Purpose
1. Determine the mass of Jupiter.
2. Gain a deeper understanding of Kepler's third
law.
3. Learn how to gather and analyze astronomical data.
2. Gain a deeper understanding of Kepler's third
law.
3. Learn how to gather and analyze astronomical data.
Equipment
-CLEA software
-Microsoft Excel
-Microsoft Excel
Procedure
1. Downloaded CLEA software onto computer.
2. Set starting date to current date. Began collecting data by day by clicking on each of Jupiter's 4 moons.
3. After over 20 days of data was collected, saved data to a spreadsheet that was put together by the CLEA software.
4. Used software to fit a sine curve to each of the r v. t graphs for the moons.
2. Set starting date to current date. Began collecting data by day by clicking on each of Jupiter's 4 moons.
3. After over 20 days of data was collected, saved data to a spreadsheet that was put together by the CLEA software.
4. Used software to fit a sine curve to each of the r v. t graphs for the moons.
Background
Copernicus hypothesized that the planets revolve in circular orbits around the sun, while others believed that everything orbited around the earth. Kepler found that there was a relationship between the period of an orbiting body and its distance from that which it orbited, but he did not know what this constant was. Newton expanded on this and derived the function:
R^3/T^2 = GM/(4pi^2) where G = 6.67 x 10^-11 Nm^2/kg^2
Using this law, we can study Jupiter's orbiting moons to find the mass of Jupiter.
R^3/T^2 = GM/(4pi^2) where G = 6.67 x 10^-11 Nm^2/kg^2
Using this law, we can study Jupiter's orbiting moons to find the mass of Jupiter.
Data/Data Analysis
Below are the r v. t sine graphs for each of Jupiter's moons. Below each graph is the period (T) and amplitude (R).
Below is the spreadsheet with the raw data collected from the CLEA software.
jupsatdata1.csv | |
File Size: | 1 kb |
File Type: | csv |
Below is the spreadsheet where the data, including R and T for each moon, is linearized and a line y = mx + b is found. The graph of the data with the line is on the spreadsheet.
y = (3 x 10^15)x + (3 x 10^25)
R^3 = (GM/(4pi^2))T^2
y = (3 x 10^15)x + (3 x 10^25)
R^3 = (GM/(4pi^2))T^2
T^2 v R^3 graph | |
File Size: | 13 kb |
File Type: | xlsx |
Below is the data analysis where the mass of Jupiter (M) was found, and percent error was calculated.
M(experimental) = 1.7748 x 10^27 kg
M(theoretical) = 1.8986 x 10^27 kg
Percent Error = 6.97 %
M(experimental) = 1.7748 x 10^27 kg
M(theoretical) = 1.8986 x 10^27 kg
Percent Error = 6.97 %
Finding M/% Error | |
File Size: | 157 kb |
File Type: |
Concluding Questions
1. Calculate the percentage error with the accepted mass of Jupiter (1.8986 × 10^27 kg).
6.97%
2. There are moons beyond the orbit of Callisto. Will they have larger or smaller periods than Callisto? Why?
They will have larger periods than Callisto because they are farther away. With a greater distance to travel, it will take a longer amount of time. Mathematically this can be seen by the relationship between T^2 and R^3 in R^3 = (GM/(4pi^2))T^2. As R^3 increases, T^2 increases.
3. 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?
A ten percent error in R would probably cause a greater error because R is cubed, while T is squared, so the error in R would be magnified more than the error in T.
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 evidence against the contemporary and officially adopted Aristotelian/Roman Catholic, geocentric view)?
Galileo observed that the moons were orbiting around Jupiter, which proved that not everything orbited around the earth. The Jupiter system provided a model that could be studied to understand the motions of the solar system, and it proved that the heliocentric model was physically possible.
6.97%
2. There are moons beyond the orbit of Callisto. Will they have larger or smaller periods than Callisto? Why?
They will have larger periods than Callisto because they are farther away. With a greater distance to travel, it will take a longer amount of time. Mathematically this can be seen by the relationship between T^2 and R^3 in R^3 = (GM/(4pi^2))T^2. As R^3 increases, T^2 increases.
3. 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?
A ten percent error in R would probably cause a greater error because R is cubed, while T is squared, so the error in R would be magnified more than the error in T.
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 evidence against the contemporary and officially adopted Aristotelian/Roman Catholic, geocentric view)?
Galileo observed that the moons were orbiting around Jupiter, which proved that not everything orbited around the earth. The Jupiter system provided a model that could be studied to understand the motions of the solar system, and it proved that the heliocentric model was physically possible.
Conclusion
The purpose of this lab was to determine the mass of Jupiter, gain a deeper understanding of Kepler's third law, and learn how to gather and analyze astronomical data. I found the mass of Jupiter to be 1.7748 x 10^27 kg, which has a 6.97% error. This error could be due to the fact that the sine curves that I fit to the data did not perfectly match the data. Also, the lack of data on cloudy days could have accounted for some error.