation: a ten percent error in "T" or a ten percent error in "r"? Why?
"R" because if there is a 10% error in this variable, it is cubed and therefor it will cause an effect of 1000%. While if there is a 10% error it is only squared to become a 100% effect.
3. 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 evidence against the contemporary and officially adopted Aristotelian/Roman Catholic, geocentric view)?
This is important because studying the moon's of Jupiter gave him a model to compare the geo and heliocentric model to. He was able to find similarities to the moons with the heliocentric, while the geocentric model did not have the same similarities with the planets that "orbit" the earth.
1. There are moons beyond the orbit of Callisto. Will they have larger or smaller periods than Callisto? Why?
They will have a larger period. According to the ratio r^3/T^2=C, if r gets bigger than T, the period, will have to get bigger in order to keep C a constant number, which it is.
2. Which do you think would cause the larger error in the mass of Jupiter calculation: a ten percent error in "T" or a ten percent error in "r"? Why?
"R" because if there is a 10% error in this variable, it is cubed and therefor it will cause an effect of 1000%. While if there is a 10% error it is only squared to become a 100% effect.
3. 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 evidence against the contemporary and officially adopted Aristotelian/Roman Catholic, geocentric view)?
This is important because studying the moon's of Jupiter gave him a model to compare the geo and heliocentric model to. He was able to find similarities to the moons with the heliocentric, while the geocentric model did not have the same similarities with the planets that "orbit" the earth.
| Jupiter's Moons |
| Graph Set 2 |
| CLEA Data |
