# Research

### Finding the Mass of Jupiter

This report is written by an Albany High School student, William Lee, on May 7, 1994 when he was a sophomore. The objective is to find the mass of Jupiter by using Hands on Universe's image processing program.

### Main Goal

The main goal of this whole assignment is to find the mass of Jupiter by getting the radius and the period of at least one of the 4 moons in the images given. It involves some knowledge of paralax and the universal law of gravation.

## Contents

You can always click on the section you want to jump to.

### Combining the Images

First of all, you are given 6 different images of Jupiter and its moon taken from the telescope in a time interval of 6 hours. The images are all one hour apart. By using the axis tool, I first find the coordinate of Jupiter of each image. Doing a little shifting can make me able to make each center of Jupiter to be the same. I then simply add them up together to form one complete image that shows the obvious moving of the 4 moons, Io, Europa, Ganymede, and Callisto. Since I don't know which moon is which, I'll mark them as A, B, C, and D. The image is shown below and the arrows are given for the direction of moving.

### Naming the Four Moons

Which moon is which? To identify each of the moons is the probably the first task I need to solve after I have gone through all the tedious work of getting the center of Jupiter at the same location. It is not hard to see Moon D is actually getting much slower and that indicates the moon is reaching the "edge" of its orbit. I get all the coordinates of Moon A, B, C, and D at different locations by using the axis tool. I also find their distances from Jupiter by using the distance formula, sqrt((x1-x2)^2+(y1-y2)^2). I graph the distance in pixels from Jupiter verses time in hours, I have something like below:

I can see the graph of Moon D is a curve. Its speed is changing rapidly. It seems like to be the closest moon of Jupiter, Io, because no other moon has a slight tendency of slowing its speed to zero at this distance. The speed of Moon D is approaching zero (the slope of the curve is 0) from approximate the 5th hour to 6th hour. I can approximate the distance at that point as the radius of the orbit of Io, that is, 193.37 pixels.

According to the graph on the data sheet, all other moons are nearly moving at a fairly constant speed. At the point when the graph of A and B meet each other (almost 200 pixel from Jupiter), I can compare their speed because they are at the same distance. The calculate the slope of A is (255.05-188.56)/(1-6)= -13.30, and B is (109.26-207.95)/(1-6)=19.74. The answer shows that B is moving faster than A at that point. I assume line A and C will meet at a certain point soon (not shown in the graph). I find the slope of C, which is (43-132.82)/(1-6)=17.96, is steeper than line A. I know B>A (">" means faster than) and C>A by now. Finally, I compare the velocity of B and C by comparing position 1 and 2 of Moon B and position 5 and 6 of Moon C. As you can see in the data sheet, the distance from Jupiter of C at position 6 is very close to the distance from Jupiter of B at position 2 (132.82 pixels and 131.09 pixels). B moves from 109.26 pixels to 131.09 in one hour, and C moves from 115.00 to 132.82. B moves 21.83 pixels/hour and C moves 17.82 pixels/hour at almost the same location. Therefore I conclude that B>C.

Here is some facts that I know:
B>A
B>C
C>A
D=Io (fastest)
Therefore: D>B>C>A

I know that Io is the closest moon, Europa is the second closest, Ganymede is the third, and Callisto is the fourth.

If you look at the relationship between the velocity and radius by using the formula, v=sqrt((GM)/r), when r increases, v decreases, and when v increases, r decreases.

So, I conclude:
D--------Io
B--------Europa
C--------Ganymede
A--------Callisto

### Orbit of Io and More

Let's focus on the data of Io:

First of all, since I only see 5 obvious dots in the image for D, I suppose the diagram is correct. I once added the last two shifted images together (aprwiaa.fts and aprwi77.fts) and found that they overlap each other.

I know the radius is 193.38 pixels, and let's call the center point of Jupiter J:

The angles should be the same. However, they are NOT according to my calculation. 13.53 degrees is definitely too large, so I throw this data away. I use the average of 8.33 and 9.5 degrees, which is 9.06 degrees in later calculation.

### The Period of Io

The formula I use to calculate the mass of Jupiter is:

Later in the lab I find that T(period)=360t/angle. The t is the time in second to go though the angle. This can make my calculation a little bit more easier. I know the angle is 9.06 degrees and the t is 1 hour in my previous calculation. Now I have the period of Io is 360*1/9.06=14.3*10^5 seconds, or 16.6 days.

### Conclusion

Given that the distance from the Earth to Jupiter is 6.63*10^11m at the time.

If you want to contact me, you can always e-mail to:

top@netcom.com for William W. Lee

Lawrence Hall of Science | © 2013 | Updated November 23, 2012