One of the observations that Galileo is famous for making is the discovery of four Moons of Jupiter [1], which these days we refer to as the "Galilean Moons."
This was one of the observations that contributed to the revolution in our understanding of the true nature of the Solar System. What Galileo personally observed was what he thought were stars near Jupiter, and night after night, he witnessed their positions change with respect to Jupiter.
I have created a Starry Night save file (.snf) [2] to let you jump to see this directly (a copy is posted in Canvas). If you would like to set it up yourself, you can do the following:
You should witness exactly what Galileo did—as you click on the forward button, each night the arrangement of the four Galilean Moons (Io, Europa, Ganymede, and Callisto) changes with respect to Jupiter.
Now, let's do something that Galileo could not. Let's look at Jupiter from above its North Pole so that we can see physically what is going on.
I have created a Starry Night save file (.snf) [3] to let you jump to see this directly (a copy is posted in Canvas), but if you would like to set it up yourself you can do the following:
What you can see in this latter view is the orbit of the moons, but what you see in the former view is what appears to be a side to side change in position of the moons. In this lab, we are going to measure that side to side motion and use that data to calculate the mass of Jupiter using Newton's version of Kepler's Third Law.
Let's talk about how the side view and top view compare. Below is an image that shows the top view (that is, as seen from Jupiter's North Pole) of the orbit of a moon:
If you study the image, you will note that when the moon is in front of Jupiter or behind it, we can describe its projected side to side distance from the planet as zero (in any units). When the moon is at a right angle from the Earth/Jupiter line, it will be seen at its maximum separation from the planet. If you consider how it will appear from Earth as it orbits and moves between these two extremes, it will appear to trace out a sine curve from maximum separation, to zero, to maximum separation, to zero, and back again. Sine curves have a few basic properties:
Below is a sample plot for a fictitious moon of Jupiter. The x-axis is labeled Julian Date, which is an easier way of sequentially marking days than relying on our calendar, which is difficult to work with on a plot because the number of days per month varies. Each tick mark on the x-axis is one day. The y-axis shows projected separation from Jupiter in units of Jupiter diameters.
If you again refer to the first image on this page, you will notice that when the moon goes from rightmost maximum separation to zero to leftmost maximum separation to zero to rightmost maximum separation again, that is the period of one orbit around Jupiter. On the curve above, the time from maximum to maximum peak is the same as the time from rightmost maximum separation to rightmost maximum separation. So, you can, therefore, estimate the period of a moon's orbit directly from one of these curves.
The amplitude of the curve illustrated above is the distance in Jupiter diameters when the moon is at either its left or rightmost maximum separation from the planet. That is a direct measurement of the semi-major axis of its orbit. Therefore, by plotting one of these curves, you can measure both P and a for a moon, which are the two quantities you need for a Kepler's third law calculation.
Recall that:
So, therefore, if you have P and a measured, you get the sum of , which is in this case the sum of .
On the next page are instructions for using a simulated observatory to take data on Jupiter's moons for the purpose of measuring the mass of Jupiter.
We're going to use the animation below to simulate taking observations of Jupiter and its Moons. For this exercise, we will be observing one of the outer moons, Ganymede. You will see Jupiter located in the center of the screen, and four dots which roughly line up horizontally across the screen (this is the plane in which the moons orbit). You cannot always see all four moons, sometimes one or two will be directly in front of or behind the planet.
Start the Revolutions of the Moons of Jupiter simulator (below) by entering the following information into the appropriate boxes:
Animation caption: Simulator that displays Jupiter and its Galilean moons as observed from Earth on a given date and a give time.
Links
[1] http://galileo.rice.edu/sci/observations/jupiter_satellites.html
[2] https://www.e-education.psu.edu/astro801/sites/www.e-education.psu.edu.astro801/files/snf/Jupiter_moons_lab_view1.snf
[3] https://www.e-education.psu.edu/astro801/sites/www.e-education.psu.edu.astro801/files/snf/Jupiter_moons_lab_view2.snf