The Period of Io and the Mass of Jupiter

Research Project by Morgan Dorman

and Adrian Steffen

Purpose: The purpose of this experiment is to calculate the mass of Jupit

In HOU_IP, select the moon that is being analyzed and find its distance from Jupiter. Use the Axes function under the "Tools" menu to find the center of Jupiter and the moon. Make a box by using the "click and drag" method encompassing the body that you want the center of. The data should be inserted into the correct column on the table. This will need to be repeated for all images taken.

 

er using the orbital radius and period of Io.

Hypothesis: We predict that the mass of Jupiter will be about 1.4 x 10^27 kilograms, assuming its density is about the same as water and its radius is 11 x Earth's radius.

Procedures:

Open the desired image. Find the time, date and location that the image was taken. Put the information into a planetarium program to change the settings to the conditions on that night. Locate the planet and its moons. Zoom in to match the pattern of the moons in the image to the pattern in the planetarium program.

Materials Needed:

1. HOU_IP

2. Astronomy program (TheSky, Sky Map, Starry Night, etc.)

3. Microsoft Excel

Design a table to hold the information while calculations are being made. It should be easily read and simple. An example is shown below.

Locate Jupiter using the time and date of the image. Then, zoom out to see the moons. Next, click on the moons to find the one that is the correct moon that is being studied.

In the "Image Info" box, the date is "DATE" and the time is "TIME-OBS" in HOU_IP. In TheSky, under "Data," click on "Site Information" and transfer the date and time to the location of a city in the area of the telescope that took the image.

Plug the information into the Excel spreadsheet titled "Jupiter Moons." If the moon moves to the left side of the planet, then put a negative sign in front of the distance.

Adjust the Amplitude, Radians, and Period so that the function (S(r-rcalc)^2) under the chart gets as close to zero as possible and no changes in Amplitude, Phase, or Period will decrease it further.

Use the following equation to solve for the mass of Jupiter:

Mass of Jupiter = (4 p^2 D^3) / (G T^2)

Our value for Io's period is T = 42.584 hr = 2555.0 s. and for its orbital radius is D = 456,417 km = 456,417,000 m

This gives a calculated mass for Jupiter of 2.39 x 10^27 kg, about 26% greater than the accepted value.

Summary of Calculations

Excel CurveFit Worksheet

Cape Fear High School Homepage

 

Our Calculated Value
Accepted Value
D%
Period of Io
42.58 hr
42.48
+ 0.24%
Orbital Radius of Io
456,417 km
422,000 km
+ 8.2%
Mass of Jupiter
2.39 x 10^27 kg
1.90 x 10^27 kg
+ 26 %
Conclusions: Our method of sine curve fitting gave a very accurate calculated value for the orbital period of Io. The value calculated for the orbital radius is slightly too large. This may mean that Io has an eliptical orbit, or that we need more images to get an accurate orbital radius. The error in the orbital radius led to a much greater error in the mass of Jupiter because the radius is cubed in calculating the planet's mass. We noticed that Io's position in the 7th and 8th images did not fit the sine curve very well. Could another moon have passed nearby and changed its position slightly?