Firstly, we are going to observe Newton's Law of Gravity and circular motion,
GMm/r² = mrw²
Our aim is to turn M into the subject of the formula:
GM/r³ = (2π/T)²
M = 4π² r³/GT²
Now we have M as the subject of the formula, we realize that need the value of the T and r to find out the mass of Jupiter. G is the gravitational constant and we can acquire this information from the internet. Also, T is obtain from the internet too. What we are looking for is the variable r.
T = Time (The time here refers to the period of Io)
r = Radius (This refers to the radius of Io)
Radius is the information we must acquire ourself though our photos. This is where our group mates trip to Desaru comes in. The reason we choose Desaru is that it has clearer skies, enabling us to take better picture to improve the quality of our results. Our trip to Desaru enables us to find out how each individual pixel is measured in kilometers.
(From now on, P = pixels)
In our trip to Desaru, we used these two pieces of equipment to gather the information required:
We took 4 photos at 15 mins intervals and this will be compiled into one set. Each of the 4 photos are taken in 2 mins, 1 mins, 2 mins intervals respectively. This is done to get a variety of photos of to narrow down our margin of error. The more pictures we get the lesser the margin of error is.
Using the pictures we have got:
Method use to find out r:
Steps used to find out r:
- Find out a and b in pixels
- Using Pythogoras’s Theorem, which states that a² + b² = r², we can find out r²
- Then, by squaring r², we would be able to get r
With this method, we would be able to get r. After getting r, we can plot it in a graph and use it to find the mass of Jupiter.
Method used to find out what each pixel is in km:
- Draw a square around the Jupiter
- Measure a
- a equals the diameter
As mass of Jupiter is constant throughout the photos, we would be able to use it to find out what each pixel equals. With that, we would be able to get pixel/km.
Data collected from 3 photos:
This will give us a plot of R₁, R₂, R₃ :
We will then use a curve fitting software to fit in the data. The equation used in question is y = A₀sin(wt+∅) when w is the value we need.
Since w = 2π/T → T = 2π/W =
Using T obtained and r obtained, we can calculate the mass of Jupiter using the equation we have previously:
M = 4π²/G x r³/T²
Now, with the values of T and r, we have the mass of Jupiter.